Subgroups — C₅₃×#22615 #27230127

Order 13568.

838 conjugacy classes of subgroups.

The subgroup lattice has 418 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#106×C₅₃	C₂	1	1	yes	maximal, characteristic
2	C₅₃×#2345	C₂	1	1	yes	maximal, characteristic
3	C₅₃×#2344	C₂	1	1	yes	maximal, characteristic
4	C₅₃×#1944	C₂	1	2	yes	maximal, complemented
5	C₅₃×#1920	C₂	1	2	yes	maximal, complemented
6	C₅₃×#1919	C₂	1	2	yes	maximal, complemented
7	C₅₃×#1824	C₂	1	1	yes	maximal, characteristic
8	C₅₃×#1823	C₂	1	1	yes	maximal, characteristic
9	C₅₃×#1808	C₂	1	1	yes	maximal, characteristic
10	C₅₃×#1776	C₂	1	2	yes	maximal, complemented
11	C₅₃×#1752	C₂	1	1	yes	maximal, characteristic
12	C₂³×C₈×C₅₃	C₂²	1	1	yes	characteristic
13	C₂×#117×C₅₃	—	2	1	no	—
14	C₂²×C₄⋊C₄×C₅₃	C₂²	1	1	yes	characteristic
15	C₂²×C₄⋊C₄×C₅₃	C₂²	1	1	yes	characteristic
16	C₂×#107×C₅₃	C₂²	1	2	yes	—
17	C₂×#106×C₅₃	C₄	1	8	yes	complemented
18	C₂×#106×C₅₃	C₂²	1	1	yes	characteristic
19	C₂×#106×C₅₃	C₂²	1	1	yes	characteristic
20	C₂×#106×C₅₃	C₂²	1	1	yes	characteristic
21	C₂×#106×C₅₃	C₂²	1	1	yes	characteristic
22	C₂×C₂²⋊C₈×C₅₃	C₂²	1	1	yes	characteristic
23	C₂×#97×C₅₃	C₂²	1	2	yes	—
24	C₅₃×#389	C₂²	1	2	yes	—
25	C₅₃×#389	C₂²	1	2	yes	—
26	C₅₃×#387	—	2	2	no	—
27	C₅₃×#387	—	2	2	no	—
28	C₅₃×#381	—	2	1	no	—
29	C₅₃×#380	—	2	1	no	—
30	C₅₃×#380	C₂²	1	1	yes	characteristic
31	C₅₃×#380	C₂²	1	1	yes	characteristic
32	C₄×C₄⋊C₄×C₅₃	C₂²	1	2	yes	—
33	C₄×C₄⋊C₄×C₅₃	C₂²	1	2	yes	—
34	C₄×C₂²⋊C₄×C₅₃	—	2	1	no	—
35	C₄×C₂²⋊C₄×C₅₃	C₂²	1	1	yes	characteristic
36	C₅₃×#340	C₂²	1	2	yes	—
37	C₅₃×#340	C₂²	1	2	yes	—
38	C₅₃×#340	C₂²	1	2	yes	—
39	C₅₃×#340	C₂²	1	2	yes	—
40	C₅₃×#340	C₂²	1	2	yes	—
41	C₅₃×#336	C₂²	1	2	yes	—
42	C₂³×C₄×C₅₃	—	2	1	no	—
43	C₂³×C₄×C₅₃	—	2	1	no	—
44	C₂³×C₄×C₅₃	C₂³	1	1	yes	characteristic
45	C₂²×C₈×C₅₃	—	2	4	no	—
46	C₂²×C₈×C₅₃	C₂×C₄	1	4	yes	—
47	C₂²×C₈×C₅₃	C₂³	1	2	yes	—
48	C₂²×C₈×C₅₃	C₂³	1	1	yes	characteristic
