Subgroups — #27367343

Order 13568.

582 conjugacy classes of subgroups.

The subgroup lattice has 402 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20285865	C₂	1	2	yes	maximal
2	#500×D₅₃	C₂	1	1	yes	maximal, characteristic
3	#20283646	C₂	1	1	yes	maximal, characteristic
4	#20274244	C₂	1	1	yes	maximal, characteristic, complemented
5	#20274243	C₂	1	1	yes	maximal, characteristic, complemented
6	#20274240	C₂	1	1	yes	maximal, characteristic, complemented
7	#20274118	C₂	1	2	yes	maximal, complemented
8	#20274117	C₂	1	2	yes	maximal, complemented
9	#20273460	C₂	1	1	yes	maximal, characteristic
10	#20273394	C₂	1	2	yes	maximal
11	C₅₃×#1696	C₂	1	1	yes	maximal, characteristic, complemented
12	#16811177	C₂²	1	1	yes	characteristic
13	C₄⋊Q₈×D₅₃	C₂²	1	2	yes	—
14	#16811028	C₂²	1	1	yes	characteristic
15	#16811027	—	2	2	no	—
16	#16811027	—	2	1	no	—
17	#16811027	—	2	1	no	—
18	#16811026	C₂²	1	2	yes	—
19	#16810972	—	2	2	no	—
20	#16810960	—	2	2	no	—
21	#16810957	—	2	2	no	—
22	#16810919	C₂²	1	2	yes	—
23	#16810918	C₂²	1	2	yes	—
24	#16810915	C₂²	1	2	yes	—
25	#16810753	C₂²	1	1	yes	characteristic
26	#16810751	—	2	1	no	—
27	Q₈×#88322	C₂²	1	1	yes	characteristic
28	#16810747	—	2	1	no	—
29	#16810747	C₂²	1	1	yes	characteristic
30	#16810675	C₂²	1	2	yes	—
31	#107×D₅₃	—	2	1	no	—
32	#107×D₅₃	—	2	1	no	—
33	#16810648	—	2	1	no	—
34	#16810645	C₂²	1	1	yes	characteristic
35	#16810580	—	2	1	no	—
36	C₄×C₈×D₅₃	C₂²	1	1	yes	characteristic
37	#16810375	C₂²	1	2	yes	—
38	#16810374	C₂²	1	1	yes	characteristic
39	#16810372	C₂²	1	2	yes	—
40	#16810371	C₂²	1	2	yes	—
41	#16810371	C₂²	1	2	yes	—
42	C₅₃×#500	C₂²	1	1	yes	characteristic
43	C₅₃×#477	C₂²	1	2	yes	—
44	C₅₃×#446	C₂²	1	1	yes	characteristic
45	C₅₃×#334	C₂²	1	1	yes	characteristic
46	C₅₃×#333	C₂²	1	2	yes	—
47	C₂×Q₈×D₅₃	—	2	2	no	—
48	C₂×#11719912	—	2	1	no	—
49	C₂×C₈×D₅₃	D₄	1	1	yes	characteristic
50	C₂×C₈×D₅₃	D₄	1	1	yes	characteristic
51	#13133217	C₂³	1	2	yes	—
52	#13133216	—	2	2	no	—
53	#13133216	—	2	2	no	—
54	#13133216	D₄	1	1	yes	characteristic
