Subgroups — C₃₁×#10791 #21467550

Order 7936.

1250 conjugacy classes of subgroups.

The subgroup lattice has 664 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₃₁×#131	C₂	1	1	yes	maximal, characteristic
2	C₂²×C₃₁×#104	C₂	1	1	yes	maximal, characteristic
3	C₂×C₃₁×#416	C₂	1	1	yes	maximal, characteristic
4	C₂×C₃₁×#341	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×C₃₁×#339	C₂	1	1	yes	maximal, characteristic, complemented
6	C₃₁×#1729	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂×C₃₁×#336	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂²×#41×C₃₁	C₂²	1	1	yes	characteristic
9	C₂×C₃₁×#131	—	2	2	no	—
10	C₂×C₃₁×#131	—	2	2	no	—
11	C₂×C₃₁×#131	—	2	2	no	—
12	C₂×C₃₁×#131	—	2	2	no	—
13	C₂×C₃₁×#131	—	2	2	no	—
14	C₂×C₃₁×#131	C₄	1	4	yes	complemented
15	C₂²×M₁₆×C₃₁	—	2	1	no	—
16	C₂²×M₁₆×C₃₁	C₂²	1	1	yes	characteristic
17	C₂²×M₁₆×C₃₁	C₂²	1	1	yes	characteristic
18	C₂³×C₈×C₃₁	—	2	1	no	—
19	C₂³×C₈×C₃₁	C₂²	1	1	yes	characteristic
20	C₂³×C₈×C₃₁	C₂²	1	1	yes	characteristic
21	C₂×C₃₁×#117	C₂²	1	1	yes	characteristic
22	C₂²×C₄²×C₃₁	C₂²	1	1	yes	characteristic
23	C₂×C₃₁×#104	—	2	2	no	—
24	C₂×C₃₁×#104	—	2	2	no	—
25	C₂×C₃₁×#104	C₄	1	4	yes	complemented
26	C₃₁×#416	—	2	2	no	—
27	C₃₁×#416	—	2	2	no	—
28	C₂×C₃₁×#103	C₄	1	2	yes	complemented
29	C₂×C₃₁×#102	C₄	1	2	yes	complemented
30	C₃₁×#341	—	2	2	no	—
31	C₃₁×#339	—	2	2	no	—
32	C₃₁×#336	—	2	2	no	—
33	C₃₁×#336	—	2	1	no	—
34	C₃₁×#336	—	2	1	no	—
35	C₂×#41×C₃₁	—	2	1	no	—
36	C₂×#41×C₃₁	—	2	1	no	—
37	C₂×#41×C₃₁	—	2	1	no	—
38	C₂×#41×C₃₁	—	2	1	no	—
39	C₂×#41×C₃₁	—	2	1	no	—
40	C₂×#41×C₃₁	—	2	1	no	—
41	C₂×#41×C₃₁	C₂×C₄	1	2	yes	—
42	C₂×#41×C₃₁	C₂×C₄	1	2	yes	—
43	C₂×#41×C₃₁	D₄	1	2	yes	—
44	C₂×#41×C₃₁	D₄	1	2	yes	—
45	C₂×#41×C₃₁	D₄	1	2	yes	—
46	C₂×#41×C₃₁	Q₈	1	2	yes	—
47	C₂²×Q₈×C₃₁	C₂×C₄	1	1	yes	characteristic
48	C₂²×D₄×C₃₁	—	2	1	no	—
49	C₂²×D₄×C₃₁	C₂×C₄	1	1	yes	characteristic
50	C₂³×C₄×C₃₁	—	2	1	no	—
51	C₂³×C₄×C₃₁	—	2	1	no	—
52	C₂³×C₄×C₃₁	C₂³	1	1	yes	characteristic
53	C₃₁×#131	—	2	8	no	—
54	C₃₁×#131	—	2	8	no	—
55	C₃₁×#131	—	2	8	no	—
56	C₃₁×#131	—	2	8	no	—
57	C₂×M₁₆×C₃₁	—	2	4	no	—
58	C₂×M₁₆×C₃₁	—	2	4	no	—
59	C₂×M₁₆×C₃₁	—	2	1	no	—
60	C₂×M₁₆×C₃₁	—	2	1	no	—
61	C₂×M₁₆×C₃₁	—	2	1	no	—
62	C₂×M₁₆×C₃₁	—	2	1	no	—
63	C₂×M₁₆×C₃₁	—	2	4	no	—
64	C₂×M₁₆×C₃₁	—	2	1	no	—
65	C₂×M₁₆×C₃₁	—	2	1	no	—
66	C₂×M₁₆×C₃₁	—	2	1	no	—
67	C₂×M₁₆×C₃₁	—	2	1	no	—
68	C₂×M₁₆×C₃₁	C₂×C₄	1	2	yes	—
69	C₂×M₁₆×C₃₁	C₂×C₄	1	2	yes	—
70	C₂×M₁₆×C₃₁	D₄	1	2	yes	—
71	C₂×M₁₆×C₃₁	D₄	1	2	yes	—
72	C₂²×C₈×C₃₁	—	2	4	no	—
73	C₂²×C₈×C₃₁	—	2	4	no	—
74	C₂²×C₈×C₃₁	—	2	1	no	—
75	C₂²×C₈×C₃₁	—	2	1	no	—
76	C₂²×C₈×C₃₁	—	2	1	no	—
77	C₂²×C₈×C₃₁	—	2	1	no	—
78	C₂²×C₈×C₃₁	—	2	1	no	—
