Subgroups — #41008200

Order 26973.

978 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	C₃₇⋊H(5,𝔽₃)	C₃	1	1	yes	maximal, characteristic
3	#22582643	C₃	1	24	yes	maximal, complemented
4	C₃×#15688516	C₃	1	12	yes	maximal, complemented
5	C₃×C₃₇×#737	C₃	1	1	yes	maximal, characteristic, complemented
6	#15688530	—	3	18	no	—
7	C₃×#11758608	C₃²	1	24	yes	complemented
8	C₃×C₃²⋊(C₃₇⋊C₃)	C₃²	1	12	yes	—
9	H(3,𝔽₃)×C₃₇⋊C₃	—	3	9	no	—
10	#15688516	C₃²	1	9	yes	complemented
11	#15688516	C₃²	1	72	yes	complemented
12	C₃₇×#737	—	3	9	no	—
13	C₃×3₋¹⁺²×C₃₇	C₃²	1	8	yes	complemented
14	C₃×H(3,𝔽₃)×C₃₇	C₃²	1	1	yes	characteristic
15	C₃²×C₉×C₃₇	C₃²	1	4	yes	complemented
16	C₃³×C₃₇	C₃³	1	4	yes	—
17	C₃²×C₃₇⋊C₃	—	3	12	no	—
18	3₋¹⁺²×C₃₇	—	3	24	no	—
19	H(3,𝔽₃)×C₃₇	H(3,𝔽₃)	1	9	yes	—
20	C₃×C₉×C₃₇	—	3	12	no	—
21	C₃×C₉×C₃₇	C₃³	1	1	yes	characteristic
22	C₃×C₉×C₃₇	C₃³	1	8	yes	—
23	C₃²⋊(C₃₇⋊C₃)	—	3	24	no	—
24	C₃²⋊(C₃₇⋊C₃)	C₃³	1	9	yes	—
25	#11758608	—	3	48	no	—
26	#11758608	C₃³	1	18	yes	—
27	C₃×#66524	—	3	24	no	—
28	#10624931	—	37	1	no	maximal, Sylow
29	C₃²×C₃₇	—	9	4	no	—
30	C₃²×C₃₇	C₃×H(3,𝔽₃)	1	12	yes	—
31	C₃²×C₃₇	C₃⁴	1	1	yes	characteristic
32	C₃×C₃₇⋊C₃	—	3	9	no	—
33	C₃×C₃₇⋊C₃	—	27	12	no	—
34	C₉×C₃₇	—	3	1	no	—
35	C₉×C₃₇	—	3	8	no	—
36	#66524	—	3	18	no	—
37	3₋¹⁺⁴	—	37	2	no	—
38	H(5,𝔽₃)	—	37	1	no	—
39	C₃×#737	C₃₇⋊C₃	1	1	yes	characteristic, complemented
40	#6910	—	37	24	no	—
41	C₃×#731	—	37	12	no	—
42	C₃×C₃₇	—	3	12	no	—
43	C₃×C₃₇	—	3	1	no	—
44	C₃×C₃₇	C₃²×H(3,𝔽₃)	1	1	yes	characteristic
45	C₃₇⋊C₃	—	9	9	no	—
46	#737	—	3	9	no	—
47	#737	—	111	18	no	—
48	C₃×3₋¹⁺²	—	37	24	no	—
49	C₃×3₋¹⁺²	C₃×C₃₇⋊C₃	1	8	yes	complemented
50	C₃×H(3,𝔽₃)	—	37	12	no	—
51	C₃×H(3,𝔽₃)	—	111	9	no	—
52	C₃×H(3,𝔽₃)	C₃×C₃₇⋊C₃	1	1	yes	characteristic
53	C₃²×C₉	C₃×C₃₇⋊C₃	1	4	yes	complemented
54	#731	—	37	9	no	—
55	#731	—	37	72	no	—
56	C₃₇	#10624931	1	1	yes	characteristic, complemented, Sylow
57	C₃³	—	111	12	no	—
58	C₃³	C₃²×C₃₇⋊C₃	1	4	yes	—
59	3₋¹⁺²	—	3	24	no	—
60	3₋¹⁺²	—	37	18	no	—
61	3₋¹⁺²	—	111	48	no	—
62	H(3,𝔽₃)	—	37	9	no	—
63	H(3,𝔽₃)	—	111	24	no	—
64	H(3,𝔽₃)	C₃²⋊(C₃₇⋊C₃)	1	9	yes	—
65	C₃×C₉	—	3	12	no	—
66	C₃×C₉	—	111	24	no	—
67	C₃×C₉	C₃²×C₃₇⋊C₃	1	1	yes	characteristic
68	C₃×C₉	C₃²×C₃₇⋊C₃	1	8	yes	—
69	C₃²	—	9	4	no	—
70	C₃²	—	111	9	no	—
71	C₃²	—	999	12	no	—
72	C₃²	C₃×C₃²⋊(C₃₇⋊C₃)	1	12	yes	—
73	C₃²	C₃³×C₃₇⋊C₃	1	1	yes	characteristic
74	C₉	—	3	1	no	—
75	C₉	—	3	8	no	—
76	C₉	—	111	18	no	—
77	C₃	—	3	1	no	—
78	C₃	—	3	12	no	—
79	C₃	—	333	9	no	—
80	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	C₃×C₃₇×#737	C₃
socle	C₃×C₃₇	C₃²×H(3,𝔽₃)
prime radical	C₃₇⋊H(5,𝔽₃)	C₃
hypercenter	C₃×#737	C₃₇⋊C₃
hypocenter	C₃₇	#10624931
LCS	G ▷ C₃²×C₃₇ ▷ C₃×C₃₇ ▷ C₃₇
UCS	C₁ ◁ C₃ ◁ C₃×H(3,𝔽₃) ◁ C₃×#737

Complements

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

N	H ≅ G/N	type
1. #22582680	C₃	semidirect
3. #22582643	C₃	semidirect
4. C₃×#15688516	C₃	semidirect
5. C₃×C₃₇×#737	C₃	semidirect
7. C₃×#11758608	C₃²	semidirect
10. #15688516	C₃²	semidirect
11. #15688516	C₃²	semidirect
13. C₃×3₋¹⁺²×C₃₇	C₃²	semidirect
15. C₃²×C₉×C₃₇	C₃²	semidirect
39. C₃×#737	C₃₇⋊C₃	semidirect
49. C₃×3₋¹⁺²	C₃×C₃₇⋊C₃	semidirect
53. C₃²×C₉	C₃×C₃₇⋊C₃	semidirect
56. C₃₇	#10624931	semidirect

Maximal subgroups

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

subgroup	index
#22582680	3
C₃₇⋊H(5,𝔽₃)	3
#22582643	3
C₃×#15688516	3
C₃×C₃₇×#737	3
#10624931	37