Subgroups — C₅₃×#22210 #27229722

Order 13568.

866 conjugacy classes of subgroups.

The subgroup lattice has 366 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₈×Q₈×C₅₃	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×C₅₃×#436	C₂	1	1	yes	maximal, characteristic
3	C₄×#124×C₅₃	C₂	1	1	yes	maximal, characteristic
4	C₅₃×#1984	C₂	1	1	yes	maximal, characteristic, complemented
5	C₅₃×#1954	C₂	1	1	yes	maximal, characteristic
6	C₅₃×#1923	C₂	1	4	yes	maximal, complemented
7	C₅₃×#1851	C₂	1	4	yes	maximal, complemented
8	C₅₃×#1842	C₂	1	1	yes	maximal, characteristic, complemented
9	C₅₃×#1744	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂²×M₁₆×C₅₃	C₂²	1	2	yes	—
11	C₂×#124×C₅₃	C₄	1	1	yes	characteristic
12	C₂×C₄×Q₈×C₅₃	C₂²	1	1	yes	characteristic
13	C₂×C₄×D₄×C₅₃	C₂²	1	1	yes	characteristic
14	C₈×Q₈×C₅₃	—	2	4	no	—
15	C₅₃×#436	—	2	4	no	—
16	C₂×C₄⋊C₈×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	C₂×C₄⋊C₈×C₅₃	C₂²	1	4	yes	—
20	C₂×C₂²⋊C₈×C₅₃	C₂²	1	2	yes	—
21	C₂×C₂²⋊C₈×C₅₃	C₂²	1	4	yes	—
22	C₂×C₄×C₈×C₅₃	C₂²	1	2	yes	—
23	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
24	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
25	C₅₃×#391	C₄	1	1	yes	characteristic
26	C₅₃×#390	C₄	1	1	yes	characteristic
27	C₅₃×#388	C₄	1	4	yes	—
28	C₅₃×#382	C₄	1	1	yes	characteristic
29	C₄×C₂²⋊C₄×C₅₃	C₂²	1	4	yes	—
30	C₄³×C₅₃	C₂²	1	1	yes	characteristic
31	C₅₃×#336	C₂²	1	2	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×D₄×C₅₃	C₂×C₄	1	1	yes	characteristic
36	C₂³×C₄×C₅₃	C₂³	1	2	yes	—
37	C₂×M₁₆×C₅₃	—	2	4	no	—
38	C₂×M₁₆×C₅₃	—	4	4	no	—
39	C₂²×C₈×C₅₃	C₂³	1	2	yes	—
40	C₂²×C₈×C₅₃	C₂³	1	2	yes	—
41	C₂²×C₈×C₅₃	C₂³	1	2	yes	—
42	C₂²×C₈×C₅₃	C₂³	1	1	yes	characteristic
43	C₂²×C₈×C₅₃	C₂³	1	1	yes	characteristic
44	#124×C₅₃	—	2	4	no	—
45	C₄×Q₈×C₅₃	—	2	2	no	—
46	C₄×Q₈×C₅₃	—	2	2	no	—
47	C₄×D₄×C₅₃	—	2	2	no	—
48	C₄×D₄×C₅₃	—	2	2	no	—
49	C₂×C₄⋊C₄×C₅₃	C₂×C₄	1	1	yes	characteristic
50	C₂×C₄⋊C₄×C₅₃	C₂×C₄	1	1	yes	characteristic
51	C₂×C₄⋊C₄×C₅₃	C₂×C₄	1	2	yes	—