79	C₂²×C₈×C₃₁	—	2	4	no	—
80	C₂²×C₈×C₃₁	—	2	1	no	—
81	C₂²×C₈×C₃₁	—	2	1	no	—
82	C₂²×C₈×C₃₁	—	2	1	no	—
83	C₂²×C₈×C₃₁	—	2	1	no	—
84	C₂²×C₈×C₃₁	—	2	1	no	—
85	C₂²×C₈×C₃₁	—	2	1	no	—
86	C₂²×C₈×C₃₁	—	2	1	no	—
87	C₂²×C₈×C₃₁	C₂×C₄	1	2	yes	—
88	C₂²×C₈×C₃₁	C₂×C₄	1	2	yes	—
89	C₂²×C₈×C₃₁	D₄	1	2	yes	—
90	C₂²×C₈×C₃₁	D₄	1	1	yes	characteristic
91	C₂²×C₈×C₃₁	D₄	1	1	yes	characteristic
92	C₃₁×#117	—	2	1	no	—
93	C₃₁×#117	—	2	1	no	—
94	C₃₁×#117	—	2	1	no	—
95	C₃₁×#117	—	2	1	no	—
96	C₂×C₄⋊C₄×C₃₁	C₂×C₄	1	2	yes	—
97	C₂×C₂²⋊C₄×C₃₁	—	2	1	no	—
98	C₂×C₄²×C₃₁	—	2	1	no	—
99	C₂×C₄²×C₃₁	—	2	1	no	—
100	C₂×C₄²×C₃₁	—	2	1	no	—
101	C₂×C₄²×C₃₁	—	2	1	no	—
102	C₂×C₄²×C₃₁	—	2	1	no	—
103	C₂×C₄²×C₃₁	—	2	1	no	—
104	C₂×C₄²×C₃₁	C₂×C₄	1	2	yes	—
105	C₃₁×#104	—	2	8	no	—
106	C₃₁×#103	—	2	4	no	—
107	C₃₁×#102	—	2	4	no	—
108	C₂⁴×C₃₁	—	2	1	no	—
109	C₂⁴×C₃₁	#41	1	1	yes	characteristic
110	#41×C₃₁	—	2	2	no	—
111	#41×C₃₁	—	2	2	no	—
112	#41×C₃₁	—	2	2	no	—
113	#41×C₃₁	—	2	2	no	—
114	#41×C₃₁	—	2	2	no	—
115	#41×C₃₁	—	2	2	no	—
116	#41×C₃₁	—	2	2	no	—
117	#41×C₃₁	—	2	2	no	—
118	#41×C₃₁	—	2	2	no	—
119	#41×C₃₁	—	2	2	no	—
120	#41×C₃₁	—	2	2	no	—
121	#41×C₃₁	—	2	2	no	—
122	#41×C₃₁	C₂²⋊C₄	1	4	yes	—
123	#41×C₃₁	C₂²⋊C₄	1	4	yes	—
124	#41×C₃₁	C₄⋊C₄	1	4	yes	—
125	#41×C₃₁	C₄⋊C₄	1	4	yes	—
126	C₂×Q₈×C₃₁	—	2	1	no	—
127	C₂×Q₈×C₃₁	—	2	1	no	—
128	C₂×Q₈×C₃₁	—	2	1	no	—
129	C₂×Q₈×C₃₁	C₄²	1	2	yes	—
130	C₂×Q₈×C₃₁	C₂²⋊C₄	1	2	yes	—
131	C₂×Q₈×C₃₁	C₄⋊C₄	1	2	yes	—
132	C₂×D₄×C₃₁	—	2	2	no	—
133	C₂×D₄×C₃₁	—	2	2	no	—
134	C₂×D₄×C₃₁	—	2	2	no	—
135	C₂×D₄×C₃₁	—	2	2	no	—
136	C₂×D₄×C₃₁	—	2	2	no	—
137	C₂×D₄×C₃₁	—	2	2	no	—
138	C₂×D₄×C₃₁	—	2	1	no	—
139	C₂×D₄×C₃₁	—	2	1	no	—
140	C₂×D₄×C₃₁	—	2	1	no	—
141	C₂×D₄×C₃₁	C₄²	1	2	yes	—
142	C₂×D₄×C₃₁	C₂²⋊C₄	1	2	yes	—
143	C₂×D₄×C₃₁	C₄⋊C₄	1	2	yes	—
144	C₂²×C₄×C₃₁	—	2	1	no	—
145	C₂²×C₄×C₃₁	—	2	2	no	—
146	C₂²×C₄×C₃₁	—	2	2	no	—
147	C₂²×C₄×C₃₁	—	2	2	no	—
148	C₂²×C₄×C₃₁	—	2	2	no	—
149	C₂²×C₄×C₃₁	—	2	2	no	—
150	C₂²×C₄×C₃₁	—	2	2	no	—
151	C₂²×C₄×C₃₁	—	2	4	no	—
152	C₂²×C₄×C₃₁	—	2	2	no	—
153	C₂²×C₄×C₃₁	—	2	2	no	—
154	C₂²×C₄×C₃₁	—	4	2	no	—
155	C₂²×C₄×C₃₁	—	4	1	no	—
156	C₂²×C₄×C₃₁	—	4	1	no	—
157	C₂²×C₄×C₃₁	C₂²×C₄	1	1	yes	characteristic
158	C₂²×C₄×C₃₁	C₂²×C₄	1	1	yes	characteristic
159	C₂²×C₄×C₃₁	C₂²×C₄	1	1	yes	characteristic
160	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
161	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
162	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
163	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
164	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
165	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
166	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
