Subgroups — C₃₁×#34405 #21491164

Order 7936.

1990 conjugacy classes of subgroups.

The subgroup lattice has 608 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×#2382	C₂	1	2	yes	maximal, complemented
3	C₃₁×#2379	C₂	1	2	yes	maximal, complemented
4	C₃₁×#2373	C₂	1	4	yes	maximal, complemented
5	C₃₁×#2356	C₂	1	2	yes	maximal, complemented
6	C₃₁×#2353	C₂	1	2	yes	maximal, complemented
7	C₃₁×#2347	C₂	1	2	yes	maximal, complemented
8	C₃₁×#2343	C₂	1	2	yes	maximal, complemented
9	C₃₁×#2335	C₂	1	4	yes	maximal, complemented
10	C₃₁×#2333	C₂	1	2	yes	maximal
11	C₃₁×#2332	C₂	1	2	yes	maximal, complemented
12	C₃₁×#2319	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	2	yes	—
51	C₃₁×#382	C₂²	1	2	yes	—
52	C₃₁×#382	C₂²	1	2	yes	—
53	C₃₁×#382	C₂²	1	2	yes	—
54	C₃₁×#381	C₂²	1	2	yes	—
55	C₃₁×#381	C₂²	1	2	yes	—
56	C₃₁×#381	C₂²	1	2	yes	—
57	C₃₁×#380	C₂²	1	1	yes	characteristic
58	C₃₁×#380	C₂²	1	1	yes	characteristic
59	C₃₁×#380	C₂²	1	2	yes	—
60	C₃₁×#380	C₂²	1	2	yes	—
61	C₄×C₄⋊C₄×C₃₁	C₂²	1	2	yes	—
62	C₄×C₄⋊C₄×C₃₁	C₂²	1	4	yes	—
63	C₄×C₄⋊C₄×C₃₁	C₂²	1	4	yes	—
64	C₄×C₄⋊C₄×C₃₁	C₂²	1	1	yes	characteristic
65	C₄×C₄⋊C₄×C₃₁	C₂²	1	1	yes	characteristic
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	2	yes	—
69	C₄×C₂²⋊C₄×C₃₁	C₂²	1	4	yes	—
70	C₄×C₂²⋊C₄×C₃₁	C₂²	1	2	yes	—
71	C₄×C₂²⋊C₄×C₃₁	C₂²	1	4	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	2	yes	—
95	C₂×C₄⋊C₄×C₃₁	C₂³	1	2	yes	—
96	C₂×C₄⋊C₄×C₃₁	C₂³	1	2	yes	—
97	C₂×C₄⋊C₄×C₃₁	C₂³	1	2	yes	—
98	C₂×C₄⋊C₄×C₃₁	C₂³	1	4	yes	—
99	C₂×C₄⋊C₄×C₃₁	C₂³	1	4	yes	—
100	C₂×C₄⋊C₄×C₃₁	C₂³	1	4	yes	—
101	C₂×C₄⋊C₄×C₃₁	C₂³	1	2	yes	—
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×C₄⋊C₄×C₃₁	C₂³	1	1	yes	characteristic
106	C₂×C₄⋊C₄×C₃₁	C₂³	1	2	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	2	yes	—
110	C₂×C₂²⋊C₄×C₃₁	C₂³	1	4	yes	—
111	C₂×C₂²⋊C₄×C₃₁	C₂³	1	4	yes	—
112	C₂×C₂²⋊C₄×C₃₁	C₂³	1	4	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×C₂²⋊C₄×C₃₁	C₂³	1	2	yes	—
117	C₂×C₂²⋊C₄×C₃₁	C₂³	1	2	yes	—
118	C₂×C₄²×C₃₁	C₂×C₄	1	8	yes	complemented
119	C₂×C₄²×C₃₁	C₂³	1	2	yes	—
120	C₂×C₄²×C₃₁	C₂³	1	4	yes	—
121	C₂×C₄²×C₃₁	C₂³	1	2	yes	—
122	C₂×C₄²×C₃₁	C₂³	1	4	yes	—
123	C₂×C₄²×C₃₁	C₂³	1	4	yes	—
124	C₂×C₄²×C₃₁	C₂³	1	4	yes	—
