Subgroups — C₂₇×#126 #11720215

Order 864.

120 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄⋊C₄×C₂₇	C₂	1	3	yes	maximal, complemented
2	C₂²⋊C₄×C₂₇	C₂	1	3	yes	maximal
3	C₄²×C₂₇	C₂	1	1	yes	maximal, characteristic, complemented
4	C₉×#126	C₃	1	1	yes	maximal, characteristic
5	C₂³×C₂₇	C₂²	1	1	yes	characteristic
6	C₂×C₄×C₂₇	C₂²	1	3	yes	—
7	C₂×C₄×C₂₇	C₂²	1	3	yes	—
8	C₉×C₄⋊C₄	C₂×C₃	1	3	yes	—
9	C₉×C₂²⋊C₄	C₂×C₃	1	3	yes	—
10	C₄²×C₉	C₂×C₃	1	1	yes	characteristic
11	C₂²×C₂₇	—	2	3	no	—
12	C₂²×C₂₇	C₂³	1	1	yes	characteristic
13	C₄×C₂₇	—	2	3	no	—
14	C₄×C₂₇	—	2	3	no	—
15	C₃×#126	C₉	1	1	yes	characteristic
16	C₂³×C₉	C₂²×C₃	1	1	yes	characteristic
17	C₂×C₄×C₉	C₂²×C₃	1	3	yes	—
18	C₂×C₄×C₉	C₂²×C₃	1	3	yes	—
19	C₂×C₂₇	—	4	1	no	—
20	C₂×C₂₇	#41	1	3	yes	—
21	C₃×C₄⋊C₄	C₂×C₉	1	3	yes	—
22	C₃×C₂²⋊C₄	C₂×C₉	1	3	yes	—
23	C₃×C₄²	C₂×C₉	1	1	yes	characteristic
24	C₂²×C₉	—	2	3	no	—
25	C₂²×C₉	C₂³×C₃	1	1	yes	characteristic
26	C₄×C₉	—	2	3	no	—
27	C₄×C₉	—	2	3	no	—
28	#126	C₂₇	1	1	yes	characteristic, direct product factor, Sylow
29	C₂₇	#126	1	1	yes	characteristic, direct product factor, Sylow
30	C₂³×C₃	C₂²×C₉	1	1	yes	characteristic
31	C₂×C₃×C₄	C₂²×C₉	1	3	yes	—
32	C₂×C₃×C₄	C₂²×C₉	1	3	yes	—
33	C₂×C₉	—	4	1	no	—
34	C₂×C₉	C₃×#41	1	3	yes	—
35	C₄⋊C₄	C₂×C₂₇	1	3	yes	complemented
36	C₂²⋊C₄	C₂×C₂₇	1	3	yes	—
37	C₄²	C₂×C₂₇	1	1	yes	characteristic, complemented
38	C₂²×C₃	—	2	3	no	—
39	C₂²×C₃	C₂³×C₉	1	1	yes	characteristic
40	C₃×C₄	—	2	3	no	—
41	C₃×C₄	—	2	3	no	—
42	C₉	C₃×#126	1	1	yes	characteristic
43	C₂³	C₂²×C₂₇	1	1	yes	characteristic
44	C₂×C₄	C₂²×C₂₇	1	3	yes	—
45	C₂×C₄	C₂²×C₂₇	1	3	yes	—
46	C₂×C₃	—	4	1	no	—
47	C₂×C₃	C₉×#41	1	3	yes	—
48	C₂²	—	2	3	no	—
49	C₂²	C₂³×C₂₇	1	1	yes	characteristic
50	C₄	—	2	3	no	—
51	C₄	—	2	3	no	—
52	C₃	C₉×#126	1	1	yes	characteristic
53	C₂	—	4	1	no	—
54	C₂	#41×C₂₇	1	3	yes	—

	subgroup	G/H
center	C₂²×C₂₇	C₂³
commutator	C₂²	C₂³×C₂₇
Frattini, cosocle	C₂²×C₉	C₂³×C₃
socle	C₂²×C₃	C₂³×C₉
prime radical	C₂³×C₃	C₂²×C₉
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₂₇ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₄⋊C₄×C₂₇	C₂	semidirect
3. C₄²×C₂₇	C₂	semidirect
28. #126	C₂₇	direct (N × H)
29. C₂₇	#126	direct (N × H)
35. C₄⋊C₄	C₂×C₂₇	semidirect
37. C₄²	C₂×C₂₇	semidirect

Maximal subgroups

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

subgroup	index
C₄⋊C₄×C₂₇	2
C₂²⋊C₄×C₂₇	2
C₄²×C₂₇	2
C₉×#126	3

Subgroups — C27×#126 #11720215

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂₇×#126 #11720215