Subgroups — #31771015

Order 17216.

90 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22575453	C₂	1	1	yes	maximal, characteristic
2	#22575386	C₂	1	2	yes	maximal
3	#22575384	C₂	1	2	yes	maximal
4	#22575364	C₂	1	1	yes	maximal, characteristic
5	#125×C₂₆₉	C₂	1	1	yes	maximal, characteristic
6	C₂×Dic₅₃₈	—	2	1	no	—
7	C₂×#14524130	C₂²	1	2	yes	—
8	C₄⋊Dic₂₆₉	—	2	1	no	—
9	C₄⋊Dic₂₆₉	C₂²	1	2	yes	—
10	C₄⋊C₄×C₂₆₉	—	2	1	no	—
11	C₄⋊C₄×C₂₆₉	—	2	1	no	—
12	C₄⋊C₄×C₂₆₉	C₂²	1	1	yes	characteristic
13	C₄⋊C₄×C₂₆₉	C₂²	1	1	yes	characteristic
14	C₄²×C₂₆₉	C₂²	1	1	yes	characteristic
15	Dic₅₃₈	—	4	2	no	—
16	C₂×Dic₂₆₉	—	2	2	no	—
17	#14524130	Q₈	1	4	yes	complemented
18	C₂×C₄×C₂₆₉	—	2	1	no	—
19	C₂×C₄×C₂₆₉	—	2	1	no	—
20	C₂×C₄×C₂₆₉	D₄	1	1	yes	characteristic
21	C₂×C₄×C₂₆₉	D₄	1	1	yes	characteristic
22	C₂×C₄×C₂₆₉	C₂³	1	1	yes	characteristic
23	C₂²×C₂₆₉	C₂×D₄	1	1	yes	characteristic
24	C₄×C₂₆₉	—	2	1	no	—
25	C₄×C₂₆₉	—	2	1	no	—
26	C₄×C₂₆₉	—	4	1	no	—
27	C₄×C₂₆₉	—	4	1	no	—
28	C₄×C₂₆₉	C₂×Q₈	1	2	yes	—
29	Dic₂₆₉	—	4	2	no	—
30	C₂×C₂₆₉	C₄⋊Q₈	1	1	yes	characteristic
31	C₂×C₂₆₉	AGL(1,ℤ/8)	1	1	yes	characteristic
32	C₂×C₂₆₉	#137	1	1	yes	characteristic
33	C₂₆₉	#501	1	1	yes	characteristic, complemented, Sylow
34	#501	—	269	1	no	maximal, Sylow
35	C₄⋊Q₈	—	269	1	no	—
36	#125	D₂₆₉	1	1	yes	characteristic
37	#107	—	269	2	no	—
38	#106	—	269	2	no	—
39	#97	—	269	1	no	—
40	C₂×Q₈	—	538	1	no	—
41	C₂×C₈	—	269	2	no	—
42	C₄⋊C₄	—	2	1	no	—
43	C₄⋊C₄	—	2	1	no	—
44	C₄⋊C₄	—	269	2	no	—
45	C₄⋊C₄	—	538	1	no	—
46	C₄⋊C₄	C₂×D₂₆₉	1	1	yes	characteristic
47	C₄⋊C₄	C₂×D₂₆₉	1	1	yes	characteristic
48	C₄²	C₂×D₂₆₉	1	1	yes	characteristic
49	Q₈	—	1076	2	no	—
50	C₂×C₄	—	2	1	no	—
51	C₂×C₄	—	2	1	no	—
52	C₂×C₄	—	538	2	no	—
53	C₂×C₄	C₂²⋊D₂₆₉	1	1	yes	characteristic
54	C₂×C₄	C₂²⋊D₂₆₉	1	1	yes	characteristic
55	C₂×C₄	C₂²×D₂₆₉	1	1	yes	characteristic
56	C₈	—	269	4	no	—
57	C₂²	C₂×C₂²⋊D₂₆₉	1	1	yes	characteristic
58	C₄	—	2	1	no	—
59	C₄	—	2	1	no	—
60	C₄	—	4	1	no	—
61	C₄	—	4	1	no	—
62	C₄	—	1076	2	no	—
63	C₄	Q₈×D₂₆₉	1	2	yes	—
64	C₂	#22575451	1	1	yes	characteristic
65	C₂	#22575478	1	1	yes	characteristic
66	C₂	#22575483	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×C₂²⋊D₂₆₉
commutator, cosocle	C₂×C₄×C₂₆₉	C₂³
Frattini	C₂×C₄	C₂²×D₂₆₉
Fitting	#125×C₂₆₉	C₂
socle, prime radical	C₂²×C₂₆₉	C₂×D₄
hypercenter	#125	D₂₆₉
hypocenter	C₂₆₉	#501
LCS	G ▷ C₂×C₄×C₂₆₉ ▷ C₂×C₂₆₉ ▷ C₂₆₉
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #125

Complements

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

N	H ≅ G/N	type
17. #14524130	Q₈	semidirect
33. C₂₆₉	#501	semidirect

Maximal subgroups

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

subgroup	index
#22575453	2
#22575386	2
#22575384	2
#22575364	2
#125×C₂₆₉	2
#501	269