Subgroups — #25079841

Order 11008.

606 conjugacy classes of subgroups.

The subgroup lattice has 416 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#19145542	C₂	1	2	yes	maximal
2	#500×D₄₃	C₂	1	1	yes	maximal, characteristic
3	#19143526	C₂	1	1	yes	maximal, characteristic
4	#19134114	C₂	1	1	yes	maximal, characteristic, complemented
5	#19134113	C₂	1	1	yes	maximal, characteristic, complemented
6	#19134110	C₂	1	1	yes	maximal, characteristic, complemented
7	#19133988	C₂	1	2	yes	maximal, complemented
8	#19133987	C₂	1	2	yes	maximal, complemented
9	#19133340	C₂	1	1	yes	maximal, characteristic
10	#19133278	C₂	1	2	yes	maximal
11	C₄₃×#1694	C₂	1	1	yes	maximal, characteristic, complemented
12	#14583453	C₂²	1	1	yes	characteristic
13	C₄⋊Q₈×D₄₃	C₂²	1	2	yes	—
14	#14583304	C₂²	1	1	yes	characteristic
15	#14583303	—	2	1	no	—
16	#14583303	—	2	1	no	—
17	#14583302	C₂²	1	2	yes	—
18	#14583277	—	2	2	no	—
19	#14583248	—	2	2	no	—
20	#14583110	C₂²	1	2	yes	—
21	#14583109	C₂²	1	2	yes	—
22	#14583106	C₂²	1	2	yes	—
23	#14583056	—	2	2	no	—
24	#14583053	—	2	2	no	—
25	#14583029	C₂²	1	1	yes	characteristic
26	#14583027	—	2	1	no	—
27	Q₈×C₄₃⋊C₈	C₂²	1	1	yes	characteristic
28	#14583023	—	2	1	no	—
29	#14583023	C₂²	1	1	yes	characteristic
30	#14582951	C₂²	1	2	yes	—
31	#107×D₄₃	—	2	1	no	—
32	#107×D₄₃	—	2	1	no	—
33	#14582924	—	2	1	no	—
34	#14582921	C₂²	1	1	yes	characteristic
35	#14582856	—	2	1	no	—
36	C₄×C₈×D₄₃	C₂²	1	1	yes	characteristic
37	#14582651	C₂²	1	2	yes	—
38	#14582650	C₂²	1	1	yes	characteristic
39	#14582646	C₂²	1	2	yes	—
40	#14582645	C₂²	1	2	yes	—
41	#14582644	C₂²	1	2	yes	—
42	C₄₃×#500	C₂²	1	1	yes	characteristic
43	C₄₃×#462	C₂²	1	2	yes	—
44	C₄₃×#446	C₂²	1	1	yes	characteristic
45	C₄₃×#334	C₂²	1	1	yes	characteristic
46	C₄₃×#332	C₂²	1	2	yes	—
47	C₂×Q₈×D₄₃	—	2	2	no	—
48	C₂×Dic₁₇₂	—	2	2	no	—
49	C₂×#10621885	—	2	2	no	—
50	C₂×#10621885	—	2	2	no	—
51	C₂×#10621872	—	2	1	no	—
52	C₂×C₈×D₄₃	D₄	1	1	yes	characteristic
53	C₂×C₈×D₄₃	D₄	1	1	yes	characteristic
54	#13063325	C₂³	1	2	yes	—
55	#13063324	—	2	2	no	—
56	#13063324	—	2	2	no	—
