Subgroups — C₄₃×#9267 #24929284

Order 11008.

492 conjugacy classes of subgroups.

The subgroup lattice has 366 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄₃×#3206	C₂	1	1	yes	maximal, characteristic
2	C₄₃×#2935	C₂	1	1	yes	maximal, characteristic
3	C₄₃×#1997	C₂	1	1	yes	maximal, characteristic, complemented
4	C₄₃×#1547	C₂	1	1	yes	maximal, characteristic
5	C₂×C₄₃×#333	C₂	1	1	yes	maximal, characteristic, complemented
6	C₄₃×#1301	C₂	1	1	yes	maximal, characteristic, complemented
7	C₄₃×#1283	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×#131×C₄₃	—	2	1	no	—
9	C₄₃×#532	C₂²	1	1	yes	characteristic
10	C₂×C₄⋊Q₈×C₄₃	C₂²	1	1	yes	characteristic
11	C₄₃×#492	—	2	1	no	—
12	C₄₃×#487	—	2	1	no	—
13	C₄₃×#461	—	2	2	no	—
14	C₄₃×#448	C₄	1	4	yes	—
15	C₄₃×#436	—	2	1	no	—
16	C₄₃×#435	—	2	2	no	—
17	C₈×D₄×C₄₃	—	2	1	no	—
18	C₄₃×#431	C₂²	1	1	yes	characteristic
19	C₄₃×#423	C₂²	1	1	yes	characteristic
20	C₂×C₄⋊C₈×C₄₃	C₂²	1	1	yes	characteristic
21	C₂×C₄⋊C₈×C₄₃	C₂²	1	1	yes	characteristic
22	C₄₃×#408	—	2	1	no	—
23	C₄×M₁₆×C₄₃	C₂²	1	1	yes	characteristic
24	C₄₃×#343	—	2	1	no	—
25	C₄₃×#343	—	2	1	no	—
26	C₄₃×#333	—	2	1	no	—
27	C₄₃×#333	—	2	1	no	—
28	C₄₃×#332	—	2	1	no	—
29	C₄₃×#331	—	2	1	no	—
30	C₂×#41×C₄₃	—	2	1	no	—
31	C₂²×Q₈×C₄₃	C₂×C₄	1	2	yes	—
32	C₂×SD₁₆×C₄₃	—	2	4	no	—
33	#131×C₄₃	—	4	2	no	—
34	#131×C₄₃	—	4	2	no	—
35	C₂×M₁₆×C₄₃	—	2	2	no	—
36	C₂×M₁₆×C₄₃	—	2	1	no	—
37	C₂×M₁₆×C₄₃	D₄	1	1	yes	characteristic
38	C₂×M₁₆×C₄₃	D₄	1	1	yes	characteristic
39	C₂²×C₈×C₄₃	—	2	2	no	—
40	C₂²×C₈×C₄₃	—	2	1	no	—
41	C₂²×C₈×C₄₃	—	2	1	no	—
42	C₄⋊Q₈×C₄₃	—	2	1	no	—
43	C₄⋊Q₈×C₄₃	—	2	1	no	—
44	C₄⋊Q₈×C₄₃	—	2	1	no	—
45	C₄⋊Q₈×C₄₃	—	2	1	no	—
46	C₄⋊Q₈×C₄₃	D₄	1	1	yes	characteristic
47	#127×C₄₃	D₄	1	1	yes	characteristic
48	#124×C₄₃	—	2	1	no	—
49	#121×C₄₃	—	2	1	no	—
50	#121×C₄₃	C₂×C₄	1	2	yes	—
51	C₄×D₄×C₄₃	—	2	1	no	—
52	C₄×D₄×C₄₃	D₄	1	2	yes	—
53	C₂×C₄⋊C₄×C₄₃	—	2	1	no	—
54	C₂×C₄⋊C₄×C₄₃	—	2	1	no	—
55	C₂×C₄²×C₄₃	C₂³	1	1	yes	characteristic
