Subgroups — C₅₃×#22291 #27229803

Order 13568.

786 conjugacy classes of subgroups.

The subgroup lattice has 632 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×C₂²⋊Q₈×C₅₃	C₂	1	1	yes	maximal, characteristic
2	C₅₃×#1985	C₂	1	1	yes	maximal, characteristic, complemented
3	C₅₃×#1984	C₂	1	1	yes	maximal, characteristic, complemented
4	C₅₃×#1940	C₂	1	1	yes	maximal, characteristic, complemented
5	C₅₃×#1939	C₂	1	1	yes	maximal, characteristic, complemented
6	C₅₃×#1917	C₂	1	1	yes	maximal, characteristic, complemented
7	C₅₃×#1917	C₂	1	1	yes	maximal, characteristic, complemented
8	C₅₃×#1916	C₂	1	1	yes	maximal, characteristic, complemented
9	C₅₃×#1916	C₂	1	1	yes	maximal, characteristic, complemented
10	C₅₃×#1850	C₂	1	1	yes	maximal, characteristic
11	C₅₃×#1850	C₂	1	1	yes	maximal, characteristic
12	C₅₃×#1797	C₂	1	1	yes	maximal, characteristic
13	C₅₃×#1752	C₂	1	1	yes	maximal, characteristic
14	C₈×C₂²⋊C₄×C₅₃	C₂	1	1	yes	maximal, characteristic
15	C₂×C₅₃×#336	C₂	1	1	yes	maximal, characteristic
16	C₂³×C₈×C₅₃	C₂²	1	1	yes	characteristic
17	C₂³×C₈×C₅₃	C₂²	1	1	yes	characteristic
18	C₂×C₂²⋊Q₈×C₅₃	C₄	1	1	yes	characteristic
19	C₂×C₄×Q₈×C₅₃	C₂²	1	1	yes	characteristic
20	C₂²×C₄²×C₅₃	C₂²	1	1	yes	characteristic
21	C₂×C₄⋊C₈×C₅₃	C₂²	1	1	yes	characteristic
22	C₂×C₄⋊C₈×C₅₃	C₂²	1	1	yes	characteristic
23	C₂×C₄⋊C₈×C₅₃	C₂²	1	1	yes	characteristic
24	C₂×C₄⋊C₈×C₅₃	C₂²	1	1	yes	characteristic
25	C₂×C₄⋊C₈×C₅₃	C₂²	1	1	yes	characteristic
26	C₂×C₄⋊C₈×C₅₃	C₂²	1	1	yes	characteristic
27	C₂×C₂²⋊C₈×C₅₃	C₂²	1	1	yes	characteristic
28	C₂×C₂²⋊C₈×C₅₃	C₂²	1	1	yes	characteristic
29	C₂×#97×C₅₃	C₂²	1	1	yes	characteristic
30	C₂×#97×C₅₃	C₂²	1	1	yes	characteristic
31	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
32	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
33	C₅₃×#391	C₄	1	1	yes	characteristic
34	C₅₃×#389	C₄	1	1	yes	characteristic
35	C₅₃×#387	C₄	1	1	yes	characteristic
36	C₅₃×#387	C₄	1	1	yes	characteristic
37	C₅₃×#385	C₄	1	1	yes	characteristic
38	C₅₃×#385	C₄	1	1	yes	characteristic
39	C₅₃×#380	C₄	1	1	yes	characteristic
40	C₄×C₄⋊C₄×C₅₃	C₂²	1	1	yes	characteristic
41	C₄×C₄⋊C₄×C₅₃	C₂²	1	1	yes	characteristic
42	C₄×C₄⋊C₄×C₅₃	C₂²	1	1	yes	characteristic
43	C₄×C₂²⋊C₄×C₅₃	C₂²	1	1	yes	characteristic
44	C₄×C₂²⋊C₄×C₅₃	C₂²	1	1	yes	characteristic
45	C₅₃×#336	C₂²	1	1	yes	characteristic
46	C₅₃×#336	C₂²	1	1	yes	characteristic
47	C₅₃×#336	C₂²	1	1	yes	characteristic
48	C₅₃×#336	C₂²	1	1	yes	characteristic
49	C₅₃×#336	C₂²	1	1	yes	characteristic
50	C₅₃×#336	C₂²	1	1	yes	characteristic
51	C₅₃×#336	C₂²	1	1	yes	characteristic
52	C₅₃×#336	C₂²	1	1	yes	characteristic
53	C₅₃×#336	C₂²	1	1	yes	characteristic
54	C₅₃×#336	C₂²	1	1	yes	characteristic
55	C₅₃×#336	C₂²	1	1	yes	characteristic
56	C₅₃×#336	C₂²	1	1	yes	characteristic
57	C₅₃×#336	C₂²	1	1	yes	characteristic
