Subgroups — #27354771

Order 13568.

560 conjugacy classes of subgroups.

The subgroup lattice has 376 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20285298	C₂	1	1	yes	maximal, characteristic
2	#20283811	C₂	1	2	yes	maximal
3	#20273622	C₂	1	2	yes	maximal, complemented
4	#20273613	C₂	1	1	yes	maximal, characteristic, complemented
5	#20273531	C₂	1	2	yes	maximal, complemented
6	Q₁₆×#88322	C₂	1	1	yes	maximal, characteristic, complemented
7	#20273521	C₂	1	1	yes	maximal, characteristic, complemented
8	#20273477	C₂	1	1	yes	maximal, characteristic
9	#20273186	C₂	1	2	yes	maximal
10	#20272997	C₂	1	1	yes	maximal, characteristic
11	C₅₃×#1591	C₂	1	1	yes	maximal, characteristic, complemented
12	#16811177	C₂²	1	2	yes	—
13	#16811027	C₄	1	1	yes	characteristic
14	#16811001	C₄	1	1	yes	characteristic
15	#16810972	C₄	1	2	yes	—
16	#16810865	C₄	1	1	yes	characteristic
17	#16810863	C₄	1	2	yes	—
18	#16810861	C₄	1	1	yes	characteristic
19	#16810753	C₂²	1	2	yes	—
20	#16810751	—	2	2	no	—
21	#16810750	—	2	2	no	—
22	Q₈×#88322	C₂²	1	2	yes	—
23	#16810747	C₂²	1	2	yes	—
24	C₄×Dic₂₁₂	C₂²	1	1	yes	characteristic
25	C₄×#11719925	C₂²	1	2	yes	—
26	#16810685	—	2	2	no	—
27	#16810683	—	2	2	no	—
28	#16810679	C₂²	1	2	yes	—
29	#16810674	C₂²	1	1	yes	characteristic
30	#16810674	C₂²	1	1	yes	characteristic
31	C₄×C₈×D₅₃	C₂²	1	1	yes	characteristic
32	#16810376	C₂²	1	1	yes	characteristic
33	#16810373	C₂²	1	2	yes	—
34	#16810355	C₂²	1	2	yes	—
35	Q₈⋊#88322	C₂²	1	2	yes	—
36	#16810353	C₂²	1	2	yes	—
37	#16810345	C₂²	1	1	yes	characteristic
38	#16810345	C₂²	1	1	yes	characteristic
39	C₈×#88322	C₂²	1	1	yes	characteristic
40	C₅₃×#446	C₂²	1	2	yes	—
41	C₄×Q₁₆×C₅₃	C₂²	1	1	yes	characteristic
42	C₅₃×#335	C₂²	1	1	yes	characteristic
43	C₅₃×Q₈⋊C₈	C₂²	1	2	yes	—
44	C₈²×C₅₃	C₂²	1	1	yes	characteristic
45	C₂×Dic₂₁₂	C₂×C₄	1	1	yes	characteristic
46	C₂×#11719925	C₂×C₄	1	2	yes	—
47	C₂×C₈×D₅₃	C₂×C₄	1	1	yes	characteristic
48	C₂×C₈×D₅₃	C₂×C₄	1	1	yes	characteristic
49	#13133216	C₂×C₄	1	2	yes	—
50	#13133215	C₂×C₄	1	2	yes	—
51	#13133207	—	2	2	no	—
52	#13133197	C₂×C₄	1	2	yes	—
53	#13133195	—	2	2	no	—
54	#13133193	C₂×C₄	1	2	yes	—
55	C₄×Dic₁₀₆	—	2	2	no	—
56	C₄×Dic₁₀₆	C₂³	1	2	yes	—
57	#13133174	—	2	2	no	—
58	C₄²×D₅₃	C₂³	1	1	yes	characteristic
59	#13133151	C₂×C₄	1	1	yes	characteristic
60	#13133150	C₂×C₄	1	2	yes	—
61	C₄⋊#88322	—	2	2	no	—
62	C₄⋊#88322	C₂³	1	2	yes	—
