Subgroups — #44535041

Order 31347.

644 conjugacy classes of subgroups.

The subgroup lattice has 238 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#23782834	C₃	1	4	yes	maximal, complemented
2	#23782834	C₃	1	2	yes	maximal, complemented
3	#23782834	C₃	1	2	yes	maximal, complemented
4	#23782828	C₃	1	2	yes	maximal, complemented
5	#23782827	C₃	1	2	yes	maximal, complemented
6	#23782826	C₃	1	4	yes	maximal, complemented
7	#23782825	C₃	1	1	yes	maximal, characteristic, complemented
8	#23782821	C₃	1	4	yes	maximal, complemented
9	#23782821	C₃	1	4	yes	maximal, complemented
10	#23782821	C₃	1	4	yes	maximal, complemented
11	#23782797	C₃	1	1	yes	maximal, characteristic
12	#23782796	C₃	1	1	yes	maximal, characteristic
13	#23782781	C₃	1	4	yes	maximal, complemented
14	#23782780	C₃	1	2	yes	maximal, complemented
15	C₃²⋊C₉×C₄₃⋊C₃	C₃	1	2	yes	maximal
16	C₄₃×#6898	C₃	1	1	yes	maximal, characteristic, complemented
17	C₃³×C₄₃⋊C₃	C₃²	1	1	yes	characteristic
18	C₃×#11923957	C₃²	1	1	yes	characteristic
19	C₃×#11923957	C₃²	1	2	yes	—
20	C₃×#11923956	C₃²	1	1	yes	characteristic
21	C₃×#11923956	C₃²	1	2	yes	—
22	C₃×C₉×C₄₃⋊C₃	C₃²	1	2	yes	—
23	C₃×C₉×C₄₃⋊C₃	C₃²	1	4	yes	—
24	#16812500	C₃²	1	4	yes	complemented
25	#16812500	C₃²	1	2	yes	complemented
26	#16812500	C₃²	1	2	yes	complemented
27	#16812499	C₃²	1	4	yes	complemented
28	#16812499	C₃²	1	2	yes	complemented
29	#16812499	C₃²	1	2	yes	complemented
30	(C₄₃⋊C₉)⋊C₉	C₃²	1	4	yes	complemented
31	(C₄₃⋊C₉)⋊C₉	C₃²	1	4	yes	complemented
32	(C₄₃⋊C₉)⋊C₉	C₃²	1	4	yes	complemented
33	(C₄₃⋊C₉)⋊C₉	C₃²	1	4	yes	complemented
34	(C₄₃⋊C₉)⋊C₉	C₃²	1	4	yes	complemented
35	(C₄₃⋊C₉)⋊C₉	C₃²	1	4	yes	complemented
36	(C₄₃⋊C₉)⋊C₉	C₃²	1	4	yes	complemented
37	(C₄₃⋊C₉)⋊C₉	C₃²	1	4	yes	complemented
38	(C₄₃⋊C₉)⋊C₉	C₃²	1	8	yes	complemented
39	(C₄₃⋊C₉)⋊C₉	C₃²	1	4	yes	complemented
40	(C₄₃⋊C₉)⋊C₉	C₃²	1	4	yes	complemented
41	C₃²⋊(C₄₃⋊C₉)	C₃²	1	4	yes	—
42	C₃²⋊(C₄₃⋊C₉)	C₃²	1	2	yes	—
43	C₃²⋊(C₄₃⋊C₉)	C₃²	1	2	yes	—
44	(C₃×C₄₃⋊C₃)⋊C₉	C₃²	1	2	yes	—
45	(C₃×C₄₃⋊C₃)⋊C₉	C₃²	1	4	yes	—
46	(C₃×C₄₃⋊C₃)⋊C₉	C₃²	1	2	yes	—
47	(C₃×C₄₃⋊C₃)⋊C₉	C₃²	1	4	yes	—
48	(C₃×C₄₃⋊C₃)⋊C₉	C₃²	1	2	yes	—
49	(C₃×C₄₃⋊C₃)⋊C₉	C₃²	1	2	yes	—
50	(C₃×C₄₃⋊C₃)⋊C₉	C₃²	1	4	yes	—
51	(C₃×C₄₃⋊C₃)⋊C₉	C₃²	1	4	yes	—
52	C₉×C₄₃⋊C₉	C₃²	1	4	yes	complemented
53	C₉×C₄₃⋊C₉	C₃²	1	2	yes	complemented
54	C₉×C₄₃⋊C₉	C₃²	1	2	yes	complemented
55	C₄₃×C₉⋊C₉	C₃²	1	4	yes	complemented
56	C₄₃×C₉⋊C₉	C₃²	1	2	yes	complemented
57	C₄₃×C₉⋊C₉	C₃²	1	2	yes	complemented
58	C₄₃×C₃²⋊C₉	C₃²	1	2	yes	—
