Subgroups — C₃₁×#25656 #21482415

Order 7936.

1012 conjugacy classes of subgroups.

The subgroup lattice has 468 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃₁×#556	C₂	1	1	yes	maximal, characteristic
2	C₂×C₃₁×#491	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×C₃₁×#478	C₂	1	1	yes	maximal, characteristic
4	C₃₁×#2584	C₂	1	1	yes	maximal, characteristic, complemented
5	C₃₁×#2042	C₂	1	2	yes	maximal
6	C₃₁×#2042	C₂	1	2	yes	maximal
7	C₃₁×#1985	C₂	1	1	yes	maximal, characteristic, complemented
8	C₃₁×#1981	C₂	1	1	yes	maximal, characteristic, complemented
9	C₃₁×#1865	C₂	1	2	yes	maximal, complemented
10	C₃₁×#1863	C₂	1	2	yes	maximal, complemented
11	C₃₁×#1762	C₂	1	1	yes	maximal, characteristic
12	C₃₁×#556	—	2	4	no	—
13	C₃₁×#556	—	2	4	no	—
14	C₂×C₃₁×C₄⋊Q₈	C₂²	1	1	yes	characteristic
15	C₂×C₃₁×#126	—	2	2	no	—
16	C₂×C₃₁×#125	C₂²	1	1	yes	characteristic
17	C₂×C₃₁×#124	C₂²	1	2	yes	—
18	C₂×C₃₁×C₂²⋊Q₈	—	2	1	no	—
19	C₂×C₄×Q₈×C₃₁	C₂²	1	2	yes	—
20	C₂×C₄×Q₈×C₃₁	C₂²	1	1	yes	characteristic
21	C₂×C₄×D₄×C₃₁	C₂²	1	1	yes	characteristic
22	C₂×C₃₁×#117	—	2	1	no	—
23	C₃₁×#491	—	2	4	no	—
24	C₃₁×#478	—	2	4	no	—
25	C₂×C₃₁×#107	C₂²	1	1	yes	characteristic
26	C₂×C₃₁×#106	C₂²	1	1	yes	characteristic
27	C₂×C₃₁×C₄⋊C₈	C₂²	1	1	yes	characteristic
28	C₂×C₃₁×#103	C₂²	1	2	yes	—
29	C₂×C₃₁×#103	C₂²	1	2	yes	—
30	C₂×C₃₁×#103	C₂²	1	2	yes	—
31	C₂×C₃₁×#103	C₂²	1	2	yes	—
32	C₂×C₃₁×#102	C₂²	1	2	yes	—
33	C₂×C₃₁×#102	C₂²	1	1	yes	characteristic
34	C₂×C₃₁×#102	C₂²	1	1	yes	characteristic
35	C₂×C₃₁×#97	C₂²	1	1	yes	characteristic
36	C₃₁×#398	—	2	1	no	—
37	C₃₁×#395	—	2	1	no	—
38	C₃₁×#389	—	2	2	no	—
39	C₃₁×#389	C₂²	1	2	yes	—
40	C₃₁×#389	C₂²	1	2	yes	—
41	C₄×C₄⋊C₄×C₃₁	C₂²	1	1	yes	characteristic
42	C₃₁×#340	C₂²	1	2	yes	—
43	C₃₁×#336	C₂²	1	2	yes	—
44	C₃₁×#336	C₂²	1	2	yes	—
45	C₂²×Q₈×C₃₁	D₄	1	1	yes	characteristic
46	C₂²×Q₈×C₃₁	C₂³	1	2	yes	—
47	C₂²×D₄×C₃₁	C₂³	1	1	yes	characteristic
48	C₂³×C₄×C₃₁	—	2	1	no	—
49	C₂²×C₈×C₃₁	C₂³	1	2	yes	—
50	C₂²×C₈×C₃₁	C₂³	1	1	yes	characteristic
51	C₂²×C₈×C₃₁	C₂³	1	1	yes	characteristic
52	C₃₁×C₄⋊Q₈	—	2	2	no	—
