Subgroups — #41008193

Order 26973.

1118 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22582680	C₃	1	2	yes	maximal, complemented
2	C₃×H(3,𝔽₃)×C₃₇⋊C₃	C₃	1	1	yes	maximal, characteristic, complemented
3	#22582637	C₃	1	24	yes	maximal, complemented
4	C₃×#15688512	C₃	1	12	yes	maximal, complemented
5	C₃₇×H(5,𝔽₃)	C₃	1	1	yes	maximal, characteristic, complemented
6	C₃³×C₃₇⋊C₃	C₃²	1	4	yes	complemented
7	#15688530	—	3	18	no	—
8	C₃×#11758608	C₃²	1	24	yes	complemented
9	C₃×C₃²⋊(C₃₇⋊C₃)	C₃²	1	8	yes	complemented
10	H(3,𝔽₃)×C₃₇⋊C₃	—	3	9	no	—
11	#15688512	C₃²	1	9	yes	complemented
12	#15688512	C₃²	1	72	yes	complemented
13	C₃×H(3,𝔽₃)×C₃₇	—	3	9	no	—
14	C₃×H(3,𝔽₃)×C₃₇	C₃²	1	1	yes	characteristic
15	C₃×H(3,𝔽₃)×C₃₇	C₃²	1	12	yes	complemented
16	C₃³×C₃₇	—	3	12	no	—
17	C₃³×C₃₇	C₃³	1	4	yes	—
18	C₃²×C₃₇⋊C₃	—	3	12	no	—
19	C₃²×C₃₇⋊C₃	—	9	12	no	—
20	C₃²×C₃₇⋊C₃	C₃³	1	1	yes	characteristic
21	C₃²×C₃₇⋊C₃	C₃³	1	8	yes	—
22	H(3,𝔽₃)×C₃₇	—	3	24	no	—
23	H(3,𝔽₃)×C₃₇	H(3,𝔽₃)	1	9	yes	—
24	H(3,𝔽₃)×C₃₇	C₃³	1	9	yes	—
25	C₃²⋊(C₃₇⋊C₃)	—	3	24	no	—
26	#11758608	—	3	48	no	—
27	#11758608	C₃³	1	18	yes	—
28	C₃×#66524	—	3	24	no	—
29	#10624930	—	37	1	no	maximal, Sylow
30	C₃²×C₃₇	—	3	9	no	—
31	C₃²×C₃₇	—	9	4	no	—
32	C₃²×C₃₇	—	27	12	no	—
33	C₃²×C₃₇	C₃×H(3,𝔽₃)	1	12	yes	—
34	C₃²×C₃₇	C₃⁴	1	1	yes	characteristic
35	C₃×C₃₇⋊C₃	—	3	1	no	—
36	C₃×C₃₇⋊C₃	—	3	3	no	—
37	C₃×C₃₇⋊C₃	—	3	8	no	—
38	C₃×C₃₇⋊C₃	—	9	36	no	—
39	C₃×C₃₇⋊C₃	—	9	8	no	—
40	#66524	—	3	18	no	—
41	3₋¹⁺⁴	—	37	2	no	—
42	H(5,𝔽₃)	C₃₇⋊C₃	1	1	yes	characteristic, complemented
43	C₃²×H(3,𝔽₃)	—	37	1	no	—
44	#6909	—	37	24	no	—
45	C₃×AΣL(1,𝔽₂₇)	—	37	12	no	—
46	C₃×C₃₇	—	3	1	no	—
47	C₃×C₃₇	—	3	12	no	—
48	C₃×C₃₇	—	9	9	no	—
49	C₃×C₃₇	C₃²×H(3,𝔽₃)	1	1	yes	characteristic
50	C₃₇⋊C₃	—	3	3	no	—
51	C₃₇⋊C₃	—	9	8	no	—
52	C₃⁴	—	37	4	no	—
53	#737	—	111	18	no	—
54	C₃×3₋¹⁺²	—	37	24	no	—
55	C₃×H(3,𝔽₃)	—	3	9	no	—
56	C₃×H(3,𝔽₃)	—	37	8	no	—
57	C₃×H(3,𝔽₃)	—	111	9	no	—
58	C₃×H(3,𝔽₃)	C₃×C₃₇⋊C₃	1	1	yes	characteristic
59	C₃×H(3,𝔽₃)	C₃×C₃₇⋊C₃	1	12	yes	complemented
60	AΣL(1,𝔽₂₇)	—	37	9	no	—
61	AΣL(1,𝔽₂₇)	—	37	72	no	—
62	C₃₇	#10624930	1	1	yes	characteristic, complemented, Sylow
63	C₃³	—	3	12	no	—
64	C₃³	—	37	8	no	—
65	C₃³	—	37	1	no	—
66	C₃³	—	333	12	no	—
67	C₃³	—	111	12	no	—
68	C₃³	C₃²×C₃₇⋊C₃	1	4	yes	—
69	3₋¹⁺²	—	37	18	no	—
70	3₋¹⁺²	—	111	48	no	—
71	H(3,𝔽₃)	—	3	24	no	—
72	H(3,𝔽₃)	—	111	24	no	—
73	H(3,𝔽₃)	C₃²⋊(C₃₇⋊C₃)	1	9	yes	—
74	H(3,𝔽₃)	C₃²×C₃₇⋊C₃	1	9	yes	—
75	C₃×C₉	—	111	24	no	—
76	C₃²	—	3	9	no	—
77	C₃²	—	9	4	no	—
78	C₃²	—	27	12	no	—
79	C₃²	—	333	8	no	—
80	C₃²	—	333	36	no	—
81	C₃²	—	111	3	no	—
82	C₃²	—	111	8	no	—
83	C₃²	—	111	1	no	—
84	C₃²	C₃×C₃²⋊(C₃₇⋊C₃)	1	12	yes	—
85	C₃²	C₃³×C₃₇⋊C₃	1	1	yes	characteristic
86	C₉	—	111	18	no	—
87	C₃	—	3	1	no	—
88	C₃	—	3	12	no	—
89	C₃	—	9	9	no	—
90	C₃	—	333	8	no	—
91	C₃	—	111	3	no	—
92	C₃	C₃²×C₃²⋊(C₃₇⋊C₃)	1	1	yes	characteristic

