Subgroups — #41007839

Order 26973.

740 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	#22582599	C₃	1	6	yes	maximal, complemented
3	#22582596	C₃	1	3	yes	maximal, complemented
4	#22582583	C₃	1	6	yes	maximal, complemented
5	#22582581	C₃	1	3	yes	maximal, complemented
6	#22582574	C₃	1	2	yes	maximal, complemented
7	#22582574	C₃	1	2	yes	maximal, complemented
8	#22582571	C₃	1	2	yes	maximal, complemented
9	#22582567	C₃	1	6	yes	maximal, complemented
10	#22582553	C₃	1	1	yes	maximal, characteristic
11	#22582553	C₃	1	1	yes	maximal, characteristic
12	3₋¹⁺²×#66524	C₃	1	2	yes	maximal, complemented
13	C₉×#11758605	C₃	1	3	yes	maximal, complemented
14	C₃×#15688502	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×#11758605	C₃²	1	3	yes	—
20	C₃×#11758605	C₃²	1	3	yes	—
21	C₃×#11758606	C₃²	1	3	yes	—
22	3₋¹⁺²×C₃₇⋊C₃	—	3	9	no	—
23	C₃²×#66524	C₃²	1	2	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	#15688505	C₃²	1	6	yes	complemented
28	#15688505	C₃²	1	6	yes	complemented
29	#15688505	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	#15688504	C₃²	1	6	yes	complemented
34	#15688504	C₃²	1	6	yes	complemented
35	#15688504	C₃²	1	6	yes	complemented
36	C₃²⋊#66524	C₃²	1	2	yes	—
37	C₃²⋊#66524	C₃²	1	2	yes	—
38	#15688502	C₃²	1	1	yes	characteristic
39	#15688502	C₃²	1	2	yes	—
40	#15688502	C₃²	1	1	yes	characteristic
41	#15688502	C₃²	1	2	yes	—
42	#15688502	C₃²	1	1	yes	characteristic
43	#15688502	C₃²	1	2	yes	—
44	#15688502	C₃²	1	3	yes	—
45	#15688502	C₃²	1	3	yes	—
46	#15688502	C₃²	1	3	yes	—
47	#15688502	C₃²	1	3	yes	—
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	—	3	3	no	—
57	C₃²×C₃₇⋊C₃	—	3	3	no	—
58	C₃²×C₃₇⋊C₃	—	9	1	no	—
59	C₃²×C₃₇⋊C₃	—	9	1	no	—
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	3₋¹⁺²	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	—	3	6	no	—
71	C₃×#66524	—	9	2	no	—
72	C₃×#66524	C₃³	1	2	yes	—
73	C₃×#66524	C₃³	1	6	yes	—
74	C₃×#66524	C₃³	1	6	yes	—
75	C₃×#66524	C₃³	1	2	yes	—
76	C₃×#66524	C₃³	1	2	yes	—
77	C₃×#66524	C₃³	1	6	yes	—
78	#11758606	—	3	9	no	—
79	#11758605	—	3	9	no	—
80	#11758605	—	3	9	no	—
81	#11758605	—	9	3	no	—
82	C₉×C₃₇⋊C₃	—	3	9	no	—
83	#10624899	—	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	C₃×3₋¹⁺²	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	—	3	1	no	—
99	C₃×C₃₇⋊C₃	—	3	1	no	—
100	C₃×C₃₇⋊C₃	—	9	9	no	—
101	C₃×C₃₇⋊C₃	#737	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	C₃×3₋¹⁺²	1	9	yes	—
107	#66524	—	3	6	no	—
108	#66524	—	3	6	no	—
109	#66524	—	3	6	no	—
110	#66524	—	3	2	no	—
111	#66524	—	3	2	no	—
112	#66524	—	3	2	no	—
113	C₃²×3₋¹⁺²	—	37	1	no	—
114	#6899	—	37	6	no	—
115	#6899	—	37	3	no	—
116	#6896	—	37	6	no	—
117	#6896	—	37	3	no	—
118	#6894	—	37	2	no	—
119	#6894	—	37	6	no	—
120	#6894	—	37	2	no	—
121	#6894	—	37	2	no	—
122	#6891	—	37	1	no	—
123	#6891	—	37	1	no	—
124	C₉×3₋¹⁺²	—	37	2	no	—
125	C₉×3₋¹⁺²	—	37	3	no	—
126	C₉×3₋¹⁺²	C₃₇⋊C₃	1	1	yes	characteristic, complemented
127	C₃×C₃²⋊C₉	—	37	1	no	—
128	C₃×C₃₇	—	3	3	no	—
129	C₃×C₃₇	C₃²×3₋¹⁺²	1	1	yes	characteristic
130	C₃×C₃₇	C₃×#737	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	—	9	1	no	—
135	C₃₇⋊C₃	—	9	1	no	—
136	C₃⁴	—	37	1	no	—
137	C₃×3₋¹⁺²	—	37	2	no	—
138	C₃×3₋¹⁺²	—	37	3	no	—
139	C₃×3₋¹⁺²	—	37	3	no	—
140	C₃×3₋¹⁺²	—	37	3	no	—
141	C₃×3₋¹⁺²	—	37	3	no	—
142	C₃×3₋¹⁺²	—	111	9	no	—
143	C₃×3₋¹⁺²	C₃×C₃₇⋊C₃	1	1	yes	characteristic
