Subgroups — C₂×C₃₁×#3105 #21517291

Order 7936.

2602 conjugacy classes of subgroups.

The subgroup lattice has 622 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃₁×#529	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂²×C₄×D₄×C₃₁	C₂	1	1	yes	maximal, characteristic, complemented
3	C₃₁×#3105	C₂	1	16	yes	maximal, direct product factor
4	C₂×C₃₁×#484	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×C₃₁×#481	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×C₃₁×#478	C₂	1	2	yes	maximal, complemented
7	C₂×C₃₁×#463	C₂	1	2	yes	maximal, complemented
8	C₂×C₃₁×#450	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂×C₃₁×#447	C₂	1	1	yes	maximal, characteristic
10	C₂×C₄×SD₁₆×C₃₁	C₂	1	2	yes	maximal, complemented
11	C₂×C₄×D₈×C₃₁	C₂	1	2	yes	maximal, complemented
12	C₂×C₃₁×#431	C₂	1	1	yes	maximal, characteristic, complemented
13	C₂³×D₄×C₃₁	C₂²	1	1	yes	characteristic
14	C₂⁴×C₄×C₃₁	—	2	1	no	—
15	C₂²×SD₁₆×C₃₁	C₂²	1	2	yes	—
16	C₂²×D₈×C₃₁	C₂²	1	2	yes	—
17	C₃₁×#529	—	2	4	no	—
18	C₃₁×#529	C₂²	1	8	yes	complemented
19	C₂×C₃₁×#126	—	2	1	no	—
20	C₂×C₃₁×#125	C₂²	1	1	yes	characteristic, complemented
21	C₂×C₃₁×#124	C₂²	1	1	yes	characteristic, complemented
22	C₂×C₃₁×#123	—	2	1	no	—
23	C₂×C₃₁×C₂²⋊Q₈	C₂²	1	1	yes	characteristic
24	C₂×C₃₁×#121	C₂²	1	1	yes	characteristic
25	C₂×C₄×Q₈×C₃₁	C₂²	1	1	yes	characteristic, complemented
26	C₂×C₄×D₄×C₃₁	—	2	1	no	—
27	C₂×C₄×D₄×C₃₁	—	2	1	no	—
28	C₂×C₄×D₄×C₃₁	—	2	4	no	—
29	C₂×C₄×D₄×C₃₁	—	2	1	no	—
30	C₂×C₄×D₄×C₃₁	C₂²	1	2	yes	—
31	C₂×C₄×D₄×C₃₁	C₂²	1	2	yes	—
32	C₂×C₄×D₄×C₃₁	C₂²	1	8	yes	complemented
33	C₂×C₄×D₄×C₃₁	C₂²	1	1	yes	characteristic, complemented
34	C₂×C₃₁×#117	—	2	1	no	—
35	C₂²×C₄⋊C₄×C₃₁	C₂²	1	1	yes	characteristic
36	C₂²×C₂²⋊C₄×C₃₁	—	2	1	no	—
37	C₂²×C₄²×C₃₁	C₂²	1	1	yes	characteristic
38	C₃₁×#484	C₂²	1	8	yes	complemented
39	C₃₁×#481	C₂²	1	8	yes	complemented
40	C₃₁×#478	C₂²	1	16	yes	complemented
41	C₃₁×#463	C₂²	1	16	yes	complemented
42	C₃₁×#450	C₂²	1	8	yes	complemented
43	C₃₁×#447	C₂²	1	8	yes	—
44	C₄×SD₁₆×C₃₁	C₂²	1	16	yes	complemented
45	C₄×D₈×C₃₁	C₂²	1	16	yes	complemented
46	C₃₁×#431	C₂²	1	8	yes	complemented
47	C₂×C₃₁×#107	C₂²	1	2	yes	complemented
48	C₂×C₃₁×#106	C₂²	1	2	yes	complemented
49	C₂×C₃₁×C₄⋊C₈	C₂²	1	2	yes	complemented
50	C₂×C₃₁×#103	C₂²	1	2	yes	complemented
51	C₂×C₃₁×#102	C₂²	1	2	yes	—
52	C₂×C₃₁×#102	C₂²	1	2	yes	—
53	C₂×C₃₁×#102	C₂²	1	2	yes	complemented
54	C₂×C₃₁×C₂²⋊C₈	C₂²	1	1	yes	characteristic
55	C₂×C₃₁×C₂²⋊C₈	C₂²	1	1	yes	characteristic
56	C₂×C₄×C₈×C₃₁	C₂²	1	2	yes	complemented
57	C₂⁵×C₃₁	—	2	1	no	—
58	C₂²×Q₈×C₃₁	C₂³	1	1	yes	characteristic
59	C₂²×D₄×C₃₁	—	2	8	no	—
60	C₂²×D₄×C₃₁	—	2	1	no	—
61	C₂²×D₄×C₃₁	—	2	1	no	—
62	C₂²×D₄×C₃₁	—	2	2	no	—
63	C₂²×D₄×C₃₁	C₂³	1	1	yes	characteristic
64	C₂²×D₄×C₃₁	C₂³	1	4	yes	—
65	C₂²×D₄×C₃₁	C₂³	1	2	yes	—
