Subgroups — C₂×C₃₁×#3063 #21517211

Order 7936.

2742 conjugacy classes of subgroups.

The subgroup lattice has 634 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₃₁×#137	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂²×C₃₁×#131	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×C₃₁×#537	C₂	1	1	yes	maximal, characteristic
4	C₂×C₃₁×#536	C₂	1	1	yes	maximal, characteristic, complemented
5	C₃₁×#3063	C₂	1	16	yes	maximal, direct product factor
6	C₂×C₃₁×#470	C₂	1	2	yes	maximal, complemented
7	C₂×C₃₁×#468	C₂	1	2	yes	maximal, complemented
8	C₂×C₃₁×#467	C₂	1	2	yes	maximal, complemented
9	C₂×C₃₁×#465	C₂	1	2	yes	maximal, complemented
10	C₂×C₃₁×#428	C₂	1	1	yes	maximal, characteristic, complemented
11	C₂×C₃₁×#416	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂²×C₃₁×#103	C₂	1	1	yes	maximal, characteristic, complemented
13	C₂²×#41×C₃₁	C₂²	1	1	yes	characteristic
14	C₂²×#41×C₃₁	C₂²	1	1	yes	characteristic
15	C₂³×Q₈×C₃₁	C₂²	1	1	yes	characteristic
16	C₂×C₃₁×#137	—	2	8	no	—
17	C₂×C₃₁×#137	C₂²	1	8	yes	complemented
18	C₂²×Q₁₆×C₃₁	C₂²	1	2	yes	—
19	C₂²×SD₁₆×C₃₁	C₂²	1	2	yes	—
20	C₂×C₃₁×#131	—	2	8	no	—
21	C₂×C₃₁×#131	C₂²	1	8	yes	complemented
22	C₂²×M₁₆×C₃₁	C₂²	1	2	yes	—
23	C₂²×M₁₆×C₃₁	C₂²	1	1	yes	characteristic
24	C₂³×C₈×C₃₁	C₂²	1	1	yes	characteristic
25	C₂³×C₈×C₃₁	C₂²	1	2	yes	—
26	C₃₁×#537	—	2	4	no	—
27	C₃₁×#537	C₂²	1	8	yes	—
28	C₃₁×#536	—	2	4	no	—
29	C₃₁×#536	C₂²	1	8	yes	complemented
30	C₂×C₃₁×C₄⋊Q₈	—	2	1	no	—
31	C₂×C₃₁×#124	—	2	1	no	—
32	C₂×C₃₁×#123	—	2	2	no	—
33	C₂×C₃₁×C₂²⋊Q₈	—	2	1	no	—
34	C₂×C₃₁×C₂²⋊Q₈	C₂²	1	2	yes	—
35	C₂×C₃₁×C₂²⋊Q₈	C₂²	1	2	yes	—
36	C₂×C₃₁×C₂²⋊Q₈	C₂²	1	2	yes	—
37	C₂×C₃₁×#121	—	2	1	no	—
38	C₂×C₃₁×#121	—	2	2	no	—
39	C₂×C₃₁×#121	C₂²	1	2	yes	—
40	C₂×C₃₁×#117	C₂²	1	1	yes	characteristic
41	C₂²×C₄⋊C₄×C₃₁	C₂²	1	1	yes	characteristic
42	C₃₁×#470	C₂²	1	16	yes	complemented
43	C₃₁×#468	C₂²	1	16	yes	complemented
44	C₃₁×#467	C₂²	1	16	yes	complemented
45	C₃₁×#465	C₂²	1	16	yes	complemented
46	C₃₁×#428	C₂²	1	8	yes	complemented
47	C₂×C₃₁×#107	C₂²	1	2	yes	—
48	C₂×C₃₁×#106	C₂²	1	2	yes	—
49	C₃₁×#416	C₂²	1	8	yes	complemented
50	C₂×C₃₁×#103	C₂²	1	2	yes	—
51	C₂×C₃₁×#103	C₂²	1	2	yes	—
52	C₂×C₃₁×#103	C₂²	1	1	yes	characteristic
53	C₂×C₃₁×#103	C₂²	1	1	yes	characteristic
54	C₂×C₃₁×#103	C₂²	1	8	yes	complemented
55	C₂×C₃₁×#102	C₂²	1	1	yes	characteristic
56	C₂×C₃₁×#102	C₂²	1	1	yes	characteristic
57	C₂×#41×C₃₁	—	2	4	no	—
58	C₂×#41×C₃₁	—	2	2	no	—
59	C₂×#41×C₃₁	—	2	4	no	—
60	C₂×#41×C₃₁	—	2	4	no	—
61	C₂×#41×C₃₁	—	2	4	no	—
62	C₂×#41×C₃₁	D₄	1	4	yes	—
63	C₂×#41×C₃₁	C₂³	1	4	yes	—
64	C₂×#41×C₃₁	C₂³	1	4	yes	—
