Subgroups — #41006335

Order 26973.

198 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	#22582538	C₃	1	1	yes	maximal, characteristic
3	#22582492	C₃	1	9	yes	maximal, complemented
4	C₃×C₃₇×C₂₇⋊C₃	C₃	1	1	yes	maximal, characteristic
5	#15688511	—	3	2	no	—
6	#15688511	—	3	2	no	—
7	#15688511	C₃²	1	6	yes	—
8	C₃×#11758601	—	3	2	no	—
9	#15688506	—	3	1	no	—
10	#15688505	C₃²	1	3	yes	—
11	C₃²⋊#66524	—	3	1	no	—
12	C₉×#66524	—	3	1	no	—
13	C₃²×C₉×C₃₇	C₃²	1	1	yes	characteristic
14	C₃₇×C₂₇⋊C₃	C₉	1	6	yes	complemented
15	C₃₇×C₂₇⋊C₃	C₃²	1	3	yes	—
16	C₃×C₂₇×C₃₇	C₉	1	3	yes	complemented
17	C₃³×C₃₇	3₋¹⁺²	1	1	yes	characteristic
18	C₃×C₉×C₃₇	C₃×C₉	1	1	yes	characteristic
19	C₃×C₉×C₃₇	C₃×C₉	1	2	yes	—
20	C₃×C₉×C₃₇	3₋¹⁺²	1	1	yes	characteristic
21	C₃×C₉×C₃₇	3₋¹⁺²	1	1	yes	characteristic
22	C₃×C₉×C₃₇	3₋¹⁺²	1	2	yes	—
23	C₃×C₉×C₃₇	3₋¹⁺²	1	2	yes	—
24	C₃×C₉×C₃₇	3₋¹⁺²	1	2	yes	—
25	C₃×C₉×C₃₇	C₃³	1	1	yes	characteristic
26	C₃×#66524	—	3	3	no	—
27	C₂₇×C₃₇	C₃×C₉	1	9	yes	—
28	#11758601	—	3	6	no	—
29	#10624709	—	37	1	no	maximal, Sylow
30	C₃²×C₃₇	—	9	1	no	—
31	C₃²×C₃₇	C₉⋊C₉	1	1	yes	characteristic
32	C₃²×C₃₇	C₉⋊C₉	1	2	yes	—
33	C₃²×C₃₇	C₃×3₋¹⁺²	1	1	yes	characteristic
34	C₉×C₃₇	C₉⋊C₉	1	1	yes	characteristic
35	C₉×C₃₇	C₉⋊C₉	1	2	yes	—
36	C₉×C₃₇	C₉⋊C₉	1	2	yes	—
37	C₉×C₃₇	C₉⋊C₉	1	1	yes	characteristic
38	C₉×C₃₇	C₃²×C₉	1	1	yes	characteristic
39	C₉×C₃₇	C₃×3₋¹⁺²	1	2	yes	—
40	#66524	—	9	3	no	—
41	C₃×C₂₇⋊C₃	—	37	2	no	—
42	C₃×C₂₇⋊C₃	C₃₇⋊C₃	1	1	yes	characteristic
43	#6887	—	37	1	no	—
44	#6875	—	37	9	no	—
45	C₃×C₃₇	—	3	1	no	—
46	C₃×C₃₇	—	3	1	no	—
47	C₃×C₃₇	—	3	2	no	—
48	C₃×C₃₇	C₃×C₉⋊C₉	1	1	yes	characteristic
49	C₃²×C₉	C₃×C₃₇⋊C₃	1	1	yes	characteristic
50	C₂₇⋊C₃	—	37	6	no	—
51	C₂₇⋊C₃	—	111	2	no	—
52	C₂₇⋊C₃	—	111	2	no	—
53	C₂₇⋊C₃	#66524	1	6	yes	complemented
54	C₂₇⋊C₃	C₃×C₃₇⋊C₃	1	3	yes	—
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	C₃²⋊C₉	—	111	1	no	—
60	C₉²	—	111	1	no	—
61	C₃₇	#10624709	1	1	yes	characteristic, complemented, Sylow
62	C₃³	#11758605	1	1	yes	characteristic
63	C₃×C₉	—	111	3	no	—
64	C₃×C₉	#11758605	1	1	yes	characteristic
65	C₃×C₉	#11758605	1	1	yes	characteristic
66	C₃×C₉	#11758605	1	2	yes	—
67	C₃×C₉	#11758605	1	2	yes	—
68	C₃×C₉	#11758605	1	2	yes	—
69	C₃×C₉	C₃×#66524	1	2	yes	—
70	C₃×C₉	C₃×#66524	1	1	yes	characteristic
71	C₃×C₉	C₃²×C₃₇⋊C₃	1	1	yes	characteristic
72	C₂₇	—	111	6	no	—
73	C₂₇	C₃×#66524	1	9	yes	—
74	C₃²	—	9	1	no	—
75	C₃²	#15688505	1	1	yes	characteristic
76	C₃²	#15688505	1	2	yes	—
77	C₃²	C₃×#11758605	1	1	yes	characteristic
78	C₉	—	333	3	no	—
79	C₉	#15688505	1	1	yes	characteristic
80	C₉	#15688505	1	2	yes	—
81	C₉	#15688505	1	2	yes	—
82	C₉	#15688505	1	1	yes	characteristic
83	C₉	C₃²×#66524	1	1	yes	characteristic
84	C₉	C₃×#11758605	1	2	yes	—
85	C₃	—	3	1	no	—
86	C₃	—	3	1	no	—
87	C₃	—	3	2	no	—
88	C₃	C₃×#15688505	1	1	yes	characteristic

	subgroup	G/H
center	C₉	#15688505
commutator	C₉×C₃₇	C₃²×C₉
Frattini	C₃×C₉	C₃²×C₃₇⋊C₃
Fitting	C₃×C₃₇×C₂₇⋊C₃	C₃
socle	C₃×C₃₇	C₃×C₉⋊C₉
prime radical	C₃³×C₃₇	3₋¹⁺²
cosocle	C₃×C₉×C₃₇	C₃³
hypercenter	C₃×C₂₇⋊C₃	C₃₇⋊C₃
hypocenter	C₃₇	#10624709
LCS	G ▷ C₉×C₃₇ ▷ C₃×C₃₇ ▷ C₃₇
UCS	C₁ ◁ C₉ ◁ C₃²×C₉ ◁ C₃×C₂₇⋊C₃

Complements

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

N	H ≅ G/N	type
3. #22582492	C₃	semidirect
14. C₃₇×C₂₇⋊C₃	C₉	semidirect
16. C₃×C₂₇×C₃₇	C₉	semidirect
53. C₂₇⋊C₃	#66524	semidirect
56. C₃×C₂₇	#66524	semidirect
61. C₃₇	#10624709	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
#22582538	3
#22582492	3
C₃×C₃₇×C₂₇⋊C₃	3
#10624709	37