Subgroups — C₅₃×#22721 #27230233

Order 13568.

760 conjugacy classes of subgroups.

The subgroup lattice has 426 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₅₃×#446	C₂	1	1	yes	maximal, characteristic
2	C₂×C₈×Q₈×C₅₃	C₂	1	1	yes	maximal, characteristic
3	C₅₃×#2295	C₂	1	1	yes	maximal, characteristic
4	C₅₃×#1940	C₂	1	2	yes	maximal
5	C₅₃×#1939	C₂	1	2	yes	maximal
6	C₅₃×#1939	C₂	1	1	yes	maximal, characteristic
7	C₅₃×#1939	C₂	1	1	yes	maximal, characteristic
8	C₅₃×#1843	C₂	1	2	yes	maximal
9	C₅₃×#1842	C₂	1	1	yes	maximal, characteristic
10	C₅₃×#1768	C₂	1	2	yes	maximal
11	C₄×C₄⋊C₈×C₅₃	C₂	1	1	yes	maximal, characteristic
12	C₂×C₄×Q₈×C₅₃	C₂²	1	1	yes	characteristic
13	C₅₃×#446	—	2	4	no	—
14	C₈×Q₈×C₅₃	—	2	4	no	—
15	C₂×C₄⋊C₈×C₅₃	C₂²	1	1	yes	characteristic
16	C₂×C₄⋊C₈×C₅₃	C₂²	1	2	yes	—
17	C₂×C₄⋊C₈×C₅₃	C₂²	1	1	yes	characteristic
18	C₂×C₄⋊C₈×C₅₃	C₂²	1	2	yes	—
19	C₂×C₄⋊C₈×C₅₃	C₂²	1	1	yes	characteristic
20	C₂×C₄⋊C₈×C₅₃	C₂²	1	2	yes	—
21	C₂×C₄⋊C₈×C₅₃	C₂²	1	2	yes	—
22	C₂×C₄⋊C₈×C₅₃	C₂²	1	2	yes	—
23	C₂×C₄⋊C₈×C₅₃	C₂²	1	2	yes	—
24	C₂×C₄⋊C₈×C₅₃	C₂²	1	1	yes	characteristic
25	C₂×#97×C₅₃	C₂²	1	2	yes	—
26	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
27	C₂×C₄×C₈×C₅₃	C₂²	1	2	yes	—
28	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
29	C₅₃×#391	C₄	1	1	yes	characteristic
30	C₅₃×#391	C₄	1	1	yes	characteristic
31	C₅₃×#389	C₄	1	1	yes	characteristic
32	C₅₃×#389	C₄	1	1	yes	characteristic
33	C₅₃×#387	C₄	1	2	yes	—
34	C₅₃×#387	C₄	1	2	yes	—
35	C₄×C₄⋊C₄×C₅₃	C₂²	1	1	yes	characteristic
36	C₄×C₄⋊C₄×C₅₃	C₂²	1	2	yes	—
37	C₅₃×#376	C₂²	1	2	yes	—
38	C₄³×C₅₃	C₂²	1	1	yes	characteristic
39	C₅₃×#336	C₂²	1	1	yes	characteristic
40	C₅₃×#336	C₂²	1	2	yes	—
41	C₅₃×#336	C₂²	1	2	yes	—
42	C₅₃×#336	C₂²	1	1	yes	characteristic
43	C₂²×Q₈×C₅₃	C₂×C₄	1	1	yes	characteristic
44	C₂²×C₈×C₅₃	C₂³	1	1	yes	characteristic
45	C₂²×C₈×C₅₃	C₂³	1	2	yes	—
46	C₂²×C₈×C₅₃	C₂³	1	1	yes	characteristic
47	C₂²×C₈×C₅₃	C₂³	1	2	yes	—
48	C₂²×C₈×C₅₃	C₂³	1	1	yes	characteristic
49	C₂²×C₈×C₅₃	C₂³	1	1	yes	characteristic
50	C₄×Q₈×C₅₃	D₄	1	4	yes	—
51	C₄×Q₈×C₅₃	Q₈	1	4	yes	—