57	#13063324	D₄	1	1	yes	characteristic
58	#13063323	—	2	2	no	—
59	#13063323	D₄	1	1	yes	characteristic
60	#13063315	—	2	1	no	—
61	#13063305	D₄	1	1	yes	characteristic
62	#13063303	—	2	1	no	—
63	#13063301	D₄	1	1	yes	characteristic
64	C₄×Dic₈₆	—	2	1	no	—
65	C₄×Dic₈₆	C₂³	1	1	yes	characteristic
66	#13063282	—	2	1	no	—
67	C₄⋊C₄×D₄₃	—	2	2	no	—
68	C₄⋊C₄×D₄₃	—	2	2	no	—
69	C₄²×D₄₃	C₂³	1	1	yes	characteristic
70	#13063259	—	2	1	no	—
71	#13063259	—	2	1	no	—
72	#13063258	—	2	2	no	—
73	#13063258	—	2	2	no	—
74	C₄⋊(C₄₃⋊C₈)	—	2	1	no	—
75	C₄⋊(C₄₃⋊C₈)	C₂³	1	2	yes	—
76	C₄⋊(C₄₃⋊C₈)	C₂³	1	1	yes	characteristic
77	Dic₄₃⋊C₈	—	2	1	no	—
78	Dic₄₃⋊C₈	—	2	1	no	—
79	Dic₄₃⋊C₈	—	2	1	no	—
80	#13063250	—	2	2	no	—
81	#13063249	—	2	2	no	—
82	#13063248	—	2	2	no	—
83	(C₂×D₄₃)⋊C₈	—	2	2	no	—
84	#13063235	—	2	1	no	—
85	C₄×C₄₃⋊C₈	—	2	1	no	—
86	C₄×C₄₃⋊C₈	C₂³	1	1	yes	characteristic
87	C₈×Dic₄₃	—	2	2	no	—
88	C₈×Dic₄₃	—	2	1	no	—
89	C₈×Dic₄₃	D₄	1	1	yes	characteristic
90	C₈×Dic₄₃	D₄	1	1	yes	characteristic
91	C₂×Q₁₆×C₄₃	—	2	2	no	—
92	C₄⋊Q₈×C₄₃	C₂³	1	2	yes	—
93	C₄×Q₈×C₄₃	C₂³	1	1	yes	characteristic
94	#107×C₄₃	—	2	1	no	—
95	#107×C₄₃	—	2	1	no	—
96	C₄⋊C₈×C₄₃	—	2	1	no	—
97	C₄⋊C₈×C₄₃	C₂³	1	2	yes	—
98	C₄⋊C₈×C₄₃	C₂³	1	1	yes	characteristic
99	#103×C₄₃	—	2	2	no	—
100	#97×C₄₃	—	2	1	no	—
101	C₄×C₈×C₄₃	C₂³	1	1	yes	characteristic
102	C₂×Q₈×C₄₃	—	2	2	no	—
103	C₂×Q₈×C₄₃	C₂×D₄	1	1	yes	characteristic
104	Q₈×D₄₃	—	8	2	no	—
105	Q₈×D₄₃	—	8	2	no	—
106	C₂×C₄×D₄₃	—	4	2	no	—
107	C₂×C₄×D₄₃	C₂×D₄	1	1	yes	characteristic
108	C₂×C₄×D₄₃	C₂×D₄	1	1	yes	characteristic
109	C₂×C₄×D₄₃	#41	1	1	yes	characteristic
110	C₂×Dic₈₆	—	2	1	no	—
111	C₂×Dic₈₆	—	2	2	no	—
112	C₂×Dic₈₆	—	4	2	no	—
113	C₂×Dic₈₆	C₂×D₄	1	1	yes	characteristic
114	Q₁₆×C₄₃	—	8	2	no	—
115	C₂×C₈×C₄₃	—	2	1	no	—
116	C₂×C₈×C₄₃	—	2	2	no	—
117	C₂×C₈×C₄₃	C₂×D₄	1	1	yes	characteristic
118	C₂×C₈×C₄₃	C₂×D₄	1	1	yes	characteristic
119	C₄⋊C₄×C₄₃	—	2	1	no	—
120	C₄⋊C₄×C₄₃	—	2	2	no	—
121	C₄⋊C₄×C₄₃	—	2	2	no	—
122	C₄⋊C₄×C₄₃	C₂×D₄	1	1	yes	characteristic
123	C₄²×C₄₃	—	2	1	no	—
124	C₄²×C₄₃	C₂⁴	1	1	yes	characteristic
