Subgroups — C₃₁×#62001 #21518760

Order 7936.

2216 conjugacy classes of subgroups.

The subgroup lattice has 532 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃₁×#3498	C₂	1	1	yes	maximal, characteristic, complemented
2	C₃₁×#3483	C₂	1	1	yes	maximal, characteristic, complemented
3	C₃₁×#3391	C₂	1	2	yes	maximal, complemented
4	C₃₁×#3362	C₂	1	2	yes	maximal, complemented
5	C₃₁×#3325	C₂	1	6	yes	maximal, complemented
6	C₃₁×#3274	C₂	1	6	yes	maximal, complemented
7	C₃₁×#3253	C₂	1	3	yes	maximal, complemented
8	C₃₁×#3205	C₂	1	3	yes	maximal, complemented
9	C₃₁×#3106	C₂	1	3	yes	maximal, complemented
10	C₃₁×#3038	C₂	1	3	yes	maximal, complemented
11	C₃₁×#2991	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×C₃₁×H(5,𝔽₂)	—	2	1	no	—
13	C₂×C₃₁×AGL(1,ℤ/8)	C₂²	1	6	yes	complemented
14	Q₈²×C₃₁	C₂²	1	1	yes	characteristic, complemented
15	C₃₁×#555	—	2	3	no	—
16	C₃₁×#555	C₂²	1	3	yes	complemented
17	C₃₁×#550	—	2	1	no	—
18	C₃₁×#550	—	2	1	no	—
19	C₃₁×#550	C₂²	1	2	yes	—
20	C₃₁×#550	C₂²	1	3	yes	complemented
21	C₃₁×#550	C₂²	1	1	yes	characteristic, complemented
22	D₄²×C₃₁	—	2	3	no	—
23	D₄²×C₃₁	C₂²	1	6	yes	—
24	C₃₁×#535	—	2	3	no	—
25	C₃₁×#532	—	2	3	no	—
26	C₃₁×#532	C₂²	1	3	yes	—
27	C₃₁×#532	C₂²	1	3	yes	complemented
28	C₂×C₃₁×#127	C₂²	1	3	yes	—
29	C₄×#41×C₃₁	—	2	1	no	—
30	C₄×#41×C₃₁	C₂²	1	1	yes	characteristic
31	C₄×#41×C₃₁	C₂²	1	3	yes	complemented
32	C₃₁×#493	C₂²	1	6	yes	complemented
33	C₃₁×#486	C₂²	1	6	yes	complemented
34	C₃₁×#482	C₂²	1	6	yes	complemented
35	C₃₁×#477	C₂²	1	1	yes	characteristic, complemented
36	C₃₁×#476	C₂²	1	3	yes	complemented
37	C₃₁×#476	C₂²	1	6	yes	complemented
38	C₃₁×#469	C₂²	1	12	yes	complemented
39	C₃₁×#460	C₂²	1	3	yes	complemented
40	C₃₁×#460	C₂²	1	6	yes	complemented
41	C₃₁×#459	C₂²	1	3	yes	complemented
42	C₃₁×#459	C₂²	1	6	yes	complemented
43	C₃₁×#459	C₂²	1	6	yes	complemented
44	C₃₁×#459	C₂²	1	6	yes	complemented
45	C₃₁×#449	C₂²	1	6	yes	complemented
46	C₃₁×#447	C₂²	1	12	yes	complemented
47	C₈×Q₈×C₃₁	C₂²	1	2	yes	complemented
48	C₄×SD₁₆×C₃₁	C₂²	1	6	yes	complemented
49	C₄×D₈×C₃₁	C₂²	1	6	yes	complemented
50	C₃₁×#436	C₂²	1	6	yes	complemented
51	C₃₁×#431	C₂²	1	3	yes	complemented
52	C₃₁×#423	C₂²	1	3	yes	complemented
53	C₃₁×#418	C₂²	1	6	yes	complemented
54	C₃₁×H(5,𝔽₂)	—	4	8	no	—
55	C₂×#41×C₃₁	—	2	1	no	—
