Subgroups — C₃₁×#34395 #21491154

Order 7936.

1990 conjugacy classes of subgroups.

The subgroup lattice has 586 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃₁×#529	C₂	1	1	yes	maximal, characteristic
2	C₃₁×#2378	C₂	1	2	yes	maximal, complemented
3	C₃₁×#2377	C₂	1	2	yes	maximal
4	C₃₁×#2372	C₂	1	2	yes	maximal, complemented
5	C₃₁×#2371	C₂	1	2	yes	maximal, complemented
6	C₃₁×#2357	C₂	1	2	yes	maximal, complemented
7	C₃₁×#2353	C₂	1	2	yes	maximal, complemented
8	C₃₁×#2347	C₂	1	2	yes	maximal, complemented
9	C₃₁×#2343	C₂	1	2	yes	maximal, complemented
10	C₃₁×#2335	C₂	1	4	yes	maximal, complemented
11	C₃₁×#2334	C₂	1	4	yes	maximal, complemented
12	C₃₁×#2320	C₂	1	2	yes	maximal, complemented
13	C₃₁×#2318	C₂	1	2	yes	maximal, complemented
14	C₃₁×#2283	C₂	1	1	yes	maximal, characteristic, complemented
15	C₃₁×#2283	C₂	1	1	yes	maximal, characteristic, complemented
16	C₃₁×#529	C₄	1	16	yes	complemented
17	C₂×C₃₁×#126	C₂²	1	2	yes	—
18	C₂×C₃₁×#125	C₂²	1	1	yes	characteristic
19	C₂×C₃₁×#124	C₂²	1	1	yes	characteristic
20	C₂×C₃₁×#123	C₂²	1	2	yes	—
21	C₂×C₃₁×C₂²⋊Q₈	C₂²	1	1	yes	characteristic
22	C₂×C₃₁×#121	C₂²	1	1	yes	characteristic
23	C₂×C₄×Q₈×C₃₁	C₂²	1	1	yes	characteristic
24	C₂×C₄×D₄×C₃₁	C₂²	1	1	yes	characteristic
25	C₂×C₄×D₄×C₃₁	C₂²	1	2	yes	—
26	C₂×C₃₁×#117	C₂²	1	2	yes	—
27	C₂²×C₄²×C₃₁	C₂²	1	1	yes	characteristic
28	C₃₁×#391	C₂²	1	4	yes	—
29	C₃₁×#390	C₂²	1	4	yes	—
30	C₃₁×#389	C₂²	1	4	yes	—
31	C₃₁×#389	C₂²	1	4	yes	—
32	C₃₁×#388	C₂²	1	4	yes	—
33	C₃₁×#388	C₂²	1	4	yes	—
34	C₃₁×#388	C₂²	1	4	yes	—
35	C₃₁×#388	C₂²	1	4	yes	—
36	C₃₁×#388	C₂²	1	4	yes	—
37	C₃₁×#388	C₂²	1	4	yes	—
38	C₃₁×#387	C₂²	1	4	yes	—
39	C₃₁×#387	C₂²	1	4	yes	—
40	C₃₁×#387	C₂²	1	4	yes	—
41	C₃₁×#387	C₂²	1	4	yes	—
42	C₃₁×#387	C₂²	1	4	yes	—
43	C₃₁×#387	C₂²	1	4	yes	—
44	C₃₁×#386	C₂²	1	4	yes	—
45	C₃₁×#386	C₂²	1	4	yes	—
46	C₃₁×#385	C₂²	1	4	yes	—
47	C₃₁×#385	C₂²	1	4	yes	—
48	C₃₁×#383	C₂²	1	2	yes	—
49	C₃₁×#383	C₂²	1	2	yes	—
50	C₃₁×#383	C₂²	1	1	yes	characteristic
51	C₃₁×#383	C₂²	1	1	yes	characteristic
52	C₃₁×#382	C₂²	1	2	yes	—
53	C₃₁×#382	C₂²	1	2	yes	—
54	C₃₁×#382	C₂²	1	2	yes	—
55	C₃₁×#381	C₂²	1	2	yes	—
56	C₃₁×#381	C₂²	1	1	yes	characteristic
57	C₃₁×#381	C₂²	1	1	yes	characteristic
58	C₃₁×#381	C₂²	1	2	yes	—
59	C₃₁×#380	C₂²	1	2	yes	—
60	C₃₁×#380	C₂²	1	2	yes	—
61	C₃₁×#380	C₂²	1	2	yes	—
62	C₄×C₄⋊C₄×C₃₁	C₂²	1	2	yes	—
63	C₄×C₄⋊C₄×C₃₁	C₂²	1	2	yes	—
64	C₄×C₄⋊C₄×C₃₁	C₂²	1	4	yes	—
65	C₄×C₄⋊C₄×C₃₁	C₂²	1	4	yes	—
66	C₄×C₄⋊C₄×C₃₁	C₂²	1	2	yes	—
67	C₄×C₂²⋊C₄×C₃₁	C₂²	1	2	yes	—
