Subgroups — #24987132

Order 11008.

400 conjugacy classes of subgroups.

The subgroup lattice has 368 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#14582961	C₂	1	1	yes	maximal, characteristic
2	#19132907	C₂	1	1	yes	maximal, characteristic
3	#19132907	C₂	1	1	yes	maximal, characteristic
4	#19132888	C₂	1	1	yes	maximal, characteristic, complemented
5	#19132879	C₂	1	1	yes	maximal, characteristic, complemented
6	#19132440	C₂	1	1	yes	maximal, characteristic, complemented
7	C₄₃×#1281	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×#13063282	C₂²	1	1	yes	characteristic
9	#14582961	—	2	1	no	—
10	#14582961	—	2	1	no	—
11	#14582961	—	2	1	no	—
12	#14582961	—	2	1	no	—
13	#14582952	C₂²	1	1	yes	characteristic
14	#14582952	C₂²	1	1	yes	characteristic
15	C₂×Dic₄₃⋊C₈	C₄	1	1	yes	characteristic
16	C₂×Dic₄₃⋊C₈	C₄	1	1	yes	characteristic
17	C₂×(C₂×D₄₃)⋊C₈	C₄	1	1	yes	characteristic
18	C₂×(C₂×D₄₃)⋊C₈	C₄	1	1	yes	characteristic
19	C₂×C₄×C₄₃⋊C₈	C₂²	1	1	yes	characteristic
20	#14582624	C₄	1	2	yes	—
21	#14582624	C₄	1	2	yes	—
22	#14582622	C₄	1	2	yes	—
23	#14582622	C₄	1	2	yes	—
24	#14582621	—	2	1	no	—
25	#14582621	—	2	1	no	—
26	#14582620	—	2	1	no	—
27	C₈×C₄₃⋊C₈	—	2	1	no	—
28	C₄₃×#431	C₂²	1	1	yes	characteristic
29	C₄₃×#431	C₂²	1	1	yes	characteristic
30	C₂×#97×C₄₃	C₂²	1	1	yes	characteristic
31	C₂²×C₄×D₄₃	C₂×C₄	1	1	yes	characteristic
32	C₂²×C₄₃⋊C₈	C₂×C₄	1	1	yes	characteristic
33	C₂²×C₄₃⋊C₈	C₂×C₄	1	1	yes	characteristic
34	#13063282	—	2	1	no	—
35	#13063282	—	2	1	no	—
36	#13063282	—	2	1	no	—
37	#13063282	—	2	1	no	—
38	C₂×Dic₄₃⋊C₄	C₂×C₄	1	1	yes	characteristic
39	C₂×Dic₄₃⋊C₄	C₂×C₄	1	1	yes	characteristic
40	C₂×(C₂×D₄₃)⋊C₄	C₂×C₄	1	1	yes	characteristic
41	C₂×(C₂×D₄₃)⋊C₄	C₂×C₄	1	1	yes	characteristic
42	C₂×C₄×Dic₄₃	C₂×C₄	1	1	yes	characteristic
43	C₄⋊(C₄₃⋊C₈)	—	2	1	no	—
44	C₄⋊(C₄₃⋊C₈)	—	2	1	no	—
45	Dic₄₃⋊C₈	—	2	1	no	—
46	Dic₄₃⋊C₈	—	2	1	no	—
47	Dic₄₃⋊C₈	—	2	1	no	—
48	Dic₄₃⋊C₈	—	2	1	no	—