66	C₂³×C₄×C₃₁	—	2	1	no	—
67	C₂³×C₄×C₃₁	—	2	2	no	—
68	C₂³×C₄×C₃₁	—	2	8	no	—
69	C₂³×C₄×C₃₁	—	2	1	no	—
70	C₂³×C₄×C₃₁	—	2	2	no	—
71	C₂³×C₄×C₃₁	—	4	8	no	—
72	C₂³×C₄×C₃₁	D₄	1	1	yes	characteristic
73	C₂³×C₄×C₃₁	D₄	1	1	yes	characteristic
74	C₂³×C₄×C₃₁	C₂³	1	1	yes	characteristic
75	C₂×SD₁₆×C₃₁	—	2	8	no	—
76	C₂×SD₁₆×C₃₁	C₂³	1	8	yes	—
77	C₂×D₈×C₃₁	—	2	8	no	—
78	C₂×D₈×C₃₁	C₂³	1	8	yes	—
79	C₂²×C₈×C₃₁	C₂³	1	2	yes	—
80	C₂²×C₈×C₃₁	C₂³	1	2	yes	—
81	C₃₁×#126	—	2	8	no	—
82	C₃₁×#125	—	2	2	no	—
83	C₃₁×#125	C₂³	1	4	yes	complemented
84	C₃₁×#124	—	2	2	no	—
85	C₃₁×#124	C₂³	1	4	yes	complemented
86	C₃₁×#123	—	2	8	no	—
87	C₃₁×C₂²⋊Q₈	—	2	2	no	—
88	C₃₁×C₂²⋊Q₈	C₂³	1	4	yes	—
89	C₃₁×#121	—	2	2	no	—
90	C₃₁×#121	C₂³	1	4	yes	—
91	C₄×Q₈×C₃₁	—	2	2	no	—
92	C₄×Q₈×C₃₁	C₂³	1	4	yes	complemented
93	C₄×D₄×C₃₁	—	2	8	no	—
94	C₄×D₄×C₃₁	—	2	4	no	—
95	C₄×D₄×C₃₁	—	2	4	no	—
96	C₄×D₄×C₃₁	—	2	8	no	—
97	C₄×D₄×C₃₁	—	2	8	no	—
98	C₄×D₄×C₃₁	—	2	2	no	—
99	C₄×D₄×C₃₁	C₂³	1	8	yes	—
100	C₄×D₄×C₃₁	C₂³	1	8	yes	—
101	C₄×D₄×C₃₁	C₂³	1	4	yes	complemented
102	C₃₁×#117	—	2	8	no	—
103	C₂×C₄⋊C₄×C₃₁	—	2	1	no	—
104	C₂×C₄⋊C₄×C₃₁	—	2	2	no	—
105	C₂×C₄⋊C₄×C₃₁	—	2	1	no	—
106	C₂×C₄⋊C₄×C₃₁	—	2	1	no	—
107	C₂×C₄⋊C₄×C₃₁	—	2	1	no	—
108	C₂×C₄⋊C₄×C₃₁	—	2	1	no	—
109	C₂×C₄⋊C₄×C₃₁	C₂³	1	1	yes	characteristic
110	C₂×C₄⋊C₄×C₃₁	C₂³	1	1	yes	characteristic
111	C₂×C₄⋊C₄×C₃₁	C₂³	1	2	yes	—
112	C₂×C₄⋊C₄×C₃₁	C₂³	1	4	yes	—
113	C₂×C₂²⋊C₄×C₃₁	—	2	1	no	—
114	C₂×C₂²⋊C₄×C₃₁	—	2	2	no	—
115	C₂×C₂²⋊C₄×C₃₁	—	2	2	no	—
116	C₂×C₂²⋊C₄×C₃₁	—	2	8	no	—
117	C₂×C₂²⋊C₄×C₃₁	—	2	1	no	—
118	C₂×C₂²⋊C₄×C₃₁	—	2	1	no	—
119	C₂×C₂²⋊C₄×C₃₁	—	2	1	no	—
120	C₂×C₂²⋊C₄×C₃₁	—	2	1	no	—
121	C₂×C₄²×C₃₁	—	2	1	no	—
122	C₂×C₄²×C₃₁	D₄	1	1	yes	characteristic
123	C₂×C₄²×C₃₁	D₄	1	4	yes	—
124	C₂×C₄²×C₃₁	D₄	1	1	yes	characteristic
125	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
126	C₂×C₄²×C₃₁	C₂³	1	4	yes	—
127	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
128	C₃₁×#107	C₂³	1	8	yes	complemented
129	C₃₁×#106	C₂³	1	8	yes	complemented
130	C₃₁×C₄⋊C₈	C₂³	1	8	yes	complemented
131	C₃₁×#103	C₂³	1	8	yes	complemented
132	C₃₁×#102	C₂³	1	8	yes	—
133	C₃₁×#102	C₂³	1	8	yes	—
134	C₃₁×#102	C₂³	1	8	yes	complemented
135	C₃₁×C₂²⋊C₈	C₂³	1	4	yes	—
136	C₃₁×C₂²⋊C₈	C₂³	1	4	yes	—
137	C₄×C₈×C₃₁	C₂³	1	8	yes	complemented
138	C₂⁴×C₃₁	—	2	8	no	—
139	C₂⁴×C₃₁	—	2	4	no	—
140	C₂⁴×C₃₁	—	2	2	no	—
141	C₂⁴×C₃₁	—	2	1	no	—
142	C₂⁴×C₃₁	—	4	4	no	—
143	C₂⁴×C₃₁	—	4	4	no	—
144	C₂⁴×C₃₁	C₂×D₄	1	1	yes	characteristic
145	C₂×Q₈×C₃₁	—	2	1	no	—
146	C₂×Q₈×C₃₁	—	2	4	no	—
147	C₂×Q₈×C₃₁	C₂⁴	1	2	yes	—
148	C₂×D₄×C₃₁	—	2	8	no	—
149	C₂×D₄×C₃₁	—	2	16	no	—
150	C₂×D₄×C₃₁	—	2	16	no	—
151	C₂×D₄×C₃₁	—	2	4	no	—
152	C₂×D₄×C₃₁	—	2	4	no	—
153	C₂×D₄×C₃₁	—	2	2	no	—
154	C₂×D₄×C₃₁	—	2	1	no	—
155	C₂×D₄×C₃₁	—	2	4	no	—
156	C₂×D₄×C₃₁	—	2	4	no	—
157	C₂×D₄×C₃₁	—	4	4	no	—
