Subgroups — C₃₁×#62150 #21518909

Order 7936.

1996 conjugacy classes of subgroups.

The subgroup lattice has 512 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃₁×#3499	C₂	1	1	yes	maximal, characteristic, complemented
2	C₃₁×#3483	C₂	1	1	yes	maximal, characteristic, complemented
3	Q₈×SD₁₆×C₃₁	C₂	1	2	yes	maximal, complemented
4	C₃₁×#3358	C₂	1	2	yes	maximal, complemented
5	C₃₁×#3354	C₂	1	6	yes	maximal, complemented
6	C₃₁×#3332	C₂	1	6	yes	maximal, complemented
7	C₃₁×#3253	C₂	1	3	yes	maximal, complemented
8	C₃₁×#3227	C₂	1	3	yes	maximal, complemented
9	C₃₁×#3106	C₂	1	3	yes	maximal, complemented
10	C₃₁×#3079	C₂	1	3	yes	maximal, complemented
11	C₃₁×#2991	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×C₃₁×2₋¹⁺⁴	—	2	1	no	—
13	Q₈²×C₃₁	C₂²	1	1	yes	characteristic
14	C₃₁×#555	—	2	3	no	—
15	C₃₁×#555	C₂²	1	3	yes	—
16	C₃₁×#550	—	2	1	no	—
17	C₃₁×#550	C₂²	1	3	yes	—
18	C₃₁×#550	C₂²	1	1	yes	characteristic
19	D₄×Q₈×C₃₁	—	2	1	no	—
20	D₄×Q₈×C₃₁	C₂²	1	2	yes	—
21	C₃₁×#547	—	2	3	no	—
22	C₃₁×#547	C₂²	1	6	yes	—
23	C₃₁×#536	—	2	3	no	—
24	C₃₁×#532	—	2	3	no	—
25	C₃₁×#532	C₂²	1	3	yes	—
26	C₃₁×#532	C₂²	1	3	yes	—
27	C₂×C₃₁×C₄⋊Q₈	C₂²	1	3	yes	—
28	C₄×#41×C₃₁	—	2	1	no	—
29	C₄×#41×C₃₁	C₂²	1	3	yes	—
30	C₄×#41×C₃₁	C₂²	1	1	yes	characteristic
31	C₃₁×#498	C₂²	1	6	yes	—
32	C₃₁×#486	C₂²	1	6	yes	—
33	C₃₁×#482	C₂²	1	6	yes	—
34	C₃₁×#481	C₂²	1	12	yes	—
35	C₃₁×#477	C₂²	1	3	yes	—
36	C₃₁×#476	C₂²	1	6	yes	—
37	C₃₁×#476	C₂²	1	6	yes	—
38	C₃₁×#476	C₂²	1	3	yes	—
39	C₃₁×#475	C₂²	1	6	yes	—
40	C₃₁×#474	C₂²	1	6	yes	—
41	C₃₁×#469	C₂²	1	12	yes	—
42	C₃₁×#460	C₂²	1	1	yes	characteristic
43	C₃₁×#459	C₂²	1	6	yes	—
44	C₃₁×#459	C₂²	1	3	yes	—
45	C₃₁×#449	C₂²	1	6	yes	—
46	C₈×Q₈×C₃₁	C₂²	1	2	yes	—
47	C₄×SD₁₆×C₃₁	C₂²	1	6	yes	—
48	C₄×D₈×C₃₁	C₂²	1	6	yes	—
49	C₃₁×#436	C₂²	1	6	yes	—
50	C₃₁×#431	C₂²	1	3	yes	—
51	C₃₁×#428	C₂²	1	6	yes	—
52	C₃₁×#423	C₂²	1	3	yes	—
53	C₃₁×#417	C₂²	1	6	yes	—
54	C₃₁×2₋¹⁺⁴	—	4	8	no	—