125	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
126	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
127	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
128	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
129	C₂×C₄²×C₃₁	C₂³	1	2	yes	—
130	C₃₁×#95	C₂³	1	2	yes	—
131	C₃₁×#95	C₂³	1	2	yes	—
132	C₃₁×#95	C₂³	1	4	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	4	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	4	yes	—
143	C₃₁×#95	C₂³	1	4	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	1	no	—
149	C₂×Q₈×C₃₁	—	4	1	no	—
150	C₂×Q₈×C₃₁	C₂²×C₄	1	4	yes	—
151	C₂×D₄×C₃₁	—	4	4	no	—
152	C₂×D₄×C₃₁	—	4	1	no	—
153	C₂×D₄×C₃₁	—	4	1	no	—
154	C₂×D₄×C₃₁	C₂²×C₄	1	4	yes	—
155	C₂×D₄×C₃₁	C₂²×C₄	1	8	yes	—
156	C₂²×C₄×C₃₁	—	4	2	no	—
157	C₂²×C₄×C₃₁	—	4	1	no	—
158	C₂²×C₄×C₃₁	—	4	1	no	—
159	C₂²×C₄×C₃₁	—	4	4	no	—
160	C₂²×C₄×C₃₁	—	4	2	no	—
161	C₂²×C₄×C₃₁	—	4	4	no	—
162	C₂²×C₄×C₃₁	—	4	4	no	—
163	C₂²×C₄×C₃₁	—	4	4	no	—
164	C₂²×C₄×C₃₁	C₂²×C₄	1	4	yes	—
165	C₂²×C₄×C₃₁	C₂²×C₄	1	4	yes	—
166	C₂²×C₄×C₃₁	C₂²×C₄	1	4	yes	—
167	C₂²×C₄×C₃₁	C₂²×C₄	1	8	yes	—
168	C₂²×C₄×C₃₁	C₂⁴	1	4	yes	—
169	C₂²×C₄×C₃₁	C₂⁴	1	4	yes	—
170	C₂²×C₄×C₃₁	C₂⁴	1	4	yes	—
171	C₂²×C₄×C₃₁	C₂⁴	1	4	yes	—
172	C₂²×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
173	C₂²×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
174	C₂²×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
175	C₂²×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
176	C₂²×C₄×C₃₁	C₂⁴	1	2	yes	—
177	C₂²×C₄×C₃₁	C₂⁴	1	2	yes	—
178	C₂²×C₄×C₃₁	C₂⁴	1	2	yes	—
179	C₂²×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
180	C₂²×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
181	C₄⋊C₄×C₃₁	—	4	4	no	—
182	C₄⋊C₄×C₃₁	—	4	4	no	—
183	C₄⋊C₄×C₃₁	—	4	2	no	—
184	C₄⋊C₄×C₃₁	—	4	2	no	—
185	C₄⋊C₄×C₃₁	—	4	2	no	—
186	C₄⋊C₄×C₃₁	—	4	2	no	—
187	C₄⋊C₄×C₃₁	—	4	4	no	—
188	C₄⋊C₄×C₃₁	C₂²×C₄	1	8	yes	—
189	C₄⋊C₄×C₃₁	C₂²×C₄	1	8	yes	—
190	C₄⋊C₄×C₃₁	C₂²×C₄	1	4	yes	—
191	C₄⋊C₄×C₃₁	C₂²×C₄	1	8	yes	—
192	C₄⋊C₄×C₃₁	C₂²×C₄	1	4	yes	—
193	C₄⋊C₄×C₃₁	C₂²×C₄	1	8	yes	—
194	C₂²⋊C₄×C₃₁	—	4	4	no	—
195	C₂²⋊C₄×C₃₁	—	4	2	no	—
196	C₂²⋊C₄×C₃₁	—	4	4	no	—
197	C₂²⋊C₄×C₃₁	—	4	4	no	—
198	C₂²⋊C₄×C₃₁	—	4	4	no	—
