Subgroups — #41004813

Order 26973.

170 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×#15688511	C₃	1	2	yes	maximal
2	C₃×#15688506	C₃	1	1	yes	maximal, characteristic
3	C₃×C₃₇×C₂₇⋊C₃	C₃	1	1	yes	maximal, characteristic
4	C₃²×#66524	—	3	1	no	—
5	#15688511	—	3	2	no	—
6	#15688511	—	3	2	no	—
7	#15688511	C₉	1	6	yes	complemented
8	C₃×#11758601	—	3	2	no	—
9	#15688506	—	3	1	no	—
10	#15688506	—	3	1	no	—
11	#15688506	C₉	1	3	yes	—
12	C₃²×C₉×C₃₇	C₃²	1	1	yes	characteristic
13	C₃₇×C₂₇⋊C₃	—	3	1	no	—
14	C₃₇×C₂₇⋊C₃	—	3	2	no	—
15	C₃×C₂₇×C₃₇	C₉	1	3	yes	complemented
16	C₃³×C₃₇	3₋¹⁺²	1	1	yes	characteristic
17	C₃×C₉×C₃₇	C₃×C₉	1	2	yes	—
18	C₃×C₉×C₃₇	C₃×C₉	1	1	yes	characteristic
19	C₃×C₉×C₃₇	C₃×C₉	1	1	yes	characteristic
20	C₃×C₉×C₃₇	H(3,𝔽₃)	1	1	yes	characteristic
21	C₃×C₉×C₃₇	3₋¹⁺²	1	2	yes	—
22	C₃×C₉×C₃₇	3₋¹⁺²	1	1	yes	characteristic
23	C₃×C₉×C₃₇	3₋¹⁺²	1	2	yes	—
24	C₃×C₉×C₃₇	3₋¹⁺²	1	2	yes	—
25	C₃×#66524	—	3	3	no	—
26	C₃×#66524	—	9	3	no	—
27	C₂₇×C₃₇	—	3	3	no	—
28	#11758601	—	3	6	no	—
29	#10624472	—	37	1	no	maximal, Sylow
30	C₃²×C₃₇	—	3	3	no	—
31	C₃²×C₃₇	C₃²⋊C₉	1	1	yes	characteristic
32	C₃²×C₃₇	C₉⋊C₉	1	2	yes	—
33	C₃²×C₃₇	C₉⋊C₉	1	1	yes	characteristic
34	C₉×C₃₇	C₉²	1	1	yes	characteristic
35	C₉×C₃₇	C₃²⋊C₉	1	2	yes	—
36	C₉×C₃₇	C₃²⋊C₉	1	1	yes	characteristic
37	C₉×C₃₇	C₉⋊C₉	1	2	yes	—
38	C₉×C₃₇	C₉⋊C₉	1	2	yes	—
39	C₉×C₃₇	C₉⋊C₉	1	1	yes	characteristic
40	#66524	—	9	3	no	—
41	C₃×C₂₇⋊C₃	—	37	2	no	—
42	C₃×C₂₇⋊C₃	C₃₇⋊C₃	1	1	yes	characteristic
43	C₃×C₉⋊C₉	—	37	1	no	—
44	C₃×C₃₇	—	3	1	no	—
45	C₃×C₃₇	—	3	2	no	—
46	C₃×C₃₇	#6855	1	1	yes	characteristic
47	C₃×C₃₇	#6875	1	3	yes	—
48	C₃²×C₉	—	111	1	no	—
49	C₃²×C₉	C₃×C₃₇⋊C₃	1	1	yes	characteristic
50	C₂₇⋊C₃	—	3	2	no	—
51	C₂₇⋊C₃	—	3	1	no	—
52	C₂₇⋊C₃	—	37	6	no	—
53	C₂₇⋊C₃	—	111	2	no	—
54	C₂₇⋊C₃	—	111	2	no	—
55	C₃×C₂₇	—	111	2	no	—
56	C₃×C₂₇	#66524	1	3	yes	complemented
57	C₉⋊C₉	—	37	3	no	—
58	C₉⋊C₉	—	111	1	no	—
59	C₉⋊C₉	—	111	1	no	—
60	C₃₇	#10624472	1	1	yes	characteristic, complemented, Sylow
61	C₃³	#11758606	1	1	yes	characteristic
62	C₃×C₉	—	333	3	no	—
63	C₃×C₉	—	111	3	no	—
64	C₃×C₉	C₉×C₃₇⋊C₃	1	2	yes	—
65	C₃×C₉	C₉×C₃₇⋊C₃	1	1	yes	characteristic
66	C₃×C₉	#11758605	1	2	yes	—
67	C₃×C₉	#11758605	1	1	yes	characteristic
68	C₃×C₉	#11758606	1	2	yes	—
69	C₃×C₉	C₃×#66524	1	1	yes	characteristic
70	C₃×C₉	#11758608	1	2	yes	—
71	C₃×C₉	C₃²⋊(C₃₇⋊C₃)	1	1	yes	characteristic
72	C₂₇	—	3	3	no	—
73	C₂₇	—	111	6	no	—
74	C₃²	—	3	3	no	—
75	C₃²	#15688502	1	1	yes	characteristic
76	C₃²	#15688504	1	2	yes	—
77	C₃²	#15688506	1	1	yes	characteristic
78	C₉	—	333	3	no	—
79	C₉	C₉×#66524	1	1	yes	characteristic
80	C₉	#15688502	1	2	yes	—
81	C₉	C₃²⋊#66524	1	1	yes	characteristic
82	C₉	#15688504	1	2	yes	—
83	C₉	#15688504	1	2	yes	—
84	C₉	#15688505	1	1	yes	characteristic
85	C₃	—	3	1	no	—
86	C₃	—	3	2	no	—
87	C₃	#22582441	1	1	yes	characteristic
88	C₃	#22582493	1	3	yes	—

	subgroup	G/H
center	C₃²	#15688506
commutator	C₉×C₃₇	C₉²
Frattini	C₃²×C₉	C₃×C₃₇⋊C₃
Fitting	C₃×C₃₇×C₂₇⋊C₃	C₃
socle	C₃²×C₃₇	C₉⋊C₉
prime radical	C₃³×C₃₇	3₋¹⁺²
cosocle	C₃²×C₉×C₃₇	C₃²
hypercenter	C₃×C₂₇⋊C₃	C₃₇⋊C₃
hypocenter	C₃₇	#10624472
LCS	G ▷ C₉×C₃₇ ▷ C₃×C₃₇ ▷ C₃₇
UCS	C₁ ◁ C₃² ◁ C₃²×C₉ ◁ C₃×C₂₇⋊C₃

Complements

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

N	H ≅ G/N	type
7. #15688511	C₉	semidirect
15. C₃×C₂₇×C₃₇	C₉	semidirect
56. C₃×C₂₇	#66524	semidirect
60. C₃₇	#10624472	semidirect

Maximal subgroups

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

subgroup	index
C₃×#15688511	3
C₃×#15688506	3
C₃×C₃₇×C₂₇⋊C₃	3
#10624472	37