55	C₂×#41×C₃₁	—	2	1	no	—
56	C₂×#41×C₃₁	—	2	6	no	—
57	C₂×#41×C₃₁	—	2	3	no	—
58	C₂×#41×C₃₁	—	2	3	no	—
59	C₂×#41×C₃₁	D₄	1	1	yes	characteristic
60	C₂×#41×C₃₁	C₂³	1	3	yes	—
61	C₂²×Q₈×C₃₁	—	2	2	no	—
62	C₂²×Q₈×C₃₁	—	2	3	no	—
63	C₂×SD₁₆×C₃₁	C₂³	1	2	yes	—
64	C₂×D₈×C₃₁	C₂³	1	6	yes	—
65	C₂×M₁₆×C₃₁	C₂³	1	6	yes	—
66	C₃₁×C₄⋊Q₈	—	2	3	no	—
67	C₃₁×C₄⋊Q₈	—	2	3	no	—
68	C₃₁×C₄⋊Q₈	—	2	3	no	—
69	C₃₁×C₄⋊Q₈	C₂³	1	6	yes	—
70	C₃₁×C₄⋊Q₈	C₂³	1	3	yes	—
71	C₃₁×C₄⋊Q₈	C₂³	1	3	yes	—
72	C₃₁×#127	—	2	3	no	—
73	C₃₁×#127	C₂³	1	1	yes	characteristic
74	C₃₁×#127	C₂³	1	3	yes	—
75	C₃₁×#124	—	2	6	no	—
76	C₃₁×#124	—	2	3	no	—
77	C₃₁×#124	—	2	3	no	—
78	C₃₁×#123	—	2	12	no	—
79	C₃₁×C₂²⋊Q₈	—	2	6	no	—
80	C₃₁×C₂²⋊Q₈	—	2	6	no	—
81	C₃₁×#121	—	2	3	no	—
82	C₃₁×#121	—	2	3	no	—
83	C₃₁×#121	—	2	3	no	—
84	C₃₁×#121	—	2	3	no	—
85	C₃₁×#121	—	2	6	no	—
86	C₃₁×#121	C₂³	1	6	yes	—
87	C₃₁×#121	C₂³	1	6	yes	—
88	C₄×Q₈×C₃₁	—	2	1	no	—
89	C₄×Q₈×C₃₁	D₄	1	1	yes	characteristic
90	C₄×Q₈×C₃₁	C₂³	1	3	yes	—
91	C₄×Q₈×C₃₁	C₂³	1	1	yes	characteristic
92	C₄×Q₈×C₃₁	C₂³	1	1	yes	characteristic
93	C₄×D₄×C₃₁	—	2	3	no	—
94	C₄×D₄×C₃₁	—	2	3	no	—
95	C₄×D₄×C₃₁	—	2	3	no	—
96	C₄×D₄×C₃₁	—	2	3	no	—
97	C₄×D₄×C₃₁	—	2	3	no	—
98	C₄×D₄×C₃₁	D₄	1	3	yes	—
99	C₄×D₄×C₃₁	C₂³	1	3	yes	—
100	C₄×D₄×C₃₁	C₂³	1	6	yes	—
101	C₄×D₄×C₃₁	C₂³	1	3	yes	—
102	C₄×D₄×C₃₁	C₂³	1	3	yes	—
103	C₃₁×#117	—	2	3	no	—
104	C₃₁×#117	—	2	3	no	—
105	C₃₁×#117	D₄	1	3	yes	—
106	C₃₁×#117	C₂³	1	3	yes	—
107	C₂×C₄⋊C₄×C₃₁	—	2	3	no	—
108	C₂×C₄⋊C₄×C₃₁	C₂³	1	6	yes	—
109	C₂×C₄²×C₃₁	—	2	3	no	—
110	C₂×C₄²×C₃₁	C₂³	1	3	yes	—
111	C₃₁×#107	C₂³	1	6	yes	—
112	C₃₁×#106	C₂³	1	12	yes	—
113	C₃₁×#106	C₂³	1	6	yes	—
114	C₃₁×C₄⋊C₈	C₂³	1	3	yes	—
115	C₃₁×C₄⋊C₈	C₂³	1	6	yes	—
116	C₃₁×C₄⋊C₈	C₂³	1	1	yes	characteristic
117	C₃₁×#103	C₂³	1	6	yes	—
118	C₃₁×#102	C₂³	1	12	yes	—
119	C₃₁×#102	C₂³	1	2	yes	—
120	C₃₁×#102	C₂³	1	6	yes	—
121	C₃₁×#102	C₂³	1	6	yes	—
122	C₃₁×C₂²⋊C₈	C₂³	1	6	yes	—
123	C₄×C₈×C₃₁	C₂³	1	6	yes	—
124	#41×C₃₁	—	2	4	no	—
