Subgroups — C₂×C₃₁×#3515 #21519538

Order 7936.

7142 conjugacy classes of subgroups.

The subgroup lattice has 868 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃₁×#3515	C₂	1	32	yes	maximal, direct product factor
2	C₂×C₃₁×#557	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×C₃₁×#554	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×C₃₁×#554	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×C₃₁×#553	C₂	1	2	yes	maximal, complemented
6	C₂×C₃₁×#552	C₂	1	2	yes	maximal, complemented
7	C₂×C₃₁×#552	C₂	1	4	yes	maximal, complemented
8	C₂×C₃₁×#552	C₂	1	2	yes	maximal, complemented
9	C₂×C₃₁×#547	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂×C₃₁×#547	C₂	1	2	yes	maximal, complemented
11	C₂×C₃₁×#541	C₂	1	2	yes	maximal, complemented
12	C₂×C₃₁×#539	C₂	1	4	yes	maximal, complemented
13	C₂×C₃₁×#533	C₂	1	1	yes	maximal, characteristic, complemented
14	C₂×C₃₁×#529	C₂	1	2	yes	maximal, complemented
15	C₂×C₃₁×#529	C₂	1	1	yes	maximal, characteristic, complemented
16	C₂²×C₃₁×#125	C₂	1	2	yes	maximal, complemented
17	C₂×C₃₁×#520	C₂	1	2	yes	maximal, complemented
18	C₂×C₄×#41×C₃₁	C₂	1	1	yes	maximal, characteristic, complemented
19	C₂²×#41×C₃₁	C₂²	1	1	yes	characteristic
20	C₂²×#41×C₃₁	C₂²	1	2	yes	—
21	C₃₁×#557	C₂²	1	16	yes	complemented
22	C₃₁×#554	C₂²	1	16	yes	complemented
23	C₃₁×#554	C₂²	1	16	yes	complemented
24	C₃₁×#553	C₂²	1	32	yes	complemented
25	C₃₁×#552	C₂²	1	32	yes	complemented
26	C₃₁×#552	C₂²	1	64	yes	complemented
27	C₃₁×#552	C₂²	1	32	yes	complemented
28	C₃₁×#547	C₂²	1	16	yes	complemented
29	C₃₁×#547	C₂²	1	32	yes	complemented
30	C₃₁×#541	C₂²	1	32	yes	complemented
31	C₃₁×#539	C₂²	1	64	yes	complemented
32	C₃₁×#533	C₂²	1	16	yes	complemented
33	C₂×C₃₁×C₄⋊Q₈	C₂²	1	1	yes	characteristic
34	C₂×C₃₁×C₄⋊Q₈	C₂²	1	1	yes	characteristic
35	C₂×C₃₁×C₄⋊Q₈	C₂²	1	2	yes	—
36	C₃₁×#529	C₂²	1	32	yes	complemented
37	C₃₁×#529	C₂²	1	16	yes	complemented
38	C₂×C₃₁×#126	C₂²	1	4	yes	—
39	C₂×C₃₁×#126	C₂²	1	2	yes	—
40	C₂×C₃₁×#126	C₂²	1	4	yes	—
41	C₂×C₃₁×#126	C₂²	1	4	yes	—
42	C₂×C₃₁×#125	C₂²	1	1	yes	characteristic
43	C₂×C₃₁×#125	C₂²	1	2	yes	—
44	C₂×C₃₁×#125	C₂²	1	4	yes	—
45	C₂×C₃₁×#125	C₂²	1	4	yes	—
46	C₂×C₃₁×#125	C₂²	1	2	yes	—
47	C₂×C₃₁×#125	C₂²	1	2	yes	—
48	C₂×C₃₁×#125	C₂²	1	4	yes	—
49	C₂×C₃₁×#125	C₂²	1	4	yes	—
50	C₂×C₃₁×#125	C₂²	1	2	yes	—
51	C₂×C₃₁×#125	C₂²	1	32	yes	complemented
52	C₂×C₃₁×#124	C₂²	1	1	yes	characteristic
53	C₂×C₃₁×#123	C₂²	1	2	yes	—
54	C₂×C₃₁×#123	C₂²	1	4	yes	—
55	C₂×C₃₁×#123	C₂²	1	4	yes	—
56	C₂×C₃₁×#123	C₂²	1	4	yes	—
57	C₂×C₃₁×#123	C₂²	1	4	yes	—
58	C₂×C₃₁×C₂²⋊Q₈	C₂²	1	1	yes	characteristic
59	C₂×C₃₁×C₂²⋊Q₈	C₂²	1	2	yes	—
60	C₂×C₃₁×C₂²⋊Q₈	C₂²	1	2	yes	—
61	C₂×C₃₁×C₂²⋊Q₈	C₂²	1	2	yes	—
62	C₂×C₃₁×C₂²⋊Q₈	C₂²	1	4	yes	—
63	C₂×C₃₁×C₂²⋊Q₈	C₂²	1	4	yes	—
64	C₂×C₃₁×C₂²⋊Q₈	C₂²	1	2	yes	—
65	C₂×C₃₁×C₂²⋊Q₈	C₂²	1	2	yes	—
66	C₂×C₃₁×#121	C₂²	1	2	yes	—
67	C₂×C₃₁×#121	C₂²	1	1	yes	characteristic
68	C₂×C₃₁×#121	C₂²	1	4	yes	—
69	C₃₁×#520	C₂²	1	32	yes	complemented
70	C₄×#41×C₃₁	C₂²	1	16	yes	complemented
71	C₂×C₄×Q₈×C₃₁	C₂²	1	1	yes	characteristic
72	C₂×C₄×Q₈×C₃₁	C₂²	1	1	yes	characteristic
73	C₂×C₄×Q₈×C₃₁	C₂²	1	2	yes	—
74	C₂×C₄×Q₈×C₃₁	C₂²	1	2	yes	—
75	C₂×C₄×Q₈×C₃₁	C₂²	1	2	yes	—
76	C₂×C₄×Q₈×C₃₁	C₂²	1	1	yes	characteristic
77	C₂×C₄×Q₈×C₃₁	C₂²	1	2	yes	—
78	C₂×C₄×D₄×C₃₁	C₂²	1	1	yes	characteristic
79	C₂×C₄×D₄×C₃₁	C₂²	1	2	yes	—
80	C₂×C₄×D₄×C₃₁	C₂²	1	1	yes	characteristic
81	C₂×C₄×D₄×C₃₁	C₂²	1	2	yes	—
82	C₂×C₄×D₄×C₃₁	C₂²	1	4	yes	—
83	C₂×C₄×D₄×C₃₁	C₂²	1	2	yes	—
84	C₂×C₄×D₄×C₃₁	C₂²	1	4	yes	—
85	C₂×C₄×D₄×C₃₁	C₂²	1	2	yes	—
86	C₂×C₄×D₄×C₃₁	C₂²	1	1	yes	characteristic
87	C₂×C₄×D₄×C₃₁	C₂²	1	2	yes	—
88	C₂×C₃₁×#117	C₂²	1	1	yes	characteristic
89	C₂×C₃₁×#117	C₂²	1	2	yes	—
90	C₂×C₃₁×#117	C₂²	1	2	yes	—
91	C₂×C₃₁×#117	C₂²	1	4	yes	—
92	C₂×C₃₁×#117	C₂²	1	4	yes	—
93	C₂×C₃₁×#117	C₂²	1	2	yes	—
