Subgroups — #28412386

Order 14656.

86 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20301047	C₂	1	2	yes	maximal, complemented
2	#20300985	C₂	1	1	yes	maximal, characteristic
3	C₂²⋊#11745021	—	2	2	no	—
4	C₂×D₉₁₆	—	2	2	no	—
5	C₂×D₉₁₆	C₄	1	1	yes	characteristic, complemented
6	C₂×D₉₁₆	C₂²	1	1	yes	characteristic
7	C₄²×C₂₂₉	C₄	1	1	yes	characteristic, complemented
8	C₂×#11745021	—	4	2	no	—
9	C₂²×D₂₂₉	—	2	1	no	—
10	C₂²×D₂₂₉	D₄	1	2	yes	—
11	D₉₁₆	—	4	1	no	—
12	D₉₁₆	—	4	1	no	—
13	D₉₁₆	—	4	4	no	—
14	C₂×C₄×C₂₂₉	—	2	1	no	—
15	C₂×C₄×C₂₂₉	C₂×C₄	1	1	yes	characteristic
16	C₂²×C₂₂₉	C₂²⋊C₄	1	1	yes	characteristic
17	C₂×D₂₂₉	—	2	2	no	—
18	C₂×D₂₂₉	—	4	2	no	—
19	C₂×D₂₂₉	—	4	2	no	—
20	C₂×D₂₂₉	—	4	1	no	—
21	#11745021	—	8	2	no	—
22	C₄×C₂₂₉	—	2	2	no	—
23	C₄×C₂₂₉	—	2	1	no	—
24	C₂×C₂₂₉	—	2	1	no	—
25	C₂×C₂₂₉	#99	1	1	yes	characteristic
26	D₂₂₉	—	4	2	no	—
27	D₂₂₉	—	8	1	no	—
28	C₂₂₉	#353	1	1	yes	characteristic, complemented, Sylow
29	#353	—	229	1	no	maximal, Sylow
30	#127	—	229	1	no	—
31	#99	—	229	2	no	—
32	C₂×D₄	—	458	2	no	—
33	C₂×D₄	—	229	1	no	—
34	C₂×D₄	—	229	1	no	—
35	C₂²⋊C₄	—	458	2	no	—
36	C₄²	#11745021	1	1	yes	characteristic, complemented
37	C₂³	—	458	1	no	—
38	C₂³	—	229	2	no	—
39	D₄	—	916	1	no	—
40	D₄	—	916	4	no	—
41	D₄	—	916	1	no	—
42	C₂×C₄	—	2	1	no	—
43	C₂×C₄	—	916	2	no	—
44	C₂×C₄	C₂×#11745021	1	1	yes	characteristic
45	C₂²	—	916	2	no	—
46	C₂²	—	916	2	no	—
47	C₂²	—	916	1	no	—
48	C₂²	—	458	2	no	—
49	C₂²	C₂²⋊#11745021	1	1	yes	characteristic
50	C₄	—	2	2	no	—
51	C₄	—	2	1	no	—
52	C₄	—	1832	2	no	—
53	C₂	—	2	1	no	—
54	C₂	—	1832	1	no	—
55	C₂	—	916	2	no	—
56	C₂	#20301047	1	1	yes	characteristic

	subgroup	G/H
center	C₂	#20301047
commutator	C₂×C₄×C₂₂₉	C₂×C₄
Frattini	C₂×C₄	C₂×#11745021
Fitting	C₄²×C₂₂₉	C₄
socle	C₂×C₂₂₉	#99
prime radical	#20300985	C₂
cosocle	C₂×D₉₁₆	C₂²
hypercenter	C₄²	#11745021
hypocenter	C₂₂₉	#353
LCS	G ▷ C₂×C₄×C₂₂₉ ▷ C₂²×C₂₂₉ ▷ C₂×C₂₂₉ ▷ C₂₂₉
UCS	C₁ ◁ C₂ ◁ C₂² ◁ C₂×C₄ ◁ C₄²

Complements

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

N	H ≅ G/N	type
1. #20301047	C₂	semidirect
5. C₂×D₉₁₆	C₄	semidirect
7. C₄²×C₂₂₉	C₄	semidirect
28. C₂₂₉	#353	semidirect
36. C₄²	#11745021	semidirect

Maximal subgroups

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

subgroup	index
#20301047	2
#20300985	2
#353	229