Subgroups — C₃₁×#35085 #21491844

Order 7936.

2710 conjugacy classes of subgroups.

The subgroup lattice has 670 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃₁×#525	C₂	1	1	yes	maximal, characteristic
2	C₂²×C₃₁×C₂²⋊Q₈	C₂	1	1	yes	maximal, characteristic, complemented
3	C₃₁×#2857	C₂	1	2	yes	maximal, complemented
4	C₃₁×#2850	C₂	1	2	yes	maximal, complemented
5	C₃₁×#2663	C₂	1	4	yes	maximal, complemented
6	C₃₁×#2564	C₂	1	4	yes	maximal, complemented
7	C₃₁×#2503	C₂	1	1	yes	maximal, characteristic, complemented
8	C₃₁×#2501	C₂	1	1	yes	maximal, characteristic, complemented
9	C₃₁×#2499	C₂	1	2	yes	maximal, complemented
10	C₃₁×#2436	C₂	1	1	yes	maximal, characteristic, complemented
11	C₃₁×#2434	C₂	1	1	yes	maximal, characteristic, complemented
12	C₃₁×#2432	C₂	1	2	yes	maximal, complemented
13	C₃₁×#2412	C₂	1	4	yes	maximal, complemented
14	C₃₁×#2322	C₂	1	4	yes	maximal, complemented
15	C₂×C₄×C₂²⋊C₄×C₃₁	C₂	1	1	yes	maximal, characteristic, complemented
16	C₂²×#41×C₃₁	C₂²	1	1	yes	characteristic
17	C₂³×Q₈×C₃₁	C₂²	1	1	yes	characteristic
18	C₂⁴×C₄×C₃₁	C₂²	1	1	yes	characteristic
19	C₂×C₃₁×C₄⋊Q₈	C₂²	1	2	yes	complemented
20	C₂×C₃₁×#124	C₂²	1	2	yes	complemented
21	C₃₁×#525	—	2	8	no	—
22	C₂×C₃₁×#123	C₂²	1	4	yes	—
23	C₂×C₃₁×#123	C₂²	1	2	yes	—
24	C₂×C₃₁×C₂²⋊Q₈	—	2	8	no	—
25	C₂×C₃₁×C₂²⋊Q₈	C₂²	1	2	yes	—
26	C₂×C₃₁×C₂²⋊Q₈	C₂²	1	2	yes	—
27	C₂×C₃₁×C₂²⋊Q₈	C₂²	1	1	yes	characteristic
28	C₂×C₃₁×C₂²⋊Q₈	C₂²	1	2	yes	—
29	C₂×C₃₁×C₂²⋊Q₈	C₂²	1	2	yes	—
30	C₂×C₃₁×C₂²⋊Q₈	C₂²	1	1	yes	characteristic
31	C₂×C₃₁×#121	C₂²	1	1	yes	characteristic
32	C₂×C₃₁×#121	C₂²	1	1	yes	characteristic
33	C₂×C₃₁×#120	C₂²	1	2	yes	—
34	C₂×C₄×D₄×C₃₁	C₂²	1	2	yes	—
35	C₂×C₄×D₄×C₃₁	C₂²	1	2	yes	—
36	C₂×C₃₁×#117	C₂²	1	1	yes	characteristic
37	C₂²×C₄⋊C₄×C₃₁	C₂²	1	2	yes	—
38	C₂²×C₄⋊C₄×C₃₁	C₂²	1	1	yes	characteristic
39	C₂²×C₂²⋊C₄×C₃₁	C₂²	1	2	yes	—
40	C₂²×C₂²⋊C₄×C₃₁	C₂²	1	2	yes	—
41	C₂²×C₄²×C₃₁	C₂²	1	2	yes	—
42	C₃₁×#400	C₂²	1	4	yes	complemented
43	C₃₁×#399	C₂²	1	4	yes	complemented
44	C₃₁×#397	C₂²	1	8	yes	—
45	C₃₁×#395	C₂²	1	4	yes	complemented
46	C₃₁×#394	C₂²	1	8	yes	—
47	C₃₁×#393	C₂²	1	4	yes	complemented
48	C₃₁×#391	C₂²	1	2	yes	complemented
49	C₃₁×#391	C₂²	1	2	yes	complemented
50	C₃₁×#391	C₂²	1	2	yes	complemented
51	C₃₁×#390	C₂²	1	2	yes	complemented
52	C₃₁×#389	C₂²	1	8	yes	complemented
53	C₃₁×#388	C₂²	1	8	yes	complemented
54	C₃₁×#387	C₂²	1	8	yes	complemented
55	C₃₁×#386	C₂²	1	8	yes	—
56	C₃₁×#386	C₂²	1	4	yes	—
57	C₃₁×#385	C₂²	1	4	yes	—
58	C₃₁×#385	C₂²	1	8	yes	complemented
59	C₃₁×#384	C₂²	1	2	yes	complemented
60	C₃₁×#382	C₂²	1	2	yes	complemented
61	C₃₁×#381	C₂²	1	4	yes	—
62	C₃₁×#380	C₂²	1	8	yes	—
63	C₄×C₂²⋊C₄×C₃₁	C₂²	1	2	yes	—
64	C₄×C₂²⋊C₄×C₃₁	C₂²	1	2	yes	—
65	C₄×C₂²⋊C₄×C₃₁	C₂²	1	2	yes	—
66	C₄×C₂²⋊C₄×C₃₁	C₂²	1	1	yes	characteristic
67	C₄×C₂²⋊C₄×C₃₁	C₂²	1	1	yes	characteristic
