Subgroups — D₄₃×#1331 #25006326

Order 11008.

470 conjugacy classes of subgroups.

The subgroup lattice has 346 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×D₈×D₄₃	C₂	1	1	yes	maximal, characteristic
2	C₄×C₁₆×D₄₃	C₂	1	1	yes	maximal, characteristic, complemented
3	#335×D₄₃	C₂	1	1	yes	maximal, characteristic, complemented
4	#19132557	C₂	1	1	yes	maximal, characteristic, complemented
5	#19132556	C₂	1	1	yes	maximal, characteristic, complemented
6	#19132555	C₂	1	1	yes	maximal, characteristic, complemented
7	C₄₃×#1331	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×D₈×D₄₃	C₄	1	1	yes	characteristic
9	C₄×D₄×D₄₃	—	2	1	no	—
10	C₂×C₁₆×D₄₃	—	2	1	no	—
11	C₄×D₃₄₄	C₂²	1	1	yes	characteristic
12	D₈×Dic₄₃	C₄	1	1	yes	characteristic
13	#14582981	C₄	1	1	yes	characteristic
14	C₄×#10621882	—	2	1	no	—
15	#14582979	—	2	1	no	—
16	#107×D₄₃	C₄	1	1	yes	characteristic
17	C₄⋊C₈×D₄₃	—	2	1	no	—
18	#102×D₄₃	—	2	1	no	—
19	C₄×C₈×D₄₃	C₂²	1	1	yes	characteristic
20	C₄×C₄₃⋊C₁₆	C₂²	1	1	yes	characteristic, complemented
21	C₁₆×Dic₄₃	—	2	1	no	—
22	#14582652	C₂²	1	1	yes	characteristic, complemented
23	#14582651	—	2	1	no	—
24	C₄×D₈×C₄₃	C₂²	1	1	yes	characteristic
25	C₄×C₁₆×C₄₃	C₂²	1	1	yes	characteristic, complemented
26	C₄₃×#335	C₂²	1	1	yes	characteristic, complemented
27	C₂×D₄×D₄₃	—	2	1	no	—
28	C₂²×C₄×D₄₃	—	4	1	no	—
29	C₂×D₃₄₄	C₂×C₄	1	1	yes	characteristic
30	C₂×#10621882	—	2	1	no	—
31	D₈×D₄₃	—	2	2	no	—
32	D₈×D₄₃	C₈	1	4	yes	complemented
33	C₂×C₈×D₄₃	—	4	1	no	—
34	C₂×C₈×D₄₃	C₂×C₄	1	1	yes	characteristic
35	C₂×C₈×D₄₃	D₄	1	1	yes	characteristic
36	C₄×D₁₇₂	—	2	1	no	—
37	D₄×Dic₄₃	—	2	1	no	—
38	#13063287	—	2	1	no	—
39	C₄×C₂²⋊D₄₃	—	4	1	no	—
40	(C₂²⋊D₄₃)⋊C₄	—	4	1	no	—
41	C₄⋊C₄×D₄₃	—	2	1	no	—
42	C₂²⋊C₄×D₄₃	—	4	1	no	—
43	C₄²×D₄₃	D₄	1	1	yes	characteristic
44	C₂×C₄₃⋊C₁₆	—	2	1	no	—
45	C₁₆×D₄₃	—	2	4	no	—
46	#13063259	C₂×C₄	1	1	yes	characteristic
47	#13063258	—	2	1	no	—
48	C₄⋊(C₄₃⋊C₈)	—	2	1	no	—
49	Dic₄₃⋊C₈	—	4	1	no	—
50	#13063247	—	2	1	no	—
51	#13063246	—	2	1	no	—
52	#13063245	—	2	1	no	—
53	C₄×C₄₃⋊C₈	D₄	1	1	yes	characteristic
54	C₈×Dic₄₃	C₂×C₄	1	1	yes	characteristic
55	C₈×Dic₄₃	D₄	1	1	yes	characteristic
56	C₂×D₈×C₄₃	C₂×C₄	1	1	yes	characteristic
57	C₄×D₄×C₄₃	—	2	1	no	—
58	C₂×C₁₆×C₄₃	—	2	1	no	—
59	#107×C₄₃	C₂×C₄	1	1	yes	characteristic
60	C₄⋊C₈×C₄₃	—	2	1	no	—
61	#102×C₄₃	—	2	1	no	—
62	C₄×C₈×C₄₃	C₂³	1	1	yes	characteristic
63	C₂³×D₄₃	—	4	1	no	—
64	C₂×D₄×C₄₃	—	2	1	no	—
65	C₂²×C₄×C₄₃	—	4	1	no	—
66	C₂×C₂²⋊D₄₃	—	4	1	no	—
67	C₂²×Dic₄₃	—	4	1	no	—
68	D₄×D₄₃	—	2	4	no	—
69	D₄×D₄₃	—	4	2	no	—
70	C₂×D₁₇₂	—	2	1	no	—
71	C₂×C₄×D₄₃	—	4	1	no	—
