Subgroups — #27276800

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	#20274867	C₂	1	1	yes	maximal, characteristic
2	C₄×C₄⋊#88322	C₂	1	1	yes	maximal, characteristic
3	C₂×Q₈⋊#88322	C₂	1	2	yes	maximal
4	#20272556	C₂	1	2	yes	maximal
5	C₅₃×#2295	C₂	1	1	yes	maximal, characteristic
6	C₂×C₄⋊#88322	—	2	1	no	—
7	C₂×C₄⋊#88322	—	2	1	no	—
8	C₂×C₄⋊#88322	—	2	1	no	—
9	C₂×C₄⋊#88322	C₂²	1	2	yes	—
10	C₂×C₄×#88322	C₂²	1	2	yes	—
11	#16810378	—	2	1	no	—
12	Q₈⋊#88322	—	2	4	no	—
13	C₂×C₄×Q₈×C₅₃	C₂²	1	1	yes	characteristic
14	C₅₃×#391	C₄	1	1	yes	characteristic
15	C₅₃×#391	C₄	1	1	yes	characteristic
16	C₅₃×#389	C₄	1	1	yes	characteristic
17	C₅₃×#389	C₄	1	1	yes	characteristic
18	C₅₃×#387	—	2	1	no	—
19	C₅₃×#387	—	2	1	no	—
20	C₄×C₄⋊C₄×C₅₃	—	2	1	no	—
21	C₄×C₄⋊C₄×C₅₃	C₂²	1	1	yes	characteristic
22	C₅₃×#376	—	2	1	no	—
23	C₄³×C₅₃	C₂²	1	1	yes	characteristic
24	C₂²×#88322	—	2	2	no	—
25	C₂²×#88322	—	2	2	no	—
26	C₄⋊#88322	—	2	4	no	—
27	C₄⋊#88322	—	2	2	no	—
28	C₄⋊#88322	—	2	2	no	—
29	C₄⋊#88322	—	4	2	no	—
30	C₄⋊#88322	—	4	2	no	—
31	C₄×#88322	—	2	2	no	—
32	C₄×#88322	—	2	2	no	—
33	C₂²×Q₈×C₅₃	C₂×C₄	1	1	yes	characteristic
34	C₄×Q₈×C₅₃	—	2	1	no	—
35	C₄×Q₈×C₅₃	—	2	1	no	—
36	C₄×Q₈×C₅₃	D₄	1	2	yes	—
37	C₄×Q₈×C₅₃	Q₈	1	2	yes	—
38	C₂×C₄⋊C₄×C₅₃	—	2	1	no	—
39	C₂×C₄⋊C₄×C₅₃	—	2	1	no	—
40	C₂×C₄⋊C₄×C₅₃	—	2	1	no	—
41	C₂×C₄⋊C₄×C₅₃	C₂×C₄	1	1	yes	characteristic
42	C₂×C₄⋊C₄×C₅₃	C₂×C₄	1	1	yes	characteristic
43	C₂×C₄⋊C₄×C₅₃	C₂×C₄	1	1	yes	characteristic
44	C₂×C₄²×C₅₃	—	2	1	no	—
45	C₂×C₄²×C₅₃	—	2	1	no	—
46	C₂×C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
47	C₂×C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
48	C₂×C₄²×C₅₃	D₄	1	1	yes	characteristic
49	C₂×C₄²×C₅₃	D₄	1	1	yes	characteristic
50	C₂×C₄²×C₅₃	D₄	1	1	yes	characteristic
51	C₂×C₄²×C₅₃	D₄	1	1	yes	characteristic
52	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
53	#95×C₅₃	—	2	1	no	—
54	#95×C₅₃	—	2	1	no	—
55	#95×C₅₃	—	2	1	no	—
56	#95×C₅₃	—	2	1	no	—
57	#95×C₅₃	—	2	1	no	—
58	#95×C₅₃	—	2	1	no	—
59	#95×C₅₃	—	2	1	no	—
60	C₂×Q₈×C₅₃	—	2	1	no	—
61	C₂×Q₈×C₅₃	—	2	2	no	—
62	C₂×Q₈×C₅₃	—	2	2	no	—
63	C₂×Q₈×C₅₃	C₄⋊C₄	1	2	yes	—
64	C₂²×C₄×C₅₃	—	2	1	no	—
65	C₂²×C₄×C₅₃	—	2	1	no	—
66	C₂²×C₄×C₅₃	—	2	1	no	—
67	C₂²×C₄×C₅₃	—	2	1	no	—
68	C₂²×C₄×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
69	C₂²×C₄×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
70	C₂²×C₄×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
71	C₂²×C₄×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
72	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
