Subgroups — C₅₃×#22627 #27230139

Order 13568.

868 conjugacy classes of subgroups.

The subgroup lattice has 534 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	C₅₃×#1940	C₂	1	2	yes	maximal, complemented
3	C₅₃×#1917	C₂	1	2	yes	maximal, complemented
4	C₅₃×#1916	C₂	1	2	yes	maximal, complemented
5	C₅₃×#1797	C₂	1	1	yes	maximal, characteristic
6	C₅₃×#1797	C₂	1	1	yes	maximal, characteristic
7	C₅₃×#1797	C₂	1	1	yes	maximal, characteristic
8	C₅₃×#1771	C₂	1	2	yes	maximal, complemented
9	C₅₃×#1749	C₂	1	1	yes	maximal, characteristic
10	C₅₃×#1749	C₂	1	1	yes	maximal, characteristic
11	C₄×C₂²⋊C₈×C₅₃	C₂	1	1	yes	maximal, characteristic
12	C₂²×C₄⋊C₄×C₅₃	C₄	1	1	yes	characteristic
13	C₂²×C₄⋊C₄×C₅₃	C₄	1	1	yes	characteristic
14	C₂²×C₄²×C₅₃	C₂²	1	1	yes	characteristic
15	C₂²×C₄²×C₅₃	C₂²	1	1	yes	characteristic
16	C₂²×C₄²×C₅₃	C₂²	1	1	yes	characteristic
17	C₂×C₄⋊C₈×C₅₃	C₂²	1	2	yes	—
18	C₂×C₄⋊C₈×C₅₃	C₂²	1	2	yes	—
19	C₂×C₄⋊C₈×C₅₃	C₂²	1	2	yes	—
20	C₂×C₂²⋊C₈×C₅₃	C₂²	1	1	yes	characteristic
21	C₂×C₂²⋊C₈×C₅₃	C₂²	1	1	yes	characteristic
22	C₂×C₂²⋊C₈×C₅₃	C₂²	1	1	yes	characteristic
23	C₂×C₂²⋊C₈×C₅₃	C₂²	1	1	yes	characteristic
24	C₂×#97×C₅₃	C₂²	1	2	yes	—
25	C₂×#97×C₅₃	C₂²	1	2	yes	—
26	C₂×C₄×C₈×C₅₃	C₂²	1	2	yes	—
27	C₄×C₄⋊C₄×C₅₃	C₄	1	1	yes	characteristic
28	C₄×C₄⋊C₄×C₅₃	C₄	1	1	yes	characteristic
29	C₄×C₄⋊C₄×C₅₃	C₄	1	1	yes	characteristic
30	C₄×C₄⋊C₄×C₅₃	C₄	1	1	yes	characteristic
31	C₄×C₄⋊C₄×C₅₃	C₂²	1	1	yes	characteristic
32	C₄×C₄⋊C₄×C₅₃	C₂²	1	1	yes	characteristic
33	C₄×C₄⋊C₄×C₅₃	C₂²	1	1	yes	characteristic
34	C₄×C₄⋊C₄×C₅₃	C₂²	1	1	yes	characteristic
35	C₂×#95×C₅₃	C₄	1	1	yes	characteristic
36	C₂×#95×C₅₃	C₄	1	1	yes	characteristic
37	C₅₃×#336	C₂²	1	2	yes	—
38	C₅₃×#336	C₂²	1	2	yes	—
39	C₅₃×#336	C₂²	1	2	yes	—
40	C₅₃×#336	C₂²	1	2	yes	—
41	C₅₃×#336	C₂²	1	2	yes	—
42	C₅₃×#336	C₂²	1	2	yes	—
43	C₂³×C₄×C₅₃	C₂×C₄	1	1	yes	characteristic
44	C₂³×C₄×C₅₃	C₂×C₄	1	1	yes	characteristic
45	C₂³×C₄×C₅₃	C₂×C₄	1	1	yes	characteristic
46	C₂³×C₄×C₅₃	C₂×C₄	1	1	yes	characteristic
47	C₂³×C₄×C₅₃	C₂×C₄	1	1	yes	characteristic
48	C₂³×C₄×C₅₃	C₂×C₄	1	1	yes	characteristic
49	C₂³×C₄×C₅₃	C₂³	1	1	yes	characteristic
50	C₂²×C₈×C₅₃	C₂³	1	2	yes	—
51	C₂²×C₈×C₅₃	C₂³	1	2	yes	—
