Subgroups — #28411265

Order 14656.

56 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20300928	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×#16814641	C₂	1	1	yes	maximal, characteristic, complemented
3	#131×C₂₂₉	C₂	1	1	yes	maximal, characteristic
4	#16814641	—	2	1	no	—
5	#16814641	—	2	1	no	—
6	#41×C₂₂₉	C₄	1	1	yes	characteristic
7	M₁₆×C₂₂₉	—	2	1	no	—
8	M₁₆×C₂₂₉	C₄	1	1	yes	characteristic
9	C₂×C₈×C₂₂₉	—	2	1	no	—
10	C₂×C₈×C₂₂₉	C₂²	1	1	yes	characteristic
11	Q₈×C₂₂₉	C₈	1	1	yes	characteristic
12	D₄×C₂₂₉	—	2	1	no	—
13	D₄×C₂₂₉	C₈	1	1	yes	characteristic
14	C₂×C₄×C₂₂₉	—	2	1	no	—
15	C₂×C₄×C₂₂₉	C₂×C₄	1	1	yes	characteristic
16	C₈×C₂₂₉	—	2	1	no	—
17	C₈×C₂₂₉	D₄	1	1	yes	characteristic
18	C₈×C₂₂₉	D₄	1	1	yes	characteristic
19	C₂²×C₂₂₉	—	2	1	no	—
20	C₂²×C₂₂₉	M₁₆	1	1	yes	characteristic
21	C₄×C₂₂₉	—	2	1	no	—
22	C₄×C₂₂₉	C₂²⋊C₄	1	1	yes	characteristic
23	C₄×C₂₂₉	C₂×C₈	1	1	yes	characteristic
24	C₂×C₂₂₉	—	2	1	no	—
25	C₂×C₂₂₉	—	4	1	no	—
26	C₂×C₂₂₉	C₂²⋊C₈	1	1	yes	characteristic
27	C₂₂₉	#350	1	1	yes	characteristic, complemented, Sylow
28	#350	—	229	1	no	maximal, Sylow
29	#131	D₂₂₉	1	1	yes	characteristic
30	M₃₂	—	229	1	no	—
31	C₂×C₁₆	—	229	1	no	—
32	#41	Dic₂₂₉	1	1	yes	characteristic
33	M₁₆	—	2	1	no	—
34	M₁₆	Dic₂₂₉	1	1	yes	characteristic
35	C₂×C₈	—	2	1	no	—
36	C₂×C₈	C₂×D₂₂₉	1	1	yes	characteristic
37	C₁₆	—	458	1	no	—
38	C₁₆	—	458	1	no	—
39	Q₈	#14270080	1	1	yes	characteristic
40	D₄	—	2	1	no	—
41	D₄	#14270080	1	1	yes	characteristic
42	C₂×C₄	—	2	1	no	—
43	C₂×C₄	C₂×Dic₂₂₉	1	1	yes	characteristic
44	C₈	—	2	1	no	—
45	C₈	C₂²⋊D₂₂₉	1	1	yes	characteristic
46	C₈	C₂²⋊D₂₂₉	1	1	yes	characteristic
47	C₂²	—	2	1	no	—
48	C₂²	#16814650	1	1	yes	characteristic
49	C₄	—	2	1	no	—
50	C₄	C₂²⋊Dic₂₂₉	1	1	yes	characteristic
51	C₄	C₂×#14270080	1	1	yes	characteristic
52	C₂	—	2	1	no	—
53	C₂	—	4	1	no	—
54	C₂	C₂²⋊#14270080	1	1	yes	characteristic

	subgroup	G/H
center	C₈	C₂²⋊D₂₂₉
commutator	C₄×C₂₂₉	C₂×C₈
Frattini	C₂×C₈	C₂×D₂₂₉
Fitting	#131×C₂₂₉	C₂
socle	C₂×C₂₂₉	C₂²⋊C₈
prime radical	#41×C₂₂₉	C₄
cosocle	C₂×C₈×C₂₂₉	C₂²
hypercenter	#131	D₂₂₉
hypocenter	C₂₂₉	#350
LCS	G ▷ C₄×C₂₂₉ ▷ C₂×C₂₂₉ ▷ C₂₂₉
UCS	C₁ ◁ C₈ ◁ C₂×C₈ ◁ #131

Complements

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

N	H ≅ G/N	type
1. #20300928	C₂	semidirect
2. C₂×#16814641	C₂	semidirect
27. C₂₂₉	#350	semidirect

Maximal subgroups

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

subgroup	index
#20300928	2
C₂×#16814641	2
#131×C₂₂₉	2
#350	229