65	C₂²×Q₈×C₃₁	—	2	1	no	—
66	C₂²×Q₈×C₃₁	—	2	2	no	—
67	C₂²×Q₈×C₃₁	—	2	4	no	—
68	C₂²×Q₈×C₃₁	—	2	4	no	—
69	C₂²×Q₈×C₃₁	D₄	1	1	yes	characteristic
70	C₂²×Q₈×C₃₁	C₂³	1	2	yes	—
71	C₂²×Q₈×C₃₁	C₂³	1	1	yes	characteristic
72	C₂²×Q₈×C₃₁	C₂³	1	4	yes	—
73	C₂²×D₄×C₃₁	—	2	1	no	—
74	C₂²×D₄×C₃₁	—	2	2	no	—
75	C₂²×D₄×C₃₁	D₄	1	2	yes	—
76	C₂²×D₄×C₃₁	D₄	1	1	yes	characteristic
77	C₂²×D₄×C₃₁	C₂³	1	1	yes	characteristic
78	C₂³×C₄×C₃₁	—	2	1	no	—
79	C₂³×C₄×C₃₁	—	2	1	no	—
80	C₂³×C₄×C₃₁	—	2	1	no	—
81	C₂³×C₄×C₃₁	C₂³	1	2	yes	—
82	C₂³×C₄×C₃₁	C₂³	1	1	yes	characteristic
83	C₃₁×#137	—	2	32	no	—
84	C₂×Q₁₆×C₃₁	—	2	8	no	—
85	C₂×Q₁₆×C₃₁	C₂³	1	8	yes	—
86	C₂×SD₁₆×C₃₁	—	2	8	no	—
87	C₂×SD₁₆×C₃₁	C₂³	1	8	yes	—
88	C₃₁×#131	—	2	32	no	—
89	C₂×M₁₆×C₃₁	—	2	8	no	—
90	C₂×M₁₆×C₃₁	D₄	1	8	yes	complemented
91	C₂×M₁₆×C₃₁	C₂³	1	8	yes	—
92	C₂×M₁₆×C₃₁	C₂³	1	4	yes	—
93	C₂²×C₈×C₃₁	—	2	4	no	—
94	C₂²×C₈×C₃₁	—	2	8	no	—
95	C₂²×C₈×C₃₁	C₂³	1	1	yes	characteristic
96	C₂²×C₈×C₃₁	C₂³	1	1	yes	characteristic
97	C₂²×C₈×C₃₁	C₂³	1	4	yes	—
98	C₂²×C₈×C₃₁	C₂³	1	8	yes	—
99	C₂²×C₈×C₃₁	C₂³	1	2	yes	—
100	C₃₁×C₄⋊Q₈	—	2	8	no	—
101	C₃₁×#124	—	2	8	no	—
102	C₃₁×#123	—	2	16	no	—
103	C₃₁×C₂²⋊Q₈	—	2	8	no	—
104	C₃₁×C₂²⋊Q₈	—	2	4	no	—
105	C₃₁×C₂²⋊Q₈	—	2	4	no	—
106	C₃₁×C₂²⋊Q₈	—	2	4	no	—
107	C₃₁×C₂²⋊Q₈	C₂³	1	8	yes	—
108	C₃₁×C₂²⋊Q₈	C₂³	1	8	yes	—
109	C₃₁×C₂²⋊Q₈	C₂³	1	8	yes	—
110	C₃₁×#121	—	2	8	no	—
111	C₃₁×#121	—	2	16	no	—
112	C₃₁×#121	—	2	4	no	—
113	C₃₁×#121	C₂³	1	8	yes	—
114	C₃₁×#117	—	2	2	no	—
115	C₃₁×#117	C₂³	1	4	yes	—
116	C₂×C₄⋊C₄×C₃₁	—	2	1	no	—
117	C₂×C₄⋊C₄×C₃₁	—	2	2	no	—
118	C₂×C₄⋊C₄×C₃₁	—	2	2	no	—
119	C₂×C₄⋊C₄×C₃₁	—	2	2	no	—
120	C₂×C₄⋊C₄×C₃₁	—	2	2	no	—
121	C₂×C₄⋊C₄×C₃₁	C₂³	1	1	yes	characteristic
122	C₂×C₄⋊C₄×C₃₁	C₂³	1	1	yes	characteristic
123	C₂×C₄⋊C₄×C₃₁	C₂³	1	2	yes	—
124	C₂×C₄⋊C₄×C₃₁	C₂³	1	4	yes	—
125	C₂×C₂²⋊C₄×C₃₁	—	2	1	no	—
126	C₂×C₂²⋊C₄×C₃₁	—	2	2	no	—
127	C₂×C₂²⋊C₄×C₃₁	—	2	2	no	—
128	C₂×C₂²⋊C₄×C₃₁	—	2	2	no	—
129	C₂×C₂²⋊C₄×C₃₁	—	2	2	no	—
130	C₂×C₄²×C₃₁	—	2	1	no	—
131	C₃₁×#107	C₂³	1	8	yes	—
132	C₃₁×#106	C₂³	1	8	yes	—
133	C₃₁×#103	C₂³	1	8	yes	—
134	C₃₁×#103	C₂³	1	8	yes	—
135	C₃₁×#103	C₂³	1	4	yes	—
136	C₃₁×#103	C₂³	1	4	yes	—
137	C₃₁×#102	C₂³	1	4	yes	—
138	C₃₁×#102	C₂³	1	4	yes	—
139	C₂⁴×C₃₁	—	2	1	no	—
140	C₂⁴×C₃₁	C₂×D₄	1	2	yes	—
141	C₂⁴×C₃₁	C₂×D₄	1	1	yes	characteristic
142	#41×C₃₁	—	2	16	no	—
143	#41×C₃₁	—	2	8	no	—
144	#41×C₃₁	—	2	8	no	—
145	#41×C₃₁	—	2	8	no	—
146	#41×C₃₁	—	2	8	no	—
147	#41×C₃₁	—	4	8	no	—
148	C₂×Q₈×C₃₁	—	2	4	no	—
149	C₂×Q₈×C₃₁	—	2	2	no	—
150	C₂×Q₈×C₃₁	—	2	1	no	—
151	C₂×Q₈×C₃₁	—	2	4	no	—
152	C₂×Q₈×C₃₁	—	2	4	no	—
153	C₂×Q₈×C₃₁	—	2	2	no	—
