Subgroups — C₅₃×#17324 #27224836

Order 13568.

742 conjugacy classes of subgroups.

The subgroup lattice has 378 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₅₃×#2920	C₂	1	1	yes	maximal, characteristic
2	C₅₃×#2920	C₂	1	1	yes	maximal, characteristic
3	C₂×C₅₃×#431	C₂	1	1	yes	maximal, characteristic
4	C₅₃×#1454	C₂	1	4	yes	maximal, complemented
5	C₅₃×#1453	C₂	1	4	yes	maximal, complemented
6	C₅₃×#1451	C₂	1	2	yes	maximal, complemented
7	C₈×M₁₆×C₅₃	C₂	1	2	yes	maximal, complemented
8	C₂²×C₄²×C₅₃	C₂²	1	1	yes	characteristic
9	C₅₃×#432	C₄	1	2	yes	—
10	C₅₃×#432	C₄	1	2	yes	—
11	C₅₃×#432	C₄	1	2	yes	—
12	C₅₃×#432	C₄	1	2	yes	—
13	C₅₃×#431	C₄	1	2	yes	—
14	C₅₃×#431	C₄	1	2	yes	—
15	C₅₃×#431	C₂²	1	2	yes	—
16	C₅₃×#431	C₂²	1	2	yes	—
17	C₅₃×#431	C₂²	1	2	yes	—
18	C₅₃×#431	C₂²	1	2	yes	—
19	C₅₃×#431	C₂²	1	2	yes	—
20	C₅₃×#431	C₂²	1	2	yes	—
21	C₅₃×#423	C₄	1	2	yes	—
22	C₅₃×#423	C₄	1	2	yes	—
23	C₂×C₄⋊C₈×C₅₃	C₄	1	2	yes	—
24	C₅₃×#407	C₄	1	2	yes	—
25	C₅₃×#407	C₄	1	2	yes	—
26	C₂×C₂²⋊C₈×C₅₃	C₄	1	2	yes	—
27	C₄×M₁₆×C₅₃	C₂²	1	2	yes	—
28	C₄×M₁₆×C₅₃	C₂²	1	2	yes	—
29	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
30	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
31	C₅₃×#322	C₂²	1	4	yes	complemented
32	C₅₃×#322	C₂²	1	8	yes	complemented
33	C₈²×C₅₃	C₂²	1	4	yes	complemented
34	C₂³×C₄×C₅₃	C₂×C₄	1	1	yes	characteristic
35	C₂³×C₄×C₅₃	C₂×C₄	1	1	yes	characteristic
36	C₂³×C₄×C₅₃	C₂×C₄	1	1	yes	characteristic
37	C₂×M₁₆×C₅₃	C₂×C₄	1	4	yes	—
38	C₂×M₁₆×C₅₃	C₂×C₄	1	4	yes	—
39	C₂²×C₈×C₅₃	C₂×C₄	1	2	yes	—
40	C₂²×C₈×C₅₃	C₂×C₄	1	2	yes	—
41	C₂×C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
42	C₂×C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
43	C₂×C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
44	C₂×C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
45	C₂×C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
46	C₂×C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
47	C₂×C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
