Subgroups — C₅₃×#21492 #27229004

Order 13568.

1058 conjugacy classes of subgroups.

The subgroup lattice has 518 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₅₃×#435	C₂	1	2	yes	maximal, complemented
2	C₂×C₈×D₄×C₅₃	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×C₅₃×#405	C₂	1	1	yes	maximal, characteristic, complemented
4	C₅₃×#2392	C₂	1	1	yes	maximal, characteristic
5	C₅₃×#1955	C₂	1	1	yes	maximal, characteristic, complemented
6	C₅₃×#1940	C₂	1	1	yes	maximal, characteristic, complemented
7	C₅₃×#1923	C₂	1	2	yes	maximal, complemented
8	C₅₃×#1869	C₂	1	2	yes	maximal, complemented
9	C₅₃×#1798	C₂	1	1	yes	maximal, characteristic, complemented
10	C₅₃×#1770	C₂	1	1	yes	maximal, characteristic, complemented
11	C₅₃×#1752	C₂	1	2	yes	maximal, complemented
12	C₂²×M₁₆×C₅₃	C₂²	1	1	yes	characteristic
13	C₂³×C₈×C₅₃	C₂²	1	2	yes	—
14	C₂³×C₈×C₅₃	C₂²	1	1	yes	characteristic
15	C₂×C₄×D₄×C₅₃	C₂²	1	1	yes	characteristic
16	C₂×C₄×D₄×C₅₃	C₂²	1	2	yes	—
17	C₂×#117×C₅₃	C₂²	1	1	yes	characteristic
18	C₅₃×#435	—	2	8	no	—
19	C₈×D₄×C₅₃	—	2	4	no	—
20	C₂×C₄⋊C₈×C₅₃	C₂²	1	2	yes	—
21	C₂×C₄⋊C₈×C₅₃	C₂²	1	1	yes	characteristic
22	C₂×C₄⋊C₈×C₅₃	C₂²	1	1	yes	characteristic
23	C₂×C₂²⋊C₈×C₅₃	C₂²	1	2	yes	—
24	C₂×C₂²⋊C₈×C₅₃	C₂²	1	2	yes	—
25	C₂×C₂²⋊C₈×C₅₃	C₂²	1	2	yes	—
26	C₂×C₂²⋊C₈×C₅₃	C₂²	1	2	yes	—
27	C₅₃×#405	—	2	4	no	—
28	C₂×#97×C₅₃	C₂²	1	1	yes	characteristic
29	C₂×#97×C₅₃	C₂²	1	1	yes	characteristic
30	C₂×#97×C₅₃	C₂²	1	2	yes	—
31	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
32	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
33	C₅₃×#398	C₄	1	1	yes	characteristic
34	C₅₃×#397	C₄	1	1	yes	characteristic
35	C₅₃×#396	C₄	1	2	yes	—
36	C₅₃×#394	C₄	1	2	yes	—
37	C₅₃×#394	C₄	1	1	yes	characteristic
38	C₅₃×#392	C₄	1	1	yes	characteristic
39	C₄×C₄⋊C₄×C₅₃	C₂²	1	1	yes	characteristic
40	C₄×C₂²⋊C₄×C₅₃	C₂²	1	2	yes	—
41	C₅₃×#336	C₂²	1	2	yes	—
42	C₅₃×#336	C₂²	1	2	yes	—
43	C₅₃×#336	C₂²	1	2	yes	—
44	C₂²×D₄×C₅₃	C₂×C₄	1	1	yes	characteristic
45	C₂²×D₄×C₅₃	C₂×C₄	1	2	yes	—
46	C₂³×C₄×C₅₃	C₂³	1	2	yes	—
47	C₂³×C₄×C₅₃	C₂³	1	1	yes	characteristic
48	C₂×M₁₆×C₅₃	—	2	2	no	—
49	C₂×M₁₆×C₅₃	—	2	2	no	—
50	C₂×M₁₆×C₅₃	D₄	1	4	yes	complemented