72	C₂×C₄×D₄₃	—	4	1	no	—
73	C₂×C₄×D₄₃	—	4	1	no	—
74	C₂×C₄×D₄₃	—	8	2	no	—
75	C₂×C₄×D₄₃	—	8	2	no	—
76	C₂×C₄×D₄₃	C₂²⋊C₄	1	1	yes	characteristic
77	C₂×C₄×D₄₃	D₈	1	1	yes	characteristic
78	C₂×C₄×D₄₃	SD₁₆	1	1	yes	characteristic
79	D₈×C₄₃	—	2	1	no	—
80	D₈×C₄₃	C₂×C₈	1	2	yes	complemented
81	C₂×C₈×C₄₃	—	4	1	no	—
82	C₂×C₈×C₄₃	C₂²×C₄	1	1	yes	characteristic
83	C₂×C₈×C₄₃	C₂×D₄	1	1	yes	characteristic
84	C₄⋊C₄×C₄₃	—	2	1	no	—
85	C₂²⋊C₄×C₄₃	—	4	1	no	—
86	C₄²×C₄₃	C₂×D₄	1	1	yes	characteristic
87	C₂²⋊Dic₄₃	—	4	1	no	—
88	#10621882	—	2	4	no	—
89	(C₂×D₄₃)⋊C₄	—	4	1	no	—
90	(C₂×D₄₃)⋊C₄	—	4	1	no	—
91	C₄⋊Dic₄₃	—	2	1	no	—
92	Dic₄₃⋊C₄	—	4	1	no	—
93	C₄×Dic₄₃	C₂²⋊C₄	1	1	yes	characteristic
94	C₄×Dic₄₃	D₈	1	1	yes	characteristic
95	C₄×Dic₄₃	SD₁₆	1	1	yes	characteristic
96	C₂×C₄₃⋊C₈	—	4	1	no	—
97	C₂×C₄₃⋊C₈	C₂²⋊C₄	1	1	yes	characteristic
98	C₂×C₄₃⋊C₈	C₂²⋊C₄	1	1	yes	characteristic
99	D₃₄₄	—	2	1	no	—
100	D₃₄₄	C₂×C₈	1	2	yes	complemented
101	C₈×D₄₃	—	2	2	no	—
102	C₈×D₄₃	—	8	2	no	—
103	C₈×D₄₃	C₂×C₈	1	2	yes	—
104	C₈×D₄₃	M₁₆	1	2	yes	—
105	C₁₆×C₄₃	—	2	2	no	—
106	C₄₃⋊C₁₆	—	2	2	no	—
107	C₂³×C₄₃	—	4	1	no	—
108	C₂²×D₄₃	—	4	1	no	—
109	C₂²×D₄₃	—	4	4	no	—
110	C₂²×D₄₃	—	8	2	no	—
111	C₂²×D₄₃	—	8	2	no	—
112	C₂²×D₄₃	#102	1	1	yes	characteristic
113	D₄×C₄₃	—	2	2	no	—
114	D₄×C₄₃	—	4	1	no	—
115	C₂×C₄×C₄₃	—	4	1	no	—
116	C₂×C₄×C₄₃	—	8	1	no	—
117	C₂×C₄×C₄₃	—	8	1	no	—
118	C₂×C₄×C₄₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
119	C₂×C₄×C₄₃	C₂×D₈	1	1	yes	characteristic
120	C₂×C₄×C₄₃	C₂×SD₁₆	1	1	yes	characteristic
121	C₂²⋊D₄₃	—	4	4	no	—
122	C₂×Dic₄₃	—	4	1	no	—
123	C₂×Dic₄₃	—	8	2	no	—
124	C₂×Dic₄₃	#102	1	1	yes	characteristic
125	C₂×Dic₄₃	#102	1	1	yes	characteristic
126	C₂×Dic₄₃	#102	1	1	yes	characteristic
127	D₁₇₂	—	2	2	no	—
128	D₁₇₂	—	4	1	no	—
129	C₄×D₄₃	—	2	2	no	—
130	C₄×D₄₃	—	8	2	no	—
131	C₄×D₄₃	—	8	1	no	—
132	C₄×D₄₃	—	8	1	no	—
133	C₄×D₄₃	—	8	2	no	—
134	C₄×D₄₃	C₂²⋊C₈	1	2	yes	—
135	C₄×D₄₃	#104	1	2	yes	—
136	C₄×D₄₃	D₁₆	1	2	yes	—
137	C₄×D₄₃	SD₃₂	1	2	yes	—
138	C₈×C₄₃	—	2	1	no	—
139	C₈×C₄₃	—	8	1	no	—
140	C₈×C₄₃	C₂²×C₈	1	1	yes	characteristic
141	C₈×C₄₃	C₂×M₁₆	1	1	yes	characteristic
142	C₄₃⋊C₈	—	2	1	no	—
143	C₄₃⋊C₈	—	8	1	no	—
144	C₄₃⋊C₈	C₂²⋊C₈	1	2	yes	—
145	C₂×#1331	—	43	1	no	maximal, Sylow
146	C₂²×C₄₃	—	4	2	no	—
147	C₂²×C₄₃	—	8	1	no	—
148	C₂²×C₄₃	—	8	1	no	—
149	C₂²×C₄₃	C₂×#102	1	1	yes	characteristic
150	C₂×D₄₃	—	4	2	no	—
151	C₂×D₄₃	—	8	1	no	—
152	C₂×D₄₃	—	8	1	no	—
153	C₂×D₄₃	—	8	8	no	—
154	C₂×D₄₃	D₄⋊C₈	1	2	yes	—
155	C₂×D₄₃	#357	1	2	yes	—
156	C₂×D₄₃	#359	1	2	yes	—
157	C₄×C₄₃	—	2	1	no	—
158	C₄×C₄₃	—	8	1	no	—
159	C₄×C₄₃	C₂×C₂²⋊C₈	1	1	yes	characteristic
