Subgroups — C₃₁×#34244 #21491003

Order 7936.

2186 conjugacy classes of subgroups.

The subgroup lattice has 634 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×#2374	C₂	1	4	yes	maximal, complemented
3	C₃₁×#2372	C₂	1	2	yes	maximal, complemented
4	C₃₁×#2371	C₂	1	2	yes	maximal, complemented
5	C₃₁×#2355	C₂	1	2	yes	maximal, complemented
6	C₃₁×#2354	C₂	1	2	yes	maximal, complemented
7	C₃₁×#2352	C₂	1	2	yes	maximal, complemented
8	C₃₁×#2352	C₂	1	2	yes	maximal, complemented
9	C₃₁×#2333	C₂	1	2	yes	maximal
10	C₃₁×#2332	C₂	1	2	yes	maximal, complemented
11	C₃₁×#2314	C₂	1	2	yes	maximal, complemented
12	C₄×C₃₁×#126	C₂	1	4	yes	maximal, complemented
13	C₃₁×#2283	C₂	1	1	yes	maximal, characteristic, complemented
14	C₃₁×#2274	C₂	1	2	yes	maximal, complemented
15	C₄×C₃₁×#117	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	C₂×C₃₁×#117	C₂²	1	2	yes	—
28	C₂×C₃₁×#117	C₂²	1	2	yes	—
29	C₂²×C₄⋊C₄×C₃₁	C₂²	1	2	yes	—
30	C₂²×C₄²×C₃₁	C₂²	1	1	yes	characteristic
31	C₃₁×#391	C₂²	1	4	yes	—
32	C₃₁×#390	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×#388	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×#387	C₂²	1	4	yes	—
45	C₃₁×#386	C₂²	1	4	yes	—
46	C₃₁×#386	C₂²	1	4	yes	—
47	C₃₁×#384	C₂²	1	2	yes	—
48	C₃₁×#383	C₂²	1	2	yes	—
49	C₃₁×#382	C₂²	1	2	yes	—
50	C₃₁×#382	C₂²	1	2	yes	—
51	C₃₁×#381	C₂²	1	2	yes	—
52	C₃₁×#380	C₂²	1	2	yes	—
53	C₃₁×#380	C₂²	1	2	yes	—
54	C₃₁×#380	C₂²	1	1	yes	characteristic
55	C₃₁×#380	C₂²	1	1	yes	characteristic
56	C₄×C₄⋊C₄×C₃₁	C₂²	1	1	yes	characteristic
57	C₄×C₄⋊C₄×C₃₁	C₂²	1	2	yes	—
58	C₄×C₄⋊C₄×C₃₁	C₂²	1	4	yes	—
59	C₄×C₄⋊C₄×C₃₁	C₂²	1	2	yes	—
60	C₄×C₄⋊C₄×C₃₁	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	4	yes	—
65	C₄×C₄⋊C₄×C₃₁	C₂²	1	1	yes	characteristic
66	C₄×C₂²⋊C₄×C₃₁	C₂²	1	2	yes	—
67	C₄×C₂²⋊C₄×C₃₁	C₂²	1	4	yes	—
68	C₄×C₂²⋊C₄×C₃₁	C₂²	1	4	yes	—
69	C₄×C₂²⋊C₄×C₃₁	C₂²	1	4	yes	—
70	C₄×C₂²⋊C₄×C₃₁	C₂²	1	4	yes	—
71	C₄×C₂²⋊C₄×C₃₁	C₂²	1	2	yes	—
72	C₄×C₂²⋊C₄×C₃₁	C₂²	1	2	yes	—
73	C₃₁×#376	C₂²	1	2	yes	—
74	C₃₁×#376	C₂²	1	2	yes	—
75	C₄³×C₃₁	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	1	yes	characteristic
80	C₂³×C₄×C₃₁	C₂³	1	2	yes	—
81	C₂³×C₄×C₃₁	C₂³	1	4	yes	—
82	C₂³×C₄×C₃₁	C₂³	1	1	yes	characteristic
83	C₂³×C₄×C₃₁	C₂³	1	1	yes	characteristic
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	—	2	8	no	—
94	C₃₁×#117	—	2	8	no	—
95	C₃₁×#117	C₂×C₄	1	16	yes	complemented
96	C₂×C₄⋊C₄×C₃₁	—	2	8	no	—
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	2	yes	—
