Subgroups — #46780976

Order 33664.

656 conjugacy classes of subgroups.

The subgroup lattice has 242 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₄⋊Dic₂₆₃	C₂	1	1	yes	maximal, characteristic
2	C₂²×C₄×Dic₂₆₃	C₂	1	1	yes	maximal, characteristic
3	C₂×C₂₆₃⋊#95	C₂	1	4	yes	maximal
4	C₂²×C₄⋊C₄×C₂₆₃	C₂	1	1	yes	maximal, characteristic
5	C₂³×Dic₂₆₃	C₂²	1	2	yes	—
6	C₂³×Dic₂₆₃	C₂²	1	2	yes	—
7	C₂×C₄⋊Dic₂₆₃	—	2	2	no	—
8	C₂×C₄⋊Dic₂₆₃	—	2	2	no	—
9	C₂×C₄⋊Dic₂₆₃	C₄	1	4	yes	complemented
10	C₂×C₄×Dic₂₆₃	—	2	2	no	—
11	C₂×C₄×Dic₂₆₃	—	2	2	no	—
12	C₂×C₄×Dic₂₆₃	C₄	1	4	yes	complemented
13	C₂₆₃⋊#95	—	2	8	no	—
14	C₂³×C₄×C₂₆₃	C₂²	1	2	yes	—
15	C₂³×C₄×C₂₆₃	C₂²	1	1	yes	characteristic
16	C₂×C₄⋊C₄×C₂₆₃	—	2	2	no	—
17	C₂×C₄⋊C₄×C₂₆₃	—	2	2	no	—
18	C₂×C₄⋊C₄×C₂₆₃	C₄	1	4	yes	complemented
19	C₂²×Dic₂₆₃	—	2	2	no	—
20	C₂²×Dic₂₆₃	—	2	2	no	—
21	C₂²×Dic₂₆₃	—	2	4	no	—
22	C₂²×Dic₂₆₃	—	2	2	no	—
23	C₂²×Dic₂₆₃	—	2	2	no	—
24	C₂²×Dic₂₆₃	—	2	2	no	—
25	C₂²×Dic₂₆₃	—	2	2	no	—
26	C₂²×Dic₂₆₃	—	2	4	no	—
27	C₂²×Dic₂₆₃	C₂×C₄	1	4	yes	—
28	C₂²×Dic₂₆₃	C₂×C₄	1	4	yes	—
29	C₂²×Dic₂₆₃	D₄	1	4	yes	—
30	C₂²×Dic₂₆₃	Q₈	1	4	yes	—
31	C₄⋊Dic₂₆₃	—	2	8	no	—
32	C₄×Dic₂₆₃	—	2	8	no	—
33	C₂⁴×C₂₆₃	C₂³	1	1	yes	characteristic
34	C₂²×C₄×C₂₆₃	—	2	4	no	—
35	C₂²×C₄×C₂₆₃	—	2	2	no	—
36	C₂²×C₄×C₂₆₃	—	2	2	no	—
37	C₂²×C₄×C₂₆₃	—	2	4	no	—
38	C₂²×C₄×C₂₆₃	C₂×C₄	1	2	yes	—
39	C₂²×C₄×C₂₆₃	C₂×C₄	1	2	yes	—
40	C₂²×C₄×C₂₆₃	C₂×C₄	1	4	yes	—
41	C₂²×C₄×C₂₆₃	C₂×C₄	1	2	yes	—
42	C₂²×C₄×C₂₆₃	D₄	1	2	yes	—
43	C₂²×C₄×C₂₆₃	D₄	1	2	yes	—
44	C₂²×C₄×C₂₆₃	D₄	1	2	yes	—
45	C₂²×C₄×C₂₆₃	Q₈	1	2	yes	—
46	C₄⋊C₄×C₂₆₃	—	2	8	no	—
47	C₂×Dic₂₆₃	—	2	4	no	—
48	C₂×Dic₂₆₃	—	2	4	no	—
49	C₂×Dic₂₆₃	—	2	4	no	—
50	C₂×Dic₂₆₃	—	2	4	no	—
51	C₂×Dic₂₆₃	—	2	4	no	—
52	C₂×Dic₂₆₃	—	2	4	no	—
53	C₂×Dic₂₆₃	—	2	4	no	—
54	C₂×Dic₂₆₃	—	2	8	no	—
55	C₂×Dic₂₆₃	—	4	4	no	—
56	C₂×Dic₂₆₃	—	4	2	no	—
57	C₂×Dic₂₆₃	—	4	2	no	—
58	C₂×Dic₂₆₃	C₄⋊C₄	1	8	yes	complemented
59	C₂³×C₂₆₃	C₂²×C₄	1	1	yes	characteristic
60	C₂³×C₂₆₃	C₂²×C₄	1	1	yes	characteristic
61	C₂³×C₂₆₃	C₂²×C₄	1	1	yes	characteristic
62	C₂³×C₂₆₃	C₂×D₄	1	2	yes	—
63	C₂³×C₂₆₃	C₂×D₄	1	1	yes	characteristic
64	C₂³×C₂₆₃	C₂×D₄	1	1	yes	characteristic
