Subgroups — C₃₁×#25914 #21482673

Order 7936.

1010 conjugacy classes of subgroups.

The subgroup lattice has 494 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃₁×#544	C₂	1	1	yes	maximal, characteristic
2	C₂×C₃₁×#491	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×C₃₁×#487	C₂	1	1	yes	maximal, characteristic, complemented
4	C₃₁×#2621	C₂	1	1	yes	maximal, characteristic
5	C₃₁×#2026	C₂	1	2	yes	maximal, complemented
6	C₃₁×#2024	C₂	1	2	yes	maximal, complemented
7	C₃₁×#1993	C₂	1	1	yes	maximal, characteristic, complemented
8	C₃₁×#1991	C₂	1	1	yes	maximal, characteristic, complemented
9	C₃₁×#1873	C₂	1	2	yes	maximal
10	C₃₁×#1872	C₂	1	2	yes	maximal
11	C₃₁×#1750	C₂	1	1	yes	maximal, characteristic
12	C₃₁×#544	—	2	4	no	—
13	C₃₁×#544	—	2	4	no	—
14	C₂×C₃₁×C₄⋊Q₈	—	2	1	no	—
15	C₂×C₃₁×C₄⋊Q₈	C₂²	1	1	yes	characteristic
16	C₂×C₃₁×C₄⋊Q₈	C₂²	1	1	yes	characteristic
17	C₂×C₃₁×C₄⋊Q₈	C₂²	1	1	yes	characteristic
18	C₂×C₃₁×#125	—	2	1	no	—
19	C₂×C₃₁×#125	C₂²	1	1	yes	characteristic
20	C₂×C₃₁×#125	C₂²	1	1	yes	characteristic
21	C₂×C₃₁×C₂²⋊Q₈	—	2	1	no	—
22	C₂×C₃₁×C₂²⋊Q₈	—	2	1	no	—
23	C₂×C₃₁×C₂²⋊Q₈	C₂²	1	2	yes	—
24	C₂×C₃₁×#117	C₂²	1	1	yes	characteristic
25	C₂×C₃₁×#117	C₂²	1	1	yes	characteristic
26	C₂²×C₄⋊C₄×C₃₁	C₂²	1	1	yes	characteristic
27	C₃₁×#491	—	2	4	no	—
28	C₃₁×#487	—	2	4	no	—
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×#103	C₂²	1	2	yes	—
33	C₂×C₃₁×C₂²⋊C₈	C₂²	1	2	yes	—
34	C₂×C₃₁×#97	C₂²	1	1	yes	characteristic
35	C₂×C₄×C₈×C₃₁	C₂²	1	1	yes	characteristic
36	C₃₁×#398	—	2	1	no	—
37	C₃₁×#398	—	2	1	no	—
38	C₃₁×#397	—	2	1	no	—
39	C₃₁×#397	—	2	1	no	—
40	C₃₁×#391	C₂²	1	2	yes	—
41	C₃₁×#382	C₂²	1	1	yes	characteristic
42	C₃₁×#340	C₂²	1	2	yes	—
43	C₃₁×#340	C₂²	1	2	yes	—
44	C₃₁×#340	C₂²	1	2	yes	—
45	C₃₁×#340	C₂²	1	2	yes	—
46	C₃₁×#336	C₂²	1	2	yes	—
47	C₂²×Q₈×C₃₁	—	2	1	no	—
48	C₂²×Q₈×C₃₁	—	2	1	no	—
49	C₂²×Q₈×C₃₁	C₂³	1	2	yes	—
50	C₂³×C₄×C₃₁	—	2	1	no	—
51	C₂³×C₄×C₃₁	C₂³	1	1	yes	characteristic