53	C₃₁×C₄⋊Q₈	D₄	1	4	yes	—
54	C₃₁×#126	—	2	16	no	—
55	C₃₁×#125	—	2	2	no	—
56	C₃₁×#125	—	2	2	no	—
57	C₃₁×#124	—	2	4	no	—
58	C₃₁×#124	—	2	4	no	—
59	C₃₁×C₂²⋊Q₈	—	2	8	no	—
60	C₄×Q₈×C₃₁	—	2	4	no	—
61	C₄×Q₈×C₃₁	—	2	4	no	—
62	C₄×Q₈×C₃₁	—	2	2	no	—
63	C₄×Q₈×C₃₁	—	2	2	no	—
64	C₄×D₄×C₃₁	—	2	2	no	—
65	C₄×D₄×C₃₁	D₄	1	4	yes	—
66	C₃₁×#117	—	2	8	no	—
67	C₂×C₄⋊C₄×C₃₁	—	2	1	no	—
68	C₂×C₄⋊C₄×C₃₁	—	2	2	no	—
69	C₂×C₄⋊C₄×C₃₁	—	2	1	no	—
70	C₂×C₄⋊C₄×C₃₁	—	2	1	no	—
71	C₂×C₄⋊C₄×C₃₁	—	2	2	no	—
72	C₂×C₄⋊C₄×C₃₁	—	2	1	no	—
73	C₂×C₄⋊C₄×C₃₁	—	2	2	no	—
74	C₂×C₄⋊C₄×C₃₁	D₄	1	1	yes	characteristic
75	C₂×C₄⋊C₄×C₃₁	D₄	1	2	yes	—
76	C₂×C₄⋊C₄×C₃₁	C₂³	1	1	yes	characteristic
77	C₂×C₄⋊C₄×C₃₁	C₂³	1	1	yes	characteristic
78	C₂×C₄⋊C₄×C₃₁	C₂³	1	1	yes	characteristic
79	C₂×C₄⋊C₄×C₃₁	C₂³	1	2	yes	—
80	C₂×C₂²⋊C₄×C₃₁	—	2	2	no	—
81	C₂×C₂²⋊C₄×C₃₁	—	2	1	no	—
82	C₂×C₂²⋊C₄×C₃₁	—	2	2	no	—
83	C₂×C₄²×C₃₁	—	2	2	no	—
84	C₂×C₄²×C₃₁	—	2	1	no	—
85	C₂×C₄²×C₃₁	C₂³	1	2	yes	—
86	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
87	C₃₁×#107	—	2	2	no	—
88	C₃₁×#106	—	2	2	no	—
89	C₃₁×C₄⋊C₈	—	2	2	no	—
90	C₃₁×#103	—	2	4	no	—
91	C₃₁×#103	—	2	4	no	—
92	C₃₁×#103	—	2	4	no	—
93	C₃₁×#103	—	2	4	no	—
94	C₃₁×#102	—	2	2	no	—
95	C₃₁×#102	—	2	2	no	—
96	C₃₁×#102	—	2	4	no	—
97	C₃₁×#97	—	2	2	no	—
98	C₃₁×#95	—	2	1	no	—
99	C₃₁×#95	—	2	2	no	—
100	C₃₁×#95	—	2	2	no	—
101	C₂⁴×C₃₁	—	2	1	no	—
102	C₂×Q₈×C₃₁	—	2	2	no	—
103	C₂×Q₈×C₃₁	—	2	2	no	—
104	C₂×Q₈×C₃₁	—	2	2	no	—
105	C₂×Q₈×C₃₁	—	2	2	no	—
106	C₂×Q₈×C₃₁	—	2	2	no	—
107	C₂×Q₈×C₃₁	—	2	2	no	—
108	C₂×Q₈×C₃₁	—	4	2	no	—
109	C₂×Q₈×C₃₁	C₂×D₄	1	4	yes	—
110	C₂×D₄×C₃₁	—	2	1	no	—
111	C₂×D₄×C₃₁	—	2	2	no	—
112	C₂×D₄×C₃₁	—	2	2	no	—
113	C₂×D₄×C₃₁	C₂×D₄	1	2	yes	—
114	C₂²×C₄×C₃₁	—	2	2	no	—
115	C₂²×C₄×C₃₁	—	2	1	no	—
116	C₂²×C₄×C₃₁	—	2	2	no	—
117	C₂²×C₄×C₃₁	—	2	4	no	—
118	C₂²×C₄×C₃₁	—	2	1	no	—
119	C₂²×C₄×C₃₁	—	2	1	no	—
120	C₂²×C₄×C₃₁	—	4	4	no	—
121	C₂²×C₄×C₃₁	C₂×D₄	1	2	yes	—
122	C₂²×C₄×C₃₁	C₂×D₄	1	2	yes	—
123	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
124	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
