Subgroups — C₅₃×#13121 #27220633

Order 13568.

382 conjugacy classes of subgroups.

The subgroup lattice has 342 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₅₃×C₄⋊C₁₆	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×C₅₃×C₂²⋊C₁₆	C₂	1	1	yes	maximal, characteristic
3	C₂×C₅₃×C₂²⋊C₁₆	C₂	1	1	yes	maximal, characteristic
4	C₂×C₅₃×#346	C₂	1	1	yes	maximal, characteristic, complemented
5	C₈×C₂²⋊C₄×C₅₃	C₂	1	1	yes	maximal, characteristic
6	C₅₃×#1374	C₂	1	1	yes	maximal, characteristic, complemented
7	C₅₃×#1374	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂³×C₈×C₅₃	C₂²	1	1	yes	characteristic
9	C₂²×C₁₆×C₅₃	C₂²	1	1	yes	characteristic
10	C₂²×C₁₆×C₅₃	C₂²	1	1	yes	characteristic
11	C₂²×C₁₆×C₅₃	C₂²	1	1	yes	characteristic
12	C₂²×C₁₆×C₅₃	C₂²	1	1	yes	characteristic
13	C₂×C₂²⋊C₈×C₅₃	C₄	1	1	yes	characteristic
14	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
15	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
16	C₄×C₂²⋊C₄×C₅₃	C₄	1	1	yes	characteristic
17	C₅₃×C₄⋊C₁₆	—	2	2	no	—
18	C₅₃×C₂²⋊C₁₆	—	2	2	no	—
19	C₅₃×C₂²⋊C₁₆	—	2	2	no	—
20	C₅₃×#346	—	2	2	no	—
21	C₅₃×#336	C₄	1	1	yes	characteristic
22	C₅₃×#336	C₄	1	1	yes	characteristic
23	C₂³×C₄×C₅₃	C₂×C₄	1	1	yes	characteristic
24	C₂²×C₈×C₅₃	—	2	1	no	—
25	C₂²×C₈×C₅₃	—	2	1	no	—
26	C₂²×C₈×C₅₃	—	2	1	no	—
27	C₂²×C₈×C₅₃	—	2	1	no	—
28	C₂²×C₈×C₅₃	—	2	1	no	—
29	C₂²×C₈×C₅₃	—	2	1	no	—
30	C₂²×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
31	C₂²×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
32	C₂²×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
33	C₂²×C₈×C₅₃	C₂³	1	1	yes	characteristic
34	C₂×C₂²⋊C₄×C₅₃	C₈	1	1	yes	characteristic
35	C₂×C₂²⋊C₄×C₅₃	C₈	1	1	yes	characteristic
36	C₂×C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
37	C₂×C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
38	C₂×C₁₆×C₅₃	—	2	1	no	—
39	C₂×C₁₆×C₅₃	—	2	1	no	—
40	C₂×C₁₆×C₅₃	—	2	1	no	—
41	C₂×C₁₆×C₅₃	—	2	1	no	—
42	C₂×C₁₆×C₅₃	—	2	1	no	—
43	C₂×C₁₆×C₅₃	—	2	1	no	—
44	C₂×C₁₆×C₅₃	—	2	1	no	—
45	C₂×C₁₆×C₅₃	—	2	1	no	—
46	C₂×C₁₆×C₅₃	—	2	1	no	—
47	C₂×C₁₆×C₅₃	—	2	1	no	—
48	C₂×C₁₆×C₅₃	D₄	1	2	yes	—
49	C₂×C₁₆×C₅₃	D₄	1	2	yes	—
50	C₂²⋊C₈×C₅₃	—	2	2	no	—
51	C₄×C₈×C₅₃	—	2	1	no	—
52	C₄×C₈×C₅₃	—	2	1	no	—
53	C₄×C₈×C₅₃	—	2	1	no	—
54	C₄×C₈×C₅₃	—	2	1	no	—
55	#95×C₅₃	C₈	1	1	yes	characteristic
56	#95×C₅₃	C₈	1	1	yes	characteristic
