Subgroups — C₄₃×#8898 #24928915

Order 11008.

390 conjugacy classes of subgroups.

The subgroup lattice has 336 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄₃×#2990	C₂	1	1	yes	maximal, characteristic
2	C₄₃×#1848	C₂	1	1	yes	maximal, characteristic, complemented
3	C₄₃×#1574	C₂	1	1	yes	maximal, characteristic, complemented
4	C₄₃×#1489	C₂	1	1	yes	maximal, characteristic
5	C₄₃×#1476	C₂	1	1	yes	maximal, characteristic
6	C₄₃×#1299	C₂	1	1	yes	maximal, characteristic, complemented
7	C₄₃×#1278	C₂	1	1	yes	maximal, characteristic, complemented
8	C₄×#41×C₄₃	C₂²	1	1	yes	characteristic
9	C₄₃×#446	—	2	1	no	—
10	C₄₃×#435	—	2	2	no	—
11	C₈×D₄×C₄₃	—	2	1	no	—
12	C₄₃×#432	—	2	1	no	—
13	C₄₃×#431	C₂²	1	1	yes	characteristic
14	C₄₃×#431	C₂²	1	1	yes	characteristic
15	C₂×#108×C₄₃	—	2	2	no	—
16	C₄₃×#423	C₂²	1	1	yes	characteristic
17	C₄₃×#423	C₂²	1	1	yes	characteristic
18	C₂×C₄⋊C₈×C₄₃	—	2	1	no	—
19	C₄×M₁₆×C₄₃	C₂²	1	1	yes	characteristic
20	C₂×C₄×C₈×C₄₃	C₂²	1	1	yes	characteristic
21	C₄₃×#335	—	2	1	no	—
22	C₄₃×#334	—	2	1	no	—
23	C₄₃×Q₈⋊C₈	—	2	1	no	—
24	C₄₃×Q₈⋊C₈	C₄	1	1	yes	characteristic
25	C₄₃×Q₈⋊C₈	C₄	1	1	yes	characteristic
26	C₄₃×D₄⋊C₈	—	2	1	no	—
27	C₄₃×D₄⋊C₈	C₄	1	1	yes	characteristic
28	C₄₃×D₄⋊C₈	C₄	1	1	yes	characteristic
29	C₂×#41×C₄₃	Q₈	1	1	yes	characteristic
30	C₂×M₁₆×C₄₃	—	2	1	no	—
31	C₂×M₁₆×C₄₃	—	2	1	no	—
32	C₂×M₁₆×C₄₃	D₄	1	2	yes	—
33	C₂²×C₈×C₄₃	—	2	1	no	—
34	C₂²×C₈×C₄₃	—	2	2	no	—
35	C₄×Q₈×C₄₃	C₂×C₄	1	1	yes	characteristic
36	C₄×Q₈×C₄₃	D₄	1	1	yes	characteristic
37	C₄×D₄×C₄₃	—	2	1	no	—
38	C₄×D₄×C₄₃	—	2	1	no	—
39	C₄×D₄×C₄₃	C₂×C₄	1	1	yes	characteristic
40	C₄×D₄×C₄₃	D₄	1	1	yes	characteristic
41	#117×C₄₃	—	2	1	no	—
42	#117×C₄₃	Q₈	1	1	yes	characteristic
43	C₂×C₄²×C₄₃	—	2	1	no	—
44	C₂×C₄²×C₄₃	C₂³	1	1	yes	characteristic
45	#108×C₄₃	—	4	4	no	—
46	C₄⋊C₈×C₄₃	—	2	2	no	—
47	C₄⋊C₈×C₄₃	—	2	1	no	—
48	C₄⋊C₈×C₄₃	—	2	1	no	—
49	C₄⋊C₈×C₄₃	—	2	1	no	—
50	C₄⋊C₈×C₄₃	—	2	1	no	—
51	C₄⋊C₈×C₄₃	C₂×C₄	1	1	yes	characteristic
52	C₄⋊C₈×C₄₃	C₂×C₄	1	1	yes	characteristic
53	C₄⋊C₈×C₄₃	D₄	1	1	yes	characteristic
54	C₄⋊C₈×C₄₃	D₄	1	1	yes	characteristic
55	C₂²⋊C₈×C₄₃	—	2	2	no	—
56	C₂²⋊C₈×C₄₃	—	2	1	no	—
57	C₂²⋊C₈×C₄₃	—	2	1	no	—
58	#97×C₄₃	—	2	2	no	—
59	#97×C₄₃	—	2	1	no	—
60	C₄×C₈×C₄₃	—	2	1	no	—
