Subgroups — C₃₁×#19220 #21475979

Order 7936.

562 conjugacy classes of subgroups.

The subgroup lattice has 422 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃₁×#3233	C₂	1	2	yes	maximal
2	C₃₁×#3170	C₂	1	1	yes	maximal, characteristic, complemented
3	C₃₁×#2966	C₂	1	1	yes	maximal, characteristic
4	C₃₁×#1704	C₂	1	2	yes	maximal, complemented
5	C₃₁×#1703	C₂	1	2	yes	maximal, complemented
6	C₃₁×#1676	C₂	1	1	yes	maximal, characteristic, complemented
7	C₃₁×#1675	C₂	1	1	yes	maximal, characteristic, complemented
8	C₃₁×#1670	C₂	1	1	yes	maximal, characteristic, complemented
9	C₃₁×#1669	C₂	1	1	yes	maximal, characteristic, complemented
10	C₃₁×#1564	C₂	1	1	yes	maximal, characteristic, complemented
11	C₃₁×#1554	C₂	1	2	yes	maximal, complemented
12	C₃₁×#533	C₂²	1	2	yes	—
13	C₃₁×#529	C₂²	1	1	yes	characteristic
14	C₃₁×#500	C₂²	1	1	yes	characteristic
15	C₃₁×#499	—	2	2	no	—
16	C₃₁×#494	C₂²	1	1	yes	characteristic
17	C₃₁×#488	—	2	2	no	—
18	C₃₁×#487	C₂²	1	2	yes	—
19	C₃₁×#485	—	2	2	no	—
20	C₃₁×#482	—	2	2	no	—
21	C₃₁×#477	C₂²	1	2	yes	—
22	C₃₁×#476	C₂²	1	2	yes	—
23	C₃₁×#475	C₂²	1	2	yes	—
24	C₃₁×#474	C₂²	1	2	yes	—
25	C₃₁×#467	—	2	1	no	—
26	C₃₁×#467	—	2	1	no	—
27	C₃₁×#446	C₂²	1	1	yes	characteristic
28	C₈×Q₈×C₃₁	C₂²	1	1	yes	characteristic
29	C₄×Q₁₆×C₃₁	—	2	1	no	—
30	C₄×Q₁₆×C₃₁	—	2	1	no	—
31	C₃₁×#436	C₂²	1	1	yes	characteristic
32	C₃₁×#435	—	2	1	no	—
33	C₈×D₄×C₃₁	—	2	1	no	—
34	C₈×D₄×C₃₁	C₂²	1	1	yes	characteristic
35	C₃₁×#433	—	2	1	no	—
36	C₃₁×#432	C₂²	1	1	yes	characteristic
37	C₃₁×#431	C₂²	1	2	yes	—
38	C₃₁×#405	—	2	1	no	—
39	C₂×C₄×C₈×C₃₁	C₂²	1	1	yes	characteristic
40	C₃₁×#335	C₂²	1	1	yes	characteristic
41	C₃₁×#335	C₂²	1	1	yes	characteristic
42	C₃₁×#334	C₂²	1	2	yes	—
43	C₃₁×#333	C₂²	1	2	yes	—
44	C₃₁×#333	C₂²	1	2	yes	—
45	C₃₁×#332	C₂²	1	2	yes	—
46	C₃₁×#332	C₂²	1	2	yes	—
47	C₂×Q₁₆×C₃₁	—	2	1	no	—
48	C₂×Q₁₆×C₃₁	—	2	1	no	—
49	C₂×M₁₆×C₃₁	—	2	1	no	—
50	C₂²×C₈×C₃₁	—	2	1	no	—
51	C₂²×C₈×C₃₁	D₄	1	1	yes	characteristic
52	C₂²×C₈×C₃₁	D₄	1	1	yes	characteristic
