Subgroups — C₃₁×#21441 #21478200

Order 7936.

1584 conjugacy classes of subgroups.

The subgroup lattice has 580 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂³×M₁₆×C₃₁	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×C₃₁×#436	C₂	1	2	yes	maximal, complemented
3	C₂×C₃₁×#435	C₂	1	1	yes	maximal, characteristic, complemented
4	C₄×C₃₁×#120	C₂	1	1	yes	maximal, characteristic
5	C₃₁×#1870	C₂	1	2	yes	maximal, complemented
6	C₃₁×#1869	C₂	1	1	yes	maximal, characteristic, complemented
7	C₃₁×#1851	C₂	1	2	yes	maximal, complemented
8	C₃₁×#1850	C₂	1	1	yes	maximal, characteristic, complemented
9	C₃₁×#1779	C₂	1	1	yes	maximal, characteristic, complemented
10	C₃₁×#1753	C₂	1	2	yes	maximal, 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	—	2	8	no	—
14	C₂²×M₁₆×C₃₁	—	2	4	no	—
15	C₂²×M₁₆×C₃₁	C₂²	1	2	yes	—
16	C₂²×M₁₆×C₃₁	C₂²	1	2	yes	—
17	C₂²×M₁₆×C₃₁	C₂²	1	1	yes	characteristic, complemented
18	C₂³×C₈×C₃₁	C₂²	1	1	yes	characteristic
19	C₂³×C₈×C₃₁	C₂²	1	1	yes	characteristic
20	C₂×C₃₁×#120	C₄	1	1	yes	characteristic
21	C₂×C₄×D₄×C₃₁	C₂²	1	1	yes	characteristic
22	C₂×C₄×D₄×C₃₁	C₂²	1	2	yes	—
23	C₃₁×#436	—	2	8	no	—
24	C₃₁×#435	—	2	4	no	—
25	C₂×C₃₁×C₄⋊C₈	C₂²	1	2	yes	complemented
26	C₂×C₃₁×C₄⋊C₈	C₂²	1	1	yes	characteristic, complemented
27	C₂×C₃₁×C₂²⋊C₈	C₂²	1	1	yes	characteristic
28	C₂×C₃₁×C₂²⋊C₈	C₂²	1	2	yes	—
29	C₂×C₃₁×C₂²⋊C₈	C₂²	1	1	yes	characteristic
30	C₂×C₃₁×C₂²⋊C₈	C₂²	1	2	yes	—
31	C₂×C₃₁×C₂²⋊C₈	C₂²	1	2	yes	complemented
32	C₂×C₃₁×C₂²⋊C₈	C₂²	1	1	yes	characteristic, complemented
33	C₂×C₃₁×#97	C₂²	1	1	yes	characteristic, complemented
34	C₂×C₄×C₈×C₃₁	C₂²	1	2	yes	complemented
35	C₃₁×#386	C₄	1	1	yes	characteristic
36	C₃₁×#386	C₄	1	2	yes	—
37	C₃₁×#385	C₄	1	2	yes	—
38	C₃₁×#385	C₄	1	1	yes	characteristic
39	C₃₁×#379	C₄	1	1	yes	characteristic
40	C₄×C₂²⋊C₄×C₃₁	C₂²	1	1	yes	characteristic
41	C₄×C₂²⋊C₄×C₃₁	C₂²	1	2	yes	—
42	C₃₁×#336	C₂²	1	2	yes	complemented
43	C₃₁×#336	C₂²	1	2	yes	complemented
44	C₃₁×#336	C₂²	1	2	yes	complemented
45	C₂⁵×C₃₁	C₂×C₄	1	1	yes	characteristic
46	C₂²×D₄×C₃₁	C₂×C₄	1	1	yes	characteristic
47	C₂²×D₄×C₃₁	C₂×C₄	1	2	yes	—
48	C₂³×C₄×C₃₁	—	2	4	no	—
49	C₂³×C₄×C₃₁	—	2	2	no	—
50	C₂³×C₄×C₃₁	—	2	4	no	—
51	C₂³×C₄×C₃₁	—	2	2	no	—
52	C₂³×C₄×C₃₁	C₂×C₄	1	2	yes	—
53	C₂³×C₄×C₃₁	C₂×C₄	1	1	yes	characteristic
54	C₂³×C₄×C₃₁	C₂³	1	1	yes	characteristic
55	C₂³×C₄×C₃₁	C₂³	1	1	yes	characteristic
56	C₂³×C₄×C₃₁	C₂³	1	2	yes	—
57	C₂×M₁₆×C₃₁	—	2	32	no	—
58	C₂×M₁₆×C₃₁	—	2	4	no	—
