Subgroups — C₃₁×#20882 #21477641

Order 7936.

1458 conjugacy classes of subgroups.

The subgroup lattice has 508 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃₁×#520	C₂	1	1	yes	maximal, characteristic
2	C₂²×C₃₁×#106	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×C₃₁×#417	C₂	1	1	yes	maximal, characteristic
4	C₂×C₃₁×#416	C₂	1	1	yes	maximal, characteristic
5	C₃₁×#2317	C₂	1	1	yes	maximal, characteristic, complemented
6	C₃₁×#1865	C₂	1	4	yes	maximal, complemented
7	C₃₁×#1864	C₂	1	4	yes	maximal, complemented
8	C₃₁×#1736	C₂	1	1	yes	maximal, characteristic, complemented
9	C₃₁×#1729	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂²×#41×C₃₁	C₂²	1	1	yes	characteristic
11	C₂²×M₁₆×C₃₁	C₂²	1	1	yes	characteristic
12	C₂³×C₈×C₃₁	C₂²	1	1	yes	characteristic
13	C₃₁×#520	—	2	4	no	—
14	C₃₁×#520	—	2	4	no	—
15	C₂×C₄×Q₈×C₃₁	C₂²	1	2	yes	—
16	C₂×C₄×D₄×C₃₁	—	2	2	no	—
17	C₂×C₄×D₄×C₃₁	C₂²	1	2	yes	—
18	C₂×C₃₁×#117	—	2	1	no	—
19	C₂×C₃₁×#117	C₂²	1	1	yes	characteristic
20	C₂×C₃₁×#117	C₂²	1	1	yes	characteristic
21	C₂²×C₄⋊C₄×C₃₁	—	2	1	no	—
22	C₂²×C₄⋊C₄×C₃₁	C₂²	1	1	yes	characteristic
23	C₂²×C₄⋊C₄×C₃₁	C₂²	1	1	yes	characteristic
24	C₂×C₃₁×#106	—	2	4	no	—
25	C₂×C₃₁×#106	C₂²	1	4	yes	—
26	C₃₁×#417	—	2	4	no	—
27	C₃₁×#416	C₄	1	8	yes	complemented
28	C₂×C₃₁×#103	C₂²	1	2	yes	—
29	C₂×C₃₁×#103	C₂²	1	2	yes	—
30	C₂×C₃₁×#102	C₂²	1	2	yes	—
31	C₂×C₃₁×#102	C₂²	1	2	yes	—
32	C₃₁×#389	C₂²	1	4	yes	—
33	C₃₁×#385	—	2	2	no	—
34	C₃₁×#384	—	2	1	no	—
35	C₃₁×#382	—	2	1	no	—
36	C₃₁×#340	C₂²	1	4	yes	—
37	C₃₁×#336	C₂²	1	4	yes	—
38	C₂×#41×C₃₁	—	2	2	no	—
39	C₂×#41×C₃₁	—	2	2	no	—
40	C₂×#41×C₃₁	—	2	2	no	—
41	C₂×#41×C₃₁	C₂×C₄	1	4	yes	—
42	C₂×#41×C₃₁	D₄	1	4	yes	—
43	C₂×#41×C₃₁	Q₈	1	4	yes	—
44	C₂²×Q₈×C₃₁	C₂³	1	1	yes	characteristic
45	C₂²×D₄×C₃₁	—	2	1	no	—
46	C₂²×D₄×C₃₁	C₂³	1	1	yes	characteristic
47	C₂³×C₄×C₃₁	—	2	1	no	—
48	C₂³×C₄×C₃₁	—	2	1	no	—
49	C₂³×C₄×C₃₁	—	2	2	no	—
50	C₂³×C₄×C₃₁	—	2	1	no	—
51	C₂³×C₄×C₃₁	C₂³	1	1	yes	characteristic
52	C₂×M₁₆×C₃₁	—	2	2	no	—
53	C₂×M₁₆×C₃₁	—	2	2	no	—
54	C₂×M₁₆×C₃₁	—	2	2	no	—
