Subgroups — C₃₁×#21486 #21478245

Order 7936.

1110 conjugacy classes of subgroups.

The subgroup lattice has 560 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	2	yes	maximal, complemented
2	C₂×C₈×D₄×C₃₁	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×C₄×M₁₆×C₃₁	C₂	1	1	yes	maximal, characteristic, complemented
4	C₄×C₃₁×#121	C₂	1	1	yes	maximal, characteristic
5	C₃₁×#1954	C₂	1	1	yes	maximal, characteristic, complemented
6	C₃₁×#1939	C₂	1	1	yes	maximal, characteristic, complemented
7	C₃₁×#1924	C₂	1	2	yes	maximal, complemented
8	C₃₁×#1870	C₂	1	2	yes	maximal, complemented
9	C₃₁×#1797	C₂	1	1	yes	maximal, characteristic, complemented
10	C₃₁×#1770	C₂	1	1	yes	maximal, characteristic, complemented
11	C₃₁×#1752	C₂	1	2	yes	maximal, complemented
12	C₂²×M₁₆×C₃₁	C₂²	1	1	yes	characteristic
13	C₂²×M₁₆×C₃₁	C₂²	1	1	yes	characteristic
14	C₂³×C₈×C₃₁	C₂²	1	2	yes	—
15	C₂×C₃₁×#121	C₄	1	1	yes	characteristic
16	C₂×C₄×D₄×C₃₁	C₂²	1	2	yes	—
17	C₂×C₄×D₄×C₃₁	C₂²	1	1	yes	characteristic
18	C₂²×C₄²×C₃₁	C₂²	1	1	yes	characteristic
19	C₃₁×#435	—	2	8	no	—
20	C₈×D₄×C₃₁	—	2	4	no	—
21	C₂×C₃₁×C₄⋊C₈	C₂²	1	1	yes	characteristic
22	C₂×C₃₁×C₄⋊C₈	C₂²	1	1	yes	characteristic
23	C₂×C₃₁×C₄⋊C₈	C₂²	1	2	yes	—
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	2	yes	—
27	C₂×C₃₁×C₂²⋊C₈	C₂²	1	2	yes	—
28	C₄×M₁₆×C₃₁	—	2	4	no	—
29	C₂×C₃₁×#97	C₂²	1	1	yes	characteristic
30	C₂×C₃₁×#97	C₂²	1	1	yes	characteristic
31	C₂×C₃₁×#97	C₂²	1	2	yes	—
32	C₂×C₄×C₈×C₃₁	C₂²	1	1	yes	characteristic
33	C₂×C₄×C₈×C₃₁	C₂²	1	1	yes	characteristic
34	C₃₁×#390	C₄	1	1	yes	characteristic
35	C₃₁×#389	C₄	1	1	yes	characteristic
36	C₃₁×#388	C₄	1	2	yes	—
37	C₃₁×#386	C₄	1	2	yes	—
38	C₃₁×#380	C₄	1	1	yes	characteristic
39	C₄×C₄⋊C₄×C₃₁	C₂²	1	1	yes	characteristic
40	C₄×C₂²⋊C₄×C₃₁	C₂²	1	2	yes	—
41	C₃₁×#336	C₂²	1	2	yes	—
42	C₃₁×#336	C₂²	1	2	yes	—
43	C₃₁×#336	C₂²	1	2	yes	—
44	C₂²×D₄×C₃₁	C₂×C₄	1	2	yes	—
45	C₂²×D₄×C₃₁	C₂×C₄	1	1	yes	characteristic
46	C₂³×C₄×C₃₁	C₂×C₄	1	1	yes	characteristic
47	C₂³×C₄×C₃₁	C₂×C₄	1	1	yes	characteristic
48	C₂³×C₄×C₃₁	C₂³	1	2	yes	—
49	C₂³×C₄×C₃₁	C₂³	1	1	yes	characteristic
50	C₂×M₁₆×C₃₁	—	2	2	no	—
51	C₂×M₁₆×C₃₁	—	2	2	no	—
52	C₂×M₁₆×C₃₁	—	2	2	no	—
53	C₂×M₁₆×C₃₁	—	2	2	no	—
54	C₂×M₁₆×C₃₁	—	2	2	no	—
55	C₂×M₁₆×C₃₁	D₄	1	4	yes	complemented
