Subgroups — #41006359

Order 26973.

292 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃²×#11758601	C₃	1	1	yes	maximal, characteristic, complemented
2	C₃×C₂₇×C₃₇⋊C₃	C₃	1	3	yes	maximal, complemented
3	C₉×C₃²⋊(C₃₇⋊C₃)	C₃	1	1	yes	maximal, characteristic
4	C₃²⋊#11758601	C₃	1	1	yes	maximal, characteristic, complemented
5	#22582465	C₃	1	6	yes	maximal, complemented
6	C₃₇×C₃²⋊C₂₇	C₃	1	1	yes	maximal, characteristic, complemented
7	C₃×C₃²⋊(C₃₇⋊C₃)	C₉	1	1	yes	characteristic
8	C₃×C₉×C₃₇⋊C₃	C₃²	1	3	yes	—
9	C₃×#11758601	—	3	3	no	—
10	C₃×#11758601	C₃²	1	3	yes	—
11	C₃×#11758601	C₃²	1	2	yes	—
12	C₃×#11758601	C₃²	1	1	yes	characteristic
13	C₂₇×C₃₇⋊C₃	—	3	9	no	—
14	C₃²⋊#66524	C₉	1	2	yes	—
15	#15688502	C₉	1	6	yes	—
16	C₃²×C₉×C₃₇	C₃²	1	1	yes	characteristic
17	C₃×C₂₇×C₃₇	C₃²	1	3	yes	—
18	C₃³×C₃₇	C₃×C₉	1	1	yes	characteristic
19	C₃²×C₃₇⋊C₃	C₃×C₉	1	3	yes	—
20	C₃×C₉×C₃₇	—	3	1	no	—
21	C₃×C₉×C₃₇	—	3	2	no	—
22	C₃×C₉×C₃₇	C₃×C₉	1	2	yes	—
23	C₃×C₉×C₃₇	C₃³	1	1	yes	characteristic
24	C₃²⋊(C₃₇⋊C₃)	C₂₇	1	9	yes	complemented
25	C₃×#66524	C₃×C₉	1	6	yes	—
26	C₉×C₃₇⋊C₃	—	3	3	no	—
27	C₉×C₃₇⋊C₃	—	3	6	no	—
28	C₂₇×C₃₇	—	3	3	no	—
29	#11758601	—	3	3	no	—
30	#11758601	—	3	2	no	—
31	#11758601	H(3,𝔽₃)	1	3	yes	complemented
32	C₂₇×H(3,𝔽₃)	—	37	1	no	maximal, Sylow
33	C₃²×C₃₇	—	3	1	no	—
34	C₃²×C₃₇	—	3	2	no	—
35	C₃²×C₃₇	C₃×C₂₇	1	3	yes	complemented
36	C₃²×C₃₇	C₃²×C₉	1	1	yes	characteristic
37	C₃×C₃₇⋊C₃	—	3	3	no	—
38	C₃×C₃₇⋊C₃	—	3	6	no	—
39	C₃×C₃₇⋊C₃	C₃×C₂₇	1	9	yes	complemented
40	C₉×C₃₇	—	3	2	no	—
41	C₉×C₃₇	C₃×H(3,𝔽₃)	1	1	yes	characteristic
42	C₉×C₃₇	#737	1	2	yes	—
43	#66524	—	3	6	no	—
44	C₃²×C₂₇	—	37	1	no	—
45	C₃²×C₂₇	—	37	3	no	—
46	C₉×H(3,𝔽₃)	—	37	1	no	—
47	C₃²⋊C₂₇	—	37	1	no	—
48	C₃²⋊C₂₇	—	37	6	no	—
49	C₃²⋊C₂₇	C₃₇⋊C₃	1	1	yes	characteristic, complemented
50	C₃×C₃₇	—	3	3	no	—
51	C₃×C₃₇	C₉×H(3,𝔽₃)	1	1	yes	characteristic
52	C₃×C₃₇	C₃²×C₂₇	1	1	yes	characteristic
