Subgroups — #41007640

Order 26973.

1718 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃²×C₃²⋊(C₃₇⋊C₃)	C₃	1	3	yes	maximal, complemented
2	#22582550	C₃	1	9	yes	maximal, complemented
3	(C₃×C₃₇⋊C₃)⋊H(3,𝔽₃)	C₃	1	27	yes	maximal, complemented
4	C₃²×H(3,𝔽₃)×C₃₇	C₃	1	1	yes	maximal, characteristic, complemented
5	C₃³×C₃₇⋊C₃	C₃²	1	9	yes	complemented
6	C₃×C₃²⋊(C₃₇⋊C₃)	—	3	27	no	—
7	C₃×C₃²⋊(C₃₇⋊C₃)	C₃²	1	18	yes	complemented
8	C₃×C₃²⋊(C₃₇⋊C₃)	C₃²	1	9	yes	complemented
9	C₃×C₃²⋊(C₃₇⋊C₃)	C₃²	1	81	yes	complemented
10	C₃⁴×C₃₇	C₃²	1	1	yes	characteristic
11	C₃⁴×C₃₇	C₃²	1	3	yes	complemented
12	C₃×H(3,𝔽₃)×C₃₇	—	3	9	no	—
13	C₃×H(3,𝔽₃)×C₃₇	C₃²	1	9	yes	complemented
14	C₃³×C₃₇	—	3	9	no	—
15	C₃³×C₃₇	—	9	3	no	—
16	C₃³×C₃₇	—	9	9	no	—
17	C₃³×C₃₇	H(3,𝔽₃)	1	9	yes	complemented
18	C₃³×C₃₇	C₃³	1	9	yes	—
19	C₃³×C₃₇	C₃³	1	3	yes	—
20	C₃³×C₃₇	C₃³	1	1	yes	characteristic
21	C₃²×C₃₇⋊C₃	—	3	27	no	—
22	C₃²×C₃₇⋊C₃	—	9	27	no	—
23	C₃²×C₃₇⋊C₃	C₃³	1	27	yes	—
24	H(3,𝔽₃)×C₃₇	—	3	27	no	—
25	C₃²⋊(C₃₇⋊C₃)	—	3	27	no	—
26	C₃²⋊(C₃₇⋊C₃)	—	3	54	no	—
27	C₃²⋊(C₃₇⋊C₃)	—	9	81	no	—
28	#10624880	—	37	1	no	maximal, Sylow
29	C₃²×C₃₇	—	3	9	no	—
30	C₃²×C₃₇	—	3	3	no	—
31	C₃²×C₃₇	—	3	9	no	—
32	C₃²×C₃₇	—	3	27	no	—
33	C₃²×C₃₇	—	9	9	no	—
34	C₃²×C₃₇	—	9	27	no	—
35	C₃²×C₃₇	C₃×H(3,𝔽₃)	1	3	yes	—
36	C₃²×C₃₇	C₃×H(3,𝔽₃)	1	9	yes	—
37	C₃²×C₃₇	C₃⁴	1	1	yes	characteristic
38	C₃×C₃₇⋊C₃	—	3	27	no	—
39	C₃×C₃₇⋊C₃	—	3	81	no	—
40	C₃×C₃₇⋊C₃	—	9	81	no	—
41	C₃²×H(3,𝔽₃)	—	37	3	no	—
42	C₃²×H(3,𝔽₃)	C₃₇⋊C₃	1	1	yes	characteristic, complemented
43	C₃²⋊H(3,𝔽₃)	—	37	9	no	—
44	C₃²⋊H(3,𝔽₃)	—	37	27	no	—
45	C₃×C₃₇	—	3	9	no	—
46	C₃×C₃₇	—	3	3	no	—
47	C₃×C₃₇	—	9	9	no	—
48	C₃×C₃₇	C₃²×H(3,𝔽₃)	1	1	yes	characteristic
49	C₃×C₃₇	H(5,𝔽₃)	1	3	yes	—
50	C₃₇⋊C₃	—	9	27	no	—
51	C₃⁴	—	37	9	no	—
52	C₃⁴	C₃×C₃₇⋊C₃	1	1	yes	characteristic
53	C₃⁴	C₃×C₃₇⋊C₃	1	3	yes	complemented
54	C₃×H(3,𝔽₃)	—	3	9	no	—
55	C₃×H(3,𝔽₃)	—	37	81	no	—
56	C₃×H(3,𝔽₃)	—	37	9	no	—
57	C₃×H(3,𝔽₃)	—	37	18	no	—
58	C₃×H(3,𝔽₃)	—	111	27	no	—
59	C₃×H(3,𝔽₃)	C₃×C₃₇⋊C₃	1	9	yes	complemented
60	C₃₇	#10624880	1	1	yes	characteristic, complemented, Sylow
61	C₃³	—	3	9	no	—
62	C₃³	—	9	3	no	—
63	C₃³	—	9	9	no	—
64	C₃³	—	37	27	no	—
65	C₃³	—	333	27	no	—
66	C₃³	—	111	27	no	—
67	C₃³	C₃²⋊(C₃₇⋊C₃)	1	9	yes	complemented
68	C₃³	C₃²×C₃₇⋊C₃	1	9	yes	—
69	C₃³	C₃²×C₃₇⋊C₃	1	1	yes	characteristic
70	C₃³	C₃²×C₃₇⋊C₃	1	3	yes	—
71	H(3,𝔽₃)	—	3	27	no	—
72	H(3,𝔽₃)	—	333	81	no	—
73	H(3,𝔽₃)	—	111	54	no	—
74	H(3,𝔽₃)	—	111	27	no	—
75	C₃²	—	3	3	no	—
76	C₃²	—	3	9	no	—
77	C₃²	—	3	9	no	—
78	C₃²	—	3	27	no	—
79	C₃²	—	9	9	no	—
80	C₃²	—	9	27	no	—
81	C₃²	—	333	81	no	—
82	C₃²	—	111	27	no	—
83	C₃²	—	111	81	no	—
84	C₃²	C₃×C₃²⋊(C₃₇⋊C₃)	1	9	yes	—
85	C₃²	C₃×C₃²⋊(C₃₇⋊C₃)	1	3	yes	—
86	C₃²	C₃³×C₃₇⋊C₃	1	1	yes	characteristic
87	C₃	—	3	3	no	—
88	C₃	—	3	9	no	—
89	C₃	—	9	9	no	—
90	C₃	—	333	27	no	—
91	C₃	C₃²×C₃²⋊(C₃₇⋊C₃)	1	1	yes	characteristic
92	C₃	C₃₇⋊H(5,𝔽₃)	1	3	yes	—

