Subgroups — C₃₁×#9223 #21465982

Order 7936.

546 conjugacy classes of subgroups.

The subgroup lattice has 414 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃₁×#476	C₂	1	1	yes	maximal, characteristic
2	C₂×C₃₁×#436	C₂	1	1	yes	maximal, characteristic
3	C₃₁×#1990	C₂	1	1	yes	maximal, characteristic, complemented
4	C₃₁×#1802	C₂	1	1	yes	maximal, characteristic
5	C₂×C₃₁×#333	C₂	1	1	yes	maximal, characteristic, complemented
6	C₃₁×#1296	C₂	1	1	yes	maximal, characteristic, complemented
7	C₃₁×#1277	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂²×M₁₆×C₃₁	—	2	1	no	—
9	C₂×C₃₁×C₄⋊Q₈	C₂²	1	1	yes	characteristic
10	C₂×C₄×D₄×C₃₁	C₂²	1	1	yes	characteristic
11	C₃₁×#476	—	2	2	no	—
12	C₃₁×#476	C₄	1	4	yes	complemented
13	C₃₁×#436	—	2	2	no	—
14	C₃₁×#436	—	2	2	no	—
15	C₂×C₃₁×#106	—	2	1	no	—
16	C₂×C₃₁×C₄⋊C₈	C₂²	1	1	yes	characteristic
17	C₂×C₃₁×C₄⋊C₈	C₂²	1	1	yes	characteristic
18	C₂×C₃₁×C₄⋊C₈	C₂²	1	1	yes	characteristic
19	C₂×C₃₁×#102	—	2	1	no	—
20	C₂×C₃₁×#102	—	2	1	no	—
21	C₂×C₃₁×C₂²⋊C₈	—	2	1	no	—
22	C₂×C₄×C₈×C₃₁	C₂²	1	1	yes	characteristic
23	C₃₁×#384	C₂²	1	1	yes	characteristic
24	C₃₁×#340	—	2	1	no	—
25	C₃₁×#340	—	2	1	no	—
26	C₃₁×#340	—	2	1	no	—
27	C₃₁×#333	—	2	2	no	—
28	C₂²×Q₈×C₃₁	—	2	1	no	—
29	C₂²×D₄×C₃₁	D₄	1	1	yes	characteristic
30	C₂³×C₄×C₃₁	—	2	1	no	—
31	C₂×M₁₆×C₃₁	—	2	4	no	—
32	C₂×M₁₆×C₃₁	—	4	2	no	—
33	C₂×M₁₆×C₃₁	—	4	2	no	—
34	C₂²×C₈×C₃₁	—	2	1	no	—
35	C₂²×C₈×C₃₁	—	2	1	no	—
36	C₂²×C₈×C₃₁	—	2	1	no	—
37	C₂²×C₈×C₃₁	D₄	1	1	yes	characteristic
38	C₂²×C₈×C₃₁	D₄	1	1	yes	characteristic
39	C₃₁×C₄⋊Q₈	—	2	1	no	—
40	C₃₁×C₄⋊Q₈	—	2	1	no	—
41	C₃₁×C₄⋊Q₈	—	2	1	no	—
42	C₃₁×C₄⋊Q₈	C₂×C₄	1	2	yes	—
43	C₄×D₄×C₃₁	—	2	1	no	—
44	C₄×D₄×C₃₁	—	2	1	no	—
45	C₄×D₄×C₃₁	C₂×C₄	1	2	yes	—
46	C₄×D₄×C₃₁	D₄	1	2	yes	—
47	C₂×C₄⋊C₄×C₃₁	—	2	1	no	—
48	C₂×C₄⋊C₄×C₃₁	—	2	1	no	—
49	C₂×C₄⋊C₄×C₃₁	—	2	1	no	—
50	C₂×C₄⋊C₄×C₃₁	—	2	1	no	—
51	C₂×C₄⋊C₄×C₃₁	—	2	1	no	—
52	C₂×C₄⋊C₄×C₃₁	D₄	1	1	yes	characteristic
