Subgroups — C₅₃×#22759 #27230271

Order 13568.

874 conjugacy classes of subgroups.

The subgroup lattice has 484 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄×SD₁₆×C₅₃	C₂	1	1	yes	maximal, characteristic
2	C₅₃×#2354	C₂	1	1	yes	maximal, characteristic
3	Q₈×C₄⋊C₄×C₅₃	C₂	1	1	yes	maximal, characteristic, complemented
4	C₅₃×#1936	C₂	1	2	yes	maximal
5	C₅₃×#1919	C₂	1	2	yes	maximal, complemented
6	C₅₃×#1880	C₂	1	2	yes	maximal
7	C₅₃×#1834	C₂	1	1	yes	maximal, characteristic, complemented
8	C₅₃×#1803	C₂	1	1	yes	maximal, characteristic, complemented
9	C₅₃×#1802	C₂	1	1	yes	maximal, characteristic
10	C₅₃×#1770	C₂	1	1	yes	maximal, characteristic, complemented
11	C₅₃×#1763	C₂	1	2	yes	maximal, complemented
12	C₂²×SD₁₆×C₅₃	C₂²	1	1	yes	characteristic
13	C₂×C₄×Q₈×C₅₃	C₂²	1	2	yes	—
14	C₂×C₄×Q₈×C₅₃	C₂²	1	1	yes	characteristic
15	C₂×C₄×D₄×C₅₃	C₂²	1	1	yes	characteristic
16	C₂×#117×C₅₃	—	2	1	no	—
17	C₄×SD₁₆×C₅₃	C₄	1	8	yes	complemented
18	C₂×#106×C₅₃	C₂²	1	1	yes	characteristic
19	C₂×C₄⋊C₈×C₅₃	C₂²	1	1	yes	characteristic
20	C₂×#103×C₅₃	C₂²	1	2	yes	—
21	C₂×#103×C₅₃	C₂²	1	1	yes	characteristic
22	C₂×#102×C₅₃	C₂²	1	2	yes	—
23	C₂×#102×C₅₃	C₂²	1	1	yes	characteristic
24	C₂×#97×C₅₃	C₂²	1	2	yes	—
25	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
26	C₅₃×#390	C₂²	1	2	yes	—
27	C₅₃×#389	—	2	2	no	—
28	C₅₃×#389	C₂²	1	2	yes	—
29	C₅₃×#388	—	2	2	no	—
30	C₅₃×#384	—	2	1	no	—
31	C₅₃×#384	C₂²	1	1	yes	characteristic
32	C₅₃×#384	C₂²	1	1	yes	characteristic
33	C₅₃×#380	—	2	1	no	—
34	C₄×C₄⋊C₄×C₅₃	—	2	1	no	—
35	C₄×C₄⋊C₄×C₅₃	C₂²	1	2	yes	—
36	C₄×C₄⋊C₄×C₅₃	C₂²	1	1	yes	characteristic
37	C₅₃×#340	C₂²	1	2	yes	—
38	C₅₃×#340	C₂²	1	2	yes	—
39	C₅₃×#340	C₂²	1	2	yes	—
40	C₅₃×#340	C₂²	1	2	yes	—
41	C₅₃×#336	C₂²	1	2	yes	—
42	C₂²×Q₈×C₅₃	C₂³	1	1	yes	characteristic
43	C₂²×D₄×C₅₃	C₂³	1	1	yes	characteristic
44	C₂³×C₄×C₅₃	—	2	1	no	—
45	C₂×SD₁₆×C₅₃	—	2	2	no	—
46	C₂×SD₁₆×C₅₃	—	2	2	no	—
47	C₂×SD₁₆×C₅₃	C₂×C₄	1	4	yes	—
48	C₂²×C₈×C₅₃	C₂³	1	2	yes	—
