Subgroups — C₄₃×#18221 #24938238

Order 11008.

890 conjugacy classes of subgroups.

The subgroup lattice has 352 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄₃×#3502	C₂	1	1	yes	maximal, characteristic
2	C₄₃×#3104	C₂	1	2	yes	maximal
3	C₄₃×#2581	C₂	1	1	yes	maximal, characteristic
4	C₄₃×#1617	C₂	1	4	yes	maximal
5	C₄₃×#1605	C₂	1	4	yes	maximal
6	C₄₃×#1518	C₂	1	2	yes	maximal
7	C₄₃×#1461	C₂	1	1	yes	maximal, characteristic
8	Q₈²×C₄₃	—	2	2	no	—
9	C₄₃×#557	—	2	2	no	—
10	C₄₃×#557	—	2	4	no	—
11	C₄₃×#544	—	2	1	no	—
12	C₄₃×#544	C₂²	1	4	yes	—
13	C₄₃×#533	—	2	2	no	—
14	C₄₃×#533	C₂²	1	2	yes	—
15	C₄₃×#519	—	2	1	no	—
16	C₂×C₄×Q₈×C₄₃	C₂²	1	1	yes	characteristic
17	C₂×C₄×Q₈×C₄₃	C₂²	1	1	yes	characteristic
18	C₄₃×#484	C₂²	1	4	yes	—
19	C₄₃×#477	—	2	2	no	—
20	C₄₃×#462	—	2	2	no	—
21	C₄₃×#451	C₂²	1	4	yes	—
22	C₄×Q₁₆×C₄₃	—	2	4	no	—
23	C₄₃×#431	C₂²	1	2	yes	—
24	C₄₃×#400	C₂²	1	2	yes	—
25	C₄₃×#395	C₂²	1	2	yes	—
26	C₄₃×#391	—	2	1	no	—
27	C₄₃×#389	—	2	1	no	—
28	C₄₃×#387	—	2	2	no	—
29	C₄₃×#382	C₂²	1	1	yes	characteristic
30	C₄₃×#328	C₂²	1	8	yes	—
31	C₄₃×#324	C₂²	1	4	yes	—
32	C₂²×Q₈×C₄₃	D₄	1	1	yes	characteristic
33	C₂²×Q₈×C₄₃	C₂³	1	1	yes	characteristic
34	C₂×Q₁₆×C₄₃	—	2	4	no	—
35	C₄⋊Q₈×C₄₃	—	2	2	no	—
36	C₄⋊Q₈×C₄₃	—	2	2	no	—
37	C₄⋊Q₈×C₄₃	—	2	4	no	—
38	C₄⋊Q₈×C₄₃	—	2	8	no	—
39	C₄⋊Q₈×C₄₃	D₄	1	2	yes	—
40	C₄⋊Q₈×C₄₃	D₄	1	2	yes	—
41	#125×C₄₃	—	2	4	no	—
42	#125×C₄₃	—	2	4	no	—
43	#125×C₄₃	—	2	8	no	—
44	#125×C₄₃	—	2	2	no	—
45	C₂²⋊Q₈×C₄₃	—	2	2	no	—
46	C₂²⋊Q₈×C₄₃	—	2	4	no	—
47	C₂²⋊Q₈×C₄₃	C₂³	1	4	yes	—
48	C₄×Q₈×C₄₃	—	2	4	no	—
49	C₄×Q₈×C₄₃	—	2	4	no	—
50	C₄×Q₈×C₄₃	—	2	4	no	—
51	C₄×Q₈×C₄₃	—	2	4	no	—
52	C₄×Q₈×C₄₃	—	2	2	no	—
53	C₄×Q₈×C₄₃	—	2	2	no	—
54	C₄×Q₈×C₄₃	—	2	2	no	—
55	C₄×Q₈×C₄₃	—	2	2	no	—
56	C₄×Q₈×C₄₃	—	2	2	no	—
57	C₄×Q₈×C₄₃	D₄	1	4	yes	—
58	#117×C₄₃	—	2	4	no	—
59	#117×C₄₃	—	2	2	no	—
60	C₂×C₄⋊C₄×C₄₃	—	2	1	no	—
61	C₂×C₄⋊C₄×C₄₃	—	2	1	no	—
62	C₂×C₄⋊C₄×C₄₃	—	2	2	no	—
63	C₂×C₄⋊C₄×C₄₃	D₄	1	1	yes	characteristic
64	C₂×C₄⋊C₄×C₄₃	C₂³	1	1	yes	characteristic
65	C₂×C₄⋊C₄×C₄₃	C₂³	1	2	yes	—
