Subgroups — C₅₃×#21129 #27228641

Order 13568.

938 conjugacy classes of subgroups.

The subgroup lattice has 450 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄×D₈×C₅₃	C₂	1	1	yes	maximal, characteristic
2	C₅₃×#2319	C₂	1	1	yes	maximal, characteristic, complemented
3	C₅₃×#2275	C₂	1	1	yes	maximal, characteristic, complemented
4	C₅₃×#1925	C₂	1	2	yes	maximal, complemented
5	C₅₃×#1921	C₂	1	2	yes	maximal, complemented
6	C₅₃×#1879	C₂	1	2	yes	maximal
7	C₅₃×#1862	C₂	1	2	yes	maximal, complemented
8	C₄×#107×C₅₃	C₂	1	1	yes	maximal, characteristic, complemented
9	C₅₃×#1762	C₂	1	1	yes	maximal, characteristic, complemented
10	C₅₃×#1762	C₂	1	1	yes	maximal, characteristic, complemented
11	C₅₃×#1746	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂²×D₈×C₅₃	C₂²	1	1	yes	characteristic
13	C₂×C₄×D₄×C₅₃	C₂²	1	2	yes	—
14	C₂×C₄×D₄×C₅₃	C₂²	1	1	yes	characteristic
15	C₂×C₄×D₄×C₅₃	C₂²	1	1	yes	characteristic
16	C₄×D₈×C₅₃	C₄	1	8	yes	complemented
17	C₂×#107×C₅₃	C₂²	1	2	yes	—
18	C₂×#107×C₅₃	C₂²	1	1	yes	characteristic
19	C₂×#102×C₅₃	C₂²	1	2	yes	—
20	C₂×#102×C₅₃	C₂²	1	2	yes	—
21	C₂×#102×C₅₃	C₂²	1	1	yes	characteristic
22	C₂×#102×C₅₃	C₂²	1	1	yes	characteristic
23	C₂×#97×C₅₃	C₂²	1	1	yes	characteristic
24	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
25	C₅₃×#390	C₂²	1	2	yes	—
26	C₅₃×#389	C₂²	1	2	yes	—
27	C₅₃×#388	—	2	2	no	—
28	C₅₃×#381	—	2	1	no	—
29	C₅₃×#380	—	2	1	no	—
30	C₄×C₄⋊C₄×C₅₃	C₂²	1	2	yes	—
31	C₄×C₄⋊C₄×C₅₃	C₂²	1	1	yes	characteristic
32	C₄×C₄⋊C₄×C₅₃	C₂²	1	1	yes	characteristic
33	C₄×C₂²⋊C₄×C₅₃	—	2	1	no	—
34	C₄×C₂²⋊C₄×C₅₃	—	2	2	no	—
35	C₅₃×#376	C₂²	1	1	yes	characteristic
36	C₂×#95×C₅₃	—	2	1	no	—
37	C₅₃×#340	C₂²	1	2	yes	—
38	C₅₃×#340	C₂²	1	2	yes	—
39	C₅₃×#340	C₂²	1	2	yes	—
40	C₅₃×#336	C₂²	1	2	yes	—
41	C₅₃×#336	C₂²	1	2	yes	—
42	C₂²×D₄×C₅₃	C₂³	1	1	yes	characteristic
43	C₂²×D₄×C₅₃	C₂³	1	1	yes	characteristic
44	C₂³×C₄×C₅₃	—	2	2	no	—
45	C₂³×C₄×C₅₃	—	2	1	no	—
46	C₂³×C₄×C₅₃	—	2	1	no	—
47	C₂×D₈×C₅₃	—	2	4	no	—
48	C₂×D₈×C₅₃	C₂×C₄	1	4	yes	—
49	C₂²×C₈×C₅₃	C₂³	1	2	yes	—
50	C₂²×C₈×C₅₃	C₂³	1	1	yes	characteristic
