Subgroups — C₂×C₃₁×#3493 #21519535

Order 7936.

7302 conjugacy classes of subgroups.

The subgroup lattice has 580 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃₁×#3493	C₂	1	32	yes	maximal, direct product factor
2	C₂×C₃₁×#556	C₂	1	4	yes	maximal, complemented
3	C₂×C₃₁×#555	C₂	1	4	yes	maximal, complemented
4	C₂×C₃₁×#553	C₂	1	4	yes	maximal, complemented
5	C₂×C₃₁×#552	C₂	1	4	yes	maximal, complemented
6	C₂×C₃₁×#544	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂×C₃₁×#538	C₂	1	4	yes	maximal, complemented
8	C₂×C₃₁×#537	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂×C₃₁×#529	C₂	1	4	yes	maximal, complemented
10	C₂×C₃₁×#525	C₂	1	2	yes	maximal, complemented
11	C₂×C₃₁×#520	C₂	1	2	yes	maximal, complemented
12	C₂²×C₃₁×#117	C₂	1	1	yes	maximal, characteristic, complemented
13	C₂²×#41×C₃₁	C₂²	1	2	yes	—
14	C₂⁴×C₄×C₃₁	C₂²	1	1	yes	characteristic
15	C₃₁×#556	C₂²	1	64	yes	complemented
16	C₃₁×#555	C₂²	1	64	yes	complemented
17	C₃₁×#553	C₂²	1	64	yes	complemented
18	C₃₁×#552	C₂²	1	64	yes	complemented
19	C₃₁×#544	C₂²	1	16	yes	complemented
20	C₃₁×#538	C₂²	1	64	yes	complemented
21	C₃₁×#537	C₂²	1	16	yes	complemented
22	C₂×C₃₁×C₄⋊Q₈	C₂²	1	4	yes	complemented
23	C₃₁×#529	C₂²	1	64	yes	complemented
24	C₂×C₃₁×#126	C₂²	1	8	yes	—
25	C₂×C₃₁×#126	C₂²	1	8	yes	complemented
26	C₂×C₃₁×#125	C₂²	1	4	yes	complemented
27	C₂×C₃₁×#125	C₂²	1	4	yes	complemented
28	C₂×C₃₁×#124	C₂²	1	8	yes	—
29	C₂×C₃₁×#124	C₂²	1	4	yes	complemented
30	C₃₁×#525	C₂²	1	32	yes	complemented
31	C₂×C₃₁×#123	C₂²	1	4	yes	—
32	C₂×C₃₁×#123	C₂²	1	8	yes	—
33	C₂×C₃₁×C₂²⋊Q₈	C₂²	1	2	yes	—
34	C₂×C₃₁×C₂²⋊Q₈	C₂²	1	2	yes	—
35	C₂×C₃₁×C₂²⋊Q₈	C₂²	1	4	yes	—
36	C₂×C₃₁×C₂²⋊Q₈	C₂²	1	4	yes	complemented
37	C₂×C₃₁×#121	C₂²	1	2	yes	—
38	C₂×C₃₁×#121	C₂²	1	2	yes	—
39	C₂×C₃₁×#121	C₂²	1	4	yes	—
40	C₂×C₃₁×#121	C₂²	1	8	yes	—
41	C₂×C₃₁×#121	C₂²	1	4	yes	complemented
42	C₂×C₃₁×#120	C₂²	1	4	yes	—
43	C₃₁×#520	C₂²	1	32	yes	complemented
44	C₂×C₄×Q₈×C₃₁	C₂²	1	4	yes	complemented
45	C₂×C₄×Q₈×C₃₁	C₂²	1	4	yes	complemented
46	C₂×C₄×D₄×C₃₁	C₂²	1	8	yes	—
47	C₂×C₄×D₄×C₃₁	C₂²	1	4	yes	—
48	C₂×C₄×D₄×C₃₁	C₂²	1	4	yes	—
49	C₂×C₄×D₄×C₃₁	C₂²	1	4	yes	complemented
50	C₂×C₄×D₄×C₃₁	C₂²	1	4	yes	complemented
51	C₂×C₃₁×#117	C₂²	1	4	yes	—
52	C₂×C₃₁×#117	C₂²	1	4	yes	—
53	C₂×C₃₁×#117	C₂²	1	2	yes	—
54	C₂×C₃₁×#117	C₂²	1	16	yes	complemented
55	C₂×C₃₁×#117	C₂²	1	4	yes	complemented
56	C₂×C₃₁×#117	C₂²	1	8	yes	complemented
57	C₂²×C₄⋊C₄×C₃₁	C₂²	1	1	yes	characteristic
58	C₂²×C₄⋊C₄×C₃₁	C₂²	1	1	yes	characteristic
59	C₂²×C₄⋊C₄×C₃₁	C₂²	1	4	yes	complemented
60	C₂²×C₂²⋊C₄×C₃₁	C₂²	1	2	yes	—
61	C₂²×C₄²×C₃₁	C₂²	1	2	yes	—
62	C₂⁵×C₃₁	C₂³	1	1	yes	characteristic
63	C₂×#41×C₃₁	—	2	16	no	—
64	C₂×#41×C₃₁	C₂³	1	16	yes	—
65	C₂²×Q₈×C₃₁	C₂³	1	2	yes	—