52	C₂×C₂²⋊C₄×C₅₃	C₂×C₄	1	2	yes	—
53	C₂×C₂²⋊C₄×C₅₃	C₂×C₄	1	4	yes	—
54	C₂×C₂²⋊C₄×C₅₃	C₂×C₄	1	4	yes	—
55	C₂×C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
56	C₂×C₄²×C₅₃	C₂×C₄	1	2	yes	—
57	C₂×C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
58	C₂×C₄²×C₅₃	C₂³	1	2	yes	—
59	C₂×C₄²×C₅₃	C₂³	1	2	yes	—
60	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
61	C₄⋊C₈×C₅₃	—	2	4	no	—
62	C₄⋊C₈×C₅₃	—	2	2	no	—
63	C₄⋊C₈×C₅₃	—	2	2	no	—
64	C₄⋊C₈×C₅₃	—	4	4	no	—
65	C₂²⋊C₈×C₅₃	—	2	4	no	—
66	C₂²⋊C₈×C₅₃	—	4	4	no	—
67	C₄×C₈×C₅₃	—	2	4	no	—
68	C₄×C₈×C₅₃	—	2	2	no	—
69	C₄×C₈×C₅₃	D₄	1	4	yes	—
70	#95×C₅₃	C₂×C₄	1	4	yes	—
71	#95×C₅₃	C₂×C₄	1	4	yes	—
72	C₂⁴×C₅₃	C₂²×C₄	1	2	yes	—
73	C₂×Q₈×C₅₃	—	2	2	no	—
74	C₂×Q₈×C₅₃	—	2	2	no	—
75	C₂×Q₈×C₅₃	M₁₆	1	4	yes	complemented
76	C₂×D₄×C₅₃	—	2	2	no	—
77	C₂×D₄×C₅₃	—	2	2	no	—
78	C₂×D₄×C₅₃	—	2	2	no	—
79	C₂²×C₄×C₅₃	—	2	2	no	—
80	C₂²×C₄×C₅₃	—	2	2	no	—
81	C₂²×C₄×C₅₃	—	2	4	no	—
82	C₂²×C₄×C₅₃	—	2	4	no	—
83	C₂²×C₄×C₅₃	C₂²×C₄	1	2	yes	—
84	C₂²×C₄×C₅₃	C₂²×C₄	1	2	yes	—
85	C₂²×C₄×C₅₃	C₂²×C₄	1	2	yes	—
86	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
87	C₂²×C₄×C₅₃	C₂²×C₄	1	2	yes	—
88	C₂²×C₄×C₅₃	C₂²×C₄	1	2	yes	—
89	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
90	C₂²×C₄×C₅₃	C₂⁴	1	1	yes	characteristic
91	M₁₆×C₅₃	—	4	8	no	—
92	C₂×C₈×C₅₃	—	2	2	no	—
93	C₂×C₈×C₅₃	—	2	2	no	—
94	C₂×C₈×C₅₃	—	2	2	no	—
95	C₂×C₈×C₅₃	—	2	2	no	—
96	C₂×C₈×C₅₃	—	2	2	no	—
97	C₂×C₈×C₅₃	—	2	2	no	—
98	C₂×C₈×C₅₃	—	2	4	no	—
99	C₂×C₈×C₅₃	—	2	4	no	—
100	C₂×C₈×C₅₃	—	2	2	no	—
101	C₂×C₈×C₅₃	C₂×D₄	1	2	yes	—
102	C₂×C₈×C₅₃	C₂×D₄	1	2	yes	—
103	C₄⋊C₄×C₅₃	—	2	2	no	—
104	C₄⋊C₄×C₅₃	—	2	4	no	—
105	C₄⋊C₄×C₅₃	—	2	2	no	—
106	C₂²⋊C₄×C₅₃	—	2	4	no	—
107	C₂²⋊C₄×C₅₃	—	2	8	no	—
108	C₂²⋊C₄×C₅₃	—	2	8	no	—
109	C₄²×C₅₃	—	2	2	no	—
110	C₄²×C₅₃	—	2	2	no	—
111	C₄²×C₅₃	—	2	4	no	—
112	C₄²×C₅₃	—	2	2	no	—
113	C₄²×C₅₃	—	2	2	no	—
114	C₄²×C₅₃	—	2	2	no	—
115	C₄²×C₅₃	—	2	2	no	—
116	C₄²×C₅₃	C₂×D₄	1	2	yes	—
117	C₄²×C₅₃	#41	1	2	yes	—
118	C₂³×C₅₃	—	2	2	no	—
119	C₂³×C₅₃	—	2	4	no	—
120	C₂³×C₅₃	—	2	2	no	—