58	C₅₃×#336	C₂²	1	1	yes	characteristic
59	C₂²×Q₈×C₅₃	C₂×C₄	1	1	yes	characteristic
60	C₂³×C₄×C₅₃	C₂×C₄	1	1	yes	characteristic
61	C₂³×C₄×C₅₃	C₂×C₄	1	1	yes	characteristic
62	C₂³×C₄×C₅₃	C₂³	1	1	yes	characteristic
63	C₂²×C₈×C₅₃	—	2	2	no	—
64	C₂²×C₈×C₅₃	—	4	1	no	—
65	C₂²×C₈×C₅₃	—	4	1	no	—
66	C₂²×C₈×C₅₃	—	4	1	no	—
67	C₂²×C₈×C₅₃	—	4	1	no	—
68	C₂²×C₈×C₅₃	Q₈	1	4	yes	complemented
69	C₂²×C₈×C₅₃	C₂³	1	1	yes	characteristic
70	C₂²×C₈×C₅₃	C₂³	1	1	yes	characteristic
71	C₂²×C₈×C₅₃	C₂³	1	1	yes	characteristic
72	C₂²×C₈×C₅₃	C₂³	1	1	yes	characteristic
73	C₂²×C₈×C₅₃	C₂³	1	1	yes	characteristic
74	C₂²×C₈×C₅₃	C₂³	1	1	yes	characteristic
75	C₂²×C₈×C₅₃	C₂³	1	1	yes	characteristic
76	C₂²×C₈×C₅₃	C₂³	1	1	yes	characteristic
77	C₂²⋊Q₈×C₅₃	—	2	4	no	—
78	C₄×Q₈×C₅₃	—	4	1	no	—
79	C₄×Q₈×C₅₃	—	4	1	no	—
80	C₂×C₄⋊C₄×C₅₃	C₂×C₄	1	1	yes	characteristic
81	C₂×C₄⋊C₄×C₅₃	C₂×C₄	1	1	yes	characteristic
82	C₂×C₄⋊C₄×C₅₃	C₂×C₄	1	1	yes	characteristic
83	C₂×C₄⋊C₄×C₅₃	C₂×C₄	1	1	yes	characteristic
84	C₂×C₄⋊C₄×C₅₃	C₂×C₄	1	1	yes	characteristic
85	C₂×C₄⋊C₄×C₅₃	C₂×C₄	1	1	yes	characteristic
86	C₂×C₄⋊C₄×C₅₃	C₂×C₄	1	1	yes	characteristic
87	C₂×C₄⋊C₄×C₅₃	C₂×C₄	1	1	yes	characteristic
88	C₂×C₄⋊C₄×C₅₃	C₂×C₄	1	1	yes	characteristic
89	C₂×C₂²⋊C₄×C₅₃	C₂×C₄	1	1	yes	characteristic
90	C₂×C₂²⋊C₄×C₅₃	C₂×C₄	1	1	yes	characteristic
91	C₂×C₂²⋊C₄×C₅₃	C₂×C₄	1	1	yes	characteristic
92	C₂×C₂²⋊C₄×C₅₃	C₂×C₄	1	1	yes	characteristic
93	C₂×C₄²×C₅₃	—	4	1	no	—
94	C₂×C₄²×C₅₃	—	4	1	no	—
95	C₂×C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
96	C₂×C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
97	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
98	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
99	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
100	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
101	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
102	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
103	C₄⋊C₈×C₅₃	—	4	1	no	—
104	C₄⋊C₈×C₅₃	—	4	1	no	—
105	C₄⋊C₈×C₅₃	—	4	1	no	—
106	C₄⋊C₈×C₅₃	—	4	1	no	—
107	C₄⋊C₈×C₅₃	—	4	1	no	—
108	C₄⋊C₈×C₅₃	—	4	1	no	—
109	C₂²⋊C₈×C₅₃	—	2	2	no	—
110	C₂²⋊C₈×C₅₃	—	2	2	no	—
111	#97×C₅₃	—	4	1	no	—
112	#97×C₅₃	—	4	1	no	—
113	C₄×C₈×C₅₃	—	4	1	no	—
114	C₄×C₈×C₅₃	—	4	1	no	—
115	#95×C₅₃	C₂×C₄	1	1	yes	characteristic
116	#95×C₅₃	C₂×C₄	1	1	yes	characteristic
117	#95×C₅₃	C₂×C₄	1	1	yes	characteristic
118	#95×C₅₃	C₂×C₄	1	1	yes	characteristic
119	#95×C₅₃	C₂×C₄	1	1	yes	characteristic
120	#95×C₅₃	C₂×C₄	1	1	yes	characteristic
121	#95×C₅₃	C₂×C₄	1	1	yes	characteristic
122	#95×C₅₃	C₂×C₄	1	1	yes	characteristic
123	#95×C₅₃	C₂×C₄	1	1	yes	characteristic
124	#95×C₅₃	C₂×C₄	1	1	yes	characteristic
125	C₂⁴×C₅₃	C₂²×C₄	1	1	yes	characteristic
126	C₂×Q₈×C₅₃	—	2	2	no	—
127	C₂×Q₈×C₅₃	—	2	2	no	—
128	C₂×Q₈×C₅₃	—	4	1	no	—