59	C₄₃×C₃²⋊C₉	C₃²	1	1	yes	characteristic
60	C₄₃×C₃²⋊C₉	C₃²	1	1	yes	characteristic
61	C₉²×C₄₃	C₃²	1	1	yes	characteristic, complemented
62	C₃²×C₄₃⋊C₃	—	3	6	no	—
63	C₃²×C₄₃⋊C₃	—	9	1	no	—
64	C₃²×C₄₃⋊C₃	—	9	1	no	—
65	C₃²×C₄₃⋊C₃	C₃³	1	1	yes	characteristic
66	C₃²×C₄₃⋊C₃	C₃³	1	2	yes	—
67	#11923957	—	9	1	no	—
68	#11923957	—	9	2	no	—
69	#11923956	—	9	1	no	—
70	#11923956	—	9	2	no	—
71	C₃×C₄₃⋊C₉	C₃³	1	4	yes	—
72	C₃×C₄₃⋊C₉	C₃³	1	4	yes	—
73	C₃×C₄₃⋊C₉	C₃³	1	4	yes	—
74	C₃×C₄₃⋊C₉	C₃³	1	2	yes	—
75	C₃×C₄₃⋊C₉	C₃³	1	2	yes	—
76	C₃×C₄₃⋊C₉	C₃³	1	4	yes	—
77	C₃×C₄₃⋊C₉	C₃³	1	4	yes	—
78	C₉×C₄₃⋊C₃	—	3	6	no	—
79	C₉×C₄₃⋊C₃	—	3	12	no	—
80	C₃³×C₄₃	C₃³	1	1	yes	characteristic
81	C₃×C₉×C₄₃	C₃³	1	2	yes	—
82	C₃×C₉×C₄₃	C₃³	1	1	yes	characteristic
83	C₃×C₉×C₄₃	C₃³	1	1	yes	characteristic
84	C₃×C₉×C₄₃	C₃³	1	2	yes	—
85	C₃×C₉×C₄₃	C₃³	1	2	yes	—
86	C₃×C₉×C₄₃	C₃³	1	4	yes	—
87	#10624902	—	43	1	no	maximal, Sylow
88	C₃²×C₄₃	—	3	2	no	—
89	C₃²×C₄₃	—	3	1	no	—
90	C₃²×C₄₃	—	3	1	no	—
91	C₃²×C₄₃	C₃⁴	1	1	yes	characteristic
92	C₃×C₄₃⋊C₃	—	3	2	no	—
93	C₃×C₄₃⋊C₃	—	3	2	no	—
94	C₃×C₄₃⋊C₃	—	3	2	no	—
95	C₃×C₄₃⋊C₃	—	3	2	no	—
96	C₃×C₄₃⋊C₃	—	3	1	no	—
97	C₃×C₄₃⋊C₃	—	3	1	no	—
98	C₃×C₄₃⋊C₃	—	9	9	no	—
99	C₃×C₄₃⋊C₃	#737	1	6	yes	—
100	C₉×C₄₃	—	3	2	no	—
101	C₉×C₄₃	—	3	4	no	—
102	C₉×C₄₃	—	3	1	no	—
103	C₉×C₄₃	—	3	2	no	—
104	C₉×C₄₃	—	3	1	no	—
105	C₉×C₄₃	—	3	2	no	—
106	C₄₃⋊C₉	—	3	4	no	—
107	C₄₃⋊C₉	—	3	4	no	—
108	C₄₃⋊C₉	—	3	4	no	—
109	C₄₃⋊C₉	—	3	4	no	—
110	C₄₃⋊C₉	—	3	2	no	—
111	C₄₃⋊C₉	—	3	4	no	—
112	C₄₃⋊C₉	—	3	2	no	—
113	#6898	—	43	2	no	—
114	#6898	—	43	2	no	—
115	#6898	—	43	4	no	—
116	#6898	C₄₃⋊C₃	1	1	yes	characteristic, complemented
117	#6896	—	43	4	no	—
118	#6896	—	43	2	no	—
119	#6896	—	43	2	no	—
120	#6896	—	43	1	no	—
121	#6895	—	43	4	no	—
122	#6895	—	43	4	no	—
123	#6895	—	43	4	no	—
124	#6891	—	43	1	no	—
125	#6891	—	43	1	no	—
126	#6887	—	43	4	no	—
127	#6887	—	43	2	no	—
128	C₃×C₃²⋊C₉	—	43	2	no	—
129	C₃×C₄₃	—	9	1	no	—
130	C₃×C₄₃	C₃×#737	1	2	yes	—
131	C₃×C₄₃	3₋¹⁺⁴	1	1	yes	characteristic
132	C₃×C₄₃	3₋¹⁺⁴	1	1	yes	characteristic
133	C₄₃⋊C₃	—	3	6	no	—
134	C₄₃⋊C₃	—	9	1	no	—
135	C₃⁴	—	43	1	no	—
136	C₃×3₋¹⁺²	—	43	2	no	—
137	C₃×3₋¹⁺²	—	43	1	no	—
138	C₃×3₋¹⁺²	—	43	2	no	—
139	C₃×3₋¹⁺²	—	43	1	no	—
140	C₃²×C₉	—	43	4	no	—
141	C₃²×C₉	—	43	2	no	—
142	C₉⋊C₉	—	43	4	no	—
143	C₉⋊C₉	—	43	4	no	—
144	C₉⋊C₉	—	43	4	no	—
145	C₉⋊C₉	—	43	4	no	—
146	C₉⋊C₉	—	43	4	no	—
147	C₉⋊C₉	—	43	4	no	—
148	C₉⋊C₉	—	43	8	no	—
149	C₉⋊C₉	—	43	4	no	—
150	C₉⋊C₉	—	43	4	no	—
151	C₉⋊C₉	—	43	4	no	—
