Subgroups — #25105374

Order 11008.

750 conjugacy classes of subgroups.

The subgroup lattice has 394 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#14583303	C₂	1	1	yes	maximal, characteristic
2	#19137342	C₂	1	2	yes	maximal
3	#19135566	C₂	1	2	yes	maximal
4	#19135330	C₂	1	2	yes	maximal
5	#19135328	C₂	1	1	yes	maximal, characteristic
6	#19134758	C₂	1	1	yes	maximal, characteristic
7	#19134491	C₂	1	2	yes	maximal
8	#107×Dic₄₃	C₂	1	1	yes	maximal, characteristic
9	#107×Dic₄₃	C₂	1	1	yes	maximal, characteristic
10	#19134281	C₂	1	1	yes	maximal, characteristic
11	C₄₃×#1942	C₂	1	1	yes	maximal, characteristic
12	C₂×#13063324	C₂²	1	2	yes	—
13	#14583303	C₄	1	8	yes	complemented
14	C₂×#13063259	C₂²	1	1	yes	characteristic
15	C₂×#13063259	C₂²	1	1	yes	characteristic
16	C₂×#13063258	C₂²	1	2	yes	—
17	C₂×#13063256	C₂²	1	2	yes	—
18	C₂×#13063236	C₂²	1	1	yes	characteristic
19	C₂×#13063235	C₂²	1	1	yes	characteristic
20	C₂×#13063235	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×#107×C₄₃	C₂²	1	1	yes	characteristic
37	C₂×C₄×C₈×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	#13063259	C₂×C₄	1	4	yes	—
55	#13063258	C₂×C₄	1	8	yes	—
56	#13063256	—	2	4	no	—
57	#13063236	—	2	2	no	—
58	#13063235	—	2	2	no	—
59	#13063235	Q₈	1	4	yes	complemented
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	1	yes	characteristic
68	C₄₃⋊#95	D₄	1	1	yes	characteristic
69	C₄₃⋊#95	D₄	1	1	yes	characteristic
70	C₄₃⋊#95	D₄	1	1	yes	characteristic
71	C₂²×C₈×C₄₃	C₂³	1	1	yes	characteristic
72	C₂²×C₈×C₄₃	C₂³	1	1	yes	characteristic
73	C₂×C₄⋊C₄×C₄₃	—	2	2	no	—
74	C₂×C₄⋊C₄×C₄₃	C₂³	1	2	yes	—
75	C₂×C₄⋊C₄×C₄₃	C₂³	1	2	yes	—
76	C₂×C₄²×C₄₃	C₂³	1	1	yes	characteristic
77	#107×C₄₃	—	2	2	no	—
78	#107×C₄₃	C₂×C₄	1	4	yes	—
79	C₄×C₈×C₄₃	—	2	2	no	—
80	#95×C₄₃	—	2	2	no	—
81	C₂²×C₄×C₄₃	—	2	2	no	—
82	C₂²×C₄×C₄₃	—	2	2	no	—
83	C₂²×C₄×C₄₃	C₂×D₄	1	1	yes	characteristic
84	C₂²×C₄×C₄₃	C₂×D₄	1	1	yes	characteristic
85	C₂²×C₄×C₄₃	C₂⁴	1	1	yes	characteristic
86	C₂²×Dic₄₃	—	2	2	no	—
87	C₂²×Dic₄₃	—	2	2	no	—
88	C₂²×Dic₄₃	C₂×D₄	1	1	yes	characteristic
89	C₂²×Dic₄₃	C₂×D₄	1	1	yes	characteristic
90	C₂²×Dic₄₃	C₂×D₄	1	1	yes	characteristic
91	C₂²×Dic₄₃	C₂×D₄	1	1	yes	characteristic
92	C₂×Dic₈₆	—	2	8	no	—
93	C₂×Dic₈₆	—	8	2	no	—
