Subgroups — C₅₃×#24812 #27232324

Order 13568.

942 conjugacy classes of subgroups.

The subgroup lattice has 530 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₅₃×#476	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×C₄×D₈×C₅₃	C₂	1	1	yes	maximal, characteristic
3	C₅₃×#2563	C₂	1	1	yes	maximal, characteristic, complemented
4	D₄×C₄⋊C₄×C₅₃	C₂	1	1	yes	maximal, characteristic, complemented
5	C₅₃×#2085	C₂	1	2	yes	maximal, complemented
6	C₅₃×#2065	C₂	1	2	yes	maximal, complemented
7	C₅₃×#1990	C₂	1	1	yes	maximal, characteristic, complemented
8	C₅₃×#1939	C₂	1	1	yes	maximal, characteristic, complemented
9	C₅₃×#1879	C₂	1	2	yes	maximal
10	C₅₃×#1804	C₂	1	2	yes	maximal, complemented
11	C₅₃×#1802	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂²×D₈×C₅₃	C₂²	1	1	yes	characteristic
13	C₂×C₄⋊Q₈×C₅₃	C₂²	1	1	yes	characteristic
14	C₂×C₄×D₄×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	C₂²×C₄⋊C₄×C₅₃	—	2	1	no	—
18	C₅₃×#476	—	2	4	no	—
19	C₄×D₈×C₅₃	—	2	4	no	—
20	C₂×#107×C₅₃	C₂²	1	1	yes	characteristic
21	C₂×#106×C₅₃	C₂²	1	2	yes	—
22	C₂×C₄⋊C₈×C₅₃	C₂²	1	1	yes	characteristic
23	C₂×C₄⋊C₈×C₅₃	C₂²	1	2	yes	—
24	C₂×#102×C₅₃	C₂²	1	1	yes	characteristic
25	C₂×#102×C₅₃	C₂²	1	1	yes	characteristic
26	C₂×#102×C₅₃	C₂²	1	2	yes	—
27	C₂×#102×C₅₃	C₂²	1	2	yes	—
28	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
29	C₅₃×#393	—	2	2	no	—
30	C₅₃×#390	C₂²	1	2	yes	—
31	C₅₃×#389	C₂²	1	2	yes	—
32	C₅₃×#385	—	2	2	no	—
33	C₅₃×#384	C₂²	1	1	yes	characteristic
34	C₅₃×#382	C₂²	1	2	yes	—
35	C₅₃×#380	—	2	1	no	—
36	C₅₃×#380	—	2	1	no	—
37	C₅₃×#380	—	2	1	no	—
38	C₄×C₄⋊C₄×C₅₃	C₂²	1	1	yes	characteristic
39	C₅₃×#340	C₂²	1	2	yes	—
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	—	2	1	no	—
44	C₂²×D₄×C₅₃	C₂³	1	1	yes	characteristic
45	C₂²×D₄×C₅₃	C₂³	1	1	yes	characteristic
46	C₂³×C₄×C₅₃	—	2	1	no	—
47	C₂³×C₄×C₅₃	—	2	1	no	—
48	C₂³×C₄×C₅₃	—	2	2	no	—
49	C₂×D₈×C₅₃	—	2	2	no	—
50	C₂×D₈×C₅₃	—	2	2	no	—
51	C₂×D₈×C₅₃	Q₈	1	4	yes	complemented
52	C₂²×C₈×C₅₃	C₂³	1	2	yes	—
53	C₂²×C₈×C₅₃	C₂³	1	1	yes	characteristic
