Subgroups — C₂²×A₄×C₂₇ #13048145

Order 1296.

216 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×A₄×C₂₇	C₂	1	3	yes	maximal, direct product factor
2	C₂²×C₉×A₄	C₃	1	1	yes	maximal, characteristic
3	C₂⁴×C₂₇	C₃	1	1	yes	maximal, characteristic, complemented
4	C₂²×C₂²⋊C₂₇	C₃	1	2	yes	maximal, complemented
5	C₂²×C₃×C₂₇	—	4	1	no	maximal
6	A₄×C₂₇	C₂²	1	1	yes	characteristic, direct product factor
7	C₂×C₉×A₄	C₂×C₃	1	3	yes	—
8	C₂³×C₂₇	—	3	1	no	—
9	C₂³×C₂₇	—	3	3	no	—
10	C₂³×C₂₇	C₂×C₃	1	3	yes	complemented
11	C₂×C₂²⋊C₂₇	C₂×C₃	1	6	yes	complemented
12	C₂×C₃×C₂₇	—	4	3	no	—
13	C₂²×C₃×A₄	C₉	1	1	yes	characteristic
14	C₂⁴×C₉	C₃²	1	1	yes	characteristic
15	C₂²×C₂²⋊C₉	C₉	1	2	yes	—
16	C₂²×C₃×C₉	—	4	1	no	—
17	C₉×A₄	C₂²×C₃	1	1	yes	characteristic
18	C₂²×C₂₇	—	3	2	no	—
19	C₂²×C₂₇	—	3	3	no	—
20	C₂²×C₂₇	—	3	3	no	—
21	C₂²×C₂₇	—	3	3	no	—
22	C₂²×C₂₇	—	4	2	no	—
23	C₂²×C₂₇	A₄	1	1	yes	characteristic, direct product factor
24	C₂²×C₂₇	C₂²×C₃	1	1	yes	characteristic, complemented
25	C₂²⋊C₂₇	C₂²×C₃	1	2	yes	complemented
26	C₃×C₂₇	—	4	1	no	Sylow
27	C₂×C₃×A₄	C₂×C₉	1	3	yes	—
28	C₂³×C₉	—	3	3	no	—
29	C₂³×C₉	—	3	1	no	—
30	C₂³×C₉	C₂×C₃²	1	3	yes	—
31	C₂×C₂²⋊C₉	C₂×C₉	1	6	yes	—
32	C₂×C₃×C₉	—	4	3	no	—
33	C₂×C₂₇	—	3	3	no	—
34	C₂×C₂₇	—	3	1	no	—
35	C₂×C₂₇	—	4	6	no	—
36	C₂×C₂₇	C₂×A₄	1	3	yes	direct product factor
37	C₂⁴×C₃	C₃×C₉	1	1	yes	characteristic
38	C₂²×A₄	C₂₇	1	3	yes	direct product factor
39	C₂²×C₃²	—	4	1	no	—
40	C₃×A₄	C₂²×C₉	1	1	yes	characteristic
41	C₂²×C₉	—	3	3	no	—
42	C₂²×C₉	—	3	2	no	—
43	C₂²×C₉	—	3	3	no	—
44	C₂²×C₉	—	3	3	no	—
45	C₂²×C₉	—	4	2	no	—
46	C₂²×C₉	C₃×A₄	1	1	yes	characteristic
47	C₂²×C₉	C₂²×C₃²	1	1	yes	characteristic
48	C₂²⋊C₉	C₂²×C₉	1	2	yes	—
49	C₃×C₉	—	4	1	no	—
50	C₂₇	—	4	2	no	—
51	C₂₇	C₂²×A₄	1	1	yes	characteristic, direct product factor
52	C₂³×C₃	—	3	3	no	—
53	C₂³×C₃	—	3	1	no	—
54	C₂³×C₃	C₂×C₃×C₉	1	3	yes	—
55	C₂×A₄	C₂×C₂₇	1	9	yes	direct product factor
56	C₂×C₃²	—	4	3	no	—
57	C₂×C₉	—	3	3	no	—
58	C₂×C₉	—	3	1	no	—
59	C₂×C₉	—	4	6	no	—
60	C₂×C₉	C₂×C₃×A₄	1	3	yes	—
61	C₂⁴	C₃×C₂₇	1	1	yes	characteristic, complemented, Sylow
62	C₂²×C₃	—	3	3	no	—
63	C₂²×C₃	—	3	2	no	—
64	C₂²×C₃	—	3	3	no	—
65	C₂²×C₃	—	3	3	no	—
66	C₂²×C₃	—	4	3	no	—
67	C₂²×C₃	C₉×A₄	1	1	yes	characteristic
68	C₂²×C₃	C₂²×C₃×C₉	1	1	yes	characteristic
69	A₄	C₂²×C₂₇	1	3	yes	direct product factor
70	C₃²	—	4	1	no	—
71	C₉	—	4	2	no	—
72	C₉	C₂²×C₃×A₄	1	1	yes	characteristic
73	C₂³	—	3	3	no	—
74	C₂³	—	3	1	no	—
75	C₂³	C₂×C₃×C₂₇	1	3	yes	complemented
76	C₂×C₃	—	3	1	no	—
77	C₂×C₃	—	3	3	no	—
78	C₂×C₃	—	4	9	no	—
79	C₂×C₃	C₂×C₉×A₄	1	3	yes	—
80	C₂²	—	3	3	no	—
81	C₂²	—	3	2	no	—
82	C₂²	—	3	3	no	—
83	C₂²	—	3	3	no	—
84	C₂²	A₄×C₂₇	1	1	yes	characteristic, direct product factor
85	C₂²	C₂²×C₃×C₂₇	1	1	yes	characteristic, complemented
86	C₃	—	4	3	no	—
87	C₃	C₂²×C₉×A₄	1	1	yes	characteristic
88	C₂	—	3	1	no	—
89	C₂	—	3	3	no	—
90	C₂	C₂×A₄×C₂₇	1	3	yes	direct product factor

