Subgroups — C₅₃×#23371 #27230883

Order 13568.

604 conjugacy classes of subgroups.

The subgroup lattice has 376 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×#125×C₅₃	C₂	1	1	yes	maximal, characteristic
2	C₅₃×#1940	C₂	1	2	yes	maximal
3	C₅₃×#1939	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×#1831	C₂	1	1	yes	maximal, characteristic
9	C₅₃×#1831	C₂	1	1	yes	maximal, characteristic
10	C₂×#125×C₅₃	C₄	1	1	yes	characteristic
11	C₂×C₄⋊C₈×C₅₃	C₂²	1	1	yes	characteristic
12	C₂×C₄⋊C₈×C₅₃	C₂²	1	2	yes	—
13	C₂×C₄⋊C₈×C₅₃	C₂²	1	1	yes	characteristic
14	C₂×C₄⋊C₈×C₅₃	C₂²	1	2	yes	—
15	C₂×C₄⋊C₈×C₅₃	C₂²	1	1	yes	characteristic
16	C₂×C₄⋊C₈×C₅₃	C₂²	1	1	yes	characteristic
17	C₂×#97×C₅₃	C₂²	1	2	yes	—
18	C₂×C₄×C₈×C₅₃	C₂²	1	2	yes	—
19	C₅₃×#389	C₄	1	1	yes	characteristic
20	C₅₃×#389	C₄	1	1	yes	characteristic
21	C₅₃×#387	C₄	1	2	yes	—
22	C₅₃×#387	C₄	1	2	yes	—
23	C₅₃×#382	C₄	1	1	yes	characteristic
24	C₄×C₄⋊C₄×C₅₃	C₂²	1	1	yes	characteristic
25	C₄×C₄⋊C₄×C₅₃	C₂²	1	2	yes	—
26	C₄×C₄⋊C₄×C₅₃	C₂²	1	1	yes	characteristic
27	C₄×C₄⋊C₄×C₅₃	C₂²	1	2	yes	—
28	C₄³×C₅₃	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	2	yes	—
36	C₅₃×#336	C₂²	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	#125×C₅₃	—	2	4	no	—
42	C₂×C₄⋊C₄×C₅₃	C₂×C₄	1	2	yes	—
43	C₂×C₄⋊C₄×C₅₃	C₂×C₄	1	2	yes	—
44	C₂×C₄⋊C₄×C₅₃	C₂×C₄	1	1	yes	characteristic
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	1	yes	characteristic
48	C₂×C₄⋊C₄×C₅₃	C₂×C₄	1	1	yes	characteristic
49	C₂×C₄⋊C₄×C₅₃	C₂×C₄	1	1	yes	characteristic
50	C₂×C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
51	C₂×C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
52	C₂×C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
53	C₂×C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
54	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
55	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
56	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
57	C₂×C₄²×C₅₃	C₂³	1	2	yes	—
58	C₂×C₄²×C₅₃	C₂³	1	2	yes	—
59	C₄⋊C₈×C₅₃	—	4	2	no	—
60	C₄⋊C₈×C₅₃	—	4	2	no	—
61	C₄⋊C₈×C₅₃	—	4	1	no	—
62	C₄⋊C₈×C₅₃	—	4	1	no	—
63	C₄⋊C₈×C₅₃	—	4	1	no	—
64	C₄⋊C₈×C₅₃	—	4	1	no	—
65	#97×C₅₃	—	4	2	no	—
66	C₄×C₈×C₅₃	—	4	2	no	—
67	#95×C₅₃	C₂×C₄	1	2	yes	—
68	#95×C₅₃	C₂×C₄	1	2	yes	—
69	#95×C₅₃	C₂×C₄	1	2	yes	—
