Subgroups — C₃₁×#33674 #21490433

Order 7936.

612 conjugacy classes of subgroups.

The subgroup lattice has 408 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃₁×#3168	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×C₄×D₈×C₃₁	C₂	1	1	yes	maximal, characteristic, complemented
3	C₃₁×#2235	C₂	1	1	yes	maximal, characteristic, complemented
4	C₃₁×#2232	C₂	1	1	yes	maximal, characteristic, complemented
5	C₃₁×#2228	C₂	1	2	yes	maximal, complemented
6	C₃₁×#2210	C₂	1	2	yes	maximal, complemented
7	C₃₁×#2187	C₂	1	1	yes	maximal, characteristic, complemented
8	C₃₁×#2184	C₂	1	1	yes	maximal, characteristic
9	C₃₁×#2177	C₂	1	2	yes	maximal, complemented
10	C₃₁×#2175	C₂	1	2	yes	maximal, complemented
11	C₃₁×#2163	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂²×D₈×C₃₁	C₂²	1	1	yes	characteristic
13	C₃₁×#529	—	2	1	no	—
14	C₂×C₄×D₄×C₃₁	—	2	1	no	—
15	C₂×C₃₁×SD₃₂	C₂²	1	2	yes	—
16	C₂×C₃₁×D₁₆	C₂²	1	2	yes	—
17	C₃₁×#499	C₂²	1	1	yes	characteristic, complemented
18	C₃₁×#495	C₂²	1	1	yes	characteristic, complemented
19	C₃₁×#488	—	2	1	no	—
20	C₃₁×#467	C₂²	1	1	yes	characteristic
21	C₃₁×#466	C₂²	1	1	yes	characteristic
22	C₃₁×#465	—	2	1	no	—
23	C₄×Q₁₆×C₃₁	C₂²	1	1	yes	characteristic, complemented
24	C₄×SD₁₆×C₃₁	—	2	1	no	—
25	C₄×D₈×C₃₁	—	2	1	no	—
26	C₄×D₈×C₃₁	—	2	1	no	—
27	C₄×D₈×C₃₁	C₂²	1	2	yes	—
28	C₄×D₈×C₃₁	C₂²	1	2	yes	—
29	C₄×D₈×C₃₁	C₂²	1	1	yes	characteristic, complemented
30	C₂×C₃₁×#107	C₂²	1	1	yes	characteristic
31	C₂×C₃₁×#102	—	2	1	no	—
32	C₂×C₄×C₈×C₃₁	C₂²	1	1	yes	characteristic
33	C₃₁×#367	C₂²	1	2	yes	complemented
34	C₃₁×#366	C₂²	1	2	yes	complemented
35	C₃₁×C₄⋊C₁₆	C₂²	1	2	yes	complemented
36	C₃₁×#358	C₂²	1	2	yes	complemented
37	C₃₁×#357	C₂²	1	2	yes	—
38	C₃₁×#357	C₂²	1	2	yes	—
39	C₃₁×#357	C₂²	1	2	yes	complemented
40	C₃₁×C₂²⋊C₁₆	C₂²	1	1	yes	characteristic
41	C₃₁×C₂²⋊C₁₆	C₂²	1	1	yes	characteristic
42	C₃₁×#346	C₂²	1	2	yes	complemented
43	C₂²×D₄×C₃₁	—	2	1	no	—
44	C₂³×C₄×C₃₁	—	4	1	no	—
45	C₂×Q₁₆×C₃₁	C₂³	1	1	yes	characteristic
46	C₂×SD₁₆×C₃₁	—	2	1	no	—
