Subgroups — C₂×C₂²⋊(C₇⋊C₂₇) #13094195

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, direct product factor
2	C₂³×C₇×C₉	C₃	1	1	yes	maximal, characteristic
3	C₂×C₇⋊C₂₇	—	4	1	no	maximal
4	C₂²×C₇×C₉	—	3	1	no	—
5	C₂²×C₇×C₉	—	3	1	no	—
6	C₂²×C₇×C₉	C₂×C₃	1	1	yes	characteristic
7	C₂×C₂²⋊C₂₇	—	7	1	no	maximal, Hall
8	C₇⋊C₂₇	—	4	1	no	Hall
9	C₂³×C₃×C₇	C₉	1	1	yes	characteristic
10	C₂×C₇×C₉	—	3	1	no	—
11	C₂×C₇×C₉	—	3	1	no	—
12	C₂×C₇×C₉	A₄	1	1	yes	characteristic
13	C₂²⋊C₂₇	—	7	1	no	—
14	C₂²×C₃×C₇	—	3	1	no	—
15	C₂²×C₃×C₇	—	3	1	no	—
16	C₂²×C₃×C₇	C₂×C₉	1	1	yes	characteristic
17	C₂³×C₉	ΓL(1,𝔽₈)	1	1	yes	characteristic
18	C₇×C₉	C₂×A₄	1	1	yes	characteristic
19	C₂³×C₇	C₂₇	1	1	yes	characteristic, complemented, Hall
20	C₂×C₂₇	—	28	1	no	—
21	C₂×C₃×C₇	—	3	1	no	—
22	C₂×C₃×C₇	—	3	1	no	—
23	C₂×C₃×C₇	C₂²⋊C₉	1	1	yes	characteristic
24	C₂²×C₉	—	3	1	no	—
25	C₂²×C₉	—	3	1	no	—
26	C₂²×C₉	C₂×ΓL(1,𝔽₈)	1	1	yes	characteristic
27	C₂²×C₇	—	3	1	no	—
28	C₂²×C₇	—	3	1	no	—
29	C₂²×C₇	C₂×C₂₇	1	1	yes	characteristic, complemented
30	C₂₇	—	28	1	no	Sylow
31	C₂³×C₃	C₇⋊C₉	1	1	yes	characteristic
32	C₃×C₇	C₂×C₂²⋊C₉	1	1	yes	characteristic
33	C₂×C₉	—	3	1	no	—
34	C₂×C₉	—	3	1	no	—
35	C₂×C₉	C₂²⋊ΓL(1,𝔽₈)	1	1	yes	characteristic
36	C₂×C₇	—	3	1	no	—
37	C₂×C₇	—	3	1	no	—
38	C₂×C₇	C₂²⋊C₂₇	1	1	yes	characteristic, complemented
39	C₂²×C₃	—	3	1	no	—
40	C₂²×C₃	—	3	1	no	—
41	C₂²×C₃	C₂×C₇⋊C₉	1	1	yes	characteristic
42	C₉	C₂×C₂²⋊ΓL(1,𝔽₈)	1	1	yes	characteristic
43	C₂³	C₇⋊C₂₇	1	1	yes	characteristic, complemented, Sylow
44	C₇	C₂×C₂²⋊C₂₇	1	1	yes	characteristic, complemented, Sylow
45	C₂×C₃	—	3	1	no	—
46	C₂×C₃	—	3	1	no	—
47	C₂×C₃	C₂²⋊(C₇⋊C₉)	1	1	yes	characteristic
48	C₂²	—	3	1	no	—
49	C₂²	—	3	1	no	—
50	C₂²	C₂×C₇⋊C₂₇	1	1	yes	characteristic, complemented
51	C₃	C₂×C₂²⋊(C₇⋊C₉)	1	1	yes	characteristic
52	C₂	—	3	1	no	—
53	C₂	—	3	1	no	—
54	C₂	C₂²⋊(C₇⋊C₂₇)	1	1	yes	characteristic, direct product factor

	subgroup	G/H
center, hypercenter	C₂×C₉	C₂²⋊ΓL(1,𝔽₈)
commutator, hypocenter	C₂²×C₇	C₂×C₂₇
Frattini	C₉	C₂×C₂²⋊ΓL(1,𝔽₈)
Fitting	C₂³×C₇×C₉	C₃
socle, prime radical	C₂³×C₃×C₇	C₉
cosocle	C₂²×C₇×C₉	C₂×C₃
LCS	G ▷ C₂²×C₇
UCS	C₁ ◁ C₂×C₉

Complements

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

N	H ≅ G/N	type
1. C₂²⋊(C₇⋊C₂₇)	C₂	direct (N × H)
19. C₂³×C₇	C₂₇	semidirect
29. C₂²×C₇	C₂×C₂₇	semidirect
38. C₂×C₇	C₂²⋊C₂₇	semidirect
43. C₂³	C₇⋊C₂₇	semidirect
44. C₇	C₂×C₂²⋊C₂₇	semidirect
50. C₂²	C₂×C₇⋊C₂₇	semidirect
54. C₂	C₂²⋊(C₇⋊C₂₇)	direct (N × H)

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
C₂³×C₇×C₉	3
C₂×C₇⋊C₂₇	4
C₂×C₂²⋊C₂₇	7

Subgroups — C2×C22⋊(C7⋊C27) #13094195

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₂²⋊(C₇⋊C₂₇) #13094195