65	C₂³×C₂₆₃	C₂×D₄	1	1	yes	characteristic
66	C₂³×C₂₆₃	C₂×Q₈	1	2	yes	—
67	C₂³×C₂₆₃	C₂×Q₈	1	1	yes	characteristic
68	C₂³×C₂₆₃	#41	1	2	yes	—
69	C₂³×C₂₆₃	#41	1	2	yes	—
70	C₂×C₄×C₂₆₃	—	2	4	no	—
71	C₂×C₄×C₂₆₃	—	2	8	no	—
72	C₂×C₄×C₂₆₃	—	4	4	no	—
73	C₂×C₄×C₂₆₃	—	4	2	no	—
74	C₂×C₄×C₂₆₃	—	4	2	no	—
75	C₂×C₄×C₂₆₃	C₄²	1	4	yes	—
76	C₂×C₄×C₂₆₃	C₂²⋊C₄	1	4	yes	—
77	C₂×C₄×C₂₆₃	C₂²⋊C₄	1	4	yes	—
78	C₂×C₄×C₂₆₃	C₂²⋊C₄	1	4	yes	—
79	C₂×C₄×C₂₆₃	C₄⋊C₄	1	4	yes	—
80	C₂×C₄×C₂₆₃	C₄⋊C₄	1	4	yes	—
81	C₂×C₄×C₂₆₃	C₄⋊C₄	1	4	yes	—
82	C₂²×C₂₆₃	C₂×C₄²	1	1	yes	characteristic
83	C₂²×C₂₆₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
84	C₂²×C₂₆₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
85	C₂²×C₂₆₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
86	C₂²×C₂₆₃	C₂×C₄⋊C₄	1	2	yes	—
87	C₂²×C₂₆₃	C₂×C₄⋊C₄	1	1	yes	characteristic
88	C₂²×C₂₆₃	C₂×C₄⋊C₄	1	1	yes	characteristic
89	C₂²×C₂₆₃	C₂×C₄⋊C₄	1	1	yes	characteristic
90	C₂²×C₂₆₃	#117	1	2	yes	—
91	C₂²×C₂₆₃	C₄×D₄	1	2	yes	—
92	C₂²×C₂₆₃	C₄×D₄	1	2	yes	—
93	C₂²×C₂₆₃	C₄×Q₈	1	2	yes	—
94	C₂²×C₂₆₃	C₄×Q₈	1	2	yes	—
95	C₂²×C₂₆₃	#121	1	2	yes	—
96	C₂²×C₂₆₃	#121	1	2	yes	—
97	C₂²×C₂₆₃	C₂²⋊Q₈	1	2	yes	—
98	C₂²×C₂₆₃	C₂²⋊Q₈	1	2	yes	—
99	C₂²×C₂₆₃	#124	1	1	yes	characteristic
100	C₂²×C₂₆₃	#124	1	1	yes	characteristic
101	C₂²×C₂₆₃	#125	1	2	yes	—
102	C₂²×C₂₆₃	#127	1	1	yes	characteristic
103	C₂²×C₂₆₃	C₄⋊Q₈	1	1	yes	characteristic
104	C₂²×C₂₆₃	C₄⋊Q₈	1	2	yes	—
105	C₄×C₂₆₃	—	4	4	no	—
106	C₄×C₂₆₃	#95	1	8	yes	complemented
107	Dic₂₆₃	—	2	8	no	—
108	Dic₂₆₃	—	4	4	no	—
109	C₂×C₂₆₃	C₂×#95	1	1	yes	characteristic
110	C₂×C₂₆₃	C₄×C₄⋊C₄	1	2	yes	—
111	C₂×C₂₆₃	#380	1	2	yes	—
112	C₂×C₂₆₃	#382	1	1	yes	characteristic
113	C₂×C₂₆₃	#384	1	1	yes	characteristic
114	C₂×C₂₆₃	#389	1	2	yes	—
115	C₂×C₂₆₃	#389	1	2	yes	—
116	C₂×C₂₆₃	#390	1	1	yes	characteristic
117	C₂×C₂₆₃	#390	1	1	yes	characteristic
118	C₂×C₂₆₃	#391	1	1	yes	characteristic
119	C₂×C₂₆₃	#391	1	1	yes	characteristic
120	C₂₆₃	C₄⋊#95	1	1	yes	characteristic, complemented, Sylow
121	C₄⋊#95	—	263	1	no	maximal, Sylow
122	C₂²×C₄⋊C₄	—	263	1	no	—
123	C₂²×C₄⋊C₄	D₂₆₃	1	1	yes	characteristic
124	C₂²×C₄²	—	263	1	no	—
