Subgroups — #11757773

Order 972.

123 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	2	yes	maximal
3	C₃²⋊(C₂²⋊C₉)	C₃	1	1	yes	maximal, characteristic
4	C₉⋊(C₂²⋊C₉)	C₃	1	2	yes	maximal, complemented
5	(C₂²⋊C₉)⋊C₉	C₃	1	4	yes	maximal, complemented
6	(C₂²⋊C₉)⋊C₉	C₃	1	2	yes	maximal, complemented
7	C₉×C₂²⋊C₉	C₃	1	1	yes	maximal, characteristic, complemented
8	#6898	—	4	1	no	maximal, Sylow
9	C₂×C₃²⋊C₉	—	3	1	no	—
10	C₂²×C₃³	C₃²	1	1	yes	characteristic
11	C₂²×C₃×C₉	C₃²	1	1	yes	characteristic
12	C₂²×C₃×C₉	C₃²	1	2	yes	—
13	C₃×C₂²⋊C₉	C₃²	1	2	yes	—
14	C₃×C₂²⋊C₉	C₃²	1	1	yes	characteristic
15	C₃×C₂²⋊C₉	C₃²	1	2	yes	—
16	C₃×C₂²⋊C₉	C₃²	1	4	yes	—
17	C₉⋊C₉	—	4	4	no	—
18	C₉⋊C₉	—	4	2	no	—
19	C₉⋊C₉	—	4	2	no	—
20	C₃²⋊C₉	—	4	1	no	—
21	C₃²⋊C₉	—	4	2	no	—
22	C₃²⋊C₉	A₄	1	1	yes	characteristic
23	C₉²	—	4	1	no	—
24	C₂×C₃³	—	3	1	no	—
25	C₂×C₃×C₉	—	3	1	no	—
26	C₂×C₃×C₉	—	3	2	no	—
27	C₂²×C₃²	—	3	1	no	—
28	C₂²×C₃²	—	3	2	no	—
29	C₂²×C₃²	—	3	1	no	—
30	C₂²×C₃²	C₃³	1	1	yes	characteristic
31	C₂²×C₉	—	3	1	no	—
32	C₂²×C₉	—	3	2	no	—
33	C₂²⋊C₉	—	3	1	no	—
34	C₂²⋊C₉	—	3	2	no	—
35	C₂²⋊C₉	—	3	2	no	—
36	C₂²⋊C₉	—	3	4	no	—
37	C₃³	C₃×A₄	1	1	yes	characteristic
38	C₃×C₉	—	4	1	no	—
39	C₃×C₉	—	4	2	no	—
40	C₃×C₉	—	4	2	no	—
41	C₃×C₉	—	4	4	no	—
42	C₃×C₉	C₃×A₄	1	1	yes	characteristic
43	C₃×C₉	C₃×A₄	1	2	yes	—
44	C₂×C₃²	—	3	1	no	—
45	C₂×C₃²	—	3	3	no	—
46	C₂×C₃²	—	9	1	no	—
47	C₂×C₃²	—	9	2	no	—
48	C₂×C₉	—	9	1	no	—
49	C₂×C₉	—	9	2	no	—
50	C₂²×C₃	—	9	1	no	—
51	C₂²×C₃	#737	1	1	yes	characteristic
52	C₂²×C₃	#737	1	2	yes	—
53	C₂²×C₃	#737	1	1	yes	characteristic
54	C₃²	—	3	1	no	—
55	C₃²	—	3	1	no	—
56	C₃²	—	3	2	no	—
57	C₃²	C₃²×A₄	1	1	yes	characteristic
58	C₉	—	3	1	no	—
59	C₉	—	3	2	no	—
60	C₉	—	12	1	no	—
61	C₉	—	12	2	no	—
62	C₉	—	12	2	no	—
63	C₉	—	12	4	no	—
64	C₂×C₃	—	3	1	no	—
65	C₂×C₃	—	3	1	no	—
66	C₂×C₃	—	3	2	no	—
67	C₂×C₃	—	9	3	no	—
68	C₂²	#6898	1	1	yes	characteristic, complemented, Sylow
69	C₃	—	9	1	no	—
70	C₃	#66436	1	1	yes	characteristic
71	C₃	#66437	1	1	yes	characteristic
72	C₃	#66440	1	2	yes	—
73	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	#6898
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
6. (C₂²⋊C₉)⋊C₉	C₃	semidirect
7. C₉×C₂²⋊C₉	C₃	semidirect
68. C₂²	#6898	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
(C₂²⋊C₉)⋊C₉	3
C₉×C₂²⋊C₉	3
#6898	4