Subgroups — #41007334

Order 26973.

226 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22582631	C₃	1	3	yes	maximal, complemented
2	C₉²×C₃₇⋊C₃	C₃	1	1	yes	maximal, characteristic, complemented
3	#22582510	C₃	1	2	yes	maximal, complemented
4	#22582501	C₃	1	6	yes	maximal, complemented
5	C₃₇×#6879	C₃	1	1	yes	maximal, characteristic, complemented
6	#15688530	—	3	3	no	—
7	C₃×C₉×C₃₇⋊C₃	—	3	1	no	—
8	C₃×C₉×C₃₇⋊C₃	C₃²	1	1	yes	characteristic
9	#15688521	C₃²	1	6	yes	—
10	#15688515	—	3	3	no	—
11	#15688512	—	3	3	no	—
12	C₉×#66524	—	3	1	no	—
13	C₉×#66524	—	3	1	no	—
14	C₉×#66524	C₃²	1	2	yes	—
15	C₃₇×#732	C₃²	1	3	yes	—
16	C₉²×C₃₇	C₃²	1	1	yes	characteristic
17	C₃²×C₃₇⋊C₃	H(3,𝔽₃)	1	1	yes	characteristic
18	H(3,𝔽₃)×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	#11758608	—	9	6	no	—
23	C₃×#66524	—	3	2	no	—
24	C₃×#66524	—	9	3	no	—
25	C₃×#66524	H(3,𝔽₃)	1	1	yes	characteristic
26	C₃×#66524	H(3,𝔽₃)	1	1	yes	characteristic
27	C₉×C₃₇⋊C₃	—	3	9	no	—
28	C₉×C₃₇⋊C₃	—	3	1	no	—
29	C₉×C₃₇⋊C₃	—	3	2	no	—
30	#10624837	—	37	1	no	maximal, Sylow
31	C₃²×C₃₇	—	9	3	no	—
32	C₃²×C₃₇	C₃×H(3,𝔽₃)	1	1	yes	characteristic
33	C₃×C₃₇⋊C₃	—	3	1	no	—
34	C₃×C₃₇⋊C₃	—	3	3	no	—
35	C₉×C₃₇	—	3	3	no	—
36	C₉×C₃₇	—	3	1	no	—
37	#66524	—	3	6	no	—
38	#66524	—	3	1	no	—
39	#66524	—	9	6	no	—
40	#66524	#732	1	1	yes	characteristic
41	#66524	#732	1	2	yes	—
42	#6908	—	37	3	no	—
43	C₃×C₉²	—	37	1	no	—
44	#6880	—	37	2	no	—
45	#6879	C₃₇⋊C₃	1	1	yes	characteristic, complemented
46	#6878	—	37	6	no	—
47	C₃×C₃₇	—	3	1	no	—
48	C₃×C₃₇	—	27	3	no	—
49	C₃×C₃₇	C₃×#732	1	1	yes	characteristic
50	C₃₇⋊C₃	—	3	3	no	—
51	#737	—	111	3	no	—
52	C₃²×C₉	—	37	1	no	—
53	C₃²×C₉	—	111	1	no	—
54	#733	—	37	6	no	—
55	#732	C₃×C₃₇⋊C₃	1	3	yes	—
56	#731	—	111	3	no	—
57	AΣL(1,𝔽₂₇)	—	111	3	no	—
58	C₉²	—	37	2	no	—
59	C₉²	—	111	1	no	—
60	C₉²	—	111	1	no	—
61	C₉²	C₃×C₃₇⋊C₃	1	1	yes	characteristic
62	C₃₇	#10624837	1	1	yes	characteristic, complemented, Sylow
63	C₃³	—	37	1	no	—
64	3₋¹⁺²	—	333	6	no	—
65	3₋¹⁺²	—	111	6	no	—
66	H(3,𝔽₃)	—	3	3	no	—
67	C₃×C₉	—	3	1	no	—
68	C₃×C₉	—	37	1	no	—
69	C₃×C₉	—	37	1	no	—
70	C₃×C₉	—	333	3	no	—
71	C₃×C₉	—	111	1	no	—
72	C₃×C₉	—	111	2	no	—
73	C₃×C₉	—	111	9	no	—
74	C₃×C₉	—	111	2	no	—
75	C₃×C₉	C₃²×C₃₇⋊C₃	1	1	yes	characteristic
76	C₃²	—	9	3	no	—
77	C₃²	—	111	3	no	—
78	C₃²	—	111	1	no	—
79	C₃²	H(3,𝔽₃)×C₃₇⋊C₃	1	1	yes	characteristic
80	C₉	—	3	3	no	—
81	C₉	—	3	1	no	—
82	C₉	—	37	1	no	—
83	C₉	—	37	2	no	—
84	C₉	—	333	6	no	—
85	C₉	—	111	6	no	—
86	C₉	—	111	1	no	—
87	C₃	—	3	1	no	—
88	C₃	—	27	3	no	—
89	C₃	—	111	3	no	—
90	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×#6879	C₃
socle	C₃×C₃₇	C₃×#732
hypercenter	#6879	C₃₇⋊C₃
hypocenter	C₃₇	#10624837
LCS	G ▷ C₃×C₉×C₃₇ ▷ C₃²×C₃₇ ▷ C₃×C₃₇ ▷ C₃₇
UCS	C₁ ◁ C₃ ◁ C₃² ◁ C₃×C₉ ◁ #6879

Complements

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

N	H ≅ G/N	type
1. #22582631	C₃	semidirect
2. C₉²×C₃₇⋊C₃	C₃	semidirect
3. #22582510	C₃	semidirect
4. #22582501	C₃	semidirect
5. C₃₇×#6879	C₃	semidirect
45. #6879	C₃₇⋊C₃	semidirect
62. C₃₇	#10624837	semidirect

Maximal subgroups

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

subgroup	index
#22582631	3
C₉²×C₃₇⋊C₃	3
#22582510	3
#22582501	3
C₃₇×#6879	3
#10624837	37