Subgroups — #27270682

Order 13568.

490 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	#20274695	C₂	1	1	yes	maximal, characteristic
2	#20272945	C₂	1	1	yes	maximal, characteristic
3	C₅₃×#1797	C₂	1	1	yes	maximal, characteristic
4	C₂²×C₄⋊Dic₅₃	—	2	1	no	—
5	C₂×C₄⋊#88322	—	2	1	no	—
6	C₂×C₂²⋊#88322	—	2	1	no	—
7	C₂×#13133124	—	2	1	no	—
8	C₂×#13133124	C₄	1	2	yes	—
9	#16810378	C₄	1	2	yes	complemented
10	C₂²×C₄²×C₅₃	C₂²	1	1	yes	characteristic
11	C₂×C₄⋊C₈×C₅₃	C₄	1	2	yes	complemented
12	C₂×C₂²⋊C₈×C₅₃	—	2	1	no	—
13	C₅₃×#336	—	2	1	no	—
14	C₂³×Dic₅₃	—	2	2	no	—
15	C₂²×#88322	—	2	2	no	—
16	C₂×C₄⋊Dic₅₃	—	2	4	no	—
17	C₂×C₄⋊Dic₅₃	—	4	2	no	—
18	C₂×C₄⋊Dic₅₃	—	4	1	no	—
19	C₂×C₄⋊Dic₅₃	—	4	1	no	—
20	C₄⋊#88322	—	4	2	no	—
21	C₂²⋊#88322	—	4	1	no	—
22	C₂²⋊#88322	—	4	1	no	—
23	#13133124	—	4	1	no	—
24	#13133124	—	4	1	no	—
25	#13133124	—	4	2	no	—
26	C₂³×C₄×C₅₃	C₂×C₄	1	1	yes	characteristic
27	C₂³×C₄×C₅₃	D₄	1	1	yes	characteristic
28	C₂³×C₄×C₅₃	D₄	1	1	yes	characteristic
29	C₂²×C₈×C₅₃	—	2	2	no	—
30	C₂×C₄²×C₅₃	—	4	2	no	—
31	C₂×C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
32	C₂×C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
33	C₂×C₄²×C₅₃	D₄	1	1	yes	characteristic
34	C₂×C₄²×C₅₃	Q₈	1	1	yes	characteristic
35	C₄⋊C₈×C₅₃	—	4	2	no	—
36	C₂²⋊C₈×C₅₃	—	4	1	no	—
37	C₂²⋊C₈×C₅₃	—	4	1	no	—
38	C₂⁴×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
39	C₂²×C₄×C₅₃	—	2	2	no	—
40	C₂²×C₄×C₅₃	—	2	2	no	—
41	C₂²×C₄×C₅₃	—	2	1	no	—
42	C₂²×C₄×C₅₃	—	2	1	no	—
43	C₂²×C₄×C₅₃	—	4	1	no	—
44	C₂²×C₄×C₅₃	—	4	1	no	—
45	C₂²×C₄×C₅₃	—	4	1	no	—
46	C₂²×C₄×C₅₃	—	4	1	no	—
47	C₂²×C₄×C₅₃	C₄²	1	1	yes	characteristic
48	C₂²×C₄×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
49	C₂²×C₄×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
50	C₂²×C₄×C₅₃	C₄⋊C₄	1	1	yes	characteristic
51	C₂²×C₄×C₅₃	C₄⋊C₄	1	1	yes	characteristic
52	C₂²×C₄×C₅₃	C₄⋊C₄	1	1	yes	characteristic
53	C₂²×C₄×C₅₃	D₈	1	1	yes	characteristic
54	C₂²×C₄×C₅₃	D₈	1	1	yes	characteristic
55	C₂²×C₄×C₅₃	SD₁₆	1	1	yes	characteristic
56	C₂²×C₄×C₅₃	SD₁₆	1	1	yes	characteristic
57	C₂²×C₄×C₅₃	SD₁₆	1	1	yes	characteristic
58	C₂²×C₄×C₅₃	SD₁₆	1	1	yes	characteristic
59	C₂²×C₄×C₅₃	Q₁₆	1	1	yes	characteristic
60	C₂²×C₄×C₅₃	Q₁₆	1	1	yes	characteristic
61	C₂²×Dic₅₃	—	2	4	no	—
62	C₂²×Dic₅₃	—	4	2	no	—
