Subgroups — #41008268

Order 26973.

316 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃²⋊#11758603	C₃	1	3	yes	maximal, complemented
2	C₃³×#66524	C₃	1	1	yes	maximal, characteristic
3	C₃×#11758603	—	3	3	no	—
4	C₃²×#66524	—	3	1	no	—
5	C₃²×#66524	—	3	2	no	—
6	C₃²×#66524	—	3	9	no	—
7	C₃²×#66524	C₉	1	2	yes	—
8	C₃²×#66524	C₃²	1	1	yes	characteristic
9	C₃⁴×C₃₇	C₉	1	1	yes	characteristic
10	C₃³×C₃₇	—	3	6	no	—
11	C₃³×C₃₇	—	3	3	no	—
12	C₃³×C₃₇	—	3	1	no	—
13	C₃³×C₃₇	—	3	2	no	—
14	C₃³×C₃₇	C₂₇	1	3	yes	complemented
15	C₃³×C₃₇	C₃×C₉	1	1	yes	characteristic
16	C₃×#66524	—	3	18	no	—
17	C₃×#66524	—	3	2	no	—
18	C₃×#66524	—	3	9	no	—
19	C₃×#66524	—	3	3	no	—
20	C₃×#66524	—	3	6	no	—
21	C₃×#66524	H(3,𝔽₃)	1	1	yes	characteristic
22	C₃×#66524	3₋¹⁺²	1	2	yes	—
23	#11758603	—	9	3	no	—
24	#10624444	—	37	1	no	maximal, Sylow
25	C₃²×C₃₇	—	3	9	no	—
26	C₃²×C₃₇	—	3	18	no	—
27	C₃²×C₃₇	—	3	3	no	—
28	C₃²×C₃₇	—	3	3	no	—
29	C₃²×C₃₇	—	3	1	no	—
30	C₃²×C₃₇	—	3	6	no	—
31	C₃²×C₃₇	—	3	2	no	—
32	C₃²×C₃₇	C₃²⋊C₉	1	1	yes	characteristic
33	C₃²×C₃₇	C₃×C₂₇	1	1	yes	characteristic
34	C₃²×C₃₇	C₂₇⋊C₃	1	2	yes	—
35	#66524	—	3	6	no	—
36	#66524	—	3	2	no	—
37	#66524	AΣL(1,𝔽₂₇)	1	3	yes	—
38	C₃³×C₉	—	37	1	no	—
39	C₃²⋊C₂₇	—	37	3	no	—
40	C₃×C₃₇	—	3	9	no	—
41	C₃×C₃₇	—	3	1	no	—
42	C₃×C₃₇	—	3	2	no	—
43	C₃×C₃₇	C₃²⋊C₂₇	1	1	yes	characteristic
44	C₃×C₃₇	#6866	1	1	yes	characteristic
45	C₃×C₃₇	#6869	1	2	yes	—
46	C₃⁴	#66526	1	1	yes	characteristic
47	C₃²×C₉	—	37	1	no	—
48	C₃²×C₉	—	37	2	no	—
49	C₃²×C₉	—	111	1	no	—
50	C₃²×C₉	—	111	2	no	—
51	C₃²×C₉	—	111	9	no	—
52	C₃×C₂₇	—	111	3	no	—
53	C₃₇	#10624444	1	1	yes	characteristic, complemented, Sylow
54	C₃³	—	3	3	no	—
55	C₃³	—	3	6	no	—
56	C₃³	—	3	1	no	—
57	C₃³	—	3	2	no	—
58	C₃³	#11758603	1	3	yes	complemented
59	C₃³	C₃×#66526	1	1	yes	characteristic
60	C₃×C₉	—	37	1	no	—
61	C₃×C₉	—	37	2	no	—
62	C₃×C₉	—	111	3	no	—
63	C₃×C₉	—	111	9	no	—
64	C₃×C₉	—	111	18	no	—
65	C₃×C₉	—	111	2	no	—
66	C₃×C₉	—	111	6	no	—
67	C₂₇	—	333	3	no	—
68	C₃²	—	3	9	no	—
69	C₃²	—	3	18	no	—
70	C₃²	—	3	3	no	—
71	C₃²	—	3	1	no	—
72	C₃²	—	3	3	no	—
73	C₃²	—	3	2	no	—
74	C₃²	—	3	6	no	—
75	C₃²	C₃²⋊#66526	1	1	yes	characteristic
76	C₃²	C₃×#11758603	1	1	yes	characteristic
77	C₃²	#15688541	1	2	yes	—
78	C₉	—	37	3	no	—
79	C₉	—	111	2	no	—
80	C₉	—	111	6	no	—
81	C₃	—	3	1	no	—
82	C₃	—	3	2	no	—
83	C₃	—	3	9	no	—
84	C₃	C₃²⋊#11758603	1	1	yes	characteristic
85	C₃	#22582691	1	1	yes	characteristic
86	C₃	#22582694	1	2	yes	—

	subgroup	G/H
center	C₃²	C₃²⋊#66526
commutator	C₃²×C₃₇	C₃×C₂₇
Frattini	C₃³	C₃×#66526
Fitting, prime radical	C₃⁴×C₃₇	C₉
socle	C₃²×C₃₇	C₃²⋊C₉
cosocle	C₃²×#66524	C₃²
hypercenter	C₃⁴	#66526
hypocenter	C₃₇	#10624444
LCS	G ▷ C₃²×C₃₇ ▷ C₃×C₃₇ ▷ C₃₇
UCS	C₁ ◁ C₃² ◁ C₃³ ◁ C₃⁴

Complements

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

N	H ≅ G/N	type
1. C₃²⋊#11758603	C₃	semidirect
14. C₃³×C₃₇	C₂₇	semidirect
53. C₃₇	#10624444	semidirect
58. C₃³	#11758603	semidirect

Maximal subgroups

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

subgroup	index
C₃²⋊#11758603	3
C₃³×#66524	3
#10624444	37