Subgroups — C₂×C₃₁×#2626 #21491314

Order 7936.

4142 conjugacy classes of subgroups.

The subgroup lattice has 472 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₃₁×C₂²⋊Q₈	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂³×C₂²⋊C₄×C₃₁	C₂	1	1	yes	maximal, characteristic
3	C₃₁×#2626	C₂	1	16	yes	maximal, direct product factor
4	C₂×C₃₁×#399	C₂	1	2	yes	maximal, complemented
5	C₂×C₃₁×#396	C₂	1	4	yes	maximal, complemented
6	C₂×C₃₁×#393	C₂	1	2	yes	maximal, complemented
7	C₂×C₃₁×#385	C₂	1	2	yes	maximal, complemented
8	C₂×C₃₁×#380	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂×C₃₁×#379	C₂	1	2	yes	maximal
10	C₂⁶×C₃₁	C₂²	1	1	yes	characteristic
11	C₂³×Q₈×C₃₁	C₂²	1	1	yes	characteristic, complemented
12	C₂⁴×C₄×C₃₁	C₂²	1	1	yes	characteristic
13	C₂⁴×C₄×C₃₁	C₂²	1	1	yes	characteristic
14	C₂×C₃₁×C₂²⋊Q₈	—	2	8	no	—
15	C₂×C₃₁×C₂²⋊Q₈	C₂²	1	8	yes	complemented
16	C₂²×C₄⋊C₄×C₃₁	C₂²	1	2	yes	complemented
17	C₂²×C₄⋊C₄×C₃₁	C₂²	1	1	yes	characteristic, complemented
18	C₂²×C₄⋊C₄×C₃₁	C₂²	1	2	yes	complemented
19	C₂²×C₄⋊C₄×C₃₁	C₂²	1	4	yes	complemented
20	C₂²×C₂²⋊C₄×C₃₁	—	2	8	no	—
21	C₂²×C₂²⋊C₄×C₃₁	C₂²	1	8	yes	—
22	C₂²×C₂²⋊C₄×C₃₁	C₂²	1	2	yes	—
23	C₂²×C₂²⋊C₄×C₃₁	C₂²	1	2	yes	—
24	C₂²×C₂²⋊C₄×C₃₁	C₂²	1	2	yes	—
25	C₂²×C₂²⋊C₄×C₃₁	C₂²	1	2	yes	—
26	C₂²×C₂²⋊C₄×C₃₁	C₂²	1	4	yes	—
27	C₂²×C₂²⋊C₄×C₃₁	C₂²	1	4	yes	—
28	C₃₁×#399	C₂²	1	16	yes	complemented
29	C₃₁×#396	C₂²	1	32	yes	complemented
30	C₃₁×#393	C₂²	1	16	yes	complemented
31	C₃₁×#385	C₂²	1	16	yes	complemented
32	C₃₁×#380	C₂²	1	8	yes	complemented
33	C₃₁×#379	C₂²	1	16	yes	—
34	C₂×C₃₁×#95	C₂²	1	4	yes	complemented
35	C₂×C₃₁×#95	C₂²	1	2	yes	complemented
36	C₂⁵×C₃₁	—	2	4	no	—
37	C₂⁵×C₃₁	—	2	4	no	—
38	C₂⁵×C₃₁	—	4	4	no	—
39	C₂⁵×C₃₁	—	4	4	no	—
40	C₂⁵×C₃₁	Q₈	1	8	yes	complemented
41	C₂⁵×C₃₁	C₂³	1	2	yes	—
42	C₂⁵×C₃₁	C₂³	1	1	yes	characteristic
43	C₂⁵×C₃₁	C₂³	1	4	yes	—
44	C₂²×Q₈×C₃₁	—	2	4	no	—
45	C₂²×Q₈×C₃₁	—	4	4	no	—
46	C₂²×Q₈×C₃₁	C₂³	1	4	yes	complemented
47	C₂³×C₄×C₃₁	—	2	4	no	—
48	C₂³×C₄×C₃₁	—	4	4	no	—
49	C₂³×C₄×C₃₁	—	4	4	no	—
50	C₂³×C₄×C₃₁	D₄	1	8	yes	—
