Subgroups — C₃²×#15688518 #41008136

Order 26973.

1376 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃²×C₃²⋊(C₃₇⋊C₃)	C₃	1	2	yes	maximal, complemented
2	C₃³×#66524	C₃	1	1	yes	maximal, characteristic, complemented
3	C₃×#15688518	C₃	1	36	yes	maximal, direct product factor
4	C₃²×H(3,𝔽₃)×C₃₇	C₃	1	1	yes	maximal, characteristic, complemented
5	C₃³×C₃₇⋊C₃	—	3	2	no	—
6	C₃×C₃²⋊(C₃₇⋊C₃)	—	3	16	no	—
7	C₃×C₃²⋊(C₃₇⋊C₃)	C₃²	1	24	yes	complemented
8	C₃²×#66524	—	3	8	no	—
9	C₃²×#66524	—	3	1	no	—
10	C₃²×#66524	C₃²	1	12	yes	complemented
11	#15688518	C₃²	1	81	yes	direct product factor
12	C₃⁴×C₃₇	—	3	1	no	—
13	C₃⁴×C₃₇	C₃²	1	1	yes	characteristic
14	C₃×H(3,𝔽₃)×C₃₇	—	3	8	no	—
15	C₃×H(3,𝔽₃)×C₃₇	C₃²	1	12	yes	complemented
16	C₃³×C₃₇	—	3	9	no	—
17	C₃³×C₃₇	—	3	12	no	—
18	C₃³×C₃₇	—	9	9	no	—
19	C₃³×C₃₇	H(3,𝔽₃)	1	8	yes	—
20	C₃³×C₃₇	H(3,𝔽₃)	1	1	yes	characteristic
21	C₃³×C₃₇	C₃³	1	4	yes	—
22	C₃²×C₃₇⋊C₃	—	3	24	no	—
23	C₃²×C₃₇⋊C₃	—	9	18	no	—
24	H(3,𝔽₃)×C₃₇	—	3	24	no	—
25	H(3,𝔽₃)×C₃₇	C₃³	1	9	yes	complemented
26	C₃²⋊(C₃₇⋊C₃)	—	3	48	no	—
27	C₃²⋊(C₃₇⋊C₃)	C₃³	1	18	yes	complemented
28	C₃×#66524	—	3	24	no	—
29	C₃×#66524	—	3	12	no	—
30	C₃×#66524	C₃³	1	9	yes	complemented
31	C₃²×#732	—	37	1	no	maximal, Sylow
32	C₃²×C₃₇	—	3	24	no	—
33	C₃²×C₃₇	—	3	12	no	—
34	C₃²×C₃₇	—	3	9	no	—
35	C₃²×C₃₇	—	9	36	no	—
36	C₃²×C₃₇	#732	1	9	yes	complemented
37	C₃²×C₃₇	C₃×H(3,𝔽₃)	1	8	yes	—
38	C₃²×C₃₇	C₃×H(3,𝔽₃)	1	4	yes	—
39	C₃²×C₃₇	C₃⁴	1	1	yes	characteristic
40	C₃×C₃₇⋊C₃	—	3	18	no	—
41	C₃×C₃₇⋊C₃	—	9	72	no	—
42	#66524	—	3	9	no	—
43	C₃²×H(3,𝔽₃)	—	37	2	no	—
44	C₃²×H(3,𝔽₃)	C₃₇⋊C₃	1	1	yes	characteristic, complemented
45	C₃³×C₉	—	37	1	no	—
46	C₃×#732	—	37	36	no	—
47	C₃×C₃₇	—	3	8	no	—
48	C₃×C₃₇	—	3	1	no	—
49	C₃×C₃₇	—	9	9	no	—
50	C₃×C₃₇	C₃×#732	1	12	yes	complemented
51	C₃×C₃₇	C₃²×H(3,𝔽₃)	1	1	yes	characteristic
52	C₃₇⋊C₃	—	9	18	no	—
53	C₃⁴	—	3	1	no	—
54	C₃⁴	—	111	2	no	—
55	C₃⁴	C₃×C₃₇⋊C₃	1	1	yes	characteristic
56	C₃×H(3,𝔽₃)	—	3	8	no	—
57	C₃×H(3,𝔽₃)	—	37	24	no	—
58	C₃×H(3,𝔽₃)	—	111	16	no	—
59	C₃×H(3,𝔽₃)	C₃×C₃₇⋊C₃	1	12	yes	complemented
60	C₃²×C₉	—	37	12	no	—
61	C₃²×C₉	—	111	8	no	—
62	C₃²×C₉	—	111	1	no	—
63	#732	—	37	81	no	—
64	C₃₇	C₃²×#732	1	1	yes	characteristic, complemented, Sylow
65	C₃³	—	3	9	no	—
66	C₃³	—	3	12	no	—
67	C₃³	—	9	9	no	—
68	C₃³	—	333	18	no	—
69	C₃³	—	111	24	no	—
70	C₃³	C₃²⋊(C₃₇⋊C₃)	1	8	yes	—
71	C₃³	C₃²⋊(C₃₇⋊C₃)	1	1	yes	characteristic
72	C₃³	C₃²×C₃₇⋊C₃	1	4	yes	—
73	H(3,𝔽₃)	—	3	24	no	—
74	H(3,𝔽₃)	—	37	18	no	—
75	H(3,𝔽₃)	—	111	48	no	—
76	H(3,𝔽₃)	C₃²×C₃₇⋊C₃	1	9	yes	complemented
77	C₃×C₉	—	37	9	no	—
78	C₃×C₉	—	111	24	no	—
79	C₃×C₉	—	111	12	no	—
80	C₃²	—	3	24	no	—
81	C₃²	—	3	12	no	—
82	C₃²	—	3	9	no	—
83	C₃²	—	9	36	no	—
84	C₃²	—	333	72	no	—
85	C₃²	—	111	18	no	—
86	C₃²	#15688518	1	9	yes	direct product factor
87	C₃²	C₃×C₃²⋊(C₃₇⋊C₃)	1	8	yes	—
88	C₃²	C₃×C₃²⋊(C₃₇⋊C₃)	1	4	yes	—
89	C₃²	C₃³×C₃₇⋊C₃	1	1	yes	characteristic
90	C₉	—	111	9	no	—
91	C₃	—	3	8	no	—
92	C₃	—	3	1	no	—
93	C₃	—	9	9	no	—
94	C₃	—	333	18	no	—
95	C₃	C₃×#15688518	1	12	yes	direct product factor
96	C₃	C₃²×C₃²⋊(C₃₇⋊C₃)	1	1	yes	characteristic

