Subgroups — #41007528

Order 26973.

926 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	3₋¹⁺²×#66524	C₃	1	24	yes	maximal, complemented
2	H(3,𝔽₃)×#66524	C₃	1	3	yes	maximal
3	#22582538	C₃	1	8	yes	maximal, complemented
4	C₃×#15688506	C₃	1	4	yes	maximal, complemented
5	C₃×C₃₇×#737	C₃	1	1	yes	maximal, characteristic
6	C₃²×#66524	C₃²	1	12	yes	—
7	#15688506	C₃²	1	1	yes	characteristic
8	#15688506	C₃²	1	8	yes	—
9	#15688506	C₃²	1	24	yes	—
10	#15688505	C₃²	1	24	yes	—
11	C₃²⋊#66524	C₃²	1	24	yes	—
12	C₉×#66524	C₃²	1	24	yes	—
13	C₃₇×#737	C₉	1	27	yes	complemented
14	C₃×3₋¹⁺²×C₃₇	C₃²	1	8	yes	—
15	C₃×H(3,𝔽₃)×C₃₇	C₃²	1	1	yes	characteristic
16	C₃²×C₉×C₃₇	C₃²	1	4	yes	—
17	C₃³×C₃₇	C₃³	1	4	yes	—
18	3₋¹⁺²×C₃₇	C₃×C₉	1	72	yes	complemented
19	H(3,𝔽₃)×C₃₇	C₃×C₉	1	9	yes	—
20	C₃×C₉×C₃₇	C₃×C₉	1	36	yes	complemented
21	C₃×C₉×C₃₇	C₃³	1	1	yes	characteristic
22	C₃×C₉×C₃₇	C₃³	1	8	yes	—
23	C₃×#66524	—	9	12	no	—
24	C₃×#66524	C₃³	1	24	yes	—
25	C₃×#66524	C₃³	1	3	yes	—
26	#10624864	—	37	1	no	maximal, Sylow
27	C₃²×C₃₇	—	3	12	no	—
28	C₃²×C₃₇	C₃²×C₉	1	12	yes	—
29	C₃²×C₃₇	C₃⁴	1	1	yes	characteristic
30	C₉×C₃₇	C₃²×C₉	1	3	yes	—
31	C₉×C₃₇	C₃²×C₉	1	24	yes	—
32	#66524	—	3	3	no	—
33	#66524	—	3	24	no	—
34	C₃×#737	C₃₇⋊C₃	1	1	yes	characteristic
35	C₉×3₋¹⁺²	—	37	24	no	—
36	C₉×H(3,𝔽₃)	—	37	3	no	—
37	#6887	—	37	8	no	—
38	C₃×C₉⋊C₉	—	37	4	no	—
39	C₃×C₃₇	—	3	12	no	—
40	C₃×C₃₇	C₃³×C₉	1	1	yes	characteristic
41	C₃×C₃₇	3₋¹⁺⁴	1	3	yes	—
42	#737	#66524	1	27	yes	complemented
43	C₃×3₋¹⁺²	C₃×C₃₇⋊C₃	1	8	yes	—
44	C₃×H(3,𝔽₃)	C₃×C₃₇⋊C₃	1	1	yes	characteristic
45	C₃²×C₉	—	37	12	no	—
46	C₃²×C₉	C₃×C₃₇⋊C₃	1	4	yes	—
47	C₉⋊C₉	—	37	24	no	—
48	C₉⋊C₉	—	37	1	no	—
49	C₉⋊C₉	—	37	8	no	—
50	C₉⋊C₉	—	37	24	no	—
51	C₃²⋊C₉	—	37	24	no	—
52	C₉²	—	37	24	no	—
53	C₃₇	#10624864	1	1	yes	characteristic, complemented, Sylow
54	C₃³	C₃²×C₃₇⋊C₃	1	4	yes	—
55	3₋¹⁺²	C₃×#66524	1	72	yes	complemented
56	H(3,𝔽₃)	C₃×#66524	1	9	yes	—
57	C₃×C₉	—	37	24	no	—
58	C₃×C₉	—	37	3	no	—
59	C₃×C₉	—	333	12	no	—
60	C₃×C₉	C₃×#66524	1	36	yes	complemented
61	C₃×C₉	C₃²×C₃₇⋊C₃	1	8	yes	—
62	C₃×C₉	C₃²×C₃₇⋊C₃	1	1	yes	characteristic
63	C₃²	—	3	12	no	—
64	C₃²	C₃²×#66524	1	12	yes	—
65	C₃²	C₃³×C₃₇⋊C₃	1	1	yes	characteristic
66	C₉	—	111	3	no	—
67	C₉	—	111	24	no	—
68	C₉	C₃²×#66524	1	3	yes	—
69	C₉	C₃²×#66524	1	24	yes	—
70	C₃	—	3	12	no	—
71	C₃	C₃³×#66524	1	1	yes	characteristic
72	C₃	#22582677	1	3	yes	—

	subgroup	G/H
center, Frattini	C₃²	C₃³×C₃₇⋊C₃
commutator	C₃×C₃₇	C₃³×C₉
Fitting	C₃×C₃₇×#737	C₃
socle, cosocle	C₃²×C₃₇	C₃⁴
prime radical	C₃×H(3,𝔽₃)×C₃₇	C₃²
hypercenter	C₃×#737	C₃₇⋊C₃
hypocenter	C₃₇	#10624864
LCS	G ▷ C₃×C₃₇ ▷ C₃₇
UCS	C₁ ◁ C₃² ◁ C₃×#737

Complements

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

N	H ≅ G/N	type
1. 3₋¹⁺²×#66524	C₃	semidirect
3. #22582538	C₃	semidirect
4. C₃×#15688506	C₃	semidirect
13. C₃₇×#737	C₉	semidirect
18. 3₋¹⁺²×C₃₇	C₃×C₉	semidirect
20. C₃×C₉×C₃₇	C₃×C₉	semidirect
42. #737	#66524	semidirect
53. C₃₇	#10624864	semidirect
55. 3₋¹⁺²	C₃×#66524	semidirect
60. C₃×C₉	C₃×#66524	semidirect

Maximal subgroups

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

subgroup	index
3₋¹⁺²×#66524	3
H(3,𝔽₃)×#66524	3
#22582538	3
C₃×#15688506	3
C₃×C₃₇×#737	3
#10624864	37