Subgroups — D₅₃×#1450 #27349668

Order 13568.

982 conjugacy classes of subgroups.

The subgroup lattice has 328 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	C₂×C₄×C₈×D₅₃	C₂	1	1	yes	maximal, characteristic
3	#20273000	C₂	1	2	yes	maximal, complemented
4	#20272999	C₂	1	2	yes	maximal, complemented
5	#20272998	C₂	1	1	yes	maximal, characteristic, complemented
6	#20272997	C₂	1	2	yes	maximal, complemented
7	#322×D₅₃	C₂	1	2	yes	maximal, complemented
8	C₈²×D₅₃	C₂	1	2	yes	maximal, complemented
9	C₅₃×#1450	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂²×C₈×D₅₃	C₄	1	2	yes	—
11	C₂×C₄²×D₅₃	C₂²	1	1	yes	characteristic
12	#16810676	C₂²	1	2	yes	—
13	#16810675	C₄	1	4	yes	—
14	#16810674	C₂²	1	4	yes	—
15	#16810673	C₄	1	4	yes	—
16	C₄⋊C₈×D₅₃	C₄	1	4	yes	—
17	C₂²⋊C₈×D₅₃	C₄	1	4	yes	—
18	C₂×C₄×#88322	C₂²	1	1	yes	characteristic
19	C₂×C₈×Dic₅₃	C₄	1	2	yes	—
20	C₄×C₈×D₅₃	C₄	1	2	yes	—
21	C₄×C₈×D₅₃	C₄	1	2	yes	—
22	C₄×C₈×D₅₃	C₂²	1	1	yes	characteristic
23	C₄×C₈×D₅₃	C₂²	1	4	yes	—
24	C₄×C₈×D₅₃	C₂²	1	1	yes	characteristic
25	C₄×C₈×D₅₃	C₂²	1	1	yes	characteristic
26	C₄×C₈×D₅₃	C₂²	1	1	yes	characteristic
27	#16810345	C₂²	1	4	yes	complemented
28	#16810344	C₂²	1	2	yes	complemented
29	C₈×#88322	C₂²	1	2	yes	complemented
30	C₈×#88322	C₂²	1	4	yes	complemented
31	C₅₃×#431	C₂²	1	2	yes	—
32	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
33	C₅₃×#322	C₂²	1	2	yes	complemented
34	C₈²×C₅₃	C₂²	1	2	yes	complemented
35	C₂²×C₄×D₅₃	C₂×C₄	1	2	yes	—
36	C₂²×C₄×D₅₃	C₂×C₄	1	1	yes	characteristic
37	C₂²×#88322	C₂×C₄	1	2	yes	—
38	C₂×C₈×D₅₃	C₈	1	16	yes	complemented
39	C₂×C₈×D₅₃	C₂×C₄	1	8	yes	—
40	C₂×C₈×D₅₃	C₂×C₄	1	2	yes	—
41	C₂×C₈×D₅₃	C₂×C₄	1	2	yes	—
42	C₂×C₈×D₅₃	C₂×C₄	1	2	yes	—
43	C₂×C₈×D₅₃	C₂×C₄	1	2	yes	—
44	C₂×C₄×Dic₅₃	C₂×C₄	1	1	yes	characteristic
45	C₂×C₄×Dic₅₃	C₂×C₄	1	2	yes	—
46	C₄²×D₅₃	C₂×C₄	1	1	yes	characteristic
