Subgroups — #28383393

Order 14272.

62 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20297244	C₂	1	1	yes	maximal, characteristic, complemented
2	#20297240	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×D₈×C₂₂₃	C₂	1	1	yes	maximal, characteristic
4	#16813154	—	2	1	no	—
5	C₂₂₃⋊C₁₆	—	2	1	no	—
6	C₂×D₄×C₂₂₃	—	2	1	no	—
7	D₈×C₂₂₃	—	2	1	no	—
8	D₈×C₂₂₃	C₄	1	2	yes	—
9	C₂×C₈×C₂₂₃	C₂²	1	1	yes	characteristic
10	C₂₂₃⋊C₈	—	4	1	no	—
11	C₂³×C₂₂₃	—	4	1	no	—
12	D₄×C₂₂₃	—	2	2	no	—
13	D₄×C₂₂₃	—	4	1	no	—
14	C₂×C₄×C₂₂₃	D₄	1	1	yes	characteristic
15	C₈×C₂₂₃	C₂×C₄	1	1	yes	characteristic
16	C₈×C₂₂₃	D₄	1	1	yes	characteristic
17	C₂²×C₂₂₃	—	4	2	no	—
18	C₂²×C₂₂₃	—	8	2	no	—
19	C₂²×C₂₂₃	SD₁₆	1	1	yes	characteristic
20	C₄×C₂₂₃	C₂²⋊C₄	1	1	yes	characteristic
21	C₄×C₂₂₃	D₈	1	1	yes	characteristic
22	C₂×C₂₂₃	—	2	1	no	—
23	C₂×C₂₂₃	—	8	2	no	—
24	C₂×C₂₂₃	#102	1	1	yes	characteristic
25	C₂₂₃	#361	1	1	yes	characteristic, complemented, Sylow
26	#361	—	223	1	no	maximal, Sylow
27	C₂×D₈	D₂₂₃	1	1	yes	characteristic
28	M₃₂	—	223	1	no	—
29	#108	—	223	1	no	—
30	C₂×D₄	—	2	1	no	—
31	D₈	—	2	1	no	—
32	D₈	Dic₂₂₃	1	2	yes	—
33	M₁₆	—	446	1	no	—
34	C₂×C₈	C₂×D₂₂₃	1	1	yes	characteristic
35	C₁₆	—	446	1	no	—
36	C₂³	—	4	1	no	—
37	D₄	—	2	2	no	—
38	D₄	—	4	1	no	—
39	C₂×C₄	C₂²⋊D₂₂₃	1	1	yes	characteristic
40	C₈	—	892	1	no	—
41	C₈	C₂×Dic₂₂₃	1	1	yes	characteristic
42	C₈	C₂²⋊D₂₂₃	1	1	yes	characteristic
43	C₂²	—	4	2	no	—
44	C₂²	—	8	2	no	—
45	C₂²	#16813158	1	1	yes	characteristic
46	C₄	C₂²⋊Dic₂₂₃	1	1	yes	characteristic
47	C₄	#16813155	1	1	yes	characteristic
48	C₂	—	2	1	no	—
49	C₂	—	8	2	no	—
50	C₂	#20297225	1	1	yes	characteristic

	subgroup	G/H
center	C₂	#20297225
commutator	C₈×C₂₂₃	C₂×C₄
Frattini	C₂×C₈	C₂×D₂₂₃
Fitting, prime radical	C₂×D₈×C₂₂₃	C₂
socle	C₂×C₂₂₃	#102
cosocle	C₂×C₈×C₂₂₃	C₂²
hypercenter	C₂×D₈	D₂₂₃
hypocenter	C₂₂₃	#361
LCS	G ▷ C₈×C₂₂₃ ▷ C₄×C₂₂₃ ▷ C₂×C₂₂₃ ▷ C₂₂₃
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₂×C₈ ◁ C₂×D₈

Complements

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

N	H ≅ G/N	type
1. #20297244	C₂	semidirect
2. #20297240	C₂	semidirect
25. C₂₂₃	#361	semidirect

Maximal subgroups

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

subgroup	index
#20297244	2
#20297240	2
C₂×D₈×C₂₂₃	2
#361	223