47	C₂×D₈×C₃₁	—	2	4	no	—
48	C₂×D₈×C₃₁	—	2	1	no	—
49	C₂×D₈×C₃₁	C₂³	1	1	yes	characteristic
50	C₂×D₈×C₃₁	C₂³	1	2	yes	—
51	C₂²×C₈×C₃₁	D₄	1	1	yes	characteristic
52	C₂²×C₈×C₃₁	C₂³	1	1	yes	characteristic
53	C₃₁×#126	—	4	1	no	—
54	C₃₁×#125	—	2	1	no	—
55	C₃₁×#124	—	2	1	no	—
56	C₃₁×#123	—	4	1	no	—
57	C₃₁×C₂²⋊Q₈	—	2	1	no	—
58	C₃₁×#121	—	2	1	no	—
59	C₄×Q₈×C₃₁	—	2	1	no	—
60	C₄×D₄×C₃₁	—	2	2	no	—
61	C₄×D₄×C₃₁	—	2	2	no	—
62	C₄×D₄×C₃₁	—	2	1	no	—
63	C₄×D₄×C₃₁	—	4	1	no	—
64	C₄×D₄×C₃₁	—	4	1	no	—
65	C₄×D₄×C₃₁	—	4	1	no	—
66	C₃₁×#117	—	4	1	no	—
67	C₂×C₄⋊C₄×C₃₁	—	2	1	no	—
68	C₂×C₂²⋊C₄×C₃₁	—	4	1	no	—
69	C₂×C₄²×C₃₁	D₄	1	1	yes	characteristic
70	C₃₁×SD₃₂	—	2	4	no	—
71	C₃₁×D₁₆	—	2	4	no	—
72	C₂×C₁₆×C₃₁	C₂³	1	2	yes	—
73	C₂×C₁₆×C₃₁	C₂³	1	2	yes	—
74	C₃₁×#107	—	2	1	no	—
75	C₃₁×#107	C₂³	1	2	yes	—
76	C₃₁×#107	C₂³	1	1	yes	characteristic
77	C₃₁×#107	C₂³	1	1	yes	characteristic
78	C₃₁×#106	—	2	1	no	—
79	C₃₁×#103	—	2	1	no	—
80	C₃₁×#103	—	2	1	no	—
81	C₃₁×#102	—	2	2	no	—
82	C₃₁×#102	—	2	2	no	—
83	C₃₁×#102	—	2	1	no	—
84	C₃₁×#102	—	2	1	no	—
85	C₄×C₈×C₃₁	D₄	1	1	yes	characteristic
86	C₄×C₈×C₃₁	D₄	1	1	yes	characteristic
87	C₄×C₈×C₃₁	C₂³	1	1	yes	characteristic
88	C₄×C₈×C₃₁	C₂³	1	1	yes	characteristic
89	C₂⁴×C₃₁	—	4	1	no	—
90	C₂×Q₈×C₃₁	—	2	1	no	—
91	C₂×D₄×C₃₁	—	2	4	no	—
92	C₂×D₄×C₃₁	—	2	2	no	—
93	C₂×D₄×C₃₁	—	2	1	no	—
94	C₂×D₄×C₃₁	—	4	2	no	—
95	C₂×D₄×C₃₁	—	4	1	no	—
96	C₂×D₄×C₃₁	—	4	1	no	—
97	C₂²×C₄×C₃₁	—	4	1	no	—
98	C₂²×C₄×C₃₁	—	4	1	no	—
99	C₂²×C₄×C₃₁	—	4	2	no	—
100	C₂²×C₄×C₃₁	—	4	2	no	—
101	C₂²×C₄×C₃₁	—	8	4	no	—
102	C₂²×C₄×C₃₁	D₈	1	1	yes	characteristic
103	C₂²×C₄×C₃₁	D₈	1	1	yes	characteristic
104	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
105	Q₁₆×C₃₁	—	2	2	no	—
106	SD₁₆×C₃₁	—	4	2	no	—
107	D₈×C₃₁	—	2	4	no	—
108	D₈×C₃₁	—	2	2	no	—
109	D₈×C₃₁	—	2	2	no	—
110	D₈×C₃₁	#41	1	4	yes	—
111	C₂×C₈×C₃₁	—	2	2	no	—
112	C₂×C₈×C₃₁	—	4	1	no	—
113	C₂×C₈×C₃₁	C₂×D₄	1	1	yes	characteristic
114	C₂×C₈×C₃₁	C₂×D₄	1	1	yes	characteristic
115	C₂×C₈×C₃₁	C₂×D₄	1	1	yes	characteristic
116	C₂×C₈×C₃₁	C₂⁴	1	1	yes	characteristic
