Subgroups — C₅₃×#22245 #27229757

Order 13568.

686 conjugacy classes of subgroups.

The subgroup lattice has 324 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₅₃×#446	C₂	1	2	yes	maximal
2	C₅₃×#2399	C₂	1	1	yes	maximal, characteristic
3	C₅₃×#1984	C₂	1	2	yes	maximal
4	C₅₃×#1917	C₂	1	2	yes	maximal
5	C₅₃×#1917	C₂	1	2	yes	maximal
6	C₅₃×#1916	C₂	1	2	yes	maximal
7	C₅₃×#1916	C₂	1	2	yes	maximal
8	C₅₃×#1832	C₂	1	1	yes	maximal, characteristic
9	C₅₃×#1746	C₂	1	1	yes	maximal, characteristic
10	C₂×C₄×Q₈×C₅₃	C₂²	1	2	yes	—
11	C₅₃×#446	—	2	8	no	—
12	C₂×C₄⋊C₈×C₅₃	C₂²	1	2	yes	—
13	C₂×C₄⋊C₈×C₅₃	C₂²	1	2	yes	—
14	C₂×C₄⋊C₈×C₅₃	C₂²	1	2	yes	—
15	C₂×#97×C₅₃	C₂²	1	2	yes	—
16	C₂×#97×C₅₃	C₂²	1	1	yes	characteristic
17	C₂×C₄×C₈×C₅₃	C₂²	1	2	yes	—
18	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
19	C₅₃×#398	C₄	1	2	yes	—
20	C₅₃×#398	C₄	1	2	yes	—
21	C₅₃×#395	C₄	1	2	yes	—
22	C₅₃×#395	C₄	1	2	yes	—
23	C₄×C₄⋊C₄×C₅₃	C₂²	1	1	yes	characteristic
24	C₄×C₄⋊C₄×C₅₃	C₂²	1	2	yes	—
25	C₄×C₄⋊C₄×C₅₃	C₂²	1	1	yes	characteristic
26	C₅₃×#376	C₂²	1	1	yes	characteristic
27	C₅₃×#336	C₂²	1	2	yes	—
28	C₅₃×#336	C₂²	1	1	yes	characteristic
29	C₅₃×#336	C₂²	1	2	yes	—
30	C₅₃×#336	C₂²	1	2	yes	—
31	C₅₃×#336	C₂²	1	2	yes	—
32	C₅₃×#336	C₂²	1	2	yes	—
33	C₅₃×#336	C₂²	1	2	yes	—
34	C₅₃×#336	C₂²	1	2	yes	—
35	C₅₃×#336	C₂²	1	1	yes	characteristic
36	C₂²×Q₈×C₅₃	C₂×C₄	1	2	yes	—
37	C₂²×C₈×C₅₃	C₂³	1	2	yes	—
38	C₂²×C₈×C₅₃	C₂³	1	2	yes	—
39	C₂²×C₈×C₅₃	C₂³	1	2	yes	—
40	C₂²×C₈×C₅₃	C₂³	1	2	yes	—
41	C₄×Q₈×C₅₃	—	2	4	no	—
42	C₄×Q₈×C₅₃	—	2	4	no	—
43	C₂×C₄⋊C₄×C₅₃	C₂×C₄	1	2	yes	—
44	C₂×C₄⋊C₄×C₅₃	C₂×C₄	1	2	yes	—
45	C₂×C₄⋊C₄×C₅₃	C₂×C₄	1	2	yes	—
46	C₂×C₄⋊C₄×C₅₃	C₂×C₄	1	2	yes	—
47	C₂×C₄⋊C₄×C₅₃	C₂×C₄	1	2	yes	—
48	C₂×C₄⋊C₄×C₅₃	C₂×C₄	1	2	yes	—
49	C₂×C₄⋊C₄×C₅₃	C₂×C₄	1	2	yes	—
50	C₂×C₄²×C₅₃	C₂³	1	2	yes	—
51	C₂×C₄²×C₅₃	C₂³	1	2	yes	—
52	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
53	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
54	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
55	C₄⋊C₈×C₅₃	—	2	4	no	—
56	C₄⋊C₈×C₅₃	—	2	4	no	—