68	C₂×C₃₁×#95	C₂²	1	2	yes	—
69	C₂⁵×C₃₁	C₂³	1	1	yes	characteristic
70	C₂×#41×C₃₁	—	2	4	no	—
71	C₂×#41×C₃₁	—	4	4	no	—
72	C₂²×Q₈×C₃₁	—	2	4	no	—
73	C₂²×Q₈×C₃₁	—	4	4	no	—
74	C₂²×Q₈×C₃₁	C₂³	1	2	yes	—
75	C₂²×Q₈×C₃₁	C₂³	1	1	yes	characteristic
76	C₂²×Q₈×C₃₁	C₂³	1	2	yes	—
77	C₂²×D₄×C₃₁	C₂³	1	2	yes	—
78	C₂²×D₄×C₃₁	C₂³	1	2	yes	—
79	C₂²×D₄×C₃₁	C₂³	1	1	yes	characteristic
80	C₂²×D₄×C₃₁	C₂³	1	2	yes	—
81	C₂³×C₄×C₃₁	—	4	4	no	—
82	C₂³×C₄×C₃₁	D₄	1	8	yes	—
83	C₂³×C₄×C₃₁	C₂³	1	2	yes	—
84	C₂³×C₄×C₃₁	C₂³	1	4	yes	—
85	C₂³×C₄×C₃₁	C₂³	1	1	yes	characteristic
86	C₂³×C₄×C₃₁	C₂³	1	1	yes	characteristic
87	C₂³×C₄×C₃₁	C₂³	1	1	yes	characteristic
88	C₂³×C₄×C₃₁	C₂³	1	2	yes	—
89	C₂³×C₄×C₃₁	C₂³	1	2	yes	—
90	C₂³×C₄×C₃₁	C₂³	1	1	yes	characteristic
91	C₂³×C₄×C₃₁	C₂³	1	1	yes	characteristic
92	C₂³×C₄×C₃₁	C₂³	1	1	yes	characteristic
93	C₃₁×C₄⋊Q₈	—	4	4	no	—
94	C₃₁×#124	—	4	4	no	—
95	C₃₁×#123	—	2	8	no	—
96	C₃₁×#123	—	2	16	no	—
97	C₃₁×C₂²⋊Q₈	—	2	8	no	—
98	C₃₁×C₂²⋊Q₈	—	2	8	no	—
99	C₃₁×C₂²⋊Q₈	—	2	8	no	—
100	C₃₁×C₂²⋊Q₈	—	2	8	no	—
101	C₃₁×C₂²⋊Q₈	—	2	4	no	—
102	C₃₁×C₂²⋊Q₈	—	2	4	no	—
103	C₃₁×#121	—	2	4	no	—
104	C₃₁×#121	—	2	4	no	—
105	C₃₁×#120	—	2	8	no	—
106	C₄×D₄×C₃₁	—	2	8	no	—
107	C₄×D₄×C₃₁	—	2	8	no	—
108	C₃₁×#117	—	2	4	no	—
109	C₂×C₄⋊C₄×C₃₁	—	2	4	no	—
110	C₂×C₄⋊C₄×C₃₁	—	2	8	no	—
111	C₂×C₄⋊C₄×C₃₁	C₂³	1	2	yes	—
112	C₂×C₄⋊C₄×C₃₁	C₂³	1	8	yes	—
113	C₂×C₄⋊C₄×C₃₁	C₂³	1	2	yes	—
114	C₂×C₄⋊C₄×C₃₁	C₂³	1	2	yes	—
115	C₂×C₄⋊C₄×C₃₁	C₂³	1	4	yes	—
116	C₂×C₄⋊C₄×C₃₁	C₂³	1	4	yes	—
117	C₂×C₄⋊C₄×C₃₁	C₂³	1	1	yes	characteristic
118	C₂×C₄⋊C₄×C₃₁	C₂³	1	1	yes	characteristic
119	C₂×C₄⋊C₄×C₃₁	C₂³	1	2	yes	—
120	C₂×C₄⋊C₄×C₃₁	C₂³	1	8	yes	—
121	C₂×C₂²⋊C₄×C₃₁	—	2	8	no	—
122	C₂×C₂²⋊C₄×C₃₁	—	2	8	no	—
123	C₂×C₂²⋊C₄×C₃₁	C₂³	1	2	yes	—
124	C₂×C₂²⋊C₄×C₃₁	C₂³	1	4	yes	—
125	C₂×C₂²⋊C₄×C₃₁	C₂³	1	4	yes	—
126	C₂×C₂²⋊C₄×C₃₁	C₂³	1	2	yes	—
127	C₂×C₂²⋊C₄×C₃₁	C₂³	1	8	yes	—
128	C₂×C₂²⋊C₄×C₃₁	C₂³	1	4	yes	—
129	C₂×C₂²⋊C₄×C₃₁	C₂³	1	2	yes	—
130	C₂×C₂²⋊C₄×C₃₁	C₂³	1	2	yes	—
131	C₂×C₂²⋊C₄×C₃₁	C₂³	1	2	yes	—
132	C₂×C₂²⋊C₄×C₃₁	C₂³	1	2	yes	—
133	C₂×C₂²⋊C₄×C₃₁	C₂³	1	8	yes	—
134	C₂×C₂²⋊C₄×C₃₁	C₂³	1	2	yes	—
135	C₂×C₄²×C₃₁	—	4	4	no	—
136	C₂×C₄²×C₃₁	C₂³	1	2	yes	—
137	C₂×C₄²×C₃₁	C₂³	1	2	yes	—
138	C₂×C₄²×C₃₁	C₂³	1	2	yes	—
139	C₂×C₄²×C₃₁	C₂³	1	4	yes	—
140	C₃₁×#95	C₂³	1	2	yes	—
141	C₃₁×#95	C₂³	1	8	yes	—
142	C₃₁×#95	C₂³	1	8	yes	—
143	C₃₁×#95	C₂³	1	8	yes	—
144	C₃₁×#95	C₂³	1	2	yes	—
145	C₃₁×#95	C₂³	1	8	yes	—
146	C₃₁×#95	C₂³	1	4	yes	—
147	C₂⁴×C₃₁	—	2	4	no	—
148	C₂⁴×C₃₁	—	2	4	no	—
149	C₂⁴×C₃₁	—	4	2	no	—
150	C₂⁴×C₃₁	C₂×D₄	1	4	yes	—
151	C₂⁴×C₃₁	C₂⁴	1	2	yes	—
152	C₂⁴×C₃₁	C₂⁴	1	1	yes	characteristic
153	C₂⁴×C₃₁	C₂⁴	1	1	yes	characteristic
154	C₂⁴×C₃₁	C₂⁴	1	1	yes	characteristic
155	#41×C₃₁	—	4	16	no	—
156	C₂×Q₈×C₃₁	—	2	4	no	—
157	C₂×Q₈×C₃₁	—	2	4	no	—
158	C₂×Q₈×C₃₁	—	2	2	no	—
159	C₂×Q₈×C₃₁	—	2	4	no	—