154	C₂×Q₈×C₃₁	—	2	8	no	—
155	C₂×Q₈×C₃₁	—	2	4	no	—
156	C₂×Q₈×C₃₁	—	2	8	no	—
157	C₂×Q₈×C₃₁	—	2	2	no	—
158	C₂×Q₈×C₃₁	—	4	4	no	—
159	C₂×Q₈×C₃₁	—	4	8	no	—
160	C₂×Q₈×C₃₁	C₂×D₄	1	4	yes	—
161	C₂×Q₈×C₃₁	C₂⁴	1	4	yes	—
162	C₂×Q₈×C₃₁	C₂⁴	1	2	yes	—
163	C₂×D₄×C₃₁	—	2	8	no	—
164	C₂×D₄×C₃₁	—	2	4	no	—
165	C₂×D₄×C₃₁	—	2	8	no	—
166	C₂×D₄×C₃₁	—	2	4	no	—
167	C₂×D₄×C₃₁	—	2	2	no	—
168	C₂×D₄×C₃₁	—	2	1	no	—
169	C₂×D₄×C₃₁	—	2	2	no	—
170	C₂×D₄×C₃₁	—	4	4	no	—
171	C₂×D₄×C₃₁	—	8	4	no	—
172	C₂×D₄×C₃₁	C₂×D₄	1	8	yes	—
173	C₂×D₄×C₃₁	C₂×D₄	1	4	yes	—
174	C₂×D₄×C₃₁	C₂⁴	1	2	yes	—
175	C₂²×C₄×C₃₁	—	2	4	no	—
176	C₂²×C₄×C₃₁	—	2	4	no	—
177	C₂²×C₄×C₃₁	—	2	4	no	—
178	C₂²×C₄×C₃₁	—	2	1	no	—
179	C₂²×C₄×C₃₁	—	2	1	no	—
180	C₂²×C₄×C₃₁	—	2	2	no	—
181	C₂²×C₄×C₃₁	—	2	1	no	—
182	C₂²×C₄×C₃₁	—	2	2	no	—
183	C₂²×C₄×C₃₁	—	2	4	no	—
184	C₂²×C₄×C₃₁	—	2	4	no	—
185	C₂²×C₄×C₃₁	—	4	4	no	—
186	C₂²×C₄×C₃₁	—	4	4	no	—
187	C₂²×C₄×C₃₁	—	8	2	no	—
188	C₂²×C₄×C₃₁	C₂×D₄	1	4	yes	—
189	C₂²×C₄×C₃₁	C₂×D₄	1	2	yes	—
190	C₂²×C₄×C₃₁	C₂×D₄	1	2	yes	—
191	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
192	C₂²×C₄×C₃₁	C₂×D₄	1	4	yes	—
193	C₂²×C₄×C₃₁	#41	1	4	yes	—
194	C₂²×C₄×C₃₁	C₂⁴	1	4	yes	—
195	C₂²×C₄×C₃₁	C₂⁴	1	2	yes	—
196	C₂²×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
197	Q₁₆×C₃₁	—	2	16	no	—
198	SD₁₆×C₃₁	—	2	16	no	—
199	M₁₆×C₃₁	—	2	16	no	—
200	M₁₆×C₃₁	C₂×D₄	1	16	yes	complemented
201	C₂×C₈×C₃₁	—	2	2	no	—
202	C₂×C₈×C₃₁	—	2	2	no	—
203	C₂×C₈×C₃₁	—	2	8	no	—
204	C₂×C₈×C₃₁	—	2	16	no	—
205	C₂×C₈×C₃₁	C₂×D₄	1	8	yes	—
206	C₂×C₈×C₃₁	C₂⁴	1	2	yes	—
207	C₂×C₈×C₃₁	C₂⁴	1	2	yes	—
208	C₂×C₈×C₃₁	C₂⁴	1	4	yes	—
209	C₄⋊C₄×C₃₁	—	2	4	no	—
210	C₄⋊C₄×C₃₁	—	2	2	no	—
211	C₄⋊C₄×C₃₁	—	2	8	no	—
212	C₄⋊C₄×C₃₁	—	2	8	no	—
213	C₄⋊C₄×C₃₁	—	2	8	no	—
214	C₄⋊C₄×C₃₁	—	2	1	no	—
215	C₄⋊C₄×C₃₁	—	2	1	no	—
216	C₄⋊C₄×C₃₁	C₂⁴	1	2	yes	—
217	C₄⋊C₄×C₃₁	C₂⁴	1	2	yes	—
218	C₄⋊C₄×C₃₁	C₂⁴	1	4	yes	—
219	C₂²⋊C₄×C₃₁	—	2	4	no	—
220	C₂²⋊C₄×C₃₁	—	2	8	no	—
221	C₂²⋊C₄×C₃₁	—	2	8	no	—
222	C₂²⋊C₄×C₃₁	—	2	8	no	—
223	C₂²⋊C₄×C₃₁	—	2	8	no	—
224	C₄²×C₃₁	—	2	4	no	—
225	C₂×#3063	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
226	C₂³×C₃₁	—	2	2	no	—
227	C₂³×C₃₁	—	2	2	no	—
228	C₂³×C₃₁	—	2	4	no	—
229	C₂³×C₃₁	—	2	4	no	—
230	C₂³×C₃₁	—	2	4	no	—
231	C₂³×C₃₁	—	2	2	no	—
232	C₂³×C₃₁	—	4	2	no	—
233	C₂³×C₃₁	—	4	2	no	—
234	C₂³×C₃₁	—	4	2	no	—
235	C₂³×C₃₁	#121	1	4	yes	—
236	C₂³×C₃₁	C₂²×D₄	1	4	yes	—
237	C₂³×C₃₁	C₂²×D₄	1	2	yes	—
238	C₂³×C₃₁	C₂²×D₄	1	1	yes	characteristic
239	Q₈×C₃₁	—	2	8	no	—
240	Q₈×C₃₁	—	2	4	no	—
241	Q₈×C₃₁	—	2	2	no	—
242	Q₈×C₃₁	—	2	4	no	—
243	Q₈×C₃₁	—	2	2	no	—
244	Q₈×C₃₁	—	4	4	no	—
245	Q₈×C₃₁	—	4	8	no	—
