Subgroups — C₃₁×#23735 #21480494

Order 7936.

1210 conjugacy classes of subgroups.

The subgroup lattice has 506 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃₁×#428	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×C₃₁×#427	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×C₃₁×#418	C₂	1	1	yes	maximal, characteristic
4	C₂²×C₃₁×#102	C₂	1	1	yes	maximal, characteristic
5	C₃₁×#2317	C₂	1	1	yes	maximal, characteristic, complemented
6	C₃₁×#2309	C₂	1	1	yes	maximal, characteristic
7	C₃₁×#2094	C₂	1	2	yes	maximal, complemented
8	C₃₁×#2093	C₂	1	2	yes	maximal, complemented
9	C₃₁×#2092	C₂	1	2	yes	maximal, complemented
10	C₃₁×#2092	C₂	1	2	yes	maximal, complemented
11	C₃₁×#1729	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂³×D₄×C₃₁	C₂²	1	1	yes	characteristic
13	C₂²×M₁₆×C₃₁	C₂²	1	1	yes	characteristic
14	C₂³×C₈×C₃₁	C₂²	1	1	yes	characteristic
15	C₂×C₃₁×#117	C₂²	1	1	yes	characteristic
16	C₂×C₃₁×#117	C₂²	1	1	yes	characteristic
17	C₂×C₃₁×#117	C₂²	1	1	yes	characteristic
18	C₂²×C₄⋊C₄×C₃₁	C₂²	1	1	yes	characteristic
19	C₃₁×#428	—	2	4	no	—
20	C₃₁×#427	—	2	4	no	—
21	C₂×C₃₁×#107	C₂²	1	1	yes	characteristic, complemented
22	C₂×C₃₁×#107	C₂²	1	2	yes	complemented
23	C₂×C₃₁×#107	C₂²	1	1	yes	characteristic, complemented
24	C₂×C₃₁×#106	C₂²	1	1	yes	characteristic, complemented
25	C₂×C₃₁×#106	C₂²	1	2	yes	complemented
26	C₂×C₃₁×#106	C₂²	1	1	yes	characteristic, complemented
27	C₃₁×#418	—	2	4	no	—
28	C₂×C₃₁×#102	—	2	4	no	—
29	C₂×C₃₁×#102	C₂²	1	2	yes	—
30	C₂×C₃₁×#102	C₂²	1	2	yes	—
31	C₂×C₃₁×#102	C₂²	1	2	yes	—
32	C₂×C₃₁×#102	C₂²	1	2	yes	—
33	C₃₁×#390	—	2	2	no	—
34	C₃₁×#390	C₂²	1	2	yes	—
35	C₃₁×#390	C₂²	1	2	yes	—
36	C₃₁×#389	—	2	2	no	—
37	C₃₁×#385	—	2	2	no	—
38	C₃₁×#384	C₂²	1	2	yes	complemented
39	C₃₁×#382	C₂²	1	2	yes	complemented
40	C₃₁×#379	—	2	2	no	—
41	C₃₁×#336	C₂²	1	2	yes	complemented
42	C₃₁×#336	C₂²	1	2	yes	complemented
43	C₂⁵×C₃₁	—	2	1	no	—
44	C₂²×D₄×C₃₁	—	2	2	no	—
45	C₂²×D₄×C₃₁	—	2	2	no	—
46	C₂²×D₄×C₃₁	—	2	1	no	—
47	C₂²×D₄×C₃₁	—	2	2	no	—
48	C₂²×D₄×C₃₁	—	2	2	no	—
49	C₂²×D₄×C₃₁	—	2	2	no	—