167	C₂²×C₄×C₃₁	C₂×Q₈	1	1	yes	characteristic
168	C₂²×C₄×C₃₁	#41	1	1	yes	characteristic
169	C₂²×C₄×C₃₁	#41	1	1	yes	characteristic
170	C₂²×C₄×C₃₁	#41	1	1	yes	characteristic
171	M₁₆×C₃₁	—	2	16	no	—
172	M₁₆×C₃₁	—	2	2	no	—
173	M₁₆×C₃₁	—	2	2	no	—
174	M₁₆×C₃₁	—	2	2	no	—
175	M₁₆×C₃₁	—	2	2	no	—
176	M₁₆×C₃₁	—	2	2	no	—
177	M₁₆×C₃₁	—	4	1	no	—
178	M₁₆×C₃₁	—	4	1	no	—
179	M₁₆×C₃₁	C₂²⋊C₄	1	4	yes	—
180	C₂×C₈×C₃₁	—	2	16	no	—
181	C₂×C₈×C₃₁	—	2	2	no	—
182	C₂×C₈×C₃₁	—	2	2	no	—
183	C₂×C₈×C₃₁	—	2	2	no	—
184	C₂×C₈×C₃₁	—	2	1	no	—
185	C₂×C₈×C₃₁	—	2	1	no	—
186	C₂×C₈×C₃₁	—	2	2	no	—
187	C₂×C₈×C₃₁	—	2	2	no	—
188	C₂×C₈×C₃₁	—	2	1	no	—
189	C₂×C₈×C₃₁	—	2	1	no	—
190	C₂×C₈×C₃₁	—	2	2	no	—
191	C₂×C₈×C₃₁	—	2	2	no	—
192	C₂×C₈×C₃₁	—	2	2	no	—
193	C₂×C₈×C₃₁	—	2	2	no	—
194	C₂×C₈×C₃₁	—	2	2	no	—
195	C₂×C₈×C₃₁	—	2	2	no	—
196	C₂×C₈×C₃₁	—	4	1	no	—
197	C₂×C₈×C₃₁	—	4	1	no	—
198	C₂×C₈×C₃₁	—	4	1	no	—
199	C₂×C₈×C₃₁	—	4	1	no	—
200	C₂×C₈×C₃₁	—	4	1	no	—
201	C₂×C₈×C₃₁	—	4	1	no	—
202	C₂×C₈×C₃₁	C₂²⋊C₄	1	2	yes	—
203	C₂×C₈×C₃₁	C₂²⋊C₄	1	2	yes	—
204	C₄⋊C₄×C₃₁	—	2	2	no	—
205	C₄⋊C₄×C₃₁	—	2	2	no	—
206	C₂²⋊C₄×C₃₁	—	4	1	no	—
207	C₂²⋊C₄×C₃₁	—	4	1	no	—
208	C₄²×C₃₁	—	2	2	no	—
209	C₄²×C₃₁	—	2	2	no	—
210	C₄²×C₃₁	—	4	1	no	—
211	C₄²×C₃₁	—	4	1	no	—
212	C₄²×C₃₁	—	4	1	no	—
213	C₄²×C₃₁	—	4	1	no	—
214	#10791	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
215	C₂³×C₃₁	—	2	2	no	—
216	C₂³×C₃₁	—	2	2	no	—
217	C₂³×C₃₁	—	2	2	no	—
218	C₂³×C₃₁	—	2	2	no	—
219	C₂³×C₃₁	—	2	2	no	—
220	C₂³×C₃₁	—	4	2	no	—
221	C₂³×C₃₁	—	4	2	no	—
222	C₂³×C₃₁	#117	1	1	yes	characteristic
223	C₂³×C₃₁	C₄×D₄	1	1	yes	characteristic
224	C₂³×C₃₁	C₄×D₄	1	1	yes	characteristic
225	C₂³×C₃₁	#121	1	1	yes	characteristic
226	C₂³×C₃₁	C₂²⋊Q₈	1	1	yes	characteristic
227	C₂³×C₃₁	#123	1	1	yes	characteristic
228	C₂³×C₃₁	#123	1	1	yes	characteristic
229	Q₈×C₃₁	—	2	2	no	—
230	Q₈×C₃₁	—	2	2	no	—
231	Q₈×C₃₁	—	2	2	no	—
232	Q₈×C₃₁	#95	1	4	yes	—
233	D₄×C₃₁	—	2	4	no	—
234	D₄×C₃₁	—	2	4	no	—
235	D₄×C₃₁	—	2	4	no	—
236	D₄×C₃₁	—	2	4	no	—
237	D₄×C₃₁	—	2	2	no	—
238	D₄×C₃₁	—	2	2	no	—
239	D₄×C₃₁	—	2	2	no	—
240	D₄×C₃₁	#95	1	4	yes	—
241	C₂×C₄×C₃₁	—	2	2	no	—
242	C₂×C₄×C₃₁	—	2	2	no	—
243	C₂×C₄×C₃₁	—	2	2	no	—
244	C₂×C₄×C₃₁	—	2	4	no	—
245	C₂×C₄×C₃₁	—	2	4	no	—
246	C₂×C₄×C₃₁	—	2	4	no	—
247	C₂×C₄×C₃₁	—	2	4	no	—
248	C₂×C₄×C₃₁	—	2	4	no	—
249	C₂×C₄×C₃₁	—	4	4	no	—
250	C₂×C₄×C₃₁	—	4	4	no	—
251	C₂×C₄×C₃₁	—	4	2	no	—
252	C₂×C₄×C₃₁	—	4	2	no	—
253	C₂×C₄×C₃₁	—	4	2	no	—
254	C₂×C₄×C₃₁	—	4	2	no	—
255	C₂×C₄×C₃₁	—	4	2	no	—
256	C₂×C₄×C₃₁	C₂×C₄²	1	1	yes	characteristic
257	C₂×C₄×C₃₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
258	C₂×C₄×C₃₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