121	C₂³×C₅₃	—	2	2	no	—
122	C₂³×C₅₃	—	2	4	no	—
123	C₂³×C₅₃	C₂³×C₄	1	1	yes	characteristic
124	Q₈×C₅₃	—	2	8	no	—
125	D₄×C₅₃	—	4	4	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	—	2	2	no	—
132	C₂×C₄×C₅₃	—	2	2	no	—
133	C₂×C₄×C₅₃	—	2	4	no	—
134	C₂×C₄×C₅₃	—	2	2	no	—
135	C₂×C₄×C₅₃	—	2	2	no	—
136	C₂×C₄×C₅₃	—	2	2	no	—
137	C₂×C₄×C₅₃	—	2	4	no	—
138	C₂×C₄×C₅₃	—	4	2	no	—
139	C₂×C₄×C₅₃	—	4	2	no	—
140	C₂×C₄×C₅₃	—	4	4	no	—
141	C₂×C₄×C₅₃	C₄×D₄	1	2	yes	—
142	C₂×C₄×C₅₃	C₄×D₄	1	2	yes	—
143	C₂×C₄×C₅₃	C₂×M₁₆	1	4	yes	—
144	C₂×C₄×C₅₃	C₂×M₁₆	1	2	yes	—
145	C₂×C₄×C₅₃	#131	1	4	yes	—
146	C₂×C₄×C₅₃	#131	1	2	yes	—
147	C₂×C₄×C₅₃	C₂²×D₄	1	1	yes	characteristic
148	C₂×C₄×C₅₃	C₂×#41	1	1	yes	characteristic
149	C₂×C₄×C₅₃	H(5,𝔽₂)	1	2	yes	—
150	C₂×C₄×C₅₃	2₋¹⁺⁴	1	2	yes	—
151	C₈×C₅₃	—	2	2	no	—
152	C₈×C₅₃	—	2	2	no	—
153	C₈×C₅₃	—	4	2	no	—
154	C₈×C₅₃	—	4	2	no	—
155	C₈×C₅₃	—	4	2	no	—
156	#22210	C₅₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
157	C₂²×C₅₃	—	4	2	no	—
158	C₂²×C₅₃	—	4	2	no	—
159	C₂²×C₅₃	—	4	2	no	—
160	C₂²×C₅₃	—	4	4	no	—
161	C₂²×C₅₃	—	4	4	no	—
162	C₂²×C₅₃	C₂×C₄×D₄	1	1	yes	characteristic
163	C₂²×C₅₃	#520	1	2	yes	—
164	C₂²×C₅₃	C₂²×M₁₆	1	1	yes	characteristic
165	C₂²×C₅₃	C₂×#131	1	1	yes	characteristic
166	C₂²×C₅₃	C₂×#131	1	2	yes	—
167	C₄×C₅₃	—	2	4	no	—
168	C₄×C₅₃	—	2	2	no	—
169	C₄×C₅₃	—	2	4	no	—
170	C₄×C₅₃	—	4	2	no	—
171	C₄×C₅₃	—	4	2	no	—
172	C₄×C₅₃	—	4	2	no	—
173	C₄×C₅₃	#435	1	4	yes	—
174	C₄×C₅₃	#535	1	1	yes	characteristic
175	C₄×C₅₃	#536	1	1	yes	characteristic
176	C₄×C₅₃	#542	1	2	yes	—
177	C₂×C₈×Q₈	C₂×C₅₃	1	1	yes	characteristic, complemented
178	C₂×#436	C₂×C₅₃	1	1	yes	characteristic
179	C₄×#124	C₂×C₅₃	1	1	yes	characteristic
180	#1984	C₂×C₅₃	1	1	yes	characteristic, complemented
181	#1954	C₂×C₅₃	1	1	yes	characteristic
182	#1923	C₂×C₅₃	1	4	yes	complemented
183	#1851	C₂×C₅₃	1	4	yes	complemented
184	#1842	C₂×C₅₃	1	1	yes	characteristic, complemented
185	#1744	C₂×C₅₃	1	1	yes	characteristic, complemented
186	C₂×C₅₃	—	8	2	no	—
187	C₂×C₅₃	#2326	1	1	yes	characteristic