59	C₂×M₁₆×C₃₁	—	2	4	no	—
60	C₂×M₁₆×C₃₁	—	2	4	no	—
61	C₂×M₁₆×C₃₁	—	2	4	no	—
62	C₂×M₁₆×C₃₁	—	2	4	no	—
63	C₂×M₁₆×C₃₁	—	2	2	no	—
64	C₂×M₁₆×C₃₁	—	2	4	no	—
65	C₂×M₁₆×C₃₁	D₄	1	8	yes	complemented
66	C₂²×C₈×C₃₁	—	2	2	no	—
67	C₂²×C₈×C₃₁	—	2	4	no	—
68	C₂²×C₈×C₃₁	—	2	4	no	—
69	C₂²×C₈×C₃₁	D₄	1	4	yes	complemented
70	C₂²×C₈×C₃₁	C₂³	1	2	yes	—
71	C₂²×C₈×C₃₁	C₂³	1	1	yes	characteristic
72	C₂²×C₈×C₃₁	C₂³	1	1	yes	characteristic
73	C₂²×C₈×C₃₁	C₂³	1	2	yes	—
74	C₂²×C₈×C₃₁	C₂³	1	1	yes	characteristic
75	C₂²×C₈×C₃₁	C₂³	1	1	yes	characteristic
76	C₃₁×#120	—	2	4	no	—
77	C₄×D₄×C₃₁	—	2	2	no	—
78	C₄×D₄×C₃₁	—	2	4	no	—
79	C₄×D₄×C₃₁	—	2	2	no	—
80	C₄×D₄×C₃₁	—	2	4	no	—
81	C₂×C₄⋊C₄×C₃₁	C₂×C₄	1	1	yes	characteristic
82	C₂×C₄⋊C₄×C₃₁	C₂×C₄	1	2	yes	—
83	C₂×C₂²⋊C₄×C₃₁	C₂×C₄	1	2	yes	—
84	C₂×C₂²⋊C₄×C₃₁	C₂×C₄	1	1	yes	characteristic
85	C₂×C₂²⋊C₄×C₃₁	C₂×C₄	1	1	yes	characteristic
86	C₂×C₂²⋊C₄×C₃₁	C₂×C₄	1	2	yes	—
87	C₂×C₂²⋊C₄×C₃₁	C₂×C₄	1	2	yes	—
88	C₂×C₂²⋊C₄×C₃₁	C₂×C₄	1	1	yes	characteristic
89	C₂×C₂²⋊C₄×C₃₁	C₂×C₄	1	2	yes	—
90	C₂×C₂²⋊C₄×C₃₁	C₂×C₄	1	1	yes	characteristic
91	C₂×C₄²×C₃₁	C₂³	1	2	yes	—
92	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
93	C₃₁×C₄⋊C₈	—	2	2	no	—
94	C₃₁×C₄⋊C₈	—	2	4	no	—
95	C₃₁×C₂²⋊C₈	—	2	2	no	—
96	C₃₁×C₂²⋊C₈	—	2	4	no	—
97	C₃₁×C₂²⋊C₈	—	2	2	no	—
98	C₃₁×C₂²⋊C₈	—	2	4	no	—
99	C₃₁×C₂²⋊C₈	—	2	2	no	—
100	C₃₁×C₂²⋊C₈	—	2	4	no	—
101	C₃₁×#97	—	2	2	no	—
102	C₄×C₈×C₃₁	—	2	4	no	—
103	C₃₁×#95	C₂×C₄	1	2	yes	—
104	C₃₁×#95	C₂×C₄	1	2	yes	—
105	C₃₁×#95	C₂×C₄	1	2	yes	—
106	C₂⁴×C₃₁	—	2	4	no	—
107	C₂⁴×C₃₁	—	2	2	no	—
108	C₂⁴×C₃₁	—	2	2	no	—
109	C₂⁴×C₃₁	—	2	2	no	—
110	C₂⁴×C₃₁	—	4	2	no	—
111	C₂⁴×C₃₁	C₂²×C₄	1	1	yes	characteristic
112	C₂⁴×C₃₁	C₂²×C₄	1	2	yes	—
113	C₂⁴×C₃₁	C₂²×C₄	1	1	yes	characteristic
114	C₂×D₄×C₃₁	—	2	4	no	—
115	C₂×D₄×C₃₁	—	2	2	no	—
116	C₂×D₄×C₃₁	—	2	2	no	—
117	C₂×D₄×C₃₁	—	2	2	no	—
118	C₂×D₄×C₃₁	—	2	4	no	—
119	C₂×D₄×C₃₁	—	4	2	no	—
120	C₂²×C₄×C₃₁	—	2	8	no	—
121	C₂²×C₄×C₃₁	—	2	8	no	—
122	C₂²×C₄×C₃₁	—	2	16	no	—
123	C₂²×C₄×C₃₁	—	2	4	no	—
124	C₂²×C₄×C₃₁	—	2	2	no	—
125	C₂²×C₄×C₃₁	—	2	4	no	—
126	C₂²×C₄×C₃₁	—	2	4	no	—
127	C₂²×C₄×C₃₁	—	2	4	no	—
128	C₂²×C₄×C₃₁	—	2	2	no	—
129	C₂²×C₄×C₃₁	—	2	2	no	—
130	C₂²×C₄×C₃₁	—	2	2	no	—
131	C₂²×C₄×C₃₁	—	2	2	no	—
132	C₂²×C₄×C₃₁	—	2	2	no	—
133	C₂²×C₄×C₃₁	—	2	2	no	—
134	C₂²×C₄×C₃₁	—	2	1	no	—
135	C₂²×C₄×C₃₁	—	2	2	no	—
136	C₂²×C₄×C₃₁	—	2	1	no	—
137	C₂²×C₄×C₃₁	—	2	2	no	—
138	C₂²×C₄×C₃₁	C₂²×C₄	1	2	yes	—
139	C₂²×C₄×C₃₁	C₂²×C₄	1	1	yes	characteristic
