Subgroups — D₅₃×#2153 #27537089

Order 13568.

698 conjugacy classes of subgroups.

The subgroup lattice has 362 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×M₃₂×D₅₃	C₂	1	2	yes	maximal
2	#405×D₅₃	C₂	1	1	yes	maximal, characteristic
3	#20276481	C₂	1	1	yes	maximal, characteristic, complemented
4	#20276480	C₂	1	2	yes	maximal, complemented
5	#20276479	C₂	1	4	yes	maximal, complemented
6	#364×D₅₃	C₂	1	4	yes	maximal, complemented
7	C₅₃×#2153	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×M₁₆×D₅₃	C₄	1	1	yes	characteristic
9	C₂²×C₈×D₅₃	C₂²	1	1	yes	characteristic
10	#117×D₅₃	C₄	1	1	yes	characteristic
11	C₂×#13133157	C₂²	1	2	yes	—
12	C₂×#13133156	—	2	2	no	—
13	M₃₂×D₅₃	—	2	2	no	—
14	M₃₂×D₅₃	—	2	2	no	—
15	M₃₂×D₅₃	C₂²	1	4	yes	—
16	M₃₂×D₅₃	C₂²	1	4	yes	—
17	C₂×C₁₆×D₅₃	—	2	2	no	—
18	#16810581	C₄	1	1	yes	characteristic
19	#16810580	C₄	1	2	yes	—
20	#16810579	C₄	1	1	yes	characteristic
21	#16810578	C₂²	1	1	yes	characteristic
22	#16810577	C₂²	1	2	yes	—
23	#97×D₅₃	C₄	1	2	yes	—
24	C₄×C₈×D₅₃	C₂²	1	2	yes	—
25	#16810444	C₂²	1	4	yes	complemented
26	#16810443	C₂²	1	8	yes	complemented
27	C₂×M₃₂×C₅₃	C₂²	1	2	yes	—
28	C₅₃×#405	C₂²	1	1	yes	characteristic
29	C₅₃×#364	C₂²	1	4	yes	complemented
30	C₂²×C₄×D₅₃	C₂×C₄	1	1	yes	characteristic
31	C₂×#11719917	C₂×C₄	1	1	yes	characteristic
32	C₂×#11719912	C₂×C₄	1	2	yes	—
33	M₁₆×D₅₃	C₈	1	4	yes	—
34	M₁₆×D₅₃	C₈	1	4	yes	—
35	C₂²×#88322	C₂×C₄	1	1	yes	characteristic
36	C₂×C₈×D₅₃	C₂×C₄	1	1	yes	characteristic
37	C₂×C₈×D₅₃	C₂×C₄	1	1	yes	characteristic
38	C₂×C₈×D₅₃	C₂×C₄	1	2	yes	—
39	C₂×C₈×D₅₃	D₄	1	2	yes	—
40	C₂×C₈×D₅₃	D₄	1	2	yes	—
41	C₂×C₈×D₅₃	Q₈	1	2	yes	—
42	C₂×C₈×D₅₃	Q₈	1	2	yes	—
43	C₂×C₈×D₅₃	C₂³	1	1	yes	characteristic
44	C₂×C₈×D₅₃	C₂³	1	1	yes	characteristic
45	#13133176	C₂×C₄	1	1	yes	characteristic
46	#13133175	—	2	1	no	—
47	#13133174	C₂×C₄	1	2	yes	—
48	#13133173	—	2	1	no	—
49	C₄⋊C₄×D₅₃	—	2	1	no	—
50	C₂²⋊C₄×D₅₃	—	2	1	no	—
51	C₄²×D₅₃	C₂×C₄	1	2	yes	—
52	#13133157	—	2	2	no	—
53	#13133157	C₂³	1	4	yes	—
54	#13133156	—	2	4	no	—
55	#13133156	—	2	4	no	—
56	C₂×#11719908	—	2	2	no	—
57	C₁₆×D₅₃	—	2	4	no	—
58	C₁₆×D₅₃	—	2	4	no	—
59	#13133128	C₂×C₄	1	2	yes	—
