Subgroups — C₅₃×#22226 #27229738

Order 13568.

926 conjugacy classes of subgroups.

The subgroup lattice has 492 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₅₃×#490	C₂	1	1	yes	maximal, characteristic
2	C₄×#127×C₅₃	C₂	1	1	yes	maximal, characteristic
3	C₄×#125×C₅₃	C₂	1	1	yes	maximal, characteristic, complemented
4	C₅₃×#1992	C₂	1	1	yes	maximal, characteristic, complemented
5	C₅₃×#1961	C₂	1	1	yes	maximal, characteristic
6	C₅₃×#1926	C₂	1	2	yes	maximal, complemented
7	C₅₃×#1925	C₂	1	2	yes	maximal, complemented
8	C₅₃×#1864	C₂	1	2	yes	maximal, complemented
9	C₅₃×#1862	C₂	1	2	yes	maximal, complemented
10	C₅₃×#1847	C₂	1	1	yes	maximal, characteristic, complemented
11	C₅₃×#1745	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×#127×C₅₃	C₂²	1	1	yes	characteristic
13	C₂×#125×C₅₃	C₂²	1	1	yes	characteristic
14	C₂×C₄×D₄×C₅₃	—	2	1	no	—
15	C₂×C₄×D₄×C₅₃	—	2	1	no	—
16	C₂×C₄×D₄×C₅₃	C₂²	1	2	yes	—
17	C₅₃×#490	C₄	1	8	yes	complemented
18	C₂×#107×C₅₃	C₂²	1	2	yes	—
19	C₂×#106×C₅₃	C₂²	1	2	yes	—
20	C₂×#102×C₅₃	C₂²	1	2	yes	—
21	C₂×#102×C₅₃	C₂²	1	2	yes	—
22	C₂×#102×C₅₃	C₂²	1	2	yes	—
23	C₂×#102×C₅₃	C₂²	1	2	yes	—
24	C₂×#97×C₅₃	C₂²	1	1	yes	characteristic
25	C₂×#97×C₅₃	C₂²	1	1	yes	characteristic
26	C₅₃×#390	—	2	1	no	—
27	C₅₃×#390	—	2	1	no	—
28	C₅₃×#390	C₂²	1	2	yes	—
29	C₅₃×#389	C₂²	1	1	yes	characteristic
30	C₅₃×#389	C₂²	1	1	yes	characteristic
31	C₅₃×#387	—	2	1	no	—
32	C₅₃×#387	—	2	1	no	—
33	C₅₃×#384	C₂²	1	1	yes	characteristic
34	C₅₃×#382	C₂²	1	1	yes	characteristic
35	C₄×C₄⋊C₄×C₅₃	—	2	1	no	—
36	C₄×C₄⋊C₄×C₅₃	—	2	1	no	—
37	C₄×C₄⋊C₄×C₅₃	C₂²	1	1	yes	characteristic
38	C₄×C₄⋊C₄×C₅₃	C₂²	1	1	yes	characteristic
39	C₄³×C₅₃	C₂²	1	1	yes	characteristic
40	C₅₃×#340	C₂²	1	2	yes	—
41	C₅₃×#340	C₂²	1	2	yes	—
42	C₅₃×#336	C₂²	1	2	yes	—
43	C₅₃×#336	C₂²	1	2	yes	—
44	C₂²×D₄×C₅₃	—	2	1	no	—
45	C₂²×D₄×C₅₃	—	2	1	no	—
46	C₂²×D₄×C₅₃	C₂³	1	2	yes	—
47	C₂³×C₄×C₅₃	—	2	2	no	—
48	C₂²×C₈×C₅₃	C₂³	1	2	yes	—
49	C₂²×C₈×C₅₃	C₂³	1	2	yes	—
50	#127×C₅₃	—	2	2	no	—
51	#127×C₅₃	C₂×C₄	1	4	yes	—
