Subgroups — C₃₁×#35131 #21491890

Order 7936.

2670 conjugacy classes of subgroups.

The subgroup lattice has 650 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, complemented
2	C₂²×C₃₁×#124	C₂	1	1	yes	maximal, characteristic, complemented
3	C₃₁×#2861	C₂	1	2	yes	maximal, complemented
4	C₃₁×#2858	C₂	1	2	yes	maximal, complemented
5	C₃₁×#2662	C₂	1	4	yes	maximal, complemented
6	C₃₁×#2610	C₂	1	4	yes	maximal, complemented
7	C₃₁×#2545	C₂	1	2	yes	maximal, complemented
8	C₃₁×#2544	C₂	1	2	yes	maximal, complemented
9	C₃₁×#2436	C₂	1	1	yes	maximal, characteristic, complemented
10	C₃₁×#2435	C₂	1	1	yes	maximal, characteristic, complemented
11	C₃₁×#2434	C₂	1	1	yes	maximal, characteristic, complemented
12	C₃₁×#2433	C₂	1	1	yes	maximal, characteristic
13	C₃₁×#2411	C₂	1	4	yes	maximal, complemented
14	C₃₁×#2321	C₂	1	4	yes	maximal, complemented
15	C₄×C₃₁×#117	C₂	1	1	yes	maximal, characteristic, complemented
16	C₂³×Q₈×C₃₁	C₂²	1	1	yes	characteristic
17	C₂³×D₄×C₃₁	C₂²	1	1	yes	characteristic
18	C₂×C₃₁×C₄⋊Q₈	C₂²	1	2	yes	complemented
19	C₂×C₃₁×#127	C₂²	1	2	yes	—
20	C₃₁×#529	—	2	8	no	—
21	C₂×C₃₁×#126	C₂²	1	2	yes	complemented
22	C₂×C₃₁×#125	C₂²	1	1	yes	characteristic, complemented
23	C₂×C₃₁×#124	—	2	8	no	—
24	C₂×C₃₁×#124	C₂²	1	2	yes	—
25	C₂×C₃₁×#124	C₂²	1	2	yes	—
26	C₂×C₃₁×#124	C₂²	1	2	yes	—
27	C₂×C₃₁×#124	C₂²	1	2	yes	—
28	C₂×C₃₁×#124	C₂²	1	2	yes	—
29	C₂×C₃₁×#124	C₂²	1	1	yes	characteristic, complemented
30	C₂×C₃₁×#124	C₂²	1	2	yes	complemented
31	C₂×C₃₁×#123	C₂²	1	2	yes	—
32	C₂×C₃₁×C₂²⋊Q₈	C₂²	1	1	yes	characteristic
33	C₂×C₃₁×#121	C₂²	1	1	yes	characteristic
34	C₂×C₃₁×#120	C₂²	1	4	yes	—
35	C₂×C₄×Q₈×C₃₁	C₂²	1	1	yes	characteristic, complemented
36	C₂×C₄×D₄×C₃₁	C₂²	1	2	yes	—
37	C₂×C₄×D₄×C₃₁	C₂²	1	1	yes	characteristic, complemented
38	C₂×C₃₁×#117	C₂²	1	2	yes	—
39	C₂×C₃₁×#117	C₂²	1	2	yes	complemented
40	C₂²×C₂²⋊C₄×C₃₁	C₂²	1	2	yes	—
41	C₂²×C₂²⋊C₄×C₃₁	C₂²	1	2	yes	—
42	C₂²×C₄²×C₃₁	C₂²	1	1	yes	characteristic
43	C₃₁×#396	C₂²	1	8	yes	complemented
44	C₃₁×#395	C₂²	1	4	yes	complemented
45	C₃₁×#394	C₂²	1	4	yes	—
46	C₃₁×#394	C₂²	1	8	yes	complemented
47	C₃₁×#393	C₂²	1	4	yes	complemented
48	C₃₁×#392	C₂²	1	4	yes	—
49	C₃₁×#391	C₂²	1	2	yes	complemented
50	C₃₁×#391	C₂²	1	2	yes	complemented
51	C₃₁×#390	C₂²	1	2	yes	—
52	C₃₁×#390	C₂²	1	2	yes	—
53	C₃₁×#390	C₂²	1	8	yes	complemented
54	C₃₁×#390	C₂²	1	8	yes	complemented
55	C₃₁×#388	C₂²	1	8	yes	complemented
56	C₃₁×#388	C₂²	1	8	yes	complemented
57	C₃₁×#388	C₂²	1	8	yes	complemented
58	C₃₁×#386	C₂²	1	4	yes	—
59	C₃₁×#386	C₂²	1	8	yes	complemented
60	C₃₁×#385	C₂²	1	4	yes	—
61	C₃₁×#379	C₂²	1	4	yes	—
62	C₄×C₄⋊C₄×C₃₁	C₂²	1	2	yes	complemented
63	C₄×C₄⋊C₄×C₃₁	C₂²	1	2	yes	complemented
64	C₄×C₂²⋊C₄×C₃₁	C₂²	1	2	yes	—
65	C₄×C₂²⋊C₄×C₃₁	C₂²	1	1	yes	characteristic
66	C₄×C₂²⋊C₄×C₃₁	C₂²	1	1	yes	characteristic
