Subgroups — #41006154

Order 26973.

236 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22582554	C₃	1	1	yes	maximal, characteristic
2	#22582554	C₃	1	2	yes	maximal
3	C₃₇⋊#6862	C₃	1	9	yes	maximal, complemented
4	C₃₇×#6891	C₃	1	1	yes	maximal, characteristic
5	C₃×#11758608	—	3	1	no	—
6	C₃×#11758608	—	3	2	no	—
7	C₃²⋊#66524	—	3	2	no	—
8	C₃²⋊#66524	—	3	4	no	—
9	C₃²⋊#66524	C₃²	1	3	yes	—
10	C₃²⋊#66524	C₃²	1	6	yes	—
11	C₃⁴×C₃₇	C₃²	1	1	yes	characteristic
12	C₃×3₋¹⁺²×C₃₇	—	3	1	no	—
13	C₃₇×C₃²⋊C₉	—	3	2	no	—
14	C₃₇×C₃²⋊C₉	C₃²	1	3	yes	—
15	C₃³×C₃₇	—	9	1	no	—
16	C₃³×C₃₇	—	9	2	no	—
17	C₃³×C₃₇	—	9	1	no	—
18	C₃³×C₃₇	H(3,𝔽₃)	1	2	yes	—
19	C₃³×C₃₇	H(3,𝔽₃)	1	1	yes	characteristic
20	C₃³×C₃₇	C₃³	1	1	yes	characteristic
21	3₋¹⁺²×C₃₇	—	9	3	no	—
22	C₃×C₉×C₃₇	—	3	3	no	—
23	#11758608	—	9	3	no	—
24	#11758608	—	9	6	no	—
25	C₃×#66524	—	3	3	no	—
26	C₃×#66524	—	3	6	no	—
27	#10624675	—	37	1	no	maximal, Sylow
28	C₃²×C₃₇	—	3	1	no	—
29	C₃²×C₃₇	—	3	2	no	—
30	C₃²×C₃₇	—	3	1	no	—
31	C₃²×C₃₇	—	3	2	no	—
32	C₃²×C₃₇	—	3	2	no	—
33	C₃²×C₃₇	—	3	4	no	—
34	C₃²×C₃₇	—	3	1	no	—
35	C₃²×C₃₇	—	3	1	no	—
36	C₃²×C₃₇	—	3	2	no	—
37	C₃²×C₃₇	—	9	9	no	—
38	C₃²×C₃₇	C₃×H(3,𝔽₃)	1	1	yes	characteristic
39	C₉×C₃₇	—	9	3	no	—
40	#66524	—	9	3	no	—
41	#66524	—	9	6	no	—
42	#6891	—	37	1	no	—
43	#6891	—	37	2	no	—
44	#6891	C₃₇⋊C₃	1	1	yes	characteristic
45	#6862	—	37	9	no	—
46	C₃×C₃₇	—	9	1	no	—
47	C₃×C₃₇	—	9	2	no	—
48	C₃×C₃₇	—	9	1	no	—
49	C₃×C₃₇	#6910	1	1	yes	characteristic
50	C₃×C₃₇	#6910	1	1	yes	characteristic
51	C₃×C₃₇	#6910	1	2	yes	—
52	C₃⁴	C₃×C₃₇⋊C₃	1	1	yes	characteristic
53	C₃×3₋¹⁺²	—	3	1	no	—
54	C₃×3₋¹⁺²	—	111	1	no	—
55	C₃×3₋¹⁺²	—	111	2	no	—
56	C₃²⋊C₉	—	3	2	no	—
57	C₃²⋊C₉	—	37	3	no	—
58	C₃²⋊C₉	—	37	6	no	—
59	C₃²⋊C₉	—	111	2	no	—
60	C₃²⋊C₉	—	111	4	no	—
61	C₃²⋊C₉	C₃×C₃₇⋊C₃	1	3	yes	—
62	C₃₇	#10624675	1	1	yes	characteristic, complemented, Sylow
63	C₃³	—	9	1	no	—
64	C₃³	—	9	1	no	—
65	C₃³	—	9	2	no	—
66	C₃³	C₃²⋊(C₃₇⋊C₃)	1	2	yes	—
67	C₃³	C₃²⋊(C₃₇⋊C₃)	1	1	yes	characteristic
68	C₃³	C₃²×C₃₇⋊C₃	1	1	yes	characteristic
69	3₋¹⁺²	—	9	3	no	—
70	3₋¹⁺²	—	333	3	no	—
71	3₋¹⁺²	—	333	6	no	—
72	C₃×C₉	—	3	3	no	—
73	C₃×C₉	—	111	3	no	—
74	C₃×C₉	—	111	6	no	—
75	C₃²	—	3	1	no	—
76	C₃²	—	3	2	no	—
77	C₃²	—	3	1	no	—
78	C₃²	—	3	1	no	—
79	C₃²	—	3	2	no	—
80	C₃²	—	3	1	no	—
81	C₃²	—	3	2	no	—
82	C₃²	—	3	4	no	—
83	C₃²	—	3	2	no	—
84	C₃²	—	9	9	no	—
85	C₃²	C₃×C₃²⋊(C₃₇⋊C₃)	1	1	yes	characteristic
86	C₉	—	9	3	no	—
87	C₉	—	333	3	no	—
88	C₉	—	333	6	no	—
89	C₃	—	9	1	no	—
90	C₃	—	9	2	no	—
91	C₃	—	9	1	no	—
92	C₃	#22582642	1	1	yes	characteristic
93	C₃	#22582643	1	1	yes	characteristic
94	C₃	#22582644	1	2	yes	—

	subgroup	G/H
center	C₃²	C₃×C₃²⋊(C₃₇⋊C₃)
commutator, cosocle	C₃³×C₃₇	C₃³
Frattini	C₃³	C₃²×C₃₇⋊C₃
Fitting	C₃₇×#6891	C₃
socle	C₃²×C₃₇	C₃×H(3,𝔽₃)
prime radical	C₃⁴×C₃₇	C₃²
hypercenter	#6891	C₃₇⋊C₃
hypocenter	C₃₇	#10624675
LCS	G ▷ C₃³×C₃₇ ▷ C₃²×C₃₇ ▷ C₃₇
UCS	C₁ ◁ C₃² ◁ C₃⁴ ◁ #6891

Complements

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

N	H ≅ G/N	type
3. C₃₇⋊#6862	C₃	semidirect
62. C₃₇	#10624675	semidirect

Maximal subgroups

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

subgroup	index
#22582554	3
C₃₇⋊#6862	3
C₃₇×#6891	3
#10624675	37