Subgroups — D₅₃×#1453 #27349708

Order 13568.

866 conjugacy classes of subgroups.

The subgroup lattice has 366 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#431×D₅₃	C₂	1	2	yes	maximal
2	C₄×M₁₆×D₅₃	C₂	1	1	yes	maximal, characteristic
3	#20273013	C₂	1	1	yes	maximal, characteristic, complemented
4	#20273012	C₂	1	1	yes	maximal, characteristic, complemented
5	#20273011	C₂	1	2	yes	maximal, complemented
6	#20273010	C₂	1	2	yes	maximal, complemented
7	#20273009	C₂	1	1	yes	maximal, characteristic, complemented
8	#322×D₅₃	C₂	1	2	yes	maximal, complemented
9	#322×D₅₃	C₂	1	1	yes	maximal, characteristic, complemented
10	C₈²×D₅₃	C₂	1	1	yes	maximal, characteristic, complemented
11	C₅₃×#1453	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×M₁₆×D₅₃	C₄	1	2	yes	—
13	C₂×C₄²×D₅₃	C₂²	1	1	yes	characteristic
14	#16810676	C₂²	1	2	yes	—
15	#16810675	C₄	1	4	yes	—
16	#16810674	C₂²	1	2	yes	—
17	#16810674	C₂²	1	2	yes	—
18	#16810673	C₄	1	4	yes	—
19	C₄⋊C₈×D₅₃	C₄	1	4	yes	—
20	C₂²⋊C₈×D₅₃	C₄	1	4	yes	—
21	C₄×#11719917	C₂²	1	1	yes	characteristic
22	#16810575	C₄	1	2	yes	—
23	M₁₆×Dic₅₃	C₄	1	2	yes	—
24	C₄×#11719912	C₂²	1	1	yes	characteristic
25	C₄×#11719912	C₂²	1	1	yes	characteristic
26	#97×D₅₃	C₄	1	2	yes	—
27	C₄×C₈×D₅₃	C₂²	1	2	yes	—
28	C₄×C₈×D₅₃	C₂²	1	1	yes	characteristic
29	C₄×C₈×D₅₃	C₂²	1	2	yes	—
30	C₄×C₈×D₅₃	C₂²	1	1	yes	characteristic
31	#16810345	C₂²	1	2	yes	complemented
32	#16810345	C₂²	1	2	yes	complemented
33	#16810345	C₂²	1	2	yes	complemented
34	#16810344	C₂²	1	1	yes	characteristic, complemented
35	#16810344	C₂²	1	2	yes	complemented
36	C₈×#88322	C₂²	1	2	yes	complemented
37	C₈×#88322	C₂²	1	1	yes	characteristic, complemented
38	C₅₃×#431	C₂²	1	2	yes	—
39	C₄×M₁₆×C₅₃	C₂²	1	1	yes	characteristic
40	C₅₃×#322	C₂²	1	2	yes	complemented
41	C₅₃×#322	C₂²	1	1	yes	characteristic, complemented
42	C₈²×C₅₃	C₂²	1	1	yes	characteristic, complemented
43	C₂²×C₄×D₅₃	C₂×C₄	1	2	yes	—
44	C₂²×C₄×D₅₃	C₂×C₄	1	1	yes	characteristic
45	C₂×#11719917	C₂×C₄	1	2	yes	—
46	C₂×#11719912	C₂×C₄	1	2	yes	—
47	C₂×#11719912	C₂×C₄	1	2	yes	—
48	M₁₆×D₅₃	—	2	8	no	—
49	C₂×C₈×D₅₃	C₂×C₄	1	2	yes	—
50	C₂×C₈×D₅₃	C₂×C₄	1	4	yes	—
51	C₂×C₈×D₅₃	C₂×C₄	1	4	yes	—
