Subgroups — #31771523

Order 17216.

62 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22575528	C₂	1	1	yes	maximal, characteristic, complemented
2	#22575523	C₂	1	1	yes	maximal, characteristic, complemented
3	D₈×D₂₆₉	C₂	1	1	yes	maximal, characteristic
4	#16864412	—	2	1	no	—
5	#16864405	—	2	1	no	—
6	D₄×D₂₆₉	—	2	1	no	—
7	D₂₁₅₂	C₄	1	1	yes	characteristic
8	#16864387	—	2	1	no	—
9	C₈×D₂₆₉	C₂²	1	1	yes	characteristic
10	D₈×C₂₆₉	C₄	1	1	yes	characteristic
11	#14524137	—	4	1	no	—
12	C₂²×D₂₆₉	—	4	1	no	—
13	D₁₀₇₆	—	2	1	no	—
14	C₂²⋊D₂₆₉	—	4	1	no	—
15	C₄×D₂₆₉	D₄	1	1	yes	characteristic
16	#14524130	D₄	1	1	yes	characteristic
17	D₄×C₂₆₉	—	2	1	no	—
18	C₈×C₂₆₉	C₂×C₄	1	1	yes	characteristic
19	C₂²×C₂₆₉	—	4	1	no	—
20	C₂×D₂₆₉	—	4	1	no	—
21	C₂×D₂₆₉	—	8	2	no	—
22	C₂×D₂₆₉	SD₁₆	1	1	yes	characteristic
23	C₄×C₂₆₉	C₂²⋊C₄	1	1	yes	characteristic
24	Dic₂₆₉	D₈	1	1	yes	characteristic
25	C₂×C₂₆₉	—	8	1	no	—
26	C₂×C₂₆₉	#102	1	1	yes	characteristic
27	D₂₆₉	—	2	1	no	—
28	D₂₆₉	—	8	1	no	—
29	C₂₆₉	#361	1	1	yes	characteristic, complemented, Sylow
30	#361	—	269	1	no	maximal, Sylow
31	C₂×D₈	—	269	1	no	—
32	M₃₂	—	269	1	no	—
33	#108	—	269	1	no	—
34	C₂×D₄	—	538	1	no	—
35	D₈	—	269	1	no	—
36	D₈	—	538	1	no	—
37	D₈	#11761640	1	1	yes	characteristic
38	M₁₆	—	538	1	no	—
39	C₂×C₈	—	269	1	no	—
40	C₁₆	—	538	1	no	—
41	C₂³	—	1076	1	no	—
42	D₄	—	2	1	no	—
43	D₄	—	538	1	no	—
44	D₄	—	1076	1	no	—
45	C₂×C₄	—	269	1	no	—
46	C₈	—	269	1	no	—
47	C₈	—	1076	1	no	—
48	C₈	C₂×#11761640	1	1	yes	characteristic
49	C₂²	—	4	1	no	—
50	C₂²	—	269	1	no	—
51	C₂²	—	1076	1	no	—
52	C₂²	—	2152	2	no	—
53	C₄	—	269	1	no	—
54	C₄	C₂²⋊#11761640	1	1	yes	characteristic
55	C₂	—	8	1	no	—
56	C₂	—	538	1	no	—
57	C₂	—	2152	1	no	—
58	C₂	#22575515	1	1	yes	characteristic

	subgroup	G/H
center	C₂	#22575515
commutator	C₈×C₂₆₉	C₂×C₄
Frattini	C₈	C₂×#11761640
Fitting	D₈×C₂₆₉	C₄
socle	C₂×C₂₆₉	#102
prime radical	D₈×D₂₆₉	C₂
cosocle	C₈×D₂₆₉	C₂²
hypercenter	D₈	#11761640
hypocenter	C₂₆₉	#361
LCS	G ▷ C₈×C₂₆₉ ▷ C₄×C₂₆₉ ▷ C₂×C₂₆₉ ▷ C₂₆₉
UCS	C₁ ◁ C₂ ◁ C₄ ◁ C₈ ◁ D₈

Complements

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

N	H ≅ G/N	type
1. #22575528	C₂	semidirect
2. #22575523	C₂	semidirect
29. C₂₆₉	#361	semidirect

Maximal subgroups

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

subgroup	index
#22575528	2
#22575523	2
D₈×D₂₆₉	2
#361	269