Subgroups — D₄₃×#1707 #25084242

Order 11008.

814 conjugacy classes of subgroups.

The subgroup lattice has 350 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#500×D₄₃	C₂	1	1	yes	maximal, characteristic
2	#498×D₄₃	C₂	1	2	yes	maximal
3	#494×D₄₃	C₂	1	1	yes	maximal, characteristic
4	#487×D₄₃	C₂	1	2	yes	maximal
5	#19134193	C₂	1	1	yes	maximal, characteristic, complemented
6	#19134192	C₂	1	1	yes	maximal, characteristic, complemented
7	#19134191	C₂	1	2	yes	maximal, complemented
8	#19134190	C₂	1	2	yes	maximal, complemented
9	#19134189	C₂	1	1	yes	maximal, characteristic, complemented
10	C₈²×D₄₃	C₂	1	1	yes	maximal, characteristic
11	C₄₃×#1707	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×Q₁₆×D₄₃	—	2	2	no	—
13	C₄⋊Q₈×D₄₃	C₂²	1	2	yes	—
14	C₄⋊Q₈×D₄₃	C₂²	1	2	yes	—
15	#14583304	C₂²	1	1	yes	characteristic
16	#14583303	—	2	2	no	—
17	#14583302	C₂²	1	2	yes	—
18	#14583297	C₂²	1	2	yes	—
19	#14583296	—	2	4	no	—
20	#14583295	C₂²	1	2	yes	—
21	#14583295	C₂²	1	2	yes	—
22	#14583279	C₂²	1	1	yes	characteristic
23	#14583278	C₂²	1	2	yes	—
24	#14583277	—	2	2	no	—
25	#14583251	C₂²	1	2	yes	—
26	#14583250	C₂²	1	2	yes	—
27	#14583249	C₂²	1	2	yes	—
28	#14583248	—	2	4	no	—
29	#107×D₄₃	—	2	2	no	—
30	#106×D₄₃	—	2	4	no	—
31	#103×D₄₃	—	2	4	no	—
32	C₄×C₈×D₄₃	C₂²	1	1	yes	characteristic
33	C₄×C₈×D₄₃	C₂²	1	1	yes	characteristic
34	C₄×C₈×D₄₃	C₂²	1	1	yes	characteristic
35	C₈×C₄₃⋊C₈	C₂²	1	1	yes	characteristic
36	C₈×C₄₃⋊C₈	C₂²	1	1	yes	characteristic
37	C₈×C₄₃⋊C₈	C₂²	1	1	yes	characteristic
38	C₄₃×#500	C₂²	1	1	yes	characteristic
39	C₄₃×#498	C₂²	1	2	yes	—
40	C₄₃×#494	C₂²	1	1	yes	characteristic
41	C₄₃×#487	C₂²	1	2	yes	—
42	C₈²×C₄₃	C₂²	1	1	yes	characteristic
43	C₂×Q₈×D₄₃	—	2	2	no	—
44	C₂×Q₈×D₄₃	—	2	2	no	—
45	C₂×Dic₁₇₂	—	2	2	no	—
46	C₂×#10621885	—	2	4	no	—
47	Q₁₆×D₄₃	—	4	8	no	—
48	C₂×C₈×D₄₃	D₄	1	2	yes	—
49	C₂×C₈×D₄₃	D₄	1	2	yes	—
50	C₂×C₈×D₄₃	D₄	1	2	yes	—
51	#13063325	C₂³	1	2	yes	—
52	#13063325	C₂³	1	2	yes	—
53	#13063324	—	2	2	no	—
54	#13063324	—	2	2	no	—
55	#13063324	—	2	2	no	—
56	#13063324	—	2	2	no	—
57	#13063323	—	2	2	no	—
58	#13063323	—	2	2	no	—
59	C₄⋊C₄×D₄₃	—	2	2	no	—
60	C₄⋊C₄×D₄₃	—	2	2	no	—
61	C₄⋊C₄×D₄₃	—	2	2	no	—
62	C₄⋊C₄×D₄₃	—	2	2	no	—