52	C₂×C₄⋊C₄×C₅₃	C₂×C₄	1	2	yes	—
53	C₂×C₄⋊C₄×C₅₃	C₂×C₄	1	1	yes	characteristic
54	C₂×C₄⋊C₄×C₅₃	C₂×C₄	1	1	yes	characteristic
55	C₂×C₄⋊C₄×C₅₃	C₂×C₄	1	2	yes	—
56	C₂×C₄⋊C₄×C₅₃	C₂×C₄	1	2	yes	—
57	C₂×C₄⋊C₄×C₅₃	C₂×C₄	1	1	yes	characteristic
58	C₂×C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
59	C₂×C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
60	C₂×C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
61	C₂×C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
62	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
63	C₂×C₄²×C₅₃	C₂³	1	2	yes	—
64	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
65	C₂×C₄²×C₅₃	C₂³	1	2	yes	—
66	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
67	C₄⋊C₈×C₅₃	—	2	2	no	—
68	C₄⋊C₈×C₅₃	—	2	4	no	—
69	C₄⋊C₈×C₅₃	—	2	2	no	—
70	C₄⋊C₈×C₅₃	—	2	4	no	—
71	C₄⋊C₈×C₅₃	—	2	2	no	—
72	C₄⋊C₈×C₅₃	—	2	4	no	—
73	C₄⋊C₈×C₅₃	—	4	2	no	—
74	C₄⋊C₈×C₅₃	—	4	2	no	—
75	C₄⋊C₈×C₅₃	—	4	2	no	—
76	C₄⋊C₈×C₅₃	Q₈	1	4	yes	complemented
77	#97×C₅₃	—	2	4	no	—
78	C₄×C₈×C₅₃	—	2	2	no	—
79	C₄×C₈×C₅₃	—	2	4	no	—
80	C₄×C₈×C₅₃	—	2	2	no	—
81	#95×C₅₃	C₂×C₄	1	2	yes	—
82	#95×C₅₃	C₂×C₄	1	2	yes	—
83	#95×C₅₃	C₂×C₄	1	2	yes	—
84	#95×C₅₃	C₂×C₄	1	2	yes	—
85	#95×C₅₃	C₂×C₄	1	2	yes	—
86	#95×C₅₃	C₂×C₄	1	2	yes	—
87	#95×C₅₃	C₂×C₄	1	2	yes	—
88	C₂×Q₈×C₅₃	—	2	2	no	—
89	C₂×Q₈×C₅₃	—	2	2	no	—
90	C₂×Q₈×C₅₃	C₄⋊C₄	1	4	yes	—
91	C₂²×C₄×C₅₃	C₂²×C₄	1	2	yes	—
92	C₂²×C₄×C₅₃	C₂²×C₄	1	2	yes	—
93	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
94	C₂²×C₄×C₅₃	C₂²×C₄	1	2	yes	—
95	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
96	C₂²×C₄×C₅₃	C₂²×C₄	1	2	yes	—
97	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
98	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
99	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
100	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
101	C₂²×C₄×C₅₃	C₂⁴	1	1	yes	characteristic
102	C₂×C₈×C₅₃	—	2	1	no	—
103	C₂×C₈×C₅₃	—	2	1	no	—
104	C₂×C₈×C₅₃	—	2	1	no	—
105	C₂×C₈×C₅₃	—	2	1	no	—
106	C₂×C₈×C₅₃	—	2	1	no	—
107	C₂×C₈×C₅₃	—	2	2	no	—
108	C₂×C₈×C₅₃	—	2	1	no	—
109	C₂×C₈×C₅₃	—	2	2	no	—
