Subgroups — #21419321

Order 7533.

156 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×#11719686	C₃	1	1	yes	maximal, characteristic, complemented
2	C₃×#11719687	C₃	1	1	yes	maximal, characteristic, complemented
3	C₃×C₃²⋊(C₃₁⋊C₃)	C₃	1	1	yes	maximal, characteristic
4	C₃×C₉×C₃₁⋊C₃	C₃	1	1	yes	maximal, characteristic, complemented
5	C₃²⋊(C₃₁⋊C₉)	C₃	1	2	yes	maximal, complemented
6	(C₃×C₃₁⋊C₃)⋊C₉	C₃	1	2	yes	maximal, complemented
7	(C₃×C₃₁⋊C₃)⋊C₉	C₃	1	2	yes	maximal, complemented
8	(C₃×C₃₁⋊C₃)⋊C₉	C₃	1	2	yes	maximal, complemented
9	C₃₁×C₃²⋊C₉	C₃	1	1	yes	maximal, characteristic, complemented
10	C₃³×C₃₁	C₃²	1	1	yes	characteristic
11	C₃²×C₃₁⋊C₃	C₃²	1	2	yes	—
12	C₃²×C₃₁⋊C₃	C₃²	1	1	yes	characteristic
13	C₃×C₉×C₃₁	C₃²	1	1	yes	characteristic
14	C₃×C₉×C₃₁	C₃²	1	1	yes	characteristic
15	C₃×C₉×C₃₁	C₃²	1	1	yes	characteristic
16	C₃²⋊(C₃₁⋊C₃)	—	3	3	no	—
17	C₃×C₃₁⋊C₉	C₃²	1	2	yes	—
18	C₃×C₃₁⋊C₉	C₃²	1	2	yes	—
19	C₃×C₃₁⋊C₉	C₃²	1	2	yes	—
20	#11719687	—	3	3	no	—
21	#11719686	—	3	3	no	—
22	C₉×C₃₁⋊C₃	—	3	3	no	—
23	C₃²×C₃₁	—	3	1	no	—
24	C₃²×C₃₁	—	3	1	no	—
25	C₃²×C₃₁	—	3	2	no	—
26	C₃²×C₃₁	C₃³	1	1	yes	characteristic
27	C₃×C₃₁⋊C₃	—	3	1	no	—
28	C₃×C₃₁⋊C₃	—	3	2	no	—
29	C₃×C₃₁⋊C₃	—	3	2	no	—
30	C₃×C₃₁⋊C₃	—	3	2	no	—
31	C₃×C₃₁⋊C₃	—	3	4	no	—
32	C₃×C₃₁⋊C₃	H(3,𝔽₃)	1	3	yes	—
33	C₉×C₃₁	—	3	1	no	—
34	C₉×C₃₁	—	3	1	no	—
35	C₉×C₃₁	—	3	1	no	—
36	C₃₁⋊C₉	—	3	2	no	—
37	C₃₁⋊C₉	—	3	2	no	—
38	C₃₁⋊C₉	—	3	2	no	—
39	#6893	—	31	1	no	maximal, Sylow
40	C₃×C₃₁	—	9	1	no	—
41	C₃×C₃₁	C₃×H(3,𝔽₃)	1	1	yes	characteristic
42	C₃×C₃₁	#737	1	1	yes	characteristic
43	C₃×C₃₁	#737	1	2	yes	—
44	C₃₁⋊C₃	—	3	3	no	—
45	C₃₁⋊C₃	—	9	2	no	—
46	C₃×3₋¹⁺²	—	31	1	no	—
47	C₃×3₋¹⁺²	—	31	1	no	—
48	C₃×H(3,𝔽₃)	—	31	1	no	—
49	C₃²×C₉	—	31	1	no	—
50	C₃²⋊C₉	—	31	2	no	—
51	C₃²⋊C₉	—	31	2	no	—
52	C₃²⋊C₉	—	31	2	no	—
53	C₃²⋊C₉	—	31	2	no	—
54	C₃²⋊C₉	C₃₁⋊C₃	1	1	yes	characteristic, complemented
55	C₃₁	#6893	1	1	yes	characteristic, complemented, Sylow
56	C₃³	—	31	1	no	—
57	C₃³	—	31	2	no	—
58	C₃³	C₃×C₃₁⋊C₃	1	1	yes	characteristic
59	3₋¹⁺²	—	93	3	no	—
60	3₋¹⁺²	—	93	3	no	—
61	H(3,𝔽₃)	—	93	3	no	—
62	C₃×C₉	—	31	2	no	—
63	C₃×C₉	—	31	2	no	—
64	C₃×C₉	—	31	2	no	—
65	C₃×C₉	—	93	3	no	—
66	C₃×C₉	C₃×C₃₁⋊C₃	1	1	yes	characteristic
67	C₃×C₉	C₃×C₃₁⋊C₃	1	1	yes	characteristic
68	C₃×C₉	C₃×C₃₁⋊C₃	1	1	yes	characteristic
69	C₃²	—	3	1	no	—
70	C₃²	—	3	1	no	—
71	C₃²	—	3	2	no	—
72	C₃²	—	31	3	no	—
73	C₃²	—	93	2	no	—
74	C₃²	—	93	2	no	—
75	C₃²	—	93	4	no	—
76	C₃²	—	93	1	no	—
77	C₃²	—	93	2	no	—
78	C₃²	C₃²×C₃₁⋊C₃	1	1	yes	characteristic
79	C₉	—	3	1	no	—
80	C₉	—	3	1	no	—
81	C₉	—	3	1	no	—
82	C₉	—	93	2	no	—
83	C₉	—	93	2	no	—
84	C₉	—	93	2	no	—
85	C₃	—	9	1	no	—
86	C₃	—	279	2	no	—
87	C₃	—	93	3	no	—
88	C₃	H(3,𝔽₃)×C₃₁⋊C₃	1	1	yes	characteristic
89	C₃	#14578912	1	1	yes	characteristic
90	C₃	#14578913	1	2	yes	—

	subgroup	G/H
center, Frattini	C₃²	C₃²×C₃₁⋊C₃
commutator, socle	C₃²×C₃₁	C₃³
Fitting	C₃₁×C₃²⋊C₉	C₃
hypercenter	C₃²⋊C₉	C₃₁⋊C₃
hypocenter	C₃₁	#6893
LCS	G ▷ C₃²×C₃₁ ▷ C₃₁
UCS	C₁ ◁ C₃² ◁ C₃²⋊C₉

Complements

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

N	H ≅ G/N	type
1. C₃×#11719686	C₃	semidirect
2. C₃×#11719687	C₃	semidirect
4. C₃×C₉×C₃₁⋊C₃	C₃	semidirect
5. C₃²⋊(C₃₁⋊C₉)	C₃	semidirect
6. (C₃×C₃₁⋊C₃)⋊C₉	C₃	semidirect
7. (C₃×C₃₁⋊C₃)⋊C₉	C₃	semidirect
8. (C₃×C₃₁⋊C₃)⋊C₉	C₃	semidirect
9. C₃₁×C₃²⋊C₉	C₃	semidirect
54. C₃²⋊C₉	C₃₁⋊C₃	semidirect
55. C₃₁	#6893	semidirect