Subgroups — C₅₃×#21027 #27228539

Order 13568.

906 conjugacy classes of subgroups.

The subgroup lattice has 370 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₅₃×#423	C₂	1	1	yes	maximal, characteristic
2	C₂×C₅₃×#405	C₂	1	1	yes	maximal, characteristic
3	C₅₃×#2296	C₂	1	1	yes	maximal, characteristic
4	C₅₃×#1917	C₂	1	4	yes	maximal, complemented
5	C₅₃×#1916	C₂	1	4	yes	maximal, complemented
6	C₅₃×#1729	C₂	1	1	yes	maximal, characteristic
7	C₅₃×#1729	C₂	1	1	yes	maximal, characteristic
8	C₅₃×#1729	C₂	1	1	yes	maximal, characteristic
9	C₂×C₅₃×#336	C₂	1	1	yes	maximal, characteristic
10	C₂²×M₁₆×C₅₃	C₂²	1	1	yes	characteristic
11	C₂²×M₁₆×C₅₃	C₂²	1	1	yes	characteristic
12	C₂³×C₈×C₅₃	C₂²	1	1	yes	characteristic
13	C₂³×C₈×C₅₃	C₂²	1	1	yes	characteristic
14	C₂×#117×C₅₃	C₂²	1	1	yes	characteristic
15	C₂×#117×C₅₃	C₂²	1	1	yes	characteristic
16	C₂²×C₄²×C₅₃	C₂²	1	1	yes	characteristic
17	C₅₃×#423	—	2	4	no	—
18	C₂×C₄⋊C₈×C₅₃	C₂²	1	4	yes	—
19	C₅₃×#405	—	2	4	no	—
20	C₂×#97×C₅₃	C₂²	1	2	yes	—
21	C₂×C₄×C₈×C₅₃	C₂²	1	2	yes	—
22	C₅₃×#389	C₄	1	2	yes	—
23	C₅₃×#389	C₄	1	2	yes	—
24	C₅₃×#385	C₄	1	4	yes	—
25	C₄×C₄⋊C₄×C₅₃	C₂²	1	4	yes	—
26	C₅₃×#336	C₂²	1	2	yes	—
27	C₅₃×#336	C₂²	1	2	yes	—
28	C₅₃×#336	C₂²	1	4	yes	—
29	C₅₃×#336	C₂²	1	4	yes	—
30	C₅₃×#336	C₂²	1	2	yes	—
31	C₅₃×#336	C₂²	1	2	yes	—
32	C₂³×C₄×C₅₃	C₂×C₄	1	2	yes	—
33	C₂³×C₄×C₅₃	C₂³	1	1	yes	characteristic
34	C₂×M₁₆×C₅₃	—	2	2	no	—
35	C₂×M₁₆×C₅₃	—	2	2	no	—
36	C₂×M₁₆×C₅₃	—	2	2	no	—
37	C₂×M₁₆×C₅₃	—	2	2	no	—
38	C₂×M₁₆×C₅₃	—	2	2	no	—
39	C₂×M₁₆×C₅₃	D₄	1	4	yes	—
40	C₂²×C₈×C₅₃	—	2	2	no	—
41	C₂²×C₈×C₅₃	—	2	2	no	—
42	C₂²×C₈×C₅₃	—	2	2	no	—
43	C₂²×C₈×C₅₃	—	2	2	no	—
44	C₂²×C₈×C₅₃	—	2	2	no	—
45	C₂²×C₈×C₅₃	Q₈	1	4	yes	—
46	C₂²×C₈×C₅₃	C₂³	1	2	yes	—
47	C₂²×C₈×C₅₃	C₂³	1	2	yes	—
48	C₂²×C₈×C₅₃	C₂³	1	2	yes	—
49	C₂²×C₈×C₅₃	C₂³	1	1	yes	characteristic
50	C₂²×C₈×C₅₃	C₂³	1	1	yes	characteristic
51	#117×C₅₃	—	2	2	no	—
52	#117×C₅₃	—	2	2	no	—
53	#117×C₅₃	—	2	2	no	—
54	#117×C₅₃	—	2	2	no	—
