Subgroups — C₄₃×#8247 #24928264

Order 11008.

444 conjugacy classes of subgroups.

The subgroup lattice has 344 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄₃×#2966	C₂	1	1	yes	maximal, characteristic
2	C₄₃×#1726	C₂	1	1	yes	maximal, characteristic, complemented
3	C₄₃×#1524	C₂	1	1	yes	maximal, characteristic
4	C₄₃×#1500	C₂	1	1	yes	maximal, characteristic
5	C₈×M₁₆×C₄₃	C₂	1	1	yes	maximal, characteristic, complemented
6	C₄₃×#1302	C₂	1	1	yes	maximal, characteristic, complemented
7	C₄₃×#1300	C₂	1	1	yes	maximal, characteristic, complemented
8	C₄₃×#529	C₂²	1	1	yes	characteristic
9	C₄₃×#446	C₄	1	1	yes	characteristic, complemented
10	C₈×Q₈×C₄₃	C₄	1	1	yes	characteristic, complemented
11	C₄₃×#436	C₄	1	1	yes	characteristic, complemented
12	C₄₃×#435	—	2	1	no	—
13	C₈×D₄×C₄₃	—	2	1	no	—
14	C₈×D₄×C₄₃	C₄	1	1	yes	characteristic, complemented
15	C₄₃×#433	—	2	1	no	—
16	C₄₃×#432	C₂²	1	1	yes	characteristic
17	C₄₃×#431	C₂²	1	1	yes	characteristic
18	C₄₃×#405	—	2	1	no	—
19	C₄×M₁₆×C₄₃	C₂²	1	1	yes	characteristic
20	C₂×#97×C₄₃	—	2	1	no	—
21	C₂×#97×C₄₃	C₂²	1	1	yes	characteristic
22	C₂×C₄×C₈×C₄₃	—	2	1	no	—
23	C₂×C₄×C₈×C₄₃	C₂²	1	1	yes	characteristic
24	C₄₃×#376	C₂²	1	1	yes	characteristic
25	C₄₃×#330	C₄	1	2	yes	complemented
26	C₄₃×#329	C₄	1	2	yes	complemented
27	C₄₃×#328	C₄	1	2	yes	—
28	C₄₃×#327	C₄	1	2	yes	—
29	C₄₃×#322	—	2	1	no	—
30	C₈²×C₄₃	—	2	1	no	—
31	C₂×M₁₆×C₄₃	—	2	1	no	—
32	C₂×M₁₆×C₄₃	—	2	1	no	—
33	C₂²×C₈×C₄₃	—	2	1	no	—
34	C₂²×C₈×C₄₃	—	2	1	no	—
35	C₂²×C₈×C₄₃	D₄	1	1	yes	characteristic
36	C₂²×C₈×C₄₃	D₄	1	1	yes	characteristic
37	#126×C₄₃	—	2	1	no	—
38	#125×C₄₃	C₂×C₄	1	1	yes	characteristic
39	#124×C₄₃	C₂×C₄	1	1	yes	characteristic
40	#123×C₄₃	—	2	1	no	—
41	C₂²⋊Q₈×C₄₃	C₂×C₄	1	1	yes	characteristic
42	#121×C₄₃	C₂×C₄	1	1	yes	characteristic
43	C₄×Q₈×C₄₃	C₂×C₄	1	1	yes	characteristic
44	C₄×D₄×C₄₃	—	2	1	no	—
45	C₄×D₄×C₄₃	C₂×C₄	1	1	yes	characteristic
46	#117×C₄₃	—	2	1	no	—
47	C₂×C₄²×C₄₃	—	2	1	no	—
48	C₂×C₄²×C₄₃	C₂³	1	1	yes	characteristic
49	C₄⋊C₈×C₄₃	—	2	1	no	—
