Subgroups — C₅₃×#8043 #27215555

Order 13568.

790 conjugacy classes of subgroups.

The subgroup lattice has 372 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₅₃×#431	C₂	1	1	yes	maximal, characteristic
2	C₂×C₅₃×#423	C₂	1	1	yes	maximal, characteristic
3	C₂×C₅₃×#384	C₂	1	1	yes	maximal, characteristic
4	C₅₃×#1296	C₂	1	4	yes	maximal, complemented
5	C₂²×M₁₆×C₅₃	—	2	1	no	—
6	C₂²×C₄⋊C₄×C₅₃	—	2	1	no	—
7	C₂²×C₄⋊C₄×C₅₃	—	2	1	no	—
8	C₂²×C₄⋊C₄×C₅₃	C₄	1	2	yes	—
9	C₂²×C₄²×C₅₃	C₂²	1	1	yes	characteristic
10	C₅₃×#431	—	2	2	no	—
11	C₅₃×#431	C₄	1	4	yes	complemented
12	C₅₃×#423	—	2	2	no	—
13	C₅₃×#423	C₄	1	4	yes	complemented
14	C₂×C₄⋊C₈×C₅₃	—	2	1	no	—
15	C₂×C₄⋊C₈×C₅₃	C₂²	1	2	yes	—
16	C₂×C₄⋊C₈×C₅₃	C₂²	1	2	yes	—
17	C₂×C₂²⋊C₈×C₅₃	—	2	1	no	—
18	C₂×C₄×C₈×C₅₃	—	2	1	no	—
19	C₅₃×#384	—	2	1	no	—
20	C₅₃×#384	—	2	1	no	—
21	C₅₃×#384	C₄	1	2	yes	—
22	C₅₃×#384	C₂²	1	2	yes	—
23	C₂³×C₄×C₅₃	—	2	2	no	—
24	C₂³×C₄×C₅₃	C₂×C₄	1	1	yes	characteristic
25	C₂³×C₄×C₅₃	D₄	1	1	yes	characteristic
26	C₂³×C₄×C₅₃	D₄	1	1	yes	characteristic
27	C₂×M₁₆×C₅₃	—	2	4	no	—
28	C₂×M₁₆×C₅₃	—	4	2	no	—
29	C₂×M₁₆×C₅₃	—	4	2	no	—
30	C₂²×C₈×C₅₃	—	2	2	no	—
31	C₂²×C₈×C₅₃	—	2	2	no	—
32	C₂×C₄⋊C₄×C₅₃	—	2	2	no	—
33	C₂×C₄⋊C₄×C₅₃	—	2	2	no	—
34	C₂×C₄⋊C₄×C₅₃	—	2	2	no	—
35	C₂×C₄⋊C₄×C₅₃	—	2	2	no	—
36	C₂×C₄⋊C₄×C₅₃	—	2	2	no	—
37	C₂×C₄⋊C₄×C₅₃	—	4	4	no	—
38	C₂×C₄⋊C₄×C₅₃	—	4	4	no	—
39	C₂×C₄⋊C₄×C₅₃	—	4	2	no	—
40	C₂×C₄⋊C₄×C₅₃	—	4	2	no	—
41	C₂×C₄⋊C₄×C₅₃	C₂×C₄	1	4	yes	—
42	C₂×C₄²×C₅₃	C₂×C₄	1	2	yes	—
43	C₂×C₄²×C₅₃	C₂×C₄	1	2	yes	—
44	C₂×C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
45	C₂×C₄²×C₅₃	D₄	1	2	yes	—
46	C₂×C₄²×C₅₃	D₄	1	1	yes	characteristic
47	C₂×C₄²×C₅₃	D₄	1	1	yes	characteristic
48	C₂×C₄²×C₅₃	Q₈	1	2	yes	—
49	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
50	C₄⋊C₈×C₅₃	—	2	4	no	—
51	C₄⋊C₈×C₅₃	—	2	2	no	—
52	C₄⋊C₈×C₅₃	—	2	2	no	—
53	C₄⋊C₈×C₅₃	C₂×C₄	1	4	yes	complemented
54	C₄⋊C₈×C₅₃	C₂×C₄	1	4	yes	complemented
55	C₂²⋊C₈×C₅₃	—	2	4	no	—
56	C₄×C₈×C₅₃	—	2	4	no	—
57	C₂⁴×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
58	C₂²×C₄×C₅₃	—	2	2	no	—
59	C₂²×C₄×C₅₃	—	2	2	no	—
60	C₂²×C₄×C₅₃	—	2	2	no	—
61	C₂²×C₄×C₅₃	—	2	2	no	—
