Subgroups — C₃₁×#36631 #21493390

Order 7936.

2670 conjugacy classes of subgroups.

The subgroup lattice has 604 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	2	yes	maximal, complemented
2	C₃₁×#2861	C₂	1	2	yes	maximal, complemented
3	C₃₁×#2860	C₂	1	4	yes	maximal, complemented
4	C₃₁×#2859	C₂	1	1	yes	maximal, characteristic, complemented
5	C₃₁×#2858	C₂	1	1	yes	maximal, characteristic, complemented
6	C₃₁×#2857	C₂	1	1	yes	maximal, characteristic, complemented
7	C₃₁×#2856	C₂	1	1	yes	maximal, characteristic
8	C₃₁×#2854	C₂	1	4	yes	maximal, complemented
9	C₃₁×#2850	C₂	1	2	yes	maximal, complemented
10	C₄×C₃₁×C₄⋊Q₈	C₂	1	1	yes	maximal, characteristic, complemented
11	C₄×C₃₁×#127	C₂	1	1	yes	maximal, characteristic, complemented
12	C₄×C₃₁×#124	C₂	1	2	yes	maximal, complemented
13	C₄×C₃₁×#121	C₂	1	4	yes	maximal, complemented
14	C₃₁×#2293	C₂	1	4	yes	maximal, complemented
15	C₂×C₄³×C₃₁	C₂	1	1	yes	maximal, characteristic, complemented
16	C₂×C₃₁×C₄⋊Q₈	C₂²	1	1	yes	characteristic
17	C₂×C₃₁×#127	C₂²	1	1	yes	characteristic
18	C₃₁×#529	—	2	16	no	—
19	C₂×C₃₁×#126	C₂²	1	4	yes	—
20	C₂×C₃₁×#125	C₂²	1	2	yes	—
21	C₂×C₃₁×#124	C₂²	1	2	yes	—
22	C₂×C₃₁×#124	C₂²	1	2	yes	—
23	C₂×C₃₁×#123	C₂²	1	4	yes	—
24	C₂×C₃₁×C₂²⋊Q₈	C₂²	1	2	yes	—
25	C₂×C₃₁×#121	C₂²	1	2	yes	—
26	C₂×C₃₁×#121	C₂²	1	4	yes	—
27	C₂×C₄×Q₈×C₃₁	C₂²	1	2	yes	—
28	C₂×C₄×D₄×C₃₁	C₂²	1	4	yes	—
29	C₂×C₄×D₄×C₃₁	C₂²	1	2	yes	—
30	C₂×C₄×D₄×C₃₁	C₂²	1	4	yes	—
31	C₂×C₃₁×#117	C₂²	1	4	yes	—
32	C₂²×C₄²×C₃₁	C₂²	1	1	yes	characteristic
33	C₂²×C₄²×C₃₁	C₂²	1	2	yes	—
34	C₂²×C₄²×C₃₁	C₂²	1	1	yes	characteristic
35	C₂²×C₄²×C₃₁	C₂²	1	2	yes	—
36	C₂²×C₄²×C₃₁	C₂²	1	1	yes	characteristic
37	C₃₁×#391	C₂²	1	2	yes	—
38	C₃₁×#391	C₂²	1	2	yes	—
39	C₃₁×#390	C₂²	1	2	yes	—
40	C₃₁×#390	C₂²	1	4	yes	—
41	C₃₁×#390	C₂²	1	4	yes	—
42	C₃₁×#390	C₂²	1	4	yes	—
43	C₃₁×#390	C₂²	1	2	yes	—
44	C₃₁×#390	C₂²	1	4	yes	—
45	C₃₁×#389	C₂²	1	4	yes	—
46	C₃₁×#389	C₂²	1	4	yes	—
47	C₃₁×#389	C₂²	1	4	yes	—
48	C₃₁×#389	C₂²	1	4	yes	—
49	C₃₁×#388	C₂²	1	8	yes	—
50	C₃₁×#388	C₂²	1	8	yes	—
51	C₃₁×#388	C₂²	1	8	yes	—
52	C₃₁×#388	C₂²	1	8	yes	—
53	C₃₁×#386	C₂²	1	8	yes	—
54	C₃₁×#384	C₂²	1	1	yes	characteristic
55	C₃₁×#384	C₂²	1	1	yes	characteristic
56	C₃₁×#382	C₂²	1	2	yes	—
57	C₃₁×#380	C₂²	1	4	yes	—
58	C₄×C₄⋊C₄×C₃₁	C₂²	1	4	yes	—
59	C₄×C₂²⋊C₄×C₃₁	C₂²	1	8	yes	—
60	C₄³×C₃₁	C₂²	1	1	yes	characteristic
61	C₄³×C₃₁	C₂²	1	1	yes	characteristic
62	C₄³×C₃₁	C₂²	1	2	yes	—