56	C₂²×C₈×C₃₁	—	2	4	no	—
57	C₂²×C₈×C₃₁	—	2	4	no	—
58	C₂²×C₈×C₃₁	—	2	4	no	—
59	C₂²×C₈×C₃₁	C₂³	1	2	yes	—
60	C₂²×C₈×C₃₁	C₂³	1	2	yes	—
61	C₂²×C₈×C₃₁	C₂³	1	1	yes	characteristic
62	C₂²×C₈×C₃₁	C₂³	1	1	yes	characteristic
63	C₂²×C₈×C₃₁	C₂³	1	1	yes	characteristic
64	C₂²×C₈×C₃₁	C₂³	1	1	yes	characteristic
65	C₃₁×#121	—	2	4	no	—
66	C₄×D₄×C₃₁	—	2	4	no	—
67	C₄×D₄×C₃₁	—	2	4	no	—
68	C₄×D₄×C₃₁	—	2	2	no	—
69	C₄×D₄×C₃₁	—	2	2	no	—
70	C₂×C₄⋊C₄×C₃₁	C₂×C₄	1	1	yes	characteristic
71	C₂×C₄⋊C₄×C₃₁	C₂×C₄	1	2	yes	—
72	C₂×C₄⋊C₄×C₃₁	C₂×C₄	1	1	yes	characteristic
73	C₂×C₄⋊C₄×C₃₁	C₂×C₄	1	1	yes	characteristic
74	C₂×C₂²⋊C₄×C₃₁	C₂×C₄	1	2	yes	—
75	C₂×C₂²⋊C₄×C₃₁	C₂×C₄	1	2	yes	—
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	2	yes	—
79	C₂×C₄²×C₃₁	—	2	2	no	—
80	C₂×C₄²×C₃₁	—	2	2	no	—
81	C₂×C₄²×C₃₁	C₂×C₄	1	1	yes	characteristic
82	C₂×C₄²×C₃₁	C₂×C₄	1	1	yes	characteristic
83	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
84	C₂×C₄²×C₃₁	C₂³	1	2	yes	—
85	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
86	C₃₁×C₄⋊C₈	—	2	4	no	—
87	C₃₁×C₄⋊C₈	—	2	2	no	—
88	C₃₁×C₄⋊C₈	—	2	2	no	—
89	C₃₁×C₂²⋊C₈	—	2	4	no	—
90	C₃₁×C₂²⋊C₈	—	2	4	no	—
91	C₃₁×C₂²⋊C₈	—	2	4	no	—
92	C₃₁×C₂²⋊C₈	—	2	4	no	—
93	C₃₁×#97	—	2	2	no	—
94	C₃₁×#97	—	2	2	no	—
95	C₃₁×#97	—	2	4	no	—
96	C₄×C₈×C₃₁	—	2	2	no	—
97	C₄×C₈×C₃₁	D₄	1	4	yes	complemented
98	C₃₁×#95	C₂×C₄	1	2	yes	—
99	C₃₁×#95	C₂×C₄	1	2	yes	—
100	C₃₁×#95	C₂×C₄	1	2	yes	—
101	C₂⁴×C₃₁	C₂²×C₄	1	1	yes	characteristic
102	C₂⁴×C₃₁	C₂²×C₄	1	2	yes	—
103	C₂×D₄×C₃₁	—	2	4	no	—
104	C₂×D₄×C₃₁	—	2	4	no	—
105	C₂×D₄×C₃₁	—	2	2	no	—
106	C₂×D₄×C₃₁	—	2	2	no	—
107	C₂×D₄×C₃₁	—	4	2	no	—
108	C₂×D₄×C₃₁	M₁₆	1	4	yes	complemented
109	C₂²×C₄×C₃₁	—	2	1	no	—
110	C₂²×C₄×C₃₁	—	2	1	no	—
111	C₂²×C₄×C₃₁	—	2	1	no	—
112	C₂²×C₄×C₃₁	—	2	1	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	2	no	—
117	C₂²×C₄×C₃₁	—	2	2	no	—
118	C₂²×C₄×C₃₁	—	2	2	no	—
119	C₂²×C₄×C₃₁	—	2	2	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	—	4	1	no	—
124	C₂²×C₄×C₃₁	—	4	1	no	—
125	C₂²×C₄×C₃₁	C₂²×C₄	1	1	yes	characteristic
126	C₂²×C₄×C₃₁	C₂²×C₄	1	1	yes	characteristic
127	C₂²×C₄×C₃₁	C₂²×C₄	1	2	yes	—
128	C₂²×C₄×C₃₁	C₂²×C₄	1	2	yes	—
129	C₂²×C₄×C₃₁	C₂²×C₄	1	2	yes	—
130	C₂²×C₄×C₃₁	C₂²×C₄	1	1	yes	characteristic
131	C₂²×C₄×C₃₁	C₂²×C₄	1	1	yes	characteristic