49	C₂²⋊(C₄₃⋊C₈)	C₂×C₄	1	2	yes	—
50	C₂²⋊(C₄₃⋊C₈)	C₂×C₄	1	2	yes	—
51	(C₂×D₄₃)⋊C₈	—	2	1	no	—
52	(C₂×D₄₃)⋊C₈	—	2	1	no	—
53	(C₂×D₄₃)⋊C₈	—	2	1	no	—
54	(C₂×D₄₃)⋊C₈	—	2	1	no	—
55	C₄×C₄₃⋊C₈	—	2	1	no	—
56	C₄×C₄₃⋊C₈	—	2	1	no	—
57	C₄×C₄₃⋊C₈	D₄	1	1	yes	characteristic
58	C₄×C₄₃⋊C₈	D₄	1	1	yes	characteristic
59	C₄×C₄₃⋊C₈	D₄	1	1	yes	characteristic
60	C₄×C₄₃⋊C₈	D₄	1	1	yes	characteristic
61	C₂²×C₈×C₄₃	C₂×C₄	1	1	yes	characteristic
62	C₂²×C₈×C₄₃	C₂×C₄	1	1	yes	characteristic
63	C₂×C₄²×C₄₃	C₂³	1	1	yes	characteristic
64	C₄⋊C₈×C₄₃	—	2	1	no	—
65	C₄⋊C₈×C₄₃	—	2	1	no	—
66	C₂²⋊C₈×C₄₃	C₂×C₄	1	2	yes	—
67	C₂²⋊C₈×C₄₃	C₂×C₄	1	2	yes	—
68	#97×C₄₃	—	2	1	no	—
69	#97×C₄₃	—	2	1	no	—
70	C₄×C₈×C₄₃	—	2	1	no	—
71	C₄×C₈×C₄₃	—	2	1	no	—
72	C₂³×D₄₃	C₄²	1	1	yes	characteristic
73	C₂²×C₄×C₄₃	C₂²×C₄	1	1	yes	characteristic
74	C₂²×C₄×C₄₃	C₂²×C₄	1	1	yes	characteristic
75	C₂²×C₄×C₄₃	C₂²×C₄	1	1	yes	characteristic
76	C₂²×Dic₄₃	C₄²	1	1	yes	characteristic
77	C₂²×Dic₄₃	C₄²	1	1	yes	characteristic
78	C₂²×Dic₄₃	C₄²	1	1	yes	characteristic
79	C₂×C₄×D₄₃	—	2	1	no	—
80	C₂×C₄×D₄₃	—	2	1	no	—
81	C₂×C₄×D₄₃	—	2	1	no	—
82	C₂×C₄×D₄₃	—	2	1	no	—
83	C₂×C₄×D₄₃	—	2	1	no	—
84	C₂×C₄×D₄₃	—	2	1	no	—
85	C₂×C₈×C₄₃	—	2	1	no	—
86	C₂×C₈×C₄₃	—	2	1	no	—
87	C₂×C₈×C₄₃	—	2	2	no	—
88	C₂×C₈×C₄₃	—	2	2	no	—
89	C₂×C₈×C₄₃	—	2	1	no	—
90	C₂×C₈×C₄₃	—	2	1	no	—
91	C₂×C₈×C₄₃	—	2	1	no	—
92	C₂×C₈×C₄₃	—	2	1	no	—
93	C₄²×C₄₃	C₂×D₄	1	1	yes	characteristic
94	C₄²×C₄₃	C₂×D₄	1	1	yes	characteristic
95	C₄²×C₄₃	#41	1	1	yes	characteristic
96	C₄²×C₄₃	#41	1	1	yes	characteristic
97	(C₂×D₄₃)⋊C₄	—	2	1	no	—
98	(C₂×D₄₃)⋊C₄	—	2	1	no	—
99	(C₂×D₄₃)⋊C₄	—	2	1	no	—
100	(C₂×D₄₃)⋊C₄	—	2	1	no	—
101	Dic₄₃⋊C₄	—	2	1	no	—
102	Dic₄₃⋊C₄	—	2	1	no	—
103	Dic₄₃⋊C₄	—	2	1	no	—
104	Dic₄₃⋊C₄	—	2	1	no	—
105	C₄×Dic₄₃	—	2	1	no	—
106	C₄×Dic₄₃	—	2	1	no	—
107	C₂×C₄₃⋊C₈	—	2	1	no	—
108	C₂×C₄₃⋊C₈	—	2	1	no	—
109	C₂×C₄₃⋊C₈	—	2	1	no	—
110	C₂×C₄₃⋊C₈	—	2	1	no	—
111	C₂×C₄₃⋊C₈	—	2	2	no	—
112	C₂×C₄₃⋊C₈	—	2	2	no	—