50	C₂²×D₄×C₃₁	Q₈	1	4	yes	—
51	C₂²×D₄×C₃₁	C₂³	1	2	yes	—
52	C₂³×C₄×C₃₁	—	2	1	no	—
53	C₂³×C₄×C₃₁	C₂³	1	1	yes	characteristic
54	C₂×M₁₆×C₃₁	—	2	2	no	—
55	C₂×M₁₆×C₃₁	—	2	2	no	—
56	C₂×M₁₆×C₃₁	D₄	1	4	yes	—
57	C₂²×C₈×C₃₁	—	2	2	no	—
58	C₂²×C₈×C₃₁	—	2	2	no	—
59	C₂²×C₈×C₃₁	D₄	1	4	yes	—
60	C₂²×C₈×C₃₁	C₂³	1	1	yes	characteristic
61	C₂²×C₈×C₃₁	C₂³	1	2	yes	—
62	C₂²×C₈×C₃₁	C₂³	1	1	yes	characteristic
63	C₃₁×#117	—	2	2	no	—
64	C₃₁×#117	—	2	2	no	—
65	C₃₁×#117	—	2	2	no	—
66	C₃₁×#117	—	2	2	no	—
67	C₃₁×#117	—	2	2	no	—
68	C₃₁×#117	—	2	2	no	—
69	C₂×C₄⋊C₄×C₃₁	—	2	2	no	—
70	C₂×C₄⋊C₄×C₃₁	—	2	2	no	—
71	C₂×C₄⋊C₄×C₃₁	—	2	1	no	—
72	C₂×C₄⋊C₄×C₃₁	—	2	4	no	—
73	C₂×C₄⋊C₄×C₃₁	D₄	1	2	yes	—
74	C₂×C₄⋊C₄×C₃₁	C₂³	1	2	yes	—
75	C₂×C₄⋊C₄×C₃₁	C₂³	1	1	yes	characteristic
76	C₂×C₄⋊C₄×C₃₁	C₂³	1	1	yes	characteristic
77	C₂×C₄⋊C₄×C₃₁	C₂³	1	1	yes	characteristic
78	C₂×C₄⋊C₄×C₃₁	C₂³	1	1	yes	characteristic
79	C₂×C₂²⋊C₄×C₃₁	—	2	1	no	—
80	C₂×C₂²⋊C₄×C₃₁	—	2	1	no	—
81	C₂×C₂²⋊C₄×C₃₁	—	2	4	no	—
82	C₂×C₂²⋊C₄×C₃₁	—	2	4	no	—
83	C₂×C₂²⋊C₄×C₃₁	D₄	1	2	yes	—
84	C₂×C₄²×C₃₁	—	2	1	no	—
85	C₂×C₄²×C₃₁	—	2	1	no	—
86	C₂×C₄²×C₃₁	C₂³	1	2	yes	—
87	C₃₁×#107	—	2	2	no	—
88	C₃₁×#107	—	2	2	no	—
89	C₃₁×#107	—	2	4	no	—
90	C₃₁×#106	—	2	2	no	—
91	C₃₁×#106	—	2	2	no	—
92	C₃₁×#106	—	2	4	no	—
93	C₃₁×#102	—	2	4	no	—
94	C₃₁×#102	—	2	4	no	—
95	C₃₁×#102	—	2	4	no	—
96	C₃₁×#102	—	2	4	no	—
97	C₃₁×#95	—	2	4	no	—
98	C₂⁴×C₃₁	—	2	2	no	—
99	C₂⁴×C₃₁	—	2	4	no	—
100	C₂⁴×C₃₁	—	4	2	no	—
101	C₂⁴×C₃₁	—	4	2	no	—
102	C₂⁴×C₃₁	—	8	2	no	—
103	C₂⁴×C₃₁	—	8	2	no	—
104	C₂⁴×C₃₁	C₂×D₄	1	1	yes	characteristic
105	C₂×D₄×C₃₁	—	2	8	no	—
106	C₂×D₄×C₃₁	—	2	8	no	—
107	C₂×D₄×C₃₁	—	2	4	no	—
108	C₂×D₄×C₃₁	—	2	2	no	—
109	C₂×D₄×C₃₁	—	2	2	no	—
110	C₂×D₄×C₃₁	—	2	2	no	—
111	C₂×D₄×C₃₁	—	2	2	no	—
112	C₂×D₄×C₃₁	—	2	2	no	—
113	C₂×D₄×C₃₁	—	4	4	no	—
114	C₂×D₄×C₃₁	—	4	4	no	—
115	C₂×D₄×C₃₁	—	4	2	no	—
116	C₂×D₄×C₃₁	—	4	2	no	—
117	C₂×D₄×C₃₁	C₂×Q₈	1	4	yes	—
118	C₂²×C₄×C₃₁	—	2	1	no	—
119	C₂²×C₄×C₃₁	—	2	2	no	—
120	C₂²×C₄×C₃₁	—	2	1	no	—
121	C₂²×C₄×C₃₁	—	2	1	no	—
122	C₂²×C₄×C₃₁	—	2	1	no	—
123	C₂²×C₄×C₃₁	—	4	2	no	—