125	#41×C₃₁	—	2	6	no	—
126	#41×C₃₁	—	4	3	no	—
127	#41×C₃₁	—	4	12	no	—
128	#41×C₃₁	—	4	24	no	—
129	#41×C₃₁	—	4	4	no	—
130	#41×C₃₁	—	8	6	no	—
131	C₂×Q₈×C₃₁	—	2	6	no	—
132	C₂×Q₈×C₃₁	—	2	3	no	—
133	C₂×Q₈×C₃₁	—	2	3	no	—
134	C₂×Q₈×C₃₁	—	4	12	no	—
135	C₂×Q₈×C₃₁	—	4	8	no	—
136	C₂×Q₈×C₃₁	C₂×D₄	1	1	yes	characteristic
137	C₂×Q₈×C₃₁	C₂⁴	1	1	yes	characteristic
138	C₂×D₄×C₃₁	—	2	3	no	—
139	C₂×D₄×C₃₁	—	2	6	no	—
140	C₂×D₄×C₃₁	—	2	6	no	—
141	C₂×D₄×C₃₁	—	2	1	no	—
142	C₂×D₄×C₃₁	—	2	3	no	—
143	C₂×D₄×C₃₁	—	2	1	no	—
144	C₂×D₄×C₃₁	—	2	3	no	—
145	C₂×D₄×C₃₁	C₂×D₄	1	3	yes	—
146	C₂×D₄×C₃₁	C₂⁴	1	3	yes	—
147	C₂×D₄×C₃₁	C₂⁴	1	2	yes	—
148	C₂²×C₄×C₃₁	—	2	6	no	—
149	C₂²×C₄×C₃₁	—	2	3	no	—
150	C₂²×C₄×C₃₁	—	2	3	no	—
151	C₂²×C₄×C₃₁	—	2	3	no	—
152	C₂²×C₄×C₃₁	—	2	6	no	—
153	C₂²×C₄×C₃₁	C₂×D₄	1	3	yes	—
154	C₂²×C₄×C₃₁	C₂×D₄	1	3	yes	—
155	C₂²×C₄×C₃₁	C₂⁴	1	3	yes	—
156	SD₁₆×C₃₁	—	2	4	no	—
157	D₈×C₃₁	—	4	6	no	—
158	M₁₆×C₃₁	—	4	6	no	—
159	C₂×C₈×C₃₁	C₂⁴	1	2	yes	—
160	C₂×C₈×C₃₁	C₂⁴	1	6	yes	—
161	C₄⋊C₄×C₃₁	—	2	3	no	—
162	C₄⋊C₄×C₃₁	—	2	6	no	—
163	C₄⋊C₄×C₃₁	—	2	6	no	—
164	C₄⋊C₄×C₃₁	—	2	1	no	—
165	C₄⋊C₄×C₃₁	—	2	3	no	—
166	C₄⋊C₄×C₃₁	—	2	3	no	—
167	C₄⋊C₄×C₃₁	C₂×D₄	1	3	yes	—
168	C₄⋊C₄×C₃₁	C₂×D₄	1	3	yes	—
169	C₄⋊C₄×C₃₁	C₂⁴	1	6	yes	—
170	C₄⋊C₄×C₃₁	C₂⁴	1	3	yes	—
171	C₄⋊C₄×C₃₁	C₂⁴	1	1	yes	characteristic
172	C₂²⋊C₄×C₃₁	—	2	3	no	—
173	C₂²⋊C₄×C₃₁	—	2	6	no	—
174	C₂²⋊C₄×C₃₁	—	2	3	no	—
175	C₂²⋊C₄×C₃₁	—	2	3	no	—
176	C₂²⋊C₄×C₃₁	—	2	6	no	—
177	C₂²⋊C₄×C₃₁	—	2	6	no	—
178	C₂²⋊C₄×C₃₁	C₂×D₄	1	6	yes	—
179	C₄²×C₃₁	—	2	1	no	—
180	C₄²×C₃₁	—	2	3	no	—
181	C₄²×C₃₁	—	2	3	no	—
182	C₄²×C₃₁	C₂×D₄	1	3	yes	—
183	C₄²×C₃₁	C₂×D₄	1	3	yes	—
184	C₄²×C₃₁	C₂⁴	1	3	yes	—
185	C₄²×C₃₁	C₂⁴	1	1	yes	characteristic
186	#62150	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
187	C₂³×C₃₁	—	2	2	no	—
188	C₂³×C₃₁	—	2	3	no	—
189	C₂³×C₃₁	C₂²×D₄	1	3	yes	—
190	Q₈×C₃₁	—	2	1	no	—
191	Q₈×C₃₁	—	4	6	no	—
192	Q₈×C₃₁	—	4	12	no	—
193	Q₈×C₃₁	—	8	3	no	—
