Subgroups — #25066512

Order 11008.

698 conjugacy classes of subgroups.

The subgroup lattice has 382 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#19144603	C₂	1	1	yes	maximal, characteristic
2	#19143525	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×D₄×C₄₃⋊C₈	C₂	1	1	yes	maximal, characteristic
4	#19133449	C₂	1	4	yes	maximal, complemented
5	#19133404	C₂	1	4	yes	maximal, complemented
6	#19133371	C₂	1	1	yes	maximal, characteristic, complemented
7	#19132999	C₂	1	1	yes	maximal, characteristic, complemented
8	#19132888	C₂	1	1	yes	maximal, characteristic, complemented
9	C₄₃×#1486	C₂	1	1	yes	maximal, characteristic
10	C₂³×C₄₃⋊C₈	—	2	1	no	—
11	#14583456	C₂²	1	1	yes	characteristic
12	#14583229	C₄	1	2	yes	—
13	#14583189	C₄	1	2	yes	—
14	#14583104	C₄	1	2	yes	—
15	#14583050	C₄	1	2	yes	—
16	#14583026	—	2	1	no	—
17	#14583026	C₂²	1	2	yes	complemented
18	#14583023	—	2	1	no	—
19	#14583023	C₂²	1	2	yes	complemented
20	C₄×#10621883	C₂²	1	4	yes	—
21	D₄×C₄₃⋊C₈	—	2	1	no	—
22	D₄×C₄₃⋊C₈	—	2	1	no	—
23	D₄×C₄₃⋊C₈	C₂²	1	2	yes	—
24	D₄×C₄₃⋊C₈	C₂²	1	2	yes	—
25	#14582952	C₂²	1	1	yes	characteristic
26	#14582927	—	2	1	no	—
27	C₂×C₄⋊(C₄₃⋊C₈)	C₂²	1	1	yes	characteristic
28	C₂×C₂²⋊(C₄₃⋊C₈)	—	2	1	no	—
29	#14582857	—	2	1	no	—
30	C₂×C₄×C₄₃⋊C₈	C₂²	1	1	yes	characteristic
31	#14582651	C₂²	1	4	yes	complemented
32	#14582631	C₂²	1	4	yes	complemented
33	#14582621	C₂²	1	2	yes	complemented
34	#14582620	C₂²	1	2	yes	complemented
35	C₂×C₄×D₄×C₄₃	C₂²	1	1	yes	characteristic
36	C₄₃×#431	C₂²	1	1	yes	characteristic
37	C₄₃×#423	C₂²	1	1	yes	characteristic
38	C₄₃×D₄⋊C₈	C₂²	1	4	yes	—
39	C₂×#10621883	C₂×C₄	1	4	yes	—
40	C₂²×C₄₃⋊C₈	—	2	1	no	—
41	C₂²×C₄₃⋊C₈	—	2	2	no	—
42	C₂²×C₄₃⋊C₈	—	2	2	no	—
43	C₂²×C₄₃⋊C₈	—	4	4	no	—
44	C₂²×C₄₃⋊C₈	D₄	1	1	yes	characteristic
45	C₂²×C₄₃⋊C₈	D₄	1	1	yes	characteristic
46	#13063325	C₂×C₄	1	1	yes	characteristic
47	#13063325	C₂×C₄	1	1	yes	characteristic
48	#13063317	—	2	1	no	—
49	#13063304	—	2	1	no	—
50	#13063304	C₂×C₄	1	2	yes	—
51	C₄×Dic₈₆	—	2	1	no	—
