Subgroups — #36345996

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	C₂×C₄×D₆₈₃	C₂	1	1	yes	maximal, characteristic
3	C₄⋊Dic₆₈₃	C₂	1	2	yes	maximal, complemented
4	(C₂×D₆₈₃)⋊C₄	C₂	1	2	yes	maximal
5	C₄⋊C₄×C₆₈₃	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂²×D₆₈₃	C₂²	1	1	yes	characteristic
7	Dic₁₃₆₆	—	2	2	no	—
8	C₂×Dic₆₈₃	C₂²	1	2	yes	—
9	C₂×Dic₆₈₃	C₂²	1	1	yes	characteristic
10	C₄×D₆₈₃	—	2	2	no	—
11	C₂×C₄×C₆₈₃	C₂²	1	2	yes	—
12	C₂×C₄×C₆₈₃	C₂²	1	1	yes	characteristic
13	C₂²×C₆₈₃	C₂³	1	1	yes	characteristic
14	C₂×D₆₈₃	—	2	1	no	—
15	C₂×D₆₈₃	—	2	1	no	—
16	C₂×D₆₈₃	Q₈	1	2	yes	complemented
17	C₄×C₆₈₃	—	2	2	no	—
18	C₄×C₆₈₃	D₄	1	2	yes	—
19	Dic₆₈₃	—	2	1	no	—
20	Dic₆₈₃	—	2	2	no	—
21	C₂×C₆₈₃	C₂×D₄	1	1	yes	characteristic
22	C₂×C₆₈₃	C₂×Q₈	1	1	yes	characteristic
23	C₂×C₆₈₃	#41	1	1	yes	characteristic
24	D₆₈₃	—	2	2	no	—
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	—	683	1	no	—
29	C₄⋊C₄	—	683	2	no	—
30	C₄⋊C₄	D₆₈₃	1	1	yes	characteristic, complemented
31	C₂²⋊C₄	—	683	2	no	—
32	C₂³	—	683	1	no	—
33	Q₈	—	1366	2	no	—
34	C₂×C₄	—	1366	2	no	—
35	C₂×C₄	—	683	2	no	—
36	C₂×C₄	—	683	1	no	—
37	C₂×C₄	C₂×D₆₈₃	1	2	yes	—
38	C₂×C₄	C₂×D₆₈₃	1	1	yes	characteristic
39	C₂²	—	1366	1	no	—
40	C₂²	—	1366	1	no	—
41	C₂²	—	683	2	no	—
42	C₂²	C₂²×D₆₈₃	1	1	yes	characteristic
43	C₄	—	2	2	no	—
44	C₄	—	1366	1	no	—
45	C₄	—	1366	2	no	—
46	C₄	D₂₇₃₂	1	2	yes	—
47	C₂	—	1366	2	no	—
48	C₂	C₂×D₂₇₃₂	1	1	yes	characteristic
49	C₂	Q₈×D₆₈₃	1	1	yes	characteristic
50	C₂	#24926420	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂²	C₂²×D₆₈₃
commutator, socle, cosocle	C₂²×C₆₈₃	C₂³
Fitting	C₄⋊C₄×C₆₈₃	C₂
prime radical	C₂²×D₆₈₃	C₂²
hypercenter	C₄⋊C₄	D₆₈₃
hypocenter	C₆₈₃	C₂²⋊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
1. C₂×Dic₁₃₆₆	C₂	semidirect
3. C₄⋊Dic₆₈₃	C₂	semidirect
5. C₄⋊C₄×C₆₈₃	C₂	semidirect
16. C₂×D₆₈₃	Q₈	semidirect
25. C₆₈₃	C₂²⋊Q₈	semidirect
30. C₄⋊C₄	D₆₈₃	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
C₂×C₄×D₆₈₃	2
C₄⋊Dic₆₈₃	2
(C₂×D₆₈₃)⋊C₄	2
C₄⋊C₄×C₆₈₃	2
C₂²⋊Q₈	683