Subgroups — #25105367

Order 11008.

750 conjugacy classes of subgroups.

The subgroup lattice has 358 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#14583296	C₂	1	1	yes	maximal, characteristic
2	#19137342	C₂	1	2	yes	maximal
3	#19135566	C₂	1	2	yes	maximal
4	#19135338	C₂	1	1	yes	maximal, characteristic
5	#19135320	C₂	1	2	yes	maximal
6	#19134758	C₂	1	1	yes	maximal, characteristic
7	#19134495	C₂	1	1	yes	maximal, characteristic
8	#19134494	C₂	1	1	yes	maximal, characteristic
9	#19134483	C₂	1	2	yes	maximal
10	#19134283	C₂	1	1	yes	maximal, characteristic
11	C₄₃×#1944	C₂	1	1	yes	maximal, characteristic
12	C₂×#13063324	C₂²	1	2	yes	—
13	#14583296	C₄	1	8	yes	complemented
14	C₂×#13063259	C₂²	1	1	yes	characteristic
15	C₂×#13063257	C₂²	1	1	yes	characteristic
16	C₂×#13063256	C₂²	1	2	yes	—
17	C₂×#13063256	C₂²	1	2	yes	—
18	C₂×#13063235	C₂²	1	1	yes	characteristic
19	C₂×#13063235	C₂²	1	1	yes	characteristic
20	C₂×C₄×C₄₃⋊C₈	C₂²	1	1	yes	characteristic
21	C₂×C₈×Dic₄₃	C₂²	1	1	yes	characteristic
22	C₂×C₈×Dic₄₃	C₂²	1	1	yes	characteristic
23	#14582804	—	2	2	no	—
24	#14582803	—	2	2	no	—
25	#14582797	C₂²	1	2	yes	—
26	#14582781	—	2	2	no	—
27	#14582780	—	2	2	no	—
28	#14582764	C₂²	1	2	yes	—
29	C₄⋊C₄×Dic₄₃	C₂²	1	2	yes	—
30	C₄⋊C₄×Dic₄₃	C₂²	1	2	yes	—
31	#14582745	C₂²	1	1	yes	characteristic
32	#14582662	C₂²	1	2	yes	—
33	#14582661	C₂²	1	2	yes	—
34	#14582661	C₂²	1	2	yes	—
35	C₂×#107×C₄₃	C₂²	1	1	yes	characteristic
36	C₂×#106×C₄₃	C₂²	1	1	yes	characteristic
37	C₂×#97×C₄₃	C₂²	1	1	yes	characteristic
38	C₄₃×#389	C₂²	1	2	yes	—
39	C₄₃×#340	C₂²	1	2	yes	—
40	C₂²×Dic₈₆	—	2	2	no	—
41	C₂²×C₄₃⋊C₈	C₂³	1	1	yes	characteristic
42	C₂²×C₄₃⋊C₈	C₂³	1	1	yes	characteristic
43	#13063324	—	2	4	no	—
44	#13063324	C₂×C₄	1	8	yes	—
45	C₂×C₄⋊Dic₄₃	C₂³	1	2	yes	—
46	C₂×C₄⋊Dic₄₃	C₂³	1	2	yes	—
47	C₂×Dic₄₃⋊C₄	—	2	2	no	—
48	C₂×Dic₄₃⋊C₄	—	2	2	no	—
49	C₂×C₄×Dic₄₃	—	2	2	no	—
50	C₂×C₄×Dic₄₃	—	2	2	no	—
51	C₂×C₄×Dic₄₃	C₂³	1	1	yes	characteristic
52	C₂×C₄×Dic₄₃	C₂³	1	1	yes	characteristic
53	#13063259	—	2	2	no	—
54	#13063257	C₂×C₄	1	4	yes	—
55	#13063256	—	2	4	no	—
56	#13063256	C₂×C₄	1	8	yes	—
57	#13063235	—	2	2	no	—
58	#13063235	Q₈	1	4	yes	complemented
59	C₄×C₄₃⋊C₈	—	2	2	no	—
60	C₈×Dic₄₃	C₂×C₄	1	4	yes	—
