Subgroups — #36382136

Order 22599.

210 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×#14578902	C₃	1	1	yes	maximal, characteristic, complemented
2	C₃×#14578900	C₃	1	2	yes	maximal, complemented
3	#21419271	C₃	1	9	yes	maximal, complemented
4	C₃₁×#6887	C₃	1	1	yes	maximal, characteristic, complemented
5	C₃×#11719689	—	3	1	no	—
6	C₃×C₃²⋊(C₃₁⋊C₃)	—	3	2	no	—
7	#14578902	—	3	1	no	—
8	#14578902	—	3	1	no	—
9	#14578902	C₃²	1	3	yes	complemented
10	#14578900	—	3	2	no	—
11	#14578900	—	3	2	no	—
12	#14578900	C₃²	1	6	yes	complemented
13	C₃²×C₉×C₃₁	C₃²	1	1	yes	characteristic
14	C₃₁×C₉⋊C₉	—	3	1	no	—
15	C₃₁×C₉⋊C₉	C₃²	1	3	yes	complemented
16	C₃₁×C₃²⋊C₉	—	3	1	no	—
17	C₉²×C₃₁	—	3	1	no	—
18	C₃³×C₃₁	H(3,𝔽₃)	1	1	yes	characteristic
19	C₃²×C₃₁⋊C₃	—	9	2	no	—
20	C₃×C₉×C₃₁	—	3	1	no	—
21	C₃×C₉×C₃₁	—	3	2	no	—
22	C₃×C₉×C₃₁	—	3	3	no	—
23	C₃×C₉×C₃₁	H(3,𝔽₃)	1	1	yes	characteristic
24	C₃×C₉×C₃₁	H(3,𝔽₃)	1	1	yes	characteristic
25	C₃×C₉×C₃₁	C₃³	1	1	yes	characteristic
26	C₃²⋊(C₃₁⋊C₃)	—	3	6	no	—
27	C₃²⋊(C₃₁⋊C₃)	—	9	2	no	—
28	C₃²⋊(C₃₁⋊C₃)	—	9	2	no	—
29	#11719689	—	3	3	no	—
30	#11719689	—	9	2	no	—
31	C₃×C₃₁⋊C₉	—	9	1	no	—
32	#10624846	—	31	1	no	maximal, Sylow
33	C₃²×C₃₁	—	9	1	no	—
34	C₃²×C₃₁	#732	1	1	yes	characteristic
35	C₃²×C₃₁	#732	1	2	yes	—
36	C₃²×C₃₁	C₃×H(3,𝔽₃)	1	1	yes	characteristic
37	C₃×C₃₁⋊C₃	—	9	6	no	—
38	C₃×C₃₁⋊C₃	—	27	6	no	—
39	C₉×C₃₁	—	3	1	no	—
40	C₉×C₃₁	—	3	1	no	—
41	C₉×C₃₁	—	3	1	no	—
42	C₉×C₃₁	—	9	3	no	—
43	C₃₁⋊C₉	—	9	3	no	—
44	C₃×#733	—	31	1	no	—
45	C₃×#732	—	31	2	no	—
46	#6887	C₃₁⋊C₃	1	1	yes	characteristic, complemented
47	#6881	—	31	9	no	—
48	C₃×C₃₁	—	3	1	no	—
49	C₃×C₃₁	—	3	2	no	—
50	C₃×C₃₁	—	3	1	no	—
51	C₃×C₃₁	C₃×#732	1	1	yes	characteristic
52	C₃₁⋊C₃	—	27	6	no	—
53	C₃×3₋¹⁺²	—	93	1	no	—
54	C₃×H(3,𝔽₃)	—	93	2	no	—
55	C₃²×C₉	C₃×C₃₁⋊C₃	1	1	yes	characteristic
56	#733	—	31	3	no	—
57	#733	—	93	1	no	—
58	#733	—	93	1	no	—
59	#732	—	31	6	no	—
60	#732	—	93	2	no	—
61	#732	—	93	2	no	—
62	C₉⋊C₉	—	3	1	no	—
63	C₉⋊C₉	C₃×C₃₁⋊C₃	1	3	yes	complemented
64	C₃²⋊C₉	—	3	1	no	—
65	C₉²	—	3	1	no	—
66	C₃₁	#10624846	1	1	yes	characteristic, complemented, Sylow
67	C₃³	—	279	2	no	—
68	C₃³	C₃²⋊(C₃₁⋊C₃)	1	1	yes	characteristic
69	3₋¹⁺²	—	279	2	no	—
70	3₋¹⁺²	—	93	3	no	—
71	H(3,𝔽₃)	—	279	2	no	—
72	H(3,𝔽₃)	—	279	2	no	—
73	H(3,𝔽₃)	—	93	6	no	—
74	C₃×C₉	—	3	1	no	—
75	C₃×C₉	—	3	2	no	—
76	C₃×C₉	—	3	3	no	—
77	C₃×C₉	—	279	1	no	—
78	C₃×C₉	C₃²⋊(C₃₁⋊C₃)	1	1	yes	characteristic
79	C₃×C₉	C₃²⋊(C₃₁⋊C₃)	1	1	yes	characteristic
80	C₃×C₉	C₃²×C₃₁⋊C₃	1	1	yes	characteristic
81	C₃²	—	9	1	no	—
82	C₃²	—	279	6	no	—
83	C₃²	—	837	6	no	—
84	C₃²	#14578900	1	1	yes	characteristic
85	C₃²	#14578900	1	2	yes	—
86	C₃²	C₃×C₃²⋊(C₃₁⋊C₃)	1	1	yes	characteristic
87	C₉	—	3	1	no	—
88	C₉	—	3	1	no	—
89	C₉	—	3	1	no	—
90	C₉	—	9	3	no	—
91	C₉	—	279	3	no	—
92	C₃	—	3	1	no	—
93	C₃	—	3	2	no	—
94	C₃	—	3	1	no	—
95	C₃	—	837	6	no	—
96	C₃	C₃×#14578900	1	1	yes	characteristic

	subgroup	G/H
center	C₃	C₃×#14578900
commutator, cosocle	C₃×C₉×C₃₁	C₃³
Frattini	C₃×C₉	C₃²×C₃₁⋊C₃
Fitting	C₃₁×#6887	C₃
socle	C₃×C₃₁	C₃×#732
hypercenter	#6887	C₃₁⋊C₃
hypocenter	C₃₁	#10624846
LCS	G ▷ C₃×C₉×C₃₁ ▷ C₃²×C₃₁ ▷ C₃×C₃₁ ▷ C₃₁
UCS	C₁ ◁ C₃ ◁ C₃³ ◁ C₃²×C₉ ◁ #6887

Complements

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

N	H ≅ G/N	type
1. C₃×#14578902	C₃	semidirect
2. C₃×#14578900	C₃	semidirect
3. #21419271	C₃	semidirect
4. C₃₁×#6887	C₃	semidirect
9. #14578902	C₃²	semidirect
12. #14578900	C₃²	semidirect
15. C₃₁×C₉⋊C₉	C₃²	semidirect
46. #6887	C₃₁⋊C₃	semidirect
63. C₉⋊C₉	C₃×C₃₁⋊C₃	semidirect
66. C₃₁	#10624846	semidirect

Maximal subgroups

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

subgroup	index
C₃×#14578902	3
C₃×#14578900	3
#21419271	3
C₃₁×#6887	3
#10624846	31