51	C₂³×C₄×C₃₁	C₂³	1	2	yes	—
52	C₂³×C₄×C₃₁	C₂³	1	4	yes	—
53	C₂³×C₄×C₃₁	C₂³	1	2	yes	—
54	C₂³×C₄×C₃₁	C₂³	1	2	yes	—
55	C₂³×C₄×C₃₁	C₂³	1	1	yes	characteristic
56	C₂³×C₄×C₃₁	C₂³	1	1	yes	characteristic
57	C₂³×C₄×C₃₁	C₂³	1	4	yes	—
58	C₂³×C₄×C₃₁	C₂³	1	4	yes	—
59	C₃₁×C₂²⋊Q₈	—	2	32	no	—
60	C₂×C₄⋊C₄×C₃₁	—	2	4	no	—
61	C₂×C₄⋊C₄×C₃₁	—	2	8	no	—
62	C₂×C₄⋊C₄×C₃₁	—	4	4	no	—
63	C₂×C₄⋊C₄×C₃₁	—	4	8	no	—
64	C₂×C₄⋊C₄×C₃₁	C₂³	1	8	yes	complemented
65	C₂×C₄⋊C₄×C₃₁	C₂³	1	4	yes	complemented
66	C₂×C₄⋊C₄×C₃₁	C₂³	1	8	yes	complemented
67	C₂×C₄⋊C₄×C₃₁	C₂³	1	16	yes	complemented
68	C₂×C₂²⋊C₄×C₃₁	—	2	8	no	—
69	C₂×C₂²⋊C₄×C₃₁	—	2	8	no	—
70	C₂×C₂²⋊C₄×C₃₁	—	2	8	no	—
71	C₂×C₂²⋊C₄×C₃₁	—	2	8	no	—
72	C₂×C₂²⋊C₄×C₃₁	—	2	32	no	—
73	C₂×C₂²⋊C₄×C₃₁	—	4	8	no	—
74	C₂×C₂²⋊C₄×C₃₁	—	4	8	no	—
75	C₂×C₂²⋊C₄×C₃₁	C₂³	1	8	yes	—
76	C₂×C₂²⋊C₄×C₃₁	C₂³	1	8	yes	—
77	C₂×C₂²⋊C₄×C₃₁	C₂³	1	8	yes	—
78	C₂×C₂²⋊C₄×C₃₁	C₂³	1	8	yes	—
79	C₂×C₂²⋊C₄×C₃₁	C₂³	1	16	yes	—
80	C₂×C₂²⋊C₄×C₃₁	C₂³	1	16	yes	—
81	C₃₁×#95	C₂³	1	16	yes	complemented
82	C₃₁×#95	C₂³	1	8	yes	complemented
83	C₂⁴×C₃₁	—	2	4	no	—
84	C₂⁴×C₃₁	—	2	4	no	—
85	C₂⁴×C₃₁	—	2	32	no	—
86	C₂⁴×C₃₁	—	2	4	no	—
87	C₂⁴×C₃₁	—	2	8	no	—
88	C₂⁴×C₃₁	—	2	4	no	—
89	C₂⁴×C₃₁	—	2	8	no	—
90	C₂⁴×C₃₁	—	2	8	no	—
91	C₂⁴×C₃₁	—	2	8	no	—
92	C₂⁴×C₃₁	—	4	32	no	—
93	C₂⁴×C₃₁	—	4	8	no	—
94	C₂⁴×C₃₁	—	4	8	no	—
95	C₂⁴×C₃₁	—	4	16	no	—
96	C₂⁴×C₃₁	—	4	16	no	—
97	C₂⁴×C₃₁	—	4	32	no	—
98	C₂⁴×C₃₁	C₂×D₄	1	4	yes	—
99	C₂⁴×C₃₁	C₂×Q₈	1	8	yes	—
100	C₂⁴×C₃₁	C₂×Q₈	1	4	yes	—
101	C₂⁴×C₃₁	C₂×Q₈	1	16	yes	complemented
102	C₂⁴×C₃₁	#41	1	4	yes	—
103	C₂⁴×C₃₁	C₂⁴	1	4	yes	—
104	C₂⁴×C₃₁	C₂⁴	1	2	yes	—
105	C₂⁴×C₃₁	C₂⁴	1	1	yes	characteristic
106	C₂×Q₈×C₃₁	—	2	8	no	—
107	C₂×Q₈×C₃₁	—	4	8	no	—
108	C₂×Q₈×C₃₁	—	4	16	no	—
109	C₂²×C₄×C₃₁	—	2	4	no	—
110	C₂²×C₄×C₃₁	—	2	4	no	—
111	C₂²×C₄×C₃₁	—	2	8	no	—
112	C₂²×C₄×C₃₁	—	2	4	no	—
113	C₂²×C₄×C₃₁	—	2	4	no	—
114	C₂²×C₄×C₃₁	—	2	4	no	—
115	C₂²×C₄×C₃₁	—	2	8	no	—
116	C₂²×C₄×C₃₁	—	2	8	no	—
117	C₂²×C₄×C₃₁	—	2	8	no	—
118	C₂²×C₄×C₃₁	—	2	8	no	—
119	C₂²×C₄×C₃₁	—	2	8	no	—
120	C₂²×C₄×C₃₁	—	2	8	no	—
121	C₂²×C₄×C₃₁	—	4	16	no	—