50	C₄⋊C₈×C₄₃	—	2	1	no	—
51	C₄⋊C₈×C₄₃	—	2	1	no	—
52	C₄⋊C₈×C₄₃	C₂×C₄	1	1	yes	characteristic
53	C₄⋊C₈×C₄₃	C₂×C₄	1	1	yes	characteristic
54	C₂²⋊C₈×C₄₃	—	2	1	no	—
55	C₂²⋊C₈×C₄₃	—	2	1	no	—
56	C₂²⋊C₈×C₄₃	—	2	1	no	—
57	C₂²⋊C₈×C₄₃	C₂×C₄	1	2	yes	—
58	#97×C₄₃	—	2	4	no	—
59	#97×C₄₃	—	2	1	no	—
60	#97×C₄₃	—	2	1	no	—
61	#97×C₄₃	—	2	1	no	—
62	#97×C₄₃	C₂×C₄	1	2	yes	—
63	#97×C₄₃	C₂×C₄	1	2	yes	—
64	C₄×C₈×C₄₃	—	2	4	no	—
65	C₄×C₈×C₄₃	—	2	1	no	—
66	C₄×C₈×C₄₃	—	2	1	no	—
67	C₄×C₈×C₄₃	—	2	1	no	—
68	C₄×C₈×C₄₃	C₂×C₄	1	1	yes	characteristic
69	C₄×C₈×C₄₃	C₂×C₄	1	1	yes	characteristic
70	C₄×C₈×C₄₃	D₄	1	1	yes	characteristic
71	C₄×C₈×C₄₃	D₄	1	1	yes	characteristic
72	#95×C₄₃	—	2	1	no	—
73	#95×C₄₃	C₂×C₄	1	2	yes	—
74	C₂×Q₈×C₄₃	C₄²	1	1	yes	characteristic
75	C₂×D₄×C₄₃	—	2	1	no	—
76	C₂×D₄×C₄₃	C₄²	1	1	yes	characteristic
77	C₂²×C₄×C₄₃	—	2	1	no	—
78	C₂²×C₄×C₄₃	—	2	2	no	—
79	C₂²×C₄×C₄₃	C₂²×C₄	1	1	yes	characteristic
80	C₂²×C₄×C₄₃	C₂×D₄	1	1	yes	characteristic
81	C₂²×C₄×C₄₃	#41	1	1	yes	characteristic
82	M₁₆×C₄₃	—	2	4	no	—
83	M₁₆×C₄₃	—	4	2	no	—
84	C₂×C₈×C₄₃	—	2	2	no	—
85	C₂×C₈×C₄₃	—	2	2	no	—
86	C₂×C₈×C₄₃	—	2	2	no	—
87	C₂×C₈×C₄₃	—	2	2	no	—
88	C₂×C₈×C₄₃	—	2	2	no	—
89	C₂×C₈×C₄₃	—	2	2	no	—
90	C₂×C₈×C₄₃	—	2	1	no	—
91	C₂×C₈×C₄₃	—	2	1	no	—
92	C₂×C₈×C₄₃	—	2	2	no	—
93	C₂×C₈×C₄₃	—	4	2	no	—
94	C₂×C₈×C₄₃	C₂²⋊C₄	1	1	yes	characteristic
95	C₂×C₈×C₄₃	C₂²⋊C₄	1	1	yes	characteristic
96	C₂×C₈×C₄₃	C₂²⋊C₄	1	1	yes	characteristic
97	C₂×C₈×C₄₃	C₂²⋊C₄	1	1	yes	characteristic
98	C₄⋊C₄×C₄₃	—	2	1	no	—
99	C₄⋊C₄×C₄₃	—	2	1	no	—
100	C₄⋊C₄×C₄₃	—	2	1	no	—
101	C₄⋊C₄×C₄₃	—	2	1	no	—
102	C₄⋊C₄×C₄₃	C₄²	1	1	yes	characteristic
103	C₄⋊C₄×C₄₃	C₄²	1	1	yes	characteristic
104	C₂²⋊C₄×C₄₃	—	2	1	no	—
105	C₂²⋊C₄×C₄₃	—	2	1	no	—
106	C₂²⋊C₄×C₄₃	—	2	1	no	—
107	C₂²⋊C₄×C₄₃	—	2	1	no	—
108	C₂²⋊C₄×C₄₃	—	2	1	no	—
109	C₄²×C₄₃	—	2	2	no	—
110	C₄²×C₄₃	—	2	2	no	—
111	C₄²×C₄₃	—	2	1	no	—
112	C₄²×C₄₃	C₂²×C₄	1	1	yes	characteristic
113	C₄²×C₄₃	C₂²×C₄	1	1	yes	characteristic
114	C₄²×C₄₃	C₂×D₄	1	1	yes	characteristic
115	C₄²×C₄₃	#41	1	1	yes	characteristic
