Subgroups — #41007518

Order 26973.

944 conjugacy classes of subgroups.

The subgroup lattice has 242 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	#22582548	C₃	1	2	yes	maximal, complemented
3	#22582548	C₃	1	2	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	#22582537	C₃	1	6	yes	maximal, complemented
11	#22582536	C₃	1	3	yes	maximal, complemented
12	C₃×#15688502	C₃	1	1	yes	maximal, characteristic
13	C₃²⋊C₉×C₃₇⋊C₃	C₃	1	1	yes	maximal, characteristic
14	C₃²⋊C₉×C₃₇⋊C₃	C₃	1	1	yes	maximal, characteristic
15	C₉×3₋¹⁺²×C₃₇	C₃	1	1	yes	maximal, characteristic, complemented
16	C₃³×C₃₇⋊C₃	C₃²	1	1	yes	characteristic
17	C₃×#11758608	C₃²	1	2	yes	—
18	C₃×#11758605	C₃²	1	3	yes	—
19	C₃×#11758606	C₃²	1	3	yes	—
20	3₋¹⁺²×C₃₇⋊C₃	C₉	1	27	yes	complemented
21	C₃²×#66524	C₃²	1	2	yes	—
22	C₃×C₉×C₃₇⋊C₃	C₃²	1	3	yes	—
23	C₃×C₉×C₃₇⋊C₃	C₃²	1	3	yes	—
24	C₃×C₉×C₃₇⋊C₃	C₃²	1	3	yes	—
25	#15688506	C₃²	1	6	yes	complemented
26	#15688505	C₃²	1	6	yes	complemented
27	#15688504	C₃²	1	6	yes	complemented
28	#15688504	C₃²	1	6	yes	complemented
29	#15688504	C₃²	1	6	yes	complemented
30	#15688504	C₃²	1	6	yes	complemented
31	#15688504	C₃²	1	6	yes	complemented
32	#15688504	C₃²	1	6	yes	complemented
33	C₃²⋊#66524	C₃²	1	2	yes	—
34	C₃²⋊#66524	C₃²	1	2	yes	—
35	#15688502	C₃²	1	3	yes	—
36	#15688502	C₃²	1	3	yes	—
37	#15688502	C₃²	1	3	yes	—
38	#15688502	C₃²	1	3	yes	—
39	#15688502	C₃²	1	1	yes	characteristic
40	#15688502	C₃²	1	2	yes	—
41	#15688502	C₃²	1	2	yes	—
42	#15688502	C₃²	1	2	yes	—
43	#15688502	C₃²	1	1	yes	characteristic
44	#15688502	C₃²	1	1	yes	characteristic
45	C₉×#66524	C₃²	1	6	yes	complemented
46	C₉×#66524	C₃²	1	6	yes	complemented
47	C₉×#66524	C₃²	1	6	yes	complemented
48	C₉×#66524	C₃²	1	6	yes	complemented
49	C₃×3₋¹⁺²×C₃₇	C₃²	1	1	yes	characteristic
50	C₃²×C₉×C₃₇	C₃²	1	1	yes	characteristic
51	C₃₇×C₉⋊C₉	C₃²	1	6	yes	complemented
52	C₃₇×C₃²⋊C₉	C₃²	1	1	yes	characteristic
53	C₃₇×C₃²⋊C₉	C₃²	1	1	yes	characteristic
54	C₉²×C₃₇	C₃²	1	3	yes	complemented
55	C₃³×C₃₇	C₃³	1	1	yes	characteristic
56	C₃²×C₃₇⋊C₃	—	9	1	no	—
57	C₃²×C₃₇⋊C₃	—	9	1	no	—
58	C₃²×C₃₇⋊C₃	—	9	1	no	—
59	C₃²×C₃₇⋊C₃	C₃×C₉	1	9	yes	—
60	C₃²×C₃₇⋊C₃	C₃³	1	1	yes	characteristic
61	C₃²×C₃₇⋊C₃	C₃³	1	1	yes	characteristic
62	C₃²×C₃₇⋊C₃	C₃³	1	1	yes	characteristic
63	3₋¹⁺²×C₃₇	C₃×C₉	1	9	yes	complemented
64	C₃×C₉×C₃₇	—	9	1	no	—
65	C₃×C₉×C₃₇	C₃³	1	1	yes	characteristic
66	C₃×C₉×C₃₇	C₃³	1	3	yes	—
67	C₃×C₉×C₃₇	C₃³	1	3	yes	—