124	C₂²×C₄×C₃₁	—	8	2	no	—
125	C₂²×C₄×C₃₁	C₂×D₄	1	4	yes	—
126	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
127	C₂²×C₄×C₃₁	C₂×D₄	1	2	yes	—
128	C₂²×C₄×C₃₁	C₂×D₄	1	2	yes	—
129	C₂²×C₄×C₃₁	C₂×Q₈	1	2	yes	—
130	C₂²×C₄×C₃₁	#41	1	2	yes	—
131	C₂²×C₄×C₃₁	#41	1	2	yes	—
132	C₂²×C₄×C₃₁	#41	1	2	yes	—
133	C₂²×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
134	M₁₆×C₃₁	—	2	8	no	—
135	C₂×C₈×C₃₁	—	2	8	no	—
136	C₂×C₈×C₃₁	—	2	1	no	—
137	C₂×C₈×C₃₁	—	2	1	no	—
138	C₂×C₈×C₃₁	—	2	1	no	—
139	C₂×C₈×C₃₁	—	2	1	no	—
140	C₂×C₈×C₃₁	—	2	2	no	—
141	C₂×C₈×C₃₁	—	2	2	no	—
142	C₂×C₈×C₃₁	C₂×D₄	1	2	yes	—
143	C₂×C₈×C₃₁	C₂×D₄	1	4	yes	—
144	C₂×C₈×C₃₁	C₂×D₄	1	2	yes	—
145	C₄⋊C₄×C₃₁	—	2	4	no	—
146	C₄⋊C₄×C₃₁	—	2	2	no	—
147	C₄⋊C₄×C₃₁	—	2	1	no	—
148	C₄⋊C₄×C₃₁	—	2	1	no	—
149	C₄⋊C₄×C₃₁	—	2	1	no	—
150	C₄⋊C₄×C₃₁	—	2	2	no	—
151	C₄⋊C₄×C₃₁	—	2	1	no	—
152	C₄⋊C₄×C₃₁	—	2	1	no	—
153	C₄⋊C₄×C₃₁	—	2	1	no	—
154	C₄⋊C₄×C₃₁	—	2	1	no	—
155	C₄⋊C₄×C₃₁	—	2	1	no	—
156	C₄⋊C₄×C₃₁	—	4	2	no	—
157	C₄⋊C₄×C₃₁	—	8	4	no	—
158	C₂²⋊C₄×C₃₁	—	2	4	no	—
159	C₂²⋊C₄×C₃₁	—	4	2	no	—
160	C₂²⋊C₄×C₃₁	—	4	2	no	—
161	C₂²⋊C₄×C₃₁	—	8	4	no	—
162	C₂²⋊C₄×C₃₁	—	8	4	no	—
163	C₄²×C₃₁	—	2	2	no	—
164	C₄²×C₃₁	—	2	2	no	—
165	C₄²×C₃₁	—	4	2	no	—
166	C₄²×C₃₁	—	4	2	no	—
167	#23735	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	4	no	—
171	C₂³×C₃₁	—	4	2	no	—
172	C₂³×C₃₁	—	4	2	no	—
173	C₂³×C₃₁	—	4	4	no	—
174	C₂³×C₃₁	—	4	4	no	—
175	C₂³×C₃₁	—	4	8	no	—
176	C₂³×C₃₁	—	8	8	no	—
177	C₂³×C₃₁	—	8	8	no	—
178	C₂³×C₃₁	—	8	4	no	—
179	C₂³×C₃₁	—	8	4	no	—
180	C₂³×C₃₁	C₂²⋊Q₈	1	2	yes	—
181	C₂³×C₃₁	#123	1	2	yes	—
182	C₂³×C₃₁	#127	1	2	yes	—
183	C₂³×C₃₁	C₂²×D₄	1	1	yes	characteristic
184	D₄×C₃₁	—	2	4	no	—
185	D₄×C₃₁	—	2	4	no	—
186	D₄×C₃₁	—	4	2	no	—
187	D₄×C₃₁	—	4	2	no	—
188	D₄×C₃₁	—	4	8	no	—
189	C₂×C₄×C₃₁	—	2	4	no	—
190	C₂×C₄×C₃₁	—	2	4	no	—
191	C₂×C₄×C₃₁	—	2	4	no	—
192	C₂×C₄×C₃₁	—	4	1	no	—
193	C₂×C₄×C₃₁	—	4	1	no	—
194	C₂×C₄×C₃₁	—	4	1	no	—
195	C₂×C₄×C₃₁	—	4	1	no	—
196	C₂×C₄×C₃₁	—	4	2	no	—
197	C₂×C₄×C₃₁	—	4	2	no	—
198	C₂×C₄×C₃₁	—	4	2	no	—
199	C₂×C₄×C₃₁	—	4	4	no	—
200	C₂×C₄×C₃₁	—	4	2	no	—
