Subgroups — #34089702

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	4	yes	maximal
3	C₄²×C₆₁₇	C₂	1	1	yes	maximal, characteristic
4	Dic₁₂₃₄	—	2	4	no	—
5	C₂×Dic₆₁₇	C₂²	1	4	yes	—
6	C₂×C₄×C₆₁₇	C₂²	1	1	yes	characteristic
7	C₂×C₄×C₆₁₇	C₂²	1	2	yes	—
8	C₂²×C₆₁₇	C₂³	1	1	yes	characteristic
9	C₄×C₆₁₇	D₄	1	2	yes	—
10	C₄×C₆₁₇	Q₈	1	4	yes	complemented
11	Dic₆₁₇	—	2	4	no	—
12	C₂×C₆₁₇	C₂×D₄	1	1	yes	characteristic
13	C₂×C₆₁₇	C₂×Q₈	1	2	yes	—
14	C₆₁₇	C₄⋊Q₈	1	1	yes	characteristic, complemented, Sylow
15	C₄⋊Q₈	—	617	1	no	maximal, Sylow
16	C₂×Q₈	—	617	2	no	—
17	C₄⋊C₄	—	617	4	no	—
18	C₄²	D₆₁₇	1	1	yes	characteristic
19	Q₈	—	1234	4	no	—
20	C₂×C₄	—	617	4	no	—
21	C₂×C₄	C₂×D₆₁₇	1	1	yes	characteristic
22	C₂×C₄	C₂×D₆₁₇	1	2	yes	—
23	C₂²	C₂²×D₆₁₇	1	1	yes	characteristic
24	C₄	—	1234	4	no	—
25	C₄	D₂₄₆₈	1	2	yes	—
26	C₄	Dic₁₂₃₄	1	4	yes	complemented
27	C₂	C₂×D₂₄₆₈	1	1	yes	characteristic
28	C₂	C₂×Dic₁₂₃₄	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	C₄⋊Q₈
LCS	G ▷ C₂²×C₆₁₇ ▷ C₆₁₇
UCS	C₁ ◁ C₂² ◁ C₄²

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

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