Subgroups — C₃₁×#61370 #21518129

Order 7936.

1818 conjugacy classes of subgroups.

The subgroup lattice has 524 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃₁×#3501	C₂	1	2	yes	maximal, complemented
2	C₃₁×#3389	C₂	1	4	yes	maximal, complemented
3	C₃₁×#3384	C₂	1	4	yes	maximal, complemented
4	C₃₁×#3380	C₂	1	2	yes	maximal, complemented
5	C₃₁×#3380	C₂	1	4	yes	maximal, complemented
6	Q₈×D₈×C₃₁	C₂	1	2	yes	maximal, complemented
7	C₃₁×#3164	C₂	1	2	yes	maximal, complemented
8	C₃₁×#3160	C₂	1	1	yes	maximal, characteristic, complemented
9	C₃₁×#3081	C₂	1	4	yes	maximal, complemented
10	C₃₁×#3077	C₂	1	2	yes	maximal, complemented
11	C₃₁×#2960	C₂	1	1	yes	maximal, characteristic, complemented
12	C₃₁×#2948	C₂	1	2	yes	maximal
13	C₂×C₄×D₈×C₃₁	C₂	1	1	yes	maximal, characteristic
14	C₂²×D₈×C₃₁	C₂²	1	1	yes	characteristic
15	C₃₁×#554	—	2	2	no	—
16	C₃₁×#554	—	2	4	no	—
17	C₃₁×#554	C₂²	1	4	yes	complemented
18	C₃₁×#554	C₂²	1	8	yes	complemented
19	D₄×Q₈×C₃₁	—	2	2	no	—
20	D₄×Q₈×C₃₁	C₂²	1	4	yes	complemented
21	C₃₁×#544	C₂²	1	4	yes	—
22	C₃₁×#543	—	2	4	no	—
23	C₃₁×#533	C₂²	1	2	yes	—
24	C₃₁×#533	C₂²	1	1	yes	characteristic
25	C₂×C₃₁×C₂²⋊Q₈	—	2	4	no	—
26	C₃₁×#518	C₂²	1	4	yes	—
27	C₂×C₄×D₄×C₃₁	C₂²	1	2	yes	—
28	C₃₁×#501	C₂²	1	8	yes	complemented
29	C₃₁×#500	C₂²	1	1	yes	characteristic, complemented
30	C₃₁×#500	C₂²	1	4	yes	complemented
31	C₃₁×#500	C₂²	1	1	yes	characteristic, complemented
32	C₃₁×#499	C₂²	1	2	yes	complemented
33	C₃₁×#499	C₂²	1	4	yes	complemented
34	C₃₁×#499	C₂²	1	2	yes	complemented
35	C₃₁×#498	C₂²	1	2	yes	complemented
36	C₃₁×#478	C₂²	1	8	yes	complemented
37	C₃₁×#478	C₂²	1	8	yes	complemented
38	C₃₁×#478	C₂²	1	8	yes	complemented
39	C₃₁×#476	C₂²	1	4	yes	complemented
40	C₃₁×#476	C₂²	1	8	yes	complemented
41	C₃₁×#474	C₂²	1	4	yes	complemented
42	C₃₁×#474	C₂²	1	8	yes	complemented
43	C₃₁×#446	C₂²	1	2	yes	complemented
44	C₃₁×#446	C₂²	1	2	yes	complemented
45	C₈×Q₈×C₃₁	C₂²	1	2	yes	complemented
46	C₈×Q₈×C₃₁	C₂²	1	2	yes	complemented
47	C₃₁×#442	C₂²	1	8	yes	—
48	C₄×D₈×C₃₁	C₂²	1	2	yes	—
49	C₄×D₈×C₃₁	C₂²	1	4	yes	—
50	C₄×D₈×C₃₁	C₂²	1	2	yes	—
51	C₄×D₈×C₃₁	C₂²	1	4	yes	—
52	C₄×D₈×C₃₁	C₂²	1	2	yes	—
53	C₄×D₈×C₃₁	C₂²	1	2	yes	—
54	C₃₁×#428	C₂²	1	4	yes	—
55	C₃₁×#427	C₂²	1	2	yes	—
56	C₃₁×#427	C₂²	1	2	yes	—
57	C₂×C₃₁×#107	C₂²	1	1	yes	characteristic
58	C₃₁×#424	C₂²	1	2	yes	—
59	C₃₁×#423	C₂²	1	1	yes	characteristic
60	C₃₁×#418	C₂²	1	4	yes	—
61	C₂×C₃₁×#102	C₂²	1	2	yes	—