47	C₄²×D₅₃	C₂×C₄	1	1	yes	characteristic
48	C₄²×D₅₃	C₂×C₄	1	2	yes	—
49	C₄²×D₅₃	C₂×C₄	1	2	yes	—
50	C₄²×D₅₃	C₂³	1	1	yes	characteristic
51	C₄²×D₅₃	C₂³	1	1	yes	characteristic
52	C₄⋊#88322	C₂×C₄	1	4	yes	—
53	#13133146	C₂×C₄	1	8	yes	—
54	C₂²⋊#88322	C₂×C₄	1	4	yes	—
55	#13133129	C₂×C₄	1	8	yes	—
56	C₄×#88322	C₂×C₄	1	2	yes	—
57	C₄×#88322	C₂³	1	1	yes	characteristic
58	C₄×#88322	C₂³	1	1	yes	characteristic
59	C₄×#88322	C₂³	1	4	yes	—
60	C₈×Dic₅₃	C₂×C₄	1	2	yes	—
61	C₈×Dic₅₃	C₂×C₄	1	2	yes	—
62	C₈×Dic₅₃	C₂×C₄	1	2	yes	—
63	C₈×Dic₅₃	C₂×C₄	1	2	yes	—
64	C₈×Dic₅₃	C₂×C₄	1	8	yes	—
65	C₂²×C₈×C₅₃	C₂×C₄	1	2	yes	—
66	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
67	C₄⋊C₈×C₅₃	C₂×C₄	1	4	yes	—
68	C₂²⋊C₈×C₅₃	C₂×C₄	1	4	yes	—
69	C₄×C₈×C₅₃	C₂×C₄	1	2	yes	—
70	C₄×C₈×C₅₃	C₂³	1	1	yes	characteristic
71	C₄×C₈×C₅₃	C₂³	1	4	yes	—
72	C₄×C₈×C₅₃	C₂³	1	1	yes	characteristic
73	C₂³×D₅₃	C₄²	1	1	yes	characteristic
74	C₂²×C₄×C₅₃	C₂²×C₄	1	2	yes	—
75	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
76	C₂²×Dic₅₃	C₄²	1	2	yes	—
77	C₂²×Dic₅₃	C₄²	1	1	yes	characteristic
78	C₂×C₄×D₅₃	—	2	8	no	—
79	C₂×C₄×D₅₃	C₄²	1	2	yes	—
80	C₂×C₄×D₅₃	C₄²	1	2	yes	—
81	C₂×C₄×D₅₃	C₄²	1	1	yes	characteristic
82	C₂×C₄×D₅₃	C₄²	1	1	yes	characteristic
83	C₂×C₄×D₅₃	C₂×C₈	1	8	yes	—
84	C₂×C₄×D₅₃	C₂²×C₄	1	2	yes	—
85	C₂×C₄×D₅₃	C₂²×C₄	1	1	yes	characteristic
86	C₂×C₄×D₅₃	C₂²×C₄	1	1	yes	characteristic
87	C₂×C₄×D₅₃	C₂²×C₄	1	2	yes	—
88	C₂×C₈×C₅₃	C₂×C₈	1	8	yes	complemented
89	C₂×C₈×C₅₃	C₂²×C₄	1	8	yes	—
90	C₂×C₈×C₅₃	C₂²×C₄	1	2	yes	—
91	C₂×C₈×C₅₃	C₂²×C₄	1	2	yes	—
92	C₄²×C₅₃	C₂²×C₄	1	1	yes	characteristic
93	C₄²×C₅₃	C₂²×C₄	1	2	yes	—
94	C₄²×C₅₃	C₂⁴	1	1	yes	characteristic
95	C₄×Dic₅₃	C₄²	1	2	yes	—
96	C₄×Dic₅₃	C₄²	1	2	yes	—
97	C₄×Dic₅₃	C₄²	1	2	yes	—
98	C₄×Dic₅₃	C₂²×C₄	1	1	yes	characteristic
99	C₄×Dic₅₃	C₂²×C₄	1	2	yes	—
100	C₄×Dic₅₃	C₂²×C₄	1	1	yes	characteristic
101	C₄×Dic₅₃	C₂²×C₄	1	2	yes	—
