Subgroups — #25079855

Order 11008.

570 conjugacy classes of subgroups.

The subgroup lattice has 404 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#19145938	C₂	1	2	yes	maximal
2	#19145106	C₂	1	1	yes	maximal, characteristic
3	#19143526	C₂	1	1	yes	maximal, characteristic
4	#19134124	C₂	1	1	yes	maximal, characteristic, complemented
5	#19134123	C₂	1	1	yes	maximal, characteristic, complemented
6	#19134110	C₂	1	1	yes	maximal, characteristic, complemented
7	#19133976	C₂	1	2	yes	maximal, complemented
8	#19133975	C₂	1	2	yes	maximal, complemented
9	#334×D₄₃	C₂	1	1	yes	maximal, characteristic
10	#19133284	C₂	1	2	yes	maximal
11	C₄₃×#1694	C₂	1	1	yes	maximal, characteristic, complemented
12	#14583454	C₂²	1	2	yes	—
13	#14583453	C₂²	1	1	yes	characteristic
14	#14583304	C₂²	1	1	yes	characteristic
15	#14583300	—	2	1	no	—
16	#14583300	—	2	1	no	—
17	#14583295	C₂²	1	2	yes	—
18	#14583195	C₂²	1	2	yes	—
19	#14583194	C₂²	1	2	yes	—
20	#14583192	—	2	2	no	—
21	#14583139	—	2	2	no	—
22	#14583137	—	2	2	no	—
23	#14583107	—	2	2	no	—
24	#14583106	C₂²	1	2	yes	—
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	#14582936	—	2	1	no	—
31	#14582936	—	2	1	no	—
32	#14582924	—	2	1	no	—
33	#14582921	C₂²	1	1	yes	characteristic
34	C₄⋊C₈×D₄₃	C₂²	1	2	yes	—
35	#14582856	—	2	1	no	—
36	C₄×C₈×D₄₃	C₂²	1	1	yes	characteristic
37	#14582650	C₂²	1	1	yes	characteristic
38	#14582649	C₂²	1	2	yes	—
39	#14582647	C₂²	1	2	yes	—
40	#14582647	C₂²	1	2	yes	—
41	#14582645	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×#10621885	—	2	2	no	—
48	C₂×#10621872	—	2	1	no	—
49	C₂×C₈×D₄₃	—	2	2	no	—
50	C₂×C₈×D₄₃	D₄	1	1	yes	characteristic
51	C₂×C₈×D₄₃	D₄	1	1	yes	characteristic
52	#13063325	C₂³	1	2	yes	—
53	#13063324	D₄	1	1	yes	characteristic
54	#13063323	D₄	1	1	yes	characteristic
55	#13063316	—	2	2	no	—
56	#13063315	—	2	1	no	—
57	#13063305	—	2	2	no	—
58	#13063305	D₄	1	1	yes	characteristic
59	#13063303	—	2	1	no	—
60	#13063301	—	2	2	no	—
61	#13063301	D₄	1	1	yes	characteristic
62	Q₈×Dic₄₃	—	2	2	no	—
63	C₄×Dic₈₆	—	2	1	no	—
64	C₄×Dic₈₆	C₂³	1	1	yes	characteristic
65	#13063290	—	2	2	no	—
66	#13063283	—	2	2	no	—
67	#13063282	—	2	1	no	—
68	C₄²×D₄₃	C₂³	1	1	yes	characteristic
69	#13063259	—	2	1	no	—
70	#13063259	—	2	1	no	—
