Subgroups — C₃₁×#34404 #21491163

Order 7936.

2586 conjugacy classes of subgroups.

The subgroup lattice has 760 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃₁×#525	C₂	1	1	yes	maximal, characteristic
2	C₃₁×#2376	C₂	1	4	yes	maximal, complemented
3	C₃₁×#2361	C₂	1	2	yes	maximal, complemented
4	C₃₁×#2360	C₂	1	2	yes	maximal, complemented
5	C₃₁×#2353	C₂	1	2	yes	maximal, complemented
6	C₃₁×#2349	C₂	1	2	yes	maximal, complemented
7	C₃₁×#2347	C₂	1	1	yes	maximal, characteristic, complemented
8	C₃₁×#2346	C₂	1	1	yes	maximal, characteristic, complemented
9	C₃₁×#2341	C₂	1	1	yes	maximal, characteristic, complemented
10	C₃₁×#2339	C₂	1	1	yes	maximal, characteristic, complemented
11	C₃₁×#2324	C₂	1	2	yes	maximal, complemented
12	C₃₁×#2323	C₂	1	2	yes	maximal, complemented
13	C₃₁×#2321	C₂	1	4	yes	maximal, complemented
14	C₃₁×#2319	C₂	1	2	yes	maximal, complemented
15	C₃₁×#2318	C₂	1	2	yes	maximal, complemented
16	C₃₁×#2280	C₂	1	1	yes	maximal, characteristic, complemented
17	C₂×C₃₁×#380	C₂	1	1	yes	maximal, characteristic, complemented
18	C₂²×#41×C₃₁	C₂²	1	1	yes	characteristic
19	C₂⁴×C₄×C₃₁	C₂²	1	1	yes	characteristic
20	C₃₁×#525	C₄	1	16	yes	complemented
21	C₂×C₃₁×#123	C₂²	1	2	yes	—
22	C₂×C₃₁×C₂²⋊Q₈	C₂²	1	1	yes	characteristic
23	C₂×C₃₁×C₂²⋊Q₈	C₂²	1	1	yes	characteristic
24	C₂×C₃₁×#121	C₂²	1	1	yes	characteristic
25	C₂×C₃₁×#121	C₂²	1	1	yes	characteristic
26	C₂×C₃₁×#120	C₂²	1	2	yes	—
27	C₂×C₄×Q₈×C₃₁	C₂²	1	2	yes	complemented
28	C₂×C₄×D₄×C₃₁	C₂²	1	4	yes	—
29	C₂×C₄×D₄×C₃₁	C₂²	1	2	yes	—
30	C₂×C₄×D₄×C₃₁	C₂²	1	2	yes	—
31	C₂×C₄×D₄×C₃₁	C₂²	1	2	yes	complemented
32	C₂×C₃₁×#117	C₂²	1	2	yes	—
33	C₂×C₃₁×#117	C₂²	1	1	yes	characteristic
34	C₂²×C₄⋊C₄×C₃₁	C₂²	1	2	yes	—
35	C₂²×C₂²⋊C₄×C₃₁	C₂²	1	2	yes	—
36	C₃₁×#391	C₂²	1	2	yes	complemented
37	C₃₁×#390	C₂²	1	4	yes	—
38	C₃₁×#390	C₂²	1	4	yes	—
39	C₃₁×#390	C₂²	1	2	yes	complemented
40	C₃₁×#389	C₂²	1	4	yes	complemented
41	C₃₁×#389	C₂²	1	2	yes	complemented
42	C₃₁×#389	C₂²	1	4	yes	complemented
43	C₃₁×#389	C₂²	1	2	yes	complemented
44	C₃₁×#388	C₂²	1	4	yes	—
45	C₃₁×#388	C₂²	1	4	yes	—
46	C₃₁×#388	C₂²	1	4	yes	—
47	C₃₁×#388	C₂²	1	4	yes	—
48	C₃₁×#388	C₂²	1	4	yes	complemented
49	C₃₁×#388	C₂²	1	4	yes	complemented
50	C₃₁×#387	C₂²	1	4	yes	complemented
51	C₃₁×#387	C₂²	1	4	yes	complemented
52	C₃₁×#386	C₂²	1	4	yes	—
53	C₃₁×#386	C₂²	1	4	yes	—
54	C₃₁×#386	C₂²	1	4	yes	—
55	C₃₁×#385	C₂²	1	4	yes	—
56	C₃₁×#385	C₂²	1	4	yes	complemented
57	C₃₁×#385	C₂²	1	4	yes	complemented
58	C₃₁×#383	C₂²	1	2	yes	complemented
59	C₃₁×#382	C₂²	1	2	yes	complemented
60	C₃₁×#381	C₂²	1	1	yes	characteristic
61	C₃₁×#381	C₂²	1	1	yes	characteristic
62	C₃₁×#381	C₂²	1	2	yes	complemented
63	C₃₁×#380	C₂²	1	2	yes	—
64	C₃₁×#380	C₂²	1	2	yes	—
65	C₃₁×#380	C₂²	1	1	yes	characteristic
66	C₃₁×#380	C₂²	1	1	yes	characteristic
67	C₃₁×#380	C₂²	1	1	yes	characteristic
68	C₃₁×#380	C₂²	1	1	yes	characteristic
69	C₃₁×#380	C₂²	1	2	yes	—
70	C₃₁×#380	C₂²	1	2	yes	—
71	C₃₁×#380	C₂²	1	2	yes	complemented
72	C₃₁×#379	C₂²	1	2	yes	—
73	C₄×C₄⋊C₄×C₃₁	C₂²	1	2	yes	complemented
74	C₄×C₄⋊C₄×C₃₁	C₂²	1	2	yes	complemented
75	C₄×C₄⋊C₄×C₃₁	C₂²	1	2	yes	complemented
76	C₄×C₂²⋊C₄×C₃₁	C₂²	1	4	yes	—
77	C₄×C₂²⋊C₄×C₃₁	C₂²	1	1	yes	characteristic
78	C₄×C₂²⋊C₄×C₃₁	C₂²	1	1	yes	characteristic
79	C₄×C₂²⋊C₄×C₃₁	C₂²	1	2	yes	—
80	C₄×C₂²⋊C₄×C₃₁	C₂²	1	2	yes	complemented
81	C₃₁×#376	C₂²	1	2	yes	complemented
82	C₂×C₃₁×#95	C₂²	1	2	yes	—
