Subgroups — C₅₃×#21068 #27228580

Order 13568.

1126 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	C₂²×#106×C₅₃	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×C₅₃×#417	C₂	1	1	yes	maximal, characteristic
3	C₂×C₅₃×#416	C₂	1	1	yes	maximal, characteristic
4	C₅₃×#2311	C₂	1	1	yes	maximal, characteristic
5	C₅₃×#2274	C₂	1	1	yes	maximal, characteristic, complemented
6	C₅₃×#1927	C₂	1	4	yes	maximal, complemented
7	C₅₃×#1926	C₂	1	4	yes	maximal, complemented
8	C₅₃×#1736	C₂	1	1	yes	maximal, characteristic, complemented
9	C₅₃×#1729	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂²×#41×C₅₃	C₂²	1	1	yes	characteristic
11	C₂²×M₁₆×C₅₃	C₂²	1	1	yes	characteristic
12	C₂³×C₈×C₅₃	C₂²	1	1	yes	characteristic
13	C₂×#117×C₅₃	C₂²	1	1	yes	characteristic
14	C₂×#117×C₅₃	C₂²	1	1	yes	characteristic
15	C₂²×C₄⋊C₄×C₅₃	C₂²	1	1	yes	characteristic
16	C₂²×C₄⋊C₄×C₅₃	C₂²	1	1	yes	characteristic
17	C₂×#106×C₅₃	—	2	4	no	—
18	C₂×#106×C₅₃	C₂²	1	4	yes	—
19	C₅₃×#417	—	2	4	no	—
20	C₅₃×#416	C₄	1	8	yes	complemented
21	C₂×#103×C₅₃	C₂²	1	2	yes	—
22	C₂×#103×C₅₃	C₂²	1	2	yes	—
23	C₂×#102×C₅₃	C₂²	1	2	yes	—
24	C₂×#102×C₅₃	C₂²	1	2	yes	—
25	C₅₃×#391	C₂²	1	2	yes	—
26	C₅₃×#390	C₂²	1	2	yes	—
27	C₅₃×#386	—	2	2	no	—
28	C₅₃×#380	—	2	2	no	—
29	C₄×C₄⋊C₄×C₅₃	—	2	2	no	—
30	C₄×C₄⋊C₄×C₅₃	C₂²	1	4	yes	—
31	C₄×C₂²⋊C₄×C₅₃	—	2	2	no	—
32	C₅₃×#340	C₂²	1	4	yes	—
33	C₅₃×#336	C₂²	1	4	yes	—
34	C₂×#41×C₅₃	—	2	2	no	—
35	C₂×#41×C₅₃	—	2	2	no	—
36	C₂×#41×C₅₃	—	2	2	no	—
37	C₂×#41×C₅₃	—	2	2	no	—
38	C₂×#41×C₅₃	—	2	2	no	—
39	C₂×#41×C₅₃	C₂×C₄	1	4	yes	—
40	C₂²×Q₈×C₅₃	C₂³	1	1	yes	characteristic
41	C₂²×D₄×C₅₃	—	2	1	no	—
42	C₂²×D₄×C₅₃	C₂³	1	1	yes	characteristic
43	C₂³×C₄×C₅₃	—	2	1	no	—
44	C₂³×C₄×C₅₃	—	2	1	no	—
45	C₂³×C₄×C₅₃	—	2	1	no	—
46	C₂³×C₄×C₅₃	C₂³	1	1	yes	characteristic
47	C₂×M₁₆×C₅₃	—	2	2	no	—
48	C₂×M₁₆×C₅₃	—	2	2	no	—
49	C₂×M₁₆×C₅₃	—	2	2	no	—
50	C₂²×C₈×C₅₃	—	2	2	no	—
51	C₂²×C₈×C₅₃	—	2	2	no	—
52	C₂²×C₈×C₅₃	C₂×C₄	1	4	yes	—
53	C₂²×C₈×C₅₃	C₂³	1	2	yes	—
54	C₂²×C₈×C₅₃	C₂³	1	2	yes	—
55	#117×C₅₃	—	2	2	no	—