53	C₂×C₂²⋊C₄×C₃₁	—	2	1	no	—
54	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
55	C₃₁×#106	—	2	4	no	—
56	C₃₁×C₄⋊C₈	—	2	1	no	—
57	C₃₁×C₄⋊C₈	—	2	1	no	—
58	C₃₁×C₄⋊C₈	—	2	1	no	—
59	C₃₁×C₄⋊C₈	—	2	1	no	—
60	C₃₁×C₄⋊C₈	C₂×C₄	1	2	yes	—
61	C₃₁×C₄⋊C₈	D₄	1	2	yes	—
62	C₃₁×#102	—	2	4	no	—
63	C₃₁×#102	—	4	2	no	—
64	C₃₁×C₂²⋊C₈	—	2	4	no	—
65	C₄×C₈×C₃₁	—	2	1	no	—
66	C₄×C₈×C₃₁	—	2	1	no	—
67	C₂⁴×C₃₁	—	2	1	no	—
68	C₂×Q₈×C₃₁	—	2	2	no	—
69	C₂×Q₈×C₃₁	—	4	1	no	—
70	C₂×Q₈×C₃₁	—	8	1	no	—
71	C₂×Q₈×C₃₁	—	8	1	no	—
72	C₂×D₄×C₃₁	—	2	1	no	—
73	C₂×D₄×C₃₁	—	2	1	no	—
74	C₂×D₄×C₃₁	—	2	1	no	—
75	C₂×D₄×C₃₁	—	2	1	no	—
76	C₂×D₄×C₃₁	—	2	1	no	—
77	C₂×D₄×C₃₁	C₂²⋊C₄	1	2	yes	—
78	C₂²×C₄×C₃₁	—	2	1	no	—
79	C₂²×C₄×C₃₁	—	2	2	no	—
80	C₂²×C₄×C₃₁	—	2	2	no	—
81	C₂²×C₄×C₃₁	—	4	1	no	—
82	C₂²×C₄×C₃₁	—	4	1	no	—
83	C₂²×C₄×C₃₁	—	4	1	no	—
84	C₂²×C₄×C₃₁	—	4	1	no	—
85	C₂²×C₄×C₃₁	—	4	1	no	—
86	C₂²×C₄×C₃₁	—	4	1	no	—
87	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
88	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
89	C₂²×C₄×C₃₁	#41	1	1	yes	characteristic
90	M₁₆×C₃₁	—	4	8	no	—
91	C₂×C₈×C₃₁	—	2	1	no	—
92	C₂×C₈×C₃₁	—	2	1	no	—
93	C₂×C₈×C₃₁	—	2	2	no	—
94	C₂×C₈×C₃₁	—	2	1	no	—
95	C₂×C₈×C₃₁	—	2	2	no	—
96	C₂×C₈×C₃₁	—	2	1	no	—
97	C₂×C₈×C₃₁	—	2	1	no	—
98	C₂×C₈×C₃₁	—	4	1	no	—
99	C₂×C₈×C₃₁	—	4	1	no	—
100	C₂×C₈×C₃₁	—	4	1	no	—
101	C₂×C₈×C₃₁	—	4	1	no	—
102	C₂×C₈×C₃₁	—	4	1	no	—
103	C₂×C₈×C₃₁	—	4	1	no	—
104	C₂×C₈×C₃₁	—	4	1	no	—
105	C₂×C₈×C₃₁	SD₁₆	1	2	yes	—
106	C₄⋊C₄×C₃₁	—	2	1	no	—
107	C₄⋊C₄×C₃₁	—	2	2	no	—
108	C₄⋊C₄×C₃₁	—	2	2	no	—
109	C₄⋊C₄×C₃₁	—	4	1	no	—
110	C₄⋊C₄×C₃₁	—	4	1	no	—
111	C₄⋊C₄×C₃₁	—	4	1	no	—
112	C₄⋊C₄×C₃₁	—	4	1	no	—
113	C₄⋊C₄×C₃₁	—	4	1	no	—
114	C₄⋊C₄×C₃₁	—	4	1	no	—
115	C₄⋊C₄×C₃₁	—	8	1	no	—
116	C₄⋊C₄×C₃₁	C₂²⋊C₄	1	2	yes	—
117	C₂²⋊C₄×C₃₁	—	2	2	no	—
118	C₂²⋊C₄×C₃₁	—	2	2	no	—
119	C₄²×C₃₁	C₂²×C₄	1	1	yes	characteristic
120	C₄²×C₃₁	C₂×D₄	1	1	yes	characteristic
121	C₄²×C₃₁	C₂×D₄	1	1	yes	characteristic
