Subgroups — #25104622

Order 11008.

778 conjugacy classes of subgroups.

The subgroup lattice has 386 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#19137344	C₂	1	1	yes	maximal, characteristic
2	#19137315	C₂	1	1	yes	maximal, characteristic
3	#19135569	C₂	1	1	yes	maximal, characteristic
4	#19135416	C₂	1	1	yes	maximal, characteristic
5	#19135287	C₂	1	2	yes	maximal
6	#19135273	C₂	1	2	yes	maximal
7	#19134726	C₂	1	1	yes	maximal, characteristic
8	#19134427	C₂	1	2	yes	maximal
9	#103×Dic₄₃	C₂	1	2	yes	maximal
10	#19134281	C₂	1	1	yes	maximal, characteristic
11	C₂×C₄₃×#487	C₂	1	1	yes	maximal, characteristic
12	C₂×Q₈×Dic₄₃	C₂²	1	2	yes	—
13	C₂×#13063249	C₂²	1	2	yes	—
14	C₂×#13063249	C₂²	1	2	yes	—
15	C₂×#13063236	C₂²	1	1	yes	characteristic
16	C₂×#13063235	C₂²	1	1	yes	characteristic
17	C₂×#13063235	C₂²	1	1	yes	characteristic
18	C₂×C₈×Dic₄₃	C₂²	1	1	yes	characteristic
19	C₂×C₈×Dic₄₃	C₂²	1	1	yes	characteristic
20	#14582806	—	2	2	no	—
21	#14582806	C₂²	1	2	yes	—
22	#14582798	C₂²	1	2	yes	—
23	#14582785	—	2	2	no	—
24	#14582785	—	2	2	no	—
25	#14582785	—	2	2	no	—
26	#14582765	C₂²	1	1	yes	characteristic
27	#14582765	C₂²	1	1	yes	characteristic
28	C₄⋊C₄×Dic₄₃	C₂²	1	2	yes	—
29	#14582745	C₂²	1	1	yes	characteristic
30	#14582662	C₂²	1	2	yes	—
31	#14582662	C₂²	1	2	yes	—
32	#14582661	C₂²	1	2	yes	—
33	#14582661	C₂²	1	2	yes	—
34	C₂×C₄⋊Q₈×C₄₃	C₂²	1	1	yes	characteristic
35	C₂×C₄⋊Q₈×C₄₃	C₂²	1	1	yes	characteristic
36	C₄₃×#487	C₄	1	8	yes	complemented
37	C₂×#103×C₄₃	C₂²	1	2	yes	—
38	C₂×#103×C₄₃	C₂²	1	2	yes	—
39	C₂×C₄×C₈×C₄₃	C₂²	1	1	yes	characteristic
40	C₂²×C₄₃⋊C₈	C₂³	1	1	yes	characteristic
41	C₂²×C₄₃⋊C₈	C₂³	1	1	yes	characteristic
42	Q₈×Dic₄₃	—	2	4	no	—
43	Q₈×Dic₄₃	—	2	4	no	—
44	C₂×C₄⋊Dic₄₃	—	2	2	no	—
45	C₂×C₄⋊Dic₄₃	—	2	2	no	—
46	C₂×C₄⋊Dic₄₃	C₂³	1	2	yes	—
47	C₂×C₄⋊Dic₄₃	C₂³	1	2	yes	—
48	C₂×C₄×Dic₄₃	—	2	2	no	—
49	C₂×C₄×Dic₄₃	—	2	2	no	—
50	C₂×C₄×Dic₄₃	C₂³	1	1	yes	characteristic