100	C₂×C₄⋊C₄×C₃₁	C₂³	1	2	yes	—
101	C₂×C₄⋊C₄×C₃₁	C₂³	1	4	yes	—
102	C₂×C₄⋊C₄×C₃₁	C₂³	1	4	yes	—
103	C₂×C₄⋊C₄×C₃₁	C₂³	1	2	yes	—
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×C₄⋊C₄×C₃₁	C₂³	1	1	yes	characteristic
108	C₂×C₄⋊C₄×C₃₁	C₂³	1	2	yes	—
109	C₂×C₄⋊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	2	yes	—
113	C₂×C₂²⋊C₄×C₃₁	C₂³	1	2	yes	—
114	C₂×C₂²⋊C₄×C₃₁	C₂³	1	4	yes	—
115	C₂×C₂²⋊C₄×C₃₁	C₂³	1	4	yes	—
116	C₂×C₂²⋊C₄×C₃₁	C₂³	1	2	yes	—
117	C₂×C₂²⋊C₄×C₃₁	C₂³	1	2	yes	—
118	C₂×C₂²⋊C₄×C₃₁	C₂³	1	2	yes	—
119	C₂×C₂²⋊C₄×C₃₁	C₂³	1	2	yes	—
120	C₂×C₂²⋊C₄×C₃₁	C₂³	1	2	yes	—
121	C₂×C₄²×C₃₁	C₂×C₄	1	8	yes	complemented
122	C₂×C₄²×C₃₁	C₂³	1	4	yes	—
123	C₂×C₄²×C₃₁	C₂³	1	4	yes	—
124	C₂×C₄²×C₃₁	C₂³	1	2	yes	—
125	C₂×C₄²×C₃₁	C₂³	1	4	yes	—
126	C₂×C₄²×C₃₁	C₂³	1	2	yes	—
127	C₂×C₄²×C₃₁	C₂³	1	4	yes	—
128	C₂×C₄²×C₃₁	C₂³	1	4	yes	—
129	C₂×C₄²×C₃₁	C₂³	1	4	yes	—
130	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
131	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
132	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
133	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
134	C₂×C₄²×C₃₁	C₂³	1	2	yes	—
135	C₃₁×#95	C₂³	1	2	yes	—
136	C₃₁×#95	C₂³	1	2	yes	—
137	C₃₁×#95	C₂³	1	4	yes	—
138	C₃₁×#95	C₂³	1	2	yes	—
139	C₃₁×#95	C₂³	1	2	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	C₃₁×#95	C₂³	1	4	yes	—
147	C₃₁×#95	C₂³	1	4	yes	—
148	C₃₁×#95	C₂³	1	4	yes	—
149	C₂⁴×C₃₁	C₂⁴	1	1	yes	characteristic
150	C₂⁴×C₃₁	C₂⁴	1	2	yes	—
151	C₂×Q₈×C₃₁	—	4	1	no	—
152	C₂×Q₈×C₃₁	—	4	1	no	—
153	C₂×Q₈×C₃₁	C₂²×C₄	1	4	yes	—
154	C₂×D₄×C₃₁	—	4	2	no	—
155	C₂×D₄×C₃₁	—	4	2	no	—
156	C₂×D₄×C₃₁	—	4	1	no	—
157	C₂×D₄×C₃₁	—	4	1	no	—
158	C₂×D₄×C₃₁	C₂²×C₄	1	4	yes	—
159	C₂×D₄×C₃₁	C₂²×C₄	1	8	yes	—
160	C₂²×C₄×C₃₁	—	2	4	no	—
161	C₂²×C₄×C₃₁	—	2	4	no	—
162	C₂²×C₄×C₃₁	—	2	4	no	—
163	C₂²×C₄×C₃₁	—	2	8	no	—
164	C₂²×C₄×C₃₁	—	4	2	no	—
165	C₂²×C₄×C₃₁	—	4	1	no	—
166	C₂²×C₄×C₃₁	—	4	1	no	—
167	C₂²×C₄×C₃₁	—	4	4	no	—
168	C₂²×C₄×C₃₁	—	4	4	no	—
169	C₂²×C₄×C₃₁	C₂²×C₄	1	4	yes	—
170	C₂²×C₄×C₃₁	C₂²×C₄	1	8	yes	—
171	C₂²×C₄×C₃₁	C₂²×C₄	1	4	yes	—
172	C₂²×C₄×C₃₁	C₂²×C₄	1	4	yes	—
173	C₂²×C₄×C₃₁	C₂⁴	1	4	yes	—
174	C₂²×C₄×C₃₁	C₂⁴	1	4	yes	—
175	C₂²×C₄×C₃₁	C₂⁴	1	4	yes	—
176	C₂²×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
177	C₂²×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
178	C₂²×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
179	C₂²×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