122	C₄²×C₃₁	#41	1	1	yes	characteristic
123	#9223	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
124	C₂³×C₃₁	—	2	2	no	—
125	C₂³×C₃₁	—	4	1	no	—
126	C₂³×C₃₁	—	4	1	no	—
127	C₂³×C₃₁	—	4	1	no	—
128	C₂³×C₃₁	—	4	1	no	—
129	C₂³×C₃₁	—	4	1	no	—
130	C₂³×C₃₁	—	4	1	no	—
131	C₂³×C₃₁	#121	1	1	yes	characteristic
132	Q₈×C₃₁	—	8	2	no	—
133	Q₈×C₃₁	—	8	2	no	—
134	D₄×C₃₁	—	2	2	no	—
135	D₄×C₃₁	—	2	2	no	—
136	D₄×C₃₁	—	4	1	no	—
137	D₄×C₃₁	—	4	1	no	—
138	C₂×C₄×C₃₁	—	2	1	no	—
139	C₂×C₄×C₃₁	—	2	1	no	—
140	C₂×C₄×C₃₁	—	2	2	no	—
141	C₂×C₄×C₃₁	—	4	1	no	—
142	C₂×C₄×C₃₁	—	4	1	no	—
143	C₂×C₄×C₃₁	—	4	2	no	—
144	C₂×C₄×C₃₁	—	4	2	no	—
145	C₂×C₄×C₃₁	—	4	2	no	—
146	C₂×C₄×C₃₁	—	4	2	no	—
147	C₂×C₄×C₃₁	—	4	2	no	—
148	C₂×C₄×C₃₁	—	8	1	no	—
149	C₂×C₄×C₃₁	—	8	1	no	—
150	C₂×C₄×C₃₁	—	8	1	no	—
151	C₂×C₄×C₃₁	—	8	1	no	—
152	C₂×C₄×C₃₁	—	8	1	no	—
153	C₂×C₄×C₃₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
154	C₂×C₄×C₃₁	C₄×D₄	1	1	yes	characteristic
155	C₂×C₄×C₃₁	C₄×D₄	1	1	yes	characteristic
156	C₂×C₄×C₃₁	#120	1	1	yes	characteristic
157	C₂×C₄×C₃₁	#121	1	1	yes	characteristic
158	C₂×C₄×C₃₁	#123	1	1	yes	characteristic
159	C₂×C₄×C₃₁	C₂×SD₁₆	1	1	yes	characteristic
160	C₂×C₄×C₃₁	#135	1	1	yes	characteristic
161	C₂×C₄×C₃₁	AGL(1,ℤ/8)	1	1	yes	characteristic
162	C₂×C₄×C₃₁	AGL(1,ℤ/8)	1	1	yes	characteristic
163	C₂×C₄×C₃₁	#137	1	1	yes	characteristic
164	C₂×C₄×C₃₁	#137	1	1	yes	characteristic
165	C₂×C₄×C₃₁	#137	1	1	yes	characteristic
166	C₂×C₄×C₃₁	#137	1	1	yes	characteristic
167	C₈×C₃₁	—	2	2	no	—
168	C₈×C₃₁	—	4	2	no	—
169	C₈×C₃₁	—	4	2	no	—
170	C₈×C₃₁	—	4	1	no	—
171	C₈×C₃₁	—	8	1	no	—
172	C₂×#476	C₂×C₃₁	1	1	yes	characteristic
173	C₂×#436	C₂×C₃₁	1	1	yes	characteristic
174	#1990	C₂×C₃₁	1	1	yes	characteristic, complemented
175	#1802	C₂×C₃₁	1	1	yes	characteristic
176	C₂×#333	C₂×C₃₁	1	1	yes	characteristic, complemented
177	#1296	C₂×C₃₁	1	1	yes	characteristic, complemented
178	#1277	C₂×C₃₁	1	1	yes	characteristic, complemented