70	#95×C₅₃	C₂×C₄	1	2	yes	—
71	#95×C₅₃	C₂×C₄	1	2	yes	—
72	#95×C₅₃	C₂×C₄	1	2	yes	—
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	2	yes	—
76	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
77	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
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	1	yes	characteristic
81	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
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	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	—	2	2	no	—
94	C₂×C₈×C₅₃	—	2	2	no	—
95	C₂×C₈×C₅₃	—	2	2	no	—
96	C₄⋊C₄×C₅₃	—	2	4	no	—
97	C₄⋊C₄×C₅₃	—	2	4	no	—
98	C₄⋊C₄×C₅₃	—	2	2	no	—
99	C₄⋊C₄×C₅₃	—	2	4	no	—
100	C₄⋊C₄×C₅₃	—	2	4	no	—
101	C₄⋊C₄×C₅₃	—	4	1	no	—
102	C₄⋊C₄×C₅₃	—	4	1	no	—
103	C₄⋊C₄×C₅₃	M₁₆	1	4	yes	—
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	1	no	—
110	C₄²×C₅₃	—	2	1	no	—
111	C₄²×C₅₃	—	2	1	no	—
112	C₄²×C₅₃	—	2	1	no	—
113	C₄²×C₅₃	—	2	1	no	—
114	C₄²×C₅₃	—	2	1	no	—
115	C₄²×C₅₃	—	2	2	no	—
116	C₄²×C₅₃	—	2	2	no	—
117	C₄²×C₅₃	—	4	1	no	—
118	C₂³×C₅₃	C₂³×C₄	1	1	yes	characteristic
119	C₂×C₄×C₅₃	—	2	1	no	—
120	C₂×C₄×C₅₃	—	2	2	no	—
121	C₂×C₄×C₅₃	—	2	1	no	—
122	C₂×C₄×C₅₃	—	2	1	no	—
123	C₂×C₄×C₅₃	—	2	2	no	—
124	C₂×C₄×C₅₃	—	2	1	no	—
125	C₂×C₄×C₅₃	—	2	1	no	—
126	C₂×C₄×C₅₃	—	2	1	no	—
127	C₂×C₄×C₅₃	—	2	1	no	—
128	C₂×C₄×C₅₃	—	2	1	no	—
129	C₂×C₄×C₅₃	—	2	1	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	1	no	—
134	C₂×C₄×C₅₃	—	2	1	no	—
135	C₂×C₄×C₅₃	—	2	2	no	—
136	C₂×C₄×C₅₃	—	2	1	no	—
137	C₂×C₄×C₅₃	—	2	1	no	—
138	C₂×C₄×C₅₃	—	2	1	no	—
139	C₂×C₄×C₅₃	—	2	1	no	—
140	C₂×C₄×C₅₃	—	2	2	no	—
141	C₂×C₄×C₅₃	—	2	2	no	—
142	C₂×C₄×C₅₃	—	2	2	no	—
143	C₂×C₄×C₅₃	—	2	2	no	—
144	C₂×C₄×C₅₃	—	2	2	no	—
145	C₂×C₄×C₅₃	C₂×M₁₆	1	2	yes	—
146	C₂×C₄×C₅₃	C₂×M₁₆	1	4	yes	—
147	C₂×C₄×C₅₃	#131	1	2	yes	—
148	C₂×C₄×C₅₃	#131	1	2	yes	—
149	C₂×C₄×C₅₃	H(5,𝔽₂)	1	1	yes	characteristic
150	C₂×C₄×C₅₃	H(5,𝔽₂)	1	1	yes	characteristic
151	C₂×C₄×C₅₃	2₋¹⁺⁴	1	1	yes	characteristic
152	C₂×C₄×C₅₃	2₋¹⁺⁴	1	1	yes	characteristic
153	C₂×C₄×C₅₃	2₋¹⁺⁴	1	1	yes	characteristic
154	C₂×C₄×C₅₃	2₋¹⁺⁴	1	1	yes	characteristic
155	C₈×C₅₃	—	4	2	no	—
156	C₈×C₅₃	—	4	2	no	—
157	C₈×C₅₃	—	4	2	no	—
158	C₈×C₅₃	—	4	2	no	—
159	#23371	C₅₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