63	C₄²×D₄₃	C₂³	1	1	yes	characteristic
64	#13063259	—	2	2	no	—
65	#13063258	—	2	4	no	—
66	#13063257	—	2	4	no	—
67	#13063256	—	2	4	no	—
68	#13063256	—	2	4	no	—
69	#13063250	—	2	4	no	—
70	#13063249	—	2	4	no	—
71	#13063248	—	2	4	no	—
72	C₄×C₄₃⋊C₈	C₂³	1	1	yes	characteristic
73	C₄×C₄₃⋊C₈	C₂³	1	1	yes	characteristic
74	C₄×C₄₃⋊C₈	C₂³	1	1	yes	characteristic
75	C₈×Dic₄₃	D₄	1	2	yes	—
76	C₈×Dic₄₃	D₄	1	2	yes	—
77	C₈×Dic₄₃	D₄	1	2	yes	—
78	C₂×Q₁₆×C₄₃	—	2	2	no	—
79	C₄⋊Q₈×C₄₃	C₂³	1	2	yes	—
80	C₄⋊Q₈×C₄₃	C₂³	1	2	yes	—
81	#107×C₄₃	—	2	2	no	—
82	#106×C₄₃	—	2	4	no	—
83	#103×C₄₃	—	2	4	no	—
84	C₄×C₈×C₄₃	C₂³	1	1	yes	characteristic
85	C₄×C₈×C₄₃	C₂³	1	1	yes	characteristic
86	C₄×C₈×C₄₃	C₂³	1	1	yes	characteristic
87	C₂×Q₈×C₄₃	—	2	2	no	—
88	C₂×Q₈×C₄₃	—	2	2	no	—
89	Q₈×D₄₃	—	4	8	no	—
90	Q₈×D₄₃	—	4	8	no	—
91	C₂×C₄×D₄₃	—	4	2	no	—
92	C₂×C₄×D₄₃	—	4	2	no	—
93	C₂×C₄×D₄₃	C₂×D₄	1	1	yes	characteristic
94	C₂×C₄×D₄₃	C₂×D₄	1	1	yes	characteristic
95	C₂×C₄×D₄₃	C₂×D₄	1	1	yes	characteristic
96	C₂×Dic₈₆	—	2	2	no	—
97	C₂×Dic₈₆	—	2	2	no	—
98	C₂×Dic₈₆	—	4	2	no	—
99	C₂×Dic₈₆	—	4	2	no	—
100	Q₁₆×C₄₃	—	4	4	no	—
101	C₂×C₈×C₄₃	C₂×D₄	1	2	yes	—
102	C₂×C₈×C₄₃	C₂×D₄	1	2	yes	—
103	C₂×C₈×C₄₃	C₂×D₄	1	2	yes	—
104	C₄⋊C₄×C₄₃	—	2	2	no	—
105	C₄⋊C₄×C₄₃	—	2	2	no	—
106	C₄⋊C₄×C₄₃	—	2	2	no	—
107	C₄⋊C₄×C₄₃	—	2	2	no	—
108	C₄²×C₄₃	C₂⁴	1	1	yes	characteristic
109	#10621885	—	4	8	no	—
110	C₄⋊Dic₄₃	—	2	2	no	—
111	C₄⋊Dic₄₃	—	2	2	no	—
112	C₄⋊Dic₄₃	—	2	2	no	—
113	C₄⋊Dic₄₃	—	2	2	no	—
114	Dic₄₃⋊C₄	—	4	2	no	—
115	Dic₄₃⋊C₄	—	4	2	no	—
116	Dic₄₃⋊C₄	—	4	2	no	—
117	Dic₄₃⋊C₄	—	4	2	no	—
118	C₄×Dic₄₃	C₂×D₄	1	1	yes	characteristic
119	C₄×Dic₄₃	C₂×D₄	1	1	yes	characteristic
120	C₄×Dic₄₃	C₂×D₄	1	1	yes	characteristic
121	C₂×C₄₃⋊C₈	C₂×D₄	1	2	yes	—
122	C₂×C₄₃⋊C₈	C₂×D₄	1	2	yes	—
123	C₂×C₄₃⋊C₈	C₂×D₄	1	2	yes	—
124	Dic₁₇₂	—	4	4	no	—
125	C₈×D₄₃	—	2	4	no	—
126	C₈×D₄₃	SD₁₆	1	8	yes	—
127	C₈×D₄₃	Q₁₆	1	8	yes	complemented
128	C₂²×D₄₃	#127	1	1	yes	characteristic
129	Q₈×C₄₃	—	4	4	no	—
130	Q₈×C₄₃	—	4	4	no	—
131	C₂×C₄×C₄₃	—	4	2	no	—
132	C₂×C₄×C₄₃	—	4	2	no	—
133	C₂×C₄×C₄₃	C₂²×D₄	1	1	yes	characteristic
134	C₂×C₄×C₄₃	C₂²×D₄	1	1	yes	characteristic
135	C₂×C₄×C₄₃	C₂²×D₄	1	1	yes	characteristic
136	C₂×Dic₄₃	—	4	2	no	—