68	C₄×C₂²⋊C₄×C₃₁	C₂²	1	4	yes	—
69	C₄×C₂²⋊C₄×C₃₁	C₂²	1	2	yes	—
70	C₄×C₂²⋊C₄×C₃₁	C₂²	1	4	yes	—
71	C₄×C₂²⋊C₄×C₃₁	C₂²	1	2	yes	—
72	C₃₁×#376	C₂²	1	2	yes	—
73	C₃₁×#376	C₂²	1	2	yes	—
74	C₃₁×#376	C₂²	1	2	yes	—
75	C₂×C₃₁×#95	C₂²	1	2	yes	—
76	C₂²×Q₈×C₃₁	C₂³	1	1	yes	characteristic
77	C₂²×D₄×C₃₁	C₂³	1	2	yes	—
78	C₂²×D₄×C₃₁	C₂³	1	1	yes	characteristic
79	C₂³×C₄×C₃₁	C₂³	1	4	yes	—
80	C₂³×C₄×C₃₁	C₂³	1	1	yes	characteristic
81	C₂³×C₄×C₃₁	C₂³	1	1	yes	characteristic
82	C₂³×C₄×C₃₁	C₂³	1	1	yes	characteristic
83	C₂³×C₄×C₃₁	C₂³	1	2	yes	—
84	C₃₁×#126	C₂×C₄	1	16	yes	complemented
85	C₃₁×#125	C₂×C₄	1	8	yes	complemented
86	C₃₁×#124	C₂×C₄	1	8	yes	—
87	C₃₁×#123	C₂×C₄	1	16	yes	complemented
88	C₃₁×C₂²⋊Q₈	C₂×C₄	1	8	yes	complemented
89	C₃₁×#121	C₂×C₄	1	8	yes	—
90	C₄×Q₈×C₃₁	C₂×C₄	1	8	yes	complemented
91	C₄×D₄×C₃₁	C₂×C₄	1	8	yes	—
92	C₄×D₄×C₃₁	C₂×C₄	1	16	yes	complemented
93	C₃₁×#117	C₂×C₄	1	16	yes	complemented
94	C₂×C₄⋊C₄×C₃₁	C₂³	1	4	yes	—
95	C₂×C₄⋊C₄×C₃₁	C₂³	1	4	yes	—
96	C₂×C₄⋊C₄×C₃₁	C₂³	1	4	yes	—
97	C₂×C₄⋊C₄×C₃₁	C₂³	1	4	yes	—
98	C₂×C₄⋊C₄×C₃₁	C₂³	1	4	yes	—
99	C₂×C₄⋊C₄×C₃₁	C₂³	1	2	yes	—
100	C₂×C₄⋊C₄×C₃₁	C₂³	1	2	yes	—
101	C₂×C₄⋊C₄×C₃₁	C₂³	1	1	yes	characteristic
102	C₂×C₄⋊C₄×C₃₁	C₂³	1	2	yes	—
103	C₂×C₄⋊C₄×C₃₁	C₂³	1	1	yes	characteristic
104	C₂×C₄⋊C₄×C₃₁	C₂³	1	2	yes	—
105	C₂×C₂²⋊C₄×C₃₁	C₂³	1	2	yes	—
106	C₂×C₂²⋊C₄×C₃₁	C₂³	1	4	yes	—
107	C₂×C₂²⋊C₄×C₃₁	C₂³	1	2	yes	—
108	C₂×C₂²⋊C₄×C₃₁	C₂³	1	4	yes	—
109	C₂×C₂²⋊C₄×C₃₁	C₂³	1	4	yes	—
110	C₂×C₂²⋊C₄×C₃₁	C₂³	1	4	yes	—
111	C₂×C₂²⋊C₄×C₃₁	C₂³	1	2	yes	—
112	C₂×C₂²⋊C₄×C₃₁	C₂³	1	2	yes	—
113	C₂×C₂²⋊C₄×C₃₁	C₂³	1	2	yes	—
114	C₂×C₂²⋊C₄×C₃₁	C₂³	1	2	yes	—
115	C₂×C₂²⋊C₄×C₃₁	C₂³	1	2	yes	—
116	C₂×C₄²×C₃₁	C₂×C₄	1	8	yes	complemented
117	C₂×C₄²×C₃₁	C₂³	1	4	yes	—
118	C₂×C₄²×C₃₁	C₂³	1	4	yes	—
119	C₂×C₄²×C₃₁	C₂³	1	4	yes	—
120	C₂×C₄²×C₃₁	C₂³	1	4	yes	—
121	C₂×C₄²×C₃₁	C₂³	1	4	yes	—
122	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
123	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
124	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
125	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
126	C₂×C₄²×C₃₁	C₂³	1	2	yes	—
127	C₃₁×#95	C₂³	1	2	yes	—
128	C₃₁×#95	C₂³	1	2	yes	—
129	C₃₁×#95	C₂³	1	4	yes	—
130	C₃₁×#95	C₂³	1	2	yes	—
131	C₃₁×#95	C₂³	1	2	yes	—
132	C₃₁×#95	C₂³	1	2	yes	—
133	C₃₁×#95	C₂³	1	2	yes	—
134	C₃₁×#95	C₂³	1	4	yes	—
135	C₃₁×#95	C₂³	1	4	yes	—
136	C₃₁×#95	C₂³	1	4	yes	—
137	C₃₁×#95	C₂³	1	2	yes	—
138	C₃₁×#95	C₂³	1	2	yes	—
139	C₃₁×#95	C₂³	1	4	yes	—
140	C₃₁×#95	C₂³	1	4	yes	—
141	C₃₁×#95	C₂³	1	4	yes	—
142	C₃₁×#95	C₂³	1	2	yes	—
