Subgroups — C₃₁×#22793 #21479552

Order 7936.

804 conjugacy classes of subgroups.

The subgroup lattice has 672 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃₁×#2290	C₂	1	1	yes	maximal, characteristic
2	C₃₁×#1924	C₂	1	1	yes	maximal, characteristic, complemented
3	C₃₁×#1923	C₂	1	1	yes	maximal, characteristic, complemented
4	C₃₁×#1916	C₂	1	1	yes	maximal, characteristic, complemented
5	C₃₁×#1916	C₂	1	1	yes	maximal, characteristic, complemented
6	C₃₁×#1870	C₂	1	1	yes	maximal, characteristic
7	C₃₁×#1869	C₂	1	1	yes	maximal, characteristic
8	C₃₁×#1840	C₂	1	1	yes	maximal, characteristic, complemented
9	C₃₁×#1817	C₂	1	1	yes	maximal, characteristic, complemented
10	C₃₁×#1798	C₂	1	1	yes	maximal, characteristic, complemented
11	C₃₁×#1753	C₂	1	1	yes	maximal, characteristic, complemented
12	C₈×C₂²⋊C₄×C₃₁	C₂	1	1	yes	maximal, characteristic, complemented
13	C₄×C₃₁×C₂²⋊C₈	C₂	1	1	yes	maximal, characteristic, complemented
14	C₃₁×#1746	C₂	1	1	yes	maximal, characteristic, complemented
15	C₃₁×#1729	C₂	1	1	yes	maximal, characteristic, complemented
16	C₂²×M₁₆×C₃₁	C₂²	1	1	yes	characteristic
17	C₂³×C₈×C₃₁	C₂²	1	1	yes	characteristic
18	C₂×C₄×D₄×C₃₁	C₂²	1	1	yes	characteristic
19	C₂×C₃₁×#117	C₂²	1	1	yes	characteristic
20	C₂²×C₄²×C₃₁	C₂²	1	1	yes	characteristic
21	C₂×C₃₁×C₄⋊C₈	C₂²	1	1	yes	characteristic
22	C₂×C₃₁×C₄⋊C₈	C₂²	1	1	yes	characteristic
23	C₂×C₃₁×C₂²⋊C₈	C₂²	1	1	yes	characteristic
24	C₂×C₃₁×C₂²⋊C₈	C₂²	1	1	yes	characteristic
25	C₂×C₃₁×C₂²⋊C₈	C₂²	1	1	yes	characteristic
26	C₂×C₃₁×C₂²⋊C₈	C₂²	1	1	yes	characteristic
27	C₂×C₃₁×C₂²⋊C₈	C₂²	1	1	yes	characteristic
28	C₂×C₃₁×C₂²⋊C₈	C₂²	1	1	yes	characteristic
29	C₂×C₃₁×#97	C₂²	1	1	yes	characteristic
30	C₂×C₄×C₈×C₃₁	C₂²	1	1	yes	characteristic
31	C₂×C₄×C₈×C₃₁	C₂²	1	1	yes	characteristic
32	C₂×C₄×C₈×C₃₁	C₂²	1	1	yes	characteristic
33	C₃₁×#388	C₄	1	1	yes	characteristic
34	C₃₁×#388	C₄	1	1	yes	characteristic
35	C₃₁×#387	C₄	1	1	yes	characteristic
36	C₃₁×#387	C₄	1	1	yes	characteristic
37	C₃₁×#386	C₄	1	1	yes	characteristic
38	C₃₁×#386	C₄	1	1	yes	characteristic
39	C₃₁×#385	C₄	1	1	yes	characteristic
40	C₃₁×#385	C₄	1	1	yes	characteristic
41	C₄×C₄⋊C₄×C₃₁	C₂²	1	1	yes	characteristic
42	C₄×C₂²⋊C₄×C₃₁	C₂²	1	1	yes	characteristic
43	C₄×C₂²⋊C₄×C₃₁	C₂²	1	1	yes	characteristic
44	C₃₁×#376	C₂²	1	1	yes	characteristic
45	C₃₁×#336	C₂²	1	1	yes	characteristic
46	C₃₁×#336	C₂²	1	1	yes	characteristic
47	C₃₁×#336	C₂²	1	1	yes	characteristic
48	C₃₁×#336	C₂²	1	1	yes	characteristic
49	C₃₁×#336	C₂²	1	1	yes	characteristic
50	C₃₁×#336	C₂²	1	1	yes	characteristic
51	C₃₁×#336	C₂²	1	1	yes	characteristic
52	C₃₁×#336	C₂²	1	1	yes	characteristic
53	C₃₁×#336	C₂²	1	1	yes	characteristic
54	C₃₁×#336	C₂²	1	1	yes	characteristic
55	C₃₁×#336	C₂²	1	1	yes	characteristic
56	C₃₁×#336	C₂²	1	1	yes	characteristic
57	C₃₁×#336	C₂²	1	1	yes	characteristic
58	C₃₁×#336	C₂²	1	1	yes	characteristic
59	C₂²×D₄×C₃₁	C₂×C₄	1	1	yes	characteristic