51	C₂²×C₈×C₅₃	C₂³	1	1	yes	characteristic
52	C₄×D₄×C₅₃	—	2	4	no	—
53	C₄×D₄×C₅₃	—	2	2	no	—
54	C₄×D₄×C₅₃	—	2	2	no	—
55	C₄×D₄×C₅₃	—	2	4	no	—
56	C₄×D₄×C₅₃	C₂×C₄	1	4	yes	—
57	C₄×D₄×C₅₃	C₂×C₄	1	4	yes	complemented
58	C₂×C₄⋊C₄×C₅₃	—	2	2	no	—
59	C₂×C₄⋊C₄×C₅₃	C₂³	1	2	yes	—
60	C₂×C₄⋊C₄×C₅₃	C₂³	1	1	yes	characteristic
61	C₂×C₄⋊C₄×C₅₃	C₂³	1	1	yes	characteristic
62	C₂×C₄⋊C₄×C₅₃	C₂³	1	2	yes	—
63	C₂×C₂²⋊C₄×C₅₃	—	2	2	no	—
64	C₂×C₂²⋊C₄×C₅₃	—	2	1	no	—
65	C₂×C₂²⋊C₄×C₅₃	—	2	1	no	—
66	C₂×C₂²⋊C₄×C₅₃	—	2	2	no	—
67	C₂×C₂²⋊C₄×C₅₃	—	2	2	no	—
68	C₂×C₄²×C₅₃	—	2	2	no	—
69	C₂×C₄²×C₅₃	—	2	1	no	—
70	C₂×C₄²×C₅₃	—	2	1	no	—
71	C₂×C₄²×C₅₃	C₂³	1	2	yes	—
72	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
73	#107×C₅₃	—	2	4	no	—
74	#107×C₅₃	C₂×C₄	1	4	yes	complemented
75	#102×C₅₃	—	2	4	no	—
76	#102×C₅₃	—	2	4	no	—
77	#102×C₅₃	C₂×C₄	1	4	yes	—
78	#102×C₅₃	C₂×C₄	1	4	yes	complemented
79	#97×C₅₃	—	2	2	no	—
80	C₄×C₈×C₅₃	C₂×C₄	1	4	yes	complemented
81	#95×C₅₃	—	2	2	no	—
82	#95×C₅₃	—	2	1	no	—
83	#95×C₅₃	—	2	1	no	—
84	#95×C₅₃	—	2	2	no	—
85	#95×C₅₃	D₄	1	1	yes	characteristic
86	#95×C₅₃	D₄	1	1	yes	characteristic
87	#95×C₅₃	D₄	1	2	yes	—
88	C₂⁴×C₅₃	—	2	1	no	—
89	C₂⁴×C₅₃	—	2	1	no	—
90	C₂×D₄×C₅₃	—	2	2	no	—
91	C₂×D₄×C₅₃	—	2	1	no	—
92	C₂×D₄×C₅₃	—	2	2	no	—
93	C₂×D₄×C₅₃	—	2	2	no	—
94	C₂×D₄×C₅₃	—	2	1	no	—
95	C₂×D₄×C₅₃	C₂²×C₄	1	2	yes	—
96	C₂×D₄×C₅₃	C₂²×C₄	1	2	yes	—
97	C₂×D₄×C₅₃	#41	1	4	yes	—
98	C₂²×C₄×C₅₃	—	2	1	no	—
99	C₂²×C₄×C₅₃	—	2	2	no	—
100	C₂²×C₄×C₅₃	—	2	1	no	—
101	C₂²×C₄×C₅₃	—	2	8	no	—
102	C₂²×C₄×C₅₃	—	2	4	no	—
103	C₂²×C₄×C₅₃	—	2	4	no	—
104	C₂²×C₄×C₅₃	—	2	2	no	—
105	C₂²×C₄×C₅₃	—	8	2	no	—
106	C₂²×C₄×C₅₃	—	8	2	no	—
107	C₂²×C₄×C₅₃	—	8	2	no	—
108	C₂²×C₄×C₅₃	—	8	2	no	—
109	C₂²×C₄×C₅₃	C₂×D₄	1	2	yes	—
110	C₂²×C₄×C₅₃	C₂×D₄	1	2	yes	—
111	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
112	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
113	C₂²×C₄×C₅₃	C₂⁴	1	1	yes	characteristic
114	D₈×C₅₃	—	2	8	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	C₂²×C₄	1	2	yes	—
121	C₂×C₈×C₅₃	C₂²×C₄	1	2	yes	—
122	C₄⋊C₄×C₅₃	—	2	1	no	—