116	C₂³×C₄₃	—	2	1	no	—
117	C₂³×C₄₃	C₄×D₄	1	1	yes	characteristic
118	Q₈×C₄₃	—	4	1	no	—
119	D₄×C₄₃	—	4	1	no	—
120	D₄×C₄₃	—	4	2	no	—
121	C₂×C₄×C₄₃	—	2	2	no	—
122	C₂×C₄×C₄₃	—	2	1	no	—
123	C₂×C₄×C₄₃	—	2	1	no	—
124	C₂×C₄×C₄₃	—	2	4	no	—
125	C₂×C₄×C₄₃	—	2	1	no	—
126	C₂×C₄×C₄₃	—	2	1	no	—
127	C₂×C₄×C₄₃	—	2	1	no	—
128	C₂×C₄×C₄₃	—	4	1	no	—
129	C₂×C₄×C₄₃	—	4	2	no	—
130	C₂×C₄×C₄₃	—	4	2	no	—
131	C₂×C₄×C₄₃	—	4	1	no	—
132	C₂×C₄×C₄₃	—	4	1	no	—
133	C₂×C₄×C₄₃	C₂×C₄²	1	1	yes	characteristic
134	C₂×C₄×C₄₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
135	C₂×C₄×C₄₃	#117	1	1	yes	characteristic
136	C₂×C₄×C₄₃	C₄×D₄	1	1	yes	characteristic
137	C₂×C₄×C₄₃	C₄×D₄	1	1	yes	characteristic
138	C₂×C₄×C₄₃	C₄×D₄	1	1	yes	characteristic
139	C₈×C₄₃	—	2	4	no	—
140	C₈×C₄₃	—	2	4	no	—
141	C₈×C₄₃	—	2	1	no	—
142	C₈×C₄₃	—	2	1	no	—
143	C₈×C₄₃	—	4	2	no	—
144	C₈×C₄₃	—	4	1	no	—
145	C₈×C₄₃	—	4	1	no	—
146	C₈×C₄₃	#104	1	2	yes	—
147	C₈×C₄₃	#104	1	2	yes	—
148	#8247	C₄₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
149	C₂²×C₄₃	—	2	1	no	—
150	C₂²×C₄₃	—	2	1	no	—
151	C₂²×C₄₃	—	4	1	no	—
152	C₂²×C₄₃	—	4	1	no	—
153	C₂²×C₄₃	—	4	1	no	—
154	C₂²×C₄₃	C₄×C₂²⋊C₄	1	1	yes	characteristic
155	C₂²×C₄₃	#443	1	2	yes	—
156	C₄×C₄₃	—	2	1	no	—
157	C₄×C₄₃	—	2	1	no	—
158	C₄×C₄₃	—	2	1	no	—
159	C₄×C₄₃	—	2	1	no	—
160	C₄×C₄₃	—	2	1	no	—
161	C₄×C₄₃	—	4	4	no	—
162	C₄×C₄₃	—	4	1	no	—
163	C₄×C₄₃	—	4	1	no	—
164	C₄×C₄₃	—	4	1	no	—
165	C₄×C₄₃	C₂×#104	1	1	yes	characteristic
166	C₄×C₄₃	C₂×#104	1	1	yes	characteristic
167	#2966	C₂×C₄₃	1	1	yes	characteristic
168	#1726	C₂×C₄₃	1	1	yes	characteristic, complemented
169	#1524	C₂×C₄₃	1	1	yes	characteristic
170	#1500	C₂×C₄₃	1	1	yes	characteristic
171	C₈×M₁₆	C₂×C₄₃	1	1	yes	characteristic, complemented
172	#1302	C₂×C₄₃	1	1	yes	characteristic, complemented
173	#1300	C₂×C₄₃	1	1	yes	characteristic, complemented
174	C₂×C₄₃	—	2	2	no	—
175	C₂×C₄₃	—	8	1	no	—
176	C₂×C₄₃	#1757	1	1	yes	characteristic
177	C₂×C₄₃	C₄×#104	1	1	yes	characteristic
178	C₂×C₄₃	#1764	1	1	yes	characteristic