122	C₂²×C₄×C₃₁	—	4	8	no	—
123	C₂²×C₄×C₃₁	—	4	8	no	—
124	C₂²×C₄×C₃₁	—	4	16	no	—
125	C₂²×C₄×C₃₁	C₂×D₄	1	4	yes	—
126	C₂²×C₄×C₃₁	C₂×D₄	1	4	yes	—
127	C₂²×C₄×C₃₁	C₂×D₄	1	16	yes	—
128	C₂²×C₄×C₃₁	C₂⁴	1	8	yes	—
129	C₂²×C₄×C₃₁	C₂⁴	1	4	yes	—
130	C₂²×C₄×C₃₁	C₂⁴	1	4	yes	—
131	C₂²×C₄×C₃₁	C₂⁴	1	4	yes	—
132	C₂²×C₄×C₃₁	C₂⁴	1	2	yes	—
133	C₂²×C₄×C₃₁	C₂⁴	1	2	yes	—
134	C₄⋊C₄×C₃₁	—	2	8	no	—
135	C₄⋊C₄×C₃₁	—	2	16	no	—
136	C₄⋊C₄×C₃₁	—	4	8	no	—
137	C₄⋊C₄×C₃₁	—	4	16	no	—
138	C₂²⋊C₄×C₃₁	—	2	16	no	—
139	C₂²⋊C₄×C₃₁	—	2	16	no	—
140	C₂²⋊C₄×C₃₁	—	2	16	no	—
141	C₂²⋊C₄×C₃₁	—	2	16	no	—
142	C₂²⋊C₄×C₃₁	—	4	16	no	—
143	C₂²⋊C₄×C₃₁	—	4	16	no	—
144	C₂×#2626	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
145	C₂³×C₃₁	—	2	8	no	—
146	C₂³×C₃₁	—	2	8	no	—
147	C₂³×C₃₁	—	2	16	no	—
148	C₂³×C₃₁	—	2	16	no	—
149	C₂³×C₃₁	—	2	4	no	—
150	C₂³×C₃₁	—	2	4	no	—
151	C₂³×C₃₁	—	2	16	no	—
152	C₂³×C₃₁	—	2	32	no	—
153	C₂³×C₃₁	—	2	32	no	—
154	C₂³×C₃₁	—	2	4	no	—
155	C₂³×C₃₁	—	2	4	no	—
156	C₂³×C₃₁	—	2	8	no	—
157	C₂³×C₃₁	—	2	8	no	—
158	C₂³×C₃₁	—	4	32	no	—
159	C₂³×C₃₁	—	4	128	no	—
160	C₂³×C₃₁	—	4	16	no	—
161	C₂³×C₃₁	—	4	16	no	—
162	C₂³×C₃₁	—	4	32	no	—
163	C₂³×C₃₁	—	4	8	no	—
164	C₂³×C₃₁	—	4	8	no	—
165	C₂³×C₃₁	—	4	16	no	—
166	C₂³×C₃₁	C₂²⋊Q₈	1	16	yes	—
167	C₂³×C₃₁	C₂²×D₄	1	4	yes	—
168	C₂³×C₃₁	C₂²×D₄	1	2	yes	—
169	C₂³×C₃₁	C₂²×Q₈	1	8	yes	—
170	C₂³×C₃₁	C₂²×Q₈	1	4	yes	—
171	C₂³×C₃₁	C₂²×Q₈	1	2	yes	—
172	C₂³×C₃₁	C₂×#41	1	4	yes	—
173	C₂³×C₃₁	C₂×#41	1	2	yes	—
174	C₂³×C₃₁	H(5,𝔽₂)	1	4	yes	—
175	C₂³×C₃₁	H(5,𝔽₂)	1	4	yes	—
176	C₂³×C₃₁	C₂⁵	1	1	yes	characteristic
177	Q₈×C₃₁	—	4	16	no	—
178	C₂×C₄×C₃₁	—	2	8	no	—
179	C₂×C₄×C₃₁	—	2	8	no	—
180	C₂×C₄×C₃₁	—	2	4	no	—
181	C₂×C₄×C₃₁	—	2	4	no	—
182	C₂×C₄×C₃₁	—	2	4	no	—
183	C₂×C₄×C₃₁	—	2	4	no	—
184	C₂×C₄×C₃₁	—	2	4	no	—
185	C₂×C₄×C₃₁	—	2	8	no	—
186	C₂×C₄×C₃₁	—	2	8	no	—
187	C₂×C₄×C₃₁	—	2	8	no	—
188	C₂×C₄×C₃₁	—	2	8	no	—
189	C₂×C₄×C₃₁	—	2	8	no	—
190	C₂×C₄×C₃₁	—	2	8	no	—
191	C₂×C₄×C₃₁	—	4	16	no	—
192	C₂×C₄×C₃₁	—	4	16	no	—
193	C₂×C₄×C₃₁	—	4	8	no	—
194	C₂×C₄×C₃₁	—	4	8	no	—
195	C₂×C₄×C₃₁	—	4	8	no	—
196	C₂×C₄×C₃₁	—	4	16	no	—
