Subgroups — #30681663

Order 16832.

78 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×Dic₂₆₃⋊C₄	C₂	1	1	yes	maximal, characteristic
2	C₂²⋊(C₂₆₃⋊C₈)	C₂	1	1	yes	maximal, characteristic
3	C₂²⋊C₈×C₂₆₃	C₂	1	1	yes	maximal, characteristic
4	C₂²×Dic₂₆₃	C₄	1	1	yes	characteristic
5	C₂²×Dic₂₆₃	C₄	1	1	yes	characteristic
6	C₂×C₂₆₃⋊C₈	—	2	1	no	—
7	Dic₂₆₃⋊C₄	—	2	1	no	—
8	Dic₂₆₃⋊C₄	—	2	1	no	—
9	C₂²×C₄×C₂₆₃	C₂²	1	1	yes	characteristic
10	C₂×C₈×C₂₆₃	—	2	1	no	—
11	C₂×Dic₂₆₃	—	2	1	no	—
12	C₂×Dic₂₆₃	—	2	1	no	—
13	C₂×Dic₂₆₃	—	2	1	no	—
14	C₂×Dic₂₆₃	—	4	1	no	—
15	C₂×Dic₂₆₃	C₈	1	2	yes	complemented
16	C₂₆₃⋊C₈	—	4	1	no	—
17	C₂³×C₂₆₃	C₂×C₄	1	1	yes	characteristic
18	C₂×C₄×C₂₆₃	—	4	1	no	—
19	C₂×C₄×C₂₆₃	D₄	1	1	yes	characteristic
20	C₂×C₄×C₂₆₃	D₄	1	1	yes	characteristic
21	C₈×C₂₆₃	—	4	1	no	—
22	C₂²×C₂₆₃	—	2	1	no	—
23	C₂²×C₂₆₃	—	2	1	no	—
24	C₂²×C₂₆₃	C₂²⋊C₄	1	1	yes	characteristic
25	C₂²×C₂₆₃	C₂×C₈	1	1	yes	characteristic
26	C₂²×C₂₆₃	M₁₆	1	1	yes	characteristic
27	C₄×C₂₆₃	—	2	1	no	—
28	C₄×C₂₆₃	—	2	1	no	—
29	Dic₂₆₃	—	2	2	no	—
30	Dic₂₆₃	—	4	1	no	—
31	C₂×C₂₆₃	—	2	1	no	—
32	C₂×C₂₆₃	—	2	1	no	—
33	C₂×C₂₆₃	C₂²⋊C₈	1	1	yes	characteristic
34	C₂×C₂₆₃	#99	1	1	yes	characteristic
35	C₂×C₂₆₃	#101	1	1	yes	characteristic
36	C₂₆₃	#324	1	1	yes	characteristic, complemented, Sylow
37	#324	—	263	1	no	maximal, Sylow
38	C₂×C₄⋊C₄	—	263	1	no	—
39	C₂²⋊C₈	—	263	1	no	—
40	C₂²⋊C₈	D₂₆₃	1	1	yes	characteristic
41	C₂²×C₄	—	263	1	no	—
42	C₂²×C₄	—	263	1	no	—
43	C₂²×C₄	C₂×D₂₆₃	1	1	yes	characteristic
44	C₂×C₈	—	2	1	no	—
45	C₂×C₈	—	526	1	no	—
46	C₄⋊C₄	—	526	1	no	—
47	C₄⋊C₄	—	526	1	no	—
48	C₂³	C₄×D₂₆₃	1	1	yes	characteristic
49	C₂×C₄	—	4	1	no	—
50	C₂×C₄	—	263	2	no	—
51	C₂×C₄	—	526	1	no	—
52	C₂×C₄	—	526	1	no	—
53	C₂×C₄	—	526	1	no	—
54	C₂×C₄	—	1052	1	no	—
55	C₂×C₄	C₂²⋊D₂₆₃	1	1	yes	characteristic
56	C₂×C₄	D₁₀₅₂	1	1	yes	characteristic
57	C₈	—	4	1	no	—
58	C₈	—	1052	1	no	—
59	C₂²	—	2	1	no	—
60	C₂²	—	2	1	no	—
61	C₂²	(C₂×D₂₆₃)⋊C₄	1	1	yes	characteristic
62	C₂²	C₈×D₂₆₃	1	1	yes	characteristic
63	C₂²	#16862359	1	1	yes	characteristic
64	C₄	—	2	1	no	—
65	C₄	—	2	1	no	—
66	C₄	—	526	2	no	—
67	C₄	—	1052	1	no	—
68	C₂	—	2	1	no	—
69	C₂	—	2	1	no	—
70	C₂	(C₂×D₂₆₃)⋊C₈	1	1	yes	characteristic
71	C₂	#22554384	1	1	yes	characteristic
72	C₂	#22554388	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	(C₂×D₂₆₃)⋊C₄
commutator	C₂²×C₂₆₃	C₂×C₈
Frattini	C₂²×C₄	C₂×D₂₆₃
Fitting	C₂²⋊C₈×C₂₆₃	C₂
socle	C₂²×C₂₆₃	C₂²⋊C₄
prime radical	C₂³×C₂₆₃	C₂×C₄
cosocle	C₂²×C₄×C₂₆₃	C₂²
hypercenter	C₂²⋊C₈	D₂₆₃
hypocenter	C₂₆₃	#324
LCS	G ▷ C₂²×C₂₆₃ ▷ C₂×C₂₆₃ ▷ C₂₆₃
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂²⋊C₈

Complements

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

N	H ≅ G/N	type
15. C₂×Dic₂₆₃	C₈	semidirect
36. C₂₆₃	#324	semidirect

Maximal subgroups

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

subgroup	index
C₂×Dic₂₆₃⋊C₄	2
C₂²⋊(C₂₆₃⋊C₈)	2
C₂²⋊C₈×C₂₆₃	2
#324	263