132	C₂²×C₄×C₃₁	C₂²×C₄	1	1	yes	characteristic
133	C₂²×C₄×C₃₁	C₂×D₄	1	2	yes	—
134	C₂²×C₄×C₃₁	#41	1	2	yes	—
135	C₂²×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
136	M₁₆×C₃₁	—	2	8	no	—
137	M₁₆×C₃₁	—	2	8	no	—
138	C₂×C₈×C₃₁	—	2	1	no	—
139	C₂×C₈×C₃₁	—	2	1	no	—
140	C₂×C₈×C₃₁	—	2	1	no	—
141	C₂×C₈×C₃₁	—	2	1	no	—
142	C₂×C₈×C₃₁	—	2	1	no	—
143	C₂×C₈×C₃₁	—	2	1	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	—	2	2	no	—
148	C₂×C₈×C₃₁	—	2	2	no	—
149	C₂×C₈×C₃₁	—	2	2	no	—
150	C₂×C₈×C₃₁	—	4	8	no	—
151	C₂×C₈×C₃₁	C₂×D₄	1	2	yes	—
152	C₂×C₈×C₃₁	C₂×D₄	1	2	yes	—
153	C₂×C₈×C₃₁	C₂×D₄	1	2	yes	—
154	C₂×C₈×C₃₁	C₂×D₄	1	2	yes	—
155	C₂×C₈×C₃₁	#41	1	2	yes	—
156	C₂×C₈×C₃₁	#41	1	2	yes	—
157	C₄⋊C₄×C₃₁	—	2	4	no	—
158	C₄⋊C₄×C₃₁	—	2	2	no	—
159	C₄⋊C₄×C₃₁	—	2	2	no	—
160	C₄⋊C₄×C₃₁	—	2	2	no	—
161	C₂²⋊C₄×C₃₁	—	2	4	no	—
162	C₂²⋊C₄×C₃₁	—	2	4	no	—
163	C₂²⋊C₄×C₃₁	—	2	4	no	—
164	C₂²⋊C₄×C₃₁	—	2	4	no	—
165	C₂²⋊C₄×C₃₁	—	4	2	no	—
166	C₄²×C₃₁	—	2	2	no	—
167	C₄²×C₃₁	—	2	2	no	—
168	C₄²×C₃₁	—	2	1	no	—
169	C₄²×C₃₁	—	2	2	no	—
170	C₄²×C₃₁	—	2	1	no	—
171	C₄²×C₃₁	—	2	2	no	—
172	C₄²×C₃₁	C₂×D₄	1	2	yes	—
173	C₄²×C₃₁	#41	1	2	yes	—
174	#21486	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	1	no	—
178	C₂³×C₃₁	—	2	2	no	—
179	C₂³×C₃₁	—	2	1	no	—
180	C₂³×C₃₁	—	2	1	no	—
181	C₂³×C₃₁	—	2	1	no	—
182	C₂³×C₃₁	—	2	2	no	—
183	C₂³×C₃₁	—	2	2	no	—
184	C₂³×C₃₁	—	2	1	no	—
185	C₂³×C₃₁	—	2	1	no	—
186	C₂³×C₃₁	—	2	2	no	—
187	C₂³×C₃₁	C₄×D₄	1	2	yes	—
188	C₂³×C₃₁	C₂×M₁₆	1	4	yes	—
189	C₂³×C₃₁	C₂³×C₄	1	1	yes	characteristic
190	D₄×C₃₁	—	2	8	no	—
191	D₄×C₃₁	—	4	8	no	—
192	C₂×C₄×C₃₁	—	2	1	no	—
193	C₂×C₄×C₃₁	—	2	1	no	—
194	C₂×C₄×C₃₁	—	2	1	no	—
195	C₂×C₄×C₃₁	—	2	1	no	—
196	C₂×C₄×C₃₁	—	2	2	no	—
197	C₂×C₄×C₃₁	—	2	2	no	—
198	C₂×C₄×C₃₁	—	2	2	no	—
199	C₂×C₄×C₃₁	—	2	2	no	—
200	C₂×C₄×C₃₁	—	2	1	no	—
201	C₂×C₄×C₃₁	—	2	1	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	2	no	—
206	C₂×C₄×C₃₁	—	2	2	no	—
207	C₂×C₄×C₃₁	—	2	1	no	—
208	C₂×C₄×C₃₁	—	2	2	no	—
209	C₂×C₄×C₃₁	—	2	2	no	—
210	C₂×C₄×C₃₁	—	2	1	no	—
211	C₂×C₄×C₃₁	—	2	1	no	—
212	C₂×C₄×C₃₁	—	2	2	no	—
213	C₂×C₄×C₃₁	—	2	1	no	—
214	C₂×C₄×C₃₁	—	2	2	no	—
215	C₂×C₄×C₃₁	—	4	4	no	—
216	C₂×C₄×C₃₁	—	4	2	no	—
217	C₂×C₄×C₃₁	—	4	2	no	—
218	C₂×C₄×C₃₁	—	4	2	no	—
219	C₂×C₄×C₃₁	—	4	2	no	—
220	C₂×C₄×C₃₁	—	4	4	no	—
221	C₂×C₄×C₃₁	C₄×D₄	1	2	yes	—