60	C₂³×C₄×C₃₁	C₂×C₄	1	1	yes	characteristic
61	C₂³×C₄×C₃₁	C₂×C₄	1	1	yes	characteristic
62	C₂³×C₄×C₃₁	C₂³	1	1	yes	characteristic
63	C₂³×C₄×C₃₁	C₂³	1	1	yes	characteristic
64	C₂×M₁₆×C₃₁	—	2	2	no	—
65	C₂×M₁₆×C₃₁	—	2	2	no	—
66	C₂×M₁₆×C₃₁	—	4	1	no	—
67	C₂²×C₈×C₃₁	—	2	2	no	—
68	C₂²×C₈×C₃₁	—	2	2	no	—
69	C₂²×C₈×C₃₁	—	4	1	no	—
70	C₂²×C₈×C₃₁	C₂³	1	1	yes	characteristic
71	C₂²×C₈×C₃₁	C₂³	1	1	yes	characteristic
72	C₂²×C₈×C₃₁	C₂³	1	1	yes	characteristic
73	C₂²×C₈×C₃₁	C₂³	1	1	yes	characteristic
74	C₂²×C₈×C₃₁	C₂³	1	1	yes	characteristic
75	C₂²×C₈×C₃₁	C₂³	1	1	yes	characteristic
76	C₂²×C₈×C₃₁	C₂³	1	1	yes	characteristic
77	C₂²×C₈×C₃₁	C₂³	1	1	yes	characteristic
78	C₄×D₄×C₃₁	—	2	2	no	—
79	C₄×D₄×C₃₁	—	2	2	no	—
80	C₃₁×#117	—	4	1	no	—
81	C₃₁×#117	—	4	1	no	—
82	C₂×C₄⋊C₄×C₃₁	C₂×C₄	1	1	yes	characteristic
83	C₂×C₄⋊C₄×C₃₁	C₂×C₄	1	1	yes	characteristic
84	C₂×C₄⋊C₄×C₃₁	C₂×C₄	1	1	yes	characteristic
85	C₂×C₄⋊C₄×C₃₁	C₂×C₄	1	1	yes	characteristic
86	C₂×C₄⋊C₄×C₃₁	C₂×C₄	1	1	yes	characteristic
87	C₂×C₂²⋊C₄×C₃₁	C₂×C₄	1	1	yes	characteristic
88	C₂×C₂²⋊C₄×C₃₁	C₂×C₄	1	1	yes	characteristic
89	C₂×C₂²⋊C₄×C₃₁	C₂×C₄	1	1	yes	characteristic
90	C₂×C₂²⋊C₄×C₃₁	C₂×C₄	1	1	yes	characteristic
91	C₂×C₂²⋊C₄×C₃₁	C₂×C₄	1	1	yes	characteristic
92	C₂×C₂²⋊C₄×C₃₁	C₂×C₄	1	1	yes	characteristic
93	C₂×C₂²⋊C₄×C₃₁	C₂×C₄	1	1	yes	characteristic
94	C₂×C₂²⋊C₄×C₃₁	C₂×C₄	1	1	yes	characteristic
95	C₂×C₄²×C₃₁	—	4	1	no	—
96	C₂×C₄²×C₃₁	—	4	1	no	—
97	C₂×C₄²×C₃₁	C₂×C₄	1	1	yes	characteristic
98	C₂×C₄²×C₃₁	C₂×C₄	1	1	yes	characteristic
99	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
100	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
101	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
102	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
103	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
104	C₃₁×C₄⋊C₈	—	4	1	no	—
105	C₃₁×C₄⋊C₈	—	4	1	no	—
106	C₃₁×C₂²⋊C₈	—	2	2	no	—
107	C₃₁×C₂²⋊C₈	—	2	2	no	—
108	C₃₁×C₂²⋊C₈	—	2	2	no	—
109	C₃₁×C₂²⋊C₈	—	2	2	no	—
110	C₃₁×C₂²⋊C₈	—	4	1	no	—
111	C₃₁×C₂²⋊C₈	—	4	1	no	—
112	C₃₁×#97	—	2	2	no	—
113	C₄×C₈×C₃₁	—	2	2	no	—
114	C₄×C₈×C₃₁	—	4	1	no	—
115	C₄×C₈×C₃₁	—	4	1	no	—
116	C₃₁×#95	C₂×C₄	1	1	yes	characteristic
117	C₃₁×#95	C₂×C₄	1	1	yes	characteristic
118	C₃₁×#95	C₂×C₄	1	1	yes	characteristic
119	C₃₁×#95	C₂×C₄	1	1	yes	characteristic
120	C₃₁×#95	C₂×C₄	1	1	yes	characteristic
121	C₃₁×#95	C₂×C₄	1	1	yes	characteristic
122	C₃₁×#95	C₂×C₄	1	1	yes	characteristic
123	C₃₁×#95	C₂×C₄	1	1	yes	characteristic
124	C₃₁×#95	C₂×C₄	1	1	yes	characteristic
125	C₃₁×#95	C₂×C₄	1	1	yes	characteristic
126	C₂⁴×C₃₁	C₂²×C₄	1	1	yes	characteristic
127	C₂⁴×C₃₁	C₂²×C₄	1	1	yes	characteristic
128	C₂×D₄×C₃₁	—	2	2	no	—
129	C₂×D₄×C₃₁	—	2	2	no	—
130	C₂×D₄×C₃₁	—	4	1	no	—
131	C₂²×C₄×C₃₁	—	2	1	no	—
132	C₂²×C₄×C₃₁	—	2	1	no	—
133	C₂²×C₄×C₃₁	—	2	1	no	—
134	C₂²×C₄×C₃₁	—	2	1	no	—
135	C₂²×C₄×C₃₁	—	2	2	no	—
136	C₂²×C₄×C₃₁	—	2	2	no	—
137	C₂²×C₄×C₃₁	—	2	2	no	—
138	C₂²×C₄×C₃₁	—	2	1	no	—