113	C₂×C₄₃⋊C₈	C₂²⋊C₄	1	1	yes	characteristic
114	C₂×C₄₃⋊C₈	C₂²⋊C₄	1	1	yes	characteristic
115	C₂×C₄₃⋊C₈	C₂²⋊C₄	1	1	yes	characteristic
116	C₂×C₄₃⋊C₈	C₂²⋊C₄	1	1	yes	characteristic
117	C₂³×C₄₃	C₂×C₄²	1	1	yes	characteristic
118	C₂²×D₄₃	—	2	1	no	—
119	C₂²×D₄₃	—	2	1	no	—
120	C₂²×D₄₃	—	2	1	no	—
121	C₂²×D₄₃	—	4	1	no	—
122	C₂²×D₄₃	—	4	1	no	—
123	C₂×C₄×C₄₃	—	4	1	no	—
124	C₂×C₄×C₄₃	—	4	1	no	—
125	C₂×C₄×C₄₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
126	C₂×C₄×C₄₃	#117	1	1	yes	characteristic
127	C₂×C₄×C₄₃	C₄×D₄	1	1	yes	characteristic
128	C₂×C₄×C₄₃	C₄×D₄	1	1	yes	characteristic
129	C₂×C₄×C₄₃	C₄×D₄	1	1	yes	characteristic
130	C₂×C₄×C₄₃	C₄×D₄	1	1	yes	characteristic
131	C₂×C₄×C₄₃	#131	1	1	yes	characteristic
132	C₂×C₄×C₄₃	#131	1	1	yes	characteristic
133	C₂×C₄×C₄₃	#131	1	1	yes	characteristic
134	C₂×C₄×C₄₃	#131	1	1	yes	characteristic
135	C₂×Dic₄₃	—	2	1	no	—
136	C₂×Dic₄₃	—	2	1	no	—
137	C₂×Dic₄₃	—	2	1	no	—
138	C₂×Dic₄₃	—	2	1	no	—
139	C₂×Dic₄₃	—	2	1	no	—
140	C₂×Dic₄₃	—	4	1	no	—
141	C₂×Dic₄₃	—	4	1	no	—
142	C₄×D₄₃	—	4	1	no	—
143	C₄×D₄₃	—	4	1	no	—
144	C₄×D₄₃	—	4	1	no	—
145	C₄×D₄₃	—	4	1	no	—
146	C₈×C₄₃	—	4	2	no	—
147	C₈×C₄₃	—	4	2	no	—
148	C₈×C₄₃	—	4	1	no	—
149	C₈×C₄₃	—	4	1	no	—
150	C₄₃⋊C₈	—	2	1	no	—
151	C₄₃⋊C₈	—	2	1	no	—
152	C₄₃⋊C₈	—	2	1	no	—
153	C₄₃⋊C₈	—	2	1	no	—
154	C₄₃⋊C₈	—	4	2	no	—
155	C₄₃⋊C₈	—	4	2	no	—
156	#8158	—	43	1	no	maximal, Sylow
157	C₂²×C₄₃	—	2	1	no	—
158	C₂²×C₄₃	—	2	1	no	—
159	C₂²×C₄₃	C₄×C₂²⋊C₄	1	1	yes	characteristic
160	C₂²×C₄₃	#405	1	1	yes	characteristic
161	C₂²×C₄₃	#405	1	1	yes	characteristic
162	C₂×D₄₃	—	4	1	no	—
163	C₂×D₄₃	—	4	1	no	—
164	C₂×D₄₃	—	4	1	no	—
165	C₂×D₄₃	—	8	1	no	—
166	C₂×D₄₃	—	8	1	no	—
167	C₄×C₄₃	—	2	1	no	—
168	C₄×C₄₃	—	2	1	no	—
169	C₄×C₄₃	—	2	1	no	—
170	C₄×C₄₃	—	2	1	no	—
171	C₄×C₄₃	#408	1	1	yes	characteristic
172	C₄×C₄₃	#408	1	1	yes	characteristic
173	C₄×C₄₃	#433	1	1	yes	characteristic
174	C₄×C₄₃	#433	1	1	yes	characteristic
175	Dic₄₃	—	4	1	no	—