67	C₃₁×#376	C₂²	1	2	yes	complemented
68	C₄³×C₃₁	C₂²	1	2	yes	complemented
69	C₂⁵×C₃₁	C₂³	1	2	yes	—
70	C₂²×Q₈×C₃₁	—	2	4	no	—
71	C₂²×Q₈×C₃₁	—	4	4	no	—
72	C₂²×Q₈×C₃₁	C₂³	1	1	yes	characteristic
73	C₂²×Q₈×C₃₁	C₂³	1	2	yes	—
74	C₂²×Q₈×C₃₁	C₂³	1	2	yes	—
75	C₂²×D₄×C₃₁	—	2	4	no	—
76	C₂²×D₄×C₃₁	—	4	4	no	—
77	C₂²×D₄×C₃₁	C₂³	1	2	yes	—
78	C₂²×D₄×C₃₁	C₂³	1	1	yes	characteristic
79	C₂²×D₄×C₃₁	C₂³	1	2	yes	—
80	C₂²×D₄×C₃₁	C₂³	1	8	yes	—
81	C₂²×D₄×C₃₁	C₂³	1	2	yes	—
82	C₂³×C₄×C₃₁	C₂³	1	1	yes	characteristic
83	C₂³×C₄×C₃₁	C₂³	1	1	yes	characteristic
84	C₂³×C₄×C₃₁	C₂³	1	2	yes	—
85	C₂³×C₄×C₃₁	C₂³	1	2	yes	—
86	C₂³×C₄×C₃₁	C₂³	1	1	yes	characteristic
87	C₃₁×C₄⋊Q₈	—	4	4	no	—
88	C₃₁×#127	—	4	4	no	—
89	C₃₁×#126	—	2	8	no	—
90	C₃₁×#125	—	2	4	no	—
91	C₃₁×#124	—	2	8	no	—
92	C₃₁×#124	—	2	8	no	—
93	C₃₁×#124	—	2	8	no	—
94	C₃₁×#124	—	2	8	no	—
95	C₃₁×#124	—	2	4	no	—
96	C₃₁×#124	—	4	4	no	—
97	C₃₁×#124	—	4	4	no	—
98	C₃₁×#123	—	2	8	no	—
99	C₃₁×C₂²⋊Q₈	—	2	4	no	—
100	C₃₁×#121	—	2	4	no	—
101	C₃₁×#120	—	2	16	no	—
102	C₄×Q₈×C₃₁	—	2	4	no	—
103	C₄×D₄×C₃₁	—	2	8	no	—
104	C₄×D₄×C₃₁	—	2	4	no	—
105	C₃₁×#117	—	2	8	no	—
106	C₃₁×#117	—	2	8	no	—
107	C₂×C₄⋊C₄×C₃₁	C₂³	1	2	yes	—
108	C₂×C₄⋊C₄×C₃₁	C₂³	1	2	yes	—
109	C₂×C₄⋊C₄×C₃₁	C₂³	1	4	yes	—
110	C₂×C₄⋊C₄×C₃₁	C₂³	1	2	yes	—
111	C₂×C₄⋊C₄×C₃₁	C₂³	1	2	yes	—
112	C₂×C₄⋊C₄×C₃₁	C₂³	1	1	yes	characteristic
113	C₂×C₄⋊C₄×C₃₁	C₂³	1	1	yes	characteristic
114	C₂×C₄⋊C₄×C₃₁	C₂³	1	8	yes	—
115	C₂×C₂²⋊C₄×C₃₁	—	2	8	no	—
116	C₂×C₂²⋊C₄×C₃₁	—	2	8	no	—
117	C₂×C₂²⋊C₄×C₃₁	C₂³	1	4	yes	—
118	C₂×C₂²⋊C₄×C₃₁	C₂³	1	8	yes	—
119	C₂×C₂²⋊C₄×C₃₁	C₂³	1	4	yes	—
120	C₂×C₂²⋊C₄×C₃₁	C₂³	1	4	yes	—
121	C₂×C₂²⋊C₄×C₃₁	C₂³	1	4	yes	—
122	C₂×C₂²⋊C₄×C₃₁	C₂³	1	2	yes	—
123	C₂×C₂²⋊C₄×C₃₁	C₂³	1	2	yes	—
124	C₂×C₂²⋊C₄×C₃₁	C₂³	1	2	yes	—
125	C₂×C₂²⋊C₄×C₃₁	C₂³	1	2	yes	—
126	C₂×C₂²⋊C₄×C₃₁	C₂³	1	2	yes	—
127	C₂×C₂²⋊C₄×C₃₁	C₂³	1	2	yes	—
128	C₂×C₂²⋊C₄×C₃₁	C₂³	1	8	yes	—
129	C₂×C₂²⋊C₄×C₃₁	C₂³	1	8	yes	—
130	C₂×C₂²⋊C₄×C₃₁	C₂³	1	8	yes	—
131	C₂×C₂²⋊C₄×C₃₁	C₂³	1	2	yes	—
132	C₂×C₄²×C₃₁	D₄	1	8	yes	complemented
133	C₂×C₄²×C₃₁	C₂³	1	4	yes	—
134	C₂×C₄²×C₃₁	C₂³	1	2	yes	—
135	C₂×C₄²×C₃₁	C₂³	1	2	yes	—
136	C₂×C₄²×C₃₁	C₂³	1	2	yes	—
137	C₂×C₄²×C₃₁	C₂³	1	4	yes	—
138	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
139	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
140	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
141	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
142	C₃₁×#95	C₂³	1	2	yes	—
143	C₃₁×#95	C₂³	1	8	yes	—
144	C₃₁×#95	C₂³	1	8	yes	—
145	C₃₁×#95	C₂³	1	4	yes	—
146	C₃₁×#95	C₂³	1	2	yes	—
147	C₂⁴×C₃₁	—	2	4	no	—
148	C₂⁴×C₃₁	—	2	8	no	—
149	C₂⁴×C₃₁	—	4	4	no	—
150	C₂⁴×C₃₁	—	4	4	no	—
151	C₂⁴×C₃₁	C₂⁴	1	4	yes	—
152	C₂⁴×C₃₁	C₂⁴	1	2	yes	—
153	C₂⁴×C₃₁	C₂⁴	1	1	yes	characteristic
154	C₂×Q₈×C₃₁	—	2	2	no	—
155	C₂×Q₈×C₃₁	—	2	4	no	—
156	C₂×Q₈×C₃₁	—	2	4	no	—
157	C₂×Q₈×C₃₁	—	2	4	no	—
158	C₂×Q₈×C₃₁	—	2	4	no	—
