Subgroups — C₅₃×#22673 #27230185

Order 13568.

974 conjugacy classes of subgroups.

The subgroup lattice has 462 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₅₃×#440	C₂	1	2	yes	maximal
2	C₂×C₄×SD₁₆×C₅₃	C₂	1	1	yes	maximal, characteristic
3	C₅₃×#2351	C₂	1	1	yes	maximal, characteristic, complemented
4	C₅₃×#2349	C₂	1	1	yes	maximal, characteristic
5	C₅₃×#1944	C₂	1	2	yes	maximal, complemented
6	C₅₃×#1865	C₂	1	2	yes	maximal, complemented
7	C₅₃×#1864	C₂	1	2	yes	maximal
8	C₅₃×#1846	C₂	1	1	yes	maximal, characteristic, complemented
9	C₅₃×#1805	C₂	1	1	yes	maximal, characteristic, complemented
10	C₅₃×#1804	C₂	1	1	yes	maximal, characteristic
11	C₅₃×#1770	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂²×SD₁₆×C₅₃	C₂²	1	1	yes	characteristic
13	C₂×C₄×Q₈×C₅₃	C₂²	1	2	yes	—
14	C₂×C₄×Q₈×C₅₃	C₂²	1	1	yes	characteristic
15	C₂×C₄×D₄×C₅₃	C₂²	1	2	yes	—
16	C₂×C₄×D₄×C₅₃	C₂²	1	1	yes	characteristic
17	C₅₃×#440	—	2	8	no	—
18	C₄×SD₁₆×C₅₃	C₄	1	8	yes	complemented
19	C₂×#107×C₅₃	C₂²	1	2	yes	—
20	C₂×#106×C₅₃	C₂²	1	2	yes	—
21	C₂×#106×C₅₃	C₂²	1	1	yes	characteristic
22	C₂×C₄⋊C₈×C₅₃	C₂²	1	1	yes	characteristic
23	C₂×#103×C₅₃	C₂²	1	2	yes	—
24	C₂×#103×C₅₃	C₂²	1	1	yes	characteristic
25	C₂×#102×C₅₃	C₂²	1	2	yes	—
26	C₂×#102×C₅₃	C₂²	1	1	yes	characteristic
27	C₂×#97×C₅₃	C₂²	1	2	yes	—
28	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
29	C₅₃×#389	—	2	2	no	—
30	C₅₃×#389	C₂²	1	2	yes	—
31	C₅₃×#389	C₂²	1	2	yes	—
32	C₅₃×#385	—	2	2	no	—
33	C₅₃×#382	—	2	1	no	—
34	C₅₃×#382	C₂²	1	1	yes	characteristic
35	C₅₃×#382	C₂²	1	1	yes	characteristic
36	C₅₃×#380	—	2	1	no	—
37	C₄×C₄⋊C₄×C₅₃	—	2	1	no	—
38	C₄×C₄⋊C₄×C₅₃	C₂²	1	1	yes	characteristic
39	C₂×#95×C₅₃	—	2	1	no	—
40	C₅₃×#340	C₂²	1	2	yes	—
41	C₅₃×#340	C₂²	1	2	yes	—
42	C₅₃×#336	C₂²	1	2	yes	—
43	C₂²×Q₈×C₅₃	C₂³	1	1	yes	characteristic
44	C₂²×D₄×C₅₃	C₂³	1	1	yes	characteristic
45	C₂³×C₄×C₅₃	—	2	1	no	—