246	Q₈×C₃₁	—	4	2	no	—
247	D₄×C₃₁	—	2	16	no	—
248	D₄×C₃₁	—	2	8	no	—
249	D₄×C₃₁	—	2	2	no	—
250	D₄×C₃₁	—	2	2	no	—
251	D₄×C₃₁	—	4	8	no	—
252	D₄×C₃₁	—	4	2	no	—
253	D₄×C₃₁	—	8	8	no	—
254	C₂×C₄×C₃₁	—	2	4	no	—
255	C₂×C₄×C₃₁	—	2	8	no	—
256	C₂×C₄×C₃₁	—	2	4	no	—
257	C₂×C₄×C₃₁	—	2	4	no	—
258	C₂×C₄×C₃₁	—	2	2	no	—
259	C₂×C₄×C₃₁	—	2	2	no	—
260	C₂×C₄×C₃₁	—	2	4	no	—
261	C₂×C₄×C₃₁	—	2	2	no	—
262	C₂×C₄×C₃₁	—	2	4	no	—
263	C₂×C₄×C₃₁	—	4	2	no	—
264	C₂×C₄×C₃₁	—	4	8	no	—
265	C₂×C₄×C₃₁	—	4	4	no	—
266	C₂×C₄×C₃₁	—	4	2	no	—
267	C₂×C₄×C₃₁	—	4	4	no	—
268	C₂×C₄×C₃₁	—	4	2	no	—
269	C₂×C₄×C₃₁	—	4	8	no	—
270	C₂×C₄×C₃₁	—	8	4	no	—
271	C₂×C₄×C₃₁	#121	1	4	yes	—
272	C₂×C₄×C₃₁	#121	1	8	yes	—
273	C₂×C₄×C₃₁	C₂²×D₄	1	4	yes	—
274	C₂×C₄×C₃₁	C₂²×D₄	1	2	yes	—
275	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
276	C₂×C₄×C₃₁	C₂²×D₄	1	2	yes	—
277	C₂×C₄×C₃₁	C₂²×D₄	1	4	yes	—
278	C₂×C₄×C₃₁	C₂×#41	1	2	yes	—
279	C₂×C₄×C₃₁	C₂×#41	1	4	yes	—
280	C₂×C₄×C₃₁	C₂⁵	1	1	yes	characteristic
281	C₈×C₃₁	—	2	2	no	—
282	C₈×C₃₁	—	2	2	no	—
283	C₈×C₃₁	C₂²×D₄	1	8	yes	—
284	C₂²×#137	C₂×C₃₁	1	1	yes	characteristic, complemented
285	C₂²×#131	C₂×C₃₁	1	1	yes	characteristic, complemented
286	C₂×#537	C₂×C₃₁	1	1	yes	characteristic
287	C₂×#536	C₂×C₃₁	1	1	yes	characteristic, complemented
288	#3063	C₂×C₃₁	1	16	yes	direct product factor
289	C₂×#470	C₂×C₃₁	1	2	yes	complemented
290	C₂×#468	C₂×C₃₁	1	2	yes	complemented
291	C₂×#467	C₂×C₃₁	1	2	yes	complemented
292	C₂×#465	C₂×C₃₁	1	2	yes	complemented
293	C₂×#428	C₂×C₃₁	1	1	yes	characteristic, complemented
294	C₂×#416	C₂×C₃₁	1	1	yes	characteristic, complemented
295	C₂²×#103	C₂×C₃₁	1	1	yes	characteristic, complemented
296	C₂²×C₃₁	—	2	8	no	—
297	C₂²×C₃₁	—	2	2	no	—
298	C₂²×C₃₁	—	2	2	no	—
299	C₂²×C₃₁	—	2	4	no	—
300	C₂²×C₃₁	—	2	4	no	—
301	C₂²×C₃₁	—	2	4	no	—
302	C₂²×C₃₁	—	4	8	no	—
303	C₂²×C₃₁	—	4	2	no	—
304	C₂²×C₃₁	—	4	2	no	—
305	C₂²×C₃₁	—	4	2	no	—
306	C₂²×C₃₁	—	8	4	no	—
307	C₂²×C₃₁	C₂×#121	1	4	yes	—
308	C₂²×C₃₁	C₂×#121	1	2	yes	—
309	C₂²×C₃₁	#577	1	2	yes	—
310	C₂²×C₃₁	#578	1	2	yes	—
311	C₂²×C₃₁	C₂³×D₄	1	1	yes	characteristic
312	C₄×C₃₁	—	2	4	no	—
313	C₄×C₃₁	—	4	2	no	—
314	C₄×C₃₁	—	4	4	no	—
315	C₄×C₃₁	—	4	2	no	—
316	C₄×C₃₁	—	4	4	no	—
317	C₄×C₃₁	—	4	2	no	—
318	C₄×C₃₁	C₂×#121	1	2	yes	—
319	C₄×C₃₁	C₂×#121	1	4	yes	—
320	C₄×C₃₁	C₂³×D₄	1	1	yes	characteristic
321	C₄×C₃₁	C₂²×#41	1	1	yes	characteristic
322	C₂²×#41	C₂²×C₃₁	1	1	yes	characteristic
323	C₂²×#41	C₂²×C₃₁	1	1	yes	characteristic
324	C₂³×Q₈	C₂²×C₃₁	1	1	yes	characteristic
325	C₂×#137	—	2	8	no	—
326	C₂×#137	C₂²×C₃₁	1	8	yes	complemented
327	C₂²×Q₁₆	C₂²×C₃₁	1	2	yes	—
328	C₂²×SD₁₆	C₂²×C₃₁	1	2	yes	—