201	C₂×C₄×C₃₁	—	4	2	no	—
202	C₂×C₄×C₃₁	—	8	4	no	—
203	C₂×C₄×C₃₁	C₂²⋊Q₈	1	4	yes	—
204	C₂×C₄×C₃₁	C₂²⋊Q₈	1	4	yes	—
205	C₂×C₄×C₃₁	C₂²⋊Q₈	1	2	yes	—
206	C₂×C₄×C₃₁	#123	1	4	yes	—
207	C₂×C₄×C₃₁	#123	1	4	yes	—
208	C₂×C₄×C₃₁	#123	1	2	yes	—
209	C₂×C₄×C₃₁	#127	1	2	yes	—
210	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
211	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
212	C₂×C₄×C₃₁	C₂²×Q₈	1	1	yes	characteristic
213	C₂×C₄×C₃₁	C₂×#41	1	1	yes	characteristic
214	C₂×C₄×C₃₁	C₂×#41	1	1	yes	characteristic
215	C₂×C₄×C₃₁	C₂×#41	1	1	yes	characteristic
216	C₈×C₃₁	—	2	2	no	—
217	C₈×C₃₁	—	2	2	no	—
218	C₈×C₃₁	—	2	4	no	—
219	C₂×#428	C₂×C₃₁	1	1	yes	characteristic, complemented
220	C₂×#427	C₂×C₃₁	1	1	yes	characteristic, complemented
221	C₂×#418	C₂×C₃₁	1	1	yes	characteristic
222	C₂²×#102	C₂×C₃₁	1	1	yes	characteristic
223	#2317	C₂×C₃₁	1	1	yes	characteristic, complemented
224	#2309	C₂×C₃₁	1	1	yes	characteristic
225	#2094	C₂×C₃₁	1	2	yes	complemented
226	#2093	C₂×C₃₁	1	2	yes	complemented
227	#2092	C₂×C₃₁	1	2	yes	complemented
228	#2092	C₂×C₃₁	1	2	yes	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	—	4	8	no	—
235	C₂²×C₃₁	—	4	4	no	—
236	C₂²×C₃₁	—	8	16	no	—
237	C₂²×C₃₁	—	8	4	no	—
238	C₂²×C₃₁	—	8	2	no	—
239	C₂²×C₃₁	—	8	2	no	—
240	C₂²×C₃₁	#399	1	4	yes	—
241	C₂²×C₃₁	C₂×C₂²⋊Q₈	1	1	yes	characteristic
242	C₂²×C₃₁	C₂×#123	1	1	yes	characteristic
243	C₂²×C₃₁	C₂×#127	1	1	yes	characteristic
244	C₂²×C₃₁	#576	1	2	yes	—
245	C₂²×C₃₁	#577	1	2	yes	—
246	C₄×C₃₁	—	4	4	no	—
247	C₄×C₃₁	—	8	1	no	—
248	C₄×C₃₁	—	8	1	no	—
249	C₄×C₃₁	—	8	1	no	—
250	C₄×C₃₁	—	8	1	no	—
251	C₄×C₃₁	—	8	2	no	—
252	C₄×C₃₁	#399	1	4	yes	—
253	C₄×C₃₁	C₂×C₂²⋊Q₈	1	1	yes	characteristic
254	C₄×C₃₁	C₂×C₂²⋊Q₈	1	1	yes	characteristic
255	C₄×C₃₁	C₂×#123	1	1	yes	characteristic
256	C₄×C₃₁	C₂×#123	1	1	yes	characteristic
257	C₂³×D₄	C₂²×C₃₁	1	1	yes	characteristic
258	C₂²×M₁₆	C₂²×C₃₁	1	1	yes	characteristic
259	C₂³×C₈	C₂²×C₃₁	1	1	yes	characteristic
260	C₂×#117	C₂²×C₃₁	1	1	yes	characteristic
261	C₂×#117	C₂²×C₃₁	1	1	yes	characteristic
