Subgroups — C₅₃×#24283 #27231795

Order 13568.

1304 conjugacy classes of subgroups.

The subgroup lattice has 760 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#137×C₅₃	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×C₅₃×#473	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×C₅₃×#471	C₂	1	1	yes	maximal, characteristic, complemented
4	C₅₃×#3005	C₂	1	1	yes	maximal, characteristic
5	C₅₃×#3004	C₂	1	1	yes	maximal, characteristic, complemented
6	C₅₃×#2930	C₂	1	1	yes	maximal, characteristic, complemented
7	C₅₃×#2047	C₂	1	1	yes	maximal, characteristic, complemented
8	C₅₃×#2046	C₂	1	1	yes	maximal, characteristic, complemented
9	C₅₃×#2020	C₂	1	1	yes	maximal, characteristic, complemented
10	C₅₃×#2019	C₂	1	1	yes	maximal, characteristic, complemented
11	C₅₃×#1979	C₂	1	1	yes	maximal, characteristic, complemented
12	C₅₃×#1977	C₂	1	1	yes	maximal, characteristic, complemented
13	C₅₃×#1811	C₂	1	1	yes	maximal, characteristic, complemented
14	C₅₃×#1785	C₂	1	1	yes	maximal, characteristic, complemented
15	C₅₃×#1784	C₂	1	1	yes	maximal, characteristic, complemented
16	C₂²×#41×C₅₃	C₂²	1	1	yes	characteristic
17	C₂³×Q₈×C₅₃	C₂²	1	1	yes	characteristic
18	C₂×#137×C₅₃	—	2	4	no	—
19	C₂×#137×C₅₃	—	2	4	no	—
20	C₂×#137×C₅₃	—	2	2	no	—
21	C₂×#137×C₅₃	—	2	2	no	—
22	C₂×#137×C₅₃	C₂²	1	1	yes	characteristic
23	C₂×#137×C₅₃	C₂²	1	1	yes	characteristic
24	C₂×#137×C₅₃	C₂²	1	1	yes	characteristic
25	C₂×AGL(1,ℤ/8)×C₅₃	C₂²	1	1	yes	characteristic
26	C₂²×Q₁₆×C₅₃	C₂²	1	1	yes	characteristic
27	C₂²×Q₁₆×C₅₃	C₂²	1	1	yes	characteristic
28	C₂²×SD₁₆×C₅₃	C₂²	1	1	yes	characteristic
29	C₂²×SD₁₆×C₅₃	C₂²	1	1	yes	characteristic
30	C₂×#131×C₅₃	C₂²	1	1	yes	characteristic
31	C₂²×M₁₆×C₅₃	C₂²	1	1	yes	characteristic
32	C₅₃×#525	C₂²	1	1	yes	characteristic
33	C₅₃×#473	—	2	4	no	—
34	C₅₃×#471	—	2	4	no	—
35	C₅₃×#452	—	2	2	no	—
36	C₅₃×#451	—	2	2	no	—
37	C₅₃×#449	—	2	2	no	—
38	C₅₃×#448	—	2	2	no	—
39	C₅₃×#436	—	2	2	no	—
40	C₅₃×#435	—	2	2	no	—
41	C₂×#108×C₅₃	C₂²	1	1	yes	characteristic
42	C₂×#108×C₅₃	C₂²	1	1	yes	characteristic
43	C₅₃×#424	C₂²	1	1	yes	characteristic
44	C₅₃×#419	C₂²	1	1	yes	characteristic
45	C₅₃×#419	C₂²	1	1	yes	characteristic
46	C₅₃×#419	C₂²	1	1	yes	characteristic
47	C₅₃×#418	C₂²	1	1	yes	characteristic
48	C₂×#101×C₅₃	C₂²	1	1	yes	characteristic
49	C₂×#101×C₅₃	C₂²	1	1	yes	characteristic
50	C₂×#101×C₅₃	C₂²	1	1	yes	characteristic
51	C₂×#101×C₅₃	C₂²	1	1	yes	characteristic
52	C₂×#101×C₅₃	C₂²	1	1	yes	characteristic
53	C₂×#101×C₅₃	C₂²	1	1	yes	characteristic
54	C₂×#100×C₅₃	C₂²	1	1	yes	characteristic
55	C₂×#100×C₅₃	C₂²	1	1	yes	characteristic
56	C₅₃×#408	C₂²	1	1	yes	characteristic
57	C₅₃×#407	C₂²	1	1	yes	characteristic
58	C₅₃×#407	C₂²	1	1	yes	characteristic
59	C₅₃×#407	C₂²	1	1	yes	characteristic
60	C₅₃×#405	C₂²	1	1	yes	characteristic
61	C₅₃×#341	C₂²	1	1	yes	characteristic
62	C₅₃×#341	C₂²	1	1	yes	characteristic
63	C₂×#41×C₅₃	—	2	2	no	—
64	C₂×#41×C₅₃	—	2	2	no	—
