Subgroups — #25104702

Order 11008.

770 conjugacy classes of subgroups.

The subgroup lattice has 374 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#19137337	C₂	1	1	yes	maximal, characteristic
2	#19137315	C₂	1	1	yes	maximal, characteristic
3	#19135561	C₂	1	1	yes	maximal, characteristic
4	#19135398	C₂	1	1	yes	maximal, characteristic
5	#19135287	C₂	1	2	yes	maximal
6	#19135273	C₂	1	2	yes	maximal
7	#19134729	C₂	1	1	yes	maximal, characteristic
8	#19134427	C₂	1	2	yes	maximal
9	#103×Dic₄₃	C₂	1	2	yes	maximal
10	#19134283	C₂	1	1	yes	maximal, characteristic
11	C₂×C₄₃×#491	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×#13063235	C₂²	1	1	yes	characteristic
16	C₂×#13063235	C₂²	1	1	yes	characteristic
17	C₂×C₄×C₄₃⋊C₈	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	C₂²	1	2	yes	—
21	#14582798	—	2	2	no	—
22	#14582798	C₂²	1	2	yes	—
23	#14582785	—	2	2	no	—
24	#14582785	—	2	2	no	—
25	#14582785	—	2	2	no	—
26	#14582769	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×#125×C₄₃	C₂²	1	1	yes	characteristic
36	C₄₃×#491	C₄	1	8	yes	complemented
37	C₂×#103×C₄₃	C₂²	1	2	yes	—
38	C₂×#103×C₄₃	C₂²	1	2	yes	—
39	C₂×#97×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	—	2	2	no	—
47	C₂×C₄⋊Dic₄₃	C₂³	1	2	yes	—
48	C₂×C₄⋊Dic₄₃	C₂³	1	2	yes	—
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	#13063249	—	2	4	no	—
54	#13063249	—	2	4	no	—
55	#13063235	—	2	2	no	—
56	#13063235	—	2	2	no	—
57	C₄×C₄₃⋊C₈	—	2	2	no	—
58	C₈×Dic₄₃	—	2	2	no	—
59	C₈×Dic₄₃	—	2	2	no	—
60	C₄₃⋊#95	—	2	2	no	—
61	C₄₃⋊#95	—	2	2	no	—
62	C₄₃⋊#95	—	2	2	no	—
63	C₄₃⋊#95	—	2	2	no	—
64	C₄₃⋊#95	—	2	2	no	—
65	C₄₃⋊#95	D₄	1	2	yes	—
66	C₄₃⋊#95	D₄	1	2	yes	—
67	C₂²×Q₈×C₄₃	C₂³	1	2	yes	—
68	C₂²×C₈×C₄₃	C₂³	1	1	yes	characteristic
69	C₂²×C₈×C₄₃	C₂³	1	1	yes	characteristic
70	C₄⋊Q₈×C₄₃	—	2	2	no	—
71	C₄⋊Q₈×C₄₃	C₂×C₄	1	4	yes	—
72	#125×C₄₃	—	2	2	no	—
73	#125×C₄₃	C₂×C₄	1	4	yes	—
74	C₂×C₄⋊C₄×C₄₃	—	2	2	no	—
75	C₂×C₄⋊C₄×C₄₃	—	2	2	no	—
76	C₂×C₄⋊C₄×C₄₃	C₂³	1	2	yes	—
77	C₂×C₄²×C₄₃	C₂³	1	1	yes	characteristic
78	#103×C₄₃	C₂×C₄	1	8	yes	—
79	#103×C₄₃	C₂×C₄	1	8	yes	—
80	#97×C₄₃	C₂×C₄	1	4	yes	—