52	C₂²×C₈×C₅₃	C₂³	1	2	yes	—
53	C₂²×C₈×C₅₃	C₂³	1	2	yes	—
54	C₂×C₄⋊C₄×C₅₃	—	2	4	no	—
55	C₂×C₄⋊C₄×C₅₃	—	2	4	no	—
56	C₂×C₄⋊C₄×C₅₃	C₂×C₄	1	1	yes	characteristic
57	C₂×C₄⋊C₄×C₅₃	C₂×C₄	1	1	yes	characteristic
58	C₂×C₄⋊C₄×C₅₃	C₂×C₄	1	1	yes	characteristic
59	C₂×C₄⋊C₄×C₅₃	C₂×C₄	1	1	yes	characteristic
60	C₂×C₄⋊C₄×C₅₃	C₂×C₄	1	1	yes	characteristic
61	C₂×C₄⋊C₄×C₅₃	C₂×C₄	1	1	yes	characteristic
62	C₂×C₄⋊C₄×C₅₃	C₂×C₄	1	1	yes	characteristic
63	C₂×C₄⋊C₄×C₅₃	C₂×C₄	1	1	yes	characteristic
64	C₂×C₄²×C₅₃	—	2	4	no	—
65	C₂×C₄²×C₅₃	—	4	2	no	—
66	C₂×C₄²×C₅₃	—	4	2	no	—
67	C₂×C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
68	C₂×C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
69	C₂×C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
70	C₂×C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
71	C₂×C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
72	C₂×C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
73	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
74	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
75	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
76	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
77	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
78	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
79	C₄⋊C₈×C₅₃	—	2	4	no	—
80	C₄⋊C₈×C₅₃	—	4	2	no	—
81	C₄⋊C₈×C₅₃	—	4	2	no	—
82	C₂²⋊C₈×C₅₃	—	2	2	no	—
83	C₂²⋊C₈×C₅₃	—	2	2	no	—
84	C₂²⋊C₈×C₅₃	—	2	2	no	—
85	C₂²⋊C₈×C₅₃	Q₈	1	4	yes	—
86	#97×C₅₃	—	2	4	no	—
87	#97×C₅₃	—	4	2	no	—
88	C₄×C₈×C₅₃	—	4	2	no	—
89	#95×C₅₃	C₂×C₄	1	1	yes	characteristic
90	#95×C₅₃	C₂×C₄	1	1	yes	characteristic
91	#95×C₅₃	C₂×C₄	1	1	yes	characteristic
92	#95×C₅₃	C₂×C₄	1	1	yes	characteristic
93	#95×C₅₃	C₂×C₄	1	1	yes	characteristic
94	#95×C₅₃	C₂×C₄	1	1	yes	characteristic
95	#95×C₅₃	C₂×C₄	1	1	yes	characteristic
96	#95×C₅₃	C₂×C₄	1	1	yes	characteristic
97	C₂⁴×C₅₃	C₂²×C₄	1	1	yes	characteristic
98	C₂²×C₄×C₅₃	—	2	2	no	—
99	C₂²×C₄×C₅₃	—	2	2	no	—
100	C₂²×C₄×C₅₃	—	2	2	no	—
101	C₂²×C₄×C₅₃	—	2	2	no	—
102	C₂²×C₄×C₅₃	—	2	2	no	—
103	C₂²×C₄×C₅₃	—	2	2	no	—
104	C₂²×C₄×C₅₃	—	2	2	no	—
105	C₂²×C₄×C₅₃	—	2	2	no	—
106	C₂²×C₄×C₅₃	—	2	2	no	—
107	C₂²×C₄×C₅₃	—	2	2	no	—
108	C₂²×C₄×C₅₃	—	2	2	no	—
