Subgroups — C₅₃×#22240 #27229752

Order 13568.

750 conjugacy classes of subgroups.

The subgroup lattice has 334 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	1	yes	maximal, characteristic
2	C₂×C₈×Q₈×C₅₃	C₂	1	1	yes	maximal, characteristic
3	C₄×C₄⋊Q₈×C₅₃	C₂	1	1	yes	maximal, characteristic
4	C₅₃×#1984	C₂	1	1	yes	maximal, characteristic
5	C₅₃×#1984	C₂	1	1	yes	maximal, characteristic
6	C₅₃×#1917	C₂	1	4	yes	maximal
7	C₅₃×#1916	C₂	1	4	yes	maximal
8	C₅₃×#1831	C₂	1	1	yes	maximal, characteristic
9	C₅₃×#1744	C₂	1	1	yes	maximal, characteristic
10	C₂×C₄⋊Q₈×C₅₃	C₄	1	1	yes	characteristic
11	C₂×C₄×Q₈×C₅₃	C₂²	1	1	yes	characteristic
12	C₂×C₄×Q₈×C₅₃	C₂²	1	1	yes	characteristic
13	C₅₃×#446	—	2	4	no	—
14	C₈×Q₈×C₅₃	—	2	4	no	—
15	C₂×C₄⋊C₈×C₅₃	C₂²	1	2	yes	—
16	C₂×C₄⋊C₈×C₅₃	C₂²	1	2	yes	—
17	C₂×C₄⋊C₈×C₅₃	C₂²	1	1	yes	characteristic
18	C₂×C₄⋊C₈×C₅₃	C₂²	1	1	yes	characteristic
19	C₂×#97×C₅₃	C₂²	1	2	yes	—
20	C₂×C₄×C₈×C₅₃	C₂²	1	2	yes	—
21	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
22	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
23	C₅₃×#391	C₄	1	1	yes	characteristic
24	C₅₃×#391	C₄	1	1	yes	characteristic
25	C₅₃×#389	C₄	1	4	yes	—
26	C₅₃×#384	C₄	1	1	yes	characteristic
27	C₄×C₄⋊C₄×C₅₃	C₂²	1	4	yes	—
28	C₄³×C₅₃	C₂²	1	1	yes	characteristic
29	C₅₃×#336	C₂²	1	2	yes	—
30	C₅₃×#336	C₂²	1	4	yes	—
31	C₅₃×#336	C₂²	1	4	yes	—
32	C₅₃×#336	C₂²	1	4	yes	—
33	C₅₃×#336	C₂²	1	2	yes	—
34	C₂²×Q₈×C₅₃	C₂×C₄	1	1	yes	characteristic
35	C₂²×Q₈×C₅₃	C₂×C₄	1	1	yes	characteristic
36	C₂²×C₈×C₅₃	C₂³	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	1	yes	characteristic
40	C₂²×C₈×C₅₃	C₂³	1	1	yes	characteristic
41	C₄⋊Q₈×C₅₃	—	2	4	no	—
42	C₄×Q₈×C₅₃	—	2	2	no	—
43	C₄×Q₈×C₅₃	—	2	2	no	—
44	C₄×Q₈×C₅₃	—	2	2	no	—
45	C₄×Q₈×C₅₃	—	2	2	no	—
46	C₂×C₄⋊C₄×C₅₃	C₂×C₄	1	1	yes	characteristic
47	C₂×C₄⋊C₄×C₅₃	C₂×C₄	1	1	yes	characteristic
48	C₂×C₄⋊C₄×C₅₃	C₂×C₄	1	2	yes	—
49	C₂×C₄⋊C₄×C₅₃	C₂×C₄	1	2	yes	—
50	C₂×C₄⋊C₄×C₅₃	C₂×C₄	1	4	yes	—
51	C₂×C₄⋊C₄×C₅₃	C₂×C₄	1	4	yes	—