68	C₃×C₉×C₃₇	C₃³	1	2	yes	—
69	C₃×C₉×C₃₇	C₃³	1	3	yes	—
70	#11758608	C₃×C₉	1	18	yes	complemented
71	C₃×#66524	—	9	2	no	—
72	C₃×#66524	C₃³	1	2	yes	—
73	C₃×#66524	C₃³	1	2	yes	—
74	C₃×#66524	C₃³	1	2	yes	—
75	C₃×#66524	C₃³	1	6	yes	—
76	C₃×#66524	C₃³	1	6	yes	—
77	C₃×#66524	C₃³	1	6	yes	—
78	#11758606	C₃×C₉	1	27	yes	complemented
79	#11758605	C₃×C₉	1	27	yes	complemented
80	C₉×C₃₇⋊C₃	—	9	3	no	—
81	C₉×C₃₇⋊C₃	—	9	3	no	—
82	C₉×C₃₇⋊C₃	C₃×C₉	1	27	yes	complemented
83	#10624863	—	37	1	no	maximal, Sylow
84	C₃²×C₃₇	—	3	1	no	—
85	C₃²×C₃₇	—	3	1	no	—
86	C₃²×C₃₇	—	3	1	no	—
87	C₃²×C₃₇	C₃²×C₉	1	3	yes	—
88	C₃²×C₃₇	C₃⁴	1	1	yes	characteristic
89	C₃×C₃₇⋊C₃	—	3	1	no	—
90	C₃×C₃₇⋊C₃	—	3	1	no	—
91	C₃×C₃₇⋊C₃	—	3	1	no	—
92	C₃×C₃₇⋊C₃	—	3	1	no	—
93	C₃×C₃₇⋊C₃	—	3	1	no	—
94	C₃×C₃₇⋊C₃	—	3	1	no	—
95	C₃×C₃₇⋊C₃	—	3	1	no	—
96	C₃×C₃₇⋊C₃	—	3	1	no	—
97	C₃×C₃₇⋊C₃	—	3	1	no	—
98	C₃×C₃₇⋊C₃	—	9	9	no	—
99	C₃×C₃₇⋊C₃	C₃²×C₉	1	3	yes	—
100	C₃×C₃₇⋊C₃	C₃²×C₉	1	3	yes	—
101	C₃×C₃₇⋊C₃	C₃²×C₉	1	3	yes	—
102	C₉×C₃₇	—	3	3	no	—
103	C₉×C₃₇	—	3	3	no	—
104	C₉×C₃₇	—	3	1	no	—
105	C₉×C₃₇	—	3	2	no	—
106	C₉×C₃₇	C₃²×C₉	1	9	yes	—
107	#66524	—	3	6	no	—
108	#66524	—	3	6	no	—
109	#66524	—	3	2	no	—
110	#66524	—	3	2	no	—
111	#66524	—	3	2	no	—
112	#66524	C₃²×C₉	1	18	yes	—
113	C₃²×3₋¹⁺²	—	37	1	no	—
114	C₉×3₋¹⁺²	—	37	2	no	—
115	C₉×3₋¹⁺²	—	37	2	no	—
116	C₉×3₋¹⁺²	—	37	6	no	—
117	C₉×3₋¹⁺²	—	37	6	no	—
118	C₉×3₋¹⁺²	—	37	2	no	—
119	C₉×3₋¹⁺²	—	37	2	no	—
120	C₉×3₋¹⁺²	—	37	3	no	—
121	C₉×3₋¹⁺²	—	37	3	no	—
122	C₉×3₋¹⁺²	C₃₇⋊C₃	1	1	yes	characteristic, complemented
123	#6887	—	37	6	no	—
124	#6887	—	37	3	no	—
125	C₃×C₃²⋊C₉	—	37	1	no	—
126	C₃×C₃²⋊C₉	—	37	1	no	—
127	C₃×C₃²⋊C₉	—	37	1	no	—
128	C₃×C₃₇	—	3	3	no	—
129	C₃×C₃₇	C₃³×C₉	1	1	yes	characteristic
130	C₃×C₃₇	3₋¹⁺⁴	1	1	yes	characteristic
131	C₃×C₃₇	3₋¹⁺⁴	1	1	yes	characteristic
132	C₃×C₃₇	3₋¹⁺⁴	1	1	yes	characteristic
133	C₃₇⋊C₃	—	3	3	no	—
134	C₃₇⋊C₃	—	3	3	no	—
135	C₃₇⋊C₃	—	3	3	no	—
136	C₃⁴	—	37	1	no	—
137	C₃×3₋¹⁺²	—	37	2	no	—
138	C₃×3₋¹⁺²	—	37	27	no	—
139	C₃×3₋¹⁺²	—	37	3	no	—
140	C₃×3₋¹⁺²	—	37	3	no	—
141	C₃×3₋¹⁺²	C₃×C₃₇⋊C₃	1	1	yes	characteristic
142	C₃²×C₉	—	37	3	no	—
143	C₃²×C₉	—	37	2	no	—
144	C₃²×C₉	—	37	3	no	—
145	C₃²×C₉	—	37	3	no	—
146	C₃²×C₉	C₃×C₃₇⋊C₃	1	1	yes	characteristic
147	C₉⋊C₉	—	37	6	no	—
148	C₉⋊C₉	—	37	6	no	—
149	C₉⋊C₉	—	37	6	no	—
150	C₉⋊C₉	—	37	6	no	—
151	C₉⋊C₉	—	37	6	no	—
152	C₉⋊C₉	—	37	6	no	—
153	C₉⋊C₉	—	37	6	no	—
154	C₉⋊C₉	—	37	6	no	—