52	C₂²×C₈×C₃₁	C₂³	1	2	yes	—
53	C₂²×C₈×C₃₁	C₂³	1	2	yes	—
54	C₃₁×C₄⋊Q₈	—	2	8	no	—
55	C₃₁×C₄⋊Q₈	—	2	2	no	—
56	C₃₁×C₄⋊Q₈	—	2	2	no	—
57	C₃₁×C₄⋊Q₈	—	2	2	no	—
58	C₃₁×C₄⋊Q₈	—	2	2	no	—
59	C₃₁×C₄⋊Q₈	—	2	2	no	—
60	C₃₁×C₄⋊Q₈	D₄	1	4	yes	—
61	C₃₁×#125	—	2	8	no	—
62	C₃₁×#125	—	2	2	no	—
63	C₃₁×#125	—	2	2	no	—
64	C₃₁×#125	—	2	2	no	—
65	C₃₁×#125	D₄	1	4	yes	—
66	C₃₁×C₂²⋊Q₈	—	2	8	no	—
67	C₃₁×C₂²⋊Q₈	—	2	8	no	—
68	C₃₁×C₂²⋊Q₈	—	2	8	no	—
69	C₃₁×#117	—	2	2	no	—
70	C₃₁×#117	—	2	2	no	—
71	C₃₁×#117	—	2	2	no	—
72	C₃₁×#117	—	2	2	no	—
73	C₂×C₄⋊C₄×C₃₁	—	2	1	no	—
74	C₂×C₄⋊C₄×C₃₁	—	2	1	no	—
75	C₂×C₄⋊C₄×C₃₁	—	2	1	no	—
76	C₂×C₄⋊C₄×C₃₁	—	2	1	no	—
77	C₂×C₄⋊C₄×C₃₁	—	2	1	no	—
78	C₂×C₄⋊C₄×C₃₁	—	2	1	no	—
79	C₂×C₄⋊C₄×C₃₁	—	2	1	no	—
80	C₂×C₄⋊C₄×C₃₁	—	2	2	no	—
81	C₂×C₄⋊C₄×C₃₁	—	2	2	no	—
82	C₂×C₄⋊C₄×C₃₁	—	2	1	no	—
83	C₂×C₄⋊C₄×C₃₁	—	2	1	no	—
84	C₂×C₄⋊C₄×C₃₁	—	2	2	no	—
85	C₂×C₄⋊C₄×C₃₁	D₄	1	1	yes	characteristic
86	C₂×C₄⋊C₄×C₃₁	D₄	1	1	yes	characteristic
87	C₂×C₄⋊C₄×C₃₁	C₂³	1	2	yes	—
88	C₂×C₄⋊C₄×C₃₁	C₂³	1	1	yes	characteristic
89	C₂×C₄⋊C₄×C₃₁	C₂³	1	1	yes	characteristic
90	C₂×C₄⋊C₄×C₃₁	C₂³	1	1	yes	characteristic
91	C₂×C₄⋊C₄×C₃₁	C₂³	1	1	yes	characteristic
92	C₂×C₂²⋊C₄×C₃₁	—	2	1	no	—
93	C₂×C₂²⋊C₄×C₃₁	—	2	2	no	—
94	C₂×C₂²⋊C₄×C₃₁	D₄	1	1	yes	characteristic
95	C₂×C₂²⋊C₄×C₃₁	D₄	1	1	yes	characteristic
96	C₂×C₄²×C₃₁	—	2	1	no	—
97	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
98	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
99	C₃₁×#103	—	2	4	no	—
100	C₃₁×#103	—	2	4	no	—
101	C₃₁×#103	—	2	4	no	—
102	C₃₁×#103	—	2	4	no	—
103	C₃₁×C₂²⋊C₈	—	2	4	no	—
104	C₃₁×#97	—	2	2	no	—
105	C₄×C₈×C₃₁	—	2	2	no	—
106	C₃₁×#95	—	2	1	no	—
107	C₃₁×#95	—	2	1	no	—
108	C₃₁×#95	—	2	2	no	—
109	C₃₁×#95	—	2	2	no	—
110	C₂⁴×C₃₁	C₂×D₄	1	1	yes	characteristic
111	C₂×Q₈×C₃₁	—	2	4	no	—
112	C₂×Q₈×C₃₁	—	2	4	no	—
113	C₂×Q₈×C₃₁	—	2	2	no	—
114	C₂×Q₈×C₃₁	—	2	2	no	—
115	C₂×Q₈×C₃₁	—	2	4	no	—
116	C₂×Q₈×C₃₁	—	2	4	no	—
