Subgroups — #41006509

Order 26973.

596 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×#15688513	C₃	1	9	yes	maximal, complemented
2	#22582554	C₃	1	2	yes	maximal, complemented
3	#22582550	C₃	1	1	yes	maximal, characteristic, complemented
4	C₃²×H(3,𝔽₃)×C₃₇	C₃	1	1	yes	maximal, characteristic, complemented
5	C₃×#11758608	C₃²	1	6	yes	—
6	C₃×C₃²⋊(C₃₇⋊C₃)	—	3	3	no	—
7	C₃×C₃²⋊(C₃₇⋊C₃)	C₃²	1	3	yes	—
8	#15688513	—	3	27	no	—
9	C₃²⋊#66524	—	3	6	no	—
10	C₃⁴×C₃₇	C₃²	1	1	yes	characteristic
11	C₃⁴×C₃₇	C₃²	1	3	yes	—
12	C₃×H(3,𝔽₃)×C₃₇	—	3	3	no	—
13	C₃×H(3,𝔽₃)×C₃₇	—	3	9	no	—
14	C₃³×C₃₇	—	3	6	no	—
15	C₃³×C₃₇	—	3	3	no	—
16	C₃³×C₃₇	—	3	3	no	—
17	C₃³×C₃₇	—	9	3	no	—
18	C₃³×C₃₇	—	9	9	no	—
19	C₃³×C₃₇	H(3,𝔽₃)	1	3	yes	—
20	C₃³×C₃₇	H(3,𝔽₃)	1	9	yes	complemented
21	C₃³×C₃₇	C₃³	1	1	yes	characteristic
22	C₃²×C₃₇⋊C₃	—	3	3	no	—
23	H(3,𝔽₃)×C₃₇	—	3	27	no	—
24	C₃²⋊(C₃₇⋊C₃)	—	3	9	no	—
25	C₃²⋊(C₃₇⋊C₃)	—	9	9	no	—
26	#11758608	—	3	18	no	—
27	C₃×#66524	—	3	6	no	—
28	#10624735	—	37	1	no	maximal, Sylow
29	C₃²×C₃₇	—	3	3	no	—
30	C₃²×C₃₇	—	3	3	no	—
31	C₃²×C₃₇	—	3	9	no	—
32	C₃²×C₃₇	—	3	3	no	—
33	C₃²×C₃₇	—	3	6	no	—
34	C₃²×C₃₇	—	9	9	no	—
35	C₃²×C₃₇	—	9	9	no	—
36	C₃²×C₃₇	—	9	18	no	—
37	C₃²×C₃₇	—	9	9	no	—
38	C₃²×C₃₇	C₃×H(3,𝔽₃)	1	3	yes	—
39	C₃²×C₃₇	C₃×H(3,𝔽₃)	1	1	yes	characteristic
40	C₃×C₃₇⋊C₃	—	9	3	no	—
41	C₃×C₃₇⋊C₃	—	9	9	no	—
42	#66524	—	9	6	no	—
43	C₃²×H(3,𝔽₃)	C₃₇⋊C₃	1	1	yes	characteristic, complemented
44	C₃×AΣL(1,𝔽₂₇)	—	37	9	no	—
45	#6891	—	37	2	no	—
46	C₃²⋊H(3,𝔽₃)	—	37	1	no	—
47	C₃×C₃₇	—	3	3	no	—
48	C₃×C₃₇	—	9	3	no	—
49	C₃×C₃₇	—	9	9	no	—
50	C₃×C₃₇	C₃²⋊H(3,𝔽₃)	1	1	yes	characteristic
51	C₃×C₃₇	#6909	1	3	yes	—
52	C₃₇⋊C₃	—	27	3	no	—
53	C₃⁴	C₃×C₃₇⋊C₃	1	3	yes	—
54	C₃⁴	C₃×C₃₇⋊C₃	1	1	yes	characteristic
55	C₃×3₋¹⁺²	—	37	6	no	—
56	C₃×H(3,𝔽₃)	—	3	3	no	—
57	C₃×H(3,𝔽₃)	—	3	9	no	—
58	C₃×H(3,𝔽₃)	—	37	3	no	—
59	C₃×H(3,𝔽₃)	—	111	3	no	—
60	AΣL(1,𝔽₂₇)	—	111	27	no	—
61	C₃²⋊C₉	—	111	6	no	—
62	C₃₇	#10624735	1	1	yes	characteristic, complemented, Sylow
63	C₃³	—	3	6	no	—
64	C₃³	—	3	3	no	—
65	C₃³	—	3	3	no	—
66	C₃³	—	9	3	no	—
67	C₃³	—	9	9	no	—
68	C₃³	—	111	3	no	—
69	C₃³	C₃²⋊(C₃₇⋊C₃)	1	3	yes	—
70	C₃³	C₃²⋊(C₃₇⋊C₃)	1	9	yes	complemented
71	C₃³	C₃²×C₃₇⋊C₃	1	1	yes	characteristic
72	3₋¹⁺²	—	111	18	no	—
73	H(3,𝔽₃)	—	3	27	no	—
74	H(3,𝔽₃)	—	333	9	no	—
75	H(3,𝔽₃)	—	111	9	no	—
76	C₃×C₉	—	111	6	no	—
77	C₃²	—	3	3	no	—
78	C₃²	—	3	3	no	—
79	C₃²	—	3	9	no	—
80	C₃²	—	3	3	no	—
81	C₃²	—	3	6	no	—
82	C₃²	—	9	9	no	—
83	C₃²	—	9	9	no	—
84	C₃²	—	9	18	no	—
85	C₃²	—	9	9	no	—
86	C₃²	—	333	3	no	—
87	C₃²	—	333	9	no	—
88	C₃²	C₃×C₃²⋊(C₃₇⋊C₃)	1	3	yes	—
89	C₃²	C₃×C₃²⋊(C₃₇⋊C₃)	1	1	yes	characteristic
90	C₉	—	333	6	no	—
91	C₃	—	3	3	no	—
92	C₃	—	9	3	no	—
93	C₃	—	9	9	no	—
94	C₃	—	999	3	no	—
95	C₃	#22582550	1	1	yes	characteristic
96	C₃	#22582638	1	3	yes	—

	subgroup	G/H
center	C₃²	C₃×C₃²⋊(C₃₇⋊C₃)
commutator, cosocle	C₃³×C₃₇	C₃³
Frattini	C₃³	C₃²×C₃₇⋊C₃
Fitting	C₃²×H(3,𝔽₃)×C₃₇	C₃
socle	C₃²×C₃₇	C₃×H(3,𝔽₃)
hypercenter	C₃²×H(3,𝔽₃)	C₃₇⋊C₃
hypocenter	C₃₇	#10624735
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. C₃×#15688513	C₃	semidirect
2. #22582554	C₃	semidirect
3. #22582550	C₃	semidirect
4. C₃²×H(3,𝔽₃)×C₃₇	C₃	semidirect
20. C₃³×C₃₇	H(3,𝔽₃)	semidirect
43. C₃²×H(3,𝔽₃)	C₃₇⋊C₃	semidirect
62. C₃₇	#10624735	semidirect
70. 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
C₃×#15688513	3
#22582554	3
#22582550	3
C₃²×H(3,𝔽₃)×C₃₇	3
#10624735	37