160	C₂×Q₈×C₃₁	—	2	4	no	—
161	C₂×Q₈×C₃₁	—	2	4	no	—
162	C₂×Q₈×C₃₁	—	4	4	no	—
163	C₂×Q₈×C₃₁	—	4	16	no	—
164	C₂×D₄×C₃₁	—	2	4	no	—
165	C₂×D₄×C₃₁	—	2	4	no	—
166	C₂×D₄×C₃₁	—	2	4	no	—
167	C₂×D₄×C₃₁	—	2	8	no	—
168	C₂×D₄×C₃₁	—	2	8	no	—
169	C₂×D₄×C₃₁	—	2	2	no	—
170	C₂×D₄×C₃₁	—	2	4	no	—
171	C₂×D₄×C₃₁	—	4	4	no	—
172	C₂²×C₄×C₃₁	—	2	4	no	—
173	C₂²×C₄×C₃₁	—	2	4	no	—
174	C₂²×C₄×C₃₁	—	2	4	no	—
175	C₂²×C₄×C₃₁	—	2	4	no	—
176	C₂²×C₄×C₃₁	—	2	4	no	—
177	C₂²×C₄×C₃₁	—	2	8	no	—
178	C₂²×C₄×C₃₁	—	2	8	no	—
179	C₂²×C₄×C₃₁	—	2	4	no	—
180	C₂²×C₄×C₃₁	—	2	8	no	—
181	C₂²×C₄×C₃₁	—	4	16	no	—
182	C₂²×C₄×C₃₁	—	4	4	no	—
183	C₂²×C₄×C₃₁	—	4	2	no	—
184	C₂²×C₄×C₃₁	—	4	2	no	—
185	C₂²×C₄×C₃₁	—	4	2	no	—
186	C₂²×C₄×C₃₁	—	4	4	no	—
187	C₂²×C₄×C₃₁	—	4	4	no	—
188	C₂²×C₄×C₃₁	C₂×D₄	1	4	yes	—
189	C₂²×C₄×C₃₁	C₂×D₄	1	4	yes	—
190	C₂²×C₄×C₃₁	C₂×D₄	1	4	yes	—
191	C₂²×C₄×C₃₁	C₂×D₄	1	4	yes	—
192	C₂²×C₄×C₃₁	C₂×D₄	1	4	yes	—
193	C₂²×C₄×C₃₁	C₂×D₄	1	4	yes	—
194	C₂²×C₄×C₃₁	C₂⁴	1	4	yes	—
195	C₂²×C₄×C₃₁	C₂⁴	1	2	yes	—
196	C₂²×C₄×C₃₁	C₂⁴	1	2	yes	—
197	C₂²×C₄×C₃₁	C₂⁴	1	4	yes	—
198	C₂²×C₄×C₃₁	C₂⁴	1	4	yes	—
199	C₂²×C₄×C₃₁	C₂⁴	1	2	yes	—
200	C₂²×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
201	C₂²×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
202	C₂²×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
203	C₂²×C₄×C₃₁	C₂⁴	1	4	yes	—
204	C₂²×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
205	C₄⋊C₄×C₃₁	—	2	4	no	—
206	C₄⋊C₄×C₃₁	—	2	4	no	—
207	C₄⋊C₄×C₃₁	—	2	8	no	—
208	C₄⋊C₄×C₃₁	—	2	8	no	—
209	C₄⋊C₄×C₃₁	—	2	2	no	—
210	C₄⋊C₄×C₃₁	—	2	2	no	—
211	C₄⋊C₄×C₃₁	—	2	4	no	—
212	C₄⋊C₄×C₃₁	—	2	4	no	—
213	C₄⋊C₄×C₃₁	—	2	16	no	—
214	C₄⋊C₄×C₃₁	—	4	8	no	—
215	C₂²⋊C₄×C₃₁	—	2	8	no	—
216	C₂²⋊C₄×C₃₁	—	2	8	no	—
217	C₂²⋊C₄×C₃₁	—	2	4	no	—
218	C₂²⋊C₄×C₃₁	—	2	4	no	—
219	C₂²⋊C₄×C₃₁	—	2	4	no	—
220	C₂²⋊C₄×C₃₁	—	2	4	no	—
221	C₂²⋊C₄×C₃₁	—	2	4	no	—
222	C₂²⋊C₄×C₃₁	—	2	4	no	—
223	C₂²⋊C₄×C₃₁	—	2	8	no	—
224	C₂²⋊C₄×C₃₁	—	2	16	no	—
225	C₂²⋊C₄×C₃₁	—	4	8	no	—
226	C₂²⋊C₄×C₃₁	#41	1	8	yes	complemented
227	C₄²×C₃₁	—	2	4	no	—
228	C₄²×C₃₁	—	2	4	no	—
229	C₄²×C₃₁	—	2	4	no	—
230	C₄²×C₃₁	—	4	4	no	—
231	#35085	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
232	C₂³×C₃₁	—	2	2	no	—
233	C₂³×C₃₁	—	2	2	no	—
234	C₂³×C₃₁	—	2	2	no	—
235	C₂³×C₃₁	—	2	2	no	—
236	C₂³×C₃₁	—	2	8	no	—
237	C₂³×C₃₁	—	2	8	no	—
238	C₂³×C₃₁	—	2	2	no	—
239	C₂³×C₃₁	—	2	2	no	—
240	C₂³×C₃₁	—	2	2	no	—
241	C₂³×C₃₁	—	2	2	no	—
242	C₂³×C₃₁	—	2	4	no	—
243	C₂³×C₃₁	—	2	4	no	—
244	C₂³×C₃₁	—	2	4	no	—
245	C₂³×C₃₁	—	4	4	no	—
246	C₂³×C₃₁	—	4	16	no	—
247	C₂³×C₃₁	C₂²×D₄	1	2	yes	—
248	C₂³×C₃₁	C₂²×D₄	1	2	yes	—
249	C₂³×C₃₁	C₂²×D₄	1	2	yes	—
250	C₂³×C₃₁	C₂×#41	1	4	yes	—
251	C₂³×C₃₁	2₋¹⁺⁴	1	4	yes	—
252	C₂³×C₃₁	C₂⁵	1	1	yes	characteristic
253	Q₈×C₃₁	—	4	8	no	—
254	Q₈×C₃₁	—	4	8	no	—
255	Q₈×C₃₁	—	4	4	no	—
256	D₄×C₃₁	—	4	8	no	—
257	D₄×C₃₁	—	4	8	no	—