52	C₂×C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
53	C₂×C₄²×C₅₃	C₂×C₄	1	2	yes	—
54	C₂×C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
55	C₂×C₄²×C₅₃	C₂³	1	2	yes	—
56	C₂×C₄²×C₅₃	C₂³	1	2	yes	—
57	C₂×C₄²×C₅₃	C₂³	1	2	yes	—
58	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
59	C₄⋊C₈×C₅₃	—	2	4	no	—
60	C₄⋊C₈×C₅₃	—	2	4	no	—
61	C₄⋊C₈×C₅₃	—	2	2	no	—
62	C₄⋊C₈×C₅₃	—	2	2	no	—
63	#97×C₅₃	—	2	4	no	—
64	C₄×C₈×C₅₃	—	2	4	no	—
65	C₄×C₈×C₅₃	—	2	2	no	—
66	C₄×C₈×C₅₃	D₄	1	4	yes	—
67	#95×C₅₃	C₂×C₄	1	4	yes	—
68	#95×C₅₃	C₂×C₄	1	4	yes	—
69	C₂×Q₈×C₅₃	—	2	2	no	—
70	C₂×Q₈×C₅₃	—	2	2	no	—
71	C₂×Q₈×C₅₃	—	2	2	no	—
72	C₂×Q₈×C₅₃	—	2	2	no	—
73	C₂×Q₈×C₅₃	—	2	2	no	—
74	C₂×Q₈×C₅₃	M₁₆	1	4	yes	—
75	C₂²×C₄×C₅₃	C₂²×C₄	1	2	yes	—
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	1	yes	characteristic
80	C₂²×C₄×C₅₃	C₂²×C₄	1	2	yes	—
81	C₂²×C₄×C₅₃	C₂²×C₄	1	2	yes	—
82	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
83	C₂²×C₄×C₅₃	C₂⁴	1	1	yes	characteristic
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	4	no	—
90	C₂×C₈×C₅₃	—	2	4	no	—
91	C₂×C₈×C₅₃	—	2	2	no	—
92	C₂×C₈×C₅₃	—	2	2	no	—
93	C₂×C₈×C₅₃	C₂×D₄	1	2	yes	—
94	C₂×C₈×C₅₃	C₂×D₄	1	2	yes	—
95	C₄⋊C₄×C₅₃	—	2	4	no	—
96	C₄⋊C₄×C₅₃	—	2	4	no	—
97	C₄⋊C₄×C₅₃	—	2	2	no	—
98	C₄⋊C₄×C₅₃	—	2	2	no	—
99	C₄⋊C₄×C₅₃	—	2	8	no	—
100	C₄⋊C₄×C₅₃	—	2	8	no	—
101	C₄²×C₅₃	—	2	2	no	—
102	C₄²×C₅₃	—	2	4	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	—	2	2	no	—
108	C₄²×C₅₃	—	2	2	no	—
109	C₄²×C₅₃	—	2	2	no	—
110	C₄²×C₅₃	C₂×D₄	1	2	yes	—
111	C₄²×C₅₃	#41	1	2	yes	—
112	C₂³×C₅₃	C₂³×C₄	1	1	yes	characteristic
113	Q₈×C₅₃	—	2	8	no	—
114	Q₈×C₅₃	—	4	4	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	2	no	—
120	C₂×C₄×C₅₃	—	2	2	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	4	no	—
129	C₂×C₄×C₅₃	—	2	4	no	—
130	C₂×C₄×C₅₃	C₄×D₄	1	2	yes	—
131	C₂×C₄×C₅₃	C₄×D₄	1	2	yes	—
132	C₂×C₄×C₅₃	C₂×M₁₆	1	4	yes	—
133	C₂×C₄×C₅₃	C₂×M₁₆	1	2	yes	—
134	C₂×C₄×C₅₃	#131	1	4	yes	—
135	C₂×C₄×C₅₃	#131	1	2	yes	—
136	C₂×C₄×C₅₃	C₂²×D₄	1	1	yes	characteristic