52	C₂×C₈×D₅₃	C₂×C₄	1	2	yes	—
53	C₂×C₄×Dic₅₃	C₂×C₄	1	1	yes	characteristic
54	C₂×C₄×Dic₅₃	C₂×C₄	1	2	yes	—
55	C₄²×D₅₃	C₂×C₄	1	1	yes	characteristic
56	C₄²×D₅₃	C₂×C₄	1	1	yes	characteristic
57	C₄²×D₅₃	C₂×C₄	1	2	yes	—
58	C₄²×D₅₃	C₂×C₄	1	2	yes	—
59	C₄²×D₅₃	C₂³	1	1	yes	characteristic
60	C₄²×D₅₃	C₂³	1	1	yes	characteristic
61	C₄⋊#88322	C₂×C₄	1	4	yes	—
62	#13133146	C₂×C₄	1	4	yes	—
63	#13133146	C₂×C₄	1	4	yes	—
64	C₂²⋊#88322	C₂×C₄	1	4	yes	—
65	#13133129	C₂×C₄	1	4	yes	—
66	#13133129	C₂×C₄	1	4	yes	—
67	#13133128	C₂×C₄	1	2	yes	—
68	#13133127	C₂×C₄	1	2	yes	—
69	#13133127	C₂×C₄	1	2	yes	—
70	C₄×#88322	C₂³	1	1	yes	characteristic
71	C₄×#88322	C₂³	1	1	yes	characteristic
72	C₄×#88322	C₂³	1	2	yes	—
73	C₄×#88322	C₂³	1	2	yes	—
74	C₈×Dic₅₃	C₂×C₄	1	2	yes	—
75	C₈×Dic₅₃	C₂×C₄	1	2	yes	—
76	C₈×Dic₅₃	C₂×C₄	1	4	yes	—
77	C₈×Dic₅₃	C₂×C₄	1	4	yes	—
78	C₂×M₁₆×C₅₃	C₂×C₄	1	2	yes	—
79	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
80	C₄⋊C₈×C₅₃	C₂×C₄	1	4	yes	—
81	C₂²⋊C₈×C₅₃	C₂×C₄	1	4	yes	—
82	#97×C₅₃	C₂×C₄	1	2	yes	—
83	C₄×C₈×C₅₃	C₂³	1	1	yes	characteristic
84	C₄×C₈×C₅₃	C₂³	1	1	yes	characteristic
85	C₄×C₈×C₅₃	C₂³	1	2	yes	—
86	C₄×C₈×C₅₃	C₂³	1	2	yes	—
87	C₂³×D₅₃	C₄²	1	1	yes	characteristic
88	C₂²×C₄×C₅₃	C₂²×C₄	1	2	yes	—
89	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
90	C₂²×Dic₅₃	C₄²	1	1	yes	characteristic
91	C₂²×Dic₅₃	C₄²	1	2	yes	—
92	C₂×C₄×D₅₃	—	2	4	no	—
93	C₂×C₄×D₅₃	—	2	8	no	—
94	C₂×C₄×D₅₃	C₄²	1	1	yes	characteristic
95	C₂×C₄×D₅₃	C₄²	1	2	yes	—
96	C₂×C₄×D₅₃	C₄²	1	1	yes	characteristic
97	C₂×C₄×D₅₃	C₄²	1	2	yes	—
98	C₂×C₄×D₅₃	C₂²×C₄	1	2	yes	—
99	C₂×C₄×D₅₃	C₂²×C₄	1	1	yes	characteristic
100	C₂×C₄×D₅₃	C₂²×C₄	1	1	yes	characteristic
101	C₂×C₄×D₅₃	C₂²×C₄	1	2	yes	—
102	M₁₆×C₅₃	—	2	4	no	—
103	C₂×C₈×C₅₃	C₂²×C₄	1	2	yes	—
104	C₂×C₈×C₅₃	C₂²×C₄	1	4	yes	—
105	C₂×C₈×C₅₃	C₂²×C₄	1	4	yes	—
106	C₂×C₈×C₅₃	C₂²×C₄	1	2	yes	—
107	C₄²×C₅₃	C₂²×C₄	1	1	yes	characteristic
108	C₄²×C₅₃	C₂²×C₄	1	2	yes	—
109	C₄²×C₅₃	C₂⁴	1	1	yes	characteristic
110	C₄×Dic₅₃	C₄²	1	2	yes	—
111	C₄×Dic₅₃	C₄²	1	2	yes	—
112	C₄×Dic₅₃	C₄²	1	2	yes	—
113	C₄×Dic₅₃	C₂²×C₄	1	2	yes	—
114	C₄×Dic₅₃	C₂²×C₄	1	1	yes	characteristic
115	C₄×Dic₅₃	C₂²×C₄	1	1	yes	characteristic
