Subgroups — #41007356

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	#22582510	C₃	1	2	yes	maximal, complemented
3	#22582507	C₃	1	1	yes	maximal, characteristic, complemented
4	#22582503	C₃	1	3	yes	maximal, complemented
5	#22582502	C₃	1	3	yes	maximal, complemented
6	C₃₇×#6887	C₃	1	1	yes	maximal, characteristic, complemented
7	#15688531	—	3	3	no	—
8	#15688521	C₃²	1	6	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	—	3	1	no	—
15	C₃₇×C₃²⋊C₉	—	3	1	no	—
16	C₉²×C₃₇	C₃²	1	1	yes	characteristic
17	C₉²×C₃₇	C₃²	1	1	yes	characteristic
18	C₉²×C₃₇	C₃²	1	1	yes	characteristic
19	C₃³×C₃₇	H(3,𝔽₃)	1	1	yes	characteristic
20	C₃×C₉×C₃₇	—	3	3	no	—
21	C₃×C₉×C₃₇	—	3	1	no	—
22	C₃×C₉×C₃₇	—	3	1	no	—
23	C₃×C₉×C₃₇	—	3	1	no	—
24	C₃×C₉×C₃₇	H(3,𝔽₃)	1	1	yes	characteristic
25	C₃×C₉×C₃₇	H(3,𝔽₃)	1	1	yes	characteristic
26	C₃×C₉×C₃₇	C₃³	1	1	yes	characteristic
27	C₃²⋊(C₃₇⋊C₃)	—	3	3	no	—
28	#11758608	—	3	6	no	—
29	#11758606	—	9	3	no	—
30	#11758605	—	9	3	no	—
31	C₉×C₃₇⋊C₃	—	9	3	no	—
32	#10624841	—	37	1	no	maximal, Sylow
33	C₃²×C₃₇	—	3	1	no	—
34	C₃²×C₃₇	—	9	1	no	—
35	C₃²×C₃₇	C₃×H(3,𝔽₃)	1	1	yes	characteristic
36	C₃×C₃₇⋊C₃	—	9	3	no	—
37	C₉×C₃₇	—	3	1	no	—
38	C₉×C₃₇	—	3	1	no	—
39	C₉×C₃₇	—	9	1	no	—
40	C₉×C₃₇	—	9	1	no	—
41	C₉×C₃₇	—	9	1	no	—
42	C₉×C₃₇	#732	1	3	yes	—
43	#66524	—	9	6	no	—
44	#6908	—	37	3	no	—
45	#6887	C₃₇⋊C₃	1	1	yes	characteristic, complemented
46	#6880	—	37	2	no	—
47	#6879	—	37	1	no	—
48	#6878	—	37	3	no	—
49	#6878	—	37	3	no	—
50	C₃×C₃₇	—	3	1	no	—
51	C₃×C₃₇	—	9	1	no	—
52	C₃×C₃₇	C₃×#732	1	1	yes	characteristic
53	C₃₇⋊C₃	—	27	3	no	—
54	#737	—	111	3	no	—
55	C₃²×C₉	C₃×C₃₇⋊C₃	1	1	yes	characteristic
56	#733	—	37	6	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	—	3	1	no	—
62	C₃²⋊C₉	—	3	1	no	—
63	C₉²	C₃×C₃₇⋊C₃	1	1	yes	characteristic
64	C₉²	C₃×C₃₇⋊C₃	1	1	yes	characteristic
65	C₉²	C₃×C₃₇⋊C₃	1	1	yes	characteristic
66	C₃₇	#10624841	1	1	yes	characteristic, complemented, Sylow
67	C₃³	C₃²⋊(C₃₇⋊C₃)	1	1	yes	characteristic
68	3₋¹⁺²	—	333	3	no	—
69	3₋¹⁺²	—	333	3	no	—
70	3₋¹⁺²	—	111	6	no	—
71	H(3,𝔽₃)	—	111	3	no	—
72	C₃×C₉	—	3	3	no	—
73	C₃×C₉	—	3	1	no	—
74	C₃×C₉	—	3	1	no	—
75	C₃×C₉	—	3	1	no	—
76	C₃×C₉	—	333	3	no	—
77	C₃×C₉	C₃²⋊(C₃₇⋊C₃)	1	1	yes	characteristic
78	C₃×C₉	C₃²⋊(C₃₇⋊C₃)	1	1	yes	characteristic
79	C₃×C₉	C₃²×C₃₇⋊C₃	1	1	yes	characteristic
80	C₃²	—	3	1	no	—
81	C₃²	—	9	1	no	—
82	C₃²	—	333	3	no	—
83	C₃²	C₃×C₃²⋊(C₃₇⋊C₃)	1	1	yes	characteristic
84	C₉	—	3	1	no	—
85	C₉	—	3	1	no	—
86	C₉	—	9	1	no	—
87	C₉	—	9	1	no	—
88	C₉	—	9	1	no	—
89	C₉	—	333	6	no	—
90	C₉	#15688519	1	3	yes	—
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	C₃	C₃×#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	#10624841
LCS	G ▷ C₃×C₉×C₃₇ ▷ C₃²×C₃₇ ▷ C₃×C₃₇ ▷ C₃₇
UCS	C₁ ◁ C₃ ◁ 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. #22582510	C₃	semidirect
3. #22582507	C₃	semidirect
4. #22582503	C₃	semidirect
5. #22582502	C₃	semidirect
6. C₃₇×#6887	C₃	semidirect
45. #6887	C₃₇⋊C₃	semidirect
66. C₃₇	#10624841	semidirect

Maximal subgroups

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

subgroup	index
#22582633	3
#22582510	3
#22582507	3
#22582503	3
#22582502	3
C₃₇×#6887	3
#10624841	37