66	C₂²×Q₈×C₃₁	C₂³	1	4	yes	—
67	C₂²×D₄×C₃₁	C₂³	1	4	yes	—
68	C₂²×D₄×C₃₁	C₂³	1	2	yes	—
69	C₂²×D₄×C₃₁	C₂³	1	4	yes	—
70	C₂²×D₄×C₃₁	C₂³	1	4	yes	—
71	C₂²×D₄×C₃₁	C₂³	1	4	yes	—
72	C₂³×C₄×C₃₁	—	2	8	no	—
73	C₂³×C₄×C₃₁	C₂³	1	1	yes	characteristic
74	C₂³×C₄×C₃₁	C₂³	1	4	yes	—
75	C₂³×C₄×C₃₁	C₂³	1	2	yes	—
76	C₂³×C₄×C₃₁	C₂³	1	2	yes	—
77	C₂³×C₄×C₃₁	C₂³	1	8	yes	—
78	C₂³×C₄×C₃₁	C₂³	1	2	yes	—
79	C₂³×C₄×C₃₁	C₂³	1	2	yes	—
80	C₂³×C₄×C₃₁	C₂³	1	8	yes	—
81	C₂³×C₄×C₃₁	C₂³	1	1	yes	characteristic
82	C₃₁×C₄⋊Q₈	C₂³	1	32	yes	complemented
83	C₃₁×#126	C₂³	1	64	yes	—
84	C₃₁×#126	C₂³	1	64	yes	complemented
85	C₃₁×#125	C₂³	1	32	yes	complemented
86	C₃₁×#125	C₂³	1	32	yes	complemented
87	C₃₁×#124	C₂³	1	64	yes	—
88	C₃₁×#124	C₂³	1	32	yes	complemented
89	C₃₁×#123	C₂³	1	32	yes	—
90	C₃₁×#123	C₂³	1	64	yes	—
91	C₃₁×C₂²⋊Q₈	C₂³	1	16	yes	—
92	C₃₁×C₂²⋊Q₈	C₂³	1	16	yes	—
93	C₃₁×C₂²⋊Q₈	C₂³	1	32	yes	—
94	C₃₁×C₂²⋊Q₈	C₂³	1	32	yes	complemented
95	C₃₁×#121	C₂³	1	16	yes	—
96	C₃₁×#121	C₂³	1	16	yes	—
97	C₃₁×#121	C₂³	1	32	yes	—
98	C₃₁×#121	C₂³	1	64	yes	—
99	C₃₁×#121	C₂³	1	32	yes	complemented
100	C₃₁×#120	C₂³	1	32	yes	—
101	C₄×Q₈×C₃₁	C₂³	1	32	yes	complemented
102	C₄×Q₈×C₃₁	C₂³	1	32	yes	complemented
103	C₄×D₄×C₃₁	C₂³	1	64	yes	—
104	C₄×D₄×C₃₁	C₂³	1	32	yes	—
105	C₄×D₄×C₃₁	C₂³	1	32	yes	—
106	C₄×D₄×C₃₁	C₂³	1	32	yes	complemented
107	C₄×D₄×C₃₁	C₂³	1	32	yes	complemented
108	C₃₁×#117	C₂³	1	32	yes	—
109	C₃₁×#117	C₂³	1	32	yes	—
110	C₃₁×#117	C₂³	1	16	yes	—
111	C₃₁×#117	C₂³	1	32	yes	complemented
112	C₃₁×#117	C₂³	1	64	yes	complemented
113	C₂×C₄⋊C₄×C₃₁	C₂³	1	4	yes	—
114	C₂×C₄⋊C₄×C₃₁	C₂³	1	4	yes	—
115	C₂×C₄⋊C₄×C₃₁	C₂³	1	8	yes	—
116	C₂×C₄⋊C₄×C₃₁	C₂³	1	8	yes	—
117	C₂×C₄⋊C₄×C₃₁	C₂³	1	2	yes	—
118	C₂×C₄⋊C₄×C₃₁	C₂³	1	8	yes	—
119	C₂×C₄⋊C₄×C₃₁	C₂³	1	2	yes	—
120	C₂×C₄⋊C₄×C₃₁	C₂³	1	4	yes	—
121	C₂×C₄⋊C₄×C₃₁	C₂³	1	8	yes	—
122	C₂×C₄⋊C₄×C₃₁	C₂³	1	4	yes	—
123	C₂×C₄⋊C₄×C₃₁	C₂³	1	4	yes	—
124	C₂×C₄⋊C₄×C₃₁	C₂³	1	4	yes	—
125	C₂×C₄⋊C₄×C₃₁	C₂³	1	32	yes	complemented
126	C₂×C₂²⋊C₄×C₃₁	C₂³	1	8	yes	—
127	C₂×C₂²⋊C₄×C₃₁	C₂³	1	16	yes	—
128	C₂×C₂²⋊C₄×C₃₁	C₂³	1	4	yes	—
129	C₂×C₂²⋊C₄×C₃₁	C₂³	1	4	yes	—
130	C₂×C₂²⋊C₄×C₃₁	C₂³	1	8	yes	—
131	C₂×C₂²⋊C₄×C₃₁	C₂³	1	8	yes	—
132	C₂×C₂²⋊C₄×C₃₁	C₂³	1	4	yes	—
133	C₂×C₂²⋊C₄×C₃₁	C₂³	1	4	yes	—
134	C₂×C₂²⋊C₄×C₃₁	C₂³	1	4	yes	—
135	C₂×C₄²×C₃₁	C₂³	1	4	yes	—
136	C₂×C₄²×C₃₁	C₂³	1	4	yes	—
137	C₂×C₄²×C₃₁	C₂³	1	16	yes	—
138	C₂×C₄²×C₃₁	C₂³	1	4	yes	—
139	C₂×C₄²×C₃₁	C₂³	1	8	yes	—
140	C₂⁴×C₃₁	—	2	4	no	—
141	C₂⁴×C₃₁	—	4	2	no	—
142	C₂⁴×C₃₁	—	4	2	no	—
143	C₂⁴×C₃₁	C₂⁴	1	2	yes	—
144	C₂⁴×C₃₁	C₂⁴	1	4	yes	—
145	C₂⁴×C₃₁	C₂⁴	1	1	yes	characteristic
146	C₂⁴×C₃₁	C₂⁴	1	4	yes	—
147	#41×C₃₁	—	2	64	no	—
148	C₂×Q₈×C₃₁	—	2	8	no	—
149	C₂×Q₈×C₃₁	—	4	8	no	—
150	C₂×Q₈×C₃₁	C₂⁴	1	8	yes	—
151	C₂×Q₈×C₃₁	C₂⁴	1	16	yes	—
152	C₂×D₄×C₃₁	—	2	16	no	—