83	C₂×C₃₁×#95	C₂²	1	2	yes	complemented
84	C₂⁵×C₃₁	C₂³	1	1	yes	characteristic
85	C₂×#41×C₃₁	—	4	4	no	—
86	C₂×#41×C₃₁	C₂×C₄	1	8	yes	complemented
87	C₂²×Q₈×C₃₁	C₂³	1	1	yes	characteristic
88	C₂²×D₄×C₃₁	C₂³	1	2	yes	—
89	C₂²×D₄×C₃₁	C₂³	1	1	yes	characteristic
90	C₂²×D₄×C₃₁	C₂³	1	2	yes	—
91	C₂²×D₄×C₃₁	C₂³	1	2	yes	—
92	C₂³×C₄×C₃₁	—	4	4	no	—
93	C₂³×C₄×C₃₁	C₂×C₄	1	8	yes	complemented
94	C₂³×C₄×C₃₁	C₂³	1	1	yes	characteristic
95	C₂³×C₄×C₃₁	C₂³	1	1	yes	characteristic
96	C₂³×C₄×C₃₁	C₂³	1	1	yes	characteristic
97	C₂³×C₄×C₃₁	C₂³	1	4	yes	—
98	C₂³×C₄×C₃₁	C₂³	1	4	yes	—
99	C₂³×C₄×C₃₁	C₂³	1	1	yes	characteristic
100	C₂³×C₄×C₃₁	C₂³	1	1	yes	characteristic
101	C₂³×C₄×C₃₁	C₂³	1	2	yes	—
102	C₂³×C₄×C₃₁	C₂³	1	1	yes	characteristic
103	C₃₁×#123	C₂×C₄	1	16	yes	complemented
104	C₃₁×C₂²⋊Q₈	C₂×C₄	1	8	yes	complemented
105	C₃₁×C₂²⋊Q₈	C₂×C₄	1	8	yes	complemented
106	C₃₁×#121	C₂×C₄	1	8	yes	—
107	C₃₁×#121	C₂×C₄	1	8	yes	complemented
108	C₃₁×#120	C₂×C₄	1	16	yes	complemented
109	C₄×Q₈×C₃₁	—	4	4	no	—
110	C₄×D₄×C₃₁	—	4	4	no	—
111	C₄×D₄×C₃₁	—	4	8	no	—
112	C₄×D₄×C₃₁	C₂×C₄	1	16	yes	complemented
113	C₄×D₄×C₃₁	C₂×C₄	1	16	yes	complemented
114	C₃₁×#117	—	4	4	no	—
115	C₃₁×#117	C₂×C₄	1	8	yes	complemented
116	C₂×C₄⋊C₄×C₃₁	—	4	4	no	—
117	C₂×C₄⋊C₄×C₃₁	C₂³	1	2	yes	—
118	C₂×C₄⋊C₄×C₃₁	C₂³	1	2	yes	—
119	C₂×C₄⋊C₄×C₃₁	C₂³	1	4	yes	—
120	C₂×C₄⋊C₄×C₃₁	C₂³	1	2	yes	—
121	C₂×C₄⋊C₄×C₃₁	C₂³	1	2	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	1	yes	characteristic
126	C₂×C₄⋊C₄×C₃₁	C₂³	1	1	yes	characteristic
127	C₂×C₄⋊C₄×C₃₁	C₂³	1	2	yes	—
128	C₂×C₄⋊C₄×C₃₁	C₂³	1	2	yes	—
129	C₂×C₂²⋊C₄×C₃₁	—	4	4	no	—
130	C₂×C₂²⋊C₄×C₃₁	C₂³	1	2	yes	—
131	C₂×C₂²⋊C₄×C₃₁	C₂³	1	2	yes	—
132	C₂×C₂²⋊C₄×C₃₁	C₂³	1	2	yes	—
133	C₂×C₂²⋊C₄×C₃₁	C₂³	1	4	yes	—
134	C₂×C₂²⋊C₄×C₃₁	C₂³	1	2	yes	—
135	C₂×C₂²⋊C₄×C₃₁	C₂³	1	2	yes	—
136	C₂×C₂²⋊C₄×C₃₁	C₂³	1	2	yes	—
137	C₂×C₂²⋊C₄×C₃₁	C₂³	1	4	yes	—
138	C₂×C₂²⋊C₄×C₃₁	C₂³	1	4	yes	—
139	C₂×C₂²⋊C₄×C₃₁	C₂³	1	2	yes	—
140	C₂×C₂²⋊C₄×C₃₁	C₂³	1	2	yes	—
141	C₂×C₂²⋊C₄×C₃₁	C₂³	1	4	yes	—
142	C₂×C₂²⋊C₄×C₃₁	C₂³	1	2	yes	—
143	C₂×C₂²⋊C₄×C₃₁	C₂³	1	2	yes	—
144	C₂×C₂²⋊C₄×C₃₁	C₂³	1	2	yes	—
145	C₂×C₂²⋊C₄×C₃₁	C₂³	1	2	yes	—
146	C₂×C₂²⋊C₄×C₃₁	C₂³	1	2	yes	—
147	C₂×C₄²×C₃₁	C₂³	1	2	yes	—
148	C₂×C₄²×C₃₁	C₂³	1	2	yes	—
149	C₂×C₄²×C₃₁	C₂³	1	4	yes	—
150	C₂×C₄²×C₃₁	C₂³	1	4	yes	—
151	C₂×C₄²×C₃₁	C₂³	1	4	yes	—
152	C₂×C₄²×C₃₁	C₂³	1	2	yes	—
153	C₃₁×#95	C₂³	1	2	yes	—
154	C₃₁×#95	C₂³	1	2	yes	—
155	C₃₁×#95	C₂³	1	4	yes	—
156	C₃₁×#95	C₂³	1	1	yes	characteristic
157	C₃₁×#95	C₂³	1	4	yes	—
158	C₃₁×#95	C₂³	1	1	yes	characteristic
159	C₃₁×#95	C₂³	1	4	yes	—
160	C₃₁×#95	C₂³	1	1	yes	characteristic
161	C₃₁×#95	C₂³	1	1	yes	characteristic
162	C₃₁×#95	C₂³	1	2	yes	—
163	C₃₁×#95	C₂³	1	2	yes	—
164	C₃₁×#95	C₂³	1	4	yes	—
165	C₃₁×#95	C₂³	1	4	yes	—
166	C₃₁×#95	C₂³	1	4	yes	—
167	C₃₁×#95	C₂³	1	4	yes	—
168	C₂⁴×C₃₁	—	2	2	no	—
169	C₂⁴×C₃₁	—	2	2	no	—
170	C₂⁴×C₃₁	—	4	2	no	—
171	C₂⁴×C₃₁	—	4	2	no	—
172	C₂⁴×C₃₁	C₂²×C₄	1	4	yes	—
173	C₂⁴×C₃₁	C₂⁴	1	1	yes	characteristic
174	C₂⁴×C₃₁	C₂⁴	1	1	yes	characteristic
175	C₂⁴×C₃₁	C₂⁴	1	2	yes	—
176	C₂⁴×C₃₁	C₂⁴	1	1	yes	characteristic
177	#41×C₃₁	—	4	16	no	—
178	C₂×Q₈×C₃₁	—	2	4	no	—
179	C₂×Q₈×C₃₁	C₂²×C₄	1	4	yes	—
180	C₂×D₄×C₃₁	—	2	4	no	—
181	C₂×D₄×C₃₁	—	2	4	no	—
182	C₂×D₄×C₃₁	—	2	4	no	—
183	C₂×D₄×C₃₁	—	4	4	no	—
184	C₂×D₄×C₃₁	—	4	4	no	—
185	C₂×D₄×C₃₁	C₂²×C₄	1	8	yes	—
186	C₂×D₄×C₃₁	C₂²×C₄	1	4	yes	—
187	C₂×D₄×C₃₁	C₂²×C₄	1	8	yes	—
188	C₂×D₄×C₃₁	C₂²×C₄	1	8	yes	—
