Subgroups — #28412397

Order 14656.

58 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20301063	C₂	1	1	yes	maximal, characteristic, complemented
2	#20301056	C₂	1	1	yes	maximal, characteristic, complemented
3	#20301007	C₂	1	1	yes	maximal, characteristic
4	C₄⋊#11745021	—	2	1	no	—
5	#16814669	—	2	1	no	—
6	#16814665	—	2	1	no	—
7	Dic₉₁₆	C₄	1	1	yes	characteristic, complemented
8	#16814654	—	2	1	no	—
9	C₈×D₂₂₉	C₂²	1	1	yes	characteristic
10	D₈×C₂₂₉	C₄	1	1	yes	characteristic, complemented
11	C₂×#11745021	—	4	1	no	—
12	Dic₄₅₈	—	2	1	no	—
13	C₂²⋊D₂₂₉	—	4	1	no	—
14	C₂×Dic₂₂₉	—	4	1	no	—
15	C₄×D₂₂₉	D₄	1	1	yes	characteristic
16	#14270080	D₄	1	1	yes	characteristic
17	D₄×C₂₂₉	—	2	1	no	—
18	C₈×C₂₂₉	C₂×C₄	1	1	yes	characteristic
19	C₂²×C₂₂₉	—	4	1	no	—
20	C₂×D₂₂₉	D₈	1	1	yes	characteristic
21	#11745021	—	8	1	no	—
22	C₄×C₂₂₉	C₂²⋊C₄	1	1	yes	characteristic
23	Dic₂₂₉	—	4	1	no	—
24	Dic₂₂₉	SD₁₆	1	1	yes	characteristic
25	C₂×C₂₂₉	—	8	1	no	—
26	C₂×C₂₂₉	#102	1	1	yes	characteristic
27	D₂₂₉	—	2	1	no	—
28	C₂₂₉	#360	1	1	yes	characteristic, complemented, Sylow
29	#360	—	229	1	no	maximal, Sylow
30	#135	—	229	1	no	—
31	M₃₂	—	229	1	no	—
32	#106	—	229	1	no	—
33	#41	—	458	1	no	—
34	Q₁₆	—	229	1	no	—
35	SD₁₆	—	458	1	no	—
36	D₈	#11745021	1	1	yes	characteristic, complemented
37	C₂×C₈	—	229	1	no	—
38	C₄⋊C₄	—	458	1	no	—
39	C₁₆	—	458	1	no	—
40	Q₈	—	458	1	no	—
41	D₄	—	2	1	no	—
42	D₄	—	916	1	no	—
43	C₂×C₄	—	916	1	no	—
44	C₂×C₄	—	916	1	no	—
45	C₂×C₄	—	229	1	no	—
46	C₈	—	229	1	no	—
47	C₈	C₂×#11745021	1	1	yes	characteristic
48	C₂²	—	4	1	no	—
49	C₂²	—	229	1	no	—
50	C₄	—	1832	1	no	—
51	C₄	—	916	1	no	—
52	C₄	—	229	1	no	—
53	C₄	C₂²⋊#11745021	1	1	yes	characteristic
54	C₂	—	8	1	no	—
55	C₂	—	458	1	no	—
56	C₂	#20301050	1	1	yes	characteristic

	subgroup	G/H
center	C₂	#20301050
commutator	C₈×C₂₂₉	C₂×C₄
Frattini	C₈	C₂×#11745021
Fitting	D₈×C₂₂₉	C₄
socle	C₂×C₂₂₉	#102
prime radical	#20301007	C₂
cosocle	C₈×D₂₂₉	C₂²
hypercenter	D₈	#11745021
hypocenter	C₂₂₉	#360
LCS	G ▷ C₈×C₂₂₉ ▷ C₄×C₂₂₉ ▷ C₂×C₂₂₉ ▷ C₂₂₉
UCS	C₁ ◁ C₂ ◁ C₄ ◁ C₈ ◁ D₈

Complements

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

N	H ≅ G/N	type
1. #20301063	C₂	semidirect
2. #20301056	C₂	semidirect
7. Dic₉₁₆	C₄	semidirect
10. D₈×C₂₂₉	C₄	semidirect
28. C₂₂₉	#360	semidirect
36. D₈	#11745021	semidirect

Maximal subgroups

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

subgroup	index
#20301063	2
#20301056	2
#20301007	2
#360	229