180	C₂²×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
181	C₂²×C₄×C₃₁	C₂⁴	1	2	yes	—
182	C₂²×C₄×C₃₁	C₂⁴	1	2	yes	—
183	C₂²×C₄×C₃₁	C₂⁴	1	2	yes	—
184	C₂²×C₄×C₃₁	C₂⁴	1	4	yes	—
185	C₂²×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
186	C₄⋊C₄×C₃₁	—	2	8	no	—
187	C₄⋊C₄×C₃₁	—	2	4	no	—
188	C₄⋊C₄×C₃₁	—	2	4	no	—
189	C₄⋊C₄×C₃₁	—	2	4	no	—
190	C₄⋊C₄×C₃₁	—	4	4	no	—
191	C₄⋊C₄×C₃₁	—	4	4	no	—
192	C₄⋊C₄×C₃₁	C₂²×C₄	1	8	yes	—
193	C₄⋊C₄×C₃₁	C₂²×C₄	1	8	yes	—
194	C₄⋊C₄×C₃₁	C₂²×C₄	1	4	yes	—
195	C₄⋊C₄×C₃₁	C₂²×C₄	1	8	yes	—
196	C₄⋊C₄×C₃₁	C₂²×C₄	1	4	yes	—
197	C₄⋊C₄×C₃₁	C₂²×C₄	1	8	yes	—
198	C₄⋊C₄×C₃₁	#41	1	8	yes	—
199	C₂²⋊C₄×C₃₁	—	2	8	no	—
200	C₂²⋊C₄×C₃₁	—	2	8	no	—
201	C₂²⋊C₄×C₃₁	—	4	4	no	—
202	C₂²⋊C₄×C₃₁	—	4	4	no	—
203	C₂²⋊C₄×C₃₁	—	4	2	no	—
204	C₂²⋊C₄×C₃₁	—	4	2	no	—
205	C₂²⋊C₄×C₃₁	C₂²×C₄	1	8	yes	—
206	C₂²⋊C₄×C₃₁	C₂²×C₄	1	8	yes	—
207	C₂²⋊C₄×C₃₁	C₂²×C₄	1	8	yes	—
208	C₂²⋊C₄×C₃₁	C₂²×C₄	1	8	yes	—
209	C₂²⋊C₄×C₃₁	C₂²×C₄	1	8	yes	—
210	C₄²×C₃₁	—	2	8	no	—
211	C₄²×C₃₁	—	2	8	no	—
212	C₄²×C₃₁	—	4	2	no	—
213	C₄²×C₃₁	—	4	4	no	—
214	C₄²×C₃₁	—	4	4	no	—
215	C₄²×C₃₁	—	4	4	no	—
216	C₄²×C₃₁	—	4	4	no	—
217	C₄²×C₃₁	—	4	2	no	—
218	C₄²×C₃₁	C₂²×C₄	1	4	yes	—
219	C₄²×C₃₁	C₂²×C₄	1	4	yes	—
220	C₄²×C₃₁	C₂²×C₄	1	4	yes	—
221	C₄²×C₃₁	C₂²×C₄	1	4	yes	—
222	C₄²×C₃₁	C₂²×C₄	1	8	yes	—
223	#34244	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
224	C₂³×C₃₁	—	2	2	no	—
225	C₂³×C₃₁	—	2	2	no	—
226	C₂³×C₃₁	—	2	2	no	—
227	C₂³×C₃₁	—	2	1	no	—
228	C₂³×C₃₁	—	2	1	no	—
229	C₂³×C₃₁	—	2	2	no	—
230	C₂³×C₃₁	—	2	2	no	—
231	C₂³×C₃₁	—	2	2	no	—
232	C₂³×C₃₁	—	2	2	no	—
233	C₂³×C₃₁	—	2	2	no	—
234	C₂³×C₃₁	C₂³×C₄	1	4	yes	—
235	C₂³×C₃₁	C₂³×C₄	1	2	yes	—
236	C₂³×C₃₁	C₂⁵	1	1	yes	characteristic
237	Q₈×C₃₁	—	4	4	no	—
238	D₄×C₃₁	—	4	4	no	—
239	D₄×C₃₁	—	4	8	no	—
240	C₂×C₄×C₃₁	—	2	2	no	—
241	C₂×C₄×C₃₁	—	2	2	no	—
242	C₂×C₄×C₃₁	—	2	2	no	—
243	C₂×C₄×C₃₁	—	2	1	no	—
244	C₂×C₄×C₃₁	—	2	2	no	—
245	C₂×C₄×C₃₁	—	2	2	no	—
246	C₂×C₄×C₃₁	—	2	1	no	—
247	C₂×C₄×C₃₁	—	2	4	no	—
248	C₂×C₄×C₃₁	—	2	4	no	—
249	C₂×C₄×C₃₁	—	2	2	no	—
250	C₂×C₄×C₃₁	—	2	2	no	—
251	C₂×C₄×C₃₁	—	2	2	no	—
252	C₂×C₄×C₃₁	—	2	2	no	—
253	C₂×C₄×C₃₁	—	2	1	no	—
254	C₂×C₄×C₃₁	—	2	1	no	—
255	C₂×C₄×C₃₁	—	2	16	no	—
256	C₂×C₄×C₃₁	—	2	8	no	—
257	C₂×C₄×C₃₁	—	2	4	no	—
258	C₂×C₄×C₃₁	—	2	4	no	—
259	C₂×C₄×C₃₁	—	2	4	no	—
260	C₂×C₄×C₃₁	—	2	4	no	—
261	C₂×C₄×C₃₁	—	2	4	no	—
262	C₂×C₄×C₃₁	—	2	4	no	—
263	C₂×C₄×C₃₁	—	2	4	no	—
264	C₂×C₄×C₃₁	—	2	4	no	—
265	C₂×C₄×C₃₁	—	2	4	no	—
266	C₂×C₄×C₃₁	—	4	4	no	—
267	C₂×C₄×C₃₁	—	4	4	no	—
268	C₂×C₄×C₃₁	—	8	8	no	—
