Subgroups — C₂×#23754585 #34089712

Order 19744.

76 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×D₂₄₆₈	C₂	1	1	yes	maximal, characteristic, complemented
2	#23754585	C₂	1	4	yes	maximal, direct product factor
3	C₂×#19096921	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×D₄×C₆₁₇	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂²×D₆₁₇	—	2	1	no	—
6	D₂₄₆₈	—	2	1	no	—
7	D₂₄₆₈	C₂²	1	2	yes	complemented
8	#19096921	C₂²	1	2	yes	complemented
9	C₂³×C₆₁₇	—	2	1	no	—
10	D₄×C₆₁₇	—	2	1	no	—
11	D₄×C₆₁₇	C₂²	1	2	yes	complemented
12	C₂×C₄×C₆₁₇	C₂²	1	1	yes	characteristic
13	C₂²×C₆₁₇	—	2	2	no	—
14	C₂²×C₆₁₇	—	4	2	no	—
15	C₂²×C₆₁₇	D₄	1	1	yes	characteristic
16	C₂×D₆₁₇	—	2	2	no	—
17	C₂×D₆₁₇	—	4	2	no	—
18	C₄×C₆₁₇	D₄	1	1	yes	characteristic
19	C₄×C₆₁₇	C₂³	1	1	yes	characteristic
20	C₂×C₆₁₇	—	4	2	no	—
21	C₂×C₆₁₇	D₈	1	2	yes	complemented
22	C₂×C₆₁₇	C₂×D₄	1	1	yes	characteristic
23	D₆₁₇	—	4	2	no	—
24	C₆₁₇	C₂×D₈	1	1	yes	characteristic, complemented, Sylow
25	C₂×D₈	—	617	1	no	maximal, Sylow
26	C₂×D₄	—	617	1	no	—
27	C₂×D₄	D₆₁₇	1	1	yes	characteristic, complemented
28	D₈	—	617	4	no	—
29	C₂×C₈	—	617	1	no	—
30	C₂³	—	2	1	no	—
31	C₂³	—	1234	1	no	—
32	D₄	—	2	1	no	—
33	D₄	—	617	2	no	—
34	D₄	—	1234	1	no	—
35	D₄	C₂×D₆₁₇	1	2	yes	complemented
36	C₂×C₄	C₂×D₆₁₇	1	1	yes	characteristic
37	C₈	—	617	2	no	—
38	C₂²	—	2	2	no	—
39	C₂²	—	4	2	no	—
40	C₂²	—	2468	2	no	—
41	C₂²	—	1234	2	no	—
42	C₂²	C₂²⋊D₆₁₇	1	1	yes	characteristic
43	C₄	C₂²⋊D₆₁₇	1	1	yes	characteristic
44	C₄	C₂²×D₆₁₇	1	1	yes	characteristic
45	C₂	—	4	2	no	—
46	C₂	—	2468	2	no	—
47	C₂	#23754585	1	2	yes	direct product factor
48	C₂	C₂×C₂²⋊D₆₁₇	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂²⋊D₆₁₇
commutator, cosocle	C₄×C₆₁₇	C₂³
Frattini	C₄	C₂²×D₆₁₇
Fitting	C₂×D₄×C₆₁₇	C₂
socle	C₂²×C₆₁₇	D₄
hypercenter	C₂×D₄	D₆₁₇
hypocenter	C₆₁₇	C₂×D₈
LCS	G ▷ C₄×C₆₁₇ ▷ C₂×C₆₁₇ ▷ C₆₁₇
UCS	C₁ ◁ C₂² ◁ C₂×C₄ ◁ C₂×D₄

Complements

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

N	H ≅ G/N	type
1. C₂×D₂₄₆₈	C₂	semidirect
2. #23754585	C₂	direct (N × H)
3. C₂×#19096921	C₂	semidirect
4. C₂×D₄×C₆₁₇	C₂	semidirect
7. D₂₄₆₈	C₂²	semidirect
8. #19096921	C₂²	semidirect
11. D₄×C₆₁₇	C₂²	semidirect
21. C₂×C₆₁₇	D₈	semidirect
24. C₆₁₇	C₂×D₈	semidirect
27. C₂×D₄	D₆₁₇	semidirect
35. D₄	C₂×D₆₁₇	semidirect
47. C₂	#23754585	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×D₂₄₆₈	2
#23754585	2
C₂×#19096921	2
C₂×D₄×C₆₁₇	2
C₂×D₈	617

Subgroups — C2×#23754585 #34089712

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#23754585 #34089712