123	C₄⋊C₄×C₅₃	—	2	1	no	—
124	C₄⋊C₄×C₅₃	—	2	2	no	—
125	C₄⋊C₄×C₅₃	—	2	2	no	—
126	C₄⋊C₄×C₅₃	—	2	2	no	—
127	C₄⋊C₄×C₅₃	—	8	2	no	—
128	C₄⋊C₄×C₅₃	C₂²×C₄	1	2	yes	—
129	C₄⋊C₄×C₅₃	C₂²×C₄	1	2	yes	—
130	C₄⋊C₄×C₅₃	#41	1	4	yes	—
131	C₂²⋊C₄×C₅₃	—	2	8	no	—
132	C₂²⋊C₄×C₅₃	—	2	4	no	—
133	C₂²⋊C₄×C₅₃	—	2	4	no	—
134	C₂²⋊C₄×C₅₃	—	8	2	no	—
135	C₂²⋊C₄×C₅₃	—	8	2	no	—
136	C₄²×C₅₃	—	2	2	no	—
137	C₄²×C₅₃	—	2	2	no	—
138	C₄²×C₅₃	—	8	1	no	—
139	C₄²×C₅₃	—	8	1	no	—
140	C₄²×C₅₃	—	8	2	no	—
141	C₄²×C₅₃	C₂²×C₄	1	2	yes	—
142	C₄²×C₅₃	#41	1	2	yes	—
143	C₂³×C₅₃	—	2	2	no	—
144	C₂³×C₅₃	—	2	4	no	—
145	C₂³×C₅₃	—	2	2	no	—
146	C₂³×C₅₃	—	4	1	no	—
147	C₂³×C₅₃	—	4	2	no	—
148	C₂³×C₅₃	—	4	1	no	—
149	C₂³×C₅₃	—	4	2	no	—
150	C₂³×C₅₃	—	4	2	no	—
151	C₂³×C₅₃	—	4	1	no	—
152	C₂³×C₅₃	—	4	1	no	—
153	C₂³×C₅₃	C₂²×D₄	1	1	yes	characteristic
154	D₄×C₅₃	—	2	2	no	—
155	D₄×C₅₃	—	2	2	no	—
156	D₄×C₅₃	—	2	2	no	—
157	D₄×C₅₃	—	2	4	no	—
158	D₄×C₅₃	—	4	2	no	—
159	D₄×C₅₃	#117	1	4	yes	—
160	C₂×C₄×C₅₃	—	2	2	no	—
161	C₂×C₄×C₅₃	—	2	1	no	—
162	C₂×C₄×C₅₃	—	2	1	no	—
163	C₂×C₄×C₅₃	—	2	2	no	—
164	C₂×C₄×C₅₃	—	2	2	no	—
165	C₂×C₄×C₅₃	—	2	2	no	—
166	C₂×C₄×C₅₃	—	2	4	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	—	4	2	no	—
173	C₂×C₄×C₅₃	—	4	2	no	—
174	C₂×C₄×C₅₃	—	4	2	no	—
175	C₂×C₄×C₅₃	—	4	2	no	—
176	C₂×C₄×C₅₃	—	4	2	no	—
177	C₂×C₄×C₅₃	—	4	2	no	—
178	C₂×C₄×C₅₃	—	4	2	no	—
179	C₂×C₄×C₅₃	—	8	4	no	—
180	C₂×C₄×C₅₃	—	8	4	no	—
181	C₂×C₄×C₅₃	—	8	8	no	—
182	C₂×C₄×C₅₃	C₄×D₄	1	2	yes	—
183	C₂×C₄×C₅₃	C₄×D₄	1	2	yes	—
184	C₂×C₄×C₅₃	C₂³×C₄	1	1	yes	characteristic
185	C₂×C₄×C₅₃	C₂×#41	1	2	yes	—
186	C₂×C₄×C₅₃	C₂×#41	1	2	yes	—
187	C₂×C₄×C₅₃	C₂×#41	1	1	yes	characteristic
188	C₈×C₅₃	—	2	2	no	—
189	C₈×C₅₃	—	2	2	no	—
190	C₈×C₅₃	—	4	2	no	—
191	#21129	C₅₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
192	C₂²×C₅₃	—	2	4	no	—
193	C₂²×C₅₃	—	4	4	no	—
194	C₂²×C₅₃	—	4	4	no	—
195	C₂²×C₅₃	—	4	2	no	—
196	C₂²×C₅₃	—	4	2	no	—
197	C₂²×C₅₃	—	4	2	no	—
198	C₂²×C₅₃	—	4	2	no	—
199	C₂²×C₅₃	—	4	2	no	—
200	C₂²×C₅₃	—	8	2	no	—
201	C₂²×C₅₃	—	8	2	no	—
202	C₂²×C₅₃	C₂×C₄×D₄	1	1	yes	characteristic
