Subgroups — #25966275

Order 11008.

1834 conjugacy classes of subgroups.

The subgroup lattice has 374 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#19148408	C₂	1	2	yes	maximal
2	#19145139	C₂	1	6	yes	maximal, complemented
3	#19145127	C₂	1	3	yes	maximal, complemented
4	#19145125	C₂	1	6	yes	maximal, complemented
5	#19145123	C₂	1	3	yes	maximal, complemented
6	#19145107	C₂	1	3	yes	maximal, complemented
7	C₂×#14583296	C₂	1	3	yes	maximal, complemented
8	#19143190	C₂	1	1	yes	maximal, characteristic
9	#19143188	C₂	1	2	yes	maximal
10	#131×Dic₄₃	C₂	1	1	yes	maximal, characteristic
11	C₄₃×#2912	C₂	1	1	yes	maximal, characteristic
12	C₂×#13063332	C₂²	1	1	yes	characteristic
13	Q₈×Dic₈₆	—	2	2	no	—
14	#14583625	—	2	2	no	—
15	#14583608	—	2	6	no	—
16	D₄×Dic₈₆	—	2	6	no	—
17	#14583530	C₂²	1	6	yes	—
18	#14583527	C₂²	1	6	yes	—
19	#14583452	C₂²	1	6	yes	—
20	C₂×#13063324	C₂²	1	6	yes	—
21	#14583382	C₂²	1	2	yes	—
22	#41×Dic₄₃	C₂²	1	1	yes	characteristic
23	#14583307	C₂²	1	12	yes	—
24	#14583306	C₂²	1	12	yes	—
25	#14583303	C₂²	1	6	yes	—
26	#14583300	C₂²	1	6	yes	—
27	#14583299	C₂²	1	3	yes	—
28	#14583299	C₂²	1	3	yes	—
29	#14583296	C₂²	1	3	yes	—
30	#14583296	C₂²	1	6	yes	—
31	#14583296	C₂²	1	6	yes	—
32	#14583296	C₂²	1	6	yes	—
33	#14583296	C₂²	1	1	yes	characteristic
34	#14582942	C₂²	1	6	yes	—
35	#14582941	C₂²	1	6	yes	—
36	#14582939	C₂²	1	6	yes	—
37	#14582937	C₂²	1	3	yes	—
38	#14582935	C₂²	1	6	yes	—
39	C₂×#13063257	C₂²	1	3	yes	—
40	C₂×#13063256	C₂²	1	6	yes	—
41	#14582857	C₂²	1	3	yes	—
42	#14582854	C₂²	1	3	yes	—
43	M₁₆×Dic₄₃	C₂²	1	3	yes	—
44	C₂×C₈×Dic₄₃	C₂²	1	3	yes	—
45	C₂×#131×C₄₃	C₂²	1	1	yes	characteristic
46	C₄₃×#520	C₂²	1	2	yes	—
47	C₄₃×#428	C₂²	1	6	yes	—
48	C₄₃×#427	C₂²	1	3	yes	—
49	C₂×#106×C₄₃	C₂²	1	3	yes	—
50	C₂²×Dic₈₆	—	2	6	no	—
51	#13063332	Q₈	1	8	yes	complemented
52	C₂×#10621877	C₂³	1	3	yes	—
53	C₂²×C₄₃⋊C₈	C₂³	1	3	yes	—
54	#13063325	—	2	6	no	—
