Subgroups — C₃₁×#19395 #21476154

Order 7936.

628 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	C₃₁×#3231	C₂	1	1	yes	maximal, characteristic, complemented
2	C₃₁×#3110	C₂	1	1	yes	maximal, characteristic
3	C₃₁×#3101	C₂	1	1	yes	maximal, characteristic
4	C₃₁×#2972	C₂	1	1	yes	maximal, characteristic
5	C₃₁×#1683	C₂	1	2	yes	maximal, complemented
6	C₃₁×#1679	C₂	1	2	yes	maximal, complemented
7	C₃₁×#1673	C₂	1	2	yes	maximal, complemented
8	C₃₁×#1671	C₂	1	2	yes	maximal, complemented
9	C₃₁×#1572	C₂	1	1	yes	maximal, characteristic, complemented
10	C₃₁×#1554	C₂	1	1	yes	maximal, characteristic, complemented
11	C₃₁×#1544	C₂	1	1	yes	maximal, characteristic, complemented
12	C₃₁×#533	C₂²	1	1	yes	characteristic
13	C₃₁×#533	C₂²	1	1	yes	characteristic
14	C₂×C₄×D₄×C₃₁	C₂²	1	1	yes	characteristic
15	C₃₁×#500	C₂²	1	1	yes	characteristic, complemented
16	C₃₁×#498	C₂²	1	1	yes	characteristic, complemented
17	C₃₁×#487	C₂²	1	1	yes	characteristic, complemented
18	C₃₁×#487	C₂²	1	1	yes	characteristic, complemented
19	C₃₁×#485	—	2	2	no	—
20	C₃₁×#484	—	2	1	no	—
21	C₃₁×#483	—	2	1	no	—
22	C₃₁×#479	—	2	2	no	—
23	C₃₁×#476	C₂²	1	2	yes	—
24	C₃₁×#474	C₂²	1	2	yes	—
25	C₃₁×#461	C₂²	1	2	yes	—
26	C₃₁×#461	C₂²	1	2	yes	—
27	C₃₁×#450	—	2	1	no	—
28	C₃₁×#450	—	2	1	no	—
29	C₃₁×#440	—	2	2	no	—
30	C₄×SD₁₆×C₃₁	—	2	2	no	—
31	C₈×D₄×C₃₁	—	2	2	no	—
32	C₈×D₄×C₃₁	C₂²	1	2	yes	—
33	C₈×D₄×C₃₁	C₂²	1	2	yes	—
34	C₃₁×#432	C₂²	1	1	yes	characteristic
35	C₃₁×#431	C₂²	1	1	yes	characteristic
36	C₃₁×#431	C₂²	1	1	yes	characteristic
37	C₃₁×#431	C₂²	1	1	yes	characteristic
38	C₂×C₃₁×C₂²⋊C₈	—	2	2	no	—
39	C₃₁×#335	C₂²	1	2	yes	complemented
40	C₃₁×#334	C₂²	1	2	yes	complemented
41	C₃₁×#333	C₂²	1	2	yes	complemented
42	C₃₁×#332	C₂²	1	2	yes	complemented
43	C₃₁×#332	C₂²	1	2	yes	complemented
44	C₃₁×#332	C₂²	1	2	yes	complemented
45	C₂²×D₄×C₃₁	D₄	1	1	yes	characteristic
46	C₂³×C₄×C₃₁	—	2	1	no	—
47	C₂×SD₁₆×C₃₁	—	2	2	no	—
48	C₂×SD₁₆×C₃₁	—	2	2	no	—
49	C₂²×C₈×C₃₁	—	2	2	no	—
50	C₂²×C₈×C₃₁	—	2	2	no	—
51	C₃₁×C₄⋊Q₈	C₂³	1	1	yes	characteristic
52	C₃₁×C₄⋊Q₈	C₂³	1	1	yes	characteristic
53	C₃₁×C₄⋊Q₈	C₂³	1	1	yes	characteristic
54	C₃₁×C₄⋊Q₈	C₂³	1	1	yes	characteristic
55	C₃₁×#125	—	2	1	no	—
56	C₃₁×#125	—	2	1	no	—
57	C₃₁×C₂²⋊Q₈	—	2	1	no	—
