Subgroups — #41007215

Order 26973.

208 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22582578	C₃	1	1	yes	maximal, characteristic, complemented
2	#22582564	C₃	1	1	yes	maximal, characteristic
3	C₃×#15688505	C₃	1	1	yes	maximal, characteristic, complemented
4	#22582481	C₃	1	3	yes	maximal, complemented
5	#22582478	C₃	1	3	yes	maximal, complemented
6	#22582475	C₃	1	3	yes	maximal
7	C₃₇×#6895	C₃	1	1	yes	maximal, characteristic, complemented
8	C₃×#11758608	C₃²	1	1	yes	characteristic
9	C₃×#11758608	C₃²	1	1	yes	characteristic
10	C₃×C₃²⋊(C₃₇⋊C₃)	C₃²	1	1	yes	characteristic
11	C₃²×#66524	C₃²	1	3	yes	—
12	#15688505	—	3	3	no	—
13	#15688504	—	3	3	no	—
14	C₃²⋊#66524	C₃²	1	3	yes	—
15	#15688502	—	3	3	no	—
16	C₃²×C₉×C₃₇	C₃²	1	1	yes	characteristic
17	C₃₇×C₉⋊C₉	—	3	3	no	—
18	C₃₇×C₃²⋊C₉	C₃²	1	3	yes	—
19	C₃³×C₃₇	C₃³	1	1	yes	characteristic
20	C₃²×C₃₇⋊C₃	—	3	1	no	—
21	C₃×C₉×C₃₇	—	3	3	no	—
22	C₃×C₉×C₃₇	—	3	3	no	—
23	C₃×C₉×C₃₇	H(3,𝔽₃)	1	3	yes	—
24	C₃²⋊(C₃₇⋊C₃)	—	3	3	no	—
25	#11758608	—	3	3	no	—
26	#11758608	—	3	3	no	—
27	C₃×#66524	—	3	1	no	—
28	C₃×#66524	—	3	1	no	—
29	C₃×#66524	—	3	3	no	—
30	C₃×#66524	—	3	3	no	—
31	C₃×#66524	—	9	3	no	—
32	#10624817	—	37	1	no	maximal, Sylow
33	C₃²×C₃₇	—	3	3	no	—
34	C₃²×C₃₇	C₃×H(3,𝔽₃)	1	1	yes	characteristic
35	C₃²×C₃₇	#737	1	3	yes	—
36	C₃×C₃₇⋊C₃	—	9	3	no	—
37	C₃×C₃₇⋊C₃	—	9	1	no	—
38	C₉×C₃₇	—	3	3	no	—
39	C₉×C₃₇	—	9	3	no	—
40	#66524	—	9	3	no	—
41	#66524	—	9	3	no	—
42	#66524	—	9	1	no	—
43	#66524	—	9	1	no	—
44	#6895	—	37	1	no	—
45	#6895	C₃₇⋊C₃	1	1	yes	characteristic, complemented
46	#6893	—	37	1	no	—
47	C₃×C₉⋊C₉	—	37	1	no	—
48	3₋¹⁺²⋊C₉	—	37	3	no	—
49	3₋¹⁺²⋊C₉	—	37	3	no	—
50	H(3,𝔽₃)⋊C₉	—	37	3	no	—
51	C₃×C₃₇	—	3	3	no	—
52	C₃×C₃₇	#6893	1	1	yes	characteristic
53	C₃×C₃₇	#6913	1	3	yes	—
54	C₃₇⋊C₃	—	27	1	no	—
55	C₃×3₋¹⁺²	—	37	1	no	—
56	C₃×3₋¹⁺²	—	37	1	no	—
57	C₃×H(3,𝔽₃)	—	37	1	no	—
58	C₃²×C₉	—	37	3	no	—
59	C₃²×C₉	C₃×C₃₇⋊C₃	1	1	yes	characteristic
60	C₉⋊C₉	—	3	3	no	—
61	C₉⋊C₉	—	111	3	no	—
62	C₉⋊C₉	—	111	3	no	—
63	C₃²⋊C₉	—	37	3	no	—
64	C₃²⋊C₉	—	111	3	no	—
65	C₃²⋊C₉	C₃×C₃₇⋊C₃	1	3	yes	—
66	C₃₇	#10624817	1	1	yes	characteristic, complemented, Sylow
67	C₃³	—	111	1	no	—
68	C₃³	C₃²×C₃₇⋊C₃	1	1	yes	characteristic
69	3₋¹⁺²	—	111	3	no	—
70	3₋¹⁺²	—	111	3	no	—
71	H(3,𝔽₃)	—	111	3	no	—
72	C₃×C₉	—	3	3	no	—
73	C₃×C₉	—	3	3	no	—
74	C₃×C₉	—	333	3	no	—
75	C₃×C₉	—	111	3	no	—
76	C₃×C₉	—	111	1	no	—
77	C₃×C₉	—	111	1	no	—
78	C₃×C₉	—	111	3	no	—
79	C₃×C₉	C₃²⋊(C₃₇⋊C₃)	1	3	yes	—
80	C₃²	—	3	3	no	—
81	C₃²	—	333	1	no	—
82	C₃²	—	333	3	no	—
83	C₃²	C₃×C₃²⋊(C₃₇⋊C₃)	1	1	yes	characteristic
84	C₃²	#15688532	1	3	yes	—
85	C₉	—	3	3	no	—
86	C₉	—	9	3	no	—
87	C₉	—	333	1	no	—
88	C₉	—	333	1	no	—
89	C₉	—	333	3	no	—
90	C₉	—	333	3	no	—
91	C₃	—	3	3	no	—
92	C₃	—	999	1	no	—
93	C₃	#22582565	1	1	yes	characteristic
94	C₃	#22582655	1	3	yes	—

	subgroup	G/H
center	C₃²	C₃×C₃²⋊(C₃₇⋊C₃)
commutator, cosocle	C₃³×C₃₇	C₃³
Frattini	C₃³	C₃²×C₃₇⋊C₃
Fitting	C₃₇×#6895	C₃
socle	C₃²×C₃₇	C₃×H(3,𝔽₃)
prime radical	C₃×C₃²⋊(C₃₇⋊C₃)	C₃²
hypercenter	#6895	C₃₇⋊C₃
hypocenter	C₃₇	#10624817
LCS	G ▷ C₃³×C₃₇ ▷ C₃×C₃₇ ▷ C₃₇
UCS	C₁ ◁ C₃² ◁ C₃²×C₉ ◁ #6895

Complements

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

N	H ≅ G/N	type
1. #22582578	C₃	semidirect
3. C₃×#15688505	C₃	semidirect
4. #22582481	C₃	semidirect
5. #22582478	C₃	semidirect
7. C₃₇×#6895	C₃	semidirect
45. #6895	C₃₇⋊C₃	semidirect
66. C₃₇	#10624817	semidirect

Maximal subgroups

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

subgroup	index
#22582578	3
#22582564	3
C₃×#15688505	3
#22582481	3
#22582478	3
#22582475	3
C₃₇×#6895	3
#10624817	37