Subgroups — C₅₃×#8208 #27215720

Order 13568.

538 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×#436	C₂	1	1	yes	maximal, characteristic
2	C₅₃×#1770	C₂	1	1	yes	maximal, characteristic, complemented
3	C₄×#102×C₅₃	C₂	1	1	yes	maximal, characteristic
4	C₂×C₅₃×D₄⋊C₈	C₂	1	1	yes	maximal, characteristic
5	C₂×C₅₃×#322	C₂	1	1	yes	maximal, characteristic, complemented
6	C₅₃×#1279	C₂	1	1	yes	maximal, characteristic, complemented
7	C₅₃×#1277	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂²×M₁₆×C₅₃	—	2	1	no	—
9	C₂×C₄×D₄×C₅₃	C₂²	1	1	yes	characteristic
10	C₅₃×#436	—	2	2	no	—
11	C₅₃×#436	C₄	1	4	yes	complemented
12	C₂×C₄⋊C₈×C₅₃	C₂²	1	1	yes	characteristic
13	C₂×C₄⋊C₈×C₅₃	C₂²	1	1	yes	characteristic
14	C₂×#102×C₅₃	C₄	1	2	yes	—
15	C₂×C₂²⋊C₈×C₅₃	—	2	1	no	—
16	C₂×#97×C₅₃	—	2	1	no	—
17	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
18	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
19	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
20	C₄×C₄⋊C₄×C₅₃	C₂²	1	1	yes	characteristic
21	C₅₃×#340	C₄	1	2	yes	—
22	C₅₃×#336	—	2	1	no	—
23	C₅₃×D₄⋊C₈	C₄	1	4	yes	complemented
24	C₅₃×#322	—	2	2	no	—
25	C₂²×D₄×C₅₃	C₂×C₄	1	1	yes	characteristic
26	C₂³×C₄×C₅₃	—	2	1	no	—
27	C₂×M₁₆×C₅₃	—	2	4	no	—
28	C₂×M₁₆×C₅₃	—	4	2	no	—
29	C₂×M₁₆×C₅₃	—	4	2	no	—
30	C₂²×C₈×C₅₃	—	2	1	no	—
31	C₂²×C₈×C₅₃	—	2	1	no	—
32	C₂²×C₈×C₅₃	—	2	1	no	—
33	C₂²×C₈×C₅₃	C₂×C₄	1	2	yes	—
34	C₂²×C₈×C₅₃	D₄	1	1	yes	characteristic
35	C₂²×C₈×C₅₃	D₄	1	1	yes	characteristic
36	C₄×D₄×C₅₃	—	2	1	no	—
37	C₄×D₄×C₅₃	—	2	1	no	—
38	C₄×D₄×C₅₃	C₂×C₄	1	2	yes	—
39	C₄×D₄×C₅₃	C₂×C₄	1	2	yes	—
40	C₂×C₄⋊C₄×C₅₃	C₂×C₄	1	2	yes	—
41	C₂×C₄⋊C₄×C₅₃	C₂×C₄	1	1	yes	characteristic
42	C₂×C₂²⋊C₄×C₅₃	—	2	1	no	—
43	C₂×C₄²×C₅₃	—	2	1	no	—
44	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
45	C₄⋊C₈×C₅₃	—	2	1	no	—
46	C₄⋊C₈×C₅₃	—	2	1	no	—
47	C₄⋊C₈×C₅₃	C₂×C₄	1	2	yes	—
48	C₄⋊C₈×C₅₃	C₂×C₄	1	2	yes	—
49	#102×C₅₃	—	2	4	no	—
50	C₂²⋊C₈×C₅₃	—	2	4	no	—