197	C₂×C₄×C₃₁	C₂²×D₄	1	4	yes	—
198	C₂×C₄×C₃₁	C₂²×D₄	1	4	yes	—
199	C₂²×C₂²⋊Q₈	C₂×C₃₁	1	1	yes	characteristic, complemented
200	C₂³×C₂²⋊C₄	C₂×C₃₁	1	1	yes	characteristic
201	#2626	C₂×C₃₁	1	16	yes	direct product factor
202	C₂×#399	C₂×C₃₁	1	2	yes	complemented
203	C₂×#396	C₂×C₃₁	1	4	yes	complemented
204	C₂×#393	C₂×C₃₁	1	2	yes	complemented
205	C₂×#385	C₂×C₃₁	1	2	yes	complemented
206	C₂×#380	C₂×C₃₁	1	1	yes	characteristic, complemented
207	C₂×#379	C₂×C₃₁	1	2	yes	—
208	C₂²×C₃₁	—	2	32	no	—
209	C₂²×C₃₁	—	2	4	no	—
210	C₂²×C₃₁	—	2	4	no	—
211	C₂²×C₃₁	—	2	8	no	—
212	C₂²×C₃₁	—	2	8	no	—
213	C₂²×C₃₁	—	2	8	no	—
214	C₂²×C₃₁	—	2	16	no	—
215	C₂²×C₃₁	—	4	64	no	—
216	C₂²×C₃₁	—	4	32	no	—
217	C₂²×C₃₁	—	4	4	no	—
218	C₂²×C₃₁	—	4	4	no	—
219	C₂²×C₃₁	—	4	8	no	—
220	C₂²×C₃₁	C₂×C₂²⋊Q₈	1	8	yes	—
221	C₂²×C₃₁	C₂×C₂²⋊Q₈	1	4	yes	—
222	C₂²×C₃₁	#534	1	4	yes	—
223	C₂²×C₃₁	#535	1	4	yes	—
224	C₂²×C₃₁	#538	1	8	yes	—
225	C₂²×C₃₁	#543	1	8	yes	—
226	C₂²×C₃₁	C₂³×D₄	1	1	yes	characteristic
227	C₂²×C₃₁	C₂³×Q₈	1	1	yes	characteristic
228	C₂²×C₃₁	C₂²×#41	1	1	yes	characteristic
229	C₂²×C₃₁	C₂×H(5,𝔽₂)	1	2	yes	—
230	C₂²×C₃₁	C₂×H(5,𝔽₂)	1	2	yes	—
231	C₄×C₃₁	—	2	4	no	—
232	C₄×C₃₁	—	2	4	no	—
233	C₄×C₃₁	—	4	4	no	—
234	C₄×C₃₁	—	4	4	no	—
235	C₄×C₃₁	—	4	4	no	—
236	C₄×C₃₁	—	4	8	no	—
237	C₂⁶	C₂²×C₃₁	1	1	yes	characteristic
238	C₂³×Q₈	C₂²×C₃₁	1	1	yes	characteristic, complemented
239	C₂⁴×C₄	C₂²×C₃₁	1	1	yes	characteristic
240	C₂⁴×C₄	C₂²×C₃₁	1	1	yes	characteristic
241	C₂×C₂²⋊Q₈	—	2	8	no	—
242	C₂×C₂²⋊Q₈	C₂²×C₃₁	1	8	yes	complemented
243	C₂²×C₄⋊C₄	C₂²×C₃₁	1	2	yes	complemented
244	C₂²×C₄⋊C₄	C₂²×C₃₁	1	1	yes	characteristic, complemented
245	C₂²×C₄⋊C₄	C₂²×C₃₁	1	2	yes	complemented
246	C₂²×C₄⋊C₄	C₂²×C₃₁	1	4	yes	complemented
247	C₂²×C₂²⋊C₄	—	2	8	no	—
248	C₂²×C₂²⋊C₄	C₂²×C₃₁	1	8	yes	—
249	C₂²×C₂²⋊C₄	C₂²×C₃₁	1	2	yes	—
250	C₂²×C₂²⋊C₄	C₂²×C₃₁	1	2	yes	—
251	C₂²×C₂²⋊C₄	C₂²×C₃₁	1	2	yes	—
252	C₂²×C₂²⋊C₄	C₂²×C₃₁	1	2	yes	—
253	C₂²×C₂²⋊C₄	C₂²×C₃₁	1	4	yes	—
254	C₂²×C₂²⋊C₄	C₂²×C₃₁	1	4	yes	—
255	#399	C₂²×C₃₁	1	16	yes	complemented
256	#396	C₂²×C₃₁	1	32	yes	complemented
