Subgroups — #41006447

Order 26973.

216 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22582613	C₃	1	3	yes	maximal, complemented
2	C₃×#15688511	C₃	1	1	yes	maximal, characteristic, complemented
3	#22582486	C₃	1	3	yes	maximal, complemented
4	#22582483	C₃	1	3	yes	maximal, complemented
5	#22582474	C₃	1	2	yes	maximal, complemented
6	C₃₇×#6908	C₃	1	1	yes	maximal, characteristic
7	#15688511	—	3	3	no	—
8	#15688511	—	3	6	no	—
9	#15688511	C₃²	1	6	yes	—
10	C₃×#11758601	—	3	3	no	—
11	C₃×#11758601	C₃²	1	1	yes	characteristic
12	C₃×#11758601	C₃²	1	1	yes	characteristic
13	C₃×#11758601	C₃²	1	1	yes	characteristic
14	C₃₇×#737	C₃²	1	3	yes	—
15	C₃²×C₉×C₃₇	C₃²	1	1	yes	characteristic
16	C₃₇×#733	C₉	1	2	yes	—
17	C₃₇×#732	C₉	1	1	yes	characteristic
18	C₃₇×#731	C₉	1	3	yes	—
19	C₃₇×AΣL(1,𝔽₂₇)	C₉	1	3	yes	—
20	C₃³×C₃₇	C₃×C₉	1	1	yes	characteristic
21	3₋¹⁺²×C₃₇	—	3	6	no	—
22	3₋¹⁺²×C₃₇	C₃×C₉	1	6	yes	—
23	H(3,𝔽₃)×C₃₇	C₃×C₉	1	3	yes	—
24	C₃×C₉×C₃₇	—	3	1	no	—
25	C₃×C₉×C₃₇	—	3	3	no	—
26	C₃×C₉×C₃₇	—	3	2	no	—
27	C₃×C₉×C₃₇	C₃×C₉	1	1	yes	characteristic
28	C₃×C₉×C₃₇	C₃×C₉	1	1	yes	characteristic
29	C₃×C₉×C₃₇	C₃³	1	1	yes	characteristic
30	#11758601	—	3	6	no	—
31	#11758601	—	3	1	no	—
32	#11758601	—	3	1	no	—
33	#11758601	H(3,𝔽₃)	1	3	yes	—
34	#10624725	—	37	1	no	maximal, Sylow
35	C₃²×C₃₇	—	3	1	no	—
36	C₃²×C₃₇	—	3	3	no	—
37	C₃²×C₃₇	—	9	1	no	—
38	C₃²×C₃₇	C₃²×C₉	1	1	yes	characteristic
39	C₉×C₃₇	—	3	1	no	—
40	C₉×C₃₇	—	3	1	no	—
41	C₉×C₃₇	—	3	6	no	—
42	C₉×C₃₇	C₃×H(3,𝔽₃)	1	1	yes	characteristic
43	C₉×C₃₇	#737	1	2	yes	—
44	#6908	C₃₇⋊C₃	1	1	yes	characteristic
45	#6903	—	37	3	no	—
46	C₃×C₂₇⋊C₃	—	37	1	no	—
47	#6873	—	37	3	no	—
48	#6872	—	37	3	no	—
49	#6869	—	37	2	no	—
50	C₃×C₃₇	—	3	1	no	—
51	C₃×C₃₇	—	9	3	no	—
52	C₃×C₃₇	—	9	1	no	—
53	C₃×C₃₇	C₉×H(3,𝔽₃)	1	1	yes	characteristic
54	#737	C₃×C₃₇⋊C₃	1	3	yes	—
55	C₃²×C₉	C₃×C₃₇⋊C₃	1	1	yes	characteristic
56	#733	#66524	1	2	yes	—
57	#732	#66524	1	1	yes	characteristic
58	#731	#66524	1	3	yes	—
59	AΣL(1,𝔽₂₇)	#66524	1	3	yes	—
60	C₂₇⋊C₃	—	37	6	no	—
61	C₂₇⋊C₃	—	111	3	no	—
62	C₂₇⋊C₃	—	111	6	no	—
63	C₃×C₂₇	—	37	1	no	—
64	C₃×C₂₇	—	37	1	no	—
65	C₃×C₂₇	—	37	1	no	—
66	C₃×C₂₇	—	111	3	no	—
67	C₃₇	#10624725	1	1	yes	characteristic, complemented, Sylow
68	C₃³	C₃×#66524	1	1	yes	characteristic
69	3₋¹⁺²	—	3	6	no	—
70	3₋¹⁺²	C₃×#66524	1	6	yes	—
71	H(3,𝔽₃)	C₃×#66524	1	3	yes	—
72	C₃×C₉	—	3	3	no	—
73	C₃×C₉	—	3	1	no	—
74	C₃×C₉	—	3	2	no	—
75	C₃×C₉	C₃×#66524	1	1	yes	characteristic
76	C₃×C₉	C₃×#66524	1	1	yes	characteristic
77	C₃×C₉	C₃²×C₃₇⋊C₃	1	1	yes	characteristic
78	C₂₇	—	37	3	no	—
79	C₂₇	—	111	1	no	—
80	C₂₇	—	111	1	no	—
81	C₂₇	—	111	6	no	—
82	C₃²	—	3	1	no	—
83	C₃²	—	3	3	no	—
84	C₃²	—	9	1	no	—
85	C₃²	C₃²×#66524	1	1	yes	characteristic
86	C₉	—	3	1	no	—
87	C₉	—	3	1	no	—
88	C₉	—	3	6	no	—
89	C₉	H(3,𝔽₃)×C₃₇⋊C₃	1	1	yes	characteristic
90	C₉	#15688530	1	2	yes	—
91	C₃	—	3	1	no	—
92	C₃	—	9	1	no	—
93	C₃	—	9	3	no	—
94	C₃	H(3,𝔽₃)×#66524	1	1	yes	characteristic

	subgroup	G/H
center	C₉	H(3,𝔽₃)×C₃₇⋊C₃
commutator	C₃²×C₃₇	C₃²×C₉
Frattini	C₃×C₉	C₃²×C₃₇⋊C₃
Fitting, prime radical	C₃₇×#6908	C₃
socle	C₃×C₃₇	C₉×H(3,𝔽₃)
cosocle	C₃×C₉×C₃₇	C₃³
hypercenter	#6908	C₃₇⋊C₃
hypocenter	C₃₇	#10624725
LCS	G ▷ C₃²×C₃₇ ▷ C₃×C₃₇ ▷ C₃₇
UCS	C₁ ◁ C₉ ◁ C₃×C₉ ◁ #6908

Complements

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

N	H ≅ G/N	type
1. #22582613	C₃	semidirect
2. C₃×#15688511	C₃	semidirect
3. #22582486	C₃	semidirect
4. #22582483	C₃	semidirect
5. #22582474	C₃	semidirect
67. C₃₇	#10624725	semidirect

Maximal subgroups

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

subgroup	index
#22582613	3
C₃×#15688511	3
#22582486	3
#22582483	3
#22582474	3
C₃₇×#6908	3
#10624725	37