179	C₂²×C₃₁	—	4	2	no	—
180	C₂²×C₃₁	—	4	2	no	—
181	C₂²×C₃₁	—	4	2	no	—
182	C₂²×C₃₁	—	8	1	no	—
183	C₂²×C₃₁	—	8	1	no	—
184	C₂²×C₃₁	—	8	1	no	—
185	C₂²×C₃₁	—	8	1	no	—
186	C₂²×C₃₁	#386	1	1	yes	characteristic
187	C₂²×C₃₁	#461	1	1	yes	characteristic
188	C₂²×C₃₁	#463	1	1	yes	characteristic
189	C₂²×C₃₁	#468	1	1	yes	characteristic
190	C₂²×C₃₁	#470	1	1	yes	characteristic
191	C₂²×C₃₁	#471	1	1	yes	characteristic
192	C₂²×C₃₁	#472	1	1	yes	characteristic
193	C₄×C₃₁	—	2	1	no	—
194	C₄×C₃₁	—	2	1	no	—
195	C₄×C₃₁	—	4	2	no	—
196	C₄×C₃₁	—	8	2	no	—
197	C₄×C₃₁	—	16	1	no	—
198	C₄×C₃₁	#417	1	1	yes	characteristic
199	C₄×C₃₁	#419	1	1	yes	characteristic
200	C₄×C₃₁	C₄×SD₁₆	1	1	yes	characteristic
201	C₄×C₃₁	#440	1	1	yes	characteristic
202	C₄×C₃₁	#450	1	1	yes	characteristic
203	C₄×C₃₁	#452	1	1	yes	characteristic
204	C₄×C₃₁	#468	1	1	yes	characteristic
205	C₄×C₃₁	#470	1	1	yes	characteristic
206	C₂²×M₁₆	—	2	1	no	—
207	C₂×C₄⋊Q₈	C₂²×C₃₁	1	1	yes	characteristic
208	C₂×C₄×D₄	C₂²×C₃₁	1	1	yes	characteristic
209	#476	—	2	2	no	—
210	#476	C₄×C₃₁	1	4	yes	complemented
211	#436	—	2	2	no	—
212	#436	—	2	2	no	—
213	C₂×#106	—	2	1	no	—
214	C₂×C₄⋊C₈	C₂²×C₃₁	1	1	yes	characteristic
215	C₂×C₄⋊C₈	C₂²×C₃₁	1	1	yes	characteristic
216	C₂×C₄⋊C₈	C₂²×C₃₁	1	1	yes	characteristic
217	C₂×#102	—	2	1	no	—
218	C₂×#102	—	2	1	no	—
219	C₂×C₂²⋊C₈	—	2	1	no	—
220	C₂×C₄×C₈	C₂²×C₃₁	1	1	yes	characteristic
221	#384	C₂²×C₃₁	1	1	yes	characteristic
222	#340	—	2	1	no	—
223	#340	—	2	1	no	—
224	#340	—	2	1	no	—
225	#333	—	2	2	no	—
226	C₂×C₃₁	—	8	2	no	—
227	C₂×C₃₁	#1691	1	1	yes	characteristic
228	C₂×C₃₁	#1695	1	1	yes	characteristic
229	C₂×C₃₁	#1697	1	1	yes	characteristic
230	C₂×C₃₁	#1703	1	1	yes	characteristic
231	C₂×C₃₁	#1877	1	1	yes	characteristic
232	C₂×C₃₁	#1894	1	1	yes	characteristic
233	C₂×C₃₁	#1907	1	1	yes	characteristic
234	C₂²×Q₈	—	2	1	no	—
235	C₂²×D₄	D₄×C₃₁	1	1	yes	characteristic
236	C₂³×C₄	—	2	1	no	—
237	C₂×M₁₆	—	2	4	no	—