49	C₂²×C₈×C₅₃	C₂³	1	1	yes	characteristic
50	C₂²×C₈×C₅₃	C₂³	1	1	yes	characteristic
51	C₄×Q₈×C₅₃	—	2	2	no	—
52	C₄×Q₈×C₅₃	—	2	4	no	—
53	C₄×Q₈×C₅₃	—	4	2	no	—
54	C₄×Q₈×C₅₃	C₂×C₄	1	4	yes	—
55	C₄×D₄×C₅₃	—	2	2	no	—
56	C₄×D₄×C₅₃	C₂×C₄	1	4	yes	—
57	#117×C₅₃	—	4	4	no	—
58	C₂×C₄⋊C₄×C₅₃	—	2	1	no	—
59	C₂×C₄⋊C₄×C₅₃	—	2	1	no	—
60	C₂×C₄⋊C₄×C₅₃	—	2	2	no	—
61	C₂×C₄⋊C₄×C₅₃	—	2	1	no	—
62	C₂×C₄⋊C₄×C₅₃	—	2	2	no	—
63	C₂×C₄⋊C₄×C₅₃	—	2	1	no	—
64	C₂×C₄⋊C₄×C₅₃	—	2	1	no	—
65	C₂×C₄⋊C₄×C₅₃	D₄	1	1	yes	characteristic
66	C₂×C₄⋊C₄×C₅₃	D₄	1	1	yes	characteristic
67	C₂×C₄⋊C₄×C₅₃	C₂³	1	2	yes	—
68	C₂×C₄⋊C₄×C₅₃	C₂³	1	2	yes	—
69	C₂×C₄⋊C₄×C₅₃	C₂³	1	1	yes	characteristic
70	C₂×C₄⋊C₄×C₅₃	C₂³	1	1	yes	characteristic
71	C₂×C₂²⋊C₄×C₅₃	—	2	1	no	—
72	C₂×C₂²⋊C₄×C₅₃	—	2	2	no	—
73	C₂×C₂²⋊C₄×C₅₃	—	2	2	no	—
74	C₂×C₄²×C₅₃	—	2	1	no	—
75	C₂×C₄²×C₅₃	—	2	2	no	—
76	C₂×C₄²×C₅₃	—	2	2	no	—
77	C₂×C₄²×C₅₃	C₂³	1	2	yes	—
78	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
79	#106×C₅₃	C₂×C₄	1	4	yes	—
80	C₄⋊C₈×C₅₃	—	2	2	no	—
81	#103×C₅₃	—	2	4	no	—
82	#103×C₅₃	C₂×C₄	1	4	yes	—
83	#102×C₅₃	—	4	2	no	—
84	#102×C₅₃	C₂×C₄	1	4	yes	—
85	#97×C₅₃	—	4	2	no	—
86	C₄×C₈×C₅₃	C₂×C₄	1	4	yes	—
87	#95×C₅₃	—	2	2	no	—
88	#95×C₅₃	—	2	2	no	—
89	#95×C₅₃	D₄	1	2	yes	—
90	C₂⁴×C₅₃	—	2	1	no	—
91	C₂×Q₈×C₅₃	—	2	1	no	—
92	C₂×Q₈×C₅₃	—	2	1	no	—
93	C₂×Q₈×C₅₃	—	2	1	no	—
94	C₂×Q₈×C₅₃	C₂²×C₄	1	2	yes	—
95	C₂×Q₈×C₅₃	#41	1	2	yes	—
96	C₂×Q₈×C₅₃	#41	1	2	yes	—
97	C₂×D₄×C₅₃	—	2	1	no	—
98	C₂×D₄×C₅₃	—	2	1	no	—
99	C₂×D₄×C₅₃	—	2	1	no	—
100	C₂×D₄×C₅₃	—	2	1	no	—
101	C₂×D₄×C₅₃	—	2	1	no	—
102	C₂×D₄×C₅₃	C₂²×C₄	1	2	yes	—
103	C₂²×C₄×C₅₃	—	2	1	no	—
104	C₂²×C₄×C₅₃	—	2	1	no	—
105	C₂²×C₄×C₅₃	—	2	1	no	—
106	C₂²×C₄×C₅₃	—	2	4	no	—
107	C₂²×C₄×C₅₃	—	2	2	no	—
108	C₂²×C₄×C₅₃	—	2	2	no	—
109	C₂²×C₄×C₅₃	—	4	2	no	—
110	C₂²×C₄×C₅₃	—	4	2	no	—
111	C₂²×C₄×C₅₃	C₂×D₄	1	2	yes	—
112	C₂²×C₄×C₅₃	C₂×D₄	1	2	yes	—
113	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
114	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
