Subgroups — C₃₁×#22032 #21478791

Order 7936.

1196 conjugacy classes of subgroups.

The subgroup lattice has 562 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃₁×#435	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂²×C₃₁×C₂²⋊C₈	C₂	1	1	yes	maximal, characteristic, complemented
3	C₄×C₃₁×#120	C₂	1	1	yes	maximal, characteristic
4	C₃₁×#1954	C₂	1	2	yes	maximal, complemented
5	C₃₁×#1924	C₂	1	2	yes	maximal, complemented
6	C₃₁×#1923	C₂	1	2	yes	maximal, complemented
7	C₃₁×#1870	C₂	1	2	yes	maximal, complemented
8	C₃₁×#1869	C₂	1	1	yes	maximal, characteristic, complemented
9	C₃₁×#1850	C₂	1	1	yes	maximal, characteristic, complemented
10	C₃₁×#1779	C₂	1	1	yes	maximal, characteristic, complemented
11	C₈×C₂²⋊C₄×C₃₁	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂⁴×C₄×C₃₁	C₂²	1	1	yes	characteristic
13	C₂²×M₁₆×C₃₁	C₂²	1	1	yes	characteristic, complemented
14	C₂³×C₈×C₃₁	C₂²	1	1	yes	characteristic
15	C₂³×C₈×C₃₁	C₂²	1	1	yes	characteristic
16	C₂×C₃₁×#120	C₄	1	1	yes	characteristic
17	C₂×C₄×D₄×C₃₁	C₂²	1	1	yes	characteristic
18	C₂×C₄×D₄×C₃₁	C₂²	1	2	yes	—
19	C₃₁×#435	—	2	4	no	—
20	C₂×C₃₁×C₄⋊C₈	C₂²	1	1	yes	characteristic, complemented
21	C₂×C₃₁×C₄⋊C₈	C₂²	1	2	yes	complemented
22	C₂×C₃₁×C₂²⋊C₈	—	2	4	no	—
23	C₂×C₃₁×C₂²⋊C₈	C₂²	1	1	yes	characteristic
24	C₂×C₃₁×C₂²⋊C₈	C₂²	1	2	yes	—
25	C₂×C₃₁×C₂²⋊C₈	C₂²	1	2	yes	—
26	C₂×C₃₁×C₂²⋊C₈	C₂²	1	1	yes	characteristic
27	C₂×C₃₁×C₂²⋊C₈	C₂²	1	2	yes	—
28	C₂×C₃₁×C₂²⋊C₈	C₂²	1	2	yes	—
29	C₂×C₃₁×C₂²⋊C₈	C₂²	1	1	yes	characteristic, complemented
30	C₂×C₃₁×C₂²⋊C₈	C₂²	1	2	yes	complemented
31	C₂×C₃₁×#97	C₂²	1	1	yes	characteristic, complemented
32	C₂×C₄×C₈×C₃₁	C₂²	1	2	yes	complemented
33	C₃₁×#386	C₄	1	2	yes	—
34	C₃₁×#386	C₄	1	1	yes	characteristic
35	C₃₁×#385	C₄	1	1	yes	characteristic
36	C₃₁×#385	C₄	1	2	yes	—
37	C₃₁×#379	C₄	1	1	yes	characteristic
38	C₄×C₂²⋊C₄×C₃₁	C₂²	1	1	yes	characteristic
39	C₄×C₂²⋊C₄×C₃₁	C₂²	1	2	yes	—
40	C₃₁×#336	C₂²	1	2	yes	complemented
41	C₃₁×#336	C₂²	1	2	yes	complemented
42	C₃₁×#336	C₂²	1	2	yes	complemented
43	C₂⁵×C₃₁	C₂×C₄	1	1	yes	characteristic
44	C₂²×D₄×C₃₁	C₂×C₄	1	1	yes	characteristic
45	C₂²×D₄×C₃₁	C₂×C₄	1	2	yes	—
46	C₂³×C₄×C₃₁	—	2	2	no	—
47	C₂³×C₄×C₃₁	—	2	2	no	—
48	C₂³×C₄×C₃₁	—	4	2	no	—
49	C₂³×C₄×C₃₁	—	4	2	no	—
50	C₂³×C₄×C₃₁	C₂×C₄	1	1	yes	characteristic
51	C₂³×C₄×C₃₁	C₂×C₄	1	2	yes	—
52	C₂³×C₄×C₃₁	C₂³	1	2	yes	—