222	C₂×C₄×C₃₁	C₄×D₄	1	2	yes	—
223	C₂×C₄×C₃₁	C₄×D₄	1	2	yes	—
224	C₂×C₄×C₃₁	C₂×M₁₆	1	2	yes	—
225	C₂×C₄×C₃₁	#131	1	2	yes	—
226	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
227	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
228	C₂×C₄×C₃₁	C₂×#41	1	1	yes	characteristic
229	C₂×C₄×C₃₁	C₂×#41	1	1	yes	characteristic
230	C₂×C₄×C₃₁	C₂×#41	1	1	yes	characteristic
231	C₂×C₄×C₃₁	C₂×#41	1	1	yes	characteristic
232	C₈×C₃₁	—	2	2	no	—
233	C₈×C₃₁	—	2	2	no	—
234	C₈×C₃₁	—	2	2	no	—
235	C₈×C₃₁	—	4	2	no	—
236	C₈×C₃₁	—	4	2	no	—
237	C₈×C₃₁	#121	1	4	yes	complemented
238	C₂×#435	C₂×C₃₁	1	2	yes	complemented
239	C₂×C₈×D₄	C₂×C₃₁	1	1	yes	characteristic, complemented
240	C₂×C₄×M₁₆	C₂×C₃₁	1	1	yes	characteristic, complemented
241	C₄×#121	C₂×C₃₁	1	1	yes	characteristic
242	#1954	C₂×C₃₁	1	1	yes	characteristic, complemented
243	#1939	C₂×C₃₁	1	1	yes	characteristic, complemented
244	#1924	C₂×C₃₁	1	2	yes	complemented
245	#1870	C₂×C₃₁	1	2	yes	complemented
246	#1797	C₂×C₃₁	1	1	yes	characteristic, complemented
247	#1770	C₂×C₃₁	1	1	yes	characteristic, complemented
248	#1752	C₂×C₃₁	1	2	yes	complemented
249	C₂²×C₃₁	—	2	4	no	—
250	C₂²×C₃₁	—	2	2	no	—
251	C₂²×C₃₁	—	2	4	no	—
252	C₂²×C₃₁	—	2	2	no	—
253	C₂²×C₃₁	—	2	4	no	—
254	C₂²×C₃₁	—	2	2	no	—
255	C₂²×C₃₁	—	4	1	no	—
256	C₂²×C₃₁	—	4	1	no	—
257	C₂²×C₃₁	—	4	2	no	—
258	C₂²×C₃₁	—	4	2	no	—
259	C₂²×C₃₁	—	4	1	no	—
260	C₂²×C₃₁	—	4	2	no	—
261	C₂²×C₃₁	—	4	1	no	—
262	C₂²×C₃₁	—	4	2	no	—
263	C₂²×C₃₁	C₂×C₄×D₄	1	1	yes	characteristic
264	C₂²×C₃₁	C₂×C₄×D₄	1	1	yes	characteristic
265	C₂²×C₃₁	C₄×#41	1	1	yes	characteristic
266	C₂²×C₃₁	C₂²×M₁₆	1	1	yes	characteristic
267	C₂²×C₃₁	C₂×#131	1	1	yes	characteristic
268	C₂²×C₃₁	#568	1	1	yes	characteristic
269	C₂²×C₃₁	#568	1	1	yes	characteristic
270	C₄×C₃₁	—	2	2	no	—
271	C₄×C₃₁	—	2	2	no	—
272	C₄×C₃₁	—	4	1	no	—
273	C₄×C₃₁	—	4	1	no	—
274	C₄×C₃₁	—	4	2	no	—
275	C₄×C₃₁	—	4	2	no	—
276	C₄×C₃₁	—	4	2	no	—
277	C₄×C₃₁	#435	1	4	yes	complemented
278	C₄×C₃₁	C₂×#121	1	1	yes	characteristic
279	C₄×C₃₁	#525	1	1	yes	characteristic
280	C₄×C₃₁	#529	1	1	yes	characteristic
281	C₄×C₃₁	#532	1	1	yes	characteristic
282	C₂²×M₁₆	C₂²×C₃₁	1	1	yes	characteristic
283	C₂²×M₁₆	C₂²×C₃₁	1	1	yes	characteristic
284	C₂³×C₈	C₂²×C₃₁	1	2	yes	—
285	C₂×#121	C₄×C₃₁	1	1	yes	characteristic