55	#13133215	—	2	2	no	—
56	#13133215	D₄	1	1	yes	characteristic
57	#13133207	—	2	1	no	—
58	#13133197	D₄	1	1	yes	characteristic
59	#13133195	—	2	1	no	—
60	#13133193	D₄	1	1	yes	characteristic
61	C₄×Dic₁₀₆	—	2	1	no	—
62	C₄×Dic₁₀₆	C₂³	1	1	yes	characteristic
63	#13133174	—	2	1	no	—
64	C₄⋊C₄×D₅₃	—	2	2	no	—
65	C₄⋊C₄×D₅₃	—	2	2	no	—
66	C₄²×D₅₃	C₂³	1	1	yes	characteristic
67	#13133151	—	2	1	no	—
68	#13133151	—	2	1	no	—
69	#13133151	—	2	2	no	—
70	#13133150	—	2	2	no	—
71	#13133150	—	2	2	no	—
72	#13133150	—	2	2	no	—
73	#13133150	—	2	2	no	—
74	C₄⋊#88322	—	2	1	no	—
75	C₄⋊#88322	C₂³	1	1	yes	characteristic
76	C₄⋊#88322	C₂³	1	2	yes	—
77	#13133146	—	2	1	no	—
78	#13133146	—	2	1	no	—
79	#13133146	—	2	1	no	—
80	#13133142	—	2	2	no	—
81	#13133141	—	2	2	no	—
82	#13133140	—	2	2	no	—
83	#13133129	—	2	2	no	—
84	#13133127	—	2	1	no	—
85	C₄×#88322	—	2	1	no	—
86	C₄×#88322	C₂³	1	1	yes	characteristic
87	C₈×Dic₅₃	—	2	2	no	—
88	C₈×Dic₅₃	—	2	1	no	—
89	C₈×Dic₅₃	D₄	1	1	yes	characteristic
90	C₈×Dic₅₃	D₄	1	1	yes	characteristic
91	C₄⋊Q₈×C₅₃	C₂³	1	2	yes	—
92	C₄×Q₈×C₅₃	C₂³	1	1	yes	characteristic
93	#107×C₅₃	—	2	1	no	—
94	#107×C₅₃	—	2	1	no	—
95	#107×C₅₃	—	2	2	no	—
96	C₄⋊C₈×C₅₃	—	2	1	no	—
97	C₄⋊C₈×C₅₃	C₂³	1	1	yes	characteristic
98	C₄⋊C₈×C₅₃	C₂³	1	2	yes	—
99	#103×C₅₃	—	2	2	no	—
100	#97×C₅₃	—	2	1	no	—
101	C₄×C₈×C₅₃	C₂³	1	1	yes	characteristic
102	C₂×Q₈×C₅₃	—	2	2	no	—
103	C₂×Q₈×C₅₃	C₂×D₄	1	1	yes	characteristic
104	Q₈×D₅₃	—	8	4	no	—
105	C₂×C₄×D₅₃	—	4	2	no	—
106	C₂×C₄×D₅₃	C₂×D₄	1	1	yes	characteristic
107	C₂×C₄×D₅₃	C₂×D₄	1	1	yes	characteristic
108	C₂×C₄×D₅₃	#41	1	1	yes	characteristic
109	C₂×Dic₁₀₆	—	2	1	no	—
110	C₂×Dic₁₀₆	—	2	2	no	—
111	C₂×Dic₁₀₆	—	4	2	no	—
112	C₂×Dic₁₀₆	C₂×D₄	1	1	yes	characteristic
113	C₂×C₈×C₅₃	—	2	1	no	—
114	C₂×C₈×C₅₃	—	2	2	no	—
115	C₂×C₈×C₅₃	C₂×D₄	1	1	yes	characteristic
116	C₂×C₈×C₅₃	C₂×D₄	1	1	yes	characteristic
117	C₄⋊C₄×C₅₃	—	2	1	no	—
118	C₄⋊C₄×C₅₃	—	2	2	no	—
119	C₄⋊C₄×C₅₃	—	2	2	no	—
120	C₄⋊C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
121	C₄²×C₅₃	—	2	1	no	—