159	C₂×Q₈×C₃₁	—	2	4	no	—
160	C₂×Q₈×C₃₁	—	4	4	no	—
161	C₂×Q₈×C₃₁	—	4	16	no	—
162	C₂×D₄×C₃₁	—	2	4	no	—
163	C₂×D₄×C₃₁	—	2	8	no	—
164	C₂×D₄×C₃₁	—	2	2	no	—
165	C₂×D₄×C₃₁	—	2	4	no	—
166	C₂×D₄×C₃₁	—	2	4	no	—
167	C₂×D₄×C₃₁	—	2	16	no	—
168	C₂×D₄×C₃₁	—	2	4	no	—
169	C₂×D₄×C₃₁	—	2	4	no	—
170	C₂×D₄×C₃₁	—	2	4	no	—
171	C₂×D₄×C₃₁	—	4	4	no	—
172	C₂×D₄×C₃₁	—	4	16	no	—
173	C₂×D₄×C₃₁	—	4	8	no	—
174	C₂×D₄×C₃₁	—	4	8	no	—
175	C₂²×C₄×C₃₁	—	2	4	no	—
176	C₂²×C₄×C₃₁	—	2	4	no	—
177	C₂²×C₄×C₃₁	—	2	4	no	—
178	C₂²×C₄×C₃₁	—	2	4	no	—
179	C₂²×C₄×C₃₁	—	4	2	no	—
180	C₂²×C₄×C₃₁	—	4	4	no	—
181	C₂²×C₄×C₃₁	—	4	4	no	—
182	C₂²×C₄×C₃₁	C₂×D₄	1	4	yes	—
183	C₂²×C₄×C₃₁	C₂×D₄	1	4	yes	—
184	C₂²×C₄×C₃₁	C₂×D₄	1	4	yes	—
185	C₂²×C₄×C₃₁	C₂⁴	1	4	yes	—
186	C₂²×C₄×C₃₁	C₂⁴	1	2	yes	—
187	C₂²×C₄×C₃₁	C₂⁴	1	2	yes	—
188	C₂²×C₄×C₃₁	C₂⁴	1	4	yes	—
189	C₂²×C₄×C₃₁	C₂⁴	1	2	yes	—
190	C₂²×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
191	C₂²×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
192	C₂²×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
193	C₂²×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
194	C₂²×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
195	C₂²×C₄×C₃₁	C₂⁴	1	4	yes	—
196	C₂²×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
197	C₄⋊C₄×C₃₁	—	2	4	no	—
198	C₄⋊C₄×C₃₁	—	2	4	no	—
199	C₄⋊C₄×C₃₁	—	2	2	no	—
200	C₄⋊C₄×C₃₁	—	2	4	no	—
201	C₄⋊C₄×C₃₁	—	2	2	no	—
202	C₄⋊C₄×C₃₁	—	2	4	no	—
203	C₄⋊C₄×C₃₁	—	2	8	no	—
204	C₄⋊C₄×C₃₁	—	4	8	no	—
205	C₂²⋊C₄×C₃₁	—	2	4	no	—
206	C₂²⋊C₄×C₃₁	—	2	4	no	—
207	C₂²⋊C₄×C₃₁	—	2	8	no	—
208	C₂²⋊C₄×C₃₁	—	2	8	no	—
209	C₂²⋊C₄×C₃₁	—	2	8	no	—
210	C₂²⋊C₄×C₃₁	—	2	8	no	—
211	C₂²⋊C₄×C₃₁	—	2	4	no	—
212	C₂²⋊C₄×C₃₁	—	2	4	no	—
213	C₂²⋊C₄×C₃₁	—	2	4	no	—
214	C₂²⋊C₄×C₃₁	—	2	4	no	—
215	C₂²⋊C₄×C₃₁	—	2	16	no	—
216	C₂²⋊C₄×C₃₁	—	4	8	no	—
217	C₂²⋊C₄×C₃₁	—	4	8	no	—
218	C₂²⋊C₄×C₃₁	—	4	8	no	—
219	C₂²⋊C₄×C₃₁	#41	1	8	yes	—
220	C₄²×C₃₁	—	2	4	no	—
221	C₄²×C₃₁	—	2	8	no	—
222	C₄²×C₃₁	—	2	4	no	—
223	C₄²×C₃₁	—	2	4	no	—
224	C₄²×C₃₁	—	4	4	no	—
225	C₄²×C₃₁	C₂×D₄	1	4	yes	—
226	C₄²×C₃₁	C₂×D₄	1	4	yes	—
227	C₄²×C₃₁	C₂×D₄	1	4	yes	—
228	C₄²×C₃₁	C₂×D₄	1	4	yes	—
229	#35131	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
230	C₂³×C₃₁	—	2	2	no	—
231	C₂³×C₃₁	—	2	2	no	—
232	C₂³×C₃₁	—	2	2	no	—
233	C₂³×C₃₁	—	2	4	no	—
234	C₂³×C₃₁	—	2	8	no	—
235	C₂³×C₃₁	—	2	4	no	—
236	C₂³×C₃₁	—	2	4	no	—
237	C₂³×C₃₁	—	2	4	no	—
238	C₂³×C₃₁	—	2	4	no	—
239	C₂³×C₃₁	—	2	8	no	—
240	C₂³×C₃₁	—	2	4	no	—
241	C₂³×C₃₁	—	4	8	no	—
242	C₂³×C₃₁	—	4	8	no	—
243	C₂³×C₃₁	—	4	8	no	—
244	C₂³×C₃₁	—	8	16	no	—
245	C₂³×C₃₁	C₂²×D₄	1	2	yes	—
246	C₂³×C₃₁	C₂×#41	1	4	yes	—
247	C₂³×C₃₁	C₂⁵	1	1	yes	characteristic
248	Q₈×C₃₁	—	4	8	no	—
249	Q₈×C₃₁	—	4	4	no	—
250	Q₈×C₃₁	—	4	8	no	—
251	D₄×C₃₁	—	4	8	no	—
252	D₄×C₃₁	—	4	8	no	—
253	D₄×C₃₁	—	4	4	no	—
254	D₄×C₃₁	—	4	8	no	—
255	D₄×C₃₁	—	8	16	no	—
256	C₂×C₄×C₃₁	—	2	8	no	—
