Subgroups — #41006421

Order 26973.

304 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	H(3,𝔽₃)×#66524	C₃	1	3	yes	maximal, complemented
2	C₃×#15688505	C₃	1	1	yes	maximal, characteristic, complemented
3	H(3,𝔽₃)⋊#66524	C₃	1	6	yes	maximal, complemented
4	#22582472	C₃	1	2	yes	maximal, complemented
5	C₃×C₃₇×#732	C₃	1	1	yes	maximal, characteristic
6	C₃²×#66524	—	3	3	no	—
7	C₃²×#66524	C₃²	1	2	yes	—
8	C₃²×#66524	C₃²	1	1	yes	characteristic
9	#15688505	—	3	3	no	—
10	C₃²⋊#66524	—	3	6	no	—
11	C₃²⋊#66524	C₃²	1	6	yes	—
12	C₃×H(3,𝔽₃)×C₃₇	C₃²	1	3	yes	—
13	C₃²×C₉×C₃₇	C₃²	1	1	yes	characteristic
14	C₃₇×#732	C₉	1	9	yes	complemented
15	C₃³×C₃₇	—	3	3	no	—
16	C₃³×C₃₇	C₃³	1	1	yes	characteristic
17	H(3,𝔽₃)×C₃₇	—	3	6	no	—
18	H(3,𝔽₃)×C₃₇	C₃×C₉	1	9	yes	complemented
19	C₃×C₉×C₃₇	—	3	1	no	—
20	C₃×C₉×C₃₇	—	3	2	no	—
21	C₃×C₉×C₃₇	C₃×C₉	1	3	yes	complemented
22	C₃×#66524	—	3	6	no	—
23	C₃×#66524	—	3	2	no	—
24	C₃×#66524	—	9	1	no	—
25	C₃×#66524	—	9	9	no	—
26	C₃×#66524	—	9	2	no	—
27	C₃×#66524	H(3,𝔽₃)	1	3	yes	—
28	#10624722	—	37	1	no	maximal, Sylow
29	C₃²×C₃₇	—	3	2	no	—
30	C₃²×C₃₇	—	3	1	no	—
31	C₃²×C₃₇	—	3	9	no	—
32	C₃²×C₃₇	—	9	6	no	—
33	C₃²×C₃₇	—	9	3	no	—
34	C₃²×C₃₇	C₃²×C₉	1	1	yes	characteristic
35	C₃²×C₃₇	C₃×H(3,𝔽₃)	1	1	yes	characteristic
36	C₃²×C₃₇	#737	1	2	yes	—
37	C₉×C₃₇	—	3	3	no	—
38	#66524	—	3	3	no	—
39	#66524	—	9	2	no	—
40	#66524	—	9	6	no	—
41	C₃×#732	C₃₇⋊C₃	1	1	yes	characteristic
42	C₉×H(3,𝔽₃)	—	37	3	no	—
43	C₃×C₉⋊C₉	—	37	1	no	—
44	H(3,𝔽₃)⋊C₉	—	37	6	no	—
45	#6868	—	37	2	no	—
46	C₃×C₃₇	—	3	1	no	—
47	C₃×C₃₇	—	3	2	no	—
48	C₃×C₃₇	—	9	9	no	—
49	C₃×C₃₇	C₉×H(3,𝔽₃)	1	1	yes	characteristic
50	C₃×C₃₇	#6911	1	1	yes	characteristic
51	C₃×C₃₇	#6913	1	2	yes	—
52	C₃×H(3,𝔽₃)	C₃×C₃₇⋊C₃	1	3	yes	—
53	C₃²×C₉	—	37	1	no	—
54	C₃²×C₉	—	37	2	no	—
55	C₃²×C₉	—	111	3	no	—
56	C₃²×C₉	C₃×C₃₇⋊C₃	1	1	yes	characteristic
57	#732	#66524	1	9	yes	complemented
58	C₉⋊C₉	—	111	3	no	—
59	C₃²⋊C₉	—	37	6	no	—
60	C₃²⋊C₉	—	111	6	no	—
61	C₃₇	#10624722	1	1	yes	characteristic, complemented, Sylow
62	C₃³	—	3	3	no	—
63	C₃³	C₃²×C₃₇⋊C₃	1	1	yes	characteristic
64	H(3,𝔽₃)	—	3	6	no	—
65	H(3,𝔽₃)	C₃×#66524	1	9	yes	complemented
66	C₃×C₉	—	3	1	no	—
67	C₃×C₉	—	3	2	no	—
68	C₃×C₉	—	37	3	no	—
69	C₃×C₉	—	333	9	no	—
70	C₃×C₉	—	333	2	no	—
71	C₃×C₉	—	333	1	no	—
72	C₃×C₉	—	111	6	no	—
73	C₃×C₉	—	111	2	no	—
74	C₃×C₉	C₃×#66524	1	3	yes	complemented
75	C₃²	—	3	1	no	—
76	C₃²	—	3	2	no	—
77	C₃²	—	3	9	no	—
78	C₃²	—	9	3	no	—
79	C₃²	—	9	6	no	—
80	C₃²	C₃²×#66524	1	1	yes	characteristic
81	C₃²	H(3,𝔽₃)×C₃₇⋊C₃	1	1	yes	characteristic
82	C₃²	#15688530	1	2	yes	—
83	C₉	—	3	3	no	—
84	C₉	—	333	6	no	—
85	C₉	—	333	2	no	—
86	C₉	—	111	3	no	—
87	C₃	—	3	1	no	—
88	C₃	—	3	2	no	—
89	C₃	—	9	9	no	—
90	C₃	H(3,𝔽₃)×#66524	1	1	yes	characteristic
91	C₃	#22582645	1	1	yes	characteristic
92	C₃	#22582657	1	2	yes	—

	subgroup	G/H
center	C₃²	H(3,𝔽₃)×C₃₇⋊C₃
commutator	C₃²×C₃₇	C₃²×C₉
Frattini	C₃³	C₃²×C₃₇⋊C₃
Fitting, prime radical	C₃×C₃₇×#732	C₃
socle	C₃²×C₃₇	C₃×H(3,𝔽₃)
cosocle	C₃³×C₃₇	C₃³
hypercenter	C₃×#732	C₃₇⋊C₃
hypocenter	C₃₇	#10624722
LCS	G ▷ C₃²×C₃₇ ▷ C₃×C₃₇ ▷ C₃₇
UCS	C₁ ◁ C₃² ◁ C₃³ ◁ C₃×#732

Complements

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

N	H ≅ G/N	type
1. H(3,𝔽₃)×#66524	C₃	semidirect
2. C₃×#15688505	C₃	semidirect
3. H(3,𝔽₃)⋊#66524	C₃	semidirect
4. #22582472	C₃	semidirect
14. C₃₇×#732	C₉	semidirect
18. H(3,𝔽₃)×C₃₇	C₃×C₉	semidirect
21. C₃×C₉×C₃₇	C₃×C₉	semidirect
57. #732	#66524	semidirect
61. C₃₇	#10624722	semidirect
65. H(3,𝔽₃)	C₃×#66524	semidirect
74. 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
H(3,𝔽₃)×#66524	3
C₃×#15688505	3
H(3,𝔽₃)⋊#66524	3
#22582472	3
C₃×C₃₇×#732	3
#10624722	37