Subgroups — C₅₃×#10807 #27218319

Order 13568.

914 conjugacy classes of subgroups.

The subgroup lattice has 442 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄×C₄⋊C₄×C₅₃	C₂	1	1	yes	maximal, characteristic
2	C₂×C₅₃×#340	C₂	1	2	yes	maximal
3	C₂×C₅₃×#340	C₂	1	2	yes	maximal
4	C₂×C₅₃×#336	C₂	1	1	yes	maximal, characteristic
5	C₅₃×C₄⋊#95	C₂	1	1	yes	maximal, characteristic
6	C₂³×C₈×C₅₃	C₂²	1	2	yes	—
7	C₂²×C₄⋊C₄×C₅₃	C₂²	1	2	yes	—
8	C₂²×C₄⋊C₄×C₅₃	C₂²	1	1	yes	characteristic
9	C₂²×C₄⋊C₄×C₅₃	C₂²	1	1	yes	characteristic
10	C₂²×C₄²×C₅₃	—	2	1	no	—
11	C₂²×C₄²×C₅₃	C₂²	1	1	yes	characteristic
12	C₄×C₄⋊C₄×C₅₃	—	2	2	no	—
13	C₄×C₄⋊C₄×C₅₃	C₄	1	4	yes	complemented
14	C₂×#95×C₅₃	—	2	1	no	—
15	C₂×#95×C₅₃	—	2	1	no	—
16	C₂×#95×C₅₃	—	2	1	no	—
17	C₅₃×#340	—	2	4	no	—
18	C₅₃×#340	C₄	1	8	yes	complemented
19	C₅₃×#336	—	2	2	no	—
20	C₂³×C₄×C₅₃	—	2	1	no	—
21	C₂³×C₄×C₅₃	—	2	1	no	—
22	C₂³×C₄×C₅₃	—	2	2	no	—
23	C₂³×C₄×C₅₃	D₄	1	1	yes	characteristic
24	C₂³×C₄×C₅₃	D₄	1	1	yes	characteristic
25	C₂³×C₄×C₅₃	C₂³	1	1	yes	characteristic
26	C₂²×C₈×C₅₃	—	2	2	no	—
27	C₂²×C₈×C₅₃	—	2	2	no	—
28	C₂²×C₈×C₅₃	—	2	2	no	—
29	C₂²×C₈×C₅₃	—	2	2	no	—
30	C₂²×C₈×C₅₃	—	2	2	no	—
31	C₂²×C₈×C₅₃	—	2	2	no	—
32	C₂²×C₈×C₅₃	C₂×C₄	1	4	yes	—
33	C₂×C₄⋊C₄×C₅₃	—	2	1	no	—
34	C₂×C₄⋊C₄×C₅₃	—	2	1	no	—
35	C₂×C₄⋊C₄×C₅₃	—	2	2	no	—
36	C₂×C₄⋊C₄×C₅₃	—	2	1	no	—
37	C₂×C₄⋊C₄×C₅₃	—	2	1	no	—
38	C₂×C₄⋊C₄×C₅₃	—	2	2	no	—
39	C₂×C₄⋊C₄×C₅₃	—	2	2	no	—
40	C₂×C₄⋊C₄×C₅₃	—	2	2	no	—
41	C₂×C₄⋊C₄×C₅₃	—	2	2	no	—
42	C₂×C₄⋊C₄×C₅₃	—	2	2	no	—
43	C₂×C₄⋊C₄×C₅₃	C₂×C₄	1	4	yes	—
44	C₂×C₄⋊C₄×C₅₃	C₂×C₄	1	4	yes	—
45	C₂×C₄⋊C₄×C₅₃	C₂×C₄	1	2	yes	—
46	C₂×C₄⋊C₄×C₅₃	C₂×C₄	1	2	yes	—
47	C₂×C₄⋊C₄×C₅₃	D₄	1	2	yes	—
48	C₂×C₄⋊C₄×C₅₃	Q₈	1	2	yes	—
49	C₂×C₄²×C₅₃	—	2	4	no	—
50	C₂×C₄²×C₅₃	—	2	2	no	—
51	C₂×C₄²×C₅₃	—	2	2	no	—
52	C₂×C₄²×C₅₃	—	4	4	no	—