257	C₂×C₄×C₃₁	—	2	2	no	—
258	C₂×C₄×C₃₁	—	2	2	no	—
259	C₂×C₄×C₃₁	—	2	4	no	—
260	C₂×C₄×C₃₁	—	2	2	no	—
261	C₂×C₄×C₃₁	—	2	2	no	—
262	C₂×C₄×C₃₁	—	2	4	no	—
263	C₂×C₄×C₃₁	—	2	2	no	—
264	C₂×C₄×C₃₁	—	2	8	no	—
265	C₂×C₄×C₃₁	—	2	2	no	—
266	C₂×C₄×C₃₁	—	2	2	no	—
267	C₂×C₄×C₃₁	—	2	4	no	—
268	C₂×C₄×C₃₁	—	2	2	no	—
269	C₂×C₄×C₃₁	—	2	4	no	—
270	C₂×C₄×C₃₁	—	2	4	no	—
271	C₂×C₄×C₃₁	—	2	4	no	—
272	C₂×C₄×C₃₁	—	2	4	no	—
273	C₂×C₄×C₃₁	—	2	4	no	—
274	C₂×C₄×C₃₁	—	2	4	no	—
275	C₂×C₄×C₃₁	—	2	4	no	—
276	C₂×C₄×C₃₁	—	2	4	no	—
277	C₂×C₄×C₃₁	—	4	4	no	—
278	C₂×C₄×C₃₁	—	4	8	no	—
279	C₂×C₄×C₃₁	—	4	8	no	—
280	C₂×C₄×C₃₁	C₂²×D₄	1	2	yes	—
281	C₂×C₄×C₃₁	C₂²×D₄	1	2	yes	—
282	C₂×C₄×C₃₁	C₂²×D₄	1	2	yes	—
283	C₂×C₄×C₃₁	C₂²×D₄	1	2	yes	—
284	C₂×C₄×C₃₁	C₂²×D₄	1	2	yes	—
285	C₂×C₄×C₃₁	C₂²×D₄	1	2	yes	—
286	C₂×C₄×C₃₁	C₂×#41	1	8	yes	—
287	C₂×C₄×C₃₁	H(5,𝔽₂)	1	4	yes	—
288	C₂×#529	C₂×C₃₁	1	1	yes	characteristic, complemented
289	C₂²×#124	C₂×C₃₁	1	1	yes	characteristic, complemented
290	#2861	C₂×C₃₁	1	2	yes	complemented
291	#2858	C₂×C₃₁	1	2	yes	complemented
292	#2662	C₂×C₃₁	1	4	yes	complemented
293	#2610	C₂×C₃₁	1	4	yes	complemented
294	#2545	C₂×C₃₁	1	2	yes	complemented
295	#2544	C₂×C₃₁	1	2	yes	complemented
296	#2436	C₂×C₃₁	1	1	yes	characteristic, complemented
297	#2435	C₂×C₃₁	1	1	yes	characteristic, complemented
298	#2434	C₂×C₃₁	1	1	yes	characteristic, complemented
299	#2433	C₂×C₃₁	1	1	yes	characteristic
300	#2411	C₂×C₃₁	1	4	yes	complemented
301	#2321	C₂×C₃₁	1	4	yes	complemented
302	C₄×#117	C₂×C₃₁	1	1	yes	characteristic, complemented
303	C₂²×C₃₁	—	2	2	no	—
304	C₂²×C₃₁	—	2	2	no	—
305	C₂²×C₃₁	—	2	8	no	—
306	C₂²×C₃₁	—	4	2	no	—
307	C₂²×C₃₁	—	4	4	no	—
308	C₂²×C₃₁	—	4	4	no	—
309	C₂²×C₃₁	—	4	2	no	—
310	C₂²×C₃₁	—	4	2	no	—
311	C₂²×C₃₁	—	4	4	no	—
312	C₂²×C₃₁	—	4	8	no	—
313	C₂²×C₃₁	—	4	8	no	—
314	C₂²×C₃₁	—	4	8	no	—
315	C₂²×C₃₁	—	8	16	no	—
316	C₂²×C₃₁	#532	1	4	yes	—
317	C₂²×C₃₁	C₂³×D₄	1	1	yes	characteristic
318	C₂²×C₃₁	C₂²×#41	1	2	yes	—
319	C₂²×C₃₁	C₂×H(5,𝔽₂)	1	2	yes	—
320	C₂²×C₃₁	#585	1	2	yes	—
321	C₄×C₃₁	—	2	2	no	—
322	C₄×C₃₁	—	2	2	no	—
323	C₄×C₃₁	—	2	2	no	—
324	C₄×C₃₁	—	2	2	no	—
325	C₄×C₃₁	—	2	2	no	—
326	C₄×C₃₁	—	2	8	no	—
327	C₄×C₃₁	—	4	2	no	—
328	C₄×C₃₁	—	4	4	no	—
329	C₄×C₃₁	—	4	2	no	—
330	C₄×C₃₁	—	4	2	no	—
331	C₄×C₃₁	—	4	4	no	—
332	C₄×C₃₁	#535	1	4	yes	—
333	C₂³×Q₈	C₂²×C₃₁	1	1	yes	characteristic
334	C₂³×D₄	C₂²×C₃₁	1	1	yes	characteristic
335	C₂×C₄⋊Q₈	C₂²×C₃₁	1	2	yes	complemented
336	C₂×#127	C₂²×C₃₁	1	2	yes	—
337	#529	—	2	8	no	—
338	C₂×#126	C₂²×C₃₁	1	2	yes	complemented
339	C₂×#125	C₂²×C₃₁	1	1	yes	characteristic, complemented
340	C₂×#124	—	2	8	no	—
341	C₂×#124	C₂²×C₃₁	1	2	yes	—
342	C₂×#124	C₂²×C₃₁	1	2	yes	—
343	C₂×#124	C₂²×C₃₁	1	2	yes	—
344	C₂×#124	C₂²×C₃₁	1	2	yes	—
345	C₂×#124	C₂²×C₃₁	1	2	yes	—
346	C₂×#124	C₂²×C₃₁	1	1	yes	characteristic, complemented
