Subgroups — #44534779

Order 31347.

758 conjugacy classes of subgroups.

The subgroup lattice has 254 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×#16812526	C₃	1	1	yes	maximal, characteristic
2	#23782847	C₃	1	1	yes	maximal, characteristic, complemented
3	#23782846	C₃	1	1	yes	maximal, characteristic, complemented
4	#23782844	C₃	1	2	yes	maximal, complemented
5	#23782843	C₃	1	2	yes	maximal, complemented
6	#23782838	C₃	1	2	yes	maximal, complemented
7	#23782828	C₃	1	3	yes	maximal, complemented
8	#23782827	C₃	1	3	yes	maximal, complemented
9	#23782824	C₃	1	3	yes	maximal, complemented
10	#23782811	C₃	1	3	yes	maximal, complemented
11	#23782810	C₃	1	3	yes	maximal, complemented
12	#23782799	C₃	1	3	yes	maximal
13	3₋¹⁺²×C₄₃⋊C₉	C₃	1	3	yes	maximal, complemented
14	#11923957⋊C₉	C₃	1	3	yes	maximal, complemented
15	#11923956⋊C₉	C₃	1	3	yes	maximal, complemented
16	#23782781	C₃	1	2	yes	maximal, complemented
17	C₉²×C₄₃⋊C₃	C₃	1	1	yes	maximal, characteristic, complemented
18	C₄₃×#6900	C₃	1	1	yes	maximal, characteristic, complemented
19	#16812526	—	3	9	no	—
20	C₃×#11923958	C₃²	1	1	yes	characteristic
21	C₃×#11923957	C₃²	1	1	yes	characteristic
22	C₃×#11923957	C₃²	1	1	yes	characteristic
23	C₃×#11923957	C₃²	1	1	yes	characteristic
24	C₃×#11923957	C₃²	1	3	yes	—
25	C₃×#11923956	C₃²	1	1	yes	characteristic
26	C₃×#11923956	C₃²	1	1	yes	characteristic
27	C₃×#11923956	C₃²	1	1	yes	characteristic
28	C₃×#11923956	C₃²	1	3	yes	—
29	C₃×C₃²⋊(C₄₃⋊C₃)	C₃²	1	1	yes	characteristic
30	C₃²×C₄₃⋊C₉	C₃²	1	1	yes	characteristic
31	C₃×C₉×C₄₃⋊C₃	C₃²	1	1	yes	characteristic
32	C₃×C₉×C₄₃⋊C₃	C₃²	1	1	yes	characteristic
33	C₃×C₉×C₄₃⋊C₃	C₃²	1	3	yes	—
34	C₃×C₉×C₄₃⋊C₃	C₃²	1	1	yes	characteristic
35	#16812500	C₃²	1	2	yes	—
36	#16812500	C₃²	1	3	yes	—
37	#16812500	C₃²	1	6	yes	—
38	#16812500	C₃²	1	3	yes	—
39	#16812499	C₃²	1	2	yes	—
40	#16812499	C₃²	1	3	yes	—
41	#16812499	C₃²	1	6	yes	—
42	#16812499	C₃²	1	3	yes	—
43	(C₄₃⋊C₉)⋊C₉	C₃²	1	6	yes	—
44	(C₄₃⋊C₉)⋊C₉	C₃²	1	6	yes	—
45	(C₄₃⋊C₉)⋊C₉	C₃²	1	6	yes	—
46	(C₄₃⋊C₉)⋊C₉	C₃²	1	6	yes	—
47	(C₄₃⋊C₉)⋊C₉	C₃²	1	6	yes	—
48	C₃²⋊(C₄₃⋊C₉)	C₃²	1	6	yes	—
49	(C₃×C₄₃⋊C₃)⋊C₉	C₃²	1	6	yes	—
50	(C₃×C₄₃⋊C₃)⋊C₉	C₃²	1	6	yes	—
51	(C₃×C₄₃⋊C₃)⋊C₉	C₃²	1	6	yes	—
52	C₉×C₄₃⋊C₉	C₃²	1	3	yes	—
53	C₉×C₄₃⋊C₉	C₃²	1	3	yes	—
54	C₉×C₄₃⋊C₉	C₃²	1	2	yes	—
55	C₉×C₄₃⋊C₉	C₃²	1	6	yes	—
56	C₃×3₋¹⁺²×C₄₃	C₃²	1	1	yes	characteristic
57	C₃×3₋¹⁺²×C₄₃	C₃²	1	3	yes	—
58	C₄₃×C₉⋊C₉	C₃²	1	2	yes	—
59	C₄₃×C₉⋊C₉	C₃²	1	3	yes	—
60	C₄₃×C₉⋊C₉	C₃²	1	3	yes	—
61	C₉²×C₄₃	C₃²	1	1	yes	characteristic
62	C₃²×C₄₃⋊C₃	C₃³	1	1	yes	characteristic
