Subgroups — C₃₁×#62151 #21518910

Order 7936.

1986 conjugacy classes of subgroups.

The subgroup lattice has 648 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃₁×#3518	C₂	1	1	yes	maximal, characteristic, complemented
2	D₄×#41×C₃₁	C₂	1	1	yes	maximal, characteristic, complemented
3	Q₈×SD₁₆×C₃₁	C₂	1	1	yes	maximal, characteristic, complemented
4	Q₈×D₈×C₃₁	C₂	1	1	yes	maximal, characteristic, complemented
5	C₃₁×#3362	C₂	1	1	yes	maximal, characteristic, complemented
6	C₃₁×#3358	C₂	1	1	yes	maximal, characteristic, complemented
7	C₃₁×#3354	C₂	1	3	yes	maximal, complemented
8	C₃₁×#3353	C₂	1	3	yes	maximal, complemented
9	C₃₁×#3325	C₂	1	3	yes	maximal, complemented
10	C₃₁×#3324	C₂	1	3	yes	maximal, complemented
11	C₃₁×#3251	C₂	1	3	yes	maximal, complemented
12	C₃₁×#3228	C₂	1	3	yes	maximal, complemented
13	C₃₁×#3105	C₂	1	3	yes	maximal, complemented
14	C₃₁×#3077	C₂	1	3	yes	maximal, complemented
15	C₃₁×#2991	C₂	1	1	yes	maximal, characteristic, complemented
16	C₂²×#41×C₃₁	—	2	1	no	—
17	Q₈²×C₃₁	C₂²	1	1	yes	characteristic, complemented
18	C₃₁×#556	—	2	3	no	—
19	C₃₁×#556	C₂²	1	3	yes	complemented
20	C₃₁×#555	C₂²	1	3	yes	complemented
21	C₃₁×#550	C₂²	1	1	yes	characteristic
22	D₄×Q₈×C₃₁	C₂²	1	1	yes	characteristic
23	D₄×Q₈×C₃₁	C₂²	1	1	yes	characteristic, complemented
24	C₃₁×#548	—	2	1	no	—
25	C₃₁×#548	—	2	1	no	—
26	C₃₁×#547	C₂²	1	3	yes	—
27	C₃₁×#546	—	2	3	no	—
28	D₄²×C₃₁	C₂²	1	3	yes	—
29	C₃₁×#542	—	2	3	no	—
30	C₃₁×#533	C₂²	1	3	yes	complemented
31	C₃₁×#532	C₂²	1	3	yes	—
32	C₃₁×#529	C₂²	1	3	yes	complemented
33	C₃₁×#525	—	2	3	no	—
34	C₃₁×#519	—	2	1	no	—
35	C₄×#41×C₃₁	C₂²	1	1	yes	characteristic
36	C₂×C₄×D₄×C₃₁	C₂²	1	3	yes	—
37	C₃₁×#500	C₂²	1	3	yes	complemented
38	C₃₁×#498	C₂²	1	3	yes	complemented
39	C₃₁×#486	C₂²	1	3	yes	complemented
40	C₃₁×#486	C₂²	1	3	yes	complemented
41	C₃₁×#484	C₂²	1	3	yes	complemented
42	C₃₁×#482	C₂²	1	6	yes	complemented
43	C₃₁×#482	C₂²	1	6	yes	complemented
44	C₃₁×#481	C₂²	1	3	yes	complemented
45	C₃₁×#478	C₂²	1	3	yes	—
46	C₃₁×#478	C₂²	1	6	yes	complemented
47	C₃₁×#478	C₂²	1	3	yes	complemented
48	C₃₁×#476	C₂²	1	3	yes	—
49	C₃₁×#476	C₂²	1	3	yes	complemented
50	C₃₁×#475	C₂²	1	3	yes	complemented
51	C₃₁×#474	C₂²	1	3	yes	—
52	C₃₁×#474	C₂²	1	3	yes	complemented
53	C₃₁×#474	C₂²	1	3	yes	complemented
54	C₃₁×#469	C₂²	1	6	yes	complemented
55	C₃₁×#468	C₂²	1	6	yes	complemented
56	C₃₁×#463	C₂²	1	1	yes	characteristic
57	C₃₁×#463	C₂²	1	3	yes	complemented
58	C₃₁×#460	C₂²	1	3	yes	complemented
59	C₃₁×#459	C₂²	1	3	yes	complemented
60	C₃₁×#450	C₂²	1	3	yes	—
61	C₃₁×#447	C₂²	1	3	yes	—
62	C₈×Q₈×C₃₁	C₂²	1	1	yes	characteristic, complemented
63	C₈×Q₈×C₃₁	C₂²	1	1	yes	characteristic, complemented
64	C₄×SD₁₆×C₃₁	C₂²	1	3	yes	—
65	C₄×SD₁₆×C₃₁	C₂²	1	3	yes	complemented
66	C₄×D₈×C₃₁	C₂²	1	3	yes	—
67	C₄×D₈×C₃₁	C₂²	1	3	yes	complemented
68	C₃₁×#436	C₂²	1	3	yes	complemented