140	C₂²×C₄×C₃₁	C₂²×C₄	1	1	yes	characteristic
141	C₂²×C₄×C₃₁	C₂²×C₄	1	1	yes	characteristic
142	C₂²×C₄×C₃₁	C₂²×C₄	1	1	yes	characteristic
143	C₂²×C₄×C₃₁	C₂²×C₄	1	2	yes	—
144	C₂²×C₄×C₃₁	C₂²×C₄	1	2	yes	—
145	C₂²×C₄×C₃₁	C₂×D₄	1	2	yes	—
146	C₂²×C₄×C₃₁	C₂×D₄	1	4	yes	—
147	C₂²×C₄×C₃₁	#41	1	4	yes	—
148	C₂²×C₄×C₃₁	#41	1	2	yes	—
149	C₂²×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
150	M₁₆×C₃₁	—	2	16	no	—
151	M₁₆×C₃₁	—	2	16	no	—
152	M₁₆×C₃₁	—	4	4	no	—
153	C₂×C₈×C₃₁	—	2	8	no	—
154	C₂×C₈×C₃₁	—	2	2	no	—
155	C₂×C₈×C₃₁	—	2	2	no	—
156	C₂×C₈×C₃₁	—	2	2	no	—
157	C₂×C₈×C₃₁	—	2	8	no	—
158	C₂×C₈×C₃₁	—	2	2	no	—
159	C₂×C₈×C₃₁	—	2	1	no	—
160	C₂×C₈×C₃₁	—	2	1	no	—
161	C₂×C₈×C₃₁	—	2	2	no	—
162	C₂×C₈×C₃₁	—	2	2	no	—
163	C₂×C₈×C₃₁	—	2	2	no	—
164	C₂×C₈×C₃₁	—	2	2	no	—
165	C₂×C₈×C₃₁	C₂×D₄	1	4	yes	—
166	C₂×C₈×C₃₁	C₂×D₄	1	2	yes	—
167	C₂×C₈×C₃₁	C₂×D₄	1	4	yes	—
168	C₂×C₈×C₃₁	C₂×D₄	1	2	yes	—
169	C₄⋊C₄×C₃₁	—	2	4	no	—
170	C₄⋊C₄×C₃₁	—	2	2	no	—
171	C₂²⋊C₄×C₃₁	—	2	4	no	—
172	C₂²⋊C₄×C₃₁	—	2	2	no	—
173	C₂²⋊C₄×C₃₁	—	2	2	no	—
174	C₂²⋊C₄×C₃₁	—	2	2	no	—
175	C₂²⋊C₄×C₃₁	—	2	4	no	—
176	C₂²⋊C₄×C₃₁	—	2	4	no	—
177	C₂²⋊C₄×C₃₁	—	4	2	no	—
178	C₂²⋊C₄×C₃₁	M₁₆	1	4	yes	complemented
179	C₄²×C₃₁	—	2	2	no	—
180	C₄²×C₃₁	—	2	1	no	—
181	C₄²×C₃₁	—	2	2	no	—
182	C₄²×C₃₁	—	2	1	no	—
183	#21441	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
184	C₂³×C₃₁	—	2	2	no	—
185	C₂³×C₃₁	—	2	2	no	—
186	C₂³×C₃₁	—	2	8	no	—
187	C₂³×C₃₁	—	2	2	no	—
188	C₂³×C₃₁	—	2	2	no	—
189	C₂³×C₃₁	—	2	2	no	—
190	C₂³×C₃₁	—	2	2	no	—
191	C₂³×C₃₁	—	2	2	no	—
192	C₂³×C₃₁	—	2	2	no	—
193	C₂³×C₃₁	—	2	1	no	—
194	C₂³×C₃₁	—	2	2	no	—
195	C₂³×C₃₁	—	2	1	no	—
196	C₂³×C₃₁	—	2	1	no	—
197	C₂³×C₃₁	—	2	2	no	—
198	C₂³×C₃₁	—	4	8	no	—
199	C₂³×C₃₁	—	4	8	no	—
200	C₂³×C₃₁	—	4	4	no	—
201	C₂³×C₃₁	—	4	4	no	—
202	C₂³×C₃₁	C₄×D₄	1	4	yes	—
203	C₂³×C₃₁	C₄×D₄	1	2	yes	—
204	C₂³×C₃₁	C₂×M₁₆	1	2	yes	—
205	C₂³×C₃₁	#131	1	2	yes	—
206	C₂³×C₃₁	C₂³×C₄	1	1	yes	characteristic
207	D₄×C₃₁	—	4	8	no	—
208	D₄×C₃₁	—	4	4	no	—
209	C₂×C₄×C₃₁	—	2	8	no	—
210	C₂×C₄×C₃₁	—	2	16	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	2	no	—
215	C₂×C₄×C₃₁	—	2	2	no	—
216	C₂×C₄×C₃₁	—	2	2	no	—
217	C₂×C₄×C₃₁	—	2	4	no	—
218	C₂×C₄×C₃₁	—	2	2	no	—
219	C₂×C₄×C₃₁	—	2	4	no	—
220	C₂×C₄×C₃₁	—	2	4	no	—
221	C₂×C₄×C₃₁	—	2	4	no	—
222	C₂×C₄×C₃₁	—	2	4	no	—
223	C₂×C₄×C₃₁	—	2	1	no	—
224	C₂×C₄×C₃₁	—	2	2	no	—
225	C₂×C₄×C₃₁	—	2	1	no	—
226	C₂×C₄×C₃₁	—	2	1	no	—
227	C₂×C₄×C₃₁	—	2	2	no	—