62	C₃₁×#405	C₂²	1	2	yes	—
63	C₂×C₄×C₈×C₃₁	C₂²	1	1	yes	characteristic
64	C₂²×Q₈×C₃₁	—	2	4	no	—
65	C₂²×D₄×C₃₁	C₂³	1	2	yes	—
66	C₂³×C₄×C₃₁	—	2	2	no	—
67	C₂×D₈×C₃₁	Q₈	1	8	yes	complemented
68	C₂×D₈×C₃₁	C₂³	1	2	yes	—
69	C₂×D₈×C₃₁	C₂³	1	2	yes	—
70	C₂×M₁₆×C₃₁	C₂³	1	2	yes	—
71	C₂²×C₈×C₃₁	C₂³	1	1	yes	characteristic
72	C₂²×C₈×C₃₁	C₂³	1	1	yes	characteristic
73	C₃₁×C₄⋊Q₈	—	2	4	no	—
74	C₃₁×C₄⋊Q₈	D₄	1	1	yes	characteristic
75	C₃₁×C₄⋊Q₈	D₄	1	1	yes	characteristic
76	C₃₁×C₄⋊Q₈	C₂³	1	2	yes	—
77	C₃₁×C₄⋊Q₈	C₂³	1	8	yes	—
78	C₃₁×C₄⋊Q₈	C₂³	1	2	yes	—
79	C₃₁×#125	—	2	4	no	—
80	C₃₁×#125	D₄	1	2	yes	—
81	C₃₁×#125	C₂³	1	4	yes	—
82	C₃₁×#125	C₂³	1	8	yes	—
83	C₃₁×C₂²⋊Q₈	—	2	4	no	—
84	C₃₁×C₂²⋊Q₈	—	2	8	no	—
85	C₃₁×C₂²⋊Q₈	—	2	2	no	—
86	C₃₁×C₂²⋊Q₈	—	2	2	no	—
87	C₃₁×C₂²⋊Q₈	—	2	4	no	—
88	C₃₁×C₂²⋊Q₈	—	2	8	no	—
89	C₃₁×C₂²⋊Q₈	—	2	8	no	—
90	C₃₁×C₂²⋊Q₈	—	2	8	no	—
91	C₃₁×C₂²⋊Q₈	—	2	8	no	—
92	C₃₁×C₂²⋊Q₈	—	2	8	no	—
93	C₃₁×C₂²⋊Q₈	D₄	1	2	yes	—
94	C₃₁×C₂²⋊Q₈	D₄	1	2	yes	—
95	C₄×Q₈×C₃₁	—	2	2	no	—
96	C₄×Q₈×C₃₁	—	2	2	no	—
97	C₄×Q₈×C₃₁	C₂³	1	2	yes	—
98	C₄×Q₈×C₃₁	C₂³	1	2	yes	—
99	C₄×D₄×C₃₁	—	2	4	no	—
100	C₄×D₄×C₃₁	—	2	4	no	—
101	C₄×D₄×C₃₁	—	2	2	no	—
102	C₄×D₄×C₃₁	—	2	2	no	—
103	C₄×D₄×C₃₁	C₂³	1	4	yes	—
104	C₄×D₄×C₃₁	C₂³	1	4	yes	—
105	C₄×D₄×C₃₁	C₂³	1	8	yes	—
106	C₄×D₄×C₃₁	C₂³	1	8	yes	—
107	C₃₁×#117	C₂³	1	4	yes	—
108	C₃₁×#117	C₂³	1	2	yes	—
109	C₂×C₄⋊C₄×C₃₁	—	2	8	no	—
110	C₂×C₄⋊C₄×C₃₁	—	2	4	no	—
111	C₂×C₄⋊C₄×C₃₁	C₂³	1	2	yes	—
112	C₂×C₂²⋊C₄×C₃₁	—	2	2	no	—
113	C₂×C₂²⋊C₄×C₃₁	—	2	8	no	—
114	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
115	C₃₁×#107	C₂³	1	2	yes	—
116	C₃₁×#107	C₂³	1	2	yes	—
117	C₃₁×#107	C₂³	1	2	yes	—
118	C₃₁×#107	C₂³	1	2	yes	—
119	C₃₁×#107	C₂³	1	2	yes	—
120	C₃₁×#107	C₂³	1	2	yes	—
121	C₃₁×#107	C₂³	1	8	yes	—
122	C₃₁×#106	C₂³	1	4	yes	—
123	C₃₁×#106	C₂³	1	4	yes	—
124	C₃₁×#106	C₂³	1	8	yes	—
125	C₃₁×C₄⋊C₈	C₂³	1	2	yes	—
126	C₃₁×C₄⋊C₈	C₂³	1	2	yes	—
127	C₃₁×C₄⋊C₈	C₂³	1	4	yes	—
128	C₃₁×C₄⋊C₈	C₂³	1	4	yes	—
129	C₃₁×#102	C₂³	1	4	yes	—
130	C₃₁×#102	C₂³	1	4	yes	—
131	C₃₁×#102	C₂³	1	8	yes	—
132	C₃₁×#102	C₂³	1	8	yes	—
133	C₃₁×#97	C₂³	1	4	yes	—
134	C₄×C₈×C₃₁	C₂³	1	1	yes	characteristic
135	C₄×C₈×C₃₁	C₂³	1	1	yes	characteristic
136	C₄×C₈×C₃₁	C₂³	1	1	yes	characteristic
137	C₄×C₈×C₃₁	C₂³	1	1	yes	characteristic
138	C₄×C₈×C₃₁	C₂³	1	4	yes	—
139	C₂⁴×C₃₁	—	2	2	no	—
140	C₂×Q₈×C₃₁	—	2	2	no	—
141	C₂×Q₈×C₃₁	—	2	2	no	—
142	C₂×Q₈×C₃₁	—	2	8	no	—
143	C₂×Q₈×C₃₁	—	8	8	no	—
144	C₂×Q₈×C₃₁	C₂×D₄	1	2	yes	—