69	C₃₁×#436	C₂²	1	3	yes	complemented
70	C₃₁×#431	C₂²	1	3	yes	complemented
71	C₃₁×#428	C₂²	1	6	yes	complemented
72	C₃₁×#423	C₂²	1	3	yes	complemented
73	C₃₁×#418	C₂²	1	6	yes	—
74	C₂×#41×C₃₁	—	2	1	no	—
75	C₂×#41×C₃₁	—	2	3	no	—
76	C₂×#41×C₃₁	—	2	3	no	—
77	C₂×#41×C₃₁	—	4	8	no	—
78	C₂×#41×C₃₁	D₄	1	1	yes	characteristic
79	C₂²×Q₈×C₃₁	—	2	1	no	—
80	C₂²×Q₈×C₃₁	—	2	1	no	—
81	C₂²×D₄×C₃₁	—	2	3	no	—
82	C₂²×D₄×C₃₁	C₂³	1	3	yes	—
83	C₂³×C₄×C₃₁	—	2	3	no	—
84	C₂×SD₁₆×C₃₁	C₂³	1	1	yes	characteristic
85	C₂×SD₁₆×C₃₁	C₂³	1	3	yes	—
86	C₂×D₈×C₃₁	C₂³	1	1	yes	characteristic
87	C₂×D₈×C₃₁	C₂³	1	3	yes	—
88	C₂×M₁₆×C₃₁	C₂³	1	6	yes	—
89	C₃₁×C₄⋊Q₈	—	2	3	no	—
90	C₃₁×C₄⋊Q₈	C₂³	1	3	yes	—
91	C₃₁×C₄⋊Q₈	C₂³	1	3	yes	—
92	C₃₁×C₄⋊Q₈	C₂³	1	3	yes	—
93	C₃₁×#127	C₂³	1	3	yes	—
94	C₃₁×#126	—	2	6	no	—
95	C₃₁×#126	—	2	3	no	—
96	C₃₁×#125	C₂³	1	3	yes	—
97	C₃₁×#124	—	2	3	no	—
98	C₃₁×#124	—	2	6	no	—
99	C₃₁×#124	—	2	3	no	—
100	C₃₁×#124	C₂³	1	1	yes	characteristic
101	C₃₁×#123	—	2	3	no	—
102	C₃₁×#123	—	2	6	no	—
103	C₃₁×C₂²⋊Q₈	—	2	3	no	—
104	C₃₁×C₂²⋊Q₈	—	2	3	no	—
105	C₃₁×C₂²⋊Q₈	—	2	3	no	—
106	C₃₁×C₂²⋊Q₈	—	2	3	no	—
107	C₃₁×C₂²⋊Q₈	—	2	3	no	—
108	C₃₁×C₂²⋊Q₈	C₂³	1	3	yes	—
109	C₃₁×#121	—	2	3	no	—
110	C₃₁×#121	—	2	3	no	—
111	C₃₁×#121	—	2	3	no	—
112	C₃₁×#121	—	2	3	no	—
113	C₃₁×#121	C₂³	1	3	yes	—
114	C₃₁×#121	C₂³	1	6	yes	—
115	C₃₁×#120	—	2	6	no	—
116	C₄×Q₈×C₃₁	—	2	1	no	—
117	C₄×Q₈×C₃₁	—	2	3	no	—
118	C₄×Q₈×C₃₁	—	2	3	no	—
119	C₄×Q₈×C₃₁	—	2	3	no	—
120	C₄×Q₈×C₃₁	D₄	1	1	yes	characteristic
121	C₄×Q₈×C₃₁	C₂³	1	3	yes	—
122	C₄×Q₈×C₃₁	C₂³	1	1	yes	characteristic
123	C₄×D₄×C₃₁	—	2	3	no	—
124	C₄×D₄×C₃₁	—	2	3	no	—
125	C₄×D₄×C₃₁	—	2	3	no	—
126	C₄×D₄×C₃₁	—	2	3	no	—
127	C₄×D₄×C₃₁	D₄	1	3	yes	—
128	C₄×D₄×C₃₁	C₂³	1	3	yes	—
129	C₄×D₄×C₃₁	C₂³	1	3	yes	—
130	C₄×D₄×C₃₁	C₂³	1	3	yes	—
131	C₄×D₄×C₃₁	C₂³	1	1	yes	characteristic
132	C₄×D₄×C₃₁	C₂³	1	3	yes	—
133	C₄×D₄×C₃₁	C₂³	1	3	yes	—
134	C₃₁×#117	—	2	3	no	—
135	C₃₁×#117	—	2	3	no	—
136	C₃₁×#117	D₄	1	3	yes	—
137	C₃₁×#117	C₂³	1	6	yes	—
138	C₂×C₄⋊C₄×C₃₁	C₂³	1	3	yes	—
139	C₂×C₂²⋊C₄×C₃₁	—	2	3	no	—
140	C₂×C₄²×C₃₁	C₂³	1	3	yes	—
141	C₃₁×#107	C₂³	1	3	yes	—
142	C₃₁×#107	C₂³	1	6	yes	—
143	C₃₁×#107	C₂³	1	3	yes	—
144	C₃₁×#106	C₂³	1	3	yes	—
145	C₃₁×#106	C₂³	1	6	yes	—
146	C₃₁×#106	C₂³	1	3	yes	—
147	C₃₁×C₄⋊C₈	C₂³	1	3	yes	—
148	C₃₁×C₄⋊C₈	C₂³	1	3	yes	—
149	C₃₁×C₄⋊C₈	C₂³	1	3	yes	—
150	C₃₁×C₄⋊C₈	C₂³	1	1	yes	characteristic
151	C₃₁×#103	C₂³	1	3	yes	—
152	C₃₁×#103	C₂³	1	1	yes	characteristic
153	C₃₁×#102	C₂³	1	3	yes	—
154	C₃₁×#102	C₂³	1	3	yes	—
155	C₃₁×#102	C₂³	1	6	yes	—
156	C₃₁×#102	C₂³	1	6	yes	—
157	C₃₁×#102	C₂³	1	3	yes	—
158	C₃₁×#102	C₂³	1	1	yes	characteristic
159	C₃₁×#102	C₂³	1	3	yes	—
160	C₃₁×#102	C₂³	1	3	yes	—
161	C₃₁×C₂²⋊C₈	C₂³	1	3	yes	—
162	C₃₁×C₂²⋊C₈	C₂³	1	3	yes	—
163	C₄×C₈×C₃₁	C₂³	1	3	yes	—
164	C₄×C₈×C₃₁	C₂³	1	3	yes	—
165	C₂⁴×C₃₁	—	2	3	no	—
166	#41×C₃₁	—	2	4	no	—
