Subgroups — #34089677

Order 19744.

94 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×D₂₄₆₈	C₂	1	2	yes	maximal, complemented
2	C₂×D₂₄₆₈	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×C₄×D₆₁₇	C₂	1	1	yes	maximal, characteristic, complemented
4	#23754577	C₂	1	2	yes	maximal, complemented
5	C₄⋊C₄×C₆₁₇	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂²×D₆₁₇	C₂²	1	2	yes	—
7	C₂²×D₆₁₇	C₂²	1	1	yes	characteristic
8	D₂₄₆₈	—	2	4	no	—
9	D₂₄₆₈	—	2	2	no	—
10	C₂×Dic₆₁₇	C₂²	1	1	yes	characteristic
11	C₄×D₆₁₇	—	2	2	no	—
12	C₂×C₄×C₆₁₇	C₂²	1	2	yes	—
13	C₂×C₄×C₆₁₇	C₂²	1	1	yes	characteristic
14	C₂²×C₆₁₇	C₂³	1	1	yes	characteristic
15	C₂×D₆₁₇	—	2	1	no	—
16	C₂×D₆₁₇	—	2	1	no	—
17	C₂×D₆₁₇	—	2	2	no	—
18	C₂×D₆₁₇	—	2	2	no	—
19	C₂×D₆₁₇	—	2	2	no	—
20	C₂×D₆₁₇	D₄	1	2	yes	complemented
21	C₄×C₆₁₇	—	2	2	no	—
22	C₄×C₆₁₇	D₄	1	2	yes	complemented
23	Dic₆₁₇	—	2	1	no	—
24	C₂×C₆₁₇	C₂×D₄	1	1	yes	characteristic
25	C₂×C₆₁₇	C₂×D₄	1	1	yes	characteristic
26	C₂×C₆₁₇	#41	1	1	yes	characteristic
27	D₆₁₇	—	2	2	no	—
28	D₆₁₇	—	4	2	no	—
29	C₆₁₇	#121	1	1	yes	characteristic, complemented, Sylow
30	#121	—	617	1	no	maximal, Sylow
31	C₂×D₄	—	617	2	no	—
32	C₂×D₄	—	617	1	no	—
33	C₂²×C₄	—	617	1	no	—
34	C₄⋊C₄	D₆₁₇	1	1	yes	characteristic, complemented
35	C₂²⋊C₄	—	617	2	no	—
36	C₂³	—	617	1	no	—
37	C₂³	—	617	2	no	—
38	D₄	—	1234	4	no	—
39	D₄	—	1234	2	no	—
40	C₂×C₄	—	617	1	no	—
41	C₂×C₄	—	1234	2	no	—
42	C₂×C₄	C₂×D₆₁₇	1	2	yes	—
43	C₂×C₄	C₂×D₆₁₇	1	1	yes	characteristic
44	C₂²	—	617	2	no	—
45	C₂²	—	1234	2	no	—
46	C₂²	—	1234	2	no	—
47	C₂²	—	1234	2	no	—
48	C₂²	—	1234	1	no	—
49	C₂²	—	1234	1	no	—
50	C₂²	C₂²×D₆₁₇	1	1	yes	characteristic
51	C₄	—	2	2	no	—
52	C₄	—	1234	1	no	—
53	C₄	D₂₄₆₈	1	2	yes	complemented
54	C₂	—	2468	2	no	—
55	C₂	—	1234	2	no	—
56	C₂	D₄×D₆₁₇	1	1	yes	characteristic
57	C₂	C₂×D₂₄₆₈	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
hypercenter	C₄⋊C₄	D₆₁₇
hypocenter	C₆₁₇	#121
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. C₂×D₂₄₆₈	C₂	semidirect
2. C₂×D₂₄₆₈	C₂	semidirect
3. C₂×C₄×D₆₁₇	C₂	semidirect
4. #23754577	C₂	semidirect
5. C₄⋊C₄×C₆₁₇	C₂	semidirect
20. C₂×D₆₁₇	D₄	semidirect
22. C₄×C₆₁₇	D₄	semidirect
29. C₆₁₇	#121	semidirect
34. C₄⋊C₄	D₆₁₇	semidirect
53. C₄	D₂₄₆₈	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×C₄×D₆₁₇	2
#23754577	2
C₄⋊C₄×C₆₁₇	2
#121	617