143	C₃₁×#95	C₂³	1	2	yes	—
144	C₃₁×#95	C₂³	1	4	yes	—
145	C₃₁×#95	C₂³	1	4	yes	—
146	C₂⁴×C₃₁	C₂⁴	1	1	yes	characteristic
147	C₂⁴×C₃₁	C₂⁴	1	2	yes	—
148	C₂×Q₈×C₃₁	—	4	2	no	—
149	C₂×Q₈×C₃₁	C₂²×C₄	1	4	yes	—
150	C₂×D₄×C₃₁	—	4	2	no	—
151	C₂×D₄×C₃₁	—	4	2	no	—
152	C₂×D₄×C₃₁	—	4	2	no	—
153	C₂×D₄×C₃₁	C₂²×C₄	1	4	yes	—
154	C₂×D₄×C₃₁	C₂²×C₄	1	8	yes	—
155	C₂²×C₄×C₃₁	—	4	2	no	—
156	C₂²×C₄×C₃₁	—	4	2	no	—
157	C₂²×C₄×C₃₁	—	4	4	no	—
158	C₂²×C₄×C₃₁	—	4	2	no	—
159	C₂²×C₄×C₃₁	—	4	4	no	—
160	C₂²×C₄×C₃₁	—	4	4	no	—
161	C₂²×C₄×C₃₁	—	4	4	no	—
162	C₂²×C₄×C₃₁	C₂²×C₄	1	4	yes	—
163	C₂²×C₄×C₃₁	C₂²×C₄	1	4	yes	—
164	C₂²×C₄×C₃₁	C₂²×C₄	1	4	yes	—
165	C₂²×C₄×C₃₁	C₂²×C₄	1	8	yes	—
166	C₂²×C₄×C₃₁	C₂⁴	1	4	yes	—
167	C₂²×C₄×C₃₁	C₂⁴	1	4	yes	—
168	C₂²×C₄×C₃₁	C₂⁴	1	4	yes	—
169	C₂²×C₄×C₃₁	C₂⁴	1	4	yes	—
170	C₂²×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
171	C₂²×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
172	C₂²×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
173	C₂²×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
174	C₂²×C₄×C₃₁	C₂⁴	1	2	yes	—
175	C₂²×C₄×C₃₁	C₂⁴	1	2	yes	—
176	C₂²×C₄×C₃₁	C₂⁴	1	2	yes	—
177	C₂²×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
178	C₂²×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
179	C₄⋊C₄×C₃₁	—	4	4	no	—
180	C₄⋊C₄×C₃₁	—	4	4	no	—
181	C₄⋊C₄×C₃₁	—	4	4	no	—
182	C₄⋊C₄×C₃₁	—	4	4	no	—
183	C₄⋊C₄×C₃₁	—	4	4	no	—
184	C₄⋊C₄×C₃₁	C₂²×C₄	1	8	yes	—
185	C₄⋊C₄×C₃₁	C₂²×C₄	1	8	yes	—
186	C₄⋊C₄×C₃₁	C₂²×C₄	1	4	yes	—
187	C₄⋊C₄×C₃₁	C₂²×C₄	1	8	yes	—
188	C₄⋊C₄×C₃₁	C₂²×C₄	1	4	yes	—
189	C₄⋊C₄×C₃₁	C₂²×C₄	1	8	yes	—
190	C₂²⋊C₄×C₃₁	—	4	2	no	—
191	C₂²⋊C₄×C₃₁	—	4	4	no	—
192	C₂²⋊C₄×C₃₁	—	4	4	no	—
193	C₂²⋊C₄×C₃₁	—	4	4	no	—
194	C₂²⋊C₄×C₃₁	—	4	4	no	—
195	C₂²⋊C₄×C₃₁	—	4	2	no	—
196	C₂²⋊C₄×C₃₁	C₂²×C₄	1	8	yes	—
197	C₂²⋊C₄×C₃₁	C₂²×C₄	1	8	yes	—
198	C₂²⋊C₄×C₃₁	C₂²×C₄	1	8	yes	—
199	C₂²⋊C₄×C₃₁	C₂²×C₄	1	8	yes	—
200	C₂²⋊C₄×C₃₁	C₂²×C₄	1	8	yes	—
201	C₄²×C₃₁	—	4	4	no	—
202	C₄²×C₃₁	—	4	4	no	—
203	C₄²×C₃₁	—	4	4	no	—
204	C₄²×C₃₁	—	4	4	no	—
205	C₄²×C₃₁	—	4	4	no	—
206	C₄²×C₃₁	C₂²×C₄	1	4	yes	—
207	C₄²×C₃₁	C₂²×C₄	1	4	yes	—
208	C₄²×C₃₁	C₂²×C₄	1	4	yes	—
209	C₄²×C₃₁	C₂²×C₄	1	4	yes	—
210	C₄²×C₃₁	C₂²×C₄	1	8	yes	—
211	#34395	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
212	C₂³×C₃₁	—	2	2	no	—
213	C₂³×C₃₁	—	2	2	no	—
214	C₂³×C₃₁	—	2	2	no	—
215	C₂³×C₃₁	—	2	4	no	—
216	C₂³×C₃₁	—	2	4	no	—
217	C₂³×C₃₁	—	2	2	no	—
218	C₂³×C₃₁	—	2	2	no	—
219	C₂³×C₃₁	C₂³×C₄	1	4	yes	—
220	C₂³×C₃₁	C₂³×C₄	1	2	yes	—
221	C₂³×C₃₁	C₂⁵	1	1	yes	characteristic