203	C₂²×C₅₃	#525	1	2	yes	—
204	C₂²×C₅₃	#576	1	1	yes	characteristic
205	C₂²×C₅₃	#576	1	2	yes	—
206	C₂²×C₅₃	#578	1	1	yes	characteristic
207	C₄×C₅₃	—	2	2	no	—
208	C₄×C₅₃	—	2	2	no	—
209	C₄×C₅₃	—	4	2	no	—
210	C₄×C₅₃	—	4	4	no	—
211	C₄×C₅₃	—	4	2	no	—
212	C₄×C₅₃	—	4	2	no	—
213	C₄×C₅₃	—	4	2	no	—
214	C₄×C₅₃	—	8	2	no	—
215	C₄×C₅₃	C₂×#117	1	1	yes	characteristic
216	C₄×C₅₃	C₄×#41	1	1	yes	characteristic
217	C₄×C₅₃	#529	1	2	yes	—
218	C₂×C₄×D₈	C₂×C₅₃	1	1	yes	characteristic
219	#2319	C₂×C₅₃	1	1	yes	characteristic, complemented
220	#2275	C₂×C₅₃	1	1	yes	characteristic, complemented
221	#1925	C₂×C₅₃	1	2	yes	complemented
222	#1921	C₂×C₅₃	1	2	yes	complemented
223	#1879	C₂×C₅₃	1	2	yes	—
224	#1862	C₂×C₅₃	1	2	yes	complemented
225	C₄×#107	C₂×C₅₃	1	1	yes	characteristic, complemented
226	#1762	C₂×C₅₃	1	1	yes	characteristic, complemented
227	#1762	C₂×C₅₃	1	1	yes	characteristic, complemented
228	#1746	C₂×C₅₃	1	1	yes	characteristic, complemented
229	C₂×C₅₃	—	4	4	no	—
230	C₂×C₅₃	—	8	2	no	—
231	C₂×C₅₃	#2290	1	1	yes	characteristic
232	C₂×C₅₃	#2948	1	1	yes	characteristic
233	C₂×C₅₃	#2951	1	1	yes	characteristic
234	C₂×C₅₃	#3122	1	2	yes	—
235	C₂×C₅₃	#3124	1	2	yes	—
236	C₂²×D₈	C₂²×C₅₃	1	1	yes	characteristic
237	C₂×C₄×D₄	C₂²×C₅₃	1	2	yes	—
238	C₂×C₄×D₄	C₂²×C₅₃	1	1	yes	characteristic
239	C₂×C₄×D₄	C₂²×C₅₃	1	1	yes	characteristic
240	C₄×D₈	C₄×C₅₃	1	8	yes	complemented
241	C₂×#107	C₂²×C₅₃	1	2	yes	—
242	C₂×#107	C₂²×C₅₃	1	1	yes	characteristic
243	C₂×#102	C₂²×C₅₃	1	2	yes	—
244	C₂×#102	C₂²×C₅₃	1	2	yes	—
245	C₂×#102	C₂²×C₅₃	1	1	yes	characteristic
246	C₂×#102	C₂²×C₅₃	1	1	yes	characteristic
247	C₂×#97	C₂²×C₅₃	1	1	yes	characteristic
248	C₂×C₄×C₈	C₂²×C₅₃	1	1	yes	characteristic
249	#390	C₂²×C₅₃	1	2	yes	—
250	#389	C₂²×C₅₃	1	2	yes	—
251	#388	—	2	2	no	—
252	#381	—	2	1	no	—
253	#380	—	2	1	no	—
254	C₄×C₄⋊C₄	C₂²×C₅₃	1	1	yes	characteristic
255	C₄×C₄⋊C₄	C₂²×C₅₃	1	2	yes	—
256	C₄×C₄⋊C₄	C₂²×C₅₃	1	1	yes	characteristic
257	C₄×C₂²⋊C₄	—	2	1	no	—
258	C₄×C₂²⋊C₄	—	2	2	no	—
259	#376	C₂²×C₅₃	1	1	yes	characteristic
260	C₂×#95	—	2	1	no	—
261	#340	C₂²×C₅₃	1	2	yes	—