238	C₂×M₁₆	—	4	2	no	—
239	C₂×M₁₆	—	4	2	no	—
240	C₂²×C₈	—	2	1	no	—
241	C₂²×C₈	—	2	1	no	—
242	C₂²×C₈	—	2	1	no	—
243	C₂²×C₈	D₄×C₃₁	1	1	yes	characteristic
244	C₂²×C₈	D₄×C₃₁	1	1	yes	characteristic
245	C₄⋊Q₈	—	2	1	no	—
246	C₄⋊Q₈	—	2	1	no	—
247	C₄⋊Q₈	—	2	1	no	—
248	C₄⋊Q₈	C₂×C₄×C₃₁	1	2	yes	—
249	C₄×D₄	—	2	1	no	—
250	C₄×D₄	—	2	1	no	—
251	C₄×D₄	C₂×C₄×C₃₁	1	2	yes	—
252	C₄×D₄	D₄×C₃₁	1	2	yes	—
253	C₂×C₄⋊C₄	—	2	1	no	—
254	C₂×C₄⋊C₄	—	2	1	no	—
255	C₂×C₄⋊C₄	—	2	1	no	—
256	C₂×C₄⋊C₄	—	2	1	no	—
257	C₂×C₄⋊C₄	—	2	1	no	—
258	C₂×C₄⋊C₄	D₄×C₃₁	1	1	yes	characteristic
259	C₂×C₂²⋊C₄	—	2	1	no	—
260	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
261	#106	—	2	4	no	—
262	C₄⋊C₈	—	2	1	no	—
263	C₄⋊C₈	—	2	1	no	—
264	C₄⋊C₈	—	2	1	no	—
265	C₄⋊C₈	—	2	1	no	—
266	C₄⋊C₈	C₂×C₄×C₃₁	1	2	yes	—
267	C₄⋊C₈	D₄×C₃₁	1	2	yes	—
268	#102	—	2	4	no	—
269	#102	—	4	2	no	—
270	C₂²⋊C₈	—	2	4	no	—
271	C₄×C₈	—	2	1	no	—
272	C₄×C₈	—	2	1	no	—
273	C₃₁	#9223	1	1	yes	characteristic, direct product factor, Sylow
274	C₂⁴	—	2	1	no	—
275	C₂×Q₈	—	2	2	no	—
276	C₂×Q₈	—	4	1	no	—
277	C₂×Q₈	—	8	1	no	—
278	C₂×Q₈	—	8	1	no	—
279	C₂×D₄	—	2	1	no	—
280	C₂×D₄	—	2	1	no	—
281	C₂×D₄	—	2	1	no	—
282	C₂×D₄	—	2	1	no	—
283	C₂×D₄	—	2	1	no	—
284	C₂×D₄	C₂²⋊C₄×C₃₁	1	2	yes	—
285	C₂²×C₄	—	2	1	no	—
286	C₂²×C₄	—	2	2	no	—
287	C₂²×C₄	—	2	2	no	—
288	C₂²×C₄	—	4	1	no	—
289	C₂²×C₄	—	4	1	no	—
290	C₂²×C₄	—	4	1	no	—
291	C₂²×C₄	—	4	1	no	—
292	C₂²×C₄	—	4	1	no	—
293	C₂²×C₄	—	4	1	no	—
294	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
295	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
296	C₂²×C₄	#41×C₃₁	1	1	yes	characteristic
297	M₁₆	—	4	8	no	—
298	C₂×C₈	—	2	1	no	—
299	C₂×C₈	—	2	1	no	—
300	C₂×C₈	—	2	2	no	—
301	C₂×C₈	—	2	1	no	—
302	C₂×C₈	—	2	2	no	—
303	C₂×C₈	—	2	1	no	—
304	C₂×C₈	—	2	1	no	—
305	C₂×C₈	—	4	1	no	—
306	C₂×C₈	—	4	1	no	—
307	C₂×C₈	—	4	1	no	—
308	C₂×C₈	—	4	1	no	—
309	C₂×C₈	—	4	1	no	—
310	C₂×C₈	—	4	1	no	—
311	C₂×C₈	—	4	1	no	—
