Subgroups — #41004994

Order 26973.

134 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×C₉×#66524	C₃	1	1	yes	maximal, characteristic
2	#22582472	C₃	1	1	yes	maximal, characteristic
3	#22582472	C₃	1	1	yes	maximal, characteristic
4	C₃₇×#6868	C₃	1	1	yes	maximal, characteristic
5	C₃²×#66524	—	3	1	no	—
6	C₃²⋊#66524	—	3	1	no	—
7	C₃²⋊#66524	—	3	1	no	—
8	C₉×#66524	—	3	1	no	—
9	C₉×#66524	—	3	1	no	—
10	C₉×#66524	C₉	1	3	yes	complemented
11	C₃²×C₉×C₃₇	C₃²	1	1	yes	characteristic
12	C₃₇×C₃²⋊C₉	—	3	1	no	—
13	C₃³×C₃₇	H(3,𝔽₃)	1	1	yes	characteristic
14	C₃×C₉×C₃₇	—	3	1	no	—
15	C₃×C₉×C₃₇	—	3	2	no	—
16	C₃×C₉×C₃₇	—	9	1	no	—
17	C₃×C₉×C₃₇	C₃×C₉	1	1	yes	characteristic
18	C₃×C₉×C₃₇	3₋¹⁺²	1	1	yes	characteristic
19	C₃×C₉×C₃₇	3₋¹⁺²	1	1	yes	characteristic
20	C₃×#66524	—	3	3	no	—
21	C₃×#66524	—	3	3	no	—
22	C₃×#66524	—	3	3	no	—
23	C₃×#66524	—	3	3	no	—
24	C₃×#66524	—	9	1	no	—
25	C₃×#66524	—	9	1	no	—
26	#10624518	—	37	1	no	maximal, Sylow
27	C₃²×C₃₇	—	3	1	no	—
28	C₃²×C₃₇	—	3	1	no	—
29	C₃²×C₃₇	—	3	1	no	—
30	C₃²×C₃₇	C₃²⋊C₉	1	1	yes	characteristic
31	C₃²×C₃₇	#732	1	1	yes	characteristic
32	C₃²×C₃₇	#733	1	2	yes	—
33	C₉×C₃₇	—	3	1	no	—
34	C₉×C₃₇	—	3	1	no	—
35	C₉×C₃₇	—	3	1	no	—
36	C₉×C₃₇	—	27	1	no	—
37	#66524	—	3	9	no	—
38	#66524	—	27	1	no	—
39	#66524	—	27	1	no	—
40	C₃×C₉²	—	37	1	no	—
41	#6868	—	37	1	no	—
42	#6868	—	37	1	no	—
43	#6868	C₃₇⋊C₃	1	1	yes	characteristic
44	C₃×C₃₇	—	3	1	no	—
45	C₃×C₃₇	—	3	2	no	—
46	C₃×C₃₇	#6868	1	1	yes	characteristic
47	C₃×C₃₇	#6878	1	1	yes	characteristic
48	C₃×C₃₇	#6878	1	1	yes	characteristic
49	C₃×C₃₇	#6878	1	1	yes	characteristic
50	C₃²×C₉	—	111	1	no	—
51	C₃²×C₉	C₃×C₃₇⋊C₃	1	1	yes	characteristic
52	C₃²⋊C₉	—	3	1	no	—
53	C₃²⋊C₉	—	111	1	no	—
54	C₃²⋊C₉	—	111	1	no	—
55	C₉²	—	37	3	no	—
56	C₉²	—	111	1	no	—
57	C₉²	—	111	1	no	—
58	C₃₇	#10624518	1	1	yes	characteristic, complemented, Sylow
59	C₃³	C₃²⋊(C₃₇⋊C₃)	1	1	yes	characteristic
60	C₃×C₉	—	3	1	no	—
61	C₃×C₉	—	3	2	no	—
62	C₃×C₉	—	9	1	no	—
63	C₃×C₉	—	333	1	no	—
64	C₃×C₉	—	333	1	no	—
65	C₃×C₉	—	111	3	no	—
66	C₃×C₉	—	111	3	no	—
67	C₃×C₉	—	111	3	no	—
68	C₃×C₉	—	111	3	no	—
69	C₃×C₉	C₉×C₃₇⋊C₃	1	1	yes	characteristic
70	C₃×C₉	#11758605	1	1	yes	characteristic
71	C₃×C₉	#11758606	1	1	yes	characteristic
72	C₃²	—	3	1	no	—
73	C₃²	—	3	1	no	—
74	C₃²	—	3	1	no	—
75	C₃²	#15688502	1	1	yes	characteristic
76	C₃²	#15688518	1	1	yes	characteristic
77	C₃²	#15688520	1	2	yes	—
78	C₉	—	3	1	no	—
79	C₉	—	3	1	no	—
80	C₉	—	3	1	no	—
81	C₉	—	27	1	no	—
82	C₉	—	111	9	no	—
83	C₉	—	999	1	no	—
84	C₉	—	999	1	no	—
85	C₃	—	3	1	no	—
86	C₃	—	3	2	no	—
87	C₃	#22582471	1	1	yes	characteristic
88	C₃	#22582501	1	1	yes	characteristic
89	C₃	#22582504	1	1	yes	characteristic
90	C₃	#22582505	1	1	yes	characteristic

	subgroup	G/H
center	C₃²	#15688518
commutator	C₃×C₉×C₃₇	C₃×C₉
Frattini	C₃²×C₉	C₃×C₃₇⋊C₃
Fitting	C₃₇×#6868	C₃
socle	C₃²×C₃₇	#732
prime radical	C₃³×C₃₇	H(3,𝔽₃)
cosocle	C₃²×C₉×C₃₇	C₃²
hypercenter	#6868	C₃₇⋊C₃
hypocenter	C₃₇	#10624518
LCS	G ▷ C₃×C₉×C₃₇ ▷ C₃²×C₃₇ ▷ C₃×C₃₇ ▷ C₃₇
UCS	C₁ ◁ C₃² ◁ C₃³ ◁ C₃²×C₉ ◁ #6868

Complements

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

N	H ≅ G/N	type
10. C₉×#66524	C₉	semidirect
58. C₃₇	#10624518	semidirect

Maximal subgroups

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

subgroup	index
C₃×C₉×#66524	3
#22582472	3
C₃₇×#6868	3
#10624518	37