Subgroups — C₃₁×#24134 #21480893

Order 7936.

1120 conjugacy classes of subgroups.

The subgroup lattice has 522 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃₁×#554	C₂	1	1	yes	maximal, characteristic
2	C₂×C₃₁×#475	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×C₃₁×#441	C₂	1	1	yes	maximal, characteristic, complemented
4	C₃₁×#2536	C₂	1	1	yes	maximal, characteristic
5	C₃₁×#2035	C₂	1	2	yes	maximal, complemented
6	C₃₁×#2001	C₂	1	2	yes	maximal, complemented
7	C₃₁×#1993	C₂	1	1	yes	maximal, characteristic, complemented
8	C₃₁×#1955	C₂	1	1	yes	maximal, characteristic
9	C₃₁×#1872	C₂	1	2	yes	maximal, complemented
10	C₃₁×#1853	C₂	1	2	yes	maximal, complemented
11	C₃₁×#1805	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂²×Q₁₆×C₃₁	C₂²	1	1	yes	characteristic
13	C₃₁×#554	—	2	4	no	—
14	C₃₁×#554	—	2	4	no	—
15	C₂×C₃₁×C₄⋊Q₈	C₂²	1	1	yes	characteristic
16	C₂×C₃₁×#125	—	2	1	no	—
17	C₂×C₃₁×C₂²⋊Q₈	—	2	2	no	—
18	C₂×C₃₁×C₂²⋊Q₈	C₂²	1	2	yes	—
19	C₂×C₃₁×C₂²⋊Q₈	C₂²	1	2	yes	—
20	C₂×C₄×Q₈×C₃₁	C₂²	1	1	yes	characteristic
21	C₂×C₄×Q₈×C₃₁	C₂²	1	1	yes	characteristic
22	C₂×C₄×D₄×C₃₁	—	2	1	no	—
23	C₂×C₄×D₄×C₃₁	C₂²	1	1	yes	characteristic
24	C₂²×C₄⋊C₄×C₃₁	C₂²	1	2	yes	—
25	C₃₁×#475	—	2	4	no	—
26	C₃₁×#441	—	2	4	no	—
27	C₂×C₃₁×#106	C₂²	1	1	yes	characteristic
28	C₂×C₃₁×#106	C₂²	1	2	yes	—
29	C₂×C₃₁×C₄⋊C₈	C₂²	1	1	yes	characteristic
30	C₂×C₃₁×#103	C₂²	1	1	yes	characteristic
31	C₂×C₃₁×#103	C₂²	1	2	yes	—
32	C₂×C₃₁×#103	C₂²	1	1	yes	characteristic
33	C₂×C₃₁×#103	C₂²	1	2	yes	—
34	C₂×C₃₁×C₂²⋊C₈	C₂²	1	2	yes	—
35	C₂×C₃₁×C₂²⋊C₈	C₂²	1	2	yes	—
36	C₂×C₃₁×#97	C₂²	1	1	yes	characteristic
37	C₃₁×#398	—	2	1	no	—
38	C₃₁×#394	—	2	1	no	—
39	C₃₁×#388	—	2	2	no	—
40	C₃₁×#382	C₂²	1	1	yes	characteristic
41	C₃₁×#380	C₂²	1	2	yes	—
42	C₃₁×#340	C₂²	1	2	yes	—
43	C₃₁×#340	C₂²	1	2	yes	—
44	C₃₁×#340	C₂²	1	2	yes	—
45	C₂²×Q₈×C₃₁	—	2	1	no	—
46	C₂²×Q₈×C₃₁	C₂³	1	1	yes	characteristic
47	C₂²×Q₈×C₃₁	C₂³	1	1	yes	characteristic
48	C₂²×D₄×C₃₁	D₄	1	1	yes	characteristic
49	C₂³×C₄×C₃₁	—	2	2	no	—
50	C₂³×C₄×C₃₁	C₂³	1	2	yes	—