94	C₂²×C₄⋊C₄×C₃₁	C₂²	1	4	yes	—
95	C₂²×C₄⋊C₄×C₃₁	C₂²	1	2	yes	—
96	C₂²×C₄⋊C₄×C₃₁	C₂²	1	2	yes	—
97	C₂²×C₄⋊C₄×C₃₁	C₂²	1	4	yes	—
98	C₂²×C₄⋊C₄×C₃₁	C₂²	1	2	yes	—
99	C₂²×C₄²×C₃₁	C₂²	1	2	yes	—
100	C₂²×C₄²×C₃₁	C₂²	1	1	yes	characteristic
101	C₂×#41×C₃₁	—	2	8	no	—
102	C₂×#41×C₃₁	—	2	16	no	—
103	C₂×#41×C₃₁	C₂³	1	8	yes	—
104	C₂×#41×C₃₁	C₂³	1	16	yes	—
105	C₂²×Q₈×C₃₁	C₂³	1	1	yes	characteristic
106	C₂²×Q₈×C₃₁	C₂³	1	1	yes	characteristic
107	C₂²×Q₈×C₃₁	C₂³	1	2	yes	—
108	C₂²×Q₈×C₃₁	C₂³	1	2	yes	—
109	C₂²×D₄×C₃₁	C₂³	1	1	yes	characteristic
110	C₂²×D₄×C₃₁	C₂³	1	2	yes	—
111	C₂²×D₄×C₃₁	C₂³	1	4	yes	—
112	C₂²×D₄×C₃₁	C₂³	1	2	yes	—
113	C₂²×D₄×C₃₁	C₂³	1	1	yes	characteristic
114	C₂³×C₄×C₃₁	C₂³	1	2	yes	—
115	C₂³×C₄×C₃₁	C₂³	1	2	yes	—
116	C₂³×C₄×C₃₁	C₂³	1	1	yes	characteristic
117	C₂³×C₄×C₃₁	C₂³	1	1	yes	characteristic
118	C₂³×C₄×C₃₁	C₂³	1	4	yes	—
119	C₂³×C₄×C₃₁	C₂³	1	2	yes	—
120	C₂³×C₄×C₃₁	C₂³	1	2	yes	—
121	C₂³×C₄×C₃₁	C₂³	1	1	yes	characteristic
122	C₂³×C₄×C₃₁	C₂³	1	4	yes	—
123	C₂³×C₄×C₃₁	C₂³	1	4	yes	—
124	C₃₁×C₄⋊Q₈	C₂³	1	8	yes	—
125	C₃₁×C₄⋊Q₈	C₂³	1	8	yes	—
126	C₃₁×C₄⋊Q₈	C₂³	1	16	yes	—
127	C₃₁×#126	C₂³	1	32	yes	—
128	C₃₁×#126	C₂³	1	16	yes	—
129	C₃₁×#126	C₂³	1	32	yes	—
130	C₃₁×#126	C₂³	1	32	yes	—
131	C₃₁×#125	C₂³	1	8	yes	—
132	C₃₁×#125	C₂³	1	16	yes	—
133	C₃₁×#125	C₂³	1	32	yes	—
134	C₃₁×#125	C₂³	1	32	yes	—
135	C₃₁×#125	C₂³	1	16	yes	—
136	C₃₁×#125	C₂³	1	16	yes	—
137	C₃₁×#125	C₂³	1	32	yes	—
138	C₃₁×#125	C₂³	1	32	yes	—
139	C₃₁×#125	C₂³	1	16	yes	—
140	C₃₁×#124	C₂³	1	8	yes	—
141	C₃₁×#123	C₂³	1	16	yes	—
142	C₃₁×#123	C₂³	1	32	yes	—
143	C₃₁×#123	C₂³	1	32	yes	—
144	C₃₁×#123	C₂³	1	32	yes	—
145	C₃₁×#123	C₂³	1	32	yes	—
146	C₃₁×C₂²⋊Q₈	C₂³	1	8	yes	—
147	C₃₁×C₂²⋊Q₈	C₂³	1	16	yes	—
148	C₃₁×C₂²⋊Q₈	C₂³	1	16	yes	—
149	C₃₁×C₂²⋊Q₈	C₂³	1	16	yes	—
150	C₃₁×C₂²⋊Q₈	C₂³	1	32	yes	—
151	C₃₁×C₂²⋊Q₈	C₂³	1	32	yes	—
152	C₃₁×C₂²⋊Q₈	C₂³	1	16	yes	—
153	C₃₁×C₂²⋊Q₈	C₂³	1	16	yes	—
154	C₃₁×#121	C₂³	1	16	yes	—
155	C₃₁×#121	C₂³	1	8	yes	—
156	C₃₁×#121	C₂³	1	32	yes	—
157	C₄×Q₈×C₃₁	C₂³	1	8	yes	—
158	C₄×Q₈×C₃₁	C₂³	1	8	yes	—
159	C₄×Q₈×C₃₁	C₂³	1	16	yes	—
160	C₄×Q₈×C₃₁	C₂³	1	16	yes	—
161	C₄×Q₈×C₃₁	C₂³	1	16	yes	—
162	C₄×Q₈×C₃₁	C₂³	1	8	yes	—
163	C₄×Q₈×C₃₁	C₂³	1	16	yes	—
164	C₄×D₄×C₃₁	C₂³	1	8	yes	—
165	C₄×D₄×C₃₁	C₂³	1	16	yes	—
166	C₄×D₄×C₃₁	C₂³	1	8	yes	—
167	C₄×D₄×C₃₁	C₂³	1	16	yes	—
168	C₄×D₄×C₃₁	C₂³	1	32	yes	—
169	C₄×D₄×C₃₁	C₂³	1	16	yes	—
170	C₄×D₄×C₃₁	C₂³	1	32	yes	—
171	C₄×D₄×C₃₁	C₂³	1	16	yes	—
172	C₄×D₄×C₃₁	C₂³	1	8	yes	—
173	C₄×D₄×C₃₁	C₂³	1	16	yes	—
174	C₃₁×#117	C₂³	1	8	yes	—
175	C₃₁×#117	C₂³	1	16	yes	—
176	C₃₁×#117	C₂³	1	16	yes	—
177	C₃₁×#117	C₂³	1	32	yes	—
178	C₃₁×#117	C₂³	1	32	yes	—
179	C₃₁×#117	C₂³	1	16	yes	—
180	C₂×C₄⋊C₄×C₃₁	C₂³	1	1	yes	characteristic
181	C₂×C₄⋊C₄×C₃₁	C₂³	1	4	yes	—
182	C₂×C₄⋊C₄×C₃₁	C₂³	1	2	yes	—
183	C₂×C₄⋊C₄×C₃₁	C₂³	1	1	yes	characteristic
184	C₂×C₄⋊C₄×C₃₁	C₂³	1	2	yes	—
185	C₂×C₄⋊C₄×C₃₁	C₂³	1	1	yes	characteristic
186	C₂×C₄⋊C₄×C₃₁	C₂³	1	2	yes	—
187	C₂×C₄⋊C₄×C₃₁	C₂³	1	2	yes	—
188	C₂×C₄⋊C₄×C₃₁	C₂³	1	4	yes	—
189	C₂×C₄⋊C₄×C₃₁	C₂³	1	2	yes	—
190	C₂×C₄⋊C₄×C₃₁	C₂³	1	4	yes	—
191	C₂×C₄⋊C₄×C₃₁	C₂³	1	4	yes	—
192	C₂×C₄⋊C₄×C₃₁	C₂³	1	32	yes	—
193	C₂×C₄⋊C₄×C₃₁	C₂³	1	16	yes	—
194	C₂×C₄⋊C₄×C₃₁	C₂³	1	4	yes	—
195	C₂×C₄⋊C₄×C₃₁	C₂³	1	4	yes	—
196	C₂×C₄⋊C₄×C₃₁	C₂³	1	4	yes	—
197	C₂×C₄⋊C₄×C₃₁	C₂³	1	4	yes	—
198	C₂×C₄⋊C₄×C₃₁	C₂³	1	16	yes	—
199	C₂×C₄⋊C₄×C₃₁	C₂³	1	32	yes	—
200	C₂×C₄⋊C₄×C₃₁	C₂³	1	16	yes	—
201	C₂×C₄⋊C₄×C₃₁	C₂³	1	2	yes	—
202	C₂×C₄⋊C₄×C₃₁	C₂³	1	2	yes	—
203	C₂×C₄⋊C₄×C₃₁	C₂³	1	1	yes	characteristic
204	C₂×C₄⋊C₄×C₃₁	C₂³	1	2	yes	—
205	C₂×C₄⋊C₄×C₃₁	C₂³	1	4	yes	—
206	C₂×C₄⋊C₄×C₃₁	C₂³	1	2	yes	—
207	C₂×C₄⋊C₄×C₃₁	C₂³	1	2	yes	—
208	C₂×C₄⋊C₄×C₃₁	C₂³	1	4	yes	—
209	C₂×C₂²⋊C₄×C₃₁	C₂³	1	2	yes	—
210	C₂×C₂²⋊C₄×C₃₁	C₂³	1	2	yes	—
211	C₂×C₂²⋊C₄×C₃₁	C₂³	1	2	yes	—
212	C₂×C₂²⋊C₄×C₃₁	C₂³	1	4	yes	—
213	C₂×C₂²⋊C₄×C₃₁	C₂³	1	2	yes	—