71	#13063259	—	2	2	no	—
72	#13063258	—	2	2	no	—
73	#13063256	—	2	2	no	—
74	#13063256	—	2	2	no	—
75	C₄⋊(C₄₃⋊C₈)	—	2	1	no	—
76	C₄⋊(C₄₃⋊C₈)	C₂³	1	2	yes	—
77	C₄⋊(C₄₃⋊C₈)	C₂³	1	1	yes	characteristic
78	Dic₄₃⋊C₈	—	2	1	no	—
79	Dic₄₃⋊C₈	—	2	2	no	—
80	Dic₄₃⋊C₈	—	2	1	no	—
81	Dic₄₃⋊C₈	—	2	1	no	—
82	#13063250	—	2	2	no	—
83	#13063249	—	2	2	no	—
84	#13063249	—	2	2	no	—
85	#13063235	—	2	1	no	—
86	C₄×C₄₃⋊C₈	—	2	1	no	—
87	C₄×C₄₃⋊C₈	C₂³	1	1	yes	characteristic
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	1	yes	characteristic
98	C₄⋊C₈×C₄₃	C₂³	1	2	yes	—
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	C₂×C₄×D₄₃	C₂×D₄	1	1	yes	characteristic
105	C₂×C₄×D₄₃	C₂×D₄	1	1	yes	characteristic
106	C₂×C₄×D₄₃	#41	1	1	yes	characteristic
107	C₂×Dic₈₆	—	2	1	no	—
108	C₂×Dic₈₆	—	2	2	no	—
109	C₂×Dic₈₆	C₂×D₄	1	1	yes	characteristic
110	Q₁₆×C₄₃	—	8	2	no	—
111	C₂×C₈×C₄₃	—	2	1	no	—
112	C₂×C₈×C₄₃	—	2	2	no	—
113	C₂×C₈×C₄₃	C₂×D₄	1	1	yes	characteristic
114	C₂×C₈×C₄₃	C₂×D₄	1	1	yes	characteristic
115	C₄⋊C₄×C₄₃	—	2	1	no	—
116	C₄⋊C₄×C₄₃	—	2	2	no	—
117	C₄⋊C₄×C₄₃	—	2	2	no	—
118	C₄⋊C₄×C₄₃	C₂×D₄	1	1	yes	characteristic
119	C₄²×C₄₃	—	2	1	no	—
120	C₄²×C₄₃	C₂⁴	1	1	yes	characteristic
121	#10621885	—	8	2	no	—
122	(C₂×D₄₃)⋊C₄	—	2	1	no	—
123	(C₂×D₄₃)⋊C₄	—	2	1	no	—
124	(C₂×D₄₃)⋊C₄	—	4	2	no	—
125	C₄⋊Dic₄₃	—	2	1	no	—
126	C₄⋊Dic₄₃	—	2	2	no	—
127	C₄⋊Dic₄₃	—	2	2	no	—
128	C₄⋊Dic₄₃	—	4	2	no	—
129	C₄⋊Dic₄₃	C₂×D₄	1	1	yes	characteristic
130	Dic₄₃⋊C₄	—	2	1	no	—
131	Dic₄₃⋊C₄	—	2	1	no	—
132	Dic₄₃⋊C₄	—	2	1	no	—
133	Dic₄₃⋊C₄	—	2	1	no	—
134	Dic₄₃⋊C₄	—	2	1	no	—
135	Dic₄₃⋊C₄	—	4	2	no	—
136	C₄×Dic₄₃	—	2	1	no	—
137	C₄×Dic₄₃	—	4	2	no	—
138	C₄×Dic₄₃	C₂×D₄	1	1	yes	characteristic
139	C₄×Dic₄₃	C₂×D₄	1	1	yes	characteristic
140	C₄×Dic₄₃	#41	1	1	yes	characteristic
141	C₂×C₄₃⋊C₈	—	2	1	no	—
142	C₂×C₄₃⋊C₈	—	2	2	no	—
143	C₂×C₄₃⋊C₈	C₂×D₄	1	1	yes	characteristic
144	C₂×C₄₃⋊C₈	C₂×D₄	1	1	yes	characteristic
145	#10621872	—	4	2	no	—
146	C₈×D₄₃	—	2	2	no	—
147	C₈×D₄₃	—	8	2	no	—
148	C₈×D₄₃	Q₁₆	1	4	yes	complemented
149	C₂²×D₄₃	#121	1	1	yes	characteristic
150	Q₈×C₄₃	—	2	2	no	—
151	Q₈×C₄₃	—	4	2	no	—
152	Q₈×C₄₃	—	4	2	no	—
153	C₂×C₄×C₄₃	—	2	1	no	—
154	C₂×C₄×C₄₃	—	2	1	no	—
155	C₂×C₄×C₄₃	—	4	2	no	—
156	C₂×C₄×C₄₃	C₂²×D₄	1	1	yes	characteristic