53	C₃×C₃₇	#6903	1	2	yes	—
54	C₃₇⋊C₃	—	3	9	no	—
55	C₃×H(3,𝔽₃)	—	37	1	no	—
56	C₃²×C₉	—	37	3	no	—
57	C₃²×C₉	C₃×C₃₇⋊C₃	1	1	yes	characteristic
58	C₃×C₂₇	—	37	1	no	—
59	C₃×C₂₇	—	37	3	no	—
60	C₃×C₂₇	—	37	2	no	—
61	C₃×C₂₇	—	111	9	no	—
62	C₃×C₂₇	—	111	3	no	—
63	C₃×C₂₇	C₃×C₃₇⋊C₃	1	3	yes	—
64	C₃²⋊C₉	—	37	2	no	—
65	C₃²⋊C₉	—	37	6	no	—
66	C₃₇	C₂₇×H(3,𝔽₃)	1	1	yes	characteristic, complemented, Sylow
67	C₃³	—	37	3	no	—
68	C₃³	C₉×C₃₇⋊C₃	1	1	yes	characteristic
69	H(3,𝔽₃)	—	37	9	no	—
70	C₃×C₉	—	3	1	no	—
71	C₃×C₉	—	3	2	no	—
72	C₃×C₉	—	37	6	no	—
73	C₃×C₉	—	111	3	no	—
74	C₃×C₉	—	111	6	no	—
75	C₃×C₉	C₉×C₃₇⋊C₃	1	2	yes	—
76	C₃×C₉	C₃²×C₃₇⋊C₃	1	1	yes	characteristic
77	C₂₇	—	3	3	no	—
78	C₂₇	—	37	3	no	—
79	C₂₇	—	111	3	no	—
80	C₂₇	—	111	2	no	—
81	C₃²	—	3	1	no	—
82	C₃²	—	3	2	no	—
83	C₃²	—	37	9	no	—
84	C₃²	—	111	3	no	—
85	C₃²	—	111	6	no	—
86	C₃²	C₂₇×C₃₇⋊C₃	1	3	yes	complemented
87	C₃²	C₃×C₉×C₃₇⋊C₃	1	1	yes	characteristic
88	C₉	—	3	2	no	—
89	C₉	—	111	6	no	—
90	C₉	H(3,𝔽₃)×C₃₇⋊C₃	1	1	yes	characteristic
91	C₉	#15688532	1	2	yes	—
92	C₃	—	3	3	no	—
93	C₃	—	111	9	no	—
94	C₃	#22582539	1	1	yes	characteristic
95	C₃	C₃×C₂₇×C₃₇⋊C₃	1	1	yes	characteristic
96	C₃	#22582614	1	2	yes	—

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

Complements

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

N	H ≅ G/N	type
1. C₃²×#11758601	C₃	semidirect
2. C₃×C₂₇×C₃₇⋊C₃	C₃	semidirect
4. C₃²⋊#11758601	C₃	semidirect
5. #22582465	C₃	semidirect
6. C₃₇×C₃²⋊C₂₇	C₃	semidirect
24. C₃²⋊(C₃₇⋊C₃)	C₂₇	semidirect
31. #11758601	H(3,𝔽₃)	semidirect
35. C₃²×C₃₇	C₃×C₂₇	semidirect
39. C₃×C₃₇⋊C₃	C₃×C₂₇	semidirect
49. C₃²⋊C₂₇	C₃₇⋊C₃	semidirect
66. C₃₇	C₂₇×H(3,𝔽₃)	semidirect
86. C₃²	C₂₇×C₃₇⋊C₃	semidirect

Maximal subgroups

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

subgroup	index
C₃²×#11758601	3
C₃×C₂₇×C₃₇⋊C₃	3
C₉×C₃²⋊(C₃₇⋊C₃)	3
C₃²⋊#11758601	3
#22582465	3
C₃₇×C₃²⋊C₂₇	3
C₂₇×H(3,𝔽₃)	37