129	C₂²×C₄×C₅₃	—	2	1	no	—
130	C₂²×C₄×C₅₃	—	2	1	no	—
131	C₂²×C₄×C₅₃	—	2	1	no	—
132	C₂²×C₄×C₅₃	—	2	1	no	—
133	C₂²×C₄×C₅₃	—	2	2	no	—
134	C₂²×C₄×C₅₃	—	2	2	no	—
135	C₂²×C₄×C₅₃	—	2	2	no	—
136	C₂²×C₄×C₅₃	—	2	2	no	—
137	C₂²×C₄×C₅₃	—	4	1	no	—
138	C₂²×C₄×C₅₃	—	4	1	no	—
139	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
140	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
141	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
142	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
143	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
144	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
145	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
146	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
147	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
148	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
149	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
150	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
151	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
152	C₂²×C₄×C₅₃	C₂×Q₈	1	2	yes	—
153	C₂²×C₄×C₅₃	#41	1	2	yes	—
154	C₂²×C₄×C₅₃	C₂⁴	1	1	yes	characteristic
155	C₂×C₈×C₅₃	—	2	1	no	—
156	C₂×C₈×C₅₃	—	2	1	no	—
157	C₂×C₈×C₅₃	—	2	1	no	—
158	C₂×C₈×C₅₃	—	2	1	no	—
159	C₂×C₈×C₅₃	—	2	1	no	—
160	C₂×C₈×C₅₃	—	2	1	no	—
161	C₂×C₈×C₅₃	—	2	1	no	—
162	C₂×C₈×C₅₃	—	2	1	no	—
163	C₂×C₈×C₅₃	—	2	1	no	—
164	C₂×C₈×C₅₃	—	2	1	no	—
165	C₂×C₈×C₅₃	—	2	1	no	—
166	C₂×C₈×C₅₃	—	2	1	no	—
167	C₂×C₈×C₅₃	—	2	1	no	—
168	C₂×C₈×C₅₃	—	2	1	no	—
169	C₂×C₈×C₅₃	—	2	1	no	—
170	C₂×C₈×C₅₃	—	2	1	no	—
171	C₂×C₈×C₅₃	—	2	1	no	—
172	C₂×C₈×C₅₃	—	2	1	no	—
173	C₂×C₈×C₅₃	—	2	1	no	—
174	C₂×C₈×C₅₃	—	2	1	no	—
175	C₂×C₈×C₅₃	—	2	1	no	—
176	C₂×C₈×C₅₃	—	2	1	no	—
177	C₂×C₈×C₅₃	—	4	4	no	—
178	C₂×C₈×C₅₃	—	4	4	no	—
179	C₂×C₈×C₅₃	C₂×Q₈	1	2	yes	—
180	C₂×C₈×C₅₃	C₂×Q₈	1	2	yes	—
181	C₄⋊C₄×C₅₃	—	2	2	no	—
182	C₄⋊C₄×C₅₃	—	2	2	no	—
183	C₄⋊C₄×C₅₃	—	2	2	no	—
184	C₄⋊C₄×C₅₃	—	2	2	no	—
185	C₄⋊C₄×C₅₃	—	2	2	no	—
186	C₄⋊C₄×C₅₃	—	2	2	no	—
187	C₄⋊C₄×C₅₃	—	4	1	no	—
188	C₄⋊C₄×C₅₃	—	4	1	no	—
189	C₄⋊C₄×C₅₃	—	4	1	no	—
190	C₂²⋊C₄×C₅₃	—	2	2	no	—
191	C₂²⋊C₄×C₅₃	—	2	2	no	—
192	C₂²⋊C₄×C₅₃	—	2	2	no	—
193	C₂²⋊C₄×C₅₃	M₁₆	1	4	yes	—
194	C₄²×C₅₃	—	2	1	no	—
195	C₄²×C₅₃	—	2	1	no	—
196	C₄²×C₅₃	—	2	1	no	—
197	C₄²×C₅₃	—	2	1	no	—
198	C₄²×C₅₃	—	2	1	no	—
199	C₄²×C₅₃	—	2	1	no	—
200	C₄²×C₅₃	—	2	1	no	—
201	C₄²×C₅₃	—	2	1	no	—
202	C₄²×C₅₃	—	2	1	no	—
203	C₄²×C₅₃	—	2	1	no	—
204	C₄²×C₅₃	—	2	1	no	—
205	C₄²×C₅₃	—	2	1	no	—
206	C₄²×C₅₃	—	4	1	no	—
207	C₄²×C₅₃	—	4	1	no	—
208	C₂³×C₅₃	—	2	1	no	—
209	C₂³×C₅₃	—	2	1	no	—
210	C₂³×C₅₃	—	2	1	no	—
211	C₂³×C₅₃	—	2	1	no	—
212	C₂³×C₅₃	C₄×Q₈	1	2	yes	—
213	C₂³×C₅₃	C₂×M₁₆	1	2	yes	—
214	C₂³×C₅₃	#131	1	2	yes	—
215	C₂³×C₅₃	C₂³×C₄	1	1	yes	characteristic
216	Q₈×C₅₃	—	4	4	no	—
217	C₂×C₄×C₅₃	—	2	1	no	—
218	C₂×C₄×C₅₃	—	2	1	no	—
