Subgroups — C₅₃×#22835 #27230347

Order 13568.

924 conjugacy classes of subgroups.

The subgroup lattice has 398 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄²×D₄×C₅₃	C₂	1	1	yes	maximal, characteristic
2	C₅₃×#1954	C₂	1	2	yes	maximal
3	C₅₃×#1923	C₂	1	4	yes	maximal, complemented
4	C₅₃×#1831	C₂	1	1	yes	maximal, characteristic, complemented
5	C₅₃×#1817	C₂	1	2	yes	maximal, complemented
6	C₈×C₄⋊C₄×C₅₃	C₂	1	2	yes	maximal, complemented
7	C₄×C₄⋊C₈×C₅₃	C₂	1	1	yes	maximal, characteristic, complemented
8	C₄×C₂²⋊C₈×C₅₃	C₂	1	2	yes	maximal, complemented
9	C₂×C₄×D₄×C₅₃	C₄	1	1	yes	characteristic
10	C₂×C₄×D₄×C₅₃	C₄	1	1	yes	characteristic
11	C₂×C₄×D₄×C₅₃	C₂²	1	1	yes	characteristic
12	C₂²×C₄²×C₅₃	C₂²	1	2	yes	—
13	C₂×C₄⋊C₈×C₅₃	C₂²	1	2	yes	—
14	C₂×C₄⋊C₈×C₅₃	C₂²	1	2	yes	—
15	C₂×C₂²⋊C₈×C₅₃	C₂²	1	4	yes	—
16	C₂×C₂²⋊C₈×C₅₃	C₂²	1	4	yes	—
17	C₂×C₄×C₈×C₅₃	C₂²	1	2	yes	—
18	C₂×C₄×C₈×C₅₃	C₂²	1	4	yes	—
19	C₄×C₄⋊C₄×C₅₃	C₄	1	1	yes	characteristic
20	C₄×C₄⋊C₄×C₅₃	C₄	1	1	yes	characteristic
21	C₄×C₄⋊C₄×C₅₃	C₂²	1	1	yes	characteristic
22	C₄×C₂²⋊C₄×C₅₃	C₄	1	2	yes	—
23	C₄×C₂²⋊C₄×C₅₃	C₄	1	2	yes	—
24	C₄×C₂²⋊C₄×C₅₃	C₂²	1	2	yes	—
25	C₄³×C₅₃	C₂²	1	1	yes	characteristic
26	C₅₃×#336	C₂²	1	2	yes	—
27	C₅₃×#336	C₂²	1	4	yes	—
28	C₅₃×#336	C₂²	1	1	yes	characteristic
29	C₅₃×#336	C₂²	1	2	yes	—
30	C₅₃×#336	C₂²	1	1	yes	characteristic
31	C₂²×D₄×C₅₃	C₂×C₄	1	1	yes	characteristic
32	C₂³×C₄×C₅₃	C₂×C₄	1	2	yes	—
33	C₂³×C₄×C₅₃	C₂×C₄	1	2	yes	—
34	C₂³×C₄×C₅₃	C₂³	1	2	yes	—
35	C₂²×C₈×C₅₃	C₂³	1	2	yes	—
36	C₂²×C₈×C₅₃	C₂³	1	4	yes	—
37	C₂²×C₈×C₅₃	C₂³	1	2	yes	—
38	C₄×D₄×C₅₃	—	2	4	no	—
39	C₄×D₄×C₅₃	—	2	2	no	—
40	C₄×D₄×C₅₃	—	2	2	no	—
41	C₄×D₄×C₅₃	C₈	1	8	yes	complemented
42	C₂×C₄⋊C₄×C₅₃	C₂×C₄	1	1	yes	characteristic
43	C₂×C₄⋊C₄×C₅₃	C₂×C₄	1	1	yes	characteristic
44	C₂×C₄⋊C₄×C₅₃	C₂×C₄	1	1	yes	characteristic
45	C₂×C₂²⋊C₄×C₅₃	C₂×C₄	1	2	yes	—
46	C₂×C₂²⋊C₄×C₅₃	C₂×C₄	1	2	yes	—
47	C₂×C₂²⋊C₄×C₅₃	C₂×C₄	1	2	yes	—
48	C₂×C₄²×C₅₃	—	4	4	no	—
49	C₂×C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
50	C₂×C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