60	#13133127	C₂×C₄	1	2	yes	—
61	#13133127	C₂×C₄	1	2	yes	—
62	C₄×#88322	C₂×C₄	1	2	yes	—
63	C₈×Dic₅₃	C₂×C₄	1	2	yes	—
64	C₈×Dic₅₃	C₂³	1	2	yes	—
65	C₂×M₁₆×C₅₃	C₂×C₄	1	1	yes	characteristic
66	C₂²×C₈×C₅₃	C₂³	1	1	yes	characteristic
67	#117×C₅₃	C₂×C₄	1	1	yes	characteristic
68	M₃₂×C₅₃	—	2	2	no	—
69	M₃₂×C₅₃	C₂³	1	4	yes	—
70	C₂×C₁₆×C₅₃	—	2	2	no	—
71	#97×C₅₃	C₂×C₄	1	2	yes	—
72	C₄×C₈×C₅₃	C₂³	1	2	yes	—
73	C₂³×D₅₃	M₁₆	1	1	yes	characteristic
74	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
75	C₂²×Dic₅₃	M₁₆	1	1	yes	characteristic
76	C₂×C₄×D₅₃	—	2	2	no	—
77	C₂×C₄×D₅₃	C₄⋊C₄	1	2	yes	—
78	C₂×C₄×D₅₃	C₄⋊C₄	1	2	yes	—
79	C₂×C₄×D₅₃	C₂×C₈	1	2	yes	—
80	C₂×C₄×D₅₃	C₂×C₈	1	2	yes	—
81	C₂×C₄×D₅₃	M₁₆	1	1	yes	characteristic
82	C₂×C₄×D₅₃	M₁₆	1	1	yes	characteristic
83	C₂×C₄×D₅₃	C₂²×C₄	1	1	yes	characteristic
84	C₂×C₄×D₅₃	C₂²×C₄	1	1	yes	characteristic
85	M₁₆×C₅₃	C₂×C₈	1	4	yes	—
86	C₂×C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
87	C₂×C₈×C₅₃	C₂²×C₄	1	2	yes	—
88	C₂×C₈×C₅₃	C₂×D₄	1	2	yes	—
89	C₂×C₈×C₅₃	C₂×Q₈	1	2	yes	—
90	C₂×C₈×C₅₃	C₂⁴	1	1	yes	characteristic
91	C₄⋊C₄×C₅₃	—	2	1	no	—
92	C₂²⋊C₄×C₅₃	—	2	1	no	—
93	C₄²×C₅₃	C₂²×C₄	1	2	yes	—
94	C₂²⋊Dic₅₃	—	2	1	no	—
95	#11719921	—	2	2	no	—
96	C₄⋊Dic₅₃	—	2	1	no	—
97	#11719919	—	2	2	no	—
98	C₄×Dic₅₃	—	2	2	no	—
99	C₄×Dic₅₃	C₂²×C₄	1	2	yes	—
100	#11719917	C₂×C₈	1	4	yes	—
101	C₂×#88322	C₄⋊C₄	1	2	yes	—
102	C₂×#88322	C₄⋊C₄	1	2	yes	—
103	C₂×#88322	C₂²×C₄	1	1	yes	characteristic
104	C₂×#88322	C₂²×C₄	1	1	yes	characteristic
105	C₂×#88322	C₂²×C₄	1	2	yes	—
106	#11719912	C₂×C₈	1	8	yes	—
107	C₈×D₅₃	C₄⋊C₄	1	2	yes	—
108	C₈×D₅₃	C₄⋊C₄	1	2	yes	—
109	C₈×D₅₃	C₄⋊C₄	1	2	yes	—
110	C₈×D₅₃	C₄⋊C₄	1	2	yes	—
111	C₈×D₅₃	C₂×C₈	1	4	yes	—
112	C₈×D₅₃	C₂×C₈	1	4	yes	—
113	C₈×D₅₃	C₂×D₄	1	1	yes	characteristic
114	C₈×D₅₃	C₂×D₄	1	2	yes	—
115	C₈×D₅₃	C₂×D₄	1	1	yes	characteristic
116	C₈×D₅₃	C₂×Q₈	1	1	yes	characteristic
117	C₈×D₅₃	C₂×Q₈	1	1	yes	characteristic
118	C₈×D₅₃	C₂×Q₈	1	2	yes	—
119	C₁₆×C₅₃	—	2	4	no	—
120	#11719908	—	2	4	no	—
121	C₂³×C₅₃	C₂×M₁₆	1	1	yes	characteristic
122	C₂²×D₅₃	—	4	2	no	—
123	C₂²×D₅₃	C₄⋊C₈	1	2	yes	—
124	C₂²×D₅₃	C₄⋊C₈	1	2	yes	—
125	C₂²×D₅₃	C₂×M₁₆	1	1	yes	characteristic
