Subgroups — #25080030

Order 11008.

562 conjugacy classes of subgroups.

The subgroup lattice has 414 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#19145772	C₂	1	2	yes	maximal
2	#19145179	C₂	1	1	yes	maximal, characteristic, complemented
3	#19143573	C₂	1	1	yes	maximal, characteristic
4	#19134124	C₂	1	1	yes	maximal, characteristic, complemented
5	#19134118	C₂	1	1	yes	maximal, characteristic, complemented
6	#19134086	C₂	1	1	yes	maximal, characteristic, complemented
7	#19134052	C₂	1	2	yes	maximal, complemented
8	#19134044	C₂	1	2	yes	maximal, complemented
9	#19133340	C₂	1	1	yes	maximal, characteristic, complemented
10	#19133302	C₂	1	2	yes	maximal, complemented
11	C₄₃×#1689	C₂	1	1	yes	maximal, characteristic, complemented
12	#14583454	C₂²	1	2	yes	—
13	#14583431	C₂²	1	1	yes	characteristic
14	#14583304	C₂²	1	1	yes	characteristic
15	#14583300	—	2	2	no	—
16	#14583252	—	2	2	no	—
17	#14583249	C₂²	1	2	yes	—
18	#14583242	—	2	2	no	—
19	#14583221	—	2	2	no	—
20	#14583195	C₂²	1	2	yes	—
21	#14583189	C₂²	1	2	yes	—
22	#14583181	C₂²	1	2	yes	—
23	#14583137	—	2	1	no	—
24	#14583137	—	2	1	no	—
25	#14583029	C₂²	1	1	yes	characteristic
26	#14583023	C₂²	1	1	yes	characteristic
27	Q₁₆×Dic₄₃	—	2	1	no	—
28	Q₁₆×Dic₄₃	—	2	1	no	—
29	#14582968	—	2	1	no	—
30	D₄×C₄₃⋊C₈	C₂²	1	1	yes	characteristic
31	(C₂²⋊D₄₃)⋊C₈	—	2	1	no	—
32	#14582957	—	2	1	no	—
33	#14582955	C₂²	1	1	yes	characteristic
34	#14582951	C₂²	1	2	yes	—
35	#14582857	—	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	#14582647	C₂²	1	2	yes	—
40	#14582647	C₂²	1	2	yes	—
41	#14582645	C₂²	1	2	yes	—
42	C₄₃×#494	C₂²	1	1	yes	characteristic
43	C₄₃×#474	C₂²	1	2	yes	—
44	C₄₃×#436	C₂²	1	1	yes	characteristic
45	C₄₃×#334	C₂²	1	1	yes	characteristic
46	C₄₃×#332	C₂²	1	2	yes	—
47	C₂²×C₄₃⋊C₈	—	2	1	no	—
48	C₂×C₈×D₄₃	D₄	1	1	yes	characteristic
49	C₂×C₈×D₄₃	D₄	1	1	yes	characteristic
50	#13063325	C₂³	1	2	yes	—
51	#13063318	—	2	1	no	—
52	#13063316	—	2	2	no	—
53	#13063316	D₄	1	1	yes	characteristic
