Subgroups — #41006206

Order 26973.

290 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×#15688511	C₃	1	1	yes	maximal, characteristic
2	C₃²×#11758601	C₃	1	1	yes	maximal, characteristic
3	#22582538	C₃	1	1	yes	maximal, characteristic
4	#22582463	C₃	1	3	yes	maximal, complemented
5	#22582462	C₃	1	3	yes	maximal, complemented
6	C₂₇×#66524	C₃	1	3	yes	maximal, complemented
7	C₃×C₃₇×C₂₇⋊C₃	C₃	1	1	yes	maximal, characteristic
8	#15688511	C₉	1	9	yes	complemented
9	C₃×#11758601	C₉	1	9	yes	complemented
10	C₃×#11758601	C₃²	1	1	yes	characteristic
11	C₃×#11758601	C₃²	1	1	yes	characteristic
12	C₃×#11758601	C₃²	1	1	yes	characteristic
13	C₃×#11758601	C₃²	1	3	yes	—
14	#15688506	C₉	1	3	yes	—
15	#15688505	C₉	1	3	yes	—
16	C₃²⋊#66524	C₉	1	3	yes	—
17	C₉×#66524	C₃²	1	3	yes	—
18	C₃²×C₉×C₃₇	C₃²	1	1	yes	characteristic
19	C₃₇×C₂₇⋊C₃	C₉	1	9	yes	complemented
20	C₃×C₂₇×C₃₇	C₃²	1	3	yes	—
21	C₃³×C₃₇	C₃×C₉	1	1	yes	characteristic
22	C₃×C₉×C₃₇	C₃×C₉	1	1	yes	characteristic
23	C₃×C₉×C₃₇	C₃×C₉	1	1	yes	characteristic
24	C₃×C₉×C₃₇	C₃×C₉	1	3	yes	—
25	C₃×C₉×C₃₇	C₃×C₉	1	3	yes	—
26	C₃×C₉×C₃₇	C₃×C₉	1	3	yes	—
27	C₃×C₉×C₃₇	C₃³	1	1	yes	characteristic
28	C₃×#66524	C₃×C₉	1	9	yes	—
29	C₂₇×C₃₇	C₃×C₉	1	9	yes	—
30	#11758601	C₃×C₉	1	3	yes	—
31	#11758601	C₃×C₉	1	3	yes	—
32	#11758601	C₃×C₉	1	3	yes	—
33	#11758601	C₃×C₉	1	9	yes	—
34	#10624681	—	37	1	no	maximal, Sylow
35	C₃²×C₃₇	—	3	3	no	—
36	C₃²×C₃₇	C₉²	1	3	yes	—
37	C₃²×C₃₇	C₃²×C₉	1	1	yes	characteristic
38	C₉×C₃₇	C₉²	1	3	yes	—
39	C₉×C₃₇	C₉²	1	3	yes	—
40	C₉×C₃₇	C₃²×C₉	1	1	yes	characteristic
41	C₉×C₃₇	C₃²×C₉	1	1	yes	characteristic
42	C₉×C₃₇	C₃²×C₉	1	1	yes	characteristic
43	#66524	—	3	9	no	—
44	C₃×C₂₇⋊C₃	—	37	1	no	—
45	C₃×C₂₇⋊C₃	C₃₇⋊C₃	1	1	yes	characteristic
46	C₃²×C₂₇	—	37	1	no	—
47	#6887	—	37	1	no	—
48	#6864	—	37	3	no	—
49	#6864	—	37	3	no	—
50	C₉×C₂₇	—	37	3	no	—
51	C₃×C₃₇	—	3	3	no	—
52	C₃×C₃₇	C₃×C₉²	1	1	yes	characteristic
53	C₃×C₃₇	#6903	1	3	yes	—
54	C₃²×C₉	C₃×C₃₇⋊C₃	1	1	yes	characteristic
55	C₂₇⋊C₃	—	37	9	no	—
56	C₂₇⋊C₃	#66524	1	9	yes	complemented
57	C₃×C₂₇	—	37	9	no	—
58	C₃×C₂₇	—	37	1	no	—
59	C₃×C₂₇	—	37	3	no	—
60	C₃×C₂₇	—	37	1	no	—
61	C₃×C₂₇	—	37	1	no	—
62	C₃×C₂₇	C₃×C₃₇⋊C₃	1	3	yes	—
63	C₉⋊C₉	—	37	3	no	—
64	C₉⋊C₉	—	37	3	no	—
65	C₃²⋊C₉	—	37	3	no	—
66	C₉²	—	37	3	no	—
67	C₃₇	#10624681	1	1	yes	characteristic, complemented, Sylow
68	C₃³	C₉×C₃₇⋊C₃	1	1	yes	characteristic
69	C₃×C₉	—	37	9	no	—
70	C₃×C₉	C₉×C₃₇⋊C₃	1	1	yes	characteristic
71	C₃×C₉	C₉×C₃₇⋊C₃	1	1	yes	characteristic
72	C₃×C₉	C₉×C₃₇⋊C₃	1	3	yes	—
73	C₃×C₉	C₉×C₃₇⋊C₃	1	3	yes	—
74	C₃×C₉	C₃×#66524	1	3	yes	—
75	C₃×C₉	C₃²×C₃₇⋊C₃	1	1	yes	characteristic
76	C₂₇	—	37	3	no	—
77	C₂₇	—	37	3	no	—
78	C₂₇	—	37	3	no	—
79	C₂₇	—	37	9	no	—
80	C₂₇	C₃×#66524	1	9	yes	—
81	C₃²	—	3	3	no	—
82	C₃²	C₉×#66524	1	3	yes	—
83	C₃²	C₃×C₉×C₃₇⋊C₃	1	1	yes	characteristic
84	C₉	—	111	9	no	—
85	C₉	C₉×#66524	1	3	yes	—
86	C₉	C₉×#66524	1	3	yes	—
87	C₉	C₃×C₉×C₃₇⋊C₃	1	1	yes	characteristic
88	C₉	C₃×C₉×C₃₇⋊C₃	1	1	yes	characteristic
89	C₉	C₃²×#66524	1	1	yes	characteristic
90	C₃	—	3	3	no	—
91	C₃	C₃×C₉×#66524	1	1	yes	characteristic
92	C₃	#22582614	1	3	yes	—

