Subgroups — C₂⁴×D₂₇⋊C₃ #72276255

Order 2592.

3126 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂⁴×C₂₇⋊C₃	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂³×D₂₇⋊C₃	C₂	1	30	yes	maximal, direct product factor
3	C₂⁴×C₃×D₉	—	3	1	no	maximal
4	C₂⁴×D₂₇	C₃	1	1	yes	maximal, characteristic, complemented
5	C₂³×C₂₇⋊C₃	C₂²	1	15	yes	complemented
6	C₂²×D₂₇⋊C₃	C₂²	1	140	yes	direct product factor
7	C₂⁴×C₃×C₉	D₃	1	1	yes	characteristic
8	C₂³×C₃×D₉	—	3	30	no	—
9	C₂⁴×C₂₇	—	2	1	no	—
10	C₂⁴×C₂₇	C₂×C₃	1	1	yes	characteristic, complemented
11	C₂³×D₂₇	C₂×C₃	1	30	yes	complemented
12	C₂²×C₂₇⋊C₃	C₂³	1	35	yes	complemented
13	C₂×D₂₇⋊C₃	C₂³	1	120	yes	direct product factor
14	C₂⁴×C₃×D₃	—	9	1	no	—
15	C₂⁴×D₉	—	3	1	no	—
16	C₂³×C₃×C₉	C₂×D₃	1	15	yes	—
17	C₂²×C₃×D₉	—	3	140	no	—
18	C₂³×C₂₇	—	2	15	no	—
19	C₂³×C₂₇	C₂²×C₃	1	15	yes	complemented
20	C₂²×D₂₇	C₂²×C₃	1	140	yes	complemented
21	C₂×C₂₇⋊C₃	C₂⁴	1	15	yes	complemented
22	D₂₇⋊C₃	C₂⁴	1	16	yes	direct product factor
23	C₂⁴×C₃²	D₉	1	1	yes	characteristic
24	C₂³×C₃×D₃	—	9	30	no	—
25	C₂⁴×C₉	—	2	1	no	—
26	C₂⁴×C₉	C₃×D₃	1	1	yes	characteristic
27	C₂³×D₉	—	3	30	no	—
28	C₂²×C₃×C₉	C₂²×D₃	1	35	yes	—
29	C₂×C₃×D₉	—	3	120	no	—
30	C₂²×C₂₇	—	2	35	no	—
31	C₂²×C₂₇	C₂³×C₃	1	35	yes	complemented
32	C₂×D₂₇	C₂³×C₃	1	120	yes	complemented
33	C₂⁵×C₃	—	27	1	no	—
34	C₂⁴×D₃	—	9	1	no	—
35	C₂₇⋊C₃	C₂⁵	1	1	yes	characteristic, complemented, Sylow
36	C₂³×C₃²	C₂×D₉	1	15	yes	—
37	C₂²×C₃×D₃	—	9	140	no	—
38	C₂³×C₉	—	2	15	no	—
39	C₂³×C₉	C₂×C₃×D₃	1	15	yes	—
40	C₂²×D₉	—	3	140	no	—
41	C₂×C₃×C₉	C₂³×D₃	1	15	yes	—
42	C₃×D₉	—	3	16	no	—
43	C₂×C₂₇	—	2	15	no	—
44	C₂×C₂₇	C₂⁴×C₃	1	15	yes	complemented
45	D₂₇	C₂⁴×C₃	1	16	yes	complemented
46	C₂⁴×C₃	—	3	1	no	—
47	C₂⁴×C₃	—	27	30	no	—
48	C₂⁴×C₃	C₃×D₉	1	1	yes	characteristic
49	C₂³×D₃	—	9	30	no	—
50	C₂²×C₃²	C₂²×D₉	1	35	yes	—
51	C₂×C₃×D₃	—	9	120	no	—
52	C₂²×C₉	—	2	35	no	—
53	C₂²×C₉	C₂²×C₃×D₃	1	35	yes	—
54	C₂×D₉	—	3	120	no	—
55	C₂⁵	—	27	1	no	Sylow
56	C₃×C₉	C₂⁴×D₃	1	1	yes	characteristic
57	C₂₇	—	2	1	no	—
58	C₂₇	C₂⁵×C₃	1	1	yes	characteristic, complemented
59	C₂³×C₃	—	3	15	no	—
60	C₂³×C₃	—	27	140	no	—
61	C₂³×C₃	C₂×C₃×D₉	1	15	yes	—
62	C₂²×D₃	—	9	140	no	—
63	C₂×C₃²	C₂³×D₉	1	15	yes	—
64	C₃×D₃	—	9	16	no	—
65	C₂×C₉	—	2	15	no	—
66	C₂×C₉	C₂³×C₃×D₃	1	15	yes	—
67	D₉	—	3	16	no	—
68	C₂⁴	—	27	30	no	—
69	C₂⁴	D₂₇⋊C₃	1	1	yes	characteristic, direct product factor
70	C₂²×C₃	—	3	35	no	—
71	C₂²×C₃	—	27	120	no	—
72	C₂²×C₃	C₂²×C₃×D₉	1	35	yes	—
73	C₂×D₃	—	9	120	no	—
74	C₃²	C₂⁴×D₉	1	1	yes	characteristic
75	C₉	—	2	1	no	—
76	C₉	C₂⁴×C₃×D₃	1	1	yes	characteristic
77	C₂³	—	27	140	no	—
78	C₂³	C₂×D₂₇⋊C₃	1	15	yes	direct product factor
79	C₂×C₃	—	3	15	no	—
80	C₂×C₃	—	27	16	no	—
81	C₂×C₃	C₂³×C₃×D₉	1	15	yes	—
82	D₃	—	9	16	no	—
83	C₂²	—	27	120	no	—
84	C₂²	C₂²×D₂₇⋊C₃	1	35	yes	direct product factor
85	C₃	—	3	1	no	—
86	C₃	C₂⁴×C₃×D₉	1	1	yes	characteristic
87	C₂	—	27	16	no	—
88	C₂	C₂³×D₂₇⋊C₃	1	15	yes	direct product factor