137	C₂×Dic₄₃	—	4	2	no	—
138	C₂×Dic₄₃	#127	1	1	yes	characteristic
139	C₂×Dic₄₃	#127	1	1	yes	characteristic
140	C₂×Dic₄₃	#127	1	1	yes	characteristic
141	C₄×D₄₃	—	8	4	no	—
142	C₄×D₄₃	—	8	4	no	—
143	C₄×D₄₃	C₂×SD₁₆	1	4	yes	—
144	C₄×D₄₃	C₂×Q₁₆	1	4	yes	—
145	C₄×D₄₃	#135	1	4	yes	—
146	Dic₈₆	—	4	4	no	—
147	Dic₈₆	—	4	4	no	—
148	Dic₈₆	—	8	4	no	—
149	Dic₈₆	—	8	4	no	—
150	C₈×C₄₃	—	2	2	no	—
151	C₈×C₄₃	C₂×SD₁₆	1	4	yes	—
152	C₈×C₄₃	C₂×Q₁₆	1	4	yes	complemented
153	C₄₃⋊C₈	—	2	2	no	—
154	C₄₃⋊C₈	C₂×SD₁₆	1	4	yes	—
155	C₄₃⋊C₈	C₂×Q₁₆	1	4	yes	complemented
156	C₂×#1707	—	43	1	no	maximal, Sylow
157	C₂²×C₄₃	C₂×#127	1	1	yes	characteristic
158	C₂×D₄₃	#492	1	2	yes	—
159	C₂×D₄₃	#494	1	2	yes	—
160	C₂×D₄₃	#495	1	2	yes	—
161	C₄×C₄₃	—	8	2	no	—
162	C₄×C₄₃	—	8	2	no	—
163	C₄×C₄₃	C₂²×SD₁₆	1	2	yes	—
164	C₄×C₄₃	C₂²×Q₁₆	1	2	yes	—
165	C₄×C₄₃	C₂×#135	1	2	yes	—
166	Dic₄₃	—	8	2	no	—
167	Dic₄₃	—	8	2	no	—
168	Dic₄₃	#492	1	2	yes	—
169	Dic₄₃	#494	1	2	yes	—
170	Dic₄₃	#495	1	2	yes	—
171	C₂×#500	—	43	1	no	—
172	C₂×#498	—	43	2	no	—
173	C₂×#494	—	43	1	no	—
174	C₂×#487	—	43	2	no	—
175	#1707	—	43	1	no	—
176	#1707	—	43	2	no	—
177	#1707	—	43	1	no	—
178	#1707	—	43	2	no	—
179	#1707	—	43	1	no	—
180	#1707	D₄₃	1	1	yes	characteristic, direct product factor
181	C₂×C₈²	—	43	1	no	—
182	C₂×C₄₃	C₂×#492	1	1	yes	characteristic
183	C₂×C₄₃	C₂×#494	1	1	yes	characteristic
184	C₂×C₄₃	C₂×#495	1	1	yes	characteristic
185	D₄₃	#1707	1	4	yes	direct product factor
186	C₂²×Q₁₆	—	86	2	no	—
187	C₂×C₄⋊Q₈	—	43	2	no	—
188	C₂×C₄⋊Q₈	—	43	2	no	—
189	#500	—	43	2	no	—
190	#500	—	43	1	no	—
191	#500	—	86	2	no	—
192	#500	C₂×D₄₃	1	1	yes	characteristic
193	#498	—	43	2	no	—
194	#498	—	43	2	no	—
195	#498	—	43	2	no	—
196	#498	—	86	4	no	—
197	#498	C₂×D₄₃	1	2	yes	—
198	#494	—	43	2	no	—
199	#494	—	43	1	no	—
200	#494	—	86	2	no	—
201	#494	C₂×D₄₃	1	1	yes	characteristic
202	#487	—	43	2	no	—
203	#487	—	43	2	no	—
204	#487	—	43	2	no	—
205	#487	—	86	4	no	—
206	#487	C₂×D₄₃	1	2	yes	—
207	C₂×#107	—	86	2	no	—
208	C₂×#106	—	86	4	no	—
209	C₂×#103	—	86	4	no	—
210	C₂×C₄×C₈	—	43	1	no	—
211	C₂×C₄×C₈	—	43	1	no	—
212	C₂×C₄×C₈	—	43	1	no	—
213	C₈²	—	43	1	no	—
214	C₈²	—	43	1	no	—
215	C₈²	—	43	1	no	—
