Subgroups — C₉×3₋¹⁺²×C₃₇⋊C₃ #41007502

Order 26973.

1148 conjugacy classes of subgroups.

The subgroup lattice has 212 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×3₋¹⁺²×C₃₇⋊C₃	C₃	1	1	yes	maximal, characteristic
2	C₃²×C₉×C₃₇⋊C₃	C₃	1	1	yes	maximal, characteristic
3	#22582548	C₃	1	4	yes	maximal, complemented
4	3₋¹⁺²×#66524	C₃	1	2	yes	maximal, complemented
5	C₉×#11758606	C₃	1	3	yes	maximal, complemented
6	C₉×#11758605	C₃	1	3	yes	maximal, complemented
7	C₉×#11758608	C₃	1	2	yes	maximal, complemented
8	#22582543	C₃	1	6	yes	maximal, complemented
9	#22582542	C₃	1	6	yes	maximal, complemented
10	C₉⋊C₉×C₃₇⋊C₃	C₃	1	6	yes	maximal, complemented
11	C₃²⋊C₉×C₃₇⋊C₃	C₃	1	2	yes	maximal
12	C₉²×C₃₇⋊C₃	C₃	1	3	yes	maximal, complemented
13	C₉×3₋¹⁺²×C₃₇	C₃	1	1	yes	maximal, characteristic, complemented
14	C₃³×C₃₇⋊C₃	C₃²	1	1	yes	characteristic
15	C₃×#11758608	C₃²	1	2	yes	—
16	C₃×#11758605	C₃²	1	3	yes	—
17	C₃×#11758606	C₃²	1	3	yes	—
18	3₋¹⁺²×C₃₇⋊C₃	C₉	1	27	yes	direct product factor
19	C₃²×#66524	C₃²	1	2	yes	—
20	C₃×C₉×C₃₇⋊C₃	—	3	9	no	—
21	C₃×C₉×C₃₇⋊C₃	C₃²	1	6	yes	—
22	C₃×C₉×C₃₇⋊C₃	C₃²	1	2	yes	—
23	C₃×C₉×C₃₇⋊C₃	C₃²	1	2	yes	—
24	C₃×C₉×C₃₇⋊C₃	C₃²	1	2	yes	—
25	C₃×C₉×C₃₇⋊C₃	C₃²	1	1	yes	characteristic
26	C₃×C₉×C₃₇⋊C₃	C₃²	1	1	yes	characteristic
27	C₃×C₉×C₃₇⋊C₃	C₃²	1	3	yes	—
28	C₃×C₉×C₃₇⋊C₃	C₃²	1	1	yes	characteristic
29	#15688506	C₃²	1	6	yes	complemented
30	#15688505	C₃²	1	6	yes	complemented
31	#15688504	C₃²	1	12	yes	complemented
32	#15688504	C₃²	1	12	yes	complemented
33	#15688504	C₃²	1	12	yes	complemented
34	C₃²⋊#66524	C₃²	1	4	yes	—
35	#15688502	C₃²	1	6	yes	—
36	#15688502	C₃²	1	6	yes	—
37	C₉×#66524	C₃²	1	6	yes	complemented
38	C₉×#66524	C₃²	1	12	yes	complemented
39	C₉×#66524	C₃²	1	6	yes	complemented
40	C₃×3₋¹⁺²×C₃₇	C₃²	1	1	yes	characteristic
41	C₃²×C₉×C₃₇	C₃²	1	1	yes	characteristic
42	C₃₇×C₉⋊C₉	C₃²	1	6	yes	complemented
43	C₃₇×C₃²⋊C₉	C₃²	1	2	yes	—
44	C₉²×C₃₇	C₃²	1	3	yes	complemented
45	C₃³×C₃₇	C₃³	1	1	yes	characteristic
46	C₃²×C₃₇⋊C₃	—	3	3	no	—
47	C₃²×C₃₇⋊C₃	—	3	6	no	—
48	C₃²×C₃₇⋊C₃	C₃×C₉	1	9	yes	—
49	C₃²×C₃₇⋊C₃	C₃³	1	1	yes	characteristic
50	C₃²×C₃₇⋊C₃	C₃³	1	1	yes	characteristic
51	C₃²×C₃₇⋊C₃	C₃³	1	1	yes	characteristic
52	3₋¹⁺²×C₃₇	C₃×C₉	1	9	yes	complemented