54	#13063312	D₄	1	1	yes	characteristic
55	#13063307	—	2	1	no	—
56	#13063305	—	2	2	no	—
57	#13063305	D₄	1	1	yes	characteristic
58	#13063301	—	2	2	no	—
59	#13063296	D₄	1	1	yes	characteristic
60	Q₈×Dic₄₃	—	2	2	no	—
61	C₄×Dic₈₆	C₂³	1	1	yes	characteristic
62	#13063290	—	2	2	no	—
63	C₄×C₂²⋊D₄₃	—	2	1	no	—
64	#13063284	—	2	1	no	—
65	#13063283	—	2	2	no	—
66	C₄²×D₄₃	C₂³	1	1	yes	characteristic
67	#13063259	—	2	1	no	—
68	#13063259	—	2	1	no	—
69	#13063259	—	2	2	no	—
70	#13063256	—	2	2	no	—
71	#13063256	—	2	2	no	—
72	C₄⋊(C₄₃⋊C₈)	C₂³	1	1	yes	characteristic
73	C₄⋊(C₄₃⋊C₈)	C₂³	1	2	yes	—
74	Dic₄₃⋊C₈	—	2	1	no	—
75	Dic₄₃⋊C₈	—	2	1	no	—
76	#13063250	—	2	2	no	—
77	#13063249	—	2	2	no	—
78	#13063249	—	2	2	no	—
79	#13063248	—	2	2	no	—
80	#13063246	—	2	2	no	—
81	C₂²⋊(C₄₃⋊C₈)	—	2	1	no	—
82	(C₂×D₄₃)⋊C₈	—	2	2	no	—
83	(C₂×D₄₃)⋊C₈	—	2	1	no	—
84	#13063235	—	2	1	no	—
85	#13063235	—	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	1	no	—
92	C₂×Q₁₆×C₄₃	—	2	1	no	—
93	C₂×M₁₆×C₄₃	—	2	1	no	—
94	C₄⋊Q₈×C₄₃	C₂³	1	2	yes	—
95	C₄×D₄×C₄₃	C₂³	1	1	yes	characteristic
96	#107×C₄₃	—	2	2	no	—
97	C₄⋊C₈×C₄₃	C₂³	1	1	yes	characteristic
98	C₄⋊C₈×C₄₃	C₂³	1	2	yes	—
99	#102×C₄₃	—	2	2	no	—
100	C₂²⋊C₈×C₄₃	—	2	1	no	—
101	C₄×C₈×C₄₃	C₂³	1	1	yes	characteristic
102	C₂×Q₈×C₄₃	—	2	2	no	—
103	C₂×D₄×C₄₃	C₂×D₄	1	1	yes	characteristic
104	C₂²×C₄×C₄₃	—	2	1	no	—
105	C₂×C₂²⋊D₄₃	—	2	1	no	—
106	C₂×C₄×D₄₃	C₂×D₄	1	1	yes	characteristic
107	C₂×C₄×D₄₃	C₂×D₄	1	1	yes	characteristic
108	C₂×C₄×D₄₃	#41	1	1	yes	characteristic
109	C₂×Dic₈₆	—	2	2	no	—
110	C₂×Dic₈₆	C₂×D₄	1	1	yes	characteristic
111	Q₁₆×C₄₃	—	4	2	no	—
112	Q₁₆×C₄₃	—	4	2	no	—
113	M₁₆×C₄₃	—	4	2	no	—
114	C₂×C₈×C₄₃	—	2	1	no	—
115	C₂×C₈×C₄₃	—	2	2	no	—
116	C₂×C₈×C₄₃	C₂×D₄	1	1	yes	characteristic
117	C₂×C₈×C₄₃	C₂×D₄	1	1	yes	characteristic
118	C₄⋊C₄×C₄₃	—	2	2	no	—
119	C₄⋊C₄×C₄₃	—	2	2	no	—
120	C₄⋊C₄×C₄₃	C₂×D₄	1	1	yes	characteristic
121	C₂²⋊C₄×C₄₃	—	2	1	no	—
122	C₄²×C₄₃	C₂⁴	1	1	yes	characteristic
123	C₂²⋊Dic₄₃	—	2	1	no	—
124	C₂²⋊Dic₄₃	—	2	1	no	—