53	C₃₁×C₄⋊Q₈	C₂³	1	2	yes	—
54	C₃₁×C₄⋊Q₈	C₂³	1	2	yes	—
55	C₃₁×#126	—	2	1	no	—
56	C₃₁×#125	—	2	2	no	—
57	C₃₁×#125	D₄	1	1	yes	characteristic
58	C₃₁×#124	D₄	1	1	yes	characteristic
59	C₃₁×#123	—	2	1	no	—
60	C₃₁×C₂²⋊Q₈	—	2	2	no	—
61	C₃₁×C₂²⋊Q₈	—	2	2	no	—
62	C₃₁×C₂²⋊Q₈	D₄	1	1	yes	characteristic
63	C₃₁×#121	D₄	1	1	yes	characteristic
64	C₄×Q₈×C₃₁	—	2	2	no	—
65	C₄×Q₈×C₃₁	—	2	2	no	—
66	C₄×Q₈×C₃₁	C₂³	1	1	yes	characteristic
67	C₄×D₄×C₃₁	—	2	1	no	—
68	C₄×D₄×C₃₁	C₂³	1	1	yes	characteristic
69	C₃₁×#117	—	2	1	no	—
70	C₃₁×#117	—	2	2	no	—
71	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
72	C₃₁×#107	—	2	1	no	—
73	C₃₁×#107	—	2	1	no	—
74	C₃₁×#107	—	2	2	no	—
75	C₃₁×#107	—	2	2	no	—
76	C₃₁×#106	—	2	2	no	—
77	C₃₁×#106	—	2	2	no	—
78	C₃₁×C₄⋊C₈	—	2	1	no	—
79	C₃₁×C₄⋊C₈	—	2	1	no	—
80	C₃₁×C₄⋊C₈	C₂³	1	2	yes	—
81	C₃₁×C₄⋊C₈	C₂³	1	2	yes	—
82	C₃₁×C₄⋊C₈	C₂³	1	1	yes	characteristic
83	C₃₁×C₄⋊C₈	C₂³	1	1	yes	characteristic
84	C₃₁×#103	—	2	2	no	—
85	C₃₁×#103	—	2	2	no	—
86	C₃₁×#103	—	2	2	no	—
87	C₃₁×#103	—	2	2	no	—
88	C₃₁×#102	—	2	2	no	—
89	C₃₁×#102	—	2	2	no	—
90	C₃₁×C₂²⋊C₈	—	2	2	no	—
91	C₃₁×C₂²⋊C₈	—	2	1	no	—
92	C₃₁×C₂²⋊C₈	—	2	1	no	—
93	C₃₁×C₂²⋊C₈	—	2	1	no	—
94	C₃₁×#97	—	2	1	no	—
95	C₃₁×#97	—	2	1	no	—
96	C₄×C₈×C₃₁	—	2	1	no	—
97	C₄×C₈×C₃₁	—	2	2	no	—
98	C₄×C₈×C₃₁	D₄	1	1	yes	characteristic
99	C₄×C₈×C₃₁	D₄	1	1	yes	characteristic
100	C₄×C₈×C₃₁	C₂³	1	1	yes	characteristic
101	C₄×C₈×C₃₁	C₂³	1	1	yes	characteristic
102	C₂×Q₈×C₃₁	—	2	2	no	—
103	C₂×Q₈×C₃₁	—	2	2	no	—
104	C₂×Q₈×C₃₁	C₂×D₄	1	1	yes	characteristic
105	C₂×D₄×C₃₁	—	2	1	no	—
106	C₂×D₄×C₃₁	C₂×D₄	1	1	yes	characteristic
107	C₂²×C₄×C₃₁	—	2	1	no	—
108	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
109	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
110	C₂²×C₄×C₃₁	#41	1	1	yes	characteristic
111	Q₁₆×C₃₁	—	4	2	no	—
112	Q₁₆×C₃₁	—	4	2	no	—
113	M₁₆×C₃₁	—	4	2	no	—
114	C₂×C₈×C₃₁	—	2	2	no	—
115	C₂×C₈×C₃₁	—	2	1	no	—
116	C₂×C₈×C₃₁	—	2	2	no	—
117	C₂×C₈×C₃₁	—	2	2	no	—
118	C₂×C₈×C₃₁	—	2	1	no	—
119	C₂×C₈×C₃₁	—	4	1	no	—