56	C₂×#41×C₃₁	—	2	2	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×D₄×C₃₁	—	2	6	no	—
62	C₂²×D₄×C₃₁	—	2	3	no	—
63	C₂²×D₄×C₃₁	—	2	3	no	—
64	C₂²×D₄×C₃₁	C₂³	1	6	yes	—
65	C₃₁×AGL(1,ℤ/8)	—	4	12	no	—
66	C₂×SD₁₆×C₃₁	C₂³	1	6	yes	—
67	C₂×D₈×C₃₁	C₂³	1	6	yes	—
68	C₂×D₈×C₃₁	C₂³	1	12	yes	—
69	C₂×M₁₆×C₃₁	C₂³	1	6	yes	—
70	C₃₁×C₄⋊Q₈	—	2	3	no	—
71	C₃₁×C₄⋊Q₈	C₂³	1	1	yes	characteristic
72	C₃₁×C₄⋊Q₈	C₂³	1	3	yes	—
73	C₃₁×#127	—	2	3	no	—
74	C₃₁×#127	—	2	3	no	—
75	C₃₁×#127	—	2	3	no	—
76	C₃₁×#127	C₂³	1	3	yes	—
77	C₃₁×#127	C₂³	1	3	yes	—
78	C₃₁×#127	C₂³	1	6	yes	—
79	C₃₁×#124	—	2	6	no	—
80	C₃₁×#124	—	2	3	no	—
81	C₃₁×#124	—	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	3	no	—
86	C₃₁×#121	—	2	6	no	—
87	C₃₁×#121	—	2	6	no	—
88	C₃₁×#121	—	2	6	no	—
89	C₃₁×#121	C₂³	1	6	yes	—
90	C₃₁×#121	C₂³	1	6	yes	—
91	C₃₁×#120	—	2	12	no	—
92	C₄×Q₈×C₃₁	—	2	1	no	—
93	C₄×Q₈×C₃₁	D₄	1	1	yes	characteristic
94	C₄×Q₈×C₃₁	C₂³	1	3	yes	—
95	C₄×Q₈×C₃₁	C₂³	1	1	yes	characteristic
96	C₄×Q₈×C₃₁	C₂³	1	1	yes	characteristic
97	C₄×D₄×C₃₁	—	2	3	no	—
98	C₄×D₄×C₃₁	—	2	3	no	—
99	C₄×D₄×C₃₁	—	2	3	no	—
100	C₄×D₄×C₃₁	—	2	3	no	—
101	C₄×D₄×C₃₁	—	2	3	no	—
102	C₄×D₄×C₃₁	D₄	1	3	yes	—
103	C₄×D₄×C₃₁	C₂³	1	3	yes	—
104	C₄×D₄×C₃₁	C₂³	1	6	yes	—
105	C₄×D₄×C₃₁	C₂³	1	3	yes	—
106	C₄×D₄×C₃₁	C₂³	1	3	yes	—
107	C₃₁×#117	—	2	3	no	—
108	C₃₁×#117	—	2	3	no	—
109	C₃₁×#117	D₄	1	3	yes	—
110	C₃₁×#117	C₂³	1	3	yes	—
111	C₂×C₄²×C₃₁	—	2	3	no	—
112	C₂×C₄²×C₃₁	C₂³	1	3	yes	—
113	C₃₁×#107	C₂³	1	2	yes	—
114	C₃₁×#106	C₂³	1	6	yes	—
115	C₃₁×C₄⋊C₈	C₂³	1	3	yes	—
116	C₃₁×C₄⋊C₈	C₂³	1	6	yes	—
117	C₃₁×C₄⋊C₈	C₂³	1	1	yes	characteristic
118	C₃₁×#103	C₂³	1	2	yes	—
119	C₃₁×#102	C₂³	1	6	yes	—
120	C₃₁×#102	C₂³	1	12	yes	—
121	C₃₁×#102	C₂³	1	6	yes	—
122	C₃₁×#102	C₂³	1	6	yes	—
123	C₃₁×C₂²⋊C₈	C₂³	1	6	yes	—
124	C₄×C₈×C₃₁	C₂³	1	6	yes	—
125	C₂⁴×C₃₁	—	2	6	no	—
126	#41×C₃₁	—	2	4	no	—
127	#41×C₃₁	—	2	6	no	—
128	#41×C₃₁	—	4	3	no	—
129	#41×C₃₁	—	4	12	no	—
130	#41×C₃₁	—	4	4	no	—
131	#41×C₃₁	—	4	8	no	—
132	#41×C₃₁	—	8	6	no	—
133	C₂×Q₈×C₃₁	—	2	1	no	—
134	C₂×Q₈×C₃₁	—	2	3	no	—
135	C₂×Q₈×C₃₁	C₂×D₄	1	1	yes	characteristic
136	C₂×Q₈×C₃₁	C₂⁴	1	1	yes	characteristic
137	C₂×D₄×C₃₁	—	2	3	no	—
138	C₂×D₄×C₃₁	—	2	6	no	—
139	C₂×D₄×C₃₁	—	2	6	no	—
140	C₂×D₄×C₃₁	—	2	3	no	—
141	C₂×D₄×C₃₁	—	2	3	no	—
142	C₂×D₄×C₃₁	—	2	6	no	—