125	#10621885	—	8	2	no	—
126	#10621885	—	8	2	no	—
127	(C₂×D₄₃)⋊C₄	—	2	1	no	—
128	(C₂×D₄₃)⋊C₄	—	2	1	no	—
129	C₄⋊Dic₄₃	—	2	1	no	—
130	C₄⋊Dic₄₃	—	2	2	no	—
131	C₄⋊Dic₄₃	—	2	2	no	—
132	C₄⋊Dic₄₃	C₂×D₄	1	1	yes	characteristic
133	Dic₄₃⋊C₄	—	2	1	no	—
134	Dic₄₃⋊C₄	—	2	1	no	—
135	Dic₄₃⋊C₄	—	2	1	no	—
136	Dic₄₃⋊C₄	—	2	1	no	—
137	Dic₄₃⋊C₄	—	2	1	no	—
138	Dic₄₃⋊C₄	—	4	2	no	—
139	Dic₄₃⋊C₄	—	4	2	no	—
140	C₄×Dic₄₃	—	2	1	no	—
141	C₄×Dic₄₃	C₂×D₄	1	1	yes	characteristic
142	C₄×Dic₄₃	C₂×D₄	1	1	yes	characteristic
143	C₄×Dic₄₃	#41	1	1	yes	characteristic
144	C₂×C₄₃⋊C₈	—	2	2	no	—
145	C₂×C₄₃⋊C₈	—	2	1	no	—
146	C₂×C₄₃⋊C₈	C₂×D₄	1	1	yes	characteristic
147	C₂×C₄₃⋊C₈	C₂×D₄	1	1	yes	characteristic
148	Dic₁₇₂	—	8	2	no	—
149	#10621872	—	4	2	no	—
150	C₈×D₄₃	—	2	2	no	—
151	C₈×D₄₃	Q₁₆	1	4	yes	complemented
152	C₂²×D₄₃	#121	1	1	yes	characteristic
153	Q₈×C₄₃	—	2	2	no	—
154	Q₈×C₄₃	—	4	2	no	—
155	Q₈×C₄₃	—	4	2	no	—
156	C₂×C₄×C₄₃	—	2	1	no	—
157	C₂×C₄×C₄₃	—	2	1	no	—
158	C₂×C₄×C₄₃	—	4	2	no	—
159	C₂×C₄×C₄₃	C₂²×D₄	1	1	yes	characteristic
160	C₂×C₄×C₄₃	C₂²×D₄	1	1	yes	characteristic
161	C₂×C₄×C₄₃	C₂×#41	1	1	yes	characteristic
162	C₂×Dic₄₃	—	2	1	no	—
163	C₂×Dic₄₃	—	2	1	no	—
164	C₂×Dic₄₃	—	4	2	no	—
165	C₂×Dic₄₃	#121	1	1	yes	characteristic
166	C₂×Dic₄₃	#121	1	1	yes	characteristic
167	C₂×Dic₄₃	#121	1	1	yes	characteristic
168	C₄×D₄₃	—	2	2	no	—
169	C₄×D₄₃	—	2	2	no	—
170	C₄×D₄₃	—	8	4	no	—
171	C₄×D₄₃	C₂×Q₁₆	1	2	yes	—
172	C₄×D₄₃	#137	1	2	yes	—
173	Dic₈₆	—	2	2	no	—
174	Dic₈₆	—	4	2	no	—
175	Dic₈₆	—	4	2	no	—
176	Dic₈₆	—	4	2	no	—
177	Dic₈₆	—	8	4	no	—
178	C₈×C₄₃	—	2	1	no	—
179	C₈×C₄₃	—	4	1	no	—
180	C₈×C₄₃	—	4	2	no	—
181	C₈×C₄₃	C₂×Q₁₆	1	2	yes	—
182	C₄₃⋊C₈	—	2	1	no	—
183	C₄₃⋊C₈	—	4	1	no	—
184	C₄₃⋊C₈	—	4	2	no	—
185	C₄₃⋊C₈	C₂×Q₁₆	1	2	yes	—
186	#19208	—	43	1	no	maximal, Sylow
187	C₂²×C₄₃	C₂×#121	1	1	yes	characteristic
188	C₂×D₄₃	—	2	1	no	—
189	C₂×D₄₃	—	2	1	no	—
190	C₂×D₄₃	#462	1	2	yes	—
191	C₄×C₄₃	—	2	1	no	—
192	C₄×C₄₃	—	4	1	no	—
193	C₄×C₄₃	—	4	1	no	—
194	C₄×C₄₃	—	8	2	no	—
195	C₄×C₄₃	C₂²×Q₁₆	1	1	yes	characteristic
196	C₄×C₄₃	C₂×#137	1	1	yes	characteristic