189	C₂²×C₄×C₃₁	—	2	4	no	—
190	C₂²×C₄×C₃₁	—	2	4	no	—
191	C₂²×C₄×C₃₁	—	2	4	no	—
192	C₂²×C₄×C₃₁	—	2	8	no	—
193	C₂²×C₄×C₃₁	—	4	16	no	—
194	C₂²×C₄×C₃₁	—	4	2	no	—
195	C₂²×C₄×C₃₁	—	4	2	no	—
196	C₂²×C₄×C₃₁	—	4	2	no	—
197	C₂²×C₄×C₃₁	—	4	4	no	—
198	C₂²×C₄×C₃₁	—	4	4	no	—
199	C₂²×C₄×C₃₁	—	4	4	no	—
200	C₂²×C₄×C₃₁	—	4	4	no	—
201	C₂²×C₄×C₃₁	—	4	4	no	—
202	C₂²×C₄×C₃₁	—	4	4	no	—
203	C₂²×C₄×C₃₁	C₂²×C₄	1	4	yes	—
204	C₂²×C₄×C₃₁	C₂²×C₄	1	4	yes	—
205	C₂²×C₄×C₃₁	C₂²×C₄	1	4	yes	—
206	C₂²×C₄×C₃₁	C₂²×C₄	1	4	yes	—
207	C₂²×C₄×C₃₁	C₂²×C₄	1	4	yes	—
208	C₂²×C₄×C₃₁	C₂²×C₄	1	8	yes	—
209	C₂²×C₄×C₃₁	C₂²×C₄	1	4	yes	—
210	C₂²×C₄×C₃₁	C₂⁴	1	2	yes	—
211	C₂²×C₄×C₃₁	C₂⁴	1	2	yes	—
212	C₂²×C₄×C₃₁	C₂⁴	1	2	yes	—
213	C₂²×C₄×C₃₁	C₂⁴	1	4	yes	—
214	C₂²×C₄×C₃₁	C₂⁴	1	4	yes	—
215	C₂²×C₄×C₃₁	C₂⁴	1	2	yes	—
216	C₂²×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
217	C₂²×C₄×C₃₁	C₂⁴	1	2	yes	—
218	C₂²×C₄×C₃₁	C₂⁴	1	2	yes	—
219	C₂²×C₄×C₃₁	C₂⁴	1	2	yes	—
220	C₂²×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
221	C₂²×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
222	C₂²×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
223	C₄⋊C₄×C₃₁	—	2	4	no	—
224	C₄⋊C₄×C₃₁	—	2	4	no	—
225	C₄⋊C₄×C₃₁	—	4	4	no	—
226	C₄⋊C₄×C₃₁	—	4	4	no	—
227	C₄⋊C₄×C₃₁	—	4	2	no	—
228	C₄⋊C₄×C₃₁	—	4	2	no	—
229	C₄⋊C₄×C₃₁	—	4	4	no	—
230	C₄⋊C₄×C₃₁	—	4	4	no	—
231	C₄⋊C₄×C₃₁	C₂²×C₄	1	4	yes	—
232	C₄⋊C₄×C₃₁	C₂²×C₄	1	4	yes	—
233	C₄⋊C₄×C₃₁	C₂²×C₄	1	8	yes	—
234	C₄⋊C₄×C₃₁	C₂²×C₄	1	8	yes	—
235	C₂²⋊C₄×C₃₁	—	2	4	no	—
236	C₂²⋊C₄×C₃₁	—	2	4	no	—
237	C₂²⋊C₄×C₃₁	—	4	2	no	—
238	C₂²⋊C₄×C₃₁	—	4	4	no	—
239	C₂²⋊C₄×C₃₁	—	4	2	no	—
240	C₂²⋊C₄×C₃₁	—	4	2	no	—
241	C₂²⋊C₄×C₃₁	—	4	4	no	—
242	C₂²⋊C₄×C₃₁	—	4	2	no	—
243	C₂²⋊C₄×C₃₁	—	4	2	no	—
244	C₂²⋊C₄×C₃₁	—	4	2	no	—
245	C₂²⋊C₄×C₃₁	—	4	4	no	—
246	C₂²⋊C₄×C₃₁	—	4	4	no	—
247	C₂²⋊C₄×C₃₁	C₂²×C₄	1	8	yes	—
248	C₂²⋊C₄×C₃₁	C₂²×C₄	1	8	yes	—
249	C₂²⋊C₄×C₃₁	C₂²×C₄	1	8	yes	—
250	C₂²⋊C₄×C₃₁	C₂²×C₄	1	8	yes	—
251	C₂²⋊C₄×C₃₁	C₂²×C₄	1	8	yes	—
252	C₄²×C₃₁	—	2	4	no	—
253	C₄²×C₃₁	—	2	4	no	—
254	C₄²×C₃₁	—	4	4	no	—
255	C₄²×C₃₁	—	4	4	no	—
256	C₄²×C₃₁	—	4	4	no	—
257	C₄²×C₃₁	C₂²×C₄	1	8	yes	—
258	#34404	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
259	C₂³×C₃₁	—	2	2	no	—
260	C₂³×C₃₁	—	2	2	no	—
261	C₂³×C₃₁	—	2	8	no	—
262	C₂³×C₃₁	—	2	1	no	—
263	C₂³×C₃₁	—	2	1	no	—
264	C₂³×C₃₁	—	2	2	no	—
265	C₂³×C₃₁	—	2	2	no	—
266	C₂³×C₃₁	—	2	4	no	—
267	C₂³×C₃₁	—	2	2	no	—
268	C₂³×C₃₁	—	2	2	no	—
269	C₂³×C₃₁	—	2	1	no	—
270	C₂³×C₃₁	—	2	1	no	—
271	C₂³×C₃₁	—	2	2	no	—
272	C₂³×C₃₁	—	2	4	no	—
273	C₂³×C₃₁	—	2	2	no	—
274	C₂³×C₃₁	—	4	4	no	—
275	C₂³×C₃₁	—	4	4	no	—
276	C₂³×C₃₁	—	4	8	no	—
277	C₂³×C₃₁	—	4	8	no	—
278	C₂³×C₃₁	C₂³×C₄	1	2	yes	—
279	C₂³×C₃₁	C₂³×C₄	1	2	yes	—
280	C₂³×C₃₁	C₂³×C₄	1	4	yes	—
281	C₂³×C₃₁	C₂³×C₄	1	2	yes	—
282	C₂³×C₃₁	H(5,𝔽₂)	1	2	yes	—
283	C₂³×C₃₁	2₋¹⁺⁴	1	2	yes	—
284	C₂³×C₃₁	C₂⁵	1	1	yes	characteristic
285	Q₈×C₃₁	—	2	8	no	—
286	D₄×C₃₁	—	2	16	no	—
287	D₄×C₃₁	—	2	8	no	—
288	D₄×C₃₁	—	4	8	no	—
289	D₄×C₃₁	—	4	8	no	—
290	C₂×C₄×C₃₁	—	2	2	no	—
291	C₂×C₄×C₃₁	—	2	4	no	—
292	C₂×C₄×C₃₁	—	2	2	no	—
293	C₂×C₄×C₃₁	—	2	2	no	—
294	C₂×C₄×C₃₁	—	2	2	no	—
295	C₂×C₄×C₃₁	—	2	2	no	—
296	C₂×C₄×C₃₁	—	2	4	no	—
297	C₂×C₄×C₃₁	—	2	8	no	—
298	C₂×C₄×C₃₁	—	2	8	no	—
299	C₂×C₄×C₃₁	—	2	8	no	—
300	C₂×C₄×C₃₁	—	2	2	no	—
301	C₂×C₄×C₃₁	—	2	2	no	—
302	C₂×C₄×C₃₁	—	2	2	no	—
303	C₂×C₄×C₃₁	—	2	2	no	—
304	C₂×C₄×C₃₁	—	2	2	no	—
305	C₂×C₄×C₃₁	—	2	4	no	—
306	C₂×C₄×C₃₁	—	2	4	no	—