228	C₂×C₄×C₃₁	—	2	1	no	—
229	C₂×C₄×C₃₁	—	2	2	no	—
230	C₂×C₄×C₃₁	—	2	1	no	—
231	C₂×C₄×C₃₁	—	2	2	no	—
232	C₂×C₄×C₃₁	—	2	2	no	—
233	C₂×C₄×C₃₁	—	4	1	no	—
234	C₂×C₄×C₃₁	—	4	1	no	—
235	C₂×C₄×C₃₁	—	4	2	no	—
236	C₂×C₄×C₃₁	C₄×D₄	1	2	yes	—
237	C₂×C₄×C₃₁	C₄×D₄	1	4	yes	—
238	C₂×C₄×C₃₁	C₂×M₁₆	1	4	yes	complemented
239	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
240	C₂×C₄×C₃₁	C₂²×D₄	1	2	yes	—
241	C₂×C₄×C₃₁	C₂×#41	1	1	yes	characteristic
242	C₂×C₄×C₃₁	C₂×#41	1	2	yes	—
243	C₈×C₃₁	—	2	4	no	—
244	C₈×C₃₁	—	2	2	no	—
245	C₈×C₃₁	—	4	1	no	—
246	C₈×C₃₁	—	4	2	no	—
247	C₈×C₃₁	—	4	1	no	—
248	C₈×C₃₁	#120	1	4	yes	complemented
249	C₂³×M₁₆	C₂×C₃₁	1	1	yes	characteristic, complemented
250	C₂×#436	C₂×C₃₁	1	2	yes	complemented
251	C₂×#435	C₂×C₃₁	1	1	yes	characteristic, complemented
252	C₄×#120	C₂×C₃₁	1	1	yes	characteristic
253	#1870	C₂×C₃₁	1	2	yes	complemented
254	#1869	C₂×C₃₁	1	1	yes	characteristic, complemented
255	#1851	C₂×C₃₁	1	2	yes	complemented
256	#1850	C₂×C₃₁	1	1	yes	characteristic, complemented
257	#1779	C₂×C₃₁	1	1	yes	characteristic, complemented
258	#1753	C₂×C₃₁	1	2	yes	complemented
259	C₈×C₂²⋊C₄	C₂×C₃₁	1	1	yes	characteristic, complemented
260	C₂²×C₃₁	—	2	2	no	—
261	C₂²×C₃₁	—	2	4	no	—
262	C₂²×C₃₁	—	2	4	no	—
263	C₂²×C₃₁	—	2	2	no	—
264	C₂²×C₃₁	—	2	4	no	—
265	C₂²×C₃₁	—	2	4	no	—
266	C₂²×C₃₁	—	2	4	no	—
267	C₂²×C₃₁	—	4	16	no	—
268	C₂²×C₃₁	—	4	2	no	—
269	C₂²×C₃₁	—	4	2	no	—
270	C₂²×C₃₁	—	4	2	no	—
271	C₂²×C₃₁	—	4	2	no	—
272	C₂²×C₃₁	—	4	1	no	—
273	C₂²×C₃₁	—	4	1	no	—
274	C₂²×C₃₁	—	4	2	no	—
275	C₂²×C₃₁	—	4	2	no	—
276	C₂²×C₃₁	—	4	1	no	—
277	C₂²×C₃₁	#436	1	4	yes	complemented
278	C₂²×C₃₁	C₂×C₄×D₄	1	1	yes	characteristic
279	C₂²×C₃₁	C₂×C₄×D₄	1	2	yes	—
280	C₂²×C₃₁	C₂²×M₁₆	1	1	yes	characteristic
281	C₂²×C₃₁	C₂×#131	1	1	yes	characteristic
282	C₂²×C₃₁	#568	1	2	yes	—
283	C₄×C₃₁	—	2	4	no	—
284	C₄×C₃₁	—	2	2	no	—
285	C₄×C₃₁	—	2	4	no	—
286	C₄×C₃₁	—	4	1	no	—
287	C₄×C₃₁	—	4	1	no	—
288	C₄×C₃₁	—	4	2	no	—
289	C₄×C₃₁	—	4	1	no	—
290	C₄×C₃₁	C₂×#120	1	1	yes	characteristic
291	C₄×C₃₁	#525	1	2	yes	—
292	C₄×C₃₁	#525	1	1	yes	characteristic
293	C₂⁴×C₄	C₂²×C₃₁	1	1	yes	characteristic
294	C₂²×M₁₆	—	2	8	no	—
295	C₂²×M₁₆	—	2	4	no	—
296	C₂²×M₁₆	C₂²×C₃₁	1	2	yes	—
297	C₂²×M₁₆	C₂²×C₃₁	1	2	yes	—
298	C₂²×M₁₆	C₂²×C₃₁	1	1	yes	characteristic, complemented
299	C₂³×C₈	C₂²×C₃₁	1	1	yes	characteristic
300	C₂³×C₈	C₂²×C₃₁	1	1	yes	characteristic
301	C₂×#120	C₄×C₃₁	1	1	yes	characteristic
302	C₂×C₄×D₄	C₂²×C₃₁	1	1	yes	characteristic
