Subgroups — C₂×C₃₁×#3091 #21517265

Order 7936.

2378 conjugacy classes of subgroups.

The subgroup lattice has 638 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃₁×#525	C₂	1	1	yes	maximal, characteristic
2	C₂²×C₃₁×#117	C₂	1	1	yes	maximal, characteristic, complemented
3	C₃₁×#3091	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×#485	C₂	1	2	yes	maximal, complemented
7	C₂×C₃₁×#482	C₂	1	2	yes	maximal, complemented
8	C₂×C₃₁×#483	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂×C₃₁×#480	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂×C₃₁×#427	C₂	1	2	yes	maximal, complemented
11	C₂×C₃₁×#416	C₂	1	2	yes	maximal, complemented
12	C₂²×C₃₁×C₂²⋊C₈	C₂	1	1	yes	maximal, characteristic, complemented
13	C₂²×#41×C₃₁	C₂²	1	1	yes	characteristic
14	C₂⁴×C₄×C₃₁	C₂²	1	1	yes	characteristic
15	C₂³×C₈×C₃₁	C₂²	1	2	yes	—
16	C₃₁×#525	—	2	4	no	—
17	C₃₁×#525	C₂²	1	8	yes	—
18	C₂×C₃₁×#123	—	2	1	no	—
19	C₂×C₃₁×C₂²⋊Q₈	C₂²	1	1	yes	characteristic
20	C₂×C₃₁×C₂²⋊Q₈	C₂²	1	1	yes	characteristic
21	C₂×C₃₁×#121	C₂²	1	1	yes	characteristic
22	C₂×C₃₁×#121	C₂²	1	1	yes	characteristic
23	C₂×C₃₁×#120	—	2	1	no	—
24	C₂×C₄×D₄×C₃₁	—	2	1	no	—
25	C₂×C₄×D₄×C₃₁	—	2	1	no	—
26	C₂×C₃₁×#117	—	2	1	no	—
27	C₂×C₃₁×#117	—	2	4	no	—
28	C₂×C₃₁×#117	—	2	1	no	—
29	C₂×C₃₁×#117	C₂²	1	2	yes	—
30	C₂×C₃₁×#117	C₂²	1	2	yes	—
31	C₂×C₃₁×#117	C₂²	1	1	yes	characteristic
32	C₂×C₃₁×#117	C₂²	1	8	yes	complemented
33	C₂²×C₄⋊C₄×C₃₁	C₂²	1	1	yes	characteristic
34	C₂²×C₄⋊C₄×C₃₁	C₂²	1	1	yes	characteristic
35	C₂²×C₂²⋊C₄×C₃₁	—	2	1	no	—
36	C₂²×C₄²×C₃₁	—	2	1	no	—
37	C₃₁×#485	C₂²	1	16	yes	complemented
38	C₃₁×#484	C₂²	1	8	yes	complemented
39	C₃₁×#483	C₂²	1	8	yes	complemented
40	C₃₁×#482	C₂²	1	16	yes	complemented
41	C₃₁×#481	C₂²	1	8	yes	complemented
42	C₃₁×#480	C₂²	1	8	yes	complemented
43	C₃₁×#427	C₂²	1	16	yes	complemented
44	C₂×C₃₁×#107	C₂²	1	2	yes	complemented
45	C₂×C₃₁×#107	C₂²	1	2	yes	complemented
46	C₂×C₃₁×#106	C₂²	1	2	yes	complemented
47	C₂×C₃₁×#106	C₂²	1	2	yes	complemented
48	C₃₁×#416	C₂²	1	16	yes	complemented
49	C₂×C₃₁×#103	C₂²	1	2	yes	complemented
50	C₂×C₃₁×#103	C₂²	1	2	yes	complemented
51	C₂×C₃₁×#102	C₂²	1	2	yes	complemented
52	C₂×C₃₁×#102	C₂²	1	2	yes	complemented
53	C₂×C₃₁×C₂²⋊C₈	C₂²	1	1	yes	characteristic
54	C₂×C₃₁×C₂²⋊C₈	C₂²	1	1	yes	characteristic
55	C₂×C₃₁×C₂²⋊C₈	C₂²	1	2	yes	—
56	C₂×C₃₁×C₂²⋊C₈	C₂²	1	8	yes	complemented
57	C₂⁵×C₃₁	D₄	1	1	yes	characteristic
58	C₂×#41×C₃₁	—	2	2	no	—
59	C₂×#41×C₃₁	—	2	8	no	—
60	C₂×#41×C₃₁	C₂³	1	4	yes	—
61	C₂²×Q₈×C₃₁	C₂³	1	1	yes	characteristic
62	C₂²×D₄×C₃₁	—	2	1	no	—
63	C₂²×D₄×C₃₁	—	2	1	no	—
64	C₂²×D₄×C₃₁	—	2	1	no	—
65	C₂²×D₄×C₃₁	C₂³	1	1	yes	characteristic
66	C₂³×C₄×C₃₁	—	2	8	no	—
67	C₂³×C₄×C₃₁	—	2	1	no	—