63	C₄³×C₃₁	C₂²	1	1	yes	characteristic
64	C₄³×C₃₁	C₂²	1	2	yes	—
65	C₄³×C₃₁	C₂²	1	1	yes	characteristic
66	C₂²×Q₈×C₃₁	C₂³	1	2	yes	—
67	C₂²×D₄×C₃₁	C₂³	1	4	yes	—
68	C₂²×D₄×C₃₁	C₂³	1	2	yes	—
69	C₂²×D₄×C₃₁	C₂³	1	4	yes	—
70	C₂³×C₄×C₃₁	C₂³	1	2	yes	—
71	C₂³×C₄×C₃₁	C₂³	1	4	yes	—
72	C₂³×C₄×C₃₁	C₂³	1	1	yes	characteristic
73	C₂³×C₄×C₃₁	C₂³	1	1	yes	characteristic
74	C₂³×C₄×C₃₁	C₂³	1	1	yes	characteristic
75	C₂³×C₄×C₃₁	C₂³	1	2	yes	—
76	C₃₁×C₄⋊Q₈	—	2	4	no	—
77	C₃₁×#127	—	2	4	no	—
78	C₃₁×#126	—	2	16	no	—
79	C₃₁×#125	—	2	8	no	—
80	C₃₁×#124	—	2	8	no	—
81	C₃₁×#124	—	2	8	no	—
82	C₃₁×#123	—	2	16	no	—
83	C₃₁×C₂²⋊Q₈	—	2	8	no	—
84	C₃₁×#121	—	2	8	no	—
85	C₃₁×#121	—	2	16	no	—
86	C₄×Q₈×C₃₁	—	2	8	no	—
87	C₄×D₄×C₃₁	—	2	16	no	—
88	C₄×D₄×C₃₁	—	2	8	no	—
89	C₄×D₄×C₃₁	—	2	16	no	—
90	C₃₁×#117	—	2	16	no	—
91	C₂×C₄⋊C₄×C₃₁	C₂³	1	4	yes	—
92	C₂×C₄⋊C₄×C₃₁	C₂³	1	4	yes	—
93	C₂×C₄⋊C₄×C₃₁	C₂³	1	2	yes	—
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	2	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	4	yes	—
100	C₂×C₂²⋊C₄×C₃₁	C₂³	1	4	yes	—
101	C₂×C₂²⋊C₄×C₃₁	C₂³	1	4	yes	—
102	C₂×C₂²⋊C₄×C₃₁	C₂³	1	4	yes	—
103	C₂×C₂²⋊C₄×C₃₁	C₂³	1	4	yes	—
104	C₂×C₂²⋊C₄×C₃₁	C₂³	1	4	yes	—
105	C₂×C₂²⋊C₄×C₃₁	C₂³	1	8	yes	—
106	C₂×C₂²⋊C₄×C₃₁	C₂³	1	8	yes	—
107	C₂×C₂²⋊C₄×C₃₁	C₂³	1	8	yes	—
108	C₂×C₄²×C₃₁	—	2	8	no	—
109	C₂×C₄²×C₃₁	—	2	8	no	—
110	C₂×C₄²×C₃₁	—	2	4	no	—
111	C₂×C₄²×C₃₁	—	2	4	no	—
112	C₂×C₄²×C₃₁	D₄	1	8	yes	complemented
113	C₂×C₄²×C₃₁	C₂³	1	2	yes	—
114	C₂×C₄²×C₃₁	C₂³	1	2	yes	—
115	C₂×C₄²×C₃₁	C₂³	1	2	yes	—
116	C₂×C₄²×C₃₁	C₂³	1	2	yes	—
117	C₂×C₄²×C₃₁	C₂³	1	2	yes	—
118	C₂×C₄²×C₃₁	C₂³	1	2	yes	—
119	C₂×C₄²×C₃₁	C₂³	1	2	yes	—
120	C₂×C₄²×C₃₁	C₂³	1	2	yes	—
121	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
122	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
123	C₂×C₄²×C₃₁	C₂³	1	2	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	1	yes	characteristic
130	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
131	C₂×C₄²×C₃₁	C₂³	1	2	yes	—
132	C₃₁×#95	C₂³	1	8	yes	—
133	C₃₁×#95	C₂³	1	8	yes	—
134	C₃₁×#95	C₂³	1	8	yes	—
135	C₂⁴×C₃₁	C₂⁴	1	4	yes	—
136	C₂⁴×C₃₁	C₂⁴	1	1	yes	characteristic
137	C₂×Q₈×C₃₁	—	2	4	no	—
138	C₂×Q₈×C₃₁	—	2	4	no	—
139	C₂×Q₈×C₃₁	—	2	4	no	—
140	C₂×D₄×C₃₁	—	2	8	no	—
141	C₂×D₄×C₃₁	—	2	16	no	—
142	C₂×D₄×C₃₁	—	2	4	no	—
143	C₂×D₄×C₃₁	—	2	8	no	—
144	C₂×D₄×C₃₁	—	2	8	no	—
145	C₂×D₄×C₃₁	—	2	4	no	—
146	C₂×D₄×C₃₁	—	2	8	no	—
147	C₂×D₄×C₃₁	—	2	4	no	—