57	C₄⋊C₈×C₅₃	—	2	4	no	—
58	#97×C₅₃	—	2	4	no	—
59	#97×C₅₃	D₄	1	4	yes	—
60	C₄×C₈×C₅₃	—	2	4	no	—
61	C₄×C₈×C₅₃	—	2	2	no	—
62	#95×C₅₃	C₂×C₄	1	2	yes	—
63	#95×C₅₃	C₂×C₄	1	2	yes	—
64	#95×C₅₃	C₂×C₄	1	1	yes	characteristic
65	#95×C₅₃	C₂×C₄	1	2	yes	—
66	#95×C₅₃	C₂×C₄	1	1	yes	characteristic
67	#95×C₅₃	C₂×C₄	1	1	yes	characteristic
68	#95×C₅₃	C₂×C₄	1	2	yes	—
69	#95×C₅₃	C₂×C₄	1	1	yes	characteristic
70	C₂×Q₈×C₅₃	—	2	4	no	—
71	C₂×Q₈×C₅₃	—	2	4	no	—
72	C₂×Q₈×C₅₃	—	4	2	no	—
73	C₂²×C₄×C₅₃	C₂²×C₄	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×C₄×C₅₃	C₂²×C₄	1	2	yes	—
77	C₂²×C₄×C₅₃	C₂²×C₄	1	2	yes	—
78	C₂²×C₄×C₅₃	C₂²×C₄	1	2	yes	—
79	C₂²×C₄×C₅₃	C₂²×C₄	1	2	yes	—
80	C₂²×C₄×C₅₃	C₂²×C₄	1	2	yes	—
81	C₂²×C₄×C₅₃	C₂⁴	1	1	yes	characteristic
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	—	2	2	no	—
86	C₂×C₈×C₅₃	—	2	2	no	—
87	C₂×C₈×C₅₃	—	2	2	no	—
88	C₂×C₈×C₅₃	—	2	2	no	—
89	C₂×C₈×C₅₃	—	2	2	no	—
90	C₂×C₈×C₅₃	—	2	2	no	—
91	C₂×C₈×C₅₃	—	2	2	no	—
92	C₂×C₈×C₅₃	—	2	2	no	—
93	C₂×C₈×C₅₃	C₂×D₄	1	4	yes	—
94	C₄⋊C₄×C₅₃	—	2	4	no	—
95	C₄⋊C₄×C₅₃	—	2	4	no	—
96	C₄⋊C₄×C₅₃	—	2	4	no	—
97	C₄⋊C₄×C₅₃	—	2	4	no	—
98	C₄⋊C₄×C₅₃	—	2	4	no	—
99	C₄⋊C₄×C₅₃	—	4	2	no	—
100	C₄⋊C₄×C₅₃	—	4	2	no	—
101	C₄²×C₅₃	—	2	2	no	—
102	C₄²×C₅₃	—	2	2	no	—
103	C₄²×C₅₃	—	2	2	no	—
104	C₄²×C₅₃	—	2	2	no	—
105	C₄²×C₅₃	—	2	2	no	—
106	C₄²×C₅₃	—	2	2	no	—
107	C₄²×C₅₃	C₂×D₄	1	2	yes	—
108	C₄²×C₅₃	#41	1	2	yes	—
109	C₂³×C₅₃	C₂³×C₄	1	1	yes	characteristic
110	Q₈×C₅₃	—	4	8	no	—
111	C₂×C₄×C₅₃	—	2	2	no	—
112	C₂×C₄×C₅₃	—	2	2	no	—
113	C₂×C₄×C₅₃	—	2	2	no	—
114	C₂×C₄×C₅₃	—	2	2	no	—
115	C₂×C₄×C₅₃	—	2	2	no	—
116	C₂×C₄×C₅₃	—	2	2	no	—
117	C₂×C₄×C₅₃	—	2	2	no	—
118	C₂×C₄×C₅₃	—	2	2	no	—
119	C₂×C₄×C₅₃	—	2	1	no	—
120	C₂×C₄×C₅₃	—	2	1	no	—
121	C₂×C₄×C₅₃	—	2	2	no	—
122	C₂×C₄×C₅₃	—	2	2	no	—
123	C₂×C₄×C₅₃	—	2	2	no	—
124	C₂×C₄×C₅₃	—	2	2	no	—
125	C₂×C₄×C₅₃	—	2	2	no	—
126	C₂×C₄×C₅₃	—	2	2	no	—
127	C₂×C₄×C₅₃	—	2	2	no	—
128	C₂×C₄×C₅₃	—	2	2	no	—
129	C₂×C₄×C₅₃	—	2	2	no	—
130	C₂×C₄×C₅₃	—	2	2	no	—
131	C₂×C₄×C₅₃	C₄×D₄	1	4	yes	—
132	C₂×C₄×C₅₃	#131	1	4	yes	—