262	C₂×#117	C₂²×C₃₁	1	1	yes	characteristic
263	C₂²×C₄⋊C₄	C₂²×C₃₁	1	1	yes	characteristic
264	#428	—	2	4	no	—
265	#427	—	2	4	no	—
266	C₂×#107	C₂²×C₃₁	1	1	yes	characteristic, complemented
267	C₂×#107	C₂²×C₃₁	1	2	yes	complemented
268	C₂×#107	C₂²×C₃₁	1	1	yes	characteristic, complemented
269	C₂×#106	C₂²×C₃₁	1	1	yes	characteristic, complemented
270	C₂×#106	C₂²×C₃₁	1	2	yes	complemented
271	C₂×#106	C₂²×C₃₁	1	1	yes	characteristic, complemented
272	#418	—	2	4	no	—
273	C₂×#102	—	2	4	no	—
274	C₂×#102	C₂²×C₃₁	1	2	yes	—
275	C₂×#102	C₂²×C₃₁	1	2	yes	—
276	C₂×#102	C₂²×C₃₁	1	2	yes	—
277	C₂×#102	C₂²×C₃₁	1	2	yes	—
278	#390	—	2	2	no	—
279	#390	C₂²×C₃₁	1	2	yes	—
280	#390	C₂²×C₃₁	1	2	yes	—
281	#389	—	2	2	no	—
282	#385	—	2	2	no	—
283	#384	C₂²×C₃₁	1	2	yes	complemented
284	#382	C₂²×C₃₁	1	2	yes	complemented
285	#379	—	2	2	no	—
286	#336	C₂²×C₃₁	1	2	yes	complemented
287	#336	C₂²×C₃₁	1	2	yes	complemented
288	C₂×C₃₁	—	2	4	no	—
289	C₂×C₃₁	—	8	4	no	—
290	C₂×C₃₁	C₂×#399	1	1	yes	characteristic
291	C₂×C₃₁	#3081	1	2	yes	—
292	C₂×C₃₁	#3096	1	2	yes	—
293	C₂×C₃₁	#3154	1	1	yes	characteristic
294	C₂×C₃₁	#3155	1	1	yes	characteristic
295	C₂⁵	—	2	1	no	—
296	C₂²×D₄	—	2	2	no	—
297	C₂²×D₄	—	2	2	no	—
298	C₂²×D₄	—	2	1	no	—
299	C₂²×D₄	—	2	2	no	—
300	C₂²×D₄	—	2	2	no	—
301	C₂²×D₄	—	2	2	no	—
302	C₂²×D₄	Q₈×C₃₁	1	4	yes	—
303	C₂²×D₄	C₂³×C₃₁	1	2	yes	—
304	C₂³×C₄	—	2	1	no	—
305	C₂³×C₄	C₂³×C₃₁	1	1	yes	characteristic
306	C₂×M₁₆	—	2	2	no	—
307	C₂×M₁₆	—	2	2	no	—
308	C₂×M₁₆	D₄×C₃₁	1	4	yes	—
309	C₂²×C₈	—	2	2	no	—
310	C₂²×C₈	—	2	2	no	—
311	C₂²×C₈	D₄×C₃₁	1	4	yes	—
312	C₂²×C₈	C₂³×C₃₁	1	1	yes	characteristic
313	C₂²×C₈	C₂³×C₃₁	1	1	yes	characteristic
314	C₂²×C₈	C₂³×C₃₁	1	2	yes	—
315	#117	—	2	2	no	—
316	#117	—	2	2	no	—
317	#117	—	2	2	no	—
318	#117	—	2	2	no	—
319	#117	—	2	2	no	—
320	#117	—	2	2	no	—
321	C₂×C₄⋊C₄	—	2	2	no	—
322	C₂×C₄⋊C₄	—	2	2	no	—
323	C₂×C₄⋊C₄	—	2	1	no	—
324	C₂×C₄⋊C₄	—	2	4	no	—
325	C₂×C₄⋊C₄	D₄×C₃₁	1	2	yes	—
326	C₂×C₄⋊C₄	C₂³×C₃₁	1	2	yes	—
327	C₂×C₄⋊C₄	C₂³×C₃₁	1	1	yes	characteristic
328	C₂×C₄⋊C₄	C₂³×C₃₁	1	1	yes	characteristic
