Subgroups — C₅₃×#18075 #27225587

Order 13568.

592 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	C₅₃×#3168	C₂	1	1	yes	maximal, characteristic
2	C₅₃×#2969	C₂	1	2	yes	maximal, complemented
3	C₅₃×#1594	C₂	1	1	yes	maximal, characteristic, complemented
4	C₅₃×#1593	C₂	1	1	yes	maximal, characteristic, complemented
5	C₅₃×#1588	C₂	1	2	yes	maximal, complemented
6	C₅₃×#1585	C₂	1	2	yes	maximal, complemented
7	C₅₃×#1580	C₂	1	1	yes	maximal, characteristic, complemented
8	C₅₃×#1579	C₂	1	1	yes	maximal, characteristic, complemented
9	C₅₃×#1564	C₂	1	1	yes	maximal, characteristic, complemented
10	C₅₃×#1482	C₂	1	2	yes	maximal, complemented
11	C₅₃×#1451	C₂	1	1	yes	maximal, characteristic, complemented
12	C₅₃×#529	C₂²	1	2	yes	—
13	C₅₃×#499	C₄	1	1	yes	characteristic
14	C₅₃×#495	C₄	1	1	yes	characteristic
15	C₅₃×#488	C₄	1	2	yes	—
16	C₅₃×#467	C₄	1	1	yes	characteristic
17	C₅₃×#466	C₄	1	1	yes	characteristic
18	C₅₃×#465	C₄	1	2	yes	—
19	C₅₃×#446	C₂²	1	2	yes	—
20	C₈×Q₈×C₅₃	C₂²	1	2	yes	—
21	C₄×Q₁₆×C₅₃	C₂²	1	1	yes	characteristic
22	C₄×SD₁₆×C₅₃	C₂²	1	2	yes	—
23	C₄×D₈×C₅₃	C₂²	1	1	yes	characteristic
24	C₅₃×#436	—	2	2	no	—
25	C₅₃×#436	C₂²	1	2	yes	—
26	C₅₃×#435	—	2	2	no	—
27	C₈×D₄×C₅₃	C₂²	1	2	yes	—
28	C₅₃×#433	—	2	2	no	—
29	C₅₃×#431	C₂²	1	2	yes	—
30	C₅₃×#431	C₂²	1	1	yes	characteristic
31	C₅₃×#405	—	2	2	no	—
32	C₄×M₁₆×C₅₃	C₂²	1	1	yes	characteristic
33	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
34	C₅₃×#335	C₂²	1	2	yes	—
35	C₅₃×#334	C₂²	1	1	yes	characteristic
36	C₅₃×#334	C₂²	1	1	yes	characteristic
37	C₅₃×Q₈⋊C₈	C₂²	1	2	yes	—
38	C₅₃×Q₈⋊C₈	C₂²	1	2	yes	—
39	C₅₃×D₄⋊C₈	C₂²	1	2	yes	—
40	C₅₃×D₄⋊C₈	C₂²	1	2	yes	—
41	C₅₃×#322	C₂²	1	1	yes	characteristic
42	C₅₃×#322	C₂²	1	1	yes	characteristic
43	C₅₃×#322	C₂²	1	1	yes	characteristic
44	C₅₃×#322	C₂²	1	1	yes	characteristic
45	C₂×Q₁₆×C₅₃	C₂×C₄	1	1	yes	characteristic
46	C₂×SD₁₆×C₅₃	C₂×C₄	1	2	yes	—
47	C₂×D₈×C₅₃	C₂×C₄	1	1	yes	characteristic
48	C₂×M₁₆×C₅₃	—	2	2	no	—
49	C₂×M₁₆×C₅₃	D₄	1	2	yes	—
50	C₂²×C₈×C₅₃	—	2	2	no	—
51	C₂²×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
52	C₂²×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
53	#126×C₅₃	—	2	2	no	—
54	#125×C₅₃	C₂×C₄	1	2	yes	—
55	#124×C₅₃	C₂×C₄	1	2	yes	—