286	C₂×C₄×D₄	C₂²×C₃₁	1	2	yes	—
287	C₂×C₄×D₄	C₂²×C₃₁	1	1	yes	characteristic
288	C₂²×C₄²	C₂²×C₃₁	1	1	yes	characteristic
289	#435	—	2	8	no	—
290	C₈×D₄	—	2	4	no	—
291	C₂×C₄⋊C₈	C₂²×C₃₁	1	1	yes	characteristic
292	C₂×C₄⋊C₈	C₂²×C₃₁	1	1	yes	characteristic
293	C₂×C₄⋊C₈	C₂²×C₃₁	1	2	yes	—
294	C₂×C₂²⋊C₈	C₂²×C₃₁	1	2	yes	—
295	C₂×C₂²⋊C₈	C₂²×C₃₁	1	2	yes	—
296	C₂×C₂²⋊C₈	C₂²×C₃₁	1	2	yes	—
297	C₂×C₂²⋊C₈	C₂²×C₃₁	1	2	yes	—
298	C₄×M₁₆	—	2	4	no	—
299	C₂×#97	C₂²×C₃₁	1	1	yes	characteristic
300	C₂×#97	C₂²×C₃₁	1	1	yes	characteristic
301	C₂×#97	C₂²×C₃₁	1	2	yes	—
302	C₂×C₄×C₈	C₂²×C₃₁	1	1	yes	characteristic
303	C₂×C₄×C₈	C₂²×C₃₁	1	1	yes	characteristic
304	#390	C₄×C₃₁	1	1	yes	characteristic
305	#389	C₄×C₃₁	1	1	yes	characteristic
306	#388	C₄×C₃₁	1	2	yes	—
307	#386	C₄×C₃₁	1	2	yes	—
308	#380	C₄×C₃₁	1	1	yes	characteristic
309	C₄×C₄⋊C₄	C₂²×C₃₁	1	1	yes	characteristic
310	C₄×C₂²⋊C₄	C₂²×C₃₁	1	2	yes	—
311	#336	C₂²×C₃₁	1	2	yes	—
312	#336	C₂²×C₃₁	1	2	yes	—
313	#336	C₂²×C₃₁	1	2	yes	—
314	C₂×C₃₁	—	4	4	no	—
315	C₂×C₃₁	—	4	2	no	—
316	C₂×C₃₁	C₄×#121	1	1	yes	characteristic
317	C₂×C₃₁	C₂×#435	1	1	yes	characteristic
318	C₂×C₃₁	#2930	1	1	yes	characteristic
319	C₂×C₃₁	D₄×M₁₆	1	1	yes	characteristic
320	C₂×C₃₁	#2935	1	1	yes	characteristic
321	C₂×C₃₁	#2969	1	1	yes	characteristic
322	C₂×C₃₁	#2970	1	1	yes	characteristic
323	C₂²×D₄	C₂×C₄×C₃₁	1	2	yes	—
324	C₂²×D₄	C₂×C₄×C₃₁	1	1	yes	characteristic
325	C₂³×C₄	C₂×C₄×C₃₁	1	1	yes	characteristic
326	C₂³×C₄	C₂×C₄×C₃₁	1	1	yes	characteristic
327	C₂³×C₄	C₂³×C₃₁	1	2	yes	—
328	C₂³×C₄	C₂³×C₃₁	1	1	yes	characteristic
329	C₂×M₁₆	—	2	2	no	—
330	C₂×M₁₆	—	2	2	no	—
331	C₂×M₁₆	—	2	2	no	—
332	C₂×M₁₆	—	2	2	no	—
333	C₂×M₁₆	—	2	2	no	—
334	C₂×M₁₆	D₄×C₃₁	1	4	yes	complemented
335	C₂²×C₈	—	2	4	no	—
336	C₂²×C₈	—	2	4	no	—
337	C₂²×C₈	—	2	4	no	—
338	C₂²×C₈	C₂³×C₃₁	1	2	yes	—
339	C₂²×C₈	C₂³×C₃₁	1	2	yes	—
340	C₂²×C₈	C₂³×C₃₁	1	1	yes	characteristic
341	C₂²×C₈	C₂³×C₃₁	1	1	yes	characteristic
342	C₂²×C₈	C₂³×C₃₁	1	1	yes	characteristic
343	C₂²×C₈	C₂³×C₃₁	1	1	yes	characteristic
344	#121	—	2	4	no	—
345	C₄×D₄	—	2	4	no	—
346	C₄×D₄	—	2	4	no	—
347	C₄×D₄	—	2	2	no	—
348	C₄×D₄	—	2	2	no	—
349	C₂×C₄⋊C₄	C₂×C₄×C₃₁	1	1	yes	characteristic
350	C₂×C₄⋊C₄	C₂×C₄×C₃₁	1	2	yes	—
351	C₂×C₄⋊C₄	C₂×C₄×C₃₁	1	1	yes	characteristic
352	C₂×C₄⋊C₄	C₂×C₄×C₃₁	1	1	yes	characteristic