125	C₂²×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
126	C₂×C₈×C₃₁	—	2	2	no	—
127	C₂×C₈×C₃₁	—	2	1	no	—
128	C₂×C₈×C₃₁	—	2	1	no	—
129	C₂×C₈×C₃₁	—	2	2	no	—
130	C₂×C₈×C₃₁	—	2	2	no	—
131	C₂×C₈×C₃₁	—	2	2	no	—
132	C₂×C₈×C₃₁	—	2	2	no	—
133	C₄⋊C₄×C₃₁	—	2	8	no	—
134	C₄⋊C₄×C₃₁	—	2	4	no	—
135	C₄⋊C₄×C₃₁	—	2	4	no	—
136	C₄⋊C₄×C₃₁	—	2	2	no	—
137	C₄⋊C₄×C₃₁	—	2	4	no	—
138	C₄⋊C₄×C₃₁	—	2	1	no	—
139	C₄⋊C₄×C₃₁	—	2	2	no	—
140	C₄⋊C₄×C₃₁	—	2	1	no	—
141	C₄⋊C₄×C₃₁	—	2	1	no	—
142	C₄⋊C₄×C₃₁	—	2	1	no	—
143	C₄⋊C₄×C₃₁	—	2	1	no	—
144	C₄⋊C₄×C₃₁	—	4	4	no	—
145	C₄⋊C₄×C₃₁	—	4	2	no	—
146	C₄⋊C₄×C₃₁	—	4	2	no	—
147	C₄⋊C₄×C₃₁	—	8	2	no	—
148	C₄⋊C₄×C₃₁	C₂×D₄	1	2	yes	—
149	C₄⋊C₄×C₃₁	#41	1	4	yes	—
150	C₂²⋊C₄×C₃₁	—	2	8	no	—
151	C₂²⋊C₄×C₃₁	—	2	8	no	—
152	C₂²⋊C₄×C₃₁	—	2	4	no	—
153	C₄²×C₃₁	—	2	8	no	—
154	C₄²×C₃₁	—	2	2	no	—
155	C₄²×C₃₁	—	2	2	no	—
156	C₄²×C₃₁	—	8	1	no	—
157	C₄²×C₃₁	C₂×D₄	1	2	yes	—
158	C₄²×C₃₁	C₂×D₄	1	2	yes	—
159	#25656	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
160	C₂³×C₃₁	—	2	2	no	—
161	C₂³×C₃₁	—	4	2	no	—
162	C₂³×C₃₁	—	4	2	no	—
163	C₂³×C₃₁	—	4	1	no	—
164	C₂³×C₃₁	—	4	1	no	—
165	C₂³×C₃₁	C₂²×D₄	1	1	yes	characteristic
166	Q₈×C₃₁	—	2	4	no	—
167	Q₈×C₃₁	—	2	4	no	—
168	Q₈×C₃₁	—	4	4	no	—
169	Q₈×C₃₁	—	4	2	no	—
170	Q₈×C₃₁	—	4	2	no	—
171	D₄×C₃₁	—	2	2	no	—
172	D₄×C₃₁	—	2	2	no	—
173	D₄×C₃₁	—	4	2	no	—
174	C₂×C₄×C₃₁	—	2	2	no	—
175	C₂×C₄×C₃₁	—	2	1	no	—
176	C₂×C₄×C₃₁	—	2	1	no	—
177	C₂×C₄×C₃₁	—	2	2	no	—
178	C₂×C₄×C₃₁	—	2	2	no	—
179	C₂×C₄×C₃₁	—	2	4	no	—
180	C₂×C₄×C₃₁	—	2	2	no	—
181	C₂×C₄×C₃₁	—	2	4	no	—
182	C₂×C₄×C₃₁	—	2	2	no	—
183	C₂×C₄×C₃₁	—	2	2	no	—
184	C₂×C₄×C₃₁	—	2	2	no	—
185	C₂×C₄×C₃₁	—	2	2	no	—
186	C₂×C₄×C₃₁	—	4	8	no	—
187	C₂×C₄×C₃₁	—	4	2	no	—
188	C₂×C₄×C₃₁	—	4	2	no	—
189	C₂×C₄×C₃₁	—	4	2	no	—
190	C₂×C₄×C₃₁	—	4	2	no	—
191	C₂×C₄×C₃₁	—	4	2	no	—
192	C₂×C₄×C₃₁	—	4	1	no	—
193	C₂×C₄×C₃₁	—	4	1	no	—
194	C₂×C₄×C₃₁	—	4	2	no	—
195	C₂×C₄×C₃₁	—	4	1	no	—
196	C₂×C₄×C₃₁	—	4	1	no	—
197	C₂×C₄×C₃₁	#120	1	2	yes	—
198	C₂×C₄×C₃₁	#120	1	2	yes	—
199	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
200	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