62	C₂²×C₄×C₅₃	—	2	4	no	—
63	C₂²×C₄×C₅₃	—	4	4	no	—
64	C₂²×C₄×C₅₃	—	4	4	no	—
65	C₂²×C₄×C₅₃	—	4	4	no	—
66	C₂²×C₄×C₅₃	C₄²	1	2	yes	—
67	C₂²×C₄×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
68	C₂²×C₄×C₅₃	C₂²⋊C₄	1	2	yes	—
69	C₂²×C₄×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
70	C₂²×C₄×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
71	C₂²×C₄×C₅₃	C₂²⋊C₄	1	2	yes	—
72	C₂²×C₄×C₅₃	C₄⋊C₄	1	2	yes	—
73	C₂²×C₄×C₅₃	C₄⋊C₄	1	2	yes	—
74	C₂²×C₄×C₅₃	C₄⋊C₄	1	2	yes	—
75	C₂²×C₄×C₅₃	D₈	1	2	yes	—
76	C₂²×C₄×C₅₃	SD₁₆	1	2	yes	—
77	C₂²×C₄×C₅₃	SD₁₆	1	2	yes	—
78	C₂²×C₄×C₅₃	Q₁₆	1	2	yes	—
79	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
80	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
81	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
82	M₁₆×C₅₃	—	4	8	no	—
83	C₂×C₈×C₅₃	—	2	4	no	—
84	C₂×C₈×C₅₃	—	2	4	no	—
85	C₂×C₈×C₅₃	—	4	2	no	—
86	C₂×C₈×C₅₃	—	4	2	no	—
87	C₂×C₈×C₅₃	—	4	2	no	—
88	C₂×C₈×C₅₃	—	4	2	no	—
89	C₄⋊C₄×C₅₃	—	4	4	no	—
90	C₄⋊C₄×C₅₃	—	4	4	no	—
91	C₄⋊C₄×C₅₃	—	4	4	no	—
92	C₄⋊C₄×C₅₃	—	4	4	no	—
93	C₄⋊C₄×C₅₃	—	4	2	no	—
94	C₄⋊C₄×C₅₃	—	4	2	no	—
95	C₄⋊C₄×C₅₃	—	4	4	no	—
96	C₄²×C₅₃	C₄²	1	2	yes	—
97	C₄²×C₅₃	C₂²⋊C₄	1	2	yes	—
98	C₄²×C₅₃	C₂²⋊C₄	1	2	yes	—
99	C₄²×C₅₃	C₄⋊C₄	1	2	yes	—
100	C₄²×C₅₃	C₄⋊C₄	1	2	yes	—
101	C₄²×C₅₃	C₄⋊C₄	1	2	yes	—
102	C₄²×C₅₃	C₂²×C₄	1	1	yes	characteristic
103	C₄²×C₅₃	C₂²×C₄	1	1	yes	characteristic
104	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
105	C₄²×C₅₃	C₂×Q₈	1	1	yes	characteristic
106	C₂³×C₅₃	—	2	1	no	—
107	C₂³×C₅₃	—	2	1	no	—
108	C₂³×C₅₃	—	2	1	no	—
109	C₂³×C₅₃	—	2	1	no	—
110	C₂³×C₅₃	#95	1	2	yes	—
111	C₂³×C₅₃	#102	1	2	yes	—
112	C₂³×C₅₃	#103	1	2	yes	—
113	C₂³×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
114	C₂×C₄×C₅₃	—	2	1	no	—
115	C₂×C₄×C₅₃	—	2	1	no	—
116	C₂×C₄×C₅₃	—	2	1	no	—
117	C₂×C₄×C₅₃	—	2	1	no	—
118	C₂×C₄×C₅₃	—	2	4	no	—
119	C₂×C₄×C₅₃	—	2	4	no	—
120	C₂×C₄×C₅₃	—	2	4	no	—
121	C₂×C₄×C₅₃	—	2	4	no	—
122	C₂×C₄×C₅₃	—	4	4	no	—
123	C₂×C₄×C₅₃	—	4	4	no	—
124	C₂×C₄×C₅₃	—	4	4	no	—
125	C₂×C₄×C₅₃	—	8	8	no	—
126	C₂×C₄×C₅₃	#95	1	2	yes	—
127	C₂×C₄×C₅₃	#95	1	2	yes	—
128	C₂×C₄×C₅₃	#95	1	2	yes	—