73	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
74	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
75	C₄⋊C₄×C₅₃	—	2	1	no	—
76	C₄⋊C₄×C₅₃	—	2	2	no	—
77	C₄⋊C₄×C₅₃	—	2	2	no	—
78	C₄⋊C₄×C₅₃	—	2	2	no	—
79	C₄⋊C₄×C₅₃	—	2	2	no	—
80	C₄⋊C₄×C₅₃	—	4	2	no	—
81	C₄⋊C₄×C₅₃	—	4	2	no	—
82	C₄⋊C₄×C₅₃	C₄⋊C₄	1	2	yes	—
83	C₄²×C₅₃	—	2	2	no	—
84	C₄²×C₅₃	—	2	2	no	—
85	C₄²×C₅₃	—	2	2	no	—
86	C₄²×C₅₃	—	2	2	no	—
87	C₄²×C₅₃	—	2	2	no	—
88	C₄²×C₅₃	—	2	1	no	—
89	C₄²×C₅₃	—	2	1	no	—
90	C₄²×C₅₃	—	4	1	no	—
91	C₄²×C₅₃	—	4	1	no	—
92	C₄²×C₅₃	—	4	1	no	—
93	C₄²×C₅₃	SD₁₆	1	2	yes	—
94	C₄²×C₅₃	Q₁₆	1	2	yes	—
95	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
96	C₄²×C₅₃	C₂×Q₈	1	1	yes	characteristic
97	C₄²×C₅₃	#41	1	1	yes	characteristic
98	C₄²×C₅₃	#41	1	1	yes	characteristic
99	C₂×#88322	—	2	4	no	—
100	C₂×#88322	—	4	2	no	—
101	C₂×#88322	—	4	2	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	Q₈×C₅₃	—	2	4	no	—
107	Q₈×C₅₃	—	4	2	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	2	no	—
117	C₂×C₄×C₅₃	—	2	2	no	—
118	C₂×C₄×C₅₃	—	2	1	no	—
119	C₂×C₄×C₅₃	—	2	1	no	—
120	C₂×C₄×C₅₃	—	2	1	no	—
121	C₂×C₄×C₅₃	—	4	1	no	—
122	C₂×C₄×C₅₃	—	4	1	no	—
123	C₂×C₄×C₅₃	—	4	1	no	—
124	C₂×C₄×C₅₃	—	4	1	no	—
125	C₂×C₄×C₅₃	—	4	1	no	—
126	C₂×C₄×C₅₃	—	4	1	no	—
127	C₂×C₄×C₅₃	—	4	1	no	—
128	C₂×C₄×C₅₃	—	4	1	no	—
129	C₂×C₄×C₅₃	—	4	1	no	—
130	C₂×C₄×C₅₃	—	4	1	no	—
131	C₂×C₄×C₅₃	#103	1	2	yes	—
132	C₂×C₄×C₅₃	#103	1	2	yes	—
133	C₂×C₄×C₅₃	#104	1	2	yes	—
134	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
135	C₂×C₄×C₅₃	#117	1	1	yes	characteristic
136	C₂×C₄×C₅₃	#121	1	1	yes	characteristic
137	C₂×C₄×C₅₃	#121	1	1	yes	characteristic
138	C₂×C₄×C₅₃	C₂²⋊Q₈	1	1	yes	characteristic
139	C₂×C₄×C₅₃	C₂²⋊Q₈	1	1	yes	characteristic
140	C₂×C₄×C₅₃	#131	1	2	yes	—
141	C₂×C₄×C₅₃	#131	1	2	yes	—
142	C₂×C₄×C₅₃	C₂×SD₁₆	1	1	yes	characteristic
143	C₂×C₄×C₅₃	C₂×Q₁₆	1	1	yes	characteristic
144	C₂×C₄×C₅₃	AGL(1,ℤ/8)	1	1	yes	characteristic
145	C₂×C₄×C₅₃	#137	1	1	yes	characteristic
146	#88322	—	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	2	no	—
162	C₄×C₅₃	—	4	2	no	—
163	C₄×C₅₃	—	4	1	no	—
164	C₄×C₅₃	—	8	1	no	—
165	C₄×C₅₃	—	8	1	no	—
166	C₄×C₅₃	#424	1	2	yes	—
167	C₄×C₅₃	#433	1	2	yes	—
168	C₄×C₅₃	#460	1	1	yes	characteristic
169	C₄×C₅₃	#462	1	1	yes	characteristic
170	C₄×C₅₃	#475	1	1	yes	characteristic
171	C₄×C₅₃	#477	1	1	yes	characteristic
172	#2295	D₅₃	1	1	yes	characteristic
173	#1842	—	53	1	no	—
174	C₄×C₄⋊C₈	—	53	1	no	—
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	C₂×D₅₃	1	1	yes	characteristic
185	C₂×C₄⋊C₈	—	53	2	no	—
186	C₂×C₄⋊C₈	—	106	1	no	—