63	C₃²×C₄₃⋊C₃	C₃³	1	1	yes	characteristic
64	C₃²×C₄₃⋊C₃	C₃³	1	1	yes	characteristic
65	#11923958	—	3	3	no	—
66	#11923957	—	3	3	no	—
67	#11923957	—	3	3	no	—
68	#11923957	—	3	9	no	—
69	#11923957	—	3	3	no	—
70	#11923956	—	3	3	no	—
71	#11923956	—	3	3	no	—
72	#11923956	—	3	9	no	—
73	#11923956	—	3	3	no	—
74	C₃²⋊(C₄₃⋊C₃)	—	3	3	no	—
75	C₃×C₄₃⋊C₉	—	3	3	no	—
76	C₃×C₄₃⋊C₉	C₃³	1	6	yes	—
77	C₃×C₄₃⋊C₉	C₃³	1	6	yes	—
78	C₃×C₄₃⋊C₉	C₃³	1	6	yes	—
79	C₃×C₄₃⋊C₉	C₃³	1	1	yes	characteristic
80	C₃×C₄₃⋊C₉	C₃³	1	1	yes	characteristic
81	C₃×C₄₃⋊C₉	C₃³	1	1	yes	characteristic
82	C₃×C₄₃⋊C₉	C₃³	1	1	yes	characteristic
83	C₃×C₄₃⋊C₉	C₃³	1	1	yes	characteristic
84	C₃×C₄₃⋊C₉	C₃³	1	1	yes	characteristic
85	C₉×C₄₃⋊C₃	—	3	9	no	—
86	C₉×C₄₃⋊C₃	—	3	3	no	—
87	C₉×C₄₃⋊C₃	—	3	3	no	—
88	C₉×C₄₃⋊C₃	3₋¹⁺²	1	9	yes	complemented
89	C₃³×C₄₃	C₃³	1	1	yes	characteristic
90	3₋¹⁺²×C₄₃	—	3	3	no	—
91	3₋¹⁺²×C₄₃	—	3	9	no	—
92	C₃×C₉×C₄₃	C₃³	1	6	yes	—
93	C₃×C₉×C₄₃	C₃³	1	1	yes	characteristic
94	C₃×C₉×C₄₃	C₃³	1	1	yes	characteristic
95	C₃×C₉×C₄₃	C₃³	1	1	yes	characteristic
96	C₃×C₉×C₄₃	C₃³	1	3	yes	—
97	#10624879	—	43	1	no	maximal, Sylow
98	C₃²×C₄₃	—	3	1	no	—
99	C₃²×C₄₃	—	3	3	no	—
100	C₃²×C₄₃	C₃⁴	1	1	yes	characteristic
101	C₃×C₄₃⋊C₃	—	3	3	no	—
102	C₃×C₄₃⋊C₃	—	3	1	no	—
103	C₃×C₄₃⋊C₃	—	3	1	no	—
104	C₃×C₄₃⋊C₃	—	3	3	no	—
105	C₃×C₄₃⋊C₃	—	3	3	no	—
106	C₃×C₄₃⋊C₃	C₃×3₋¹⁺²	1	3	yes	—
107	C₉×C₄₃	—	3	1	no	—
108	C₉×C₄₃	—	3	1	no	—
109	C₉×C₄₃	—	3	6	no	—
110	C₉×C₄₃	C₃×3₋¹⁺²	1	3	yes	—
111	C₉×C₄₃	#737	1	9	yes	complemented
112	C₄₃⋊C₉	—	3	1	no	—
113	C₄₃⋊C₉	—	3	1	no	—
114	C₄₃⋊C₉	—	3	1	no	—
115	C₄₃⋊C₉	—	3	1	no	—
116	C₄₃⋊C₉	—	3	6	no	—
117	C₄₃⋊C₉	—	3	6	no	—
118	C₄₃⋊C₉	—	3	6	no	—
119	C₄₃⋊C₉	C₃×3₋¹⁺²	1	3	yes	—
120	C₄₃⋊C₉	C₃×3₋¹⁺²	1	3	yes	—
121	C₃×#737	—	43	1	no	—
122	#6900	—	43	1	no	—
123	#6900	—	43	1	no	—
124	#6900	C₄₃⋊C₃	1	1	yes	characteristic, complemented
125	#6899	—	43	2	no	—
126	#6899	—	43	2	no	—
127	#6899	—	43	2	no	—
128	#6896	—	43	3	no	—
129	#6896	—	43	3	no	—
130	#6896	—	43	3	no	—
131	#6894	—	43	3	no	—
132	#6894	—	43	3	no	—
133	C₉⋊H(3,𝔽₃)	—	43	3	no	—
134	C₉×3₋¹⁺²	—	43	3	no	—
135	C₉×3₋¹⁺²	—	43	3	no	—
136	C₉×3₋¹⁺²	—	43	3	no	—
137	#6887	—	43	2	no	—
138	C₃×C₉²	—	43	1	no	—
139	C₃×C₄₃	—	9	1	no	—
140	C₃×C₄₃	C₃²×3₋¹⁺²	1	1	yes	characteristic
141	C₃×C₄₃	C₃×#737	1	3	yes	—
142	C₄₃⋊C₃	—	3	3	no	—
143	C₄₃⋊C₃	—	9	1	no	—
144	C₄₃⋊C₃	—	9	1	no	—
145	#737	—	129	9	no	—
146	C₃×3₋¹⁺²	—	43	1	no	—
147	C₃×3₋¹⁺²	—	43	1	no	—
148	C₃×3₋¹⁺²	—	43	1	no	—
149	C₃×3₋¹⁺²	—	43	1	no	—
150	C₃×3₋¹⁺²	—	43	1	no	—
151	C₃×3₋¹⁺²	—	43	1	no	—