63	C₂²×Dic₅₃	—	4	2	no	—
64	C₂²×Dic₅₃	—	4	2	no	—
65	C₂²×Dic₅₃	—	4	2	no	—
66	C₂²×Dic₅₃	—	4	2	no	—
67	C₂²×Dic₅₃	—	4	2	no	—
68	C₂×C₈×C₅₃	—	4	2	no	—
69	C₂×C₈×C₅₃	—	4	2	no	—
70	C₂×C₈×C₅₃	—	4	2	no	—
71	C₄²×C₅₃	—	2	2	no	—
72	C₄²×C₅₃	—	2	2	no	—
73	C₄²×C₅₃	—	4	1	no	—
74	C₄²×C₅₃	—	4	1	no	—
75	C₄⋊Dic₅₃	—	4	4	no	—
76	C₄⋊Dic₅₃	—	4	4	no	—
77	C₂×#88322	—	4	2	no	—
78	C₂×#88322	—	4	2	no	—
79	C₂×#88322	—	4	2	no	—
80	C₂³×C₅₃	—	2	1	no	—
81	C₂³×C₅₃	—	2	1	no	—
82	C₂³×C₅₃	—	2	1	no	—
83	C₂³×C₅₃	—	2	1	no	—
84	C₂³×C₅₃	#95	1	1	yes	characteristic
85	C₂³×C₅₃	#100	1	1	yes	characteristic
86	C₂³×C₅₃	#101	1	1	yes	characteristic
87	C₂³×C₅₃	#102	1	1	yes	characteristic
88	C₂³×C₅₃	#102	1	1	yes	characteristic
89	C₂³×C₅₃	#103	1	1	yes	characteristic
90	C₂³×C₅₃	#103	1	1	yes	characteristic
91	C₂×C₄×C₅₃	—	2	1	no	—
92	C₂×C₄×C₅₃	—	2	1	no	—
93	C₂×C₄×C₅₃	—	2	1	no	—
94	C₂×C₄×C₅₃	—	2	1	no	—
95	C₂×C₄×C₅₃	—	2	1	no	—
96	C₂×C₄×C₅₃	—	2	1	no	—
97	C₂×C₄×C₅₃	—	2	2	no	—
98	C₂×C₄×C₅₃	—	2	2	no	—
99	C₂×C₄×C₅₃	—	2	2	no	—
100	C₂×C₄×C₅₃	—	2	2	no	—
101	C₂×C₄×C₅₃	—	2	1	no	—
102	C₂×C₄×C₅₃	—	2	1	no	—
103	C₂×C₄×C₅₃	—	2	1	no	—
104	C₂×C₄×C₅₃	—	2	1	no	—
105	C₂×C₄×C₅₃	—	2	1	no	—
106	C₂×C₄×C₅₃	—	2	1	no	—
107	C₂×C₄×C₅₃	—	4	1	no	—
108	C₂×C₄×C₅₃	—	4	1	no	—
109	C₂×C₄×C₅₃	—	4	1	no	—
110	C₂×C₄×C₅₃	—	4	1	no	—
111	C₂×C₄×C₅₃	—	4	2	no	—
112	C₂×C₄×C₅₃	—	4	2	no	—
113	C₂×C₄×C₅₃	#99	1	2	yes	—
114	C₂×C₄×C₅₃	#102	1	1	yes	characteristic
115	C₂×C₄×C₅₃	#102	1	1	yes	characteristic
116	C₂×C₄×C₅₃	#103	1	1	yes	characteristic
117	C₂×C₄×C₅₃	#103	1	1	yes	characteristic
118	C₂×C₄×C₅₃	#104	1	2	yes	—
119	C₂×C₄×C₅₃	#106	1	1	yes	characteristic
120	C₂×C₄×C₅₃	#106	1	1	yes	characteristic
121	C₂×C₄×C₅₃	#107	1	1	yes	characteristic
122	C₂×C₄×C₅₃	#107	1	1	yes	characteristic
123	C₂×Dic₅₃	—	4	4	no	—
124	C₂×Dic₅₃	—	4	4	no	—
125	C₂×Dic₅₃	—	4	4	no	—
126	C₂×Dic₅₃	—	8	2	no	—
127	C₂×Dic₅₃	—	8	2	no	—
128	C₂×Dic₅₃	—	8	2	no	—
129	C₂×Dic₅₃	—	8	2	no	—
130	C₈×C₅₃	—	8	2	no	—
131	#88322	—	8	2	no	—
132	#7039	—	53	1	no	maximal, Sylow
133	C₂²×C₅₃	—	2	1	no	—
134	C₂²×C₅₃	—	2	1	no	—
135	C₂²×C₅₃	—	2	1	no	—
136	C₂²×C₅₃	—	2	1	no	—
137	C₂²×C₅₃	—	2	1	no	—
138	C₂²×C₅₃	—	2	1	no	—
139	C₂²×C₅₃	—	2	1	no	—
140	C₂²×C₅₃	—	2	1	no	—
141	C₂²×C₅₃	—	2	1	no	—
142	C₂²×C₅₃	—	2	1	no	—
143	C₂²×C₅₃	—	2	1	no	—
144	C₂²×C₅₃	—	2	1	no	—
145	C₂²×C₅₃	#331	1	1	yes	characteristic