145	C₂×Q₈×C₃₁	C₂×D₄	1	2	yes	—
146	C₂×D₄×C₃₁	—	2	2	no	—
147	C₂×D₄×C₃₁	—	2	4	no	—
148	C₂×D₄×C₃₁	C₂×Q₈	1	8	yes	complemented
149	C₂×D₄×C₃₁	C₂⁴	1	4	yes	—
150	C₂²×C₄×C₃₁	—	2	2	no	—
151	C₂²×C₄×C₃₁	—	2	4	no	—
152	C₂²×C₄×C₃₁	—	2	8	no	—
153	C₂²×C₄×C₃₁	—	2	8	no	—
154	C₂²×C₄×C₃₁	—	2	4	no	—
155	C₂²×C₄×C₃₁	—	8	4	no	—
156	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
157	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
158	C₂²×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
159	D₈×C₃₁	C₂×Q₈	1	8	yes	complemented
160	D₈×C₃₁	C₂×Q₈	1	8	yes	complemented
161	M₁₆×C₃₁	—	2	4	no	—
162	C₂×C₈×C₃₁	—	2	2	no	—
163	C₂×C₈×C₃₁	C₂×Q₈	1	4	yes	complemented
164	C₂×C₈×C₃₁	C₂⁴	1	1	yes	characteristic
165	C₂×C₈×C₃₁	C₂⁴	1	1	yes	characteristic
166	C₂×C₈×C₃₁	C₂⁴	1	4	yes	—
167	C₂×C₈×C₃₁	C₂⁴	1	1	yes	characteristic
168	C₂×C₈×C₃₁	C₂⁴	1	1	yes	characteristic
169	C₄⋊C₄×C₃₁	—	2	4	no	—
170	C₄⋊C₄×C₃₁	—	2	2	no	—
171	C₄⋊C₄×C₃₁	—	2	8	no	—
172	C₄⋊C₄×C₃₁	—	2	8	no	—
173	C₄⋊C₄×C₃₁	—	2	8	no	—
174	C₄⋊C₄×C₃₁	—	2	2	no	—
175	C₄⋊C₄×C₃₁	—	2	4	no	—
176	C₄⋊C₄×C₃₁	—	2	2	no	—
177	C₄⋊C₄×C₃₁	C₂×D₄	1	4	yes	—
178	C₄⋊C₄×C₃₁	C₂×D₄	1	2	yes	—
179	C₄⋊C₄×C₃₁	C₂×D₄	1	2	yes	—
180	C₄⋊C₄×C₃₁	C₂×D₄	1	2	yes	—
181	C₄⋊C₄×C₃₁	C₂×D₄	1	4	yes	—
182	C₄⋊C₄×C₃₁	C₂×D₄	1	2	yes	—
183	C₄⋊C₄×C₃₁	C₂⁴	1	4	yes	—
184	C₄⋊C₄×C₃₁	C₂⁴	1	4	yes	—
185	C₄⋊C₄×C₃₁	C₂⁴	1	4	yes	—
186	C₂²⋊C₄×C₃₁	—	2	4	no	—
187	C₂²⋊C₄×C₃₁	—	2	8	no	—
188	C₂²⋊C₄×C₃₁	—	2	8	no	—
189	C₂²⋊C₄×C₃₁	—	2	4	no	—
190	C₂²⋊C₄×C₃₁	—	2	4	no	—
191	C₂²⋊C₄×C₃₁	C₂×D₄	1	4	yes	—
192	C₄²×C₃₁	—	2	4	no	—
193	C₄²×C₃₁	C₂×D₄	1	1	yes	characteristic
194	C₄²×C₃₁	C₂×D₄	1	1	yes	characteristic
195	C₄²×C₃₁	C₂⁴	1	1	yes	characteristic
196	C₄²×C₃₁	C₂⁴	1	1	yes	characteristic
197	C₄²×C₃₁	C₂⁴	1	4	yes	—
198	#61370	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
199	C₂³×C₃₁	—	2	4	no	—
200	C₂³×C₃₁	—	2	8	no	—
201	C₂³×C₃₁	—	8	2	no	—
202	C₂³×C₃₁	—	8	2	no	—
203	C₂³×C₃₁	C₂²×D₄	1	1	yes	characteristic
204	Q₈×C₃₁	—	2	4	no	—
205	Q₈×C₃₁	—	4	2	no	—
206	Q₈×C₃₁	—	4	4	no	—
207	Q₈×C₃₁	—	8	2	no	—
208	Q₈×C₃₁	—	8	8	no	—
209	D₄×C₃₁	—	2	8	no	—
210	D₄×C₃₁	—	2	4	no	—
211	D₄×C₃₁	C₂²×Q₈	1	8	yes	—
212	C₂×C₄×C₃₁	—	2	2	no	—
213	C₂×C₄×C₃₁	—	2	4	no	—
214	C₂×C₄×C₃₁	—	2	4	no	—
215	C₂×C₄×C₃₁	—	2	4	no	—
216	C₂×C₄×C₃₁	—	4	1	no	—
217	C₂×C₄×C₃₁	—	4	8	no	—
218	C₂×C₄×C₃₁	—	8	8	no	—
219	C₂×C₄×C₃₁	—	8	4	no	—
220	C₂×C₄×C₃₁	—	8	8	no	—
221	C₂×C₄×C₃₁	—	8	2	no	—
222	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