129	C₂×C₄×C₅₃	#100	1	2	yes	—
130	C₂×C₄×C₅₃	#101	1	2	yes	—
131	C₂×C₄×C₅₃	#102	1	2	yes	—
132	C₂×C₄×C₅₃	#102	1	4	yes	—
133	C₂×C₄×C₅₃	#103	1	2	yes	—
134	C₂×C₄×C₅₃	#103	1	4	yes	—
135	C₂×C₄×C₅₃	#106	1	4	yes	—
136	C₂×C₄×C₅₃	#107	1	4	yes	—
137	C₂×C₄×C₅₃	C₂×C₄²	1	1	yes	characteristic
138	C₂×C₄×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
139	C₂×C₄×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
140	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
141	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
142	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
143	C₂×C₄×C₅₃	C₂×D₈	1	1	yes	characteristic
144	C₂×C₄×C₅₃	C₂×SD₁₆	1	1	yes	characteristic
145	C₂×C₄×C₅₃	C₂×SD₁₆	1	1	yes	characteristic
146	C₂×C₄×C₅₃	C₂×Q₁₆	1	1	yes	characteristic
147	C₂×C₄×C₅₃	AGL(1,ℤ/8)	1	2	yes	—
148	C₂×C₄×C₅₃	#137	1	2	yes	—
149	C₈×C₅₃	—	4	4	no	—
150	C₈×C₅₃	—	4	4	no	—
151	#8043	C₅₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
152	C₂²×C₅₃	—	2	2	no	—
153	C₂²×C₅₃	—	2	2	no	—
154	C₂²×C₅₃	—	2	2	no	—
155	C₂²×C₅₃	—	2	2	no	—
156	C₂²×C₅₃	—	2	2	no	—
157	C₂²×C₅₃	—	2	2	no	—
158	C₂²×C₅₃	#340	1	4	yes	—
159	C₂²×C₅₃	C₂×#95	1	1	yes	characteristic
160	C₂²×C₅₃	C₂×#100	1	1	yes	characteristic
161	C₂²×C₅₃	C₂×#101	1	1	yes	characteristic
162	C₂²×C₅₃	C₂×#102	1	1	yes	characteristic
163	C₂²×C₅₃	C₂×#103	1	1	yes	characteristic
164	C₂²×C₅₃	#418	1	1	yes	characteristic
165	C₂²×C₅₃	#419	1	1	yes	characteristic
166	C₄×C₅₃	—	2	2	no	—
167	C₄×C₅₃	—	2	1	no	—
168	C₄×C₅₃	—	2	1	no	—
169	C₄×C₅₃	—	8	4	no	—
170	C₄×C₅₃	#340	1	4	yes	—
171	C₄×C₅₃	#343	1	4	yes	—
172	C₄×C₅₃	C₂×#102	1	1	yes	characteristic
173	C₄×C₅₃	C₂×#103	1	1	yes	characteristic
174	C₄×C₅₃	#417	1	2	yes	—
175	C₄×C₅₃	C₂×#106	1	1	yes	characteristic
176	C₄×C₅₃	C₂×#107	1	1	yes	characteristic
177	C₄×C₅₃	#428	1	2	yes	—
178	C₂×#431	C₂×C₅₃	1	1	yes	characteristic
179	C₂×#423	C₂×C₅₃	1	1	yes	characteristic
180	C₂×#384	C₂×C₅₃	1	1	yes	characteristic
181	#1296	C₂×C₅₃	1	4	yes	complemented
182	C₂×C₅₃	—	2	4	no	—
183	C₂×C₅₃	#1545	1	1	yes	characteristic
184	C₂×C₅₃	#1546	1	1	yes	characteristic
185	C₂×C₅₃	#1548	1	1	yes	characteristic