353	C₂×C₂²⋊C₄	C₂×C₄×C₃₁	1	2	yes	—
354	C₂×C₂²⋊C₄	C₂×C₄×C₃₁	1	2	yes	—
355	C₂×C₂²⋊C₄	C₂×C₄×C₃₁	1	2	yes	—
356	C₂×C₂²⋊C₄	C₂×C₄×C₃₁	1	2	yes	—
357	C₂×C₂²⋊C₄	C₂×C₄×C₃₁	1	2	yes	—
358	C₂×C₄²	—	2	2	no	—
359	C₂×C₄²	—	2	2	no	—
360	C₂×C₄²	C₂×C₄×C₃₁	1	1	yes	characteristic
361	C₂×C₄²	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	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
365	C₄⋊C₈	—	2	4	no	—
366	C₄⋊C₈	—	2	2	no	—
367	C₄⋊C₈	—	2	2	no	—
368	C₂²⋊C₈	—	2	4	no	—
369	C₂²⋊C₈	—	2	4	no	—
370	C₂²⋊C₈	—	2	4	no	—
371	C₂²⋊C₈	—	2	4	no	—
372	#97	—	2	2	no	—
373	#97	—	2	2	no	—
374	#97	—	2	4	no	—
375	C₄×C₈	—	2	2	no	—
376	C₄×C₈	D₄×C₃₁	1	4	yes	complemented
377	#95	C₂×C₄×C₃₁	1	2	yes	—
378	#95	C₂×C₄×C₃₁	1	2	yes	—
379	#95	C₂×C₄×C₃₁	1	2	yes	—
380	C₃₁	#21486	1	1	yes	characteristic, direct product factor, Sylow
381	C₂⁴	C₂²×C₄×C₃₁	1	1	yes	characteristic
382	C₂⁴	C₂²×C₄×C₃₁	1	2	yes	—
383	C₂×D₄	—	2	4	no	—
384	C₂×D₄	—	2	4	no	—
385	C₂×D₄	—	2	2	no	—
386	C₂×D₄	—	2	2	no	—
387	C₂×D₄	—	4	2	no	—
388	C₂×D₄	M₁₆×C₃₁	1	4	yes	complemented
389	C₂²×C₄	—	2	1	no	—
390	C₂²×C₄	—	2	1	no	—
391	C₂²×C₄	—	2	1	no	—
392	C₂²×C₄	—	2	1	no	—
393	C₂²×C₄	—	2	2	no	—
394	C₂²×C₄	—	2	2	no	—
395	C₂²×C₄	—	2	2	no	—
396	C₂²×C₄	—	2	2	no	—
397	C₂²×C₄	—	2	2	no	—
398	C₂²×C₄	—	2	2	no	—
399	C₂²×C₄	—	2	2	no	—
400	C₂²×C₄	—	2	2	no	—
401	C₂²×C₄	—	2	2	no	—
402	C₂²×C₄	—	2	2	no	—
403	C₂²×C₄	—	4	1	no	—
404	C₂²×C₄	—	4	1	no	—
405	C₂²×C₄	C₂²×C₄×C₃₁	1	1	yes	characteristic
406	C₂²×C₄	C₂²×C₄×C₃₁	1	1	yes	characteristic
407	C₂²×C₄	C₂²×C₄×C₃₁	1	2	yes	—
408	C₂²×C₄	C₂²×C₄×C₃₁	1	2	yes	—
409	C₂²×C₄	C₂²×C₄×C₃₁	1	2	yes	—
410	C₂²×C₄	C₂²×C₄×C₃₁	1	1	yes	characteristic
411	C₂²×C₄	C₂²×C₄×C₃₁	1	1	yes	characteristic
412	C₂²×C₄	C₂²×C₄×C₃₁	1	1	yes	characteristic
413	C₂²×C₄	C₂×D₄×C₃₁	1	2	yes	—
414	C₂²×C₄	#41×C₃₁	1	2	yes	—
415	C₂²×C₄	C₂⁴×C₃₁	1	1	yes	characteristic
416	M₁₆	—	2	8	no	—
417	M₁₆	—	2	8	no	—
418	C₂×C₈	—	2	1	no	—
419	C₂×C₈	—	2	1	no	—
420	C₂×C₈	—	2	1	no	—
421	C₂×C₈	—	2	1	no	—
422	C₂×C₈	—	2	1	no	—
423	C₂×C₈	—	2	1	no	—
424	C₂×C₈	—	2	2	no	—
425	C₂×C₈	—	2	2	no	—
426	C₂×C₈	—	2	2	no	—
427	C₂×C₈	—	2	2	no	—
428	C₂×C₈	—	2	2	no	—
429	C₂×C₈	—	2	2	no	—
430	C₂×C₈	—	4	8	no	—
431	C₂×C₈	C₂×D₄×C₃₁	1	2	yes	—
432	C₂×C₈	C₂×D₄×C₃₁	1	2	yes	—
433	C₂×C₈	C₂×D₄×C₃₁	1	2	yes	—