158	C₂×D₄×C₃₁	#41	1	8	yes	—
159	C₂×D₄×C₃₁	C₂⁴	1	2	yes	—
160	C₂×D₄×C₃₁	C₂⁴	1	4	yes	—
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	1	no	—
165	C₂²×C₄×C₃₁	—	2	1	no	—
166	C₂²×C₄×C₃₁	—	2	1	no	—
167	C₂²×C₄×C₃₁	—	2	4	no	—
168	C₂²×C₄×C₃₁	—	2	2	no	—
169	C₂²×C₄×C₃₁	—	2	4	no	—
170	C₂²×C₄×C₃₁	—	2	8	no	—
171	C₂²×C₄×C₃₁	—	2	8	no	—
172	C₂²×C₄×C₃₁	—	4	8	no	—
173	C₂²×C₄×C₃₁	—	4	32	no	—
174	C₂²×C₄×C₃₁	—	4	4	no	—
175	C₂²×C₄×C₃₁	—	4	4	no	—
176	C₂²×C₄×C₃₁	—	4	8	no	—
177	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
178	C₂²×C₄×C₃₁	C₂×D₄	1	2	yes	—
179	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
180	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
181	C₂²×C₄×C₃₁	C₂×D₄	1	4	yes	—
182	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
183	C₂²×C₄×C₃₁	C₂×D₄	1	4	yes	—
184	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
185	C₂²×C₄×C₃₁	C₂⁴	1	2	yes	—
186	C₂²×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
187	SD₁₆×C₃₁	—	2	16	no	—
188	D₈×C₃₁	—	2	16	no	—
189	C₂×C₈×C₃₁	—	2	4	no	—
190	C₂×C₈×C₃₁	—	2	4	no	—
191	C₂×C₈×C₃₁	C₂⁴	1	4	yes	—
192	C₂×C₈×C₃₁	C₂⁴	1	4	yes	—
193	C₄⋊C₄×C₃₁	—	2	4	no	—
194	C₄⋊C₄×C₃₁	—	2	2	no	—
195	C₄⋊C₄×C₃₁	—	2	4	no	—
196	C₄⋊C₄×C₃₁	—	2	4	no	—
197	C₄⋊C₄×C₃₁	—	2	4	no	—
198	C₄⋊C₄×C₃₁	—	2	4	no	—
199	C₄⋊C₄×C₃₁	—	2	1	no	—
200	C₄⋊C₄×C₃₁	—	2	1	no	—
201	C₄⋊C₄×C₃₁	C₂⁴	1	2	yes	—
202	C₄⋊C₄×C₃₁	C₂⁴	1	2	yes	—
203	C₄⋊C₄×C₃₁	C₂⁴	1	4	yes	—
204	C₂²⋊C₄×C₃₁	—	2	4	no	—
205	C₂²⋊C₄×C₃₁	—	2	8	no	—
206	C₂²⋊C₄×C₃₁	—	2	8	no	—
207	C₂²⋊C₄×C₃₁	—	2	4	no	—
208	C₂²⋊C₄×C₃₁	—	2	4	no	—
209	C₂²⋊C₄×C₃₁	—	2	4	no	—
210	C₂²⋊C₄×C₃₁	—	2	4	no	—
211	C₄²×C₃₁	—	2	4	no	—
212	C₄²×C₃₁	C₂×D₄	1	4	yes	—
213	C₄²×C₃₁	C₂×D₄	1	2	yes	—
214	C₄²×C₃₁	C₂×D₄	1	2	yes	—
215	C₄²×C₃₁	C₂×D₄	1	4	yes	—
216	C₄²×C₃₁	C₂⁴	1	2	yes	—
217	C₄²×C₃₁	C₂⁴	1	2	yes	—
218	C₂×#3105	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
219	C₂³×C₃₁	—	2	2	no	—
220	C₂³×C₃₁	—	2	2	no	—
221	C₂³×C₃₁	—	2	16	no	—
222	C₂³×C₃₁	—	2	8	no	—
223	C₂³×C₃₁	—	2	4	no	—
224	C₂³×C₃₁	—	2	2	no	—
225	C₂³×C₃₁	—	4	8	no	—
226	C₂³×C₃₁	—	4	32	no	—
227	C₂³×C₃₁	—	4	8	no	—
228	C₂³×C₃₁	—	4	4	no	—
229	C₂³×C₃₁	—	4	4	no	—
230	C₂³×C₃₁	—	4	2	no	—
231	C₂³×C₃₁	—	4	2	no	—
232	C₂³×C₃₁	—	4	2	no	—
233	C₂³×C₃₁	#135	1	4	yes	—
234	C₂³×C₃₁	C₂²×D₄	1	2	yes	—
235	C₂³×C₃₁	C₂²×D₄	1	1	yes	characteristic
236	Q₈×C₃₁	—	2	4	no	—
237	Q₈×C₃₁	—	4	2	no	—
238	D₄×C₃₁	—	2	16	no	—
239	D₄×C₃₁	—	2	8	no	—
240	D₄×C₃₁	—	2	4	no	—
241	D₄×C₃₁	—	2	4	no	—
242	D₄×C₃₁	—	4	8	no	—
243	D₄×C₃₁	—	4	2	no	—
244	D₄×C₃₁	C₂×#41	1	8	yes	—
245	C₂×C₄×C₃₁	—	2	2	no	—
246	C₂×C₄×C₃₁	—	2	8	no	—
247	C₂×C₄×C₃₁	—	2	2	no	—
248	C₂×C₄×C₃₁	—	2	2	no	—
249	C₂×C₄×C₃₁	—	2	2	no	—
250	C₂×C₄×C₃₁	—	2	4	no	—
251	C₂×C₄×C₃₁	—	2	8	no	—
252	C₂×C₄×C₃₁	—	2	4	no	—
253	C₂×C₄×C₃₁	—	2	2	no	—
254	C₂×C₄×C₃₁	—	2	2	no	—
255	C₂×C₄×C₃₁	—	2	2	no	—
256	C₂×C₄×C₃₁	—	2	4	no	—