55	C₂×C₄⋊C₄×C₅₃	C₂×C₄	1	2	yes	—
56	C₂×C₄⋊C₄×C₅₃	C₂×C₄	1	2	yes	—
57	C₂×C₄⋊C₄×C₅₃	C₂×C₄	1	4	yes	—
58	C₂×C₄⋊C₄×C₅₃	C₂×C₄	1	4	yes	—
59	C₂×C₂²⋊C₄×C₅₃	C₂×C₄	1	2	yes	—
60	C₂×C₂²⋊C₄×C₅₃	C₂×C₄	1	2	yes	—
61	C₂×C₄²×C₅₃	—	2	2	no	—
62	C₂×C₄²×C₅₃	—	2	2	no	—
63	C₂×C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
64	C₂×C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
65	C₂×C₄²×C₅₃	C₂³	1	2	yes	—
66	C₂×C₄²×C₅₃	C₂³	1	2	yes	—
67	C₂×C₄²×C₅₃	C₂³	1	2	yes	—
68	C₄⋊C₈×C₅₃	—	2	8	no	—
69	#97×C₅₃	—	2	4	no	—
70	C₄×C₈×C₅₃	—	2	4	no	—
71	#95×C₅₃	C₂×C₄	1	4	yes	—
72	#95×C₅₃	C₂×C₄	1	4	yes	—
73	C₂⁴×C₅₃	C₂²×C₄	1	1	yes	characteristic
74	C₂²×C₄×C₅₃	—	2	4	no	—
75	C₂²×C₄×C₅₃	—	2	4	no	—
76	C₂²×C₄×C₅₃	—	4	2	no	—
77	C₂²×C₄×C₅₃	C₂²×C₄	1	2	yes	—
78	C₂²×C₄×C₅₃	C₂²×C₄	1	4	yes	—
79	C₂²×C₄×C₅₃	C₂²×C₄	1	4	yes	—
80	C₂²×C₄×C₅₃	C₂²×C₄	1	2	yes	—
81	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
82	C₂²×C₄×C₅₃	C₂×D₄	1	2	yes	—
83	C₂²×C₄×C₅₃	C₂×Q₈	1	2	yes	—
84	C₂²×C₄×C₅₃	#41	1	2	yes	—
85	C₂²×C₄×C₅₃	#41	1	2	yes	—
86	C₂²×C₄×C₅₃	#41	1	2	yes	—
87	C₂²×C₄×C₅₃	#41	1	2	yes	—
88	C₂²×C₄×C₅₃	C₂⁴	1	1	yes	characteristic
89	M₁₆×C₅₃	—	2	8	no	—
90	M₁₆×C₅₃	—	4	4	no	—
91	C₂×C₈×C₅₃	—	2	8	no	—
92	C₂×C₈×C₅₃	—	2	1	no	—
93	C₂×C₈×C₅₃	—	2	1	no	—
94	C₂×C₈×C₅₃	—	2	1	no	—
95	C₂×C₈×C₅₃	—	2	1	no	—
96	C₂×C₈×C₅₃	—	2	2	no	—
97	C₂×C₈×C₅₃	—	2	2	no	—
98	C₂×C₈×C₅₃	—	2	2	no	—
99	C₂×C₈×C₅₃	—	2	2	no	—
100	C₂×C₈×C₅₃	—	2	2	no	—
101	C₂×C₈×C₅₃	—	2	2	no	—
102	C₂×C₈×C₅₃	—	2	2	no	—
103	C₂×C₈×C₅₃	—	2	2	no	—
104	C₂×C₈×C₅₃	—	4	4	no	—
105	C₂×C₈×C₅₃	C₂×D₄	1	4	yes	—
106	C₂×C₈×C₅₃	C₂×Q₈	1	2	yes	—
107	C₂×C₈×C₅₃	C₂×Q₈	1	2	yes	—
108	C₄⋊C₄×C₅₃	—	2	4	no	—
109	C₄⋊C₄×C₅₃	—	2	4	no	—
110	C₄⋊C₄×C₅₃	—	2	8	no	—
111	C₄⋊C₄×C₅₃	—	4	4	no	—
112	C₂²⋊C₄×C₅₃	—	2	4	no	—
113	C₂²⋊C₄×C₅₃	—	2	4	no	—
114	C₄²×C₅₃	—	2	2	no	—
115	C₄²×C₅₃	—	2	2	no	—
116	C₄²×C₅₃	—	2	2	no	—
117	C₄²×C₅₃	—	2	2	no	—
118	C₄²×C₅₃	—	2	2	no	—
119	C₄²×C₅₃	—	2	2	no	—
120	C₄²×C₅₃	—	2	2	no	—
121	C₄²×C₅₃	—	2	2	no	—
122	C₂³×C₅₃	—	2	2	no	—