55	C₂²×C₈×C₃₁	—	2	2	no	—
56	C₂²×C₈×C₃₁	—	2	2	no	—
57	C₂²×C₈×C₃₁	C₂×C₄	1	4	yes	complemented
58	C₂²×C₈×C₃₁	C₂³	1	2	yes	—
59	C₂²×C₈×C₃₁	C₂³	1	2	yes	—
60	C₄×Q₈×C₃₁	—	2	4	no	—
61	C₄×Q₈×C₃₁	—	2	4	no	—
62	C₄×D₄×C₃₁	—	2	16	no	—
63	C₄×D₄×C₃₁	—	2	4	no	—
64	C₄×D₄×C₃₁	—	2	4	no	—
65	C₃₁×#117	—	2	8	no	—
66	C₃₁×#117	—	2	2	no	—
67	C₃₁×#117	—	2	2	no	—
68	C₃₁×#117	—	2	2	no	—
69	C₃₁×#117	C₂×C₄	1	4	yes	complemented
70	C₂×C₄⋊C₄×C₃₁	—	2	8	no	—
71	C₂×C₄⋊C₄×C₃₁	—	2	1	no	—
72	C₂×C₄⋊C₄×C₃₁	—	2	2	no	—
73	C₂×C₄⋊C₄×C₃₁	—	2	2	no	—
74	C₂×C₄⋊C₄×C₃₁	—	2	2	no	—
75	C₂×C₄⋊C₄×C₃₁	—	2	2	no	—
76	C₂×C₄⋊C₄×C₃₁	—	2	1	no	—
77	C₂×C₄⋊C₄×C₃₁	—	2	4	no	—
78	C₂×C₄⋊C₄×C₃₁	D₄	1	1	yes	characteristic
79	C₂×C₄⋊C₄×C₃₁	D₄	1	1	yes	characteristic
80	C₂×C₄⋊C₄×C₃₁	D₄	1	4	yes	—
81	C₂×C₄⋊C₄×C₃₁	C₂³	1	2	yes	—
82	C₂×C₄⋊C₄×C₃₁	C₂³	1	2	yes	—
83	C₂×C₄⋊C₄×C₃₁	C₂³	1	2	yes	—
84	C₂×C₂²⋊C₄×C₃₁	—	2	1	no	—
85	C₂×C₂²⋊C₄×C₃₁	—	2	2	no	—
86	C₂×C₂²⋊C₄×C₃₁	D₄	1	2	yes	—
87	C₂×C₄²×C₃₁	—	2	2	no	—
88	C₂×C₄²×C₃₁	—	2	1	no	—
89	C₂×C₄²×C₃₁	C₂³	1	2	yes	—
90	C₃₁×#106	—	2	8	no	—
91	C₃₁×#103	—	2	4	no	—
92	C₃₁×#103	C₂×C₄	1	8	yes	complemented
93	C₃₁×#102	—	2	4	no	—
94	C₃₁×#102	C₂×C₄	1	8	yes	complemented
95	C₃₁×#95	—	2	4	no	—
96	C₂⁴×C₃₁	—	2	1	no	—
97	C₂⁴×C₃₁	C₂×D₄	1	1	yes	characteristic
98	#41×C₃₁	—	2	8	no	—
99	#41×C₃₁	—	2	8	no	—
100	#41×C₃₁	—	2	8	no	—
101	#41×C₃₁	C₄⋊C₄	1	16	yes	complemented
102	C₂×Q₈×C₃₁	—	2	1	no	—
103	C₂×Q₈×C₃₁	—	2	1	no	—
104	C₂×Q₈×C₃₁	—	2	1	no	—
105	C₂×Q₈×C₃₁	C₂²×C₄	1	2	yes	—
106	C₂×Q₈×C₃₁	C₂×D₄	1	2	yes	—
107	C₂×Q₈×C₃₁	C₂×Q₈	1	2	yes	—
108	C₂×D₄×C₃₁	—	2	4	no	—
109	C₂×D₄×C₃₁	—	2	4	no	—
110	C₂×D₄×C₃₁	—	2	4	no	—
111	C₂×D₄×C₃₁	—	2	1	no	—
112	C₂×D₄×C₃₁	—	2	1	no	—
113	C₂×D₄×C₃₁	—	2	1	no	—
114	C₂×D₄×C₃₁	C₂²×C₄	1	2	yes	—
115	C₂×D₄×C₃₁	C₂×D₄	1	2	yes	—
116	C₂×D₄×C₃₁	C₂×Q₈	1	2	yes	—
117	C₂²×C₄×C₃₁	—	2	1	no	—
118	C₂²×C₄×C₃₁	—	2	4	no	—
119	C₂²×C₄×C₃₁	—	2	4	no	—
120	C₂²×C₄×C₃₁	—	2	4	no	—
121	C₂²×C₄×C₃₁	—	2	2	no	—
122	C₂²×C₄×C₃₁	—	2	2	no	—
123	C₂²×C₄×C₃₁	—	2	8	no	—
124	C₂²×C₄×C₃₁	—	2	4	no	—
125	C₂²×C₄×C₃₁	—	2	2	no	—