55	#13063324	—	2	6	no	—
56	#13063324	—	2	6	no	—
57	#13063324	—	2	6	no	—
58	#13063324	C₂³	1	6	yes	—
59	#13063324	C₂³	1	12	yes	—
60	#13063324	C₂³	1	2	yes	—
61	#13063317	—	2	6	no	—
62	#13063316	—	2	6	no	—
63	#13063316	C₂³	1	6	yes	—
64	#13063316	C₂³	1	6	yes	—
65	#13063304	—	2	12	no	—
66	#13063302	—	2	6	no	—
67	#13063302	—	2	6	no	—
68	#13063302	—	2	12	no	—
69	Q₈×Dic₄₃	—	2	2	no	—
70	Q₈×Dic₄₃	D₄	1	2	yes	—
71	C₄×Dic₈₆	—	2	6	no	—
72	C₄×Dic₈₆	—	2	6	no	—
73	D₄×Dic₄₃	—	2	6	no	—
74	D₄×Dic₄₃	D₄	1	6	yes	—
75	#13063284	C₂³	1	6	yes	—
76	#13063284	C₂³	1	3	yes	—
77	C₂×C₄⋊Dic₄₃	C₂³	1	6	yes	—
78	C₂×Dic₄₃⋊C₄	—	2	6	no	—
79	C₂×C₄×Dic₄₃	C₂³	1	3	yes	—
80	#13063259	C₂³	1	6	yes	—
81	#13063258	C₂³	1	6	yes	—
82	#13063258	C₂³	1	6	yes	—
83	#13063257	C₂³	1	3	yes	—
84	#13063257	C₂³	1	6	yes	—
85	#13063257	C₂³	1	1	yes	characteristic
86	#13063256	C₂³	1	6	yes	—
87	#13063256	C₂³	1	12	yes	—
88	#13063256	C₂³	1	2	yes	—
89	#13063235	C₂³	1	6	yes	—
90	C₈×Dic₄₃	C₂³	1	1	yes	characteristic
91	C₈×Dic₄₃	C₂³	1	3	yes	—
92	C₈×Dic₄₃	C₂³	1	3	yes	—
93	C₈×Dic₄₃	C₂³	1	3	yes	—
94	C₂×#41×C₄₃	C₂³	1	1	yes	characteristic
95	#131×C₄₃	Q₈	1	8	yes	complemented
96	C₂×M₁₆×C₄₃	C₂³	1	3	yes	—
97	C₂²×C₈×C₄₃	C₂³	1	3	yes	—
98	C₄×Q₈×C₄₃	—	2	2	no	—
99	C₄×D₄×C₄₃	—	2	6	no	—
100	#117×C₄₃	C₂³	1	6	yes	—
101	C₂×C₄⋊C₄×C₄₃	C₂³	1	6	yes	—
102	#107×C₄₃	C₂³	1	6	yes	—
103	#106×C₄₃	C₂³	1	3	yes	—
104	#106×C₄₃	C₂³	1	6	yes	—
105	#106×C₄₃	C₂³	1	1	yes	characteristic
106	#41×C₄₃	C₂×Q₈	1	4	yes	—
107	#41×C₄₃	C₂×Q₈	1	4	yes	—
108	C₂×Q₈×C₄₃	C₂×D₄	1	1	yes	characteristic
109	C₂×D₄×C₄₃	C₂×D₄	1	3	yes	—
110	C₂²×C₄×C₄₃	—	2	6	no	—
111	C₂²×C₄×C₄₃	C₂⁴	1	3	yes	—
112	C₂²×Dic₄₃	—	2	6	no	—
113	C₂²×Dic₄₃	C₂×D₄	1	6	yes	—
114	C₂×Dic₈₆	—	2	2	no	—
115	C₂×Dic₈₆	—	2	6	no	—
116	C₂×Dic₈₆	—	2	12	no	—
117	C₂×Dic₈₆	—	8	12	no	—
118	M₁₆×C₄₃	C₂×Q₈	1	12	yes	complemented
119	C₂×C₈×C₄₃	C₂×Q₈	1	12	yes	complemented
120	C₂×C₈×C₄₃	C₂⁴	1	1	yes	characteristic