125	C₂²⋊Dic₄₃	—	2	1	no	—
126	(C₂×D₄₃)⋊C₄	—	2	1	no	—
127	(C₂×D₄₃)⋊C₄	—	4	2	no	—
128	C₄⋊Dic₄₃	—	2	1	no	—
129	C₄⋊Dic₄₃	—	2	1	no	—
130	C₄⋊Dic₄₃	—	2	2	no	—
131	C₄⋊Dic₄₃	—	2	2	no	—
132	C₄⋊Dic₄₃	—	4	2	no	—
133	C₄⋊Dic₄₃	C₂×D₄	1	1	yes	characteristic
134	Dic₄₃⋊C₄	—	2	1	no	—
135	Dic₄₃⋊C₄	—	2	1	no	—
136	Dic₄₃⋊C₄	—	4	2	no	—
137	C₄×Dic₄₃	—	2	1	no	—
138	C₄×Dic₄₃	—	4	2	no	—
139	C₄×Dic₄₃	C₂×D₄	1	1	yes	characteristic
140	C₄×Dic₄₃	C₂×D₄	1	1	yes	characteristic
141	C₄×Dic₄₃	#41	1	1	yes	characteristic
142	C₂×C₄₃⋊C₈	—	2	2	no	—
143	C₂×C₄₃⋊C₈	—	2	1	no	—
144	C₂×C₄₃⋊C₈	—	4	2	no	—
145	C₂×C₄₃⋊C₈	C₂×D₄	1	1	yes	characteristic
146	C₂×C₄₃⋊C₈	C₂×D₄	1	1	yes	characteristic
147	C₈×D₄₃	—	2	2	no	—
148	C₈×D₄₃	—	4	1	no	—
149	C₂³×C₄₃	—	2	1	no	—
150	C₂²×D₄₃	#121	1	1	yes	characteristic
151	Q₈×C₄₃	—	4	2	no	—
152	Q₈×C₄₃	—	4	2	no	—
153	D₄×C₄₃	—	2	2	no	—
154	C₂×C₄×C₄₃	—	2	1	no	—
155	C₂×C₄×C₄₃	—	4	1	no	—
156	C₂×C₄×C₄₃	—	4	1	no	—
157	C₂×C₄×C₄₃	—	4	2	no	—
158	C₂×C₄×C₄₃	C₂²×D₄	1	1	yes	characteristic
159	C₂×C₄×C₄₃	C₂²×D₄	1	1	yes	characteristic
160	C₂×C₄×C₄₃	C₂×#41	1	1	yes	characteristic
161	C₂²⋊D₄₃	—	4	2	no	—
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	1	no	—
169	C₄×D₄₃	—	2	1	no	—
170	C₄×D₄₃	—	2	2	no	—
171	C₄×D₄₃	—	4	1	no	—
172	Dic₈₆	—	2	2	no	—
173	Dic₈₆	—	8	2	no	—
174	C₈×C₄₃	—	2	1	no	—
175	C₈×C₄₃	—	4	1	no	—
176	C₈×C₄₃	—	4	2	no	—
177	C₈×C₄₃	#135	1	2	yes	—
178	C₄₃⋊C₈	—	2	1	no	—
179	C₄₃⋊C₈	—	2	1	no	—
180	C₄₃⋊C₈	—	4	1	no	—
181	C₄₃⋊C₈	—	4	2	no	—
182	#19226	—	43	1	no	maximal, Sylow
183	C₂²×C₄₃	—	4	1	no	—
184	C₂²×C₄₃	—	4	1	no	—
185	C₂²×C₄₃	—	4	1	no	—
186	C₂²×C₄₃	C₂×#121	1	1	yes	characteristic
187	C₂×D₄₃	—	2	1	no	—
188	C₂×D₄₃	—	2	1	no	—
189	C₂×D₄₃	—	2	1	no	—
190	C₄×C₄₃	—	2	1	no	—
191	C₄×C₄₃	—	4	1	no	—
192	C₄×C₄₃	—	8	2	no	—
193	C₄×C₄₃	C₂×#135	1	1	yes	characteristic
194	C₄×C₄₃	#575	1	1	yes	characteristic
195	C₄×C₄₃	#578	1	2	yes	—
196	Dic₄₃	—	2	1	no	—
197	Dic₄₃	—	2	1	no	—
198	Dic₄₃	—	2	1	no	—
199	Dic₄₃	—	4	1	no	—
200	Dic₄₃	—	4	1	no	—
201	Dic₄₃	—	8	2	no	—
202	#3233	—	43	2	no	—