137	C₂×C₄×C₅₃	C₂×#41	1	1	yes	characteristic
138	C₂×C₄×C₅₃	2₋¹⁺⁴	1	2	yes	—
139	C₂×C₄×C₅₃	2₋¹⁺⁴	1	2	yes	—
140	C₈×C₅₃	—	2	2	no	—
141	C₈×C₅₃	—	2	2	no	—
142	C₈×C₅₃	—	4	2	no	—
143	C₈×C₅₃	—	4	2	no	—
144	C₈×C₅₃	—	4	2	no	—
145	#22240	C₅₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
146	C₂²×C₅₃	C₂×C₄×D₄	1	1	yes	characteristic
147	C₂²×C₅₃	#519	1	2	yes	—
148	C₂²×C₅₃	C₂²×M₁₆	1	1	yes	characteristic
149	C₂²×C₅₃	C₂×#131	1	1	yes	characteristic
150	C₂²×C₅₃	C₂×#131	1	2	yes	—
151	C₄×C₅₃	—	2	4	no	—
152	C₄×C₅₃	—	2	2	no	—
153	C₄×C₅₃	—	2	4	no	—
154	C₄×C₅₃	—	4	2	no	—
155	C₄×C₅₃	—	4	2	no	—
156	C₄×C₅₃	—	4	2	no	—
157	C₄×C₅₃	—	4	2	no	—
158	C₄×C₅₃	#435	1	4	yes	—
159	C₄×C₅₃	#536	1	1	yes	characteristic
160	C₄×C₅₃	#536	1	1	yes	characteristic
161	C₄×C₅₃	#541	1	2	yes	—
162	C₂×#446	C₂×C₅₃	1	1	yes	characteristic
163	C₂×C₈×Q₈	C₂×C₅₃	1	1	yes	characteristic
164	C₄×C₄⋊Q₈	C₂×C₅₃	1	1	yes	characteristic
165	#1984	C₂×C₅₃	1	1	yes	characteristic
166	#1984	C₂×C₅₃	1	1	yes	characteristic
167	#1917	C₂×C₅₃	1	4	yes	—
168	#1916	C₂×C₅₃	1	4	yes	—
169	#1831	C₂×C₅₃	1	1	yes	characteristic
170	#1744	C₂×C₅₃	1	1	yes	characteristic
171	C₂×C₅₃	#2327	1	1	yes	characteristic
172	C₂×C₅₃	C₂×#435	1	1	yes	characteristic
173	C₂×C₅₃	#2931	1	1	yes	characteristic
174	C₂×C₅₃	#2984	1	2	yes	—
175	C₂×C₅₃	#2986	1	2	yes	—
176	C₂×C₄⋊Q₈	C₄×C₅₃	1	1	yes	characteristic
177	C₂×C₄×Q₈	C₂²×C₅₃	1	1	yes	characteristic
178	C₂×C₄×Q₈	C₂²×C₅₃	1	1	yes	characteristic
179	#446	—	2	4	no	—
180	C₈×Q₈	—	2	4	no	—
181	C₂×C₄⋊C₈	C₂²×C₅₃	1	1	yes	characteristic
182	C₂×C₄⋊C₈	C₂²×C₅₃	1	1	yes	characteristic
183	C₂×C₄⋊C₈	C₂²×C₅₃	1	2	yes	—
184	C₂×C₄⋊C₈	C₂²×C₅₃	1	2	yes	—
185	C₂×#97	C₂²×C₅₃	1	2	yes	—
186	C₂×C₄×C₈	C₂²×C₅₃	1	2	yes	—
187	C₂×C₄×C₈	C₂²×C₅₃	1	1	yes	characteristic
188	C₂×C₄×C₈	C₂²×C₅₃	1	1	yes	characteristic
189	#391	C₄×C₅₃	1	1	yes	characteristic
190	#391	C₄×C₅₃	1	1	yes	characteristic
191	#389	C₄×C₅₃	1	4	yes	—
192	#384	C₄×C₅₃	1	1	yes	characteristic
193	C₄×C₄⋊C₄	C₂²×C₅₃	1	4	yes	—
194	C₄³	C₂²×C₅₃	1	1	yes	characteristic