56	#123×C₅₃	—	2	2	no	—
57	C₂²⋊Q₈×C₅₃	C₂×C₄	1	2	yes	—
58	#121×C₅₃	C₂×C₄	1	2	yes	—
59	C₄×Q₈×C₅₃	C₂³	1	2	yes	—
60	C₄×D₄×C₅₃	—	2	2	no	—
61	C₄×D₄×C₅₃	C₂³	1	2	yes	—
62	#117×C₅₃	—	2	2	no	—
63	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
64	#107×C₅₃	C₂×C₄	1	1	yes	characteristic
65	#107×C₅₃	C₂×C₄	1	1	yes	characteristic
66	#106×C₅₃	C₂×C₄	1	2	yes	—
67	C₄⋊C₈×C₅₃	—	2	2	no	—
68	C₄⋊C₈×C₅₃	—	2	2	no	—
69	C₄⋊C₈×C₅₃	—	2	1	no	—
70	C₄⋊C₈×C₅₃	C₂³	1	2	yes	—
71	C₄⋊C₈×C₅₃	C₂³	1	2	yes	—
72	#103×C₅₃	C₂×C₄	1	2	yes	—
73	#103×C₅₃	C₂×C₄	1	2	yes	—
74	#102×C₅₃	C₂×C₄	1	2	yes	—
75	#102×C₅₃	C₂×C₄	1	2	yes	—
76	C₂²⋊C₈×C₅₃	—	2	2	no	—
77	C₂²⋊C₈×C₅₃	—	2	2	no	—
78	C₂²⋊C₈×C₅₃	—	2	2	no	—
79	C₂²⋊C₈×C₅₃	—	2	1	no	—
80	#97×C₅₃	—	2	2	no	—
81	#97×C₅₃	D₄	1	2	yes	—
82	C₄×C₈×C₅₃	—	2	2	no	—
83	C₄×C₈×C₅₃	—	2	2	no	—
84	C₄×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
85	C₄×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
86	C₄×C₈×C₅₃	C₂³	1	1	yes	characteristic
87	C₄×C₈×C₅₃	C₂³	1	1	yes	characteristic
88	C₄×C₈×C₅₃	C₂³	1	1	yes	characteristic
89	C₄×C₈×C₅₃	C₂³	1	1	yes	characteristic
90	C₄×C₈×C₅₃	C₂³	1	1	yes	characteristic
91	C₄×C₈×C₅₃	C₂³	1	1	yes	characteristic
92	C₂×Q₈×C₅₃	C₂²×C₄	1	2	yes	—
93	C₂×D₄×C₅₃	—	2	2	no	—
94	C₂×D₄×C₅₃	C₂²×C₄	1	2	yes	—
95	C₂²×C₄×C₅₃	—	2	2	no	—
96	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
97	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
98	C₂²×C₄×C₅₃	#41	1	1	yes	characteristic
99	Q₁₆×C₅₃	—	2	2	no	—
100	SD₁₆×C₅₃	—	4	2	no	—
101	D₈×C₅₃	—	2	2	no	—
102	M₁₆×C₅₃	—	4	2	no	—
103	M₁₆×C₅₃	—	4	4	no	—
104	C₂×C₈×C₅₃	—	2	2	no	—
105	C₂×C₈×C₅₃	—	2	2	no	—
106	C₂×C₈×C₅₃	—	2	2	no	—
107	C₂×C₈×C₅₃	—	2	1	no	—
108	C₂×C₈×C₅₃	—	2	1	no	—
109	C₂×C₈×C₅₃	—	4	1	no	—
110	C₂×C₈×C₅₃	—	4	4	no	—
111	C₂×C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
112	C₂×C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
113	C₂×C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
114	C₂×C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
115	C₂×C₈×C₅₃	C₂×D₄	1	2	yes	—
116	C₂×C₈×C₅₃	C₂×D₄	1	2	yes	—
117	C₄⋊C₄×C₅₃	—	2	2	no	—
118	C₄⋊C₄×C₅₃	—	2	2	no	—
119	C₄⋊C₄×C₅₃	—	2	2	no	—
120	C₄⋊C₄×C₅₃	—	2	2	no	—
121	C₄⋊C₄×C₅₃	C₂²×C₄	1	2	yes	—
122	C₄⋊C₄×C₅₃	C₂²×C₄	1	2	yes	—
