Subgroups — C₂²×C₃×Dic₂₇ #13047395

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×C₃×Dic₂₇	C₂	1	6	yes	maximal, direct product factor
3	C₂²×C₃×Dic₉	—	3	1	no	maximal
4	C₂²×Dic₂₇	C₃	1	1	yes	maximal, characteristic, direct product factor
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	—	2	1	no	—
11	C₂³×C₂₇	C₂×C₃	1	1	yes	characteristic
12	C₂×Dic₂₇	C₂×C₃	1	6	yes	direct product factor
13	C₂×C₃×C₂₇	C₂×C₄	1	6	yes	complemented
14	C₂×C₃×C₂₇	C₂³	1	1	yes	characteristic
15	C₂²×C₃×Dic₃	—	9	1	no	—
16	C₂²×Dic₉	—	3	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	—	2	4	no	—
21	C₂²×C₂₇	—	2	3	no	—
22	C₂²×C₂₇	C₃×C₄	1	4	yes	complemented
23	C₂²×C₂₇	C₂²×C₃	1	3	yes	—
24	Dic₂₇	C₂²×C₃	1	4	yes	direct product factor
25	C₃×C₂₇	C₂²×C₄	1	1	yes	characteristic, complemented, Sylow
26	C₂³×C₃²	D₉	1	1	yes	characteristic
27	C₂×C₃×Dic₃	—	9	6	no	—
28	C₂³×C₉	—	2	1	no	—
29	C₂³×C₉	C₃×D₃	1	1	yes	characteristic
30	C₂×Dic₉	—	3	6	no	—
31	C₂×C₃×C₉	C₂×Dic₃	1	6	yes	—
32	C₂×C₃×C₉	C₂²×D₃	1	1	yes	characteristic
33	C₂×C₂₇	—	2	6	no	—
34	C₂×C₂₇	—	2	1	no	—
35	C₂×C₂₇	C₂×C₃×C₄	1	6	yes	complemented
36	C₂×C₂₇	C₂³×C₃	1	1	yes	characteristic
37	C₂²×C₃×C₄	—	27	1	no	—
38	C₂²×Dic₃	—	9	1	no	—
39	C₂²×C₃²	Dic₉	1	4	yes	—
40	C₂²×C₃²	C₂×D₉	1	3	yes	—
41	C₃×Dic₃	—	9	4	no	—
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	Dic₉	—	3	4	no	—
47	C₃×C₉	C₂²×Dic₃	1	1	yes	characteristic
48	C₂₇	—	2	1	no	—
49	C₂₇	C₂²×C₃×C₄	1	1	yes	characteristic, complemented
50	C₂³×C₃	—	2	1	no	—
51	C₂³×C₃	D₂₇	1	1	yes	characteristic
52	C₂³×C₃	C₃×D₉	1	1	yes	characteristic
53	C₂×C₃×C₄	—	27	6	no	—
54	C₂×Dic₃	—	9	6	no	—
55	C₂×C₃²	C₂×Dic₉	1	6	yes	—
56	C₂×C₃²	C₂²×D₉	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	—	27	1	no	Sylow
62	C₂²×C₃	—	2	4	no	—
63	C₂²×C₃	—	2	3	no	—
64	C₂²×C₃	Dic₂₇	1	4	yes	direct product factor
65	C₂²×C₃	C₂×D₂₇	1	3	yes	—
66	C₂²×C₃	C₃×Dic₉	1	4	yes	—
67	C₂²×C₃	C₂×C₃×D₉	1	3	yes	—
68	C₃×C₄	—	27	4	no	—
69	Dic₃	—	9	4	no	—
70	C₃²	C₂²×Dic₉	1	1	yes	characteristic
71	C₉	—	2	1	no	—
72	C₉	C₂²×C₃×Dic₃	1	1	yes	characteristic
73	C₂³	C₃×D₂₇	1	1	yes	characteristic
74	C₂×C₄	—	27	6	no	—
75	C₂×C₃	—	2	1	no	—
76	C₂×C₃	—	2	6	no	—
77	C₂×C₃	C₂×Dic₂₇	1	6	yes	direct product factor
78	C₂×C₃	C₂²×D₂₇	1	1	yes	characteristic
79	C₂×C₃	C₂×C₃×Dic₉	1	6	yes	—
80	C₂×C₃	C₂²×C₃×D₉	1	1	yes	characteristic
81	C₂²	C₃×Dic₂₇	1	4	yes	direct product factor
82	C₂²	C₂×C₃×D₂₇	1	3	yes	—
83	C₄	—	27	4	no	—
84	C₃	—	2	1	no	—
85	C₃	C₂²×Dic₂₇	1	1	yes	characteristic, direct product factor
86	C₃	C₂²×C₃×Dic₉	1	1	yes	characteristic
87	C₂	C₂×C₃×Dic₂₇	1	6	yes	direct product factor
88	C₂	C₂²×C₃×D₂₇	1	1	yes	characteristic

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

Complements

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

N	H ≅ G/N	type
2. C₂×C₃×Dic₂₇	C₂	direct (N × H)
4. C₂²×Dic₂₇	C₃	direct (N × H)
5. C₂²×C₃×C₂₇	C₄	semidirect
7. C₃×Dic₂₇	C₂²	direct (N × H)
12. C₂×Dic₂₇	C₂×C₃	direct (N × H)
13. C₂×C₃×C₂₇	C₂×C₄	semidirect
22. C₂²×C₂₇	C₃×C₄	semidirect
24. Dic₂₇	C₂²×C₃	direct (N × H)
25. C₃×C₂₇	C₂²×C₄	semidirect
35. C₂×C₂₇	C₂×C₃×C₄	semidirect
49. C₂₇	C₂²×C₃×C₄	semidirect
64. C₂²×C₃	Dic₂₇	direct (N × H)
77. C₂×C₃	C₂×Dic₂₇	direct (N × H)
81. C₂²	C₃×Dic₂₇	direct (N × H)
85. C₃	C₂²×Dic₂₇	direct (N × H)
87. 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 #13047395

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×C₃×Dic₂₇ #13047395