Subgroups — #31771462

Order 17216.

60 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22575508	C₂	1	1	yes	maximal, characteristic
2	#22575508	C₂	1	1	yes	maximal, characteristic
3	#22575444	C₂	1	1	yes	maximal, characteristic
4	C₂×#14524137	—	2	1	no	—
5	C₂×#14524137	—	2	1	no	—
6	C₄⋊Dic₂₆₉	—	2	1	no	—
7	C₄⋊Dic₂₆₉	C₄	1	1	yes	characteristic
8	#16864381	—	2	1	no	—
9	C₄×Dic₂₆₉	C₂²	1	1	yes	characteristic
10	C₄⋊C₄×C₂₆₉	C₄	1	1	yes	characteristic
11	#14524137	—	2	2	no	—
12	#14524137	—	2	2	no	—
13	C₂×Dic₂₆₉	—	2	1	no	—
14	C₂×Dic₂₆₉	D₄	1	1	yes	characteristic
15	C₂×Dic₂₆₉	D₄	1	1	yes	characteristic
16	C₂×C₄×C₂₆₉	—	2	1	no	—
17	C₂×C₄×C₂₆₉	C₂×C₄	1	1	yes	characteristic
18	C₂²×C₂₆₉	C₂²⋊C₄	1	1	yes	characteristic
19	C₄×C₂₆₉	—	2	1	no	—
20	C₄×C₂₆₉	—	4	1	no	—
21	Dic₂₆₉	—	2	1	no	—
22	Dic₂₆₉	—	2	1	no	—
23	Dic₂₆₉	—	4	1	no	—
24	C₂×C₂₆₉	#101	1	1	yes	characteristic
25	C₂×C₂₆₉	#104	1	1	yes	characteristic
26	C₂×C₂₆₉	#104	1	1	yes	characteristic
27	C₂₆₉	#330	1	1	yes	characteristic, complemented, Sylow
28	#330	—	269	1	no	maximal, Sylow
29	#125	—	269	1	no	—
30	#97	—	269	1	no	—
31	#97	—	269	1	no	—
32	C₂×C₈	—	538	1	no	—
33	C₂×C₈	—	538	1	no	—
34	C₄⋊C₄	—	269	1	no	—
35	C₄⋊C₄	—	538	1	no	—
36	C₄⋊C₄	—	538	1	no	—
37	C₄⋊C₄	#11761640	1	1	yes	characteristic
38	C₄²	—	269	1	no	—
39	C₂×C₄	—	2	1	no	—
40	C₂×C₄	—	269	1	no	—
41	C₂×C₄	—	269	1	no	—
42	C₂×C₄	—	538	1	no	—
43	C₂×C₄	C₂×#11761640	1	1	yes	characteristic
44	C₈	—	538	2	no	—
45	C₈	—	538	2	no	—
46	C₂²	C₂²⋊#11761640	1	1	yes	characteristic
47	C₄	—	2	1	no	—
48	C₄	—	4	1	no	—
49	C₄	—	538	1	no	—
50	C₄	—	538	1	no	—
51	C₄	—	1076	1	no	—
52	C₂	#22575514	1	1	yes	characteristic
53	C₂	#22575517	1	1	yes	characteristic
54	C₂	#22575518	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂²⋊#11761640
commutator	C₂×C₄×C₂₆₉	C₂×C₄
Frattini	C₂×C₄	C₂×#11761640
Fitting	C₄⋊C₄×C₂₆₉	C₄
socle, prime radical	C₂²×C₂₆₉	C₂²⋊C₄
cosocle	C₄×Dic₂₆₉	C₂²
hypercenter	C₄⋊C₄	#11761640
hypocenter	C₂₆₉	#330
LCS	G ▷ C₂×C₄×C₂₆₉ ▷ C₂²×C₂₆₉ ▷ C₂₆₉
UCS	C₁ ◁ C₂² ◁ C₂×C₄ ◁ C₄⋊C₄

Complements

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

N	H ≅ G/N	type
27. C₂₆₉	#330	semidirect

Maximal subgroups

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

subgroup	index
#22575508	2
#22575444	2
#330	269