187	C₂×C₄⋊C₈	—	106	1	no	—
188	C₂×C₄⋊C₈	—	106	1	no	—
189	C₂×C₄×C₈	—	53	2	no	—
190	#391	Dic₅₃	1	1	yes	characteristic
191	#391	Dic₅₃	1	1	yes	characteristic
192	#389	Dic₅₃	1	1	yes	characteristic
193	#389	Dic₅₃	1	1	yes	characteristic
194	#387	—	2	1	no	—
195	#387	—	2	1	no	—
196	C₄×C₄⋊C₄	—	2	1	no	—
197	C₄×C₄⋊C₄	C₂×D₅₃	1	1	yes	characteristic
198	#376	—	2	1	no	—
199	C₄³	C₂×D₅₃	1	1	yes	characteristic
200	#336	—	106	1	no	—
201	Q₈⋊C₈	—	106	4	no	—
202	C₅₃	#8568	1	1	yes	characteristic, complemented, Sylow
203	C₂²×Q₈	C₂×Dic₅₃	1	1	yes	characteristic
204	C₂²×C₈	—	106	2	no	—
205	C₂²×C₈	—	106	2	no	—
206	C₄×Q₈	—	2	1	no	—
207	C₄×Q₈	—	2	1	no	—
208	C₄×Q₈	Dic₁₀₆	1	2	yes	—
209	C₄×Q₈	D₂₁₂	1	2	yes	—
210	C₂×C₄⋊C₄	—	2	1	no	—
211	C₂×C₄⋊C₄	—	2	1	no	—
212	C₂×C₄⋊C₄	—	2	1	no	—
213	C₂×C₄⋊C₄	C₂×Dic₅₃	1	1	yes	characteristic
214	C₂×C₄⋊C₄	C₂×Dic₅₃	1	1	yes	characteristic
215	C₂×C₄⋊C₄	C₂×Dic₅₃	1	1	yes	characteristic
216	C₂×C₄²	—	2	1	no	—
217	C₂×C₄²	—	2	1	no	—
218	C₂×C₄²	C₂×Dic₅₃	1	1	yes	characteristic
219	C₂×C₄²	C₂×Dic₅₃	1	1	yes	characteristic
220	C₂×C₄²	C₂²⋊D₅₃	1	1	yes	characteristic
221	C₂×C₄²	C₂²⋊D₅₃	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	—	106	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	—	106	2	no	—
231	C₄×C₈	—	106	2	no	—
232	#95	—	2	1	no	—
233	#95	—	2	1	no	—
234	#95	—	2	1	no	—
235	#95	—	2	1	no	—
236	#95	—	2	1	no	—
237	#95	—	2	1	no	—
238	#95	—	2	1	no	—
239	C₂×Q₈	—	2	1	no	—
240	C₂×Q₈	—	2	2	no	—
241	C₂×Q₈	—	2	2	no	—
242	C₂×Q₈	C₄⋊Dic₅₃	1	2	yes	—
243	C₂²×C₄	—	2	1	no	—
244	C₂²×C₄	—	2	1	no	—
245	C₂²×C₄	—	2	1	no	—
246	C₂²×C₄	—	2	1	no	—
247	C₂²×C₄	C₂²⋊Dic₅₃	1	1	yes	characteristic
248	C₂²×C₄	C₂²⋊Dic₅₃	1	1	yes	characteristic
249	C₂²×C₄	C₂²⋊Dic₅₃	1	1	yes	characteristic
250	C₂²×C₄	C₂²⋊Dic₅₃	1	1	yes	characteristic
251	C₂²×C₄	C₂²×Dic₅₃	1	1	yes	characteristic
252	C₂²×C₄	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
253	C₂²×C₄	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
254	C₂×C₈	—	106	4	no	—
255	C₂×C₈	—	212	2	no	—
256	C₂×C₈	—	212	2	no	—
257	C₂×C₈	—	212	2	no	—
258	C₂×C₈	—	212	2	no	—
259	C₂×C₈	—	212	2	no	—
260	C₄⋊C₄	—	2	1	no	—
261	C₄⋊C₄	—	2	2	no	—
262	C₄⋊C₄	—	2	2	no	—
263	C₄⋊C₄	—	2	2	no	—
264	C₄⋊C₄	—	2	2	no	—
265	C₄⋊C₄	—	4	2	no	—
266	C₄⋊C₄	—	4	2	no	—
267	C₄⋊C₄	C₄⋊Dic₅₃	1	2	yes	—
268	C₄²	—	2	2	no	—
269	C₄²	—	2	2	no	—
270	C₄²	—	2	2	no	—
271	C₄²	—	2	2	no	—
272	C₄²	—	2	2	no	—
273	C₄²	—	2	1	no	—
274	C₄²	—	2	1	no	—
275	C₄²	—	4	1	no	—
276	C₄²	—	4	1	no	—
277	C₄²	—	4	1	no	—
278	C₄²	#11719924	1	2	yes	—
279	C₄²	#11719925	1	2	yes	—
280	C₄²	C₂×Dic₁₀₆	1	1	yes	characteristic