51	C₂²×C₈×C₅₃	—	2	2	no	—
52	C₂²×C₈×C₅₃	—	2	2	no	—
53	C₂²×C₈×C₅₃	—	2	4	no	—
54	C₂²×C₈×C₅₃	—	2	4	no	—
55	C₂²×C₈×C₅₃	—	2	4	no	—
56	C₂²×C₈×C₅₃	—	2	2	no	—
57	C₂²×C₈×C₅₃	C₂³	1	2	yes	—
58	C₂²×C₈×C₅₃	C₂³	1	2	yes	—
59	C₂²×C₈×C₅₃	C₂³	1	1	yes	characteristic
60	C₂²×C₈×C₅₃	C₂³	1	1	yes	characteristic
61	C₂²×C₈×C₅₃	C₂³	1	1	yes	characteristic
62	C₂²×C₈×C₅₃	C₂³	1	1	yes	characteristic
63	C₄×D₄×C₅₃	—	2	2	no	—
64	C₄×D₄×C₅₃	—	2	2	no	—
65	C₄×D₄×C₅₃	—	2	4	no	—
66	C₄×D₄×C₅₃	—	2	4	no	—
67	#117×C₅₃	—	2	2	no	—
68	#117×C₅₃	—	2	2	no	—
69	C₂×C₄⋊C₄×C₅₃	C₂×C₄	1	1	yes	characteristic
70	C₂×C₄⋊C₄×C₅₃	C₂×C₄	1	1	yes	characteristic
71	C₂×C₄⋊C₄×C₅₃	C₂×C₄	1	2	yes	—
72	C₂×C₄⋊C₄×C₅₃	C₂×C₄	1	1	yes	characteristic
73	C₂×C₄⋊C₄×C₅₃	C₂×C₄	1	2	yes	—
74	C₂×C₂²⋊C₄×C₅₃	C₂×C₄	1	1	yes	characteristic
75	C₂×C₂²⋊C₄×C₅₃	C₂×C₄	1	1	yes	characteristic
76	C₂×C₂²⋊C₄×C₅₃	C₂×C₄	1	2	yes	—
77	C₂×C₂²⋊C₄×C₅₃	C₂×C₄	1	2	yes	—
78	C₂×C₂²⋊C₄×C₅₃	C₂×C₄	1	2	yes	—
79	C₂×C₂²⋊C₄×C₅₃	C₂×C₄	1	2	yes	—
80	C₂×C₂²⋊C₄×C₅₃	C₂×C₄	1	2	yes	—
81	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
82	C₂×C₄²×C₅₃	C₂³	1	2	yes	—
83	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
84	C₄⋊C₈×C₅₃	—	2	4	no	—
85	C₄⋊C₈×C₅₃	—	2	2	no	—
86	C₄⋊C₈×C₅₃	—	2	2	no	—
87	C₂²⋊C₈×C₅₃	—	2	4	no	—
88	C₂²⋊C₈×C₅₃	—	2	4	no	—
89	C₂²⋊C₈×C₅₃	—	2	4	no	—
90	C₂²⋊C₈×C₅₃	—	2	4	no	—
91	#97×C₅₃	—	2	2	no	—
92	#97×C₅₃	—	2	4	no	—
93	#97×C₅₃	—	2	2	no	—
94	C₄×C₈×C₅₃	—	2	2	no	—
95	C₄×C₈×C₅₃	D₄	1	4	yes	complemented
96	#95×C₅₃	C₂×C₄	1	2	yes	—
97	#95×C₅₃	C₂×C₄	1	2	yes	—
98	#95×C₅₃	C₂×C₄	1	1	yes	characteristic
99	#95×C₅₃	C₂×C₄	1	1	yes	characteristic
100	C₂⁴×C₅₃	C₂²×C₄	1	1	yes	characteristic
101	C₂⁴×C₅₃	C₂²×C₄	1	2	yes	—
102	C₂×D₄×C₅₃	—	2	4	no	—
103	C₂×D₄×C₅₃	—	2	4	no	—
104	C₂×D₄×C₅₃	—	2	4	no	—
105	C₂×D₄×C₅₃	—	2	2	no	—
106	C₂×D₄×C₅₃	—	4	2	no	—
107	C₂²×C₄×C₅₃	—	2	1	no	—
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	2	no	—
112	C₂²×C₄×C₅₃	—	2	2	no	—
113	C₂²×C₄×C₅₃	—	2	2	no	—
114	C₂²×C₄×C₅₃	—	2	2	no	—
115	C₂²×C₄×C₅₃	—	2	2	no	—
116	C₂²×C₄×C₅₃	—	2	2	no	—
117	C₂²×C₄×C₅₃	C₂²×C₄	1	2	yes	—
118	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
119	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
