Subgroups — C₅₃×#21104 #27228616

Order 13568.

880 conjugacy classes of subgroups.

The subgroup lattice has 440 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₅₃×#433	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×C₄×M₁₆×C₅₃	C₂	1	1	yes	maximal, characteristic, complemented
3	C₅₃×#2290	C₂	1	1	yes	maximal, characteristic
4	C₅₃×#1924	C₂	1	2	yes	maximal, complemented
5	C₅₃×#1917	C₂	1	2	yes	maximal, complemented
6	C₅₃×#1870	C₂	1	2	yes	maximal
7	C₅₃×#1851	C₂	1	2	yes	maximal, complemented
8	C₅₃×#1768	C₂	1	1	yes	maximal, characteristic, complemented
9	C₅₃×#1750	C₂	1	1	yes	maximal, characteristic, complemented
10	C₅₃×#1749	C₂	1	1	yes	maximal, characteristic, complemented
11	C₅₃×#1746	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂²×M₁₆×C₅₃	C₂²	1	2	yes	—
13	C₂×C₄×D₄×C₅₃	C₂²	1	1	yes	characteristic
14	C₂×#117×C₅₃	C₂²	1	1	yes	characteristic
15	C₂²×C₄²×C₅₃	C₂²	1	1	yes	characteristic
16	C₅₃×#433	—	2	4	no	—
17	C₂×C₄⋊C₈×C₅₃	C₂²	1	2	yes	—
18	C₂×C₄⋊C₈×C₅₃	C₂²	1	2	yes	—
19	C₂×C₂²⋊C₈×C₅₃	C₂²	1	2	yes	—
20	C₂×C₂²⋊C₈×C₅₃	C₂²	1	2	yes	—
21	C₂×C₂²⋊C₈×C₅₃	C₂²	1	2	yes	—
22	C₄×M₁₆×C₅₃	—	2	4	no	—
23	C₂×#97×C₅₃	C₂²	1	2	yes	—
24	C₂×#97×C₅₃	C₂²	1	1	yes	characteristic
25	C₂×#97×C₅₃	C₂²	1	1	yes	characteristic
26	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
27	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
28	C₅₃×#388	C₄	1	2	yes	—
29	C₅₃×#387	C₄	1	2	yes	—
30	C₅₃×#386	C₄	1	2	yes	—
31	C₅₃×#385	C₄	1	2	yes	—
32	C₄×C₄⋊C₄×C₅₃	C₂²	1	1	yes	characteristic
33	C₄×C₂²⋊C₄×C₅₃	C₂²	1	1	yes	characteristic
34	C₄×C₂²⋊C₄×C₅₃	C₂²	1	1	yes	characteristic
35	C₅₃×#376	C₂²	1	1	yes	characteristic
36	C₅₃×#336	C₂²	1	2	yes	—
37	C₅₃×#336	C₂²	1	2	yes	—
38	C₅₃×#336	C₂²	1	2	yes	—
39	C₅₃×#336	C₂²	1	2	yes	—
40	C₅₃×#336	C₂²	1	2	yes	—
41	C₂²×D₄×C₅₃	C₂×C₄	1	1	yes	characteristic
42	C₂³×C₄×C₅₃	C₂×C₄	1	2	yes	—
43	C₂³×C₄×C₅₃	C₂³	1	1	yes	characteristic
44	C₂³×C₄×C₅₃	C₂³	1	1	yes	characteristic
45	C₂×M₁₆×C₅₃	—	2	4	no	—
46	C₂×M₁₆×C₅₃	—	2	4	no	—
47	C₂×M₁₆×C₅₃	—	2	4	no	—
48	C₂²×C₈×C₅₃	C₂³	1	2	yes	—
