Subgroups — #34089694

Order 19744.

56 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄⋊Dic₆₁₇	C₂	1	2	yes	maximal
2	#23754579	C₂	1	4	yes	maximal
3	C₄²×C₆₁₇	C₂	1	1	yes	maximal, characteristic
4	C₂×Dic₆₁₇	C₂²	1	4	yes	—
5	C₂×C₄×C₆₁₇	C₂²	1	2	yes	—
6	C₂×C₄×C₆₁₇	C₂²	1	1	yes	characteristic
7	C₂²×C₆₁₇	C₂³	1	1	yes	characteristic
8	C₄×C₆₁₇	—	2	2	no	—
9	C₄×C₆₁₇	Q₈	1	2	yes	—
10	Dic₆₁₇	—	2	4	no	—
11	C₂×C₆₁₇	C₂×Q₈	1	1	yes	characteristic
12	C₂×C₆₁₇	#41	1	2	yes	—
13	C₆₁₇	#125	1	1	yes	characteristic, complemented, Sylow
14	#125	—	617	1	no	maximal, Sylow
15	C₄⋊C₄	—	617	4	no	—
16	C₄⋊C₄	—	617	2	no	—
17	C₄²	D₆₁₇	1	1	yes	characteristic
18	C₂×C₄	—	617	4	no	—
19	C₂×C₄	C₂×D₆₁₇	1	2	yes	—
20	C₂×C₄	C₂×D₆₁₇	1	1	yes	characteristic
21	C₂²	C₂²×D₆₁₇	1	1	yes	characteristic
22	C₄	—	2	2	no	—
23	C₄	—	1234	4	no	—
24	C₄	Dic₁₂₃₄	1	2	yes	—
25	C₂	C₂×Dic₁₂₃₄	1	1	yes	characteristic
26	C₂	#23754600	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂²	C₂²×D₆₁₇
commutator, socle, prime radical, cosocle	C₂²×C₆₁₇	C₂³
Fitting	C₄²×C₆₁₇	C₂
hypercenter	C₄²	D₆₁₇
hypocenter	C₆₁₇	#125
LCS	G ▷ C₂²×C₆₁₇ ▷ C₆₁₇
UCS	C₁ ◁ C₂² ◁ C₄²

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

N	H ≅ G/N	type
13. C₆₁₇	#125	semidirect

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