Subgroups — C₃×#21419294 #36382245

Order 22599.

1052 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃²×C₉×C₃₁⋊C₃	C₃	1	1	yes	maximal, characteristic
2	#21419294	C₃	1	27	yes	maximal, direct product factor
3	C₃×(C₃₁⋊C₉)⋊C₉	C₃	1	6	yes	maximal, complemented
4	C₃×(C₃×C₃₁⋊C₃)⋊C₉	C₃	1	3	yes	maximal
5	C₃×C₉×C₃₁⋊C₉	C₃	1	2	yes	maximal, complemented
6	C₃×C₉²×C₃₁	C₃	1	1	yes	maximal, characteristic, complemented
7	C₃³×C₃₁⋊C₃	C₃²	1	1	yes	characteristic
8	C₃²×C₃₁⋊C₉	C₃²	1	6	yes	—
9	C₃²×C₃₁⋊C₉	C₃²	1	2	yes	—
10	C₃×C₉×C₃₁⋊C₃	C₉	1	27	yes	complemented
11	C₃×C₉×C₃₁⋊C₃	C₃²	1	9	yes	—
12	(C₃₁⋊C₉)⋊C₉	C₃²	1	54	yes	complemented
13	(C₃×C₃₁⋊C₃)⋊C₉	C₃²	1	27	yes	—
14	C₉×C₃₁⋊C₉	C₃²	1	18	yes	complemented
15	C₃²×C₉×C₃₁	C₃²	1	3	yes	—
16	C₃²×C₉×C₃₁	C₃²	1	1	yes	characteristic
17	C₉²×C₃₁	C₃²	1	9	yes	complemented
18	C₃³×C₃₁	C₃³	1	1	yes	characteristic
19	C₃²×C₃₁⋊C₃	—	3	9	no	—
20	C₃²×C₃₁⋊C₃	C₃×C₉	1	9	yes	—
21	C₃²×C₃₁⋊C₃	C₃³	1	3	yes	—
22	C₃×C₉×C₃₁	—	3	9	no	—
23	C₃×C₉×C₃₁	C₃×C₉	1	9	yes	—
24	C₃×C₉×C₃₁	C₃³	1	9	yes	—
25	C₃×C₉×C₃₁	C₃³	1	3	yes	—
26	C₃×C₃₁⋊C₉	—	3	18	no	—
27	C₃×C₃₁⋊C₉	C₃×C₉	1	18	yes	—
28	C₃×C₃₁⋊C₉	C₃³	1	18	yes	—
29	C₃×C₃₁⋊C₉	C₃³	1	6	yes	—
30	C₉×C₃₁⋊C₃	C₃×C₉	1	81	yes	complemented
31	C₃×#6887	—	31	1	no	maximal, Sylow
32	C₃²×C₃₁	C₃²×C₉	1	3	yes	—
33	C₃²×C₃₁	#737	1	9	yes	—
34	C₃²×C₃₁	C₃⁴	1	1	yes	characteristic
35	C₃×C₃₁⋊C₃	—	3	9	no	—
36	C₃×C₃₁⋊C₃	—	3	27	no	—
37	C₃×C₃₁⋊C₃	C₃²×C₉	1	9	yes	—
38	C₉×C₃₁	—	3	9	no	—
39	C₉×C₃₁	C₃²×C₉	1	9	yes	—
40	C₃₁⋊C₉	—	3	18	no	—
41	C₃₁⋊C₉	C₃²×C₉	1	18	yes	—
42	C₃³×C₉	—	31	1	no	—
43	#6887	—	31	27	no	—
44	C₃×C₉⋊C₉	—	31	6	no	—
45	C₃×C₃²⋊C₉	—	31	3	no	—
46	C₃×C₉²	—	31	2	no	—
47	C₃×C₉²	C₃₁⋊C₃	1	1	yes	characteristic, complemented
48	C₃×C₃₁	#6887	1	9	yes	complemented
49	C₃×C₃₁	C₃³×C₉	1	1	yes	characteristic
50	C₃×C₃₁	C₃×#737	1	3	yes	—
51	C₃₁⋊C₃	—	3	9	no	—
52	C₃⁴	—	31	1	no	—
53	C₃²×C₉	—	31	2	no	—
54	C₃²×C₉	—	31	27	no	—
55	C₃²×C₉	—	31	9	no	—
56	C₃²×C₉	—	31	6	no	—
57	C₃²×C₉	C₃×C₃₁⋊C₃	1	3	yes	—
58	C₃²×C₉	C₃×C₃₁⋊C₃	1	1	yes	characteristic
59	C₉⋊C₉	—	31	54	no	—
60	C₃²⋊C₉	—	31	27	no	—
61	C₉²	—	31	18	no	—
62	C₉²	C₃×C₃₁⋊C₃	1	9	yes	complemented
63	C₃₁	C₃×#6887	1	1	yes	characteristic, complemented, Sylow
64	C₃³	—	31	9	no	—
65	C₃³	—	31	3	no	—
66	C₃³	—	93	9	no	—
67	C₃³	C₃²×C₃₁⋊C₃	1	1	yes	characteristic
68	C₃×C₉	—	3	9	no	—
69	C₃×C₉	—	31	18	no	—
70	C₃×C₉	—	31	6	no	—
71	C₃×C₉	—	31	81	no	—
72	C₃×C₉	—	31	18	no	—
73	C₃×C₉	—	93	18	no	—
74	C₃×C₉	C₉×C₃₁⋊C₃	1	9	yes	—
75	C₃×C₉	C₃²×C₃₁⋊C₃	1	9	yes	—
76	C₃×C₉	C₃²×C₃₁⋊C₃	1	3	yes	—
77	C₃²	—	31	9	no	—
78	C₃²	—	93	27	no	—
79	C₃²	—	93	9	no	—
80	C₃²	C₃×C₉×C₃₁⋊C₃	1	3	yes	—
81	C₃²	#14578912	1	9	yes	—
82	C₃²	C₃³×C₃₁⋊C₃	1	1	yes	characteristic
83	C₉	—	3	9	no	—
84	C₉	—	31	18	no	—
85	C₉	—	93	18	no	—
86	C₉	C₃×C₉×C₃₁⋊C₃	1	9	yes	—
87	C₃	—	93	9	no	—
88	C₃	#21419294	1	9	yes	direct product factor
89	C₃	C₃²×C₉×C₃₁⋊C₃	1	1	yes	characteristic
90	C₃	C₃×#14578912	1	3	yes	—