46	C₂³×C₄×C₅₃	—	2	2	no	—
47	C₂×SD₁₆×C₅₃	C₂×C₄	1	4	yes	—
48	C₂×SD₁₆×C₅₃	D₄	1	4	yes	—
49	C₂×SD₁₆×C₅₃	Q₈	1	4	yes	—
50	C₂²×C₈×C₅₃	C₂³	1	2	yes	—
51	C₂²×C₈×C₅₃	C₂³	1	1	yes	characteristic
52	C₂²×C₈×C₅₃	C₂³	1	1	yes	characteristic
53	C₄×Q₈×C₅₃	—	2	2	no	—
54	C₄×Q₈×C₅₃	—	2	4	no	—
55	C₄×Q₈×C₅₃	—	4	2	no	—
56	C₄×Q₈×C₅₃	C₂×C₄	1	4	yes	complemented
57	C₄×D₄×C₅₃	—	2	2	no	—
58	C₄×D₄×C₅₃	—	2	4	no	—
59	C₄×D₄×C₅₃	—	4	2	no	—
60	C₄×D₄×C₅₃	C₂×C₄	1	4	yes	—
61	C₂×C₄⋊C₄×C₅₃	—	2	1	no	—
62	C₂×C₄⋊C₄×C₅₃	—	2	2	no	—
63	C₂×C₄⋊C₄×C₅₃	—	2	2	no	—
64	C₂×C₄⋊C₄×C₅₃	—	2	2	no	—
65	C₂×C₄⋊C₄×C₅₃	D₄	1	2	yes	—
66	C₂×C₄⋊C₄×C₅₃	C₂³	1	2	yes	—
67	C₂×C₄⋊C₄×C₅₃	C₂³	1	2	yes	—
68	C₂×C₄⋊C₄×C₅₃	C₂³	1	1	yes	characteristic
69	C₂×C₄⋊C₄×C₅₃	C₂³	1	1	yes	characteristic
70	C₂×C₂²⋊C₄×C₅₃	—	2	1	no	—
71	C₂×C₂²⋊C₄×C₅₃	—	2	2	no	—
72	C₂×C₄²×C₅₃	—	2	1	no	—
73	C₂×C₄²×C₅₃	—	2	2	no	—
74	C₂×C₄²×C₅₃	C₂³	1	2	yes	—
75	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
76	#107×C₅₃	—	2	4	no	—
77	#106×C₅₃	—	4	2	no	—
78	#106×C₅₃	C₂×C₄	1	4	yes	complemented
79	C₄⋊C₈×C₅₃	—	2	2	no	—
80	#103×C₅₃	—	2	4	no	—
81	#103×C₅₃	C₂×C₄	1	4	yes	complemented
82	#102×C₅₃	—	2	4	no	—
83	#102×C₅₃	C₂×C₄	1	4	yes	—
84	#97×C₅₃	—	2	4	no	—
85	C₄×C₈×C₅₃	C₂×C₄	1	4	yes	complemented
86	#95×C₅₃	—	2	1	no	—
87	#95×C₅₃	—	2	2	no	—
88	#95×C₅₃	—	2	1	no	—
89	#95×C₅₃	—	2	2	no	—
90	#95×C₅₃	—	2	1	no	—
91	#95×C₅₃	—	2	1	no	—
92	#95×C₅₃	D₄	1	1	yes	characteristic
93	#95×C₅₃	D₄	1	1	yes	characteristic
94	C₂⁴×C₅₃	—	2	1	no	—
95	C₂×Q₈×C₅₃	—	2	1	no	—
96	C₂×Q₈×C₅₃	—	2	1	no	—
97	C₂×Q₈×C₅₃	—	2	1	no	—
98	C₂×Q₈×C₅₃	C₂²×C₄	1	2	yes	—
99	C₂×Q₈×C₅₃	C₂×D₄	1	2	yes	—
100	C₂×Q₈×C₅₃	C₂×Q₈	1	2	yes	—
101	C₂×D₄×C₅₃	—	2	1	no	—
102	C₂×D₄×C₅₃	—	2	1	no	—
103	C₂×D₄×C₅₃	—	2	1	no	—
104	C₂×D₄×C₅₃	C₂²×C₄	1	2	yes	—
105	C₂×D₄×C₅₃	C₂×D₄	1	2	yes	—
106	C₂×D₄×C₅₃	C₂×Q₈	1	2	yes	—
107	C₂²×C₄×C₅₃	—	2	1	no	—
108	C₂²×C₄×C₅₃	—	2	1	no	—
109	C₂²×C₄×C₅₃	—	2	1	no	—
110	C₂²×C₄×C₅₃	—	2	4	no	—