160	C₂²×C₅₃	#519	1	1	yes	characteristic
161	C₂²×C₅₃	#520	1	1	yes	characteristic
162	C₂²×C₅₃	#520	1	1	yes	characteristic
163	C₂²×C₅₃	C₂²×M₁₆	1	1	yes	characteristic
164	C₂²×C₅₃	C₂×#131	1	1	yes	characteristic
165	C₂²×C₅₃	C₂×#131	1	1	yes	characteristic
166	C₂²×C₅₃	#568	1	1	yes	characteristic
167	C₄×C₅₃	—	2	2	no	—
168	C₄×C₅₃	—	2	4	no	—
169	C₄×C₅₃	—	2	2	no	—
170	C₄×C₅₃	—	2	2	no	—
171	C₄×C₅₃	—	4	1	no	—
172	C₄×C₅₃	—	4	1	no	—
173	C₄×C₅₃	—	4	1	no	—
174	C₄×C₅₃	—	4	2	no	—
175	C₄×C₅₃	—	4	2	no	—
176	C₄×C₅₃	—	4	2	no	—
177	C₄×C₅₃	#562	1	1	yes	characteristic
178	C₄×C₅₃	#563	1	1	yes	characteristic
179	C₄×C₅₃	#563	1	1	yes	characteristic
180	C₄×C₅₃	#564	1	1	yes	characteristic
181	C₄×#125	C₂×C₅₃	1	1	yes	characteristic
182	#1940	C₂×C₅₃	1	2	yes	—
183	#1939	C₂×C₅₃	1	2	yes	—
184	#1917	C₂×C₅₃	1	2	yes	—
185	#1917	C₂×C₅₃	1	2	yes	—
186	#1916	C₂×C₅₃	1	2	yes	—
187	#1916	C₂×C₅₃	1	2	yes	—
188	#1831	C₂×C₅₃	1	1	yes	characteristic
189	#1831	C₂×C₅₃	1	1	yes	characteristic
190	C₂×C₅₃	#2382	1	1	yes	characteristic
191	C₂×C₅₃	#2983	1	1	yes	characteristic
192	C₂×C₅₃	#2986	1	1	yes	characteristic
193	C₂×C₅₃	#2990	1	1	yes	characteristic
194	C₂×C₅₃	#2991	1	1	yes	characteristic
195	C₂×C₅₃	#2992	1	1	yes	characteristic
196	C₂×C₅₃	#2992	1	1	yes	characteristic
197	C₂×#125	C₄×C₅₃	1	1	yes	characteristic
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	1	yes	characteristic
201	C₂×C₄⋊C₈	C₂²×C₅₃	1	1	yes	characteristic
202	C₂×C₄⋊C₈	C₂²×C₅₃	1	2	yes	—
203	C₂×C₄⋊C₈	C₂²×C₅₃	1	2	yes	—
204	C₂×#97	C₂²×C₅₃	1	2	yes	—
205	C₂×C₄×C₈	C₂²×C₅₃	1	2	yes	—
206	#389	C₄×C₅₃	1	1	yes	characteristic
207	#389	C₄×C₅₃	1	1	yes	characteristic
208	#387	C₄×C₅₃	1	2	yes	—
209	#387	C₄×C₅₃	1	2	yes	—
210	#382	C₄×C₅₃	1	1	yes	characteristic
211	C₄×C₄⋊C₄	C₂²×C₅₃	1	1	yes	characteristic
212	C₄×C₄⋊C₄	C₂²×C₅₃	1	2	yes	—
213	C₄×C₄⋊C₄	C₂²×C₅₃	1	1	yes	characteristic
214	C₄×C₄⋊C₄	C₂²×C₅₃	1	2	yes	—
215	C₄³	C₂²×C₅₃	1	1	yes	characteristic
216	#336	C₂²×C₅₃	1	2	yes	—
217	#336	C₂²×C₅₃	1	2	yes	—
218	#336	C₂²×C₅₃	1	2	yes	—
219	#336	C₂²×C₅₃	1	2	yes	—
220	#336	C₂²×C₅₃	1	2	yes	—
221	#336	C₂²×C₅₃	1	2	yes	—
222	#336	C₂²×C₅₃	1	2	yes	—
223	#336	C₂²×C₅₃	1	2	yes	—
224	C₅₃	#23371	1	1	yes	characteristic, direct product factor, Sylow
225	C₂²×C₈	C₂³×C₅₃	1	2	yes	—
226	C₂²×C₈	C₂³×C₅₃	1	2	yes	—