52	C₄×Dic₈₆	C₂³	1	2	yes	—
53	#13063284	—	2	1	no	—
54	#13063256	C₂×C₄	1	4	yes	—
55	C₄⋊(C₄₃⋊C₈)	—	2	1	no	—
56	C₄⋊(C₄₃⋊C₈)	C₂×C₄	1	2	yes	—
57	C₄⋊(C₄₃⋊C₈)	C₂³	1	2	yes	—
58	Dic₄₃⋊C₈	—	2	2	no	—
59	Dic₄₃⋊C₈	—	2	2	no	—
60	#13063248	C₂×C₄	1	4	yes	—
61	#13063246	C₂×C₄	1	4	yes	—
62	C₂²⋊(C₄₃⋊C₈)	—	2	2	no	—
63	C₂²⋊(C₄₃⋊C₈)	—	2	2	no	—
64	C₂²⋊(C₄₃⋊C₈)	C₂×C₄	1	2	yes	—
65	#13063235	—	2	2	no	—
66	C₄×C₄₃⋊C₈	D₄	1	1	yes	characteristic
67	C₄×C₄₃⋊C₈	D₄	1	1	yes	characteristic
68	C₄×C₄₃⋊C₈	C₂³	1	2	yes	—
69	C₄×C₄₃⋊C₈	C₂³	1	1	yes	characteristic
70	C₄×C₄₃⋊C₈	C₂³	1	1	yes	characteristic
71	C₈×Dic₄₃	—	2	2	no	—
72	C₂²×D₄×C₄₃	C₂×C₄	1	1	yes	characteristic
73	C₂³×C₄×C₄₃	—	2	1	no	—
74	C₂×M₁₆×C₄₃	—	2	1	no	—
75	C₄×D₄×C₄₃	—	2	1	no	—
76	C₄×D₄×C₄₃	—	2	1	no	—
77	C₄×D₄×C₄₃	C₂×C₄	1	2	yes	—
78	C₄×D₄×C₄₃	C₂³	1	2	yes	—
79	C₂×C₄⋊C₄×C₄₃	C₂×C₄	1	1	yes	characteristic
80	C₂×C₂²⋊C₄×C₄₃	—	2	1	no	—
81	C₂×C₄²×C₄₃	C₂³	1	1	yes	characteristic
82	C₄⋊C₈×C₄₃	—	2	1	no	—
83	C₄⋊C₈×C₄₃	—	2	1	no	—
84	C₄⋊C₈×C₄₃	C₂³	1	2	yes	—
85	C₂²⋊C₈×C₄₃	—	2	1	no	—
86	C₄×C₈×C₄₃	C₂³	1	2	yes	—
87	C₂⁴×C₄₃	—	2	1	no	—
88	C₂×D₄×C₄₃	—	2	4	no	—
89	C₂×D₄×C₄₃	—	2	1	no	—
90	C₂×D₄×C₄₃	C₂²×C₄	1	2	yes	—
91	C₂²×C₄×C₄₃	—	2	2	no	—
92	C₂²×C₄×C₄₃	—	2	2	no	—
93	C₂²×C₄×C₄₃	—	2	1	no	—
94	C₂²×C₄×C₄₃	—	4	2	no	—
95	C₂²×C₄×C₄₃	—	4	2	no	—
96	C₂²×C₄×C₄₃	C₂²×C₄	1	1	yes	characteristic
97	C₂²×C₄×C₄₃	C₂×D₄	1	1	yes	characteristic
98	C₂²×C₄×C₄₃	#41	1	1	yes	characteristic
99	C₂×Dic₈₆	—	2	1	no	—
100	C₂×Dic₈₆	C₂²×C₄	1	2	yes	—
101	M₁₆×C₄₃	—	4	2	no	—
102	C₂×C₈×C₄₃	—	2	2	no	—
103	C₂×C₈×C₄₃	—	2	2	no	—
104	C₄⋊C₄×C₄₃	—	2	1	no	—
105	C₄⋊C₄×C₄₃	C₂²×C₄	1	2	yes	—
106	C₂²⋊C₄×C₄₃	—	2	2	no	—
107	C₂²⋊C₄×C₄₃	—	2	2	no	—
108	C₄²×C₄₃	C₂²×C₄	1	1	yes	characteristic
109	C₄²×C₄₃	C₂×D₄	1	1	yes	characteristic
110	C₄²×C₄₃	#41	1	1	yes	characteristic
111	C₄²×C₄₃	C₂⁴	1	1	yes	characteristic
112	C₂²⋊Dic₄₃	—	2	2	no	—
113	#10621883	—	2	8	no	—
114	C₄⋊Dic₄₃	—	2	1	no	—
115	C₄⋊Dic₄₃	—	2	1	no	—
116	C₄⋊Dic₄₃	—	2	1	no	—
117	C₄⋊Dic₄₃	C₂²×C₄	1	2	yes	—
118	Dic₄₃⋊C₄	—	2	2	no	—