153	C₂×D₄×C₃₁	—	2	8	no	—
154	C₂×D₄×C₃₁	—	4	8	no	—
155	C₂×D₄×C₃₁	—	4	8	no	—
156	C₂×D₄×C₃₁	—	4	8	no	—
157	C₂×D₄×C₃₁	C₂⁴	1	16	yes	—
158	C₂×D₄×C₃₁	C₂⁴	1	8	yes	—
159	C₂×D₄×C₃₁	C₂⁴	1	16	yes	—
160	C₂×D₄×C₃₁	C₂⁴	1	16	yes	—
161	C₂×D₄×C₃₁	C₂⁴	1	16	yes	—
162	C₂²×C₄×C₃₁	—	2	4	no	—
163	C₂²×C₄×C₃₁	—	2	8	no	—
164	C₂²×C₄×C₃₁	—	2	8	no	—
165	C₂²×C₄×C₃₁	—	2	32	no	—
166	C₂²×C₄×C₃₁	—	2	16	no	—
167	C₂²×C₄×C₃₁	—	4	4	no	—
168	C₂²×C₄×C₃₁	—	4	16	no	—
169	C₂²×C₄×C₃₁	—	4	4	no	—
170	C₂²×C₄×C₃₁	#41	1	8	yes	—
171	C₂²×C₄×C₃₁	C₂⁴	1	4	yes	—
172	C₂²×C₄×C₃₁	C₂⁴	1	16	yes	—
173	C₂²×C₄×C₃₁	C₂⁴	1	8	yes	—
174	C₂²×C₄×C₃₁	C₂⁴	1	8	yes	—
175	C₂²×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
176	C₂²×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
177	C₂²×C₄×C₃₁	C₂⁴	1	4	yes	—
178	C₂²×C₄×C₃₁	C₂⁴	1	4	yes	—
179	C₂²×C₄×C₃₁	C₂⁴	1	4	yes	—
180	C₂²×C₄×C₃₁	C₂⁴	1	4	yes	—
181	C₂²×C₄×C₃₁	C₂⁴	1	4	yes	—
182	C₂²×C₄×C₃₁	C₂⁴	1	8	yes	—
183	C₂²×C₄×C₃₁	C₂⁴	1	8	yes	—
184	C₂²×C₄×C₃₁	C₂⁴	1	32	yes	—
185	C₂²×C₄×C₃₁	C₂⁴	1	4	yes	—
186	C₂²×C₄×C₃₁	C₂⁴	1	2	yes	—
187	C₄⋊C₄×C₃₁	C₂⁴	1	16	yes	—
188	C₄⋊C₄×C₃₁	C₂⁴	1	16	yes	—
189	C₄⋊C₄×C₃₁	C₂⁴	1	8	yes	—
190	C₄⋊C₄×C₃₁	C₂⁴	1	32	yes	—
191	C₄⋊C₄×C₃₁	C₂⁴	1	8	yes	—
192	C₄⋊C₄×C₃₁	C₂⁴	1	16	yes	—
193	C₄⋊C₄×C₃₁	C₂⁴	1	32	yes	—
194	C₄⋊C₄×C₃₁	C₂⁴	1	16	yes	—
195	C₄⋊C₄×C₃₁	C₂⁴	1	16	yes	—
196	C₄⋊C₄×C₃₁	C₂⁴	1	16	yes	—
197	C₂²⋊C₄×C₃₁	C₂⁴	1	32	yes	—
198	C₂²⋊C₄×C₃₁	C₂⁴	1	16	yes	—
199	C₂²⋊C₄×C₃₁	C₂⁴	1	16	yes	—
200	C₂²⋊C₄×C₃₁	C₂⁴	1	32	yes	—
201	C₂²⋊C₄×C₃₁	C₂⁴	1	32	yes	—
202	C₂²⋊C₄×C₃₁	C₂⁴	1	16	yes	—
203	C₂²⋊C₄×C₃₁	C₂⁴	1	16	yes	—
204	C₂²⋊C₄×C₃₁	C₂⁴	1	16	yes	—
205	C₄²×C₃₁	C₂⁴	1	16	yes	—
206	C₄²×C₃₁	C₂⁴	1	16	yes	—
207	C₄²×C₃₁	C₂⁴	1	16	yes	—
208	C₄²×C₃₁	C₂⁴	1	32	yes	—
209	C₂×#3493	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
210	C₂³×C₃₁	—	2	2	no	—
211	C₂³×C₃₁	—	2	2	no	—
212	C₂³×C₃₁	—	2	8	no	—
213	C₂³×C₃₁	—	2	4	no	—
214	C₂³×C₃₁	—	2	8	no	—
215	C₂³×C₃₁	—	2	8	no	—
216	C₂³×C₃₁	—	2	8	no	—
217	C₂³×C₃₁	—	2	4	no	—
218	C₂³×C₃₁	—	2	4	no	—
219	C₂³×C₃₁	—	4	4	no	—
220	C₂³×C₃₁	—	4	4	no	—
221	C₂³×C₃₁	—	4	16	no	—
222	C₂³×C₃₁	C₂×#41	1	4	yes	—
223	C₂³×C₃₁	C₂⁵	1	4	yes	—
224	C₂³×C₃₁	C₂⁵	1	8	yes	—
225	C₂³×C₃₁	C₂⁵	1	2	yes	—
226	C₂³×C₃₁	C₂⁵	1	1	yes	characteristic
227	Q₈×C₃₁	—	2	16	no	—
228	Q₈×C₃₁	—	4	16	no	—
229	D₄×C₃₁	—	2	32	no	—
230	D₄×C₃₁	—	2	16	no	—
231	D₄×C₃₁	—	4	16	no	—
232	D₄×C₃₁	—	4	16	no	—
233	D₄×C₃₁	—	4	16	no	—
234	C₂×C₄×C₃₁	—	2	8	no	—
235	C₂×C₄×C₃₁	—	2	2	no	—
236	C₂×C₄×C₃₁	—	2	2	no	—
237	C₂×C₄×C₃₁	—	2	8	no	—
238	C₂×C₄×C₃₁	—	2	8	no	—
239	C₂×C₄×C₃₁	—	2	8	no	—
240	C₂×C₄×C₃₁	—	2	8	no	—
241	C₂×C₄×C₃₁	—	2	8	no	—
242	C₂×C₄×C₃₁	—	2	16	no	—
243	C₂×C₄×C₃₁	—	2	16	no	—
244	C₂×C₄×C₃₁	—	2	32	no	—
245	C₂×C₄×C₃₁	—	4	8	no	—
246	C₂×C₄×C₃₁	—	4	32	no	—
247	C₂×C₄×C₃₁	—	4	8	no	—