176	Dic₄₃	—	4	1	no	—
177	Dic₄₃	—	4	1	no	—
178	C₂×#433	—	43	1	no	—
179	#1458	—	43	1	no	—
180	#1458	—	43	1	no	—
181	#1452	—	43	1	no	—
182	#1450	—	43	1	no	—
183	#1281	—	43	1	no	—
184	#1281	D₄₃	1	1	yes	characteristic, complemented
185	C₂×C₄₃	—	2	1	no	—
186	C₂×C₄₃	—	2	1	no	—
187	C₂×C₄₃	#1750	1	1	yes	characteristic
188	C₂×C₄₃	#1757	1	1	yes	characteristic
189	C₂×C₄₃	#1757	1	1	yes	characteristic
190	D₄₃	—	8	1	no	—
191	C₂×#117	—	43	1	no	—
192	#433	—	86	1	no	—
193	#433	—	86	1	no	—
194	#433	—	86	1	no	—
195	#433	—	86	1	no	—
196	#431	—	43	1	no	—
197	#431	—	43	1	no	—
198	#431	C₂×D₄₃	1	1	yes	characteristic
199	#431	C₂×D₄₃	1	1	yes	characteristic
200	C₂×C₄⋊C₈	—	43	1	no	—
201	C₂×C₄⋊C₈	—	43	1	no	—
202	C₂×C₂²⋊C₈	—	43	1	no	—
203	C₂×C₂²⋊C₈	—	43	1	no	—
204	C₂×#97	C₂×D₄₃	1	1	yes	characteristic
205	C₂×C₄×C₈	—	43	1	no	—
206	#324	—	43	2	no	—
207	#324	—	43	2	no	—
208	#323	—	43	2	no	—
209	#323	—	43	2	no	—
210	#322	—	86	1	no	—
211	#322	—	86	1	no	—
212	#322	—	86	1	no	—
213	C₈²	—	86	1	no	—
214	C₄₃	#8158	1	1	yes	characteristic, complemented, Sylow
215	C₂³×C₄	—	43	1	no	—
216	C₂²×C₈	—	43	1	no	—
217	C₂²×C₈	—	43	1	no	—
218	C₂²×C₈	C₄×D₄₃	1	1	yes	characteristic
219	C₂²×C₈	C₄×D₄₃	1	1	yes	characteristic
220	#117	—	86	1	no	—
221	#117	—	86	1	no	—
222	#117	—	86	1	no	—
223	#117	—	86	1	no	—
224	C₂×C₄⋊C₄	—	43	1	no	—
225	C₂×C₄⋊C₄	—	43	1	no	—
226	C₂×C₂²⋊C₄	—	43	1	no	—
227	C₂×C₂²⋊C₄	—	43	1	no	—
228	C₂×C₄²	—	43	1	no	—
229	C₂×C₄²	C₂²×D₄₃	1	1	yes	characteristic
230	C₄⋊C₈	—	2	1	no	—
231	C₄⋊C₈	—	2	1	no	—
232	C₄⋊C₈	—	86	1	no	—
233	C₄⋊C₈	—	86	1	no	—
234	C₄⋊C₈	—	86	1	no	—
235	C₄⋊C₈	—	86	1	no	—
236	C₄⋊C₈	—	86	1	no	—
237	C₄⋊C₈	—	86	1	no	—
238	C₂²⋊C₈	—	43	2	no	—
239	C₂²⋊C₈	—	43	2	no	—
240	C₂²⋊C₈	—	86	1	no	—
241	C₂²⋊C₈	—	86	1	no	—
242	C₂²⋊C₈	—	86	1	no	—
243	C₂²⋊C₈	—	86	1	no	—
244	C₂²⋊C₈	C₄×D₄₃	1	2	yes	—
245	C₂²⋊C₈	C₄×D₄₃	1	2	yes	—
246	#97	—	2	1	no	—
247	#97	—	2	1	no	—
248	C₄×C₈	—	2	1	no	—
249	C₄×C₈	—	2	1	no	—
