Subgroups — C₂²×Dic₃×C₂₇ #13047396

Order 1296.

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

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

Complements

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

N	H ≅ G/N	type
2. C₂×Dic₃×C₂₇	C₂	direct (N × H)
5. C₂²×C₃×C₂₇	C₄	semidirect
7. Dic₃×C₂₇	C₂²	direct (N × H)
13. C₂×C₃×C₂₇	C₂×C₄	semidirect
22. C₂²×C₂₇	Dic₃	direct (N × H)
25. C₃×C₂₇	C₂²×C₄	semidirect
35. C₂×C₂₇	C₂×Dic₃	direct (N × H)
38. C₂²×Dic₃	C₂₇	direct (N × H)
49. C₂₇	C₂²×Dic₃	direct (N × H)
54. C₂×Dic₃	C₂×C₂₇	direct (N × H)
64. C₂²×C₃	C₄×C₂₇	semidirect
69. Dic₃	C₂²×C₂₇	direct (N × H)
77. C₂×C₃	C₂×C₄×C₂₇	semidirect
81. C₂²	Dic₃×C₂₇	direct (N × H)
85. C₃	C₂²×C₄×C₂₇	semidirect
87. C₂	C₂×Dic₃×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
C₂³×C₃×C₂₇	2
C₂×Dic₃×C₂₇	2
C₂²×C₉×Dic₃	3
C₂²×C₄×C₂₇	3

Subgroups — C22×Dic3×C27 #13047396

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×Dic₃×C₂₇ #13047396