Subgroups — #41004929

Order 26973.

92 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22582472	C₃	1	1	yes	maximal, characteristic
2	#22582472	C₃	1	1	yes	maximal, characteristic
3	#22582441	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	C₃²×C₉×C₃₇	C₃²	1	1	yes	characteristic
9	C₃₇×C₃²⋊C₉	—	3	1	no	—
10	C₃³×C₃₇	H(3,𝔽₃)	1	1	yes	characteristic
11	C₃×C₉×C₃₇	—	3	1	no	—
12	C₃×C₉×C₃₇	—	3	1	no	—
13	C₃×C₉×C₃₇	—	3	1	no	—
14	C₃×C₉×C₃₇	—	3	1	no	—
15	C₃×C₉×C₃₇	—	9	1	no	—
16	C₃×#66524	—	3	3	no	—
17	C₃×#66524	—	9	1	no	—
18	C₃×#66524	—	9	1	no	—
19	C₃×#66524	—	9	1	no	—
20	C₃×#66524	—	9	1	no	—
21	C₃×#66524	—	9	1	no	—
22	#10624495	—	37	1	no	maximal, Sylow
23	C₃²×C₃₇	—	3	1	no	—
24	C₃²×C₃₇	—	3	1	no	—
25	C₃²×C₃₇	—	3	1	no	—
26	C₃²×C₃₇	#731	1	1	yes	characteristic
27	C₃²×C₃₇	#732	1	1	yes	characteristic
28	C₃²×C₃₇	#733	1	1	yes	characteristic
29	C₃²×C₃₇	#733	1	1	yes	characteristic
30	C₉×C₃₇	—	9	1	no	—
31	C₉×C₃₇	—	27	1	no	—
32	#66524	—	9	3	no	—
33	#66524	—	27	1	no	—
34	#66524	—	27	1	no	—
35	#6868	—	37	1	no	—
36	#6868	—	37	1	no	—
37	#6868	C₃₇⋊C₃	1	1	yes	characteristic
38	#6855	—	37	1	no	—
39	C₃×C₃₇	—	3	1	no	—
40	C₃×C₃₇	—	3	1	no	—
41	C₃×C₃₇	—	3	1	no	—
42	C₃×C₃₇	#6861	1	1	yes	characteristic
43	C₃×C₃₇	#6880	1	1	yes	characteristic
44	C₃×C₃₇	#6881	1	1	yes	characteristic
45	C₃×C₃₇	#6883	1	1	yes	characteristic
46	C₃²×C₉	—	111	1	no	—
47	C₃²×C₉	C₃×C₃₇⋊C₃	1	1	yes	characteristic
48	C₃²⋊C₉	—	3	1	no	—
49	C₃²⋊C₉	—	111	1	no	—
50	C₃²⋊C₉	—	111	1	no	—
51	C₃₇	#10624495	1	1	yes	characteristic, complemented, Sylow
52	C₃³	C₃²⋊(C₃₇⋊C₃)	1	1	yes	characteristic
53	C₃×C₉	—	3	1	no	—
54	C₃×C₉	—	3	1	no	—
55	C₃×C₉	—	3	1	no	—
56	C₃×C₉	—	3	1	no	—
57	C₃×C₉	—	9	1	no	—
58	C₃×C₉	—	333	1	no	—
59	C₃×C₉	—	333	1	no	—
60	C₃×C₉	—	333	1	no	—
61	C₃×C₉	—	333	1	no	—
62	C₃×C₉	—	333	1	no	—
63	C₃×C₉	—	111	3	no	—
64	C₃²	—	3	1	no	—
65	C₃²	—	3	1	no	—
66	C₃²	—	3	1	no	—
67	C₃²	#15688517	1	1	yes	characteristic
68	C₃²	#15688518	1	1	yes	characteristic
69	C₃²	#15688520	1	1	yes	characteristic
70	C₃²	#15688521	1	1	yes	characteristic
71	C₉	—	9	1	no	—
72	C₉	—	27	1	no	—
73	C₉	—	333	3	no	—
74	C₉	—	999	1	no	—
75	C₉	—	999	1	no	—
76	C₃	—	3	1	no	—
77	C₃	—	3	1	no	—
78	C₃	—	3	1	no	—
79	C₃	#22582458	1	1	yes	characteristic
80	C₃	#22582509	1	1	yes	characteristic
81	C₃	#22582511	1	1	yes	characteristic
82	C₃	#22582522	1	1	yes	characteristic

	subgroup	G/H
center	C₃²	#15688518
commutator, cosocle	C₃²×C₉×C₃₇	C₃²
Frattini	C₃²×C₉	C₃×C₃₇⋊C₃
Fitting	C₃₇×#6868	C₃
socle	C₃²×C₃₇	#732
prime radical	C₃³×C₃₇	H(3,𝔽₃)
hypercenter	#6868	C₃₇⋊C₃
hypocenter	C₃₇	#10624495
LCS	G ▷ C₃²×C₉×C₃₇ ▷ C₃³×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
51. C₃₇	#10624495	semidirect

Maximal subgroups

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

subgroup	index
#22582472	3
#22582441	3
C₃₇×#6868	3
#10624495	37