123	C₂²⋊C₄×C₅₃	—	2	2	no	—
124	C₂²⋊C₄×C₅₃	—	2	2	no	—
125	C₂²⋊C₄×C₅₃	—	2	2	no	—
126	C₂²⋊C₄×C₅₃	—	2	2	no	—
127	C₂²⋊C₄×C₅₃	—	2	2	no	—
128	C₄²×C₅₃	—	2	2	no	—
129	C₄²×C₅₃	C₂²×C₄	1	1	yes	characteristic
130	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
131	C₄²×C₅₃	#41	1	1	yes	characteristic
132	C₄²×C₅₃	C₂⁴	1	1	yes	characteristic
133	C₂³×C₅₃	—	2	2	no	—
134	C₂³×C₅₃	C₄×D₄	1	1	yes	characteristic
135	Q₈×C₅₃	—	2	2	no	—
136	Q₈×C₅₃	—	2	2	no	—
137	D₄×C₅₃	—	2	2	no	—
138	D₄×C₅₃	—	2	2	no	—
139	D₄×C₅₃	—	8	2	no	—
140	C₂×C₄×C₅₃	—	2	1	no	—
141	C₂×C₄×C₅₃	—	2	1	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	—	2	2	no	—
146	C₂×C₄×C₅₃	—	4	1	no	—
147	C₂×C₄×C₅₃	—	4	2	no	—
148	C₂×C₄×C₅₃	—	4	2	no	—
149	C₂×C₄×C₅₃	C₄×D₄	1	1	yes	characteristic
150	C₂×C₄×C₅₃	C₄×D₄	1	1	yes	characteristic
151	C₂×C₄×C₅₃	C₄×D₄	1	1	yes	characteristic
152	C₂×C₄×C₅₃	C₂³×C₄	1	1	yes	characteristic
153	C₂×C₄×C₅₃	C₂²×D₄	1	1	yes	characteristic
154	C₂×C₄×C₅₃	C₂×#41	1	1	yes	characteristic
155	C₈×C₅₃	—	2	1	no	—
156	C₈×C₅₃	—	2	1	no	—
157	C₈×C₅₃	—	2	1	no	—
158	C₈×C₅₃	—	2	2	no	—
159	C₈×C₅₃	—	2	2	no	—
160	C₈×C₅₃	—	4	1	no	—
161	C₈×C₅₃	—	4	1	no	—
162	C₈×C₅₃	—	4	2	no	—
163	C₈×C₅₃	—	4	2	no	—
164	C₈×C₅₃	#131	1	2	yes	—
165	#18075	C₅₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
166	C₂²×C₅₃	—	2	1	no	—
167	C₂²×C₅₃	—	2	1	no	—
168	C₂²×C₅₃	—	2	1	no	—
169	C₂²×C₅₃	—	4	2	no	—
170	C₂²×C₅₃	—	4	2	no	—
171	C₂²×C₅₃	—	4	2	no	—
172	C₂²×C₅₃	C₂×C₄×D₄	1	1	yes	characteristic
173	C₄×C₅₃	—	2	1	no	—
174	C₄×C₅₃	—	2	1	no	—
175	C₄×C₅₃	—	2	1	no	—
176	C₄×C₅₃	—	2	1	no	—
177	C₄×C₅₃	—	4	2	no	—
178	C₄×C₅₃	—	4	2	no	—
179	C₄×C₅₃	—	4	2	no	—
180	C₄×C₅₃	C₂×#131	1	1	yes	characteristic
181	C₄×C₅₃	#568	1	1	yes	characteristic
182	C₄×C₅₃	#575	1	2	yes	—
183	#3168	C₂×C₅₃	1	1	yes	characteristic
184	#2969	C₂×C₅₃	1	2	yes	complemented
185	#1594	C₂×C₅₃	1	1	yes	characteristic, complemented
186	#1593	C₂×C₅₃	1	1	yes	characteristic, complemented
187	#1588	C₂×C₅₃	1	2	yes	complemented
188	#1585	C₂×C₅₃	1	2	yes	complemented
189	#1580	C₂×C₅₃	1	1	yes	characteristic, complemented
190	#1579	C₂×C₅₃	1	1	yes	characteristic, complemented
191	#1564	C₂×C₅₃	1	1	yes	characteristic, complemented
192	#1482	C₂×C₅₃	1	2	yes	complemented
