Subgroups — C₃₁×#61837 #21518596

Order 7936.

1928 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	C₃₁×#3523	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×D₄×Q₈×C₃₁	C₂	1	1	yes	maximal, characteristic, complemented
3	C₃₁×#3386	C₂	1	6	yes	maximal, complemented
4	Q₈×D₈×C₃₁	C₂	1	2	yes	maximal, complemented
5	C₃₁×#3367	C₂	1	6	yes	maximal, complemented
6	C₃₁×#3359	C₂	1	2	yes	maximal, complemented
7	C₃₁×#3239	C₂	1	3	yes	maximal, complemented
8	C₃₁×#3230	C₂	1	3	yes	maximal, complemented
9	C₃₁×#3110	C₂	1	3	yes	maximal, complemented
10	C₃₁×#3105	C₂	1	3	yes	maximal
11	C₃₁×#2984	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂³×Q₈×C₃₁	—	2	1	no	—
13	C₃₁×#557	C₂²	1	1	yes	characteristic, complemented
14	C₃₁×#557	C₂²	1	3	yes	complemented
15	C₃₁×#556	C₂²	1	3	yes	complemented
16	C₃₁×#552	—	2	3	no	—
17	D₄×Q₈×C₃₁	—	2	1	no	—
18	D₄×Q₈×C₃₁	—	2	1	no	—
19	D₄×Q₈×C₃₁	—	2	3	no	—
20	D₄×Q₈×C₃₁	C₂²	1	2	yes	—
21	D₄×Q₈×C₃₁	C₂²	1	6	yes	—
22	D₄×Q₈×C₃₁	C₂²	1	1	yes	characteristic, complemented
23	C₃₁×#541	—	2	3	no	—
24	C₃₁×#533	C₂²	1	3	yes	—
25	C₂×C₃₁×C₄⋊Q₈	C₂²	1	3	yes	—
26	C₃₁×#529	C₂²	1	3	yes	—
27	C₂×C₃₁×C₂²⋊Q₈	—	2	3	no	—
28	C₃₁×#519	—	2	1	no	—
29	C₂×C₄×Q₈×C₃₁	C₂²	1	1	yes	characteristic
30	C₂×C₄×D₄×C₃₁	C₂²	1	3	yes	—
31	C₃₁×#501	C₂²	1	6	yes	complemented
32	C₃₁×#500	C₂²	1	6	yes	complemented
33	C₃₁×#489	C₂²	1	6	yes	complemented
34	C₃₁×#487	C₂²	1	6	yes	complemented
35	C₃₁×#484	C₂²	1	3	yes	—
36	C₃₁×#484	C₂²	1	6	yes	—
37	C₃₁×#481	C₂²	1	3	yes	—
38	C₃₁×#478	C₂²	1	6	yes	—
39	C₃₁×#477	C₂²	1	12	yes	complemented
40	C₃₁×#475	C₂²	1	6	yes	complemented
41	C₃₁×#474	C₂²	1	6	yes	—
42	C₃₁×#474	C₂²	1	6	yes	complemented
43	C₃₁×#463	C₂²	1	6	yes	—
44	C₃₁×#461	C₂²	1	6	yes	—
45	C₃₁×#450	C₂²	1	3	yes	—
46	C₃₁×#447	C₂²	1	1	yes	characteristic
47	C₃₁×#446	C₂²	1	6	yes	complemented
48	C₈×Q₈×C₃₁	C₂²	1	2	yes	complemented
49	C₃₁×#440	C₂²	1	12	yes	—
50	C₄×SD₁₆×C₃₁	C₂²	1	6	yes	—
51	C₄×D₈×C₃₁	C₂²	1	6	yes	—
52	C₃₁×#432	C₂²	1	3	yes	—
53	C₃₁×#431	C₂²	1	3	yes	—
54	C₂²×Q₈×C₃₁	—	2	1	no	—
