Subgroups — #25105394

Order 11008.

750 conjugacy classes of subgroups.

The subgroup lattice has 356 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#14583298	C₂	1	1	yes	maximal, characteristic
2	C₄×#13063317	C₂	1	1	yes	maximal, characteristic
3	#19135561	C₂	1	1	yes	maximal, characteristic
4	#19135559	C₂	1	1	yes	maximal, characteristic
5	#19135335	C₂	1	2	yes	maximal
6	#19135330	C₂	1	2	yes	maximal
7	#19134757	C₂	1	1	yes	maximal, characteristic
8	#19134493	C₂	1	2	yes	maximal
9	#19134493	C₂	1	2	yes	maximal
10	C₄×#13063236	C₂	1	1	yes	maximal, characteristic
11	C₄×#125×C₄₃	C₂	1	1	yes	maximal, characteristic
12	C₂×#13063317	C₂²	1	1	yes	characteristic
13	#14583298	C₄	1	8	yes	complemented
14	C₂×#13063258	C₂²	1	2	yes	—
15	C₂×#13063258	C₂²	1	2	yes	—
16	C₂×#13063256	C₂²	1	2	yes	—
17	C₂×#13063256	C₂²	1	2	yes	—
18	C₂×#13063236	C₂²	1	1	yes	characteristic
19	C₂×#13063236	C₂²	1	1	yes	characteristic
20	C₂×#13063236	C₂²	1	1	yes	characteristic
21	C₂×#13063236	C₂²	1	1	yes	characteristic
22	C₂×C₄×C₄₃⋊C₈	C₂²	1	1	yes	characteristic
23	C₂×C₄×C₄₃⋊C₈	C₂²	1	1	yes	characteristic
24	#14582797	C₂²	1	2	yes	—
25	#14582783	—	2	2	no	—
26	#14582765	C₂²	1	1	yes	characteristic
27	C₄×Dic₄₃⋊C₄	—	2	2	no	—
28	C₄×C₄⋊Dic₄₃	C₂²	1	2	yes	—
29	#14582661	C₂²	1	2	yes	—
30	#14582661	C₂²	1	2	yes	—
31	#14582661	C₂²	1	2	yes	—
32	#14582661	C₂²	1	2	yes	—
33	C₂×#125×C₄₃	C₂²	1	1	yes	characteristic
34	C₄₃×#389	C₂²	1	2	yes	—
35	C₄₃×#387	—	2	2	no	—
36	C₄₃×#382	C₂²	1	1	yes	characteristic
37	C₄×C₄⋊C₄×C₄₃	—	2	2	no	—
38	C₄×C₄⋊C₄×C₄₃	C₂²	1	2	yes	—
39	C₄³×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	C₂²×C₄₃⋊C₈	C₂³	1	1	yes	characteristic
43	C₂²×C₄₃⋊C₈	C₂³	1	1	yes	characteristic
44	#13063317	—	2	2	no	—
45	#13063317	C₂×C₄	1	4	yes	—
46	C₂×C₄⋊Dic₄₃	C₂³	1	2	yes	—
47	C₂×C₄⋊Dic₄₃	C₂³	1	2	yes	—
48	C₂×Dic₄₃⋊C₄	—	2	2	no	—
49	C₂×Dic₄₃⋊C₄	—	2	2	no	—
50	C₂×C₄×Dic₄₃	—	2	2	no	—
51	#13063258	—	2	4	no	—
52	#13063258	C₂×C₄	1	8	yes	—
53	#13063256	—	2	4	no	—
54	#13063256	C₂×C₄	1	8	yes	—
55	#13063236	—	2	2	no	—
56	#13063236	—	2	2	no	—
57	#13063236	Q₈	1	4	yes	—
58	#13063236	Q₈	1	4	yes	—
59	C₄×C₄₃⋊C₈	—	2	2	no	—