201	C₂×C₄×C₃₁	C₂×#41	1	2	yes	—
202	C₂×C₄×C₃₁	2₋¹⁺⁴	1	2	yes	—
203	C₈×C₃₁	—	4	1	no	—
204	C₈×C₃₁	—	4	1	no	—
205	C₈×C₃₁	—	4	2	no	—
206	C₂×#556	C₂×C₃₁	1	1	yes	characteristic
207	C₂×#491	C₂×C₃₁	1	1	yes	characteristic, complemented
208	C₂×#478	C₂×C₃₁	1	1	yes	characteristic
209	#2584	C₂×C₃₁	1	1	yes	characteristic, complemented
210	#2042	C₂×C₃₁	1	2	yes	—
211	#2042	C₂×C₃₁	1	2	yes	—
212	#1985	C₂×C₃₁	1	1	yes	characteristic, complemented
213	#1981	C₂×C₃₁	1	1	yes	characteristic, complemented
214	#1865	C₂×C₃₁	1	2	yes	complemented
215	#1863	C₂×C₃₁	1	2	yes	complemented
216	#1762	C₂×C₃₁	1	1	yes	characteristic
217	C₂²×C₃₁	—	4	2	no	—
218	C₂²×C₃₁	—	4	2	no	—
219	C₂²×C₃₁	—	4	2	no	—
220	C₂²×C₃₁	—	8	2	no	—
221	C₂²×C₃₁	—	8	2	no	—
222	C₂²×C₃₁	C₂×#120	1	1	yes	characteristic
223	C₂²×C₃₁	#548	1	2	yes	—
224	C₂²×C₃₁	#575	1	2	yes	—
225	C₂²×C₃₁	#577	1	1	yes	characteristic
226	C₂²×C₃₁	#578	1	1	yes	characteristic
227	C₄×C₃₁	—	2	2	no	—
228	C₄×C₃₁	—	2	2	no	—
229	C₄×C₃₁	—	4	4	no	—
230	C₄×C₃₁	—	4	4	no	—
231	C₄×C₃₁	—	4	2	no	—
232	C₄×C₃₁	—	4	2	no	—
233	C₄×C₃₁	—	4	2	no	—
234	C₄×C₃₁	—	8	1	no	—
235	C₄×C₃₁	—	8	1	no	—
236	C₄×C₃₁	#532	1	1	yes	characteristic
237	C₄×C₃₁	#536	1	1	yes	characteristic
238	C₄×C₃₁	#547	1	2	yes	—
239	#556	—	2	4	no	—
240	#556	—	2	4	no	—
241	C₂×C₄⋊Q₈	C₂²×C₃₁	1	1	yes	characteristic
242	C₂×#126	—	2	2	no	—
243	C₂×#125	C₂²×C₃₁	1	1	yes	characteristic
244	C₂×#124	C₂²×C₃₁	1	2	yes	—
245	C₂×C₂²⋊Q₈	—	2	1	no	—
246	C₂×C₄×Q₈	C₂²×C₃₁	1	2	yes	—
247	C₂×C₄×Q₈	C₂²×C₃₁	1	1	yes	characteristic
248	C₂×C₄×D₄	C₂²×C₃₁	1	1	yes	characteristic
249	C₂×#117	—	2	1	no	—
250	#491	—	2	4	no	—
251	#478	—	2	4	no	—
252	C₂×#107	C₂²×C₃₁	1	1	yes	characteristic
253	C₂×#106	C₂²×C₃₁	1	1	yes	characteristic
254	C₂×C₄⋊C₈	C₂²×C₃₁	1	1	yes	characteristic
255	C₂×#103	C₂²×C₃₁	1	2	yes	—
256	C₂×#103	C₂²×C₃₁	1	2	yes	—
257	C₂×#103	C₂²×C₃₁	1	2	yes	—
258	C₂×#103	C₂²×C₃₁	1	2	yes	—
259	C₂×#102	C₂²×C₃₁	1	1	yes	characteristic
260	C₂×#102	C₂²×C₃₁	1	1	yes	characteristic
261	C₂×#102	C₂²×C₃₁	1	2	yes	—
262	C₂×#97	C₂²×C₃₁	1	1	yes	characteristic
263	#398	—	2	1	no	—