68	C₂³×C₄×C₃₁	—	2	1	no	—
69	C₂³×C₄×C₃₁	—	2	1	no	—
70	C₂³×C₄×C₃₁	D₄	1	1	yes	characteristic
71	C₂³×C₄×C₃₁	D₄	1	4	yes	—
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	C₂³×C₄×C₃₁	C₂³	1	1	yes	characteristic
76	C₂³×C₄×C₃₁	C₂³	1	4	yes	—
77	C₂³×C₄×C₃₁	C₂³	1	1	yes	characteristic
78	C₂²×C₈×C₃₁	—	2	8	no	—
79	C₂²×C₈×C₃₁	C₂³	1	2	yes	—
80	C₂²×C₈×C₃₁	C₂³	1	2	yes	—
81	C₂²×C₈×C₃₁	C₂³	1	8	yes	—
82	C₃₁×#123	—	2	8	no	—
83	C₃₁×C₂²⋊Q₈	—	2	2	no	—
84	C₃₁×C₂²⋊Q₈	—	2	2	no	—
85	C₃₁×C₂²⋊Q₈	C₂³	1	4	yes	—
86	C₃₁×C₂²⋊Q₈	C₂³	1	4	yes	—
87	C₃₁×#121	—	2	2	no	—
88	C₃₁×#121	—	2	2	no	—
89	C₃₁×#121	C₂³	1	4	yes	—
90	C₃₁×#121	C₂³	1	4	yes	—
91	C₃₁×#120	—	2	8	no	—
92	C₄×D₄×C₃₁	—	2	8	no	—
93	C₄×D₄×C₃₁	—	2	8	no	—
94	C₃₁×#117	—	2	4	no	—
95	C₃₁×#117	—	2	4	no	—
96	C₃₁×#117	—	2	8	no	—
97	C₃₁×#117	—	2	8	no	—
98	C₃₁×#117	—	2	2	no	—
99	C₃₁×#117	C₂³	1	8	yes	—
100	C₃₁×#117	C₂³	1	4	yes	—
101	C₃₁×#117	C₂³	1	8	yes	—
102	C₂×C₄⋊C₄×C₃₁	—	2	1	no	—
103	C₂×C₄⋊C₄×C₃₁	—	2	1	no	—
104	C₂×C₄⋊C₄×C₃₁	—	2	2	no	—
105	C₂×C₄⋊C₄×C₃₁	—	2	2	no	—
106	C₂×C₄⋊C₄×C₃₁	—	2	1	no	—
107	C₂×C₄⋊C₄×C₃₁	—	2	1	no	—
108	C₂×C₄⋊C₄×C₃₁	C₂³	1	2	yes	—
109	C₂×C₄⋊C₄×C₃₁	C₂³	1	2	yes	—
110	C₂×C₄⋊C₄×C₃₁	C₂³	1	1	yes	characteristic
111	C₂×C₄⋊C₄×C₃₁	C₂³	1	1	yes	characteristic
112	C₂×C₄⋊C₄×C₃₁	C₂³	1	4	yes	—
113	C₂×C₄⋊C₄×C₃₁	C₂³	1	4	yes	—
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×C₂²⋊C₄×C₃₁	—	2	1	no	—
122	C₂×C₄²×C₃₁	—	2	2	no	—
123	C₂×C₄²×C₃₁	—	2	8	no	—
124	C₂×C₄²×C₃₁	—	2	2	no	—
125	C₂×C₄²×C₃₁	—	2	1	no	—
126	C₃₁×#107	C₂³	1	8	yes	complemented
127	C₃₁×#107	C₂³	1	8	yes	complemented
128	C₃₁×#106	C₂³	1	8	yes	complemented
129	C₃₁×#106	C₂³	1	8	yes	complemented
130	C₃₁×#103	C₂³	1	8	yes	complemented
131	C₃₁×#103	C₂³	1	8	yes	complemented
132	C₃₁×#102	C₂³	1	8	yes	complemented
133	C₃₁×#102	C₂³	1	8	yes	complemented
134	C₃₁×C₂²⋊C₈	C₂³	1	4	yes	—
135	C₃₁×C₂²⋊C₈	C₂³	1	4	yes	—
136	C₃₁×C₂²⋊C₈	C₂³	1	8	yes	—
137	C₂⁴×C₃₁	—	2	4	no	—
138	C₂⁴×C₃₁	—	2	4	no	—
139	C₂⁴×C₃₁	—	2	1	no	—
140	C₂⁴×C₃₁	—	4	2	no	—
141	C₂⁴×C₃₁	C₂×D₄	1	1	yes	characteristic
142	C₂⁴×C₃₁	C₂×D₄	1	4	yes	—
143	C₂⁴×C₃₁	C₂×D₄	1	1	yes	characteristic
144	C₂⁴×C₃₁	C₂×D₄	1	1	yes	characteristic
145	#41×C₃₁	—	2	16	no	—
146	#41×C₃₁	—	4	8	no	—
147	C₂×Q₈×C₃₁	—	2	1	no	—
148	C₂×Q₈×C₃₁	—	2	4	no	—
149	C₂×Q₈×C₃₁	C₂⁴	1	2	yes	—
150	C₂×D₄×C₃₁	—	2	4	no	—
151	C₂×D₄×C₃₁	—	2	4	no	—
152	C₂×D₄×C₃₁	—	2	4	no	—
153	C₂×D₄×C₃₁	—	2	1	no	—
154	C₂×D₄×C₃₁	—	2	4	no	—
155	C₂×D₄×C₃₁	—	4	4	no	—
156	C₂×D₄×C₃₁	—	4	4	no	—
157	C₂×D₄×C₃₁	—	4	4	no	—
158	C₂×D₄×C₃₁	C₂⁴	1	2	yes	—
159	C₂²×C₄×C₃₁	—	2	4	no	—
160	C₂²×C₄×C₃₁	—	2	4	no	—
161	C₂²×C₄×C₃₁	—	2	4	no	—
162	C₂²×C₄×C₃₁	—	2	16	no	—
163	C₂²×C₄×C₃₁	—	2	4	no	—
164	C₂²×C₄×C₃₁	—	2	4	no	—
165	C₂²×C₄×C₃₁	—	2	16	no	—
166	C₂²×C₄×C₃₁	—	2	4	no	—
