Subgroups — #34089697

Order 19744.

60 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄⋊Dic₆₁₇	C₂	1	1	yes	maximal, characteristic, complemented
2	#23754579	C₂	1	1	yes	maximal, characteristic, complemented
3	#23754577	C₂	1	1	yes	maximal, characteristic
4	#23754577	C₂	1	1	yes	maximal, characteristic
5	#23754577	C₂	1	1	yes	maximal, characteristic
6	C₄×Dic₆₁₇	C₂	1	1	yes	maximal, characteristic, complemented
7	C₄⋊C₄×C₆₁₇	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂²×D₆₁₇	C₂²	1	1	yes	characteristic
9	C₂×Dic₆₁₇	C₂²	1	1	yes	characteristic
10	C₂×Dic₆₁₇	C₂²	1	1	yes	characteristic
11	C₂×Dic₆₁₇	C₂²	1	1	yes	characteristic
12	C₂×C₄×C₆₁₇	C₂²	1	1	yes	characteristic
13	C₂×C₄×C₆₁₇	C₂²	1	1	yes	characteristic
14	C₂×C₄×C₆₁₇	C₂²	1	1	yes	characteristic
15	C₂²×C₆₁₇	C₂³	1	1	yes	characteristic
16	C₂×D₆₁₇	—	2	1	no	—
17	C₂×D₆₁₇	—	2	1	no	—
18	C₂×D₆₁₇	—	2	1	no	—
19	C₄×C₆₁₇	—	2	1	no	—
20	C₄×C₆₁₇	—	2	1	no	—
21	C₄×C₆₁₇	—	2	1	no	—
22	Dic₆₁₇	—	2	1	no	—
23	Dic₆₁₇	—	2	1	no	—
24	Dic₆₁₇	—	2	1	no	—
25	C₂×C₆₁₇	#41	1	1	yes	characteristic
26	C₂×C₆₁₇	#41	1	1	yes	characteristic
27	C₂×C₆₁₇	#41	1	1	yes	characteristic
28	D₆₁₇	—	4	1	no	—
29	C₆₁₇	#126	1	1	yes	characteristic, complemented, Sylow
30	#126	—	617	1	no	maximal, Sylow
31	C₄⋊C₄	—	617	1	no	—
32	C₄⋊C₄	—	617	1	no	—
33	C₄⋊C₄	D₆₁₇	1	1	yes	characteristic, complemented
34	C₂²⋊C₄	—	617	1	no	—
35	C₂²⋊C₄	—	617	1	no	—
36	C₂²⋊C₄	—	617	1	no	—
37	C₄²	—	617	1	no	—
38	C₂³	—	617	1	no	—
39	C₂×C₄	—	617	1	no	—
40	C₂×C₄	—	617	1	no	—
41	C₂×C₄	—	617	1	no	—
42	C₂×C₄	C₂×D₆₁₇	1	1	yes	characteristic
43	C₂×C₄	C₂×D₆₁₇	1	1	yes	characteristic
44	C₂×C₄	C₂×D₆₁₇	1	1	yes	characteristic
45	C₂²	—	1234	1	no	—
46	C₂²	—	1234	1	no	—
47	C₂²	—	1234	1	no	—
48	C₂²	C₂²×D₆₁₇	1	1	yes	characteristic
49	C₄	—	2	1	no	—
50	C₄	—	2	1	no	—
51	C₄	—	2	1	no	—
52	C₄	—	1234	1	no	—
53	C₄	—	1234	1	no	—
54	C₄	—	1234	1	no	—
55	C₂	—	2468	1	no	—
56	C₂	#23754599	1	1	yes	characteristic
57	C₂	#23754600	1	1	yes	characteristic
58	C₂	#23754601	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂²	C₂²×D₆₁₇
commutator, socle, cosocle	C₂²×C₆₁₇	C₂³
Fitting	C₄⋊C₄×C₆₁₇	C₂
prime radical	C₂²×D₆₁₇	C₂²
hypercenter	C₄⋊C₄	D₆₁₇
hypocenter	C₆₁₇	#126
LCS	G ▷ C₂²×C₆₁₇ ▷ C₆₁₇
UCS	C₁ ◁ C₂² ◁ C₄⋊C₄

Complements

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

N	H ≅ G/N	type
1. C₄⋊Dic₆₁₇	C₂	semidirect
2. #23754579	C₂	semidirect
6. C₄×Dic₆₁₇	C₂	semidirect
7. C₄⋊C₄×C₆₁₇	C₂	semidirect
29. C₆₁₇	#126	semidirect
33. C₄⋊C₄	D₆₁₇	semidirect

Maximal subgroups

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

subgroup	index
C₄⋊Dic₆₁₇	2
#23754579	2
#23754577	2
C₄×Dic₆₁₇	2
C₄⋊C₄×C₆₁₇	2
#126	617