	subgroup	G/H
center, Frattini	C₃×C₉	C₃²×C₃₇⋊C₃
commutator	C₃×C₃₇	C₃×C₉²
Fitting	C₃×C₃₇×C₂₇⋊C₃	C₃
socle	C₃²×C₃₇	C₃²×C₉
prime radical	C₃³×C₃₇	C₃×C₉
cosocle	C₃×C₉×C₃₇	C₃³
hypercenter	C₃×C₂₇⋊C₃	C₃₇⋊C₃
hypocenter	C₃₇	#10624681
LCS	G ▷ C₃×C₃₇ ▷ C₃₇
UCS	C₁ ◁ C₃×C₉ ◁ C₃×C₂₇⋊C₃

Complements

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

N	H ≅ G/N	type
4. #22582463	C₃	semidirect
5. #22582462	C₃	semidirect
6. C₂₇×#66524	C₃	semidirect
8. #15688511	C₉	semidirect
9. C₃×#11758601	C₉	semidirect
19. C₃₇×C₂₇⋊C₃	C₉	semidirect
56. C₂₇⋊C₃	#66524	semidirect
67. C₃₇	#10624681	semidirect

Maximal subgroups

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

subgroup	index
C₃×#15688511	3
C₃²×#11758601	3
#22582538	3
#22582463	3
#22582462	3
C₂₇×#66524	3
C₃×C₃₇×C₂₇⋊C₃	3
#10624681	37