56	#117×C₅₃	—	2	2	no	—
57	#117×C₅₃	—	2	2	no	—
58	#117×C₅₃	C₂×C₄	1	4	yes	—
59	C₂×C₄⋊C₄×C₅₃	—	2	1	no	—
60	C₂×C₄⋊C₄×C₅₃	—	2	1	no	—
61	C₂×C₄⋊C₄×C₅₃	—	2	2	no	—
62	C₂×C₄⋊C₄×C₅₃	—	2	2	no	—
63	C₂×C₄⋊C₄×C₅₃	—	2	2	no	—
64	C₂×C₄⋊C₄×C₅₃	D₄	1	4	yes	—
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	2	yes	—
69	C₂×C₂²⋊C₄×C₅₃	—	2	1	no	—
70	C₂×C₂²⋊C₄×C₅₃	—	2	4	no	—
71	C₂×C₂²⋊C₄×C₅₃	D₄	1	2	yes	—
72	C₂×C₄²×C₅₃	—	2	1	no	—
73	C₂×C₄²×C₅₃	—	2	4	no	—
74	C₂×C₄²×C₅₃	C₂³	1	2	yes	—
75	#106×C₅₃	—	2	8	no	—
76	#103×C₅₃	—	2	4	no	—
77	#103×C₅₃	C₂×C₄	1	8	yes	complemented
78	#102×C₅₃	—	2	4	no	—
79	#102×C₅₃	C₂×C₄	1	8	yes	complemented
80	#95×C₅₃	—	2	4	no	—
81	#95×C₅₃	—	2	4	no	—
82	C₂⁴×C₅₃	—	2	1	no	—
83	C₂⁴×C₅₃	C₂×D₄	1	1	yes	characteristic
84	#41×C₅₃	—	2	8	no	—
85	#41×C₅₃	—	2	8	no	—
86	#41×C₅₃	—	4	4	no	—
87	#41×C₅₃	—	4	4	no	—
88	C₂×Q₈×C₅₃	—	2	1	no	—
89	C₂×Q₈×C₅₃	—	2	1	no	—
90	C₂×Q₈×C₅₃	—	2	1	no	—
91	C₂×Q₈×C₅₃	—	2	1	no	—
92	C₂×Q₈×C₅₃	—	2	1	no	—
93	C₂×Q₈×C₅₃	C₂²×C₄	1	2	yes	—
94	C₂×D₄×C₅₃	—	2	4	no	—
95	C₂×D₄×C₅₃	—	2	1	no	—
96	C₂×D₄×C₅₃	—	2	1	no	—
97	C₂×D₄×C₅₃	—	2	1	no	—
98	C₂×D₄×C₅₃	—	2	1	no	—
99	C₂×D₄×C₅₃	—	2	1	no	—
100	C₂×D₄×C₅₃	—	4	2	no	—
101	C₂×D₄×C₅₃	—	4	2	no	—
102	C₂×D₄×C₅₃	C₂²×C₄	1	2	yes	—
103	C₂²×C₄×C₅₃	—	2	1	no	—
104	C₂²×C₄×C₅₃	—	2	2	no	—
105	C₂²×C₄×C₅₃	—	2	2	no	—
106	C₂²×C₄×C₅₃	—	2	4	no	—
107	C₂²×C₄×C₅₃	—	2	4	no	—
108	C₂²×C₄×C₅₃	—	2	2	no	—
109	C₂²×C₄×C₅₃	—	4	2	no	—
110	C₂²×C₄×C₅₃	—	4	2	no	—
111	C₂²×C₄×C₅₃	—	4	2	no	—
112	C₂²×C₄×C₅₃	—	8	2	no	—
113	C₂²×C₄×C₅₃	—	8	2	no	—
114	C₂²×C₄×C₅₃	C₂²×C₄	1	2	yes	—
115	C₂²×C₄×C₅₃	C₂×D₄	1	4	yes	—
116	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
117	C₂²×C₄×C₅₃	C₂×D₄	1	2	yes	—
118	C₂²×C₄×C₅₃	#41	1	2	yes	—
119	C₂²×C₄×C₅₃	#41	1	2	yes	—
120	C₂²×C₄×C₅₃	#41	1	2	yes	—
121	C₂²×C₄×C₅₃	#41	1	2	yes	—
122	C₂²×C₄×C₅₃	C₂⁴	1	1	yes	characteristic
123	M₁₆×C₅₃	—	4	4	no	—
124	C₂×C₈×C₅₃	—	2	8	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	—	2	2	no	—
129	C₂×C₈×C₅₃	—	2	2	no	—
130	C₂×C₈×C₅₃	C₂²×C₄	1	4	yes	—
131	C₄⋊C₄×C₅₃	—	2	4	no	—