52	#125×C₅₃	—	2	2	no	—
53	#125×C₅₃	C₂×C₄	1	4	yes	complemented
54	C₄×D₄×C₅₃	—	4	2	no	—
55	C₄×D₄×C₅₃	—	4	2	no	—
56	C₄×D₄×C₅₃	—	4	4	no	—
57	C₄×D₄×C₅₃	—	4	4	no	—
58	C₂×C₄⋊C₄×C₅₃	—	2	1	no	—
59	C₂×C₄⋊C₄×C₅₃	—	2	1	no	—
60	C₂×C₄⋊C₄×C₅₃	—	2	1	no	—
61	C₂×C₄⋊C₄×C₅₃	—	2	1	no	—
62	C₂×C₄⋊C₄×C₅₃	—	2	1	no	—
63	C₂×C₄⋊C₄×C₅₃	—	2	1	no	—
64	C₂×C₄⋊C₄×C₅₃	C₂³	1	2	yes	—
65	C₂×C₄⋊C₄×C₅₃	C₂³	1	1	yes	characteristic
66	C₂×C₄⋊C₄×C₅₃	C₂³	1	1	yes	characteristic
67	C₂×C₄⋊C₄×C₅₃	C₂³	1	1	yes	characteristic
68	C₂×C₄⋊C₄×C₅₃	C₂³	1	1	yes	characteristic
69	C₂×C₂²⋊C₄×C₅₃	—	2	2	no	—
70	C₂×C₂²⋊C₄×C₅₃	—	2	2	no	—
71	C₂×C₂²⋊C₄×C₅₃	—	2	2	no	—
72	C₂×C₄²×C₅₃	—	2	1	no	—
73	C₂×C₄²×C₅₃	—	2	1	no	—
74	C₂×C₄²×C₅₃	D₄	1	1	yes	characteristic
75	C₂×C₄²×C₅₃	D₄	1	1	yes	characteristic
76	C₂×C₄²×C₅₃	D₄	1	1	yes	characteristic
77	C₂×C₄²×C₅₃	D₄	1	1	yes	characteristic
78	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
79	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
80	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
81	#107×C₅₃	—	4	2	no	—
82	#106×C₅₃	—	4	2	no	—
83	#102×C₅₃	—	4	2	no	—
84	#102×C₅₃	—	4	2	no	—
85	#102×C₅₃	C₂×C₄	1	8	yes	complemented
86	#102×C₅₃	C₂×C₄	1	8	yes	complemented
87	#97×C₅₃	—	2	2	no	—
88	#97×C₅₃	C₂×C₄	1	4	yes	complemented
89	#95×C₅₃	—	2	1	no	—
90	#95×C₅₃	—	2	1	no	—
91	#95×C₅₃	—	2	1	no	—
92	#95×C₅₃	—	2	1	no	—
93	#95×C₅₃	—	2	1	no	—
94	#95×C₅₃	—	2	1	no	—
95	C₂⁴×C₅₃	—	2	2	no	—
96	C₂×D₄×C₅₃	—	2	4	no	—
97	C₂×D₄×C₅₃	—	2	4	no	—
98	C₂×D₄×C₅₃	—	2	2	no	—
99	C₂×D₄×C₅₃	—	2	2	no	—
100	C₂×D₄×C₅₃	—	2	2	no	—
101	C₂×D₄×C₅₃	—	2	2	no	—
102	C₂×D₄×C₅₃	—	2	2	no	—
103	C₂×D₄×C₅₃	—	4	2	no	—
104	C₂×D₄×C₅₃	—	4	2	no	—
105	C₂×D₄×C₅₃	—	4	2	no	—
106	C₂×D₄×C₅₃	—	4	2	no	—
107	C₂×D₄×C₅₃	C₂²×C₄	1	4	yes	—
108	C₂²×C₄×C₅₃	—	2	1	no	—
109	C₂²×C₄×C₅₃	—	2	1	no	—
110	C₂²×C₄×C₅₃	—	2	1	no	—
111	C₂²×C₄×C₅₃	—	2	1	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	—	4	4	no	—
116	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
117	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