160	C₄×C₄₃	C₂×#104	1	1	yes	characteristic
161	C₄×C₄₃	C₂×D₁₆	1	1	yes	characteristic
162	C₄×C₄₃	C₂×SD₃₂	1	1	yes	characteristic
163	Dic₄₃	—	2	1	no	—
164	Dic₄₃	—	8	1	no	—
165	Dic₄₃	D₄⋊C₈	1	2	yes	—
166	Dic₄₃	#357	1	2	yes	—
167	C₂×C₄×D₈	—	43	1	no	—
168	C₂×C₄×C₁₆	—	43	1	no	—
169	C₂×#335	—	43	1	no	—
170	#1331	—	43	1	no	—
171	#1331	—	43	1	no	—
172	#1331	—	43	1	no	—
173	#1331	D₄₃	1	1	yes	characteristic, direct product factor
174	C₂×C₄₃	—	8	2	no	—
175	C₂×C₄₃	C₂×D₄⋊C₈	1	1	yes	characteristic
176	C₂×C₄₃	C₂×#357	1	1	yes	characteristic
177	C₂×C₄₃	C₂×#359	1	1	yes	characteristic
178	D₄₃	—	8	2	no	—
179	D₄₃	#1331	1	4	yes	direct product factor
180	C₂²×D₈	—	43	1	no	—
181	C₂×C₄×D₄	—	86	1	no	—
182	C₂²×C₁₆	—	86	1	no	—
183	C₄×D₈	—	43	1	no	—
184	C₄×D₈	—	43	1	no	—
185	C₄×D₈	—	43	1	no	—
186	C₄×D₈	—	86	1	no	—
187	C₄×D₈	—	86	1	no	—
188	C₄×D₈	C₂×D₄₃	1	1	yes	characteristic
189	C₂×#107	—	43	1	no	—
190	C₂×C₄⋊C₈	—	86	1	no	—
191	C₂×#102	—	86	1	no	—
192	C₂×C₄×C₈	—	43	1	no	—
193	C₄×C₁₆	—	43	1	no	—
194	C₄×C₁₆	—	86	1	no	—
195	C₄×C₁₆	C₂×D₄₃	1	1	yes	characteristic, complemented
196	#335	—	43	1	no	—
197	#335	—	86	1	no	—
198	#335	C₂×D₄₃	1	1	yes	characteristic, complemented
199	C₄₃	C₂×#1331	1	1	yes	characteristic, complemented, Sylow
200	C₂²×D₄	—	86	1	no	—
201	C₂³×C₄	—	172	1	no	—
202	C₂×D₈	—	43	4	no	—
203	C₂×D₈	—	43	1	no	—
204	C₂×D₈	—	86	2	no	—
205	C₂×D₈	—	86	1	no	—
206	C₂×D₈	C₄×D₄₃	1	1	yes	characteristic
207	C₂²×C₈	—	43	1	no	—
208	C₂²×C₈	—	43	1	no	—
209	C₂²×C₈	—	172	1	no	—
210	C₄×D₄	—	2	1	no	—
211	C₄×D₄	—	86	1	no	—
212	C₄×D₄	—	86	1	no	—
213	C₄×D₄	—	86	1	no	—
214	C₄×D₄	—	172	1	no	—
215	C₄×D₄	—	172	1	no	—
216	C₂×C₄⋊C₄	—	86	1	no	—
217	C₂×C₂²⋊C₄	—	172	1	no	—
218	C₂×C₄²	—	43	1	no	—
219	C₂×C₁₆	—	2	1	no	—
220	C₂×C₁₆	—	86	4	no	—
221	C₂×C₁₆	—	86	1	no	—
222	#107	—	43	1	no	—
223	#107	—	86	1	no	—
224	#107	C₄×D₄₃	1	1	yes	characteristic
225	C₄⋊C₈	—	2	1	no	—
226	C₄⋊C₈	—	86	1	no	—
227	C₄⋊C₈	—	172	1	no	—
228	#102	—	2	1	no	—
229	#102	—	86	1	no	—
230	#102	—	86	1	no	—
231	#102	—	86	1	no	—
232	C₄×C₈	—	43	1	no	—
233	C₄×C₈	—	43	1	no	—
234	C₄×C₈	—	43	1	no	—
235	C₄×C₈	C₂²×D₄₃	1	1	yes	characteristic
236	C₂⁴	—	172	1	no	—
237	C₂×D₄	—	2	1	no	—
238	C₂×D₄	—	86	4	no	—
239	C₂×D₄	—	86	1	no	—
240	C₂×D₄	—	172	2	no	—
241	C₂×D₄	—	172	1	no	—
242	C₂²×C₄	—	4	1	no	—
243	C₂²×C₄	—	43	1	no	—
244	C₂²×C₄	—	43	1	no	—
245	C₂²×C₄	—	43	1	no	—
246	C₂²×C₄	—	344	2	no	—
247	C₂²×C₄	—	344	2	no	—
248	C₂²×C₄	—	172	1	no	—
249	C₂²×C₄	—	172	1	no	—
250	C₂²×C₄	—	172	1	no	—
251	C₂²×C₄	—	172	1	no	—
252	D₈	—	2	1	no	—
253	D₈	—	43	2	no	—
254	D₈	—	86	4	no	—