188	C₂×C₅₃	C₂×#435	1	1	yes	characteristic
189	C₂×C₅₃	#2931	1	1	yes	characteristic
190	C₂×C₅₃	#2991	1	2	yes	—
191	C₂×C₅₃	#2994	1	2	yes	—
192	C₂²×M₁₆	C₂²×C₅₃	1	2	yes	—
193	C₂×#124	C₄×C₅₃	1	1	yes	characteristic
194	C₂×C₄×Q₈	C₂²×C₅₃	1	1	yes	characteristic
195	C₂×C₄×D₄	C₂²×C₅₃	1	1	yes	characteristic
196	C₈×Q₈	—	2	4	no	—
197	#436	—	2	4	no	—
198	C₂×C₄⋊C₈	C₂²×C₅₃	1	1	yes	characteristic
199	C₂×C₄⋊C₈	C₂²×C₅₃	1	1	yes	characteristic
200	C₂×C₄⋊C₈	C₂²×C₅₃	1	2	yes	—
201	C₂×C₄⋊C₈	C₂²×C₅₃	1	4	yes	—
202	C₂×C₂²⋊C₈	C₂²×C₅₃	1	2	yes	—
203	C₂×C₂²⋊C₈	C₂²×C₅₃	1	4	yes	—
204	C₂×C₄×C₈	C₂²×C₅₃	1	2	yes	—
205	C₂×C₄×C₈	C₂²×C₅₃	1	1	yes	characteristic
206	C₂×C₄×C₈	C₂²×C₅₃	1	1	yes	characteristic
207	#391	C₄×C₅₃	1	1	yes	characteristic
208	#390	C₄×C₅₃	1	1	yes	characteristic
209	#388	C₄×C₅₃	1	4	yes	—
210	#382	C₄×C₅₃	1	1	yes	characteristic
211	C₄×C₂²⋊C₄	C₂²×C₅₃	1	4	yes	—
212	C₄³	C₂²×C₅₃	1	1	yes	characteristic
213	#336	C₂²×C₅₃	1	2	yes	—
214	#336	C₂²×C₅₃	1	4	yes	—
215	#336	C₂²×C₅₃	1	2	yes	—
216	C₅₃	#22210	1	1	yes	characteristic, direct product factor, Sylow
217	C₂²×Q₈	C₂×C₄×C₅₃	1	1	yes	characteristic
218	C₂²×D₄	C₂×C₄×C₅₃	1	1	yes	characteristic
219	C₂³×C₄	C₂³×C₅₃	1	2	yes	—
220	C₂×M₁₆	—	2	4	no	—
221	C₂×M₁₆	—	4	4	no	—
222	C₂²×C₈	C₂³×C₅₃	1	2	yes	—
223	C₂²×C₈	C₂³×C₅₃	1	2	yes	—
224	C₂²×C₈	C₂³×C₅₃	1	2	yes	—
225	C₂²×C₈	C₂³×C₅₃	1	1	yes	characteristic
226	C₂²×C₈	C₂³×C₅₃	1	1	yes	characteristic
227	#124	—	2	4	no	—
228	C₄×Q₈	—	2	2	no	—
229	C₄×Q₈	—	2	2	no	—
230	C₄×D₄	—	2	2	no	—
231	C₄×D₄	—	2	2	no	—
232	C₂×C₄⋊C₄	C₂×C₄×C₅₃	1	1	yes	characteristic
233	C₂×C₄⋊C₄	C₂×C₄×C₅₃	1	1	yes	characteristic
234	C₂×C₄⋊C₄	C₂×C₄×C₅₃	1	2	yes	—
235	C₂×C₂²⋊C₄	C₂×C₄×C₅₃	1	2	yes	—
236	C₂×C₂²⋊C₄	C₂×C₄×C₅₃	1	4	yes	—
237	C₂×C₂²⋊C₄	C₂×C₄×C₅₃	1	4	yes	—
238	C₂×C₄²	C₂×C₄×C₅₃	1	1	yes	characteristic
239	C₂×C₄²	C₂×C₄×C₅₃	1	2	yes	—
240	C₂×C₄²	C₂×C₄×C₅₃	1	1	yes	characteristic
241	C₂×C₄²	C₂³×C₅₃	1	2	yes	—
242	C₂×C₄²	C₂³×C₅₃	1	2	yes	—
243	C₂×C₄²	C₂³×C₅₃	1	1	yes	characteristic
244	C₄⋊C₈	—	2	4	no	—