	subgroup	G/H
center, hypercenter	C₂²×C₂₇	A₄
commutator, hypocenter	C₂²	C₂²×C₃×C₂₇
Frattini	C₉	C₂²×C₃×A₄
Fitting	C₂⁴×C₂₇	C₃
socle	C₂⁴×C₃	C₃×C₉
prime radical	C₂²×C₃×A₄	C₉
cosocle	C₂²×C₉	C₂²×C₃²
LCS	G ▷ C₂²
UCS	C₁ ◁ C₂²×C₂₇

Complements

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

N	H ≅ G/N	type
1. C₂×A₄×C₂₇	C₂	direct (N × H)
3. C₂⁴×C₂₇	C₃	semidirect
4. C₂²×C₂²⋊C₂₇	C₃	semidirect
6. A₄×C₂₇	C₂²	direct (N × H)
10. C₂³×C₂₇	C₂×C₃	semidirect
11. C₂×C₂²⋊C₂₇	C₂×C₃	semidirect
23. C₂²×C₂₇	A₄	direct (N × H)
24. C₂²×C₂₇	C₂²×C₃	semidirect
25. C₂²⋊C₂₇	C₂²×C₃	semidirect
36. C₂×C₂₇	C₂×A₄	direct (N × H)
38. C₂²×A₄	C₂₇	direct (N × H)
51. C₂₇	C₂²×A₄	direct (N × H)
55. C₂×A₄	C₂×C₂₇	direct (N × H)
61. C₂⁴	C₃×C₂₇	semidirect
69. A₄	C₂²×C₂₇	direct (N × H)
75. C₂³	C₂×C₃×C₂₇	semidirect
84. C₂²	A₄×C₂₇	direct (N × H)
85. C₂²	C₂²×C₃×C₂₇	semidirect
90. C₂	C₂×A₄×C₂₇	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×A₄×C₂₇	2
C₂²×C₉×A₄	3
C₂⁴×C₂₇	3
C₂²×C₂²⋊C₂₇	3
C₂²×C₃×C₂₇	4

Subgroups — C22×A4×C27 #13048145

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×A₄×C₂₇ #13048145