329	C₂×#131	—	2	8	no	—
330	C₂×#131	C₂²×C₃₁	1	8	yes	complemented
331	C₂²×M₁₆	C₂²×C₃₁	1	2	yes	—
332	C₂²×M₁₆	C₂²×C₃₁	1	1	yes	characteristic
333	C₂³×C₈	C₂²×C₃₁	1	1	yes	characteristic
334	C₂³×C₈	C₂²×C₃₁	1	2	yes	—
335	#537	—	2	4	no	—
336	#537	C₂²×C₃₁	1	8	yes	—
337	#536	—	2	4	no	—
338	#536	C₂²×C₃₁	1	8	yes	complemented
339	C₂×C₄⋊Q₈	—	2	1	no	—
340	C₂×#124	—	2	1	no	—
341	C₂×#123	—	2	2	no	—
342	C₂×C₂²⋊Q₈	—	2	1	no	—
343	C₂×C₂²⋊Q₈	C₂²×C₃₁	1	2	yes	—
344	C₂×C₂²⋊Q₈	C₂²×C₃₁	1	2	yes	—
345	C₂×C₂²⋊Q₈	C₂²×C₃₁	1	2	yes	—
346	C₂×#121	—	2	1	no	—
347	C₂×#121	—	2	2	no	—
348	C₂×#121	C₂²×C₃₁	1	2	yes	—
349	C₂×#117	C₂²×C₃₁	1	1	yes	characteristic
350	C₂²×C₄⋊C₄	C₂²×C₃₁	1	1	yes	characteristic
351	#470	C₂²×C₃₁	1	16	yes	complemented
352	#468	C₂²×C₃₁	1	16	yes	complemented
353	#467	C₂²×C₃₁	1	16	yes	complemented
354	#465	C₂²×C₃₁	1	16	yes	complemented
355	#428	C₂²×C₃₁	1	8	yes	complemented
356	C₂×#107	C₂²×C₃₁	1	2	yes	—
357	C₂×#106	C₂²×C₃₁	1	2	yes	—
358	#416	C₂²×C₃₁	1	8	yes	complemented
359	C₂×#103	C₂²×C₃₁	1	2	yes	—
360	C₂×#103	C₂²×C₃₁	1	2	yes	—
361	C₂×#103	C₂²×C₃₁	1	1	yes	characteristic
362	C₂×#103	C₂²×C₃₁	1	1	yes	characteristic
363	C₂×#103	C₂²×C₃₁	1	8	yes	complemented
364	C₂×#102	C₂²×C₃₁	1	1	yes	characteristic
365	C₂×#102	C₂²×C₃₁	1	1	yes	characteristic
366	C₂×C₃₁	—	2	4	no	—
367	C₂×C₃₁	—	4	4	no	—
368	C₂×C₃₁	—	8	2	no	—
369	C₂×C₃₁	#3063	1	4	yes	direct product factor
370	C₂×C₃₁	C₂²×#121	1	1	yes	characteristic
371	C₂×C₃₁	C₂×#577	1	1	yes	characteristic
372	C₂×C₃₁	C₂×#578	1	1	yes	characteristic
373	C₂×#41	—	2	4	no	—
374	C₂×#41	—	2	2	no	—
375	C₂×#41	—	2	4	no	—
376	C₂×#41	—	2	4	no	—
377	C₂×#41	—	2	4	no	—
378	C₂×#41	D₄×C₃₁	1	4	yes	—
379	C₂×#41	C₂³×C₃₁	1	4	yes	—
380	C₂×#41	C₂³×C₃₁	1	4	yes	—
381	C₂²×Q₈	—	2	1	no	—
382	C₂²×Q₈	—	2	2	no	—
383	C₂²×Q₈	—	2	4	no	—
384	C₂²×Q₈	—	2	4	no	—
385	C₂²×Q₈	D₄×C₃₁	1	1	yes	characteristic
386	C₂²×Q₈	C₂³×C₃₁	1	2	yes	—
387	C₂²×Q₈	C₂³×C₃₁	1	1	yes	characteristic
388	C₂²×Q₈	C₂³×C₃₁	1	4	yes	—
389	C₂²×D₄	—	2	1	no	—
390	C₂²×D₄	—	2	2	no	—
391	C₂²×D₄	D₄×C₃₁	1	2	yes	—
392	C₂²×D₄	D₄×C₃₁	1	1	yes	characteristic
393	C₂²×D₄	C₂³×C₃₁	1	1	yes	characteristic
394	C₂³×C₄	—	2	1	no	—
395	C₂³×C₄	—	2	1	no	—
396	C₂³×C₄	—	2	1	no	—
397	C₂³×C₄	C₂³×C₃₁	1	2	yes	—
398	C₂³×C₄	C₂³×C₃₁	1	1	yes	characteristic
399	#137	—	2	32	no	—
400	C₂×Q₁₆	—	2	8	no	—
401	C₂×Q₁₆	C₂³×C₃₁	1	8	yes	—
402	C₂×SD₁₆	—	2	8	no	—
403	C₂×SD₁₆	C₂³×C₃₁	1	8	yes	—
404	#131	—	2	32	no	—
405	C₂×M₁₆	—	2	8	no	—
406	C₂×M₁₆	D₄×C₃₁	1	8	yes	complemented
407	C₂×M₁₆	C₂³×C₃₁	1	8	yes	—
408	C₂×M₁₆	C₂³×C₃₁	1	4	yes	—
409	C₂²×C₈	—	2	4	no	—
410	C₂²×C₈	—	2	8	no	—