58	C₃₁×C₂²⋊Q₈	—	2	1	no	—
59	C₃₁×C₂²⋊Q₈	—	2	1	no	—
60	C₃₁×C₂²⋊Q₈	—	2	1	no	—
61	C₄×Q₈×C₃₁	—	2	1	no	—
62	C₄×Q₈×C₃₁	—	2	1	no	—
63	C₄×Q₈×C₃₁	—	2	1	no	—
64	C₄×Q₈×C₃₁	—	2	1	no	—
65	C₄×D₄×C₃₁	—	2	1	no	—
66	C₄×D₄×C₃₁	—	2	1	no	—
67	C₄×D₄×C₃₁	D₄	1	2	yes	—
68	C₄×D₄×C₃₁	C₂³	1	2	yes	—
69	C₃₁×#117	—	2	1	no	—
70	C₃₁×#117	—	2	1	no	—
71	C₂×C₄⋊C₄×C₃₁	D₄	1	1	yes	characteristic
72	C₂×C₂²⋊C₄×C₃₁	—	2	1	no	—
73	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
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	D₄	1	2	yes	—
79	C₃₁×C₄⋊C₈	C₂³	1	2	yes	—
80	C₃₁×C₄⋊C₈	C₂³	1	2	yes	—
81	C₃₁×C₄⋊C₈	C₂³	1	2	yes	—
82	C₃₁×#103	—	2	2	no	—
83	C₃₁×#103	—	2	2	no	—
84	C₃₁×#103	—	2	2	no	—
85	C₃₁×#103	—	2	2	no	—
86	C₃₁×#102	—	2	2	no	—
87	C₃₁×#102	—	2	2	no	—
88	C₃₁×C₂²⋊C₈	—	2	1	no	—
89	C₃₁×C₂²⋊C₈	—	2	1	no	—
90	C₃₁×C₂²⋊C₈	—	2	1	no	—
91	C₃₁×C₂²⋊C₈	—	2	2	no	—
92	C₃₁×C₂²⋊C₈	—	2	2	no	—
93	C₃₁×C₂²⋊C₈	D₄	1	2	yes	—
94	C₃₁×#97	—	2	1	no	—
95	C₄×C₈×C₃₁	—	2	1	no	—
96	C₄×C₈×C₃₁	—	2	1	no	—
97	C₄×C₈×C₃₁	C₂³	1	1	yes	characteristic
98	C₄×C₈×C₃₁	C₂³	1	1	yes	characteristic
99	C₂⁴×C₃₁	—	2	1	no	—
100	C₂×Q₈×C₃₁	—	2	1	no	—
101	C₂×Q₈×C₃₁	—	2	1	no	—
102	C₂×Q₈×C₃₁	—	2	1	no	—
103	C₂×Q₈×C₃₁	—	2	1	no	—
104	C₂×D₄×C₃₁	—	2	4	no	—
105	C₂×D₄×C₃₁	—	2	1	no	—
106	C₂×D₄×C₃₁	C₂×D₄	1	2	yes	—
107	C₂²×C₄×C₃₁	—	2	1	no	—
108	C₂²×C₄×C₃₁	—	2	2	no	—
109	C₂²×C₄×C₃₁	—	2	2	no	—
110	C₂²×C₄×C₃₁	—	8	2	no	—
111	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
112	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
113	C₂²×C₄×C₃₁	#41	1	1	yes	characteristic
114	SD₁₆×C₃₁	—	4	4	no	—
115	SD₁₆×C₃₁	—	4	4	no	—
116	C₂×C₈×C₃₁	—	2	2	no	—
117	C₂×C₈×C₃₁	—	2	2	no	—
118	C₂×C₈×C₃₁	—	2	2	no	—
119	C₂×C₈×C₃₁	—	8	2	no	—
120	C₂×C₈×C₃₁	—	8	2	no	—
121	C₂×C₈×C₃₁	C₂×D₄	1	2	yes	—
122	C₂×C₈×C₃₁	C₂×D₄	1	2	yes	—
123	C₄⋊C₄×C₃₁	—	2	1	no	—
124	C₄⋊C₄×C₃₁	—	2	1	no	—
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	1	no	—
130	C₄⋊C₄×C₃₁	—	2	1	no	—
131	C₄⋊C₄×C₃₁	—	2	1	no	—
132	C₄⋊C₄×C₃₁	—	4	1	no	—
133	C₄⋊C₄×C₃₁	—	4	1	no	—
134	C₄⋊C₄×C₃₁	—	4	1	no	—
135	C₄⋊C₄×C₃₁	—	4	1	no	—
136	C₄⋊C₄×C₃₁	C₂×D₄	1	2	yes	—
137	C₂²⋊C₄×C₃₁	—	2	2	no	—
138	C₂²⋊C₄×C₃₁	—	2	2	no	—
139	C₂²⋊C₄×C₃₁	—	4	1	no	—
140	C₂²⋊C₄×C₃₁	—	4	1	no	—
141	C₄²×C₃₁	—	4	1	no	—
142	C₄²×C₃₁	—	4	1	no	—