126	C₂²×D₅₃	C₂×M₁₆	1	1	yes	characteristic
127	C₂×C₄×C₅₃	—	2	2	no	—
128	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	2	yes	—
129	C₂×C₄×C₅₃	C₂²×C₈	1	2	yes	—
130	C₂×C₄×C₅₃	C₂×M₁₆	1	1	yes	characteristic
131	C₂×C₄×C₅₃	C₂³×C₄	1	1	yes	characteristic
132	C₂×Dic₅₃	—	2	2	no	—
133	C₂×Dic₅₃	C₄⋊C₈	1	2	yes	—
134	C₂×Dic₅₃	C₄⋊C₈	1	2	yes	—
135	C₂×Dic₅₃	C₂×M₁₆	1	1	yes	characteristic
136	C₂×Dic₅₃	C₂×M₁₆	1	1	yes	characteristic
137	C₄×D₅₃	—	4	4	no	—
138	C₄×D₅₃	C₄⋊C₈	1	2	yes	—
139	C₄×D₅₃	C₄⋊C₈	1	2	yes	—
140	C₄×D₅₃	C₄⋊C₈	1	2	yes	—
141	C₄×D₅₃	C₄⋊C₈	1	2	yes	—
142	C₄×D₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
143	C₄×D₅₃	C₂×C₄⋊C₄	1	2	yes	—
144	C₄×D₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
145	C₄×D₅₃	C₂²×C₈	1	1	yes	characteristic
146	C₄×D₅₃	C₂²×C₈	1	1	yes	characteristic
147	C₄×D₅₃	C₂²×C₈	1	2	yes	—
148	C₈×C₅₃	C₂×C₄⋊C₄	1	2	yes	—
149	C₈×C₅₃	C₂²×C₈	1	4	yes	—
150	C₈×C₅₃	C₂²×D₄	1	1	yes	characteristic
151	C₈×C₅₃	C₂²×Q₈	1	1	yes	characteristic
152	#88322	C₂×C₄⋊C₄	1	1	yes	characteristic
153	#88322	C₂×C₄⋊C₄	1	2	yes	—
154	#88322	C₂×C₄⋊C₄	1	1	yes	characteristic
155	#88322	C₂²×C₈	1	4	yes	—
156	C₂×#2153	—	53	1	no	maximal, Sylow
157	C₂²×C₅₃	—	4	1	no	—
158	C₂²×C₅₃	C₂×C₄⋊C₈	1	2	yes	—
159	C₂²×C₅₃	C₂²×M₁₆	1	1	yes	characteristic
160	C₂×D₅₃	—	2	2	no	—
161	C₂×D₅₃	—	2	4	no	—
162	C₂×D₅₃	—	4	2	no	—
163	C₂×D₅₃	—	4	1	no	—
164	C₂×D₅₃	C₂×C₄⋊C₈	1	1	yes	characteristic
165	C₂×D₅₃	C₂×C₄⋊C₈	1	2	yes	—
166	C₂×D₅₃	C₂×C₄⋊C₈	1	1	yes	characteristic
167	C₄×C₅₃	—	4	2	no	—
168	C₄×C₅₃	C₂×C₄⋊C₈	1	2	yes	—
169	C₄×C₅₃	C₂²×C₄⋊C₄	1	1	yes	characteristic
170	C₄×C₅₃	C₂³×C₈	1	1	yes	characteristic
171	Dic₅₃	—	4	2	no	—
172	Dic₅₃	C₂×C₄⋊C₈	1	1	yes	characteristic
173	Dic₅₃	C₂×C₄⋊C₈	1	1	yes	characteristic
174	Dic₅₃	C₂×C₄⋊C₈	1	2	yes	—
175	C₂²×M₃₂	—	53	2	no	—
176	C₂×#405	—	53	1	no	—
177	#2153	—	53	2	no	—
178	#2153	—	53	1	no	—
179	#2153	—	53	4	no	—
180	#2153	D₅₃	1	1	yes	characteristic, direct product factor
181	C₂×#364	—	53	4	no	—
182	C₂×C₅₃	—	2	1	no	—
183	C₂×C₅₃	—	2	2	no	—
184	C₂×C₅₃	C₂²×C₄⋊C₈	1	1	yes	characteristic
185	D₅₃	—	2	1	no	—
186	D₅₃	—	2	2	no	—
187	D₅₃	#2153	1	2	yes	direct product factor
188	C₂²×M₁₆	—	53	1	no	—