51	C₂×Q₁₆×C₃₁	—	2	2	no	—
52	C₂×Q₁₆×C₃₁	—	2	2	no	—
53	C₂×Q₁₆×C₃₁	—	2	2	no	—
54	C₂²×C₈×C₃₁	C₂³	1	1	yes	characteristic
55	C₂²×C₈×C₃₁	C₂³	1	2	yes	—
56	C₂²×C₈×C₃₁	C₂³	1	1	yes	characteristic
57	C₃₁×C₄⋊Q₈	—	2	2	no	—
58	C₃₁×C₄⋊Q₈	D₄	1	4	yes	—
59	C₃₁×#125	—	2	8	no	—
60	C₃₁×C₂²⋊Q₈	—	2	16	no	—
61	C₃₁×C₂²⋊Q₈	—	2	4	no	—
62	C₃₁×C₂²⋊Q₈	—	2	4	no	—
63	C₃₁×C₂²⋊Q₈	—	2	4	no	—
64	C₃₁×C₂²⋊Q₈	—	4	2	no	—
65	C₄×Q₈×C₃₁	—	2	2	no	—
66	C₄×Q₈×C₃₁	—	2	2	no	—
67	C₄×Q₈×C₃₁	—	2	2	no	—
68	C₄×Q₈×C₃₁	D₄	1	4	yes	—
69	C₄×D₄×C₃₁	—	2	8	no	—
70	C₄×D₄×C₃₁	—	2	2	no	—
71	C₄×D₄×C₃₁	—	2	2	no	—
72	C₂×C₄⋊C₄×C₃₁	—	2	2	no	—
73	C₂×C₄⋊C₄×C₃₁	—	2	1	no	—
74	C₂×C₄⋊C₄×C₃₁	—	2	1	no	—
75	C₂×C₄⋊C₄×C₃₁	—	2	2	no	—
76	C₂×C₄⋊C₄×C₃₁	—	2	4	no	—
77	C₂×C₄⋊C₄×C₃₁	—	2	4	no	—
78	C₂×C₄⋊C₄×C₃₁	—	2	1	no	—
79	C₂×C₄⋊C₄×C₃₁	—	2	2	no	—
80	C₂×C₄⋊C₄×C₃₁	D₄	1	1	yes	characteristic
81	C₂×C₄⋊C₄×C₃₁	C₂³	1	1	yes	characteristic
82	C₂×C₄⋊C₄×C₃₁	C₂³	1	2	yes	—
83	C₂×C₄⋊C₄×C₃₁	C₂³	1	1	yes	characteristic
84	C₂×C₄⋊C₄×C₃₁	C₂³	1	2	yes	—
85	C₂×C₂²⋊C₄×C₃₁	—	2	2	no	—
86	C₂×C₂²⋊C₄×C₃₁	—	2	2	no	—
87	C₂×C₂²⋊C₄×C₃₁	—	2	2	no	—
88	C₂×C₂²⋊C₄×C₃₁	D₄	1	2	yes	—
89	C₂×C₄²×C₃₁	—	2	1	no	—
90	C₂×C₄²×C₃₁	—	2	1	no	—
91	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
92	C₃₁×#106	—	2	2	no	—
93	C₃₁×#106	—	2	4	no	—
94	C₃₁×C₄⋊C₈	—	2	2	no	—
95	C₃₁×#103	—	2	2	no	—
96	C₃₁×#103	—	2	2	no	—
97	C₃₁×#103	—	2	4	no	—
98	C₃₁×#103	—	4	2	no	—
99	C₃₁×C₂²⋊C₈	—	2	4	no	—
100	C₃₁×C₂²⋊C₈	—	4	2	no	—
101	C₃₁×#97	—	2	2	no	—
102	C₃₁×#95	—	2	2	no	—
103	C₃₁×#95	—	2	1	no	—
104	C₃₁×#95	—	2	1	no	—
105	C₂⁴×C₃₁	C₂×D₄	1	2	yes	—
106	C₂×Q₈×C₃₁	—	2	4	no	—
107	C₂×Q₈×C₃₁	—	2	1	no	—
108	C₂×Q₈×C₃₁	—	2	1	no	—
109	C₂×Q₈×C₃₁	—	2	1	no	—
110	C₂×Q₈×C₃₁	—	2	1	no	—
111	C₂×Q₈×C₃₁	—	2	1	no	—
112	C₂×Q₈×C₃₁	—	2	1	no	—
113	C₂×Q₈×C₃₁	—	2	1	no	—
114	C₂×Q₈×C₃₁	—	2	1	no	—
115	C₂×Q₈×C₃₁	—	2	1	no	—
116	C₂×Q₈×C₃₁	—	2	1	no	—
117	C₂×Q₈×C₃₁	—	8	2	no	—
118	C₂×Q₈×C₃₁	C₂×D₄	1	2	yes	—
119	C₂×Q₈×C₃₁	#41	1	2	yes	—
120	C₂×D₄×C₃₁	—	2	4	no	—
121	C₂×D₄×C₃₁	—	2	2	no	—
122	C₂²×C₄×C₃₁	—	2	1	no	—
123	C₂²×C₄×C₃₁	—	2	1	no	—
