Subgroups — #25252178

Order 11008.

510 conjugacy classes of subgroups.

The subgroup lattice has 382 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#19145179	C₂	1	1	yes	maximal, characteristic, complemented
2	#19145162	C₂	1	1	yes	maximal, characteristic
3	#19136881	C₂	1	1	yes	maximal, characteristic, complemented
4	#19136840	C₂	1	1	yes	maximal, characteristic, complemented
5	#19136831	C₂	1	1	yes	maximal, characteristic, complemented
6	#19136705	C₂	1	2	yes	maximal
7	#19136700	C₂	1	2	yes	maximal, complemented
8	Q₃₂×Dic₄₃	C₂	1	2	yes	maximal, complemented
9	SD₃₂×Dic₄₃	C₂	1	2	yes	maximal, complemented
10	#19136188	C₂	1	1	yes	maximal, characteristic, complemented
11	C₄₃×#2251	C₂	1	1	yes	maximal, characteristic, complemented
12	#14583454	—	2	1	no	—
13	#14583428	—	2	1	no	—
14	C₂×#13063264	C₂²	1	2	yes	—
15	#14583304	C₂²	1	1	yes	characteristic
16	#14583301	C₂²	1	1	yes	characteristic
17	#14583253	—	2	1	no	—
18	#14583249	—	2	1	no	—
19	#14583137	—	2	1	no	—
20	#14583137	C₂²	1	1	yes	characteristic
21	#14583137	C₂²	1	2	yes	—
22	#14583132	C₂²	1	1	yes	characteristic
23	#14583125	—	2	1	no	—
24	Q₁₆×Dic₄₃	—	2	1	no	—
25	Q₁₆×Dic₄₃	C₂²	1	2	yes	—
26	Q₁₆×Dic₄₃	C₂²	1	1	yes	characteristic
27	SD₁₆×Dic₄₃	—	2	1	no	—
28	D₈×Dic₄₃	C₂²	1	1	yes	characteristic
29	C₄×C₈×D₄₃	C₂²	1	1	yes	characteristic
30	#14582727	C₂²	1	2	yes	—
31	#14582725	C₂²	1	2	yes	—
32	#14582703	C₂²	1	2	yes	—
33	#14582703	C₂²	1	2	yes	—
34	#14582703	C₂²	1	2	yes	—
35	#14582700	C₂²	1	2	yes	—
36	#14582675	C₂²	1	1	yes	characteristic
37	C₁₆×Dic₄₃	C₂²	1	2	yes	—
38	C₂×Q₃₂×C₄₃	C₂²	1	2	yes	—
39	C₂×SD₃₂×C₄₃	C₂²	1	2	yes	—
40	C₄₃×#495	C₂²	1	1	yes	characteristic
41	C₄₃×#494	C₂²	1	1	yes	characteristic
42	C₄₃×#346	C₂²	1	1	yes	characteristic
43	C₂×C₈×D₄₃	D₄	1	1	yes	characteristic
44	C₂×C₈×D₄₃	C₂³	1	1	yes	characteristic
45	#13063325	—	2	1	no	—
46	#13063318	—	4	1	no	—
47	#13063317	—	2	1	no	—
48	#13063316	—	4	1	no	—