179	#529	C₂²×C₄₃	1	1	yes	characteristic
180	#446	C₄×C₄₃	1	1	yes	characteristic, complemented
181	C₈×Q₈	C₄×C₄₃	1	1	yes	characteristic, complemented
182	#436	C₄×C₄₃	1	1	yes	characteristic, complemented
183	#435	—	2	1	no	—
184	C₈×D₄	—	2	1	no	—
185	C₈×D₄	C₄×C₄₃	1	1	yes	characteristic, complemented
186	#433	—	2	1	no	—
187	#432	C₂²×C₄₃	1	1	yes	characteristic
188	#431	C₂²×C₄₃	1	1	yes	characteristic
189	#405	—	2	1	no	—
190	C₄×M₁₆	C₂²×C₄₃	1	1	yes	characteristic
191	C₂×#97	—	2	1	no	—
192	C₂×#97	C₂²×C₄₃	1	1	yes	characteristic
193	C₂×C₄×C₈	—	2	1	no	—
194	C₂×C₄×C₈	C₂²×C₄₃	1	1	yes	characteristic
195	#376	C₂²×C₄₃	1	1	yes	characteristic
196	#330	C₄×C₄₃	1	2	yes	complemented
197	#329	C₄×C₄₃	1	2	yes	complemented
198	#328	C₄×C₄₃	1	2	yes	—
199	#327	C₄×C₄₃	1	2	yes	—
200	#322	—	2	1	no	—
201	C₈²	—	2	1	no	—
202	C₄₃	#8247	1	1	yes	characteristic, direct product factor, Sylow
203	C₂×M₁₆	—	2	1	no	—
204	C₂×M₁₆	—	2	1	no	—
205	C₂²×C₈	—	2	1	no	—
206	C₂²×C₈	—	2	1	no	—
207	C₂²×C₈	D₄×C₄₃	1	1	yes	characteristic
208	C₂²×C₈	D₄×C₄₃	1	1	yes	characteristic
209	#126	—	2	1	no	—
210	#125	C₂×C₄×C₄₃	1	1	yes	characteristic
211	#124	C₂×C₄×C₄₃	1	1	yes	characteristic
212	#123	—	2	1	no	—
213	C₂²⋊Q₈	C₂×C₄×C₄₃	1	1	yes	characteristic
214	#121	C₂×C₄×C₄₃	1	1	yes	characteristic
215	C₄×Q₈	C₂×C₄×C₄₃	1	1	yes	characteristic
216	C₄×D₄	—	2	1	no	—
217	C₄×D₄	C₂×C₄×C₄₃	1	1	yes	characteristic
218	#117	—	2	1	no	—
219	C₂×C₄²	—	2	1	no	—
220	C₂×C₄²	C₂³×C₄₃	1	1	yes	characteristic
221	C₄⋊C₈	—	2	1	no	—
222	C₄⋊C₈	—	2	1	no	—
223	C₄⋊C₈	—	2	1	no	—
224	C₄⋊C₈	C₂×C₄×C₄₃	1	1	yes	characteristic
225	C₄⋊C₈	C₂×C₄×C₄₃	1	1	yes	characteristic
226	C₂²⋊C₈	—	2	1	no	—
227	C₂²⋊C₈	—	2	1	no	—
228	C₂²⋊C₈	—	2	1	no	—
229	C₂²⋊C₈	C₂×C₄×C₄₃	1	2	yes	—
230	#97	—	2	4	no	—
231	#97	—	2	1	no	—
232	#97	—	2	1	no	—
233	#97	—	2	1	no	—
234	#97	C₂×C₄×C₄₃	1	2	yes	—
235	#97	C₂×C₄×C₄₃	1	2	yes	—
236	C₄×C₈	—	2	4	no	—
237	C₄×C₈	—	2	1	no	—
238	C₄×C₈	—	2	1	no	—
239	C₄×C₈	—	2	1	no	—
240	C₄×C₈	C₂×C₄×C₄₃	1	1	yes	characteristic