303	C₂×C₄×D₄	C₂²×C₃₁	1	2	yes	—
304	#436	—	2	8	no	—
305	#435	—	2	4	no	—
306	C₂×C₄⋊C₈	C₂²×C₃₁	1	2	yes	complemented
307	C₂×C₄⋊C₈	C₂²×C₃₁	1	1	yes	characteristic, complemented
308	C₂×C₂²⋊C₈	C₂²×C₃₁	1	1	yes	characteristic
309	C₂×C₂²⋊C₈	C₂²×C₃₁	1	2	yes	—
310	C₂×C₂²⋊C₈	C₂²×C₃₁	1	1	yes	characteristic
311	C₂×C₂²⋊C₈	C₂²×C₃₁	1	2	yes	—
312	C₂×C₂²⋊C₈	C₂²×C₃₁	1	2	yes	complemented
313	C₂×C₂²⋊C₈	C₂²×C₃₁	1	1	yes	characteristic, complemented
314	C₂×#97	C₂²×C₃₁	1	1	yes	characteristic, complemented
315	C₂×C₄×C₈	C₂²×C₃₁	1	2	yes	complemented
316	#386	C₄×C₃₁	1	1	yes	characteristic
317	#386	C₄×C₃₁	1	2	yes	—
318	#385	C₄×C₃₁	1	2	yes	—
319	#385	C₄×C₃₁	1	1	yes	characteristic
320	#379	C₄×C₃₁	1	1	yes	characteristic
321	C₄×C₂²⋊C₄	C₂²×C₃₁	1	1	yes	characteristic
322	C₄×C₂²⋊C₄	C₂²×C₃₁	1	2	yes	—
323	#336	C₂²×C₃₁	1	2	yes	complemented
324	#336	C₂²×C₃₁	1	2	yes	complemented
325	#336	C₂²×C₃₁	1	2	yes	complemented
326	C₂×C₃₁	—	2	4	no	—
327	C₂×C₃₁	—	4	4	no	—
328	C₂×C₃₁	—	8	1	no	—
329	C₂×C₃₁	C₄×#120	1	1	yes	characteristic
330	C₂×C₃₁	C₂×#436	1	1	yes	characteristic
331	C₂×C₃₁	#2930	1	1	yes	characteristic
332	C₂×C₃₁	#2933	1	2	yes	—
333	C₂×C₃₁	D₄×M₁₆	1	2	yes	—
334	C₂⁵	C₂×C₄×C₃₁	1	1	yes	characteristic
335	C₂²×D₄	C₂×C₄×C₃₁	1	1	yes	characteristic
336	C₂²×D₄	C₂×C₄×C₃₁	1	2	yes	—
337	C₂³×C₄	—	2	4	no	—
338	C₂³×C₄	—	2	2	no	—
339	C₂³×C₄	—	2	4	no	—
340	C₂³×C₄	—	2	2	no	—
341	C₂³×C₄	C₂×C₄×C₃₁	1	2	yes	—
342	C₂³×C₄	C₂×C₄×C₃₁	1	1	yes	characteristic
343	C₂³×C₄	C₂³×C₃₁	1	1	yes	characteristic
344	C₂³×C₄	C₂³×C₃₁	1	1	yes	characteristic
345	C₂³×C₄	C₂³×C₃₁	1	2	yes	—
346	C₂×M₁₆	—	2	32	no	—
347	C₂×M₁₆	—	2	4	no	—
348	C₂×M₁₆	—	2	4	no	—
349	C₂×M₁₆	—	2	4	no	—
350	C₂×M₁₆	—	2	4	no	—
351	C₂×M₁₆	—	2	4	no	—
352	C₂×M₁₆	—	2	2	no	—
353	C₂×M₁₆	—	2	4	no	—
354	C₂×M₁₆	D₄×C₃₁	1	8	yes	complemented
355	C₂²×C₈	—	2	2	no	—
356	C₂²×C₈	—	2	4	no	—
357	C₂²×C₈	—	2	4	no	—
358	C₂²×C₈	D₄×C₃₁	1	4	yes	complemented
359	C₂²×C₈	C₂³×C₃₁	1	1	yes	characteristic
360	C₂²×C₈	C₂³×C₃₁	1	2	yes	—
361	C₂²×C₈	C₂³×C₃₁	1	1	yes	characteristic
362	C₂²×C₈	C₂³×C₃₁	1	2	yes	—
363	C₂²×C₈	C₂³×C₃₁	1	1	yes	characteristic
364	C₂²×C₈	C₂³×C₃₁	1	1	yes	characteristic
365	#120	—	2	4	no	—
366	C₄×D₄	—	2	2	no	—
367	C₄×D₄	—	2	4	no	—
368	C₄×D₄	—	2	2	no	—
369	C₄×D₄	—	2	4	no	—
370	C₂×C₄⋊C₄	C₂×C₄×C₃₁	1	1	yes	characteristic
371	C₂×C₄⋊C₄	C₂×C₄×C₃₁	1	2	yes	—
372	C₂×C₂²⋊C₄	C₂×C₄×C₃₁	1	2	yes	—
373	C₂×C₂²⋊C₄	C₂×C₄×C₃₁	1	1	yes	characteristic
374	C₂×C₂²⋊C₄	C₂×C₄×C₃₁	1	1	yes	characteristic