56	C₄⋊C₈×C₄₃	—	2	1	no	—
57	C₄⋊C₈×C₄₃	—	2	1	no	—
58	C₄⋊C₈×C₄₃	—	2	1	no	—
59	C₄⋊C₈×C₄₃	—	2	1	no	—
60	C₄⋊C₈×C₄₃	—	2	1	no	—
61	C₄⋊C₈×C₄₃	—	2	1	no	—
62	C₄⋊C₈×C₄₃	D₄	1	1	yes	characteristic
63	C₄⋊C₈×C₄₃	D₄	1	1	yes	characteristic
64	#103×C₄₃	—	2	4	no	—
65	C₂²⋊C₈×C₄₃	—	2	2	no	—
66	C₂²⋊C₈×C₄₃	—	2	2	no	—
67	C₂²⋊C₈×C₄₃	C₂×C₄	1	2	yes	—
68	#97×C₄₃	—	2	1	no	—
69	C₄×C₈×C₄₃	—	2	1	no	—
70	C₄×C₈×C₄₃	—	2	1	no	—
71	#41×C₄₃	—	4	1	no	—
72	#41×C₄₃	—	4	1	no	—
73	#41×C₄₃	—	4	1	no	—
74	#41×C₄₃	—	4	1	no	—
75	C₂×Q₈×C₄₃	—	2	1	no	—
76	C₂×Q₈×C₄₃	—	2	2	no	—
77	C₂×Q₈×C₄₃	—	2	2	no	—
78	C₂×Q₈×C₄₃	—	8	2	no	—
79	C₂×D₄×C₄₃	—	2	1	no	—
80	C₂×D₄×C₄₃	—	2	1	no	—
81	C₂×D₄×C₄₃	—	2	2	no	—
82	C₂×D₄×C₄₃	C₂²⋊C₄	1	2	yes	—
83	C₂²×C₄×C₄₃	—	2	2	no	—
84	C₂²×C₄×C₄₃	—	2	2	no	—
85	C₂²×C₄×C₄₃	C₂²×C₄	1	1	yes	characteristic
86	C₂²×C₄×C₄₃	C₂×D₄	1	1	yes	characteristic
87	C₂²×C₄×C₄₃	#41	1	1	yes	characteristic
88	SD₁₆×C₄₃	—	8	4	no	—
89	M₁₆×C₄₃	—	2	1	no	—
90	M₁₆×C₄₃	—	2	1	no	—
91	M₁₆×C₄₃	—	2	1	no	—
92	M₁₆×C₄₃	—	4	4	no	—
93	M₁₆×C₄₃	—	4	1	no	—
94	M₁₆×C₄₃	—	4	1	no	—
95	M₁₆×C₄₃	SD₁₆	1	2	yes	complemented
96	C₂×C₈×C₄₃	—	2	1	no	—
97	C₂×C₈×C₄₃	—	2	2	no	—
98	C₂×C₈×C₄₃	—	2	1	no	—
99	C₂×C₈×C₄₃	—	2	1	no	—
100	C₂×C₈×C₄₃	—	2	1	no	—
101	C₂×C₈×C₄₃	—	4	4	no	—
102	C₂×C₈×C₄₃	—	4	1	no	—
103	C₂×C₈×C₄₃	—	4	1	no	—
104	C₂×C₈×C₄₃	—	4	1	no	—
105	C₂×C₈×C₄₃	—	4	1	no	—
106	C₂×C₈×C₄₃	—	8	1	no	—
107	C₄⋊C₄×C₄₃	—	2	1	no	—
108	C₄⋊C₄×C₄₃	—	4	1	no	—
109	C₄⋊C₄×C₄₃	—	4	1	no	—
110	C₄⋊C₄×C₄₃	—	4	1	no	—
111	C₄⋊C₄×C₄₃	—	4	1	no	—
112	C₄⋊C₄×C₄₃	C₂²⋊C₄	1	2	yes	—
113	C₂²⋊C₄×C₄₃	—	2	2	no	—
114	C₂²⋊C₄×C₄₃	—	2	2	no	—
115	C₄²×C₄₃	C₂×D₄	1	1	yes	characteristic
116	C₄²×C₄₃	C₂×D₄	1	1	yes	characteristic
117	C₄²×C₄₃	C₂×D₄	1	1	yes	characteristic
118	C₄²×C₄₃	#41	1	1	yes	characteristic
119	C₂³×C₄₃	—	2	2	no	—
120	C₂³×C₄₃	C₄×D₄	1	1	yes	characteristic
121	Q₈×C₄₃	—	4	4	no	—
122	Q₈×C₄₃	—	4	1	no	—
123	Q₈×C₄₃	—	4	1	no	—
124	Q₈×C₄₃	—	8	2	no	—
125	D₄×C₄₃	—	4	2	no	—
126	D₄×C₄₃	—	4	2	no	—