121	C₂×C₈×C₄₃	C₂⁴	1	3	yes	—
122	C₄⋊C₄×C₄₃	—	2	6	no	—
123	C₄⋊C₄×C₄₃	C₂⁴	1	6	yes	—
124	C₄⋊C₄×C₄₃	C₂⁴	1	2	yes	—
125	C₂²⋊C₄×C₄₃	—	2	6	no	—
126	C₄²×C₄₃	—	2	6	no	—
127	C₂²⋊Dic₄₃	—	2	6	no	—
128	C₂²⋊Dic₄₃	C₂×D₄	1	6	yes	—
129	C₄⋊Dic₄₃	—	2	6	no	—
130	C₄⋊Dic₄₃	—	2	6	no	—
131	C₄⋊Dic₄₃	—	2	6	no	—
132	C₄⋊Dic₄₃	C₂×D₄	1	6	yes	—
133	C₄⋊Dic₄₃	C₂⁴	1	6	yes	—
134	C₄⋊Dic₄₃	C₂⁴	1	2	yes	—
135	Dic₄₃⋊C₄	—	2	6	no	—
136	Dic₄₃⋊C₄	—	2	12	no	—
137	Dic₄₃⋊C₄	—	2	2	no	—
138	Dic₄₃⋊C₄	—	2	6	no	—
139	Dic₄₃⋊C₄	—	2	6	no	—
140	C₄×Dic₄₃	—	2	6	no	—
141	C₄×Dic₄₃	C₂×D₄	1	6	yes	—
142	C₄×Dic₄₃	C₂⁴	1	1	yes	characteristic
143	C₄×Dic₄₃	C₂⁴	1	3	yes	—
144	#10621877	C₂×Q₈	1	12	yes	complemented
145	C₂×C₄₃⋊C₈	C₂×Q₈	1	12	yes	complemented
146	C₂×C₄₃⋊C₈	C₂⁴	1	1	yes	characteristic
147	C₂×C₄₃⋊C₈	C₂⁴	1	3	yes	—
148	C₂³×C₄₃	C₂²×D₄	1	3	yes	—
149	Q₈×C₄₃	C₄⋊Q₈	1	4	yes	—
150	D₄×C₄₃	C₄⋊Q₈	1	12	yes	—
151	C₂×C₄×C₄₃	—	2	2	no	—
152	C₂×C₄×C₄₃	—	2	6	no	—
153	C₂×C₄×C₄₃	—	8	6	no	—
154	C₂×C₄×C₄₃	C₂²×D₄	1	3	yes	—
155	C₂×C₄×C₄₃	C₂²×Q₈	1	6	yes	—
156	C₂×C₄×C₄₃	C₂²×Q₈	1	6	yes	—
157	C₂×C₄×C₄₃	C₂⁵	1	1	yes	characteristic
158	C₂×Dic₄₃	—	2	12	no	—
159	C₂×Dic₄₃	—	2	2	no	—
160	C₂×Dic₄₃	—	2	6	no	—
161	C₂×Dic₄₃	—	8	6	no	—
162	C₂×Dic₄₃	C₂²×D₄	1	6	yes	—
163	C₂×Dic₄₃	C₂²×D₄	1	2	yes	—
164	Dic₈₆	—	4	4	no	—
165	Dic₈₆	—	4	12	no	—
166	Dic₈₆	—	8	12	no	—
167	C₈×C₄₃	C₂²×Q₈	1	2	yes	—
168	C₈×C₄₃	C₂²×Q₈	1	6	yes	—
169	C₄₃⋊C₈	C₂²×Q₈	1	2	yes	—
170	C₄₃⋊C₈	C₂²×Q₈	1	6	yes	—
171	#60665	—	43	1	no	maximal, Sylow
172	C₂²×C₄₃	—	2	6	no	—
173	C₂²×C₄₃	C₂×C₄⋊Q₈	1	6	yes	—
174	C₂²×C₄₃	C₂³×D₄	1	1	yes	characteristic
175	C₄×C₄₃	—	4	6	no	—
176	C₄×C₄₃	—	4	2	no	—
177	C₄×C₄₃	C₂×C₄⋊Q₈	1	6	yes	—
178	C₄×C₄₃	C₂³×Q₈	1	1	yes	characteristic
179	C₄×C₄₃	C₂³×Q₈	1	1	yes	characteristic
180	Dic₄₃	—	2	6	no	—
181	Dic₄₃	—	4	6	no	—
182	Dic₄₃	—	4	2	no	—
183	Dic₄₃	#577	1	4	yes	—
184	#3503	—	43	2	no	—
185	#3165	—	43	6	no	—
186	#3163	—	43	6	no	—
187	#3163	—	43	3	no	—
188	#3163	—	43	3	no	—