124	C₂²×C₄×C₃₁	—	2	1	no	—
125	C₂²×C₄×C₃₁	—	2	2	no	—
126	C₂²×C₄×C₃₁	—	2	8	no	—
127	C₂²×C₄×C₃₁	—	2	2	no	—
128	C₂²×C₄×C₃₁	—	2	2	no	—
129	C₂²×C₄×C₃₁	—	2	2	no	—
130	C₂²×C₄×C₃₁	—	2	2	no	—
131	C₂²×C₄×C₃₁	—	2	2	no	—
132	C₂²×C₄×C₃₁	—	2	2	no	—
133	C₂²×C₄×C₃₁	—	2	2	no	—
134	C₂²×C₄×C₃₁	—	2	1	no	—
135	C₂²×C₄×C₃₁	—	8	2	no	—
136	C₂²×C₄×C₃₁	—	8	2	no	—
137	C₂²×C₄×C₃₁	C₂×D₄	1	2	yes	—
138	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
139	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
140	C₂²×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
141	Q₁₆×C₃₁	—	2	8	no	—
142	C₂×C₈×C₃₁	—	2	1	no	—
143	C₂×C₈×C₃₁	—	2	1	no	—
144	C₂×C₈×C₃₁	—	2	1	no	—
145	C₂×C₈×C₃₁	—	2	2	no	—
146	C₂×C₈×C₃₁	—	2	2	no	—
147	C₂×C₈×C₃₁	—	2	1	no	—
148	C₂×C₈×C₃₁	—	2	2	no	—
149	C₂×C₈×C₃₁	—	2	1	no	—
150	C₂×C₈×C₃₁	#41	1	2	yes	—
151	C₄⋊C₄×C₃₁	—	2	8	no	—
152	C₄⋊C₄×C₃₁	—	2	8	no	—
153	C₄⋊C₄×C₃₁	—	2	4	no	—
154	C₄⋊C₄×C₃₁	—	2	4	no	—
155	C₄⋊C₄×C₃₁	—	2	2	no	—
156	C₄⋊C₄×C₃₁	—	2	1	no	—
157	C₄⋊C₄×C₃₁	—	2	2	no	—
158	C₄⋊C₄×C₃₁	—	2	1	no	—
159	C₄⋊C₄×C₃₁	—	2	1	no	—
160	C₄⋊C₄×C₃₁	—	2	2	no	—
161	C₄⋊C₄×C₃₁	—	2	2	no	—
162	C₄⋊C₄×C₃₁	—	4	4	no	—
163	C₄⋊C₄×C₃₁	—	4	2	no	—
164	C₄⋊C₄×C₃₁	C₂×D₄	1	2	yes	—
165	C₄⋊C₄×C₃₁	C₂×D₄	1	4	yes	—
166	C₂²⋊C₄×C₃₁	—	2	8	no	—
167	C₂²⋊C₄×C₃₁	—	2	4	no	—
168	C₂²⋊C₄×C₃₁	—	4	4	no	—
169	C₂²⋊C₄×C₃₁	—	4	4	no	—
170	C₄²×C₃₁	—	2	4	no	—
171	C₄²×C₃₁	—	4	2	no	—
172	C₄²×C₃₁	C₂×D₄	1	2	yes	—
173	C₄²×C₃₁	C₂×D₄	1	2	yes	—
174	#24134	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
175	C₂³×C₃₁	—	2	2	no	—
176	C₂³×C₃₁	—	2	2	no	—
177	C₂³×C₃₁	—	2	2	no	—
178	C₂³×C₃₁	—	2	2	no	—
179	C₂³×C₃₁	—	2	2	no	—
180	C₂³×C₃₁	—	2	2	no	—
181	C₂³×C₃₁	#137	1	4	yes	—
182	C₂³×C₃₁	C₂²×D₄	1	1	yes	characteristic
183	Q₈×C₃₁	—	2	2	no	—
184	Q₈×C₃₁	—	2	2	no	—
185	Q₈×C₃₁	—	2	2	no	—
186	Q₈×C₃₁	—	2	2	no	—
187	Q₈×C₃₁	—	4	1	no	—
188	Q₈×C₃₁	—	4	1	no	—
189	Q₈×C₃₁	—	4	1	no	—
190	Q₈×C₃₁	—	4	1	no	—
191	Q₈×C₃₁	—	8	4	no	—
192	D₄×C₃₁	—	2	8	no	—
193	C₂×C₄×C₃₁	—	2	2	no	—
194	C₂×C₄×C₃₁	—	2	2	no	—
195	C₂×C₄×C₃₁	—	2	2	no	—