51	#97×C₅₃	—	4	2	no	—
52	C₄×C₈×C₅₃	—	2	1	no	—
53	C₄×C₈×C₅₃	—	2	1	no	—
54	C₄×C₈×C₅₃	—	2	1	no	—
55	C₄×C₈×C₅₃	C₂×C₄	1	2	yes	—
56	C₄×C₈×C₅₃	C₂×C₄	1	2	yes	—
57	C₄×C₈×C₅₃	D₄	1	2	yes	—
58	#95×C₅₃	—	2	1	no	—
59	C₂⁴×C₅₃	—	2	1	no	—
60	C₂×D₄×C₅₃	—	2	1	no	—
61	C₂×D₄×C₅₃	—	2	2	no	—
62	C₂×D₄×C₅₃	—	2	2	no	—
63	C₂×D₄×C₅₃	C₄²	1	2	yes	—
64	C₂²×C₄×C₅₃	—	2	1	no	—
65	C₂²×C₄×C₅₃	—	2	2	no	—
66	C₂²×C₄×C₅₃	—	2	2	no	—
67	C₂²×C₄×C₅₃	—	2	2	no	—
68	C₂²×C₄×C₅₃	—	4	1	no	—
69	C₂²×C₄×C₅₃	—	4	1	no	—
70	C₂²×C₄×C₅₃	—	4	1	no	—
71	C₂²×C₄×C₅₃	—	4	1	no	—
72	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
73	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
74	C₂²×C₄×C₅₃	#41	1	1	yes	characteristic
75	M₁₆×C₅₃	—	4	8	no	—
76	C₂×C₈×C₅₃	—	2	2	no	—
77	C₂×C₈×C₅₃	—	2	2	no	—
78	C₂×C₈×C₅₃	—	2	1	no	—
79	C₂×C₈×C₅₃	—	2	1	no	—
80	C₂×C₈×C₅₃	—	2	1	no	—
81	C₂×C₈×C₅₃	—	2	1	no	—
82	C₂×C₈×C₅₃	—	2	2	no	—
83	C₂×C₈×C₅₃	—	2	2	no	—
84	C₂×C₈×C₅₃	—	2	2	no	—
85	C₂×C₈×C₅₃	—	4	2	no	—
86	C₂×C₈×C₅₃	—	4	1	no	—
87	C₂×C₈×C₅₃	—	4	1	no	—
88	C₂×C₈×C₅₃	—	4	1	no	—
89	C₂×C₈×C₅₃	—	4	1	no	—
90	C₂×C₈×C₅₃	C₂²⋊C₄	1	2	yes	—
91	C₂×C₈×C₅₃	C₂²⋊C₄	1	2	yes	—
92	C₂×C₈×C₅₃	M₁₆	1	4	yes	complemented
93	C₄⋊C₄×C₅₃	—	2	1	no	—
94	C₄⋊C₄×C₅₃	—	2	2	no	—
95	C₄⋊C₄×C₅₃	—	2	2	no	—
96	C₄⋊C₄×C₅₃	C₄²	1	2	yes	—
97	C₂²⋊C₄×C₅₃	—	2	2	no	—
98	C₂²⋊C₄×C₅₃	—	2	2	no	—
99	C₄²×C₅₃	—	4	1	no	—
100	C₄²×C₅₃	—	4	1	no	—
101	C₄²×C₅₃	C₂²×C₄	1	1	yes	characteristic
102	C₄²×C₅₃	C₂²×C₄	1	1	yes	characteristic
103	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
104	C₄²×C₅₃	#41	1	1	yes	characteristic
105	C₂³×C₅₃	—	2	2	no	—
106	C₂³×C₅₃	—	4	2	no	—
107	C₂³×C₅₃	—	4	1	no	—
108	C₂³×C₅₃	—	4	1	no	—
109	C₂³×C₅₃	—	4	2	no	—
110	C₂³×C₅₃	C₄×D₄	1	1	yes	characteristic
111	D₄×C₅₃	—	2	2	no	—
112	D₄×C₅₃	—	2	2	no	—
113	D₄×C₅₃	—	4	2	no	—
114	C₂×C₄×C₅₃	—	2	2	no	—
115	C₂×C₄×C₅₃	—	2	2	no	—
116	C₂×C₄×C₅₃	—	4	1	no	—
117	C₂×C₄×C₅₃	—	4	1	no	—
118	C₂×C₄×C₅₃	—	4	2	no	—
119	C₂×C₄×C₅₃	—	4	2	no	—
120	C₂×C₄×C₅₃	—	4	2	no	—