118	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
119	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
120	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
121	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
122	C₂²×C₄×C₅₃	C₂⁴	1	1	yes	characteristic
123	C₂×C₈×C₅₃	—	2	2	no	—
124	C₂×C₈×C₅₃	—	2	2	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	4	yes	—
129	C₄⋊C₄×C₅₃	—	2	4	no	—
130	C₄⋊C₄×C₅₃	—	2	4	no	—
131	C₄⋊C₄×C₅₃	—	2	1	no	—
132	C₄⋊C₄×C₅₃	—	2	1	no	—
133	C₄⋊C₄×C₅₃	—	2	2	no	—
134	C₄⋊C₄×C₅₃	—	2	2	no	—
135	C₄⋊C₄×C₅₃	—	2	2	no	—
136	C₄⋊C₄×C₅₃	—	2	2	no	—
137	C₄⋊C₄×C₅₃	—	4	2	no	—
138	C₄⋊C₄×C₅₃	—	4	2	no	—
139	C₄⋊C₄×C₅₃	—	4	2	no	—
140	C₄⋊C₄×C₅₃	—	4	2	no	—
141	C₄⋊C₄×C₅₃	C₂²×C₄	1	2	yes	—
142	C₄⋊C₄×C₅₃	C₂²×C₄	1	2	yes	—
143	C₂²⋊C₄×C₅₃	—	4	4	no	—
144	C₂²⋊C₄×C₅₃	—	4	4	no	—
145	C₂²⋊C₄×C₅₃	—	4	4	no	—
146	C₄²×C₅₃	—	2	2	no	—
147	C₄²×C₅₃	—	2	2	no	—
148	C₄²×C₅₃	—	2	2	no	—
149	C₄²×C₅₃	—	2	2	no	—
150	C₄²×C₅₃	—	4	1	no	—
151	C₄²×C₅₃	—	4	1	no	—
152	C₄²×C₅₃	—	4	2	no	—
153	C₄²×C₅₃	—	4	2	no	—
154	C₄²×C₅₃	C₂²×C₄	1	2	yes	—
155	C₄²×C₅₃	#41	1	2	yes	—
156	C₂³×C₅₃	—	2	4	no	—
157	C₂³×C₅₃	—	4	2	no	—
158	C₂³×C₅₃	—	4	2	no	—
159	C₂³×C₅₃	—	4	2	no	—
160	C₂³×C₅₃	—	4	2	no	—
161	C₂³×C₅₃	—	4	2	no	—
162	C₂³×C₅₃	—	4	2	no	—
163	C₂³×C₅₃	C₂²×D₄	1	1	yes	characteristic
164	D₄×C₅₃	—	2	4	no	—
165	D₄×C₅₃	—	2	4	no	—
166	D₄×C₅₃	—	4	8	no	—
167	D₄×C₅₃	—	4	8	no	—
168	D₄×C₅₃	—	4	2	no	—
169	D₄×C₅₃	—	4	2	no	—
170	C₂×C₄×C₅₃	—	2	1	no	—
171	C₂×C₄×C₅₃	—	2	1	no	—
172	C₂×C₄×C₅₃	—	2	1	no	—
173	C₂×C₄×C₅₃	—	2	1	no	—
174	C₂×C₄×C₅₃	—	2	1	no	—
175	C₂×C₄×C₅₃	—	2	1	no	—
176	C₂×C₄×C₅₃	—	2	1	no	—
177	C₂×C₄×C₅₃	—	2	1	no	—
178	C₂×C₄×C₅₃	—	2	2	no	—
179	C₂×C₄×C₅₃	—	2	2	no	—
180	C₂×C₄×C₅₃	—	2	1	no	—
181	C₂×C₄×C₅₃	—	2	1	no	—
182	C₂×C₄×C₅₃	—	2	1	no	—
183	C₂×C₄×C₅₃	—	2	1	no	—
184	C₂×C₄×C₅₃	—	2	1	no	—
185	C₂×C₄×C₅₃	—	2	1	no	—
186	C₂×C₄×C₅₃	—	4	1	no	—
187	C₂×C₄×C₅₃	—	4	1	no	—
188	C₂×C₄×C₅₃	—	4	1	no	—
189	C₂×C₄×C₅₃	—	4	1	no	—
190	C₂×C₄×C₅₃	—	4	2	no	—
191	C₂×C₄×C₅₃	—	4	1	no	—
192	C₂×C₄×C₅₃	—	4	1	no	—
