Subgroups — #28383333

Order 14272.

54 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄⋊(C₂₂₃⋊C₈)	C₂	1	2	yes	maximal
2	C₄×C₈×C₂₂₃	C₂	1	1	yes	maximal, characteristic
3	C₂×C₂₂₃⋊C₈	—	2	2	no	—
4	C₂×C₈×C₂₂₃	C₄	1	1	yes	characteristic
5	C₂×C₈×C₂₂₃	C₄	1	1	yes	characteristic
6	C₄²×C₂₂₃	C₂²	1	1	yes	characteristic
7	C₂₂₃⋊C₈	—	4	2	no	—
8	C₂×C₄×C₂₂₃	C₂×C₄	1	1	yes	characteristic
9	C₂×C₄×C₂₂₃	D₄	1	1	yes	characteristic
10	C₂×C₄×C₂₂₃	Q₈	1	1	yes	characteristic
11	C₈×C₂₂₃	—	2	1	no	—
12	C₈×C₂₂₃	C₈	1	2	yes	complemented
13	C₂²×C₂₂₃	C₄⋊C₄	1	1	yes	characteristic
14	C₄×C₂₂₃	—	2	1	no	—
15	C₄×C₂₂₃	C₂×C₈	1	1	yes	characteristic
16	C₄×C₂₂₃	M₁₆	1	1	yes	characteristic
17	C₄×C₂₂₃	D₈	1	1	yes	characteristic
18	C₄×C₂₂₃	Q₁₆	1	1	yes	characteristic
19	C₂×C₂₂₃	C₄⋊C₈	1	1	yes	characteristic
20	C₂×C₂₂₃	#107	1	1	yes	characteristic
21	C₂×C₂₂₃	#108	1	1	yes	characteristic
22	C₂₂₃	#335	1	1	yes	characteristic, complemented, Sylow
23	#335	—	223	1	no	maximal, Sylow
24	C₄⋊C₈	—	223	2	no	—
25	C₄×C₈	D₂₂₃	1	1	yes	characteristic
26	C₂×C₈	—	446	2	no	—
27	C₂×C₈	Dic₂₂₃	1	1	yes	characteristic
28	C₂×C₈	Dic₂₂₃	1	1	yes	characteristic
29	C₄²	C₂×D₂₂₃	1	1	yes	characteristic
30	C₂×C₄	C₂×Dic₂₂₃	1	1	yes	characteristic
31	C₂×C₄	D₈₉₂	1	1	yes	characteristic
32	C₂×C₄	Dic₄₄₆	1	1	yes	characteristic
33	C₈	—	2	1	no	—
34	C₈	—	892	2	no	—
35	C₈	C₂₂₃⋊C₈	1	2	yes	complemented
36	C₂²	C₄⋊Dic₂₂₃	1	1	yes	characteristic
37	C₄	—	2	1	no	—
38	C₄	C₂×C₂₂₃⋊C₈	1	1	yes	characteristic
39	C₄	#16813154	1	1	yes	characteristic
40	C₄	D₁₇₈₄	1	1	yes	characteristic
41	C₄	Dic₈₉₂	1	1	yes	characteristic
42	C₂	C₄⋊(C₂₂₃⋊C₈)	1	1	yes	characteristic
43	C₂	#20297238	1	1	yes	characteristic
44	C₂	#20297240	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	D₈₉₂
commutator	C₄×C₂₂₃	C₂×C₈
Frattini	C₄²	C₂×D₂₂₃
Fitting	C₄×C₈×C₂₂₃	C₂
socle, prime radical	C₂²×C₂₂₃	C₄⋊C₄
cosocle	C₄²×C₂₂₃	C₂²
hypercenter	C₄×C₈	D₂₂₃
hypocenter	C₂₂₃	#335
LCS	G ▷ C₄×C₂₂₃ ▷ C₂×C₂₂₃ ▷ C₂₂₃
UCS	C₁ ◁ C₂×C₄ ◁ C₄² ◁ C₄×C₈

Complements

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

N	H ≅ G/N	type
12. C₈×C₂₂₃	C₈	semidirect
22. C₂₂₃	#335	semidirect
35. C₈	C₂₂₃⋊C₈	semidirect

Maximal subgroups

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

subgroup	index
C₄⋊(C₂₂₃⋊C₈)	2
C₄×C₈×C₂₂₃	2
#335	223