143	C₄²×C₃₁	C₂×D₄	1	1	yes	characteristic
144	C₄²×C₃₁	C₂×D₄	1	1	yes	characteristic
145	C₄²×C₃₁	#41	1	1	yes	characteristic
146	C₄²×C₃₁	C₂⁴	1	1	yes	characteristic
147	#19395	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
148	C₂³×C₃₁	—	2	4	no	—
149	C₂³×C₃₁	—	2	2	no	—
150	C₂³×C₃₁	—	8	1	no	—
151	C₂³×C₃₁	—	8	1	no	—
152	C₂³×C₃₁	#121	1	1	yes	characteristic
153	Q₈×C₃₁	—	4	2	no	—
154	Q₈×C₃₁	—	4	2	no	—
155	Q₈×C₃₁	—	4	2	no	—
156	Q₈×C₃₁	—	8	1	no	—
157	D₄×C₃₁	—	2	4	no	—
158	D₄×C₃₁	—	2	2	no	—
159	D₄×C₃₁	#135	1	4	yes	—
160	C₂×C₄×C₃₁	—	2	2	no	—
161	C₂×C₄×C₃₁	—	2	2	no	—
162	C₂×C₄×C₃₁	—	2	2	no	—
163	C₂×C₄×C₃₁	—	4	1	no	—
164	C₂×C₄×C₃₁	—	4	2	no	—
165	C₂×C₄×C₃₁	—	4	2	no	—
166	C₂×C₄×C₃₁	—	4	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	—	8	2	no	—
171	C₂×C₄×C₃₁	—	8	1	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	2	no	—
179	C₈×C₃₁	—	2	2	no	—
180	C₈×C₃₁	—	4	2	no	—
181	C₈×C₃₁	—	4	2	no	—
182	C₈×C₃₁	—	4	2	no	—
183	#3231	C₂×C₃₁	1	1	yes	characteristic, complemented
184	#3110	C₂×C₃₁	1	1	yes	characteristic
185	#3101	C₂×C₃₁	1	1	yes	characteristic
186	#2972	C₂×C₃₁	1	1	yes	characteristic
187	#1683	C₂×C₃₁	1	2	yes	complemented
188	#1679	C₂×C₃₁	1	2	yes	complemented
189	#1673	C₂×C₃₁	1	2	yes	complemented
190	#1671	C₂×C₃₁	1	2	yes	complemented
191	#1572	C₂×C₃₁	1	1	yes	characteristic, complemented
192	#1554	C₂×C₃₁	1	1	yes	characteristic, complemented
193	#1544	C₂×C₃₁	1	1	yes	characteristic, complemented
194	C₂²×C₃₁	—	2	1	no	—
195	C₂²×C₃₁	—	2	1	no	—
196	C₂²×C₃₁	—	2	4	no	—
197	C₂²×C₃₁	—	2	1	no	—
198	C₂²×C₃₁	—	4	2	no	—
199	C₂²×C₃₁	—	4	2	no	—
200	C₂²×C₃₁	—	8	4	no	—
201	C₂²×C₃₁	C₂×#121	1	1	yes	characteristic
202	C₄×C₃₁	—	2	1	no	—
203	C₄×C₃₁	—	2	1	no	—
204	C₄×C₃₁	—	2	1	no	—
205	C₄×C₃₁	—	2	1	no	—
206	C₄×C₃₁	—	4	2	no	—
207	C₄×C₃₁	—	8	1	no	—
208	C₄×C₃₁	—	8	1	no	—
209	C₄×C₃₁	—	8	1	no	—
210	C₄×C₃₁	—	8	1	no	—
211	C₄×C₃₁	C₂×#135	1	2	yes	—
212	C₄×C₃₁	#577	1	2	yes	—
213	#533	C₂²×C₃₁	1	1	yes	characteristic
214	#533	C₂²×C₃₁	1	1	yes	characteristic
215	C₂×C₄×D₄	C₂²×C₃₁	1	1	yes	characteristic
216	#500	C₂²×C₃₁	1	1	yes	characteristic, complemented
217	#498	C₂²×C₃₁	1	1	yes	characteristic, complemented
218	#487	C₂²×C₃₁	1	1	yes	characteristic, complemented
219	#487	C₂²×C₃₁	1	1	yes	characteristic, complemented