143	C₂×D₄×C₃₁	—	2	3	no	—
144	C₂×D₄×C₃₁	—	2	6	no	—
145	C₂×D₄×C₃₁	—	2	6	no	—
146	C₂×D₄×C₃₁	—	2	3	no	—
147	C₂×D₄×C₃₁	—	4	6	no	—
148	C₂×D₄×C₃₁	—	4	6	no	—
149	C₂×D₄×C₃₁	—	4	12	no	—
150	C₂×D₄×C₃₁	—	4	24	no	—
151	C₂×D₄×C₃₁	—	8	6	no	—
152	C₂×D₄×C₃₁	C₂×D₄	1	3	yes	—
153	C₂×D₄×C₃₁	C₂⁴	1	6	yes	—
154	C₂×D₄×C₃₁	C₂⁴	1	3	yes	—
155	C₂²×C₄×C₃₁	—	2	3	no	—
156	C₂²×C₄×C₃₁	—	2	3	no	—
157	C₂²×C₄×C₃₁	—	2	3	no	—
158	C₂²×C₄×C₃₁	—	2	6	no	—
159	C₂²×C₄×C₃₁	C₂×D₄	1	3	yes	—
160	C₂²×C₄×C₃₁	C₂×D₄	1	3	yes	—
161	C₂²×C₄×C₃₁	C₂⁴	1	3	yes	—
162	SD₁₆×C₃₁	—	2	12	no	—
163	D₈×C₃₁	—	4	6	no	—
164	D₈×C₃₁	—	4	12	no	—
165	M₁₆×C₃₁	—	4	6	no	—
166	C₂×C₈×C₃₁	C₂⁴	1	2	yes	—
167	C₂×C₈×C₃₁	C₂⁴	1	6	yes	—
168	C₄⋊C₄×C₃₁	—	2	1	no	—
169	C₄⋊C₄×C₃₁	—	2	3	no	—
170	C₄⋊C₄×C₃₁	—	2	3	no	—
171	C₄⋊C₄×C₃₁	—	2	1	no	—
172	C₄⋊C₄×C₃₁	C₂×D₄	1	3	yes	—
173	C₄⋊C₄×C₃₁	C₂×D₄	1	3	yes	—
174	C₄⋊C₄×C₃₁	C₂⁴	1	3	yes	—
175	C₄⋊C₄×C₃₁	C₂⁴	1	1	yes	characteristic
176	C₄⋊C₄×C₃₁	C₂⁴	1	2	yes	—
177	C₂²⋊C₄×C₃₁	—	2	3	no	—
178	C₂²⋊C₄×C₃₁	—	2	6	no	—
179	C₂²⋊C₄×C₃₁	—	2	3	no	—
180	C₂²⋊C₄×C₃₁	—	2	3	no	—
181	C₂²⋊C₄×C₃₁	—	2	6	no	—
182	C₂²⋊C₄×C₃₁	—	2	6	no	—
183	C₂²⋊C₄×C₃₁	C₂×D₄	1	6	yes	—
184	C₄²×C₃₁	—	2	1	no	—
185	C₄²×C₃₁	—	2	3	no	—
186	C₄²×C₃₁	—	2	3	no	—
187	C₄²×C₃₁	C₂×D₄	1	3	yes	—
188	C₄²×C₃₁	C₂×D₄	1	3	yes	—
189	C₄²×C₃₁	C₂⁴	1	3	yes	—
190	C₄²×C₃₁	C₂⁴	1	1	yes	characteristic
191	#62001	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
192	C₂³×C₃₁	—	2	3	no	—
193	C₂³×C₃₁	—	2	6	no	—
194	C₂³×C₃₁	—	4	12	no	—
195	C₂³×C₃₁	—	8	6	no	—
196	C₂³×C₃₁	—	8	6	no	—
197	C₂³×C₃₁	C₂²×D₄	1	3	yes	—
198	Q₈×C₃₁	—	2	1	no	—
199	Q₈×C₃₁	—	4	2	no	—
200	Q₈×C₃₁	—	8	3	no	—
201	Q₈×C₃₁	AGL(1,ℤ/8)	1	4	yes	complemented
202	Q₈×C₃₁	H(5,𝔽₂)	1	2	yes	complemented
203	D₄×C₃₁	—	2	6	no	—
204	D₄×C₃₁	—	2	3	no	—
205	D₄×C₃₁	—	2	6	no	—
206	D₄×C₃₁	—	2	3	no	—
207	D₄×C₃₁	—	2	6	no	—
208	D₄×C₃₁	—	4	6	no	—
209	D₄×C₃₁	—	4	6	no	—
210	D₄×C₃₁	—	4	12	no	—
211	D₄×C₃₁	—	4	12	no	—
212	D₄×C₃₁	—	4	6	no	—
213	D₄×C₃₁	—	4	6	no	—
214	D₄×C₃₁	—	8	6	no	—
215	D₄×C₃₁	—	8	6	no	—
216	D₄×C₃₁	—	8	3	no	—
217	D₄×C₃₁	—	8	6	no	—
218	C₂×C₄×C₃₁	—	2	1	no	—
219	C₂×C₄×C₃₁	—	2	6	no	—
220	C₂×C₄×C₃₁	—	2	6	no	—
221	C₂×C₄×C₃₁	—	2	3	no	—
222	C₂×C₄×C₃₁	—	2	3	no	—
223	C₂×C₄×C₃₁	—	2	1	no	—
224	C₂×C₄×C₃₁	—	2	3	no	—
225	C₂×C₄×C₃₁	—	2	2	no	—
226	C₂×C₄×C₃₁	—	4	6	no	—
227	C₂×C₄×C₃₁	—	4	12	no	—