347	C₂×#124	C₂²×C₃₁	1	2	yes	complemented
348	C₂×#123	C₂²×C₃₁	1	2	yes	—
349	C₂×C₂²⋊Q₈	C₂²×C₃₁	1	1	yes	characteristic
350	C₂×#121	C₂²×C₃₁	1	1	yes	characteristic
351	C₂×#120	C₂²×C₃₁	1	4	yes	—
352	C₂×C₄×Q₈	C₂²×C₃₁	1	1	yes	characteristic, complemented
353	C₂×C₄×D₄	C₂²×C₃₁	1	2	yes	—
354	C₂×C₄×D₄	C₂²×C₃₁	1	1	yes	characteristic, complemented
355	C₂×#117	C₂²×C₃₁	1	2	yes	—
356	C₂×#117	C₂²×C₃₁	1	2	yes	complemented
357	C₂²×C₂²⋊C₄	C₂²×C₃₁	1	2	yes	—
358	C₂²×C₂²⋊C₄	C₂²×C₃₁	1	2	yes	—
359	C₂²×C₄²	C₂²×C₃₁	1	1	yes	characteristic
360	#396	C₂²×C₃₁	1	8	yes	complemented
361	#395	C₂²×C₃₁	1	4	yes	complemented
362	#394	C₂²×C₃₁	1	4	yes	—
363	#394	C₂²×C₃₁	1	8	yes	complemented
364	#393	C₂²×C₃₁	1	4	yes	complemented
365	#392	C₂²×C₃₁	1	4	yes	—
366	#391	C₂²×C₃₁	1	2	yes	complemented
367	#391	C₂²×C₃₁	1	2	yes	complemented
368	#390	C₂²×C₃₁	1	2	yes	—
369	#390	C₂²×C₃₁	1	2	yes	—
370	#390	C₂²×C₃₁	1	8	yes	complemented
371	#390	C₂²×C₃₁	1	8	yes	complemented
372	#388	C₂²×C₃₁	1	8	yes	complemented
373	#388	C₂²×C₃₁	1	8	yes	complemented
374	#388	C₂²×C₃₁	1	8	yes	complemented
375	#386	C₂²×C₃₁	1	4	yes	—
376	#386	C₂²×C₃₁	1	8	yes	complemented
377	#385	C₂²×C₃₁	1	4	yes	—
378	#379	C₂²×C₃₁	1	4	yes	—
379	C₄×C₄⋊C₄	C₂²×C₃₁	1	2	yes	complemented
380	C₄×C₄⋊C₄	C₂²×C₃₁	1	2	yes	complemented
381	C₄×C₂²⋊C₄	C₂²×C₃₁	1	2	yes	—
382	C₄×C₂²⋊C₄	C₂²×C₃₁	1	1	yes	characteristic
383	C₄×C₂²⋊C₄	C₂²×C₃₁	1	1	yes	characteristic
384	#376	C₂²×C₃₁	1	2	yes	complemented
385	C₄³	C₂²×C₃₁	1	2	yes	complemented
386	C₂×C₃₁	—	2	4	no	—
387	C₂×C₃₁	—	8	2	no	—
388	C₂×C₃₁	—	8	4	no	—
389	C₂×C₃₁	C₂×#532	1	1	yes	characteristic
390	C₂×C₃₁	C₂×#535	1	1	yes	characteristic
391	C₂×C₃₁	#3449	1	1	yes	characteristic
392	C₂×C₃₁	#3514	1	4	yes	—
393	C₂⁵	C₂³×C₃₁	1	2	yes	—
394	C₂²×Q₈	—	2	4	no	—
395	C₂²×Q₈	—	4	4	no	—
396	C₂²×Q₈	C₂³×C₃₁	1	1	yes	characteristic
397	C₂²×Q₈	C₂³×C₃₁	1	2	yes	—
398	C₂²×Q₈	C₂³×C₃₁	1	2	yes	—
399	C₂²×D₄	—	2	4	no	—
400	C₂²×D₄	—	4	4	no	—
401	C₂²×D₄	C₂³×C₃₁	1	2	yes	—
402	C₂²×D₄	C₂³×C₃₁	1	1	yes	characteristic
403	C₂²×D₄	C₂³×C₃₁	1	2	yes	—
404	C₂²×D₄	C₂³×C₃₁	1	8	yes	—
405	C₂²×D₄	C₂³×C₃₁	1	2	yes	—
406	C₂³×C₄	C₂³×C₃₁	1	1	yes	characteristic
407	C₂³×C₄	C₂³×C₃₁	1	1	yes	characteristic
408	C₂³×C₄	C₂³×C₃₁	1	2	yes	—
409	C₂³×C₄	C₂³×C₃₁	1	2	yes	—
410	C₂³×C₄	C₂³×C₃₁	1	1	yes	characteristic
411	C₄⋊Q₈	—	4	4	no	—
412	#127	—	4	4	no	—
413	#126	—	2	8	no	—
414	#125	—	2	4	no	—
415	#124	—	2	8	no	—
416	#124	—	2	8	no	—
417	#124	—	2	8	no	—
418	#124	—	2	8	no	—
419	#124	—	2	4	no	—
420	#124	—	4	4	no	—
421	#124	—	4	4	no	—
422	#123	—	2	8	no	—
423	C₂²⋊Q₈	—	2	4	no	—
424	#121	—	2	4	no	—
425	#120	—	2	16	no	—
426	C₄×Q₈	—	2	4	no	—
427	C₄×D₄	—	2	8	no	—
428	C₄×D₄	—	2	4	no	—
429	#117	—	2	8	no	—
430	#117	—	2	8	no	—
431	C₂×C₄⋊C₄	C₂³×C₃₁	1	4	yes	—
432	C₂×C₄⋊C₄	C₂³×C₃₁	1	2	yes	—
433	C₂×C₄⋊C₄	C₂³×C₃₁	1	2	yes	—
434	C₂×C₄⋊C₄	C₂³×C₃₁	1	2	yes	—