65	C₂×#41×C₅₃	—	2	2	no	—
66	C₂×#41×C₅₃	—	2	2	no	—
67	C₂×#41×C₅₃	—	4	1	no	—
68	C₂×#41×C₅₃	—	4	1	no	—
69	C₂×#41×C₅₃	C₂³	1	1	yes	characteristic
70	C₂²×Q₈×C₅₃	—	2	1	no	—
71	C₂²×Q₈×C₅₃	—	2	2	no	—
72	C₂²×Q₈×C₅₃	—	2	2	no	—
73	C₂²×Q₈×C₅₃	—	2	2	no	—
74	C₂²×Q₈×C₅₃	—	2	2	no	—
75	C₂²×Q₈×C₅₃	—	4	1	no	—
76	C₂²×Q₈×C₅₃	—	4	1	no	—
77	C₂²×Q₈×C₅₃	C₂³	1	1	yes	characteristic
78	C₂²×Q₈×C₅₃	C₂³	1	1	yes	characteristic
79	C₂²×Q₈×C₅₃	C₂³	1	1	yes	characteristic
80	C₂²×D₄×C₅₃	—	2	1	no	—
81	C₂²×D₄×C₅₃	C₂³	1	1	yes	characteristic
82	C₂³×C₄×C₅₃	—	2	1	no	—
83	C₂³×C₄×C₅₃	—	2	1	no	—
84	C₂³×C₄×C₅₃	C₂³	1	1	yes	characteristic
85	#137×C₅₃	—	2	16	no	—
86	#137×C₅₃	—	2	16	no	—
87	#137×C₅₃	—	2	2	no	—
88	#137×C₅₃	—	2	2	no	—
89	#137×C₅₃	—	2	2	no	—
90	#137×C₅₃	—	4	1	no	—
91	#137×C₅₃	—	4	1	no	—
92	#137×C₅₃	—	4	1	no	—
93	AGL(1,ℤ/8)×C₅₃	—	2	2	no	—
94	AGL(1,ℤ/8)×C₅₃	—	4	1	no	—
95	C₂×Q₁₆×C₅₃	—	2	2	no	—
96	C₂×Q₁₆×C₅₃	—	2	1	no	—
97	C₂×Q₁₆×C₅₃	—	2	1	no	—
98	C₂×Q₁₆×C₅₃	—	2	2	no	—
99	C₂×Q₁₆×C₅₃	—	2	2	no	—
100	C₂×Q₁₆×C₅₃	—	2	2	no	—
101	C₂×Q₁₆×C₅₃	—	2	1	no	—
102	C₂×Q₁₆×C₅₃	—	2	1	no	—
103	C₂×Q₁₆×C₅₃	—	2	1	no	—
104	C₂×Q₁₆×C₅₃	D₄	1	4	yes	complemented
105	C₂×SD₁₆×C₅₃	—	2	2	no	—
106	C₂×SD₁₆×C₅₃	—	2	1	no	—
107	C₂×SD₁₆×C₅₃	—	2	1	no	—
108	C₂×SD₁₆×C₅₃	—	2	2	no	—
109	C₂×SD₁₆×C₅₃	—	2	2	no	—
110	C₂×SD₁₆×C₅₃	—	2	1	no	—
111	C₂×SD₁₆×C₅₃	—	2	2	no	—
112	C₂×SD₁₆×C₅₃	—	2	1	no	—
113	C₂×SD₁₆×C₅₃	—	2	1	no	—
114	C₂×SD₁₆×C₅₃	—	2	1	no	—
115	C₂×SD₁₆×C₅₃	D₄	1	4	yes	complemented
116	C₂×D₈×C₅₃	—	2	1	no	—
117	#131×C₅₃	—	2	2	no	—
118	#131×C₅₃	—	2	2	no	—
119	C₂×M₁₆×C₅₃	—	2	2	no	—
120	C₂×M₁₆×C₅₃	—	2	2	no	—
121	C₂×M₁₆×C₅₃	—	2	1	no	—
122	C₂×M₁₆×C₅₃	D₄	1	2	yes	—
123	C₂×M₁₆×C₅₃	D₄	1	2	yes	—
124	C₂×M₁₆×C₅₃	C₂³	1	1	yes	characteristic
125	C₂×M₁₆×C₅₃	C₂³	1	1	yes	characteristic
126	C₂×M₁₆×C₅₃	C₂³	1	1	yes	characteristic
127	C₂×M₁₆×C₅₃	C₂³	1	1	yes	characteristic
128	C₂×M₁₆×C₅₃	C₂³	1	1	yes	characteristic
129	C₂×M₁₆×C₅₃	C₂³	1	1	yes	characteristic
130	C₂²×C₈×C₅₃	—	2	1	no	—
131	C₂²×C₈×C₅₃	C₂³	1	1	yes	characteristic
132	C₂²×C₈×C₅₃	C₂³	1	1	yes	characteristic
133	#123×C₅₃	—	2	1	no	—
134	C₂²⋊Q₈×C₅₃	D₄	1	2	yes	—
135	#121×C₅₃	D₄	1	2	yes	—
136	#120×C₅₃	—	2	1	no	—
137	C₄×D₄×C₅₃	—	2	2	no	—
138	#117×C₅₃	C₂³	1	1	yes	characteristic
139	#108×C₅₃	—	2	2	no	—
140	#108×C₅₃	—	2	2	no	—
141	C₄⋊C₈×C₅₃	—	2	2	no	—
142	#103×C₅₃	—	2	2	no	—
143	#103×C₅₃	—	2	2	no	—
144	#103×C₅₃	—	2	2	no	—
145	#102×C₅₃	—	2	2	no	—
146	#101×C₅₃	—	2	2	no	—
147	#101×C₅₃	—	2	2	no	—
148	#101×C₅₃	—	2	2	no	—
149	#101×C₅₃	—	4	1	no	—
150	#101×C₅₃	—	4	1	no	—
151	#101×C₅₃	—	4	1	no	—
152	#100×C₅₃	—	2	2	no	—
153	#100×C₅₃	—	4	1	no	—
154	C₂²⋊C₈×C₅₃	—	2	2	no	—
155	C₂²⋊C₈×C₅₃	—	2	2	no	—
156	C₂²⋊C₈×C₅₃	—	2	2	no	—
157	C₂²⋊C₈×C₅₃	—	2	2	no	—
158	#97×C₅₃	—	2	1	no	—
159	C₄×C₈×C₅₃	—	2	1	no	—
160	C₂⁴×C₅₃	—	2	1	no	—
161	C₂⁴×C₅₃	#41	1	1	yes	characteristic
162	#41×C₅₃	—	2	8	no	—
163	#41×C₅₃	—	2	1	no	—
164	#41×C₅₃	—	2	1	no	—
165	#41×C₅₃	—	2	1	no	—
166	#41×C₅₃	—	2	1	no	—
167	#41×C₅₃	—	4	4	no	—