146	C₂²×C₅₃	#332	1	1	yes	characteristic
147	C₂²×C₅₃	#332	1	1	yes	characteristic
148	C₂²×C₅₃	#333	1	1	yes	characteristic
149	C₂²×C₅₃	H(3,ℤ/4)	1	1	yes	characteristic
150	C₂²×C₅₃	#339	1	1	yes	characteristic
151	C₂²×C₅₃	#340	1	1	yes	characteristic
152	C₂²×C₅₃	#340	1	1	yes	characteristic
153	C₂²×C₅₃	#342	1	1	yes	characteristic
154	C₂²×C₅₃	#343	1	1	yes	characteristic
155	C₂²×C₅₃	#344	1	1	yes	characteristic
156	C₄×C₅₃	—	2	1	no	—
157	C₄×C₅₃	—	2	1	no	—
158	C₄×C₅₃	—	2	1	no	—
159	C₄×C₅₃	—	2	1	no	—
160	C₄×C₅₃	—	2	1	no	—
161	C₄×C₅₃	—	2	1	no	—
162	C₄×C₅₃	—	4	1	no	—
163	C₄×C₅₃	—	4	1	no	—
164	C₄×C₅₃	#327	1	2	yes	—
165	C₄×C₅₃	#328	1	2	yes	—
166	Dic₅₃	—	8	4	no	—
167	#1797	—	53	1	no	—
168	#1797	D₅₃	1	1	yes	characteristic
169	C₄⋊#95	—	53	1	no	—
170	C₂×C₅₃	—	2	1	no	—
171	C₂×C₅₃	—	2	1	no	—
172	C₂×C₅₃	—	2	1	no	—
173	C₂×C₅₃	—	2	1	no	—
174	C₂×C₅₃	#1285	1	1	yes	characteristic
175	C₂×C₅₃	#1286	1	1	yes	characteristic
176	C₂×C₅₃	#1289	1	1	yes	characteristic
177	C₂×C₅₃	#1294	1	1	yes	characteristic
178	C₂×C₅₃	#1296	1	1	yes	characteristic
179	C₂×C₅₃	#1303	1	1	yes	characteristic
180	C₂×C₅₃	#1305	1	1	yes	characteristic
181	C₂²×C₄⋊C₄	—	106	1	no	—
182	C₂²×C₄²	C₂×D₅₃	1	1	yes	characteristic
183	C₂×C₄⋊C₈	—	106	1	no	—
184	C₂×C₄⋊C₈	Dic₅₃	1	2	yes	complemented
185	C₂×C₂²⋊C₈	—	2	1	no	—
186	C₂×C₂²⋊C₈	—	106	1	no	—
187	C₂×#95	—	53	2	no	—
188	C₂×#95	—	106	1	no	—
189	#336	—	2	1	no	—
190	#336	—	53	2	no	—
191	C₅₃	#7039	1	1	yes	characteristic, complemented, Sylow
192	C₂³×C₄	—	106	2	no	—
193	C₂³×C₄	C₄×D₅₃	1	1	yes	characteristic
194	C₂³×C₄	D₂₁₂	1	1	yes	characteristic
195	C₂³×C₄	C₂²⋊D₅₃	1	1	yes	characteristic
196	C₂²×C₈	—	2	2	no	—
197	C₂²×C₈	—	106	2	no	—
198	C₂×C₄⋊C₄	—	106	4	no	—
199	C₂×C₄⋊C₄	—	212	2	no	—
200	C₂×C₄⋊C₄	—	212	1	no	—
201	C₂×C₄⋊C₄	—	212	1	no	—
202	C₂×C₄²	—	4	2	no	—
203	C₂×C₄²	Dic₁₀₆	1	1	yes	characteristic
204	C₂×C₄²	C₄×D₅₃	1	1	yes	characteristic
205	C₂×C₄²	C₂×Dic₅₃	1	1	yes	characteristic
206	C₂×C₄²	C₂²⋊D₅₃	1	1	yes	characteristic
207	C₄⋊C₈	—	4	2	no	—
208	C₄⋊C₈	—	212	2	no	—
209	C₂²⋊C₈	—	4	1	no	—
210	C₂²⋊C₈	—	4	1	no	—
211	C₂²⋊C₈	—	212	1	no	—
212	C₂²⋊C₈	—	212	1	no	—
213	#95	—	212	2	no	—
214	#95	—	212	1	no	—
215	#95	—	212	1	no	—
216	C₂⁴	#11719921	1	1	yes	characteristic
217	C₂²×C₄	—	2	2	no	—
218	C₂²×C₄	—	2	2	no	—
219	C₂²×C₄	—	2	1	no	—