258	D₄×C₃₁	—	4	8	no	—
259	D₄×C₃₁	—	4	4	no	—
260	C₂×C₄×C₃₁	—	2	4	no	—
261	C₂×C₄×C₃₁	—	2	2	no	—
262	C₂×C₄×C₃₁	—	2	2	no	—
263	C₂×C₄×C₃₁	—	2	2	no	—
264	C₂×C₄×C₃₁	—	2	4	no	—
265	C₂×C₄×C₃₁	—	2	8	no	—
266	C₂×C₄×C₃₁	—	2	8	no	—
267	C₂×C₄×C₃₁	—	2	8	no	—
268	C₂×C₄×C₃₁	—	2	2	no	—
269	C₂×C₄×C₃₁	—	2	2	no	—
270	C₂×C₄×C₃₁	—	2	4	no	—
271	C₂×C₄×C₃₁	—	2	2	no	—
272	C₂×C₄×C₃₁	—	2	4	no	—
273	C₂×C₄×C₃₁	—	2	4	no	—
274	C₂×C₄×C₃₁	—	2	8	no	—
275	C₂×C₄×C₃₁	—	2	4	no	—
276	C₂×C₄×C₃₁	—	2	4	no	—
277	C₂×C₄×C₃₁	—	2	4	no	—
278	C₂×C₄×C₃₁	—	2	4	no	—
279	C₂×C₄×C₃₁	—	2	4	no	—
280	C₂×C₄×C₃₁	—	2	4	no	—
281	C₂×C₄×C₃₁	—	2	4	no	—
282	C₂×C₄×C₃₁	—	4	4	no	—
283	C₂×C₄×C₃₁	—	4	4	no	—
284	C₂×C₄×C₃₁	—	4	4	no	—
285	C₂×C₄×C₃₁	—	4	16	no	—
286	C₂×C₄×C₃₁	—	4	8	no	—
287	C₂×C₄×C₃₁	—	4	8	no	—
288	C₂×C₄×C₃₁	—	4	8	no	—
289	C₂×C₄×C₃₁	C₂²×D₄	1	2	yes	—
290	C₂×C₄×C₃₁	C₂²×D₄	1	2	yes	—
291	C₂×C₄×C₃₁	C₂²×D₄	1	2	yes	—
292	C₂×C₄×C₃₁	C₂²×D₄	1	2	yes	—
293	C₂×C₄×C₃₁	C₂×#41	1	8	yes	—
294	C₂×C₄×C₃₁	H(5,𝔽₂)	1	4	yes	—
295	C₂×#525	C₂×C₃₁	1	1	yes	characteristic
296	C₂²×C₂²⋊Q₈	C₂×C₃₁	1	1	yes	characteristic, complemented
297	#2857	C₂×C₃₁	1	2	yes	complemented
298	#2850	C₂×C₃₁	1	2	yes	complemented
299	#2663	C₂×C₃₁	1	4	yes	complemented
300	#2564	C₂×C₃₁	1	4	yes	complemented
301	#2503	C₂×C₃₁	1	1	yes	characteristic, complemented
302	#2501	C₂×C₃₁	1	1	yes	characteristic, complemented
303	#2499	C₂×C₃₁	1	2	yes	complemented
304	#2436	C₂×C₃₁	1	1	yes	characteristic, complemented
305	#2434	C₂×C₃₁	1	1	yes	characteristic, complemented
306	#2432	C₂×C₃₁	1	2	yes	complemented
307	#2412	C₂×C₃₁	1	4	yes	complemented
308	#2322	C₂×C₃₁	1	4	yes	complemented
309	C₂×C₄×C₂²⋊C₄	C₂×C₃₁	1	1	yes	characteristic, complemented
310	C₂²×C₃₁	—	2	2	no	—
311	C₂²×C₃₁	—	2	2	no	—
312	C₂²×C₃₁	—	2	2	no	—
313	C₂²×C₃₁	—	2	2	no	—
314	C₂²×C₃₁	—	2	8	no	—
315	C₂²×C₃₁	—	2	8	no	—
316	C₂²×C₃₁	—	2	2	no	—
317	C₂²×C₃₁	—	2	2	no	—
318	C₂²×C₃₁	—	2	2	no	—
319	C₂²×C₃₁	—	4	16	no	—
320	C₂²×C₃₁	—	4	4	no	—
321	C₂²×C₃₁	—	4	2	no	—
322	C₂²×C₃₁	—	4	2	no	—
323	C₂²×C₃₁	—	4	2	no	—
324	C₂²×C₃₁	—	4	8	no	—
325	C₂²×C₃₁	#532	1	4	yes	—
326	C₂²×C₃₁	#536	1	4	yes	—
327	C₂²×C₃₁	C₂³×D₄	1	1	yes	characteristic
328	C₂²×C₃₁	C₂²×#41	1	2	yes	—
329	C₂²×C₃₁	C₂×H(5,𝔽₂)	1	2	yes	—
330	C₂²×C₃₁	C₂×2₋¹⁺⁴	1	2	yes	—
331	C₄×C₃₁	—	2	2	no	—
332	C₄×C₃₁	—	2	2	no	—
333	C₄×C₃₁	—	2	2	no	—
334	C₄×C₃₁	—	2	8	no	—
335	C₄×C₃₁	—	4	4	no	—
336	C₄×C₃₁	—	4	2	no	—
337	C₄×C₃₁	—	4	2	no	—
338	C₄×C₃₁	—	4	4	no	—
339	C₄×C₃₁	—	4	2	no	—
340	C₄×C₃₁	—	4	4	no	—
341	C₄×C₃₁	#534	1	4	yes	—
342	C₂²×#41	C₂²×C₃₁	1	1	yes	characteristic
343	C₂³×Q₈	C₂²×C₃₁	1	1	yes	characteristic
344	C₂⁴×C₄	C₂²×C₃₁	1	1	yes	characteristic
345	C₂×C₄⋊Q₈	C₂²×C₃₁	1	2	yes	complemented
346	C₂×#124	C₂²×C₃₁	1	2	yes	complemented
347	#525	—	2	8	no	—
348	C₂×#123	C₂²×C₃₁	1	4	yes	—
349	C₂×#123	C₂²×C₃₁	1	2	yes	—