167	C₂²×C₄×C₃₁	—	2	4	no	—
168	C₂²×C₄×C₃₁	—	2	2	no	—
169	C₂²×C₄×C₃₁	—	2	2	no	—
170	C₂²×C₄×C₃₁	—	2	1	no	—
171	C₂²×C₄×C₃₁	—	2	1	no	—
172	C₂²×C₄×C₃₁	—	2	1	no	—
173	C₂²×C₄×C₃₁	—	2	4	no	—
174	C₂²×C₄×C₃₁	—	4	4	no	—
175	C₂²×C₄×C₃₁	—	4	4	no	—
176	C₂²×C₄×C₃₁	—	4	4	no	—
177	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
178	C₂²×C₄×C₃₁	C₂×D₄	1	4	yes	—
179	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
180	C₂²×C₄×C₃₁	C₂×D₄	1	2	yes	—
181	C₂²×C₄×C₃₁	C₂×D₄	1	4	yes	—
182	C₂²×C₄×C₃₁	C₂×D₄	1	2	yes	—
183	C₂²×C₄×C₃₁	C₂×D₄	1	4	yes	—
184	C₂²×C₄×C₃₁	C₂×D₄	1	2	yes	—
185	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
186	C₂²×C₄×C₃₁	#41	1	8	yes	—
187	C₂²×C₄×C₃₁	C₂⁴	1	2	yes	—
188	C₂²×C₄×C₃₁	C₂⁴	1	2	yes	—
189	C₂²×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
190	C₂²×C₄×C₃₁	C₂⁴	1	2	yes	—
191	C₂×C₈×C₃₁	—	2	4	no	—
192	C₂×C₈×C₃₁	—	2	4	no	—
193	C₂×C₈×C₃₁	—	2	16	no	—
194	C₂×C₈×C₃₁	C₂⁴	1	4	yes	—
195	C₂×C₈×C₃₁	C₂⁴	1	4	yes	—
196	C₄⋊C₄×C₃₁	—	2	4	no	—
197	C₄⋊C₄×C₃₁	—	2	4	no	—
198	C₄⋊C₄×C₃₁	—	2	2	no	—
199	C₄⋊C₄×C₃₁	—	2	2	no	—
200	C₄⋊C₄×C₃₁	—	2	4	no	—
201	C₄⋊C₄×C₃₁	—	2	4	no	—
202	C₄⋊C₄×C₃₁	—	2	1	no	—
203	C₄⋊C₄×C₃₁	—	2	1	no	—
204	C₄⋊C₄×C₃₁	C₂⁴	1	4	yes	—
205	C₄⋊C₄×C₃₁	C₂⁴	1	4	yes	—
206	C₄⋊C₄×C₃₁	C₂⁴	1	2	yes	—
207	C₄⋊C₄×C₃₁	C₂⁴	1	2	yes	—
208	C₂²⋊C₄×C₃₁	—	2	8	no	—
209	C₂²⋊C₄×C₃₁	—	2	8	no	—
210	C₂²⋊C₄×C₃₁	—	2	4	no	—
211	C₂²⋊C₄×C₃₁	—	2	4	no	—
212	C₂²⋊C₄×C₃₁	—	2	4	no	—
213	C₂²⋊C₄×C₃₁	—	2	4	no	—
214	C₂²⋊C₄×C₃₁	—	2	4	no	—
215	C₄²×C₃₁	—	2	8	no	—
216	C₄²×C₃₁	—	2	8	no	—
217	C₄²×C₃₁	—	2	4	no	—
218	C₂×#3091	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	2	no	—
222	C₂³×C₃₁	—	2	2	no	—
223	C₂³×C₃₁	—	2	8	no	—
224	C₂³×C₃₁	—	2	2	no	—
225	C₂³×C₃₁	—	2	8	no	—
226	C₂³×C₃₁	—	2	2	no	—
227	C₂³×C₃₁	—	2	2	no	—
228	C₂³×C₃₁	—	2	2	no	—
229	C₂³×C₃₁	—	4	4	no	—
230	C₂³×C₃₁	—	4	16	no	—
231	C₂³×C₃₁	—	4	2	no	—
232	C₂³×C₃₁	—	4	2	no	—
233	C₂³×C₃₁	—	4	2	no	—
234	C₂³×C₃₁	#123	1	4	yes	—
235	C₂³×C₃₁	#135	1	4	yes	—
236	C₂³×C₃₁	C₂²×D₄	1	2	yes	—
237	C₂³×C₃₁	C₂²×D₄	1	2	yes	—
238	C₂³×C₃₁	C₂²×D₄	1	2	yes	—
239	C₂³×C₃₁	C₂²×D₄	1	1	yes	characteristic
240	Q₈×C₃₁	—	2	4	no	—
241	Q₈×C₃₁	—	4	2	no	—
242	D₄×C₃₁	—	2	4	no	—
243	D₄×C₃₁	—	4	8	no	—
244	D₄×C₃₁	—	4	8	no	—
245	D₄×C₃₁	—	4	8	no	—
246	D₄×C₃₁	—	4	2	no	—
247	C₂×C₄×C₃₁	—	2	8	no	—
248	C₂×C₄×C₃₁	—	2	2	no	—
249	C₂×C₄×C₃₁	—	2	2	no	—
250	C₂×C₄×C₃₁	—	2	8	no	—
251	C₂×C₄×C₃₁	—	2	8	no	—
252	C₂×C₄×C₃₁	—	2	4	no	—
253	C₂×C₄×C₃₁	—	2	4	no	—
254	C₂×C₄×C₃₁	—	2	4	no	—
255	C₂×C₄×C₃₁	—	2	4	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	2	no	—
261	C₂×C₄×C₃₁	—	4	8	no	—
262	C₂×C₄×C₃₁	—	4	2	no	—
263	C₂×C₄×C₃₁	—	4	8	no	—
264	C₂×C₄×C₃₁	—	4	4	no	—
265	C₂×C₄×C₃₁	—	4	2	no	—
266	C₂×C₄×C₃₁	—	4	4	no	—