269	C₂×C₄×C₃₁	C₂³×C₄	1	2	yes	—
270	C₂×C₄×C₃₁	C₂³×C₄	1	2	yes	—
271	C₂×C₄×C₃₁	C₂³×C₄	1	2	yes	—
272	C₂×C₄×C₃₁	C₂³×C₄	1	2	yes	—
273	C₂×C₄×C₃₁	C₂³×C₄	1	2	yes	—
274	C₂×C₄×C₃₁	C₂³×C₄	1	4	yes	—
275	C₂×C₄×C₃₁	C₂³×C₄	1	4	yes	—
276	C₂×C₄×C₃₁	C₂³×C₄	1	4	yes	—
277	C₂×C₄×C₃₁	C₂³×C₄	1	2	yes	—
278	C₂×C₄×C₃₁	C₂×#41	1	8	yes	—
279	C₂×C₄×C₃₁	C₂×#41	1	4	yes	—
280	C₂×#529	C₂×C₃₁	1	1	yes	characteristic
281	#2374	C₂×C₃₁	1	4	yes	complemented
282	#2372	C₂×C₃₁	1	2	yes	complemented
283	#2371	C₂×C₃₁	1	2	yes	complemented
284	#2355	C₂×C₃₁	1	2	yes	complemented
285	#2354	C₂×C₃₁	1	2	yes	complemented
286	#2352	C₂×C₃₁	1	2	yes	complemented
287	#2352	C₂×C₃₁	1	2	yes	complemented
288	#2333	C₂×C₃₁	1	2	yes	—
289	#2332	C₂×C₃₁	1	2	yes	complemented
290	#2314	C₂×C₃₁	1	2	yes	complemented
291	C₄×#126	C₂×C₃₁	1	4	yes	complemented
292	#2283	C₂×C₃₁	1	1	yes	characteristic, complemented
293	#2274	C₂×C₃₁	1	2	yes	complemented
294	C₄×#117	C₂×C₃₁	1	1	yes	characteristic, complemented
295	C₂²×C₃₁	—	2	2	no	—
296	C₂²×C₃₁	—	2	4	no	—
297	C₂²×C₃₁	—	2	2	no	—
298	C₂²×C₃₁	—	2	2	no	—
299	C₂²×C₃₁	—	2	4	no	—
300	C₂²×C₃₁	—	2	4	no	—
301	C₂²×C₃₁	—	4	2	no	—
302	C₂²×C₃₁	—	4	2	no	—
303	C₂²×C₃₁	—	4	2	no	—
304	C₂²×C₃₁	—	4	2	no	—
305	C₂²×C₃₁	—	4	2	no	—
306	C₂²×C₃₁	—	4	2	no	—
307	C₂²×C₃₁	C₂⁴×C₄	1	1	yes	characteristic
308	C₂²×C₃₁	C₂²×#41	1	2	yes	—
309	C₂²×C₃₁	#585	1	1	yes	characteristic
310	C₂²×C₃₁	#585	1	2	yes	—
311	C₂²×C₃₁	#585	1	1	yes	characteristic
312	C₄×C₃₁	—	2	2	no	—
313	C₄×C₃₁	—	2	4	no	—
314	C₄×C₃₁	—	2	2	no	—
315	C₄×C₃₁	—	2	2	no	—
316	C₄×C₃₁	—	2	2	no	—
317	C₄×C₃₁	—	2	2	no	—
318	C₄×C₃₁	—	2	4	no	—
319	C₄×C₃₁	—	2	4	no	—
320	C₄×C₃₁	—	2	8	no	—
321	C₄×C₃₁	—	4	4	no	—
322	C₄×C₃₁	—	4	4	no	—
323	C₄×C₃₁	—	4	4	no	—
324	C₄×C₃₁	C₄×#41	1	4	yes	—
325	#529	C₄×C₃₁	1	16	yes	complemented
326	C₂×#126	C₂²×C₃₁	1	2	yes	—
327	C₂×#125	C₂²×C₃₁	1	1	yes	characteristic
328	C₂×#124	C₂²×C₃₁	1	1	yes	characteristic
329	C₂×#123	C₂²×C₃₁	1	2	yes	—
330	C₂×C₂²⋊Q₈	C₂²×C₃₁	1	1	yes	characteristic
331	C₂×#121	C₂²×C₃₁	1	1	yes	characteristic
332	C₂×C₄×Q₈	C₂²×C₃₁	1	1	yes	characteristic
333	C₂×C₄×D₄	C₂²×C₃₁	1	1	yes	characteristic
334	C₂×C₄×D₄	C₂²×C₃₁	1	2	yes	—
335	C₂×#117	C₂²×C₃₁	1	2	yes	—
336	C₂×#117	C₂²×C₃₁	1	2	yes	—
337	C₂×#117	C₂²×C₃₁	1	2	yes	—
338	C₂²×C₄⋊C₄	C₂²×C₃₁	1	2	yes	—
339	C₂²×C₄²	C₂²×C₃₁	1	1	yes	characteristic
340	#391	C₂²×C₃₁	1	4	yes	—
341	#390	C₂²×C₃₁	1	4	yes	—
342	#388	C₂²×C₃₁	1	4	yes	—
343	#388	C₂²×C₃₁	1	4	yes	—
344	#388	C₂²×C₃₁	1	4	yes	—