126	C₂²×C₄×C₃₁	—	4	2	no	—
127	C₂²×C₄×C₃₁	—	8	4	no	—
128	C₂²×C₄×C₃₁	—	8	2	no	—
129	C₂²×C₄×C₃₁	—	8	2	no	—
130	C₂²×C₄×C₃₁	C₂²×C₄	1	2	yes	—
131	C₂²×C₄×C₃₁	C₂×D₄	1	2	yes	—
132	C₂²×C₄×C₃₁	C₂×D₄	1	2	yes	—
133	C₂²×C₄×C₃₁	C₂×D₄	1	2	yes	—
134	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
135	C₂²×C₄×C₃₁	C₂×D₄	1	2	yes	—
136	C₂²×C₄×C₃₁	C₂×Q₈	1	2	yes	—
137	C₂²×C₄×C₃₁	#41	1	2	yes	—
138	C₂²×C₄×C₃₁	#41	1	2	yes	—
139	C₂²×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
140	M₁₆×C₃₁	—	4	4	no	—
141	C₂×C₈×C₃₁	—	2	8	no	—
142	C₂×C₈×C₃₁	—	2	2	no	—
143	C₂×C₈×C₃₁	—	2	2	no	—
144	C₂×C₈×C₃₁	—	2	2	no	—
145	C₂×C₈×C₃₁	—	2	2	no	—
146	C₂×C₈×C₃₁	—	2	2	no	—
147	C₂×C₈×C₃₁	C₂²×C₄	1	4	yes	—
148	C₄⋊C₄×C₃₁	—	2	8	no	—
149	C₄⋊C₄×C₃₁	—	2	2	no	—
150	C₄⋊C₄×C₃₁	—	2	4	no	—
151	C₄⋊C₄×C₃₁	—	2	2	no	—
152	C₄⋊C₄×C₃₁	—	2	2	no	—
153	C₄⋊C₄×C₃₁	—	2	2	no	—
154	C₄⋊C₄×C₃₁	—	2	2	no	—
155	C₄⋊C₄×C₃₁	—	2	2	no	—
156	C₄⋊C₄×C₃₁	—	4	2	no	—
157	C₄⋊C₄×C₃₁	—	8	4	no	—
158	C₄⋊C₄×C₃₁	C₂²×C₄	1	4	yes	—
159	C₄⋊C₄×C₃₁	C₂×D₄	1	4	yes	—
160	C₂²⋊C₄×C₃₁	—	2	8	no	—
161	C₂²⋊C₄×C₃₁	—	2	4	no	—
162	C₂²⋊C₄×C₃₁	—	2	4	no	—
163	C₄²×C₃₁	—	2	8	no	—
164	C₄²×C₃₁	—	2	4	no	—
165	C₄²×C₃₁	—	2	2	no	—
166	C₄²×C₃₁	—	2	2	no	—
167	#20882	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
168	C₂³×C₃₁	—	2	2	no	—
169	C₂³×C₃₁	—	2	2	no	—
170	C₂³×C₃₁	—	2	2	no	—
171	C₂³×C₃₁	—	2	2	no	—
172	C₂³×C₃₁	—	2	2	no	—
173	C₂³×C₃₁	—	4	2	no	—
174	C₂³×C₃₁	—	4	2	no	—
175	C₂³×C₃₁	C₄×D₄	1	2	yes	—
176	C₂³×C₃₁	#120	1	2	yes	—
177	C₂³×C₃₁	C₂²⋊Q₈	1	2	yes	—
178	C₂³×C₃₁	C₂²×D₄	1	1	yes	characteristic
179	Q₈×C₃₁	—	2	2	no	—
180	Q₈×C₃₁	—	2	2	no	—
181	Q₈×C₃₁	—	2	2	no	—
182	Q₈×C₃₁	C₂×C₄⋊C₄	1	4	yes	—
183	D₄×C₃₁	—	2	16	no	—
184	D₄×C₃₁	—	2	2	no	—
185	D₄×C₃₁	—	2	2	no	—
186	D₄×C₃₁	—	2	2	no	—
187	D₄×C₃₁	C₂×C₄⋊C₄	1	4	yes	—
188	C₂×C₄×C₃₁	—	2	2	no	—
189	C₂×C₄×C₃₁	—	2	2	no	—
190	C₂×C₄×C₃₁	—	2	2	no	—
191	C₂×C₄×C₃₁	—	2	16	no	—
192	C₂×C₄×C₃₁	—	2	4	no	—
193	C₂×C₄×C₃₁	—	2	2	no	—
194	C₂×C₄×C₃₁	—	2	2	no	—
195	C₂×C₄×C₃₁	—	2	4	no	—
196	C₂×C₄×C₃₁	—	2	4	no	—
197	C₂×C₄×C₃₁	—	2	4	no	—