148	C₂²×C₄×C₃₁	—	2	4	no	—
149	C₂²×C₄×C₃₁	—	2	4	no	—
150	C₂²×C₄×C₃₁	—	2	4	no	—
151	C₂²×C₄×C₃₁	—	2	4	no	—
152	C₂²×C₄×C₃₁	—	2	4	no	—
153	C₂²×C₄×C₃₁	—	2	4	no	—
154	C₂²×C₄×C₃₁	—	2	4	no	—
155	C₂²×C₄×C₃₁	—	2	2	no	—
156	C₂²×C₄×C₃₁	—	2	2	no	—
157	C₂²×C₄×C₃₁	—	2	8	no	—
158	C₂²×C₄×C₃₁	—	2	16	no	—
159	C₂²×C₄×C₃₁	C₂×D₄	1	4	yes	—
160	C₂²×C₄×C₃₁	C₂×D₄	1	4	yes	—
161	C₂²×C₄×C₃₁	C₂×D₄	1	4	yes	—
162	C₂²×C₄×C₃₁	#41	1	8	yes	—
163	C₂²×C₄×C₃₁	C₂⁴	1	4	yes	—
164	C₂²×C₄×C₃₁	C₂⁴	1	4	yes	—
165	C₂²×C₄×C₃₁	C₂⁴	1	4	yes	—
166	C₂²×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
167	C₂²×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
168	C₂²×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
169	C₂²×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
170	C₂²×C₄×C₃₁	C₂⁴	1	2	yes	—
171	C₂²×C₄×C₃₁	C₂⁴	1	2	yes	—
172	C₂²×C₄×C₃₁	C₂⁴	1	2	yes	—
173	C₂²×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
174	C₂²×C₄×C₃₁	C₂⁴	1	2	yes	—
175	C₂²×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
176	C₄⋊C₄×C₃₁	—	2	8	no	—
177	C₄⋊C₄×C₃₁	—	2	8	no	—
178	C₄⋊C₄×C₃₁	—	2	4	no	—
179	C₄⋊C₄×C₃₁	—	2	8	no	—
180	C₄⋊C₄×C₃₁	—	2	4	no	—
181	C₄⋊C₄×C₃₁	—	2	8	no	—
182	C₄⋊C₄×C₃₁	—	2	8	no	—
183	C₄⋊C₄×C₃₁	—	2	8	no	—
184	C₄⋊C₄×C₃₁	—	2	8	no	—
185	C₂²⋊C₄×C₃₁	—	2	8	no	—
186	C₂²⋊C₄×C₃₁	—	2	8	no	—
187	C₂²⋊C₄×C₃₁	—	2	8	no	—
188	C₂²⋊C₄×C₃₁	—	2	8	no	—
189	C₂²⋊C₄×C₃₁	—	2	8	no	—
190	C₂²⋊C₄×C₃₁	—	2	16	no	—
191	C₂²⋊C₄×C₃₁	—	2	16	no	—
192	C₂²⋊C₄×C₃₁	—	2	16	no	—
193	C₄²×C₃₁	—	2	8	no	—
194	C₄²×C₃₁	—	2	4	no	—
195	C₄²×C₃₁	—	2	4	no	—
196	C₄²×C₃₁	—	2	4	no	—
197	C₄²×C₃₁	—	2	4	no	—
198	C₄²×C₃₁	—	2	4	no	—
199	C₄²×C₃₁	—	2	4	no	—
200	C₄²×C₃₁	—	2	4	no	—
201	C₄²×C₃₁	—	2	4	no	—
202	C₄²×C₃₁	—	2	4	no	—
203	C₄²×C₃₁	—	2	2	no	—
204	C₄²×C₃₁	—	2	2	no	—
205	C₄²×C₃₁	C₂×D₄	1	4	yes	—
206	C₄²×C₃₁	C₂×D₄	1	4	yes	—
207	C₄²×C₃₁	C₂×D₄	1	4	yes	—
208	C₄²×C₃₁	C₂×D₄	1	4	yes	—
209	C₄²×C₃₁	#41	1	8	yes	—
210	C₄²×C₃₁	#41	1	4	yes	—
211	C₄²×C₃₁	#41	1	4	yes	—
212	#36631	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
213	C₂³×C₃₁	—	2	1	no	—
214	C₂³×C₃₁	—	2	2	no	—
215	C₂³×C₃₁	—	2	1	no	—
216	C₂³×C₃₁	—	2	4	no	—
217	C₂³×C₃₁	—	2	4	no	—
218	C₂³×C₃₁	—	2	8	no	—
219	C₂³×C₃₁	—	2	8	no	—
220	C₂³×C₃₁	—	2	4	no	—
221	C₂³×C₃₁	C₂²×D₄	1	2	yes	—
222	C₂³×C₃₁	C₂×#41	1	4	yes	—
223	C₂³×C₃₁	C₂⁵	1	1	yes	characteristic
224	Q₈×C₃₁	—	4	8	no	—
225	D₄×C₃₁	—	4	16	no	—
226	D₄×C₃₁	—	4	16	no	—
227	D₄×C₃₁	—	4	8	no	—
228	C₂×C₄×C₃₁	—	2	16	no	—