248	C₂×C₄×C₃₁	C₂×#41	1	8	yes	—
249	C₂×C₄×C₃₁	C₂×#41	1	16	yes	—
250	C₂×C₄×C₃₁	C₂⁵	1	2	yes	—
251	C₂×C₄×C₃₁	C₂⁵	1	2	yes	—
252	C₂×C₄×C₃₁	C₂⁵	1	8	yes	—
253	C₂×C₄×C₃₁	C₂⁵	1	8	yes	—
254	C₂×C₄×C₃₁	C₂⁵	1	8	yes	—
255	C₂×C₄×C₃₁	C₂⁵	1	8	yes	—
256	C₂×C₄×C₃₁	C₂⁵	1	8	yes	—
257	C₂×C₄×C₃₁	C₂⁵	1	4	yes	—
258	#3493	C₂×C₃₁	1	32	yes	direct product factor
259	C₂×#556	C₂×C₃₁	1	4	yes	complemented
260	C₂×#555	C₂×C₃₁	1	4	yes	complemented
261	C₂×#553	C₂×C₃₁	1	4	yes	complemented
262	C₂×#552	C₂×C₃₁	1	4	yes	complemented
263	C₂×#544	C₂×C₃₁	1	1	yes	characteristic, complemented
264	C₂×#538	C₂×C₃₁	1	4	yes	complemented
265	C₂×#537	C₂×C₃₁	1	1	yes	characteristic, complemented
266	C₂×#529	C₂×C₃₁	1	4	yes	complemented
267	C₂×#525	C₂×C₃₁	1	2	yes	complemented
268	C₂×#520	C₂×C₃₁	1	2	yes	complemented
269	C₂²×#117	C₂×C₃₁	1	1	yes	characteristic, complemented
270	C₂²×C₃₁	—	2	2	no	—
271	C₂²×C₃₁	—	2	2	no	—
272	C₂²×C₃₁	—	2	8	no	—
273	C₂²×C₃₁	—	2	4	no	—
274	C₂²×C₃₁	—	2	8	no	—
275	C₂²×C₃₁	—	2	8	no	—
276	C₂²×C₃₁	—	2	8	no	—
277	C₂²×C₃₁	—	2	4	no	—
278	C₂²×C₃₁	—	2	4	no	—
279	C₂²×C₃₁	—	4	16	no	—
280	C₂²×C₃₁	—	4	16	no	—
281	C₂²×C₃₁	—	4	8	no	—
282	C₂²×C₃₁	C₂²×#41	1	4	yes	—
283	C₂²×C₃₁	C₂²×#41	1	2	yes	—
284	C₂²×C₃₁	#585	1	2	yes	—
285	C₂²×C₃₁	#585	1	2	yes	—
286	C₂²×C₃₁	C₂⁶	1	1	yes	characteristic
287	C₄×C₃₁	—	2	2	no	—
288	C₄×C₃₁	—	2	2	no	—
289	C₄×C₃₁	—	2	8	no	—
290	C₄×C₃₁	—	2	8	no	—
291	C₄×C₃₁	—	2	8	no	—
292	C₄×C₃₁	—	2	8	no	—
293	C₄×C₃₁	—	2	8	no	—
294	C₄×C₃₁	C₂²×#41	1	8	yes	—
295	C₂²×#41	C₂²×C₃₁	1	2	yes	—
296	C₂⁴×C₄	C₂²×C₃₁	1	1	yes	characteristic
297	#556	C₂²×C₃₁	1	64	yes	complemented
298	#555	C₂²×C₃₁	1	64	yes	complemented
299	#553	C₂²×C₃₁	1	64	yes	complemented
300	#552	C₂²×C₃₁	1	64	yes	complemented
301	#544	C₂²×C₃₁	1	16	yes	complemented
302	#538	C₂²×C₃₁	1	64	yes	complemented
303	#537	C₂²×C₃₁	1	16	yes	complemented
304	C₂×C₄⋊Q₈	C₂²×C₃₁	1	4	yes	complemented
305	#529	C₂²×C₃₁	1	64	yes	complemented
306	C₂×#126	C₂²×C₃₁	1	8	yes	—
307	C₂×#126	C₂²×C₃₁	1	8	yes	complemented
308	C₂×#125	C₂²×C₃₁	1	4	yes	complemented
309	C₂×#125	C₂²×C₃₁	1	4	yes	complemented
310	C₂×#124	C₂²×C₃₁	1	8	yes	—
311	C₂×#124	C₂²×C₃₁	1	4	yes	complemented
312	#525	C₂²×C₃₁	1	32	yes	complemented
313	C₂×#123	C₂²×C₃₁	1	4	yes	—
314	C₂×#123	C₂²×C₃₁	1	8	yes	—
315	C₂×C₂²⋊Q₈	C₂²×C₃₁	1	2	yes	—
316	C₂×C₂²⋊Q₈	C₂²×C₃₁	1	2	yes	—
317	C₂×C₂²⋊Q₈	C₂²×C₃₁	1	4	yes	—
318	C₂×C₂²⋊Q₈	C₂²×C₃₁	1	4	yes	complemented
319	C₂×#121	C₂²×C₃₁	1	2	yes	—
320	C₂×#121	C₂²×C₃₁	1	2	yes	—
321	C₂×#121	C₂²×C₃₁	1	4	yes	—
322	C₂×#121	C₂²×C₃₁	1	8	yes	—
323	C₂×#121	C₂²×C₃₁	1	4	yes	complemented
324	C₂×#120	C₂²×C₃₁	1	4	yes	—
325	#520	C₂²×C₃₁	1	32	yes	complemented
326	C₂×C₄×Q₈	C₂²×C₃₁	1	4	yes	complemented
327	C₂×C₄×Q₈	C₂²×C₃₁	1	4	yes	complemented
328	C₂×C₄×D₄	C₂²×C₃₁	1	8	yes	—