157	C₂×C₄×C₄₃	C₂²×D₄	1	1	yes	characteristic
158	C₂×C₄×C₄₃	C₂×#41	1	1	yes	characteristic
159	C₂×Dic₄₃	—	2	1	no	—
160	C₂×Dic₄₃	—	2	1	no	—
161	C₂×Dic₄₃	—	4	2	no	—
162	C₂×Dic₄₃	#121	1	1	yes	characteristic
163	C₂×Dic₄₃	#121	1	1	yes	characteristic
164	C₂×Dic₄₃	#121	1	1	yes	characteristic
165	C₄×D₄₃	—	2	2	no	—
166	C₄×D₄₃	—	4	1	no	—
167	C₄×D₄₃	C₂×Q₁₆	1	2	yes	—
168	C₄×D₄₃	#137	1	2	yes	—
169	Dic₈₆	—	2	2	no	—
170	Dic₈₆	—	4	2	no	—
171	Dic₈₆	—	8	2	no	—
172	C₈×C₄₃	—	2	1	no	—
173	C₈×C₄₃	—	4	1	no	—
174	C₈×C₄₃	—	4	2	no	—
175	C₈×C₄₃	C₂×Q₁₆	1	2	yes	—
176	C₄₃⋊C₈	—	2	1	no	—
177	C₄₃⋊C₈	—	4	1	no	—
178	C₄₃⋊C₈	—	4	2	no	—
179	C₄₃⋊C₈	C₂×Q₁₆	1	2	yes	—
180	#19210	—	43	1	no	maximal, Sylow
181	C₂²×C₄₃	C₂×#121	1	1	yes	characteristic
182	C₂×D₄₃	—	2	1	no	—
183	C₂×D₄₃	—	2	1	no	—
184	C₂×D₄₃	#462	1	2	yes	—
185	C₄×C₄₃	—	2	1	no	—
186	C₄×C₄₃	—	4	1	no	—
187	C₄×C₄₃	—	4	1	no	—
188	C₄×C₄₃	—	8	2	no	—
189	C₄×C₄₃	C₂²×Q₁₆	1	1	yes	characteristic
190	C₄×C₄₃	C₂×#137	1	1	yes	characteristic
191	C₄×C₄₃	#577	1	2	yes	—
192	Dic₄₃	—	2	1	no	—
193	Dic₄₃	—	2	1	no	—
194	Dic₄₃	—	4	1	no	—
195	Dic₄₃	—	4	1	no	—
196	Dic₄₃	—	8	2	no	—
197	Dic₄₃	#462	1	2	yes	—
198	#3249	—	43	2	no	—
199	#3160	—	43	1	no	—
200	#2960	—	43	1	no	—
201	#1696	—	43	1	no	—
202	#1696	—	43	1	no	—
203	#1694	—	43	1	no	—
204	#1694	D₄₃	1	1	yes	characteristic, complemented
205	#1672	—	43	2	no	—
206	#1672	—	43	2	no	—
207	C₂×#334	—	43	1	no	—
208	#1550	—	43	2	no	—
209	C₂×C₄₃	C₂×#462	1	1	yes	characteristic
210	C₂×C₄₃	#3060	1	1	yes	characteristic
211	C₂×C₄₃	#3067	1	1	yes	characteristic
212	D₄₃	—	2	2	no	—
213	#533	—	43	1	no	—
214	#533	—	43	2	no	—
215	#500	—	43	1	no	—
216	#500	C₂×D₄₃	1	1	yes	characteristic
217	#499	—	86	1	no	—
218	#499	—	86	1	no	—
219	#498	—	43	2	no	—
220	#477	—	43	2	no	—
221	#477	—	43	2	no	—
222	#477	—	86	2	no	—
223	#467	—	86	2	no	—
224	#467	—	86	2	no	—
225	#462	—	43	2	no	—
226	#462	—	86	2	no	—
227	#462	C₂×D₄₃	1	2	yes	—
228	#446	—	43	1	no	—
229	#446	—	86	1	no	—
230	#446	C₂×D₄₃	1	1	yes	characteristic
231	C₈×Q₈	—	43	1	no	—
232	C₈×Q₈	—	43	1	no	—
233	C₈×Q₈	—	86	1	no	—
234	#427	—	86	1	no	—
235	#427	—	86	1	no	—
236	#424	—	86	1	no	—
237	#423	—	43	1	no	—
238	C₂×C₄⋊C₈	—	43	2	no	—
239	#405	—	86	1	no	—
240	C₂×C₄×C₈	—	43	1	no	—
241	#334	—	43	2	no	—