262	#340	C₂²×C₅₃	1	2	yes	—
263	#340	C₂²×C₅₃	1	2	yes	—
264	#336	C₂²×C₅₃	1	2	yes	—
265	#336	C₂²×C₅₃	1	2	yes	—
266	C₅₃	#21129	1	1	yes	characteristic, direct product factor, Sylow
267	C₂²×D₄	C₂³×C₅₃	1	1	yes	characteristic
268	C₂²×D₄	C₂³×C₅₃	1	1	yes	characteristic
269	C₂³×C₄	—	2	2	no	—
270	C₂³×C₄	—	2	1	no	—
271	C₂³×C₄	—	2	1	no	—
272	C₂×D₈	—	2	4	no	—
273	C₂×D₈	C₂×C₄×C₅₃	1	4	yes	—
274	C₂²×C₈	C₂³×C₅₃	1	2	yes	—
275	C₂²×C₈	C₂³×C₅₃	1	1	yes	characteristic
276	C₂²×C₈	C₂³×C₅₃	1	1	yes	characteristic
277	C₄×D₄	—	2	4	no	—
278	C₄×D₄	—	2	4	no	—
279	C₄×D₄	—	2	2	no	—
280	C₄×D₄	—	2	2	no	—
281	C₄×D₄	C₂×C₄×C₅₃	1	4	yes	—
282	C₄×D₄	C₂×C₄×C₅₃	1	4	yes	complemented
283	C₂×C₄⋊C₄	—	2	2	no	—
284	C₂×C₄⋊C₄	C₂³×C₅₃	1	2	yes	—
285	C₂×C₄⋊C₄	C₂³×C₅₃	1	1	yes	characteristic
286	C₂×C₄⋊C₄	C₂³×C₅₃	1	1	yes	characteristic
287	C₂×C₄⋊C₄	C₂³×C₅₃	1	2	yes	—
288	C₂×C₂²⋊C₄	—	2	2	no	—
289	C₂×C₂²⋊C₄	—	2	1	no	—
290	C₂×C₂²⋊C₄	—	2	1	no	—
291	C₂×C₂²⋊C₄	—	2	2	no	—
292	C₂×C₂²⋊C₄	—	2	2	no	—
293	C₂×C₄²	—	2	2	no	—
294	C₂×C₄²	—	2	1	no	—
295	C₂×C₄²	—	2	1	no	—
296	C₂×C₄²	C₂³×C₅₃	1	2	yes	—
297	C₂×C₄²	C₂³×C₅₃	1	1	yes	characteristic
298	#107	—	2	4	no	—
299	#107	C₂×C₄×C₅₃	1	4	yes	complemented
300	#102	—	2	4	no	—
301	#102	—	2	4	no	—
302	#102	C₂×C₄×C₅₃	1	4	yes	—
303	#102	C₂×C₄×C₅₃	1	4	yes	complemented
304	#97	—	2	2	no	—
305	C₄×C₈	C₂×C₄×C₅₃	1	4	yes	complemented
306	#95	—	2	2	no	—
307	#95	—	2	1	no	—
308	#95	—	2	1	no	—
309	#95	—	2	2	no	—
310	#95	D₄×C₅₃	1	1	yes	characteristic
311	#95	D₄×C₅₃	1	1	yes	characteristic
312	#95	D₄×C₅₃	1	2	yes	—
313	C₂⁴	—	2	1	no	—
314	C₂⁴	—	2	1	no	—
315	C₂×D₄	—	2	2	no	—
316	C₂×D₄	—	2	1	no	—
317	C₂×D₄	—	2	2	no	—
318	C₂×D₄	—	2	2	no	—
319	C₂×D₄	—	2	1	no	—
320	C₂×D₄	C₂²×C₄×C₅₃	1	2	yes	—
321	C₂×D₄	C₂²×C₄×C₅₃	1	2	yes	—
322	C₂×D₄	#41×C₅₃	1	4	yes	—
323	C₂²×C₄	—	2	1	no	—
324	C₂²×C₄	—	2	2	no	—
325	C₂²×C₄	—	2	1	no	—
326	C₂²×C₄	—	2	8	no	—
327	C₂²×C₄	—	2	4	no	—
328	C₂²×C₄	—	2	4	no	—
329	C₂²×C₄	—	2	2	no	—
330	C₂²×C₄	—	8	2	no	—
331	C₂²×C₄	—	8	2	no	—
332	C₂²×C₄	—	8	2	no	—
333	C₂²×C₄	—	8	2	no	—
334	C₂²×C₄	C₂×D₄×C₅₃	1	2	yes	—
335	C₂²×C₄	C₂×D₄×C₅₃	1	2	yes	—