53	C₃×C₉×C₃₇	—	3	3	no	—
54	C₃×C₉×C₃₇	C₃³	1	6	yes	—
55	C₃×C₉×C₃₇	C₃³	1	2	yes	—
56	C₃×C₉×C₃₇	C₃³	1	3	yes	—
57	C₃×C₉×C₃₇	C₃³	1	1	yes	characteristic
58	#11758608	C₃×C₉	1	18	yes	complemented
59	C₃×#66524	—	3	6	no	—
60	C₃×#66524	C₃³	1	12	yes	—
61	C₃×#66524	C₃³	1	2	yes	—
62	C₃×#66524	C₃³	1	2	yes	—
63	C₃×#66524	C₃³	1	6	yes	—
64	C₃×#66524	C₃³	1	2	yes	—
65	#11758606	C₃×C₉	1	27	yes	complemented
66	#11758605	C₃×C₉	1	27	yes	complemented
67	C₉×C₃₇⋊C₃	—	3	6	no	—
68	C₉×C₃₇⋊C₃	—	3	18	no	—
69	C₉×C₃₇⋊C₃	—	3	3	no	—
70	C₉×C₃₇⋊C₃	—	3	3	no	—
71	C₉×C₃₇⋊C₃	—	3	6	no	—
72	C₉×C₃₇⋊C₃	—	3	6	no	—
73	C₉×C₃₇⋊C₃	C₃×C₉	1	27	yes	complemented
74	C₉×C₃₇⋊C₃	3₋¹⁺²	1	9	yes	direct product factor
75	C₃×C₉×3₋¹⁺²	—	37	1	no	maximal, Sylow
76	C₃²×C₃₇	—	3	1	no	—
77	C₃²×C₃₇	—	3	2	no	—
78	C₃²×C₃₇	C₃²×C₉	1	3	yes	—
79	C₃²×C₃₇	C₃⁴	1	1	yes	characteristic
80	C₃×C₃₇⋊C₃	—	3	1	no	—
81	C₃×C₃₇⋊C₃	—	3	1	no	—
82	C₃×C₃₇⋊C₃	—	3	2	no	—
83	C₃×C₃₇⋊C₃	—	3	2	no	—
84	C₃×C₃₇⋊C₃	—	3	27	no	—
85	C₃×C₃₇⋊C₃	C₃²×C₉	1	3	yes	—
86	C₃×C₃₇⋊C₃	C₃²×C₉	1	3	yes	—
87	C₃×C₃₇⋊C₃	C₃²×C₉	1	3	yes	—
88	C₃×C₃₇⋊C₃	C₃×3₋¹⁺²	1	3	yes	—
89	C₃×C₃₇⋊C₃	#737	1	6	yes	—
90	C₉×C₃₇	—	3	2	no	—
91	C₉×C₃₇	—	3	6	no	—
92	C₉×C₃₇	C₃²×C₉	1	9	yes	complemented
93	C₉×C₃₇	C₃×3₋¹⁺²	1	3	yes	complemented
94	#66524	—	3	2	no	—
95	#66524	—	3	2	no	—
96	#66524	—	3	12	no	—
97	#66524	C₃²×C₉	1	18	yes	complemented
98	#66524	C₃×3₋¹⁺²	1	6	yes	complemented
99	C₃²×3₋¹⁺²	—	37	1	no	—
100	C₃³×C₉	—	37	1	no	—
101	C₉×3₋¹⁺²	—	37	2	no	—
102	C₉×3₋¹⁺²	—	37	2	no	—
103	C₉×3₋¹⁺²	—	37	6	no	—
104	C₉×3₋¹⁺²	—	37	6	no	—
105	C₉×3₋¹⁺²	—	37	4	no	—
106	C₉×3₋¹⁺²	—	37	3	no	—
107	C₉×3₋¹⁺²	—	37	3	no	—
108	C₉×3₋¹⁺²	C₃₇⋊C₃	1	1	yes	characteristic, direct product factor
109	C₃×C₉⋊C₉	—	37	6	no	—
110	C₃×C₃²⋊C₉	—	37	2	no	—
111	C₃×C₉²	—	37	3	no	—
112	C₃×C₃₇	—	3	3	no	—
113	C₃×C₃₇	C₃³×C₉	1	1	yes	characteristic
114	C₃×C₃₇	C₃²×3₋¹⁺²	1	1	yes	characteristic
115	C₃×C₃₇	C₃×#737	1	2	yes	—
116	C₃₇⋊C₃	—	3	3	no	—
117	C₃₇⋊C₃	—	3	3	no	—
118	C₃₇⋊C₃	C₉×3₋¹⁺²	1	9	yes	direct product factor
119	C₃⁴	—	37	1	no	—
120	C₃×3₋¹⁺²	—	37	2	no	—
121	C₃×3₋¹⁺²	—	37	27	no	—
122	C₃×3₋¹⁺²	—	37	3	no	—
123	C₃×3₋¹⁺²	—	37	3	no	—
124	C₃×3₋¹⁺²	C₃×C₃₇⋊C₃	1	1	yes	characteristic
125	C₃²×C₉	—	37	3	no	—
126	C₃²×C₉	—	37	1	no	—