242	#334	—	43	1	no	—
243	#334	C₂×D₄₃	1	1	yes	characteristic
244	#333	—	43	2	no	—
245	#333	—	43	2	no	—
246	#332	—	43	2	no	—
247	#332	C₂×D₄₃	1	2	yes	—
248	C₄₃	#19210	1	1	yes	characteristic, complemented, Sylow
249	C₂×Q₁₆	—	2	2	no	—
250	C₂×Q₁₆	—	86	2	no	—
251	C₂×M₁₆	—	86	1	no	—
252	C₂²×C₈	—	43	1	no	—
253	C₂²×C₈	—	43	1	no	—
254	C₂²×C₈	—	86	2	no	—
255	C₄⋊Q₈	—	43	1	no	—
256	C₄⋊Q₈	—	43	1	no	—
257	C₄⋊Q₈	—	43	2	no	—
258	C₄⋊Q₈	C₂²×D₄₃	1	2	yes	—
259	#125	—	86	1	no	—
260	#125	—	86	2	no	—
261	C₂²⋊Q₈	—	43	1	no	—
262	C₂²⋊Q₈	—	43	1	no	—
263	C₂²⋊Q₈	—	86	1	no	—
264	C₂²⋊Q₈	—	86	2	no	—
265	C₂²⋊Q₈	—	86	2	no	—
266	C₄×Q₈	—	43	1	no	—
267	C₄×Q₈	—	86	1	no	—
268	C₄×Q₈	—	86	2	no	—
269	C₄×Q₈	—	86	2	no	—
270	C₄×Q₈	C₂²×D₄₃	1	1	yes	characteristic
271	#117	—	86	1	no	—
272	#117	—	86	2	no	—
273	C₂×C₄²	—	43	1	no	—
274	#107	—	2	1	no	—
275	#107	—	2	1	no	—
276	#107	—	86	2	no	—
277	#107	—	86	1	no	—
278	#107	—	86	1	no	—
279	#107	—	86	2	no	—
280	#106	—	86	2	no	—
281	#106	—	86	2	no	—
282	C₄⋊C₈	—	2	1	no	—
283	C₄⋊C₈	—	43	1	no	—
284	C₄⋊C₈	—	43	2	no	—
285	C₄⋊C₈	—	86	1	no	—
286	C₄⋊C₈	—	86	2	no	—
287	C₄⋊C₈	—	86	1	no	—
288	C₄⋊C₈	—	86	1	no	—
289	C₄⋊C₈	—	86	1	no	—
290	C₄⋊C₈	C₂²×D₄₃	1	1	yes	characteristic
291	C₄⋊C₈	C₂²×D₄₃	1	2	yes	—
292	#103	—	2	2	no	—
293	#103	—	86	2	no	—
294	#103	—	86	2	no	—
295	#103	—	86	2	no	—
296	#97	—	2	1	no	—
297	#97	—	86	1	no	—
298	C₄×C₈	—	43	1	no	—
299	C₄×C₈	—	43	1	no	—
300	C₄×C₈	—	43	1	no	—
301	C₄×C₈	—	86	1	no	—
302	C₄×C₈	—	86	1	no	—
303	C₄×C₈	C₂²×D₄₃	1	1	yes	characteristic
304	C₂×Q₈	—	2	2	no	—
305	C₂×Q₈	—	43	1	no	—
306	C₂×Q₈	—	86	1	no	—
307	C₂×Q₈	—	86	2	no	—
308	C₂×Q₈	D₄×D₄₃	1	1	yes	characteristic
309	C₂²×C₄	—	43	1	no	—
310	C₂²×C₄	—	43	1	no	—
311	C₂²×C₄	—	43	1	no	—
312	Q₁₆	—	8	2	no	—
313	Q₁₆	—	344	2	no	—
314	M₁₆	—	172	2	no	—
315	C₂×C₈	—	2	1	no	—
316	C₂×C₈	—	2	2	no	—
317	C₂×C₈	—	43	1	no	—
318	C₂×C₈	—	43	4	no	—
319	C₂×C₈	—	43	1	no	—
320	C₂×C₈	—	86	2	no	—
321	C₂×C₈	—	86	1	no	—
322	C₂×C₈	—	86	2	no	—
323	C₂×C₈	—	344	2	no	—
324	C₂×C₈	D₄×D₄₃	1	1	yes	characteristic
325	C₂×C₈	D₄×D₄₃	1	1	yes	characteristic
326	C₄⋊C₄	—	2	1	no	—
327	C₄⋊C₄	—	2	2	no	—
328	C₄⋊C₄	—	2	2	no	—
329	C₄⋊C₄	—	43	1	no	—
330	C₄⋊C₄	—	86	1	no	—
331	C₄⋊C₄	—	86	1	no	—
332	C₄⋊C₄	—	86	1	no	—
333	C₄⋊C₄	—	86	1	no	—
334	C₄⋊C₄	—	86	1	no	—
