Subgroups — C₅₃×#22262 #27229774

Order 13568.

1058 conjugacy classes of subgroups.

The subgroup lattice has 408 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₅₃×#436	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×C₈×D₄×C₅₃	C₂	1	1	yes	maximal, characteristic, complemented
3	C₄×#127×C₅₃	C₂	1	1	yes	maximal, characteristic
4	C₅₃×#1954	C₂	1	1	yes	maximal, characteristic, complemented
5	C₅₃×#1954	C₂	1	1	yes	maximal, characteristic, complemented
6	C₅₃×#1870	C₂	1	4	yes	maximal, complemented
7	C₅₃×#1869	C₂	1	4	yes	maximal, complemented
8	C₅₃×#1831	C₂	1	1	yes	maximal, characteristic, complemented
9	C₅₃×#1744	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂²×M₁₆×C₅₃	C₂²	1	2	yes	—
11	C₂³×C₈×C₅₃	C₂²	1	2	yes	—
12	C₂×#127×C₅₃	C₄	1	1	yes	characteristic
13	C₂×C₄×D₄×C₅₃	C₂²	1	1	yes	characteristic
14	C₂×C₄×D₄×C₅₃	C₂²	1	1	yes	characteristic
15	C₂×C₄×D₄×C₅₃	C₂²	1	4	yes	—
16	C₅₃×#436	—	2	4	no	—
17	C₈×D₄×C₅₃	—	2	4	no	—
18	C₂×C₄⋊C₈×C₅₃	C₂²	1	1	yes	characteristic
19	C₂×C₄⋊C₈×C₅₃	C₂²	1	1	yes	characteristic
20	C₂×C₂²⋊C₈×C₅₃	C₂²	1	2	yes	—
21	C₂×C₂²⋊C₈×C₅₃	C₂²	1	2	yes	—
22	C₂×C₂²⋊C₈×C₅₃	C₂²	1	4	yes	—
23	C₂×C₂²⋊C₈×C₅₃	C₂²	1	4	yes	—
24	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
25	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
26	C₅₃×#390	C₄	1	4	yes	—
27	C₅₃×#390	C₄	1	1	yes	characteristic
28	C₅₃×#390	C₄	1	1	yes	characteristic
29	C₅₃×#384	C₄	1	1	yes	characteristic
30	C₄³×C₅₃	C₂²	1	1	yes	characteristic
31	C₅₃×#336	C₂²	1	2	yes	—
32	C₅₃×#336	C₂²	1	2	yes	—
33	C₅₃×#336	C₂²	1	4	yes	—
34	C₂²×D₄×C₅₃	C₂×C₄	1	1	yes	characteristic
35	C₂²×D₄×C₅₃	C₂×C₄	1	4	yes	—
36	C₂²×D₄×C₅₃	C₂×C₄	1	1	yes	characteristic
37	C₂³×C₄×C₅₃	C₂³	1	2	yes	—
38	C₂³×C₄×C₅₃	C₂³	1	2	yes	—
39	C₂×M₁₆×C₅₃	—	2	4	no	—
40	C₂×M₁₆×C₅₃	—	4	4	no	—
41	C₂²×C₈×C₅₃	—	2	4	no	—
42	C₂²×C₈×C₅₃	—	4	4	no	—
43	C₂²×C₈×C₅₃	C₂³	1	2	yes	—
44	C₂²×C₈×C₅₃	C₂³	1	2	yes	—