214	C₂×C₂²⋊C₄×C₃₁	C₂³	1	2	yes	—
215	C₂×C₂²⋊C₄×C₃₁	C₂³	1	2	yes	—
216	C₂×C₂²⋊C₄×C₃₁	C₂³	1	4	yes	—
217	C₂×C₂²⋊C₄×C₃₁	C₂³	1	4	yes	—
218	C₂×C₂²⋊C₄×C₃₁	C₂³	1	4	yes	—
219	C₂×C₂²⋊C₄×C₃₁	C₂³	1	2	yes	—
220	C₂×C₂²⋊C₄×C₃₁	C₂³	1	2	yes	—
221	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
222	C₂×C₄²×C₃₁	C₂³	1	2	yes	—
223	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
224	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
225	C₂×C₄²×C₃₁	C₂³	1	2	yes	—
226	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
227	C₂×C₄²×C₃₁	C₂³	1	2	yes	—
228	C₂×C₄²×C₃₁	C₂³	1	16	yes	—
229	C₂×C₄²×C₃₁	C₂³	1	8	yes	—
230	C₂×C₄²×C₃₁	C₂³	1	4	yes	—
231	C₂×C₄²×C₃₁	C₂³	1	4	yes	—
232	C₂×C₄²×C₃₁	C₂³	1	2	yes	—
233	C₂⁴×C₃₁	C₂⁴	1	1	yes	characteristic
234	C₂⁴×C₃₁	C₂⁴	1	2	yes	—
235	C₂⁴×C₃₁	C₂⁴	1	2	yes	—
236	#41×C₃₁	—	2	32	no	—
237	#41×C₃₁	—	2	64	no	—
238	C₂×Q₈×C₃₁	—	2	4	no	—
239	C₂×Q₈×C₃₁	—	2	8	no	—
240	C₂×Q₈×C₃₁	—	2	4	no	—
241	C₂×Q₈×C₃₁	—	4	4	no	—
242	C₂×Q₈×C₃₁	C₂⁴	1	4	yes	—
243	C₂×Q₈×C₃₁	C₂⁴	1	4	yes	—
244	C₂×Q₈×C₃₁	C₂⁴	1	8	yes	—
245	C₂×Q₈×C₃₁	C₂⁴	1	8	yes	—
246	C₂×D₄×C₃₁	—	2	8	no	—
247	C₂×D₄×C₃₁	—	2	16	no	—
248	C₂×D₄×C₃₁	—	2	4	no	—
249	C₂×D₄×C₃₁	—	4	4	no	—
250	C₂×D₄×C₃₁	#41	1	8	yes	—
251	C₂×D₄×C₃₁	C₂⁴	1	4	yes	—
252	C₂×D₄×C₃₁	C₂⁴	1	8	yes	—
253	C₂×D₄×C₃₁	C₂⁴	1	16	yes	—
254	C₂×D₄×C₃₁	C₂⁴	1	8	yes	—
255	C₂×D₄×C₃₁	C₂⁴	1	4	yes	—
256	C₂²×C₄×C₃₁	—	2	8	no	—
257	C₂²×C₄×C₃₁	—	2	16	no	—
258	C₂²×C₄×C₃₁	—	2	4	no	—
259	C₂²×C₄×C₃₁	—	2	8	no	—
260	C₂²×C₄×C₃₁	—	2	8	no	—
261	C₂²×C₄×C₃₁	—	2	4	no	—
262	C₂²×C₄×C₃₁	—	4	4	no	—
263	C₂²×C₄×C₃₁	—	4	8	no	—
264	C₂²×C₄×C₃₁	—	4	2	no	—
265	C₂²×C₄×C₃₁	—	4	8	no	—
266	C₂²×C₄×C₃₁	C₂⁴	1	8	yes	—
267	C₂²×C₄×C₃₁	C₂⁴	1	8	yes	—
268	C₂²×C₄×C₃₁	C₂⁴	1	4	yes	—
269	C₂²×C₄×C₃₁	C₂⁴	1	4	yes	—
270	C₂²×C₄×C₃₁	C₂⁴	1	16	yes	—
271	C₂²×C₄×C₃₁	C₂⁴	1	8	yes	—
272	C₂²×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
273	C₂²×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
274	C₂²×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
275	C₂²×C₄×C₃₁	C₂⁴	1	2	yes	—
276	C₂²×C₄×C₃₁	C₂⁴	1	2	yes	—
277	C₂²×C₄×C₃₁	C₂⁴	1	2	yes	—
278	C₂²×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
279	C₂²×C₄×C₃₁	C₂⁴	1	2	yes	—
280	C₂²×C₄×C₃₁	C₂⁴	1	4	yes	—
281	C₂²×C₄×C₃₁	C₂⁴	1	4	yes	—
282	C₂²×C₄×C₃₁	C₂⁴	1	2	yes	—
283	C₂²×C₄×C₃₁	C₂⁴	1	2	yes	—
284	C₂²×C₄×C₃₁	C₂⁴	1	8	yes	—
285	C₂²×C₄×C₃₁	C₂⁴	1	4	yes	—
286	C₂²×C₄×C₃₁	C₂⁴	1	16	yes	—
287	C₂²×C₄×C₃₁	C₂⁴	1	16	yes	—
288	C₂²×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
289	C₂²×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
290	C₄⋊C₄×C₃₁	C₂⁴	1	4	yes	—
291	C₄⋊C₄×C₃₁	C₂⁴	1	16	yes	—
292	C₄⋊C₄×C₃₁	C₂⁴	1	8	yes	—
293	C₄⋊C₄×C₃₁	C₂⁴	1	4	yes	—
294	C₄⋊C₄×C₃₁	C₂⁴	1	8	yes	—
295	C₄⋊C₄×C₃₁	C₂⁴	1	4	yes	—
296	C₄⋊C₄×C₃₁	C₂⁴	1	8	yes	—
297	C₄⋊C₄×C₃₁	C₂⁴	1	8	yes	—
298	C₄⋊C₄×C₃₁	C₂⁴	1	16	yes	—
299	C₄⋊C₄×C₃₁	C₂⁴	1	8	yes	—
300	C₄⋊C₄×C₃₁	C₂⁴	1	16	yes	—
301	C₄⋊C₄×C₃₁	C₂⁴	1	16	yes	—
302	C₄⋊C₄×C₃₁	C₂⁴	1	16	yes	—
303	C₄⋊C₄×C₃₁	C₂⁴	1	16	yes	—
304	C₄⋊C₄×C₃₁	C₂⁴	1	16	yes	—
305	C₄⋊C₄×C₃₁	C₂⁴	1	16	yes	—
306	C₄⋊C₄×C₃₁	C₂⁴	1	8	yes	—
307	C₄⋊C₄×C₃₁	C₂⁴	1	8	yes	—
308	C₄⋊C₄×C₃₁	C₂⁴	1	4	yes	—
309	C₄⋊C₄×C₃₁	C₂⁴	1	8	yes	—
310	C₄⋊C₄×C₃₁	C₂⁴	1	16	yes	—
311	C₄⋊C₄×C₃₁	C₂⁴	1	8	yes	—
312	C₄⋊C₄×C₃₁	C₂⁴	1	8	yes	—
313	C₄⋊C₄×C₃₁	C₂⁴	1	16	yes	—
314	C₂²⋊C₄×C₃₁	C₂⁴	1	8	yes	—
315	C₂²⋊C₄×C₃₁	C₂⁴	1	8	yes	—
316	C₂²⋊C₄×C₃₁	C₂⁴	1	8	yes	—
317	C₂²⋊C₄×C₃₁	C₂⁴	1	16	yes	—
318	C₂²⋊C₄×C₃₁	C₂⁴	1	8	yes	—
319	C₂²⋊C₄×C₃₁	C₂⁴	1	8	yes	—
320	C₂²⋊C₄×C₃₁	C₂⁴	1	8	yes	—
321	C₂²⋊C₄×C₃₁	C₂⁴	1	16	yes	—
322	C₂²⋊C₄×C₃₁	C₂⁴	1	16	yes	—
323	C₂²⋊C₄×C₃₁	C₂⁴	1	16	yes	—
324	C₂²⋊C₄×C₃₁	C₂⁴	1	8	yes	—
325	C₂²⋊C₄×C₃₁	C₂⁴	1	8	yes	—
326	C₄²×C₃₁	C₂⁴	1	4	yes	—
327	C₄²×C₃₁	C₂⁴	1	8	yes	—
328	C₄²×C₃₁	C₂⁴	1	4	yes	—
329	C₄²×C₃₁	C₂⁴	1	4	yes	—
330	C₄²×C₃₁	C₂⁴	1	8	yes	—
331	C₄²×C₃₁	C₂⁴	1	4	yes	—
332	C₄²×C₃₁	C₂⁴	1	8	yes	—
333	C₄²×C₃₁	C₂⁴	1	16	yes	—