219	C₂×C₄×C₅₃	—	2	1	no	—
220	C₂×C₄×C₅₃	—	2	1	no	—
221	C₂×C₄×C₅₃	—	2	1	no	—
222	C₂×C₄×C₅₃	—	2	1	no	—
223	C₂×C₄×C₅₃	—	2	1	no	—
224	C₂×C₄×C₅₃	—	2	1	no	—
225	C₂×C₄×C₅₃	—	2	1	no	—
226	C₂×C₄×C₅₃	—	2	1	no	—
227	C₂×C₄×C₅₃	—	2	2	no	—
228	C₂×C₄×C₅₃	—	2	2	no	—
229	C₂×C₄×C₅₃	—	2	2	no	—
230	C₂×C₄×C₅₃	—	2	2	no	—
231	C₂×C₄×C₅₃	—	2	1	no	—
232	C₂×C₄×C₅₃	—	2	1	no	—
233	C₂×C₄×C₅₃	—	2	1	no	—
234	C₂×C₄×C₅₃	—	2	1	no	—
235	C₂×C₄×C₅₃	—	2	1	no	—
236	C₂×C₄×C₅₃	—	2	1	no	—
237	C₂×C₄×C₅₃	—	2	1	no	—
238	C₂×C₄×C₅₃	—	2	1	no	—
239	C₂×C₄×C₅₃	—	2	1	no	—
240	C₂×C₄×C₅₃	—	2	1	no	—
241	C₂×C₄×C₅₃	—	2	1	no	—
242	C₂×C₄×C₅₃	—	2	1	no	—
243	C₂×C₄×C₅₃	—	2	1	no	—
244	C₂×C₄×C₅₃	—	2	1	no	—
245	C₂×C₄×C₅₃	—	2	1	no	—
246	C₂×C₄×C₅₃	—	2	1	no	—
247	C₂×C₄×C₅₃	—	2	1	no	—
248	C₂×C₄×C₅₃	—	2	1	no	—
249	C₂×C₄×C₅₃	—	2	1	no	—
250	C₂×C₄×C₅₃	—	2	1	no	—
251	C₂×C₄×C₅₃	—	2	1	no	—
252	C₂×C₄×C₅₃	—	2	1	no	—
253	C₂×C₄×C₅₃	—	2	1	no	—
254	C₂×C₄×C₅₃	—	2	1	no	—
255	C₂×C₄×C₅₃	—	4	4	no	—
256	C₂×C₄×C₅₃	—	4	4	no	—
257	C₂×C₄×C₅₃	C₄×Q₈	1	2	yes	—
258	C₂×C₄×C₅₃	C₂×M₁₆	1	2	yes	—
259	C₂×C₄×C₅₃	C₂×M₁₆	1	2	yes	—
260	C₂×C₄×C₅₃	#131	1	2	yes	—
261	C₂×C₄×C₅₃	#131	1	2	yes	—
262	C₂×C₄×C₅₃	C₂²×Q₈	1	1	yes	characteristic
263	C₂×C₄×C₅₃	C₂×#41	1	1	yes	characteristic
264	C₂×C₄×C₅₃	H(5,𝔽₂)	1	1	yes	characteristic
265	C₂×C₄×C₅₃	H(5,𝔽₂)	1	1	yes	characteristic
266	C₂×C₄×C₅₃	H(5,𝔽₂)	1	1	yes	characteristic
267	C₂×C₄×C₅₃	2₋¹⁺⁴	1	1	yes	characteristic
268	C₈×C₅₃	—	2	2	no	—
269	C₈×C₅₃	—	2	2	no	—
270	C₈×C₅₃	—	4	1	no	—
271	C₈×C₅₃	—	4	1	no	—
272	C₈×C₅₃	—	4	1	no	—
273	C₈×C₅₃	—	4	1	no	—
274	C₈×C₅₃	—	4	1	no	—
275	C₈×C₅₃	—	4	1	no	—
276	#22291	C₅₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
277	C₂²×C₅₃	—	2	2	no	—
278	C₂²×C₅₃	—	2	2	no	—
279	C₂²×C₅₃	—	2	2	no	—
280	C₂²×C₅₃	—	2	2	no	—
281	C₂²×C₅₃	—	2	2	no	—
282	C₂²×C₅₃	—	2	2	no	—
283	C₂²×C₅₃	#446	1	4	yes	—
284	C₂²×C₅₃	C₂×C₄×Q₈	1	1	yes	characteristic
285	C₂²×C₅₃	#518	1	1	yes	characteristic
286	C₂²×C₅₃	#520	1	1	yes	characteristic
287	C₂²×C₅₃	C₂²×M₁₆	1	1	yes	characteristic
288	C₂²×C₅₃	C₂×#131	1	1	yes	characteristic
289	C₂²×C₅₃	C₂×#131	1	1	yes	characteristic
290	C₂²×C₅₃	C₂×#131	1	1	yes	characteristic
291	C₄×C₅₃	—	2	2	no	—
292	C₄×C₅₃	—	2	2	no	—
293	C₄×C₅₃	—	2	2	no	—
294	C₄×C₅₃	—	2	2	no	—
295	C₄×C₅₃	—	2	2	no	—
296	C₄×C₅₃	—	4	1	no	—
297	C₄×C₅₃	—	4	1	no	—
298	C₄×C₅₃	—	4	1	no	—
299	C₄×C₅₃	—	4	1	no	—
300	C₄×C₅₃	—	4	1	no	—
301	C₄×C₅₃	—	4	1	no	—
302	C₄×C₅₃	—	4	1	no	—
303	C₄×C₅₃	—	4	1	no	—
304	C₄×C₅₃	#538	1	1	yes	characteristic
305	C₄×C₅₃	#539	1	1	yes	characteristic
306	C₄×C₅₃	#543	1	1	yes	characteristic
307	C₄×C₅₃	#544	1	1	yes	characteristic
308	C₄×C₂²⋊Q₈	C₂×C₅₃	1	1	yes	characteristic
309	#1985	C₂×C₅₃	1	1	yes	characteristic, complemented
310	#1984	C₂×C₅₃	1	1	yes	characteristic, complemented
311	#1940	C₂×C₅₃	1	1	yes	characteristic, complemented