198	C₂×C₄×C₃₁	—	4	2	no	—
199	C₂×C₄×C₃₁	—	4	2	no	—
200	C₂×C₄×C₃₁	—	4	4	no	—
201	C₂×C₄×C₃₁	—	4	2	no	—
202	C₂×C₄×C₃₁	—	4	4	no	—
203	C₂×C₄×C₃₁	—	8	4	no	—
204	C₂×C₄×C₃₁	—	8	8	no	—
205	C₂×C₄×C₃₁	—	8	4	no	—
206	C₂×C₄×C₃₁	C₂×C₄⋊C₄	1	4	yes	—
207	C₂×C₄×C₃₁	C₄×D₄	1	4	yes	—
208	C₂×C₄×C₃₁	C₄×D₄	1	2	yes	—
209	C₂×C₄×C₃₁	#120	1	2	yes	—
210	C₂×C₄×C₃₁	C₂²⋊Q₈	1	4	yes	—
211	C₂×C₄×C₃₁	C₂²⋊Q₈	1	2	yes	—
212	C₂×C₄×C₃₁	#123	1	4	yes	—
213	C₂×C₄×C₃₁	C₂³×C₄	1	1	yes	characteristic
214	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
215	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
216	C₂×C₄×C₃₁	C₂²×Q₈	1	1	yes	characteristic
217	C₂×C₄×C₃₁	C₂×#41	1	1	yes	characteristic
218	C₂×C₄×C₃₁	C₂×#41	1	1	yes	characteristic
219	C₈×C₃₁	—	2	4	no	—
220	C₈×C₃₁	—	4	2	no	—
221	C₂×#520	C₂×C₃₁	1	1	yes	characteristic
222	C₂²×#106	C₂×C₃₁	1	1	yes	characteristic, complemented
223	C₂×#417	C₂×C₃₁	1	1	yes	characteristic
224	C₂×#416	C₂×C₃₁	1	1	yes	characteristic
225	#2317	C₂×C₃₁	1	1	yes	characteristic, complemented
226	#1865	C₂×C₃₁	1	4	yes	complemented
227	#1864	C₂×C₃₁	1	4	yes	complemented
228	#1736	C₂×C₃₁	1	1	yes	characteristic, complemented
229	#1729	C₂×C₃₁	1	1	yes	characteristic, complemented
230	C₂²×C₃₁	—	2	4	no	—
231	C₂²×C₃₁	—	2	4	no	—
232	C₂²×C₃₁	—	2	2	no	—
233	C₂²×C₃₁	—	2	2	no	—
234	C₂²×C₃₁	—	2	4	no	—
235	C₂²×C₃₁	—	4	4	no	—
236	C₂²×C₃₁	—	4	4	no	—
237	C₂²×C₃₁	—	4	2	no	—
238	C₂²×C₃₁	—	4	2	no	—
239	C₂²×C₃₁	#385	1	4	yes	—
240	C₂²×C₃₁	C₂×C₄×D₄	1	1	yes	characteristic
241	C₂²×C₃₁	C₂×#120	1	1	yes	characteristic
242	C₂²×C₃₁	C₂×C₂²⋊Q₈	1	1	yes	characteristic
243	C₂²×C₃₁	#577	1	2	yes	—
244	C₂²×C₃₁	#577	1	2	yes	—
245	C₄×C₃₁	—	2	4	no	—
246	C₄×C₃₁	—	4	4	no	—
247	C₄×C₃₁	—	4	2	no	—
248	C₄×C₃₁	—	4	4	no	—
249	C₄×C₃₁	—	4	2	no	—
250	C₄×C₃₁	—	8	2	no	—
251	C₄×C₃₁	#385	1	4	yes	—
252	C₄×C₃₁	C₂²×C₄⋊C₄	1	1	yes	characteristic
253	C₄×C₃₁	C₂×C₄×D₄	1	1	yes	characteristic
254	C₄×C₃₁	C₂×C₂²⋊Q₈	1	1	yes	characteristic
255	C₄×C₃₁	C₂×#123	1	1	yes	characteristic
256	C₂²×#41	C₂²×C₃₁	1	1	yes	characteristic
257	C₂²×M₁₆	C₂²×C₃₁	1	1	yes	characteristic