111	C₂²×C₄×C₅₃	—	2	2	no	—
112	C₂²×C₄×C₅₃	—	2	2	no	—
113	C₂²×C₄×C₅₃	—	4	4	no	—
114	C₂²×C₄×C₅₃	—	4	4	no	—
115	C₂²×C₄×C₅₃	—	8	2	no	—
116	C₂²×C₄×C₅₃	—	8	2	no	—
117	C₂²×C₄×C₅₃	C₂×D₄	1	2	yes	—
118	C₂²×C₄×C₅₃	C₂×D₄	1	2	yes	—
119	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
120	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
121	C₂²×C₄×C₅₃	C₂⁴	1	1	yes	characteristic
122	SD₁₆×C₅₃	C₄⋊C₄	1	16	yes	complemented
123	C₂×C₈×C₅₃	—	2	1	no	—
124	C₂×C₈×C₅₃	—	2	1	no	—
125	C₂×C₈×C₅₃	—	2	2	no	—
126	C₂×C₈×C₅₃	—	2	2	no	—
127	C₂×C₈×C₅₃	—	2	2	no	—
128	C₂×C₈×C₅₃	C₂²×C₄	1	2	yes	—
129	C₂×C₈×C₅₃	C₂²×C₄	1	2	yes	—
130	C₂×C₈×C₅₃	C₂×D₄	1	2	yes	—
131	C₂×C₈×C₅₃	C₂×Q₈	1	2	yes	—
132	C₄⋊C₄×C₅₃	—	2	4	no	—
133	C₄⋊C₄×C₅₃	—	2	1	no	—
134	C₄⋊C₄×C₅₃	—	2	1	no	—
135	C₄⋊C₄×C₅₃	—	2	2	no	—
136	C₄⋊C₄×C₅₃	—	2	2	no	—
137	C₄⋊C₄×C₅₃	—	2	2	no	—
138	C₄⋊C₄×C₅₃	—	2	2	no	—
139	C₄⋊C₄×C₅₃	—	4	2	no	—
140	C₄⋊C₄×C₅₃	—	4	4	no	—
141	C₄⋊C₄×C₅₃	—	4	4	no	—
142	C₄⋊C₄×C₅₃	—	8	2	no	—
143	C₄⋊C₄×C₅₃	C₂²×C₄	1	2	yes	—
144	C₄⋊C₄×C₅₃	C₂²×C₄	1	2	yes	—
145	C₂²⋊C₄×C₅₃	—	2	4	no	—
146	C₂²⋊C₄×C₅₃	—	4	4	no	—
147	C₄²×C₅₃	—	2	4	no	—
148	C₄²×C₅₃	—	2	2	no	—
149	C₄²×C₅₃	—	2	2	no	—
150	C₄²×C₅₃	—	4	4	no	—
151	C₄²×C₅₃	C₂²×C₄	1	2	yes	—
152	C₄²×C₅₃	#41	1	2	yes	—
153	C₂³×C₅₃	—	2	2	no	—
154	C₂³×C₅₃	—	2	2	no	—
155	C₂³×C₅₃	—	2	2	no	—
156	C₂³×C₅₃	—	4	2	no	—
157	C₂³×C₅₃	—	4	2	no	—
158	C₂³×C₅₃	C₂²×D₄	1	1	yes	characteristic
159	Q₈×C₅₃	—	2	2	no	—
160	Q₈×C₅₃	—	2	2	no	—
161	Q₈×C₅₃	—	2	2	no	—
162	Q₈×C₅₃	C₂×C₄⋊C₄	1	4	yes	—
163	D₄×C₅₃	—	2	2	no	—
164	D₄×C₅₃	—	2	2	no	—
165	D₄×C₅₃	—	2	2	no	—
166	D₄×C₅₃	C₂×C₄⋊C₄	1	4	yes	—
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	—	2	2	no	—
175	C₂×C₄×C₅₃	—	2	2	no	—
176	C₂×C₄×C₅₃	—	2	2	no	—
177	C₂×C₄×C₅₃	—	2	2	no	—
178	C₂×C₄×C₅₃	—	2	2	no	—
179	C₂×C₄×C₅₃	—	2	2	no	—
180	C₂×C₄×C₅₃	—	4	2	no	—
181	C₂×C₄×C₅₃	—	4	2	no	—
182	C₂×C₄×C₅₃	—	4	2	no	—