49	C₂²×C₈×C₅₃	C₂³	1	1	yes	characteristic
50	#117×C₅₃	—	4	4	no	—
51	C₂×C₄⋊C₄×C₅₃	—	2	1	no	—
52	C₂×C₄⋊C₄×C₅₃	—	2	1	no	—
53	C₂×C₄⋊C₄×C₅₃	—	2	2	no	—
54	C₂×C₄⋊C₄×C₅₃	—	2	2	no	—
55	C₂×C₄⋊C₄×C₅₃	—	2	2	no	—
56	C₂×C₄⋊C₄×C₅₃	—	2	2	no	—
57	C₂×C₄⋊C₄×C₅₃	C₂×C₄	1	4	yes	—
58	C₂×C₄⋊C₄×C₅₃	C₂×C₄	1	4	yes	—
59	C₂×C₄⋊C₄×C₅₃	C₂³	1	2	yes	—
60	C₂×C₄⋊C₄×C₅₃	C₂³	1	2	yes	—
61	C₂×C₄⋊C₄×C₅₃	C₂³	1	1	yes	characteristic
62	C₂×C₄⋊C₄×C₅₃	C₂³	1	1	yes	characteristic
63	C₂×C₄⋊C₄×C₅₃	C₂³	1	1	yes	characteristic
64	C₂×C₄⋊C₄×C₅₃	C₂³	1	1	yes	characteristic
65	C₂×C₂²⋊C₄×C₅₃	—	2	1	no	—
66	C₂×C₂²⋊C₄×C₅₃	—	2	1	no	—
67	C₂×C₂²⋊C₄×C₅₃	D₄	1	1	yes	characteristic
68	C₂×C₂²⋊C₄×C₅₃	D₄	1	1	yes	characteristic
69	C₂×C₄²×C₅₃	—	2	2	no	—
70	C₂×C₄²×C₅₃	—	2	2	no	—
71	C₂×C₄²×C₅₃	C₂³	1	2	yes	—
72	#107×C₅₃	—	2	4	no	—
73	#106×C₅₃	C₂×C₄	1	4	yes	—
74	#106×C₅₃	C₂×C₄	1	4	yes	—
75	#106×C₅₃	C₂×C₄	1	4	yes	—
76	#106×C₅₃	C₂×C₄	1	4	yes	—
77	C₂²⋊C₈×C₅₃	—	2	2	no	—
78	#97×C₅₃	—	2	4	no	—
79	#95×C₅₃	—	2	2	no	—
80	#95×C₅₃	—	2	2	no	—
81	#95×C₅₃	—	2	2	no	—
82	#95×C₅₃	—	2	1	no	—
83	#95×C₅₃	—	2	1	no	—
84	#95×C₅₃	—	2	2	no	—
85	#95×C₅₃	D₄	1	2	yes	—
86	C₂⁴×C₅₃	C₂×D₄	1	1	yes	characteristic
87	C₂²×C₄×C₅₃	—	2	2	no	—
88	C₂²×C₄×C₅₃	—	2	2	no	—
89	C₂²×C₄×C₅₃	—	2	4	no	—
90	C₂²×C₄×C₅₃	—	2	1	no	—
91	C₂²×C₄×C₅₃	—	2	1	no	—
92	C₂²×C₄×C₅₃	—	2	1	no	—
93	C₂²×C₄×C₅₃	—	2	1	no	—
94	C₂²×C₄×C₅₃	—	2	4	no	—
95	C₂²×C₄×C₅₃	—	2	2	no	—
96	C₂²×C₄×C₅₃	—	2	2	no	—
97	C₂²×C₄×C₅₃	—	4	2	no	—
98	C₂²×C₄×C₅₃	—	4	2	no	—
99	C₂²×C₄×C₅₃	—	8	2	no	—
100	C₂²×C₄×C₅₃	C₂²×C₄	1	2	yes	—
101	C₂²×C₄×C₅₃	C₂×D₄	1	2	yes	—
102	C₂²×C₄×C₅₃	C₂×D₄	1	2	yes	—
103	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
104	C₂²×C₄×C₅₃	#41	1	2	yes	—
105	C₂²×C₄×C₅₃	C₂⁴	1	1	yes	characteristic
106	C₂×C₈×C₅₃	—	2	2	no	—
107	C₂×C₈×C₅₃	—	2	8	no	—
108	C₂×C₈×C₅₃	—	2	2	no	—
109	C₂×C₈×C₅₃	—	2	2	no	—
110	C₂×C₈×C₅₃	—	2	2	no	—
111	C₂×C₈×C₅₃	C₂²×C₄	1	2	yes	—
112	C₂×C₈×C₅₃	C₂²×C₄	1	2	yes	—
