Subgroups — C₂³×C₃⋊D₂₇ #13048198

Order 1296.

938 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂³×C₃×C₂₇	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂²×C₃⋊D₂₇	C₂	1	14	yes	maximal, direct product factor
3	C₂³×C₃⋊D₉	—	3	1	no	maximal
4	C₂³×D₂₇	—	3	3	no	maximal
5	C₂²×C₃×C₂₇	C₂²	1	7	yes	complemented
6	C₂×C₃⋊D₂₇	C₂²	1	28	yes	direct product factor
7	C₂³×C₃×C₉	D₃	1	1	yes	characteristic
8	C₂²×C₃⋊D₉	—	3	14	no	—
9	C₂³×C₂₇	D₃	1	3	yes	complemented
10	C₂²×D₂₇	—	3	42	no	—
11	C₂×C₃×C₂₇	C₂³	1	7	yes	complemented
12	C₃⋊D₂₇	C₂³	1	8	yes	direct product factor
13	C₂³×C₃⋊D₃	—	9	1	no	—
14	C₂³×D₉	—	9	2	no	—
15	C₂³×D₉	—	9	1	no	—
16	C₂²×C₃×C₉	C₂×D₃	1	7	yes	—
17	C₂×C₃⋊D₉	—	3	28	no	—
18	C₂²×C₂₇	C₂×D₃	1	21	yes	complemented
19	C₂×D₂₇	—	3	84	no	—
20	C₃×C₂₇	C₂⁴	1	1	yes	characteristic, complemented, Sylow
21	C₂³×C₃²	D₉	1	1	yes	characteristic
22	C₂²×C₃⋊D₃	—	9	14	no	—
23	C₂³×C₉	D₉	1	2	yes	—
24	C₂³×C₉	C₃⋊D₃	1	1	yes	characteristic
25	C₂²×D₉	—	9	28	no	—
26	C₂²×D₉	—	9	14	no	—
27	C₂×C₃×C₉	C₂²×D₃	1	7	yes	—
28	C₃⋊D₉	—	3	8	no	—
29	C₂×C₂₇	C₂²×D₃	1	21	yes	complemented
30	D₂₇	—	3	24	no	—
31	C₂³×D₃	—	27	3	no	—
32	C₂³×D₃	—	27	1	no	—
33	C₂²×C₃²	C₂×D₉	1	7	yes	—
34	C₂×C₃⋊D₃	—	9	28	no	—
35	C₂²×C₉	C₂×D₉	1	14	yes	—
36	C₂²×C₉	C₂×C₃⋊D₃	1	7	yes	—
37	C₂×D₉	—	9	56	no	—
38	C₂×D₉	—	9	28	no	—
39	C₃×C₉	C₂³×D₃	1	1	yes	characteristic
40	C₂₇	C₂³×D₃	1	3	yes	complemented
41	C₂³×C₃	D₂₇	1	3	yes	complemented
42	C₂³×C₃	C₃⋊D₉	1	1	yes	characteristic
43	C₂²×D₃	—	27	42	no	—
44	C₂²×D₃	—	27	14	no	—
45	C₂×C₃²	C₂²×D₉	1	7	yes	—
46	C₃⋊D₃	—	9	8	no	—
47	C₂×C₉	C₂²×D₉	1	14	yes	—
48	C₂×C₉	C₂²×C₃⋊D₃	1	7	yes	—
49	D₉	—	9	8	no	—
50	D₉	—	9	16	no	—
51	C₂⁴	—	81	1	no	Sylow
52	C₂²×C₃	C₂×D₂₇	1	21	yes	complemented
53	C₂²×C₃	C₂×C₃⋊D₉	1	7	yes	—
54	C₂×D₃	—	27	84	no	—
55	C₂×D₃	—	27	28	no	—
56	C₃²	C₂³×D₉	1	1	yes	characteristic
57	C₉	C₂³×D₉	1	2	yes	—
58	C₉	C₂³×C₃⋊D₃	1	1	yes	characteristic
59	C₂³	—	81	14	no	—
60	C₂³	C₃⋊D₂₇	1	1	yes	characteristic, direct product factor
61	C₂×C₃	C₂²×D₂₇	1	21	yes	complemented
62	C₂×C₃	C₂²×C₃⋊D₉	1	7	yes	—
63	D₃	—	27	24	no	—
64	D₃	—	27	8	no	—
65	C₂²	—	81	28	no	—
66	C₂²	C₂×C₃⋊D₂₇	1	7	yes	direct product factor
67	C₃	C₂³×D₂₇	1	3	yes	complemented
68	C₃	C₂³×C₃⋊D₉	1	1	yes	characteristic
69	C₂	—	81	8	no	—
70	C₂	C₂²×C₃⋊D₂₇	1	7	yes	direct product factor

	subgroup	G/H
center, hypercenter	C₂³	C₃⋊D₂₇
commutator, cosocle, hypocenter	C₃×C₂₇	C₂⁴
Frattini	C₉	C₂³×C₃⋊D₃
Fitting	C₂³×C₃×C₂₇	C₂
socle	C₂³×C₃²	D₉
LCS	G ▷ C₃×C₂₇
UCS	C₁ ◁ C₂³

Complements

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

N	H ≅ G/N	type
1. C₂³×C₃×C₂₇	C₂	semidirect
2. C₂²×C₃⋊D₂₇	C₂	direct (N × H)
5. C₂²×C₃×C₂₇	C₂²	semidirect
6. C₂×C₃⋊D₂₇	C₂²	direct (N × H)
9. C₂³×C₂₇	D₃	semidirect
11. C₂×C₃×C₂₇	C₂³	semidirect
12. C₃⋊D₂₇	C₂³	direct (N × H)
18. C₂²×C₂₇	C₂×D₃	semidirect
20. C₃×C₂₇	C₂⁴	semidirect
29. C₂×C₂₇	C₂²×D₃	semidirect
40. C₂₇	C₂³×D₃	semidirect
41. C₂³×C₃	D₂₇	semidirect
52. C₂²×C₃	C₂×D₂₇	semidirect
60. C₂³	C₃⋊D₂₇	direct (N × H)
61. C₂×C₃	C₂²×D₂₇	semidirect
66. C₂²	C₂×C₃⋊D₂₇	direct (N × H)
67. C₃	C₂³×D₂₇	semidirect
70. C₂	C₂²×C₃⋊D₂₇	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂³×C₃×C₂₇	2
C₂²×C₃⋊D₂₇	2
C₂³×C₃⋊D₉	3
C₂³×D₂₇	3

Subgroups — C23×C3⋊D27 #13048198

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂³×C₃⋊D₂₇ #13048198