94	C₂×Dic₈₆	—	8	2	no	—
95	C₂×C₈×C₄₃	—	2	1	no	—
96	C₂×C₈×C₄₃	—	2	1	no	—
97	C₂×C₈×C₄₃	C₂²×C₄	1	2	yes	—
98	C₂×C₈×C₄₃	C₂×Q₈	1	2	yes	—
99	C₂×C₈×C₄₃	C₂×Q₈	1	2	yes	—
100	C₂×C₈×C₄₃	#41	1	2	yes	—
101	C₄⋊C₄×C₄₃	—	2	2	no	—
102	C₄⋊C₄×C₄₃	—	2	2	no	—
103	C₄⋊C₄×C₄₃	—	2	2	no	—
104	C₄⋊C₄×C₄₃	—	8	2	no	—
105	C₄⋊C₄×C₄₃	C₂²×C₄	1	4	yes	—
106	C₄²×C₄₃	—	2	1	no	—
107	C₄²×C₄₃	—	2	1	no	—
108	C₄⋊Dic₄₃	—	2	2	no	—
109	C₄⋊Dic₄₃	—	2	2	no	—
110	C₄⋊Dic₄₃	—	2	2	no	—
111	C₄⋊Dic₄₃	C₂²×C₄	1	4	yes	—
112	Dic₄₃⋊C₄	—	2	8	no	—
113	Dic₄₃⋊C₄	—	8	2	no	—
114	C₄×Dic₄₃	—	8	2	no	—
115	C₄×Dic₄₃	—	8	2	no	—
116	C₄×Dic₄₃	C₂²×C₄	1	2	yes	—
117	C₄×Dic₄₃	C₂×Q₈	1	2	yes	—
118	C₄×Dic₄₃	C₂×Q₈	1	2	yes	—
119	C₄×Dic₄₃	#41	1	2	yes	—
120	C₂×C₄₃⋊C₈	—	2	1	no	—
121	C₂×C₄₃⋊C₈	—	2	1	no	—
122	C₂×C₄₃⋊C₈	C₂²×C₄	1	2	yes	—
123	C₂×C₄₃⋊C₈	C₂×Q₈	1	2	yes	—
124	C₂×C₄₃⋊C₈	C₂×Q₈	1	2	yes	—
125	C₂×C₄₃⋊C₈	#41	1	2	yes	—
126	C₂³×C₄₃	C₂²×D₄	1	1	yes	characteristic
127	C₂×C₄×C₄₃	—	2	4	no	—
128	C₂×C₄×C₄₃	—	2	1	no	—
129	C₂×C₄×C₄₃	—	2	1	no	—
130	C₂×C₄×C₄₃	—	2	1	no	—
131	C₂×C₄×C₄₃	—	2	1	no	—
132	C₂×C₄×C₄₃	—	2	1	no	—
133	C₂×C₄×C₄₃	—	2	1	no	—
134	C₂×C₄×C₄₃	—	4	2	no	—
135	C₂×C₄×C₄₃	—	4	2	no	—
136	C₂×C₄×C₄₃	—	4	2	no	—
137	C₂×C₄×C₄₃	—	4	2	no	—
138	C₂×C₄×C₄₃	—	4	2	no	—
139	C₂×C₄×C₄₃	C₂³×C₄	1	1	yes	characteristic
140	C₂×C₄×C₄₃	C₂²×Q₈	1	1	yes	characteristic
141	C₂×C₄×C₄₃	C₂²×Q₈	1	1	yes	characteristic
142	C₂×C₄×C₄₃	C₂×#41	1	1	yes	characteristic
143	C₂×C₄×C₄₃	C₂×#41	1	1	yes	characteristic
144	C₂×C₄×C₄₃	C₂×#41	1	1	yes	characteristic
145	C₂×Dic₄₃	—	2	1	no	—
146	C₂×Dic₄₃	—	2	1	no	—
147	C₂×Dic₄₃	—	2	1	no	—
148	C₂×Dic₄₃	—	2	1	no	—
149	C₂×Dic₄₃	—	2	1	no	—
150	C₂×Dic₄₃	—	2	1	no	—
151	C₂×Dic₄₃	—	2	1	no	—
152	C₂×Dic₄₃	—	2	1	no	—
153	C₂×Dic₄₃	—	2	4	no	—
154	C₂×Dic₄₃	—	4	2	no	—
155	C₂×Dic₄₃	—	4	2	no	—
156	C₂×Dic₄₃	—	4	2	no	—
157	C₂×Dic₄₃	—	4	2	no	—
158	C₂×Dic₄₃	—	4	2	no	—
159	C₂×Dic₄₃	C₄×D₄	1	2	yes	—
160	C₂×Dic₄₃	C₄×D₄	1	2	yes	—
161	C₂×Dic₄₃	C₄⋊Q₈	1	2	yes	—
162	C₂×Dic₄₃	C₄⋊Q₈	1	2	yes	—
163	Dic₈₆	—	8	8	no	—
164	C₈×C₄₃	—	2	2	no	—
165	C₈×C₄₃	C₄×Q₈	1	4	yes	—
166	C₄₃⋊C₈	—	2	2	no	—
167	C₄₃⋊C₈	C₄×Q₈	1	4	yes	—
168	#21725	—	43	1	no	maximal, Sylow
169	C₂²×C₄₃	C₂×C₄×D₄	1	1	yes	characteristic
170	C₂²×C₄₃	C₂×C₄⋊Q₈	1	1	yes	characteristic
171	C₂²×C₄₃	#532	1	1	yes	characteristic