120	C₂×C₈×C₃₁	—	4	2	no	—
121	C₂×C₈×C₃₁	C₂×D₄	1	1	yes	characteristic
122	C₂×C₈×C₃₁	C₂×D₄	1	1	yes	characteristic
123	C₂×C₈×C₃₁	C₂×D₄	1	1	yes	characteristic
124	C₂×C₈×C₃₁	C₂×D₄	1	1	yes	characteristic
125	C₄⋊C₄×C₃₁	—	2	1	no	—
126	C₄⋊C₄×C₃₁	—	2	1	no	—
127	C₄⋊C₄×C₃₁	—	2	1	no	—
128	C₄⋊C₄×C₃₁	—	2	1	no	—
129	C₄⋊C₄×C₃₁	—	2	2	no	—
130	C₄⋊C₄×C₃₁	—	2	2	no	—
131	C₄⋊C₄×C₃₁	—	2	2	no	—
132	C₄⋊C₄×C₃₁	—	2	2	no	—
133	C₄⋊C₄×C₃₁	—	4	2	no	—
134	C₄⋊C₄×C₃₁	—	4	2	no	—
135	C₄⋊C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
136	C₄⋊C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
137	C₂²⋊C₄×C₃₁	—	2	1	no	—
138	C₂²⋊C₄×C₃₁	—	2	1	no	—
139	C₂²⋊C₄×C₃₁	—	2	1	no	—
140	C₂²⋊C₄×C₃₁	—	2	1	no	—
141	C₂²⋊C₄×C₃₁	—	2	1	no	—
142	C₂²⋊C₄×C₃₁	—	4	2	no	—
143	C₄²×C₃₁	—	2	1	no	—
144	C₄²×C₃₁	—	4	2	no	—
145	C₄²×C₃₁	C₂×D₄	1	1	yes	characteristic
146	C₄²×C₃₁	C₂×D₄	1	1	yes	characteristic
147	C₄²×C₃₁	#41	1	1	yes	characteristic
148	C₄²×C₃₁	C₂⁴	1	1	yes	characteristic
149	#19220	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
150	C₂³×C₃₁	—	2	1	no	—
151	C₂³×C₃₁	#121	1	1	yes	characteristic
152	Q₈×C₃₁	—	2	1	no	—
153	Q₈×C₃₁	—	2	1	no	—
154	Q₈×C₃₁	—	4	2	no	—
155	Q₈×C₃₁	—	4	2	no	—
156	Q₈×C₃₁	—	8	2	no	—
157	D₄×C₃₁	—	2	1	no	—
158	D₄×C₃₁	—	2	1	no	—
159	D₄×C₃₁	—	4	2	no	—
160	C₂×C₄×C₃₁	—	2	1	no	—
161	C₂×C₄×C₃₁	—	2	1	no	—
162	C₂×C₄×C₃₁	—	2	1	no	—
163	C₂×C₄×C₃₁	—	2	1	no	—
164	C₂×C₄×C₃₁	—	2	1	no	—
165	C₂×C₄×C₃₁	—	2	1	no	—
166	C₂×C₄×C₃₁	—	2	1	no	—
167	C₂×C₄×C₃₁	—	4	1	no	—
168	C₂×C₄×C₃₁	—	4	1	no	—
169	C₂×C₄×C₃₁	—	4	1	no	—
170	C₂×C₄×C₃₁	—	4	2	no	—
171	C₂×C₄×C₃₁	—	4	2	no	—
172	C₂×C₄×C₃₁	#121	1	1	yes	characteristic
173	C₂×C₄×C₃₁	#121	1	1	yes	characteristic
174	C₂×C₄×C₃₁	#121	1	1	yes	characteristic
175	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
176	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
177	C₂×C₄×C₃₁	C₂×#41	1	1	yes	characteristic
178	C₈×C₃₁	—	2	1	no	—
179	C₈×C₃₁	—	2	1	no	—
180	C₈×C₃₁	—	2	1	no	—
181	C₈×C₃₁	—	4	1	no	—
182	C₈×C₃₁	—	4	1	no	—