61	C₄×C₈×C₄₃	—	2	1	no	—
62	C₄×C₈×C₄₃	—	2	1	no	—
63	C₄×C₈×C₄₃	—	2	1	no	—
64	C₄×C₈×C₄₃	C₂×C₄	1	1	yes	characteristic
65	C₄×C₈×C₄₃	C₂×C₄	1	1	yes	characteristic
66	#41×C₄₃	—	2	2	no	—
67	#41×C₄₃	—	4	1	no	—
68	C₂×Q₈×C₄₃	C₄⋊C₄	1	1	yes	characteristic
69	C₂×D₄×C₄₃	—	2	1	no	—
70	C₂×D₄×C₄₃	C₄⋊C₄	1	1	yes	characteristic
71	C₂²×C₄×C₄₃	—	2	1	no	—
72	C₂²×C₄×C₄₃	—	2	1	no	—
73	C₂²×C₄×C₄₃	—	2	1	no	—
74	C₂²×C₄×C₄₃	C₂×D₄	1	1	yes	characteristic
75	C₂²×C₄×C₄₃	C₂×Q₈	1	1	yes	characteristic
76	C₂²×C₄×C₄₃	#41	1	1	yes	characteristic
77	M₁₆×C₄₃	—	2	2	no	—
78	M₁₆×C₄₃	—	4	1	no	—
79	M₁₆×C₄₃	—	4	2	no	—
80	M₁₆×C₄₃	—	4	2	no	—
81	C₂×C₈×C₄₃	—	2	2	no	—
82	C₂×C₈×C₄₃	—	2	2	no	—
83	C₂×C₈×C₄₃	—	2	1	no	—
84	C₂×C₈×C₄₃	—	2	1	no	—
85	C₂×C₈×C₄₃	—	2	1	no	—
86	C₂×C₈×C₄₃	—	2	1	no	—
87	C₂×C₈×C₄₃	—	2	1	no	—
88	C₂×C₈×C₄₃	—	2	1	no	—
89	C₂×C₈×C₄₃	—	2	2	no	—
90	C₂×C₈×C₄₃	—	4	1	no	—
91	C₂×C₈×C₄₃	—	4	1	no	—
92	C₂×C₈×C₄₃	—	8	2	no	—
93	C₄⋊C₄×C₄₃	—	2	1	no	—
94	C₄⋊C₄×C₄₃	—	2	1	no	—
95	C₄⋊C₄×C₄₃	C₄⋊C₄	1	1	yes	characteristic
96	C₄⋊C₄×C₄₃	C₄⋊C₄	1	1	yes	characteristic
97	C₂²⋊C₄×C₄₃	—	2	1	no	—
98	C₂²⋊C₄×C₄₃	—	2	1	no	—
99	C₂²⋊C₄×C₄₃	—	2	1	no	—
100	C₄²×C₄₃	—	2	1	no	—
101	C₄²×C₄₃	—	2	1	no	—
102	C₄²×C₄₃	—	2	1	no	—
103	C₄²×C₄₃	C₂²×C₄	1	1	yes	characteristic
104	C₄²×C₄₃	C₂×D₄	1	1	yes	characteristic
105	C₄²×C₄₃	C₂×D₄	1	1	yes	characteristic
106	C₄²×C₄₃	#41	1	1	yes	characteristic
107	C₂³×C₄₃	—	2	1	no	—
108	C₂³×C₄₃	C₂²⋊Q₈	1	1	yes	characteristic
109	Q₈×C₄₃	—	2	1	no	—
110	Q₈×C₄₃	—	2	1	no	—
111	D₄×C₄₃	—	2	1	no	—
112	D₄×C₄₃	—	4	2	no	—
113	D₄×C₄₃	#108	1	2	yes	complemented
114	C₂×C₄×C₄₃	—	2	1	no	—
115	C₂×C₄×C₄₃	—	2	1	no	—
116	C₂×C₄×C₄₃	—	2	1	no	—
117	C₂×C₄×C₄₃	—	2	1	no	—
118	C₂×C₄×C₄₃	—	2	1	no	—
119	C₂×C₄×C₄₃	—	2	1	no	—
120	C₂×C₄×C₄₃	—	2	1	no	—
121	C₂×C₄×C₄₃	—	2	1	no	—
122	C₂×C₄×C₄₃	—	2	1	no	—
123	C₂×C₄×C₄₃	—	4	2	no	—
124	C₂×C₄×C₄₃	—	4	1	no	—
125	C₂×C₄×C₄₃	—	4	1	no	—
126	C₂×C₄×C₄₃	—	4	1	no	—
127	C₂×C₄×C₄₃	—	4	1	no	—
128	C₂×C₄×C₄₃	C₂×C₄⋊C₄	1	1	yes	characteristic
129	C₂×C₄×C₄₃	C₄×D₄	1	1	yes	characteristic
130	C₂×C₄×C₄₃	C₄×D₄	1	1	yes	characteristic
131	C₂×C₄×C₄₃	#120	1	1	yes	characteristic
132	C₂×C₄×C₄₃	C₂²⋊Q₈	1	1	yes	characteristic
133	C₂×C₄×C₄₃	#123	1	1	yes	characteristic
134	C₈×C₄₃	—	2	2	no	—
135	C₈×C₄₃	—	2	2	no	—
136	C₈×C₄₃	—	4	2	no	—
