Subgroups — #27276801

Order 13568.

492 conjugacy classes of subgroups.

The subgroup lattice has 358 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20277192	C₂	1	1	yes	maximal, characteristic
2	#20274867	C₂	1	1	yes	maximal, characteristic
3	C₂×#16810355	C₂	1	2	yes	maximal
4	#20272555	C₂	1	2	yes	maximal
5	C₄×C₄⋊C₈×C₅₃	C₂	1	1	yes	maximal, characteristic
6	C₂×C₄×Dic₁₀₆	C₂²	1	1	yes	characteristic
7	C₂×C₄⋊#88322	—	2	1	no	—
8	C₂×C₄⋊#88322	—	2	1	no	—
9	C₂×C₄⋊#88322	C₂²	1	2	yes	—
10	#16810529	C₄	1	1	yes	characteristic
11	#16810529	C₄	1	1	yes	characteristic
12	#16810521	C₄	1	1	yes	characteristic
13	#16810521	C₄	1	1	yes	characteristic
14	#16810507	—	2	1	no	—
15	#16810507	—	2	1	no	—
16	C₄×#11719919	—	2	1	no	—
17	C₄×C₄⋊Dic₅₃	C₂²	1	1	yes	characteristic
18	#16810469	—	2	1	no	—
19	#16810355	—	2	4	no	—
20	C₂×C₄⋊C₈×C₅₃	—	2	1	no	—
21	C₂×C₄×C₈×C₅₃	C₂²	1	2	yes	—
22	C₄³×C₅₃	C₂²	1	1	yes	characteristic
23	C₅₃×#336	—	2	1	no	—
24	C₂²×Dic₁₀₆	C₂×C₄	1	1	yes	characteristic
25	C₂²×#88322	—	2	2	no	—
26	C₄×Dic₁₀₆	—	2	1	no	—
27	C₄×Dic₁₀₆	—	2	1	no	—
28	C₄×Dic₁₀₆	D₄	1	2	yes	—
29	C₄×Dic₁₀₆	Q₈	1	2	yes	—
30	C₂×C₄⋊Dic₅₃	C₂×C₄	1	1	yes	characteristic
31	C₂×C₄⋊Dic₅₃	C₂×C₄	1	1	yes	characteristic
32	C₂×C₄⋊Dic₅₃	C₂×C₄	1	1	yes	characteristic
33	C₂×#11719919	—	2	1	no	—
34	C₂×#11719919	—	2	1	no	—
35	C₂×#11719919	—	2	1	no	—
36	C₂×C₄×Dic₅₃	—	2	1	no	—
37	C₂×C₄×Dic₅₃	—	2	1	no	—
38	C₄⋊#88322	—	2	2	no	—
39	C₄⋊#88322	—	2	2	no	—
40	C₄⋊#88322	—	4	2	no	—
41	C₄⋊#88322	—	4	2	no	—
42	#13133124	—	2	1	no	—
43	#13133124	—	2	1	no	—
44	#13133124	—	2	1	no	—
45	#13133124	—	2	1	no	—
46	#13133124	—	2	1	no	—
47	#13133124	—	2	1	no	—
48	#13133124	—	2	1	no	—
49	C₂²×C₈×C₅₃	—	2	2	no	—
50	C₂×C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
51	C₂×C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
52	C₂×C₄²×C₅₃	D₄	1	1	yes	characteristic
53	C₂×C₄²×C₅₃	D₄	1	1	yes	characteristic
54	C₂×C₄²×C₅₃	D₄	1	1	yes	characteristic
55	C₂×C₄²×C₅₃	D₄	1	1	yes	characteristic
56	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
57	C₄⋊C₈×C₅₃	—	2	4	no	—
58	C₄×C₈×C₅₃	—	2	2	no	—
59	C₄×C₈×C₅₃	—	2	2	no	—
60	C₂²×C₄×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
61	C₂²×C₄×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
62	C₂²×C₄×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
63	C₂²×C₄×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