51	C₂×C₄×Dic₄₃	C₂³	1	1	yes	characteristic
52	#13063249	—	2	4	no	—
53	#13063249	—	2	4	no	—
54	#13063236	—	2	2	no	—
55	#13063235	—	2	2	no	—
56	#13063235	—	2	2	no	—
57	C₈×Dic₄₃	—	2	2	no	—
58	C₈×Dic₄₃	—	2	2	no	—
59	C₄₃⋊#95	—	2	1	no	—
60	C₄₃⋊#95	—	2	1	no	—
61	C₄₃⋊#95	—	2	1	no	—
62	C₄₃⋊#95	—	2	1	no	—
63	C₄₃⋊#95	—	2	4	no	—
64	C₄₃⋊#95	—	2	2	no	—
65	C₄₃⋊#95	—	2	2	no	—
66	C₄₃⋊#95	D₄	1	2	yes	—
67	C₄₃⋊#95	D₄	1	2	yes	—
68	C₂²×Q₈×C₄₃	—	2	1	no	—
69	C₂²×Q₈×C₄₃	C₂³	1	2	yes	—
70	C₂²×C₈×C₄₃	C₂³	1	1	yes	characteristic
71	C₂²×C₈×C₄₃	C₂³	1	1	yes	characteristic
72	C₄⋊Q₈×C₄₃	—	2	2	no	—
73	C₄⋊Q₈×C₄₃	—	2	2	no	—
74	C₄⋊Q₈×C₄₃	C₂×C₄	1	4	yes	—
75	C₄⋊Q₈×C₄₃	C₂×C₄	1	4	yes	—
76	C₂×C₄⋊C₄×C₄₃	—	2	1	no	—
77	C₂×C₄⋊C₄×C₄₃	—	2	1	no	—
78	C₂×C₄⋊C₄×C₄₃	—	2	1	no	—
79	C₂×C₄⋊C₄×C₄₃	C₂³	1	2	yes	—
80	C₂×C₄²×C₄₃	C₂³	1	1	yes	characteristic
81	#103×C₄₃	C₂×C₄	1	8	yes	—
82	#103×C₄₃	C₂×C₄	1	8	yes	—
83	C₄×C₈×C₄₃	C₂×C₄	1	4	yes	—
84	C₂×Q₈×C₄₃	—	2	4	no	—
85	C₂×Q₈×C₄₃	—	2	2	no	—
86	C₂×Q₈×C₄₃	—	2	2	no	—
87	C₂×Q₈×C₄₃	—	2	2	no	—
88	C₂×Q₈×C₄₃	—	2	2	no	—
89	C₂×Q₈×C₄₃	—	2	2	no	—
90	C₂×Q₈×C₄₃	—	8	2	no	—
91	C₂×Q₈×C₄₃	C₂²×C₄	1	4	yes	—
92	C₂²×C₄×C₄₃	—	2	2	no	—
93	C₂²×C₄×C₄₃	—	2	2	no	—
94	C₂²×C₄×C₄₃	C₂×D₄	1	1	yes	characteristic
95	C₂²×C₄×C₄₃	C₂×D₄	1	1	yes	characteristic
96	C₂²×C₄×C₄₃	C₂⁴	1	1	yes	characteristic
97	C₂²×Dic₄₃	—	2	2	no	—
98	C₂²×Dic₄₃	—	2	2	no	—
99	C₂²×Dic₄₃	C₂×D₄	1	2	yes	—
100	C₂²×Dic₄₃	C₂×D₄	1	2	yes	—
101	C₂×C₈×C₄₃	—	2	1	no	—
102	C₂×C₈×C₄₃	—	2	1	no	—
103	C₂×C₈×C₄₃	—	2	1	no	—
104	C₂×C₈×C₄₃	—	2	1	no	—
105	C₂×C₈×C₄₃	C₂²×C₄	1	2	yes	—
106	C₂×C₈×C₄₃	C₂²×C₄	1	2	yes	—
107	C₄⋊C₄×C₄₃	—	2	4	no	—
108	C₄⋊C₄×C₄₃	—	2	4	no	—
109	C₄⋊C₄×C₄₃	—	2	4	no	—
110	C₄⋊C₄×C₄₃	—	2	2	no	—
111	C₄⋊C₄×C₄₃	C₂²×C₄	1	4	yes	—
112	C₄²×C₄₃	C₂²×C₄	1	2	yes	—
113	C₄²×C₄₃	#41	1	2	yes	—
114	C₄⋊Dic₄₃	—	2	2	no	—
115	C₄⋊Dic₄₃	—	2	2	no	—
116	C₄⋊Dic₄₃	—	2	2	no	—
117	C₄⋊Dic₄₃	—	2	2	no	—
118	C₄⋊Dic₄₃	—	4	4	no	—
119	C₄⋊Dic₄₃	—	8	2	no	—
120	C₄×Dic₄₃	—	2	1	no	—
121	C₄×Dic₄₃	—	2	1	no	—