102	C₂×#88322	C₂×C₈	1	8	yes	complemented
103	C₂×#88322	C₂²×C₄	1	8	yes	—
104	C₂×#88322	C₂²×C₄	1	2	yes	—
105	C₂×#88322	C₂²×C₄	1	2	yes	—
106	C₈×D₅₃	—	2	16	no	—
107	C₈×D₅₃	C₂×C₈	1	8	yes	—
108	C₈×D₅₃	C₂×C₈	1	8	yes	—
109	C₈×D₅₃	C₂×C₈	1	8	yes	complemented
110	C₈×D₅₃	C₂×C₈	1	8	yes	complemented
111	C₂³×C₅₃	C₂×C₄²	1	1	yes	characteristic
112	C₂²×D₅₃	—	2	4	no	—
113	C₂²×D₅₃	C₄×C₈	1	4	yes	—
114	C₂²×D₅₃	C₂×C₄²	1	1	yes	characteristic
115	C₂²×D₅₃	C₂×C₄²	1	1	yes	characteristic
116	C₂×C₄×C₅₃	—	2	4	no	—
117	C₂×C₄×C₅₃	C₂×C₄²	1	2	yes	—
118	C₂×C₄×C₅₃	C₂×C₄²	1	1	yes	characteristic
119	C₂×C₄×C₅₃	C₂²×C₈	1	4	yes	—
120	C₂×C₄×C₅₃	C₂³×C₄	1	1	yes	characteristic
121	C₂×C₄×C₅₃	C₂³×C₄	1	2	yes	—
122	C₂×Dic₅₃	—	2	4	no	—
123	C₂×Dic₅₃	C₄×C₈	1	4	yes	—
124	C₂×Dic₅₃	C₂×C₄²	1	2	yes	—
125	C₂×Dic₅₃	C₂×C₄²	1	1	yes	characteristic
126	C₂×Dic₅₃	C₂×C₄²	1	1	yes	characteristic
127	C₂×Dic₅₃	C₂×C₄²	1	2	yes	—
128	C₄×D₅₃	—	2	4	no	—
129	C₄×D₅₃	—	2	4	no	—
130	C₄×D₅₃	—	2	4	no	—
131	C₄×D₅₃	C₄×C₈	1	4	yes	—
132	C₄×D₅₃	C₄×C₈	1	4	yes	—
133	C₄×D₅₃	C₂²×C₈	1	4	yes	—
134	C₄×D₅₃	C₂²×C₈	1	4	yes	—
135	C₄×D₅₃	#131	1	8	yes	—
136	C₈×C₅₃	—	2	8	no	—
137	C₈×C₅₃	C₂²×C₈	1	4	yes	—
138	C₈×C₅₃	C₂²×C₈	1	4	yes	—
139	#88322	—	2	8	no	—
140	#88322	C₂²×C₈	1	4	yes	—
141	#88322	C₂²×C₈	1	4	yes	—
142	C₂×#1450	—	53	1	no	maximal, Sylow
143	C₂²×C₅₃	—	2	2	no	—
144	C₂²×C₅₃	C₂×C₄×C₈	1	2	yes	—
145	C₂²×C₅₃	C₂²×C₄²	1	1	yes	characteristic
146	C₂×D₅₃	—	2	2	no	—
147	C₂×D₅₃	—	2	8	no	—
148	C₂×D₅₃	C₂×C₄×C₈	1	2	yes	—
149	C₂×D₅₃	C₂×C₄×C₈	1	2	yes	—
150	C₂×D₅₃	#405	1	4	yes	—
151	C₄×C₅₃	—	2	2	no	—
152	C₄×C₅₃	C₂×C₄×C₈	1	2	yes	—
153	C₄×C₅₃	C₂³×C₈	1	2	yes	—
154	C₄×C₅₃	C₂×#131	1	4	yes	—
155	Dic₅₃	—	2	2	no	—
156	Dic₅₃	C₂×C₄×C₈	1	2	yes	—
157	Dic₅₃	C₂×C₄×C₈	1	2	yes	—
158	Dic₅₃	#405	1	4	yes	—
159	C₂×#431	—	53	2	no	—
160	C₂²×C₄×C₈	—	53	1	no	—
161	#1450	—	53	2	no	—
162	#1450	—	53	1	no	—