	subgroup	G/H
center	C₃²×C₉	C₃×C₃₁⋊C₃
commutator	C₃×C₃₁	C₃³×C₉
Frattini	C₃²	C₃³×C₃₁⋊C₃
Fitting	C₃×C₉²×C₃₁	C₃
socle	C₃³×C₃₁	C₃³
prime radical	C₃³×C₃₁⋊C₃	C₃²
cosocle	C₃²×C₃₁	C₃⁴
hypercenter	C₃×C₉²	C₃₁⋊C₃
hypocenter	C₃₁	C₃×#6887
LCS	G ▷ C₃×C₃₁ ▷ C₃₁
UCS	C₁ ◁ C₃²×C₉ ◁ C₃×C₉²

Complements

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

N	H ≅ G/N	type
2. #21419294	C₃	direct (N × H)
3. C₃×(C₃₁⋊C₉)⋊C₉	C₃	semidirect
5. C₃×C₉×C₃₁⋊C₉	C₃	semidirect
6. C₃×C₉²×C₃₁	C₃	semidirect
10. C₃×C₉×C₃₁⋊C₃	C₉	semidirect
12. (C₃₁⋊C₉)⋊C₉	C₃²	semidirect
14. C₉×C₃₁⋊C₉	C₃²	semidirect
17. C₉²×C₃₁	C₃²	semidirect
30. C₉×C₃₁⋊C₃	C₃×C₉	semidirect
47. C₃×C₉²	C₃₁⋊C₃	semidirect
48. C₃×C₃₁	#6887	semidirect
62. C₉²	C₃×C₃₁⋊C₃	semidirect
63. C₃₁	C₃×#6887	semidirect
88. C₃	#21419294	direct (N × H)

Maximal subgroups

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

subgroup	index
C₃²×C₉×C₃₁⋊C₃	3
#21419294	3
C₃×(C₃₁⋊C₉)⋊C₉	3
C₃×(C₃×C₃₁⋊C₃)⋊C₉	3
C₃×C₉×C₃₁⋊C₉	3
C₃×C₉²×C₃₁	3
C₃×#6887	31

Subgroups — C3×#21419294 #36382245

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×#21419294 #36382245