196	C₂×C₄×C₃₁	—	2	2	no	—
197	C₂×C₄×C₃₁	—	2	2	no	—
198	C₂×C₄×C₃₁	—	2	4	no	—
199	C₂×C₄×C₃₁	—	2	2	no	—
200	C₂×C₄×C₃₁	—	2	2	no	—
201	C₂×C₄×C₃₁	—	2	2	no	—
202	C₂×C₄×C₃₁	—	4	1	no	—
203	C₂×C₄×C₃₁	—	4	1	no	—
204	C₂×C₄×C₃₁	—	4	1	no	—
205	C₂×C₄×C₃₁	—	4	1	no	—
206	C₂×C₄×C₃₁	—	4	2	no	—
207	C₂×C₄×C₃₁	—	4	2	no	—
208	C₂×C₄×C₃₁	—	4	2	no	—
209	C₂×C₄×C₃₁	—	4	2	no	—
210	C₂×C₄×C₃₁	—	4	2	no	—
211	C₂×C₄×C₃₁	—	4	2	no	—
212	C₂×C₄×C₃₁	—	4	2	no	—
213	C₂×C₄×C₃₁	—	4	1	no	—
214	C₂×C₄×C₃₁	—	4	2	no	—
215	C₂×C₄×C₃₁	—	4	2	no	—
216	C₂×C₄×C₃₁	—	4	2	no	—
217	C₂×C₄×C₃₁	—	4	2	no	—
218	C₂×C₄×C₃₁	—	8	8	no	—
219	C₂×C₄×C₃₁	#120	1	2	yes	—
220	C₂×C₄×C₃₁	#120	1	2	yes	—
221	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
222	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
223	C₂×C₄×C₃₁	C₂×#41	1	1	yes	characteristic
224	C₂×C₄×C₃₁	C₂×#41	1	1	yes	characteristic
225	C₂×C₄×C₃₁	H(5,𝔽₂)	1	1	yes	characteristic
226	C₂×C₄×C₃₁	2₋¹⁺⁴	1	1	yes	characteristic
227	C₈×C₃₁	—	2	2	no	—
228	C₈×C₃₁	—	4	1	no	—
229	C₈×C₃₁	—	4	2	no	—
230	C₂×#554	C₂×C₃₁	1	1	yes	characteristic
231	C₂×#475	C₂×C₃₁	1	1	yes	characteristic, complemented
232	C₂×#441	C₂×C₃₁	1	1	yes	characteristic, complemented
233	#2536	C₂×C₃₁	1	1	yes	characteristic
234	#2035	C₂×C₃₁	1	2	yes	complemented
235	#2001	C₂×C₃₁	1	2	yes	complemented
236	#1993	C₂×C₃₁	1	1	yes	characteristic, complemented
237	#1955	C₂×C₃₁	1	1	yes	characteristic
238	#1872	C₂×C₃₁	1	2	yes	complemented
239	#1853	C₂×C₃₁	1	2	yes	complemented
240	#1805	C₂×C₃₁	1	1	yes	characteristic, complemented
241	C₂²×C₃₁	—	2	4	no	—
242	C₂²×C₃₁	—	2	4	no	—
243	C₂²×C₃₁	—	2	4	no	—
244	C₂²×C₃₁	—	4	2	no	—
245	C₂²×C₃₁	—	4	2	no	—
246	C₂²×C₃₁	—	4	2	no	—
247	C₂²×C₃₁	—	4	2	no	—
248	C₂²×C₃₁	C₂×#120	1	1	yes	characteristic
249	C₂²×C₃₁	#546	1	1	yes	characteristic
250	C₂²×C₃₁	#547	1	1	yes	characteristic
251	C₂²×C₃₁	C₂×#137	1	2	yes	—
252	C₂²×C₃₁	#577	1	2	yes	—
253	C₄×C₃₁	—	2	2	no	—
254	C₄×C₃₁	—	2	2	no	—
255	C₄×C₃₁	—	4	2	no	—
256	C₄×C₃₁	—	4	2	no	—
257	C₄×C₃₁	—	4	2	no	—
258	C₄×C₃₁	—	4	2	no	—
259	C₄×C₃₁	—	4	4	no	—