186	C₂×C₅₃	#1549	1	1	yes	characteristic
187	C₂×C₅₃	C₂×#340	1	1	yes	characteristic
188	C₂×C₅₃	#1736	1	1	yes	characteristic
189	C₂×C₅₃	C₂×#343	1	1	yes	characteristic
190	C₂²×M₁₆	—	2	1	no	—
191	C₂²×C₄⋊C₄	—	2	1	no	—
192	C₂²×C₄⋊C₄	—	2	1	no	—
193	C₂²×C₄⋊C₄	C₄×C₅₃	1	2	yes	—
194	C₂²×C₄²	C₂²×C₅₃	1	1	yes	characteristic
195	#431	—	2	2	no	—
196	#431	C₄×C₅₃	1	4	yes	complemented
197	#423	—	2	2	no	—
198	#423	C₄×C₅₃	1	4	yes	complemented
199	C₂×C₄⋊C₈	—	2	1	no	—
200	C₂×C₄⋊C₈	C₂²×C₅₃	1	2	yes	—
201	C₂×C₄⋊C₈	C₂²×C₅₃	1	2	yes	—
202	C₂×C₂²⋊C₈	—	2	1	no	—
203	C₂×C₄×C₈	—	2	1	no	—
204	#384	—	2	1	no	—
205	#384	—	2	1	no	—
206	#384	C₄×C₅₃	1	2	yes	—
207	#384	C₂²×C₅₃	1	2	yes	—
208	C₅₃	#8043	1	1	yes	characteristic, direct product factor, Sylow
209	C₂³×C₄	—	2	2	no	—
210	C₂³×C₄	C₂×C₄×C₅₃	1	1	yes	characteristic
211	C₂³×C₄	D₄×C₅₃	1	1	yes	characteristic
212	C₂³×C₄	D₄×C₅₃	1	1	yes	characteristic
213	C₂×M₁₆	—	2	4	no	—
214	C₂×M₁₆	—	4	2	no	—
215	C₂×M₁₆	—	4	2	no	—
216	C₂²×C₈	—	2	2	no	—
217	C₂²×C₈	—	2	2	no	—
218	C₂×C₄⋊C₄	—	2	2	no	—
219	C₂×C₄⋊C₄	—	2	2	no	—
220	C₂×C₄⋊C₄	—	2	2	no	—
221	C₂×C₄⋊C₄	—	2	2	no	—
222	C₂×C₄⋊C₄	—	2	2	no	—
223	C₂×C₄⋊C₄	—	4	4	no	—
224	C₂×C₄⋊C₄	—	4	2	no	—
225	C₂×C₄⋊C₄	—	4	2	no	—
226	C₂×C₄⋊C₄	—	4	4	no	—
227	C₂×C₄⋊C₄	C₂×C₄×C₅₃	1	4	yes	—
228	C₂×C₄²	C₂×C₄×C₅₃	1	2	yes	—
229	C₂×C₄²	C₂×C₄×C₅₃	1	2	yes	—
230	C₂×C₄²	C₂×C₄×C₅₃	1	1	yes	characteristic
231	C₂×C₄²	D₄×C₅₃	1	2	yes	—
232	C₂×C₄²	D₄×C₅₃	1	1	yes	characteristic
233	C₂×C₄²	D₄×C₅₃	1	1	yes	characteristic
234	C₂×C₄²	Q₈×C₅₃	1	2	yes	—
235	C₂×C₄²	C₂³×C₅₃	1	1	yes	characteristic
236	C₄⋊C₈	—	2	2	no	—
237	C₄⋊C₈	—	2	2	no	—
238	C₄⋊C₈	—	2	4	no	—
239	C₄⋊C₈	C₂×C₄×C₅₃	1	4	yes	complemented
240	C₄⋊C₈	C₂×C₄×C₅₃	1	4	yes	complemented
241	C₂²⋊C₈	—	2	4	no	—
242	C₄×C₈	—	2	4	no	—
243	C₂⁴	C₂²⋊C₄×C₅₃	1	1	yes	characteristic
244	C₂²×C₄	—	2	2	no	—
245	C₂²×C₄	—	2	2	no	—
246	C₂²×C₄	—	2	2	no	—
247	C₂²×C₄	—	2	2	no	—
248	C₂²×C₄	—	2	4	no	—
249	C₂²×C₄	—	4	4	no	—
250	C₂²×C₄	—	4	4	no	—
251	C₂²×C₄	—	4	4	no	—
252	C₂²×C₄	C₄²×C₅₃	1	2	yes	—
253	C₂²×C₄	C₂²⋊C₄×C₅₃	1	1	yes	characteristic
254	C₂²×C₄	C₂²⋊C₄×C₅₃	1	2	yes	—
255	C₂²×C₄	C₂²⋊C₄×C₅₃	1	1	yes	characteristic