53	C₂³×C₄×C₃₁	C₂³	1	1	yes	characteristic
54	C₂³×C₄×C₃₁	C₂³	1	1	yes	characteristic
55	C₂×M₁₆×C₃₁	—	2	2	no	—
56	C₂×M₁₆×C₃₁	—	4	2	no	—
57	C₂²×C₈×C₃₁	—	2	2	no	—
58	C₂²×C₈×C₃₁	—	4	2	no	—
59	C₂²×C₈×C₃₁	—	4	2	no	—
60	C₂²×C₈×C₃₁	D₄	1	4	yes	complemented
61	C₂²×C₈×C₃₁	C₂³	1	2	yes	—
62	C₂²×C₈×C₃₁	C₂³	1	1	yes	characteristic
63	C₂²×C₈×C₃₁	C₂³	1	1	yes	characteristic
64	C₂²×C₈×C₃₁	C₂³	1	2	yes	—
65	C₂²×C₈×C₃₁	C₂³	1	1	yes	characteristic
66	C₂²×C₈×C₃₁	C₂³	1	1	yes	characteristic
67	C₃₁×#120	—	2	4	no	—
68	C₄×D₄×C₃₁	—	2	2	no	—
69	C₄×D₄×C₃₁	—	2	2	no	—
70	C₄×D₄×C₃₁	—	4	2	no	—
71	C₄×D₄×C₃₁	—	4	2	no	—
72	C₂×C₄⋊C₄×C₃₁	C₂×C₄	1	1	yes	characteristic
73	C₂×C₄⋊C₄×C₃₁	C₂×C₄	1	2	yes	—
74	C₂×C₂²⋊C₄×C₃₁	C₂×C₄	1	1	yes	characteristic
75	C₂×C₂²⋊C₄×C₃₁	C₂×C₄	1	1	yes	characteristic
76	C₂×C₂²⋊C₄×C₃₁	C₂×C₄	1	2	yes	—
77	C₂×C₂²⋊C₄×C₃₁	C₂×C₄	1	2	yes	—
78	C₂×C₂²⋊C₄×C₃₁	C₂×C₄	1	1	yes	characteristic
79	C₂×C₂²⋊C₄×C₃₁	C₂×C₄	1	2	yes	—
80	C₂×C₂²⋊C₄×C₃₁	C₂×C₄	1	2	yes	—
81	C₂×C₂²⋊C₄×C₃₁	C₂×C₄	1	1	yes	characteristic
82	C₂×C₄²×C₃₁	C₂³	1	2	yes	—
83	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
84	C₃₁×C₄⋊C₈	—	2	2	no	—
85	C₃₁×C₄⋊C₈	—	4	2	no	—
86	C₃₁×C₂²⋊C₈	—	2	2	no	—
87	C₃₁×C₂²⋊C₈	—	2	2	no	—
88	C₃₁×C₂²⋊C₈	—	2	4	no	—
89	C₃₁×C₂²⋊C₈	—	2	4	no	—
90	C₃₁×C₂²⋊C₈	—	2	2	no	—
91	C₃₁×C₂²⋊C₈	—	4	2	no	—
92	C₃₁×C₂²⋊C₈	—	4	2	no	—
93	C₃₁×C₂²⋊C₈	—	4	2	no	—
94	C₃₁×#97	—	2	2	no	—
95	C₄×C₈×C₃₁	—	4	2	no	—
96	C₃₁×#95	C₂×C₄	1	2	yes	—
97	C₃₁×#95	C₂×C₄	1	2	yes	—
98	C₃₁×#95	C₂×C₄	1	2	yes	—
99	C₂⁴×C₃₁	—	2	2	no	—
100	C₂⁴×C₃₁	—	2	4	no	—
101	C₂⁴×C₃₁	—	2	2	no	—
102	C₂⁴×C₃₁	—	2	2	no	—
103	C₂⁴×C₃₁	—	4	2	no	—
104	C₂⁴×C₃₁	C₂²×C₄	1	1	yes	characteristic
105	C₂⁴×C₃₁	C₂²×C₄	1	2	yes	—
106	C₂⁴×C₃₁	C₂²×C₄	1	1	yes	characteristic
107	C₂×D₄×C₃₁	—	2	2	no	—
108	C₂×D₄×C₃₁	—	2	2	no	—
109	C₂×D₄×C₃₁	—	2	4	no	—
110	C₂×D₄×C₃₁	—	2	2	no	—
111	C₂×D₄×C₃₁	—	2	4	no	—
112	C₂×D₄×C₃₁	—	4	2	no	—
113	C₂²×C₄×C₃₁	—	2	2	no	—
114	C₂²×C₄×C₃₁	—	2	2	no	—
115	C₂²×C₄×C₃₁	—	2	2	no	—
116	C₂²×C₄×C₃₁	—	2	4	no	—
117	C₂²×C₄×C₃₁	—	2	2	no	—
118	C₂²×C₄×C₃₁	—	2	1	no	—
119	C₂²×C₄×C₃₁	—	2	1	no	—
120	C₂²×C₄×C₃₁	—	2	2	no	—
121	C₂²×C₄×C₃₁	—	2	2	no	—
122	C₂²×C₄×C₃₁	—	2	2	no	—
123	C₂²×C₄×C₃₁	—	2	2	no	—
124	C₂²×C₄×C₃₁	—	2	2	no	—
125	C₂²×C₄×C₃₁	—	2	2	no	—
126	C₂²×C₄×C₃₁	—	2	2	no	—
127	C₂²×C₄×C₃₁	—	4	2	no	—
128	C₂²×C₄×C₃₁	—	4	2	no	—
129	C₂²×C₄×C₃₁	—	4	2	no	—