167	#41×C₃₁	—	4	12	no	—
168	#41×C₃₁	—	4	12	no	—
169	#41×C₃₁	—	4	4	no	—
170	C₂×Q₈×C₃₁	—	2	3	no	—
171	C₂×Q₈×C₃₁	—	2	3	no	—
172	C₂×Q₈×C₃₁	—	2	3	no	—
173	C₂×Q₈×C₃₁	—	4	4	no	—
174	C₂×Q₈×C₃₁	—	4	4	no	—
175	C₂×Q₈×C₃₁	C₂×D₄	1	1	yes	characteristic
176	C₂×Q₈×C₃₁	C₂⁴	1	1	yes	characteristic
177	C₂×D₄×C₃₁	—	2	6	no	—
178	C₂×D₄×C₃₁	—	2	6	no	—
179	C₂×D₄×C₃₁	—	2	3	no	—
180	C₂×D₄×C₃₁	—	2	3	no	—
181	C₂×D₄×C₃₁	—	2	3	no	—
182	C₂×D₄×C₃₁	—	2	3	no	—
183	C₂×D₄×C₃₁	—	4	3	no	—
184	C₂×D₄×C₃₁	—	4	12	no	—
185	C₂×D₄×C₃₁	C₂×D₄	1	3	yes	—
186	C₂×D₄×C₃₁	C₂⁴	1	3	yes	—
187	C₂×D₄×C₃₁	C₂⁴	1	1	yes	characteristic
188	C₂×D₄×C₃₁	C₂⁴	1	1	yes	characteristic
189	C₂²×C₄×C₃₁	—	2	3	no	—
190	C₂²×C₄×C₃₁	—	2	3	no	—
191	C₂²×C₄×C₃₁	—	2	1	no	—
192	C₂²×C₄×C₃₁	—	2	3	no	—
193	C₂²×C₄×C₃₁	—	2	3	no	—
194	C₂²×C₄×C₃₁	—	2	3	no	—
195	C₂²×C₄×C₃₁	—	4	12	no	—
196	C₂²×C₄×C₃₁	C₂×D₄	1	3	yes	—
197	C₂²×C₄×C₃₁	C₂×D₄	1	3	yes	—
198	C₂²×C₄×C₃₁	C₂⁴	1	3	yes	—
199	SD₁₆×C₃₁	—	2	2	no	—
200	SD₁₆×C₃₁	—	4	3	no	—
201	D₈×C₃₁	—	2	2	no	—
202	D₈×C₃₁	—	4	3	no	—
203	M₁₆×C₃₁	—	4	6	no	—
204	C₂×C₈×C₃₁	C₂⁴	1	1	yes	characteristic
205	C₂×C₈×C₃₁	C₂⁴	1	1	yes	characteristic
206	C₂×C₈×C₃₁	C₂⁴	1	3	yes	—
207	C₂×C₈×C₃₁	C₂⁴	1	3	yes	—
208	C₄⋊C₄×C₃₁	—	2	3	no	—
209	C₄⋊C₄×C₃₁	—	2	3	no	—
210	C₄⋊C₄×C₃₁	—	2	3	no	—
211	C₄⋊C₄×C₃₁	—	2	3	no	—
212	C₄⋊C₄×C₃₁	—	2	3	no	—
213	C₄⋊C₄×C₃₁	—	2	3	no	—
214	C₄⋊C₄×C₃₁	—	2	3	no	—
215	C₄⋊C₄×C₃₁	—	2	3	no	—
216	C₄⋊C₄×C₃₁	C₂×D₄	1	3	yes	—
217	C₄⋊C₄×C₃₁	C₂×D₄	1	3	yes	—
218	C₄⋊C₄×C₃₁	C₂⁴	1	3	yes	—
219	C₄⋊C₄×C₃₁	C₂⁴	1	3	yes	—
220	C₄⋊C₄×C₃₁	C₂⁴	1	1	yes	characteristic
221	C₄⋊C₄×C₃₁	C₂⁴	1	3	yes	—
222	C₂²⋊C₄×C₃₁	—	2	6	no	—
223	C₂²⋊C₄×C₃₁	—	2	3	no	—
224	C₂²⋊C₄×C₃₁	—	2	1	no	—
225	C₂²⋊C₄×C₃₁	—	2	3	no	—
226	C₂²⋊C₄×C₃₁	—	2	3	no	—
227	C₂²⋊C₄×C₃₁	—	2	3	no	—
228	C₂²⋊C₄×C₃₁	—	2	3	no	—
229	C₂²⋊C₄×C₃₁	—	2	1	no	—
230	C₂²⋊C₄×C₃₁	—	2	1	no	—
231	C₂²⋊C₄×C₃₁	—	2	6	no	—
232	C₂²⋊C₄×C₃₁	—	2	3	no	—
233	C₂²⋊C₄×C₃₁	C₂×D₄	1	6	yes	—
234	C₄²×C₃₁	—	2	6	no	—
235	C₄²×C₃₁	—	2	3	no	—
236	C₄²×C₃₁	C₂×D₄	1	3	yes	—
237	C₄²×C₃₁	C₂×D₄	1	3	yes	—
238	C₄²×C₃₁	C₂⁴	1	3	yes	—
239	C₄²×C₃₁	C₂⁴	1	1	yes	characteristic
240	#62151	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
241	C₂³×C₃₁	—	2	1	no	—
242	C₂³×C₃₁	—	2	1	no	—
243	C₂³×C₃₁	—	2	3	no	—
244	C₂³×C₃₁	—	4	6	no	—
245	C₂³×C₃₁	—	4	6	no	—
246	C₂³×C₃₁	—	8	3	no	—
247	C₂³×C₃₁	C₂²×D₄	1	3	yes	—
248	Q₈×C₃₁	—	2	1	no	—
249	Q₈×C₃₁	—	2	1	no	—
250	Q₈×C₃₁	—	4	6	no	—
251	Q₈×C₃₁	—	4	6	no	—
252	Q₈×C₃₁	—	8	3	no	—
253	Q₈×C₃₁	AGL(1,ℤ/8)	1	4	yes	—
254	D₄×C₃₁	—	2	6	no	—
255	D₄×C₃₁	—	2	1	no	—
256	D₄×C₃₁	—	2	3	no	—
257	D₄×C₃₁	—	2	1	no	—
258	D₄×C₃₁	—	2	1	no	—
259	D₄×C₃₁	—	2	3	no	—
260	D₄×C₃₁	—	4	6	no	—
261	D₄×C₃₁	—	4	12	no	—
262	D₄×C₃₁	—	4	6	no	—
263	D₄×C₃₁	—	8	3	no	—
264	D₄×C₃₁	—	8	3	no	—
265	D₄×C₃₁	2₋¹⁺⁴	1	2	yes	—
266	C₂×C₄×C₃₁	—	2	6	no	—
267	C₂×C₄×C₃₁	—	2	3	no	—
268	C₂×C₄×C₃₁	—	2	3	no	—
269	C₂×C₄×C₃₁	—	2	1	no	—