199	C₂²⋊C₄×C₃₁	—	4	2	no	—
200	C₂²⋊C₄×C₃₁	C₂²×C₄	1	8	yes	—
201	C₂²⋊C₄×C₃₁	C₂²×C₄	1	8	yes	—
202	C₂²⋊C₄×C₃₁	C₂²×C₄	1	8	yes	—
203	C₂²⋊C₄×C₃₁	C₂²×C₄	1	8	yes	—
204	C₂²⋊C₄×C₃₁	C₂²×C₄	1	8	yes	—
205	C₄²×C₃₁	—	4	2	no	—
206	C₄²×C₃₁	—	4	4	no	—
207	C₄²×C₃₁	—	4	2	no	—
208	C₄²×C₃₁	—	4	4	no	—
209	C₄²×C₃₁	—	4	4	no	—
210	C₄²×C₃₁	—	4	4	no	—
211	C₄²×C₃₁	C₂²×C₄	1	4	yes	—
212	C₄²×C₃₁	C₂²×C₄	1	4	yes	—
213	C₄²×C₃₁	C₂²×C₄	1	4	yes	—
214	C₄²×C₃₁	C₂²×C₄	1	4	yes	—
215	C₄²×C₃₁	C₂²×C₄	1	8	yes	—
216	#34405	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
217	C₂³×C₃₁	—	2	1	no	—
218	C₂³×C₃₁	—	2	2	no	—
219	C₂³×C₃₁	—	2	1	no	—
220	C₂³×C₃₁	—	2	2	no	—
221	C₂³×C₃₁	—	2	4	no	—
222	C₂³×C₃₁	—	2	4	no	—
223	C₂³×C₃₁	—	2	2	no	—
224	C₂³×C₃₁	—	2	2	no	—
225	C₂³×C₃₁	C₂³×C₄	1	4	yes	—
226	C₂³×C₃₁	C₂³×C₄	1	2	yes	—
227	C₂³×C₃₁	C₂⁵	1	1	yes	characteristic
228	Q₈×C₃₁	—	4	4	no	—
229	D₄×C₃₁	—	4	4	no	—
230	D₄×C₃₁	—	4	8	no	—
231	C₂×C₄×C₃₁	—	2	1	no	—
232	C₂×C₄×C₃₁	—	2	2	no	—
233	C₂×C₄×C₃₁	—	2	1	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	2	no	—
238	C₂×C₄×C₃₁	—	2	2	no	—
239	C₂×C₄×C₃₁	—	2	2	no	—
240	C₂×C₄×C₃₁	—	2	2	no	—
241	C₂×C₄×C₃₁	—	2	1	no	—
242	C₂×C₄×C₃₁	—	2	1	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	—	2	4	no	—
248	C₂×C₄×C₃₁	—	2	4	no	—
249	C₂×C₄×C₃₁	—	2	4	no	—
250	C₂×C₄×C₃₁	—	2	4	no	—
251	C₂×C₄×C₃₁	—	2	4	no	—
252	C₂×C₄×C₃₁	—	2	4	no	—
253	C₂×C₄×C₃₁	—	2	4	no	—
254	C₂×C₄×C₃₁	—	2	4	no	—
255	C₂×C₄×C₃₁	—	4	4	no	—
256	C₂×C₄×C₃₁	—	4	8	no	—
257	C₂×C₄×C₃₁	—	4	4	no	—
258	C₂×C₄×C₃₁	—	4	4	no	—
259	C₂×C₄×C₃₁	—	8	8	no	—
260	C₂×C₄×C₃₁	C₂³×C₄	1	2	yes	—
261	C₂×C₄×C₃₁	C₂³×C₄	1	2	yes	—
262	C₂×C₄×C₃₁	C₂³×C₄	1	2	yes	—
263	C₂×C₄×C₃₁	C₂³×C₄	1	2	yes	—
264	C₂×C₄×C₃₁	C₂³×C₄	1	2	yes	—
265	C₂×C₄×C₃₁	C₂³×C₄	1	2	yes	—
266	C₂×C₄×C₃₁	C₂³×C₄	1	4	yes	—
267	C₂×C₄×C₃₁	C₂³×C₄	1	4	yes	—
268	C₂×C₄×C₃₁	C₂³×C₄	1	4	yes	—
269	C₂×#529	C₂×C₃₁	1	1	yes	characteristic
270	#2382	C₂×C₃₁	1	2	yes	complemented
271	#2379	C₂×C₃₁	1	2	yes	complemented
272	#2373	C₂×C₃₁	1	4	yes	complemented
273	#2356	C₂×C₃₁	1	2	yes	complemented
274	#2353	C₂×C₃₁	1	2	yes	complemented
275	#2347	C₂×C₃₁	1	2	yes	complemented
276	#2343	C₂×C₃₁	1	2	yes	complemented
277	#2335	C₂×C₃₁	1	4	yes	complemented
278	#2333	C₂×C₃₁	1	2	yes	—
