Subgroups — #41008575

Order 26973.

256 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	C₃²×#11758603	C₃	1	1	yes	maximal, characteristic, complemented
3	C₃²⋊#11758603	C₃	1	2	yes	maximal, complemented
4	C₃²⋊#11758603	C₃	1	6	yes	maximal, complemented
5	H(3,𝔽₃)×#66524	C₃	1	1	yes	maximal, characteristic
6	#15688541	—	3	9	no	—
7	C₃×#11758603	—	3	3	no	—
8	C₃×#11758603	C₃²	1	2	yes	—
9	C₃×#11758603	C₃²	1	6	yes	—
10	C₃×#11758603	C₃²	1	1	yes	characteristic
11	C₃²×#66524	C₃²	1	1	yes	characteristic
12	C₃²×#66524	C₃²	1	3	yes	—
13	C₃²⋊#66524	C₉	1	6	yes	—
14	C₃²⋊#66524	C₉	1	2	yes	—
15	C₃×H(3,𝔽₃)×C₃₇	C₉	1	1	yes	characteristic
16	C₃³×C₃₇	C₃×C₉	1	3	yes	—
17	C₃³×C₃₇	C₃×C₉	1	1	yes	characteristic
18	H(3,𝔽₃)×C₃₇	—	3	3	no	—
19	C₃×#66524	—	3	1	no	—
20	C₃×#66524	—	3	3	no	—
21	C₃×#66524	—	3	2	no	—
22	C₃×#66524	—	3	6	no	—
23	C₃×#66524	C₃×C₉	1	2	yes	—
24	C₃×#66524	C₃×C₉	1	6	yes	—
25	C₃×#66524	C₃³	1	1	yes	characteristic
26	#11758603	—	3	2	no	—
27	#11758603	—	3	6	no	—
28	#11758603	H(3,𝔽₃)	1	3	yes	complemented
29	C₂₇⋊H(3,𝔽₃)	—	37	1	no	maximal, Sylow
30	C₃²×C₃₇	—	3	1	no	—
31	C₃²×C₃₇	—	3	2	no	—
32	C₃²×C₃₇	—	3	3	no	—
33	C₃²×C₃₇	—	3	3	no	—
34	C₃²×C₃₇	—	3	6	no	—
35	C₃²×C₃₇	C₂₇⋊C₃	1	3	yes	complemented
36	C₃²×C₃₇	C₃²×C₉	1	1	yes	characteristic
37	#66524	—	3	2	no	—
38	#66524	—	3	6	no	—
39	#66524	C₃×H(3,𝔽₃)	1	1	yes	characteristic
40	#66524	#737	1	2	yes	—
41	C₃×C₂₇⋊C₃	—	37	3	no	—
42	C₃²×C₂₇	—	37	1	no	—
43	C₉×H(3,𝔽₃)	—	37	1	no	—
44	C₃²⋊C₂₇	—	37	6	no	—
45	C₃²⋊C₂₇	—	37	2	no	—
46	C₃×C₃₇	—	3	3	no	—
47	C₃×C₃₇	—	9	3	no	—
48	C₃×C₃₇	C₉×H(3,𝔽₃)	1	1	yes	characteristic
49	C₃×C₃₇	C₃×C₂₇⋊C₃	1	1	yes	characteristic
50	C₃×C₃₇	#6903	1	2	yes	—
51	C₃×H(3,𝔽₃)	#66526	1	1	yes	characteristic
52	C₃²×C₉	—	37	1	no	—
53	C₃²×C₉	—	37	3	no	—
54	C₂₇⋊C₃	—	111	9	no	—
55	C₃×C₂₇	—	37	1	no	—
56	C₃×C₂₇	—	37	2	no	—
57	C₃×C₂₇	—	37	6	no	—
58	C₃×C₂₇	—	111	3	no	—
59	C₃²⋊C₉	—	37	6	no	—
60	C₃²⋊C₉	—	37	2	no	—
61	C₃₇	C₂₇⋊H(3,𝔽₃)	1	1	yes	characteristic, complemented, Sylow
62	C₃³	C₃×#66526	1	3	yes	—
63	C₃³	C₃×#66526	1	1	yes	characteristic
64	H(3,𝔽₃)	—	3	3	no	—
65	C₃×C₉	—	37	1	no	—
66	C₃×C₉	—	37	2	no	—
67	C₃×C₉	—	37	6	no	—
68	C₃×C₉	—	111	1	no	—
69	C₃×C₉	—	111	3	no	—
70	C₃×C₉	—	111	2	no	—
71	C₃×C₉	—	111	6	no	—
72	C₂₇	—	37	3	no	—
73	C₂₇	—	111	2	no	—
74	C₂₇	—	111	6	no	—
75	C₃²	—	3	1	no	—
76	C₃²	—	3	2	no	—
77	C₃²	—	3	3	no	—
78	C₃²	—	3	3	no	—
79	C₃²	—	3	6	no	—
80	C₃²	#15688541	1	3	yes	complemented
81	C₃²	C₃²×#66526	1	1	yes	characteristic
82	C₉	—	37	1	no	—
83	C₉	—	37	2	no	—
84	C₉	—	111	2	no	—
85	C₉	—	111	6	no	—
86	C₃	—	3	3	no	—
87	C₃	—	9	3	no	—
88	C₃	H(3,𝔽₃)×#66526	1	1	yes	characteristic
89	C₃	C₃×#15688541	1	1	yes	characteristic
90	C₃	#22582731	1	2	yes	—

	subgroup	G/H
center, Frattini	C₃²	C₃²×#66526
commutator, socle	C₃²×C₃₇	C₃²×C₉
Fitting, prime radical	C₃×H(3,𝔽₃)×C₃₇	C₉
cosocle	C₃×#66524	C₃³
hypercenter	C₃×H(3,𝔽₃)	#66526
hypocenter	C₃₇	C₂₇⋊H(3,𝔽₃)
LCS	G ▷ C₃²×C₃₇ ▷ C₃₇
UCS	C₁ ◁ C₃² ◁ 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. C₃²×#11758603	C₃	semidirect
3. C₃²⋊#11758603	C₃	semidirect
4. C₃²⋊#11758603	C₃	semidirect
28. #11758603	H(3,𝔽₃)	semidirect
35. C₃²×C₃₇	C₂₇⋊C₃	semidirect
61. C₃₇	C₂₇⋊H(3,𝔽₃)	semidirect
80. C₃²	#15688541	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
C₃²×#11758603	3
C₃²⋊#11758603	3
H(3,𝔽₃)×#66524	3
C₂₇⋊H(3,𝔽₃)	37