Subgroups — C₃×#126⋊C₉ #11721349

Order 864.

108 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×C₄²⋊C₉	C₂	1	1	yes	maximal, characteristic, complemented
2	C₃²×#126	C₃	1	1	yes	maximal, characteristic
3	#126⋊C₉	C₃	1	3	yes	maximal, direct product factor
4	C₂×C₃×C₂²⋊C₉	—	4	1	no	maximal
5	C₃²×C₄⋊C₄	—	3	1	no	—
6	C₃²×C₂²⋊C₄	—	3	1	no	—
7	C₃²×C₄²	C₂×C₃	1	1	yes	characteristic
8	C₄²⋊C₉	C₂×C₃	1	3	yes	complemented
9	C₃×C₂²⋊C₉	—	4	1	no	—
10	C₃×#126	C₉	1	3	yes	complemented
11	C₃×#126	C₃²	1	1	yes	characteristic
12	C₂³×C₃²	A₄	1	1	yes	characteristic
13	C₂×C₃²×C₄	—	3	1	no	—
14	C₂×C₃²×C₄	—	3	1	no	—
15	C₂×C₂²⋊C₉	—	4	3	no	—
16	C₂×C₃×C₉	—	16	1	no	—
17	C₃×C₄⋊C₄	—	3	3	no	—
18	C₃×C₄⋊C₄	—	3	1	no	—
19	C₃×C₂²⋊C₄	—	3	3	no	—
20	C₃×C₂²⋊C₄	—	3	1	no	—
21	C₃×C₄²	C₂×C₉	1	3	yes	complemented
22	C₃×C₄²	C₂×C₃²	1	1	yes	characteristic
23	C₂²×C₃²	—	6	1	no	—
24	C₂²×C₃²	C₂×A₄	1	1	yes	characteristic
25	C₃²×C₄	—	6	1	no	—
26	C₃²×C₄	—	6	1	no	—
27	C₂²⋊C₉	—	4	3	no	—
28	#126	C₃×C₉	1	1	yes	characteristic, complemented, Sylow
29	C₃×C₉	—	16	1	no	Sylow
30	C₂³×C₃	C₂²⋊C₉	1	3	yes	—
31	C₂³×C₃	C₃×A₄	1	1	yes	characteristic
32	C₂×C₃×C₄	—	3	3	no	—
33	C₂×C₃×C₄	—	3	3	no	—
34	C₂×C₃×C₄	—	3	1	no	—
35	C₂×C₃×C₄	—	3	1	no	—
36	C₂×C₃²	—	3	1	no	—
37	C₂×C₃²	—	4	1	no	—
38	C₂×C₉	—	16	3	no	—
39	C₄⋊C₄	—	3	1	no	—
40	C₂²⋊C₄	—	3	1	no	—
41	C₄²	C₂×C₃×C₉	1	1	yes	characteristic, complemented
42	C₂²×C₃	—	6	3	no	—
43	C₂²×C₃	—	6	1	no	—
44	C₂²×C₃	C₂×C₂²⋊C₉	1	3	yes	—
45	C₂²×C₃	C₂×C₃×A₄	1	1	yes	characteristic
46	C₃×C₄	—	6	3	no	—
47	C₃×C₄	—	6	3	no	—
48	C₃×C₄	—	6	1	no	—
49	C₃×C₄	—	6	1	no	—
50	C₃²	#126⋊C₃	1	1	yes	characteristic
51	C₉	—	16	3	no	—
52	C₂³	C₃×C₂²⋊C₉	1	1	yes	characteristic
53	C₂×C₄	—	3	1	no	—
54	C₂×C₄	—	3	1	no	—
55	C₂×C₃	—	3	3	no	—
56	C₂×C₃	—	3	1	no	—
57	C₂×C₃	—	4	3	no	—
58	C₂×C₃	—	4	1	no	—
59	C₂²	—	6	1	no	—
60	C₂²	C₂×C₃×C₂²⋊C₉	1	1	yes	characteristic
61	C₄	—	6	1	no	—
62	C₄	—	6	1	no	—
63	C₃	#126⋊C₉	1	3	yes	direct product factor
64	C₃	C₃×#126⋊C₃	1	1	yes	characteristic
65	C₂	—	3	1	no	—
66	C₂	—	4	1	no	—

	subgroup	G/H
center, hypercenter	C₃²	#126⋊C₃
commutator, hypocenter	C₄²	C₂×C₃×C₉
Frattini	C₂²×C₃	C₂×C₃×A₄
Fitting	C₃²×#126	C₃
socle	C₂²×C₃²	C₂×A₄
prime radical	C₂³×C₃²	A₄
cosocle	C₃×C₄²	C₂×C₃²
LCS	G ▷ C₄²
UCS	C₁ ◁ C₃²

Complements

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

N	H ≅ G/N	type
1. C₃×C₄²⋊C₉	C₂	semidirect
3. #126⋊C₉	C₃	direct (N × H)
8. C₄²⋊C₉	C₂×C₃	semidirect
10. C₃×#126	C₉	semidirect
21. C₃×C₄²	C₂×C₉	semidirect
28. #126	C₃×C₉	semidirect
41. C₄²	C₂×C₃×C₉	semidirect
63. C₃	#126⋊C₉	direct (N × H)

Maximal subgroups

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

subgroup	index
C₃×C₄²⋊C₉	2
C₃²×#126	3
#126⋊C₉	3
C₂×C₃×C₂²⋊C₉	4

Subgroups — C3×#126⋊C9 #11721349

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×#126⋊C₉ #11721349