51	C₂×C₄²×C₅₃	C₂×C₄	1	2	yes	—
52	C₂×C₄²×C₅₃	C₂×C₄	1	2	yes	—
53	C₂×C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
54	C₂×C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
55	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
56	C₂×C₄²×C₅₃	C₂³	1	2	yes	—
57	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
58	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
59	C₄⋊C₈×C₅₃	—	2	4	no	—
60	C₄⋊C₈×C₅₃	—	2	4	no	—
61	C₂²⋊C₈×C₅₃	—	2	8	no	—
62	C₂²⋊C₈×C₅₃	—	4	4	no	—
63	C₄×C₈×C₅₃	—	2	4	no	—
64	C₄×C₈×C₅₃	—	4	4	no	—
65	#95×C₅₃	C₂×C₄	1	2	yes	—
66	#95×C₅₃	C₂×C₄	1	2	yes	—
67	#95×C₅₃	C₂×C₄	1	2	yes	—
68	C₂⁴×C₅₃	C₂²×C₄	1	2	yes	—
69	C₂×D₄×C₅₃	—	2	2	no	—
70	C₂×D₄×C₅₃	—	2	2	no	—
71	C₂×D₄×C₅₃	C₂×C₈	1	4	yes	—
72	C₂²×C₄×C₅₃	—	2	4	no	—
73	C₂²×C₄×C₅₃	—	2	4	no	—
74	C₂²×C₄×C₅₃	—	2	4	no	—
75	C₂²×C₄×C₅₃	—	2	2	no	—
76	C₂²×C₄×C₅₃	—	2	2	no	—
77	C₂²×C₄×C₅₃	—	2	4	no	—
78	C₂²×C₄×C₅₃	—	4	2	no	—
79	C₂²×C₄×C₅₃	—	4	2	no	—
80	C₂²×C₄×C₅₃	—	4	2	no	—
81	C₂²×C₄×C₅₃	C₂×C₈	1	8	yes	—
82	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
83	C₂²×C₄×C₅₃	C₂²×C₄	1	2	yes	—
84	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
85	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
86	C₂²×C₄×C₅₃	C₂²×C₄	1	2	yes	—
87	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
88	C₂²×C₄×C₅₃	C₂²×C₄	1	2	yes	—
89	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
90	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
91	C₂²×C₄×C₅₃	C₂⁴	1	1	yes	characteristic
92	C₂×C₈×C₅₃	—	2	2	no	—
93	C₂×C₈×C₅₃	—	2	2	no	—
94	C₂×C₈×C₅₃	—	2	2	no	—
95	C₂×C₈×C₅₃	—	2	4	no	—
96	C₂×C₈×C₅₃	—	2	2	no	—
97	C₂×C₈×C₅₃	—	2	4	no	—
98	C₂×C₈×C₅₃	—	2	4	no	—
99	C₂×C₈×C₅₃	—	2	2	no	—
100	C₂×C₈×C₅₃	#41	1	4	yes	—
101	C₄⋊C₄×C₅₃	—	2	2	no	—
102	C₄⋊C₄×C₅₃	C₂×C₈	1	4	yes	—
103	C₄⋊C₄×C₅₃	M₁₆	1	4	yes	—
104	C₂²⋊C₄×C₅₃	—	2	4	no	—
105	C₂²⋊C₄×C₅₃	—	2	4	no	—
106	C₂²⋊C₄×C₅₃	C₂×C₈	1	8	yes	—
107	C₄²×C₅₃	—	2	2	no	—
108	C₄²×C₅₃	—	2	2	no	—
109	C₄²×C₅₃	—	2	2	no	—
110	C₄²×C₅₃	—	2	4	no	—
111	C₄²×C₅₃	—	2	2	no	—
112	C₄²×C₅₃	—	4	2	no	—
