Subgroups — #41004741

Order 26973.

136 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×C₉×#66524	C₃	1	1	yes	maximal, characteristic
2	C₃²⋊#11758601	C₃	1	2	yes	maximal
3	C₃₇×C₃²⋊C₂₇	C₃	1	1	yes	maximal, characteristic
4	C₃²×#66524	C₉	1	2	yes	—
5	C₃²×#66524	C₉	1	1	yes	characteristic
6	C₃×#11758601	—	3	2	no	—
7	C₉×#66524	—	3	1	no	—
8	C₉×#66524	—	3	1	no	—
9	C₉×#66524	—	3	1	no	—
10	C₃²×C₉×C₃₇	C₃²	1	1	yes	characteristic
11	C₃×C₂₇×C₃₇	—	3	1	no	—
12	C₃³×C₃₇	C₃×C₉	1	1	yes	characteristic
13	C₃×C₉×C₃₇	—	3	1	no	—
14	C₃×C₉×C₃₇	—	3	2	no	—
15	C₃×C₉×C₃₇	H(3,𝔽₃)	1	1	yes	characteristic
16	C₃×C₉×C₃₇	3₋¹⁺²	1	1	yes	characteristic
17	C₃×C₉×C₃₇	3₋¹⁺²	1	1	yes	characteristic
18	C₃×#66524	—	3	1	no	—
19	C₃×#66524	—	3	2	no	—
20	C₃×#66524	—	3	1	no	—
21	C₃×#66524	—	3	1	no	—
22	C₃×#66524	—	3	6	no	—
23	C₃×#66524	C₂₇	1	3	yes	complemented
24	C₂₇×C₃₇	—	9	1	no	—
25	#11758601	—	9	2	no	—
26	#10624445	—	37	1	no	maximal, Sylow
27	C₃²×C₃₇	—	3	1	no	—
28	C₃²×C₃₇	—	3	2	no	—
29	C₃²×C₃₇	C₃²⋊C₉	1	1	yes	characteristic
30	C₃²×C₃₇	C₃×C₂₇	1	1	yes	characteristic
31	C₃²×C₃₇	C₂₇⋊C₃	1	1	yes	characteristic
32	C₃²×C₃₇	C₂₇⋊C₃	1	1	yes	characteristic
33	C₉×C₃₇	—	3	1	no	—
34	C₉×C₃₇	—	3	1	no	—
35	C₉×C₃₇	#731	1	1	yes	characteristic
36	C₉×C₃₇	#731	1	2	yes	—
37	#66524	—	3	3	no	—
38	#66524	—	3	6	no	—
39	C₃×C₉²	—	37	1	no	—
40	C₃²⋊C₂₇	—	37	2	no	—
41	C₃²⋊C₂₇	C₃₇⋊C₃	1	1	yes	characteristic
42	C₃×C₃₇	—	3	1	no	—
43	C₃×C₃₇	—	3	1	no	—
44	C₃×C₃₇	—	3	1	no	—
45	C₃×C₃₇	C₃²⋊C₂₇	1	1	yes	characteristic
46	C₃×C₃₇	#6867	1	1	yes	characteristic
47	C₃×C₃₇	#6869	1	2	yes	—
48	C₃²×C₉	—	37	1	no	—
49	C₃²×C₉	—	37	2	no	—
50	C₃²×C₉	C₃×C₃₇⋊C₃	1	1	yes	characteristic
51	C₃×C₂₇	—	3	1	no	—
52	C₃×C₂₇	—	111	2	no	—
53	C₉²	—	111	1	no	—
54	C₉²	—	111	1	no	—
55	C₉²	—	111	1	no	—
56	C₃₇	#10624445	1	1	yes	characteristic, complemented, Sylow
57	C₃³	C₉×C₃₇⋊C₃	1	1	yes	characteristic
58	C₃×C₉	—	3	1	no	—
59	C₃×C₉	—	3	2	no	—
60	C₃×C₉	—	37	3	no	—
61	C₃×C₉	—	111	1	no	—
62	C₃×C₉	—	111	2	no	—
63	C₃×C₉	—	111	1	no	—
64	C₃×C₉	—	111	1	no	—
65	C₃×C₉	—	111	6	no	—
66	C₃×C₉	#11758605	1	1	yes	characteristic
67	C₃×C₉	#11758606	1	1	yes	characteristic
68	C₃×C₉	C₃²⋊(C₃₇⋊C₃)	1	1	yes	characteristic
69	C₂₇	—	9	1	no	—
70	C₂₇	—	333	2	no	—
71	C₃²	—	3	1	no	—
72	C₃²	—	3	2	no	—
73	C₃²	#15688502	1	1	yes	characteristic
74	C₃²	C₂₇×C₃₇⋊C₃	1	1	yes	characteristic
75	C₃²	#15688509	1	1	yes	characteristic
76	C₃²	#15688510	1	1	yes	characteristic
77	C₉	—	3	1	no	—
78	C₉	—	3	1	no	—
79	C₉	—	111	3	no	—
80	C₉	—	111	6	no	—
81	C₉	#15688515	1	1	yes	characteristic
82	C₉	#15688517	1	2	yes	—
83	C₃	—	3	1	no	—
84	C₃	—	3	1	no	—
85	C₃	—	3	1	no	—
86	C₃	#22582465	1	1	yes	characteristic
87	C₃	#22582469	1	1	yes	characteristic
88	C₃	#22582473	1	2	yes	—

	subgroup	G/H
center	C₃×C₉	C₃²⋊(C₃₇⋊C₃)
commutator	C₃²×C₃₇	C₃×C₂₇
Frattini	C₃²×C₉	C₃×C₃₇⋊C₃
Fitting	C₃₇×C₃²⋊C₂₇	C₃
socle	C₃²×C₃₇	C₃²⋊C₉
prime radical	C₃³×C₃₇	C₃×C₉
cosocle	C₃²×C₉×C₃₇	C₃²
hypercenter	C₃²⋊C₂₇	C₃₇⋊C₃
hypocenter	C₃₇	#10624445
LCS	G ▷ C₃²×C₃₇ ▷ C₃×C₃₇ ▷ C₃₇
UCS	C₁ ◁ C₃×C₉ ◁ C₃²×C₉ ◁ C₃²⋊C₂₇

Complements

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

N	H ≅ G/N	type
23. C₃×#66524	C₂₇	semidirect
56. C₃₇	#10624445	semidirect

Maximal subgroups

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

subgroup	index
C₃×C₉×#66524	3
C₃²⋊#11758601	3
C₃₇×C₃²⋊C₂₇	3
#10624445	37