	subgroup	G/H
center	C₃³	C₃²⋊(C₃₇⋊C₃)
commutator, cosocle	C₃²×C₃₇	C₃⁴
Frattini	C₃²	C₃³×C₃₇⋊C₃
Fitting	C₃²×H(3,𝔽₃)×C₃₇	C₃
socle	C₃³×C₃₇	H(3,𝔽₃)
hypercenter	C₃²×H(3,𝔽₃)	C₃₇⋊C₃
hypocenter	C₃₇	C₃²×#732
LCS	G ▷ C₃²×C₃₇ ▷ C₃×C₃₇ ▷ C₃₇
UCS	C₁ ◁ C₃³ ◁ C₃⁴ ◁ 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. C₃³×#66524	C₃	semidirect
3. C₃×#15688518	C₃	direct (N × H)
4. C₃²×H(3,𝔽₃)×C₃₇	C₃	semidirect
7. C₃×C₃²⋊(C₃₇⋊C₃)	C₃²	semidirect
10. C₃²×#66524	C₃²	semidirect
11. #15688518	C₃²	direct (N × H)
15. C₃×H(3,𝔽₃)×C₃₇	C₃²	semidirect
25. H(3,𝔽₃)×C₃₇	C₃³	semidirect
27. C₃²⋊(C₃₇⋊C₃)	C₃³	semidirect
30. C₃×#66524	C₃³	semidirect
36. C₃²×C₃₇	#732	semidirect
44. C₃²×H(3,𝔽₃)	C₃₇⋊C₃	semidirect
50. C₃×C₃₇	C₃×#732	semidirect
59. C₃×H(3,𝔽₃)	C₃×C₃₇⋊C₃	semidirect
64. C₃₇	C₃²×#732	semidirect
76. H(3,𝔽₃)	C₃²×C₃₇⋊C₃	semidirect
86. C₃²	#15688518	direct (N × H)
95. C₃	C₃×#15688518	direct (N × H)

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
C₃³×#66524	3
C₃×#15688518	3
C₃²×H(3,𝔽₃)×C₃₇	3
C₃²×#732	37

Subgroups — C32×#15688518 #41008136

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃²×#15688518 #41008136