122	C₄×Dic₄₃	—	4	4	no	—
123	C₄×Dic₄₃	—	8	2	no	—
124	C₄×Dic₄₃	#41	1	2	yes	—
125	C₄×Dic₄₃	#41	1	2	yes	—
126	C₂×C₄₃⋊C₈	—	2	1	no	—
127	C₂×C₄₃⋊C₈	—	2	1	no	—
128	C₂×C₄₃⋊C₈	—	2	1	no	—
129	C₂×C₄₃⋊C₈	—	2	1	no	—
130	C₂×C₄₃⋊C₈	#41	1	2	yes	—
131	C₂×C₄₃⋊C₈	#41	1	2	yes	—
132	C₂³×C₄₃	C₂²×D₄	1	1	yes	characteristic
133	Q₈×C₄₃	—	2	4	no	—
134	Q₈×C₄₃	—	2	4	no	—
135	Q₈×C₄₃	—	4	2	no	—
136	Q₈×C₄₃	—	4	2	no	—
137	Q₈×C₄₃	—	8	4	no	—
138	C₂×C₄×C₄₃	—	2	2	no	—
139	C₂×C₄×C₄₃	—	2	2	no	—
140	C₂×C₄×C₄₃	—	2	4	no	—
141	C₂×C₄×C₄₃	—	2	4	no	—
142	C₂×C₄×C₄₃	—	4	2	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	C₄×D₄	1	2	yes	—
147	C₂×C₄×C₄₃	C₄×D₄	1	2	yes	—
148	C₂×C₄×C₄₃	C₂³×C₄	1	1	yes	characteristic
149	C₂×C₄×C₄₃	C₂×#41	1	1	yes	characteristic
150	C₂×C₄×C₄₃	C₂×#41	1	1	yes	characteristic
151	C₂×C₄×C₄₃	C₂×#41	1	1	yes	characteristic
152	C₂×C₄×C₄₃	C₂×#41	1	1	yes	characteristic
153	C₂×C₄×C₄₃	C₂×#41	1	1	yes	characteristic
154	C₂×Dic₄₃	—	2	2	no	—
155	C₂×Dic₄₃	—	2	2	no	—
156	C₂×Dic₄₃	—	2	2	no	—
157	C₂×Dic₄₃	—	2	2	no	—
158	C₂×Dic₄₃	—	2	2	no	—
159	C₂×Dic₄₃	—	4	2	no	—
160	C₂×Dic₄₃	—	4	2	no	—
161	C₂×Dic₄₃	—	4	2	no	—
162	C₂×Dic₄₃	—	4	2	no	—
163	C₂×Dic₄₃	—	4	2	no	—
164	C₂×Dic₄₃	—	4	2	no	—
165	C₂×Dic₄₃	#124	1	4	yes	—
166	C₈×C₄₃	—	2	2	no	—
167	C₈×C₄₃	—	2	2	no	—
168	C₄₃⋊C₈	—	2	2	no	—
169	C₄₃⋊C₈	—	2	2	no	—
170	#21645	—	43	1	no	maximal, Sylow
171	C₂²×C₄₃	C₂×C₄×D₄	1	1	yes	characteristic
172	C₂²×C₄₃	C₂×#124	1	1	yes	characteristic
173	C₂²×C₄₃	#532	1	1	yes	characteristic
174	C₂²×C₄₃	#577	1	2	yes	—
175	C₂²×C₄₃	#578	1	2	yes	—
176	C₄×C₄₃	—	2	2	no	—
177	C₄×C₄₃	—	2	2	no	—
178	C₄×C₄₃	—	4	4	no	—
179	C₄×C₄₃	—	4	4	no	—
180	C₄×C₄₃	C₄×#41	1	1	yes	characteristic
181	C₄×C₄₃	C₄×#41	1	1	yes	characteristic
182	C₄×C₄₃	#529	1	1	yes	characteristic
183	C₄×C₄₃	#529	1	1	yes	characteristic
184	Dic₄₃	—	2	4	no	—
185	Dic₄₃	—	4	2	no	—
186	Dic₄₃	—	8	2	no	—
187	Dic₄₃	—	8	2	no	—
188	C₂×#487	D₄₃	1	1	yes	characteristic
189	#2331	—	43	1	no	—
