Subgroups — #41006339

Order 26973.

230 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	#22582488	C₃	1	6	yes	maximal, complemented
3	C₃²⋊#11758601	C₃	1	2	yes	maximal
4	C₂₇×#66524	C₃	1	3	yes	maximal, complemented
5	C₃₇×C₃²⋊C₂₇	C₃	1	1	yes	maximal, characteristic
6	C₃²×#66524	C₉	1	2	yes	—
7	C₃²×#66524	C₉	1	1	yes	characteristic
8	C₃×#11758601	C₃²	1	6	yes	—
9	C₉×#66524	C₉	1	2	yes	—
10	C₉×#66524	C₉	1	4	yes	—
11	C₉×#66524	C₃²	1	1	yes	characteristic
12	C₉×#66524	C₃²	1	2	yes	—
13	C₃²×C₉×C₃₇	C₃²	1	1	yes	characteristic
14	C₃×C₂₇×C₃₇	C₃²	1	3	yes	—
15	C₃³×C₃₇	C₃×C₉	1	1	yes	characteristic
16	C₃×C₉×C₃₇	—	3	1	no	—
17	C₃×C₉×C₃₇	—	3	2	no	—
18	C₃×C₉×C₃₇	C₃×C₉	1	2	yes	—
19	C₃×C₉×C₃₇	C₃³	1	1	yes	characteristic
20	C₃×#66524	—	3	6	no	—
21	C₃×#66524	C₂₇	1	9	yes	complemented
22	C₃×#66524	C₃×C₉	1	4	yes	—
23	C₃×#66524	C₃×C₉	1	1	yes	characteristic
24	C₃×#66524	C₃×C₉	1	2	yes	—
25	C₃×#66524	C₃×C₉	1	2	yes	—
26	C₂₇×C₃₇	—	3	3	no	—
27	#11758601	—	3	6	no	—
28	#10624710	—	37	1	no	maximal, Sylow
29	C₃²×C₃₇	—	3	1	no	—
30	C₃²×C₃₇	—	3	2	no	—
31	C₃²×C₃₇	C₃×C₂₇	1	3	yes	—
32	C₃²×C₃₇	C₃²×C₉	1	1	yes	characteristic
33	C₉×C₃₇	—	3	2	no	—
34	C₉×C₃₇	#737	1	1	yes	characteristic
35	C₉×C₃₇	#737	1	2	yes	—
36	#66524	—	3	4	no	—
37	#66524	C₃×C₂₇	1	3	yes	—
38	#66524	C₃×C₂₇	1	6	yes	—
39	#66524	C₂₇⋊C₃	1	6	yes	—
40	C₃×C₉²	—	37	1	no	—
41	C₉⋊C₂₇	—	37	6	no	—
42	C₃²⋊C₂₇	—	37	2	no	—
43	C₃²⋊C₂₇	C₃₇⋊C₃	1	1	yes	characteristic
44	C₉×C₂₇	—	37	3	no	—
45	C₃×C₃₇	—	3	3	no	—
46	C₃×C₃₇	#6887	1	1	yes	characteristic
47	C₃×C₃₇	C₃²×C₂₇	1	1	yes	characteristic
48	C₃×C₃₇	C₃×C₂₇⋊C₃	1	2	yes	—
49	C₃²×C₉	—	37	1	no	—
50	C₃²×C₉	—	37	2	no	—
51	C₃²×C₉	C₃×C₃₇⋊C₃	1	1	yes	characteristic
52	C₃×C₂₇	—	37	6	no	—
53	C₃×C₂₇	C₃×C₃₇⋊C₃	1	3	yes	—
54	C₉²	—	37	4	no	—
55	C₉²	—	37	2	no	—
56	C₉²	—	37	1	no	—
57	C₉²	—	37	2	no	—
58	C₃₇	#10624710	1	1	yes	characteristic, complemented, Sylow
59	C₃³	C₉×C₃₇⋊C₃	1	1	yes	characteristic
60	C₃×C₉	—	3	1	no	—
61	C₃×C₉	—	3	2	no	—
62	C₃×C₉	—	37	9	no	—
63	C₃×C₉	—	37	1	no	—
64	C₃×C₉	—	37	2	no	—
65	C₃×C₉	—	37	2	no	—
66	C₃×C₉	—	37	4	no	—
67	C₃×C₉	—	111	6	no	—
68	C₃×C₉	C₉×C₃₇⋊C₃	1	2	yes	—
69	C₃×C₉	C₃²×C₃₇⋊C₃	1	1	yes	characteristic
70	C₂₇	—	3	3	no	—
71	C₂₇	—	111	6	no	—
72	C₃²	—	3	1	no	—
73	C₃²	—	3	2	no	—
74	C₃²	C₂₇×C₃₇⋊C₃	1	3	yes	—
75	C₃²	C₃×C₉×C₃₇⋊C₃	1	1	yes	characteristic
76	C₉	—	3	2	no	—
77	C₉	—	37	3	no	—
78	C₉	—	37	6	no	—
79	C₉	—	37	6	no	—
80	C₉	—	111	4	no	—
81	C₉	#15688530	1	1	yes	characteristic
82	C₉	#15688532	1	2	yes	—
83	C₃	—	3	3	no	—
84	C₃	#22582535	1	1	yes	characteristic
85	C₃	C₃×C₂₇×C₃₇⋊C₃	1	1	yes	characteristic
86	C₃	C₂₇⋊C₃×C₃₇⋊C₃	1	2	yes	—

	subgroup	G/H
center, Frattini	C₃×C₉	C₃²×C₃₇⋊C₃
commutator	C₃×C₃₇	C₃²×C₂₇
Fitting	C₃₇×C₃²⋊C₂₇	C₃
socle	C₃²×C₃₇	C₃²×C₉
prime radical	C₃³×C₃₇	C₃×C₉
cosocle	C₃×C₉×C₃₇	C₃³
hypercenter	C₃²⋊C₂₇	C₃₇⋊C₃
hypocenter	C₃₇	#10624710
LCS	G ▷ C₃×C₃₇ ▷ C₃₇
UCS	C₁ ◁ C₃×C₉ ◁ C₃²⋊C₂₇

Complements

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

N	H ≅ G/N	type
2. #22582488	C₃	semidirect
4. C₂₇×#66524	C₃	semidirect
21. C₃×#66524	C₂₇	semidirect
58. C₃₇	#10624710	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
#22582488	3
C₃²⋊#11758601	3
C₂₇×#66524	3
C₃₇×C₃²⋊C₂₇	3
#10624710	37