434	C₂×C₈	C₂×D₄×C₃₁	1	2	yes	—
435	C₂×C₈	#41×C₃₁	1	2	yes	—
436	C₂×C₈	#41×C₃₁	1	2	yes	—
437	C₄⋊C₄	—	2	4	no	—
438	C₄⋊C₄	—	2	2	no	—
439	C₄⋊C₄	—	2	2	no	—
440	C₄⋊C₄	—	2	2	no	—
441	C₂²⋊C₄	—	2	4	no	—
442	C₂²⋊C₄	—	2	4	no	—
443	C₂²⋊C₄	—	2	4	no	—
444	C₂²⋊C₄	—	2	4	no	—
445	C₂²⋊C₄	—	4	2	no	—
446	C₄²	—	2	2	no	—
447	C₄²	—	2	2	no	—
448	C₄²	—	2	1	no	—
449	C₄²	—	2	2	no	—
450	C₄²	—	2	1	no	—
451	C₄²	—	2	2	no	—
452	C₄²	C₂×D₄×C₃₁	1	2	yes	—
453	C₄²	#41×C₃₁	1	2	yes	—
454	C₂³	—	2	2	no	—
455	C₂³	—	2	2	no	—
456	C₂³	—	2	1	no	—
457	C₂³	—	2	2	no	—
458	C₂³	—	2	1	no	—
459	C₂³	—	2	1	no	—
460	C₂³	—	2	1	no	—
461	C₂³	—	2	2	no	—
462	C₂³	—	2	2	no	—
463	C₂³	—	2	1	no	—
464	C₂³	—	2	1	no	—
465	C₂³	—	2	2	no	—
466	C₂³	C₄×D₄×C₃₁	1	2	yes	—
467	C₂³	C₂×M₁₆×C₃₁	1	4	yes	—
468	C₂³	C₂³×C₄×C₃₁	1	1	yes	characteristic
469	D₄	—	2	8	no	—
470	D₄	—	4	8	no	—
471	C₂×C₄	—	2	1	no	—
472	C₂×C₄	—	2	1	no	—
473	C₂×C₄	—	2	1	no	—
474	C₂×C₄	—	2	1	no	—
475	C₂×C₄	—	2	2	no	—
476	C₂×C₄	—	2	2	no	—
477	C₂×C₄	—	2	2	no	—
478	C₂×C₄	—	2	2	no	—
479	C₂×C₄	—	2	1	no	—
480	C₂×C₄	—	2	1	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	2	no	—
486	C₂×C₄	—	2	1	no	—
487	C₂×C₄	—	2	2	no	—
488	C₂×C₄	—	2	2	no	—
489	C₂×C₄	—	2	1	no	—
490	C₂×C₄	—	2	1	no	—
491	C₂×C₄	—	2	2	no	—
492	C₂×C₄	—	2	1	no	—
493	C₂×C₄	—	2	2	no	—
494	C₂×C₄	—	4	4	no	—
495	C₂×C₄	—	4	4	no	—
496	C₂×C₄	—	4	2	no	—
497	C₂×C₄	—	4	2	no	—
498	C₂×C₄	—	4	2	no	—
499	C₂×C₄	—	4	2	no	—
500	C₂×C₄	C₄×D₄×C₃₁	1	2	yes	—
501	C₂×C₄	C₄×D₄×C₃₁	1	2	yes	—
502	C₂×C₄	C₄×D₄×C₃₁	1	2	yes	—
503	C₂×C₄	C₂×M₁₆×C₃₁	1	2	yes	—
504	C₂×C₄	C₃₁×#131	1	2	yes	—
505	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
506	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
507	C₂×C₄	C₂×#41×C₃₁	1	1	yes	characteristic
508	C₂×C₄	C₂×#41×C₃₁	1	1	yes	characteristic
509	C₂×C₄	C₂×#41×C₃₁	1	1	yes	characteristic
510	C₂×C₄	C₂×#41×C₃₁	1	1	yes	characteristic
511	C₈	—	2	2	no	—
512	C₈	—	2	2	no	—
513	C₈	—	2	2	no	—
514	C₈	—	4	2	no	—
515	C₈	—	4	2	no	—
516	C₈	C₃₁×#121	1	4	yes	complemented
517	C₂²	—	2	4	no	—
518	C₂²	—	2	2	no	—
519	C₂²	—	2	4	no	—
520	C₂²	—	2	2	no	—
521	C₂²	—	2	4	no	—
522	C₂²	—	2	2	no	—
523	C₂²	—	4	1	no	—
524	C₂²	—	4	1	no	—
525	C₂²	—	4	2	no	—
526	C₂²	—	4	2	no	—
527	C₂²	—	4	1	no	—
528	C₂²	—	4	2	no	—
529	C₂²	—	4	1	no	—
530	C₂²	—	4	2	no	—