48	C₂×C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
49	C₂×C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
50	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
51	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
52	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
53	C₄⋊C₈×C₅₃	C₂×C₄	1	4	yes	—
54	C₄⋊C₈×C₅₃	C₂×C₄	1	4	yes	—
55	C₄⋊C₈×C₅₃	C₂×C₄	1	4	yes	—
56	C₄⋊C₈×C₅₃	C₂×C₄	1	4	yes	—
57	C₄⋊C₈×C₅₃	C₂×C₄	1	4	yes	—
58	C₄⋊C₈×C₅₃	C₂×C₄	1	4	yes	—
59	C₂²⋊C₈×C₅₃	C₂×C₄	1	4	yes	—
60	C₂²⋊C₈×C₅₃	C₂×C₄	1	4	yes	—
61	C₂²⋊C₈×C₅₃	C₂×C₄	1	4	yes	—
62	C₂²⋊C₈×C₅₃	C₂×C₄	1	4	yes	—
63	C₂²⋊C₈×C₅₃	C₂×C₄	1	4	yes	—
64	C₂²⋊C₈×C₅₃	C₂×C₄	1	4	yes	—
65	#97×C₅₃	C₂×C₄	1	4	yes	—
66	#97×C₅₃	C₂×C₄	1	4	yes	—
67	C₄×C₈×C₅₃	C₂×C₄	1	2	yes	—
68	C₄×C₈×C₅₃	C₂×C₄	1	2	yes	—
69	C₄×C₈×C₅₃	C₂³	1	2	yes	—
70	C₄×C₈×C₅₃	C₂³	1	4	yes	—
71	C₄×C₈×C₅₃	C₂³	1	4	yes	—
72	C₄×C₈×C₅₃	C₂³	1	2	yes	—
73	C₂⁴×C₅₃	C₄²	1	1	yes	characteristic
74	C₂²×C₄×C₅₃	—	2	4	no	—
75	C₂²×C₄×C₅₃	—	4	2	no	—
76	C₂²×C₄×C₅₃	—	4	2	no	—
77	C₂²×C₄×C₅₃	C₄²	1	1	yes	characteristic
78	C₂²×C₄×C₅₃	C₄²	1	1	yes	characteristic
79	C₂²×C₄×C₅₃	C₄²	1	1	yes	characteristic
80	C₂²×C₄×C₅₃	C₄²	1	1	yes	characteristic
81	C₂²×C₄×C₅₃	C₄²	1	1	yes	characteristic
82	C₂²×C₄×C₅₃	C₄²	1	1	yes	characteristic
83	C₂²×C₄×C₅₃	C₄²	1	1	yes	characteristic
84	C₂²×C₄×C₅₃	C₄²	1	1	yes	characteristic
85	C₂²×C₄×C₅₃	C₄²	1	1	yes	characteristic
86	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
87	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
88	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
89	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
90	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
91	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
92	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
93	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
94	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
95	M₁₆×C₅₃	—	2	8	no	—
96	M₁₆×C₅₃	—	4	4	no	—
97	C₂×C₈×C₅₃	—	4	2	no	—
98	C₂×C₈×C₅₃	—	4	2	no	—
99	C₂×C₈×C₅₃	C₂²×C₄	1	4	yes	—
