Subgroups — #41007347

Order 26973.

186 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#733×C₃₇⋊C₃	C₃	1	3	yes	maximal
2	#22582536	C₃	1	1	yes	maximal, characteristic
3	#22582510	C₃	1	2	yes	maximal, complemented
4	#22582509	C₃	1	3	yes	maximal, complemented
5	#22582508	C₃	1	3	yes	maximal, complemented
6	C₃₇×#6880	C₃	1	1	yes	maximal, characteristic, complemented
7	3₋¹⁺²×C₃₇⋊C₃	—	3	3	no	—
8	C₃×C₉×C₃₇⋊C₃	C₃²	1	1	yes	characteristic
9	#15688521	C₃²	1	6	yes	—
10	#15688520	—	3	6	no	—
11	#15688504	—	3	2	no	—
12	#15688502	—	3	1	no	—
13	C₉×#66524	C₃²	1	1	yes	characteristic
14	C₉×#66524	C₃²	1	1	yes	characteristic
15	C₃₇×#733	C₃²	1	3	yes	—
16	C₉²×C₃₇	C₃²	1	1	yes	characteristic
17	C₃²×C₃₇⋊C₃	H(3,𝔽₃)	1	1	yes	characteristic
18	3₋¹⁺²×C₃₇	—	3	3	no	—
19	C₃×C₉×C₃₇	—	3	1	no	—
20	C₃×C₉×C₃₇	C₃³	1	1	yes	characteristic
21	#11758608	—	3	6	no	—
22	C₃×#66524	—	3	1	no	—
23	C₃×#66524	—	3	1	no	—
24	C₃×#66524	H(3,𝔽₃)	1	2	yes	—
25	#11758606	—	9	3	no	—
26	#11758605	—	9	3	no	—
27	C₉×C₃₇⋊C₃	—	3	3	no	—
28	C₉×C₃₇⋊C₃	—	9	3	no	—
29	#10624840	—	37	1	no	maximal, Sylow
30	C₃²×C₃₇	C₃×H(3,𝔽₃)	1	1	yes	characteristic
31	C₃×C₃₇⋊C₃	—	9	1	no	—
32	C₃×C₃₇⋊C₃	#732	1	3	yes	—
33	C₉×C₃₇	—	3	1	no	—
34	C₉×C₃₇	—	9	1	no	—
35	C₉×C₃₇	—	9	3	no	—
36	#66524	—	3	2	no	—
37	#66524	—	9	1	no	—
38	#66524	—	9	1	no	—
39	#66524	—	9	6	no	—
40	C₃×#733	—	37	3	no	—
41	#6887	—	37	1	no	—
42	#6880	—	37	3	no	—
43	#6880	—	37	3	no	—
44	#6880	—	37	2	no	—
45	#6880	C₃₇⋊C₃	1	1	yes	characteristic, complemented
46	C₃×C₃₇	—	3	1	no	—
47	C₃×C₃₇	C₃×#732	1	1	yes	characteristic
48	C₃₇⋊C₃	—	3	3	no	—
49	C₃×3₋¹⁺²	—	111	3	no	—
50	C₃²×C₉	—	37	1	no	—
51	#733	—	37	6	no	—
52	#733	—	111	6	no	—
53	#733	C₃×C₃₇⋊C₃	1	3	yes	—
54	C₉⋊C₉	—	111	2	no	—
55	C₃²⋊C₉	—	111	1	no	—
56	C₉²	—	37	1	no	—
57	C₉²	—	37	1	no	—
58	C₉²	C₃×C₃₇⋊C₃	1	1	yes	characteristic
59	C₃₇	#10624840	1	1	yes	characteristic, complemented, Sylow
60	C₃³	—	37	1	no	—
61	3₋¹⁺²	—	3	3	no	—
62	3₋¹⁺²	—	333	3	no	—
63	3₋¹⁺²	—	333	3	no	—
64	3₋¹⁺²	—	111	6	no	—
65	C₃×C₉	—	3	1	no	—
66	C₃×C₉	—	37	2	no	—
67	C₃×C₉	—	333	3	no	—
68	C₃×C₉	—	111	3	no	—
69	C₃×C₉	—	111	1	no	—
70	C₃×C₉	—	111	1	no	—
71	C₃×C₉	C₃²×C₃₇⋊C₃	1	1	yes	characteristic
72	C₃²	—	37	3	no	—
73	C₃²	—	333	1	no	—
74	C₃²	H(3,𝔽₃)×C₃₇⋊C₃	1	1	yes	characteristic
75	C₉	—	3	1	no	—
76	C₉	—	9	1	no	—
77	C₉	—	9	3	no	—
78	C₉	—	333	1	no	—
79	C₉	—	333	1	no	—
80	C₉	—	333	6	no	—
81	C₉	—	111	2	no	—
82	C₃	—	3	1	no	—
83	C₃	—	111	3	no	—
84	C₃	#732×C₃₇⋊C₃	1	1	yes	characteristic

	subgroup	G/H
center	C₃	#732×C₃₇⋊C₃
commutator, cosocle	C₃×C₉×C₃₇	C₃³
Frattini	C₃×C₉	C₃²×C₃₇⋊C₃
Fitting	C₃₇×#6880	C₃
socle	C₃×C₃₇	C₃×#732
prime radical	C₃²×C₃₇⋊C₃	H(3,𝔽₃)
hypercenter	#6880	C₃₇⋊C₃
hypocenter	C₃₇	#10624840
LCS	G ▷ C₃×C₉×C₃₇ ▷ C₃²×C₃₇ ▷ C₃×C₃₇ ▷ C₃₇
UCS	C₁ ◁ C₃ ◁ C₃² ◁ C₃×C₉ ◁ #6880

Complements

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

N	H ≅ G/N	type
3. #22582510	C₃	semidirect
4. #22582509	C₃	semidirect
5. #22582508	C₃	semidirect
6. C₃₇×#6880	C₃	semidirect
45. #6880	C₃₇⋊C₃	semidirect
59. C₃₇	#10624840	semidirect

Maximal subgroups

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

subgroup	index
#733×C₃₇⋊C₃	3
#22582536	3
#22582510	3
#22582509	3
#22582508	3
C₃₇×#6880	3
#10624840	37