115	C₂²×C₄×C₅₃	C₂⁴	1	1	yes	characteristic
116	SD₁₆×C₅₃	—	2	8	no	—
117	C₂×C₈×C₅₃	—	2	1	no	—
118	C₂×C₈×C₅₃	—	2	1	no	—
119	C₂×C₈×C₅₃	—	2	1	no	—
120	C₂×C₈×C₅₃	—	2	1	no	—
121	C₂×C₈×C₅₃	—	2	2	no	—
122	C₂×C₈×C₅₃	—	2	2	no	—
123	C₂×C₈×C₅₃	—	2	2	no	—
124	C₂×C₈×C₅₃	C₂²×C₄	1	2	yes	—
125	C₂×C₈×C₅₃	C₂²×C₄	1	2	yes	—
126	C₄⋊C₄×C₅₃	—	2	4	no	—
127	C₄⋊C₄×C₅₃	—	2	2	no	—
128	C₄⋊C₄×C₅₃	—	2	2	no	—
129	C₄⋊C₄×C₅₃	—	2	1	no	—
130	C₄⋊C₄×C₅₃	—	2	1	no	—
131	C₄⋊C₄×C₅₃	—	2	2	no	—
132	C₄⋊C₄×C₅₃	—	2	2	no	—
133	C₄⋊C₄×C₅₃	—	2	2	no	—
134	C₄⋊C₄×C₅₃	—	2	2	no	—
135	C₄⋊C₄×C₅₃	—	4	4	no	—
136	C₄⋊C₄×C₅₃	—	4	2	no	—
137	C₄⋊C₄×C₅₃	—	4	2	no	—
138	C₄⋊C₄×C₅₃	—	8	1	no	—
139	C₄⋊C₄×C₅₃	—	8	1	no	—
140	C₄⋊C₄×C₅₃	—	8	2	no	—
141	C₄⋊C₄×C₅₃	C₂²×C₄	1	2	yes	—
142	C₄⋊C₄×C₅₃	C₂²×C₄	1	2	yes	—
143	C₂²⋊C₄×C₅₃	—	2	4	no	—
144	C₂²⋊C₄×C₅₃	—	4	4	no	—
145	C₂²⋊C₄×C₅₃	—	8	2	no	—
146	C₄²×C₅₃	—	2	4	no	—
147	C₄²×C₅₃	—	2	2	no	—
148	C₄²×C₅₃	—	2	2	no	—
149	C₄²×C₅₃	—	4	4	no	—
150	C₄²×C₅₃	—	4	4	no	—
151	C₄²×C₅₃	C₂²×C₄	1	2	yes	—
152	C₄²×C₅₃	#41	1	2	yes	—
153	C₂³×C₅₃	—	2	2	no	—
154	C₂³×C₅₃	—	4	1	no	—
155	C₂³×C₅₃	—	4	1	no	—
156	C₂³×C₅₃	—	4	1	no	—
157	C₂³×C₅₃	—	4	1	no	—
158	C₂³×C₅₃	—	4	1	no	—
159	C₂³×C₅₃	—	4	1	no	—
160	C₂³×C₅₃	C₂²×D₄	1	1	yes	characteristic
161	Q₈×C₅₃	—	2	2	no	—
162	Q₈×C₅₃	—	2	2	no	—
163	Q₈×C₅₃	—	2	2	no	—
164	Q₈×C₅₃	#117	1	4	yes	complemented
165	D₄×C₅₃	—	2	2	no	—
166	D₄×C₅₃	—	2	2	no	—
167	D₄×C₅₃	—	4	1	no	—
168	D₄×C₅₃	—	4	1	no	—
169	C₂×C₄×C₅₃	—	2	2	no	—
170	C₂×C₄×C₅₃	—	2	2	no	—
171	C₂×C₄×C₅₃	—	2	2	no	—
172	C₂×C₄×C₅₃	—	2	1	no	—
173	C₂×C₄×C₅₃	—	2	1	no	—
174	C₂×C₄×C₅₃	—	2	2	no	—
175	C₂×C₄×C₅₃	—	2	2	no	—
176	C₂×C₄×C₅₃	—	2	2	no	—
177	C₂×C₄×C₅₃	—	2	2	no	—
178	C₂×C₄×C₅₃	—	2	2	no	—
179	C₂×C₄×C₅₃	—	2	2	no	—
180	C₂×C₄×C₅₃	—	2	2	no	—
181	C₂×C₄×C₅₃	—	2	2	no	—
182	C₂×C₄×C₅₃	—	4	1	no	—
183	C₂×C₄×C₅₃	—	4	1	no	—
184	C₂×C₄×C₅₃	—	4	1	no	—
185	C₂×C₄×C₅₃	—	4	1	no	—
186	C₂×C₄×C₅₃	—	4	8	no	—
187	C₂×C₄×C₅₃	—	4	2	no	—
188	C₂×C₄×C₅₃	—	4	2	no	—
189	C₂×C₄×C₅₃	—	4	2	no	—