260	C₄×C₃₁	—	8	1	no	—
261	C₄×C₃₁	—	8	2	no	—
262	C₄×C₃₁	#525	1	1	yes	characteristic
263	C₄×C₃₁	#537	1	1	yes	characteristic
264	C₄×C₃₁	#548	1	1	yes	characteristic
265	C₄×C₃₁	#550	1	1	yes	characteristic
266	C₂²×Q₁₆	C₂²×C₃₁	1	1	yes	characteristic
267	#554	—	2	4	no	—
268	#554	—	2	4	no	—
269	C₂×C₄⋊Q₈	C₂²×C₃₁	1	1	yes	characteristic
270	C₂×#125	—	2	1	no	—
271	C₂×C₂²⋊Q₈	—	2	2	no	—
272	C₂×C₂²⋊Q₈	C₂²×C₃₁	1	2	yes	—
273	C₂×C₂²⋊Q₈	C₂²×C₃₁	1	2	yes	—
274	C₂×C₄×Q₈	C₂²×C₃₁	1	1	yes	characteristic
275	C₂×C₄×Q₈	C₂²×C₃₁	1	1	yes	characteristic
276	C₂×C₄×D₄	—	2	1	no	—
277	C₂×C₄×D₄	C₂²×C₃₁	1	1	yes	characteristic
278	C₂²×C₄⋊C₄	C₂²×C₃₁	1	2	yes	—
279	#475	—	2	4	no	—
280	#441	—	2	4	no	—
281	C₂×#106	C₂²×C₃₁	1	2	yes	—
282	C₂×#106	C₂²×C₃₁	1	1	yes	characteristic
283	C₂×C₄⋊C₈	C₂²×C₃₁	1	1	yes	characteristic
284	C₂×#103	C₂²×C₃₁	1	2	yes	—
285	C₂×#103	C₂²×C₃₁	1	1	yes	characteristic
286	C₂×#103	C₂²×C₃₁	1	1	yes	characteristic
287	C₂×#103	C₂²×C₃₁	1	2	yes	—
288	C₂×C₂²⋊C₈	C₂²×C₃₁	1	2	yes	—
289	C₂×C₂²⋊C₈	C₂²×C₃₁	1	2	yes	—
290	C₂×#97	C₂²×C₃₁	1	1	yes	characteristic
291	#398	—	2	1	no	—
292	#394	—	2	1	no	—
293	#388	—	2	2	no	—
294	#382	C₂²×C₃₁	1	1	yes	characteristic
295	#380	C₂²×C₃₁	1	2	yes	—
296	#340	C₂²×C₃₁	1	2	yes	—
297	#340	C₂²×C₃₁	1	2	yes	—
298	#340	C₂²×C₃₁	1	2	yes	—
299	C₂×C₃₁	—	4	4	no	—
300	C₂×C₃₁	#2407	1	1	yes	characteristic
301	C₂×C₃₁	#3004	1	1	yes	characteristic
302	C₂×C₃₁	#3009	1	1	yes	characteristic
303	C₂×C₃₁	#3316	1	2	yes	—
304	C₂×C₃₁	#3344	1	2	yes	—
305	C₂²×Q₈	—	2	1	no	—
306	C₂²×Q₈	C₂³×C₃₁	1	1	yes	characteristic
307	C₂²×Q₈	C₂³×C₃₁	1	1	yes	characteristic
308	C₂²×D₄	D₄×C₃₁	1	1	yes	characteristic
309	C₂³×C₄	—	2	2	no	—
310	C₂³×C₄	C₂³×C₃₁	1	2	yes	—
311	C₂×Q₁₆	—	2	2	no	—
312	C₂×Q₁₆	—	2	2	no	—
313	C₂×Q₁₆	—	2	2	no	—
314	C₂²×C₈	C₂³×C₃₁	1	1	yes	characteristic
315	C₂²×C₈	C₂³×C₃₁	1	2	yes	—
316	C₂²×C₈	C₂³×C₃₁	1	1	yes	characteristic
317	C₄⋊Q₈	—	2	2	no	—
318	C₄⋊Q₈	D₄×C₃₁	1	4	yes	—
319	#125	—	2	8	no	—
320	C₂²⋊Q₈	—	2	16	no	—
321	C₂²⋊Q₈	—	2	4	no	—
322	C₂²⋊Q₈	—	2	4	no	—
323	C₂²⋊Q₈	—	2	4	no	—