81	C₂×Q₈×C₄₃	—	2	2	no	—
82	C₂×Q₈×C₄₃	—	2	2	no	—
83	C₂×Q₈×C₄₃	—	2	2	no	—
84	C₂×Q₈×C₄₃	—	2	2	no	—
85	C₂×Q₈×C₄₃	—	2	2	no	—
86	C₂×Q₈×C₄₃	C₂²×C₄	1	4	yes	—
87	C₂²×C₄×C₄₃	—	2	2	no	—
88	C₂²×C₄×C₄₃	—	2	2	no	—
89	C₂²×C₄×C₄₃	C₂×D₄	1	2	yes	—
90	C₂²×C₄×C₄₃	C₂⁴	1	1	yes	characteristic
91	C₂²×Dic₄₃	—	2	2	no	—
92	C₂²×Dic₄₃	—	2	2	no	—
93	C₂²×Dic₄₃	C₂×D₄	1	1	yes	characteristic
94	C₂²×Dic₄₃	C₂×D₄	1	1	yes	characteristic
95	C₂²×Dic₄₃	C₂×D₄	1	2	yes	—
96	C₂×C₈×C₄₃	—	2	1	no	—
97	C₂×C₈×C₄₃	—	2	1	no	—
98	C₂×C₈×C₄₃	—	2	1	no	—
99	C₂×C₈×C₄₃	—	2	1	no	—
100	C₂×C₈×C₄₃	C₂²×C₄	1	2	yes	—
101	C₂×C₈×C₄₃	C₂²×C₄	1	2	yes	—
102	C₄⋊C₄×C₄₃	—	2	8	no	—
103	C₄⋊C₄×C₄₃	—	2	8	no	—
104	C₄⋊C₄×C₄₃	—	2	2	no	—
105	C₄⋊C₄×C₄₃	C₂²×C₄	1	4	yes	—
106	C₄²×C₄₃	C₂²×C₄	1	2	yes	—
107	C₄²×C₄₃	#41	1	2	yes	—
108	C₄⋊Dic₄₃	—	2	2	no	—
109	C₄⋊Dic₄₃	—	2	2	no	—
110	C₄⋊Dic₄₃	—	2	2	no	—
111	C₄⋊Dic₄₃	—	2	2	no	—
112	C₄⋊Dic₄₃	—	4	4	no	—
113	C₄⋊Dic₄₃	—	8	2	no	—
114	C₄⋊Dic₄₃	—	8	2	no	—
115	C₄×Dic₄₃	—	2	1	no	—
116	C₄×Dic₄₃	—	2	1	no	—
117	C₄×Dic₄₃	—	4	4	no	—
118	C₄×Dic₄₃	—	8	2	no	—
119	C₄×Dic₄₃	#41	1	2	yes	—
120	C₄×Dic₄₃	#41	1	2	yes	—
121	C₂×C₄₃⋊C₈	—	2	1	no	—
122	C₂×C₄₃⋊C₈	—	2	1	no	—
123	C₂×C₄₃⋊C₈	—	2	1	no	—
124	C₂×C₄₃⋊C₈	—	2	1	no	—
125	C₂×C₄₃⋊C₈	#41	1	2	yes	—
126	C₂×C₄₃⋊C₈	#41	1	2	yes	—
127	C₂³×C₄₃	C₂²×D₄	1	1	yes	characteristic
128	Q₈×C₄₃	—	2	4	no	—
129	Q₈×C₄₃	—	2	4	no	—
130	Q₈×C₄₃	—	4	2	no	—
131	Q₈×C₄₃	—	4	2	no	—
132	C₂×C₄×C₄₃	—	2	2	no	—
133	C₂×C₄×C₄₃	—	2	2	no	—
134	C₂×C₄×C₄₃	—	2	4	no	—
135	C₂×C₄×C₄₃	—	2	4	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	—	4	2	no	—
140	C₂×C₄×C₄₃	C₄×D₄	1	4	yes	—
141	C₂×C₄×C₄₃	C₂³×C₄	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×C₄×C₄₃	C₂×#41	1	1	yes	characteristic
146	C₂×C₄×C₄₃	C₂×#41	1	1	yes	characteristic
147	C₂×Dic₄₃	—	2	2	no	—
148	C₂×Dic₄₃	—	2	2	no	—
149	C₂×Dic₄₃	—	2	1	no	—
150	C₂×Dic₄₃	—	2	1	no	—
151	C₂×Dic₄₃	—	2	2	no	—
152	C₂×Dic₄₃	—	2	2	no	—
153	C₂×Dic₄₃	—	4	2	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	#124	1	4	yes	—
160	C₈×C₄₃	—	2	2	no	—
161	C₈×C₄₃	—	2	2	no	—
162	C₄₃⋊C₈	—	2	2	no	—
163	C₄₃⋊C₈	—	2	2	no	—