49	#13063307	—	4	1	no	—
50	#13063305	—	4	1	no	—
51	#13063301	—	2	1	no	—
52	#13063301	—	2	2	no	—
53	#13063296	—	2	1	no	—
54	Q₈×Dic₄₃	—	2	1	no	—
55	Q₈×Dic₄₃	—	2	2	no	—
56	#13063290	—	4	1	no	—
57	D₄×Dic₄₃	—	2	1	no	—
58	(C₂²⋊D₄₃)⋊C₄	—	4	1	no	—
59	#13063283	—	4	1	no	—
60	#13063281	—	4	1	no	—
61	C₄²×D₄₃	D₄	1	1	yes	characteristic
62	#13063264	—	2	4	no	—
63	C₂×C₄₃⋊C₁₆	C₂³	1	2	yes	—
64	#13063259	—	2	1	no	—
65	#13063259	C₂³	1	1	yes	characteristic
66	#13063259	C₂³	1	1	yes	characteristic
67	#13063259	C₂³	1	2	yes	—
68	#13063257	—	2	1	no	—
69	#13063249	—	2	1	no	—
70	#13063249	—	2	2	no	—
71	#13063249	—	2	1	no	—
72	#13063249	—	2	2	no	—
73	#13063246	—	2	1	no	—
74	#13063246	—	2	1	no	—
75	C₄×C₄₃⋊C₈	D₄	1	1	yes	characteristic
76	C₈×Dic₄₃	D₄	1	1	yes	characteristic
77	C₈×Dic₄₃	C₂³	1	1	yes	characteristic
78	C₂×Q₁₆×C₄₃	—	2	1	no	—
79	C₂×Q₁₆×C₄₃	C₂³	1	2	yes	—
80	C₂×Q₁₆×C₄₃	C₂³	1	1	yes	characteristic
81	C₂×SD₁₆×C₄₃	—	2	1	no	—
82	C₂×D₈×C₄₃	C₂³	1	1	yes	characteristic
83	C₄⋊Q₈×C₄₃	—	2	1	no	—
84	#124×C₄₃	—	2	1	no	—
85	Q₃₂×C₄₃	—	2	4	no	—
86	SD₃₂×C₄₃	—	2	4	no	—
87	C₂×C₁₆×C₄₃	C₂³	1	2	yes	—
88	C₄×C₈×C₄₃	C₂³	1	1	yes	characteristic
89	C₂×Q₈×C₄₃	—	2	1	no	—
90	C₂×Q₈×C₄₃	—	2	2	no	—
91	C₂×D₄×C₄₃	—	2	1	no	—
92	C₂×C₂²⋊D₄₃	—	4	1	no	—
93	C₂²×Dic₄₃	—	4	1	no	—
94	C₂×C₄×D₄₃	D₈	1	1	yes	characteristic
95	C₂×C₄×D₄₃	D₈	1	1	yes	characteristic
96	C₂×C₄×D₄₃	C₂×D₄	1	1	yes	characteristic
97	C₂×Dic₈₆	—	4	1	no	—
98	Q₁₆×C₄₃	—	2	2	no	—
99	Q₁₆×C₄₃	—	2	2	no	—
100	Q₁₆×C₄₃	—	2	2	no	—
101	Q₁₆×C₄₃	—	4	2	no	—
102	SD₁₆×C₄₃	—	4	2	no	—
103	D₈×C₄₃	—	2	2	no	—
104	C₂×C₈×C₄₃	C₂×D₄	1	1	yes	characteristic
105	C₂×C₈×C₄₃	C₂⁴	1	1	yes	characteristic
106	C₄⋊C₄×C₄₃	—	4	1	no	—
107	C₄⋊C₄×C₄₃	—	4	1	no	—
108	C₂²⋊C₄×C₄₃	—	4	1	no	—
109	C₂²⋊C₄×C₄₃	—	4	1	no	—
110	C₄²×C₄₃	C₂×D₄	1	1	yes	characteristic
111	C₂²⋊Dic₄₃	—	4	1	no	—
112	C₂²⋊Dic₄₃	—	4	1	no	—
113	(C₂×D₄₃)⋊C₄	—	4	2	no	—
114	(C₂×D₄₃)⋊C₄	—	4	1	no	—
115	C₄⋊Dic₄₃	—	2	1	no	—