435	C₂×C₄⋊C₄	C₂³×C₃₁	1	1	yes	characteristic
436	C₂×C₄⋊C₄	C₂³×C₃₁	1	1	yes	characteristic
437	C₂×C₄⋊C₄	C₂³×C₃₁	1	2	yes	—
438	C₂×C₄⋊C₄	C₂³×C₃₁	1	8	yes	—
439	C₂×C₂²⋊C₄	—	2	8	no	—
440	C₂×C₂²⋊C₄	—	2	8	no	—
441	C₂×C₂²⋊C₄	C₂³×C₃₁	1	4	yes	—
442	C₂×C₂²⋊C₄	C₂³×C₃₁	1	4	yes	—
443	C₂×C₂²⋊C₄	C₂³×C₃₁	1	4	yes	—
444	C₂×C₂²⋊C₄	C₂³×C₃₁	1	4	yes	—
445	C₂×C₂²⋊C₄	C₂³×C₃₁	1	2	yes	—
446	C₂×C₂²⋊C₄	C₂³×C₃₁	1	2	yes	—
447	C₂×C₂²⋊C₄	C₂³×C₃₁	1	2	yes	—
448	C₂×C₂²⋊C₄	C₂³×C₃₁	1	2	yes	—
449	C₂×C₂²⋊C₄	C₂³×C₃₁	1	2	yes	—
450	C₂×C₂²⋊C₄	C₂³×C₃₁	1	2	yes	—
451	C₂×C₂²⋊C₄	C₂³×C₃₁	1	8	yes	—
452	C₂×C₂²⋊C₄	C₂³×C₃₁	1	8	yes	—
453	C₂×C₂²⋊C₄	C₂³×C₃₁	1	8	yes	—
454	C₂×C₂²⋊C₄	C₂³×C₃₁	1	8	yes	—
455	C₂×C₂²⋊C₄	C₂³×C₃₁	1	2	yes	—
456	C₂×C₄²	D₄×C₃₁	1	8	yes	complemented
457	C₂×C₄²	C₂³×C₃₁	1	4	yes	—
458	C₂×C₄²	C₂³×C₃₁	1	2	yes	—
459	C₂×C₄²	C₂³×C₃₁	1	2	yes	—
460	C₂×C₄²	C₂³×C₃₁	1	2	yes	—
461	C₂×C₄²	C₂³×C₃₁	1	4	yes	—
462	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
463	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
464	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
465	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
466	#95	C₂³×C₃₁	1	4	yes	—
467	#95	C₂³×C₃₁	1	2	yes	—
468	#95	C₂³×C₃₁	1	2	yes	—
469	#95	C₂³×C₃₁	1	8	yes	—
470	#95	C₂³×C₃₁	1	8	yes	—
471	C₃₁	#35131	1	1	yes	characteristic, direct product factor, Sylow
472	C₂⁴	—	2	4	no	—
473	C₂⁴	—	2	8	no	—
474	C₂⁴	—	4	4	no	—
475	C₂⁴	—	4	4	no	—
476	C₂⁴	C₂⁴×C₃₁	1	4	yes	—
477	C₂⁴	C₂⁴×C₃₁	1	2	yes	—
478	C₂⁴	C₂⁴×C₃₁	1	1	yes	characteristic
479	C₂×Q₈	—	2	2	no	—
480	C₂×Q₈	—	2	4	no	—
481	C₂×Q₈	—	2	4	no	—
482	C₂×Q₈	—	2	4	no	—
483	C₂×Q₈	—	2	4	no	—
484	C₂×Q₈	—	2	4	no	—
485	C₂×Q₈	—	4	4	no	—
486	C₂×Q₈	—	4	16	no	—
487	C₂×D₄	—	2	4	no	—
488	C₂×D₄	—	2	8	no	—
489	C₂×D₄	—	2	2	no	—
490	C₂×D₄	—	2	4	no	—
491	C₂×D₄	—	2	4	no	—
492	C₂×D₄	—	2	16	no	—
493	C₂×D₄	—	2	4	no	—
494	C₂×D₄	—	2	4	no	—
495	C₂×D₄	—	2	4	no	—
496	C₂×D₄	—	4	4	no	—
497	C₂×D₄	—	4	16	no	—
498	C₂×D₄	—	4	8	no	—
499	C₂×D₄	—	4	8	no	—
500	C₂²×C₄	—	2	4	no	—
501	C₂²×C₄	—	2	4	no	—
502	C₂²×C₄	—	2	4	no	—
503	C₂²×C₄	—	2	4	no	—
504	C₂²×C₄	—	4	2	no	—
505	C₂²×C₄	—	4	4	no	—
506	C₂²×C₄	—	4	4	no	—
507	C₂²×C₄	C₂×D₄×C₃₁	1	4	yes	—
508	C₂²×C₄	C₂×D₄×C₃₁	1	4	yes	—
509	C₂²×C₄	C₂×D₄×C₃₁	1	4	yes	—
510	C₂²×C₄	C₂⁴×C₃₁	1	4	yes	—
511	C₂²×C₄	C₂⁴×C₃₁	1	2	yes	—
512	C₂²×C₄	C₂⁴×C₃₁	1	2	yes	—
513	C₂²×C₄	C₂⁴×C₃₁	1	4	yes	—
514	C₂²×C₄	C₂⁴×C₃₁	1	2	yes	—
515	C₂²×C₄	C₂⁴×C₃₁	1	1	yes	characteristic
516	C₂²×C₄	C₂⁴×C₃₁	1	1	yes	characteristic
517	C₂²×C₄	C₂⁴×C₃₁	1	1	yes	characteristic
518	C₂²×C₄	C₂⁴×C₃₁	1	1	yes	characteristic
519	C₂²×C₄	C₂⁴×C₃₁	1	1	yes	characteristic
520	C₂²×C₄	C₂⁴×C₃₁	1	4	yes	—
521	C₂²×C₄	C₂⁴×C₃₁	1	1	yes	characteristic
522	C₄⋊C₄	—	2	4	no	—
523	C₄⋊C₄	—	2	4	no	—
524	C₄⋊C₄	—	2	2	no	—
525	C₄⋊C₄	—	2	4	no	—
526	C₄⋊C₄	—	2	2	no	—
527	C₄⋊C₄	—	2	4	no	—
528	C₄⋊C₄	—	2	8	no	—
529	C₄⋊C₄	—	4	8	no	—
530	C₂²⋊C₄	—	2	4	no	—