64	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
65	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
66	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
67	C₂²×Dic₅₃	—	2	1	no	—
68	C₂²×Dic₅₃	—	2	1	no	—
69	C₂²×Dic₅₃	—	2	1	no	—
70	C₂²×Dic₅₃	—	2	1	no	—
71	C₂×Dic₁₀₆	—	2	2	no	—
72	C₂×Dic₁₀₆	—	2	2	no	—
73	C₂×Dic₁₀₆	—	2	1	no	—
74	C₂×Dic₁₀₆	C₄⋊C₄	1	2	yes	—
75	C₂×C₈×C₅₃	—	2	4	no	—
76	C₂×C₈×C₅₃	—	4	2	no	—
77	C₂×C₈×C₅₃	—	4	2	no	—
78	C₄²×C₅₃	—	2	2	no	—
79	C₄²×C₅₃	—	2	2	no	—
80	C₄²×C₅₃	—	2	2	no	—
81	C₄²×C₅₃	—	2	1	no	—
82	C₄²×C₅₃	—	2	1	no	—
83	C₄²×C₅₃	—	4	1	no	—
84	C₄²×C₅₃	SD₁₆	1	2	yes	—
85	C₄²×C₅₃	Q₁₆	1	2	yes	—
86	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
87	C₄²×C₅₃	C₂×Q₈	1	1	yes	characteristic
88	C₄²×C₅₃	#41	1	1	yes	characteristic
89	C₄²×C₅₃	#41	1	1	yes	characteristic
90	C₄⋊Dic₅₃	—	2	1	no	—
91	C₄⋊Dic₅₃	—	2	2	no	—
92	C₄⋊Dic₅₃	—	2	2	no	—
93	C₄⋊Dic₅₃	C₄⋊C₄	1	2	yes	—
94	#11719919	—	2	2	no	—
95	#11719919	—	2	2	no	—
96	#11719919	—	4	2	no	—
97	#11719919	—	4	2	no	—
98	C₄×Dic₅₃	—	2	2	no	—
99	C₄×Dic₅₃	—	2	2	no	—
100	C₄×Dic₅₃	—	4	1	no	—
101	C₄×Dic₅₃	—	4	1	no	—
102	C₂×#88322	—	4	2	no	—
103	C₂×#88322	—	4	2	no	—
104	C₂×#88322	—	4	2	no	—
105	C₂³×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
106	C₂×C₄×C₅₃	—	2	1	no	—
107	C₂×C₄×C₅₃	—	2	1	no	—
108	C₂×C₄×C₅₃	—	2	1	no	—
109	C₂×C₄×C₅₃	—	2	1	no	—
110	C₂×C₄×C₅₃	—	2	1	no	—
111	C₂×C₄×C₅₃	—	2	1	no	—
112	C₂×C₄×C₅₃	—	2	1	no	—
113	C₂×C₄×C₅₃	—	2	1	no	—
114	C₂×C₄×C₅₃	—	2	1	no	—
115	C₂×C₄×C₅₃	—	2	1	no	—
116	C₂×C₄×C₅₃	—	2	1	no	—
117	C₂×C₄×C₅₃	#103	1	2	yes	—
118	C₂×C₄×C₅₃	#103	1	2	yes	—
119	C₂×C₄×C₅₃	#104	1	2	yes	—
120	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
121	C₂×C₄×C₅₃	#117	1	1	yes	characteristic
122	C₂×C₄×C₅₃	#121	1	1	yes	characteristic
123	C₂×C₄×C₅₃	#121	1	1	yes	characteristic
124	C₂×C₄×C₅₃	C₂²⋊Q₈	1	1	yes	characteristic
125	C₂×C₄×C₅₃	C₂²⋊Q₈	1	1	yes	characteristic
126	C₂×C₄×C₅₃	#131	1	2	yes	—
127	C₂×C₄×C₅₃	#131	1	2	yes	—
128	C₂×C₄×C₅₃	C₂×SD₁₆	1	1	yes	characteristic
129	C₂×C₄×C₅₃	C₂×Q₁₆	1	1	yes	characteristic
130	C₂×C₄×C₅₃	AGL(1,ℤ/8)	1	1	yes	characteristic
131	C₂×C₄×C₅₃	#137	1	1	yes	characteristic
132	C₂×Dic₅₃	—	2	2	no	—
133	C₂×Dic₅₃	—	2	2	no	—
134	C₂×Dic₅₃	—	4	1	no	—
135	C₂×Dic₅₃	—	4	1	no	—
136	C₂×Dic₅₃	—	4	1	no	—
137	C₂×Dic₅₃	—	4	1	no	—
138	C₂×Dic₅₃	—	4	1	no	—