45	C₂²×C₈×C₅₃	C₂³	1	2	yes	—
46	C₂²×C₈×C₅₃	C₂³	1	1	yes	characteristic
47	C₂²×C₈×C₅₃	C₂³	1	1	yes	characteristic
48	#127×C₅₃	—	2	4	no	—
49	C₄×D₄×C₅₃	—	2	2	no	—
50	C₄×D₄×C₅₃	—	2	2	no	—
51	C₄×D₄×C₅₃	—	2	2	no	—
52	C₄×D₄×C₅₃	—	2	2	no	—
53	C₄×D₄×C₅₃	—	4	4	no	—
54	C₄×D₄×C₅₃	—	4	4	no	—
55	C₂×C₄⋊C₄×C₅₃	C₂×C₄	1	1	yes	characteristic
56	C₂×C₄⋊C₄×C₅₃	C₂×C₄	1	1	yes	characteristic
57	C₂×C₄⋊C₄×C₅₃	C₂×C₄	1	4	yes	—
58	C₂×C₂²⋊C₄×C₅₃	C₂×C₄	1	2	yes	—
59	C₂×C₂²⋊C₄×C₅₃	C₂×C₄	1	2	yes	—
60	C₂×C₂²⋊C₄×C₅₃	C₂×C₄	1	4	yes	—
61	C₂×C₂²⋊C₄×C₅₃	C₂×C₄	1	4	yes	—
62	C₂×C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
63	C₂×C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
64	C₂×C₄²×C₅₃	C₂×C₄	1	2	yes	—
65	C₂×C₄²×C₅₃	C₂³	1	2	yes	—
66	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
67	C₄⋊C₈×C₅₃	—	2	2	no	—
68	C₄⋊C₈×C₅₃	—	2	2	no	—
69	C₂²⋊C₈×C₅₃	—	2	4	no	—
70	C₂²⋊C₈×C₅₃	—	2	4	no	—
71	C₂²⋊C₈×C₅₃	—	4	4	no	—
72	C₂²⋊C₈×C₅₃	—	4	4	no	—
73	C₄×C₈×C₅₃	—	2	2	no	—
74	C₄×C₈×C₅₃	D₄	1	4	yes	complemented
75	C₂⁴×C₅₃	C₂²×C₄	1	2	yes	—
76	C₂⁴×C₅₃	C₂²×C₄	1	2	yes	—
77	C₂×D₄×C₅₃	—	2	8	no	—
78	C₂×D₄×C₅₃	—	2	2	no	—
79	C₂×D₄×C₅₃	—	2	2	no	—
80	C₂×D₄×C₅₃	—	2	2	no	—
81	C₂×D₄×C₅₃	—	2	2	no	—
82	C₂×D₄×C₅₃	—	2	8	no	—
83	C₂×D₄×C₅₃	—	2	2	no	—
84	C₂×D₄×C₅₃	—	4	4	no	—
85	C₂×D₄×C₅₃	M₁₆	1	4	yes	complemented
86	C₂²×C₄×C₅₃	—	2	2	no	—
87	C₂²×C₄×C₅₃	—	2	2	no	—
88	C₂²×C₄×C₅₃	—	2	2	no	—
89	C₂²×C₄×C₅₃	—	2	2	no	—
90	C₂²×C₄×C₅₃	—	2	4	no	—
91	C₂²×C₄×C₅₃	—	2	4	no	—
92	C₂²×C₄×C₅₃	—	2	4	no	—
93	C₂²×C₄×C₅₃	—	2	4	no	—
94	C₂²×C₄×C₅₃	C₂²×C₄	1	2	yes	—
95	C₂²×C₄×C₅₃	C₂²×C₄	1	2	yes	—
96	C₂²×C₄×C₅₃	C₂²×C₄	1	2	yes	—
97	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
98	C₂²×C₄×C₅₃	C₂²×C₄	1	2	yes	—
99	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
100	C₂²×C₄×C₅₃	C₂⁴	1	1	yes	characteristic
101	M₁₆×C₅₃	—	4	8	no	—
102	C₂×C₈×C₅₃	—	2	2	no	—
103	C₂×C₈×C₅₃	—	2	2	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	2	no	—