133	C₂×C₄×C₅₃	C₂²×D₄	1	1	yes	characteristic
134	C₂×C₄×C₅₃	C₂×#41	1	1	yes	characteristic
135	C₂×C₄×C₅₃	2₋¹⁺⁴	1	2	yes	—
136	C₂×C₄×C₅₃	2₋¹⁺⁴	1	2	yes	—
137	C₈×C₅₃	—	2	4	no	—
138	C₈×C₅₃	—	4	2	no	—
139	C₈×C₅₃	—	4	2	no	—
140	C₈×C₅₃	—	4	2	no	—
141	#22245	C₅₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
142	C₂²×C₅₃	C₂×C₄×D₄	1	1	yes	characteristic
143	C₂²×C₅₃	#519	1	1	yes	characteristic
144	C₂²×C₅₃	#519	1	1	yes	characteristic
145	C₂²×C₅₃	C₂×#131	1	2	yes	—
146	C₂²×C₅₃	#568	1	2	yes	—
147	C₄×C₅₃	—	2	4	no	—
148	C₄×C₅₃	—	2	4	no	—
149	C₄×C₅₃	—	4	2	no	—
150	C₄×C₅₃	—	4	2	no	—
151	C₄×C₅₃	—	4	2	no	—
152	C₄×C₅₃	—	4	2	no	—
153	C₄×C₅₃	—	4	1	no	—
154	C₄×C₅₃	—	4	1	no	—
155	C₄×C₅₃	#536	1	2	yes	—
156	C₄×C₅₃	#541	1	2	yes	—
157	C₂×#446	C₂×C₅₃	1	2	yes	—
158	#2399	C₂×C₅₃	1	1	yes	characteristic
159	#1984	C₂×C₅₃	1	2	yes	—
160	#1917	C₂×C₅₃	1	2	yes	—
161	#1917	C₂×C₅₃	1	2	yes	—
162	#1916	C₂×C₅₃	1	2	yes	—
163	#1916	C₂×C₅₃	1	2	yes	—
164	#1832	C₂×C₅₃	1	1	yes	characteristic
165	#1746	C₂×C₅₃	1	1	yes	characteristic
166	C₂×C₅₃	#2327	1	1	yes	characteristic
167	C₂×C₅₃	#2933	1	2	yes	—
168	C₂×C₅₃	#2985	1	2	yes	—
169	C₂×C₅₃	#2986	1	1	yes	characteristic
170	C₂×C₅₃	#2987	1	1	yes	characteristic
171	C₂×C₄×Q₈	C₂²×C₅₃	1	2	yes	—
172	#446	—	2	8	no	—
173	C₂×C₄⋊C₈	C₂²×C₅₃	1	2	yes	—
174	C₂×C₄⋊C₈	C₂²×C₅₃	1	2	yes	—
175	C₂×C₄⋊C₈	C₂²×C₅₃	1	2	yes	—
176	C₂×#97	C₂²×C₅₃	1	2	yes	—
177	C₂×#97	C₂²×C₅₃	1	1	yes	characteristic
178	C₂×C₄×C₈	C₂²×C₅₃	1	2	yes	—
179	C₂×C₄×C₈	C₂²×C₅₃	1	1	yes	characteristic
180	#398	C₄×C₅₃	1	2	yes	—
181	#398	C₄×C₅₃	1	2	yes	—
182	#395	C₄×C₅₃	1	2	yes	—
183	#395	C₄×C₅₃	1	2	yes	—
184	C₄×C₄⋊C₄	C₂²×C₅₃	1	1	yes	characteristic
185	C₄×C₄⋊C₄	C₂²×C₅₃	1	2	yes	—
186	C₄×C₄⋊C₄	C₂²×C₅₃	1	1	yes	characteristic
187	#376	C₂²×C₅₃	1	1	yes	characteristic
188	#336	C₂²×C₅₃	1	1	yes	characteristic
189	#336	C₂²×C₅₃	1	2	yes	—
190	#336	C₂²×C₅₃	1	1	yes	characteristic
191	#336	C₂²×C₅₃	1	2	yes	—
192	#336	C₂²×C₅₃	1	2	yes	—
193	#336	C₂²×C₅₃	1	2	yes	—
194	#336	C₂²×C₅₃	1	2	yes	—
195	#336	C₂²×C₅₃	1	2	yes	—
196	#336	C₂²×C₅₃	1	2	yes	—
