Subgroups — #41008640

Order 26973.

1178 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	H(3,𝔽₃)×#66526	C₃	1	27	yes	maximal, complemented
2	C₃×C₃²⋊#66526	C₃	1	12	yes	maximal, complemented
3	C₃×H(3,𝔽₃)×C₃₇⋊C₃	C₃	1	1	yes	maximal, characteristic
4	C₃²×#66526	C₃²	1	36	yes	complemented
5	C₃²⋊#66526	C₃²	1	9	yes	—
6	C₃²⋊#66526	C₃²	1	72	yes	complemented
7	C₃³×C₃₇⋊C₃	C₃²	1	4	yes	—
8	C₃×C₃²⋊(C₃₇⋊C₃)	C₉	1	8	yes	complemented
9	H(3,𝔽₃)×C₃₇⋊C₃	C₉	1	18	yes	complemented
10	H(3,𝔽₃)×C₃₇⋊C₃	C₃²	1	9	yes	—
11	C₃×H(3,𝔽₃)×C₃₇	C₉	1	1	yes	characteristic, complemented
12	C₃³×C₃₇	C₃×C₉	1	4	yes	complemented
13	C₃²×C₃₇⋊C₃	—	9	4	no	—
14	C₃²×C₃₇⋊C₃	—	9	8	no	—
15	C₃²×C₃₇⋊C₃	C₃×C₉	1	24	yes	complemented
16	C₃²×C₃₇⋊C₃	C₃×C₉	1	8	yes	complemented
17	C₃²×C₃₇⋊C₃	C₃³	1	12	yes	—
18	C₃²×C₃₇⋊C₃	C₃³	1	1	yes	characteristic
19	C₃×#66526	—	9	36	no	—
20	C₃×#66526	C₃³	1	27	yes	—
21	H(3,𝔽₃)×C₃₇	C₃×C₉	1	9	yes	complemented
22	C₃²⋊(C₃₇⋊C₃)	C₃×C₉	1	72	yes	complemented
23	#10624862	—	37	1	no	maximal, Sylow
24	C₃²×C₃₇	—	9	4	no	—
25	C₃²×C₃₇	C₃²×C₉	1	12	yes	—
26	C₃²×C₃₇	C₃²×C₉	1	1	yes	characteristic
27	C₃×C₃₇⋊C₃	—	3	12	no	—
28	C₃×C₃₇⋊C₃	—	3	1	no	—
29	C₃×C₃₇⋊C₃	—	3	2	no	—
30	C₃×C₃₇⋊C₃	—	3	24	no	—
31	C₃×C₃₇⋊C₃	—	9	24	no	—
32	C₃×C₃₇⋊C₃	—	9	8	no	—
33	C₃×C₃₇⋊C₃	C₃²×C₉	1	24	yes	—
34	C₃×C₃₇⋊C₃	C₃²×C₉	1	2	yes	—
35	C₃×C₃₇⋊C₃	C₃⁴	1	1	yes	characteristic
36	#66526	—	3	27	no	—
37	C₃²×H(3,𝔽₃)	—	37	1	no	—
38	C₉×H(3,𝔽₃)	—	37	27	no	—
39	C₃×C₃²⋊C₉	—	37	12	no	—
40	C₃×C₃₇	—	3	1	no	—
41	C₃×C₃₇	—	3	12	no	—
42	C₃×C₃₇	C₃³×C₉	1	1	yes	characteristic
43	C₃₇⋊C₃	—	3	2	no	—
44	C₃₇⋊C₃	—	3	24	no	—
45	C₃₇⋊C₃	H(5,𝔽₃)	1	1	yes	characteristic
46	C₃₇⋊C₃	3₋¹⁺⁴	1	2	yes	—
47	C₃⁴	—	37	4	no	—
48	C₃×H(3,𝔽₃)	—	37	8	no	—
49	C₃×H(3,𝔽₃)	—	37	18	no	—
50	C₃×H(3,𝔽₃)	—	37	9	no	—
51	C₃×H(3,𝔽₃)	#66526	1	1	yes	characteristic, complemented
52	C₃²×C₉	—	37	36	no	—
53	C₃²⋊C₉	—	37	9	no	—
54	C₃²⋊C₉	—	37	72	no	—
55	C₃₇	#10624862	1	1	yes	characteristic, complemented, Sylow
56	C₃³	—	37	8	no	—
57	C₃³	—	37	24	no	—
58	C₃³	—	37	1	no	—
59	C₃³	—	37	12	no	—
60	C₃³	—	333	4	no	—
61	C₃³	—	333	8	no	—
62	C₃³	C₃×#66526	1	4	yes	complemented
63	H(3,𝔽₃)	—	37	72	no	—
64	H(3,𝔽₃)	C₃×#66526	1	9	yes	complemented
65	C₃×C₉	—	37	27	no	—
66	C₃×C₉	—	333	36	no	—
67	C₃²	—	9	4	no	—
68	C₃²	—	37	1	no	—
69	C₃²	—	37	24	no	—
70	C₃²	—	37	2	no	—
71	C₃²	—	333	8	no	—
72	C₃²	—	333	24	no	—
73	C₃²	—	111	12	no	—
74	C₃²	—	111	1	no	—
75	C₃²	—	111	2	no	—
76	C₃²	—	111	24	no	—
77	C₃²	C₃²×#66526	1	12	yes	—
78	C₃²	C₃²×#66526	1	1	yes	characteristic
79	C₉	—	111	27	no	—
80	C₃	—	3	1	no	—
81	C₃	—	3	12	no	—
82	C₃	—	37	1	no	—
83	C₃	—	37	2	no	—
84	C₃	—	111	2	no	—
85	C₃	—	111	24	no	—
86	C₃	C₃³×#66526	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₃	C₃³×#66526
commutator, socle	C₃×C₃₇	C₃³×C₉
Fitting	C₃×H(3,𝔽₃)×C₃₇	C₉
prime radical	C₃×H(3,𝔽₃)×C₃₇⋊C₃	C₃
cosocle	C₃×C₃₇⋊C₃	C₃⁴
hypercenter	C₃×H(3,𝔽₃)	#66526
hypocenter	C₃₇	#10624862
LCS	G ▷ C₃×C₃₇ ▷ C₃₇
UCS	C₁ ◁ C₃ ◁ C₃×H(3,𝔽₃)

