Subgroups — #28412393

Order 14656.

66 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	D₂₂₉⋊C₁₆	C₂	1	1	yes	maximal, characteristic, complemented
2	#20301057	C₂	1	1	yes	maximal, characteristic, complemented
3	D₈×D₂₂₉	C₂	1	1	yes	maximal, characteristic
4	C₄⋊#11745021	—	2	1	no	—
5	#16814669	—	2	1	no	—
6	D₄×D₂₂₉	—	2	1	no	—
7	D₁₈₃₂	C₄	1	1	yes	characteristic, complemented
8	#16814651	—	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	C₂²×D₂₂₉	—	4	1	no	—
13	D₉₁₆	—	2	1	no	—
14	C₂²⋊D₂₂₉	—	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	—	4	1	no	—
21	C₂×D₂₂₉	—	8	1	no	—
22	C₂×D₂₂₉	—	8	1	no	—
23	C₂×D₂₂₉	D₈	1	1	yes	characteristic
24	#11745021	—	8	1	no	—
25	C₄×C₂₂₉	C₂²⋊C₄	1	1	yes	characteristic
26	Dic₂₂₉	SD₁₆	1	1	yes	characteristic
27	C₂×C₂₂₉	—	8	1	no	—
28	C₂×C₂₂₉	#102	1	1	yes	characteristic
29	D₂₂₉	—	8	1	no	—
30	D₂₂₉	D₁₆	1	1	yes	characteristic
31	D₂₂₉	SD₃₂	1	1	yes	characteristic
32	C₂₂₉	#357	1	1	yes	characteristic, complemented, Sylow
33	#357	—	229	1	no	maximal, Sylow
34	C₂×D₈	—	229	1	no	—
35	C₂×C₁₆	—	229	1	no	—
36	#107	—	229	1	no	—
37	C₂×D₄	—	458	1	no	—
38	D₈	—	458	1	no	—
39	D₈	—	229	1	no	—
40	D₈	#11745021	1	1	yes	characteristic, complemented
41	C₂×C₈	—	229	1	no	—
42	C₄⋊C₄	—	458	1	no	—
43	C₁₆	—	458	1	no	—
44	C₂³	—	916	1	no	—
45	D₄	—	2	1	no	—
46	D₄	—	916	1	no	—
47	D₄	—	458	1	no	—
48	C₂×C₄	—	916	1	no	—
49	C₂×C₄	—	229	1	no	—
50	C₈	—	229	1	no	—
51	C₈	C₂×#11745021	1	1	yes	characteristic
52	C₂²	—	4	1	no	—
53	C₂²	—	1832	1	no	—
54	C₂²	—	1832	1	no	—
55	C₂²	—	916	1	no	—
56	C₂²	—	229	1	no	—
57	C₄	—	1832	1	no	—
58	C₄	—	229	1	no	—
59	C₄	C₂²⋊#11745021	1	1	yes	characteristic
60	C₂	—	8	1	no	—
61	C₂	—	1832	1	no	—
62	C₂	—	229	1	no	—
63	C₂	—	229	1	no	—
64	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	D₈×D₂₂₉	C₂
cosocle	C₈×D₂₂₉	C₂²
hypercenter	D₈	#11745021
hypocenter	C₂₂₉	#357
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. D₂₂₉⋊C₁₆	C₂	semidirect
2. #20301057	C₂	semidirect
7. D₁₈₃₂	C₄	semidirect
10. D₈×C₂₂₉	C₄	semidirect
32. C₂₂₉	#357	semidirect
40. D₈	#11745021	semidirect

Maximal subgroups

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

subgroup	index
D₂₂₉⋊C₁₆	2
#20301057	2
D₈×D₂₂₉	2
#357	229