Subgroups — D₅₃×#1846 #27384495

Order 13568.

1182 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	C₂×#107×D₅₃	C₂	1	2	yes	maximal
2	C₂×#106×D₅₃	C₂	1	2	yes	maximal
3	C₂×C₄×C₈×D₅₃	C₂	1	1	yes	maximal, characteristic
4	#382×D₅₃	C₂	1	2	yes	maximal
5	#20274879	C₂	1	1	yes	maximal, characteristic, complemented
6	#20274878	C₂	1	2	yes	maximal, complemented
7	#20274877	C₂	1	2	yes	maximal, complemented
8	#20274876	C₂	1	2	yes	maximal, complemented
9	C₅₃×#1846	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂²×C₈×D₅₃	C₂²	1	1	yes	characteristic
11	C₂²×C₈×D₅₃	C₂²	1	1	yes	characteristic
12	C₂×C₄⋊C₄×D₅₃	C₂²	1	4	yes	—
13	C₂×C₄²×D₅₃	C₂²	1	1	yes	characteristic
14	C₂×#13133151	C₂²	1	2	yes	—
15	C₂×#13133150	C₂²	1	4	yes	—
16	#107×D₅₃	—	2	8	no	—
17	C₂×#13133149	C₂²	1	2	yes	—
18	C₂×#13133148	C₂²	1	4	yes	—
19	#106×D₅₃	—	2	8	no	—
20	C₂×C₄×#88322	C₂²	1	1	yes	characteristic
21	C₂×C₈×Dic₅₃	C₂²	1	1	yes	characteristic
22	C₂×C₈×Dic₅₃	C₂²	1	1	yes	characteristic
23	C₄×C₈×D₅₃	C₄	1	8	yes	complemented
24	#16810489	C₂²	1	2	yes	—
25	#16810488	C₂²	1	4	yes	—
26	#16810487	—	2	4	no	—
27	#95×D₅₃	—	2	4	no	—
28	C₂×#107×C₅₃	C₂²	1	2	yes	—
29	C₂×#106×C₅₃	C₂²	1	2	yes	—
30	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
31	C₅₃×#382	C₂²	1	2	yes	—
32	C₂²×C₄×D₅₃	—	2	4	no	—
33	C₂²×C₄×D₅₃	D₄	1	2	yes	—
34	C₂²×C₄×D₅₃	C₂³	1	1	yes	characteristic
35	C₂²×#88322	C₂³	1	1	yes	characteristic
36	C₂²×#88322	C₂³	1	1	yes	characteristic
37	C₂×C₈×D₅₃	C₂×C₄	1	4	yes	—
38	C₂×C₈×D₅₃	C₂×C₄	1	4	yes	—
39	C₂×C₈×D₅₃	D₄	1	4	yes	—
40	C₂×C₈×D₅₃	D₄	1	4	yes	—
41	C₂×C₈×D₅₃	Q₈	1	4	yes	—
42	C₂×C₈×D₅₃	Q₈	1	4	yes	—
43	C₂×C₄⋊Dic₅₃	C₂³	1	4	yes	—
44	C₂×#11719919	—	2	4	no	—
45	C₄⋊C₄×D₅₃	—	2	8	no	—
46	C₄⋊C₄×D₅₃	—	2	8	no	—
47	C₂×C₄×Dic₅₃	D₄	1	2	yes	—
48	C₂×C₄×Dic₅₃	C₂³	1	1	yes	characteristic
49	C₄²×D₅₃	C₂×C₄	1	4	yes	—
50	C₄²×D₅₃	Q₈	1	4	yes	—
51	#13133151	—	2	4	no	—
52	#13133150	—	2	8	no	—
53	#13133149	—	2	4	no	—
54	#13133148	—	2	8	no	—
55	C₄×#88322	C₂×C₄	1	4	yes	complemented
56	C₈×Dic₅₃	C₂×C₄	1	4	yes	complemented
57	C₈×Dic₅₃	C₂×C₄	1	4	yes	complemented
58	#13133124	—	2	4	no	—
59	#13133124	—	2	8	no	—
60	C₂²×C₈×C₅₃	C₂³	1	1	yes	characteristic
61	C₂²×C₈×C₅₃	C₂³	1	1	yes	characteristic
62	C₂×C₄⋊C₄×C₅₃	C₂³	1	4	yes	—
63	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
64	#107×C₅₃	—	2	4	no	—
65	#106×C₅₃	—	2	4	no	—
66	C₄×C₈×C₅₃	C₂×C₄	1	4	yes	complemented
67	#95×C₅₃	—	2	4	no	—
68	C₂³×D₅₃	C₂×D₄	1	1	yes	characteristic
