Subgroups — #41006495

Order 26973.

644 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃³×#66524	C₃	1	1	yes	maximal, characteristic, complemented
2	C₃×#15688518	C₃	1	9	yes	maximal, complemented
3	#22582550	C₃	1	2	yes	maximal, complemented
4	C₃₇×C₃²⋊H(3,𝔽₃)	C₃	1	1	yes	maximal, characteristic, complemented
5	C₃×C₃²⋊(C₃₇⋊C₃)	—	3	6	no	—
6	C₃×C₃²⋊(C₃₇⋊C₃)	C₃²	1	6	yes	—
7	C₃²×#66524	—	3	3	no	—
8	C₃²×#66524	—	3	9	no	—
9	C₃²×#66524	C₃²	1	3	yes	—
10	#15688518	—	3	27	no	—
11	C₃⁴×C₃₇	C₃²	1	1	yes	characteristic
12	C₃×H(3,𝔽₃)×C₃₇	—	3	3	no	—
13	C₃×H(3,𝔽₃)×C₃₇	C₃²	1	3	yes	—
14	C₃³×C₃₇	—	3	9	no	—
15	C₃³×C₃₇	—	3	3	no	—
16	C₃³×C₃₇	—	3	3	no	—
17	C₃³×C₃₇	H(3,𝔽₃)	1	3	yes	—
18	C₃³×C₃₇	C₃³	1	1	yes	characteristic
19	C₃²×C₃₇⋊C₃	—	3	6	no	—
20	H(3,𝔽₃)×C₃₇	—	3	9	no	—
21	H(3,𝔽₃)×C₃₇	—	9	9	no	—
22	C₃²⋊(C₃₇⋊C₃)	—	3	18	no	—
23	C₃²⋊(C₃₇⋊C₃)	—	9	18	no	—
24	C₃×#66524	—	3	27	no	—
25	C₃×#66524	—	3	6	no	—
26	C₃×#66524	—	3	3	no	—
27	C₃×#66524	H(3,𝔽₃)	1	9	yes	complemented
28	#10624732	—	37	1	no	maximal, Sylow
29	C₃²×C₃₇	—	3	27	no	—
30	C₃²×C₃₇	—	3	3	no	—
31	C₃²×C₃₇	—	3	6	no	—
32	C₃²×C₃₇	—	3	3	no	—
33	C₃²×C₃₇	—	9	3	no	—
34	C₃²×C₃₇	—	9	9	no	—
35	C₃²×C₃₇	#732	1	9	yes	complemented
36	C₃²×C₃₇	C₃×H(3,𝔽₃)	1	3	yes	—
37	C₃²×C₃₇	C₃×H(3,𝔽₃)	1	1	yes	characteristic
38	C₃×C₃₇⋊C₃	—	9	6	no	—
39	C₃×C₃₇⋊C₃	—	9	18	no	—
40	#66524	—	3	9	no	—
41	C₃³×C₉	—	37	1	no	—
42	C₃×#732	—	37	9	no	—
43	C₃²⋊H(3,𝔽₃)	—	37	2	no	—
44	C₃²⋊H(3,𝔽₃)	C₃₇⋊C₃	1	1	yes	characteristic, complemented
45	C₃×C₃₇	—	3	9	no	—
46	C₃×C₃₇	—	3	3	no	—
47	C₃×C₃₇	—	27	3	no	—
48	C₃×C₃₇	C₃²⋊H(3,𝔽₃)	1	1	yes	characteristic
49	C₃×C₃₇	C₃×#732	1	3	yes	—
50	C₃₇⋊C₃	—	27	6	no	—
51	C₃⁴	C₃×C₃₇⋊C₃	1	1	yes	characteristic
52	C₃×H(3,𝔽₃)	—	3	3	no	—
53	C₃×H(3,𝔽₃)	—	37	6	no	—
54	C₃×H(3,𝔽₃)	—	111	6	no	—
55	C₃×H(3,𝔽₃)	C₃×C₃₇⋊C₃	1	3	yes	—
56	C₃²×C₉	—	37	3	no	—
57	C₃²×C₉	—	111	9	no	—
58	C₃²×C₉	—	111	3	no	—
59	#732	—	111	27	no	—
60	C₃₇	#10624732	1	1	yes	characteristic, complemented, Sylow
61	C₃³	—	3	9	no	—
62	C₃³	—	3	3	no	—
63	C₃³	—	3	3	no	—
64	C₃³	—	111	6	no	—
65	C₃³	C₃²⋊(C₃₇⋊C₃)	1	3	yes	—
66	C₃³	C₃²×C₃₇⋊C₃	1	1	yes	characteristic
67	H(3,𝔽₃)	—	3	9	no	—
68	H(3,𝔽₃)	—	9	9	no	—
69	H(3,𝔽₃)	—	333	18	no	—
70	H(3,𝔽₃)	—	111	18	no	—
71	C₃×C₉	—	37	9	no	—
72	C₃×C₉	—	111	27	no	—
73	C₃×C₉	—	111	6	no	—
74	C₃×C₉	—	111	3	no	—
75	C₃²	—	3	27	no	—
76	C₃²	—	3	3	no	—
77	C₃²	—	3	6	no	—
78	C₃²	—	3	3	no	—
79	C₃²	—	9	3	no	—
80	C₃²	—	9	9	no	—
81	C₃²	—	333	6	no	—
82	C₃²	—	333	18	no	—
83	C₃²	#15688518	1	9	yes	complemented
84	C₃²	H(3,𝔽₃)×C₃₇⋊C₃	1	3	yes	—
85	C₃²	C₃×C₃²⋊(C₃₇⋊C₃)	1	1	yes	characteristic
86	C₉	—	111	9	no	—
87	C₃	—	3	9	no	—
88	C₃	—	3	3	no	—
89	C₃	—	27	3	no	—
90	C₃	—	999	6	no	—
91	C₃	(C₃×C₃₇⋊C₃)⋊H(3,𝔽₃)	1	1	yes	characteristic
92	C₃	C₃×#15688518	1	3	yes	—

	subgroup	G/H
center	C₃²	C₃×C₃²⋊(C₃₇⋊C₃)
commutator, cosocle	C₃³×C₃₇	C₃³
Frattini	C₃³	C₃²×C₃₇⋊C₃
Fitting	C₃₇×C₃²⋊H(3,𝔽₃)	C₃
socle	C₃²×C₃₇	C₃×H(3,𝔽₃)
hypercenter	C₃²⋊H(3,𝔽₃)	C₃₇⋊C₃
hypocenter	C₃₇	#10624732
LCS	G ▷ C₃³×C₃₇ ▷ C₃×C₃₇ ▷ C₃₇
UCS	C₁ ◁ C₃² ◁ C₃⁴ ◁ C₃²⋊H(3,𝔽₃)

Complements

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

N	H ≅ G/N	type
1. C₃³×#66524	C₃	semidirect
2. C₃×#15688518	C₃	semidirect
3. #22582550	C₃	semidirect
4. C₃₇×C₃²⋊H(3,𝔽₃)	C₃	semidirect
27. C₃×#66524	H(3,𝔽₃)	semidirect
35. C₃²×C₃₇	#732	semidirect
44. C₃²⋊H(3,𝔽₃)	C₃₇⋊C₃	semidirect
60. C₃₇	#10624732	semidirect
83. C₃²	#15688518	semidirect

Maximal subgroups

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

subgroup	index
C₃³×#66524	3
C₃×#15688518	3
#22582550	3
C₃₇×C₃²⋊H(3,𝔽₃)	3
#10624732	37