123	C₂³×C₅₃	—	2	2	no	—
124	C₂³×C₅₃	#117	1	2	yes	—
125	C₂³×C₅₃	C₄×D₄	1	2	yes	—
126	C₂³×C₅₃	C₄×Q₈	1	2	yes	—
127	C₂³×C₅₃	C₂³×C₄	1	1	yes	characteristic
128	C₂×C₄×C₅₃	—	2	2	no	—
129	C₂×C₄×C₅₃	—	2	2	no	—
130	C₂×C₄×C₅₃	—	2	4	no	—
131	C₂×C₄×C₅₃	—	2	2	no	—
132	C₂×C₄×C₅₃	—	2	4	no	—
133	C₂×C₄×C₅₃	—	2	2	no	—
134	C₂×C₄×C₅₃	—	2	4	no	—
135	C₂×C₄×C₅₃	—	2	4	no	—
136	C₂×C₄×C₅₃	—	2	2	no	—
137	C₂×C₄×C₅₃	—	2	4	no	—
138	C₂×C₄×C₅₃	—	2	4	no	—
139	C₂×C₄×C₅₃	—	4	8	no	—
140	C₂×C₄×C₅₃	#117	1	2	yes	—
141	C₂×C₄×C₅₃	C₄×D₄	1	2	yes	—
142	C₂×C₄×C₅₃	C₄×Q₈	1	2	yes	—
143	C₂×C₄×C₅₃	C₂²⋊Q₈	1	4	yes	—
144	C₂×C₄×C₅₃	#123	1	4	yes	—
145	C₂×C₄×C₅₃	#125	1	4	yes	—
146	C₂×C₄×C₅₃	#126	1	4	yes	—
147	C₂×C₄×C₅₃	#131	1	4	yes	—
148	C₂×C₄×C₅₃	C₂²×D₄	1	1	yes	characteristic
149	C₂×C₄×C₅₃	C₂²×Q₈	1	1	yes	characteristic
150	C₂×C₄×C₅₃	C₂×#41	1	1	yes	characteristic
151	C₂×C₄×C₅₃	C₂×#41	1	1	yes	characteristic
152	C₂×C₄×C₅₃	C₂×#41	1	1	yes	characteristic
153	C₂×C₄×C₅₃	C₂×#41	1	1	yes	characteristic
154	C₈×C₅₃	—	2	2	no	—
155	C₈×C₅₃	—	2	2	no	—
156	C₈×C₅₃	—	2	4	no	—
157	C₈×C₅₃	—	4	2	no	—
158	C₈×C₅₃	—	4	2	no	—
159	#21027	C₅₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
160	C₂²×C₅₃	—	2	4	no	—
161	C₂²×C₅₃	—	2	4	no	—
162	C₂²×C₅₃	—	2	2	no	—
163	C₂²×C₅₃	—	2	2	no	—
164	C₂²×C₅₃	#387	1	4	yes	—
165	C₂²×C₅₃	C₂×#117	1	1	yes	characteristic
166	C₂²×C₅₃	C₂×C₄×D₄	1	1	yes	characteristic
167	C₂²×C₅₃	C₂×C₄×Q₈	1	1	yes	characteristic
168	C₂²×C₅₃	C₂×#131	1	2	yes	—
169	C₂²×C₅₃	#568	1	2	yes	—
170	C₄×C₅₃	—	2	4	no	—
171	C₄×C₅₃	—	4	4	no	—
172	C₄×C₅₃	—	4	4	no	—
173	C₄×C₅₃	—	4	2	no	—
174	C₄×C₅₃	#387	1	4	yes	—
175	C₄×C₅₃	C₂×C₂²⋊Q₈	1	1	yes	characteristic
176	C₄×C₅₃	C₂×#123	1	1	yes	characteristic
177	C₄×C₅₃	C₂×#125	1	1	yes	characteristic
178	C₄×C₅₃	C₂×#126	1	1	yes	characteristic
179	C₂×#423	C₂×C₅₃	1	1	yes	characteristic
180	C₂×#405	C₂×C₅₃	1	1	yes	characteristic
181	#2296	C₂×C₅₃	1	1	yes	characteristic
182	#1917	C₂×C₅₃	1	4	yes	complemented
183	#1916	C₂×C₅₃	1	4	yes	complemented