255	D₈	—	86	1	no	—
256	D₈	C₈×D₄₃	1	2	yes	complemented
257	C₂×C₈	—	4	1	no	—
258	C₂×C₈	—	43	1	no	—
259	C₂×C₈	—	43	2	no	—
260	C₂×C₈	—	43	2	no	—
261	C₂×C₈	—	43	1	no	—
262	C₂×C₈	—	86	2	no	—
263	C₂×C₈	—	344	2	no	—
264	C₂×C₈	—	172	1	no	—
265	C₂×C₈	C₂×C₄×D₄₃	1	1	yes	characteristic
266	C₂×C₈	D₄×D₄₃	1	1	yes	characteristic
267	C₄⋊C₄	—	2	1	no	—
268	C₄⋊C₄	—	86	1	no	—
269	C₄⋊C₄	—	172	1	no	—
270	C₂²⋊C₄	—	4	1	no	—
271	C₂²⋊C₄	—	172	1	no	—
272	C₂²⋊C₄	—	172	1	no	—
273	C₂²⋊C₄	—	172	1	no	—
274	C₄²	—	43	1	no	—
275	C₄²	—	43	1	no	—
276	C₄²	—	43	1	no	—
277	C₄²	D₄×D₄₃	1	1	yes	characteristic
278	C₁₆	—	2	2	no	—
279	C₁₆	—	86	2	no	—
280	C₂³	—	4	1	no	—
281	C₂³	—	43	1	no	—
282	C₂³	—	344	2	no	—
283	C₂³	—	344	2	no	—
284	C₂³	—	172	4	no	—
285	C₂³	—	172	1	no	—
286	D₄	—	2	2	no	—
287	D₄	—	4	1	no	—
288	D₄	—	86	2	no	—
289	D₄	—	172	4	no	—
290	D₄	—	172	1	no	—
291	C₂×C₄	—	4	1	no	—
292	C₂×C₄	—	8	1	no	—
293	C₂×C₄	—	8	1	no	—
294	C₂×C₄	—	43	1	no	—
295	C₂×C₄	—	43	1	no	—
296	C₂×C₄	—	43	2	no	—
297	C₂×C₄	—	43	2	no	—
298	C₂×C₄	—	43	2	no	—
299	C₂×C₄	—	43	2	no	—
300	C₂×C₄	—	43	1	no	—
301	C₂×C₄	—	86	2	no	—
302	C₂×C₄	—	344	2	no	—
303	C₂×C₄	—	344	2	no	—
304	C₂×C₄	—	344	2	no	—
305	C₂×C₄	—	344	1	no	—
306	C₂×C₄	—	344	1	no	—
307	C₂×C₄	—	172	1	no	—
308	C₂×C₄	C₂²⋊C₄×D₄₃	1	1	yes	characteristic
309	C₂×C₄	D₈×D₄₃	1	1	yes	characteristic
310	C₂×C₄	SD₁₆×D₄₃	1	1	yes	characteristic
311	C₈	—	2	1	no	—
312	C₈	—	8	1	no	—
313	C₈	—	43	2	no	—
314	C₈	—	86	1	no	—
315	C₈	—	344	1	no	—
316	C₈	C₂×C₈×D₄₃	1	1	yes	characteristic
317	C₈	M₁₆×D₄₃	1	1	yes	characteristic
318	C₂²	—	4	2	no	—
319	C₂²	—	8	1	no	—
320	C₂²	—	8	1	no	—
321	C₂²	—	43	2	no	—
322	C₂²	—	43	2	no	—
323	C₂²	—	43	2	no	—
324	C₂²	—	344	8	no	—
325	C₂²	—	344	1	no	—
326	C₂²	—	344	1	no	—
327	C₂²	—	172	2	no	—
328	C₂²	#102×D₄₃	1	1	yes	characteristic
329	C₄	—	2	1	no	—
330	C₄	—	8	1	no	—
331	C₄	—	43	2	no	—
332	C₄	—	43	2	no	—
333	C₄	—	86	1	no	—
334	C₄	—	344	1	no	—
335	C₄	C₂²⋊C₈×D₄₃	1	1	yes	characteristic
336	C₄	#104×D₄₃	1	1	yes	characteristic
337	C₄	D₁₆×D₄₃	1	1	yes	characteristic
338	C₄	SD₃₂×D₄₃	1	1	yes	characteristic
339	C₂	—	8	2	no	—
340	C₂	—	43	4	no	—
341	C₂	—	344	2	no	—
342	C₂	D₄⋊C₈×D₄₃	1	1	yes	characteristic
343	C₂	#357×D₄₃	1	1	yes	characteristic
344	C₂	#359×D₄₃	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	D₈×D₄₃
commutator	C₈×C₄₃	C₂²×C₈
Frattini	C₄×C₈	C₂²×D₄₃
Fitting	C₄₃×#1331	C₂
socle	C₂²×C₄₃	C₂×#102
prime radical	C₂×D₈×D₄₃	C₄
cosocle	C₄×C₈×C₄₃	C₂³
hypercenter	#1331	D₄₃
hypocenter	C₄₃	C₂×#1331
LCS	G ▷ C₈×C₄₃ ▷ C₄×C₄₃ ▷ C₂×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₂×C₄ ◁ C₄² ◁ C₄×C₈ ◁ #1331