227	C₂²×C₈	C₂³×C₅₃	1	2	yes	—
228	C₂²×C₈	C₂³×C₅₃	1	2	yes	—
229	#125	—	2	4	no	—
230	C₂×C₄⋊C₄	C₂×C₄×C₅₃	1	2	yes	—
231	C₂×C₄⋊C₄	C₂×C₄×C₅₃	1	2	yes	—
232	C₂×C₄⋊C₄	C₂×C₄×C₅₃	1	1	yes	characteristic
233	C₂×C₄⋊C₄	C₂×C₄×C₅₃	1	2	yes	—
234	C₂×C₄⋊C₄	C₂×C₄×C₅₃	1	2	yes	—
235	C₂×C₄⋊C₄	C₂×C₄×C₅₃	1	1	yes	characteristic
236	C₂×C₄⋊C₄	C₂×C₄×C₅₃	1	1	yes	characteristic
237	C₂×C₄⋊C₄	C₂×C₄×C₅₃	1	1	yes	characteristic
238	C₂×C₄²	C₂×C₄×C₅₃	1	1	yes	characteristic
239	C₂×C₄²	C₂×C₄×C₅₃	1	1	yes	characteristic
240	C₂×C₄²	C₂×C₄×C₅₃	1	1	yes	characteristic
241	C₂×C₄²	C₂×C₄×C₅₃	1	1	yes	characteristic
242	C₂×C₄²	C₂³×C₅₃	1	1	yes	characteristic
243	C₂×C₄²	C₂³×C₅₃	1	1	yes	characteristic
244	C₂×C₄²	C₂³×C₅₃	1	1	yes	characteristic
245	C₂×C₄²	C₂³×C₅₃	1	2	yes	—
246	C₂×C₄²	C₂³×C₅₃	1	2	yes	—
247	C₄⋊C₈	—	4	2	no	—
248	C₄⋊C₈	—	4	2	no	—
249	C₄⋊C₈	—	4	1	no	—
250	C₄⋊C₈	—	4	1	no	—
251	C₄⋊C₈	—	4	1	no	—
252	C₄⋊C₈	—	4	1	no	—
253	#97	—	4	2	no	—
254	C₄×C₈	—	4	2	no	—
255	#95	C₂×C₄×C₅₃	1	2	yes	—
256	#95	C₂×C₄×C₅₃	1	2	yes	—
257	#95	C₂×C₄×C₅₃	1	2	yes	—
258	#95	C₂×C₄×C₅₃	1	2	yes	—
259	#95	C₂×C₄×C₅₃	1	2	yes	—
260	#95	C₂×C₄×C₅₃	1	2	yes	—
261	C₂²×C₄	C₂²×C₄×C₅₃	1	1	yes	characteristic
262	C₂²×C₄	C₂²×C₄×C₅₃	1	2	yes	—
263	C₂²×C₄	C₂²×C₄×C₅₃	1	2	yes	—
264	C₂²×C₄	C₂²×C₄×C₅₃	1	1	yes	characteristic
265	C₂²×C₄	C₂²×C₄×C₅₃	1	1	yes	characteristic
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	1	yes	characteristic
269	C₂²×C₄	C₂²×C₄×C₅₃	1	1	yes	characteristic
270	C₂²×C₄	C₂²×C₄×C₅₃	1	1	yes	characteristic
271	C₂²×C₄	C₂⁴×C₅₃	1	1	yes	characteristic
272	C₂×C₈	—	2	2	no	—
273	C₂×C₈	—	2	2	no	—
274	C₂×C₈	—	2	2	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	2	no	—
282	C₂×C₈	—	2	2	no	—
283	C₂×C₈	—	2	2	no	—
284	C₄⋊C₄	—	2	4	no	—
285	C₄⋊C₄	—	2	4	no	—
286	C₄⋊C₄	—	2	2	no	—
287	C₄⋊C₄	—	2	4	no	—
288	C₄⋊C₄	—	2	4	no	—
289	C₄⋊C₄	—	4	1	no	—
290	C₄⋊C₄	—	4	1	no	—
291	C₄⋊C₄	M₁₆×C₅₃	1	4	yes	—
292	C₄²	—	2	2	no	—
293	C₄²	—	2	2	no	—
294	C₄²	—	2	2	no	—
295	C₄²	—	2	2	no	—
296	C₄²	—	2	2	no	—
297	C₄²	—	2	1	no	—
298	C₄²	—	2	1	no	—
299	C₄²	—	2	1	no	—
300	C₄²	—	2	1	no	—
301	C₄²	—	2	1	no	—
302	C₄²	—	2	1	no	—
303	C₄²	—	2	2	no	—
304	C₄²	—	2	2	no	—
305	C₄²	—	4	1	no	—
306	C₂³	C₂³×C₄×C₅₃	1	1	yes	characteristic
307	C₂×C₄	—	2	1	no	—
308	C₂×C₄	—	2	2	no	—
309	C₂×C₄	—	2	1	no	—