183	C₈×C₃₁	—	4	2	no	—
184	C₈×C₃₁	—	4	2	no	—
185	C₈×C₃₁	#135	1	2	yes	—
186	#3233	C₂×C₃₁	1	2	yes	—
187	#3170	C₂×C₃₁	1	1	yes	characteristic, complemented
188	#2966	C₂×C₃₁	1	1	yes	characteristic
189	#1704	C₂×C₃₁	1	2	yes	complemented
190	#1703	C₂×C₃₁	1	2	yes	complemented
191	#1676	C₂×C₃₁	1	1	yes	characteristic, complemented
192	#1675	C₂×C₃₁	1	1	yes	characteristic, complemented
193	#1670	C₂×C₃₁	1	1	yes	characteristic, complemented
194	#1669	C₂×C₃₁	1	1	yes	characteristic, complemented
195	#1564	C₂×C₃₁	1	1	yes	characteristic, complemented
196	#1554	C₂×C₃₁	1	2	yes	complemented
197	C₂²×C₃₁	—	2	1	no	—
198	C₂²×C₃₁	—	2	1	no	—
199	C₂²×C₃₁	—	2	1	no	—
200	C₂²×C₃₁	—	4	1	no	—
201	C₂²×C₃₁	—	4	1	no	—
202	C₂²×C₃₁	—	4	1	no	—
203	C₂²×C₃₁	C₂×#121	1	1	yes	characteristic
204	C₄×C₃₁	—	2	1	no	—
205	C₄×C₃₁	—	2	1	no	—
206	C₄×C₃₁	—	2	1	no	—
207	C₄×C₃₁	—	2	1	no	—
208	C₄×C₃₁	—	4	1	no	—
209	C₄×C₃₁	—	4	1	no	—
210	C₄×C₃₁	—	4	1	no	—
211	C₄×C₃₁	—	8	2	no	—
212	C₄×C₃₁	—	8	2	no	—
213	C₄×C₃₁	C₂×#135	1	1	yes	characteristic
214	C₄×C₃₁	#575	1	1	yes	characteristic
215	C₄×C₃₁	#578	1	1	yes	characteristic
216	C₄×C₃₁	#578	1	1	yes	characteristic
217	#533	C₂²×C₃₁	1	2	yes	—
218	#529	C₂²×C₃₁	1	1	yes	characteristic
219	#500	C₂²×C₃₁	1	1	yes	characteristic
220	#499	—	2	2	no	—
221	#494	C₂²×C₃₁	1	1	yes	characteristic
222	#488	—	2	2	no	—
223	#487	C₂²×C₃₁	1	2	yes	—
224	#485	—	2	2	no	—
225	#482	—	2	2	no	—
226	#477	C₂²×C₃₁	1	2	yes	—
227	#476	C₂²×C₃₁	1	2	yes	—
228	#475	C₂²×C₃₁	1	2	yes	—
229	#474	C₂²×C₃₁	1	2	yes	—
230	#467	—	2	1	no	—
231	#467	—	2	1	no	—
232	#446	C₂²×C₃₁	1	1	yes	characteristic
233	C₈×Q₈	C₂²×C₃₁	1	1	yes	characteristic
234	C₄×Q₁₆	—	2	1	no	—
235	C₄×Q₁₆	—	2	1	no	—
236	#436	C₂²×C₃₁	1	1	yes	characteristic
237	#435	—	2	1	no	—
238	C₈×D₄	—	2	1	no	—
239	C₈×D₄	C₂²×C₃₁	1	1	yes	characteristic
240	#433	—	2	1	no	—
241	#432	C₂²×C₃₁	1	1	yes	characteristic
242	#431	C₂²×C₃₁	1	2	yes	—
243	#405	—	2	1	no	—
244	C₂×C₄×C₈	C₂²×C₃₁	1	1	yes	characteristic
245	#335	C₂²×C₃₁	1	1	yes	characteristic