109	C₂²×C₄×C₅₃	—	2	2	no	—
110	C₂²×C₄×C₅₃	—	4	1	no	—
111	C₂²×C₄×C₅₃	—	4	1	no	—
112	C₂²×C₄×C₅₃	—	4	1	no	—
113	C₂²×C₄×C₅₃	—	4	1	no	—
114	C₂²×C₄×C₅₃	—	4	1	no	—
115	C₂²×C₄×C₅₃	—	4	1	no	—
116	C₂²×C₄×C₅₃	—	4	1	no	—
117	C₂²×C₄×C₅₃	M₁₆	1	4	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	1	yes	characteristic
121	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
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	C₂²×C₄	1	1	yes	characteristic
126	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
127	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
128	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
129	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
130	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
131	C₂²×C₄×C₅₃	C₂×Q₈	1	2	yes	—
132	C₂²×C₄×C₅₃	#41	1	2	yes	—
133	C₂²×C₄×C₅₃	C₂⁴	1	1	yes	characteristic
134	C₂×C₈×C₅₃	—	2	2	no	—
135	C₂×C₈×C₅₃	—	2	2	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	—	2	2	no	—
143	C₂×C₈×C₅₃	—	2	2	no	—
144	C₂×C₈×C₅₃	—	2	2	no	—
145	C₂×C₈×C₅₃	C₂×Q₈	1	4	yes	—
146	C₄⋊C₄×C₅₃	—	2	2	no	—
147	C₄⋊C₄×C₅₃	—	2	2	no	—
148	C₄⋊C₄×C₅₃	—	2	2	no	—
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	2	no	—
153	C₄⋊C₄×C₅₃	M₁₆	1	4	yes	—
154	C₄²×C₅₃	—	2	2	no	—
155	C₄²×C₅₃	—	2	2	no	—
156	C₄²×C₅₃	—	2	2	no	—
157	C₄²×C₅₃	—	2	2	no	—
158	C₄²×C₅₃	—	2	2	no	—
159	C₄²×C₅₃	—	2	2	no	—
160	C₄²×C₅₃	—	2	2	no	—
161	C₄²×C₅₃	—	4	1	no	—
162	C₄²×C₅₃	—	4	1	no	—
163	C₄²×C₅₃	—	4	1	no	—
164	C₄²×C₅₃	—	4	1	no	—
165	C₄²×C₅₃	#41	1	4	yes	—
166	C₂³×C₅₃	—	2	1	no	—
167	C₂³×C₅₃	—	2	1	no	—
168	C₂³×C₅₃	—	2	1	no	—
169	C₂³×C₅₃	—	2	1	no	—
170	C₂³×C₅₃	C₄×Q₈	1	2	yes	—
171	C₂³×C₅₃	C₂×M₁₆	1	2	yes	—
172	C₂³×C₅₃	#131	1	2	yes	—
173	C₂³×C₅₃	C₂³×C₄	1	1	yes	characteristic
174	C₂×C₄×C₅₃	—	2	8	no	—
175	C₂×C₄×C₅₃	—	2	1	no	—
176	C₂×C₄×C₅₃	—	2	1	no	—
177	C₂×C₄×C₅₃	—	2	1	no	—
178	C₂×C₄×C₅₃	—	2	1	no	—
179	C₂×C₄×C₅₃	—	2	1	no	—
180	C₂×C₄×C₅₃	—	2	1	no	—
181	C₂×C₄×C₅₃	—	2	1	no	—
182	C₂×C₄×C₅₃	—	2	1	no	—
183	C₂×C₄×C₅₃	—	2	1	no	—
184	C₂×C₄×C₅₃	—	2	1	no	—
185	C₂×C₄×C₅₃	—	2	1	no	—
186	C₂×C₄×C₅₃	—	2	1	no	—
187	C₂×C₄×C₅₃	—	2	1	no	—
188	C₂×C₄×C₅₃	—	2	1	no	—
189	C₂×C₄×C₅₃	—	2	1	no	—
190	C₂×C₄×C₅₃	—	2	1	no	—
191	C₂×C₄×C₅₃	—	2	4	no	—