223	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
224	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
225	C₂×C₄×C₃₁	C₂²×D₄	1	4	yes	—
226	C₂×C₄×C₃₁	C₂²×D₄	1	4	yes	—
227	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
228	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
229	C₂×C₄×C₃₁	C₂²×Q₈	1	2	yes	—
230	C₂×C₄×C₃₁	2₋¹⁺⁴	1	2	yes	—
231	C₂×C₄×C₃₁	C₂⁵	1	1	yes	characteristic
232	C₈×C₃₁	—	2	1	no	—
233	C₈×C₃₁	—	2	1	no	—
234	C₈×C₃₁	—	2	4	no	—
235	C₈×C₃₁	C₂²×Q₈	1	2	yes	—
236	C₈×C₃₁	C₂²×Q₈	1	2	yes	—
237	#3501	C₂×C₃₁	1	2	yes	complemented
238	#3389	C₂×C₃₁	1	4	yes	complemented
239	#3384	C₂×C₃₁	1	4	yes	complemented
240	#3380	C₂×C₃₁	1	2	yes	complemented
241	#3380	C₂×C₃₁	1	4	yes	complemented
242	Q₈×D₈	C₂×C₃₁	1	2	yes	complemented
243	#3164	C₂×C₃₁	1	2	yes	complemented
244	#3160	C₂×C₃₁	1	1	yes	characteristic, complemented
245	#3081	C₂×C₃₁	1	4	yes	complemented
246	#3077	C₂×C₃₁	1	2	yes	complemented
247	#2960	C₂×C₃₁	1	1	yes	characteristic, complemented
248	#2948	C₂×C₃₁	1	2	yes	—
249	C₂×C₄×D₈	C₂×C₃₁	1	1	yes	characteristic
250	C₂²×C₃₁	—	2	1	no	—
251	C₂²×C₃₁	—	2	1	no	—
252	C₂²×C₃₁	—	2	8	no	—
253	C₂²×C₃₁	—	4	4	no	—
254	C₂²×C₃₁	—	4	4	no	—
255	C₂²×C₃₁	—	8	8	no	—
256	C₂²×C₃₁	D₄×Q₈	1	2	yes	—
257	C₂²×C₃₁	C₂³×D₄	1	1	yes	characteristic
258	C₄×C₃₁	—	2	1	no	—
259	C₄×C₃₁	—	2	1	no	—
260	C₄×C₃₁	—	2	4	no	—
261	C₄×C₃₁	—	2	1	no	—
262	C₄×C₃₁	—	2	4	no	—
263	C₄×C₃₁	—	4	4	no	—
264	C₄×C₃₁	—	4	4	no	—
265	C₄×C₃₁	—	4	4	no	—
266	C₄×C₃₁	D₄×Q₈	1	2	yes	—
267	C₄×C₃₁	#576	1	2	yes	—
268	C₄×C₃₁	C₂³×Q₈	1	1	yes	characteristic
269	C₄×C₃₁	C₂×2₋¹⁺⁴	1	1	yes	characteristic
270	C₂²×D₈	C₂²×C₃₁	1	1	yes	characteristic
271	#554	—	2	2	no	—
272	#554	—	2	4	no	—
273	#554	C₂²×C₃₁	1	4	yes	complemented
274	#554	C₂²×C₃₁	1	8	yes	complemented
275	D₄×Q₈	—	2	2	no	—
276	D₄×Q₈	C₂²×C₃₁	1	4	yes	complemented
277	#544	C₂²×C₃₁	1	4	yes	—
278	#543	—	2	4	no	—
279	#533	C₂²×C₃₁	1	2	yes	—
280	#533	C₂²×C₃₁	1	1	yes	characteristic
281	C₂×C₂²⋊Q₈	—	2	4	no	—
282	#518	C₂²×C₃₁	1	4	yes	—
283	C₂×C₄×D₄	C₂²×C₃₁	1	2	yes	—
284	#501	C₂²×C₃₁	1	8	yes	complemented
285	#500	C₂²×C₃₁	1	1	yes	characteristic, complemented
286	#500	C₂²×C₃₁	1	4	yes	complemented
287	#500	C₂²×C₃₁	1	1	yes	characteristic, complemented
288	#499	C₂²×C₃₁	1	2	yes	complemented
289	#499	C₂²×C₃₁	1	4	yes	complemented
290	#499	C₂²×C₃₁	1	2	yes	complemented
291	#498	C₂²×C₃₁	1	2	yes	complemented
292	#478	C₂²×C₃₁	1	8	yes	complemented
