Subgroups — D₄×C₇⋊C₂₇ #13094123

Order 1512.

64 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₇⋊C₂₇	C₂	1	2	yes	maximal, complemented
2	C₄×C₇⋊C₂₇	C₂	1	1	yes	maximal, characteristic, complemented
3	C₇×D₄×C₉	C₃	1	1	yes	maximal, characteristic
4	C₂×C₇⋊C₂₇	—	2	2	no	—
5	C₂×C₇⋊C₂₇	C₂²	1	1	yes	characteristic
6	C₂²×C₇×C₉	C₂×C₃	1	2	yes	—
7	C₄×C₇×C₉	C₂×C₃	1	1	yes	characteristic
8	D₄×C₂₇	—	7	1	no	maximal, Hall
9	C₇⋊C₂₇	D₄	1	1	yes	characteristic, direct product factor, Hall
10	C₃×C₇×D₄	C₉	1	1	yes	characteristic
11	C₂×C₇×C₉	—	2	2	no	—
12	C₂×C₇×C₉	C₂²×C₃	1	1	yes	characteristic
13	C₂²×C₂₇	—	7	2	no	—
14	C₄×C₂₇	—	7	1	no	—
15	C₂²×C₃×C₇	C₂×C₉	1	2	yes	—
16	C₃×C₄×C₇	C₂×C₉	1	1	yes	characteristic
17	D₄×C₉	ΓL(1,𝔽₈)	1	1	yes	characteristic
18	C₇×C₉	C₃×D₄	1	1	yes	characteristic
19	C₇×D₄	C₂₇	1	1	yes	characteristic, complemented, Hall
20	C₂×C₂₇	—	7	1	no	—
21	C₂×C₂₇	—	14	2	no	—
22	C₂×C₃×C₇	—	2	2	no	—
23	C₂×C₃×C₇	C₂²×C₉	1	1	yes	characteristic
24	C₂²×C₉	C₂×ΓL(1,𝔽₈)	1	2	yes	—
25	C₄×C₉	C₂×ΓL(1,𝔽₈)	1	1	yes	characteristic
26	C₂²×C₇	C₂×C₂₇	1	2	yes	complemented
27	C₄×C₇	C₂×C₂₇	1	1	yes	characteristic, complemented
28	C₂₇	—	7	1	no	Sylow
29	C₃×D₄	C₇⋊C₉	1	1	yes	characteristic
30	C₃×C₇	D₄×C₉	1	1	yes	characteristic
31	C₂×C₉	—	2	2	no	—
32	C₂×C₉	C₂²×ΓL(1,𝔽₈)	1	1	yes	characteristic
33	C₂×C₇	—	2	2	no	—
34	C₂×C₇	C₂²×C₂₇	1	1	yes	characteristic
35	C₂²×C₃	C₂×C₇⋊C₉	1	2	yes	—
36	C₃×C₄	C₂×C₇⋊C₉	1	1	yes	characteristic
37	C₉	D₄×ΓL(1,𝔽₈)	1	1	yes	characteristic
38	D₄	C₇⋊C₂₇	1	1	yes	characteristic, direct product factor, Sylow
39	C₇	D₄×C₂₇	1	1	yes	characteristic, complemented, Sylow
40	C₂×C₃	—	2	2	no	—
41	C₂×C₃	C₂²×C₇⋊C₉	1	1	yes	characteristic
42	C₂²	C₂×C₇⋊C₂₇	1	2	yes	complemented
43	C₄	C₂×C₇⋊C₂₇	1	1	yes	characteristic, complemented
44	C₃	D₄×C₇⋊C₉	1	1	yes	characteristic
45	C₂	—	2	2	no	—
46	C₂	C₂²×C₇⋊C₂₇	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂×C₉	C₂²×ΓL(1,𝔽₈)
commutator	C₂×C₇	C₂²×C₂₇
Fitting	C₇×D₄×C₉	C₃
socle	C₂×C₃×C₇	C₂²×C₉
prime radical	C₃×C₇×D₄	C₉
cosocle	C₂×C₇×C₉	C₂²×C₃
hypercenter	D₄×C₉	ΓL(1,𝔽₈)
hypocenter	C₇	D₄×C₂₇
LCS	G ▷ C₂×C₇ ▷ C₇
UCS	C₁ ◁ C₂×C₉ ◁ D₄×C₉

Complements

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

N	H ≅ G/N	type
1. C₂²×C₇⋊C₂₇	C₂	semidirect
2. C₄×C₇⋊C₂₇	C₂	semidirect
9. C₇⋊C₂₇	D₄	direct (N × H)
19. C₇×D₄	C₂₇	semidirect
26. C₂²×C₇	C₂×C₂₇	semidirect
27. C₄×C₇	C₂×C₂₇	semidirect
38. D₄	C₇⋊C₂₇	direct (N × H)
39. C₇	D₄×C₂₇	semidirect
42. C₂²	C₂×C₇⋊C₂₇	semidirect
43. C₄	C₂×C₇⋊C₂₇	semidirect

Maximal subgroups

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

subgroup	index
C₂²×C₇⋊C₂₇	2
C₄×C₇⋊C₂₇	2
C₇×D₄×C₉	3
D₄×C₂₇	7

Subgroups — D4×C7⋊C27 #13094123

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — D₄×C₇⋊C₂₇ #13094123