195	#336	C₂²×C₅₃	1	4	yes	—
196	#336	C₂²×C₅₃	1	2	yes	—
197	#336	C₂²×C₅₃	1	4	yes	—
198	#336	C₂²×C₅₃	1	4	yes	—
199	#336	C₂²×C₅₃	1	2	yes	—
200	C₅₃	#22240	1	1	yes	characteristic, direct product factor, Sylow
201	C₂²×Q₈	C₂×C₄×C₅₃	1	1	yes	characteristic
202	C₂²×Q₈	C₂×C₄×C₅₃	1	1	yes	characteristic
203	C₂²×C₈	C₂³×C₅₃	1	2	yes	—
204	C₂²×C₈	C₂³×C₅₃	1	2	yes	—
205	C₂²×C₈	C₂³×C₅₃	1	2	yes	—
206	C₂²×C₈	C₂³×C₅₃	1	1	yes	characteristic
207	C₂²×C₈	C₂³×C₅₃	1	1	yes	characteristic
208	C₄⋊Q₈	—	2	4	no	—
209	C₄×Q₈	—	2	2	no	—
210	C₄×Q₈	—	2	2	no	—
211	C₄×Q₈	—	2	2	no	—
212	C₄×Q₈	—	2	2	no	—
213	C₂×C₄⋊C₄	C₂×C₄×C₅₃	1	1	yes	characteristic
214	C₂×C₄⋊C₄	C₂×C₄×C₅₃	1	1	yes	characteristic
215	C₂×C₄⋊C₄	C₂×C₄×C₅₃	1	2	yes	—
216	C₂×C₄⋊C₄	C₂×C₄×C₅₃	1	2	yes	—
217	C₂×C₄⋊C₄	C₂×C₄×C₅₃	1	4	yes	—
218	C₂×C₄⋊C₄	C₂×C₄×C₅₃	1	4	yes	—
219	C₂×C₄²	C₂×C₄×C₅₃	1	1	yes	characteristic
220	C₂×C₄²	C₂×C₄×C₅₃	1	2	yes	—
221	C₂×C₄²	C₂×C₄×C₅₃	1	1	yes	characteristic
222	C₂×C₄²	C₂³×C₅₃	1	2	yes	—
223	C₂×C₄²	C₂³×C₅₃	1	2	yes	—
224	C₂×C₄²	C₂³×C₅₃	1	2	yes	—
225	C₂×C₄²	C₂³×C₅₃	1	1	yes	characteristic
226	C₄⋊C₈	—	2	4	no	—
227	C₄⋊C₈	—	2	4	no	—
228	C₄⋊C₈	—	2	2	no	—
229	C₄⋊C₈	—	2	2	no	—
230	#97	—	2	4	no	—
231	C₄×C₈	—	2	4	no	—
232	C₄×C₈	—	2	2	no	—
233	C₄×C₈	D₄×C₅₃	1	4	yes	—
234	#95	C₂×C₄×C₅₃	1	4	yes	—
235	#95	C₂×C₄×C₅₃	1	4	yes	—
236	C₂×Q₈	—	2	2	no	—
237	C₂×Q₈	—	2	2	no	—
238	C₂×Q₈	—	2	2	no	—
239	C₂×Q₈	—	2	2	no	—
240	C₂×Q₈	—	2	2	no	—
241	C₂×Q₈	M₁₆×C₅₃	1	4	yes	—
242	C₂²×C₄	C₂²×C₄×C₅₃	1	2	yes	—
243	C₂²×C₄	C₂²×C₄×C₅₃	1	2	yes	—
244	C₂²×C₄	C₂²×C₄×C₅₃	1	2	yes	—
245	C₂²×C₄	C₂²×C₄×C₅₃	1	2	yes	—
246	C₂²×C₄	C₂²×C₄×C₅₃	1	1	yes	characteristic
247	C₂²×C₄	C₂²×C₄×C₅₃	1	2	yes	—
248	C₂²×C₄	C₂²×C₄×C₅₃	1	2	yes	—
249	C₂²×C₄	C₂²×C₄×C₅₃	1	1	yes	characteristic
250	C₂²×C₄	C₂⁴×C₅₃	1	1	yes	characteristic
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	—	2	2	no	—
256	C₂×C₈	—	2	4	no	—
257	C₂×C₈	—	2	4	no	—
258	C₂×C₈	—	2	2	no	—
259	C₂×C₈	—	2	2	no	—
260	C₂×C₈	C₂×D₄×C₅₃	1	2	yes	—
261	C₂×C₈	C₂×D₄×C₅₃	1	2	yes	—
262	C₄⋊C₄	—	2	4	no	—