189	#3160	—	43	3	no	—
190	C₂×#498	—	43	3	no	—
191	#2912	—	43	1	no	—
192	#2912	—	43	2	no	—
193	#2912	D₄₃	1	1	yes	characteristic
194	C₄×#131	—	43	1	no	—
195	C₂×C₄₃	—	2	6	no	—
196	C₂×C₄₃	C₂²×C₄⋊Q₈	1	1	yes	characteristic
197	C₂×C₄₃	C₂×#577	1	2	yes	—
198	C₂×#131	—	43	1	no	—
199	C₂×#131	C₂×D₄₃	1	1	yes	characteristic
200	Q₈²	—	86	2	no	—
201	#557	—	86	2	no	—
202	#554	—	86	6	no	—
203	D₄×Q₈	—	86	6	no	—
204	#544	—	43	6	no	—
205	#544	—	43	6	no	—
206	#533	—	43	6	no	—
207	C₂×C₄⋊Q₈	—	43	6	no	—
208	#520	—	43	2	no	—
209	#520	C₂×D₄₃	1	2	yes	—
210	C₄×#41	—	43	1	no	—
211	#501	—	43	12	no	—
212	#501	—	43	12	no	—
213	#500	—	43	6	no	—
214	#499	—	43	6	no	—
215	#499	—	43	3	no	—
216	#499	—	43	3	no	—
217	#498	—	43	3	no	—
218	#498	—	43	6	no	—
219	#498	—	43	6	no	—
220	#498	—	43	6	no	—
221	#498	—	43	1	no	—
222	#428	—	43	6	no	—
223	#428	—	43	6	no	—
224	#428	—	43	6	no	—
225	#428	C₂×D₄₃	1	6	yes	—
226	#427	—	43	3	no	—
227	#427	—	43	6	no	—
228	#427	C₂×D₄₃	1	3	yes	—
229	C₂×#106	—	43	3	no	—
230	C₂×#106	—	43	6	no	—
231	C₂×#106	C₂×D₄₃	1	3	yes	—
232	#405	—	43	3	no	—
233	#405	—	43	3	no	—
234	C₄×M₁₆	—	43	3	no	—
235	C₂×C₄×C₈	—	43	3	no	—
236	C₄₃	#60665	1	1	yes	characteristic, complemented, Sylow
237	C₂×#41	C₂²×D₄₃	1	1	yes	characteristic
238	C₂²×Q₈	—	86	6	no	—
239	#131	—	43	8	no	—
240	#131	Dic₈₆	1	8	yes	complemented
241	C₂×M₁₆	—	43	3	no	—
242	C₂×M₁₆	C₂²×D₄₃	1	3	yes	—
243	C₂²×C₈	—	43	3	no	—
244	C₂²×C₈	C₂²×D₄₃	1	3	yes	—
245	C₄⋊Q₈	—	43	6	no	—
246	C₄⋊Q₈	—	43	12	no	—
247	C₄⋊Q₈	—	43	2	no	—
248	C₄⋊Q₈	—	86	6	no	—
249	C₄⋊Q₈	—	86	6	no	—
250	C₄⋊Q₈	—	86	6	no	—
251	C₄⋊Q₈	—	86	6	no	—
252	#125	—	43	6	no	—
253	#125	—	43	6	no	—
254	#125	—	86	6	no	—
255	#125	—	86	6	no	—
256	C₂²⋊Q₈	—	86	12	no	—
257	C₂²⋊Q₈	—	86	6	no	—
258	C₂²⋊Q₈	—	86	6	no	—
259	C₂²⋊Q₈	—	86	12	no	—
260	C₄×Q₈	—	2	2	no	—
261	C₄×Q₈	—	43	2	no	—
262	C₄×Q₈	—	86	6	no	—
263	C₄×Q₈	—	86	6	no	—
264	C₄×Q₈	—	86	2	no	—
265	C₄×D₄	—	2	6	no	—