334	C₄²×C₃₁	C₂⁴	1	16	yes	—
335	C₄²×C₃₁	C₂⁴	1	8	yes	—
336	C₂×#3515	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
337	C₂³×C₃₁	—	2	4	no	—
338	C₂³×C₃₁	—	2	4	no	—
339	C₂³×C₃₁	—	2	2	no	—
340	C₂³×C₃₁	—	2	4	no	—
341	C₂³×C₃₁	—	2	2	no	—
342	C₂³×C₃₁	—	2	4	no	—
343	C₂³×C₃₁	—	2	4	no	—
344	C₂³×C₃₁	—	2	2	no	—
345	C₂³×C₃₁	C₂×#41	1	8	yes	—
346	C₂³×C₃₁	C₂⁵	1	2	yes	—
347	C₂³×C₃₁	C₂⁵	1	4	yes	—
348	C₂³×C₃₁	C₂⁵	1	4	yes	—
349	C₂³×C₃₁	C₂⁵	1	1	yes	characteristic
350	Q₈×C₃₁	—	2	8	no	—
351	Q₈×C₃₁	—	2	16	no	—
352	Q₈×C₃₁	—	2	8	no	—
353	Q₈×C₃₁	—	4	8	no	—
354	D₄×C₃₁	—	2	16	no	—
355	D₄×C₃₁	—	2	32	no	—
356	D₄×C₃₁	—	2	8	no	—
357	D₄×C₃₁	—	4	8	no	—
358	D₄×C₃₁	C₂×#41	1	16	yes	—
359	C₂×C₄×C₃₁	—	2	2	no	—
360	C₂×C₄×C₃₁	—	2	4	no	—
361	C₂×C₄×C₃₁	—	2	4	no	—
362	C₂×C₄×C₃₁	—	2	8	no	—
363	C₂×C₄×C₃₁	—	2	16	no	—
364	C₂×C₄×C₃₁	—	2	32	no	—
365	C₂×C₄×C₃₁	—	2	2	no	—
366	C₂×C₄×C₃₁	—	2	4	no	—
367	C₂×C₄×C₃₁	—	2	4	no	—
368	C₂×C₄×C₃₁	—	2	2	no	—
369	C₂×C₄×C₃₁	—	2	4	no	—
370	C₂×C₄×C₃₁	—	2	4	no	—
371	C₂×C₄×C₃₁	—	2	8	no	—
372	C₂×C₄×C₃₁	—	2	2	no	—
373	C₂×C₄×C₃₁	—	2	8	no	—
374	C₂×C₄×C₃₁	—	2	16	no	—
375	C₂×C₄×C₃₁	—	2	16	no	—
376	C₂×C₄×C₃₁	—	2	8	no	—
377	C₂×C₄×C₃₁	—	4	8	no	—
378	C₂×C₄×C₃₁	—	4	16	no	—
379	C₂×C₄×C₃₁	—	4	4	no	—
380	C₂×C₄×C₃₁	—	4	16	no	—
381	C₂×C₄×C₃₁	C₂×#41	1	4	yes	—
382	C₂×C₄×C₃₁	2₋¹⁺⁴	1	4	yes	—
383	C₂×C₄×C₃₁	C₂⁵	1	2	yes	—
384	C₂×C₄×C₃₁	C₂⁵	1	2	yes	—
385	C₂×C₄×C₃₁	C₂⁵	1	2	yes	—
386	C₂×C₄×C₃₁	C₂⁵	1	4	yes	—
387	C₂×C₄×C₃₁	C₂⁵	1	4	yes	—
388	C₂×C₄×C₃₁	C₂⁵	1	4	yes	—
389	C₂×C₄×C₃₁	C₂⁵	1	2	yes	—
390	C₂×C₄×C₃₁	C₂⁵	1	4	yes	—
391	C₂×C₄×C₃₁	C₂⁵	1	8	yes	—
392	C₂×C₄×C₃₁	C₂⁵	1	8	yes	—
393	C₂×C₄×C₃₁	C₂⁵	1	4	yes	—
394	C₂×C₄×C₃₁	C₂⁵	1	4	yes	—
395	C₂×C₄×C₃₁	C₂⁵	1	2	yes	—
396	C₂×C₄×C₃₁	C₂⁵	1	2	yes	—
397	#3515	C₂×C₃₁	1	32	yes	direct product factor
398	C₂×#557	C₂×C₃₁	1	1	yes	characteristic, complemented
399	C₂×#554	C₂×C₃₁	1	1	yes	characteristic, complemented
400	C₂×#554	C₂×C₃₁	1	1	yes	characteristic, complemented
401	C₂×#553	C₂×C₃₁	1	2	yes	complemented
402	C₂×#552	C₂×C₃₁	1	2	yes	complemented
403	C₂×#552	C₂×C₃₁	1	4	yes	complemented
404	C₂×#552	C₂×C₃₁	1	2	yes	complemented
405	C₂×#547	C₂×C₃₁	1	1	yes	characteristic, complemented
406	C₂×#547	C₂×C₃₁	1	2	yes	complemented
407	C₂×#541	C₂×C₃₁	1	2	yes	complemented
408	C₂×#539	C₂×C₃₁	1	4	yes	complemented
409	C₂×#533	C₂×C₃₁	1	1	yes	characteristic, complemented
410	C₂×#529	C₂×C₃₁	1	1	yes	characteristic, complemented
411	C₂×#529	C₂×C₃₁	1	2	yes	complemented
412	C₂²×#125	C₂×C₃₁	1	2	yes	complemented
413	C₂×#520	C₂×C₃₁	1	2	yes	complemented
414	C₂×C₄×#41	C₂×C₃₁	1	1	yes	characteristic, complemented
415	C₂²×C₃₁	—	2	4	no	—
416	C₂²×C₃₁	—	2	4	no	—
417	C₂²×C₃₁	—	2	16	no	—
418	C₂²×C₃₁	—	2	2	no	—
419	C₂²×C₃₁	—	2	4	no	—
420	C₂²×C₃₁	—	2	2	no	—
421	C₂²×C₃₁	—	2	4	no	—
422	C₂²×C₃₁	—	2	4	no	—
423	C₂²×C₃₁	—	2	2	no	—
424	C₂²×C₃₁	—	4	4	no	—
425	C₂²×C₃₁	—	4	8	no	—
426	C₂²×C₃₁	C₂²×#41	1	8	yes	—
427	C₂²×C₃₁	C₂²×#41	1	2	yes	—
428	C₂²×C₃₁	C₂×2₋¹⁺⁴	1	2	yes	—
429	C₂²×C₃₁	#585	1	2	yes	—
430	C₂²×C₃₁	C₂⁶	1	1	yes	characteristic
431	C₄×C₃₁	—	2	2	no	—
432	C₄×C₃₁	—	2	4	no	—
433	C₄×C₃₁	—	2	4	no	—
434	C₄×C₃₁	—	2	8	no	—
435	C₄×C₃₁	—	2	2	no	—
436	C₄×C₃₁	—	2	4	no	—
437	C₄×C₃₁	—	2	4	no	—
438	C₄×C₃₁	—	2	2	no	—
439	C₄×C₃₁	—	2	4	no	—
440	C₄×C₃₁	—	2	4	no	—
441	C₄×C₃₁	—	2	8	no	—
442	C₄×C₃₁	—	2	2	no	—
443	C₄×C₃₁	C₂²×#41	1	4	yes	—
444	C₄×C₃₁	C₂×2₋¹⁺⁴	1	4	yes	—
445	C₂²×#41	C₂²×C₃₁	1	1	yes	characteristic
446	C₂²×#41	C₂²×C₃₁	1	2	yes	—
447	#557	C₂²×C₃₁	1	16	yes	complemented
448	#554	C₂²×C₃₁	1	16	yes	complemented
449	#554	C₂²×C₃₁	1	16	yes	complemented
450	#553	C₂²×C₃₁	1	32	yes	complemented
451	#552	C₂²×C₃₁	1	32	yes	complemented
452	#552	C₂²×C₃₁	1	64	yes	complemented
453	#552	C₂²×C₃₁	1	32	yes	complemented
454	#547	C₂²×C₃₁	1	16	yes	complemented
455	#547	C₂²×C₃₁	1	32	yes	complemented
456	#541	C₂²×C₃₁	1	32	yes	complemented