155	C₉⋊C₉	C₃×C₃₇⋊C₃	1	6	yes	complemented
156	C₃²⋊C₉	—	37	2	no	—
157	C₃²⋊C₉	—	37	2	no	—
158	C₃²⋊C₉	—	37	2	no	—
159	C₃²⋊C₉	—	37	3	no	—
160	C₃²⋊C₉	—	37	3	no	—
161	C₃²⋊C₉	—	37	1	no	—
162	C₃²⋊C₉	—	37	2	no	—
163	C₃²⋊C₉	—	37	2	no	—
164	C₃²⋊C₉	—	37	3	no	—
165	C₃²⋊C₉	—	37	3	no	—
166	C₃²⋊C₉	—	37	1	no	—
167	C₃²⋊C₉	—	37	1	no	—
168	C₃²⋊C₉	C₃×C₃₇⋊C₃	1	1	yes	characteristic
169	C₃²⋊C₉	C₃×C₃₇⋊C₃	1	1	yes	characteristic
170	C₉²	—	37	6	no	—
171	C₉²	—	37	6	no	—
172	C₉²	—	37	6	no	—
173	C₉²	—	37	6	no	—
174	C₉²	C₃×C₃₇⋊C₃	1	3	yes	complemented
175	C₃₇	#10624863	1	1	yes	characteristic, complemented, Sylow
176	C₃³	—	37	9	no	—
177	C₃³	—	37	1	no	—
178	C₃³	—	37	1	no	—
179	C₃³	—	37	1	no	—
180	C₃³	—	333	1	no	—
181	C₃³	—	333	1	no	—
182	C₃³	—	333	1	no	—
183	C₃³	C₃²×C₃₇⋊C₃	1	1	yes	characteristic
184	3₋¹⁺²	—	37	18	no	—
185	3₋¹⁺²	—	37	27	no	—
186	3₋¹⁺²	—	37	27	no	—
187	3₋¹⁺²	C₉×C₃₇⋊C₃	1	9	yes	complemented
188	C₃×C₉	—	9	1	no	—
189	C₃×C₉	—	37	6	no	—
190	C₃×C₉	—	37	27	no	—
191	C₃×C₉	—	37	6	no	—
192	C₃×C₉	—	37	6	no	—
193	C₃×C₉	—	37	2	no	—
194	C₃×C₉	—	37	2	no	—
195	C₃×C₉	—	37	2	no	—
196	C₃×C₉	—	333	3	no	—
197	C₃×C₉	—	333	3	no	—
198	C₃×C₉	—	333	2	no	—
199	C₃×C₉	C₃²×C₃₇⋊C₃	1	2	yes	—
200	C₃×C₉	C₃²×C₃₇⋊C₃	1	3	yes	—
201	C₃×C₉	C₃²×C₃₇⋊C₃	1	3	yes	—
202	C₃×C₉	C₃²×C₃₇⋊C₃	1	1	yes	characteristic
203	C₃×C₉	C₃²×C₃₇⋊C₃	1	3	yes	—
204	C₃²	—	3	1	no	—
205	C₃²	—	3	1	no	—
206	C₃²	—	3	1	no	—
207	C₃²	—	37	3	no	—
208	C₃²	—	37	3	no	—
209	C₃²	—	37	3	no	—
210	C₃²	—	333	9	no	—
211	C₃²	—	111	1	no	—
212	C₃²	—	111	1	no	—
213	C₃²	—	111	1	no	—
214	C₃²	—	111	1	no	—
215	C₃²	—	111	1	no	—
216	C₃²	—	111	1	no	—
217	C₃²	—	111	1	no	—
218	C₃²	—	111	1	no	—
219	C₃²	—	111	1	no	—
220	C₃²	C₃×C₉×C₃₇⋊C₃	1	3	yes	—
221	C₃²	C₃³×C₃₇⋊C₃	1	1	yes	characteristic
222	C₉	—	3	3	no	—
223	C₉	—	3	3	no	—
224	C₉	—	3	2	no	—
225	C₉	—	3	1	no	—
226	C₉	—	37	18	no	—
227	C₉	—	111	6	no	—
228	C₉	—	111	6	no	—
229	C₉	—	111	2	no	—
230	C₉	—	111	2	no	—
231	C₉	—	111	2	no	—
232	C₉	C₃×C₉×C₃₇⋊C₃	1	9	yes	—
233	C₃	—	3	3	no	—
234	C₃	—	111	3	no	—
235	C₃	—	111	3	no	—
236	C₃	—	111	3	no	—
237	C₃	C₃²×C₉×C₃₇⋊C₃	1	1	yes	characteristic
238	C₃	#22582677	1	1	yes	characteristic
239	C₃	#22582678	1	1	yes	characteristic
240	C₃	#22582679	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₃²	C₃³×C₃₇⋊C₃
commutator	C₃×C₃₇	C₃³×C₉
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	#10624863
LCS	G ▷ C₃×C₃₇ ▷ C₃₇
UCS	C₁ ◁ C₃² ◁ C₉×3₋¹⁺²