350	C₂×C₂²⋊Q₈	—	2	8	no	—
351	C₂×C₂²⋊Q₈	C₂²×C₃₁	1	2	yes	—
352	C₂×C₂²⋊Q₈	C₂²×C₃₁	1	2	yes	—
353	C₂×C₂²⋊Q₈	C₂²×C₃₁	1	1	yes	characteristic
354	C₂×C₂²⋊Q₈	C₂²×C₃₁	1	2	yes	—
355	C₂×C₂²⋊Q₈	C₂²×C₃₁	1	2	yes	—
356	C₂×C₂²⋊Q₈	C₂²×C₃₁	1	1	yes	characteristic
357	C₂×#121	C₂²×C₃₁	1	1	yes	characteristic
358	C₂×#121	C₂²×C₃₁	1	1	yes	characteristic
359	C₂×#120	C₂²×C₃₁	1	2	yes	—
360	C₂×C₄×D₄	C₂²×C₃₁	1	2	yes	—
361	C₂×C₄×D₄	C₂²×C₃₁	1	2	yes	—
362	C₂×#117	C₂²×C₃₁	1	1	yes	characteristic
363	C₂²×C₄⋊C₄	C₂²×C₃₁	1	2	yes	—
364	C₂²×C₄⋊C₄	C₂²×C₃₁	1	1	yes	characteristic
365	C₂²×C₂²⋊C₄	C₂²×C₃₁	1	2	yes	—
366	C₂²×C₂²⋊C₄	C₂²×C₃₁	1	2	yes	—
367	C₂²×C₄²	C₂²×C₃₁	1	2	yes	—
368	#400	C₂²×C₃₁	1	4	yes	complemented
369	#399	C₂²×C₃₁	1	4	yes	complemented
370	#397	C₂²×C₃₁	1	8	yes	—
371	#395	C₂²×C₃₁	1	4	yes	complemented
372	#394	C₂²×C₃₁	1	8	yes	—
373	#393	C₂²×C₃₁	1	4	yes	complemented
374	#391	C₂²×C₃₁	1	2	yes	complemented
375	#391	C₂²×C₃₁	1	2	yes	complemented
376	#391	C₂²×C₃₁	1	2	yes	complemented
377	#390	C₂²×C₃₁	1	2	yes	complemented
378	#389	C₂²×C₃₁	1	8	yes	complemented
379	#388	C₂²×C₃₁	1	8	yes	complemented
380	#387	C₂²×C₃₁	1	8	yes	complemented
381	#386	C₂²×C₃₁	1	8	yes	—
382	#386	C₂²×C₃₁	1	4	yes	—
383	#385	C₂²×C₃₁	1	4	yes	—
384	#385	C₂²×C₃₁	1	8	yes	complemented
385	#384	C₂²×C₃₁	1	2	yes	complemented
386	#382	C₂²×C₃₁	1	2	yes	complemented
387	#381	C₂²×C₃₁	1	4	yes	—
388	#380	C₂²×C₃₁	1	8	yes	—
389	C₄×C₂²⋊C₄	C₂²×C₃₁	1	2	yes	—
390	C₄×C₂²⋊C₄	C₂²×C₃₁	1	2	yes	—
391	C₄×C₂²⋊C₄	C₂²×C₃₁	1	2	yes	—
392	C₄×C₂²⋊C₄	C₂²×C₃₁	1	1	yes	characteristic
393	C₄×C₂²⋊C₄	C₂²×C₃₁	1	1	yes	characteristic
394	C₂×#95	C₂²×C₃₁	1	2	yes	—
395	C₂×C₃₁	—	2	4	no	—
396	C₂×C₃₁	—	2	4	no	—
397	C₂×C₃₁	—	4	2	no	—
398	C₂×C₃₁	—	8	2	no	—
399	C₂×C₃₁	C₂×#532	1	1	yes	characteristic
400	C₂×C₃₁	C₂×#534	1	1	yes	characteristic
401	C₂×C₃₁	C₂×#536	1	1	yes	characteristic
402	C₂×C₃₁	#3520	1	4	yes	—
403	C₂⁵	C₂³×C₃₁	1	1	yes	characteristic
404	C₂×#41	—	2	4	no	—
405	C₂×#41	—	4	4	no	—
406	C₂²×Q₈	—	2	4	no	—
407	C₂²×Q₈	—	4	4	no	—
408	C₂²×Q₈	C₂³×C₃₁	1	2	yes	—
409	C₂²×Q₈	C₂³×C₃₁	1	1	yes	characteristic
410	C₂²×Q₈	C₂³×C₃₁	1	2	yes	—
411	C₂²×D₄	C₂³×C₃₁	1	2	yes	—
412	C₂²×D₄	C₂³×C₃₁	1	2	yes	—
413	C₂²×D₄	C₂³×C₃₁	1	1	yes	characteristic
414	C₂²×D₄	C₂³×C₃₁	1	2	yes	—
415	C₂³×C₄	—	4	4	no	—
416	C₂³×C₄	D₄×C₃₁	1	8	yes	—
417	C₂³×C₄	C₂³×C₃₁	1	2	yes	—
418	C₂³×C₄	C₂³×C₃₁	1	4	yes	—
419	C₂³×C₄	C₂³×C₃₁	1	1	yes	characteristic
420	C₂³×C₄	C₂³×C₃₁	1	1	yes	characteristic
421	C₂³×C₄	C₂³×C₃₁	1	1	yes	characteristic
422	C₂³×C₄	C₂³×C₃₁	1	2	yes	—
423	C₂³×C₄	C₂³×C₃₁	1	2	yes	—
424	C₂³×C₄	C₂³×C₃₁	1	1	yes	characteristic
425	C₂³×C₄	C₂³×C₃₁	1	1	yes	characteristic
426	C₂³×C₄	C₂³×C₃₁	1	1	yes	characteristic
427	C₄⋊Q₈	—	4	4	no	—
428	#124	—	4	4	no	—
429	#123	—	2	8	no	—
430	#123	—	2	16	no	—
431	C₂²⋊Q₈	—	2	8	no	—
432	C₂²⋊Q₈	—	2	8	no	—
433	C₂²⋊Q₈	—	2	8	no	—