345	#388	C₂²×C₃₁	1	4	yes	—
346	#388	C₂²×C₃₁	1	4	yes	—
347	#388	C₂²×C₃₁	1	4	yes	—
348	#387	C₂²×C₃₁	1	4	yes	—
349	#387	C₂²×C₃₁	1	4	yes	—
350	#387	C₂²×C₃₁	1	4	yes	—
351	#387	C₂²×C₃₁	1	4	yes	—
352	#387	C₂²×C₃₁	1	4	yes	—
353	#387	C₂²×C₃₁	1	4	yes	—
354	#386	C₂²×C₃₁	1	4	yes	—
355	#386	C₂²×C₃₁	1	4	yes	—
356	#384	C₂²×C₃₁	1	2	yes	—
357	#383	C₂²×C₃₁	1	2	yes	—
358	#382	C₂²×C₃₁	1	2	yes	—
359	#382	C₂²×C₃₁	1	2	yes	—
360	#381	C₂²×C₃₁	1	2	yes	—
361	#380	C₂²×C₃₁	1	2	yes	—
362	#380	C₂²×C₃₁	1	2	yes	—
363	#380	C₂²×C₃₁	1	1	yes	characteristic
364	#380	C₂²×C₃₁	1	1	yes	characteristic
365	C₄×C₄⋊C₄	C₂²×C₃₁	1	4	yes	—
366	C₄×C₄⋊C₄	C₂²×C₃₁	1	4	yes	—
367	C₄×C₄⋊C₄	C₂²×C₃₁	1	4	yes	—
368	C₄×C₄⋊C₄	C₂²×C₃₁	1	1	yes	characteristic
369	C₄×C₄⋊C₄	C₂²×C₃₁	1	2	yes	—
370	C₄×C₄⋊C₄	C₂²×C₃₁	1	2	yes	—
371	C₄×C₄⋊C₄	C₂²×C₃₁	1	2	yes	—
372	C₄×C₄⋊C₄	C₂²×C₃₁	1	2	yes	—
373	C₄×C₄⋊C₄	C₂²×C₃₁	1	4	yes	—
374	C₄×C₄⋊C₄	C₂²×C₃₁	1	1	yes	characteristic
375	C₄×C₂²⋊C₄	C₂²×C₃₁	1	4	yes	—
376	C₄×C₂²⋊C₄	C₂²×C₃₁	1	4	yes	—
377	C₄×C₂²⋊C₄	C₂²×C₃₁	1	4	yes	—
378	C₄×C₂²⋊C₄	C₂²×C₃₁	1	2	yes	—
379	C₄×C₂²⋊C₄	C₂²×C₃₁	1	4	yes	—
380	C₄×C₂²⋊C₄	C₂²×C₃₁	1	2	yes	—
381	C₄×C₂²⋊C₄	C₂²×C₃₁	1	2	yes	—
382	#376	C₂²×C₃₁	1	2	yes	—
383	#376	C₂²×C₃₁	1	2	yes	—
384	C₄³	C₂²×C₃₁	1	2	yes	—
385	C₂×C₃₁	—	4	2	no	—
386	C₂×C₃₁	—	4	4	no	—
387	C₂×C₃₁	C₂×C₄×#41	1	1	yes	characteristic
388	C₂×C₃₁	#3428	1	1	yes	characteristic
389	C₂×C₃₁	#3428	1	1	yes	characteristic
390	C₂×C₃₁	#3521	1	2	yes	—
391	C₂×C₃₁	#3524	1	2	yes	—
392	C₂²×Q₈	C₂³×C₃₁	1	1	yes	characteristic
393	C₂²×D₄	C₂³×C₃₁	1	2	yes	—
394	C₂²×D₄	C₂³×C₃₁	1	1	yes	characteristic
395	C₂³×C₄	C₂³×C₃₁	1	2	yes	—
396	C₂³×C₄	C₂³×C₃₁	1	1	yes	characteristic
397	C₂³×C₄	C₂³×C₃₁	1	4	yes	—
398	C₂³×C₄	C₂³×C₃₁	1	1	yes	characteristic
399	C₂³×C₄	C₂³×C₃₁	1	1	yes	characteristic
400	#126	C₂×C₄×C₃₁	1	16	yes	complemented
401	#125	C₂×C₄×C₃₁	1	8	yes	complemented
402	#124	C₂×C₄×C₃₁	1	8	yes	—
403	#123	C₂×C₄×C₃₁	1	16	yes	complemented
404	C₂²⋊Q₈	C₂×C₄×C₃₁	1	8	yes	complemented
405	#121	C₂×C₄×C₃₁	1	8	yes	—
406	C₄×Q₈	C₂×C₄×C₃₁	1	8	yes	complemented
407	C₄×D₄	C₂×C₄×C₃₁	1	8	yes	—
408	C₄×D₄	C₂×C₄×C₃₁	1	16	yes	complemented
409	#117	—	2	8	no	—
410	#117	—	2	8	no	—
411	#117	C₂×C₄×C₃₁	1	16	yes	complemented
412	C₂×C₄⋊C₄	—	2	8	no	—
413	C₂×C₄⋊C₄	C₂³×C₃₁	1	2	yes	—
414	C₂×C₄⋊C₄	C₂³×C₃₁	1	4	yes	—
415	C₂×C₄⋊C₄	C₂³×C₃₁	1	2	yes	—
416	C₂×C₄⋊C₄	C₂³×C₃₁	1	2	yes	—
417	C₂×C₄⋊C₄	C₂³×C₃₁	1	4	yes	—
418	C₂×C₄⋊C₄	C₂³×C₃₁	1	4	yes	—
419	C₂×C₄⋊C₄	C₂³×C₃₁	1	2	yes	—
420	C₂×C₄⋊C₄	C₂³×C₃₁	1	2	yes	—