183	C₂×C₄×C₅₃	—	4	2	no	—
184	C₂×C₄×C₅₃	—	4	2	no	—
185	C₂×C₄×C₅₃	—	4	2	no	—
186	C₂×C₄×C₅₃	—	4	2	no	—
187	C₂×C₄×C₅₃	—	4	2	no	—
188	C₂×C₄×C₅₃	—	8	4	no	—
189	C₂×C₄×C₅₃	—	8	8	no	—
190	C₂×C₄×C₅₃	C₄×D₄	1	2	yes	—
191	C₂×C₄×C₅₃	C₄×D₄	1	2	yes	—
192	C₂×C₄×C₅₃	C₂³×C₄	1	1	yes	characteristic
193	C₂×C₄×C₅₃	C₂²×D₄	1	1	yes	characteristic
194	C₂×C₄×C₅₃	C₂²×Q₈	1	1	yes	characteristic
195	C₂×C₄×C₅₃	C₂×#41	1	1	yes	characteristic
196	C₂×C₄×C₅₃	H(5,𝔽₂)	1	1	yes	characteristic
197	C₂×C₄×C₅₃	2₋¹⁺⁴	1	1	yes	characteristic
198	C₈×C₅₃	—	2	2	no	—
199	C₈×C₅₃	—	4	2	no	—
200	C₈×C₅₃	C₂×C₄⋊C₄	1	4	yes	—
201	#22673	C₅₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
202	C₂²×C₅₃	—	2	4	no	—
203	C₂²×C₅₃	—	4	4	no	—
204	C₂²×C₅₃	—	4	2	no	—
205	C₂²×C₅₃	—	4	2	no	—
206	C₂²×C₅₃	—	4	4	no	—
207	C₂²×C₅₃	C₂×C₄×D₄	1	1	yes	characteristic
208	C₂²×C₅₃	D₄²	1	1	yes	characteristic
209	C₂²×C₅₃	D₄×Q₈	1	1	yes	characteristic
210	C₂²×C₅₃	#575	1	2	yes	—
211	C₂²×C₅₃	#577	1	2	yes	—
212	C₄×C₅₃	—	2	4	no	—
213	C₄×C₅₃	—	2	2	no	—
214	C₄×C₅₃	—	2	2	no	—
215	C₄×C₅₃	—	4	2	no	—
216	C₄×C₅₃	—	4	2	no	—
217	C₄×C₅₃	—	4	2	no	—
218	C₄×C₅₃	—	4	2	no	—
219	C₄×C₅₃	—	8	2	no	—
220	C₄×C₅₃	—	8	2	no	—
221	C₄×C₅₃	C₂²×C₄⋊C₄	1	1	yes	characteristic
222	C₄×C₅₃	#520	1	1	yes	characteristic
223	C₄×C₅₃	#547	1	1	yes	characteristic
224	C₄×C₅₃	#554	1	1	yes	characteristic
225	C₂×#440	C₂×C₅₃	1	2	yes	—
226	C₂×C₄×SD₁₆	C₂×C₅₃	1	1	yes	characteristic
227	#2351	C₂×C₅₃	1	1	yes	characteristic, complemented
228	#2349	C₂×C₅₃	1	1	yes	characteristic
229	#1944	C₂×C₅₃	1	2	yes	complemented
230	#1865	C₂×C₅₃	1	2	yes	complemented
231	#1864	C₂×C₅₃	1	2	yes	—
232	#1846	C₂×C₅₃	1	1	yes	characteristic, complemented
233	#1805	C₂×C₅₃	1	1	yes	characteristic, complemented
234	#1804	C₂×C₅₃	1	1	yes	characteristic
235	#1770	C₂×C₅₃	1	1	yes	characteristic, complemented
236	C₂×C₅₃	—	4	4	no	—
237	C₂×C₅₃	D₄×C₄⋊C₄	1	1	yes	characteristic