193	C₂×C₄×C₅₃	—	4	2	no	—
194	C₂×C₄×C₅₃	—	4	2	no	—
195	C₂×C₄×C₅₃	—	4	1	no	—
196	C₂×C₄×C₅₃	—	4	1	no	—
197	C₂×C₄×C₅₃	—	4	1	no	—
198	C₂×C₄×C₅₃	—	4	1	no	—
199	C₂×C₄×C₅₃	—	8	8	no	—
200	C₂×C₄×C₅₃	C₄×D₄	1	2	yes	—
201	C₂×C₄×C₅₃	C₄×D₄	1	2	yes	—
202	C₂×C₄×C₅₃	AGL(1,ℤ/8)	1	2	yes	—
203	C₂×C₄×C₅₃	AGL(1,ℤ/8)	1	2	yes	—
204	C₂×C₄×C₅₃	C₂³×C₄	1	1	yes	characteristic
205	C₂×C₄×C₅₃	C₂×#41	1	1	yes	characteristic
206	C₂×C₄×C₅₃	H(5,𝔽₂)	1	2	yes	—
207	C₂×C₄×C₅₃	2₋¹⁺⁴	1	2	yes	—
208	C₈×C₅₃	—	2	4	no	—
209	C₈×C₅₃	—	4	2	no	—
210	#22226	C₅₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
211	C₂²×C₅₃	—	4	4	no	—
212	C₂²×C₅₃	—	4	4	no	—
213	C₂²×C₅₃	—	4	4	no	—
214	C₂²×C₅₃	—	8	2	no	—
215	C₂²×C₅₃	—	8	2	no	—
216	C₂²×C₅₃	—	8	2	no	—
217	C₂²×C₅₃	—	8	2	no	—
218	C₂²×C₅₃	C₂×C₄×D₄	1	1	yes	characteristic
219	C₂²×C₅₃	#535	1	1	yes	characteristic
220	C₂²×C₅₃	#536	1	1	yes	characteristic
221	C₂²×C₅₃	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
222	C₂²×C₅₃	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
223	C₂²×C₅₃	#575	1	1	yes	characteristic
224	C₂²×C₅₃	#575	1	1	yes	characteristic
225	C₄×C₅₃	—	2	2	no	—
226	C₄×C₅₃	—	2	2	no	—
227	C₄×C₅₃	—	2	2	no	—
228	C₄×C₅₃	—	2	2	no	—
229	C₄×C₅₃	—	4	2	no	—
230	C₄×C₅₃	—	4	2	no	—
231	C₄×C₅₃	—	4	1	no	—
232	C₄×C₅₃	—	4	1	no	—
233	C₄×C₅₃	—	8	1	no	—
234	C₄×C₅₃	—	8	2	no	—
235	C₄×C₅₃	—	8	1	no	—
236	C₄×C₅₃	#520	1	2	yes	—
237	C₄×C₅₃	#542	1	2	yes	—
238	C₂×#490	C₂×C₅₃	1	1	yes	characteristic
239	C₄×#127	C₂×C₅₃	1	1	yes	characteristic
240	C₄×#125	C₂×C₅₃	1	1	yes	characteristic, complemented
241	#1992	C₂×C₅₃	1	1	yes	characteristic, complemented
242	#1961	C₂×C₅₃	1	1	yes	characteristic
243	#1926	C₂×C₅₃	1	2	yes	complemented
244	#1925	C₂×C₅₃	1	2	yes	complemented
245	#1864	C₂×C₅₃	1	2	yes	complemented
246	#1862	C₂×C₅₃	1	2	yes	complemented
247	#1847	C₂×C₅₃	1	1	yes	characteristic, complemented
248	#1745	C₂×C₅₃	1	1	yes	characteristic, complemented
249	C₂×C₅₃	—	8	4	no	—
250	C₂×C₅₃	#2326	1	1	yes	characteristic
251	C₂×C₅₃	C₄×AGL(1,ℤ/8)	1	1	yes	characteristic
