Subgroups — #11757795

Order 972.

124 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#66436	C₃	1	1	yes	maximal, characteristic
2	C₂²×C₃×C₂₇	C₃	1	1	yes	maximal, characteristic, complemented
3	#66353	C₃	1	2	yes	maximal, complemented
4	C₂₇⋊A₄	C₃	1	6	yes	maximal, complemented
5	A₄×C₂₇	C₃	1	3	yes	maximal, complemented
6	#6903	—	4	1	no	maximal, Sylow
7	C₂×C₃×C₂₇	—	3	1	no	—
8	C₂²×C₃×C₉	C₃²	1	1	yes	characteristic
9	C₂²⋊H(3,𝔽₃)	C₉	1	1	yes	characteristic
10	#1094	C₉	1	2	yes	—
11	C₉⋊A₄	C₉	1	6	yes	—
12	C₉×A₄	C₃²	1	3	yes	—
13	C₂²×C₂₇	C₃²	1	1	yes	characteristic
14	C₂²×C₂₇	C₃²	1	2	yes	—
15	C₂²⋊C₂₇	C₃²	1	6	yes	—
16	#737	—	4	1	no	—
17	C₂₇⋊C₃	—	4	2	no	—
18	C₂₇⋊C₃	—	4	6	no	—
19	C₃×C₂₇	—	4	3	no	—
20	C₃×C₂₇	A₄	1	1	yes	characteristic, complemented
21	C₂×C₃×C₉	—	3	1	no	—
22	C₂×C₂₇	—	3	2	no	—
23	C₂×C₂₇	—	3	1	no	—
24	C₂²×C₃²	C₃×C₉	1	1	yes	characteristic
25	C₃×A₄	C₃×C₉	1	3	yes	—
26	C₂²×C₉	C₃×C₉	1	2	yes	—
27	C₂²×C₉	C₃³	1	1	yes	characteristic
28	C₂²⋊C₉	C₃×C₉	1	6	yes	—
29	3₋¹⁺²	—	4	2	no	—
30	3₋¹⁺²	—	4	6	no	—
31	H(3,𝔽₃)	—	4	1	no	—
32	C₃×C₉	—	4	3	no	—
33	C₃×C₉	C₃×A₄	1	1	yes	characteristic
34	C₂₇	—	4	6	no	—
35	C₂₇	C₃×A₄	1	2	yes	—
36	C₂₇	C₃×A₄	1	1	yes	characteristic
37	C₂×C₃²	—	3	1	no	—
38	C₂×C₉	—	3	2	no	—
39	C₂×C₉	—	3	1	no	—
40	C₂²×C₃	—	3	1	no	—
41	C₂²×C₃	C₃²×C₉	1	1	yes	characteristic
42	A₄	—	3	3	no	—
43	C₃²	—	4	3	no	—
44	C₃²	C₉×A₄	1	1	yes	characteristic
45	C₉	—	4	6	no	—
46	C₉	C₉×A₄	1	2	yes	—
47	C₉	C₃²×A₄	1	1	yes	characteristic
48	C₂×C₃	—	3	3	no	—
49	C₂×C₃	—	3	1	no	—
50	C₂²	#6903	1	1	yes	characteristic, complemented, Sylow
51	C₃	—	3	1	no	—
52	C₃	—	12	3	no	—
53	C₃	C₃×C₉×A₄	1	1	yes	characteristic
54	C₂	—	3	1	no	—

	subgroup	G/H
center	C₂₇	C₃×A₄
commutator, socle	C₂²×C₃	C₃²×C₉
Frattini	C₉	C₃²×A₄
Fitting	C₂²×C₃×C₂₇	C₃
prime radical	C₂²⋊H(3,𝔽₃)	C₉
cosocle	C₂²×C₉	C₃³
hypercenter	C₃×C₂₇	A₄
hypocenter	C₂²	#6903
LCS	G ▷ C₂²×C₃ ▷ C₂²
UCS	C₁ ◁ C₂₇ ◁ C₃×C₂₇

Complements

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

N	H ≅ G/N	type
2. C₂²×C₃×C₂₇	C₃	semidirect
3. #66353	C₃	semidirect
4. C₂₇⋊A₄	C₃	semidirect
5. A₄×C₂₇	C₃	semidirect
20. C₃×C₂₇	A₄	semidirect
50. C₂²	#6903	semidirect

Maximal subgroups

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

subgroup	index
#66436	3
C₂²×C₃×C₂₇	3
#66353	3
C₂₇⋊A₄	3
A₄×C₂₇	3
#6903	4