108	C₂×C₈×C₅₃	—	2	2	no	—
109	C₂×C₈×C₅₃	—	2	4	no	—
110	C₂×C₈×C₅₃	—	2	4	no	—
111	C₂×C₈×C₅₃	—	4	8	no	—
112	C₂×C₈×C₅₃	C₂×D₄	1	2	yes	—
113	C₂×C₈×C₅₃	C₂×D₄	1	2	yes	—
114	C₄⋊C₄×C₅₃	—	2	2	no	—
115	C₄⋊C₄×C₅₃	—	2	2	no	—
116	C₄⋊C₄×C₅₃	—	4	4	no	—
117	C₂²⋊C₄×C₅₃	—	2	4	no	—
118	C₂²⋊C₄×C₅₃	—	2	4	no	—
119	C₂²⋊C₄×C₅₃	—	4	4	no	—
120	C₂²⋊C₄×C₅₃	—	4	4	no	—
121	C₄²×C₅₃	—	2	2	no	—
122	C₄²×C₅₃	—	2	4	no	—
123	C₄²×C₅₃	—	2	2	no	—
124	C₄²×C₅₃	—	2	2	no	—
125	C₄²×C₅₃	—	2	2	no	—
126	C₄²×C₅₃	C₂×D₄	1	2	yes	—
127	C₄²×C₅₃	#41	1	2	yes	—
128	C₂³×C₅₃	—	2	2	no	—
129	C₂³×C₅₃	—	2	4	no	—
130	C₂³×C₅₃	—	2	2	no	—
131	C₂³×C₅₃	—	2	4	no	—
132	C₂³×C₅₃	—	2	2	no	—
133	C₂³×C₅₃	—	2	4	no	—
134	C₂³×C₅₃	—	2	2	no	—
135	C₂³×C₅₃	—	2	2	no	—
136	C₂³×C₅₃	—	2	4	no	—
137	C₂³×C₅₃	C₂×M₁₆	1	4	yes	—
138	C₂³×C₅₃	C₂³×C₄	1	1	yes	characteristic
139	D₄×C₅₃	—	2	8	no	—
140	D₄×C₅₃	—	4	16	no	—
141	D₄×C₅₃	—	4	4	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	—	2	2	no	—
147	C₂×C₄×C₅₃	—	2	4	no	—
148	C₂×C₄×C₅₃	—	2	4	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	—	4	2	no	—
153	C₂×C₄×C₅₃	—	4	2	no	—
154	C₂×C₄×C₅₃	—	4	2	no	—
155	C₂×C₄×C₅₃	—	4	2	no	—
156	C₂×C₄×C₅₃	—	4	4	no	—
157	C₂×C₄×C₅₃	—	4	4	no	—
158	C₂×C₄×C₅₃	C₄×D₄	1	2	yes	—
159	C₂×C₄×C₅₃	C₄×D₄	1	2	yes	—
160	C₂×C₄×C₅₃	C₂×M₁₆	1	2	yes	—
161	C₂×C₄×C₅₃	#131	1	4	yes	—
162	C₂×C₄×C₅₃	#131	1	2	yes	—
163	C₂×C₄×C₅₃	C₂²×D₄	1	1	yes	characteristic
164	C₂×C₄×C₅₃	C₂×#41	1	1	yes	characteristic
165	C₂×C₄×C₅₃	H(5,𝔽₂)	1	2	yes	—
166	C₂×C₄×C₅₃	H(5,𝔽₂)	1	2	yes	—
167	C₈×C₅₃	—	2	2	no	—
168	C₈×C₅₃	—	2	2	no	—
169	C₈×C₅₃	—	4	2	no	—
170	C₈×C₅₃	—	4	2	no	—
171	C₈×C₅₃	—	4	2	no	—
172	#22262	C₅₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
173	C₂²×C₅₃	—	2	4	no	—
174	C₂²×C₅₃	—	2	8	no	—
175	C₂²×C₅₃	—	4	2	no	—
176	C₂²×C₅₃	—	4	2	no	—
177	C₂²×C₅₃	—	4	4	no	—
178	C₂²×C₅₃	—	4	2	no	—
179	C₂²×C₅₃	—	4	2	no	—
180	C₂²×C₅₃	—	4	2	no	—
181	C₂²×C₅₃	—	4	4	no	—