190	C₂×C₄×C₅₃	—	4	2	no	—
191	C₂×C₄×C₅₃	—	4	2	no	—
192	C₂×C₄×C₅₃	—	4	2	no	—
193	C₂×C₄×C₅₃	C₄×D₄	1	2	yes	—
194	C₂×C₄×C₅₃	C₄×D₄	1	2	yes	—
195	C₂×C₄×C₅₃	AGL(1,ℤ/8)	1	2	yes	—
196	C₂×C₄×C₅₃	#137	1	2	yes	—
197	C₂×C₄×C₅₃	C₂³×C₄	1	1	yes	characteristic
198	C₂×C₄×C₅₃	C₂×#41	1	1	yes	characteristic
199	C₂×C₄×C₅₃	C₂×#41	1	1	yes	characteristic
200	C₂×C₄×C₅₃	C₂×#41	1	1	yes	characteristic
201	C₂×C₄×C₅₃	H(5,𝔽₂)	1	1	yes	characteristic
202	C₂×C₄×C₅₃	2₋¹⁺⁴	1	1	yes	characteristic
203	C₈×C₅₃	—	2	2	no	—
204	C₈×C₅₃	—	2	2	no	—
205	C₈×C₅₃	—	4	2	no	—
206	#22759	C₅₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
207	C₂²×C₅₃	—	4	2	no	—
208	C₂²×C₅₃	—	4	2	no	—
209	C₂²×C₅₃	—	4	2	no	—
210	C₂²×C₅₃	—	8	1	no	—
211	C₂²×C₅₃	—	8	1	no	—
212	C₂²×C₅₃	—	8	1	no	—
213	C₂²×C₅₃	—	8	1	no	—
214	C₂²×C₅₃	C₂×C₄×D₄	1	1	yes	characteristic
215	C₂²×C₅₃	#546	1	1	yes	characteristic
216	C₂²×C₅₃	#548	1	1	yes	characteristic
217	C₂²×C₅₃	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
218	C₂²×C₅₃	C₂×#137	1	1	yes	characteristic
219	C₂²×C₅₃	#577	1	1	yes	characteristic
220	C₂²×C₅₃	#577	1	1	yes	characteristic
221	C₄×C₅₃	—	2	4	no	—
222	C₄×C₅₃	—	2	2	no	—
223	C₄×C₅₃	—	4	2	no	—
224	C₄×C₅₃	—	4	2	no	—
225	C₄×C₅₃	—	4	2	no	—
226	C₄×C₅₃	—	4	2	no	—
227	C₄×C₅₃	—	8	2	no	—
228	C₄×C₅₃	—	8	2	no	—
229	C₄×C₅₃	#440	1	4	yes	—
230	C₄×C₅₃	C₂×#117	1	1	yes	characteristic
231	C₄×C₅₃	#520	1	1	yes	characteristic
232	C₄×C₅₃	#552	1	1	yes	characteristic
233	C₄×C₅₃	#553	1	1	yes	characteristic
234	C₂×C₄×SD₁₆	C₂×C₅₃	1	1	yes	characteristic
235	#2354	C₂×C₅₃	1	1	yes	characteristic
236	Q₈×C₄⋊C₄	C₂×C₅₃	1	1	yes	characteristic, complemented
237	#1936	C₂×C₅₃	1	2	yes	—
238	#1919	C₂×C₅₃	1	2	yes	complemented
239	#1880	C₂×C₅₃	1	2	yes	—
240	#1834	C₂×C₅₃	1	1	yes	characteristic, complemented
241	#1803	C₂×C₅₃	1	1	yes	characteristic, complemented
242	#1802	C₂×C₅₃	1	1	yes	characteristic
243	#1770	C₂×C₅₃	1	1	yes	characteristic, complemented
244	#1763	C₂×C₅₃	1	2	yes	complemented