55	C₂²×Q₈×C₃₁	—	2	6	no	—
56	C₂²×Q₈×C₃₁	—	2	2	no	—
57	C₂²×Q₈×C₃₁	—	2	3	no	—
58	C₂²×Q₈×C₃₁	—	4	8	no	—
59	C₂²×Q₈×C₃₁	D₄	1	1	yes	characteristic
60	C₂²×D₄×C₃₁	C₂³	1	1	yes	characteristic
61	C₂³×C₄×C₃₁	—	2	3	no	—
62	C₂×SD₁₆×C₃₁	C₂³	1	6	yes	—
63	C₂×D₈×C₃₁	C₂³	1	2	yes	—
64	C₃₁×C₄⋊Q₈	—	2	3	no	—
65	C₃₁×C₄⋊Q₈	—	2	3	no	—
66	C₃₁×C₄⋊Q₈	—	2	3	no	—
67	C₃₁×C₄⋊Q₈	C₂³	1	3	yes	—
68	C₃₁×C₄⋊Q₈	C₂³	1	3	yes	—
69	C₃₁×C₄⋊Q₈	C₂³	1	6	yes	—
70	C₃₁×C₄⋊Q₈	C₂³	1	6	yes	—
71	C₃₁×#126	—	2	6	no	—
72	C₃₁×#126	—	2	3	no	—
73	C₃₁×#125	—	2	6	no	—
74	C₃₁×#125	—	2	3	no	—
75	C₃₁×#125	—	2	3	no	—
76	C₃₁×#125	C₂³	1	3	yes	—
77	C₃₁×#124	C₂³	1	3	yes	—
78	C₃₁×#123	—	2	3	no	—
79	C₃₁×C₂²⋊Q₈	—	2	3	no	—
80	C₃₁×C₂²⋊Q₈	—	2	3	no	—
81	C₃₁×C₂²⋊Q₈	—	2	6	no	—
82	C₃₁×C₂²⋊Q₈	—	2	6	no	—
83	C₃₁×C₂²⋊Q₈	—	2	12	no	—
84	C₃₁×C₂²⋊Q₈	—	2	3	no	—
85	C₃₁×C₂²⋊Q₈	C₂³	1	3	yes	—
86	C₃₁×#121	C₂³	1	1	yes	characteristic
87	C₄×Q₈×C₃₁	—	2	1	no	—
88	C₄×Q₈×C₃₁	—	2	3	no	—
89	C₄×Q₈×C₃₁	—	2	3	no	—
90	C₄×Q₈×C₃₁	D₄	1	1	yes	characteristic
91	C₄×Q₈×C₃₁	D₄	1	3	yes	—
92	C₄×Q₈×C₃₁	C₂³	1	3	yes	—
93	C₄×Q₈×C₃₁	C₂³	1	6	yes	—
94	C₄×Q₈×C₃₁	C₂³	1	1	yes	characteristic
95	C₄×Q₈×C₃₁	C₂³	1	3	yes	—
96	C₄×D₄×C₃₁	—	2	3	no	—
97	C₄×D₄×C₃₁	—	2	3	no	—
98	C₄×D₄×C₃₁	—	2	3	no	—
99	C₄×D₄×C₃₁	C₂³	1	3	yes	—
100	C₄×D₄×C₃₁	C₂³	1	6	yes	—
101	C₄×D₄×C₃₁	C₂³	1	6	yes	—
102	C₃₁×#117	—	2	3	no	—
103	C₃₁×#117	—	2	3	no	—
104	C₃₁×#117	—	2	3	no	—
105	C₂×C₄⋊C₄×C₃₁	—	2	3	no	—
106	C₂×C₄⋊C₄×C₃₁	D₄	1	3	yes	—
107	C₂×C₄⋊C₄×C₃₁	C₂³	1	3	yes	—
108	C₂×C₂²⋊C₄×C₃₁	—	2	3	no	—
109	C₂×C₄²×C₃₁	C₂³	1	3	yes	—
110	C₃₁×#107	C₂³	1	6	yes	—
111	C₃₁×#107	C₂³	1	12	yes	—
112	C₃₁×#106	C₂³	1	6	yes	—
113	C₃₁×C₄⋊C₈	C₂³	1	6	yes	—
114	C₃₁×C₄⋊C₈	C₂³	1	6	yes	—
115	C₃₁×#103	C₂³	1	6	yes	—
116	C₃₁×#103	C₂³	1	12	yes	—
117	C₃₁×#102	C₂³	1	6	yes	—
118	C₃₁×#102	C₂³	1	6	yes	—
119	C₃₁×#102	C₂³	1	2	yes	—
120	C₃₁×C₂²⋊C₈	C₂³	1	1	yes	characteristic
121	C₃₁×C₂²⋊C₈	C₂³	1	3	yes	—
122	C₃₁×#97	C₂³	1	6	yes	—
123	C₄×C₈×C₃₁	C₂³	1	6	yes	—
124	C₂⁴×C₃₁	—	2	1	no	—
125	C₂×Q₈×C₃₁	—	2	4	no	—
126	C₂×Q₈×C₃₁	—	2	3	no	—