127	D₄×C₄₃	—	4	2	no	—
128	D₄×C₄₃	—	4	1	no	—
129	D₄×C₄₃	—	4	1	no	—
130	D₄×C₄₃	—	4	1	no	—
131	D₄×C₄₃	—	4	1	no	—
132	C₂×C₄×C₄₃	—	2	1	no	—
133	C₂×C₄×C₄₃	—	2	1	no	—
134	C₂×C₄×C₄₃	—	2	2	no	—
135	C₂×C₄×C₄₃	—	4	1	no	—
136	C₂×C₄×C₄₃	—	4	2	no	—
137	C₂×C₄×C₄₃	—	4	2	no	—
138	C₂×C₄×C₄₃	—	4	2	no	—
139	C₂×C₄×C₄₃	—	8	2	no	—
140	C₂×C₄×C₄₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
141	C₂×C₄×C₄₃	C₄×D₄	1	1	yes	characteristic
142	C₂×C₄×C₄₃	#120	1	1	yes	characteristic
143	C₂×C₄×C₄₃	#121	1	1	yes	characteristic
144	C₂×C₄×C₄₃	#121	1	1	yes	characteristic
145	C₂×C₄×C₄₃	#123	1	1	yes	characteristic
146	C₂×C₄×C₄₃	C₂×SD₁₆	1	1	yes	characteristic
147	C₂×C₄×C₄₃	#135	1	1	yes	characteristic
148	C₂×C₄×C₄₃	AGL(1,ℤ/8)	1	1	yes	characteristic
149	C₂×C₄×C₄₃	#137	1	1	yes	characteristic
150	C₈×C₄₃	—	2	2	no	—
151	C₈×C₄₃	—	4	2	no	—
152	C₈×C₄₃	—	4	1	no	—
153	C₈×C₄₃	—	4	1	no	—
154	C₈×C₄₃	—	4	1	no	—
155	C₈×C₄₃	—	8	1	no	—
156	C₈×C₄₃	—	8	1	no	—
157	#9267	C₄₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
158	C₂²×C₄₃	—	2	1	no	—
159	C₂²×C₄₃	—	2	1	no	—
160	C₂²×C₄₃	—	4	2	no	—
161	C₂²×C₄₃	—	4	2	no	—
162	C₂²×C₄₃	—	4	2	no	—
163	C₂²×C₄₃	#386	1	1	yes	characteristic
164	C₂²×C₄₃	C₄×SD₁₆	1	1	yes	characteristic
165	C₂²×C₄₃	#440	1	1	yes	characteristic
166	C₄×C₄₃	—	2	1	no	—
167	C₄×C₄₃	—	2	1	no	—
168	C₄×C₄₃	—	2	1	no	—
169	C₄×C₄₃	—	4	2	no	—
170	C₄×C₄₃	—	8	2	no	—
171	C₄×C₄₃	#450	1	1	yes	characteristic
172	C₄×C₄₃	#452	1	1	yes	characteristic
173	C₄×C₄₃	#461	1	1	yes	characteristic
174	C₄×C₄₃	#463	1	1	yes	characteristic
175	C₄×C₄₃	#471	1	1	yes	characteristic
176	C₄×C₄₃	#473	1	1	yes	characteristic
177	#3206	C₂×C₄₃	1	1	yes	characteristic
178	#2935	C₂×C₄₃	1	1	yes	characteristic
179	#1997	C₂×C₄₃	1	1	yes	characteristic, complemented
180	#1547	C₂×C₄₃	1	1	yes	characteristic
181	C₂×#333	C₂×C₄₃	1	1	yes	characteristic, complemented
182	#1301	C₂×C₄₃	1	1	yes	characteristic, complemented
183	#1283	C₂×C₄₃	1	1	yes	characteristic, complemented