61	C₈×Dic₄₃	Q₈	1	4	yes	complemented
62	C₄₃⋊#95	—	2	2	no	—
63	C₄₃⋊#95	—	2	2	no	—
64	C₄₃⋊#95	—	2	2	no	—
65	C₄₃⋊#95	—	2	2	no	—
66	C₄₃⋊#95	—	2	2	no	—
67	C₄₃⋊#95	D₄	1	2	yes	—
68	C₄₃⋊#95	D₄	1	2	yes	—
69	C₂²×C₈×C₄₃	C₂³	1	1	yes	characteristic
70	C₂²×C₈×C₄₃	C₂³	1	1	yes	characteristic
71	C₂×C₄⋊C₄×C₄₃	—	2	2	no	—
72	C₂×C₄⋊C₄×C₄₃	C₂³	1	2	yes	—
73	C₂×C₄⋊C₄×C₄₃	C₂³	1	2	yes	—
74	C₂×C₄²×C₄₃	C₂³	1	1	yes	characteristic
75	#107×C₄₃	—	2	2	no	—
76	#106×C₄₃	C₂×C₄	1	4	yes	—
77	#97×C₄₃	—	2	2	no	—
78	#95×C₄₃	—	2	2	no	—
79	C₂²×C₄×C₄₃	—	2	2	no	—
80	C₂²×C₄×C₄₃	—	2	2	no	—
81	C₂²×C₄×C₄₃	C₂×D₄	1	1	yes	characteristic
82	C₂²×C₄×C₄₃	C₂×D₄	1	1	yes	characteristic
83	C₂²×C₄×C₄₃	C₂⁴	1	1	yes	characteristic
84	C₂²×Dic₄₃	—	2	2	no	—
85	C₂²×Dic₄₃	—	2	2	no	—
86	C₂²×Dic₄₃	C₂×D₄	1	2	yes	—
87	C₂²×Dic₄₃	C₂×D₄	1	2	yes	—
88	C₂×Dic₈₆	—	2	8	no	—
89	C₂×Dic₈₆	—	8	2	no	—
90	C₂×Dic₈₆	—	8	2	no	—
91	C₂×C₈×C₄₃	—	2	1	no	—
92	C₂×C₈×C₄₃	—	2	1	no	—
93	C₂×C₈×C₄₃	C₂²×C₄	1	2	yes	—
94	C₂×C₈×C₄₃	C₂×Q₈	1	2	yes	—
95	C₂×C₈×C₄₃	C₂×Q₈	1	2	yes	—
96	C₂×C₈×C₄₃	#41	1	2	yes	—
97	C₄⋊C₄×C₄₃	—	2	2	no	—
98	C₄⋊C₄×C₄₃	—	2	2	no	—
99	C₄⋊C₄×C₄₃	—	2	2	no	—
100	C₄⋊C₄×C₄₃	—	8	2	no	—
101	C₄⋊C₄×C₄₃	C₂²×C₄	1	4	yes	—
102	C₄²×C₄₃	—	2	1	no	—
103	C₄²×C₄₃	—	2	1	no	—
104	C₄⋊Dic₄₃	—	2	2	no	—
105	C₄⋊Dic₄₃	—	2	2	no	—
106	C₄⋊Dic₄₃	—	2	2	no	—
107	C₄⋊Dic₄₃	C₂²×C₄	1	4	yes	—
108	Dic₄₃⋊C₄	—	2	8	no	—
109	Dic₄₃⋊C₄	—	8	2	no	—
110	C₄×Dic₄₃	—	8	2	no	—
111	C₄×Dic₄₃	—	8	2	no	—
112	C₄×Dic₄₃	C₂²×C₄	1	2	yes	—
113	C₄×Dic₄₃	C₂×Q₈	1	2	yes	—
114	C₄×Dic₄₃	C₂×Q₈	1	2	yes	—
115	C₄×Dic₄₃	#41	1	2	yes	—
116	C₂×C₄₃⋊C₈	—	2	1	no	—
117	C₂×C₄₃⋊C₈	—	2	1	no	—
118	C₂×C₄₃⋊C₈	C₂²×C₄	1	2	yes	—
119	C₂×C₄₃⋊C₈	C₂×Q₈	1	2	yes	—
120	C₂×C₄₃⋊C₈	C₂×Q₈	1	2	yes	—
121	C₂×C₄₃⋊C₈	#41	1	2	yes	—
122	C₂³×C₄₃	C₂²×D₄	1	1	yes	characteristic
123	C₂×C₄×C₄₃	—	2	4	no	—
124	C₂×C₄×C₄₃	—	2	1	no	—
125	C₂×C₄×C₄₃	—	2	1	no	—
126	C₂×C₄×C₄₃	—	2	2	no	—
127	C₂×C₄×C₄₃	—	2	2	no	—
128	C₂×C₄×C₄₃	—	4	2	no	—
129	C₂×C₄×C₄₃	—	4	2	no	—
130	C₂×C₄×C₄₃	—	4	2	no	—
131	C₂×C₄×C₄₃	—	4	2	no	—
132	C₂×C₄×C₄₃	—	4	2	no	—
133	C₂×C₄×C₄₃	C₂³×C₄	1	1	yes	characteristic
134	C₂×C₄×C₄₃	C₂²×Q₈	1	1	yes	characteristic
135	C₂×C₄×C₄₃	C₂²×Q₈	1	1	yes	characteristic
136	C₂×C₄×C₄₃	C₂×#41	1	1	yes	characteristic