421	C₂×C₄⋊C₄	C₂³×C₃₁	1	1	yes	characteristic
422	C₂×C₄⋊C₄	C₂³×C₃₁	1	2	yes	—
423	C₂×C₄⋊C₄	C₂³×C₃₁	1	1	yes	characteristic
424	C₂×C₄⋊C₄	C₂³×C₃₁	1	2	yes	—
425	C₂×C₄⋊C₄	C₂³×C₃₁	1	2	yes	—
426	C₂×C₂²⋊C₄	C₂³×C₃₁	1	4	yes	—
427	C₂×C₂²⋊C₄	C₂³×C₃₁	1	4	yes	—
428	C₂×C₂²⋊C₄	C₂³×C₃₁	1	2	yes	—
429	C₂×C₂²⋊C₄	C₂³×C₃₁	1	2	yes	—
430	C₂×C₂²⋊C₄	C₂³×C₃₁	1	4	yes	—
431	C₂×C₂²⋊C₄	C₂³×C₃₁	1	4	yes	—
432	C₂×C₂²⋊C₄	C₂³×C₃₁	1	2	yes	—
433	C₂×C₂²⋊C₄	C₂³×C₃₁	1	2	yes	—
434	C₂×C₂²⋊C₄	C₂³×C₃₁	1	2	yes	—
435	C₂×C₂²⋊C₄	C₂³×C₃₁	1	2	yes	—
436	C₂×C₂²⋊C₄	C₂³×C₃₁	1	2	yes	—
437	C₂×C₄²	C₂×C₄×C₃₁	1	8	yes	complemented
438	C₂×C₄²	C₂³×C₃₁	1	4	yes	—
439	C₂×C₄²	C₂³×C₃₁	1	4	yes	—
440	C₂×C₄²	C₂³×C₃₁	1	2	yes	—
441	C₂×C₄²	C₂³×C₃₁	1	4	yes	—
442	C₂×C₄²	C₂³×C₃₁	1	2	yes	—
443	C₂×C₄²	C₂³×C₃₁	1	4	yes	—
444	C₂×C₄²	C₂³×C₃₁	1	4	yes	—
445	C₂×C₄²	C₂³×C₃₁	1	4	yes	—
446	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
447	C₂×C₄²	C₂³×C₃₁	1	2	yes	—
448	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
449	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
450	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
451	#95	C₂³×C₃₁	1	2	yes	—
452	#95	C₂³×C₃₁	1	2	yes	—
453	#95	C₂³×C₃₁	1	4	yes	—
454	#95	C₂³×C₃₁	1	2	yes	—
455	#95	C₂³×C₃₁	1	2	yes	—
456	#95	C₂³×C₃₁	1	4	yes	—
457	#95	C₂³×C₃₁	1	4	yes	—
458	#95	C₂³×C₃₁	1	4	yes	—
459	#95	C₂³×C₃₁	1	4	yes	—
460	#95	C₂³×C₃₁	1	4	yes	—
461	#95	C₂³×C₃₁	1	4	yes	—
462	#95	C₂³×C₃₁	1	4	yes	—
463	#95	C₂³×C₃₁	1	4	yes	—
464	#95	C₂³×C₃₁	1	4	yes	—
465	C₃₁	#34244	1	1	yes	characteristic, direct product factor, Sylow
466	C₂⁴	C₂⁴×C₃₁	1	1	yes	characteristic
467	C₂⁴	C₂⁴×C₃₁	1	2	yes	—
468	C₂×Q₈	—	4	1	no	—
469	C₂×Q₈	—	4	1	no	—
470	C₂×Q₈	C₂²×C₄×C₃₁	1	4	yes	—
471	C₂×D₄	—	4	2	no	—
472	C₂×D₄	—	4	2	no	—
473	C₂×D₄	—	4	1	no	—
474	C₂×D₄	—	4	1	no	—
475	C₂×D₄	C₂²×C₄×C₃₁	1	4	yes	—
476	C₂×D₄	C₂²×C₄×C₃₁	1	8	yes	—
477	C₂²×C₄	—	2	4	no	—
478	C₂²×C₄	—	2	4	no	—
479	C₂²×C₄	—	2	4	no	—
480	C₂²×C₄	—	2	8	no	—
481	C₂²×C₄	—	4	2	no	—
482	C₂²×C₄	—	4	1	no	—
483	C₂²×C₄	—	4	1	no	—
484	C₂²×C₄	—	4	4	no	—
485	C₂²×C₄	—	4	4	no	—
486	C₂²×C₄	C₂²×C₄×C₃₁	1	4	yes	—
487	C₂²×C₄	C₂²×C₄×C₃₁	1	8	yes	—
488	C₂²×C₄	C₂²×C₄×C₃₁	1	4	yes	—
489	C₂²×C₄	C₂²×C₄×C₃₁	1	4	yes	—
490	C₂²×C₄	C₂⁴×C₃₁	1	4	yes	—
491	C₂²×C₄	C₂⁴×C₃₁	1	4	yes	—
492	C₂²×C₄	C₂⁴×C₃₁	1	4	yes	—
493	C₂²×C₄	C₂⁴×C₃₁	1	1	yes	characteristic
494	C₂²×C₄	C₂⁴×C₃₁	1	1	yes	characteristic
495	C₂²×C₄	C₂⁴×C₃₁	1	1	yes	characteristic
496	C₂²×C₄	C₂⁴×C₃₁	1	1	yes	characteristic
497	C₂²×C₄	C₂⁴×C₃₁	1	1	yes	characteristic