216	C₈²	C₂×D₄₃	1	1	yes	characteristic
217	C₄₃	C₂×#1707	1	1	yes	characteristic, complemented, Sylow
218	C₂²×Q₈	—	86	2	no	—
219	C₂²×Q₈	—	86	2	no	—
220	C₂×Q₁₆	—	2	2	no	—
221	C₂×Q₁₆	—	86	4	no	—
222	C₂×Q₁₆	—	86	2	no	—
223	C₂×Q₁₆	—	172	8	no	—
224	C₂²×C₈	—	43	2	no	—
225	C₂²×C₈	—	43	2	no	—
226	C₂²×C₈	—	43	2	no	—
227	C₄⋊Q₈	—	43	2	no	—
228	C₄⋊Q₈	—	43	2	no	—
229	C₄⋊Q₈	—	86	2	no	—
230	C₄⋊Q₈	—	86	2	no	—
231	C₄⋊Q₈	—	86	2	no	—
232	C₄⋊Q₈	—	86	2	no	—
233	C₄⋊Q₈	—	86	2	no	—
234	C₄⋊Q₈	—	86	2	no	—
235	C₄⋊Q₈	C₂²×D₄₃	1	2	yes	—
236	C₄⋊Q₈	C₂²×D₄₃	1	2	yes	—
237	C₂×C₄⋊C₄	—	86	2	no	—
238	C₂×C₄⋊C₄	—	86	2	no	—
239	C₂×C₄⋊C₄	—	86	2	no	—
240	C₂×C₄⋊C₄	—	86	2	no	—
241	C₂×C₄²	—	43	1	no	—
242	#107	—	2	2	no	—
243	#107	—	86	4	no	—
244	#107	—	86	2	no	—
245	#106	—	2	4	no	—
246	#106	—	86	4	no	—
247	#106	—	86	4	no	—
248	#106	—	86	4	no	—
249	#103	—	2	4	no	—
250	#103	—	86	4	no	—
251	#103	—	86	4	no	—
252	#103	—	86	4	no	—
253	C₄×C₈	—	43	2	no	—
254	C₄×C₈	—	43	2	no	—
255	C₄×C₈	—	43	2	no	—
256	C₄×C₈	—	43	1	no	—
257	C₄×C₈	—	43	1	no	—
258	C₄×C₈	—	43	1	no	—
259	C₄×C₈	C₂²×D₄₃	1	1	yes	characteristic
260	C₄×C₈	C₂²×D₄₃	1	1	yes	characteristic
261	C₄×C₈	C₂²×D₄₃	1	1	yes	characteristic
262	C₂×Q₈	—	2	2	no	—
263	C₂×Q₈	—	2	2	no	—
264	C₂×Q₈	—	86	2	no	—
265	C₂×Q₈	—	86	2	no	—
266	C₂×Q₈	—	172	2	no	—
267	C₂×Q₈	—	172	2	no	—
268	C₂×Q₈	—	172	8	no	—
269	C₂×Q₈	—	172	8	no	—
270	C₂²×C₄	—	43	1	no	—
271	C₂²×C₄	—	43	1	no	—
272	C₂²×C₄	—	43	1	no	—
273	C₂²×C₄	—	172	2	no	—
274	C₂²×C₄	—	172	2	no	—
275	Q₁₆	—	4	4	no	—
276	Q₁₆	—	172	8	no	—
277	Q₁₆	—	172	4	no	—
278	C₂×C₈	—	43	2	no	—
279	C₂×C₈	—	43	2	no	—
280	C₂×C₈	—	43	8	no	—
281	C₂×C₈	—	43	8	no	—
282	C₂×C₈	—	43	2	no	—
283	C₂×C₈	—	86	4	no	—
284	C₂×C₈	D₄×D₄₃	1	2	yes	—
285	C₂×C₈	D₄×D₄₃	1	2	yes	—
286	C₂×C₈	D₄×D₄₃	1	2	yes	—
287	C₄⋊C₄	—	2	2	no	—
288	C₄⋊C₄	—	2	2	no	—
289	C₄⋊C₄	—	2	2	no	—
290	C₄⋊C₄	—	2	2	no	—
291	C₄⋊C₄	—	86	2	no	—
292	C₄⋊C₄	—	86	2	no	—
293	C₄⋊C₄	—	86	2	no	—
294	C₄⋊C₄	—	86	2	no	—
295	C₄⋊C₄	—	172	2	no	—
296	C₄⋊C₄	—	172	2	no	—
297	C₄⋊C₄	—	172	2	no	—
298	C₄⋊C₄	—	172	2	no	—
299	C₄²	—	43	1	no	—
300	C₄²	—	43	1	no	—
301	C₄²	—	43	1	no	—
302	C₄²	C₂³×D₄₃	1	1	yes	characteristic
303	C₂³	—	43	1	no	—
304	Q₈	—	4	4	no	—
305	Q₈	—	4	4	no	—
306	Q₈	—	344	4	no	—