127	C₂×Q₈×C₃₁	—	2	6	no	—
128	C₂×Q₈×C₃₁	—	2	6	no	—
129	C₂×Q₈×C₃₁	—	4	12	no	—
130	C₂×Q₈×C₃₁	—	4	24	no	—
131	C₂×Q₈×C₃₁	—	4	4	no	—
132	C₂×Q₈×C₃₁	—	4	8	no	—
133	C₂×Q₈×C₃₁	C₂×D₄	1	3	yes	—
134	C₂×Q₈×C₃₁	C₂×D₄	1	1	yes	characteristic
135	C₂×Q₈×C₃₁	C₂⁴	1	3	yes	—
136	C₂×D₄×C₃₁	—	2	4	no	—
137	C₂×D₄×C₃₁	—	2	1	no	—
138	C₂×D₄×C₃₁	—	2	1	no	—
139	C₂×D₄×C₃₁	C₂⁴	1	1	yes	characteristic
140	C₂×D₄×C₃₁	C₂⁴	1	2	yes	—
141	C₂²×C₄×C₃₁	—	2	3	no	—
142	C₂²×C₄×C₃₁	—	2	3	no	—
143	C₂²×C₄×C₃₁	—	2	6	no	—
144	C₂²×C₄×C₃₁	—	2	6	no	—
145	C₂²×C₄×C₃₁	—	4	12	no	—
146	C₂²×C₄×C₃₁	C₂×D₄	1	3	yes	—
147	C₂²×C₄×C₃₁	C₂×D₄	1	3	yes	—
148	C₂²×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
149	SD₁₆×C₃₁	—	2	12	no	—
150	D₈×C₃₁	—	2	4	no	—
151	C₂×C₈×C₃₁	C₂⁴	1	2	yes	—
152	C₂×C₈×C₃₁	C₂⁴	1	6	yes	—
153	C₄⋊C₄×C₃₁	—	2	6	no	—
154	C₄⋊C₄×C₃₁	—	2	6	no	—
155	C₄⋊C₄×C₃₁	—	2	3	no	—
156	C₄⋊C₄×C₃₁	—	2	3	no	—
157	C₄⋊C₄×C₃₁	—	2	3	no	—
158	C₄⋊C₄×C₃₁	—	2	3	no	—
159	C₄⋊C₄×C₃₁	—	2	3	no	—
160	C₄⋊C₄×C₃₁	—	2	3	no	—
161	C₄⋊C₄×C₃₁	—	2	3	no	—
162	C₄⋊C₄×C₃₁	—	2	6	no	—
163	C₄⋊C₄×C₃₁	—	2	3	no	—
164	C₄⋊C₄×C₃₁	C₂×D₄	1	3	yes	—
165	C₄⋊C₄×C₃₁	C₂×D₄	1	3	yes	—
166	C₄⋊C₄×C₃₁	C₂×D₄	1	6	yes	—
167	C₄⋊C₄×C₃₁	C₂⁴	1	1	yes	characteristic
168	C₄⋊C₄×C₃₁	C₂⁴	1	3	yes	—
169	C₄⋊C₄×C₃₁	C₂⁴	1	6	yes	—
170	C₂²⋊C₄×C₃₁	—	2	3	no	—
171	C₂²⋊C₄×C₃₁	—	2	6	no	—
172	C₂²⋊C₄×C₃₁	—	2	6	no	—
173	C₂²⋊C₄×C₃₁	—	2	1	no	—
174	C₂²⋊C₄×C₃₁	—	2	3	no	—
175	C₂²⋊C₄×C₃₁	—	2	3	no	—
176	C₂²⋊C₄×C₃₁	—	2	3	no	—
177	C₄²×C₃₁	—	2	6	no	—
178	C₄²×C₃₁	—	2	3	no	—
179	C₄²×C₃₁	C₂×D₄	1	3	yes	—
180	C₄²×C₃₁	C₂×D₄	1	3	yes	—
181	C₄²×C₃₁	C₂⁴	1	3	yes	—
182	C₄²×C₃₁	C₂⁴	1	3	yes	—
183	#61837	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
184	C₂³×C₃₁	—	2	4	no	—
185	C₂³×C₃₁	—	2	2	no	—
186	C₂³×C₃₁	—	2	1	no	—
187	C₂³×C₃₁	—	4	2	no	—
188	C₂³×C₃₁	—	4	2	no	—
189	C₂³×C₃₁	C₂²×D₄	1	1	yes	characteristic
190	Q₈×C₃₁	—	2	6	no	—
191	Q₈×C₃₁	—	2	6	no	—
192	Q₈×C₃₁	—	4	12	no	—
193	Q₈×C₃₁	—	4	6	no	—
194	Q₈×C₃₁	—	4	12	no	—
195	Q₈×C₃₁	AGL(1,ℤ/8)	1	4	yes	—
196	D₄×C₃₁	—	2	4	no	—
197	D₄×C₃₁	—	2	2	no	—