122	C₄²×C₅₃	C₂⁴	1	1	yes	characteristic
123	#11719921	—	2	1	no	—
124	#11719921	—	2	1	no	—
125	C₄⋊Dic₅₃	—	2	1	no	—
126	C₄⋊Dic₅₃	—	2	2	no	—
127	C₄⋊Dic₅₃	—	2	2	no	—
128	C₄⋊Dic₅₃	C₂×D₄	1	1	yes	characteristic
129	#11719919	—	2	1	no	—
130	#11719919	—	2	1	no	—
131	#11719919	—	2	1	no	—
132	#11719919	—	2	1	no	—
133	#11719919	—	2	1	no	—
134	#11719919	—	4	2	no	—
135	#11719919	—	4	2	no	—
136	C₄×Dic₅₃	—	2	1	no	—
137	C₄×Dic₅₃	C₂×D₄	1	1	yes	characteristic
138	C₄×Dic₅₃	C₂×D₄	1	1	yes	characteristic
139	C₄×Dic₅₃	#41	1	1	yes	characteristic
140	C₂×#88322	—	2	2	no	—
141	C₂×#88322	—	2	1	no	—
142	C₂×#88322	C₂×D₄	1	1	yes	characteristic
143	C₂×#88322	C₂×D₄	1	1	yes	characteristic
144	#11719912	—	4	2	no	—
145	C₈×D₅₃	—	2	2	no	—
146	C₈×D₅₃	Q₁₆	1	4	yes	—
147	C₂²×D₅₃	#121	1	1	yes	characteristic
148	Q₈×C₅₃	—	2	2	no	—
149	Q₈×C₅₃	—	8	2	no	—
150	C₂×C₄×C₅₃	—	2	1	no	—
151	C₂×C₄×C₅₃	—	2	1	no	—
152	C₂×C₄×C₅₃	—	4	2	no	—
153	C₂×C₄×C₅₃	C₂²×D₄	1	1	yes	characteristic
154	C₂×C₄×C₅₃	C₂²×D₄	1	1	yes	characteristic
155	C₂×C₄×C₅₃	C₂×#41	1	1	yes	characteristic
156	C₂×Dic₅₃	—	2	1	no	—
157	C₂×Dic₅₃	—	2	1	no	—
158	C₂×Dic₅₃	—	4	2	no	—
159	C₂×Dic₅₃	#121	1	1	yes	characteristic
160	C₂×Dic₅₃	#121	1	1	yes	characteristic
161	C₂×Dic₅₃	#121	1	1	yes	characteristic
162	C₄×D₅₃	—	2	2	no	—
163	C₄×D₅₃	—	2	2	no	—
164	C₄×D₅₃	—	8	4	no	—
165	C₄×D₅₃	C₂×Q₁₆	1	2	yes	—
166	C₄×D₅₃	#137	1	2	yes	—
167	Dic₁₀₆	—	2	2	no	—
168	Dic₁₀₆	—	4	2	no	—
169	Dic₁₀₆	—	8	4	no	—
170	Dic₁₀₆	—	8	2	no	—
171	C₈×C₅₃	—	2	1	no	—
172	C₈×C₅₃	—	4	1	no	—
173	C₈×C₅₃	—	4	2	no	—
174	C₈×C₅₃	C₂×Q₁₆	1	2	yes	—
175	#88322	—	2	1	no	—
176	#88322	—	4	1	no	—
177	#88322	—	4	2	no	—
178	#88322	C₂×Q₁₆	1	2	yes	—
179	#19209	—	53	1	no	maximal, Sylow
180	C₂²×C₅₃	C₂×#121	1	1	yes	characteristic
181	C₂×D₅₃	—	2	1	no	—
182	C₂×D₅₃	—	2	1	no	—
183	C₂×D₅₃	#462	1	2	yes	—
184	C₄×C₅₃	—	2	1	no	—
185	C₄×C₅₃	—	4	1	no	—
186	C₄×C₅₃	—	4	1	no	—
187	C₄×C₅₃	—	8	2	no	—
188	C₄×C₅₃	C₂²×Q₁₆	1	1	yes	characteristic
189	C₄×C₅₃	C₂×#137	1	1	yes	characteristic
190	C₄×C₅₃	#578	1	2	yes	—