246	#335	C₂²×C₃₁	1	1	yes	characteristic
247	#334	C₂²×C₃₁	1	2	yes	—
248	#333	C₂²×C₃₁	1	2	yes	—
249	#333	C₂²×C₃₁	1	2	yes	—
250	#332	C₂²×C₃₁	1	2	yes	—
251	#332	C₂²×C₃₁	1	2	yes	—
252	C₂×C₃₁	—	4	1	no	—
253	C₂×C₃₁	—	8	1	no	—
254	C₂×C₃₁	#3035	1	1	yes	characteristic
255	C₂×C₃₁	#3059	1	1	yes	characteristic
256	C₂×C₃₁	#3067	1	1	yes	characteristic
257	C₂×Q₁₆	—	2	1	no	—
258	C₂×Q₁₆	—	2	1	no	—
259	C₂×M₁₆	—	2	1	no	—
260	C₂²×C₈	—	2	1	no	—
261	C₂²×C₈	D₄×C₃₁	1	1	yes	characteristic
262	C₂²×C₈	D₄×C₃₁	1	1	yes	characteristic
263	C₄⋊Q₈	C₂³×C₃₁	1	2	yes	—
264	C₄⋊Q₈	C₂³×C₃₁	1	2	yes	—
265	#126	—	2	1	no	—
266	#125	—	2	2	no	—
267	#125	D₄×C₃₁	1	1	yes	characteristic
268	#124	D₄×C₃₁	1	1	yes	characteristic
269	#123	—	2	1	no	—
270	C₂²⋊Q₈	—	2	2	no	—
271	C₂²⋊Q₈	—	2	2	no	—
272	C₂²⋊Q₈	D₄×C₃₁	1	1	yes	characteristic
273	#121	D₄×C₃₁	1	1	yes	characteristic
274	C₄×Q₈	—	2	2	no	—
275	C₄×Q₈	—	2	2	no	—
276	C₄×Q₈	C₂³×C₃₁	1	1	yes	characteristic
277	C₄×D₄	—	2	1	no	—
278	C₄×D₄	C₂³×C₃₁	1	1	yes	characteristic
279	#117	—	2	1	no	—
280	#117	—	2	2	no	—
281	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
282	#107	—	2	1	no	—
283	#107	—	2	1	no	—
284	#107	—	2	2	no	—
285	#107	—	2	2	no	—
286	#106	—	2	2	no	—
287	#106	—	2	2	no	—
288	C₄⋊C₈	—	2	1	no	—
289	C₄⋊C₈	—	2	1	no	—
290	C₄⋊C₈	C₂³×C₃₁	1	1	yes	characteristic
291	C₄⋊C₈	C₂³×C₃₁	1	1	yes	characteristic
292	C₄⋊C₈	C₂³×C₃₁	1	2	yes	—
293	C₄⋊C₈	C₂³×C₃₁	1	2	yes	—
294	#103	—	2	2	no	—
295	#103	—	2	2	no	—
296	#103	—	2	2	no	—
297	#103	—	2	2	no	—
298	#102	—	2	2	no	—
299	#102	—	2	2	no	—
300	C₂²⋊C₈	—	2	2	no	—
301	C₂²⋊C₈	—	2	1	no	—
302	C₂²⋊C₈	—	2	1	no	—
303	C₂²⋊C₈	—	2	1	no	—
304	#97	—	2	1	no	—
305	#97	—	2	1	no	—
306	C₄×C₈	—	2	1	no	—
307	C₄×C₈	—	2	2	no	—
308	C₄×C₈	D₄×C₃₁	1	1	yes	characteristic
309	C₄×C₈	D₄×C₃₁	1	1	yes	characteristic
310	C₄×C₈	C₂³×C₃₁	1	1	yes	characteristic
311	C₄×C₈	C₂³×C₃₁	1	1	yes	characteristic
312	C₃₁	#19220	1	1	yes	characteristic, direct product factor, Sylow
313	C₂×Q₈	—	2	2	no	—
314	C₂×Q₈	—	2	2	no	—
315	C₂×Q₈	C₂×D₄×C₃₁	1	1	yes	characteristic