100	C₂×C₈×C₅₃	C₂²×C₄	1	4	yes	—
101	C₂×C₈×C₅₃	C₂²×C₄	1	8	yes	—
102	C₂×C₈×C₅₃	C₂²×C₄	1	8	yes	—
103	C₄²×C₅₃	C₄²	1	1	yes	characteristic
104	C₄²×C₅₃	C₄²	1	1	yes	characteristic
105	C₄²×C₅₃	C₄²	1	1	yes	characteristic
106	C₄²×C₅₃	C₄²	1	1	yes	characteristic
107	C₄²×C₅₃	C₄²	1	1	yes	characteristic
108	C₄²×C₅₃	C₄²	1	1	yes	characteristic
109	C₄²×C₅₃	C₂²×C₄	1	1	yes	characteristic
110	C₄²×C₅₃	C₂²×C₄	1	1	yes	characteristic
111	C₄²×C₅₃	C₂²×C₄	1	1	yes	characteristic
112	C₄²×C₅₃	C₂²×C₄	1	1	yes	characteristic
113	C₄²×C₅₃	C₂²×C₄	1	1	yes	characteristic
114	C₄²×C₅₃	C₂²×C₄	1	1	yes	characteristic
115	C₄²×C₅₃	C₂²×C₄	1	1	yes	characteristic
116	C₄²×C₅₃	C₂²×C₄	1	1	yes	characteristic
117	C₄²×C₅₃	C₂²×C₄	1	1	yes	characteristic
118	C₄²×C₅₃	C₂⁴	1	1	yes	characteristic
119	C₂³×C₅₃	—	2	2	no	—
120	C₂³×C₅₃	—	4	1	no	—
121	C₂³×C₅₃	—	4	1	no	—
122	C₂³×C₅₃	C₂×C₄²	1	1	yes	characteristic
123	C₂³×C₅₃	C₂×C₄²	1	1	yes	characteristic
124	C₂³×C₅₃	C₂×C₄²	1	1	yes	characteristic
125	C₂×C₄×C₅₃	—	2	2	no	—
126	C₂×C₄×C₅₃	—	2	2	no	—
127	C₂×C₄×C₅₃	—	2	2	no	—
128	C₂×C₄×C₅₃	—	2	2	no	—
129	C₂×C₄×C₅₃	—	2	2	no	—
130	C₂×C₄×C₅₃	—	2	2	no	—
131	C₂×C₄×C₅₃	—	2	2	no	—
132	C₂×C₄×C₅₃	—	2	2	no	—
133	C₂×C₄×C₅₃	—	2	2	no	—
134	C₂×C₄×C₅₃	—	2	2	no	—
135	C₂×C₄×C₅₃	—	2	2	no	—
136	C₂×C₄×C₅₃	—	4	1	no	—
137	C₂×C₄×C₅₃	—	4	1	no	—
138	C₂×C₄×C₅₃	—	4	1	no	—
139	C₂×C₄×C₅₃	—	4	1	no	—
140	C₂×C₄×C₅₃	—	4	1	no	—
141	C₂×C₄×C₅₃	—	4	1	no	—
142	C₂×C₄×C₅₃	C₂×C₄²	1	1	yes	characteristic
143	C₂×C₄×C₅₃	C₂×C₄²	1	1	yes	characteristic
144	C₂×C₄×C₅₃	C₂×C₄²	1	1	yes	characteristic
145	C₂×C₄×C₅₃	C₂×C₄²	1	1	yes	characteristic
146	C₂×C₄×C₅₃	C₂×C₄²	1	1	yes	characteristic
147	C₂×C₄×C₅₃	C₂×C₄²	1	1	yes	characteristic
148	C₂×C₄×C₅₃	C₂×C₄²	1	1	yes	characteristic
149	C₂×C₄×C₅₃	C₂×C₄²	1	1	yes	characteristic
150	C₂×C₄×C₅₃	C₂×C₄²	1	1	yes	characteristic
151	C₂×C₄×C₅₃	#131	1	4	yes	—
152	C₂×C₄×C₅₃	C₂³×C₄	1	1	yes	characteristic
153	C₂×C₄×C₅₃	C₂³×C₄	1	1	yes	characteristic
154	C₂×C₄×C₅₃	C₂³×C₄	1	1	yes	characteristic
155	C₈×C₅₃	—	2	8	no	—