127	C₃²×C₉	—	37	2	no	—
128	C₃²×C₉	—	37	2	no	—
129	C₃²×C₉	—	37	2	no	—
130	C₃²×C₉	—	37	2	no	—
131	C₃²×C₉	—	37	6	no	—
132	C₃²×C₉	—	37	1	no	—
133	C₃²×C₉	—	37	1	no	—
134	C₃²×C₉	—	111	9	no	—
135	C₃²×C₉	C₃×C₃₇⋊C₃	1	1	yes	characteristic
136	C₉⋊C₉	—	37	12	no	—
137	C₉⋊C₉	—	37	12	no	—
138	C₉⋊C₉	—	37	12	no	—
139	C₉⋊C₉	—	37	6	no	—
140	C₉⋊C₉	—	37	6	no	—
141	C₉⋊C₉	C₃×C₃₇⋊C₃	1	6	yes	complemented
142	C₃²⋊C₉	—	37	4	no	—
143	C₃²⋊C₉	—	37	6	no	—
144	C₃²⋊C₉	—	37	6	no	—
145	C₃²⋊C₉	C₃×C₃₇⋊C₃	1	2	yes	—
146	C₉²	—	37	12	no	—
147	C₉²	—	37	6	no	—
148	C₉²	—	37	6	no	—
149	C₉²	C₃×C₃₇⋊C₃	1	3	yes	complemented
150	C₃₇	C₃×C₉×3₋¹⁺²	1	1	yes	characteristic, complemented, Sylow
151	C₃³	—	37	1	no	—
152	C₃³	—	37	9	no	—
153	C₃³	—	37	1	no	—
154	C₃³	—	37	1	no	—
155	C₃³	—	111	3	no	—
156	C₃³	—	111	6	no	—
157	C₃³	C₃²×C₃₇⋊C₃	1	1	yes	characteristic
158	3₋¹⁺²	—	37	18	no	—
159	3₋¹⁺²	—	37	27	no	—
160	3₋¹⁺²	—	37	27	no	—
161	3₋¹⁺²	C₉×C₃₇⋊C₃	1	9	yes	direct product factor
162	C₃×C₉	—	3	3	no	—
163	C₃×C₉	—	37	2	no	—
164	C₃×C₉	—	37	6	no	—
165	C₃×C₉	—	37	27	no	—
166	C₃×C₉	—	37	9	no	—
167	C₃×C₉	—	37	12	no	—
168	C₃×C₉	—	37	2	no	—
169	C₃×C₉	—	37	2	no	—
170	C₃×C₉	—	111	6	no	—
171	C₃×C₉	—	111	6	no	—
172	C₃×C₉	—	111	18	no	—
173	C₃×C₉	—	111	6	no	—
174	C₃×C₉	—	111	6	no	—
175	C₃×C₉	—	111	3	no	—
176	C₃×C₉	—	111	3	no	—
177	C₃×C₉	C₃²×C₃₇⋊C₃	1	6	yes	—
178	C₃×C₉	C₃²×C₃₇⋊C₃	1	2	yes	—
179	C₃×C₉	C₃²×C₃₇⋊C₃	1	3	yes	—
180	C₃×C₉	C₃²×C₃₇⋊C₃	1	1	yes	characteristic
181	C₃²	—	3	1	no	—
182	C₃²	—	3	2	no	—
183	C₃²	—	37	3	no	—
184	C₃²	—	37	3	no	—
185	C₃²	—	37	6	no	—
186	C₃²	—	37	3	no	—
187	C₃²	—	37	3	no	—
188	C₃²	—	111	27	no	—
189	C₃²	—	111	1	no	—
190	C₃²	—	111	1	no	—
191	C₃²	—	111	2	no	—
192	C₃²	—	111	2	no	—
193	C₃²	C₃×C₉×C₃₇⋊C₃	1	3	yes	—
194	C₃²	C₃³×C₃₇⋊C₃	1	1	yes	characteristic
195	C₉	—	3	2	no	—
196	C₉	—	3	6	no	—
197	C₉	—	37	18	no	—
198	C₉	—	37	6	no	—
199	C₉	—	111	2	no	—
200	C₉	—	111	2	no	—
201	C₉	—	111	12	no	—
202	C₉	C₃×C₉×C₃₇⋊C₃	1	9	yes	complemented
203	C₉	3₋¹⁺²×C₃₇⋊C₃	1	3	yes	direct product factor
204	C₃	—	3	3	no	—
205	C₃	—	37	9	no	—
206	C₃	—	111	3	no	—
207	C₃	—	111	3	no	—
208	C₃	C₃²×C₉×C₃₇⋊C₃	1	1	yes	characteristic
209	C₃	C₃×3₋¹⁺²×C₃₇⋊C₃	1	1	yes	characteristic
210	C₃	#737×C₃₇⋊C₃	1	2	yes	—