324	C₂²⋊Q₈	—	4	2	no	—
325	C₄×Q₈	—	2	2	no	—
326	C₄×Q₈	—	2	2	no	—
327	C₄×Q₈	—	2	2	no	—
328	C₄×Q₈	D₄×C₃₁	1	4	yes	—
329	C₄×D₄	—	2	8	no	—
330	C₄×D₄	—	2	2	no	—
331	C₄×D₄	—	2	2	no	—
332	C₂×C₄⋊C₄	—	2	2	no	—
333	C₂×C₄⋊C₄	—	2	1	no	—
334	C₂×C₄⋊C₄	—	2	1	no	—
335	C₂×C₄⋊C₄	—	2	2	no	—
336	C₂×C₄⋊C₄	—	2	4	no	—
337	C₂×C₄⋊C₄	—	2	4	no	—
338	C₂×C₄⋊C₄	—	2	1	no	—
339	C₂×C₄⋊C₄	—	2	2	no	—
340	C₂×C₄⋊C₄	D₄×C₃₁	1	1	yes	characteristic
341	C₂×C₄⋊C₄	C₂³×C₃₁	1	1	yes	characteristic
342	C₂×C₄⋊C₄	C₂³×C₃₁	1	2	yes	—
343	C₂×C₄⋊C₄	C₂³×C₃₁	1	1	yes	characteristic
344	C₂×C₄⋊C₄	C₂³×C₃₁	1	2	yes	—
345	C₂×C₂²⋊C₄	—	2	2	no	—
346	C₂×C₂²⋊C₄	—	2	2	no	—
347	C₂×C₂²⋊C₄	—	2	2	no	—
348	C₂×C₂²⋊C₄	D₄×C₃₁	1	2	yes	—
349	C₂×C₄²	—	2	1	no	—
350	C₂×C₄²	—	2	1	no	—
351	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
352	#106	—	2	2	no	—
353	#106	—	2	4	no	—
354	C₄⋊C₈	—	2	2	no	—
355	#103	—	2	2	no	—
356	#103	—	2	2	no	—
357	#103	—	2	4	no	—
358	#103	—	4	2	no	—
359	C₂²⋊C₈	—	2	4	no	—
360	C₂²⋊C₈	—	4	2	no	—
361	#97	—	2	2	no	—
362	#95	—	2	2	no	—
363	#95	—	2	1	no	—
364	#95	—	2	1	no	—
365	C₃₁	#24134	1	1	yes	characteristic, direct product factor, Sylow
366	C₂⁴	C₂×D₄×C₃₁	1	2	yes	—
367	C₂×Q₈	—	2	4	no	—
368	C₂×Q₈	—	2	1	no	—
369	C₂×Q₈	—	2	1	no	—
370	C₂×Q₈	—	2	1	no	—
371	C₂×Q₈	—	2	1	no	—
372	C₂×Q₈	—	2	1	no	—
373	C₂×Q₈	—	2	1	no	—
374	C₂×Q₈	—	2	1	no	—
375	C₂×Q₈	—	2	1	no	—
376	C₂×Q₈	—	2	1	no	—
377	C₂×Q₈	—	2	1	no	—
378	C₂×Q₈	—	8	2	no	—
379	C₂×Q₈	C₂×D₄×C₃₁	1	2	yes	—
380	C₂×Q₈	#41×C₃₁	1	2	yes	—
381	C₂×D₄	—	2	4	no	—
382	C₂×D₄	—	2	2	no	—
383	C₂²×C₄	—	2	1	no	—
384	C₂²×C₄	—	2	1	no	—
385	C₂²×C₄	—	2	1	no	—
386	C₂²×C₄	—	2	2	no	—
387	C₂²×C₄	—	2	8	no	—
388	C₂²×C₄	—	2	2	no	—
389	C₂²×C₄	—	2	2	no	—
390	C₂²×C₄	—	2	2	no	—
391	C₂²×C₄	—	2	2	no	—
392	C₂²×C₄	—	2	2	no	—
393	C₂²×C₄	—	2	2	no	—
394	C₂²×C₄	—	2	2	no	—
395	C₂²×C₄	—	2	1	no	—
396	C₂²×C₄	—	8	2	no	—
397	C₂²×C₄	—	8	2	no	—
398	C₂²×C₄	C₂×D₄×C₃₁	1	2	yes	—
399	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
400	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