152	C₉⋊C₉	—	43	4	no	—
153	C₉⋊C₉	—	43	4	no	—
154	C₉⋊C₉	—	43	2	no	—
155	C₉⋊C₉	—	43	2	no	—
156	C₉⋊C₉	—	43	4	no	—
157	C₉⋊C₉	—	43	2	no	—
158	C₉⋊C₉	—	43	2	no	—
159	C₉⋊C₉	C₃×C₄₃⋊C₃	1	4	yes	complemented
160	C₉⋊C₉	C₃×C₄₃⋊C₃	1	2	yes	complemented
161	C₉⋊C₉	C₃×C₄₃⋊C₃	1	2	yes	complemented
162	C₃²⋊C₉	—	43	4	no	—
163	C₃²⋊C₉	—	43	2	no	—
164	C₃²⋊C₉	—	43	2	no	—
165	C₃²⋊C₉	—	43	2	no	—
166	C₃²⋊C₉	—	43	4	no	—
167	C₃²⋊C₉	—	43	4	no	—
168	C₃²⋊C₉	—	43	2	no	—
169	C₃²⋊C₉	—	43	4	no	—
170	C₃²⋊C₉	—	43	4	no	—
171	C₃²⋊C₉	—	43	2	no	—
172	C₃²⋊C₉	—	43	2	no	—
173	C₃²⋊C₉	C₃×C₄₃⋊C₃	1	2	yes	—
174	C₃²⋊C₉	C₃×C₄₃⋊C₃	1	1	yes	characteristic
175	C₃²⋊C₉	C₃×C₄₃⋊C₃	1	1	yes	characteristic
176	C₉²	—	43	2	no	—
177	C₉²	—	43	2	no	—
178	C₉²	—	43	4	no	—
179	C₉²	C₃×C₄₃⋊C₃	1	1	yes	characteristic, complemented
180	C₄₃	#10624902	1	1	yes	characteristic, complemented, Sylow
181	C₃³	—	43	2	no	—
182	C₃³	—	43	1	no	—
183	C₃³	—	129	6	no	—
184	C₃³	—	387	1	no	—
185	C₃³	—	387	1	no	—
186	C₃³	C₃²×C₄₃⋊C₃	1	1	yes	characteristic
187	3₋¹⁺²	—	387	2	no	—
188	3₋¹⁺²	—	387	1	no	—
189	3₋¹⁺²	—	387	2	no	—
190	3₋¹⁺²	—	387	1	no	—
191	C₃×C₉	—	43	4	no	—
192	C₃×C₉	—	43	4	no	—
193	C₃×C₉	—	43	2	no	—
194	C₃×C₉	—	43	4	no	—
195	C₃×C₉	—	43	2	no	—
196	C₃×C₉	—	43	4	no	—
197	C₃×C₉	—	43	4	no	—
198	C₃×C₉	—	129	6	no	—
199	C₃×C₉	—	129	12	no	—
200	C₃×C₉	C₃²×C₄₃⋊C₃	1	2	yes	—
201	C₃×C₉	C₃²×C₄₃⋊C₃	1	2	yes	—
202	C₃×C₉	C₃²×C₄₃⋊C₃	1	2	yes	—
203	C₃×C₉	C₃²×C₄₃⋊C₃	1	1	yes	characteristic
204	C₃×C₉	C₃²×C₄₃⋊C₃	1	1	yes	characteristic
205	C₃×C₉	C₃²×C₄₃⋊C₃	1	4	yes	—
206	C₃²	—	3	2	no	—
207	C₃²	—	3	1	no	—
208	C₃²	—	3	1	no	—
209	C₃²	—	43	6	no	—
210	C₃²	—	129	2	no	—
211	C₃²	—	129	2	no	—
212	C₃²	—	129	2	no	—
213	C₃²	—	129	2	no	—
214	C₃²	—	129	1	no	—
215	C₃²	—	129	1	no	—
216	C₃²	—	387	9	no	—
217	C₃²	C₃³×C₄₃⋊C₃	1	1	yes	characteristic
218	C₉	—	3	2	no	—
219	C₉	—	3	4	no	—
220	C₉	—	3	2	no	—
221	C₉	—	3	1	no	—
222	C₉	—	3	2	no	—
223	C₉	—	3	1	no	—
224	C₉	—	129	4	no	—
225	C₉	—	129	4	no	—
226	C₉	—	129	4	no	—
227	C₉	—	129	4	no	—
228	C₉	—	129	2	no	—
229	C₉	—	129	4	no	—
230	C₉	—	129	2	no	—
231	C₃	—	9	1	no	—
232	C₃	—	129	6	no	—
233	C₃	—	387	1	no	—
234	C₃	#737×C₄₃⋊C₃	1	2	yes	—
235	C₃	#23782921	1	1	yes	characteristic
236	C₃	#23782922	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₃²	C₃³×C₄₃⋊C₃
commutator, socle, cosocle	C₃²×C₄₃	C₃⁴
Fitting	C₄₃×#6898	C₃
prime radical	C₃³×C₄₃⋊C₃	C₃²
hypercenter	#6898	C₄₃⋊C₃
hypocenter	C₄₃	#10624902
LCS	G ▷ C₃²×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₃² ◁ #6898

