Subgroups — #11757822

Order 972.

188 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₃²×C₉	C₃	1	1	yes	maximal, characteristic
2	C₃²⋊(C₂²⋊C₉)	C₃	1	3	yes	maximal
3	C₉⋊(C₂²⋊C₉)	C₃	1	6	yes	maximal, complemented
4	C₉×C₂²⋊C₉	C₃	1	3	yes	maximal, complemented
5	#6887	—	4	1	no	maximal, Sylow
6	C₂×C₃²×C₉	—	3	1	no	—
7	C₂²×C₃³	C₃²	1	1	yes	characteristic
8	C₂²×C₃×C₉	C₉	1	9	yes	complemented
9	C₂²×C₃×C₉	C₃²	1	1	yes	characteristic
10	C₂²×C₃×C₉	C₃²	1	2	yes	—
11	C₃×C₂²⋊C₉	C₃²	1	9	yes	—
12	C₃²×C₉	A₄	1	1	yes	characteristic
13	C₉⋊C₉	—	4	6	no	—
14	C₃²⋊C₉	—	4	3	no	—
15	C₉²	—	4	3	no	—
16	C₂×C₃³	—	3	1	no	—
17	C₂×C₃×C₉	—	3	9	no	—
18	C₂×C₃×C₉	—	3	2	no	—
19	C₂×C₃×C₉	—	3	1	no	—
20	C₂²×C₃²	—	3	3	no	—
21	C₂²×C₃²	C₃×C₉	1	3	yes	—
22	C₂²×C₃²	C₃³	1	1	yes	characteristic
23	C₂²×C₉	C₃×C₉	1	3	yes	—
24	C₂²×C₉	C₃×C₉	1	6	yes	—
25	C₂²⋊C₉	—	3	9	no	—
26	C₃³	C₃×A₄	1	1	yes	characteristic
27	C₃×C₉	—	4	9	no	—
28	C₃×C₉	C₂²⋊C₉	1	9	yes	complemented
29	C₃×C₉	C₃×A₄	1	2	yes	—
30	C₃×C₉	C₃×A₄	1	1	yes	characteristic
31	C₂×C₃²	—	3	9	no	—
32	C₂×C₃²	—	3	3	no	—
33	C₂×C₃²	—	3	1	no	—
34	C₂×C₉	—	3	6	no	—
35	C₂×C₉	—	3	3	no	—
36	C₂²×C₃	—	3	3	no	—
37	C₂²×C₃	C₃²×C₉	1	1	yes	characteristic
38	C₂²×C₃	#737	1	3	yes	—
39	C₃²	—	3	3	no	—
40	C₃²	C₃×C₂²⋊C₉	1	3	yes	—
41	C₃²	C₃²×A₄	1	1	yes	characteristic
42	C₉	—	12	9	no	—
43	C₉	C₃×C₂²⋊C₉	1	6	yes	—
44	C₉	C₃×C₂²⋊C₉	1	3	yes	—
45	C₂×C₃	—	3	9	no	—
46	C₂×C₃	—	3	3	no	—
47	C₂×C₃	—	3	1	no	—
48	C₂²	#6887	1	1	yes	characteristic, complemented, Sylow
49	C₃	—	3	3	no	—
50	C₃	#66436	1	3	yes	—
51	C₃	C₃²×C₂²⋊C₉	1	1	yes	characteristic
52	C₂	—	3	1	no	—

	subgroup	G/H
center	C₃×C₉	C₃×A₄
commutator	C₂²×C₃	C₃²×C₉
Frattini	C₃²	C₃²×A₄
Fitting	C₂²×C₃²×C₉	C₃
socle, cosocle	C₂²×C₃²	C₃³
prime radical	C₂²×C₃³	C₃²
hypercenter	C₃²×C₉	A₄
hypocenter	C₂²	#6887
LCS	G ▷ C₂²×C₃ ▷ C₂²
UCS	C₁ ◁ C₃×C₉ ◁ C₃²×C₉

Complements

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

N	H ≅ G/N	type
3. C₉⋊(C₂²⋊C₉)	C₃	semidirect
4. C₉×C₂²⋊C₉	C₃	semidirect
8. C₂²×C₃×C₉	C₉	semidirect
28. C₃×C₉	C₂²⋊C₉	semidirect
48. C₂²	#6887	semidirect

Maximal subgroups

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

subgroup	index
C₂²×C₃²×C₉	3
C₃²⋊(C₂²⋊C₉)	3
C₉⋊(C₂²⋊C₉)	3
C₉×C₂²⋊C₉	3
#6887	4