270	C₂×C₄×C₃₁	—	2	3	no	—
271	C₂×C₄×C₃₁	—	2	3	no	—
272	C₂×C₄×C₃₁	—	2	1	no	—
273	C₂×C₄×C₃₁	—	2	3	no	—
274	C₂×C₄×C₃₁	—	4	3	no	—
275	C₂×C₄×C₃₁	—	4	2	no	—
276	C₂×C₄×C₃₁	—	4	6	no	—
277	C₂×C₄×C₃₁	—	4	6	no	—
278	C₂×C₄×C₃₁	—	4	6	no	—
279	C₂×C₄×C₃₁	—	4	6	no	—
280	C₂×C₄×C₃₁	—	4	6	no	—
281	C₂×C₄×C₃₁	C₂²×D₄	1	3	yes	—
282	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
283	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
284	C₂×C₄×C₃₁	C₂²×D₄	1	3	yes	—
285	C₂×C₄×C₃₁	C₂²×D₄	1	3	yes	—
286	C₂×C₄×C₃₁	C₂⁵	1	1	yes	characteristic
287	C₈×C₃₁	—	2	1	no	—
288	C₈×C₃₁	—	2	1	no	—
289	C₈×C₃₁	—	2	3	no	—
290	C₈×C₃₁	—	2	3	no	—
291	#3518	C₂×C₃₁	1	1	yes	characteristic, complemented
292	D₄×#41	C₂×C₃₁	1	1	yes	characteristic, complemented
293	Q₈×SD₁₆	C₂×C₃₁	1	1	yes	characteristic, complemented
294	Q₈×D₈	C₂×C₃₁	1	1	yes	characteristic, complemented
295	#3362	C₂×C₃₁	1	1	yes	characteristic, complemented
296	#3358	C₂×C₃₁	1	1	yes	characteristic, complemented
297	#3354	C₂×C₃₁	1	3	yes	complemented
298	#3353	C₂×C₃₁	1	3	yes	complemented
299	#3325	C₂×C₃₁	1	3	yes	complemented
300	#3324	C₂×C₃₁	1	3	yes	complemented
301	#3251	C₂×C₃₁	1	3	yes	complemented
302	#3228	C₂×C₃₁	1	3	yes	complemented
303	#3105	C₂×C₃₁	1	3	yes	complemented
304	#3077	C₂×C₃₁	1	3	yes	complemented
305	#2991	C₂×C₃₁	1	1	yes	characteristic, complemented
306	C₂²×C₃₁	—	2	3	no	—
307	C₂²×C₃₁	—	2	3	no	—
308	C₂²×C₃₁	—	2	2	no	—
309	C₂²×C₃₁	—	2	3	no	—
310	C₂²×C₃₁	—	4	1	no	—
311	C₂²×C₃₁	—	4	1	no	—
312	C₂²×C₃₁	—	4	1	no	—
313	C₂²×C₃₁	—	4	3	no	—
314	C₂²×C₃₁	—	4	3	no	—
315	C₂²×C₃₁	—	4	1	no	—
316	C₂²×C₃₁	—	4	1	no	—
317	C₂²×C₃₁	—	4	3	no	—
318	C₂²×C₃₁	—	8	12	no	—
319	C₂²×C₃₁	C₂³×D₄	1	1	yes	characteristic
320	C₄×C₃₁	—	2	1	no	—
321	C₄×C₃₁	—	2	3	no	—
322	C₄×C₃₁	—	2	3	no	—
323	C₄×C₃₁	—	2	1	no	—
324	C₄×C₃₁	—	4	3	no	—
325	C₄×C₃₁	—	4	1	no	—
326	C₄×C₃₁	—	4	1	no	—
327	C₄×C₃₁	—	4	3	no	—
328	C₄×C₃₁	—	4	3	no	—
329	C₄×C₃₁	C₂×AGL(1,ℤ/8)	1	6	yes	—
330	C₄×C₃₁	C₂×2₋¹⁺⁴	1	1	yes	characteristic
331	C₄×C₃₁	#585	1	1	yes	characteristic
332	C₂²×#41	—	2	1	no	—
333	Q₈²	C₂²×C₃₁	1	1	yes	characteristic, complemented
334	#556	—	2	3	no	—
335	#556	C₂²×C₃₁	1	3	yes	complemented
336	#555	C₂²×C₃₁	1	3	yes	complemented
337	#550	C₂²×C₃₁	1	1	yes	characteristic
338	D₄×Q₈	C₂²×C₃₁	1	1	yes	characteristic
339	D₄×Q₈	C₂²×C₃₁	1	1	yes	characteristic, complemented
340	#548	—	2	1	no	—
341	#548	—	2	1	no	—
342	#547	C₂²×C₃₁	1	3	yes	—
343	#546	—	2	3	no	—
344	D₄²	C₂²×C₃₁	1	3	yes	—
345	#542	—	2	3	no	—
346	#533	C₂²×C₃₁	1	3	yes	complemented
347	#532	C₂²×C₃₁	1	3	yes	—
348	#529	C₂²×C₃₁	1	3	yes	complemented
349	#525	—	2	3	no	—
350	#519	—	2	1	no	—
351	C₄×#41	C₂²×C₃₁	1	1	yes	characteristic
352	C₂×C₄×D₄	C₂²×C₃₁	1	3	yes	—
353	#500	C₂²×C₃₁	1	3	yes	complemented
354	#498	C₂²×C₃₁	1	3	yes	complemented
355	#486	C₂²×C₃₁	1	3	yes	complemented
356	#486	C₂²×C₃₁	1	3	yes	complemented