132	C₄⋊C₄×C₅₃	—	2	4	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	C₂²×C₄	1	4	yes	—
139	C₄⋊C₄×C₅₃	C₂×D₄	1	4	yes	—
140	C₂²⋊C₄×C₅₃	—	2	4	no	—
141	C₂²⋊C₄×C₅₃	—	2	4	no	—
142	C₂²⋊C₄×C₅₃	—	8	4	no	—
143	C₄²×C₅₃	—	2	4	no	—
144	C₄²×C₅₃	—	2	2	no	—
145	C₄²×C₅₃	—	2	2	no	—
146	C₄²×C₅₃	—	8	4	no	—
147	C₂³×C₅₃	—	2	2	no	—
148	C₂³×C₅₃	—	2	2	no	—
149	C₂³×C₅₃	—	2	2	no	—
150	C₂³×C₅₃	—	4	1	no	—
151	C₂³×C₅₃	—	4	1	no	—
152	C₂³×C₅₃	—	4	2	no	—
153	C₂³×C₅₃	—	4	2	no	—
154	C₂³×C₅₃	C₄×D₄	1	2	yes	—
155	C₂³×C₅₃	#121	1	2	yes	—
156	C₂³×C₅₃	#123	1	2	yes	—
157	C₂³×C₅₃	C₂²×D₄	1	1	yes	characteristic
158	Q₈×C₅₃	—	2	2	no	—
159	Q₈×C₅₃	—	2	2	no	—
160	Q₈×C₅₃	—	4	1	no	—
161	Q₈×C₅₃	—	4	1	no	—
162	D₄×C₅₃	—	2	2	no	—
163	D₄×C₅₃	—	2	2	no	—
164	D₄×C₅₃	—	4	8	no	—
165	D₄×C₅₃	—	4	1	no	—
166	D₄×C₅₃	—	4	1	no	—
167	C₂×C₄×C₅₃	—	2	2	no	—
168	C₂×C₄×C₅₃	—	2	4	no	—
169	C₂×C₄×C₅₃	—	2	4	no	—
170	C₂×C₄×C₅₃	—	2	4	no	—
171	C₂×C₄×C₅₃	—	2	4	no	—
172	C₂×C₄×C₅₃	—	2	4	no	—
173	C₂×C₄×C₅₃	—	4	1	no	—
174	C₂×C₄×C₅₃	—	4	1	no	—
175	C₂×C₄×C₅₃	—	4	2	no	—
176	C₂×C₄×C₅₃	—	4	4	no	—
177	C₂×C₄×C₅₃	—	4	2	no	—
178	C₂×C₄×C₅₃	—	4	8	no	—
179	C₂×C₄×C₅₃	—	4	2	no	—
180	C₂×C₄×C₅₃	—	4	4	no	—
181	C₂×C₄×C₅₃	—	8	4	no	—
182	C₂×C₄×C₅₃	—	8	4	no	—
183	C₂×C₄×C₅₃	#117	1	4	yes	—
184	C₂×C₄×C₅₃	C₄×D₄	1	4	yes	—
185	C₂×C₄×C₅₃	C₄×D₄	1	2	yes	—
186	C₂×C₄×C₅₃	#121	1	2	yes	—
187	C₂×C₄×C₅₃	#123	1	2	yes	—
188	C₂×C₄×C₅₃	#124	1	4	yes	—
189	C₂×C₄×C₅₃	#126	1	4	yes	—
190	C₂×C₄×C₅₃	C₂³×C₄	1	1	yes	characteristic
191	C₂×C₄×C₅₃	C₂²×D₄	1	1	yes	characteristic
192	C₂×C₄×C₅₃	C₂×#41	1	1	yes	characteristic
193	C₂×C₄×C₅₃	C₂×#41	1	1	yes	characteristic
194	C₂×C₄×C₅₃	C₂×#41	1	1	yes	characteristic
195	C₂×C₄×C₅₃	C₂×#41	1	1	yes	characteristic
196	C₈×C₅₃	—	2	4	no	—
197	C₈×C₅₃	—	4	2	no	—
198	#21068	C₅₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
199	C₂²×C₅₃	—	2	4	no	—
200	C₂²×C₅₃	—	2	4	no	—
201	C₂²×C₅₃	—	2	2	no	—
202	C₂²×C₅₃	—	2	2	no	—
203	C₂²×C₅₃	—	4	4	no	—
204	C₂²×C₅₃	—	4	2	no	—
205	C₂²×C₅₃	—	8	2	no	—
206	C₂²×C₅₃	—	8	1	no	—
207	C₂²×C₅₃	—	8	1	no	—
208	C₂²×C₅₃	#388	1	4	yes	—
