Subgroups — D₇⋊(C₂²⋊C₂₇) #13094191

Order 1512.

56 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²⋊(C₇⋊C₂₇)	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂²×C₉×D₇	C₃	1	1	yes	maximal, characteristic
3	D₇⋊C₂₇	—	4	1	no	maximal
4	C₂²×C₇×C₉	C₂×C₃	1	1	yes	characteristic
5	C₂×C₉×D₇	—	3	1	no	—
6	C₂×C₉×D₇	—	3	1	no	—
7	C₂×C₂²⋊C₂₇	—	7	1	no	maximal, Hall
8	C₇⋊C₂₇	—	4	1	no	Hall
9	C₂²×C₃×D₇	C₉	1	1	yes	characteristic
10	C₂×C₇×C₉	—	3	1	no	—
11	C₉×D₇	—	3	1	no	—
12	C₉×D₇	A₄	1	1	yes	characteristic
13	C₂²⋊C₂₇	—	7	1	no	—
14	C₂²×C₃×C₇	C₂×C₉	1	1	yes	characteristic
15	C₂×C₃×D₇	—	3	1	no	—
16	C₂×C₃×D₇	—	3	1	no	—
17	C₂³×C₉	—	7	1	no	—
18	C₇×C₉	C₂×A₄	1	1	yes	characteristic
19	C₂²×D₇	C₂₇	1	1	yes	characteristic, complemented, Hall
20	C₂×C₂₇	—	28	1	no	—
21	C₂×C₃×C₇	—	3	1	no	—
22	C₃×D₇	—	3	1	no	—
23	C₃×D₇	C₂²⋊C₉	1	1	yes	characteristic
24	C₂²×C₉	—	21	1	no	—
25	C₂²×C₉	—	21	1	no	—
26	C₂²×C₉	AGL(1,𝔽₇)	1	1	yes	characteristic
27	C₂²×C₇	C₂×C₂₇	1	1	yes	characteristic, complemented
28	C₂×D₇	—	3	1	no	—
29	C₂×D₇	—	3	1	no	—
30	C₂₇	—	28	1	no	Sylow
31	C₂³×C₃	—	7	1	no	—
32	C₃×C₇	C₂×C₂²⋊C₉	1	1	yes	characteristic
33	C₂×C₉	—	3	1	no	—
34	C₂×C₉	—	7	1	no	—
35	C₂×C₉	—	21	1	no	—
36	C₂×C₇	—	3	1	no	—
37	D₇	—	3	1	no	—
38	D₇	C₂²⋊C₂₇	1	1	yes	characteristic, complemented
39	C₂²×C₃	—	21	1	no	—
40	C₂²×C₃	—	21	1	no	—
41	C₂²×C₃	D₇⋊C₉	1	1	yes	characteristic
42	C₉	D₇⋊A₄	1	1	yes	characteristic
43	C₂³	—	7	1	no	Sylow
44	C₇	C₂×C₂²⋊C₂₇	1	1	yes	characteristic, complemented, Sylow
45	C₂×C₃	—	3	1	no	—
46	C₂×C₃	—	7	1	no	—
47	C₂×C₃	—	21	1	no	—
48	C₂²	—	21	1	no	—
49	C₂²	—	21	1	no	—
50	C₂²	D₇⋊C₂₇	1	1	yes	characteristic, complemented
51	C₃	D₇⋊(C₂²⋊C₉)	1	1	yes	characteristic
52	C₂	—	3	1	no	—
53	C₂	—	7	1	no	—
54	C₂	—	21	1	no	—

	subgroup	G/H
center, Frattini, hypercenter	C₉	D₇⋊A₄
commutator, hypocenter	C₂²×C₇	C₂×C₂₇
Fitting, cosocle	C₂²×C₇×C₉	C₂×C₃
socle	C₂²×C₃×C₇	C₂×C₉
prime radical	C₂²×C₃×D₇	C₉
LCS	G ▷ C₂²×C₇
UCS	C₁ ◁ C₉

Complements

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

N	H ≅ G/N	type
1. C₂²⋊(C₇⋊C₂₇)	C₂	semidirect
19. C₂²×D₇	C₂₇	semidirect
27. C₂²×C₇	C₂×C₂₇	semidirect
38. D₇	C₂²⋊C₂₇	semidirect
44. C₇	C₂×C₂²⋊C₂₇	semidirect
50. C₂²	D₇⋊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
C₂²×C₉×D₇	3
D₇⋊C₂₇	4
C₂×C₂²⋊C₂₇	7

Subgroups — D7⋊(C22⋊C27) #13094191

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — D₇⋊(C₂²⋊C₂₇) #13094191