139	C₂×Dic₅₃	—	4	1	no	—
140	C₂×Dic₅₃	—	4	1	no	—
141	C₂×Dic₅₃	—	4	1	no	—
142	C₂×Dic₅₃	—	4	1	no	—
143	C₂×Dic₅₃	—	4	1	no	—
144	Dic₁₀₆	—	2	4	no	—
145	Dic₁₀₆	—	4	2	no	—
146	C₈×C₅₃	—	4	4	no	—
147	#88322	—	8	2	no	—
148	#8568	—	53	1	no	maximal, Sylow
149	C₂²×C₅₃	#380	1	1	yes	characteristic
150	C₂²×C₅₃	#408	1	1	yes	characteristic
151	C₂²×C₅₃	#408	1	1	yes	characteristic
152	C₂²×C₅₃	C₂×#103	1	1	yes	characteristic
153	C₂²×C₅₃	#417	1	1	yes	characteristic
154	C₂²×C₅₃	C₂×#104	1	1	yes	characteristic
155	C₂²×C₅₃	#421	1	1	yes	characteristic
156	C₄×C₅₃	—	2	1	no	—
157	C₄×C₅₃	—	2	1	no	—
158	C₄×C₅₃	—	2	2	no	—
159	C₄×C₅₃	—	2	2	no	—
160	C₄×C₅₃	—	2	2	no	—
161	C₄×C₅₃	—	4	1	no	—
162	C₄×C₅₃	#424	1	2	yes	—
163	C₄×C₅₃	#433	1	2	yes	—
164	C₄×C₅₃	#460	1	1	yes	characteristic
165	C₄×C₅₃	#462	1	1	yes	characteristic
166	C₄×C₅₃	#475	1	1	yes	characteristic
167	C₄×C₅₃	#477	1	1	yes	characteristic
168	Dic₅₃	—	4	2	no	—
169	Dic₅₃	—	4	2	no	—
170	Dic₅₃	—	8	1	no	—
171	Dic₅₃	—	8	1	no	—
172	#2295	—	53	1	no	—
173	#1842	—	53	1	no	—
174	C₄×C₄⋊C₈	D₅₃	1	1	yes	characteristic
175	C₂×Q₈⋊C₈	—	53	2	no	—
176	#1279	—	53	2	no	—
177	C₂×C₅₃	#1481	1	1	yes	characteristic
178	C₂×C₅₃	#1483	1	1	yes	characteristic
179	C₂×C₅₃	#1493	1	1	yes	characteristic
180	C₂×C₅₃	#1495	1	1	yes	characteristic
181	C₂×C₅₃	#1798	1	1	yes	characteristic
182	C₂×C₅₃	#1803	1	1	yes	characteristic
183	C₂×C₅₃	#1806	1	1	yes	characteristic
184	C₂×C₄×Q₈	—	53	1	no	—
185	C₂×C₄⋊C₈	—	2	1	no	—
186	C₂×C₄⋊C₈	—	53	2	no	—
187	C₂×C₄⋊C₈	—	106	1	no	—
188	C₂×C₄⋊C₈	—	106	1	no	—
189	C₂×C₄×C₈	C₂×D₅₃	1	2	yes	—
190	#391	—	53	1	no	—
191	#391	—	53	1	no	—
192	#389	—	53	1	no	—
193	#389	—	53	1	no	—
194	#387	—	106	1	no	—
195	#387	—	106	1	no	—
196	C₄×C₄⋊C₄	—	53	1	no	—
197	C₄×C₄⋊C₄	—	106	1	no	—
198	#376	—	106	1	no	—
199	C₄³	C₂×D₅₃	1	1	yes	characteristic
200	#336	—	2	1	no	—
201	Q₈⋊C₈	—	106	4	no	—
202	C₅₃	#8568	1	1	yes	characteristic, complemented, Sylow
203	C₂²×Q₈	—	53	1	no	—
204	C₂²×C₈	—	2	2	no	—
205	C₂²×C₈	—	106	2	no	—
206	C₄×Q₈	—	53	2	no	—
207	C₄×Q₈	—	53	2	no	—
208	C₄×Q₈	—	106	1	no	—
209	C₄×Q₈	—	106	1	no	—
210	C₂×C₄⋊C₄	—	53	1	no	—
211	C₂×C₄⋊C₄	—	53	1	no	—
212	C₂×C₄⋊C₄	—	53	1	no	—
213	C₂×C₄⋊C₄	—	106	1	no	—
214	C₂×C₄⋊C₄	—	106	1	no	—
215	C₂×C₄⋊C₄	—	106	1	no	—
216	C₂×C₄²	—	106	1	no	—
217	C₂×C₄²	—	106	1	no	—
218	C₂×C₄²	C₄×D₅₃	1	1	yes	characteristic