266	C₄×D₄	—	43	6	no	—
267	C₄×D₄	—	86	6	no	—
268	#117	—	43	3	no	—
269	#117	—	43	6	no	—
270	#117	C₂²×D₄₃	1	6	yes	—
271	C₂×C₄⋊C₄	—	43	6	no	—
272	C₂×C₄⋊C₄	—	86	6	no	—
273	C₂×C₄⋊C₄	C₂²×D₄₃	1	6	yes	—
274	C₂×C₄²	—	43	3	no	—
275	#107	—	43	6	no	—
276	#107	—	43	6	no	—
277	#107	—	43	6	no	—
278	#107	C₂²×D₄₃	1	6	yes	—
279	#106	—	43	3	no	—
280	#106	—	43	6	no	—
281	#106	—	43	1	no	—
282	#106	—	43	6	no	—
283	#106	—	43	12	no	—
284	#106	—	43	2	no	—
285	#106	C₂²×D₄₃	1	3	yes	—
286	#106	C₂²×D₄₃	1	6	yes	—
287	#106	C₂²×D₄₃	1	1	yes	characteristic
288	#97	—	43	6	no	—
289	C₄×C₈	—	43	1	no	—
290	C₄×C₈	—	43	3	no	—
291	C₄×C₈	—	43	3	no	—
292	C₄×C₈	—	43	3	no	—
293	#41	C₂×Dic₈₆	1	4	yes	—
294	#41	Q₈×D₄₃	1	4	yes	—
295	C₂×Q₈	—	86	12	no	—
296	C₂×Q₈	—	86	2	no	—
297	C₂×Q₈	—	86	6	no	—
298	C₂×Q₈	—	344	12	no	—
299	C₂×Q₈	D₄×D₄₃	1	1	yes	characteristic
300	C₂×D₄	D₄×D₄₃	1	3	yes	—
301	C₂²×C₄	—	2	6	no	—
302	C₂²×C₄	—	43	6	no	—
303	C₂²×C₄	—	86	6	no	—
304	C₂²×C₄	C₂³×D₄₃	1	3	yes	—
305	M₁₆	—	43	12	no	—
306	M₁₆	C₂×Dic₈₆	1	12	yes	complemented
307	C₂×C₈	—	43	12	no	—
308	C₂×C₈	—	43	1	no	—
309	C₂×C₈	—	43	3	no	—
310	C₂×C₈	C₂×Dic₈₆	1	12	yes	complemented
311	C₂×C₈	C₂³×D₄₃	1	1	yes	characteristic
312	C₂×C₈	C₂³×D₄₃	1	3	yes	—
313	C₄⋊C₄	—	2	6	no	—
314	C₄⋊C₄	—	43	6	no	—
315	C₄⋊C₄	—	43	6	no	—
316	C₄⋊C₄	—	43	2	no	—
317	C₄⋊C₄	—	86	6	no	—
318	C₄⋊C₄	—	86	12	no	—
319	C₄⋊C₄	—	86	2	no	—
320	C₄⋊C₄	—	86	6	no	—
321	C₄⋊C₄	—	86	6	no	—
322	C₄⋊C₄	—	86	6	no	—
323	C₄⋊C₄	—	86	6	no	—
324	C₄⋊C₄	—	86	6	no	—
325	C₄⋊C₄	C₂³×D₄₃	1	6	yes	—
326	C₄⋊C₄	C₂³×D₄₃	1	2	yes	—
327	C₂²⋊C₄	—	2	6	no	—
328	C₂²⋊C₄	—	43	6	no	—
329	C₂²⋊C₄	—	86	6	no	—
330	C₄²	—	2	6	no	—
331	C₄²	—	43	6	no	—
332	C₄²	—	43	1	no	—
333	C₄²	—	43	3	no	—
334	C₄²	—	86	6	no	—
335	C₂³	C₂×D₄×D₄₃	1	3	yes	—
336	Q₈	—	344	12	no	—
337	Q₈	—	172	12	no	—
338	Q₈	—	172	4	no	—
339	Q₈	#13063324	1	4	yes	—
340	D₄	#13063324	1	12	yes	—
341	C₂×C₄	—	2	2	no	—
342	C₂×C₄	—	2	6	no	—