119	Dic₄₃⋊C₄	—	2	2	no	—
120	C₄×Dic₄₃	—	2	2	no	—
121	C₂×C₄₃⋊C₈	—	2	2	no	—
122	C₂×C₄₃⋊C₈	—	2	2	no	—
123	C₂×C₄₃⋊C₈	—	2	2	no	—
124	C₂×C₄₃⋊C₈	—	4	4	no	—
125	C₂×C₄₃⋊C₈	—	4	4	no	—
126	C₂×C₄₃⋊C₈	—	4	2	no	—
127	C₂×C₄₃⋊C₈	C₂²×C₄	1	4	yes	—
128	C₂×C₄₃⋊C₈	C₂×D₄	1	1	yes	characteristic
129	C₂×C₄₃⋊C₈	C₂×D₄	1	1	yes	characteristic
130	C₂×C₄₃⋊C₈	C₂×D₄	1	1	yes	characteristic
131	C₂×C₄₃⋊C₈	C₂×D₄	1	1	yes	characteristic
132	C₂³×C₄₃	—	2	4	no	—
133	C₂³×C₄₃	—	2	2	no	—
134	C₂³×C₄₃	—	4	2	no	—
135	C₂³×C₄₃	—	4	2	no	—
136	C₂³×C₄₃	C₄×D₄	1	1	yes	characteristic
137	D₄×C₄₃	—	2	4	no	—
138	D₄×C₄₃	—	2	2	no	—
139	D₄×C₄₃	#131	1	4	yes	—
140	C₂×C₄×C₄₃	—	2	2	no	—
141	C₂×C₄×C₄₃	—	2	2	no	—
142	C₂×C₄×C₄₃	—	4	1	no	—
143	C₂×C₄×C₄₃	—	4	2	no	—
144	C₂×C₄×C₄₃	—	4	2	no	—
145	C₂×C₄×C₄₃	—	4	2	no	—
146	C₂×C₄×C₄₃	—	4	4	no	—
147	C₂×C₄×C₄₃	C₄×D₄	1	1	yes	characteristic
148	C₂×C₄×C₄₃	C₄×D₄	1	1	yes	characteristic
149	C₂×C₄×C₄₃	C₄×D₄	1	1	yes	characteristic
150	C₂×C₄×C₄₃	AGL(1,ℤ/8)	1	2	yes	—
151	C₂×C₄×C₄₃	#137	1	2	yes	—
152	C₂×C₄×C₄₃	C₂³×C₄	1	1	yes	characteristic
153	C₂×C₄×C₄₃	C₂²×D₄	1	1	yes	characteristic
154	C₂×C₄×C₄₃	C₂×#41	1	1	yes	characteristic
155	C₂×Dic₄₃	—	2	2	no	—
156	C₂×Dic₄₃	—	2	2	no	—
157	Dic₈₆	—	2	4	no	—
158	Dic₈₆	—	4	2	no	—
159	C₈×C₄₃	—	4	2	no	—
160	C₈×C₄₃	—	4	2	no	—
161	C₄₃⋊C₈	—	2	4	no	—
162	C₄₃⋊C₈	—	2	1	no	—
163	C₄₃⋊C₈	—	2	1	no	—
164	C₄₃⋊C₈	—	2	1	no	—
165	C₄₃⋊C₈	—	2	1	no	—
166	C₄₃⋊C₈	—	4	2	no	—
167	#17905	—	43	1	no	maximal, Sylow
168	C₂²×C₄₃	—	2	1	no	—
169	C₂²×C₄₃	—	2	1	no	—
170	C₂²×C₄₃	—	2	4	no	—
171	C₂²×C₄₃	—	4	8	no	—
172	C₂²×C₄₃	—	4	2	no	—
173	C₂²×C₄₃	—	4	2	no	—
174	C₂²×C₄₃	#440	1	2	yes	—
175	C₂²×C₄₃	C₂×C₄×D₄	1	1	yes	characteristic
176	C₄×C₄₃	—	2	1	no	—
177	C₄×C₄₃	—	2	1	no	—
178	C₄×C₄₃	—	2	1	no	—
179	C₄×C₄₃	—	4	2	no	—
180	C₄×C₄₃	#440	1	2	yes	—
181	C₄×C₄₃	C₂×#131	1	2	yes	—
182	C₄×C₄₃	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
183	C₄×C₄₃	C₂×#137	1	1	yes	characteristic
184	Dic₄₃	—	4	2	no	—
185	Dic₄₃	—	4	2	no	—
186	#3101	—	43	1	no	—