190	#2327	—	43	1	no	—
191	#1993	—	43	1	no	—
192	#1960	—	43	1	no	—
193	#1935	—	43	2	no	—
194	#1933	—	43	2	no	—
195	#1835	—	43	1	no	—
196	#1763	—	43	2	no	—
197	C₄×#103	—	43	2	no	—
198	#1726	—	43	1	no	—
199	C₂×C₄₃	C₄×#124	1	1	yes	characteristic
200	C₂×C₄₃	#2949	1	1	yes	characteristic
201	C₂×C₄₃	#2950	1	1	yes	characteristic
202	C₂×C₄₃	#3137	1	2	yes	—
203	C₂×C₄₃	#3178	1	2	yes	—
204	C₂×C₄⋊Q₈	C₂×D₄₃	1	1	yes	characteristic
205	C₂×C₄⋊Q₈	C₂×D₄₃	1	1	yes	characteristic
206	C₂×C₄×Q₈	—	43	2	no	—
207	#487	Dic₄₃	1	8	yes	complemented
208	C₂×#103	—	43	2	no	—
209	C₂×#103	—	43	2	no	—
210	C₂×#103	C₂×D₄₃	1	2	yes	—
211	C₂×#103	C₂×D₄₃	1	2	yes	—
212	C₂×#97	—	43	1	no	—
213	C₂×#97	—	43	1	no	—
214	C₂×#97	—	43	1	no	—
215	C₂×C₄×C₈	—	43	1	no	—
216	C₂×C₄×C₈	—	43	1	no	—
217	C₂×C₄×C₈	C₂×D₄₃	1	1	yes	characteristic
218	#391	—	43	2	no	—
219	#391	—	86	2	no	—
220	#389	—	43	2	no	—
221	#387	—	86	2	no	—
222	#387	—	86	2	no	—
223	#387	—	86	2	no	—
224	#382	—	43	1	no	—
225	#382	—	43	1	no	—
226	C₄×C₄⋊C₄	—	43	2	no	—
227	#376	—	43	1	no	—
228	#340	—	43	2	no	—
229	#340	—	43	2	no	—
230	#340	—	43	2	no	—
231	#340	—	43	2	no	—
232	C₄₃	#21645	1	1	yes	characteristic, complemented, Sylow
233	C₂²×Q₈	—	2	1	no	—
234	C₂²×Q₈	C₂²×D₄₃	1	2	yes	—
235	C₂²×C₈	—	43	1	no	—
236	C₂²×C₈	—	43	1	no	—
237	C₂²×C₈	C₂²×D₄₃	1	1	yes	characteristic
238	C₂²×C₈	C₂²×D₄₃	1	1	yes	characteristic
239	C₄⋊Q₈	—	2	2	no	—
240	C₄⋊Q₈	—	2	2	no	—
241	C₄⋊Q₈	C₂×Dic₄₃	1	4	yes	—
242	C₄⋊Q₈	C₂×Dic₄₃	1	4	yes	—
243	C₄×Q₈	—	86	4	no	—
244	C₄×Q₈	—	86	4	no	—
245	C₂×C₄⋊C₄	—	2	1	no	—
246	C₂×C₄⋊C₄	—	2	1	no	—
247	C₂×C₄⋊C₄	—	2	1	no	—
248	C₂×C₄⋊C₄	—	43	2	no	—
249	C₂×C₄⋊C₄	—	43	2	no	—
250	C₂×C₄⋊C₄	—	86	2	no	—
251	C₂×C₄⋊C₄	—	86	2	no	—
252	C₂×C₄⋊C₄	C₂²×D₄₃	1	2	yes	—
253	C₂×C₄²	—	43	1	no	—
254	C₂×C₄²	—	43	1	no	—
255	C₂×C₄²	—	86	2	no	—
256	C₂×C₄²	—	86	2	no	—
257	C₂×C₄²	C₂²×D₄₃	1	1	yes	characteristic
258	#103	—	86	4	no	—
259	#103	—	86	4	no	—
260	#103	C₂×Dic₄₃	1	8	yes	—
261	#103	C₂×Dic₄₃	1	8	yes	—