220	#485	—	2	2	no	—
221	#484	—	2	1	no	—
222	#483	—	2	1	no	—
223	#479	—	2	2	no	—
224	#476	C₂²×C₃₁	1	2	yes	—
225	#474	C₂²×C₃₁	1	2	yes	—
226	#461	C₂²×C₃₁	1	2	yes	—
227	#461	C₂²×C₃₁	1	2	yes	—
228	#450	—	2	1	no	—
229	#450	—	2	1	no	—
230	#440	—	2	2	no	—
231	C₄×SD₁₆	—	2	2	no	—
232	C₈×D₄	—	2	2	no	—
233	C₈×D₄	C₂²×C₃₁	1	2	yes	—
234	C₈×D₄	C₂²×C₃₁	1	2	yes	—
235	#432	C₂²×C₃₁	1	1	yes	characteristic
236	#431	C₂²×C₃₁	1	1	yes	characteristic
237	#431	C₂²×C₃₁	1	1	yes	characteristic
238	#431	C₂²×C₃₁	1	1	yes	characteristic
239	C₂×C₂²⋊C₈	—	2	2	no	—
240	#335	C₂²×C₃₁	1	2	yes	complemented
241	#334	C₂²×C₃₁	1	2	yes	complemented
242	#333	C₂²×C₃₁	1	2	yes	complemented
243	#332	C₂²×C₃₁	1	2	yes	complemented
244	#332	C₂²×C₃₁	1	2	yes	complemented
245	#332	C₂²×C₃₁	1	2	yes	complemented
246	C₂×C₃₁	—	4	1	no	—
247	C₂×C₃₁	—	4	4	no	—
248	C₂×C₃₁	#3042	1	1	yes	characteristic
249	C₂×C₃₁	#3050	1	1	yes	characteristic
250	C₂×C₃₁	#3069	1	1	yes	characteristic
251	C₂²×D₄	D₄×C₃₁	1	1	yes	characteristic
252	C₂³×C₄	—	2	1	no	—
253	C₂×SD₁₆	—	2	2	no	—
254	C₂×SD₁₆	—	2	2	no	—
255	C₂²×C₈	—	2	2	no	—
256	C₂²×C₈	—	2	2	no	—
257	C₄⋊Q₈	C₂³×C₃₁	1	1	yes	characteristic
258	C₄⋊Q₈	C₂³×C₃₁	1	1	yes	characteristic
259	C₄⋊Q₈	C₂³×C₃₁	1	1	yes	characteristic
260	C₄⋊Q₈	C₂³×C₃₁	1	1	yes	characteristic
261	#125	—	2	1	no	—
262	#125	—	2	1	no	—
263	C₂²⋊Q₈	—	2	1	no	—
264	C₂²⋊Q₈	—	2	1	no	—
265	C₂²⋊Q₈	—	2	1	no	—
266	C₂²⋊Q₈	—	2	1	no	—
267	C₄×Q₈	—	2	1	no	—
268	C₄×Q₈	—	2	1	no	—
269	C₄×Q₈	—	2	1	no	—
270	C₄×Q₈	—	2	1	no	—
271	C₄×D₄	—	2	1	no	—
272	C₄×D₄	—	2	1	no	—
273	C₄×D₄	D₄×C₃₁	1	2	yes	—
274	C₄×D₄	C₂³×C₃₁	1	2	yes	—
275	#117	—	2	1	no	—
276	#117	—	2	1	no	—
277	C₂×C₄⋊C₄	D₄×C₃₁	1	1	yes	characteristic
278	C₂×C₂²⋊C₄	—	2	1	no	—
279	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
280	#107	—	2	2	no	—
281	#107	—	2	2	no	—
282	#106	—	2	2	no	—
283	#106	—	2	2	no	—
284	C₄⋊C₈	D₄×C₃₁	1	2	yes	—
285	C₄⋊C₈	C₂³×C₃₁	1	2	yes	—
286	C₄⋊C₈	C₂³×C₃₁	1	2	yes	—
287	C₄⋊C₈	C₂³×C₃₁	1	2	yes	—
288	#103	—	2	2	no	—
289	#103	—	2	2	no	—
290	#103	—	2	2	no	—
291	#103	—	2	2	no	—
292	#102	—	2	2	no	—
293	#102	—	2	2	no	—
294	C₂²⋊C₈	—	2	1	no	—
295	C₂²⋊C₈	—	2	1	no	—
296	C₂²⋊C₈	—	2	1	no	—
297	C₂²⋊C₈	—	2	2	no	—
298	C₂²⋊C₈	—	2	2	no	—
299	C₂²⋊C₈	D₄×C₃₁	1	2	yes	—
300	#97	—	2	1	no	—
