Subgroups — #10619501

Order 648.

70 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×AΣL(1,𝔽₂₇)	C₂	1	1	yes	maximal, characteristic
2	C₃⋊(C₃⋊(C₃⋊C₈))	C₃	1	1	yes	maximal, characteristic, complemented
3	(C₃⋊(C₃⋊C₈))⋊C₃	—	3	1	no	maximal
4	C₂×AΣL(1,𝔽₂₇)	C₄	1	1	yes	characteristic
5	C₃³×C₄	C₂×C₃	1	1	yes	characteristic
6	C₄×3₋¹⁺²	—	2	1	no	—
7	C₄×H(3,𝔽₃)	D₃	1	1	yes	characteristic
8	AΣL(1,𝔽₂₇)	C₈	1	1	yes	characteristic, complemented, Sylow
9	C₃⋊(C₃⋊C₈)	—	3	1	no	—
10	C₃⋊(C₃⋊C₈)	—	9	1	no	—
11	C₃⋊(C₃⋊C₈)	—	9	3	no	—
12	C₃×C₃⋊C₈	—	9	1	no	—
13	C₂×C₃³	C₃×C₄	1	1	yes	characteristic
14	C₂×3₋¹⁺²	—	2	1	no	—
15	C₂×H(3,𝔽₃)	Dic₃	1	1	yes	characteristic
16	C₃²×C₄	—	3	1	no	—
17	C₃²×C₄	—	3	3	no	—
18	C₃²×C₄	—	3	1	no	—
19	C₃²×C₄	C₃×D₃	1	1	yes	characteristic
20	C₄×C₉	—	6	1	no	—
21	C₃³	C₃×C₈	1	1	yes	characteristic, complemented
22	3₋¹⁺²	—	2	1	no	—
23	H(3,𝔽₃)	C₃⋊C₈	1	1	yes	characteristic, complemented
24	C₃×C₈	—	27	1	no	—
25	C₃⋊C₈	—	9	1	no	—
26	C₃⋊C₈	—	27	1	no	—
27	C₃⋊C₈	—	27	3	no	—
28	C₂×C₃²	—	3	1	no	—
29	C₂×C₃²	—	3	3	no	—
30	C₂×C₃²	—	3	1	no	—
31	C₂×C₃²	C₃×Dic₃	1	1	yes	characteristic
32	C₂×C₉	—	6	1	no	—
33	C₃×C₄	—	3	3	no	—
34	C₃×C₄	—	3	1	no	—
35	C₃×C₄	—	18	1	no	—
36	C₃×C₄	(C₃⋊D₃)⋊C₃	1	1	yes	characteristic
37	C₃²	—	3	3	no	—
38	C₃²	—	3	1	no	—
39	C₃²	—	3	1	no	—
40	C₃²	C₃×C₃⋊C₈	1	1	yes	characteristic
41	C₉	—	6	1	no	—
42	C₈	—	27	1	no	Sylow
43	C₂×C₃	—	3	3	no	—
44	C₂×C₃	—	3	1	no	—
45	C₂×C₃	—	18	1	no	—
46	C₂×C₃	(C₃⋊Dic₃)⋊C₃	1	1	yes	characteristic
47	C₄	#4175	1	1	yes	characteristic
48	C₃	—	3	3	no	—
49	C₃	—	3	1	no	—
50	C₃	—	18	1	no	—
51	C₃	(C₃⋊(C₃⋊C₈))⋊C₃	1	1	yes	characteristic
52	C₂	#66322	1	1	yes	characteristic

	subgroup	G/H
center, hypercenter	C₄	#4175
commutator, hypocenter	C₃³	C₃×C₈
Frattini	C₃²×C₄	C₃×D₃
Fitting	C₄×AΣL(1,𝔽₂₇)	C₂
socle	C₂×C₃	(C₃⋊Dic₃)⋊C₃
prime radical	C₂×AΣL(1,𝔽₂₇)	C₄
cosocle	C₃³×C₄	C₂×C₃
LCS	G ▷ C₃³
UCS	C₁ ◁ C₄

Complements

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

N	H ≅ G/N	type
2. C₃⋊(C₃⋊(C₃⋊C₈))	C₃	semidirect
8. AΣL(1,𝔽₂₇)	C₈	semidirect
21. C₃³	C₃×C₈	semidirect
23. H(3,𝔽₃)	C₃⋊C₈	semidirect

Maximal subgroups

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

subgroup	index
C₄×AΣL(1,𝔽₂₇)	2
C₃⋊(C₃⋊(C₃⋊C₈))	3
(C₃⋊(C₃⋊C₈))⋊C₃	3