125	C₂×#95	—	263	4	no	—
126	C₂³×C₄	—	263	2	no	—
127	C₂³×C₄	—	263	2	no	—
128	C₂³×C₄	C₂×D₂₆₃	1	2	yes	—
129	C₂³×C₄	C₂×D₂₆₃	1	1	yes	characteristic
130	C₂×C₄⋊C₄	—	2	2	no	—
131	C₂×C₄⋊C₄	—	2	2	no	—
132	C₂×C₄⋊C₄	—	263	4	no	—
133	C₂×C₄⋊C₄	—	526	2	no	—
134	C₂×C₄⋊C₄	—	526	2	no	—
135	C₂×C₄⋊C₄	Dic₂₆₃	1	4	yes	complemented
136	C₂×C₄²	—	263	4	no	—
137	C₂×C₄²	—	526	2	no	—
138	C₂×C₄²	—	526	2	no	—
139	#95	—	526	8	no	—
140	C₂⁴	C₂²×D₂₆₃	1	1	yes	characteristic
141	C₂²×C₄	—	2	4	no	—
142	C₂²×C₄	—	2	2	no	—
143	C₂²×C₄	—	2	2	no	—
144	C₂²×C₄	—	2	4	no	—
145	C₂²×C₄	—	263	4	no	—
146	C₂²×C₄	—	263	4	no	—
147	C₂²×C₄	—	263	4	no	—
148	C₂²×C₄	—	263	4	no	—
149	C₂²×C₄	—	526	2	no	—
150	C₂²×C₄	—	526	2	no	—
151	C₂²×C₄	—	526	2	no	—
152	C₂²×C₄	—	526	2	no	—
153	C₂²×C₄	—	526	4	no	—
154	C₂²×C₄	—	526	2	no	—
155	C₂²×C₄	—	526	2	no	—
156	C₂²×C₄	—	526	4	no	—
157	C₂²×C₄	C₄×D₂₆₃	1	2	yes	—
158	C₂²×C₄	C₄×D₂₆₃	1	2	yes	—
159	C₂²×C₄	C₂×Dic₂₆₃	1	4	yes	—
160	C₂²×C₄	C₂×Dic₂₆₃	1	2	yes	—
161	C₂²×C₄	C₂²⋊D₂₆₃	1	2	yes	—
162	C₂²×C₄	C₂²⋊D₂₆₃	1	2	yes	—
163	C₂²×C₄	D₁₀₅₂	1	2	yes	—
164	C₂²×C₄	Dic₅₂₆	1	2	yes	—
165	C₄⋊C₄	—	2	8	no	—
166	C₄⋊C₄	—	526	8	no	—
167	C₄²	—	526	8	no	—
168	C₂³	C₂×C₄×D₂₆₃	1	1	yes	characteristic
169	C₂³	C₂×C₄×D₂₆₃	1	1	yes	characteristic
170	C₂³	C₂²×Dic₂₆₃	1	1	yes	characteristic
171	C₂³	D₄×D₂₆₃	1	2	yes	—
172	C₂³	C₂×C₂²⋊D₂₆₃	1	1	yes	characteristic
173	C₂³	C₂×C₂²⋊D₂₆₃	1	1	yes	characteristic
174	C₂³	C₂×D₁₀₅₂	1	1	yes	characteristic
175	C₂³	Q₈×D₂₆₃	1	2	yes	—
176	C₂³	C₂×Dic₅₂₆	1	1	yes	characteristic
177	C₂³	#16862375	1	2	yes	—
178	C₂³	#16862377	1	2	yes	—
179	C₂×C₄	—	2	4	no	—
180	C₂×C₄	—	2	8	no	—
181	C₂×C₄	—	4	4	no	—
182	C₂×C₄	—	4	2	no	—
183	C₂×C₄	—	4	2	no	—
184	C₂×C₄	—	263	8	no	—
185	C₂×C₄	—	526	4	no	—
186	C₂×C₄	—	526	4	no	—
187	C₂×C₄	—	526	4	no	—
188	C₂×C₄	—	526	4	no	—
189	C₂×C₄	—	526	4	no	—
190	C₂×C₄	—	526	4	no	—
191	C₂×C₄	—	526	4	no	—
192	C₂×C₄	—	526	8	no	—
193	C₂×C₄	—	1052	4	no	—
194	C₂×C₄	—	1052	2	no	—
195	C₂×C₄	—	1052	2	no	—
196	C₂×C₄	C₄×Dic₂₆₃	1	4	yes	—
197	C₂×C₄	(C₂×D₂₆₃)⋊C₄	1	4	yes	—
198	C₂×C₄	(C₂×D₂₆₃)⋊C₄	1	4	yes	—
199	C₂×C₄	C₂²⋊Dic₂₆₃	1	4	yes	—
200	C₂×C₄	Dic₂₆₃⋊C₄	1	4	yes	—
201	C₂×C₄	Dic₂₆₃⋊C₄	1	4	yes	—