117	C₂×Q₈×C₃₁	—	8	2	no	—
118	C₂×Q₈×C₃₁	—	8	2	no	—
119	C₂²×C₄×C₃₁	—	2	1	no	—
120	C₂²×C₄×C₃₁	—	2	1	no	—
121	C₂²×C₄×C₃₁	—	2	2	no	—
122	C₂²×C₄×C₃₁	—	2	1	no	—
123	C₂²×C₄×C₃₁	—	2	1	no	—
124	C₂²×C₄×C₃₁	—	2	1	no	—
125	C₂²×C₄×C₃₁	—	2	1	no	—
126	C₂²×C₄×C₃₁	—	2	4	no	—
127	C₂²×C₄×C₃₁	—	2	2	no	—
128	C₂²×C₄×C₃₁	—	2	2	no	—
129	C₂²×C₄×C₃₁	—	8	2	no	—
130	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
131	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
132	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
133	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
134	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
135	C₂²×C₄×C₃₁	#41	1	4	yes	—
136	C₂²×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
137	C₂×C₈×C₃₁	—	2	2	no	—
138	C₂×C₈×C₃₁	—	2	2	no	—
139	C₂×C₈×C₃₁	—	2	2	no	—
140	C₂×C₈×C₃₁	—	2	2	no	—
141	C₂×C₈×C₃₁	—	2	2	no	—
142	C₂×C₈×C₃₁	—	2	2	no	—
143	C₄⋊C₄×C₃₁	—	2	4	no	—
144	C₄⋊C₄×C₃₁	—	2	4	no	—
145	C₄⋊C₄×C₃₁	—	2	4	no	—
146	C₄⋊C₄×C₃₁	—	2	4	no	—
147	C₄⋊C₄×C₃₁	—	2	4	no	—
148	C₄⋊C₄×C₃₁	—	2	4	no	—
149	C₄⋊C₄×C₃₁	—	2	4	no	—
150	C₄⋊C₄×C₃₁	—	2	2	no	—
151	C₄⋊C₄×C₃₁	—	2	2	no	—
152	C₄⋊C₄×C₃₁	—	2	1	no	—
153	C₄⋊C₄×C₃₁	—	2	1	no	—
154	C₄⋊C₄×C₃₁	—	2	1	no	—
155	C₄⋊C₄×C₃₁	—	2	1	no	—
156	C₄⋊C₄×C₃₁	—	2	4	no	—
157	C₄⋊C₄×C₃₁	—	4	2	no	—
158	C₄⋊C₄×C₃₁	—	4	2	no	—
159	C₄⋊C₄×C₃₁	—	4	4	no	—
160	C₄⋊C₄×C₃₁	C₂×D₄	1	2	yes	—
161	C₄⋊C₄×C₃₁	C₂×D₄	1	2	yes	—
162	C₄⋊C₄×C₃₁	C₂×D₄	1	2	yes	—
163	C₄⋊C₄×C₃₁	C₂×D₄	1	2	yes	—
164	C₂²⋊C₄×C₃₁	—	2	4	no	—
165	C₂²⋊C₄×C₃₁	—	2	2	no	—
166	C₂²⋊C₄×C₃₁	—	2	2	no	—
167	C₂²⋊C₄×C₃₁	—	4	4	no	—
168	C₄²×C₃₁	—	2	4	no	—
169	C₄²×C₃₁	—	2	1	no	—
170	C₄²×C₃₁	—	2	1	no	—
171	C₄²×C₃₁	C₂×D₄	1	2	yes	—
172	C₄²×C₃₁	C₂×D₄	1	2	yes	—
173	#25914	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
174	C₂³×C₃₁	—	2	1	no	—
175	C₂³×C₃₁	—	2	1	no	—
176	C₂³×C₃₁	—	2	2	no	—
177	C₂³×C₃₁	—	2	2	no	—
178	C₂³×C₃₁	#124	1	2	yes	—
179	C₂³×C₃₁	C₂²×D₄	1	1	yes	characteristic
180	Q₈×C₃₁	—	4	4	no	—