238	C₂×C₅₃	#2943	1	1	yes	characteristic
239	C₂×C₅₃	#2949	1	1	yes	characteristic
240	C₂×C₅₃	#3275	1	2	yes	—
241	C₂×C₅₃	#3388	1	2	yes	—
242	C₂²×SD₁₆	C₂²×C₅₃	1	1	yes	characteristic
243	C₂×C₄×Q₈	C₂²×C₅₃	1	2	yes	—
244	C₂×C₄×Q₈	C₂²×C₅₃	1	1	yes	characteristic
245	C₂×C₄×D₄	C₂²×C₅₃	1	2	yes	—
246	C₂×C₄×D₄	C₂²×C₅₃	1	1	yes	characteristic
247	#440	—	2	8	no	—
248	C₄×SD₁₆	C₄×C₅₃	1	8	yes	complemented
249	C₂×#107	C₂²×C₅₃	1	2	yes	—
250	C₂×#106	C₂²×C₅₃	1	2	yes	—
251	C₂×#106	C₂²×C₅₃	1	1	yes	characteristic
252	C₂×C₄⋊C₈	C₂²×C₅₃	1	1	yes	characteristic
253	C₂×#103	C₂²×C₅₃	1	2	yes	—
254	C₂×#103	C₂²×C₅₃	1	1	yes	characteristic
255	C₂×#102	C₂²×C₅₃	1	2	yes	—
256	C₂×#102	C₂²×C₅₃	1	1	yes	characteristic
257	C₂×#97	C₂²×C₅₃	1	2	yes	—
258	C₂×C₄×C₈	C₂²×C₅₃	1	1	yes	characteristic
259	#389	—	2	2	no	—
260	#389	C₂²×C₅₃	1	2	yes	—
261	#389	C₂²×C₅₃	1	2	yes	—
262	#385	—	2	2	no	—
263	#382	—	2	1	no	—
264	#382	C₂²×C₅₃	1	1	yes	characteristic
265	#382	C₂²×C₅₃	1	1	yes	characteristic
266	#380	—	2	1	no	—
267	C₄×C₄⋊C₄	—	2	1	no	—
268	C₄×C₄⋊C₄	C₂²×C₅₃	1	1	yes	characteristic
269	C₂×#95	—	2	1	no	—
270	#340	C₂²×C₅₃	1	2	yes	—
271	#340	C₂²×C₅₃	1	2	yes	—
272	#336	C₂²×C₅₃	1	2	yes	—
273	C₅₃	#22673	1	1	yes	characteristic, direct product factor, Sylow
274	C₂²×Q₈	C₂³×C₅₃	1	1	yes	characteristic
275	C₂²×D₄	C₂³×C₅₃	1	1	yes	characteristic
276	C₂³×C₄	—	2	1	no	—
277	C₂³×C₄	—	2	2	no	—
278	C₂×SD₁₆	C₂×C₄×C₅₃	1	4	yes	—
279	C₂×SD₁₆	D₄×C₅₃	1	4	yes	—
280	C₂×SD₁₆	Q₈×C₅₃	1	4	yes	—
281	C₂²×C₈	C₂³×C₅₃	1	2	yes	—
282	C₂²×C₈	C₂³×C₅₃	1	1	yes	characteristic
283	C₂²×C₈	C₂³×C₅₃	1	1	yes	characteristic
284	C₄×Q₈	—	2	2	no	—
285	C₄×Q₈	—	2	4	no	—
286	C₄×Q₈	—	4	2	no	—
287	C₄×Q₈	C₂×C₄×C₅₃	1	4	yes	complemented
288	C₄×D₄	—	2	4	no	—
289	C₄×D₄	—	2	2	no	—
290	C₄×D₄	—	4	2	no	—
291	C₄×D₄	C₂×C₄×C₅₃	1	4	yes	—
292	C₂×C₄⋊C₄	—	2	1	no	—
293	C₂×C₄⋊C₄	—	2	2	no	—
294	C₂×C₄⋊C₄	—	2	2	no	—
295	C₂×C₄⋊C₄	—	2	2	no	—
296	C₂×C₄⋊C₄	D₄×C₅₃	1	2	yes	—