187	#2960	—	43	1	no	—
188	C₂×C₈×D₄	—	43	1	no	—
189	#1585	—	43	4	no	—
190	SD₁₆⋊C₈	—	43	4	no	—
191	#1571	—	43	1	no	—
192	#1486	D₄₃	1	1	yes	characteristic
193	#1481	—	43	1	no	—
194	#1452	—	43	1	no	—
195	C₂×C₄₃	—	2	2	no	—
196	C₂×C₄₃	—	4	4	no	—
197	C₂×C₄₃	#2929	1	1	yes	characteristic
198	C₂×C₄₃	C₂×#440	1	1	yes	characteristic
199	C₂×C₄₃	#2955	1	1	yes	characteristic
200	C₂³×C₈	—	86	1	no	—
201	#533	—	43	1	no	—
202	C₂×C₄×D₄	C₂×D₄₃	1	1	yes	characteristic
203	#483	—	43	2	no	—
204	#476	—	43	2	no	—
205	#461	—	43	2	no	—
206	#450	—	43	2	no	—
207	#446	—	43	2	no	—
208	#446	—	86	1	no	—
209	C₈×Q₈	—	43	2	no	—
210	C₈×Q₈	—	86	1	no	—
211	C₄×SD₁₆	—	43	4	no	—
212	C₈×D₄	—	43	2	no	—
213	C₈×D₄	—	43	2	no	—
214	C₈×D₄	—	86	1	no	—
215	C₈×D₄	—	86	1	no	—
216	#431	—	43	1	no	—
217	#431	C₂×D₄₃	1	1	yes	characteristic
218	#424	—	86	1	no	—
219	#423	C₂×D₄₃	1	1	yes	characteristic
220	C₂×C₄⋊C₈	—	43	1	no	—
221	C₂×C₂²⋊C₈	—	86	1	no	—
222	#405	—	86	1	no	—
223	C₂×C₄×C₈	—	43	1	no	—
224	#335	—	43	4	no	—
225	Q₈⋊C₈	—	43	4	no	—
226	D₄⋊C₈	C₂×D₄₃	1	4	yes	—
227	#322	—	43	2	no	—
228	#322	—	43	2	no	—
229	C₄₃	#17905	1	1	yes	characteristic, complemented, Sylow
230	C₂²×D₄	C₄×D₄₃	1	1	yes	characteristic
231	C₂³×C₄	—	2	1	no	—
232	C₂×SD₁₆	—	43	4	no	—
233	C₂×M₁₆	—	2	1	no	—
234	C₂²×C₈	—	43	1	no	—
235	C₂²×C₈	—	43	1	no	—
236	C₂²×C₈	—	86	2	no	—
237	C₂²×C₈	—	86	2	no	—
238	C₂²×C₈	—	86	1	no	—
239	C₂²×C₈	—	172	4	no	—
240	C₄⋊Q₈	—	43	1	no	—
241	C₄⋊Q₈	—	43	1	no	—
242	#125	—	86	1	no	—
243	C₂²⋊Q₈	—	43	2	no	—
244	C₂²⋊Q₈	—	86	1	no	—
245	C₄×Q₈	—	43	2	no	—
246	C₄×Q₈	—	86	1	no	—
247	C₄×D₄	—	2	1	no	—
248	C₄×D₄	—	2	1	no	—
249	C₄×D₄	C₄×D₄₃	1	2	yes	—
250	C₄×D₄	C₂²×D₄₃	1	2	yes	—
251	#117	—	86	1	no	—
252	C₂×C₄⋊C₄	C₄×D₄₃	1	1	yes	characteristic
253	C₂×C₂²⋊C₄	—	2	1	no	—
254	C₂×C₄²	C₂²×D₄₃	1	1	yes	characteristic
255	#106	—	43	4	no	—
256	C₄⋊C₈	—	2	1	no	—
257	C₄⋊C₈	—	2	1	no	—
258	C₄⋊C₈	—	43	2	no	—
259	C₄⋊C₈	—	43	2	no	—
260	C₄⋊C₈	—	86	2	no	—
261	C₄⋊C₈	—	86	2	no	—
262	C₄⋊C₈	—	86	1	no	—
263	C₄⋊C₈	C₂²×D₄₃	1	2	yes	—