329	C₂×C₄×D₄	C₂²×C₃₁	1	4	yes	—
330	C₂×C₄×D₄	C₂²×C₃₁	1	4	yes	—
331	C₂×C₄×D₄	C₂²×C₃₁	1	4	yes	complemented
332	C₂×C₄×D₄	C₂²×C₃₁	1	4	yes	complemented
333	C₂×#117	C₂²×C₃₁	1	4	yes	—
334	C₂×#117	C₂²×C₃₁	1	4	yes	—
335	C₂×#117	C₂²×C₃₁	1	2	yes	—
336	C₂×#117	C₂²×C₃₁	1	16	yes	complemented
337	C₂×#117	C₂²×C₃₁	1	4	yes	complemented
338	C₂×#117	C₂²×C₃₁	1	8	yes	complemented
339	C₂²×C₄⋊C₄	C₂²×C₃₁	1	1	yes	characteristic
340	C₂²×C₄⋊C₄	C₂²×C₃₁	1	1	yes	characteristic
341	C₂²×C₄⋊C₄	C₂²×C₃₁	1	4	yes	complemented
342	C₂²×C₂²⋊C₄	C₂²×C₃₁	1	2	yes	—
343	C₂²×C₄²	C₂²×C₃₁	1	2	yes	—
344	C₂×C₃₁	—	2	4	no	—
345	C₂×C₃₁	—	4	8	no	—
346	C₂×C₃₁	—	4	4	no	—
347	C₂×C₃₁	#3493	1	4	yes	direct product factor
348	C₂×C₃₁	C₂³×#41	1	1	yes	characteristic
349	C₂×C₃₁	C₂×#585	1	1	yes	characteristic
350	C₂×C₃₁	C₂×#585	1	1	yes	characteristic
351	C₂⁵	C₂³×C₃₁	1	1	yes	characteristic
352	C₂×#41	—	2	16	no	—
353	C₂×#41	C₂³×C₃₁	1	16	yes	—
354	C₂²×Q₈	C₂³×C₃₁	1	2	yes	—
355	C₂²×Q₈	C₂³×C₃₁	1	4	yes	—
356	C₂²×D₄	C₂³×C₃₁	1	4	yes	—
357	C₂²×D₄	C₂³×C₃₁	1	2	yes	—
358	C₂²×D₄	C₂³×C₃₁	1	4	yes	—
359	C₂²×D₄	C₂³×C₃₁	1	4	yes	—
360	C₂²×D₄	C₂³×C₃₁	1	4	yes	—
361	C₂³×C₄	—	2	8	no	—
362	C₂³×C₄	C₂³×C₃₁	1	4	yes	—
363	C₂³×C₄	C₂³×C₃₁	1	2	yes	—
364	C₂³×C₄	C₂³×C₃₁	1	2	yes	—
365	C₂³×C₄	C₂³×C₃₁	1	8	yes	—
366	C₂³×C₄	C₂³×C₃₁	1	1	yes	characteristic
367	C₂³×C₄	C₂³×C₃₁	1	2	yes	—
368	C₂³×C₄	C₂³×C₃₁	1	2	yes	—
369	C₂³×C₄	C₂³×C₃₁	1	8	yes	—
370	C₂³×C₄	C₂³×C₃₁	1	1	yes	characteristic
371	C₄⋊Q₈	C₂³×C₃₁	1	32	yes	complemented
372	#126	C₂³×C₃₁	1	64	yes	—
373	#126	C₂³×C₃₁	1	64	yes	complemented
374	#125	C₂³×C₃₁	1	32	yes	complemented
375	#125	C₂³×C₃₁	1	32	yes	complemented
376	#124	C₂³×C₃₁	1	64	yes	—
377	#124	C₂³×C₃₁	1	32	yes	complemented
378	#123	C₂³×C₃₁	1	32	yes	—
379	#123	C₂³×C₃₁	1	64	yes	—
380	C₂²⋊Q₈	C₂³×C₃₁	1	16	yes	—
381	C₂²⋊Q₈	C₂³×C₃₁	1	16	yes	—
382	C₂²⋊Q₈	C₂³×C₃₁	1	32	yes	—
383	C₂²⋊Q₈	C₂³×C₃₁	1	32	yes	complemented
384	#121	C₂³×C₃₁	1	16	yes	—
385	#121	C₂³×C₃₁	1	16	yes	—
386	#121	C₂³×C₃₁	1	32	yes	—
387	#121	C₂³×C₃₁	1	64	yes	—
388	#121	C₂³×C₃₁	1	32	yes	complemented
389	#120	C₂³×C₃₁	1	32	yes	—
390	C₄×Q₈	C₂³×C₃₁	1	32	yes	complemented
391	C₄×Q₈	C₂³×C₃₁	1	32	yes	complemented
392	C₄×D₄	C₂³×C₃₁	1	64	yes	—
393	C₄×D₄	C₂³×C₃₁	1	32	yes	—
394	C₄×D₄	C₂³×C₃₁	1	32	yes	—
395	C₄×D₄	C₂³×C₃₁	1	32	yes	complemented
396	C₄×D₄	C₂³×C₃₁	1	32	yes	complemented
397	#117	C₂³×C₃₁	1	32	yes	—
398	#117	C₂³×C₃₁	1	32	yes	—
399	#117	C₂³×C₃₁	1	16	yes	—
400	#117	C₂³×C₃₁	1	32	yes	complemented
401	#117	C₂³×C₃₁	1	64	yes	complemented
402	C₂×C₄⋊C₄	C₂³×C₃₁	1	4	yes	—
403	C₂×C₄⋊C₄	C₂³×C₃₁	1	4	yes	—
404	C₂×C₄⋊C₄	C₂³×C₃₁	1	8	yes	—
405	C₂×C₄⋊C₄	C₂³×C₃₁	1	8	yes	—
406	C₂×C₄⋊C₄	C₂³×C₃₁	1	2	yes	—
407	C₂×C₄⋊C₄	C₂³×C₃₁	1	8	yes	—
408	C₂×C₄⋊C₄	C₂³×C₃₁	1	2	yes	—