264	#395	—	2	1	no	—
265	#389	—	2	2	no	—
266	#389	C₂²×C₃₁	1	2	yes	—
267	#389	C₂²×C₃₁	1	2	yes	—
268	C₄×C₄⋊C₄	C₂²×C₃₁	1	1	yes	characteristic
269	#340	C₂²×C₃₁	1	2	yes	—
270	#336	C₂²×C₃₁	1	2	yes	—
271	#336	C₂²×C₃₁	1	2	yes	—
272	C₂×C₃₁	—	8	2	no	—
273	C₂×C₃₁	#2432	1	1	yes	characteristic
274	C₂×C₃₁	#3005	1	1	yes	characteristic
275	C₂×C₃₁	#3013	1	1	yes	characteristic
276	C₂×C₃₁	#3376	1	2	yes	—
277	C₂×C₃₁	#3377	1	2	yes	—
278	C₂²×Q₈	D₄×C₃₁	1	1	yes	characteristic
279	C₂²×Q₈	C₂³×C₃₁	1	2	yes	—
280	C₂²×D₄	C₂³×C₃₁	1	1	yes	characteristic
281	C₂³×C₄	—	2	1	no	—
282	C₂²×C₈	C₂³×C₃₁	1	2	yes	—
283	C₂²×C₈	C₂³×C₃₁	1	1	yes	characteristic
284	C₂²×C₈	C₂³×C₃₁	1	1	yes	characteristic
285	C₄⋊Q₈	—	2	2	no	—
286	C₄⋊Q₈	D₄×C₃₁	1	4	yes	—
287	#126	—	2	16	no	—
288	#125	—	2	2	no	—
289	#125	—	2	2	no	—
290	#124	—	2	4	no	—
291	#124	—	2	4	no	—
292	C₂²⋊Q₈	—	2	8	no	—
293	C₄×Q₈	—	2	4	no	—
294	C₄×Q₈	—	2	4	no	—
295	C₄×Q₈	—	2	2	no	—
296	C₄×Q₈	—	2	2	no	—
297	C₄×D₄	—	2	2	no	—
298	C₄×D₄	D₄×C₃₁	1	4	yes	—
299	#117	—	2	8	no	—
300	C₂×C₄⋊C₄	—	2	1	no	—
301	C₂×C₄⋊C₄	—	2	2	no	—
302	C₂×C₄⋊C₄	—	2	1	no	—
303	C₂×C₄⋊C₄	—	2	1	no	—
304	C₂×C₄⋊C₄	—	2	1	no	—
305	C₂×C₄⋊C₄	—	2	2	no	—
306	C₂×C₄⋊C₄	—	2	2	no	—
307	C₂×C₄⋊C₄	D₄×C₃₁	1	1	yes	characteristic
308	C₂×C₄⋊C₄	D₄×C₃₁	1	2	yes	—
309	C₂×C₄⋊C₄	C₂³×C₃₁	1	1	yes	characteristic
310	C₂×C₄⋊C₄	C₂³×C₃₁	1	1	yes	characteristic
311	C₂×C₄⋊C₄	C₂³×C₃₁	1	1	yes	characteristic
312	C₂×C₄⋊C₄	C₂³×C₃₁	1	2	yes	—
313	C₂×C₂²⋊C₄	—	2	2	no	—
314	C₂×C₂²⋊C₄	—	2	1	no	—
315	C₂×C₂²⋊C₄	—	2	2	no	—
316	C₂×C₄²	—	2	2	no	—
317	C₂×C₄²	—	2	1	no	—
318	C₂×C₄²	C₂³×C₃₁	1	2	yes	—
319	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
320	#107	—	2	2	no	—
321	#106	—	2	2	no	—
322	C₄⋊C₈	—	2	2	no	—
323	#103	—	2	4	no	—
324	#103	—	2	4	no	—
325	#103	—	2	4	no	—
326	#103	—	2	4	no	—
327	#102	—	2	2	no	—
328	#102	—	2	2	no	—
329	#102	—	2	4	no	—
330	#97	—	2	2	no	—
331	#95	—	2	1	no	—
332	#95	—	2	2	no	—
333	#95	—	2	2	no	—
334	C₃₁	#25656	1	1	yes	characteristic, direct product factor, Sylow
335	C₂⁴	—	2	1	no	—
336	C₂×Q₈	—	2	2	no	—
337	C₂×Q₈	—	2	2	no	—