49	C₂²×C₈×C₅₃	C₂³	1	2	yes	—
50	C₂²×C₈×C₅₃	C₂³	1	2	yes	—
51	C₂²×C₈×C₅₃	C₂³	1	2	yes	—
52	C₄×D₄×C₅₃	—	2	4	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	1	yes	characteristic
57	C₂×C₄⋊C₄×C₅₃	C₂×C₄	1	2	yes	—
58	C₂×C₂²⋊C₄×C₅₃	C₂×C₄	1	2	yes	—
59	C₂×C₂²⋊C₄×C₅₃	C₂×C₄	1	1	yes	characteristic
60	C₂×C₂²⋊C₄×C₅₃	C₂×C₄	1	1	yes	characteristic
61	C₂×C₂²⋊C₄×C₅₃	C₂×C₄	1	2	yes	—
62	C₂×C₂²⋊C₄×C₅₃	C₂×C₄	1	2	yes	—
63	C₂×C₄²×C₅₃	—	2	2	no	—
64	C₂×C₄²×C₅₃	—	2	2	no	—
65	C₂×C₄²×C₅₃	C₂×C₄	1	2	yes	—
66	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
67	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
68	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
69	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
70	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
71	C₄⋊C₈×C₅₃	—	2	4	no	—
72	C₄⋊C₈×C₅₃	—	2	4	no	—
73	C₂²⋊C₈×C₅₃	—	2	4	no	—
74	C₂²⋊C₈×C₅₃	—	2	4	no	—
75	C₂²⋊C₈×C₅₃	—	2	4	no	—
76	#97×C₅₃	—	2	4	no	—
77	#97×C₅₃	—	2	2	no	—
78	#97×C₅₃	—	2	2	no	—
79	C₄×C₈×C₅₃	—	2	2	no	—
80	C₄×C₈×C₅₃	D₄	1	4	yes	complemented
81	#95×C₅₃	C₂×C₄	1	2	yes	—
82	#95×C₅₃	C₂×C₄	1	2	yes	—
83	#95×C₅₃	C₂×C₄	1	2	yes	—
84	#95×C₅₃	C₂×C₄	1	2	yes	—
85	#95×C₅₃	C₂×C₄	1	2	yes	—
86	C₂⁴×C₅₃	C₂²×C₄	1	1	yes	characteristic
87	C₂⁴×C₅₃	C₂²×C₄	1	1	yes	characteristic
88	C₂×D₄×C₅₃	—	2	2	no	—
89	C₂×D₄×C₅₃	—	4	2	no	—
90	C₂²×C₄×C₅₃	—	2	1	no	—
91	C₂²×C₄×C₅₃	—	2	1	no	—
92	C₂²×C₄×C₅₃	—	2	2	no	—
93	C₂²×C₄×C₅₃	—	2	4	no	—
94	C₂²×C₄×C₅₃	—	2	1	no	—
95	C₂²×C₄×C₅₃	—	2	2	no	—
96	C₂²×C₄×C₅₃	—	2	1	no	—
97	C₂²×C₄×C₅₃	—	2	2	no	—
98	C₂²×C₄×C₅₃	—	4	2	no	—
99	C₂²×C₄×C₅₃	—	4	2	no	—
100	C₂²×C₄×C₅₃	C₂²×C₄	1	2	yes	—
101	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
102	C₂²×C₄×C₅₃	C₂²×C₄	1	2	yes	—
103	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
104	C₂²×C₄×C₅₃	C₂²×C₄	1	2	yes	—
105	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
106	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
107	C₂²×C₄×C₅₃	C₂²×C₄	1	2	yes	—
108	C₂²×C₄×C₅₃	#41	1	4	yes	—
109	C₂²×C₄×C₅₃	C₂⁴	1	1	yes	characteristic
110	M₁₆×C₅₃	—	2	16	no	—
111	C₂×C₈×C₅₃	—	2	2	no	—