184	C₂×C₄₃	—	2	1	no	—
185	C₂×C₄₃	—	2	1	no	—
186	C₂×C₄₃	—	8	2	no	—
187	C₂×C₄₃	#1877	1	1	yes	characteristic
188	C₂×C₄₃	#1908	1	1	yes	characteristic
189	C₂×C₄₃	#1914	1	1	yes	characteristic
190	C₂×#131	—	2	1	no	—
191	#532	C₂²×C₄₃	1	1	yes	characteristic
192	C₂×C₄⋊Q₈	C₂²×C₄₃	1	1	yes	characteristic
193	#492	—	2	1	no	—
194	#487	—	2	1	no	—
195	#461	—	2	2	no	—
196	#448	C₄×C₄₃	1	4	yes	—
197	#436	—	2	1	no	—
198	#435	—	2	2	no	—
199	C₈×D₄	—	2	1	no	—
200	#431	C₂²×C₄₃	1	1	yes	characteristic
201	#423	C₂²×C₄₃	1	1	yes	characteristic
202	C₂×C₄⋊C₈	C₂²×C₄₃	1	1	yes	characteristic
203	C₂×C₄⋊C₈	C₂²×C₄₃	1	1	yes	characteristic
204	#408	—	2	1	no	—
205	C₄×M₁₆	C₂²×C₄₃	1	1	yes	characteristic
206	#343	—	2	1	no	—
207	#343	—	2	1	no	—
208	#333	—	2	1	no	—
209	#333	—	2	1	no	—
210	#332	—	2	1	no	—
211	#331	—	2	1	no	—
212	C₄₃	#9267	1	1	yes	characteristic, direct product factor, Sylow
213	C₂×#41	—	2	1	no	—
214	C₂²×Q₈	C₂×C₄×C₄₃	1	2	yes	—
215	C₂×SD₁₆	—	2	4	no	—
216	#131	—	4	2	no	—
217	#131	—	4	2	no	—
218	C₂×M₁₆	—	2	2	no	—
219	C₂×M₁₆	—	2	1	no	—
220	C₂×M₁₆	D₄×C₄₃	1	1	yes	characteristic
221	C₂×M₁₆	D₄×C₄₃	1	1	yes	characteristic
222	C₂²×C₈	—	2	2	no	—
223	C₂²×C₈	—	2	1	no	—
224	C₂²×C₈	—	2	1	no	—
225	C₄⋊Q₈	—	2	1	no	—
226	C₄⋊Q₈	—	2	1	no	—
227	C₄⋊Q₈	—	2	1	no	—
228	C₄⋊Q₈	—	2	1	no	—
229	C₄⋊Q₈	D₄×C₄₃	1	1	yes	characteristic
230	#127	D₄×C₄₃	1	1	yes	characteristic
231	#124	—	2	1	no	—
232	#121	—	2	1	no	—
233	#121	C₂×C₄×C₄₃	1	2	yes	—
234	C₄×D₄	—	2	1	no	—
235	C₄×D₄	D₄×C₄₃	1	2	yes	—
236	C₂×C₄⋊C₄	—	2	1	no	—
237	C₂×C₄⋊C₄	—	2	1	no	—
238	C₂×C₄²	C₂³×C₄₃	1	1	yes	characteristic
239	C₄⋊C₈	—	2	1	no	—
240	C₄⋊C₈	—	2	1	no	—
241	C₄⋊C₈	—	2	1	no	—
242	C₄⋊C₈	—	2	1	no	—
243	C₄⋊C₈	—	2	1	no	—
244	C₄⋊C₈	—	2	1	no	—
245	C₄⋊C₈	D₄×C₄₃	1	1	yes	characteristic
246	C₄⋊C₈	D₄×C₄₃	1	1	yes	characteristic
247	#103	—	2	4	no	—
248	C₂²⋊C₈	—	2	2	no	—
249	C₂²⋊C₈	—	2	2	no	—
250	C₂²⋊C₈	C₂×C₄×C₄₃	1	2	yes	—
251	#97	—	2	1	no	—
252	C₄×C₈	—	2	1	no	—
253	C₄×C₈	—	2	1	no	—