245	C₄⋊C₈	—	2	2	no	—
246	C₄⋊C₈	—	2	2	no	—
247	C₄⋊C₈	—	4	4	no	—
248	C₂²⋊C₈	—	2	4	no	—
249	C₂²⋊C₈	—	4	4	no	—
250	C₄×C₈	—	2	4	no	—
251	C₄×C₈	—	2	2	no	—
252	C₄×C₈	D₄×C₅₃	1	4	yes	—
253	#95	C₂×C₄×C₅₃	1	4	yes	—
254	#95	C₂×C₄×C₅₃	1	4	yes	—
255	C₂⁴	C₂²×C₄×C₅₃	1	2	yes	—
256	C₂×Q₈	—	2	2	no	—
257	C₂×Q₈	—	2	2	no	—
258	C₂×Q₈	M₁₆×C₅₃	1	4	yes	complemented
259	C₂×D₄	—	2	2	no	—
260	C₂×D₄	—	2	2	no	—
261	C₂×D₄	—	2	2	no	—
262	C₂²×C₄	—	2	2	no	—
263	C₂²×C₄	—	2	2	no	—
264	C₂²×C₄	—	2	4	no	—
265	C₂²×C₄	—	2	4	no	—
266	C₂²×C₄	C₂²×C₄×C₅₃	1	2	yes	—
267	C₂²×C₄	C₂²×C₄×C₅₃	1	2	yes	—
268	C₂²×C₄	C₂²×C₄×C₅₃	1	2	yes	—
269	C₂²×C₄	C₂²×C₄×C₅₃	1	1	yes	characteristic
270	C₂²×C₄	C₂²×C₄×C₅₃	1	2	yes	—
271	C₂²×C₄	C₂²×C₄×C₅₃	1	2	yes	—
272	C₂²×C₄	C₂²×C₄×C₅₃	1	1	yes	characteristic
273	C₂²×C₄	C₂⁴×C₅₃	1	1	yes	characteristic
274	M₁₆	—	4	8	no	—
275	C₂×C₈	—	2	2	no	—
276	C₂×C₈	—	2	2	no	—
277	C₂×C₈	—	2	2	no	—
278	C₂×C₈	—	2	2	no	—
279	C₂×C₈	—	2	2	no	—
280	C₂×C₈	—	2	2	no	—
281	C₂×C₈	—	2	4	no	—
282	C₂×C₈	—	2	4	no	—
283	C₂×C₈	—	2	2	no	—
284	C₂×C₈	C₂×D₄×C₅₃	1	2	yes	—
285	C₂×C₈	C₂×D₄×C₅₃	1	2	yes	—
286	C₄⋊C₄	—	2	2	no	—
287	C₄⋊C₄	—	2	4	no	—
288	C₄⋊C₄	—	2	2	no	—
289	C₂²⋊C₄	—	2	4	no	—
290	C₂²⋊C₄	—	2	8	no	—
291	C₂²⋊C₄	—	2	8	no	—
292	C₄²	—	2	2	no	—
293	C₄²	—	2	2	no	—
294	C₄²	—	2	4	no	—
295	C₄²	—	2	2	no	—
296	C₄²	—	2	2	no	—
297	C₄²	—	2	2	no	—
298	C₄²	—	2	2	no	—
299	C₄²	C₂×D₄×C₅₃	1	2	yes	—
300	C₄²	#41×C₅₃	1	2	yes	—
301	C₂³	—	2	2	no	—
302	C₂³	—	2	4	no	—
303	C₂³	—	2	2	no	—
304	C₂³	—	2	4	no	—
305	C₂³	—	2	2	no	—
306	C₂³	C₂³×C₄×C₅₃	1	1	yes	characteristic
307	Q₈	—	2	8	no	—
308	D₄	—	4	4	no	—
309	C₂×C₄	—	2	2	no	—
310	C₂×C₄	—	2	2	no	—
311	C₂×C₄	—	2	2	no	—
312	C₂×C₄	—	2	2	no	—
313	C₂×C₄	—	2	2	no	—
314	C₂×C₄	—	2	2	no	—
315	C₂×C₄	—	2	2	no	—
316	C₂×C₄	—	2	2	no	—
317	C₂×C₄	—	2	4	no	—
318	C₂×C₄	—	2	2	no	—
319	C₂×C₄	—	2	2	no	—
320	C₂×C₄	—	2	4	no	—
321	C₂×C₄	—	4	2	no	—
322	C₂×C₄	—	4	2	no	—
323	C₂×C₄	—	4	4	no	—
324	C₂×C₄	C₄×D₄×C₅₃	1	2	yes	—
325	C₂×C₄	C₄×D₄×C₅₃	1	2	yes	—
326	C₂×C₄	C₂×M₁₆×C₅₃	1	4	yes	—
