Subgroups — #41008283

Order 26973.

132 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×#15688541	C₃	1	1	yes	maximal, characteristic, complemented
2	C₃²⋊#11758603	C₃	1	2	yes	maximal, complemented
3	H(3,𝔽₃)×#66524	C₃	1	1	yes	maximal, characteristic
4	#15688541	—	3	3	no	—
5	C₃×#11758603	—	3	1	no	—
6	C₃×#11758603	—	3	2	no	—
7	C₃²×#66524	—	3	1	no	—
8	C₃²×#66524	C₃²	1	1	yes	characteristic
9	C₃²⋊#66524	—	3	2	no	—
10	C₃²⋊#66524	C₉	1	2	yes	—
11	C₃×H(3,𝔽₃)×C₃₇	C₉	1	1	yes	characteristic
12	C₃³×C₃₇	—	3	1	no	—
13	C₃³×C₃₇	C₃×C₉	1	1	yes	characteristic
14	H(3,𝔽₃)×C₃₇	—	3	3	no	—
15	C₃×#66524	—	3	1	no	—
16	C₃×#66524	—	3	1	no	—
17	C₃×#66524	—	3	1	no	—
18	C₃×#66524	—	3	2	no	—
19	C₃×#66524	—	9	1	no	—
20	C₃×#66524	—	9	1	no	—
21	C₃×#66524	—	9	1	no	—
22	C₃×#66524	H(3,𝔽₃)	1	1	yes	characteristic
23	C₃×#66524	3₋¹⁺²	1	2	yes	—
24	#11758603	—	3	3	no	—
25	#11758603	—	9	2	no	—
26	#10624453	—	37	1	no	maximal, Sylow
27	C₃²×C₃₇	—	3	1	no	—
28	C₃²×C₃₇	—	3	3	no	—
29	C₃²×C₃₇	—	3	1	no	—
30	C₃²×C₃₇	—	3	1	no	—
31	C₃²×C₃₇	—	3	1	no	—
32	C₃²×C₃₇	—	9	1	no	—
33	C₃²×C₃₇	—	9	1	no	—
34	C₃²×C₃₇	—	9	1	no	—
35	C₃²×C₃₇	C₃²⋊C₉	1	1	yes	characteristic
36	#66524	—	3	2	no	—
37	#66524	—	9	2	no	—
38	#66524	AΣL(1,𝔽₂₇)	1	1	yes	characteristic
39	#66524	#731	1	1	yes	characteristic
40	#66524	#732	1	1	yes	characteristic
41	C₃×C₂₇⋊C₃	—	37	1	no	—
42	C₉×H(3,𝔽₃)	—	37	1	no	—
43	C₃²⋊C₂₇	—	37	2	no	—
44	C₃×C₃₇	—	9	1	no	—
45	C₃×C₃₇	—	9	3	no	—
46	C₃×C₃₇	#6869	1	1	yes	characteristic
47	C₃×C₃₇	H(3,𝔽₃)⋊C₉	1	1	yes	characteristic
48	C₃×C₃₇	#6872	1	1	yes	characteristic
49	C₃×C₃₇	#6873	1	1	yes	characteristic
50	C₃×H(3,𝔽₃)	#66526	1	1	yes	characteristic
51	C₃²×C₉	—	37	1	no	—
52	C₃²×C₉	—	111	1	no	—
53	C₂₇⋊C₃	—	111	3	no	—
54	C₃×C₂₇	—	111	2	no	—
55	C₃×C₂₇	—	111	1	no	—
56	C₃²⋊C₉	—	37	2	no	—
57	C₃²⋊C₉	—	111	2	no	—
58	C₃₇	#10624453	1	1	yes	characteristic, complemented, Sylow
59	C₃³	—	3	1	no	—
60	C₃³	C₃×#66526	1	1	yes	characteristic
61	H(3,𝔽₃)	—	3	3	no	—
62	C₃×C₉	—	37	1	no	—
63	C₃×C₉	—	37	2	no	—
64	C₃×C₉	—	333	1	no	—
65	C₃×C₉	—	333	1	no	—
66	C₃×C₉	—	333	1	no	—
67	C₃×C₉	—	111	1	no	—
68	C₃×C₉	—	111	1	no	—
69	C₃×C₉	—	111	1	no	—
70	C₃×C₉	—	111	2	no	—
71	C₂₇	—	333	2	no	—
72	C₂₇	—	111	3	no	—
73	C₃²	—	3	1	no	—
74	C₃²	—	3	3	no	—
75	C₃²	—	3	1	no	—
76	C₃²	—	3	1	no	—
77	C₃²	—	3	1	no	—
78	C₃²	—	9	1	no	—
79	C₃²	—	9	1	no	—
80	C₃²	—	9	1	no	—
81	C₃²	C₃²⋊#66526	1	1	yes	characteristic
82	C₉	—	37	1	no	—
83	C₉	—	37	1	no	—
84	C₉	—	37	1	no	—
85	C₉	—	333	2	no	—
86	C₉	—	111	2	no	—
87	C₃	—	9	1	no	—
88	C₃	—	9	3	no	—
89	C₃	#22582694	1	1	yes	characteristic
90	C₃	H(3,𝔽₃)⋊#66526	1	1	yes	characteristic
91	C₃	#22582699	1	1	yes	characteristic
92	C₃	#22582701	1	1	yes	characteristic

	subgroup	G/H
center	C₃²	C₃²⋊#66526
commutator	C₃³×C₃₇	C₃×C₉
Frattini	C₃³	C₃×#66526
Fitting, prime radical	C₃×H(3,𝔽₃)×C₃₇	C₉
socle	C₃²×C₃₇	C₃²⋊C₉
cosocle	C₃²×#66524	C₃²
hypercenter	C₃×H(3,𝔽₃)	#66526
hypocenter	C₃₇	#10624453
LCS	G ▷ C₃³×C₃₇ ▷ C₃²×C₃₇ ▷ C₃₇
UCS	C₁ ◁ C₃² ◁ C₃³ ◁ C₃×H(3,𝔽₃)

Complements

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

N	H ≅ G/N	type
1. C₃×#15688541	C₃	semidirect
2. C₃²⋊#11758603	C₃	semidirect
58. C₃₇	#10624453	semidirect

Maximal subgroups

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

subgroup	index
C₃×#15688541	3
C₃²⋊#11758603	3
H(3,𝔽₃)×#66524	3
#10624453	37