219	C₂×C₄²	C₄×D₅₃	1	1	yes	characteristic
220	C₂×C₄²	D₂₁₂	1	1	yes	characteristic
221	C₂×C₄²	D₂₁₂	1	1	yes	characteristic
222	C₂×C₄²	C₂²⋊D₅₃	1	1	yes	characteristic
223	C₂×C₄²	C₂²⋊D₅₃	1	1	yes	characteristic
224	C₂×C₄²	C₂²×D₅₃	1	1	yes	characteristic
225	C₄⋊C₈	—	2	4	no	—
226	C₄⋊C₈	—	106	2	no	—
227	C₄⋊C₈	—	106	2	no	—
228	C₄⋊C₈	—	212	2	no	—
229	C₄⋊C₈	—	212	2	no	—
230	C₄×C₈	—	2	2	no	—
231	C₄×C₈	—	2	2	no	—
232	#95	—	106	1	no	—
233	#95	—	106	1	no	—
234	#95	—	106	1	no	—
235	#95	—	106	1	no	—
236	#95	—	106	1	no	—
237	#95	—	106	1	no	—
238	#95	—	106	1	no	—
239	C₂×Q₈	—	53	2	no	—
240	C₂×Q₈	—	106	1	no	—
241	C₂×Q₈	—	106	2	no	—
242	C₂×Q₈	—	106	2	no	—
243	C₂²×C₄	—	106	1	no	—
244	C₂²×C₄	—	106	1	no	—
245	C₂²×C₄	—	106	1	no	—
246	C₂²×C₄	—	106	1	no	—
247	C₂²×C₄	#11719921	1	1	yes	characteristic
248	C₂²×C₄	#11719921	1	1	yes	characteristic
249	C₂²×C₄	#11719921	1	1	yes	characteristic
250	C₂²×C₄	#11719921	1	1	yes	characteristic
251	C₂²×C₄	C₂×C₄×D₅₃	1	1	yes	characteristic
252	C₂²×C₄	C₂×D₂₁₂	1	1	yes	characteristic
253	C₂²×C₄	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
254	C₂×C₈	—	2	4	no	—
255	C₂×C₈	—	4	2	no	—
256	C₂×C₈	—	4	2	no	—
257	C₂×C₈	—	212	2	no	—
258	C₂×C₈	—	212	2	no	—
259	C₂×C₈	—	212	2	no	—
260	C₄⋊C₄	—	53	2	no	—
261	C₄⋊C₄	—	106	1	no	—
262	C₄⋊C₄	—	106	2	no	—
263	C₄⋊C₄	—	106	2	no	—
264	C₄⋊C₄	—	106	2	no	—
265	C₄⋊C₄	—	106	2	no	—
266	C₄⋊C₄	—	212	2	no	—
267	C₄⋊C₄	—	212	2	no	—
268	C₄²	—	2	2	no	—
269	C₄²	—	2	2	no	—
270	C₄²	—	2	2	no	—
271	C₄²	—	2	1	no	—
272	C₄²	—	2	1	no	—
273	C₄²	—	4	1	no	—
274	C₄²	—	106	2	no	—
275	C₄²	—	106	2	no	—
276	C₄²	—	212	1	no	—
277	C₄²	—	212	1	no	—
278	C₄²	#11719913	1	2	yes	—
279	C₄²	Dic₂₁₂	1	2	yes	—
280	C₄²	D₄×D₅₃	1	1	yes	characteristic
281	C₄²	#11719947	1	1	yes	characteristic
282	C₄²	Q₈×D₅₃	1	1	yes	characteristic
283	C₄²	#11719949	1	1	yes	characteristic
284	C₂³	C₂×#11719921	1	1	yes	characteristic
285	Q₈	—	106	4	no	—
286	Q₈	—	212	2	no	—
287	C₂×C₄	—	2	1	no	—
288	C₂×C₄	—	2	1	no	—
289	C₂×C₄	—	2	1	no	—
290	C₂×C₄	—	2	1	no	—
291	C₂×C₄	—	2	1	no	—
292	C₂×C₄	—	2	1	no	—
293	C₂×C₄	—	2	1	no	—
294	C₂×C₄	—	2	1	no	—
295	C₂×C₄	—	2	1	no	—
296	C₂×C₄	—	2	1	no	—
297	C₂×C₄	—	2	1	no	—
298	C₂×C₄	—	106	2	no	—
299	C₂×C₄	—	106	2	no	—
300	C₂×C₄	—	212	1	no	—
301	C₂×C₄	—	212	1	no	—
302	C₂×C₄	—	212	1	no	—
303	C₂×C₄	—	212	1	no	—
304	C₂×C₄	—	212	1	no	—
305	C₂×C₄	—	212	1	no	—
306	C₂×C₄	—	212	1	no	—