229	C₂×C₄×C₃₁	—	2	8	no	—
230	C₂×C₄×C₃₁	—	2	2	no	—
231	C₂×C₄×C₃₁	—	2	2	no	—
232	C₂×C₄×C₃₁	—	2	1	no	—
233	C₂×C₄×C₃₁	—	2	2	no	—
234	C₂×C₄×C₃₁	—	2	2	no	—
235	C₂×C₄×C₃₁	—	2	2	no	—
236	C₂×C₄×C₃₁	—	2	1	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	16	no	—
242	C₂×C₄×C₃₁	—	2	8	no	—
243	C₂×C₄×C₃₁	—	2	8	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	8	no	—
249	C₂×C₄×C₃₁	—	2	8	no	—
250	C₂×C₄×C₃₁	—	2	4	no	—
251	C₂×C₄×C₃₁	—	4	16	no	—
252	C₂×C₄×C₃₁	C₂²×D₄	1	2	yes	—
253	C₂×C₄×C₃₁	C₂²×D₄	1	2	yes	—
254	C₂×C₄×C₃₁	C₂²×D₄	1	2	yes	—
255	C₂×C₄×C₃₁	C₂²×D₄	1	2	yes	—
256	C₂×C₄×C₃₁	C₂²×D₄	1	2	yes	—
257	C₂×C₄×C₃₁	C₂²×D₄	1	2	yes	—
258	C₂×C₄×C₃₁	C₂×#41	1	4	yes	—
259	C₂×C₄×C₃₁	C₂×#41	1	4	yes	—
260	C₂×C₄×C₃₁	C₂×#41	1	4	yes	—
261	C₂×C₄×C₃₁	C₂×#41	1	4	yes	—
262	C₂×C₄×C₃₁	C₂×#41	1	4	yes	—
263	C₂×C₄×C₃₁	C₂×#41	1	4	yes	—
264	C₂×C₄×C₃₁	C₂×#41	1	4	yes	—
265	C₂×C₄×C₃₁	C₂×#41	1	2	yes	—
266	C₂×C₄×C₃₁	C₂×#41	1	2	yes	—
267	C₂×#529	C₂×C₃₁	1	2	yes	complemented
268	#2861	C₂×C₃₁	1	2	yes	complemented
269	#2860	C₂×C₃₁	1	4	yes	complemented
270	#2859	C₂×C₃₁	1	1	yes	characteristic, complemented
271	#2858	C₂×C₃₁	1	1	yes	characteristic, complemented
272	#2857	C₂×C₃₁	1	1	yes	characteristic, complemented
273	#2856	C₂×C₃₁	1	1	yes	characteristic
274	#2854	C₂×C₃₁	1	4	yes	complemented
275	#2850	C₂×C₃₁	1	2	yes	complemented
276	C₄×C₄⋊Q₈	C₂×C₃₁	1	1	yes	characteristic, complemented
277	C₄×#127	C₂×C₃₁	1	1	yes	characteristic, complemented
278	C₄×#124	C₂×C₃₁	1	2	yes	complemented
279	C₄×#121	C₂×C₃₁	1	4	yes	complemented
280	#2293	C₂×C₃₁	1	4	yes	complemented
281	C₂×C₄³	C₂×C₃₁	1	1	yes	characteristic, complemented
282	C₂²×C₃₁	—	2	4	no	—
283	C₂²×C₃₁	—	2	2	no	—
284	C₂²×C₃₁	—	2	2	no	—
285	C₂²×C₃₁	—	2	4	no	—
286	C₂²×C₃₁	—	4	8	no	—
287	C₂²×C₃₁	—	4	4	no	—
288	C₂²×C₃₁	—	4	4	no	—
289	C₂²×C₃₁	—	4	4	no	—
290	C₂²×C₃₁	—	4	8	no	—
291	C₂²×C₃₁	#532	1	4	yes	—
292	C₂²×C₃₁	C₂³×D₄	1	1	yes	characteristic
293	C₂²×C₃₁	C₂²×#41	1	1	yes	characteristic
294	C₂²×C₃₁	C₂²×#41	1	2	yes	—
295	C₂²×C₃₁	C₂²×#41	1	2	yes	—
296	C₂²×C₃₁	C₂²×#41	1	1	yes	characteristic
297	C₄×C₃₁	—	2	4	no	—
298	C₄×C₃₁	—	2	2	no	—
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	4	no	—
304	C₄×C₃₁	—	4	4	no	—
305	C₄×C₃₁	—	4	4	no	—
306	C₄×C₃₁	—	4	4	no	—
307	C₄×C₃₁	#529	1	8	yes	complemented
308	C₄×C₃₁	#532	1	4	yes	—
309	C₄×C₃₁	#532	1	4	yes	—
310	C₂×C₄⋊Q₈	C₂²×C₃₁	1	1	yes	characteristic
311	C₂×#127	C₂²×C₃₁	1	1	yes	characteristic