257	C₂×C₄×C₃₁	—	4	16	no	—
258	C₂×C₄×C₃₁	—	4	8	no	—
259	C₂×C₄×C₃₁	—	4	2	no	—
260	C₂×C₄×C₃₁	—	4	2	no	—
261	C₂×C₄×C₃₁	—	4	2	no	—
262	C₂×C₄×C₃₁	—	4	4	no	—
263	C₂×C₄×C₃₁	—	4	8	no	—
264	C₂×C₄×C₃₁	—	4	16	no	—
265	C₂×C₄×C₃₁	AGL(1,ℤ/8)	1	4	yes	—
266	C₂×C₄×C₃₁	C₂²×D₄	1	2	yes	—
267	C₂×C₄×C₃₁	C₂²×D₄	1	2	yes	—
268	C₂×C₄×C₃₁	C₂²×D₄	1	2	yes	—
269	C₂×C₄×C₃₁	C₂²×D₄	1	2	yes	—
270	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
271	C₂×C₄×C₃₁	C₂²×D₄	1	2	yes	—
272	C₂×C₄×C₃₁	C₂×#41	1	4	yes	—
273	C₂×C₄×C₃₁	C₂⁵	1	1	yes	characteristic
274	C₈×C₃₁	—	2	4	no	—
275	C₈×C₃₁	—	2	4	no	—
276	C₂×#529	C₂×C₃₁	1	1	yes	characteristic, complemented
277	C₂²×C₄×D₄	C₂×C₃₁	1	1	yes	characteristic, complemented
278	#3105	C₂×C₃₁	1	16	yes	direct product factor
279	C₂×#484	C₂×C₃₁	1	1	yes	characteristic, complemented
280	C₂×#481	C₂×C₃₁	1	1	yes	characteristic, complemented
281	C₂×#478	C₂×C₃₁	1	2	yes	complemented
282	C₂×#463	C₂×C₃₁	1	2	yes	complemented
283	C₂×#450	C₂×C₃₁	1	1	yes	characteristic, complemented
284	C₂×#447	C₂×C₃₁	1	1	yes	characteristic
285	C₂×C₄×SD₁₆	C₂×C₃₁	1	2	yes	complemented
286	C₂×C₄×D₈	C₂×C₃₁	1	2	yes	complemented
287	C₂×#431	C₂×C₃₁	1	1	yes	characteristic, complemented
288	C₂²×C₃₁	—	2	8	no	—
289	C₂²×C₃₁	—	2	2	no	—
290	C₂²×C₃₁	—	2	2	no	—
291	C₂²×C₃₁	—	2	8	no	—
292	C₂²×C₃₁	—	4	32	no	—
293	C₂²×C₃₁	—	4	16	no	—
294	C₂²×C₃₁	—	4	4	no	—
295	C₂²×C₃₁	—	4	4	no	—
296	C₂²×C₃₁	—	4	2	no	—
297	C₂²×C₃₁	—	4	2	no	—
298	C₂²×C₃₁	—	4	2	no	—
299	C₂²×C₃₁	—	8	4	no	—
300	C₂²×C₃₁	#525	1	2	yes	—
301	C₂²×C₃₁	C₂×#135	1	4	yes	—
302	C₂²×C₃₁	C₂×#135	1	2	yes	—
303	C₂²×C₃₁	C₂×AGL(1,ℤ/8)	1	2	yes	—
304	C₂²×C₃₁	C₂³×D₄	1	1	yes	characteristic
305	C₄×C₃₁	—	2	2	no	—
306	C₄×C₃₁	—	2	2	no	—
307	C₄×C₃₁	—	2	2	no	—
308	C₄×C₃₁	—	2	2	no	—
309	C₄×C₃₁	—	4	2	no	—
310	C₄×C₃₁	—	4	2	no	—
311	C₄×C₃₁	—	4	2	no	—
312	C₄×C₃₁	—	4	4	no	—
313	C₄×C₃₁	#525	1	2	yes	—
314	C₄×C₃₁	C₂×AGL(1,ℤ/8)	1	4	yes	—
315	C₄×C₃₁	C₂²×#41	1	2	yes	—
316	C₂³×D₄	C₂²×C₃₁	1	1	yes	characteristic
317	C₂⁴×C₄	—	2	1	no	—
318	C₂²×SD₁₆	C₂²×C₃₁	1	2	yes	—
319	C₂²×D₈	C₂²×C₃₁	1	2	yes	—
320	#529	—	2	4	no	—
321	#529	C₂²×C₃₁	1	8	yes	complemented
322	C₂×#126	—	2	1	no	—
323	C₂×#125	C₂²×C₃₁	1	1	yes	characteristic, complemented
324	C₂×#124	C₂²×C₃₁	1	1	yes	characteristic, complemented
325	C₂×#123	—	2	1	no	—
326	C₂×C₂²⋊Q₈	C₂²×C₃₁	1	1	yes	characteristic
327	C₂×#121	C₂²×C₃₁	1	1	yes	characteristic
328	C₂×C₄×Q₈	C₂²×C₃₁	1	1	yes	characteristic, complemented
329	C₂×C₄×D₄	—	2	1	no	—
330	C₂×C₄×D₄	—	2	1	no	—
331	C₂×C₄×D₄	—	2	1	no	—
332	C₂×C₄×D₄	—	2	4	no	—
333	C₂×C₄×D₄	C₂²×C₃₁	1	2	yes	—
334	C₂×C₄×D₄	C₂²×C₃₁	1	2	yes	—
335	C₂×C₄×D₄	C₂²×C₃₁	1	8	yes	complemented
336	C₂×C₄×D₄	C₂²×C₃₁	1	1	yes	characteristic, complemented
337	C₂×#117	—	2	1	no	—
338	C₂²×C₄⋊C₄	C₂²×C₃₁	1	1	yes	characteristic
339	C₂²×C₂²⋊C₄	—	2	1	no	—