259	C₂×C₄×C₃₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
260	C₂×C₄×C₃₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
261	C₂×C₄×C₃₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
262	C₂×C₄×C₃₁	C₂×C₄⋊C₄	1	1	yes	characteristic
263	C₂×C₄×C₃₁	C₂×C₄⋊C₄	1	1	yes	characteristic
264	C₂×C₄×C₃₁	C₂×C₄⋊C₄	1	1	yes	characteristic
265	C₂×C₄×C₃₁	#117	1	1	yes	characteristic
266	C₂×C₄×C₃₁	C₄×D₄	1	1	yes	characteristic
267	C₂×C₄×C₃₁	C₄×D₄	1	1	yes	characteristic
268	C₂×C₄×C₃₁	C₄×D₄	1	1	yes	characteristic
269	C₂×C₄×C₃₁	C₄×D₄	1	1	yes	characteristic
270	C₂×C₄×C₃₁	C₄×D₄	1	1	yes	characteristic
271	C₂×C₄×C₃₁	C₄×D₄	1	1	yes	characteristic
272	C₂×C₄×C₃₁	#120	1	1	yes	characteristic
273	C₂×C₄×C₃₁	#120	1	1	yes	characteristic
274	C₂×C₄×C₃₁	#120	1	1	yes	characteristic
275	C₂×C₄×C₃₁	#120	1	1	yes	characteristic
276	C₂×C₄×C₃₁	#121	1	1	yes	characteristic
277	C₂×C₄×C₃₁	#121	1	1	yes	characteristic
278	C₂×C₄×C₃₁	#121	1	1	yes	characteristic
279	C₂×C₄×C₃₁	C₂²⋊Q₈	1	1	yes	characteristic
280	C₂×C₄×C₃₁	C₂²⋊Q₈	1	1	yes	characteristic
281	C₂×C₄×C₃₁	C₂²⋊Q₈	1	1	yes	characteristic
282	C₂×C₄×C₃₁	#123	1	1	yes	characteristic
283	C₂×C₄×C₃₁	#123	1	1	yes	characteristic
284	C₈×C₃₁	—	2	2	no	—
285	C₈×C₃₁	—	2	4	no	—
286	C₈×C₃₁	—	2	2	no	—
287	C₈×C₃₁	—	4	2	no	—
288	C₈×C₃₁	—	4	2	no	—
289	C₂²×#131	C₂×C₃₁	1	1	yes	characteristic
290	C₂²×#104	C₂×C₃₁	1	1	yes	characteristic
291	C₂×#416	C₂×C₃₁	1	1	yes	characteristic
292	C₂×#341	C₂×C₃₁	1	1	yes	characteristic, complemented
293	C₂×#339	C₂×C₃₁	1	1	yes	characteristic, complemented
294	#1729	C₂×C₃₁	1	1	yes	characteristic, complemented
295	C₂×#336	C₂×C₃₁	1	1	yes	characteristic, complemented
296	C₂²×C₃₁	—	2	2	no	—
297	C₂²×C₃₁	—	2	2	no	—
298	C₂²×C₃₁	—	2	2	no	—
299	C₂²×C₃₁	—	2	2	no	—
300	C₂²×C₃₁	—	2	1	no	—
301	C₂²×C₃₁	—	2	1	no	—
302	C₂²×C₃₁	—	2	1	no	—
303	C₂²×C₃₁	—	2	1	no	—
304	C₂²×C₃₁	—	2	4	no	—
305	C₂²×C₃₁	—	4	4	no	—
306	C₂²×C₃₁	—	4	4	no	—
307	C₂²×C₃₁	—	4	2	no	—
308	C₂²×C₃₁	—	4	2	no	—
309	C₂²×C₃₁	C₄×C₂²⋊C₄	1	1	yes	characteristic
310	C₂²×C₃₁	#380	1	1	yes	characteristic
311	C₂²×C₃₁	#381	1	1	yes	characteristic
312	C₂²×C₃₁	#385	1	1	yes	characteristic
313	C₂²×C₃₁	#385	1	1	yes	characteristic
314	C₂²×C₃₁	#386	1	1	yes	characteristic
315	C₂²×C₃₁	#386	1	1	yes	characteristic
316	C₂²×C₃₁	#443	1	2	yes	—
317	C₂²×C₃₁	#444	1	2	yes	—
318	C₄×C₃₁	—	2	4	no	—
319	C₄×C₃₁	—	4	4	no	—
320	C₄×C₃₁	—	8	2	no	—
321	C₄×C₃₁	C₂×#95	1	1	yes	characteristic
322	C₄×C₃₁	C₄×C₂²⋊C₄	1	1	yes	characteristic
323	C₄×C₃₁	#379	1	1	yes	characteristic
324	C₄×C₃₁	#380	1	1	yes	characteristic
325	C₄×C₃₁	#385	1	1	yes	characteristic
326	C₄×C₃₁	#385	1	1	yes	characteristic
327	C₄×C₃₁	#386	1	1	yes	characteristic