434	C₂²⋊Q₈	—	2	8	no	—
435	C₂²⋊Q₈	—	2	4	no	—
436	C₂²⋊Q₈	—	2	4	no	—
437	#121	—	2	4	no	—
438	#121	—	2	4	no	—
439	#120	—	2	8	no	—
440	C₄×D₄	—	2	8	no	—
441	C₄×D₄	—	2	8	no	—
442	#117	—	2	4	no	—
443	C₂×C₄⋊C₄	—	2	4	no	—
444	C₂×C₄⋊C₄	—	2	8	no	—
445	C₂×C₄⋊C₄	C₂³×C₃₁	1	2	yes	—
446	C₂×C₄⋊C₄	C₂³×C₃₁	1	4	yes	—
447	C₂×C₄⋊C₄	C₂³×C₃₁	1	4	yes	—
448	C₂×C₄⋊C₄	C₂³×C₃₁	1	2	yes	—
449	C₂×C₄⋊C₄	C₂³×C₃₁	1	1	yes	characteristic
450	C₂×C₄⋊C₄	C₂³×C₃₁	1	1	yes	characteristic
451	C₂×C₄⋊C₄	C₂³×C₃₁	1	2	yes	—
452	C₂×C₄⋊C₄	C₂³×C₃₁	1	2	yes	—
453	C₂×C₄⋊C₄	C₂³×C₃₁	1	8	yes	—
454	C₂×C₄⋊C₄	C₂³×C₃₁	1	8	yes	—
455	C₂×C₂²⋊C₄	—	2	8	no	—
456	C₂×C₂²⋊C₄	—	2	8	no	—
457	C₂×C₂²⋊C₄	C₂³×C₃₁	1	4	yes	—
458	C₂×C₂²⋊C₄	C₂³×C₃₁	1	4	yes	—
459	C₂×C₂²⋊C₄	C₂³×C₃₁	1	2	yes	—
460	C₂×C₂²⋊C₄	C₂³×C₃₁	1	4	yes	—
461	C₂×C₂²⋊C₄	C₂³×C₃₁	1	2	yes	—
462	C₂×C₂²⋊C₄	C₂³×C₃₁	1	2	yes	—
463	C₂×C₂²⋊C₄	C₂³×C₃₁	1	2	yes	—
464	C₂×C₂²⋊C₄	C₂³×C₃₁	1	2	yes	—
465	C₂×C₂²⋊C₄	C₂³×C₃₁	1	2	yes	—
466	C₂×C₂²⋊C₄	C₂³×C₃₁	1	8	yes	—
467	C₂×C₂²⋊C₄	C₂³×C₃₁	1	8	yes	—
468	C₂×C₂²⋊C₄	C₂³×C₃₁	1	2	yes	—
469	C₂×C₄²	—	4	4	no	—
470	C₂×C₄²	C₂³×C₃₁	1	2	yes	—
471	C₂×C₄²	C₂³×C₃₁	1	2	yes	—
472	C₂×C₄²	C₂³×C₃₁	1	2	yes	—
473	C₂×C₄²	C₂³×C₃₁	1	4	yes	—
474	#95	C₂³×C₃₁	1	4	yes	—
475	#95	C₂³×C₃₁	1	2	yes	—
476	#95	C₂³×C₃₁	1	2	yes	—
477	#95	C₂³×C₃₁	1	8	yes	—
478	#95	C₂³×C₃₁	1	8	yes	—
479	#95	C₂³×C₃₁	1	8	yes	—
480	#95	C₂³×C₃₁	1	8	yes	—
481	C₃₁	#35085	1	1	yes	characteristic, direct product factor, Sylow
482	C₂⁴	—	2	4	no	—
483	C₂⁴	—	2	4	no	—
484	C₂⁴	—	4	2	no	—
485	C₂⁴	C₂×D₄×C₃₁	1	4	yes	—
486	C₂⁴	C₂⁴×C₃₁	1	2	yes	—
487	C₂⁴	C₂⁴×C₃₁	1	1	yes	characteristic
488	C₂⁴	C₂⁴×C₃₁	1	1	yes	characteristic
489	C₂⁴	C₂⁴×C₃₁	1	1	yes	characteristic
490	#41	—	4	16	no	—
491	C₂×Q₈	—	2	4	no	—
492	C₂×Q₈	—	2	4	no	—
493	C₂×Q₈	—	2	2	no	—
494	C₂×Q₈	—	2	4	no	—
495	C₂×Q₈	—	2	4	no	—
496	C₂×Q₈	—	2	4	no	—
497	C₂×Q₈	—	4	4	no	—
498	C₂×Q₈	—	4	16	no	—
499	C₂×D₄	—	2	4	no	—
500	C₂×D₄	—	2	4	no	—
501	C₂×D₄	—	2	4	no	—
502	C₂×D₄	—	2	8	no	—
503	C₂×D₄	—	2	8	no	—
504	C₂×D₄	—	2	2	no	—
505	C₂×D₄	—	2	4	no	—
506	C₂×D₄	—	4	4	no	—
507	C₂²×C₄	—	2	4	no	—
508	C₂²×C₄	—	2	4	no	—
509	C₂²×C₄	—	2	4	no	—
510	C₂²×C₄	—	2	4	no	—
511	C₂²×C₄	—	2	4	no	—
512	C₂²×C₄	—	2	8	no	—
513	C₂²×C₄	—	2	8	no	—
514	C₂²×C₄	—	2	4	no	—
515	C₂²×C₄	—	2	8	no	—
516	C₂²×C₄	—	4	16	no	—
517	C₂²×C₄	—	4	4	no	—
518	C₂²×C₄	—	4	2	no	—
519	C₂²×C₄	—	4	2	no	—
520	C₂²×C₄	—	4	2	no	—
521	C₂²×C₄	—	4	4	no	—
522	C₂²×C₄	—	4	4	no	—
523	C₂²×C₄	C₂×D₄×C₃₁	1	4	yes	—
524	C₂²×C₄	C₂×D₄×C₃₁	1	4	yes	—
525	C₂²×C₄	C₂×D₄×C₃₁	1	4	yes	—
526	C₂²×C₄	C₂×D₄×C₃₁	1	4	yes	—
527	C₂²×C₄	C₂×D₄×C₃₁	1	4	yes	—
528	C₂²×C₄	C₂×D₄×C₃₁	1	4	yes	—
529	C₂²×C₄	C₂⁴×C₃₁	1	4	yes	—
530	C₂²×C₄	C₂⁴×C₃₁	1	2	yes	—
531	C₂²×C₄	C₂⁴×C₃₁	1	2	yes	—
532	C₂²×C₄	C₂⁴×C₃₁	1	4	yes	—
533	C₂²×C₄	C₂⁴×C₃₁	1	4	yes	—
534	C₂²×C₄	C₂⁴×C₃₁	1	2	yes	—
535	C₂²×C₄	C₂⁴×C₃₁	1	1	yes	characteristic