228	C₂×C₄×C₃₁	—	4	6	no	—
229	C₂×C₄×C₃₁	—	4	6	no	—
230	C₂×C₄×C₃₁	C₂²×D₄	1	3	yes	—
231	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
232	C₂×C₄×C₃₁	C₂²×D₄	1	3	yes	—
233	C₂×C₄×C₃₁	C₂²×D₄	1	3	yes	—
234	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
235	C₂×C₄×C₃₁	C₂⁵	1	1	yes	characteristic
236	C₈×C₃₁	—	2	6	no	—
237	C₈×C₃₁	—	2	2	no	—
238	#3498	C₂×C₃₁	1	1	yes	characteristic, complemented
239	#3483	C₂×C₃₁	1	1	yes	characteristic, complemented
240	#3391	C₂×C₃₁	1	2	yes	complemented
241	#3362	C₂×C₃₁	1	2	yes	complemented
242	#3325	C₂×C₃₁	1	6	yes	complemented
243	#3274	C₂×C₃₁	1	6	yes	complemented
244	#3253	C₂×C₃₁	1	3	yes	complemented
245	#3205	C₂×C₃₁	1	3	yes	complemented
246	#3106	C₂×C₃₁	1	3	yes	complemented
247	#3038	C₂×C₃₁	1	3	yes	complemented
248	#2991	C₂×C₃₁	1	1	yes	characteristic, complemented
249	C₂²×C₃₁	—	2	3	no	—
250	C₂²×C₃₁	—	2	3	no	—
251	C₂²×C₃₁	—	2	3	no	—
252	C₂²×C₃₁	—	4	3	no	—
253	C₂²×C₃₁	—	4	6	no	—
254	C₂²×C₃₁	—	4	6	no	—
255	C₂²×C₃₁	—	4	3	no	—
256	C₂²×C₃₁	—	4	3	no	—
257	C₂²×C₃₁	—	4	6	no	—
258	C₂²×C₃₁	—	16	12	no	—
259	C₂²×C₃₁	C₂³×D₄	1	1	yes	characteristic
260	C₄×C₃₁	—	2	3	no	—
261	C₄×C₃₁	—	2	2	no	—
262	C₄×C₃₁	—	2	1	no	—
263	C₄×C₃₁	—	2	3	no	—
264	C₄×C₃₁	—	4	1	no	—
265	C₄×C₃₁	—	4	2	no	—
266	C₄×C₃₁	—	4	3	no	—
267	C₄×C₃₁	C₂×AGL(1,ℤ/8)	1	6	yes	—
268	C₄×C₃₁	#576	1	2	yes	—
269	C₄×C₃₁	C₂×H(5,𝔽₂)	1	1	yes	characteristic
270	C₄×C₃₁	#585	1	1	yes	characteristic
271	C₂×H(5,𝔽₂)	—	2	1	no	—
272	C₂×AGL(1,ℤ/8)	C₂²×C₃₁	1	6	yes	complemented
273	Q₈²	C₂²×C₃₁	1	1	yes	characteristic, complemented
274	#555	—	2	3	no	—
275	#555	C₂²×C₃₁	1	3	yes	complemented
276	#550	—	2	1	no	—
277	#550	—	2	1	no	—
278	#550	C₂²×C₃₁	1	2	yes	—
279	#550	C₂²×C₃₁	1	3	yes	complemented
280	#550	C₂²×C₃₁	1	1	yes	characteristic, complemented
281	D₄²	—	2	3	no	—
282	D₄²	C₂²×C₃₁	1	6	yes	—
283	#535	—	2	3	no	—
284	#532	—	2	3	no	—
285	#532	C₂²×C₃₁	1	3	yes	—
286	#532	C₂²×C₃₁	1	3	yes	complemented
287	C₂×#127	C₂²×C₃₁	1	3	yes	—
288	C₄×#41	—	2	1	no	—
289	C₄×#41	C₂²×C₃₁	1	1	yes	characteristic
290	C₄×#41	C₂²×C₃₁	1	3	yes	complemented
291	#493	C₂²×C₃₁	1	6	yes	complemented
292	#486	C₂²×C₃₁	1	6	yes	complemented
293	#482	C₂²×C₃₁	1	6	yes	complemented
294	#477	C₂²×C₃₁	1	1	yes	characteristic, complemented
295	#476	C₂²×C₃₁	1	3	yes	complemented