307	Q₈	—	344	4	no	—
308	Q₈	—	172	4	no	—
309	Q₈	—	172	4	no	—
310	C₂×C₄	—	4	2	no	—
311	C₂×C₄	—	4	2	no	—
312	C₂×C₄	—	43	1	no	—
313	C₂×C₄	—	43	1	no	—
314	C₂×C₄	—	43	1	no	—
315	C₂×C₄	—	43	4	no	—
316	C₂×C₄	—	43	4	no	—
317	C₂×C₄	—	43	4	no	—
318	C₂×C₄	—	344	4	no	—
319	C₂×C₄	—	344	4	no	—
320	C₂×C₄	—	172	2	no	—
321	C₂×C₄	—	172	2	no	—
322	C₂×C₄	C₂×D₄×D₄₃	1	1	yes	characteristic
323	C₂×C₄	C₂×D₄×D₄₃	1	1	yes	characteristic
324	C₂×C₄	C₂×D₄×D₄₃	1	1	yes	characteristic
325	C₈	—	2	2	no	—
326	C₈	—	43	4	no	—
327	C₈	—	43	4	no	—
328	C₈	—	86	2	no	—
329	C₈	SD₁₆×D₄₃	1	4	yes	—
330	C₈	Q₁₆×D₄₃	1	4	yes	complemented
331	C₂²	—	43	2	no	—
332	C₂²	—	43	2	no	—
333	C₂²	—	43	2	no	—
334	C₂²	#127×D₄₃	1	1	yes	characteristic
335	C₄	—	8	2	no	—
336	C₄	—	8	2	no	—
337	C₄	—	43	2	no	—
338	C₄	—	43	2	no	—
339	C₄	—	43	2	no	—
340	C₄	—	344	2	no	—
341	C₄	—	344	2	no	—
342	C₄	C₂×SD₁₆×D₄₃	1	2	yes	—
343	C₄	C₂×Q₁₆×D₄₃	1	2	yes	—
344	C₄	#135×D₄₃	1	2	yes	—
345	C₂	—	43	4	no	—
346	C₂	#492×D₄₃	1	1	yes	characteristic
347	C₂	#494×D₄₃	1	1	yes	characteristic
348	C₂	#495×D₄₃	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	#127×D₄₃
commutator, cosocle	C₄²×C₄₃	C₂⁴
Frattini	C₄²	C₂³×D₄₃
Fitting	C₄₃×#1707	C₂
socle	C₂²×C₄₃	C₂×#127
prime radical	C₂²×D₄₃	#127
hypercenter	#1707	D₄₃
hypocenter	C₄₃	C₂×#1707
LCS	G ▷ C₄²×C₄₃ ▷ C₂²×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #1707

Complements

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

N	H ≅ G/N	type
5. #19134193	C₂	semidirect
6. #19134192	C₂	semidirect
7. #19134191	C₂	semidirect
8. #19134190	C₂	semidirect
9. #19134189	C₂	semidirect
11. C₄₃×#1707	C₂	semidirect
127. C₈×D₄₃	Q₁₆	semidirect
152. C₈×C₄₃	C₂×Q₁₆	semidirect
155. C₄₃⋊C₈	C₂×Q₁₆	semidirect
180. #1707	D₄₃	direct (N × H)
185. D₄₃	#1707	direct (N × H)
217. C₄₃	C₂×#1707	semidirect
330. C₈	Q₁₆×D₄₃	semidirect

Maximal subgroups

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

subgroup	index
#500×D₄₃	2
#498×D₄₃	2
#494×D₄₃	2
#487×D₄₃	2
#19134193	2
#19134192	2
#19134191	2
#19134190	2
#19134189	2
C₈²×D₄₃	2
C₄₃×#1707	2
C₂×#1707	43

Subgroups — D43×#1707 #25084242

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — D₄₃×#1707 #25084242