54	C₂²×C₈×C₅₃	C₂³	1	1	yes	characteristic
55	C₄⋊Q₈×C₅₃	—	2	2	no	—
56	C₄⋊Q₈×C₅₃	—	2	2	no	—
57	C₄×D₄×C₅₃	—	2	2	no	—
58	C₄×D₄×C₅₃	—	2	2	no	—
59	C₄×D₄×C₅₃	—	2	2	no	—
60	C₄×D₄×C₅₃	—	2	4	no	—
61	C₄×D₄×C₅₃	—	4	2	no	—
62	C₄×D₄×C₅₃	D₄	1	4	yes	—
63	C₂×C₄⋊C₄×C₅₃	—	2	1	no	—
64	C₂×C₄⋊C₄×C₅₃	—	2	2	no	—
65	C₂×C₄⋊C₄×C₅₃	—	2	1	no	—
66	C₂×C₄⋊C₄×C₅₃	—	2	1	no	—
67	C₂×C₄⋊C₄×C₅₃	—	4	4	no	—
68	C₂×C₄⋊C₄×C₅₃	D₄	1	1	yes	characteristic
69	C₂×C₄⋊C₄×C₅₃	D₄	1	1	yes	characteristic
70	C₂×C₄⋊C₄×C₅₃	C₂³	1	2	yes	—
71	C₂×C₄⋊C₄×C₅₃	C₂³	1	1	yes	characteristic
72	C₂×C₄⋊C₄×C₅₃	C₂³	1	2	yes	—
73	C₂×C₄⋊C₄×C₅₃	C₂³	1	1	yes	characteristic
74	C₂×C₂²⋊C₄×C₅₃	—	2	1	no	—
75	C₂×C₂²⋊C₄×C₅₃	—	2	2	no	—
76	C₂×C₂²⋊C₄×C₅₃	—	2	1	no	—
77	C₂×C₂²⋊C₄×C₅₃	—	2	2	no	—
78	C₂×C₂²⋊C₄×C₅₃	—	2	2	no	—
79	C₂×C₄²×C₅₃	C₂³	1	2	yes	—
80	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
81	#107×C₅₃	—	2	2	no	—
82	#106×C₅₃	—	2	4	no	—
83	C₄⋊C₈×C₅₃	—	2	2	no	—
84	C₄⋊C₈×C₅₃	—	4	2	no	—
85	#102×C₅₃	—	2	2	no	—
86	#102×C₅₃	—	2	2	no	—
87	#102×C₅₃	—	2	4	no	—
88	#102×C₅₃	—	4	2	no	—
89	C₄×C₈×C₅₃	—	2	2	no	—
90	#95×C₅₃	—	2	2	no	—
91	#95×C₅₃	—	2	2	no	—
92	#95×C₅₃	—	2	2	no	—
93	#95×C₅₃	D₄	1	2	yes	—
94	C₂⁴×C₅₃	—	2	1	no	—
95	C₂⁴×C₅₃	—	2	1	no	—
96	C₂×Q₈×C₅₃	—	4	2	no	—
97	C₂×Q₈×C₅₃	—	8	1	no	—
98	C₂×Q₈×C₅₃	—	8	1	no	—
99	C₂×D₄×C₅₃	—	2	1	no	—
100	C₂×D₄×C₅₃	—	2	1	no	—
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	—	2	1	no	—
105	C₂×D₄×C₅₃	—	2	1	no	—
106	C₂×D₄×C₅₃	—	2	1	no	—
107	C₂×D₄×C₅₃	C₂×D₄	1	2	yes	—
108	C₂×D₄×C₅₃	C₂×Q₈	1	2	yes	—
109	C₂×D₄×C₅₃	C₂×Q₈	1	2	yes	—
110	C₂×D₄×C₅₃	#41	1	2	yes	—
111	C₂²×C₄×C₅₃	—	2	1	no	—
112	C₂²×C₄×C₅₃	—	2	4	no	—
113	C₂²×C₄×C₅₃	—	2	1	no	—
114	C₂²×C₄×C₅₃	—	2	2	no	—
115	C₂²×C₄×C₅₃	—	2	2	no	—
116	C₂²×C₄×C₅₃	—	4	4	no	—
117	C₂²×C₄×C₅₃	—	4	4	no	—
118	C₂²×C₄×C₅₃	—	4	2	no	—
119	C₂²×C₄×C₅₃	—	4	2	no	—
120	C₂²×C₄×C₅₃	—	4	2	no	—
121	C₂²×C₄×C₅₃	—	4	2	no	—
122	C₂²×C₄×C₅₃	—	4	2	no	—
123	C₂²×C₄×C₅₃	—	8	2	no	—
124	C₂²×C₄×C₅₃	C₂×D₄	1	2	yes	—
125	C₂²×C₄×C₅₃	C₂×D₄	1	2	yes	—
126	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
127	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