194	Q₈×C₃₁	AGL(1,ℤ/8)	1	4	yes	—
195	Q₈×C₃₁	2₋¹⁺⁴	1	2	yes	complemented
196	D₄×C₃₁	—	2	6	no	—
197	D₄×C₃₁	—	2	2	no	—
198	D₄×C₃₁	—	2	3	no	—
199	D₄×C₃₁	—	2	2	no	—
200	D₄×C₃₁	—	2	3	no	—
201	D₄×C₃₁	—	4	6	no	—
202	D₄×C₃₁	—	4	2	no	—
203	D₄×C₃₁	—	4	12	no	—
204	D₄×C₃₁	—	4	6	no	—
205	D₄×C₃₁	—	4	2	no	—
206	D₄×C₃₁	—	8	3	no	—
207	D₄×C₃₁	—	8	6	no	—
208	C₂×C₄×C₃₁	—	2	1	no	—
209	C₂×C₄×C₃₁	—	2	6	no	—
210	C₂×C₄×C₃₁	—	2	6	no	—
211	C₂×C₄×C₃₁	—	2	3	no	—
212	C₂×C₄×C₃₁	—	2	3	no	—
213	C₂×C₄×C₃₁	—	2	1	no	—
214	C₂×C₄×C₃₁	—	2	3	no	—
215	C₂×C₄×C₃₁	—	2	6	no	—
216	C₂×C₄×C₃₁	—	4	6	no	—
217	C₂×C₄×C₃₁	—	4	12	no	—
218	C₂×C₄×C₃₁	—	4	12	no	—
219	C₂×C₄×C₃₁	—	4	6	no	—
220	C₂×C₄×C₃₁	—	4	6	no	—
221	C₂×C₄×C₃₁	C₂²×D₄	1	3	yes	—
222	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
223	C₂×C₄×C₃₁	C₂²×D₄	1	3	yes	—
224	C₂×C₄×C₃₁	C₂²×D₄	1	3	yes	—
225	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
226	C₂×C₄×C₃₁	C₂⁵	1	1	yes	characteristic
227	C₈×C₃₁	—	2	2	no	—
228	C₈×C₃₁	—	2	6	no	—
229	#3499	C₂×C₃₁	1	1	yes	characteristic, complemented
230	#3483	C₂×C₃₁	1	1	yes	characteristic, complemented
231	Q₈×SD₁₆	C₂×C₃₁	1	2	yes	complemented
232	#3358	C₂×C₃₁	1	2	yes	complemented
233	#3354	C₂×C₃₁	1	6	yes	complemented
234	#3332	C₂×C₃₁	1	6	yes	complemented
235	#3253	C₂×C₃₁	1	3	yes	complemented
236	#3227	C₂×C₃₁	1	3	yes	complemented
237	#3106	C₂×C₃₁	1	3	yes	complemented
238	#3079	C₂×C₃₁	1	3	yes	complemented
239	#2991	C₂×C₃₁	1	1	yes	characteristic, complemented
240	C₂²×C₃₁	—	2	3	no	—
241	C₂²×C₃₁	—	2	3	no	—
242	C₂²×C₃₁	—	2	3	no	—
243	C₂²×C₃₁	—	4	3	no	—
244	C₂²×C₃₁	—	4	2	no	—
245	C₂²×C₃₁	—	4	2	no	—
246	C₂²×C₃₁	—	4	3	no	—
247	C₂²×C₃₁	—	4	3	no	—
248	C₂²×C₃₁	—	4	2	no	—
249	C₂²×C₃₁	C₂³×D₄	1	1	yes	characteristic
250	C₄×C₃₁	—	2	3	no	—
251	C₄×C₃₁	—	2	2	no	—
252	C₄×C₃₁	—	2	1	no	—
253	C₄×C₃₁	—	2	3	no	—
254	C₄×C₃₁	—	4	1	no	—
255	C₄×C₃₁	—	4	6	no	—
256	C₄×C₃₁	—	4	3	no	—
257	C₄×C₃₁	C₂×AGL(1,ℤ/8)	1	6	yes	—
258	C₄×C₃₁	#577	1	2	yes	—