113	C₂×C₈×C₅₃	#41	1	4	yes	—
114	C₄⋊C₄×C₅₃	—	2	4	no	—
115	C₄⋊C₄×C₅₃	—	2	4	no	—
116	C₄⋊C₄×C₅₃	—	2	1	no	—
117	C₄⋊C₄×C₅₃	—	2	1	no	—
118	C₄⋊C₄×C₅₃	—	2	1	no	—
119	C₄⋊C₄×C₅₃	—	2	1	no	—
120	C₄⋊C₄×C₅₃	—	2	2	no	—
121	C₄⋊C₄×C₅₃	—	2	2	no	—
122	C₄⋊C₄×C₅₃	—	2	2	no	—
123	C₄⋊C₄×C₅₃	—	2	2	no	—
124	C₄⋊C₄×C₅₃	—	4	4	no	—
125	C₄⋊C₄×C₅₃	—	8	2	no	—
126	C₄⋊C₄×C₅₃	C₂²×C₄	1	2	yes	—
127	C₄⋊C₄×C₅₃	C₂²×C₄	1	2	yes	—
128	C₄⋊C₄×C₅₃	C₂²×C₄	1	2	yes	—
129	C₄⋊C₄×C₅₃	C₂²×C₄	1	2	yes	—
130	C₂²⋊C₄×C₅₃	—	2	2	no	—
131	C₂²⋊C₄×C₅₃	—	2	2	no	—
132	C₂²⋊C₄×C₅₃	—	4	2	no	—
133	C₂²⋊C₄×C₅₃	—	4	2	no	—
134	C₄²×C₅₃	—	2	2	no	—
135	C₄²×C₅₃	—	2	2	no	—
136	C₄²×C₅₃	—	4	4	no	—
137	C₄²×C₅₃	—	8	2	no	—
138	C₂³×C₅₃	—	2	2	no	—
139	C₂³×C₅₃	—	2	2	no	—
140	C₂³×C₅₃	C₄×D₄	1	2	yes	—
141	C₂³×C₅₃	AGL(1,ℤ/8)	1	2	yes	—
142	C₂³×C₅₃	#137	1	2	yes	—
143	C₂³×C₅₃	C₂²×D₄	1	1	yes	characteristic
144	C₂×C₄×C₅₃	—	2	2	no	—
145	C₂×C₄×C₅₃	—	2	2	no	—
146	C₂×C₄×C₅₃	—	2	2	no	—
147	C₂×C₄×C₅₃	—	2	4	no	—
148	C₂×C₄×C₅₃	—	2	2	no	—
149	C₂×C₄×C₅₃	—	2	2	no	—
150	C₂×C₄×C₅₃	—	2	2	no	—
151	C₂×C₄×C₅₃	—	2	2	no	—
152	C₂×C₄×C₅₃	—	2	2	no	—
153	C₂×C₄×C₅₃	—	2	2	no	—
154	C₂×C₄×C₅₃	—	2	2	no	—
155	C₂×C₄×C₅₃	—	2	2	no	—
156	C₂×C₄×C₅₃	—	2	2	no	—
157	C₂×C₄×C₅₃	—	2	2	no	—
158	C₂×C₄×C₅₃	—	4	1	no	—
159	C₂×C₄×C₅₃	—	4	1	no	—
160	C₂×C₄×C₅₃	—	4	1	no	—
161	C₂×C₄×C₅₃	—	4	1	no	—
162	C₂×C₄×C₅₃	—	4	2	no	—
163	C₂×C₄×C₅₃	—	4	2	no	—
164	C₂×C₄×C₅₃	—	4	8	no	—
165	C₂×C₄×C₅₃	—	4	2	no	—
166	C₂×C₄×C₅₃	—	4	2	no	—
167	C₂×C₄×C₅₃	—	4	2	no	—
168	C₂×C₄×C₅₃	—	4	2	no	—
169	C₂×C₄×C₅₃	—	4	2	no	—
170	C₂×C₄×C₅₃	—	4	2	no	—
171	C₂×C₄×C₅₃	—	8	4	no	—
172	C₂×C₄×C₅₃	C₄×D₄	1	2	yes	—
173	C₂×C₄×C₅₃	C₂³×C₄	1	1	yes	characteristic
174	C₂×C₄×C₅₃	C₂×#41	1	2	yes	—
175	C₂×C₄×C₅₃	C₂×#41	1	1	yes	characteristic
176	C₂×C₄×C₅₃	2₋¹⁺⁴	1	2	yes	—
177	C₈×C₅₃	—	2	2	no	—
178	C₈×C₅₃	—	4	2	no	—
179	C₈×C₅₃	#117	1	4	yes	complemented
180	#22615	C₅₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
181	C₂²×C₅₃	—	2	4	no	—