156	C₈×C₅₃	—	2	4	no	—
157	C₈×C₅₃	—	2	4	no	—
158	C₈×C₅₃	—	2	8	no	—
159	#17324	C₅₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
160	C₂²×C₅₃	—	2	1	no	—
161	C₂²×C₅₃	—	2	1	no	—
162	C₂²×C₅₃	—	2	1	no	—
163	C₂²×C₅₃	—	2	1	no	—
164	C₂²×C₅₃	—	2	1	no	—
165	C₂²×C₅₃	—	2	1	no	—
166	C₂²×C₅₃	—	2	1	no	—
167	C₂²×C₅₃	—	2	1	no	—
168	C₂²×C₅₃	—	4	4	no	—
169	C₂²×C₅₃	#405	1	2	yes	—
170	C₂²×C₅₃	C₂²×C₄²	1	1	yes	characteristic
171	C₄×C₅₃	—	2	1	no	—
172	C₄×C₅₃	—	2	1	no	—
173	C₄×C₅₃	—	2	1	no	—
174	C₄×C₅₃	—	2	1	no	—
175	C₄×C₅₃	—	2	1	no	—
176	C₄×C₅₃	—	2	1	no	—
177	C₄×C₅₃	—	2	1	no	—
178	C₄×C₅₃	—	2	1	no	—
179	C₄×C₅₃	#405	1	2	yes	—
180	C₄×C₅₃	C₂×#131	1	2	yes	—
181	C₄×C₅₃	#568	1	2	yes	—
182	C₄×C₅₃	#568	1	2	yes	—
183	#2920	C₂×C₅₃	1	1	yes	characteristic
184	#2920	C₂×C₅₃	1	1	yes	characteristic
185	C₂×#431	C₂×C₅₃	1	1	yes	characteristic
186	#1454	C₂×C₅₃	1	4	yes	complemented
187	#1453	C₂×C₅₃	1	4	yes	complemented
188	#1451	C₂×C₅₃	1	2	yes	complemented
189	C₈×M₁₆	C₂×C₅₃	1	2	yes	complemented
190	C₂×C₅₃	—	2	2	no	—
191	C₂×C₅₃	—	4	1	no	—
192	C₂×C₅₃	—	4	1	no	—
193	C₂×C₅₃	C₂×#405	1	1	yes	characteristic
194	C₂×C₅₃	#2873	1	1	yes	characteristic
195	C₂×C₅₃	#2873	1	1	yes	characteristic
196	C₂²×C₄²	C₂²×C₅₃	1	1	yes	characteristic
197	#432	C₄×C₅₃	1	2	yes	—
198	#432	C₄×C₅₃	1	2	yes	—
199	#432	C₄×C₅₃	1	2	yes	—
200	#432	C₄×C₅₃	1	2	yes	—
201	#431	C₄×C₅₃	1	2	yes	—
202	#431	C₄×C₅₃	1	2	yes	—
203	#431	C₂²×C₅₃	1	2	yes	—
204	#431	C₂²×C₅₃	1	2	yes	—
205	#431	C₂²×C₅₃	1	2	yes	—
206	#431	C₂²×C₅₃	1	2	yes	—
207	#431	C₂²×C₅₃	1	2	yes	—
208	#431	C₂²×C₅₃	1	2	yes	—
209	#423	C₄×C₅₃	1	2	yes	—
210	#423	C₄×C₅₃	1	2	yes	—
211	C₂×C₄⋊C₈	C₄×C₅₃	1	2	yes	—
212	#407	C₄×C₅₃	1	2	yes	—
213	#407	C₄×C₅₃	1	2	yes	—
214	C₂×C₂²⋊C₈	C₄×C₅₃	1	2	yes	—
215	C₄×M₁₆	C₂²×C₅₃	1	2	yes	—
216	C₄×M₁₆	C₂²×C₅₃	1	2	yes	—
217	C₂×C₄×C₈	C₂²×C₅₃	1	1	yes	characteristic
218	C₂×C₄×C₈	C₂²×C₅₃	1	1	yes	characteristic
219	#322	C₂²×C₅₃	1	4	yes	complemented
220	#322	C₂²×C₅₃	1	8	yes	complemented
221	C₈²	C₂²×C₅₃	1	4	yes	complemented