259	C₄×C₃₁	C₂×2₋¹⁺⁴	1	1	yes	characteristic
260	C₄×C₃₁	#585	1	1	yes	characteristic
261	C₂×2₋¹⁺⁴	—	2	1	no	—
262	Q₈²	C₂²×C₃₁	1	1	yes	characteristic
263	#555	—	2	3	no	—
264	#555	C₂²×C₃₁	1	3	yes	—
265	#550	—	2	1	no	—
266	#550	C₂²×C₃₁	1	3	yes	—
267	#550	C₂²×C₃₁	1	1	yes	characteristic
268	D₄×Q₈	—	2	1	no	—
269	D₄×Q₈	C₂²×C₃₁	1	2	yes	—
270	#547	—	2	3	no	—
271	#547	C₂²×C₃₁	1	6	yes	—
272	#536	—	2	3	no	—
273	#532	—	2	3	no	—
274	#532	C₂²×C₃₁	1	3	yes	—
275	#532	C₂²×C₃₁	1	3	yes	—
276	C₂×C₄⋊Q₈	C₂²×C₃₁	1	3	yes	—
277	C₄×#41	—	2	1	no	—
278	C₄×#41	C₂²×C₃₁	1	3	yes	—
279	C₄×#41	C₂²×C₃₁	1	1	yes	characteristic
280	#498	C₂²×C₃₁	1	6	yes	—
281	#486	C₂²×C₃₁	1	6	yes	—
282	#482	C₂²×C₃₁	1	6	yes	—
283	#481	C₂²×C₃₁	1	12	yes	—
284	#477	C₂²×C₃₁	1	3	yes	—
285	#476	C₂²×C₃₁	1	6	yes	—
286	#476	C₂²×C₃₁	1	6	yes	—
287	#476	C₂²×C₃₁	1	3	yes	—
288	#475	C₂²×C₃₁	1	6	yes	—
289	#474	C₂²×C₃₁	1	6	yes	—
290	#469	C₂²×C₃₁	1	12	yes	—
291	#460	C₂²×C₃₁	1	1	yes	characteristic
292	#459	C₂²×C₃₁	1	6	yes	—
293	#459	C₂²×C₃₁	1	3	yes	—
294	#449	C₂²×C₃₁	1	6	yes	—
295	C₈×Q₈	C₂²×C₃₁	1	2	yes	—
296	C₄×SD₁₆	C₂²×C₃₁	1	6	yes	—
297	C₄×D₈	C₂²×C₃₁	1	6	yes	—
298	#436	C₂²×C₃₁	1	6	yes	—
299	#431	C₂²×C₃₁	1	3	yes	—
300	#428	C₂²×C₃₁	1	6	yes	—
301	#423	C₂²×C₃₁	1	3	yes	—
302	#417	C₂²×C₃₁	1	6	yes	—
303	C₂×C₃₁	—	4	3	no	—
304	C₂×C₃₁	—	8	3	no	—
305	C₂×C₃₁	—	8	2	no	—
306	C₂×C₃₁	#3489	1	1	yes	characteristic
307	C₂×C₃₁	C₂²×AGL(1,ℤ/8)	1	1	yes	characteristic
308	C₂×C₃₁	C₂×#577	1	1	yes	characteristic
309	2₋¹⁺⁴	—	4	8	no	—
310	C₂×#41	—	2	1	no	—
311	C₂×#41	—	2	6	no	—
312	C₂×#41	—	2	3	no	—
313	C₂×#41	—	2	3	no	—
314	C₂×#41	D₄×C₃₁	1	1	yes	characteristic
315	C₂×#41	C₂³×C₃₁	1	3	yes	—
316	C₂²×Q₈	—	2	2	no	—
317	C₂²×Q₈	—	2	3	no	—
318	C₂×SD₁₆	C₂³×C₃₁	1	2	yes	—
319	C₂×D₈	C₂³×C₃₁	1	6	yes	—
320	C₂×M₁₆	C₂³×C₃₁	1	6	yes	—
321	C₄⋊Q₈	—	2	3	no	—
322	C₄⋊Q₈	—	2	3	no	—
323	C₄⋊Q₈	—	2	3	no	—
324	C₄⋊Q₈	C₂³×C₃₁	1	3	yes	—
325	C₄⋊Q₈	C₂³×C₃₁	1	6	yes	—
326	C₄⋊Q₈	C₂³×C₃₁	1	3	yes	—