130	C₂²×C₄×C₃₁	—	4	4	no	—
131	C₂²×C₄×C₃₁	—	4	4	no	—
132	C₂²×C₄×C₃₁	—	4	8	no	—
133	C₂²×C₄×C₃₁	C₂²×C₄	1	1	yes	characteristic
134	C₂²×C₄×C₃₁	C₂²×C₄	1	2	yes	—
135	C₂²×C₄×C₃₁	C₂²×C₄	1	1	yes	characteristic
136	C₂²×C₄×C₃₁	C₂²×C₄	1	2	yes	—
137	C₂²×C₄×C₃₁	C₂²×C₄	1	1	yes	characteristic
138	C₂²×C₄×C₃₁	C₂²×C₄	1	1	yes	characteristic
139	C₂²×C₄×C₃₁	C₂²×C₄	1	2	yes	—
140	C₂²×C₄×C₃₁	C₂×D₄	1	2	yes	—
141	C₂²×C₄×C₃₁	#41	1	2	yes	—
142	C₂²×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
143	M₁₆×C₃₁	—	4	4	no	—
144	C₂×C₈×C₃₁	—	2	2	no	—
145	C₂×C₈×C₃₁	—	2	2	no	—
146	C₂×C₈×C₃₁	—	2	1	no	—
147	C₂×C₈×C₃₁	—	2	2	no	—
148	C₂×C₈×C₃₁	—	2	1	no	—
149	C₂×C₈×C₃₁	—	2	2	no	—
150	C₂×C₈×C₃₁	—	2	2	no	—
151	C₂×C₈×C₃₁	—	2	2	no	—
152	C₂×C₈×C₃₁	—	2	2	no	—
153	C₂×C₈×C₃₁	—	2	2	no	—
154	C₂×C₈×C₃₁	—	2	2	no	—
155	C₂×C₈×C₃₁	—	2	2	no	—
156	C₂×C₈×C₃₁	—	4	4	no	—
157	C₂×C₈×C₃₁	—	4	4	no	—
158	C₂×C₈×C₃₁	C₂×D₄	1	2	yes	—
159	C₂×C₈×C₃₁	C₂×D₄	1	2	yes	—
160	C₄⋊C₄×C₃₁	—	2	2	no	—
161	C₄⋊C₄×C₃₁	—	4	2	no	—
162	C₂²⋊C₄×C₃₁	—	2	2	no	—
163	C₂²⋊C₄×C₃₁	—	2	4	no	—
164	C₂²⋊C₄×C₃₁	—	2	2	no	—
165	C₂²⋊C₄×C₃₁	—	2	2	no	—
166	C₂²⋊C₄×C₃₁	—	2	4	no	—
167	C₂²⋊C₄×C₃₁	—	4	2	no	—
168	C₂²⋊C₄×C₃₁	—	4	2	no	—
169	C₂²⋊C₄×C₃₁	M₁₆	1	4	yes	complemented
170	C₄²×C₃₁	—	2	1	no	—
171	C₄²×C₃₁	—	2	1	no	—
172	C₄²×C₃₁	—	2	2	no	—
173	C₄²×C₃₁	—	2	2	no	—
174	#22032	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	8	no	—
178	C₂³×C₃₁	—	2	2	no	—
179	C₂³×C₃₁	—	2	2	no	—
180	C₂³×C₃₁	—	2	2	no	—
181	C₂³×C₃₁	—	2	2	no	—
182	C₂³×C₃₁	—	2	2	no	—
183	C₂³×C₃₁	—	2	2	no	—
184	C₂³×C₃₁	—	2	1	no	—
185	C₂³×C₃₁	—	2	2	no	—
186	C₂³×C₃₁	—	2	1	no	—
187	C₂³×C₃₁	—	2	1	no	—
188	C₂³×C₃₁	—	2	2	no	—
189	C₂³×C₃₁	—	4	8	no	—
190	C₂³×C₃₁	—	4	8	no	—
191	C₂³×C₃₁	—	4	4	no	—
192	C₂³×C₃₁	—	4	4	no	—
193	C₂³×C₃₁	C₄×D₄	1	2	yes	—
194	C₂³×C₃₁	C₂×M₁₆	1	2	yes	—
195	C₂³×C₃₁	#131	1	4	yes	—
196	C₂³×C₃₁	#131	1	2	yes	—
197	C₂³×C₃₁	C₂³×C₄	1	1	yes	characteristic
198	D₄×C₃₁	—	4	8	no	—
199	D₄×C₃₁	—	4	4	no	—
200	C₂×C₄×C₃₁	—	2	2	no	—
201	C₂×C₄×C₃₁	—	2	2	no	—
202	C₂×C₄×C₃₁	—	2	2	no	—
203	C₂×C₄×C₃₁	—	2	2	no	—
204	C₂×C₄×C₃₁	—	2	2	no	—
205	C₂×C₄×C₃₁	—	2	4	no	—
206	C₂×C₄×C₃₁	—	2	1	no	—
207	C₂×C₄×C₃₁	—	2	1	no	—
208	C₂×C₄×C₃₁	—	2	1	no	—
209	C₂×C₄×C₃₁	—	2	2	no	—
210	C₂×C₄×C₃₁	—	2	2	no	—
211	C₂×C₄×C₃₁	—	2	2	no	—
212	C₂×C₄×C₃₁	—	2	2	no	—
213	C₂×C₄×C₃₁	—	2	2	no	—
214	C₂×C₄×C₃₁	—	2	1	no	—
215	C₂×C₄×C₃₁	—	2	2	no	—
