Subgroups — #34089722

Order 19744.

62 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#23754600	C₂	1	1	yes	maximal, characteristic, complemented
2	#23754590	C₂	1	1	yes	maximal, characteristic, complemented
3	#23754588	C₂	1	1	yes	maximal, characteristic, complemented
4	#23754587	C₂	1	1	yes	maximal, characteristic, complemented
5	#23754585	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×#19096921	C₂	1	1	yes	maximal, characteristic, complemented
7	#41×C₆₁₇	C₂	1	1	yes	maximal, characteristic, complemented
8	Dic₁₂₃₄	C₂²	1	1	yes	characteristic
9	D₂₄₆₈	C₂²	1	1	yes	characteristic
10	C₂²⋊D₆₁₇	—	2	1	no	—
11	C₄×D₆₁₇	—	2	1	no	—
12	#19096921	C₂²	1	1	yes	characteristic
13	#19096921	C₂²	1	1	yes	characteristic
14	Q₈×C₆₁₇	C₂²	1	1	yes	characteristic
15	D₄×C₆₁₇	—	2	1	no	—
16	D₄×C₆₁₇	C₂²	1	1	yes	characteristic
17	C₂×C₄×C₆₁₇	—	2	1	no	—
18	C₂×C₄×C₆₁₇	C₂²	1	1	yes	characteristic
19	C₂²×C₆₁₇	—	2	1	no	—
20	C₂²×C₆₁₇	D₄	1	1	yes	characteristic
21	C₂×D₆₁₇	—	2	1	no	—
22	C₄×C₆₁₇	—	2	1	no	—
23	C₄×C₆₁₇	D₄	1	1	yes	characteristic
24	C₄×C₆₁₇	C₂³	1	1	yes	characteristic
25	Dic₆₁₇	—	2	1	no	—
26	C₂×C₆₁₇	—	2	1	no	—
27	C₂×C₆₁₇	—	4	1	no	—
28	C₂×C₆₁₇	C₂×D₄	1	1	yes	characteristic
29	D₆₁₇	—	4	1	no	—
30	C₆₁₇	#135	1	1	yes	characteristic, complemented, Sylow
31	#135	—	617	1	no	maximal, Sylow
32	#41	—	617	1	no	—
33	#41	D₆₁₇	1	1	yes	characteristic, complemented
34	Q₁₆	—	617	1	no	—
35	SD₁₆	—	617	1	no	—
36	SD₁₆	—	617	1	no	—
37	D₈	—	617	1	no	—
38	C₂×C₈	—	617	1	no	—
39	Q₈	—	617	1	no	—
40	Q₈	C₂×D₆₁₇	1	1	yes	characteristic
41	D₄	—	2	1	no	—
42	D₄	—	617	1	no	—
43	D₄	—	1234	1	no	—
44	D₄	C₂×D₆₁₇	1	1	yes	characteristic
45	C₂×C₄	—	2	1	no	—
46	C₂×C₄	—	1234	1	no	—
47	C₂×C₄	C₂×D₆₁₇	1	1	yes	characteristic
48	C₈	—	617	1	no	—
49	C₈	—	617	1	no	—
50	C₂²	—	2	1	no	—
51	C₂²	—	1234	1	no	—
52	C₂²	C₂²⋊D₆₁₇	1	1	yes	characteristic
53	C₄	—	2	1	no	—
54	C₄	—	1234	1	no	—
55	C₄	C₂²⋊D₆₁₇	1	1	yes	characteristic
56	C₄	C₂²×D₆₁₇	1	1	yes	characteristic
57	C₂	—	2	1	no	—
58	C₂	—	4	1	no	—
59	C₂	—	2468	1	no	—
60	C₂	C₂×C₂²⋊D₆₁₇	1	1	yes	characteristic

	subgroup	G/H
center	C₄	C₂²⋊D₆₁₇
commutator, cosocle	C₄×C₆₁₇	C₂³
Frattini	C₄	C₂²×D₆₁₇
Fitting	#41×C₆₁₇	C₂
socle	C₂×C₆₁₇	C₂×D₄
hypercenter	#41	D₆₁₇
hypocenter	C₆₁₇	#135
LCS	G ▷ C₄×C₆₁₇ ▷ C₂×C₆₁₇ ▷ C₆₁₇
UCS	C₁ ◁ C₄ ◁ C₂×C₄ ◁ #41

Complements

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

N	H ≅ G/N	type
1. #23754600	C₂	semidirect
2. #23754590	C₂	semidirect
3. #23754588	C₂	semidirect
4. #23754587	C₂	semidirect
5. #23754585	C₂	semidirect
6. C₂×#19096921	C₂	semidirect
7. #41×C₆₁₇	C₂	semidirect
30. C₆₁₇	#135	semidirect
33. #41	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
#23754590	2
#23754588	2
#23754587	2
#23754585	2
C₂×#19096921	2
#41×C₆₁₇	2
#135	617