Complements

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

N	H ≅ G/N	type
2. C₄×C₁₆×D₄₃	C₂	semidirect
3. #335×D₄₃	C₂	semidirect
4. #19132557	C₂	semidirect
5. #19132556	C₂	semidirect
6. #19132555	C₂	semidirect
7. C₄₃×#1331	C₂	semidirect
20. C₄×C₄₃⋊C₁₆	C₂²	semidirect
22. #14582652	C₂²	semidirect
25. C₄×C₁₆×C₄₃	C₂²	semidirect
26. C₄₃×#335	C₂²	semidirect
32. D₈×D₄₃	C₈	semidirect
80. D₈×C₄₃	C₂×C₈	semidirect
100. D₃₄₄	C₂×C₈	semidirect
173. #1331	D₄₃	direct (N × H)
179. D₄₃	#1331	direct (N × H)
195. C₄×C₁₆	C₂×D₄₃	semidirect
198. #335	C₂×D₄₃	semidirect
199. C₄₃	C₂×#1331	semidirect
256. D₈	C₈×D₄₃	semidirect

Maximal subgroups

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

subgroup	index
C₄×D₈×D₄₃	2
C₄×C₁₆×D₄₃	2
#335×D₄₃	2
#19132557	2
#19132556	2
#19132555	2
C₄₃×#1331	2
C₂×#1331	43

Subgroups — D43×#1331 #25006326

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — D₄₃×#1331 #25006326