312	#529	—	2	16	no	—
313	C₂×#126	C₂²×C₃₁	1	4	yes	—
314	C₂×#125	C₂²×C₃₁	1	2	yes	—
315	C₂×#124	C₂²×C₃₁	1	2	yes	—
316	C₂×#124	C₂²×C₃₁	1	2	yes	—
317	C₂×#123	C₂²×C₃₁	1	4	yes	—
318	C₂×C₂²⋊Q₈	C₂²×C₃₁	1	2	yes	—
319	C₂×#121	C₂²×C₃₁	1	2	yes	—
320	C₂×#121	C₂²×C₃₁	1	4	yes	—
321	C₂×C₄×Q₈	C₂²×C₃₁	1	2	yes	—
322	C₂×C₄×D₄	C₂²×C₃₁	1	2	yes	—
323	C₂×C₄×D₄	C₂²×C₃₁	1	4	yes	—
324	C₂×C₄×D₄	C₂²×C₃₁	1	4	yes	—
325	C₂×#117	C₂²×C₃₁	1	4	yes	—
326	C₂²×C₄²	C₂²×C₃₁	1	1	yes	characteristic
327	C₂²×C₄²	C₂²×C₃₁	1	2	yes	—
328	C₂²×C₄²	C₂²×C₃₁	1	1	yes	characteristic
329	C₂²×C₄²	C₂²×C₃₁	1	2	yes	—
330	C₂²×C₄²	C₂²×C₃₁	1	1	yes	characteristic
331	#391	C₂²×C₃₁	1	2	yes	—
332	#391	C₂²×C₃₁	1	2	yes	—
333	#390	C₂²×C₃₁	1	2	yes	—
334	#390	C₂²×C₃₁	1	4	yes	—
335	#390	C₂²×C₃₁	1	4	yes	—
336	#390	C₂²×C₃₁	1	4	yes	—
337	#390	C₂²×C₃₁	1	2	yes	—
338	#390	C₂²×C₃₁	1	4	yes	—
339	#389	C₂²×C₃₁	1	4	yes	—
340	#389	C₂²×C₃₁	1	4	yes	—
341	#389	C₂²×C₃₁	1	4	yes	—
342	#389	C₂²×C₃₁	1	4	yes	—
343	#388	C₂²×C₃₁	1	8	yes	—
344	#388	C₂²×C₃₁	1	8	yes	—
345	#388	C₂²×C₃₁	1	8	yes	—
346	#388	C₂²×C₃₁	1	8	yes	—
347	#386	C₂²×C₃₁	1	8	yes	—
348	#384	C₂²×C₃₁	1	1	yes	characteristic
349	#384	C₂²×C₃₁	1	1	yes	characteristic
350	#382	C₂²×C₃₁	1	2	yes	—
351	#380	C₂²×C₃₁	1	4	yes	—
352	C₄×C₄⋊C₄	C₂²×C₃₁	1	4	yes	—
353	C₄×C₂²⋊C₄	C₂²×C₃₁	1	8	yes	—
354	C₄³	C₂²×C₃₁	1	1	yes	characteristic
355	C₄³	C₂²×C₃₁	1	1	yes	characteristic
356	C₄³	C₂²×C₃₁	1	2	yes	—
357	C₄³	C₂²×C₃₁	1	1	yes	characteristic
358	C₄³	C₂²×C₃₁	1	2	yes	—
359	C₄³	C₂²×C₃₁	1	1	yes	characteristic
360	C₂×C₃₁	—	2	4	no	—
361	C₂×C₃₁	—	8	4	no	—
362	C₂×C₃₁	C₂×#529	1	2	yes	—
363	C₂×C₃₁	C₂×#532	1	1	yes	characteristic
364	C₂×C₃₁	C₂×#532	1	1	yes	characteristic
365	C₂×C₃₁	C₂×#532	1	1	yes	characteristic
366	C₂×C₃₁	#3444	1	2	yes	—
367	C₂²×Q₈	C₂³×C₃₁	1	2	yes	—
368	C₂²×D₄	C₂³×C₃₁	1	4	yes	—
369	C₂²×D₄	C₂³×C₃₁	1	2	yes	—
370	C₂²×D₄	C₂³×C₃₁	1	4	yes	—
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	C₂³×C₄	C₂³×C₃₁	1	2	yes	—
377	C₄⋊Q₈	—	2	4	no	—
378	#127	—	2	4	no	—
379	#126	—	2	16	no	—
380	#125	—	2	8	no	—
381	#124	—	2	8	no	—
382	#124	—	2	8	no	—
383	#123	—	2	16	no	—
384	C₂²⋊Q₈	—	2	8	no	—
385	#121	—	2	8	no	—
386	#121	—	2	16	no	—
387	C₄×Q₈	—	2	8	no	—
388	C₄×D₄	—	2	16	no	—
389	C₄×D₄	—	2	8	no	—
390	C₄×D₄	—	2	16	no	—
391	#117	—	2	16	no	—
392	C₂×C₄⋊C₄	C₂³×C₃₁	1	4	yes	—
393	C₂×C₄⋊C₄	C₂³×C₃₁	1	4	yes	—
394	C₂×C₄⋊C₄	C₂³×C₃₁	1	2	yes	—
395	C₂×C₄⋊C₄	C₂³×C₃₁	1	4	yes	—