245	C₂×C₅₃	—	8	2	no	—
246	C₂×C₅₃	#2352	1	1	yes	characteristic
247	C₂×C₅₃	C₂×#440	1	1	yes	characteristic
248	C₂×C₅₃	#2952	1	1	yes	characteristic
249	C₂×C₅₃	#3329	1	1	yes	characteristic
250	C₂×C₅₃	#3332	1	1	yes	characteristic
251	C₂×C₅₃	#3368	1	1	yes	characteristic
252	C₂×C₅₃	#3369	1	1	yes	characteristic
253	C₂²×SD₁₆	C₂²×C₅₃	1	1	yes	characteristic
254	C₂×C₄×Q₈	C₂²×C₅₃	1	2	yes	—
255	C₂×C₄×Q₈	C₂²×C₅₃	1	1	yes	characteristic
256	C₂×C₄×D₄	C₂²×C₅₃	1	1	yes	characteristic
257	C₂×#117	—	2	1	no	—
258	C₄×SD₁₆	C₄×C₅₃	1	8	yes	complemented
259	C₂×#106	C₂²×C₅₃	1	1	yes	characteristic
260	C₂×C₄⋊C₈	C₂²×C₅₃	1	1	yes	characteristic
261	C₂×#103	C₂²×C₅₃	1	2	yes	—
262	C₂×#103	C₂²×C₅₃	1	1	yes	characteristic
263	C₂×#102	C₂²×C₅₃	1	2	yes	—
264	C₂×#102	C₂²×C₅₃	1	1	yes	characteristic
265	C₂×#97	C₂²×C₅₃	1	2	yes	—
266	C₂×C₄×C₈	C₂²×C₅₃	1	1	yes	characteristic
267	#390	C₂²×C₅₃	1	2	yes	—
268	#389	—	2	2	no	—
269	#389	C₂²×C₅₃	1	2	yes	—
270	#388	—	2	2	no	—
271	#384	—	2	1	no	—
272	#384	C₂²×C₅₃	1	1	yes	characteristic
273	#384	C₂²×C₅₃	1	1	yes	characteristic
274	#380	—	2	1	no	—
275	C₄×C₄⋊C₄	—	2	1	no	—
276	C₄×C₄⋊C₄	C₂²×C₅₃	1	2	yes	—
277	C₄×C₄⋊C₄	C₂²×C₅₃	1	1	yes	characteristic
278	#340	C₂²×C₅₃	1	2	yes	—
279	#340	C₂²×C₅₃	1	2	yes	—
280	#340	C₂²×C₅₃	1	2	yes	—
281	#340	C₂²×C₅₃	1	2	yes	—
282	#336	C₂²×C₅₃	1	2	yes	—
283	C₅₃	#22759	1	1	yes	characteristic, direct product factor, Sylow
284	C₂²×Q₈	C₂³×C₅₃	1	1	yes	characteristic
285	C₂²×D₄	C₂³×C₅₃	1	1	yes	characteristic
286	C₂³×C₄	—	2	1	no	—
287	C₂×SD₁₆	—	2	2	no	—
288	C₂×SD₁₆	—	2	2	no	—
289	C₂×SD₁₆	C₂×C₄×C₅₃	1	4	yes	—
290	C₂²×C₈	C₂³×C₅₃	1	2	yes	—
291	C₂²×C₈	C₂³×C₅₃	1	1	yes	characteristic
292	C₂²×C₈	C₂³×C₅₃	1	1	yes	characteristic
293	C₄×Q₈	—	2	2	no	—
294	C₄×Q₈	—	2	4	no	—
295	C₄×Q₈	—	4	2	no	—
296	C₄×Q₈	C₂×C₄×C₅₃	1	4	yes	—
297	C₄×D₄	—	2	2	no	—
298	C₄×D₄	C₂×C₄×C₅₃	1	4	yes	—
299	#117	—	4	4	no	—
300	C₂×C₄⋊C₄	—	2	1	no	—
301	C₂×C₄⋊C₄	—	2	1	no	—
302	C₂×C₄⋊C₄	—	2	2	no	—
303	C₂×C₄⋊C₄	—	2	1	no	—