250	C₄×C₈	—	43	1	no	—
251	C₄×C₈	—	43	1	no	—
252	C₄×C₈	—	43	1	no	—
253	C₄×C₈	—	43	1	no	—
254	C₄×C₈	—	86	1	no	—
255	C₄×C₈	—	86	1	no	—
256	C₂⁴	—	43	1	no	—
257	C₂²×C₄	—	43	1	no	—
258	C₂²×C₄	—	43	1	no	—
259	C₂²×C₄	—	43	1	no	—
260	C₂²×C₄	—	86	1	no	—
261	C₂²×C₄	—	86	1	no	—
262	C₂²×C₄	—	86	1	no	—
263	C₂²×C₄	—	86	1	no	—
264	C₂²×C₄	—	86	1	no	—
265	C₂²×C₄	—	86	1	no	—
266	C₂²×C₄	C₂×C₄×D₄₃	1	1	yes	characteristic
267	C₂²×C₄	C₂×C₄×D₄₃	1	1	yes	characteristic
268	C₂²×C₄	C₂×C₄×D₄₃	1	1	yes	characteristic
269	C₂×C₈	—	2	1	no	—
270	C₂×C₈	—	2	1	no	—
271	C₂×C₈	—	2	2	no	—
272	C₂×C₈	—	2	2	no	—
273	C₂×C₈	—	2	1	no	—
274	C₂×C₈	—	2	1	no	—
275	C₂×C₈	—	2	1	no	—
276	C₂×C₈	—	2	1	no	—
277	C₂×C₈	—	43	1	no	—
278	C₂×C₈	—	43	1	no	—
279	C₂×C₈	—	43	1	no	—
280	C₂×C₈	—	43	1	no	—
281	C₂×C₈	—	86	1	no	—
282	C₂×C₈	—	86	1	no	—
283	C₂×C₈	—	86	1	no	—
284	C₂×C₈	—	86	1	no	—
285	C₂×C₈	—	86	2	no	—
286	C₂×C₈	—	86	2	no	—
287	C₄⋊C₄	—	86	1	no	—
288	C₄⋊C₄	—	86	1	no	—
289	C₄⋊C₄	—	86	1	no	—
290	C₄⋊C₄	—	86	1	no	—
291	C₂²⋊C₄	—	86	1	no	—
292	C₂²⋊C₄	—	86	1	no	—
293	C₂²⋊C₄	—	86	1	no	—
294	C₂²⋊C₄	—	86	1	no	—
295	C₄²	—	86	1	no	—
296	C₄²	—	86	1	no	—
297	C₄²	D₄×D₄₃	1	1	yes	characteristic
298	C₄²	D₄×D₄₃	1	1	yes	characteristic
299	C₄²	#10621900	1	1	yes	characteristic
300	C₄²	#10621902	1	1	yes	characteristic
301	C₂³	—	86	1	no	—
302	C₂³	—	86	1	no	—
303	C₂³	—	86	1	no	—
304	C₂³	—	172	1	no	—
305	C₂³	—	172	1	no	—
306	C₂³	C₄²×D₄₃	1	1	yes	characteristic
307	C₂×C₄	—	4	1	no	—
308	C₂×C₄	—	4	1	no	—
309	C₂×C₄	—	86	1	no	—
310	C₂×C₄	—	86	1	no	—
311	C₂×C₄	—	86	1	no	—
312	C₂×C₄	—	86	1	no	—
313	C₂×C₄	—	86	1	no	—
314	C₂×C₄	—	172	1	no	—
315	C₂×C₄	—	172	1	no	—
316	C₂×C₄	—	172	1	no	—
317	C₂×C₄	—	172	1	no	—
318	C₂×C₄	—	172	1	no	—
319	C₂×C₄	—	172	1	no	—
320	C₂×C₄	C₂²⋊C₄×D₄₃	1	1	yes	characteristic
321	C₂×C₄	#13063283	1	1	yes	characteristic
322	C₂×C₄	(C₂²⋊D₄₃)⋊C₄	1	1	yes	characteristic
323	C₂×C₄	(C₂²⋊D₄₃)⋊C₄	1	1	yes	characteristic
