Subgroups — #41006318

Order 26973.

248 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×#15688511	C₃	1	1	yes	maximal, characteristic
2	C₃×#15688511	C₃	1	1	yes	maximal, characteristic
3	C₃×C₉×#66524	C₃	1	1	yes	maximal, characteristic
4	#22582488	C₃	1	9	yes	maximal, complemented
5	C₃×C₃₇×C₂₇⋊C₃	C₃	1	1	yes	maximal, characteristic
6	C₃²×#66524	C₉	1	3	yes	—
7	#15688511	—	3	3	no	—
8	#15688511	—	3	3	no	—
9	C₃×#11758601	C₃²	1	3	yes	—
10	C₃×#11758601	C₃²	1	3	yes	—
11	C₉×#66524	C₉	1	2	yes	—
12	C₉×#66524	C₉	1	3	yes	—
13	C₉×#66524	C₉	1	1	yes	characteristic
14	C₉×#66524	C₃²	1	3	yes	—
15	C₃²×C₉×C₃₇	C₃²	1	1	yes	characteristic
16	C₃₇×C₂₇⋊C₃	—	3	3	no	—
17	C₃×C₂₇×C₃₇	C₃²	1	3	yes	—
18	C₃³×C₃₇	C₃×C₉	1	1	yes	characteristic
19	C₃×C₉×C₃₇	C₃×C₉	1	1	yes	characteristic
20	C₃×C₉×C₃₇	C₃×C₉	1	1	yes	characteristic
21	C₃×C₉×C₃₇	3₋¹⁺²	1	3	yes	—
22	C₃×C₉×C₃₇	3₋¹⁺²	1	3	yes	—
23	C₃×C₉×C₃₇	3₋¹⁺²	1	3	yes	—
24	C₃×C₉×C₃₇	C₃³	1	1	yes	characteristic
25	C₃×#66524	—	3	9	no	—
26	C₃×#66524	C₃×C₉	1	6	yes	—
27	C₃×#66524	C₃×C₉	1	3	yes	—
28	C₂₇×C₃₇	—	3	3	no	—
29	#11758601	—	3	3	no	—
30	#11758601	—	3	3	no	—
31	#10624706	—	37	1	no	maximal, Sylow
32	C₃²×C₃₇	—	3	3	no	—
33	C₃²×C₃₇	C₉⋊C₉	1	3	yes	—
34	C₃²×C₃₇	C₃²×C₉	1	1	yes	characteristic
35	C₉×C₃₇	C₉⋊C₉	1	3	yes	—
36	C₉×C₃₇	C₉⋊C₉	1	3	yes	—
37	C₉×C₃₇	C₃×3₋¹⁺²	1	1	yes	characteristic
38	C₉×C₃₇	C₃×3₋¹⁺²	1	1	yes	characteristic
39	C₉×C₃₇	C₃×3₋¹⁺²	1	1	yes	characteristic
40	#66524	—	3	6	no	—
41	#66524	C₂₇⋊C₃	1	9	yes	complemented
42	C₃×C₂₇⋊C₃	—	37	1	no	—
43	C₃×C₂₇⋊C₃	—	37	1	no	—
44	C₃×C₂₇⋊C₃	C₃₇⋊C₃	1	1	yes	characteristic
45	C₃×C₉²	—	37	1	no	—
46	C₉⋊C₂₇	—	37	9	no	—
47	C₃×C₃₇	—	3	3	no	—
48	C₃×C₃₇	C₃×C₉⋊C₉	1	1	yes	characteristic
49	C₃×C₃₇	C₃×C₂₇⋊C₃	1	3	yes	—
50	C₃²×C₉	—	37	3	no	—
51	C₃²×C₉	C₃×C₃₇⋊C₃	1	1	yes	characteristic
52	C₂₇⋊C₃	—	3	3	no	—
53	C₂₇⋊C₃	—	111	3	no	—
54	C₂₇⋊C₃	—	111	3	no	—
55	C₃×C₂₇	—	37	3	no	—
56	C₃×C₂₇	—	37	3	no	—
57	C₃×C₂₇	C₃×C₃₇⋊C₃	1	3	yes	—
58	C₉²	—	37	1	no	—
59	C₉²	—	37	3	no	—
60	C₉²	—	37	2	no	—
61	C₉²	—	37	3	no	—
62	C₃₇	#10624706	1	1	yes	characteristic, complemented, Sylow
63	C₃³	C₉×C₃₇⋊C₃	1	1	yes	characteristic
64	C₃×C₉	—	37	3	no	—
65	C₃×C₉	—	37	6	no	—
66	C₃×C₉	—	111	9	no	—
67	C₃×C₉	C₉×C₃₇⋊C₃	1	1	yes	characteristic
68	C₃×C₉	C₉×C₃₇⋊C₃	1	1	yes	characteristic
69	C₃×C₉	#11758605	1	3	yes	—
70	C₃×C₉	#11758606	1	3	yes	—
71	C₃×C₉	#11758608	1	3	yes	—
72	C₃×C₉	C₃²×C₃₇⋊C₃	1	1	yes	characteristic
73	C₂₇	—	3	3	no	—
74	C₂₇	—	111	3	no	—
75	C₂₇	—	111	3	no	—
76	C₃²	—	3	3	no	—
77	C₃²	#15688504	1	3	yes	—
78	C₃²	C₃×C₉×C₃₇⋊C₃	1	1	yes	characteristic
79	C₉	—	37	9	no	—
80	C₉	—	111	6	no	—
81	C₉	#15688504	1	3	yes	—
82	C₉	#15688504	1	3	yes	—
83	C₉	C₃×#11758606	1	1	yes	characteristic
84	C₉	C₃×#11758605	1	1	yes	characteristic
85	C₉	C₃×#11758608	1	1	yes	characteristic
86	C₃	—	3	3	no	—
87	C₃	C₃×#15688504	1	1	yes	characteristic
88	C₃	C₂₇⋊C₃×C₃₇⋊C₃	1	3	yes	—

	subgroup	G/H
center, Frattini	C₃×C₉	C₃²×C₃₇⋊C₃
commutator, socle	C₃²×C₃₇	C₃²×C₉
Fitting	C₃×C₃₇×C₂₇⋊C₃	C₃
prime radical	C₃³×C₃₇	C₃×C₉
cosocle	C₃×C₉×C₃₇	C₃³
hypercenter	C₃×C₂₇⋊C₃	C₃₇⋊C₃
hypocenter	C₃₇	#10624706
LCS	G ▷ C₃²×C₃₇ ▷ C₃₇
UCS	C₁ ◁ C₃×C₉ ◁ C₃×C₂₇⋊C₃

Complements

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

N	H ≅ G/N	type
4. #22582488	C₃	semidirect
41. #66524	C₂₇⋊C₃	semidirect
62. C₃₇	#10624706	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×C₉×#66524	3
#22582488	3
C₃×C₃₇×C₂₇⋊C₃	3
#10624706	37