307	C₂×C₄×C₃₁	—	2	2	no	—
308	C₂×C₄×C₃₁	—	2	4	no	—
309	C₂×C₄×C₃₁	—	2	4	no	—
310	C₂×C₄×C₃₁	—	2	2	no	—
311	C₂×C₄×C₃₁	—	2	2	no	—
312	C₂×C₄×C₃₁	—	2	2	no	—
313	C₂×C₄×C₃₁	—	2	4	no	—
314	C₂×C₄×C₃₁	—	2	2	no	—
315	C₂×C₄×C₃₁	—	2	2	no	—
316	C₂×C₄×C₃₁	—	2	2	no	—
317	C₂×C₄×C₃₁	—	2	4	no	—
318	C₂×C₄×C₃₁	—	4	4	no	—
319	C₂×C₄×C₃₁	—	4	4	no	—
320	C₂×C₄×C₃₁	—	4	4	no	—
321	C₂×C₄×C₃₁	—	4	8	no	—
322	C₂×C₄×C₃₁	—	8	8	no	—
323	C₂×C₄×C₃₁	—	8	8	no	—
324	C₂×C₄×C₃₁	C₂³×C₄	1	2	yes	—
325	C₂×C₄×C₃₁	C₂³×C₄	1	4	yes	—
326	C₂×C₄×C₃₁	C₂³×C₄	1	4	yes	—
327	C₂×C₄×C₃₁	C₂³×C₄	1	2	yes	—
328	C₂×C₄×C₃₁	C₂³×C₄	1	2	yes	—
329	C₂×C₄×C₃₁	C₂³×C₄	1	4	yes	—
330	C₂×C₄×C₃₁	C₂³×C₄	1	2	yes	—
331	C₂×C₄×C₃₁	H(5,𝔽₂)	1	4	yes	—
332	C₂×#525	C₂×C₃₁	1	1	yes	characteristic
333	#2376	C₂×C₃₁	1	4	yes	complemented
334	#2361	C₂×C₃₁	1	2	yes	complemented
335	#2360	C₂×C₃₁	1	2	yes	complemented
336	#2353	C₂×C₃₁	1	2	yes	complemented
337	#2349	C₂×C₃₁	1	2	yes	complemented
338	#2347	C₂×C₃₁	1	1	yes	characteristic, complemented
339	#2346	C₂×C₃₁	1	1	yes	characteristic, complemented
340	#2341	C₂×C₃₁	1	1	yes	characteristic, complemented
341	#2339	C₂×C₃₁	1	1	yes	characteristic, complemented
342	#2324	C₂×C₃₁	1	2	yes	complemented
343	#2323	C₂×C₃₁	1	2	yes	complemented
344	#2321	C₂×C₃₁	1	4	yes	complemented
345	#2319	C₂×C₃₁	1	2	yes	complemented
346	#2318	C₂×C₃₁	1	2	yes	complemented
347	#2280	C₂×C₃₁	1	1	yes	characteristic, complemented
348	C₂×#380	C₂×C₃₁	1	1	yes	characteristic, complemented
349	C₂²×C₃₁	—	2	2	no	—
350	C₂²×C₃₁	—	2	2	no	—
351	C₂²×C₃₁	—	2	4	no	—
352	C₂²×C₃₁	—	2	4	no	—
353	C₂²×C₃₁	—	2	2	no	—
354	C₂²×C₃₁	—	2	2	no	—
355	C₂²×C₃₁	—	2	4	no	—
356	C₂²×C₃₁	—	2	4	no	—
357	C₂²×C₃₁	—	2	2	no	—
358	C₂²×C₃₁	—	2	2	no	—
359	C₂²×C₃₁	—	2	2	no	—
360	C₂²×C₃₁	—	2	4	no	—
361	C₂²×C₃₁	—	2	2	no	—
362	C₂²×C₃₁	—	4	16	no	—
363	C₂²×C₃₁	—	4	4	no	—
364	C₂²×C₃₁	—	4	2	no	—
365	C₂²×C₃₁	—	4	2	no	—
366	C₂²×C₃₁	—	4	4	no	—
367	C₂²×C₃₁	—	4	2	no	—
368	C₂²×C₃₁	—	4	2	no	—
369	C₂²×C₃₁	#520	1	4	yes	—
370	C₂²×C₃₁	C₂⁴×C₄	1	1	yes	characteristic
371	C₂²×C₃₁	C₂×H(5,𝔽₂)	1	2	yes	—
372	C₂²×C₃₁	C₂×H(5,𝔽₂)	1	1	yes	characteristic
373	C₂²×C₃₁	C₂×2₋¹⁺⁴	1	1	yes	characteristic
374	C₂²×C₃₁	#585	1	2	yes	—
375	C₄×C₃₁	—	2	2	no	—
376	C₄×C₃₁	—	2	2	no	—
377	C₄×C₃₁	—	2	2	no	—
378	C₄×C₃₁	—	2	4	no	—
379	C₄×C₃₁	—	2	4	no	—
380	C₄×C₃₁	—	2	4	no	—
381	C₄×C₃₁	—	4	2	no	—
382	C₄×C₃₁	—	4	2	no	—
383	C₄×C₃₁	—	4	2	no	—
384	C₄×C₃₁	—	4	4	no	—
385	C₄×C₃₁	—	4	2	no	—
386	C₄×C₃₁	—	4	4	no	—
387	C₄×C₃₁	#518	1	4	yes	—
388	C₂²×#41	C₂²×C₃₁	1	1	yes	characteristic
389	C₂⁴×C₄	C₂²×C₃₁	1	1	yes	characteristic
390	#525	C₄×C₃₁	1	16	yes	complemented
391	C₂×#123	C₂²×C₃₁	1	2	yes	—
392	C₂×C₂²⋊Q₈	C₂²×C₃₁	1	1	yes	characteristic
393	C₂×C₂²⋊Q₈	C₂²×C₃₁	1	1	yes	characteristic
394	C₂×#121	C₂²×C₃₁	1	1	yes	characteristic
395	C₂×#121	C₂²×C₃₁	1	1	yes	characteristic
396	C₂×#120	C₂²×C₃₁	1	2	yes	—
397	C₂×C₄×Q₈	C₂²×C₃₁	1	2	yes	complemented
398	C₂×C₄×D₄	C₂²×C₃₁	1	4	yes	—
399	C₂×C₄×D₄	C₂²×C₃₁	1	2	yes	—
400	C₂×C₄×D₄	C₂²×C₃₁	1	2	yes	—
401	C₂×C₄×D₄	C₂²×C₃₁	1	2	yes	complemented
402	C₂×#117	C₂²×C₃₁	1	2	yes	—
403	C₂×#117	C₂²×C₃₁	1	1	yes	characteristic
404	C₂²×C₄⋊C₄	C₂²×C₃₁	1	2	yes	—
405	C₂²×C₂²⋊C₄	C₂²×C₃₁	1	2	yes	—
406	#391	C₂²×C₃₁	1	2	yes	complemented
407	#390	C₂²×C₃₁	1	4	yes	—
408	#390	C₂²×C₃₁	1	4	yes	—
409	#390	C₂²×C₃₁	1	2	yes	complemented
410	#389	C₂²×C₃₁	1	4	yes	complemented
411	#389	C₂²×C₃₁	1	2	yes	complemented
412	#389	C₂²×C₃₁	1	4	yes	complemented
