Subgroups — #31771006

Order 17216.

86 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22575444	C₂	1	2	yes	maximal
2	#22575387	C₂	1	1	yes	maximal, characteristic
3	#22575384	C₂	1	2	yes	maximal
4	C₈×Dic₂₆₉	C₂	1	1	yes	maximal, characteristic
5	#107×C₂₆₉	C₂	1	1	yes	maximal, characteristic
6	C₂×#14524130	C₂²	1	1	yes	characteristic
7	C₄⋊Dic₂₆₉	—	2	2	no	—
8	C₄⋊Dic₂₆₉	C₂²	1	2	yes	—
9	#16864381	—	2	2	no	—
10	C₄×Dic₂₆₉	C₂²	1	1	yes	characteristic
11	C₂×C₈×C₂₆₉	C₂²	1	1	yes	characteristic
12	C₄⋊C₄×C₂₆₉	C₂²	1	2	yes	—
13	C₂×Dic₂₆₉	—	2	2	no	—
14	C₂×Dic₂₆₉	D₄	1	1	yes	characteristic
15	C₂×Dic₂₆₉	D₄	1	1	yes	characteristic
16	#14524130	Q₈	1	2	yes	—
17	C₂×C₄×C₂₆₉	—	2	2	no	—
18	C₂×C₄×C₂₆₉	C₂³	1	1	yes	characteristic
19	C₈×C₂₆₉	Q₈	1	2	yes	—
20	C₂²×C₂₆₉	C₂×D₄	1	1	yes	characteristic
21	C₄×C₂₆₉	—	4	2	no	—
22	C₄×C₂₆₉	C₂×Q₈	1	1	yes	characteristic
23	C₄×C₂₆₉	C₂×Q₈	1	1	yes	characteristic
24	Dic₂₆₉	—	2	1	no	—
25	Dic₂₆₉	—	2	1	no	—
26	Dic₂₆₉	—	4	2	no	—
27	C₂×C₂₆₉	C₄⋊Q₈	1	1	yes	characteristic
28	C₂×C₂₆₉	#135	1	1	yes	characteristic
29	C₂×C₂₆₉	#135	1	1	yes	characteristic
30	C₂₆₉	#499	1	1	yes	characteristic, complemented, Sylow
31	#499	—	269	1	no	maximal, Sylow
32	#125	—	269	2	no	—
33	#107	—	269	1	no	—
34	#107	D₂₆₉	1	1	yes	characteristic
35	#106	—	269	2	no	—
36	C₄×C₈	—	269	1	no	—
37	C₂×C₈	—	269	1	no	—
38	C₂×C₈	C₂×D₂₆₉	1	1	yes	characteristic
39	C₄⋊C₄	—	269	2	no	—
40	C₄⋊C₄	—	538	2	no	—
41	C₄⋊C₄	—	538	2	no	—
42	C₄⋊C₄	C₂×D₂₆₉	1	2	yes	—
43	C₄²	—	269	1	no	—
44	C₂×C₄	—	2	2	no	—
45	C₂×C₄	—	269	1	no	—
46	C₂×C₄	—	269	1	no	—
47	C₂×C₄	—	538	2	no	—
48	C₂×C₄	C₂²×D₂₆₉	1	1	yes	characteristic
49	C₈	—	269	2	no	—
50	C₈	Dic₅₃₈	1	2	yes	—
51	C₂²	D₄×D₂₆₉	1	1	yes	characteristic
52	C₄	—	4	2	no	—
53	C₄	—	538	1	no	—
54	C₄	—	538	1	no	—
55	C₄	—	1076	2	no	—
56	C₄	Q₈×D₂₆₉	1	1	yes	characteristic
57	C₄	C₂×Dic₅₃₈	1	1	yes	characteristic
58	C₂	#22575452	1	1	yes	characteristic
59	C₂	#22575472	1	1	yes	characteristic
60	C₂	#22575476	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	D₄×D₂₆₉
commutator, cosocle	C₂×C₄×C₂₆₉	C₂³
Frattini	C₂×C₄	C₂²×D₂₆₉
Fitting	#107×C₂₆₉	C₂
socle, prime radical	C₂²×C₂₆₉	C₂×D₄
hypercenter	#107	D₂₆₉
hypocenter	C₂₆₉	#499
LCS	G ▷ C₂×C₄×C₂₆₉ ▷ C₂×C₂₆₉ ▷ C₂₆₉
UCS	C₁ ◁ C₂² ◁ C₂×C₄ ◁ #107

Complements

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

N	H ≅ G/N	type
30. C₂₆₉	#499	semidirect

Maximal subgroups

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

subgroup	index
#22575444	2
#22575387	2
#22575384	2
C₈×Dic₂₆₉	2
#107×C₂₆₉	2
#499	269