110	C₂×C₈×C₅₃	—	2	1	no	—
111	C₂×C₈×C₅₃	—	2	1	no	—
112	C₂×C₈×C₅₃	—	2	2	no	—
113	C₂×C₈×C₅₃	—	2	2	no	—
114	C₂×C₈×C₅₃	—	2	1	no	—
115	C₂×C₈×C₅₃	—	2	2	no	—
116	C₂×C₈×C₅₃	—	2	1	no	—
117	C₂×C₈×C₅₃	—	2	2	no	—
118	C₂×C₈×C₅₃	C₂×Q₈	1	2	yes	—
119	C₂×C₈×C₅₃	C₂×Q₈	1	2	yes	—
120	C₄⋊C₄×C₅₃	—	2	4	no	—
121	C₄⋊C₄×C₅₃	—	2	2	no	—
122	C₄⋊C₄×C₅₃	—	2	2	no	—
123	C₄⋊C₄×C₅₃	—	4	2	no	—
124	C₄⋊C₄×C₅₃	C₄⋊C₄	1	8	yes	—
125	C₄⋊C₄×C₅₃	C₄⋊C₄	1	4	yes	—
126	C₄²×C₅₃	—	2	2	no	—
127	C₄²×C₅₃	—	2	2	no	—
128	C₄²×C₅₃	—	2	2	no	—
129	C₄²×C₅₃	—	2	1	no	—
130	C₄²×C₅₃	—	2	2	no	—
131	C₄²×C₅₃	—	2	1	no	—
132	C₄²×C₅₃	—	2	2	no	—
133	C₄²×C₅₃	—	4	1	no	—
134	C₄²×C₅₃	C₂×D₄	1	2	yes	—
135	C₄²×C₅₃	C₂×D₄	1	4	yes	—
136	C₄²×C₅₃	C₂×Q₈	1	4	yes	—
137	C₄²×C₅₃	C₂×Q₈	1	2	yes	—
138	C₄²×C₅₃	C₂×Q₈	1	2	yes	—
139	C₄²×C₅₃	#41	1	2	yes	—
140	C₂³×C₅₃	C₂³×C₄	1	1	yes	characteristic
141	Q₈×C₅₃	—	2	8	no	—
142	C₂×C₄×C₅₃	—	2	1	no	—
143	C₂×C₄×C₅₃	—	2	2	no	—
144	C₂×C₄×C₅₃	—	2	2	no	—
145	C₂×C₄×C₅₃	—	2	1	no	—
146	C₂×C₄×C₅₃	—	2	1	no	—
147	C₂×C₄×C₅₃	—	2	1	no	—
148	C₂×C₄×C₅₃	—	2	1	no	—
149	C₂×C₄×C₅₃	—	2	1	no	—
150	C₂×C₄×C₅₃	—	2	2	no	—
151	C₂×C₄×C₅₃	—	2	1	no	—
152	C₂×C₄×C₅₃	—	2	1	no	—
153	C₂×C₄×C₅₃	—	2	2	no	—
154	C₂×C₄×C₅₃	—	2	2	no	—
155	C₂×C₄×C₅₃	—	2	1	no	—
156	C₂×C₄×C₅₃	—	2	2	no	—
157	C₂×C₄×C₅₃	—	2	1	no	—
158	C₂×C₄×C₅₃	—	2	2	no	—
159	C₂×C₄×C₅₃	—	2	2	no	—
160	C₂×C₄×C₅₃	—	2	2	no	—
161	C₂×C₄×C₅₃	—	2	1	no	—
162	C₂×C₄×C₅₃	—	2	2	no	—
163	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	4	yes	—
164	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	2	yes	—
165	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	4	yes	—
166	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	2	yes	—
167	C₂×C₄×C₅₃	C₄×Q₈	1	2	yes	—
168	C₂×C₄×C₅₃	C₄×Q₈	1	2	yes	—
169	C₂×C₄×C₅₃	#131	1	2	yes	—
170	C₂×C₄×C₅₃	#131	1	2	yes	—
171	C₂×C₄×C₅₃	#131	1	2	yes	—
172	C₂×C₄×C₅₃	C₂²×D₄	1	1	yes	characteristic
173	C₂×C₄×C₅₃	C₂²×Q₈	1	1	yes	characteristic
174	C₂×C₄×C₅₃	C₂²×Q₈	1	1	yes	characteristic
175	C₂×C₄×C₅₃	C₂×#41	1	1	yes	characteristic
176	C₂×C₄×C₅₃	H(5,𝔽₂)	1	1	yes	characteristic
177	C₂×C₄×C₅₃	2₋¹⁺⁴	1	1	yes	characteristic