197	C₄×C₄₃	#576	1	2	yes	—
198	Dic₄₃	—	2	1	no	—
199	Dic₄₃	—	4	1	no	—
200	Dic₄₃	—	4	1	no	—
201	Dic₄₃	—	8	2	no	—
202	Dic₄₃	#462	1	2	yes	—
203	Dic₄₃	#473	1	2	yes	—
204	#3209	—	43	2	no	—
205	C₂×#500	—	43	1	no	—
206	#2960	—	43	1	no	—
207	#1694	—	43	1	no	—
208	#1694	—	43	1	no	—
209	#1694	—	43	1	no	—
210	#1694	D₄₃	1	1	yes	characteristic, complemented
211	#1674	—	43	2	no	—
212	#1674	—	43	2	no	—
213	#1564	—	43	1	no	—
214	#1549	—	43	2	no	—
215	C₂×C₄₃	C₂×#462	1	1	yes	characteristic
216	C₂×C₄₃	#3058	1	1	yes	characteristic
217	C₂×C₄₃	C₂×#473	1	1	yes	characteristic
218	D₄₃	—	2	2	no	—
219	#533	—	43	1	no	—
220	C₂×C₄⋊Q₈	—	43	2	no	—
221	#500	—	43	2	no	—
222	#500	—	43	1	no	—
223	#500	—	86	1	no	—
224	#500	—	86	1	no	—
225	#500	C₂×D₄₃	1	1	yes	characteristic
226	#494	—	86	2	no	—
227	#487	—	86	2	no	—
228	#462	—	43	2	no	—
229	#462	—	43	2	no	—
230	#462	—	43	2	no	—
231	#462	C₂×D₄₃	1	2	yes	—
232	#451	—	86	2	no	—
233	#451	—	86	2	no	—
234	#446	—	43	1	no	—
235	#446	—	86	1	no	—
236	#446	C₂×D₄₃	1	1	yes	characteristic
237	C₈×Q₈	—	43	1	no	—
238	C₈×Q₈	—	43	1	no	—
239	C₈×Q₈	—	86	1	no	—
240	#431	—	43	2	no	—
241	C₂×#107	—	86	1	no	—
242	C₂×#107	—	86	1	no	—
243	#424	—	86	1	no	—
244	#423	—	43	1	no	—
245	#405	—	86	1	no	—
246	C₂×C₄×C₈	—	43	1	no	—
247	#335	—	43	2	no	—
248	#334	—	43	1	no	—
249	#334	C₂×D₄₃	1	1	yes	characteristic
250	#332	—	43	2	no	—
251	#332	—	43	2	no	—
252	#332	—	43	2	no	—
253	#332	C₂×D₄₃	1	2	yes	—
254	C₄₃	#19208	1	1	yes	characteristic, complemented, Sylow
255	C₂²×Q₈	—	86	2	no	—
256	C₂×Q₁₆	—	2	2	no	—
257	C₂×Q₁₆	—	86	2	no	—
258	C₂×Q₁₆	—	86	2	no	—
259	C₂×Q₁₆	—	86	2	no	—
260	C₂×M₁₆	—	86	1	no	—
261	C₂²×C₈	—	43	1	no	—
262	C₂²×C₈	—	43	1	no	—
263	C₄⋊Q₈	—	43	1	no	—
264	C₄⋊Q₈	—	43	1	no	—
265	C₄⋊Q₈	—	43	2	no	—
266	C₄⋊Q₈	—	86	2	no	—
267	C₄⋊Q₈	—	86	2	no	—
268	C₄⋊Q₈	—	86	2	no	—
269	C₄⋊Q₈	C₂²×D₄₃	1	2	yes	—
270	#125	—	86	1	no	—
271	C₂²⋊Q₈	—	43	1	no	—
272	C₂²⋊Q₈	—	43	1	no	—
273	C₂²⋊Q₈	—	86	1	no	—
274	C₄×Q₈	—	43	1	no	—
275	C₄×Q₈	—	86	1	no	—
276	C₄×Q₈	C₂²×D₄₃	1	1	yes	characteristic
277	#117	—	86	1	no	—
278	C₂×C₄⋊C₄	—	86	2	no	—