116	C₄×Dic₅₃	C₂²×C₄	1	2	yes	—
117	#11719917	—	2	4	no	—
118	C₂×#88322	C₂²×C₄	1	2	yes	—
119	C₂×#88322	C₂²×C₄	1	4	yes	—
120	C₂×#88322	C₂²×C₄	1	4	yes	—
121	C₂×#88322	C₂²×C₄	1	2	yes	—
122	#11719912	—	2	4	no	—
123	#11719912	—	2	4	no	—
124	C₈×D₅₃	—	2	4	no	—
125	C₈×D₅₃	—	2	8	no	—
126	C₈×D₅₃	—	2	8	no	—
127	C₈×D₅₃	M₁₆	1	8	yes	—
128	C₂³×C₅₃	C₂×C₄²	1	1	yes	characteristic
129	C₂²×D₅₃	—	2	2	no	—
130	C₂²×D₅₃	—	2	4	no	—
131	C₂²×D₅₃	C₂×C₄²	1	1	yes	characteristic
132	C₂²×D₅₃	C₂×C₄²	1	1	yes	characteristic
133	C₂×C₄×C₅₃	—	2	2	no	—
134	C₂×C₄×C₅₃	—	2	4	no	—
135	C₂×C₄×C₅₃	C₂×C₄²	1	1	yes	characteristic
136	C₂×C₄×C₅₃	C₂×C₄²	1	2	yes	—
137	C₂×C₄×C₅₃	C₂³×C₄	1	1	yes	characteristic
138	C₂×C₄×C₅₃	C₂³×C₄	1	2	yes	—
139	C₂×Dic₅₃	—	2	2	no	—
140	C₂×Dic₅₃	—	2	4	no	—
141	C₂×Dic₅₃	C₂×C₄²	1	1	yes	characteristic
142	C₂×Dic₅₃	C₂×C₄²	1	2	yes	—
143	C₂×Dic₅₃	C₂×C₄²	1	1	yes	characteristic
144	C₂×Dic₅₃	C₂×C₄²	1	2	yes	—
145	C₄×D₅₃	—	2	2	no	—
146	C₄×D₅₃	—	2	4	no	—
147	C₄×D₅₃	—	2	2	no	—
148	C₄×D₅₃	—	2	4	no	—
149	C₄×D₅₃	—	2	2	no	—
150	C₄×D₅₃	—	2	4	no	—
151	C₄×D₅₃	C₂×M₁₆	1	4	yes	—
152	C₄×D₅₃	#131	1	8	yes	—
153	C₈×C₅₃	—	2	2	no	—
154	C₈×C₅₃	—	2	4	no	—
155	C₈×C₅₃	—	2	4	no	—
156	C₈×C₅₃	C₂×M₁₆	1	4	yes	—
157	#88322	—	2	2	no	—
158	#88322	—	2	4	no	—
159	#88322	—	2	4	no	—
160	#88322	C₂×M₁₆	1	4	yes	—
161	C₂×#1453	—	53	1	no	maximal, Sylow
162	C₂²×C₅₃	—	2	1	no	—
163	C₂²×C₅₃	—	2	2	no	—
164	C₂²×C₅₃	C₂²×C₄²	1	1	yes	characteristic
165	C₂×D₅₃	—	2	1	no	—
166	C₂×D₅₃	—	2	2	no	—
167	C₂×D₅₃	—	4	4	no	—
168	C₂×D₅₃	C₄×M₁₆	1	2	yes	—
169	C₂×D₅₃	#405	1	4	yes	—
170	C₄×C₅₃	—	2	1	no	—
171	C₄×C₅₃	—	2	2	no	—
172	C₄×C₅₃	C₂²×M₁₆	1	2	yes	—
173	C₄×C₅₃	C₂×#131	1	4	yes	—
174	Dic₅₃	—	2	1	no	—
175	Dic₅₃	—	2	2	no	—
176	Dic₅₃	C₄×M₁₆	1	2	yes	—
177	Dic₅₃	#405	1	4	yes	—
178	C₂×#431	—	53	2	no	—
179	C₂×C₄×M₁₆	—	53	1	no	—
180	#1453	—	53	2	no	—
181	#1453	—	53	1	no	—
182	#1453	—	53	2	no	—
183	#1453	—	53	1	no	—
184	#1453	—	53	1	no	—
185	#1453	D₅₃	1	1	yes	characteristic, direct product factor
186	C₂×#322	—	53	1	no	—
187	C₂×#322	—	53	2	no	—
188	C₂×C₈²	—	53	1	no	—
189	C₂×C₅₃	—	4	1	no	—
190	C₂×C₅₃	C₂×C₄×M₁₆	1	1	yes	characteristic
191	C₂×C₅₃	C₂×#405	1	2	yes	—
192	D₅₃	—	4	1	no	—