182	C₂²×C₅₃	—	2	4	no	—
183	C₂²×C₅₃	—	2	2	no	—
184	C₂²×C₅₃	—	2	2	no	—
185	C₂²×C₅₃	#440	1	4	yes	—
186	C₂²×C₅₃	C₂×C₄×D₄	1	1	yes	characteristic
187	C₂²×C₅₃	#547	1	2	yes	—
188	C₂²×C₅₃	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
189	C₂²×C₅₃	C₂×#137	1	1	yes	characteristic
190	C₂²×C₅₃	#577	1	2	yes	—
191	C₄×C₅₃	—	2	2	no	—
192	C₄×C₅₃	—	4	2	no	—
193	C₄×C₅₃	—	4	2	no	—
194	C₄×C₅₃	—	4	2	no	—
195	C₄×C₅₃	—	4	2	no	—
196	C₄×C₅₃	—	4	2	no	—
197	C₄×C₅₃	—	4	2	no	—
198	C₄×C₅₃	—	8	2	no	—
199	C₄×C₅₃	—	8	2	no	—
200	C₄×C₅₃	C₂×#117	1	1	yes	characteristic
201	C₄×C₅₃	#519	1	1	yes	characteristic
202	C₄×C₅₃	#552	1	2	yes	—
203	C₂²×#106	C₂×C₅₃	1	1	yes	characteristic
204	#2345	C₂×C₅₃	1	1	yes	characteristic
205	#2344	C₂×C₅₃	1	1	yes	characteristic
206	#1944	C₂×C₅₃	1	2	yes	complemented
207	#1920	C₂×C₅₃	1	2	yes	complemented
208	#1919	C₂×C₅₃	1	2	yes	complemented
209	#1824	C₂×C₅₃	1	1	yes	characteristic
210	#1823	C₂×C₅₃	1	1	yes	characteristic
211	#1808	C₂×C₅₃	1	1	yes	characteristic
212	#1776	C₂×C₅₃	1	2	yes	complemented
213	#1752	C₂×C₅₃	1	1	yes	characteristic
214	C₂×C₅₃	—	2	4	no	—
215	C₂×C₅₃	#2344	1	1	yes	characteristic
216	C₂×C₅₃	C₂×#440	1	1	yes	characteristic
217	C₂×C₅₃	#2949	1	1	yes	characteristic
218	C₂×C₅₃	#3344	1	2	yes	—
219	C₂×C₅₃	#3345	1	2	yes	—
220	C₂³×C₈	C₂²×C₅₃	1	1	yes	characteristic
221	C₂×#117	—	2	1	no	—
222	C₂²×C₄⋊C₄	C₂²×C₅₃	1	1	yes	characteristic
223	C₂²×C₄⋊C₄	C₂²×C₅₃	1	1	yes	characteristic
224	C₂×#107	C₂²×C₅₃	1	2	yes	—
225	C₂×#106	C₄×C₅₃	1	8	yes	complemented
226	C₂×#106	C₂²×C₅₃	1	1	yes	characteristic
227	C₂×#106	C₂²×C₅₃	1	1	yes	characteristic
228	C₂×#106	C₂²×C₅₃	1	1	yes	characteristic
229	C₂×#106	C₂²×C₅₃	1	1	yes	characteristic
230	C₂×C₂²⋊C₈	C₂²×C₅₃	1	1	yes	characteristic
231	C₂×#97	C₂²×C₅₃	1	2	yes	—
232	#389	C₂²×C₅₃	1	2	yes	—
233	#389	C₂²×C₅₃	1	2	yes	—
234	#387	—	2	2	no	—
235	#387	—	2	2	no	—
236	#381	—	2	1	no	—
237	#380	—	2	1	no	—