262	#97	—	86	2	no	—
263	#97	—	86	2	no	—
264	#97	—	86	2	no	—
265	C₄×C₈	—	86	2	no	—
266	C₄×C₈	—	86	2	no	—
267	C₄×C₈	C₂×Dic₄₃	1	4	yes	—
268	#95	—	43	2	no	—
269	#95	—	43	2	no	—
270	#95	—	86	2	no	—
271	#95	—	86	4	no	—
272	#95	—	86	2	no	—
273	#95	—	86	1	no	—
274	#95	—	86	1	no	—
275	#95	—	86	1	no	—
276	#95	—	86	1	no	—
277	C₂×Q₈	—	2	4	no	—
278	C₂×Q₈	—	2	2	no	—
279	C₂×Q₈	—	2	2	no	—
280	C₂×Q₈	—	2	2	no	—
281	C₂×Q₈	—	2	2	no	—
282	C₂×Q₈	—	2	2	no	—
283	C₂×Q₈	—	8	2	no	—
284	C₂×Q₈	C₂²×Dic₄₃	1	4	yes	—
285	C₂²×C₄	—	2	2	no	—
286	C₂²×C₄	—	2	2	no	—
287	C₂²×C₄	—	43	2	no	—
288	C₂²×C₄	—	43	2	no	—
289	C₂²×C₄	—	86	2	no	—
290	C₂²×C₄	—	86	2	no	—
291	C₂²×C₄	D₄×D₄₃	1	1	yes	characteristic
292	C₂²×C₄	D₄×D₄₃	1	1	yes	characteristic
293	C₂²×C₄	C₂³×D₄₃	1	1	yes	characteristic
294	C₂×C₈	—	2	1	no	—
295	C₂×C₈	—	2	1	no	—
296	C₂×C₈	—	2	1	no	—
297	C₂×C₈	—	2	1	no	—
298	C₂×C₈	—	43	2	no	—
299	C₂×C₈	—	43	2	no	—
300	C₂×C₈	—	86	1	no	—
301	C₂×C₈	—	86	1	no	—
302	C₂×C₈	—	86	1	no	—
303	C₂×C₈	—	86	1	no	—
304	C₂×C₈	C₂²×Dic₄₃	1	2	yes	—
305	C₂×C₈	C₂²×Dic₄₃	1	2	yes	—
306	C₄⋊C₄	—	2	4	no	—
307	C₄⋊C₄	—	2	4	no	—
308	C₄⋊C₄	—	2	4	no	—
309	C₄⋊C₄	—	2	2	no	—
310	C₄⋊C₄	—	86	2	no	—
311	C₄⋊C₄	—	86	2	no	—
312	C₄⋊C₄	—	86	2	no	—
313	C₄⋊C₄	—	86	2	no	—
314	C₄⋊C₄	—	344	2	no	—
315	C₄⋊C₄	—	172	4	no	—
316	C₄⋊C₄	C₂²×Dic₄₃	1	4	yes	—
317	C₄²	—	43	2	no	—
318	C₄²	—	43	2	no	—
319	C₄²	—	86	1	no	—
320	C₄²	—	86	1	no	—
321	C₄²	—	344	2	no	—
322	C₄²	—	172	4	no	—
323	C₄²	#10621900	1	2	yes	—
324	C₄²	C₂²×Dic₄₃	1	2	yes	—
325	C₂³	C₂×D₄×D₄₃	1	1	yes	characteristic
326	Q₈	—	2	4	no	—
327	Q₈	—	2	4	no	—
328	Q₈	—	4	2	no	—
329	Q₈	—	4	2	no	—
330	Q₈	—	8	4	no	—
331	C₂×C₄	—	2	2	no	—
332	C₂×C₄	—	2	2	no	—
333	C₂×C₄	—	2	4	no	—
334	C₂×C₄	—	2	4	no	—
335	C₂×C₄	—	4	2	no	—
336	C₂×C₄	—	4	2	no	—
337	C₂×C₄	—	4	2	no	—
338	C₂×C₄	—	4	2	no	—
339	C₂×C₄	—	43	4	no	—
340	C₂×C₄	—	86	2	no	—
341	C₂×C₄	—	86	2	no	—
342	C₂×C₄	—	86	2	no	—
343	C₂×C₄	—	86	2	no	—
344	C₂×C₄	—	86	2	no	—