69	C₂²×C₄×C₅₃	—	2	4	no	—
70	C₂²×C₄×C₅₃	C₂×D₄	1	2	yes	—
71	C₂²×C₄×C₅₃	C₂⁴	1	1	yes	characteristic
72	C₂²×Dic₅₃	—	2	4	no	—
73	C₂²×Dic₅₃	C₂×D₄	1	2	yes	—
74	C₂²×Dic₅₃	C₂×D₄	1	1	yes	characteristic
75	C₂×C₄×D₅₃	—	2	8	no	—
76	C₂×C₄×D₅₃	—	4	8	no	—
77	C₂×C₄×D₅₃	—	4	16	no	—
78	C₂×C₄×D₅₃	C₄⋊C₄	1	8	yes	—
79	C₂×C₄×D₅₃	C₂²×C₄	1	2	yes	—
80	C₂×C₄×D₅₃	C₂×D₄	1	2	yes	—
81	C₂×C₄×D₅₃	C₂×D₄	1	2	yes	—
82	C₂×C₄×D₅₃	C₂×Q₈	1	2	yes	—
83	C₂×C₄×D₅₃	C₂×Q₈	1	2	yes	—
84	C₂×C₄×D₅₃	C₂×Q₈	1	2	yes	—
85	C₂×C₈×C₅₃	C₂²×C₄	1	2	yes	—
86	C₂×C₈×C₅₃	C₂²×C₄	1	2	yes	—
87	C₂×C₈×C₅₃	C₂×D₄	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	C₂×Q₈	1	2	yes	—
91	C₄⋊C₄×C₅₃	—	2	4	no	—
92	C₄⋊C₄×C₅₃	—	2	4	no	—
93	C₄²×C₅₃	C₂²×C₄	1	2	yes	—
94	C₄²×C₅₃	C₂×Q₈	1	2	yes	—
95	C₄⋊Dic₅₃	—	2	4	no	—
96	C₄⋊Dic₅₃	—	2	4	no	—
97	#11719919	—	4	8	no	—
98	C₄×Dic₅₃	C₄⋊C₄	1	8	yes	—
99	C₄×Dic₅₃	C₂²×C₄	1	2	yes	—
100	C₄×Dic₅₃	C₂×Q₈	1	2	yes	—
101	C₂×#88322	C₂²×C₄	1	2	yes	—
102	C₂×#88322	C₂²×C₄	1	2	yes	—
103	C₂×#88322	C₂×D₄	1	2	yes	—
104	C₂×#88322	C₂×D₄	1	2	yes	—
105	C₂×#88322	C₂×Q₈	1	2	yes	—
106	C₂×#88322	C₂×Q₈	1	2	yes	—
107	C₈×D₅₃	C₄⋊C₄	1	16	yes	—
108	C₈×D₅₃	C₄⋊C₄	1	16	yes	—
109	C₂³×C₅₃	C₂²×D₄	1	1	yes	characteristic
110	C₂²×D₅₃	C₂×C₄⋊C₄	1	2	yes	—
111	C₂²×D₅₃	#127	1	2	yes	—
112	C₂²×D₅₃	C₄⋊Q₈	1	2	yes	—
113	C₂²×D₅₃	#135	1	8	yes	—
114	C₂×C₄×C₅₃	—	2	4	no	—
115	C₂×C₄×C₅₃	—	4	4	no	—
116	C₂×C₄×C₅₃	—	4	8	no	—
117	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	4	yes	—
118	C₂×C₄×C₅₃	C₂³×C₄	1	1	yes	characteristic
119	C₂×C₄×C₅₃	C₂²×D₄	1	1	yes	characteristic
120	C₂×C₄×C₅₃	C₂²×D₄	1	1	yes	characteristic
121	C₂×C₄×C₅₃	C₂²×Q₈	1	1	yes	characteristic
122	C₂×C₄×C₅₃	C₂²×Q₈	1	1	yes	characteristic
123	C₂×C₄×C₅₃	C₂²×Q₈	1	1	yes	characteristic
124	C₂×Dic₅₃	—	2	4	no	—
125	C₂×Dic₅₃	—	4	4	no	—
126	C₂×Dic₅₃	—	4	8	no	—
127	C₂×Dic₅₃	C₂×C₄⋊C₄	1	4	yes	—
128	C₂×Dic₅₃	C₂×C₄⋊C₄	1	2	yes	—
129	C₂×Dic₅₃	#127	1	2	yes	—
130	C₂×Dic₅₃	C₄⋊Q₈	1	2	yes	—
131	C₄×D₅₃	—	2	16	no	—
132	C₄×D₅₃	—	8	16	no	—
133	C₄×D₅₃	C₂×C₄⋊C₄	1	4	yes	—
134	C₄×D₅₃	C₂×C₄⋊C₄	1	4	yes	—
135	C₄×D₅₃	C₄⋊Q₈	1	4	yes	—
136	C₄×D₅₃	C₄⋊Q₈	1	4	yes	—
137	C₈×C₅₃	C₂×C₄⋊C₄	1	4	yes	—
138	C₈×C₅₃	C₂×C₄⋊C₄	1	4	yes	—
139	#88322	C₂×C₄⋊C₄	1	4	yes	—
140	#88322	C₂×C₄⋊C₄	1	4	yes	—
141	C₂×#1846	—	53	1	no	maximal, Sylow
142	C₂²×C₅₃	C₂²×C₄⋊C₄	1	1	yes	characteristic
143	C₂²×C₅₃	C₂×#127	1	1	yes	characteristic
144	C₂²×C₅₃	C₂×C₄⋊Q₈	1	1	yes	characteristic
145	C₂²×C₅₃	C₂×#135	1	4	yes	—