152	C₃×3₋¹⁺²	—	43	1	no	—
153	C₃×3₋¹⁺²	—	43	3	no	—
154	C₃×3₋¹⁺²	—	43	3	no	—
155	C₃×3₋¹⁺²	C₃×C₄₃⋊C₃	1	1	yes	characteristic
156	C₃×3₋¹⁺²	C₃×C₄₃⋊C₃	1	3	yes	—
157	C₃×H(3,𝔽₃)	—	43	1	no	—
158	C₃²×C₉	—	43	1	no	—
159	C₃²×C₉	—	43	1	no	—
160	C₃²×C₉	—	43	1	no	—
161	C₃²×C₉	—	43	1	no	—
162	C₃²×C₉	—	43	3	no	—
163	C₉⋊C₉	—	43	6	no	—
164	C₉⋊C₉	—	43	6	no	—
165	C₉⋊C₉	—	43	6	no	—
166	C₉⋊C₉	—	43	6	no	—
167	C₉⋊C₉	—	43	6	no	—
168	C₉⋊C₉	—	43	2	no	—
169	C₉⋊C₉	—	43	6	no	—
170	C₉⋊C₉	—	43	3	no	—
171	C₉⋊C₉	—	43	3	no	—
172	C₉⋊C₉	—	43	2	no	—
173	C₉⋊C₉	—	43	6	no	—
174	C₉⋊C₉	—	43	3	no	—
175	C₉⋊C₉	—	43	3	no	—
176	C₉⋊C₉	C₃×C₄₃⋊C₃	1	2	yes	—
177	C₉⋊C₉	C₃×C₄₃⋊C₃	1	3	yes	—
178	C₉⋊C₉	C₃×C₄₃⋊C₃	1	3	yes	—
179	C₃²⋊C₉	—	43	6	no	—
180	C₃²⋊C₉	—	43	6	no	—
181	C₃²⋊C₉	—	43	6	no	—
182	C₃²⋊C₉	—	43	6	no	—
183	C₉²	—	43	3	no	—
184	C₉²	—	43	3	no	—
185	C₉²	—	43	2	no	—
186	C₉²	—	43	6	no	—
187	C₉²	C₃×C₄₃⋊C₃	1	1	yes	characteristic
188	C₄₃	#10624879	1	1	yes	characteristic, complemented, Sylow
189	C₃³	—	43	1	no	—
190	C₃³	—	43	1	no	—
191	C₃³	—	43	1	no	—
192	C₃³	C₃²×C₄₃⋊C₃	1	1	yes	characteristic
193	3₋¹⁺²	—	3	3	no	—
194	3₋¹⁺²	—	3	9	no	—
195	3₋¹⁺²	—	129	3	no	—
196	3₋¹⁺²	—	129	3	no	—
197	3₋¹⁺²	—	129	3	no	—
198	3₋¹⁺²	—	129	3	no	—
199	3₋¹⁺²	—	129	3	no	—
200	3₋¹⁺²	—	129	3	no	—
201	3₋¹⁺²	—	129	3	no	—
202	3₋¹⁺²	—	129	9	no	—
203	3₋¹⁺²	—	129	9	no	—
204	H(3,𝔽₃)	—	129	3	no	—
205	C₃×C₉	—	43	1	no	—
206	C₃×C₉	—	43	1	no	—
207	C₃×C₉	—	43	9	no	—
208	C₃×C₉	—	43	6	no	—
209	C₃×C₉	—	43	6	no	—
210	C₃×C₉	—	43	6	no	—
211	C₃×C₉	—	43	1	no	—
212	C₃×C₉	—	43	1	no	—
213	C₃×C₉	—	43	1	no	—
214	C₃×C₉	—	43	1	no	—
215	C₃×C₉	—	129	3	no	—
216	C₃×C₉	—	129	3	no	—
217	C₃×C₉	—	129	3	no	—
218	C₃×C₉	—	129	9	no	—
219	C₃×C₉	C₃²×C₄₃⋊C₃	1	6	yes	—
220	C₃×C₉	C₃²×C₄₃⋊C₃	1	1	yes	characteristic
221	C₃×C₉	C₃²×C₄₃⋊C₃	1	1	yes	characteristic
222	C₃×C₉	C₃²×C₄₃⋊C₃	1	1	yes	characteristic
223	C₃×C₉	C₃²×C₄₃⋊C₃	1	3	yes	—
224	C₃²	—	3	1	no	—
225	C₃²	—	3	3	no	—
226	C₃²	—	43	3	no	—
227	C₃²	—	129	3	no	—
228	C₃²	—	129	1	no	—
229	C₃²	—	129	1	no	—
230	C₃²	—	129	3	no	—
231	C₃²	—	129	3	no	—
232	C₃²	C₃³×C₄₃⋊C₃	1	1	yes	characteristic
233	C₉	—	3	1	no	—
234	C₉	—	3	1	no	—
235	C₉	—	3	6	no	—
236	C₉	—	43	3	no	—
237	C₉	—	43	3	no	—
238	C₉	—	129	1	no	—
239	C₉	—	129	1	no	—
240	C₉	—	129	1	no	—
241	C₉	—	129	1	no	—
242	C₉	—	129	6	no	—
243	C₉	—	129	6	no	—
244	C₉	—	129	6	no	—
245	C₉	3₋¹⁺²×C₄₃⋊C₃	1	3	yes	—
246	C₉	#16812526	1	9	yes	complemented
247	C₃	—	9	1	no	—
248	C₃	—	129	3	no	—
249	C₃	—	387	1	no	—
250	C₃	—	387	1	no	—
251	C₃	C₃×3₋¹⁺²×C₄₃⋊C₃	1	1	yes	characteristic
252	C₃	C₃×#16812526	1	3	yes	—

