Subgroups — C₂³×C₃×D₂₇ #13048187

Order 1296.

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

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

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)
4. C₂³×D₂₇	C₃	direct (N × H)
5. C₂²×C₃×C₂₇	C₂²	semidirect
6. C₂×C₃×D₂₇	C₂²	direct (N × H)
10. C₂³×C₂₇	C₂×C₃	semidirect
11. C₂²×D₂₇	C₂×C₃	direct (N × H)
12. C₂×C₃×C₂₇	C₂³	semidirect
13. C₃×D₂₇	C₂³	direct (N × H)
19. C₂²×C₂₇	C₂²×C₃	semidirect
20. C₂×D₂₇	C₂²×C₃	direct (N × H)
21. C₃×C₂₇	C₂⁴	semidirect
30. C₂×C₂₇	C₂³×C₃	semidirect
31. D₂₇	C₂³×C₃	direct (N × H)
41. C₂₇	C₂⁴×C₃	semidirect
44. C₂³×C₃	D₂₇	direct (N × H)
55. C₂²×C₃	C₂×D₂₇	direct (N × H)
62. C₂³	C₃×D₂₇	direct (N × H)
65. C₂×C₃	C₂²×D₂₇	direct (N × H)
69. C₂²	C₂×C₃×D₂₇	direct (N × H)
71. C₃	C₂³×D₂₇	direct (N × H)
74. 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 #13048187

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂³×C₃×D₂₇ #13048187