256	C₂²×C₄	C₂²⋊C₄×C₅₃	1	1	yes	characteristic
257	C₂²×C₄	C₂²⋊C₄×C₅₃	1	2	yes	—
258	C₂²×C₄	C₄⋊C₄×C₅₃	1	2	yes	—
259	C₂²×C₄	C₄⋊C₄×C₅₃	1	2	yes	—
260	C₂²×C₄	C₄⋊C₄×C₅₃	1	2	yes	—
261	C₂²×C₄	D₈×C₅₃	1	2	yes	—
262	C₂²×C₄	SD₁₆×C₅₃	1	2	yes	—
263	C₂²×C₄	SD₁₆×C₅₃	1	2	yes	—
264	C₂²×C₄	Q₁₆×C₅₃	1	2	yes	—
265	C₂²×C₄	C₂²×C₄×C₅₃	1	1	yes	characteristic
266	C₂²×C₄	C₂×D₄×C₅₃	1	1	yes	characteristic
267	C₂²×C₄	C₂×D₄×C₅₃	1	1	yes	characteristic
268	M₁₆	—	4	8	no	—
269	C₂×C₈	—	2	4	no	—
270	C₂×C₈	—	2	4	no	—
271	C₂×C₈	—	4	2	no	—
272	C₂×C₈	—	4	2	no	—
273	C₂×C₈	—	4	2	no	—
274	C₂×C₈	—	4	2	no	—
275	C₄⋊C₄	—	4	4	no	—
276	C₄⋊C₄	—	4	4	no	—
277	C₄⋊C₄	—	4	4	no	—
278	C₄⋊C₄	—	4	4	no	—
279	C₄⋊C₄	—	4	2	no	—
280	C₄⋊C₄	—	4	2	no	—
281	C₄⋊C₄	—	4	4	no	—
282	C₄²	C₄²×C₅₃	1	2	yes	—
283	C₄²	C₂²⋊C₄×C₅₃	1	2	yes	—
284	C₄²	C₂²⋊C₄×C₅₃	1	2	yes	—
285	C₄²	C₄⋊C₄×C₅₃	1	2	yes	—
286	C₄²	C₄⋊C₄×C₅₃	1	2	yes	—
287	C₄²	C₄⋊C₄×C₅₃	1	2	yes	—
288	C₄²	C₂²×C₄×C₅₃	1	1	yes	characteristic
289	C₄²	C₂²×C₄×C₅₃	1	1	yes	characteristic
290	C₄²	C₂×D₄×C₅₃	1	1	yes	characteristic
291	C₄²	C₂×Q₈×C₅₃	1	1	yes	characteristic
292	C₂³	—	2	1	no	—
293	C₂³	—	2	1	no	—
294	C₂³	—	2	1	no	—
295	C₂³	—	2	1	no	—
296	C₂³	#95×C₅₃	1	2	yes	—
297	C₂³	#102×C₅₃	1	2	yes	—
298	C₂³	#103×C₅₃	1	2	yes	—
299	C₂³	C₂×C₂²⋊C₄×C₅₃	1	1	yes	characteristic
300	C₂×C₄	—	2	1	no	—
301	C₂×C₄	—	2	1	no	—
302	C₂×C₄	—	2	1	no	—
303	C₂×C₄	—	2	1	no	—
304	C₂×C₄	—	2	4	no	—
305	C₂×C₄	—	2	4	no	—
306	C₂×C₄	—	2	4	no	—
307	C₂×C₄	—	2	4	no	—
308	C₂×C₄	—	4	4	no	—
309	C₂×C₄	—	4	4	no	—
310	C₂×C₄	—	4	4	no	—
311	C₂×C₄	—	8	8	no	—
312	C₂×C₄	#95×C₅₃	1	2	yes	—
313	C₂×C₄	#95×C₅₃	1	2	yes	—
314	C₂×C₄	#95×C₅₃	1	2	yes	—
315	C₂×C₄	#100×C₅₃	1	2	yes	—
316	C₂×C₄	#101×C₅₃	1	2	yes	—
317	C₂×C₄	#102×C₅₃	1	2	yes	—
318	C₂×C₄	#102×C₅₃	1	4	yes	—
319	C₂×C₄	#103×C₅₃	1	2	yes	—
320	C₂×C₄	#103×C₅₃	1	4	yes	—
321	C₂×C₄	#106×C₅₃	1	4	yes	—
322	C₂×C₄	#107×C₅₃	1	4	yes	—
323	C₂×C₄	C₂×C₄²×C₅₃	1	1	yes	characteristic
324	C₂×C₄	C₂×C₂²⋊C₄×C₅₃	1	1	yes	characteristic
325	C₂×C₄	C₂×C₂²⋊C₄×C₅₃	1	1	yes	characteristic
326	C₂×C₄	C₂×C₄⋊C₄×C₅₃	1	1	yes	characteristic
327	C₂×C₄	C₂×C₄⋊C₄×C₅₃	1	1	yes	characteristic
328	C₂×C₄	C₂×C₄⋊C₄×C₅₃	1	1	yes	characteristic
