Subgroups — #11757409

Order 972.

117 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×C₂²⋊H(3,𝔽₃)	C₃	1	1	yes	maximal, characteristic
2	C₃²×C₂²⋊C₉	C₃	1	1	yes	maximal, characteristic, complemented
3	C₂²×C₃²⋊C₉	C₃	1	1	yes	maximal, characteristic, complemented
4	C₃²⋊(C₂²⋊C₉)	C₃	1	1	yes	maximal, characteristic, complemented
5	H(3,𝔽₃)⋊C₉	—	4	1	no	maximal, Sylow
6	C₂×C₃²⋊C₉	—	3	1	no	—
7	C₂²×C₃³	C₃²	1	1	yes	characteristic
8	C₃²×A₄	—	3	1	no	—
9	C₂²×C₃×C₉	—	3	1	no	—
10	C₂²⋊H(3,𝔽₃)	—	3	2	no	—
11	C₂²⋊H(3,𝔽₃)	C₉	1	3	yes	complemented
12	C₃×C₂²⋊C₉	—	3	1	no	—
13	C₃×C₂²⋊C₉	—	3	3	no	—
14	C₃×C₂²⋊C₉	—	3	1	no	—
15	C₃×H(3,𝔽₃)	—	4	1	no	—
16	C₃²×C₉	—	4	1	no	—
17	C₃²⋊C₉	—	4	1	no	—
18	C₃²⋊C₉	A₄	1	1	yes	characteristic, complemented
19	C₂×C₃³	—	3	1	no	—
20	C₂×C₃×C₉	—	3	3	no	—
21	C₂²×C₃²	—	3	1	no	—
22	C₂²×C₃²	—	3	1	no	—
23	C₂²×C₃²	—	3	1	no	—
24	C₂²×C₃²	C₃×C₉	1	1	yes	characteristic
25	C₂²×C₃²	H(3,𝔽₃)	1	1	yes	characteristic
26	C₂²×C₃²	3₋¹⁺²	1	2	yes	—
27	C₃×A₄	—	3	3	no	—
28	C₃×A₄	—	9	1	no	—
29	C₃×A₄	—	9	1	no	—
30	C₃×A₄	—	9	1	no	—
31	C₂²×C₉	—	9	1	no	—
32	C₂²⋊C₉	—	3	3	no	—
33	C₂²⋊C₉	—	9	1	no	—
34	C₃³	—	12	1	no	—
35	C₃³	C₃×A₄	1	1	yes	characteristic
36	H(3,𝔽₃)	—	4	3	no	—
37	H(3,𝔽₃)	—	12	2	no	—
38	C₃×C₉	—	3	1	no	—
39	C₃×C₉	—	12	1	no	—
40	C₃×C₉	—	12	3	no	—
41	C₃×C₉	—	12	1	no	—
42	C₂×C₃²	—	3	1	no	—
43	C₂×C₃²	—	3	1	no	—
44	C₂×C₃²	—	3	2	no	—
45	C₂×C₃²	—	9	1	no	—
46	C₂×C₃²	—	9	1	no	—
47	C₂×C₃²	—	9	1	no	—
48	C₂×C₉	—	9	3	no	—
49	C₂²×C₃	—	3	1	no	—
50	C₂²×C₃	—	3	2	no	—
51	C₂²×C₃	C₃²⋊C₉	1	1	yes	characteristic
52	C₂²×C₃	AΣL(1,𝔽₂₇)	1	1	yes	characteristic
53	C₂²×C₃	#731	1	1	yes	characteristic
54	C₂²×C₃	#732	1	1	yes	characteristic
55	A₄	—	9	3	no	—
56	C₃²	—	3	1	no	—
57	C₃²	—	3	1	no	—
58	C₃²	—	3	1	no	—
59	C₃²	—	12	3	no	—
60	C₃²	—	36	1	no	—
61	C₃²	—	36	1	no	—
62	C₃²	—	36	1	no	—
63	C₃²	C₉×A₄	1	1	yes	characteristic
64	C₃²	C₉⋊A₄	1	2	yes	—
65	C₃²	C₂²⋊H(3,𝔽₃)	1	1	yes	characteristic
66	C₉	—	9	1	no	—
67	C₉	—	12	3	no	—
68	C₉	—	36	1	no	—
69	C₂×C₃	—	3	1	no	—
70	C₂×C₃	—	3	1	no	—
71	C₂×C₃	—	3	1	no	—
72	C₂×C₃	—	3	1	no	—
73	C₂×C₃	—	9	1	no	—
74	C₂×C₃	—	9	2	no	—
75	C₂²	H(3,𝔽₃)⋊C₉	1	1	yes	characteristic, complemented, Sylow
76	C₃	—	3	1	no	—
77	C₃	—	3	2	no	—
78	C₃	—	36	3	no	—
79	C₃	(C₃×A₄)⋊C₉	1	1	yes	characteristic
80	C₃	#66362	1	1	yes	characteristic
81	C₃	#66365	1	1	yes	characteristic
82	C₃	#66366	1	1	yes	characteristic
83	C₂	—	3	1	no	—

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

Complements

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

N	H ≅ G/N	type
2. C₃²×C₂²⋊C₉	C₃	semidirect
3. C₂²×C₃²⋊C₉	C₃	semidirect
4. C₃²⋊(C₂²⋊C₉)	C₃	semidirect
11. C₂²⋊H(3,𝔽₃)	C₉	semidirect
18. C₃²⋊C₉	A₄	semidirect
75. C₂²	H(3,𝔽₃)⋊C₉	semidirect

Maximal subgroups

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

subgroup	index
C₃×C₂²⋊H(3,𝔽₃)	3
C₃²×C₂²⋊C₉	3
C₂²×C₃²⋊C₉	3
C₃²⋊(C₂²⋊C₉)	3
H(3,𝔽₃)⋊C₉	4