172	C₂²×C₄₃	#576	1	1	yes	characteristic
173	C₂²×C₄₃	#576	1	1	yes	characteristic
174	C₂²×C₄₃	#578	1	1	yes	characteristic
175	C₂²×C₄₃	#578	1	1	yes	characteristic
176	C₄×C₄₃	—	4	1	no	—
177	C₄×C₄₃	—	4	1	no	—
178	C₄×C₄₃	—	4	4	no	—
179	C₄×C₄₃	—	8	2	no	—
180	C₄×C₄₃	C₂×C₄×Q₈	1	1	yes	characteristic
181	C₄×C₄₃	C₂×C₄×Q₈	1	1	yes	characteristic
182	C₄×C₄₃	#533	1	1	yes	characteristic
183	C₄×C₄₃	#533	1	1	yes	characteristic
184	Dic₄₃	—	2	2	no	—
185	Dic₄₃	—	2	2	no	—
186	Dic₄₃	—	2	2	no	—
187	Dic₄₃	—	2	2	no	—
188	Dic₄₃	—	4	4	no	—
189	Dic₄₃	—	8	2	no	—
190	C₂×#500	—	43	1	no	—
191	#2331	—	43	2	no	—
192	#1993	—	43	2	no	—
193	#1943	—	43	2	no	—
194	#1942	—	43	1	no	—
195	#1942	D₄₃	1	1	yes	characteristic
196	#1846	—	43	1	no	—
197	#1776	—	43	2	no	—
198	C₄×#107	—	43	1	no	—
199	C₄×#107	—	43	1	no	—
200	#1726	—	43	1	no	—
201	C₂×C₄₃	C₄×C₄⋊Q₈	1	1	yes	characteristic
202	C₂×C₄₃	#2948	1	1	yes	characteristic
203	C₂×C₄₃	#2950	1	1	yes	characteristic
204	C₂×C₄₃	#3164	1	1	yes	characteristic
205	C₂×C₄₃	#3164	1	1	yes	characteristic
206	C₂×C₄₃	#3175	1	1	yes	characteristic
207	C₂×C₄₃	#3175	1	1	yes	characteristic
208	C₂×C₄⋊Q₈	—	43	2	no	—
209	#500	—	43	8	no	—
210	C₂×#107	—	43	1	no	—
211	C₂×#107	—	43	2	no	—
212	C₂×#107	—	43	1	no	—
213	C₂×#107	C₂×D₄₃	1	1	yes	characteristic
214	C₂×#107	C₂×D₄₃	1	1	yes	characteristic
215	C₂×#106	—	43	2	no	—
216	C₂×#97	—	43	1	no	—
217	C₂×#97	—	43	1	no	—
218	C₂×#97	—	43	1	no	—
219	C₂×C₄×C₈	—	43	1	no	—
220	C₂×C₄×C₈	—	43	1	no	—
221	C₂×C₄×C₈	C₂×D₄₃	1	1	yes	characteristic
222	#391	—	86	2	no	—
223	#391	—	86	2	no	—
224	#389	—	43	2	no	—
225	#389	C₂×D₄₃	1	2	yes	—
226	#387	—	86	2	no	—
227	#387	—	86	2	no	—
228	#382	—	43	2	no	—
229	C₄×C₄⋊C₄	—	43	2	no	—
230	C₄×C₄⋊C₄	—	43	2	no	—
231	#376	—	43	1	no	—
232	#340	—	43	2	no	—
233	#340	—	43	2	no	—
234	#340	—	43	2	no	—
235	#340	C₂×D₄₃	1	2	yes	—
236	C₄₃	#21725	1	1	yes	characteristic, complemented, Sylow
237	C₂²×Q₈	—	86	2	no	—
238	C₂²×C₈	—	43	1	no	—
239	C₂²×C₈	—	43	1	no	—
240	C₂²×C₈	C₂²×D₄₃	1	1	yes	characteristic
241	C₂²×C₈	C₂²×D₄₃	1	1	yes	characteristic
242	C₄⋊Q₈	—	43	8	no	—
243	C₄⋊Q₈	—	86	4	no	—
244	C₂×C₄⋊C₄	—	2	2	no	—
245	C₂×C₄⋊C₄	—	43	2	no	—
246	C₂×C₄⋊C₄	—	43	2	no	—
247	C₂×C₄⋊C₄	—	86	2	no	—
248	C₂×C₄⋊C₄	—	86	2	no	—