340	C₂²×C₄²	C₂²×C₃₁	1	1	yes	characteristic
341	#484	C₂²×C₃₁	1	8	yes	complemented
342	#481	C₂²×C₃₁	1	8	yes	complemented
343	#478	C₂²×C₃₁	1	16	yes	complemented
344	#463	C₂²×C₃₁	1	16	yes	complemented
345	#450	C₂²×C₃₁	1	8	yes	complemented
346	#447	C₂²×C₃₁	1	8	yes	—
347	C₄×SD₁₆	C₂²×C₃₁	1	16	yes	complemented
348	C₄×D₈	C₂²×C₃₁	1	16	yes	complemented
349	#431	C₂²×C₃₁	1	8	yes	complemented
350	C₂×#107	C₂²×C₃₁	1	2	yes	complemented
351	C₂×#106	C₂²×C₃₁	1	2	yes	complemented
352	C₂×C₄⋊C₈	C₂²×C₃₁	1	2	yes	complemented
353	C₂×#103	C₂²×C₃₁	1	2	yes	complemented
354	C₂×#102	C₂²×C₃₁	1	2	yes	—
355	C₂×#102	C₂²×C₃₁	1	2	yes	—
356	C₂×#102	C₂²×C₃₁	1	2	yes	complemented
357	C₂×C₂²⋊C₈	C₂²×C₃₁	1	1	yes	characteristic
358	C₂×C₂²⋊C₈	C₂²×C₃₁	1	1	yes	characteristic
359	C₂×C₄×C₈	C₂²×C₃₁	1	2	yes	complemented
360	C₂×C₃₁	—	2	4	no	—
361	C₂×C₃₁	—	4	8	no	—
362	C₂×C₃₁	—	8	2	no	—
363	C₂×C₃₁	#3105	1	4	yes	direct product factor
364	C₂×C₃₁	C₂×#525	1	1	yes	characteristic
365	C₂×C₃₁	C₂²×#135	1	1	yes	characteristic
366	C₂×C₃₁	C₂²×AGL(1,ℤ/8)	1	1	yes	characteristic
367	C₂⁵	—	2	1	no	—
368	C₂²×Q₈	C₂³×C₃₁	1	1	yes	characteristic
369	C₂²×D₄	—	2	8	no	—
370	C₂²×D₄	—	2	1	no	—
371	C₂²×D₄	—	2	1	no	—
372	C₂²×D₄	—	2	2	no	—
373	C₂²×D₄	C₂³×C₃₁	1	4	yes	—
374	C₂²×D₄	C₂³×C₃₁	1	1	yes	characteristic
375	C₂²×D₄	C₂³×C₃₁	1	2	yes	—
376	C₂³×C₄	—	2	1	no	—
377	C₂³×C₄	—	2	1	no	—
378	C₂³×C₄	—	2	2	no	—
379	C₂³×C₄	—	2	2	no	—
380	C₂³×C₄	—	2	8	no	—
381	C₂³×C₄	—	4	8	no	—
382	C₂³×C₄	D₄×C₃₁	1	1	yes	characteristic
383	C₂³×C₄	D₄×C₃₁	1	1	yes	characteristic
384	C₂³×C₄	C₂³×C₃₁	1	1	yes	characteristic
385	C₂×SD₁₆	—	2	8	no	—
386	C₂×SD₁₆	C₂³×C₃₁	1	8	yes	—
387	C₂×D₈	—	2	8	no	—
388	C₂×D₈	C₂³×C₃₁	1	8	yes	—
389	C₂²×C₈	C₂³×C₃₁	1	2	yes	—
390	C₂²×C₈	C₂³×C₃₁	1	2	yes	—
391	#126	—	2	8	no	—
392	#125	—	2	2	no	—
393	#125	C₂³×C₃₁	1	4	yes	complemented
394	#124	—	2	2	no	—
395	#124	C₂³×C₃₁	1	4	yes	complemented
396	#123	—	2	8	no	—
397	C₂²⋊Q₈	—	2	2	no	—
398	C₂²⋊Q₈	C₂³×C₃₁	1	4	yes	—
399	#121	—	2	2	no	—
400	#121	C₂³×C₃₁	1	4	yes	—
401	C₄×Q₈	—	2	2	no	—
402	C₄×Q₈	C₂³×C₃₁	1	4	yes	complemented
403	C₄×D₄	—	2	8	no	—
404	C₄×D₄	—	2	4	no	—
405	C₄×D₄	—	2	4	no	—
406	C₄×D₄	—	2	8	no	—
407	C₄×D₄	—	2	8	no	—
408	C₄×D₄	—	2	2	no	—
409	C₄×D₄	C₂³×C₃₁	1	8	yes	—
410	C₄×D₄	C₂³×C₃₁	1	8	yes	—
411	C₄×D₄	C₂³×C₃₁	1	4	yes	complemented
412	#117	—	2	8	no	—
413	C₂×C₄⋊C₄	—	2	1	no	—
414	C₂×C₄⋊C₄	—	2	2	no	—
415	C₂×C₄⋊C₄	—	2	1	no	—
416	C₂×C₄⋊C₄	—	2	1	no	—
417	C₂×C₄⋊C₄	—	2	1	no	—
418	C₂×C₄⋊C₄	—	2	1	no	—
419	C₂×C₄⋊C₄	C₂³×C₃₁	1	2	yes	—
420	C₂×C₄⋊C₄	C₂³×C₃₁	1	4	yes	—
421	C₂×C₄⋊C₄	C₂³×C₃₁	1	1	yes	characteristic
422	C₂×C₄⋊C₄	C₂³×C₃₁	1	1	yes	characteristic
423	C₂×C₂²⋊C₄	—	2	1	no	—
424	C₂×C₂²⋊C₄	—	2	2	no	—
425	C₂×C₂²⋊C₄	—	2	2	no	—
426	C₂×C₂²⋊C₄	—	2	8	no	—
427	C₂×C₂²⋊C₄	—	2	1	no	—
428	C₂×C₂²⋊C₄	—	2	1	no	—
429	C₂×C₂²⋊C₄	—	2	1	no	—