60	C₄×C₄₃⋊C₈	C₂×C₄	1	4	yes	—
61	C₄₃⋊#95	—	2	2	no	—
62	C₄₃⋊#95	—	2	2	no	—
63	C₄₃⋊#95	—	2	2	no	—
64	#125×C₄₃	—	2	2	no	—
65	#125×C₄₃	C₂×C₄	1	4	yes	—
66	C₂×C₄⋊C₄×C₄₃	—	2	2	no	—
67	C₂×C₄⋊C₄×C₄₃	—	2	2	no	—
68	C₂×C₄⋊C₄×C₄₃	C₂³	1	2	yes	—
69	C₂×C₄⋊C₄×C₄₃	C₂³	1	2	yes	—
70	C₂×C₄²×C₄₃	—	2	2	no	—
71	C₂×C₄²×C₄₃	D₄	1	2	yes	—
72	C₂×C₄²×C₄₃	D₄	1	2	yes	—
73	C₂×C₄²×C₄₃	C₂³	1	1	yes	characteristic
74	C₂×C₄²×C₄₃	C₂³	1	1	yes	characteristic
75	C₂×C₄²×C₄₃	C₂³	1	1	yes	characteristic
76	#95×C₄₃	—	2	2	no	—
77	#95×C₄₃	—	2	2	no	—
78	#95×C₄₃	—	2	2	no	—
79	C₂²×C₄×C₄₃	—	2	2	no	—
80	C₂²×C₄×C₄₃	—	2	2	no	—
81	C₂²×C₄×C₄₃	C₂×D₄	1	2	yes	—
82	C₂²×C₄×C₄₃	C₂×D₄	1	2	yes	—
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	C₄⋊C₄×C₄₃	—	2	8	no	—
89	C₄⋊C₄×C₄₃	—	2	2	no	—
90	C₄⋊C₄×C₄₃	—	2	2	no	—
91	C₄⋊C₄×C₄₃	—	2	2	no	—
92	C₄⋊C₄×C₄₃	—	4	4	no	—
93	C₄⋊C₄×C₄₃	C₂²×C₄	1	4	yes	—
94	C₄²×C₄₃	—	2	1	no	—
95	C₄²×C₄₃	—	2	1	no	—
96	C₄²×C₄₃	—	4	2	no	—
97	C₄²×C₄₃	—	4	2	no	—
98	C₄²×C₄₃	—	4	4	no	—
99	C₄²×C₄₃	C₂²×C₄	1	2	yes	—
100	C₄²×C₄₃	C₂×Q₈	1	2	yes	—
101	C₄²×C₄₃	C₂×Q₈	1	2	yes	—
102	C₄²×C₄₃	#41	1	2	yes	—
103	C₄⋊Dic₄₃	—	2	2	no	—
104	C₄⋊Dic₄₃	—	2	2	no	—
105	C₄⋊Dic₄₃	—	2	2	no	—
106	C₄⋊Dic₄₃	C₂²×C₄	1	4	yes	—
107	Dic₄₃⋊C₄	—	2	8	no	—
108	Dic₄₃⋊C₄	—	4	4	no	—
109	C₄×Dic₄₃	—	4	4	no	—
110	C₂×C₄₃⋊C₈	—	2	1	no	—
111	C₂×C₄₃⋊C₈	—	2	1	no	—
112	C₂×C₄₃⋊C₈	—	2	1	no	—
113	C₂×C₄₃⋊C₈	—	2	1	no	—
114	C₂×C₄₃⋊C₈	C₂²×C₄	1	2	yes	—
115	C₂×C₄₃⋊C₈	C₂²×C₄	1	2	yes	—
116	C₂×C₄₃⋊C₈	C₂×Q₈	1	2	yes	—
117	C₂×C₄₃⋊C₈	C₂×Q₈	1	2	yes	—
118	C₂×C₄₃⋊C₈	C₂×Q₈	1	2	yes	—
119	C₂×C₄₃⋊C₈	C₂×Q₈	1	2	yes	—
120	C₂×C₄₃⋊C₈	#41	1	2	yes	—
121	C₂×C₄₃⋊C₈	#41	1	2	yes	—
122	C₂³×C₄₃	C₂²×D₄	1	1	yes	characteristic
123	C₂×C₄×C₄₃	—	2	2	no	—
124	C₂×C₄×C₄₃	—	2	2	no	—
125	C₂×C₄×C₄₃	—	2	2	no	—
126	C₂×C₄×C₄₃	—	2	2	no	—
127	C₂×C₄×C₄₃	—	2	4	no	—
128	C₂×C₄×C₄₃	—	2	2	no	—
129	C₂×C₄×C₄₃	—	2	2	no	—
130	C₂×C₄×C₄₃	—	2	2	no	—
131	C₂×C₄×C₄₃	—	4	2	no	—
132	C₂×C₄×C₄₃	—	4	2	no	—
133	C₂×C₄×C₄₃	—	4	2	no	—
134	C₂×C₄×C₄₃	—	4	2	no	—
135	C₂×C₄×C₄₃	—	4	2	no	—
136	C₂×C₄×C₄₃	C₄×D₄	1	4	yes	—
137	C₂×C₄×C₄₃	C₄⋊Q₈	1	2	yes	—
138	C₂×C₄×C₄₃	C₄⋊Q₈	1	2	yes	—
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	4	no	—
