Subgroups — #41008480

Order 26973.

428 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×#15688541	C₃	1	3	yes	maximal, complemented
2	C₃²⋊#11758603	C₃	1	9	yes	maximal, complemented
3	C₃³×#66524	C₃	1	1	yes	maximal, characteristic
4	#15688541	—	3	9	no	—
5	C₃×#11758603	C₃²	1	9	yes	complemented
6	C₃²×#66524	—	3	3	no	—
7	C₃²×#66524	—	3	6	no	—
8	C₃²×#66524	C₉	1	2	yes	—
9	C₃²×#66524	C₉	1	6	yes	—
10	C₃²×#66524	C₃²	1	3	yes	—
11	C₃²×#66524	C₃²	1	1	yes	characteristic
12	C₃⁴×C₃₇	C₉	1	1	yes	characteristic
13	C₃³×C₃₇	—	3	3	no	—
14	C₃³×C₃₇	—	3	9	no	—
15	C₃³×C₃₇	C₃×C₉	1	1	yes	characteristic
16	C₃³×C₃₇	C₃×C₉	1	3	yes	—
17	C₃×#66524	—	3	6	no	—
18	C₃×#66524	—	3	18	no	—
19	C₃×#66524	—	3	3	no	—
20	C₃×#66524	—	3	6	no	—
21	C₃×#66524	C₃×C₉	1	6	yes	—
22	C₃×#66524	C₃×C₉	1	2	yes	—
23	C₃×#66524	H(3,𝔽₃)	1	3	yes	—
24	C₃×#66524	3₋¹⁺²	1	6	yes	—
25	C₃×#66524	C₃³	1	1	yes	characteristic
26	#11758603	—	3	9	no	—
27	#10624683	—	37	1	no	maximal, Sylow
28	C₃²×C₃₇	—	3	27	no	—
29	C₃²×C₃₇	—	3	3	no	—
30	C₃²×C₃₇	—	3	3	no	—
31	C₃²×C₃₇	—	3	6	no	—
32	C₃²×C₃₇	C₃²⋊C₉	1	3	yes	—
33	C₃²×C₃₇	C₂₇⋊C₃	1	9	yes	complemented
34	C₃²×C₃₇	C₃²×C₉	1	1	yes	characteristic
35	#66524	—	3	6	no	—
36	#66524	C₃²⋊C₉	1	6	yes	—
37	#66524	C₃×H(3,𝔽₃)	1	1	yes	characteristic
38	#66524	C₃×3₋¹⁺²	1	2	yes	—
39	C₃³×C₉	—	37	1	no	—
40	C₃×C₂₇⋊C₃	—	37	3	no	—
41	C₃²⋊C₂₇	—	37	9	no	—
42	C₃×C₃₇	—	3	3	no	—
43	C₃×C₃₇	—	3	9	no	—
44	C₃×C₃₇	C₃×C₃²⋊C₉	1	1	yes	characteristic
45	C₃×C₃₇	C₃×C₂₇⋊C₃	1	3	yes	—
46	C₃⁴	#66526	1	1	yes	characteristic
47	C₃²×C₉	—	37	1	no	—
48	C₃²×C₉	—	37	6	no	—
49	C₃²×C₉	—	37	2	no	—
50	C₃²×C₉	—	37	3	no	—
51	C₃²×C₉	—	111	3	no	—
52	C₃²×C₉	—	111	6	no	—
53	C₂₇⋊C₃	—	111	9	no	—
54	C₃×C₂₇	—	37	9	no	—
55	C₃₇	#10624683	1	1	yes	characteristic, complemented, Sylow
56	C₃³	—	3	3	no	—
57	C₃³	—	3	9	no	—
58	C₃³	C₃×#66526	1	1	yes	characteristic
59	C₃³	C₃×#66526	1	3	yes	—
60	C₃×C₉	—	37	3	no	—
61	C₃×C₉	—	37	6	no	—
62	C₃×C₉	—	37	2	no	—
63	C₃×C₉	—	37	1	no	—
64	C₃×C₉	—	37	6	no	—
65	C₃×C₉	—	111	3	no	—
66	C₃×C₉	—	111	18	no	—
67	C₃×C₉	—	111	6	no	—
68	C₃×C₉	—	111	6	no	—
69	C₂₇	—	111	9	no	—
70	C₃²	—	3	27	no	—
71	C₃²	—	3	3	no	—
72	C₃²	—	3	3	no	—
73	C₃²	—	3	6	no	—
74	C₃²	C₃²⋊#66526	1	3	yes	—
75	C₃²	#15688541	1	9	yes	complemented
76	C₃²	C₃²×#66526	1	1	yes	characteristic
77	C₉	—	37	1	no	—
78	C₉	—	37	6	no	—
79	C₉	—	37	2	no	—
80	C₉	—	111	6	no	—
81	C₃	—	3	3	no	—
82	C₃	—	3	9	no	—
83	C₃	C₃×C₃²⋊#66526	1	1	yes	characteristic
84	C₃	C₃×#15688541	1	3	yes	—

	subgroup	G/H
center, Frattini	C₃²	C₃²×#66526
commutator, socle	C₃²×C₃₇	C₃²×C₉
Fitting, prime radical	C₃⁴×C₃₇	C₉
cosocle	C₃×#66524	C₃³
hypercenter	C₃⁴	#66526
hypocenter	C₃₇	#10624683
LCS	G ▷ C₃²×C₃₇ ▷ C₃₇
UCS	C₁ ◁ C₃² ◁ C₃⁴

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
5. C₃×#11758603	C₃²	semidirect
33. C₃²×C₃₇	C₂₇⋊C₃	semidirect
55. C₃₇	#10624683	semidirect
75. C₃²	#15688541	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
C₃³×#66524	3
#10624683	37