57	C₂⁴×C₅₃	C₂×C₈	1	1	yes	characteristic
58	C₂²×C₄×C₅₃	—	2	1	no	—
59	C₂²×C₄×C₅₃	—	2	1	no	—
60	C₂²×C₄×C₅₃	—	2	1	no	—
61	C₂²×C₄×C₅₃	—	2	1	no	—
62	C₂²×C₄×C₅₃	—	2	1	no	—
63	C₂²×C₄×C₅₃	—	2	1	no	—
64	C₂²×C₄×C₅₃	C₂×C₈	1	1	yes	characteristic
65	C₂²×C₄×C₅₃	C₂×C₈	1	1	yes	characteristic
66	C₂²×C₄×C₅₃	C₂×C₈	1	1	yes	characteristic
67	C₂²×C₄×C₅₃	C₂×C₈	1	1	yes	characteristic
68	C₂²×C₄×C₅₃	C₂×C₈	1	1	yes	characteristic
69	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
70	C₂×C₈×C₅₃	—	2	1	no	—
71	C₂×C₈×C₅₃	—	2	1	no	—
72	C₂×C₈×C₅₃	—	2	1	no	—
73	C₂×C₈×C₅₃	—	2	1	no	—
74	C₂×C₈×C₅₃	—	2	1	no	—
75	C₂×C₈×C₅₃	—	2	1	no	—
76	C₂×C₈×C₅₃	—	2	1	no	—
77	C₂×C₈×C₅₃	—	2	1	no	—
78	C₂×C₈×C₅₃	—	4	1	no	—
79	C₂×C₈×C₅₃	—	4	1	no	—
80	C₂×C₈×C₅₃	—	4	1	no	—
81	C₂×C₈×C₅₃	—	4	1	no	—
82	C₂×C₈×C₅₃	M₁₆	1	2	yes	—
83	C₂×C₈×C₅₃	C₂×D₄	1	1	yes	characteristic
84	C₂×C₈×C₅₃	C₂×D₄	1	1	yes	characteristic
85	C₂×C₈×C₅₃	#41	1	1	yes	characteristic
86	C₂×C₈×C₅₃	#41	1	1	yes	characteristic
87	C₂×C₈×C₅₃	#41	1	1	yes	characteristic
88	C₂×C₈×C₅₃	#41	1	1	yes	characteristic
89	C₂²⋊C₄×C₅₃	—	2	2	no	—
90	C₂²⋊C₄×C₅₃	—	2	2	no	—
91	C₄²×C₅₃	—	2	1	no	—
92	C₄²×C₅₃	—	2	1	no	—
93	C₄²×C₅₃	—	2	1	no	—
94	C₄²×C₅₃	—	2	1	no	—
95	C₁₆×C₅₃	—	2	2	no	—
96	C₁₆×C₅₃	—	2	2	no	—
97	C₁₆×C₅₃	—	4	1	no	—
98	C₁₆×C₅₃	—	4	1	no	—
99	C₂³×C₅₃	—	2	1	no	—
100	C₂³×C₅₃	—	2	1	no	—
101	C₂³×C₅₃	—	2	1	no	—
102	C₂³×C₅₃	—	2	1	no	—
103	C₂³×C₅₃	—	2	1	no	—
104	C₂³×C₅₃	—	2	1	no	—
105	C₂³×C₅₃	M₃₂	1	2	yes	—
106	C₂³×C₅₃	C₂²×C₈	1	1	yes	characteristic
107	C₂×C₄×C₅₃	—	2	1	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	1	no	—
117	C₂×C₄×C₅₃	—	2	1	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	M₃₂	1	2	yes	—
126	C₂×C₄×C₅₃	#117	1	1	yes	characteristic
127	C₂×C₄×C₅₃	C₄×D₄	1	1	yes	characteristic
128	C₂×C₄×C₅₃	C₄×D₄	1	1	yes	characteristic
129	C₂×C₄×C₅₃	C₂×M₁₆	1	1	yes	characteristic
130	C₂×C₄×C₅₃	#131	1	1	yes	characteristic
131	C₂×C₄×C₅₃	#131	1	1	yes	characteristic
132	C₈×C₅₃	—	2	2	no	—
133	C₈×C₅₃	—	4	1	no	—
134	C₈×C₅₃	—	4	1	no	—
135	C₈×C₅₃	#121	1	1	yes	characteristic
136	C₈×C₅₃	#123	1	1	yes	characteristic
137	C₈×C₅₃	#124	1	1	yes	characteristic
138	C₈×C₅₃	#126	1	1	yes	characteristic
139	#13121	C₅₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
140	C₂²×C₅₃	—	2	2	no	—
141	C₂²×C₅₃	—	2	2	no	—
142	C₂²×C₅₃	—	2	2	no	—
143	C₂²×C₅₃	—	4	1	no	—
