Subgroups — #34089726

Order 19744.

68 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#23754599	C₂	1	1	yes	maximal, characteristic, complemented
2	D₄×D₆₁₇	C₂	1	1	yes	maximal, characteristic, complemented
3	#23754589	C₂	1	1	yes	maximal, characteristic, complemented
4	#23754588	C₂	1	1	yes	maximal, characteristic, complemented
5	#23754585	C₂	1	1	yes	maximal, characteristic, complemented
6	#23754583	C₂	1	1	yes	maximal, characteristic, complemented
7	D₈×C₆₁₇	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂²×D₆₁₇	—	2	1	no	—
9	Dic₁₂₃₄	C₂²	1	1	yes	characteristic, complemented
10	D₂₄₆₈	C₂²	1	1	yes	characteristic
11	C₂²⋊D₆₁₇	—	2	1	no	—
12	C₂²⋊D₆₁₇	—	2	1	no	—
13	C₂×Dic₆₁₇	—	2	1	no	—
14	C₄×D₆₁₇	C₂²	1	1	yes	characteristic
15	#19096921	C₂²	1	1	yes	characteristic, complemented
16	D₄×C₆₁₇	C₂²	1	1	yes	characteristic
17	D₄×C₆₁₇	C₂²	1	1	yes	characteristic, complemented
18	C₈×C₆₁₇	C₂²	1	1	yes	characteristic, complemented
19	C₂²×C₆₁₇	—	2	1	no	—
20	C₂²×C₆₁₇	—	2	1	no	—
21	C₂×D₆₁₇	—	2	1	no	—
22	C₂×D₆₁₇	—	4	2	no	—
23	C₂×D₆₁₇	D₄	1	1	yes	characteristic
24	C₄×C₆₁₇	C₂³	1	1	yes	characteristic
25	Dic₆₁₇	—	2	1	no	—
26	Dic₆₁₇	D₄	1	1	yes	characteristic
27	C₂×C₆₁₇	—	4	1	no	—
28	C₂×C₆₁₇	—	4	1	no	—
29	C₂×C₆₁₇	C₂×D₄	1	1	yes	characteristic
30	D₆₁₇	—	2	1	no	—
31	D₆₁₇	—	4	1	no	—
32	C₆₁₇	AGL(1,ℤ/8)	1	1	yes	characteristic, complemented, Sylow
33	AGL(1,ℤ/8)	—	617	1	no	maximal, Sylow
34	#41	—	617	1	no	—
35	C₂×D₄	—	617	1	no	—
36	SD₁₆	—	617	1	no	—
37	SD₁₆	—	617	1	no	—
38	D₈	—	617	1	no	—
39	D₈	D₆₁₇	1	1	yes	characteristic, complemented
40	M₁₆	—	617	1	no	—
41	C₂³	—	1234	1	no	—
42	Q₈	—	617	1	no	—
43	D₄	—	617	1	no	—
44	D₄	—	1234	1	no	—
45	D₄	—	1234	1	no	—
46	D₄	C₂×D₆₁₇	1	1	yes	characteristic
47	D₄	C₂×D₆₁₇	1	1	yes	characteristic, complemented
48	C₂×C₄	—	617	1	no	—
49	C₂×C₄	—	1234	1	no	—
50	C₈	—	617	1	no	—
51	C₈	C₂×D₆₁₇	1	1	yes	characteristic, complemented
52	C₂²	—	2	1	no	—
53	C₂²	—	2	1	no	—
54	C₂²	—	617	1	no	—
55	C₂²	—	2468	2	no	—
56	C₂²	—	1234	1	no	—
57	C₄	—	617	1	no	—
58	C₄	—	1234	1	no	—
59	C₄	C₂²×D₆₁₇	1	1	yes	characteristic
60	C₂	—	4	1	no	—
61	C₂	—	4	1	no	—
62	C₂	—	2468	1	no	—
63	C₂	—	1234	1	no	—
64	C₂	D₄×D₆₁₇	1	1	yes	characteristic

	subgroup	G/H
center	C₂	D₄×D₆₁₇
commutator, cosocle	C₄×C₆₁₇	C₂³
Frattini	C₄	C₂²×D₆₁₇
Fitting	D₈×C₆₁₇	C₂
socle	C₂×C₆₁₇	C₂×D₄
hypercenter	D₈	D₆₁₇
hypocenter	C₆₁₇	AGL(1,ℤ/8)
LCS	G ▷ C₄×C₆₁₇ ▷ C₂×C₆₁₇ ▷ C₆₁₇
UCS	C₁ ◁ C₂ ◁ C₄ ◁ D₈

Complements

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

N	H ≅ G/N	type
1. #23754599	C₂	semidirect
2. D₄×D₆₁₇	C₂	semidirect
3. #23754589	C₂	semidirect
4. #23754588	C₂	semidirect
5. #23754585	C₂	semidirect
6. #23754583	C₂	semidirect
7. D₈×C₆₁₇	C₂	semidirect
9. Dic₁₂₃₄	C₂²	semidirect
15. #19096921	C₂²	semidirect
17. D₄×C₆₁₇	C₂²	semidirect
18. C₈×C₆₁₇	C₂²	semidirect
32. C₆₁₇	AGL(1,ℤ/8)	semidirect
39. D₈	D₆₁₇	semidirect
47. D₄	C₂×D₆₁₇	semidirect
51. C₈	C₂×D₆₁₇	semidirect

Maximal subgroups

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

subgroup	index
#23754599	2
D₄×D₆₁₇	2
#23754589	2
#23754588	2
#23754585	2
#23754583	2
D₈×C₆₁₇	2
AGL(1,ℤ/8)	617