257	#393	C₂²×C₃₁	1	16	yes	complemented
258	#385	C₂²×C₃₁	1	16	yes	complemented
259	#380	C₂²×C₃₁	1	8	yes	complemented
260	#379	C₂²×C₃₁	1	16	yes	—
261	C₂×#95	C₂²×C₃₁	1	4	yes	complemented
262	C₂×#95	C₂²×C₃₁	1	2	yes	complemented
263	C₂×C₃₁	—	2	8	no	—
264	C₂×C₃₁	—	4	8	no	—
265	C₂×C₃₁	#2626	1	8	yes	direct product factor
266	C₂×C₃₁	C₂²×C₂²⋊Q₈	1	1	yes	characteristic
267	C₂×C₃₁	C₂×#534	1	1	yes	characteristic
268	C₂×C₃₁	C₂×#535	1	1	yes	characteristic
269	C₂×C₃₁	C₂×#538	1	2	yes	—
270	C₂×C₃₁	C₂×#543	1	2	yes	—
271	C₂⁵	—	2	4	no	—
272	C₂⁵	—	2	4	no	—
273	C₂⁵	—	4	4	no	—
274	C₂⁵	—	4	4	no	—
275	C₂⁵	Q₈×C₃₁	1	8	yes	complemented
276	C₂⁵	C₂³×C₃₁	1	2	yes	—
277	C₂⁵	C₂³×C₃₁	1	1	yes	characteristic
278	C₂⁵	C₂³×C₃₁	1	4	yes	—
279	C₂²×Q₈	—	2	4	no	—
280	C₂²×Q₈	—	4	4	no	—
281	C₂²×Q₈	C₂³×C₃₁	1	4	yes	complemented
282	C₂³×C₄	—	2	4	no	—
283	C₂³×C₄	—	4	4	no	—
284	C₂³×C₄	—	4	4	no	—
285	C₂³×C₄	D₄×C₃₁	1	8	yes	—
286	C₂³×C₄	C₂³×C₃₁	1	2	yes	—
287	C₂³×C₄	C₂³×C₃₁	1	4	yes	—
288	C₂³×C₄	C₂³×C₃₁	1	2	yes	—
289	C₂³×C₄	C₂³×C₃₁	1	2	yes	—
290	C₂³×C₄	C₂³×C₃₁	1	1	yes	characteristic
291	C₂³×C₄	C₂³×C₃₁	1	1	yes	characteristic
292	C₂³×C₄	C₂³×C₃₁	1	4	yes	—
293	C₂³×C₄	C₂³×C₃₁	1	4	yes	—
294	C₂²⋊Q₈	—	2	32	no	—
295	C₂×C₄⋊C₄	—	2	4	no	—
296	C₂×C₄⋊C₄	—	2	8	no	—
297	C₂×C₄⋊C₄	—	4	4	no	—
298	C₂×C₄⋊C₄	—	4	8	no	—
299	C₂×C₄⋊C₄	C₂³×C₃₁	1	8	yes	complemented
300	C₂×C₄⋊C₄	C₂³×C₃₁	1	4	yes	complemented
301	C₂×C₄⋊C₄	C₂³×C₃₁	1	8	yes	complemented
302	C₂×C₄⋊C₄	C₂³×C₃₁	1	16	yes	complemented
303	C₂×C₂²⋊C₄	—	2	8	no	—
304	C₂×C₂²⋊C₄	—	2	8	no	—
305	C₂×C₂²⋊C₄	—	2	8	no	—
306	C₂×C₂²⋊C₄	—	2	8	no	—
307	C₂×C₂²⋊C₄	—	2	32	no	—
308	C₂×C₂²⋊C₄	—	4	8	no	—
309	C₂×C₂²⋊C₄	—	4	8	no	—
310	C₂×C₂²⋊C₄	C₂³×C₃₁	1	8	yes	—
311	C₂×C₂²⋊C₄	C₂³×C₃₁	1	8	yes	—
312	C₂×C₂²⋊C₄	C₂³×C₃₁	1	8	yes	—
313	C₂×C₂²⋊C₄	C₂³×C₃₁	1	8	yes	—
314	C₂×C₂²⋊C₄	C₂³×C₃₁	1	16	yes	—
315	C₂×C₂²⋊C₄	C₂³×C₃₁	1	16	yes	—
316	#95	C₂³×C₃₁	1	8	yes	complemented
317	#95	C₂³×C₃₁	1	16	yes	complemented
318	C₃₁	C₂×#2626	1	1	yes	characteristic, direct product factor, Sylow
319	C₂⁴	—	2	4	no	—
320	C₂⁴	—	2	4	no	—