189	C₂³×C₈	—	53	1	no	—
190	C₂×#117	—	53	1	no	—
191	C₂×M₃₂	—	53	2	no	—
192	C₂×M₃₂	—	53	4	no	—
193	C₂×M₃₂	—	53	4	no	—
194	C₂×M₃₂	—	106	2	no	—
195	C₂×M₃₂	—	106	2	no	—
196	C₂×M₃₂	—	106	2	no	—
197	C₂×M₃₂	C₂×D₅₃	1	2	yes	—
198	C₂²×C₁₆	—	106	2	no	—
199	#405	—	53	1	no	—
200	#405	—	53	1	no	—
201	#405	—	53	1	no	—
202	#405	—	53	2	no	—
203	#405	—	53	2	no	—
204	#405	C₂×D₅₃	1	1	yes	characteristic
205	C₂×#97	—	53	2	no	—
206	C₂×C₄×C₈	—	53	2	no	—
207	#364	—	53	8	no	—
208	#364	—	53	4	no	—
209	#364	C₂×D₅₃	1	4	yes	complemented
210	C₅₃	C₂×#2153	1	1	yes	characteristic, complemented, Sylow
211	C₂³×C₄	—	53	1	no	—
212	C₂×M₁₆	—	53	1	no	—
213	C₂×M₁₆	—	53	4	no	—
214	C₂×M₁₆	—	53	4	no	—
215	C₂×M₁₆	—	53	2	no	—
216	C₂×M₁₆	C₄×D₅₃	1	1	yes	characteristic
217	C₂²×C₈	—	53	1	no	—
218	C₂²×C₈	—	53	1	no	—
219	C₂²×C₈	—	53	1	no	—
220	C₂²×C₈	—	53	2	no	—
221	C₂²×C₈	—	53	2	no	—
222	C₂²×C₈	—	53	2	no	—
223	C₂²×C₈	—	53	2	no	—
224	C₂²×C₈	—	53	2	no	—
225	C₂²×C₈	—	53	1	no	—
226	C₂²×C₈	—	53	1	no	—
227	C₂²×C₈	C₂²×D₅₃	1	1	yes	characteristic
228	#117	—	53	1	no	—
229	#117	—	53	2	no	—
230	#117	—	106	1	no	—
231	#117	—	106	1	no	—
232	#117	C₄×D₅₃	1	1	yes	characteristic
233	C₂×C₄⋊C₄	—	106	1	no	—
234	C₂×C₂²⋊C₄	—	106	1	no	—
235	C₂×C₄²	—	53	2	no	—
236	M₃₂	—	2	2	no	—
237	M₃₂	—	53	4	no	—
238	M₃₂	—	106	4	no	—
239	M₃₂	—	106	4	no	—
240	M₃₂	—	106	2	no	—
241	M₃₂	C₂²×D₅₃	1	4	yes	—
242	C₂×C₁₆	—	2	2	no	—
243	C₂×C₁₆	—	106	4	no	—
244	C₂×C₁₆	—	106	4	no	—
245	C₂×C₁₆	—	106	2	no	—
246	#97	—	53	2	no	—
247	#97	—	53	2	no	—
248	#97	—	53	2	no	—
249	#97	C₄×D₅₃	1	2	yes	—
250	C₄×C₈	—	53	2	no	—
251	C₄×C₈	—	53	2	no	—
252	C₄×C₈	—	53	2	no	—
253	C₄×C₈	C₂²×D₅₃	1	2	yes	—
254	C₂⁴	—	53	1	no	—
255	C₂²×C₄	—	53	1	no	—
256	C₂²×C₄	—	53	2	no	—
257	C₂²×C₄	—	53	2	no	—
258	C₂²×C₄	—	53	2	no	—
259	C₂²×C₄	—	53	2	no	—
260	C₂²×C₄	—	53	1	no	—
261	C₂²×C₄	—	53	1	no	—
262	C₂²×C₄	—	53	1	no	—
263	C₂²×C₄	—	53	1	no	—
264	C₂²×C₄	—	106	2	no	—
265	C₂²×C₄	C₂×C₄×D₅₃	1	1	yes	characteristic
266	M₁₆	—	53	4	no	—
267	M₁₆	—	53	8	no	—
268	M₁₆	C₈×D₅₃	1	4	yes	—
269	C₂×C₈	—	53	2	no	—
270	C₂×C₈	—	53	2	no	—
271	C₂×C₈	—	53	1	no	—
272	C₂×C₈	—	53	2	no	—
273	C₂×C₈	—	53	1	no	—
274	C₂×C₈	—	53	2	no	—