279	#2332	C₂×C₃₁	1	2	yes	complemented
280	#2319	C₂×C₃₁	1	2	yes	complemented
281	#2318	C₂×C₃₁	1	2	yes	complemented
282	#2283	C₂×C₃₁	1	1	yes	characteristic, complemented
283	#2283	C₂×C₃₁	1	1	yes	characteristic, complemented
284	C₂²×C₃₁	—	2	2	no	—
285	C₂²×C₃₁	—	2	2	no	—
286	C₂²×C₃₁	—	2	2	no	—
287	C₂²×C₃₁	—	2	4	no	—
288	C₂²×C₃₁	—	2	4	no	—
289	C₂²×C₃₁	—	2	4	no	—
290	C₂²×C₃₁	—	4	2	no	—
291	C₂²×C₃₁	—	4	4	no	—
292	C₂²×C₃₁	—	4	4	no	—
293	C₂²×C₃₁	—	4	2	no	—
294	C₂²×C₃₁	C₂⁴×C₄	1	1	yes	characteristic
295	C₂²×C₃₁	#585	1	1	yes	characteristic
296	C₂²×C₃₁	#585	1	2	yes	—
297	C₂²×C₃₁	#585	1	2	yes	—
298	C₂²×C₃₁	#585	1	1	yes	characteristic
299	C₄×C₃₁	—	2	2	no	—
300	C₄×C₃₁	—	2	2	no	—
301	C₄×C₃₁	—	2	2	no	—
302	C₄×C₃₁	—	2	4	no	—
303	C₄×C₃₁	—	2	2	no	—
304	C₄×C₃₁	—	2	2	no	—
305	C₄×C₃₁	—	2	2	no	—
306	C₄×C₃₁	—	2	4	no	—
307	C₄×C₃₁	—	2	4	no	—
308	C₄×C₃₁	—	4	4	no	—
309	C₄×C₃₁	—	4	4	no	—
310	C₄×C₃₁	—	4	4	no	—
311	C₄×C₃₁	—	4	4	no	—
312	#529	C₄×C₃₁	1	16	yes	complemented
313	C₂×#126	C₂²×C₃₁	1	2	yes	—
314	C₂×#125	C₂²×C₃₁	1	1	yes	characteristic
315	C₂×#124	C₂²×C₃₁	1	1	yes	characteristic
316	C₂×#123	C₂²×C₃₁	1	2	yes	—
317	C₂×C₂²⋊Q₈	C₂²×C₃₁	1	1	yes	characteristic
318	C₂×#121	C₂²×C₃₁	1	1	yes	characteristic
319	C₂×C₄×Q₈	C₂²×C₃₁	1	1	yes	characteristic
320	C₂×C₄×D₄	C₂²×C₃₁	1	1	yes	characteristic
321	C₂×C₄×D₄	C₂²×C₃₁	1	2	yes	—
322	C₂×#117	C₂²×C₃₁	1	2	yes	—
323	C₂²×C₄²	C₂²×C₃₁	1	1	yes	characteristic
324	#391	C₂²×C₃₁	1	4	yes	—
325	#390	C₂²×C₃₁	1	4	yes	—
326	#389	C₂²×C₃₁	1	4	yes	—
327	#389	C₂²×C₃₁	1	4	yes	—
328	#388	C₂²×C₃₁	1	4	yes	—
329	#388	C₂²×C₃₁	1	4	yes	—
330	#388	C₂²×C₃₁	1	4	yes	—
331	#388	C₂²×C₃₁	1	4	yes	—
332	#388	C₂²×C₃₁	1	4	yes	—
333	#388	C₂²×C₃₁	1	4	yes	—
334	#387	C₂²×C₃₁	1	4	yes	—
335	#387	C₂²×C₃₁	1	4	yes	—
336	#387	C₂²×C₃₁	1	4	yes	—
337	#387	C₂²×C₃₁	1	4	yes	—
338	#387	C₂²×C₃₁	1	4	yes	—
339	#387	C₂²×C₃₁	1	4	yes	—
340	#386	C₂²×C₃₁	1	4	yes	—
341	#386	C₂²×C₃₁	1	4	yes	—
342	#385	C₂²×C₃₁	1	4	yes	—
343	#385	C₂²×C₃₁	1	4	yes	—
344	#383	C₂²×C₃₁	1	2	yes	—
345	#383	C₂²×C₃₁	1	2	yes	—
346	#383	C₂²×C₃₁	1	2	yes	—
347	#382	C₂²×C₃₁	1	2	yes	—
348	#382	C₂²×C₃₁	1	2	yes	—
349	#382	C₂²×C₃₁	1	2	yes	—
350	#381	C₂²×C₃₁	1	2	yes	—
351	#381	C₂²×C₃₁	1	2	yes	—
352	#381	C₂²×C₃₁	1	2	yes	—