409	C₂×C₄⋊C₄	C₂³×C₃₁	1	4	yes	—
410	C₂×C₄⋊C₄	C₂³×C₃₁	1	8	yes	—
411	C₂×C₄⋊C₄	C₂³×C₃₁	1	4	yes	—
412	C₂×C₄⋊C₄	C₂³×C₃₁	1	4	yes	—
413	C₂×C₄⋊C₄	C₂³×C₃₁	1	4	yes	—
414	C₂×C₄⋊C₄	C₂³×C₃₁	1	32	yes	complemented
415	C₂×C₂²⋊C₄	C₂³×C₃₁	1	8	yes	—
416	C₂×C₂²⋊C₄	C₂³×C₃₁	1	16	yes	—
417	C₂×C₂²⋊C₄	C₂³×C₃₁	1	4	yes	—
418	C₂×C₂²⋊C₄	C₂³×C₃₁	1	4	yes	—
419	C₂×C₂²⋊C₄	C₂³×C₃₁	1	8	yes	—
420	C₂×C₂²⋊C₄	C₂³×C₃₁	1	8	yes	—
421	C₂×C₂²⋊C₄	C₂³×C₃₁	1	4	yes	—
422	C₂×C₂²⋊C₄	C₂³×C₃₁	1	4	yes	—
423	C₂×C₂²⋊C₄	C₂³×C₃₁	1	4	yes	—
424	C₂×C₄²	C₂³×C₃₁	1	4	yes	—
425	C₂×C₄²	C₂³×C₃₁	1	4	yes	—
426	C₂×C₄²	C₂³×C₃₁	1	4	yes	—
427	C₂×C₄²	C₂³×C₃₁	1	8	yes	—
428	C₂×C₄²	C₂³×C₃₁	1	16	yes	—
429	C₃₁	C₂×#3493	1	1	yes	characteristic, direct product factor, Sylow
430	C₂⁴	—	2	4	no	—
431	C₂⁴	—	4	2	no	—
432	C₂⁴	—	4	2	no	—
433	C₂⁴	C₂⁴×C₃₁	1	2	yes	—
434	C₂⁴	C₂⁴×C₃₁	1	4	yes	—
435	C₂⁴	C₂⁴×C₃₁	1	1	yes	characteristic
436	C₂⁴	C₂⁴×C₃₁	1	4	yes	—
437	#41	—	2	64	no	—
438	C₂×Q₈	—	2	8	no	—
439	C₂×Q₈	—	4	8	no	—
440	C₂×Q₈	C₂⁴×C₃₁	1	8	yes	—
441	C₂×Q₈	C₂⁴×C₃₁	1	16	yes	—
442	C₂×D₄	—	2	16	no	—
443	C₂×D₄	—	2	8	no	—
444	C₂×D₄	—	4	8	no	—
445	C₂×D₄	—	4	8	no	—
446	C₂×D₄	—	4	8	no	—
447	C₂×D₄	C₂⁴×C₃₁	1	16	yes	—
448	C₂×D₄	C₂⁴×C₃₁	1	8	yes	—
449	C₂×D₄	C₂⁴×C₃₁	1	16	yes	—
450	C₂×D₄	C₂⁴×C₃₁	1	16	yes	—
451	C₂×D₄	C₂⁴×C₃₁	1	16	yes	—
452	C₂²×C₄	—	2	4	no	—
453	C₂²×C₄	—	2	8	no	—
454	C₂²×C₄	—	2	8	no	—
455	C₂²×C₄	—	2	32	no	—
456	C₂²×C₄	—	2	16	no	—
457	C₂²×C₄	—	4	4	no	—
458	C₂²×C₄	—	4	16	no	—
459	C₂²×C₄	—	4	4	no	—
460	C₂²×C₄	#41×C₃₁	1	8	yes	—
461	C₂²×C₄	C₂⁴×C₃₁	1	1	yes	characteristic
462	C₂²×C₄	C₂⁴×C₃₁	1	1	yes	characteristic
463	C₂²×C₄	C₂⁴×C₃₁	1	4	yes	—
464	C₂²×C₄	C₂⁴×C₃₁	1	4	yes	—
465	C₂²×C₄	C₂⁴×C₃₁	1	4	yes	—
466	C₂²×C₄	C₂⁴×C₃₁	1	4	yes	—
467	C₂²×C₄	C₂⁴×C₃₁	1	4	yes	—
468	C₂²×C₄	C₂⁴×C₃₁	1	4	yes	—
469	C₂²×C₄	C₂⁴×C₃₁	1	16	yes	—
470	C₂²×C₄	C₂⁴×C₃₁	1	8	yes	—
471	C₂²×C₄	C₂⁴×C₃₁	1	8	yes	—
472	C₂²×C₄	C₂⁴×C₃₁	1	8	yes	—
473	C₂²×C₄	C₂⁴×C₃₁	1	8	yes	—
474	C₂²×C₄	C₂⁴×C₃₁	1	32	yes	—
475	C₂²×C₄	C₂⁴×C₃₁	1	2	yes	—
476	C₂²×C₄	C₂⁴×C₃₁	1	4	yes	—
477	C₄⋊C₄	C₂⁴×C₃₁	1	16	yes	—
478	C₄⋊C₄	C₂⁴×C₃₁	1	16	yes	—
479	C₄⋊C₄	C₂⁴×C₃₁	1	8	yes	—
480	C₄⋊C₄	C₂⁴×C₃₁	1	32	yes	—
481	C₄⋊C₄	C₂⁴×C₃₁	1	8	yes	—
482	C₄⋊C₄	C₂⁴×C₃₁	1	16	yes	—
483	C₄⋊C₄	C₂⁴×C₃₁	1	32	yes	—
484	C₄⋊C₄	C₂⁴×C₃₁	1	16	yes	—
485	C₄⋊C₄	C₂⁴×C₃₁	1	16	yes	—
486	C₄⋊C₄	C₂⁴×C₃₁	1	16	yes	—
487	C₂²⋊C₄	C₂⁴×C₃₁	1	32	yes	—
488	C₂²⋊C₄	C₂⁴×C₃₁	1	16	yes	—
489	C₂²⋊C₄	C₂⁴×C₃₁	1	16	yes	—
490	C₂²⋊C₄	C₂⁴×C₃₁	1	32	yes	—
491	C₂²⋊C₄	C₂⁴×C₃₁	1	32	yes	—
492	C₂²⋊C₄	C₂⁴×C₃₁	1	16	yes	—
493	C₂²⋊C₄	C₂⁴×C₃₁	1	16	yes	—
494	C₂²⋊C₄	C₂⁴×C₃₁	1	16	yes	—
495	C₄²	C₂⁴×C₃₁	1	16	yes	—
496	C₄²	C₂⁴×C₃₁	1	16	yes	—
497	C₄²	C₂⁴×C₃₁	1	16	yes	—
498	C₄²	C₂⁴×C₃₁	1	32	yes	—
499	C₂³	—	2	2	no	—
500	C₂³	—	2	2	no	—
501	C₂³	—	2	8	no	—
502	C₂³	—	2	4	no	—
503	C₂³	—	2	8	no	—