396	C₂×C₄⋊C₄	C₂³×C₃₁	1	4	yes	—
397	C₂×C₄⋊C₄	C₂³×C₃₁	1	2	yes	—
398	C₂×C₄⋊C₄	C₂³×C₃₁	1	4	yes	—
399	C₂×C₄⋊C₄	C₂³×C₃₁	1	4	yes	—
400	C₂×C₄⋊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	4	yes	—
404	C₂×C₂²⋊C₄	C₂³×C₃₁	1	4	yes	—
405	C₂×C₂²⋊C₄	C₂³×C₃₁	1	4	yes	—
406	C₂×C₂²⋊C₄	C₂³×C₃₁	1	8	yes	—
407	C₂×C₂²⋊C₄	C₂³×C₃₁	1	8	yes	—
408	C₂×C₂²⋊C₄	C₂³×C₃₁	1	8	yes	—
409	C₂×C₄²	—	2	8	no	—
410	C₂×C₄²	—	2	8	no	—
411	C₂×C₄²	—	2	4	no	—
412	C₂×C₄²	—	2	4	no	—
413	C₂×C₄²	D₄×C₃₁	1	8	yes	complemented
414	C₂×C₄²	C₂³×C₃₁	1	4	yes	—
415	C₂×C₄²	C₂³×C₃₁	1	2	yes	—
416	C₂×C₄²	C₂³×C₃₁	1	2	yes	—
417	C₂×C₄²	C₂³×C₃₁	1	2	yes	—
418	C₂×C₄²	C₂³×C₃₁	1	2	yes	—
419	C₂×C₄²	C₂³×C₃₁	1	2	yes	—
420	C₂×C₄²	C₂³×C₃₁	1	2	yes	—
421	C₂×C₄²	C₂³×C₃₁	1	2	yes	—
422	C₂×C₄²	C₂³×C₃₁	1	2	yes	—
423	C₂×C₄²	C₂³×C₃₁	1	2	yes	—
424	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
425	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
426	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
427	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
428	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
429	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
430	C₂×C₄²	C₂³×C₃₁	1	2	yes	—
431	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
432	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
433	#95	C₂³×C₃₁	1	8	yes	—
434	#95	C₂³×C₃₁	1	8	yes	—
435	#95	C₂³×C₃₁	1	8	yes	—
436	C₃₁	#36631	1	1	yes	characteristic, direct product factor, Sylow
437	C₂⁴	C₂⁴×C₃₁	1	4	yes	—
438	C₂⁴	C₂⁴×C₃₁	1	1	yes	characteristic
439	C₂×Q₈	—	2	4	no	—
440	C₂×Q₈	—	2	4	no	—
441	C₂×Q₈	—	2	4	no	—
442	C₂×D₄	—	2	8	no	—
443	C₂×D₄	—	2	16	no	—
444	C₂×D₄	—	2	4	no	—
445	C₂×D₄	—	2	8	no	—
446	C₂×D₄	—	2	8	no	—
447	C₂×D₄	—	2	4	no	—
448	C₂×D₄	—	2	8	no	—
449	C₂×D₄	—	2	4	no	—
450	C₂²×C₄	—	2	4	no	—
451	C₂²×C₄	—	2	4	no	—
452	C₂²×C₄	—	2	4	no	—
453	C₂²×C₄	—	2	4	no	—
454	C₂²×C₄	—	2	4	no	—
455	C₂²×C₄	—	2	4	no	—
456	C₂²×C₄	—	2	4	no	—
457	C₂²×C₄	—	2	2	no	—
458	C₂²×C₄	—	2	2	no	—
459	C₂²×C₄	—	2	8	no	—
460	C₂²×C₄	—	2	16	no	—
461	C₂²×C₄	C₂×D₄×C₃₁	1	4	yes	—
462	C₂²×C₄	C₂×D₄×C₃₁	1	4	yes	—
463	C₂²×C₄	C₂×D₄×C₃₁	1	4	yes	—
464	C₂²×C₄	#41×C₃₁	1	8	yes	—
465	C₂²×C₄	C₂⁴×C₃₁	1	4	yes	—
466	C₂²×C₄	C₂⁴×C₃₁	1	4	yes	—
467	C₂²×C₄	C₂⁴×C₃₁	1	4	yes	—
468	C₂²×C₄	C₂⁴×C₃₁	1	1	yes	characteristic
469	C₂²×C₄	C₂⁴×C₃₁	1	1	yes	characteristic
470	C₂²×C₄	C₂⁴×C₃₁	1	1	yes	characteristic
471	C₂²×C₄	C₂⁴×C₃₁	1	1	yes	characteristic
472	C₂²×C₄	C₂⁴×C₃₁	1	2	yes	—