241	C₄×C₈	C₂×C₄×C₄₃	1	1	yes	characteristic
242	C₄×C₈	D₄×C₄₃	1	1	yes	characteristic
243	C₄×C₈	D₄×C₄₃	1	1	yes	characteristic
244	#95	—	2	1	no	—
245	#95	C₂×C₄×C₄₃	1	2	yes	—
246	C₂×Q₈	C₄²×C₄₃	1	1	yes	characteristic
247	C₂×D₄	—	2	1	no	—
248	C₂×D₄	C₄²×C₄₃	1	1	yes	characteristic
249	C₂²×C₄	—	2	2	no	—
250	C₂²×C₄	—	2	1	no	—
251	C₂²×C₄	C₂²×C₄×C₄₃	1	1	yes	characteristic
252	C₂²×C₄	C₂×D₄×C₄₃	1	1	yes	characteristic
253	C₂²×C₄	#41×C₄₃	1	1	yes	characteristic
254	M₁₆	—	2	4	no	—
255	M₁₆	—	4	2	no	—
256	C₂×C₈	—	2	2	no	—
257	C₂×C₈	—	2	2	no	—
258	C₂×C₈	—	2	2	no	—
259	C₂×C₈	—	2	2	no	—
260	C₂×C₈	—	2	2	no	—
261	C₂×C₈	—	2	2	no	—
262	C₂×C₈	—	2	1	no	—
263	C₂×C₈	—	2	1	no	—
264	C₂×C₈	—	2	2	no	—
265	C₂×C₈	—	4	2	no	—
266	C₂×C₈	C₂²⋊C₄×C₄₃	1	1	yes	characteristic
267	C₂×C₈	C₂²⋊C₄×C₄₃	1	1	yes	characteristic
268	C₂×C₈	C₂²⋊C₄×C₄₃	1	1	yes	characteristic
269	C₂×C₈	C₂²⋊C₄×C₄₃	1	1	yes	characteristic
270	C₄⋊C₄	—	2	1	no	—
271	C₄⋊C₄	—	2	1	no	—
272	C₄⋊C₄	—	2	1	no	—
273	C₄⋊C₄	—	2	1	no	—
274	C₄⋊C₄	C₄²×C₄₃	1	1	yes	characteristic
275	C₄⋊C₄	C₄²×C₄₃	1	1	yes	characteristic
276	C₂²⋊C₄	—	2	1	no	—
277	C₂²⋊C₄	—	2	1	no	—
278	C₂²⋊C₄	—	2	1	no	—
279	C₂²⋊C₄	—	2	1	no	—
280	C₂²⋊C₄	—	2	1	no	—
281	C₄²	—	2	2	no	—
282	C₄²	—	2	2	no	—
283	C₄²	—	2	1	no	—
284	C₄²	C₂²×C₄×C₄₃	1	1	yes	characteristic
285	C₄²	C₂²×C₄×C₄₃	1	1	yes	characteristic
286	C₄²	C₂×D₄×C₄₃	1	1	yes	characteristic
287	C₄²	#41×C₄₃	1	1	yes	characteristic
288	C₂³	—	2	1	no	—
289	C₂³	C₄×D₄×C₄₃	1	1	yes	characteristic
290	Q₈	—	4	1	no	—
291	D₄	—	4	1	no	—
292	D₄	—	4	2	no	—
293	C₂×C₄	—	2	2	no	—
294	C₂×C₄	—	2	1	no	—
295	C₂×C₄	—	2	1	no	—
296	C₂×C₄	—	2	4	no	—
297	C₂×C₄	—	2	1	no	—
298	C₂×C₄	—	2	1	no	—
299	C₂×C₄	—	2	1	no	—
300	C₂×C₄	—	4	1	no	—
301	C₂×C₄	—	4	2	no	—
302	C₂×C₄	—	4	2	no	—
303	C₂×C₄	—	4	1	no	—
304	C₂×C₄	—	4	1	no	—
305	C₂×C₄	C₂×C₄²×C₄₃	1	1	yes	characteristic
306	C₂×C₄	C₂×C₂²⋊C₄×C₄₃	1	1	yes	characteristic
307	C₂×C₄	#117×C₄₃	1	1	yes	characteristic
308	C₂×C₄	C₄×D₄×C₄₃	1	1	yes	characteristic
309	C₂×C₄	C₄×D₄×C₄₃	1	1	yes	characteristic
310	C₂×C₄	C₄×D₄×C₄₃	1	1	yes	characteristic