121	C₂×C₄×C₅₃	—	4	2	no	—
122	C₂×C₄×C₅₃	—	4	2	no	—
123	C₂×C₄×C₅₃	—	4	2	no	—
124	C₂×C₄×C₅₃	—	4	2	no	—
125	C₂×C₄×C₅₃	C₂×C₄²	1	1	yes	characteristic
126	C₂×C₄×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
127	C₂×C₄×C₅₃	#117	1	1	yes	characteristic
128	C₂×C₄×C₅₃	C₄×D₄	1	1	yes	characteristic
129	C₂×C₄×C₅₃	C₄×D₄	1	1	yes	characteristic
130	C₂×C₄×C₅₃	C₄×D₄	1	1	yes	characteristic
131	C₂×C₄×C₅₃	C₂×M₁₆	1	2	yes	—
132	C₂×C₄×C₅₃	#131	1	2	yes	—
133	C₂×C₄×C₅₃	AGL(1,ℤ/8)	1	1	yes	characteristic
134	C₂×C₄×C₅₃	AGL(1,ℤ/8)	1	1	yes	characteristic
135	C₂×C₄×C₅₃	AGL(1,ℤ/8)	1	1	yes	characteristic
136	C₂×C₄×C₅₃	#137	1	1	yes	characteristic
137	C₈×C₅₃	—	2	2	no	—
138	C₈×C₅₃	—	2	2	no	—
139	C₈×C₅₃	—	2	4	no	—
140	C₈×C₅₃	—	4	2	no	—
141	C₈×C₅₃	—	4	2	no	—
142	C₈×C₅₃	—	4	2	no	—
143	#8208	C₅₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
144	C₂²×C₅₃	—	4	2	no	—
145	C₂²×C₅₃	—	4	2	no	—
146	C₂²×C₅₃	—	4	2	no	—
147	C₂²×C₅₃	—	8	2	no	—
148	C₂²×C₅₃	—	8	2	no	—
149	C₂²×C₅₃	C₄×C₂²⋊C₄	1	1	yes	characteristic
150	C₂²×C₅₃	#436	1	2	yes	—
151	C₂²×C₅₃	#440	1	1	yes	characteristic
152	C₂²×C₅₃	#442	1	1	yes	characteristic
153	C₂²×C₅₃	#444	1	2	yes	—
154	C₄×C₅₃	—	2	1	no	—
155	C₄×C₅₃	—	2	1	no	—
156	C₄×C₅₃	—	4	2	no	—
157	C₄×C₅₃	—	8	2	no	—
158	C₄×C₅₃	C₄×M₁₆	1	1	yes	characteristic
159	C₄×C₅₃	#405	1	1	yes	characteristic
160	C₄×C₅₃	#417	1	1	yes	characteristic
161	C₄×C₅₃	#418	1	1	yes	characteristic
162	C₄×C₅₃	#435	1	2	yes	—
163	C₄×C₅₃	#440	1	1	yes	characteristic
164	C₄×C₅₃	#442	1	1	yes	characteristic
165	C₂×#436	C₂×C₅₃	1	1	yes	characteristic
166	#1770	C₂×C₅₃	1	1	yes	characteristic, complemented
167	C₄×#102	C₂×C₅₃	1	1	yes	characteristic
168	C₂×D₄⋊C₈	C₂×C₅₃	1	1	yes	characteristic
169	C₂×#322	C₂×C₅₃	1	1	yes	characteristic, complemented
170	#1279	C₂×C₅₃	1	1	yes	characteristic, complemented
171	#1277	C₂×C₅₃	1	1	yes	characteristic, complemented
172	C₂×C₅₃	—	8	2	no	—
173	C₂×C₅₃	#1592	1	2	yes	—
174	C₂×C₅₃	#1594	1	2	yes	—