193	D₅₃	#1453	1	4	yes	direct product factor
194	C₂²×M₁₆	—	53	2	no	—
195	C₂²×C₄²	—	53	1	no	—
196	#431	—	53	2	no	—
197	#431	—	53	4	no	—
198	#431	—	53	4	no	—
199	#431	—	53	2	no	—
200	#431	—	53	2	no	—
201	#431	C₂×D₅₃	1	2	yes	—
202	C₂×C₄⋊C₈	—	53	4	no	—
203	C₂×C₂²⋊C₈	—	53	4	no	—
204	C₄×M₁₆	—	53	2	no	—
205	C₄×M₁₆	—	53	1	no	—
206	C₄×M₁₆	—	53	2	no	—
207	C₄×M₁₆	—	53	1	no	—
208	C₄×M₁₆	—	53	1	no	—
209	C₄×M₁₆	C₂×D₅₃	1	1	yes	characteristic
210	C₂×#97	—	53	2	no	—
211	C₂×C₄×C₈	—	53	1	no	—
212	C₂×C₄×C₈	—	53	1	no	—
213	C₂×C₄×C₈	—	53	2	no	—
214	C₂×C₄×C₈	—	53	2	no	—
215	#322	—	53	2	no	—
216	#322	—	53	1	no	—
217	#322	—	53	2	no	—
218	#322	—	53	2	no	—
219	#322	—	53	2	no	—
220	#322	C₂×D₅₃	1	1	yes	characteristic, complemented
221	#322	C₂×D₅₃	1	2	yes	complemented
222	C₈²	—	53	2	no	—
223	C₈²	—	53	1	no	—
224	C₈²	C₂×D₅₃	1	1	yes	characteristic, complemented
225	C₅₃	C₂×#1453	1	1	yes	characteristic, complemented, Sylow
226	C₂³×C₄	—	53	1	no	—
227	C₂³×C₄	—	53	2	no	—
228	C₂×M₁₆	—	53	2	no	—
229	C₂×M₁₆	—	53	2	no	—
230	C₂×M₁₆	—	53	2	no	—
231	C₂×M₁₆	—	106	8	no	—
232	C₂×M₁₆	C₄×D₅₃	1	2	yes	—
233	C₂²×C₈	—	53	2	no	—
234	C₂²×C₈	—	53	2	no	—
235	C₂²×C₈	—	53	4	no	—
236	C₂²×C₈	—	53	4	no	—
237	C₂×C₄²	—	53	1	no	—
238	C₂×C₄²	—	53	2	no	—
239	C₂×C₄²	—	53	1	no	—
240	C₂×C₄²	—	53	1	no	—
241	C₂×C₄²	—	53	2	no	—
242	C₂×C₄²	—	53	2	no	—
243	C₂×C₄²	—	53	1	no	—
244	C₂×C₄²	—	53	1	no	—
245	C₂×C₄²	C₂²×D₅₃	1	1	yes	characteristic
246	C₄⋊C₈	—	53	4	no	—
247	C₄⋊C₈	—	53	4	no	—
248	C₄⋊C₈	—	53	4	no	—
249	C₄⋊C₈	C₄×D₅₃	1	4	yes	—
250	C₂²⋊C₈	—	53	4	no	—
251	C₂²⋊C₈	—	53	4	no	—
252	C₂²⋊C₈	—	53	4	no	—
253	C₂²⋊C₈	C₄×D₅₃	1	4	yes	—
254	#97	—	53	2	no	—
255	#97	—	53	2	no	—
256	#97	—	53	2	no	—
257	#97	C₄×D₅₃	1	2	yes	—
258	C₄×C₈	—	53	2	no	—
259	C₄×C₈	—	53	2	no	—
260	C₄×C₈	—	53	4	no	—
261	C₄×C₈	—	53	4	no	—
262	C₄×C₈	—	53	1	no	—
263	C₄×C₈	—	53	2	no	—
264	C₄×C₈	—	53	1	no	—
265	C₄×C₈	—	53	2	no	—
266	C₄×C₈	C₂²×D₅₃	1	1	yes	characteristic
267	C₄×C₈	C₂²×D₅₃	1	2	yes	—
268	C₄×C₈	C₂²×D₅₃	1	1	yes	characteristic
269	C₄×C₈	C₂²×D₅₃	1	2	yes	—
270	C₂⁴	—	53	1	no	—
271	C₂²×C₄	—	53	1	no	—
272	C₂²×C₄	—	53	2	no	—
273	C₂²×C₄	—	53	2	no	—
274	C₂²×C₄	—	53	1	no	—
275	C₂²×C₄	—	53	2	no	—