	subgroup	G/H
center, hypercenter	C₂⁴	D₂₇⋊C₃
commutator, cosocle, hypocenter	C₂₇	C₂⁵×C₃
Frattini	C₉	C₂⁴×C₃×D₃
Fitting	C₂⁴×C₂₇⋊C₃	C₂
socle	C₂⁴×C₃	C₃×D₉
LCS	G ▷ C₂₇
UCS	C₁ ◁ C₂⁴

Complements

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

N	H ≅ G/N	type
1. C₂⁴×C₂₇⋊C₃	C₂	semidirect
2. C₂³×D₂₇⋊C₃	C₂	direct (N × H)
4. C₂⁴×D₂₇	C₃	semidirect
5. C₂³×C₂₇⋊C₃	C₂²	semidirect
6. C₂²×D₂₇⋊C₃	C₂²	direct (N × H)
10. C₂⁴×C₂₇	C₂×C₃	semidirect
11. C₂³×D₂₇	C₂×C₃	semidirect
12. C₂²×C₂₇⋊C₃	C₂³	semidirect
13. C₂×D₂₇⋊C₃	C₂³	direct (N × H)
19. C₂³×C₂₇	C₂²×C₃	semidirect
20. C₂²×D₂₇	C₂²×C₃	semidirect
21. C₂×C₂₇⋊C₃	C₂⁴	semidirect
22. D₂₇⋊C₃	C₂⁴	direct (N × H)
31. C₂²×C₂₇	C₂³×C₃	semidirect
32. C₂×D₂₇	C₂³×C₃	semidirect
35. C₂₇⋊C₃	C₂⁵	semidirect
44. C₂×C₂₇	C₂⁴×C₃	semidirect
45. D₂₇	C₂⁴×C₃	semidirect
58. C₂₇	C₂⁵×C₃	semidirect
69. C₂⁴	D₂₇⋊C₃	direct (N × H)
78. C₂³	C₂×D₂₇⋊C₃	direct (N × H)
84. C₂²	C₂²×D₂₇⋊C₃	direct (N × H)
88. C₂	C₂³×D₂₇⋊C₃	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂⁴×C₂₇⋊C₃	2
C₂³×D₂₇⋊C₃	2
C₂⁴×C₃×D₉	3
C₂⁴×D₂₇	3

Subgroups — C24×D27⋊C3 #72276255

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂⁴×D₂₇⋊C₃ #72276255