353	#380	C₂²×C₃₁	1	1	yes	characteristic
354	#380	C₂²×C₃₁	1	1	yes	characteristic
355	#380	C₂²×C₃₁	1	2	yes	—
356	#380	C₂²×C₃₁	1	2	yes	—
357	C₄×C₄⋊C₄	C₂²×C₃₁	1	2	yes	—
358	C₄×C₄⋊C₄	C₂²×C₃₁	1	4	yes	—
359	C₄×C₄⋊C₄	C₂²×C₃₁	1	4	yes	—
360	C₄×C₄⋊C₄	C₂²×C₃₁	1	1	yes	characteristic
361	C₄×C₄⋊C₄	C₂²×C₃₁	1	1	yes	characteristic
362	C₄×C₄⋊C₄	C₂²×C₃₁	1	2	yes	—
363	C₄×C₂²⋊C₄	C₂²×C₃₁	1	2	yes	—
364	C₄×C₂²⋊C₄	C₂²×C₃₁	1	2	yes	—
365	C₄×C₂²⋊C₄	C₂²×C₃₁	1	4	yes	—
366	C₄×C₂²⋊C₄	C₂²×C₃₁	1	2	yes	—
367	C₄×C₂²⋊C₄	C₂²×C₃₁	1	4	yes	—
368	#376	C₂²×C₃₁	1	2	yes	—
369	#376	C₂²×C₃₁	1	2	yes	—
370	#376	C₂²×C₃₁	1	2	yes	—
371	C₂×#95	C₂²×C₃₁	1	2	yes	—
372	C₂×C₃₁	—	4	2	no	—
373	C₂×C₃₁	—	4	4	no	—
374	C₂×C₃₁	#3428	1	1	yes	characteristic
375	C₂×C₃₁	#3428	1	1	yes	characteristic
376	C₂×C₃₁	#3428	1	1	yes	characteristic
377	C₂×C₃₁	#3567	1	2	yes	—
378	C₂×C₃₁	#3568	1	2	yes	—
379	C₂²×Q₈	C₂³×C₃₁	1	1	yes	characteristic
380	C₂²×D₄	C₂³×C₃₁	1	2	yes	—
381	C₂²×D₄	C₂³×C₃₁	1	1	yes	characteristic
382	C₂³×C₄	C₂³×C₃₁	1	4	yes	—
383	C₂³×C₄	C₂³×C₃₁	1	2	yes	—
384	C₂³×C₄	C₂³×C₃₁	1	1	yes	characteristic
385	C₂³×C₄	C₂³×C₃₁	1	1	yes	characteristic
386	C₂³×C₄	C₂³×C₃₁	1	1	yes	characteristic
387	#126	C₂×C₄×C₃₁	1	16	yes	complemented
388	#125	C₂×C₄×C₃₁	1	8	yes	complemented
389	#124	C₂×C₄×C₃₁	1	8	yes	—
390	#123	C₂×C₄×C₃₁	1	16	yes	complemented
391	C₂²⋊Q₈	C₂×C₄×C₃₁	1	8	yes	complemented
392	#121	C₂×C₄×C₃₁	1	8	yes	—
393	C₄×Q₈	C₂×C₄×C₃₁	1	8	yes	complemented
394	C₄×D₄	C₂×C₄×C₃₁	1	8	yes	—
395	C₄×D₄	C₂×C₄×C₃₁	1	16	yes	complemented
396	#117	C₂×C₄×C₃₁	1	16	yes	complemented
397	C₂×C₄⋊C₄	C₂³×C₃₁	1	2	yes	—
398	C₂×C₄⋊C₄	C₂³×C₃₁	1	2	yes	—
399	C₂×C₄⋊C₄	C₂³×C₃₁	1	2	yes	—
400	C₂×C₄⋊C₄	C₂³×C₃₁	1	2	yes	—
401	C₂×C₄⋊C₄	C₂³×C₃₁	1	4	yes	—
402	C₂×C₄⋊C₄	C₂³×C₃₁	1	4	yes	—
403	C₂×C₄⋊C₄	C₂³×C₃₁	1	4	yes	—
404	C₂×C₄⋊C₄	C₂³×C₃₁	1	2	yes	—
405	C₂×C₄⋊C₄	C₂³×C₃₁	1	2	yes	—
406	C₂×C₄⋊C₄	C₂³×C₃₁	1	1	yes	characteristic
407	C₂×C₄⋊C₄	C₂³×C₃₁	1	2	yes	—
408	C₂×C₄⋊C₄	C₂³×C₃₁	1	1	yes	characteristic
409	C₂×C₄⋊C₄	C₂³×C₃₁	1	2	yes	—
410	C₂×C₂²⋊C₄	C₂³×C₃₁	1	2	yes	—
411	C₂×C₂²⋊C₄	C₂³×C₃₁	1	4	yes	—
412	C₂×C₂²⋊C₄	C₂³×C₃₁	1	2	yes	—
413	C₂×C₂²⋊C₄	C₂³×C₃₁	1	4	yes	—
414	C₂×C₂²⋊C₄	C₂³×C₃₁	1	4	yes	—
415	C₂×C₂²⋊C₄	C₂³×C₃₁	1	4	yes	—