137	C₈×C₄₃	—	4	1	no	—
138	C₈×C₄₃	—	4	1	no	—
139	C₈×C₄₃	—	4	1	no	—
140	C₈×C₄₃	—	4	2	no	—
141	C₈×C₄₃	—	4	1	no	—
142	C₈×C₄₃	—	8	1	no	—
143	#8898	C₄₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
144	C₂²×C₄₃	—	2	2	no	—
145	C₂²×C₄₃	—	2	1	no	—
146	C₂²×C₄₃	—	2	1	no	—
147	C₂²×C₄₃	—	2	1	no	—
148	C₂²×C₄₃	—	4	1	no	—
149	C₂²×C₄₃	—	4	1	no	—
150	C₂²×C₄₃	#385	1	1	yes	characteristic
151	C₄×C₄₃	—	2	1	no	—
152	C₄×C₄₃	—	2	1	no	—
153	C₄×C₄₃	—	2	1	no	—
154	C₄×C₄₃	—	4	1	no	—
155	C₄×C₄₃	—	4	1	no	—
156	C₄×C₄₃	—	4	1	no	—
157	C₄×C₄₃	C₂×#108	1	1	yes	characteristic
158	C₄×C₄₃	#430	1	1	yes	characteristic
159	C₄×C₄₃	#443	1	1	yes	characteristic
160	C₄×C₄₃	#444	1	1	yes	characteristic
161	C₄×C₄₃	#453	1	1	yes	characteristic
162	C₄×C₄₃	#456	1	1	yes	characteristic
163	#2990	C₂×C₄₃	1	1	yes	characteristic
164	#1848	C₂×C₄₃	1	1	yes	characteristic, complemented
165	#1574	C₂×C₄₃	1	1	yes	characteristic, complemented
166	#1489	C₂×C₄₃	1	1	yes	characteristic
167	#1476	C₂×C₄₃	1	1	yes	characteristic
168	#1299	C₂×C₄₃	1	1	yes	characteristic, complemented
169	#1278	C₂×C₄₃	1	1	yes	characteristic, complemented
170	C₂×C₄₃	—	4	1	no	—
171	C₂×C₄₃	—	4	2	no	—
172	C₂×C₄₃	#1855	1	1	yes	characteristic
173	C₂×C₄₃	#1859	1	1	yes	characteristic
174	C₂×C₄₃	#1866	1	1	yes	characteristic
175	C₄×#41	C₂²×C₄₃	1	1	yes	characteristic
176	#446	—	2	1	no	—
177	#435	—	2	2	no	—
178	C₈×D₄	—	2	1	no	—
179	#432	—	2	1	no	—
180	#431	C₂²×C₄₃	1	1	yes	characteristic
181	#431	C₂²×C₄₃	1	1	yes	characteristic
182	C₂×#108	—	2	2	no	—
183	#423	C₂²×C₄₃	1	1	yes	characteristic
184	#423	C₂²×C₄₃	1	1	yes	characteristic
185	C₂×C₄⋊C₈	—	2	1	no	—
186	C₄×M₁₆	C₂²×C₄₃	1	1	yes	characteristic
187	C₂×C₄×C₈	C₂²×C₄₃	1	1	yes	characteristic
188	#335	—	2	1	no	—
189	#334	—	2	1	no	—
190	Q₈⋊C₈	—	2	1	no	—
191	Q₈⋊C₈	C₄×C₄₃	1	1	yes	characteristic
192	Q₈⋊C₈	C₄×C₄₃	1	1	yes	characteristic
193	D₄⋊C₈	—	2	1	no	—
194	D₄⋊C₈	C₄×C₄₃	1	1	yes	characteristic
195	D₄⋊C₈	C₄×C₄₃	1	1	yes	characteristic
196	C₄₃	#8898	1	1	yes	characteristic, direct product factor, Sylow
197	C₂×#41	Q₈×C₄₃	1	1	yes	characteristic
198	C₂×M₁₆	—	2	1	no	—
199	C₂×M₁₆	—	2	1	no	—
200	C₂×M₁₆	D₄×C₄₃	1	2	yes	—
201	C₂²×C₈	—	2	1	no	—
202	C₂²×C₈	—	2	2	no	—
203	C₄×Q₈	C₂×C₄×C₄₃	1	1	yes	characteristic
