Subgroups — #41008574

Order 26973.

292 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×(C₃₇⋊C₃)⋊C₂₇	C₃	1	3	yes	maximal, complemented
2	C₃²×#11758603	C₃	1	1	yes	maximal, characteristic, complemented
3	#22582690	C₃	1	6	yes	maximal, complemented
4	C₃²⋊#11758603	C₃	1	2	yes	maximal, complemented
5	#22582539	C₃	1	1	yes	maximal, characteristic
6	(C₃₇⋊C₃)⋊C₂₇	—	3	9	no	—
7	C₃×#11758603	—	3	3	no	—
8	C₃×#11758603	C₃²	1	6	yes	—
9	C₃×#11758603	C₃²	1	2	yes	—
10	C₃×#11758603	C₃²	1	1	yes	characteristic
11	C₃×C₃²⋊(C₃₇⋊C₃)	C₉	1	1	yes	characteristic
12	C₃²×#66524	C₃²	1	1	yes	characteristic
13	C₃×C₉×C₃₇⋊C₃	C₃²	1	3	yes	—
14	C₃²⋊#66524	C₉	1	1	yes	characteristic
15	#15688502	C₉	1	6	yes	—
16	C₃₇×C₃²⋊C₉	C₉	1	1	yes	characteristic
17	C₃³×C₃₇	C₃×C₉	1	1	yes	characteristic
18	C₃²×C₃₇⋊C₃	C₃×C₉	1	3	yes	—
19	C₃×C₉×C₃₇	C₃×C₉	1	3	yes	—
20	C₃²⋊(C₃₇⋊C₃)	C₂₇	1	9	yes	complemented
21	C₃×#66524	—	3	1	no	—
22	C₃×#66524	—	3	2	no	—
23	C₃×#66524	C₃×C₉	1	2	yes	—
24	C₃×#66524	C₃×C₉	1	3	yes	—
25	C₃×#66524	C₃³	1	1	yes	characteristic
26	C₉×C₃₇⋊C₃	—	3	3	no	—
27	C₉×C₃₇⋊C₃	—	3	6	no	—
28	#11758603	—	3	2	no	—
29	#11758603	—	3	6	no	—
30	#11758603	H(3,𝔽₃)	1	3	yes	complemented
31	C₂₇×H(3,𝔽₃)	—	37	1	no	maximal, Sylow
32	C₃²×C₃₇	—	3	1	no	—
33	C₃²×C₃₇	—	3	2	no	—
34	C₃²×C₃₇	C₃×C₂₇	1	3	yes	complemented
35	C₃²×C₃₇	C₃²×C₉	1	1	yes	characteristic
36	C₃×C₃₇⋊C₃	—	3	3	no	—
37	C₃×C₃₇⋊C₃	—	3	6	no	—
38	C₃×C₃₇⋊C₃	C₃×C₂₇	1	9	yes	complemented
39	C₉×C₃₇	—	3	3	no	—
40	#66524	—	3	3	no	—
41	#66524	—	3	2	no	—
42	#66524	C₃×H(3,𝔽₃)	1	1	yes	characteristic
43	#66524	#737	1	2	yes	—
44	C₃²×C₂₇	—	37	1	no	—
45	C₃²×C₂₇	—	37	3	no	—
46	C₉×H(3,𝔽₃)	—	37	1	no	—
47	C₃²⋊C₂₇	—	37	2	no	—
48	C₃²⋊C₂₇	—	37	6	no	—
49	C₃×C₃₇	—	3	3	no	—
50	C₃×C₃₇	C₉×H(3,𝔽₃)	1	1	yes	characteristic
51	C₃×C₃₇	C₃²×C₂₇	1	1	yes	characteristic
52	C₃×C₃₇	#6903	1	2	yes	—
53	C₃₇⋊C₃	—	3	9	no	—
54	C₃×H(3,𝔽₃)	—	37	1	no	—
55	C₃²×C₉	—	37	1	no	—
56	C₃²×C₉	—	37	3	no	—
57	C₃×C₂₇	—	37	1	no	—
58	C₃×C₂₇	—	37	6	no	—
59	C₃×C₂₇	—	37	2	no	—
60	C₃×C₂₇	—	111	3	no	—
61	C₃×C₂₇	—	111	9	no	—
62	C₃²⋊C₉	—	37	1	no	—
63	C₃²⋊C₉	—	37	6	no	—
64	C₃²⋊C₉	#66526	1	1	yes	characteristic
65	C₃₇	C₂₇×H(3,𝔽₃)	1	1	yes	characteristic, complemented, Sylow
66	C₃³	—	37	3	no	—
67	C₃³	C₃×#66526	1	1	yes	characteristic
68	H(3,𝔽₃)	—	37	9	no	—
69	C₃×C₉	—	37	1	no	—
70	C₃×C₉	—	37	2	no	—
71	C₃×C₉	—	37	3	no	—
72	C₃×C₉	—	111	3	no	—
73	C₃×C₉	—	111	6	no	—
74	C₃×C₉	—	111	1	no	—
75	C₃×C₉	—	111	2	no	—
76	C₃×C₉	C₃×#66526	1	3	yes	—
77	C₂₇	—	37	3	no	—
78	C₂₇	—	111	2	no	—
79	C₂₇	—	111	6	no	—
80	C₃²	—	3	1	no	—
81	C₃²	—	3	2	no	—
82	C₃²	—	37	9	no	—
83	C₃²	—	111	3	no	—
84	C₃²	—	111	6	no	—
85	C₃²	(C₃₇⋊C₃)⋊C₂₇	1	3	yes	complemented
86	C₃²	C₃²×#66526	1	1	yes	characteristic
87	C₉	—	3	3	no	—
88	C₉	—	37	1	no	—
89	C₉	—	37	2	no	—
90	C₉	—	111	3	no	—
91	C₉	—	111	2	no	—
92	C₃	—	3	3	no	—
93	C₃	—	111	9	no	—
94	C₃	H(3,𝔽₃)×#66526	1	1	yes	characteristic
95	C₃	C₃×(C₃₇⋊C₃)⋊C₂₇	1	1	yes	characteristic
96	C₃	#22582732	1	2	yes	—