137	C₂×C₄×C₄₃	C₂×#41	1	1	yes	characteristic
138	C₂×C₄×C₄₃	C₂×#41	1	1	yes	characteristic
139	C₂×Dic₄₃	—	2	2	no	—
140	C₂×Dic₄₃	—	2	2	no	—
141	C₂×Dic₄₃	—	2	2	no	—
142	C₂×Dic₄₃	—	2	2	no	—
143	C₂×Dic₄₃	—	2	4	no	—
144	C₂×Dic₄₃	—	4	2	no	—
145	C₂×Dic₄₃	—	4	2	no	—
146	C₂×Dic₄₃	—	4	2	no	—
147	C₂×Dic₄₃	—	4	2	no	—
148	C₂×Dic₄₃	—	4	2	no	—
149	C₂×Dic₄₃	C₄×D₄	1	4	yes	—
150	C₂×Dic₄₃	C₄⋊Q₈	1	4	yes	—
151	Dic₈₆	—	8	8	no	—
152	C₈×C₄₃	—	2	2	no	—
153	C₈×C₄₃	C₄×Q₈	1	4	yes	—
154	C₄₃⋊C₈	—	2	2	no	—
155	C₄₃⋊C₈	C₄×Q₈	1	4	yes	—
156	#21724	—	43	1	no	maximal, Sylow
157	C₂²×C₄₃	C₂×C₄×D₄	1	1	yes	characteristic
158	C₂²×C₄₃	C₂×C₄⋊Q₈	1	1	yes	characteristic
159	C₂²×C₄₃	#532	1	1	yes	characteristic
160	C₂²×C₄₃	#577	1	2	yes	—
161	C₂²×C₄₃	#577	1	2	yes	—
162	C₄×C₄₃	—	4	1	no	—
163	C₄×C₄₃	—	4	1	no	—
164	C₄×C₄₃	—	4	4	no	—
165	C₄×C₄₃	—	8	2	no	—
166	C₄×C₄₃	C₂×C₄×Q₈	1	1	yes	characteristic
167	C₄×C₄₃	C₂×C₄×Q₈	1	1	yes	characteristic
168	C₄×C₄₃	#533	1	1	yes	characteristic
169	C₄×C₄₃	#533	1	1	yes	characteristic
170	Dic₄₃	—	2	4	no	—
171	Dic₄₃	—	2	4	no	—
172	Dic₄₃	—	4	4	no	—
173	Dic₄₃	—	8	2	no	—
174	C₂×#498	—	43	1	no	—
175	#2331	—	43	2	no	—
176	#1993	—	43	2	no	—
177	#1944	—	43	1	no	—
178	#1944	D₄₃	1	1	yes	characteristic
179	#1941	—	43	2	no	—
180	#1846	—	43	1	no	—
181	#1776	—	43	1	no	—
182	#1776	—	43	1	no	—
183	C₄×#106	—	43	2	no	—
184	#1726	—	43	1	no	—
185	C₂×C₄₃	C₄×C₄⋊Q₈	1	1	yes	characteristic
186	C₂×C₄₃	#2949	1	2	yes	—
187	C₂×C₄₃	#3163	1	2	yes	—
188	C₂×C₄₃	#3173	1	1	yes	characteristic
189	C₂×C₄₃	#3173	1	1	yes	characteristic
190	C₂×C₄⋊Q₈	—	43	2	no	—
191	#498	—	43	8	no	—
192	C₂×#107	—	43	1	no	—
193	C₂×#107	C₂×D₄₃	1	1	yes	characteristic
194	C₂×#106	—	43	1	no	—
195	C₂×#106	—	43	2	no	—
196	C₂×#106	—	43	2	no	—
197	C₂×#106	C₂×D₄₃	1	1	yes	characteristic
198	C₂×#97	—	43	1	no	—
199	C₂×#97	—	43	1	no	—
200	C₂×#97	C₂×D₄₃	1	1	yes	characteristic
201	C₂×C₄×C₈	—	43	1	no	—
202	C₂×C₄×C₈	—	43	1	no	—
203	C₂×C₄×C₈	—	43	1	no	—
204	#391	—	86	2	no	—
205	#391	—	86	2	no	—
206	#389	—	43	2	no	—
207	#389	C₂×D₄₃	1	2	yes	—
208	#387	—	86	2	no	—
209	#387	—	86	2	no	—
210	#382	—	43	2	no	—
211	C₄×C₄⋊C₄	—	43	2	no	—
212	C₄×C₄⋊C₄	—	43	2	no	—
213	#376	—	43	1	no	—
214	#340	—	43	2	no	—
215	#340	—	43	2	no	—
216	#340	—	43	2	no	—