66	C₂×C₄²×C₄₃	—	2	1	no	—
67	C₂×C₄²×C₄₃	D₄	1	2	yes	—
68	C₂×C₄²×C₄₃	C₂³	1	1	yes	characteristic
69	#107×C₄₃	—	2	4	no	—
70	C₄⋊C₈×C₄₃	—	2	2	no	—
71	#103×C₄₃	—	2	4	no	—
72	#103×C₄₃	—	2	4	no	—
73	C₂²⋊C₈×C₄₃	C₂³	1	4	yes	—
74	C₄×C₈×C₄₃	—	2	2	no	—
75	#95×C₄₃	—	2	1	no	—
76	#95×C₄₃	—	2	2	no	—
77	#95×C₄₃	—	2	2	no	—
78	#95×C₄₃	—	2	2	no	—
79	#95×C₄₃	C₂³	1	2	yes	—
80	C₂×Q₈×C₄₃	—	2	4	no	—
81	C₂×Q₈×C₄₃	—	2	2	no	—
82	C₂×Q₈×C₄₃	—	2	2	no	—
83	C₂×Q₈×C₄₃	—	2	2	no	—
84	C₂×Q₈×C₄₃	—	2	2	no	—
85	C₂×Q₈×C₄₃	—	4	2	no	—
86	C₂×Q₈×C₄₃	Q₁₆	1	4	yes	complemented
87	C₂×Q₈×C₄₃	C₂×D₄	1	4	yes	—
88	C₂²×C₄×C₄₃	—	2	1	no	—
89	C₂²×C₄×C₄₃	—	2	2	no	—
90	C₂²×C₄×C₄₃	—	2	1	no	—
91	C₂²×C₄×C₄₃	C₂×D₄	1	2	yes	—
92	C₂²×C₄×C₄₃	C₂×D₄	1	2	yes	—
93	C₂²×C₄×C₄₃	C₂×D₄	1	1	yes	characteristic
94	C₂²×C₄×C₄₃	C₂×D₄	1	1	yes	characteristic
95	C₂²×C₄×C₄₃	C₂⁴	1	1	yes	characteristic
96	Q₁₆×C₄₃	—	4	8	no	—
97	C₂×C₈×C₄₃	—	2	4	no	—
98	C₄⋊C₄×C₄₃	—	2	4	no	—
99	C₄⋊C₄×C₄₃	—	2	4	no	—
100	C₄⋊C₄×C₄₃	—	2	4	no	—
101	C₄⋊C₄×C₄₃	—	2	8	no	—
102	C₄⋊C₄×C₄₃	—	2	2	no	—
103	C₄⋊C₄×C₄₃	—	2	2	no	—
104	C₄⋊C₄×C₄₃	—	2	2	no	—
105	C₄⋊C₄×C₄₃	—	2	2	no	—
106	C₄⋊C₄×C₄₃	—	2	4	no	—
107	C₄⋊C₄×C₄₃	—	2	4	no	—
108	C₄⋊C₄×C₄₃	—	2	4	no	—
109	C₄⋊C₄×C₄₃	—	2	2	no	—
110	C₄⋊C₄×C₄₃	—	4	4	no	—
111	C₄⋊C₄×C₄₃	—	4	1	no	—
112	C₄⋊C₄×C₄₃	—	4	1	no	—
113	C₄⋊C₄×C₄₃	—	4	2	no	—
114	C₄⋊C₄×C₄₃	C₂×D₄	1	4	yes	—
115	C₂²⋊C₄×C₄₃	—	2	4	no	—
116	C₄²×C₄₃	—	2	8	no	—
117	C₄²×C₄₃	—	2	2	no	—
118	C₄²×C₄₃	—	2	2	no	—
119	C₄²×C₄₃	—	2	4	no	—
120	C₄²×C₄₃	—	4	2	no	—
121	C₄²×C₄₃	C₂×D₄	1	2	yes	—
122	C₄²×C₄₃	C₂×D₄	1	2	yes	—
123	C₂³×C₄₃	C₂²×D₄	1	1	yes	characteristic
124	Q₈×C₄₃	—	2	8	no	—
125	Q₈×C₄₃	—	2	4	no	—
126	Q₈×C₄₃	—	4	4	no	—
127	Q₈×C₄₃	—	4	4	no	—
128	Q₈×C₄₃	—	4	2	no	—
129	Q₈×C₄₃	—	8	4	no	—
130	C₂×C₄×C₄₃	—	2	2	no	—
131	C₂×C₄×C₄₃	—	2	2	no	—
132	C₂×C₄×C₄₃	—	2	2	no	—
133	C₂×C₄×C₄₃	—	2	4	no	—
134	C₂×C₄×C₄₃	—	2	4	no	—
135	C₂×C₄×C₄₃	—	2	2	no	—
136	C₂×C₄×C₄₃	—	4	1	no	—
137	C₂×C₄×C₄₃	—	4	1	no	—
138	C₂×C₄×C₄₃	—	4	1	no	—
139	C₂×C₄×C₄₃	—	4	1	no	—
140	C₂×C₄×C₄₃	—	4	2	no	—
141	C₂×C₄×C₄₃	—	4	2	no	—
142	C₂×C₄×C₄₃	—	4	2	no	—