216	C₂×C₄×C₃₁	—	2	2	no	—
217	C₂×C₄×C₃₁	—	2	2	no	—
218	C₂×C₄×C₃₁	—	2	1	no	—
219	C₂×C₄×C₃₁	—	4	8	no	—
220	C₂×C₄×C₃₁	—	4	4	no	—
221	C₂×C₄×C₃₁	—	4	1	no	—
222	C₂×C₄×C₃₁	—	4	2	no	—
223	C₂×C₄×C₃₁	—	4	1	no	—
224	C₂×C₄×C₃₁	—	4	2	no	—
225	C₂×C₄×C₃₁	—	4	2	no	—
226	C₂×C₄×C₃₁	—	4	2	no	—
227	C₂×C₄×C₃₁	—	4	2	no	—
228	C₂×C₄×C₃₁	C₄×D₄	1	2	yes	—
229	C₂×C₄×C₃₁	C₂×M₁₆	1	4	yes	—
230	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
231	C₂×C₄×C₃₁	C₂×#41	1	1	yes	characteristic
232	C₂×C₄×C₃₁	H(5,𝔽₂)	1	2	yes	—
233	C₂×C₄×C₃₁	H(5,𝔽₂)	1	2	yes	—
234	C₈×C₃₁	—	2	2	no	—
235	C₈×C₃₁	—	2	2	no	—
236	C₈×C₃₁	—	4	1	no	—
237	C₈×C₃₁	—	4	1	no	—
238	C₈×C₃₁	—	4	2	no	—
239	C₈×C₃₁	—	4	2	no	—
240	C₂×#435	C₂×C₃₁	1	1	yes	characteristic, complemented
241	C₂²×C₂²⋊C₈	C₂×C₃₁	1	1	yes	characteristic, complemented
242	C₄×#120	C₂×C₃₁	1	1	yes	characteristic
243	#1954	C₂×C₃₁	1	2	yes	complemented
244	#1924	C₂×C₃₁	1	2	yes	complemented
245	#1923	C₂×C₃₁	1	2	yes	complemented
246	#1870	C₂×C₃₁	1	2	yes	complemented
247	#1869	C₂×C₃₁	1	1	yes	characteristic, complemented
248	#1850	C₂×C₃₁	1	1	yes	characteristic, complemented
249	#1779	C₂×C₃₁	1	1	yes	characteristic, complemented
250	C₈×C₂²⋊C₄	C₂×C₃₁	1	1	yes	characteristic, complemented
251	C₂²×C₃₁	—	2	2	no	—
252	C₂²×C₃₁	—	2	2	no	—
253	C₂²×C₃₁	—	2	4	no	—
254	C₂²×C₃₁	—	2	4	no	—
255	C₂²×C₃₁	—	2	4	no	—
256	C₂²×C₃₁	—	2	4	no	—
257	C₂²×C₃₁	—	2	4	no	—
258	C₂²×C₃₁	—	4	16	no	—
259	C₂²×C₃₁	—	4	2	no	—
260	C₂²×C₃₁	—	4	2	no	—
261	C₂²×C₃₁	—	4	2	no	—
262	C₂²×C₃₁	—	4	2	no	—
263	C₂²×C₃₁	—	4	1	no	—
264	C₂²×C₃₁	—	4	1	no	—
265	C₂²×C₃₁	—	4	1	no	—
266	C₂²×C₃₁	—	4	2	no	—
267	C₂²×C₃₁	—	4	2	no	—
268	C₂²×C₃₁	#436	1	4	yes	—
269	C₂²×C₃₁	C₂×C₄×D₄	1	1	yes	characteristic
270	C₂²×C₃₁	#518	1	2	yes	—
271	C₂²×C₃₁	C₂²×M₁₆	1	1	yes	characteristic
272	C₂²×C₃₁	C₂×#131	1	1	yes	characteristic
273	C₂²×C₃₁	C₂×#131	1	2	yes	—
274	C₄×C₃₁	—	2	2	no	—
275	C₄×C₃₁	—	2	4	no	—
276	C₄×C₃₁	—	4	1	no	—
277	C₄×C₃₁	—	4	2	no	—
278	C₄×C₃₁	—	4	1	no	—
279	C₄×C₃₁	—	4	1	no	—
280	C₄×C₃₁	—	4	2	no	—
281	C₄×C₃₁	#534	1	1	yes	characteristic
282	C₄×C₃₁	#535	1	1	yes	characteristic
283	C₄×C₃₁	#538	1	2	yes	—
284	C₂⁴×C₄	C₂²×C₃₁	1	1	yes	characteristic
285	C₂²×M₁₆	C₂²×C₃₁	1	1	yes	characteristic, complemented
286	C₂³×C₈	C₂²×C₃₁	1	1	yes	characteristic
287	C₂³×C₈	C₂²×C₃₁	1	1	yes	characteristic
288	C₂×#120	C₄×C₃₁	1	1	yes	characteristic