411	C₂²×C₈	C₂³×C₃₁	1	1	yes	characteristic
412	C₂²×C₈	C₂³×C₃₁	1	1	yes	characteristic
413	C₂²×C₈	C₂³×C₃₁	1	4	yes	—
414	C₂²×C₈	C₂³×C₃₁	1	8	yes	—
415	C₂²×C₈	C₂³×C₃₁	1	2	yes	—
416	C₄⋊Q₈	—	2	8	no	—
417	#124	—	2	8	no	—
418	#123	—	2	16	no	—
419	C₂²⋊Q₈	—	2	8	no	—
420	C₂²⋊Q₈	—	2	4	no	—
421	C₂²⋊Q₈	—	2	4	no	—
422	C₂²⋊Q₈	—	2	4	no	—
423	C₂²⋊Q₈	C₂³×C₃₁	1	8	yes	—
424	C₂²⋊Q₈	C₂³×C₃₁	1	8	yes	—
425	C₂²⋊Q₈	C₂³×C₃₁	1	8	yes	—
426	#121	—	2	8	no	—
427	#121	—	2	16	no	—
428	#121	—	2	4	no	—
429	#121	C₂³×C₃₁	1	8	yes	—
430	#117	—	2	2	no	—
431	#117	C₂³×C₃₁	1	4	yes	—
432	C₂×C₄⋊C₄	—	2	1	no	—
433	C₂×C₄⋊C₄	—	2	2	no	—
434	C₂×C₄⋊C₄	—	2	2	no	—
435	C₂×C₄⋊C₄	—	2	2	no	—
436	C₂×C₄⋊C₄	—	2	2	no	—
437	C₂×C₄⋊C₄	C₂³×C₃₁	1	2	yes	—
438	C₂×C₄⋊C₄	C₂³×C₃₁	1	4	yes	—
439	C₂×C₄⋊C₄	C₂³×C₃₁	1	1	yes	characteristic
440	C₂×C₄⋊C₄	C₂³×C₃₁	1	1	yes	characteristic
441	C₂×C₂²⋊C₄	—	2	1	no	—
442	C₂×C₂²⋊C₄	—	2	2	no	—
443	C₂×C₂²⋊C₄	—	2	2	no	—
444	C₂×C₂²⋊C₄	—	2	2	no	—
445	C₂×C₂²⋊C₄	—	2	2	no	—
446	C₂×C₄²	—	2	1	no	—
447	#107	C₂³×C₃₁	1	8	yes	—
448	#106	C₂³×C₃₁	1	8	yes	—
449	#103	C₂³×C₃₁	1	8	yes	—
450	#103	C₂³×C₃₁	1	8	yes	—
451	#103	C₂³×C₃₁	1	4	yes	—
452	#103	C₂³×C₃₁	1	4	yes	—
453	#102	C₂³×C₃₁	1	4	yes	—
454	#102	C₂³×C₃₁	1	4	yes	—
455	C₃₁	C₂×#3063	1	1	yes	characteristic, direct product factor, Sylow
456	C₂⁴	—	2	1	no	—
457	C₂⁴	C₂×D₄×C₃₁	1	2	yes	—
458	C₂⁴	C₂×D₄×C₃₁	1	1	yes	characteristic
459	#41	—	2	16	no	—
460	#41	—	2	8	no	—
461	#41	—	2	8	no	—
462	#41	—	2	8	no	—
463	#41	—	2	8	no	—
464	#41	—	4	8	no	—
465	C₂×Q₈	—	2	4	no	—
466	C₂×Q₈	—	2	2	no	—
467	C₂×Q₈	—	2	1	no	—
468	C₂×Q₈	—	2	4	no	—
469	C₂×Q₈	—	2	4	no	—
470	C₂×Q₈	—	2	2	no	—
471	C₂×Q₈	—	2	8	no	—
472	C₂×Q₈	—	2	4	no	—
473	C₂×Q₈	—	2	8	no	—
474	C₂×Q₈	—	2	2	no	—
475	C₂×Q₈	—	4	4	no	—
476	C₂×Q₈	—	4	8	no	—
477	C₂×Q₈	C₂×D₄×C₃₁	1	4	yes	—
478	C₂×Q₈	C₂⁴×C₃₁	1	4	yes	—
479	C₂×Q₈	C₂⁴×C₃₁	1	2	yes	—
480	C₂×D₄	—	2	8	no	—
481	C₂×D₄	—	2	4	no	—
482	C₂×D₄	—	2	8	no	—
483	C₂×D₄	—	2	4	no	—
484	C₂×D₄	—	2	2	no	—
485	C₂×D₄	—	2	1	no	—
486	C₂×D₄	—	2	2	no	—
487	C₂×D₄	—	4	4	no	—
488	C₂×D₄	—	8	4	no	—
489	C₂×D₄	C₂×D₄×C₃₁	1	8	yes	—
490	C₂×D₄	C₂×D₄×C₃₁	1	4	yes	—
491	C₂×D₄	C₂⁴×C₃₁	1	2	yes	—
492	C₂²×C₄	—	2	4	no	—
493	C₂²×C₄	—	2	4	no	—
494	C₂²×C₄	—	2	1	no	—
495	C₂²×C₄	—	2	1	no	—
496	C₂²×C₄	—	2	2	no	—
497	C₂²×C₄	—	2	1	no	—
498	C₂²×C₄	—	2	2	no	—
499	C₂²×C₄	—	2	4	no	—
500	C₂²×C₄	—	2	4	no	—
501	C₂²×C₄	—	2	4	no	—
502	C₂²×C₄	—	4	4	no	—
503	C₂²×C₄	—	4	4	no	—
504	C₂²×C₄	—	8	2	no	—
505	C₂²×C₄	C₂×D₄×C₃₁	1	2	yes	—
506	C₂²×C₄	C₂×D₄×C₃₁	1	4	yes	—
507	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
508	C₂²×C₄	C₂×D₄×C₃₁	1	2	yes	—
509	C₂²×C₄	C₂×D₄×C₃₁	1	4	yes	—