181	Q₈×C₃₁	—	4	4	no	—
182	Q₈×C₃₁	—	8	4	no	—
183	Q₈×C₃₁	—	8	4	no	—
184	C₂×C₄×C₃₁	—	2	1	no	—
185	C₂×C₄×C₃₁	—	2	1	no	—
186	C₂×C₄×C₃₁	—	2	2	no	—
187	C₂×C₄×C₃₁	—	2	2	no	—
188	C₂×C₄×C₃₁	—	2	4	no	—
189	C₂×C₄×C₃₁	—	2	4	no	—
190	C₂×C₄×C₃₁	—	2	1	no	—
191	C₂×C₄×C₃₁	—	2	2	no	—
192	C₂×C₄×C₃₁	—	2	1	no	—
193	C₂×C₄×C₃₁	—	2	2	no	—
194	C₂×C₄×C₃₁	—	2	2	no	—
195	C₂×C₄×C₃₁	—	2	2	no	—
196	C₂×C₄×C₃₁	—	2	2	no	—
197	C₂×C₄×C₃₁	—	2	2	no	—
198	C₂×C₄×C₃₁	—	4	2	no	—
199	C₂×C₄×C₃₁	—	4	2	no	—
200	C₂×C₄×C₃₁	—	4	2	no	—
201	C₂×C₄×C₃₁	—	4	2	no	—
202	C₂×C₄×C₃₁	—	4	2	no	—
203	C₂×C₄×C₃₁	—	4	2	no	—
204	C₂×C₄×C₃₁	—	4	2	no	—
205	C₂×C₄×C₃₁	—	4	2	no	—
206	C₂×C₄×C₃₁	—	4	2	no	—
207	C₂×C₄×C₃₁	—	4	2	no	—
208	C₂×C₄×C₃₁	—	8	4	no	—
209	C₂×C₄×C₃₁	#120	1	2	yes	—
210	C₂×C₄×C₃₁	#120	1	2	yes	—
211	C₂×C₄×C₃₁	#124	1	2	yes	—
212	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
213	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
214	C₂×C₄×C₃₁	C₂×#41	1	2	yes	—
215	C₂×C₄×C₃₁	2₋¹⁺⁴	1	2	yes	—
216	C₈×C₃₁	—	4	2	no	—
217	C₈×C₃₁	—	4	2	no	—
218	C₂×#544	C₂×C₃₁	1	1	yes	characteristic
219	C₂×#491	C₂×C₃₁	1	1	yes	characteristic, complemented
220	C₂×#487	C₂×C₃₁	1	1	yes	characteristic, complemented
221	#2621	C₂×C₃₁	1	1	yes	characteristic
222	#2026	C₂×C₃₁	1	2	yes	complemented
223	#2024	C₂×C₃₁	1	2	yes	complemented
224	#1993	C₂×C₃₁	1	1	yes	characteristic, complemented
225	#1991	C₂×C₃₁	1	1	yes	characteristic, complemented
226	#1873	C₂×C₃₁	1	2	yes	—
227	#1872	C₂×C₃₁	1	2	yes	—
228	#1750	C₂×C₃₁	1	1	yes	characteristic
229	C₂²×C₃₁	—	2	4	no	—
230	C₂²×C₃₁	—	2	2	no	—
231	C₂²×C₃₁	—	4	2	no	—
232	C₂²×C₃₁	—	4	1	no	—
233	C₂²×C₃₁	—	4	1	no	—
234	C₂²×C₃₁	C₂×#120	1	1	yes	characteristic
235	C₂²×C₃₁	C₂×#124	1	1	yes	characteristic
236	C₂²×C₃₁	#536	1	1	yes	characteristic
237	C₂²×C₃₁	#577	1	2	yes	—
238	C₂²×C₃₁	#578	1	2	yes	—
239	C₄×C₃₁	—	2	2	no	—
240	C₄×C₃₁	—	2	2	no	—
241	C₄×C₃₁	—	2	2	no	—
242	C₄×C₃₁	—	4	1	no	—
243	C₄×C₃₁	—	4	2	no	—