144	C₃²×C₉	—	37	2	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	—	37	6	no	—
156	C₉⋊C₉	—	37	6	no	—
157	C₉⋊C₉	—	37	6	no	—
158	C₉⋊C₉	C₃×C₃₇⋊C₃	1	6	yes	complemented
159	C₃²⋊C₉	—	37	2	no	—
160	C₃²⋊C₉	—	37	2	no	—
161	C₃²⋊C₉	—	37	1	no	—
162	C₃²⋊C₉	—	37	2	no	—
163	C₃²⋊C₉	—	37	2	no	—
164	C₃²⋊C₉	—	37	1	no	—
165	C₃²⋊C₉	—	37	2	no	—
166	C₃²⋊C₉	—	37	1	no	—
167	C₃²⋊C₉	—	37	3	no	—
168	C₃²⋊C₉	—	37	3	no	—
169	C₃²⋊C₉	—	37	3	no	—
170	C₃²⋊C₉	—	37	3	no	—
171	C₃²⋊C₉	C₃×C₃₇⋊C₃	1	1	yes	characteristic
172	C₃²⋊C₉	C₃×C₃₇⋊C₃	1	1	yes	characteristic
173	C₉²	—	37	6	no	—
174	C₉²	C₃×C₃₇⋊C₃	1	3	yes	complemented
175	C₃₇	#10624899	1	1	yes	characteristic, complemented, Sylow
176	C₃³	—	37	1	no	—
177	C₃³	—	37	1	no	—
178	C₃³	—	37	1	no	—
179	C₃³	—	333	1	no	—
180	C₃³	—	333	1	no	—
181	C₃³	—	111	3	no	—
182	C₃³	—	111	3	no	—
183	C₃³	C₃²×C₃₇⋊C₃	1	1	yes	characteristic
184	3₋¹⁺²	—	333	3	no	—
185	3₋¹⁺²	—	111	9	no	—
186	3₋¹⁺²	—	111	6	no	—
187	3₋¹⁺²	—	111	9	no	—
188	3₋¹⁺²	—	111	9	no	—
189	3₋¹⁺²	#11758605	1	9	yes	complemented
190	C₃×C₉	—	9	1	no	—
191	C₃×C₉	—	37	2	no	—
192	C₃×C₉	—	37	6	no	—
193	C₃×C₉	—	37	6	no	—
194	C₃×C₉	—	37	2	no	—
195	C₃×C₉	—	37	2	no	—
196	C₃×C₉	—	37	6	no	—
197	C₃×C₉	—	333	2	no	—
198	C₃×C₉	—	111	9	no	—
199	C₃×C₉	C₃²×C₃₇⋊C₃	1	1	yes	characteristic
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	2	yes	—
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	—	333	9	no	—
209	C₃²	—	111	1	no	—
210	C₃²	—	111	1	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×#11758605	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	1	no	—
225	C₉	—	3	2	no	—
226	C₉	—	111	6	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×#11758605	1	9	yes	—
233	C₃	—	3	3	no	—
234	C₃	—	333	1	no	—
235	C₃	—	333	1	no	—
236	C₃	—	111	3	no	—
237	C₃	C₃²×#11758605	1	1	yes	characteristic
238	C₃	#737×C₃₇⋊C₃	1	1	yes	characteristic
239	C₃	#22582677	1	1	yes	characteristic
240	C₃	#22582679	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₃²	C₃³×C₃₇⋊C₃
commutator, socle, cosocle	C₃²×C₃₇	C₃⁴
Fitting	C₉×3₋¹⁺²×C₃₇	C₃
prime radical	C₃³×C₃₇⋊C₃	C₃²
hypercenter	C₉×3₋¹⁺²	C₃₇⋊C₃
hypocenter	C₃₇	#10624899
LCS	G ▷ C₃²×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. #22582599	C₃	semidirect
3. #22582596	C₃	semidirect
4. #22582583	C₃	semidirect
5. #22582581	C₃	semidirect
6. #22582574	C₃	semidirect
7. #22582574	C₃	semidirect
8. #22582571	C₃	semidirect
9. #22582567	C₃	semidirect
12. 3₋¹⁺²×#66524	C₃	semidirect
13. C₉×#11758605	C₃	semidirect
15. C₉×3₋¹⁺²×C₃₇	C₃	semidirect
25. #15688506	C₃²	semidirect
26. #15688505	C₃²	semidirect
27. #15688505	C₃²	semidirect
28. #15688505	C₃²	semidirect
29. #15688505	C₃²	semidirect
30. #15688504	C₃²	semidirect
31. #15688504	C₃²	semidirect
32. #15688504	C₃²	semidirect
33. #15688504	C₃²	semidirect
34. #15688504	C₃²	semidirect
35. #15688504	C₃²	semidirect
48. C₉×#66524	C₃²	semidirect
51. C₃₇×C₉⋊C₉	C₃²	semidirect
54. C₉²×C₃₇	C₃²	semidirect
63. 3₋¹⁺²×C₃₇	3₋¹⁺²	semidirect
126. C₉×3₋¹⁺²	C₃₇⋊C₃	semidirect
158. C₉⋊C₉	C₃×C₃₇⋊C₃	semidirect
174. C₉²	C₃×C₃₇⋊C₃	semidirect
175. C₃₇	#10624899	semidirect
189. 3₋¹⁺²	#11758605	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
#22582599	3
#22582596	3
#22582583	3
#22582581	3
#22582574	3
#22582571	3
#22582567	3
#22582553	3
3₋¹⁺²×#66524	3
C₉×#11758605	3
C₃×#15688502	3
C₉×3₋¹⁺²×C₃₇	3
#10624899	37