112	C₂×C₈×C₅₃	—	2	2	no	—
113	C₂×C₈×C₅₃	—	2	2	no	—
114	C₂×C₈×C₅₃	—	2	2	no	—
115	C₂×C₈×C₅₃	—	2	2	no	—
116	C₂×C₈×C₅₃	—	2	2	no	—
117	C₂×C₈×C₅₃	—	2	2	no	—
118	C₂×C₈×C₅₃	—	2	2	no	—
119	C₂×C₈×C₅₃	—	2	2	no	—
120	C₂×C₈×C₅₃	—	2	2	no	—
121	C₂×C₈×C₅₃	C₂×D₄	1	4	yes	—
122	C₂×C₈×C₅₃	#41	1	4	yes	—
123	C₄⋊C₄×C₅₃	—	2	2	no	—
124	C₄⋊C₄×C₅₃	—	2	4	no	—
125	C₄⋊C₄×C₅₃	—	4	2	no	—
126	C₂²⋊C₄×C₅₃	—	2	2	no	—
127	C₂²⋊C₄×C₅₃	—	2	2	no	—
128	C₂²⋊C₄×C₅₃	—	2	4	no	—
129	C₂²⋊C₄×C₅₃	—	4	2	no	—
130	C₂²⋊C₄×C₅₃	—	4	2	no	—
131	C₄²×C₅₃	—	2	4	no	—
132	C₄²×C₅₃	—	2	1	no	—
133	C₄²×C₅₃	—	2	1	no	—
134	C₄²×C₅₃	—	2	2	no	—
135	C₄²×C₅₃	—	2	1	no	—
136	C₄²×C₅₃	—	2	1	no	—
137	C₄²×C₅₃	—	2	2	no	—
138	C₄²×C₅₃	C₂×D₄	1	2	yes	—
139	C₄²×C₅₃	#41	1	2	yes	—
140	C₂³×C₅₃	—	2	1	no	—
141	C₂³×C₅₃	—	2	1	no	—
142	C₂³×C₅₃	—	2	2	no	—
143	C₂³×C₅₃	—	2	2	no	—
144	C₂³×C₅₃	—	2	1	no	—
145	C₂³×C₅₃	—	2	1	no	—
146	C₂³×C₅₃	—	2	1	no	—
147	C₂³×C₅₃	—	2	2	no	—
148	C₂³×C₅₃	—	2	2	no	—
149	C₂³×C₅₃	#117	1	2	yes	—
150	C₂³×C₅₃	C₂³×C₄	1	1	yes	characteristic
151	D₄×C₅₃	—	4	4	no	—
152	C₂×C₄×C₅₃	—	2	1	no	—
153	C₂×C₄×C₅₃	—	2	1	no	—
154	C₂×C₄×C₅₃	—	2	2	no	—
155	C₂×C₄×C₅₃	—	2	2	no	—
156	C₂×C₄×C₅₃	—	2	4	no	—
157	C₂×C₄×C₅₃	—	2	4	no	—
158	C₂×C₄×C₅₃	—	2	1	no	—
159	C₂×C₄×C₅₃	—	2	1	no	—
160	C₂×C₄×C₅₃	—	2	1	no	—
161	C₂×C₄×C₅₃	—	2	2	no	—
162	C₂×C₄×C₅₃	—	2	1	no	—
163	C₂×C₄×C₅₃	—	2	2	no	—
164	C₂×C₄×C₅₃	—	2	1	no	—
165	C₂×C₄×C₅₃	—	2	2	no	—
166	C₂×C₄×C₅₃	—	2	2	no	—
167	C₂×C₄×C₅₃	—	2	2	no	—
168	C₂×C₄×C₅₃	—	2	2	no	—
169	C₂×C₄×C₅₃	—	2	2	no	—
170	C₂×C₄×C₅₃	—	2	2	no	—
171	C₂×C₄×C₅₃	—	2	2	no	—
172	C₂×C₄×C₅₃	—	2	2	no	—
173	C₂×C₄×C₅₃	—	2	2	no	—
174	C₂×C₄×C₅₃	—	4	8	no	—
175	C₂×C₄×C₅₃	—	4	2	no	—
176	C₂×C₄×C₅₃	—	4	2	no	—
177	C₂×C₄×C₅₃	#117	1	2	yes	—
178	C₂×C₄×C₅₃	C₄×D₄	1	4	yes	—
179	C₂×C₄×C₅₃	C₂²×D₄	1	1	yes	characteristic
180	C₂×C₄×C₅₃	C₂×#41	1	2	yes	—
181	C₂×C₄×C₅₃	C₂×#41	1	2	yes	—
182	C₂×C₄×C₅₃	C₂×#41	1	1	yes	characteristic
183	C₈×C₅₃	—	2	4	no	—
184	C₈×C₅₃	—	2	4	no	—
185	C₈×C₅₃	—	4	2	no	—
186	C₈×C₅₃	—	4	2	no	—
187	#21104	C₅₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
188	C₂²×C₅₃	—	2	2	no	—