244	C₄×C₃₁	—	4	2	no	—
245	C₄×C₃₁	—	4	2	no	—
246	C₄×C₃₁	—	4	1	no	—
247	C₄×C₃₁	—	4	2	no	—
248	C₄×C₃₁	—	8	2	no	—
249	C₄×C₃₁	—	8	2	no	—
250	C₄×C₃₁	#548	1	2	yes	—
251	C₄×C₃₁	#548	1	2	yes	—
252	#544	—	2	4	no	—
253	#544	—	2	4	no	—
254	C₂×C₄⋊Q₈	—	2	1	no	—
255	C₂×C₄⋊Q₈	C₂²×C₃₁	1	1	yes	characteristic
256	C₂×C₄⋊Q₈	C₂²×C₃₁	1	1	yes	characteristic
257	C₂×C₄⋊Q₈	C₂²×C₃₁	1	1	yes	characteristic
258	C₂×#125	—	2	1	no	—
259	C₂×#125	C₂²×C₃₁	1	1	yes	characteristic
260	C₂×#125	C₂²×C₃₁	1	1	yes	characteristic
261	C₂×C₂²⋊Q₈	—	2	1	no	—
262	C₂×C₂²⋊Q₈	—	2	1	no	—
263	C₂×C₂²⋊Q₈	C₂²×C₃₁	1	2	yes	—
264	C₂×#117	C₂²×C₃₁	1	1	yes	characteristic
265	C₂×#117	C₂²×C₃₁	1	1	yes	characteristic
266	C₂²×C₄⋊C₄	C₂²×C₃₁	1	1	yes	characteristic
267	#491	—	2	4	no	—
268	#487	—	2	4	no	—
269	C₂×#103	C₂²×C₃₁	1	2	yes	—
270	C₂×#103	C₂²×C₃₁	1	2	yes	—
271	C₂×#103	C₂²×C₃₁	1	2	yes	—
272	C₂×#103	C₂²×C₃₁	1	2	yes	—
273	C₂×C₂²⋊C₈	C₂²×C₃₁	1	2	yes	—
274	C₂×#97	C₂²×C₃₁	1	1	yes	characteristic
275	C₂×C₄×C₈	C₂²×C₃₁	1	1	yes	characteristic
276	#398	—	2	1	no	—
277	#398	—	2	1	no	—
278	#397	—	2	1	no	—
279	#397	—	2	1	no	—
280	#391	C₂²×C₃₁	1	2	yes	—
281	#382	C₂²×C₃₁	1	1	yes	characteristic
282	#340	C₂²×C₃₁	1	2	yes	—
283	#340	C₂²×C₃₁	1	2	yes	—
284	#340	C₂²×C₃₁	1	2	yes	—
285	#340	C₂²×C₃₁	1	2	yes	—
286	#336	C₂²×C₃₁	1	2	yes	—
287	C₂×C₃₁	—	4	2	no	—
288	C₂×C₃₁	#2436	1	1	yes	characteristic
289	C₂×C₃₁	#3009	1	1	yes	characteristic
290	C₂×C₃₁	#3011	1	1	yes	characteristic
291	C₂×C₃₁	#3137	1	2	yes	—
292	C₂×C₃₁	#3242	1	2	yes	—
293	C₂²×Q₈	—	2	1	no	—
294	C₂²×Q₈	—	2	1	no	—
295	C₂²×Q₈	C₂³×C₃₁	1	2	yes	—
296	C₂³×C₄	—	2	1	no	—
297	C₂³×C₄	C₂³×C₃₁	1	1	yes	characteristic
298	C₂²×C₈	C₂³×C₃₁	1	2	yes	—
299	C₂²×C₈	C₂³×C₃₁	1	2	yes	—
300	C₄⋊Q₈	—	2	8	no	—
301	C₄⋊Q₈	—	2	2	no	—
302	C₄⋊Q₈	—	2	2	no	—
303	C₄⋊Q₈	—	2	2	no	—
304	C₄⋊Q₈	—	2	2	no	—
305	C₄⋊Q₈	—	2	2	no	—
306	C₄⋊Q₈	D₄×C₃₁	1	4	yes	—
307	#125	—	2	8	no	—
308	#125	—	2	2	no	—
309	#125	—	2	2	no	—