Complements

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

N	H ≅ G/N	type
2. #22582548	C₃	semidirect
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. #22582537	C₃	semidirect
11. #22582536	C₃	semidirect
15. C₉×3₋¹⁺²×C₃₇	C₃	semidirect
20. 3₋¹⁺²×C₃₇⋊C₃	C₉	semidirect
25. #15688506	C₃²	semidirect
26. #15688505	C₃²	semidirect
27. #15688504	C₃²	semidirect
28. #15688504	C₃²	semidirect
29. #15688504	C₃²	semidirect
30. #15688504	C₃²	semidirect
31. #15688504	C₃²	semidirect
32. #15688504	C₃²	semidirect
45. C₉×#66524	C₃²	semidirect
46. C₉×#66524	C₃²	semidirect
47. C₉×#66524	C₃²	semidirect
48. C₉×#66524	C₃²	semidirect
51. C₃₇×C₉⋊C₉	C₃²	semidirect
54. C₉²×C₃₇	C₃²	semidirect
63. 3₋¹⁺²×C₃₇	C₃×C₉	semidirect
70. #11758608	C₃×C₉	semidirect
78. #11758606	C₃×C₉	semidirect
79. #11758605	C₃×C₉	semidirect
82. C₉×C₃₇⋊C₃	C₃×C₉	semidirect
122. C₉×3₋¹⁺²	C₃₇⋊C₃	semidirect
155. C₉⋊C₉	C₃×C₃₇⋊C₃	semidirect
174. C₉²	C₃×C₃₇⋊C₃	semidirect
175. C₃₇	#10624863	semidirect
187. 3₋¹⁺²	C₉×C₃₇⋊C₃	semidirect

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
#22582548	3
3₋¹⁺²×#66524	3
C₉×#11758606	3
C₉×#11758605	3
C₉×#11758608	3
#22582543	3
#22582542	3
#22582537	3
#22582536	3
C₃×#15688502	3
C₃²⋊C₉×C₃₇⋊C₃	3
C₉×3₋¹⁺²×C₃₇	3
#10624863	37