312	C₂×C₈	SD₁₆×C₃₁	1	2	yes	—
313	C₄⋊C₄	—	2	1	no	—
314	C₄⋊C₄	—	2	2	no	—
315	C₄⋊C₄	—	2	2	no	—
316	C₄⋊C₄	—	4	1	no	—
317	C₄⋊C₄	—	4	1	no	—
318	C₄⋊C₄	—	4	1	no	—
319	C₄⋊C₄	—	4	1	no	—
320	C₄⋊C₄	—	4	1	no	—
321	C₄⋊C₄	—	4	1	no	—
322	C₄⋊C₄	—	8	1	no	—
323	C₄⋊C₄	C₂²⋊C₄×C₃₁	1	2	yes	—
324	C₂²⋊C₄	—	2	2	no	—
325	C₂²⋊C₄	—	2	2	no	—
326	C₄²	C₂²×C₄×C₃₁	1	1	yes	characteristic
327	C₄²	C₂×D₄×C₃₁	1	1	yes	characteristic
328	C₄²	C₂×D₄×C₃₁	1	1	yes	characteristic
329	C₄²	#41×C₃₁	1	1	yes	characteristic
330	C₂³	—	2	2	no	—
331	C₂³	—	4	1	no	—
332	C₂³	—	4	1	no	—
333	C₂³	—	4	1	no	—
334	C₂³	—	4	1	no	—
335	C₂³	—	4	1	no	—
336	C₂³	—	4	1	no	—
337	C₂³	C₃₁×#121	1	1	yes	characteristic
338	Q₈	—	8	2	no	—
339	Q₈	—	8	2	no	—
340	D₄	—	2	2	no	—
341	D₄	—	2	2	no	—
342	D₄	—	4	1	no	—
343	D₄	—	4	1	no	—
344	C₂×C₄	—	2	1	no	—
345	C₂×C₄	—	2	1	no	—
346	C₂×C₄	—	2	2	no	—
347	C₂×C₄	—	4	1	no	—
348	C₂×C₄	—	4	1	no	—
349	C₂×C₄	—	4	2	no	—
350	C₂×C₄	—	4	2	no	—
351	C₂×C₄	—	4	2	no	—
352	C₂×C₄	—	4	2	no	—
353	C₂×C₄	—	4	2	no	—
354	C₂×C₄	—	8	1	no	—
355	C₂×C₄	—	8	1	no	—
356	C₂×C₄	—	8	1	no	—
357	C₂×C₄	—	8	1	no	—
358	C₂×C₄	—	8	1	no	—
359	C₂×C₄	C₂×C₂²⋊C₄×C₃₁	1	1	yes	characteristic
360	C₂×C₄	C₄×D₄×C₃₁	1	1	yes	characteristic
361	C₂×C₄	C₄×D₄×C₃₁	1	1	yes	characteristic
362	C₂×C₄	C₃₁×#120	1	1	yes	characteristic
363	C₂×C₄	C₃₁×#121	1	1	yes	characteristic
364	C₂×C₄	C₃₁×#123	1	1	yes	characteristic
365	C₂×C₄	C₂×SD₁₆×C₃₁	1	1	yes	characteristic
366	C₂×C₄	C₃₁×#135	1	1	yes	characteristic
367	C₂×C₄	C₃₁×AGL(1,ℤ/8)	1	1	yes	characteristic
368	C₂×C₄	C₃₁×AGL(1,ℤ/8)	1	1	yes	characteristic
369	C₂×C₄	C₃₁×#137	1	1	yes	characteristic
370	C₂×C₄	C₃₁×#137	1	1	yes	characteristic
371	C₂×C₄	C₃₁×#137	1	1	yes	characteristic
372	C₂×C₄	C₃₁×#137	1	1	yes	characteristic
373	C₈	—	2	2	no	—
374	C₈	—	4	2	no	—
375	C₈	—	4	2	no	—
376	C₈	—	4	1	no	—
377	C₈	—	8	1	no	—
378	C₂²	—	4	2	no	—
379	C₂²	—	4	2	no	—
380	C₂²	—	4	2	no	—
381	C₂²	—	8	1	no	—
382	C₂²	—	8	1	no	—