191	Dic₅₃	—	2	1	no	—
192	Dic₅₃	—	4	1	no	—
193	Dic₅₃	—	4	1	no	—
194	Dic₅₃	—	8	2	no	—
195	Dic₅₃	#462	1	2	yes	—
196	Dic₅₃	#472	1	2	yes	—
197	#3230	—	53	2	no	—
198	C₂×#500	—	53	1	no	—
199	#2960	—	53	1	no	—
200	#1696	—	53	1	no	—
201	#1696	—	53	1	no	—
202	#1696	—	53	1	no	—
203	#1696	D₅₃	1	1	yes	characteristic, complemented
204	#1676	—	53	2	no	—
205	#1676	—	53	2	no	—
206	#1564	—	53	1	no	—
207	#1548	—	53	2	no	—
208	C₂×C₅₃	C₂×#462	1	1	yes	characteristic
209	C₂×C₅₃	#3059	1	1	yes	characteristic
210	C₂×C₅₃	C₂×#472	1	1	yes	characteristic
211	D₅₃	—	2	2	no	—
212	#533	—	53	1	no	—
213	C₂×C₄⋊Q₈	—	53	2	no	—
214	#500	—	53	2	no	—
215	#500	—	53	1	no	—
216	#500	—	106	1	no	—
217	#500	—	106	1	no	—
218	#500	—	106	2	no	—
219	#500	C₂×D₅₃	1	1	yes	characteristic
220	#487	—	106	2	no	—
221	#484	—	106	2	no	—
222	#484	—	106	2	no	—
223	#477	—	53	2	no	—
224	#477	—	53	2	no	—
225	#477	—	53	2	no	—
226	#477	C₂×D₅₃	1	2	yes	—
227	#446	—	53	1	no	—
228	#446	—	106	1	no	—
229	#446	C₂×D₅₃	1	1	yes	characteristic
230	C₈×Q₈	—	53	1	no	—
231	C₈×Q₈	—	53	1	no	—
232	C₈×Q₈	—	106	1	no	—
233	#431	—	53	2	no	—
234	C₂×#107	—	106	1	no	—
235	C₂×#107	—	106	1	no	—
236	#424	—	106	1	no	—
237	#423	—	53	1	no	—
238	#405	—	106	1	no	—
239	C₂×C₄×C₈	—	53	1	no	—
240	#335	—	53	2	no	—
241	#334	—	53	1	no	—
242	#334	C₂×D₅₃	1	1	yes	characteristic
243	#333	—	53	2	no	—
244	#333	—	53	2	no	—
245	#333	—	53	2	no	—
246	#333	C₂×D₅₃	1	2	yes	—
247	C₅₃	#19209	1	1	yes	characteristic, complemented, Sylow
248	C₂²×Q₈	—	106	2	no	—
249	C₂×M₁₆	—	106	1	no	—
250	C₂²×C₈	—	53	1	no	—
251	C₂²×C₈	—	53	1	no	—
252	C₄⋊Q₈	—	53	1	no	—
253	C₄⋊Q₈	—	53	1	no	—
254	C₄⋊Q₈	—	53	2	no	—
255	C₄⋊Q₈	—	106	2	no	—
256	C₄⋊Q₈	—	106	2	no	—
257	C₄⋊Q₈	—	106	2	no	—
258	C₄⋊Q₈	C₂²×D₅₃	1	2	yes	—
259	#125	—	106	1	no	—
260	C₂²⋊Q₈	—	53	1	no	—
261	C₂²⋊Q₈	—	53	1	no	—
262	C₂²⋊Q₈	—	106	1	no	—
263	C₄×Q₈	—	53	1	no	—
264	C₄×Q₈	—	106	1	no	—
265	C₄×Q₈	C₂²×D₅₃	1	1	yes	characteristic
266	#117	—	106	1	no	—
267	C₂×C₄⋊C₄	—	106	2	no	—
268	C₂×C₄⋊C₄	—	106	2	no	—
269	C₂×C₄²	—	53	1	no	—
