Subgroups — #36382084

Order 22599.

226 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#21419389	C₃	1	3	yes	maximal, complemented
2	C₉²×C₃₁⋊C₃	C₃	1	1	yes	maximal, characteristic, complemented
3	#21419268	C₃	1	2	yes	maximal, complemented
4	#21419259	C₃	1	6	yes	maximal, complemented
5	C₃₁×#6879	C₃	1	1	yes	maximal, characteristic, complemented
6	#14578911	—	3	3	no	—
7	C₃×C₉×C₃₁⋊C₃	—	3	1	no	—
8	C₃×C₉×C₃₁⋊C₃	C₃²	1	1	yes	characteristic
9	#14578902	C₃²	1	6	yes	—
10	#14578896	—	3	3	no	—
11	#14578893	—	3	3	no	—
12	C₉×C₃₁⋊C₉	—	3	1	no	—
13	C₉×C₃₁⋊C₉	—	3	1	no	—
14	C₉×C₃₁⋊C₉	C₃²	1	2	yes	—
15	C₃₁×#732	C₃²	1	3	yes	—
16	C₉²×C₃₁	C₃²	1	1	yes	characteristic
17	C₃²×C₃₁⋊C₃	H(3,𝔽₃)	1	1	yes	characteristic
18	H(3,𝔽₃)×C₃₁	—	3	3	no	—
19	C₃×C₉×C₃₁	—	3	1	no	—
20	C₃×C₉×C₃₁	C₃³	1	1	yes	characteristic
21	#11719689	—	3	6	no	—
22	#11719689	—	9	6	no	—
23	C₃×C₃₁⋊C₉	—	3	2	no	—
24	C₃×C₃₁⋊C₉	—	9	3	no	—
25	C₃×C₃₁⋊C₉	H(3,𝔽₃)	1	1	yes	characteristic
26	C₃×C₃₁⋊C₉	H(3,𝔽₃)	1	1	yes	characteristic
27	C₉×C₃₁⋊C₃	—	3	9	no	—
28	C₉×C₃₁⋊C₃	—	3	1	no	—
29	C₉×C₃₁⋊C₃	—	3	2	no	—
30	#10624837	—	31	1	no	maximal, Sylow
31	C₃²×C₃₁	—	9	3	no	—
32	C₃²×C₃₁	C₃×H(3,𝔽₃)	1	1	yes	characteristic
33	C₃×C₃₁⋊C₃	—	3	1	no	—
34	C₃×C₃₁⋊C₃	—	3	3	no	—
35	C₉×C₃₁	—	3	3	no	—
36	C₉×C₃₁	—	3	1	no	—
37	C₃₁⋊C₉	—	3	6	no	—
38	C₃₁⋊C₉	—	3	1	no	—
39	C₃₁⋊C₉	—	9	6	no	—
40	C₃₁⋊C₉	#732	1	1	yes	characteristic
41	C₃₁⋊C₉	#732	1	2	yes	—
42	#6908	—	31	3	no	—
43	C₃×C₉²	—	31	1	no	—
44	#6880	—	31	2	no	—
45	#6879	C₃₁⋊C₃	1	1	yes	characteristic, complemented
46	#6878	—	31	6	no	—
47	C₃×C₃₁	—	3	1	no	—
48	C₃×C₃₁	—	27	3	no	—
49	C₃×C₃₁	C₃×#732	1	1	yes	characteristic
50	C₃₁⋊C₃	—	3	3	no	—
51	#737	—	93	3	no	—
52	C₃²×C₉	—	31	1	no	—
53	C₃²×C₉	—	93	1	no	—
54	#733	—	31	6	no	—
55	#732	C₃×C₃₁⋊C₃	1	3	yes	—
56	#731	—	93	3	no	—
57	AΣL(1,𝔽₂₇)	—	93	3	no	—
58	C₉²	—	31	2	no	—
59	C₉²	—	93	1	no	—
60	C₉²	—	93	1	no	—
61	C₉²	C₃×C₃₁⋊C₃	1	1	yes	characteristic
62	C₃₁	#10624837	1	1	yes	characteristic, complemented, Sylow
63	C₃³	—	31	1	no	—
64	3₋¹⁺²	—	279	6	no	—
65	3₋¹⁺²	—	93	6	no	—
66	H(3,𝔽₃)	—	3	3	no	—
67	C₃×C₉	—	3	1	no	—
68	C₃×C₉	—	279	3	no	—
69	C₃×C₉	—	31	1	no	—
70	C₃×C₉	—	31	1	no	—
71	C₃×C₉	—	93	1	no	—
72	C₃×C₉	—	93	2	no	—
73	C₃×C₉	—	93	9	no	—
74	C₃×C₉	—	93	2	no	—
75	C₃×C₉	C₃²×C₃₁⋊C₃	1	1	yes	characteristic
76	C₃²	—	9	3	no	—
77	C₃²	—	93	3	no	—
78	C₃²	—	93	1	no	—
79	C₃²	H(3,𝔽₃)×C₃₁⋊C₃	1	1	yes	characteristic
80	C₉	—	3	3	no	—
81	C₉	—	3	1	no	—
82	C₉	—	279	6	no	—
83	C₉	—	31	1	no	—
84	C₉	—	31	2	no	—
85	C₉	—	93	6	no	—
86	C₉	—	93	1	no	—
87	C₃	—	3	1	no	—
88	C₃	—	27	3	no	—
89	C₃	—	93	3	no	—
90	C₃	#732×C₃₁⋊C₃	1	1	yes	characteristic

	subgroup	G/H
center	C₃	#732×C₃₁⋊C₃
commutator, cosocle	C₃×C₉×C₃₁	C₃³
Frattini	C₃×C₉	C₃²×C₃₁⋊C₃
Fitting	C₃₁×#6879	C₃
socle	C₃×C₃₁	C₃×#732
hypercenter	#6879	C₃₁⋊C₃
hypocenter	C₃₁	#10624837
LCS	G ▷ C₃×C₉×C₃₁ ▷ C₃²×C₃₁ ▷ C₃×C₃₁ ▷ C₃₁
UCS	C₁ ◁ C₃ ◁ C₃² ◁ C₃×C₉ ◁ #6879

Complements

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

N	H ≅ G/N	type
1. #21419389	C₃	semidirect
2. C₉²×C₃₁⋊C₃	C₃	semidirect
3. #21419268	C₃	semidirect
4. #21419259	C₃	semidirect
5. C₃₁×#6879	C₃	semidirect
45. #6879	C₃₁⋊C₃	semidirect
62. C₃₁	#10624837	semidirect

Maximal subgroups

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

subgroup	index
#21419389	3
C₉²×C₃₁⋊C₃	3
#21419268	3
#21419259	3
C₃₁×#6879	3
#10624837	31