143	C₂×C₄×C₄₃	—	4	2	no	—
144	C₂×C₄×C₄₃	—	4	1	no	—
145	C₂×C₄×C₄₃	—	4	2	no	—
146	C₂×C₄×C₄₃	#120	1	1	yes	characteristic
147	C₂×C₄×C₄₃	#120	1	1	yes	characteristic
148	C₂×C₄×C₄₃	#120	1	2	yes	—
149	C₂×C₄×C₄₃	C₂×Q₁₆	1	4	yes	—
150	C₂×C₄×C₄₃	C₂×Q₁₆	1	2	yes	—
151	C₂×C₄×C₄₃	#137	1	2	yes	—
152	C₂×C₄×C₄₃	C₂²×D₄	1	2	yes	—
153	C₈×C₄₃	—	4	4	no	—
154	#18221	C₄₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
155	C₂²×C₄₃	—	2	1	no	—
156	C₂²×C₄₃	—	2	1	no	—
157	C₂²×C₄₃	C₂×#120	1	1	yes	characteristic
158	C₂²×C₄₃	C₂²×Q₁₆	1	1	yes	characteristic
159	C₂²×C₄₃	C₂×#137	1	1	yes	characteristic
160	C₄×C₄₃	—	2	4	no	—
161	C₄×C₄₃	—	2	1	no	—
162	C₄×C₄₃	—	2	1	no	—
163	C₄×C₄₃	—	2	2	no	—
164	C₄×C₄₃	—	4	4	no	—
165	C₄×C₄₃	—	4	2	no	—
166	C₄×C₄₃	—	4	2	no	—
167	C₄×C₄₃	—	4	4	no	—
168	C₄×C₄₃	—	8	2	no	—
169	C₄×C₄₃	—	8	1	no	—
170	C₄×C₄₃	#451	1	4	yes	—
171	#3502	C₂×C₄₃	1	1	yes	characteristic
172	#3104	C₂×C₄₃	1	2	yes	—
173	#2581	C₂×C₄₃	1	1	yes	characteristic
174	#1617	C₂×C₄₃	1	4	yes	—
175	#1605	C₂×C₄₃	1	4	yes	—
176	#1518	C₂×C₄₃	1	2	yes	—
177	#1461	C₂×C₄₃	1	1	yes	characteristic
178	C₂×C₄₃	—	2	1	no	—
179	C₂×C₄₃	—	2	1	no	—
180	C₂×C₄₃	C₂×#451	1	1	yes	characteristic
181	C₂×C₄₃	#3020	1	1	yes	characteristic
182	C₂×C₄₃	#3025	1	1	yes	characteristic
183	Q₈²	—	2	2	no	—
184	#557	—	2	2	no	—
185	#557	—	2	4	no	—
186	#544	—	2	1	no	—
187	#544	C₂²×C₄₃	1	4	yes	—
188	#533	—	2	2	no	—
189	#533	C₂²×C₄₃	1	2	yes	—
190	#519	—	2	1	no	—
191	C₂×C₄×Q₈	C₂²×C₄₃	1	1	yes	characteristic
192	C₂×C₄×Q₈	C₂²×C₄₃	1	1	yes	characteristic
193	#484	C₂²×C₄₃	1	4	yes	—
194	#477	—	2	2	no	—
195	#462	—	2	2	no	—
196	#451	C₂²×C₄₃	1	4	yes	—
197	C₄×Q₁₆	—	2	4	no	—
198	#431	C₂²×C₄₃	1	2	yes	—
199	#400	C₂²×C₄₃	1	2	yes	—
200	#395	C₂²×C₄₃	1	2	yes	—
201	#391	—	2	1	no	—
202	#389	—	2	1	no	—
203	#387	—	2	2	no	—
204	#382	C₂²×C₄₃	1	1	yes	characteristic
205	#328	C₂²×C₄₃	1	8	yes	—
206	#324	C₂²×C₄₃	1	4	yes	—
207	C₄₃	#18221	1	1	yes	characteristic, direct product factor, Sylow
208	C₂²×Q₈	D₄×C₄₃	1	1	yes	characteristic
209	C₂²×Q₈	C₂³×C₄₃	1	1	yes	characteristic
