Subgroups — #41008397

Order 26973.

110 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22582693	C₃	1	2	yes	maximal
2	#22582693	C₃	1	1	yes	maximal, characteristic
3	C₉⋊C₉×C₃₇⋊C₃	C₃	1	1	yes	maximal, characteristic
4	C₃²⋊#66526	—	3	2	no	—
5	C₃²⋊#66526	—	3	1	no	—
6	C₃×C₉×C₃₇⋊C₃	—	3	1	no	—
7	C₃×C₉×C₃₇⋊C₃	C₃²	1	1	yes	characteristic
8	#15688506	C₉	1	1	yes	characteristic, complemented
9	#15688505	C₉	1	1	yes	characteristic, complemented
10	#15688504	—	3	2	no	—
11	C₃₇×C₉⋊C₉	C₉	1	1	yes	characteristic, complemented
12	C₃²×C₃₇⋊C₃	H(3,𝔽₃)	1	1	yes	characteristic
13	C₃×#66526	—	9	2	no	—
14	C₃×#66526	—	9	1	no	—
15	C₃×C₉×C₃₇	—	3	1	no	—
16	C₃×C₉×C₃₇	C₃×C₉	1	1	yes	characteristic
17	C₃×#66524	—	3	1	no	—
18	C₃×#66524	—	3	1	no	—
19	C₃×#66524	3₋¹⁺²	1	2	yes	—
20	C₉×C₃₇⋊C₃	—	3	1	no	—
21	C₉×C₃₇⋊C₃	—	3	2	no	—
22	C₉×C₃₇⋊C₃	—	9	1	no	—
23	C₉×C₃₇⋊C₃	—	9	1	no	—
24	C₉×C₃₇⋊C₃	—	9	1	no	—
25	#10624521	—	37	1	no	maximal, Sylow
26	C₃²×C₃₇	C₃²⋊C₉	1	1	yes	characteristic
27	C₃×C₃₇⋊C₃	—	3	1	no	—
28	C₃×C₃₇⋊C₃	—	3	1	no	—
29	C₃×C₃₇⋊C₃	—	3	1	no	—
30	C₃×C₃₇⋊C₃	#732	1	1	yes	characteristic
31	C₃×C₃₇⋊C₃	#733	1	2	yes	—
32	#66526	—	27	2	no	—
33	#66526	—	27	1	no	—
34	C₉×C₃₇	—	3	1	no	—
35	C₉×C₃₇	—	9	1	no	—
36	#66524	—	3	2	no	—
37	#66524	—	9	1	no	—
38	#66524	—	9	1	no	—
39	C₃×C₉⋊C₉	—	37	1	no	—
40	#6868	—	37	2	no	—
41	#6868	—	37	1	no	—
42	C₃×C₃₇	—	3	1	no	—
43	C₃×C₃₇	#6868	1	1	yes	characteristic
44	C₃₇⋊C₃	—	3	2	no	—
45	C₃₇⋊C₃	#6881	1	1	yes	characteristic
46	C₃₇⋊C₃	#6882	1	1	yes	characteristic
47	C₃₇⋊C₃	#6883	1	1	yes	characteristic
48	C₃²×C₉	—	37	1	no	—
49	C₃²×C₉	—	111	1	no	—
50	C₉⋊C₉	—	37	1	no	—
51	C₉⋊C₉	—	37	1	no	—
52	C₉⋊C₉	—	111	2	no	—
53	C₉⋊C₉	#66526	1	1	yes	characteristic, complemented
54	C₃²⋊C₉	—	111	2	no	—
55	C₃²⋊C₉	—	111	1	no	—
56	C₃₇	#10624521	1	1	yes	characteristic, complemented, Sylow
57	C₃³	—	37	1	no	—
58	C₃×C₉	—	3	1	no	—
59	C₃×C₉	—	37	2	no	—
60	C₃×C₉	—	333	1	no	—
61	C₃×C₉	—	333	1	no	—
62	C₃×C₉	—	333	1	no	—
63	C₃×C₉	—	333	2	no	—
64	C₃×C₉	—	333	1	no	—
65	C₃×C₉	—	111	1	no	—
66	C₃×C₉	—	111	2	no	—
67	C₃×C₉	—	111	1	no	—
68	C₃×C₉	—	111	1	no	—
69	C₃×C₉	C₃×#66526	1	1	yes	characteristic
70	C₃²	—	37	1	no	—
71	C₃²	—	37	2	no	—
72	C₃²	—	111	1	no	—
73	C₃²	—	111	1	no	—
74	C₃²	—	111	1	no	—
75	C₃²	C₃²⋊#66526	1	1	yes	characteristic
76	C₉	—	3	1	no	—
77	C₉	—	9	1	no	—
78	C₉	—	333	1	no	—
79	C₉	—	333	1	no	—
80	C₉	—	111	2	no	—
81	C₉	—	999	2	no	—
82	C₉	—	999	1	no	—
83	C₃	—	3	1	no	—
84	C₃	—	37	1	no	—
85	C₃	—	37	1	no	—
86	C₃	—	37	1	no	—
87	C₃	—	111	2	no	—
88	C₃	#22582693	1	1	yes	characteristic

	subgroup	G/H
center	C₃	#22582693
commutator	C₃×C₉×C₃₇	C₃×C₉
Frattini	C₃×C₉	C₃×#66526
Fitting	C₃₇×C₉⋊C₉	C₉
socle	C₃×C₃₇	#6868
prime radical	C₃²×C₃₇⋊C₃	H(3,𝔽₃)
cosocle	C₃×C₉×C₃₇⋊C₃	C₃²
hypercenter	C₉⋊C₉	#66526
hypocenter	C₃₇	#10624521
LCS	G ▷ C₃×C₉×C₃₇ ▷ C₃²×C₃₇ ▷ C₃×C₃₇ ▷ C₃₇
UCS	C₁ ◁ C₃ ◁ C₃² ◁ C₃×C₉ ◁ C₉⋊C₉

Complements

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

N	H ≅ G/N	type
8. #15688506	C₉	semidirect
9. #15688505	C₉	semidirect
11. C₃₇×C₉⋊C₉	C₉	semidirect
53. C₉⋊C₉	#66526	semidirect
56. C₃₇	#10624521	semidirect

Maximal subgroups

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

subgroup	index
#22582693	3
C₉⋊C₉×C₃₇⋊C₃	3
#10624521	37