275	C₂×C₈	—	53	2	no	—
276	C₂×C₈	—	53	2	no	—
277	C₂×C₈	—	53	2	no	—
278	C₂×C₈	—	53	4	no	—
279	C₂×C₈	—	53	4	no	—
280	C₂×C₈	—	53	1	no	—
281	C₂×C₈	—	53	2	no	—
282	C₂×C₈	—	53	1	no	—
283	C₂×C₈	—	53	1	no	—
284	C₂×C₈	—	53	1	no	—
285	C₂×C₈	—	53	2	no	—
286	C₂×C₈	C₂×C₄×D₅₃	1	1	yes	characteristic
287	C₂×C₈	C₂×C₄×D₅₃	1	2	yes	—
288	C₂×C₈	D₄×D₅₃	1	2	yes	—
289	C₂×C₈	Q₈×D₅₃	1	2	yes	—
290	C₂×C₈	C₂³×D₅₃	1	1	yes	characteristic
291	C₄⋊C₄	—	2	1	no	—
292	C₄⋊C₄	—	106	2	no	—
293	C₄⋊C₄	—	106	1	no	—
294	C₂²⋊C₄	—	2	1	no	—
295	C₂²⋊C₄	—	106	2	no	—
296	C₂²⋊C₄	—	106	1	no	—
297	C₄²	—	53	2	no	—
298	C₄²	—	106	2	no	—
299	C₄²	C₂×C₄×D₅₃	1	2	yes	—
300	C₁₆	—	2	4	no	—
301	C₁₆	—	106	4	no	—
302	C₂³	—	53	1	no	—
303	C₂³	—	53	2	no	—
304	C₂³	—	53	1	no	—
305	C₂³	—	53	2	no	—
306	C₂³	—	212	2	no	—
307	C₂³	M₁₆×D₅₃	1	1	yes	characteristic
308	C₂×C₄	—	2	2	no	—
309	C₂×C₄	—	53	1	no	—
310	C₂×C₄	—	53	1	no	—
311	C₂×C₄	—	53	2	no	—
312	C₂×C₄	—	53	2	no	—
313	C₂×C₄	—	53	1	no	—
314	C₂×C₄	—	53	2	no	—
315	C₂×C₄	—	53	1	no	—
316	C₂×C₄	—	53	1	no	—
317	C₂×C₄	—	53	1	no	—
318	C₂×C₄	—	53	2	no	—
319	C₂×C₄	—	53	2	no	—
320	C₂×C₄	—	53	2	no	—
321	C₂×C₄	—	53	2	no	—
322	C₂×C₄	—	53	2	no	—
323	C₂×C₄	—	106	2	no	—
324	C₂×C₄	—	212	4	no	—
325	C₂×C₄	C₄⋊C₄×D₅₃	1	2	yes	—
326	C₂×C₄	C₂×C₈×D₅₃	1	2	yes	—
327	C₂×C₄	M₁₆×D₅₃	1	1	yes	characteristic
328	C₂×C₄	C₂²×C₄×D₅₃	1	1	yes	characteristic
329	C₈	—	53	1	no	—
330	C₈	—	53	1	no	—
331	C₈	—	53	2	no	—
332	C₈	—	53	4	no	—
333	C₈	C₄⋊C₄×D₅₃	1	2	yes	—
334	C₈	C₂×C₈×D₅₃	1	4	yes	—
335	C₈	C₂×D₄×D₅₃	1	1	yes	characteristic
336	C₈	C₂×Q₈×D₅₃	1	1	yes	characteristic
337	C₂²	—	4	1	no	—
338	C₂²	—	53	1	no	—
339	C₂²	—	53	1	no	—
340	C₂²	—	53	2	no	—
341	C₂²	—	106	2	no	—
342	C₂²	—	106	4	no	—
343	C₂²	—	212	1	no	—
344	C₂²	—	212	2	no	—
345	C₂²	C₄⋊C₈×D₅₃	1	2	yes	—
346	C₂²	C₂×M₁₆×D₅₃	1	1	yes	characteristic
347	C₄	—	4	2	no	—
348	C₄	—	53	1	no	—
349	C₄	—	53	1	no	—
350	C₄	—	53	2	no	—
351	C₄	—	212	2	no	—
352	C₄	C₄⋊C₈×D₅₃	1	2	yes	—
353	C₄	C₂×C₄⋊C₄×D₅₃	1	1	yes	characteristic
354	C₄	C₂²×C₈×D₅₃	1	1	yes	characteristic
355	C₂	—	2	1	no	—
356	C₂	—	2	2	no	—
357	C₂	—	53	2	no	—
358	C₂	—	106	1	no	—
359	C₂	—	106	2	no	—
360	C₂	C₂×C₄⋊C₈×D₅₃	1	1	yes	characteristic