209	C₂²×C₅₃	C₂×C₄×D₄	1	1	yes	characteristic
210	C₂²×C₅₃	C₂×#121	1	1	yes	characteristic
211	C₂²×C₅₃	C₂×#123	1	1	yes	characteristic
212	C₂²×C₅₃	#577	1	2	yes	—
213	C₂²×C₅₃	#577	1	2	yes	—
214	C₄×C₅₃	—	4	4	no	—
215	C₄×C₅₃	—	4	4	no	—
216	C₄×C₅₃	—	4	4	no	—
217	C₄×C₅₃	—	4	2	no	—
218	C₄×C₅₃	—	8	2	no	—
219	C₄×C₅₃	#388	1	4	yes	—
220	C₄×C₅₃	C₂×#117	1	1	yes	characteristic
221	C₄×C₅₃	C₂×C₄×D₄	1	1	yes	characteristic
222	C₄×C₅₃	C₂×#124	1	1	yes	characteristic
223	C₄×C₅₃	C₂×#126	1	1	yes	characteristic
224	C₂²×#106	C₂×C₅₃	1	1	yes	characteristic, complemented
225	C₂×#417	C₂×C₅₃	1	1	yes	characteristic
226	C₂×#416	C₂×C₅₃	1	1	yes	characteristic
227	#2311	C₂×C₅₃	1	1	yes	characteristic
228	#2274	C₂×C₅₃	1	1	yes	characteristic, complemented
229	#1927	C₂×C₅₃	1	4	yes	complemented
230	#1926	C₂×C₅₃	1	4	yes	complemented
231	#1736	C₂×C₅₃	1	1	yes	characteristic, complemented
232	#1729	C₂×C₅₃	1	1	yes	characteristic, complemented
233	C₂×C₅₃	—	2	4	no	—
234	C₂×C₅₃	—	8	2	no	—
235	C₂×C₅₃	C₂×#388	1	1	yes	characteristic
236	C₂×C₅₃	#2952	1	2	yes	—
237	C₂×C₅₃	#3042	1	1	yes	characteristic
238	C₂×C₅₃	#3043	1	1	yes	characteristic
239	C₂×C₅₃	#3099	1	2	yes	—
240	C₂²×#41	C₂²×C₅₃	1	1	yes	characteristic
241	C₂²×M₁₆	C₂²×C₅₃	1	1	yes	characteristic
242	C₂³×C₈	C₂²×C₅₃	1	1	yes	characteristic
243	C₂×#117	C₂²×C₅₃	1	1	yes	characteristic
244	C₂×#117	C₂²×C₅₃	1	1	yes	characteristic
245	C₂²×C₄⋊C₄	C₂²×C₅₃	1	1	yes	characteristic
246	C₂²×C₄⋊C₄	C₂²×C₅₃	1	1	yes	characteristic
247	C₂×#106	—	2	4	no	—
248	C₂×#106	C₂²×C₅₃	1	4	yes	—
249	#417	—	2	4	no	—
250	#416	C₄×C₅₃	1	8	yes	complemented
251	C₂×#103	C₂²×C₅₃	1	2	yes	—
252	C₂×#103	C₂²×C₅₃	1	2	yes	—
253	C₂×#102	C₂²×C₅₃	1	2	yes	—
254	C₂×#102	C₂²×C₅₃	1	2	yes	—
255	#391	C₂²×C₅₃	1	2	yes	—
256	#390	C₂²×C₅₃	1	2	yes	—
257	#386	—	2	2	no	—
258	#380	—	2	2	no	—
259	C₄×C₄⋊C₄	—	2	2	no	—
260	C₄×C₄⋊C₄	C₂²×C₅₃	1	4	yes	—
261	C₄×C₂²⋊C₄	—	2	2	no	—
262	#340	C₂²×C₅₃	1	4	yes	—
263	#336	C₂²×C₅₃	1	4	yes	—
264	C₅₃	#21068	1	1	yes	characteristic, direct product factor, Sylow