457	#539	C₂²×C₃₁	1	64	yes	complemented
458	#533	C₂²×C₃₁	1	16	yes	complemented
459	C₂×C₄⋊Q₈	C₂²×C₃₁	1	1	yes	characteristic
460	C₂×C₄⋊Q₈	C₂²×C₃₁	1	1	yes	characteristic
461	C₂×C₄⋊Q₈	C₂²×C₃₁	1	2	yes	—
462	#529	C₂²×C₃₁	1	32	yes	complemented
463	#529	C₂²×C₃₁	1	16	yes	complemented
464	C₂×#126	C₂²×C₃₁	1	4	yes	—
465	C₂×#126	C₂²×C₃₁	1	2	yes	—
466	C₂×#126	C₂²×C₃₁	1	4	yes	—
467	C₂×#126	C₂²×C₃₁	1	4	yes	—
468	C₂×#125	C₂²×C₃₁	1	1	yes	characteristic
469	C₂×#125	C₂²×C₃₁	1	2	yes	—
470	C₂×#125	C₂²×C₃₁	1	4	yes	—
471	C₂×#125	C₂²×C₃₁	1	4	yes	—
472	C₂×#125	C₂²×C₃₁	1	2	yes	—
473	C₂×#125	C₂²×C₃₁	1	2	yes	—
474	C₂×#125	C₂²×C₃₁	1	4	yes	—
475	C₂×#125	C₂²×C₃₁	1	4	yes	—
476	C₂×#125	C₂²×C₃₁	1	2	yes	—
477	C₂×#125	C₂²×C₃₁	1	32	yes	complemented
478	C₂×#124	C₂²×C₃₁	1	1	yes	characteristic
479	C₂×#123	C₂²×C₃₁	1	2	yes	—
480	C₂×#123	C₂²×C₃₁	1	4	yes	—
481	C₂×#123	C₂²×C₃₁	1	4	yes	—
482	C₂×#123	C₂²×C₃₁	1	4	yes	—
483	C₂×#123	C₂²×C₃₁	1	4	yes	—
484	C₂×C₂²⋊Q₈	C₂²×C₃₁	1	1	yes	characteristic
485	C₂×C₂²⋊Q₈	C₂²×C₃₁	1	2	yes	—
486	C₂×C₂²⋊Q₈	C₂²×C₃₁	1	2	yes	—
487	C₂×C₂²⋊Q₈	C₂²×C₃₁	1	2	yes	—
488	C₂×C₂²⋊Q₈	C₂²×C₃₁	1	4	yes	—
489	C₂×C₂²⋊Q₈	C₂²×C₃₁	1	4	yes	—
490	C₂×C₂²⋊Q₈	C₂²×C₃₁	1	2	yes	—
491	C₂×C₂²⋊Q₈	C₂²×C₃₁	1	2	yes	—
492	C₂×#121	C₂²×C₃₁	1	2	yes	—
493	C₂×#121	C₂²×C₃₁	1	1	yes	characteristic
494	C₂×#121	C₂²×C₃₁	1	4	yes	—
495	#520	C₂²×C₃₁	1	32	yes	complemented
496	C₄×#41	C₂²×C₃₁	1	16	yes	complemented
497	C₂×C₄×Q₈	C₂²×C₃₁	1	1	yes	characteristic
498	C₂×C₄×Q₈	C₂²×C₃₁	1	1	yes	characteristic
499	C₂×C₄×Q₈	C₂²×C₃₁	1	2	yes	—
500	C₂×C₄×Q₈	C₂²×C₃₁	1	2	yes	—
501	C₂×C₄×Q₈	C₂²×C₃₁	1	2	yes	—
502	C₂×C₄×Q₈	C₂²×C₃₁	1	1	yes	characteristic
503	C₂×C₄×Q₈	C₂²×C₃₁	1	2	yes	—
504	C₂×C₄×D₄	C₂²×C₃₁	1	1	yes	characteristic
505	C₂×C₄×D₄	C₂²×C₃₁	1	2	yes	—
506	C₂×C₄×D₄	C₂²×C₃₁	1	1	yes	characteristic
507	C₂×C₄×D₄	C₂²×C₃₁	1	2	yes	—
508	C₂×C₄×D₄	C₂²×C₃₁	1	4	yes	—
509	C₂×C₄×D₄	C₂²×C₃₁	1	2	yes	—
510	C₂×C₄×D₄	C₂²×C₃₁	1	4	yes	—
511	C₂×C₄×D₄	C₂²×C₃₁	1	2	yes	—
512	C₂×C₄×D₄	C₂²×C₃₁	1	1	yes	characteristic
513	C₂×C₄×D₄	C₂²×C₃₁	1	2	yes	—
514	C₂×#117	C₂²×C₃₁	1	1	yes	characteristic
515	C₂×#117	C₂²×C₃₁	1	2	yes	—
516	C₂×#117	C₂²×C₃₁	1	2	yes	—
517	C₂×#117	C₂²×C₃₁	1	4	yes	—
518	C₂×#117	C₂²×C₃₁	1	4	yes	—
519	C₂×#117	C₂²×C₃₁	1	2	yes	—
520	C₂²×C₄⋊C₄	C₂²×C₃₁	1	4	yes	—
521	C₂²×C₄⋊C₄	C₂²×C₃₁	1	2	yes	—
522	C₂²×C₄⋊C₄	C₂²×C₃₁	1	2	yes	—
523	C₂²×C₄⋊C₄	C₂²×C₃₁	1	4	yes	—
524	C₂²×C₄⋊C₄	C₂²×C₃₁	1	2	yes	—
525	C₂²×C₄²	C₂²×C₃₁	1	2	yes	—
526	C₂²×C₄²	C₂²×C₃₁	1	1	yes	characteristic
527	C₂×C₃₁	—	2	8	no	—
528	C₂×C₃₁	—	4	2	no	—
529	C₂×C₃₁	—	4	4	no	—
530	C₂×C₃₁	#3515	1	4	yes	direct product factor
531	C₂×C₃₁	C₂³×#41	1	1	yes	characteristic
532	C₂×C₃₁	C₂²×2₋¹⁺⁴	1	1	yes	characteristic
533	C₂×C₃₁	C₂×#585	1	1	yes	characteristic
534	C₂×#41	—	2	8	no	—
535	C₂×#41	—	2	16	no	—
536	C₂×#41	C₂³×C₃₁	1	8	yes	—
537	C₂×#41	C₂³×C₃₁	1	16	yes	—
538	C₂²×Q₈	C₂³×C₃₁	1	1	yes	characteristic
539	C₂²×Q₈	C₂³×C₃₁	1	1	yes	characteristic
540	C₂²×Q₈	C₂³×C₃₁	1	2	yes	—
541	C₂²×Q₈	C₂³×C₃₁	1	2	yes	—
542	C₂²×D₄	C₂³×C₃₁	1	1	yes	characteristic
543	C₂²×D₄	C₂³×C₃₁	1	2	yes	—
544	C₂²×D₄	C₂³×C₃₁	1	4	yes	—
545	C₂²×D₄	C₂³×C₃₁	1	2	yes	—
546	C₂²×D₄	C₂³×C₃₁	1	1	yes	characteristic
547	C₂³×C₄	C₂³×C₃₁	1	4	yes	—
548	C₂³×C₄	C₂³×C₃₁	1	2	yes	—
549	C₂³×C₄	C₂³×C₃₁	1	2	yes	—
550	C₂³×C₄	C₂³×C₃₁	1	2	yes	—
551	C₂³×C₄	C₂³×C₃₁	1	1	yes	characteristic
552	C₂³×C₄	C₂³×C₃₁	1	1	yes	characteristic
553	C₂³×C₄	C₂³×C₃₁	1	4	yes	—
554	C₂³×C₄	C₂³×C₃₁	1	2	yes	—
555	C₂³×C₄	C₂³×C₃₁	1	4	yes	—
556	C₂³×C₄	C₂³×C₃₁	1	1	yes	characteristic
557	C₄⋊Q₈	C₂³×C₃₁	1	8	yes	—
558	C₄⋊Q₈	C₂³×C₃₁	1	8	yes	—
559	C₄⋊Q₈	C₂³×C₃₁	1	16	yes	—
560	#126	C₂³×C₃₁	1	32	yes	—
561	#126	C₂³×C₃₁	1	16	yes	—
562	#126	C₂³×C₃₁	1	32	yes	—
563	#126	C₂³×C₃₁	1	32	yes	—
564	#125	C₂³×C₃₁	1	8	yes	—
565	#125	C₂³×C₃₁	1	16	yes	—
566	#125	C₂³×C₃₁	1	32	yes	—
567	#125	C₂³×C₃₁	1	32	yes	—