128	C₂²×C₄×C₅₃	C₂⁴	1	1	yes	characteristic
129	D₈×C₅₃	—	2	8	no	—
130	C₂×C₈×C₅₃	—	2	1	no	—
131	C₂×C₈×C₅₃	—	2	1	no	—
132	C₂×C₈×C₅₃	—	2	1	no	—
133	C₂×C₈×C₅₃	—	2	2	no	—
134	C₂×C₈×C₅₃	—	2	2	no	—
135	C₂×C₈×C₅₃	—	2	1	no	—
136	C₂×C₈×C₅₃	—	2	2	no	—
137	C₂×C₈×C₅₃	—	2	1	no	—
138	C₂×C₈×C₅₃	C₂×Q₈	1	2	yes	—
139	C₄⋊C₄×C₅₃	—	2	2	no	—
140	C₄⋊C₄×C₅₃	—	2	2	no	—
141	C₄⋊C₄×C₅₃	—	2	1	no	—
142	C₄⋊C₄×C₅₃	—	2	2	no	—
143	C₄⋊C₄×C₅₃	—	2	1	no	—
144	C₄⋊C₄×C₅₃	—	2	1	no	—
145	C₄⋊C₄×C₅₃	—	2	2	no	—
146	C₄⋊C₄×C₅₃	—	2	2	no	—
147	C₄⋊C₄×C₅₃	—	2	2	no	—
148	C₄⋊C₄×C₅₃	—	4	4	no	—
149	C₄⋊C₄×C₅₃	—	4	2	no	—
150	C₄⋊C₄×C₅₃	—	4	2	no	—
151	C₄⋊C₄×C₅₃	—	8	1	no	—
152	C₄⋊C₄×C₅₃	C₂×D₄	1	2	yes	—
153	C₂²⋊C₄×C₅₃	—	2	4	no	—
154	C₂²⋊C₄×C₅₃	—	4	4	no	—
155	C₂²⋊C₄×C₅₃	—	4	2	no	—
156	C₂²⋊C₄×C₅₃	—	8	2	no	—
157	C₂²⋊C₄×C₅₃	—	8	2	no	—
158	C₄²×C₅₃	—	2	2	no	—
159	C₄²×C₅₃	—	2	2	no	—
160	C₄²×C₅₃	C₂×D₄	1	2	yes	—
161	C₄²×C₅₃	#41	1	2	yes	—
162	C₂³×C₅₃	—	2	2	no	—
163	C₂³×C₅₃	—	2	2	no	—
164	C₂³×C₅₃	—	2	2	no	—
165	C₂³×C₅₃	—	2	2	no	—
166	C₂³×C₅₃	—	4	1	no	—
167	C₂³×C₅₃	—	4	1	no	—
168	C₂³×C₅₃	—	4	1	no	—
169	C₂³×C₅₃	—	4	1	no	—
170	C₂³×C₅₃	—	4	1	no	—
171	C₂³×C₅₃	—	4	1	no	—
172	C₂³×C₅₃	—	4	1	no	—
173	C₂³×C₅₃	—	4	1	no	—
174	C₂³×C₅₃	—	4	1	no	—
175	C₂³×C₅₃	—	4	1	no	—
176	C₂³×C₅₃	C₂²×D₄	1	1	yes	characteristic
177	Q₈×C₅₃	—	8	4	no	—
178	D₄×C₅₃	—	2	2	no	—
179	D₄×C₅₃	—	2	2	no	—
180	D₄×C₅₃	—	2	2	no	—
181	D₄×C₅₃	—	2	2	no	—
182	D₄×C₅₃	—	2	2	no	—
183	D₄×C₅₃	—	4	1	no	—
184	D₄×C₅₃	—	4	1	no	—
185	D₄×C₅₃	C₂²⋊Q₈	1	4	yes	—
186	C₂×C₄×C₅₃	—	2	1	no	—
187	C₂×C₄×C₅₃	—	2	1	no	—
188	C₂×C₄×C₅₃	—	2	2	no	—
189	C₂×C₄×C₅₃	—	2	2	no	—
190	C₂×C₄×C₅₃	—	2	2	no	—
191	C₂×C₄×C₅₃	—	2	2	no	—
192	C₂×C₄×C₅₃	—	2	2	no	—
193	C₂×C₄×C₅₃	—	2	2	no	—
194	C₂×C₄×C₅₃	—	2	2	no	—
195	C₂×C₄×C₅₃	—	4	1	no	—
196	C₂×C₄×C₅₃	—	4	1	no	—
197	C₂×C₄×C₅₃	—	4	8	no	—
198	C₂×C₄×C₅₃	—	4	1	no	—
199	C₂×C₄×C₅₃	—	4	2	no	—
200	C₂×C₄×C₅₃	—	4	2	no	—
201	C₂×C₄×C₅₃	—	4	1	no	—
202	C₂×C₄×C₅₃	—	4	1	no	—
203	C₂×C₄×C₅₃	—	4	2	no	—
204	C₂×C₄×C₅₃	—	4	2	no	—
205	C₂×C₄×C₅₃	—	4	2	no	—
206	C₂×C₄×C₅₃	—	4	2	no	—
207	C₂×C₄×C₅₃	—	8	8	no	—