163	#1450	—	53	2	no	—
164	#1450	—	53	2	no	—
165	#1450	D₅₃	1	1	yes	characteristic, direct product factor
166	C₂×#322	—	53	2	no	—
167	C₂×C₈²	—	53	2	no	—
168	C₂×C₅₃	—	2	2	no	—
169	C₂×C₅₃	C₂²×C₄×C₈	1	1	yes	characteristic
170	C₂×C₅₃	C₂×#405	1	2	yes	—
171	D₅₃	—	2	2	no	—
172	D₅₃	#1450	1	4	yes	direct product factor
173	C₂³×C₈	—	53	2	no	—
174	C₂²×C₄²	—	53	1	no	—
175	#431	—	53	2	no	—
176	#431	—	53	4	no	—
177	#431	—	53	4	no	—
178	#431	—	53	4	no	—
179	#431	C₂×D₅₃	1	2	yes	—
180	C₂×C₄⋊C₈	—	53	4	no	—
181	C₂×C₂²⋊C₈	—	53	4	no	—
182	C₂×C₄×C₈	—	53	2	no	—
183	C₂×C₄×C₈	—	53	1	no	—
184	C₂×C₄×C₈	—	53	2	no	—
185	C₂×C₄×C₈	—	53	2	no	—
186	C₂×C₄×C₈	—	53	1	no	—
187	C₂×C₄×C₈	—	53	1	no	—
188	C₂×C₄×C₈	—	53	4	no	—
189	C₂×C₄×C₈	—	53	1	no	—
190	C₂×C₄×C₈	—	53	1	no	—
191	C₂×C₄×C₈	C₂×D₅₃	1	1	yes	characteristic
192	#322	—	53	2	no	—
193	#322	—	53	4	no	—
194	#322	C₂×D₅₃	1	2	yes	complemented
195	C₈²	—	53	4	no	—
196	C₈²	—	53	2	no	—
197	C₈²	C₂×D₅₃	1	2	yes	complemented
198	C₅₃	C₂×#1450	1	1	yes	characteristic, complemented, Sylow
199	C₂³×C₄	—	53	1	no	—
200	C₂³×C₄	—	53	2	no	—
201	C₂²×C₈	—	53	2	no	—
202	C₂²×C₈	—	53	2	no	—
203	C₂²×C₈	—	53	2	no	—
204	C₂²×C₈	—	53	2	no	—
205	C₂²×C₈	—	53	2	no	—
206	C₂²×C₈	—	53	16	no	—
207	C₂²×C₈	—	53	8	no	—
208	C₂²×C₈	C₄×D₅₃	1	2	yes	—
209	C₂×C₄²	—	53	1	no	—
210	C₂×C₄²	—	53	2	no	—
211	C₂×C₄²	—	53	1	no	—
212	C₂×C₄²	—	53	1	no	—
213	C₂×C₄²	—	53	2	no	—
214	C₂×C₄²	—	53	2	no	—
215	C₂×C₄²	—	53	1	no	—
216	C₂×C₄²	—	53	1	no	—
217	C₂×C₄²	C₂²×D₅₃	1	1	yes	characteristic
218	C₄⋊C₈	—	53	8	no	—
219	C₄⋊C₈	—	53	4	no	—
220	C₄⋊C₈	C₄×D₅₃	1	4	yes	—
221	C₂²⋊C₈	—	53	4	no	—
222	C₂²⋊C₈	—	53	8	no	—
223	C₂²⋊C₈	C₄×D₅₃	1	4	yes	—
224	C₄×C₈	—	53	2	no	—
225	C₄×C₈	—	53	2	no	—
226	C₄×C₈	—	53	2	no	—
227	C₄×C₈	—	53	2	no	—
228	C₄×C₈	—	53	8	no	—
229	C₄×C₈	—	53	2	no	—
230	C₄×C₈	—	53	1	no	—
231	C₄×C₈	—	53	4	no	—
232	C₄×C₈	—	53	1	no	—
233	C₄×C₈	C₄×D₅₃	1	2	yes	—