192	C₂×C₄×C₅₃	—	2	4	no	—
193	C₂×C₄×C₅₃	—	2	1	no	—
194	C₂×C₄×C₅₃	—	2	1	no	—
195	C₂×C₄×C₅₃	—	2	1	no	—
196	C₂×C₄×C₅₃	—	2	1	no	—
197	C₂×C₄×C₅₃	—	2	1	no	—
198	C₂×C₄×C₅₃	—	2	1	no	—
199	C₂×C₄×C₅₃	—	2	1	no	—
200	C₂×C₄×C₅₃	—	2	1	no	—
201	C₂×C₄×C₅₃	—	2	1	no	—
202	C₂×C₄×C₅₃	—	2	1	no	—
203	C₂×C₄×C₅₃	—	2	1	no	—
204	C₂×C₄×C₅₃	—	2	1	no	—
205	C₂×C₄×C₅₃	—	2	1	no	—
206	C₂×C₄×C₅₃	—	2	1	no	—
207	C₂×C₄×C₅₃	—	2	1	no	—
208	C₂×C₄×C₅₃	—	4	4	no	—
209	C₂×C₄×C₅₃	—	4	4	no	—
210	C₂×C₄×C₅₃	—	4	4	no	—
211	C₂×C₄×C₅₃	—	4	4	no	—
212	C₂×C₄×C₅₃	—	4	4	no	—
213	C₂×C₄×C₅₃	#117	1	2	yes	—
214	C₂×C₄×C₅₃	#117	1	2	yes	—
215	C₂×C₄×C₅₃	C₄×Q₈	1	2	yes	—
216	C₂×C₄×C₅₃	C₂×M₁₆	1	2	yes	—
217	C₂×C₄×C₅₃	C₂×M₁₆	1	2	yes	—
218	C₂×C₄×C₅₃	C₂×M₁₆	1	2	yes	—
219	C₂×C₄×C₅₃	C₂×M₁₆	1	2	yes	—
220	C₂×C₄×C₅₃	C₂×M₁₆	1	2	yes	—
221	C₂×C₄×C₅₃	C₂²×Q₈	1	1	yes	characteristic
222	C₂×C₄×C₅₃	C₂×#41	1	1	yes	characteristic
223	C₂×C₄×C₅₃	C₂×#41	1	2	yes	—
224	C₂×C₄×C₅₃	H(5,𝔽₂)	1	2	yes	—
225	C₈×C₅₃	—	2	4	no	—
226	C₈×C₅₃	—	4	2	no	—
227	C₈×C₅₃	—	4	2	no	—
228	C₈×C₅₃	—	4	2	no	—
229	#22627	C₅₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
230	C₂²×C₅₃	—	2	2	no	—
231	C₂²×C₅₃	—	2	2	no	—
232	C₂²×C₅₃	—	2	2	no	—
233	C₂²×C₅₃	—	2	2	no	—
234	C₂²×C₅₃	—	2	2	no	—
235	C₂²×C₅₃	—	2	2	no	—
236	C₂²×C₅₃	#446	1	4	yes	—
237	C₂²×C₅₃	C₂×#117	1	1	yes	characteristic
238	C₂²×C₅₃	C₂×C₄×Q₈	1	1	yes	characteristic
239	C₂²×C₅₃	#518	1	1	yes	characteristic
240	C₂²×C₅₃	C₂²×M₁₆	1	1	yes	characteristic
241	C₂²×C₅₃	C₂²×M₁₆	1	1	yes	characteristic
242	C₂²×C₅₃	C₂×#131	1	1	yes	characteristic
243	C₂²×C₅₃	#568	1	1	yes	characteristic
244	C₄×C₅₃	—	2	2	no	—
245	C₄×C₅₃	—	2	2	no	—
246	C₄×C₅₃	—	2	2	no	—
247	C₄×C₅₃	—	2	2	no	—
248	C₄×C₅₃	—	2	2	no	—
249	C₄×C₅₃	—	2	2	no	—
250	C₄×C₅₃	—	2	2	no	—
251	C₄×C₅₃	—	2	2	no	—
252	C₄×C₅₃	—	4	1	no	—
253	C₄×C₅₃	—	4	1	no	—
254	C₄×C₅₃	—	4	1	no	—
255	C₄×C₅₃	—	4	1	no	—
256	C₄×C₅₃	—	4	1	no	—
257	C₄×C₅₃	#553	1	2	yes	—
258	C₄×C₅₃	#554	1	2	yes	—
259	C₂×C₄×C₄⋊C₄	C₂×C₅₃	1	1	yes	characteristic
260	#1940	C₂×C₅₃	1	2	yes	complemented
261	#1917	C₂×C₅₃	1	2	yes	complemented
262	#1916	C₂×C₅₃	1	2	yes	complemented
263	#1797	C₂×C₅₃	1	1	yes	characteristic
264	#1797	C₂×C₅₃	1	1	yes	characteristic