321	C₂⁴	—	2	32	no	—
322	C₂⁴	—	2	4	no	—
323	C₂⁴	—	2	8	no	—
324	C₂⁴	—	2	4	no	—
325	C₂⁴	—	2	8	no	—
326	C₂⁴	—	2	8	no	—
327	C₂⁴	—	2	8	no	—
328	C₂⁴	—	4	32	no	—
329	C₂⁴	—	4	8	no	—
330	C₂⁴	—	4	8	no	—
331	C₂⁴	—	4	16	no	—
332	C₂⁴	—	4	16	no	—
333	C₂⁴	—	4	32	no	—
334	C₂⁴	C₂×D₄×C₃₁	1	4	yes	—
335	C₂⁴	C₂×Q₈×C₃₁	1	8	yes	—
336	C₂⁴	C₂×Q₈×C₃₁	1	4	yes	—
337	C₂⁴	C₂×Q₈×C₃₁	1	16	yes	complemented
338	C₂⁴	#41×C₃₁	1	4	yes	—
339	C₂⁴	C₂⁴×C₃₁	1	4	yes	—
340	C₂⁴	C₂⁴×C₃₁	1	2	yes	—
341	C₂⁴	C₂⁴×C₃₁	1	1	yes	characteristic
342	C₂×Q₈	—	2	8	no	—
343	C₂×Q₈	—	4	8	no	—
344	C₂×Q₈	—	4	16	no	—
345	C₂²×C₄	—	2	4	no	—
346	C₂²×C₄	—	2	4	no	—
347	C₂²×C₄	—	2	8	no	—
348	C₂²×C₄	—	2	4	no	—
349	C₂²×C₄	—	2	4	no	—
350	C₂²×C₄	—	2	4	no	—
351	C₂²×C₄	—	2	8	no	—
352	C₂²×C₄	—	2	8	no	—
353	C₂²×C₄	—	2	8	no	—
354	C₂²×C₄	—	2	8	no	—
355	C₂²×C₄	—	2	8	no	—
356	C₂²×C₄	—	2	8	no	—
357	C₂²×C₄	—	4	16	no	—
358	C₂²×C₄	—	4	8	no	—
359	C₂²×C₄	—	4	8	no	—
360	C₂²×C₄	—	4	16	no	—
361	C₂²×C₄	C₂×D₄×C₃₁	1	4	yes	—
362	C₂²×C₄	C₂×D₄×C₃₁	1	4	yes	—
363	C₂²×C₄	C₂×D₄×C₃₁	1	16	yes	—
364	C₂²×C₄	C₂⁴×C₃₁	1	8	yes	—
365	C₂²×C₄	C₂⁴×C₃₁	1	4	yes	—
366	C₂²×C₄	C₂⁴×C₃₁	1	4	yes	—
367	C₂²×C₄	C₂⁴×C₃₁	1	4	yes	—
368	C₂²×C₄	C₂⁴×C₃₁	1	2	yes	—
369	C₂²×C₄	C₂⁴×C₃₁	1	2	yes	—
370	C₄⋊C₄	—	2	8	no	—
371	C₄⋊C₄	—	2	16	no	—
372	C₄⋊C₄	—	4	8	no	—
373	C₄⋊C₄	—	4	16	no	—
374	C₂²⋊C₄	—	2	16	no	—
375	C₂²⋊C₄	—	2	16	no	—
376	C₂²⋊C₄	—	2	16	no	—
377	C₂²⋊C₄	—	2	16	no	—
378	C₂²⋊C₄	—	4	16	no	—
379	C₂²⋊C₄	—	4	16	no	—
380	C₂³	—	2	8	no	—
381	C₂³	—	2	8	no	—
382	C₂³	—	2	16	no	—
383	C₂³	—	2	16	no	—
384	C₂³	—	2	4	no	—
385	C₂³	—	2	4	no	—
386	C₂³	—	2	16	no	—
387	C₂³	—	2	32	no	—
388	C₂³	—	2	32	no	—
389	C₂³	—	2	4	no	—
390	C₂³	—	2	4	no	—
391	C₂³	—	2	8	no	—
392	C₂³	—	2	8	no	—
393	C₂³	—	4	32	no	—
394	C₂³	—	4	128	no	—
395	C₂³	—	4	16	no	—
396	C₂³	—	4	16	no	—
397	C₂³	—	4	32	no	—
398	C₂³	—	4	8	no	—
399	C₂³	—	4	8	no	—
400	C₂³	—	4	16	no	—
401	C₂³	C₃₁×C₂²⋊Q₈	1	16	yes	—
402	C₂³	C₂²×D₄×C₃₁	1	4	yes	—
403	C₂³	C₂²×D₄×C₃₁	1	2	yes	—
404	C₂³	C₂²×Q₈×C₃₁	1	8	yes	—
405	C₂³	C₂²×Q₈×C₃₁	1	4	yes	—
406	C₂³	C₂²×Q₈×C₃₁	1	2	yes	—
407	C₂³	C₂×#41×C₃₁	1	4	yes	—