222	C₅₃	#17324	1	1	yes	characteristic, direct product factor, Sylow
223	C₂³×C₄	C₂×C₄×C₅₃	1	1	yes	characteristic
224	C₂³×C₄	C₂×C₄×C₅₃	1	1	yes	characteristic
225	C₂³×C₄	C₂×C₄×C₅₃	1	1	yes	characteristic
226	C₂×M₁₆	C₂×C₄×C₅₃	1	4	yes	—
227	C₂×M₁₆	C₂×C₄×C₅₃	1	4	yes	—
228	C₂²×C₈	C₂×C₄×C₅₃	1	2	yes	—
229	C₂²×C₈	C₂×C₄×C₅₃	1	2	yes	—
230	C₂×C₄²	C₂×C₄×C₅₃	1	1	yes	characteristic
231	C₂×C₄²	C₂×C₄×C₅₃	1	1	yes	characteristic
232	C₂×C₄²	C₂×C₄×C₅₃	1	1	yes	characteristic
233	C₂×C₄²	C₂×C₄×C₅₃	1	1	yes	characteristic
234	C₂×C₄²	C₂×C₄×C₅₃	1	1	yes	characteristic
235	C₂×C₄²	C₂×C₄×C₅₃	1	1	yes	characteristic
236	C₂×C₄²	C₂×C₄×C₅₃	1	1	yes	characteristic
237	C₂×C₄²	C₂×C₄×C₅₃	1	1	yes	characteristic
238	C₂×C₄²	C₂×C₄×C₅₃	1	1	yes	characteristic
239	C₂×C₄²	C₂³×C₅₃	1	1	yes	characteristic
240	C₂×C₄²	C₂³×C₅₃	1	1	yes	characteristic
241	C₂×C₄²	C₂³×C₅₃	1	1	yes	characteristic
242	C₄⋊C₈	C₂×C₄×C₅₃	1	4	yes	—
243	C₄⋊C₈	C₂×C₄×C₅₃	1	4	yes	—
244	C₄⋊C₈	C₂×C₄×C₅₃	1	4	yes	—
245	C₄⋊C₈	C₂×C₄×C₅₃	1	4	yes	—
246	C₄⋊C₈	C₂×C₄×C₅₃	1	4	yes	—
247	C₄⋊C₈	C₂×C₄×C₅₃	1	4	yes	—
248	C₂²⋊C₈	C₂×C₄×C₅₃	1	4	yes	—
249	C₂²⋊C₈	C₂×C₄×C₅₃	1	4	yes	—
250	C₂²⋊C₈	C₂×C₄×C₅₃	1	4	yes	—
251	C₂²⋊C₈	C₂×C₄×C₅₃	1	4	yes	—
252	C₂²⋊C₈	C₂×C₄×C₅₃	1	4	yes	—
253	C₂²⋊C₈	C₂×C₄×C₅₃	1	4	yes	—
254	#97	C₂×C₄×C₅₃	1	4	yes	—
255	#97	C₂×C₄×C₅₃	1	4	yes	—
256	C₄×C₈	C₂×C₄×C₅₃	1	2	yes	—
257	C₄×C₈	C₂×C₄×C₅₃	1	2	yes	—
258	C₄×C₈	C₂³×C₅₃	1	2	yes	—
259	C₄×C₈	C₂³×C₅₃	1	4	yes	—
260	C₄×C₈	C₂³×C₅₃	1	4	yes	—
261	C₄×C₈	C₂³×C₅₃	1	2	yes	—
262	C₂⁴	C₄²×C₅₃	1	1	yes	characteristic
263	C₂²×C₄	—	2	4	no	—
264	C₂²×C₄	—	4	2	no	—
265	C₂²×C₄	—	4	2	no	—
266	C₂²×C₄	C₄²×C₅₃	1	1	yes	characteristic
267	C₂²×C₄	C₄²×C₅₃	1	1	yes	characteristic
268	C₂²×C₄	C₄²×C₅₃	1	1	yes	characteristic
269	C₂²×C₄	C₄²×C₅₃	1	1	yes	characteristic
270	C₂²×C₄	C₄²×C₅₃	1	1	yes	characteristic
271	C₂²×C₄	C₄²×C₅₃	1	1	yes	characteristic
272	C₂²×C₄	C₄²×C₅₃	1	1	yes	characteristic
273	C₂²×C₄	C₄²×C₅₃	1	1	yes	characteristic
274	C₂²×C₄	C₄²×C₅₃	1	1	yes	characteristic
275	C₂²×C₄	C₂²×C₄×C₅₃	1	1	yes	characteristic