304	C₂×C₄⋊C₄	—	2	2	no	—
305	C₂×C₄⋊C₄	—	2	1	no	—
306	C₂×C₄⋊C₄	—	2	1	no	—
307	C₂×C₄⋊C₄	D₄×C₅₃	1	1	yes	characteristic
308	C₂×C₄⋊C₄	D₄×C₅₃	1	1	yes	characteristic
309	C₂×C₄⋊C₄	C₂³×C₅₃	1	2	yes	—
310	C₂×C₄⋊C₄	C₂³×C₅₃	1	2	yes	—
311	C₂×C₄⋊C₄	C₂³×C₅₃	1	1	yes	characteristic
312	C₂×C₄⋊C₄	C₂³×C₅₃	1	1	yes	characteristic
313	C₂×C₂²⋊C₄	—	2	1	no	—
314	C₂×C₂²⋊C₄	—	2	2	no	—
315	C₂×C₂²⋊C₄	—	2	2	no	—
316	C₂×C₄²	—	2	1	no	—
317	C₂×C₄²	—	2	2	no	—
318	C₂×C₄²	—	2	2	no	—
319	C₂×C₄²	C₂³×C₅₃	1	2	yes	—
320	C₂×C₄²	C₂³×C₅₃	1	1	yes	characteristic
321	#106	C₂×C₄×C₅₃	1	4	yes	—
322	C₄⋊C₈	—	2	2	no	—
323	#103	—	2	4	no	—
324	#103	C₂×C₄×C₅₃	1	4	yes	—
325	#102	—	4	2	no	—
326	#102	C₂×C₄×C₅₃	1	4	yes	—
327	#97	—	4	2	no	—
328	C₄×C₈	C₂×C₄×C₅₃	1	4	yes	—
329	#95	—	2	2	no	—
330	#95	—	2	2	no	—
331	#95	D₄×C₅₃	1	2	yes	—
332	C₂⁴	—	2	1	no	—
333	C₂×Q₈	—	2	1	no	—
334	C₂×Q₈	—	2	1	no	—
335	C₂×Q₈	—	2	1	no	—
336	C₂×Q₈	C₂²×C₄×C₅₃	1	2	yes	—
337	C₂×Q₈	#41×C₅₃	1	2	yes	—
338	C₂×Q₈	#41×C₅₃	1	2	yes	—
339	C₂×D₄	—	2	1	no	—
340	C₂×D₄	—	2	1	no	—
341	C₂×D₄	—	2	1	no	—
342	C₂×D₄	—	2	1	no	—
343	C₂×D₄	—	2	1	no	—
344	C₂×D₄	C₂²×C₄×C₅₃	1	2	yes	—
345	C₂²×C₄	—	2	1	no	—
346	C₂²×C₄	—	2	1	no	—
347	C₂²×C₄	—	2	1	no	—
348	C₂²×C₄	—	2	4	no	—
349	C₂²×C₄	—	2	2	no	—
350	C₂²×C₄	—	2	2	no	—
351	C₂²×C₄	—	4	2	no	—
352	C₂²×C₄	—	4	2	no	—
353	C₂²×C₄	C₂×D₄×C₅₃	1	2	yes	—
354	C₂²×C₄	C₂×D₄×C₅₃	1	2	yes	—
355	C₂²×C₄	C₂×D₄×C₅₃	1	1	yes	characteristic
356	C₂²×C₄	C₂×D₄×C₅₃	1	1	yes	characteristic
357	C₂²×C₄	C₂⁴×C₅₃	1	1	yes	characteristic
358	SD₁₆	—	2	8	no	—
359	C₂×C₈	—	2	1	no	—
360	C₂×C₈	—	2	1	no	—
361	C₂×C₈	—	2	1	no	—
362	C₂×C₈	—	2	1	no	—
363	C₂×C₈	—	2	2	no	—
364	C₂×C₈	—	2	2	no	—
365	C₂×C₈	—	2	2	no	—
366	C₂×C₈	C₂²×C₄×C₅₃	1	2	yes	—
367	C₂×C₈	C₂²×C₄×C₅₃	1	2	yes	—
368	C₄⋊C₄	—	2	4	no	—
369	C₄⋊C₄	—	2	2	no	—
370	C₄⋊C₄	—	2	2	no	—
371	C₄⋊C₄	—	2	1	no	—
372	C₄⋊C₄	—	2	1	no	—
373	C₄⋊C₄	—	2	2	no	—
374	C₄⋊C₄	—	2	2	no	—
375	C₄⋊C₄	—	2	2	no	—
376	C₄⋊C₄	—	2	2	no	—
377	C₄⋊C₄	—	4	4	no	—
378	C₄⋊C₄	—	4	2	no	—