146	C₂×D₅₃	#384	1	4	yes	—
147	C₂×D₅₃	#427	1	8	yes	—
148	C₂×D₅₃	#495	1	8	yes	—
149	C₂×D₅₃	#499	1	8	yes	—
150	C₄×C₅₃	—	2	4	no	—
151	C₄×C₅₃	—	8	4	no	—
152	C₄×C₅₃	C₂²×C₄⋊C₄	1	1	yes	characteristic
153	C₄×C₅₃	C₂²×C₄⋊C₄	1	1	yes	characteristic
154	C₄×C₅₃	C₂×C₄⋊Q₈	1	1	yes	characteristic
155	C₄×C₅₃	C₂×C₄⋊Q₈	1	1	yes	characteristic
156	Dic₅₃	—	2	4	no	—
157	Dic₅₃	—	8	4	no	—
158	Dic₅₃	#384	1	4	yes	—
159	C₂²×#107	—	53	2	no	—
160	C₂²×#106	—	53	2	no	—
161	C₂²×C₄×C₈	—	53	1	no	—
162	C₂×#382	—	53	2	no	—
163	#1846	—	53	2	no	—
164	#1846	—	53	2	no	—
165	#1846	—	53	1	no	—
166	#1846	—	53	2	no	—
167	#1846	D₅₃	1	1	yes	characteristic, direct product factor
168	C₂×C₅₃	C₂×#384	1	1	yes	characteristic
169	C₂×C₅₃	C₂×#427	1	2	yes	—
170	C₂×C₅₃	C₂×#495	1	2	yes	—
171	C₂×C₅₃	C₂×#499	1	2	yes	—
172	D₅₃	#1846	1	8	yes	direct product factor
173	C₂³×C₈	—	53	1	no	—
174	C₂³×C₈	—	53	1	no	—
175	C₂²×C₄⋊C₄	—	53	4	no	—
176	C₂²×C₄²	—	53	1	no	—
177	C₂×#107	—	53	2	no	—
178	C₂×#107	—	53	4	no	—
179	C₂×#107	—	106	8	no	—
180	C₂×#107	C₂×D₅₃	1	2	yes	—
181	C₂×#106	—	53	2	no	—
182	C₂×#106	—	53	4	no	—
183	C₂×#106	—	106	8	no	—
184	C₂×#106	C₂×D₅₃	1	2	yes	—
185	C₂×C₄×C₈	—	53	1	no	—
186	C₂×C₄×C₈	—	53	1	no	—
187	C₂×C₄×C₈	—	53	1	no	—
188	C₂×C₄×C₈	—	53	8	no	—
189	C₂×C₄×C₈	C₂×D₅₃	1	1	yes	characteristic
190	#382	—	53	4	no	—
191	#382	—	53	2	no	—
192	#382	—	106	4	no	—
193	#382	C₂×D₅₃	1	2	yes	—
194	C₂×#95	—	106	4	no	—
195	C₅₃	C₂×#1846	1	1	yes	characteristic, complemented, Sylow
196	C₂³×C₄	—	53	1	no	—
197	C₂³×C₄	—	53	2	no	—
198	C₂³×C₄	—	106	4	no	—
199	C₂²×C₈	—	53	1	no	—
200	C₂²×C₈	—	53	1	no	—
201	C₂²×C₈	—	53	4	no	—
202	C₂²×C₈	—	53	4	no	—
203	C₂²×C₈	—	53	4	no	—
204	C₂²×C₈	—	53	4	no	—
205	C₂²×C₈	—	53	4	no	—
206	C₂²×C₈	—	53	4	no	—
207	C₂²×C₈	C₂²×D₅₃	1	1	yes	characteristic
208	C₂²×C₈	C₂²×D₅₃	1	1	yes	characteristic
209	C₂×C₄⋊C₄	—	53	4	no	—
210	C₂×C₄⋊C₄	—	106	8	no	—
211	C₂×C₄⋊C₄	—	106	8	no	—
212	C₂×C₄⋊C₄	—	106	4	no	—
213	C₂×C₄⋊C₄	C₂²×D₅₃	1	4	yes	—
214	C₂×C₄²	—	53	2	no	—
215	C₂×C₄²	—	53	1	no	—
216	C₂×C₄²	—	53	4	no	—
217	C₂×C₄²	—	53	4	no	—
218	C₂×C₄²	C₂²×D₅₃	1	1	yes	characteristic
219	#107	—	2	4	no	—
220	#107	—	106	8	no	—
221	#107	—	106	4	no	—
222	#106	—	2	4	no	—
223	#106	—	106	8	no	—
224	#106	—	106	4	no	—
225	C₄×C₈	—	53	4	no	—
226	C₄×C₈	—	53	4	no	—
227	C₄×C₈	—	53	4	no	—
228	C₄×C₈	C₄×D₅₃	1	4	yes	complemented
229	#95	—	2	4	no	—
230	#95	—	106	8	no	—
231	#95	—	106	4	no	—
232	C₂⁴	—	53	1	no	—