204	C₄×Q₈	D₄×C₄₃	1	1	yes	characteristic
205	C₄×D₄	—	2	1	no	—
206	C₄×D₄	—	2	1	no	—
207	C₄×D₄	C₂×C₄×C₄₃	1	1	yes	characteristic
208	C₄×D₄	D₄×C₄₃	1	1	yes	characteristic
209	#117	—	2	1	no	—
210	#117	Q₈×C₄₃	1	1	yes	characteristic
211	C₂×C₄²	—	2	1	no	—
212	C₂×C₄²	C₂³×C₄₃	1	1	yes	characteristic
213	#108	—	4	4	no	—
214	C₄⋊C₈	—	2	2	no	—
215	C₄⋊C₈	—	2	1	no	—
216	C₄⋊C₈	—	2	1	no	—
217	C₄⋊C₈	—	2	1	no	—
218	C₄⋊C₈	—	2	1	no	—
219	C₄⋊C₈	C₂×C₄×C₄₃	1	1	yes	characteristic
220	C₄⋊C₈	C₂×C₄×C₄₃	1	1	yes	characteristic
221	C₄⋊C₈	D₄×C₄₃	1	1	yes	characteristic
222	C₄⋊C₈	D₄×C₄₃	1	1	yes	characteristic
223	C₂²⋊C₈	—	2	2	no	—
224	C₂²⋊C₈	—	2	1	no	—
225	C₂²⋊C₈	—	2	1	no	—
226	#97	—	2	2	no	—
227	#97	—	2	1	no	—
228	C₄×C₈	—	2	1	no	—
229	C₄×C₈	—	2	1	no	—
230	C₄×C₈	—	2	1	no	—
231	C₄×C₈	—	2	1	no	—
232	C₄×C₈	C₂×C₄×C₄₃	1	1	yes	characteristic
233	C₄×C₈	C₂×C₄×C₄₃	1	1	yes	characteristic
234	#41	—	2	2	no	—
235	#41	—	4	1	no	—
236	C₂×Q₈	C₄⋊C₄×C₄₃	1	1	yes	characteristic
237	C₂×D₄	—	2	1	no	—
238	C₂×D₄	C₄⋊C₄×C₄₃	1	1	yes	characteristic
239	C₂²×C₄	—	2	1	no	—
240	C₂²×C₄	—	2	1	no	—
241	C₂²×C₄	—	2	1	no	—
242	C₂²×C₄	C₂×D₄×C₄₃	1	1	yes	characteristic
243	C₂²×C₄	C₂×Q₈×C₄₃	1	1	yes	characteristic
244	C₂²×C₄	#41×C₄₃	1	1	yes	characteristic
245	M₁₆	—	2	2	no	—
246	M₁₆	—	4	1	no	—
247	M₁₆	—	4	2	no	—
248	M₁₆	—	4	2	no	—
249	C₂×C₈	—	2	2	no	—
250	C₂×C₈	—	2	2	no	—
251	C₂×C₈	—	2	1	no	—
252	C₂×C₈	—	2	1	no	—
253	C₂×C₈	—	2	1	no	—
254	C₂×C₈	—	2	1	no	—
255	C₂×C₈	—	2	1	no	—
256	C₂×C₈	—	2	1	no	—
257	C₂×C₈	—	2	2	no	—
258	C₂×C₈	—	4	1	no	—
259	C₂×C₈	—	4	1	no	—
260	C₂×C₈	—	8	2	no	—
261	C₄⋊C₄	—	2	1	no	—
262	C₄⋊C₄	—	2	1	no	—
263	C₄⋊C₄	C₄⋊C₄×C₄₃	1	1	yes	characteristic
264	C₄⋊C₄	C₄⋊C₄×C₄₃	1	1	yes	characteristic
265	C₂²⋊C₄	—	2	1	no	—
266	C₂²⋊C₄	—	2	1	no	—
267	C₂²⋊C₄	—	2	1	no	—
268	C₄²	—	2	1	no	—
269	C₄²	—	2	1	no	—
270	C₄²	—	2	1	no	—
271	C₄²	C₂²×C₄×C₄₃	1	1	yes	characteristic
272	C₄²	C₂×D₄×C₄₃	1	1	yes	characteristic
273	C₄²	C₂×D₄×C₄₃	1	1	yes	characteristic
274	C₄²	#41×C₄₃	1	1	yes	characteristic
275	C₂³	—	2	1	no	—
276	C₂³	C₂²⋊Q₈×C₄₃	1	1	yes	characteristic
277	Q₈	—	2	1	no	—
278	Q₈	—	2	1	no	—
279	D₄	—	2	1	no	—
280	D₄	—	4	2	no	—
281	D₄	#108×C₄₃	1	2	yes	complemented
282	C₂×C₄	—	2	1	no	—
283	C₂×C₄	—	2	1	no	—
284	C₂×C₄	—	2	1	no	—
285	C₂×C₄	—	2	1	no	—