416	C₂×C₂²⋊C₄	C₂³×C₃₁	1	2	yes	—
417	C₂×C₂²⋊C₄	C₂³×C₃₁	1	2	yes	—
418	C₂×C₂²⋊C₄	C₂³×C₃₁	1	2	yes	—
419	C₂×C₂²⋊C₄	C₂³×C₃₁	1	2	yes	—
420	C₂×C₂²⋊C₄	C₂³×C₃₁	1	2	yes	—
421	C₂×C₄²	C₂×C₄×C₃₁	1	8	yes	complemented
422	C₂×C₄²	C₂³×C₃₁	1	2	yes	—
423	C₂×C₄²	C₂³×C₃₁	1	4	yes	—
424	C₂×C₄²	C₂³×C₃₁	1	2	yes	—
425	C₂×C₄²	C₂³×C₃₁	1	4	yes	—
426	C₂×C₄²	C₂³×C₃₁	1	4	yes	—
427	C₂×C₄²	C₂³×C₃₁	1	4	yes	—
428	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
429	C₂×C₄²	C₂³×C₃₁	1	2	yes	—
430	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
431	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
432	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
433	#95	C₂³×C₃₁	1	2	yes	—
434	#95	C₂³×C₃₁	1	2	yes	—
435	#95	C₂³×C₃₁	1	4	yes	—
436	#95	C₂³×C₃₁	1	2	yes	—
437	#95	C₂³×C₃₁	1	4	yes	—
438	#95	C₂³×C₃₁	1	4	yes	—
439	#95	C₂³×C₃₁	1	4	yes	—
440	#95	C₂³×C₃₁	1	4	yes	—
441	#95	C₂³×C₃₁	1	2	yes	—
442	#95	C₂³×C₃₁	1	4	yes	—
443	#95	C₂³×C₃₁	1	4	yes	—
444	#95	C₂³×C₃₁	1	4	yes	—
445	#95	C₂³×C₃₁	1	4	yes	—
446	#95	C₂³×C₃₁	1	4	yes	—
447	#95	C₂³×C₃₁	1	4	yes	—
448	#95	C₂³×C₃₁	1	4	yes	—
449	C₃₁	#34405	1	1	yes	characteristic, direct product factor, Sylow
450	C₂⁴	C₂⁴×C₃₁	1	1	yes	characteristic
451	C₂⁴	C₂⁴×C₃₁	1	2	yes	—
452	C₂×Q₈	—	4	1	no	—
453	C₂×Q₈	—	4	1	no	—
454	C₂×Q₈	C₂²×C₄×C₃₁	1	4	yes	—
455	C₂×D₄	—	4	4	no	—
456	C₂×D₄	—	4	1	no	—
457	C₂×D₄	—	4	1	no	—
458	C₂×D₄	C₂²×C₄×C₃₁	1	4	yes	—
459	C₂×D₄	C₂²×C₄×C₃₁	1	8	yes	—
460	C₂²×C₄	—	4	2	no	—
461	C₂²×C₄	—	4	1	no	—
462	C₂²×C₄	—	4	1	no	—
463	C₂²×C₄	—	4	4	no	—
464	C₂²×C₄	—	4	2	no	—
465	C₂²×C₄	—	4	4	no	—
466	C₂²×C₄	—	4	4	no	—
467	C₂²×C₄	—	4	4	no	—
468	C₂²×C₄	C₂²×C₄×C₃₁	1	4	yes	—
469	C₂²×C₄	C₂²×C₄×C₃₁	1	4	yes	—
470	C₂²×C₄	C₂²×C₄×C₃₁	1	4	yes	—
471	C₂²×C₄	C₂²×C₄×C₃₁	1	8	yes	—
472	C₂²×C₄	C₂⁴×C₃₁	1	4	yes	—
473	C₂²×C₄	C₂⁴×C₃₁	1	4	yes	—
474	C₂²×C₄	C₂⁴×C₃₁	1	4	yes	—
475	C₂²×C₄	C₂⁴×C₃₁	1	4	yes	—
476	C₂²×C₄	C₂⁴×C₃₁	1	1	yes	characteristic
477	C₂²×C₄	C₂⁴×C₃₁	1	1	yes	characteristic
478	C₂²×C₄	C₂⁴×C₃₁	1	1	yes	characteristic
479	C₂²×C₄	C₂⁴×C₃₁	1	1	yes	characteristic
480	C₂²×C₄	C₂⁴×C₃₁	1	2	yes	—
481	C₂²×C₄	C₂⁴×C₃₁	1	2	yes	—
482	C₂²×C₄	C₂⁴×C₃₁	1	2	yes	—
483	C₂²×C₄	C₂⁴×C₃₁	1	1	yes	characteristic
484	C₂²×C₄	C₂⁴×C₃₁	1	1	yes	characteristic