531	C₂²	C₂×C₄×D₄×C₃₁	1	1	yes	characteristic
532	C₂²	C₂×C₄×D₄×C₃₁	1	1	yes	characteristic
533	C₂²	C₄×#41×C₃₁	1	1	yes	characteristic
534	C₂²	C₂²×M₁₆×C₃₁	1	1	yes	characteristic
535	C₂²	C₂×C₃₁×#131	1	1	yes	characteristic
536	C₂²	C₃₁×#568	1	1	yes	characteristic
537	C₂²	C₃₁×#568	1	1	yes	characteristic
538	C₄	—	2	2	no	—
539	C₄	—	2	2	no	—
540	C₄	—	4	1	no	—
541	C₄	—	4	1	no	—
542	C₄	—	4	2	no	—
543	C₄	—	4	2	no	—
544	C₄	—	4	2	no	—
545	C₄	C₃₁×#435	1	4	yes	complemented
546	C₄	C₂×C₃₁×#121	1	1	yes	characteristic
547	C₄	C₃₁×#525	1	1	yes	characteristic
548	C₄	C₃₁×#529	1	1	yes	characteristic
549	C₄	C₃₁×#532	1	1	yes	characteristic
550	C₂	—	4	4	no	—
551	C₂	—	4	2	no	—
552	C₂	C₄×C₃₁×#121	1	1	yes	characteristic
553	C₂	C₂×C₃₁×#435	1	1	yes	characteristic
554	C₂	C₃₁×#2930	1	1	yes	characteristic
555	C₂	D₄×M₁₆×C₃₁	1	1	yes	characteristic
556	C₂	C₃₁×#2935	1	1	yes	characteristic
557	C₂	C₃₁×#2969	1	1	yes	characteristic
558	C₂	C₃₁×#2970	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	C₂×C₃₁×#121	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. C₂×C₈×D₄×C₃₁	C₂	semidirect
3. C₂×C₄×M₁₆×C₃₁	C₂	semidirect
5. C₃₁×#1954	C₂	semidirect
6. C₃₁×#1939	C₂	semidirect
7. C₃₁×#1924	C₂	semidirect
8. C₃₁×#1870	C₂	semidirect
9. C₃₁×#1797	C₂	semidirect
10. C₃₁×#1770	C₂	semidirect
11. C₃₁×#1752	C₂	semidirect
55. C₂×M₁₆×C₃₁	D₄	semidirect
97. C₄×C₈×C₃₁	D₄	semidirect
108. C₂×D₄×C₃₁	M₁₆	semidirect
174. #21486	C₃₁	direct (N × H)
237. C₈×C₃₁	#121	semidirect
238. C₂×#435	C₂×C₃₁	semidirect
239. C₂×C₈×D₄	C₂×C₃₁	semidirect
240. C₂×C₄×M₁₆	C₂×C₃₁	semidirect
242. #1954	C₂×C₃₁	semidirect
243. #1939	C₂×C₃₁	semidirect
244. #1924	C₂×C₃₁	semidirect
245. #1870	C₂×C₃₁	semidirect
246. #1797	C₂×C₃₁	semidirect
247. #1770	C₂×C₃₁	semidirect
248. #1752	C₂×C₃₁	semidirect
277. C₄×C₃₁	#435	semidirect
334. C₂×M₁₆	D₄×C₃₁	semidirect
376. C₄×C₈	D₄×C₃₁	semidirect
380. C₃₁	#21486	direct (N × H)
388. C₂×D₄	M₁₆×C₃₁	semidirect
516. C₈	C₃₁×#121	semidirect
545. C₄	C₃₁×#435	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
C₂×C₈×D₄×C₃₁	2
C₂×C₄×M₁₆×C₃₁	2
C₄×C₃₁×#121	2
C₃₁×#1954	2
C₃₁×#1939	2
C₃₁×#1924	2
C₃₁×#1870	2
C₃₁×#1797	2
C₃₁×#1770	2
C₃₁×#1752	2
#21486	31

Subgroups — C31×#21486 #21478245

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#21486 #21478245