117	C₄⋊C₄×C₃₁	—	2	2	no	—
118	C₄⋊C₄×C₃₁	—	2	1	no	—
119	C₄⋊C₄×C₃₁	—	2	1	no	—
120	C₄⋊C₄×C₃₁	—	4	1	no	—
121	C₄⋊C₄×C₃₁	—	4	1	no	—
122	C₄⋊C₄×C₃₁	—	4	1	no	—
123	C₄⋊C₄×C₃₁	—	4	1	no	—
124	C₄⋊C₄×C₃₁	—	4	1	no	—
125	C₂²⋊C₄×C₃₁	—	4	1	no	—
126	C₂²⋊C₄×C₃₁	—	4	2	no	—
127	C₂²⋊C₄×C₃₁	—	4	2	no	—
128	C₂²⋊C₄×C₃₁	—	4	1	no	—
129	C₂²⋊C₄×C₃₁	—	4	1	no	—
130	C₂²⋊C₄×C₃₁	—	4	1	no	—
131	C₂²⋊C₄×C₃₁	—	4	1	no	—
132	C₄²×C₃₁	—	4	1	no	—
133	C₄²×C₃₁	D₈	1	1	yes	characteristic
134	C₄²×C₃₁	D₈	1	1	yes	characteristic
135	C₄²×C₃₁	C₂×D₄	1	1	yes	characteristic
136	C₄²×C₃₁	C₂×D₄	1	1	yes	characteristic
137	C₁₆×C₃₁	—	2	2	no	—
138	C₁₆×C₃₁	—	2	2	no	—
139	#33674	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
140	C₂³×C₃₁	—	4	4	no	—
141	C₂³×C₃₁	—	4	2	no	—
142	C₂³×C₃₁	—	4	1	no	—
143	C₂³×C₃₁	—	8	2	no	—
144	C₂³×C₃₁	—	8	2	no	—
145	C₂³×C₃₁	C₂×D₈	1	1	yes	characteristic
146	Q₈×C₃₁	—	4	1	no	—
147	Q₈×C₃₁	—	4	1	no	—
148	D₄×C₃₁	—	2	4	no	—
149	D₄×C₃₁	—	4	4	no	—
150	D₄×C₃₁	—	4	1	no	—
151	D₄×C₃₁	—	4	2	no	—
152	D₄×C₃₁	—	4	1	no	—
153	D₄×C₃₁	—	8	2	no	—
154	C₂×C₄×C₃₁	—	2	2	no	—
155	C₂×C₄×C₃₁	—	2	1	no	—
156	C₂×C₄×C₃₁	—	2	1	no	—
157	C₂×C₄×C₃₁	—	4	1	no	—
158	C₂×C₄×C₃₁	—	4	1	no	—
159	C₂×C₄×C₃₁	—	4	1	no	—
160	C₂×C₄×C₃₁	—	4	1	no	—
161	C₂×C₄×C₃₁	—	4	2	no	—
162	C₂×C₄×C₃₁	—	8	2	no	—
163	C₂×C₄×C₃₁	—	8	4	no	—
164	C₂×C₄×C₃₁	—	8	4	no	—
165	C₂×C₄×C₃₁	—	8	2	no	—
166	C₂×C₄×C₃₁	C₂×D₈	1	1	yes	characteristic
167	C₂×C₄×C₃₁	C₂×D₈	1	1	yes	characteristic
168	C₂×C₄×C₃₁	C₂×D₈	1	1	yes	characteristic
169	C₂×C₄×C₃₁	C₂×D₈	1	1	yes	characteristic
170	C₂×C₄×C₃₁	C₂×D₈	1	1	yes	characteristic
171	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
172	C₈×C₃₁	—	2	1	no	—
173	C₈×C₃₁	—	2	1	no	—
174	C₈×C₃₁	—	2	1	no	—
175	C₈×C₃₁	C₂×#41	1	2	yes	—
176	#3168	C₂×C₃₁	1	1	yes	characteristic, complemented
177	C₂×C₄×D₈	C₂×C₃₁	1	1	yes	characteristic, complemented
178	#2235	C₂×C₃₁	1	1	yes	characteristic, complemented
179	#2232	C₂×C₃₁	1	1	yes	characteristic, complemented
180	#2228	C₂×C₃₁	1	2	yes	complemented
181	#2210	C₂×C₃₁	1	2	yes	complemented
