Subgroups — #36346014

Order 21856.

68 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×Dic₁₃₆₆	C₂	1	1	yes	maximal, characteristic
2	Dic₆₈₃⋊C₄	C₂	1	4	yes	maximal
3	C₄×Dic₆₈₃	C₂	1	1	yes	maximal, characteristic
4	C₂×Q₈×C₆₈₃	C₂	1	1	yes	maximal, characteristic
5	Dic₁₃₆₆	—	2	2	no	—
6	C₂×Dic₆₈₃	C₂²	1	2	yes	—
7	C₂×Dic₆₈₃	C₂²	1	2	yes	—
8	Q₈×C₆₈₃	—	2	2	no	—
9	C₂×C₄×C₆₈₃	C₂²	1	2	yes	—
10	C₂×C₄×C₆₈₃	C₂²	1	1	yes	characteristic
11	C₂²×C₆₈₃	C₂³	1	1	yes	characteristic
12	C₄×C₆₈₃	—	2	2	no	—
13	C₄×C₆₈₃	D₄	1	2	yes	—
14	Dic₆₈₃	—	2	2	no	—
15	Dic₆₈₃	Q₈	1	4	yes	complemented
16	C₂×C₆₈₃	C₂×D₄	1	1	yes	characteristic
17	C₂×C₆₈₃	C₂×Q₈	1	2	yes	—
18	C₆₈₃	C₄⋊Q₈	1	1	yes	characteristic, complemented, Sylow
19	C₄⋊Q₈	—	683	1	no	maximal, Sylow
20	C₂×Q₈	—	683	1	no	—
21	C₂×Q₈	D₆₈₃	1	1	yes	characteristic
22	C₄⋊C₄	—	683	4	no	—
23	C₄²	—	683	1	no	—
24	Q₈	—	2	2	no	—
25	Q₈	—	1366	2	no	—
26	C₂×C₄	—	683	2	no	—
27	C₂×C₄	—	683	2	no	—
28	C₂×C₄	C₂×D₆₈₃	1	2	yes	—
29	C₂×C₄	C₂×D₆₈₃	1	1	yes	characteristic
30	C₂²	C₂²×D₆₈₃	1	1	yes	characteristic
31	C₄	—	2	2	no	—
32	C₄	—	1366	2	no	—
33	C₄	—	683	4	no	—
34	C₄	C₂²⋊D₆₈₃	1	2	yes	—
35	C₂	C₂×C₂²⋊D₆₈₃	1	1	yes	characteristic
36	C₂	Q₈×D₆₈₃	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂²	C₂²×D₆₈₃
commutator, socle, prime radical, cosocle	C₂²×C₆₈₃	C₂³
Fitting	C₂×Q₈×C₆₈₃	C₂
hypercenter	C₂×Q₈	D₆₈₃
hypocenter	C₆₈₃	C₄⋊Q₈
LCS	G ▷ C₂²×C₆₈₃ ▷ C₆₈₃
UCS	C₁ ◁ C₂² ◁ C₂×Q₈

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

N	H ≅ G/N	type
15. Dic₆₈₃	Q₈	semidirect
18. C₆₈₃	C₄⋊Q₈	semidirect

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