430	C₂×C₂²⋊C₄	—	2	1	no	—
431	C₂×C₄²	—	2	1	no	—
432	C₂×C₄²	D₄×C₃₁	1	1	yes	characteristic
433	C₂×C₄²	D₄×C₃₁	1	1	yes	characteristic
434	C₂×C₄²	D₄×C₃₁	1	4	yes	—
435	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
436	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
437	C₂×C₄²	C₂³×C₃₁	1	4	yes	—
438	#107	C₂³×C₃₁	1	8	yes	complemented
439	#106	C₂³×C₃₁	1	8	yes	complemented
440	C₄⋊C₈	C₂³×C₃₁	1	8	yes	complemented
441	#103	C₂³×C₃₁	1	8	yes	complemented
442	#102	C₂³×C₃₁	1	8	yes	—
443	#102	C₂³×C₃₁	1	8	yes	—
444	#102	C₂³×C₃₁	1	8	yes	complemented
445	C₂²⋊C₈	C₂³×C₃₁	1	4	yes	—
446	C₂²⋊C₈	C₂³×C₃₁	1	4	yes	—
447	C₄×C₈	C₂³×C₃₁	1	8	yes	complemented
448	C₃₁	C₂×#3105	1	1	yes	characteristic, direct product factor, Sylow
449	C₂⁴	—	2	8	no	—
450	C₂⁴	—	2	4	no	—
451	C₂⁴	—	2	2	no	—
452	C₂⁴	—	2	1	no	—
453	C₂⁴	—	4	4	no	—
454	C₂⁴	—	4	4	no	—
455	C₂⁴	C₂×D₄×C₃₁	1	1	yes	characteristic
456	C₂×Q₈	—	2	1	no	—
457	C₂×Q₈	—	2	4	no	—
458	C₂×Q₈	C₂⁴×C₃₁	1	2	yes	—
459	C₂×D₄	—	2	8	no	—
460	C₂×D₄	—	2	16	no	—
461	C₂×D₄	—	2	16	no	—
462	C₂×D₄	—	2	4	no	—
463	C₂×D₄	—	2	4	no	—
464	C₂×D₄	—	2	2	no	—
465	C₂×D₄	—	2	1	no	—
466	C₂×D₄	—	2	4	no	—
467	C₂×D₄	—	2	4	no	—
468	C₂×D₄	—	4	4	no	—
469	C₂×D₄	#41×C₃₁	1	8	yes	—
470	C₂×D₄	C₂⁴×C₃₁	1	2	yes	—
471	C₂×D₄	C₂⁴×C₃₁	1	4	yes	—
472	C₂²×C₄	—	2	4	no	—
473	C₂²×C₄	—	2	4	no	—
474	C₂²×C₄	—	2	4	no	—
475	C₂²×C₄	—	2	1	no	—
476	C₂²×C₄	—	2	1	no	—
477	C₂²×C₄	—	2	1	no	—
478	C₂²×C₄	—	2	4	no	—
479	C₂²×C₄	—	2	2	no	—
480	C₂²×C₄	—	2	4	no	—
481	C₂²×C₄	—	2	8	no	—
482	C₂²×C₄	—	2	8	no	—
483	C₂²×C₄	—	4	8	no	—
484	C₂²×C₄	—	4	32	no	—
485	C₂²×C₄	—	4	4	no	—
486	C₂²×C₄	—	4	4	no	—
487	C₂²×C₄	—	4	8	no	—
488	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
489	C₂²×C₄	C₂×D₄×C₃₁	1	2	yes	—
490	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
491	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
492	C₂²×C₄	C₂×D₄×C₃₁	1	4	yes	—
493	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
494	C₂²×C₄	C₂×D₄×C₃₁	1	4	yes	—
495	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
496	C₂²×C₄	C₂⁴×C₃₁	1	2	yes	—
497	C₂²×C₄	C₂⁴×C₃₁	1	1	yes	characteristic
498	SD₁₆	—	2	16	no	—
499	D₈	—	2	16	no	—
500	C₂×C₈	—	2	4	no	—
501	C₂×C₈	—	2	4	no	—
502	C₂×C₈	C₂⁴×C₃₁	1	4	yes	—
503	C₂×C₈	C₂⁴×C₃₁	1	4	yes	—
504	C₄⋊C₄	—	2	4	no	—
505	C₄⋊C₄	—	2	2	no	—
506	C₄⋊C₄	—	2	4	no	—
507	C₄⋊C₄	—	2	4	no	—
508	C₄⋊C₄	—	2	4	no	—
509	C₄⋊C₄	—	2	4	no	—
510	C₄⋊C₄	—	2	1	no	—
511	C₄⋊C₄	—	2	1	no	—
512	C₄⋊C₄	C₂⁴×C₃₁	1	2	yes	—
513	C₄⋊C₄	C₂⁴×C₃₁	1	2	yes	—
514	C₄⋊C₄	C₂⁴×C₃₁	1	4	yes	—
515	C₂²⋊C₄	—	2	4	no	—
516	C₂²⋊C₄	—	2	8	no	—
517	C₂²⋊C₄	—	2	8	no	—
518	C₂²⋊C₄	—	2	4	no	—
519	C₂²⋊C₄	—	2	4	no	—
520	C₂²⋊C₄	—	2	4	no	—
521	C₂²⋊C₄	—	2	4	no	—
522	C₄²	—	2	4	no	—
523	C₄²	C₂×D₄×C₃₁	1	4	yes	—
524	C₄²	C₂×D₄×C₃₁	1	2	yes	—
525	C₄²	C₂×D₄×C₃₁	1	2	yes	—
526	C₄²	C₂×D₄×C₃₁	1	4	yes	—