193	#1451	C₂×C₅₃	1	1	yes	characteristic, complemented
194	C₂×C₅₃	—	4	1	no	—
195	C₂×C₅₃	—	8	2	no	—
196	C₂×C₅₃	#2935	1	1	yes	characteristic
197	C₂×C₅₃	#2943	1	1	yes	characteristic
198	C₂×C₅₃	#2957	1	1	yes	characteristic
199	#529	C₂²×C₅₃	1	2	yes	—
200	#499	C₄×C₅₃	1	1	yes	characteristic
201	#495	C₄×C₅₃	1	1	yes	characteristic
202	#488	C₄×C₅₃	1	2	yes	—
203	#467	C₄×C₅₃	1	1	yes	characteristic
204	#466	C₄×C₅₃	1	1	yes	characteristic
205	#465	C₄×C₅₃	1	2	yes	—
206	#446	C₂²×C₅₃	1	2	yes	—
207	C₈×Q₈	C₂²×C₅₃	1	2	yes	—
208	C₄×Q₁₆	C₂²×C₅₃	1	1	yes	characteristic
209	C₄×SD₁₆	C₂²×C₅₃	1	2	yes	—
210	C₄×D₈	C₂²×C₅₃	1	1	yes	characteristic
211	#436	—	2	2	no	—
212	#436	C₂²×C₅₃	1	2	yes	—
213	#435	—	2	2	no	—
214	C₈×D₄	C₂²×C₅₃	1	2	yes	—
215	#433	—	2	2	no	—
216	#431	C₂²×C₅₃	1	2	yes	—
217	#431	C₂²×C₅₃	1	1	yes	characteristic
218	#405	—	2	2	no	—
219	C₄×M₁₆	C₂²×C₅₃	1	1	yes	characteristic
220	C₂×C₄×C₈	C₂²×C₅₃	1	1	yes	characteristic
221	#335	C₂²×C₅₃	1	2	yes	—
222	#334	C₂²×C₅₃	1	1	yes	characteristic
223	#334	C₂²×C₅₃	1	1	yes	characteristic
224	Q₈⋊C₈	C₂²×C₅₃	1	2	yes	—
225	Q₈⋊C₈	C₂²×C₅₃	1	2	yes	—
226	D₄⋊C₈	C₂²×C₅₃	1	2	yes	—
227	D₄⋊C₈	C₂²×C₅₃	1	2	yes	—
228	#322	C₂²×C₅₃	1	1	yes	characteristic
229	#322	C₂²×C₅₃	1	1	yes	characteristic
230	#322	C₂²×C₅₃	1	1	yes	characteristic
231	#322	C₂²×C₅₃	1	1	yes	characteristic
232	C₅₃	#18075	1	1	yes	characteristic, direct product factor, Sylow
233	C₂×Q₁₆	C₂×C₄×C₅₃	1	1	yes	characteristic
234	C₂×SD₁₆	C₂×C₄×C₅₃	1	2	yes	—
235	C₂×D₈	C₂×C₄×C₅₃	1	1	yes	characteristic
236	C₂×M₁₆	—	2	2	no	—
237	C₂×M₁₆	D₄×C₅₃	1	2	yes	—
238	C₂²×C₈	—	2	2	no	—
239	C₂²×C₈	C₂×C₄×C₅₃	1	1	yes	characteristic
240	C₂²×C₈	C₂×C₄×C₅₃	1	1	yes	characteristic
241	#126	—	2	2	no	—
242	#125	C₂×C₄×C₅₃	1	2	yes	—
243	#124	C₂×C₄×C₅₃	1	2	yes	—
244	#123	—	2	2	no	—
245	C₂²⋊Q₈	C₂×C₄×C₅₃	1	2	yes	—
246	#121	C₂×C₄×C₅₃	1	2	yes	—
247	C₄×Q₈	C₂³×C₅₃	1	2	yes	—
248	C₄×D₄	—	2	2	no	—
249	C₄×D₄	C₂³×C₅₃	1	2	yes	—
250	#117	—	2	2	no	—
251	C₂×C₄²	C₂³×C₅₃	1	1	yes	characteristic
252	#107	C₂×C₄×C₅₃	1	1	yes	characteristic
253	#107	C₂×C₄×C₅₃	1	1	yes	characteristic
254	#106	C₂×C₄×C₅₃	1	2	yes	—
255	C₄⋊C₈	—	2	2	no	—
256	C₄⋊C₈	—	2	2	no	—
257	C₄⋊C₈	—	2	1	no	—
258	C₄⋊C₈	C₂³×C₅₃	1	2	yes	—
259	C₄⋊C₈	C₂³×C₅₃	1	2	yes	—