120	C₂²×C₄×C₅₃	C₂²×C₄	1	2	yes	—
121	C₂²×C₄×C₅₃	C₂²×C₄	1	2	yes	—
122	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
123	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
124	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
125	C₂²×C₄×C₅₃	C₂×D₄	1	2	yes	—
126	C₂²×C₄×C₅₃	#41	1	2	yes	—
127	C₂²×C₄×C₅₃	C₂⁴	1	1	yes	characteristic
128	M₁₆×C₅₃	—	2	8	no	—
129	C₂×C₈×C₅₃	—	2	8	no	—
130	C₂×C₈×C₅₃	—	2	1	no	—
131	C₂×C₈×C₅₃	—	2	1	no	—
132	C₂×C₈×C₅₃	—	2	1	no	—
133	C₂×C₈×C₅₃	—	2	1	no	—
134	C₂×C₈×C₅₃	—	2	1	no	—
135	C₂×C₈×C₅₃	—	2	1	no	—
136	C₂×C₈×C₅₃	—	2	2	no	—
137	C₂×C₈×C₅₃	—	2	2	no	—
138	C₂×C₈×C₅₃	—	2	2	no	—
139	C₂×C₈×C₅₃	—	2	2	no	—
140	C₂×C₈×C₅₃	—	2	2	no	—
141	C₂×C₈×C₅₃	—	2	2	no	—
142	C₂×C₈×C₅₃	—	4	8	no	—
143	C₂×C₈×C₅₃	C₂×D₄	1	2	yes	—
144	C₂×C₈×C₅₃	C₂×D₄	1	2	yes	—
145	C₂×C₈×C₅₃	C₂×D₄	1	2	yes	—
146	C₂×C₈×C₅₃	C₂×D₄	1	2	yes	—
147	C₂×C₈×C₅₃	#41	1	2	yes	—
148	C₂×C₈×C₅₃	#41	1	2	yes	—
149	C₄⋊C₄×C₅₃	—	2	2	no	—
150	C₄⋊C₄×C₅₃	—	2	2	no	—
151	C₄⋊C₄×C₅₃	—	2	2	no	—
152	C₄⋊C₄×C₅₃	—	2	4	no	—
153	C₄⋊C₄×C₅₃	—	4	2	no	—
154	C₂²⋊C₄×C₅₃	—	2	4	no	—
155	C₂²⋊C₄×C₅₃	—	2	4	no	—
156	C₂²⋊C₄×C₅₃	—	2	2	no	—
157	C₂²⋊C₄×C₅₃	—	2	2	no	—
158	C₂²⋊C₄×C₅₃	—	2	4	no	—
159	C₂²⋊C₄×C₅₃	—	2	4	no	—
160	C₂²⋊C₄×C₅₃	—	4	2	no	—
161	C₄²×C₅₃	—	2	1	no	—
162	C₄²×C₅₃	—	2	2	no	—
163	C₄²×C₅₃	—	2	1	no	—
164	C₄²×C₅₃	—	2	2	no	—
165	C₄²×C₅₃	C₂×D₄	1	2	yes	—
166	C₄²×C₅₃	#41	1	2	yes	—
167	C₂³×C₅₃	—	2	2	no	—
168	C₂³×C₅₃	—	2	2	no	—
169	C₂³×C₅₃	—	2	1	no	—
170	C₂³×C₅₃	—	2	2	no	—
171	C₂³×C₅₃	—	2	1	no	—
172	C₂³×C₅₃	—	2	2	no	—
173	C₂³×C₅₃	—	2	2	no	—
174	C₂³×C₅₃	—	2	2	no	—
175	C₂³×C₅₃	—	2	2	no	—
176	C₂³×C₅₃	—	2	2	no	—
177	C₂³×C₅₃	C₄×D₄	1	2	yes	—
178	C₂³×C₅₃	#131	1	4	yes	—
179	C₂³×C₅₃	C₂³×C₄	1	1	yes	characteristic
180	D₄×C₅₃	—	2	8	no	—
181	D₄×C₅₃	—	4	8	no	—
182	C₂×C₄×C₅₃	—	2	1	no	—
183	C₂×C₄×C₅₃	—	2	1	no	—
184	C₂×C₄×C₅₃	—	2	2	no	—
185	C₂×C₄×C₅₃	—	2	2	no	—
186	C₂×C₄×C₅₃	—	2	2	no	—
187	C₂×C₄×C₅₃	—	2	2	no	—
188	C₂×C₄×C₅₃	—	2	2	no	—
189	C₂×C₄×C₅₃	—	2	2	no	—
190	C₂×C₄×C₅₃	—	2	2	no	—
191	C₂×C₄×C₅₃	—	2	2	no	—
192	C₂×C₄×C₅₃	—	2	2	no	—
193	C₂×C₄×C₅₃	—	2	2	no	—
194	C₂×C₄×C₅₃	—	2	1	no	—
195	C₂×C₄×C₅₃	—	2	1	no	—