113	C₄²×C₅₃	—	4	2	no	—
114	C₄²×C₅₃	—	4	1	no	—
115	C₄²×C₅₃	C₂×C₈	1	4	yes	—
116	C₄²×C₅₃	#41	1	2	yes	—
117	C₄²×C₅₃	#41	1	2	yes	—
118	C₂³×C₅₃	—	2	2	no	—
119	C₂³×C₅₃	—	2	2	no	—
120	C₂³×C₅₃	—	2	2	no	—
121	C₂³×C₅₃	—	2	2	no	—
122	C₂³×C₅₃	—	2	2	no	—
123	C₂³×C₅₃	—	2	2	no	—
124	C₂³×C₅₃	C₂²×C₈	1	4	yes	—
125	C₂³×C₅₃	C₂³×C₄	1	1	yes	characteristic
126	D₄×C₅₃	—	2	8	no	—
127	C₂×C₄×C₅₃	—	2	1	no	—
128	C₂×C₄×C₅₃	—	2	1	no	—
129	C₂×C₄×C₅₃	—	2	1	no	—
130	C₂×C₄×C₅₃	—	2	2	no	—
131	C₂×C₄×C₅₃	—	2	2	no	—
132	C₂×C₄×C₅₃	—	2	1	no	—
133	C₂×C₄×C₅₃	—	2	2	no	—
134	C₂×C₄×C₅₃	—	2	1	no	—
135	C₂×C₄×C₅₃	—	2	1	no	—
136	C₂×C₄×C₅₃	—	2	1	no	—
137	C₂×C₄×C₅₃	—	2	2	no	—
138	C₂×C₄×C₅₃	—	2	1	no	—
139	C₂×C₄×C₅₃	—	2	2	no	—
140	C₂×C₄×C₅₃	—	2	2	no	—
141	C₂×C₄×C₅₃	—	2	1	no	—
142	C₂×C₄×C₅₃	—	2	2	no	—
143	C₂×C₄×C₅₃	—	2	1	no	—
144	C₂×C₄×C₅₃	—	2	1	no	—
145	C₂×C₄×C₅₃	—	4	8	no	—
146	C₂×C₄×C₅₃	—	4	8	no	—
147	C₂×C₄×C₅₃	—	4	4	no	—
148	C₂×C₄×C₅₃	—	4	4	no	—
149	C₂×C₄×C₅₃	#117	1	2	yes	—
150	C₂×C₄×C₅₃	#117	1	2	yes	—
151	C₂×C₄×C₅₃	C₂²×C₈	1	2	yes	—
152	C₂×C₄×C₅₃	C₂²×C₈	1	4	yes	—
153	C₂×C₄×C₅₃	C₂²×C₈	1	2	yes	—
154	C₂×C₄×C₅₃	C₂²×C₈	1	2	yes	—
155	C₂×C₄×C₅₃	C₂×M₁₆	1	4	yes	—
156	C₂×C₄×C₅₃	C₂×M₁₆	1	2	yes	—
157	C₂×C₄×C₅₃	#131	1	2	yes	—
158	C₂×C₄×C₅₃	C₂×#41	1	1	yes	characteristic
159	C₂×C₄×C₅₃	C₂×#41	1	2	yes	—
160	C₂×C₄×C₅₃	C₂×#41	1	1	yes	characteristic
161	C₂×C₄×C₅₃	H(5,𝔽₂)	1	2	yes	—
162	C₈×C₅₃	—	2	4	no	—
163	C₈×C₅₃	—	4	2	no	—
164	C₈×C₅₃	—	4	4	no	—
165	#22835	C₅₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
166	C₂²×C₅₃	—	2	4	no	—
167	C₂²×C₅₃	—	2	4	no	—
168	C₂²×C₅₃	—	2	4	no	—
169	C₂²×C₅₃	—	4	2	no	—
170	C₂²×C₅₃	—	4	2	no	—
171	C₂²×C₅₃	—	4	2	no	—
172	C₂²×C₅₃	—	4	2	no	—
173	C₂²×C₅₃	C₂×#117	1	1	yes	characteristic
174	C₂²×C₅₃	C₄×#41	1	1	yes	characteristic
175	C₂²×C₅₃	#518	1	1	yes	characteristic
176	C₂²×C₅₃	C₂³×C₈	1	1	yes	characteristic
177	C₂²×C₅₃	C₂²×M₁₆	1	1	yes	characteristic
178	C₂²×C₅₃	C₂×#131	1	1	yes	characteristic
179	C₂²×C₅₃	#568	1	1	yes	characteristic
180	C₄×C₅₃	—	2	2	no	—
181	C₄×C₅₃	—	2	2	no	—
182	C₄×C₅₃	—	2	2	no	—