220	C₂²×C₄	—	2	1	no	—
221	C₂²×C₄	—	4	1	no	—
222	C₂²×C₄	—	4	1	no	—
223	C₂²×C₄	—	4	1	no	—
224	C₂²×C₄	—	4	1	no	—
225	C₂²×C₄	—	106	4	no	—
226	C₂²×C₄	—	212	2	no	—
227	C₂²×C₄	—	212	2	no	—
228	C₂²×C₄	—	212	2	no	—
229	C₂²×C₄	—	212	2	no	—
230	C₂²×C₄	—	212	2	no	—
231	C₂²×C₄	—	212	2	no	—
232	C₂²×C₄	#11719913	1	1	yes	characteristic
233	C₂²×C₄	#11719913	1	1	yes	characteristic
234	C₂²×C₄	D₄₂₄	1	1	yes	characteristic
235	C₂²×C₄	Dic₂₁₂	1	1	yes	characteristic
236	C₂²×C₄	C₄×Dic₅₃	1	1	yes	characteristic
237	C₂²×C₄	#11719919	1	1	yes	characteristic
238	C₂²×C₄	#11719919	1	1	yes	characteristic
239	C₂²×C₄	C₄⋊Dic₅₃	1	1	yes	characteristic
240	C₂²×C₄	#11719921	1	1	yes	characteristic
241	C₂²×C₄	#11719922	1	1	yes	characteristic
242	C₂²×C₄	#11719923	1	1	yes	characteristic
243	C₂²×C₄	#11719924	1	1	yes	characteristic
244	C₂²×C₄	#11719925	1	1	yes	characteristic
245	C₂²×C₄	C₂²⋊Dic₅₃	1	1	yes	characteristic
246	C₂×C₈	—	4	2	no	—
247	C₂×C₈	—	4	2	no	—
248	C₂×C₈	—	4	2	no	—
249	C₂×C₈	—	212	2	no	—
250	C₂×C₈	—	212	2	no	—
251	C₂×C₈	—	212	2	no	—
252	C₄⋊C₄	—	212	4	no	—
253	C₄⋊C₄	—	212	4	no	—
254	C₄²	—	2	2	no	—
255	C₄²	—	2	2	no	—
256	C₄²	—	4	1	no	—
257	C₄²	—	4	1	no	—
258	C₂³	—	2	1	no	—
259	C₂³	—	2	1	no	—
260	C₂³	—	2	1	no	—
261	C₂³	—	2	1	no	—
262	C₂³	#13133124	1	1	yes	characteristic
263	C₂³	#13133133	1	1	yes	characteristic
264	C₂³	#13133135	1	1	yes	characteristic
265	C₂³	#13133137	1	1	yes	characteristic
266	C₂³	#13133139	1	1	yes	characteristic
267	C₂³	#13133140	1	1	yes	characteristic
268	C₂³	#13133142	1	1	yes	characteristic
269	C₂×C₄	—	2	1	no	—
270	C₂×C₄	—	2	1	no	—
271	C₂×C₄	—	2	1	no	—
272	C₂×C₄	—	2	1	no	—
273	C₂×C₄	—	2	1	no	—
274	C₂×C₄	—	2	1	no	—
275	C₂×C₄	—	2	2	no	—
276	C₂×C₄	—	2	2	no	—
277	C₂×C₄	—	2	2	no	—
278	C₂×C₄	—	2	2	no	—
279	C₂×C₄	—	2	1	no	—
280	C₂×C₄	—	2	1	no	—
281	C₂×C₄	—	2	1	no	—
282	C₂×C₄	—	2	1	no	—
283	C₂×C₄	—	2	1	no	—
284	C₂×C₄	—	2	1	no	—
285	C₂×C₄	—	4	1	no	—
286	C₂×C₄	—	4	1	no	—
287	C₂×C₄	—	4	1	no	—
288	C₂×C₄	—	4	1	no	—
289	C₂×C₄	—	4	2	no	—
290	C₂×C₄	—	4	2	no	—
291	C₂×C₄	—	424	2	no	—
292	C₂×C₄	—	424	2	no	—
293	C₂×C₄	—	424	2	no	—
294	C₂×C₄	—	424	2	no	—
295	C₂×C₄	—	212	4	no	—
296	C₂×C₄	—	212	4	no	—
297	C₂×C₄	—	212	4	no	—
298	C₂×C₄	#13133132	1	2	yes	—
299	C₂×C₄	#13133138	1	1	yes	characteristic
300	C₂×C₄	#13133139	1	1	yes	characteristic
