Subgroups — #36345993

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	C₂²×Dic₆₈₃	C₂	1	1	yes	maximal, characteristic
3	C₄⋊Dic₆₈₃	C₂	1	1	yes	maximal, characteristic, complemented
4	Dic₆₈₃⋊C₄	C₂	1	1	yes	maximal, characteristic, complemented
5	Dic₆₈₃⋊C₄	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂²⋊Dic₆₈₃	C₂	1	1	yes	maximal, characteristic
7	C₂²⋊C₄×C₆₈₃	C₂	1	1	yes	maximal, characteristic
8	Dic₁₃₆₆	—	2	2	no	—
9	C₂×Dic₆₈₃	—	2	2	no	—
10	C₂×Dic₆₈₃	C₂²	1	1	yes	characteristic
11	C₂×Dic₆₈₃	C₂²	1	1	yes	characteristic
12	C₂×Dic₆₈₃	C₂²	1	1	yes	characteristic
13	C₂×Dic₆₈₃	C₂²	1	1	yes	characteristic
14	C₂³×C₆₈₃	C₂²	1	1	yes	characteristic
15	C₂×C₄×C₆₈₃	C₂²	1	1	yes	characteristic
16	C₂×C₄×C₆₈₃	C₂²	1	1	yes	characteristic
17	C₂²×C₆₈₃	—	2	1	no	—
18	C₂²×C₆₈₃	—	2	1	no	—
19	C₂²×C₆₈₃	Q₈	1	2	yes	complemented
20	C₂²×C₆₈₃	C₂³	1	1	yes	characteristic
21	C₄×C₆₈₃	—	2	1	no	—
22	C₄×C₆₈₃	—	2	1	no	—
23	Dic₆₈₃	—	2	1	no	—
24	Dic₆₈₃	—	2	1	no	—
25	Dic₆₈₃	—	2	1	no	—
26	Dic₆₈₃	D₄	1	2	yes	—
27	C₂×C₆₈₃	—	2	2	no	—
28	C₂×C₆₈₃	C₂×D₄	1	1	yes	characteristic
29	C₂×C₆₈₃	C₂×Q₈	1	1	yes	characteristic
30	C₂×C₆₈₃	#41	1	1	yes	characteristic
31	C₆₈₃	C₂²⋊Q₈	1	1	yes	characteristic, complemented, Sylow
32	C₂²⋊Q₈	—	683	1	no	maximal, Sylow
33	C₂×Q₈	—	683	1	no	—
34	C₂²×C₄	—	683	1	no	—
35	C₄⋊C₄	—	683	1	no	—
36	C₄⋊C₄	—	683	1	no	—
37	C₄⋊C₄	—	683	1	no	—
38	C₂²⋊C₄	—	683	1	no	—
39	C₂²⋊C₄	D₆₈₃	1	1	yes	characteristic
40	C₂³	C₂×D₆₈₃	1	1	yes	characteristic
41	Q₈	—	1366	2	no	—
42	C₂×C₄	—	1366	2	no	—
43	C₂×C₄	—	683	1	no	—
44	C₂×C₄	—	683	1	no	—
45	C₂×C₄	—	683	1	no	—
46	C₂×C₄	—	683	1	no	—
47	C₂×C₄	C₂×D₆₈₃	1	1	yes	characteristic
48	C₂×C₄	C₂×D₆₈₃	1	1	yes	characteristic
49	C₂²	—	2	1	no	—
50	C₂²	—	2	1	no	—
51	C₂²	Dic₁₃₆₆	1	2	yes	complemented
52	C₂²	C₂²×D₆₈₃	1	1	yes	characteristic
53	C₄	—	2	1	no	—
54	C₄	—	2	1	no	—
55	C₄	—	1366	1	no	—
56	C₄	—	1366	1	no	—
57	C₄	—	1366	1	no	—
58	C₄	—	683	2	no	—
59	C₂	—	2	2	no	—
60	C₂	D₄×D₆₈₃	1	1	yes	characteristic
61	C₂	C₂×Dic₁₃₆₆	1	1	yes	characteristic
62	C₂	#24926420	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
3. C₄⋊Dic₆₈₃	C₂	semidirect
4. Dic₆₈₃⋊C₄	C₂	semidirect
5. Dic₆₈₃⋊C₄	C₂	semidirect
19. C₂²×C₆₈₃	Q₈	semidirect
31. C₆₈₃	C₂²⋊Q₈	semidirect
51. 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
C₂²×Dic₆₈₃	2
C₄⋊Dic₆₈₃	2
Dic₆₈₃⋊C₄	2
C₂²⋊Dic₆₈₃	2
C₂²⋊C₄×C₆₈₃	2
C₂²⋊Q₈	683