276	C₂²×C₄	—	53	1	no	—
277	C₂²×C₄	—	53	2	no	—
278	C₂²×C₄	—	53	1	no	—
279	C₂²×C₄	—	53	2	no	—
280	C₂²×C₄	—	53	1	no	—
281	C₂²×C₄	—	106	4	no	—
282	C₂²×C₄	—	106	8	no	—
283	C₂²×C₄	C₂×C₄×D₅₃	1	2	yes	—
284	C₂²×C₄	C₂×C₄×D₅₃	1	1	yes	characteristic
285	M₁₆	—	2	4	no	—
286	M₁₆	—	106	4	no	—
287	M₁₆	—	106	4	no	—
288	M₁₆	—	106	4	no	—
289	C₂×C₈	—	53	2	no	—
290	C₂×C₈	—	53	8	no	—
291	C₂×C₈	—	53	2	no	—
292	C₂×C₈	—	53	4	no	—
293	C₂×C₈	—	53	4	no	—
294	C₂×C₈	—	106	4	no	—
295	C₂×C₈	—	106	8	no	—
296	C₂×C₈	—	106	8	no	—
297	C₂×C₈	C₂×C₄×D₅₃	1	2	yes	—
298	C₂×C₈	C₂×C₄×D₅₃	1	4	yes	—
299	C₂×C₈	C₂×C₄×D₅₃	1	4	yes	—
300	C₂×C₈	C₂×C₄×D₅₃	1	2	yes	—
301	C₄²	—	53	2	no	—
302	C₄²	—	53	2	no	—
303	C₄²	—	53	2	no	—
304	C₄²	—	53	1	no	—
305	C₄²	—	53	2	no	—
306	C₄²	—	53	1	no	—
307	C₄²	—	53	2	no	—
308	C₄²	C₂×C₄×D₅₃	1	1	yes	characteristic
309	C₄²	C₂×C₄×D₅₃	1	2	yes	—
310	C₄²	C₂³×D₅₃	1	1	yes	characteristic
311	C₂³	—	53	1	no	—
312	C₂³	—	53	1	no	—
313	C₂³	—	106	2	no	—
314	C₂³	—	106	4	no	—
315	C₂³	C₄²×D₅₃	1	1	yes	characteristic
316	C₂×C₄	—	2	2	no	—
317	C₂×C₄	—	2	4	no	—
318	C₂×C₄	—	53	1	no	—
319	C₂×C₄	—	53	2	no	—
320	C₂×C₄	—	53	4	no	—
321	C₂×C₄	—	53	8	no	—
322	C₂×C₄	—	53	1	no	—
323	C₂×C₄	—	53	2	no	—
324	C₂×C₄	—	106	2	no	—
325	C₂×C₄	—	106	4	no	—
326	C₂×C₄	—	106	2	no	—
327	C₂×C₄	—	106	4	no	—
328	C₂×C₄	—	106	2	no	—
329	C₂×C₄	—	106	4	no	—
330	C₂×C₄	—	106	2	no	—
331	C₂×C₄	—	106	4	no	—
332	C₂×C₄	C₄²×D₅₃	1	1	yes	characteristic
333	C₂×C₄	C₄²×D₅₃	1	2	yes	—
334	C₂×C₄	C₂²×C₄×D₅₃	1	1	yes	characteristic
335	C₂×C₄	C₂²×C₄×D₅₃	1	2	yes	—
336	C₈	—	2	2	no	—
337	C₈	—	2	4	no	—
338	C₈	—	2	4	no	—
339	C₈	—	53	4	no	—
340	C₈	—	106	2	no	—
341	C₈	—	106	4	no	—
342	C₈	—	106	4	no	—
343	C₈	M₁₆×D₅₃	1	4	yes	—
344	C₂²	—	2	1	no	—
345	C₂²	—	2	2	no	—
346	C₂²	—	53	2	no	—
347	C₂²	—	53	4	no	—
348	C₂²	—	106	1	no	—
349	C₂²	—	106	2	no	—
350	C₂²	—	212	4	no	—
351	C₂²	C₂×C₄²×D₅₃	1	1	yes	characteristic
352	C₄	—	2	1	no	—
353	C₄	—	2	2	no	—
354	C₄	—	53	2	no	—
355	C₄	—	53	4	no	—
356	C₄	—	106	1	no	—
357	C₄	—	106	2	no	—
358	C₄	C₂×M₁₆×D₅₃	1	2	yes	—
359	C₄	#131×D₅₃	1	4	yes	—
360	C₂	—	4	1	no	—
361	C₂	—	53	4	no	—
362	C₂	—	212	1	no	—
363	C₂	C₄×M₁₆×D₅₃	1	1	yes	characteristic
364	C₂	#405×D₅₃	1	2	yes	—