293	#478	C₂²×C₃₁	1	8	yes	complemented
294	#478	C₂²×C₃₁	1	8	yes	complemented
295	#476	C₂²×C₃₁	1	4	yes	complemented
296	#476	C₂²×C₃₁	1	8	yes	complemented
297	#474	C₂²×C₃₁	1	4	yes	complemented
298	#474	C₂²×C₃₁	1	8	yes	complemented
299	#446	C₂²×C₃₁	1	2	yes	complemented
300	#446	C₂²×C₃₁	1	2	yes	complemented
301	C₈×Q₈	C₂²×C₃₁	1	2	yes	complemented
302	C₈×Q₈	C₂²×C₃₁	1	2	yes	complemented
303	#442	C₂²×C₃₁	1	8	yes	—
304	C₄×D₈	C₂²×C₃₁	1	2	yes	—
305	C₄×D₈	C₂²×C₃₁	1	4	yes	—
306	C₄×D₈	C₂²×C₃₁	1	2	yes	—
307	C₄×D₈	C₂²×C₃₁	1	4	yes	—
308	C₄×D₈	C₂²×C₃₁	1	2	yes	—
309	C₄×D₈	C₂²×C₃₁	1	2	yes	—
310	#428	C₂²×C₃₁	1	4	yes	—
311	#427	C₂²×C₃₁	1	2	yes	—
312	#427	C₂²×C₃₁	1	2	yes	—
313	C₂×#107	C₂²×C₃₁	1	1	yes	characteristic
314	#424	C₂²×C₃₁	1	2	yes	—
315	#423	C₂²×C₃₁	1	1	yes	characteristic
316	#418	C₂²×C₃₁	1	4	yes	—
317	C₂×#102	C₂²×C₃₁	1	2	yes	—
318	#405	C₂²×C₃₁	1	2	yes	—
319	C₂×C₄×C₈	C₂²×C₃₁	1	1	yes	characteristic
320	C₂×C₃₁	—	2	2	no	—
321	C₂×C₃₁	—	4	8	no	—
322	C₂×C₃₁	C₂×D₄×Q₈	1	1	yes	characteristic
323	C₂×C₃₁	C₂×#576	1	1	yes	characteristic
324	C₂×C₃₁	#3584	1	1	yes	characteristic
325	C₂²×Q₈	—	2	4	no	—
326	C₂²×D₄	C₂³×C₃₁	1	2	yes	—
327	C₂³×C₄	—	2	2	no	—
328	C₂×D₈	Q₈×C₃₁	1	8	yes	complemented
329	C₂×D₈	C₂³×C₃₁	1	2	yes	—
330	C₂×D₈	C₂³×C₃₁	1	2	yes	—
331	C₂×M₁₆	C₂³×C₃₁	1	2	yes	—
332	C₂²×C₈	C₂³×C₃₁	1	1	yes	characteristic
333	C₂²×C₈	C₂³×C₃₁	1	1	yes	characteristic
334	C₄⋊Q₈	—	2	4	no	—
335	C₄⋊Q₈	D₄×C₃₁	1	1	yes	characteristic
336	C₄⋊Q₈	D₄×C₃₁	1	1	yes	characteristic
337	C₄⋊Q₈	C₂³×C₃₁	1	2	yes	—
338	C₄⋊Q₈	C₂³×C₃₁	1	2	yes	—
339	C₄⋊Q₈	C₂³×C₃₁	1	8	yes	—
340	#125	—	2	4	no	—
341	#125	D₄×C₃₁	1	2	yes	—
342	#125	C₂³×C₃₁	1	4	yes	—
343	#125	C₂³×C₃₁	1	8	yes	—
344	C₂²⋊Q₈	—	2	4	no	—
345	C₂²⋊Q₈	—	2	8	no	—
346	C₂²⋊Q₈	—	2	2	no	—
347	C₂²⋊Q₈	—	2	2	no	—
348	C₂²⋊Q₈	—	2	4	no	—
349	C₂²⋊Q₈	—	2	8	no	—
350	C₂²⋊Q₈	—	2	8	no	—
351	C₂²⋊Q₈	—	2	8	no	—
352	C₂²⋊Q₈	—	2	8	no	—
353	C₂²⋊Q₈	—	2	8	no	—
354	C₂²⋊Q₈	D₄×C₃₁	1	2	yes	—
355	C₂²⋊Q₈	D₄×C₃₁	1	2	yes	—
356	C₄×Q₈	—	2	2	no	—
357	C₄×Q₈	—	2	2	no	—
358	C₄×Q₈	C₂³×C₃₁	1	2	yes	—
359	C₄×Q₈	C₂³×C₃₁	1	2	yes	—
360	C₄×D₄	—	2	4	no	—
361	C₄×D₄	—	2	4	no	—
362	C₄×D₄	—	2	2	no	—
363	C₄×D₄	—	2	2	no	—
364	C₄×D₄	C₂³×C₃₁	1	4	yes	—
365	C₄×D₄	C₂³×C₃₁	1	4	yes	—
366	C₄×D₄	C₂³×C₃₁	1	8	yes	—
367	C₄×D₄	C₂³×C₃₁	1	8	yes	—
368	#117	C₂³×C₃₁	1	4	yes	—
369	#117	C₂³×C₃₁	1	2	yes	—
370	C₂×C₄⋊C₄	—	2	8	no	—
371	C₂×C₄⋊C₄	—	2	4	no	—
372	C₂×C₄⋊C₄	C₂³×C₃₁	1	2	yes	—
373	C₂×C₂²⋊C₄	—	2	2	no	—