208	C₂×C₄×C₅₃	—	8	4	no	—
209	C₂×C₄×C₅₃	#121	1	2	yes	—
210	C₂×C₄×C₅₃	#121	1	2	yes	—
211	C₂×C₄×C₅₃	#135	1	2	yes	—
212	C₂×C₄×C₅₃	AGL(1,ℤ/8)	1	2	yes	—
213	C₂×C₄×C₅₃	C₂²×D₄	1	1	yes	characteristic
214	C₂×C₄×C₅₃	C₂²×Q₈	1	1	yes	characteristic
215	C₂×C₄×C₅₃	C₂×#41	1	1	yes	characteristic
216	C₂×C₄×C₅₃	C₂×#41	1	1	yes	characteristic
217	C₂×C₄×C₅₃	H(5,𝔽₂)	1	1	yes	characteristic
218	C₂×C₄×C₅₃	2₋¹⁺⁴	1	1	yes	characteristic
219	C₈×C₅₃	—	2	2	no	—
220	C₈×C₅₃	—	4	1	no	—
221	C₈×C₅₃	—	4	2	no	—
222	#24812	C₅₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
223	C₂²×C₅₃	—	2	4	no	—
224	C₂²×C₅₃	—	4	2	no	—
225	C₂²×C₅₃	—	4	2	no	—
226	C₂²×C₅₃	—	4	2	no	—
227	C₂²×C₅₃	—	4	2	no	—
228	C₂²×C₅₃	—	4	2	no	—
229	C₂²×C₅₃	—	4	2	no	—
230	C₂²×C₅₃	—	4	2	no	—
231	C₂²×C₅₃	—	4	2	no	—
232	C₂²×C₅₃	—	4	2	no	—
233	C₂²×C₅₃	—	8	1	no	—
234	C₂²×C₅₃	—	8	1	no	—
235	C₂²×C₅₃	—	8	1	no	—
236	C₂²×C₅₃	—	8	1	no	—
237	C₂²×C₅₃	C₂×#121	1	1	yes	characteristic
238	C₂²×C₅₃	#546	1	1	yes	characteristic
239	C₂²×C₅₃	D₄×Q₈	1	1	yes	characteristic
240	C₂²×C₅₃	C₂×#135	1	1	yes	characteristic
241	C₂²×C₅₃	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
242	C₂²×C₅₃	#576	1	1	yes	characteristic
243	C₂²×C₅₃	#577	1	1	yes	characteristic
244	C₄×C₅₃	—	2	2	no	—
245	C₄×C₅₃	—	4	2	no	—
246	C₄×C₅₃	—	4	2	no	—
247	C₄×C₅₃	—	4	2	no	—
248	C₄×C₅₃	—	8	1	no	—
249	C₄×C₅₃	—	8	2	no	—
250	C₄×C₅₃	—	8	2	no	—
251	C₄×C₅₃	#463	1	4	yes	—
252	C₄×C₅₃	C₂×C₂²⋊Q₈	1	1	yes	characteristic
253	C₄×C₅₃	#537	1	1	yes	characteristic
254	C₄×C₅₃	#552	1	1	yes	characteristic
255	C₄×C₅₃	#554	1	1	yes	characteristic
256	C₂×#476	C₂×C₅₃	1	1	yes	characteristic, complemented
257	C₂×C₄×D₈	C₂×C₅₃	1	1	yes	characteristic
258	#2563	C₂×C₅₃	1	1	yes	characteristic, complemented
259	D₄×C₄⋊C₄	C₂×C₅₃	1	1	yes	characteristic, complemented
260	#2085	C₂×C₅₃	1	2	yes	complemented
261	#2065	C₂×C₅₃	1	2	yes	complemented
262	#1990	C₂×C₅₃	1	1	yes	characteristic, complemented
263	#1939	C₂×C₅₃	1	1	yes	characteristic, complemented
264	#1879	C₂×C₅₃	1	2	yes	—
265	#1804	C₂×C₅₃	1	2	yes	complemented
266	#1802	C₂×C₅₃	1	1	yes	characteristic, complemented