53	C₂×C₄²×C₅₃	C₂×C₄	1	2	yes	—
54	C₂×C₄²×C₅₃	D₄	1	2	yes	—
55	#95×C₅₃	—	2	4	no	—
56	#95×C₅₃	—	4	2	no	—
57	#95×C₅₃	—	4	2	no	—
58	C₂⁴×C₅₃	C₂×D₄	1	1	yes	characteristic
59	C₂²×C₄×C₅₃	—	2	2	no	—
60	C₂²×C₄×C₅₃	—	2	2	no	—
61	C₂²×C₄×C₅₃	—	2	2	no	—
62	C₂²×C₄×C₅₃	—	2	2	no	—
63	C₂²×C₄×C₅₃	—	2	4	no	—
64	C₂²×C₄×C₅₃	—	2	2	no	—
65	C₂²×C₄×C₅₃	—	2	2	no	—
66	C₂²×C₄×C₅₃	—	2	2	no	—
67	C₂²×C₄×C₅₃	—	2	2	no	—
68	C₂²×C₄×C₅₃	—	2	1	no	—
69	C₂²×C₄×C₅₃	—	2	1	no	—
70	C₂²×C₄×C₅₃	—	2	4	no	—
71	C₂²×C₄×C₅₃	—	4	2	no	—
72	C₂²×C₄×C₅₃	—	4	2	no	—
73	C₂²×C₄×C₅₃	—	4	2	no	—
74	C₂²×C₄×C₅₃	—	4	1	no	—
75	C₂²×C₄×C₅₃	—	4	1	no	—
76	C₂²×C₄×C₅₃	—	4	2	no	—
77	C₂²×C₄×C₅₃	—	4	2	no	—
78	C₂²×C₄×C₅₃	—	4	2	no	—
79	C₂²×C₄×C₅₃	—	4	2	no	—
80	C₂²×C₄×C₅₃	—	4	2	no	—
81	C₂²×C₄×C₅₃	—	4	2	no	—
82	C₂²×C₄×C₅₃	C₂²⋊C₄	1	2	yes	—
83	C₂²×C₄×C₅₃	C₂²⋊C₄	1	2	yes	—
84	C₂²×C₄×C₅₃	SD₁₆	1	2	yes	—
85	C₂²×C₄×C₅₃	Q₁₆	1	2	yes	—
86	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
87	C₂²×C₄×C₅₃	C₂²×C₄	1	2	yes	—
88	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
89	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
90	C₂²×C₄×C₅₃	C₂×Q₈	1	1	yes	characteristic
91	C₂²×C₄×C₅₃	#41	1	1	yes	characteristic
92	C₂²×C₄×C₅₃	#41	1	1	yes	characteristic
93	C₂²×C₄×C₅₃	#41	1	2	yes	—
94	C₂²×C₄×C₅₃	#41	1	2	yes	—
95	C₂²×C₄×C₅₃	#41	1	2	yes	—
96	C₂×C₈×C₅₃	—	2	4	no	—
97	C₂×C₈×C₅₃	—	2	4	no	—
98	C₂×C₈×C₅₃	—	2	4	no	—
99	C₂×C₈×C₅₃	—	4	2	no	—
100	C₂×C₈×C₅₃	—	4	2	no	—
101	C₂×C₈×C₅₃	—	4	2	no	—
102	C₂×C₈×C₅₃	—	4	2	no	—
103	C₄⋊C₄×C₅₃	—	2	2	no	—
104	C₄⋊C₄×C₅₃	—	2	2	no	—
105	C₄⋊C₄×C₅₃	—	2	4	no	—
106	C₄⋊C₄×C₅₃	—	2	2	no	—
107	C₄⋊C₄×C₅₃	—	2	2	no	—
108	C₄⋊C₄×C₅₃	—	2	4	no	—
109	C₄⋊C₄×C₅₃	—	2	4	no	—
110	C₄⋊C₄×C₅₃	—	4	2	no	—
111	C₄⋊C₄×C₅₃	—	4	2	no	—
112	C₄⋊C₄×C₅₃	C₄²	1	4	yes	—
113	C₄⋊C₄×C₅₃	C₄⋊C₄	1	4	yes	—
