Subgroups — #36346015

Order 21856.

68 conjugacy classes of subgroups.

Conjugacy classes of subgroups

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

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

Complements

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

N	H ≅ G/N	type
14. C₄×C₆₈₃	Q₈	semidirect
17. Dic₆₈₃	Q₈	semidirect
21. C₆₈₃	C₄⋊Q₈	semidirect
39. C₄	Dic₁₃₆₆	semidirect

Maximal subgroups

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

subgroup	index
C₂×Dic₁₃₆₆	2
C₄⋊Dic₆₈₃	2
Dic₆₈₃⋊C₄	2
C₄×Dic₆₈₃	2
C₄⋊C₄×C₆₈₃	2
C₄⋊Q₈	683