473	C₂²×C₄	C₂⁴×C₃₁	1	2	yes	—
474	C₂²×C₄	C₂⁴×C₃₁	1	2	yes	—
475	C₂²×C₄	C₂⁴×C₃₁	1	1	yes	characteristic
476	C₂²×C₄	C₂⁴×C₃₁	1	1	yes	characteristic
477	C₂²×C₄	C₂⁴×C₃₁	1	2	yes	—
478	C₄⋊C₄	—	2	8	no	—
479	C₄⋊C₄	—	2	8	no	—
480	C₄⋊C₄	—	2	4	no	—
481	C₄⋊C₄	—	2	8	no	—
482	C₄⋊C₄	—	2	4	no	—
483	C₄⋊C₄	—	2	8	no	—
484	C₄⋊C₄	—	2	8	no	—
485	C₄⋊C₄	—	2	8	no	—
486	C₄⋊C₄	—	2	8	no	—
487	C₂²⋊C₄	—	2	8	no	—
488	C₂²⋊C₄	—	2	8	no	—
489	C₂²⋊C₄	—	2	8	no	—
490	C₂²⋊C₄	—	2	8	no	—
491	C₂²⋊C₄	—	2	8	no	—
492	C₂²⋊C₄	—	2	16	no	—
493	C₂²⋊C₄	—	2	16	no	—
494	C₂²⋊C₄	—	2	16	no	—
495	C₄²	—	2	8	no	—
496	C₄²	—	2	4	no	—
497	C₄²	—	2	4	no	—
498	C₄²	—	2	4	no	—
499	C₄²	—	2	4	no	—
500	C₄²	—	2	4	no	—
501	C₄²	—	2	4	no	—
502	C₄²	—	2	4	no	—
503	C₄²	—	2	4	no	—
504	C₄²	—	2	4	no	—
505	C₄²	—	2	2	no	—
506	C₄²	—	2	2	no	—
507	C₄²	C₂×D₄×C₃₁	1	4	yes	—
508	C₄²	C₂×D₄×C₃₁	1	4	yes	—
509	C₄²	C₂×D₄×C₃₁	1	4	yes	—
510	C₄²	C₂×D₄×C₃₁	1	4	yes	—
511	C₄²	#41×C₃₁	1	8	yes	—
512	C₄²	#41×C₃₁	1	4	yes	—
513	C₄²	#41×C₃₁	1	4	yes	—
514	C₂³	—	2	1	no	—
515	C₂³	—	2	2	no	—
516	C₂³	—	2	1	no	—
517	C₂³	—	2	4	no	—
518	C₂³	—	2	4	no	—
519	C₂³	—	2	8	no	—
520	C₂³	—	2	8	no	—
521	C₂³	—	2	4	no	—
522	C₂³	C₂²×D₄×C₃₁	1	2	yes	—
523	C₂³	C₂×#41×C₃₁	1	4	yes	—
524	C₂³	C₂⁵×C₃₁	1	1	yes	characteristic
525	Q₈	—	4	8	no	—
526	D₄	—	4	16	no	—
527	D₄	—	4	16	no	—
528	D₄	—	4	8	no	—
529	C₂×C₄	—	2	16	no	—
530	C₂×C₄	—	2	8	no	—
531	C₂×C₄	—	2	2	no	—
532	C₂×C₄	—	2	2	no	—
533	C₂×C₄	—	2	1	no	—
534	C₂×C₄	—	2	2	no	—
535	C₂×C₄	—	2	2	no	—
536	C₂×C₄	—	2	2	no	—
537	C₂×C₄	—	2	1	no	—
538	C₂×C₄	—	2	2	no	—
539	C₂×C₄	—	2	2	no	—
540	C₂×C₄	—	2	2	no	—
541	C₂×C₄	—	2	2	no	—
542	C₂×C₄	—	2	16	no	—
543	C₂×C₄	—	2	8	no	—
544	C₂×C₄	—	2	8	no	—
545	C₂×C₄	—	2	4	no	—
546	C₂×C₄	—	2	4	no	—
547	C₂×C₄	—	2	4	no	—
548	C₂×C₄	—	2	4	no	—
549	C₂×C₄	—	2	8	no	—
550	C₂×C₄	—	2	8	no	—
551	C₂×C₄	—	2	4	no	—
552	C₂×C₄	—	4	16	no	—
553	C₂×C₄	C₂²×D₄×C₃₁	1	2	yes	—
554	C₂×C₄	C₂²×D₄×C₃₁	1	2	yes	—
555	C₂×C₄	C₂²×D₄×C₃₁	1	2	yes	—
556	C₂×C₄	C₂²×D₄×C₃₁	1	2	yes	—
557	C₂×C₄	C₂²×D₄×C₃₁	1	2	yes	—
558	C₂×C₄	C₂²×D₄×C₃₁	1	2	yes	—
559	C₂×C₄	C₂×#41×C₃₁	1	4	yes	—
560	C₂×C₄	C₂×#41×C₃₁	1	4	yes	—
561	C₂×C₄	C₂×#41×C₃₁	1	4	yes	—
562	C₂×C₄	C₂×#41×C₃₁	1	4	yes	—
563	C₂×C₄	C₂×#41×C₃₁	1	4	yes	—
564	C₂×C₄	C₂×#41×C₃₁	1	4	yes	—
565	C₂×C₄	C₂×#41×C₃₁	1	4	yes	—
566	C₂×C₄	C₂×#41×C₃₁	1	2	yes	—
567	C₂×C₄	C₂×#41×C₃₁	1	2	yes	—
568	C₂²	—	2	4	no	—
569	C₂²	—	2	2	no	—
570	C₂²	—	2	2	no	—
571	C₂²	—	2	4	no	—
572	C₂²	—	4	8	no	—