184	#1729	C₂×C₅₃	1	1	yes	characteristic
185	#1729	C₂×C₅₃	1	1	yes	characteristic
186	#1729	C₂×C₅₃	1	1	yes	characteristic
187	C₂×#336	C₂×C₅₃	1	1	yes	characteristic
188	C₂×C₅₃	—	2	4	no	—
189	C₂×C₅₃	C₂×#387	1	1	yes	characteristic
190	C₂×C₅₃	#2925	1	2	yes	—
191	C₂×C₅₃	#2933	1	2	yes	—
192	C₂×C₅₃	#2960	1	1	yes	characteristic
193	C₂×C₅₃	#2961	1	1	yes	characteristic
194	C₂²×M₁₆	C₂²×C₅₃	1	1	yes	characteristic
195	C₂²×M₁₆	C₂²×C₅₃	1	1	yes	characteristic
196	C₂³×C₈	C₂²×C₅₃	1	1	yes	characteristic
197	C₂³×C₈	C₂²×C₅₃	1	1	yes	characteristic
198	C₂×#117	C₂²×C₅₃	1	1	yes	characteristic
199	C₂×#117	C₂²×C₅₃	1	1	yes	characteristic
200	C₂²×C₄²	C₂²×C₅₃	1	1	yes	characteristic
201	#423	—	2	4	no	—
202	C₂×C₄⋊C₈	C₂²×C₅₃	1	4	yes	—
203	#405	—	2	4	no	—
204	C₂×#97	C₂²×C₅₃	1	2	yes	—
205	C₂×C₄×C₈	C₂²×C₅₃	1	2	yes	—
206	#389	C₄×C₅₃	1	2	yes	—
207	#389	C₄×C₅₃	1	2	yes	—
208	#385	C₄×C₅₃	1	4	yes	—
209	C₄×C₄⋊C₄	C₂²×C₅₃	1	4	yes	—
210	#336	C₂²×C₅₃	1	2	yes	—
211	#336	C₂²×C₅₃	1	2	yes	—
212	#336	C₂²×C₅₃	1	4	yes	—
213	#336	C₂²×C₅₃	1	4	yes	—
214	#336	C₂²×C₅₃	1	2	yes	—
215	#336	C₂²×C₅₃	1	2	yes	—
216	C₅₃	#21027	1	1	yes	characteristic, direct product factor, Sylow
217	C₂³×C₄	C₂×C₄×C₅₃	1	2	yes	—
218	C₂³×C₄	C₂³×C₅₃	1	1	yes	characteristic
219	C₂×M₁₆	—	2	2	no	—
220	C₂×M₁₆	—	2	2	no	—
221	C₂×M₁₆	—	2	2	no	—
222	C₂×M₁₆	—	2	2	no	—
223	C₂×M₁₆	—	2	2	no	—
224	C₂×M₁₆	D₄×C₅₃	1	4	yes	—
225	C₂²×C₈	—	2	2	no	—
226	C₂²×C₈	—	2	2	no	—
227	C₂²×C₈	—	2	2	no	—
228	C₂²×C₈	—	2	2	no	—
229	C₂²×C₈	—	2	2	no	—
230	C₂²×C₈	Q₈×C₅₃	1	4	yes	—
231	C₂²×C₈	C₂³×C₅₃	1	2	yes	—
232	C₂²×C₈	C₂³×C₅₃	1	2	yes	—
233	C₂²×C₈	C₂³×C₅₃	1	2	yes	—
234	C₂²×C₈	C₂³×C₅₃	1	1	yes	characteristic
235	C₂²×C₈	C₂³×C₅₃	1	1	yes	characteristic
236	#117	—	2	2	no	—
237	#117	—	2	2	no	—
238	#117	—	2	2	no	—
239	#117	—	2	2	no	—
240	C₂×C₄⋊C₄	C₂×C₄×C₅₃	1	2	yes	—
241	C₂×C₄⋊C₄	C₂×C₄×C₅₃	1	2	yes	—
242	C₂×C₄⋊C₄	C₂×C₄×C₅₃	1	4	yes	—
243	C₂×C₄⋊C₄	C₂×C₄×C₅₃	1	4	yes	—
244	C₂×C₂²⋊C₄	C₂×C₄×C₅₃	1	2	yes	—
245	C₂×C₂²⋊C₄	C₂×C₄×C₅₃	1	2	yes	—