198	D₄×C₃₁	—	2	2	no	—
199	D₄×C₃₁	—	4	2	no	—
200	D₄×C₃₁	2₋¹⁺⁴	1	4	yes	—
201	C₂×C₄×C₃₁	—	2	6	no	—
202	C₂×C₄×C₃₁	—	2	6	no	—
203	C₂×C₄×C₃₁	—	2	2	no	—
204	C₂×C₄×C₃₁	—	2	3	no	—
205	C₂×C₄×C₃₁	—	2	1	no	—
206	C₂×C₄×C₃₁	—	2	3	no	—
207	C₂×C₄×C₃₁	—	2	6	no	—
208	C₂×C₄×C₃₁	—	4	12	no	—
209	C₂×C₄×C₃₁	—	4	6	no	—
210	C₂×C₄×C₃₁	—	4	12	no	—
211	C₂×C₄×C₃₁	—	4	6	no	—
212	C₂×C₄×C₃₁	C₂²×D₄	1	3	yes	—
213	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
214	C₂×C₄×C₃₁	C₂²×D₄	1	3	yes	—
215	C₂×C₄×C₃₁	C₂²×D₄	1	3	yes	—
216	C₂×C₄×C₃₁	C₂²×D₄	1	3	yes	—
217	C₂×C₄×C₃₁	C₂⁵	1	1	yes	characteristic
218	C₈×C₃₁	—	2	2	no	—
219	C₈×C₃₁	—	2	6	no	—
220	#3523	C₂×C₃₁	1	1	yes	characteristic, complemented
221	C₂×D₄×Q₈	C₂×C₃₁	1	1	yes	characteristic, complemented
222	#3386	C₂×C₃₁	1	6	yes	complemented
223	Q₈×D₈	C₂×C₃₁	1	2	yes	complemented
224	#3367	C₂×C₃₁	1	6	yes	complemented
225	#3359	C₂×C₃₁	1	2	yes	complemented
226	#3239	C₂×C₃₁	1	3	yes	complemented
227	#3230	C₂×C₃₁	1	3	yes	complemented
228	#3110	C₂×C₃₁	1	3	yes	complemented
229	#3105	C₂×C₃₁	1	3	yes	—
230	#2984	C₂×C₃₁	1	1	yes	characteristic, complemented
231	C₂²×C₃₁	—	2	1	no	—
232	C₂²×C₃₁	—	2	1	no	—
233	C₂²×C₃₁	—	2	4	no	—
234	C₂²×C₃₁	—	4	8	no	—
235	C₂²×C₃₁	—	4	2	no	—
236	C₂²×C₃₁	—	4	2	no	—
237	C₂²×C₃₁	—	4	1	no	—
238	C₂²×C₃₁	—	4	1	no	—
239	C₂²×C₃₁	—	4	1	no	—
240	C₂²×C₃₁	#578	1	2	yes	—
241	C₂²×C₃₁	C₂³×D₄	1	1	yes	characteristic
242	C₄×C₃₁	—	2	1	no	—
243	C₄×C₃₁	—	2	3	no	—
244	C₄×C₃₁	—	2	3	no	—
245	C₄×C₃₁	—	2	3	no	—
246	C₄×C₃₁	—	4	3	no	—
247	C₄×C₃₁	—	4	6	no	—
248	C₄×C₃₁	—	4	3	no	—
249	C₄×C₃₁	—	4	1	no	—
250	C₄×C₃₁	C₂×AGL(1,ℤ/8)	1	6	yes	—
251	C₄×C₃₁	C₂×2₋¹⁺⁴	1	2	yes	—
252	C₂³×Q₈	—	2	1	no	—
253	#557	C₂²×C₃₁	1	1	yes	characteristic, complemented
254	#557	C₂²×C₃₁	1	3	yes	complemented
255	#556	C₂²×C₃₁	1	3	yes	complemented
256	#552	—	2	3	no	—
257	D₄×Q₈	—	2	1	no	—
258	D₄×Q₈	—	2	1	no	—
259	D₄×Q₈	—	2	3	no	—
260	D₄×Q₈	C₂²×C₃₁	1	2	yes	—
261	D₄×Q₈	C₂²×C₃₁	1	6	yes	—
262	D₄×Q₈	C₂²×C₃₁	1	1	yes	characteristic, complemented
263	#541	—	2	3	no	—