383	C₂²	—	8	1	no	—
384	C₂²	—	8	1	no	—
385	C₂²	C₃₁×#386	1	1	yes	characteristic
386	C₂²	C₃₁×#461	1	1	yes	characteristic
387	C₂²	C₃₁×#463	1	1	yes	characteristic
388	C₂²	C₃₁×#468	1	1	yes	characteristic
389	C₂²	C₃₁×#470	1	1	yes	characteristic
390	C₂²	C₃₁×#471	1	1	yes	characteristic
391	C₂²	C₃₁×#472	1	1	yes	characteristic
392	C₄	—	2	1	no	—
393	C₄	—	2	1	no	—
394	C₄	—	4	2	no	—
395	C₄	—	8	2	no	—
396	C₄	—	16	1	no	—
397	C₄	C₃₁×#417	1	1	yes	characteristic
398	C₄	C₃₁×#419	1	1	yes	characteristic
399	C₄	C₄×SD₁₆×C₃₁	1	1	yes	characteristic
400	C₄	C₃₁×#440	1	1	yes	characteristic
401	C₄	C₃₁×#450	1	1	yes	characteristic
402	C₄	C₃₁×#452	1	1	yes	characteristic
403	C₄	C₃₁×#468	1	1	yes	characteristic
404	C₄	C₃₁×#470	1	1	yes	characteristic
405	C₂	—	8	2	no	—
406	C₂	C₃₁×#1691	1	1	yes	characteristic
407	C₂	C₃₁×#1695	1	1	yes	characteristic
408	C₂	C₃₁×#1697	1	1	yes	characteristic
409	C₂	C₃₁×#1703	1	1	yes	characteristic
410	C₂	C₃₁×#1877	1	1	yes	characteristic
411	C₂	C₃₁×#1894	1	1	yes	characteristic
412	C₂	C₃₁×#1907	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂³×C₃₁	#121
commutator	C₄²	C₂²×C₄×C₃₁
Frattini, cosocle	C₂×C₄²	C₂³×C₃₁
prime radical	C₂²×D₄×C₃₁	D₄
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³×C₃₁ ◁ C₂×C₄²×C₃₁ ◁ G

Complements

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

N	H ≅ G/N	type
3. C₃₁×#1990	C₂	semidirect
5. C₂×C₃₁×#333	C₂	semidirect
6. C₃₁×#1296	C₂	semidirect
7. C₃₁×#1277	C₂	semidirect
12. C₃₁×#476	C₄	semidirect
123. #9223	C₃₁	direct (N × H)
174. #1990	C₂×C₃₁	semidirect
176. C₂×#333	C₂×C₃₁	semidirect
177. #1296	C₂×C₃₁	semidirect
178. #1277	C₂×C₃₁	semidirect
210. #476	C₄×C₃₁	semidirect
273. C₃₁	#9223	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×#476	2
C₂×C₃₁×#436	2
C₃₁×#1990	2
C₃₁×#1802	2
C₂×C₃₁×#333	2
C₃₁×#1296	2
C₃₁×#1277	2
#9223	31

Subgroups — C31×#9223 #21465982

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#9223 #21465982