Subgroups — #10619539

Order 648.

57 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃⋊Dic₂₇	C₂	1	1	yes	maximal, characteristic
2	Dic₃×C₂₇	C₂	1	1	yes	maximal, characteristic
3	C₃×Dic₂₇	C₂	1	1	yes	maximal, characteristic
4	#6189	—	3	1	no	maximal
5	Dic₅₄	—	3	1	no	maximal
6	C₂×C₃×C₂₇	C₂²	1	1	yes	characteristic
7	C₃⋊Dic₉	—	3	1	no	—
8	C₉×Dic₃	D₃	1	1	yes	characteristic
9	C₃×Dic₉	—	3	1	no	—
10	C₄×C₂₇	—	3	1	no	—
11	Dic₂₇	—	3	1	no	—
12	Dic₂₇	—	6	1	no	—
13	Dic₂₇	D₃	1	1	yes	characteristic
14	C₃×C₂₇	Q₈	1	1	yes	characteristic, complemented, Sylow
15	#627	—	9	1	no	—
16	Dic₁₈	—	9	1	no	—
17	C₂×C₃×C₉	C₂×D₃	1	1	yes	characteristic
18	C₂×C₂₇	—	2	1	no	—
19	C₂×C₂₇	C₂×D₃	1	1	yes	characteristic
20	C₃⋊Dic₃	—	9	1	no	—
21	C₃×Dic₃	—	9	1	no	—
22	C₃×Dic₃	D₉	1	1	yes	characteristic
23	C₄×C₉	—	3	1	no	—
24	Dic₉	—	3	1	no	—
25	Dic₉	—	9	1	no	—
26	Dic₉	—	18	1	no	—
27	C₃×C₉	Dic₆	1	1	yes	characteristic
28	C₂₇	—	2	1	no	—
29	C₂₇	Dic₆	1	1	yes	characteristic, complemented
30	Dic₆	—	27	1	no	—
31	Dic₆	—	27	1	no	—
32	C₂×C₃²	C₂×D₉	1	1	yes	characteristic
33	C₂×C₉	—	2	1	no	—
34	C₂×C₉	D₃²	1	1	yes	characteristic
35	C₃×C₄	—	3	1	no	—
36	C₃×C₄	—	27	1	no	—
37	Dic₃	—	9	1	no	—
38	Dic₃	—	27	1	no	—
39	Dic₃	—	27	1	no	—
40	Dic₃	—	54	1	no	—
41	Dic₃	D₂₇	1	1	yes	characteristic
42	C₃²	Dic₁₈	1	1	yes	characteristic
43	C₉	—	2	1	no	—
44	C₉	#627	1	1	yes	characteristic
45	Q₈	—	81	1	no	Sylow
46	C₂×C₃	—	2	1	no	—
47	C₂×C₃	C₂×D₂₇	1	1	yes	characteristic
48	C₂×C₃	D₃×D₉	1	1	yes	characteristic
49	C₄	—	3	1	no	—
50	C₄	—	27	1	no	—
51	C₄	—	81	1	no	—
52	C₃	—	2	1	no	—
53	C₃	Dic₅₄	1	1	yes	characteristic, complemented
54	C₃	#6189	1	1	yes	characteristic
55	C₂	D₃×D₂₇	1	1	yes	characteristic

	subgroup	G/H
center, hypercenter	C₂	D₃×D₂₇
commutator, Fitting, cosocle	C₂×C₃×C₂₇	C₂²
Frattini	C₂×C₉	D₃²
socle, prime radical	C₂×C₃²	C₂×D₉
hypocenter	C₃×C₂₇	Q₈
LCS	G ▷ C₂×C₃×C₂₇ ▷ C₃×C₂₇
UCS	C₁ ◁ C₂

Complements

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

N	H ≅ G/N	type
14. C₃×C₂₇	Q₈	semidirect
29. C₂₇	Dic₆	semidirect
53. C₃	Dic₅₄	semidirect

Maximal subgroups

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

subgroup	index
C₃⋊Dic₂₇	2
Dic₃×C₂₇	2
C₃×Dic₂₇	2
#6189	3
Dic₅₄	3