527	C₄²	C₂⁴×C₃₁	1	2	yes	—
528	C₄²	C₂⁴×C₃₁	1	2	yes	—
529	C₂³	—	2	2	no	—
530	C₂³	—	2	2	no	—
531	C₂³	—	2	16	no	—
532	C₂³	—	2	8	no	—
533	C₂³	—	2	4	no	—
534	C₂³	—	2	2	no	—
535	C₂³	—	4	8	no	—
536	C₂³	—	4	32	no	—
537	C₂³	—	4	8	no	—
538	C₂³	—	4	4	no	—
539	C₂³	—	4	4	no	—
540	C₂³	—	4	2	no	—
541	C₂³	—	4	2	no	—
542	C₂³	—	4	2	no	—
543	C₂³	C₃₁×#135	1	4	yes	—
544	C₂³	C₂²×D₄×C₃₁	1	2	yes	—
545	C₂³	C₂²×D₄×C₃₁	1	1	yes	characteristic
546	Q₈	—	2	4	no	—
547	Q₈	—	4	2	no	—
548	D₄	—	2	16	no	—
549	D₄	—	2	8	no	—
550	D₄	—	2	4	no	—
551	D₄	—	2	4	no	—
552	D₄	—	4	8	no	—
553	D₄	—	4	2	no	—
554	D₄	C₂×#41×C₃₁	1	8	yes	—
555	C₂×C₄	—	2	2	no	—
556	C₂×C₄	—	2	8	no	—
557	C₂×C₄	—	2	2	no	—
558	C₂×C₄	—	2	2	no	—
559	C₂×C₄	—	2	2	no	—
560	C₂×C₄	—	2	4	no	—
561	C₂×C₄	—	2	8	no	—
562	C₂×C₄	—	2	4	no	—
563	C₂×C₄	—	2	2	no	—
564	C₂×C₄	—	2	2	no	—
565	C₂×C₄	—	2	2	no	—
566	C₂×C₄	—	2	4	no	—
567	C₂×C₄	—	4	16	no	—
568	C₂×C₄	—	4	8	no	—
569	C₂×C₄	—	4	2	no	—
570	C₂×C₄	—	4	2	no	—
571	C₂×C₄	—	4	2	no	—
572	C₂×C₄	—	4	4	no	—
573	C₂×C₄	—	4	8	no	—
574	C₂×C₄	—	4	16	no	—
575	C₂×C₄	C₃₁×AGL(1,ℤ/8)	1	4	yes	—
576	C₂×C₄	C₂²×D₄×C₃₁	1	2	yes	—
577	C₂×C₄	C₂²×D₄×C₃₁	1	2	yes	—
578	C₂×C₄	C₂²×D₄×C₃₁	1	2	yes	—
579	C₂×C₄	C₂²×D₄×C₃₁	1	2	yes	—
580	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
581	C₂×C₄	C₂²×D₄×C₃₁	1	2	yes	—
582	C₂×C₄	C₂×#41×C₃₁	1	4	yes	—
583	C₂×C₄	C₂⁵×C₃₁	1	1	yes	characteristic
584	C₈	—	2	4	no	—
585	C₈	—	2	4	no	—
586	C₂²	—	2	8	no	—
587	C₂²	—	2	2	no	—
588	C₂²	—	2	2	no	—
589	C₂²	—	2	8	no	—
590	C₂²	—	4	32	no	—
591	C₂²	—	4	16	no	—
592	C₂²	—	4	4	no	—
593	C₂²	—	4	4	no	—
594	C₂²	—	4	2	no	—
595	C₂²	—	4	2	no	—
596	C₂²	—	4	2	no	—
597	C₂²	—	8	4	no	—
598	C₂²	C₃₁×#525	1	2	yes	—
599	C₂²	C₂×C₃₁×#135	1	4	yes	—
600	C₂²	C₂×C₃₁×#135	1	2	yes	—
601	C₂²	C₂×C₃₁×AGL(1,ℤ/8)	1	2	yes	—
602	C₂²	C₂³×D₄×C₃₁	1	1	yes	characteristic
603	C₄	—	2	2	no	—
604	C₄	—	2	2	no	—
605	C₄	—	2	2	no	—
606	C₄	—	2	2	no	—
607	C₄	—	4	2	no	—
608	C₄	—	4	2	no	—
609	C₄	—	4	4	no	—
610	C₄	—	4	2	no	—
611	C₄	C₃₁×#525	1	2	yes	—
612	C₄	C₂×C₃₁×AGL(1,ℤ/8)	1	4	yes	—
613	C₄	C₂²×#41×C₃₁	1	2	yes	—
614	C₂	—	2	4	no	—
615	C₂	—	4	8	no	—
616	C₂	—	8	2	no	—
617	C₂	C₃₁×#3105	1	4	yes	direct product factor
618	C₂	C₂×C₃₁×#525	1	1	yes	characteristic
619	C₂	C₂²×C₃₁×#135	1	1	yes	characteristic
620	C₂	C₂²×C₃₁×AGL(1,ℤ/8)	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄×C₃₁	C₂×D₄
commutator, Frattini, cosocle	C₂×C₄	C₂⁵×C₃₁
socle	C₂³×C₃₁	C₂²×D₄
prime radical	C₂×C₃₁×#447	C₂
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄×C₃₁ ◁ C₂²×C₄²×C₃₁ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂×C₃₁×#529	C₂	semidirect
2. C₂²×C₄×D₄×C₃₁	C₂	semidirect
3. C₃₁×#3105	C₂	direct (N × H)
4. C₂×C₃₁×#484	C₂	semidirect
5. C₂×C₃₁×#481	C₂	semidirect
6. C₂×C₃₁×#478	C₂	semidirect
7. C₂×C₃₁×#463	C₂	semidirect
8. C₂×C₃₁×#450	C₂	semidirect
