Subgroups — #31770643

Order 17216.

114 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22575412	C₂	1	2	yes	maximal
2	#22575387	C₂	1	2	yes	maximal, complemented
3	#22575385	C₂	1	2	yes	maximal, complemented
4	C₂²×C₈×C₂₆₉	C₂	1	1	yes	maximal, characteristic
5	C₄⋊Dic₂₆₉	C₂²	1	4	yes	—
6	C₂²⋊Dic₂₆₉	—	2	2	no	—
7	C₄×Dic₂₆₉	—	2	2	no	—
8	C₂²×C₄×C₂₆₉	C₂²	1	1	yes	characteristic
9	C₂×C₈×C₂₆₉	C₄	1	4	yes	complemented
10	C₂×C₈×C₂₆₉	C₂²	1	1	yes	characteristic
11	C₂×C₈×C₂₆₉	C₂²	1	1	yes	characteristic
12	C₂×Dic₂₆₉	—	2	4	no	—
13	C₂³×C₂₆₉	D₄	1	1	yes	characteristic
14	C₂×C₄×C₂₆₉	C₂×C₄	1	2	yes	—
15	C₂×C₄×C₂₆₉	D₄	1	1	yes	characteristic
16	C₂×C₄×C₂₆₉	Q₈	1	2	yes	—
17	C₂×C₄×C₂₆₉	C₂³	1	1	yes	characteristic
18	C₈×C₂₆₉	C₂×C₄	1	2	yes	—
19	C₈×C₂₆₉	C₂×C₄	1	2	yes	—
20	C₂²×C₂₆₉	—	2	2	no	—
21	C₂²×C₂₆₉	C₄⋊C₄	1	2	yes	—
22	C₂²×C₂₆₉	C₂×D₄	1	1	yes	characteristic
23	C₄×C₂₆₉	C₄⋊C₄	1	2	yes	—
24	C₄×C₂₆₉	C₂²×C₄	1	1	yes	characteristic
25	C₄×C₂₆₉	C₂×Q₈	1	1	yes	characteristic
26	Dic₂₆₉	—	4	4	no	—
27	C₂×C₂₆₉	—	2	2	no	—
28	C₂×C₂₆₉	C₂×C₄⋊C₄	1	1	yes	characteristic
29	C₂×C₂₆₉	#135	1	2	yes	—
30	C₂₆₉	#427	1	1	yes	characteristic, complemented, Sylow
31	#427	—	269	1	no	maximal, Sylow
32	C₂²×C₈	D₂₆₉	1	1	yes	characteristic
33	#117	—	269	2	no	—
34	#107	—	269	2	no	—
35	#106	—	269	2	no	—
36	C₂²×C₄	C₂×D₂₆₉	1	1	yes	characteristic
37	C₂×C₈	Dic₂₆₉	1	4	yes	complemented
38	C₂×C₈	C₂×D₂₆₉	1	1	yes	characteristic
39	C₂×C₈	C₂×D₂₆₉	1	1	yes	characteristic
40	C₄⋊C₄	—	269	4	no	—
41	C₂²⋊C₄	—	538	2	no	—
42	C₄²	—	538	2	no	—
43	C₂³	D₁₀₇₆	1	1	yes	characteristic
44	C₂×C₄	—	538	4	no	—
45	C₂×C₄	C₂×Dic₂₆₉	1	2	yes	—
46	C₂×C₄	D₁₀₇₆	1	1	yes	characteristic
47	C₂×C₄	Dic₅₃₈	1	2	yes	—
48	C₂×C₄	C₂²×D₂₆₉	1	1	yes	characteristic
49	C₈	C₂×Dic₂₆₉	1	2	yes	—
50	C₈	C₂×Dic₂₆₉	1	2	yes	—
51	C₂²	—	2	2	no	—
52	C₂²	C₄⋊Dic₂₆₉	1	2	yes	—
53	C₂²	C₂×D₁₀₇₆	1	1	yes	characteristic
54	C₄	—	1076	4	no	—
55	C₄	C₄⋊Dic₂₆₉	1	2	yes	—
56	C₄	C₂²×Dic₂₆₉	1	1	yes	characteristic
57	C₄	C₂×Dic₅₃₈	1	1	yes	characteristic
58	C₂	—	2	2	no	—
59	C₂	C₂×C₄⋊Dic₂₆₉	1	1	yes	characteristic
60	C₂	#22575475	1	2	yes	—

	subgroup	G/H
center	C₂×C₄	D₁₀₇₆
commutator	C₄×C₂₆₉	C₂²×C₄
Frattini	C₂×C₄	C₂²×D₂₆₉
Fitting	C₂²×C₈×C₂₆₉	C₂
socle	C₂²×C₂₆₉	C₂×D₄
prime radical	C₂³×C₂₆₉	D₄
cosocle	C₂×C₄×C₂₆₉	C₂³
hypercenter	C₂²×C₈	D₂₆₉
hypocenter	C₂₆₉	#427
LCS	G ▷ C₄×C₂₆₉ ▷ C₂×C₂₆₉ ▷ C₂₆₉
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄ ◁ C₂²×C₈

Complements

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

N	H ≅ G/N	type
2. #22575387	C₂	semidirect
3. #22575385	C₂	semidirect
9. C₂×C₈×C₂₆₉	C₄	semidirect
30. C₂₆₉	#427	semidirect
37. C₂×C₈	Dic₂₆₉	semidirect

Maximal subgroups

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

subgroup	index
#22575412	2
#22575387	2
#22575385	2
C₂²×C₈×C₂₆₉	2
#427	269