336	C₂²×C₄	C₂×D₄×C₅₃	1	1	yes	characteristic
337	C₂²×C₄	C₂×D₄×C₅₃	1	1	yes	characteristic
338	C₂²×C₄	C₂⁴×C₅₃	1	1	yes	characteristic
339	D₈	—	2	8	no	—
340	C₂×C₈	—	2	2	no	—
341	C₂×C₈	—	2	2	no	—
342	C₂×C₈	—	2	2	no	—
343	C₂×C₈	—	2	2	no	—
344	C₂×C₈	—	2	2	no	—
345	C₂×C₈	C₂²×C₄×C₅₃	1	2	yes	—
346	C₂×C₈	C₂²×C₄×C₅₃	1	2	yes	—
347	C₄⋊C₄	—	2	1	no	—
348	C₄⋊C₄	—	2	1	no	—
349	C₄⋊C₄	—	2	2	no	—
350	C₄⋊C₄	—	2	2	no	—
351	C₄⋊C₄	—	2	2	no	—
352	C₄⋊C₄	—	8	2	no	—
353	C₄⋊C₄	C₂²×C₄×C₅₃	1	2	yes	—
354	C₄⋊C₄	C₂²×C₄×C₅₃	1	2	yes	—
355	C₄⋊C₄	#41×C₅₃	1	4	yes	—
356	C₂²⋊C₄	—	2	8	no	—
357	C₂²⋊C₄	—	2	4	no	—
358	C₂²⋊C₄	—	2	4	no	—
359	C₂²⋊C₄	—	8	2	no	—
360	C₂²⋊C₄	—	8	2	no	—
361	C₄²	—	2	2	no	—
362	C₄²	—	2	2	no	—
363	C₄²	—	8	1	no	—
364	C₄²	—	8	1	no	—
365	C₄²	—	8	2	no	—
366	C₄²	C₂²×C₄×C₅₃	1	2	yes	—
367	C₄²	#41×C₅₃	1	2	yes	—
368	C₂³	—	2	2	no	—
369	C₂³	—	2	4	no	—
370	C₂³	—	2	2	no	—
371	C₂³	—	4	2	no	—
372	C₂³	—	4	1	no	—
373	C₂³	—	4	1	no	—
374	C₂³	—	4	2	no	—
375	C₂³	—	4	1	no	—
376	C₂³	—	4	2	no	—
377	C₂³	—	4	1	no	—
378	C₂³	C₂²×D₄×C₅₃	1	1	yes	characteristic
379	D₄	—	2	2	no	—
380	D₄	—	2	2	no	—
381	D₄	—	2	2	no	—
382	D₄	—	2	4	no	—
383	D₄	—	4	2	no	—
384	D₄	#117×C₅₃	1	4	yes	—
385	C₂×C₄	—	2	1	no	—
386	C₂×C₄	—	2	2	no	—
387	C₂×C₄	—	2	1	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	4	no	—
393	C₂×C₄	—	2	2	no	—
394	C₂×C₄	—	2	2	no	—
395	C₂×C₄	—	2	2	no	—
396	C₂×C₄	—	2	2	no	—
397	C₂×C₄	—	4	2	no	—
398	C₂×C₄	—	4	2	no	—
399	C₂×C₄	—	4	2	no	—
400	C₂×C₄	—	4	2	no	—
401	C₂×C₄	—	4	2	no	—
402	C₂×C₄	—	4	2	no	—
403	C₂×C₄	—	4	2	no	—
404	C₂×C₄	—	8	4	no	—
405	C₂×C₄	—	8	4	no	—
406	C₂×C₄	—	8	8	no	—
407	C₂×C₄	C₄×D₄×C₅₃	1	2	yes	—
408	C₂×C₄	C₄×D₄×C₅₃	1	2	yes	—
409	C₂×C₄	C₂³×C₄×C₅₃	1	1	yes	characteristic
410	C₂×C₄	C₂×#41×C₅₃	1	2	yes	—
411	C₂×C₄	C₂×#41×C₅₃	1	2	yes	—
412	C₂×C₄	C₂×#41×C₅₃	1	1	yes	characteristic
413	C₈	—	2	2	no	—
414	C₈	—	2	2	no	—
415	C₈	—	4	2	no	—
416	C₂²	—	2	4	no	—
417	C₂²	—	4	2	no	—
418	C₂²	—	4	2	no	—
419	C₂²	—	4	4	no	—
420	C₂²	—	4	4	no	—
421	C₂²	—	4	2	no	—
422	C₂²	—	4	2	no	—
423	C₂²	—	4	2	no	—
424	C₂²	—	8	2	no	—
