Subgroups — #30682668

Order 16832.

212 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22554469	C₂	1	4	yes	maximal
2	#22554468	C₂	1	4	yes	maximal
3	C₄×Dic₅₂₆	C₂	1	2	yes	maximal
4	#22554435	C₂	1	4	yes	maximal
5	C₄⋊Q₈×C₂₆₃	C₂	1	1	yes	maximal, characteristic
6	C₂×Dic₅₂₆	C₂²	1	4	yes	—
7	C₂×Dic₅₂₆	C₂²	1	2	yes	—
8	C₄⋊Dic₂₆₃	C₂²	1	2	yes	—
9	Dic₂₆₃⋊C₄	C₂²	1	8	yes	—
10	Dic₂₆₃⋊C₄	C₂²	1	4	yes	—
11	C₄×Dic₂₆₃	C₂²	1	4	yes	—
12	C₄×Dic₂₆₃	C₂²	1	4	yes	—
13	C₂×Q₈×C₂₆₃	C₂²	1	2	yes	—
14	C₄⋊C₄×C₂₆₃	C₂²	1	4	yes	—
15	C₄²×C₂₆₃	C₂²	1	1	yes	characteristic
16	Dic₅₂₆	—	4	4	no	—
17	Dic₅₂₆	Q₈	1	8	yes	complemented
18	C₂×Dic₂₆₃	C₂³	1	4	yes	—
19	C₂×Dic₂₆₃	C₂³	1	4	yes	—
20	Q₈×C₂₆₃	—	4	2	no	—
21	C₂×C₄×C₂₆₃	C₂³	1	1	yes	characteristic
22	C₂×C₄×C₂₆₃	C₂³	1	4	yes	—
23	C₂×C₄×C₂₆₃	C₂³	1	2	yes	—
24	C₂²×C₂₆₃	C₂⁴	1	1	yes	characteristic
25	C₄×C₂₆₃	—	2	1	no	—
26	C₄×C₂₆₃	—	2	4	no	—
27	C₄×C₂₆₃	C₂×Q₈	1	4	yes	—
28	Dic₂₆₃	—	2	4	no	—
29	Dic₂₆₃	C₂×Q₈	1	8	yes	—
30	C₂×C₂₆₃	C₂²×Q₈	1	2	yes	—
31	C₂×C₂₆₃	H(5,𝔽₂)	1	1	yes	characteristic
32	C₂₆₃	Q₈²	1	1	yes	characteristic, complemented, Sylow
33	Q₈²	—	263	1	no	maximal, Sylow
34	C₄⋊Q₈	—	263	4	no	—
35	C₄⋊Q₈	—	263	4	no	—
36	C₄⋊Q₈	D₂₆₃	1	1	yes	characteristic
37	C₄×Q₈	—	263	4	no	—
38	C₄×Q₈	—	263	2	no	—
39	C₂×Q₈	—	263	2	no	—
40	C₂×Q₈	—	263	4	no	—
41	C₂×Q₈	C₂×D₂₆₃	1	2	yes	—
42	C₄⋊C₄	—	263	8	no	—
43	C₄⋊C₄	—	263	4	no	—
44	C₄⋊C₄	—	263	2	no	—
45	C₄⋊C₄	C₂×D₂₆₃	1	4	yes	—
46	C₄²	—	263	4	no	—
47	C₄²	—	263	4	no	—
48	C₄²	C₂×D₂₆₃	1	1	yes	characteristic
49	Q₈	—	4	2	no	—
50	Q₈	—	263	8	no	—
51	Q₈	—	1052	4	no	—
52	C₂×C₄	—	263	4	no	—
53	C₂×C₄	—	263	4	no	—
54	C₂×C₄	C₂²×D₂₆₃	1	4	yes	—
55	C₂×C₄	C₂²×D₂₆₃	1	1	yes	characteristic
56	C₂×C₄	C₂²×D₂₆₃	1	2	yes	—
57	C₂²	C₂³×D₂₆₃	1	1	yes	characteristic
58	C₄	—	2	1	no	—
59	C₄	—	2	4	no	—
60	C₄	—	263	8	no	—
61	C₄	—	526	4	no	—
62	C₄	Q₈×D₂₆₃	1	4	yes	—
63	C₂	C₂×Q₈×D₂₆₃	1	2	yes	—
64	C₂	#22554515	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂²	C₂³×D₂₆₃
commutator, socle, prime radical, cosocle	C₂²×C₂₆₃	C₂⁴
Fitting	C₄⋊Q₈×C₂₆₃	C₂
hypercenter	C₄⋊Q₈	D₂₆₃
hypocenter	C₂₆₃	Q₈²
LCS	G ▷ C₂²×C₂₆₃ ▷ C₂₆₃
UCS	C₁ ◁ C₂² ◁ C₄⋊Q₈

Complements

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

N	H ≅ G/N	type
17. Dic₅₂₆	Q₈	semidirect
32. C₂₆₃	Q₈²	semidirect

Maximal subgroups

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

subgroup	index
#22554469	2
#22554468	2
C₄×Dic₅₂₆	2
#22554435	2
C₄⋊Q₈×C₂₆₃	2
Q₈²	263