265	C₂×#41	—	2	2	no	—
266	C₂×#41	—	2	2	no	—
267	C₂×#41	—	2	2	no	—
268	C₂×#41	—	2	2	no	—
269	C₂×#41	—	2	2	no	—
270	C₂×#41	C₂×C₄×C₅₃	1	4	yes	—
271	C₂²×Q₈	C₂³×C₅₃	1	1	yes	characteristic
272	C₂²×D₄	—	2	1	no	—
273	C₂²×D₄	C₂³×C₅₃	1	1	yes	characteristic
274	C₂³×C₄	—	2	1	no	—
275	C₂³×C₄	—	2	1	no	—
276	C₂³×C₄	—	2	1	no	—
277	C₂³×C₄	C₂³×C₅₃	1	1	yes	characteristic
278	C₂×M₁₆	—	2	2	no	—
279	C₂×M₁₆	—	2	2	no	—
280	C₂×M₁₆	—	2	2	no	—
281	C₂²×C₈	—	2	2	no	—
282	C₂²×C₈	—	2	2	no	—
283	C₂²×C₈	C₂×C₄×C₅₃	1	4	yes	—
284	C₂²×C₈	C₂³×C₅₃	1	2	yes	—
285	C₂²×C₈	C₂³×C₅₃	1	2	yes	—
286	#117	—	2	2	no	—
287	#117	—	2	2	no	—
288	#117	—	2	2	no	—
289	#117	C₂×C₄×C₅₃	1	4	yes	—
290	C₂×C₄⋊C₄	—	2	1	no	—
291	C₂×C₄⋊C₄	—	2	2	no	—
292	C₂×C₄⋊C₄	—	2	2	no	—
293	C₂×C₄⋊C₄	—	2	2	no	—
294	C₂×C₄⋊C₄	—	2	1	no	—
295	C₂×C₄⋊C₄	D₄×C₅₃	1	2	yes	—
296	C₂×C₄⋊C₄	D₄×C₅₃	1	4	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	2	yes	—
300	C₂×C₂²⋊C₄	—	2	1	no	—
301	C₂×C₂²⋊C₄	—	2	4	no	—
302	C₂×C₂²⋊C₄	D₄×C₅₃	1	2	yes	—
303	C₂×C₄²	—	2	1	no	—
304	C₂×C₄²	—	2	4	no	—
305	C₂×C₄²	C₂³×C₅₃	1	2	yes	—
306	#106	—	2	8	no	—
307	#103	—	2	4	no	—
308	#103	C₂×C₄×C₅₃	1	8	yes	complemented
309	#102	—	2	4	no	—
310	#102	C₂×C₄×C₅₃	1	8	yes	complemented
311	#95	—	2	4	no	—
312	#95	—	2	4	no	—
313	C₂⁴	—	2	1	no	—
314	C₂⁴	C₂×D₄×C₅₃	1	1	yes	characteristic
315	#41	—	2	8	no	—
316	#41	—	2	8	no	—
317	#41	—	4	4	no	—
318	#41	—	4	4	no	—
319	C₂×Q₈	—	2	1	no	—
320	C₂×Q₈	—	2	1	no	—
321	C₂×Q₈	—	2	1	no	—
322	C₂×Q₈	—	2	1	no	—
323	C₂×Q₈	—	2	1	no	—
324	C₂×Q₈	C₂²×C₄×C₅₃	1	2	yes	—
325	C₂×D₄	—	2	4	no	—
326	C₂×D₄	—	2	1	no	—
327	C₂×D₄	—	2	1	no	—
328	C₂×D₄	—	2	1	no	—
329	C₂×D₄	—	2	1	no	—
330	C₂×D₄	—	2	1	no	—
331	C₂×D₄	—	4	2	no	—
332	C₂×D₄	—	4	2	no	—
333	C₂×D₄	C₂²×C₄×C₅₃	1	2	yes	—
334	C₂²×C₄	—	2	1	no	—
335	C₂²×C₄	—	2	2	no	—
336	C₂²×C₄	—	2	2	no	—
337	C₂²×C₄	—	2	4	no	—
338	C₂²×C₄	—	2	4	no	—
339	C₂²×C₄	—	2	2	no	—
340	C₂²×C₄	—	4	2	no	—
341	C₂²×C₄	—	4	2	no	—
342	C₂²×C₄	—	4	2	no	—
343	C₂²×C₄	—	8	2	no	—
344	C₂²×C₄	—	8	2	no	—
345	C₂²×C₄	C₂²×C₄×C₅₃	1	2	yes	—