276	C₂²×C₄	C₂²×C₄×C₅₃	1	1	yes	characteristic
277	C₂²×C₄	C₂²×C₄×C₅₃	1	1	yes	characteristic
278	C₂²×C₄	C₂²×C₄×C₅₃	1	1	yes	characteristic
279	C₂²×C₄	C₂²×C₄×C₅₃	1	1	yes	characteristic
280	C₂²×C₄	C₂²×C₄×C₅₃	1	1	yes	characteristic
281	C₂²×C₄	C₂²×C₄×C₅₃	1	1	yes	characteristic
282	C₂²×C₄	C₂²×C₄×C₅₃	1	1	yes	characteristic
283	C₂²×C₄	C₂²×C₄×C₅₃	1	1	yes	characteristic
284	M₁₆	—	2	8	no	—
285	M₁₆	—	4	4	no	—
286	C₂×C₈	—	4	2	no	—
287	C₂×C₈	—	4	2	no	—
288	C₂×C₈	C₂²×C₄×C₅₃	1	4	yes	—
289	C₂×C₈	C₂²×C₄×C₅₃	1	4	yes	—
290	C₂×C₈	C₂²×C₄×C₅₃	1	8	yes	—
291	C₂×C₈	C₂²×C₄×C₅₃	1	8	yes	—
292	C₄²	C₄²×C₅₃	1	1	yes	characteristic
293	C₄²	C₄²×C₅₃	1	1	yes	characteristic
294	C₄²	C₄²×C₅₃	1	1	yes	characteristic
295	C₄²	C₄²×C₅₃	1	1	yes	characteristic
296	C₄²	C₄²×C₅₃	1	1	yes	characteristic
297	C₄²	C₄²×C₅₃	1	1	yes	characteristic
298	C₄²	C₂²×C₄×C₅₃	1	1	yes	characteristic
299	C₄²	C₂²×C₄×C₅₃	1	1	yes	characteristic
300	C₄²	C₂²×C₄×C₅₃	1	1	yes	characteristic
301	C₄²	C₂²×C₄×C₅₃	1	1	yes	characteristic
302	C₄²	C₂²×C₄×C₅₃	1	1	yes	characteristic
303	C₄²	C₂²×C₄×C₅₃	1	1	yes	characteristic
304	C₄²	C₂²×C₄×C₅₃	1	1	yes	characteristic
305	C₄²	C₂²×C₄×C₅₃	1	1	yes	characteristic
306	C₄²	C₂²×C₄×C₅₃	1	1	yes	characteristic
307	C₄²	C₂⁴×C₅₃	1	1	yes	characteristic
308	C₂³	—	2	2	no	—
309	C₂³	—	4	1	no	—
310	C₂³	—	4	1	no	—
311	C₂³	C₂×C₄²×C₅₃	1	1	yes	characteristic
312	C₂³	C₂×C₄²×C₅₃	1	1	yes	characteristic
313	C₂³	C₂×C₄²×C₅₃	1	1	yes	characteristic
314	C₂×C₄	—	2	2	no	—
315	C₂×C₄	—	2	2	no	—
316	C₂×C₄	—	2	2	no	—
317	C₂×C₄	—	2	2	no	—
318	C₂×C₄	—	2	2	no	—
319	C₂×C₄	—	2	2	no	—
320	C₂×C₄	—	2	2	no	—
321	C₂×C₄	—	2	2	no	—
322	C₂×C₄	—	2	2	no	—
323	C₂×C₄	—	2	2	no	—
324	C₂×C₄	—	2	2	no	—
325	C₂×C₄	—	4	1	no	—
326	C₂×C₄	—	4	1	no	—
327	C₂×C₄	—	4	1	no	—
328	C₂×C₄	—	4	1	no	—
329	C₂×C₄	—	4	1	no	—
330	C₂×C₄	—	4	1	no	—
331	C₂×C₄	C₂×C₄²×C₅₃	1	1	yes	characteristic
332	C₂×C₄	C₂×C₄²×C₅₃	1	1	yes	characteristic
333	C₂×C₄	C₂×C₄²×C₅₃	1	1	yes	characteristic
334	C₂×C₄	C₂×C₄²×C₅₃	1	1	yes	characteristic
335	C₂×C₄	C₂×C₄²×C₅₃	1	1	yes	characteristic