297	C₂×C₄⋊C₄	C₂³×C₅₃	1	2	yes	—
298	C₂×C₄⋊C₄	C₂³×C₅₃	1	2	yes	—
299	C₂×C₄⋊C₄	C₂³×C₅₃	1	1	yes	characteristic
300	C₂×C₄⋊C₄	C₂³×C₅₃	1	1	yes	characteristic
301	C₂×C₂²⋊C₄	—	2	1	no	—
302	C₂×C₂²⋊C₄	—	2	2	no	—
303	C₂×C₄²	—	2	1	no	—
304	C₂×C₄²	—	2	2	no	—
305	C₂×C₄²	C₂³×C₅₃	1	2	yes	—
306	C₂×C₄²	C₂³×C₅₃	1	1	yes	characteristic
307	#107	—	2	4	no	—
308	#106	—	4	2	no	—
309	#106	C₂×C₄×C₅₃	1	4	yes	complemented
310	C₄⋊C₈	—	2	2	no	—
311	#103	—	2	4	no	—
312	#103	C₂×C₄×C₅₃	1	4	yes	complemented
313	#102	—	2	4	no	—
314	#102	C₂×C₄×C₅₃	1	4	yes	—
315	#97	—	2	4	no	—
316	C₄×C₈	C₂×C₄×C₅₃	1	4	yes	complemented
317	#95	—	2	1	no	—
318	#95	—	2	2	no	—
319	#95	—	2	1	no	—
320	#95	—	2	2	no	—
321	#95	—	2	1	no	—
322	#95	—	2	1	no	—
323	#95	D₄×C₅₃	1	1	yes	characteristic
324	#95	D₄×C₅₃	1	1	yes	characteristic
325	C₂⁴	—	2	1	no	—
326	C₂×Q₈	—	2	1	no	—
327	C₂×Q₈	—	2	1	no	—
328	C₂×Q₈	—	2	1	no	—
329	C₂×Q₈	C₂²×C₄×C₅₃	1	2	yes	—
330	C₂×Q₈	C₂×D₄×C₅₃	1	2	yes	—
331	C₂×Q₈	C₂×Q₈×C₅₃	1	2	yes	—
332	C₂×D₄	—	2	1	no	—
333	C₂×D₄	—	2	1	no	—
334	C₂×D₄	—	2	1	no	—
335	C₂×D₄	C₂²×C₄×C₅₃	1	2	yes	—
336	C₂×D₄	C₂×D₄×C₅₃	1	2	yes	—
337	C₂×D₄	C₂×Q₈×C₅₃	1	2	yes	—
338	C₂²×C₄	—	2	1	no	—
339	C₂²×C₄	—	2	1	no	—
340	C₂²×C₄	—	2	1	no	—
341	C₂²×C₄	—	2	4	no	—
342	C₂²×C₄	—	2	2	no	—
343	C₂²×C₄	—	2	2	no	—
344	C₂²×C₄	—	4	4	no	—
345	C₂²×C₄	—	4	4	no	—
346	C₂²×C₄	—	8	2	no	—
347	C₂²×C₄	—	8	2	no	—
348	C₂²×C₄	C₂×D₄×C₅₃	1	2	yes	—
349	C₂²×C₄	C₂×D₄×C₅₃	1	2	yes	—
350	C₂²×C₄	C₂×D₄×C₅₃	1	1	yes	characteristic
351	C₂²×C₄	C₂×D₄×C₅₃	1	1	yes	characteristic
352	C₂²×C₄	C₂⁴×C₅₃	1	1	yes	characteristic
353	SD₁₆	C₄⋊C₄×C₅₃	1	16	yes	complemented
354	C₂×C₈	—	2	1	no	—
355	C₂×C₈	—	2	1	no	—
356	C₂×C₈	—	2	2	no	—
357	C₂×C₈	—	2	2	no	—
358	C₂×C₈	—	2	2	no	—
359	C₂×C₈	C₂²×C₄×C₅₃	1	2	yes	—
360	C₂×C₈	C₂²×C₄×C₅₃	1	2	yes	—
361	C₂×C₈	C₂×D₄×C₅₃	1	2	yes	—
362	C₂×C₈	C₂×Q₈×C₅₃	1	2	yes	—
363	C₄⋊C₄	—	2	4	no	—
364	C₄⋊C₄	—	2	1	no	—