258	C₂³×C₈	C₂²×C₃₁	1	1	yes	characteristic
259	#520	—	2	4	no	—
260	#520	—	2	4	no	—
261	C₂×C₄×Q₈	C₂²×C₃₁	1	2	yes	—
262	C₂×C₄×D₄	—	2	2	no	—
263	C₂×C₄×D₄	C₂²×C₃₁	1	2	yes	—
264	C₂×#117	—	2	1	no	—
265	C₂×#117	C₂²×C₃₁	1	1	yes	characteristic
266	C₂×#117	C₂²×C₃₁	1	1	yes	characteristic
267	C₂²×C₄⋊C₄	—	2	1	no	—
268	C₂²×C₄⋊C₄	C₂²×C₃₁	1	1	yes	characteristic
269	C₂²×C₄⋊C₄	C₂²×C₃₁	1	1	yes	characteristic
270	C₂×#106	—	2	4	no	—
271	C₂×#106	C₂²×C₃₁	1	4	yes	—
272	#417	—	2	4	no	—
273	#416	C₄×C₃₁	1	8	yes	complemented
274	C₂×#103	C₂²×C₃₁	1	2	yes	—
275	C₂×#103	C₂²×C₃₁	1	2	yes	—
276	C₂×#102	C₂²×C₃₁	1	2	yes	—
277	C₂×#102	C₂²×C₃₁	1	2	yes	—
278	#389	C₂²×C₃₁	1	4	yes	—
279	#385	—	2	2	no	—
280	#384	—	2	1	no	—
281	#382	—	2	1	no	—
282	#340	C₂²×C₃₁	1	4	yes	—
283	#336	C₂²×C₃₁	1	4	yes	—
284	C₂×C₃₁	—	2	4	no	—
285	C₂×C₃₁	—	4	4	no	—
286	C₂×C₃₁	C₂×#385	1	1	yes	characteristic
287	C₂×C₃₁	#2952	1	2	yes	—
288	C₂×C₃₁	#3009	1	1	yes	characteristic
289	C₂×C₃₁	#3010	1	1	yes	characteristic
290	C₂×C₃₁	#3084	1	2	yes	—
291	C₂×#41	—	2	2	no	—
292	C₂×#41	—	2	2	no	—
293	C₂×#41	—	2	2	no	—
294	C₂×#41	C₂×C₄×C₃₁	1	4	yes	—
295	C₂×#41	D₄×C₃₁	1	4	yes	—
296	C₂×#41	Q₈×C₃₁	1	4	yes	—
297	C₂²×Q₈	C₂³×C₃₁	1	1	yes	characteristic
298	C₂²×D₄	—	2	1	no	—
299	C₂²×D₄	C₂³×C₃₁	1	1	yes	characteristic
300	C₂³×C₄	—	2	1	no	—
301	C₂³×C₄	—	2	1	no	—
302	C₂³×C₄	—	2	2	no	—
303	C₂³×C₄	—	2	1	no	—
304	C₂³×C₄	C₂³×C₃₁	1	1	yes	characteristic
305	C₂×M₁₆	—	2	2	no	—
306	C₂×M₁₆	—	2	2	no	—
307	C₂×M₁₆	—	2	2	no	—
308	C₂²×C₈	—	2	2	no	—
309	C₂²×C₈	—	2	2	no	—
310	C₂²×C₈	C₂×C₄×C₃₁	1	4	yes	complemented
311	C₂²×C₈	C₂³×C₃₁	1	2	yes	—
312	C₂²×C₈	C₂³×C₃₁	1	2	yes	—
313	C₄×Q₈	—	2	4	no	—
314	C₄×Q₈	—	2	4	no	—
315	C₄×D₄	—	2	16	no	—
316	C₄×D₄	—	2	4	no	—
317	C₄×D₄	—	2	4	no	—
318	#117	—	2	8	no	—
319	#117	—	2	2	no	—
320	#117	—	2	2	no	—
321	#117	—	2	2	no	—
322	#117	C₂×C₄×C₃₁	1	4	yes	complemented
323	C₂×C₄⋊C₄	—	2	8	no	—
324	C₂×C₄⋊C₄	—	2	1	no	—
325	C₂×C₄⋊C₄	—	2	2	no	—
326	C₂×C₄⋊C₄	—	2	2	no	—
327	C₂×C₄⋊C₄	—	2	2	no	—
328	C₂×C₄⋊C₄	—	2	2	no	—