	subgroup	G/H
center, Frattini	C₃²	C₃³×C₃₇⋊C₃
commutator, socle, cosocle	C₃²×C₃₇	C₃⁴
Fitting	C₃²×H(3,𝔽₃)×C₃₇	C₃
hypercenter	C₃²×H(3,𝔽₃)	C₃₇⋊C₃
hypocenter	C₃₇	#10624880
LCS	G ▷ C₃²×C₃₇ ▷ C₃₇
UCS	C₁ ◁ C₃² ◁ C₃²×H(3,𝔽₃)

Complements

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

N	H ≅ G/N	type
1. C₃²×C₃²⋊(C₃₇⋊C₃)	C₃	semidirect
2. #22582550	C₃	semidirect
3. (C₃×C₃₇⋊C₃)⋊H(3,𝔽₃)	C₃	semidirect
4. C₃²×H(3,𝔽₃)×C₃₇	C₃	semidirect
5. C₃³×C₃₇⋊C₃	C₃²	semidirect
7. C₃×C₃²⋊(C₃₇⋊C₃)	C₃²	semidirect
8. C₃×C₃²⋊(C₃₇⋊C₃)	C₃²	semidirect
9. C₃×C₃²⋊(C₃₇⋊C₃)	C₃²	semidirect
11. C₃⁴×C₃₇	C₃²	semidirect
13. C₃×H(3,𝔽₃)×C₃₇	C₃²	semidirect
17. C₃³×C₃₇	H(3,𝔽₃)	semidirect
42. C₃²×H(3,𝔽₃)	C₃₇⋊C₃	semidirect
53. C₃⁴	C₃×C₃₇⋊C₃	semidirect
59. C₃×H(3,𝔽₃)	C₃×C₃₇⋊C₃	semidirect
60. C₃₇	#10624880	semidirect
67. C₃³	C₃²⋊(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×C₃²⋊(C₃₇⋊C₃)	3
#22582550	3
(C₃×C₃₇⋊C₃)⋊H(3,𝔽₃)	3
C₃²×H(3,𝔽₃)×C₃₇	3
#10624880	37