168	#41×C₅₃	—	4	4	no	—
169	#41×C₅₃	—	4	4	no	—
170	C₂×Q₈×C₅₃	—	2	1	no	—
171	C₂×Q₈×C₅₃	—	2	1	no	—
172	C₂×Q₈×C₅₃	—	2	1	no	—
173	C₂×Q₈×C₅₃	—	2	8	no	—
174	C₂×Q₈×C₅₃	—	2	1	no	—
175	C₂×Q₈×C₅₃	—	2	1	no	—
176	C₂×Q₈×C₅₃	—	2	1	no	—
177	C₂×Q₈×C₅₃	—	2	1	no	—
178	C₂×Q₈×C₅₃	—	2	1	no	—
179	C₂×Q₈×C₅₃	—	2	1	no	—
180	C₂×Q₈×C₅₃	—	2	1	no	—
181	C₂×Q₈×C₅₃	—	2	1	no	—
182	C₂×Q₈×C₅₃	—	2	1	no	—
183	C₂×Q₈×C₅₃	—	2	1	no	—
184	C₂×Q₈×C₅₃	—	2	1	no	—
185	C₂×Q₈×C₅₃	—	2	1	no	—
186	C₂×Q₈×C₅₃	—	4	2	no	—
187	C₂×Q₈×C₅₃	—	4	2	no	—
188	C₂×Q₈×C₅₃	—	4	4	no	—
189	C₂×Q₈×C₅₃	—	4	4	no	—
190	C₂×Q₈×C₅₃	—	4	4	no	—
191	C₂×Q₈×C₅₃	—	4	2	no	—
192	C₂×Q₈×C₅₃	C₂×D₄	1	2	yes	—
193	C₂×Q₈×C₅₃	C₂×D₄	1	2	yes	—
194	C₂×Q₈×C₅₃	C₂×D₄	1	2	yes	—
195	C₂×Q₈×C₅₃	C₂×D₄	1	1	yes	characteristic
196	C₂×D₄×C₅₃	—	2	1	no	—
197	C₂×D₄×C₅₃	—	2	2	no	—
198	C₂×D₄×C₅₃	—	2	1	no	—
199	C₂×D₄×C₅₃	—	2	1	no	—
200	C₂×D₄×C₅₃	—	2	1	no	—
201	C₂×D₄×C₅₃	—	2	1	no	—
202	C₂×D₄×C₅₃	—	2	1	no	—
203	C₂×D₄×C₅₃	—	4	2	no	—
204	C₂×D₄×C₅₃	—	4	2	no	—
205	C₂×D₄×C₅₃	—	4	2	no	—
206	C₂×D₄×C₅₃	C₂×D₄	1	2	yes	—
207	C₂×D₄×C₅₃	C₂×D₄	1	1	yes	characteristic
208	C₂²×C₄×C₅₃	—	2	1	no	—
209	C₂²×C₄×C₅₃	—	2	1	no	—
210	C₂²×C₄×C₅₃	—	2	1	no	—
211	C₂²×C₄×C₅₃	—	2	1	no	—
212	C₂²×C₄×C₅₃	—	2	1	no	—
213	C₂²×C₄×C₅₃	—	2	1	no	—
214	C₂²×C₄×C₅₃	—	2	1	no	—
215	C₂²×C₄×C₅₃	—	2	1	no	—
216	C₂²×C₄×C₅₃	—	4	2	no	—
217	C₂²×C₄×C₅₃	—	4	2	no	—
218	C₂²×C₄×C₅₃	—	4	2	no	—
219	C₂²×C₄×C₅₃	—	4	2	no	—
220	C₂²×C₄×C₅₃	—	4	2	no	—
221	C₂²×C₄×C₅₃	—	4	2	no	—
222	C₂²×C₄×C₅₃	C₂×D₄	1	2	yes	—
223	C₂²×C₄×C₅₃	C₂×D₄	1	2	yes	—
224	C₂²×C₄×C₅₃	#41	1	1	yes	characteristic
225	C₂²×C₄×C₅₃	C₂⁴	1	1	yes	characteristic
226	Q₁₆×C₅₃	—	2	8	no	—
227	Q₁₆×C₅₃	—	2	4	no	—
228	Q₁₆×C₅₃	—	2	4	no	—
229	Q₁₆×C₅₃	—	4	2	no	—
230	Q₁₆×C₅₃	—	4	2	no	—
231	Q₁₆×C₅₃	—	4	2	no	—
232	SD₁₆×C₅₃	—	2	8	no	—
233	SD₁₆×C₅₃	—	2	4	no	—
234	SD₁₆×C₅₃	—	2	4	no	—
235	SD₁₆×C₅₃	—	4	2	no	—
236	SD₁₆×C₅₃	—	4	2	no	—
237	SD₁₆×C₅₃	—	4	2	no	—
238	SD₁₆×C₅₃	—	4	2	no	—
239	D₈×C₅₃	—	4	2	no	—
240	M₁₆×C₅₃	—	2	4	no	—
241	M₁₆×C₅₃	—	2	4	no	—
242	M₁₆×C₅₃	—	2	1	no	—
243	M₁₆×C₅₃	—	2	1	no	—
244	M₁₆×C₅₃	—	2	1	no	—
245	M₁₆×C₅₃	—	2	1	no	—
246	M₁₆×C₅₃	—	2	1	no	—
247	M₁₆×C₅₃	—	2	1	no	—
248	M₁₆×C₅₃	—	2	1	no	—
249	M₁₆×C₅₃	—	2	1	no	—
250	M₁₆×C₅₃	—	2	1	no	—
251	M₁₆×C₅₃	—	2	1	no	—
252	M₁₆×C₅₃	—	2	1	no	—
253	M₁₆×C₅₃	—	2	1	no	—
254	M₁₆×C₅₃	—	4	2	no	—
255	C₂×C₈×C₅₃	—	2	1	no	—
256	C₂×C₈×C₅₃	—	2	1	no	—
257	C₂×C₈×C₅₃	—	2	1	no	—
258	C₂×C₈×C₅₃	—	2	1	no	—
259	C₂×C₈×C₅₃	—	2	1	no	—
260	C₂×C₈×C₅₃	—	2	1	no	—
261	C₂×C₈×C₅₃	—	2	1	no	—
262	C₂×C₈×C₅₃	—	2	1	no	—
263	C₂×C₈×C₅₃	—	4	2	no	—
264	C₂×C₈×C₅₃	C₂×D₄	1	2	yes	—
265	C₂×C₈×C₅₃	C₂×D₄	1	1	yes	characteristic
266	C₂×C₈×C₅₃	C₂×D₄	1	1	yes	characteristic
267	C₂×C₈×C₅₃	C₂×D₄	1	2	yes	—
268	C₂×C₈×C₅₃	C₂×D₄	1	2	yes	—
269	C₄⋊C₄×C₅₃	—	2	2	no	—
270	C₄⋊C₄×C₅₃	C₂×D₄	1	2	yes	—
271	C₂²⋊C₄×C₅₃	—	2	2	no	—
272	C₂²⋊C₄×C₅₃	—	2	1	no	—
273	C₂²⋊C₄×C₅₃	—	2	2	no	—
274	C₄²×C₅₃	—	2	1	no	—
275	C₂³×C₅₃	—	2	1	no	—
276	C₂³×C₅₃	—	2	1	no	—
277	C₂³×C₅₃	—	2	1	no	—