267	C₂×C₄×C₃₁	—	4	2	no	—
268	C₂×C₄×C₃₁	—	4	8	no	—
269	C₂×C₄×C₃₁	#123	1	4	yes	—
270	C₂×C₄×C₃₁	#123	1	8	yes	—
271	C₂×C₄×C₃₁	C₂²×D₄	1	2	yes	—
272	C₂×C₄×C₃₁	C₂²×D₄	1	2	yes	—
273	C₂×C₄×C₃₁	C₂²×D₄	1	2	yes	—
274	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
275	C₂×C₄×C₃₁	C₂×#41	1	4	yes	—
276	C₂×C₄×C₃₁	C₂×#41	1	8	yes	—
277	C₂×C₄×C₃₁	C₂⁵	1	1	yes	characteristic
278	C₈×C₃₁	—	2	4	no	—
279	C₈×C₃₁	—	2	4	no	—
280	C₂×#525	C₂×C₃₁	1	1	yes	characteristic
281	C₂²×#117	C₂×C₃₁	1	1	yes	characteristic, complemented
282	#3091	C₂×C₃₁	1	16	yes	direct product factor
283	C₂×#484	C₂×C₃₁	1	1	yes	characteristic, complemented
284	C₂×#481	C₂×C₃₁	1	1	yes	characteristic, complemented
285	C₂×#485	C₂×C₃₁	1	2	yes	complemented
286	C₂×#482	C₂×C₃₁	1	2	yes	complemented
287	C₂×#483	C₂×C₃₁	1	1	yes	characteristic, complemented
288	C₂×#480	C₂×C₃₁	1	1	yes	characteristic, complemented
289	C₂×#427	C₂×C₃₁	1	2	yes	complemented
290	C₂×#416	C₂×C₃₁	1	2	yes	complemented
291	C₂²×C₂²⋊C₈	C₂×C₃₁	1	1	yes	characteristic, complemented
292	C₂²×C₃₁	—	2	8	no	—
293	C₂²×C₃₁	—	2	8	no	—
294	C₂²×C₃₁	—	2	2	no	—
295	C₂²×C₃₁	—	2	2	no	—
296	C₂²×C₃₁	—	2	2	no	—
297	C₂²×C₃₁	—	2	2	no	—
298	C₂²×C₃₁	—	2	2	no	—
299	C₂²×C₃₁	—	2	2	no	—
300	C₂²×C₃₁	—	2	2	no	—
301	C₂²×C₃₁	—	4	16	no	—
302	C₂²×C₃₁	—	4	4	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×#123	1	4	yes	—
308	C₂²×C₃₁	C₂×#123	1	2	yes	—
309	C₂²×C₃₁	C₂×#135	1	4	yes	—
310	C₂²×C₃₁	C₂×#135	1	2	yes	—
311	C₂²×C₃₁	#575	1	2	yes	—
312	C₂²×C₃₁	C₂³×D₄	1	1	yes	characteristic
313	C₄×C₃₁	—	2	2	no	—
314	C₄×C₃₁	—	2	2	no	—
315	C₄×C₃₁	—	4	2	no	—
316	C₄×C₃₁	—	4	4	no	—
317	C₄×C₃₁	—	4	2	no	—
318	C₄×C₃₁	—	4	4	no	—
319	C₄×C₃₁	—	4	2	no	—
320	C₄×C₃₁	C₂×#123	1	4	yes	—
321	C₄×C₃₁	C₂×#123	1	2	yes	—
322	C₄×C₃₁	C₂²×#41	1	2	yes	—
323	C₂²×#41	C₂²×C₃₁	1	1	yes	characteristic
324	C₂⁴×C₄	C₂²×C₃₁	1	1	yes	characteristic
325	C₂³×C₈	C₂²×C₃₁	1	2	yes	—
326	#525	—	2	4	no	—
327	#525	C₂²×C₃₁	1	8	yes	—
328	C₂×#123	—	2	1	no	—
329	C₂×C₂²⋊Q₈	C₂²×C₃₁	1	1	yes	characteristic
330	C₂×C₂²⋊Q₈	C₂²×C₃₁	1	1	yes	characteristic
331	C₂×#121	C₂²×C₃₁	1	1	yes	characteristic
332	C₂×#121	C₂²×C₃₁	1	1	yes	characteristic
333	C₂×#120	—	2	1	no	—
334	C₂×C₄×D₄	—	2	1	no	—
335	C₂×C₄×D₄	—	2	1	no	—
336	C₂×#117	—	2	1	no	—
337	C₂×#117	—	2	1	no	—
338	C₂×#117	—	2	4	no	—
339	C₂×#117	C₂²×C₃₁	1	2	yes	—
340	C₂×#117	C₂²×C₃₁	1	2	yes	—
341	C₂×#117	C₂²×C₃₁	1	1	yes	characteristic
342	C₂×#117	C₂²×C₃₁	1	8	yes	complemented
343	C₂²×C₄⋊C₄	C₂²×C₃₁	1	1	yes	characteristic
344	C₂²×C₄⋊C₄	C₂²×C₃₁	1	1	yes	characteristic
345	C₂²×C₂²⋊C₄	—	2	1	no	—
346	C₂²×C₄²	—	2	1	no	—
347	#485	C₂²×C₃₁	1	16	yes	complemented
348	#484	C₂²×C₃₁	1	8	yes	complemented
349	#483	C₂²×C₃₁	1	8	yes	complemented