267	C₂×C₅₃	—	4	4	no	—
268	C₂×C₅₃	—	8	2	no	—
269	C₂×C₅₃	#2417	1	1	yes	characteristic
270	C₂×C₅₃	C₂×#463	1	1	yes	characteristic
271	C₂×C₅₃	#3045	1	1	yes	characteristic
272	C₂×C₅₃	#3303	1	1	yes	characteristic
273	C₂×C₅₃	#3325	1	1	yes	characteristic
274	C₂×C₅₃	#3380	1	1	yes	characteristic
275	C₂×C₅₃	#3384	1	1	yes	characteristic
276	C₂²×D₈	C₂²×C₅₃	1	1	yes	characteristic
277	C₂×C₄⋊Q₈	C₂²×C₅₃	1	1	yes	characteristic
278	C₂×C₄×D₄	C₂²×C₅₃	1	1	yes	characteristic
279	C₂×C₄×D₄	C₂²×C₅₃	1	2	yes	—
280	C₂×C₄×D₄	C₂²×C₅₃	1	1	yes	characteristic
281	C₂²×C₄⋊C₄	—	2	1	no	—
282	#476	—	2	4	no	—
283	C₄×D₈	—	2	4	no	—
284	C₂×#107	C₂²×C₅₃	1	1	yes	characteristic
285	C₂×#106	C₂²×C₅₃	1	2	yes	—
286	C₂×C₄⋊C₈	C₂²×C₅₃	1	1	yes	characteristic
287	C₂×C₄⋊C₈	C₂²×C₅₃	1	2	yes	—
288	C₂×#102	C₂²×C₅₃	1	1	yes	characteristic
289	C₂×#102	C₂²×C₅₃	1	2	yes	—
290	C₂×#102	C₂²×C₅₃	1	1	yes	characteristic
291	C₂×#102	C₂²×C₅₃	1	2	yes	—
292	C₂×C₄×C₈	C₂²×C₅₃	1	1	yes	characteristic
293	#393	—	2	2	no	—
294	#390	C₂²×C₅₃	1	2	yes	—
295	#389	C₂²×C₅₃	1	2	yes	—
296	#385	—	2	2	no	—
297	#384	C₂²×C₅₃	1	1	yes	characteristic
298	#382	C₂²×C₅₃	1	2	yes	—
299	#380	—	2	1	no	—
300	#380	—	2	1	no	—
301	#380	—	2	1	no	—
302	C₄×C₄⋊C₄	C₂²×C₅₃	1	1	yes	characteristic
303	#340	C₂²×C₅₃	1	2	yes	—
304	#340	C₂²×C₅₃	1	2	yes	—
305	#340	C₂²×C₅₃	1	2	yes	—
306	#336	C₂²×C₅₃	1	2	yes	—
307	C₅₃	#24812	1	1	yes	characteristic, direct product factor, Sylow
308	C₂²×Q₈	—	2	1	no	—
309	C₂²×D₄	C₂³×C₅₃	1	1	yes	characteristic
310	C₂²×D₄	C₂³×C₅₃	1	1	yes	characteristic
311	C₂³×C₄	—	2	1	no	—
312	C₂³×C₄	—	2	1	no	—
313	C₂³×C₄	—	2	2	no	—
314	C₂×D₈	—	2	2	no	—
315	C₂×D₈	—	2	2	no	—
316	C₂×D₈	Q₈×C₅₃	1	4	yes	complemented
317	C₂²×C₈	C₂³×C₅₃	1	2	yes	—
318	C₂²×C₈	C₂³×C₅₃	1	1	yes	characteristic
319	C₂²×C₈	C₂³×C₅₃	1	1	yes	characteristic
320	C₄⋊Q₈	—	2	2	no	—
321	C₄⋊Q₈	—	2	2	no	—
322	C₄×D₄	—	2	4	no	—
323	C₄×D₄	—	2	2	no	—
324	C₄×D₄	—	2	2	no	—
325	C₄×D₄	—	2	2	no	—
326	C₄×D₄	—	4	2	no	—
327	C₄×D₄	D₄×C₅₃	1	4	yes	—
328	C₂×C₄⋊C₄	—	2	1	no	—
329	C₂×C₄⋊C₄	—	2	2	no	—
330	C₂×C₄⋊C₄	—	2	1	no	—
