Subgroups — #11757821

Order 972.

174 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	C₃×#1094	C₃	1	1	yes	maximal, characteristic
3	C₃²⋊(C₂²⋊C₉)	C₃	1	2	yes	maximal
4	C₉⋊(C₂²⋊C₉)	C₃	1	3	yes	maximal, complemented
5	(C₂²⋊C₉)⋊C₉	C₃	1	6	yes	maximal, complemented
6	#6894	—	4	1	no	maximal, Sylow
7	C₂×C₃²×C₉	—	3	1	no	—
8	C₂²×C₃³	C₃²	1	1	yes	characteristic
9	C₂²×C₃×C₉	—	3	3	no	—
10	C₂²×C₃×C₉	C₃²	1	2	yes	—
11	C₂²×C₃×C₉	C₃²	1	1	yes	characteristic
12	#1094	—	3	3	no	—
13	C₃×C₂²⋊C₉	C₃²	1	6	yes	—
14	C₃×C₂²⋊C₉	C₃²	1	3	yes	—
15	C₃×3₋¹⁺²	—	4	1	no	—
16	C₃²×C₉	A₄	1	1	yes	characteristic
17	C₉⋊C₉	—	4	6	no	—
18	C₉⋊C₉	—	4	3	no	—
19	C₃²⋊C₉	—	4	2	no	—
20	C₂×C₃³	—	3	1	no	—
21	C₂×C₃×C₉	—	3	9	no	—
22	C₂×C₃×C₉	—	3	2	no	—
23	C₂×C₃×C₉	—	3	1	no	—
24	C₂²×C₃²	—	3	1	no	—
25	C₂²×C₃²	—	3	2	no	—
26	C₂²×C₃²	3₋¹⁺²	1	3	yes	—
27	C₂²×C₃²	C₃³	1	1	yes	characteristic
28	C₂²×C₉	—	3	2	no	—
29	C₂²×C₉	3₋¹⁺²	1	3	yes	complemented
30	C₂²⋊C₉	—	3	3	no	—
31	C₂²⋊C₉	—	3	6	no	—
32	C₃³	C₃×A₄	1	1	yes	characteristic
33	3₋¹⁺²	—	12	3	no	—
34	C₃×C₉	—	3	3	no	—
35	C₃×C₉	—	4	6	no	—
36	C₃×C₉	—	4	3	no	—
37	C₃×C₉	C₃×A₄	1	2	yes	—
38	C₃×C₉	C₃×A₄	1	1	yes	characteristic
39	C₂×C₃²	—	3	3	no	—
40	C₂×C₃²	—	3	6	no	—
41	C₂×C₃²	—	3	3	no	—
42	C₂×C₃²	—	3	1	no	—
43	C₂×C₉	—	3	6	no	—
44	C₂×C₉	—	3	3	no	—
45	C₂²×C₃	—	3	3	no	—
46	C₂²×C₃	C₃×3₋¹⁺²	1	1	yes	characteristic
47	C₂²×C₃	C₃×3₋¹⁺²	1	1	yes	characteristic
48	C₂²×C₃	#737	1	2	yes	—
49	C₃²	—	3	2	no	—
50	C₃²	—	3	1	no	—
51	C₃²	C₉⋊A₄	1	3	yes	—
52	C₃²	C₃²×A₄	1	1	yes	characteristic
53	C₉	—	3	2	no	—
54	C₉	—	12	3	no	—
55	C₉	—	12	6	no	—
56	C₉	#1094	1	3	yes	complemented
57	C₂×C₃	—	3	9	no	—
58	C₂×C₃	—	3	1	no	—
59	C₂×C₃	—	3	2	no	—
60	C₂×C₃	—	3	1	no	—
61	C₂²	#6894	1	1	yes	characteristic, complemented, Sylow
62	C₃	—	3	3	no	—
63	C₃	C₃×C₉⋊A₄	1	1	yes	characteristic
64	C₃	#66436	1	2	yes	—
65	C₃	C₃×#1094	1	1	yes	characteristic
66	C₂	—	3	1	no	—

	subgroup	G/H
center, Frattini	C₃²	C₃²×A₄
commutator, socle, cosocle	C₂²×C₃²	C₃³
Fitting	C₂²×C₃²×C₉	C₃
prime radical	C₂²×C₃³	C₃²
hypercenter	C₃²×C₉	A₄
hypocenter	C₂²	#6894
LCS	G ▷ C₂²×C₃² ▷ C₂²
UCS	C₁ ◁ C₃² ◁ C₃²×C₉

Complements

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

N	H ≅ G/N	type
4. C₉⋊(C₂²⋊C₉)	C₃	semidirect
5. (C₂²⋊C₉)⋊C₉	C₃	semidirect
29. C₂²×C₉	3₋¹⁺²	semidirect
56. C₉	#1094	semidirect
61. C₂²	#6894	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
C₃×#1094	3
C₃²⋊(C₂²⋊C₉)	3
C₉⋊(C₂²⋊C₉)	3
(C₂²⋊C₉)⋊C₉	3
#6894	4