327	#127	—	2	3	no	—
328	#127	C₂³×C₃₁	1	1	yes	characteristic
329	#127	C₂³×C₃₁	1	3	yes	—
330	#124	—	2	6	no	—
331	#124	—	2	3	no	—
332	#124	—	2	3	no	—
333	#123	—	2	12	no	—
334	C₂²⋊Q₈	—	2	6	no	—
335	C₂²⋊Q₈	—	2	6	no	—
336	#121	—	2	3	no	—
337	#121	—	2	3	no	—
338	#121	—	2	3	no	—
339	#121	—	2	3	no	—
340	#121	—	2	6	no	—
341	#121	C₂³×C₃₁	1	6	yes	—
342	#121	C₂³×C₃₁	1	6	yes	—
343	C₄×Q₈	—	2	1	no	—
344	C₄×Q₈	D₄×C₃₁	1	1	yes	characteristic
345	C₄×Q₈	C₂³×C₃₁	1	1	yes	characteristic
346	C₄×Q₈	C₂³×C₃₁	1	3	yes	—
347	C₄×Q₈	C₂³×C₃₁	1	1	yes	characteristic
348	C₄×D₄	—	2	3	no	—
349	C₄×D₄	—	2	3	no	—
350	C₄×D₄	—	2	3	no	—
351	C₄×D₄	—	2	3	no	—
352	C₄×D₄	—	2	3	no	—
353	C₄×D₄	D₄×C₃₁	1	3	yes	—
354	C₄×D₄	C₂³×C₃₁	1	3	yes	—
355	C₄×D₄	C₂³×C₃₁	1	3	yes	—
356	C₄×D₄	C₂³×C₃₁	1	6	yes	—
357	C₄×D₄	C₂³×C₃₁	1	3	yes	—
358	#117	—	2	3	no	—
359	#117	—	2	3	no	—
360	#117	D₄×C₃₁	1	3	yes	—
361	#117	C₂³×C₃₁	1	3	yes	—
362	C₂×C₄⋊C₄	—	2	3	no	—
363	C₂×C₄⋊C₄	C₂³×C₃₁	1	6	yes	—
364	C₂×C₄²	—	2	3	no	—
365	C₂×C₄²	C₂³×C₃₁	1	3	yes	—
366	#107	C₂³×C₃₁	1	6	yes	—
367	#106	C₂³×C₃₁	1	12	yes	—
368	#106	C₂³×C₃₁	1	6	yes	—
369	C₄⋊C₈	C₂³×C₃₁	1	3	yes	—
370	C₄⋊C₈	C₂³×C₃₁	1	6	yes	—
371	C₄⋊C₈	C₂³×C₃₁	1	1	yes	characteristic
372	#103	C₂³×C₃₁	1	6	yes	—
373	#102	C₂³×C₃₁	1	12	yes	—
374	#102	C₂³×C₃₁	1	2	yes	—
375	#102	C₂³×C₃₁	1	6	yes	—
376	#102	C₂³×C₃₁	1	6	yes	—
377	C₂²⋊C₈	C₂³×C₃₁	1	6	yes	—
378	C₄×C₈	C₂³×C₃₁	1	6	yes	—
379	C₃₁	#62150	1	1	yes	characteristic, direct product factor, Sylow
380	#41	—	2	4	no	—
381	#41	—	2	6	no	—
382	#41	—	4	3	no	—
383	#41	—	4	12	no	—
384	#41	—	4	24	no	—
385	#41	—	4	4	no	—
386	#41	—	8	6	no	—
387	C₂×Q₈	—	2	6	no	—
388	C₂×Q₈	—	2	3	no	—
389	C₂×Q₈	—	2	3	no	—
390	C₂×Q₈	—	4	12	no	—
391	C₂×Q₈	—	4	8	no	—
392	C₂×Q₈	C₂×D₄×C₃₁	1	1	yes	characteristic
393	C₂×Q₈	C₂⁴×C₃₁	1	1	yes	characteristic
394	C₂×D₄	—	2	3	no	—
395	C₂×D₄	—	2	6	no	—
396	C₂×D₄	—	2	6	no	—
397	C₂×D₄	—	2	1	no	—
398	C₂×D₄	—	2	3	no	—
399	C₂×D₄	—	2	1	no	—