238	#380	C₂²×C₅₃	1	1	yes	characteristic
239	#380	C₂²×C₅₃	1	1	yes	characteristic
240	C₄×C₄⋊C₄	C₂²×C₅₃	1	2	yes	—
241	C₄×C₄⋊C₄	C₂²×C₅₃	1	2	yes	—
242	C₄×C₂²⋊C₄	—	2	1	no	—
243	C₄×C₂²⋊C₄	C₂²×C₅₃	1	1	yes	characteristic
244	#340	C₂²×C₅₃	1	2	yes	—
245	#340	C₂²×C₅₃	1	2	yes	—
246	#340	C₂²×C₅₃	1	2	yes	—
247	#340	C₂²×C₅₃	1	2	yes	—
248	#340	C₂²×C₅₃	1	2	yes	—
249	#336	C₂²×C₅₃	1	2	yes	—
250	C₅₃	#22615	1	1	yes	characteristic, direct product factor, Sylow
251	C₂³×C₄	—	2	1	no	—
252	C₂³×C₄	—	2	1	no	—
253	C₂³×C₄	C₂³×C₅₃	1	1	yes	characteristic
254	C₂²×C₈	—	2	4	no	—
255	C₂²×C₈	C₂×C₄×C₅₃	1	4	yes	—
256	C₂²×C₈	C₂³×C₅₃	1	2	yes	—
257	C₂²×C₈	C₂³×C₅₃	1	1	yes	characteristic
258	C₂²×C₈	C₂³×C₅₃	1	1	yes	characteristic
259	#117	—	4	4	no	—
260	C₂×C₄⋊C₄	—	2	1	no	—
261	C₂×C₄⋊C₄	—	2	1	no	—
262	C₂×C₄⋊C₄	—	2	2	no	—
263	C₂×C₄⋊C₄	—	2	2	no	—
264	C₂×C₄⋊C₄	—	2	2	no	—
265	C₂×C₄⋊C₄	—	2	2	no	—
266	C₂×C₄⋊C₄	C₂×C₄×C₅₃	1	4	yes	—
267	C₂×C₄⋊C₄	C₂×C₄×C₅₃	1	4	yes	—
268	C₂×C₄⋊C₄	C₂³×C₅₃	1	2	yes	—
269	C₂×C₄⋊C₄	C₂³×C₅₃	1	2	yes	—
270	C₂×C₄⋊C₄	C₂³×C₅₃	1	1	yes	characteristic
271	C₂×C₄⋊C₄	C₂³×C₅₃	1	1	yes	characteristic
272	C₂×C₄⋊C₄	C₂³×C₅₃	1	1	yes	characteristic
273	C₂×C₄⋊C₄	C₂³×C₅₃	1	1	yes	characteristic
274	C₂×C₂²⋊C₄	—	2	1	no	—
275	C₂×C₂²⋊C₄	—	2	1	no	—
276	C₂×C₂²⋊C₄	D₄×C₅₃	1	1	yes	characteristic
277	C₂×C₂²⋊C₄	D₄×C₅₃	1	1	yes	characteristic
278	C₂×C₄²	—	2	2	no	—
279	C₂×C₄²	—	2	2	no	—
280	C₂×C₄²	C₂³×C₅₃	1	2	yes	—
281	#107	—	2	4	no	—
282	#106	C₂×C₄×C₅₃	1	4	yes	—
283	#106	C₂×C₄×C₅₃	1	4	yes	—
284	#106	C₂×C₄×C₅₃	1	4	yes	—
285	#106	C₂×C₄×C₅₃	1	4	yes	—
286	C₂²⋊C₈	—	2	2	no	—
287	#97	—	2	4	no	—
288	#95	—	2	2	no	—
289	#95	—	2	2	no	—
290	#95	—	2	2	no	—
291	#95	—	2	1	no	—
292	#95	—	2	1	no	—
293	#95	—	2	2	no	—
294	#95	D₄×C₅₃	1	2	yes	—
295	C₂⁴	C₂×D₄×C₅₃	1	1	yes	characteristic
296	C₂²×C₄	—	2	2	no	—
297	C₂²×C₄	—	2	2	no	—
298	C₂²×C₄	—	2	4	no	—
299	C₂²×C₄	—	2	1	no	—
300	C₂²×C₄	—	2	1	no	—
301	C₂²×C₄	—	2	1	no	—
302	C₂²×C₄	—	2	1	no	—