343	C₂×C₄	—	8	6	no	—
344	C₂×C₄	—	43	2	no	—
345	C₂×C₄	—	43	6	no	—
346	C₂×C₄	—	86	12	no	—
347	C₂×C₄	—	86	2	no	—
348	C₂×C₄	—	86	6	no	—
349	C₂×C₄	—	344	6	no	—
350	C₂×C₄	C₂×D₄×D₄₃	1	3	yes	—
351	C₂×C₄	C₂×Q₈×D₄₃	1	6	yes	—
352	C₂×C₄	C₂²×Dic₈₆	1	6	yes	—
353	C₂×C₄	C₂⁴×D₄₃	1	1	yes	characteristic
354	C₈	—	43	2	no	—
355	C₈	—	43	6	no	—
356	C₈	C₂²×Dic₈₆	1	2	yes	—
357	C₈	C₂²×Dic₈₆	1	6	yes	—
358	C₂²	—	2	6	no	—
359	C₂²	C₂×#13063324	1	6	yes	—
360	C₂²	C₂²×D₄×D₄₃	1	1	yes	characteristic
361	C₄	—	4	6	no	—
362	C₄	—	4	2	no	—
363	C₄	—	43	4	no	—
364	C₄	—	86	6	no	—
365	C₄	—	172	6	no	—
366	C₄	—	172	2	no	—
367	C₄	C₂×#13063324	1	6	yes	—
368	C₄	C₂²×Q₈×D₄₃	1	1	yes	characteristic
369	C₄	C₂³×Dic₈₆	1	1	yes	characteristic
370	C₂	—	2	6	no	—
371	C₂	C₂²×#13063324	1	1	yes	characteristic
372	C₂	#577×D₄₃	1	2	yes	—

	subgroup	G/H
center	C₂²	C₂²×D₄×D₄₃
commutator, cosocle	C₂×C₄×C₄₃	C₂⁵
Frattini	C₂×C₄	C₂⁴×D₄₃
Fitting	C₄₃×#2912	C₂
socle	C₂²×C₄₃	C₂³×D₄
prime radical	C₂×#41×C₄₃	C₂³
hypercenter	#2912	D₄₃
hypocenter	C₄₃	#60665
LCS	G ▷ C₂×C₄×C₄₃ ▷ C₂×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₂² ◁ C₂×#41 ◁ #2912

Complements

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

N	H ≅ G/N	type
2. #19145139	C₂	semidirect
3. #19145127	C₂	semidirect
4. #19145125	C₂	semidirect
5. #19145123	C₂	semidirect
6. #19145107	C₂	semidirect
7. C₂×#14583296	C₂	semidirect
51. #13063332	Q₈	semidirect
95. #131×C₄₃	Q₈	semidirect
118. M₁₆×C₄₃	C₂×Q₈	semidirect
119. C₂×C₈×C₄₃	C₂×Q₈	semidirect
144. #10621877	C₂×Q₈	semidirect
145. C₂×C₄₃⋊C₈	C₂×Q₈	semidirect
236. C₄₃	#60665	semidirect
240. #131	Dic₈₆	semidirect
306. M₁₆	C₂×Dic₈₆	semidirect
310. C₂×C₈	C₂×Dic₈₆	semidirect

Maximal subgroups

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

subgroup	index
#19148408	2
#19145139	2
#19145127	2
#19145125	2
#19145123	2
#19145107	2
C₂×#14583296	2
#19143190	2
#19143188	2
#131×Dic₄₃	2
C₄₃×#2912	2
#60665	43