265	#1797	C₂×C₅₃	1	1	yes	characteristic
266	#1771	C₂×C₅₃	1	2	yes	complemented
267	#1749	C₂×C₅₃	1	1	yes	characteristic
268	#1749	C₂×C₅₃	1	1	yes	characteristic
269	C₄×C₂²⋊C₈	C₂×C₅₃	1	1	yes	characteristic
270	C₂×C₅₃	—	2	4	no	—
271	C₂×C₅₃	#2345	1	1	yes	characteristic
272	C₂×C₅₃	#2920	1	1	yes	characteristic
273	C₂×C₅₃	#2924	1	1	yes	characteristic
274	C₂×C₅₃	C₂×#446	1	1	yes	characteristic
275	C₂×C₅₃	Q₈×M₁₆	1	1	yes	characteristic
276	C₂×C₅₃	#2973	1	1	yes	characteristic
277	C₂×C₅₃	#2974	1	1	yes	characteristic
278	C₂²×C₄⋊C₄	C₄×C₅₃	1	1	yes	characteristic
279	C₂²×C₄⋊C₄	C₄×C₅₃	1	1	yes	characteristic
280	C₂²×C₄²	C₂²×C₅₃	1	1	yes	characteristic
281	C₂²×C₄²	C₂²×C₅₃	1	1	yes	characteristic
282	C₂²×C₄²	C₂²×C₅₃	1	1	yes	characteristic
283	C₂×C₄⋊C₈	C₂²×C₅₃	1	2	yes	—
284	C₂×C₄⋊C₈	C₂²×C₅₃	1	2	yes	—
285	C₂×C₄⋊C₈	C₂²×C₅₃	1	2	yes	—
286	C₂×C₂²⋊C₈	C₂²×C₅₃	1	1	yes	characteristic
287	C₂×C₂²⋊C₈	C₂²×C₅₃	1	1	yes	characteristic
288	C₂×C₂²⋊C₈	C₂²×C₅₃	1	1	yes	characteristic
289	C₂×C₂²⋊C₈	C₂²×C₅₃	1	1	yes	characteristic
290	C₂×#97	C₂²×C₅₃	1	2	yes	—
291	C₂×#97	C₂²×C₅₃	1	2	yes	—
292	C₂×C₄×C₈	C₂²×C₅₃	1	2	yes	—
293	C₄×C₄⋊C₄	C₄×C₅₃	1	1	yes	characteristic
294	C₄×C₄⋊C₄	C₄×C₅₃	1	1	yes	characteristic
295	C₄×C₄⋊C₄	C₄×C₅₃	1	1	yes	characteristic
296	C₄×C₄⋊C₄	C₄×C₅₃	1	1	yes	characteristic
297	C₄×C₄⋊C₄	C₂²×C₅₃	1	1	yes	characteristic
298	C₄×C₄⋊C₄	C₂²×C₅₃	1	1	yes	characteristic
299	C₄×C₄⋊C₄	C₂²×C₅₃	1	1	yes	characteristic
300	C₄×C₄⋊C₄	C₂²×C₅₃	1	1	yes	characteristic
301	C₂×#95	C₄×C₅₃	1	1	yes	characteristic
302	C₂×#95	C₄×C₅₃	1	1	yes	characteristic
303	#336	C₂²×C₅₃	1	2	yes	—
304	#336	C₂²×C₅₃	1	2	yes	—
305	#336	C₂²×C₅₃	1	2	yes	—
306	#336	C₂²×C₅₃	1	2	yes	—
307	#336	C₂²×C₅₃	1	2	yes	—
308	#336	C₂²×C₅₃	1	2	yes	—
309	C₅₃	#22627	1	1	yes	characteristic, direct product factor, Sylow
310	C₂³×C₄	C₂×C₄×C₅₃	1	1	yes	characteristic
311	C₂³×C₄	C₂×C₄×C₅₃	1	1	yes	characteristic
312	C₂³×C₄	C₂×C₄×C₅₃	1	1	yes	characteristic
313	C₂³×C₄	C₂×C₄×C₅₃	1	1	yes	characteristic
314	C₂³×C₄	C₂×C₄×C₅₃	1	1	yes	characteristic
315	C₂³×C₄	C₂×C₄×C₅₃	1	1	yes	characteristic
316	C₂³×C₄	C₂³×C₅₃	1	1	yes	characteristic
317	C₂²×C₈	C₂³×C₅₃	1	2	yes	—
318	C₂²×C₈	C₂³×C₅₃	1	2	yes	—
319	C₂²×C₈	C₂³×C₅₃	1	2	yes	—
320	C₂²×C₈	C₂³×C₅₃	1	2	yes	—