531	C₂²⋊C₄	—	2	4	no	—
532	C₂²⋊C₄	—	2	8	no	—
533	C₂²⋊C₄	—	2	8	no	—
534	C₂²⋊C₄	—	2	8	no	—
535	C₂²⋊C₄	—	2	8	no	—
536	C₂²⋊C₄	—	2	4	no	—
537	C₂²⋊C₄	—	2	4	no	—
538	C₂²⋊C₄	—	2	4	no	—
539	C₂²⋊C₄	—	2	4	no	—
540	C₂²⋊C₄	—	2	16	no	—
541	C₂²⋊C₄	—	4	8	no	—
542	C₂²⋊C₄	—	4	8	no	—
543	C₂²⋊C₄	—	4	8	no	—
544	C₂²⋊C₄	#41×C₃₁	1	8	yes	—
545	C₄²	—	2	4	no	—
546	C₄²	—	2	8	no	—
547	C₄²	—	2	4	no	—
548	C₄²	—	2	4	no	—
549	C₄²	—	4	4	no	—
550	C₄²	C₂×D₄×C₃₁	1	4	yes	—
551	C₄²	C₂×D₄×C₃₁	1	4	yes	—
552	C₄²	C₂×D₄×C₃₁	1	4	yes	—
553	C₄²	C₂×D₄×C₃₁	1	4	yes	—
554	C₂³	—	2	2	no	—
555	C₂³	—	2	2	no	—
556	C₂³	—	2	2	no	—
557	C₂³	—	2	4	no	—
558	C₂³	—	2	8	no	—
559	C₂³	—	2	4	no	—
560	C₂³	—	2	4	no	—
561	C₂³	—	2	4	no	—
562	C₂³	—	2	4	no	—
563	C₂³	—	2	8	no	—
564	C₂³	—	2	4	no	—
565	C₂³	—	4	8	no	—
566	C₂³	—	4	8	no	—
567	C₂³	—	4	8	no	—
568	C₂³	—	8	16	no	—
569	C₂³	C₂²×D₄×C₃₁	1	2	yes	—
570	C₂³	C₂×#41×C₃₁	1	4	yes	—
571	C₂³	C₂⁵×C₃₁	1	1	yes	characteristic
572	Q₈	—	4	8	no	—
573	Q₈	—	4	4	no	—
574	Q₈	—	4	8	no	—
575	D₄	—	4	8	no	—
576	D₄	—	4	8	no	—
577	D₄	—	4	4	no	—
578	D₄	—	4	8	no	—
579	D₄	—	8	16	no	—
580	C₂×C₄	—	2	8	no	—
581	C₂×C₄	—	2	2	no	—
582	C₂×C₄	—	2	2	no	—
583	C₂×C₄	—	2	4	no	—
584	C₂×C₄	—	2	2	no	—
585	C₂×C₄	—	2	2	no	—
586	C₂×C₄	—	2	4	no	—
587	C₂×C₄	—	2	2	no	—
588	C₂×C₄	—	2	8	no	—
589	C₂×C₄	—	2	2	no	—
590	C₂×C₄	—	2	2	no	—
591	C₂×C₄	—	2	4	no	—
592	C₂×C₄	—	2	2	no	—
593	C₂×C₄	—	2	4	no	—
594	C₂×C₄	—	2	4	no	—
595	C₂×C₄	—	2	4	no	—
596	C₂×C₄	—	2	4	no	—
597	C₂×C₄	—	2	4	no	—
598	C₂×C₄	—	2	4	no	—
599	C₂×C₄	—	2	4	no	—
600	C₂×C₄	—	2	4	no	—
601	C₂×C₄	—	4	4	no	—
602	C₂×C₄	—	4	8	no	—
603	C₂×C₄	—	4	8	no	—
604	C₂×C₄	C₂²×D₄×C₃₁	1	2	yes	—
605	C₂×C₄	C₂²×D₄×C₃₁	1	2	yes	—
606	C₂×C₄	C₂²×D₄×C₃₁	1	2	yes	—
607	C₂×C₄	C₂²×D₄×C₃₁	1	2	yes	—
608	C₂×C₄	C₂²×D₄×C₃₁	1	2	yes	—
609	C₂×C₄	C₂²×D₄×C₃₁	1	2	yes	—
610	C₂×C₄	C₂×#41×C₃₁	1	8	yes	—
611	C₂×C₄	C₃₁×H(5,𝔽₂)	1	4	yes	—
612	C₂²	—	2	2	no	—
613	C₂²	—	2	2	no	—
614	C₂²	—	2	8	no	—
615	C₂²	—	4	2	no	—
616	C₂²	—	4	4	no	—
617	C₂²	—	4	4	no	—
618	C₂²	—	4	2	no	—
619	C₂²	—	4	2	no	—
620	C₂²	—	4	4	no	—
621	C₂²	—	4	8	no	—
622	C₂²	—	4	8	no	—
623	C₂²	—	4	8	no	—
624	C₂²	—	8	16	no	—
625	C₂²	C₃₁×#532	1	4	yes	—
626	C₂²	C₂³×D₄×C₃₁	1	1	yes	characteristic
627	C₂²	C₂²×#41×C₃₁	1	2	yes	—
628	C₂²	C₂×C₃₁×H(5,𝔽₂)	1	2	yes	—
629	C₂²	C₃₁×#585	1	2	yes	—
630	C₄	—	2	2	no	—
631	C₄	—	2	2	no	—
632	C₄	—	2	2	no	—
633	C₄	—	2	2	no	—
634	C₄	—	2	2	no	—
635	C₄	—	2	8	no	—
636	C₄	—	4	2	no	—
637	C₄	—	4	4	no	—
638	C₄	—	4	2	no	—
639	C₄	—	4	2	no	—
640	C₄	—	4	4	no	—
641	C₄	C₃₁×#535	1	4	yes	—
642	C₂	—	2	4	no	—
643	C₂	—	8	4	no	—
644	C₂	—	8	2	no	—
645	C₂	C₂×C₃₁×#532	1	1	yes	characteristic
646	C₂	C₂×C₃₁×#535	1	1	yes	characteristic
647	C₂	C₃₁×#3449	1	1	yes	characteristic
648	C₂	C₃₁×#3514	1	4	yes	—

	subgroup	G/H
center, socle	C₂³×C₃₁	C₂⁵
commutator, Frattini, cosocle	C₂³	C₂⁵×C₃₁