289	C₂×C₄×D₄	C₂²×C₃₁	1	1	yes	characteristic
290	C₂×C₄×D₄	C₂²×C₃₁	1	2	yes	—
291	#435	—	2	4	no	—
292	C₂×C₄⋊C₈	C₂²×C₃₁	1	1	yes	characteristic, complemented
293	C₂×C₄⋊C₈	C₂²×C₃₁	1	2	yes	complemented
294	C₂×C₂²⋊C₈	—	2	4	no	—
295	C₂×C₂²⋊C₈	C₂²×C₃₁	1	1	yes	characteristic
296	C₂×C₂²⋊C₈	C₂²×C₃₁	1	2	yes	—
297	C₂×C₂²⋊C₈	C₂²×C₃₁	1	2	yes	—
298	C₂×C₂²⋊C₈	C₂²×C₃₁	1	1	yes	characteristic
299	C₂×C₂²⋊C₈	C₂²×C₃₁	1	2	yes	—
300	C₂×C₂²⋊C₈	C₂²×C₃₁	1	2	yes	—
301	C₂×C₂²⋊C₈	C₂²×C₃₁	1	1	yes	characteristic, complemented
302	C₂×C₂²⋊C₈	C₂²×C₃₁	1	2	yes	complemented
303	C₂×#97	C₂²×C₃₁	1	1	yes	characteristic, complemented
304	C₂×C₄×C₈	C₂²×C₃₁	1	2	yes	complemented
305	#386	C₄×C₃₁	1	2	yes	—
306	#386	C₄×C₃₁	1	1	yes	characteristic
307	#385	C₄×C₃₁	1	1	yes	characteristic
308	#385	C₄×C₃₁	1	2	yes	—
309	#379	C₄×C₃₁	1	1	yes	characteristic
310	C₄×C₂²⋊C₄	C₂²×C₃₁	1	1	yes	characteristic
311	C₄×C₂²⋊C₄	C₂²×C₃₁	1	2	yes	—
312	#336	C₂²×C₃₁	1	2	yes	complemented
313	#336	C₂²×C₃₁	1	2	yes	complemented
314	#336	C₂²×C₃₁	1	2	yes	complemented
315	C₂×C₃₁	—	2	4	no	—
316	C₂×C₃₁	—	4	4	no	—
317	C₂×C₃₁	—	8	1	no	—
318	C₂×C₃₁	#2321	1	1	yes	characteristic
319	C₂×C₃₁	C₂×#436	1	1	yes	characteristic
320	C₂×C₃₁	#2929	1	1	yes	characteristic
321	C₂×C₃₁	#2974	1	2	yes	—
322	C₂×C₃₁	#2977	1	2	yes	—
323	C₂⁵	C₂×C₄×C₃₁	1	1	yes	characteristic
324	C₂²×D₄	C₂×C₄×C₃₁	1	1	yes	characteristic
325	C₂²×D₄	C₂×C₄×C₃₁	1	2	yes	—
326	C₂³×C₄	—	2	2	no	—
327	C₂³×C₄	—	2	2	no	—
328	C₂³×C₄	—	4	2	no	—
329	C₂³×C₄	—	4	2	no	—
330	C₂³×C₄	C₂×C₄×C₃₁	1	1	yes	characteristic
331	C₂³×C₄	C₂×C₄×C₃₁	1	2	yes	—
332	C₂³×C₄	C₂³×C₃₁	1	1	yes	characteristic
333	C₂³×C₄	C₂³×C₃₁	1	2	yes	—
334	C₂³×C₄	C₂³×C₃₁	1	1	yes	characteristic
335	C₂×M₁₆	—	2	2	no	—
336	C₂×M₁₆	—	4	2	no	—
337	C₂²×C₈	—	2	2	no	—
338	C₂²×C₈	—	4	2	no	—
339	C₂²×C₈	—	4	2	no	—
340	C₂²×C₈	D₄×C₃₁	1	4	yes	complemented
341	C₂²×C₈	C₂³×C₃₁	1	2	yes	—
342	C₂²×C₈	C₂³×C₃₁	1	1	yes	characteristic
343	C₂²×C₈	C₂³×C₃₁	1	2	yes	—
344	C₂²×C₈	C₂³×C₃₁	1	1	yes	characteristic
345	C₂²×C₈	C₂³×C₃₁	1	1	yes	characteristic
346	C₂²×C₈	C₂³×C₃₁	1	1	yes	characteristic
347	#120	—	2	4	no	—
348	C₄×D₄	—	2	2	no	—
349	C₄×D₄	—	2	2	no	—
350	C₄×D₄	—	4	2	no	—
351	C₄×D₄	—	4	2	no	—
352	C₂×C₄⋊C₄	C₂×C₄×C₃₁	1	1	yes	characteristic
353	C₂×C₄⋊C₄	C₂×C₄×C₃₁	1	2	yes	—
354	C₂×C₂²⋊C₄	C₂×C₄×C₃₁	1	1	yes	characteristic
355	C₂×C₂²⋊C₄	C₂×C₄×C₃₁	1	1	yes	characteristic