328	C₄×C₃₁	#386	1	1	yes	characteristic
329	C₂²×#41	C₂²×C₃₁	1	1	yes	characteristic
330	C₂×#131	—	2	2	no	—
331	C₂×#131	—	2	2	no	—
332	C₂×#131	—	2	2	no	—
333	C₂×#131	—	2	2	no	—
334	C₂×#131	—	2	2	no	—
335	C₂×#131	C₄×C₃₁	1	4	yes	complemented
336	C₂²×M₁₆	—	2	1	no	—
337	C₂²×M₁₆	C₂²×C₃₁	1	1	yes	characteristic
338	C₂²×M₁₆	C₂²×C₃₁	1	1	yes	characteristic
339	C₂³×C₈	—	2	1	no	—
340	C₂³×C₈	C₂²×C₃₁	1	1	yes	characteristic
341	C₂³×C₈	C₂²×C₃₁	1	1	yes	characteristic
342	C₂×#117	C₂²×C₃₁	1	1	yes	characteristic
343	C₂²×C₄²	C₂²×C₃₁	1	1	yes	characteristic
344	C₂×#104	—	2	2	no	—
345	C₂×#104	—	2	2	no	—
346	C₂×#104	C₄×C₃₁	1	4	yes	complemented
347	#416	—	2	2	no	—
348	#416	—	2	2	no	—
349	C₂×#103	C₄×C₃₁	1	2	yes	complemented
350	C₂×#102	C₄×C₃₁	1	2	yes	complemented
351	#341	—	2	2	no	—
352	#339	—	2	2	no	—
353	#336	—	2	2	no	—
354	#336	—	2	1	no	—
355	#336	—	2	1	no	—
356	C₂×C₃₁	—	2	1	no	—
357	C₂×C₃₁	—	2	1	no	—
358	C₂×C₃₁	—	2	1	no	—
359	C₂×C₃₁	—	2	1	no	—
360	C₂×C₃₁	—	4	4	no	—
361	C₂×C₃₁	C₂²⋊#95	1	1	yes	characteristic
362	C₂×C₃₁	#1764	1	1	yes	characteristic
363	C₂×C₃₁	#1765	1	1	yes	characteristic
364	C₂×C₃₁	#1866	1	2	yes	—
365	C₂×C₃₁	#1876	1	2	yes	—
366	C₂×#41	—	2	1	no	—
367	C₂×#41	—	2	1	no	—
368	C₂×#41	—	2	1	no	—
369	C₂×#41	—	2	1	no	—
370	C₂×#41	—	2	1	no	—
371	C₂×#41	—	2	1	no	—
372	C₂×#41	C₂×C₄×C₃₁	1	2	yes	—
373	C₂×#41	C₂×C₄×C₃₁	1	2	yes	—
374	C₂×#41	D₄×C₃₁	1	2	yes	—
375	C₂×#41	D₄×C₃₁	1	2	yes	—
376	C₂×#41	D₄×C₃₁	1	2	yes	—
377	C₂×#41	Q₈×C₃₁	1	2	yes	—
378	C₂²×Q₈	C₂×C₄×C₃₁	1	1	yes	characteristic
379	C₂²×D₄	—	2	1	no	—
380	C₂²×D₄	C₂×C₄×C₃₁	1	1	yes	characteristic
381	C₂³×C₄	—	2	1	no	—
382	C₂³×C₄	—	2	1	no	—
383	C₂³×C₄	C₂³×C₃₁	1	1	yes	characteristic
384	#131	—	2	8	no	—
385	#131	—	2	8	no	—
386	#131	—	2	8	no	—
387	#131	—	2	8	no	—
388	C₂×M₁₆	—	2	4	no	—
389	C₂×M₁₆	—	2	4	no	—
390	C₂×M₁₆	—	2	1	no	—
391	C₂×M₁₆	—	2	1	no	—
392	C₂×M₁₆	—	2	1	no	—
393	C₂×M₁₆	—	2	1	no	—
394	C₂×M₁₆	—	2	4	no	—
395	C₂×M₁₆	—	2	1	no	—
396	C₂×M₁₆	—	2	1	no	—
397	C₂×M₁₆	—	2	1	no	—
398	C₂×M₁₆	—	2	1	no	—
399	C₂×M₁₆	C₂×C₄×C₃₁	1	2	yes	—
400	C₂×M₁₆	C₂×C₄×C₃₁	1	2	yes	—
401	C₂×M₁₆	D₄×C₃₁	1	2	yes	—
402	C₂×M₁₆	D₄×C₃₁	1	2	yes	—
403	C₂²×C₈	—	2	4	no	—
404	C₂²×C₈	—	2	4	no	—
405	C₂²×C₈	—	2	1	no	—
406	C₂²×C₈	—	2	1	no	—
407	C₂²×C₈	—	2	1	no	—
408	C₂²×C₈	—	2	1	no	—
409	C₂²×C₈	—	2	1	no	—
410	C₂²×C₈	—	2	4	no	—
411	C₂²×C₈	—	2	1	no	—
412	C₂²×C₈	—	2	1	no	—
413	C₂²×C₈	—	2	1	no	—
414	C₂²×C₈	—	2	1	no	—
415	C₂²×C₈	—	2	1	no	—
416	C₂²×C₈	—	2	1	no	—
417	C₂²×C₈	—	2	1	no	—
418	C₂²×C₈	C₂×C₄×C₃₁	1	2	yes	—
419	C₂²×C₈	C₂×C₄×C₃₁	1	2	yes	—
420	C₂²×C₈	D₄×C₃₁	1	1	yes	characteristic
421	C₂²×C₈	D₄×C₃₁	1	1	yes	characteristic
422	C₂²×C₈	D₄×C₃₁	1	2	yes	—
