Subgroups — #41007340

Order 26973.

198 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×#15688521	C₃	1	2	yes	maximal, complemented
2	C₃×#15688519	C₃	1	1	yes	maximal, characteristic
3	#22582502	C₃	1	9	yes	maximal, complemented
4	C₃₇×#6887	C₃	1	1	yes	maximal, characteristic, complemented
5	C₃×#11758608	—	3	2	no	—
6	C₃×C₃²⋊(C₃₇⋊C₃)	—	3	1	no	—
7	#15688521	—	3	4	no	—
8	#15688521	C₃²	1	6	yes	complemented
9	#15688519	—	3	2	no	—
10	#15688519	C₃²	1	3	yes	—
11	C₃²×C₉×C₃₇	C₃²	1	1	yes	characteristic
12	C₃₇×C₉⋊C₉	—	3	2	no	—
13	C₃₇×C₃²⋊C₉	—	3	1	no	—
14	C₉²×C₃₇	C₃²	1	3	yes	complemented
15	C₃³×C₃₇	H(3,𝔽₃)	1	1	yes	characteristic
16	C₃²×C₃₇⋊C₃	—	9	1	no	—
17	C₃×C₉×C₃₇	—	3	2	no	—
18	C₃×C₉×C₃₇	—	3	1	no	—
19	C₃×C₉×C₃₇	—	3	3	no	—
20	C₃×C₉×C₃₇	H(3,𝔽₃)	1	2	yes	—
21	C₃×C₉×C₃₇	C₃³	1	1	yes	characteristic
22	C₃²⋊(C₃₇⋊C₃)	—	3	3	no	—
23	C₃²⋊(C₃₇⋊C₃)	—	9	2	no	—
24	#11758608	—	3	6	no	—
25	#11758608	—	9	2	no	—
26	#11758608	—	9	2	no	—
27	C₃×#66524	—	9	2	no	—
28	#10624838	—	37	1	no	maximal, Sylow
29	C₃²×C₃₇	—	9	1	no	—
30	C₃²×C₃₇	#732	1	2	yes	—
31	C₃²×C₃₇	#732	1	1	yes	characteristic
32	C₃²×C₃₇	C₃×H(3,𝔽₃)	1	1	yes	characteristic
33	C₃×C₃₇⋊C₃	—	9	3	no	—
34	C₃×C₃₇⋊C₃	—	27	3	no	—
35	C₉×C₃₇	—	3	2	no	—
36	C₉×C₃₇	—	3	1	no	—
37	C₉×C₃₇	—	9	3	no	—
38	#66524	—	9	6	no	—
39	C₃×#733	—	37	2	no	—
40	C₃×#732	—	37	1	no	—
41	#6887	C₃₇⋊C₃	1	1	yes	characteristic, complemented
42	#6878	—	37	9	no	—
43	C₃×C₃₇	—	3	1	no	—
44	C₃×C₃₇	—	3	1	no	—
45	C₃×C₃₇	—	3	2	no	—
46	C₃×C₃₇	C₃×#732	1	1	yes	characteristic
47	C₃₇⋊C₃	—	27	3	no	—
48	C₃×3₋¹⁺²	—	111	2	no	—
49	C₃×H(3,𝔽₃)	—	111	1	no	—
50	C₃²×C₉	C₃×C₃₇⋊C₃	1	1	yes	characteristic
51	#733	—	37	6	no	—
52	#733	—	111	4	no	—
53	#732	—	37	3	no	—
54	#732	—	111	2	no	—
55	C₉⋊C₉	—	3	2	no	—
56	C₃²⋊C₉	—	3	1	no	—
57	C₉²	C₃×C₃₇⋊C₃	1	3	yes	complemented
58	C₃₇	#10624838	1	1	yes	characteristic, complemented, Sylow
59	C₃³	—	333	1	no	—
60	C₃³	C₃²⋊(C₃₇⋊C₃)	1	1	yes	characteristic
61	3₋¹⁺²	—	333	2	no	—
62	3₋¹⁺²	—	333	2	no	—
63	3₋¹⁺²	—	111	6	no	—
64	H(3,𝔽₃)	—	333	2	no	—
65	H(3,𝔽₃)	—	111	3	no	—
66	C₃×C₉	—	3	2	no	—
67	C₃×C₉	—	3	1	no	—
68	C₃×C₉	—	3	3	no	—
69	C₃×C₉	—	333	2	no	—
70	C₃×C₉	C₃²⋊(C₃₇⋊C₃)	1	2	yes	—
71	C₃×C₉	C₃²×C₃₇⋊C₃	1	1	yes	characteristic
72	C₃²	—	9	1	no	—
73	C₃²	—	333	3	no	—
74	C₃²	—	999	3	no	—
75	C₃²	#15688519	1	2	yes	—
76	C₃²	#15688519	1	1	yes	characteristic
77	C₃²	C₃×C₃²⋊(C₃₇⋊C₃)	1	1	yes	characteristic
78	C₉	—	3	2	no	—
79	C₉	—	3	1	no	—
80	C₉	—	9	3	no	—
81	C₉	—	333	6	no	—
82	C₃	—	3	1	no	—
83	C₃	—	3	1	no	—
84	C₃	—	3	2	no	—
85	C₃	—	999	3	no	—
86	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	C₃₇×#6887	C₃
socle	C₃×C₃₇	C₃×#732
prime radical	C₃×#15688519	C₃
hypercenter	#6887	C₃₇⋊C₃
hypocenter	C₃₇	#10624838
LCS	G ▷ C₃×C₉×C₃₇ ▷ C₃²×C₃₇ ▷ C₃×C₃₇ ▷ C₃₇
UCS	C₁ ◁ C₃ ◁ C₃³ ◁ C₃²×C₉ ◁ #6887

Complements

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

N	H ≅ G/N	type
1. C₃×#15688521	C₃	semidirect
3. #22582502	C₃	semidirect
4. C₃₇×#6887	C₃	semidirect
8. #15688521	C₃²	semidirect
14. C₉²×C₃₇	C₃²	semidirect
41. #6887	C₃₇⋊C₃	semidirect
57. C₉²	C₃×C₃₇⋊C₃	semidirect
58. C₃₇	#10624838	semidirect

Maximal subgroups

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

subgroup	index
C₃×#15688521	3
C₃×#15688519	3
#22582502	3
C₃₇×#6887	3
#10624838	37