Complements

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

N	H ≅ G/N	type
1. #23782834	C₃	semidirect
2. #23782834	C₃	semidirect
3. #23782834	C₃	semidirect
4. #23782828	C₃	semidirect
5. #23782827	C₃	semidirect
6. #23782826	C₃	semidirect
7. #23782825	C₃	semidirect
8. #23782821	C₃	semidirect
9. #23782821	C₃	semidirect
10. #23782821	C₃	semidirect
13. #23782781	C₃	semidirect
14. #23782780	C₃	semidirect
16. C₄₃×#6898	C₃	semidirect
24. #16812500	C₃²	semidirect
25. #16812500	C₃²	semidirect
26. #16812500	C₃²	semidirect
27. #16812499	C₃²	semidirect
28. #16812499	C₃²	semidirect
29. #16812499	C₃²	semidirect
30. (C₄₃⋊C₉)⋊C₉	C₃²	semidirect
31. (C₄₃⋊C₉)⋊C₉	C₃²	semidirect
32. (C₄₃⋊C₉)⋊C₉	C₃²	semidirect
33. (C₄₃⋊C₉)⋊C₉	C₃²	semidirect
34. (C₄₃⋊C₉)⋊C₉	C₃²	semidirect
35. (C₄₃⋊C₉)⋊C₉	C₃²	semidirect
36. (C₄₃⋊C₉)⋊C₉	C₃²	semidirect
37. (C₄₃⋊C₉)⋊C₉	C₃²	semidirect
38. (C₄₃⋊C₉)⋊C₉	C₃²	semidirect
39. (C₄₃⋊C₉)⋊C₉	C₃²	semidirect
40. (C₄₃⋊C₉)⋊C₉	C₃²	semidirect
52. C₉×C₄₃⋊C₉	C₃²	semidirect
53. C₉×C₄₃⋊C₉	C₃²	semidirect
54. C₉×C₄₃⋊C₉	C₃²	semidirect
55. C₄₃×C₉⋊C₉	C₃²	semidirect
56. C₄₃×C₉⋊C₉	C₃²	semidirect
57. C₄₃×C₉⋊C₉	C₃²	semidirect
61. C₉²×C₄₃	C₃²	semidirect
116. #6898	C₄₃⋊C₃	semidirect
159. C₉⋊C₉	C₃×C₄₃⋊C₃	semidirect
160. C₉⋊C₉	C₃×C₄₃⋊C₃	semidirect
161. C₉⋊C₉	C₃×C₄₃⋊C₃	semidirect
179. C₉²	C₃×C₄₃⋊C₃	semidirect
180. C₄₃	#10624902	semidirect

Maximal subgroups

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

subgroup	index
#23782834	3
#23782828	3
#23782827	3
#23782826	3
#23782825	3
#23782821	3
#23782797	3
#23782796	3
#23782781	3
#23782780	3
C₃²⋊C₉×C₄₃⋊C₃	3
C₄₃×#6898	3
#10624902	43