210	C₂×Q₁₆	—	2	4	no	—
211	C₄⋊Q₈	—	2	2	no	—
212	C₄⋊Q₈	—	2	2	no	—
213	C₄⋊Q₈	—	2	4	no	—
214	C₄⋊Q₈	—	2	8	no	—
215	C₄⋊Q₈	D₄×C₄₃	1	2	yes	—
216	C₄⋊Q₈	D₄×C₄₃	1	2	yes	—
217	#125	—	2	4	no	—
218	#125	—	2	4	no	—
219	#125	—	2	8	no	—
220	#125	—	2	2	no	—
221	C₂²⋊Q₈	—	2	2	no	—
222	C₂²⋊Q₈	—	2	4	no	—
223	C₂²⋊Q₈	C₂³×C₄₃	1	4	yes	—
224	C₄×Q₈	—	2	4	no	—
225	C₄×Q₈	—	2	4	no	—
226	C₄×Q₈	—	2	4	no	—
227	C₄×Q₈	—	2	4	no	—
228	C₄×Q₈	—	2	2	no	—
229	C₄×Q₈	—	2	2	no	—
230	C₄×Q₈	—	2	2	no	—
231	C₄×Q₈	—	2	2	no	—
232	C₄×Q₈	—	2	2	no	—
233	C₄×Q₈	D₄×C₄₃	1	4	yes	—
234	#117	—	2	4	no	—
235	#117	—	2	2	no	—
236	C₂×C₄⋊C₄	—	2	1	no	—
237	C₂×C₄⋊C₄	—	2	1	no	—
238	C₂×C₄⋊C₄	—	2	2	no	—
239	C₂×C₄⋊C₄	D₄×C₄₃	1	1	yes	characteristic
240	C₂×C₄⋊C₄	C₂³×C₄₃	1	1	yes	characteristic
241	C₂×C₄⋊C₄	C₂³×C₄₃	1	2	yes	—
242	C₂×C₄²	—	2	1	no	—
243	C₂×C₄²	D₄×C₄₃	1	2	yes	—
244	C₂×C₄²	C₂³×C₄₃	1	1	yes	characteristic
245	#107	—	2	4	no	—
246	C₄⋊C₈	—	2	2	no	—
247	#103	—	2	4	no	—
248	#103	—	2	4	no	—
249	C₂²⋊C₈	C₂³×C₄₃	1	4	yes	—
250	C₄×C₈	—	2	2	no	—
251	#95	—	2	1	no	—
252	#95	—	2	2	no	—
253	#95	—	2	2	no	—
254	#95	—	2	2	no	—
255	#95	C₂³×C₄₃	1	2	yes	—
256	C₂×Q₈	—	2	4	no	—
257	C₂×Q₈	—	2	2	no	—
258	C₂×Q₈	—	2	2	no	—
259	C₂×Q₈	—	2	2	no	—
260	C₂×Q₈	—	2	2	no	—
261	C₂×Q₈	—	4	2	no	—
262	C₂×Q₈	Q₁₆×C₄₃	1	4	yes	complemented
263	C₂×Q₈	C₂×D₄×C₄₃	1	4	yes	—
264	C₂²×C₄	—	2	1	no	—
265	C₂²×C₄	—	2	2	no	—
266	C₂²×C₄	—	2	1	no	—
267	C₂²×C₄	C₂×D₄×C₄₃	1	2	yes	—
268	C₂²×C₄	C₂×D₄×C₄₃	1	2	yes	—
269	C₂²×C₄	C₂×D₄×C₄₃	1	1	yes	characteristic
270	C₂²×C₄	C₂×D₄×C₄₃	1	1	yes	characteristic
271	C₂²×C₄	C₂⁴×C₄₃	1	1	yes	characteristic
272	Q₁₆	—	4	8	no	—
273	C₂×C₈	—	2	4	no	—
274	C₄⋊C₄	—	2	4	no	—
275	C₄⋊C₄	—	2	4	no	—
276	C₄⋊C₄	—	2	4	no	—
277	C₄⋊C₄	—	2	8	no	—
278	C₄⋊C₄	—	2	2	no	—
279	C₄⋊C₄	—	2	2	no	—
280	C₄⋊C₄	—	2	2	no	—
281	C₄⋊C₄	—	2	2	no	—
282	C₄⋊C₄	—	2	4	no	—
283	C₄⋊C₄	—	2	4	no	—
284	C₄⋊C₄	—	2	4	no	—
285	C₄⋊C₄	—	2	2	no	—
286	C₄⋊C₄	—	4	4	no	—
287	C₄⋊C₄	—	4	1	no	—
288	C₄⋊C₄	—	4	1	no	—
289	C₄⋊C₄	—	4	2	no	—
290	C₄⋊C₄	C₂×D₄×C₄₃	1	4	yes	—
291	C₂²⋊C₄	—	2	4	no	—
292	C₄²	—	2	8	no	—