331	C₂×C₄⋊C₄	—	2	1	no	—
332	C₂×C₄⋊C₄	—	4	4	no	—
333	C₂×C₄⋊C₄	D₄×C₅₃	1	1	yes	characteristic
334	C₂×C₄⋊C₄	D₄×C₅₃	1	1	yes	characteristic
335	C₂×C₄⋊C₄	C₂³×C₅₃	1	2	yes	—
336	C₂×C₄⋊C₄	C₂³×C₅₃	1	1	yes	characteristic
337	C₂×C₄⋊C₄	C₂³×C₅₃	1	2	yes	—
338	C₂×C₄⋊C₄	C₂³×C₅₃	1	1	yes	characteristic
339	C₂×C₂²⋊C₄	—	2	1	no	—
340	C₂×C₂²⋊C₄	—	2	2	no	—
341	C₂×C₂²⋊C₄	—	2	1	no	—
342	C₂×C₂²⋊C₄	—	2	2	no	—
343	C₂×C₂²⋊C₄	—	2	2	no	—
344	C₂×C₄²	C₂³×C₅₃	1	2	yes	—
345	C₂×C₄²	C₂³×C₅₃	1	1	yes	characteristic
346	#107	—	2	2	no	—
347	#106	—	2	4	no	—
348	C₄⋊C₈	—	2	2	no	—
349	C₄⋊C₈	—	4	2	no	—
350	#102	—	2	2	no	—
351	#102	—	2	4	no	—
352	#102	—	2	2	no	—
353	#102	—	4	2	no	—
354	C₄×C₈	—	2	2	no	—
355	#95	—	2	2	no	—
356	#95	—	2	2	no	—
357	#95	—	2	2	no	—
358	#95	D₄×C₅₃	1	2	yes	—
359	C₂⁴	—	2	1	no	—
360	C₂⁴	—	2	1	no	—
361	C₂×Q₈	—	4	2	no	—
362	C₂×Q₈	—	8	1	no	—
363	C₂×Q₈	—	8	1	no	—
364	C₂×D₄	—	2	1	no	—
365	C₂×D₄	—	2	1	no	—
366	C₂×D₄	—	2	1	no	—
367	C₂×D₄	—	2	1	no	—
368	C₂×D₄	—	2	1	no	—
369	C₂×D₄	—	2	1	no	—
370	C₂×D₄	—	2	1	no	—
371	C₂×D₄	—	2	1	no	—
372	C₂×D₄	C₂×D₄×C₅₃	1	2	yes	—
373	C₂×D₄	C₂×Q₈×C₅₃	1	2	yes	—
374	C₂×D₄	C₂×Q₈×C₅₃	1	2	yes	—
375	C₂×D₄	#41×C₅₃	1	2	yes	—
376	C₂²×C₄	—	2	1	no	—
377	C₂²×C₄	—	2	4	no	—
378	C₂²×C₄	—	2	1	no	—
379	C₂²×C₄	—	2	2	no	—
380	C₂²×C₄	—	2	2	no	—
381	C₂²×C₄	—	4	4	no	—
382	C₂²×C₄	—	4	4	no	—
383	C₂²×C₄	—	4	2	no	—
384	C₂²×C₄	—	4	2	no	—
385	C₂²×C₄	—	4	2	no	—
386	C₂²×C₄	—	4	2	no	—
387	C₂²×C₄	—	4	2	no	—
388	C₂²×C₄	—	8	2	no	—
389	C₂²×C₄	C₂×D₄×C₅₃	1	2	yes	—
390	C₂²×C₄	C₂×D₄×C₅₃	1	2	yes	—
391	C₂²×C₄	C₂×D₄×C₅₃	1	1	yes	characteristic
392	C₂²×C₄	C₂×D₄×C₅₃	1	1	yes	characteristic
393	C₂²×C₄	C₂⁴×C₅₃	1	1	yes	characteristic
394	D₈	—	2	8	no	—
395	C₂×C₈	—	2	1	no	—
396	C₂×C₈	—	2	1	no	—
397	C₂×C₈	—	2	1	no	—
398	C₂×C₈	—	2	2	no	—
399	C₂×C₈	—	2	2	no	—
400	C₂×C₈	—	2	1	no	—
401	C₂×C₈	—	2	2	no	—
402	C₂×C₈	—	2	1	no	—
403	C₂×C₈	C₂×Q₈×C₅₃	1	2	yes	—
404	C₄⋊C₄	—	2	2	no	—
405	C₄⋊C₄	—	2	2	no	—
406	C₄⋊C₄	—	2	1	no	—
407	C₄⋊C₄	—	2	2	no	—
408	C₄⋊C₄	—	2	1	no	—
409	C₄⋊C₄	—	2	1	no	—
410	C₄⋊C₄	—	2	2	no	—
411	C₄⋊C₄	—	2	2	no	—
412	C₄⋊C₄	—	2	2	no	—