312	#1939	C₂×C₅₃	1	1	yes	characteristic, complemented
313	#1917	C₂×C₅₃	1	1	yes	characteristic, complemented
314	#1917	C₂×C₅₃	1	1	yes	characteristic, complemented
315	#1916	C₂×C₅₃	1	1	yes	characteristic, complemented
316	#1916	C₂×C₅₃	1	1	yes	characteristic, complemented
317	#1850	C₂×C₅₃	1	1	yes	characteristic
318	#1850	C₂×C₅₃	1	1	yes	characteristic
319	#1797	C₂×C₅₃	1	1	yes	characteristic
320	#1752	C₂×C₅₃	1	1	yes	characteristic
321	C₈×C₂²⋊C₄	C₂×C₅₃	1	1	yes	characteristic
322	C₂×#336	C₂×C₅₃	1	1	yes	characteristic
323	C₂×C₅₃	—	2	4	no	—
324	C₂×C₅₃	#2329	1	1	yes	characteristic
325	C₂×C₅₃	C₂×#446	1	1	yes	characteristic
326	C₂×C₅₃	#2960	1	1	yes	characteristic
327	C₂×C₅₃	#2974	1	1	yes	characteristic
328	C₂×C₅₃	#2977	1	1	yes	characteristic
329	C₂×C₅₃	#2990	1	1	yes	characteristic
330	C₂×C₅₃	#2994	1	1	yes	characteristic
331	C₂³×C₈	C₂²×C₅₃	1	1	yes	characteristic
332	C₂³×C₈	C₂²×C₅₃	1	1	yes	characteristic
333	C₂×C₂²⋊Q₈	C₄×C₅₃	1	1	yes	characteristic
334	C₂×C₄×Q₈	C₂²×C₅₃	1	1	yes	characteristic
335	C₂²×C₄²	C₂²×C₅₃	1	1	yes	characteristic
336	C₂×C₄⋊C₈	C₂²×C₅₃	1	1	yes	characteristic
337	C₂×C₄⋊C₈	C₂²×C₅₃	1	1	yes	characteristic
338	C₂×C₄⋊C₈	C₂²×C₅₃	1	1	yes	characteristic
339	C₂×C₄⋊C₈	C₂²×C₅₃	1	1	yes	characteristic
340	C₂×C₄⋊C₈	C₂²×C₅₃	1	1	yes	characteristic
341	C₂×C₄⋊C₈	C₂²×C₅₃	1	1	yes	characteristic
342	C₂×C₂²⋊C₈	C₂²×C₅₃	1	1	yes	characteristic
343	C₂×C₂²⋊C₈	C₂²×C₅₃	1	1	yes	characteristic
344	C₂×#97	C₂²×C₅₃	1	1	yes	characteristic
345	C₂×#97	C₂²×C₅₃	1	1	yes	characteristic
346	C₂×C₄×C₈	C₂²×C₅₃	1	1	yes	characteristic
347	C₂×C₄×C₈	C₂²×C₅₃	1	1	yes	characteristic
348	#391	C₄×C₅₃	1	1	yes	characteristic
349	#389	C₄×C₅₃	1	1	yes	characteristic
350	#387	C₄×C₅₃	1	1	yes	characteristic
351	#387	C₄×C₅₃	1	1	yes	characteristic
352	#385	C₄×C₅₃	1	1	yes	characteristic
353	#385	C₄×C₅₃	1	1	yes	characteristic
354	#380	C₄×C₅₃	1	1	yes	characteristic
355	C₄×C₄⋊C₄	C₂²×C₅₃	1	1	yes	characteristic
356	C₄×C₄⋊C₄	C₂²×C₅₃	1	1	yes	characteristic
357	C₄×C₄⋊C₄	C₂²×C₅₃	1	1	yes	characteristic
358	C₄×C₂²⋊C₄	C₂²×C₅₃	1	1	yes	characteristic
359	C₄×C₂²⋊C₄	C₂²×C₅₃	1	1	yes	characteristic
360	#336	C₂²×C₅₃	1	1	yes	characteristic
361	#336	C₂²×C₅₃	1	1	yes	characteristic
362	#336	C₂²×C₅₃	1	1	yes	characteristic
363	#336	C₂²×C₅₃	1	1	yes	characteristic
364	#336	C₂²×C₅₃	1	1	yes	characteristic
365	#336	C₂²×C₅₃	1	1	yes	characteristic
366	#336	C₂²×C₅₃	1	1	yes	characteristic
367	#336	C₂²×C₅₃	1	1	yes	characteristic
368	#336	C₂²×C₅₃	1	1	yes	characteristic
369	#336	C₂²×C₅₃	1	1	yes	characteristic
370	#336	C₂²×C₅₃	1	1	yes	characteristic
371	#336	C₂²×C₅₃	1	1	yes	characteristic
372	#336	C₂²×C₅₃	1	1	yes	characteristic
373	#336	C₂²×C₅₃	1	1	yes	characteristic
374	C₅₃	#22291	1	1	yes	characteristic, direct product factor, Sylow
375	C₂²×Q₈	C₂×C₄×C₅₃	1	1	yes	characteristic