264	#533	C₂²×C₃₁	1	3	yes	—
265	C₂×C₄⋊Q₈	C₂²×C₃₁	1	3	yes	—
266	#529	C₂²×C₃₁	1	3	yes	—
267	C₂×C₂²⋊Q₈	—	2	3	no	—
268	#519	—	2	1	no	—
269	C₂×C₄×Q₈	C₂²×C₃₁	1	1	yes	characteristic
270	C₂×C₄×D₄	C₂²×C₃₁	1	3	yes	—
271	#501	C₂²×C₃₁	1	6	yes	complemented
272	#500	C₂²×C₃₁	1	6	yes	complemented
273	#489	C₂²×C₃₁	1	6	yes	complemented
274	#487	C₂²×C₃₁	1	6	yes	complemented
275	#484	C₂²×C₃₁	1	3	yes	—
276	#484	C₂²×C₃₁	1	6	yes	—
277	#481	C₂²×C₃₁	1	3	yes	—
278	#478	C₂²×C₃₁	1	6	yes	—
279	#477	C₂²×C₃₁	1	12	yes	complemented
280	#475	C₂²×C₃₁	1	6	yes	complemented
281	#474	C₂²×C₃₁	1	6	yes	—
282	#474	C₂²×C₃₁	1	6	yes	complemented
283	#463	C₂²×C₃₁	1	6	yes	—
284	#461	C₂²×C₃₁	1	6	yes	—
285	#450	C₂²×C₃₁	1	3	yes	—
286	#447	C₂²×C₃₁	1	1	yes	characteristic
287	#446	C₂²×C₃₁	1	6	yes	complemented
288	C₈×Q₈	C₂²×C₃₁	1	2	yes	complemented
289	#440	C₂²×C₃₁	1	12	yes	—
290	C₄×SD₁₆	C₂²×C₃₁	1	6	yes	—
291	C₄×D₈	C₂²×C₃₁	1	6	yes	—
292	#432	C₂²×C₃₁	1	3	yes	—
293	#431	C₂²×C₃₁	1	3	yes	—
294	C₂×C₃₁	—	2	2	no	—
295	C₂×C₃₁	—	4	4	no	—
296	C₂×C₃₁	—	8	1	no	—
297	C₂×C₃₁	#3486	1	1	yes	characteristic
298	C₂×C₃₁	C₂²×AGL(1,ℤ/8)	1	1	yes	characteristic
299	C₂×C₃₁	C₂×#578	1	1	yes	characteristic
300	C₂²×Q₈	—	2	1	no	—
301	C₂²×Q₈	—	2	6	no	—
302	C₂²×Q₈	—	2	2	no	—
303	C₂²×Q₈	—	2	3	no	—
304	C₂²×Q₈	—	4	8	no	—
305	C₂²×Q₈	D₄×C₃₁	1	1	yes	characteristic
306	C₂²×D₄	C₂³×C₃₁	1	1	yes	characteristic
307	C₂³×C₄	—	2	3	no	—
308	C₂×SD₁₆	C₂³×C₃₁	1	6	yes	—
309	C₂×D₈	C₂³×C₃₁	1	2	yes	—
310	C₄⋊Q₈	—	2	3	no	—
311	C₄⋊Q₈	—	2	3	no	—
312	C₄⋊Q₈	—	2	3	no	—
313	C₄⋊Q₈	C₂³×C₃₁	1	3	yes	—
314	C₄⋊Q₈	C₂³×C₃₁	1	6	yes	—
315	C₄⋊Q₈	C₂³×C₃₁	1	6	yes	—
316	C₄⋊Q₈	C₂³×C₃₁	1	3	yes	—
317	#126	—	2	6	no	—
318	#126	—	2	3	no	—
319	#125	—	2	6	no	—
320	#125	—	2	3	no	—
321	#125	—	2	3	no	—
322	#125	C₂³×C₃₁	1	3	yes	—
323	#124	C₂³×C₃₁	1	3	yes	—
324	#123	—	2	3	no	—
325	C₂²⋊Q₈	—	2	3	no	—
326	C₂²⋊Q₈	—	2	3	no	—
327	C₂²⋊Q₈	—	2	6	no	—
328	C₂²⋊Q₈	—	2	6	no	—
329	C₂²⋊Q₈	—	2	12	no	—
330	C₂²⋊Q₈	—	2	3	no	—
331	C₂²⋊Q₈	C₂³×C₃₁	1	3	yes	—
332	#121	C₂³×C₃₁	1	1	yes	characteristic
