Subgroups — #41008499

Order 26973.

312 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×#15688541	C₃	1	3	yes	maximal, complemented
2	#22582694	C₃	1	9	yes	maximal, complemented
3	C₃×#15688530	C₃	1	1	yes	maximal, characteristic
4	#15688541	—	3	6	no	—
5	#15688541	C₃²	1	9	yes	—
6	C₃×#11758603	—	3	3	no	—
7	#15688530	—	3	3	no	—
8	#15688530	—	3	6	no	—
9	C₃×#11758608	C₉	1	2	yes	—
10	C₃×#11758608	C₉	1	6	yes	—
11	C₃²×#66524	C₃²	1	3	yes	—
12	C₃²×#66524	C₃²	1	1	yes	characteristic
13	C₃×H(3,𝔽₃)×C₃₇	C₉	1	1	yes	characteristic
14	C₃³×C₃₇	C₃×C₉	1	3	yes	—
15	C₃³×C₃₇	C₃×C₉	1	1	yes	characteristic
16	H(3,𝔽₃)×C₃₇	—	3	3	no	—
17	#11758608	—	3	6	no	—
18	#11758608	—	3	18	no	—
19	C₃×#66524	—	3	3	no	—
20	C₃×#66524	—	3	6	no	—
21	C₃×#66524	C₃×C₉	1	6	yes	—
22	C₃×#66524	C₃×C₉	1	2	yes	—
23	C₃×#66524	H(3,𝔽₃)	1	3	yes	—
24	C₃×#66524	3₋¹⁺²	1	6	yes	—
25	C₃×#66524	C₃³	1	1	yes	characteristic
26	#11758603	—	3	9	no	—
27	#10624694	—	37	1	no	maximal, Sylow
28	C₃²×C₃₇	—	3	3	no	—
29	C₃²×C₃₇	—	9	1	no	—
30	C₃²×C₃₇	—	9	3	no	—
31	C₃²×C₃₇	C₃²⋊C₉	1	3	yes	—
32	C₃²×C₃₇	C₃²×C₉	1	1	yes	characteristic
33	#66524	—	3	6	no	—
34	#66524	C₃²⋊C₉	1	6	yes	—
35	#66524	C₃×H(3,𝔽₃)	1	1	yes	characteristic
36	#66524	C₃×3₋¹⁺²	1	2	yes	—
37	C₃×#737	—	37	1	no	—
38	C₃×C₂₇⋊C₃	—	37	3	no	—
39	#6869	—	37	9	no	—
40	C₃×C₃₇	—	3	1	no	—
41	C₃×C₃₇	—	3	3	no	—
42	C₃×C₃₇	—	9	3	no	—
43	C₃×C₃₇	C₃×C₃²⋊C₉	1	1	yes	characteristic
44	#737	—	111	3	no	—
45	#737	—	111	6	no	—
46	C₃×3₋¹⁺²	—	37	6	no	—
47	C₃×3₋¹⁺²	—	37	2	no	—
48	C₃×H(3,𝔽₃)	#66526	1	1	yes	characteristic
49	C₃²×C₉	—	37	1	no	—
50	C₃²×C₉	—	37	3	no	—
51	C₂₇⋊C₃	—	37	9	no	—
52	C₂₇⋊C₃	—	111	6	no	—
53	C₃×C₂₇	—	111	3	no	—
54	C₃₇	#10624694	1	1	yes	characteristic, complemented, Sylow
55	C₃³	C₃×#66526	1	3	yes	—
56	C₃³	C₃×#66526	1	1	yes	characteristic
57	3₋¹⁺²	—	111	18	no	—
58	3₋¹⁺²	—	111	6	no	—
59	H(3,𝔽₃)	—	3	3	no	—
60	C₃×C₉	—	37	3	no	—
61	C₃×C₉	—	37	6	no	—
62	C₃×C₉	—	37	2	no	—
63	C₃×C₉	—	37	1	no	—
64	C₃×C₉	—	37	6	no	—
65	C₃×C₉	—	111	3	no	—
66	C₃×C₉	—	111	6	no	—
67	C₂₇	—	111	9	no	—
68	C₃²	—	3	3	no	—
69	C₃²	—	9	1	no	—
70	C₃²	—	9	3	no	—
71	C₃²	C₃²⋊#66526	1	3	yes	—
72	C₃²	C₃²×#66526	1	1	yes	characteristic
73	C₉	—	37	1	no	—
74	C₉	—	37	2	no	—
75	C₉	—	37	6	no	—
76	C₉	—	111	6	no	—
77	C₃	—	3	1	no	—
78	C₃	—	3	3	no	—
79	C₃	—	9	3	no	—
80	C₃	C₃×C₃²⋊#66526	1	1	yes	characteristic

	subgroup	G/H
center	C₃	C₃×C₃²⋊#66526
commutator	C₃²×C₃₇	C₃²×C₉
Frattini	C₃²	C₃²×#66526
Fitting, prime radical	C₃×H(3,𝔽₃)×C₃₇	C₉
socle	C₃×C₃₇	C₃×C₃²⋊C₉
cosocle	C₃×#66524	C₃³
hypercenter	C₃×H(3,𝔽₃)	#66526
hypocenter	C₃₇	#10624694
LCS	G ▷ C₃²×C₃₇ ▷ C₃×C₃₇ ▷ C₃₇
UCS	C₁ ◁ C₃ ◁ C₃³ ◁ C₃×H(3,𝔽₃)

Complements

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

N	H ≅ G/N	type
1. C₃×#15688541	C₃	semidirect
2. #22582694	C₃	semidirect
54. C₃₇	#10624694	semidirect

Maximal subgroups

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

subgroup	index
C₃×#15688541	3
#22582694	3
C₃×#15688530	3
#10624694	37