338	C₂×Q₈	—	2	2	no	—
339	C₂×Q₈	—	2	2	no	—
340	C₂×Q₈	—	2	2	no	—
341	C₂×Q₈	—	2	2	no	—
342	C₂×Q₈	—	4	2	no	—
343	C₂×Q₈	C₂×D₄×C₃₁	1	4	yes	—
344	C₂×D₄	—	2	1	no	—
345	C₂×D₄	—	2	2	no	—
346	C₂×D₄	—	2	2	no	—
347	C₂×D₄	C₂×D₄×C₃₁	1	2	yes	—
348	C₂²×C₄	—	2	2	no	—
349	C₂²×C₄	—	2	1	no	—
350	C₂²×C₄	—	2	2	no	—
351	C₂²×C₄	—	2	4	no	—
352	C₂²×C₄	—	2	1	no	—
353	C₂²×C₄	—	2	1	no	—
354	C₂²×C₄	—	4	4	no	—
355	C₂²×C₄	C₂×D₄×C₃₁	1	2	yes	—
356	C₂²×C₄	C₂×D₄×C₃₁	1	2	yes	—
357	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
358	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
359	C₂²×C₄	C₂⁴×C₃₁	1	1	yes	characteristic
360	C₂×C₈	—	2	2	no	—
361	C₂×C₈	—	2	1	no	—
362	C₂×C₈	—	2	1	no	—
363	C₂×C₈	—	2	2	no	—
364	C₂×C₈	—	2	2	no	—
365	C₂×C₈	—	2	2	no	—
366	C₂×C₈	—	2	2	no	—
367	C₄⋊C₄	—	2	8	no	—
368	C₄⋊C₄	—	2	4	no	—
369	C₄⋊C₄	—	2	4	no	—
370	C₄⋊C₄	—	2	2	no	—
371	C₄⋊C₄	—	2	4	no	—
372	C₄⋊C₄	—	2	1	no	—
373	C₄⋊C₄	—	2	2	no	—
374	C₄⋊C₄	—	2	1	no	—
375	C₄⋊C₄	—	2	1	no	—
376	C₄⋊C₄	—	2	1	no	—
377	C₄⋊C₄	—	2	1	no	—
378	C₄⋊C₄	—	4	4	no	—
379	C₄⋊C₄	—	4	2	no	—
380	C₄⋊C₄	—	4	2	no	—
381	C₄⋊C₄	—	8	2	no	—
382	C₄⋊C₄	C₂×D₄×C₃₁	1	2	yes	—
383	C₄⋊C₄	#41×C₃₁	1	4	yes	—
384	C₂²⋊C₄	—	2	8	no	—
385	C₂²⋊C₄	—	2	8	no	—
386	C₂²⋊C₄	—	2	4	no	—
387	C₄²	—	2	8	no	—
388	C₄²	—	2	2	no	—
389	C₄²	—	2	2	no	—
390	C₄²	—	8	1	no	—
391	C₄²	C₂×D₄×C₃₁	1	2	yes	—
392	C₄²	C₂×D₄×C₃₁	1	2	yes	—
393	C₂³	—	2	2	no	—
394	C₂³	—	4	2	no	—
395	C₂³	—	4	2	no	—
396	C₂³	—	4	1	no	—
397	C₂³	—	4	1	no	—
398	C₂³	C₂²×D₄×C₃₁	1	1	yes	characteristic
399	Q₈	—	2	4	no	—
400	Q₈	—	2	4	no	—
401	Q₈	—	4	4	no	—
402	Q₈	—	4	2	no	—
403	Q₈	—	4	2	no	—
404	D₄	—	2	2	no	—
405	D₄	—	2	2	no	—
406	D₄	—	4	2	no	—
407	C₂×C₄	—	2	2	no	—
408	C₂×C₄	—	2	1	no	—
409	C₂×C₄	—	2	1	no	—
410	C₂×C₄	—	2	2	no	—
411	C₂×C₄	—	2	2	no	—
412	C₂×C₄	—	2	4	no	—
413	C₂×C₄	—	2	2	no	—
414	C₂×C₄	—	2	4	no	—
415	C₂×C₄	—	2	2	no	—
416	C₂×C₄	—	2	2	no	—
417	C₂×C₄	—	2	2	no	—
418	C₂×C₄	—	2	2	no	—
419	C₂×C₄	—	4	8	no	—
420	C₂×C₄	—	4	2	no	—
421	C₂×C₄	—	4	2	no	—
422	C₂×C₄	—	4	2	no	—
423	C₂×C₄	—	4	2	no	—
424	C₂×C₄	—	4	2	no	—
425	C₂×C₄	—	4	1	no	—
426	C₂×C₄	—	4	1	no	—
