Subgroups — #36346012

Order 21856.

108 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×D₂₇₃₂	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×C₂²⋊D₆₈₃	C₂	1	4	yes	maximal, complemented
3	C₄×Dic₆₈₃	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×D₄×C₆₈₃	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂²×D₆₈₃	C₂²	1	2	yes	—
6	D₂₇₃₂	—	2	2	no	—
7	C₂²⋊D₆₈₃	—	2	8	no	—
8	C₂×Dic₆₈₃	C₂²	1	2	yes	—
9	C₂³×C₆₈₃	C₂²	1	2	yes	—
10	D₄×C₆₈₃	—	2	2	no	—
11	C₂×C₄×C₆₈₃	C₂²	1	1	yes	characteristic
12	C₂²×C₆₈₃	—	2	2	no	—
13	C₂²×C₆₈₃	—	2	4	no	—
14	C₂²×C₆₈₃	C₂³	1	1	yes	characteristic
15	C₂×D₆₈₃	—	2	4	no	—
16	C₂×D₆₈₃	—	2	2	no	—
17	C₄×C₆₈₃	D₄	1	2	yes	complemented
18	Dic₆₈₃	D₄	1	4	yes	complemented
19	C₂×C₆₈₃	—	4	2	no	—
20	C₂×C₆₈₃	C₂×D₄	1	2	yes	—
21	C₂×C₆₈₃	C₂×D₄	1	1	yes	characteristic
22	D₆₈₃	—	4	2	no	—
23	C₆₈₃	#127	1	1	yes	characteristic, complemented, Sylow
24	#127	—	683	1	no	maximal, Sylow
25	C₂×D₄	—	683	4	no	—
26	C₂×D₄	—	683	1	no	—
27	C₂×D₄	D₆₈₃	1	1	yes	characteristic, complemented
28	C₄²	—	683	1	no	—
29	C₂³	—	683	2	no	—
30	C₂³	C₂×D₆₈₃	1	2	yes	—
31	D₄	—	2	2	no	—
32	D₄	—	1366	8	no	—
33	D₄	—	1366	2	no	—
34	C₂×C₄	—	683	2	no	—
35	C₂×C₄	C₂×D₆₈₃	1	1	yes	characteristic
36	C₂²	—	2	4	no	—
37	C₂²	—	2	2	no	—
38	C₂²	—	1366	4	no	—
39	C₂²	—	1366	2	no	—
40	C₂²	C₂²×D₆₈₃	1	1	yes	characteristic
41	C₄	—	683	4	no	—
42	C₄	C₂²⋊D₆₈₃	1	2	yes	complemented
43	C₂	—	4	2	no	—
44	C₂	—	2732	2	no	—
45	C₂	D₄×D₆₈₃	1	2	yes	—
46	C₂	C₂×C₂²⋊D₆₈₃	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂²	C₂²×D₆₈₃
commutator, socle, cosocle	C₂²×C₆₈₃	C₂³
Fitting	C₂×D₄×C₆₈₃	C₂
hypercenter	C₂×D₄	D₆₈₃
hypocenter	C₆₈₃	#127
LCS	G ▷ C₂²×C₆₈₃ ▷ C₆₈₃
UCS	C₁ ◁ C₂² ◁ C₂×D₄

Complements

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

N	H ≅ G/N	type
1. C₂×D₂₇₃₂	C₂	semidirect
2. C₂×C₂²⋊D₆₈₃	C₂	semidirect
3. C₄×Dic₆₈₃	C₂	semidirect
4. C₂×D₄×C₆₈₃	C₂	semidirect
17. C₄×C₆₈₃	D₄	semidirect
18. Dic₆₈₃	D₄	semidirect
23. C₆₈₃	#127	semidirect
27. C₂×D₄	D₆₈₃	semidirect
42. C₄	C₂²⋊D₆₈₃	semidirect

Maximal subgroups

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

subgroup	index
C₂×D₂₇₃₂	2
C₂×C₂²⋊D₆₈₃	2
C₄×Dic₆₈₃	2
C₂×D₄×C₆₈₃	2
#127	683