329	C₂×C₄⋊C₄	—	2	1	no	—
330	C₂×C₄⋊C₄	—	2	4	no	—
331	C₂×C₄⋊C₄	D₄×C₃₁	1	1	yes	characteristic
332	C₂×C₄⋊C₄	D₄×C₃₁	1	1	yes	characteristic
333	C₂×C₄⋊C₄	D₄×C₃₁	1	4	yes	—
334	C₂×C₄⋊C₄	C₂³×C₃₁	1	2	yes	—
335	C₂×C₄⋊C₄	C₂³×C₃₁	1	2	yes	—
336	C₂×C₄⋊C₄	C₂³×C₃₁	1	2	yes	—
337	C₂×C₂²⋊C₄	—	2	1	no	—
338	C₂×C₂²⋊C₄	—	2	2	no	—
339	C₂×C₂²⋊C₄	D₄×C₃₁	1	2	yes	—
340	C₂×C₄²	—	2	2	no	—
341	C₂×C₄²	—	2	1	no	—
342	C₂×C₄²	C₂³×C₃₁	1	2	yes	—
343	#106	—	2	8	no	—
344	#103	—	2	4	no	—
345	#103	C₂×C₄×C₃₁	1	8	yes	complemented
346	#102	—	2	4	no	—
347	#102	C₂×C₄×C₃₁	1	8	yes	complemented
348	#95	—	2	4	no	—
349	C₃₁	#20882	1	1	yes	characteristic, direct product factor, Sylow
350	C₂⁴	—	2	1	no	—
351	C₂⁴	C₂×D₄×C₃₁	1	1	yes	characteristic
352	#41	—	2	8	no	—
353	#41	—	2	8	no	—
354	#41	—	2	8	no	—
355	#41	C₄⋊C₄×C₃₁	1	16	yes	complemented
356	C₂×Q₈	—	2	1	no	—
357	C₂×Q₈	—	2	1	no	—
358	C₂×Q₈	—	2	1	no	—
359	C₂×Q₈	C₂²×C₄×C₃₁	1	2	yes	—
360	C₂×Q₈	C₂×D₄×C₃₁	1	2	yes	—
361	C₂×Q₈	C₂×Q₈×C₃₁	1	2	yes	—
362	C₂×D₄	—	2	4	no	—
363	C₂×D₄	—	2	4	no	—
364	C₂×D₄	—	2	4	no	—
365	C₂×D₄	—	2	1	no	—
366	C₂×D₄	—	2	1	no	—
367	C₂×D₄	—	2	1	no	—
368	C₂×D₄	C₂²×C₄×C₃₁	1	2	yes	—
369	C₂×D₄	C₂×D₄×C₃₁	1	2	yes	—
370	C₂×D₄	C₂×Q₈×C₃₁	1	2	yes	—
371	C₂²×C₄	—	2	1	no	—
372	C₂²×C₄	—	2	4	no	—
373	C₂²×C₄	—	2	4	no	—
374	C₂²×C₄	—	2	4	no	—
375	C₂²×C₄	—	2	2	no	—
376	C₂²×C₄	—	2	2	no	—
377	C₂²×C₄	—	2	8	no	—
378	C₂²×C₄	—	2	4	no	—
379	C₂²×C₄	—	2	2	no	—
380	C₂²×C₄	—	4	2	no	—
381	C₂²×C₄	—	8	4	no	—
382	C₂²×C₄	—	8	2	no	—
383	C₂²×C₄	—	8	2	no	—
384	C₂²×C₄	C₂²×C₄×C₃₁	1	2	yes	—
385	C₂²×C₄	C₂×D₄×C₃₁	1	2	yes	—
386	C₂²×C₄	C₂×D₄×C₃₁	1	2	yes	—
387	C₂²×C₄	C₂×D₄×C₃₁	1	2	yes	—
388	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
389	C₂²×C₄	C₂×D₄×C₃₁	1	2	yes	—
390	C₂²×C₄	C₂×Q₈×C₃₁	1	2	yes	—
391	C₂²×C₄	#41×C₃₁	1	2	yes	—
392	C₂²×C₄	#41×C₃₁	1	2	yes	—
393	C₂²×C₄	C₂⁴×C₃₁	1	1	yes	characteristic
394	M₁₆	—	4	4	no	—
395	C₂×C₈	—	2	8	no	—
396	C₂×C₈	—	2	2	no	—
397	C₂×C₈	—	2	2	no	—
398	C₂×C₈	—	2	2	no	—
399	C₂×C₈	—	2	2	no	—
400	C₂×C₈	—	2	2	no	—
401	C₂×C₈	C₂²×C₄×C₃₁	1	4	yes	—
402	C₄⋊C₄	—	2	8	no	—