254	#41	—	4	1	no	—
255	#41	—	4	1	no	—
256	#41	—	4	1	no	—
257	#41	—	4	1	no	—
258	C₂×Q₈	—	2	1	no	—
259	C₂×Q₈	—	2	2	no	—
260	C₂×Q₈	—	2	2	no	—
261	C₂×Q₈	—	8	2	no	—
262	C₂×D₄	—	2	2	no	—
263	C₂×D₄	—	2	1	no	—
264	C₂×D₄	—	2	1	no	—
265	C₂×D₄	C₂²⋊C₄×C₄₃	1	2	yes	—
266	C₂²×C₄	—	2	2	no	—
267	C₂²×C₄	—	2	2	no	—
268	C₂²×C₄	C₂²×C₄×C₄₃	1	1	yes	characteristic
269	C₂²×C₄	C₂×D₄×C₄₃	1	1	yes	characteristic
270	C₂²×C₄	#41×C₄₃	1	1	yes	characteristic
271	SD₁₆	—	8	4	no	—
272	M₁₆	—	2	1	no	—
273	M₁₆	—	2	1	no	—
274	M₁₆	—	2	1	no	—
275	M₁₆	—	4	4	no	—
276	M₁₆	—	4	1	no	—
277	M₁₆	—	4	1	no	—
278	M₁₆	SD₁₆×C₄₃	1	2	yes	complemented
279	C₂×C₈	—	2	1	no	—
280	C₂×C₈	—	2	2	no	—
281	C₂×C₈	—	2	1	no	—
282	C₂×C₈	—	2	1	no	—
283	C₂×C₈	—	2	1	no	—
284	C₂×C₈	—	4	4	no	—
285	C₂×C₈	—	4	1	no	—
286	C₂×C₈	—	4	1	no	—
287	C₂×C₈	—	4	1	no	—
288	C₂×C₈	—	4	1	no	—
289	C₂×C₈	—	8	1	no	—
290	C₄⋊C₄	—	2	1	no	—
291	C₄⋊C₄	—	4	1	no	—
292	C₄⋊C₄	—	4	1	no	—
293	C₄⋊C₄	—	4	1	no	—
294	C₄⋊C₄	—	4	1	no	—
295	C₄⋊C₄	C₂²⋊C₄×C₄₃	1	2	yes	—
296	C₂²⋊C₄	—	2	2	no	—
297	C₂²⋊C₄	—	2	2	no	—
298	C₄²	C₂×D₄×C₄₃	1	1	yes	characteristic
299	C₄²	C₂×D₄×C₄₃	1	1	yes	characteristic
300	C₄²	C₂×D₄×C₄₃	1	1	yes	characteristic
301	C₄²	#41×C₄₃	1	1	yes	characteristic
302	C₂³	—	2	2	no	—
303	C₂³	C₄×D₄×C₄₃	1	1	yes	characteristic
304	Q₈	—	4	4	no	—
305	Q₈	—	4	1	no	—
306	Q₈	—	4	1	no	—
307	Q₈	—	8	2	no	—
308	D₄	—	4	2	no	—
309	D₄	—	4	2	no	—
310	D₄	—	4	2	no	—
311	D₄	—	4	1	no	—
312	D₄	—	4	1	no	—
313	D₄	—	4	1	no	—
314	D₄	—	4	1	no	—
315	C₂×C₄	—	2	1	no	—
316	C₂×C₄	—	2	1	no	—
317	C₂×C₄	—	2	2	no	—
318	C₂×C₄	—	4	1	no	—
319	C₂×C₄	—	4	2	no	—
320	C₂×C₄	—	4	2	no	—
321	C₂×C₄	—	4	2	no	—
322	C₂×C₄	—	8	2	no	—
323	C₂×C₄	C₂×C₂²⋊C₄×C₄₃	1	1	yes	characteristic
324	C₂×C₄	C₄×D₄×C₄₃	1	1	yes	characteristic
325	C₂×C₄	#120×C₄₃	1	1	yes	characteristic
326	C₂×C₄	#121×C₄₃	1	1	yes	characteristic
327	C₂×C₄	#121×C₄₃	1	1	yes	characteristic
328	C₂×C₄	#123×C₄₃	1	1	yes	characteristic
329	C₂×C₄	C₂×SD₁₆×C₄₃	1	1	yes	characteristic
