Subgroups — #41008604

Order 26973.

220 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃²×#11758603	C₃	1	1	yes	maximal, characteristic
2	#22582706	C₃	1	3	yes	maximal, complemented
3	#22582705	C₃	1	3	yes	maximal, complemented
4	C₃²⋊#11758603	C₃	1	2	yes	maximal
5	C₉×#11758603	C₃	1	3	yes	maximal, complemented
6	#22582548	C₃	1	1	yes	maximal, characteristic
7	C₃×#11758603	—	3	3	no	—
8	C₃×#11758603	C₃²	1	6	yes	—
9	C₃×#11758603	C₃²	1	1	yes	characteristic
10	C₃×#11758603	C₃²	1	1	yes	characteristic
11	C₃×#11758603	C₃²	1	1	yes	characteristic
12	C₃×#11758608	C₉	1	1	yes	characteristic
13	C₃²×#66524	C₃²	1	1	yes	characteristic
14	#15688504	C₉	1	6	yes	—
15	C₃²⋊#66524	C₉	1	1	yes	characteristic
16	C₉×#66524	C₃²	1	3	yes	—
17	C₃₇×C₃²⋊C₉	C₉	1	1	yes	characteristic
18	C₃³×C₃₇	C₃×C₉	1	1	yes	characteristic
19	C₃×C₉×C₃₇	C₃×C₉	1	3	yes	—
20	#11758608	C₂₇	1	9	yes	complemented
21	C₃×#66524	—	3	1	no	—
22	C₃×#66524	—	3	2	no	—
23	C₃×#66524	C₃×C₉	1	3	yes	—
24	C₃×#66524	C₃×C₉	1	2	yes	—
25	C₃×#66524	C₃×C₉	1	3	yes	—
26	C₃×#66524	C₃³	1	1	yes	characteristic
27	#11758603	—	3	1	no	—
28	#11758603	—	3	1	no	—
29	#11758603	—	3	6	no	—
30	#11758603	3₋¹⁺²	1	3	yes	complemented
31	C₂₇×3₋¹⁺²	—	37	1	no	maximal, Sylow
32	C₃²×C₃₇	—	3	1	no	—
33	C₃²×C₃₇	—	3	1	no	—
34	C₃²×C₃₇	—	3	1	no	—
35	C₃²×C₃₇	C₃×C₂₇	1	3	yes	—
36	C₃²×C₃₇	C₃²×C₉	1	1	yes	characteristic
37	C₉×C₃₇	—	3	3	no	—
38	#66524	—	3	2	no	—
39	#66524	—	3	3	no	—
40	#66524	C₃×C₂₇	1	9	yes	complemented
41	#66524	C₃×3₋¹⁺²	1	1	yes	characteristic
42	#66524	#737	1	2	yes	—
43	C₃²×C₂₇	—	37	1	no	—
44	C₉×3₋¹⁺²	—	37	1	no	—
45	C₉⋊C₂₇	—	37	3	no	—
46	C₉⋊C₂₇	—	37	3	no	—
47	C₃²⋊C₂₇	—	37	2	no	—
48	C₉×C₂₇	—	37	3	no	—
49	C₃×C₃₇	—	3	3	no	—
50	C₃×C₃₇	C₉×3₋¹⁺²	1	1	yes	characteristic
51	C₃×C₃₇	C₃²×C₂₇	1	1	yes	characteristic
52	C₃×C₃₇	#6903	1	1	yes	characteristic
53	C₃×C₃₇	#6903	1	1	yes	characteristic
54	C₃×3₋¹⁺²	—	37	1	no	—
55	C₃²×C₉	—	37	1	no	—
56	C₃×C₂₇	—	37	1	no	—
57	C₃×C₂₇	—	37	6	no	—
58	C₃×C₂₇	—	37	1	no	—
59	C₃×C₂₇	—	37	1	no	—
60	C₃×C₂₇	—	111	3	no	—
61	C₉⋊C₉	—	37	6	no	—
62	C₃²⋊C₉	—	37	1	no	—
63	C₃²⋊C₉	#66526	1	1	yes	characteristic
64	C₉²	—	37	3	no	—
65	C₃₇	C₂₇×3₋¹⁺²	1	1	yes	characteristic, complemented, Sylow
66	C₃³	C₃×#66526	1	1	yes	characteristic
67	3₋¹⁺²	—	37	9	no	—
68	C₃×C₉	—	37	3	no	—
69	C₃×C₉	—	37	1	no	—
70	C₃×C₉	—	37	3	no	—
71	C₃×C₉	—	37	2	no	—
72	C₃×C₉	—	111	1	no	—
73	C₃×C₉	—	111	2	no	—
74	C₃×C₉	C₃×#66526	1	3	yes	—
75	C₂₇	—	37	3	no	—
76	C₂₇	—	111	6	no	—
77	C₂₇	—	111	1	no	—
78	C₂₇	—	111	1	no	—
79	C₃²	—	3	1	no	—
80	C₃²	—	3	1	no	—
81	C₃²	—	3	1	no	—
82	C₃²	(C₃₇⋊C₃)⋊C₂₇	1	3	yes	—
83	C₃²	C₃²×#66526	1	1	yes	characteristic
84	C₉	—	3	3	no	—
85	C₉	—	37	1	no	—
86	C₉	—	37	2	no	—
87	C₉	—	37	9	no	—
88	C₉	—	111	2	no	—
89	C₉	—	111	3	no	—
90	C₃	—	3	3	no	—
91	C₃	3₋¹⁺²×#66526	1	1	yes	characteristic
92	C₃	C₃×(C₃₇⋊C₃)⋊C₂₇	1	1	yes	characteristic
93	C₃	#22582731	1	1	yes	characteristic
94	C₃	#22582732	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₃²	C₃²×#66526
commutator	C₃×C₃₇	C₃²×C₂₇
Fitting	C₃₇×C₃²⋊C₉	C₉
socle	C₃²×C₃₇	C₃²×C₉
prime radical	C₃³×C₃₇	C₃×C₉
cosocle	C₃×#66524	C₃³
hypercenter	C₃²⋊C₉	#66526
hypocenter	C₃₇	C₂₇×3₋¹⁺²
LCS	G ▷ C₃×C₃₇ ▷ C₃₇
UCS	C₁ ◁ C₃² ◁ C₃²⋊C₉

Complements

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

N	H ≅ G/N	type
2. #22582706	C₃	semidirect
3. #22582705	C₃	semidirect
5. C₉×#11758603	C₃	semidirect
20. #11758608	C₂₇	semidirect
30. #11758603	3₋¹⁺²	semidirect
40. #66524	C₃×C₂₇	semidirect
65. C₃₇	C₂₇×3₋¹⁺²	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
#22582706	3
#22582705	3
C₃²⋊#11758603	3
C₉×#11758603	3
#22582548	3
C₂₇×3₋¹⁺²	37