116	C₄⋊Dic₄₃	—	2	1	no	—
117	C₄⋊Dic₄₃	—	2	2	no	—
118	C₄⋊Dic₄₃	—	4	1	no	—
119	C₄⋊Dic₄₃	—	4	2	no	—
120	C₄⋊Dic₄₃	—	4	1	no	—
121	Dic₄₃⋊C₄	—	4	1	no	—
122	Dic₄₃⋊C₄	—	4	1	no	—
123	Dic₄₃⋊C₄	—	4	1	no	—
124	Dic₄₃⋊C₄	—	4	2	no	—
125	C₄×Dic₄₃	—	4	1	no	—
126	C₄×Dic₄₃	—	4	2	no	—
127	C₄×Dic₄₃	D₈	1	1	yes	characteristic
128	C₄×Dic₄₃	D₈	1	1	yes	characteristic
129	C₄×Dic₄₃	C₂×D₄	1	1	yes	characteristic
130	C₂×C₄₃⋊C₈	C₂×D₄	1	1	yes	characteristic
131	C₂×C₄₃⋊C₈	C₂×D₄	1	1	yes	characteristic
132	C₈×D₄₃	—	2	2	no	—
133	C₈×D₄₃	—	2	2	no	—
134	C₁₆×C₄₃	#41	1	4	yes	—
135	C₄₃⋊C₁₆	—	2	2	no	—
136	C₂³×C₄₃	—	4	1	no	—
137	C₂²×D₄₃	C₂×D₈	1	1	yes	characteristic
138	Q₈×C₄₃	—	4	2	no	—
139	Q₈×C₄₃	—	4	1	no	—
140	Q₈×C₄₃	—	4	1	no	—
141	Q₈×C₄₃	—	4	2	no	—
142	D₄×C₄₃	—	4	1	no	—
143	D₄×C₄₃	—	4	1	no	—
144	C₂×C₄×C₄₃	—	4	1	no	—
145	C₂×C₄×C₄₃	—	4	2	no	—
146	C₂×C₄×C₄₃	C₂×D₈	1	1	yes	characteristic
147	C₂×C₄×C₄₃	C₂×D₈	1	1	yes	characteristic
148	C₂×C₄×C₄₃	C₂²×D₄	1	1	yes	characteristic
149	C₂²⋊D₄₃	—	8	2	no	—
150	C₂×Dic₄₃	—	4	1	no	—
151	C₂×Dic₄₃	—	4	2	no	—
152	C₂×Dic₄₃	—	4	1	no	—
153	C₂×Dic₄₃	—	8	2	no	—
154	C₂×Dic₄₃	C₂×D₈	1	1	yes	characteristic
155	C₂×Dic₄₃	C₂×D₈	1	1	yes	characteristic
156	C₂×Dic₄₃	C₂×D₈	1	1	yes	characteristic
157	C₄×D₄₃	—	2	1	no	—
158	C₄×D₄₃	—	2	1	no	—
159	C₄×D₄₃	—	2	2	no	—
160	C₄×D₄₃	—	4	1	no	—
161	Dic₈₆	—	8	2	no	—
162	C₈×C₄₃	#135	1	2	yes	—
163	C₈×C₄₃	C₂×#41	1	2	yes	—
164	C₄₃⋊C₈	—	2	1	no	—
165	C₄₃⋊C₈	—	2	1	no	—
166	#33738	—	43	1	no	maximal, Sylow
167	C₂²×C₄₃	—	8	1	no	—
168	C₂²×C₄₃	—	8	1	no	—
169	C₂²×C₄₃	—	8	1	no	—
170	C₂²×C₄₃	C₂²×D₈	1	1	yes	characteristic
171	C₂×D₄₃	—	2	1	no	—
172	C₂×D₄₃	—	2	1	no	—
173	C₂×D₄₃	—	2	1	no	—
174	C₄×C₄₃	—	2	1	no	—
175	C₄×C₄₃	—	2	1	no	—
176	C₄×C₄₃	—	8	1	no	—
177	C₄×C₄₃	—	8	2	no	—
178	C₄×C₄₃	#532	1	1	yes	characteristic
179	C₄×C₄₃	C₂×#135	1	1	yes	characteristic
180	Dic₄₃	—	2	1	no	—
181	Dic₄₃	—	2	1	no	—
182	Dic₄₃	—	8	1	no	—
183	Dic₄₃	—	8	2	no	—
184	Dic₄₃	—	8	1	no	—