217	#340	C₂×D₄₃	1	2	yes	—
218	C₄₃	#21724	1	1	yes	characteristic, complemented, Sylow
219	C₂²×Q₈	—	86	2	no	—
220	C₂²×C₈	—	43	1	no	—
221	C₂²×C₈	—	43	1	no	—
222	C₂²×C₈	C₂²×D₄₃	1	1	yes	characteristic
223	C₂²×C₈	C₂²×D₄₃	1	1	yes	characteristic
224	C₄⋊Q₈	—	43	8	no	—
225	C₄⋊Q₈	—	86	4	no	—
226	C₂×C₄⋊C₄	—	2	2	no	—
227	C₂×C₄⋊C₄	—	43	2	no	—
228	C₂×C₄⋊C₄	—	43	2	no	—
229	C₂×C₄⋊C₄	—	86	2	no	—
230	C₂×C₄⋊C₄	—	86	2	no	—
231	C₂×C₄⋊C₄	C₂²×D₄₃	1	2	yes	—
232	C₂×C₄⋊C₄	C₂²×D₄₃	1	2	yes	—
233	C₂×C₄²	—	43	1	no	—
234	C₂×C₄²	—	43	1	no	—
235	C₂×C₄²	—	86	2	no	—
236	C₂×C₄²	—	86	2	no	—
237	C₂×C₄²	C₂²×D₄₃	1	1	yes	characteristic
238	#107	—	2	2	no	—
239	#107	—	86	2	no	—
240	#106	—	43	4	no	—
241	#106	—	43	8	no	—
242	#106	—	86	4	no	—
243	#106	C₄×D₄₃	1	4	yes	—
244	#97	—	2	2	no	—
245	#97	—	43	4	no	—
246	#97	—	86	2	no	—
247	C₄×C₈	—	43	4	no	—
248	C₄×C₈	—	43	4	no	—
249	C₄×C₈	—	86	2	no	—
250	#95	—	2	2	no	—
251	#95	—	43	2	no	—
252	#95	—	43	2	no	—
253	#95	—	86	2	no	—
254	#95	—	86	2	no	—
255	#95	—	86	2	no	—
256	#95	—	86	2	no	—
257	#95	—	86	2	no	—
258	C₂×Q₈	—	86	8	no	—
259	C₂×Q₈	—	344	2	no	—
260	C₂×Q₈	—	344	2	no	—
261	C₂²×C₄	—	2	2	no	—
262	C₂²×C₄	—	2	2	no	—
263	C₂²×C₄	—	43	2	no	—
264	C₂²×C₄	—	43	2	no	—
265	C₂²×C₄	—	86	2	no	—
266	C₂²×C₄	—	86	2	no	—
267	C₂²×C₄	D₄×D₄₃	1	1	yes	characteristic
268	C₂²×C₄	D₄×D₄₃	1	1	yes	characteristic
269	C₂²×C₄	C₂³×D₄₃	1	1	yes	characteristic
270	C₂×C₈	—	2	1	no	—
271	C₂×C₈	—	2	1	no	—
272	C₂×C₈	—	43	2	no	—
273	C₂×C₈	—	43	2	no	—
274	C₂×C₈	—	43	2	no	—
275	C₂×C₈	—	43	2	no	—
276	C₂×C₈	—	86	1	no	—
277	C₂×C₈	—	86	1	no	—
278	C₂×C₈	C₂×C₄×D₄₃	1	2	yes	—
279	C₂×C₈	#10621900	1	2	yes	—
280	C₂×C₈	Q₈×D₄₃	1	2	yes	—
281	C₂×C₈	Q₈×D₄₃	1	2	yes	—
282	C₄⋊C₄	—	2	2	no	—
283	C₄⋊C₄	—	2	2	no	—
284	C₄⋊C₄	—	2	2	no	—
285	C₄⋊C₄	—	8	2	no	—
286	C₄⋊C₄	—	43	4	no	—
287	C₄⋊C₄	—	86	8	no	—
288	C₄⋊C₄	—	86	2	no	—
289	C₄⋊C₄	—	86	2	no	—
290	C₄⋊C₄	—	86	2	no	—
291	C₄⋊C₄	—	344	2	no	—
292	C₄⋊C₄	C₂×C₄×D₄₃	1	4	yes	—
293	C₄²	—	2	1	no	—
294	C₄²	—	2	1	no	—
295	C₄²	—	43	2	no	—
296	C₄²	—	43	2	no	—
297	C₄²	—	43	2	no	—
298	C₄²	—	43	2	no	—
299	C₄²	—	344	2	no	—
300	C₄²	—	344	2	no	—
301	C₂³	C₂×D₄×D₄₃	1	1	yes	characteristic
302	Q₈	—	344	8	no	—
303	C₂×C₄	—	2	4	no	—
304	C₂×C₄	—	2	1	no	—
305	C₂×C₄	—	2	1	no	—