252	C₂×C₅₃	C₄×AGL(1,ℤ/8)	1	1	yes	characteristic
253	C₂×C₅₃	#3217	1	1	yes	characteristic
254	C₂×C₅₃	#3219	1	1	yes	characteristic
255	C₂×C₅₃	#3234	1	1	yes	characteristic
256	C₂×C₅₃	#3235	1	1	yes	characteristic
257	C₂×#127	C₂²×C₅₃	1	1	yes	characteristic
258	C₂×#125	C₂²×C₅₃	1	1	yes	characteristic
259	C₂×C₄×D₄	—	2	1	no	—
260	C₂×C₄×D₄	—	2	1	no	—
261	C₂×C₄×D₄	C₂²×C₅₃	1	2	yes	—
262	#490	C₄×C₅₃	1	8	yes	complemented
263	C₂×#107	C₂²×C₅₃	1	2	yes	—
264	C₂×#106	C₂²×C₅₃	1	2	yes	—
265	C₂×#102	C₂²×C₅₃	1	2	yes	—
266	C₂×#102	C₂²×C₅₃	1	2	yes	—
267	C₂×#102	C₂²×C₅₃	1	2	yes	—
268	C₂×#102	C₂²×C₅₃	1	2	yes	—
269	C₂×#97	C₂²×C₅₃	1	1	yes	characteristic
270	C₂×#97	C₂²×C₅₃	1	1	yes	characteristic
271	#390	—	2	1	no	—
272	#390	—	2	1	no	—
273	#390	C₂²×C₅₃	1	2	yes	—
274	#389	C₂²×C₅₃	1	1	yes	characteristic
275	#389	C₂²×C₅₃	1	1	yes	characteristic
276	#387	—	2	1	no	—
277	#387	—	2	1	no	—
278	#384	C₂²×C₅₃	1	1	yes	characteristic
279	#382	C₂²×C₅₃	1	1	yes	characteristic
280	C₄×C₄⋊C₄	—	2	1	no	—
281	C₄×C₄⋊C₄	—	2	1	no	—
282	C₄×C₄⋊C₄	C₂²×C₅₃	1	1	yes	characteristic
283	C₄×C₄⋊C₄	C₂²×C₅₃	1	1	yes	characteristic
284	C₄³	C₂²×C₅₃	1	1	yes	characteristic
285	#340	C₂²×C₅₃	1	2	yes	—
286	#340	C₂²×C₅₃	1	2	yes	—
287	#336	C₂²×C₅₃	1	2	yes	—
288	#336	C₂²×C₅₃	1	2	yes	—
289	C₅₃	#22226	1	1	yes	characteristic, direct product factor, Sylow
290	C₂²×D₄	—	2	1	no	—
291	C₂²×D₄	—	2	1	no	—
292	C₂²×D₄	C₂³×C₅₃	1	2	yes	—
293	C₂³×C₄	—	2	2	no	—
294	C₂²×C₈	C₂³×C₅₃	1	2	yes	—
295	C₂²×C₈	C₂³×C₅₃	1	2	yes	—
296	#127	—	2	2	no	—
297	#127	C₂×C₄×C₅₃	1	4	yes	—
298	#125	—	2	2	no	—
299	#125	C₂×C₄×C₅₃	1	4	yes	complemented
300	C₄×D₄	—	4	2	no	—
301	C₄×D₄	—	4	2	no	—
302	C₄×D₄	—	4	4	no	—
303	C₄×D₄	—	4	4	no	—
304	C₂×C₄⋊C₄	—	2	1	no	—
305	C₂×C₄⋊C₄	—	2	1	no	—
306	C₂×C₄⋊C₄	—	2	1	no	—
307	C₂×C₄⋊C₄	—	2	1	no	—
308	C₂×C₄⋊C₄	—	2	1	no	—
309	C₂×C₄⋊C₄	—	2	1	no	—
310	C₂×C₄⋊C₄	C₂³×C₅₃	1	2	yes	—
311	C₂×C₄⋊C₄	C₂³×C₅₃	1	1	yes	characteristic
312	C₂×C₄⋊C₄	C₂³×C₅₃	1	1	yes	characteristic
313	C₂×C₄⋊C₄	C₂³×C₅₃	1	1	yes	characteristic