260	#103	C₂×C₄×C₅₃	1	2	yes	—
261	#103	C₂×C₄×C₅₃	1	2	yes	—
262	#102	C₂×C₄×C₅₃	1	2	yes	—
263	#102	C₂×C₄×C₅₃	1	2	yes	—
264	C₂²⋊C₈	—	2	2	no	—
265	C₂²⋊C₈	—	2	1	no	—
266	C₂²⋊C₈	—	2	2	no	—
267	C₂²⋊C₈	—	2	2	no	—
268	#97	—	2	2	no	—
269	#97	D₄×C₅₃	1	2	yes	—
270	C₄×C₈	—	2	2	no	—
271	C₄×C₈	—	2	2	no	—
272	C₄×C₈	C₂×C₄×C₅₃	1	1	yes	characteristic
273	C₄×C₈	C₂×C₄×C₅₃	1	1	yes	characteristic
274	C₄×C₈	C₂³×C₅₃	1	1	yes	characteristic
275	C₄×C₈	C₂³×C₅₃	1	1	yes	characteristic
276	C₄×C₈	C₂³×C₅₃	1	1	yes	characteristic
277	C₄×C₈	C₂³×C₅₃	1	1	yes	characteristic
278	C₄×C₈	C₂³×C₅₃	1	1	yes	characteristic
279	C₄×C₈	C₂³×C₅₃	1	1	yes	characteristic
280	C₂×Q₈	C₂²×C₄×C₅₃	1	2	yes	—
281	C₂×D₄	—	2	2	no	—
282	C₂×D₄	C₂²×C₄×C₅₃	1	2	yes	—
283	C₂²×C₄	—	2	2	no	—
284	C₂²×C₄	C₂²×C₄×C₅₃	1	1	yes	characteristic
285	C₂²×C₄	C₂×D₄×C₅₃	1	1	yes	characteristic
286	C₂²×C₄	#41×C₅₃	1	1	yes	characteristic
287	Q₁₆	—	2	2	no	—
288	SD₁₆	—	4	2	no	—
289	D₈	—	2	2	no	—
290	M₁₆	—	4	2	no	—
291	M₁₆	—	4	4	no	—
292	C₂×C₈	—	2	2	no	—
293	C₂×C₈	—	2	2	no	—
294	C₂×C₈	—	2	2	no	—
295	C₂×C₈	—	2	1	no	—
296	C₂×C₈	—	2	1	no	—
297	C₂×C₈	—	4	1	no	—
298	C₂×C₈	—	4	4	no	—
299	C₂×C₈	C₂²×C₄×C₅₃	1	1	yes	characteristic
300	C₂×C₈	C₂²×C₄×C₅₃	1	1	yes	characteristic
301	C₂×C₈	C₂²×C₄×C₅₃	1	1	yes	characteristic
302	C₂×C₈	C₂²×C₄×C₅₃	1	1	yes	characteristic
303	C₂×C₈	C₂×D₄×C₅₃	1	2	yes	—
304	C₂×C₈	C₂×D₄×C₅₃	1	2	yes	—
305	C₄⋊C₄	—	2	2	no	—
306	C₄⋊C₄	—	2	2	no	—
307	C₄⋊C₄	—	2	2	no	—
308	C₄⋊C₄	—	2	2	no	—
309	C₄⋊C₄	C₂²×C₄×C₅₃	1	2	yes	—
310	C₄⋊C₄	C₂²×C₄×C₅₃	1	2	yes	—
311	C₂²⋊C₄	—	2	2	no	—
312	C₂²⋊C₄	—	2	2	no	—
313	C₂²⋊C₄	—	2	2	no	—
314	C₂²⋊C₄	—	2	2	no	—
315	C₂²⋊C₄	—	2	2	no	—
316	C₄²	—	2	2	no	—
317	C₄²	C₂²×C₄×C₅₃	1	1	yes	characteristic
318	C₄²	C₂×D₄×C₅₃	1	1	yes	characteristic
319	C₄²	#41×C₅₃	1	1	yes	characteristic
320	C₄²	C₂⁴×C₅₃	1	1	yes	characteristic
321	C₂³	—	2	2	no	—
322	C₂³	C₄×D₄×C₅₃	1	1	yes	characteristic
323	Q₈	—	2	2	no	—
324	Q₈	—	2	2	no	—
325	D₄	—	2	2	no	—
326	D₄	—	2	2	no	—
327	D₄	—	8	2	no	—
328	C₂×C₄	—	2	1	no	—
329	C₂×C₄	—	2	1	no	—
330	C₂×C₄	—	2	2	no	—
331	C₂×C₄	—	2	2	no	—
332	C₂×C₄	—	2	2	no	—
333	C₂×C₄	—	2	2	no	—
334	C₂×C₄	—	4	1	no	—
335	C₂×C₄	—	4	2	no	—
336	C₂×C₄	—	4	2	no	—
337	C₂×C₄	C₄×D₄×C₅₃	1	1	yes	characteristic