222	Q₈×C₃₁	—	4	4	no	—
223	D₄×C₃₁	—	4	4	no	—
224	D₄×C₃₁	—	4	8	no	—
225	C₂×C₄×C₃₁	—	2	2	no	—
226	C₂×C₄×C₃₁	—	2	2	no	—
227	C₂×C₄×C₃₁	—	2	2	no	—
228	C₂×C₄×C₃₁	—	2	4	no	—
229	C₂×C₄×C₃₁	—	2	4	no	—
230	C₂×C₄×C₃₁	—	2	2	no	—
231	C₂×C₄×C₃₁	—	2	2	no	—
232	C₂×C₄×C₃₁	—	2	2	no	—
233	C₂×C₄×C₃₁	—	2	2	no	—
234	C₂×C₄×C₃₁	—	2	2	no	—
235	C₂×C₄×C₃₁	—	2	4	no	—
236	C₂×C₄×C₃₁	—	2	4	no	—
237	C₂×C₄×C₃₁	—	2	4	no	—
238	C₂×C₄×C₃₁	—	2	4	no	—
239	C₂×C₄×C₃₁	—	2	4	no	—
240	C₂×C₄×C₃₁	—	2	4	no	—
241	C₂×C₄×C₃₁	—	2	4	no	—
242	C₂×C₄×C₃₁	—	2	4	no	—
243	C₂×C₄×C₃₁	—	2	4	no	—
244	C₂×C₄×C₃₁	—	2	4	no	—
245	C₂×C₄×C₃₁	—	2	4	no	—
246	C₂×C₄×C₃₁	—	2	4	no	—
247	C₂×C₄×C₃₁	—	4	4	no	—
248	C₂×C₄×C₃₁	—	4	8	no	—
249	C₂×C₄×C₃₁	—	4	4	no	—
250	C₂×C₄×C₃₁	—	4	4	no	—
251	C₂×C₄×C₃₁	—	8	8	no	—
252	C₂×C₄×C₃₁	C₂³×C₄	1	2	yes	—
253	C₂×C₄×C₃₁	C₂³×C₄	1	2	yes	—
254	C₂×C₄×C₃₁	C₂³×C₄	1	2	yes	—
255	C₂×C₄×C₃₁	C₂³×C₄	1	2	yes	—
256	C₂×C₄×C₃₁	C₂³×C₄	1	2	yes	—
257	C₂×C₄×C₃₁	C₂³×C₄	1	2	yes	—
258	C₂×C₄×C₃₁	C₂³×C₄	1	4	yes	—
259	C₂×C₄×C₃₁	C₂³×C₄	1	4	yes	—
260	C₂×C₄×C₃₁	C₂³×C₄	1	4	yes	—
261	C₂×#529	C₂×C₃₁	1	1	yes	characteristic
262	#2378	C₂×C₃₁	1	2	yes	complemented
263	#2377	C₂×C₃₁	1	2	yes	—
264	#2372	C₂×C₃₁	1	2	yes	complemented
265	#2371	C₂×C₃₁	1	2	yes	complemented
266	#2357	C₂×C₃₁	1	2	yes	complemented
267	#2353	C₂×C₃₁	1	2	yes	complemented
268	#2347	C₂×C₃₁	1	2	yes	complemented
269	#2343	C₂×C₃₁	1	2	yes	complemented
270	#2335	C₂×C₃₁	1	4	yes	complemented
271	#2334	C₂×C₃₁	1	4	yes	complemented
272	#2320	C₂×C₃₁	1	2	yes	complemented
273	#2318	C₂×C₃₁	1	2	yes	complemented
274	#2283	C₂×C₃₁	1	1	yes	characteristic, complemented
275	#2283	C₂×C₃₁	1	1	yes	characteristic, complemented
276	C₂²×C₃₁	—	2	2	no	—
277	C₂²×C₃₁	—	2	2	no	—
278	C₂²×C₃₁	—	2	2	no	—
279	C₂²×C₃₁	—	2	4	no	—
280	C₂²×C₃₁	—	2	4	no	—
281	C₂²×C₃₁	—	2	4	no	—
282	C₂²×C₃₁	—	4	4	no	—
283	C₂²×C₃₁	—	4	4	no	—
284	C₂²×C₃₁	—	4	4	no	—
285	C₂²×C₃₁	C₂⁴×C₄	1	1	yes	characteristic
286	C₂²×C₃₁	#585	1	2	yes	—
287	C₂²×C₃₁	#585	1	2	yes	—
288	C₂²×C₃₁	#585	1	2	yes	—
289	C₄×C₃₁	—	2	2	no	—
290	C₄×C₃₁	—	2	2	no	—
291	C₄×C₃₁	—	2	2	no	—
292	C₄×C₃₁	—	2	4	no	—
293	C₄×C₃₁	—	2	2	no	—
294	C₄×C₃₁	—	2	2	no	—
295	C₄×C₃₁	—	2	2	no	—
296	C₄×C₃₁	—	2	4	no	—
297	C₄×C₃₁	—	2	4	no	—
298	C₄×C₃₁	—	4	4	no	—
299	C₄×C₃₁	—	4	4	no	—
300	C₄×C₃₁	—	4	4	no	—
301	C₄×C₃₁	—	4	4	no	—
302	#529	C₄×C₃₁	1	16	yes	complemented
303	C₂×#126	C₂²×C₃₁	1	2	yes	—
304	C₂×#125	C₂²×C₃₁	1	1	yes	characteristic
305	C₂×#124	C₂²×C₃₁	1	1	yes	characteristic
306	C₂×#123	C₂²×C₃₁	1	2	yes	—
307	C₂×C₂²⋊Q₈	C₂²×C₃₁	1	1	yes	characteristic