185	Dic₄₃	#510	1	2	yes	—
186	#3170	—	43	1	no	—
187	#3168	—	43	1	no	—
188	#2255	—	43	1	no	—
189	#2251	D₄₃	1	1	yes	characteristic, complemented
190	#2245	—	43	1	no	—
191	#2243	—	43	1	no	—
192	#2220	—	43	2	no	—
193	#2219	—	43	2	no	—
194	C₄×Q₃₂	—	43	2	no	—
195	C₄×SD₃₂	—	43	2	no	—
196	C₄×M₃₂	—	43	1	no	—
197	C₂×C₄₃	—	16	1	no	—
198	C₂×C₄₃	#3169	1	1	yes	characteristic
199	C₂×C₄₃	C₂×#510	1	1	yes	characteristic
200	C₂×C₄₃	#3414	1	1	yes	characteristic
201	D₄₃	—	4	1	no	—
202	#533	—	86	1	no	—
203	#529	—	86	1	no	—
204	C₂×Q₃₂	C₂×D₄₃	1	2	yes	—
205	C₂×SD₃₂	C₂×D₄₃	1	2	yes	—
206	C₂×M₃₂	—	43	2	no	—
207	#500	—	43	1	no	—
208	#499	—	43	1	no	—
209	#495	C₂×D₄₃	1	1	yes	characteristic
210	#494	C₂×D₄₃	1	1	yes	characteristic
211	#488	—	86	1	no	—
212	#487	—	86	1	no	—
213	#467	—	43	1	no	—
214	#467	—	43	2	no	—
215	#467	—	86	1	no	—
216	#466	—	43	1	no	—
217	#465	—	86	1	no	—
218	C₄×Q₁₆	—	43	1	no	—
219	C₄×Q₁₆	—	43	2	no	—
220	C₄×Q₁₆	—	86	1	no	—
221	C₄×SD₁₆	—	86	1	no	—
222	C₄×D₈	—	43	1	no	—
223	C₂×C₄×C₈	—	43	1	no	—
224	#367	—	43	2	no	—
225	#366	—	43	2	no	—
226	#358	—	43	2	no	—
227	#358	—	43	2	no	—
228	#358	—	43	2	no	—
229	#357	—	43	2	no	—
230	#346	—	43	1	no	—
231	#346	C₂×D₄₃	1	1	yes	characteristic
232	C₄×C₁₆	—	43	2	no	—
233	C₄₃	#33738	1	1	yes	characteristic, complemented, Sylow
234	C₂×Q₁₆	—	2	1	no	—
235	C₂×Q₁₆	C₂²×D₄₃	1	2	yes	—
236	C₂×Q₁₆	C₂²×D₄₃	1	1	yes	characteristic
237	C₂×SD₁₆	—	2	1	no	—
238	C₂×D₈	C₂²×D₄₃	1	1	yes	characteristic
239	C₂²×C₈	—	43	1	no	—
240	C₂²×C₈	—	43	1	no	—
241	C₄⋊Q₈	—	2	1	no	—
242	C₄⋊Q₈	—	86	1	no	—
243	#126	—	172	1	no	—
244	#125	—	86	1	no	—
245	#125	—	172	1	no	—
246	#124	—	2	1	no	—
247	#123	—	172	1	no	—
248	C₂²⋊Q₈	—	86	1	no	—
249	C₂²⋊Q₈	—	86	2	no	—
250	C₂²⋊Q₈	—	172	1	no	—
251	#121	—	86	1	no	—
252	C₄×Q₈	—	86	1	no	—
253	C₄×Q₈	—	86	2	no	—
254	C₄×Q₈	—	172	1	no	—
255	C₄×D₄	—	86	1	no	—
256	C₄×D₄	—	172	1	no	—
257	#117	—	172	1	no	—
258	#117	—	172	1	no	—