346	C₂²×C₄	C₂×D₄×C₅₃	1	4	yes	—
347	C₂²×C₄	C₂×D₄×C₅₃	1	1	yes	characteristic
348	C₂²×C₄	C₂×D₄×C₅₃	1	2	yes	—
349	C₂²×C₄	#41×C₅₃	1	2	yes	—
350	C₂²×C₄	#41×C₅₃	1	2	yes	—
351	C₂²×C₄	#41×C₅₃	1	2	yes	—
352	C₂²×C₄	#41×C₅₃	1	2	yes	—
353	C₂²×C₄	C₂⁴×C₅₃	1	1	yes	characteristic
354	M₁₆	—	4	4	no	—
355	C₂×C₈	—	2	8	no	—
356	C₂×C₈	—	2	2	no	—
357	C₂×C₈	—	2	2	no	—
358	C₂×C₈	—	2	2	no	—
359	C₂×C₈	—	2	2	no	—
360	C₂×C₈	—	2	2	no	—
361	C₂×C₈	C₂²×C₄×C₅₃	1	4	yes	—
362	C₄⋊C₄	—	2	4	no	—
363	C₄⋊C₄	—	2	4	no	—
364	C₄⋊C₄	—	2	2	no	—
365	C₄⋊C₄	—	2	2	no	—
366	C₄⋊C₄	—	2	2	no	—
367	C₄⋊C₄	—	2	2	no	—
368	C₄⋊C₄	—	4	2	no	—
369	C₄⋊C₄	C₂²×C₄×C₅₃	1	4	yes	—
370	C₄⋊C₄	C₂×D₄×C₅₃	1	4	yes	—
371	C₂²⋊C₄	—	2	4	no	—
372	C₂²⋊C₄	—	2	4	no	—
373	C₂²⋊C₄	—	8	4	no	—
374	C₄²	—	2	4	no	—
375	C₄²	—	2	2	no	—
376	C₄²	—	2	2	no	—
377	C₄²	—	8	4	no	—
378	C₂³	—	2	2	no	—
379	C₂³	—	2	2	no	—
380	C₂³	—	2	2	no	—
381	C₂³	—	4	1	no	—
382	C₂³	—	4	1	no	—
383	C₂³	—	4	2	no	—
384	C₂³	—	4	2	no	—
385	C₂³	C₄×D₄×C₅₃	1	2	yes	—
386	C₂³	#121×C₅₃	1	2	yes	—
387	C₂³	#123×C₅₃	1	2	yes	—
388	C₂³	C₂²×D₄×C₅₃	1	1	yes	characteristic
389	Q₈	—	2	2	no	—
390	Q₈	—	2	2	no	—
391	Q₈	—	4	1	no	—
392	Q₈	—	4	1	no	—
393	D₄	—	2	2	no	—
394	D₄	—	2	2	no	—
395	D₄	—	4	8	no	—
396	D₄	—	4	1	no	—
397	D₄	—	4	1	no	—
398	C₂×C₄	—	2	4	no	—
399	C₂×C₄	—	2	2	no	—
400	C₂×C₄	—	2	4	no	—
401	C₂×C₄	—	2	4	no	—
402	C₂×C₄	—	2	4	no	—
403	C₂×C₄	—	2	4	no	—
404	C₂×C₄	—	4	1	no	—
405	C₂×C₄	—	4	1	no	—
406	C₂×C₄	—	4	2	no	—
407	C₂×C₄	—	4	4	no	—
408	C₂×C₄	—	4	2	no	—
409	C₂×C₄	—	4	8	no	—
410	C₂×C₄	—	4	2	no	—
411	C₂×C₄	—	4	4	no	—
412	C₂×C₄	—	8	4	no	—
413	C₂×C₄	—	8	4	no	—
414	C₂×C₄	#117×C₅₃	1	4	yes	—
415	C₂×C₄	C₄×D₄×C₅₃	1	4	yes	—
416	C₂×C₄	C₄×D₄×C₅₃	1	2	yes	—
417	C₂×C₄	#121×C₅₃	1	2	yes	—
418	C₂×C₄	#123×C₅₃	1	2	yes	—
419	C₂×C₄	#124×C₅₃	1	4	yes	—
420	C₂×C₄	#126×C₅₃	1	4	yes	—
421	C₂×C₄	C₂³×C₄×C₅₃	1	1	yes	characteristic
422	C₂×C₄	C₂²×D₄×C₅₃	1	1	yes	characteristic
423	C₂×C₄	C₂×#41×C₅₃	1	1	yes	characteristic
424	C₂×C₄	C₂×#41×C₅₃	1	1	yes	characteristic
425	C₂×C₄	C₂×#41×C₅₃	1	1	yes	characteristic