408	C₂³	C₂×#41×C₃₁	1	2	yes	—
409	C₂³	C₃₁×H(5,𝔽₂)	1	4	yes	—
410	C₂³	C₃₁×H(5,𝔽₂)	1	4	yes	—
411	C₂³	C₂⁵×C₃₁	1	1	yes	characteristic
412	Q₈	—	4	16	no	—
413	C₂×C₄	—	2	8	no	—
414	C₂×C₄	—	2	8	no	—
415	C₂×C₄	—	2	4	no	—
416	C₂×C₄	—	2	4	no	—
417	C₂×C₄	—	2	4	no	—
418	C₂×C₄	—	2	4	no	—
419	C₂×C₄	—	2	4	no	—
420	C₂×C₄	—	2	8	no	—
421	C₂×C₄	—	2	8	no	—
422	C₂×C₄	—	2	8	no	—
423	C₂×C₄	—	2	8	no	—
424	C₂×C₄	—	2	8	no	—
425	C₂×C₄	—	2	8	no	—
426	C₂×C₄	—	4	16	no	—
427	C₂×C₄	—	4	16	no	—
428	C₂×C₄	—	4	8	no	—
429	C₂×C₄	—	4	8	no	—
430	C₂×C₄	—	4	8	no	—
431	C₂×C₄	—	4	16	no	—
432	C₂×C₄	C₂²×D₄×C₃₁	1	4	yes	—
433	C₂×C₄	C₂²×D₄×C₃₁	1	4	yes	—
434	C₂²	—	2	32	no	—
435	C₂²	—	2	4	no	—
436	C₂²	—	2	4	no	—
437	C₂²	—	2	8	no	—
438	C₂²	—	2	8	no	—
439	C₂²	—	2	8	no	—
440	C₂²	—	2	16	no	—
441	C₂²	—	4	64	no	—
442	C₂²	—	4	32	no	—
443	C₂²	—	4	4	no	—
444	C₂²	—	4	4	no	—
445	C₂²	—	4	8	no	—
446	C₂²	C₂×C₃₁×C₂²⋊Q₈	1	8	yes	—
447	C₂²	C₂×C₃₁×C₂²⋊Q₈	1	4	yes	—
448	C₂²	C₃₁×#534	1	4	yes	—
449	C₂²	C₃₁×#535	1	4	yes	—
450	C₂²	C₃₁×#538	1	8	yes	—
451	C₂²	C₃₁×#543	1	8	yes	—
452	C₂²	C₂³×D₄×C₃₁	1	1	yes	characteristic
453	C₂²	C₂³×Q₈×C₃₁	1	1	yes	characteristic
454	C₂²	C₂²×#41×C₃₁	1	1	yes	characteristic
455	C₂²	C₂×C₃₁×H(5,𝔽₂)	1	2	yes	—
456	C₂²	C₂×C₃₁×H(5,𝔽₂)	1	2	yes	—
457	C₄	—	2	4	no	—
458	C₄	—	2	4	no	—
459	C₄	—	4	4	no	—
460	C₄	—	4	4	no	—
461	C₄	—	4	4	no	—
462	C₄	—	4	8	no	—
463	C₂	—	2	8	no	—
464	C₂	—	4	8	no	—
465	C₂	C₃₁×#2626	1	8	yes	direct product factor
466	C₂	C₂²×C₃₁×C₂²⋊Q₈	1	1	yes	characteristic
467	C₂	C₂×C₃₁×#534	1	1	yes	characteristic
468	C₂	C₂×C₃₁×#535	1	1	yes	characteristic
469	C₂	C₂×C₃₁×#538	1	2	yes	—
470	C₂	C₂×C₃₁×#543	1	2	yes	—

	subgroup	G/H
center, socle	C₂⁴×C₃₁	C₂⁴
commutator, Frattini, cosocle	C₂³	C₂⁵×C₃₁
prime radical	C₂⁶×C₃₁	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂⁴×C₃₁ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂²×C₃₁×C₂²⋊Q₈	C₂	semidirect
3. C₃₁×#2626	C₂	direct (N × H)
4. C₂×C₃₁×#399	C₂	semidirect
5. C₂×C₃₁×#396	C₂	semidirect
6. C₂×C₃₁×#393	C₂	semidirect
7. C₂×C₃₁×#385	C₂	semidirect
