Subgroups — #34089701

Order 19744.

68 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×Dic₁₂₃₄	C₂	1	2	yes	maximal
2	C₄⋊Dic₆₁₇	C₂	1	1	yes	maximal, characteristic
3	#23754579	C₂	1	2	yes	maximal
4	C₄×Dic₆₁₇	C₂	1	1	yes	maximal, characteristic
5	C₄⋊C₄×C₆₁₇	C₂	1	1	yes	maximal, characteristic
6	Dic₁₂₃₄	—	2	4	no	—
7	C₂×Dic₆₁₇	C₂²	1	2	yes	—
8	C₂×Dic₆₁₇	C₂²	1	1	yes	characteristic
9	C₂×Dic₆₁₇	C₂²	1	1	yes	characteristic
10	C₂×C₄×C₆₁₇	C₂²	1	2	yes	—
11	C₂×C₄×C₆₁₇	C₂²	1	1	yes	characteristic
12	C₂²×C₆₁₇	C₂³	1	1	yes	characteristic
13	C₄×C₆₁₇	—	2	2	no	—
14	C₄×C₆₁₇	Q₈	1	2	yes	complemented
15	Dic₆₁₇	—	2	2	no	—
16	Dic₆₁₇	D₄	1	2	yes	—
17	Dic₆₁₇	Q₈	1	2	yes	complemented
18	C₂×C₆₁₇	C₂×D₄	1	1	yes	characteristic
19	C₂×C₆₁₇	C₂×Q₈	1	1	yes	characteristic
20	C₂×C₆₁₇	C₂×Q₈	1	1	yes	characteristic
21	C₆₁₇	C₄⋊Q₈	1	1	yes	characteristic, complemented, Sylow
22	C₄⋊Q₈	—	617	1	no	maximal, Sylow
23	C₂×Q₈	—	617	2	no	—
24	C₄⋊C₄	—	617	2	no	—
25	C₄⋊C₄	—	617	1	no	—
26	C₄⋊C₄	D₆₁₇	1	1	yes	characteristic
27	C₄²	—	617	1	no	—
28	Q₈	—	1234	4	no	—
29	C₂×C₄	—	617	1	no	—
30	C₂×C₄	—	617	1	no	—
31	C₂×C₄	—	617	2	no	—
32	C₂×C₄	C₂×D₆₁₇	1	2	yes	—
33	C₂×C₄	C₂×D₆₁₇	1	1	yes	characteristic
34	C₂²	C₂²×D₆₁₇	1	1	yes	characteristic
35	C₄	—	2	2	no	—
36	C₄	—	617	2	no	—
37	C₄	—	617	2	no	—
38	C₄	—	1234	2	no	—
39	C₄	Dic₁₂₃₄	1	2	yes	complemented
40	C₂	D₄×D₆₁₇	1	1	yes	characteristic
41	C₂	Q₈×D₆₁₇	1	1	yes	characteristic
42	C₂	C₂×Dic₁₂₃₄	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂²	C₂²×D₆₁₇
commutator, socle, prime radical, cosocle	C₂²×C₆₁₇	C₂³
Fitting	C₄⋊C₄×C₆₁₇	C₂
hypercenter	C₄⋊C₄	D₆₁₇
hypocenter	C₆₁₇	C₄⋊Q₈
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
14. C₄×C₆₁₇	Q₈	semidirect
17. Dic₆₁₇	Q₈	semidirect
21. C₆₁₇	C₄⋊Q₈	semidirect
39. C₄	Dic₁₂₃₄	semidirect

Maximal subgroups

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

subgroup	index
C₂×Dic₁₂₃₄	2
C₄⋊Dic₆₁₇	2
#23754579	2
C₄×Dic₆₁₇	2
C₄⋊C₄×C₆₁₇	2
C₄⋊Q₈	617