175	C₂×C₅₃	#1753	1	1	yes	characteristic
176	C₂×C₅₃	#1762	1	1	yes	characteristic
177	C₂×C₅₃	#1765	1	1	yes	characteristic
178	C₂²×M₁₆	—	2	1	no	—
179	C₂×C₄×D₄	C₂²×C₅₃	1	1	yes	characteristic
180	#436	—	2	2	no	—
181	#436	C₄×C₅₃	1	4	yes	complemented
182	C₂×C₄⋊C₈	C₂²×C₅₃	1	1	yes	characteristic
183	C₂×C₄⋊C₈	C₂²×C₅₃	1	1	yes	characteristic
184	C₂×#102	C₄×C₅₃	1	2	yes	—
185	C₂×C₂²⋊C₈	—	2	1	no	—
186	C₂×#97	—	2	1	no	—
187	C₂×C₄×C₈	C₂²×C₅₃	1	1	yes	characteristic
188	C₂×C₄×C₈	C₂²×C₅₃	1	1	yes	characteristic
189	C₂×C₄×C₈	C₂²×C₅₃	1	1	yes	characteristic
190	C₄×C₄⋊C₄	C₂²×C₅₃	1	1	yes	characteristic
191	#340	C₄×C₅₃	1	2	yes	—
192	#336	—	2	1	no	—
193	D₄⋊C₈	C₄×C₅₃	1	4	yes	complemented
194	#322	—	2	2	no	—
195	C₅₃	#8208	1	1	yes	characteristic, direct product factor, Sylow
196	C₂²×D₄	C₂×C₄×C₅₃	1	1	yes	characteristic
197	C₂³×C₄	—	2	1	no	—
198	C₂×M₁₆	—	2	4	no	—
199	C₂×M₁₆	—	4	2	no	—
200	C₂×M₁₆	—	4	2	no	—
201	C₂²×C₈	—	2	1	no	—
202	C₂²×C₈	—	2	1	no	—
203	C₂²×C₈	—	2	1	no	—
204	C₂²×C₈	C₂×C₄×C₅₃	1	2	yes	—
205	C₂²×C₈	D₄×C₅₃	1	1	yes	characteristic
206	C₂²×C₈	D₄×C₅₃	1	1	yes	characteristic
207	C₄×D₄	—	2	1	no	—
208	C₄×D₄	—	2	1	no	—
209	C₄×D₄	C₂×C₄×C₅₃	1	2	yes	—
210	C₄×D₄	C₂×C₄×C₅₃	1	2	yes	—
211	C₂×C₄⋊C₄	C₂×C₄×C₅₃	1	2	yes	—
212	C₂×C₄⋊C₄	C₂×C₄×C₅₃	1	1	yes	characteristic
213	C₂×C₂²⋊C₄	—	2	1	no	—
214	C₂×C₄²	—	2	1	no	—
215	C₂×C₄²	C₂³×C₅₃	1	1	yes	characteristic
216	C₄⋊C₈	—	2	1	no	—
217	C₄⋊C₈	—	2	1	no	—
218	C₄⋊C₈	C₂×C₄×C₅₃	1	2	yes	—
219	C₄⋊C₈	C₂×C₄×C₅₃	1	2	yes	—
220	#102	—	2	4	no	—
221	C₂²⋊C₈	—	2	4	no	—
222	#97	—	4	2	no	—
223	C₄×C₈	—	2	1	no	—
224	C₄×C₈	—	2	1	no	—
225	C₄×C₈	—	2	1	no	—
226	C₄×C₈	C₂×C₄×C₅₃	1	2	yes	—
227	C₄×C₈	C₂×C₄×C₅₃	1	2	yes	—
228	C₄×C₈	D₄×C₅₃	1	2	yes	—
229	#95	—	2	1	no	—
230	C₂⁴	—	2	1	no	—
231	C₂×D₄	—	2	1	no	—
232	C₂×D₄	—	2	2	no	—
233	C₂×D₄	—	2	2	no	—
234	C₂×D₄	C₄²×C₅₃	1	2	yes	—
235	C₂²×C₄	—	2	1	no	—
236	C₂²×C₄	—	2	2	no	—
237	C₂²×C₄	—	2	2	no	—
238	C₂²×C₄	—	2	2	no	—
239	C₂²×C₄	—	4	1	no	—