356	C₂×C₂²⋊C₄	C₂×C₄×C₃₁	1	2	yes	—
357	C₂×C₂²⋊C₄	C₂×C₄×C₃₁	1	2	yes	—
358	C₂×C₂²⋊C₄	C₂×C₄×C₃₁	1	1	yes	characteristic
359	C₂×C₂²⋊C₄	C₂×C₄×C₃₁	1	2	yes	—
360	C₂×C₂²⋊C₄	C₂×C₄×C₃₁	1	2	yes	—
361	C₂×C₂²⋊C₄	C₂×C₄×C₃₁	1	1	yes	characteristic
362	C₂×C₄²	C₂³×C₃₁	1	2	yes	—
363	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
364	C₄⋊C₈	—	2	2	no	—
365	C₄⋊C₈	—	4	2	no	—
366	C₂²⋊C₈	—	2	2	no	—
367	C₂²⋊C₈	—	2	2	no	—
368	C₂²⋊C₈	—	2	4	no	—
369	C₂²⋊C₈	—	2	4	no	—
370	C₂²⋊C₈	—	2	2	no	—
371	C₂²⋊C₈	—	4	2	no	—
372	C₂²⋊C₈	—	4	2	no	—
373	C₂²⋊C₈	—	4	2	no	—
374	#97	—	2	2	no	—
375	C₄×C₈	—	4	2	no	—
376	#95	C₂×C₄×C₃₁	1	2	yes	—
377	#95	C₂×C₄×C₃₁	1	2	yes	—
378	#95	C₂×C₄×C₃₁	1	2	yes	—
379	C₃₁	#22032	1	1	yes	characteristic, direct product factor, Sylow
380	C₂⁴	—	2	2	no	—
381	C₂⁴	—	2	4	no	—
382	C₂⁴	—	2	2	no	—
383	C₂⁴	—	2	2	no	—
384	C₂⁴	—	4	2	no	—
385	C₂⁴	C₂²×C₄×C₃₁	1	1	yes	characteristic
386	C₂⁴	C₂²×C₄×C₃₁	1	2	yes	—
387	C₂⁴	C₂²×C₄×C₃₁	1	1	yes	characteristic
388	C₂×D₄	—	2	2	no	—
389	C₂×D₄	—	2	2	no	—
390	C₂×D₄	—	2	4	no	—
391	C₂×D₄	—	2	2	no	—
392	C₂×D₄	—	2	4	no	—
393	C₂×D₄	—	4	2	no	—
394	C₂²×C₄	—	2	2	no	—
395	C₂²×C₄	—	2	2	no	—
396	C₂²×C₄	—	2	2	no	—
397	C₂²×C₄	—	2	4	no	—
398	C₂²×C₄	—	2	2	no	—
399	C₂²×C₄	—	2	1	no	—
400	C₂²×C₄	—	2	1	no	—
401	C₂²×C₄	—	2	2	no	—
402	C₂²×C₄	—	2	2	no	—
403	C₂²×C₄	—	2	2	no	—
404	C₂²×C₄	—	2	2	no	—
405	C₂²×C₄	—	2	2	no	—
406	C₂²×C₄	—	2	2	no	—
407	C₂²×C₄	—	2	2	no	—
408	C₂²×C₄	—	4	2	no	—
409	C₂²×C₄	—	4	2	no	—
410	C₂²×C₄	—	4	2	no	—
411	C₂²×C₄	—	4	4	no	—
412	C₂²×C₄	—	4	4	no	—
413	C₂²×C₄	—	4	8	no	—
414	C₂²×C₄	C₂²×C₄×C₃₁	1	1	yes	characteristic
415	C₂²×C₄	C₂²×C₄×C₃₁	1	2	yes	—
416	C₂²×C₄	C₂²×C₄×C₃₁	1	1	yes	characteristic
417	C₂²×C₄	C₂²×C₄×C₃₁	1	2	yes	—
418	C₂²×C₄	C₂²×C₄×C₃₁	1	1	yes	characteristic
419	C₂²×C₄	C₂²×C₄×C₃₁	1	1	yes	characteristic
420	C₂²×C₄	C₂²×C₄×C₃₁	1	2	yes	—
421	C₂²×C₄	C₂×D₄×C₃₁	1	2	yes	—
422	C₂²×C₄	#41×C₃₁	1	2	yes	—
423	C₂²×C₄	C₂⁴×C₃₁	1	1	yes	characteristic
424	M₁₆	—	4	4	no	—
425	C₂×C₈	—	2	2	no	—
426	C₂×C₈	—	2	2	no	—
427	C₂×C₈	—	2	1	no	—
428	C₂×C₈	—	2	2	no	—
429	C₂×C₈	—	2	1	no	—
430	C₂×C₈	—	2	2	no	—
431	C₂×C₈	—	2	2	no	—
432	C₂×C₈	—	2	2	no	—
433	C₂×C₈	—	2	2	no	—
434	C₂×C₈	—	2	2	no	—
435	C₂×C₈	—	2	2	no	—
436	C₂×C₈	—	2	2	no	—
437	C₂×C₈	—	4	4	no	—
438	C₂×C₈	—	4	4	no	—
439	C₂×C₈	C₂×D₄×C₃₁	1	2	yes	—
440	C₂×C₈	C₂×D₄×C₃₁	1	2	yes	—
441	C₄⋊C₄	—	2	2	no	—