279	C₂×C₄⋊C₄	—	86	2	no	—
280	C₂×C₄²	—	43	1	no	—
281	#107	—	2	1	no	—
282	#107	—	2	1	no	—
283	#107	—	86	2	no	—
284	#107	—	86	2	no	—
285	#107	—	86	1	no	—
286	#107	—	86	1	no	—
287	C₄⋊C₈	—	2	1	no	—
288	C₄⋊C₈	—	43	1	no	—
289	C₄⋊C₈	—	43	2	no	—
290	C₄⋊C₈	—	86	1	no	—
291	C₄⋊C₈	—	86	1	no	—
292	C₄⋊C₈	—	86	1	no	—
293	C₄⋊C₈	—	86	1	no	—
294	C₄⋊C₈	C₂²×D₄₃	1	1	yes	characteristic
295	C₄⋊C₈	C₂²×D₄₃	1	2	yes	—
296	#103	—	2	2	no	—
297	#103	—	86	2	no	—
298	#103	—	86	2	no	—
299	#103	—	86	2	no	—
300	C₂²⋊C₈	—	86	2	no	—
301	#97	—	2	1	no	—
302	#97	—	86	1	no	—
303	C₄×C₈	—	43	1	no	—
304	C₄×C₈	—	43	1	no	—
305	C₄×C₈	—	43	1	no	—
306	C₄×C₈	—	86	1	no	—
307	C₄×C₈	—	86	2	no	—
308	C₄×C₈	—	86	1	no	—
309	C₄×C₈	C₂²×D₄₃	1	1	yes	characteristic
310	C₂×Q₈	—	2	2	no	—
311	C₂×Q₈	—	43	1	no	—
312	C₂×Q₈	—	86	1	no	—
313	C₂×Q₈	—	86	2	no	—
314	C₂×Q₈	—	344	2	no	—
315	C₂×Q₈	—	344	2	no	—
316	C₂×Q₈	—	172	2	no	—
317	C₂×Q₈	D₄×D₄₃	1	1	yes	characteristic
318	C₂²×C₄	—	43	1	no	—
319	C₂²×C₄	—	43	1	no	—
320	C₂²×C₄	—	43	1	no	—
321	C₂²×C₄	—	172	2	no	—
322	Q₁₆	—	8	2	no	—
323	Q₁₆	—	344	2	no	—
324	Q₁₆	—	344	2	no	—
325	Q₁₆	—	344	2	no	—
326	M₁₆	—	172	2	no	—
327	C₂×C₈	—	2	1	no	—
328	C₂×C₈	—	2	2	no	—
329	C₂×C₈	—	43	1	no	—
330	C₂×C₈	—	43	4	no	—
331	C₂×C₈	—	43	1	no	—
332	C₂×C₈	—	86	2	no	—
333	C₂×C₈	—	86	1	no	—
334	C₂×C₈	—	86	2	no	—
335	C₂×C₈	D₄×D₄₃	1	1	yes	characteristic
336	C₂×C₈	D₄×D₄₃	1	1	yes	characteristic
337	C₄⋊C₄	—	2	1	no	—
338	C₄⋊C₄	—	2	2	no	—
339	C₄⋊C₄	—	2	2	no	—
340	C₄⋊C₄	—	43	1	no	—
341	C₄⋊C₄	—	86	1	no	—
342	C₄⋊C₄	—	86	1	no	—
343	C₄⋊C₄	—	86	1	no	—
344	C₄⋊C₄	—	86	1	no	—
345	C₄⋊C₄	—	86	1	no	—
346	C₄⋊C₄	—	86	1	no	—
347	C₄⋊C₄	—	86	2	no	—
348	C₄⋊C₄	—	86	2	no	—
349	C₄⋊C₄	—	172	2	no	—
350	C₄⋊C₄	—	172	2	no	—
351	C₄⋊C₄	D₄×D₄₃	1	1	yes	characteristic
352	C₂²⋊C₄	—	86	1	no	—
353	C₂²⋊C₄	—	86	1	no	—
354	C₄²	—	2	1	no	—
355	C₄²	—	43	1	no	—
356	C₄²	—	43	1	no	—
357	C₄²	—	43	1	no	—
358	C₄²	—	86	1	no	—
359	C₄²	C₂³×D₄₃	1	1	yes	characteristic
360	C₂³	—	43	1	no	—
361	Q₈	—	2	2	no	—
362	Q₈	—	4	2	no	—
363	Q₈	—	4	2	no	—
364	Q₈	—	86	2	no	—
365	Q₈	—	344	4	no	—
366	Q₈	—	172	2	no	—