307	C₂×C₄	—	212	1	no	—
308	C₂×C₄	—	212	1	no	—
309	C₂×C₄	—	212	1	no	—
310	C₂×C₄	#13133142	1	2	yes	—
311	C₂×C₄	#13133142	1	2	yes	—
312	C₂×C₄	#13133145	1	2	yes	—
313	C₂×C₄	C₄⋊C₄×D₅₃	1	1	yes	characteristic
314	C₂×C₄	#13133175	1	1	yes	characteristic
315	C₂×C₄	#13133188	1	1	yes	characteristic
316	C₂×C₄	#13133192	1	1	yes	characteristic
317	C₂×C₄	#13133193	1	1	yes	characteristic
318	C₂×C₄	#13133197	1	1	yes	characteristic
319	C₂×C₄	#13133223	1	2	yes	—
320	C₂×C₄	#13133225	1	2	yes	—
321	C₂×C₄	C₂×#11719913	1	1	yes	characteristic
322	C₂×C₄	C₂×Dic₂₁₂	1	1	yes	characteristic
323	C₂×C₄	#13133245	1	1	yes	characteristic
324	C₂×C₄	#13133250	1	1	yes	characteristic
325	C₈	—	4	4	no	—
326	C₈	—	424	2	no	—
327	C₂²	#16810480	1	1	yes	characteristic
328	C₂²	#16810589	1	1	yes	characteristic
329	C₂²	#16810591	1	1	yes	characteristic
330	C₂²	C₂×#13133142	1	1	yes	characteristic
331	C₂²	#16810625	1	1	yes	characteristic
332	C₂²	C₂×#13133145	1	1	yes	characteristic
333	C₂²	#16810641	1	1	yes	characteristic
334	C₄	—	2	2	no	—
335	C₄	—	2	1	no	—
336	C₄	—	2	1	no	—
337	C₄	—	2	2	no	—
338	C₄	—	2	2	no	—
339	C₄	—	4	1	no	—
340	C₄	—	424	1	no	—
341	C₄	—	424	1	no	—
342	C₄	—	212	2	no	—
343	C₄	—	212	2	no	—
344	C₄	#16810648	1	2	yes	—
345	C₄	#16810683	1	2	yes	—
346	C₄	#16810824	1	1	yes	characteristic
347	C₄	#16810834	1	1	yes	characteristic
348	C₄	#16810909	1	1	yes	characteristic
349	C₄	#16810919	1	1	yes	characteristic
350	C₂	#20273119	1	1	yes	characteristic
351	C₂	#20273129	1	1	yes	characteristic
352	C₂	#20273171	1	1	yes	characteristic
353	C₂	#20273181	1	1	yes	characteristic
354	C₂	#20274696	1	1	yes	characteristic
355	C₂	#20274721	1	1	yes	characteristic
356	C₂	#20274736	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄	C₂×D₂₁₂
commutator	C₂²×C₄×C₅₃	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₅₃
Fitting	C₄×C₄⋊C₈×C₅₃	C₂
socle, prime radical	C₂³×C₅₃	C₂×C₂²⋊C₄
cosocle	C₂×C₄²×C₅₃	C₂³
hypercenter	C₄×C₄⋊C₈	D₅₃
hypocenter	C₅₃	#8568
LCS	G ▷ C₂²×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂²×C₄ ◁ C₄³ ◁ C₄×C₄⋊C₈

Complements

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

N	H ≅ G/N	type
202. C₅₃	#8568	semidirect

Maximal subgroups

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

subgroup	index
#20277192	2
#20274867	2
C₂×#16810355	2
#20272555	2
C₄×C₄⋊C₈×C₅₃	2
#8568	53