365	C₄⋊C₄	—	2	1	no	—
366	C₄⋊C₄	—	2	2	no	—
367	C₄⋊C₄	—	2	2	no	—
368	C₄⋊C₄	—	2	2	no	—
369	C₄⋊C₄	—	2	2	no	—
370	C₄⋊C₄	—	4	2	no	—
371	C₄⋊C₄	—	4	4	no	—
372	C₄⋊C₄	—	4	4	no	—
373	C₄⋊C₄	—	8	2	no	—
374	C₄⋊C₄	C₂²×C₄×C₅₃	1	2	yes	—
375	C₄⋊C₄	C₂²×C₄×C₅₃	1	2	yes	—
376	C₂²⋊C₄	—	2	4	no	—
377	C₂²⋊C₄	—	4	4	no	—
378	C₄²	—	2	4	no	—
379	C₄²	—	2	2	no	—
380	C₄²	—	2	2	no	—
381	C₄²	—	4	4	no	—
382	C₄²	C₂²×C₄×C₅₃	1	2	yes	—
383	C₄²	#41×C₅₃	1	2	yes	—
384	C₂³	—	2	2	no	—
385	C₂³	—	2	2	no	—
386	C₂³	—	2	2	no	—
387	C₂³	—	4	2	no	—
388	C₂³	—	4	2	no	—
389	C₂³	C₂²×D₄×C₅₃	1	1	yes	characteristic
390	Q₈	—	2	2	no	—
391	Q₈	—	2	2	no	—
392	Q₈	—	2	2	no	—
393	Q₈	C₂×C₄⋊C₄×C₅₃	1	4	yes	—
394	D₄	—	2	2	no	—
395	D₄	—	2	2	no	—
396	D₄	—	2	2	no	—
397	D₄	C₂×C₄⋊C₄×C₅₃	1	4	yes	—
398	C₂×C₄	—	2	2	no	—
399	C₂×C₄	—	2	2	no	—
400	C₂×C₄	—	2	2	no	—
401	C₂×C₄	—	2	2	no	—
402	C₂×C₄	—	2	2	no	—
403	C₂×C₄	—	2	2	no	—
404	C₂×C₄	—	2	2	no	—
405	C₂×C₄	—	2	2	no	—
406	C₂×C₄	—	2	2	no	—
407	C₂×C₄	—	2	2	no	—
408	C₂×C₄	—	2	2	no	—
409	C₂×C₄	—	2	2	no	—
410	C₂×C₄	—	2	2	no	—
411	C₂×C₄	—	4	2	no	—
412	C₂×C₄	—	4	2	no	—
413	C₂×C₄	—	4	2	no	—
414	C₂×C₄	—	4	2	no	—
415	C₂×C₄	—	4	2	no	—
416	C₂×C₄	—	4	2	no	—
417	C₂×C₄	—	4	2	no	—
418	C₂×C₄	—	4	2	no	—
419	C₂×C₄	—	8	4	no	—
420	C₂×C₄	—	8	8	no	—
421	C₂×C₄	C₄×D₄×C₅₃	1	2	yes	—
422	C₂×C₄	C₄×D₄×C₅₃	1	2	yes	—
423	C₂×C₄	C₂³×C₄×C₅₃	1	1	yes	characteristic
424	C₂×C₄	C₂²×D₄×C₅₃	1	1	yes	characteristic
425	C₂×C₄	C₂²×Q₈×C₅₃	1	1	yes	characteristic
426	C₂×C₄	C₂×#41×C₅₃	1	1	yes	characteristic
427	C₂×C₄	H(5,𝔽₂)×C₅₃	1	1	yes	characteristic
428	C₂×C₄	2₋¹⁺⁴×C₅₃	1	1	yes	characteristic
429	C₈	—	2	2	no	—
430	C₈	—	4	2	no	—
431	C₈	C₂×C₄⋊C₄×C₅₃	1	4	yes	—
432	C₂²	—	2	4	no	—
433	C₂²	—	4	2	no	—
434	C₂²	—	4	2	no	—
435	C₂²	—	4	4	no	—
436	C₂²	—	4	4	no	—
437	C₂²	C₂×C₄×D₄×C₅₃	1	1	yes	characteristic
438	C₂²	D₄²×C₅₃	1	1	yes	characteristic
439	C₂²	D₄×Q₈×C₅₃	1	1	yes	characteristic
