Subgroups — C₂²×C₃⋊Dic₂₇ #13047473

Order 1296.

378 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
2	C₂×C₃⋊Dic₂₇	C₂	1	6	yes	maximal, direct product factor
3	C₂²×C₃⋊Dic₉	—	3	1	no	maximal
4	C₂²×Dic₂₇	—	3	3	no	maximal
5	C₂²×C₃×C₂₇	C₄	1	4	yes	complemented
6	C₂²×C₃×C₂₇	C₂²	1	3	yes	—
7	C₃⋊Dic₂₇	C₂²	1	4	yes	direct product factor
8	C₂³×C₃×C₉	D₃	1	1	yes	characteristic
9	C₂×C₃⋊Dic₉	—	3	6	no	—
10	C₂³×C₂₇	D₃	1	3	yes	—
11	C₂×Dic₂₇	—	3	18	no	—
12	C₂×C₃×C₂₇	C₂×C₄	1	6	yes	complemented
13	C₂×C₃×C₂₇	C₂³	1	1	yes	characteristic
14	C₂²×C₃⋊Dic₃	—	9	1	no	—
15	C₂²×Dic₉	—	9	2	no	—
16	C₂²×Dic₉	—	9	1	no	—
17	C₂²×C₃×C₉	Dic₃	1	4	yes	—
18	C₂²×C₃×C₉	C₂×D₃	1	3	yes	—
19	C₃⋊Dic₉	—	3	4	no	—
20	C₂²×C₂₇	Dic₃	1	12	yes	complemented
21	C₂²×C₂₇	C₂×D₃	1	9	yes	—
22	Dic₂₇	—	3	12	no	—
23	C₃×C₂₇	C₂²×C₄	1	1	yes	characteristic, complemented, Sylow
24	C₂³×C₃²	D₉	1	1	yes	characteristic
25	C₂×C₃⋊Dic₃	—	9	6	no	—
26	C₂³×C₉	D₉	1	2	yes	—
27	C₂³×C₉	C₃⋊D₃	1	1	yes	characteristic
28	C₂×Dic₉	—	9	12	no	—
29	C₂×Dic₉	—	9	6	no	—
30	C₂×C₃×C₉	C₂×Dic₃	1	6	yes	—
31	C₂×C₃×C₉	C₂²×D₃	1	1	yes	characteristic
32	C₂×C₂₇	C₂×Dic₃	1	18	yes	complemented
33	C₂×C₂₇	C₂²×D₃	1	3	yes	—
34	C₂²×Dic₃	—	27	3	no	—
35	C₂²×Dic₃	—	27	1	no	—
36	C₂²×C₃²	Dic₉	1	4	yes	—
37	C₂²×C₃²	C₂×D₉	1	3	yes	—
38	C₃⋊Dic₃	—	9	4	no	—
39	C₂²×C₉	Dic₉	1	8	yes	—
40	C₂²×C₉	C₂×D₉	1	6	yes	—
41	C₂²×C₉	C₃⋊Dic₃	1	4	yes	—
42	C₂²×C₉	C₂×C₃⋊D₃	1	3	yes	—
43	Dic₉	—	9	8	no	—
44	Dic₉	—	9	4	no	—
45	C₃×C₉	C₂²×Dic₃	1	1	yes	characteristic
46	C₂₇	C₂²×Dic₃	1	3	yes	complemented
47	C₂³×C₃	D₂₇	1	3	yes	—
48	C₂³×C₃	C₃⋊D₉	1	1	yes	characteristic
49	C₂×Dic₃	—	27	18	no	—
50	C₂×Dic₃	—	27	6	no	—
51	C₂×C₃²	C₂×Dic₉	1	6	yes	—
52	C₂×C₃²	C₂²×D₉	1	1	yes	characteristic
53	C₂×C₉	C₂×Dic₉	1	12	yes	—
54	C₂×C₉	C₂²×D₉	1	2	yes	—
55	C₂×C₉	C₂×C₃⋊Dic₃	1	6	yes	—
56	C₂×C₉	C₂²×C₃⋊D₃	1	1	yes	characteristic
57	C₂²×C₄	—	81	1	no	Sylow
58	C₂²×C₃	Dic₂₇	1	12	yes	complemented
59	C₂²×C₃	C₂×D₂₇	1	9	yes	—
60	C₂²×C₃	C₃⋊Dic₉	1	4	yes	—
61	C₂²×C₃	C₂×C₃⋊D₉	1	3	yes	—
62	Dic₃	—	27	12	no	—
63	Dic₃	—	27	4	no	—
64	C₃²	C₂²×Dic₉	1	1	yes	characteristic
65	C₉	C₂²×Dic₉	1	2	yes	—
66	C₉	C₂²×C₃⋊Dic₃	1	1	yes	characteristic
67	C₂³	C₃⋊D₂₇	1	1	yes	characteristic
68	C₂×C₄	—	81	6	no	—
69	C₂×C₃	C₂×Dic₂₇	1	18	yes	complemented
70	C₂×C₃	C₂²×D₂₇	1	3	yes	—
71	C₂×C₃	C₂×C₃⋊Dic₉	1	6	yes	—
72	C₂×C₃	C₂²×C₃⋊D₉	1	1	yes	characteristic
73	C₂²	C₃⋊Dic₂₇	1	4	yes	direct product factor
74	C₂²	C₂×C₃⋊D₂₇	1	3	yes	—
75	C₄	—	81	4	no	—
76	C₃	C₂²×Dic₂₇	1	3	yes	complemented
77	C₃	C₂²×C₃⋊Dic₉	1	1	yes	characteristic
78	C₂	C₂×C₃⋊Dic₂₇	1	6	yes	direct product factor
79	C₂	C₂²×C₃⋊D₂₇	1	1	yes	characteristic

	subgroup	G/H
center, hypercenter	C₂³	C₃⋊D₂₇
commutator, hypocenter	C₃×C₂₇	C₂²×C₄
Frattini	C₂×C₉	C₂²×C₃⋊D₃
Fitting	C₂³×C₃×C₂₇	C₂
socle, prime radical	C₂³×C₃²	D₉
cosocle	C₂×C₃×C₂₇	C₂³
LCS	G ▷ C₃×C₂₇
UCS	C₁ ◁ C₂³

Complements

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

N	H ≅ G/N	type
2. C₂×C₃⋊Dic₂₇	C₂	direct (N × H)
5. C₂²×C₃×C₂₇	C₄	semidirect
7. C₃⋊Dic₂₇	C₂²	direct (N × H)
12. C₂×C₃×C₂₇	C₂×C₄	semidirect
20. C₂²×C₂₇	Dic₃	semidirect
23. C₃×C₂₇	C₂²×C₄	semidirect
32. C₂×C₂₇	C₂×Dic₃	semidirect
46. C₂₇	C₂²×Dic₃	semidirect
58. C₂²×C₃	Dic₂₇	semidirect
69. C₂×C₃	C₂×Dic₂₇	semidirect
73. C₂²	C₃⋊Dic₂₇	direct (N × H)
76. C₃	C₂²×Dic₂₇	semidirect
78. C₂	C₂×C₃⋊Dic₂₇	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
C₂×C₃⋊Dic₂₇	2
C₂²×C₃⋊Dic₉	3
C₂²×Dic₂₇	3

Subgroups — C22×C3⋊Dic27 #13047473

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×C₃⋊Dic₂₇ #13047473