63	#13133146	—	2	2	no	—
64	#13133146	—	2	1	no	—
65	#13133146	—	2	2	no	—
66	#13133146	D₄	1	1	yes	characteristic
67	#13133146	D₄	1	1	yes	characteristic
68	#13133142	C₂×C₄	1	2	yes	—
69	#13133141	C₂×C₄	1	2	yes	—
70	#13133140	C₂×C₄	1	2	yes	—
71	#13133129	—	2	2	no	—
72	#13133129	—	2	1	no	—
73	#13133129	D₄	1	1	yes	characteristic
74	#13133129	D₄	1	1	yes	characteristic
75	#13133127	—	2	2	no	—
76	#13133127	—	2	2	no	—
77	C₄×#88322	—	2	2	no	—
78	C₄×#88322	C₂³	1	1	yes	characteristic
79	C₄×#88322	C₂³	1	1	yes	characteristic
80	C₄×#88322	C₂³	1	1	yes	characteristic
81	C₈×Dic₅₃	—	2	2	no	—
82	C₈×Dic₅₃	C₂×C₄	1	1	yes	characteristic
83	C₈×Dic₅₃	C₂×C₄	1	1	yes	characteristic
84	C₂×Q₁₆×C₅₃	C₂×C₄	1	1	yes	characteristic
85	C₄×Q₈×C₅₃	C₂³	1	2	yes	—
86	#107×C₅₃	C₂×C₄	1	1	yes	characteristic
87	C₄⋊C₈×C₅₃	—	2	2	no	—
88	C₄⋊C₈×C₅₃	C₂³	1	2	yes	—
89	#103×C₅₃	C₂×C₄	1	2	yes	—
90	#97×C₅₃	—	2	2	no	—
91	C₄×C₈×C₅₃	C₂³	1	1	yes	characteristic
92	C₄×C₈×C₅₃	C₂³	1	1	yes	characteristic
93	C₄×C₈×C₅₃	C₂³	1	1	yes	characteristic
94	C₂×Q₈×C₅₃	C₂²×C₄	1	2	yes	—
95	C₂×C₄×D₅₃	C₂²×C₄	1	1	yes	characteristic
96	C₂×C₄×D₅₃	C₂×D₄	1	1	yes	characteristic
97	C₂×C₄×D₅₃	#41	1	1	yes	characteristic
98	C₂×Dic₁₀₆	—	2	2	no	—
99	C₂×Dic₁₀₆	C₂²×C₄	1	2	yes	—
100	Q₁₆×C₅₃	—	2	2	no	—
101	C₂×C₈×C₅₃	—	2	2	no	—
102	C₂×C₈×C₅₃	—	2	1	no	—
103	C₂×C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
104	C₂×C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
105	C₂×C₈×C₅₃	C₂×D₄	1	1	yes	characteristic
106	C₂×C₈×C₅₃	C₂×D₄	1	1	yes	characteristic
107	C₄⋊C₄×C₅₃	—	2	2	no	—
108	C₄⋊C₄×C₅₃	C₂²×C₄	1	2	yes	—
109	C₄²×C₅₃	—	2	2	no	—
110	C₄²×C₅₃	C₂⁴	1	1	yes	characteristic
111	#11719925	—	2	4	no	—
112	#11719921	—	2	2	no	—
113	#11719921	—	2	2	no	—
114	C₄⋊Dic₅₃	—	2	2	no	—
115	C₄⋊Dic₅₃	C₂²×C₄	1	2	yes	—
116	#11719919	—	2	2	no	—
117	#11719919	—	2	2	no	—
118	#11719919	—	2	2	no	—
119	#11719919	—	2	2	no	—
120	#11719919	—	2	2	no	—
121	C₄×Dic₅₃	—	2	2	no	—
122	C₄×Dic₅₃	C₂²×C₄	1	1	yes	characteristic
123	C₄×Dic₅₃	C₂×D₄	1	1	yes	characteristic
124	C₄×Dic₅₃	#41	1	1	yes	characteristic
125	C₂×#88322	—	2	2	no	—
126	C₂×#88322	—	2	1	no	—
127	C₂×#88322	C₂²×C₄	1	1	yes	characteristic
128	C₂×#88322	C₂²×C₄	1	1	yes	characteristic
129	C₂×#88322	C₂×D₄	1	1	yes	characteristic
130	C₂×#88322	C₂×D₄	1	1	yes	characteristic
131	Dic₂₁₂	—	2	2	no	—
132	C₈×D₅₃	—	2	2	no	—
133	C₈×D₅₃	—	2	2	no	—
