Subgroups — #34089724

Order 19744.

62 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#23754600	C₂	1	2	yes	maximal, complemented
2	Dic₂₄₆₈	C₂	1	1	yes	maximal, characteristic, complemented
3	#23754589	C₂	1	2	yes	maximal, complemented
4	D₄₉₃₆	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×C₈×C₆₁₇	C₂	1	1	yes	maximal, characteristic, complemented
6	Dic₁₂₃₄	C₂²	1	2	yes	—
7	D₂₄₆₈	C₂²	1	2	yes	—
8	C₂²⋊D₆₁₇	—	2	2	no	—
9	C₄×D₆₁₇	—	2	2	no	—
10	C₂×C₄×C₆₁₇	C₂²	1	1	yes	characteristic
11	C₈×C₆₁₇	C₂²	1	1	yes	characteristic
12	C₈×C₆₁₇	C₂²	1	1	yes	characteristic
13	C₂²×C₆₁₇	D₄	1	1	yes	characteristic
14	C₂×D₆₁₇	—	2	2	no	—
15	C₄×C₆₁₇	D₄	1	1	yes	characteristic
16	C₄×C₆₁₇	C₂³	1	1	yes	characteristic
17	Dic₆₁₇	—	2	2	no	—
18	C₂×C₆₁₇	—	2	1	no	—
19	C₂×C₆₁₇	C₂×D₄	1	1	yes	characteristic
20	D₆₁₇	—	4	2	no	—
21	C₆₁₇	#135	1	1	yes	characteristic, complemented, Sylow
22	#135	—	617	1	no	maximal, Sylow
23	#41	—	617	2	no	—
24	Q₁₆	—	617	1	no	—
25	SD₁₆	—	617	2	no	—
26	D₈	—	617	1	no	—
27	C₂×C₈	D₆₁₇	1	1	yes	characteristic, complemented
28	Q₈	—	617	2	no	—
29	D₄	—	617	2	no	—
30	D₄	—	1234	2	no	—
31	C₂×C₄	—	1234	2	no	—
32	C₂×C₄	C₂×D₆₁₇	1	1	yes	characteristic
33	C₈	C₂×D₆₁₇	1	1	yes	characteristic
34	C₈	C₂×D₆₁₇	1	1	yes	characteristic
35	C₂²	—	1234	2	no	—
36	C₂²	D₂₄₆₈	1	1	yes	characteristic
37	C₄	—	1234	2	no	—
38	C₄	D₂₄₆₈	1	1	yes	characteristic
39	C₄	C₂²×D₆₁₇	1	1	yes	characteristic
40	C₂	—	2	1	no	—
41	C₂	—	2468	2	no	—
42	C₂	C₂×D₂₄₆₈	1	1	yes	characteristic

	subgroup	G/H
center	C₄	D₂₄₆₈
commutator, cosocle	C₄×C₆₁₇	C₂³
Frattini	C₄	C₂²×D₆₁₇
Fitting	C₂×C₈×C₆₁₇	C₂
socle	C₂×C₆₁₇	C₂×D₄
hypercenter	C₂×C₈	D₆₁₇
hypocenter	C₆₁₇	#135
LCS	G ▷ C₄×C₆₁₇ ▷ C₂×C₆₁₇ ▷ C₆₁₇
UCS	C₁ ◁ C₄ ◁ C₂×C₄ ◁ C₂×C₈

Complements

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

N	H ≅ G/N	type
1. #23754600	C₂	semidirect
2. Dic₂₄₆₈	C₂	semidirect
3. #23754589	C₂	semidirect
4. D₄₉₃₆	C₂	semidirect
5. C₂×C₈×C₆₁₇	C₂	semidirect
21. C₆₁₇	#135	semidirect
27. C₂×C₈	D₆₁₇	semidirect

Maximal subgroups

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

subgroup	index
#23754600	2
Dic₂₄₆₈	2
#23754589	2
D₄₉₃₆	2
C₂×C₈×C₆₁₇	2
#135	617