139	C₂²×C₄×C₃₁	—	2	1	no	—
140	C₂²×C₄×C₃₁	—	2	1	no	—
141	C₂²×C₄×C₃₁	—	2	1	no	—
142	C₂²×C₄×C₃₁	—	2	1	no	—
143	C₂²×C₄×C₃₁	—	2	1	no	—
144	C₂²×C₄×C₃₁	—	4	1	no	—
145	C₂²×C₄×C₃₁	—	4	1	no	—
146	C₂²×C₄×C₃₁	—	4	1	no	—
147	C₂²×C₄×C₃₁	C₂²×C₄	1	1	yes	characteristic
148	C₂²×C₄×C₃₁	C₂²×C₄	1	1	yes	characteristic
149	C₂²×C₄×C₃₁	C₂²×C₄	1	1	yes	characteristic
150	C₂²×C₄×C₃₁	C₂²×C₄	1	1	yes	characteristic
151	C₂²×C₄×C₃₁	C₂²×C₄	1	1	yes	characteristic
152	C₂²×C₄×C₃₁	C₂²×C₄	1	1	yes	characteristic
153	C₂²×C₄×C₃₁	C₂²×C₄	1	1	yes	characteristic
154	C₂²×C₄×C₃₁	C₂²×C₄	1	1	yes	characteristic
155	C₂²×C₄×C₃₁	C₂²×C₄	1	1	yes	characteristic
156	C₂²×C₄×C₃₁	C₂²×C₄	1	1	yes	characteristic
157	C₂²×C₄×C₃₁	C₂²×C₄	1	1	yes	characteristic
158	C₂²×C₄×C₃₁	C₂²×C₄	1	1	yes	characteristic
159	C₂²×C₄×C₃₁	#41	1	2	yes	—
160	C₂²×C₄×C₃₁	#41	1	2	yes	—
161	C₂²×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
162	M₁₆×C₃₁	—	4	4	no	—
163	C₂×C₈×C₃₁	—	2	1	no	—
164	C₂×C₈×C₃₁	—	2	1	no	—
165	C₂×C₈×C₃₁	—	2	1	no	—
166	C₂×C₈×C₃₁	—	2	1	no	—
167	C₂×C₈×C₃₁	—	2	1	no	—
168	C₂×C₈×C₃₁	—	2	1	no	—
169	C₂×C₈×C₃₁	—	2	1	no	—
170	C₂×C₈×C₃₁	—	2	1	no	—
171	C₂×C₈×C₃₁	—	2	1	no	—
172	C₂×C₈×C₃₁	—	2	1	no	—
173	C₂×C₈×C₃₁	—	2	1	no	—
174	C₂×C₈×C₃₁	—	2	1	no	—
175	C₂×C₈×C₃₁	—	2	1	no	—
176	C₂×C₈×C₃₁	—	2	1	no	—
177	C₂×C₈×C₃₁	—	2	1	no	—
178	C₂×C₈×C₃₁	—	2	1	no	—
179	C₂×C₈×C₃₁	—	2	1	no	—
180	C₂×C₈×C₃₁	—	2	1	no	—
181	C₂×C₈×C₃₁	—	2	1	no	—
182	C₂×C₈×C₃₁	—	2	1	no	—
183	C₂×C₈×C₃₁	—	2	1	no	—
184	C₂×C₈×C₃₁	—	2	1	no	—
185	C₂×C₈×C₃₁	—	4	4	no	—
186	C₂×C₈×C₃₁	#41	1	2	yes	—
187	C₂×C₈×C₃₁	#41	1	2	yes	—
188	C₄⋊C₄×C₃₁	—	2	2	no	—
189	C₄⋊C₄×C₃₁	—	2	2	no	—
190	C₄⋊C₄×C₃₁	—	2	2	no	—
191	C₄⋊C₄×C₃₁	—	4	1	no	—
192	C₄⋊C₄×C₃₁	—	4	1	no	—
193	C₂²⋊C₄×C₃₁	—	2	2	no	—
194	C₂²⋊C₄×C₃₁	—	2	2	no	—
195	C₂²⋊C₄×C₃₁	—	2	2	no	—
196	C₂²⋊C₄×C₃₁	—	2	2	no	—
197	C₂²⋊C₄×C₃₁	—	2	2	no	—
198	C₂²⋊C₄×C₃₁	—	4	1	no	—
199	C₂²⋊C₄×C₃₁	—	4	1	no	—
200	C₂²⋊C₄×C₃₁	—	4	1	no	—
201	C₄²×C₃₁	—	2	1	no	—
202	C₄²×C₃₁	—	2	1	no	—
203	C₄²×C₃₁	—	2	1	no	—
204	C₄²×C₃₁	—	2	1	no	—
205	C₄²×C₃₁	—	2	1	no	—
206	C₄²×C₃₁	—	2	1	no	—
207	C₄²×C₃₁	—	2	1	no	—
208	C₄²×C₃₁	—	2	1	no	—
209	C₄²×C₃₁	—	2	1	no	—
210	C₄²×C₃₁	—	2	1	no	—
211	C₄²×C₃₁	—	4	1	no	—
212	C₄²×C₃₁	—	4	1	no	—
213	#22793	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
214	C₂³×C₃₁	—	2	1	no	—
215	C₂³×C₃₁	—	2	1	no	—
216	C₂³×C₃₁	—	2	1	no	—
217	C₂³×C₃₁	—	2	1	no	—
218	C₂³×C₃₁	—	2	1	no	—
219	C₂³×C₃₁	—	2	1	no	—
220	C₂³×C₃₁	—	2	1	no	—
221	C₂³×C₃₁	—	2	1	no	—
222	C₂³×C₃₁	—	2	1	no	—
223	C₂³×C₃₁	—	2	1	no	—
224	C₂³×C₃₁	—	2	1	no	—
225	C₂³×C₃₁	—	2	1	no	—
226	C₂³×C₃₁	#117	1	2	yes	—
227	C₂³×C₃₁	#131	1	2	yes	—
228	C₂³×C₃₁	C₂³×C₄	1	1	yes	characteristic
229	D₄×C₃₁	—	4	4	no	—
230	C₂×C₄×C₃₁	—	2	1	no	—
231	C₂×C₄×C₃₁	—	2	1	no	—
232	C₂×C₄×C₃₁	—	2	1	no	—
233	C₂×C₄×C₃₁	—	2	1	no	—
234	C₂×C₄×C₃₁	—	2	1	no	—
235	C₂×C₄×C₃₁	—	2	1	no	—
236	C₂×C₄×C₃₁	—	2	2	no	—
237	C₂×C₄×C₃₁	—	2	2	no	—
238	C₂×C₄×C₃₁	—	2	2	no	—
239	C₂×C₄×C₃₁	—	2	2	no	—