233	C₂²×C₄	—	2	4	no	—
234	C₂²×C₄	—	53	1	no	—
235	C₂²×C₄	—	53	2	no	—
236	C₂²×C₄	—	53	2	no	—
237	C₂²×C₄	—	53	2	no	—
238	C₂²×C₄	—	53	2	no	—
239	C₂²×C₄	—	53	2	no	—
240	C₂²×C₄	—	53	2	no	—
241	C₂²×C₄	—	53	2	no	—
242	C₂²×C₄	—	53	8	no	—
243	C₂²×C₄	—	106	8	no	—
244	C₂²×C₄	—	106	4	no	—
245	C₂²×C₄	—	212	8	no	—
246	C₂²×C₄	—	212	16	no	—
247	C₂²×C₄	D₄×D₅₃	1	2	yes	—
248	C₂²×C₄	C₂³×D₅₃	1	1	yes	characteristic
249	C₂×C₈	—	53	2	no	—
250	C₂×C₈	—	53	2	no	—
251	C₂×C₈	—	53	2	no	—
252	C₂×C₈	—	53	2	no	—
253	C₂×C₈	—	53	2	no	—
254	C₂×C₈	—	53	2	no	—
255	C₂×C₈	—	53	16	no	—
256	C₂×C₈	—	53	16	no	—
257	C₂×C₈	C₂×C₄×D₅₃	1	2	yes	—
258	C₂×C₈	C₂×C₄×D₅₃	1	2	yes	—
259	C₂×C₈	D₄×D₅₃	1	2	yes	—
260	C₂×C₈	D₄×D₅₃	1	2	yes	—
261	C₂×C₈	Q₈×D₅₃	1	2	yes	—
262	C₂×C₈	Q₈×D₅₃	1	2	yes	—
263	C₄⋊C₄	—	2	4	no	—
264	C₄⋊C₄	—	2	4	no	—
265	C₄⋊C₄	—	106	4	no	—
266	C₄⋊C₄	—	106	4	no	—
267	C₄⋊C₄	—	212	8	no	—
268	C₄²	—	53	8	no	—
269	C₄²	—	53	2	no	—
270	C₄²	—	53	2	no	—
271	C₄²	C₂×C₄×D₅₃	1	2	yes	—
272	C₄²	Q₈×D₅₃	1	2	yes	—
273	C₂³	—	53	2	no	—
274	C₂³	—	53	8	no	—
275	C₂³	—	53	2	no	—
276	C₂³	—	53	2	no	—
277	C₂³	C₂×D₄×D₅₃	1	1	yes	characteristic
278	C₂×C₄	—	2	4	no	—
279	C₂×C₄	—	4	8	no	—
280	C₂×C₄	—	4	4	no	—
281	C₂×C₄	—	53	2	no	—
282	C₂×C₄	—	53	2	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	—	106	16	no	—
290	C₂×C₄	—	106	4	no	—
291	C₂×C₄	—	424	16	no	—
292	C₂×C₄	—	212	8	no	—
293	C₂×C₄	—	212	4	no	—
294	C₂×C₄	C₄⋊C₄×D₅₃	1	4	yes	—
295	C₂×C₄	C₂²×C₄×D₅₃	1	1	yes	characteristic
296	C₂×C₄	C₂×D₄×D₅₃	1	1	yes	characteristic
297	C₂×C₄	C₂×D₄×D₅₃	1	1	yes	characteristic
298	C₂×C₄	C₂×Q₈×D₅₃	1	1	yes	characteristic
299	C₂×C₄	C₂×Q₈×D₅₃	1	1	yes	characteristic
300	C₂×C₄	C₂×Q₈×D₅₃	1	1	yes	characteristic
301	C₈	—	53	4	no	—
302	C₈	—	53	4	no	—
303	C₈	C₄⋊C₄×D₅₃	1	4	yes	—
304	C₈	C₄⋊C₄×D₅₃	1	4	yes	—
305	C₂²	—	53	8	no	—
306	C₂²	—	53	4	no	—
307	C₂²	—	53	8	no	—
308	C₂²	—	53	8	no	—
309	C₂²	C₂×C₄⋊C₄×D₅₃	1	1	yes	characteristic
310	C₂²	#127×D₅₃	1	1	yes	characteristic
311	C₂²	C₄⋊Q₈×D₅₃	1	1	yes	characteristic
312	C₂²	#135×D₅₃	1	4	yes	—
313	C₄	—	2	4	no	—
314	C₄	—	8	4	no	—
315	C₄	—	53	4	no	—
316	C₄	—	106	4	no	—
317	C₄	—	424	4	no	—
318	C₄	C₂×C₄⋊C₄×D₅₃	1	1	yes	characteristic
319	C₄	C₂×C₄⋊C₄×D₅₃	1	1	yes	characteristic
320	C₄	C₄⋊Q₈×D₅₃	1	1	yes	characteristic
321	C₄	C₄⋊Q₈×D₅₃	1	1	yes	characteristic
322	C₂	—	53	8	no	—
323	C₂	#384×D₅₃	1	1	yes	characteristic
324	C₂	#427×D₅₃	1	2	yes	—
325	C₂	#495×D₅₃	1	2	yes	—
326	C₂	#499×D₅₃	1	2	yes	—