144	C₂²×C₅₃	—	4	1	no	—
145	C₂²×C₅₃	—	4	1	no	—
146	C₂²×C₅₃	—	4	1	no	—
147	C₂²×C₅₃	#431	1	1	yes	characteristic
148	C₂²×C₅₃	C₈×D₄	1	1	yes	characteristic
149	C₂²×C₅₃	C₈×D₄	1	1	yes	characteristic
150	C₂²×C₅₃	C₂×M₃₂	1	1	yes	characteristic
151	C₂²×C₅₃	C₂×M₃₂	1	1	yes	characteristic
152	C₂²×C₅₃	#504	1	1	yes	characteristic
153	C₂²×C₅₃	#504	1	1	yes	characteristic
154	C₄×C₅₃	—	2	2	no	—
155	C₄×C₅₃	—	4	1	no	—
156	C₄×C₅₃	—	4	1	no	—
157	C₄×C₅₃	—	4	1	no	—
158	C₄×C₅₃	—	4	1	no	—
159	C₄×C₅₃	#388	1	1	yes	characteristic
160	C₄×C₅₃	#433	1	1	yes	characteristic
161	C₄×C₅₃	#435	1	1	yes	characteristic
162	C₄×C₅₃	#436	1	1	yes	characteristic
163	C₂×C₄⋊C₁₆	C₂×C₅₃	1	1	yes	characteristic, complemented
164	C₂×C₂²⋊C₁₆	C₂×C₅₃	1	1	yes	characteristic
165	C₂×C₂²⋊C₁₆	C₂×C₅₃	1	1	yes	characteristic
166	C₂×#346	C₂×C₅₃	1	1	yes	characteristic, complemented
167	C₈×C₂²⋊C₄	C₂×C₅₃	1	1	yes	characteristic
168	#1374	C₂×C₅₃	1	1	yes	characteristic, complemented
169	#1374	C₂×C₅₃	1	1	yes	characteristic, complemented
170	C₂×C₅₃	—	4	2	no	—
171	C₂×C₅₃	#1923	1	1	yes	characteristic
172	C₂×C₅₃	#2163	1	1	yes	characteristic
173	C₂×C₅₃	#2164	1	1	yes	characteristic
174	C₂×C₅₃	#2168	1	1	yes	characteristic
175	C₂×C₅₃	#2168	1	1	yes	characteristic
176	C₂×C₅₃	#2168	1	1	yes	characteristic
177	C₂×C₅₃	#2169	1	1	yes	characteristic
178	C₂³×C₈	C₂²×C₅₃	1	1	yes	characteristic
179	C₂²×C₁₆	C₂²×C₅₃	1	1	yes	characteristic
180	C₂²×C₁₆	C₂²×C₅₃	1	1	yes	characteristic
181	C₂²×C₁₆	C₂²×C₅₃	1	1	yes	characteristic
182	C₂²×C₁₆	C₂²×C₅₃	1	1	yes	characteristic
183	C₂×C₂²⋊C₈	C₄×C₅₃	1	1	yes	characteristic
184	C₂×C₄×C₈	C₂²×C₅₃	1	1	yes	characteristic
185	C₂×C₄×C₈	C₂²×C₅₃	1	1	yes	characteristic
186	C₄×C₂²⋊C₄	C₄×C₅₃	1	1	yes	characteristic
187	C₄⋊C₁₆	—	2	2	no	—
188	C₂²⋊C₁₆	—	2	2	no	—
189	C₂²⋊C₁₆	—	2	2	no	—
190	#346	—	2	2	no	—
191	#336	C₄×C₅₃	1	1	yes	characteristic
192	#336	C₄×C₅₃	1	1	yes	characteristic
193	C₅₃	#13121	1	1	yes	characteristic, direct product factor, Sylow
194	C₂³×C₄	C₂×C₄×C₅₃	1	1	yes	characteristic
195	C₂²×C₈	—	2	1	no	—
196	C₂²×C₈	—	2	1	no	—
197	C₂²×C₈	—	2	1	no	—
198	C₂²×C₈	—	2	1	no	—
199	C₂²×C₈	—	2	1	no	—
200	C₂²×C₈	—	2	1	no	—
201	C₂²×C₈	C₂×C₄×C₅₃	1	1	yes	characteristic
202	C₂²×C₈	C₂×C₄×C₅₃	1	1	yes	characteristic
203	C₂²×C₈	C₂×C₄×C₅₃	1	1	yes	characteristic
204	C₂²×C₈	C₂³×C₅₃	1	1	yes	characteristic