189	C₂²×C₅₃	—	2	2	no	—
190	C₂²×C₅₃	—	2	2	no	—
191	C₂²×C₅₃	—	4	2	no	—
192	C₂²×C₅₃	—	4	2	no	—
193	C₂²×C₅₃	—	4	1	no	—
194	C₂²×C₅₃	—	4	1	no	—
195	C₂²×C₅₃	—	4	2	no	—
196	C₂²×C₅₃	—	4	2	no	—
197	C₂²×C₅₃	—	4	1	no	—
198	C₂²×C₅₃	C₂×#117	1	1	yes	characteristic
199	C₂²×C₅₃	C₂×C₄×D₄	1	1	yes	characteristic
200	C₂²×C₅₃	C₄×#41	1	1	yes	characteristic
201	C₂²×C₅₃	#568	1	2	yes	—
202	C₂²×C₅₃	#568	1	2	yes	—
203	C₄×C₅₃	—	2	4	no	—
204	C₄×C₅₃	—	2	2	no	—
205	C₄×C₅₃	—	4	1	no	—
206	C₄×C₅₃	—	4	1	no	—
207	C₄×C₅₃	—	4	2	no	—
208	C₄×C₅₃	—	4	2	no	—
209	C₄×C₅₃	—	4	2	no	—
210	C₄×C₅₃	—	4	1	no	—
211	C₄×C₅₃	#525	1	2	yes	—
212	C₄×C₅₃	#529	1	2	yes	—
213	C₂×#433	C₂×C₅₃	1	1	yes	characteristic, complemented
214	C₂×C₄×M₁₆	C₂×C₅₃	1	1	yes	characteristic, complemented
215	#2290	C₂×C₅₃	1	1	yes	characteristic
216	#1924	C₂×C₅₃	1	2	yes	complemented
217	#1917	C₂×C₅₃	1	2	yes	complemented
218	#1870	C₂×C₅₃	1	2	yes	—
219	#1851	C₂×C₅₃	1	2	yes	complemented
220	#1768	C₂×C₅₃	1	1	yes	characteristic, complemented
221	#1750	C₂×C₅₃	1	1	yes	characteristic, complemented
222	#1749	C₂×C₅₃	1	1	yes	characteristic, complemented
223	#1746	C₂×C₅₃	1	1	yes	characteristic, complemented
224	C₂×C₅₃	—	4	2	no	—
225	C₂×C₅₃	—	8	1	no	—
226	C₂×C₅₃	#2290	1	1	yes	characteristic
227	C₂×C₅₃	#2921	1	1	yes	characteristic
228	C₂×C₅₃	#2923	1	1	yes	characteristic
229	C₂×C₅₃	#2930	1	2	yes	—
230	C₂×C₅₃	#2967	1	2	yes	—
231	C₂²×M₁₆	C₂²×C₅₃	1	2	yes	—
232	C₂×C₄×D₄	C₂²×C₅₃	1	1	yes	characteristic
233	C₂×#117	C₂²×C₅₃	1	1	yes	characteristic
234	C₂²×C₄²	C₂²×C₅₃	1	1	yes	characteristic
235	#433	—	2	4	no	—
236	C₂×C₄⋊C₈	C₂²×C₅₃	1	2	yes	—
237	C₂×C₄⋊C₈	C₂²×C₅₃	1	2	yes	—
238	C₂×C₂²⋊C₈	C₂²×C₅₃	1	2	yes	—
239	C₂×C₂²⋊C₈	C₂²×C₅₃	1	2	yes	—
240	C₂×C₂²⋊C₈	C₂²×C₅₃	1	2	yes	—
241	C₄×M₁₆	—	2	4	no	—
242	C₂×#97	C₂²×C₅₃	1	2	yes	—
243	C₂×#97	C₂²×C₅₃	1	1	yes	characteristic
244	C₂×#97	C₂²×C₅₃	1	1	yes	characteristic
245	C₂×C₄×C₈	C₂²×C₅₃	1	1	yes	characteristic
246	C₂×C₄×C₈	C₂²×C₅₃	1	1	yes	characteristic
247	#388	C₄×C₅₃	1	2	yes	—
248	#387	C₄×C₅₃	1	2	yes	—
249	#386	C₄×C₅₃	1	2	yes	—