296	#476	C₂²×C₃₁	1	6	yes	complemented
297	#469	C₂²×C₃₁	1	12	yes	complemented
298	#460	C₂²×C₃₁	1	3	yes	complemented
299	#460	C₂²×C₃₁	1	6	yes	complemented
300	#459	C₂²×C₃₁	1	3	yes	complemented
301	#459	C₂²×C₃₁	1	6	yes	complemented
302	#459	C₂²×C₃₁	1	6	yes	complemented
303	#459	C₂²×C₃₁	1	6	yes	complemented
304	#449	C₂²×C₃₁	1	6	yes	complemented
305	#447	C₂²×C₃₁	1	12	yes	complemented
306	C₈×Q₈	C₂²×C₃₁	1	2	yes	complemented
307	C₄×SD₁₆	C₂²×C₃₁	1	6	yes	complemented
308	C₄×D₈	C₂²×C₃₁	1	6	yes	complemented
309	#436	C₂²×C₃₁	1	6	yes	complemented
310	#431	C₂²×C₃₁	1	3	yes	complemented
311	#423	C₂²×C₃₁	1	3	yes	complemented
312	#418	C₂²×C₃₁	1	6	yes	complemented
313	C₂×C₃₁	—	4	3	no	—
314	C₂×C₃₁	—	8	3	no	—
315	C₂×C₃₁	—	8	6	no	—
316	C₂×C₃₁	#3488	1	1	yes	characteristic
317	C₂×C₃₁	C₂²×AGL(1,ℤ/8)	1	1	yes	characteristic
318	C₂×C₃₁	C₂×#576	1	1	yes	characteristic
319	H(5,𝔽₂)	—	4	8	no	—
320	C₂×#41	—	2	1	no	—
321	C₂×#41	—	2	2	no	—
322	C₂×#41	—	2	3	no	—
323	C₂×#41	—	2	3	no	—
324	C₂×#41	D₄×C₃₁	1	1	yes	characteristic
325	C₂×#41	C₂³×C₃₁	1	3	yes	—
326	C₂²×D₄	—	2	6	no	—
327	C₂²×D₄	—	2	3	no	—
328	C₂²×D₄	—	2	3	no	—
329	C₂²×D₄	C₂³×C₃₁	1	6	yes	—
330	AGL(1,ℤ/8)	—	4	12	no	—
331	C₂×SD₁₆	C₂³×C₃₁	1	6	yes	—
332	C₂×D₈	C₂³×C₃₁	1	12	yes	—
333	C₂×D₈	C₂³×C₃₁	1	6	yes	—
334	C₂×M₁₆	C₂³×C₃₁	1	6	yes	—
335	C₄⋊Q₈	—	2	3	no	—
336	C₄⋊Q₈	C₂³×C₃₁	1	1	yes	characteristic
337	C₄⋊Q₈	C₂³×C₃₁	1	3	yes	—
338	#127	—	2	3	no	—
339	#127	—	2	3	no	—
340	#127	—	2	3	no	—
341	#127	C₂³×C₃₁	1	3	yes	—
342	#127	C₂³×C₃₁	1	6	yes	—
343	#127	C₂³×C₃₁	1	3	yes	—
344	#124	—	2	6	no	—
345	#124	—	2	3	no	—
346	#124	—	2	3	no	—
347	#121	—	2	3	no	—
348	#121	—	2	3	no	—
349	#121	—	2	3	no	—
350	#121	—	2	3	no	—
351	#121	—	2	6	no	—
352	#121	—	2	6	no	—
353	#121	—	2	6	no	—
354	#121	C₂³×C₃₁	1	6	yes	—
355	#121	C₂³×C₃₁	1	6	yes	—
356	#120	—	2	12	no	—
357	C₄×Q₈	—	2	1	no	—
358	C₄×Q₈	D₄×C₃₁	1	1	yes	characteristic
359	C₄×Q₈	C₂³×C₃₁	1	1	yes	characteristic
360	C₄×Q₈	C₂³×C₃₁	1	3	yes	—
361	C₄×Q₈	C₂³×C₃₁	1	1	yes	characteristic
362	C₄×D₄	—	2	3	no	—
363	C₄×D₄	—	2	3	no	—
364	C₄×D₄	—	2	3	no	—
365	C₄×D₄	—	2	3	no	—
366	C₄×D₄	—	2	3	no	—
367	C₄×D₄	D₄×C₃₁	1	3	yes	—
368	C₄×D₄	C₂³×C₃₁	1	3	yes	—
369	C₄×D₄	C₂³×C₃₁	1	3	yes	—
370	C₄×D₄	C₂³×C₃₁	1	6	yes	—
371	C₄×D₄	C₂³×C₃₁	1	3	yes	—
372	#117	—	2	3	no	—
373	#117	—	2	3	no	—
374	#117	D₄×C₃₁	1	3	yes	—