442	C₄⋊C₄	—	4	2	no	—
443	C₂²⋊C₄	—	2	2	no	—
444	C₂²⋊C₄	—	2	4	no	—
445	C₂²⋊C₄	—	2	2	no	—
446	C₂²⋊C₄	—	2	2	no	—
447	C₂²⋊C₄	—	2	4	no	—
448	C₂²⋊C₄	—	4	2	no	—
449	C₂²⋊C₄	—	4	2	no	—
450	C₂²⋊C₄	M₁₆×C₃₁	1	4	yes	complemented
451	C₄²	—	2	1	no	—
452	C₄²	—	2	1	no	—
453	C₄²	—	2	2	no	—
454	C₄²	—	2	2	no	—
455	C₂³	—	2	2	no	—
456	C₂³	—	2	2	no	—
457	C₂³	—	2	8	no	—
458	C₂³	—	2	2	no	—
459	C₂³	—	2	2	no	—
460	C₂³	—	2	2	no	—
461	C₂³	—	2	2	no	—
462	C₂³	—	2	2	no	—
463	C₂³	—	2	2	no	—
464	C₂³	—	2	1	no	—
465	C₂³	—	2	2	no	—
466	C₂³	—	2	1	no	—
467	C₂³	—	2	1	no	—
468	C₂³	—	2	2	no	—
469	C₂³	—	4	8	no	—
470	C₂³	—	4	8	no	—
471	C₂³	—	4	4	no	—
472	C₂³	—	4	4	no	—
473	C₂³	C₄×D₄×C₃₁	1	2	yes	—
474	C₂³	C₂×M₁₆×C₃₁	1	2	yes	—
475	C₂³	C₃₁×#131	1	4	yes	—
476	C₂³	C₃₁×#131	1	2	yes	—
477	C₂³	C₂³×C₄×C₃₁	1	1	yes	characteristic
478	D₄	—	4	8	no	—
479	D₄	—	4	4	no	—
480	C₂×C₄	—	2	2	no	—
481	C₂×C₄	—	2	2	no	—
482	C₂×C₄	—	2	2	no	—
483	C₂×C₄	—	2	2	no	—
484	C₂×C₄	—	2	2	no	—
485	C₂×C₄	—	2	4	no	—
486	C₂×C₄	—	2	1	no	—
487	C₂×C₄	—	2	1	no	—
488	C₂×C₄	—	2	1	no	—
489	C₂×C₄	—	2	2	no	—
490	C₂×C₄	—	2	2	no	—
491	C₂×C₄	—	2	2	no	—
492	C₂×C₄	—	2	2	no	—
493	C₂×C₄	—	2	2	no	—
494	C₂×C₄	—	2	1	no	—
495	C₂×C₄	—	2	2	no	—
496	C₂×C₄	—	2	2	no	—
497	C₂×C₄	—	2	2	no	—
498	C₂×C₄	—	2	1	no	—
499	C₂×C₄	—	4	8	no	—
500	C₂×C₄	—	4	4	no	—
501	C₂×C₄	—	4	1	no	—
502	C₂×C₄	—	4	2	no	—
503	C₂×C₄	—	4	1	no	—
504	C₂×C₄	—	4	2	no	—
505	C₂×C₄	—	4	2	no	—
506	C₂×C₄	—	4	2	no	—
507	C₂×C₄	—	4	2	no	—
508	C₂×C₄	C₄×D₄×C₃₁	1	2	yes	—
509	C₂×C₄	C₂×M₁₆×C₃₁	1	4	yes	—
510	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
511	C₂×C₄	C₂×#41×C₃₁	1	1	yes	characteristic
512	C₂×C₄	C₃₁×H(5,𝔽₂)	1	2	yes	—
513	C₂×C₄	C₃₁×H(5,𝔽₂)	1	2	yes	—
514	C₈	—	2	2	no	—
515	C₈	—	2	2	no	—
516	C₈	—	4	1	no	—
517	C₈	—	4	1	no	—
518	C₈	—	4	2	no	—
519	C₈	—	4	2	no	—
520	C₂²	—	2	2	no	—
521	C₂²	—	2	2	no	—
522	C₂²	—	2	4	no	—
523	C₂²	—	2	4	no	—
524	C₂²	—	2	4	no	—
525	C₂²	—	2	4	no	—
526	C₂²	—	2	4	no	—
527	C₂²	—	4	16	no	—
528	C₂²	—	4	2	no	—
529	C₂²	—	4	2	no	—
530	C₂²	—	4	2	no	—
531	C₂²	—	4	2	no	—
532	C₂²	—	4	1	no	—
533	C₂²	—	4	1	no	—
534	C₂²	—	4	1	no	—
535	C₂²	—	4	2	no	—
536	C₂²	—	4	2	no	—
537	C₂²	C₃₁×#436	1	4	yes	—
538	C₂²	C₂×C₄×D₄×C₃₁	1	1	yes	characteristic
539	C₂²	C₃₁×#518	1	2	yes	—
540	C₂²	C₂²×M₁₆×C₃₁	1	1	yes	characteristic
541	C₂²	C₂×C₃₁×#131	1	1	yes	characteristic
542	C₂²	C₂×C₃₁×#131	1	2	yes	—