250	#385	C₄×C₅₃	1	2	yes	—
251	C₄×C₄⋊C₄	C₂²×C₅₃	1	1	yes	characteristic
252	C₄×C₂²⋊C₄	C₂²×C₅₃	1	1	yes	characteristic
253	C₄×C₂²⋊C₄	C₂²×C₅₃	1	1	yes	characteristic
254	#376	C₂²×C₅₃	1	1	yes	characteristic
255	#336	C₂²×C₅₃	1	2	yes	—
256	#336	C₂²×C₅₃	1	2	yes	—
257	#336	C₂²×C₅₃	1	2	yes	—
258	#336	C₂²×C₅₃	1	2	yes	—
259	#336	C₂²×C₅₃	1	2	yes	—
260	C₅₃	#21104	1	1	yes	characteristic, direct product factor, Sylow
261	C₂²×D₄	C₂×C₄×C₅₃	1	1	yes	characteristic
262	C₂³×C₄	C₂×C₄×C₅₃	1	2	yes	—
263	C₂³×C₄	C₂³×C₅₃	1	1	yes	characteristic
264	C₂³×C₄	C₂³×C₅₃	1	1	yes	characteristic
265	C₂×M₁₆	—	2	4	no	—
266	C₂×M₁₆	—	2	4	no	—
267	C₂×M₁₆	—	2	4	no	—
268	C₂²×C₈	C₂³×C₅₃	1	2	yes	—
269	C₂²×C₈	C₂³×C₅₃	1	2	yes	—
270	C₂²×C₈	C₂³×C₅₃	1	2	yes	—
271	C₂²×C₈	C₂³×C₅₃	1	2	yes	—
272	C₄×D₄	—	2	4	no	—
273	#117	—	2	2	no	—
274	#117	—	2	2	no	—
275	C₂×C₄⋊C₄	C₂×C₄×C₅₃	1	2	yes	—
276	C₂×C₄⋊C₄	C₂×C₄×C₅₃	1	1	yes	characteristic
277	C₂×C₄⋊C₄	C₂×C₄×C₅₃	1	2	yes	—
278	C₂×C₂²⋊C₄	C₂×C₄×C₅₃	1	2	yes	—
279	C₂×C₂²⋊C₄	C₂×C₄×C₅₃	1	1	yes	characteristic
280	C₂×C₂²⋊C₄	C₂×C₄×C₅₃	1	1	yes	characteristic
281	C₂×C₂²⋊C₄	C₂×C₄×C₅₃	1	2	yes	—
282	C₂×C₂²⋊C₄	C₂×C₄×C₅₃	1	2	yes	—
283	C₂×C₄²	—	2	2	no	—
284	C₂×C₄²	—	2	2	no	—
285	C₂×C₄²	C₂×C₄×C₅₃	1	2	yes	—
286	C₂×C₄²	C₂³×C₅₃	1	1	yes	characteristic
287	C₂×C₄²	C₂³×C₅₃	1	1	yes	characteristic
288	C₂×C₄²	C₂³×C₅₃	1	1	yes	characteristic
289	C₂×C₄²	C₂³×C₅₃	1	1	yes	characteristic
290	C₂×C₄²	C₂³×C₅₃	1	1	yes	characteristic
291	C₄⋊C₈	—	2	4	no	—
292	C₄⋊C₈	—	2	4	no	—
293	C₂²⋊C₈	—	2	4	no	—
294	C₂²⋊C₈	—	2	4	no	—
295	C₂²⋊C₈	—	2	4	no	—
296	#97	—	2	4	no	—
297	#97	—	2	2	no	—
298	#97	—	2	2	no	—
299	C₄×C₈	—	2	2	no	—
300	C₄×C₈	D₄×C₅₃	1	4	yes	complemented
301	#95	C₂×C₄×C₅₃	1	2	yes	—
302	#95	C₂×C₄×C₅₃	1	2	yes	—
303	#95	C₂×C₄×C₅₃	1	2	yes	—
304	#95	C₂×C₄×C₅₃	1	2	yes	—
305	#95	C₂×C₄×C₅₃	1	2	yes	—
306	C₂⁴	C₂²×C₄×C₅₃	1	1	yes	characteristic
307	C₂⁴	C₂²×C₄×C₅₃	1	1	yes	characteristic
308	C₂×D₄	—	2	2	no	—
309	C₂×D₄	—	4	2	no	—
310	C₂²×C₄	—	2	1	no	—
311	C₂²×C₄	—	2	1	no	—
312	C₂²×C₄	—	2	2	no	—
313	C₂²×C₄	—	2	4	no	—
314	C₂²×C₄	—	2	1	no	—