240	C₂×C₄×C₃₁	—	2	1	no	—
241	C₂×C₄×C₃₁	—	2	1	no	—
242	C₂×C₄×C₃₁	—	2	1	no	—
243	C₂×C₄×C₃₁	—	2	1	no	—
244	C₂×C₄×C₃₁	—	2	1	no	—
245	C₂×C₄×C₃₁	—	2	1	no	—
246	C₂×C₄×C₃₁	—	2	1	no	—
247	C₂×C₄×C₃₁	—	2	1	no	—
248	C₂×C₄×C₃₁	—	2	1	no	—
249	C₂×C₄×C₃₁	—	2	1	no	—
250	C₂×C₄×C₃₁	—	2	1	no	—
251	C₂×C₄×C₃₁	—	2	1	no	—
252	C₂×C₄×C₃₁	—	2	1	no	—
253	C₂×C₄×C₃₁	—	2	1	no	—
254	C₂×C₄×C₃₁	—	2	1	no	—
255	C₂×C₄×C₃₁	—	2	1	no	—
256	C₂×C₄×C₃₁	—	2	1	no	—
257	C₂×C₄×C₃₁	—	2	1	no	—
258	C₂×C₄×C₃₁	—	2	1	no	—
259	C₂×C₄×C₃₁	—	2	1	no	—
260	C₂×C₄×C₃₁	—	2	1	no	—
261	C₂×C₄×C₃₁	—	2	1	no	—
262	C₂×C₄×C₃₁	—	2	1	no	—
263	C₂×C₄×C₃₁	—	2	1	no	—
264	C₂×C₄×C₃₁	—	2	1	no	—
265	C₂×C₄×C₃₁	—	2	1	no	—
266	C₂×C₄×C₃₁	—	2	1	no	—
267	C₂×C₄×C₃₁	—	4	4	no	—
268	C₂×C₄×C₃₁	—	4	4	no	—
269	C₂×C₄×C₃₁	—	4	1	no	—
270	C₂×C₄×C₃₁	—	4	1	no	—
271	C₂×C₄×C₃₁	—	4	1	no	—
272	C₂×C₄×C₃₁	—	4	1	no	—
273	C₂×C₄×C₃₁	#117	1	2	yes	—
274	C₂×C₄×C₃₁	#131	1	2	yes	—
275	C₂×C₄×C₃₁	#131	1	2	yes	—
276	C₂×C₄×C₃₁	C₂×#41	1	1	yes	characteristic
277	C₂×C₄×C₃₁	C₂×#41	1	1	yes	characteristic
278	C₂×C₄×C₃₁	C₂×#41	1	1	yes	characteristic
279	C₂×C₄×C₃₁	C₂×#41	1	1	yes	characteristic
280	C₂×C₄×C₃₁	H(5,𝔽₂)	1	1	yes	characteristic
281	C₂×C₄×C₃₁	H(5,𝔽₂)	1	1	yes	characteristic
282	C₈×C₃₁	—	2	2	no	—
283	C₈×C₃₁	—	2	2	no	—
284	C₈×C₃₁	—	4	1	no	—
285	C₈×C₃₁	—	4	1	no	—
286	C₈×C₃₁	—	4	1	no	—
287	C₈×C₃₁	—	4	1	no	—
288	C₈×C₃₁	—	4	1	no	—
289	C₈×C₃₁	—	4	1	no	—
290	#2290	C₂×C₃₁	1	1	yes	characteristic
291	#1924	C₂×C₃₁	1	1	yes	characteristic, complemented
292	#1923	C₂×C₃₁	1	1	yes	characteristic, complemented
293	#1916	C₂×C₃₁	1	1	yes	characteristic, complemented
294	#1916	C₂×C₃₁	1	1	yes	characteristic, complemented
295	#1870	C₂×C₃₁	1	1	yes	characteristic
296	#1869	C₂×C₃₁	1	1	yes	characteristic
297	#1840	C₂×C₃₁	1	1	yes	characteristic, complemented
298	#1817	C₂×C₃₁	1	1	yes	characteristic, complemented
299	#1798	C₂×C₃₁	1	1	yes	characteristic, complemented
300	#1753	C₂×C₃₁	1	1	yes	characteristic, complemented
301	C₈×C₂²⋊C₄	C₂×C₃₁	1	1	yes	characteristic, complemented
302	C₄×C₂²⋊C₈	C₂×C₃₁	1	1	yes	characteristic, complemented
303	#1746	C₂×C₃₁	1	1	yes	characteristic, complemented
304	#1729	C₂×C₃₁	1	1	yes	characteristic, complemented
305	C₂²×C₃₁	—	2	2	no	—
306	C₂²×C₃₁	—	2	2	no	—
307	C₂²×C₃₁	—	2	2	no	—
308	C₂²×C₃₁	—	2	2	no	—
309	C₂²×C₃₁	—	2	2	no	—
310	C₂²×C₃₁	—	2	2	no	—
311	C₂²×C₃₁	—	4	1	no	—
312	C₂²×C₃₁	—	4	1	no	—
313	C₂²×C₃₁	—	4	1	no	—
314	C₂²×C₃₁	—	4	1	no	—
315	C₂²×C₃₁	—	4	1	no	—
316	C₂²×C₃₁	—	4	1	no	—
317	C₂²×C₃₁	—	4	1	no	—
318	C₂²×C₃₁	—	4	1	no	—
319	C₂²×C₃₁	C₂×#117	1	1	yes	characteristic
320	C₂²×C₃₁	C₄×#41	1	1	yes	characteristic
321	C₂²×C₃₁	#518	1	1	yes	characteristic
322	C₂²×C₃₁	C₂×#131	1	1	yes	characteristic
323	C₂²×C₃₁	C₂×#131	1	1	yes	characteristic
324	C₂²×C₃₁	C₂×#131	1	1	yes	characteristic
325	C₂²×C₃₁	#568	1	1	yes	characteristic
326	C₄×C₃₁	—	2	2	no	—
327	C₄×C₃₁	—	2	2	no	—
328	C₄×C₃₁	—	2	2	no	—
329	C₄×C₃₁	—	4	1	no	—