8. C₂×C₃₁×#380	C₂	semidirect
11. C₂³×Q₈×C₃₁	C₂²	semidirect
15. C₂×C₃₁×C₂²⋊Q₈	C₂²	semidirect
16. C₂²×C₄⋊C₄×C₃₁	C₂²	semidirect
17. C₂²×C₄⋊C₄×C₃₁	C₂²	semidirect
18. C₂²×C₄⋊C₄×C₃₁	C₂²	semidirect
19. C₂²×C₄⋊C₄×C₃₁	C₂²	semidirect
28. C₃₁×#399	C₂²	semidirect
29. C₃₁×#396	C₂²	semidirect
30. C₃₁×#393	C₂²	semidirect
31. C₃₁×#385	C₂²	semidirect
32. C₃₁×#380	C₂²	semidirect
34. C₂×C₃₁×#95	C₂²	semidirect
35. C₂×C₃₁×#95	C₂²	semidirect
40. C₂⁵×C₃₁	Q₈	semidirect
46. C₂²×Q₈×C₃₁	C₂³	semidirect
64. C₂×C₄⋊C₄×C₃₁	C₂³	semidirect
65. C₂×C₄⋊C₄×C₃₁	C₂³	semidirect
66. C₂×C₄⋊C₄×C₃₁	C₂³	semidirect
67. C₂×C₄⋊C₄×C₃₁	C₂³	semidirect
81. C₃₁×#95	C₂³	semidirect
82. C₃₁×#95	C₂³	semidirect
101. C₂⁴×C₃₁	C₂×Q₈	semidirect
144. C₂×#2626	C₃₁	direct (N × H)
199. C₂²×C₂²⋊Q₈	C₂×C₃₁	semidirect
201. #2626	C₂×C₃₁	direct (N × H)
202. C₂×#399	C₂×C₃₁	semidirect
203. C₂×#396	C₂×C₃₁	semidirect
204. C₂×#393	C₂×C₃₁	semidirect
205. C₂×#385	C₂×C₃₁	semidirect
206. C₂×#380	C₂×C₃₁	semidirect
238. C₂³×Q₈	C₂²×C₃₁	semidirect
242. C₂×C₂²⋊Q₈	C₂²×C₃₁	semidirect
243. C₂²×C₄⋊C₄	C₂²×C₃₁	semidirect
244. C₂²×C₄⋊C₄	C₂²×C₃₁	semidirect
245. C₂²×C₄⋊C₄	C₂²×C₃₁	semidirect
246. C₂²×C₄⋊C₄	C₂²×C₃₁	semidirect
255. #399	C₂²×C₃₁	semidirect
256. #396	C₂²×C₃₁	semidirect
257. #393	C₂²×C₃₁	semidirect
258. #385	C₂²×C₃₁	semidirect
259. #380	C₂²×C₃₁	semidirect
261. C₂×#95	C₂²×C₃₁	semidirect
262. C₂×#95	C₂²×C₃₁	semidirect
265. C₂×C₃₁	#2626	direct (N × H)
275. C₂⁵	Q₈×C₃₁	semidirect
281. C₂²×Q₈	C₂³×C₃₁	semidirect
299. C₂×C₄⋊C₄	C₂³×C₃₁	semidirect
300. C₂×C₄⋊C₄	C₂³×C₃₁	semidirect
301. C₂×C₄⋊C₄	C₂³×C₃₁	semidirect
302. C₂×C₄⋊C₄	C₂³×C₃₁	semidirect
316. #95	C₂³×C₃₁	semidirect
317. #95	C₂³×C₃₁	semidirect
318. C₃₁	C₂×#2626	direct (N × H)
337. C₂⁴	C₂×Q₈×C₃₁	semidirect
465. C₂	C₃₁×#2626	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×C₂²⋊Q₈	2
C₂³×C₂²⋊C₄×C₃₁	2
C₃₁×#2626	2
C₂×C₃₁×#399	2
C₂×C₃₁×#396	2
C₂×C₃₁×#393	2
C₂×C₃₁×#385	2
C₂×C₃₁×#380	2
C₂×C₃₁×#379	2
C₂×#2626	31

Subgroups — C2×C31×#2626 #21491314

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₃₁×#2626 #21491314