114	C₄²×C₅₃	—	2	2	no	—
115	C₄²×C₅₃	—	2	2	no	—
116	C₄²×C₅₃	—	2	4	no	—
117	C₄²×C₅₃	—	4	8	no	—
118	C₂³×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
119	C₂³×C₅₃	C₄×D₄	1	2	yes	—
120	C₂³×C₅₃	#121	1	1	yes	characteristic
121	C₂³×C₅₃	C₂²⋊Q₈	1	1	yes	characteristic
122	C₂³×C₅₃	#123	1	2	yes	—
123	C₂³×C₅₃	C₂×SD₁₆	1	1	yes	characteristic
124	C₂³×C₅₃	C₂×Q₁₆	1	1	yes	characteristic
125	C₂³×C₅₃	#135	1	2	yes	—
126	C₂³×C₅₃	AGL(1,ℤ/8)	1	1	yes	characteristic
127	C₂³×C₅₃	AGL(1,ℤ/8)	1	2	yes	—
128	C₂³×C₅₃	#137	1	1	yes	characteristic
129	C₂×C₄×C₅₃	—	2	2	no	—
130	C₂×C₄×C₅₃	—	2	2	no	—
131	C₂×C₄×C₅₃	—	2	4	no	—
132	C₂×C₄×C₅₃	—	2	4	no	—
133	C₂×C₄×C₅₃	—	2	4	no	—
134	C₂×C₄×C₅₃	—	2	4	no	—
135	C₂×C₄×C₅₃	—	2	2	no	—
136	C₂×C₄×C₅₃	—	2	2	no	—
137	C₂×C₄×C₅₃	—	2	2	no	—
138	C₂×C₄×C₅₃	—	4	2	no	—
139	C₂×C₄×C₅₃	—	4	2	no	—
140	C₂×C₄×C₅₃	—	4	2	no	—
141	C₂×C₄×C₅₃	—	4	2	no	—
142	C₂×C₄×C₅₃	—	4	4	no	—
143	C₂×C₄×C₅₃	—	4	4	no	—
144	C₂×C₄×C₅₃	—	4	4	no	—
145	C₂×C₄×C₅₃	—	4	4	no	—
146	C₂×C₄×C₅₃	—	4	2	no	—
147	C₂×C₄×C₅₃	—	4	2	no	—
148	C₂×C₄×C₅₃	—	4	4	no	—
149	C₂×C₄×C₅₃	—	4	4	no	—
150	C₂×C₄×C₅₃	—	4	4	no	—
151	C₂×C₄×C₅₃	—	8	2	no	—
152	C₂×C₄×C₅₃	—	8	2	no	—
153	C₂×C₄×C₅₃	—	8	2	no	—
154	C₂×C₄×C₅₃	—	8	2	no	—
155	C₂×C₄×C₅₃	#103	1	4	yes	—
156	C₂×C₄×C₅₃	C₂×C₄²	1	1	yes	characteristic
157	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
158	C₂×C₄×C₅₃	#117	1	1	yes	characteristic
159	C₂×C₄×C₅₃	#117	1	1	yes	characteristic
160	C₂×C₄×C₅₃	#117	1	2	yes	—
161	C₂×C₄×C₅₃	#117	1	2	yes	—
162	C₂×C₄×C₅₃	#117	1	1	yes	characteristic
163	C₂×C₄×C₅₃	#117	1	1	yes	characteristic
164	C₂×C₄×C₅₃	C₄×D₄	1	2	yes	—
165	C₂×C₄×C₅₃	C₄×D₄	1	2	yes	—
166	C₂×C₄×C₅₃	C₄×Q₈	1	2	yes	—
167	C₂×C₄×C₅₃	#121	1	1	yes	characteristic
168	C₂×C₄×C₅₃	C₂²⋊Q₈	1	1	yes	characteristic
169	C₂×C₄×C₅₃	#123	1	2	yes	—