196	C₂×C₄×C₅₃	—	2	2	no	—
197	C₂×C₄×C₅₃	—	2	2	no	—
198	C₂×C₄×C₅₃	—	2	2	no	—
199	C₂×C₄×C₅₃	—	2	2	no	—
200	C₂×C₄×C₅₃	—	2	2	no	—
201	C₂×C₄×C₅₃	—	2	2	no	—
202	C₂×C₄×C₅₃	—	2	2	no	—
203	C₂×C₄×C₅₃	—	4	2	no	—
204	C₂×C₄×C₅₃	—	4	2	no	—
205	C₂×C₄×C₅₃	—	4	2	no	—
206	C₂×C₄×C₅₃	—	4	2	no	—
207	C₂×C₄×C₅₃	C₄×D₄	1	2	yes	—
208	C₂×C₄×C₅₃	C₄×D₄	1	2	yes	—
209	C₂×C₄×C₅₃	C₄×D₄	1	2	yes	—
210	C₂×C₄×C₅₃	C₂²×D₄	1	1	yes	characteristic
211	C₂×C₄×C₅₃	C₂²×D₄	1	1	yes	characteristic
212	C₂×C₄×C₅₃	C₂×#41	1	1	yes	characteristic
213	C₂×C₄×C₅₃	C₂×#41	1	1	yes	characteristic
214	C₂×C₄×C₅₃	C₂×#41	1	1	yes	characteristic
215	C₂×C₄×C₅₃	C₂×#41	1	1	yes	characteristic
216	C₈×C₅₃	—	2	2	no	—
217	C₈×C₅₃	—	2	2	no	—
218	C₈×C₅₃	—	2	2	no	—
219	C₈×C₅₃	—	4	2	no	—
220	C₈×C₅₃	—	4	2	no	—
221	C₈×C₅₃	#121	1	4	yes	—
222	#21492	C₅₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
223	C₂²×C₅₃	—	2	4	no	—
224	C₂²×C₅₃	—	2	4	no	—
225	C₂²×C₅₃	—	2	4	no	—
226	C₂²×C₅₃	—	2	4	no	—
227	C₂²×C₅₃	—	2	2	no	—
228	C₂²×C₅₃	—	4	1	no	—
229	C₂²×C₅₃	—	4	2	no	—
230	C₂²×C₅₃	—	4	1	no	—
231	C₂²×C₅₃	—	4	2	no	—
232	C₂²×C₅₃	—	4	2	no	—
233	C₂²×C₅₃	—	4	2	no	—
234	C₂²×C₅₃	—	4	2	no	—
235	C₂²×C₅₃	C₂×C₄×D₄	1	1	yes	characteristic
236	C₂²×C₅₃	C₂×C₄×D₄	1	1	yes	characteristic
237	C₂²×C₅₃	C₄×#41	1	1	yes	characteristic
238	C₂²×C₅₃	C₂×#131	1	2	yes	—
239	C₂²×C₅₃	#568	1	2	yes	—
240	C₄×C₅₃	—	2	2	no	—
241	C₄×C₅₃	—	2	2	no	—
242	C₄×C₅₃	—	2	2	no	—
243	C₄×C₅₃	—	4	1	no	—
244	C₄×C₅₃	—	4	1	no	—
245	C₄×C₅₃	—	4	2	no	—
246	C₄×C₅₃	—	4	2	no	—
247	C₄×C₅₃	—	4	2	no	—
248	C₄×C₅₃	C₂×#121	1	1	yes	characteristic
249	C₄×C₅₃	#525	1	1	yes	characteristic
250	C₄×C₅₃	#529	1	1	yes	characteristic
251	C₄×C₅₃	#532	1	1	yes	characteristic
252	C₂×#435	C₂×C₅₃	1	2	yes	complemented
253	C₂×C₈×D₄	C₂×C₅₃	1	1	yes	characteristic, complemented
254	C₂×#405	C₂×C₅₃	1	1	yes	characteristic, complemented
255	#2392	C₂×C₅₃	1	1	yes	characteristic
256	#1955	C₂×C₅₃	1	1	yes	characteristic, complemented
257	#1940	C₂×C₅₃	1	1	yes	characteristic, complemented
258	#1923	C₂×C₅₃	1	2	yes	complemented
259	#1869	C₂×C₅₃	1	2	yes	complemented
260	#1798	C₂×C₅₃	1	1	yes	characteristic, complemented
261	#1770	C₂×C₅₃	1	1	yes	characteristic, complemented