413	#389	C₂²×C₃₁	1	2	yes	complemented
414	#388	C₂²×C₃₁	1	4	yes	—
415	#388	C₂²×C₃₁	1	4	yes	—
416	#388	C₂²×C₃₁	1	4	yes	—
417	#388	C₂²×C₃₁	1	4	yes	—
418	#388	C₂²×C₃₁	1	4	yes	complemented
419	#388	C₂²×C₃₁	1	4	yes	complemented
420	#387	C₂²×C₃₁	1	4	yes	complemented
421	#387	C₂²×C₃₁	1	4	yes	complemented
422	#386	C₂²×C₃₁	1	4	yes	—
423	#386	C₂²×C₃₁	1	4	yes	—
424	#386	C₂²×C₃₁	1	4	yes	—
425	#385	C₂²×C₃₁	1	4	yes	—
426	#385	C₂²×C₃₁	1	4	yes	complemented
427	#385	C₂²×C₃₁	1	4	yes	complemented
428	#383	C₂²×C₃₁	1	2	yes	complemented
429	#382	C₂²×C₃₁	1	2	yes	complemented
430	#381	C₂²×C₃₁	1	1	yes	characteristic
431	#381	C₂²×C₃₁	1	1	yes	characteristic
432	#381	C₂²×C₃₁	1	2	yes	complemented
433	#380	C₂²×C₃₁	1	2	yes	—
434	#380	C₂²×C₃₁	1	2	yes	—
435	#380	C₂²×C₃₁	1	1	yes	characteristic
436	#380	C₂²×C₃₁	1	1	yes	characteristic
437	#380	C₂²×C₃₁	1	1	yes	characteristic
438	#380	C₂²×C₃₁	1	1	yes	characteristic
439	#380	C₂²×C₃₁	1	2	yes	—
440	#380	C₂²×C₃₁	1	2	yes	—
441	#380	C₂²×C₃₁	1	2	yes	complemented
442	#379	C₂²×C₃₁	1	2	yes	—
443	C₄×C₄⋊C₄	C₂²×C₃₁	1	2	yes	complemented
444	C₄×C₄⋊C₄	C₂²×C₃₁	1	2	yes	complemented
445	C₄×C₄⋊C₄	C₂²×C₃₁	1	2	yes	complemented
446	C₄×C₂²⋊C₄	C₂²×C₃₁	1	4	yes	—
447	C₄×C₂²⋊C₄	C₂²×C₃₁	1	1	yes	characteristic
448	C₄×C₂²⋊C₄	C₂²×C₃₁	1	1	yes	characteristic
449	C₄×C₂²⋊C₄	C₂²×C₃₁	1	2	yes	—
450	C₄×C₂²⋊C₄	C₂²×C₃₁	1	2	yes	complemented
451	#376	C₂²×C₃₁	1	2	yes	complemented
452	C₂×#95	C₂²×C₃₁	1	2	yes	—
453	C₂×#95	C₂²×C₃₁	1	2	yes	complemented
454	C₂×C₃₁	—	2	4	no	—
455	C₂×C₃₁	—	4	4	no	—
456	C₂×C₃₁	—	4	2	no	—
457	C₂×C₃₁	—	4	2	no	—
458	C₂×C₃₁	C₂×#518	1	1	yes	characteristic
459	C₂×C₃₁	C₂×#520	1	1	yes	characteristic
460	C₂×C₃₁	#3428	1	1	yes	characteristic
461	C₂×C₃₁	#3541	1	2	yes	—
462	C₂×C₃₁	#3542	1	2	yes	—
463	C₂⁵	C₂³×C₃₁	1	1	yes	characteristic
464	C₂×#41	—	4	4	no	—
465	C₂×#41	C₂×C₄×C₃₁	1	8	yes	complemented
466	C₂²×Q₈	C₂³×C₃₁	1	1	yes	characteristic
467	C₂²×D₄	C₂³×C₃₁	1	2	yes	—
468	C₂²×D₄	C₂³×C₃₁	1	1	yes	characteristic
469	C₂²×D₄	C₂³×C₃₁	1	2	yes	—
470	C₂²×D₄	C₂³×C₃₁	1	2	yes	—
471	C₂³×C₄	—	4	4	no	—
472	C₂³×C₄	C₂×C₄×C₃₁	1	8	yes	complemented
473	C₂³×C₄	C₂³×C₃₁	1	1	yes	characteristic
474	C₂³×C₄	C₂³×C₃₁	1	1	yes	characteristic
475	C₂³×C₄	C₂³×C₃₁	1	1	yes	characteristic
476	C₂³×C₄	C₂³×C₃₁	1	4	yes	—
477	C₂³×C₄	C₂³×C₃₁	1	4	yes	—
478	C₂³×C₄	C₂³×C₃₁	1	1	yes	characteristic
479	C₂³×C₄	C₂³×C₃₁	1	1	yes	characteristic
480	C₂³×C₄	C₂³×C₃₁	1	2	yes	—
481	C₂³×C₄	C₂³×C₃₁	1	1	yes	characteristic
482	#123	C₂×C₄×C₃₁	1	16	yes	complemented
483	C₂²⋊Q₈	C₂×C₄×C₃₁	1	8	yes	complemented
484	C₂²⋊Q₈	C₂×C₄×C₃₁	1	8	yes	complemented
485	#121	C₂×C₄×C₃₁	1	8	yes	—
486	#121	C₂×C₄×C₃₁	1	8	yes	complemented
487	#120	C₂×C₄×C₃₁	1	16	yes	complemented
488	C₄×Q₈	—	4	4	no	—
489	C₄×D₄	—	4	4	no	—
490	C₄×D₄	—	4	8	no	—
491	C₄×D₄	C₂×C₄×C₃₁	1	16	yes	complemented
492	C₄×D₄	C₂×C₄×C₃₁	1	16	yes	complemented
493	#117	—	4	4	no	—
494	#117	C₂×C₄×C₃₁	1	8	yes	complemented
495	C₂×C₄⋊C₄	—	4	4	no	—
496	C₂×C₄⋊C₄	C₂³×C₃₁	1	2	yes	—
497	C₂×C₄⋊C₄	C₂³×C₃₁	1	2	yes	—
498	C₂×C₄⋊C₄	C₂³×C₃₁	1	4	yes	—
499	C₂×C₄⋊C₄	C₂³×C₃₁	1	2	yes	—
500	C₂×C₄⋊C₄	C₂³×C₃₁	1	2	yes	—
501	C₂×C₄⋊C₄	C₂³×C₃₁	1	4	yes	—
502	C₂×C₄⋊C₄	C₂³×C₃₁	1	4	yes	—
503	C₂×C₄⋊C₄	C₂³×C₃₁	1	4	yes	—
504	C₂×C₄⋊C₄	C₂³×C₃₁	1	1	yes	characteristic
505	C₂×C₄⋊C₄	C₂³×C₃₁	1	1	yes	characteristic
506	C₂×C₄⋊C₄	C₂³×C₃₁	1	2	yes	—
507	C₂×C₄⋊C₄	C₂³×C₃₁	1	2	yes	—
508	C₂×C₂²⋊C₄	—	4	4	no	—
509	C₂×C₂²⋊C₄	C₂³×C₃₁	1	2	yes	—