170	C₂×C₄×C₅₃	#124	1	2	yes	—
171	C₂×C₄×C₅₃	#125	1	2	yes	—
172	C₂×C₄×C₅₃	#126	1	2	yes	—
173	C₂×C₄×C₅₃	#126	1	2	yes	—
174	C₈×C₅₃	—	4	4	no	—
175	#10807	C₅₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
176	C₂²×C₅₃	C₄×C₂²⋊C₄	1	1	yes	characteristic
177	C₂²×C₅₃	#380	1	1	yes	characteristic
178	C₂²×C₅₃	#381	1	1	yes	characteristic
179	C₂²×C₅₃	#387	1	2	yes	—
180	C₂²×C₅₃	#388	1	2	yes	—
181	C₂²×C₅₃	C₂×#103	1	1	yes	characteristic
182	C₂²×C₅₃	#416	1	1	yes	characteristic
183	C₂²×C₅₃	#417	1	1	yes	characteristic
184	C₂²×C₅₃	#418	1	1	yes	characteristic
185	C₂²×C₅₃	C₄×SD₁₆	1	2	yes	—
186	C₂²×C₅₃	C₄×Q₁₆	1	2	yes	—
187	C₂²×C₅₃	#440	1	2	yes	—
188	C₂²×C₅₃	#442	1	2	yes	—
189	C₂²×C₅₃	#460	1	1	yes	characteristic
190	C₂²×C₅₃	#462	1	1	yes	characteristic
191	C₂²×C₅₃	#463	1	2	yes	—
192	C₂²×C₅₃	#475	1	1	yes	characteristic
193	C₂²×C₅₃	#477	1	1	yes	characteristic
194	C₂²×C₅₃	#478	1	2	yes	—
195	C₂²×C₅₃	#481	1	2	yes	—
196	C₂²×C₅₃	#482	1	2	yes	—
197	C₂²×C₅₃	#484	1	2	yes	—
198	C₂²×C₅₃	#485	1	2	yes	—
199	C₄×C₅₃	—	2	4	no	—
200	C₄×C₅₃	—	4	4	no	—
201	C₄×C₅₃	—	4	2	no	—
202	C₄×C₅₃	—	4	4	no	—
203	C₄×C₅₃	—	8	4	no	—
204	C₄×C₅₃	#376	1	1	yes	characteristic
205	C₄×C₅₃	C₄×C₄⋊C₄	1	1	yes	characteristic
206	C₄×C₅₃	#382	1	1	yes	characteristic
207	C₄×C₅₃	#383	1	1	yes	characteristic
208	C₄×C₅₃	#387	1	2	yes	—
209	C₄×C₅₃	#388	1	2	yes	—
210	C₂×C₄×C₄⋊C₄	C₂×C₅₃	1	1	yes	characteristic
211	C₂×#340	C₂×C₅₃	1	2	yes	—
212	C₂×#340	C₂×C₅₃	1	2	yes	—
213	C₂×#336	C₂×C₅₃	1	1	yes	characteristic
214	C₄⋊#95	C₂×C₅₃	1	1	yes	characteristic
215	C₂×C₅₃	#1434	1	1	yes	characteristic
216	C₂×C₅₃	C₄×#103	1	1	yes	characteristic
217	C₂×C₅₃	#1762	1	1	yes	characteristic
218	C₂×C₅₃	#1803	1	1	yes	characteristic
219	C₂×C₅₃	#1804	1	1	yes	characteristic
220	C₂×C₅₃	#1824	1	1	yes	characteristic