164	#21653	—	43	1	no	maximal, Sylow
165	C₂²×C₄₃	C₂×C₄×D₄	1	1	yes	characteristic
166	C₂²×C₄₃	C₂×#124	1	1	yes	characteristic
167	C₂²×C₄₃	#532	1	1	yes	characteristic
168	C₂²×C₄₃	#577	1	2	yes	—
169	C₂²×C₄₃	#578	1	2	yes	—
170	C₄×C₄₃	—	2	4	no	—
171	C₄×C₄₃	—	4	4	no	—
172	C₄×C₄₃	—	4	4	no	—
173	C₄×C₄₃	C₄×#41	1	1	yes	characteristic
174	C₄×C₄₃	C₄×#41	1	1	yes	characteristic
175	C₄×C₄₃	#529	1	1	yes	characteristic
176	C₄×C₄₃	#529	1	1	yes	characteristic
177	Dic₄₃	—	2	4	no	—
178	Dic₄₃	—	4	1	no	—
179	Dic₄₃	—	4	1	no	—
180	Dic₄₃	—	8	2	no	—
181	Dic₄₃	—	8	2	no	—
182	C₂×#491	D₄₃	1	1	yes	characteristic
183	#2330	—	43	1	no	—
184	#2327	—	43	1	no	—
185	#1991	—	43	1	no	—
186	#1956	—	43	1	no	—
187	#1935	—	43	2	no	—
188	#1933	—	43	2	no	—
189	#1836	—	43	1	no	—
190	#1763	—	43	2	no	—
191	C₄×#103	—	43	2	no	—
192	#1726	—	43	1	no	—
193	C₂×C₄₃	C₄×#124	1	1	yes	characteristic
194	C₂×C₄₃	#2947	1	2	yes	—
195	C₂×C₄₃	#3137	1	2	yes	—
196	C₂×C₄₃	#3174	1	1	yes	characteristic
197	C₂×C₄₃	#3177	1	1	yes	characteristic
198	C₂×C₄⋊Q₈	C₂×D₄₃	1	1	yes	characteristic
199	C₂×#125	C₂×D₄₃	1	1	yes	characteristic
200	C₂×C₄×Q₈	—	43	2	no	—
201	#491	Dic₄₃	1	8	yes	complemented
202	C₂×#103	—	43	2	no	—
203	C₂×#103	—	43	2	no	—
204	C₂×#103	C₂×D₄₃	1	2	yes	—
205	C₂×#103	C₂×D₄₃	1	2	yes	—
206	C₂×#97	—	43	1	no	—
207	C₂×#97	—	43	1	no	—
208	C₂×#97	C₂×D₄₃	1	1	yes	characteristic
209	C₂×C₄×C₈	—	43	1	no	—
210	C₂×C₄×C₈	—	43	1	no	—
211	C₂×C₄×C₈	—	43	1	no	—
212	#391	—	43	2	no	—
213	#389	—	43	2	no	—
214	#389	—	86	2	no	—
215	#387	—	86	2	no	—
216	#387	—	86	2	no	—
217	#387	—	86	2	no	—
218	#384	—	43	1	no	—
219	#382	—	43	1	no	—
220	C₄×C₄⋊C₄	—	43	2	no	—
221	#376	—	43	1	no	—
222	#340	—	43	2	no	—
223	#340	—	43	2	no	—
224	#340	—	43	2	no	—
225	#340	—	43	2	no	—
226	C₄₃	#21653	1	1	yes	characteristic, complemented, Sylow
227	C₂²×Q₈	C₂²×D₄₃	1	2	yes	—
228	C₂²×C₈	—	43	1	no	—
229	C₂²×C₈	—	43	1	no	—
230	C₂²×C₈	C₂²×D₄₃	1	1	yes	characteristic
231	C₂²×C₈	C₂²×D₄₃	1	1	yes	characteristic
232	C₄⋊Q₈	—	2	2	no	—
233	C₄⋊Q₈	C₂×Dic₄₃	1	4	yes	—
234	#125	—	2	2	no	—
235	#125	C₂×Dic₄₃	1	4	yes	—
236	C₄×Q₈	—	86	4	no	—
237	C₄×Q₈	—	86	4	no	—