350	#482	C₂²×C₃₁	1	16	yes	complemented
351	#481	C₂²×C₃₁	1	8	yes	complemented
352	#480	C₂²×C₃₁	1	8	yes	complemented
353	#427	C₂²×C₃₁	1	16	yes	complemented
354	C₂×#107	C₂²×C₃₁	1	2	yes	complemented
355	C₂×#107	C₂²×C₃₁	1	2	yes	complemented
356	C₂×#106	C₂²×C₃₁	1	2	yes	complemented
357	C₂×#106	C₂²×C₃₁	1	2	yes	complemented
358	#416	C₂²×C₃₁	1	16	yes	complemented
359	C₂×#103	C₂²×C₃₁	1	2	yes	complemented
360	C₂×#103	C₂²×C₃₁	1	2	yes	complemented
361	C₂×#102	C₂²×C₃₁	1	2	yes	complemented
362	C₂×#102	C₂²×C₃₁	1	2	yes	complemented
363	C₂×C₂²⋊C₈	C₂²×C₃₁	1	1	yes	characteristic
364	C₂×C₂²⋊C₈	C₂²×C₃₁	1	1	yes	characteristic
365	C₂×C₂²⋊C₈	C₂²×C₃₁	1	2	yes	—
366	C₂×C₂²⋊C₈	C₂²×C₃₁	1	8	yes	complemented
367	C₂×C₃₁	—	2	4	no	—
368	C₂×C₃₁	—	2	4	no	—
369	C₂×C₃₁	—	4	2	no	—
370	C₂×C₃₁	—	8	2	no	—
371	C₂×C₃₁	#3091	1	4	yes	direct product factor
372	C₂×C₃₁	C₂²×#123	1	1	yes	characteristic
373	C₂×C₃₁	C₂²×#135	1	1	yes	characteristic
374	C₂×C₃₁	C₂×#575	1	1	yes	characteristic
375	C₂⁵	D₄×C₃₁	1	1	yes	characteristic
376	C₂×#41	—	2	2	no	—
377	C₂×#41	—	2	8	no	—
378	C₂×#41	C₂³×C₃₁	1	4	yes	—
379	C₂²×Q₈	C₂³×C₃₁	1	1	yes	characteristic
380	C₂²×D₄	—	2	1	no	—
381	C₂²×D₄	—	2	1	no	—
382	C₂²×D₄	—	2	1	no	—
383	C₂²×D₄	C₂³×C₃₁	1	1	yes	characteristic
384	C₂³×C₄	—	2	8	no	—
385	C₂³×C₄	—	2	1	no	—
386	C₂³×C₄	—	2	1	no	—
387	C₂³×C₄	—	2	1	no	—
388	C₂³×C₄	D₄×C₃₁	1	1	yes	characteristic
389	C₂³×C₄	D₄×C₃₁	1	1	yes	characteristic
390	C₂³×C₄	D₄×C₃₁	1	4	yes	—
391	C₂³×C₄	D₄×C₃₁	1	1	yes	characteristic
392	C₂³×C₄	C₂³×C₃₁	1	1	yes	characteristic
393	C₂³×C₄	C₂³×C₃₁	1	1	yes	characteristic
394	C₂³×C₄	C₂³×C₃₁	1	4	yes	—
395	C₂³×C₄	C₂³×C₃₁	1	1	yes	characteristic
396	C₂²×C₈	—	2	8	no	—
397	C₂²×C₈	C₂³×C₃₁	1	2	yes	—
398	C₂²×C₈	C₂³×C₃₁	1	2	yes	—
399	C₂²×C₈	C₂³×C₃₁	1	8	yes	—
400	#123	—	2	8	no	—
401	C₂²⋊Q₈	—	2	2	no	—
402	C₂²⋊Q₈	—	2	2	no	—
403	C₂²⋊Q₈	C₂³×C₃₁	1	4	yes	—
404	C₂²⋊Q₈	C₂³×C₃₁	1	4	yes	—
405	#121	—	2	2	no	—
406	#121	—	2	2	no	—
407	#121	C₂³×C₃₁	1	4	yes	—
408	#121	C₂³×C₃₁	1	4	yes	—
409	#120	—	2	8	no	—
410	C₄×D₄	—	2	8	no	—
411	C₄×D₄	—	2	8	no	—
412	#117	—	2	4	no	—
413	#117	—	2	4	no	—
414	#117	—	2	8	no	—
415	#117	—	2	8	no	—
416	#117	—	2	2	no	—
417	#117	C₂³×C₃₁	1	8	yes	—
418	#117	C₂³×C₃₁	1	8	yes	—
419	#117	C₂³×C₃₁	1	4	yes	—
420	C₂×C₄⋊C₄	—	2	1	no	—
421	C₂×C₄⋊C₄	—	2	1	no	—
422	C₂×C₄⋊C₄	—	2	2	no	—
423	C₂×C₄⋊C₄	—	2	2	no	—
424	C₂×C₄⋊C₄	—	2	1	no	—
425	C₂×C₄⋊C₄	—	2	1	no	—
426	C₂×C₄⋊C₄	C₂³×C₃₁	1	2	yes	—
427	C₂×C₄⋊C₄	C₂³×C₃₁	1	2	yes	—
428	C₂×C₄⋊C₄	C₂³×C₃₁	1	4	yes	—
429	C₂×C₄⋊C₄	C₂³×C₃₁	1	4	yes	—
430	C₂×C₄⋊C₄	C₂³×C₃₁	1	1	yes	characteristic
431	C₂×C₄⋊C₄	C₂³×C₃₁	1	1	yes	characteristic
432	C₂×C₂²⋊C₄	—	2	2	no	—
433	C₂×C₂²⋊C₄	—	2	2	no	—
434	C₂×C₂²⋊C₄	—	2	8	no	—
435	C₂×C₂²⋊C₄	—	2	1	no	—
436	C₂×C₂²⋊C₄	—	2	1	no	—