178	C₈×C₅₃	—	2	2	no	—
179	C₈×C₅₃	—	2	2	no	—
180	C₈×C₅₃	—	4	1	no	—
181	C₈×C₅₃	—	4	2	no	—
182	C₈×C₅₃	—	4	1	no	—
183	C₈×C₅₃	—	4	2	no	—
184	#22721	C₅₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
185	C₂²×C₅₃	C₂²×C₄⋊C₄	1	1	yes	characteristic
186	C₂²×C₅₃	C₂×C₄×Q₈	1	1	yes	characteristic
187	C₂²×C₅₃	#520	1	1	yes	characteristic
188	C₂²×C₅₃	C₂×#131	1	1	yes	characteristic
189	C₂²×C₅₃	C₂×#131	1	1	yes	characteristic
190	C₂²×C₅₃	C₂×#131	1	1	yes	characteristic
191	C₂²×C₅₃	#568	1	1	yes	characteristic
192	C₄×C₅₃	—	2	4	no	—
193	C₄×C₅₃	—	2	2	no	—
194	C₄×C₅₃	—	2	4	no	—
195	C₄×C₅₃	—	2	2	no	—
196	C₄×C₅₃	—	2	2	no	—
197	C₄×C₅₃	—	2	2	no	—
198	C₄×C₅₃	—	4	1	no	—
199	C₄×C₅₃	—	4	2	no	—
200	C₄×C₅₃	—	4	2	no	—
201	C₄×C₅₃	#424	1	4	yes	—
202	C₄×C₅₃	D₄×Q₈	1	1	yes	characteristic
203	C₄×C₅₃	#550	1	1	yes	characteristic
204	C₄×C₅₃	#557	1	1	yes	characteristic
205	C₄×C₅₃	Q₈²	1	1	yes	characteristic
206	C₂×#446	C₂×C₅₃	1	1	yes	characteristic
207	C₂×C₈×Q₈	C₂×C₅₃	1	1	yes	characteristic
208	#2295	C₂×C₅₃	1	1	yes	characteristic
209	#1940	C₂×C₅₃	1	2	yes	—
210	#1939	C₂×C₅₃	1	2	yes	—
211	#1939	C₂×C₅₃	1	1	yes	characteristic
212	#1939	C₂×C₅₃	1	1	yes	characteristic
213	#1843	C₂×C₅₃	1	2	yes	—
214	#1842	C₂×C₅₃	1	1	yes	characteristic
215	#1768	C₂×C₅₃	1	2	yes	—
216	C₄×C₄⋊C₈	C₂×C₅₃	1	1	yes	characteristic
217	C₂×C₅₃	Q₈×C₄⋊C₄	1	1	yes	characteristic
218	C₂×C₅₃	C₂×#424	1	1	yes	characteristic
219	C₂×C₅₃	#2906	1	1	yes	characteristic
220	C₂×C₅₃	#2960	1	1	yes	characteristic
221	C₂×C₅₃	#2962	1	1	yes	characteristic
222	C₂×C₅₃	#2993	1	1	yes	characteristic
223	C₂×C₅₃	#2994	1	1	yes	characteristic
224	C₂×C₄×Q₈	C₂²×C₅₃	1	1	yes	characteristic
225	#446	—	2	4	no	—
226	C₈×Q₈	—	2	4	no	—
227	C₂×C₄⋊C₈	C₂²×C₅₃	1	2	yes	—
228	C₂×C₄⋊C₈	C₂²×C₅₃	1	1	yes	characteristic
229	C₂×C₄⋊C₈	C₂²×C₅₃	1	1	yes	characteristic
230	C₂×C₄⋊C₈	C₂²×C₅₃	1	1	yes	characteristic
231	C₂×C₄⋊C₈	C₂²×C₅₃	1	2	yes	—
232	C₂×C₄⋊C₈	C₂²×C₅₃	1	2	yes	—
233	C₂×C₄⋊C₈	C₂²×C₅₃	1	2	yes	—
234	C₂×C₄⋊C₈	C₂²×C₅₃	1	2	yes	—
235	C₂×C₄⋊C₈	C₂²×C₅₃	1	2	yes	—
236	C₂×C₄⋊C₈	C₂²×C₅₃	1	1	yes	characteristic