510	C₂²×C₄	#41×C₃₁	1	4	yes	—
511	C₂²×C₄	C₂⁴×C₃₁	1	4	yes	—
512	C₂²×C₄	C₂⁴×C₃₁	1	1	yes	characteristic
513	C₂²×C₄	C₂⁴×C₃₁	1	2	yes	—
514	Q₁₆	—	2	16	no	—
515	SD₁₆	—	2	16	no	—
516	M₁₆	—	2	16	no	—
517	M₁₆	C₂×D₄×C₃₁	1	16	yes	complemented
518	C₂×C₈	—	2	2	no	—
519	C₂×C₈	—	2	2	no	—
520	C₂×C₈	—	2	8	no	—
521	C₂×C₈	—	2	16	no	—
522	C₂×C₈	C₂×D₄×C₃₁	1	8	yes	—
523	C₂×C₈	C₂⁴×C₃₁	1	2	yes	—
524	C₂×C₈	C₂⁴×C₃₁	1	2	yes	—
525	C₂×C₈	C₂⁴×C₃₁	1	4	yes	—
526	C₄⋊C₄	—	2	4	no	—
527	C₄⋊C₄	—	2	2	no	—
528	C₄⋊C₄	—	2	8	no	—
529	C₄⋊C₄	—	2	8	no	—
530	C₄⋊C₄	—	2	8	no	—
531	C₄⋊C₄	—	2	1	no	—
532	C₄⋊C₄	—	2	1	no	—
533	C₄⋊C₄	C₂⁴×C₃₁	1	2	yes	—
534	C₄⋊C₄	C₂⁴×C₃₁	1	2	yes	—
535	C₄⋊C₄	C₂⁴×C₃₁	1	4	yes	—
536	C₂²⋊C₄	—	2	4	no	—
537	C₂²⋊C₄	—	2	8	no	—
538	C₂²⋊C₄	—	2	8	no	—
539	C₂²⋊C₄	—	2	8	no	—
540	C₂²⋊C₄	—	2	8	no	—
541	C₄²	—	2	4	no	—
542	C₂³	—	2	2	no	—
543	C₂³	—	2	2	no	—
544	C₂³	—	2	4	no	—
545	C₂³	—	2	4	no	—
546	C₂³	—	2	4	no	—
547	C₂³	—	2	2	no	—
548	C₂³	—	4	2	no	—
549	C₂³	—	4	2	no	—
550	C₂³	—	4	2	no	—
551	C₂³	C₃₁×#121	1	4	yes	—
552	C₂³	C₂²×D₄×C₃₁	1	4	yes	—
553	C₂³	C₂²×D₄×C₃₁	1	2	yes	—
554	C₂³	C₂²×D₄×C₃₁	1	1	yes	characteristic
555	Q₈	—	2	8	no	—
556	Q₈	—	2	4	no	—
557	Q₈	—	2	2	no	—
558	Q₈	—	2	4	no	—
559	Q₈	—	2	2	no	—
560	Q₈	—	4	4	no	—
561	Q₈	—	4	8	no	—
562	Q₈	—	4	2	no	—
563	D₄	—	2	16	no	—
564	D₄	—	2	8	no	—
565	D₄	—	2	2	no	—
566	D₄	—	2	2	no	—
567	D₄	—	4	8	no	—
568	D₄	—	4	2	no	—
569	D₄	—	8	8	no	—
570	C₂×C₄	—	2	4	no	—
571	C₂×C₄	—	2	8	no	—
572	C₂×C₄	—	2	4	no	—
573	C₂×C₄	—	2	4	no	—
574	C₂×C₄	—	2	2	no	—
575	C₂×C₄	—	2	2	no	—
576	C₂×C₄	—	2	4	no	—
577	C₂×C₄	—	2	2	no	—
578	C₂×C₄	—	2	4	no	—
579	C₂×C₄	—	4	2	no	—
580	C₂×C₄	—	4	8	no	—
581	C₂×C₄	—	4	4	no	—
582	C₂×C₄	—	4	2	no	—
583	C₂×C₄	—	4	4	no	—
584	C₂×C₄	—	4	2	no	—
585	C₂×C₄	—	4	8	no	—
586	C₂×C₄	—	8	4	no	—
587	C₂×C₄	C₃₁×#121	1	4	yes	—
588	C₂×C₄	C₃₁×#121	1	8	yes	—
589	C₂×C₄	C₂²×D₄×C₃₁	1	4	yes	—
590	C₂×C₄	C₂²×D₄×C₃₁	1	2	yes	—
591	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
592	C₂×C₄	C₂²×D₄×C₃₁	1	2	yes	—
593	C₂×C₄	C₂²×D₄×C₃₁	1	4	yes	—
594	C₂×C₄	C₂×#41×C₃₁	1	2	yes	—
595	C₂×C₄	C₂×#41×C₃₁	1	4	yes	—
596	C₂×C₄	C₂⁵×C₃₁	1	1	yes	characteristic
597	C₈	—	2	2	no	—
598	C₈	—	2	2	no	—
599	C₈	C₂²×D₄×C₃₁	1	8	yes	—
600	C₂²	—	2	8	no	—
601	C₂²	—	2	2	no	—
602	C₂²	—	2	2	no	—
603	C₂²	—	2	4	no	—
604	C₂²	—	2	4	no	—
605	C₂²	—	2	4	no	—
606	C₂²	—	4	8	no	—
607	C₂²	—	4	2	no	—
608	C₂²	—	4	2	no	—
609	C₂²	—	4	2	no	—
610	C₂²	—	8	4	no	—
611	C₂²	C₂×C₃₁×#121	1	4	yes	—
612	C₂²	C₂×C₃₁×#121	1	2	yes	—
613	C₂²	C₃₁×#577	1	2	yes	—
614	C₂²	C₃₁×#578	1	2	yes	—
615	C₂²	C₂³×D₄×C₃₁	1	1	yes	characteristic
616	C₄	—	2	4	no	—