536	C₂²×C₄	C₂⁴×C₃₁	1	1	yes	characteristic
537	C₂²×C₄	C₂⁴×C₃₁	1	1	yes	characteristic
538	C₂²×C₄	C₂⁴×C₃₁	1	4	yes	—
539	C₂²×C₄	C₂⁴×C₃₁	1	1	yes	characteristic
540	C₄⋊C₄	—	2	4	no	—
541	C₄⋊C₄	—	2	4	no	—
542	C₄⋊C₄	—	2	8	no	—
543	C₄⋊C₄	—	2	8	no	—
544	C₄⋊C₄	—	2	2	no	—
545	C₄⋊C₄	—	2	2	no	—
546	C₄⋊C₄	—	2	4	no	—
547	C₄⋊C₄	—	2	4	no	—
548	C₄⋊C₄	—	2	16	no	—
549	C₄⋊C₄	—	4	8	no	—
550	C₂²⋊C₄	—	2	8	no	—
551	C₂²⋊C₄	—	2	8	no	—
552	C₂²⋊C₄	—	2	4	no	—
553	C₂²⋊C₄	—	2	4	no	—
554	C₂²⋊C₄	—	2	4	no	—
555	C₂²⋊C₄	—	2	4	no	—
556	C₂²⋊C₄	—	2	4	no	—
557	C₂²⋊C₄	—	2	4	no	—
558	C₂²⋊C₄	—	2	8	no	—
559	C₂²⋊C₄	—	2	16	no	—
560	C₂²⋊C₄	—	4	8	no	—
561	C₂²⋊C₄	#41×C₃₁	1	8	yes	complemented
562	C₄²	—	2	4	no	—
563	C₄²	—	2	4	no	—
564	C₄²	—	2	4	no	—
565	C₄²	—	4	4	no	—
566	C₂³	—	2	2	no	—
567	C₂³	—	2	2	no	—
568	C₂³	—	2	2	no	—
569	C₂³	—	2	2	no	—
570	C₂³	—	2	8	no	—
571	C₂³	—	2	8	no	—
572	C₂³	—	2	2	no	—
573	C₂³	—	2	2	no	—
574	C₂³	—	2	2	no	—
575	C₂³	—	2	2	no	—
576	C₂³	—	2	4	no	—
577	C₂³	—	2	4	no	—
578	C₂³	—	2	4	no	—
579	C₂³	—	4	4	no	—
580	C₂³	—	4	16	no	—
581	C₂³	C₂²×D₄×C₃₁	1	2	yes	—
582	C₂³	C₂²×D₄×C₃₁	1	2	yes	—
583	C₂³	C₂²×D₄×C₃₁	1	2	yes	—
584	C₂³	C₂×#41×C₃₁	1	4	yes	—
585	C₂³	C₃₁×2₋¹⁺⁴	1	4	yes	—
586	C₂³	C₂⁵×C₃₁	1	1	yes	characteristic
587	Q₈	—	4	8	no	—
588	Q₈	—	4	8	no	—
589	Q₈	—	4	4	no	—
590	D₄	—	4	8	no	—
591	D₄	—	4	8	no	—
592	D₄	—	4	8	no	—
593	D₄	—	4	4	no	—
594	C₂×C₄	—	2	4	no	—
595	C₂×C₄	—	2	2	no	—
596	C₂×C₄	—	2	2	no	—
597	C₂×C₄	—	2	2	no	—
598	C₂×C₄	—	2	4	no	—
599	C₂×C₄	—	2	8	no	—
600	C₂×C₄	—	2	8	no	—
601	C₂×C₄	—	2	8	no	—
602	C₂×C₄	—	2	2	no	—
603	C₂×C₄	—	2	2	no	—
604	C₂×C₄	—	2	4	no	—
605	C₂×C₄	—	2	2	no	—
606	C₂×C₄	—	2	4	no	—
607	C₂×C₄	—	2	4	no	—
608	C₂×C₄	—	2	8	no	—
609	C₂×C₄	—	2	4	no	—
610	C₂×C₄	—	2	4	no	—
611	C₂×C₄	—	2	4	no	—
612	C₂×C₄	—	2	4	no	—
613	C₂×C₄	—	2	4	no	—
614	C₂×C₄	—	2	4	no	—
615	C₂×C₄	—	2	4	no	—
616	C₂×C₄	—	4	4	no	—
617	C₂×C₄	—	4	4	no	—
618	C₂×C₄	—	4	4	no	—
619	C₂×C₄	—	4	16	no	—
620	C₂×C₄	—	4	8	no	—
621	C₂×C₄	—	4	8	no	—
622	C₂×C₄	—	4	8	no	—
623	C₂×C₄	C₂²×D₄×C₃₁	1	2	yes	—
624	C₂×C₄	C₂²×D₄×C₃₁	1	2	yes	—
625	C₂×C₄	C₂²×D₄×C₃₁	1	2	yes	—
626	C₂×C₄	C₂²×D₄×C₃₁	1	2	yes	—
627	C₂×C₄	C₂×#41×C₃₁	1	8	yes	—
628	C₂×C₄	C₃₁×H(5,𝔽₂)	1	4	yes	—
629	C₂²	—	2	2	no	—
630	C₂²	—	2	2	no	—
631	C₂²	—	2	2	no	—
632	C₂²	—	2	2	no	—
633	C₂²	—	2	8	no	—
634	C₂²	—	2	8	no	—
635	C₂²	—	2	2	no	—
636	C₂²	—	2	2	no	—
637	C₂²	—	2	2	no	—
638	C₂²	—	4	16	no	—
639	C₂²	—	4	4	no	—
640	C₂²	—	4	2	no	—
641	C₂²	—	4	2	no	—
642	C₂²	—	4	2	no	—
643	C₂²	—	4	8	no	—
644	C₂²	C₃₁×#532	1	4	yes	—
645	C₂²	C₃₁×#536	1	4	yes	—
646	C₂²	C₂³×D₄×C₃₁	1	1	yes	characteristic
647	C₂²	C₂²×#41×C₃₁	1	2	yes	—
648	C₂²	C₂×C₃₁×H(5,𝔽₂)	1	2	yes	—
649	C₂²	C₂×C₃₁×2₋¹⁺⁴	1	2	yes	—
650	C₄	—	2	2	no	—
651	C₄	—	2	2	no	—
652	C₄	—	2	2	no	—
653	C₄	—	2	8	no	—
654	C₄	—	4	4	no	—
655	C₄	—	4	2	no	—
656	C₄	—	4	2	no	—