237	C₂×#97	C₂²×C₅₃	1	2	yes	—
238	C₂×C₄×C₈	C₂²×C₅₃	1	1	yes	characteristic
239	C₂×C₄×C₈	C₂²×C₅₃	1	2	yes	—
240	C₂×C₄×C₈	C₂²×C₅₃	1	1	yes	characteristic
241	#391	C₄×C₅₃	1	1	yes	characteristic
242	#391	C₄×C₅₃	1	1	yes	characteristic
243	#389	C₄×C₅₃	1	1	yes	characteristic
244	#389	C₄×C₅₃	1	1	yes	characteristic
245	#387	C₄×C₅₃	1	2	yes	—
246	#387	C₄×C₅₃	1	2	yes	—
247	C₄×C₄⋊C₄	C₂²×C₅₃	1	1	yes	characteristic
248	C₄×C₄⋊C₄	C₂²×C₅₃	1	2	yes	—
249	#376	C₂²×C₅₃	1	2	yes	—
250	C₄³	C₂²×C₅₃	1	1	yes	characteristic
251	#336	C₂²×C₅₃	1	1	yes	characteristic
252	#336	C₂²×C₅₃	1	2	yes	—
253	#336	C₂²×C₅₃	1	2	yes	—
254	#336	C₂²×C₅₃	1	1	yes	characteristic
255	C₅₃	#22721	1	1	yes	characteristic, direct product factor, Sylow
256	C₂²×Q₈	C₂×C₄×C₅₃	1	1	yes	characteristic
257	C₂²×C₈	C₂³×C₅₃	1	1	yes	characteristic
258	C₂²×C₈	C₂³×C₅₃	1	2	yes	—
259	C₂²×C₈	C₂³×C₅₃	1	1	yes	characteristic
260	C₂²×C₈	C₂³×C₅₃	1	2	yes	—
261	C₂²×C₈	C₂³×C₅₃	1	1	yes	characteristic
262	C₂²×C₈	C₂³×C₅₃	1	1	yes	characteristic
263	C₄×Q₈	D₄×C₅₃	1	4	yes	—
264	C₄×Q₈	Q₈×C₅₃	1	4	yes	—
265	C₂×C₄⋊C₄	C₂×C₄×C₅₃	1	2	yes	—
266	C₂×C₄⋊C₄	C₂×C₄×C₅₃	1	1	yes	characteristic
267	C₂×C₄⋊C₄	C₂×C₄×C₅₃	1	1	yes	characteristic
268	C₂×C₄⋊C₄	C₂×C₄×C₅₃	1	2	yes	—
269	C₂×C₄⋊C₄	C₂×C₄×C₅₃	1	2	yes	—
270	C₂×C₄⋊C₄	C₂×C₄×C₅₃	1	1	yes	characteristic
271	C₂×C₄²	C₂×C₄×C₅₃	1	1	yes	characteristic
272	C₂×C₄²	C₂×C₄×C₅₃	1	1	yes	characteristic
273	C₂×C₄²	C₂×C₄×C₅₃	1	1	yes	characteristic
274	C₂×C₄²	C₂×C₄×C₅₃	1	1	yes	characteristic
275	C₂×C₄²	C₂³×C₅₃	1	1	yes	characteristic
276	C₂×C₄²	C₂³×C₅₃	1	2	yes	—
277	C₂×C₄²	C₂³×C₅₃	1	1	yes	characteristic
278	C₂×C₄²	C₂³×C₅₃	1	2	yes	—
279	C₂×C₄²	C₂³×C₅₃	1	1	yes	characteristic
280	C₄⋊C₈	—	2	2	no	—
281	C₄⋊C₈	—	2	4	no	—
282	C₄⋊C₈	—	2	2	no	—
283	C₄⋊C₈	—	2	4	no	—
284	C₄⋊C₈	—	2	4	no	—
285	C₄⋊C₈	—	2	2	no	—
286	C₄⋊C₈	—	4	2	no	—
287	C₄⋊C₈	—	4	2	no	—
288	C₄⋊C₈	—	4	2	no	—
289	C₄⋊C₈	Q₈×C₅₃	1	4	yes	complemented
290	#97	—	2	4	no	—
291	C₄×C₈	—	2	2	no	—
292	C₄×C₈	—	2	4	no	—
293	C₄×C₈	—	2	2	no	—
294	#95	C₂×C₄×C₅₃	1	2	yes	—
295	#95	C₂×C₄×C₅₃	1	2	yes	—
296	#95	C₂×C₄×C₅₃	1	2	yes	—