306	C₂×C₄	—	2	2	no	—
307	C₂×C₄	—	2	2	no	—
308	C₂×C₄	—	4	2	no	—
309	C₂×C₄	—	4	2	no	—
310	C₂×C₄	—	4	2	no	—
311	C₂×C₄	—	4	2	no	—
312	C₂×C₄	—	4	2	no	—
313	C₂×C₄	—	43	4	no	—
314	C₂×C₄	—	43	4	no	—
315	C₂×C₄	—	86	2	no	—
316	C₂×C₄	—	86	2	no	—
317	C₂×C₄	—	86	2	no	—
318	C₂×C₄	—	86	2	no	—
319	C₂×C₄	—	86	4	no	—
320	C₂×C₄	—	172	2	no	—
321	C₂×C₄	—	172	2	no	—
322	C₂×C₄	—	172	2	no	—
323	C₂×C₄	—	172	2	no	—
324	C₂×C₄	—	172	2	no	—
325	C₂×C₄	C₂²×C₄×D₄₃	1	1	yes	characteristic
326	C₂×C₄	C₂×Q₈×D₄₃	1	1	yes	characteristic
327	C₂×C₄	C₂×Q₈×D₄₃	1	1	yes	characteristic
328	C₂×C₄	#41×D₄₃	1	1	yes	characteristic
329	C₂×C₄	#41×D₄₃	1	1	yes	characteristic
330	C₂×C₄	C₂×#10621900	1	1	yes	characteristic
331	C₈	—	2	2	no	—
332	C₈	—	43	4	no	—
333	C₈	—	86	2	no	—
334	C₈	#13063290	1	4	yes	—
335	C₂²	C₄×D₄×D₄₃	1	1	yes	characteristic
336	C₂²	C₄⋊Q₈×D₄₃	1	1	yes	characteristic
337	C₂²	#14583444	1	1	yes	characteristic
338	C₂²	#14583708	1	2	yes	—
339	C₂²	#14583709	1	2	yes	—
340	C₄	—	4	4	no	—
341	C₄	—	4	1	no	—
342	C₄	—	4	1	no	—
343	C₄	—	8	2	no	—
344	C₄	—	86	4	no	—
345	C₄	—	86	4	no	—
346	C₄	—	344	2	no	—
347	C₄	—	172	4	no	—
348	C₄	C₄×Q₈×D₄₃	1	1	yes	characteristic
349	C₄	C₂×#13063290	1	1	yes	characteristic
350	C₄	#14583451	1	1	yes	characteristic
351	C₄	#14583455	1	1	yes	characteristic
352	C₂	#19137166	1	1	yes	characteristic
353	C₂	#19143451	1	2	yes	—
354	C₂	#19145122	1	2	yes	—
355	C₂	#19145207	1	1	yes	characteristic
356	C₂	#19145213	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	C₂×D₄×D₄₃
commutator	C₂×C₄×C₄₃	C₂³×C₄
Frattini	C₂²×C₄	C₂³×D₄₃
Fitting	C₄₃×#1944	C₂
socle, prime radical	C₂³×C₄₃	C₂²×D₄
cosocle	C₂²×C₄×C₄₃	C₂⁴
hypercenter	#1944	D₄₃
hypocenter	C₄₃	#21724
LCS	G ▷ C₂×C₄×C₄₃ ▷ C₂×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #1944

Complements

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

N	H ≅ G/N	type
13. #14583296	C₄	semidirect
58. #13063235	Q₈	semidirect
61. C₈×Dic₄₃	Q₈	semidirect
218. C₄₃	#21724	semidirect

Maximal subgroups

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

subgroup	index
C₂×#14583296	2
#19137342	2
#19135566	2
#19135338	2
#19135320	2
#19134758	2
#19134495	2
#19134494	2
#19134483	2
#19134283	2
C₄₃×#1944	2
#21724	43