	subgroup	G/H
center	C₂³	C₂×D₄×D₅₃
commutator	C₂×C₄×C₅₃	C₂³×C₄
Frattini	C₂²×C₄	C₂³×D₅₃
Fitting	C₅₃×#1846	C₂
socle	C₂³×C₅₃	C₂²×D₄
prime radical	C₂³×D₅₃	C₂×D₄
cosocle	C₂²×C₄×C₅₃	C₂⁴
hypercenter	#1846	D₅₃
hypocenter	C₅₃	C₂×#1846
LCS	G ▷ C₂×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #1846

Complements

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

N	H ≅ G/N	type
5. #20274879	C₂	semidirect
6. #20274878	C₂	semidirect
7. #20274877	C₂	semidirect
8. #20274876	C₂	semidirect
9. C₅₃×#1846	C₂	semidirect
23. C₄×C₈×D₅₃	C₄	semidirect
55. C₄×#88322	C₂×C₄	semidirect
56. C₈×Dic₅₃	C₂×C₄	semidirect
57. C₈×Dic₅₃	C₂×C₄	semidirect
66. C₄×C₈×C₅₃	C₂×C₄	semidirect
167. #1846	D₅₃	direct (N × H)
172. D₅₃	#1846	direct (N × H)
195. C₅₃	C₂×#1846	semidirect
228. C₄×C₈	C₄×D₅₃	semidirect

Maximal subgroups

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

subgroup	index
C₂×#107×D₅₃	2
C₂×#106×D₅₃	2
C₂×C₄×C₈×D₅₃	2
#382×D₅₃	2
#20274879	2
#20274878	2
#20274877	2
#20274876	2
C₅₃×#1846	2
C₂×#1846	53

Subgroups — D53×#1846 #27384495

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — D₅₃×#1846 #27384495