Subgroups — #41007380

Order 26973.

188 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	C₃₇×#6887	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	C₃²	1	1	yes	characteristic
13	C₃₇×C₉⋊C₉	—	3	1	no	—
14	C₃₇×C₉⋊C₉	C₃²	1	3	yes	—
15	C₃₇×C₃²⋊C₉	—	3	1	no	—
16	C₉²×C₃₇	—	3	1	no	—
17	C₃³×C₃₇	H(3,𝔽₃)	1	1	yes	characteristic
18	C₃×C₉×C₃₇	—	3	1	no	—
19	C₃×C₉×C₃₇	—	3	1	no	—
20	C₃×C₉×C₃₇	—	3	1	no	—
21	C₃×C₉×C₃₇	—	3	3	no	—
22	C₃×C₉×C₃₇	H(3,𝔽₃)	1	1	yes	characteristic
23	C₃×C₉×C₃₇	H(3,𝔽₃)	1	1	yes	characteristic
24	C₃×C₉×C₃₇	C₃³	1	1	yes	characteristic
25	C₃²⋊(C₃₇⋊C₃)	—	3	3	no	—
26	#11758608	—	3	3	no	—
27	#11758608	—	3	3	no	—
28	#11758606	—	9	3	no	—
29	#11758605	—	9	3	no	—
30	C₉×C₃₇⋊C₃	—	9	3	no	—
31	#10624845	—	37	1	no	maximal, Sylow
32	C₃²×C₃₇	—	3	1	no	—
33	C₃²×C₃₇	—	9	1	no	—
34	C₃²×C₃₇	C₃×H(3,𝔽₃)	1	1	yes	characteristic
35	C₃×C₃₇⋊C₃	—	9	3	no	—
36	C₉×C₃₇	—	3	1	no	—
37	C₉×C₃₇	—	3	1	no	—
38	C₉×C₃₇	—	9	3	no	—
39	C₉×C₃₇	#732	1	1	yes	characteristic
40	C₉×C₃₇	#732	1	1	yes	characteristic
41	C₉×C₃₇	#732	1	1	yes	characteristic
42	#66524	—	9	3	no	—
43	#66524	—	9	3	no	—
44	#6908	—	37	3	no	—
45	#6887	C₃₇⋊C₃	1	1	yes	characteristic, complemented
46	#6883	—	37	3	no	—
47	#6882	—	37	3	no	—
48	#6881	—	37	3	no	—
49	C₃×C₃₇	—	3	1	no	—
50	C₃×C₃₇	—	9	1	no	—
51	C₃×C₃₇	C₃×#732	1	1	yes	characteristic
52	C₃₇⋊C₃	—	27	3	no	—
53	#737	—	111	3	no	—
54	C₃²×C₉	C₃×C₃₇⋊C₃	1	1	yes	characteristic
55	#733	—	37	3	no	—
56	#733	—	37	3	no	—
57	#732	—	37	3	no	—
58	#731	—	111	3	no	—
59	AΣL(1,𝔽₂₇)	—	111	3	no	—
60	C₉⋊C₉	—	3	1	no	—
61	C₉⋊C₉	C₃×C₃₇⋊C₃	1	3	yes	—
62	C₃²⋊C₉	—	3	1	no	—
63	C₉²	—	3	1	no	—
64	C₃₇	#10624845	1	1	yes	characteristic, complemented, Sylow
65	C₃³	C₃²⋊(C₃₇⋊C₃)	1	1	yes	characteristic
66	3₋¹⁺²	—	333	3	no	—
67	3₋¹⁺²	—	333	3	no	—
68	3₋¹⁺²	—	111	3	no	—
69	3₋¹⁺²	—	111	3	no	—
70	H(3,𝔽₃)	—	111	3	no	—
71	C₃×C₉	—	3	1	no	—
72	C₃×C₉	—	3	1	no	—
73	C₃×C₉	—	3	1	no	—
74	C₃×C₉	—	3	3	no	—
75	C₃×C₉	—	333	3	no	—
76	C₃×C₉	C₃²⋊(C₃₇⋊C₃)	1	1	yes	characteristic
77	C₃×C₉	C₃²⋊(C₃₇⋊C₃)	1	1	yes	characteristic
78	C₃×C₉	C₃²×C₃₇⋊C₃	1	1	yes	characteristic
79	C₃²	—	3	1	no	—
80	C₃²	—	9	1	no	—
81	C₃²	—	333	3	no	—
82	C₃²	C₃×C₃²⋊(C₃₇⋊C₃)	1	1	yes	characteristic
83	C₉	—	3	1	no	—
84	C₉	—	3	1	no	—
85	C₉	—	9	3	no	—
86	C₉	—	333	3	no	—
87	C₉	—	333	3	no	—
88	C₉	#15688519	1	1	yes	characteristic
89	C₉	#15688519	1	1	yes	characteristic
90	C₉	#15688519	1	1	yes	characteristic
91	C₃	—	3	1	no	—
92	C₃	—	9	1	no	—
93	C₃	—	999	3	no	—
94	C₃	C₃×#15688519	1	1	yes	characteristic

	subgroup	G/H
center	C₉	#15688519
commutator, cosocle	C₃×C₉×C₃₇	C₃³
Frattini	C₃×C₉	C₃²×C₃₇⋊C₃
Fitting	C₃₇×#6887	C₃
socle	C₃×C₃₇	C₃×#732
hypercenter	#6887	C₃₇⋊C₃
hypocenter	C₃₇	#10624845
LCS	G ▷ C₃×C₉×C₃₇ ▷ C₃²×C₃₇ ▷ C₃×C₃₇ ▷ C₃₇
UCS	C₁ ◁ C₉ ◁ C₃×C₉ ◁ C₃²×C₉ ◁ #6887

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. C₃₇×#6887	C₃	semidirect
45. #6887	C₃₇⋊C₃	semidirect
64. C₃₇	#10624845	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
C₃₇×#6887	3
#10624845	37