400	C₂×D₄	—	2	3	no	—
401	C₂×D₄	C₂×D₄×C₃₁	1	3	yes	—
402	C₂×D₄	C₂⁴×C₃₁	1	3	yes	—
403	C₂×D₄	C₂⁴×C₃₁	1	2	yes	—
404	C₂²×C₄	—	2	6	no	—
405	C₂²×C₄	—	2	3	no	—
406	C₂²×C₄	—	2	3	no	—
407	C₂²×C₄	—	2	3	no	—
408	C₂²×C₄	—	2	6	no	—
409	C₂²×C₄	C₂×D₄×C₃₁	1	3	yes	—
410	C₂²×C₄	C₂×D₄×C₃₁	1	3	yes	—
411	C₂²×C₄	C₂⁴×C₃₁	1	3	yes	—
412	SD₁₆	—	2	4	no	—
413	D₈	—	4	6	no	—
414	M₁₆	—	4	6	no	—
415	C₂×C₈	C₂⁴×C₃₁	1	2	yes	—
416	C₂×C₈	C₂⁴×C₃₁	1	6	yes	—
417	C₄⋊C₄	—	2	3	no	—
418	C₄⋊C₄	—	2	6	no	—
419	C₄⋊C₄	—	2	6	no	—
420	C₄⋊C₄	—	2	1	no	—
421	C₄⋊C₄	—	2	3	no	—
422	C₄⋊C₄	—	2	3	no	—
423	C₄⋊C₄	C₂×D₄×C₃₁	1	3	yes	—
424	C₄⋊C₄	C₂×D₄×C₃₁	1	3	yes	—
425	C₄⋊C₄	C₂⁴×C₃₁	1	6	yes	—
426	C₄⋊C₄	C₂⁴×C₃₁	1	3	yes	—
427	C₄⋊C₄	C₂⁴×C₃₁	1	1	yes	characteristic
428	C₂²⋊C₄	—	2	3	no	—
429	C₂²⋊C₄	—	2	6	no	—
430	C₂²⋊C₄	—	2	3	no	—
431	C₂²⋊C₄	—	2	3	no	—
432	C₂²⋊C₄	—	2	6	no	—
433	C₂²⋊C₄	—	2	6	no	—
434	C₂²⋊C₄	C₂×D₄×C₃₁	1	6	yes	—
435	C₄²	—	2	1	no	—
436	C₄²	—	2	3	no	—
437	C₄²	—	2	3	no	—
438	C₄²	C₂×D₄×C₃₁	1	3	yes	—
439	C₄²	C₂×D₄×C₃₁	1	3	yes	—
440	C₄²	C₂⁴×C₃₁	1	3	yes	—
441	C₄²	C₂⁴×C₃₁	1	1	yes	characteristic
442	C₂³	—	2	2	no	—
443	C₂³	—	2	3	no	—
444	C₂³	C₂²×D₄×C₃₁	1	3	yes	—
445	Q₈	—	2	1	no	—
446	Q₈	—	4	6	no	—
447	Q₈	—	4	12	no	—
448	Q₈	—	8	3	no	—
449	Q₈	C₃₁×AGL(1,ℤ/8)	1	4	yes	—
450	Q₈	C₃₁×2₋¹⁺⁴	1	2	yes	complemented
451	D₄	—	2	6	no	—
452	D₄	—	2	2	no	—
453	D₄	—	2	3	no	—
454	D₄	—	2	2	no	—
455	D₄	—	2	3	no	—
456	D₄	—	4	6	no	—
457	D₄	—	4	2	no	—
458	D₄	—	4	12	no	—
459	D₄	—	4	6	no	—
460	D₄	—	4	2	no	—
461	D₄	—	8	3	no	—
462	D₄	—	8	6	no	—
463	C₂×C₄	—	2	1	no	—
464	C₂×C₄	—	2	6	no	—
465	C₂×C₄	—	2	6	no	—
466	C₂×C₄	—	2	3	no	—
467	C₂×C₄	—	2	3	no	—
468	C₂×C₄	—	2	1	no	—
469	C₂×C₄	—	2	3	no	—
470	C₂×C₄	—	2	6	no	—
471	C₂×C₄	—	4	6	no	—
472	C₂×C₄	—	4	12	no	—
473	C₂×C₄	—	4	12	no	—
474	C₂×C₄	—	4	6	no	—
475	C₂×C₄	—	4	6	no	—
476	C₂×C₄	C₂²×D₄×C₃₁	1	3	yes	—
477	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
478	C₂×C₄	C₂²×D₄×C₃₁	1	3	yes	—