327	C₂×C₄	C₂×M₁₆×C₅₃	1	2	yes	—
328	C₂×C₄	#131×C₅₃	1	4	yes	—
329	C₂×C₄	#131×C₅₃	1	2	yes	—
330	C₂×C₄	C₂²×D₄×C₅₃	1	1	yes	characteristic
331	C₂×C₄	C₂×#41×C₅₃	1	1	yes	characteristic
332	C₂×C₄	H(5,𝔽₂)×C₅₃	1	2	yes	—
333	C₂×C₄	2₋¹⁺⁴×C₅₃	1	2	yes	—
334	C₈	—	2	2	no	—
335	C₈	—	2	2	no	—
336	C₈	—	4	2	no	—
337	C₈	—	4	2	no	—
338	C₈	—	4	2	no	—
339	C₂²	—	4	4	no	—
340	C₂²	—	4	4	no	—
341	C₂²	—	4	2	no	—
342	C₂²	—	4	2	no	—
343	C₂²	—	4	2	no	—
344	C₂²	C₂×C₄×D₄×C₅₃	1	1	yes	characteristic
345	C₂²	C₅₃×#520	1	2	yes	—
346	C₂²	C₂²×M₁₆×C₅₃	1	1	yes	characteristic
347	C₂²	C₂×#131×C₅₃	1	2	yes	—
348	C₂²	C₂×#131×C₅₃	1	1	yes	characteristic
349	C₄	—	2	4	no	—
350	C₄	—	2	4	no	—
351	C₄	—	2	2	no	—
352	C₄	—	4	2	no	—
353	C₄	—	4	2	no	—
354	C₄	—	4	2	no	—
355	C₄	C₅₃×#435	1	4	yes	—
356	C₄	C₅₃×#535	1	1	yes	characteristic
357	C₄	C₅₃×#536	1	1	yes	characteristic
358	C₄	C₅₃×#542	1	2	yes	—
359	C₂	—	8	2	no	—
360	C₂	C₅₃×#2326	1	1	yes	characteristic
361	C₂	C₂×C₅₃×#435	1	1	yes	characteristic
362	C₂	C₅₃×#2931	1	1	yes	characteristic
363	C₂	C₅₃×#2991	1	2	yes	—
364	C₂	C₅₃×#2994	1	2	yes	—

	subgroup	G/H
center	C₂²×C₄×C₅₃	C₂⁴
commutator	C₂³	C₂³×C₄×C₅₃
Frattini, cosocle	C₂²×C₄	C₂⁴×C₅₃
socle	C₂³×C₅₃	C₂³×C₄
prime radical	C₂²×D₄×C₅₃	C₂×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
1. C₂×C₈×Q₈×C₅₃	C₂	semidirect
4. C₅₃×#1984	C₂	semidirect
6. C₅₃×#1923	C₂	semidirect
7. C₅₃×#1851	C₂	semidirect
8. C₅₃×#1842	C₂	semidirect
9. C₅₃×#1744	C₂	semidirect
75. C₂×Q₈×C₅₃	M₁₆	semidirect
156. #22210	C₅₃	direct (N × H)
177. C₂×C₈×Q₈	C₂×C₅₃	semidirect
180. #1984	C₂×C₅₃	semidirect
182. #1923	C₂×C₅₃	semidirect
183. #1851	C₂×C₅₃	semidirect
184. #1842	C₂×C₅₃	semidirect
185. #1744	C₂×C₅₃	semidirect
216. C₅₃	#22210	direct (N × H)
258. C₂×Q₈	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×C₈×Q₈×C₅₃	2
C₂×C₅₃×#436	2
C₄×#124×C₅₃	2
C₅₃×#1984	2
C₅₃×#1954	2
C₅₃×#1923	2
C₅₃×#1851	2
C₅₃×#1842	2
C₅₃×#1744	2
#22210	53

Subgroups — C53×#22210 #27229722

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅₃×#22210 #27229722