	subgroup	G/H
center, Frattini	C₃²	C₃³×C₄₃⋊C₃
commutator, socle, cosocle	C₃²×C₄₃	C₃⁴
Fitting	C₄₃×#6900	C₃
prime radical	C₃×C₃²⋊(C₄₃⋊C₃)	C₃²
hypercenter	#6900	C₄₃⋊C₃
hypocenter	C₄₃	#10624879
LCS	G ▷ C₃²×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₃² ◁ #6900

Complements

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

N	H ≅ G/N	type
2. #23782847	C₃	semidirect
3. #23782846	C₃	semidirect
4. #23782844	C₃	semidirect
5. #23782843	C₃	semidirect
6. #23782838	C₃	semidirect
7. #23782828	C₃	semidirect
8. #23782827	C₃	semidirect
9. #23782824	C₃	semidirect
10. #23782811	C₃	semidirect
11. #23782810	C₃	semidirect
13. 3₋¹⁺²×C₄₃⋊C₉	C₃	semidirect
14. #11923957⋊C₉	C₃	semidirect
15. #11923956⋊C₉	C₃	semidirect
16. #23782781	C₃	semidirect
17. C₉²×C₄₃⋊C₃	C₃	semidirect
18. C₄₃×#6900	C₃	semidirect
88. C₉×C₄₃⋊C₃	3₋¹⁺²	semidirect
111. C₉×C₄₃	#737	semidirect
124. #6900	C₄₃⋊C₃	semidirect
188. C₄₃	#10624879	semidirect
246. C₉	#16812526	semidirect

Maximal subgroups

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

subgroup	index
C₃×#16812526	3
#23782847	3
#23782846	3
#23782844	3
#23782843	3
#23782838	3
#23782828	3
#23782827	3
#23782824	3
#23782811	3
#23782810	3
#23782799	3
3₋¹⁺²×C₄₃⋊C₉	3
#11923957⋊C₉	3
#11923956⋊C₉	3
#23782781	3
C₉²×C₄₃⋊C₃	3
C₄₃×#6900	3
#10624879	43