Subgroups — #41008444

Order 26973.

512 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×C₃²⋊#66526	C₃	1	3	yes	maximal
2	(C₃×#66526)⋊C₉	C₃	1	9	yes	maximal, complemented
3	C₃²×C₉×C₃₇⋊C₃	C₃	1	1	yes	maximal, characteristic
4	C₃²×#66526	C₃²	1	9	yes	—
5	C₃²⋊#66526	—	3	9	no	—
6	C₃³×C₃₇⋊C₃	C₃²	1	1	yes	characteristic
7	C₃²×#66524	C₉	1	2	yes	complemented
8	C₃×C₉×C₃₇⋊C₃	—	3	3	no	—
9	C₃×C₉×C₃₇⋊C₃	—	3	6	no	—
10	C₃×C₉×C₃₇⋊C₃	C₉	1	6	yes	complemented
11	C₃×C₉×C₃₇⋊C₃	C₃²	1	3	yes	—
12	C₃²×C₉×C₃₇	C₉	1	1	yes	characteristic, complemented
13	C₃³×C₃₇	C₃×C₉	1	1	yes	characteristic
14	C₃²×C₃₇⋊C₃	—	3	9	no	—
15	C₃²×C₃₇⋊C₃	C₃×C₉	1	2	yes	—
16	C₃²×C₃₇⋊C₃	H(3,𝔽₃)	1	3	yes	—
17	C₃²×C₃₇⋊C₃	3₋¹⁺²	1	6	yes	—
18	C₃²×C₃₇⋊C₃	C₃³	1	1	yes	characteristic
19	C₃×#66526	—	3	9	no	—
20	C₃×#66526	—	3	27	no	—
21	C₃×C₉×C₃₇	—	3	3	no	—
22	C₃×C₉×C₃₇	C₃×C₉	1	3	yes	—
23	C₃×#66524	—	3	6	no	—
24	C₃×#66524	C₃×C₉	1	6	yes	—
25	C₉×C₃₇⋊C₃	—	3	18	no	—
26	C₉×C₃₇⋊C₃	—	3	3	no	—
27	C₉×C₃₇⋊C₃	—	3	6	no	—
28	#10624549	—	37	1	no	maximal, Sylow
29	C₃²×C₃₇	—	3	3	no	—
30	C₃²×C₃₇	C₃²⋊C₉	1	3	yes	—
31	C₃²×C₃₇	C₃²×C₉	1	1	yes	characteristic
32	C₃×C₃₇⋊C₃	—	3	6	no	—
33	C₃×C₃₇⋊C₃	—	3	9	no	—
34	C₃×C₃₇⋊C₃	—	3	18	no	—
35	C₃×C₃₇⋊C₃	C₃²⋊C₉	1	6	yes	—
36	C₃×C₃₇⋊C₃	C₃×H(3,𝔽₃)	1	1	yes	characteristic
37	C₃×C₃₇⋊C₃	C₃×3₋¹⁺²	1	2	yes	—
38	C₃×C₃₇⋊C₃	#737	1	9	yes	—
39	#66526	—	9	9	no	—
40	C₉×C₃₇	—	3	3	no	—
41	#66524	—	3	6	no	—
42	C₃³×C₉	—	37	1	no	—
43	C₃×C₃²⋊C₉	—	37	3	no	—
44	#6855	—	37	9	no	—
45	C₃×C₃₇	—	3	3	no	—
46	C₃×C₃₇	C₃×C₃²⋊C₉	1	1	yes	characteristic
47	C₃×C₃₇	#6887	1	3	yes	—
48	C₃₇⋊C₃	—	3	6	no	—
49	C₃₇⋊C₃	#6893	1	3	yes	—
50	C₃₇⋊C₃	#6895	1	6	yes	—
51	C₃⁴	—	37	1	no	—
52	C₃²×C₉	—	37	2	no	—
53	C₃²×C₉	—	37	6	no	—
54	C₃²×C₉	—	37	3	no	—
55	C₃²×C₉	—	37	9	no	—
56	C₃²×C₉	—	111	3	no	—
57	C₃²×C₉	—	111	6	no	—
58	C₃²×C₉	#66526	1	1	yes	characteristic, complemented
59	C₃²⋊C₉	—	111	9	no	—
60	C₃₇	#10624549	1	1	yes	characteristic, complemented, Sylow
61	C₃³	—	37	1	no	—
62	C₃³	—	37	3	no	—
63	C₃³	—	37	6	no	—
64	C₃³	—	37	2	no	—
65	C₃³	—	111	9	no	—
66	C₃³	C₃×#66526	1	1	yes	characteristic
67	C₃×C₉	—	3	3	no	—
68	C₃×C₉	—	37	6	no	—
69	C₃×C₉	—	111	3	no	—
70	C₃×C₉	—	111	6	no	—
71	C₃×C₉	—	111	18	no	—
72	C₃×C₉	—	111	9	no	—
73	C₃×C₉	—	111	27	no	—
74	C₃×C₉	—	111	6	no	—
75	C₃×C₉	C₃×#66526	1	3	yes	—
76	C₃²	—	3	3	no	—
77	C₃²	—	37	1	no	—
78	C₃²	—	37	2	no	—
79	C₃²	—	37	9	no	—
80	C₃²	—	37	6	no	—
81	C₃²	—	111	9	no	—
82	C₃²	—	111	18	no	—
83	C₃²	—	111	6	no	—
84	C₃²	C₃²⋊#66526	1	3	yes	—
85	C₃²	C₃²×#66526	1	1	yes	characteristic
86	C₉	—	3	3	no	—
87	C₉	—	333	9	no	—
88	C₉	—	111	6	no	—
89	C₃	—	3	3	no	—
90	C₃	—	37	3	no	—
91	C₃	—	37	6	no	—
92	C₃	—	111	6	no	—
93	C₃	C₃×C₃²⋊#66526	1	1	yes	characteristic
94	C₃	#22582722	1	3	yes	—

	subgroup	G/H
center, Frattini	C₃²	C₃²×#66526
commutator, socle	C₃²×C₃₇	C₃²×C₉
Fitting	C₃²×C₉×C₃₇	C₉
prime radical	C₃³×C₃₇⋊C₃	C₃²
cosocle	C₃²×C₃₇⋊C₃	C₃³
hypercenter	C₃²×C₉	#66526
hypocenter	C₃₇	#10624549
LCS	G ▷ C₃²×C₃₇ ▷ C₃₇
UCS	C₁ ◁ C₃² ◁ C₃²×C₉

Complements

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

N	H ≅ G/N	type
2. (C₃×#66526)⋊C₉	C₃	semidirect
7. C₃²×#66524	C₉	semidirect
10. C₃×C₉×C₃₇⋊C₃	C₉	semidirect
12. C₃²×C₉×C₃₇	C₉	semidirect
58. C₃²×C₉	#66526	semidirect
60. C₃₇	#10624549	semidirect

Maximal subgroups

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

subgroup	index
C₃×C₃²⋊#66526	3
(C₃×#66526)⋊C₉	3
C₃²×C₉×C₃₇⋊C₃	3
#10624549	37