221	C₂×C₅₃	#1825	1	1	yes	characteristic
222	C₂×C₅₃	#1919	1	2	yes	—
223	C₂×C₅₃	#1921	1	2	yes	—
224	C₂×C₅₃	#1926	1	2	yes	—
225	C₂×C₅₃	#1928	1	2	yes	—
226	C₂³×C₈	C₂²×C₅₃	1	2	yes	—
227	C₂²×C₄⋊C₄	C₂²×C₅₃	1	2	yes	—
228	C₂²×C₄⋊C₄	C₂²×C₅₃	1	1	yes	characteristic
229	C₂²×C₄⋊C₄	C₂²×C₅₃	1	1	yes	characteristic
230	C₂²×C₄²	—	2	1	no	—
231	C₂²×C₄²	C₂²×C₅₃	1	1	yes	characteristic
232	C₄×C₄⋊C₄	—	2	2	no	—
233	C₄×C₄⋊C₄	C₄×C₅₃	1	4	yes	complemented
234	C₂×#95	—	2	1	no	—
235	C₂×#95	—	2	1	no	—
236	C₂×#95	—	2	1	no	—
237	#340	—	2	4	no	—
238	#340	C₄×C₅₃	1	8	yes	complemented
239	#336	—	2	2	no	—
240	C₅₃	#10807	1	1	yes	characteristic, direct product factor, Sylow
241	C₂³×C₄	—	2	1	no	—
242	C₂³×C₄	—	2	1	no	—
243	C₂³×C₄	—	2	2	no	—
244	C₂³×C₄	D₄×C₅₃	1	1	yes	characteristic
245	C₂³×C₄	D₄×C₅₃	1	1	yes	characteristic
246	C₂³×C₄	C₂³×C₅₃	1	1	yes	characteristic
247	C₂²×C₈	—	2	2	no	—
248	C₂²×C₈	—	2	2	no	—
249	C₂²×C₈	—	2	2	no	—
250	C₂²×C₈	—	2	2	no	—
251	C₂²×C₈	—	2	2	no	—
252	C₂²×C₈	—	2	2	no	—
253	C₂²×C₈	C₂×C₄×C₅₃	1	4	yes	—
254	C₂×C₄⋊C₄	—	2	1	no	—
255	C₂×C₄⋊C₄	—	2	1	no	—
256	C₂×C₄⋊C₄	—	2	2	no	—
257	C₂×C₄⋊C₄	—	2	1	no	—
258	C₂×C₄⋊C₄	—	2	1	no	—
259	C₂×C₄⋊C₄	—	2	2	no	—
260	C₂×C₄⋊C₄	—	2	2	no	—
261	C₂×C₄⋊C₄	—	2	2	no	—
262	C₂×C₄⋊C₄	—	2	2	no	—
263	C₂×C₄⋊C₄	—	2	2	no	—
264	C₂×C₄⋊C₄	C₂×C₄×C₅₃	1	4	yes	—
265	C₂×C₄⋊C₄	C₂×C₄×C₅₃	1	4	yes	—
266	C₂×C₄⋊C₄	C₂×C₄×C₅₃	1	2	yes	—
267	C₂×C₄⋊C₄	C₂×C₄×C₅₃	1	2	yes	—
268	C₂×C₄⋊C₄	D₄×C₅₃	1	2	yes	—
269	C₂×C₄⋊C₄	Q₈×C₅₃	1	2	yes	—
270	C₂×C₄²	—	2	2	no	—
271	C₂×C₄²	—	2	2	no	—
272	C₂×C₄²	—	2	4	no	—
273	C₂×C₄²	—	4	4	no	—
274	C₂×C₄²	C₂×C₄×C₅₃	1	2	yes	—
275	C₂×C₄²	D₄×C₅₃	1	2	yes	—
276	#95	—	2	4	no	—
277	#95	—	4	2	no	—
278	#95	—	4	2	no	—
279	C₂⁴	C₂×D₄×C₅₃	1	1	yes	characteristic
280	C₂²×C₄	—	2	2	no	—
281	C₂²×C₄	—	2	2	no	—
282	C₂²×C₄	—	2	2	no	—