146	C₂×Dic₄₃	—	4	2	no	—
147	C₂×Dic₄₃	—	4	2	no	—
148	C₂×Dic₄₃	—	4	2	no	—
149	C₂×Dic₄₃	—	4	2	no	—
150	C₂×Dic₄₃	—	4	2	no	—
151	C₄₃⋊C₈	—	2	2	no	—
152	C₄₃⋊C₈	—	2	2	no	—
153	C₄₃⋊C₈	C₄×Q₈	1	4	yes	—
154	C₄₃⋊C₈	C₄×Q₈	1	4	yes	—
155	#21728	—	43	1	no	maximal, Sylow
156	C₂²×C₄₃	C₂×C₄×D₄	1	1	yes	characteristic
157	C₂²×C₄₃	C₂×C₄⋊Q₈	1	1	yes	characteristic
158	C₂²×C₄₃	#532	1	1	yes	characteristic
159	C₂²×C₄₃	#575	1	2	yes	—
160	C₂²×C₄₃	#575	1	2	yes	—
161	C₄×C₄₃	—	2	2	no	—
162	C₄×C₄₃	—	2	2	no	—
163	C₄×C₄₃	—	2	4	no	—
164	C₄×C₄₃	—	4	2	no	—
165	C₄×C₄₃	—	4	4	no	—
166	C₄×C₄₃	—	8	2	no	—
167	C₄×C₄₃	C₂×C₄×Q₈	1	1	yes	characteristic
168	C₄×C₄₃	C₂×C₄×Q₈	1	1	yes	characteristic
169	C₄×C₄₃	#533	1	1	yes	characteristic
170	C₄×C₄₃	#533	1	1	yes	characteristic
171	Dic₄₃	—	4	4	no	—
172	Dic₄₃	—	8	2	no	—
173	C₂×#499	—	43	1	no	—
174	C₄×#125	—	43	1	no	—
175	C₄×#125	D₄₃	1	1	yes	characteristic
176	#1991	—	43	1	no	—
177	#1991	—	43	1	no	—
178	#1944	—	43	2	no	—
179	#1943	—	43	2	no	—
180	#1846	—	43	1	no	—
181	#1776	—	43	2	no	—
182	#1776	—	43	2	no	—
183	C₄×#97	—	43	1	no	—
184	C₂×C₄₃	C₄×C₄⋊Q₈	1	1	yes	characteristic
185	C₂×C₄₃	#2943	1	2	yes	—
186	C₂×C₄₃	#3166	1	1	yes	characteristic
187	C₂×C₄₃	#3166	1	1	yes	characteristic
188	C₂×C₄₃	#3183	1	2	yes	—
189	C₂×#125	—	43	1	no	—
190	C₂×#125	C₂×D₄₃	1	1	yes	characteristic
191	#499	—	43	8	no	—
192	C₂×#107	—	43	2	no	—
193	C₂×#107	—	43	2	no	—
194	C₂×#106	—	43	2	no	—
195	C₂×#106	—	43	2	no	—
196	C₂×#97	—	43	1	no	—
197	C₂×#97	—	43	1	no	—
198	C₂×#97	—	43	1	no	—
199	C₂×#97	—	43	1	no	—
200	C₂×C₄×C₈	—	43	1	no	—
201	C₂×C₄×C₈	—	43	1	no	—
202	#389	—	43	2	no	—
203	#389	C₂×D₄₃	1	2	yes	—
204	#387	—	2	2	no	—
205	#387	—	86	2	no	—
206	#382	—	43	1	no	—
207	#382	C₂×D₄₃	1	1	yes	characteristic
208	C₄×C₄⋊C₄	—	2	2	no	—
209	C₄×C₄⋊C₄	—	43	2	no	—
210	C₄×C₄⋊C₄	—	86	2	no	—
211	C₄×C₄⋊C₄	C₂×D₄₃	1	2	yes	—
212	C₄³	C₂×D₄₃	1	1	yes	characteristic
213	#340	—	43	2	no	—
214	#340	—	43	2	no	—
215	#340	—	43	2	no	—
216	#340	—	43	2	no	—
217	C₄₃	#21728	1	1	yes	characteristic, complemented, Sylow
218	C₂²×C₈	—	43	1	no	—
219	C₂²×C₈	—	43	1	no	—
220	C₂²×C₈	—	43	1	no	—