379	C₄⋊C₄	—	4	2	no	—
380	C₄⋊C₄	—	8	1	no	—
381	C₄⋊C₄	—	8	1	no	—
382	C₄⋊C₄	—	8	2	no	—
383	C₄⋊C₄	C₂²×C₄×C₅₃	1	2	yes	—
384	C₄⋊C₄	C₂²×C₄×C₅₃	1	2	yes	—
385	C₂²⋊C₄	—	2	4	no	—
386	C₂²⋊C₄	—	4	4	no	—
387	C₂²⋊C₄	—	8	2	no	—
388	C₄²	—	2	4	no	—
389	C₄²	—	2	2	no	—
390	C₄²	—	2	2	no	—
391	C₄²	—	4	4	no	—
392	C₄²	—	4	4	no	—
393	C₄²	C₂²×C₄×C₅₃	1	2	yes	—
394	C₄²	#41×C₅₃	1	2	yes	—
395	C₂³	—	2	2	no	—
396	C₂³	—	4	1	no	—
397	C₂³	—	4	1	no	—
398	C₂³	—	4	1	no	—
399	C₂³	—	4	1	no	—
400	C₂³	—	4	1	no	—
401	C₂³	—	4	1	no	—
402	C₂³	C₂²×D₄×C₅₃	1	1	yes	characteristic
403	Q₈	—	2	2	no	—
404	Q₈	—	2	2	no	—
405	Q₈	—	2	2	no	—
406	Q₈	#117×C₅₃	1	4	yes	complemented
407	D₄	—	2	2	no	—
408	D₄	—	2	2	no	—
409	D₄	—	4	1	no	—
410	D₄	—	4	1	no	—
411	C₂×C₄	—	2	2	no	—
412	C₂×C₄	—	2	2	no	—
413	C₂×C₄	—	2	2	no	—
414	C₂×C₄	—	2	1	no	—
415	C₂×C₄	—	2	1	no	—
416	C₂×C₄	—	2	2	no	—
417	C₂×C₄	—	2	2	no	—
418	C₂×C₄	—	2	2	no	—
419	C₂×C₄	—	2	2	no	—
420	C₂×C₄	—	2	2	no	—
421	C₂×C₄	—	2	2	no	—
422	C₂×C₄	—	2	2	no	—
423	C₂×C₄	—	2	2	no	—
424	C₂×C₄	—	4	1	no	—
425	C₂×C₄	—	4	1	no	—
426	C₂×C₄	—	4	1	no	—
427	C₂×C₄	—	4	1	no	—
428	C₂×C₄	—	4	8	no	—
429	C₂×C₄	—	4	2	no	—
430	C₂×C₄	—	4	2	no	—
431	C₂×C₄	—	4	2	no	—
432	C₂×C₄	—	4	2	no	—
433	C₂×C₄	—	4	2	no	—
434	C₂×C₄	—	4	2	no	—
435	C₂×C₄	C₄×D₄×C₅₃	1	2	yes	—
436	C₂×C₄	C₄×D₄×C₅₃	1	2	yes	—
437	C₂×C₄	AGL(1,ℤ/8)×C₅₃	1	2	yes	—
438	C₂×C₄	#137×C₅₃	1	2	yes	—
439	C₂×C₄	C₂³×C₄×C₅₃	1	1	yes	characteristic
440	C₂×C₄	C₂×#41×C₅₃	1	1	yes	characteristic
441	C₂×C₄	C₂×#41×C₅₃	1	1	yes	characteristic
442	C₂×C₄	C₂×#41×C₅₃	1	1	yes	characteristic
443	C₂×C₄	H(5,𝔽₂)×C₅₃	1	1	yes	characteristic
444	C₂×C₄	2₋¹⁺⁴×C₅₃	1	1	yes	characteristic
445	C₈	—	2	2	no	—
446	C₈	—	2	2	no	—
447	C₈	—	4	2	no	—
448	C₂²	—	4	2	no	—
449	C₂²	—	4	2	no	—
450	C₂²	—	4	2	no	—
451	C₂²	—	8	1	no	—
452	C₂²	—	8	1	no	—
453	C₂²	—	8	1	no	—
454	C₂²	—	8	1	no	—
455	C₂²	C₂×C₄×D₄×C₅₃	1	1	yes	characteristic
456	C₂²	C₅₃×#546	1	1	yes	characteristic
457	C₂²	C₅₃×#548	1	1	yes	characteristic