182	C₂²×C₅₃	—	4	4	no	—
183	C₂²×C₅₃	C₂×C₄×D₄	1	1	yes	characteristic
184	C₂²×C₅₃	#518	1	2	yes	—
185	C₂²×C₅₃	C₂²×M₁₆	1	1	yes	characteristic
186	C₂²×C₅₃	C₂×#131	1	1	yes	characteristic
187	C₂²×C₅₃	C₂×#131	1	2	yes	—
188	C₄×C₅₃	—	2	2	no	—
189	C₄×C₅₃	—	2	4	no	—
190	C₄×C₅₃	—	4	2	no	—
191	C₄×C₅₃	—	4	2	no	—
192	C₄×C₅₃	—	4	2	no	—
193	C₄×C₅₃	#435	1	4	yes	—
194	C₄×C₅₃	#535	1	1	yes	characteristic
195	C₄×C₅₃	#535	1	1	yes	characteristic
196	C₄×C₅₃	#540	1	2	yes	—
197	C₂×#436	C₂×C₅₃	1	1	yes	characteristic, complemented
198	C₂×C₈×D₄	C₂×C₅₃	1	1	yes	characteristic, complemented
199	C₄×#127	C₂×C₅₃	1	1	yes	characteristic
200	#1954	C₂×C₅₃	1	1	yes	characteristic, complemented
201	#1954	C₂×C₅₃	1	1	yes	characteristic, complemented
202	#1870	C₂×C₅₃	1	4	yes	complemented
203	#1869	C₂×C₅₃	1	4	yes	complemented
204	#1831	C₂×C₅₃	1	1	yes	characteristic, complemented
205	#1744	C₂×C₅₃	1	1	yes	characteristic, complemented
206	C₂×C₅₃	—	4	4	no	—
207	C₂×C₅₃	—	8	2	no	—
208	C₂×C₅₃	#2328	1	1	yes	characteristic
209	C₂×C₅₃	C₂×#435	1	1	yes	characteristic
210	C₂×C₅₃	#2931	1	1	yes	characteristic
211	C₂×C₅₃	#2974	1	2	yes	—
212	C₂×C₅₃	#2979	1	2	yes	—
213	C₂²×M₁₆	C₂²×C₅₃	1	2	yes	—
214	C₂³×C₈	C₂²×C₅₃	1	2	yes	—
215	C₂×#127	C₄×C₅₃	1	1	yes	characteristic
216	C₂×C₄×D₄	C₂²×C₅₃	1	1	yes	characteristic
217	C₂×C₄×D₄	C₂²×C₅₃	1	1	yes	characteristic
218	C₂×C₄×D₄	C₂²×C₅₃	1	4	yes	—
219	#436	—	2	4	no	—
220	C₈×D₄	—	2	4	no	—
221	C₂×C₄⋊C₈	C₂²×C₅₃	1	1	yes	characteristic
222	C₂×C₄⋊C₈	C₂²×C₅₃	1	1	yes	characteristic
223	C₂×C₂²⋊C₈	C₂²×C₅₃	1	2	yes	—
224	C₂×C₂²⋊C₈	C₂²×C₅₃	1	2	yes	—
225	C₂×C₂²⋊C₈	C₂²×C₅₃	1	4	yes	—
226	C₂×C₂²⋊C₈	C₂²×C₅₃	1	4	yes	—
227	C₂×C₄×C₈	C₂²×C₅₃	1	1	yes	characteristic
228	C₂×C₄×C₈	C₂²×C₅₃	1	1	yes	characteristic
229	#390	C₄×C₅₃	1	4	yes	—
230	#390	C₄×C₅₃	1	1	yes	characteristic
231	#390	C₄×C₅₃	1	1	yes	characteristic
232	#384	C₄×C₅₃	1	1	yes	characteristic