321	C₂×C₄⋊C₄	—	2	4	no	—
322	C₂×C₄⋊C₄	—	2	4	no	—
323	C₂×C₄⋊C₄	C₂×C₄×C₅₃	1	1	yes	characteristic
324	C₂×C₄⋊C₄	C₂×C₄×C₅₃	1	1	yes	characteristic
325	C₂×C₄⋊C₄	C₂×C₄×C₅₃	1	1	yes	characteristic
326	C₂×C₄⋊C₄	C₂×C₄×C₅₃	1	1	yes	characteristic
327	C₂×C₄⋊C₄	C₂×C₄×C₅₃	1	1	yes	characteristic
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	1	yes	characteristic
331	C₂×C₄²	—	2	4	no	—
332	C₂×C₄²	—	4	2	no	—
333	C₂×C₄²	—	4	2	no	—
334	C₂×C₄²	C₂×C₄×C₅₃	1	1	yes	characteristic
335	C₂×C₄²	C₂×C₄×C₅₃	1	1	yes	characteristic
336	C₂×C₄²	C₂×C₄×C₅₃	1	1	yes	characteristic
337	C₂×C₄²	C₂×C₄×C₅₃	1	1	yes	characteristic
338	C₂×C₄²	C₂×C₄×C₅₃	1	1	yes	characteristic
339	C₂×C₄²	C₂×C₄×C₅₃	1	1	yes	characteristic
340	C₂×C₄²	C₂³×C₅₃	1	1	yes	characteristic
341	C₂×C₄²	C₂³×C₅₃	1	1	yes	characteristic
342	C₂×C₄²	C₂³×C₅₃	1	1	yes	characteristic
343	C₂×C₄²	C₂³×C₅₃	1	1	yes	characteristic
344	C₂×C₄²	C₂³×C₅₃	1	1	yes	characteristic
345	C₂×C₄²	C₂³×C₅₃	1	1	yes	characteristic
346	C₄⋊C₈	—	2	4	no	—
347	C₄⋊C₈	—	4	2	no	—
348	C₄⋊C₈	—	4	2	no	—
349	C₂²⋊C₈	—	2	2	no	—
350	C₂²⋊C₈	—	2	2	no	—
351	C₂²⋊C₈	—	2	2	no	—
352	C₂²⋊C₈	Q₈×C₅₃	1	4	yes	—
353	#97	—	2	4	no	—
354	#97	—	4	2	no	—
355	C₄×C₈	—	4	2	no	—
356	#95	C₂×C₄×C₅₃	1	1	yes	characteristic
357	#95	C₂×C₄×C₅₃	1	1	yes	characteristic
358	#95	C₂×C₄×C₅₃	1	1	yes	characteristic
359	#95	C₂×C₄×C₅₃	1	1	yes	characteristic
360	#95	C₂×C₄×C₅₃	1	1	yes	characteristic
361	#95	C₂×C₄×C₅₃	1	1	yes	characteristic
362	#95	C₂×C₄×C₅₃	1	1	yes	characteristic
363	#95	C₂×C₄×C₅₃	1	1	yes	characteristic
364	C₂⁴	C₂²×C₄×C₅₃	1	1	yes	characteristic
365	C₂²×C₄	—	2	2	no	—
366	C₂²×C₄	—	2	2	no	—
367	C₂²×C₄	—	2	2	no	—
368	C₂²×C₄	—	2	2	no	—
369	C₂²×C₄	—	2	2	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	—	2	2	no	—
377	C₂²×C₄	—	4	1	no	—
378	C₂²×C₄	—	4	1	no	—
379	C₂²×C₄	—	4	1	no	—
380	C₂²×C₄	—	4	1	no	—
381	C₂²×C₄	—	4	1	no	—
382	C₂²×C₄	—	4	1	no	—
383	C₂²×C₄	—	4	1	no	—
384	C₂²×C₄	M₁₆×C₅₃	1	4	yes	—
385	C₂²×C₄	C₂²×C₄×C₅₃	1	1	yes	characteristic
386	C₂²×C₄	C₂²×C₄×C₅₃	1	1	yes	characteristic
387	C₂²×C₄	C₂²×C₄×C₅₃	1	1	yes	characteristic
388	C₂²×C₄	C₂²×C₄×C₅₃	1	1	yes	characteristic
389	C₂²×C₄	C₂²×C₄×C₅₃	1	1	yes	characteristic
390	C₂²×C₄	C₂²×C₄×C₅₃	1	1	yes	characteristic