458	C₂²	C₂×AGL(1,ℤ/8)×C₅₃	1	1	yes	characteristic
459	C₂²	C₂×#137×C₅₃	1	1	yes	characteristic
460	C₂²	C₅₃×#577	1	1	yes	characteristic
461	C₂²	C₅₃×#577	1	1	yes	characteristic
462	C₄	—	2	2	no	—
463	C₄	—	2	4	no	—
464	C₄	—	4	2	no	—
465	C₄	—	4	2	no	—
466	C₄	—	4	2	no	—
467	C₄	—	4	2	no	—
468	C₄	—	8	2	no	—
469	C₄	—	8	2	no	—
470	C₄	C₅₃×#440	1	4	yes	—
471	C₄	C₂×#117×C₅₃	1	1	yes	characteristic
472	C₄	C₅₃×#520	1	1	yes	characteristic
473	C₄	C₅₃×#552	1	1	yes	characteristic
474	C₄	C₅₃×#553	1	1	yes	characteristic
475	C₂	—	8	2	no	—
476	C₂	C₅₃×#2352	1	1	yes	characteristic
477	C₂	C₂×C₅₃×#440	1	1	yes	characteristic
478	C₂	C₅₃×#2952	1	1	yes	characteristic
479	C₂	C₅₃×#3329	1	1	yes	characteristic
480	C₂	C₅₃×#3332	1	1	yes	characteristic
481	C₂	C₅₃×#3368	1	1	yes	characteristic
482	C₂	C₅₃×#3369	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂³×C₅₃	C₂²×D₄
commutator	C₂×C₄	C₂³×C₄×C₅₃
Frattini, cosocle	C₂²×C₄	C₂⁴×C₅₃
prime radical	C₂²×D₄×C₅₃	C₂³
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₅₃ ◁ C₄×C₄⋊C₄×C₅₃ ◁ G

Complements

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

N	H ≅ G/N	type
3. Q₈×C₄⋊C₄×C₅₃	C₂	semidirect
5. C₅₃×#1919	C₂	semidirect
7. C₅₃×#1834	C₂	semidirect
8. C₅₃×#1803	C₂	semidirect
10. C₅₃×#1770	C₂	semidirect
11. C₅₃×#1763	C₂	semidirect
17. C₄×SD₁₆×C₅₃	C₄	semidirect
164. Q₈×C₅₃	#117	semidirect
206. #22759	C₅₃	direct (N × H)
236. Q₈×C₄⋊C₄	C₂×C₅₃	semidirect
238. #1919	C₂×C₅₃	semidirect
240. #1834	C₂×C₅₃	semidirect
241. #1803	C₂×C₅₃	semidirect
243. #1770	C₂×C₅₃	semidirect
244. #1763	C₂×C₅₃	semidirect
258. C₄×SD₁₆	C₄×C₅₃	semidirect
283. C₅₃	#22759	direct (N × H)
406. Q₈	#117×C₅₃	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₄×SD₁₆×C₅₃	2
C₅₃×#2354	2
Q₈×C₄⋊C₄×C₅₃	2
C₅₃×#1936	2
C₅₃×#1919	2
C₅₃×#1880	2
C₅₃×#1834	2
C₅₃×#1803	2
C₅₃×#1802	2
C₅₃×#1770	2
C₅₃×#1763	2
#22759	53

Subgroups — C53×#22759 #27230271

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅₃×#22759 #27230271