310	#125	—	2	2	no	—
311	#125	D₄×C₃₁	1	4	yes	—
312	C₂²⋊Q₈	—	2	8	no	—
313	C₂²⋊Q₈	—	2	8	no	—
314	C₂²⋊Q₈	—	2	8	no	—
315	#117	—	2	2	no	—
316	#117	—	2	2	no	—
317	#117	—	2	2	no	—
318	#117	—	2	2	no	—
319	C₂×C₄⋊C₄	—	2	1	no	—
320	C₂×C₄⋊C₄	—	2	1	no	—
321	C₂×C₄⋊C₄	—	2	1	no	—
322	C₂×C₄⋊C₄	—	2	1	no	—
323	C₂×C₄⋊C₄	—	2	1	no	—
324	C₂×C₄⋊C₄	—	2	1	no	—
325	C₂×C₄⋊C₄	—	2	1	no	—
326	C₂×C₄⋊C₄	—	2	2	no	—
327	C₂×C₄⋊C₄	—	2	2	no	—
328	C₂×C₄⋊C₄	—	2	2	no	—
329	C₂×C₄⋊C₄	—	2	1	no	—
330	C₂×C₄⋊C₄	—	2	1	no	—
331	C₂×C₄⋊C₄	D₄×C₃₁	1	1	yes	characteristic
332	C₂×C₄⋊C₄	D₄×C₃₁	1	1	yes	characteristic
333	C₂×C₄⋊C₄	C₂³×C₃₁	1	2	yes	—
334	C₂×C₄⋊C₄	C₂³×C₃₁	1	1	yes	characteristic
335	C₂×C₄⋊C₄	C₂³×C₃₁	1	1	yes	characteristic
336	C₂×C₄⋊C₄	C₂³×C₃₁	1	1	yes	characteristic
337	C₂×C₄⋊C₄	C₂³×C₃₁	1	1	yes	characteristic
338	C₂×C₂²⋊C₄	—	2	1	no	—
339	C₂×C₂²⋊C₄	—	2	2	no	—
340	C₂×C₂²⋊C₄	D₄×C₃₁	1	1	yes	characteristic
341	C₂×C₂²⋊C₄	D₄×C₃₁	1	1	yes	characteristic
342	C₂×C₄²	—	2	1	no	—
343	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
344	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
345	#103	—	2	4	no	—
346	#103	—	2	4	no	—
347	#103	—	2	4	no	—
348	#103	—	2	4	no	—
349	C₂²⋊C₈	—	2	4	no	—
350	#97	—	2	2	no	—
351	C₄×C₈	—	2	2	no	—
352	#95	—	2	1	no	—
353	#95	—	2	1	no	—
354	#95	—	2	2	no	—
355	#95	—	2	2	no	—
356	C₃₁	#25914	1	1	yes	characteristic, direct product factor, Sylow
357	C₂⁴	C₂×D₄×C₃₁	1	1	yes	characteristic
358	C₂×Q₈	—	2	4	no	—
359	C₂×Q₈	—	2	4	no	—
360	C₂×Q₈	—	2	2	no	—
361	C₂×Q₈	—	2	2	no	—
362	C₂×Q₈	—	2	4	no	—
363	C₂×Q₈	—	2	4	no	—
364	C₂×Q₈	—	8	2	no	—
365	C₂×Q₈	—	8	2	no	—
366	C₂²×C₄	—	2	1	no	—
367	C₂²×C₄	—	2	1	no	—
368	C₂²×C₄	—	2	2	no	—
369	C₂²×C₄	—	2	1	no	—
370	C₂²×C₄	—	2	1	no	—
371	C₂²×C₄	—	2	1	no	—
372	C₂²×C₄	—	2	1	no	—
373	C₂²×C₄	—	2	4	no	—
374	C₂²×C₄	—	2	2	no	—
375	C₂²×C₄	—	2	2	no	—
376	C₂²×C₄	—	8	2	no	—
377	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
378	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
