Subgroups — #31771013

Order 17216.

90 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22575452	C₂	1	1	yes	maximal, characteristic
2	#22575444	C₂	1	1	yes	maximal, characteristic
3	#22575386	C₂	1	1	yes	maximal, characteristic
4	#22575385	C₂	1	1	yes	maximal, characteristic
5	#22575384	C₂	1	1	yes	maximal, characteristic
6	#22575363	C₂	1	1	yes	maximal, characteristic
7	#107×C₂₆₉	C₂	1	1	yes	maximal, characteristic
8	C₂×Dic₅₃₈	—	2	1	no	—
9	C₂×#14524130	C₂²	1	1	yes	characteristic
10	C₄⋊Dic₂₆₉	—	2	1	no	—
11	C₄⋊Dic₂₆₉	C₂²	1	1	yes	characteristic
12	C₄⋊Dic₂₆₉	C₂²	1	1	yes	characteristic
13	#16864381	—	2	1	no	—
14	#16864381	—	2	1	no	—
15	C₄×Dic₂₆₉	C₂²	1	1	yes	characteristic
16	C₂×C₈×C₂₆₉	C₂²	1	1	yes	characteristic
17	C₄⋊C₄×C₂₆₉	C₂²	1	1	yes	characteristic
18	C₄⋊C₄×C₂₆₉	C₂²	1	1	yes	characteristic
19	Dic₅₃₈	—	4	2	no	—
20	C₂×Dic₂₆₉	—	2	1	no	—
21	C₂×Dic₂₆₉	—	2	1	no	—
22	C₂×Dic₂₆₉	D₄	1	1	yes	characteristic
23	C₂×Dic₂₆₉	D₄	1	1	yes	characteristic
24	#14524130	Q₈	1	2	yes	complemented
25	C₂×C₄×C₂₆₉	—	2	1	no	—
26	C₂×C₄×C₂₆₉	—	2	1	no	—
27	C₂×C₄×C₂₆₉	C₂³	1	1	yes	characteristic
28	C₈×C₂₆₉	Q₈	1	2	yes	complemented
29	C₂²×C₂₆₉	C₂×D₄	1	1	yes	characteristic
30	C₄×C₂₆₉	—	4	1	no	—
31	C₄×C₂₆₉	—	4	1	no	—
32	C₄×C₂₆₉	C₂×Q₈	1	1	yes	characteristic
33	C₄×C₂₆₉	C₂×Q₈	1	1	yes	characteristic
34	Dic₂₆₉	—	2	1	no	—
35	Dic₂₆₉	—	2	1	no	—
36	Dic₂₆₉	—	4	1	no	—
37	Dic₂₆₉	—	4	1	no	—
38	C₂×C₂₆₉	C₄⋊Q₈	1	1	yes	characteristic
39	C₂×C₂₆₉	AGL(1,ℤ/8)	1	1	yes	characteristic
40	C₂×C₂₆₉	#137	1	1	yes	characteristic
41	C₂₆₉	#501	1	1	yes	characteristic, complemented, Sylow
42	#501	—	269	1	no	maximal, Sylow
43	C₄⋊Q₈	—	269	1	no	—
44	#125	—	269	1	no	—
45	#107	—	269	1	no	—
46	#107	D₂₆₉	1	1	yes	characteristic
47	#106	—	269	1	no	—
48	#106	—	269	1	no	—
49	#97	—	269	1	no	—
50	C₂×Q₈	—	538	1	no	—
51	C₂×C₈	—	269	1	no	—
52	C₂×C₈	C₂×D₂₆₉	1	1	yes	characteristic
53	C₄⋊C₄	—	269	1	no	—
54	C₄⋊C₄	—	269	1	no	—
55	C₄⋊C₄	—	538	1	no	—
56	C₄⋊C₄	—	538	1	no	—
57	C₄⋊C₄	—	538	1	no	—
58	C₄⋊C₄	C₂×D₂₆₉	1	1	yes	characteristic
59	C₄⋊C₄	C₂×D₂₆₉	1	1	yes	characteristic
60	C₄²	—	269	1	no	—
61	Q₈	—	1076	2	no	—
62	C₂×C₄	—	2	1	no	—
63	C₂×C₄	—	2	1	no	—
64	C₂×C₄	—	269	1	no	—
65	C₂×C₄	—	269	1	no	—
66	C₂×C₄	—	538	1	no	—
67	C₂×C₄	—	538	1	no	—
68	C₂×C₄	C₂²×D₂₆₉	1	1	yes	characteristic
69	C₈	—	269	2	no	—
70	C₈	Dic₅₃₈	1	2	yes	complemented
71	C₂²	D₄×D₂₆₉	1	1	yes	characteristic
72	C₄	—	4	1	no	—
73	C₄	—	4	1	no	—
74	C₄	—	538	1	no	—
75	C₄	—	538	1	no	—
76	C₄	—	1076	1	no	—
77	C₄	—	1076	1	no	—
78	C₄	Q₈×D₂₆₉	1	1	yes	characteristic
79	C₄	C₂×Dic₅₃₈	1	1	yes	characteristic
80	C₂	#22575452	1	1	yes	characteristic
81	C₂	#22575477	1	1	yes	characteristic
82	C₂	#22575485	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	#501
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
24. #14524130	Q₈	semidirect
28. C₈×C₂₆₉	Q₈	semidirect
41. C₂₆₉	#501	semidirect
70. C₈	Dic₅₃₈	semidirect

Maximal subgroups

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

subgroup	index
#22575452	2
#22575444	2
#22575386	2
#22575385	2
#22575384	2
#22575363	2
#107×C₂₆₉	2
#501	269