498	C₂²×C₄	C₂⁴×C₃₁	1	2	yes	—
499	C₂²×C₄	C₂⁴×C₃₁	1	2	yes	—
500	C₂²×C₄	C₂⁴×C₃₁	1	2	yes	—
501	C₂²×C₄	C₂⁴×C₃₁	1	4	yes	—
502	C₂²×C₄	C₂⁴×C₃₁	1	1	yes	characteristic
503	C₄⋊C₄	—	2	8	no	—
504	C₄⋊C₄	—	2	4	no	—
505	C₄⋊C₄	—	2	4	no	—
506	C₄⋊C₄	—	2	4	no	—
507	C₄⋊C₄	—	4	4	no	—
508	C₄⋊C₄	—	4	4	no	—
509	C₄⋊C₄	C₂²×C₄×C₃₁	1	8	yes	—
510	C₄⋊C₄	C₂²×C₄×C₃₁	1	8	yes	—
511	C₄⋊C₄	C₂²×C₄×C₃₁	1	4	yes	—
512	C₄⋊C₄	C₂²×C₄×C₃₁	1	8	yes	—
513	C₄⋊C₄	C₂²×C₄×C₃₁	1	4	yes	—
514	C₄⋊C₄	C₂²×C₄×C₃₁	1	8	yes	—
515	C₄⋊C₄	#41×C₃₁	1	8	yes	—
516	C₂²⋊C₄	—	2	8	no	—
517	C₂²⋊C₄	—	2	8	no	—
518	C₂²⋊C₄	—	4	4	no	—
519	C₂²⋊C₄	—	4	4	no	—
520	C₂²⋊C₄	—	4	2	no	—
521	C₂²⋊C₄	—	4	2	no	—
522	C₂²⋊C₄	C₂²×C₄×C₃₁	1	8	yes	—
523	C₂²⋊C₄	C₂²×C₄×C₃₁	1	8	yes	—
524	C₂²⋊C₄	C₂²×C₄×C₃₁	1	8	yes	—
525	C₂²⋊C₄	C₂²×C₄×C₃₁	1	8	yes	—
526	C₂²⋊C₄	C₂²×C₄×C₃₁	1	8	yes	—
527	C₄²	—	2	8	no	—
528	C₄²	—	2	8	no	—
529	C₄²	—	4	2	no	—
530	C₄²	—	4	4	no	—
531	C₄²	—	4	4	no	—
532	C₄²	—	4	4	no	—
533	C₄²	—	4	4	no	—
534	C₄²	—	4	2	no	—
535	C₄²	C₂²×C₄×C₃₁	1	4	yes	—
536	C₄²	C₂²×C₄×C₃₁	1	4	yes	—
537	C₄²	C₂²×C₄×C₃₁	1	4	yes	—
538	C₄²	C₂²×C₄×C₃₁	1	4	yes	—
539	C₄²	C₂²×C₄×C₃₁	1	8	yes	—
540	C₂³	—	2	2	no	—
541	C₂³	—	2	2	no	—
542	C₂³	—	2	2	no	—
543	C₂³	—	2	1	no	—
544	C₂³	—	2	1	no	—
545	C₂³	—	2	2	no	—
546	C₂³	—	2	2	no	—
547	C₂³	—	2	2	no	—
548	C₂³	—	2	2	no	—
549	C₂³	—	2	2	no	—
550	C₂³	C₂³×C₄×C₃₁	1	4	yes	—
551	C₂³	C₂³×C₄×C₃₁	1	2	yes	—
552	C₂³	C₂⁵×C₃₁	1	1	yes	characteristic
553	Q₈	—	4	4	no	—
554	D₄	—	4	4	no	—
555	D₄	—	4	8	no	—
556	C₂×C₄	—	2	2	no	—
557	C₂×C₄	—	2	2	no	—
558	C₂×C₄	—	2	2	no	—
559	C₂×C₄	—	2	1	no	—
560	C₂×C₄	—	2	2	no	—
561	C₂×C₄	—	2	2	no	—
562	C₂×C₄	—	2	1	no	—
563	C₂×C₄	—	2	4	no	—
564	C₂×C₄	—	2	4	no	—
565	C₂×C₄	—	2	2	no	—
566	C₂×C₄	—	2	2	no	—
567	C₂×C₄	—	2	2	no	—
568	C₂×C₄	—	2	2	no	—
569	C₂×C₄	—	2	1	no	—
570	C₂×C₄	—	2	1	no	—
571	C₂×C₄	—	2	16	no	—
572	C₂×C₄	—	2	8	no	—
573	C₂×C₄	—	2	4	no	—
574	C₂×C₄	—	2	4	no	—
575	C₂×C₄	—	2	4	no	—
576	C₂×C₄	—	2	4	no	—
577	C₂×C₄	—	2	4	no	—
578	C₂×C₄	—	2	4	no	—
579	C₂×C₄	—	2	4	no	—
580	C₂×C₄	—	2	4	no	—
581	C₂×C₄	—	2	4	no	—
582	C₂×C₄	—	4	4	no	—
583	C₂×C₄	—	4	4	no	—
584	C₂×C₄	—	8	8	no	—
585	C₂×C₄	C₂³×C₄×C₃₁	1	2	yes	—
586	C₂×C₄	C₂³×C₄×C₃₁	1	2	yes	—
587	C₂×C₄	C₂³×C₄×C₃₁	1	2	yes	—
588	C₂×C₄	C₂³×C₄×C₃₁	1	2	yes	—
589	C₂×C₄	C₂³×C₄×C₃₁	1	2	yes	—
590	C₂×C₄	C₂³×C₄×C₃₁	1	4	yes	—
591	C₂×C₄	C₂³×C₄×C₃₁	1	4	yes	—
592	C₂×C₄	C₂³×C₄×C₃₁	1	4	yes	—
593	C₂×C₄	C₂³×C₄×C₃₁	1	2	yes	—
594	C₂×C₄	C₂×#41×C₃₁	1	8	yes	—
595	C₂×C₄	C₂×#41×C₃₁	1	4	yes	—