308	C₂×#121	C₂²×C₃₁	1	1	yes	characteristic
309	C₂×C₄×Q₈	C₂²×C₃₁	1	1	yes	characteristic
310	C₂×C₄×D₄	C₂²×C₃₁	1	1	yes	characteristic
311	C₂×C₄×D₄	C₂²×C₃₁	1	2	yes	—
312	C₂×#117	C₂²×C₃₁	1	2	yes	—
313	C₂²×C₄²	C₂²×C₃₁	1	1	yes	characteristic
314	#391	C₂²×C₃₁	1	4	yes	—
315	#390	C₂²×C₃₁	1	4	yes	—
316	#389	C₂²×C₃₁	1	4	yes	—
317	#389	C₂²×C₃₁	1	4	yes	—
318	#388	C₂²×C₃₁	1	4	yes	—
319	#388	C₂²×C₃₁	1	4	yes	—
320	#388	C₂²×C₃₁	1	4	yes	—
321	#388	C₂²×C₃₁	1	4	yes	—
322	#388	C₂²×C₃₁	1	4	yes	—
323	#388	C₂²×C₃₁	1	4	yes	—
324	#387	C₂²×C₃₁	1	4	yes	—
325	#387	C₂²×C₃₁	1	4	yes	—
326	#387	C₂²×C₃₁	1	4	yes	—
327	#387	C₂²×C₃₁	1	4	yes	—
328	#387	C₂²×C₃₁	1	4	yes	—
329	#387	C₂²×C₃₁	1	4	yes	—
330	#386	C₂²×C₃₁	1	4	yes	—
331	#386	C₂²×C₃₁	1	4	yes	—
332	#385	C₂²×C₃₁	1	4	yes	—
333	#385	C₂²×C₃₁	1	4	yes	—
334	#383	C₂²×C₃₁	1	2	yes	—
335	#383	C₂²×C₃₁	1	2	yes	—
336	#383	C₂²×C₃₁	1	1	yes	characteristic
337	#383	C₂²×C₃₁	1	1	yes	characteristic
338	#382	C₂²×C₃₁	1	2	yes	—
339	#382	C₂²×C₃₁	1	2	yes	—
340	#382	C₂²×C₃₁	1	2	yes	—
341	#381	C₂²×C₃₁	1	2	yes	—
342	#381	C₂²×C₃₁	1	1	yes	characteristic
343	#381	C₂²×C₃₁	1	1	yes	characteristic
344	#381	C₂²×C₃₁	1	2	yes	—
345	#380	C₂²×C₃₁	1	2	yes	—
346	#380	C₂²×C₃₁	1	2	yes	—
347	#380	C₂²×C₃₁	1	2	yes	—
348	C₄×C₄⋊C₄	C₂²×C₃₁	1	2	yes	—
349	C₄×C₄⋊C₄	C₂²×C₃₁	1	2	yes	—
350	C₄×C₄⋊C₄	C₂²×C₃₁	1	4	yes	—
351	C₄×C₄⋊C₄	C₂²×C₃₁	1	4	yes	—
352	C₄×C₄⋊C₄	C₂²×C₃₁	1	2	yes	—
353	C₄×C₂²⋊C₄	C₂²×C₃₁	1	2	yes	—
354	C₄×C₂²⋊C₄	C₂²×C₃₁	1	4	yes	—
355	C₄×C₂²⋊C₄	C₂²×C₃₁	1	2	yes	—
356	C₄×C₂²⋊C₄	C₂²×C₃₁	1	4	yes	—
357	C₄×C₂²⋊C₄	C₂²×C₃₁	1	2	yes	—
358	#376	C₂²×C₃₁	1	2	yes	—
359	#376	C₂²×C₃₁	1	2	yes	—
360	#376	C₂²×C₃₁	1	2	yes	—
361	C₂×#95	C₂²×C₃₁	1	2	yes	—
362	C₂×C₃₁	—	4	2	no	—
363	C₂×C₃₁	—	4	4	no	—
364	C₂×C₃₁	#3428	1	1	yes	characteristic
365	C₂×C₃₁	#3428	1	1	yes	characteristic
366	C₂×C₃₁	#3428	1	1	yes	characteristic
367	C₂×C₃₁	#3566	1	4	yes	—
368	C₂²×Q₈	C₂³×C₃₁	1	1	yes	characteristic
369	C₂²×D₄	C₂³×C₃₁	1	2	yes	—
370	C₂²×D₄	C₂³×C₃₁	1	1	yes	characteristic
371	C₂³×C₄	C₂³×C₃₁	1	4	yes	—
372	C₂³×C₄	C₂³×C₃₁	1	2	yes	—
373	C₂³×C₄	C₂³×C₃₁	1	1	yes	characteristic
374	C₂³×C₄	C₂³×C₃₁	1	1	yes	characteristic
375	C₂³×C₄	C₂³×C₃₁	1	1	yes	characteristic
376	#126	C₂×C₄×C₃₁	1	16	yes	complemented
377	#125	C₂×C₄×C₃₁	1	8	yes	complemented
378	#124	C₂×C₄×C₃₁	1	8	yes	—
379	#123	C₂×C₄×C₃₁	1	16	yes	complemented
380	C₂²⋊Q₈	C₂×C₄×C₃₁	1	8	yes	complemented
381	#121	C₂×C₄×C₃₁	1	8	yes	—
382	C₄×Q₈	C₂×C₄×C₃₁	1	8	yes	complemented
383	C₄×D₄	C₂×C₄×C₃₁	1	8	yes	—
384	C₄×D₄	C₂×C₄×C₃₁	1	16	yes	complemented