485	C₄⋊C₄	—	4	4	no	—
486	C₄⋊C₄	—	4	4	no	—
487	C₄⋊C₄	—	4	2	no	—
488	C₄⋊C₄	—	4	2	no	—
489	C₄⋊C₄	—	4	2	no	—
490	C₄⋊C₄	—	4	2	no	—
491	C₄⋊C₄	—	4	4	no	—
492	C₄⋊C₄	C₂²×C₄×C₃₁	1	8	yes	—
493	C₄⋊C₄	C₂²×C₄×C₃₁	1	8	yes	—
494	C₄⋊C₄	C₂²×C₄×C₃₁	1	4	yes	—
495	C₄⋊C₄	C₂²×C₄×C₃₁	1	8	yes	—
496	C₄⋊C₄	C₂²×C₄×C₃₁	1	4	yes	—
497	C₄⋊C₄	C₂²×C₄×C₃₁	1	8	yes	—
498	C₂²⋊C₄	—	4	4	no	—
499	C₂²⋊C₄	—	4	2	no	—
500	C₂²⋊C₄	—	4	4	no	—
501	C₂²⋊C₄	—	4	4	no	—
502	C₂²⋊C₄	—	4	4	no	—
503	C₂²⋊C₄	—	4	2	no	—
504	C₂²⋊C₄	C₂²×C₄×C₃₁	1	8	yes	—
505	C₂²⋊C₄	C₂²×C₄×C₃₁	1	8	yes	—
506	C₂²⋊C₄	C₂²×C₄×C₃₁	1	8	yes	—
507	C₂²⋊C₄	C₂²×C₄×C₃₁	1	8	yes	—
508	C₂²⋊C₄	C₂²×C₄×C₃₁	1	8	yes	—
509	C₄²	—	4	2	no	—
510	C₄²	—	4	4	no	—
511	C₄²	—	4	2	no	—
512	C₄²	—	4	4	no	—
513	C₄²	—	4	4	no	—
514	C₄²	—	4	4	no	—
515	C₄²	C₂²×C₄×C₃₁	1	4	yes	—
516	C₄²	C₂²×C₄×C₃₁	1	4	yes	—
517	C₄²	C₂²×C₄×C₃₁	1	4	yes	—
518	C₄²	C₂²×C₄×C₃₁	1	4	yes	—
519	C₄²	C₂²×C₄×C₃₁	1	8	yes	—
520	C₂³	—	2	1	no	—
521	C₂³	—	2	2	no	—
522	C₂³	—	2	1	no	—
523	C₂³	—	2	2	no	—
524	C₂³	—	2	4	no	—
525	C₂³	—	2	4	no	—
526	C₂³	—	2	2	no	—
527	C₂³	—	2	2	no	—
528	C₂³	C₂³×C₄×C₃₁	1	4	yes	—
529	C₂³	C₂³×C₄×C₃₁	1	2	yes	—
530	C₂³	C₂⁵×C₃₁	1	1	yes	characteristic
531	Q₈	—	4	4	no	—
532	D₄	—	4	4	no	—
533	D₄	—	4	8	no	—
534	C₂×C₄	—	2	1	no	—
535	C₂×C₄	—	2	2	no	—
536	C₂×C₄	—	2	1	no	—
537	C₂×C₄	—	2	2	no	—
538	C₂×C₄	—	2	4	no	—
539	C₂×C₄	—	2	4	no	—
540	C₂×C₄	—	2	2	no	—
541	C₂×C₄	—	2	2	no	—
542	C₂×C₄	—	2	2	no	—
543	C₂×C₄	—	2	2	no	—
544	C₂×C₄	—	2	1	no	—
545	C₂×C₄	—	2	1	no	—
546	C₂×C₄	—	2	4	no	—
547	C₂×C₄	—	2	4	no	—
548	C₂×C₄	—	2	4	no	—
549	C₂×C₄	—	2	4	no	—
550	C₂×C₄	—	2	4	no	—
551	C₂×C₄	—	2	4	no	—
552	C₂×C₄	—	2	4	no	—
553	C₂×C₄	—	2	4	no	—
554	C₂×C₄	—	2	4	no	—
555	C₂×C₄	—	2	4	no	—
556	C₂×C₄	—	2	4	no	—
557	C₂×C₄	—	2	4	no	—
558	C₂×C₄	—	4	4	no	—
559	C₂×C₄	—	4	8	no	—
560	C₂×C₄	—	4	4	no	—
561	C₂×C₄	—	4	4	no	—
562	C₂×C₄	—	8	8	no	—
563	C₂×C₄	C₂³×C₄×C₃₁	1	2	yes	—
564	C₂×C₄	C₂³×C₄×C₃₁	1	2	yes	—
565	C₂×C₄	C₂³×C₄×C₃₁	1	2	yes	—
566	C₂×C₄	C₂³×C₄×C₃₁	1	2	yes	—
567	C₂×C₄	C₂³×C₄×C₃₁	1	2	yes	—
568	C₂×C₄	C₂³×C₄×C₃₁	1	2	yes	—
569	C₂×C₄	C₂³×C₄×C₃₁	1	4	yes	—
570	C₂×C₄	C₂³×C₄×C₃₁	1	4	yes	—
571	C₂×C₄	C₂³×C₄×C₃₁	1	4	yes	—
572	C₂²	—	2	2	no	—
573	C₂²	—	2	2	no	—
574	C₂²	—	2	2	no	—
575	C₂²	—	2	4	no	—