182	#2187	C₂×C₃₁	1	1	yes	characteristic, complemented
183	#2184	C₂×C₃₁	1	1	yes	characteristic
184	#2177	C₂×C₃₁	1	2	yes	complemented
185	#2175	C₂×C₃₁	1	2	yes	complemented
186	#2163	C₂×C₃₁	1	1	yes	characteristic, complemented
187	C₂²×C₃₁	—	2	1	no	—
188	C₂²×C₃₁	—	2	1	no	—
189	C₂²×C₃₁	—	4	4	no	—
190	C₂²×C₃₁	—	8	8	no	—
191	C₂²×C₃₁	—	8	2	no	—
192	C₂²×C₃₁	—	8	2	no	—
193	C₂²×C₃₁	—	8	1	no	—
194	C₂²×C₃₁	—	8	1	no	—
195	C₂²×C₃₁	—	8	1	no	—
196	C₂²×C₃₁	#509	1	2	yes	—
197	C₂²×C₃₁	C₂²×D₈	1	1	yes	characteristic
198	C₄×C₃₁	—	2	1	no	—
199	C₄×C₃₁	—	2	1	no	—
200	C₄×C₃₁	—	2	1	no	—
201	C₄×C₃₁	—	2	1	no	—
202	C₄×C₃₁	—	2	1	no	—
203	C₄×C₃₁	—	8	1	no	—
204	C₄×C₃₁	—	8	2	no	—
205	C₄×C₃₁	—	8	1	no	—
206	C₄×C₃₁	—	8	1	no	—
207	C₄×C₃₁	#525	1	1	yes	characteristic
208	C₄×C₃₁	#575	1	1	yes	characteristic
209	C₂²×D₈	C₂²×C₃₁	1	1	yes	characteristic
210	#529	—	2	1	no	—
211	C₂×C₄×D₄	—	2	1	no	—
212	C₂×SD₃₂	C₂²×C₃₁	1	2	yes	—
213	C₂×D₁₆	C₂²×C₃₁	1	2	yes	—
214	#499	C₂²×C₃₁	1	1	yes	characteristic, complemented
215	#495	C₂²×C₃₁	1	1	yes	characteristic, complemented
216	#488	—	2	1	no	—
217	#467	C₂²×C₃₁	1	1	yes	characteristic
218	#466	C₂²×C₃₁	1	1	yes	characteristic
219	#465	—	2	1	no	—
220	C₄×Q₁₆	C₂²×C₃₁	1	1	yes	characteristic, complemented
221	C₄×SD₁₆	—	2	1	no	—
222	C₄×D₈	—	2	1	no	—
223	C₄×D₈	—	2	1	no	—
224	C₄×D₈	C₂²×C₃₁	1	2	yes	—
225	C₄×D₈	C₂²×C₃₁	1	2	yes	—
226	C₄×D₈	C₂²×C₃₁	1	1	yes	characteristic, complemented
227	C₂×#107	C₂²×C₃₁	1	1	yes	characteristic
228	C₂×#102	—	2	1	no	—
229	C₂×C₄×C₈	C₂²×C₃₁	1	1	yes	characteristic
230	#367	C₂²×C₃₁	1	2	yes	complemented
231	#366	C₂²×C₃₁	1	2	yes	complemented
232	C₄⋊C₁₆	C₂²×C₃₁	1	2	yes	complemented
233	#358	C₂²×C₃₁	1	2	yes	complemented
234	#357	C₂²×C₃₁	1	2	yes	—
235	#357	C₂²×C₃₁	1	2	yes	—
236	#357	C₂²×C₃₁	1	2	yes	complemented
237	C₂²⋊C₁₆	C₂²×C₃₁	1	1	yes	characteristic
238	C₂²⋊C₁₆	C₂²×C₃₁	1	1	yes	characteristic
239	#346	C₂²×C₃₁	1	2	yes	complemented
240	C₂×C₃₁	—	2	2	no	—
241	C₂×C₃₁	—	8	4	no	—
242	C₂×C₃₁	—	16	1	no	—
243	C₂×C₃₁	#3100	1	1	yes	characteristic