259	C₂×C₄²	—	43	1	no	—
260	Q₃₂	—	2	4	no	—
261	SD₃₂	—	2	4	no	—
262	M₃₂	—	86	4	no	—
263	C₂×C₁₆	—	43	2	no	—
264	C₂×C₁₆	C₂²×D₄₃	1	2	yes	—
265	#107	—	43	2	no	—
266	#107	—	43	1	no	—
267	#107	—	43	1	no	—
268	#107	—	86	1	no	—
269	#106	—	86	1	no	—
270	#103	—	86	1	no	—
271	#103	—	86	1	no	—
272	#103	—	86	2	no	—
273	#103	—	86	2	no	—
274	#102	—	86	1	no	—
275	#102	—	86	1	no	—
276	C₄×C₈	—	43	1	no	—
277	C₄×C₈	—	43	1	no	—
278	C₄×C₈	—	43	1	no	—
279	C₄×C₈	C₂²×D₄₃	1	1	yes	characteristic
280	C₂×Q₈	—	2	1	no	—
281	C₂×Q₈	—	2	2	no	—
282	C₂×Q₈	—	172	1	no	—
283	C₂×D₄	—	2	1	no	—
284	C₂×D₄	—	172	1	no	—
285	C₂²×C₄	—	43	1	no	—
286	C₂²×C₄	—	43	1	no	—
287	C₂²×C₄	—	43	1	no	—
288	C₂²×C₄	—	172	1	no	—
289	Q₁₆	—	2	2	no	—
290	Q₁₆	—	2	2	no	—
291	Q₁₆	—	2	2	no	—
292	Q₁₆	—	4	2	no	—
293	SD₁₆	—	4	2	no	—
294	D₈	—	2	2	no	—
295	C₂×C₈	—	43	1	no	—
296	C₂×C₈	—	43	1	no	—
297	C₂×C₈	—	86	2	no	—
298	C₂×C₈	—	86	2	no	—
299	C₂×C₈	D₄×D₄₃	1	1	yes	characteristic
300	C₂×C₈	C₂³×D₄₃	1	1	yes	characteristic
301	C₄⋊C₄	—	4	1	no	—
302	C₄⋊C₄	—	4	1	no	—
303	C₄⋊C₄	—	86	1	no	—
304	C₄⋊C₄	—	86	1	no	—
305	C₄⋊C₄	—	86	2	no	—
306	C₄⋊C₄	—	172	1	no	—
307	C₄⋊C₄	—	172	2	no	—
308	C₄⋊C₄	—	172	1	no	—
309	C₄⋊C₄	—	172	2	no	—
310	C₄⋊C₄	—	172	1	no	—
311	C₄⋊C₄	—	172	1	no	—
312	C₄⋊C₄	—	172	1	no	—
313	C₂²⋊C₄	—	4	1	no	—
314	C₂²⋊C₄	—	4	1	no	—
315	C₂²⋊C₄	—	172	2	no	—
316	C₂²⋊C₄	—	172	1	no	—
317	C₂²⋊C₄	—	172	1	no	—
318	C₂²⋊C₄	—	172	1	no	—
319	C₄²	—	43	1	no	—
320	C₄²	—	43	1	no	—
321	C₄²	—	43	1	no	—
322	C₄²	—	172	1	no	—
323	C₄²	—	172	2	no	—
324	C₄²	D₄×D₄₃	1	1	yes	characteristic
325	C₁₆	—	86	2	no	—
326	C₁₆	#10621900	1	4	yes	—
327	C₂³	—	4	1	no	—
328	C₂³	—	43	1	no	—
329	Q₈	—	4	1	no	—
330	Q₈	—	4	2	no	—
331	Q₈	—	4	2	no	—
332	Q₈	—	4	1	no	—
333	Q₈	—	344	2	no	—
334	D₄	—	4	1	no	—
335	D₄	—	4	1	no	—
336	D₄	—	344	2	no	—
337	C₂×C₄	—	4	1	no	—
338	C₂×C₄	—	4	2	no	—
339	C₂×C₄	—	43	1	no	—
340	C₂×C₄	—	43	1	no	—
341	C₂×C₄	—	43	1	no	—
342	C₂×C₄	—	86	2	no	—