238	C₂×C₄⋊C₄	—	2	2	no	—
239	C₂×C₄⋊C₄	—	2	2	no	—
240	C₂×C₄⋊C₄	—	43	2	no	—
241	C₂×C₄⋊C₄	—	43	2	no	—
242	C₂×C₄⋊C₄	—	86	2	no	—
243	C₂×C₄⋊C₄	—	86	2	no	—
244	C₂×C₄⋊C₄	—	86	2	no	—
245	C₂×C₄⋊C₄	C₂²×D₄₃	1	2	yes	—
246	C₂×C₄²	—	43	1	no	—
247	C₂×C₄²	—	43	1	no	—
248	C₂×C₄²	—	86	2	no	—
249	C₂×C₄²	—	86	2	no	—
250	C₂×C₄²	C₂²×D₄₃	1	1	yes	characteristic
251	#103	—	86	4	no	—
252	#103	—	86	4	no	—
253	#103	C₂×Dic₄₃	1	8	yes	—
254	#103	C₂×Dic₄₃	1	8	yes	—
255	#97	—	86	2	no	—
256	#97	—	86	2	no	—
257	#97	C₂×Dic₄₃	1	4	yes	—
258	C₄×C₈	—	86	2	no	—
259	C₄×C₈	—	86	2	no	—
260	C₄×C₈	—	86	2	no	—
261	#95	—	43	2	no	—
262	#95	—	43	2	no	—
263	#95	—	86	2	no	—
264	#95	—	86	2	no	—
265	#95	—	86	2	no	—
266	#95	—	86	2	no	—
267	#95	—	86	2	no	—
268	C₂×Q₈	—	2	2	no	—
269	C₂×Q₈	—	2	2	no	—
270	C₂×Q₈	—	2	2	no	—
271	C₂×Q₈	—	2	2	no	—
272	C₂×Q₈	—	2	2	no	—
273	C₂×Q₈	C₂²×Dic₄₃	1	4	yes	—
274	C₂²×C₄	—	2	2	no	—
275	C₂²×C₄	—	2	2	no	—
276	C₂²×C₄	—	43	2	no	—
277	C₂²×C₄	—	43	1	no	—
278	C₂²×C₄	—	43	1	no	—
279	C₂²×C₄	—	86	2	no	—
280	C₂²×C₄	—	86	2	no	—
281	C₂²×C₄	D₄×D₄₃	1	2	yes	—
282	C₂²×C₄	C₂³×D₄₃	1	1	yes	characteristic
283	C₂×C₈	—	2	1	no	—
284	C₂×C₈	—	2	1	no	—
285	C₂×C₈	—	2	1	no	—
286	C₂×C₈	—	2	1	no	—
287	C₂×C₈	—	43	2	no	—
288	C₂×C₈	—	43	2	no	—
289	C₂×C₈	—	86	1	no	—
290	C₂×C₈	—	86	1	no	—
291	C₂×C₈	—	86	1	no	—
292	C₂×C₈	—	86	1	no	—
293	C₂×C₈	C₂²×Dic₄₃	1	2	yes	—
294	C₂×C₈	C₂²×Dic₄₃	1	2	yes	—
295	C₄⋊C₄	—	2	8	no	—
296	C₄⋊C₄	—	2	8	no	—
297	C₄⋊C₄	—	2	2	no	—
298	C₄⋊C₄	—	86	2	no	—
299	C₄⋊C₄	—	86	2	no	—
300	C₄⋊C₄	—	86	2	no	—
301	C₄⋊C₄	—	86	2	no	—
302	C₄⋊C₄	—	344	2	no	—
303	C₄⋊C₄	—	344	2	no	—
304	C₄⋊C₄	—	172	4	no	—
305	C₄⋊C₄	C₂²×Dic₄₃	1	4	yes	—
306	C₄²	—	43	2	no	—
307	C₄²	—	43	2	no	—
308	C₄²	—	86	1	no	—
309	C₄²	—	86	1	no	—
310	C₄²	—	344	2	no	—
311	C₄²	—	172	4	no	—
312	C₄²	#10621900	1	2	yes	—
313	C₄²	C₂²×Dic₄₃	1	2	yes	—
314	C₂³	C₂×D₄×D₄₃	1	1	yes	characteristic
315	Q₈	—	2	4	no	—
316	Q₈	—	2	4	no	—
317	Q₈	—	4	2	no	—
318	Q₈	—	4	2	no	—
319	C₂×C₄	—	2	2	no	—
320	C₂×C₄	—	2	2	no	—
321	C₂×C₄	—	2	4	no	—
322	C₂×C₄	—	2	4	no	—
323	C₂×C₄	—	4	2	no	—
324	C₂×C₄	—	4	2	no	—
325	C₂×C₄	—	4	2	no	—