262	#1752	C₂×C₅₃	1	2	yes	complemented
263	C₂×C₅₃	—	4	4	no	—
264	C₂×C₅₃	—	4	2	no	—
265	C₂×C₅₃	C₄×#121	1	1	yes	characteristic
266	C₂×C₅₃	#2929	1	1	yes	characteristic
267	C₂×C₅₃	#2932	1	1	yes	characteristic
268	C₂×C₅₃	#2933	1	2	yes	—
269	C₂×C₅₃	#2968	1	2	yes	—
270	C₂²×M₁₆	C₂²×C₅₃	1	1	yes	characteristic
271	C₂³×C₈	C₂²×C₅₃	1	1	yes	characteristic
272	C₂³×C₈	C₂²×C₅₃	1	2	yes	—
273	C₂×C₄×D₄	C₂²×C₅₃	1	2	yes	—
274	C₂×C₄×D₄	C₂²×C₅₃	1	1	yes	characteristic
275	C₂×#117	C₂²×C₅₃	1	1	yes	characteristic
276	#435	—	2	8	no	—
277	C₈×D₄	—	2	4	no	—
278	C₂×C₄⋊C₈	C₂²×C₅₃	1	1	yes	characteristic
279	C₂×C₄⋊C₈	C₂²×C₅₃	1	1	yes	characteristic
280	C₂×C₄⋊C₈	C₂²×C₅₃	1	2	yes	—
281	C₂×C₂²⋊C₈	C₂²×C₅₃	1	2	yes	—
282	C₂×C₂²⋊C₈	C₂²×C₅₃	1	2	yes	—
283	C₂×C₂²⋊C₈	C₂²×C₅₃	1	2	yes	—
284	C₂×C₂²⋊C₈	C₂²×C₅₃	1	2	yes	—
285	#405	—	2	4	no	—
286	C₂×#97	C₂²×C₅₃	1	1	yes	characteristic
287	C₂×#97	C₂²×C₅₃	1	1	yes	characteristic
288	C₂×#97	C₂²×C₅₃	1	2	yes	—
289	C₂×C₄×C₈	C₂²×C₅₃	1	1	yes	characteristic
290	C₂×C₄×C₈	C₂²×C₅₃	1	1	yes	characteristic
291	#398	C₄×C₅₃	1	1	yes	characteristic
292	#397	C₄×C₅₃	1	1	yes	characteristic
293	#396	C₄×C₅₃	1	2	yes	—
294	#394	C₄×C₅₃	1	2	yes	—
295	#394	C₄×C₅₃	1	1	yes	characteristic
296	#392	C₄×C₅₃	1	1	yes	characteristic
297	C₄×C₄⋊C₄	C₂²×C₅₃	1	1	yes	characteristic
298	C₄×C₂²⋊C₄	C₂²×C₅₃	1	2	yes	—
299	#336	C₂²×C₅₃	1	2	yes	—
300	#336	C₂²×C₅₃	1	2	yes	—
301	#336	C₂²×C₅₃	1	2	yes	—
302	C₅₃	#21492	1	1	yes	characteristic, direct product factor, Sylow
303	C₂²×D₄	C₂×C₄×C₅₃	1	1	yes	characteristic
304	C₂²×D₄	C₂×C₄×C₅₃	1	2	yes	—
305	C₂³×C₄	C₂³×C₅₃	1	2	yes	—
306	C₂³×C₄	C₂³×C₅₃	1	1	yes	characteristic
307	C₂×M₁₆	—	2	2	no	—
308	C₂×M₁₆	—	2	2	no	—
309	C₂×M₁₆	D₄×C₅₃	1	4	yes	complemented
310	C₂²×C₈	—	2	2	no	—
311	C₂²×C₈	—	2	2	no	—
312	C₂²×C₈	—	2	2	no	—
313	C₂²×C₈	—	2	4	no	—
314	C₂²×C₈	—	2	4	no	—
315	C₂²×C₈	—	2	4	no	—
316	C₂²×C₈	C₂³×C₅₃	1	2	yes	—
317	C₂²×C₈	C₂³×C₅₃	1	2	yes	—
318	C₂²×C₈	C₂³×C₅₃	1	1	yes	characteristic
319	C₂²×C₈	C₂³×C₅₃	1	1	yes	characteristic
320	C₂²×C₈	C₂³×C₅₃	1	1	yes	characteristic
321	C₂²×C₈	C₂³×C₅₃	1	1	yes	characteristic
322	C₄×D₄	—	2	4	no	—
323	C₄×D₄	—	2	4	no	—
324	C₄×D₄	—	2	2	no	—