330	C₄×C₃₁	—	4	1	no	—
331	C₄×C₃₁	—	4	1	no	—
332	C₄×C₃₁	—	4	1	no	—
333	C₄×C₃₁	—	4	1	no	—
334	C₄×C₃₁	—	4	1	no	—
335	C₄×C₃₁	—	4	1	no	—
336	C₄×C₃₁	—	4	1	no	—
337	C₄×C₃₁	—	4	1	no	—
338	C₄×C₃₁	#551	1	1	yes	characteristic
339	C₄×C₃₁	#551	1	1	yes	characteristic
340	C₄×C₃₁	#553	1	1	yes	characteristic
341	C₄×C₃₁	#553	1	1	yes	characteristic
342	C₂²×M₁₆	C₂²×C₃₁	1	1	yes	characteristic
343	C₂³×C₈	C₂²×C₃₁	1	1	yes	characteristic
344	C₂×C₄×D₄	C₂²×C₃₁	1	1	yes	characteristic
345	C₂×#117	C₂²×C₃₁	1	1	yes	characteristic
346	C₂²×C₄²	C₂²×C₃₁	1	1	yes	characteristic
347	C₂×C₄⋊C₈	C₂²×C₃₁	1	1	yes	characteristic
348	C₂×C₄⋊C₈	C₂²×C₃₁	1	1	yes	characteristic
349	C₂×C₂²⋊C₈	C₂²×C₃₁	1	1	yes	characteristic
350	C₂×C₂²⋊C₈	C₂²×C₃₁	1	1	yes	characteristic
351	C₂×C₂²⋊C₈	C₂²×C₃₁	1	1	yes	characteristic
352	C₂×C₂²⋊C₈	C₂²×C₃₁	1	1	yes	characteristic
353	C₂×C₂²⋊C₈	C₂²×C₃₁	1	1	yes	characteristic
354	C₂×C₂²⋊C₈	C₂²×C₃₁	1	1	yes	characteristic
355	C₂×#97	C₂²×C₃₁	1	1	yes	characteristic
356	C₂×C₄×C₈	C₂²×C₃₁	1	1	yes	characteristic
357	C₂×C₄×C₈	C₂²×C₃₁	1	1	yes	characteristic
358	C₂×C₄×C₈	C₂²×C₃₁	1	1	yes	characteristic
359	#388	C₄×C₃₁	1	1	yes	characteristic
360	#388	C₄×C₃₁	1	1	yes	characteristic
361	#387	C₄×C₃₁	1	1	yes	characteristic
362	#387	C₄×C₃₁	1	1	yes	characteristic
363	#386	C₄×C₃₁	1	1	yes	characteristic
364	#386	C₄×C₃₁	1	1	yes	characteristic
365	#385	C₄×C₃₁	1	1	yes	characteristic
366	#385	C₄×C₃₁	1	1	yes	characteristic
367	C₄×C₄⋊C₄	C₂²×C₃₁	1	1	yes	characteristic
368	C₄×C₂²⋊C₄	C₂²×C₃₁	1	1	yes	characteristic
369	C₄×C₂²⋊C₄	C₂²×C₃₁	1	1	yes	characteristic
370	#376	C₂²×C₃₁	1	1	yes	characteristic
371	#336	C₂²×C₃₁	1	1	yes	characteristic
372	#336	C₂²×C₃₁	1	1	yes	characteristic
373	#336	C₂²×C₃₁	1	1	yes	characteristic
374	#336	C₂²×C₃₁	1	1	yes	characteristic
375	#336	C₂²×C₃₁	1	1	yes	characteristic
376	#336	C₂²×C₃₁	1	1	yes	characteristic
377	#336	C₂²×C₃₁	1	1	yes	characteristic
378	#336	C₂²×C₃₁	1	1	yes	characteristic
379	#336	C₂²×C₃₁	1	1	yes	characteristic
380	#336	C₂²×C₃₁	1	1	yes	characteristic
381	#336	C₂²×C₃₁	1	1	yes	characteristic
382	#336	C₂²×C₃₁	1	1	yes	characteristic
383	#336	C₂²×C₃₁	1	1	yes	characteristic
384	#336	C₂²×C₃₁	1	1	yes	characteristic
385	C₂×C₃₁	—	4	2	no	—
386	C₂×C₃₁	—	4	2	no	—
387	C₂×C₃₁	#2353	1	1	yes	characteristic
388	C₂×C₃₁	#2922	1	1	yes	characteristic
389	C₂×C₃₁	#2924	1	1	yes	characteristic
390	C₂×C₃₁	#2966	1	1	yes	characteristic
391	C₂×C₃₁	#2969	1	1	yes	characteristic
392	C₂×C₃₁	#2976	1	1	yes	characteristic
393	C₂×C₃₁	#2977	1	1	yes	characteristic
394	C₂²×D₄	C₂×C₄×C₃₁	1	1	yes	characteristic
395	C₂³×C₄	C₂×C₄×C₃₁	1	1	yes	characteristic
396	C₂³×C₄	C₂×C₄×C₃₁	1	1	yes	characteristic
397	C₂³×C₄	C₂³×C₃₁	1	1	yes	characteristic
398	C₂³×C₄	C₂³×C₃₁	1	1	yes	characteristic
399	C₂×M₁₆	—	2	2	no	—
400	C₂×M₁₆	—	2	2	no	—
401	C₂×M₁₆	—	4	1	no	—
402	C₂²×C₈	—	2	2	no	—
403	C₂²×C₈	—	2	2	no	—
404	C₂²×C₈	—	4	1	no	—
405	C₂²×C₈	C₂³×C₃₁	1	1	yes	characteristic