376	C₂³×C₄	C₂×C₄×C₅₃	1	1	yes	characteristic
377	C₂³×C₄	C₂×C₄×C₅₃	1	1	yes	characteristic
378	C₂³×C₄	C₂³×C₅₃	1	1	yes	characteristic
379	C₂²×C₈	—	2	2	no	—
380	C₂²×C₈	—	4	1	no	—
381	C₂²×C₈	—	4	1	no	—
382	C₂²×C₈	—	4	1	no	—
383	C₂²×C₈	—	4	1	no	—
384	C₂²×C₈	Q₈×C₅₃	1	4	yes	complemented
385	C₂²×C₈	C₂³×C₅₃	1	1	yes	characteristic
386	C₂²×C₈	C₂³×C₅₃	1	1	yes	characteristic
387	C₂²×C₈	C₂³×C₅₃	1	1	yes	characteristic
388	C₂²×C₈	C₂³×C₅₃	1	1	yes	characteristic
389	C₂²×C₈	C₂³×C₅₃	1	1	yes	characteristic
390	C₂²×C₈	C₂³×C₅₃	1	1	yes	characteristic
391	C₂²×C₈	C₂³×C₅₃	1	1	yes	characteristic
392	C₂²×C₈	C₂³×C₅₃	1	1	yes	characteristic
393	C₂²⋊Q₈	—	2	4	no	—
394	C₄×Q₈	—	4	1	no	—
395	C₄×Q₈	—	4	1	no	—
396	C₂×C₄⋊C₄	C₂×C₄×C₅₃	1	1	yes	characteristic
397	C₂×C₄⋊C₄	C₂×C₄×C₅₃	1	1	yes	characteristic
398	C₂×C₄⋊C₄	C₂×C₄×C₅₃	1	1	yes	characteristic
399	C₂×C₄⋊C₄	C₂×C₄×C₅₃	1	1	yes	characteristic
400	C₂×C₄⋊C₄	C₂×C₄×C₅₃	1	1	yes	characteristic
401	C₂×C₄⋊C₄	C₂×C₄×C₅₃	1	1	yes	characteristic
402	C₂×C₄⋊C₄	C₂×C₄×C₅₃	1	1	yes	characteristic
403	C₂×C₄⋊C₄	C₂×C₄×C₅₃	1	1	yes	characteristic
404	C₂×C₄⋊C₄	C₂×C₄×C₅₃	1	1	yes	characteristic
405	C₂×C₂²⋊C₄	C₂×C₄×C₅₃	1	1	yes	characteristic
406	C₂×C₂²⋊C₄	C₂×C₄×C₅₃	1	1	yes	characteristic
407	C₂×C₂²⋊C₄	C₂×C₄×C₅₃	1	1	yes	characteristic
408	C₂×C₂²⋊C₄	C₂×C₄×C₅₃	1	1	yes	characteristic
409	C₂×C₄²	—	4	1	no	—
410	C₂×C₄²	—	4	1	no	—
411	C₂×C₄²	C₂×C₄×C₅₃	1	1	yes	characteristic
412	C₂×C₄²	C₂×C₄×C₅₃	1	1	yes	characteristic
413	C₂×C₄²	C₂³×C₅₃	1	1	yes	characteristic
414	C₂×C₄²	C₂³×C₅₃	1	1	yes	characteristic
415	C₂×C₄²	C₂³×C₅₃	1	1	yes	characteristic
416	C₂×C₄²	C₂³×C₅₃	1	1	yes	characteristic
417	C₂×C₄²	C₂³×C₅₃	1	1	yes	characteristic
418	C₂×C₄²	C₂³×C₅₃	1	1	yes	characteristic
419	C₄⋊C₈	—	4	1	no	—
420	C₄⋊C₈	—	4	1	no	—
421	C₄⋊C₈	—	4	1	no	—
422	C₄⋊C₈	—	4	1	no	—
423	C₄⋊C₈	—	4	1	no	—
424	C₄⋊C₈	—	4	1	no	—
425	C₂²⋊C₈	—	2	2	no	—
426	C₂²⋊C₈	—	2	2	no	—
427	#97	—	4	1	no	—
428	#97	—	4	1	no	—
429	C₄×C₈	—	4	1	no	—
430	C₄×C₈	—	4	1	no	—
431	#95	C₂×C₄×C₅₃	1	1	yes	characteristic
432	#95	C₂×C₄×C₅₃	1	1	yes	characteristic
433	#95	C₂×C₄×C₅₃	1	1	yes	characteristic
434	#95	C₂×C₄×C₅₃	1	1	yes	characteristic
435	#95	C₂×C₄×C₅₃	1	1	yes	characteristic
436	#95	C₂×C₄×C₅₃	1	1	yes	characteristic
437	#95	C₂×C₄×C₅₃	1	1	yes	characteristic
438	#95	C₂×C₄×C₅₃	1	1	yes	characteristic
439	#95	C₂×C₄×C₅₃	1	1	yes	characteristic
440	#95	C₂×C₄×C₅₃	1	1	yes	characteristic
441	C₂⁴	C₂²×C₄×C₅₃	1	1	yes	characteristic
442	C₂×Q₈	—	2	2	no	—
443	C₂×Q₈	—	2	2	no	—
444	C₂×Q₈	—	4	1	no	—
445	C₂²×C₄	—	2	1	no	—
446	C₂²×C₄	—	2	1	no	—
447	C₂²×C₄	—	2	1	no	—
448	C₂²×C₄	—	2	1	no	—
449	C₂²×C₄	—	2	2	no	—
450	C₂²×C₄	—	2	2	no	—
451	C₂²×C₄	—	2	2	no	—
452	C₂²×C₄	—	2	2	no	—
453	C₂²×C₄	—	4	1	no	—
454	C₂²×C₄	—	4	1	no	—
455	C₂²×C₄	C₂²×C₄×C₅₃	1	1	yes	characteristic