576	C₂²	—	2	4	no	—
577	C₂²	—	2	4	no	—
578	C₂²	—	4	2	no	—
579	C₂²	—	4	4	no	—
580	C₂²	—	4	4	no	—
581	C₂²	—	4	2	no	—
582	C₂²	C₂⁴×C₄×C₃₁	1	1	yes	characteristic
583	C₂²	C₃₁×#585	1	1	yes	characteristic
584	C₂²	C₃₁×#585	1	2	yes	—
585	C₂²	C₃₁×#585	1	2	yes	—
586	C₂²	C₃₁×#585	1	1	yes	characteristic
587	C₄	—	2	2	no	—
588	C₄	—	2	2	no	—
589	C₄	—	2	2	no	—
590	C₄	—	2	2	no	—
591	C₄	—	2	4	no	—
592	C₄	—	2	2	no	—
593	C₄	—	2	2	no	—
594	C₄	—	2	4	no	—
595	C₄	—	2	4	no	—
596	C₄	—	4	4	no	—
597	C₄	—	4	4	no	—
598	C₄	—	4	4	no	—
599	C₄	—	4	4	no	—
600	C₂	—	4	2	no	—
601	C₂	—	4	4	no	—
602	C₂	C₃₁×#3428	1	1	yes	characteristic
603	C₂	C₃₁×#3428	1	1	yes	characteristic
604	C₂	C₃₁×#3428	1	1	yes	characteristic
605	C₂	C₃₁×#3567	1	2	yes	—
606	C₂	C₃₁×#3568	1	2	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×#2382	C₂	semidirect
3. C₃₁×#2379	C₂	semidirect
4. C₃₁×#2373	C₂	semidirect
5. C₃₁×#2356	C₂	semidirect
6. C₃₁×#2353	C₂	semidirect
7. C₃₁×#2347	C₂	semidirect
8. C₃₁×#2343	C₂	semidirect
9. C₃₁×#2335	C₂	semidirect
11. C₃₁×#2332	C₂	semidirect
12. C₃₁×#2319	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
118. C₂×C₄²×C₃₁	C₂×C₄	semidirect
216. #34405	C₃₁	direct (N × H)
270. #2382	C₂×C₃₁	semidirect
271. #2379	C₂×C₃₁	semidirect
272. #2373	C₂×C₃₁	semidirect
273. #2356	C₂×C₃₁	semidirect
274. #2353	C₂×C₃₁	semidirect
275. #2347	C₂×C₃₁	semidirect
276. #2343	C₂×C₃₁	semidirect
277. #2335	C₂×C₃₁	semidirect
279. #2332	C₂×C₃₁	semidirect
280. #2319	C₂×C₃₁	semidirect
281. #2318	C₂×C₃₁	semidirect
282. #2283	C₂×C₃₁	semidirect
283. #2283	C₂×C₃₁	semidirect
312. #529	C₄×C₃₁	semidirect
387. #126	C₂×C₄×C₃₁	semidirect
388. #125	C₂×C₄×C₃₁	semidirect
390. #123	C₂×C₄×C₃₁	semidirect
391. C₂²⋊Q₈	C₂×C₄×C₃₁	semidirect
393. C₄×Q₈	C₂×C₄×C₃₁	semidirect
395. C₄×D₄	C₂×C₄×C₃₁	semidirect
396. #117	C₂×C₄×C₃₁	semidirect
421. C₂×C₄²	C₂×C₄×C₃₁	semidirect
449. C₃₁	#34405	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×#2382	2
C₃₁×#2379	2
C₃₁×#2373	2
C₃₁×#2356	2
C₃₁×#2353	2
C₃₁×#2347	2
C₃₁×#2343	2
C₃₁×#2335	2
C₃₁×#2333	2
C₃₁×#2332	2
C₃₁×#2319	2
C₃₁×#2318	2
C₃₁×#2283	2
#34405	31

Subgroups — C31×#34405 #21491164

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#34405 #21491164