330	C₂×C₄	#135×C₄₃	1	1	yes	characteristic
331	C₂×C₄	AGL(1,ℤ/8)×C₄₃	1	1	yes	characteristic
332	C₂×C₄	#137×C₄₃	1	1	yes	characteristic
333	C₈	—	2	2	no	—
334	C₈	—	4	2	no	—
335	C₈	—	4	1	no	—
336	C₈	—	4	1	no	—
337	C₈	—	4	1	no	—
338	C₈	—	8	1	no	—
339	C₈	—	8	1	no	—
340	C₂²	—	2	1	no	—
341	C₂²	—	2	1	no	—
342	C₂²	—	4	2	no	—
343	C₂²	—	4	2	no	—
344	C₂²	—	4	2	no	—
345	C₂²	C₄₃×#386	1	1	yes	characteristic
346	C₂²	C₄×SD₁₆×C₄₃	1	1	yes	characteristic
347	C₂²	C₄₃×#440	1	1	yes	characteristic
348	C₄	—	2	1	no	—
349	C₄	—	2	1	no	—
350	C₄	—	2	1	no	—
351	C₄	—	4	2	no	—
352	C₄	—	8	2	no	—
353	C₄	C₄₃×#450	1	1	yes	characteristic
354	C₄	C₄₃×#452	1	1	yes	characteristic
355	C₄	C₄₃×#461	1	1	yes	characteristic
356	C₄	C₄₃×#463	1	1	yes	characteristic
357	C₄	C₄₃×#471	1	1	yes	characteristic
358	C₄	C₄₃×#473	1	1	yes	characteristic
359	C₂	—	2	1	no	—
360	C₂	—	2	1	no	—
361	C₂	—	8	2	no	—
362	C₂	C₄₃×#1877	1	1	yes	characteristic
363	C₂	C₄₃×#1908	1	1	yes	characteristic
364	C₂	C₄₃×#1914	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₄₃	#386
commutator	C₂²×C₄	C₂²×C₄×C₄₃
Frattini, cosocle	C₂×C₄²	C₂³×C₄₃
prime radical	C₄₃×#532	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₄₃ ◁ C₂×C₄²×C₄₃ ◁ G

Complements

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

N	H ≅ G/N	type
3. C₄₃×#1997	C₂	semidirect
5. C₂×C₄₃×#333	C₂	semidirect
6. C₄₃×#1301	C₂	semidirect
7. C₄₃×#1283	C₂	semidirect
95. M₁₆×C₄₃	SD₁₆	semidirect
157. #9267	C₄₃	direct (N × H)
179. #1997	C₂×C₄₃	semidirect
181. C₂×#333	C₂×C₄₃	semidirect
182. #1301	C₂×C₄₃	semidirect
183. #1283	C₂×C₄₃	semidirect
212. C₄₃	#9267	direct (N × H)
278. M₁₆	SD₁₆×C₄₃	semidirect

Maximal subgroups

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

subgroup	index
C₄₃×#3206	2
C₄₃×#2935	2
C₄₃×#1997	2
C₄₃×#1547	2
C₂×C₄₃×#333	2
C₄₃×#1301	2
C₄₃×#1283	2
#9267	43

Subgroups — C43×#9267 #24929284

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄₃×#9267 #24929284