357	#484	C₂²×C₃₁	1	3	yes	complemented
358	#482	C₂²×C₃₁	1	6	yes	complemented
359	#482	C₂²×C₃₁	1	6	yes	complemented
360	#481	C₂²×C₃₁	1	3	yes	complemented
361	#478	C₂²×C₃₁	1	3	yes	—
362	#478	C₂²×C₃₁	1	6	yes	complemented
363	#478	C₂²×C₃₁	1	3	yes	complemented
364	#476	C₂²×C₃₁	1	3	yes	—
365	#476	C₂²×C₃₁	1	3	yes	complemented
366	#475	C₂²×C₃₁	1	3	yes	complemented
367	#474	C₂²×C₃₁	1	3	yes	—
368	#474	C₂²×C₃₁	1	3	yes	complemented
369	#474	C₂²×C₃₁	1	3	yes	complemented
370	#469	C₂²×C₃₁	1	6	yes	complemented
371	#468	C₂²×C₃₁	1	6	yes	complemented
372	#463	C₂²×C₃₁	1	1	yes	characteristic
373	#463	C₂²×C₃₁	1	3	yes	complemented
374	#460	C₂²×C₃₁	1	3	yes	complemented
375	#459	C₂²×C₃₁	1	3	yes	complemented
376	#450	C₂²×C₃₁	1	3	yes	—
377	#447	C₂²×C₃₁	1	3	yes	—
378	C₈×Q₈	C₂²×C₃₁	1	1	yes	characteristic, complemented
379	C₈×Q₈	C₂²×C₃₁	1	1	yes	characteristic, complemented
380	C₄×SD₁₆	C₂²×C₃₁	1	3	yes	—
381	C₄×SD₁₆	C₂²×C₃₁	1	3	yes	complemented
382	C₄×D₈	C₂²×C₃₁	1	3	yes	—
383	C₄×D₈	C₂²×C₃₁	1	3	yes	complemented
384	#436	C₂²×C₃₁	1	3	yes	complemented
385	#436	C₂²×C₃₁	1	3	yes	complemented
386	#431	C₂²×C₃₁	1	3	yes	complemented
387	#428	C₂²×C₃₁	1	6	yes	complemented
388	#423	C₂²×C₃₁	1	3	yes	complemented
389	#418	C₂²×C₃₁	1	6	yes	—
390	C₂×C₃₁	—	4	3	no	—
391	C₂×C₃₁	—	4	2	no	—
392	C₂×C₃₁	—	8	1	no	—
393	C₂×C₃₁	—	8	3	no	—
394	C₂×C₃₁	#3489	1	1	yes	characteristic
395	C₂×C₃₁	C₂²×AGL(1,ℤ/8)	1	1	yes	characteristic
396	C₂×C₃₁	#3584	1	1	yes	characteristic
397	C₂×#41	—	2	1	no	—
398	C₂×#41	—	2	3	no	—
399	C₂×#41	—	2	3	no	—
400	C₂×#41	—	4	8	no	—
401	C₂×#41	D₄×C₃₁	1	1	yes	characteristic
402	C₂²×Q₈	—	2	1	no	—
403	C₂²×Q₈	—	2	1	no	—
404	C₂²×D₄	—	2	3	no	—
405	C₂²×D₄	C₂³×C₃₁	1	3	yes	—
406	C₂³×C₄	—	2	3	no	—
407	C₂×SD₁₆	C₂³×C₃₁	1	1	yes	characteristic
408	C₂×SD₁₆	C₂³×C₃₁	1	3	yes	—
409	C₂×D₈	C₂³×C₃₁	1	1	yes	characteristic
410	C₂×D₈	C₂³×C₃₁	1	3	yes	—
411	C₂×M₁₆	C₂³×C₃₁	1	6	yes	—
412	C₄⋊Q₈	—	2	3	no	—
413	C₄⋊Q₈	C₂³×C₃₁	1	3	yes	—
414	C₄⋊Q₈	C₂³×C₃₁	1	3	yes	—
415	C₄⋊Q₈	C₂³×C₃₁	1	3	yes	—
416	#127	C₂³×C₃₁	1	3	yes	—
417	#126	—	2	6	no	—
418	#126	—	2	3	no	—
419	#125	C₂³×C₃₁	1	3	yes	—
420	#124	—	2	3	no	—
421	#124	—	2	6	no	—
422	#124	—	2	3	no	—
423	#124	C₂³×C₃₁	1	1	yes	characteristic
424	#123	—	2	3	no	—
425	#123	—	2	6	no	—
426	C₂²⋊Q₈	—	2	3	no	—
427	C₂²⋊Q₈	—	2	3	no	—
428	C₂²⋊Q₈	—	2	3	no	—
429	C₂²⋊Q₈	—	2	3	no	—
430	C₂²⋊Q₈	—	2	3	no	—
431	C₂²⋊Q₈	C₂³×C₃₁	1	3	yes	—
432	#121	—	2	3	no	—
433	#121	—	2	3	no	—
434	#121	—	2	3	no	—
435	#121	—	2	3	no	—
436	#121	C₂³×C₃₁	1	3	yes	—
437	#121	C₂³×C₃₁	1	6	yes	—
438	#120	—	2	6	no	—
439	C₄×Q₈	—	2	1	no	—
440	C₄×Q₈	—	2	3	no	—
441	C₄×Q₈	—	2	3	no	—
442	C₄×Q₈	—	2	3	no	—
443	C₄×Q₈	D₄×C₃₁	1	1	yes	characteristic
444	C₄×Q₈	C₂³×C₃₁	1	3	yes	—
445	C₄×Q₈	C₂³×C₃₁	1	1	yes	characteristic
446	C₄×D₄	—	2	3	no	—
447	C₄×D₄	—	2	3	no	—
448	C₄×D₄	—	2	3	no	—
449	C₄×D₄	—	2	3	no	—