234	C₄×C₈	C₂²×D₅₃	1	4	yes	—
235	C₄×C₈	C₂²×D₅₃	1	1	yes	characteristic
236	C₄×C₈	C₂²×D₅₃	1	1	yes	characteristic
237	C₂⁴	—	53	1	no	—
238	C₂²×C₄	—	53	1	no	—
239	C₂²×C₄	—	53	1	no	—
240	C₂²×C₄	—	53	1	no	—
241	C₂²×C₄	—	53	8	no	—
242	C₂²×C₄	—	53	1	no	—
243	C₂²×C₄	—	53	1	no	—
244	C₂²×C₄	—	53	2	no	—
245	C₂²×C₄	—	53	2	no	—
246	C₂²×C₄	—	53	2	no	—
247	C₂²×C₄	—	53	2	no	—
248	C₂²×C₄	—	53	2	no	—
249	C₂²×C₄	—	106	8	no	—
250	C₂²×C₄	C₂×C₄×D₅₃	1	2	yes	—
251	C₂²×C₄	C₂×C₄×D₅₃	1	1	yes	characteristic
252	C₂×C₈	—	53	8	no	—
253	C₂×C₈	—	53	2	no	—
254	C₂×C₈	—	53	2	no	—
255	C₂×C₈	—	53	8	no	—
256	C₂×C₈	—	53	8	no	—
257	C₂×C₈	—	53	8	no	—
258	C₂×C₈	—	53	8	no	—
259	C₂×C₈	—	53	8	no	—
260	C₂×C₈	—	106	16	no	—
261	C₂×C₈	C₈×D₅₃	1	8	yes	complemented
262	C₂×C₈	C₂×C₄×D₅₃	1	8	yes	—
263	C₂×C₈	C₂×C₄×D₅₃	1	2	yes	—
264	C₂×C₈	C₂×C₄×D₅₃	1	2	yes	—
265	C₄²	—	53	2	no	—
266	C₄²	—	53	2	no	—
267	C₄²	—	53	2	no	—
268	C₄²	—	53	1	no	—
269	C₄²	—	53	2	no	—
270	C₄²	—	53	1	no	—
271	C₄²	—	53	2	no	—
272	C₄²	C₂×C₄×D₅₃	1	1	yes	characteristic
273	C₄²	C₂×C₄×D₅₃	1	2	yes	—
274	C₄²	C₂³×D₅₃	1	1	yes	characteristic
275	C₂³	—	53	1	no	—
276	C₂³	—	53	1	no	—
277	C₂³	—	53	4	no	—
278	C₂³	—	106	4	no	—
279	C₂³	C₄²×D₅₃	1	1	yes	characteristic
280	C₂×C₄	—	2	4	no	—
281	C₂×C₄	—	53	1	no	—
282	C₂×C₄	—	53	1	no	—
283	C₂×C₄	—	53	2	no	—
284	C₂×C₄	—	53	4	no	—
285	C₂×C₄	—	53	4	no	—
286	C₂×C₄	—	53	4	no	—
287	C₂×C₄	—	53	4	no	—
288	C₂×C₄	—	53	4	no	—
289	C₂×C₄	—	53	8	no	—
290	C₂×C₄	—	53	2	no	—
291	C₂×C₄	—	106	4	no	—
292	C₂×C₄	—	106	4	no	—
293	C₂×C₄	—	106	4	no	—
294	C₂×C₄	—	106	4	no	—
295	C₂×C₄	C₄²×D₅₃	1	2	yes	—
296	C₂×C₄	C₄²×D₅₃	1	1	yes	characteristic
297	C₂×C₄	C₂×C₈×D₅₃	1	4	yes	—
298	C₂×C₄	C₂²×C₄×D₅₃	1	1	yes	characteristic
299	C₂×C₄	C₂²×C₄×D₅₃	1	2	yes	—
300	C₈	—	2	8	no	—
301	C₈	—	53	4	no	—
302	C₈	—	53	4	no	—
303	C₈	—	106	8	no	—
304	C₈	C₂×C₈×D₅₃	1	4	yes	—
305	C₈	C₂×C₈×D₅₃	1	4	yes	—