283	C₂²×C₄	—	2	2	no	—
284	C₂²×C₄	—	2	4	no	—
285	C₂²×C₄	—	2	2	no	—
286	C₂²×C₄	—	2	2	no	—
287	C₂²×C₄	—	2	2	no	—
288	C₂²×C₄	—	2	2	no	—
289	C₂²×C₄	—	2	1	no	—
290	C₂²×C₄	—	2	1	no	—
291	C₂²×C₄	—	2	4	no	—
292	C₂²×C₄	—	4	2	no	—
293	C₂²×C₄	—	4	2	no	—
294	C₂²×C₄	—	4	2	no	—
295	C₂²×C₄	—	4	1	no	—
296	C₂²×C₄	—	4	1	no	—
297	C₂²×C₄	—	4	2	no	—
298	C₂²×C₄	—	4	2	no	—
299	C₂²×C₄	—	4	2	no	—
300	C₂²×C₄	—	4	2	no	—
301	C₂²×C₄	—	4	2	no	—
302	C₂²×C₄	—	4	2	no	—
303	C₂²×C₄	C₂²⋊C₄×C₅₃	1	2	yes	—
304	C₂²×C₄	C₂²⋊C₄×C₅₃	1	2	yes	—
305	C₂²×C₄	SD₁₆×C₅₃	1	2	yes	—
306	C₂²×C₄	Q₁₆×C₅₃	1	2	yes	—
307	C₂²×C₄	C₂²×C₄×C₅₃	1	1	yes	characteristic
308	C₂²×C₄	C₂²×C₄×C₅₃	1	2	yes	—
309	C₂²×C₄	C₂×D₄×C₅₃	1	1	yes	characteristic
310	C₂²×C₄	C₂×D₄×C₅₃	1	1	yes	characteristic
311	C₂²×C₄	C₂×Q₈×C₅₃	1	1	yes	characteristic
312	C₂²×C₄	#41×C₅₃	1	1	yes	characteristic
313	C₂²×C₄	#41×C₅₃	1	1	yes	characteristic
314	C₂²×C₄	#41×C₅₃	1	2	yes	—
315	C₂²×C₄	#41×C₅₃	1	2	yes	—
316	C₂²×C₄	#41×C₅₃	1	2	yes	—
317	C₂×C₈	—	2	4	no	—
318	C₂×C₈	—	2	4	no	—
319	C₂×C₈	—	2	4	no	—
320	C₂×C₈	—	4	2	no	—
321	C₂×C₈	—	4	2	no	—
322	C₂×C₈	—	4	2	no	—
323	C₂×C₈	—	4	2	no	—
324	C₄⋊C₄	—	2	2	no	—
325	C₄⋊C₄	—	2	2	no	—
326	C₄⋊C₄	—	2	4	no	—
327	C₄⋊C₄	—	2	2	no	—
328	C₄⋊C₄	—	2	2	no	—
329	C₄⋊C₄	—	2	4	no	—
330	C₄⋊C₄	—	2	4	no	—
331	C₄⋊C₄	—	4	2	no	—
332	C₄⋊C₄	—	4	2	no	—
333	C₄⋊C₄	C₄²×C₅₃	1	4	yes	—
334	C₄⋊C₄	C₄⋊C₄×C₅₃	1	4	yes	—
335	C₄²	—	2	2	no	—
336	C₄²	—	2	2	no	—
337	C₄²	—	2	4	no	—
338	C₄²	—	4	8	no	—
339	C₂³	C₂×C₂²⋊C₄×C₅₃	1	1	yes	characteristic
340	C₂³	C₄×D₄×C₅₃	1	2	yes	—
341	C₂³	#121×C₅₃	1	1	yes	characteristic
342	C₂³	C₂²⋊Q₈×C₅₃	1	1	yes	characteristic
343	C₂³	#123×C₅₃	1	2	yes	—
344	C₂³	C₂×SD₁₆×C₅₃	1	1	yes	characteristic
345	C₂³	C₂×Q₁₆×C₅₃	1	1	yes	characteristic
346	C₂³	#135×C₅₃	1	2	yes	—
347	C₂³	AGL(1,ℤ/8)×C₅₃	1	1	yes	characteristic