423	#117	—	2	1	no	—
424	#117	—	2	1	no	—
425	#117	—	2	1	no	—
426	#117	—	2	1	no	—
427	C₂×C₄⋊C₄	C₂×C₄×C₃₁	1	2	yes	—
428	C₂×C₂²⋊C₄	—	2	1	no	—
429	C₂×C₄²	—	2	1	no	—
430	C₂×C₄²	—	2	1	no	—
431	C₂×C₄²	—	2	1	no	—
432	C₂×C₄²	—	2	1	no	—
433	C₂×C₄²	—	2	1	no	—
434	C₂×C₄²	—	2	1	no	—
435	C₂×C₄²	C₂×C₄×C₃₁	1	2	yes	—
436	#104	—	2	8	no	—
437	#103	—	2	4	no	—
438	#102	—	2	4	no	—
439	C₃₁	#10791	1	1	yes	characteristic, direct product factor, Sylow
440	C₂⁴	—	2	1	no	—
441	C₂⁴	#41×C₃₁	1	1	yes	characteristic
442	#41	—	2	2	no	—
443	#41	—	2	2	no	—
444	#41	—	2	2	no	—
445	#41	—	2	2	no	—
446	#41	—	2	2	no	—
447	#41	—	2	2	no	—
448	#41	—	2	2	no	—
449	#41	—	2	2	no	—
450	#41	—	2	2	no	—
451	#41	—	2	2	no	—
452	#41	—	2	2	no	—
453	#41	—	2	2	no	—
454	#41	C₂²⋊C₄×C₃₁	1	4	yes	—
455	#41	C₂²⋊C₄×C₃₁	1	4	yes	—
456	#41	C₄⋊C₄×C₃₁	1	4	yes	—
457	#41	C₄⋊C₄×C₃₁	1	4	yes	—
458	C₂×Q₈	—	2	1	no	—
459	C₂×Q₈	—	2	1	no	—
460	C₂×Q₈	—	2	1	no	—
461	C₂×Q₈	C₄²×C₃₁	1	2	yes	—
462	C₂×Q₈	C₂²⋊C₄×C₃₁	1	2	yes	—
463	C₂×Q₈	C₄⋊C₄×C₃₁	1	2	yes	—
464	C₂×D₄	—	2	2	no	—
465	C₂×D₄	—	2	2	no	—
466	C₂×D₄	—	2	2	no	—
467	C₂×D₄	—	2	2	no	—
468	C₂×D₄	—	2	2	no	—
469	C₂×D₄	—	2	2	no	—
470	C₂×D₄	—	2	1	no	—
471	C₂×D₄	—	2	1	no	—
472	C₂×D₄	—	2	1	no	—
473	C₂×D₄	C₄²×C₃₁	1	2	yes	—
474	C₂×D₄	C₂²⋊C₄×C₃₁	1	2	yes	—
475	C₂×D₄	C₄⋊C₄×C₃₁	1	2	yes	—
476	C₂²×C₄	—	2	1	no	—
477	C₂²×C₄	—	2	2	no	—
478	C₂²×C₄	—	2	2	no	—
479	C₂²×C₄	—	2	2	no	—
480	C₂²×C₄	—	2	2	no	—
481	C₂²×C₄	—	2	2	no	—
482	C₂²×C₄	—	2	2	no	—
483	C₂²×C₄	—	2	4	no	—
484	C₂²×C₄	—	2	2	no	—
485	C₂²×C₄	—	2	2	no	—
486	C₂²×C₄	—	4	2	no	—
487	C₂²×C₄	—	4	1	no	—
488	C₂²×C₄	—	4	1	no	—
489	C₂²×C₄	C₂²×C₄×C₃₁	1	1	yes	characteristic
490	C₂²×C₄	C₂²×C₄×C₃₁	1	1	yes	characteristic
491	C₂²×C₄	C₂²×C₄×C₃₁	1	1	yes	characteristic
492	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
493	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
494	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
495	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
496	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
497	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
498	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
499	C₂²×C₄	C₂×Q₈×C₃₁	1	1	yes	characteristic
500	C₂²×C₄	#41×C₃₁	1	1	yes	characteristic
501	C₂²×C₄	#41×C₃₁	1	1	yes	characteristic
502	C₂²×C₄	#41×C₃₁	1	1	yes	characteristic
503	M₁₆	—	2	16	no	—
504	M₁₆	—	2	2	no	—
505	M₁₆	—	2	2	no	—
506	M₁₆	—	2	2	no	—
507	M₁₆	—	2	2	no	—
508	M₁₆	—	2	2	no	—
509	M₁₆	—	4	1	no	—
510	M₁₆	—	4	1	no	—
511	M₁₆	C₂²⋊C₄×C₃₁	1	4	yes	—
512	C₂×C₈	—	2	16	no	—
513	C₂×C₈	—	2	2	no	—
514	C₂×C₈	—	2	2	no	—
515	C₂×C₈	—	2	2	no	—
516	C₂×C₈	—	2	1	no	—
517	C₂×C₈	—	2	1	no	—