202	C₂×C₄	C₄⋊Dic₂₆₃	1	4	yes	—
203	C₂²	C₂×C₄×Dic₂₆₃	1	1	yes	characteristic
204	C₂²	C₂×(C₂×D₂₆₃)⋊C₄	1	1	yes	characteristic
205	C₂²	C₂×(C₂×D₂₆₃)⋊C₄	1	1	yes	characteristic
206	C₂²	C₂×C₂²⋊Dic₂₆₃	1	1	yes	characteristic
207	C₂²	C₄⋊C₄×D₂₆₃	1	2	yes	—
208	C₂²	C₂×Dic₂₆₃⋊C₄	1	1	yes	characteristic
209	C₂²	C₂×Dic₂₆₃⋊C₄	1	1	yes	characteristic
210	C₂²	C₂×C₄⋊Dic₂₆₃	1	1	yes	characteristic
211	C₂²	#22554428	1	2	yes	—
212	C₂²	#22554432	1	2	yes	—
213	C₂²	D₄×Dic₂₆₃	1	2	yes	—
214	C₂²	#22554435	1	2	yes	—
215	C₂²	Q₈×Dic₂₆₃	1	2	yes	—
216	C₂²	#22554441	1	2	yes	—
217	C₂²	#22554445	1	2	yes	—
218	C₂²	#22554446	1	2	yes	—
219	C₂²	#22554450	1	2	yes	—
220	C₂²	#22554457	1	1	yes	characteristic
221	C₂²	#22554459	1	1	yes	characteristic
222	C₂²	#22554461	1	2	yes	—
223	C₂²	#22554466	1	1	yes	characteristic
224	C₂²	#22554468	1	1	yes	characteristic
225	C₂²	#22554469	1	2	yes	—
226	C₄	—	4	4	no	—
227	C₄	—	526	8	no	—
228	C₄	—	1052	4	no	—
229	C₄	C₂₆₃⋊#95	1	8	yes	complemented
230	C₂	C₂×C₂₆₃⋊#95	1	1	yes	characteristic
231	C₂	C₄⋊C₄×Dic₂₆₃	1	2	yes	—
232	C₂	#30681795	1	2	yes	—
233	C₂	#30681803	1	1	yes	characteristic
234	C₂	#30681807	1	1	yes	characteristic
235	C₂	#30681834	1	2	yes	—
236	C₂	#30681837	1	2	yes	—
237	C₂	#30681839	1	1	yes	characteristic
238	C₂	#30681841	1	1	yes	characteristic
239	C₂	#30681843	1	1	yes	characteristic
240	C₂	#30681845	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂⁴	C₂²×D₂₆₃
commutator	C₂²×C₂₆₃	C₂×C₄²
Fitting	C₂²×C₄⋊C₄×C₂₆₃	C₂
socle, prime radical, cosocle	C₂⁴×C₂₆₃	C₂³
hypercenter	C₂²×C₄⋊C₄	D₂₆₃
hypocenter	C₂₆₃	C₄⋊#95
LCS	G ▷ C₂²×C₂₆₃ ▷ C₂₆₃
UCS	C₁ ◁ C₂⁴ ◁ C₂²×C₄⋊C₄

Complements

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

N	H ≅ G/N	type
9. C₂×C₄⋊Dic₂₆₃	C₄	semidirect
12. C₂×C₄×Dic₂₆₃	C₄	semidirect
18. C₂×C₄⋊C₄×C₂₆₃	C₄	semidirect
58. C₂×Dic₂₆₃	C₄⋊C₄	semidirect
106. C₄×C₂₆₃	#95	semidirect
120. C₂₆₃	C₄⋊#95	semidirect
135. C₂×C₄⋊C₄	Dic₂₆₃	semidirect
229. C₄	C₂₆₃⋊#95	semidirect

Maximal subgroups

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

subgroup	index
C₂²×C₄⋊Dic₂₆₃	2
C₂²×C₄×Dic₂₆₃	2
C₂×C₂₆₃⋊#95	2
C₂²×C₄⋊C₄×C₂₆₃	2
C₄⋊#95	263