345	C₂×C₄	—	172	2	no	—
346	C₂×C₄	—	172	2	no	—
347	C₂×C₄	—	172	2	no	—
348	C₂×C₄	—	172	2	no	—
349	C₂×C₄	—	172	2	no	—
350	C₂×C₄	—	172	2	no	—
351	C₂×C₄	D₄×Dic₄₃	1	2	yes	—
352	C₂×C₄	D₄×Dic₄₃	1	2	yes	—
353	C₂×C₄	C₂³×Dic₄₃	1	1	yes	characteristic
354	C₂×C₄	#41×D₄₃	1	1	yes	characteristic
355	C₂×C₄	#41×D₄₃	1	1	yes	characteristic
356	C₂×C₄	#41×D₄₃	1	1	yes	characteristic
357	C₂×C₄	#41×D₄₃	1	1	yes	characteristic
358	C₂×C₄	C₂×#10621900	1	1	yes	characteristic
359	C₈	—	2	2	no	—
360	C₈	—	2	2	no	—
361	C₈	—	86	2	no	—
362	C₈	—	86	2	no	—
363	C₂²	C₂×D₄×Dic₄₃	1	1	yes	characteristic
364	C₂²	#124×D₄₃	1	1	yes	characteristic
365	C₂²	#14583448	1	1	yes	characteristic
366	C₂²	#14583705	1	2	yes	—
367	C₂²	#14583712	1	2	yes	—
368	C₄	—	2	2	no	—
369	C₄	—	2	2	no	—
370	C₄	—	4	4	no	—
371	C₄	—	4	4	no	—
372	C₄	—	86	4	no	—
373	C₄	—	344	2	no	—
374	C₄	—	344	2	no	—
375	C₄	—	172	2	no	—
376	C₄	#41×Dic₄₃	1	1	yes	characteristic
377	C₄	#41×Dic₄₃	1	1	yes	characteristic
378	C₄	#14583428	1	1	yes	characteristic
379	C₄	#14583428	1	1	yes	characteristic
380	C₂	#124×Dic₄₃	1	1	yes	characteristic
381	C₂	#19143455	1	1	yes	characteristic
382	C₂	#19143463	1	1	yes	characteristic
383	C₂	#19144955	1	2	yes	—
384	C₂	#19145270	1	2	yes	—

	subgroup	G/H
center	C₂³	C₂×D₄×D₄₃
commutator	C₂×C₄×C₄₃	C₂³×C₄
Frattini	C₂²×C₄	C₂³×D₄₃
Fitting	C₂×C₄₃×#487	C₂
socle, prime radical	C₂³×C₄₃	C₂²×D₄
cosocle	C₂²×C₄×C₄₃	C₂⁴
hypercenter	C₂×#487	D₄₃
hypocenter	C₄₃	#21645
LCS	G ▷ C₂×C₄×C₄₃ ▷ C₂×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×#487

Complements

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

N	H ≅ G/N	type
36. C₄₃×#487	C₄	semidirect
207. #487	Dic₄₃	semidirect
232. C₄₃	#21645	semidirect

Maximal subgroups

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

subgroup	index
#19137344	2
#19137315	2
#19135569	2
#19135416	2
#19135287	2
#19135273	2
#19134726	2
#19134427	2
#103×Dic₄₃	2
#19134281	2
C₂×C₄₃×#487	2
#21645	43