413	C₄⋊C₄	—	4	4	no	—
414	C₄⋊C₄	—	4	2	no	—
415	C₄⋊C₄	—	4	2	no	—
416	C₄⋊C₄	—	8	1	no	—
417	C₄⋊C₄	C₂×D₄×C₅₃	1	2	yes	—
418	C₂²⋊C₄	—	2	4	no	—
419	C₂²⋊C₄	—	4	4	no	—
420	C₂²⋊C₄	—	4	2	no	—
421	C₂²⋊C₄	—	8	2	no	—
422	C₂²⋊C₄	—	8	2	no	—
423	C₄²	—	2	2	no	—
424	C₄²	—	2	2	no	—
425	C₄²	C₂×D₄×C₅₃	1	2	yes	—
426	C₄²	#41×C₅₃	1	2	yes	—
427	C₂³	—	2	2	no	—
428	C₂³	—	2	2	no	—
429	C₂³	—	2	2	no	—
430	C₂³	—	2	2	no	—
431	C₂³	—	4	1	no	—
432	C₂³	—	4	1	no	—
433	C₂³	—	4	1	no	—
434	C₂³	—	4	1	no	—
435	C₂³	—	4	1	no	—
436	C₂³	—	4	1	no	—
437	C₂³	—	4	1	no	—
438	C₂³	—	4	1	no	—
439	C₂³	—	4	1	no	—
440	C₂³	—	4	1	no	—
441	C₂³	C₂²×D₄×C₅₃	1	1	yes	characteristic
442	Q₈	—	8	4	no	—
443	D₄	—	2	2	no	—
444	D₄	—	2	2	no	—
445	D₄	—	2	2	no	—
446	D₄	—	2	2	no	—
447	D₄	—	2	2	no	—
448	D₄	—	4	1	no	—
449	D₄	—	4	1	no	—
450	D₄	C₂²⋊Q₈×C₅₃	1	4	yes	—
451	C₂×C₄	—	2	1	no	—
452	C₂×C₄	—	2	1	no	—
453	C₂×C₄	—	2	2	no	—
454	C₂×C₄	—	2	2	no	—
455	C₂×C₄	—	2	2	no	—
456	C₂×C₄	—	2	2	no	—
457	C₂×C₄	—	2	2	no	—
458	C₂×C₄	—	2	2	no	—
459	C₂×C₄	—	2	2	no	—
460	C₂×C₄	—	4	1	no	—
461	C₂×C₄	—	4	1	no	—
462	C₂×C₄	—	4	8	no	—
463	C₂×C₄	—	4	1	no	—
464	C₂×C₄	—	4	2	no	—
465	C₂×C₄	—	4	2	no	—
466	C₂×C₄	—	4	1	no	—
467	C₂×C₄	—	4	2	no	—
468	C₂×C₄	—	4	1	no	—
469	C₂×C₄	—	4	2	no	—
470	C₂×C₄	—	4	2	no	—
471	C₂×C₄	—	4	2	no	—
472	C₂×C₄	—	8	8	no	—
473	C₂×C₄	—	8	4	no	—
474	C₂×C₄	#121×C₅₃	1	2	yes	—
475	C₂×C₄	#121×C₅₃	1	2	yes	—
476	C₂×C₄	#135×C₅₃	1	2	yes	—
477	C₂×C₄	AGL(1,ℤ/8)×C₅₃	1	2	yes	—
478	C₂×C₄	C₂²×D₄×C₅₃	1	1	yes	characteristic
479	C₂×C₄	C₂²×Q₈×C₅₃	1	1	yes	characteristic
480	C₂×C₄	C₂×#41×C₅₃	1	1	yes	characteristic
481	C₂×C₄	C₂×#41×C₅₃	1	1	yes	characteristic
482	C₂×C₄	H(5,𝔽₂)×C₅₃	1	1	yes	characteristic
483	C₂×C₄	2₋¹⁺⁴×C₅₃	1	1	yes	characteristic
484	C₈	—	2	2	no	—
485	C₈	—	4	1	no	—
486	C₈	—	4	2	no	—
487	C₂²	—	2	4	no	—
488	C₂²	—	4	2	no	—
489	C₂²	—	4	2	no	—
490	C₂²	—	4	2	no	—
491	C₂²	—	4	2	no	—
492	C₂²	—	4	2	no	—
493	C₂²	—	4	2	no	—
494	C₂²	—	4	2	no	—
495	C₂²	—	4	2	no	—
496	C₂²	—	4	2	no	—
497	C₂²	—	8	1	no	—
498	C₂²	—	8	1	no	—
499	C₂²	—	8	1	no	—
500	C₂²	—	8	1	no	—
501	C₂²	C₂×#121×C₅₃	1	1	yes	characteristic
