Subgroups — #11757797

Order 972.

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

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

Complements

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

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