329	C₂×C₄⋊C₄	C₂³×C₃₁	1	1	yes	characteristic
330	C₂×C₄⋊C₄	C₂³×C₃₁	1	1	yes	characteristic
331	C₂×C₂²⋊C₄	—	2	1	no	—
332	C₂×C₂²⋊C₄	—	2	1	no	—
333	C₂×C₂²⋊C₄	—	2	4	no	—
334	C₂×C₂²⋊C₄	—	2	4	no	—
335	C₂×C₂²⋊C₄	D₄×C₃₁	1	2	yes	—
336	C₂×C₄²	—	2	1	no	—
337	C₂×C₄²	—	2	1	no	—
338	C₂×C₄²	C₂³×C₃₁	1	2	yes	—
339	#107	—	2	2	no	—
340	#107	—	2	2	no	—
341	#107	—	2	4	no	—
342	#106	—	2	2	no	—
343	#106	—	2	2	no	—
344	#106	—	2	4	no	—
345	#102	—	2	4	no	—
346	#102	—	2	4	no	—
347	#102	—	2	4	no	—
348	#102	—	2	4	no	—
349	#95	—	2	4	no	—
350	C₃₁	#23735	1	1	yes	characteristic, direct product factor, Sylow
351	C₂⁴	—	2	2	no	—
352	C₂⁴	—	2	4	no	—
353	C₂⁴	—	4	2	no	—
354	C₂⁴	—	4	2	no	—
355	C₂⁴	—	8	2	no	—
356	C₂⁴	—	8	2	no	—
357	C₂⁴	C₂×D₄×C₃₁	1	1	yes	characteristic
358	C₂×D₄	—	2	8	no	—
359	C₂×D₄	—	2	8	no	—
360	C₂×D₄	—	2	4	no	—
361	C₂×D₄	—	2	2	no	—
362	C₂×D₄	—	2	2	no	—
363	C₂×D₄	—	2	2	no	—
364	C₂×D₄	—	2	2	no	—
365	C₂×D₄	—	2	2	no	—
366	C₂×D₄	—	4	2	no	—
367	C₂×D₄	—	4	4	no	—
368	C₂×D₄	—	4	4	no	—
369	C₂×D₄	—	4	2	no	—
370	C₂×D₄	C₂×Q₈×C₃₁	1	4	yes	—
371	C₂²×C₄	—	2	1	no	—
372	C₂²×C₄	—	2	2	no	—
373	C₂²×C₄	—	2	1	no	—
374	C₂²×C₄	—	2	1	no	—
375	C₂²×C₄	—	2	1	no	—
376	C₂²×C₄	—	4	2	no	—
377	C₂²×C₄	—	8	2	no	—
378	C₂²×C₄	C₂×D₄×C₃₁	1	4	yes	—
379	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
380	C₂²×C₄	C₂×D₄×C₃₁	1	2	yes	—
381	C₂²×C₄	C₂×D₄×C₃₁	1	2	yes	—
382	C₂²×C₄	C₂×Q₈×C₃₁	1	2	yes	—
383	C₂²×C₄	#41×C₃₁	1	2	yes	—
384	C₂²×C₄	#41×C₃₁	1	2	yes	—
385	C₂²×C₄	#41×C₃₁	1	2	yes	—
386	C₂²×C₄	C₂⁴×C₃₁	1	1	yes	characteristic
387	M₁₆	—	2	8	no	—
388	C₂×C₈	—	2	8	no	—
389	C₂×C₈	—	2	1	no	—
390	C₂×C₈	—	2	1	no	—
391	C₂×C₈	—	2	1	no	—
392	C₂×C₈	—	2	1	no	—
393	C₂×C₈	—	2	2	no	—
394	C₂×C₈	—	2	2	no	—
395	C₂×C₈	C₂×D₄×C₃₁	1	2	yes	—
396	C₂×C₈	C₂×D₄×C₃₁	1	4	yes	—
397	C₂×C₈	C₂×D₄×C₃₁	1	2	yes	—
398	C₄⋊C₄	—	2	4	no	—
399	C₄⋊C₄	—	2	2	no	—
400	C₄⋊C₄	—	2	1	no	—
401	C₄⋊C₄	—	2	1	no	—
402	C₄⋊C₄	—	2	1	no	—
403	C₄⋊C₄	—	2	2	no	—
404	C₄⋊C₄	—	2	1	no	—
405	C₄⋊C₄	—	2	1	no	—
406	C₄⋊C₄	—	2	1	no	—
407	C₄⋊C₄	—	2	1	no	—
408	C₄⋊C₄	—	2	1	no	—
409	C₄⋊C₄	—	4	2	no	—
410	C₄⋊C₄	—	8	4	no	—