244	C₂×C₃₁	C₂×#509	1	1	yes	characteristic
245	C₂×C₃₁	#3414	1	1	yes	characteristic
246	C₂²×D₄	—	2	1	no	—
247	C₂³×C₄	—	4	1	no	—
248	C₂×Q₁₆	C₂³×C₃₁	1	1	yes	characteristic
249	C₂×SD₁₆	—	2	1	no	—
250	C₂×D₈	—	2	4	no	—
251	C₂×D₈	—	2	1	no	—
252	C₂×D₈	C₂³×C₃₁	1	1	yes	characteristic
253	C₂×D₈	C₂³×C₃₁	1	2	yes	—
254	C₂²×C₈	D₄×C₃₁	1	1	yes	characteristic
255	C₂²×C₈	C₂³×C₃₁	1	1	yes	characteristic
256	#126	—	4	1	no	—
257	#125	—	2	1	no	—
258	#124	—	2	1	no	—
259	#123	—	4	1	no	—
260	C₂²⋊Q₈	—	2	1	no	—
261	#121	—	2	1	no	—
262	C₄×Q₈	—	2	1	no	—
263	C₄×D₄	—	2	2	no	—
264	C₄×D₄	—	2	2	no	—
265	C₄×D₄	—	2	1	no	—
266	C₄×D₄	—	4	1	no	—
267	C₄×D₄	—	4	1	no	—
268	C₄×D₄	—	4	1	no	—
269	#117	—	4	1	no	—
270	C₂×C₄⋊C₄	—	2	1	no	—
271	C₂×C₂²⋊C₄	—	4	1	no	—
272	C₂×C₄²	D₄×C₃₁	1	1	yes	characteristic
273	SD₃₂	—	2	4	no	—
274	D₁₆	—	2	4	no	—
275	C₂×C₁₆	C₂³×C₃₁	1	2	yes	—
276	C₂×C₁₆	C₂³×C₃₁	1	2	yes	—
277	#107	—	2	1	no	—
278	#107	C₂³×C₃₁	1	1	yes	characteristic
279	#107	C₂³×C₃₁	1	1	yes	characteristic
280	#107	C₂³×C₃₁	1	2	yes	—
281	#106	—	2	1	no	—
282	#103	—	2	1	no	—
283	#103	—	2	1	no	—
284	#102	—	2	2	no	—
285	#102	—	2	2	no	—
286	#102	—	2	1	no	—
287	#102	—	2	1	no	—
288	C₄×C₈	D₄×C₃₁	1	1	yes	characteristic
289	C₄×C₈	D₄×C₃₁	1	1	yes	characteristic
290	C₄×C₈	C₂³×C₃₁	1	1	yes	characteristic
291	C₄×C₈	C₂³×C₃₁	1	1	yes	characteristic
292	C₃₁	#33674	1	1	yes	characteristic, direct product factor, Sylow
293	C₂⁴	—	4	1	no	—
294	C₂×Q₈	—	2	1	no	—
295	C₂×D₄	—	2	4	no	—
296	C₂×D₄	—	2	2	no	—
297	C₂×D₄	—	2	1	no	—
298	C₂×D₄	—	4	2	no	—
299	C₂×D₄	—	4	1	no	—
300	C₂×D₄	—	4	1	no	—
301	C₂²×C₄	—	4	1	no	—
302	C₂²×C₄	—	4	1	no	—
303	C₂²×C₄	—	4	2	no	—
304	C₂²×C₄	—	4	2	no	—
305	C₂²×C₄	—	8	4	no	—
306	C₂²×C₄	D₈×C₃₁	1	1	yes	characteristic
307	C₂²×C₄	D₈×C₃₁	1	1	yes	characteristic
308	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
309	Q₁₆	—	2	2	no	—
310	SD₁₆	—	4	2	no	—
311	D₈	—	2	4	no	—
312	D₈	—	2	2	no	—
313	D₈	—	2	2	no	—
314	D₈	#41×C₃₁	1	4	yes	—
315	C₂×C₈	—	2	2	no	—
316	C₂×C₈	—	4	1	no	—
317	C₂×C₈	C₂×D₄×C₃₁	1	1	yes	characteristic
