Subgroups — C₂²⋊(C₄²⋊C₂₇) #13136199

Order 1728.

285 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₄²×C₉	C₃	1	1	yes	maximal, characteristic
2	C₂²⋊(C₂²⋊C₂₇)	—	4	1	no	maximal
3	C₄²⋊C₂₇	—	4	4	no	maximal
4	C₂³×C₄×C₉	—	3	1	no	—
5	C₂×C₄²×C₉	—	3	4	no	—
6	C₂²×C₃×C₄²	C₉	1	1	yes	characteristic
7	C₂⁴×C₉	A₄	1	1	yes	characteristic
8	C₂²×C₄×C₉	—	3	6	no	—
9	C₂²×C₄×C₉	—	3	8	no	—
10	C₄²×C₉	—	3	4	no	—
11	C₄²×C₉	A₄	1	4	yes	—
12	C₂²⋊C₂₇	—	16	4	no	—
13	C₂²⋊C₂₇	—	16	1	no	—
14	C₂³×C₃×C₄	—	3	1	no	—
15	C₂×C₃×C₄²	—	3	4	no	—
16	C₂³×C₉	—	3	4	no	—
17	C₂³×C₉	—	3	1	no	—
18	C₂×C₄×C₉	—	3	24	no	—
19	C₂×C₄×C₉	—	3	4	no	—
20	C₂²×C₄²	C₂₇	1	1	yes	characteristic, complemented, Sylow
21	C₂⁴×C₃	C₂²⋊C₉	1	1	yes	characteristic
22	C₂²×C₃×C₄	—	3	6	no	—
23	C₂²×C₃×C₄	—	3	8	no	—
24	C₃×C₄²	—	3	4	no	—
25	C₃×C₄²	C₂²⋊C₉	1	4	yes	—
26	C₂²×C₉	—	3	6	no	—
27	C₂²×C₉	—	3	4	no	—
28	C₂²×C₉	C₄²⋊C₃	1	4	yes	—
29	C₂²×C₉	C₂²⋊A₄	1	1	yes	characteristic
30	C₄×C₉	—	3	8	no	—
31	C₂³×C₄	—	3	1	no	—
32	C₂×C₄²	—	3	4	no	—
33	C₂₇	—	64	1	no	Sylow
34	C₂³×C₃	—	3	4	no	—
35	C₂³×C₃	—	3	1	no	—
36	C₂×C₃×C₄	—	3	24	no	—
37	C₂×C₃×C₄	—	3	4	no	—
38	C₂×C₉	—	3	1	no	—
39	C₂×C₉	—	3	4	no	—
40	C₂⁴	C₂²⋊C₂₇	1	1	yes	characteristic
41	C₂²×C₄	—	3	6	no	—
42	C₂²×C₄	—	3	8	no	—
43	C₄²	—	3	4	no	—
44	C₄²	C₂²⋊C₂₇	1	4	yes	complemented
45	C₂²×C₃	—	3	6	no	—
46	C₂²×C₃	—	3	4	no	—
47	C₂²×C₃	C₄²⋊C₉	1	4	yes	—
48	C₂²×C₃	C₂²⋊(C₂²⋊C₉)	1	1	yes	characteristic
49	C₃×C₄	—	3	8	no	—
50	C₉	C₂²⋊(C₄²⋊C₃)	1	1	yes	characteristic
51	C₂³	—	3	1	no	—
52	C₂³	—	3	4	no	—
53	C₂×C₄	—	3	24	no	—
54	C₂×C₄	—	3	4	no	—
55	C₂×C₃	—	3	1	no	—
56	C₂×C₃	—	3	4	no	—
57	C₂²	—	3	6	no	—
58	C₂²	—	3	4	no	—
59	C₂²	C₄²⋊C₂₇	1	4	yes	complemented
60	C₂²	C₂²⋊(C₂²⋊C₂₇)	1	1	yes	characteristic
61	C₄	—	3	8	no	—
62	C₃	C₂²⋊(C₄²⋊C₉)	1	1	yes	characteristic
63	C₂	—	3	1	no	—
64	C₂	—	3	4	no	—

	subgroup	G/H
center, hypercenter	C₉	C₂²⋊(C₄²⋊C₃)
commutator, hypocenter	C₂²×C₄²	C₂₇
Frattini	C₂²×C₉	C₂²⋊A₄
Fitting, cosocle	C₂²×C₄²×C₉	C₃
socle, prime radical	C₂⁴×C₃	C₂²⋊C₉
LCS	G ▷ C₂²×C₄²
UCS	C₁ ◁ C₉

Complements

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

N	H ≅ G/N	type
20. C₂²×C₄²	C₂₇	semidirect
44. C₄²	C₂²⋊C₂₇	semidirect
59. C₂²	C₄²⋊C₂₇	semidirect

Maximal subgroups

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

subgroup	index
C₂²×C₄²×C₉	3
C₂²⋊(C₂²⋊C₂₇)	4
C₄²⋊C₂₇	4

Subgroups — C22⋊(C42⋊C27) #13136199

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²⋊(C₄²⋊C₂₇) #13136199