303	C₂²×C₄	—	2	4	no	—
304	C₂²×C₄	—	2	2	no	—
305	C₂²×C₄	—	2	2	no	—
306	C₂²×C₄	—	4	2	no	—
307	C₂²×C₄	—	4	2	no	—
308	C₂²×C₄	—	8	2	no	—
309	C₂²×C₄	C₂²×C₄×C₅₃	1	2	yes	—
310	C₂²×C₄	C₂×D₄×C₅₃	1	2	yes	—
311	C₂²×C₄	C₂×D₄×C₅₃	1	2	yes	—
312	C₂²×C₄	C₂×D₄×C₅₃	1	1	yes	characteristic
313	C₂²×C₄	#41×C₅₃	1	2	yes	—
314	C₂²×C₄	C₂⁴×C₅₃	1	1	yes	characteristic
315	C₂×C₈	—	2	2	no	—
316	C₂×C₈	—	2	8	no	—
317	C₂×C₈	—	2	2	no	—
318	C₂×C₈	—	2	2	no	—
319	C₂×C₈	—	2	2	no	—
320	C₂×C₈	C₂²×C₄×C₅₃	1	2	yes	—
321	C₂×C₈	C₂²×C₄×C₅₃	1	2	yes	—
322	C₂×C₈	#41×C₅₃	1	4	yes	—
323	C₄⋊C₄	—	2	4	no	—
324	C₄⋊C₄	—	2	4	no	—
325	C₄⋊C₄	—	2	1	no	—
326	C₄⋊C₄	—	2	1	no	—
327	C₄⋊C₄	—	2	1	no	—
328	C₄⋊C₄	—	2	1	no	—
329	C₄⋊C₄	—	2	2	no	—
330	C₄⋊C₄	—	2	2	no	—
331	C₄⋊C₄	—	2	2	no	—
332	C₄⋊C₄	—	2	2	no	—
333	C₄⋊C₄	—	4	4	no	—
334	C₄⋊C₄	—	8	2	no	—
335	C₄⋊C₄	C₂²×C₄×C₅₃	1	2	yes	—
336	C₄⋊C₄	C₂²×C₄×C₅₃	1	2	yes	—
337	C₄⋊C₄	C₂²×C₄×C₅₃	1	2	yes	—
338	C₄⋊C₄	C₂²×C₄×C₅₃	1	2	yes	—
339	C₂²⋊C₄	—	2	2	no	—
340	C₂²⋊C₄	—	2	2	no	—
341	C₂²⋊C₄	—	4	2	no	—
342	C₂²⋊C₄	—	4	2	no	—
343	C₄²	—	2	2	no	—
344	C₄²	—	2	2	no	—
345	C₄²	—	4	4	no	—
346	C₄²	—	8	2	no	—
347	C₂³	—	2	2	no	—
348	C₂³	—	2	2	no	—
349	C₂³	C₄×D₄×C₅₃	1	2	yes	—
350	C₂³	AGL(1,ℤ/8)×C₅₃	1	2	yes	—
351	C₂³	#137×C₅₃	1	2	yes	—
352	C₂³	C₂²×D₄×C₅₃	1	1	yes	characteristic
353	C₂×C₄	—	2	2	no	—
354	C₂×C₄	—	2	2	no	—
355	C₂×C₄	—	2	2	no	—
356	C₂×C₄	—	2	4	no	—
357	C₂×C₄	—	2	2	no	—
358	C₂×C₄	—	2	2	no	—
359	C₂×C₄	—	2	2	no	—
360	C₂×C₄	—	2	2	no	—
361	C₂×C₄	—	2	2	no	—
362	C₂×C₄	—	2	2	no	—
363	C₂×C₄	—	2	2	no	—
364	C₂×C₄	—	2	2	no	—
365	C₂×C₄	—	2	2	no	—
366	C₂×C₄	—	2	2	no	—
367	C₂×C₄	—	4	1	no	—
368	C₂×C₄	—	4	1	no	—
369	C₂×C₄	—	4	1	no	—
370	C₂×C₄	—	4	1	no	—
371	C₂×C₄	—	4	2	no	—
372	C₂×C₄	—	4	2	no	—
373	C₂×C₄	—	4	8	no	—
374	C₂×C₄	—	4	2	no	—
375	C₂×C₄	—	4	2	no	—
376	C₂×C₄	—	4	2	no	—
377	C₂×C₄	—	4	2	no	—
378	C₂×C₄	—	4	2	no	—
379	C₂×C₄	—	4	2	no	—
380	C₂×C₄	—	8	4	no	—
381	C₂×C₄	C₄×D₄×C₅₃	1	2	yes	—