203	#3170	—	43	1	no	—
204	#2966	—	43	1	no	—
205	#1696	—	43	1	no	—
206	#1695	—	43	1	no	—
207	#1690	—	43	1	no	—
208	#1689	D₄₃	1	1	yes	characteristic, complemented
209	#1684	—	43	2	no	—
210	#1683	—	43	2	no	—
211	#1564	—	43	1	no	—
212	#1554	—	43	2	no	—
213	C₂×C₄₃	—	8	1	no	—
214	C₂×C₄₃	#3035	1	1	yes	characteristic
215	C₂×C₄₃	#3059	1	1	yes	characteristic
216	C₂×C₄₃	#3067	1	1	yes	characteristic
217	D₄₃	—	4	1	no	—
218	#533	—	43	2	no	—
219	#529	—	43	1	no	—
220	#500	—	43	1	no	—
221	#499	—	86	2	no	—
222	#494	C₂×D₄₃	1	1	yes	characteristic
223	#488	—	86	2	no	—
224	#487	—	43	2	no	—
225	#485	—	86	2	no	—
226	#482	—	86	2	no	—
227	#477	—	43	2	no	—
228	#476	—	43	2	no	—
229	#475	—	43	2	no	—
230	#474	C₂×D₄₃	1	2	yes	—
231	#467	—	86	1	no	—
232	#467	—	86	1	no	—
233	#446	—	43	1	no	—
234	C₈×Q₈	—	43	1	no	—
235	C₄×Q₁₆	—	86	1	no	—
236	C₄×Q₁₆	—	86	1	no	—
237	#436	C₂×D₄₃	1	1	yes	characteristic
238	#435	—	86	1	no	—
239	C₈×D₄	—	43	1	no	—
240	C₈×D₄	—	86	1	no	—
241	#433	—	86	1	no	—
242	#432	—	43	1	no	—
243	#431	—	43	2	no	—
244	#405	—	86	1	no	—
245	C₂×C₄×C₈	—	43	1	no	—
246	#335	—	43	2	no	—
247	#334	—	43	1	no	—
248	#334	C₂×D₄₃	1	1	yes	characteristic
249	#333	—	43	2	no	—
250	#333	—	43	2	no	—
251	#332	—	43	2	no	—
252	#332	C₂×D₄₃	1	2	yes	—
253	C₄₃	#19226	1	1	yes	characteristic, complemented, Sylow
254	C₂×Q₁₆	—	2	1	no	—
255	C₂×Q₁₆	—	2	1	no	—
256	C₂×M₁₆	—	2	1	no	—
257	C₂²×C₈	—	43	1	no	—
258	C₂²×C₈	—	43	1	no	—
259	C₂²×C₈	—	86	1	no	—
260	C₄⋊Q₈	—	43	2	no	—
261	C₄⋊Q₈	C₂²×D₄₃	1	2	yes	—
262	#126	—	86	1	no	—
263	#125	—	43	1	no	—
264	#125	—	86	2	no	—
265	#124	—	43	1	no	—
266	#123	—	86	1	no	—
267	C₂²⋊Q₈	—	43	1	no	—
268	C₂²⋊Q₈	—	86	2	no	—
269	C₂²⋊Q₈	—	86	2	no	—
270	#121	—	43	1	no	—
271	C₄×Q₈	—	43	1	no	—
272	C₄×Q₈	—	86	2	no	—
273	C₄×Q₈	—	86	2	no	—
274	C₄×D₄	—	86	1	no	—
275	C₄×D₄	C₂²×D₄₃	1	1	yes	characteristic
276	#117	—	86	1	no	—
277	#117	—	86	2	no	—
278	C₂×C₄²	—	43	1	no	—
279	#107	—	2	2	no	—
280	#107	—	86	2	no	—
281	#107	—	86	1	no	—
282	#107	—	86	1	no	—
283	#106	—	86	2	no	—
