Subgroups — #26015016

Order 11296.

68 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#19150736	C₂	1	1	yes	maximal, characteristic, complemented
2	D₄×D₃₅₃	C₂	1	1	yes	maximal, characteristic, complemented
3	#19150726	C₂	1	1	yes	maximal, characteristic, complemented
4	#19150725	C₂	1	1	yes	maximal, characteristic, complemented
5	#19150722	C₂	1	1	yes	maximal, characteristic, complemented
6	#19150720	C₂	1	1	yes	maximal, characteristic, complemented
7	D₈×C₃₅₃	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂²×D₃₅₃	—	2	1	no	—
9	Dic₇₀₆	C₂²	1	1	yes	characteristic, complemented
10	D₁₄₁₂	C₂²	1	1	yes	characteristic
11	C₂²⋊D₃₅₃	—	2	1	no	—
12	C₂²⋊D₃₅₃	—	2	1	no	—
13	C₂×Dic₃₅₃	—	2	1	no	—
14	C₄×D₃₅₃	C₂²	1	1	yes	characteristic
15	#15667725	C₂²	1	1	yes	characteristic, complemented
16	D₄×C₃₅₃	C₂²	1	1	yes	characteristic
17	D₄×C₃₅₃	C₂²	1	1	yes	characteristic, complemented
18	C₈×C₃₅₃	C₂²	1	1	yes	characteristic, complemented
19	C₂²×C₃₅₃	—	2	1	no	—
20	C₂²×C₃₅₃	—	2	1	no	—
21	C₂×D₃₅₃	—	2	1	no	—
22	C₂×D₃₅₃	—	4	2	no	—
23	C₂×D₃₅₃	D₄	1	1	yes	characteristic
24	C₄×C₃₅₃	C₂³	1	1	yes	characteristic
25	Dic₃₅₃	—	2	1	no	—
26	Dic₃₅₃	D₄	1	1	yes	characteristic
27	C₂×C₃₅₃	—	4	1	no	—
28	C₂×C₃₅₃	—	4	1	no	—
29	C₂×C₃₅₃	C₂×D₄	1	1	yes	characteristic
30	D₃₅₃	—	2	1	no	—
31	D₃₅₃	—	4	1	no	—
32	C₃₅₃	AGL(1,ℤ/8)	1	1	yes	characteristic, complemented, Sylow
33	AGL(1,ℤ/8)	—	353	1	no	maximal, Sylow
34	#41	—	353	1	no	—
35	C₂×D₄	—	353	1	no	—
36	SD₁₆	—	353	1	no	—
37	SD₁₆	—	353	1	no	—
38	D₈	—	353	1	no	—
39	D₈	D₃₅₃	1	1	yes	characteristic, complemented
40	M₁₆	—	353	1	no	—
41	C₂³	—	706	1	no	—
42	Q₈	—	353	1	no	—
43	D₄	—	353	1	no	—
44	D₄	—	706	1	no	—
45	D₄	—	706	1	no	—
46	D₄	C₂×D₃₅₃	1	1	yes	characteristic
47	D₄	C₂×D₃₅₃	1	1	yes	characteristic, complemented
48	C₂×C₄	—	353	1	no	—
49	C₂×C₄	—	706	1	no	—
50	C₈	—	353	1	no	—
51	C₈	C₂×D₃₅₃	1	1	yes	characteristic, complemented
52	C₂²	—	2	1	no	—
53	C₂²	—	2	1	no	—
54	C₂²	—	353	1	no	—
55	C₂²	—	1412	2	no	—
56	C₂²	—	706	1	no	—
57	C₄	—	353	1	no	—
58	C₄	—	706	1	no	—
59	C₄	C₂²×D₃₅₃	1	1	yes	characteristic
60	C₂	—	4	1	no	—
61	C₂	—	4	1	no	—
62	C₂	—	1412	1	no	—
63	C₂	—	706	1	no	—
64	C₂	D₄×D₃₅₃	1	1	yes	characteristic

	subgroup	G/H
center	C₂	D₄×D₃₅₃
commutator, cosocle	C₄×C₃₅₃	C₂³
Frattini	C₄	C₂²×D₃₅₃
Fitting	D₈×C₃₅₃	C₂
socle	C₂×C₃₅₃	C₂×D₄
hypercenter	D₈	D₃₅₃
hypocenter	C₃₅₃	AGL(1,ℤ/8)
LCS	G ▷ C₄×C₃₅₃ ▷ C₂×C₃₅₃ ▷ C₃₅₃
UCS	C₁ ◁ C₂ ◁ C₄ ◁ D₈

Complements

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

N	H ≅ G/N	type
1. #19150736	C₂	semidirect
2. D₄×D₃₅₃	C₂	semidirect
3. #19150726	C₂	semidirect
4. #19150725	C₂	semidirect
5. #19150722	C₂	semidirect
6. #19150720	C₂	semidirect
7. D₈×C₃₅₃	C₂	semidirect
9. Dic₇₀₆	C₂²	semidirect
15. #15667725	C₂²	semidirect
17. D₄×C₃₅₃	C₂²	semidirect
18. C₈×C₃₅₃	C₂²	semidirect
32. C₃₅₃	AGL(1,ℤ/8)	semidirect
39. D₈	D₃₅₃	semidirect
47. D₄	C₂×D₃₅₃	semidirect
51. C₈	C₂×D₃₅₃	semidirect

Maximal subgroups

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

subgroup	index
#19150736	2
D₄×D₃₅₃	2
#19150726	2
#19150725	2
#19150722	2
#19150720	2
D₈×C₃₅₃	2
AGL(1,ℤ/8)	353