314	C₂×C₄⋊C₄	C₂³×C₅₃	1	1	yes	characteristic
315	C₂×C₂²⋊C₄	—	2	2	no	—
316	C₂×C₂²⋊C₄	—	2	2	no	—
317	C₂×C₂²⋊C₄	—	2	2	no	—
318	C₂×C₄²	—	2	1	no	—
319	C₂×C₄²	—	2	1	no	—
320	C₂×C₄²	D₄×C₅₃	1	1	yes	characteristic
321	C₂×C₄²	D₄×C₅₃	1	1	yes	characteristic
322	C₂×C₄²	D₄×C₅₃	1	1	yes	characteristic
323	C₂×C₄²	D₄×C₅₃	1	1	yes	characteristic
324	C₂×C₄²	C₂³×C₅₃	1	1	yes	characteristic
325	C₂×C₄²	C₂³×C₅₃	1	1	yes	characteristic
326	C₂×C₄²	C₂³×C₅₃	1	1	yes	characteristic
327	#107	—	4	2	no	—
328	#106	—	4	2	no	—
329	#102	—	4	2	no	—
330	#102	—	4	2	no	—
331	#102	C₂×C₄×C₅₃	1	8	yes	complemented
332	#102	C₂×C₄×C₅₃	1	8	yes	complemented
333	#97	—	2	2	no	—
334	#97	C₂×C₄×C₅₃	1	4	yes	complemented
335	#95	—	2	1	no	—
336	#95	—	2	1	no	—
337	#95	—	2	1	no	—
338	#95	—	2	1	no	—
339	#95	—	2	1	no	—
340	#95	—	2	1	no	—
341	C₂⁴	—	2	2	no	—
342	C₂×D₄	—	2	4	no	—
343	C₂×D₄	—	2	4	no	—
344	C₂×D₄	—	2	2	no	—
345	C₂×D₄	—	2	2	no	—
346	C₂×D₄	—	2	2	no	—
347	C₂×D₄	—	2	2	no	—
348	C₂×D₄	—	2	2	no	—
349	C₂×D₄	—	4	2	no	—
350	C₂×D₄	—	4	2	no	—
351	C₂×D₄	—	4	2	no	—
352	C₂×D₄	—	4	2	no	—
353	C₂×D₄	C₂²×C₄×C₅₃	1	4	yes	—
354	C₂²×C₄	—	2	1	no	—
355	C₂²×C₄	—	2	1	no	—
356	C₂²×C₄	—	2	1	no	—
357	C₂²×C₄	—	2	1	no	—
358	C₂²×C₄	—	2	2	no	—
359	C₂²×C₄	—	4	4	no	—
360	C₂²×C₄	—	4	4	no	—
361	C₂²×C₄	—	4	4	no	—
362	C₂²×C₄	C₂×D₄×C₅₃	1	1	yes	characteristic
363	C₂²×C₄	C₂×D₄×C₅₃	1	1	yes	characteristic
364	C₂²×C₄	C₂×D₄×C₅₃	1	1	yes	characteristic
365	C₂²×C₄	C₂×D₄×C₅₃	1	1	yes	characteristic
366	C₂²×C₄	C₂×D₄×C₅₃	1	1	yes	characteristic
367	C₂²×C₄	C₂×D₄×C₅₃	1	1	yes	characteristic
368	C₂²×C₄	C₂⁴×C₅₃	1	1	yes	characteristic
369	C₂×C₈	—	2	2	no	—
370	C₂×C₈	—	2	2	no	—
371	C₂×C₈	—	2	2	no	—
372	C₂×C₈	—	2	2	no	—
373	C₂×C₈	—	2	2	no	—
374	C₂×C₈	C₂²×C₄×C₅₃	1	4	yes	—
375	C₄⋊C₄	—	2	4	no	—
376	C₄⋊C₄	—	2	4	no	—
377	C₄⋊C₄	—	2	1	no	—
378	C₄⋊C₄	—	2	1	no	—
379	C₄⋊C₄	—	2	2	no	—
380	C₄⋊C₄	—	2	2	no	—
381	C₄⋊C₄	—	2	2	no	—
382	C₄⋊C₄	—	2	2	no	—
383	C₄⋊C₄	—	4	2	no	—
384	C₄⋊C₄	—	4	2	no	—
385	C₄⋊C₄	—	4	2	no	—
386	C₄⋊C₄	—	4	2	no	—
387	C₄⋊C₄	C₂²×C₄×C₅₃	1	2	yes	—