406	C₂²×C₈	C₂³×C₃₁	1	1	yes	characteristic
407	C₂²×C₈	C₂³×C₃₁	1	1	yes	characteristic
408	C₂²×C₈	C₂³×C₃₁	1	1	yes	characteristic
409	C₂²×C₈	C₂³×C₃₁	1	1	yes	characteristic
410	C₂²×C₈	C₂³×C₃₁	1	1	yes	characteristic
411	C₂²×C₈	C₂³×C₃₁	1	1	yes	characteristic
412	C₂²×C₈	C₂³×C₃₁	1	1	yes	characteristic
413	C₄×D₄	—	2	2	no	—
414	C₄×D₄	—	2	2	no	—
415	#117	—	4	1	no	—
416	#117	—	4	1	no	—
417	C₂×C₄⋊C₄	C₂×C₄×C₃₁	1	1	yes	characteristic
418	C₂×C₄⋊C₄	C₂×C₄×C₃₁	1	1	yes	characteristic
419	C₂×C₄⋊C₄	C₂×C₄×C₃₁	1	1	yes	characteristic
420	C₂×C₄⋊C₄	C₂×C₄×C₃₁	1	1	yes	characteristic
421	C₂×C₄⋊C₄	C₂×C₄×C₃₁	1	1	yes	characteristic
422	C₂×C₂²⋊C₄	C₂×C₄×C₃₁	1	1	yes	characteristic
423	C₂×C₂²⋊C₄	C₂×C₄×C₃₁	1	1	yes	characteristic
424	C₂×C₂²⋊C₄	C₂×C₄×C₃₁	1	1	yes	characteristic
425	C₂×C₂²⋊C₄	C₂×C₄×C₃₁	1	1	yes	characteristic
426	C₂×C₂²⋊C₄	C₂×C₄×C₃₁	1	1	yes	characteristic
427	C₂×C₂²⋊C₄	C₂×C₄×C₃₁	1	1	yes	characteristic
428	C₂×C₂²⋊C₄	C₂×C₄×C₃₁	1	1	yes	characteristic
429	C₂×C₂²⋊C₄	C₂×C₄×C₃₁	1	1	yes	characteristic
430	C₂×C₄²	—	4	1	no	—
431	C₂×C₄²	—	4	1	no	—
432	C₂×C₄²	C₂×C₄×C₃₁	1	1	yes	characteristic
433	C₂×C₄²	C₂×C₄×C₃₁	1	1	yes	characteristic
434	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
435	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
436	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
437	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
438	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
439	C₄⋊C₈	—	4	1	no	—
440	C₄⋊C₈	—	4	1	no	—
441	C₂²⋊C₈	—	2	2	no	—
442	C₂²⋊C₈	—	2	2	no	—
443	C₂²⋊C₈	—	2	2	no	—
444	C₂²⋊C₈	—	2	2	no	—
445	C₂²⋊C₈	—	4	1	no	—
446	C₂²⋊C₈	—	4	1	no	—
447	#97	—	2	2	no	—
448	C₄×C₈	—	2	2	no	—
449	C₄×C₈	—	4	1	no	—
450	C₄×C₈	—	4	1	no	—
451	#95	C₂×C₄×C₃₁	1	1	yes	characteristic
452	#95	C₂×C₄×C₃₁	1	1	yes	characteristic
453	#95	C₂×C₄×C₃₁	1	1	yes	characteristic
454	#95	C₂×C₄×C₃₁	1	1	yes	characteristic
455	#95	C₂×C₄×C₃₁	1	1	yes	characteristic
456	#95	C₂×C₄×C₃₁	1	1	yes	characteristic
457	#95	C₂×C₄×C₃₁	1	1	yes	characteristic
458	#95	C₂×C₄×C₃₁	1	1	yes	characteristic
459	#95	C₂×C₄×C₃₁	1	1	yes	characteristic
460	#95	C₂×C₄×C₃₁	1	1	yes	characteristic
461	C₃₁	#22793	1	1	yes	characteristic, direct product factor, Sylow
462	C₂⁴	C₂²×C₄×C₃₁	1	1	yes	characteristic
463	C₂⁴	C₂²×C₄×C₃₁	1	1	yes	characteristic
464	C₂×D₄	—	2	2	no	—
465	C₂×D₄	—	2	2	no	—
466	C₂×D₄	—	4	1	no	—
467	C₂²×C₄	—	2	1	no	—
468	C₂²×C₄	—	2	1	no	—
469	C₂²×C₄	—	2	1	no	—
470	C₂²×C₄	—	2	1	no	—
471	C₂²×C₄	—	2	2	no	—
472	C₂²×C₄	—	2	2	no	—
473	C₂²×C₄	—	2	2	no	—
474	C₂²×C₄	—	2	1	no	—
475	C₂²×C₄	—	2	1	no	—
476	C₂²×C₄	—	2	1	no	—
477	C₂²×C₄	—	2	1	no	—
478	C₂²×C₄	—	2	1	no	—
479	C₂²×C₄	—	2	1	no	—
480	C₂²×C₄	—	4	1	no	—
481	C₂²×C₄	—	4	1	no	—
482	C₂²×C₄	—	4	1	no	—
483	C₂²×C₄	C₂²×C₄×C₃₁	1	1	yes	characteristic
484	C₂²×C₄	C₂²×C₄×C₃₁	1	1	yes	characteristic
485	C₂²×C₄	C₂²×C₄×C₃₁	1	1	yes	characteristic
486	C₂²×C₄	C₂²×C₄×C₃₁	1	1	yes	characteristic
