Subgroups — C₂²×C₂²⋊C₈₁ #13046913

Order 1296.

126 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₂²⋊C₈₁	C₂	1	3	yes	maximal, direct product factor
2	C₂⁴×C₂₇	C₃	1	1	yes	maximal, characteristic
3	C₂²×C₈₁	—	4	1	no	maximal
4	C₂²⋊C₈₁	C₂²	1	1	yes	characteristic, direct product factor
5	C₂³×C₂₇	—	3	1	no	—
6	C₂³×C₂₇	—	3	3	no	—
7	C₂³×C₂₇	C₂×C₃	1	3	yes	—
8	C₂×C₈₁	—	4	3	no	—
9	C₂⁴×C₉	C₉	1	1	yes	characteristic
10	C₂²×C₂₇	—	3	2	no	—
11	C₂²×C₂₇	—	3	3	no	—
12	C₂²×C₂₇	—	3	3	no	—
13	C₂²×C₂₇	—	3	3	no	—
14	C₂²×C₂₇	A₄	1	1	yes	characteristic
15	C₂²×C₂₇	C₂²×C₃	1	1	yes	characteristic
16	C₈₁	—	4	1	no	Sylow
17	C₂³×C₉	—	3	3	no	—
18	C₂³×C₉	—	3	1	no	—
19	C₂³×C₉	C₂×C₉	1	3	yes	—
20	C₂×C₂₇	—	3	3	no	—
21	C₂×C₂₇	—	3	1	no	—
22	C₂×C₂₇	C₂×A₄	1	3	yes	—
23	C₂⁴×C₃	C₂₇	1	1	yes	characteristic
24	C₂²×C₉	—	3	3	no	—
25	C₂²×C₉	—	3	2	no	—
26	C₂²×C₉	—	3	3	no	—
27	C₂²×C₉	—	3	3	no	—
28	C₂²×C₉	C₂²⋊C₉	1	1	yes	characteristic
29	C₂²×C₉	C₂²×C₉	1	1	yes	characteristic
30	C₂₇	C₂²×A₄	1	1	yes	characteristic
31	C₂³×C₃	—	3	3	no	—
32	C₂³×C₃	—	3	1	no	—
33	C₂³×C₃	C₂×C₂₇	1	3	yes	—
34	C₂×C₉	—	3	3	no	—
35	C₂×C₉	—	3	1	no	—
36	C₂×C₉	C₂×C₂²⋊C₉	1	3	yes	—
37	C₂⁴	C₈₁	1	1	yes	characteristic, complemented, Sylow
38	C₂²×C₃	—	3	3	no	—
39	C₂²×C₃	—	3	2	no	—
40	C₂²×C₃	—	3	3	no	—
41	C₂²×C₃	—	3	3	no	—
42	C₂²×C₃	C₂²⋊C₂₇	1	1	yes	characteristic
43	C₂²×C₃	C₂²×C₂₇	1	1	yes	characteristic
44	C₉	C₂²×C₂²⋊C₉	1	1	yes	characteristic
45	C₂³	—	3	3	no	—
46	C₂³	—	3	1	no	—
47	C₂³	C₂×C₈₁	1	3	yes	complemented
48	C₂×C₃	—	3	1	no	—
49	C₂×C₃	—	3	3	no	—
50	C₂×C₃	C₂×C₂²⋊C₂₇	1	3	yes	—
51	C₂²	—	3	3	no	—
52	C₂²	—	3	2	no	—
53	C₂²	—	3	3	no	—
54	C₂²	—	3	3	no	—
55	C₂²	C₂²⋊C₈₁	1	1	yes	characteristic, direct product factor
56	C₂²	C₂²×C₈₁	1	1	yes	characteristic, complemented
57	C₃	C₂²×C₂²⋊C₂₇	1	1	yes	characteristic
58	C₂	—	3	1	no	—
59	C₂	—	3	3	no	—
60	C₂	C₂×C₂²⋊C₈₁	1	3	yes	direct product factor

	subgroup	G/H
center, hypercenter	C₂²×C₂₇	A₄
commutator, hypocenter	C₂²	C₂²×C₈₁
Frattini	C₂₇	C₂²×A₄
Fitting	C₂⁴×C₂₇	C₃
socle, prime radical	C₂⁴×C₃	C₂₇
cosocle	C₂²×C₂₇	C₂²×C₃
LCS	G ▷ C₂²
UCS	C₁ ◁ C₂²×C₂₇

Complements

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

N	H ≅ G/N	type
1. C₂×C₂²⋊C₈₁	C₂	direct (N × H)
4. C₂²⋊C₈₁	C₂²	direct (N × H)
37. C₂⁴	C₈₁	semidirect
47. C₂³	C₂×C₈₁	semidirect
55. C₂²	C₂²⋊C₈₁	direct (N × H)
56. C₂²	C₂²×C₈₁	semidirect
60. C₂	C₂×C₂²⋊C₈₁	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×C₂²⋊C₈₁	2
C₂⁴×C₂₇	3
C₂²×C₈₁	4

Subgroups — C22×C22⋊C81 #13046913

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×C₂²⋊C₈₁ #13046913