375	C₂×C₂²⋊C₄	C₂×C₄×C₃₁	1	2	yes	—
376	C₂×C₂²⋊C₄	C₂×C₄×C₃₁	1	2	yes	—
377	C₂×C₂²⋊C₄	C₂×C₄×C₃₁	1	1	yes	characteristic
378	C₂×C₂²⋊C₄	C₂×C₄×C₃₁	1	2	yes	—
379	C₂×C₂²⋊C₄	C₂×C₄×C₃₁	1	1	yes	characteristic
380	C₂×C₄²	C₂³×C₃₁	1	2	yes	—
381	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
382	C₄⋊C₈	—	2	2	no	—
383	C₄⋊C₈	—	2	4	no	—
384	C₂²⋊C₈	—	2	2	no	—
385	C₂²⋊C₈	—	2	4	no	—
386	C₂²⋊C₈	—	2	2	no	—
387	C₂²⋊C₈	—	2	4	no	—
388	C₂²⋊C₈	—	2	2	no	—
389	C₂²⋊C₈	—	2	4	no	—
390	#97	—	2	2	no	—
391	C₄×C₈	—	2	4	no	—
392	#95	C₂×C₄×C₃₁	1	2	yes	—
393	#95	C₂×C₄×C₃₁	1	2	yes	—
394	#95	C₂×C₄×C₃₁	1	2	yes	—
395	C₃₁	#21441	1	1	yes	characteristic, direct product factor, Sylow
396	C₂⁴	—	2	4	no	—
397	C₂⁴	—	2	2	no	—
398	C₂⁴	—	2	2	no	—
399	C₂⁴	—	2	2	no	—
400	C₂⁴	—	4	2	no	—
401	C₂⁴	C₂²×C₄×C₃₁	1	1	yes	characteristic
402	C₂⁴	C₂²×C₄×C₃₁	1	2	yes	—
403	C₂⁴	C₂²×C₄×C₃₁	1	1	yes	characteristic
404	C₂×D₄	—	2	4	no	—
405	C₂×D₄	—	2	2	no	—
406	C₂×D₄	—	2	2	no	—
407	C₂×D₄	—	2	2	no	—
408	C₂×D₄	—	2	4	no	—
409	C₂×D₄	—	4	2	no	—
410	C₂²×C₄	—	2	8	no	—
411	C₂²×C₄	—	2	8	no	—
412	C₂²×C₄	—	2	16	no	—
413	C₂²×C₄	—	2	4	no	—
414	C₂²×C₄	—	2	2	no	—
415	C₂²×C₄	—	2	4	no	—
416	C₂²×C₄	—	2	4	no	—
417	C₂²×C₄	—	2	4	no	—
418	C₂²×C₄	—	2	2	no	—
419	C₂²×C₄	—	2	2	no	—
420	C₂²×C₄	—	2	2	no	—
421	C₂²×C₄	—	2	2	no	—
422	C₂²×C₄	—	2	2	no	—
423	C₂²×C₄	—	2	2	no	—
424	C₂²×C₄	—	2	1	no	—
425	C₂²×C₄	—	2	2	no	—
426	C₂²×C₄	—	2	1	no	—
427	C₂²×C₄	—	2	2	no	—
428	C₂²×C₄	C₂²×C₄×C₃₁	1	2	yes	—
429	C₂²×C₄	C₂²×C₄×C₃₁	1	1	yes	characteristic
430	C₂²×C₄	C₂²×C₄×C₃₁	1	1	yes	characteristic
431	C₂²×C₄	C₂²×C₄×C₃₁	1	1	yes	characteristic
432	C₂²×C₄	C₂²×C₄×C₃₁	1	1	yes	characteristic
433	C₂²×C₄	C₂²×C₄×C₃₁	1	2	yes	—
434	C₂²×C₄	C₂²×C₄×C₃₁	1	2	yes	—
435	C₂²×C₄	C₂×D₄×C₃₁	1	2	yes	—
436	C₂²×C₄	C₂×D₄×C₃₁	1	4	yes	—
437	C₂²×C₄	#41×C₃₁	1	4	yes	—
438	C₂²×C₄	#41×C₃₁	1	2	yes	—
439	C₂²×C₄	C₂⁴×C₃₁	1	1	yes	characteristic
440	M₁₆	—	2	16	no	—
441	M₁₆	—	2	16	no	—
442	M₁₆	—	4	4	no	—
443	C₂×C₈	—	2	8	no	—
444	C₂×C₈	—	2	2	no	—
445	C₂×C₈	—	2	2	no	—
446	C₂×C₈	—	2	2	no	—
447	C₂×C₈	—	2	8	no	—
448	C₂×C₈	—	2	2	no	—
449	C₂×C₈	—	2	1	no	—
450	C₂×C₈	—	2	1	no	—
451	C₂×C₈	—	2	2	no	—
452	C₂×C₈	—	2	2	no	—
453	C₂×C₈	—	2	2	no	—
454	C₂×C₈	—	2	2	no	—
455	C₂×C₈	C₂×D₄×C₃₁	1	4	yes	—
456	C₂×C₈	C₂×D₄×C₃₁	1	2	yes	—
457	C₂×C₈	C₂×D₄×C₃₁	1	4	yes	—
458	C₂×C₈	C₂×D₄×C₃₁	1	2	yes	—
459	C₄⋊C₄	—	2	4	no	—
460	C₄⋊C₄	—	2	2	no	—
461	C₂²⋊C₄	—	2	4	no	—
462	C₂²⋊C₄	—	2	2	no	—
463	C₂²⋊C₄	—	2	2	no	—