403	C₄⋊C₄	—	2	2	no	—
404	C₄⋊C₄	—	2	4	no	—
405	C₄⋊C₄	—	2	2	no	—
406	C₄⋊C₄	—	2	2	no	—
407	C₄⋊C₄	—	2	2	no	—
408	C₄⋊C₄	—	2	2	no	—
409	C₄⋊C₄	—	2	2	no	—
410	C₄⋊C₄	—	4	2	no	—
411	C₄⋊C₄	—	8	4	no	—
412	C₄⋊C₄	C₂²×C₄×C₃₁	1	4	yes	—
413	C₄⋊C₄	C₂×D₄×C₃₁	1	4	yes	—
414	C₂²⋊C₄	—	2	8	no	—
415	C₂²⋊C₄	—	2	4	no	—
416	C₂²⋊C₄	—	2	4	no	—
417	C₄²	—	2	8	no	—
418	C₄²	—	2	4	no	—
419	C₄²	—	2	2	no	—
420	C₄²	—	2	2	no	—
421	C₂³	—	2	2	no	—
422	C₂³	—	2	2	no	—
423	C₂³	—	2	2	no	—
424	C₂³	—	2	2	no	—
425	C₂³	—	2	2	no	—
426	C₂³	—	4	2	no	—
427	C₂³	—	4	2	no	—
428	C₂³	C₄×D₄×C₃₁	1	2	yes	—
429	C₂³	C₃₁×#120	1	2	yes	—
430	C₂³	C₃₁×C₂²⋊Q₈	1	2	yes	—
431	C₂³	C₂²×D₄×C₃₁	1	1	yes	characteristic
432	Q₈	—	2	2	no	—
433	Q₈	—	2	2	no	—
434	Q₈	—	2	2	no	—
435	Q₈	C₂×C₄⋊C₄×C₃₁	1	4	yes	—
436	D₄	—	2	16	no	—
437	D₄	—	2	2	no	—
438	D₄	—	2	2	no	—
439	D₄	—	2	2	no	—
440	D₄	C₂×C₄⋊C₄×C₃₁	1	4	yes	—
441	C₂×C₄	—	2	2	no	—
442	C₂×C₄	—	2	2	no	—
443	C₂×C₄	—	2	2	no	—
444	C₂×C₄	—	2	16	no	—
445	C₂×C₄	—	2	4	no	—
446	C₂×C₄	—	2	2	no	—
447	C₂×C₄	—	2	2	no	—
448	C₂×C₄	—	2	4	no	—
449	C₂×C₄	—	2	4	no	—
450	C₂×C₄	—	2	4	no	—
451	C₂×C₄	—	4	2	no	—
452	C₂×C₄	—	4	2	no	—
453	C₂×C₄	—	4	4	no	—
454	C₂×C₄	—	4	2	no	—
455	C₂×C₄	—	4	4	no	—
456	C₂×C₄	—	8	4	no	—
457	C₂×C₄	—	8	8	no	—
458	C₂×C₄	—	8	4	no	—
459	C₂×C₄	C₂×C₄⋊C₄×C₃₁	1	4	yes	—
460	C₂×C₄	C₄×D₄×C₃₁	1	4	yes	—
461	C₂×C₄	C₄×D₄×C₃₁	1	2	yes	—
462	C₂×C₄	C₃₁×#120	1	2	yes	—
463	C₂×C₄	C₃₁×C₂²⋊Q₈	1	4	yes	—
464	C₂×C₄	C₃₁×C₂²⋊Q₈	1	2	yes	—
465	C₂×C₄	C₃₁×#123	1	4	yes	—
466	C₂×C₄	C₂³×C₄×C₃₁	1	1	yes	characteristic
467	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
468	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
469	C₂×C₄	C₂²×Q₈×C₃₁	1	1	yes	characteristic
470	C₂×C₄	C₂×#41×C₃₁	1	1	yes	characteristic
471	C₂×C₄	C₂×#41×C₃₁	1	1	yes	characteristic
472	C₈	—	2	4	no	—
473	C₈	—	4	2	no	—
474	C₂²	—	2	4	no	—
475	C₂²	—	2	4	no	—
476	C₂²	—	2	2	no	—
477	C₂²	—	2	2	no	—
478	C₂²	—	2	4	no	—
479	C₂²	—	4	4	no	—
480	C₂²	—	4	4	no	—
481	C₂²	—	4	2	no	—
482	C₂²	—	4	2	no	—
483	C₂²	C₃₁×#385	1	4	yes	—