374	C₂×C₂²⋊C₄	—	2	8	no	—
375	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
376	#107	C₂³×C₃₁	1	2	yes	—
377	#107	C₂³×C₃₁	1	2	yes	—
378	#107	C₂³×C₃₁	1	2	yes	—
379	#107	C₂³×C₃₁	1	2	yes	—
380	#107	C₂³×C₃₁	1	2	yes	—
381	#107	C₂³×C₃₁	1	2	yes	—
382	#107	C₂³×C₃₁	1	8	yes	—
383	#106	C₂³×C₃₁	1	4	yes	—
384	#106	C₂³×C₃₁	1	4	yes	—
385	#106	C₂³×C₃₁	1	8	yes	—
386	C₄⋊C₈	C₂³×C₃₁	1	2	yes	—
387	C₄⋊C₈	C₂³×C₃₁	1	2	yes	—
388	C₄⋊C₈	C₂³×C₃₁	1	4	yes	—
389	C₄⋊C₈	C₂³×C₃₁	1	4	yes	—
390	#102	C₂³×C₃₁	1	4	yes	—
391	#102	C₂³×C₃₁	1	4	yes	—
392	#102	C₂³×C₃₁	1	8	yes	—
393	#102	C₂³×C₃₁	1	8	yes	—
394	#97	C₂³×C₃₁	1	4	yes	—
395	C₄×C₈	C₂³×C₃₁	1	1	yes	characteristic
396	C₄×C₈	C₂³×C₃₁	1	1	yes	characteristic
397	C₄×C₈	C₂³×C₃₁	1	1	yes	characteristic
398	C₄×C₈	C₂³×C₃₁	1	1	yes	characteristic
399	C₄×C₈	C₂³×C₃₁	1	4	yes	—
400	C₃₁	#61370	1	1	yes	characteristic, direct product factor, Sylow
401	C₂⁴	—	2	2	no	—
402	C₂×Q₈	—	2	2	no	—
403	C₂×Q₈	—	2	2	no	—
404	C₂×Q₈	—	2	8	no	—
405	C₂×Q₈	—	8	8	no	—
406	C₂×Q₈	C₂×D₄×C₃₁	1	2	yes	—
407	C₂×Q₈	C₂×D₄×C₃₁	1	2	yes	—
408	C₂×D₄	—	2	2	no	—
409	C₂×D₄	—	2	4	no	—
410	C₂×D₄	C₂×Q₈×C₃₁	1	8	yes	complemented
411	C₂×D₄	C₂⁴×C₃₁	1	4	yes	—
412	C₂²×C₄	—	2	2	no	—
413	C₂²×C₄	—	2	4	no	—
414	C₂²×C₄	—	2	8	no	—
415	C₂²×C₄	—	2	8	no	—
416	C₂²×C₄	—	2	4	no	—
417	C₂²×C₄	—	8	4	no	—
418	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
419	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
420	C₂²×C₄	C₂⁴×C₃₁	1	1	yes	characteristic
421	D₈	C₂×Q₈×C₃₁	1	8	yes	complemented
422	D₈	C₂×Q₈×C₃₁	1	8	yes	complemented
423	M₁₆	—	2	4	no	—
424	C₂×C₈	—	2	2	no	—
425	C₂×C₈	C₂×Q₈×C₃₁	1	4	yes	complemented
426	C₂×C₈	C₂⁴×C₃₁	1	1	yes	characteristic
427	C₂×C₈	C₂⁴×C₃₁	1	1	yes	characteristic
428	C₂×C₈	C₂⁴×C₃₁	1	4	yes	—
429	C₂×C₈	C₂⁴×C₃₁	1	1	yes	characteristic
430	C₂×C₈	C₂⁴×C₃₁	1	1	yes	characteristic
431	C₄⋊C₄	—	2	4	no	—
432	C₄⋊C₄	—	2	2	no	—
433	C₄⋊C₄	—	2	8	no	—
434	C₄⋊C₄	—	2	8	no	—
435	C₄⋊C₄	—	2	8	no	—
436	C₄⋊C₄	—	2	2	no	—
437	C₄⋊C₄	—	2	4	no	—
438	C₄⋊C₄	—	2	2	no	—
439	C₄⋊C₄	C₂×D₄×C₃₁	1	4	yes	—
440	C₄⋊C₄	C₂×D₄×C₃₁	1	2	yes	—
441	C₄⋊C₄	C₂×D₄×C₃₁	1	2	yes	—
442	C₄⋊C₄	C₂×D₄×C₃₁	1	2	yes	—
443	C₄⋊C₄	C₂×D₄×C₃₁	1	4	yes	—
444	C₄⋊C₄	C₂×D₄×C₃₁	1	2	yes	—
445	C₄⋊C₄	C₂⁴×C₃₁	1	4	yes	—
446	C₄⋊C₄	C₂⁴×C₃₁	1	4	yes	—
447	C₄⋊C₄	C₂⁴×C₃₁	1	4	yes	—
448	C₂²⋊C₄	—	2	4	no	—
449	C₂²⋊C₄	—	2	8	no	—
450	C₂²⋊C₄	—	2	8	no	—
451	C₂²⋊C₄	—	2	4	no	—
452	C₂²⋊C₄	—	2	4	no	—
453	C₂²⋊C₄	C₂×D₄×C₃₁	1	4	yes	—
454	C₄²	—	2	4	no	—