233	C₄³	C₂²×C₅₃	1	1	yes	characteristic
234	#336	C₂²×C₅₃	1	4	yes	—
235	#336	C₂²×C₅₃	1	2	yes	—
236	#336	C₂²×C₅₃	1	2	yes	—
237	C₅₃	#22262	1	1	yes	characteristic, direct product factor, Sylow
238	C₂²×D₄	C₂×C₄×C₅₃	1	1	yes	characteristic
239	C₂²×D₄	C₂×C₄×C₅₃	1	4	yes	—
240	C₂²×D₄	C₂×C₄×C₅₃	1	1	yes	characteristic
241	C₂³×C₄	C₂³×C₅₃	1	2	yes	—
242	C₂³×C₄	C₂³×C₅₃	1	2	yes	—
243	C₂×M₁₆	—	2	4	no	—
244	C₂×M₁₆	—	4	4	no	—
245	C₂²×C₈	—	2	4	no	—
246	C₂²×C₈	—	4	4	no	—
247	C₂²×C₈	C₂³×C₅₃	1	2	yes	—
248	C₂²×C₈	C₂³×C₅₃	1	2	yes	—
249	C₂²×C₈	C₂³×C₅₃	1	2	yes	—
250	C₂²×C₈	C₂³×C₅₃	1	1	yes	characteristic
251	C₂²×C₈	C₂³×C₅₃	1	1	yes	characteristic
252	#127	—	2	4	no	—
253	C₄×D₄	—	2	2	no	—
254	C₄×D₄	—	2	2	no	—
255	C₄×D₄	—	2	2	no	—
256	C₄×D₄	—	2	2	no	—
257	C₄×D₄	—	4	4	no	—
258	C₄×D₄	—	4	4	no	—
259	C₂×C₄⋊C₄	C₂×C₄×C₅₃	1	1	yes	characteristic
260	C₂×C₄⋊C₄	C₂×C₄×C₅₃	1	1	yes	characteristic
261	C₂×C₄⋊C₄	C₂×C₄×C₅₃	1	4	yes	—
262	C₂×C₂²⋊C₄	C₂×C₄×C₅₃	1	2	yes	—
263	C₂×C₂²⋊C₄	C₂×C₄×C₅₃	1	2	yes	—
264	C₂×C₂²⋊C₄	C₂×C₄×C₅₃	1	4	yes	—
265	C₂×C₂²⋊C₄	C₂×C₄×C₅₃	1	4	yes	—
266	C₂×C₄²	C₂×C₄×C₅₃	1	1	yes	characteristic
267	C₂×C₄²	C₂×C₄×C₅₃	1	1	yes	characteristic
268	C₂×C₄²	C₂×C₄×C₅₃	1	2	yes	—
269	C₂×C₄²	C₂³×C₅₃	1	2	yes	—
270	C₂×C₄²	C₂³×C₅₃	1	1	yes	characteristic
271	C₄⋊C₈	—	2	2	no	—
272	C₄⋊C₈	—	2	2	no	—
273	C₂²⋊C₈	—	2	4	no	—
274	C₂²⋊C₈	—	2	4	no	—
275	C₂²⋊C₈	—	4	4	no	—
276	C₂²⋊C₈	—	4	4	no	—
277	C₄×C₈	—	2	2	no	—
278	C₄×C₈	D₄×C₅₃	1	4	yes	complemented
279	C₂⁴	C₂²×C₄×C₅₃	1	2	yes	—
280	C₂⁴	C₂²×C₄×C₅₃	1	2	yes	—
281	C₂×D₄	—	2	8	no	—
282	C₂×D₄	—	2	2	no	—
283	C₂×D₄	—	2	2	no	—
284	C₂×D₄	—	2	2	no	—
285	C₂×D₄	—	2	2	no	—
286	C₂×D₄	—	2	8	no	—
287	C₂×D₄	—	2	2	no	—
288	C₂×D₄	—	4	4	no	—
289	C₂×D₄	M₁₆×C₅₃	1	4	yes	complemented
290	C₂²×C₄	—	2	2	no	—
291	C₂²×C₄	—	2	2	no	—
292	C₂²×C₄	—	2	2	no	—
293	C₂²×C₄	—	2	2	no	—
294	C₂²×C₄	—	2	4	no	—
295	C₂²×C₄	—	2	4	no	—
296	C₂²×C₄	—	2	4	no	—
297	C₂²×C₄	—	2	4	no	—