437	C₂×C₂²⋊C₄	—	2	1	no	—
438	C₂×C₂²⋊C₄	—	2	1	no	—
439	C₂×C₂²⋊C₄	—	2	1	no	—
440	C₂×C₄²	—	2	2	no	—
441	C₂×C₄²	—	2	2	no	—
442	C₂×C₄²	—	2	8	no	—
443	C₂×C₄²	—	2	1	no	—
444	#107	C₂³×C₃₁	1	8	yes	complemented
445	#107	C₂³×C₃₁	1	8	yes	complemented
446	#106	C₂³×C₃₁	1	8	yes	complemented
447	#106	C₂³×C₃₁	1	8	yes	complemented
448	#103	C₂³×C₃₁	1	8	yes	complemented
449	#103	C₂³×C₃₁	1	8	yes	complemented
450	#102	C₂³×C₃₁	1	8	yes	complemented
451	#102	C₂³×C₃₁	1	8	yes	complemented
452	C₂²⋊C₈	C₂³×C₃₁	1	4	yes	—
453	C₂²⋊C₈	C₂³×C₃₁	1	4	yes	—
454	C₂²⋊C₈	C₂³×C₃₁	1	8	yes	—
455	C₃₁	C₂×#3091	1	1	yes	characteristic, direct product factor, Sylow
456	C₂⁴	—	2	4	no	—
457	C₂⁴	—	2	4	no	—
458	C₂⁴	—	2	1	no	—
459	C₂⁴	—	4	2	no	—
460	C₂⁴	C₂×D₄×C₃₁	1	1	yes	characteristic
461	C₂⁴	C₂×D₄×C₃₁	1	4	yes	—
462	C₂⁴	C₂×D₄×C₃₁	1	1	yes	characteristic
463	C₂⁴	C₂×D₄×C₃₁	1	1	yes	characteristic
464	#41	—	2	16	no	—
465	#41	—	4	8	no	—
466	C₂×Q₈	—	2	1	no	—
467	C₂×Q₈	—	2	4	no	—
468	C₂×Q₈	C₂⁴×C₃₁	1	2	yes	—
469	C₂×D₄	—	2	4	no	—
470	C₂×D₄	—	2	4	no	—
471	C₂×D₄	—	2	4	no	—
472	C₂×D₄	—	2	1	no	—
473	C₂×D₄	—	2	4	no	—
474	C₂×D₄	—	4	4	no	—
475	C₂×D₄	—	4	4	no	—
476	C₂×D₄	—	4	4	no	—
477	C₂×D₄	C₂⁴×C₃₁	1	2	yes	—
478	C₂²×C₄	—	2	4	no	—
479	C₂²×C₄	—	2	4	no	—
480	C₂²×C₄	—	2	4	no	—
481	C₂²×C₄	—	2	16	no	—
482	C₂²×C₄	—	2	4	no	—
483	C₂²×C₄	—	2	4	no	—
484	C₂²×C₄	—	2	16	no	—
485	C₂²×C₄	—	2	4	no	—
486	C₂²×C₄	—	2	4	no	—
487	C₂²×C₄	—	2	2	no	—
488	C₂²×C₄	—	2	2	no	—
489	C₂²×C₄	—	2	1	no	—
490	C₂²×C₄	—	2	1	no	—
491	C₂²×C₄	—	2	1	no	—
492	C₂²×C₄	—	2	4	no	—
493	C₂²×C₄	—	4	4	no	—
494	C₂²×C₄	—	4	4	no	—
495	C₂²×C₄	—	4	4	no	—
496	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
497	C₂²×C₄	C₂×D₄×C₃₁	1	4	yes	—
498	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
499	C₂²×C₄	C₂×D₄×C₃₁	1	2	yes	—
500	C₂²×C₄	C₂×D₄×C₃₁	1	4	yes	—
501	C₂²×C₄	C₂×D₄×C₃₁	1	2	yes	—
502	C₂²×C₄	C₂×D₄×C₃₁	1	4	yes	—
503	C₂²×C₄	C₂×D₄×C₃₁	1	2	yes	—
504	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
505	C₂²×C₄	#41×C₃₁	1	8	yes	—
506	C₂²×C₄	C₂⁴×C₃₁	1	2	yes	—
507	C₂²×C₄	C₂⁴×C₃₁	1	2	yes	—
508	C₂²×C₄	C₂⁴×C₃₁	1	1	yes	characteristic
509	C₂²×C₄	C₂⁴×C₃₁	1	2	yes	—
510	C₂×C₈	—	2	4	no	—
511	C₂×C₈	—	2	4	no	—
512	C₂×C₈	—	2	16	no	—
513	C₂×C₈	C₂⁴×C₃₁	1	4	yes	—
514	C₂×C₈	C₂⁴×C₃₁	1	4	yes	—
515	C₄⋊C₄	—	2	4	no	—
516	C₄⋊C₄	—	2	4	no	—
517	C₄⋊C₄	—	2	2	no	—
518	C₄⋊C₄	—	2	2	no	—
519	C₄⋊C₄	—	2	4	no	—
520	C₄⋊C₄	—	2	4	no	—
521	C₄⋊C₄	—	2	1	no	—
522	C₄⋊C₄	—	2	1	no	—
523	C₄⋊C₄	C₂⁴×C₃₁	1	4	yes	—
524	C₄⋊C₄	C₂⁴×C₃₁	1	4	yes	—
525	C₄⋊C₄	C₂⁴×C₃₁	1	2	yes	—
526	C₄⋊C₄	C₂⁴×C₃₁	1	2	yes	—
527	C₂²⋊C₄	—	2	8	no	—
528	C₂²⋊C₄	—	2	8	no	—
529	C₂²⋊C₄	—	2	4	no	—
530	C₂²⋊C₄	—	2	4	no	—
531	C₂²⋊C₄	—	2	4	no	—
532	C₂²⋊C₄	—	2	4	no	—
533	C₂²⋊C₄	—	2	4	no	—