134	C₂²×D₅₃	C₄×D₄	1	1	yes	characteristic
135	Q₈×C₅₃	—	2	2	no	—
136	Q₈×C₅₃	—	2	2	no	—
137	C₂×C₄×C₅₃	—	2	2	no	—
138	C₂×C₄×C₅₃	—	2	2	no	—
139	C₂×C₄×C₅₃	C₂³×C₄	1	1	yes	characteristic
140	C₂×C₄×C₅₃	C₂²×D₄	1	1	yes	characteristic
141	C₂×C₄×C₅₃	C₂×#41	1	1	yes	characteristic
142	C₂×Dic₅₃	—	2	2	no	—
143	C₂×Dic₅₃	—	2	2	no	—
144	C₂×Dic₅₃	C₄×D₄	1	1	yes	characteristic
145	C₂×Dic₅₃	C₄×D₄	1	1	yes	characteristic
146	C₂×Dic₅₃	C₄×D₄	1	1	yes	characteristic
147	C₄×D₅₃	—	2	1	no	—
148	C₄×D₅₃	—	2	1	no	—
149	C₄×D₅₃	—	2	1	no	—
150	C₄×D₅₃	—	2	1	no	—
151	C₄×D₅₃	—	4	1	no	—
152	Dic₁₀₆	—	2	2	no	—
153	Dic₁₀₆	—	2	2	no	—
154	Dic₁₀₆	—	8	2	no	—
155	C₈×C₅₃	—	2	1	no	—
156	C₈×C₅₃	—	2	1	no	—
157	C₈×C₅₃	—	2	1	no	—
158	C₈×C₅₃	—	4	1	no	—
159	C₈×C₅₃	—	4	2	no	—
160	C₈×C₅₃	#131	1	2	yes	—
161	#88322	—	2	1	no	—
162	#88322	—	2	1	no	—
163	#88322	—	2	1	no	—
164	#88322	—	4	1	no	—
165	#88322	—	4	2	no	—
166	#88322	#131	1	2	yes	—
167	#17972	—	53	1	no	maximal, Sylow
168	C₂²×C₅₃	C₂×C₄×D₄	1	1	yes	characteristic
169	C₂×D₅₃	—	2	1	no	—
170	C₂×D₅₃	—	2	1	no	—
171	C₂×D₅₃	—	2	1	no	—
172	C₄×C₅₃	—	2	1	no	—
173	C₄×C₅₃	—	4	2	no	—
174	C₄×C₅₃	—	4	2	no	—
175	C₄×C₅₃	C₂×#131	1	1	yes	characteristic
176	C₄×C₅₃	C₂×#131	1	1	yes	characteristic
177	C₄×C₅₃	#578	1	1	yes	characteristic
178	C₄×C₅₃	#578	1	1	yes	characteristic
179	Dic₅₃	—	2	1	no	—
180	Dic₅₃	—	2	1	no	—
181	Dic₅₃	—	2	1	no	—
182	Dic₅₃	—	4	2	no	—
183	Dic₅₃	—	4	2	no	—
184	#3170	—	53	1	no	—
185	#2985	—	53	2	no	—
186	#1593	—	53	2	no	—
187	#1591	—	53	1	no	—
188	#1591	D₅₃	1	1	yes	characteristic, complemented
189	#1579	—	53	2	no	—
190	C₈×Q₁₆	—	53	1	no	—
191	C₈×Q₁₆	—	53	1	no	—
192	#1568	—	53	1	no	—
193	#1497	—	53	2	no	—
194	#1450	—	53	1	no	—
195	C₂×C₅₃	#2931	1	1	yes	characteristic
196	C₂×C₅₃	#2950	1	1	yes	characteristic
197	C₂×C₅₃	#2957	1	1	yes	characteristic
198	D₅₃	—	4	1	no	—
199	#533	—	53	2	no	—
200	#500	—	53	1	no	—
201	#494	—	53	1	no	—
202	#487	—	53	2	no	—
203	#467	—	53	1	no	—
204	#467	—	53	1	no	—
205	#467	—	53	2	no	—
206	#446	—	53	2	no	—
207	#446	—	106	2	no	—
208	#446	—	106	2	no	—
209	#446	C₂×D₅₃	1	2	yes	—
210	C₈×Q₈	—	53	2	no	—
211	C₈×Q₈	—	53	2	no	—
212	C₄×Q₁₆	—	53	2	no	—
213	C₄×Q₁₆	—	53	1	no	—
214	C₄×Q₁₆	C₂×D₅₃	1	1	yes	characteristic
215	#433	—	106	2	no	—
216	#433	—	106	2	no	—
217	#432	—	53	2	no	—
218	#431	—	53	1	no	—
219	#431	—	53	1	no	—
220	C₂×C₄×C₈	—	53	1	no	—
221	#335	—	53	1	no	—
222	#335	C₂×D₅₃	1	1	yes	characteristic