298	C₂²×C₄	C₂²×C₄×C₅₃	1	2	yes	—
299	C₂²×C₄	C₂²×C₄×C₅₃	1	2	yes	—
300	C₂²×C₄	C₂²×C₄×C₅₃	1	2	yes	—
301	C₂²×C₄	C₂²×C₄×C₅₃	1	1	yes	characteristic
302	C₂²×C₄	C₂²×C₄×C₅₃	1	2	yes	—
303	C₂²×C₄	C₂²×C₄×C₅₃	1	1	yes	characteristic
304	C₂²×C₄	C₂⁴×C₅₃	1	1	yes	characteristic
305	M₁₆	—	4	8	no	—
306	C₂×C₈	—	2	2	no	—
307	C₂×C₈	—	2	2	no	—
308	C₂×C₈	—	2	2	no	—
309	C₂×C₈	—	2	2	no	—
310	C₂×C₈	—	2	2	no	—
311	C₂×C₈	—	2	2	no	—
312	C₂×C₈	—	2	2	no	—
313	C₂×C₈	—	2	4	no	—
314	C₂×C₈	—	2	4	no	—
315	C₂×C₈	—	4	8	no	—
316	C₂×C₈	C₂×D₄×C₅₃	1	2	yes	—
317	C₂×C₈	C₂×D₄×C₅₃	1	2	yes	—
318	C₄⋊C₄	—	2	2	no	—
319	C₄⋊C₄	—	2	2	no	—
320	C₄⋊C₄	—	4	4	no	—
321	C₂²⋊C₄	—	2	4	no	—
322	C₂²⋊C₄	—	2	4	no	—
323	C₂²⋊C₄	—	4	4	no	—
324	C₂²⋊C₄	—	4	4	no	—
325	C₄²	—	2	2	no	—
326	C₄²	—	2	4	no	—
327	C₄²	—	2	2	no	—
328	C₄²	—	2	2	no	—
329	C₄²	—	2	2	no	—
330	C₄²	C₂×D₄×C₅₃	1	2	yes	—
331	C₄²	#41×C₅₃	1	2	yes	—
332	C₂³	—	2	2	no	—
333	C₂³	—	2	4	no	—
334	C₂³	—	2	4	no	—
335	C₂³	—	2	2	no	—
336	C₂³	—	2	2	no	—
337	C₂³	—	2	4	no	—
338	C₂³	—	2	2	no	—
339	C₂³	—	2	4	no	—
340	C₂³	—	2	2	no	—
341	C₂³	C₂×M₁₆×C₅₃	1	4	yes	—
342	C₂³	C₂³×C₄×C₅₃	1	1	yes	characteristic
343	D₄	—	2	8	no	—
344	D₄	—	4	16	no	—
345	D₄	—	4	4	no	—
346	C₂×C₄	—	2	2	no	—
347	C₂×C₄	—	2	2	no	—
348	C₂×C₄	—	2	2	no	—
349	C₂×C₄	—	2	2	no	—
350	C₂×C₄	—	2	2	no	—
351	C₂×C₄	—	2	2	no	—
352	C₂×C₄	—	2	4	no	—
353	C₂×C₄	—	2	4	no	—
354	C₂×C₄	—	2	2	no	—
355	C₂×C₄	—	2	2	no	—
356	C₂×C₄	—	4	2	no	—
357	C₂×C₄	—	4	2	no	—
358	C₂×C₄	—	4	2	no	—
359	C₂×C₄	—	4	2	no	—
360	C₂×C₄	—	4	4	no	—
361	C₂×C₄	—	4	4	no	—
362	C₂×C₄	C₄×D₄×C₅₃	1	2	yes	—
363	C₂×C₄	C₄×D₄×C₅₃	1	2	yes	—
364	C₂×C₄	C₂×M₁₆×C₅₃	1	2	yes	—
365	C₂×C₄	#131×C₅₃	1	4	yes	—
366	C₂×C₄	#131×C₅₃	1	2	yes	—
367	C₂×C₄	C₂²×D₄×C₅₃	1	1	yes	characteristic
368	C₂×C₄	C₂×#41×C₅₃	1	1	yes	characteristic
369	C₂×C₄	H(5,𝔽₂)×C₅₃	1	2	yes	—
370	C₂×C₄	H(5,𝔽₂)×C₅₃	1	2	yes	—
371	C₈	—	2	2	no	—
372	C₈	—	2	2	no	—
373	C₈	—	4	2	no	—
374	C₈	—	4	2	no	—