311	C₈	—	2	4	no	—
312	C₈	—	2	4	no	—
313	C₈	—	2	1	no	—
314	C₈	—	2	1	no	—
315	C₈	—	4	2	no	—
316	C₈	—	4	1	no	—
317	C₈	—	4	1	no	—
318	C₈	#104×C₄₃	1	2	yes	—
319	C₈	#104×C₄₃	1	2	yes	—
320	C₂²	—	2	1	no	—
321	C₂²	—	2	1	no	—
322	C₂²	—	4	1	no	—
323	C₂²	—	4	1	no	—
324	C₂²	—	4	1	no	—
325	C₂²	C₄×C₂²⋊C₄×C₄₃	1	1	yes	characteristic
326	C₂²	C₄₃×#443	1	2	yes	—
327	C₄	—	2	1	no	—
328	C₄	—	2	1	no	—
329	C₄	—	2	1	no	—
330	C₄	—	2	1	no	—
331	C₄	—	2	1	no	—
332	C₄	—	4	4	no	—
333	C₄	—	4	1	no	—
334	C₄	—	4	1	no	—
335	C₄	—	4	1	no	—
336	C₄	C₂×#104×C₄₃	1	1	yes	characteristic
337	C₄	C₂×#104×C₄₃	1	1	yes	characteristic
338	C₂	—	2	2	no	—
339	C₂	—	8	1	no	—
340	C₂	C₄₃×#1757	1	1	yes	characteristic
341	C₂	C₄×#104×C₄₃	1	1	yes	characteristic
342	C₂	C₄₃×#1764	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₈×C₄₃	C₂²⋊C₄
commutator	C₂×C₄	C₂×C₄²×C₄₃
Frattini, cosocle	C₂×C₄²	C₂³×C₄₃
socle	C₂²×C₄₃	C₄×C₂²⋊C₄
prime radical	#121×C₄₃	C₂×C₄
LCS	G ▷ C₂×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₈×C₄₃ ◁ C₂×C₄×C₈×C₄₃ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₄₃×#1726	C₂	semidirect
5. C₈×M₁₆×C₄₃	C₂	semidirect
6. C₄₃×#1302	C₂	semidirect
7. C₄₃×#1300	C₂	semidirect
9. C₄₃×#446	C₄	semidirect
10. C₈×Q₈×C₄₃	C₄	semidirect
11. C₄₃×#436	C₄	semidirect
14. C₈×D₄×C₄₃	C₄	semidirect
25. C₄₃×#330	C₄	semidirect
26. C₄₃×#329	C₄	semidirect
148. #8247	C₄₃	direct (N × H)
168. #1726	C₂×C₄₃	semidirect
171. C₈×M₁₆	C₂×C₄₃	semidirect
172. #1302	C₂×C₄₃	semidirect
173. #1300	C₂×C₄₃	semidirect
180. #446	C₄×C₄₃	semidirect
181. C₈×Q₈	C₄×C₄₃	semidirect
182. #436	C₄×C₄₃	semidirect
185. C₈×D₄	C₄×C₄₃	semidirect
196. #330	C₄×C₄₃	semidirect
197. #329	C₄×C₄₃	semidirect
202. C₄₃	#8247	direct (N × H)

Maximal subgroups

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

subgroup	index
C₄₃×#2966	2
C₄₃×#1726	2
C₄₃×#1524	2
C₄₃×#1500	2
C₈×M₁₆×C₄₃	2
C₄₃×#1302	2
C₄₃×#1300	2
#8247	43

Subgroups — C43×#8247 #24928264

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄₃×#8247 #24928264