183	C₄×C₅₃	—	2	4	no	—
184	C₄×C₅₃	—	2	2	no	—
185	C₄×C₅₃	—	2	4	no	—
186	C₄×C₅₃	—	4	2	no	—
187	C₄×C₅₃	—	4	1	no	—
188	C₄×C₅₃	#431	1	4	yes	—
189	C₄×C₅₃	#553	1	2	yes	—
190	C₄×C₅₃	#559	1	2	yes	—
191	C₄²×D₄	C₂×C₅₃	1	1	yes	characteristic
192	#1954	C₂×C₅₃	1	2	yes	—
193	#1923	C₂×C₅₃	1	4	yes	complemented
194	#1831	C₂×C₅₃	1	1	yes	characteristic, complemented
195	#1817	C₂×C₅₃	1	2	yes	complemented
196	C₈×C₄⋊C₄	C₂×C₅₃	1	2	yes	complemented
197	C₄×C₄⋊C₈	C₂×C₅₃	1	1	yes	characteristic, complemented
198	C₄×C₂²⋊C₈	C₂×C₅₃	1	2	yes	complemented
199	C₂×C₅₃	—	4	4	no	—
200	C₂×C₅₃	#2357	1	1	yes	characteristic
201	C₂×C₅₃	C₂×#431	1	1	yes	characteristic
202	C₂×C₅₃	#2925	1	1	yes	characteristic
203	C₂×C₅₃	C₈×#41	1	1	yes	characteristic
204	C₂×C₅₃	#2971	1	1	yes	characteristic
205	C₂×C₅₃	#2972	1	1	yes	characteristic
206	C₂×C₅₃	#2974	1	1	yes	characteristic
207	C₂×C₄×D₄	C₄×C₅₃	1	1	yes	characteristic
208	C₂×C₄×D₄	C₄×C₅₃	1	1	yes	characteristic
209	C₂×C₄×D₄	C₂²×C₅₃	1	1	yes	characteristic
210	C₂²×C₄²	C₂²×C₅₃	1	2	yes	—
211	C₂×C₄⋊C₈	C₂²×C₅₃	1	2	yes	—
212	C₂×C₄⋊C₈	C₂²×C₅₃	1	2	yes	—
213	C₂×C₂²⋊C₈	C₂²×C₅₃	1	4	yes	—
214	C₂×C₂²⋊C₈	C₂²×C₅₃	1	4	yes	—
215	C₂×C₄×C₈	C₂²×C₅₃	1	2	yes	—
216	C₂×C₄×C₈	C₂²×C₅₃	1	4	yes	—
217	C₄×C₄⋊C₄	C₄×C₅₃	1	1	yes	characteristic
218	C₄×C₄⋊C₄	C₄×C₅₃	1	1	yes	characteristic
219	C₄×C₄⋊C₄	C₂²×C₅₃	1	1	yes	characteristic
220	C₄×C₂²⋊C₄	C₄×C₅₃	1	2	yes	—
221	C₄×C₂²⋊C₄	C₄×C₅₃	1	2	yes	—
222	C₄×C₂²⋊C₄	C₂²×C₅₃	1	2	yes	—
223	C₄³	C₂²×C₅₃	1	1	yes	characteristic
224	#336	C₂²×C₅₃	1	1	yes	characteristic
225	#336	C₂²×C₅₃	1	2	yes	—
226	#336	C₂²×C₅₃	1	4	yes	—
227	#336	C₂²×C₅₃	1	2	yes	—
228	#336	C₂²×C₅₃	1	1	yes	characteristic
229	C₅₃	#22835	1	1	yes	characteristic, direct product factor, Sylow
230	C₂²×D₄	C₂×C₄×C₅₃	1	1	yes	characteristic
231	C₂³×C₄	C₂×C₄×C₅₃	1	2	yes	—
232	C₂³×C₄	C₂×C₄×C₅₃	1	2	yes	—
233	C₂³×C₄	C₂³×C₅₃	1	2	yes	—
234	C₂²×C₈	C₂³×C₅₃	1	2	yes	—
235	C₂²×C₈	C₂³×C₅₃	1	4	yes	—
236	C₂²×C₈	C₂³×C₅₃	1	2	yes	—
237	C₄×D₄	—	2	4	no	—
238	C₄×D₄	—	2	2	no	—
239	C₄×D₄	—	2	2	no	—