367	Q₈	—	172	2	no	—
368	Q₈	—	172	2	no	—
369	C₂×C₄	—	2	1	no	—
370	C₂×C₄	—	2	1	no	—
371	C₂×C₄	—	4	2	no	—
372	C₂×C₄	—	43	1	no	—
373	C₂×C₄	—	43	1	no	—
374	C₂×C₄	—	43	2	no	—
375	C₂×C₄	—	43	2	no	—
376	C₂×C₄	—	43	1	no	—
377	C₂×C₄	—	86	2	no	—
378	C₂×C₄	—	86	2	no	—
379	C₂×C₄	—	86	1	no	—
380	C₂×C₄	—	86	1	no	—
381	C₂×C₄	—	344	4	no	—
382	C₂×C₄	—	172	2	no	—
383	C₂×C₄	C₂×D₄×D₄₃	1	1	yes	characteristic
384	C₂×C₄	C₂×D₄×D₄₃	1	1	yes	characteristic
385	C₂×C₄	#41×D₄₃	1	1	yes	characteristic
386	C₈	—	2	1	no	—
387	C₈	—	4	1	no	—
388	C₈	—	4	2	no	—
389	C₈	—	43	2	no	—
390	C₈	—	86	1	no	—
391	C₈	—	172	1	no	—
392	C₈	—	172	2	no	—
393	C₈	Q₁₆×D₄₃	1	2	yes	—
394	C₂²	—	43	2	no	—
395	C₂²	—	86	1	no	—
396	C₂²	—	86	1	no	—
397	C₂²	#121×D₄₃	1	1	yes	characteristic
398	C₄	—	2	1	no	—
399	C₄	—	4	1	no	—
400	C₄	—	4	1	no	—
401	C₄	—	8	2	no	—
402	C₄	—	43	2	no	—
403	C₄	—	43	2	no	—
404	C₄	—	86	1	no	—
405	C₄	—	344	2	no	—
406	C₄	—	172	1	no	—
407	C₄	—	172	1	no	—
408	C₄	C₂×Q₁₆×D₄₃	1	1	yes	characteristic
409	C₄	#137×D₄₃	1	1	yes	characteristic
410	C₄	#14583703	1	2	yes	—
411	C₂	—	86	2	no	—
412	C₂	#462×D₄₃	1	1	yes	characteristic
413	C₂	#19144241	1	1	yes	characteristic
414	C₂	#473×D₄₃	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	#121×D₄₃
commutator, cosocle	C₄²×C₄₃	C₂⁴
Frattini	C₄²	C₂³×D₄₃
Fitting	C₄₃×#1694	C₂
socle	C₂²×C₄₃	C₂×#121
prime radical	C₂²×D₄₃	#121
hypercenter	#1694	D₄₃
hypocenter	C₄₃	#19208
LCS	G ▷ C₄²×C₄₃ ▷ C₂²×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #1694

Complements

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

N	H ≅ G/N	type
4. #19134114	C₂	semidirect
5. #19134113	C₂	semidirect
6. #19134110	C₂	semidirect
7. #19133988	C₂	semidirect
8. #19133987	C₂	semidirect
11. C₄₃×#1694	C₂	semidirect
151. C₈×D₄₃	Q₁₆	semidirect
210. #1694	D₄₃	semidirect
254. C₄₃	#19208	semidirect

Maximal subgroups

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

subgroup	index
#19145542	2
#500×D₄₃	2
#19143526	2
#19134114	2
#19134113	2
#19134110	2
#19133988	2
#19133987	2
#19133340	2
#19133278	2
C₄₃×#1694	2
#19208	43