301	C₄×C₈	—	2	1	no	—
302	C₄×C₈	—	2	1	no	—
303	C₄×C₈	C₂³×C₃₁	1	1	yes	characteristic
304	C₄×C₈	C₂³×C₃₁	1	1	yes	characteristic
305	C₃₁	#19395	1	1	yes	characteristic, direct product factor, Sylow
306	C₂⁴	—	2	1	no	—
307	C₂×Q₈	—	2	1	no	—
308	C₂×Q₈	—	2	1	no	—
309	C₂×Q₈	—	2	1	no	—
310	C₂×Q₈	—	2	1	no	—
311	C₂×D₄	—	2	4	no	—
312	C₂×D₄	—	2	1	no	—
313	C₂×D₄	C₂×D₄×C₃₁	1	2	yes	—
314	C₂²×C₄	—	2	1	no	—
315	C₂²×C₄	—	2	2	no	—
316	C₂²×C₄	—	2	2	no	—
317	C₂²×C₄	—	8	2	no	—
318	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
319	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
320	C₂²×C₄	#41×C₃₁	1	1	yes	characteristic
321	SD₁₆	—	4	4	no	—
322	SD₁₆	—	4	4	no	—
323	C₂×C₈	—	2	2	no	—
324	C₂×C₈	—	2	2	no	—
325	C₂×C₈	—	2	2	no	—
326	C₂×C₈	—	8	2	no	—
327	C₂×C₈	—	8	2	no	—
328	C₂×C₈	C₂×D₄×C₃₁	1	2	yes	—
329	C₂×C₈	C₂×D₄×C₃₁	1	2	yes	—
330	C₄⋊C₄	—	2	1	no	—
331	C₄⋊C₄	—	2	1	no	—
332	C₄⋊C₄	—	2	1	no	—
333	C₄⋊C₄	—	2	1	no	—
334	C₄⋊C₄	—	2	1	no	—
335	C₄⋊C₄	—	2	1	no	—
336	C₄⋊C₄	—	2	1	no	—
337	C₄⋊C₄	—	2	1	no	—
338	C₄⋊C₄	—	2	1	no	—
339	C₄⋊C₄	—	4	1	no	—
340	C₄⋊C₄	—	4	1	no	—
341	C₄⋊C₄	—	4	1	no	—
342	C₄⋊C₄	—	4	1	no	—
343	C₄⋊C₄	C₂×D₄×C₃₁	1	2	yes	—
344	C₂²⋊C₄	—	2	2	no	—
345	C₂²⋊C₄	—	2	2	no	—
346	C₂²⋊C₄	—	4	1	no	—
347	C₂²⋊C₄	—	4	1	no	—
348	C₄²	—	4	1	no	—
349	C₄²	—	4	1	no	—
350	C₄²	C₂×D₄×C₃₁	1	1	yes	characteristic
351	C₄²	C₂×D₄×C₃₁	1	1	yes	characteristic
352	C₄²	#41×C₃₁	1	1	yes	characteristic
353	C₄²	C₂⁴×C₃₁	1	1	yes	characteristic
354	C₂³	—	2	4	no	—
355	C₂³	—	2	2	no	—
356	C₂³	—	8	1	no	—
357	C₂³	—	8	1	no	—
358	C₂³	C₃₁×#121	1	1	yes	characteristic
359	Q₈	—	4	2	no	—
360	Q₈	—	4	2	no	—
361	Q₈	—	4	2	no	—
362	Q₈	—	8	1	no	—
363	D₄	—	2	4	no	—
364	D₄	—	2	2	no	—
365	D₄	C₃₁×#135	1	4	yes	—
366	C₂×C₄	—	2	2	no	—
367	C₂×C₄	—	2	2	no	—
368	C₂×C₄	—	2	2	no	—
369	C₂×C₄	—	4	1	no	—
370	C₂×C₄	—	4	2	no	—
371	C₂×C₄	—	4	2	no	—
372	C₂×C₄	—	4	1	no	—
373	C₂×C₄	—	4	1	no	—
374	C₂×C₄	—	4	1	no	—
375	C₂×C₄	—	4	1	no	—
376	C₂×C₄	—	8	2	no	—
377	C₂×C₄	—	8	1	no	—
378	C₂×C₄	C₃₁×#121	1	1	yes	characteristic
379	C₂×C₄	C₃₁×#121	1	1	yes	characteristic
380	C₂×C₄	C₃₁×#121	1	1	yes	characteristic
381	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
382	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
383	C₂×C₄	C₂×#41×C₃₁	1	1	yes	characteristic