240	C₂²×C₄	—	4	1	no	—
241	C₂²×C₄	—	4	1	no	—
242	C₂²×C₄	—	4	1	no	—
243	C₂²×C₄	C₂²×C₄×C₅₃	1	1	yes	characteristic
244	C₂²×C₄	C₂×D₄×C₅₃	1	1	yes	characteristic
245	C₂²×C₄	#41×C₅₃	1	1	yes	characteristic
246	M₁₆	—	4	8	no	—
247	C₂×C₈	—	2	2	no	—
248	C₂×C₈	—	2	2	no	—
249	C₂×C₈	—	2	1	no	—
250	C₂×C₈	—	2	1	no	—
251	C₂×C₈	—	2	1	no	—
252	C₂×C₈	—	2	1	no	—
253	C₂×C₈	—	2	2	no	—
254	C₂×C₈	—	2	2	no	—
255	C₂×C₈	—	2	2	no	—
256	C₂×C₈	—	4	2	no	—
257	C₂×C₈	—	4	1	no	—
258	C₂×C₈	—	4	1	no	—
259	C₂×C₈	—	4	1	no	—
260	C₂×C₈	—	4	1	no	—
261	C₂×C₈	C₂²⋊C₄×C₅₃	1	2	yes	—
262	C₂×C₈	C₂²⋊C₄×C₅₃	1	2	yes	—
263	C₂×C₈	M₁₆×C₅₃	1	4	yes	complemented
264	C₄⋊C₄	—	2	1	no	—
265	C₄⋊C₄	—	2	2	no	—
266	C₄⋊C₄	—	2	2	no	—
267	C₄⋊C₄	C₄²×C₅₃	1	2	yes	—
268	C₂²⋊C₄	—	2	2	no	—
269	C₂²⋊C₄	—	2	2	no	—
270	C₄²	—	4	1	no	—
271	C₄²	—	4	1	no	—
272	C₄²	C₂²×C₄×C₅₃	1	1	yes	characteristic
273	C₄²	C₂²×C₄×C₅₃	1	1	yes	characteristic
274	C₄²	C₂×D₄×C₅₃	1	1	yes	characteristic
275	C₄²	#41×C₅₃	1	1	yes	characteristic
276	C₂³	—	2	2	no	—
277	C₂³	—	4	2	no	—
278	C₂³	—	4	1	no	—
279	C₂³	—	4	1	no	—
280	C₂³	—	4	2	no	—
281	C₂³	C₄×D₄×C₅₃	1	1	yes	characteristic
282	D₄	—	2	2	no	—
283	D₄	—	2	2	no	—
284	D₄	—	4	2	no	—
285	C₂×C₄	—	2	2	no	—
286	C₂×C₄	—	2	2	no	—
287	C₂×C₄	—	4	1	no	—
288	C₂×C₄	—	4	1	no	—
289	C₂×C₄	—	4	2	no	—
290	C₂×C₄	—	4	2	no	—
291	C₂×C₄	—	4	2	no	—
292	C₂×C₄	—	4	2	no	—
293	C₂×C₄	—	4	2	no	—
294	C₂×C₄	—	4	2	no	—
295	C₂×C₄	—	4	2	no	—
296	C₂×C₄	C₂×C₄²×C₅₃	1	1	yes	characteristic
297	C₂×C₄	C₂×C₂²⋊C₄×C₅₃	1	1	yes	characteristic
298	C₂×C₄	#117×C₅₃	1	1	yes	characteristic
299	C₂×C₄	C₄×D₄×C₅₃	1	1	yes	characteristic
300	C₂×C₄	C₄×D₄×C₅₃	1	1	yes	characteristic
301	C₂×C₄	C₄×D₄×C₅₃	1	1	yes	characteristic
302	C₂×C₄	C₂×M₁₆×C₅₃	1	2	yes	—
303	C₂×C₄	#131×C₅₃	1	2	yes	—
304	C₂×C₄	AGL(1,ℤ/8)×C₅₃	1	1	yes	characteristic
305	C₂×C₄	AGL(1,ℤ/8)×C₅₃	1	1	yes	characteristic
306	C₂×C₄	AGL(1,ℤ/8)×C₅₃	1	1	yes	characteristic
307	C₂×C₄	#137×C₅₃	1	1	yes	characteristic
