Subgroups — D₇₅₆ #13094139

Order 1512.

64 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×D₁₈₉	C₂	1	2	yes	maximal, complemented
2	C₄×C₇×C₂₇	C₂	1	1	yes	maximal, characteristic, complemented
3	D₂₅₂	—	3	1	no	maximal
4	C₂×C₇×C₂₇	C₂²	1	1	yes	characteristic
5	D₁₈₉	—	2	2	no	—
6	C₂×D₆₃	—	3	2	no	—
7	C₄×C₇×C₉	D₃	1	1	yes	characteristic
8	D₁₀₈	—	7	1	no	maximal, Hall
9	C₇×C₂₇	D₄	1	1	yes	characteristic, complemented, Hall
10	D₈₄	—	9	1	no	—
11	C₂×C₇×C₉	C₂×D₃	1	1	yes	characteristic
12	D₆₃	—	6	2	no	—
13	C₂×D₂₇	—	7	2	no	—
14	C₄×C₂₇	D₇	1	1	yes	characteristic, complemented
15	C₂×D₂₁	—	9	2	no	—
16	C₃×C₄×C₇	D₉	1	1	yes	characteristic
17	D₃₆	—	21	1	no	—
18	C₇×C₉	D₁₂	1	1	yes	characteristic
19	D₂₈	—	27	1	no	Hall
20	C₂×C₂₇	C₂×D₇	1	1	yes	characteristic
21	D₂₇	—	14	2	no	—
22	C₂×C₃×C₇	C₂×D₉	1	1	yes	characteristic
23	D₂₁	—	18	2	no	—
24	C₂×D₉	—	21	2	no	—
25	C₄×C₉	D₂₁	1	1	yes	characteristic
26	C₂×D₇	—	27	2	no	—
27	C₄×C₇	D₂₇	1	1	yes	characteristic, complemented
28	C₂₇	D₂₈	1	1	yes	characteristic, complemented, Sylow
29	D₁₂	—	63	1	no	—
30	C₃×C₇	D₃₆	1	1	yes	characteristic
31	C₂×C₉	C₂×D₂₁	1	1	yes	characteristic
32	D₉	—	42	2	no	—
33	C₂×C₇	C₂×D₂₇	1	1	yes	characteristic
34	D₇	—	54	2	no	—
35	C₂×D₃	—	63	2	no	—
36	C₃×C₄	D₆₃	1	1	yes	characteristic
37	C₉	D₈₄	1	1	yes	characteristic
38	D₄	—	189	1	no	Sylow
39	C₇	D₁₀₈	1	1	yes	characteristic, complemented, Sylow
40	C₂×C₃	C₂×D₆₃	1	1	yes	characteristic
41	D₃	—	126	2	no	—
42	C₂²	—	189	2	no	—
43	C₄	D₁₈₉	1	1	yes	characteristic, complemented
44	C₃	D₂₅₂	1	1	yes	characteristic
45	C₂	—	378	2	no	—
46	C₂	C₂×D₁₈₉	1	1	yes	characteristic

	subgroup	G/H
center	C₂	C₂×D₁₈₉
commutator, cosocle	C₂×C₇×C₂₇	C₂²
Frattini	C₂×C₉	C₂×D₂₁
Fitting	C₄×C₇×C₂₇	C₂
socle	C₂×C₃×C₇	C₂×D₉
hypercenter	C₄	D₁₈₉
hypocenter	C₇×C₂₇	D₄
LCS	G ▷ C₂×C₇×C₂₇ ▷ C₇×C₂₇
UCS	C₁ ◁ C₂ ◁ C₄

Complements

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

N	H ≅ G/N	type
1. C₂×D₁₈₉	C₂	semidirect
2. C₄×C₇×C₂₇	C₂	semidirect
9. C₇×C₂₇	D₄	semidirect
14. C₄×C₂₇	D₇	semidirect
27. C₄×C₇	D₂₇	semidirect
28. C₂₇	D₂₈	semidirect
39. C₇	D₁₀₈	semidirect
43. C₄	D₁₈₉	semidirect

Maximal subgroups

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

subgroup	index
C₂×D₁₈₉	2
C₄×C₇×C₂₇	2
D₂₅₂	3
D₁₀₈	7

Subgroups — D756 #13094139

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — D₇₅₆ #13094139