450	C₄×D₄	D₄×C₃₁	1	3	yes	—
451	C₄×D₄	C₂³×C₃₁	1	1	yes	characteristic
452	C₄×D₄	C₂³×C₃₁	1	3	yes	—
453	C₄×D₄	C₂³×C₃₁	1	3	yes	—
454	C₄×D₄	C₂³×C₃₁	1	3	yes	—
455	C₄×D₄	C₂³×C₃₁	1	3	yes	—
456	C₄×D₄	C₂³×C₃₁	1	3	yes	—
457	#117	—	2	3	no	—
458	#117	—	2	3	no	—
459	#117	D₄×C₃₁	1	3	yes	—
460	#117	C₂³×C₃₁	1	6	yes	—
461	C₂×C₄⋊C₄	C₂³×C₃₁	1	3	yes	—
462	C₂×C₂²⋊C₄	—	2	3	no	—
463	C₂×C₄²	C₂³×C₃₁	1	3	yes	—
464	#107	C₂³×C₃₁	1	3	yes	—
465	#107	C₂³×C₃₁	1	6	yes	—
466	#107	C₂³×C₃₁	1	3	yes	—
467	#106	C₂³×C₃₁	1	3	yes	—
468	#106	C₂³×C₃₁	1	6	yes	—
469	#106	C₂³×C₃₁	1	3	yes	—
470	C₄⋊C₈	C₂³×C₃₁	1	3	yes	—
471	C₄⋊C₈	C₂³×C₃₁	1	3	yes	—
472	C₄⋊C₈	C₂³×C₃₁	1	3	yes	—
473	C₄⋊C₈	C₂³×C₃₁	1	1	yes	characteristic
474	#103	C₂³×C₃₁	1	3	yes	—
475	#103	C₂³×C₃₁	1	1	yes	characteristic
476	#102	C₂³×C₃₁	1	3	yes	—
477	#102	C₂³×C₃₁	1	3	yes	—
478	#102	C₂³×C₃₁	1	6	yes	—
479	#102	C₂³×C₃₁	1	6	yes	—
480	#102	C₂³×C₃₁	1	3	yes	—
481	#102	C₂³×C₃₁	1	1	yes	characteristic
482	#102	C₂³×C₃₁	1	3	yes	—
483	#102	C₂³×C₃₁	1	3	yes	—
484	C₂²⋊C₈	C₂³×C₃₁	1	3	yes	—
485	C₂²⋊C₈	C₂³×C₃₁	1	3	yes	—
486	C₄×C₈	C₂³×C₃₁	1	3	yes	—
487	C₄×C₈	C₂³×C₃₁	1	3	yes	—
488	C₃₁	#62151	1	1	yes	characteristic, direct product factor, Sylow
489	C₂⁴	—	2	3	no	—
490	#41	—	2	4	no	—
491	#41	—	4	12	no	—
492	#41	—	4	12	no	—
493	#41	—	4	4	no	—
494	C₂×Q₈	—	2	3	no	—
495	C₂×Q₈	—	2	3	no	—
496	C₂×Q₈	—	2	3	no	—
497	C₂×Q₈	—	4	4	no	—
498	C₂×Q₈	—	4	4	no	—
499	C₂×Q₈	C₂×D₄×C₃₁	1	1	yes	characteristic
500	C₂×Q₈	C₂⁴×C₃₁	1	1	yes	characteristic
501	C₂×D₄	—	2	6	no	—
502	C₂×D₄	—	2	6	no	—
503	C₂×D₄	—	2	3	no	—
504	C₂×D₄	—	2	3	no	—
505	C₂×D₄	—	2	3	no	—
506	C₂×D₄	—	2	3	no	—
507	C₂×D₄	—	4	3	no	—
508	C₂×D₄	—	4	12	no	—
509	C₂×D₄	C₂×D₄×C₃₁	1	3	yes	—
510	C₂×D₄	C₂⁴×C₃₁	1	3	yes	—
511	C₂×D₄	C₂⁴×C₃₁	1	1	yes	characteristic
512	C₂×D₄	C₂⁴×C₃₁	1	1	yes	characteristic
513	C₂²×C₄	—	2	3	no	—
514	C₂²×C₄	—	2	3	no	—
515	C₂²×C₄	—	2	1	no	—
516	C₂²×C₄	—	2	3	no	—
517	C₂²×C₄	—	2	3	no	—
518	C₂²×C₄	—	2	3	no	—
519	C₂²×C₄	—	4	12	no	—
520	C₂²×C₄	C₂×D₄×C₃₁	1	3	yes	—
521	C₂²×C₄	C₂×D₄×C₃₁	1	3	yes	—
522	C₂²×C₄	C₂⁴×C₃₁	1	3	yes	—
523	SD₁₆	—	2	2	no	—
524	SD₁₆	—	4	3	no	—
525	D₈	—	2	2	no	—
526	D₈	—	4	3	no	—
527	M₁₆	—	4	6	no	—
528	C₂×C₈	C₂⁴×C₃₁	1	1	yes	characteristic
529	C₂×C₈	C₂⁴×C₃₁	1	1	yes	characteristic
530	C₂×C₈	C₂⁴×C₃₁	1	3	yes	—
531	C₂×C₈	C₂⁴×C₃₁	1	3	yes	—
532	C₄⋊C₄	—	2	3	no	—
533	C₄⋊C₄	—	2	3	no	—
534	C₄⋊C₄	—	2	3	no	—
535	C₄⋊C₄	—	2	3	no	—
536	C₄⋊C₄	—	2	3	no	—
537	C₄⋊C₄	—	2	3	no	—
538	C₄⋊C₄	—	2	3	no	—
539	C₄⋊C₄	—	2	3	no	—
540	C₄⋊C₄	C₂×D₄×C₃₁	1	3	yes	—
541	C₄⋊C₄	C₂×D₄×C₃₁	1	3	yes	—
542	C₄⋊C₄	C₂⁴×C₃₁	1	3	yes	—
543	C₄⋊C₄	C₂⁴×C₃₁	1	3	yes	—
544	C₄⋊C₄	C₂⁴×C₃₁	1	1	yes	characteristic
545	C₄⋊C₄	C₂⁴×C₃₁	1	3	yes	—
546	C₂²⋊C₄	—	2	6	no	—
547	C₂²⋊C₄	—	2	3	no	—
548	C₂²⋊C₄	—	2	1	no	—
549	C₂²⋊C₄	—	2	3	no	—
550	C₂²⋊C₄	—	2	3	no	—
551	C₂²⋊C₄	—	2	3	no	—
552	C₂²⋊C₄	—	2	3	no	—
553	C₂²⋊C₄	—	2	1	no	—
554	C₂²⋊C₄	—	2	1	no	—