617	C₄	—	4	2	no	—
618	C₄	—	4	4	no	—
619	C₄	—	4	4	no	—
620	C₄	—	4	2	no	—
621	C₄	—	4	2	no	—
622	C₄	C₂×C₃₁×#121	1	2	yes	—
623	C₄	C₂×C₃₁×#121	1	4	yes	—
624	C₄	C₂³×D₄×C₃₁	1	1	yes	characteristic
625	C₄	C₂²×#41×C₃₁	1	1	yes	characteristic
626	C₂	—	2	4	no	—
627	C₂	—	4	4	no	—
628	C₂	—	8	2	no	—
629	C₂	C₃₁×#3063	1	4	yes	direct product factor
630	C₂	C₂²×C₃₁×#121	1	1	yes	characteristic
631	C₂	C₂×C₃₁×#577	1	1	yes	characteristic
632	C₂	C₂×C₃₁×#578	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂³×C₃₁	C₂²×D₄
commutator, Frattini, cosocle	C₂×C₄	C₂⁵×C₃₁
prime radical	C₂×C₃₁×#537	C₂
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₃₁ ◁ C₂²×#41×C₃₁ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂²×C₃₁×#137	C₂	semidirect
2. C₂²×C₃₁×#131	C₂	semidirect
4. C₂×C₃₁×#536	C₂	semidirect
5. C₃₁×#3063	C₂	direct (N × H)
6. C₂×C₃₁×#470	C₂	semidirect
7. C₂×C₃₁×#468	C₂	semidirect
8. C₂×C₃₁×#467	C₂	semidirect
9. C₂×C₃₁×#465	C₂	semidirect
10. C₂×C₃₁×#428	C₂	semidirect
11. C₂×C₃₁×#416	C₂	semidirect
12. C₂²×C₃₁×#103	C₂	semidirect
17. C₂×C₃₁×#137	C₂²	semidirect
21. C₂×C₃₁×#131	C₂²	semidirect
29. C₃₁×#536	C₂²	semidirect
42. C₃₁×#470	C₂²	semidirect
43. C₃₁×#468	C₂²	semidirect
44. C₃₁×#467	C₂²	semidirect
45. C₃₁×#465	C₂²	semidirect
46. C₃₁×#428	C₂²	semidirect
49. C₃₁×#416	C₂²	semidirect
54. C₂×C₃₁×#103	C₂²	semidirect
90. C₂×M₁₆×C₃₁	D₄	semidirect
200. M₁₆×C₃₁	C₂×D₄	semidirect
225. C₂×#3063	C₃₁	direct (N × H)
284. C₂²×#137	C₂×C₃₁	semidirect
285. C₂²×#131	C₂×C₃₁	semidirect
287. C₂×#536	C₂×C₃₁	semidirect
288. #3063	C₂×C₃₁	direct (N × H)
289. C₂×#470	C₂×C₃₁	semidirect
290. C₂×#468	C₂×C₃₁	semidirect
291. C₂×#467	C₂×C₃₁	semidirect
292. C₂×#465	C₂×C₃₁	semidirect
293. C₂×#428	C₂×C₃₁	semidirect
294. C₂×#416	C₂×C₃₁	semidirect
295. C₂²×#103	C₂×C₃₁	semidirect
326. C₂×#137	C₂²×C₃₁	semidirect
330. C₂×#131	C₂²×C₃₁	semidirect
338. #536	C₂²×C₃₁	semidirect
351. #470	C₂²×C₃₁	semidirect
352. #468	C₂²×C₃₁	semidirect
353. #467	C₂²×C₃₁	semidirect
354. #465	C₂²×C₃₁	semidirect
355. #428	C₂²×C₃₁	semidirect
358. #416	C₂²×C₃₁	semidirect
363. C₂×#103	C₂²×C₃₁	semidirect
369. C₂×C₃₁	#3063	direct (N × H)
406. C₂×M₁₆	D₄×C₃₁	semidirect
455. C₃₁	C₂×#3063	direct (N × H)
517. M₁₆	C₂×D₄×C₃₁	semidirect
629. C₂	C₃₁×#3063	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×#137	2
C₂²×C₃₁×#131	2
C₂×C₃₁×#537	2
C₂×C₃₁×#536	2
C₃₁×#3063	2
C₂×C₃₁×#470	2
C₂×C₃₁×#468	2
C₂×C₃₁×#467	2
C₂×C₃₁×#465	2
C₂×C₃₁×#428	2
C₂×C₃₁×#416	2
C₂²×C₃₁×#103	2
C₂×#3063	31

Subgroups — C2×C31×#3063 #21517211

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₃₁×#3063 #21517211