596	C₂²	—	2	2	no	—
597	C₂²	—	2	4	no	—
598	C₂²	—	2	2	no	—
599	C₂²	—	2	2	no	—
600	C₂²	—	2	4	no	—
601	C₂²	—	2	4	no	—
602	C₂²	—	4	2	no	—
603	C₂²	—	4	2	no	—
604	C₂²	—	4	2	no	—
605	C₂²	—	4	2	no	—
606	C₂²	—	4	2	no	—
607	C₂²	—	4	2	no	—
608	C₂²	C₂⁴×C₄×C₃₁	1	1	yes	characteristic
609	C₂²	C₂²×#41×C₃₁	1	2	yes	—
610	C₂²	C₃₁×#585	1	1	yes	characteristic
611	C₂²	C₃₁×#585	1	2	yes	—
612	C₂²	C₃₁×#585	1	1	yes	characteristic
613	C₄	—	2	2	no	—
614	C₄	—	2	4	no	—
615	C₄	—	2	2	no	—
616	C₄	—	2	2	no	—
617	C₄	—	2	2	no	—
618	C₄	—	2	2	no	—
619	C₄	—	2	4	no	—
620	C₄	—	2	4	no	—
621	C₄	—	2	8	no	—
622	C₄	—	4	4	no	—
623	C₄	—	4	4	no	—
624	C₄	—	4	4	no	—
625	C₄	C₄×#41×C₃₁	1	4	yes	—
626	C₂	—	4	2	no	—
627	C₂	—	4	4	no	—
628	C₂	C₂×C₄×#41×C₃₁	1	1	yes	characteristic
629	C₂	C₃₁×#3428	1	1	yes	characteristic
630	C₂	C₃₁×#3428	1	1	yes	characteristic
631	C₂	C₃₁×#3521	1	2	yes	—
632	C₂	C₃₁×#3524	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×#2374	C₂	semidirect
3. C₃₁×#2372	C₂	semidirect
4. C₃₁×#2371	C₂	semidirect
5. C₃₁×#2355	C₂	semidirect
6. C₃₁×#2354	C₂	semidirect
7. C₃₁×#2352	C₂	semidirect
8. C₃₁×#2352	C₂	semidirect
10. C₃₁×#2332	C₂	semidirect
11. C₃₁×#2314	C₂	semidirect
12. C₄×C₃₁×#126	C₂	semidirect
13. C₃₁×#2283	C₂	semidirect
14. C₃₁×#2274	C₂	semidirect
15. C₄×C₃₁×#117	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
95. C₃₁×#117	C₂×C₄	semidirect
121. C₂×C₄²×C₃₁	C₂×C₄	semidirect
223. #34244	C₃₁	direct (N × H)
281. #2374	C₂×C₃₁	semidirect
282. #2372	C₂×C₃₁	semidirect
283. #2371	C₂×C₃₁	semidirect
284. #2355	C₂×C₃₁	semidirect
285. #2354	C₂×C₃₁	semidirect
286. #2352	C₂×C₃₁	semidirect
287. #2352	C₂×C₃₁	semidirect
289. #2332	C₂×C₃₁	semidirect
290. #2314	C₂×C₃₁	semidirect
291. C₄×#126	C₂×C₃₁	semidirect
292. #2283	C₂×C₃₁	semidirect
293. #2274	C₂×C₃₁	semidirect
294. C₄×#117	C₂×C₃₁	semidirect
325. #529	C₄×C₃₁	semidirect
400. #126	C₂×C₄×C₃₁	semidirect
401. #125	C₂×C₄×C₃₁	semidirect
403. #123	C₂×C₄×C₃₁	semidirect
404. C₂²⋊Q₈	C₂×C₄×C₃₁	semidirect
406. C₄×Q₈	C₂×C₄×C₃₁	semidirect
408. C₄×D₄	C₂×C₄×C₃₁	semidirect
411. #117	C₂×C₄×C₃₁	semidirect
437. C₂×C₄²	C₂×C₄×C₃₁	semidirect
465. C₃₁	#34244	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×#2374	2
C₃₁×#2372	2
C₃₁×#2371	2
C₃₁×#2355	2
C₃₁×#2354	2
C₃₁×#2352	2
C₃₁×#2333	2
C₃₁×#2332	2
C₃₁×#2314	2
C₄×C₃₁×#126	2
C₃₁×#2283	2
C₃₁×#2274	2
C₄×C₃₁×#117	2
#34244	31

Subgroups — C31×#34244 #21491003

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#34244 #21491003