543	C₄	—	2	2	no	—
544	C₄	—	2	4	no	—
545	C₄	—	4	2	no	—
546	C₄	—	4	1	no	—
547	C₄	—	4	1	no	—
548	C₄	—	4	1	no	—
549	C₄	—	4	2	no	—
550	C₄	C₃₁×#534	1	1	yes	characteristic
551	C₄	C₃₁×#535	1	1	yes	characteristic
552	C₄	C₃₁×#538	1	2	yes	—
553	C₂	—	2	4	no	—
554	C₂	—	4	4	no	—
555	C₂	—	8	1	no	—
556	C₂	C₃₁×#2321	1	1	yes	characteristic
557	C₂	C₂×C₃₁×#436	1	1	yes	characteristic
558	C₂	C₃₁×#2929	1	1	yes	characteristic
559	C₂	C₃₁×#2974	1	2	yes	—
560	C₂	C₃₁×#2977	1	2	yes	—

	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	C₂×C₃₁×#120	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
1. C₂×C₃₁×#435	C₂	semidirect
2. C₂²×C₃₁×C₂²⋊C₈	C₂	semidirect
4. C₃₁×#1954	C₂	semidirect
5. C₃₁×#1924	C₂	semidirect
6. C₃₁×#1923	C₂	semidirect
7. C₃₁×#1870	C₂	semidirect
8. C₃₁×#1869	C₂	semidirect
9. C₃₁×#1850	C₂	semidirect
10. C₃₁×#1779	C₂	semidirect
11. C₈×C₂²⋊C₄×C₃₁	C₂	semidirect
13. C₂²×M₁₆×C₃₁	C₂²	semidirect
20. C₂×C₃₁×C₄⋊C₈	C₂²	semidirect
21. C₂×C₃₁×C₄⋊C₈	C₂²	semidirect
29. C₂×C₃₁×C₂²⋊C₈	C₂²	semidirect
30. C₂×C₃₁×C₂²⋊C₈	C₂²	semidirect
31. C₂×C₃₁×#97	C₂²	semidirect
32. C₂×C₄×C₈×C₃₁	C₂²	semidirect
40. C₃₁×#336	C₂²	semidirect
41. C₃₁×#336	C₂²	semidirect
42. C₃₁×#336	C₂²	semidirect
60. C₂²×C₈×C₃₁	D₄	semidirect
169. C₂²⋊C₄×C₃₁	M₁₆	semidirect
174. #22032	C₃₁	direct (N × H)
240. C₂×#435	C₂×C₃₁	semidirect
241. C₂²×C₂²⋊C₈	C₂×C₃₁	semidirect
243. #1954	C₂×C₃₁	semidirect
244. #1924	C₂×C₃₁	semidirect
245. #1923	C₂×C₃₁	semidirect
246. #1870	C₂×C₃₁	semidirect
247. #1869	C₂×C₃₁	semidirect
248. #1850	C₂×C₃₁	semidirect
249. #1779	C₂×C₃₁	semidirect
250. C₈×C₂²⋊C₄	C₂×C₃₁	semidirect
285. C₂²×M₁₆	C₂²×C₃₁	semidirect
292. C₂×C₄⋊C₈	C₂²×C₃₁	semidirect
293. C₂×C₄⋊C₈	C₂²×C₃₁	semidirect
301. C₂×C₂²⋊C₈	C₂²×C₃₁	semidirect
302. C₂×C₂²⋊C₈	C₂²×C₃₁	semidirect
303. C₂×#97	C₂²×C₃₁	semidirect
304. C₂×C₄×C₈	C₂²×C₃₁	semidirect
312. #336	C₂²×C₃₁	semidirect
313. #336	C₂²×C₃₁	semidirect
314. #336	C₂²×C₃₁	semidirect
340. C₂²×C₈	D₄×C₃₁	semidirect
379. C₃₁	#22032	direct (N × H)
450. C₂²⋊C₄	M₁₆×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×#435	2
C₂²×C₃₁×C₂²⋊C₈	2
C₄×C₃₁×#120	2
C₃₁×#1954	2
C₃₁×#1924	2
C₃₁×#1923	2
C₃₁×#1870	2
C₃₁×#1869	2
C₃₁×#1850	2
C₃₁×#1779	2
C₈×C₂²⋊C₄×C₃₁	2
#22032	31

Subgroups — C31×#22032 #21478791

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#22032 #21478791