246	C₂×C₄²	—	2	2	no	—
247	C₂×C₄²	—	2	2	no	—
248	C₂×C₄²	C₂×C₄×C₅₃	1	1	yes	characteristic
249	C₂×C₄²	C₂×C₄×C₅₃	1	1	yes	characteristic
250	C₂×C₄²	C₂³×C₅₃	1	2	yes	—
251	C₂×C₄²	C₂³×C₅₃	1	2	yes	—
252	C₂×C₄²	C₂³×C₅₃	1	2	yes	—
253	C₄⋊C₈	—	2	8	no	—
254	#97	—	2	4	no	—
255	C₄×C₈	—	2	4	no	—
256	#95	C₂×C₄×C₅₃	1	4	yes	—
257	#95	C₂×C₄×C₅₃	1	4	yes	—
258	C₂⁴	C₂²×C₄×C₅₃	1	1	yes	characteristic
259	C₂²×C₄	—	2	4	no	—
260	C₂²×C₄	—	2	4	no	—
261	C₂²×C₄	—	4	2	no	—
262	C₂²×C₄	C₂²×C₄×C₅₃	1	2	yes	—
263	C₂²×C₄	C₂²×C₄×C₅₃	1	4	yes	—
264	C₂²×C₄	C₂²×C₄×C₅₃	1	4	yes	—
265	C₂²×C₄	C₂²×C₄×C₅₃	1	2	yes	—
266	C₂²×C₄	C₂²×C₄×C₅₃	1	1	yes	characteristic
267	C₂²×C₄	C₂×D₄×C₅₃	1	2	yes	—
268	C₂²×C₄	C₂×Q₈×C₅₃	1	2	yes	—
269	C₂²×C₄	#41×C₅₃	1	2	yes	—
270	C₂²×C₄	#41×C₅₃	1	2	yes	—
271	C₂²×C₄	#41×C₅₃	1	2	yes	—
272	C₂²×C₄	#41×C₅₃	1	2	yes	—
273	C₂²×C₄	C₂⁴×C₅₃	1	1	yes	characteristic
274	M₁₆	—	2	8	no	—
275	M₁₆	—	4	4	no	—
276	C₂×C₈	—	2	8	no	—
277	C₂×C₈	—	2	1	no	—
278	C₂×C₈	—	2	1	no	—
279	C₂×C₈	—	2	1	no	—
280	C₂×C₈	—	2	1	no	—
281	C₂×C₈	—	2	2	no	—
282	C₂×C₈	—	2	2	no	—
283	C₂×C₈	—	2	2	no	—
284	C₂×C₈	—	2	2	no	—
285	C₂×C₈	—	2	2	no	—
286	C₂×C₈	—	2	2	no	—
287	C₂×C₈	—	2	2	no	—
288	C₂×C₈	—	2	2	no	—
289	C₂×C₈	—	4	4	no	—
290	C₂×C₈	C₂×D₄×C₅₃	1	4	yes	—
291	C₂×C₈	C₂×Q₈×C₅₃	1	2	yes	—
292	C₂×C₈	C₂×Q₈×C₅₃	1	2	yes	—
293	C₄⋊C₄	—	2	4	no	—
294	C₄⋊C₄	—	2	4	no	—
295	C₄⋊C₄	—	2	8	no	—
296	C₄⋊C₄	—	4	4	no	—
297	C₂²⋊C₄	—	2	4	no	—
298	C₂²⋊C₄	—	2	4	no	—
299	C₄²	—	2	2	no	—
300	C₄²	—	2	2	no	—
301	C₄²	—	2	2	no	—
302	C₄²	—	2	2	no	—
303	C₄²	—	2	2	no	—
304	C₄²	—	2	2	no	—
305	C₄²	—	2	2	no	—
306	C₄²	—	2	2	no	—
307	C₂³	—	2	2	no	—
308	C₂³	—	2	2	no	—
309	C₂³	#117×C₅₃	1	2	yes	—
310	C₂³	C₄×D₄×C₅₃	1	2	yes	—
311	C₂³	C₄×Q₈×C₅₃	1	2	yes	—
312	C₂³	C₂³×C₄×C₅₃	1	1	yes	characteristic
313	C₂×C₄	—	2	2	no	—
314	C₂×C₄	—	2	2	no	—
315	C₂×C₄	—	2	4	no	—
316	C₂×C₄	—	2	2	no	—
317	C₂×C₄	—	2	4	no	—
318	C₂×C₄	—	2	4	no	—
319	C₂×C₄	—	2	4	no	—
320	C₂×C₄	—	2	2	no	—
321	C₂×C₄	—	2	2	no	—
322	C₂×C₄	—	2	4	no	—