555	C₂²⋊C₄	—	2	6	no	—
556	C₂²⋊C₄	—	2	3	no	—
557	C₂²⋊C₄	C₂×D₄×C₃₁	1	6	yes	—
558	C₄²	—	2	6	no	—
559	C₄²	—	2	3	no	—
560	C₄²	C₂×D₄×C₃₁	1	3	yes	—
561	C₄²	C₂×D₄×C₃₁	1	3	yes	—
562	C₄²	C₂⁴×C₃₁	1	3	yes	—
563	C₄²	C₂⁴×C₃₁	1	1	yes	characteristic
564	C₂³	—	2	1	no	—
565	C₂³	—	2	1	no	—
566	C₂³	—	2	3	no	—
567	C₂³	—	4	6	no	—
568	C₂³	—	4	6	no	—
569	C₂³	—	8	3	no	—
570	C₂³	C₂²×D₄×C₃₁	1	3	yes	—
571	Q₈	—	2	1	no	—
572	Q₈	—	2	1	no	—
573	Q₈	—	4	6	no	—
574	Q₈	—	4	6	no	—
575	Q₈	—	8	3	no	—
576	Q₈	C₃₁×AGL(1,ℤ/8)	1	4	yes	—
577	D₄	—	2	6	no	—
578	D₄	—	2	1	no	—
579	D₄	—	2	3	no	—
580	D₄	—	2	1	no	—
581	D₄	—	2	1	no	—
582	D₄	—	2	3	no	—
583	D₄	—	4	6	no	—
584	D₄	—	4	12	no	—
585	D₄	—	4	6	no	—
586	D₄	—	8	3	no	—
587	D₄	—	8	3	no	—
588	D₄	C₃₁×2₋¹⁺⁴	1	2	yes	—
589	C₂×C₄	—	2	6	no	—
590	C₂×C₄	—	2	3	no	—
591	C₂×C₄	—	2	3	no	—
592	C₂×C₄	—	2	1	no	—
593	C₂×C₄	—	2	3	no	—
594	C₂×C₄	—	2	3	no	—
595	C₂×C₄	—	2	1	no	—
596	C₂×C₄	—	2	3	no	—
597	C₂×C₄	—	4	3	no	—
598	C₂×C₄	—	4	2	no	—
599	C₂×C₄	—	4	6	no	—
600	C₂×C₄	—	4	6	no	—
601	C₂×C₄	—	4	6	no	—
602	C₂×C₄	—	4	6	no	—
603	C₂×C₄	—	4	6	no	—
604	C₂×C₄	C₂²×D₄×C₃₁	1	3	yes	—
605	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
606	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
607	C₂×C₄	C₂²×D₄×C₃₁	1	3	yes	—
608	C₂×C₄	C₂²×D₄×C₃₁	1	3	yes	—
609	C₂×C₄	C₂⁵×C₃₁	1	1	yes	characteristic
610	C₈	—	2	1	no	—
611	C₈	—	2	1	no	—
612	C₈	—	2	3	no	—
613	C₈	—	2	3	no	—
614	C₂²	—	2	3	no	—
615	C₂²	—	2	2	no	—
616	C₂²	—	2	3	no	—
617	C₂²	—	2	3	no	—
618	C₂²	—	4	1	no	—
619	C₂²	—	4	1	no	—
620	C₂²	—	4	3	no	—
621	C₂²	—	4	1	no	—
622	C₂²	—	4	1	no	—
623	C₂²	—	4	3	no	—
624	C₂²	—	4	1	no	—
625	C₂²	—	4	3	no	—
626	C₂²	—	8	12	no	—
627	C₂²	C₂³×D₄×C₃₁	1	1	yes	characteristic
628	C₄	—	2	3	no	—
629	C₄	—	2	1	no	—
630	C₄	—	2	1	no	—
631	C₄	—	2	3	no	—
632	C₄	—	4	1	no	—
633	C₄	—	4	1	no	—
634	C₄	—	4	3	no	—
635	C₄	—	4	3	no	—
636	C₄	—	4	3	no	—
637	C₄	C₂×C₃₁×AGL(1,ℤ/8)	1	6	yes	—
638	C₄	C₂×C₃₁×2₋¹⁺⁴	1	1	yes	characteristic
639	C₄	C₃₁×#585	1	1	yes	characteristic
640	C₂	—	4	3	no	—
641	C₂	—	4	2	no	—
642	C₂	—	8	3	no	—
643	C₂	—	8	1	no	—
644	C₂	C₃₁×#3489	1	1	yes	characteristic
645	C₂	C₂²×C₃₁×AGL(1,ℤ/8)	1	1	yes	characteristic
646	C₂	C₃₁×#3584	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₃₁	C₂³×D₄
commutator, Frattini, cosocle	C₂×C₄	C₂⁵×C₃₁
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₃₁ ◁ C₄×#41×C₃₁ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₃₁×#3518	C₂	semidirect
2. D₄×#41×C₃₁	C₂	semidirect
3. Q₈×SD₁₆×C₃₁	C₂	semidirect
4. Q₈×D₈×C₃₁	C₂	semidirect
5. C₃₁×#3362	C₂	semidirect
6. C₃₁×#3358	C₂	semidirect
7. C₃₁×#3354	C₂	semidirect
8. C₃₁×#3353	C₂	semidirect
9. C₃₁×#3325	C₂	semidirect
10. C₃₁×#3324	C₂	semidirect
11. C₃₁×#3251	C₂	semidirect
12. C₃₁×#3228	C₂	semidirect
13. C₃₁×#3105	C₂	semidirect
14. C₃₁×#3077	C₂	semidirect
15. C₃₁×#2991	C₂	semidirect
17. Q₈²×C₃₁	C₂²	semidirect