286	C₂×C₄	—	2	1	no	—
287	C₂×C₄	—	2	1	no	—
288	C₂×C₄	—	2	1	no	—
289	C₂×C₄	—	2	1	no	—
290	C₂×C₄	—	2	1	no	—
291	C₂×C₄	—	4	2	no	—
292	C₂×C₄	—	4	1	no	—
293	C₂×C₄	—	4	1	no	—
294	C₂×C₄	—	4	1	no	—
295	C₂×C₄	—	4	1	no	—
296	C₂×C₄	C₂×C₄⋊C₄×C₄₃	1	1	yes	characteristic
297	C₂×C₄	C₄×D₄×C₄₃	1	1	yes	characteristic
298	C₂×C₄	C₄×D₄×C₄₃	1	1	yes	characteristic
299	C₂×C₄	#120×C₄₃	1	1	yes	characteristic
300	C₂×C₄	C₂²⋊Q₈×C₄₃	1	1	yes	characteristic
301	C₂×C₄	#123×C₄₃	1	1	yes	characteristic
302	C₈	—	2	2	no	—
303	C₈	—	2	2	no	—
304	C₈	—	4	2	no	—
305	C₈	—	4	1	no	—
306	C₈	—	4	1	no	—
307	C₈	—	4	1	no	—
308	C₈	—	4	2	no	—
309	C₈	—	4	1	no	—
310	C₈	—	8	1	no	—
311	C₂²	—	2	1	no	—
312	C₂²	—	2	2	no	—
313	C₂²	—	2	1	no	—
314	C₂²	—	2	1	no	—
315	C₂²	—	4	1	no	—
316	C₂²	—	4	1	no	—
317	C₂²	C₄₃×#385	1	1	yes	characteristic
318	C₄	—	2	1	no	—
319	C₄	—	2	1	no	—
320	C₄	—	2	1	no	—
321	C₄	—	4	1	no	—
322	C₄	—	4	1	no	—
323	C₄	—	4	1	no	—
324	C₄	C₂×#108×C₄₃	1	1	yes	characteristic
325	C₄	C₄₃×#430	1	1	yes	characteristic
326	C₄	C₄₃×#443	1	1	yes	characteristic
327	C₄	C₄₃×#444	1	1	yes	characteristic
328	C₄	C₄₃×#453	1	1	yes	characteristic
329	C₄	C₄₃×#456	1	1	yes	characteristic
330	C₂	—	4	1	no	—
331	C₂	—	4	2	no	—
332	C₂	C₄₃×#1855	1	1	yes	characteristic
333	C₂	C₄₃×#1859	1	1	yes	characteristic
334	C₂	C₄₃×#1866	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄×C₄₃	#120
commutator	C₄²	C₂²×C₄×C₄₃
Frattini, cosocle	C₂×C₄²	C₂³×C₄₃
socle	C₂²×C₄₃	#385
prime radical	C₂×#41×C₄₃	Q₈
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₄₃ ◁ C₂×C₄²×C₄₃ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₄₃×#1848	C₂	semidirect
3. C₄₃×#1574	C₂	semidirect
6. C₄₃×#1299	C₂	semidirect
7. C₄₃×#1278	C₂	semidirect
113. D₄×C₄₃	#108	semidirect
143. #8898	C₄₃	direct (N × H)
164. #1848	C₂×C₄₃	semidirect
165. #1574	C₂×C₄₃	semidirect
168. #1299	C₂×C₄₃	semidirect
169. #1278	C₂×C₄₃	semidirect
196. C₄₃	#8898	direct (N × H)
281. D₄	#108×C₄₃	semidirect

Maximal subgroups

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

subgroup	index
C₄₃×#2990	2
C₄₃×#1848	2
C₄₃×#1574	2
C₄₃×#1489	2
C₄₃×#1476	2
C₄₃×#1299	2
C₄₃×#1278	2
#8898	43

Subgroups — C43×#8898 #24928915

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄₃×#8898 #24928915