333	C₄×Q₈	—	2	1	no	—
334	C₄×Q₈	—	2	3	no	—
335	C₄×Q₈	—	2	3	no	—
336	C₄×Q₈	D₄×C₃₁	1	1	yes	characteristic
337	C₄×Q₈	D₄×C₃₁	1	3	yes	—
338	C₄×Q₈	C₂³×C₃₁	1	3	yes	—
339	C₄×Q₈	C₂³×C₃₁	1	6	yes	—
340	C₄×Q₈	C₂³×C₃₁	1	1	yes	characteristic
341	C₄×Q₈	C₂³×C₃₁	1	3	yes	—
342	C₄×D₄	—	2	3	no	—
343	C₄×D₄	—	2	3	no	—
344	C₄×D₄	—	2	3	no	—
345	C₄×D₄	C₂³×C₃₁	1	6	yes	—
346	C₄×D₄	C₂³×C₃₁	1	6	yes	—
347	C₄×D₄	C₂³×C₃₁	1	3	yes	—
348	#117	—	2	3	no	—
349	#117	—	2	3	no	—
350	#117	—	2	3	no	—
351	C₂×C₄⋊C₄	—	2	3	no	—
352	C₂×C₄⋊C₄	D₄×C₃₁	1	3	yes	—
353	C₂×C₄⋊C₄	C₂³×C₃₁	1	3	yes	—
354	C₂×C₂²⋊C₄	—	2	3	no	—
355	C₂×C₄²	C₂³×C₃₁	1	3	yes	—
356	#107	C₂³×C₃₁	1	6	yes	—
357	#107	C₂³×C₃₁	1	12	yes	—
358	#106	C₂³×C₃₁	1	6	yes	—
359	C₄⋊C₈	C₂³×C₃₁	1	6	yes	—
360	C₄⋊C₈	C₂³×C₃₁	1	6	yes	—
361	#103	C₂³×C₃₁	1	6	yes	—
362	#103	C₂³×C₃₁	1	12	yes	—
363	#102	C₂³×C₃₁	1	6	yes	—
364	#102	C₂³×C₃₁	1	6	yes	—
365	#102	C₂³×C₃₁	1	2	yes	—
366	C₂²⋊C₈	C₂³×C₃₁	1	1	yes	characteristic
367	C₂²⋊C₈	C₂³×C₃₁	1	3	yes	—
368	#97	C₂³×C₃₁	1	6	yes	—
369	C₄×C₈	C₂³×C₃₁	1	6	yes	—
370	C₃₁	#61837	1	1	yes	characteristic, direct product factor, Sylow
371	C₂⁴	—	2	1	no	—
372	C₂×Q₈	—	2	4	no	—
373	C₂×Q₈	—	2	3	no	—
374	C₂×Q₈	—	2	6	no	—
375	C₂×Q₈	—	2	6	no	—
376	C₂×Q₈	—	4	12	no	—
377	C₂×Q₈	—	4	24	no	—
378	C₂×Q₈	—	4	4	no	—
379	C₂×Q₈	—	4	8	no	—
380	C₂×Q₈	C₂×D₄×C₃₁	1	3	yes	—
381	C₂×Q₈	C₂×D₄×C₃₁	1	1	yes	characteristic
382	C₂×Q₈	C₂⁴×C₃₁	1	3	yes	—
383	C₂×D₄	—	2	4	no	—
384	C₂×D₄	—	2	1	no	—
385	C₂×D₄	—	2	1	no	—
386	C₂×D₄	C₂⁴×C₃₁	1	1	yes	characteristic
387	C₂×D₄	C₂⁴×C₃₁	1	2	yes	—
388	C₂²×C₄	—	2	3	no	—
389	C₂²×C₄	—	2	3	no	—
390	C₂²×C₄	—	2	6	no	—
391	C₂²×C₄	—	2	6	no	—
392	C₂²×C₄	—	4	12	no	—
393	C₂²×C₄	C₂×D₄×C₃₁	1	3	yes	—
394	C₂²×C₄	C₂×D₄×C₃₁	1	3	yes	—
395	C₂²×C₄	C₂⁴×C₃₁	1	1	yes	characteristic
396	SD₁₆	—	2	12	no	—
397	D₈	—	2	4	no	—
398	C₂×C₈	C₂⁴×C₃₁	1	2	yes	—
399	C₂×C₈	C₂⁴×C₃₁	1	6	yes	—
400	C₄⋊C₄	—	2	6	no	—
401	C₄⋊C₄	—	2	6	no	—
402	C₄⋊C₄	—	2	3	no	—
403	C₄⋊C₄	—	2	3	no	—
404	C₄⋊C₄	—	2	3	no	—
405	C₄⋊C₄	—	2	3	no	—
406	C₄⋊C₄	—	2	3	no	—
407	C₄⋊C₄	—	2	3	no	—