502	C₂²	C₅₃×#546	1	1	yes	characteristic
503	C₂²	D₄×Q₈×C₅₃	1	1	yes	characteristic
504	C₂²	C₂×#135×C₅₃	1	1	yes	characteristic
505	C₂²	C₂×AGL(1,ℤ/8)×C₅₃	1	1	yes	characteristic
506	C₂²	C₅₃×#576	1	1	yes	characteristic
507	C₂²	C₅₃×#577	1	1	yes	characteristic
508	C₄	—	2	2	no	—
509	C₄	—	4	2	no	—
510	C₄	—	4	2	no	—
511	C₄	—	4	2	no	—
512	C₄	—	8	1	no	—
513	C₄	—	8	2	no	—
514	C₄	—	8	2	no	—
515	C₄	C₅₃×#463	1	4	yes	—
516	C₄	C₂×C₂²⋊Q₈×C₅₃	1	1	yes	characteristic
517	C₄	C₅₃×#537	1	1	yes	characteristic
518	C₄	C₅₃×#552	1	1	yes	characteristic
519	C₄	C₅₃×#554	1	1	yes	characteristic
520	C₂	—	4	4	no	—
521	C₂	—	8	2	no	—
522	C₂	C₅₃×#2417	1	1	yes	characteristic
523	C₂	C₂×C₅₃×#463	1	1	yes	characteristic
524	C₂	C₅₃×#3045	1	1	yes	characteristic
525	C₂	C₅₃×#3303	1	1	yes	characteristic
526	C₂	C₅₃×#3325	1	1	yes	characteristic
527	C₂	C₅₃×#3380	1	1	yes	characteristic
528	C₂	C₅₃×#3384	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂³×C₅₃	C₂²×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁴×C₅₃
prime radical	C₂²×D₈×C₅₃	C₂²
LCS	G ▷ C₂²×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
1. C₂×C₅₃×#476	C₂	semidirect
3. C₅₃×#2563	C₂	semidirect
4. D₄×C₄⋊C₄×C₅₃	C₂	semidirect
5. C₅₃×#2085	C₂	semidirect
6. C₅₃×#2065	C₂	semidirect
7. C₅₃×#1990	C₂	semidirect
8. C₅₃×#1939	C₂	semidirect
10. C₅₃×#1804	C₂	semidirect
11. C₅₃×#1802	C₂	semidirect
51. C₂×D₈×C₅₃	Q₈	semidirect
222. #24812	C₅₃	direct (N × H)
256. C₂×#476	C₂×C₅₃	semidirect
258. #2563	C₂×C₅₃	semidirect
259. D₄×C₄⋊C₄	C₂×C₅₃	semidirect
260. #2085	C₂×C₅₃	semidirect
261. #2065	C₂×C₅₃	semidirect
262. #1990	C₂×C₅₃	semidirect
263. #1939	C₂×C₅₃	semidirect
265. #1804	C₂×C₅₃	semidirect
266. #1802	C₂×C₅₃	semidirect
307. C₅₃	#24812	direct (N × H)
316. C₂×D₈	Q₈×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×#476	2
C₂×C₄×D₈×C₅₃	2
C₅₃×#2563	2
D₄×C₄⋊C₄×C₅₃	2
C₅₃×#2085	2
C₅₃×#2065	2
C₅₃×#1990	2
C₅₃×#1939	2
C₅₃×#1879	2
C₅₃×#1804	2
C₅₃×#1802	2
#24812	53

Subgroups — C53×#24812 #27232324

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅₃×#24812 #27232324