264	#103	—	43	4	no	—
265	#102	—	43	4	no	—
266	C₂²⋊C₈	—	2	1	no	—
267	C₂²⋊C₈	—	43	2	no	—
268	C₂²⋊C₈	—	86	2	no	—
269	C₂²⋊C₈	—	86	2	no	—
270	#97	—	86	2	no	—
271	C₄×C₈	—	43	1	no	—
272	C₄×C₈	—	43	1	no	—
273	C₄×C₈	—	43	2	no	—
274	C₄×C₈	—	43	1	no	—
275	C₄×C₈	—	43	1	no	—
276	C₄×C₈	—	86	2	no	—
277	C₄×C₈	C₂²×D₄₃	1	2	yes	—
278	C₂⁴	—	2	1	no	—
279	C₂×Q₈	—	43	2	no	—
280	C₂×Q₈	—	86	1	no	—
281	C₂×D₄	—	2	4	no	—
282	C₂×D₄	—	2	1	no	—
283	C₂×D₄	C₂×C₄×D₄₃	1	2	yes	—
284	C₂²×C₄	—	2	1	no	—
285	C₂²×C₄	—	2	2	no	—
286	C₂²×C₄	—	2	2	no	—
287	C₂²×C₄	—	4	2	no	—
288	C₂²×C₄	—	4	2	no	—
289	C₂²×C₄	C₂×C₄×D₄₃	1	1	yes	characteristic
290	C₂²×C₄	D₄×D₄₃	1	1	yes	characteristic
291	C₂²×C₄	#10621900	1	1	yes	characteristic
292	SD₁₆	—	86	8	no	—
293	M₁₆	—	4	2	no	—
294	C₂×C₈	—	2	2	no	—
295	C₂×C₈	—	2	2	no	—
296	C₂×C₈	—	43	4	no	—
297	C₂×C₈	—	43	1	no	—
298	C₂×C₈	—	43	1	no	—
299	C₂×C₈	—	43	1	no	—
300	C₂×C₈	—	43	1	no	—
301	C₂×C₈	—	86	2	no	—
302	C₂×C₈	—	86	2	no	—
303	C₂×C₈	—	86	2	no	—
304	C₂×C₈	—	172	4	no	—
305	C₂×C₈	—	172	4	no	—
306	C₂×C₈	—	172	2	no	—
307	C₄⋊C₄	—	2	1	no	—
308	C₄⋊C₄	—	43	2	no	—
309	C₄⋊C₄	—	86	1	no	—
310	C₄⋊C₄	—	86	1	no	—
311	C₄⋊C₄	—	86	2	no	—
312	C₄⋊C₄	—	86	1	no	—
313	C₄⋊C₄	—	86	2	no	—
314	C₄⋊C₄	C₂×C₄×D₄₃	1	2	yes	—
315	C₂²⋊C₄	—	2	2	no	—
316	C₂²⋊C₄	—	2	2	no	—
317	C₂²⋊C₄	—	86	2	no	—
318	C₄²	—	86	2	no	—
319	C₄²	C₂×C₄×D₄₃	1	1	yes	characteristic
320	C₄²	D₄×D₄₃	1	1	yes	characteristic
321	C₄²	#10621900	1	1	yes	characteristic
322	C₄²	C₂³×D₄₃	1	1	yes	characteristic
323	C₂³	—	2	4	no	—
324	C₂³	—	2	2	no	—
325	C₂³	—	4	2	no	—
326	C₂³	—	4	2	no	—
327	C₂³	(C₂²⋊D₄₃)⋊C₄	1	1	yes	characteristic
328	Q₈	—	86	4	no	—
329	Q₈	—	172	2	no	—
330	D₄	—	2	4	no	—
331	D₄	—	2	2	no	—
332	D₄	#13063333	1	4	yes	—
333	C₂×C₄	—	2	2	no	—
334	C₂×C₄	—	2	2	no	—
335	C₂×C₄	—	4	1	no	—
336	C₂×C₄	—	4	2	no	—
337	C₂×C₄	—	4	2	no	—
338	C₂×C₄	—	4	2	no	—
339	C₂×C₄	—	4	4	no	—
340	C₂×C₄	—	86	2	no	—
341	C₂×C₄	—	86	2	no	—
342	C₂×C₄	(C₂²⋊D₄₃)⋊C₄	1	1	yes	characteristic
343	C₂×C₄	(C₂²⋊D₄₃)⋊C₄	1	1	yes	characteristic