384	C₈	—	2	2	no	—
385	C₈	—	2	2	no	—
386	C₈	—	4	2	no	—
387	C₈	—	4	2	no	—
388	C₈	—	4	2	no	—
389	C₂²	—	2	1	no	—
390	C₂²	—	2	1	no	—
391	C₂²	—	2	4	no	—
392	C₂²	—	2	1	no	—
393	C₂²	—	4	2	no	—
394	C₂²	—	4	2	no	—
395	C₂²	—	8	4	no	—
396	C₂²	C₂×C₃₁×#121	1	1	yes	characteristic
397	C₄	—	2	1	no	—
398	C₄	—	2	1	no	—
399	C₄	—	2	1	no	—
400	C₄	—	2	1	no	—
401	C₄	—	4	2	no	—
402	C₄	—	8	1	no	—
403	C₄	—	8	1	no	—
404	C₄	—	8	1	no	—
405	C₄	—	8	1	no	—
406	C₄	C₂×C₃₁×#135	1	2	yes	—
407	C₄	C₃₁×#577	1	2	yes	—
408	C₂	—	4	1	no	—
409	C₂	—	4	4	no	—
410	C₂	C₃₁×#3042	1	1	yes	characteristic
411	C₂	C₃₁×#3050	1	1	yes	characteristic
412	C₂	C₃₁×#3069	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₃₁	C₂×#121
commutator, Frattini, cosocle	C₄²	C₂⁴×C₃₁
prime radical	C₂²×D₄×C₃₁	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
1. C₃₁×#3231	C₂	semidirect
5. C₃₁×#1683	C₂	semidirect
6. C₃₁×#1679	C₂	semidirect
7. C₃₁×#1673	C₂	semidirect
8. C₃₁×#1671	C₂	semidirect
9. C₃₁×#1572	C₂	semidirect
10. C₃₁×#1554	C₂	semidirect
11. C₃₁×#1544	C₂	semidirect
15. C₃₁×#500	C₂²	semidirect
16. C₃₁×#498	C₂²	semidirect
17. C₃₁×#487	C₂²	semidirect
18. C₃₁×#487	C₂²	semidirect
39. C₃₁×#335	C₂²	semidirect
40. C₃₁×#334	C₂²	semidirect
41. C₃₁×#333	C₂²	semidirect
42. C₃₁×#332	C₂²	semidirect
43. C₃₁×#332	C₂²	semidirect
44. C₃₁×#332	C₂²	semidirect
147. #19395	C₃₁	direct (N × H)
183. #3231	C₂×C₃₁	semidirect
187. #1683	C₂×C₃₁	semidirect
188. #1679	C₂×C₃₁	semidirect
189. #1673	C₂×C₃₁	semidirect
190. #1671	C₂×C₃₁	semidirect
191. #1572	C₂×C₃₁	semidirect
192. #1554	C₂×C₃₁	semidirect
193. #1544	C₂×C₃₁	semidirect
216. #500	C₂²×C₃₁	semidirect
217. #498	C₂²×C₃₁	semidirect
218. #487	C₂²×C₃₁	semidirect
219. #487	C₂²×C₃₁	semidirect
240. #335	C₂²×C₃₁	semidirect
241. #334	C₂²×C₃₁	semidirect
242. #333	C₂²×C₃₁	semidirect
243. #332	C₂²×C₃₁	semidirect
244. #332	C₂²×C₃₁	semidirect
245. #332	C₂²×C₃₁	semidirect
305. C₃₁	#19395	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×#3231	2
C₃₁×#3110	2
C₃₁×#3101	2
C₃₁×#2972	2
C₃₁×#1683	2
C₃₁×#1679	2
C₃₁×#1673	2
C₃₁×#1671	2
C₃₁×#1572	2
C₃₁×#1554	2
C₃₁×#1544	2
#19395	31

Subgroups — C31×#19395 #21476154

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#19395 #21476154