338	C₂×C₄	C₄×D₄×C₅₃	1	1	yes	characteristic
339	C₂×C₄	C₄×D₄×C₅₃	1	1	yes	characteristic
340	C₂×C₄	C₂³×C₄×C₅₃	1	1	yes	characteristic
341	C₂×C₄	C₂²×D₄×C₅₃	1	1	yes	characteristic
342	C₂×C₄	C₂×#41×C₅₃	1	1	yes	characteristic
343	C₈	—	2	1	no	—
344	C₈	—	2	1	no	—
345	C₈	—	2	1	no	—
346	C₈	—	2	2	no	—
347	C₈	—	2	2	no	—
348	C₈	—	4	1	no	—
349	C₈	—	4	1	no	—
350	C₈	—	4	2	no	—
351	C₈	—	4	2	no	—
352	C₈	#131×C₅₃	1	2	yes	—
353	C₂²	—	2	1	no	—
354	C₂²	—	2	1	no	—
355	C₂²	—	2	1	no	—
356	C₂²	—	4	2	no	—
357	C₂²	—	4	2	no	—
358	C₂²	—	4	2	no	—
359	C₂²	C₂×C₄×D₄×C₅₃	1	1	yes	characteristic
360	C₄	—	2	1	no	—
361	C₄	—	2	1	no	—
362	C₄	—	2	1	no	—
363	C₄	—	2	1	no	—
364	C₄	—	4	2	no	—
365	C₄	—	4	2	no	—
366	C₄	—	4	2	no	—
367	C₄	C₂×#131×C₅₃	1	1	yes	characteristic
368	C₄	C₅₃×#568	1	1	yes	characteristic
369	C₄	C₅₃×#575	1	2	yes	—
370	C₂	—	4	1	no	—
371	C₂	—	8	2	no	—
372	C₂	C₅₃×#2935	1	1	yes	characteristic
373	C₂	C₅₃×#2943	1	1	yes	characteristic
374	C₂	C₅₃×#2957	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄×C₅₃	C₂²×D₄
commutator	C₂×C₄	C₂³×C₄×C₅₃
Frattini, cosocle	C₄²	C₂⁴×C₅₃
socle	C₂²×C₅₃	C₂×C₄×D₄
prime radical	C₅₃×#466	C₄
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₅₃ ◁ C₄×M₁₆×C₅₃ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₅₃×#2969	C₂	semidirect
3. C₅₃×#1594	C₂	semidirect
4. C₅₃×#1593	C₂	semidirect
5. C₅₃×#1588	C₂	semidirect
6. C₅₃×#1585	C₂	semidirect
7. C₅₃×#1580	C₂	semidirect
8. C₅₃×#1579	C₂	semidirect
9. C₅₃×#1564	C₂	semidirect
10. C₅₃×#1482	C₂	semidirect
11. C₅₃×#1451	C₂	semidirect
165. #18075	C₅₃	direct (N × H)
184. #2969	C₂×C₅₃	semidirect
185. #1594	C₂×C₅₃	semidirect
186. #1593	C₂×C₅₃	semidirect
187. #1588	C₂×C₅₃	semidirect
188. #1585	C₂×C₅₃	semidirect
189. #1580	C₂×C₅₃	semidirect
190. #1579	C₂×C₅₃	semidirect
191. #1564	C₂×C₅₃	semidirect
192. #1482	C₂×C₅₃	semidirect
193. #1451	C₂×C₅₃	semidirect
232. C₅₃	#18075	direct (N × H)

Maximal subgroups

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

subgroup	index
C₅₃×#3168	2
C₅₃×#2969	2
C₅₃×#1594	2
C₅₃×#1593	2
C₅₃×#1588	2
C₅₃×#1585	2
C₅₃×#1580	2
C₅₃×#1579	2
C₅₃×#1564	2
C₅₃×#1482	2
C₅₃×#1451	2
#18075	53

Subgroups — C53×#18075 #27225587

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅₃×#18075 #27225587