348	C₂³	AGL(1,ℤ/8)×C₅₃	1	2	yes	—
349	C₂³	#137×C₅₃	1	1	yes	characteristic
350	C₂×C₄	—	2	2	no	—
351	C₂×C₄	—	2	2	no	—
352	C₂×C₄	—	2	4	no	—
353	C₂×C₄	—	2	4	no	—
354	C₂×C₄	—	2	4	no	—
355	C₂×C₄	—	2	4	no	—
356	C₂×C₄	—	2	2	no	—
357	C₂×C₄	—	2	2	no	—
358	C₂×C₄	—	2	2	no	—
359	C₂×C₄	—	4	2	no	—
360	C₂×C₄	—	4	2	no	—
361	C₂×C₄	—	4	2	no	—
362	C₂×C₄	—	4	2	no	—
363	C₂×C₄	—	4	2	no	—
364	C₂×C₄	—	4	4	no	—
365	C₂×C₄	—	4	4	no	—
366	C₂×C₄	—	4	4	no	—
367	C₂×C₄	—	4	4	no	—
368	C₂×C₄	—	4	2	no	—
369	C₂×C₄	—	4	4	no	—
370	C₂×C₄	—	4	4	no	—
371	C₂×C₄	—	4	4	no	—
372	C₂×C₄	—	8	2	no	—
373	C₂×C₄	—	8	2	no	—
374	C₂×C₄	—	8	2	no	—
375	C₂×C₄	—	8	2	no	—
376	C₂×C₄	#103×C₅₃	1	4	yes	—
377	C₂×C₄	C₂×C₄²×C₅₃	1	1	yes	characteristic
378	C₂×C₄	C₂×C₄⋊C₄×C₅₃	1	1	yes	characteristic
379	C₂×C₄	#117×C₅₃	1	1	yes	characteristic
380	C₂×C₄	#117×C₅₃	1	1	yes	characteristic
381	C₂×C₄	#117×C₅₃	1	2	yes	—
382	C₂×C₄	#117×C₅₃	1	2	yes	—
383	C₂×C₄	#117×C₅₃	1	1	yes	characteristic
384	C₂×C₄	#117×C₅₃	1	1	yes	characteristic
385	C₂×C₄	C₄×D₄×C₅₃	1	2	yes	—
386	C₂×C₄	C₄×D₄×C₅₃	1	2	yes	—
387	C₂×C₄	C₄×Q₈×C₅₃	1	2	yes	—
388	C₂×C₄	#121×C₅₃	1	1	yes	characteristic
389	C₂×C₄	C₂²⋊Q₈×C₅₃	1	1	yes	characteristic
390	C₂×C₄	#123×C₅₃	1	2	yes	—
391	C₂×C₄	#124×C₅₃	1	2	yes	—
392	C₂×C₄	#125×C₅₃	1	2	yes	—
393	C₂×C₄	#126×C₅₃	1	2	yes	—
394	C₂×C₄	#126×C₅₃	1	2	yes	—
395	C₈	—	4	4	no	—
396	C₂²	C₄×C₂²⋊C₄×C₅₃	1	1	yes	characteristic
397	C₂²	C₅₃×#380	1	1	yes	characteristic
398	C₂²	C₅₃×#381	1	1	yes	characteristic
399	C₂²	C₅₃×#387	1	2	yes	—
400	C₂²	C₅₃×#388	1	2	yes	—
401	C₂²	C₂×#103×C₅₃	1	1	yes	characteristic
402	C₂²	C₅₃×#416	1	1	yes	characteristic
403	C₂²	C₅₃×#417	1	1	yes	characteristic
404	C₂²	C₅₃×#418	1	1	yes	characteristic
405	C₂²	C₄×SD₁₆×C₅₃	1	2	yes	—
406	C₂²	C₄×Q₁₆×C₅₃	1	2	yes	—
407	C₂²	C₅₃×#440	1	2	yes	—