657	C₄	—	4	4	no	—
658	C₄	—	4	2	no	—
659	C₄	—	4	4	no	—
660	C₄	C₃₁×#534	1	4	yes	—
661	C₂	—	2	4	no	—
662	C₂	—	2	4	no	—
663	C₂	—	4	2	no	—
664	C₂	—	8	2	no	—
665	C₂	C₂×C₃₁×#532	1	1	yes	characteristic
666	C₂	C₂×C₃₁×#534	1	1	yes	characteristic
667	C₂	C₂×C₃₁×#536	1	1	yes	characteristic
668	C₂	C₃₁×#3520	1	4	yes	—

	subgroup	G/H
center, socle	C₂³×C₃₁	C₂⁵
commutator, Frattini, cosocle	C₂³	C₂⁵×C₃₁
prime radical	C₂×C₃₁×#525	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³×C₃₁ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂²×C₃₁×C₂²⋊Q₈	C₂	semidirect
3. C₃₁×#2857	C₂	semidirect
4. C₃₁×#2850	C₂	semidirect
5. C₃₁×#2663	C₂	semidirect
6. C₃₁×#2564	C₂	semidirect
7. C₃₁×#2503	C₂	semidirect
8. C₃₁×#2501	C₂	semidirect
9. C₃₁×#2499	C₂	semidirect
10. C₃₁×#2436	C₂	semidirect
11. C₃₁×#2434	C₂	semidirect
12. C₃₁×#2432	C₂	semidirect
13. C₃₁×#2412	C₂	semidirect
14. C₃₁×#2322	C₂	semidirect
15. C₂×C₄×C₂²⋊C₄×C₃₁	C₂	semidirect
19. C₂×C₃₁×C₄⋊Q₈	C₂²	semidirect
20. C₂×C₃₁×#124	C₂²	semidirect
42. C₃₁×#400	C₂²	semidirect
43. C₃₁×#399	C₂²	semidirect
45. C₃₁×#395	C₂²	semidirect
47. C₃₁×#393	C₂²	semidirect
48. C₃₁×#391	C₂²	semidirect
49. C₃₁×#391	C₂²	semidirect
50. C₃₁×#391	C₂²	semidirect
51. C₃₁×#390	C₂²	semidirect
52. C₃₁×#389	C₂²	semidirect
53. C₃₁×#388	C₂²	semidirect
54. C₃₁×#387	C₂²	semidirect
58. C₃₁×#385	C₂²	semidirect
59. C₃₁×#384	C₂²	semidirect
60. C₃₁×#382	C₂²	semidirect
226. C₂²⋊C₄×C₃₁	#41	semidirect
231. #35085	C₃₁	direct (N × H)
296. C₂²×C₂²⋊Q₈	C₂×C₃₁	semidirect
297. #2857	C₂×C₃₁	semidirect
298. #2850	C₂×C₃₁	semidirect
299. #2663	C₂×C₃₁	semidirect
300. #2564	C₂×C₃₁	semidirect
301. #2503	C₂×C₃₁	semidirect
302. #2501	C₂×C₃₁	semidirect
303. #2499	C₂×C₃₁	semidirect
304. #2436	C₂×C₃₁	semidirect
305. #2434	C₂×C₃₁	semidirect
306. #2432	C₂×C₃₁	semidirect
307. #2412	C₂×C₃₁	semidirect
308. #2322	C₂×C₃₁	semidirect
309. C₂×C₄×C₂²⋊C₄	C₂×C₃₁	semidirect
345. C₂×C₄⋊Q₈	C₂²×C₃₁	semidirect
346. C₂×#124	C₂²×C₃₁	semidirect
368. #400	C₂²×C₃₁	semidirect
369. #399	C₂²×C₃₁	semidirect
371. #395	C₂²×C₃₁	semidirect
373. #393	C₂²×C₃₁	semidirect
374. #391	C₂²×C₃₁	semidirect
375. #391	C₂²×C₃₁	semidirect
376. #391	C₂²×C₃₁	semidirect
377. #390	C₂²×C₃₁	semidirect
378. #389	C₂²×C₃₁	semidirect
379. #388	C₂²×C₃₁	semidirect
380. #387	C₂²×C₃₁	semidirect
384. #385	C₂²×C₃₁	semidirect
385. #384	C₂²×C₃₁	semidirect
386. #382	C₂²×C₃₁	semidirect
481. C₃₁	#35085	direct (N × H)
561. C₂²⋊C₄	#41×C₃₁	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₃₁×#525	2
C₂²×C₃₁×C₂²⋊Q₈	2
C₃₁×#2857	2
C₃₁×#2850	2
C₃₁×#2663	2
C₃₁×#2564	2
C₃₁×#2503	2
C₃₁×#2501	2
C₃₁×#2499	2
C₃₁×#2436	2
C₃₁×#2434	2
C₃₁×#2432	2
C₃₁×#2412	2
C₃₁×#2322	2
C₂×C₄×C₂²⋊C₄×C₃₁	2
#35085	31

Subgroups — C31×#35085 #21491844

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#35085 #21491844