411	C₂²⋊C₄	—	2	4	no	—
412	C₂²⋊C₄	—	4	2	no	—
413	C₂²⋊C₄	—	4	2	no	—
414	C₂²⋊C₄	—	8	4	no	—
415	C₂²⋊C₄	—	8	4	no	—
416	C₄²	—	2	2	no	—
417	C₄²	—	2	2	no	—
418	C₄²	—	4	2	no	—
419	C₄²	—	4	2	no	—
420	C₂³	—	2	2	no	—
421	C₂³	—	2	2	no	—
422	C₂³	—	2	4	no	—
423	C₂³	—	4	8	no	—
424	C₂³	—	4	2	no	—
425	C₂³	—	4	2	no	—
426	C₂³	—	4	4	no	—
427	C₂³	—	4	4	no	—
428	C₂³	—	8	8	no	—
429	C₂³	—	8	8	no	—
430	C₂³	—	8	4	no	—
431	C₂³	—	8	4	no	—
432	C₂³	C₃₁×C₂²⋊Q₈	1	2	yes	—
433	C₂³	C₃₁×#123	1	2	yes	—
434	C₂³	C₃₁×#127	1	2	yes	—
435	C₂³	C₂²×D₄×C₃₁	1	1	yes	characteristic
436	D₄	—	2	4	no	—
437	D₄	—	2	4	no	—
438	D₄	—	4	8	no	—
439	D₄	—	4	2	no	—
440	D₄	—	4	2	no	—
441	C₂×C₄	—	2	4	no	—
442	C₂×C₄	—	2	4	no	—
443	C₂×C₄	—	2	4	no	—
444	C₂×C₄	—	4	1	no	—
445	C₂×C₄	—	4	1	no	—
446	C₂×C₄	—	4	1	no	—
447	C₂×C₄	—	4	1	no	—
448	C₂×C₄	—	4	2	no	—
449	C₂×C₄	—	4	2	no	—
450	C₂×C₄	—	4	2	no	—
451	C₂×C₄	—	4	4	no	—
452	C₂×C₄	—	4	2	no	—
453	C₂×C₄	—	4	2	no	—
454	C₂×C₄	—	8	4	no	—
455	C₂×C₄	C₃₁×C₂²⋊Q₈	1	2	yes	—
456	C₂×C₄	C₃₁×C₂²⋊Q₈	1	4	yes	—
457	C₂×C₄	C₃₁×C₂²⋊Q₈	1	4	yes	—
458	C₂×C₄	C₃₁×#123	1	2	yes	—
459	C₂×C₄	C₃₁×#123	1	4	yes	—
460	C₂×C₄	C₃₁×#123	1	4	yes	—
461	C₂×C₄	C₃₁×#127	1	2	yes	—
462	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
463	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
464	C₂×C₄	C₂²×Q₈×C₃₁	1	1	yes	characteristic
465	C₂×C₄	C₂×#41×C₃₁	1	1	yes	characteristic
466	C₂×C₄	C₂×#41×C₃₁	1	1	yes	characteristic
467	C₂×C₄	C₂×#41×C₃₁	1	1	yes	characteristic
468	C₈	—	2	2	no	—
469	C₈	—	2	2	no	—
470	C₈	—	2	4	no	—
471	C₂²	—	2	4	no	—
472	C₂²	—	2	4	no	—
473	C₂²	—	2	2	no	—
474	C₂²	—	2	2	no	—
475	C₂²	—	4	8	no	—
476	C₂²	—	4	4	no	—
477	C₂²	—	8	16	no	—
478	C₂²	—	8	4	no	—
479	C₂²	—	8	2	no	—
480	C₂²	—	8	2	no	—
481	C₂²	C₃₁×#399	1	4	yes	—
482	C₂²	C₂×C₃₁×C₂²⋊Q₈	1	1	yes	characteristic
483	C₂²	C₂×C₃₁×#123	1	1	yes	characteristic
484	C₂²	C₂×C₃₁×#127	1	1	yes	characteristic
485	C₂²	C₃₁×#576	1	2	yes	—
486	C₂²	C₃₁×#577	1	2	yes	—
487	C₄	—	4	4	no	—
488	C₄	—	8	1	no	—
489	C₄	—	8	1	no	—
490	C₄	—	8	2	no	—
491	C₄	—	8	1	no	—
492	C₄	—	8	1	no	—