	subgroup	G/H
center	C₃	C₃²×C₃²⋊(C₃₇⋊C₃)
commutator, cosocle	C₃²×C₃₇	C₃⁴
Frattini	C₃²	C₃³×C₃₇⋊C₃
Fitting	C₃₇×H(5,𝔽₃)	C₃
socle	C₃×C₃₇	C₃²×H(3,𝔽₃)
hypercenter	H(5,𝔽₃)	C₃₇⋊C₃
hypocenter	C₃₇	#10624930
LCS	G ▷ C₃²×C₃₇ ▷ C₃×C₃₇ ▷ C₃₇
UCS	C₁ ◁ C₃ ◁ C₃×H(3,𝔽₃) ◁ H(5,𝔽₃)

Complements

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

N	H ≅ G/N	type
1. #22582680	C₃	semidirect
2. C₃×H(3,𝔽₃)×C₃₇⋊C₃	C₃	semidirect
3. #22582637	C₃	semidirect
4. C₃×#15688512	C₃	semidirect
5. C₃₇×H(5,𝔽₃)	C₃	semidirect
6. C₃³×C₃₇⋊C₃	C₃²	semidirect
8. C₃×#11758608	C₃²	semidirect
9. C₃×C₃²⋊(C₃₇⋊C₃)	C₃²	semidirect
11. #15688512	C₃²	semidirect
12. #15688512	C₃²	semidirect
15. C₃×H(3,𝔽₃)×C₃₇	C₃²	semidirect
42. H(5,𝔽₃)	C₃₇⋊C₃	semidirect
59. C₃×H(3,𝔽₃)	C₃×C₃₇⋊C₃	semidirect
62. C₃₇	#10624930	semidirect

Maximal subgroups

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

subgroup	index
#22582680	3
C₃×H(3,𝔽₃)×C₃₇⋊C₃	3
#22582637	3
C₃×#15688512	3
C₃₇×H(5,𝔽₃)	3
#10624930	37