408	C₄⋊C₄	—	2	3	no	—
409	C₄⋊C₄	—	2	6	no	—
410	C₄⋊C₄	—	2	3	no	—
411	C₄⋊C₄	C₂×D₄×C₃₁	1	3	yes	—
412	C₄⋊C₄	C₂×D₄×C₃₁	1	3	yes	—
413	C₄⋊C₄	C₂×D₄×C₃₁	1	6	yes	—
414	C₄⋊C₄	C₂⁴×C₃₁	1	1	yes	characteristic
415	C₄⋊C₄	C₂⁴×C₃₁	1	3	yes	—
416	C₄⋊C₄	C₂⁴×C₃₁	1	6	yes	—
417	C₂²⋊C₄	—	2	3	no	—
418	C₂²⋊C₄	—	2	6	no	—
419	C₂²⋊C₄	—	2	6	no	—
420	C₂²⋊C₄	—	2	1	no	—
421	C₂²⋊C₄	—	2	3	no	—
422	C₂²⋊C₄	—	2	3	no	—
423	C₂²⋊C₄	—	2	3	no	—
424	C₄²	—	2	6	no	—
425	C₄²	—	2	3	no	—
426	C₄²	C₂×D₄×C₃₁	1	3	yes	—
427	C₄²	C₂×D₄×C₃₁	1	3	yes	—
428	C₄²	C₂⁴×C₃₁	1	3	yes	—
429	C₄²	C₂⁴×C₃₁	1	3	yes	—
430	C₂³	—	2	4	no	—
431	C₂³	—	2	2	no	—
432	C₂³	—	2	1	no	—
433	C₂³	—	4	2	no	—
434	C₂³	—	4	2	no	—
435	C₂³	C₂²×D₄×C₃₁	1	1	yes	characteristic
436	Q₈	—	2	6	no	—
437	Q₈	—	2	6	no	—
438	Q₈	—	4	12	no	—
439	Q₈	—	4	6	no	—
440	Q₈	—	4	12	no	—
441	Q₈	C₃₁×AGL(1,ℤ/8)	1	4	yes	—
442	D₄	—	2	4	no	—
443	D₄	—	2	2	no	—
444	D₄	—	2	2	no	—
445	D₄	—	4	2	no	—
446	D₄	C₃₁×2₋¹⁺⁴	1	4	yes	—
447	C₂×C₄	—	2	6	no	—
448	C₂×C₄	—	2	6	no	—
449	C₂×C₄	—	2	2	no	—
450	C₂×C₄	—	2	3	no	—
451	C₂×C₄	—	2	1	no	—
452	C₂×C₄	—	2	3	no	—
453	C₂×C₄	—	2	6	no	—
454	C₂×C₄	—	4	12	no	—
455	C₂×C₄	—	4	6	no	—
456	C₂×C₄	—	4	12	no	—
457	C₂×C₄	—	4	6	no	—
458	C₂×C₄	C₂²×D₄×C₃₁	1	3	yes	—
459	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
460	C₂×C₄	C₂²×D₄×C₃₁	1	3	yes	—
461	C₂×C₄	C₂²×D₄×C₃₁	1	3	yes	—
462	C₂×C₄	C₂²×D₄×C₃₁	1	3	yes	—
463	C₂×C₄	C₂⁵×C₃₁	1	1	yes	characteristic
464	C₈	—	2	2	no	—
465	C₈	—	2	6	no	—
466	C₂²	—	2	1	no	—
467	C₂²	—	2	1	no	—
468	C₂²	—	2	4	no	—
469	C₂²	—	4	8	no	—
470	C₂²	—	4	2	no	—
471	C₂²	—	4	2	no	—
472	C₂²	—	4	1	no	—
473	C₂²	—	4	1	no	—
474	C₂²	—	4	1	no	—
475	C₂²	C₃₁×#578	1	2	yes	—
476	C₂²	C₂³×D₄×C₃₁	1	1	yes	characteristic
477	C₄	—	2	1	no	—
478	C₄	—	2	3	no	—
479	C₄	—	2	3	no	—
480	C₄	—	2	3	no	—
481	C₄	—	4	3	no	—
482	C₄	—	4	6	no	—
483	C₄	—	4	3	no	—
484	C₄	—	4	1	no	—
485	C₄	C₂×C₃₁×AGL(1,ℤ/8)	1	6	yes	—
486	C₄	C₂×C₃₁×2₋¹⁺⁴	1	2	yes	—
487	C₂	—	2	2	no	—
488	C₂	—	4	4	no	—
489	C₂	—	8	1	no	—
490	C₂	C₃₁×#3486	1	1	yes	characteristic