318	C₂×C₈	C₂×D₄×C₃₁	1	1	yes	characteristic
319	C₂×C₈	C₂×D₄×C₃₁	1	1	yes	characteristic
320	C₂×C₈	C₂⁴×C₃₁	1	1	yes	characteristic
321	C₄⋊C₄	—	2	2	no	—
322	C₄⋊C₄	—	2	1	no	—
323	C₄⋊C₄	—	2	1	no	—
324	C₄⋊C₄	—	4	1	no	—
325	C₄⋊C₄	—	4	1	no	—
326	C₄⋊C₄	—	4	1	no	—
327	C₄⋊C₄	—	4	1	no	—
328	C₄⋊C₄	—	4	1	no	—
329	C₂²⋊C₄	—	4	1	no	—
330	C₂²⋊C₄	—	4	2	no	—
331	C₂²⋊C₄	—	4	2	no	—
332	C₂²⋊C₄	—	4	1	no	—
333	C₂²⋊C₄	—	4	1	no	—
334	C₂²⋊C₄	—	4	1	no	—
335	C₂²⋊C₄	—	4	1	no	—
336	C₄²	—	4	1	no	—
337	C₄²	D₈×C₃₁	1	1	yes	characteristic
338	C₄²	D₈×C₃₁	1	1	yes	characteristic
339	C₄²	C₂×D₄×C₃₁	1	1	yes	characteristic
340	C₄²	C₂×D₄×C₃₁	1	1	yes	characteristic
341	C₁₆	—	2	2	no	—
342	C₁₆	—	2	2	no	—
343	C₂³	—	4	4	no	—
344	C₂³	—	4	2	no	—
345	C₂³	—	4	1	no	—
346	C₂³	—	8	2	no	—
347	C₂³	—	8	2	no	—
348	C₂³	C₂×D₈×C₃₁	1	1	yes	characteristic
349	Q₈	—	4	1	no	—
350	Q₈	—	4	1	no	—
351	D₄	—	2	4	no	—
352	D₄	—	4	4	no	—
353	D₄	—	4	1	no	—
354	D₄	—	4	2	no	—
355	D₄	—	4	1	no	—
356	D₄	—	8	2	no	—
357	C₂×C₄	—	2	2	no	—
358	C₂×C₄	—	2	1	no	—
359	C₂×C₄	—	2	1	no	—
360	C₂×C₄	—	4	1	no	—
361	C₂×C₄	—	4	1	no	—
362	C₂×C₄	—	4	1	no	—
363	C₂×C₄	—	4	1	no	—
364	C₂×C₄	—	4	2	no	—
365	C₂×C₄	—	8	2	no	—
366	C₂×C₄	—	8	4	no	—
367	C₂×C₄	—	8	4	no	—
368	C₂×C₄	—	8	2	no	—
369	C₂×C₄	C₂×D₈×C₃₁	1	1	yes	characteristic
370	C₂×C₄	C₂×D₈×C₃₁	1	1	yes	characteristic
371	C₂×C₄	C₂×D₈×C₃₁	1	1	yes	characteristic
372	C₂×C₄	C₂×D₈×C₃₁	1	1	yes	characteristic
373	C₂×C₄	C₂×D₈×C₃₁	1	1	yes	characteristic
374	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
375	C₈	—	2	1	no	—
376	C₈	—	2	1	no	—
377	C₈	—	2	1	no	—
378	C₈	C₂×#41×C₃₁	1	2	yes	—
379	C₂²	—	2	1	no	—
380	C₂²	—	2	1	no	—
381	C₂²	—	4	4	no	—
382	C₂²	—	8	8	no	—
383	C₂²	—	8	2	no	—
384	C₂²	—	8	2	no	—
385	C₂²	—	8	1	no	—
386	C₂²	—	8	1	no	—
387	C₂²	—	8	1	no	—
388	C₂²	C₃₁×#509	1	2	yes	—
389	C₂²	C₂²×D₈×C₃₁	1	1	yes	characteristic
390	C₄	—	2	1	no	—
391	C₄	—	2	1	no	—
392	C₄	—	2	1	no	—
393	C₄	—	2	1	no	—
394	C₄	—	2	1	no	—
395	C₄	—	8	1	no	—
396	C₄	—	8	2	no	—
397	C₄	—	8	1	no	—
398	C₄	—	8	1	no	—
399	C₄	C₃₁×#525	1	1	yes	characteristic