	subgroup	G/H
center, Frattini	C₃²	C₃²×#66526
commutator	C₃×C₃₇	C₃²×C₂₇
Fitting	C₃₇×C₃²⋊C₉	C₉
socle	C₃²×C₃₇	C₃²×C₉
prime radical	C₃×C₃²⋊(C₃₇⋊C₃)	C₉
cosocle	C₃×#66524	C₃³
hypercenter	C₃²⋊C₉	#66526
hypocenter	C₃₇	C₂₇×H(3,𝔽₃)
LCS	G ▷ C₃×C₃₇ ▷ C₃₇
UCS	C₁ ◁ C₃² ◁ C₃²⋊C₉

Complements

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

N	H ≅ G/N	type
1. C₃×(C₃₇⋊C₃)⋊C₂₇	C₃	semidirect
2. C₃²×#11758603	C₃	semidirect
3. #22582690	C₃	semidirect
4. C₃²⋊#11758603	C₃	semidirect
20. C₃²⋊(C₃₇⋊C₃)	C₂₇	semidirect
30. #11758603	H(3,𝔽₃)	semidirect
34. C₃²×C₃₇	C₃×C₂₇	semidirect
38. C₃×C₃₇⋊C₃	C₃×C₂₇	semidirect
65. C₃₇	C₂₇×H(3,𝔽₃)	semidirect
85. C₃²	(C₃₇⋊C₃)⋊C₂₇	semidirect

Maximal subgroups

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

subgroup	index
C₃×(C₃₇⋊C₃)⋊C₂₇	3
C₃²×#11758603	3
#22582690	3
C₃²⋊#11758603	3
#22582539	3
C₂₇×H(3,𝔽₃)	37