385	#117	C₂×C₄×C₃₁	1	16	yes	complemented
386	C₂×C₄⋊C₄	C₂³×C₃₁	1	4	yes	—
387	C₂×C₄⋊C₄	C₂³×C₃₁	1	4	yes	—
388	C₂×C₄⋊C₄	C₂³×C₃₁	1	4	yes	—
389	C₂×C₄⋊C₄	C₂³×C₃₁	1	4	yes	—
390	C₂×C₄⋊C₄	C₂³×C₃₁	1	4	yes	—
391	C₂×C₄⋊C₄	C₂³×C₃₁	1	2	yes	—
392	C₂×C₄⋊C₄	C₂³×C₃₁	1	2	yes	—
393	C₂×C₄⋊C₄	C₂³×C₃₁	1	1	yes	characteristic
394	C₂×C₄⋊C₄	C₂³×C₃₁	1	2	yes	—
395	C₂×C₄⋊C₄	C₂³×C₃₁	1	1	yes	characteristic
396	C₂×C₄⋊C₄	C₂³×C₃₁	1	2	yes	—
397	C₂×C₂²⋊C₄	C₂³×C₃₁	1	2	yes	—
398	C₂×C₂²⋊C₄	C₂³×C₃₁	1	4	yes	—
399	C₂×C₂²⋊C₄	C₂³×C₃₁	1	2	yes	—
400	C₂×C₂²⋊C₄	C₂³×C₃₁	1	4	yes	—
401	C₂×C₂²⋊C₄	C₂³×C₃₁	1	4	yes	—
402	C₂×C₂²⋊C₄	C₂³×C₃₁	1	4	yes	—
403	C₂×C₂²⋊C₄	C₂³×C₃₁	1	2	yes	—
404	C₂×C₂²⋊C₄	C₂³×C₃₁	1	2	yes	—
405	C₂×C₂²⋊C₄	C₂³×C₃₁	1	2	yes	—
406	C₂×C₂²⋊C₄	C₂³×C₃₁	1	2	yes	—
407	C₂×C₂²⋊C₄	C₂³×C₃₁	1	2	yes	—
408	C₂×C₄²	C₂×C₄×C₃₁	1	8	yes	complemented
409	C₂×C₄²	C₂³×C₃₁	1	4	yes	—
410	C₂×C₄²	C₂³×C₃₁	1	4	yes	—
411	C₂×C₄²	C₂³×C₃₁	1	4	yes	—
412	C₂×C₄²	C₂³×C₃₁	1	4	yes	—
413	C₂×C₄²	C₂³×C₃₁	1	4	yes	—
414	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
415	C₂×C₄²	C₂³×C₃₁	1	2	yes	—
416	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
417	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
418	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
419	#95	C₂³×C₃₁	1	2	yes	—
420	#95	C₂³×C₃₁	1	2	yes	—
421	#95	C₂³×C₃₁	1	4	yes	—
422	#95	C₂³×C₃₁	1	2	yes	—
423	#95	C₂³×C₃₁	1	2	yes	—
424	#95	C₂³×C₃₁	1	2	yes	—
425	#95	C₂³×C₃₁	1	2	yes	—
426	#95	C₂³×C₃₁	1	4	yes	—
427	#95	C₂³×C₃₁	1	4	yes	—
428	#95	C₂³×C₃₁	1	4	yes	—
429	#95	C₂³×C₃₁	1	2	yes	—
430	#95	C₂³×C₃₁	1	2	yes	—
431	#95	C₂³×C₃₁	1	4	yes	—
432	#95	C₂³×C₃₁	1	4	yes	—
433	#95	C₂³×C₃₁	1	4	yes	—
434	#95	C₂³×C₃₁	1	2	yes	—
435	#95	C₂³×C₃₁	1	2	yes	—
436	#95	C₂³×C₃₁	1	4	yes	—
437	#95	C₂³×C₃₁	1	4	yes	—
438	C₃₁	#34395	1	1	yes	characteristic, direct product factor, Sylow
439	C₂⁴	C₂⁴×C₃₁	1	1	yes	characteristic
440	C₂⁴	C₂⁴×C₃₁	1	2	yes	—
441	C₂×Q₈	—	4	2	no	—
442	C₂×Q₈	C₂²×C₄×C₃₁	1	4	yes	—
443	C₂×D₄	—	4	2	no	—
444	C₂×D₄	—	4	2	no	—
445	C₂×D₄	—	4	2	no	—
446	C₂×D₄	C₂²×C₄×C₃₁	1	4	yes	—
447	C₂×D₄	C₂²×C₄×C₃₁	1	8	yes	—
448	C₂²×C₄	—	4	2	no	—
449	C₂²×C₄	—	4	2	no	—
450	C₂²×C₄	—	4	4	no	—
451	C₂²×C₄	—	4	2	no	—
452	C₂²×C₄	—	4	4	no	—
453	C₂²×C₄	—	4	4	no	—
454	C₂²×C₄	—	4	4	no	—
455	C₂²×C₄	C₂²×C₄×C₃₁	1	4	yes	—
456	C₂²×C₄	C₂²×C₄×C₃₁	1	4	yes	—
457	C₂²×C₄	C₂²×C₄×C₃₁	1	4	yes	—
458	C₂²×C₄	C₂²×C₄×C₃₁	1	8	yes	—
459	C₂²×C₄	C₂⁴×C₃₁	1	4	yes	—
460	C₂²×C₄	C₂⁴×C₃₁	1	4	yes	—
461	C₂²×C₄	C₂⁴×C₃₁	1	4	yes	—
462	C₂²×C₄	C₂⁴×C₃₁	1	4	yes	—