284	#106	—	86	2	no	—
285	C₄⋊C₈	—	43	1	no	—
286	C₄⋊C₈	—	43	2	no	—
287	C₄⋊C₈	—	86	1	no	—
288	C₄⋊C₈	—	86	1	no	—
289	C₄⋊C₈	C₂²×D₄₃	1	1	yes	characteristic
290	C₄⋊C₈	C₂²×D₄₃	1	2	yes	—
291	#103	—	86	2	no	—
292	#103	—	86	2	no	—
293	#103	—	86	2	no	—
294	#103	—	86	2	no	—
295	#102	—	2	2	no	—
296	#102	—	86	2	no	—
297	C₂²⋊C₈	—	2	1	no	—
298	C₂²⋊C₈	—	86	2	no	—
299	C₂²⋊C₈	—	86	1	no	—
300	C₂²⋊C₈	—	86	1	no	—
301	#97	—	86	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	2	no	—
307	C₄×C₈	—	86	1	no	—
308	C₄×C₈	C₂²×D₄₃	1	1	yes	characteristic
309	C₂×Q₈	—	2	2	no	—
310	C₂×Q₈	—	43	1	no	—
311	C₂×Q₈	—	86	2	no	—
312	C₂×D₄	—	86	1	no	—
313	C₂×D₄	D₄×D₄₃	1	1	yes	characteristic
314	C₂²×C₄	—	2	1	no	—
315	C₂²×C₄	—	43	1	no	—
316	C₂²×C₄	—	43	1	no	—
317	C₂²×C₄	—	43	1	no	—
318	Q₁₆	—	4	2	no	—
319	Q₁₆	—	4	2	no	—
320	M₁₆	—	4	2	no	—
321	C₂×C₈	—	2	1	no	—
322	C₂×C₈	—	2	2	no	—
323	C₂×C₈	—	43	1	no	—
324	C₂×C₈	—	43	1	no	—
325	C₂×C₈	—	86	2	no	—
326	C₂×C₈	—	86	2	no	—
327	C₂×C₈	—	86	1	no	—
328	C₂×C₈	—	172	1	no	—
329	C₂×C₈	—	172	2	no	—
330	C₂×C₈	D₄×D₄₃	1	1	yes	characteristic
331	C₂×C₈	D₄×D₄₃	1	1	yes	characteristic
332	C₄⋊C₄	—	2	2	no	—
333	C₄⋊C₄	—	2	2	no	—
334	C₄⋊C₄	—	43	1	no	—
335	C₄⋊C₄	—	86	1	no	—
336	C₄⋊C₄	—	86	1	no	—
337	C₄⋊C₄	—	86	1	no	—
338	C₄⋊C₄	—	86	1	no	—
339	C₄⋊C₄	—	86	2	no	—
340	C₄⋊C₄	—	86	2	no	—
341	C₄⋊C₄	—	172	2	no	—
342	C₄⋊C₄	—	172	2	no	—
343	C₄⋊C₄	D₄×D₄₃	1	1	yes	characteristic
344	C₂²⋊C₄	—	2	1	no	—
345	C₂²⋊C₄	—	86	1	no	—
346	C₂²⋊C₄	—	86	1	no	—
347	C₂²⋊C₄	—	86	1	no	—
348	C₂²⋊C₄	—	86	1	no	—
349	C₂²⋊C₄	—	172	2	no	—
350	C₄²	—	43	1	no	—
351	C₄²	—	43	1	no	—
352	C₄²	—	43	1	no	—
353	C₄²	—	86	1	no	—
354	C₄²	—	172	2	no	—
355	C₄²	C₂³×D₄₃	1	1	yes	characteristic
356	C₂³	—	2	1	no	—
357	C₂³	—	43	1	no	—
358	Q₈	—	4	2	no	—
359	Q₈	—	4	2	no	—
360	Q₈	—	86	2	no	—
361	Q₈	—	344	2	no	—
362	D₄	—	2	2	no	—
363	D₄	—	172	2	no	—
364	C₂×C₄	—	2	1	no	—
365	C₂×C₄	—	4	1	no	—
366	C₂×C₄	—	4	1	no	—
367	C₂×C₄	—	4	2	no	—
368	C₂×C₄	—	43	1	no	—
369	C₂×C₄	—	43	1	no	—
370	C₂×C₄	—	43	1	no	—