510	C₂×C₂²⋊C₄	C₂³×C₃₁	1	2	yes	—
511	C₂×C₂²⋊C₄	C₂³×C₃₁	1	2	yes	—
512	C₂×C₂²⋊C₄	C₂³×C₃₁	1	4	yes	—
513	C₂×C₂²⋊C₄	C₂³×C₃₁	1	2	yes	—
514	C₂×C₂²⋊C₄	C₂³×C₃₁	1	2	yes	—
515	C₂×C₂²⋊C₄	C₂³×C₃₁	1	2	yes	—
516	C₂×C₂²⋊C₄	C₂³×C₃₁	1	4	yes	—
517	C₂×C₂²⋊C₄	C₂³×C₃₁	1	4	yes	—
518	C₂×C₂²⋊C₄	C₂³×C₃₁	1	2	yes	—
519	C₂×C₂²⋊C₄	C₂³×C₃₁	1	2	yes	—
520	C₂×C₂²⋊C₄	C₂³×C₃₁	1	4	yes	—
521	C₂×C₂²⋊C₄	C₂³×C₃₁	1	2	yes	—
522	C₂×C₂²⋊C₄	C₂³×C₃₁	1	2	yes	—
523	C₂×C₂²⋊C₄	C₂³×C₃₁	1	2	yes	—
524	C₂×C₂²⋊C₄	C₂³×C₃₁	1	2	yes	—
525	C₂×C₂²⋊C₄	C₂³×C₃₁	1	2	yes	—
526	C₂×C₄²	C₂³×C₃₁	1	2	yes	—
527	C₂×C₄²	C₂³×C₃₁	1	2	yes	—
528	C₂×C₄²	C₂³×C₃₁	1	4	yes	—
529	C₂×C₄²	C₂³×C₃₁	1	4	yes	—
530	C₂×C₄²	C₂³×C₃₁	1	4	yes	—
531	C₂×C₄²	C₂³×C₃₁	1	2	yes	—
532	#95	C₂³×C₃₁	1	2	yes	—
533	#95	C₂³×C₃₁	1	2	yes	—
534	#95	C₂³×C₃₁	1	4	yes	—
535	#95	C₂³×C₃₁	1	1	yes	characteristic
536	#95	C₂³×C₃₁	1	4	yes	—
537	#95	C₂³×C₃₁	1	1	yes	characteristic
538	#95	C₂³×C₃₁	1	4	yes	—
539	#95	C₂³×C₃₁	1	1	yes	characteristic
540	#95	C₂³×C₃₁	1	1	yes	characteristic
541	#95	C₂³×C₃₁	1	2	yes	—
542	#95	C₂³×C₃₁	1	2	yes	—
543	#95	C₂³×C₃₁	1	4	yes	—
544	#95	C₂³×C₃₁	1	4	yes	—
545	#95	C₂³×C₃₁	1	4	yes	—
546	#95	C₂³×C₃₁	1	4	yes	—
547	C₃₁	#34404	1	1	yes	characteristic, direct product factor, Sylow
548	C₂⁴	—	2	2	no	—
549	C₂⁴	—	2	2	no	—
550	C₂⁴	—	4	2	no	—
551	C₂⁴	—	4	2	no	—
552	C₂⁴	C₂²×C₄×C₃₁	1	4	yes	—
553	C₂⁴	C₂⁴×C₃₁	1	1	yes	characteristic
554	C₂⁴	C₂⁴×C₃₁	1	1	yes	characteristic
555	C₂⁴	C₂⁴×C₃₁	1	2	yes	—
556	C₂⁴	C₂⁴×C₃₁	1	1	yes	characteristic
557	#41	—	4	16	no	—
558	C₂×Q₈	—	2	4	no	—
559	C₂×Q₈	C₂²×C₄×C₃₁	1	4	yes	—
560	C₂×D₄	—	2	4	no	—
561	C₂×D₄	—	2	4	no	—
562	C₂×D₄	—	2	4	no	—
563	C₂×D₄	—	4	4	no	—
564	C₂×D₄	—	4	4	no	—
565	C₂×D₄	C₂²×C₄×C₃₁	1	8	yes	—
566	C₂×D₄	C₂²×C₄×C₃₁	1	4	yes	—
567	C₂×D₄	C₂²×C₄×C₃₁	1	8	yes	—
568	C₂×D₄	C₂²×C₄×C₃₁	1	8	yes	—
569	C₂²×C₄	—	2	4	no	—
570	C₂²×C₄	—	2	4	no	—
571	C₂²×C₄	—	2	4	no	—
572	C₂²×C₄	—	2	8	no	—
573	C₂²×C₄	—	4	16	no	—
574	C₂²×C₄	—	4	2	no	—
575	C₂²×C₄	—	4	2	no	—
576	C₂²×C₄	—	4	2	no	—
577	C₂²×C₄	—	4	4	no	—
578	C₂²×C₄	—	4	4	no	—
579	C₂²×C₄	—	4	4	no	—
580	C₂²×C₄	—	4	4	no	—
581	C₂²×C₄	—	4	4	no	—
582	C₂²×C₄	—	4	4	no	—
583	C₂²×C₄	C₂²×C₄×C₃₁	1	4	yes	—
584	C₂²×C₄	C₂²×C₄×C₃₁	1	4	yes	—
585	C₂²×C₄	C₂²×C₄×C₃₁	1	4	yes	—
586	C₂²×C₄	C₂²×C₄×C₃₁	1	4	yes	—
587	C₂²×C₄	C₂²×C₄×C₃₁	1	4	yes	—
588	C₂²×C₄	C₂²×C₄×C₃₁	1	8	yes	—
589	C₂²×C₄	C₂²×C₄×C₃₁	1	4	yes	—
590	C₂²×C₄	C₂⁴×C₃₁	1	2	yes	—
591	C₂²×C₄	C₂⁴×C₃₁	1	2	yes	—
592	C₂²×C₄	C₂⁴×C₃₁	1	2	yes	—
593	C₂²×C₄	C₂⁴×C₃₁	1	4	yes	—
594	C₂²×C₄	C₂⁴×C₃₁	1	4	yes	—
595	C₂²×C₄	C₂⁴×C₃₁	1	2	yes	—
596	C₂²×C₄	C₂⁴×C₃₁	1	1	yes	characteristic
597	C₂²×C₄	C₂⁴×C₃₁	1	2	yes	—
598	C₂²×C₄	C₂⁴×C₃₁	1	2	yes	—
599	C₂²×C₄	C₂⁴×C₃₁	1	2	yes	—
600	C₂²×C₄	C₂⁴×C₃₁	1	1	yes	characteristic
601	C₂²×C₄	C₂⁴×C₃₁	1	1	yes	characteristic
602	C₂²×C₄	C₂⁴×C₃₁	1	1	yes	characteristic
603	C₄⋊C₄	—	2	4	no	—
604	C₄⋊C₄	—	2	4	no	—
605	C₄⋊C₄	—	4	4	no	—
606	C₄⋊C₄	—	4	4	no	—
607	C₄⋊C₄	—	4	2	no	—
608	C₄⋊C₄	—	4	2	no	—
609	C₄⋊C₄	—	4	4	no	—
610	C₄⋊C₄	—	4	4	no	—
611	C₄⋊C₄	C₂²×C₄×C₃₁	1	4	yes	—
612	C₄⋊C₄	C₂²×C₄×C₃₁	1	4	yes	—
613	C₄⋊C₄	C₂²×C₄×C₃₁	1	8	yes	—
614	C₄⋊C₄	C₂²×C₄×C₃₁	1	8	yes	—
615	C₂²⋊C₄	—	2	4	no	—
616	C₂²⋊C₄	—	2	4	no	—
617	C₂²⋊C₄	—	4	2	no	—
618	C₂²⋊C₄	—	4	4	no	—
619	C₂²⋊C₄	—	4	2	no	—
620	C₂²⋊C₄	—	4	2	no	—
621	C₂²⋊C₄	—	4	4	no	—
622	C₂²⋊C₄	—	4	2	no	—
623	C₂²⋊C₄	—	4	2	no	—
624	C₂²⋊C₄	—	4	2	no	—
625	C₂²⋊C₄	—	4	4	no	—
626	C₂²⋊C₄	—	4	4	no	—
627	C₂²⋊C₄	C₂²×C₄×C₃₁	1	8	yes	—