573	C₂²	—	4	4	no	—
574	C₂²	—	4	4	no	—
575	C₂²	—	4	4	no	—
576	C₂²	—	4	8	no	—
577	C₂²	C₃₁×#532	1	4	yes	—
578	C₂²	C₂³×D₄×C₃₁	1	1	yes	characteristic
579	C₂²	C₂²×#41×C₃₁	1	1	yes	characteristic
580	C₂²	C₂²×#41×C₃₁	1	2	yes	—
581	C₂²	C₂²×#41×C₃₁	1	2	yes	—
582	C₂²	C₂²×#41×C₃₁	1	1	yes	characteristic
583	C₄	—	2	4	no	—
584	C₄	—	2	2	no	—
585	C₄	—	2	2	no	—
586	C₄	—	2	2	no	—
587	C₄	—	2	2	no	—
588	C₄	—	2	4	no	—
589	C₄	—	2	4	no	—
590	C₄	—	4	4	no	—
591	C₄	—	4	4	no	—
592	C₄	—	4	4	no	—
593	C₄	C₃₁×#529	1	8	yes	complemented
594	C₄	C₃₁×#532	1	4	yes	—
595	C₄	C₃₁×#532	1	4	yes	—
596	C₂	—	2	4	no	—
597	C₂	—	8	4	no	—
598	C₂	C₂×C₃₁×#529	1	2	yes	—
599	C₂	C₂×C₃₁×#532	1	1	yes	characteristic
600	C₂	C₂×C₃₁×#532	1	1	yes	characteristic
601	C₂	C₂×C₃₁×#532	1	1	yes	characteristic
602	C₂	C₃₁×#3444	1	2	yes	—

	subgroup	G/H
center	C₂²×C₄×C₃₁	C₂⁴
commutator, Frattini, cosocle	C₂³	C₂⁵×C₃₁
socle	C₂³×C₃₁	C₂⁵
prime radical	C₃₁×#2856	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂²×C₄×C₃₁ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂×C₃₁×#529	C₂	semidirect
2. C₃₁×#2861	C₂	semidirect
3. C₃₁×#2860	C₂	semidirect
4. C₃₁×#2859	C₂	semidirect
5. C₃₁×#2858	C₂	semidirect
6. C₃₁×#2857	C₂	semidirect
8. C₃₁×#2854	C₂	semidirect
9. C₃₁×#2850	C₂	semidirect
10. C₄×C₃₁×C₄⋊Q₈	C₂	semidirect
11. C₄×C₃₁×#127	C₂	semidirect
12. C₄×C₃₁×#124	C₂	semidirect
13. C₄×C₃₁×#121	C₂	semidirect
14. C₃₁×#2293	C₂	semidirect
15. C₂×C₄³×C₃₁	C₂	semidirect
112. C₂×C₄²×C₃₁	D₄	semidirect
212. #36631	C₃₁	direct (N × H)
267. C₂×#529	C₂×C₃₁	semidirect
268. #2861	C₂×C₃₁	semidirect
269. #2860	C₂×C₃₁	semidirect
270. #2859	C₂×C₃₁	semidirect
271. #2858	C₂×C₃₁	semidirect
272. #2857	C₂×C₃₁	semidirect
274. #2854	C₂×C₃₁	semidirect
275. #2850	C₂×C₃₁	semidirect
276. C₄×C₄⋊Q₈	C₂×C₃₁	semidirect
277. C₄×#127	C₂×C₃₁	semidirect
278. C₄×#124	C₂×C₃₁	semidirect
279. C₄×#121	C₂×C₃₁	semidirect
280. #2293	C₂×C₃₁	semidirect
281. C₂×C₄³	C₂×C₃₁	semidirect
307. C₄×C₃₁	#529	semidirect
413. C₂×C₄²	D₄×C₃₁	semidirect
436. C₃₁	#36631	direct (N × H)
593. C₄	C₃₁×#529	semidirect

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×#2861	2
C₃₁×#2860	2
C₃₁×#2859	2
C₃₁×#2858	2
C₃₁×#2857	2
C₃₁×#2856	2
C₃₁×#2854	2
C₃₁×#2850	2
C₄×C₃₁×C₄⋊Q₈	2
C₄×C₃₁×#127	2
C₄×C₃₁×#124	2
C₄×C₃₁×#121	2
C₃₁×#2293	2
C₂×C₄³×C₃₁	2
#36631	31

Subgroups — C31×#36631 #21493390

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#36631 #21493390