Subgroups — #41005022

Order 26973.

112 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22582633	C₃	1	1	yes	maximal, characteristic
2	C₃²×#11758601	C₃	1	1	yes	maximal, characteristic, complemented
3	#22582474	C₃	1	1	yes	maximal, characteristic, complemented
4	C₃₇×#6869	C₃	1	1	yes	maximal, characteristic, complemented
5	#15688531	—	3	1	no	—
6	#15688521	C₉	1	2	yes	—
7	#15688519	C₉	1	1	yes	characteristic
8	#15688516	—	3	1	no	—
9	#15688513	—	3	1	no	—
10	#15688511	—	3	1	no	—
11	C₃×#11758601	—	3	1	no	—
12	C₃×#11758601	—	3	3	no	—
13	C₃²×C₉×C₃₇	C₃²	1	1	yes	characteristic
14	C₃₇×C₂₇⋊C₃	—	3	1	no	—
15	C₃³×C₃₇	3₋¹⁺²	1	1	yes	characteristic
16	C₃×C₉×C₃₇	—	3	1	no	—
17	C₃×C₉×C₃₇	—	3	1	no	—
18	C₃×C₉×C₃₇	—	3	1	no	—
19	C₃×C₉×C₃₇	C₃×C₉	1	1	yes	characteristic
20	C₃×C₉×C₃₇	H(3,𝔽₃)	1	1	yes	characteristic
21	C₃×C₉×C₃₇	3₋¹⁺²	1	1	yes	characteristic
22	C₃²⋊(C₃₇⋊C₃)	—	3	1	no	—
23	#11758608	—	3	2	no	—
24	#11758606	—	9	1	no	—
25	#11758605	—	9	1	no	—
26	C₉×C₃₇⋊C₃	—	9	1	no	—
27	C₂₇×C₃₇	—	9	1	no	—
28	#11758601	—	3	3	no	—
29	#11758601	—	9	1	no	—
30	#10624526	—	37	1	no	maximal, Sylow
31	C₃²×C₃₇	—	3	3	no	—
32	C₃²×C₃₇	—	3	1	no	—
33	C₃²×C₃₇	C₃²⋊C₉	1	1	yes	characteristic
34	C₃×C₃₇⋊C₃	—	9	1	no	—
35	C₉×C₃₇	—	3	1	no	—
36	C₉×C₃₇	—	3	1	no	—
37	C₉×C₃₇	AΣL(1,𝔽₂₇)	1	1	yes	characteristic
38	C₉×C₃₇	#731	1	1	yes	characteristic
39	C₉×C₃₇	#732	1	1	yes	characteristic
40	#66524	—	9	2	no	—
41	#6908	—	37	1	no	—
42	C₃²×C₂₇	—	37	1	no	—
43	#6869	—	37	1	no	—
44	#6869	C₃₇⋊C₃	1	1	yes	characteristic, complemented
45	C₃×C₃₇	—	3	3	no	—
46	C₃×C₃₇	—	3	1	no	—
47	C₃×C₃₇	H(3,𝔽₃)⋊C₉	1	1	yes	characteristic
48	C₃₇⋊C₃	—	27	1	no	—
49	#737	—	111	1	no	—
50	C₃²×C₉	C₃×C₃₇⋊C₃	1	1	yes	characteristic
51	#733	—	37	2	no	—
52	#732	—	37	1	no	—
53	#731	—	111	1	no	—
54	AΣL(1,𝔽₂₇)	—	111	1	no	—
55	C₂₇⋊C₃	—	3	1	no	—
56	C₂₇⋊C₃	—	111	1	no	—
57	C₃×C₂₇	—	111	1	no	—
58	C₃×C₂₇	—	111	3	no	—
59	C₃₇	#10624526	1	1	yes	characteristic, complemented, Sylow
60	C₃³	#11758606	1	1	yes	characteristic
61	3₋¹⁺²	—	333	1	no	—
62	3₋¹⁺²	—	333	1	no	—
63	3₋¹⁺²	—	111	2	no	—
64	H(3,𝔽₃)	—	111	1	no	—
65	C₃×C₉	—	3	1	no	—
66	C₃×C₉	—	3	1	no	—
67	C₃×C₉	—	3	1	no	—
68	C₃×C₉	—	333	1	no	—
69	C₃×C₉	C₉×C₃₇⋊C₃	1	1	yes	characteristic
70	C₃×C₉	#11758605	1	1	yes	characteristic
71	C₃×C₉	C₃²⋊(C₃₇⋊C₃)	1	1	yes	characteristic
72	C₂₇	—	9	1	no	—
73	C₂₇	—	333	1	no	—
74	C₂₇	—	111	3	no	—
75	C₃²	—	3	3	no	—
76	C₃²	—	3	1	no	—
77	C₃²	—	333	1	no	—
78	C₃²	#15688502	1	1	yes	characteristic
79	C₉	—	3	1	no	—
80	C₉	—	3	1	no	—
81	C₉	—	333	2	no	—
82	C₉	#15688514	1	1	yes	characteristic
83	C₉	#15688517	1	1	yes	characteristic
84	C₉	#15688518	1	1	yes	characteristic
85	C₃	—	3	1	no	—
86	C₃	—	3	3	no	—
87	C₃	—	999	1	no	—
88	C₃	#22582475	1	1	yes	characteristic

	subgroup	G/H
center	C₉	#15688518
commutator	C₃×C₉×C₃₇	C₃×C₉
Frattini	C₃²×C₉	C₃×C₃₇⋊C₃
Fitting	C₃₇×#6869	C₃
socle	C₃×C₃₇	H(3,𝔽₃)⋊C₉
prime radical	#22582633	C₃
cosocle	C₃²×C₉×C₃₇	C₃²
hypercenter	#6869	C₃₇⋊C₃
hypocenter	C₃₇	#10624526
LCS	G ▷ C₃×C₉×C₃₇ ▷ C₃²×C₃₇ ▷ C₃×C₃₇ ▷ C₃₇
UCS	C₁ ◁ C₉ ◁ C₃×C₉ ◁ C₃²×C₉ ◁ #6869

Complements

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

N	H ≅ G/N	type
2. C₃²×#11758601	C₃	semidirect
3. #22582474	C₃	semidirect
4. C₃₇×#6869	C₃	semidirect
44. #6869	C₃₇⋊C₃	semidirect
59. C₃₇	#10624526	semidirect

Maximal subgroups

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

subgroup	index
#22582633	3
C₃²×#11758601	3
#22582474	3
C₃₇×#6869	3
#10624526	37