	subgroup	G/H
center, Frattini	C₄²	C₂³×D₅₃
commutator, socle	C₂²×C₅₃	C₂²×C₄²
Fitting	C₅₃×#1453	C₂
prime radical	C₂³×D₅₃	C₄²
cosocle	C₄²×C₅₃	C₂⁴
hypercenter	#1453	D₅₃
hypocenter	C₅₃	C₂×#1453
LCS	G ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₄² ◁ #1453

Complements

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

N	H ≅ G/N	type
3. #20273013	C₂	semidirect
4. #20273012	C₂	semidirect
5. #20273011	C₂	semidirect
6. #20273010	C₂	semidirect
7. #20273009	C₂	semidirect
8. #322×D₅₃	C₂	semidirect
9. #322×D₅₃	C₂	semidirect
10. C₈²×D₅₃	C₂	semidirect
11. C₅₃×#1453	C₂	semidirect
31. #16810345	C₂²	semidirect
32. #16810345	C₂²	semidirect
33. #16810345	C₂²	semidirect
34. #16810344	C₂²	semidirect
35. #16810344	C₂²	semidirect
36. C₈×#88322	C₂²	semidirect
37. C₈×#88322	C₂²	semidirect
40. C₅₃×#322	C₂²	semidirect
41. C₅₃×#322	C₂²	semidirect
42. C₈²×C₅₃	C₂²	semidirect
185. #1453	D₅₃	direct (N × H)
193. D₅₃	#1453	direct (N × H)
220. #322	C₂×D₅₃	semidirect
221. #322	C₂×D₅₃	semidirect
224. C₈²	C₂×D₅₃	semidirect
225. C₅₃	C₂×#1453	semidirect

Maximal subgroups

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

subgroup	index
#431×D₅₃	2
C₄×M₁₆×D₅₃	2
#20273013	2
#20273012	2
#20273011	2
#20273010	2
#20273009	2
#322×D₅₃	2
C₈²×D₅₃	2
C₅₃×#1453	2
C₂×#1453	53

Subgroups — D53×#1453 #27349708

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — D₅₃×#1453 #27349708