456	C₂²×C₄	C₂²×C₄×C₅₃	1	1	yes	characteristic
457	C₂²×C₄	C₂²×C₄×C₅₃	1	1	yes	characteristic
458	C₂²×C₄	C₂²×C₄×C₅₃	1	1	yes	characteristic
459	C₂²×C₄	C₂²×C₄×C₅₃	1	1	yes	characteristic
460	C₂²×C₄	C₂²×C₄×C₅₃	1	1	yes	characteristic
461	C₂²×C₄	C₂²×C₄×C₅₃	1	1	yes	characteristic
462	C₂²×C₄	C₂²×C₄×C₅₃	1	1	yes	characteristic
463	C₂²×C₄	C₂²×C₄×C₅₃	1	1	yes	characteristic
464	C₂²×C₄	C₂²×C₄×C₅₃	1	1	yes	characteristic
465	C₂²×C₄	C₂²×C₄×C₅₃	1	1	yes	characteristic
466	C₂²×C₄	C₂²×C₄×C₅₃	1	1	yes	characteristic
467	C₂²×C₄	C₂²×C₄×C₅₃	1	1	yes	characteristic
468	C₂²×C₄	C₂×Q₈×C₅₃	1	2	yes	—
469	C₂²×C₄	#41×C₅₃	1	2	yes	—
470	C₂²×C₄	C₂⁴×C₅₃	1	1	yes	characteristic
471	C₂×C₈	—	2	1	no	—
472	C₂×C₈	—	2	1	no	—
473	C₂×C₈	—	2	1	no	—
474	C₂×C₈	—	2	1	no	—
475	C₂×C₈	—	2	1	no	—
476	C₂×C₈	—	2	1	no	—
477	C₂×C₈	—	2	1	no	—
478	C₂×C₈	—	2	1	no	—
479	C₂×C₈	—	2	1	no	—
480	C₂×C₈	—	2	1	no	—
481	C₂×C₈	—	2	1	no	—
482	C₂×C₈	—	2	1	no	—
483	C₂×C₈	—	2	1	no	—
484	C₂×C₈	—	2	1	no	—
485	C₂×C₈	—	2	1	no	—
486	C₂×C₈	—	2	1	no	—
487	C₂×C₈	—	2	1	no	—
488	C₂×C₈	—	2	1	no	—
489	C₂×C₈	—	2	1	no	—
490	C₂×C₈	—	2	1	no	—
491	C₂×C₈	—	2	1	no	—
492	C₂×C₈	—	2	1	no	—
493	C₂×C₈	—	4	4	no	—
494	C₂×C₈	—	4	4	no	—
495	C₂×C₈	C₂×Q₈×C₅₃	1	2	yes	—
496	C₂×C₈	C₂×Q₈×C₅₃	1	2	yes	—
497	C₄⋊C₄	—	2	2	no	—
498	C₄⋊C₄	—	2	2	no	—
499	C₄⋊C₄	—	2	2	no	—
500	C₄⋊C₄	—	2	2	no	—
501	C₄⋊C₄	—	2	2	no	—
502	C₄⋊C₄	—	2	2	no	—
503	C₄⋊C₄	—	4	1	no	—
504	C₄⋊C₄	—	4	1	no	—
505	C₄⋊C₄	—	4	1	no	—
506	C₂²⋊C₄	—	2	2	no	—
507	C₂²⋊C₄	—	2	2	no	—
508	C₂²⋊C₄	—	2	2	no	—
509	C₂²⋊C₄	M₁₆×C₅₃	1	4	yes	—
510	C₄²	—	2	1	no	—
511	C₄²	—	2	1	no	—
512	C₄²	—	2	1	no	—
513	C₄²	—	2	1	no	—
514	C₄²	—	2	1	no	—
515	C₄²	—	2	1	no	—
516	C₄²	—	2	1	no	—
517	C₄²	—	2	1	no	—
518	C₄²	—	2	1	no	—
519	C₄²	—	2	1	no	—
520	C₄²	—	2	1	no	—
521	C₄²	—	2	1	no	—
522	C₄²	—	4	1	no	—
523	C₄²	—	4	1	no	—
524	C₂³	—	2	1	no	—
525	C₂³	—	2	1	no	—
526	C₂³	—	2	1	no	—
527	C₂³	—	2	1	no	—
528	C₂³	C₄×Q₈×C₅₃	1	2	yes	—
529	C₂³	C₂×M₁₆×C₅₃	1	2	yes	—
530	C₂³	#131×C₅₃	1	2	yes	—
531	C₂³	C₂³×C₄×C₅₃	1	1	yes	characteristic
532	Q₈	—	4	4	no	—
533	C₂×C₄	—	2	1	no	—
534	C₂×C₄	—	2	1	no	—
535	C₂×C₄	—	2	1	no	—
536	C₂×C₄	—	2	1	no	—
537	C₂×C₄	—	2	1	no	—
538	C₂×C₄	—	2	1	no	—
539	C₂×C₄	—	2	1	no	—
540	C₂×C₄	—	2	1	no	—
541	C₂×C₄	—	2	1	no	—
542	C₂×C₄	—	2	1	no	—
543	C₂×C₄	—	2	2	no	—
544	C₂×C₄	—	2	2	no	—
545	C₂×C₄	—	2	2	no	—
546	C₂×C₄	—	2	2	no	—
547	C₂×C₄	—	2	1	no	—
548	C₂×C₄	—	2	1	no	—
549	C₂×C₄	—	2	1	no	—
550	C₂×C₄	—	2	1	no	—
551	C₂×C₄	—	2	1	no	—
552	C₂×C₄	—	2	1	no	—
553	C₂×C₄	—	2	1	no	—
554	C₂×C₄	—	2	1	no	—
555	C₂×C₄	—	2	1	no	—
556	C₂×C₄	—	2	1	no	—
557	C₂×C₄	—	2	1	no	—
558	C₂×C₄	—	2	1	no	—
559	C₂×C₄	—	2	1	no	—
560	C₂×C₄	—	2	1	no	—
561	C₂×C₄	—	2	1	no	—
562	C₂×C₄	—	2	1	no	—
563	C₂×C₄	—	2	1	no	—
564	C₂×C₄	—	2	1	no	—
565	C₂×C₄	—	2	1	no	—
566	C₂×C₄	—	2	1	no	—
567	C₂×C₄	—	2	1	no	—
568	C₂×C₄	—	2	1	no	—