323	C₂×C₄	—	2	4	no	—
324	C₂×C₄	—	4	8	no	—
325	C₂×C₄	#117×C₅₃	1	2	yes	—
326	C₂×C₄	C₄×D₄×C₅₃	1	2	yes	—
327	C₂×C₄	C₄×Q₈×C₅₃	1	2	yes	—
328	C₂×C₄	C₂²⋊Q₈×C₅₃	1	4	yes	—
329	C₂×C₄	#123×C₅₃	1	4	yes	—
330	C₂×C₄	#125×C₅₃	1	4	yes	—
331	C₂×C₄	#126×C₅₃	1	4	yes	—
332	C₂×C₄	#131×C₅₃	1	4	yes	—
333	C₂×C₄	C₂²×D₄×C₅₃	1	1	yes	characteristic
334	C₂×C₄	C₂²×Q₈×C₅₃	1	1	yes	characteristic
335	C₂×C₄	C₂×#41×C₅₃	1	1	yes	characteristic
336	C₂×C₄	C₂×#41×C₅₃	1	1	yes	characteristic
337	C₂×C₄	C₂×#41×C₅₃	1	1	yes	characteristic
338	C₂×C₄	C₂×#41×C₅₃	1	1	yes	characteristic
339	C₈	—	2	2	no	—
340	C₈	—	2	2	no	—
341	C₈	—	2	4	no	—
342	C₈	—	4	2	no	—
343	C₈	—	4	2	no	—
344	C₂²	—	2	2	no	—
345	C₂²	—	2	2	no	—
346	C₂²	—	2	4	no	—
347	C₂²	—	2	4	no	—
348	C₂²	C₅₃×#387	1	4	yes	—
349	C₂²	C₂×#117×C₅₃	1	1	yes	characteristic
350	C₂²	C₂×C₄×D₄×C₅₃	1	1	yes	characteristic
351	C₂²	C₂×C₄×Q₈×C₅₃	1	1	yes	characteristic
352	C₂²	C₂×#131×C₅₃	1	2	yes	—
353	C₂²	C₅₃×#568	1	2	yes	—
354	C₄	—	2	4	no	—
355	C₄	—	4	4	no	—
356	C₄	—	4	2	no	—
357	C₄	—	4	4	no	—
358	C₄	C₅₃×#387	1	4	yes	—
359	C₄	C₂×C₂²⋊Q₈×C₅₃	1	1	yes	characteristic
360	C₄	C₂×#123×C₅₃	1	1	yes	characteristic
361	C₄	C₂×#125×C₅₃	1	1	yes	characteristic
362	C₄	C₂×#126×C₅₃	1	1	yes	characteristic
363	C₂	—	2	4	no	—
364	C₂	C₂×C₅₃×#387	1	1	yes	characteristic
365	C₂	C₅₃×#2925	1	2	yes	—
366	C₂	C₅₃×#2933	1	2	yes	—
367	C₂	C₅₃×#2960	1	1	yes	characteristic
368	C₂	C₅₃×#2961	1	1	yes	characteristic

	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	C₂⁴×C₅₃	C₂²×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
4. C₅₃×#1917	C₂	semidirect
5. C₅₃×#1916	C₂	semidirect
159. #21027	C₅₃	direct (N × H)
182. #1917	C₂×C₅₃	semidirect
183. #1916	C₂×C₅₃	semidirect
216. C₅₃	#21027	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×#423	2
C₂×C₅₃×#405	2
C₅₃×#2296	2
C₅₃×#1917	2
C₅₃×#1916	2
C₅₃×#1729	2
C₂×C₅₃×#336	2
#21027	53

Subgroups — C53×#21027 #27228539

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅₃×#21027 #27228539