325	C₄×D₄	—	2	2	no	—
326	#117	—	2	2	no	—
327	#117	—	2	2	no	—
328	C₂×C₄⋊C₄	C₂×C₄×C₅₃	1	1	yes	characteristic
329	C₂×C₄⋊C₄	C₂×C₄×C₅₃	1	1	yes	characteristic
330	C₂×C₄⋊C₄	C₂×C₄×C₅₃	1	2	yes	—
331	C₂×C₄⋊C₄	C₂×C₄×C₅₃	1	1	yes	characteristic
332	C₂×C₄⋊C₄	C₂×C₄×C₅₃	1	2	yes	—
333	C₂×C₂²⋊C₄	C₂×C₄×C₅₃	1	1	yes	characteristic
334	C₂×C₂²⋊C₄	C₂×C₄×C₅₃	1	1	yes	characteristic
335	C₂×C₂²⋊C₄	C₂×C₄×C₅₃	1	2	yes	—
336	C₂×C₂²⋊C₄	C₂×C₄×C₅₃	1	2	yes	—
337	C₂×C₂²⋊C₄	C₂×C₄×C₅₃	1	2	yes	—
338	C₂×C₂²⋊C₄	C₂×C₄×C₅₃	1	2	yes	—
339	C₂×C₂²⋊C₄	C₂×C₄×C₅₃	1	2	yes	—
340	C₂×C₄²	C₂³×C₅₃	1	1	yes	characteristic
341	C₂×C₄²	C₂³×C₅₃	1	2	yes	—
342	C₂×C₄²	C₂³×C₅₃	1	1	yes	characteristic
343	C₄⋊C₈	—	2	4	no	—
344	C₄⋊C₈	—	2	2	no	—
345	C₄⋊C₈	—	2	2	no	—
346	C₂²⋊C₈	—	2	4	no	—
347	C₂²⋊C₈	—	2	4	no	—
348	C₂²⋊C₈	—	2	4	no	—
349	C₂²⋊C₈	—	2	4	no	—
350	#97	—	2	2	no	—
351	#97	—	2	4	no	—
352	#97	—	2	2	no	—
353	C₄×C₈	—	2	2	no	—
354	C₄×C₈	D₄×C₅₃	1	4	yes	complemented
355	#95	C₂×C₄×C₅₃	1	2	yes	—
356	#95	C₂×C₄×C₅₃	1	2	yes	—
357	#95	C₂×C₄×C₅₃	1	1	yes	characteristic
358	#95	C₂×C₄×C₅₃	1	1	yes	characteristic
359	C₂⁴	C₂²×C₄×C₅₃	1	1	yes	characteristic
360	C₂⁴	C₂²×C₄×C₅₃	1	2	yes	—
361	C₂×D₄	—	2	4	no	—
362	C₂×D₄	—	2	4	no	—
363	C₂×D₄	—	2	4	no	—
364	C₂×D₄	—	2	2	no	—
365	C₂×D₄	—	4	2	no	—
366	C₂²×C₄	—	2	1	no	—
367	C₂²×C₄	—	2	1	no	—
368	C₂²×C₄	—	2	1	no	—
369	C₂²×C₄	—	2	1	no	—
370	C₂²×C₄	—	2	2	no	—
371	C₂²×C₄	—	2	2	no	—
372	C₂²×C₄	—	2	2	no	—
373	C₂²×C₄	—	2	2	no	—
374	C₂²×C₄	—	2	2	no	—
375	C₂²×C₄	—	2	2	no	—
376	C₂²×C₄	C₂²×C₄×C₅₃	1	2	yes	—
377	C₂²×C₄	C₂²×C₄×C₅₃	1	1	yes	characteristic
378	C₂²×C₄	C₂²×C₄×C₅₃	1	1	yes	characteristic
379	C₂²×C₄	C₂²×C₄×C₅₃	1	2	yes	—
380	C₂²×C₄	C₂²×C₄×C₅₃	1	2	yes	—
381	C₂²×C₄	C₂²×C₄×C₅₃	1	1	yes	characteristic
382	C₂²×C₄	C₂²×C₄×C₅₃	1	1	yes	characteristic
383	C₂²×C₄	C₂²×C₄×C₅₃	1	1	yes	characteristic
384	C₂²×C₄	C₂×D₄×C₅₃	1	2	yes	—
385	C₂²×C₄	#41×C₅₃	1	2	yes	—
386	C₂²×C₄	C₂⁴×C₅₃	1	1	yes	characteristic
387	M₁₆	—	2	8	no	—
388	C₂×C₈	—	2	8	no	—
389	C₂×C₈	—	2	1	no	—
390	C₂×C₈	—	2	1	no	—
391	C₂×C₈	—	2	1	no	—
392	C₂×C₈	—	2	1	no	—
393	C₂×C₈	—	2	1	no	—
394	C₂×C₈	—	2	1	no	—
395	C₂×C₈	—	2	2	no	—
396	C₂×C₈	—	2	2	no	—
397	C₂×C₈	—	2	2	no	—
