Subgroups — #41005107

Order 26973.

374 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×#15688504	C₃	1	6	yes	maximal
2	C₃×C₉×#66524	C₃	1	1	yes	maximal, characteristic
3	#22582441	C₃	1	3	yes	maximal
4	#22582441	C₃	1	2	yes	maximal
5	C₃₇×#6855	C₃	1	1	yes	maximal, characteristic
6	C₃²×#66524	C₃²	1	6	yes	—
7	C₃²×#66524	C₃²	1	1	yes	characteristic
8	C₃²×#66524	C₃²	1	2	yes	—
9	#15688504	—	3	18	no	—
10	C₉×#66524	C₉	1	9	yes	complemented
11	C₃²×C₉×C₃₇	C₃²	1	3	yes	—
12	C₃²×C₉×C₃₇	C₃²	1	1	yes	characteristic
13	C₃³×C₃₇	C₃³	1	1	yes	characteristic
14	C₃×C₉×C₃₇	—	3	1	no	—
15	C₃×C₉×C₃₇	—	3	2	no	—
16	C₃×C₉×C₃₇	—	3	6	no	—
17	C₃×C₉×C₃₇	—	3	3	no	—
18	C₃×C₉×C₃₇	—	3	3	no	—
19	C₃×C₉×C₃₇	C₃×C₉	1	3	yes	—
20	C₃×#66524	—	3	3	no	—
21	C₃×#66524	—	3	12	no	—
22	C₃×#66524	—	3	6	no	—
23	C₃×#66524	—	3	6	no	—
24	C₃×#66524	C₃×C₉	1	3	yes	—
25	C₃×#66524	C₃×C₉	1	6	yes	—
26	C₃×#66524	3₋¹⁺²	1	18	yes	—
27	#10624550	—	37	1	no	maximal, Sylow
28	C₃²×C₃₇	C₃²×C₉	1	1	yes	characteristic
29	C₃²×C₃₇	C₃×3₋¹⁺²	1	6	yes	—
30	C₃²×C₃₇	#737	1	1	yes	characteristic
31	C₃²×C₃₇	#737	1	3	yes	—
32	C₃²×C₃₇	#737	1	2	yes	—
33	C₉×C₃₇	—	3	3	no	—
34	C₉×C₃₇	—	9	3	no	—
35	#66524	—	3	3	no	—
36	#66524	—	9	6	no	—
37	#66524	C₉⋊C₉	1	18	yes	complemented
38	C₃×C₉⋊C₉	—	37	6	no	—
39	C₃×C₉²	—	37	1	no	—
40	#6855	—	37	3	no	—
41	#6855	—	37	2	no	—
42	#6855	C₃₇⋊C₃	1	1	yes	characteristic
43	C₃×C₃₇	C₃×C₉⋊C₉	1	2	yes	—
44	C₃×C₃₇	#6887	1	1	yes	characteristic
45	C₃×C₃₇	#6887	1	1	yes	characteristic
46	C₃×C₃₇	#6896	1	3	yes	—
47	C₃×C₃₇	#6896	1	6	yes	—
48	C₃²×C₉	—	37	2	no	—
49	C₃²×C₉	—	37	1	no	—
50	C₃²×C₉	—	37	6	no	—
51	C₃²×C₉	C₃×C₃₇⋊C₃	1	3	yes	—
52	C₃²×C₉	C₃×C₃₇⋊C₃	1	1	yes	characteristic
53	C₉⋊C₉	—	111	18	no	—
54	C₉²	—	37	9	no	—
55	C₃₇	#10624550	1	1	yes	characteristic, complemented, Sylow
56	C₃³	C₃²×C₃₇⋊C₃	1	1	yes	characteristic
57	C₃×C₉	—	3	1	no	—
58	C₃×C₉	—	3	2	no	—
59	C₃×C₉	—	3	6	no	—
60	C₃×C₉	—	3	3	no	—
61	C₃×C₉	—	3	3	no	—
62	C₃×C₉	—	37	6	no	—
63	C₃×C₉	—	37	18	no	—
64	C₃×C₉	—	37	3	no	—
65	C₃×C₉	—	111	12	no	—
66	C₃×C₉	—	111	6	no	—
67	C₃×C₉	—	111	6	no	—
68	C₃×C₉	—	111	3	no	—
69	C₃×C₉	C₉×C₃₇⋊C₃	1	3	yes	—
70	C₃²	C₃×C₉×C₃₇⋊C₃	1	1	yes	characteristic
71	C₃²	3₋¹⁺²×C₃₇⋊C₃	1	6	yes	—
72	C₃²	#15688530	1	1	yes	characteristic
73	C₃²	#15688531	1	3	yes	—
74	C₃²	#15688532	1	2	yes	—
75	C₉	—	3	3	no	—
76	C₉	—	9	3	no	—
77	C₉	—	37	18	no	—
78	C₉	—	333	6	no	—
79	C₉	—	111	3	no	—
80	C₃	C₉⋊C₉×C₃₇⋊C₃	1	2	yes	—
81	C₃	#22582535	1	1	yes	characteristic
82	C₃	#22582536	1	1	yes	characteristic
83	C₃	#22582579	1	3	yes	—
84	C₃	#22582582	1	6	yes	—

	subgroup	G/H
center, Frattini	C₃³	C₃²×C₃₇⋊C₃
commutator	C₃²×C₃₇	C₃²×C₉
Fitting	C₃₇×#6855	C₃
socle, prime radical, cosocle	C₃³×C₃₇	C₃³
hypercenter	#6855	C₃₇⋊C₃
hypocenter	C₃₇	#10624550
LCS	G ▷ C₃²×C₃₇ ▷ C₃₇
UCS	C₁ ◁ C₃³ ◁ #6855

Complements

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

N	H ≅ G/N	type
10. C₉×#66524	C₉	semidirect
37. #66524	C₉⋊C₉	semidirect
55. C₃₇	#10624550	semidirect

Maximal subgroups

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

subgroup	index
C₃×#15688504	3
C₃×C₉×#66524	3
#22582441	3
C₃₇×#6855	3
#10624550	37