388	C₄⋊C₄	C₂²×C₄×C₅₃	1	2	yes	—
389	C₂²⋊C₄	—	4	4	no	—
390	C₂²⋊C₄	—	4	4	no	—
391	C₂²⋊C₄	—	4	4	no	—
392	C₄²	—	2	2	no	—
393	C₄²	—	2	2	no	—
394	C₄²	—	2	2	no	—
395	C₄²	—	2	2	no	—
396	C₄²	—	4	1	no	—
397	C₄²	—	4	1	no	—
398	C₄²	—	4	2	no	—
399	C₄²	—	4	2	no	—
400	C₄²	C₂²×C₄×C₅₃	1	2	yes	—
401	C₄²	#41×C₅₃	1	2	yes	—
402	C₂³	—	2	4	no	—
403	C₂³	—	4	2	no	—
404	C₂³	—	4	2	no	—
405	C₂³	—	4	2	no	—
406	C₂³	—	4	2	no	—
407	C₂³	—	4	2	no	—
408	C₂³	—	4	2	no	—
409	C₂³	C₂²×D₄×C₅₃	1	1	yes	characteristic
410	D₄	—	2	4	no	—
411	D₄	—	2	4	no	—
412	D₄	—	4	8	no	—
413	D₄	—	4	8	no	—
414	D₄	—	4	2	no	—
415	D₄	—	4	2	no	—
416	C₂×C₄	—	2	1	no	—
417	C₂×C₄	—	2	1	no	—
418	C₂×C₄	—	2	1	no	—
419	C₂×C₄	—	2	1	no	—
420	C₂×C₄	—	2	1	no	—
421	C₂×C₄	—	2	1	no	—
422	C₂×C₄	—	2	1	no	—
423	C₂×C₄	—	2	1	no	—
424	C₂×C₄	—	2	2	no	—
425	C₂×C₄	—	2	2	no	—
426	C₂×C₄	—	2	1	no	—
427	C₂×C₄	—	2	1	no	—
428	C₂×C₄	—	2	1	no	—
429	C₂×C₄	—	2	1	no	—
430	C₂×C₄	—	2	1	no	—
431	C₂×C₄	—	2	1	no	—
432	C₂×C₄	—	4	1	no	—
433	C₂×C₄	—	4	1	no	—
434	C₂×C₄	—	4	1	no	—
435	C₂×C₄	—	4	1	no	—
436	C₂×C₄	—	4	2	no	—
437	C₂×C₄	—	4	1	no	—
438	C₂×C₄	—	4	1	no	—
439	C₂×C₄	—	4	1	no	—
440	C₂×C₄	—	4	2	no	—
441	C₂×C₄	—	4	2	no	—
442	C₂×C₄	—	4	1	no	—
443	C₂×C₄	—	4	1	no	—
444	C₂×C₄	—	4	1	no	—
445	C₂×C₄	—	8	8	no	—
446	C₂×C₄	C₄×D₄×C₅₃	1	2	yes	—
447	C₂×C₄	C₄×D₄×C₅₃	1	2	yes	—
448	C₂×C₄	AGL(1,ℤ/8)×C₅₃	1	2	yes	—
449	C₂×C₄	AGL(1,ℤ/8)×C₅₃	1	2	yes	—
450	C₂×C₄	C₂³×C₄×C₅₃	1	1	yes	characteristic
451	C₂×C₄	C₂×#41×C₅₃	1	1	yes	characteristic
452	C₂×C₄	H(5,𝔽₂)×C₅₃	1	2	yes	—
453	C₂×C₄	2₋¹⁺⁴×C₅₃	1	2	yes	—
454	C₈	—	2	4	no	—
455	C₈	—	4	2	no	—
456	C₂²	—	4	4	no	—
457	C₂²	—	4	4	no	—
458	C₂²	—	4	4	no	—
459	C₂²	—	8	2	no	—
460	C₂²	—	8	2	no	—
461	C₂²	—	8	2	no	—
462	C₂²	—	8	2	no	—
463	C₂²	C₂×C₄×D₄×C₅₃	1	1	yes	characteristic
464	C₂²	C₅₃×#535	1	1	yes	characteristic
465	C₂²	C₅₃×#536	1	1	yes	characteristic
466	C₂²	C₂×AGL(1,ℤ/8)×C₅₃	1	1	yes	characteristic
467	C₂²	C₂×AGL(1,ℤ/8)×C₅₃	1	1	yes	characteristic
468	C₂²	C₅₃×#575	1	1	yes	characteristic
469	C₂²	C₅₃×#575	1	1	yes	characteristic