518	C₂×C₈	—	2	2	no	—
519	C₂×C₈	—	2	2	no	—
520	C₂×C₈	—	2	1	no	—
521	C₂×C₈	—	2	1	no	—
522	C₂×C₈	—	2	2	no	—
523	C₂×C₈	—	2	2	no	—
524	C₂×C₈	—	2	2	no	—
525	C₂×C₈	—	2	2	no	—
526	C₂×C₈	—	2	2	no	—
527	C₂×C₈	—	2	2	no	—
528	C₂×C₈	—	4	1	no	—
529	C₂×C₈	—	4	1	no	—
530	C₂×C₈	—	4	1	no	—
531	C₂×C₈	—	4	1	no	—
532	C₂×C₈	—	4	1	no	—
533	C₂×C₈	—	4	1	no	—
534	C₂×C₈	C₂²⋊C₄×C₃₁	1	2	yes	—
535	C₂×C₈	C₂²⋊C₄×C₃₁	1	2	yes	—
536	C₄⋊C₄	—	2	2	no	—
537	C₄⋊C₄	—	2	2	no	—
538	C₂²⋊C₄	—	4	1	no	—
539	C₂²⋊C₄	—	4	1	no	—
540	C₄²	—	2	2	no	—
541	C₄²	—	2	2	no	—
542	C₄²	—	4	1	no	—
543	C₄²	—	4	1	no	—
544	C₄²	—	4	1	no	—
545	C₄²	—	4	1	no	—
546	C₂³	—	2	2	no	—
547	C₂³	—	2	2	no	—
548	C₂³	—	2	2	no	—
549	C₂³	—	2	2	no	—
550	C₂³	—	2	2	no	—
551	C₂³	—	4	2	no	—
552	C₂³	—	4	2	no	—
553	C₂³	C₃₁×#117	1	1	yes	characteristic
554	C₂³	C₄×D₄×C₃₁	1	1	yes	characteristic
555	C₂³	C₄×D₄×C₃₁	1	1	yes	characteristic
556	C₂³	C₃₁×#121	1	1	yes	characteristic
557	C₂³	C₃₁×C₂²⋊Q₈	1	1	yes	characteristic
558	C₂³	C₃₁×#123	1	1	yes	characteristic
559	C₂³	C₃₁×#123	1	1	yes	characteristic
560	Q₈	—	2	2	no	—
561	Q₈	—	2	2	no	—
562	Q₈	—	2	2	no	—
563	Q₈	C₃₁×#95	1	4	yes	—
564	D₄	—	2	4	no	—
565	D₄	—	2	4	no	—
566	D₄	—	2	4	no	—
567	D₄	—	2	4	no	—
568	D₄	—	2	2	no	—
569	D₄	—	2	2	no	—
570	D₄	—	2	2	no	—
571	D₄	C₃₁×#95	1	4	yes	—
572	C₂×C₄	—	2	2	no	—
573	C₂×C₄	—	2	2	no	—
574	C₂×C₄	—	2	2	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	—	4	4	no	—
581	C₂×C₄	—	4	4	no	—
582	C₂×C₄	—	4	2	no	—
583	C₂×C₄	—	4	2	no	—
584	C₂×C₄	—	4	2	no	—
585	C₂×C₄	—	4	2	no	—
586	C₂×C₄	—	4	2	no	—
587	C₂×C₄	C₂×C₄²×C₃₁	1	1	yes	characteristic
588	C₂×C₄	C₂×C₂²⋊C₄×C₃₁	1	1	yes	characteristic
589	C₂×C₄	C₂×C₂²⋊C₄×C₃₁	1	1	yes	characteristic
590	C₂×C₄	C₂×C₂²⋊C₄×C₃₁	1	1	yes	characteristic
591	C₂×C₄	C₂×C₂²⋊C₄×C₃₁	1	1	yes	characteristic
592	C₂×C₄	C₂×C₂²⋊C₄×C₃₁	1	1	yes	characteristic
593	C₂×C₄	C₂×C₄⋊C₄×C₃₁	1	1	yes	characteristic
594	C₂×C₄	C₂×C₄⋊C₄×C₃₁	1	1	yes	characteristic
595	C₂×C₄	C₂×C₄⋊C₄×C₃₁	1	1	yes	characteristic
596	C₂×C₄	C₃₁×#117	1	1	yes	characteristic
597	C₂×C₄	C₄×D₄×C₃₁	1	1	yes	characteristic
598	C₂×C₄	C₄×D₄×C₃₁	1	1	yes	characteristic
599	C₂×C₄	C₄×D₄×C₃₁	1	1	yes	characteristic
600	C₂×C₄	C₄×D₄×C₃₁	1	1	yes	characteristic
601	C₂×C₄	C₄×D₄×C₃₁	1	1	yes	characteristic
602	C₂×C₄	C₄×D₄×C₃₁	1	1	yes	characteristic
603	C₂×C₄	C₃₁×#120	1	1	yes	characteristic
604	C₂×C₄	C₃₁×#120	1	1	yes	characteristic
605	C₂×C₄	C₃₁×#120	1	1	yes	characteristic
606	C₂×C₄	C₃₁×#120	1	1	yes	characteristic
607	C₂×C₄	C₃₁×#121	1	1	yes	characteristic
608	C₂×C₄	C₃₁×#121	1	1	yes	characteristic
609	C₂×C₄	C₃₁×#121	1	1	yes	characteristic
610	C₂×C₄	C₃₁×C₂²⋊Q₈	1	1	yes	characteristic
611	C₂×C₄	C₃₁×C₂²⋊Q₈	1	1	yes	characteristic