306	C₂²	—	2	2	no	—
307	C₂²	—	53	2	no	—
308	C₂²	—	53	4	no	—
309	C₂²	—	53	2	no	—
310	C₂²	—	106	8	no	—
311	C₂²	—	106	2	no	—
312	C₂²	C₄×C₈×D₅₃	1	2	yes	—
313	C₂²	C₂×C₄²×D₅₃	1	1	yes	characteristic
314	C₄	—	2	2	no	—
315	C₄	—	53	2	no	—
316	C₄	—	53	2	no	—
317	C₄	—	53	4	no	—
318	C₄	—	106	2	no	—
319	C₄	C₄×C₈×D₅₃	1	2	yes	—
320	C₄	C₂²×C₈×D₅₃	1	2	yes	—
321	C₄	#131×D₅₃	1	4	yes	—
322	C₂	—	2	2	no	—
323	C₂	—	53	4	no	—
324	C₂	—	106	2	no	—
325	C₂	C₂×C₄×C₈×D₅₃	1	1	yes	characteristic
326	C₂	#405×D₅₃	1	2	yes	—

	subgroup	G/H
center	C₄×C₈	C₂²×D₅₃
commutator	C₂×C₅₃	C₂²×C₄×C₈
Frattini	C₄²	C₂³×D₅₃
Fitting	C₅₃×#1450	C₂
socle	C₂²×C₅₃	C₂²×C₄²
prime radical	C₂³×D₅₃	C₄²
cosocle	C₄²×C₅₃	C₂⁴
hypercenter	#1450	D₅₃
hypocenter	C₅₃	C₂×#1450
LCS	G ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₄×C₈ ◁ #1450

Complements

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

N	H ≅ G/N	type
3. #20273000	C₂	semidirect
4. #20272999	C₂	semidirect
5. #20272998	C₂	semidirect
6. #20272997	C₂	semidirect
7. #322×D₅₃	C₂	semidirect
8. C₈²×D₅₃	C₂	semidirect
9. C₅₃×#1450	C₂	semidirect
27. #16810345	C₂²	semidirect
28. #16810344	C₂²	semidirect
29. C₈×#88322	C₂²	semidirect
30. C₈×#88322	C₂²	semidirect
33. C₅₃×#322	C₂²	semidirect
34. C₈²×C₅₃	C₂²	semidirect
38. C₂×C₈×D₅₃	C₈	semidirect
88. C₂×C₈×C₅₃	C₂×C₈	semidirect
102. C₂×#88322	C₂×C₈	semidirect
109. C₈×D₅₃	C₂×C₈	semidirect
110. C₈×D₅₃	C₂×C₈	semidirect
165. #1450	D₅₃	direct (N × H)
172. D₅₃	#1450	direct (N × H)
194. #322	C₂×D₅₃	semidirect
197. C₈²	C₂×D₅₃	semidirect
198. C₅₃	C₂×#1450	semidirect
261. C₂×C₈	C₈×D₅₃	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
C₂×C₄×C₈×D₅₃	2
#20273000	2
#20272999	2
#20272998	2
#20272997	2
#322×D₅₃	2
C₈²×D₅₃	2
C₅₃×#1450	2
C₂×#1450	53

Subgroups — D53×#1450 #27349668

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — D₅₃×#1450 #27349668