	subgroup	G/H
center	C₃×C₉	C₃²×C₃₇⋊C₃
commutator	C₃×C₃₇	C₃³×C₉
Frattini	C₃²	C₃³×C₃₇⋊C₃
Fitting	C₉×3₋¹⁺²×C₃₇	C₃
socle, cosocle	C₃²×C₃₇	C₃⁴
prime radical	C₃³×C₃₇⋊C₃	C₃²
hypercenter	C₉×3₋¹⁺²	C₃₇⋊C₃
hypocenter	C₃₇	C₃×C₉×3₋¹⁺²
LCS	G ▷ C₃×C₃₇ ▷ C₃₇
UCS	C₁ ◁ C₃×C₉ ◁ C₉×3₋¹⁺²

Complements

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

N	H ≅ G/N	type
3. #22582548	C₃	semidirect
4. 3₋¹⁺²×#66524	C₃	semidirect
5. C₉×#11758606	C₃	semidirect
6. C₉×#11758605	C₃	semidirect
7. C₉×#11758608	C₃	semidirect
8. #22582543	C₃	semidirect
9. #22582542	C₃	semidirect
10. C₉⋊C₉×C₃₇⋊C₃	C₃	semidirect
12. C₉²×C₃₇⋊C₃	C₃	semidirect
13. C₉×3₋¹⁺²×C₃₇	C₃	semidirect
18. 3₋¹⁺²×C₃₇⋊C₃	C₉	direct (N × H)
29. #15688506	C₃²	semidirect
30. #15688505	C₃²	semidirect
31. #15688504	C₃²	semidirect
32. #15688504	C₃²	semidirect
33. #15688504	C₃²	semidirect
37. C₉×#66524	C₃²	semidirect
38. C₉×#66524	C₃²	semidirect
39. C₉×#66524	C₃²	semidirect
42. C₃₇×C₉⋊C₉	C₃²	semidirect
44. C₉²×C₃₇	C₃²	semidirect
52. 3₋¹⁺²×C₃₇	C₃×C₉	semidirect
58. #11758608	C₃×C₉	semidirect
65. #11758606	C₃×C₉	semidirect
66. #11758605	C₃×C₉	semidirect
73. C₉×C₃₇⋊C₃	C₃×C₉	semidirect
74. C₉×C₃₇⋊C₃	3₋¹⁺²	direct (N × H)
92. C₉×C₃₇	C₃²×C₉	semidirect
93. C₉×C₃₇	C₃×3₋¹⁺²	semidirect
97. #66524	C₃²×C₉	semidirect
98. #66524	C₃×3₋¹⁺²	semidirect
108. C₉×3₋¹⁺²	C₃₇⋊C₃	direct (N × H)
118. C₃₇⋊C₃	C₉×3₋¹⁺²	direct (N × H)
141. C₉⋊C₉	C₃×C₃₇⋊C₃	semidirect
149. C₉²	C₃×C₃₇⋊C₃	semidirect
150. C₃₇	C₃×C₉×3₋¹⁺²	semidirect
161. 3₋¹⁺²	C₉×C₃₇⋊C₃	direct (N × H)
202. C₉	C₃×C₉×C₃₇⋊C₃	semidirect
203. C₉	3₋¹⁺²×C₃₇⋊C₃	direct (N × H)

Maximal subgroups

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

subgroup	index
C₃×3₋¹⁺²×C₃₇⋊C₃	3
C₃²×C₉×C₃₇⋊C₃	3
#22582548	3
3₋¹⁺²×#66524	3
C₉×#11758606	3
C₉×#11758605	3
C₉×#11758608	3
#22582543	3
#22582542	3
C₉⋊C₉×C₃₇⋊C₃	3
C₃²⋊C₉×C₃₇⋊C₃	3
C₉²×C₃₇⋊C₃	3
C₉×3₋¹⁺²×C₃₇	3
C₃×C₉×3₋¹⁺²	37

Subgroups — C9×3−1+2×C37⋊C3 #41007502

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₉×3₋¹⁺²×C₃₇⋊C₃ #41007502