463	C₂²×C₄	C₂⁴×C₃₁	1	1	yes	characteristic
464	C₂²×C₄	C₂⁴×C₃₁	1	1	yes	characteristic
465	C₂²×C₄	C₂⁴×C₃₁	1	1	yes	characteristic
466	C₂²×C₄	C₂⁴×C₃₁	1	1	yes	characteristic
467	C₂²×C₄	C₂⁴×C₃₁	1	2	yes	—
468	C₂²×C₄	C₂⁴×C₃₁	1	2	yes	—
469	C₂²×C₄	C₂⁴×C₃₁	1	2	yes	—
470	C₂²×C₄	C₂⁴×C₃₁	1	1	yes	characteristic
471	C₂²×C₄	C₂⁴×C₃₁	1	1	yes	characteristic
472	C₄⋊C₄	—	4	4	no	—
473	C₄⋊C₄	—	4	4	no	—
474	C₄⋊C₄	—	4	4	no	—
475	C₄⋊C₄	—	4	4	no	—
476	C₄⋊C₄	—	4	4	no	—
477	C₄⋊C₄	C₂²×C₄×C₃₁	1	8	yes	—
478	C₄⋊C₄	C₂²×C₄×C₃₁	1	8	yes	—
479	C₄⋊C₄	C₂²×C₄×C₃₁	1	4	yes	—
480	C₄⋊C₄	C₂²×C₄×C₃₁	1	8	yes	—
481	C₄⋊C₄	C₂²×C₄×C₃₁	1	4	yes	—
482	C₄⋊C₄	C₂²×C₄×C₃₁	1	8	yes	—
483	C₂²⋊C₄	—	4	2	no	—
484	C₂²⋊C₄	—	4	4	no	—
485	C₂²⋊C₄	—	4	4	no	—
486	C₂²⋊C₄	—	4	4	no	—
487	C₂²⋊C₄	—	4	4	no	—
488	C₂²⋊C₄	—	4	2	no	—
489	C₂²⋊C₄	C₂²×C₄×C₃₁	1	8	yes	—
490	C₂²⋊C₄	C₂²×C₄×C₃₁	1	8	yes	—
491	C₂²⋊C₄	C₂²×C₄×C₃₁	1	8	yes	—
492	C₂²⋊C₄	C₂²×C₄×C₃₁	1	8	yes	—
493	C₂²⋊C₄	C₂²×C₄×C₃₁	1	8	yes	—
494	C₄²	—	4	4	no	—
495	C₄²	—	4	4	no	—
496	C₄²	—	4	4	no	—
497	C₄²	—	4	4	no	—
498	C₄²	—	4	4	no	—
499	C₄²	C₂²×C₄×C₃₁	1	4	yes	—
500	C₄²	C₂²×C₄×C₃₁	1	4	yes	—
501	C₄²	C₂²×C₄×C₃₁	1	4	yes	—
502	C₄²	C₂²×C₄×C₃₁	1	4	yes	—
503	C₄²	C₂²×C₄×C₃₁	1	8	yes	—
504	C₂³	—	2	2	no	—
505	C₂³	—	2	2	no	—
506	C₂³	—	2	2	no	—
507	C₂³	—	2	4	no	—
508	C₂³	—	2	4	no	—
509	C₂³	—	2	2	no	—
510	C₂³	—	2	2	no	—
511	C₂³	C₂³×C₄×C₃₁	1	4	yes	—
512	C₂³	C₂³×C₄×C₃₁	1	2	yes	—
513	C₂³	C₂⁵×C₃₁	1	1	yes	characteristic
514	Q₈	—	4	4	no	—
515	D₄	—	4	4	no	—
516	D₄	—	4	8	no	—
517	C₂×C₄	—	2	2	no	—
518	C₂×C₄	—	2	2	no	—
519	C₂×C₄	—	2	2	no	—
520	C₂×C₄	—	2	4	no	—
521	C₂×C₄	—	2	4	no	—
522	C₂×C₄	—	2	2	no	—
523	C₂×C₄	—	2	2	no	—
524	C₂×C₄	—	2	2	no	—
525	C₂×C₄	—	2	2	no	—
526	C₂×C₄	—	2	2	no	—
527	C₂×C₄	—	2	4	no	—
528	C₂×C₄	—	2	4	no	—
529	C₂×C₄	—	2	4	no	—
530	C₂×C₄	—	2	4	no	—
531	C₂×C₄	—	2	4	no	—
532	C₂×C₄	—	2	4	no	—
533	C₂×C₄	—	2	4	no	—
534	C₂×C₄	—	2	4	no	—
535	C₂×C₄	—	2	4	no	—
536	C₂×C₄	—	2	4	no	—
537	C₂×C₄	—	2	4	no	—
538	C₂×C₄	—	2	4	no	—
539	C₂×C₄	—	4	4	no	—
540	C₂×C₄	—	4	8	no	—
541	C₂×C₄	—	4	4	no	—
542	C₂×C₄	—	4	4	no	—
543	C₂×C₄	—	8	8	no	—
544	C₂×C₄	C₂³×C₄×C₃₁	1	2	yes	—
545	C₂×C₄	C₂³×C₄×C₃₁	1	2	yes	—
546	C₂×C₄	C₂³×C₄×C₃₁	1	2	yes	—
547	C₂×C₄	C₂³×C₄×C₃₁	1	2	yes	—
548	C₂×C₄	C₂³×C₄×C₃₁	1	2	yes	—
549	C₂×C₄	C₂³×C₄×C₃₁	1	2	yes	—
550	C₂×C₄	C₂³×C₄×C₃₁	1	4	yes	—
551	C₂×C₄	C₂³×C₄×C₃₁	1	4	yes	—
552	C₂×C₄	C₂³×C₄×C₃₁	1	4	yes	—
553	C₂²	—	2	2	no	—
554	C₂²	—	2	2	no	—
555	C₂²	—	2	2	no	—
556	C₂²	—	2	4	no	—
557	C₂²	—	2	4	no	—
558	C₂²	—	2	4	no	—
559	C₂²	—	4	4	no	—
560	C₂²	—	4	4	no	—
561	C₂²	—	4	4	no	—