278	C₂³×C₅₃	—	2	2	no	—
279	C₂³×C₅₃	—	4	2	no	—
280	C₂³×C₅₃	—	4	1	no	—
281	C₂³×C₅₃	—	4	1	no	—
282	C₂³×C₅₃	—	8	2	no	—
283	C₂³×C₅₃	#121	1	2	yes	—
284	C₂³×C₅₃	C₂×#41	1	1	yes	characteristic
285	Q₈×C₅₃	—	2	2	no	—
286	Q₈×C₅₃	—	2	2	no	—
287	Q₈×C₅₃	—	2	2	no	—
288	Q₈×C₅₃	—	2	2	no	—
289	Q₈×C₅₃	—	2	2	no	—
290	Q₈×C₅₃	—	2	2	no	—
291	Q₈×C₅₃	—	2	1	no	—
292	Q₈×C₅₃	—	2	1	no	—
293	Q₈×C₅₃	—	4	8	no	—
294	Q₈×C₅₃	—	4	1	no	—
295	Q₈×C₅₃	—	4	1	no	—
296	Q₈×C₅₃	—	4	1	no	—
297	Q₈×C₅₃	—	4	1	no	—
298	Q₈×C₅₃	—	4	1	no	—
299	Q₈×C₅₃	—	4	1	no	—
300	D₄×C₅₃	—	2	2	no	—
301	D₄×C₅₃	—	2	2	no	—
302	D₄×C₅₃	—	2	1	no	—
303	D₄×C₅₃	—	2	1	no	—
304	D₄×C₅₃	—	4	8	no	—
305	D₄×C₅₃	—	4	1	no	—
306	D₄×C₅₃	—	4	1	no	—
307	D₄×C₅₃	—	4	1	no	—
308	D₄×C₅₃	—	4	1	no	—
309	D₄×C₅₃	—	8	2	no	—
310	C₂×C₄×C₅₃	—	2	1	no	—
311	C₂×C₄×C₅₃	—	2	1	no	—
312	C₂×C₄×C₅₃	—	2	2	no	—
313	C₂×C₄×C₅₃	—	2	2	no	—
314	C₂×C₄×C₅₃	—	2	2	no	—
315	C₂×C₄×C₅₃	—	2	2	no	—
316	C₂×C₄×C₅₃	—	2	1	no	—
317	C₂×C₄×C₅₃	—	2	2	no	—
318	C₂×C₄×C₅₃	—	2	2	no	—
319	C₂×C₄×C₅₃	—	2	2	no	—
320	C₂×C₄×C₅₃	—	2	2	no	—
321	C₂×C₄×C₅₃	—	4	8	no	—
322	C₂×C₄×C₅₃	—	4	8	no	—
323	C₂×C₄×C₅₃	—	4	1	no	—
324	C₂×C₄×C₅₃	—	4	1	no	—
325	C₂×C₄×C₅₃	—	4	1	no	—
326	C₂×C₄×C₅₃	—	4	1	no	—
327	C₂×C₄×C₅₃	—	4	1	no	—
328	C₂×C₄×C₅₃	—	4	1	no	—
329	C₂×C₄×C₅₃	—	4	1	no	—
330	C₂×C₄×C₅₃	—	4	1	no	—
331	C₂×C₄×C₅₃	#121	1	2	yes	—
332	C₂×C₄×C₅₃	C₂²×D₄	1	1	yes	characteristic
333	C₂×C₄×C₅₃	C₂²×D₄	1	1	yes	characteristic
334	C₂×C₄×C₅₃	C₂²×D₄	1	1	yes	characteristic
335	C₂×C₄×C₅₃	C₂²×D₄	1	1	yes	characteristic
336	C₂×C₄×C₅₃	H(5,𝔽₂)	1	1	yes	characteristic
337	C₂×C₄×C₅₃	H(5,𝔽₂)	1	1	yes	characteristic
338	C₈×C₅₃	—	2	2	no	—
339	C₈×C₅₃	—	4	1	no	—
340	C₈×C₅₃	—	4	1	no	—
341	C₈×C₅₃	—	4	1	no	—
342	C₈×C₅₃	—	4	1	no	—
343	C₈×C₅₃	—	4	1	no	—
344	C₈×C₅₃	—	4	1	no	—
345	C₈×C₅₃	#120	1	2	yes	—
346	C₈×C₅₃	#120	1	2	yes	—
347	#24283	C₅₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
348	C₂²×C₅₃	—	2	2	no	—
349	C₂²×C₅₃	—	2	2	no	—
350	C₂²×C₅₃	—	2	2	no	—
351	C₂²×C₅₃	—	2	2	no	—
352	C₂²×C₅₃	—	4	2	no	—
353	C₂²×C₅₃	—	4	2	no	—
354	C₂²×C₅₃	—	4	2	no	—
355	C₂²×C₅₃	—	4	1	no	—
356	C₂²×C₅₃	—	4	2	no	—
357	C₂²×C₅₃	—	8	2	no	—
358	C₂²×C₅₃	—	8	2	no	—
359	C₂²×C₅₃	—	8	2	no	—
360	C₂²×C₅₃	C₂×#121	1	1	yes	characteristic
361	C₂²×C₅₃	#546	1	1	yes	characteristic
362	C₂²×C₅₃	#546	1	1	yes	characteristic
363	C₄×C₅₃	—	2	2	no	—
364	C₄×C₅₃	—	4	2	no	—
365	C₄×C₅₃	—	4	2	no	—
366	C₄×C₅₃	—	4	2	no	—
367	C₄×C₅₃	—	4	1	no	—
368	C₄×C₅₃	—	4	2	no	—
369	C₄×C₅₃	C₂×#120	1	1	yes	characteristic
370	C₄×C₅₃	#535	1	1	yes	characteristic
371	C₄×C₅₃	D₄²	1	1	yes	characteristic
372	C₄×C₅₃	D₄²	1	1	yes	characteristic
373	C₂²×#137	C₂×C₅₃	1	1	yes	characteristic, complemented
374	C₂×#473	C₂×C₅₃	1	1	yes	characteristic, complemented
375	C₂×#471	C₂×C₅₃	1	1	yes	characteristic, complemented
376	#3005	C₂×C₅₃	1	1	yes	characteristic
377	#3004	C₂×C₅₃	1	1	yes	characteristic, complemented
378	#2930	C₂×C₅₃	1	1	yes	characteristic, complemented
379	#2047	C₂×C₅₃	1	1	yes	characteristic, complemented
