Subgroups — #36345995

Order 21856.

74 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, complemented
2	C₄⋊Dic₆₈₃	C₂	1	1	yes	maximal, characteristic, complemented
3	Dic₆₈₃⋊C₄	C₂	1	2	yes	maximal, complemented
4	C₂²⋊Dic₆₈₃	C₂	1	2	yes	maximal
5	C₂²×C₄×C₆₈₃	C₂	1	1	yes	maximal, characteristic
6	Dic₁₃₆₆	—	2	2	no	—
7	C₂×Dic₆₈₃	C₂²	1	2	yes	—
8	C₂×Dic₆₈₃	C₂²	1	2	yes	—
9	C₂³×C₆₈₃	C₂²	1	1	yes	characteristic
10	C₂×C₄×C₆₈₃	—	2	2	no	—
11	C₂×C₄×C₆₈₃	C₂²	1	1	yes	characteristic
12	C₂×C₄×C₆₈₃	C₂²	1	1	yes	characteristic
13	C₂²×C₆₈₃	—	2	1	no	—
14	C₂²×C₆₈₃	—	2	1	no	—
15	C₂²×C₆₈₃	Q₈	1	2	yes	complemented
16	C₂²×C₆₈₃	C₂³	1	1	yes	characteristic
17	C₄×C₆₈₃	—	2	1	no	—
18	C₄×C₆₈₃	D₄	1	2	yes	—
19	Dic₆₈₃	—	2	2	no	—
20	Dic₆₈₃	—	2	2	no	—
21	C₂×C₆₈₃	—	2	2	no	—
22	C₂×C₆₈₃	C₂×D₄	1	1	yes	characteristic
23	C₂×C₆₈₃	C₂×Q₈	1	1	yes	characteristic
24	C₂×C₆₈₃	#41	1	1	yes	characteristic
25	C₆₈₃	C₂²⋊Q₈	1	1	yes	characteristic, complemented, Sylow
26	C₂²⋊Q₈	—	683	1	no	maximal, Sylow
27	C₂×Q₈	—	683	1	no	—
28	C₂²×C₄	D₆₈₃	1	1	yes	characteristic
29	C₄⋊C₄	—	683	2	no	—
30	C₄⋊C₄	—	683	1	no	—
31	C₂²⋊C₄	—	683	2	no	—
32	C₂³	C₂×D₆₈₃	1	1	yes	characteristic
33	Q₈	—	1366	2	no	—
34	C₂×C₄	—	2	2	no	—
35	C₂×C₄	—	683	2	no	—
36	C₂×C₄	—	683	2	no	—
37	C₂×C₄	C₂×D₆₈₃	1	1	yes	characteristic
38	C₂×C₄	C₂×D₆₈₃	1	1	yes	characteristic
39	C₂²	—	2	1	no	—
40	C₂²	—	2	1	no	—
41	C₂²	Dic₁₃₆₆	1	2	yes	complemented
42	C₂²	C₂²×D₆₈₃	1	1	yes	characteristic
43	C₄	—	2	1	no	—
44	C₄	—	1366	2	no	—
45	C₄	—	1366	2	no	—
46	C₄	C₂²⋊D₆₈₃	1	2	yes	—
47	C₂	—	2	2	no	—
48	C₂	C₂×C₂²⋊D₆₈₃	1	1	yes	characteristic
49	C₂	C₂×Dic₁₃₆₆	1	1	yes	characteristic
50	C₂	#24926421	1	1	yes	characteristic

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

Complements

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

N	H ≅ G/N	type
1. C₂×Dic₁₃₆₆	C₂	semidirect
2. C₄⋊Dic₆₈₃	C₂	semidirect
3. Dic₆₈₃⋊C₄	C₂	semidirect
15. C₂²×C₆₈₃	Q₈	semidirect
25. C₆₈₃	C₂²⋊Q₈	semidirect
41. C₂²	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
C₂²⋊Dic₆₈₃	2
C₂²×C₄×C₆₈₃	2
C₂²⋊Q₈	683