375	#117	C₂³×C₃₁	1	3	yes	—
376	C₂×C₄²	—	2	3	no	—
377	C₂×C₄²	C₂³×C₃₁	1	3	yes	—
378	#107	C₂³×C₃₁	1	2	yes	—
379	#106	C₂³×C₃₁	1	6	yes	—
380	C₄⋊C₈	C₂³×C₃₁	1	3	yes	—
381	C₄⋊C₈	C₂³×C₃₁	1	6	yes	—
382	C₄⋊C₈	C₂³×C₃₁	1	1	yes	characteristic
383	#103	C₂³×C₃₁	1	2	yes	—
384	#102	C₂³×C₃₁	1	6	yes	—
385	#102	C₂³×C₃₁	1	12	yes	—
386	#102	C₂³×C₃₁	1	6	yes	—
387	#102	C₂³×C₃₁	1	6	yes	—
388	C₂²⋊C₈	C₂³×C₃₁	1	6	yes	—
389	C₄×C₈	C₂³×C₃₁	1	6	yes	—
390	C₃₁	#62001	1	1	yes	characteristic, direct product factor, Sylow
391	C₂⁴	—	2	6	no	—
392	#41	—	2	4	no	—
393	#41	—	2	6	no	—
394	#41	—	4	3	no	—
395	#41	—	4	12	no	—
396	#41	—	4	4	no	—
397	#41	—	4	8	no	—
398	#41	—	8	6	no	—
399	C₂×Q₈	—	2	1	no	—
400	C₂×Q₈	—	2	3	no	—
401	C₂×Q₈	C₂×D₄×C₃₁	1	1	yes	characteristic
402	C₂×Q₈	C₂⁴×C₃₁	1	1	yes	characteristic
403	C₂×D₄	—	2	3	no	—
404	C₂×D₄	—	2	6	no	—
405	C₂×D₄	—	2	6	no	—
406	C₂×D₄	—	2	3	no	—
407	C₂×D₄	—	2	3	no	—
408	C₂×D₄	—	2	6	no	—
409	C₂×D₄	—	2	3	no	—
410	C₂×D₄	—	2	6	no	—
411	C₂×D₄	—	2	6	no	—
412	C₂×D₄	—	2	3	no	—
413	C₂×D₄	—	4	6	no	—
414	C₂×D₄	—	4	6	no	—
415	C₂×D₄	—	4	12	no	—
416	C₂×D₄	—	4	24	no	—
417	C₂×D₄	—	8	6	no	—
418	C₂×D₄	C₂×D₄×C₃₁	1	3	yes	—
419	C₂×D₄	C₂⁴×C₃₁	1	6	yes	—
420	C₂×D₄	C₂⁴×C₃₁	1	3	yes	—
421	C₂²×C₄	—	2	3	no	—
422	C₂²×C₄	—	2	3	no	—
423	C₂²×C₄	—	2	3	no	—
424	C₂²×C₄	—	2	6	no	—
425	C₂²×C₄	C₂×D₄×C₃₁	1	3	yes	—
426	C₂²×C₄	C₂×D₄×C₃₁	1	3	yes	—
427	C₂²×C₄	C₂⁴×C₃₁	1	3	yes	—
428	SD₁₆	—	2	12	no	—
429	D₈	—	4	6	no	—
430	D₈	—	4	12	no	—
431	M₁₆	—	4	6	no	—
432	C₂×C₈	C₂⁴×C₃₁	1	2	yes	—
433	C₂×C₈	C₂⁴×C₃₁	1	6	yes	—
434	C₄⋊C₄	—	2	1	no	—
435	C₄⋊C₄	—	2	3	no	—
436	C₄⋊C₄	—	2	3	no	—
437	C₄⋊C₄	—	2	1	no	—
438	C₄⋊C₄	C₂×D₄×C₃₁	1	3	yes	—
439	C₄⋊C₄	C₂×D₄×C₃₁	1	3	yes	—
440	C₄⋊C₄	C₂⁴×C₃₁	1	3	yes	—
441	C₄⋊C₄	C₂⁴×C₃₁	1	1	yes	characteristic
442	C₄⋊C₄	C₂⁴×C₃₁	1	2	yes	—
443	C₂²⋊C₄	—	2	3	no	—
444	C₂²⋊C₄	—	2	6	no	—
445	C₂²⋊C₄	—	2	3	no	—
446	C₂²⋊C₄	—	2	3	no	—
447	C₂²⋊C₄	—	2	6	no	—
448	C₂²⋊C₄	—	2	6	no	—
449	C₂²⋊C₄	C₂×D₄×C₃₁	1	6	yes	—
450	C₄²	—	2	1	no	—
451	C₄²	—	2	3	no	—
452	C₄²	—	2	3	no	—
453	C₄²	C₂×D₄×C₃₁	1	3	yes	—
454	C₄²	C₂×D₄×C₃₁	1	3	yes	—
455	C₄²	C₂⁴×C₃₁	1	3	yes	—
456	C₄²	C₂⁴×C₃₁	1	1	yes	characteristic
457	C₂³	—	2	3	no	—
458	C₂³	—	2	6	no	—
459	C₂³	—	4	12	no	—
460	C₂³	—	8	6	no	—
461	C₂³	—	8	6	no	—
462	C₂³	C₂²×D₄×C₃₁	1	3	yes	—
463	Q₈	—	2	1	no	—
464	Q₈	—	4	2	no	—