380	#2046	C₂×C₅₃	1	1	yes	characteristic, complemented
381	#2020	C₂×C₅₃	1	1	yes	characteristic, complemented
382	#2019	C₂×C₅₃	1	1	yes	characteristic, complemented
383	#1979	C₂×C₅₃	1	1	yes	characteristic, complemented
384	#1977	C₂×C₅₃	1	1	yes	characteristic, complemented
385	#1811	C₂×C₅₃	1	1	yes	characteristic, complemented
386	#1785	C₂×C₅₃	1	1	yes	characteristic, complemented
387	#1784	C₂×C₅₃	1	1	yes	characteristic, complemented
388	C₂×C₅₃	—	2	1	no	—
389	C₂×C₅₃	—	2	1	no	—
390	C₂×C₅₃	—	4	2	no	—
391	C₂×C₅₃	—	8	1	no	—
392	C₂×C₅₃	—	8	2	no	—
393	C₂×C₅₃	#2408	1	1	yes	characteristic
394	C₂×C₅₃	#3069	1	2	yes	—
395	C₂²×#41	C₂²×C₅₃	1	1	yes	characteristic
396	C₂³×Q₈	C₂²×C₅₃	1	1	yes	characteristic
397	C₂×#137	—	2	4	no	—
398	C₂×#137	—	2	4	no	—
399	C₂×#137	—	2	2	no	—
400	C₂×#137	—	2	2	no	—
401	C₂×#137	C₂²×C₅₃	1	1	yes	characteristic
402	C₂×#137	C₂²×C₅₃	1	1	yes	characteristic
403	C₂×#137	C₂²×C₅₃	1	1	yes	characteristic
404	C₂×AGL(1,ℤ/8)	C₂²×C₅₃	1	1	yes	characteristic
405	C₂²×Q₁₆	C₂²×C₅₃	1	1	yes	characteristic
406	C₂²×Q₁₆	C₂²×C₅₃	1	1	yes	characteristic
407	C₂²×SD₁₆	C₂²×C₅₃	1	1	yes	characteristic
408	C₂²×SD₁₆	C₂²×C₅₃	1	1	yes	characteristic
409	C₂×#131	C₂²×C₅₃	1	1	yes	characteristic
410	C₂²×M₁₆	C₂²×C₅₃	1	1	yes	characteristic
411	#525	C₂²×C₅₃	1	1	yes	characteristic
412	#473	—	2	4	no	—
413	#471	—	2	4	no	—
414	#452	—	2	2	no	—
415	#451	—	2	2	no	—
416	#449	—	2	2	no	—
417	#448	—	2	2	no	—
418	#436	—	2	2	no	—
419	#435	—	2	2	no	—
420	C₂×#108	C₂²×C₅₃	1	1	yes	characteristic
421	C₂×#108	C₂²×C₅₃	1	1	yes	characteristic
422	#424	C₂²×C₅₃	1	1	yes	characteristic
423	#419	C₂²×C₅₃	1	1	yes	characteristic
424	#419	C₂²×C₅₃	1	1	yes	characteristic
425	#419	C₂²×C₅₃	1	1	yes	characteristic
426	#418	C₂²×C₅₃	1	1	yes	characteristic
427	C₂×#101	C₂²×C₅₃	1	1	yes	characteristic
428	C₂×#101	C₂²×C₅₃	1	1	yes	characteristic
429	C₂×#101	C₂²×C₅₃	1	1	yes	characteristic
430	C₂×#101	C₂²×C₅₃	1	1	yes	characteristic
431	C₂×#101	C₂²×C₅₃	1	1	yes	characteristic
432	C₂×#101	C₂²×C₅₃	1	1	yes	characteristic
433	C₂×#100	C₂²×C₅₃	1	1	yes	characteristic
434	C₂×#100	C₂²×C₅₃	1	1	yes	characteristic
435	#408	C₂²×C₅₃	1	1	yes	characteristic
436	#407	C₂²×C₅₃	1	1	yes	characteristic
437	#407	C₂²×C₅₃	1	1	yes	characteristic
438	#407	C₂²×C₅₃	1	1	yes	characteristic
439	#405	C₂²×C₅₃	1	1	yes	characteristic
440	#341	C₂²×C₅₃	1	1	yes	characteristic
441	#341	C₂²×C₅₃	1	1	yes	characteristic
442	C₅₃	#24283	1	1	yes	characteristic, direct product factor, Sylow
443	C₂×#41	—	2	2	no	—
444	C₂×#41	—	2	2	no	—
445	C₂×#41	—	2	2	no	—
446	C₂×#41	—	2	2	no	—
447	C₂×#41	—	4	1	no	—
448	C₂×#41	—	4	1	no	—
449	C₂×#41	C₂³×C₅₃	1	1	yes	characteristic
450	C₂²×Q₈	—	2	1	no	—
451	C₂²×Q₈	—	2	2	no	—
452	C₂²×Q₈	—	2	2	no	—
453	C₂²×Q₈	—	2	2	no	—
454	C₂²×Q₈	—	2	2	no	—
455	C₂²×Q₈	—	4	1	no	—
456	C₂²×Q₈	—	4	1	no	—
457	C₂²×Q₈	C₂³×C₅₃	1	1	yes	characteristic
458	C₂²×Q₈	C₂³×C₅₃	1	1	yes	characteristic
459	C₂²×Q₈	C₂³×C₅₃	1	1	yes	characteristic