464	C₂²⋊C₄	—	2	2	no	—
465	C₂²⋊C₄	—	2	4	no	—
466	C₂²⋊C₄	—	2	4	no	—
467	C₂²⋊C₄	—	4	2	no	—
468	C₂²⋊C₄	M₁₆×C₃₁	1	4	yes	complemented
469	C₄²	—	2	2	no	—
470	C₄²	—	2	1	no	—
471	C₄²	—	2	2	no	—
472	C₄²	—	2	1	no	—
473	C₂³	—	2	2	no	—
474	C₂³	—	2	2	no	—
475	C₂³	—	2	8	no	—
476	C₂³	—	2	2	no	—
477	C₂³	—	2	2	no	—
478	C₂³	—	2	2	no	—
479	C₂³	—	2	2	no	—
480	C₂³	—	2	2	no	—
481	C₂³	—	2	2	no	—
482	C₂³	—	2	1	no	—
483	C₂³	—	2	2	no	—
484	C₂³	—	2	1	no	—
485	C₂³	—	2	1	no	—
486	C₂³	—	2	2	no	—
487	C₂³	—	4	8	no	—
488	C₂³	—	4	8	no	—
489	C₂³	—	4	4	no	—
490	C₂³	—	4	4	no	—
491	C₂³	C₄×D₄×C₃₁	1	4	yes	—
492	C₂³	C₄×D₄×C₃₁	1	2	yes	—
493	C₂³	C₂×M₁₆×C₃₁	1	2	yes	—
494	C₂³	C₃₁×#131	1	2	yes	—
495	C₂³	C₂³×C₄×C₃₁	1	1	yes	characteristic
496	D₄	—	4	8	no	—
497	D₄	—	4	4	no	—
498	C₂×C₄	—	2	8	no	—
499	C₂×C₄	—	2	16	no	—
500	C₂×C₄	—	2	2	no	—
501	C₂×C₄	—	2	2	no	—
502	C₂×C₄	—	2	2	no	—
503	C₂×C₄	—	2	2	no	—
504	C₂×C₄	—	2	2	no	—
505	C₂×C₄	—	2	2	no	—
506	C₂×C₄	—	2	4	no	—
507	C₂×C₄	—	2	2	no	—
508	C₂×C₄	—	2	4	no	—
509	C₂×C₄	—	2	4	no	—
510	C₂×C₄	—	2	4	no	—
511	C₂×C₄	—	2	4	no	—
512	C₂×C₄	—	2	1	no	—
513	C₂×C₄	—	2	2	no	—
514	C₂×C₄	—	2	1	no	—
515	C₂×C₄	—	2	1	no	—
516	C₂×C₄	—	2	2	no	—
517	C₂×C₄	—	2	1	no	—
518	C₂×C₄	—	2	2	no	—
519	C₂×C₄	—	2	1	no	—
520	C₂×C₄	—	2	2	no	—
521	C₂×C₄	—	2	2	no	—
522	C₂×C₄	—	4	1	no	—
523	C₂×C₄	—	4	1	no	—
524	C₂×C₄	—	4	2	no	—
525	C₂×C₄	C₄×D₄×C₃₁	1	2	yes	—
526	C₂×C₄	C₄×D₄×C₃₁	1	4	yes	—
527	C₂×C₄	C₂×M₁₆×C₃₁	1	4	yes	complemented
528	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
529	C₂×C₄	C₂²×D₄×C₃₁	1	2	yes	—
530	C₂×C₄	C₂×#41×C₃₁	1	1	yes	characteristic
531	C₂×C₄	C₂×#41×C₃₁	1	2	yes	—
532	C₈	—	2	4	no	—
533	C₈	—	2	2	no	—
534	C₈	—	4	1	no	—
535	C₈	—	4	2	no	—
536	C₈	—	4	1	no	—
537	C₈	C₃₁×#120	1	4	yes	complemented
538	C₂²	—	2	2	no	—
539	C₂²	—	2	4	no	—
540	C₂²	—	2	4	no	—
541	C₂²	—	2	2	no	—
542	C₂²	—	2	4	no	—
543	C₂²	—	2	4	no	—
544	C₂²	—	2	4	no	—
545	C₂²	—	4	16	no	—
546	C₂²	—	4	2	no	—
547	C₂²	—	4	2	no	—
548	C₂²	—	4	2	no	—
549	C₂²	—	4	2	no	—
550	C₂²	—	4	1	no	—
551	C₂²	—	4	1	no	—
552	C₂²	—	4	2	no	—
553	C₂²	—	4	2	no	—
554	C₂²	—	4	1	no	—
555	C₂²	C₃₁×#436	1	4	yes	complemented
556	C₂²	C₂×C₄×D₄×C₃₁	1	1	yes	characteristic
557	C₂²	C₂×C₄×D₄×C₃₁	1	2	yes	—
558	C₂²	C₂²×M₁₆×C₃₁	1	1	yes	characteristic
559	C₂²	C₂×C₃₁×#131	1	1	yes	characteristic
560	C₂²	C₃₁×#568	1	2	yes	—
561	C₄	—	2	4	no	—
562	C₄	—	2	2	no	—
563	C₄	—	2	4	no	—
564	C₄	—	4	1	no	—
565	C₄	—	4	1	no	—
566	C₄	—	4	2	no	—
