Subgroups — #34089698

Order 19744.

108 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	C₂×C₂²⋊D₆₁₇	C₂	1	4	yes	maximal, complemented
3	C₄×Dic₆₁₇	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×D₄×C₆₁₇	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂²×D₆₁₇	C₂²	1	2	yes	—
6	D₂₄₆₈	—	2	2	no	—
7	C₂²⋊D₆₁₇	—	2	8	no	—
8	C₂×Dic₆₁₇	C₂²	1	2	yes	—
9	C₂³×C₆₁₇	C₂²	1	2	yes	—
10	D₄×C₆₁₇	—	2	2	no	—
11	C₂×C₄×C₆₁₇	C₂²	1	1	yes	characteristic
12	C₂²×C₆₁₇	—	2	2	no	—
13	C₂²×C₆₁₇	—	2	4	no	—
14	C₂²×C₆₁₇	C₂³	1	1	yes	characteristic
15	C₂×D₆₁₇	—	2	4	no	—
16	C₂×D₆₁₇	—	2	2	no	—
17	C₄×C₆₁₇	D₄	1	2	yes	complemented
18	Dic₆₁₇	D₄	1	4	yes	complemented
19	C₂×C₆₁₇	—	4	2	no	—
20	C₂×C₆₁₇	C₂×D₄	1	2	yes	—
21	C₂×C₆₁₇	C₂×D₄	1	1	yes	characteristic
22	D₆₁₇	—	4	2	no	—
23	C₆₁₇	#127	1	1	yes	characteristic, complemented, Sylow
24	#127	—	617	1	no	maximal, Sylow
25	C₂×D₄	—	617	4	no	—
26	C₂×D₄	—	617	1	no	—
27	C₂×D₄	D₆₁₇	1	1	yes	characteristic, complemented
28	C₄²	—	617	1	no	—
29	C₂³	—	617	2	no	—
30	C₂³	C₂×D₆₁₇	1	2	yes	—
31	D₄	—	2	2	no	—
32	D₄	—	1234	8	no	—
33	D₄	—	1234	2	no	—
34	C₂×C₄	—	617	2	no	—
35	C₂×C₄	C₂×D₆₁₇	1	1	yes	characteristic
36	C₂²	—	2	4	no	—
37	C₂²	—	2	2	no	—
38	C₂²	—	1234	4	no	—
39	C₂²	—	1234	2	no	—
40	C₂²	C₂²×D₆₁₇	1	1	yes	characteristic
41	C₄	—	617	4	no	—
42	C₄	C₂²⋊D₆₁₇	1	2	yes	complemented
43	C₂	—	4	2	no	—
44	C₂	—	2468	2	no	—
45	C₂	D₄×D₆₁₇	1	2	yes	—
46	C₂	C₂×C₂²⋊D₆₁₇	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂²	C₂²×D₆₁₇
commutator, socle, cosocle	C₂²×C₆₁₇	C₂³
Fitting	C₂×D₄×C₆₁₇	C₂
hypercenter	C₂×D₄	D₆₁₇
hypocenter	C₆₁₇	#127
LCS	G ▷ C₂²×C₆₁₇ ▷ C₆₁₇
UCS	C₁ ◁ C₂² ◁ C₂×D₄

Complements

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

N	H ≅ G/N	type
1. C₂×D₂₄₆₈	C₂	semidirect
2. C₂×C₂²⋊D₆₁₇	C₂	semidirect
3. C₄×Dic₆₁₇	C₂	semidirect
4. C₂×D₄×C₆₁₇	C₂	semidirect
17. C₄×C₆₁₇	D₄	semidirect
18. Dic₆₁₇	D₄	semidirect
23. C₆₁₇	#127	semidirect
27. C₂×D₄	D₆₁₇	semidirect
42. C₄	C₂²⋊D₆₁₇	semidirect

Maximal subgroups

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

subgroup	index
C₂×D₂₄₆₈	2
C₂×C₂²⋊D₆₁₇	2
C₄×Dic₆₁₇	2
C₂×D₄×C₆₁₇	2
#127	617