308	C₈	—	2	2	no	—
309	C₈	—	2	2	no	—
310	C₈	—	2	4	no	—
311	C₈	—	4	2	no	—
312	C₈	—	4	2	no	—
313	C₈	—	4	2	no	—
314	C₂²	—	4	2	no	—
315	C₂²	—	4	2	no	—
316	C₂²	—	4	2	no	—
317	C₂²	—	8	2	no	—
318	C₂²	—	8	2	no	—
319	C₂²	C₄×C₂²⋊C₄×C₅₃	1	1	yes	characteristic
320	C₂²	C₅₃×#436	1	2	yes	—
321	C₂²	C₅₃×#440	1	1	yes	characteristic
322	C₂²	C₅₃×#442	1	1	yes	characteristic
323	C₂²	C₅₃×#444	1	2	yes	—
324	C₄	—	2	1	no	—
325	C₄	—	2	1	no	—
326	C₄	—	4	2	no	—
327	C₄	—	8	2	no	—
328	C₄	C₄×M₁₆×C₅₃	1	1	yes	characteristic
329	C₄	C₅₃×#405	1	1	yes	characteristic
330	C₄	C₅₃×#417	1	1	yes	characteristic
331	C₄	C₅₃×#418	1	1	yes	characteristic
332	C₄	C₅₃×#435	1	2	yes	—
333	C₄	C₅₃×#440	1	1	yes	characteristic
334	C₄	C₅₃×#442	1	1	yes	characteristic
335	C₂	—	8	2	no	—
336	C₂	C₅₃×#1592	1	2	yes	—
337	C₂	C₅₃×#1594	1	2	yes	—
338	C₂	C₅₃×#1753	1	1	yes	characteristic
339	C₂	C₅₃×#1762	1	1	yes	characteristic
340	C₂	C₅₃×#1765	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄×C₅₃	C₂×D₄
commutator	C₂×C₄	C₂×C₄²×C₅₃
Frattini, cosocle	C₂×C₄²	C₂³×C₅₃
socle	C₂³×C₅₃	C₄×D₄
prime radical	C₂²×D₄×C₅₃	C₂×C₄
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄×C₅₃ ◁ C₂×C₄×C₈×C₅₃ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₅₃×#1770	C₂	semidirect
5. C₂×C₅₃×#322	C₂	semidirect
6. C₅₃×#1279	C₂	semidirect
7. C₅₃×#1277	C₂	semidirect
11. C₅₃×#436	C₄	semidirect
23. C₅₃×D₄⋊C₈	C₄	semidirect
92. C₂×C₈×C₅₃	M₁₆	semidirect
143. #8208	C₅₃	direct (N × H)
166. #1770	C₂×C₅₃	semidirect
169. C₂×#322	C₂×C₅₃	semidirect
170. #1279	C₂×C₅₃	semidirect
171. #1277	C₂×C₅₃	semidirect
181. #436	C₄×C₅₃	semidirect
193. D₄⋊C₈	C₄×C₅₃	semidirect
195. C₅₃	#8208	direct (N × H)
263. C₂×C₈	M₁₆×C₅₃	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₅₃×#436	2
C₅₃×#1770	2
C₄×#102×C₅₃	2
C₂×C₅₃×D₄⋊C₈	2
C₂×C₅₃×#322	2
C₅₃×#1279	2
C₅₃×#1277	2
#8208	53

Subgroups — C53×#8208 #27215720

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅₃×#8208 #27215720