297	#95	C₂×C₄×C₅₃	1	2	yes	—
298	#95	C₂×C₄×C₅₃	1	2	yes	—
299	#95	C₂×C₄×C₅₃	1	2	yes	—
300	#95	C₂×C₄×C₅₃	1	2	yes	—
301	C₂×Q₈	—	2	2	no	—
302	C₂×Q₈	—	2	2	no	—
303	C₂×Q₈	C₄⋊C₄×C₅₃	1	4	yes	—
304	C₂²×C₄	C₂²×C₄×C₅₃	1	2	yes	—
305	C₂²×C₄	C₂²×C₄×C₅₃	1	2	yes	—
306	C₂²×C₄	C₂²×C₄×C₅₃	1	1	yes	characteristic
307	C₂²×C₄	C₂²×C₄×C₅₃	1	2	yes	—
308	C₂²×C₄	C₂²×C₄×C₅₃	1	1	yes	characteristic
309	C₂²×C₄	C₂²×C₄×C₅₃	1	2	yes	—
310	C₂²×C₄	C₂²×C₄×C₅₃	1	1	yes	characteristic
311	C₂²×C₄	C₂²×C₄×C₅₃	1	1	yes	characteristic
312	C₂²×C₄	C₂²×C₄×C₅₃	1	1	yes	characteristic
313	C₂²×C₄	C₂²×C₄×C₅₃	1	1	yes	characteristic
314	C₂²×C₄	C₂⁴×C₅₃	1	1	yes	characteristic
315	C₂×C₈	—	2	1	no	—
316	C₂×C₈	—	2	1	no	—
317	C₂×C₈	—	2	1	no	—
318	C₂×C₈	—	2	1	no	—
319	C₂×C₈	—	2	1	no	—
320	C₂×C₈	—	2	2	no	—
321	C₂×C₈	—	2	1	no	—
322	C₂×C₈	—	2	2	no	—
323	C₂×C₈	—	2	2	no	—
324	C₂×C₈	—	2	1	no	—
325	C₂×C₈	—	2	1	no	—
326	C₂×C₈	—	2	2	no	—
327	C₂×C₈	—	2	2	no	—
328	C₂×C₈	—	2	1	no	—
329	C₂×C₈	—	2	1	no	—
330	C₂×C₈	—	2	2	no	—
331	C₂×C₈	C₂×Q₈×C₅₃	1	2	yes	—
332	C₂×C₈	C₂×Q₈×C₅₃	1	2	yes	—
333	C₄⋊C₄	—	2	4	no	—
334	C₄⋊C₄	—	2	2	no	—
335	C₄⋊C₄	—	2	2	no	—
336	C₄⋊C₄	—	4	2	no	—
337	C₄⋊C₄	C₄⋊C₄×C₅₃	1	8	yes	—
338	C₄⋊C₄	C₄⋊C₄×C₅₃	1	4	yes	—
339	C₄²	—	2	2	no	—
340	C₄²	—	2	2	no	—
341	C₄²	—	2	2	no	—
342	C₄²	—	2	1	no	—
343	C₄²	—	2	2	no	—
344	C₄²	—	2	1	no	—
345	C₄²	—	2	2	no	—
346	C₄²	—	4	1	no	—
347	C₄²	C₂×D₄×C₅₃	1	2	yes	—
348	C₄²	C₂×D₄×C₅₃	1	4	yes	—
349	C₄²	C₂×Q₈×C₅₃	1	4	yes	—
350	C₄²	C₂×Q₈×C₅₃	1	2	yes	—
351	C₄²	C₂×Q₈×C₅₃	1	2	yes	—
352	C₄²	#41×C₅₃	1	2	yes	—
353	C₂³	C₂³×C₄×C₅₃	1	1	yes	characteristic
354	Q₈	—	2	8	no	—
355	C₂×C₄	—	2	1	no	—
356	C₂×C₄	—	2	1	no	—
357	C₂×C₄	—	2	1	no	—
358	C₂×C₄	—	2	1	no	—
359	C₂×C₄	—	2	1	no	—
360	C₂×C₄	—	2	1	no	—
361	C₂×C₄	—	2	2	no	—
362	C₂×C₄	—	2	2	no	—
363	C₂×C₄	—	2	2	no	—
364	C₂×C₄	—	2	1	no	—
365	C₂×C₄	—	2	1	no	—
366	C₂×C₄	—	2	2	no	—
367	C₂×C₄	—	2	2	no	—
368	C₂×C₄	—	2	1	no	—
369	C₂×C₄	—	2	2	no	—
370	C₂×C₄	—	2	1	no	—
371	C₂×C₄	—	2	2	no	—
372	C₂×C₄	—	2	2	no	—
373	C₂×C₄	—	2	2	no	—
374	C₂×C₄	—	2	1	no	—
375	C₂×C₄	—	2	2	no	—
376	C₂×C₄	C₂×C₄⋊C₄×C₅₃	1	2	yes	—