205	C₂×C₂²⋊C₄	C₈×C₅₃	1	1	yes	characteristic
206	C₂×C₂²⋊C₄	C₈×C₅₃	1	1	yes	characteristic
207	C₂×C₄²	C₂×C₄×C₅₃	1	1	yes	characteristic
208	C₂×C₄²	C₂×C₄×C₅₃	1	1	yes	characteristic
209	C₂×C₁₆	—	2	1	no	—
210	C₂×C₁₆	—	2	1	no	—
211	C₂×C₁₆	—	2	1	no	—
212	C₂×C₁₆	—	2	1	no	—
213	C₂×C₁₆	—	2	1	no	—
214	C₂×C₁₆	—	2	1	no	—
215	C₂×C₁₆	—	2	1	no	—
216	C₂×C₁₆	—	2	1	no	—
217	C₂×C₁₆	—	2	1	no	—
218	C₂×C₁₆	—	2	1	no	—
219	C₂×C₁₆	D₄×C₅₃	1	2	yes	—
220	C₂×C₁₆	D₄×C₅₃	1	2	yes	—
221	C₂²⋊C₈	—	2	2	no	—
222	C₄×C₈	—	2	1	no	—
223	C₄×C₈	—	2	1	no	—
224	C₄×C₈	—	2	1	no	—
225	C₄×C₈	—	2	1	no	—
226	#95	C₈×C₅₃	1	1	yes	characteristic
227	#95	C₈×C₅₃	1	1	yes	characteristic
228	C₂⁴	C₂×C₈×C₅₃	1	1	yes	characteristic
229	C₂²×C₄	—	2	1	no	—
230	C₂²×C₄	—	2	1	no	—
231	C₂²×C₄	—	2	1	no	—
232	C₂²×C₄	—	2	1	no	—
233	C₂²×C₄	—	2	1	no	—
234	C₂²×C₄	—	2	1	no	—
235	C₂²×C₄	C₂×C₈×C₅₃	1	1	yes	characteristic
236	C₂²×C₄	C₂×C₈×C₅₃	1	1	yes	characteristic
237	C₂²×C₄	C₂×C₈×C₅₃	1	1	yes	characteristic
238	C₂²×C₄	C₂×C₈×C₅₃	1	1	yes	characteristic
239	C₂²×C₄	C₂×C₈×C₅₃	1	1	yes	characteristic
240	C₂²×C₄	C₂²×C₄×C₅₃	1	1	yes	characteristic
241	C₂×C₈	—	2	1	no	—
242	C₂×C₈	—	2	1	no	—
243	C₂×C₈	—	2	1	no	—
244	C₂×C₈	—	2	1	no	—
245	C₂×C₈	—	2	1	no	—
246	C₂×C₈	—	2	1	no	—
247	C₂×C₈	—	2	1	no	—
248	C₂×C₈	—	2	1	no	—
249	C₂×C₈	—	4	1	no	—
250	C₂×C₈	—	4	1	no	—
251	C₂×C₈	—	4	1	no	—
252	C₂×C₈	—	4	1	no	—
253	C₂×C₈	M₁₆×C₅₃	1	2	yes	—
254	C₂×C₈	C₂×D₄×C₅₃	1	1	yes	characteristic
255	C₂×C₈	C₂×D₄×C₅₃	1	1	yes	characteristic
256	C₂×C₈	#41×C₅₃	1	1	yes	characteristic
257	C₂×C₈	#41×C₅₃	1	1	yes	characteristic
258	C₂×C₈	#41×C₅₃	1	1	yes	characteristic
259	C₂×C₈	#41×C₅₃	1	1	yes	characteristic
260	C₂²⋊C₄	—	2	2	no	—
261	C₂²⋊C₄	—	2	2	no	—
262	C₄²	—	2	1	no	—
263	C₄²	—	2	1	no	—
264	C₄²	—	2	1	no	—
265	C₄²	—	2	1	no	—
266	C₁₆	—	2	2	no	—
267	C₁₆	—	2	2	no	—
268	C₁₆	—	4	1	no	—
269	C₁₆	—	4	1	no	—
270	C₂³	—	2	1	no	—
271	C₂³	—	2	1	no	—
272	C₂³	—	2	1	no	—
273	C₂³	—	2	1	no	—
274	C₂³	—	2	1	no	—
275	C₂³	—	2	1	no	—
276	C₂³	M₃₂×C₅₃	1	2	yes	—
277	C₂³	C₂²×C₈×C₅₃	1	1	yes	characteristic
278	C₂×C₄	—	2	1	no	—
279	C₂×C₄	—	2	1	no	—
280	C₂×C₄	—	2	1	no	—
281	C₂×C₄	—	2	1	no	—
282	C₂×C₄	—	2	1	no	—
283	C₂×C₄	—	2	1	no	—
284	C₂×C₄	—	2	1	no	—
285	C₂×C₄	—	2	1	no	—
286	C₂×C₄	—	2	1	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	—	4	1	no	—
293	C₂×C₄	—	4	1	no	—
294	C₂×C₄	—	4	1	no	—