562	C₂²	C₂⁴×C₄×C₃₁	1	1	yes	characteristic
563	C₂²	C₃₁×#585	1	2	yes	—
564	C₂²	C₃₁×#585	1	2	yes	—
565	C₂²	C₃₁×#585	1	2	yes	—
566	C₄	—	2	2	no	—
567	C₄	—	2	2	no	—
568	C₄	—	2	2	no	—
569	C₄	—	2	2	no	—
570	C₄	—	2	4	no	—
571	C₄	—	2	2	no	—
572	C₄	—	2	2	no	—
573	C₄	—	2	4	no	—
574	C₄	—	2	4	no	—
575	C₄	—	4	4	no	—
576	C₄	—	4	4	no	—
577	C₄	—	4	4	no	—
578	C₄	—	4	4	no	—
579	C₂	—	4	2	no	—
580	C₂	—	4	4	no	—
581	C₂	C₃₁×#3428	1	1	yes	characteristic
582	C₂	C₃₁×#3428	1	1	yes	characteristic
583	C₂	C₃₁×#3428	1	1	yes	characteristic
584	C₂	C₃₁×#3566	1	4	yes	—

	subgroup	G/H
center, socle	C₂³×C₃₁	C₂⁵
commutator	C₂²	C₂⁴×C₄×C₃₁
Frattini, cosocle	C₂³	C₂⁵×C₃₁
prime radical	C₂×C₃₁×#121	C₂²
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³×C₃₁ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₃₁×#2378	C₂	semidirect
4. C₃₁×#2372	C₂	semidirect
5. C₃₁×#2371	C₂	semidirect
6. C₃₁×#2357	C₂	semidirect
7. C₃₁×#2353	C₂	semidirect
8. C₃₁×#2347	C₂	semidirect
9. C₃₁×#2343	C₂	semidirect
10. C₃₁×#2335	C₂	semidirect
11. C₃₁×#2334	C₂	semidirect
12. C₃₁×#2320	C₂	semidirect
13. C₃₁×#2318	C₂	semidirect
14. C₃₁×#2283	C₂	semidirect
15. C₃₁×#2283	C₂	semidirect
16. C₃₁×#529	C₄	semidirect
84. C₃₁×#126	C₂×C₄	semidirect
85. C₃₁×#125	C₂×C₄	semidirect
87. C₃₁×#123	C₂×C₄	semidirect
88. C₃₁×C₂²⋊Q₈	C₂×C₄	semidirect
90. C₄×Q₈×C₃₁	C₂×C₄	semidirect
92. C₄×D₄×C₃₁	C₂×C₄	semidirect
93. C₃₁×#117	C₂×C₄	semidirect
116. C₂×C₄²×C₃₁	C₂×C₄	semidirect
211. #34395	C₃₁	direct (N × H)
262. #2378	C₂×C₃₁	semidirect
264. #2372	C₂×C₃₁	semidirect
265. #2371	C₂×C₃₁	semidirect
266. #2357	C₂×C₃₁	semidirect
267. #2353	C₂×C₃₁	semidirect
268. #2347	C₂×C₃₁	semidirect
269. #2343	C₂×C₃₁	semidirect
270. #2335	C₂×C₃₁	semidirect
271. #2334	C₂×C₃₁	semidirect
272. #2320	C₂×C₃₁	semidirect
273. #2318	C₂×C₃₁	semidirect
274. #2283	C₂×C₃₁	semidirect
275. #2283	C₂×C₃₁	semidirect
302. #529	C₄×C₃₁	semidirect
376. #126	C₂×C₄×C₃₁	semidirect
377. #125	C₂×C₄×C₃₁	semidirect
379. #123	C₂×C₄×C₃₁	semidirect
380. C₂²⋊Q₈	C₂×C₄×C₃₁	semidirect
382. C₄×Q₈	C₂×C₄×C₃₁	semidirect
384. C₄×D₄	C₂×C₄×C₃₁	semidirect
385. #117	C₂×C₄×C₃₁	semidirect
408. C₂×C₄²	C₂×C₄×C₃₁	semidirect
438. C₃₁	#34395	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×C₃₁×#529	2
C₃₁×#2378	2
C₃₁×#2377	2
C₃₁×#2372	2
C₃₁×#2371	2
C₃₁×#2357	2
C₃₁×#2353	2
C₃₁×#2347	2
C₃₁×#2343	2
C₃₁×#2335	2
C₃₁×#2334	2
C₃₁×#2320	2
C₃₁×#2318	2
C₃₁×#2283	2
#34395	31

Subgroups — C31×#34395 #21491154

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#34395 #21491154