316	C₂×D₄	—	2	1	no	—
317	C₂×D₄	C₂×D₄×C₃₁	1	1	yes	characteristic
318	C₂²×C₄	—	2	1	no	—
319	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
320	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
321	C₂²×C₄	#41×C₃₁	1	1	yes	characteristic
322	Q₁₆	—	4	2	no	—
323	Q₁₆	—	4	2	no	—
324	M₁₆	—	4	2	no	—
325	C₂×C₈	—	2	2	no	—
326	C₂×C₈	—	2	1	no	—
327	C₂×C₈	—	2	2	no	—
328	C₂×C₈	—	2	2	no	—
329	C₂×C₈	—	2	1	no	—
330	C₂×C₈	—	4	1	no	—
331	C₂×C₈	—	4	2	no	—
332	C₂×C₈	C₂×D₄×C₃₁	1	1	yes	characteristic
333	C₂×C₈	C₂×D₄×C₃₁	1	1	yes	characteristic
334	C₂×C₈	C₂×D₄×C₃₁	1	1	yes	characteristic
335	C₂×C₈	C₂×D₄×C₃₁	1	1	yes	characteristic
336	C₄⋊C₄	—	2	1	no	—
337	C₄⋊C₄	—	2	1	no	—
338	C₄⋊C₄	—	2	1	no	—
339	C₄⋊C₄	—	2	1	no	—
340	C₄⋊C₄	—	2	2	no	—
341	C₄⋊C₄	—	2	2	no	—
342	C₄⋊C₄	—	2	2	no	—
343	C₄⋊C₄	—	2	2	no	—
344	C₄⋊C₄	—	4	2	no	—
345	C₄⋊C₄	—	4	2	no	—
346	C₄⋊C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
347	C₄⋊C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
348	C₂²⋊C₄	—	2	1	no	—
349	C₂²⋊C₄	—	2	1	no	—
350	C₂²⋊C₄	—	2	1	no	—
351	C₂²⋊C₄	—	2	1	no	—
352	C₂²⋊C₄	—	2	1	no	—
353	C₂²⋊C₄	—	4	2	no	—
354	C₄²	—	2	1	no	—
355	C₄²	—	4	2	no	—
356	C₄²	C₂×D₄×C₃₁	1	1	yes	characteristic
357	C₄²	C₂×D₄×C₃₁	1	1	yes	characteristic
358	C₄²	#41×C₃₁	1	1	yes	characteristic
359	C₄²	C₂⁴×C₃₁	1	1	yes	characteristic
360	C₂³	—	2	1	no	—
361	C₂³	C₃₁×#121	1	1	yes	characteristic
362	Q₈	—	2	1	no	—
363	Q₈	—	2	1	no	—
364	Q₈	—	4	2	no	—
365	Q₈	—	4	2	no	—
366	Q₈	—	8	2	no	—
367	D₄	—	2	1	no	—
368	D₄	—	2	1	no	—
369	D₄	—	4	2	no	—
370	C₂×C₄	—	2	1	no	—
371	C₂×C₄	—	2	1	no	—
372	C₂×C₄	—	2	1	no	—
373	C₂×C₄	—	2	1	no	—
374	C₂×C₄	—	2	1	no	—
375	C₂×C₄	—	2	1	no	—
376	C₂×C₄	—	2	1	no	—
377	C₂×C₄	—	4	1	no	—
378	C₂×C₄	—	4	1	no	—
379	C₂×C₄	—	4	1	no	—
380	C₂×C₄	—	4	2	no	—
381	C₂×C₄	—	4	2	no	—
382	C₂×C₄	C₃₁×#121	1	1	yes	characteristic
383	C₂×C₄	C₃₁×#121	1	1	yes	characteristic
384	C₂×C₄	C₃₁×#121	1	1	yes	characteristic
385	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