534	C₄²	—	2	8	no	—
535	C₄²	—	2	8	no	—
536	C₄²	—	2	4	no	—
537	C₂³	—	2	2	no	—
538	C₂³	—	2	2	no	—
539	C₂³	—	2	2	no	—
540	C₂³	—	2	2	no	—
541	C₂³	—	2	8	no	—
542	C₂³	—	2	2	no	—
543	C₂³	—	2	8	no	—
544	C₂³	—	2	2	no	—
545	C₂³	—	2	2	no	—
546	C₂³	—	2	2	no	—
547	C₂³	—	4	4	no	—
548	C₂³	—	4	16	no	—
549	C₂³	—	4	2	no	—
550	C₂³	—	4	2	no	—
551	C₂³	—	4	2	no	—
552	C₂³	C₃₁×#123	1	4	yes	—
553	C₂³	C₃₁×#135	1	4	yes	—
554	C₂³	C₂²×D₄×C₃₁	1	2	yes	—
555	C₂³	C₂²×D₄×C₃₁	1	2	yes	—
556	C₂³	C₂²×D₄×C₃₁	1	2	yes	—
557	C₂³	C₂²×D₄×C₃₁	1	1	yes	characteristic
558	Q₈	—	2	4	no	—
559	Q₈	—	4	2	no	—
560	D₄	—	2	4	no	—
561	D₄	—	4	8	no	—
562	D₄	—	4	8	no	—
563	D₄	—	4	8	no	—
564	D₄	—	4	2	no	—
565	C₂×C₄	—	2	8	no	—
566	C₂×C₄	—	2	2	no	—
567	C₂×C₄	—	2	2	no	—
568	C₂×C₄	—	2	8	no	—
569	C₂×C₄	—	2	8	no	—
570	C₂×C₄	—	2	4	no	—
571	C₂×C₄	—	2	4	no	—
572	C₂×C₄	—	2	4	no	—
573	C₂×C₄	—	2	4	no	—
574	C₂×C₄	—	2	4	no	—
575	C₂×C₄	—	2	4	no	—
576	C₂×C₄	—	2	2	no	—
577	C₂×C₄	—	2	2	no	—
578	C₂×C₄	—	2	2	no	—
579	C₂×C₄	—	4	8	no	—
580	C₂×C₄	—	4	2	no	—
581	C₂×C₄	—	4	8	no	—
582	C₂×C₄	—	4	4	no	—
583	C₂×C₄	—	4	2	no	—
584	C₂×C₄	—	4	4	no	—
585	C₂×C₄	—	4	2	no	—
586	C₂×C₄	—	4	8	no	—
587	C₂×C₄	C₃₁×#123	1	4	yes	—
588	C₂×C₄	C₃₁×#123	1	8	yes	—
589	C₂×C₄	C₂²×D₄×C₃₁	1	2	yes	—
590	C₂×C₄	C₂²×D₄×C₃₁	1	2	yes	—
591	C₂×C₄	C₂²×D₄×C₃₁	1	2	yes	—
592	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
593	C₂×C₄	C₂×#41×C₃₁	1	4	yes	—
594	C₂×C₄	C₂×#41×C₃₁	1	8	yes	—
595	C₂×C₄	C₂⁵×C₃₁	1	1	yes	characteristic
596	C₈	—	2	4	no	—
597	C₈	—	2	4	no	—
598	C₂²	—	2	8	no	—
599	C₂²	—	2	8	no	—
600	C₂²	—	2	2	no	—
601	C₂²	—	2	2	no	—
602	C₂²	—	2	2	no	—
603	C₂²	—	2	2	no	—
604	C₂²	—	2	2	no	—
605	C₂²	—	2	2	no	—
606	C₂²	—	2	2	no	—
607	C₂²	—	4	16	no	—
608	C₂²	—	4	4	no	—
609	C₂²	—	4	2	no	—
610	C₂²	—	4	2	no	—
611	C₂²	—	4	2	no	—
612	C₂²	—	8	4	no	—
613	C₂²	C₂×C₃₁×#123	1	4	yes	—
614	C₂²	C₂×C₃₁×#123	1	2	yes	—
615	C₂²	C₂×C₃₁×#135	1	4	yes	—
616	C₂²	C₂×C₃₁×#135	1	2	yes	—
617	C₂²	C₃₁×#575	1	2	yes	—
618	C₂²	C₂³×D₄×C₃₁	1	1	yes	characteristic
619	C₄	—	2	2	no	—
620	C₄	—	2	2	no	—
621	C₄	—	4	2	no	—
622	C₄	—	4	4	no	—
623	C₄	—	4	2	no	—
624	C₄	—	4	4	no	—
625	C₄	—	4	2	no	—
626	C₄	C₂×C₃₁×#123	1	4	yes	—
627	C₄	C₂×C₃₁×#123	1	2	yes	—
628	C₄	C₂²×#41×C₃₁	1	2	yes	—
629	C₂	—	2	4	no	—
630	C₂	—	2	4	no	—
631	C₂	—	4	2	no	—
632	C₂	—	8	2	no	—
633	C₂	C₃₁×#3091	1	4	yes	direct product factor
634	C₂	C₂²×C₃₁×#123	1	1	yes	characteristic
635	C₂	C₂²×C₃₁×#135	1	1	yes	characteristic
636	C₂	C₂×C₃₁×#575	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×#525	C₂
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄×C₃₁ ◁ C₂⁴×C₄×C₃₁ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂²×C₃₁×#117	C₂	semidirect
3. C₃₁×#3091	C₂	direct (N × H)