487	C₂²×C₄	C₂²×C₄×C₃₁	1	1	yes	characteristic
488	C₂²×C₄	C₂²×C₄×C₃₁	1	1	yes	characteristic
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	C₂²×C₄×C₃₁	1	1	yes	characteristic
493	C₂²×C₄	C₂²×C₄×C₃₁	1	1	yes	characteristic
494	C₂²×C₄	C₂²×C₄×C₃₁	1	1	yes	characteristic
495	C₂²×C₄	#41×C₃₁	1	2	yes	—
496	C₂²×C₄	#41×C₃₁	1	2	yes	—
497	C₂²×C₄	C₂⁴×C₃₁	1	1	yes	characteristic
498	M₁₆	—	4	4	no	—
499	C₂×C₈	—	2	1	no	—
500	C₂×C₈	—	2	1	no	—
501	C₂×C₈	—	2	1	no	—
502	C₂×C₈	—	2	1	no	—
503	C₂×C₈	—	2	1	no	—
504	C₂×C₈	—	2	1	no	—
505	C₂×C₈	—	2	1	no	—
506	C₂×C₈	—	2	1	no	—
507	C₂×C₈	—	2	1	no	—
508	C₂×C₈	—	2	1	no	—
509	C₂×C₈	—	2	1	no	—
510	C₂×C₈	—	2	1	no	—
511	C₂×C₈	—	2	1	no	—
512	C₂×C₈	—	2	1	no	—
513	C₂×C₈	—	2	1	no	—
514	C₂×C₈	—	2	1	no	—
515	C₂×C₈	—	2	1	no	—
516	C₂×C₈	—	2	1	no	—
517	C₂×C₈	—	2	1	no	—
518	C₂×C₈	—	2	1	no	—
519	C₂×C₈	—	2	1	no	—
520	C₂×C₈	—	2	1	no	—
521	C₂×C₈	—	4	4	no	—
522	C₂×C₈	#41×C₃₁	1	2	yes	—
523	C₂×C₈	#41×C₃₁	1	2	yes	—
524	C₄⋊C₄	—	2	2	no	—
525	C₄⋊C₄	—	2	2	no	—
526	C₄⋊C₄	—	2	2	no	—
527	C₄⋊C₄	—	4	1	no	—
528	C₄⋊C₄	—	4	1	no	—
529	C₂²⋊C₄	—	2	2	no	—
530	C₂²⋊C₄	—	2	2	no	—
531	C₂²⋊C₄	—	2	2	no	—
532	C₂²⋊C₄	—	2	2	no	—
533	C₂²⋊C₄	—	2	2	no	—
534	C₂²⋊C₄	—	4	1	no	—
535	C₂²⋊C₄	—	4	1	no	—
536	C₂²⋊C₄	—	4	1	no	—
537	C₄²	—	2	1	no	—
538	C₄²	—	2	1	no	—
539	C₄²	—	2	1	no	—
540	C₄²	—	2	1	no	—
541	C₄²	—	2	1	no	—
542	C₄²	—	2	1	no	—
543	C₄²	—	2	1	no	—
544	C₄²	—	2	1	no	—
545	C₄²	—	2	1	no	—
546	C₄²	—	2	1	no	—
547	C₄²	—	4	1	no	—
548	C₄²	—	4	1	no	—
549	C₂³	—	2	1	no	—
550	C₂³	—	2	1	no	—
551	C₂³	—	2	1	no	—
552	C₂³	—	2	1	no	—
553	C₂³	—	2	1	no	—
554	C₂³	—	2	1	no	—
555	C₂³	—	2	1	no	—
556	C₂³	—	2	1	no	—
557	C₂³	—	2	1	no	—
558	C₂³	—	2	1	no	—
559	C₂³	—	2	1	no	—
560	C₂³	—	2	1	no	—
561	C₂³	C₃₁×#117	1	2	yes	—
562	C₂³	C₃₁×#131	1	2	yes	—
563	C₂³	C₂³×C₄×C₃₁	1	1	yes	characteristic
564	D₄	—	4	4	no	—
565	C₂×C₄	—	2	1	no	—
566	C₂×C₄	—	2	1	no	—
567	C₂×C₄	—	2	1	no	—
568	C₂×C₄	—	2	1	no	—
569	C₂×C₄	—	2	1	no	—
570	C₂×C₄	—	2	1	no	—
571	C₂×C₄	—	2	2	no	—
572	C₂×C₄	—	2	2	no	—
573	C₂×C₄	—	2	2	no	—
574	C₂×C₄	—	2	2	no	—
575	C₂×C₄	—	2	1	no	—
576	C₂×C₄	—	2	1	no	—
577	C₂×C₄	—	2	1	no	—
578	C₂×C₄	—	2	1	no	—
579	C₂×C₄	—	2	1	no	—
580	C₂×C₄	—	2	1	no	—
581	C₂×C₄	—	2	1	no	—
582	C₂×C₄	—	2	1	no	—
583	C₂×C₄	—	2	1	no	—
584	C₂×C₄	—	2	1	no	—
585	C₂×C₄	—	2	1	no	—
586	C₂×C₄	—	2	1	no	—
587	C₂×C₄	—	2	1	no	—
588	C₂×C₄	—	2	1	no	—
589	C₂×C₄	—	2	1	no	—
590	C₂×C₄	—	2	1	no	—
591	C₂×C₄	—	2	1	no	—
592	C₂×C₄	—	2	1	no	—
593	C₂×C₄	—	2	1	no	—
594	C₂×C₄	—	2	1	no	—
595	C₂×C₄	—	2	1	no	—
596	C₂×C₄	—	2	1	no	—
597	C₂×C₄	—	2	1	no	—
598	C₂×C₄	—	2	1	no	—
599	C₂×C₄	—	2	1	no	—
600	C₂×C₄	—	2	1	no	—
601	C₂×C₄	—	2	1	no	—
602	C₂×C₄	—	4	4	no	—
603	C₂×C₄	—	4	4	no	—
604	C₂×C₄	—	4	1	no	—
605	C₂×C₄	—	4	1	no	—
606	C₂×C₄	—	4	1	no	—
607	C₂×C₄	—	4	1	no	—
608	C₂×C₄	C₃₁×#117	1	2	yes	—
609	C₂×C₄	C₃₁×#131	1	2	yes	—
610	C₂×C₄	C₃₁×#131	1	2	yes	—
611	C₂×C₄	C₂×#41×C₃₁	1	1	yes	characteristic
612	C₂×C₄	C₂×#41×C₃₁	1	1	yes	characteristic