221	C₂²×C₈	—	43	1	no	—
222	#125	—	2	2	no	—
223	#125	—	43	4	no	—
224	#125	—	86	2	no	—
225	#125	C₄×D₄₃	1	4	yes	—
226	C₂×C₄⋊C₄	—	2	2	no	—
227	C₂×C₄⋊C₄	—	2	2	no	—
228	C₂×C₄⋊C₄	—	43	2	no	—
229	C₂×C₄⋊C₄	—	43	2	no	—
230	C₂×C₄⋊C₄	—	86	2	no	—
231	C₂×C₄⋊C₄	—	86	2	no	—
232	C₂×C₄⋊C₄	C₂²×D₄₃	1	2	yes	—
233	C₂×C₄⋊C₄	C₂²×D₄₃	1	2	yes	—
234	C₂×C₄²	—	2	2	no	—
235	C₂×C₄²	—	86	2	no	—
236	C₂×C₄²	C₂²⋊D₄₃	1	2	yes	—
237	C₂×C₄²	C₂²⋊D₄₃	1	2	yes	—
238	C₂×C₄²	C₂²×D₄₃	1	1	yes	characteristic
239	C₂×C₄²	C₂²×D₄₃	1	1	yes	characteristic
240	C₂×C₄²	C₂²×D₄₃	1	1	yes	characteristic
241	#107	—	43	8	no	—
242	#107	—	86	4	no	—
243	#106	—	43	8	no	—
244	#106	—	86	4	no	—
245	#97	—	43	4	no	—
246	#97	—	43	4	no	—
247	#97	—	86	2	no	—
248	#97	—	86	2	no	—
249	C₄×C₈	—	43	4	no	—
250	C₄×C₈	—	86	2	no	—
251	#95	—	2	2	no	—
252	#95	—	2	2	no	—
253	#95	—	2	2	no	—
254	#95	—	86	2	no	—
255	#95	—	86	2	no	—
256	#95	—	86	2	no	—
257	C₂²×C₄	—	2	2	no	—
258	C₂²×C₄	—	2	2	no	—
259	C₂²×C₄	—	86	2	no	—
260	C₂²×C₄	—	86	2	no	—
261	C₂²×C₄	C₂×C₂²⋊D₄₃	1	2	yes	—
262	C₂²×C₄	C₂×C₂²⋊D₄₃	1	2	yes	—
263	C₂²×C₄	C₂×C₂²⋊D₄₃	1	1	yes	characteristic
264	C₂²×C₄	C₂×C₂²⋊D₄₃	1	1	yes	characteristic
265	C₂²×C₄	C₂³×D₄₃	1	1	yes	characteristic
266	C₂×C₈	—	43	2	no	—
267	C₂×C₈	—	43	2	no	—
268	C₂×C₈	—	43	2	no	—
269	C₂×C₈	—	43	2	no	—
270	C₂×C₈	—	43	2	no	—
271	C₂×C₈	—	43	2	no	—
272	C₂×C₈	—	43	2	no	—
273	C₂×C₈	—	43	2	no	—
274	C₂×C₈	—	86	1	no	—
275	C₂×C₈	—	86	1	no	—
276	C₂×C₈	—	86	1	no	—
277	C₂×C₈	—	86	1	no	—
278	C₄⋊C₄	—	2	8	no	—
279	C₄⋊C₄	—	2	2	no	—
280	C₄⋊C₄	—	2	2	no	—
281	C₄⋊C₄	—	2	2	no	—
282	C₄⋊C₄	—	4	4	no	—
283	C₄⋊C₄	—	43	4	no	—
284	C₄⋊C₄	—	86	8	no	—
285	C₄⋊C₄	—	86	2	no	—
286	C₄⋊C₄	—	86	2	no	—
287	C₄⋊C₄	—	86	2	no	—
288	C₄⋊C₄	—	172	4	no	—
289	C₄⋊C₄	C₂×C₄×D₄₃	1	4	yes	—
290	C₄²	—	2	1	no	—
291	C₄²	—	2	1	no	—
292	C₄²	—	4	2	no	—
293	C₄²	—	4	2	no	—
294	C₄²	—	4	4	no	—
295	C₄²	—	172	4	no	—
296	C₄²	C₂×C₄×D₄₃	1	2	yes	—
297	C₄²	#10621898	1	2	yes	—
298	C₄²	Q₈×D₄₃	1	2	yes	—
299	C₄²	Q₈×D₄₃	1	2	yes	—
300	C₂³	C₂²×C₂²⋊D₄₃	1	1	yes	characteristic
301	C₂×C₄	—	2	2	no	—
302	C₂×C₄	—	2	2	no	—
303	C₂×C₄	—	2	2	no	—
304	C₂×C₄	—	2	2	no	—
305	C₂×C₄	—	2	4	no	—