401	C₂²×C₄	C₂⁴×C₃₁	1	1	yes	characteristic
402	Q₁₆	—	2	8	no	—
403	C₂×C₈	—	2	1	no	—
404	C₂×C₈	—	2	1	no	—
405	C₂×C₈	—	2	1	no	—
406	C₂×C₈	—	2	2	no	—
407	C₂×C₈	—	2	2	no	—
408	C₂×C₈	—	2	1	no	—
409	C₂×C₈	—	2	2	no	—
410	C₂×C₈	—	2	1	no	—
411	C₂×C₈	#41×C₃₁	1	2	yes	—
412	C₄⋊C₄	—	2	8	no	—
413	C₄⋊C₄	—	2	8	no	—
414	C₄⋊C₄	—	2	4	no	—
415	C₄⋊C₄	—	2	4	no	—
416	C₄⋊C₄	—	2	2	no	—
417	C₄⋊C₄	—	2	1	no	—
418	C₄⋊C₄	—	2	2	no	—
419	C₄⋊C₄	—	2	1	no	—
420	C₄⋊C₄	—	2	1	no	—
421	C₄⋊C₄	—	2	2	no	—
422	C₄⋊C₄	—	2	2	no	—
423	C₄⋊C₄	—	4	4	no	—
424	C₄⋊C₄	—	4	2	no	—
425	C₄⋊C₄	C₂×D₄×C₃₁	1	2	yes	—
426	C₄⋊C₄	C₂×D₄×C₃₁	1	4	yes	—
427	C₂²⋊C₄	—	2	8	no	—
428	C₂²⋊C₄	—	2	4	no	—
429	C₂²⋊C₄	—	4	4	no	—
430	C₂²⋊C₄	—	4	4	no	—
431	C₄²	—	2	4	no	—
432	C₄²	—	4	2	no	—
433	C₄²	C₂×D₄×C₃₁	1	2	yes	—
434	C₄²	C₂×D₄×C₃₁	1	2	yes	—
435	C₂³	—	2	2	no	—
436	C₂³	—	2	2	no	—
437	C₂³	—	2	2	no	—
438	C₂³	—	2	2	no	—
439	C₂³	—	2	2	no	—
440	C₂³	—	2	2	no	—
441	C₂³	C₃₁×#137	1	4	yes	—
442	C₂³	C₂²×D₄×C₃₁	1	1	yes	characteristic
443	Q₈	—	2	2	no	—
444	Q₈	—	2	2	no	—
445	Q₈	—	2	2	no	—
446	Q₈	—	2	2	no	—
447	Q₈	—	4	1	no	—
448	Q₈	—	4	1	no	—
449	Q₈	—	4	1	no	—
450	Q₈	—	4	1	no	—
451	Q₈	—	8	4	no	—
452	D₄	—	2	8	no	—
453	C₂×C₄	—	2	2	no	—
454	C₂×C₄	—	2	2	no	—
455	C₂×C₄	—	2	2	no	—
456	C₂×C₄	—	2	2	no	—
457	C₂×C₄	—	2	2	no	—
458	C₂×C₄	—	2	4	no	—
459	C₂×C₄	—	2	2	no	—
460	C₂×C₄	—	2	2	no	—
461	C₂×C₄	—	2	2	no	—
462	C₂×C₄	—	4	1	no	—
463	C₂×C₄	—	4	1	no	—
464	C₂×C₄	—	4	1	no	—
465	C₂×C₄	—	4	1	no	—
466	C₂×C₄	—	4	2	no	—
467	C₂×C₄	—	4	2	no	—
468	C₂×C₄	—	4	2	no	—
469	C₂×C₄	—	4	2	no	—
470	C₂×C₄	—	4	2	no	—
471	C₂×C₄	—	4	2	no	—
472	C₂×C₄	—	4	2	no	—
473	C₂×C₄	—	4	1	no	—
474	C₂×C₄	—	4	2	no	—
475	C₂×C₄	—	4	2	no	—
476	C₂×C₄	—	4	2	no	—
477	C₂×C₄	—	4	2	no	—
478	C₂×C₄	—	8	8	no	—
479	C₂×C₄	C₃₁×#120	1	2	yes	—
480	C₂×C₄	C₃₁×#120	1	2	yes	—
481	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
482	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
483	C₂×C₄	C₂×#41×C₃₁	1	1	yes	characteristic
484	C₂×C₄	C₂×#41×C₃₁	1	1	yes	characteristic
485	C₂×C₄	C₃₁×H(5,𝔽₂)	1	1	yes	characteristic