612	C₂×C₄	C₃₁×C₂²⋊Q₈	1	1	yes	characteristic
613	C₂×C₄	C₃₁×#123	1	1	yes	characteristic
614	C₂×C₄	C₃₁×#123	1	1	yes	characteristic
615	C₈	—	2	2	no	—
616	C₈	—	2	4	no	—
617	C₈	—	2	2	no	—
618	C₈	—	4	2	no	—
619	C₈	—	4	2	no	—
620	C₂²	—	2	2	no	—
621	C₂²	—	2	2	no	—
622	C₂²	—	2	2	no	—
623	C₂²	—	2	2	no	—
624	C₂²	—	2	1	no	—
625	C₂²	—	2	1	no	—
626	C₂²	—	2	1	no	—
627	C₂²	—	2	1	no	—
628	C₂²	—	2	4	no	—
629	C₂²	—	4	4	no	—
630	C₂²	—	4	4	no	—
631	C₂²	—	4	2	no	—
632	C₂²	—	4	2	no	—
633	C₂²	C₄×C₂²⋊C₄×C₃₁	1	1	yes	characteristic
634	C₂²	C₃₁×#380	1	1	yes	characteristic
635	C₂²	C₃₁×#381	1	1	yes	characteristic
636	C₂²	C₃₁×#385	1	1	yes	characteristic
637	C₂²	C₃₁×#385	1	1	yes	characteristic
638	C₂²	C₃₁×#386	1	1	yes	characteristic
639	C₂²	C₃₁×#386	1	1	yes	characteristic
640	C₂²	C₃₁×#443	1	2	yes	—
641	C₂²	C₃₁×#444	1	2	yes	—
642	C₄	—	2	4	no	—
643	C₄	—	4	4	no	—
644	C₄	—	8	2	no	—
645	C₄	C₂×C₃₁×#95	1	1	yes	characteristic
646	C₄	C₄×C₂²⋊C₄×C₃₁	1	1	yes	characteristic
647	C₄	C₃₁×#379	1	1	yes	characteristic
648	C₄	C₃₁×#380	1	1	yes	characteristic
649	C₄	C₃₁×#385	1	1	yes	characteristic
650	C₄	C₃₁×#385	1	1	yes	characteristic
651	C₄	C₃₁×#386	1	1	yes	characteristic
652	C₄	C₃₁×#386	1	1	yes	characteristic
653	C₂	—	2	1	no	—
654	C₂	—	2	1	no	—
655	C₂	—	2	1	no	—
656	C₂	—	2	1	no	—
657	C₂	—	4	4	no	—
658	C₂	C₃₁×C₂²⋊#95	1	1	yes	characteristic
659	C₂	C₃₁×#1764	1	1	yes	characteristic
660	C₂	C₃₁×#1765	1	1	yes	characteristic
661	C₂	C₃₁×#1866	1	2	yes	—
662	C₂	C₃₁×#1876	1	2	yes	—

	subgroup	G/H
center	C₂²×C₄×C₃₁	C₂×D₄
commutator	C₂×C₄	C₂×C₄²×C₃₁
Frattini, cosocle	C₂³×C₄	C₂³×C₃₁
socle	C₂³×C₃₁	C₄×D₄
prime radical	C₂²×#41×C₃₁	C₂²
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄×C₃₁ ◁ C₂³×C₈×C₃₁ ◁ G

Complements

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

N	H ≅ G/N	type
4. C₂×C₃₁×#341	C₂	semidirect
5. C₂×C₃₁×#339	C₂	semidirect
6. C₃₁×#1729	C₂	semidirect
7. C₂×C₃₁×#336	C₂	semidirect
14. C₂×C₃₁×#131	C₄	semidirect
25. C₂×C₃₁×#104	C₄	semidirect
28. C₂×C₃₁×#103	C₄	semidirect
29. C₂×C₃₁×#102	C₄	semidirect
214. #10791	C₃₁	direct (N × H)
292. C₂×#341	C₂×C₃₁	semidirect
293. C₂×#339	C₂×C₃₁	semidirect
294. #1729	C₂×C₃₁	semidirect
295. C₂×#336	C₂×C₃₁	semidirect
335. C₂×#131	C₄×C₃₁	semidirect
346. C₂×#104	C₄×C₃₁	semidirect
349. C₂×#103	C₄×C₃₁	semidirect
350. C₂×#102	C₄×C₃₁	semidirect
439. C₃₁	#10791	direct (N × H)

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
C₂²×C₃₁×#131	2
C₂²×C₃₁×#104	2
C₂×C₃₁×#416	2
C₂×C₃₁×#341	2
C₂×C₃₁×#339	2
C₃₁×#1729	2
C₂×C₃₁×#336	2
#10791	31

Subgroups — C31×#10791 #21467550

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#10791 #21467550