613	C₂×C₄	C₂×#41×C₃₁	1	1	yes	characteristic
614	C₂×C₄	C₂×#41×C₃₁	1	1	yes	characteristic
615	C₂×C₄	C₃₁×H(5,𝔽₂)	1	1	yes	characteristic
616	C₂×C₄	C₃₁×H(5,𝔽₂)	1	1	yes	characteristic
617	C₈	—	2	2	no	—
618	C₈	—	2	2	no	—
619	C₈	—	4	1	no	—
620	C₈	—	4	1	no	—
621	C₈	—	4	1	no	—
622	C₈	—	4	1	no	—
623	C₈	—	4	1	no	—
624	C₈	—	4	1	no	—
625	C₂²	—	2	2	no	—
626	C₂²	—	2	2	no	—
627	C₂²	—	2	2	no	—
628	C₂²	—	2	2	no	—
629	C₂²	—	2	2	no	—
630	C₂²	—	2	2	no	—
631	C₂²	—	4	1	no	—
632	C₂²	—	4	1	no	—
633	C₂²	—	4	1	no	—
634	C₂²	—	4	1	no	—
635	C₂²	—	4	1	no	—
636	C₂²	—	4	1	no	—
637	C₂²	—	4	1	no	—
638	C₂²	—	4	1	no	—
639	C₂²	C₂×C₃₁×#117	1	1	yes	characteristic
640	C₂²	C₄×#41×C₃₁	1	1	yes	characteristic
641	C₂²	C₃₁×#518	1	1	yes	characteristic
642	C₂²	C₂×C₃₁×#131	1	1	yes	characteristic
643	C₂²	C₂×C₃₁×#131	1	1	yes	characteristic
644	C₂²	C₂×C₃₁×#131	1	1	yes	characteristic
645	C₂²	C₃₁×#568	1	1	yes	characteristic
646	C₄	—	2	2	no	—
647	C₄	—	2	2	no	—
648	C₄	—	2	2	no	—
649	C₄	—	4	1	no	—
650	C₄	—	4	1	no	—
651	C₄	—	4	1	no	—
652	C₄	—	4	1	no	—
653	C₄	—	4	1	no	—
654	C₄	—	4	1	no	—
655	C₄	—	4	1	no	—
656	C₄	—	4	1	no	—
657	C₄	—	4	1	no	—
658	C₄	C₃₁×#551	1	1	yes	characteristic
659	C₄	C₃₁×#551	1	1	yes	characteristic
660	C₄	C₃₁×#553	1	1	yes	characteristic
661	C₄	C₃₁×#553	1	1	yes	characteristic
662	C₂	—	4	2	no	—
663	C₂	—	4	2	no	—
664	C₂	C₃₁×#2353	1	1	yes	characteristic
665	C₂	C₃₁×#2922	1	1	yes	characteristic
666	C₂	C₃₁×#2924	1	1	yes	characteristic
667	C₂	C₃₁×#2966	1	1	yes	characteristic
668	C₂	C₃₁×#2969	1	1	yes	characteristic
669	C₂	C₃₁×#2976	1	1	yes	characteristic
670	C₂	C₃₁×#2977	1	1	yes	characteristic

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

Complements

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

N	H ≅ G/N	type
2. C₃₁×#1924	C₂	semidirect
3. C₃₁×#1923	C₂	semidirect
4. C₃₁×#1916	C₂	semidirect
5. C₃₁×#1916	C₂	semidirect
8. C₃₁×#1840	C₂	semidirect
9. C₃₁×#1817	C₂	semidirect
10. C₃₁×#1798	C₂	semidirect
11. C₃₁×#1753	C₂	semidirect
12. C₈×C₂²⋊C₄×C₃₁	C₂	semidirect
13. C₄×C₃₁×C₂²⋊C₈	C₂	semidirect
14. C₃₁×#1746	C₂	semidirect
15. C₃₁×#1729	C₂	semidirect
213. #22793	C₃₁	direct (N × H)
291. #1924	C₂×C₃₁	semidirect
292. #1923	C₂×C₃₁	semidirect
293. #1916	C₂×C₃₁	semidirect
294. #1916	C₂×C₃₁	semidirect
297. #1840	C₂×C₃₁	semidirect
298. #1817	C₂×C₃₁	semidirect
299. #1798	C₂×C₃₁	semidirect
300. #1753	C₂×C₃₁	semidirect
301. C₈×C₂²⋊C₄	C₂×C₃₁	semidirect
302. C₄×C₂²⋊C₈	C₂×C₃₁	semidirect
303. #1746	C₂×C₃₁	semidirect
304. #1729	C₂×C₃₁	semidirect
461. C₃₁	#22793	direct (N × H)

Maximal subgroups

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

subgroup	index
C₃₁×#2290	2
C₃₁×#1924	2
C₃₁×#1923	2
C₃₁×#1916	2
C₃₁×#1870	2
C₃₁×#1869	2
C₃₁×#1840	2
C₃₁×#1817	2
C₃₁×#1798	2
C₃₁×#1753	2
C₈×C₂²⋊C₄×C₃₁	2
C₄×C₃₁×C₂²⋊C₈	2
C₃₁×#1746	2
C₃₁×#1729	2
#22793	31

Subgroups — C31×#22793 #21479552

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#22793 #21479552