315	C₂²×C₄	—	2	2	no	—
316	C₂²×C₄	—	2	1	no	—
317	C₂²×C₄	—	2	2	no	—
318	C₂²×C₄	—	4	2	no	—
319	C₂²×C₄	—	4	2	no	—
320	C₂²×C₄	C₂²×C₄×C₅₃	1	2	yes	—
321	C₂²×C₄	C₂²×C₄×C₅₃	1	1	yes	characteristic
322	C₂²×C₄	C₂²×C₄×C₅₃	1	2	yes	—
323	C₂²×C₄	C₂²×C₄×C₅₃	1	1	yes	characteristic
324	C₂²×C₄	C₂²×C₄×C₅₃	1	2	yes	—
325	C₂²×C₄	C₂²×C₄×C₅₃	1	1	yes	characteristic
326	C₂²×C₄	C₂²×C₄×C₅₃	1	1	yes	characteristic
327	C₂²×C₄	C₂²×C₄×C₅₃	1	2	yes	—
328	C₂²×C₄	#41×C₅₃	1	4	yes	—
329	C₂²×C₄	C₂⁴×C₅₃	1	1	yes	characteristic
330	M₁₆	—	2	16	no	—
331	C₂×C₈	—	2	2	no	—
332	C₂×C₈	—	2	2	no	—
333	C₂×C₈	—	2	2	no	—
334	C₂×C₈	—	2	2	no	—
335	C₂×C₈	—	2	2	no	—
336	C₂×C₈	—	2	2	no	—
337	C₂×C₈	—	2	2	no	—
338	C₂×C₈	—	2	2	no	—
339	C₂×C₈	—	2	2	no	—
340	C₂×C₈	—	2	2	no	—
341	C₂×C₈	C₂×D₄×C₅₃	1	4	yes	—
342	C₂×C₈	#41×C₅₃	1	4	yes	—
343	C₄⋊C₄	—	2	2	no	—
344	C₄⋊C₄	—	2	4	no	—
345	C₄⋊C₄	—	4	2	no	—
346	C₂²⋊C₄	—	2	2	no	—
347	C₂²⋊C₄	—	2	2	no	—
348	C₂²⋊C₄	—	2	4	no	—
349	C₂²⋊C₄	—	4	2	no	—
350	C₂²⋊C₄	—	4	2	no	—
351	C₄²	—	2	4	no	—
352	C₄²	—	2	1	no	—
353	C₄²	—	2	1	no	—
354	C₄²	—	2	2	no	—
355	C₄²	—	2	1	no	—
356	C₄²	—	2	1	no	—
357	C₄²	—	2	2	no	—
358	C₄²	C₂×D₄×C₅₃	1	2	yes	—
359	C₄²	#41×C₅₃	1	2	yes	—
360	C₂³	—	2	1	no	—
361	C₂³	—	2	1	no	—
362	C₂³	—	2	2	no	—
363	C₂³	—	2	2	no	—
364	C₂³	—	2	1	no	—
365	C₂³	—	2	1	no	—
366	C₂³	—	2	1	no	—
367	C₂³	—	2	2	no	—
368	C₂³	—	2	2	no	—
369	C₂³	#117×C₅₃	1	2	yes	—
370	C₂³	C₂³×C₄×C₅₃	1	1	yes	characteristic
371	D₄	—	4	4	no	—
372	C₂×C₄	—	2	1	no	—
373	C₂×C₄	—	2	1	no	—
374	C₂×C₄	—	2	2	no	—
375	C₂×C₄	—	2	2	no	—
376	C₂×C₄	—	2	4	no	—
377	C₂×C₄	—	2	4	no	—
378	C₂×C₄	—	2	1	no	—
379	C₂×C₄	—	2	1	no	—
380	C₂×C₄	—	2	1	no	—
381	C₂×C₄	—	2	2	no	—
382	C₂×C₄	—	2	2	no	—
383	C₂×C₄	—	2	1	no	—
384	C₂×C₄	—	2	2	no	—
385	C₂×C₄	—	2	1	no	—
386	C₂×C₄	—	2	2	no	—
387	C₂×C₄	—	2	2	no	—
388	C₂×C₄	—	2	2	no	—
389	C₂×C₄	—	2	2	no	—
390	C₂×C₄	—	2	2	no	—
391	C₂×C₄	—	2	2	no	—
392	C₂×C₄	—	2	2	no	—
393	C₂×C₄	—	2	2	no	—
394	C₂×C₄	—	4	8	no	—
395	C₂×C₄	—	4	2	no	—
396	C₂×C₄	—	4	2	no	—
397	C₂×C₄	#117×C₅₃	1	2	yes	—
398	C₂×C₄	C₄×D₄×C₅₃	1	4	yes	—
399	C₂×C₄	C₂²×D₄×C₅₃	1	1	yes	characteristic