379	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
380	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
381	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
382	C₂²×C₄	#41×C₃₁	1	4	yes	—
383	C₂²×C₄	C₂⁴×C₃₁	1	1	yes	characteristic
384	C₂×C₈	—	2	2	no	—
385	C₂×C₈	—	2	2	no	—
386	C₂×C₈	—	2	2	no	—
387	C₂×C₈	—	2	2	no	—
388	C₂×C₈	—	2	2	no	—
389	C₂×C₈	—	2	2	no	—
390	C₄⋊C₄	—	2	4	no	—
391	C₄⋊C₄	—	2	4	no	—
392	C₄⋊C₄	—	2	4	no	—
393	C₄⋊C₄	—	2	4	no	—
394	C₄⋊C₄	—	2	4	no	—
395	C₄⋊C₄	—	2	4	no	—
396	C₄⋊C₄	—	2	4	no	—
397	C₄⋊C₄	—	2	2	no	—
398	C₄⋊C₄	—	2	2	no	—
399	C₄⋊C₄	—	2	1	no	—
400	C₄⋊C₄	—	2	1	no	—
401	C₄⋊C₄	—	2	1	no	—
402	C₄⋊C₄	—	2	1	no	—
403	C₄⋊C₄	—	2	4	no	—
404	C₄⋊C₄	—	4	2	no	—
405	C₄⋊C₄	—	4	2	no	—
406	C₄⋊C₄	—	4	4	no	—
407	C₄⋊C₄	C₂×D₄×C₃₁	1	2	yes	—
408	C₄⋊C₄	C₂×D₄×C₃₁	1	2	yes	—
409	C₄⋊C₄	C₂×D₄×C₃₁	1	2	yes	—
410	C₄⋊C₄	C₂×D₄×C₃₁	1	2	yes	—
411	C₂²⋊C₄	—	2	4	no	—
412	C₂²⋊C₄	—	2	2	no	—
413	C₂²⋊C₄	—	2	2	no	—
414	C₂²⋊C₄	—	4	4	no	—
415	C₄²	—	2	4	no	—
416	C₄²	—	2	1	no	—
417	C₄²	—	2	1	no	—
418	C₄²	C₂×D₄×C₃₁	1	2	yes	—
419	C₄²	C₂×D₄×C₃₁	1	2	yes	—
420	C₂³	—	2	1	no	—
421	C₂³	—	2	1	no	—
422	C₂³	—	2	2	no	—
423	C₂³	—	2	2	no	—
424	C₂³	C₃₁×#124	1	2	yes	—
425	C₂³	C₂²×D₄×C₃₁	1	1	yes	characteristic
426	Q₈	—	4	4	no	—
427	Q₈	—	4	4	no	—
428	Q₈	—	8	4	no	—
429	Q₈	—	8	4	no	—
430	C₂×C₄	—	2	1	no	—
431	C₂×C₄	—	2	1	no	—
432	C₂×C₄	—	2	2	no	—
433	C₂×C₄	—	2	2	no	—
434	C₂×C₄	—	2	4	no	—
435	C₂×C₄	—	2	4	no	—
436	C₂×C₄	—	2	1	no	—
437	C₂×C₄	—	2	2	no	—
438	C₂×C₄	—	2	1	no	—
439	C₂×C₄	—	2	2	no	—
440	C₂×C₄	—	2	2	no	—
441	C₂×C₄	—	2	2	no	—
442	C₂×C₄	—	2	2	no	—
443	C₂×C₄	—	2	2	no	—
444	C₂×C₄	—	4	2	no	—
445	C₂×C₄	—	4	2	no	—
446	C₂×C₄	—	4	2	no	—
447	C₂×C₄	—	4	2	no	—
448	C₂×C₄	—	4	2	no	—
449	C₂×C₄	—	4	2	no	—
450	C₂×C₄	—	4	2	no	—
451	C₂×C₄	—	4	2	no	—
452	C₂×C₄	—	4	2	no	—
453	C₂×C₄	—	4	2	no	—
454	C₂×C₄	—	8	4	no	—
455	C₂×C₄	C₃₁×#120	1	2	yes	—
456	C₂×C₄	C₃₁×#120	1	2	yes	—
457	C₂×C₄	C₃₁×#124	1	2	yes	—