336	C₂×C₄	C₂×C₄²×C₅₃	1	1	yes	characteristic
337	C₂×C₄	C₂×C₄²×C₅₃	1	1	yes	characteristic
338	C₂×C₄	C₂×C₄²×C₅₃	1	1	yes	characteristic
339	C₂×C₄	C₂×C₄²×C₅₃	1	1	yes	characteristic
340	C₂×C₄	#131×C₅₃	1	4	yes	—
341	C₂×C₄	C₂³×C₄×C₅₃	1	1	yes	characteristic
342	C₂×C₄	C₂³×C₄×C₅₃	1	1	yes	characteristic
343	C₂×C₄	C₂³×C₄×C₅₃	1	1	yes	characteristic
344	C₈	—	2	8	no	—
345	C₈	—	2	4	no	—
346	C₈	—	2	4	no	—
347	C₈	—	2	8	no	—
348	C₂²	—	2	1	no	—
349	C₂²	—	2	1	no	—
350	C₂²	—	2	1	no	—
351	C₂²	—	2	1	no	—
352	C₂²	—	2	1	no	—
353	C₂²	—	2	1	no	—
354	C₂²	—	2	1	no	—
355	C₂²	—	2	1	no	—
356	C₂²	—	4	4	no	—
357	C₂²	C₅₃×#405	1	2	yes	—
358	C₂²	C₂²×C₄²×C₅₃	1	1	yes	characteristic
359	C₄	—	2	1	no	—
360	C₄	—	2	1	no	—
361	C₄	—	2	1	no	—
362	C₄	—	2	1	no	—
363	C₄	—	2	1	no	—
364	C₄	—	2	1	no	—
365	C₄	—	2	1	no	—
366	C₄	—	2	1	no	—
367	C₄	C₅₃×#405	1	2	yes	—
368	C₄	C₂×#131×C₅₃	1	2	yes	—
369	C₄	C₅₃×#568	1	2	yes	—
370	C₄	C₅₃×#568	1	2	yes	—
371	C₂	—	2	2	no	—
372	C₂	—	4	1	no	—
373	C₂	—	4	1	no	—
374	C₂	C₂×C₅₃×#405	1	1	yes	characteristic
375	C₂	C₅₃×#2873	1	1	yes	characteristic
376	C₂	C₅₃×#2873	1	1	yes	characteristic

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

Complements

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

N	H ≅ G/N	type
4. C₅₃×#1454	C₂	semidirect
5. C₅₃×#1453	C₂	semidirect
6. C₅₃×#1451	C₂	semidirect
7. C₈×M₁₆×C₅₃	C₂	semidirect
31. C₅₃×#322	C₂²	semidirect
32. C₅₃×#322	C₂²	semidirect
33. C₈²×C₅₃	C₂²	semidirect
159. #17324	C₅₃	direct (N × H)
186. #1454	C₂×C₅₃	semidirect
187. #1453	C₂×C₅₃	semidirect
188. #1451	C₂×C₅₃	semidirect
189. C₈×M₁₆	C₂×C₅₃	semidirect
219. #322	C₂²×C₅₃	semidirect
220. #322	C₂²×C₅₃	semidirect
221. C₈²	C₂²×C₅₃	semidirect
222. C₅₃	#17324	direct (N × H)

Maximal subgroups

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

subgroup	index
C₅₃×#2920	2
C₂×C₅₃×#431	2
C₅₃×#1454	2
C₅₃×#1453	2
C₅₃×#1451	2
C₈×M₁₆×C₅₃	2
#17324	53

Subgroups — C53×#17324 #27224836

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅₃×#17324 #27224836