270	#107	—	2	1	no	—
271	#107	—	2	1	no	—
272	#107	—	2	2	no	—
273	#107	—	106	2	no	—
274	#107	—	106	2	no	—
275	#107	—	106	2	no	—
276	#107	—	106	2	no	—
277	#107	—	106	1	no	—
278	#107	—	106	1	no	—
279	#107	—	106	2	no	—
280	C₄⋊C₈	—	2	1	no	—
281	C₄⋊C₈	—	53	1	no	—
282	C₄⋊C₈	—	53	2	no	—
283	C₄⋊C₈	—	106	1	no	—
284	C₄⋊C₈	—	106	1	no	—
285	C₄⋊C₈	—	106	1	no	—
286	C₄⋊C₈	—	106	1	no	—
287	C₄⋊C₈	C₂²×D₅₃	1	1	yes	characteristic
288	C₄⋊C₈	C₂²×D₅₃	1	2	yes	—
289	#103	—	2	2	no	—
290	#103	—	106	2	no	—
291	#103	—	106	2	no	—
292	#103	—	106	2	no	—
293	C₂²⋊C₈	—	106	2	no	—
294	#97	—	2	1	no	—
295	#97	—	106	1	no	—
296	C₄×C₈	—	53	1	no	—
297	C₄×C₈	—	53	1	no	—
298	C₄×C₈	—	53	1	no	—
299	C₄×C₈	—	106	1	no	—
300	C₄×C₈	—	106	2	no	—
301	C₄×C₈	—	106	1	no	—
302	C₄×C₈	C₂²×D₅₃	1	1	yes	characteristic
303	C₂×Q₈	—	2	2	no	—
304	C₂×Q₈	—	53	1	no	—
305	C₂×Q₈	—	106	1	no	—
306	C₂×Q₈	—	106	2	no	—
307	C₂×Q₈	—	424	4	no	—
308	C₂×Q₈	—	212	2	no	—
309	C₂×Q₈	D₄×D₅₃	1	1	yes	characteristic
310	C₂²×C₄	—	53	1	no	—
311	C₂²×C₄	—	53	1	no	—
312	C₂²×C₄	—	53	1	no	—
313	C₂²×C₄	—	212	2	no	—
314	M₁₆	—	212	2	no	—
315	C₂×C₈	—	2	1	no	—
316	C₂×C₈	—	2	2	no	—
317	C₂×C₈	—	53	1	no	—
318	C₂×C₈	—	53	4	no	—
319	C₂×C₈	—	53	1	no	—
320	C₂×C₈	—	106	2	no	—
321	C₂×C₈	—	106	1	no	—
322	C₂×C₈	—	106	2	no	—
323	C₂×C₈	D₄×D₅₃	1	1	yes	characteristic
324	C₂×C₈	D₄×D₅₃	1	1	yes	characteristic
325	C₄⋊C₄	—	2	1	no	—
326	C₄⋊C₄	—	2	2	no	—
327	C₄⋊C₄	—	2	2	no	—
328	C₄⋊C₄	—	53	1	no	—
329	C₄⋊C₄	—	106	1	no	—
330	C₄⋊C₄	—	106	1	no	—
331	C₄⋊C₄	—	106	1	no	—
332	C₄⋊C₄	—	106	1	no	—
333	C₄⋊C₄	—	106	1	no	—
334	C₄⋊C₄	—	106	1	no	—
335	C₄⋊C₄	—	106	2	no	—
336	C₄⋊C₄	—	106	2	no	—
337	C₄⋊C₄	—	212	2	no	—
338	C₄⋊C₄	—	212	2	no	—
339	C₄⋊C₄	D₄×D₅₃	1	1	yes	characteristic
340	C₂²⋊C₄	—	106	1	no	—
341	C₂²⋊C₄	—	106	1	no	—
342	C₄²	—	2	1	no	—
343	C₄²	—	53	1	no	—
344	C₄²	—	53	1	no	—
345	C₄²	—	53	1	no	—
346	C₄²	—	106	1	no	—
347	C₄²	C₂³×D₅₃	1	1	yes	characteristic
348	C₂³	—	53	1	no	—
349	Q₈	—	2	2	no	—
350	Q₈	—	8	2	no	—
351	Q₈	—	106	2	no	—
352	Q₈	—	424	2	no	—
353	Q₈	—	424	4	no	—