335	C₄⋊C₄	—	86	1	no	—
336	C₄⋊C₄	—	86	2	no	—
337	C₄⋊C₄	—	86	2	no	—
338	C₄⋊C₄	—	172	2	no	—
339	C₄⋊C₄	—	172	2	no	—
340	C₄⋊C₄	D₄×D₄₃	1	1	yes	characteristic
341	C₂²⋊C₄	—	86	1	no	—
342	C₂²⋊C₄	—	86	1	no	—
343	C₂²⋊C₄	—	172	2	no	—
344	C₄²	—	2	1	no	—
345	C₄²	—	43	1	no	—
346	C₄²	—	43	1	no	—
347	C₄²	—	43	1	no	—
348	C₄²	—	86	1	no	—
349	C₄²	—	172	2	no	—
350	C₄²	C₂³×D₄₃	1	1	yes	characteristic
351	C₂³	—	43	1	no	—
352	Q₈	—	2	2	no	—
353	Q₈	—	4	2	no	—
354	Q₈	—	4	2	no	—
355	Q₈	—	86	2	no	—
356	Q₈	—	344	2	no	—
357	Q₈	—	172	2	no	—
358	C₂×C₄	—	2	1	no	—
359	C₂×C₄	—	2	1	no	—
360	C₂×C₄	—	4	2	no	—
361	C₂×C₄	—	43	1	no	—
362	C₂×C₄	—	43	2	no	—
363	C₂×C₄	—	43	2	no	—
364	C₂×C₄	—	43	1	no	—
365	C₂×C₄	—	43	1	no	—
366	C₂×C₄	—	86	2	no	—
367	C₂×C₄	—	86	1	no	—
368	C₂×C₄	—	86	1	no	—
369	C₂×C₄	—	172	1	no	—
370	C₂×C₄	—	172	2	no	—
371	C₂×C₄	C₂×D₄×D₄₃	1	1	yes	characteristic
372	C₂×C₄	C₂×D₄×D₄₃	1	1	yes	characteristic
373	C₂×C₄	#41×D₄₃	1	1	yes	characteristic
374	C₈	—	2	1	no	—
375	C₈	—	4	1	no	—
376	C₈	—	4	2	no	—
377	C₈	—	43	2	no	—
378	C₈	—	86	1	no	—
379	C₈	—	172	1	no	—
380	C₈	—	172	2	no	—
381	C₈	Q₁₆×D₄₃	1	2	yes	—
382	C₂²	—	43	2	no	—
383	C₂²	—	86	1	no	—
384	C₂²	—	86	1	no	—
385	C₂²	#121×D₄₃	1	1	yes	characteristic
386	C₄	—	2	1	no	—
387	C₄	—	4	1	no	—
388	C₄	—	4	1	no	—
389	C₄	—	8	2	no	—
390	C₄	—	43	2	no	—
391	C₄	—	86	1	no	—
392	C₄	—	86	1	no	—
393	C₄	—	344	2	no	—
394	C₄	—	172	1	no	—
395	C₄	—	172	1	no	—
396	C₄	C₂×Q₁₆×D₄₃	1	1	yes	characteristic
397	C₄	#137×D₄₃	1	1	yes	characteristic
398	C₄	#14583709	1	2	yes	—
399	C₂	—	86	2	no	—
400	C₂	#462×D₄₃	1	1	yes	characteristic
401	C₂	#19144255	1	1	yes	characteristic
402	C₂	#19144317	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	#19210
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. #19134124	C₂	semidirect
5. #19134123	C₂	semidirect
6. #19134110	C₂	semidirect
7. #19133976	C₂	semidirect
8. #19133975	C₂	semidirect
11. C₄₃×#1694	C₂	semidirect
148. C₈×D₄₃	Q₁₆	semidirect
204. #1694	D₄₃	semidirect
248. C₄₃	#19210	semidirect

Maximal subgroups

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

subgroup	index
#19145938	2
#19145106	2
#19143526	2
#19134124	2
#19134123	2
#19134110	2
#19133976	2
#19133975	2
#334×D₄₃	2
#19133284	2
C₄₃×#1694	2
#19210	43