249	C₂×C₄⋊C₄	C₂²×D₄₃	1	2	yes	—
250	C₂×C₄⋊C₄	C₂²×D₄₃	1	2	yes	—
251	C₂×C₄²	—	43	1	no	—
252	C₂×C₄²	—	43	1	no	—
253	C₂×C₄²	—	86	2	no	—
254	C₂×C₄²	—	86	2	no	—
255	C₂×C₄²	C₂²×D₄₃	1	1	yes	characteristic
256	#107	—	2	2	no	—
257	#107	—	43	4	no	—
258	#107	—	43	8	no	—
259	#107	—	86	2	no	—
260	#107	C₄×D₄₃	1	4	yes	—
261	#106	—	86	4	no	—
262	#97	—	43	4	no	—
263	#97	—	86	2	no	—
264	#97	—	86	2	no	—
265	C₄×C₈	—	2	2	no	—
266	C₄×C₈	—	43	4	no	—
267	C₄×C₈	—	43	4	no	—
268	#95	—	2	2	no	—
269	#95	—	43	1	no	—
270	#95	—	43	1	no	—
271	#95	—	43	1	no	—
272	#95	—	43	1	no	—
273	#95	—	86	2	no	—
274	#95	—	86	2	no	—
275	#95	—	86	2	no	—
276	#95	—	86	2	no	—
277	#95	—	86	2	no	—
278	C₂×Q₈	—	86	8	no	—
279	C₂×Q₈	—	344	2	no	—
280	C₂×Q₈	—	344	2	no	—
281	C₂²×C₄	—	2	2	no	—
282	C₂²×C₄	—	2	2	no	—
283	C₂²×C₄	—	43	1	no	—
284	C₂²×C₄	—	43	1	no	—
285	C₂²×C₄	—	43	1	no	—
286	C₂²×C₄	—	43	1	no	—
287	C₂²×C₄	—	86	2	no	—
288	C₂²×C₄	—	86	2	no	—
289	C₂²×C₄	D₄×D₄₃	1	1	yes	characteristic
290	C₂²×C₄	D₄×D₄₃	1	1	yes	characteristic
291	C₂²×C₄	C₂³×D₄₃	1	1	yes	characteristic
292	C₂×C₈	—	2	1	no	—
293	C₂×C₈	—	2	1	no	—
294	C₂×C₈	—	43	2	no	—
295	C₂×C₈	—	43	2	no	—
296	C₂×C₈	—	43	2	no	—
297	C₂×C₈	—	43	2	no	—
298	C₂×C₈	—	86	1	no	—
299	C₂×C₈	—	86	1	no	—
300	C₂×C₈	C₂×C₄×D₄₃	1	2	yes	—
301	C₂×C₈	#10621900	1	2	yes	—
302	C₂×C₈	Q₈×D₄₃	1	2	yes	—
303	C₂×C₈	Q₈×D₄₃	1	2	yes	—
304	C₄⋊C₄	—	2	2	no	—
305	C₄⋊C₄	—	2	2	no	—
306	C₄⋊C₄	—	2	2	no	—
307	C₄⋊C₄	—	8	2	no	—
308	C₄⋊C₄	—	43	4	no	—
309	C₄⋊C₄	—	86	8	no	—
310	C₄⋊C₄	—	86	2	no	—
311	C₄⋊C₄	—	86	2	no	—
312	C₄⋊C₄	—	86	2	no	—
313	C₄⋊C₄	—	344	2	no	—
314	C₄⋊C₄	C₂×C₄×D₄₃	1	4	yes	—
315	C₄²	—	2	1	no	—
316	C₄²	—	2	1	no	—
317	C₄²	—	43	2	no	—
318	C₄²	—	43	2	no	—
319	C₄²	—	43	2	no	—
320	C₄²	—	43	2	no	—
321	C₄²	—	344	2	no	—
322	C₄²	—	344	2	no	—
323	C₂³	C₂×D₄×D₄₃	1	1	yes	characteristic
324	Q₈	—	344	8	no	—
325	C₂×C₄	—	2	4	no	—
326	C₂×C₄	—	2	1	no	—
327	C₂×C₄	—	2	1	no	—
328	C₂×C₄	—	2	1	no	—
329	C₂×C₄	—	2	1	no	—
330	C₂×C₄	—	2	1	no	—
331	C₂×C₄	—	2	1	no	—
332	C₂×C₄	—	4	2	no	—
333	C₂×C₄	—	4	2	no	—
334	C₂×C₄	—	4	2	no	—
335	C₂×C₄	—	4	2	no	—
336	C₂×C₄	—	4	2	no	—
337	C₂×C₄	—	43	2	no	—
338	C₂×C₄	—	43	2	no	—
339	C₂×C₄	—	43	2	no	—
340	C₂×C₄	—	43	2	no	—
341	C₂×C₄	—	86	1	no	—
342	C₂×C₄	—	86	1	no	—