301	C₂×C₄	#13133141	1	1	yes	characteristic
302	C₂×C₄	#13133142	1	1	yes	characteristic
303	C₂×C₄	#13133144	1	2	yes	—
304	C₂×C₄	#13133148	1	1	yes	characteristic
305	C₂×C₄	#13133149	1	1	yes	characteristic
306	C₂×C₄	#13133150	1	1	yes	characteristic
307	C₂×C₄	#13133151	1	1	yes	characteristic
308	C₈	—	8	2	no	—
309	C₈	—	424	2	no	—
310	C₂²	—	2	1	no	—
311	C₂²	—	2	1	no	—
312	C₂²	—	2	1	no	—
313	C₂²	—	2	1	no	—
314	C₂²	—	2	1	no	—
315	C₂²	—	2	1	no	—
316	C₂²	—	2	1	no	—
317	C₂²	—	2	1	no	—
318	C₂²	—	2	1	no	—
319	C₂²	—	2	1	no	—
320	C₂²	—	2	1	no	—
321	C₂²	—	2	1	no	—
322	C₂²	#16810366	1	1	yes	characteristic
323	C₂²	#16810368	1	1	yes	characteristic
324	C₂²	#16810370	1	1	yes	characteristic
325	C₂²	#16810371	1	1	yes	characteristic
326	C₂²	#16810380	1	1	yes	characteristic
327	C₂²	#16810382	1	1	yes	characteristic
328	C₂²	#16810385	1	1	yes	characteristic
329	C₂²	#16810386	1	1	yes	characteristic
330	C₂²	#16810390	1	1	yes	characteristic
331	C₂²	#16810391	1	1	yes	characteristic
332	C₂²	#16810395	1	1	yes	characteristic
333	C₄	—	2	1	no	—
334	C₄	—	2	1	no	—
335	C₄	—	2	1	no	—
336	C₄	—	2	1	no	—
337	C₄	—	2	1	no	—
338	C₄	—	2	1	no	—
339	C₄	—	4	1	no	—
340	C₄	—	4	1	no	—
341	C₄	—	424	4	no	—
342	C₄	#16810357	1	2	yes	—
343	C₄	#16810360	1	2	yes	—
344	C₂	—	2	1	no	—
345	C₂	—	2	1	no	—
346	C₂	—	2	1	no	—
347	C₂	—	2	1	no	—
348	C₂	#20272572	1	1	yes	characteristic
349	C₂	#20272575	1	1	yes	characteristic
350	C₂	#20272583	1	1	yes	characteristic
351	C₂	#20272593	1	1	yes	characteristic
352	C₂	#20272599	1	1	yes	characteristic
353	C₂	#20272617	1	1	yes	characteristic
354	C₂	#20272620	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	#13133124
commutator	C₂²×C₄×C₅₃	C₄²
Frattini	C₂²×C₄²	C₂×D₅₃
Fitting	C₅₃×#1797	C₂
socle	C₂³×C₅₃	#95
prime radical	C₂⁴×C₅₃	C₂²⋊C₄
cosocle	C₂²×C₄²×C₅₃	C₂²
hypercenter	#1797	D₅₃
hypocenter	C₅₃	#7039
LCS	G ▷ C₂²×C₄×C₅₃ ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ #1797

Complements

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

N	H ≅ G/N	type
9. #16810378	C₄	semidirect
11. C₂×C₄⋊C₈×C₅₃	C₄	semidirect
184. C₂×C₄⋊C₈	Dic₅₃	semidirect
191. C₅₃	#7039	semidirect

Maximal subgroups

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

subgroup	index
#20274695	2
#20272945	2
C₅₃×#1797	2
#7039	53