628	C₂²⋊C₄	C₂²×C₄×C₃₁	1	8	yes	—
629	C₂²⋊C₄	C₂²×C₄×C₃₁	1	8	yes	—
630	C₂²⋊C₄	C₂²×C₄×C₃₁	1	8	yes	—
631	C₂²⋊C₄	C₂²×C₄×C₃₁	1	8	yes	—
632	C₄²	—	2	4	no	—
633	C₄²	—	2	4	no	—
634	C₄²	—	4	4	no	—
635	C₄²	—	4	4	no	—
636	C₄²	—	4	4	no	—
637	C₄²	C₂²×C₄×C₃₁	1	8	yes	—
638	C₂³	—	2	2	no	—
639	C₂³	—	2	2	no	—
640	C₂³	—	2	8	no	—
641	C₂³	—	2	1	no	—
642	C₂³	—	2	1	no	—
643	C₂³	—	2	2	no	—
644	C₂³	—	2	2	no	—
645	C₂³	—	2	4	no	—
646	C₂³	—	2	2	no	—
647	C₂³	—	2	2	no	—
648	C₂³	—	2	1	no	—
649	C₂³	—	2	1	no	—
650	C₂³	—	2	2	no	—
651	C₂³	—	2	4	no	—
652	C₂³	—	2	2	no	—
653	C₂³	—	4	4	no	—
654	C₂³	—	4	4	no	—
655	C₂³	—	4	8	no	—
656	C₂³	—	4	8	no	—
657	C₂³	C₂³×C₄×C₃₁	1	2	yes	—
658	C₂³	C₂³×C₄×C₃₁	1	2	yes	—
659	C₂³	C₂³×C₄×C₃₁	1	4	yes	—
660	C₂³	C₂³×C₄×C₃₁	1	2	yes	—
661	C₂³	C₃₁×H(5,𝔽₂)	1	2	yes	—
662	C₂³	C₃₁×2₋¹⁺⁴	1	2	yes	—
663	C₂³	C₂⁵×C₃₁	1	1	yes	characteristic
664	Q₈	—	2	8	no	—
665	D₄	—	2	16	no	—
666	D₄	—	2	8	no	—
667	D₄	—	4	8	no	—
668	D₄	—	4	8	no	—
669	C₂×C₄	—	2	2	no	—
670	C₂×C₄	—	2	4	no	—
671	C₂×C₄	—	2	2	no	—
672	C₂×C₄	—	2	2	no	—
673	C₂×C₄	—	2	2	no	—
674	C₂×C₄	—	2	2	no	—
675	C₂×C₄	—	2	4	no	—
676	C₂×C₄	—	2	8	no	—
677	C₂×C₄	—	2	8	no	—
678	C₂×C₄	—	2	8	no	—
679	C₂×C₄	—	2	2	no	—
680	C₂×C₄	—	2	2	no	—
681	C₂×C₄	—	2	2	no	—
682	C₂×C₄	—	2	2	no	—
683	C₂×C₄	—	2	2	no	—
684	C₂×C₄	—	2	4	no	—
685	C₂×C₄	—	2	4	no	—
686	C₂×C₄	—	2	2	no	—
687	C₂×C₄	—	2	4	no	—
688	C₂×C₄	—	2	4	no	—
689	C₂×C₄	—	2	2	no	—
690	C₂×C₄	—	2	2	no	—
691	C₂×C₄	—	2	2	no	—
692	C₂×C₄	—	2	4	no	—
693	C₂×C₄	—	2	2	no	—
694	C₂×C₄	—	2	2	no	—
695	C₂×C₄	—	2	2	no	—
696	C₂×C₄	—	2	4	no	—
697	C₂×C₄	—	4	4	no	—
698	C₂×C₄	—	4	4	no	—
699	C₂×C₄	—	4	4	no	—
700	C₂×C₄	—	4	8	no	—
701	C₂×C₄	—	8	8	no	—
702	C₂×C₄	—	8	8	no	—
703	C₂×C₄	C₂³×C₄×C₃₁	1	2	yes	—
704	C₂×C₄	C₂³×C₄×C₃₁	1	4	yes	—
705	C₂×C₄	C₂³×C₄×C₃₁	1	4	yes	—
706	C₂×C₄	C₂³×C₄×C₃₁	1	2	yes	—
707	C₂×C₄	C₂³×C₄×C₃₁	1	2	yes	—
708	C₂×C₄	C₂³×C₄×C₃₁	1	4	yes	—
709	C₂×C₄	C₂³×C₄×C₃₁	1	2	yes	—
710	C₂×C₄	C₃₁×H(5,𝔽₂)	1	4	yes	—
711	C₂²	—	2	2	no	—
712	C₂²	—	2	2	no	—
713	C₂²	—	2	4	no	—
714	C₂²	—	2	4	no	—
715	C₂²	—	2	2	no	—
716	C₂²	—	2	2	no	—
717	C₂²	—	2	4	no	—
718	C₂²	—	2	4	no	—
719	C₂²	—	2	2	no	—
720	C₂²	—	2	2	no	—
721	C₂²	—	2	2	no	—
722	C₂²	—	2	4	no	—
723	C₂²	—	2	2	no	—
724	C₂²	—	4	16	no	—
725	C₂²	—	4	2	no	—
726	C₂²	—	4	2	no	—
727	C₂²	—	4	4	no	—
728	C₂²	—	4	4	no	—
729	C₂²	—	4	2	no	—
730	C₂²	—	4	2	no	—
731	C₂²	C₃₁×#520	1	4	yes	—
732	C₂²	C₂⁴×C₄×C₃₁	1	1	yes	characteristic
733	C₂²	C₂×C₃₁×H(5,𝔽₂)	1	2	yes	—
734	C₂²	C₂×C₃₁×H(5,𝔽₂)	1	1	yes	characteristic
735	C₂²	C₂×C₃₁×2₋¹⁺⁴	1	1	yes	characteristic
736	C₂²	C₃₁×#585	1	2	yes	—
737	C₄	—	2	2	no	—
738	C₄	—	2	2	no	—
739	C₄	—	2	2	no	—
740	C₄	—	2	4	no	—
741	C₄	—	2	4	no	—
742	C₄	—	2	4	no	—
743	C₄	—	4	2	no	—
744	C₄	—	4	2	no	—
745	C₄	—	4	2	no	—
746	C₄	—	4	4	no	—
747	C₄	—	4	4	no	—
748	C₄	—	4	2	no	—
749	C₄	C₃₁×#518	1	4	yes	—
750	C₂	—	2	4	no	—
751	C₂	—	4	2	no	—
752	C₂	—	4	2	no	—
753	C₂	—	4	4	no	—
754	C₂	C₂×C₃₁×#518	1	1	yes	characteristic
755	C₂	C₂×C₃₁×#520	1	1	yes	characteristic
756	C₂	C₃₁×#3428	1	1	yes	characteristic
757	C₂	C₃₁×#3541	1	2	yes	—
758	C₂	C₃₁×#3542	1	2	yes	—

	subgroup	G/H
center, socle	C₂³×C₃₁	C₂⁵
commutator	C₂²	C₂⁴×C₄×C₃₁
Frattini, cosocle	C₂³	C₂⁵×C₃₁
prime radical	C₂×C₃₁×#525	C₂
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
2. C₃₁×#2376	C₂	semidirect
3. C₃₁×#2361	C₂	semidirect
4. C₃₁×#2360	C₂	semidirect
5. C₃₁×#2353	C₂	semidirect
6. C₃₁×#2349	C₂	semidirect