458	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
459	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
460	C₂×C₄	C₂×#41×C₃₁	1	2	yes	—
461	C₂×C₄	C₃₁×2₋¹⁺⁴	1	2	yes	—
462	C₈	—	4	2	no	—
463	C₈	—	4	2	no	—
464	C₂²	—	2	4	no	—
465	C₂²	—	2	2	no	—
466	C₂²	—	4	2	no	—
467	C₂²	—	4	1	no	—
468	C₂²	—	4	1	no	—
469	C₂²	C₂×C₃₁×#120	1	1	yes	characteristic
470	C₂²	C₂×C₃₁×#124	1	1	yes	characteristic
471	C₂²	C₃₁×#536	1	1	yes	characteristic
472	C₂²	C₃₁×#577	1	2	yes	—
473	C₂²	C₃₁×#578	1	2	yes	—
474	C₄	—	2	2	no	—
475	C₄	—	2	2	no	—
476	C₄	—	2	2	no	—
477	C₄	—	4	1	no	—
478	C₄	—	4	2	no	—
479	C₄	—	4	2	no	—
480	C₄	—	4	2	no	—
481	C₄	—	4	1	no	—
482	C₄	—	4	2	no	—
483	C₄	—	8	2	no	—
484	C₄	—	8	2	no	—
485	C₄	C₃₁×#548	1	2	yes	—
486	C₄	C₃₁×#548	1	2	yes	—
487	C₂	—	4	2	no	—
488	C₂	C₃₁×#2436	1	1	yes	characteristic
489	C₂	C₃₁×#3009	1	1	yes	characteristic
490	C₂	C₃₁×#3011	1	1	yes	characteristic
491	C₂	C₃₁×#3137	1	2	yes	—
492	C₂	C₃₁×#3242	1	2	yes	—

	subgroup	G/H
center, socle	C₂³×C₃₁	C₂²×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁴×C₃₁
prime radical	C₂⁴×C₃₁	C₂×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₃₁ ◁ C₂×C₃₁×#117 ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂×C₃₁×#491	C₂	semidirect
3. C₂×C₃₁×#487	C₂	semidirect
5. C₃₁×#2026	C₂	semidirect
6. C₃₁×#2024	C₂	semidirect
7. C₃₁×#1993	C₂	semidirect
8. C₃₁×#1991	C₂	semidirect
173. #25914	C₃₁	direct (N × H)
219. C₂×#491	C₂×C₃₁	semidirect
220. C₂×#487	C₂×C₃₁	semidirect
222. #2026	C₂×C₃₁	semidirect
223. #2024	C₂×C₃₁	semidirect
224. #1993	C₂×C₃₁	semidirect
225. #1991	C₂×C₃₁	semidirect
356. C₃₁	#25914	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×#544	2
C₂×C₃₁×#491	2
C₂×C₃₁×#487	2
C₃₁×#2621	2
C₃₁×#2026	2
C₃₁×#2024	2
C₃₁×#1993	2
C₃₁×#1991	2
C₃₁×#1873	2
C₃₁×#1872	2
C₃₁×#1750	2
#25914	31

Subgroups — C31×#25914 #21482673

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#25914 #21482673