400	C₄	C₃₁×#575	1	1	yes	characteristic
401	C₂	—	2	2	no	—
402	C₂	—	8	4	no	—
403	C₂	—	16	1	no	—
404	C₂	C₃₁×#3100	1	1	yes	characteristic
405	C₂	C₂×C₃₁×#509	1	1	yes	characteristic
406	C₂	C₃₁×#3414	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₃₁	C₂²×D₈
commutator, Frattini, cosocle	C₂×C₈	C₂⁴×C₃₁
prime radical	C₃₁×#2184	C₂
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₃₁ ◁ C₂×C₄²×C₃₁ ◁ C₂×C₄×C₈×C₃₁ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₃₁×#3168	C₂	semidirect
2. C₂×C₄×D₈×C₃₁	C₂	semidirect
3. C₃₁×#2235	C₂	semidirect
4. C₃₁×#2232	C₂	semidirect
5. C₃₁×#2228	C₂	semidirect
6. C₃₁×#2210	C₂	semidirect
7. C₃₁×#2187	C₂	semidirect
9. C₃₁×#2177	C₂	semidirect
10. C₃₁×#2175	C₂	semidirect
11. C₃₁×#2163	C₂	semidirect
17. C₃₁×#499	C₂²	semidirect
18. C₃₁×#495	C₂²	semidirect
23. C₄×Q₁₆×C₃₁	C₂²	semidirect
29. C₄×D₈×C₃₁	C₂²	semidirect
33. C₃₁×#367	C₂²	semidirect
34. C₃₁×#366	C₂²	semidirect
35. C₃₁×C₄⋊C₁₆	C₂²	semidirect
36. C₃₁×#358	C₂²	semidirect
39. C₃₁×#357	C₂²	semidirect
42. C₃₁×#346	C₂²	semidirect
139. #33674	C₃₁	direct (N × H)
176. #3168	C₂×C₃₁	semidirect
177. C₂×C₄×D₈	C₂×C₃₁	semidirect
178. #2235	C₂×C₃₁	semidirect
179. #2232	C₂×C₃₁	semidirect
180. #2228	C₂×C₃₁	semidirect
181. #2210	C₂×C₃₁	semidirect
182. #2187	C₂×C₃₁	semidirect
184. #2177	C₂×C₃₁	semidirect
185. #2175	C₂×C₃₁	semidirect
186. #2163	C₂×C₃₁	semidirect
214. #499	C₂²×C₃₁	semidirect
215. #495	C₂²×C₃₁	semidirect
220. C₄×Q₁₆	C₂²×C₃₁	semidirect
226. C₄×D₈	C₂²×C₃₁	semidirect
230. #367	C₂²×C₃₁	semidirect
231. #366	C₂²×C₃₁	semidirect
232. C₄⋊C₁₆	C₂²×C₃₁	semidirect
233. #358	C₂²×C₃₁	semidirect
236. #357	C₂²×C₃₁	semidirect
239. #346	C₂²×C₃₁	semidirect
292. C₃₁	#33674	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×#3168	2
C₂×C₄×D₈×C₃₁	2
C₃₁×#2235	2
C₃₁×#2232	2
C₃₁×#2228	2
C₃₁×#2210	2
C₃₁×#2187	2
C₃₁×#2184	2
C₃₁×#2177	2
C₃₁×#2175	2
C₃₁×#2163	2
#33674	31

Subgroups — C31×#33674 #21490433

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#33674 #21490433