486	C₂×C₄	C₃₁×2₋¹⁺⁴	1	1	yes	characteristic
487	C₈	—	2	2	no	—
488	C₈	—	4	1	no	—
489	C₈	—	4	2	no	—
490	C₂²	—	2	4	no	—
491	C₂²	—	2	4	no	—
492	C₂²	—	2	4	no	—
493	C₂²	—	4	2	no	—
494	C₂²	—	4	2	no	—
495	C₂²	—	4	2	no	—
496	C₂²	—	4	2	no	—
497	C₂²	C₂×C₃₁×#120	1	1	yes	characteristic
498	C₂²	C₃₁×#546	1	1	yes	characteristic
499	C₂²	C₃₁×#547	1	1	yes	characteristic
500	C₂²	C₂×C₃₁×#137	1	2	yes	—
501	C₂²	C₃₁×#577	1	2	yes	—
502	C₄	—	2	2	no	—
503	C₄	—	2	2	no	—
504	C₄	—	4	2	no	—
505	C₄	—	4	2	no	—
506	C₄	—	4	2	no	—
507	C₄	—	4	2	no	—
508	C₄	—	4	4	no	—
509	C₄	—	8	1	no	—
510	C₄	—	8	2	no	—
511	C₄	C₃₁×#525	1	1	yes	characteristic
512	C₄	C₃₁×#537	1	1	yes	characteristic
513	C₄	C₃₁×#548	1	1	yes	characteristic
514	C₄	C₃₁×#550	1	1	yes	characteristic
515	C₂	—	4	4	no	—
516	C₂	C₃₁×#2407	1	1	yes	characteristic
517	C₂	C₃₁×#3004	1	1	yes	characteristic
518	C₂	C₃₁×#3009	1	1	yes	characteristic
519	C₂	C₃₁×#3316	1	2	yes	—
520	C₂	C₃₁×#3344	1	2	yes	—

	subgroup	G/H
center, socle	C₂³×C₃₁	C₂²×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁴×C₃₁
prime radical	C₂²×D₄×C₃₁	D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₃₁ ◁ C₂×C₄×D₄×C₃₁ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂×C₃₁×#475	C₂	semidirect
3. C₂×C₃₁×#441	C₂	semidirect
5. C₃₁×#2035	C₂	semidirect
6. C₃₁×#2001	C₂	semidirect
7. C₃₁×#1993	C₂	semidirect
9. C₃₁×#1872	C₂	semidirect
10. C₃₁×#1853	C₂	semidirect
11. C₃₁×#1805	C₂	semidirect
174. #24134	C₃₁	direct (N × H)
231. C₂×#475	C₂×C₃₁	semidirect
232. C₂×#441	C₂×C₃₁	semidirect
234. #2035	C₂×C₃₁	semidirect
235. #2001	C₂×C₃₁	semidirect
236. #1993	C₂×C₃₁	semidirect
238. #1872	C₂×C₃₁	semidirect
239. #1853	C₂×C₃₁	semidirect
240. #1805	C₂×C₃₁	semidirect
365. C₃₁	#24134	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×#554	2
C₂×C₃₁×#475	2
C₂×C₃₁×#441	2
C₃₁×#2536	2
C₃₁×#2035	2
C₃₁×#2001	2
C₃₁×#1993	2
C₃₁×#1955	2
C₃₁×#1872	2
C₃₁×#1853	2
C₃₁×#1805	2
#24134	31

Subgroups — C31×#24134 #21480893

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#24134 #21480893