	subgroup	G/H
center	C₈	C₂×D₄×D₅₃
commutator	C₄×C₅₃	C₂³×C₈
Frattini	C₂×C₈	C₂³×D₅₃
Fitting	C₅₃×#2153	C₂
socle	C₂×C₅₃	C₂²×C₄⋊C₈
prime radical	C₂³×D₅₃	M₁₆
cosocle	C₂×C₈×C₅₃	C₂⁴
hypercenter	#2153	D₅₃
hypocenter	C₅₃	C₂×#2153
LCS	G ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₈ ◁ C₂²×C₈ ◁ #2153

Complements

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

N	H ≅ G/N	type
3. #20276481	C₂	semidirect
4. #20276480	C₂	semidirect
5. #20276479	C₂	semidirect
6. #364×D₅₃	C₂	semidirect
7. C₅₃×#2153	C₂	semidirect
25. #16810444	C₂²	semidirect
26. #16810443	C₂²	semidirect
29. C₅₃×#364	C₂²	semidirect
180. #2153	D₅₃	direct (N × H)
187. D₅₃	#2153	direct (N × H)
209. #364	C₂×D₅₃	semidirect
210. C₅₃	C₂×#2153	semidirect

Maximal subgroups

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

subgroup	index
C₂×M₃₂×D₅₃	2
#405×D₅₃	2
#20276481	2
#20276480	2
#20276479	2
#364×D₅₃	2
C₅₃×#2153	2
C₂×#2153	53

Subgroups — D53×#2153 #27537089

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — D₅₃×#2153 #27537089