354	Q₈	—	212	2	no	—
355	C₂×C₄	—	2	1	no	—
356	C₂×C₄	—	2	1	no	—
357	C₂×C₄	—	4	2	no	—
358	C₂×C₄	—	53	1	no	—
359	C₂×C₄	—	53	1	no	—
360	C₂×C₄	—	53	2	no	—
361	C₂×C₄	—	53	2	no	—
362	C₂×C₄	—	53	1	no	—
363	C₂×C₄	—	106	2	no	—
364	C₂×C₄	—	106	2	no	—
365	C₂×C₄	—	106	1	no	—
366	C₂×C₄	—	106	1	no	—
367	C₂×C₄	—	424	4	no	—
368	C₂×C₄	—	212	2	no	—
369	C₂×C₄	C₂×D₄×D₅₃	1	1	yes	characteristic
370	C₂×C₄	C₂×D₄×D₅₃	1	1	yes	characteristic
371	C₂×C₄	#41×D₅₃	1	1	yes	characteristic
372	C₈	—	2	1	no	—
373	C₈	—	4	1	no	—
374	C₈	—	4	2	no	—
375	C₈	—	53	2	no	—
376	C₈	—	106	1	no	—
377	C₈	—	212	1	no	—
378	C₈	—	212	2	no	—
379	C₈	Q₁₆×D₅₃	1	2	yes	—
380	C₂²	—	53	2	no	—
381	C₂²	—	106	1	no	—
382	C₂²	—	106	1	no	—
383	C₂²	#121×D₅₃	1	1	yes	characteristic
384	C₄	—	2	1	no	—
385	C₄	—	4	1	no	—
386	C₄	—	4	1	no	—
387	C₄	—	8	2	no	—
388	C₄	—	53	2	no	—
389	C₄	—	53	2	no	—
390	C₄	—	106	1	no	—
391	C₄	—	424	2	no	—
392	C₄	—	212	1	no	—
393	C₄	—	212	1	no	—
394	C₄	C₂×Q₁₆×D₅₃	1	1	yes	characteristic
395	C₄	#137×D₅₃	1	1	yes	characteristic
396	C₄	#16811439	1	2	yes	—
397	C₂	—	106	2	no	—
398	C₂	#462×D₅₃	1	1	yes	characteristic
399	C₂	#20284368	1	1	yes	characteristic
400	C₂	#472×D₅₃	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	#121×D₅₃
commutator, cosocle	C₄²×C₅₃	C₂⁴
Frattini	C₄²	C₂³×D₅₃
Fitting	C₅₃×#1696	C₂
socle	C₂²×C₅₃	C₂×#121
prime radical	C₂²×D₅₃	#121
hypercenter	#1696	D₅₃
hypocenter	C₅₃	#19209
LCS	G ▷ C₄²×C₅₃ ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #1696

Complements

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

N	H ≅ G/N	type
4. #20274244	C₂	semidirect
5. #20274243	C₂	semidirect
6. #20274240	C₂	semidirect
7. #20274118	C₂	semidirect
8. #20274117	C₂	semidirect
11. C₅₃×#1696	C₂	semidirect
203. #1696	D₅₃	semidirect
247. C₅₃	#19209	semidirect

Maximal subgroups

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

subgroup	index
#20285865	2
#500×D₅₃	2
#20283646	2
#20274244	2
#20274243	2
#20274240	2
#20274118	2
#20274117	2
#20273460	2
#20273394	2
C₅₃×#1696	2
#19209	53