324	C₂×C₄	#13063287	1	1	yes	characteristic
325	C₂×C₄	#13063287	1	1	yes	characteristic
326	C₂×C₄	#13063331	1	1	yes	characteristic
327	C₂×C₄	#13063331	1	1	yes	characteristic
328	C₂×C₄	#13063333	1	1	yes	characteristic
329	C₂×C₄	#13063333	1	1	yes	characteristic
330	C₈	—	4	2	no	—
331	C₈	—	4	2	no	—
332	C₈	—	4	1	no	—
333	C₈	—	4	1	no	—
334	C₈	—	86	1	no	—
335	C₈	—	86	1	no	—
336	C₈	—	86	1	no	—
337	C₈	—	86	1	no	—
338	C₈	—	172	2	no	—
339	C₈	—	172	2	no	—
340	C₂²	—	2	1	no	—
341	C₂²	—	2	1	no	—
342	C₂²	—	344	1	no	—
343	C₂²	—	344	1	no	—
344	C₂²	—	172	1	no	—
345	C₂²	—	172	1	no	—
346	C₂²	—	172	1	no	—
347	C₂²	#14582746	1	1	yes	characteristic
348	C₂²	#14582853	1	1	yes	characteristic
349	C₂²	#14582856	1	1	yes	characteristic
350	C₄	—	2	1	no	—
351	C₄	—	2	1	no	—
352	C₄	—	2	1	no	—
353	C₄	—	2	1	no	—
354	C₄	—	172	1	no	—
355	C₄	—	172	1	no	—
356	C₄	—	172	1	no	—
357	C₄	#14582864	1	1	yes	characteristic
358	C₄	#14582864	1	1	yes	characteristic
359	C₄	#14582959	1	1	yes	characteristic
360	C₄	#14582959	1	1	yes	characteristic
361	C₂	—	2	1	no	—
362	C₂	—	2	1	no	—
363	C₂	—	344	1	no	—
364	C₂	#19134360	1	1	yes	characteristic
365	C₂	#19134394	1	1	yes	characteristic
366	C₂	#19134394	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂²⋊C₄×D₄₃
commutator	C₂³×C₄₃	C₂×C₄²
Frattini	C₂×C₄²	C₂²×D₄₃
Fitting	C₄₃×#1281	C₂
socle	C₂²×C₄₃	C₄×C₂²⋊C₄
prime radical	C₂³×D₄₃	C₄²
cosocle	C₂×C₄²×C₄₃	C₂³
hypercenter	#1281	D₄₃
hypocenter	C₄₃	#8158
LCS	G ▷ C₂³×C₄₃ ▷ C₂×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #1281

Complements

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

N	H ≅ G/N	type
4. #19132888	C₂	semidirect
5. #19132879	C₂	semidirect
6. #19132440	C₂	semidirect
7. C₄₃×#1281	C₂	semidirect
184. #1281	D₄₃	semidirect
214. C₄₃	#8158	semidirect

Maximal subgroups

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

subgroup	index
C₂×#14582961	2
#19132907	2
#19132888	2
#19132879	2
#19132440	2
C₄₃×#1281	2
#8158	43