398	C₂×C₈	—	2	2	no	—
399	C₂×C₈	—	2	2	no	—
400	C₂×C₈	—	2	2	no	—
401	C₂×C₈	—	4	8	no	—
402	C₂×C₈	C₂×D₄×C₅₃	1	2	yes	—
403	C₂×C₈	C₂×D₄×C₅₃	1	2	yes	—
404	C₂×C₈	C₂×D₄×C₅₃	1	2	yes	—
405	C₂×C₈	C₂×D₄×C₅₃	1	2	yes	—
406	C₂×C₈	#41×C₅₃	1	2	yes	—
407	C₂×C₈	#41×C₅₃	1	2	yes	—
408	C₄⋊C₄	—	2	2	no	—
409	C₄⋊C₄	—	2	2	no	—
410	C₄⋊C₄	—	2	4	no	—
411	C₄⋊C₄	—	2	2	no	—
412	C₄⋊C₄	—	4	2	no	—
413	C₂²⋊C₄	—	2	4	no	—
414	C₂²⋊C₄	—	2	4	no	—
415	C₂²⋊C₄	—	2	2	no	—
416	C₂²⋊C₄	—	2	2	no	—
417	C₂²⋊C₄	—	2	4	no	—
418	C₂²⋊C₄	—	2	4	no	—
419	C₂²⋊C₄	—	4	2	no	—
420	C₄²	—	2	1	no	—
421	C₄²	—	2	2	no	—
422	C₄²	—	2	1	no	—
423	C₄²	—	2	2	no	—
424	C₄²	C₂×D₄×C₅₃	1	2	yes	—
425	C₄²	#41×C₅₃	1	2	yes	—
426	C₂³	—	2	1	no	—
427	C₂³	—	2	2	no	—
428	C₂³	—	2	2	no	—
429	C₂³	—	2	2	no	—
430	C₂³	—	2	1	no	—
431	C₂³	—	2	2	no	—
432	C₂³	—	2	2	no	—
433	C₂³	—	2	2	no	—
434	C₂³	—	2	2	no	—
435	C₂³	—	2	2	no	—
436	C₂³	C₄×D₄×C₅₃	1	2	yes	—
437	C₂³	#131×C₅₃	1	4	yes	—
438	C₂³	C₂³×C₄×C₅₃	1	1	yes	characteristic
439	D₄	—	2	8	no	—
440	D₄	—	4	8	no	—
441	C₂×C₄	—	2	1	no	—
442	C₂×C₄	—	2	1	no	—
443	C₂×C₄	—	2	2	no	—
444	C₂×C₄	—	2	2	no	—
445	C₂×C₄	—	2	2	no	—
446	C₂×C₄	—	2	2	no	—
447	C₂×C₄	—	2	2	no	—
448	C₂×C₄	—	2	2	no	—
449	C₂×C₄	—	2	2	no	—
450	C₂×C₄	—	2	1	no	—
451	C₂×C₄	—	2	1	no	—
452	C₂×C₄	—	2	2	no	—
453	C₂×C₄	—	2	2	no	—
454	C₂×C₄	—	2	2	no	—
455	C₂×C₄	—	2	2	no	—
456	C₂×C₄	—	2	2	no	—
457	C₂×C₄	—	2	2	no	—
458	C₂×C₄	—	2	2	no	—
459	C₂×C₄	—	2	2	no	—
460	C₂×C₄	—	2	2	no	—
461	C₂×C₄	—	2	2	no	—
462	C₂×C₄	—	4	2	no	—
463	C₂×C₄	—	4	2	no	—
464	C₂×C₄	—	4	2	no	—
465	C₂×C₄	—	4	2	no	—
466	C₂×C₄	C₄×D₄×C₅₃	1	2	yes	—
467	C₂×C₄	C₄×D₄×C₅₃	1	2	yes	—
468	C₂×C₄	C₄×D₄×C₅₃	1	2	yes	—
469	C₂×C₄	C₂²×D₄×C₅₃	1	1	yes	characteristic
470	C₂×C₄	C₂²×D₄×C₅₃	1	1	yes	characteristic
471	C₂×C₄	C₂×#41×C₅₃	1	1	yes	characteristic
472	C₂×C₄	C₂×#41×C₅₃	1	1	yes	characteristic
473	C₂×C₄	C₂×#41×C₅₃	1	1	yes	characteristic
474	C₂×C₄	C₂×#41×C₅₃	1	1	yes	characteristic
475	C₈	—	2	2	no	—
476	C₈	—	2	2	no	—
477	C₈	—	2	2	no	—
478	C₈	—	4	2	no	—
479	C₈	—	4	2	no	—
480	C₈	#121×C₅₃	1	4	yes	—
481	C₂²	—	2	4	no	—
482	C₂²	—	2	2	no	—
483	C₂²	—	2	4	no	—
484	C₂²	—	2	4	no	—
485	C₂²	—	2	4	no	—
486	C₂²	—	4	2	no	—
487	C₂²	—	4	2	no	—