408	C₂²	C₅₃×#442	1	2	yes	—
409	C₂²	C₅₃×#460	1	1	yes	characteristic
410	C₂²	C₅₃×#462	1	1	yes	characteristic
411	C₂²	C₅₃×#463	1	2	yes	—
412	C₂²	C₅₃×#475	1	1	yes	characteristic
413	C₂²	C₅₃×#477	1	1	yes	characteristic
414	C₂²	C₅₃×#478	1	2	yes	—
415	C₂²	C₅₃×#481	1	2	yes	—
416	C₂²	C₅₃×#482	1	2	yes	—
417	C₂²	C₅₃×#484	1	2	yes	—
418	C₂²	C₅₃×#485	1	2	yes	—
419	C₄	—	2	4	no	—
420	C₄	—	4	4	no	—
421	C₄	—	4	4	no	—
422	C₄	—	4	2	no	—
423	C₄	—	8	4	no	—
424	C₄	C₅₃×#376	1	1	yes	characteristic
425	C₄	C₄×C₄⋊C₄×C₅₃	1	1	yes	characteristic
426	C₄	C₅₃×#382	1	1	yes	characteristic
427	C₄	C₅₃×#383	1	1	yes	characteristic
428	C₄	C₅₃×#387	1	2	yes	—
429	C₄	C₅₃×#388	1	2	yes	—
430	C₂	C₅₃×#1434	1	1	yes	characteristic
431	C₂	C₄×#103×C₅₃	1	1	yes	characteristic
432	C₂	C₅₃×#1762	1	1	yes	characteristic
433	C₂	C₅₃×#1803	1	1	yes	characteristic
434	C₂	C₅₃×#1804	1	1	yes	characteristic
435	C₂	C₅₃×#1824	1	1	yes	characteristic
436	C₂	C₅₃×#1825	1	1	yes	characteristic
437	C₂	C₅₃×#1919	1	2	yes	—
438	C₂	C₅₃×#1921	1	2	yes	—
439	C₂	C₅₃×#1926	1	2	yes	—
440	C₂	C₅₃×#1928	1	2	yes	—

	subgroup	G/H
center, socle, prime radical	C₂⁴×C₅₃	C₂×D₄
commutator	C₂×C₄	C₂×C₄²×C₅₃
Frattini, cosocle	C₂³×C₄	C₂³×C₅₃
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂⁴×C₅₃ ◁ C₂²×C₄²×C₅₃ ◁ G

Complements

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

N	H ≅ G/N	type
13. C₄×C₄⋊C₄×C₅₃	C₄	semidirect
18. C₅₃×#340	C₄	semidirect
175. #10807	C₅₃	direct (N × H)
233. C₄×C₄⋊C₄	C₄×C₅₃	semidirect
238. #340	C₄×C₅₃	semidirect
240. C₅₃	#10807	direct (N × H)

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
C₂×C₄×C₄⋊C₄×C₅₃	2
C₂×C₅₃×#340	2
C₂×C₅₃×#336	2
C₅₃×C₄⋊#95	2
#10807	53

Subgroups — C53×#10807 #27218319

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅₃×#10807 #27218319