prime radical	C₃₁×#2433	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
1. C₂×C₃₁×#529	C₂	semidirect
2. C₂²×C₃₁×#124	C₂	semidirect
3. C₃₁×#2861	C₂	semidirect
4. C₃₁×#2858	C₂	semidirect
5. C₃₁×#2662	C₂	semidirect
6. C₃₁×#2610	C₂	semidirect
7. C₃₁×#2545	C₂	semidirect
8. C₃₁×#2544	C₂	semidirect
9. C₃₁×#2436	C₂	semidirect
10. C₃₁×#2435	C₂	semidirect
11. C₃₁×#2434	C₂	semidirect
13. C₃₁×#2411	C₂	semidirect
14. C₃₁×#2321	C₂	semidirect
15. C₄×C₃₁×#117	C₂	semidirect
18. C₂×C₃₁×C₄⋊Q₈	C₂²	semidirect
21. C₂×C₃₁×#126	C₂²	semidirect
22. C₂×C₃₁×#125	C₂²	semidirect
29. C₂×C₃₁×#124	C₂²	semidirect
30. C₂×C₃₁×#124	C₂²	semidirect
35. C₂×C₄×Q₈×C₃₁	C₂²	semidirect
37. C₂×C₄×D₄×C₃₁	C₂²	semidirect
39. C₂×C₃₁×#117	C₂²	semidirect
43. C₃₁×#396	C₂²	semidirect
44. C₃₁×#395	C₂²	semidirect
46. C₃₁×#394	C₂²	semidirect
47. C₃₁×#393	C₂²	semidirect
49. C₃₁×#391	C₂²	semidirect
50. C₃₁×#391	C₂²	semidirect
53. C₃₁×#390	C₂²	semidirect
54. C₃₁×#390	C₂²	semidirect
55. C₃₁×#388	C₂²	semidirect
56. C₃₁×#388	C₂²	semidirect
57. C₃₁×#388	C₂²	semidirect
59. C₃₁×#386	C₂²	semidirect
62. C₄×C₄⋊C₄×C₃₁	C₂²	semidirect
63. C₄×C₄⋊C₄×C₃₁	C₂²	semidirect
67. C₃₁×#376	C₂²	semidirect
68. C₄³×C₃₁	C₂²	semidirect
132. C₂×C₄²×C₃₁	D₄	semidirect
229. #35131	C₃₁	direct (N × H)
288. C₂×#529	C₂×C₃₁	semidirect
289. C₂²×#124	C₂×C₃₁	semidirect
290. #2861	C₂×C₃₁	semidirect
291. #2858	C₂×C₃₁	semidirect
292. #2662	C₂×C₃₁	semidirect
293. #2610	C₂×C₃₁	semidirect
294. #2545	C₂×C₃₁	semidirect
295. #2544	C₂×C₃₁	semidirect
296. #2436	C₂×C₃₁	semidirect
297. #2435	C₂×C₃₁	semidirect
298. #2434	C₂×C₃₁	semidirect
300. #2411	C₂×C₃₁	semidirect
301. #2321	C₂×C₃₁	semidirect
302. C₄×#117	C₂×C₃₁	semidirect
335. C₂×C₄⋊Q₈	C₂²×C₃₁	semidirect
338. C₂×#126	C₂²×C₃₁	semidirect
339. C₂×#125	C₂²×C₃₁	semidirect
346. C₂×#124	C₂²×C₃₁	semidirect
347. C₂×#124	C₂²×C₃₁	semidirect
352. C₂×C₄×Q₈	C₂²×C₃₁	semidirect
354. C₂×C₄×D₄	C₂²×C₃₁	semidirect
356. C₂×#117	C₂²×C₃₁	semidirect
360. #396	C₂²×C₃₁	semidirect
361. #395	C₂²×C₃₁	semidirect
363. #394	C₂²×C₃₁	semidirect
364. #393	C₂²×C₃₁	semidirect
366. #391	C₂²×C₃₁	semidirect
367. #391	C₂²×C₃₁	semidirect
370. #390	C₂²×C₃₁	semidirect
371. #390	C₂²×C₃₁	semidirect
372. #388	C₂²×C₃₁	semidirect
373. #388	C₂²×C₃₁	semidirect
374. #388	C₂²×C₃₁	semidirect
376. #386	C₂²×C₃₁	semidirect
379. C₄×C₄⋊C₄	C₂²×C₃₁	semidirect
380. C₄×C₄⋊C₄	C₂²×C₃₁	semidirect
384. #376	C₂²×C₃₁	semidirect
385. C₄³	C₂²×C₃₁	semidirect
456. C₂×C₄²	D₄×C₃₁	semidirect
471. C₃₁	#35131	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
C₂²×C₃₁×#124	2
C₃₁×#2861	2
C₃₁×#2858	2
C₃₁×#2662	2
C₃₁×#2610	2
C₃₁×#2545	2
C₃₁×#2544	2
C₃₁×#2436	2
C₃₁×#2435	2
C₃₁×#2434	2
C₃₁×#2433	2
C₃₁×#2411	2
C₃₁×#2321	2
C₄×C₃₁×#117	2
#35131	31

Subgroups — C31×#35131 #21491890

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#35131 #21491890