Subgroups — #41005098

Order 26973.

2150 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×C₃²⋊#66524	C₃	1	12	yes	maximal, complemented
2	C₃⁵×C₃₇	C₃	1	1	yes	maximal, characteristic
3	C₃²×#66524	C₃²	1	9	yes	complemented
4	C₃²⋊#66524	—	3	36	no	—
5	C₃⁴×C₃₇	—	3	12	no	—
6	C₃⁴×C₃₇	—	3	24	no	—
7	C₃⁴×C₃₇	C₉	1	9	yes	complemented
8	C₃⁴×C₃₇	C₃²	1	4	yes	—
9	C₃³×C₃₇	—	3	27	no	—
10	C₃³×C₃₇	—	3	216	no	—
11	C₃³×C₃₇	—	3	108	no	—
12	C₃³×C₃₇	—	3	12	no	—
13	C₃³×C₃₇	—	3	24	no	—
14	C₃³×C₃₇	C₃×C₉	1	12	yes	—
15	C₃³×C₃₇	H(3,𝔽₃)	1	12	yes	—
16	C₃³×C₃₇	3₋¹⁺²	1	24	yes	—
17	C₃³×C₃₇	C₃³	1	1	yes	characteristic
18	C₃×#66524	—	3	36	no	—
19	#10624547	—	37	1	no	maximal, Sylow
20	C₃²×C₃₇	—	3	243	no	—
21	C₃²×C₃₇	—	3	36	no	—
22	C₃²×C₃₇	—	3	12	no	—
23	C₃²×C₃₇	—	3	24	no	—
24	C₃²×C₃₇	—	3	72	no	—
25	C₃²×C₃₇	C₃²⋊C₉	1	36	yes	complemented
26	C₃²×C₃₇	C₃²×C₉	1	1	yes	characteristic
27	C₃²×C₃₇	C₃×H(3,𝔽₃)	1	4	yes	—
28	C₃²×C₃₇	C₃×3₋¹⁺²	1	8	yes	—
29	#66524	—	9	9	no	—
30	C₃⁵	C₃₇⋊C₃	1	1	yes	characteristic
31	C₃×C₃²⋊C₉	—	37	12	no	—
32	C₃×C₃₇	—	3	36	no	—
33	C₃×C₃₇	C₃×C₃²⋊C₉	1	4	yes	—
34	C₃×C₃₇	C₃²⋊H(3,𝔽₃)	1	1	yes	characteristic
35	C₃×C₃₇	#6891	1	8	yes	—
36	C₃⁴	—	3	24	no	—
37	C₃⁴	—	3	12	no	—
38	C₃⁴	#66524	1	9	yes	complemented
39	C₃⁴	C₃×C₃₇⋊C₃	1	4	yes	—
40	C₃²×C₉	—	37	9	no	—
41	C₃²⋊C₉	—	111	36	no	—
42	C₃₇	#10624547	1	1	yes	characteristic, complemented, Sylow
43	C₃³	—	3	27	no	—
44	C₃³	—	3	216	no	—
45	C₃³	—	3	108	no	—
46	C₃³	—	3	12	no	—
47	C₃³	—	3	24	no	—
48	C₃³	C₃×#66524	1	12	yes	—
49	C₃³	#11758608	1	24	yes	—
50	C₃³	C₃²⋊(C₃₇⋊C₃)	1	12	yes	—
51	C₃³	C₃²×C₃₇⋊C₃	1	1	yes	characteristic
52	C₃×C₉	—	111	36	no	—
53	C₃²	—	3	243	no	—
54	C₃²	—	3	36	no	—
55	C₃²	—	3	12	no	—
56	C₃²	—	3	72	no	—
57	C₃²	—	3	24	no	—
58	C₃²	C₃²⋊#66524	1	36	yes	complemented
59	C₃²	C₃²×#66524	1	1	yes	characteristic
60	C₃²	C₃×C₃²⋊(C₃₇⋊C₃)	1	4	yes	—
61	C₃²	C₃×#11758608	1	8	yes	—
62	C₉	—	333	9	no	—
63	C₃	—	3	36	no	—
64	C₃	C₃×C₃²⋊#66524	1	4	yes	—
65	C₃	#22582550	1	1	yes	characteristic
66	C₃	#22582554	1	8	yes	—

	subgroup	G/H
center, Frattini	C₃³	C₃²×C₃₇⋊C₃
commutator	C₃²×C₃₇	C₃²×C₉
Fitting, prime radical	C₃⁵×C₃₇	C₃
socle, cosocle	C₃³×C₃₇	C₃³
hypercenter	C₃⁵	C₃₇⋊C₃
hypocenter	C₃₇	#10624547
LCS	G ▷ C₃²×C₃₇ ▷ C₃₇
UCS	C₁ ◁ C₃³ ◁ C₃⁵

Complements

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

N	H ≅ G/N	type
1. C₃×C₃²⋊#66524	C₃	semidirect
3. C₃²×#66524	C₃²	semidirect
7. C₃⁴×C₃₇	C₉	semidirect
25. C₃²×C₃₇	C₃²⋊C₉	semidirect
38. C₃⁴	#66524	semidirect
42. C₃₇	#10624547	semidirect
58. C₃²	C₃²⋊#66524	semidirect

Maximal subgroups

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

subgroup	index
C₃×C₃²⋊#66524	3
C₃⁵×C₃₇	3
#10624547	37