569	C₂×C₄	—	2	1	no	—
570	C₂×C₄	—	2	1	no	—
571	C₂×C₄	—	4	4	no	—
572	C₂×C₄	—	4	4	no	—
573	C₂×C₄	C₄×Q₈×C₅₃	1	2	yes	—
574	C₂×C₄	C₂×M₁₆×C₅₃	1	2	yes	—
575	C₂×C₄	C₂×M₁₆×C₅₃	1	2	yes	—
576	C₂×C₄	#131×C₅₃	1	2	yes	—
577	C₂×C₄	#131×C₅₃	1	2	yes	—
578	C₂×C₄	C₂²×Q₈×C₅₃	1	1	yes	characteristic
579	C₂×C₄	C₂×#41×C₅₃	1	1	yes	characteristic
580	C₂×C₄	H(5,𝔽₂)×C₅₃	1	1	yes	characteristic
581	C₂×C₄	H(5,𝔽₂)×C₅₃	1	1	yes	characteristic
582	C₂×C₄	H(5,𝔽₂)×C₅₃	1	1	yes	characteristic
583	C₂×C₄	2₋¹⁺⁴×C₅₃	1	1	yes	characteristic
584	C₈	—	2	2	no	—
585	C₈	—	2	2	no	—
586	C₈	—	4	1	no	—
587	C₈	—	4	1	no	—
588	C₈	—	4	1	no	—
589	C₈	—	4	1	no	—
590	C₈	—	4	1	no	—
591	C₈	—	4	1	no	—
592	C₂²	—	2	2	no	—
593	C₂²	—	2	2	no	—
594	C₂²	—	2	2	no	—
595	C₂²	—	2	2	no	—
596	C₂²	—	2	2	no	—
597	C₂²	—	2	2	no	—
598	C₂²	C₅₃×#446	1	4	yes	—
599	C₂²	C₂×C₄×Q₈×C₅₃	1	1	yes	characteristic
600	C₂²	C₅₃×#518	1	1	yes	characteristic
601	C₂²	C₅₃×#520	1	1	yes	characteristic
602	C₂²	C₂²×M₁₆×C₅₃	1	1	yes	characteristic
603	C₂²	C₂×#131×C₅₃	1	1	yes	characteristic
604	C₂²	C₂×#131×C₅₃	1	1	yes	characteristic
605	C₂²	C₂×#131×C₅₃	1	1	yes	characteristic
606	C₄	—	2	2	no	—
607	C₄	—	2	2	no	—
608	C₄	—	2	2	no	—
609	C₄	—	2	2	no	—
610	C₄	—	2	2	no	—
611	C₄	—	4	1	no	—
612	C₄	—	4	1	no	—
613	C₄	—	4	1	no	—
614	C₄	—	4	1	no	—
615	C₄	—	4	1	no	—
616	C₄	—	4	1	no	—
617	C₄	—	4	1	no	—
618	C₄	—	4	1	no	—
619	C₄	C₅₃×#538	1	1	yes	characteristic
620	C₄	C₅₃×#539	1	1	yes	characteristic
621	C₄	C₅₃×#543	1	1	yes	characteristic
622	C₄	C₅₃×#544	1	1	yes	characteristic
623	C₂	—	2	4	no	—
624	C₂	C₅₃×#2329	1	1	yes	characteristic
625	C₂	C₂×C₅₃×#446	1	1	yes	characteristic
626	C₂	C₅₃×#2960	1	1	yes	characteristic
627	C₂	C₅₃×#2974	1	1	yes	characteristic
628	C₂	C₅₃×#2977	1	1	yes	characteristic
629	C₂	C₅₃×#2990	1	1	yes	characteristic
630	C₂	C₅₃×#2994	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×#1985	C₂	semidirect
3. C₅₃×#1984	C₂	semidirect
4. C₅₃×#1940	C₂	semidirect
5. C₅₃×#1939	C₂	semidirect
6. C₅₃×#1917	C₂	semidirect
7. C₅₃×#1917	C₂	semidirect
8. C₅₃×#1916	C₂	semidirect
9. C₅₃×#1916	C₂	semidirect
68. C₂²×C₈×C₅₃	Q₈	semidirect
276. #22291	C₅₃	direct (N × H)
309. #1985	C₂×C₅₃	semidirect
310. #1984	C₂×C₅₃	semidirect
311. #1940	C₂×C₅₃	semidirect
312. #1939	C₂×C₅₃	semidirect
313. #1917	C₂×C₅₃	semidirect
314. #1917	C₂×C₅₃	semidirect
315. #1916	C₂×C₅₃	semidirect
316. #1916	C₂×C₅₃	semidirect
374. C₅₃	#22291	direct (N × H)
384. C₂²×C₈	Q₈×C₅₃	semidirect

Maximal subgroups

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

subgroup	index
C₄×C₂²⋊Q₈×C₅₃	2
C₅₃×#1985	2
C₅₃×#1984	2
C₅₃×#1940	2
C₅₃×#1939	2
C₅₃×#1917	2
C₅₃×#1916	2
C₅₃×#1850	2
C₅₃×#1797	2
C₅₃×#1752	2
C₈×C₂²⋊C₄×C₅₃	2
C₂×C₅₃×#336	2
#22291	53

Subgroups — C53×#22291 #27229803

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅₃×#22291 #27229803