440	C₂²	C₅₃×#575	1	2	yes	—
441	C₂²	C₅₃×#577	1	2	yes	—
442	C₄	—	2	2	no	—
443	C₄	—	2	2	no	—
444	C₄	—	2	4	no	—
445	C₄	—	4	2	no	—
446	C₄	—	4	2	no	—
447	C₄	—	4	2	no	—
448	C₄	—	4	2	no	—
449	C₄	—	8	2	no	—
450	C₄	—	8	2	no	—
451	C₄	C₂²×C₄⋊C₄×C₅₃	1	1	yes	characteristic
452	C₄	C₅₃×#520	1	1	yes	characteristic
453	C₄	C₅₃×#547	1	1	yes	characteristic
454	C₄	C₅₃×#554	1	1	yes	characteristic
455	C₂	—	4	4	no	—
456	C₂	D₄×C₄⋊C₄×C₅₃	1	1	yes	characteristic
457	C₂	C₅₃×#2943	1	1	yes	characteristic
458	C₂	C₅₃×#2949	1	1	yes	characteristic
459	C₂	C₅₃×#3275	1	2	yes	—
460	C₂	C₅₃×#3388	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	C₂²×D₄×C₅₃	C₂³
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₅₃ ◁ C₄×C₄⋊C₄×C₅₃ ◁ G

Complements

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

N	H ≅ G/N	type
3. C₅₃×#2351	C₂	semidirect
5. C₅₃×#1944	C₂	semidirect
6. C₅₃×#1865	C₂	semidirect
8. C₅₃×#1846	C₂	semidirect
9. C₅₃×#1805	C₂	semidirect
11. C₅₃×#1770	C₂	semidirect
18. C₄×SD₁₆×C₅₃	C₄	semidirect
56. C₄×Q₈×C₅₃	C₂×C₄	semidirect
78. #106×C₅₃	C₂×C₄	semidirect
81. #103×C₅₃	C₂×C₄	semidirect
85. C₄×C₈×C₅₃	C₂×C₄	semidirect
122. SD₁₆×C₅₃	C₄⋊C₄	semidirect
201. #22673	C₅₃	direct (N × H)
227. #2351	C₂×C₅₃	semidirect
229. #1944	C₂×C₅₃	semidirect
230. #1865	C₂×C₅₃	semidirect
232. #1846	C₂×C₅₃	semidirect
233. #1805	C₂×C₅₃	semidirect
235. #1770	C₂×C₅₃	semidirect
248. C₄×SD₁₆	C₄×C₅₃	semidirect
273. C₅₃	#22673	direct (N × H)
287. C₄×Q₈	C₂×C₄×C₅₃	semidirect
309. #106	C₂×C₄×C₅₃	semidirect
312. #103	C₂×C₄×C₅₃	semidirect
316. C₄×C₈	C₂×C₄×C₅₃	semidirect
353. SD₁₆	C₄⋊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×#440	2
C₂×C₄×SD₁₆×C₅₃	2
C₅₃×#2351	2
C₅₃×#2349	2
C₅₃×#1944	2
C₅₃×#1865	2
C₅₃×#1864	2
C₅₃×#1846	2
C₅₃×#1805	2
C₅₃×#1804	2
C₅₃×#1770	2
#22673	53

Subgroups — C53×#22673 #27230185

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅₃×#22673 #27230185