386	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
387	C₂×C₄	C₂×#41×C₃₁	1	1	yes	characteristic
388	C₈	—	2	1	no	—
389	C₈	—	2	1	no	—
390	C₈	—	2	1	no	—
391	C₈	—	4	1	no	—
392	C₈	—	4	1	no	—
393	C₈	—	4	2	no	—
394	C₈	—	4	2	no	—
395	C₈	C₃₁×#135	1	2	yes	—
396	C₂²	—	2	1	no	—
397	C₂²	—	2	1	no	—
398	C₂²	—	2	1	no	—
399	C₂²	—	4	1	no	—
400	C₂²	—	4	1	no	—
401	C₂²	—	4	1	no	—
402	C₂²	C₂×C₃₁×#121	1	1	yes	characteristic
403	C₄	—	2	1	no	—
404	C₄	—	2	1	no	—
405	C₄	—	2	1	no	—
406	C₄	—	2	1	no	—
407	C₄	—	4	1	no	—
408	C₄	—	4	1	no	—
409	C₄	—	4	1	no	—
410	C₄	—	8	2	no	—
411	C₄	—	8	2	no	—
412	C₄	C₂×C₃₁×#135	1	1	yes	characteristic
413	C₄	C₃₁×#575	1	1	yes	characteristic
414	C₄	C₃₁×#578	1	1	yes	characteristic
415	C₄	C₃₁×#578	1	1	yes	characteristic
416	C₂	—	4	1	no	—
417	C₂	—	8	1	no	—
418	C₂	C₃₁×#3035	1	1	yes	characteristic
419	C₂	C₃₁×#3059	1	1	yes	characteristic
420	C₂	C₃₁×#3067	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₃₁	C₂×#121
commutator, Frattini, cosocle	C₄²	C₂⁴×C₃₁
prime radical	C₃₁×#121	D₄
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₃₁ ◁ C₂×C₄²×C₃₁ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₃₁×#3170	C₂	semidirect
4. C₃₁×#1704	C₂	semidirect
5. C₃₁×#1703	C₂	semidirect
6. C₃₁×#1676	C₂	semidirect
7. C₃₁×#1675	C₂	semidirect
8. C₃₁×#1670	C₂	semidirect
9. C₃₁×#1669	C₂	semidirect
10. C₃₁×#1564	C₂	semidirect
11. C₃₁×#1554	C₂	semidirect
149. #19220	C₃₁	direct (N × H)
187. #3170	C₂×C₃₁	semidirect
189. #1704	C₂×C₃₁	semidirect
190. #1703	C₂×C₃₁	semidirect
191. #1676	C₂×C₃₁	semidirect
192. #1675	C₂×C₃₁	semidirect
193. #1670	C₂×C₃₁	semidirect
194. #1669	C₂×C₃₁	semidirect
195. #1564	C₂×C₃₁	semidirect
196. #1554	C₂×C₃₁	semidirect
312. C₃₁	#19220	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×#3233	2
C₃₁×#3170	2
C₃₁×#2966	2
C₃₁×#1704	2
C₃₁×#1703	2
C₃₁×#1676	2
C₃₁×#1675	2
C₃₁×#1670	2
C₃₁×#1669	2
C₃₁×#1564	2
C₃₁×#1554	2
#19220	31

Subgroups — C31×#19220 #21475979

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#19220 #21475979