391	C₂²×C₄	C₂²×C₄×C₅₃	1	1	yes	characteristic
392	C₂²×C₄	C₂²×C₄×C₅₃	1	1	yes	characteristic
393	C₂²×C₄	C₂²×C₄×C₅₃	1	1	yes	characteristic
394	C₂²×C₄	C₂²×C₄×C₅₃	1	1	yes	characteristic
395	C₂²×C₄	C₂²×C₄×C₅₃	1	1	yes	characteristic
396	C₂²×C₄	C₂²×C₄×C₅₃	1	1	yes	characteristic
397	C₂²×C₄	C₂²×C₄×C₅₃	1	1	yes	characteristic
398	C₂²×C₄	C₂×Q₈×C₅₃	1	2	yes	—
399	C₂²×C₄	#41×C₅₃	1	2	yes	—
400	C₂²×C₄	C₂⁴×C₅₃	1	1	yes	characteristic
401	C₂×C₈	—	2	2	no	—
402	C₂×C₈	—	2	2	no	—
403	C₂×C₈	—	2	2	no	—
404	C₂×C₈	—	2	2	no	—
405	C₂×C₈	—	2	2	no	—
406	C₂×C₈	—	2	2	no	—
407	C₂×C₈	—	2	2	no	—
408	C₂×C₈	—	2	2	no	—
409	C₂×C₈	—	2	2	no	—
410	C₂×C₈	—	2	2	no	—
411	C₂×C₈	—	2	2	no	—
412	C₂×C₈	C₂×Q₈×C₅₃	1	4	yes	—
413	C₄⋊C₄	—	2	2	no	—
414	C₄⋊C₄	—	2	2	no	—
415	C₄⋊C₄	—	2	2	no	—
416	C₄⋊C₄	—	2	2	no	—
417	C₄⋊C₄	—	2	2	no	—
418	C₄⋊C₄	—	2	2	no	—
419	C₄⋊C₄	—	2	2	no	—
420	C₄⋊C₄	M₁₆×C₅₃	1	4	yes	—
421	C₄²	—	2	2	no	—
422	C₄²	—	2	2	no	—
423	C₄²	—	2	2	no	—
424	C₄²	—	2	2	no	—
425	C₄²	—	2	2	no	—
426	C₄²	—	2	2	no	—
427	C₄²	—	2	2	no	—
428	C₄²	—	4	1	no	—
429	C₄²	—	4	1	no	—
430	C₄²	—	4	1	no	—
431	C₄²	—	4	1	no	—
432	C₄²	#41×C₅₃	1	4	yes	—
433	C₂³	—	2	1	no	—
434	C₂³	—	2	1	no	—
435	C₂³	—	2	1	no	—
436	C₂³	—	2	1	no	—
437	C₂³	C₄×Q₈×C₅₃	1	2	yes	—
438	C₂³	C₂×M₁₆×C₅₃	1	2	yes	—
439	C₂³	#131×C₅₃	1	2	yes	—
440	C₂³	C₂³×C₄×C₅₃	1	1	yes	characteristic
441	C₂×C₄	—	2	8	no	—
442	C₂×C₄	—	2	1	no	—
443	C₂×C₄	—	2	1	no	—
444	C₂×C₄	—	2	1	no	—
445	C₂×C₄	—	2	1	no	—
446	C₂×C₄	—	2	1	no	—
447	C₂×C₄	—	2	1	no	—
448	C₂×C₄	—	2	1	no	—
449	C₂×C₄	—	2	1	no	—
450	C₂×C₄	—	2	1	no	—
451	C₂×C₄	—	2	1	no	—
452	C₂×C₄	—	2	1	no	—
453	C₂×C₄	—	2	1	no	—
454	C₂×C₄	—	2	1	no	—
455	C₂×C₄	—	2	1	no	—
456	C₂×C₄	—	2	1	no	—
457	C₂×C₄	—	2	1	no	—
458	C₂×C₄	—	2	4	no	—
459	C₂×C₄	—	2	4	no	—
460	C₂×C₄	—	2	1	no	—
461	C₂×C₄	—	2	1	no	—
462	C₂×C₄	—	2	1	no	—
463	C₂×C₄	—	2	1	no	—
464	C₂×C₄	—	2	1	no	—
465	C₂×C₄	—	2	1	no	—
466	C₂×C₄	—	2	1	no	—
467	C₂×C₄	—	2	1	no	—
468	C₂×C₄	—	2	1	no	—
469	C₂×C₄	—	2	1	no	—
470	C₂×C₄	—	2	1	no	—
471	C₂×C₄	—	2	1	no	—
472	C₂×C₄	—	2	1	no	—
473	C₂×C₄	—	2	1	no	—
474	C₂×C₄	—	2	1	no	—
475	C₂×C₄	—	4	4	no	—
476	C₂×C₄	—	4	4	no	—
477	C₂×C₄	—	4	4	no	—
478	C₂×C₄	—	4	4	no	—
479	C₂×C₄	—	4	4	no	—