488	C₂²	—	4	2	no	—
489	C₂²	—	4	1	no	—
490	C₂²	—	4	1	no	—
491	C₂²	—	4	2	no	—
492	C₂²	—	4	2	no	—
493	C₂²	C₂×C₄×D₄×C₅₃	1	1	yes	characteristic
494	C₂²	C₂×C₄×D₄×C₅₃	1	1	yes	characteristic
495	C₂²	C₄×#41×C₅₃	1	1	yes	characteristic
496	C₂²	C₂×#131×C₅₃	1	2	yes	—
497	C₂²	C₅₃×#568	1	2	yes	—
498	C₄	—	2	2	no	—
499	C₄	—	2	2	no	—
500	C₄	—	2	2	no	—
501	C₄	—	4	2	no	—
502	C₄	—	4	1	no	—
503	C₄	—	4	1	no	—
504	C₄	—	4	2	no	—
505	C₄	—	4	2	no	—
506	C₄	C₂×#121×C₅₃	1	1	yes	characteristic
507	C₄	C₅₃×#525	1	1	yes	characteristic
508	C₄	C₅₃×#529	1	1	yes	characteristic
509	C₄	C₅₃×#532	1	1	yes	characteristic
510	C₂	—	4	4	no	—
511	C₂	—	4	2	no	—
512	C₂	C₄×#121×C₅₃	1	1	yes	characteristic
513	C₂	C₅₃×#2929	1	1	yes	characteristic
514	C₂	C₅₃×#2932	1	1	yes	characteristic
515	C₂	C₅₃×#2933	1	2	yes	—
516	C₂	C₅₃×#2968	1	2	yes	—

	subgroup	G/H
center	C₂²×C₄×C₅₃	C₂⁴
commutator	C₂³	C₂³×C₄×C₅₃
Frattini, cosocle	C₂²×C₄	C₂⁴×C₅₃
socle	C₂³×C₅₃	C₂³×C₄
prime radical	C₅₃×#392	C₄
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂²×C₄×C₅₃ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂×C₅₃×#435	C₂	semidirect
2. C₂×C₈×D₄×C₅₃	C₂	semidirect
3. C₂×C₅₃×#405	C₂	semidirect
5. C₅₃×#1955	C₂	semidirect
6. C₅₃×#1940	C₂	semidirect
7. C₅₃×#1923	C₂	semidirect
8. C₅₃×#1869	C₂	semidirect
9. C₅₃×#1798	C₂	semidirect
10. C₅₃×#1770	C₂	semidirect
11. C₅₃×#1752	C₂	semidirect
50. C₂×M₁₆×C₅₃	D₄	semidirect
95. C₄×C₈×C₅₃	D₄	semidirect
222. #21492	C₅₃	direct (N × H)
252. C₂×#435	C₂×C₅₃	semidirect
253. C₂×C₈×D₄	C₂×C₅₃	semidirect
254. C₂×#405	C₂×C₅₃	semidirect
256. #1955	C₂×C₅₃	semidirect
257. #1940	C₂×C₅₃	semidirect
258. #1923	C₂×C₅₃	semidirect
259. #1869	C₂×C₅₃	semidirect
260. #1798	C₂×C₅₃	semidirect
261. #1770	C₂×C₅₃	semidirect
262. #1752	C₂×C₅₃	semidirect
302. C₅₃	#21492	direct (N × H)
309. C₂×M₁₆	D₄×C₅₃	semidirect
354. C₄×C₈	D₄×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×#435	2
C₂×C₈×D₄×C₅₃	2
C₂×C₅₃×#405	2
C₅₃×#2392	2
C₅₃×#1955	2
C₅₃×#1940	2
C₅₃×#1923	2
C₅₃×#1869	2
C₅₃×#1798	2
C₅₃×#1770	2
C₅₃×#1752	2
#21492	53

Subgroups — C53×#21492 #27229004

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅₃×#21492 #27229004