465	Q₈	—	8	3	no	—
466	Q₈	C₃₁×AGL(1,ℤ/8)	1	4	yes	complemented
467	Q₈	C₃₁×H(5,𝔽₂)	1	2	yes	complemented
468	D₄	—	2	6	no	—
469	D₄	—	2	3	no	—
470	D₄	—	2	6	no	—
471	D₄	—	2	3	no	—
472	D₄	—	2	6	no	—
473	D₄	—	4	6	no	—
474	D₄	—	4	6	no	—
475	D₄	—	4	12	no	—
476	D₄	—	4	12	no	—
477	D₄	—	4	6	no	—
478	D₄	—	4	6	no	—
479	D₄	—	8	6	no	—
480	D₄	—	8	6	no	—
481	D₄	—	8	3	no	—
482	D₄	—	8	6	no	—
483	C₂×C₄	—	2	1	no	—
484	C₂×C₄	—	2	6	no	—
485	C₂×C₄	—	2	6	no	—
486	C₂×C₄	—	2	3	no	—
487	C₂×C₄	—	2	3	no	—
488	C₂×C₄	—	2	1	no	—
489	C₂×C₄	—	2	3	no	—
490	C₂×C₄	—	2	2	no	—
491	C₂×C₄	—	4	6	no	—
492	C₂×C₄	—	4	12	no	—
493	C₂×C₄	—	4	6	no	—
494	C₂×C₄	—	4	6	no	—
495	C₂×C₄	C₂²×D₄×C₃₁	1	3	yes	—
496	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
497	C₂×C₄	C₂²×D₄×C₃₁	1	3	yes	—
498	C₂×C₄	C₂²×D₄×C₃₁	1	3	yes	—
499	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
500	C₂×C₄	C₂⁵×C₃₁	1	1	yes	characteristic
501	C₈	—	2	6	no	—
502	C₈	—	2	2	no	—
503	C₂²	—	2	3	no	—
504	C₂²	—	2	3	no	—
505	C₂²	—	2	3	no	—
506	C₂²	—	4	3	no	—
507	C₂²	—	4	6	no	—
508	C₂²	—	4	6	no	—
509	C₂²	—	4	3	no	—
510	C₂²	—	4	3	no	—
511	C₂²	—	4	6	no	—
512	C₂²	—	16	12	no	—
513	C₂²	C₂³×D₄×C₃₁	1	1	yes	characteristic
514	C₄	—	2	3	no	—
515	C₄	—	2	2	no	—
516	C₄	—	2	1	no	—
517	C₄	—	2	3	no	—
518	C₄	—	4	2	no	—
519	C₄	—	4	1	no	—
520	C₄	—	4	3	no	—
521	C₄	C₂×C₃₁×AGL(1,ℤ/8)	1	6	yes	—
522	C₄	C₃₁×#576	1	2	yes	—
523	C₄	C₂×C₃₁×H(5,𝔽₂)	1	1	yes	characteristic
524	C₄	C₃₁×#585	1	1	yes	characteristic
525	C₂	—	4	3	no	—
526	C₂	—	8	6	no	—
527	C₂	—	8	3	no	—
528	C₂	C₃₁×#3488	1	1	yes	characteristic
529	C₂	C₂²×C₃₁×AGL(1,ℤ/8)	1	1	yes	characteristic
530	C₂	C₂×C₃₁×#576	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×#3498	C₂	semidirect
2. C₃₁×#3483	C₂	semidirect
3. C₃₁×#3391	C₂	semidirect
4. C₃₁×#3362	C₂	semidirect
5. C₃₁×#3325	C₂	semidirect
6. C₃₁×#3274	C₂	semidirect
7. C₃₁×#3253	C₂	semidirect
8. C₃₁×#3205	C₂	semidirect
9. C₃₁×#3106	C₂	semidirect
10. C₃₁×#3038	C₂	semidirect
11. C₃₁×#2991	C₂	semidirect
13. C₂×C₃₁×AGL(1,ℤ/8)	C₂²	semidirect
14. Q₈²×C₃₁	C₂²	semidirect
16. C₃₁×#555	C₂²	semidirect
20. C₃₁×#550	C₂²	semidirect
21. C₃₁×#550	C₂²	semidirect
27. C₃₁×#532	C₂²	semidirect
31. C₄×#41×C₃₁	C₂²	semidirect
32. C₃₁×#493	C₂²	semidirect
33. C₃₁×#486	C₂²	semidirect
34. C₃₁×#482	C₂²	semidirect