4. C₂×C₃₁×#484	C₂	semidirect
5. C₂×C₃₁×#481	C₂	semidirect
6. C₂×C₃₁×#485	C₂	semidirect
7. C₂×C₃₁×#482	C₂	semidirect
8. C₂×C₃₁×#483	C₂	semidirect
9. C₂×C₃₁×#480	C₂	semidirect
10. C₂×C₃₁×#427	C₂	semidirect
11. C₂×C₃₁×#416	C₂	semidirect
12. C₂²×C₃₁×C₂²⋊C₈	C₂	semidirect
32. C₂×C₃₁×#117	C₂²	semidirect
37. C₃₁×#485	C₂²	semidirect
38. C₃₁×#484	C₂²	semidirect
39. C₃₁×#483	C₂²	semidirect
40. C₃₁×#482	C₂²	semidirect
41. C₃₁×#481	C₂²	semidirect
42. C₃₁×#480	C₂²	semidirect
43. C₃₁×#427	C₂²	semidirect
44. C₂×C₃₁×#107	C₂²	semidirect
45. C₂×C₃₁×#107	C₂²	semidirect
46. C₂×C₃₁×#106	C₂²	semidirect
47. C₂×C₃₁×#106	C₂²	semidirect
48. C₃₁×#416	C₂²	semidirect
49. C₂×C₃₁×#103	C₂²	semidirect
50. C₂×C₃₁×#103	C₂²	semidirect
51. C₂×C₃₁×#102	C₂²	semidirect
52. C₂×C₃₁×#102	C₂²	semidirect
56. C₂×C₃₁×C₂²⋊C₈	C₂²	semidirect
126. C₃₁×#107	C₂³	semidirect
127. C₃₁×#107	C₂³	semidirect
128. C₃₁×#106	C₂³	semidirect
129. C₃₁×#106	C₂³	semidirect
130. C₃₁×#103	C₂³	semidirect
131. C₃₁×#103	C₂³	semidirect
132. C₃₁×#102	C₂³	semidirect
133. C₃₁×#102	C₂³	semidirect
218. C₂×#3091	C₃₁	direct (N × H)
281. C₂²×#117	C₂×C₃₁	semidirect
282. #3091	C₂×C₃₁	direct (N × H)
283. C₂×#484	C₂×C₃₁	semidirect
284. C₂×#481	C₂×C₃₁	semidirect
285. C₂×#485	C₂×C₃₁	semidirect
286. C₂×#482	C₂×C₃₁	semidirect
287. C₂×#483	C₂×C₃₁	semidirect
288. C₂×#480	C₂×C₃₁	semidirect
289. C₂×#427	C₂×C₃₁	semidirect
290. C₂×#416	C₂×C₃₁	semidirect
291. C₂²×C₂²⋊C₈	C₂×C₃₁	semidirect
342. C₂×#117	C₂²×C₃₁	semidirect
347. #485	C₂²×C₃₁	semidirect
348. #484	C₂²×C₃₁	semidirect
349. #483	C₂²×C₃₁	semidirect
350. #482	C₂²×C₃₁	semidirect
351. #481	C₂²×C₃₁	semidirect
352. #480	C₂²×C₃₁	semidirect
353. #427	C₂²×C₃₁	semidirect
354. C₂×#107	C₂²×C₃₁	semidirect
355. C₂×#107	C₂²×C₃₁	semidirect
356. C₂×#106	C₂²×C₃₁	semidirect
357. C₂×#106	C₂²×C₃₁	semidirect
358. #416	C₂²×C₃₁	semidirect
359. C₂×#103	C₂²×C₃₁	semidirect
360. C₂×#103	C₂²×C₃₁	semidirect
361. C₂×#102	C₂²×C₃₁	semidirect
362. C₂×#102	C₂²×C₃₁	semidirect
366. C₂×C₂²⋊C₈	C₂²×C₃₁	semidirect
371. C₂×C₃₁	#3091	direct (N × H)
444. #107	C₂³×C₃₁	semidirect
445. #107	C₂³×C₃₁	semidirect
446. #106	C₂³×C₃₁	semidirect
447. #106	C₂³×C₃₁	semidirect
448. #103	C₂³×C₃₁	semidirect
449. #103	C₂³×C₃₁	semidirect
450. #102	C₂³×C₃₁	semidirect
451. #102	C₂³×C₃₁	semidirect
455. C₃₁	C₂×#3091	direct (N × H)
633. C₂	C₃₁×#3091	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×#525	2
C₂²×C₃₁×#117	2
C₃₁×#3091	2
C₂×C₃₁×#484	2
C₂×C₃₁×#481	2
C₂×C₃₁×#485	2
C₂×C₃₁×#482	2
C₂×C₃₁×#483	2
C₂×C₃₁×#480	2
C₂×C₃₁×#427	2
C₂×C₃₁×#416	2
C₂²×C₃₁×C₂²⋊C₈	2
C₂×#3091	31

Subgroups — C2×C31×#3091 #21517265

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₃₁×#3091 #21517265