455	C₄²	C₂×D₄×C₃₁	1	1	yes	characteristic
456	C₄²	C₂×D₄×C₃₁	1	1	yes	characteristic
457	C₄²	C₂⁴×C₃₁	1	1	yes	characteristic
458	C₄²	C₂⁴×C₃₁	1	1	yes	characteristic
459	C₄²	C₂⁴×C₃₁	1	4	yes	—
460	C₂³	—	2	4	no	—
461	C₂³	—	2	8	no	—
462	C₂³	—	8	2	no	—
463	C₂³	—	8	2	no	—
464	C₂³	C₂²×D₄×C₃₁	1	1	yes	characteristic
465	Q₈	—	2	4	no	—
466	Q₈	—	4	2	no	—
467	Q₈	—	4	4	no	—
468	Q₈	—	8	2	no	—
469	Q₈	—	8	8	no	—
470	D₄	—	2	8	no	—
471	D₄	—	2	4	no	—
472	D₄	C₂²×Q₈×C₃₁	1	8	yes	—
473	C₂×C₄	—	2	2	no	—
474	C₂×C₄	—	2	4	no	—
475	C₂×C₄	—	2	4	no	—
476	C₂×C₄	—	2	4	no	—
477	C₂×C₄	—	4	1	no	—
478	C₂×C₄	—	4	8	no	—
479	C₂×C₄	—	8	8	no	—
480	C₂×C₄	—	8	4	no	—
481	C₂×C₄	—	8	8	no	—
482	C₂×C₄	—	8	2	no	—
483	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
484	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
485	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
486	C₂×C₄	C₂²×D₄×C₃₁	1	4	yes	—
487	C₂×C₄	C₂²×D₄×C₃₁	1	4	yes	—
488	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
489	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
490	C₂×C₄	C₂²×Q₈×C₃₁	1	2	yes	—
491	C₂×C₄	C₃₁×2₋¹⁺⁴	1	2	yes	—
492	C₂×C₄	C₂⁵×C₃₁	1	1	yes	characteristic
493	C₈	—	2	1	no	—
494	C₈	—	2	1	no	—
495	C₈	—	2	4	no	—
496	C₈	C₂²×Q₈×C₃₁	1	2	yes	—
497	C₈	C₂²×Q₈×C₃₁	1	2	yes	—
498	C₂²	—	2	1	no	—
499	C₂²	—	2	1	no	—
500	C₂²	—	2	8	no	—
501	C₂²	—	4	4	no	—
502	C₂²	—	4	4	no	—
503	C₂²	—	8	8	no	—
504	C₂²	D₄×Q₈×C₃₁	1	2	yes	—
505	C₂²	C₂³×D₄×C₃₁	1	1	yes	characteristic
506	C₄	—	2	1	no	—
507	C₄	—	2	4	no	—
508	C₄	—	2	1	no	—
509	C₄	—	2	1	no	—
510	C₄	—	2	4	no	—
511	C₄	—	4	4	no	—
512	C₄	—	4	4	no	—
513	C₄	—	4	4	no	—
514	C₄	D₄×Q₈×C₃₁	1	2	yes	—
515	C₄	C₃₁×#576	1	2	yes	—
516	C₄	C₂³×Q₈×C₃₁	1	1	yes	characteristic
517	C₄	C₂×C₃₁×2₋¹⁺⁴	1	1	yes	characteristic
518	C₂	—	2	2	no	—
519	C₂	—	4	8	no	—
520	C₂	C₂×D₄×Q₈×C₃₁	1	1	yes	characteristic
521	C₂	C₂×C₃₁×#576	1	1	yes	characteristic
522	C₂	C₃₁×#3584	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₃₁	C₂³×D₄
commutator, Frattini, cosocle	C₂×C₄	C₂⁵×C₃₁
prime radical	C₂²×D₈×C₃₁	C₂²
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₃₁ ◁ C₃₁×#533 ◁ G

Complements

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

N	H ≅ G/N	type
1. C₃₁×#3501	C₂	semidirect
2. C₃₁×#3389	C₂	semidirect
3. C₃₁×#3384	C₂	semidirect
4. C₃₁×#3380	C₂	semidirect
5. C₃₁×#3380	C₂	semidirect
6. Q₈×D₈×C₃₁	C₂	semidirect
7. C₃₁×#3164	C₂	semidirect
8. C₃₁×#3160	C₂	semidirect
9. C₃₁×#3081	C₂	semidirect