567	C₄	—	4	1	no	—
568	C₄	C₂×C₃₁×#120	1	1	yes	characteristic
569	C₄	C₃₁×#525	1	2	yes	—
570	C₄	C₃₁×#525	1	1	yes	characteristic
571	C₂	—	2	4	no	—
572	C₂	—	4	4	no	—
573	C₂	—	8	1	no	—
574	C₂	C₄×C₃₁×#120	1	1	yes	characteristic
575	C₂	C₂×C₃₁×#436	1	1	yes	characteristic
576	C₂	C₃₁×#2930	1	1	yes	characteristic
577	C₂	C₃₁×#2933	1	2	yes	—
578	C₂	D₄×M₁₆×C₃₁	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. C₂³×M₁₆×C₃₁	C₂	semidirect
2. C₂×C₃₁×#436	C₂	semidirect
3. C₂×C₃₁×#435	C₂	semidirect
5. C₃₁×#1870	C₂	semidirect
6. C₃₁×#1869	C₂	semidirect
7. C₃₁×#1851	C₂	semidirect
8. C₃₁×#1850	C₂	semidirect
9. C₃₁×#1779	C₂	semidirect
10. C₃₁×#1753	C₂	semidirect
11. C₈×C₂²⋊C₄×C₃₁	C₂	semidirect
17. C₂²×M₁₆×C₃₁	C₂²	semidirect
25. C₂×C₃₁×C₄⋊C₈	C₂²	semidirect
26. C₂×C₃₁×C₄⋊C₈	C₂²	semidirect
31. C₂×C₃₁×C₂²⋊C₈	C₂²	semidirect
32. C₂×C₃₁×C₂²⋊C₈	C₂²	semidirect
33. C₂×C₃₁×#97	C₂²	semidirect
34. C₂×C₄×C₈×C₃₁	C₂²	semidirect
42. C₃₁×#336	C₂²	semidirect
43. C₃₁×#336	C₂²	semidirect
44. C₃₁×#336	C₂²	semidirect
65. C₂×M₁₆×C₃₁	D₄	semidirect
69. C₂²×C₈×C₃₁	D₄	semidirect
178. C₂²⋊C₄×C₃₁	M₁₆	semidirect
183. #21441	C₃₁	direct (N × H)
238. C₂×C₄×C₃₁	C₂×M₁₆	semidirect
248. C₈×C₃₁	#120	semidirect
249. C₂³×M₁₆	C₂×C₃₁	semidirect
250. C₂×#436	C₂×C₃₁	semidirect
251. C₂×#435	C₂×C₃₁	semidirect
253. #1870	C₂×C₃₁	semidirect
254. #1869	C₂×C₃₁	semidirect
255. #1851	C₂×C₃₁	semidirect
256. #1850	C₂×C₃₁	semidirect
257. #1779	C₂×C₃₁	semidirect
258. #1753	C₂×C₃₁	semidirect
259. C₈×C₂²⋊C₄	C₂×C₃₁	semidirect
277. C₂²×C₃₁	#436	semidirect
298. C₂²×M₁₆	C₂²×C₃₁	semidirect
306. C₂×C₄⋊C₈	C₂²×C₃₁	semidirect
307. C₂×C₄⋊C₈	C₂²×C₃₁	semidirect
312. C₂×C₂²⋊C₈	C₂²×C₃₁	semidirect
313. C₂×C₂²⋊C₈	C₂²×C₃₁	semidirect
314. C₂×#97	C₂²×C₃₁	semidirect
315. C₂×C₄×C₈	C₂²×C₃₁	semidirect
323. #336	C₂²×C₃₁	semidirect
324. #336	C₂²×C₃₁	semidirect
325. #336	C₂²×C₃₁	semidirect
354. C₂×M₁₆	D₄×C₃₁	semidirect
358. C₂²×C₈	D₄×C₃₁	semidirect
395. C₃₁	#21441	direct (N × H)
468. C₂²⋊C₄	M₁₆×C₃₁	semidirect
527. C₂×C₄	C₂×M₁₆×C₃₁	semidirect
537. C₈	C₃₁×#120	semidirect
555. C₂²	C₃₁×#436	semidirect

Maximal subgroups

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

subgroup	index
C₂³×M₁₆×C₃₁	2
C₂×C₃₁×#436	2
C₂×C₃₁×#435	2
C₄×C₃₁×#120	2
C₃₁×#1870	2
C₃₁×#1869	2
C₃₁×#1851	2
C₃₁×#1850	2
C₃₁×#1779	2
C₃₁×#1753	2
C₈×C₂²⋊C₄×C₃₁	2
#21441	31

Subgroups — C31×#21441 #21478200

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#21441 #21478200