263	C₄⋊C₄	—	2	2	no	—
264	C₄⋊C₄	—	2	2	no	—
265	C₄⋊C₄	—	2	4	no	—
266	C₄⋊C₄	—	2	8	no	—
267	C₄⋊C₄	—	2	8	no	—
268	C₄²	—	2	2	no	—
269	C₄²	—	2	4	no	—
270	C₄²	—	2	2	no	—
271	C₄²	—	2	2	no	—
272	C₄²	—	2	2	no	—
273	C₄²	—	2	2	no	—
274	C₄²	—	2	2	no	—
275	C₄²	—	2	2	no	—
276	C₄²	—	2	2	no	—
277	C₄²	C₂×D₄×C₅₃	1	2	yes	—
278	C₄²	#41×C₅₃	1	2	yes	—
279	C₂³	C₂³×C₄×C₅₃	1	1	yes	characteristic
280	Q₈	—	2	8	no	—
281	Q₈	—	4	4	no	—
282	C₂×C₄	—	2	2	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	—	2	2	no	—
294	C₂×C₄	—	2	4	no	—
295	C₂×C₄	—	2	4	no	—
296	C₂×C₄	—	2	2	no	—
297	C₂×C₄	C₄×D₄×C₅₃	1	2	yes	—
298	C₂×C₄	C₄×D₄×C₅₃	1	2	yes	—
299	C₂×C₄	C₂×M₁₆×C₅₃	1	4	yes	—
300	C₂×C₄	C₂×M₁₆×C₅₃	1	2	yes	—
301	C₂×C₄	#131×C₅₃	1	4	yes	—
302	C₂×C₄	#131×C₅₃	1	2	yes	—
303	C₂×C₄	C₂²×D₄×C₅₃	1	1	yes	characteristic
304	C₂×C₄	C₂×#41×C₅₃	1	1	yes	characteristic
305	C₂×C₄	2₋¹⁺⁴×C₅₃	1	2	yes	—
306	C₂×C₄	2₋¹⁺⁴×C₅₃	1	2	yes	—
307	C₈	—	2	2	no	—
308	C₈	—	2	2	no	—
309	C₈	—	4	2	no	—
310	C₈	—	4	2	no	—
311	C₈	—	4	2	no	—
312	C₂²	C₂×C₄×D₄×C₅₃	1	1	yes	characteristic
313	C₂²	C₅₃×#519	1	2	yes	—
314	C₂²	C₂²×M₁₆×C₅₃	1	1	yes	characteristic
315	C₂²	C₂×#131×C₅₃	1	2	yes	—
316	C₂²	C₂×#131×C₅₃	1	1	yes	characteristic
317	C₄	—	2	4	no	—
318	C₄	—	2	4	no	—
319	C₄	—	2	2	no	—
320	C₄	—	4	2	no	—
321	C₄	—	4	2	no	—
322	C₄	—	4	2	no	—
323	C₄	—	4	2	no	—
324	C₄	C₅₃×#435	1	4	yes	—
325	C₄	C₅₃×#536	1	1	yes	characteristic
326	C₄	C₅₃×#536	1	1	yes	characteristic
327	C₄	C₅₃×#541	1	2	yes	—
328	C₂	C₅₃×#2327	1	1	yes	characteristic
329	C₂	C₂×C₅₃×#435	1	1	yes	characteristic
330	C₂	C₅₃×#2931	1	1	yes	characteristic
331	C₂	C₅₃×#2984	1	2	yes	—
332	C₂	C₅₃×#2986	1	2	yes	—

	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
145. #22240	C₅₃	direct (N × H)
200. C₅₃	#22240	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
C₂×C₈×Q₈×C₅₃	2
C₄×C₄⋊Q₈×C₅₃	2
C₅₃×#1984	2
C₅₃×#1917	2
C₅₃×#1916	2
C₅₃×#1831	2
C₅₃×#1744	2
#22240	53

Subgroups — C53×#22240 #27229752

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅₃×#22240 #27229752