326	C₂×C₄	—	4	2	no	—
327	C₂×C₄	—	43	4	no	—
328	C₂×C₄	—	86	2	no	—
329	C₂×C₄	—	86	2	no	—
330	C₂×C₄	—	86	1	no	—
331	C₂×C₄	—	86	1	no	—
332	C₂×C₄	—	86	2	no	—
333	C₂×C₄	—	86	2	no	—
334	C₂×C₄	—	172	2	no	—
335	C₂×C₄	—	172	2	no	—
336	C₂×C₄	—	172	2	no	—
337	C₂×C₄	—	172	2	no	—
338	C₂×C₄	—	172	2	no	—
339	C₂×C₄	—	172	2	no	—
340	C₂×C₄	D₄×Dic₄₃	1	4	yes	—
341	C₂×C₄	C₂³×Dic₄₃	1	1	yes	characteristic
342	C₂×C₄	#41×D₄₃	1	1	yes	characteristic
343	C₂×C₄	#41×D₄₃	1	1	yes	characteristic
344	C₂×C₄	#41×D₄₃	1	1	yes	characteristic
345	C₂×C₄	#41×D₄₃	1	1	yes	characteristic
346	C₂×C₄	C₂×#10621900	1	1	yes	characteristic
347	C₈	—	2	2	no	—
348	C₈	—	2	2	no	—
349	C₈	—	86	2	no	—
350	C₈	—	86	2	no	—
351	C₂²	C₂×D₄×Dic₄₃	1	1	yes	characteristic
352	C₂²	#124×D₄₃	1	1	yes	characteristic
353	C₂²	#14583448	1	1	yes	characteristic
354	C₂²	#14583708	1	2	yes	—
355	C₂²	#14583715	1	2	yes	—
356	C₄	—	2	4	no	—
357	C₄	—	4	4	no	—
358	C₄	—	4	4	no	—
359	C₄	—	86	4	no	—
360	C₄	—	344	2	no	—
361	C₄	—	344	2	no	—
362	C₄	—	172	1	no	—
363	C₄	—	172	1	no	—
364	C₄	#41×Dic₄₃	1	1	yes	characteristic
365	C₄	#41×Dic₄₃	1	1	yes	characteristic
366	C₄	#14583428	1	1	yes	characteristic
367	C₄	#14583428	1	1	yes	characteristic
368	C₂	#124×Dic₄₃	1	1	yes	characteristic
369	C₂	#19143439	1	2	yes	—
370	C₂	#19144961	1	2	yes	—
371	C₂	#19145226	1	1	yes	characteristic
372	C₂	#19145253	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	C₂×D₄×D₄₃
commutator	C₂×C₄×C₄₃	C₂³×C₄
Frattini	C₂²×C₄	C₂³×D₄₃
Fitting	C₂×C₄₃×#491	C₂
socle, prime radical	C₂³×C₄₃	C₂²×D₄
cosocle	C₂²×C₄×C₄₃	C₂⁴
hypercenter	C₂×#491	D₄₃
hypocenter	C₄₃	#21653
LCS	G ▷ C₂×C₄×C₄₃ ▷ C₂×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×#491

Complements

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

N	H ≅ G/N	type
36. C₄₃×#491	C₄	semidirect
201. #491	Dic₄₃	semidirect
226. C₄₃	#21653	semidirect

Maximal subgroups

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

subgroup	index
#19137337	2
#19137315	2
#19135561	2
#19135398	2
#19135287	2
#19135273	2
#19134729	2
#19134427	2
#103×Dic₄₃	2
#19134283	2
C₂×C₄₃×#491	2
#21653	43