Subgroups — #31771522

Order 17216.

58 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	#22575521	C₂	1	1	yes	maximal, characteristic, complemented
3	#22575476	C₂	1	1	yes	maximal, characteristic
4	C₄⋊#11761640	—	2	1	no	—
5	#16864405	—	2	1	no	—
6	#16864403	—	2	1	no	—
7	#16864389	—	2	1	no	—
8	D₂₁₅₂	C₄	1	1	yes	characteristic, complemented
9	C₈×D₂₆₉	C₂²	1	1	yes	characteristic
10	Q₁₆×C₂₆₉	C₄	1	1	yes	characteristic, complemented
11	C₂×#11761640	—	4	1	no	—
12	D₁₀₇₆	—	2	1	no	—
13	D₁₀₇₆	—	4	1	no	—
14	C₄×D₂₆₉	—	4	1	no	—
15	C₄×D₂₆₉	D₄	1	1	yes	characteristic
16	#14524130	D₄	1	1	yes	characteristic
17	Q₈×C₂₆₉	—	2	1	no	—
18	C₈×C₂₆₉	C₂×C₄	1	1	yes	characteristic
19	C₂×D₂₆₉	—	4	1	no	—
20	C₂×D₂₆₉	D₈	1	1	yes	characteristic
21	#11761640	—	8	1	no	—
22	C₄×C₂₆₉	—	4	1	no	—
23	C₄×C₂₆₉	C₂²⋊C₄	1	1	yes	characteristic
24	Dic₂₆₉	SD₁₆	1	1	yes	characteristic
25	C₂×C₂₆₉	#102	1	1	yes	characteristic
26	D₂₆₉	—	2	1	no	—
27	D₂₆₉	—	8	1	no	—
28	C₂₆₉	#360	1	1	yes	characteristic, complemented, Sylow
29	#360	—	269	1	no	maximal, Sylow
30	#135	—	269	1	no	—
31	M₃₂	—	269	1	no	—
32	#106	—	269	1	no	—
33	#41	—	538	1	no	—
34	Q₁₆	#11761640	1	1	yes	characteristic, complemented
35	SD₁₆	—	538	1	no	—
36	D₈	—	269	1	no	—
37	C₂×C₈	—	269	1	no	—
38	C₄⋊C₄	—	538	1	no	—
39	C₁₆	—	538	1	no	—
40	Q₈	—	2	1	no	—
41	D₄	—	538	1	no	—
42	D₄	—	1076	1	no	—
43	C₂×C₄	—	269	1	no	—
44	C₂×C₄	—	1076	1	no	—
45	C₂×C₄	—	1076	1	no	—
46	C₈	—	269	1	no	—
47	C₈	C₂×#11761640	1	1	yes	characteristic
48	C₂²	—	269	1	no	—
49	C₂²	—	1076	1	no	—
50	C₄	—	4	1	no	—
51	C₄	—	269	1	no	—
52	C₄	—	2152	1	no	—
53	C₄	C₂²⋊#11761640	1	1	yes	characteristic
54	C₂	—	538	1	no	—
55	C₂	—	2152	1	no	—
56	C₂	#22575515	1	1	yes	characteristic

	subgroup	G/H
center	C₂	#22575515
commutator	C₈×C₂₆₉	C₂×C₄
Frattini	C₈	C₂×#11761640
Fitting	Q₁₆×C₂₆₉	C₄
socle	C₂×C₂₆₉	#102
prime radical	#22575476	C₂
cosocle	C₈×D₂₆₉	C₂²
hypercenter	Q₁₆	#11761640
hypocenter	C₂₆₉	#360
LCS	G ▷ C₈×C₂₆₉ ▷ C₄×C₂₆₉ ▷ C₂×C₂₆₉ ▷ C₂₆₉
UCS	C₁ ◁ C₂ ◁ C₄ ◁ C₈ ◁ Q₁₆

Complements

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

N	H ≅ G/N	type
1. #22575528	C₂	semidirect
2. #22575521	C₂	semidirect
8. D₂₁₅₂	C₄	semidirect
10. Q₁₆×C₂₆₉	C₄	semidirect
28. C₂₆₉	#360	semidirect
34. Q₁₆	#11761640	semidirect

Maximal subgroups

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

subgroup	index
#22575528	2
#22575521	2
#22575476	2
#360	269