504	C₂³	—	2	8	no	—
505	C₂³	—	2	8	no	—
506	C₂³	—	2	4	no	—
507	C₂³	—	2	4	no	—
508	C₂³	—	4	4	no	—
509	C₂³	—	4	4	no	—
510	C₂³	—	4	16	no	—
511	C₂³	C₂×#41×C₃₁	1	4	yes	—
512	C₂³	C₂⁵×C₃₁	1	4	yes	—
513	C₂³	C₂⁵×C₃₁	1	8	yes	—
514	C₂³	C₂⁵×C₃₁	1	2	yes	—
515	C₂³	C₂⁵×C₃₁	1	1	yes	characteristic
516	Q₈	—	2	16	no	—
517	Q₈	—	4	16	no	—
518	D₄	—	2	32	no	—
519	D₄	—	2	16	no	—
520	D₄	—	4	16	no	—
521	D₄	—	4	16	no	—
522	D₄	—	4	16	no	—
523	C₂×C₄	—	2	8	no	—
524	C₂×C₄	—	2	2	no	—
525	C₂×C₄	—	2	2	no	—
526	C₂×C₄	—	2	8	no	—
527	C₂×C₄	—	2	8	no	—
528	C₂×C₄	—	2	8	no	—
529	C₂×C₄	—	2	8	no	—
530	C₂×C₄	—	2	8	no	—
531	C₂×C₄	—	2	16	no	—
532	C₂×C₄	—	2	16	no	—
533	C₂×C₄	—	2	32	no	—
534	C₂×C₄	—	4	8	no	—
535	C₂×C₄	—	4	32	no	—
536	C₂×C₄	—	4	8	no	—
537	C₂×C₄	C₂×#41×C₃₁	1	8	yes	—
538	C₂×C₄	C₂×#41×C₃₁	1	16	yes	—
539	C₂×C₄	C₂⁵×C₃₁	1	2	yes	—
540	C₂×C₄	C₂⁵×C₃₁	1	2	yes	—
541	C₂×C₄	C₂⁵×C₃₁	1	8	yes	—
542	C₂×C₄	C₂⁵×C₃₁	1	8	yes	—
543	C₂×C₄	C₂⁵×C₃₁	1	8	yes	—
544	C₂×C₄	C₂⁵×C₃₁	1	8	yes	—
545	C₂×C₄	C₂⁵×C₃₁	1	8	yes	—
546	C₂×C₄	C₂⁵×C₃₁	1	4	yes	—
547	C₂²	—	2	2	no	—
548	C₂²	—	2	2	no	—
549	C₂²	—	2	8	no	—
550	C₂²	—	2	4	no	—
551	C₂²	—	2	8	no	—
552	C₂²	—	2	8	no	—
553	C₂²	—	2	8	no	—
554	C₂²	—	2	4	no	—
555	C₂²	—	2	4	no	—
556	C₂²	—	4	16	no	—
557	C₂²	—	4	8	no	—
558	C₂²	—	4	16	no	—
559	C₂²	C₂²×#41×C₃₁	1	4	yes	—
560	C₂²	C₂²×#41×C₃₁	1	2	yes	—
561	C₂²	C₃₁×#585	1	2	yes	—
562	C₂²	C₃₁×#585	1	2	yes	—
563	C₂²	C₂⁶×C₃₁	1	1	yes	characteristic
564	C₄	—	2	2	no	—
565	C₄	—	2	2	no	—
566	C₄	—	2	8	no	—
567	C₄	—	2	8	no	—
568	C₄	—	2	8	no	—
569	C₄	—	2	8	no	—
570	C₄	—	2	8	no	—
571	C₄	C₂²×#41×C₃₁	1	8	yes	—
572	C₂	—	2	4	no	—
573	C₂	—	4	4	no	—
574	C₂	—	4	8	no	—
575	C₂	C₃₁×#3493	1	4	yes	direct product factor
576	C₂	C₂³×#41×C₃₁	1	1	yes	characteristic
577	C₂	C₂×C₃₁×#585	1	1	yes	characteristic
578	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×#3493	C₂	direct (N × H)
2. C₂×C₃₁×#556	C₂	semidirect
3. C₂×C₃₁×#555	C₂	semidirect
4. C₂×C₃₁×#553	C₂	semidirect
5. C₂×C₃₁×#552	C₂	semidirect
6. C₂×C₃₁×#544	C₂	semidirect
7. C₂×C₃₁×#538	C₂	semidirect
8. C₂×C₃₁×#537	C₂	semidirect
9. C₂×C₃₁×#529	C₂	semidirect
10. C₂×C₃₁×#525	C₂	semidirect
11. C₂×C₃₁×#520	C₂	semidirect
12. C₂²×C₃₁×#117	C₂	semidirect
15. C₃₁×#556	C₂²	semidirect
16. C₃₁×#555	C₂²	semidirect
17. C₃₁×#553	C₂²	semidirect
18. C₃₁×#552	C₂²	semidirect
19. C₃₁×#544	C₂²	semidirect
20. C₃₁×#538	C₂²	semidirect
21. C₃₁×#537	C₂²	semidirect
22. C₂×C₃₁×C₄⋊Q₈	C₂²	semidirect
23. C₃₁×#529	C₂²	semidirect
25. C₂×C₃₁×#126	C₂²	semidirect
26. C₂×C₃₁×#125	C₂²	semidirect
27. C₂×C₃₁×#125	C₂²	semidirect
29. C₂×C₃₁×#124	C₂²	semidirect
30. C₃₁×#525	C₂²	semidirect
36. C₂×C₃₁×C₂²⋊Q₈	C₂²	semidirect
41. C₂×C₃₁×#121	C₂²	semidirect
43. C₃₁×#520	C₂²	semidirect
44. C₂×C₄×Q₈×C₃₁	C₂²	semidirect
