Subgroups — #11757791

Order 972.

117 conjugacy classes of subgroups.

Conjugacy classes of subgroups

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

	subgroup	G/H
center, Frattini	C₃²	C₃²×A₄
commutator, socle, cosocle	C₂²×C₃²	C₃³
Fitting	C₂²×C₉⋊C₉	C₃
prime radical	C₃²×A₄	C₃²
hypercenter	C₉⋊C₉	A₄
hypocenter	C₂²	#6898
LCS	G ▷ C₂²×C₃² ▷ C₂²
UCS	C₁ ◁ C₃² ◁ C₉⋊C₉

Complements

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

N	H ≅ G/N	type
1. C₂²×C₉⋊C₉	C₃	semidirect
5. C₉⋊(C₂²⋊C₉)	C₃	semidirect
6. (C₂²⋊C₉)⋊C₉	C₃	semidirect
7. (C₂²⋊C₉)⋊C₉	C₃	semidirect
8. (C₂²⋊C₉)⋊C₉	C₃	semidirect
9. C₉×C₂²⋊C₉	C₃	semidirect
25. C₉⋊C₉	A₄	semidirect
77. 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×C₉⋊C₉	3
(C₃×A₄)⋊C₉	3
C₉⋊(C₂²⋊C₉)	3
(C₂²⋊C₉)⋊C₉	3
C₉×C₂²⋊C₉	3
#6898	4