568	#125	C₂³×C₃₁	1	16	yes	—
569	#125	C₂³×C₃₁	1	16	yes	—
570	#125	C₂³×C₃₁	1	32	yes	—
571	#125	C₂³×C₃₁	1	32	yes	—
572	#125	C₂³×C₃₁	1	16	yes	—
573	#124	C₂³×C₃₁	1	8	yes	—
574	#123	C₂³×C₃₁	1	16	yes	—
575	#123	C₂³×C₃₁	1	32	yes	—
576	#123	C₂³×C₃₁	1	32	yes	—
577	#123	C₂³×C₃₁	1	32	yes	—
578	#123	C₂³×C₃₁	1	32	yes	—
579	C₂²⋊Q₈	C₂³×C₃₁	1	8	yes	—
580	C₂²⋊Q₈	C₂³×C₃₁	1	16	yes	—
581	C₂²⋊Q₈	C₂³×C₃₁	1	16	yes	—
582	C₂²⋊Q₈	C₂³×C₃₁	1	16	yes	—
583	C₂²⋊Q₈	C₂³×C₃₁	1	32	yes	—
584	C₂²⋊Q₈	C₂³×C₃₁	1	32	yes	—
585	C₂²⋊Q₈	C₂³×C₃₁	1	16	yes	—
586	C₂²⋊Q₈	C₂³×C₃₁	1	16	yes	—
587	#121	C₂³×C₃₁	1	16	yes	—
588	#121	C₂³×C₃₁	1	8	yes	—
589	#121	C₂³×C₃₁	1	32	yes	—
590	C₄×Q₈	C₂³×C₃₁	1	8	yes	—
591	C₄×Q₈	C₂³×C₃₁	1	8	yes	—
592	C₄×Q₈	C₂³×C₃₁	1	16	yes	—
593	C₄×Q₈	C₂³×C₃₁	1	16	yes	—
594	C₄×Q₈	C₂³×C₃₁	1	16	yes	—
595	C₄×Q₈	C₂³×C₃₁	1	8	yes	—
596	C₄×Q₈	C₂³×C₃₁	1	16	yes	—
597	C₄×D₄	C₂³×C₃₁	1	8	yes	—
598	C₄×D₄	C₂³×C₃₁	1	16	yes	—
599	C₄×D₄	C₂³×C₃₁	1	8	yes	—
600	C₄×D₄	C₂³×C₃₁	1	16	yes	—
601	C₄×D₄	C₂³×C₃₁	1	32	yes	—
602	C₄×D₄	C₂³×C₃₁	1	16	yes	—
603	C₄×D₄	C₂³×C₃₁	1	32	yes	—
604	C₄×D₄	C₂³×C₃₁	1	16	yes	—
605	C₄×D₄	C₂³×C₃₁	1	8	yes	—
606	C₄×D₄	C₂³×C₃₁	1	16	yes	—
607	#117	C₂³×C₃₁	1	8	yes	—
608	#117	C₂³×C₃₁	1	16	yes	—
609	#117	C₂³×C₃₁	1	16	yes	—
610	#117	C₂³×C₃₁	1	32	yes	—
611	#117	C₂³×C₃₁	1	32	yes	—
612	#117	C₂³×C₃₁	1	16	yes	—
613	C₂×C₄⋊C₄	C₂³×C₃₁	1	1	yes	characteristic
614	C₂×C₄⋊C₄	C₂³×C₃₁	1	4	yes	—
615	C₂×C₄⋊C₄	C₂³×C₃₁	1	2	yes	—
616	C₂×C₄⋊C₄	C₂³×C₃₁	1	1	yes	characteristic
617	C₂×C₄⋊C₄	C₂³×C₃₁	1	2	yes	—
618	C₂×C₄⋊C₄	C₂³×C₃₁	1	1	yes	characteristic
619	C₂×C₄⋊C₄	C₂³×C₃₁	1	2	yes	—
620	C₂×C₄⋊C₄	C₂³×C₃₁	1	2	yes	—
621	C₂×C₄⋊C₄	C₂³×C₃₁	1	4	yes	—
622	C₂×C₄⋊C₄	C₂³×C₃₁	1	2	yes	—
623	C₂×C₄⋊C₄	C₂³×C₃₁	1	4	yes	—
624	C₂×C₄⋊C₄	C₂³×C₃₁	1	4	yes	—
625	C₂×C₄⋊C₄	C₂³×C₃₁	1	32	yes	—
626	C₂×C₄⋊C₄	C₂³×C₃₁	1	16	yes	—
627	C₂×C₄⋊C₄	C₂³×C₃₁	1	4	yes	—
628	C₂×C₄⋊C₄	C₂³×C₃₁	1	4	yes	—
629	C₂×C₄⋊C₄	C₂³×C₃₁	1	4	yes	—
630	C₂×C₄⋊C₄	C₂³×C₃₁	1	4	yes	—
631	C₂×C₄⋊C₄	C₂³×C₃₁	1	16	yes	—
632	C₂×C₄⋊C₄	C₂³×C₃₁	1	32	yes	—
633	C₂×C₄⋊C₄	C₂³×C₃₁	1	16	yes	—
634	C₂×C₄⋊C₄	C₂³×C₃₁	1	2	yes	—
635	C₂×C₄⋊C₄	C₂³×C₃₁	1	2	yes	—
636	C₂×C₄⋊C₄	C₂³×C₃₁	1	1	yes	characteristic
637	C₂×C₄⋊C₄	C₂³×C₃₁	1	2	yes	—
638	C₂×C₄⋊C₄	C₂³×C₃₁	1	4	yes	—
639	C₂×C₄⋊C₄	C₂³×C₃₁	1	2	yes	—
640	C₂×C₄⋊C₄	C₂³×C₃₁	1	2	yes	—
641	C₂×C₄⋊C₄	C₂³×C₃₁	1	4	yes	—
642	C₂×C₂²⋊C₄	C₂³×C₃₁	1	2	yes	—
643	C₂×C₂²⋊C₄	C₂³×C₃₁	1	2	yes	—
644	C₂×C₂²⋊C₄	C₂³×C₃₁	1	2	yes	—
645	C₂×C₂²⋊C₄	C₂³×C₃₁	1	4	yes	—
646	C₂×C₂²⋊C₄	C₂³×C₃₁	1	2	yes	—
647	C₂×C₂²⋊C₄	C₂³×C₃₁	1	2	yes	—
648	C₂×C₂²⋊C₄	C₂³×C₃₁	1	2	yes	—
649	C₂×C₂²⋊C₄	C₂³×C₃₁	1	4	yes	—
650	C₂×C₂²⋊C₄	C₂³×C₃₁	1	4	yes	—
651	C₂×C₂²⋊C₄	C₂³×C₃₁	1	4	yes	—
652	C₂×C₂²⋊C₄	C₂³×C₃₁	1	2	yes	—
653	C₂×C₂²⋊C₄	C₂³×C₃₁	1	2	yes	—
654	C₂×C₄²	C₂³×C₃₁	1	2	yes	—
655	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
656	C₂×C₄²	C₂³×C₃₁	1	4	yes	—
657	C₂×C₄²	C₂³×C₃₁	1	4	yes	—
658	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
659	C₂×C₄²	C₂³×C₃₁	1	2	yes	—
660	C₂×C₄²	C₂³×C₃₁	1	2	yes	—
661	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
662	C₂×C₄²	C₂³×C₃₁	1	2	yes	—
663	C₂×C₄²	C₂³×C₃₁	1	16	yes	—
664	C₂×C₄²	C₂³×C₃₁	1	8	yes	—
665	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
666	C₃₁	C₂×#3515	1	1	yes	characteristic, direct product factor, Sylow
667	C₂⁴	C₂⁴×C₃₁	1	1	yes	characteristic
668	C₂⁴	C₂⁴×C₃₁	1	2	yes	—
669	C₂⁴	C₂⁴×C₃₁	1	2	yes	—
670	#41	—	2	32	no	—
671	#41	—	2	64	no	—
672	C₂×Q₈	—	2	4	no	—
673	C₂×Q₈	—	2	8	no	—
674	C₂×Q₈	—	2	4	no	—
675	C₂×Q₈	—	4	4	no	—
676	C₂×Q₈	C₂⁴×C₃₁	1	4	yes	—
677	C₂×Q₈	C₂⁴×C₃₁	1	4	yes	—
678	C₂×Q₈	C₂⁴×C₃₁	1	8	yes	—
679	C₂×Q₈	C₂⁴×C₃₁	1	8	yes	—
680	C₂×D₄	—	2	8	no	—
681	C₂×D₄	—	2	16	no	—
682	C₂×D₄	—	2	4	no	—
683	C₂×D₄	—	4	4	no	—
684	C₂×D₄	#41×C₃₁	1	8	yes	—
685	C₂×D₄	C₂⁴×C₃₁	1	4	yes	—
686	C₂×D₄	C₂⁴×C₃₁	1	8	yes	—
687	C₂×D₄	C₂⁴×C₃₁	1	16	yes	—