281	C₄²	C₂×D₂₁₂	1	1	yes	characteristic
282	C₄²	#11719945	1	1	yes	characteristic
283	C₄²	#11719945	1	1	yes	characteristic
284	C₂³	C₂×C₂²⋊Dic₅₃	1	1	yes	characteristic
285	Q₈	—	2	4	no	—
286	Q₈	—	4	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	—	2	2	no	—
299	C₂×C₄	—	2	2	no	—
300	C₂×C₄	—	4	1	no	—
301	C₂×C₄	—	4	1	no	—
302	C₂×C₄	—	4	1	no	—
303	C₂×C₄	—	4	1	no	—
304	C₂×C₄	—	4	1	no	—
305	C₂×C₄	—	4	1	no	—
306	C₂×C₄	—	4	1	no	—
307	C₂×C₄	—	4	1	no	—
308	C₂×C₄	—	4	1	no	—
309	C₂×C₄	—	4	1	no	—
310	C₂×C₄	#13133141	1	2	yes	—
311	C₂×C₄	#13133141	1	2	yes	—
312	C₂×C₄	#13133144	1	2	yes	—
313	C₂×C₄	C₂×C₄⋊Dic₅₃	1	1	yes	characteristic
314	C₂×C₄	#13133176	1	1	yes	characteristic
315	C₂×C₄	#13133191	1	1	yes	characteristic
316	C₂×C₄	#13133191	1	1	yes	characteristic
317	C₂×C₄	#13133196	1	1	yes	characteristic
318	C₂×C₄	#13133196	1	1	yes	characteristic
319	C₂×C₄	#13133224	1	2	yes	—
320	C₂×C₄	#13133224	1	2	yes	—
321	C₂×C₄	C₂×#11719924	1	1	yes	characteristic
322	C₂×C₄	C₂×#11719925	1	1	yes	characteristic
323	C₂×C₄	#13133242	1	1	yes	characteristic
324	C₂×C₄	#13133247	1	1	yes	characteristic
325	C₈	—	424	2	no	—
326	C₈	—	212	4	no	—
327	C₂²	#16810483	1	1	yes	characteristic
328	C₂²	#16810590	1	1	yes	characteristic
329	C₂²	#16810590	1	1	yes	characteristic
330	C₂²	C₂×#13133141	1	1	yes	characteristic
331	C₂²	#16810624	1	1	yes	characteristic
332	C₂²	C₂×#13133144	1	1	yes	characteristic
333	C₂²	#16810640	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	2	no	—
340	C₄	—	4	1	no	—
341	C₄	—	4	2	no	—
342	C₄	—	8	1	no	—
343	C₄	—	8	1	no	—
344	C₄	#16810650	1	2	yes	—
345	C₄	#16810684	1	2	yes	—
346	C₄	#16810823	1	1	yes	characteristic
347	C₄	#16810833	1	1	yes	characteristic
348	C₄	#16810908	1	1	yes	characteristic
349	C₄	#16810918	1	1	yes	characteristic
350	C₂	#20273118	1	1	yes	characteristic
351	C₂	#20273128	1	1	yes	characteristic
352	C₂	#20273170	1	1	yes	characteristic
353	C₂	#20273180	1	1	yes	characteristic
354	C₂	#20274699	1	1	yes	characteristic
355	C₂	#20274720	1	1	yes	characteristic
356	C₂	#20274735	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄	C₂×C₂²⋊D₅₃
commutator	C₂²×C₄×C₅₃	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₅₃
Fitting	C₅₃×#2295	C₂
socle, prime radical	C₂³×C₅₃	C₂×C₂²⋊C₄
cosocle	C₂×C₄²×C₅₃	C₂³
hypercenter	#2295	D₅₃
hypocenter	C₅₃	#8568
LCS	G ▷ C₂²×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂²×C₄ ◁ C₄³ ◁ #2295

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
#20274867	2
C₄×C₄⋊#88322	2
C₂×Q₈⋊#88322	2
#20272556	2
C₅₃×#2295	2
#8568	53