223	#334	—	53	2	no	—
224	Q₈⋊C₈	—	53	2	no	—
225	Q₈⋊C₈	—	53	2	no	—
226	Q₈⋊C₈	—	53	2	no	—
227	Q₈⋊C₈	C₂×D₅₃	1	2	yes	—
228	#322	—	53	1	no	—
229	#322	—	53	1	no	—
230	C₈²	—	53	1	no	—
231	C₈²	C₂×D₅₃	1	1	yes	characteristic
232	C₅₃	#17972	1	1	yes	characteristic, complemented, Sylow
233	C₂×Q₁₆	—	53	2	no	—
234	C₂×Q₁₆	—	53	1	no	—
235	C₂×Q₁₆	C₄×D₅₃	1	1	yes	characteristic
236	C₂²×C₈	—	53	1	no	—
237	C₂²×C₈	—	53	1	no	—
238	C₄⋊Q₈	—	53	2	no	—
239	C₄⋊Q₈	—	53	2	no	—
240	#125	—	106	2	no	—
241	C₂²⋊Q₈	—	53	2	no	—
242	C₂²⋊Q₈	—	53	2	no	—
243	C₂²⋊Q₈	—	106	2	no	—
244	C₄×Q₈	—	53	2	no	—
245	C₄×Q₈	—	106	2	no	—
246	C₄×Q₈	C₂²×D₅₃	1	2	yes	—
247	#117	—	106	2	no	—
248	C₂×C₄²	—	53	1	no	—
249	#107	—	53	1	no	—
250	#107	—	53	2	no	—
251	#107	C₄×D₅₃	1	1	yes	characteristic
252	C₄⋊C₈	—	2	2	no	—
253	C₄⋊C₈	—	53	1	no	—
254	C₄⋊C₈	—	53	1	no	—
255	C₄⋊C₈	—	53	2	no	—
256	C₄⋊C₈	—	106	2	no	—
257	C₄⋊C₈	—	106	1	no	—
258	C₄⋊C₈	—	106	2	no	—
259	C₄⋊C₈	—	106	2	no	—
260	C₄⋊C₈	C₂²×D₅₃	1	2	yes	—
261	#103	—	53	2	no	—
262	#103	—	53	2	no	—
263	#103	—	53	2	no	—
264	#103	C₄×D₅₃	1	2	yes	—
265	C₂²⋊C₈	—	53	1	no	—
266	C₂²⋊C₈	—	53	1	no	—
267	C₂²⋊C₈	—	106	1	no	—
268	C₂²⋊C₈	—	106	2	no	—
269	#97	—	2	2	no	—
270	#97	—	106	2	no	—
271	#97	—	106	2	no	—
272	C₄×C₈	—	53	1	no	—
273	C₄×C₈	—	53	1	no	—
274	C₄×C₈	—	53	1	no	—
275	C₄×C₈	—	53	1	no	—
276	C₄×C₈	—	53	1	no	—
277	C₄×C₈	—	106	2	no	—
278	C₄×C₈	—	106	2	no	—
279	C₄×C₈	C₂²×D₅₃	1	1	yes	characteristic
280	C₄×C₈	C₂²×D₅₃	1	1	yes	characteristic
281	C₄×C₈	C₂²×D₅₃	1	1	yes	characteristic
282	C₂×Q₈	—	53	2	no	—
283	C₂×Q₈	—	106	2	no	—
284	C₂×Q₈	C₂×C₄×D₅₃	1	2	yes	—
285	C₂²×C₄	—	53	1	no	—
286	C₂²×C₄	—	53	1	no	—
287	C₂²×C₄	—	53	1	no	—
288	Q₁₆	—	2	2	no	—
289	Q₁₆	—	106	4	no	—
290	Q₁₆	—	106	2	no	—
291	C₂×C₈	—	2	2	no	—
292	C₂×C₈	—	2	1	no	—
293	C₂×C₈	—	53	1	no	—
294	C₂×C₈	—	53	1	no	—
295	C₂×C₈	—	53	1	no	—
296	C₂×C₈	—	53	1	no	—
297	C₂×C₈	—	106	2	no	—
298	C₂×C₈	—	106	2	no	—
299	C₂×C₈	—	106	2	no	—
300	C₂×C₈	—	106	1	no	—
301	C₂×C₈	C₂×C₄×D₅₃	1	1	yes	characteristic
302	C₂×C₈	C₂×C₄×D₅₃	1	1	yes	characteristic
303	C₂×C₈	D₄×D₅₃	1	1	yes	characteristic
304	C₂×C₈	D₄×D₅₃	1	1	yes	characteristic
305	C₄⋊C₄	—	2	2	no	—
306	C₄⋊C₄	—	53	2	no	—
307	C₄⋊C₄	—	106	2	no	—
308	C₄⋊C₄	—	106	2	no	—
309	C₄⋊C₄	—	106	2	no	—
310	C₄⋊C₄	—	106	2	no	—
311	C₄⋊C₄	—	106	2	no	—
312	C₄⋊C₄	—	106	2	no	—
313	C₄⋊C₄	C₂×C₄×D₅₃	1	2	yes	—
314	C₂²⋊C₄	—	106	2	no	—