10. C₃₁×#3077	C₂	semidirect
11. C₃₁×#2960	C₂	semidirect
17. C₃₁×#554	C₂²	semidirect
18. C₃₁×#554	C₂²	semidirect
20. D₄×Q₈×C₃₁	C₂²	semidirect
28. C₃₁×#501	C₂²	semidirect
29. C₃₁×#500	C₂²	semidirect
30. C₃₁×#500	C₂²	semidirect
31. C₃₁×#500	C₂²	semidirect
32. C₃₁×#499	C₂²	semidirect
33. C₃₁×#499	C₂²	semidirect
34. C₃₁×#499	C₂²	semidirect
35. C₃₁×#498	C₂²	semidirect
36. C₃₁×#478	C₂²	semidirect
37. C₃₁×#478	C₂²	semidirect
38. C₃₁×#478	C₂²	semidirect
39. C₃₁×#476	C₂²	semidirect
40. C₃₁×#476	C₂²	semidirect
41. C₃₁×#474	C₂²	semidirect
42. C₃₁×#474	C₂²	semidirect
43. C₃₁×#446	C₂²	semidirect
44. C₃₁×#446	C₂²	semidirect
45. C₈×Q₈×C₃₁	C₂²	semidirect
46. C₈×Q₈×C₃₁	C₂²	semidirect
67. C₂×D₈×C₃₁	Q₈	semidirect
148. C₂×D₄×C₃₁	C₂×Q₈	semidirect
159. D₈×C₃₁	C₂×Q₈	semidirect
160. D₈×C₃₁	C₂×Q₈	semidirect
163. C₂×C₈×C₃₁	C₂×Q₈	semidirect
198. #61370	C₃₁	direct (N × H)
237. #3501	C₂×C₃₁	semidirect
238. #3389	C₂×C₃₁	semidirect
239. #3384	C₂×C₃₁	semidirect
240. #3380	C₂×C₃₁	semidirect
241. #3380	C₂×C₃₁	semidirect
242. Q₈×D₈	C₂×C₃₁	semidirect
243. #3164	C₂×C₃₁	semidirect
244. #3160	C₂×C₃₁	semidirect
245. #3081	C₂×C₃₁	semidirect
246. #3077	C₂×C₃₁	semidirect
247. #2960	C₂×C₃₁	semidirect
273. #554	C₂²×C₃₁	semidirect
274. #554	C₂²×C₃₁	semidirect
276. D₄×Q₈	C₂²×C₃₁	semidirect
284. #501	C₂²×C₃₁	semidirect
285. #500	C₂²×C₃₁	semidirect
286. #500	C₂²×C₃₁	semidirect
287. #500	C₂²×C₃₁	semidirect
288. #499	C₂²×C₃₁	semidirect
289. #499	C₂²×C₃₁	semidirect
290. #499	C₂²×C₃₁	semidirect
291. #498	C₂²×C₃₁	semidirect
292. #478	C₂²×C₃₁	semidirect
293. #478	C₂²×C₃₁	semidirect
294. #478	C₂²×C₃₁	semidirect
295. #476	C₂²×C₃₁	semidirect
296. #476	C₂²×C₃₁	semidirect
297. #474	C₂²×C₃₁	semidirect
298. #474	C₂²×C₃₁	semidirect
299. #446	C₂²×C₃₁	semidirect
300. #446	C₂²×C₃₁	semidirect
301. C₈×Q₈	C₂²×C₃₁	semidirect
302. C₈×Q₈	C₂²×C₃₁	semidirect
328. C₂×D₈	Q₈×C₃₁	semidirect
400. C₃₁	#61370	direct (N × H)
410. C₂×D₄	C₂×Q₈×C₃₁	semidirect
421. D₈	C₂×Q₈×C₃₁	semidirect
422. D₈	C₂×Q₈×C₃₁	semidirect
425. C₂×C₈	C₂×Q₈×C₃₁	semidirect

Maximal subgroups

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

subgroup	index
C₃₁×#3501	2
C₃₁×#3389	2
C₃₁×#3384	2
C₃₁×#3380	2
Q₈×D₈×C₃₁	2
C₃₁×#3164	2
C₃₁×#3160	2
C₃₁×#3081	2
C₃₁×#3077	2
C₃₁×#2960	2
C₃₁×#2948	2
C₂×C₄×D₈×C₃₁	2
#61370	31

Subgroups — C31×#61370 #21518129

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#61370 #21518129