460	C₂²×D₄	—	2	1	no	—
461	C₂²×D₄	C₂³×C₅₃	1	1	yes	characteristic
462	C₂³×C₄	—	2	1	no	—
463	C₂³×C₄	—	2	1	no	—
464	C₂³×C₄	C₂³×C₅₃	1	1	yes	characteristic
465	#137	—	2	16	no	—
466	#137	—	2	16	no	—
467	#137	—	2	2	no	—
468	#137	—	2	2	no	—
469	#137	—	2	2	no	—
470	#137	—	4	1	no	—
471	#137	—	4	1	no	—
472	#137	—	4	1	no	—
473	AGL(1,ℤ/8)	—	2	2	no	—
474	AGL(1,ℤ/8)	—	4	1	no	—
475	C₂×Q₁₆	—	2	2	no	—
476	C₂×Q₁₆	—	2	2	no	—
477	C₂×Q₁₆	—	2	1	no	—
478	C₂×Q₁₆	—	2	1	no	—
479	C₂×Q₁₆	—	2	1	no	—
480	C₂×Q₁₆	—	2	2	no	—
481	C₂×Q₁₆	—	2	1	no	—
482	C₂×Q₁₆	—	2	1	no	—
483	C₂×Q₁₆	—	2	2	no	—
484	C₂×Q₁₆	D₄×C₅₃	1	4	yes	complemented
485	C₂×SD₁₆	—	2	2	no	—
486	C₂×SD₁₆	—	2	2	no	—
487	C₂×SD₁₆	—	2	1	no	—
488	C₂×SD₁₆	—	2	1	no	—
489	C₂×SD₁₆	—	2	1	no	—
490	C₂×SD₁₆	—	2	1	no	—
491	C₂×SD₁₆	—	2	2	no	—
492	C₂×SD₁₆	—	2	1	no	—
493	C₂×SD₁₆	—	2	1	no	—
494	C₂×SD₁₆	—	2	2	no	—
495	C₂×SD₁₆	D₄×C₅₃	1	4	yes	complemented
496	C₂×D₈	—	2	1	no	—
497	#131	—	2	2	no	—
498	#131	—	2	2	no	—
499	C₂×M₁₆	—	2	2	no	—
500	C₂×M₁₆	—	2	2	no	—
501	C₂×M₁₆	—	2	1	no	—
502	C₂×M₁₆	D₄×C₅₃	1	2	yes	—
503	C₂×M₁₆	D₄×C₅₃	1	2	yes	—
504	C₂×M₁₆	C₂³×C₅₃	1	1	yes	characteristic
505	C₂×M₁₆	C₂³×C₅₃	1	1	yes	characteristic
506	C₂×M₁₆	C₂³×C₅₃	1	1	yes	characteristic
507	C₂×M₁₆	C₂³×C₅₃	1	1	yes	characteristic
508	C₂×M₁₆	C₂³×C₅₃	1	1	yes	characteristic
509	C₂×M₁₆	C₂³×C₅₃	1	1	yes	characteristic
510	C₂²×C₈	—	2	1	no	—
511	C₂²×C₈	C₂³×C₅₃	1	1	yes	characteristic
512	C₂²×C₈	C₂³×C₅₃	1	1	yes	characteristic
513	#123	—	2	1	no	—
514	C₂²⋊Q₈	D₄×C₅₃	1	2	yes	—
515	#121	D₄×C₅₃	1	2	yes	—
516	#120	—	2	1	no	—
517	C₄×D₄	—	2	2	no	—
518	#117	C₂³×C₅₃	1	1	yes	characteristic
519	#108	—	2	2	no	—
520	#108	—	2	2	no	—
521	C₄⋊C₈	—	2	2	no	—
522	#103	—	2	2	no	—
523	#103	—	2	2	no	—
524	#103	—	2	2	no	—
525	#102	—	2	2	no	—
526	#101	—	2	2	no	—
527	#101	—	2	2	no	—
528	#101	—	2	2	no	—
529	#101	—	4	1	no	—
530	#101	—	4	1	no	—
531	#101	—	4	1	no	—
532	#100	—	2	2	no	—
533	#100	—	4	1	no	—
534	C₂²⋊C₈	—	2	2	no	—
535	C₂²⋊C₈	—	2	2	no	—
536	C₂²⋊C₈	—	2	2	no	—
537	C₂²⋊C₈	—	2	2	no	—
538	#97	—	2	1	no	—
539	C₄×C₈	—	2	1	no	—
540	C₂⁴	—	2	1	no	—
541	C₂⁴	#41×C₅₃	1	1	yes	characteristic
542	#41	—	2	8	no	—
543	#41	—	2	1	no	—
544	#41	—	2	1	no	—
545	#41	—	2	1	no	—
546	#41	—	2	1	no	—
547	#41	—	4	4	no	—
548	#41	—	4	4	no	—
549	#41	—	4	4	no	—
550	C₂×Q₈	—	2	1	no	—
551	C₂×Q₈	—	2	1	no	—
552	C₂×Q₈	—	2	1	no	—
553	C₂×Q₈	—	2	8	no	—
554	C₂×Q₈	—	2	1	no	—
555	C₂×Q₈	—	2	1	no	—
556	C₂×Q₈	—	2	1	no	—
557	C₂×Q₈	—	2	1	no	—
558	C₂×Q₈	—	2	1	no	—
559	C₂×Q₈	—	2	1	no	—
560	C₂×Q₈	—	2	1	no	—
561	C₂×Q₈	—	2	1	no	—
562	C₂×Q₈	—	2	1	no	—
563	C₂×Q₈	—	2	1	no	—
564	C₂×Q₈	—	2	1	no	—
565	C₂×Q₈	—	2	1	no	—
566	C₂×Q₈	—	4	2	no	—
567	C₂×Q₈	—	4	2	no	—
568	C₂×Q₈	—	4	4	no	—
569	C₂×Q₈	—	4	4	no	—
570	C₂×Q₈	—	4	4	no	—
571	C₂×Q₈	—	4	2	no	—
572	C₂×Q₈	C₂×D₄×C₅₃	1	2	yes	—
573	C₂×Q₈	C₂×D₄×C₅₃	1	2	yes	—
574	C₂×Q₈	C₂×D₄×C₅₃	1	2	yes	—
575	C₂×Q₈	C₂×D₄×C₅₃	1	1	yes	characteristic
576	C₂×D₄	—	2	1	no	—
577	C₂×D₄	—	2	2	no	—