470	C₄	—	2	2	no	—
471	C₄	—	2	2	no	—
472	C₄	—	2	2	no	—
473	C₄	—	2	2	no	—
474	C₄	—	4	2	no	—
475	C₄	—	4	2	no	—
476	C₄	—	4	1	no	—
477	C₄	—	4	1	no	—
478	C₄	—	8	2	no	—
479	C₄	—	8	1	no	—
480	C₄	—	8	1	no	—
481	C₄	C₅₃×#520	1	2	yes	—
482	C₄	C₅₃×#542	1	2	yes	—
483	C₂	—	8	4	no	—
484	C₂	C₅₃×#2326	1	1	yes	characteristic
485	C₂	C₄×AGL(1,ℤ/8)×C₅₃	1	1	yes	characteristic
486	C₂	C₄×AGL(1,ℤ/8)×C₅₃	1	1	yes	characteristic
487	C₂	C₅₃×#3217	1	1	yes	characteristic
488	C₂	C₅₃×#3219	1	1	yes	characteristic
489	C₂	C₅₃×#3234	1	1	yes	characteristic
490	C₂	C₅₃×#3235	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂³×C₅₃	C₂²×D₄
commutator	C₂×C₄	C₂³×C₄×C₅₃
Frattini, cosocle	C₂²×C₄	C₂⁴×C₅₃
prime radical	C₂×#127×C₅₃	C₂²
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₅₃ ◁ C₄³×C₅₃ ◁ G

Complements

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

N	H ≅ G/N	type
3. C₄×#125×C₅₃	C₂	semidirect
4. C₅₃×#1992	C₂	semidirect
6. C₅₃×#1926	C₂	semidirect
7. C₅₃×#1925	C₂	semidirect
8. C₅₃×#1864	C₂	semidirect
9. C₅₃×#1862	C₂	semidirect
10. C₅₃×#1847	C₂	semidirect
11. C₅₃×#1745	C₂	semidirect
17. C₅₃×#490	C₄	semidirect
53. #125×C₅₃	C₂×C₄	semidirect
85. #102×C₅₃	C₂×C₄	semidirect
86. #102×C₅₃	C₂×C₄	semidirect
88. #97×C₅₃	C₂×C₄	semidirect
210. #22226	C₅₃	direct (N × H)
240. C₄×#125	C₂×C₅₃	semidirect
241. #1992	C₂×C₅₃	semidirect
243. #1926	C₂×C₅₃	semidirect
244. #1925	C₂×C₅₃	semidirect
245. #1864	C₂×C₅₃	semidirect
246. #1862	C₂×C₅₃	semidirect
247. #1847	C₂×C₅₃	semidirect
248. #1745	C₂×C₅₃	semidirect
262. #490	C₄×C₅₃	semidirect
289. C₅₃	#22226	direct (N × H)
299. #125	C₂×C₄×C₅₃	semidirect
331. #102	C₂×C₄×C₅₃	semidirect
332. #102	C₂×C₄×C₅₃	semidirect
334. #97	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×#490	2
C₄×#127×C₅₃	2
C₄×#125×C₅₃	2
C₅₃×#1992	2
C₅₃×#1961	2
C₅₃×#1926	2
C₅₃×#1925	2
C₅₃×#1864	2
C₅₃×#1862	2
C₅₃×#1847	2
C₅₃×#1745	2
#22226	53

Subgroups — C53×#22226 #27229738

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅₃×#22226 #27229738