375	C₈	—	4	2	no	—
376	C₂²	—	2	8	no	—
377	C₂²	—	2	4	no	—
378	C₂²	—	4	4	no	—
379	C₂²	—	4	2	no	—
380	C₂²	—	4	2	no	—
381	C₂²	—	4	4	no	—
382	C₂²	—	4	4	no	—
383	C₂²	—	4	2	no	—
384	C₂²	—	4	2	no	—
385	C₂²	—	4	2	no	—
386	C₂²	C₂×C₄×D₄×C₅₃	1	1	yes	characteristic
387	C₂²	C₅₃×#518	1	2	yes	—
388	C₂²	C₂²×M₁₆×C₅₃	1	1	yes	characteristic
389	C₂²	C₂×#131×C₅₃	1	2	yes	—
390	C₂²	C₂×#131×C₅₃	1	1	yes	characteristic
391	C₄	—	2	4	no	—
392	C₄	—	2	2	no	—
393	C₄	—	4	2	no	—
394	C₄	—	4	2	no	—
395	C₄	—	4	2	no	—
396	C₄	C₅₃×#435	1	4	yes	—
397	C₄	C₅₃×#535	1	1	yes	characteristic
398	C₄	C₅₃×#535	1	1	yes	characteristic
399	C₄	C₅₃×#540	1	2	yes	—
400	C₂	—	4	4	no	—
401	C₂	—	8	2	no	—
402	C₂	C₅₃×#2328	1	1	yes	characteristic
403	C₂	C₂×C₅₃×#435	1	1	yes	characteristic
404	C₂	C₅₃×#2931	1	1	yes	characteristic
405	C₂	C₅₃×#2974	1	2	yes	—
406	C₂	C₅₃×#2979	1	2	yes	—

	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	C₂×#127×C₅₃	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
1. C₂×C₅₃×#436	C₂	semidirect
2. C₂×C₈×D₄×C₅₃	C₂	semidirect
4. C₅₃×#1954	C₂	semidirect
5. C₅₃×#1954	C₂	semidirect
6. C₅₃×#1870	C₂	semidirect
7. C₅₃×#1869	C₂	semidirect
8. C₅₃×#1831	C₂	semidirect
9. C₅₃×#1744	C₂	semidirect
74. C₄×C₈×C₅₃	D₄	semidirect
85. C₂×D₄×C₅₃	M₁₆	semidirect
172. #22262	C₅₃	direct (N × H)
197. C₂×#436	C₂×C₅₃	semidirect
198. C₂×C₈×D₄	C₂×C₅₃	semidirect
200. #1954	C₂×C₅₃	semidirect
201. #1954	C₂×C₅₃	semidirect
202. #1870	C₂×C₅₃	semidirect
203. #1869	C₂×C₅₃	semidirect
204. #1831	C₂×C₅₃	semidirect
205. #1744	C₂×C₅₃	semidirect
237. C₅₃	#22262	direct (N × H)
278. C₄×C₈	D₄×C₅₃	semidirect
289. C₂×D₄	M₁₆×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×#436	2
C₂×C₈×D₄×C₅₃	2
C₄×#127×C₅₃	2
C₅₃×#1954	2
C₅₃×#1870	2
C₅₃×#1869	2
C₅₃×#1831	2
C₅₃×#1744	2
#22262	53

Subgroups — C53×#22262 #27229774

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅₃×#22262 #27229774