425	C₂²	—	8	2	no	—
426	C₂²	C₂×C₄×D₄×C₅₃	1	1	yes	characteristic
427	C₂²	C₅₃×#525	1	2	yes	—
428	C₂²	C₅₃×#576	1	2	yes	—
429	C₂²	C₅₃×#576	1	1	yes	characteristic
430	C₂²	C₅₃×#578	1	1	yes	characteristic
431	C₄	—	2	2	no	—
432	C₄	—	2	2	no	—
433	C₄	—	4	2	no	—
434	C₄	—	4	2	no	—
435	C₄	—	4	2	no	—
436	C₄	—	4	4	no	—
437	C₄	—	4	2	no	—
438	C₄	—	8	2	no	—
439	C₄	C₂×#117×C₅₃	1	1	yes	characteristic
440	C₄	C₄×#41×C₅₃	1	1	yes	characteristic
441	C₄	C₅₃×#529	1	2	yes	—
442	C₂	—	4	4	no	—
443	C₂	—	8	2	no	—
444	C₂	C₅₃×#2290	1	1	yes	characteristic
445	C₂	C₅₃×#2948	1	1	yes	characteristic
446	C₂	C₅₃×#2951	1	1	yes	characteristic
447	C₂	C₅₃×#3122	1	2	yes	—
448	C₂	C₅₃×#3124	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×D₈×C₅₃	C₂²
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₅₃ ◁ C₅₃×#376 ◁ G

Complements

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

N	H ≅ G/N	type
2. C₅₃×#2319	C₂	semidirect
3. C₅₃×#2275	C₂	semidirect
4. C₅₃×#1925	C₂	semidirect
5. C₅₃×#1921	C₂	semidirect
7. C₅₃×#1862	C₂	semidirect
8. C₄×#107×C₅₃	C₂	semidirect
9. C₅₃×#1762	C₂	semidirect
10. C₅₃×#1762	C₂	semidirect
11. C₅₃×#1746	C₂	semidirect
16. C₄×D₈×C₅₃	C₄	semidirect
57. C₄×D₄×C₅₃	C₂×C₄	semidirect
74. #107×C₅₃	C₂×C₄	semidirect
78. #102×C₅₃	C₂×C₄	semidirect
80. C₄×C₈×C₅₃	C₂×C₄	semidirect
191. #21129	C₅₃	direct (N × H)
219. #2319	C₂×C₅₃	semidirect
220. #2275	C₂×C₅₃	semidirect
221. #1925	C₂×C₅₃	semidirect
222. #1921	C₂×C₅₃	semidirect
224. #1862	C₂×C₅₃	semidirect
225. C₄×#107	C₂×C₅₃	semidirect
226. #1762	C₂×C₅₃	semidirect
227. #1762	C₂×C₅₃	semidirect
228. #1746	C₂×C₅₃	semidirect
240. C₄×D₈	C₄×C₅₃	semidirect
266. C₅₃	#21129	direct (N × H)
282. C₄×D₄	C₂×C₄×C₅₃	semidirect
299. #107	C₂×C₄×C₅₃	semidirect
303. #102	C₂×C₄×C₅₃	semidirect
305. C₄×C₈	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×C₄×D₈×C₅₃	2
C₅₃×#2319	2
C₅₃×#2275	2
C₅₃×#1925	2
C₅₃×#1921	2
C₅₃×#1879	2
C₅₃×#1862	2
C₄×#107×C₅₃	2
C₅₃×#1762	2
C₅₃×#1746	2
#21129	53

Subgroups — C53×#21129 #27228641

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅₃×#21129 #27228641