688	C₂×D₄	C₂⁴×C₃₁	1	8	yes	—
689	C₂×D₄	C₂⁴×C₃₁	1	4	yes	—
690	C₂²×C₄	—	2	16	no	—
691	C₂²×C₄	—	2	8	no	—
692	C₂²×C₄	—	2	8	no	—
693	C₂²×C₄	—	2	4	no	—
694	C₂²×C₄	—	2	8	no	—
695	C₂²×C₄	—	2	4	no	—
696	C₂²×C₄	—	4	4	no	—
697	C₂²×C₄	—	4	8	no	—
698	C₂²×C₄	—	4	2	no	—
699	C₂²×C₄	—	4	8	no	—
700	C₂²×C₄	C₂⁴×C₃₁	1	1	yes	characteristic
701	C₂²×C₄	C₂⁴×C₃₁	1	1	yes	characteristic
702	C₂²×C₄	C₂⁴×C₃₁	1	2	yes	—
703	C₂²×C₄	C₂⁴×C₃₁	1	1	yes	characteristic
704	C₂²×C₄	C₂⁴×C₃₁	1	2	yes	—
705	C₂²×C₄	C₂⁴×C₃₁	1	2	yes	—
706	C₂²×C₄	C₂⁴×C₃₁	1	1	yes	characteristic
707	C₂²×C₄	C₂⁴×C₃₁	1	4	yes	—
708	C₂²×C₄	C₂⁴×C₃₁	1	2	yes	—
709	C₂²×C₄	C₂⁴×C₃₁	1	8	yes	—
710	C₂²×C₄	C₂⁴×C₃₁	1	8	yes	—
711	C₂²×C₄	C₂⁴×C₃₁	1	4	yes	—
712	C₂²×C₄	C₂⁴×C₃₁	1	4	yes	—
713	C₂²×C₄	C₂⁴×C₃₁	1	4	yes	—
714	C₂²×C₄	C₂⁴×C₃₁	1	2	yes	—
715	C₂²×C₄	C₂⁴×C₃₁	1	2	yes	—
716	C₂²×C₄	C₂⁴×C₃₁	1	16	yes	—
717	C₂²×C₄	C₂⁴×C₃₁	1	8	yes	—
718	C₂²×C₄	C₂⁴×C₃₁	1	8	yes	—
719	C₂²×C₄	C₂⁴×C₃₁	1	4	yes	—
720	C₂²×C₄	C₂⁴×C₃₁	1	16	yes	—
721	C₂²×C₄	C₂⁴×C₃₁	1	16	yes	—
722	C₂²×C₄	C₂⁴×C₃₁	1	1	yes	characteristic
723	C₂²×C₄	C₂⁴×C₃₁	1	1	yes	characteristic
724	C₄⋊C₄	C₂⁴×C₃₁	1	4	yes	—
725	C₄⋊C₄	C₂⁴×C₃₁	1	16	yes	—
726	C₄⋊C₄	C₂⁴×C₃₁	1	8	yes	—
727	C₄⋊C₄	C₂⁴×C₃₁	1	4	yes	—
728	C₄⋊C₄	C₂⁴×C₃₁	1	8	yes	—
729	C₄⋊C₄	C₂⁴×C₃₁	1	4	yes	—
730	C₄⋊C₄	C₂⁴×C₃₁	1	8	yes	—
731	C₄⋊C₄	C₂⁴×C₃₁	1	8	yes	—
732	C₄⋊C₄	C₂⁴×C₃₁	1	16	yes	—
733	C₄⋊C₄	C₂⁴×C₃₁	1	8	yes	—
734	C₄⋊C₄	C₂⁴×C₃₁	1	16	yes	—
735	C₄⋊C₄	C₂⁴×C₃₁	1	16	yes	—
736	C₄⋊C₄	C₂⁴×C₃₁	1	16	yes	—
737	C₄⋊C₄	C₂⁴×C₃₁	1	16	yes	—
738	C₄⋊C₄	C₂⁴×C₃₁	1	16	yes	—
739	C₄⋊C₄	C₂⁴×C₃₁	1	16	yes	—
740	C₄⋊C₄	C₂⁴×C₃₁	1	8	yes	—
741	C₄⋊C₄	C₂⁴×C₃₁	1	8	yes	—
742	C₄⋊C₄	C₂⁴×C₃₁	1	4	yes	—
743	C₄⋊C₄	C₂⁴×C₃₁	1	8	yes	—
744	C₄⋊C₄	C₂⁴×C₃₁	1	16	yes	—
745	C₄⋊C₄	C₂⁴×C₃₁	1	8	yes	—
746	C₄⋊C₄	C₂⁴×C₃₁	1	8	yes	—
747	C₄⋊C₄	C₂⁴×C₃₁	1	16	yes	—
748	C₂²⋊C₄	C₂⁴×C₃₁	1	8	yes	—
749	C₂²⋊C₄	C₂⁴×C₃₁	1	8	yes	—
750	C₂²⋊C₄	C₂⁴×C₃₁	1	8	yes	—
751	C₂²⋊C₄	C₂⁴×C₃₁	1	16	yes	—
752	C₂²⋊C₄	C₂⁴×C₃₁	1	8	yes	—
753	C₂²⋊C₄	C₂⁴×C₃₁	1	8	yes	—
754	C₂²⋊C₄	C₂⁴×C₃₁	1	8	yes	—
755	C₂²⋊C₄	C₂⁴×C₃₁	1	16	yes	—
756	C₂²⋊C₄	C₂⁴×C₃₁	1	16	yes	—
757	C₂²⋊C₄	C₂⁴×C₃₁	1	16	yes	—
758	C₂²⋊C₄	C₂⁴×C₃₁	1	8	yes	—
759	C₂²⋊C₄	C₂⁴×C₃₁	1	8	yes	—
760	C₄²	C₂⁴×C₃₁	1	4	yes	—
761	C₄²	C₂⁴×C₃₁	1	8	yes	—
762	C₄²	C₂⁴×C₃₁	1	4	yes	—
763	C₄²	C₂⁴×C₃₁	1	4	yes	—
764	C₄²	C₂⁴×C₃₁	1	8	yes	—
765	C₄²	C₂⁴×C₃₁	1	4	yes	—
766	C₄²	C₂⁴×C₃₁	1	8	yes	—
767	C₄²	C₂⁴×C₃₁	1	16	yes	—
768	C₄²	C₂⁴×C₃₁	1	16	yes	—
769	C₄²	C₂⁴×C₃₁	1	8	yes	—
770	C₂³	—	2	4	no	—
771	C₂³	—	2	4	no	—
772	C₂³	—	2	2	no	—
773	C₂³	—	2	4	no	—
774	C₂³	—	2	2	no	—
775	C₂³	—	2	4	no	—
776	C₂³	—	2	4	no	—
777	C₂³	—	2	2	no	—
778	C₂³	C₂×#41×C₃₁	1	8	yes	—
779	C₂³	C₂⁵×C₃₁	1	2	yes	—
780	C₂³	C₂⁵×C₃₁	1	4	yes	—
781	C₂³	C₂⁵×C₃₁	1	4	yes	—
782	C₂³	C₂⁵×C₃₁	1	1	yes	characteristic
783	Q₈	—	2	8	no	—
784	Q₈	—	2	16	no	—
785	Q₈	—	2	8	no	—
786	Q₈	—	4	8	no	—
787	D₄	—	2	16	no	—
788	D₄	—	2	32	no	—
789	D₄	—	2	8	no	—
790	D₄	—	4	8	no	—
791	D₄	C₂×#41×C₃₁	1	16	yes	—
792	C₂×C₄	—	2	2	no	—
793	C₂×C₄	—	2	4	no	—
794	C₂×C₄	—	2	8	no	—
795	C₂×C₄	—	2	4	no	—
796	C₂×C₄	—	2	16	no	—
797	C₂×C₄	—	2	32	no	—
798	C₂×C₄	—	2	2	no	—
799	C₂×C₄	—	2	4	no	—
800	C₂×C₄	—	2	4	no	—
801	C₂×C₄	—	2	2	no	—
802	C₂×C₄	—	2	4	no	—
803	C₂×C₄	—	2	4	no	—
804	C₂×C₄	—	2	8	no	—
805	C₂×C₄	—	2	2	no	—
806	C₂×C₄	—	2	8	no	—
807	C₂×C₄	—	2	16	no	—
808	C₂×C₄	—	2	16	no	—
809	C₂×C₄	—	2	8	no	—
810	C₂×C₄	—	4	8	no	—
811	C₂×C₄	—	4	16	no	—
812	C₂×C₄	—	4	4	no	—
813	C₂×C₄	—	4	16	no	—
814	C₂×C₄	C₂×#41×C₃₁	1	4	yes	—
815	C₂×C₄	C₃₁×2₋¹⁺⁴	1	4	yes	—
816	C₂×C₄	C₂⁵×C₃₁	1	2	yes	—
817	C₂×C₄	C₂⁵×C₃₁	1	2	yes	—
818	C₂×C₄	C₂⁵×C₃₁	1	2	yes	—
819	C₂×C₄	C₂⁵×C₃₁	1	4	yes	—
820	C₂×C₄	C₂⁵×C₃₁	1	4	yes	—
821	C₂×C₄	C₂⁵×C₃₁	1	4	yes	—
822	C₂×C₄	C₂⁵×C₃₁	1	2	yes	—
823	C₂×C₄	C₂⁵×C₃₁	1	4	yes	—
824	C₂×C₄	C₂⁵×C₃₁	1	8	yes	—
825	C₂×C₄	C₂⁵×C₃₁	1	8	yes	—
826	C₂×C₄	C₂⁵×C₃₁	1	4	yes	—
827	C₂×C₄	C₂⁵×C₃₁	1	4	yes	—
828	C₂×C₄	C₂⁵×C₃₁	1	2	yes	—