484	C₂²	C₂×C₄×D₄×C₃₁	1	1	yes	characteristic
485	C₂²	C₂×C₃₁×#120	1	1	yes	characteristic
486	C₂²	C₂×C₃₁×C₂²⋊Q₈	1	1	yes	characteristic
487	C₂²	C₃₁×#577	1	2	yes	—
488	C₂²	C₃₁×#577	1	2	yes	—
489	C₄	—	2	4	no	—
490	C₄	—	4	4	no	—
491	C₄	—	4	2	no	—
492	C₄	—	4	4	no	—
493	C₄	—	4	2	no	—
494	C₄	—	8	2	no	—
495	C₄	C₃₁×#385	1	4	yes	—
496	C₄	C₂²×C₄⋊C₄×C₃₁	1	1	yes	characteristic
497	C₄	C₂×C₄×D₄×C₃₁	1	1	yes	characteristic
498	C₄	C₂×C₃₁×C₂²⋊Q₈	1	1	yes	characteristic
499	C₄	C₂×C₃₁×#123	1	1	yes	characteristic
500	C₂	—	2	4	no	—
501	C₂	—	4	4	no	—
502	C₂	C₂×C₃₁×#385	1	1	yes	characteristic
503	C₂	C₃₁×#2952	1	2	yes	—
504	C₂	C₃₁×#3009	1	1	yes	characteristic
505	C₂	C₃₁×#3010	1	1	yes	characteristic
506	C₂	C₃₁×#3084	1	2	yes	—

	subgroup	G/H
center, socle	C₂³×C₃₁	C₂²×D₄
commutator	C₂×C₄	C₂³×C₄×C₃₁
Frattini, cosocle	C₂²×C₄	C₂⁴×C₃₁
prime radical	C₂²×#41×C₃₁	C₂²
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₃₁ ◁ C₂×C₃₁×#117 ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂²×C₃₁×#106	C₂	semidirect
5. C₃₁×#2317	C₂	semidirect
6. C₃₁×#1865	C₂	semidirect
7. C₃₁×#1864	C₂	semidirect
8. C₃₁×#1736	C₂	semidirect
9. C₃₁×#1729	C₂	semidirect
27. C₃₁×#416	C₄	semidirect
57. C₂²×C₈×C₃₁	C₂×C₄	semidirect
69. C₃₁×#117	C₂×C₄	semidirect
92. C₃₁×#103	C₂×C₄	semidirect
94. C₃₁×#102	C₂×C₄	semidirect
101. #41×C₃₁	C₄⋊C₄	semidirect
167. #20882	C₃₁	direct (N × H)
222. C₂²×#106	C₂×C₃₁	semidirect
225. #2317	C₂×C₃₁	semidirect
226. #1865	C₂×C₃₁	semidirect
227. #1864	C₂×C₃₁	semidirect
228. #1736	C₂×C₃₁	semidirect
229. #1729	C₂×C₃₁	semidirect
273. #416	C₄×C₃₁	semidirect
310. C₂²×C₈	C₂×C₄×C₃₁	semidirect
322. #117	C₂×C₄×C₃₁	semidirect
345. #103	C₂×C₄×C₃₁	semidirect
347. #102	C₂×C₄×C₃₁	semidirect
349. C₃₁	#20882	direct (N × H)
355. #41	C₄⋊C₄×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×#520	2
C₂²×C₃₁×#106	2
C₂×C₃₁×#417	2
C₂×C₃₁×#416	2
C₃₁×#2317	2
C₃₁×#1865	2
C₃₁×#1864	2
C₃₁×#1736	2
C₃₁×#1729	2
#20882	31

Subgroups — C31×#20882 #21477641

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#20882 #21477641