426	C₂×C₄	C₂×#41×C₅₃	1	1	yes	characteristic
427	C₈	—	2	4	no	—
428	C₈	—	4	2	no	—
429	C₂²	—	2	4	no	—
430	C₂²	—	2	2	no	—
431	C₂²	—	2	2	no	—
432	C₂²	—	2	4	no	—
433	C₂²	—	4	4	no	—
434	C₂²	—	4	2	no	—
435	C₂²	—	8	1	no	—
436	C₂²	—	8	1	no	—
437	C₂²	—	8	2	no	—
438	C₂²	C₅₃×#388	1	4	yes	—
439	C₂²	C₂×C₄×D₄×C₅₃	1	1	yes	characteristic
440	C₂²	C₂×#121×C₅₃	1	1	yes	characteristic
441	C₂²	C₂×#123×C₅₃	1	1	yes	characteristic
442	C₂²	C₅₃×#577	1	2	yes	—
443	C₂²	C₅₃×#577	1	2	yes	—
444	C₄	—	4	4	no	—
445	C₄	—	4	4	no	—
446	C₄	—	4	2	no	—
447	C₄	—	4	4	no	—
448	C₄	—	8	2	no	—
449	C₄	C₅₃×#388	1	4	yes	—
450	C₄	C₂×#117×C₅₃	1	1	yes	characteristic
451	C₄	C₂×C₄×D₄×C₅₃	1	1	yes	characteristic
452	C₄	C₂×#124×C₅₃	1	1	yes	characteristic
453	C₄	C₂×#126×C₅₃	1	1	yes	characteristic
454	C₂	—	2	4	no	—
455	C₂	—	8	2	no	—
456	C₂	C₂×C₅₃×#388	1	1	yes	characteristic
457	C₂	C₅₃×#2952	1	2	yes	—
458	C₂	C₅₃×#3042	1	1	yes	characteristic
459	C₂	C₅₃×#3043	1	1	yes	characteristic
460	C₂	C₅₃×#3099	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×#41×C₅₃	C₂²
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₅₃ ◁ C₂×#117×C₅₃ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂²×#106×C₅₃	C₂	semidirect
5. C₅₃×#2274	C₂	semidirect
6. C₅₃×#1927	C₂	semidirect
7. C₅₃×#1926	C₂	semidirect
8. C₅₃×#1736	C₂	semidirect
9. C₅₃×#1729	C₂	semidirect
20. C₅₃×#416	C₄	semidirect
77. #103×C₅₃	C₂×C₄	semidirect
79. #102×C₅₃	C₂×C₄	semidirect
198. #21068	C₅₃	direct (N × H)
224. C₂²×#106	C₂×C₅₃	semidirect
228. #2274	C₂×C₅₃	semidirect
229. #1927	C₂×C₅₃	semidirect
230. #1926	C₂×C₅₃	semidirect
231. #1736	C₂×C₅₃	semidirect
232. #1729	C₂×C₅₃	semidirect
250. #416	C₄×C₅₃	semidirect
264. C₅₃	#21068	direct (N × H)
308. #103	C₂×C₄×C₅₃	semidirect
310. #102	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
C₂²×#106×C₅₃	2
C₂×C₅₃×#417	2
C₂×C₅₃×#416	2
C₅₃×#2311	2
C₅₃×#2274	2
C₅₃×#1927	2
C₅₃×#1926	2
C₅₃×#1736	2
C₅₃×#1729	2
#21068	53

Subgroups — C53×#21068 #27228580

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅₃×#21068 #27228580