578	C₂×D₄	—	2	1	no	—
579	C₂×D₄	—	2	1	no	—
580	C₂×D₄	—	2	1	no	—
581	C₂×D₄	—	2	1	no	—
582	C₂×D₄	—	2	1	no	—
583	C₂×D₄	—	4	2	no	—
584	C₂×D₄	—	4	2	no	—
585	C₂×D₄	—	4	2	no	—
586	C₂×D₄	C₂×D₄×C₅₃	1	2	yes	—
587	C₂×D₄	C₂×D₄×C₅₃	1	1	yes	characteristic
588	C₂²×C₄	—	2	1	no	—
589	C₂²×C₄	—	2	1	no	—
590	C₂²×C₄	—	2	1	no	—
591	C₂²×C₄	—	2	1	no	—
592	C₂²×C₄	—	2	1	no	—
593	C₂²×C₄	—	2	1	no	—
594	C₂²×C₄	—	2	1	no	—
595	C₂²×C₄	—	2	1	no	—
596	C₂²×C₄	—	4	2	no	—
597	C₂²×C₄	—	4	2	no	—
598	C₂²×C₄	—	4	2	no	—
599	C₂²×C₄	—	4	2	no	—
600	C₂²×C₄	—	4	2	no	—
601	C₂²×C₄	—	4	2	no	—
602	C₂²×C₄	C₂×D₄×C₅₃	1	2	yes	—
603	C₂²×C₄	C₂×D₄×C₅₃	1	2	yes	—
604	C₂²×C₄	#41×C₅₃	1	1	yes	characteristic
605	C₂²×C₄	C₂⁴×C₅₃	1	1	yes	characteristic
606	Q₁₆	—	2	8	no	—
607	Q₁₆	—	2	4	no	—
608	Q₁₆	—	2	4	no	—
609	Q₁₆	—	4	2	no	—
610	Q₁₆	—	4	2	no	—
611	Q₁₆	—	4	2	no	—
612	SD₁₆	—	2	8	no	—
613	SD₁₆	—	2	4	no	—
614	SD₁₆	—	2	4	no	—
615	SD₁₆	—	4	2	no	—
616	SD₁₆	—	4	2	no	—
617	SD₁₆	—	4	2	no	—
618	SD₁₆	—	4	2	no	—
619	D₈	—	4	2	no	—
620	M₁₆	—	2	4	no	—
621	M₁₆	—	2	4	no	—
622	M₁₆	—	2	1	no	—
623	M₁₆	—	2	1	no	—
624	M₁₆	—	2	1	no	—
625	M₁₆	—	2	1	no	—
626	M₁₆	—	2	1	no	—
627	M₁₆	—	2	1	no	—
628	M₁₆	—	2	1	no	—
629	M₁₆	—	2	1	no	—
630	M₁₆	—	2	1	no	—
631	M₁₆	—	2	1	no	—
632	M₁₆	—	2	1	no	—
633	M₁₆	—	2	1	no	—
634	M₁₆	—	4	2	no	—
635	C₂×C₈	—	2	1	no	—
636	C₂×C₈	—	2	1	no	—
637	C₂×C₈	—	2	1	no	—
638	C₂×C₈	—	2	1	no	—
639	C₂×C₈	—	2	1	no	—
640	C₂×C₈	—	2	1	no	—
641	C₂×C₈	—	2	1	no	—
642	C₂×C₈	—	2	1	no	—
643	C₂×C₈	—	4	2	no	—
644	C₂×C₈	C₂×D₄×C₅₃	1	2	yes	—
645	C₂×C₈	C₂×D₄×C₅₃	1	1	yes	characteristic
646	C₂×C₈	C₂×D₄×C₅₃	1	1	yes	characteristic
647	C₂×C₈	C₂×D₄×C₅₃	1	2	yes	—
648	C₂×C₈	C₂×D₄×C₅₃	1	2	yes	—
649	C₄⋊C₄	—	2	2	no	—
650	C₄⋊C₄	C₂×D₄×C₅₃	1	2	yes	—
651	C₂²⋊C₄	—	2	2	no	—
652	C₂²⋊C₄	—	2	1	no	—
653	C₂²⋊C₄	—	2	2	no	—
654	C₄²	—	2	1	no	—
655	C₂³	—	2	1	no	—
656	C₂³	—	2	1	no	—
657	C₂³	—	2	1	no	—
658	C₂³	—	2	2	no	—
659	C₂³	—	4	2	no	—
660	C₂³	—	4	1	no	—
661	C₂³	—	4	1	no	—
662	C₂³	—	8	2	no	—
663	C₂³	#121×C₅₃	1	2	yes	—
664	C₂³	C₂×#41×C₅₃	1	1	yes	characteristic
665	Q₈	—	2	2	no	—
666	Q₈	—	2	2	no	—
667	Q₈	—	2	2	no	—
668	Q₈	—	2	1	no	—
669	Q₈	—	2	2	no	—
670	Q₈	—	2	2	no	—
671	Q₈	—	2	2	no	—
672	Q₈	—	2	1	no	—
673	Q₈	—	4	8	no	—
674	Q₈	—	4	1	no	—
675	Q₈	—	4	1	no	—
676	Q₈	—	4	1	no	—
677	Q₈	—	4	1	no	—
678	Q₈	—	4	1	no	—
679	Q₈	—	4	1	no	—
680	D₄	—	2	2	no	—
681	D₄	—	2	1	no	—
682	D₄	—	2	2	no	—
683	D₄	—	2	1	no	—
684	D₄	—	4	8	no	—
685	D₄	—	4	1	no	—
686	D₄	—	4	1	no	—
687	D₄	—	4	1	no	—
688	D₄	—	4	1	no	—
689	D₄	—	8	2	no	—
690	C₂×C₄	—	2	1	no	—
691	C₂×C₄	—	2	1	no	—
692	C₂×C₄	—	2	2	no	—
693	C₂×C₄	—	2	2	no	—
694	C₂×C₄	—	2	2	no	—
695	C₂×C₄	—	2	2	no	—
696	C₂×C₄	—	2	2	no	—
697	C₂×C₄	—	2	1	no	—
698	C₂×C₄	—	2	2	no	—
699	C₂×C₄	—	2	2	no	—
700	C₂×C₄	—	2	2	no	—
701	C₂×C₄	—	4	8	no	—
702	C₂×C₄	—	4	8	no	—
703	C₂×C₄	—	4	1	no	—
704	C₂×C₄	—	4	1	no	—
705	C₂×C₄	—	4	1	no	—
706	C₂×C₄	—	4	1	no	—
707	C₂×C₄	—	4	1	no	—
708	C₂×C₄	—	4	1	no	—
709	C₂×C₄	—	4	1	no	—
710	C₂×C₄	—	4	1	no	—
