Subgroups — #28411288

Order 14656.

56 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#16814641	C₂	1	1	yes	maximal, characteristic, complemented
2	#20300924	C₂	1	1	yes	maximal, characteristic, complemented
3	#135×C₂₂₉	C₂	1	1	yes	maximal, characteristic
4	#16814650	—	2	1	no	—
5	#16814641	—	2	1	no	—
6	#41×C₂₂₉	—	2	1	no	—
7	Q₁₆×C₂₂₉	C₄	1	1	yes	characteristic
8	SD₁₆×C₂₂₉	—	2	1	no	—
9	D₈×C₂₂₉	C₄	1	1	yes	characteristic
10	C₂×C₈×C₂₂₉	C₂²	1	1	yes	characteristic
11	#14270080	—	4	1	no	—
12	Q₈×C₂₂₉	—	2	1	no	—
13	D₄×C₂₂₉	—	2	1	no	—
14	D₄×C₂₂₉	—	4	1	no	—
15	C₂×C₄×C₂₂₉	—	4	1	no	—
16	C₂×C₄×C₂₂₉	D₄	1	1	yes	characteristic
17	C₈×C₂₂₉	C₂×C₄	1	1	yes	characteristic
18	C₈×C₂₂₉	D₄	1	1	yes	characteristic
19	C₂²×C₂₂₉	—	4	1	no	—
20	C₂²×C₂₂₉	SD₁₆	1	1	yes	characteristic
21	C₄×C₂₂₉	—	4	1	no	—
22	C₄×C₂₂₉	C₂²⋊C₄	1	1	yes	characteristic
23	C₄×C₂₂₉	D₈	1	1	yes	characteristic
24	C₂×C₂₂₉	—	2	1	no	—
25	C₂×C₂₂₉	—	8	1	no	—
26	C₂×C₂₂₉	#102	1	1	yes	characteristic
27	C₂₂₉	#359	1	1	yes	characteristic, complemented, Sylow
28	#359	—	229	1	no	maximal, Sylow
29	#135	D₂₂₉	1	1	yes	characteristic
30	C₂×C₁₆	—	229	1	no	—
31	#108	—	229	1	no	—
32	#41	—	2	1	no	—
33	Q₁₆	Dic₂₂₉	1	1	yes	characteristic
34	SD₁₆	—	2	1	no	—
35	D₈	Dic₂₂₉	1	1	yes	characteristic
36	M₁₆	—	458	1	no	—
37	C₂×C₈	C₂×D₂₂₉	1	1	yes	characteristic
38	C₁₆	—	458	1	no	—
39	Q₈	—	2	1	no	—
40	D₄	—	2	1	no	—
41	D₄	—	4	1	no	—
42	C₂×C₄	—	4	1	no	—
43	C₂×C₄	C₂²⋊D₂₂₉	1	1	yes	characteristic
44	C₈	—	916	1	no	—
45	C₈	C₂×Dic₂₂₉	1	1	yes	characteristic
46	C₈	C₂²⋊D₂₂₉	1	1	yes	characteristic
47	C₂²	—	4	1	no	—
48	C₂²	#16814654	1	1	yes	characteristic
49	C₄	—	4	1	no	—
50	C₄	C₂²⋊Dic₂₂₉	1	1	yes	characteristic
51	C₄	#16814651	1	1	yes	characteristic
52	C₂	—	2	1	no	—
53	C₂	—	8	1	no	—
54	C₂	#20300909	1	1	yes	characteristic

	subgroup	G/H
center	C₄	#16814651
commutator	C₈×C₂₂₉	C₂×C₄
Frattini	C₂×C₈	C₂×D₂₂₉
Fitting, prime radical	#135×C₂₂₉	C₂
socle	C₂×C₂₂₉	#102
cosocle	C₂×C₈×C₂₂₉	C₂²
hypercenter	#135	D₂₂₉
hypocenter	C₂₂₉	#359
LCS	G ▷ C₈×C₂₂₉ ▷ C₄×C₂₂₉ ▷ C₂×C₂₂₉ ▷ C₂₂₉
UCS	C₁ ◁ C₄ ◁ C₂×C₄ ◁ C₂×C₈ ◁ #135

Complements

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

N	H ≅ G/N	type
1. C₂×#16814641	C₂	semidirect
2. #20300924	C₂	semidirect
27. C₂₂₉	#359	semidirect

Maximal subgroups

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

subgroup	index
C₂×#16814641	2
#20300924	2
#135×C₂₂₉	2
#359	229