427	C₂×C₄	—	4	2	no	—
428	C₂×C₄	—	4	1	no	—
429	C₂×C₄	—	4	1	no	—
430	C₂×C₄	C₃₁×#120	1	2	yes	—
431	C₂×C₄	C₃₁×#120	1	2	yes	—
432	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
433	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
434	C₂×C₄	C₂×#41×C₃₁	1	2	yes	—
435	C₂×C₄	C₃₁×2₋¹⁺⁴	1	2	yes	—
436	C₈	—	4	1	no	—
437	C₈	—	4	1	no	—
438	C₈	—	4	2	no	—
439	C₂²	—	4	2	no	—
440	C₂²	—	4	2	no	—
441	C₂²	—	4	2	no	—
442	C₂²	—	8	2	no	—
443	C₂²	—	8	2	no	—
444	C₂²	C₂×C₃₁×#120	1	1	yes	characteristic
445	C₂²	C₃₁×#548	1	2	yes	—
446	C₂²	C₃₁×#575	1	2	yes	—
447	C₂²	C₃₁×#577	1	1	yes	characteristic
448	C₂²	C₃₁×#578	1	1	yes	characteristic
449	C₄	—	2	2	no	—
450	C₄	—	2	2	no	—
451	C₄	—	4	4	no	—
452	C₄	—	4	4	no	—
453	C₄	—	4	2	no	—
454	C₄	—	4	2	no	—
455	C₄	—	4	2	no	—
456	C₄	—	8	1	no	—
457	C₄	—	8	1	no	—
458	C₄	C₃₁×#532	1	1	yes	characteristic
459	C₄	C₃₁×#536	1	1	yes	characteristic
460	C₄	C₃₁×#547	1	2	yes	—
461	C₂	—	8	2	no	—
462	C₂	C₃₁×#2432	1	1	yes	characteristic
463	C₂	C₃₁×#3005	1	1	yes	characteristic
464	C₂	C₃₁×#3013	1	1	yes	characteristic
465	C₂	C₃₁×#3376	1	2	yes	—
466	C₂	C₃₁×#3377	1	2	yes	—

	subgroup	G/H
center, socle	C₂³×C₃₁	C₂²×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁴×C₃₁
prime radical	C₂²×D₄×C₃₁	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₃₁ ◁ C₂×C₄×Q₈×C₃₁ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂×C₃₁×#491	C₂	semidirect
4. C₃₁×#2584	C₂	semidirect
7. C₃₁×#1985	C₂	semidirect
8. C₃₁×#1981	C₂	semidirect
9. C₃₁×#1865	C₂	semidirect
10. C₃₁×#1863	C₂	semidirect
159. #25656	C₃₁	direct (N × H)
207. C₂×#491	C₂×C₃₁	semidirect
209. #2584	C₂×C₃₁	semidirect
212. #1985	C₂×C₃₁	semidirect
213. #1981	C₂×C₃₁	semidirect
214. #1865	C₂×C₃₁	semidirect
215. #1863	C₂×C₃₁	semidirect
334. C₃₁	#25656	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×#556	2
C₂×C₃₁×#491	2
C₂×C₃₁×#478	2
C₃₁×#2584	2
C₃₁×#2042	2
C₃₁×#1985	2
C₃₁×#1981	2
C₃₁×#1865	2
C₃₁×#1863	2
C₃₁×#1762	2
#25656	31

Subgroups — C31×#25656 #21482415

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#25656 #21482415