Subgroups — #30682344

Order 16832.

90 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22554462	C₂	1	1	yes	maximal, characteristic
2	#22554403	C₂	1	2	yes	maximal
3	#22554401	C₂	1	2	yes	maximal
4	#22554381	C₂	1	1	yes	maximal, characteristic
5	C₄⋊Q₈×C₂₆₃	C₂	1	1	yes	maximal, characteristic
6	C₂×C₂₆₃⋊C₈	C₂²	1	2	yes	—
7	C₄⋊Dic₂₆₃	C₂²	1	1	yes	characteristic
8	C₄⋊Dic₂₆₃	C₂²	1	1	yes	characteristic
9	Dic₂₆₃⋊C₄	—	2	1	no	—
10	Dic₂₆₃⋊C₄	—	2	1	no	—
11	C₂×Q₈×C₂₆₃	—	2	1	no	—
12	C₄⋊C₄×C₂₆₃	—	2	1	no	—
13	C₄⋊C₄×C₂₆₃	C₂²	1	2	yes	—
14	C₄²×C₂₆₃	C₂²	1	1	yes	characteristic
15	C₂×Dic₂₆₃	—	2	1	no	—
16	C₂×Dic₂₆₃	—	2	1	no	—
17	C₂₆₃⋊C₈	Q₈	1	4	yes	complemented
18	Q₈×C₂₆₃	—	4	2	no	—
19	C₂×C₄×C₂₆₃	—	2	2	no	—
20	C₂×C₄×C₂₆₃	D₄	1	1	yes	characteristic
21	C₂×C₄×C₂₆₃	D₄	1	1	yes	characteristic
22	C₂×C₄×C₂₆₃	C₂³	1	1	yes	characteristic
23	C₂²×C₂₆₃	C₂×D₄	1	1	yes	characteristic
24	C₄×C₂₆₃	—	2	1	no	—
25	C₄×C₂₆₃	—	2	1	no	—
26	C₄×C₂₆₃	—	4	2	no	—
27	C₄×C₂₆₃	C₂×Q₈	1	2	yes	—
28	Dic₂₆₃	—	4	1	no	—
29	Dic₂₆₃	—	4	1	no	—
30	C₂×C₂₆₃	C₄⋊Q₈	1	1	yes	characteristic
31	C₂×C₂₆₃	AGL(1,ℤ/8)	1	1	yes	characteristic
32	C₂×C₂₆₃	#137	1	1	yes	characteristic
33	C₂₆₃	#501	1	1	yes	characteristic, complemented, Sylow
34	#501	—	263	1	no	maximal, Sylow
35	C₄⋊Q₈	D₂₆₃	1	1	yes	characteristic
36	#125	—	263	1	no	—
37	#107	—	263	2	no	—
38	#106	—	263	2	no	—
39	#97	—	263	1	no	—
40	C₂×Q₈	—	2	1	no	—
41	C₂×C₈	—	263	2	no	—
42	C₄⋊C₄	—	2	1	no	—
43	C₄⋊C₄	—	263	1	no	—
44	C₄⋊C₄	—	263	1	no	—
45	C₄⋊C₄	—	526	1	no	—
46	C₄⋊C₄	—	526	1	no	—
47	C₄⋊C₄	C₂×D₂₆₃	1	2	yes	—
48	C₄²	C₂×D₂₆₃	1	1	yes	characteristic
49	Q₈	—	4	2	no	—
50	C₂×C₄	—	2	2	no	—
51	C₂×C₄	—	526	1	no	—
52	C₂×C₄	—	526	1	no	—
53	C₂×C₄	C₂²⋊D₂₆₃	1	1	yes	characteristic
54	C₂×C₄	C₂²⋊D₂₆₃	1	1	yes	characteristic
55	C₂×C₄	C₂²×D₂₆₃	1	1	yes	characteristic
56	C₈	—	263	4	no	—
57	C₂²	C₂×C₂²⋊D₂₆₃	1	1	yes	characteristic
58	C₄	—	2	1	no	—
59	C₄	—	2	1	no	—
60	C₄	—	4	2	no	—
61	C₄	—	1052	1	no	—
62	C₄	—	1052	1	no	—
63	C₄	Q₈×D₂₆₃	1	2	yes	—
64	C₂	#22554468	1	1	yes	characteristic
65	C₂	#22554496	1	1	yes	characteristic
66	C₂	#22554501	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×C₂²⋊D₂₆₃
commutator, cosocle	C₂×C₄×C₂₆₃	C₂³
Frattini	C₂×C₄	C₂²×D₂₆₃
Fitting	C₄⋊Q₈×C₂₆₃	C₂
socle, prime radical	C₂²×C₂₆₃	C₂×D₄
hypercenter	C₄⋊Q₈	D₂₆₃
hypocenter	C₂₆₃	#501
LCS	G ▷ C₂×C₄×C₂₆₃ ▷ C₂×C₂₆₃ ▷ C₂₆₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄⋊Q₈

Complements

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

N	H ≅ G/N	type
17. C₂₆₃⋊C₈	Q₈	semidirect
33. C₂₆₃	#501	semidirect

Maximal subgroups

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

subgroup	index
#22554462	2
#22554403	2
#22554401	2
#22554381	2
C₄⋊Q₈×C₂₆₃	2
#501	263