343	C₂×C₄	—	86	1	no	—
344	C₂×C₄	—	86	1	no	—
345	C₂×C₄	—	86	1	no	—
346	C₂×C₄	—	86	1	no	—
347	C₂×C₄	—	86	1	no	—
348	C₂×C₄	—	86	1	no	—
349	C₂×C₄	—	86	4	no	—
350	C₂×C₄	—	172	2	no	—
351	C₂×C₄	—	172	2	no	—
352	C₂×C₄	—	172	2	no	—
353	C₂×C₄	—	172	2	no	—
354	C₂×C₄	—	172	2	no	—
355	C₂×C₄	C₂²×C₄×D₄₃	1	1	yes	characteristic
356	C₂×C₄	C₂×Q₈×D₄₃	1	1	yes	characteristic
357	C₂×C₄	C₂×Q₈×D₄₃	1	1	yes	characteristic
358	C₂×C₄	#41×D₄₃	1	1	yes	characteristic
359	C₂×C₄	#41×D₄₃	1	1	yes	characteristic
360	C₂×C₄	C₂×#10621900	1	1	yes	characteristic
361	C₈	—	2	2	no	—
362	C₈	—	43	4	no	—
363	C₈	—	86	2	no	—
364	C₈	#13063290	1	4	yes	—
365	C₂²	C₄×D₄×D₄₃	1	1	yes	characteristic
366	C₂²	C₄⋊Q₈×D₄₃	1	1	yes	characteristic
367	C₂²	#14583444	1	1	yes	characteristic
368	C₂²	#14583700	1	1	yes	characteristic
369	C₂²	#14583701	1	1	yes	characteristic
370	C₂²	#14583712	1	1	yes	characteristic
371	C₂²	#14583713	1	1	yes	characteristic
372	C₄	—	4	1	no	—
373	C₄	—	4	4	no	—
374	C₄	—	4	1	no	—
375	C₄	—	8	2	no	—
376	C₄	—	86	2	no	—
377	C₄	—	86	2	no	—
378	C₄	—	86	2	no	—
379	C₄	—	86	2	no	—
380	C₄	—	344	2	no	—
381	C₄	—	172	4	no	—
382	C₄	C₄×Q₈×D₄₃	1	1	yes	characteristic
383	C₄	C₂×#13063290	1	1	yes	characteristic
384	C₄	#14583451	1	1	yes	characteristic
385	C₄	#14583455	1	1	yes	characteristic
386	C₂	#19137166	1	1	yes	characteristic
387	C₂	#19143442	1	1	yes	characteristic
388	C₂	#19143458	1	1	yes	characteristic
389	C₂	#19145128	1	1	yes	characteristic
390	C₂	#19145132	1	1	yes	characteristic
391	C₂	#19145227	1	1	yes	characteristic
392	C₂	#19145235	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×#1942	C₂
socle, prime radical	C₂³×C₄₃	C₂²×D₄
cosocle	C₂²×C₄×C₄₃	C₂⁴
hypercenter	#1942	D₄₃
hypocenter	C₄₃	#21725
LCS	G ▷ C₂×C₄×C₄₃ ▷ C₂×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #1942

Complements

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

N	H ≅ G/N	type
13. #14583303	C₄	semidirect
59. #13063235	Q₈	semidirect
61. C₈×Dic₄₃	Q₈	semidirect
236. C₄₃	#21725	semidirect

Maximal subgroups

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

subgroup	index
C₂×#14583303	2
#19137342	2
#19135566	2
#19135330	2
#19135328	2
#19134758	2
#19134491	2
#107×Dic₄₃	2
#19134281	2
C₄₃×#1942	2
#21725	43