197	C₅₃	#22245	1	1	yes	characteristic, direct product factor, Sylow
198	C₂²×Q₈	C₂×C₄×C₅₃	1	2	yes	—
199	C₂²×C₈	C₂³×C₅₃	1	2	yes	—
200	C₂²×C₈	C₂³×C₅₃	1	2	yes	—
201	C₂²×C₈	C₂³×C₅₃	1	2	yes	—
202	C₂²×C₈	C₂³×C₅₃	1	2	yes	—
203	C₄×Q₈	—	2	4	no	—
204	C₄×Q₈	—	2	4	no	—
205	C₂×C₄⋊C₄	C₂×C₄×C₅₃	1	2	yes	—
206	C₂×C₄⋊C₄	C₂×C₄×C₅₃	1	2	yes	—
207	C₂×C₄⋊C₄	C₂×C₄×C₅₃	1	2	yes	—
208	C₂×C₄⋊C₄	C₂×C₄×C₅₃	1	2	yes	—
209	C₂×C₄⋊C₄	C₂×C₄×C₅₃	1	2	yes	—
210	C₂×C₄⋊C₄	C₂×C₄×C₅₃	1	2	yes	—
211	C₂×C₄⋊C₄	C₂×C₄×C₅₃	1	2	yes	—
212	C₂×C₄²	C₂³×C₅₃	1	2	yes	—
213	C₂×C₄²	C₂³×C₅₃	1	2	yes	—
214	C₂×C₄²	C₂³×C₅₃	1	1	yes	characteristic
215	C₂×C₄²	C₂³×C₅₃	1	1	yes	characteristic
216	C₂×C₄²	C₂³×C₅₃	1	1	yes	characteristic
217	C₄⋊C₈	—	2	4	no	—
218	C₄⋊C₈	—	2	4	no	—
219	C₄⋊C₈	—	2	4	no	—
220	#97	—	2	4	no	—
221	#97	D₄×C₅₃	1	4	yes	—
222	C₄×C₈	—	2	4	no	—
223	C₄×C₈	—	2	2	no	—
224	#95	C₂×C₄×C₅₃	1	2	yes	—
225	#95	C₂×C₄×C₅₃	1	2	yes	—
226	#95	C₂×C₄×C₅₃	1	1	yes	characteristic
227	#95	C₂×C₄×C₅₃	1	2	yes	—
228	#95	C₂×C₄×C₅₃	1	1	yes	characteristic
229	#95	C₂×C₄×C₅₃	1	1	yes	characteristic
230	#95	C₂×C₄×C₅₃	1	2	yes	—
231	#95	C₂×C₄×C₅₃	1	1	yes	characteristic
232	C₂×Q₈	—	2	4	no	—
233	C₂×Q₈	—	2	4	no	—
234	C₂×Q₈	—	4	2	no	—
235	C₂²×C₄	C₂²×C₄×C₅₃	1	1	yes	characteristic
236	C₂²×C₄	C₂²×C₄×C₅₃	1	2	yes	—
237	C₂²×C₄	C₂²×C₄×C₅₃	1	1	yes	characteristic
238	C₂²×C₄	C₂²×C₄×C₅₃	1	2	yes	—
239	C₂²×C₄	C₂²×C₄×C₅₃	1	2	yes	—
240	C₂²×C₄	C₂²×C₄×C₅₃	1	2	yes	—
241	C₂²×C₄	C₂²×C₄×C₅₃	1	2	yes	—
242	C₂²×C₄	C₂²×C₄×C₅₃	1	2	yes	—
243	C₂²×C₄	C₂⁴×C₅₃	1	1	yes	characteristic
244	C₂×C₈	—	2	2	no	—
245	C₂×C₈	—	2	2	no	—
246	C₂×C₈	—	2	2	no	—
247	C₂×C₈	—	2	2	no	—
248	C₂×C₈	—	2	2	no	—
249	C₂×C₈	—	2	2	no	—
250	C₂×C₈	—	2	2	no	—
251	C₂×C₈	—	2	2	no	—
252	C₂×C₈	—	2	2	no	—
253	C₂×C₈	—	2	2	no	—
254	C₂×C₈	—	2	2	no	—
255	C₂×C₈	C₂×D₄×C₅₃	1	4	yes	—
256	C₄⋊C₄	—	2	4	no	—
257	C₄⋊C₄	—	2	4	no	—
258	C₄⋊C₄	—	2	4	no	—
259	C₄⋊C₄	—	2	4	no	—
260	C₄⋊C₄	—	2	4	no	—
261	C₄⋊C₄	—	4	2	no	—
262	C₄⋊C₄	—	4	2	no	—
263	C₄²	—	2	2	no	—
264	C₄²	—	2	2	no	—
265	C₄²	—	2	2	no	—
266	C₄²	—	2	2	no	—