329	C₂×C₄	C₂×D₈×C₅₃	1	1	yes	characteristic
330	C₂×C₄	C₂×SD₁₆×C₅₃	1	1	yes	characteristic
331	C₂×C₄	C₂×SD₁₆×C₅₃	1	1	yes	characteristic
332	C₂×C₄	C₂×Q₁₆×C₅₃	1	1	yes	characteristic
333	C₂×C₄	AGL(1,ℤ/8)×C₅₃	1	2	yes	—
334	C₂×C₄	#137×C₅₃	1	2	yes	—
335	C₈	—	4	4	no	—
336	C₈	—	4	4	no	—
337	C₂²	—	2	2	no	—
338	C₂²	—	2	2	no	—
339	C₂²	—	2	2	no	—
340	C₂²	—	2	2	no	—
341	C₂²	—	2	2	no	—
342	C₂²	—	2	2	no	—
343	C₂²	C₅₃×#340	1	4	yes	—
344	C₂²	C₂×#95×C₅₃	1	1	yes	characteristic
345	C₂²	C₂×#100×C₅₃	1	1	yes	characteristic
346	C₂²	C₂×#101×C₅₃	1	1	yes	characteristic
347	C₂²	C₂×#102×C₅₃	1	1	yes	characteristic
348	C₂²	C₂×#103×C₅₃	1	1	yes	characteristic
349	C₂²	C₅₃×#418	1	1	yes	characteristic
350	C₂²	C₅₃×#419	1	1	yes	characteristic
351	C₄	—	2	2	no	—
352	C₄	—	2	1	no	—
353	C₄	—	2	1	no	—
354	C₄	—	8	4	no	—
355	C₄	C₅₃×#340	1	4	yes	—
356	C₄	C₅₃×#343	1	4	yes	—
357	C₄	C₂×#102×C₅₃	1	1	yes	characteristic
358	C₄	C₂×#103×C₅₃	1	1	yes	characteristic
359	C₄	C₅₃×#417	1	2	yes	—
360	C₄	C₂×#106×C₅₃	1	1	yes	characteristic
361	C₄	C₂×#107×C₅₃	1	1	yes	characteristic
362	C₄	C₅₃×#428	1	2	yes	—
363	C₂	—	2	4	no	—
364	C₂	C₅₃×#1545	1	1	yes	characteristic
365	C₂	C₅₃×#1546	1	1	yes	characteristic
366	C₂	C₅₃×#1548	1	1	yes	characteristic
367	C₂	C₅₃×#1549	1	1	yes	characteristic
368	C₂	C₂×C₅₃×#340	1	1	yes	characteristic
369	C₂	C₅₃×#1736	1	1	yes	characteristic
370	C₂	C₂×C₅₃×#343	1	1	yes	characteristic

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

Complements

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

N	H ≅ G/N	type
4. C₅₃×#1296	C₂	semidirect
11. C₅₃×#431	C₄	semidirect
13. C₅₃×#423	C₄	semidirect
53. C₄⋊C₈×C₅₃	C₂×C₄	semidirect
54. C₄⋊C₈×C₅₃	C₂×C₄	semidirect
151. #8043	C₅₃	direct (N × H)
181. #1296	C₂×C₅₃	semidirect
196. #431	C₄×C₅₃	semidirect
198. #423	C₄×C₅₃	semidirect
208. C₅₃	#8043	direct (N × H)
239. C₄⋊C₈	C₂×C₄×C₅₃	semidirect
240. C₄⋊C₈	C₂×C₄×C₅₃	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₅₃×#431	2
C₂×C₅₃×#423	2
C₂×C₅₃×#384	2
C₅₃×#1296	2
#8043	53

Subgroups — C53×#8043 #27215555

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅₃×#8043 #27215555