Subgroups — #41005030

Order 26973.

142 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×#15688519	C₃	1	1	yes	maximal, characteristic
2	#22582472	C₃	1	2	yes	maximal, complemented
3	C₃×C₃₇×C₉⋊C₉	C₃	1	1	yes	maximal, characteristic, complemented
4	C₃×C₃²⋊(C₃₇⋊C₃)	—	3	1	no	—
5	#15688519	—	3	1	no	—
6	#15688519	—	3	1	no	—
7	#15688519	C₉	1	3	yes	complemented
8	C₃²⋊#66524	—	3	2	no	—
9	C₃²×C₉×C₃₇	—	3	1	no	—
10	C₃²×C₉×C₃₇	C₃²	1	1	yes	characteristic
11	C₃₇×C₉⋊C₉	—	3	3	no	—
12	C₃³×C₃₇	H(3,𝔽₃)	1	1	yes	characteristic
13	C₃²×C₃₇⋊C₃	—	9	1	no	—
14	C₃×C₉×C₃₇	—	3	1	no	—
15	C₃×C₉×C₃₇	—	3	1	no	—
16	C₃×C₉×C₃₇	—	3	1	no	—
17	C₃×C₉×C₃₇	—	3	3	no	—
18	C₃×C₉×C₃₇	—	9	2	no	—
19	C₃×C₉×C₃₇	—	9	1	no	—
20	C₃×C₉×C₃₇	C₃×C₉	1	1	yes	characteristic
21	C₃×C₉×C₃₇	3₋¹⁺²	1	1	yes	characteristic
22	C₃×C₉×C₃₇	3₋¹⁺²	1	1	yes	characteristic
23	C₃²⋊(C₃₇⋊C₃)	—	3	3	no	—
24	C₃²⋊(C₃₇⋊C₃)	—	9	1	no	—
25	C₃²⋊(C₃₇⋊C₃)	—	9	1	no	—
26	C₃×#66524	—	9	2	no	—
27	#10624527	—	37	1	no	maximal, Sylow
28	C₃²×C₃₇	—	3	2	no	—
29	C₃²×C₃₇	—	3	1	no	—
30	C₃²×C₃₇	C₃²⋊C₉	1	1	yes	characteristic
31	C₃²×C₃₇	AΣL(1,𝔽₂₇)	1	1	yes	characteristic
32	C₃²×C₃₇	#731	1	1	yes	characteristic
33	C₃²×C₃₇	#732	1	1	yes	characteristic
34	C₃×C₃₇⋊C₃	—	9	3	no	—
35	C₃×C₃₇⋊C₃	—	27	2	no	—
36	C₃×C₃₇⋊C₃	—	27	1	no	—
37	C₉×C₃₇	—	3	1	no	—
38	C₉×C₃₇	—	3	1	no	—
39	C₉×C₃₇	—	3	1	no	—
40	C₉×C₃₇	—	9	3	no	—
41	#66524	—	27	2	no	—
42	C₃×#732	—	37	1	no	—
43	C₃×C₉⋊C₉	C₃₇⋊C₃	1	1	yes	characteristic, complemented
44	#6868	—	37	2	no	—
45	C₃×C₃₇	—	3	1	no	—
46	C₃×C₃₇	—	3	1	no	—
47	C₃×C₃₇	—	3	1	no	—
48	C₃×C₃₇	H(3,𝔽₃)⋊C₉	1	1	yes	characteristic
49	C₃×C₃₇	#6881	1	2	yes	—
50	C₃×C₃₇	#6883	1	1	yes	characteristic
51	C₃₇⋊C₃	—	27	3	no	—
52	C₃×H(3,𝔽₃)	—	111	1	no	—
53	C₃²×C₉	—	3	1	no	—
54	C₃²×C₉	C₃×C₃₇⋊C₃	1	1	yes	characteristic
55	#732	—	37	3	no	—
56	#732	—	111	1	no	—
57	#732	—	111	1	no	—
58	C₉⋊C₉	—	3	3	no	—
59	C₃²⋊C₉	—	111	2	no	—
60	C₃₇	#10624527	1	1	yes	characteristic, complemented, Sylow
61	C₃³	—	333	1	no	—
62	C₃³	C₃²⋊(C₃₇⋊C₃)	1	1	yes	characteristic
63	H(3,𝔽₃)	—	333	1	no	—
64	H(3,𝔽₃)	—	333	1	no	—
65	H(3,𝔽₃)	—	111	3	no	—
66	C₃×C₉	—	3	1	no	—
67	C₃×C₉	—	3	1	no	—
68	C₃×C₉	—	3	1	no	—
69	C₃×C₉	—	3	3	no	—
70	C₃×C₉	—	9	2	no	—
71	C₃×C₉	—	9	1	no	—
72	C₃×C₉	—	333	2	no	—
73	C₃×C₉	C₉×C₃₇⋊C₃	1	1	yes	characteristic
74	C₃×C₉	#11758605	1	1	yes	characteristic
75	C₃×C₉	#11758606	1	1	yes	characteristic
76	C₃²	—	3	2	no	—
77	C₃²	—	3	1	no	—
78	C₃²	—	333	3	no	—
79	C₃²	—	999	2	no	—
80	C₃²	—	999	1	no	—
81	C₃²	#15688502	1	1	yes	characteristic
82	C₃²	#15688513	1	1	yes	characteristic
83	C₃²	#15688516	1	1	yes	characteristic
84	C₃²	#15688519	1	1	yes	characteristic
85	C₉	—	3	1	no	—
86	C₉	—	3	1	no	—
87	C₉	—	3	1	no	—
88	C₉	—	9	3	no	—
89	C₉	—	999	2	no	—
90	C₃	—	3	1	no	—
91	C₃	—	3	1	no	—
92	C₃	—	3	1	no	—
93	C₃	—	999	3	no	—
94	C₃	#22582477	1	1	yes	characteristic
95	C₃	#22582513	1	2	yes	—
96	C₃	#22582523	1	1	yes	characteristic

	subgroup	G/H
center	C₃²	#15688519
commutator	C₃×C₉×C₃₇	C₃×C₉
Frattini	C₃²×C₉	C₃×C₃₇⋊C₃
Fitting	C₃×C₃₇×C₉⋊C₉	C₃
socle	C₃²×C₃₇	#732
prime radical	C₃×#15688519	C₃
cosocle	C₃²×C₉×C₃₇	C₃²
hypercenter	C₃×C₉⋊C₉	C₃₇⋊C₃
hypocenter	C₃₇	#10624527
LCS	G ▷ C₃×C₉×C₃₇ ▷ C₃²×C₃₇ ▷ C₃×C₃₇ ▷ C₃₇
UCS	C₁ ◁ C₃² ◁ C₃³ ◁ C₃²×C₉ ◁ C₃×C₉⋊C₉

Complements

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

N	H ≅ G/N	type
2. #22582472	C₃	semidirect
3. C₃×C₃₇×C₉⋊C₉	C₃	semidirect
7. #15688519	C₉	semidirect
43. C₃×C₉⋊C₉	C₃₇⋊C₃	semidirect
60. C₃₇	#10624527	semidirect

Maximal subgroups

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

subgroup	index
C₃×#15688519	3
#22582472	3
C₃×C₃₇×C₉⋊C₉	3
#10624527	37