7. C₃₁×#2347	C₂	semidirect
8. C₃₁×#2346	C₂	semidirect
9. C₃₁×#2341	C₂	semidirect
10. C₃₁×#2339	C₂	semidirect
11. C₃₁×#2324	C₂	semidirect
12. C₃₁×#2323	C₂	semidirect
13. C₃₁×#2321	C₂	semidirect
14. C₃₁×#2319	C₂	semidirect
15. C₃₁×#2318	C₂	semidirect
16. C₃₁×#2280	C₂	semidirect
17. C₂×C₃₁×#380	C₂	semidirect
20. C₃₁×#525	C₄	semidirect
27. C₂×C₄×Q₈×C₃₁	C₂²	semidirect
31. C₂×C₄×D₄×C₃₁	C₂²	semidirect
36. C₃₁×#391	C₂²	semidirect
39. C₃₁×#390	C₂²	semidirect
40. C₃₁×#389	C₂²	semidirect
41. C₃₁×#389	C₂²	semidirect
42. C₃₁×#389	C₂²	semidirect
43. C₃₁×#389	C₂²	semidirect
48. C₃₁×#388	C₂²	semidirect
49. C₃₁×#388	C₂²	semidirect
50. C₃₁×#387	C₂²	semidirect
51. C₃₁×#387	C₂²	semidirect
56. C₃₁×#385	C₂²	semidirect
57. C₃₁×#385	C₂²	semidirect
58. C₃₁×#383	C₂²	semidirect
59. C₃₁×#382	C₂²	semidirect
62. C₃₁×#381	C₂²	semidirect
71. C₃₁×#380	C₂²	semidirect
73. C₄×C₄⋊C₄×C₃₁	C₂²	semidirect
74. C₄×C₄⋊C₄×C₃₁	C₂²	semidirect
75. C₄×C₄⋊C₄×C₃₁	C₂²	semidirect
80. C₄×C₂²⋊C₄×C₃₁	C₂²	semidirect
81. C₃₁×#376	C₂²	semidirect
83. C₂×C₃₁×#95	C₂²	semidirect
86. C₂×#41×C₃₁	C₂×C₄	semidirect
93. C₂³×C₄×C₃₁	C₂×C₄	semidirect
103. C₃₁×#123	C₂×C₄	semidirect
104. C₃₁×C₂²⋊Q₈	C₂×C₄	semidirect
105. C₃₁×C₂²⋊Q₈	C₂×C₄	semidirect
107. C₃₁×#121	C₂×C₄	semidirect
108. C₃₁×#120	C₂×C₄	semidirect
112. C₄×D₄×C₃₁	C₂×C₄	semidirect
113. C₄×D₄×C₃₁	C₂×C₄	semidirect
115. C₃₁×#117	C₂×C₄	semidirect
258. #34404	C₃₁	direct (N × H)
333. #2376	C₂×C₃₁	semidirect
334. #2361	C₂×C₃₁	semidirect
335. #2360	C₂×C₃₁	semidirect
336. #2353	C₂×C₃₁	semidirect
337. #2349	C₂×C₃₁	semidirect
338. #2347	C₂×C₃₁	semidirect
339. #2346	C₂×C₃₁	semidirect
340. #2341	C₂×C₃₁	semidirect
341. #2339	C₂×C₃₁	semidirect
342. #2324	C₂×C₃₁	semidirect
343. #2323	C₂×C₃₁	semidirect
344. #2321	C₂×C₃₁	semidirect
345. #2319	C₂×C₃₁	semidirect
346. #2318	C₂×C₃₁	semidirect
347. #2280	C₂×C₃₁	semidirect
348. C₂×#380	C₂×C₃₁	semidirect
390. #525	C₄×C₃₁	semidirect
397. C₂×C₄×Q₈	C₂²×C₃₁	semidirect
401. C₂×C₄×D₄	C₂²×C₃₁	semidirect
406. #391	C₂²×C₃₁	semidirect
409. #390	C₂²×C₃₁	semidirect
410. #389	C₂²×C₃₁	semidirect
411. #389	C₂²×C₃₁	semidirect
412. #389	C₂²×C₃₁	semidirect
413. #389	C₂²×C₃₁	semidirect
418. #388	C₂²×C₃₁	semidirect
419. #388	C₂²×C₃₁	semidirect
420. #387	C₂²×C₃₁	semidirect
421. #387	C₂²×C₃₁	semidirect
426. #385	C₂²×C₃₁	semidirect
427. #385	C₂²×C₃₁	semidirect
428. #383	C₂²×C₃₁	semidirect
429. #382	C₂²×C₃₁	semidirect
432. #381	C₂²×C₃₁	semidirect
441. #380	C₂²×C₃₁	semidirect
443. C₄×C₄⋊C₄	C₂²×C₃₁	semidirect
444. C₄×C₄⋊C₄	C₂²×C₃₁	semidirect
445. C₄×C₄⋊C₄	C₂²×C₃₁	semidirect
450. C₄×C₂²⋊C₄	C₂²×C₃₁	semidirect
451. #376	C₂²×C₃₁	semidirect
453. C₂×#95	C₂²×C₃₁	semidirect
465. C₂×#41	C₂×C₄×C₃₁	semidirect
472. C₂³×C₄	C₂×C₄×C₃₁	semidirect
482. #123	C₂×C₄×C₃₁	semidirect
483. C₂²⋊Q₈	C₂×C₄×C₃₁	semidirect
484. C₂²⋊Q₈	C₂×C₄×C₃₁	semidirect
486. #121	C₂×C₄×C₃₁	semidirect
487. #120	C₂×C₄×C₃₁	semidirect
491. C₄×D₄	C₂×C₄×C₃₁	semidirect
492. C₄×D₄	C₂×C₄×C₃₁	semidirect
494. #117	C₂×C₄×C₃₁	semidirect
547. C₃₁	#34404	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×C₃₁×#525	2
C₃₁×#2376	2
C₃₁×#2361	2
C₃₁×#2360	2
C₃₁×#2353	2
C₃₁×#2349	2
C₃₁×#2347	2
C₃₁×#2346	2
C₃₁×#2341	2
C₃₁×#2339	2
C₃₁×#2324	2
C₃₁×#2323	2
C₃₁×#2321	2
C₃₁×#2319	2
C₃₁×#2318	2
C₃₁×#2280	2
C₂×C₃₁×#380	2
#34404	31

Subgroups — C31×#34404 #21491163

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#34404 #21491163