400	C₂×C₄	C₂×#41×C₅₃	1	2	yes	—
401	C₂×C₄	C₂×#41×C₅₃	1	2	yes	—
402	C₂×C₄	C₂×#41×C₅₃	1	1	yes	characteristic
403	C₈	—	2	4	no	—
404	C₈	—	2	4	no	—
405	C₈	—	4	2	no	—
406	C₈	—	4	2	no	—
407	C₂²	—	2	2	no	—
408	C₂²	—	2	2	no	—
409	C₂²	—	2	2	no	—
410	C₂²	—	4	2	no	—
411	C₂²	—	4	2	no	—
412	C₂²	—	4	2	no	—
413	C₂²	—	4	1	no	—
414	C₂²	—	4	1	no	—
415	C₂²	—	4	1	no	—
416	C₂²	—	4	2	no	—
417	C₂²	C₂×#117×C₅₃	1	1	yes	characteristic
418	C₂²	C₂×C₄×D₄×C₅₃	1	1	yes	characteristic
419	C₂²	C₄×#41×C₅₃	1	1	yes	characteristic
420	C₂²	C₅₃×#568	1	2	yes	—
421	C₂²	C₅₃×#568	1	2	yes	—
422	C₄	—	2	2	no	—
423	C₄	—	2	4	no	—
424	C₄	—	4	2	no	—
425	C₄	—	4	1	no	—
426	C₄	—	4	1	no	—
427	C₄	—	4	1	no	—
428	C₄	—	4	2	no	—
429	C₄	—	4	2	no	—
430	C₄	C₅₃×#525	1	2	yes	—
431	C₄	C₅₃×#529	1	2	yes	—
432	C₂	—	4	2	no	—
433	C₂	—	8	1	no	—
434	C₂	C₅₃×#2290	1	1	yes	characteristic
435	C₂	C₅₃×#2921	1	1	yes	characteristic
436	C₂	C₅₃×#2923	1	1	yes	characteristic
437	C₂	C₅₃×#2930	1	2	yes	—
438	C₂	C₅₃×#2967	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	C₂²×D₄×C₅₃	C₂×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×#433	C₂	semidirect
2. C₂×C₄×M₁₆×C₅₃	C₂	semidirect
4. C₅₃×#1924	C₂	semidirect
5. C₅₃×#1917	C₂	semidirect
7. C₅₃×#1851	C₂	semidirect
8. C₅₃×#1768	C₂	semidirect
9. C₅₃×#1750	C₂	semidirect
10. C₅₃×#1749	C₂	semidirect
11. C₅₃×#1746	C₂	semidirect
80. C₄×C₈×C₅₃	D₄	semidirect
187. #21104	C₅₃	direct (N × H)
213. C₂×#433	C₂×C₅₃	semidirect
214. C₂×C₄×M₁₆	C₂×C₅₃	semidirect
216. #1924	C₂×C₅₃	semidirect
217. #1917	C₂×C₅₃	semidirect
219. #1851	C₂×C₅₃	semidirect
220. #1768	C₂×C₅₃	semidirect
221. #1750	C₂×C₅₃	semidirect
222. #1749	C₂×C₅₃	semidirect
223. #1746	C₂×C₅₃	semidirect
260. C₅₃	#21104	direct (N × H)
300. C₄×C₈	D₄×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×#433	2
C₂×C₄×M₁₆×C₅₃	2
C₅₃×#2290	2
C₅₃×#1924	2
C₅₃×#1917	2
C₅₃×#1870	2
C₅₃×#1851	2
C₅₃×#1768	2
C₅₃×#1750	2
C₅₃×#1749	2
C₅₃×#1746	2
#21104	53

Subgroups — C53×#21104 #27228616

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅₃×#21104 #27228616