479	C₂×C₄	C₂²×D₄×C₃₁	1	3	yes	—
480	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
481	C₂×C₄	C₂⁵×C₃₁	1	1	yes	characteristic
482	C₈	—	2	2	no	—
483	C₈	—	2	6	no	—
484	C₂²	—	2	3	no	—
485	C₂²	—	2	3	no	—
486	C₂²	—	2	3	no	—
487	C₂²	—	4	3	no	—
488	C₂²	—	4	2	no	—
489	C₂²	—	4	2	no	—
490	C₂²	—	4	3	no	—
491	C₂²	—	4	3	no	—
492	C₂²	—	4	2	no	—
493	C₂²	C₂³×D₄×C₃₁	1	1	yes	characteristic
494	C₄	—	2	3	no	—
495	C₄	—	2	2	no	—
496	C₄	—	2	1	no	—
497	C₄	—	2	3	no	—
498	C₄	—	4	6	no	—
499	C₄	—	4	1	no	—
500	C₄	—	4	3	no	—
501	C₄	C₂×C₃₁×AGL(1,ℤ/8)	1	6	yes	—
502	C₄	C₃₁×#577	1	2	yes	—
503	C₄	C₂×C₃₁×2₋¹⁺⁴	1	1	yes	characteristic
504	C₄	C₃₁×#585	1	1	yes	characteristic
505	C₂	—	4	3	no	—
506	C₂	—	8	3	no	—
507	C₂	—	8	2	no	—
508	C₂	C₃₁×#3489	1	1	yes	characteristic
509	C₂	C₂²×C₃₁×AGL(1,ℤ/8)	1	1	yes	characteristic
510	C₂	C₂×C₃₁×#577	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×#3499	C₂	semidirect
2. C₃₁×#3483	C₂	semidirect
3. Q₈×SD₁₆×C₃₁	C₂	semidirect
4. C₃₁×#3358	C₂	semidirect
5. C₃₁×#3354	C₂	semidirect
6. C₃₁×#3332	C₂	semidirect
7. C₃₁×#3253	C₂	semidirect
8. C₃₁×#3227	C₂	semidirect
9. C₃₁×#3106	C₂	semidirect
10. C₃₁×#3079	C₂	semidirect
11. C₃₁×#2991	C₂	semidirect
186. #62150	C₃₁	direct (N × H)
195. Q₈×C₃₁	2₋¹⁺⁴	semidirect
229. #3499	C₂×C₃₁	semidirect
230. #3483	C₂×C₃₁	semidirect
231. Q₈×SD₁₆	C₂×C₃₁	semidirect
232. #3358	C₂×C₃₁	semidirect
233. #3354	C₂×C₃₁	semidirect
234. #3332	C₂×C₃₁	semidirect
235. #3253	C₂×C₃₁	semidirect
236. #3227	C₂×C₃₁	semidirect
237. #3106	C₂×C₃₁	semidirect
238. #3079	C₂×C₃₁	semidirect
239. #2991	C₂×C₃₁	semidirect
379. C₃₁	#62150	direct (N × H)
450. Q₈	C₃₁×2₋¹⁺⁴	semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
C₃₁×#3499	2
C₃₁×#3483	2
Q₈×SD₁₆×C₃₁	2
C₃₁×#3358	2
C₃₁×#3354	2
C₃₁×#3332	2
C₃₁×#3253	2
C₃₁×#3227	2
C₃₁×#3106	2
C₃₁×#3079	2
C₃₁×#2991	2
#62150	31

Subgroups — C31×#62150 #21518909

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#62150 #21518909