Subgroups — #41007410

Order 26973.

202 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22582633	C₃	1	3	yes	maximal, complemented
2	#22582523	C₃	1	3	yes	maximal, complemented
3	#22582518	C₃	1	3	yes	maximal, complemented
4	#22582513	C₃	1	3	yes	maximal, complemented
5	C₃×C₃₇×C₉⋊C₉	C₃	1	1	yes	maximal, characteristic, complemented
6	#15688531	—	3	3	no	—
7	#15688521	C₃²	1	3	yes	—
8	#15688521	C₃²	1	3	yes	—
9	#15688519	C₃²	1	3	yes	—
10	#15688516	—	3	3	no	—
11	#15688513	—	3	3	no	—
12	C₃²×C₉×C₃₇	—	3	1	no	—
13	C₃²×C₉×C₃₇	C₃²	1	1	yes	characteristic
14	C₃₇×C₉⋊C₉	—	3	1	no	—
15	C₃₇×C₉⋊C₉	—	3	1	no	—
16	C₃₇×C₉⋊C₉	C₃²	1	3	yes	—
17	C₃³×C₃₇	H(3,𝔽₃)	1	1	yes	characteristic
18	C₃×C₉×C₃₇	—	3	3	no	—
19	C₃×C₉×C₃₇	—	3	3	no	—
20	C₃×C₉×C₃₇	—	9	3	no	—
21	C₃×C₉×C₃₇	H(3,𝔽₃)	1	1	yes	characteristic
22	C₃×C₉×C₃₇	H(3,𝔽₃)	1	1	yes	characteristic
23	C₃×C₉×C₃₇	C₃³	1	1	yes	characteristic
24	C₃²⋊(C₃₇⋊C₃)	—	3	3	no	—
25	#11758608	—	3	3	no	—
26	#11758608	—	3	3	no	—
27	#11758606	—	9	3	no	—
28	#11758605	—	9	3	no	—
29	C₉×C₃₇⋊C₃	—	9	3	no	—
30	#10624850	—	37	1	no	maximal, Sylow
31	C₃²×C₃₇	—	3	3	no	—
32	C₃²×C₃₇	—	3	1	no	—
33	C₃²×C₃₇	C₃×H(3,𝔽₃)	1	1	yes	characteristic
34	C₃×C₃₇⋊C₃	—	9	3	no	—
35	C₉×C₃₇	—	3	1	no	—
36	C₉×C₃₇	—	3	1	no	—
37	C₉×C₃₇	—	9	3	no	—
38	C₉×C₃₇	#732	1	3	yes	—
39	#66524	—	9	3	no	—
40	#66524	—	9	3	no	—
41	#6908	—	37	3	no	—
42	C₃×C₉⋊C₉	C₃₇⋊C₃	1	1	yes	characteristic, complemented
43	#6883	—	37	3	no	—
44	#6882	—	37	3	no	—
45	#6881	—	37	3	no	—
46	C₃×C₃₇	—	3	1	no	—
47	C₃×C₃₇	—	3	3	no	—
48	C₃×C₃₇	C₃×#732	1	1	yes	characteristic
49	C₃₇⋊C₃	—	27	3	no	—
50	#737	—	111	3	no	—
51	C₃²×C₉	—	3	1	no	—
52	C₃²×C₉	C₃×C₃₇⋊C₃	1	1	yes	characteristic
53	#733	—	37	3	no	—
54	#733	—	37	3	no	—
55	#732	—	37	3	no	—
56	#731	—	111	3	no	—
57	AΣL(1,𝔽₂₇)	—	111	3	no	—
58	C₉⋊C₉	—	3	1	no	—
59	C₉⋊C₉	—	3	1	no	—
60	C₉⋊C₉	C₃×C₃₇⋊C₃	1	3	yes	—
61	C₃₇	#10624850	1	1	yes	characteristic, complemented, Sylow
62	C₃³	C₃²⋊(C₃₇⋊C₃)	1	1	yes	characteristic
63	3₋¹⁺²	—	333	3	no	—
64	3₋¹⁺²	—	333	3	no	—
65	3₋¹⁺²	—	111	3	no	—
66	3₋¹⁺²	—	111	3	no	—
67	H(3,𝔽₃)	—	111	3	no	—
68	C₃×C₉	—	3	3	no	—
69	C₃×C₉	—	3	3	no	—
70	C₃×C₉	—	9	3	no	—
71	C₃×C₉	—	333	3	no	—
72	C₃×C₉	C₃²⋊(C₃₇⋊C₃)	1	1	yes	characteristic
73	C₃×C₉	C₃²⋊(C₃₇⋊C₃)	1	1	yes	characteristic
74	C₃×C₉	C₃²×C₃₇⋊C₃	1	1	yes	characteristic
75	C₃²	—	3	3	no	—
76	C₃²	—	3	1	no	—
77	C₃²	—	333	3	no	—
78	C₃²	C₃×C₃²⋊(C₃₇⋊C₃)	1	1	yes	characteristic
79	C₉	—	3	1	no	—
80	C₉	—	3	1	no	—
81	C₉	—	9	3	no	—
82	C₉	—	333	3	no	—
83	C₉	—	333	3	no	—
84	C₉	#15688519	1	3	yes	—
85	C₃	—	3	1	no	—
86	C₃	—	3	3	no	—
87	C₃	—	999	3	no	—
88	C₃	C₃×#15688519	1	1	yes	characteristic

	subgroup	G/H
center	C₃	C₃×#15688519
commutator, cosocle	C₃×C₉×C₃₇	C₃³
Frattini	C₃×C₉	C₃²×C₃₇⋊C₃
Fitting	C₃×C₃₇×C₉⋊C₉	C₃
socle	C₃×C₃₇	C₃×#732
hypercenter	C₃×C₉⋊C₉	C₃₇⋊C₃
hypocenter	C₃₇	#10624850
LCS	G ▷ C₃×C₉×C₃₇ ▷ C₃²×C₃₇ ▷ C₃×C₃₇ ▷ C₃₇
UCS	C₁ ◁ C₃ ◁ C₃×C₉ ◁ C₃²×C₉ ◁ C₃×C₉⋊C₉

Complements

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

N	H ≅ G/N	type
1. #22582633	C₃	semidirect
2. #22582523	C₃	semidirect
3. #22582518	C₃	semidirect
4. #22582513	C₃	semidirect
5. C₃×C₃₇×C₉⋊C₉	C₃	semidirect
42. C₃×C₉⋊C₉	C₃₇⋊C₃	semidirect
61. C₃₇	#10624850	semidirect

Maximal subgroups

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

subgroup	index
#22582633	3
#22582523	3
#22582518	3
#22582513	3
C₃×C₃₇×C₉⋊C₉	3
#10624850	37