382	C₂×C₄	C₂³×C₄×C₅₃	1	1	yes	characteristic
383	C₂×C₄	C₂×#41×C₅₃	1	2	yes	—
384	C₂×C₄	C₂×#41×C₅₃	1	1	yes	characteristic
385	C₂×C₄	2₋¹⁺⁴×C₅₃	1	2	yes	—
386	C₈	—	2	2	no	—
387	C₈	—	4	2	no	—
388	C₈	#117×C₅₃	1	4	yes	complemented
389	C₂²	—	2	2	no	—
390	C₂²	—	2	4	no	—
391	C₂²	—	2	4	no	—
392	C₂²	—	2	2	no	—
393	C₂²	C₅₃×#440	1	4	yes	—
394	C₂²	C₂×C₄×D₄×C₅₃	1	1	yes	characteristic
395	C₂²	C₅₃×#547	1	2	yes	—
396	C₂²	C₂×AGL(1,ℤ/8)×C₅₃	1	1	yes	characteristic
397	C₂²	C₂×#137×C₅₃	1	1	yes	characteristic
398	C₂²	C₅₃×#577	1	2	yes	—
399	C₄	—	2	2	no	—
400	C₄	—	4	2	no	—
401	C₄	—	4	2	no	—
402	C₄	—	4	2	no	—
403	C₄	—	4	2	no	—
404	C₄	—	4	2	no	—
405	C₄	—	4	2	no	—
406	C₄	—	8	2	no	—
407	C₄	—	8	2	no	—
408	C₄	C₂×#117×C₅₃	1	1	yes	characteristic
409	C₄	C₅₃×#519	1	1	yes	characteristic
410	C₄	C₅₃×#552	1	2	yes	—
411	C₂	—	2	4	no	—
412	C₂	C₅₃×#2344	1	1	yes	characteristic
413	C₂	C₂×C₅₃×#440	1	1	yes	characteristic
414	C₂	C₅₃×#2949	1	1	yes	characteristic
415	C₂	C₅₃×#3344	1	2	yes	—
416	C₂	C₅₃×#3345	1	2	yes	—

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

Complements

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

N	H ≅ G/N	type
4. C₅₃×#1944	C₂	semidirect
5. C₅₃×#1920	C₂	semidirect
6. C₅₃×#1919	C₂	semidirect
10. C₅₃×#1776	C₂	semidirect
17. C₂×#106×C₅₃	C₄	semidirect
179. C₈×C₅₃	#117	semidirect
180. #22615	C₅₃	direct (N × H)
206. #1944	C₂×C₅₃	semidirect
207. #1920	C₂×C₅₃	semidirect
208. #1919	C₂×C₅₃	semidirect
212. #1776	C₂×C₅₃	semidirect
225. C₂×#106	C₄×C₅₃	semidirect
250. C₅₃	#22615	direct (N × H)
388. C₈	#117×C₅₃	semidirect

Maximal subgroups

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

subgroup	index
C₂²×#106×C₅₃	2
C₅₃×#2345	2
C₅₃×#2344	2
C₅₃×#1944	2
C₅₃×#1920	2
C₅₃×#1919	2
C₅₃×#1824	2
C₅₃×#1823	2
C₅₃×#1808	2
C₅₃×#1776	2
C₅₃×#1752	2
#22615	53

Subgroups — C53×#22615 #27230127

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅₃×#22615 #27230127