293	C₄²	—	2	2	no	—
294	C₄²	—	2	2	no	—
295	C₄²	—	2	4	no	—
296	C₄²	—	4	2	no	—
297	C₄²	C₂×D₄×C₄₃	1	2	yes	—
298	C₄²	C₂×D₄×C₄₃	1	2	yes	—
299	C₂³	C₂²×D₄×C₄₃	1	1	yes	characteristic
300	Q₈	—	2	8	no	—
301	Q₈	—	2	4	no	—
302	Q₈	—	4	4	no	—
303	Q₈	—	4	2	no	—
304	Q₈	—	4	4	no	—
305	Q₈	—	8	4	no	—
306	C₂×C₄	—	2	2	no	—
307	C₂×C₄	—	2	2	no	—
308	C₂×C₄	—	2	2	no	—
309	C₂×C₄	—	2	4	no	—
310	C₂×C₄	—	2	4	no	—
311	C₂×C₄	—	2	2	no	—
312	C₂×C₄	—	4	1	no	—
313	C₂×C₄	—	4	1	no	—
314	C₂×C₄	—	4	1	no	—
315	C₂×C₄	—	4	1	no	—
316	C₂×C₄	—	4	2	no	—
317	C₂×C₄	—	4	2	no	—
318	C₂×C₄	—	4	2	no	—
319	C₂×C₄	—	4	2	no	—
320	C₂×C₄	—	4	1	no	—
321	C₂×C₄	—	4	2	no	—
322	C₂×C₄	#120×C₄₃	1	1	yes	characteristic
323	C₂×C₄	#120×C₄₃	1	1	yes	characteristic
324	C₂×C₄	#120×C₄₃	1	2	yes	—
325	C₂×C₄	C₂×Q₁₆×C₄₃	1	4	yes	—
326	C₂×C₄	C₂×Q₁₆×C₄₃	1	2	yes	—
327	C₂×C₄	#137×C₄₃	1	2	yes	—
328	C₂×C₄	C₂²×D₄×C₄₃	1	2	yes	—
329	C₈	—	4	4	no	—
330	C₂²	—	2	1	no	—
331	C₂²	—	2	1	no	—
332	C₂²	C₂×#120×C₄₃	1	1	yes	characteristic
333	C₂²	C₂²×Q₁₆×C₄₃	1	1	yes	characteristic
334	C₂²	C₂×#137×C₄₃	1	1	yes	characteristic
335	C₄	—	2	4	no	—
336	C₄	—	2	1	no	—
337	C₄	—	2	1	no	—
338	C₄	—	2	2	no	—
339	C₄	—	4	4	no	—
340	C₄	—	4	2	no	—
341	C₄	—	4	2	no	—
342	C₄	—	4	4	no	—
343	C₄	—	8	2	no	—
344	C₄	—	8	1	no	—
345	C₄	C₄₃×#451	1	4	yes	—
346	C₂	—	2	1	no	—
347	C₂	—	2	1	no	—
348	C₂	C₂×C₄₃×#451	1	1	yes	characteristic
349	C₂	C₄₃×#3020	1	1	yes	characteristic
350	C₂	C₄₃×#3025	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₄₃	C₂×#120
commutator, Frattini, cosocle	C₂²×C₄	C₂⁴×C₄₃
prime radical	C₂³×C₄₃	C₂²×D₄
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
86. C₂×Q₈×C₄₃	Q₁₆	semidirect
154. #18221	C₄₃	direct (N × H)
207. C₄₃	#18221	direct (N × H)
262. C₂×Q₈	Q₁₆×C₄₃	semidirect

Maximal subgroups

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

subgroup	index
C₄₃×#3502	2
C₄₃×#3104	2
C₄₃×#2581	2
C₄₃×#1617	2
C₄₃×#1605	2
C₄₃×#1518	2
C₄₃×#1461	2
#18221	43

Subgroups — C43×#18221 #24938238

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄₃×#18221 #24938238