315	C₂²⋊C₄	—	106	2	no	—
316	C₄²	—	2	2	no	—
317	C₄²	—	53	1	no	—
318	C₄²	—	53	1	no	—
319	C₄²	—	53	1	no	—
320	C₄²	—	106	2	no	—
321	C₄²	C₂³×D₅₃	1	1	yes	characteristic
322	C₂³	—	53	1	no	—
323	Q₈	—	2	2	no	—
324	Q₈	—	2	2	no	—
325	Q₈	—	106	2	no	—
326	Q₈	—	106	2	no	—
327	Q₈	—	424	2	no	—
328	C₂×C₄	—	2	2	no	—
329	C₂×C₄	—	2	2	no	—
330	C₂×C₄	—	53	1	no	—
331	C₂×C₄	—	53	1	no	—
332	C₂×C₄	—	53	1	no	—
333	C₂×C₄	—	106	1	no	—
334	C₂×C₄	—	106	1	no	—
335	C₂×C₄	—	106	1	no	—
336	C₂×C₄	—	106	1	no	—
337	C₂×C₄	—	106	2	no	—
338	C₂×C₄	—	106	2	no	—
339	C₂×C₄	—	212	1	no	—
340	C₂×C₄	C₂²×C₄×D₅₃	1	1	yes	characteristic
341	C₂×C₄	C₂×D₄×D₅₃	1	1	yes	characteristic
342	C₂×C₄	#41×D₅₃	1	1	yes	characteristic
343	C₈	—	2	1	no	—
344	C₈	—	2	1	no	—
345	C₈	—	2	1	no	—
346	C₈	—	4	1	no	—
347	C₈	—	4	2	no	—
348	C₈	—	53	2	no	—
349	C₈	—	106	1	no	—
350	C₈	—	106	1	no	—
351	C₈	—	106	1	no	—
352	C₈	—	212	1	no	—
353	C₈	—	212	2	no	—
354	C₈	#13133225	1	2	yes	—
355	C₂²	—	106	1	no	—
356	C₂²	—	106	1	no	—
357	C₂²	—	106	1	no	—
358	C₂²	C₄×D₄×D₅₃	1	1	yes	characteristic
359	C₄	—	2	1	no	—
360	C₄	—	4	2	no	—
361	C₄	—	4	2	no	—
362	C₄	—	106	1	no	—
363	C₄	—	106	1	no	—
364	C₄	—	106	1	no	—
365	C₄	—	212	2	no	—
366	C₄	—	212	2	no	—
367	C₄	#131×D₅₃	1	1	yes	characteristic
368	C₄	C₂×#13133225	1	1	yes	characteristic
369	C₄	#16811436	1	1	yes	characteristic
370	C₄	#16811437	1	1	yes	characteristic
371	C₂	—	212	1	no	—
372	C₂	#20283424	1	1	yes	characteristic
373	C₂	#20283578	1	1	yes	characteristic
374	C₂	#20283627	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂×D₄×D₅₃
commutator	C₂×C₄×C₅₃	C₂³×C₄
Frattini	C₄²	C₂³×D₅₃
Fitting	C₅₃×#1591	C₂
socle	C₂²×C₅₃	C₂×C₄×D₄
prime radical	C₂²×D₅₃	C₄×D₄
cosocle	C₄²×C₅₃	C₂⁴
hypercenter	#1591	D₅₃
hypocenter	C₅₃	#17972
LCS	G ▷ C₂×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂×C₄ ◁ C₄×C₈ ◁ #1591

Complements

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

N	H ≅ G/N	type
3. #20273622	C₂	semidirect
4. #20273613	C₂	semidirect
5. #20273531	C₂	semidirect
6. Q₁₆×#88322	C₂	semidirect
7. #20273521	C₂	semidirect
11. C₅₃×#1591	C₂	semidirect
188. #1591	D₅₃	semidirect
232. C₅₃	#17972	semidirect

Maximal subgroups

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

subgroup	index
#20285298	2
#20283811	2
#20273622	2
#20273613	2
#20273531	2
Q₁₆×#88322	2
#20273521	2
#20273477	2
#20273186	2
#20272997	2
C₅₃×#1591	2
#17972	53