Subgroups — #10619544

Order 648.

73 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×D₃×C₂₇	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×C₃×D₂₇	C₂	1	1	yes	maximal, characteristic, complemented
3	C₃⋊Dic₂₇	C₂	1	1	yes	maximal, characteristic, complemented
4	#6194	—	3	1	no	maximal
5	#6171	—	3	1	no	maximal
6	C₂×C₃×C₂₇	C₂²	1	1	yes	characteristic
7	D₃×C₂₇	—	2	1	no	—
8	C₃×D₂₇	—	2	1	no	—
9	C₂×D₃×C₉	D₃	1	1	yes	characteristic
10	C₂×C₃×D₉	—	3	1	no	—
11	C₃⋊Dic₉	—	3	1	no	—
12	C₂²×C₂₇	—	3	1	no	—
13	C₂×D₂₇	D₃	1	1	yes	characteristic, complemented
14	Dic₂₇	—	3	1	no	—
15	Dic₂₇	—	6	1	no	—
16	C₃×C₂₇	D₄	1	1	yes	characteristic, complemented, Sylow
17	#625	—	9	1	no	—
18	#611	—	9	1	no	—
19	C₂×C₃×C₉	C₂×D₃	1	1	yes	characteristic
20	D₃×C₉	—	2	1	no	—
21	C₃×D₉	—	6	1	no	—
22	C₂×C₂₇	—	2	1	no	—
23	C₂×C₂₇	—	6	1	no	—
24	C₂×C₂₇	C₂×D₃	1	1	yes	characteristic
25	D₂₇	—	2	1	no	—
26	C₂×C₃×D₃	—	9	1	no	—
27	C₂×C₃×D₃	D₉	1	1	yes	characteristic
28	C₃⋊Dic₃	—	9	1	no	—
29	C₂²×C₉	—	3	1	no	—
30	C₂×D₉	—	3	1	no	—
31	Dic₉	—	9	1	no	—
32	Dic₉	—	18	1	no	—
33	C₃×C₉	#67	1	1	yes	characteristic
34	C₂₇	—	2	1	no	—
35	C₂₇	#67	1	1	yes	characteristic, complemented
36	#67	—	27	1	no	—
37	#67	—	27	1	no	—
38	C₂×C₃²	C₂×D₉	1	1	yes	characteristic
39	C₃×D₃	—	2	1	no	—
40	C₃×D₃	—	18	1	no	—
41	C₂×C₉	—	2	1	no	—
42	C₂×C₉	—	6	1	no	—
43	C₂×C₉	D₃²	1	1	yes	characteristic
44	D₉	—	6	1	no	—
45	C₂²×C₃	—	3	1	no	—
46	C₂²×C₃	—	27	1	no	—
47	C₂×D₃	—	9	1	no	—
48	C₂×D₃	D₂₇	1	1	yes	characteristic, complemented
49	Dic₃	—	27	1	no	—
50	Dic₃	—	27	1	no	—
51	Dic₃	—	54	1	no	—
52	C₃²	#611	1	1	yes	characteristic
53	C₉	—	2	1	no	—
54	C₉	#625	1	1	yes	characteristic
55	D₄	—	81	1	no	Sylow
56	C₂×C₃	—	2	1	no	—
57	C₂×C₃	—	6	1	no	—
58	C₂×C₃	—	54	1	no	—
59	C₂×C₃	C₂×D₂₇	1	1	yes	characteristic
60	C₂×C₃	D₃×D₉	1	1	yes	characteristic
61	D₃	—	2	1	no	—
62	D₃	—	18	1	no	—
63	C₂²	—	3	1	no	—
64	C₂²	—	27	1	no	—
65	C₄	—	81	1	no	—
66	C₃	—	2	1	no	—
67	C₃	#6171	1	1	yes	characteristic, complemented
68	C₃	#6194	1	1	yes	characteristic
69	C₂	—	6	1	no	—
70	C₂	—	54	1	no	—
71	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	C₂×C₃²	C₂×D₉
hypocenter	C₃×C₂₇	D₄
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
1. C₂×D₃×C₂₇	C₂	semidirect
2. C₂×C₃×D₂₇	C₂	semidirect
3. C₃⋊Dic₂₇	C₂	semidirect
13. C₂×D₂₇	D₃	semidirect
16. C₃×C₂₇	D₄	semidirect
35. C₂₇	#67	semidirect
48. C₂×D₃	D₂₇	semidirect
67. C₃	#6171	semidirect

Maximal subgroups

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

subgroup	index
C₂×D₃×C₂₇	2
C₂×C₃×D₂₇	2
C₃⋊Dic₂₇	2
#6194	3
#6171	3