310	C₂×C₄	—	2	1	no	—
311	C₂×C₄	—	2	2	no	—
312	C₂×C₄	—	2	1	no	—
313	C₂×C₄	—	2	1	no	—
314	C₂×C₄	—	2	1	no	—
315	C₂×C₄	—	2	1	no	—
316	C₂×C₄	—	2	1	no	—
317	C₂×C₄	—	2	2	no	—
318	C₂×C₄	—	2	1	no	—
319	C₂×C₄	—	2	2	no	—
320	C₂×C₄	—	2	2	no	—
321	C₂×C₄	—	2	1	no	—
322	C₂×C₄	—	2	1	no	—
323	C₂×C₄	—	2	2	no	—
324	C₂×C₄	—	2	1	no	—
325	C₂×C₄	—	2	1	no	—
326	C₂×C₄	—	2	1	no	—
327	C₂×C₄	—	2	1	no	—
328	C₂×C₄	—	2	2	no	—
329	C₂×C₄	—	2	2	no	—
330	C₂×C₄	—	2	2	no	—
331	C₂×C₄	—	2	2	no	—
332	C₂×C₄	—	2	2	no	—
333	C₂×C₄	C₂×M₁₆×C₅₃	1	2	yes	—
334	C₂×C₄	C₂×M₁₆×C₅₃	1	4	yes	—
335	C₂×C₄	#131×C₅₃	1	2	yes	—
336	C₂×C₄	#131×C₅₃	1	2	yes	—
337	C₂×C₄	H(5,𝔽₂)×C₅₃	1	1	yes	characteristic
338	C₂×C₄	H(5,𝔽₂)×C₅₃	1	1	yes	characteristic
339	C₂×C₄	2₋¹⁺⁴×C₅₃	1	1	yes	characteristic
340	C₂×C₄	2₋¹⁺⁴×C₅₃	1	1	yes	characteristic
341	C₂×C₄	2₋¹⁺⁴×C₅₃	1	1	yes	characteristic
342	C₂×C₄	2₋¹⁺⁴×C₅₃	1	1	yes	characteristic
343	C₈	—	4	2	no	—
344	C₈	—	4	2	no	—
345	C₈	—	4	2	no	—
346	C₈	—	4	2	no	—
347	C₂²	C₅₃×#519	1	1	yes	characteristic
348	C₂²	C₅₃×#520	1	1	yes	characteristic
349	C₂²	C₅₃×#520	1	1	yes	characteristic
350	C₂²	C₂²×M₁₆×C₅₃	1	1	yes	characteristic
351	C₂²	C₂×#131×C₅₃	1	1	yes	characteristic
352	C₂²	C₂×#131×C₅₃	1	1	yes	characteristic
353	C₂²	C₅₃×#568	1	1	yes	characteristic
354	C₄	—	2	2	no	—
355	C₄	—	2	4	no	—
356	C₄	—	2	2	no	—
357	C₄	—	2	2	no	—
358	C₄	—	4	1	no	—
359	C₄	—	4	1	no	—
360	C₄	—	4	1	no	—
361	C₄	—	4	2	no	—
362	C₄	—	4	2	no	—
363	C₄	—	4	2	no	—
364	C₄	C₅₃×#562	1	1	yes	characteristic
365	C₄	C₅₃×#563	1	1	yes	characteristic
366	C₄	C₅₃×#563	1	1	yes	characteristic
367	C₄	C₅₃×#564	1	1	yes	characteristic
368	C₂	C₅₃×#2382	1	1	yes	characteristic
369	C₂	C₅₃×#2983	1	1	yes	characteristic
370	C₂	C₅₃×#2986	1	1	yes	characteristic
371	C₂	C₅₃×#2990	1	1	yes	characteristic
372	C₂	C₅₃×#2991	1	1	yes	characteristic
373	C₂	C₅₃×#2992	1	1	yes	characteristic
374	C₂	C₅₃×#2992	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
159. #23371	C₅₃	direct (N × H)
224. C₅₃	#23371	direct (N × H)

Maximal subgroups

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

subgroup	index
C₄×#125×C₅₃	2
C₅₃×#1940	2
C₅₃×#1939	2
C₅₃×#1917	2
C₅₃×#1916	2
C₅₃×#1831	2
#23371	53

Subgroups — C53×#23371 #27230883

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅₃×#23371 #27230883