491	C₂	C₂²×C₃₁×AGL(1,ℤ/8)	1	1	yes	characteristic
492	C₂	C₂×C₃₁×#578	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₃₁	C₂³×D₄
commutator, Frattini, cosocle	C₂×C₄	C₂⁵×C₃₁
prime radical	C₃₁×#447	C₂²
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₃₁ ◁ C₂×C₄×Q₈×C₃₁ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₃₁×#3523	C₂	semidirect
2. C₂×D₄×Q₈×C₃₁	C₂	semidirect
3. C₃₁×#3386	C₂	semidirect
4. Q₈×D₈×C₃₁	C₂	semidirect
5. C₃₁×#3367	C₂	semidirect
6. C₃₁×#3359	C₂	semidirect
7. C₃₁×#3239	C₂	semidirect
8. C₃₁×#3230	C₂	semidirect
9. C₃₁×#3110	C₂	semidirect
11. C₃₁×#2984	C₂	semidirect
13. C₃₁×#557	C₂²	semidirect
14. C₃₁×#557	C₂²	semidirect
15. C₃₁×#556	C₂²	semidirect
22. D₄×Q₈×C₃₁	C₂²	semidirect
31. C₃₁×#501	C₂²	semidirect
32. C₃₁×#500	C₂²	semidirect
33. C₃₁×#489	C₂²	semidirect
34. C₃₁×#487	C₂²	semidirect
39. C₃₁×#477	C₂²	semidirect
40. C₃₁×#475	C₂²	semidirect
42. C₃₁×#474	C₂²	semidirect
47. C₃₁×#446	C₂²	semidirect
48. C₈×Q₈×C₃₁	C₂²	semidirect
183. #61837	C₃₁	direct (N × H)
220. #3523	C₂×C₃₁	semidirect
221. C₂×D₄×Q₈	C₂×C₃₁	semidirect
222. #3386	C₂×C₃₁	semidirect
223. Q₈×D₈	C₂×C₃₁	semidirect
224. #3367	C₂×C₃₁	semidirect
225. #3359	C₂×C₃₁	semidirect
226. #3239	C₂×C₃₁	semidirect
227. #3230	C₂×C₃₁	semidirect
228. #3110	C₂×C₃₁	semidirect
230. #2984	C₂×C₃₁	semidirect
253. #557	C₂²×C₃₁	semidirect
254. #557	C₂²×C₃₁	semidirect
255. #556	C₂²×C₃₁	semidirect
262. D₄×Q₈	C₂²×C₃₁	semidirect
271. #501	C₂²×C₃₁	semidirect
272. #500	C₂²×C₃₁	semidirect
273. #489	C₂²×C₃₁	semidirect
274. #487	C₂²×C₃₁	semidirect
279. #477	C₂²×C₃₁	semidirect
280. #475	C₂²×C₃₁	semidirect
282. #474	C₂²×C₃₁	semidirect
287. #446	C₂²×C₃₁	semidirect
288. C₈×Q₈	C₂²×C₃₁	semidirect
370. C₃₁	#61837	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×#3523	2
C₂×D₄×Q₈×C₃₁	2
C₃₁×#3386	2
Q₈×D₈×C₃₁	2
C₃₁×#3367	2
C₃₁×#3359	2
C₃₁×#3239	2
C₃₁×#3230	2
C₃₁×#3110	2
C₃₁×#3105	2
C₃₁×#2984	2
#61837	31

Subgroups — C31×#61837 #21518596

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#61837 #21518596