371	C₂×C₄	—	86	1	no	—
372	C₂×C₄	—	86	1	no	—
373	C₂×C₄	—	86	2	no	—
374	C₂×C₄	—	86	1	no	—
375	C₂×C₄	—	86	1	no	—
376	C₂×C₄	—	172	1	no	—
377	C₂×C₄	—	172	2	no	—
378	C₂×C₄	C₂×D₄×D₄₃	1	1	yes	characteristic
379	C₂×C₄	C₂×D₄×D₄₃	1	1	yes	characteristic
380	C₂×C₄	#41×D₄₃	1	1	yes	characteristic
381	C₈	—	2	1	no	—
382	C₈	—	4	1	no	—
383	C₈	—	4	2	no	—
384	C₈	—	86	1	no	—
385	C₈	—	86	1	no	—
386	C₈	—	172	1	no	—
387	C₈	—	172	2	no	—
388	C₈	#13063344	1	2	yes	—
389	C₂²	—	4	1	no	—
390	C₂²	—	4	1	no	—
391	C₂²	—	4	1	no	—
392	C₂²	—	86	1	no	—
393	C₂²	—	86	1	no	—
394	C₂²	—	86	1	no	—
395	C₂²	#121×D₄₃	1	1	yes	characteristic
396	C₄	—	2	1	no	—
397	C₄	—	4	1	no	—
398	C₄	—	8	2	no	—
399	C₄	—	86	1	no	—
400	C₄	—	86	1	no	—
401	C₄	—	86	1	no	—
402	C₄	—	344	2	no	—
403	C₄	—	172	1	no	—
404	C₄	—	172	1	no	—
405	C₄	C₂×#13063344	1	1	yes	characteristic
406	C₄	#14583695	1	1	yes	characteristic
407	C₄	#14583715	1	2	yes	—
408	C₂	—	8	1	no	—
409	C₂	—	172	1	no	—
410	C₂	#19144043	1	1	yes	characteristic
411	C₂	#19144248	1	1	yes	characteristic
412	C₂	#19144317	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	#121×D₄₃
commutator, cosocle	C₄²×C₄₃	C₂⁴
Frattini	C₄²	C₂³×D₄₃
Fitting	C₄₃×#1689	C₂
socle	C₂²×C₄₃	C₂×#121
prime radical	#13063296	D₄
hypercenter	#1689	D₄₃
hypocenter	C₄₃	#19226
LCS	G ▷ C₄²×C₄₃ ▷ C₂²×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #1689

Complements

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

N	H ≅ G/N	type
2. #19145179	C₂	semidirect
4. #19134124	C₂	semidirect
5. #19134118	C₂	semidirect
6. #19134086	C₂	semidirect
7. #19134052	C₂	semidirect
8. #19134044	C₂	semidirect
9. #19133340	C₂	semidirect
10. #19133302	C₂	semidirect
11. C₄₃×#1689	C₂	semidirect
208. #1689	D₄₃	semidirect
253. C₄₃	#19226	semidirect

Maximal subgroups

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

subgroup	index
#19145772	2
#19145179	2
#19143573	2
#19134124	2
#19134118	2
#19134086	2
#19134052	2
#19134044	2
#19133340	2
#19133302	2
C₄₃×#1689	2
#19226	43