493	C₄	C₃₁×#399	1	4	yes	—
494	C₄	C₂×C₃₁×C₂²⋊Q₈	1	1	yes	characteristic
495	C₄	C₂×C₃₁×C₂²⋊Q₈	1	1	yes	characteristic
496	C₄	C₂×C₃₁×#123	1	1	yes	characteristic
497	C₄	C₂×C₃₁×#123	1	1	yes	characteristic
498	C₂	—	2	4	no	—
499	C₂	—	8	4	no	—
500	C₂	C₂×C₃₁×#399	1	1	yes	characteristic
501	C₂	C₃₁×#3081	1	2	yes	—
502	C₂	C₃₁×#3096	1	2	yes	—
503	C₂	C₃₁×#3154	1	1	yes	characteristic
504	C₂	C₃₁×#3155	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂³×C₃₁	C₂²×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁴×C₃₁
prime radical	C₂³×D₄×C₃₁	C₂²
LCS	G ▷ C₂²×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
1. C₂×C₃₁×#428	C₂	semidirect
2. C₂×C₃₁×#427	C₂	semidirect
5. C₃₁×#2317	C₂	semidirect
7. C₃₁×#2094	C₂	semidirect
8. C₃₁×#2093	C₂	semidirect
9. C₃₁×#2092	C₂	semidirect
10. C₃₁×#2092	C₂	semidirect
11. C₃₁×#1729	C₂	semidirect
21. C₂×C₃₁×#107	C₂²	semidirect
22. C₂×C₃₁×#107	C₂²	semidirect
23. C₂×C₃₁×#107	C₂²	semidirect
24. C₂×C₃₁×#106	C₂²	semidirect
25. C₂×C₃₁×#106	C₂²	semidirect
26. C₂×C₃₁×#106	C₂²	semidirect
38. C₃₁×#384	C₂²	semidirect
39. C₃₁×#382	C₂²	semidirect
41. C₃₁×#336	C₂²	semidirect
42. C₃₁×#336	C₂²	semidirect
167. #23735	C₃₁	direct (N × H)
219. C₂×#428	C₂×C₃₁	semidirect
220. C₂×#427	C₂×C₃₁	semidirect
223. #2317	C₂×C₃₁	semidirect
225. #2094	C₂×C₃₁	semidirect
226. #2093	C₂×C₃₁	semidirect
227. #2092	C₂×C₃₁	semidirect
228. #2092	C₂×C₃₁	semidirect
229. #1729	C₂×C₃₁	semidirect
266. C₂×#107	C₂²×C₃₁	semidirect
267. C₂×#107	C₂²×C₃₁	semidirect
268. C₂×#107	C₂²×C₃₁	semidirect
269. C₂×#106	C₂²×C₃₁	semidirect
270. C₂×#106	C₂²×C₃₁	semidirect
271. C₂×#106	C₂²×C₃₁	semidirect
283. #384	C₂²×C₃₁	semidirect
284. #382	C₂²×C₃₁	semidirect
286. #336	C₂²×C₃₁	semidirect
287. #336	C₂²×C₃₁	semidirect
350. C₃₁	#23735	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×#428	2
C₂×C₃₁×#427	2
C₂×C₃₁×#418	2
C₂²×C₃₁×#102	2
C₃₁×#2317	2
C₃₁×#2309	2
C₃₁×#2094	2
C₃₁×#2093	2
C₃₁×#2092	2
C₃₁×#1729	2
#23735	31

Subgroups — C31×#23735 #21480494

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#23735 #21480494