829	C₂×C₄	C₂⁵×C₃₁	1	2	yes	—
830	C₂²	—	2	4	no	—
831	C₂²	—	2	4	no	—
832	C₂²	—	2	16	no	—
833	C₂²	—	2	2	no	—
834	C₂²	—	2	4	no	—
835	C₂²	—	2	2	no	—
836	C₂²	—	2	4	no	—
837	C₂²	—	2	4	no	—
838	C₂²	—	2	2	no	—
839	C₂²	—	4	4	no	—
840	C₂²	—	4	8	no	—
841	C₂²	C₂²×#41×C₃₁	1	8	yes	—
842	C₂²	C₂²×#41×C₃₁	1	2	yes	—
843	C₂²	C₂×C₃₁×2₋¹⁺⁴	1	2	yes	—
844	C₂²	C₃₁×#585	1	2	yes	—
845	C₂²	C₂⁶×C₃₁	1	1	yes	characteristic
846	C₄	—	2	2	no	—
847	C₄	—	2	4	no	—
848	C₄	—	2	4	no	—
849	C₄	—	2	8	no	—
850	C₄	—	2	2	no	—
851	C₄	—	2	4	no	—
852	C₄	—	2	4	no	—
853	C₄	—	2	2	no	—
854	C₄	—	2	4	no	—
855	C₄	—	2	4	no	—
856	C₄	—	2	8	no	—
857	C₄	—	2	2	no	—
858	C₄	C₂²×#41×C₃₁	1	4	yes	—
859	C₄	C₂×C₃₁×2₋¹⁺⁴	1	4	yes	—
860	C₂	—	2	8	no	—
861	C₂	—	4	2	no	—
862	C₂	—	4	4	no	—
863	C₂	C₃₁×#3515	1	4	yes	direct product factor
864	C₂	C₂³×#41×C₃₁	1	1	yes	characteristic
865	C₂	C₂²×C₃₁×2₋¹⁺⁴	1	1	yes	characteristic
866	C₂	C₂×C₃₁×#585	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂³×C₃₁	C₂⁵
commutator, Frattini, cosocle	C₂²	C₂⁶×C₃₁
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³×C₃₁ ◁ G

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

N	H ≅ G/N	type
1. C₃₁×#3515	C₂	direct (N × H)
2. C₂×C₃₁×#557	C₂	semidirect
3. C₂×C₃₁×#554	C₂	semidirect
4. C₂×C₃₁×#554	C₂	semidirect
5. C₂×C₃₁×#553	C₂	semidirect
6. C₂×C₃₁×#552	C₂	semidirect
7. C₂×C₃₁×#552	C₂	semidirect
8. C₂×C₃₁×#552	C₂	semidirect
9. C₂×C₃₁×#547	C₂	semidirect
10. C₂×C₃₁×#547	C₂	semidirect
11. C₂×C₃₁×#541	C₂	semidirect
12. C₂×C₃₁×#539	C₂	semidirect
13. C₂×C₃₁×#533	C₂	semidirect
14. C₂×C₃₁×#529	C₂	semidirect
15. C₂×C₃₁×#529	C₂	semidirect
16. C₂²×C₃₁×#125	C₂	semidirect
17. C₂×C₃₁×#520	C₂	semidirect
18. C₂×C₄×#41×C₃₁	C₂	semidirect
21. C₃₁×#557	C₂²	semidirect
22. C₃₁×#554	C₂²	semidirect
23. C₃₁×#554	C₂²	semidirect
24. C₃₁×#553	C₂²	semidirect
25. C₃₁×#552	C₂²	semidirect
26. C₃₁×#552	C₂²	semidirect
27. C₃₁×#552	C₂²	semidirect
28. C₃₁×#547	C₂²	semidirect
29. C₃₁×#547	C₂²	semidirect
30. C₃₁×#541	C₂²	semidirect
31. C₃₁×#539	C₂²	semidirect
32. C₃₁×#533	C₂²	semidirect
36. C₃₁×#529	C₂²	semidirect
37. C₃₁×#529	C₂²	semidirect
51. C₂×C₃₁×#125	C₂²	semidirect
69. C₃₁×#520	C₂²	semidirect
70. C₄×#41×C₃₁	C₂²	semidirect
336. C₂×#3515	C₃₁	direct (N × H)
397. #3515	C₂×C₃₁	direct (N × H)
398. C₂×#557	C₂×C₃₁	semidirect
399. C₂×#554	C₂×C₃₁	semidirect
400. C₂×#554	C₂×C₃₁	semidirect
401. C₂×#553	C₂×C₃₁	semidirect
402. C₂×#552	C₂×C₃₁	semidirect
403. C₂×#552	C₂×C₃₁	semidirect
404. C₂×#552	C₂×C₃₁	semidirect
405. C₂×#547	C₂×C₃₁	semidirect
406. C₂×#547	C₂×C₃₁	semidirect
407. C₂×#541	C₂×C₃₁	semidirect
408. C₂×#539	C₂×C₃₁	semidirect
409. C₂×#533	C₂×C₃₁	semidirect
410. C₂×#529	C₂×C₃₁	semidirect
411. C₂×#529	C₂×C₃₁	semidirect
412. C₂²×#125	C₂×C₃₁	semidirect
413. C₂×#520	C₂×C₃₁	semidirect
414. C₂×C₄×#41	C₂×C₃₁	semidirect
447. #557	C₂²×C₃₁	semidirect
448. #554	C₂²×C₃₁	semidirect
449. #554	C₂²×C₃₁	semidirect
450. #553	C₂²×C₃₁	semidirect
451. #552	C₂²×C₃₁	semidirect
452. #552	C₂²×C₃₁	semidirect
453. #552	C₂²×C₃₁	semidirect
454. #547	C₂²×C₃₁	semidirect
455. #547	C₂²×C₃₁	semidirect
456. #541	C₂²×C₃₁	semidirect
457. #539	C₂²×C₃₁	semidirect
458. #533	C₂²×C₃₁	semidirect
462. #529	C₂²×C₃₁	semidirect
463. #529	C₂²×C₃₁	semidirect
477. C₂×#125	C₂²×C₃₁	semidirect
495. #520	C₂²×C₃₁	semidirect
496. C₄×#41	C₂²×C₃₁	semidirect
530. C₂×C₃₁	#3515	direct (N × H)
666. C₃₁	C₂×#3515	direct (N × H)
863. C₂	C₃₁×#3515	direct (N × H)

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
C₃₁×#3515	2
C₂×C₃₁×#557	2
C₂×C₃₁×#554	2
C₂×C₃₁×#553	2
C₂×C₃₁×#552	2
C₂×C₃₁×#547	2
C₂×C₃₁×#541	2
C₂×C₃₁×#539	2
C₂×C₃₁×#533	2
C₂×C₃₁×#529	2
C₂²×C₃₁×#125	2
C₂×C₃₁×#520	2
C₂×C₄×#41×C₃₁	2
C₂×#3515	31

Subgroups — C2×C31×#3515 #21519538

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₃₁×#3515 #21519538