377	C₂×C₄	C₂×C₄⋊C₄×C₅₃	1	4	yes	—
378	C₂×C₄	C₂×C₄⋊C₄×C₅₃	1	4	yes	—
379	C₂×C₄	C₂×C₄⋊C₄×C₅₃	1	2	yes	—
380	C₂×C₄	C₄×Q₈×C₅₃	1	2	yes	—
381	C₂×C₄	C₄×Q₈×C₅₃	1	2	yes	—
382	C₂×C₄	#131×C₅₃	1	2	yes	—
383	C₂×C₄	#131×C₅₃	1	2	yes	—
384	C₂×C₄	#131×C₅₃	1	2	yes	—
385	C₂×C₄	C₂²×D₄×C₅₃	1	1	yes	characteristic
386	C₂×C₄	C₂²×Q₈×C₅₃	1	1	yes	characteristic
387	C₂×C₄	C₂²×Q₈×C₅₃	1	1	yes	characteristic
388	C₂×C₄	C₂×#41×C₅₃	1	1	yes	characteristic
389	C₂×C₄	H(5,𝔽₂)×C₅₃	1	1	yes	characteristic
390	C₂×C₄	2₋¹⁺⁴×C₅₃	1	1	yes	characteristic
391	C₈	—	2	2	no	—
392	C₈	—	2	2	no	—
393	C₈	—	4	1	no	—
394	C₈	—	4	2	no	—
395	C₈	—	4	1	no	—
396	C₈	—	4	2	no	—
397	C₂²	C₂²×C₄⋊C₄×C₅₃	1	1	yes	characteristic
398	C₂²	C₂×C₄×Q₈×C₅₃	1	1	yes	characteristic
399	C₂²	C₅₃×#520	1	1	yes	characteristic
400	C₂²	C₂×#131×C₅₃	1	1	yes	characteristic
401	C₂²	C₂×#131×C₅₃	1	1	yes	characteristic
402	C₂²	C₂×#131×C₅₃	1	1	yes	characteristic
403	C₂²	C₅₃×#568	1	1	yes	characteristic
404	C₄	—	2	2	no	—
405	C₄	—	2	2	no	—
406	C₄	—	2	2	no	—
407	C₄	—	2	4	no	—
408	C₄	—	2	2	no	—
409	C₄	—	2	4	no	—
410	C₄	—	4	1	no	—
411	C₄	—	4	2	no	—
412	C₄	—	4	2	no	—
413	C₄	C₅₃×#424	1	4	yes	—
414	C₄	D₄×Q₈×C₅₃	1	1	yes	characteristic
415	C₄	C₅₃×#550	1	1	yes	characteristic
416	C₄	C₅₃×#557	1	1	yes	characteristic
417	C₄	Q₈²×C₅₃	1	1	yes	characteristic
418	C₂	Q₈×C₄⋊C₄×C₅₃	1	1	yes	characteristic
419	C₂	C₂×C₅₃×#424	1	1	yes	characteristic
420	C₂	C₅₃×#2906	1	1	yes	characteristic
421	C₂	C₅₃×#2960	1	1	yes	characteristic
422	C₂	C₅₃×#2962	1	1	yes	characteristic
423	C₂	C₅₃×#2993	1	1	yes	characteristic
424	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, prime radical	C₂³×C₅₃	C₂³×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
76. C₄⋊C₈×C₅₃	Q₈	semidirect
184. #22721	C₅₃	direct (N × H)
255. C₅₃	#22721	direct (N × H)
289. C₄⋊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
C₂×C₅₃×#446	2
C₂×C₈×Q₈×C₅₃	2
C₅₃×#2295	2
C₅₃×#1940	2
C₅₃×#1939	2
C₅₃×#1843	2
C₅₃×#1842	2
C₅₃×#1768	2
C₄×C₄⋊C₈×C₅₃	2
#22721	53

Subgroups — C53×#22721 #27230233

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅₃×#22721 #27230233