295	C₂×C₄	—	4	1	no	—
296	C₂×C₄	M₃₂×C₅₃	1	2	yes	—
297	C₂×C₄	#117×C₅₃	1	1	yes	characteristic
298	C₂×C₄	C₄×D₄×C₅₃	1	1	yes	characteristic
299	C₂×C₄	C₄×D₄×C₅₃	1	1	yes	characteristic
300	C₂×C₄	C₂×M₁₆×C₅₃	1	1	yes	characteristic
301	C₂×C₄	#131×C₅₃	1	1	yes	characteristic
302	C₂×C₄	#131×C₅₃	1	1	yes	characteristic
303	C₈	—	2	2	no	—
304	C₈	—	4	1	no	—
305	C₈	—	4	1	no	—
306	C₈	#121×C₅₃	1	1	yes	characteristic
307	C₈	#123×C₅₃	1	1	yes	characteristic
308	C₈	#124×C₅₃	1	1	yes	characteristic
309	C₈	#126×C₅₃	1	1	yes	characteristic
310	C₂²	—	2	2	no	—
311	C₂²	—	2	2	no	—
312	C₂²	—	2	2	no	—
313	C₂²	—	4	1	no	—
314	C₂²	—	4	1	no	—
315	C₂²	—	4	1	no	—
316	C₂²	—	4	1	no	—
317	C₂²	C₅₃×#431	1	1	yes	characteristic
318	C₂²	C₈×D₄×C₅₃	1	1	yes	characteristic
319	C₂²	C₈×D₄×C₅₃	1	1	yes	characteristic
320	C₂²	C₂×M₃₂×C₅₃	1	1	yes	characteristic
321	C₂²	C₂×M₃₂×C₅₃	1	1	yes	characteristic
322	C₂²	C₅₃×#504	1	1	yes	characteristic
323	C₂²	C₅₃×#504	1	1	yes	characteristic
324	C₄	—	2	2	no	—
325	C₄	—	4	1	no	—
326	C₄	—	4	1	no	—
327	C₄	—	4	1	no	—
328	C₄	—	4	1	no	—
329	C₄	C₅₃×#388	1	1	yes	characteristic
330	C₄	C₅₃×#433	1	1	yes	characteristic
331	C₄	C₅₃×#435	1	1	yes	characteristic
332	C₄	C₅₃×#436	1	1	yes	characteristic
333	C₂	—	4	2	no	—
334	C₂	C₅₃×#1923	1	1	yes	characteristic
335	C₂	C₅₃×#2163	1	1	yes	characteristic
336	C₂	C₅₃×#2164	1	1	yes	characteristic
337	C₂	C₅₃×#2168	1	1	yes	characteristic
338	C₂	C₅₃×#2168	1	1	yes	characteristic
339	C₂	C₅₃×#2168	1	1	yes	characteristic
340	C₂	C₅₃×#2169	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₈×C₅₃	C₂³
commutator	C₂³	C₂²×C₈×C₅₃
Frattini, cosocle	C₂²×C₈	C₂³×C₅₃
socle	C₂³×C₅₃	C₂²×C₈
prime radical	C₂⁴×C₅₃	C₂×C₈
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂²×C₈×C₅₃ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂×C₅₃×C₄⋊C₁₆	C₂	semidirect
4. C₂×C₅₃×#346	C₂	semidirect
6. C₅₃×#1374	C₂	semidirect
7. C₅₃×#1374	C₂	semidirect
139. #13121	C₅₃	direct (N × H)
163. C₂×C₄⋊C₁₆	C₂×C₅₃	semidirect
166. C₂×#346	C₂×C₅₃	semidirect
168. #1374	C₂×C₅₃	semidirect
169. #1374	C₂×C₅₃	semidirect
193. C₅₃	#13121	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×C₅₃×C₄⋊C₁₆	2
C₂×C₅₃×C₂²⋊C₁₆	2
C₂×C₅₃×#346	2
C₈×C₂²⋊C₄×C₅₃	2
C₅₃×#1374	2
#13121	53

Subgroups — C53×#13121 #27220633

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅₃×#13121 #27220633