306	C₂×C₄	—	2	2	no	—
307	C₂×C₄	—	2	2	no	—
308	C₂×C₄	—	2	2	no	—
309	C₂×C₄	—	4	2	no	—
310	C₂×C₄	—	4	2	no	—
311	C₂×C₄	—	4	2	no	—
312	C₂×C₄	—	4	2	no	—
313	C₂×C₄	—	4	2	no	—
314	C₂×C₄	—	86	4	no	—
315	C₂×C₄	—	172	2	no	—
316	C₂×C₄	—	172	2	no	—
317	C₂×C₄	—	172	2	no	—
318	C₂×C₄	—	172	2	no	—
319	C₂×C₄	—	172	2	no	—
320	C₂×C₄	C₄×C₂²⋊D₄₃	1	4	yes	—
321	C₂×C₄	#13063323	1	2	yes	—
322	C₂×C₄	#13063323	1	2	yes	—
323	C₂×C₄	C₂²×C₄×D₄₃	1	1	yes	characteristic
324	C₂×C₄	C₂×Q₈×D₄₃	1	1	yes	characteristic
325	C₂×C₄	C₂×Q₈×D₄₃	1	1	yes	characteristic
326	C₂×C₄	#41×D₄₃	1	1	yes	characteristic
327	C₂×C₄	#41×D₄₃	1	1	yes	characteristic
328	C₂×C₄	C₂×#10621898	1	1	yes	characteristic
329	C₈	—	43	4	no	—
330	C₈	—	43	4	no	—
331	C₈	—	86	2	no	—
332	C₈	—	86	2	no	—
333	C₂²	C₂×C₄×C₂²⋊D₄₃	1	1	yes	characteristic
334	C₂²	C₂×#13063323	1	1	yes	characteristic
335	C₂²	#14583445	1	1	yes	characteristic
336	C₂²	#14583697	1	2	yes	—
337	C₂²	#14583697	1	2	yes	—
338	C₄	—	2	2	no	—
339	C₄	—	2	2	no	—
340	C₄	—	2	4	no	—
341	C₄	—	4	2	no	—
342	C₄	—	4	4	no	—
343	C₄	—	8	2	no	—
344	C₄	—	344	2	no	—
345	C₄	—	172	4	no	—
346	C₄	C₄×Q₈×D₄₃	1	1	yes	characteristic
347	C₄	C₄×Q₈×D₄₃	1	1	yes	characteristic
348	C₄	#14583453	1	1	yes	characteristic
349	C₄	#14583453	1	1	yes	characteristic
350	C₂	C₄×#13063323	1	1	yes	characteristic
351	C₂	#19143393	1	2	yes	—
352	C₂	#19145144	1	1	yes	characteristic
353	C₂	#19145148	1	1	yes	characteristic
354	C₂	#19145318	1	2	yes	—

	subgroup	G/H
center	C₂³	C₂²×C₂²⋊D₄₃
commutator	C₂×C₄×C₄₃	C₂³×C₄
Frattini	C₂²×C₄	C₂³×D₄₃
Fitting	C₄×#125×C₄₃	C₂
socle, prime radical	C₂³×C₄₃	C₂²×D₄
cosocle	C₂²×C₄×C₄₃	C₂⁴
hypercenter	C₄×#125	D₄₃
hypocenter	C₄₃	#21728
LCS	G ▷ C₂×C₄×C₄₃ ▷ C₂×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ C₄×#125

Complements

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

N	H ≅ G/N	type
13. #14583298	C₄	semidirect
217. C₄₃	#21728	semidirect

Maximal subgroups

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

subgroup	index
C₂×#14583298	2
C₄×#13063317	2
#19135561	2
#19135559	2
#19135335	2
#19135330	2
#19134757	2
#19134493	2
C₄×#13063236	2
C₄×#125×C₄₃	2
#21728	43