Subgroups — #31771014

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	#22575445	C₂	1	1	yes	maximal, characteristic
3	#22575387	C₂	1	2	yes	maximal
4	#22575385	C₂	1	2	yes	maximal
5	#97×C₂₆₉	C₂	1	1	yes	maximal, characteristic
6	C₂×Dic₅₃₈	—	2	1	no	—
7	C₄⋊Dic₂₆₉	—	2	1	no	—
8	C₄⋊Dic₂₆₉	C₂²	1	1	yes	characteristic
9	C₄⋊Dic₂₆₉	C₂²	1	2	yes	—
10	C₄⋊Dic₂₆₉	C₂²	1	1	yes	characteristic
11	#16864381	—	2	1	no	—
12	#16864381	—	2	1	no	—
13	C₂×C₈×C₂₆₉	C₂²	1	2	yes	—
14	C₄²×C₂₆₉	C₂²	1	1	yes	characteristic
15	Dic₅₃₈	—	4	2	no	—
16	C₂×Dic₂₆₉	—	2	1	no	—
17	C₂×Dic₂₆₉	—	2	1	no	—
18	C₂×Dic₂₆₉	—	2	2	no	—
19	C₂×C₄×C₂₆₉	D₄	1	1	yes	characteristic
20	C₂×C₄×C₂₆₉	D₄	1	1	yes	characteristic
21	C₂×C₄×C₂₆₉	C₂³	1	1	yes	characteristic
22	C₈×C₂₆₉	Q₈	1	4	yes	complemented
23	C₂²×C₂₆₉	C₂×D₄	1	1	yes	characteristic
24	C₄×C₂₆₉	—	2	1	no	—
25	C₄×C₂₆₉	—	2	1	no	—
26	C₄×C₂₆₉	C₂×Q₈	1	2	yes	—
27	Dic₂₆₉	—	4	1	no	—
28	Dic₂₆₉	—	4	2	no	—
29	Dic₂₆₉	—	4	1	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	—	269	1	no	—
37	#107	—	269	2	no	—
38	#106	—	269	2	no	—
39	#97	D₂₆₉	1	1	yes	characteristic
40	C₂×Q₈	—	538	1	no	—
41	C₂×C₈	C₂×D₂₆₉	1	2	yes	—
42	C₄⋊C₄	—	269	1	no	—
43	C₄⋊C₄	—	269	2	no	—
44	C₄⋊C₄	—	269	1	no	—
45	C₄⋊C₄	—	538	1	no	—
46	C₄⋊C₄	—	538	1	no	—
47	C₄⋊C₄	—	538	1	no	—
48	C₄²	C₂×D₂₆₉	1	1	yes	characteristic
49	Q₈	—	1076	2	no	—
50	C₂×C₄	—	538	1	no	—
51	C₂×C₄	—	538	1	no	—
52	C₂×C₄	—	538	2	no	—
53	C₂×C₄	D₁₀₇₆	1	1	yes	characteristic
54	C₂×C₄	D₁₀₇₆	1	1	yes	characteristic
55	C₂×C₄	C₂²×D₂₆₉	1	1	yes	characteristic
56	C₈	Dic₅₃₈	1	4	yes	complemented
57	C₂²	C₂×D₁₀₇₆	1	1	yes	characteristic
58	C₄	—	2	1	no	—
59	C₄	—	2	1	no	—
60	C₄	—	1076	2	no	—
61	C₄	—	1076	1	no	—
62	C₄	—	1076	1	no	—
63	C₄	C₂×Dic₅₃₈	1	2	yes	—
64	C₂	#22575453	1	1	yes	characteristic
65	C₂	#22575481	1	1	yes	characteristic
66	C₂	#22575486	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×D₁₀₇₆
commutator, cosocle	C₂×C₄×C₂₆₉	C₂³
Frattini	C₂×C₄	C₂²×D₂₆₉
Fitting	#97×C₂₆₉	C₂
socle, prime radical	C₂²×C₂₆₉	C₂×D₄
hypercenter	#97	D₂₆₉
hypocenter	C₂₆₉	#501
LCS	G ▷ C₂×C₄×C₂₆₉ ▷ C₂×C₂₆₉ ▷ C₂₆₉
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #97

Complements

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

N	H ≅ G/N	type
22. C₈×C₂₆₉	Q₈	semidirect
33. C₂₆₉	#501	semidirect
56. C₈	Dic₅₃₈	semidirect

Maximal subgroups

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

subgroup	index
#22575453	2
#22575445	2
#22575387	2
#22575385	2
#97×C₂₆₉	2
#501	269