480	C₂×C₄	#117×C₅₃	1	2	yes	—
481	C₂×C₄	#117×C₅₃	1	2	yes	—
482	C₂×C₄	C₄×Q₈×C₅₃	1	2	yes	—
483	C₂×C₄	C₂×M₁₆×C₅₃	1	2	yes	—
484	C₂×C₄	C₂×M₁₆×C₅₃	1	2	yes	—
485	C₂×C₄	C₂×M₁₆×C₅₃	1	2	yes	—
486	C₂×C₄	C₂×M₁₆×C₅₃	1	2	yes	—
487	C₂×C₄	C₂×M₁₆×C₅₃	1	2	yes	—
488	C₂×C₄	C₂²×Q₈×C₅₃	1	1	yes	characteristic
489	C₂×C₄	C₂×#41×C₅₃	1	1	yes	characteristic
490	C₂×C₄	C₂×#41×C₅₃	1	2	yes	—
491	C₂×C₄	H(5,𝔽₂)×C₅₃	1	2	yes	—
492	C₈	—	2	4	no	—
493	C₈	—	4	2	no	—
494	C₈	—	4	2	no	—
495	C₈	—	4	2	no	—
496	C₂²	—	2	2	no	—
497	C₂²	—	2	2	no	—
498	C₂²	—	2	2	no	—
499	C₂²	—	2	2	no	—
500	C₂²	—	2	2	no	—
501	C₂²	—	2	2	no	—
502	C₂²	C₅₃×#446	1	4	yes	—
503	C₂²	C₂×#117×C₅₃	1	1	yes	characteristic
504	C₂²	C₂×C₄×Q₈×C₅₃	1	1	yes	characteristic
505	C₂²	C₅₃×#518	1	1	yes	characteristic
506	C₂²	C₂²×M₁₆×C₅₃	1	1	yes	characteristic
507	C₂²	C₂²×M₁₆×C₅₃	1	1	yes	characteristic
508	C₂²	C₂×#131×C₅₃	1	1	yes	characteristic
509	C₂²	C₅₃×#568	1	1	yes	characteristic
510	C₄	—	2	2	no	—
511	C₄	—	2	2	no	—
512	C₄	—	2	2	no	—
513	C₄	—	2	2	no	—
514	C₄	—	2	2	no	—
515	C₄	—	2	2	no	—
516	C₄	—	2	2	no	—
517	C₄	—	2	2	no	—
518	C₄	—	4	1	no	—
519	C₄	—	4	1	no	—
520	C₄	—	4	1	no	—
521	C₄	—	4	1	no	—
522	C₄	—	4	1	no	—
523	C₄	C₅₃×#553	1	2	yes	—
524	C₄	C₅₃×#554	1	2	yes	—
525	C₂	—	2	4	no	—
526	C₂	C₅₃×#2345	1	1	yes	characteristic
527	C₂	C₅₃×#2920	1	1	yes	characteristic
528	C₂	C₅₃×#2924	1	1	yes	characteristic
529	C₂	C₂×C₅₃×#446	1	1	yes	characteristic
530	C₂	Q₈×M₁₆×C₅₃	1	1	yes	characteristic
531	C₂	C₅₃×#2973	1	1	yes	characteristic
532	C₂	C₅₃×#2974	1	1	yes	characteristic

	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	C₂⁴×C₅₃	C₂²×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
2. C₅₃×#1940	C₂	semidirect
3. C₅₃×#1917	C₂	semidirect
4. C₅₃×#1916	C₂	semidirect
8. C₅₃×#1771	C₂	semidirect
229. #22627	C₅₃	direct (N × H)
260. #1940	C₂×C₅₃	semidirect
261. #1917	C₂×C₅₃	semidirect
262. #1916	C₂×C₅₃	semidirect
266. #1771	C₂×C₅₃	semidirect
309. C₅₃	#22627	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
C₅₃×#1940	2
C₅₃×#1917	2
C₅₃×#1916	2
C₅₃×#1797	2
C₅₃×#1771	2
C₅₃×#1749	2
C₄×C₂²⋊C₈×C₅₃	2
#22627	53

Subgroups — C53×#22627 #27230139

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅₃×#22627 #27230139