267	C₄²	—	2	2	no	—
268	C₄²	—	2	2	no	—
269	C₄²	C₂×D₄×C₅₃	1	2	yes	—
270	C₄²	#41×C₅₃	1	2	yes	—
271	C₂³	C₂³×C₄×C₅₃	1	1	yes	characteristic
272	Q₈	—	4	8	no	—
273	C₂×C₄	—	2	2	no	—
274	C₂×C₄	—	2	2	no	—
275	C₂×C₄	—	2	2	no	—
276	C₂×C₄	—	2	2	no	—
277	C₂×C₄	—	2	2	no	—
278	C₂×C₄	—	2	2	no	—
279	C₂×C₄	—	2	2	no	—
280	C₂×C₄	—	2	2	no	—
281	C₂×C₄	—	2	1	no	—
282	C₂×C₄	—	2	1	no	—
283	C₂×C₄	—	2	2	no	—
284	C₂×C₄	—	2	2	no	—
285	C₂×C₄	—	2	2	no	—
286	C₂×C₄	—	2	2	no	—
287	C₂×C₄	—	2	2	no	—
288	C₂×C₄	—	2	2	no	—
289	C₂×C₄	—	2	2	no	—
290	C₂×C₄	—	2	2	no	—
291	C₂×C₄	—	2	2	no	—
292	C₂×C₄	—	2	2	no	—
293	C₂×C₄	C₄×D₄×C₅₃	1	4	yes	—
294	C₂×C₄	#131×C₅₃	1	4	yes	—
295	C₂×C₄	C₂²×D₄×C₅₃	1	1	yes	characteristic
296	C₂×C₄	C₂×#41×C₅₃	1	1	yes	characteristic
297	C₂×C₄	2₋¹⁺⁴×C₅₃	1	2	yes	—
298	C₂×C₄	2₋¹⁺⁴×C₅₃	1	2	yes	—
299	C₈	—	2	4	no	—
300	C₈	—	4	2	no	—
301	C₈	—	4	2	no	—
302	C₈	—	4	2	no	—
303	C₂²	C₂×C₄×D₄×C₅₃	1	1	yes	characteristic
304	C₂²	C₅₃×#519	1	1	yes	characteristic
305	C₂²	C₅₃×#519	1	1	yes	characteristic
306	C₂²	C₂×#131×C₅₃	1	2	yes	—
307	C₂²	C₅₃×#568	1	2	yes	—
308	C₄	—	2	4	no	—
309	C₄	—	2	4	no	—
310	C₄	—	4	2	no	—
311	C₄	—	4	1	no	—
312	C₄	—	4	1	no	—
313	C₄	—	4	2	no	—
314	C₄	—	4	2	no	—
315	C₄	—	4	2	no	—
316	C₄	C₅₃×#536	1	2	yes	—
317	C₄	C₅₃×#541	1	2	yes	—
318	C₂	C₅₃×#2327	1	1	yes	characteristic
319	C₂	C₅₃×#2933	1	2	yes	—
320	C₂	C₅₃×#2985	1	2	yes	—
321	C₂	C₅₃×#2986	1	1	yes	characteristic
322	C₂	C₅₃×#2987	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄×C₅₃	C₂⁴
commutator	C₂³	C₂³×C₄×C₅₃
Frattini, cosocle	C₂²×C₄	C₂⁴×C₅₃
socle, prime radical	C₂³×C₅₃	C₂³×C₄
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂²×C₄×C₅₃ ◁ G

Complements

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

N	H ≅ G/N	type
141. #22245	C₅₃	direct (N × H)
197. C₅₃	#22245	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×#446	2
C₅₃×#2399	2
C₅₃×#1984	2
C₅₃×#1917	2
C₅₃×#1916	2
C₅₃×#1832	2
C₅₃×#1746	2
#22245	53

Subgroups — C53×#22245 #27229757

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅₃×#22245 #27229757