45. C₂×C₄×Q₈×C₃₁	C₂²	semidirect
49. C₂×C₄×D₄×C₃₁	C₂²	semidirect
50. C₂×C₄×D₄×C₃₁	C₂²	semidirect
54. C₂×C₃₁×#117	C₂²	semidirect
55. C₂×C₃₁×#117	C₂²	semidirect
56. C₂×C₃₁×#117	C₂²	semidirect
59. C₂²×C₄⋊C₄×C₃₁	C₂²	semidirect
82. C₃₁×C₄⋊Q₈	C₂³	semidirect
84. C₃₁×#126	C₂³	semidirect
85. C₃₁×#125	C₂³	semidirect
86. C₃₁×#125	C₂³	semidirect
88. C₃₁×#124	C₂³	semidirect
94. C₃₁×C₂²⋊Q₈	C₂³	semidirect
99. C₃₁×#121	C₂³	semidirect
101. C₄×Q₈×C₃₁	C₂³	semidirect
102. C₄×Q₈×C₃₁	C₂³	semidirect
106. C₄×D₄×C₃₁	C₂³	semidirect
107. C₄×D₄×C₃₁	C₂³	semidirect
111. C₃₁×#117	C₂³	semidirect
112. C₃₁×#117	C₂³	semidirect
125. C₂×C₄⋊C₄×C₃₁	C₂³	semidirect
209. C₂×#3493	C₃₁	direct (N × H)
258. #3493	C₂×C₃₁	direct (N × H)
259. C₂×#556	C₂×C₃₁	semidirect
260. C₂×#555	C₂×C₃₁	semidirect
261. C₂×#553	C₂×C₃₁	semidirect
262. C₂×#552	C₂×C₃₁	semidirect
263. C₂×#544	C₂×C₃₁	semidirect
264. C₂×#538	C₂×C₃₁	semidirect
265. C₂×#537	C₂×C₃₁	semidirect
266. C₂×#529	C₂×C₃₁	semidirect
267. C₂×#525	C₂×C₃₁	semidirect
268. C₂×#520	C₂×C₃₁	semidirect
269. C₂²×#117	C₂×C₃₁	semidirect
297. #556	C₂²×C₃₁	semidirect
298. #555	C₂²×C₃₁	semidirect
299. #553	C₂²×C₃₁	semidirect
300. #552	C₂²×C₃₁	semidirect
301. #544	C₂²×C₃₁	semidirect
302. #538	C₂²×C₃₁	semidirect
303. #537	C₂²×C₃₁	semidirect
304. C₂×C₄⋊Q₈	C₂²×C₃₁	semidirect
305. #529	C₂²×C₃₁	semidirect
307. C₂×#126	C₂²×C₃₁	semidirect
308. C₂×#125	C₂²×C₃₁	semidirect
309. C₂×#125	C₂²×C₃₁	semidirect
311. C₂×#124	C₂²×C₃₁	semidirect
312. #525	C₂²×C₃₁	semidirect
318. C₂×C₂²⋊Q₈	C₂²×C₃₁	semidirect
323. C₂×#121	C₂²×C₃₁	semidirect
325. #520	C₂²×C₃₁	semidirect
326. C₂×C₄×Q₈	C₂²×C₃₁	semidirect
327. C₂×C₄×Q₈	C₂²×C₃₁	semidirect
331. C₂×C₄×D₄	C₂²×C₃₁	semidirect
332. C₂×C₄×D₄	C₂²×C₃₁	semidirect
336. C₂×#117	C₂²×C₃₁	semidirect
337. C₂×#117	C₂²×C₃₁	semidirect
338. C₂×#117	C₂²×C₃₁	semidirect
341. C₂²×C₄⋊C₄	C₂²×C₃₁	semidirect
347. C₂×C₃₁	#3493	direct (N × H)
371. C₄⋊Q₈	C₂³×C₃₁	semidirect
373. #126	C₂³×C₃₁	semidirect
374. #125	C₂³×C₃₁	semidirect
375. #125	C₂³×C₃₁	semidirect
377. #124	C₂³×C₃₁	semidirect
383. C₂²⋊Q₈	C₂³×C₃₁	semidirect
388. #121	C₂³×C₃₁	semidirect
390. C₄×Q₈	C₂³×C₃₁	semidirect
391. C₄×Q₈	C₂³×C₃₁	semidirect
395. C₄×D₄	C₂³×C₃₁	semidirect
396. C₄×D₄	C₂³×C₃₁	semidirect
400. #117	C₂³×C₃₁	semidirect
401. #117	C₂³×C₃₁	semidirect
414. C₂×C₄⋊C₄	C₂³×C₃₁	semidirect
429. C₃₁	C₂×#3493	direct (N × H)
575. C₂	C₃₁×#3493	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×#3493	2
C₂×C₃₁×#556	2
C₂×C₃₁×#555	2
C₂×C₃₁×#553	2
C₂×C₃₁×#552	2
C₂×C₃₁×#544	2
C₂×C₃₁×#538	2
C₂×C₃₁×#537	2
C₂×C₃₁×#529	2
C₂×C₃₁×#525	2
C₂×C₃₁×#520	2
C₂²×C₃₁×#117	2
C₂×#3493	31

Subgroups — C2×C31×#3493 #21519535

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₃₁×#3493 #21519535