10. C₂×C₄×SD₁₆×C₃₁	C₂	semidirect
11. C₂×C₄×D₈×C₃₁	C₂	semidirect
12. C₂×C₃₁×#431	C₂	semidirect
18. C₃₁×#529	C₂²	semidirect
20. C₂×C₃₁×#125	C₂²	semidirect
21. C₂×C₃₁×#124	C₂²	semidirect
25. C₂×C₄×Q₈×C₃₁	C₂²	semidirect
32. C₂×C₄×D₄×C₃₁	C₂²	semidirect
33. C₂×C₄×D₄×C₃₁	C₂²	semidirect
38. C₃₁×#484	C₂²	semidirect
39. C₃₁×#481	C₂²	semidirect
40. C₃₁×#478	C₂²	semidirect
41. C₃₁×#463	C₂²	semidirect
42. C₃₁×#450	C₂²	semidirect
44. C₄×SD₁₆×C₃₁	C₂²	semidirect
45. C₄×D₈×C₃₁	C₂²	semidirect
46. C₃₁×#431	C₂²	semidirect
47. C₂×C₃₁×#107	C₂²	semidirect
48. C₂×C₃₁×#106	C₂²	semidirect
49. C₂×C₃₁×C₄⋊C₈	C₂²	semidirect
50. C₂×C₃₁×#103	C₂²	semidirect
53. C₂×C₃₁×#102	C₂²	semidirect
56. C₂×C₄×C₈×C₃₁	C₂²	semidirect
83. C₃₁×#125	C₂³	semidirect
85. C₃₁×#124	C₂³	semidirect
92. C₄×Q₈×C₃₁	C₂³	semidirect
101. C₄×D₄×C₃₁	C₂³	semidirect
128. C₃₁×#107	C₂³	semidirect
129. C₃₁×#106	C₂³	semidirect
130. C₃₁×C₄⋊C₈	C₂³	semidirect
131. C₃₁×#103	C₂³	semidirect
134. C₃₁×#102	C₂³	semidirect
137. C₄×C₈×C₃₁	C₂³	semidirect
218. C₂×#3105	C₃₁	direct (N × H)
276. C₂×#529	C₂×C₃₁	semidirect
277. C₂²×C₄×D₄	C₂×C₃₁	semidirect
278. #3105	C₂×C₃₁	direct (N × H)
279. C₂×#484	C₂×C₃₁	semidirect
280. C₂×#481	C₂×C₃₁	semidirect
281. C₂×#478	C₂×C₃₁	semidirect
282. C₂×#463	C₂×C₃₁	semidirect
283. C₂×#450	C₂×C₃₁	semidirect
285. C₂×C₄×SD₁₆	C₂×C₃₁	semidirect
286. C₂×C₄×D₈	C₂×C₃₁	semidirect
287. C₂×#431	C₂×C₃₁	semidirect
321. #529	C₂²×C₃₁	semidirect
323. C₂×#125	C₂²×C₃₁	semidirect
324. C₂×#124	C₂²×C₃₁	semidirect
328. C₂×C₄×Q₈	C₂²×C₃₁	semidirect
335. C₂×C₄×D₄	C₂²×C₃₁	semidirect
336. C₂×C₄×D₄	C₂²×C₃₁	semidirect
341. #484	C₂²×C₃₁	semidirect
342. #481	C₂²×C₃₁	semidirect
343. #478	C₂²×C₃₁	semidirect
344. #463	C₂²×C₃₁	semidirect
345. #450	C₂²×C₃₁	semidirect
347. C₄×SD₁₆	C₂²×C₃₁	semidirect
348. C₄×D₈	C₂²×C₃₁	semidirect
349. #431	C₂²×C₃₁	semidirect
350. C₂×#107	C₂²×C₃₁	semidirect
351. C₂×#106	C₂²×C₃₁	semidirect
352. C₂×C₄⋊C₈	C₂²×C₃₁	semidirect
353. C₂×#103	C₂²×C₃₁	semidirect
356. C₂×#102	C₂²×C₃₁	semidirect
359. C₂×C₄×C₈	C₂²×C₃₁	semidirect
363. C₂×C₃₁	#3105	direct (N × H)
393. #125	C₂³×C₃₁	semidirect
395. #124	C₂³×C₃₁	semidirect
402. C₄×Q₈	C₂³×C₃₁	semidirect
411. C₄×D₄	C₂³×C₃₁	semidirect
438. #107	C₂³×C₃₁	semidirect
439. #106	C₂³×C₃₁	semidirect
440. C₄⋊C₈	C₂³×C₃₁	semidirect
441. #103	C₂³×C₃₁	semidirect
444. #102	C₂³×C₃₁	semidirect
447. C₄×C₈	C₂³×C₃₁	semidirect
448. C₃₁	C₂×#3105	direct (N × H)
617. C₂	C₃₁×#3105	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×C₃₁×#529	2
C₂²×C₄×D₄×C₃₁	2
C₃₁×#3105	2
C₂×C₃₁×#484	2
C₂×C₃₁×#481	2
C₂×C₃₁×#478	2
C₂×C₃₁×#463	2
C₂×C₃₁×#450	2
C₂×C₃₁×#447	2
C₂×C₄×SD₁₆×C₃₁	2
C₂×C₄×D₈×C₃₁	2
C₂×C₃₁×#431	2
C₂×#3105	31

Subgroups — C2×C31×#3105 #21517291

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₃₁×#3105 #21517291