19. C₃₁×#556	C₂²	semidirect
20. C₃₁×#555	C₂²	semidirect
23. D₄×Q₈×C₃₁	C₂²	semidirect
30. C₃₁×#533	C₂²	semidirect
32. C₃₁×#529	C₂²	semidirect
37. C₃₁×#500	C₂²	semidirect
38. C₃₁×#498	C₂²	semidirect
39. C₃₁×#486	C₂²	semidirect
40. C₃₁×#486	C₂²	semidirect
41. C₃₁×#484	C₂²	semidirect
42. C₃₁×#482	C₂²	semidirect
43. C₃₁×#482	C₂²	semidirect
44. C₃₁×#481	C₂²	semidirect
46. C₃₁×#478	C₂²	semidirect
47. C₃₁×#478	C₂²	semidirect
49. C₃₁×#476	C₂²	semidirect
50. C₃₁×#475	C₂²	semidirect
52. C₃₁×#474	C₂²	semidirect
53. C₃₁×#474	C₂²	semidirect
54. C₃₁×#469	C₂²	semidirect
55. C₃₁×#468	C₂²	semidirect
57. C₃₁×#463	C₂²	semidirect
58. C₃₁×#460	C₂²	semidirect
59. C₃₁×#459	C₂²	semidirect
62. C₈×Q₈×C₃₁	C₂²	semidirect
63. C₈×Q₈×C₃₁	C₂²	semidirect
65. C₄×SD₁₆×C₃₁	C₂²	semidirect
67. C₄×D₈×C₃₁	C₂²	semidirect
68. C₃₁×#436	C₂²	semidirect
69. C₃₁×#436	C₂²	semidirect
70. C₃₁×#431	C₂²	semidirect
71. C₃₁×#428	C₂²	semidirect
72. C₃₁×#423	C₂²	semidirect
240. #62151	C₃₁	direct (N × H)
291. #3518	C₂×C₃₁	semidirect
292. D₄×#41	C₂×C₃₁	semidirect
293. Q₈×SD₁₆	C₂×C₃₁	semidirect
294. Q₈×D₈	C₂×C₃₁	semidirect
295. #3362	C₂×C₃₁	semidirect
296. #3358	C₂×C₃₁	semidirect
297. #3354	C₂×C₃₁	semidirect
298. #3353	C₂×C₃₁	semidirect
299. #3325	C₂×C₃₁	semidirect
300. #3324	C₂×C₃₁	semidirect
301. #3251	C₂×C₃₁	semidirect
302. #3228	C₂×C₃₁	semidirect
303. #3105	C₂×C₃₁	semidirect
304. #3077	C₂×C₃₁	semidirect
305. #2991	C₂×C₃₁	semidirect
333. Q₈²	C₂²×C₃₁	semidirect
335. #556	C₂²×C₃₁	semidirect
336. #555	C₂²×C₃₁	semidirect
339. D₄×Q₈	C₂²×C₃₁	semidirect
346. #533	C₂²×C₃₁	semidirect
348. #529	C₂²×C₃₁	semidirect
353. #500	C₂²×C₃₁	semidirect
354. #498	C₂²×C₃₁	semidirect
355. #486	C₂²×C₃₁	semidirect
356. #486	C₂²×C₃₁	semidirect
357. #484	C₂²×C₃₁	semidirect
358. #482	C₂²×C₃₁	semidirect
359. #482	C₂²×C₃₁	semidirect
360. #481	C₂²×C₃₁	semidirect
362. #478	C₂²×C₃₁	semidirect
363. #478	C₂²×C₃₁	semidirect
365. #476	C₂²×C₃₁	semidirect
366. #475	C₂²×C₃₁	semidirect
368. #474	C₂²×C₃₁	semidirect
369. #474	C₂²×C₃₁	semidirect
370. #469	C₂²×C₃₁	semidirect
371. #468	C₂²×C₃₁	semidirect
373. #463	C₂²×C₃₁	semidirect
374. #460	C₂²×C₃₁	semidirect
375. #459	C₂²×C₃₁	semidirect
378. C₈×Q₈	C₂²×C₃₁	semidirect
379. C₈×Q₈	C₂²×C₃₁	semidirect
381. C₄×SD₁₆	C₂²×C₃₁	semidirect
383. C₄×D₈	C₂²×C₃₁	semidirect
384. #436	C₂²×C₃₁	semidirect
385. #436	C₂²×C₃₁	semidirect
386. #431	C₂²×C₃₁	semidirect
387. #428	C₂²×C₃₁	semidirect
388. #423	C₂²×C₃₁	semidirect
488. C₃₁	#62151	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×#3518	2
D₄×#41×C₃₁	2
Q₈×SD₁₆×C₃₁	2
Q₈×D₈×C₃₁	2
C₃₁×#3362	2
C₃₁×#3358	2
C₃₁×#3354	2
C₃₁×#3353	2
C₃₁×#3325	2
C₃₁×#3324	2
C₃₁×#3251	2
C₃₁×#3228	2
C₃₁×#3105	2
C₃₁×#3077	2
C₃₁×#2991	2
#62151	31

Subgroups — C31×#62151 #21518910

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#62151 #21518910