Complements

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

N	H ≅ G/N	type
1. H(3,𝔽₃)×#66526	C₃	semidirect
2. C₃×C₃²⋊#66526	C₃	semidirect
4. C₃²×#66526	C₃²	semidirect
6. C₃²⋊#66526	C₃²	semidirect
8. C₃×C₃²⋊(C₃₇⋊C₃)	C₉	semidirect
9. H(3,𝔽₃)×C₃₇⋊C₃	C₉	semidirect
11. C₃×H(3,𝔽₃)×C₃₇	C₉	semidirect
12. C₃³×C₃₇	C₃×C₉	semidirect
15. C₃²×C₃₇⋊C₃	C₃×C₉	semidirect
16. C₃²×C₃₇⋊C₃	C₃×C₉	semidirect
21. H(3,𝔽₃)×C₃₇	C₃×C₉	semidirect
22. C₃²⋊(C₃₇⋊C₃)	C₃×C₉	semidirect
51. C₃×H(3,𝔽₃)	#66526	semidirect
55. C₃₇	#10624862	semidirect
62. C₃³	C₃×#66526	semidirect
64. H(3,𝔽₃)	C₃×#66526	semidirect

Maximal subgroups

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

subgroup	index
H(3,𝔽₃)×#66526	3
C₃×C₃²⋊#66526	3
C₃×H(3,𝔽₃)×C₃₇⋊C₃	3
#10624862	37