344	C₂×C₄	(C₂²⋊D₄₃)⋊C₄	1	1	yes	characteristic
345	C₂×C₄	#13063349	1	2	yes	—
346	C₂×C₄	#13063354	1	2	yes	—
347	C₂×C₄	C₂²×C₄×D₄₃	1	1	yes	characteristic
348	C₂×C₄	C₂×D₄×D₄₃	1	1	yes	characteristic
349	C₂×C₄	C₂×#10621900	1	1	yes	characteristic
350	C₈	—	4	2	no	—
351	C₈	—	4	2	no	—
352	C₈	—	86	4	no	—
353	C₈	—	86	1	no	—
354	C₈	—	86	1	no	—
355	C₈	—	86	1	no	—
356	C₈	—	86	1	no	—
357	C₈	—	172	2	no	—
358	C₂²	—	2	1	no	—
359	C₂²	—	2	1	no	—
360	C₂²	—	2	4	no	—
361	C₂²	—	4	8	no	—
362	C₂²	—	4	2	no	—
363	C₂²	—	4	2	no	—
364	C₂²	#14582996	1	2	yes	—
365	C₂²	C₂×(C₂²⋊D₄₃)⋊C₄	1	1	yes	characteristic
366	C₄	—	2	1	no	—
367	C₄	—	2	1	no	—
368	C₄	—	2	1	no	—
369	C₄	—	4	2	no	—
370	C₄	—	172	2	no	—
371	C₄	—	172	2	no	—
372	C₄	#14582996	1	2	yes	—
373	C₄	C₂×#13063333	1	2	yes	—
374	C₄	C₂×#13063349	1	1	yes	characteristic
375	C₄	C₂×#13063354	1	1	yes	characteristic
376	C₂	—	2	2	no	—
377	C₂	—	4	4	no	—
378	C₂	#19143285	1	1	yes	characteristic
379	C₂	C₂×#14582996	1	1	yes	characteristic
380	C₂	#19143494	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂×D₄×D₄₃
commutator	C₂×C₄×C₄₃	C₂³×C₄
Frattini	C₄²	C₂³×D₄₃
Fitting	C₄₃×#1486	C₂
socle	C₂²×C₄₃	C₂×C₄×D₄
prime radical	C₂²×D₄×C₄₃	C₂×C₄
cosocle	C₄²×C₄₃	C₂⁴
hypercenter	#1486	D₄₃
hypocenter	C₄₃	#17905
LCS	G ▷ C₂×C₄×C₄₃ ▷ C₂×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #1486

Complements

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

N	H ≅ G/N	type
2. #19143525	C₂	semidirect
4. #19133449	C₂	semidirect
5. #19133404	C₂	semidirect
6. #19133371	C₂	semidirect
7. #19132999	C₂	semidirect
8. #19132888	C₂	semidirect
17. #14583026	C₂²	semidirect
19. #14583023	C₂²	semidirect
31. #14582651	C₂²	semidirect
32. #14582631	C₂²	semidirect
33. #14582621	C₂²	semidirect
34. #14582620	C₂²	semidirect
229. C₄₃	#17905	semidirect

Maximal subgroups

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

subgroup	index
#19144603	2
#19143525	2
C₂×D₄×C₄₃⋊C₈	2
#19133449	2
#19133404	2
#19133371	2
#19132999	2
#19132888	2
C₄₃×#1486	2
#17905	43