35. C₃₁×#477	C₂²	semidirect
36. C₃₁×#476	C₂²	semidirect
37. C₃₁×#476	C₂²	semidirect
38. C₃₁×#469	C₂²	semidirect
39. C₃₁×#460	C₂²	semidirect
40. C₃₁×#460	C₂²	semidirect
41. C₃₁×#459	C₂²	semidirect
42. C₃₁×#459	C₂²	semidirect
43. C₃₁×#459	C₂²	semidirect
44. C₃₁×#459	C₂²	semidirect
45. C₃₁×#449	C₂²	semidirect
46. C₃₁×#447	C₂²	semidirect
47. C₈×Q₈×C₃₁	C₂²	semidirect
48. C₄×SD₁₆×C₃₁	C₂²	semidirect
49. C₄×D₈×C₃₁	C₂²	semidirect
50. C₃₁×#436	C₂²	semidirect
51. C₃₁×#431	C₂²	semidirect
52. C₃₁×#423	C₂²	semidirect
53. C₃₁×#418	C₂²	semidirect
191. #62001	C₃₁	direct (N × H)
201. Q₈×C₃₁	AGL(1,ℤ/8)	semidirect
202. Q₈×C₃₁	H(5,𝔽₂)	semidirect
238. #3498	C₂×C₃₁	semidirect
239. #3483	C₂×C₃₁	semidirect
240. #3391	C₂×C₃₁	semidirect
241. #3362	C₂×C₃₁	semidirect
242. #3325	C₂×C₃₁	semidirect
243. #3274	C₂×C₃₁	semidirect
244. #3253	C₂×C₃₁	semidirect
245. #3205	C₂×C₃₁	semidirect
246. #3106	C₂×C₃₁	semidirect
247. #3038	C₂×C₃₁	semidirect
248. #2991	C₂×C₃₁	semidirect
272. C₂×AGL(1,ℤ/8)	C₂²×C₃₁	semidirect
273. Q₈²	C₂²×C₃₁	semidirect
275. #555	C₂²×C₃₁	semidirect
279. #550	C₂²×C₃₁	semidirect
280. #550	C₂²×C₃₁	semidirect
286. #532	C₂²×C₃₁	semidirect
290. C₄×#41	C₂²×C₃₁	semidirect
291. #493	C₂²×C₃₁	semidirect
292. #486	C₂²×C₃₁	semidirect
293. #482	C₂²×C₃₁	semidirect
294. #477	C₂²×C₃₁	semidirect
295. #476	C₂²×C₃₁	semidirect
296. #476	C₂²×C₃₁	semidirect
297. #469	C₂²×C₃₁	semidirect
298. #460	C₂²×C₃₁	semidirect
299. #460	C₂²×C₃₁	semidirect
300. #459	C₂²×C₃₁	semidirect
301. #459	C₂²×C₃₁	semidirect
302. #459	C₂²×C₃₁	semidirect
303. #459	C₂²×C₃₁	semidirect
304. #449	C₂²×C₃₁	semidirect
305. #447	C₂²×C₃₁	semidirect
306. C₈×Q₈	C₂²×C₃₁	semidirect
307. C₄×SD₁₆	C₂²×C₃₁	semidirect
308. C₄×D₈	C₂²×C₃₁	semidirect
309. #436	C₂²×C₃₁	semidirect
310. #431	C₂²×C₃₁	semidirect
311. #423	C₂²×C₃₁	semidirect
312. #418	C₂²×C₃₁	semidirect
390. C₃₁	#62001	direct (N × H)
466. Q₈	C₃₁×AGL(1,ℤ/8)	semidirect
467. Q₈	C₃₁×H(5,𝔽₂)	semidirect

Maximal subgroups

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

subgroup	index
C₃₁×#3498	2
C₃₁×#3483	2
C₃₁×#3391	2
C₃₁×#3362	2
C₃₁×#3325	2
C₃₁×#3274	2
C₃₁×#3253	2
C₃₁×#3205	2
C₃₁×#3106	2
C₃₁×#3038	2
C₃₁×#2991	2
#62001	31

Subgroups — C31×#62001 #21518760

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#62001 #21518760