711	C₂×C₄	#121×C₅₃	1	2	yes	—
712	C₂×C₄	C₂²×D₄×C₅₃	1	1	yes	characteristic
713	C₂×C₄	C₂²×D₄×C₅₃	1	1	yes	characteristic
714	C₂×C₄	C₂²×D₄×C₅₃	1	1	yes	characteristic
715	C₂×C₄	C₂²×D₄×C₅₃	1	1	yes	characteristic
716	C₂×C₄	H(5,𝔽₂)×C₅₃	1	1	yes	characteristic
717	C₂×C₄	H(5,𝔽₂)×C₅₃	1	1	yes	characteristic
718	C₈	—	2	2	no	—
719	C₈	—	4	1	no	—
720	C₈	—	4	1	no	—
721	C₈	—	4	1	no	—
722	C₈	—	4	1	no	—
723	C₈	—	4	1	no	—
724	C₈	—	4	1	no	—
725	C₈	#120×C₅₃	1	2	yes	—
726	C₈	#120×C₅₃	1	2	yes	—
727	C₂²	—	2	2	no	—
728	C₂²	—	2	2	no	—
729	C₂²	—	2	2	no	—
730	C₂²	—	2	2	no	—
731	C₂²	—	4	2	no	—
732	C₂²	—	4	2	no	—
733	C₂²	—	4	2	no	—
734	C₂²	—	4	2	no	—
735	C₂²	—	4	1	no	—
736	C₂²	—	8	2	no	—
737	C₂²	—	8	2	no	—
738	C₂²	—	8	2	no	—
739	C₂²	C₂×#121×C₅₃	1	1	yes	characteristic
740	C₂²	C₅₃×#546	1	1	yes	characteristic
741	C₂²	C₅₃×#546	1	1	yes	characteristic
742	C₄	—	2	2	no	—
743	C₄	—	4	2	no	—
744	C₄	—	4	2	no	—
745	C₄	—	4	2	no	—
746	C₄	—	4	2	no	—
747	C₄	—	4	1	no	—
748	C₄	C₂×#120×C₅₃	1	1	yes	characteristic
749	C₄	C₅₃×#535	1	1	yes	characteristic
750	C₄	D₄²×C₅₃	1	1	yes	characteristic
751	C₄	D₄²×C₅₃	1	1	yes	characteristic
752	C₂	—	2	1	no	—
753	C₂	—	2	1	no	—
754	C₂	—	4	2	no	—
755	C₂	—	8	1	no	—
756	C₂	—	8	2	no	—
757	C₂	C₅₃×#2408	1	1	yes	characteristic
758	C₂	C₅₃×#3069	1	2	yes	—

	subgroup	G/H
center, socle	C₂²×C₅₃	C₂×#121
commutator, Frattini, cosocle	C₂²×C₄	C₂⁴×C₅₃
prime radical	C₅₃×#3005	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₅₃ ◁ C₂³×C₄×C₅₃ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂²×#137×C₅₃	C₂	semidirect
2. C₂×C₅₃×#473	C₂	semidirect
3. C₂×C₅₃×#471	C₂	semidirect
5. C₅₃×#3004	C₂	semidirect
6. C₅₃×#2930	C₂	semidirect
7. C₅₃×#2047	C₂	semidirect
8. C₅₃×#2046	C₂	semidirect
9. C₅₃×#2020	C₂	semidirect
10. C₅₃×#2019	C₂	semidirect
11. C₅₃×#1979	C₂	semidirect
12. C₅₃×#1977	C₂	semidirect
13. C₅₃×#1811	C₂	semidirect
14. C₅₃×#1785	C₂	semidirect
15. C₅₃×#1784	C₂	semidirect
104. C₂×Q₁₆×C₅₃	D₄	semidirect
115. C₂×SD₁₆×C₅₃	D₄	semidirect
347. #24283	C₅₃	direct (N × H)
373. C₂²×#137	C₂×C₅₃	semidirect
374. C₂×#473	C₂×C₅₃	semidirect
375. C₂×#471	C₂×C₅₃	semidirect
377. #3004	C₂×C₅₃	semidirect
378. #2930	C₂×C₅₃	semidirect
379. #2047	C₂×C₅₃	semidirect
380. #2046	C₂×C₅₃	semidirect
381. #2020	C₂×C₅₃	semidirect
382. #2019	C₂×C₅₃	semidirect
383. #1979	C₂×C₅₃	semidirect
384. #1977	C₂×C₅₃	semidirect
385. #1811	C₂×C₅₃	semidirect
386. #1785	C₂×C₅₃	semidirect
387. #1784	C₂×C₅₃	semidirect
442. C₅₃	#24283	direct (N × H)
484. C₂×Q₁₆	D₄×C₅₃	semidirect
495. C₂×SD₁₆	D₄×C₅₃	semidirect

Maximal subgroups

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

subgroup	index
C₂²×#137×C₅₃	2
C₂×C₅₃×#473	2
C₂×C₅₃×#471	2
C₅₃×#3005	2
C₅₃×#3004	2
C₅₃×#2930	2
C₅₃×#2047	2
C₅₃×#2046	2
C₅₃×#2020	2
C₅₃×#2019	2
C₅₃×#1979	2
C₅₃×#1977	2
C₅₃×#1811	2
C₅₃×#1785	2
C₅₃×#1784	2
#24283	53

Subgroups — C53×#24283 #27231795

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅₃×#24283 #27231795