343	C₂×C₄	—	86	1	no	—
344	C₂×C₄	—	86	1	no	—
345	C₂×C₄	—	344	2	no	—
346	C₂×C₄	—	172	1	no	—
347	C₂×C₄	—	172	1	no	—
348	C₂×C₄	—	172	2	no	—
349	C₂×C₄	—	172	1	no	—
350	C₂×C₄	D₈×D₄₃	1	1	yes	characteristic
351	C₂×C₄	D₈×D₄₃	1	1	yes	characteristic
352	C₂×C₄	C₂×D₄×D₄₃	1	1	yes	characteristic
353	C₈	—	86	1	no	—
354	C₈	—	86	1	no	—
355	C₈	#13063344	1	2	yes	—
356	C₈	C₂×#10621900	1	2	yes	—
357	C₂²	—	8	1	no	—
358	C₂²	—	8	1	no	—
359	C₂²	—	8	1	no	—
360	C₂²	—	86	1	no	—
361	C₂²	—	86	1	no	—
362	C₂²	—	86	1	no	—
363	C₂²	C₂×D₈×D₄₃	1	1	yes	characteristic
364	C₄	—	2	1	no	—
365	C₄	—	2	1	no	—
366	C₄	—	8	1	no	—
367	C₄	—	8	2	no	—
368	C₄	—	43	2	no	—
369	C₄	—	86	1	no	—
370	C₄	—	86	1	no	—
371	C₄	—	344	1	no	—
372	C₄	—	344	1	no	—
373	C₄	—	344	2	no	—
374	C₄	#14583447	1	1	yes	characteristic
375	C₄	C₂×#13063344	1	1	yes	characteristic
376	C₂	—	16	1	no	—
377	C₂	—	172	1	no	—
378	C₂	#19145172	1	1	yes	characteristic
379	C₂	#510×D₄₃	1	1	yes	characteristic
380	C₂	#19147730	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×D₈×D₄₃
commutator, cosocle	C₂×C₈×C₄₃	C₂⁴
Frattini	C₂×C₈	C₂³×D₄₃
Fitting	C₄₃×#2251	C₂
socle	C₂²×C₄₃	C₂²×D₈
prime radical	#14583132	C₂²
hypercenter	#2251	D₄₃
hypocenter	C₄₃	#33738
LCS	G ▷ C₂×C₈×C₄₃ ▷ C₄×C₄₃ ▷ C₂×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ #2251

Complements

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

N	H ≅ G/N	type
1. #19145179	C₂	semidirect
3. #19136881	C₂	semidirect
4. #19136840	C₂	semidirect
5. #19136831	C₂	semidirect
7. #19136700	C₂	semidirect
8. Q₃₂×Dic₄₃	C₂	semidirect
9. SD₃₂×Dic₄₃	C₂	semidirect
10. #19136188	C₂	semidirect
11. C₄₃×#2251	C₂	semidirect
189. #2251	D₄₃	semidirect
233. C₄₃	#33738	semidirect

Maximal subgroups

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

subgroup	index
#19145179	2
#19145162	2
#19136881	2
#19136840	2
#19136831	2
#19136705	2
#19136700	2
Q₃₂×Dic₄₃	2
SD₃₂×Dic₄₃	2
#19136188	2
C₄₃×#2251	2
#33738	43