240	C₄×D₄	C₈×C₅₃	1	8	yes	complemented
241	C₂×C₄⋊C₄	C₂×C₄×C₅₃	1	1	yes	characteristic
242	C₂×C₄⋊C₄	C₂×C₄×C₅₃	1	1	yes	characteristic
243	C₂×C₄⋊C₄	C₂×C₄×C₅₃	1	1	yes	characteristic
244	C₂×C₂²⋊C₄	C₂×C₄×C₅₃	1	2	yes	—
245	C₂×C₂²⋊C₄	C₂×C₄×C₅₃	1	2	yes	—
246	C₂×C₂²⋊C₄	C₂×C₄×C₅₃	1	2	yes	—
247	C₂×C₄²	—	4	4	no	—
248	C₂×C₄²	C₂×C₄×C₅₃	1	1	yes	characteristic
249	C₂×C₄²	C₂×C₄×C₅₃	1	1	yes	characteristic
250	C₂×C₄²	C₂×C₄×C₅₃	1	1	yes	characteristic
251	C₂×C₄²	C₂×C₄×C₅₃	1	2	yes	—
252	C₂×C₄²	C₂×C₄×C₅₃	1	2	yes	—
253	C₂×C₄²	C₂×C₄×C₅₃	1	1	yes	characteristic
254	C₂×C₄²	C₂³×C₅₃	1	1	yes	characteristic
255	C₂×C₄²	C₂³×C₅₃	1	1	yes	characteristic
256	C₂×C₄²	C₂³×C₅₃	1	2	yes	—
257	C₂×C₄²	C₂³×C₅₃	1	1	yes	characteristic
258	C₄⋊C₈	—	2	4	no	—
259	C₄⋊C₈	—	2	4	no	—
260	C₂²⋊C₈	—	2	8	no	—
261	C₂²⋊C₈	—	4	4	no	—
262	C₄×C₈	—	2	4	no	—
263	C₄×C₈	—	4	4	no	—
264	#95	C₂×C₄×C₅₃	1	2	yes	—
265	#95	C₂×C₄×C₅₃	1	2	yes	—
266	#95	C₂×C₄×C₅₃	1	2	yes	—
267	C₂⁴	C₂²×C₄×C₅₃	1	2	yes	—
268	C₂×D₄	—	2	2	no	—
269	C₂×D₄	—	2	2	no	—
270	C₂×D₄	C₂×C₈×C₅₃	1	4	yes	—
271	C₂²×C₄	—	2	4	no	—
272	C₂²×C₄	—	2	4	no	—
273	C₂²×C₄	—	2	4	no	—
274	C₂²×C₄	—	2	2	no	—
275	C₂²×C₄	—	2	2	no	—
276	C₂²×C₄	—	2	4	no	—
277	C₂²×C₄	—	4	2	no	—
278	C₂²×C₄	—	4	2	no	—
279	C₂²×C₄	—	4	2	no	—
280	C₂²×C₄	C₂×C₈×C₅₃	1	8	yes	—
281	C₂²×C₄	C₂²×C₄×C₅₃	1	1	yes	characteristic
282	C₂²×C₄	C₂²×C₄×C₅₃	1	2	yes	—
283	C₂²×C₄	C₂²×C₄×C₅₃	1	1	yes	characteristic
284	C₂²×C₄	C₂²×C₄×C₅₃	1	1	yes	characteristic
285	C₂²×C₄	C₂²×C₄×C₅₃	1	2	yes	—
286	C₂²×C₄	C₂²×C₄×C₅₃	1	1	yes	characteristic
287	C₂²×C₄	C₂²×C₄×C₅₃	1	2	yes	—
288	C₂²×C₄	C₂²×C₄×C₅₃	1	1	yes	characteristic
289	C₂²×C₄	C₂²×C₄×C₅₃	1	1	yes	characteristic
290	C₂²×C₄	C₂⁴×C₅₃	1	1	yes	characteristic
291	C₂×C₈	—	2	2	no	—
292	C₂×C₈	—	2	2	no	—
293	C₂×C₈	—	2	2	no	—
294	C₂×C₈	—	2	4	no	—
295	C₂×C₈	—	2	2	no	—
296	C₂×C₈	—	2	4	no	—
297	C₂×C₈	—	2	4	no	—
298	C₂×C₈	—	2	2	no	—
299	C₂×C₈	#41×C₅₃	1	4	yes	—
300	C₄⋊C₄	—	2	2	no	—
301	C₄⋊C₄	C₂×C₈×C₅₃	1	4	yes	—
302	C₄⋊C₄	M₁₆×C₅₃	1	4	yes	—
303	C₂²⋊C₄	—	2	4	no	—
304	C₂²⋊C₄	—	2	4	no	—
305	C₂²⋊C₄	C₂×C₈×C₅₃	1	8	yes	—
306	C₄²	—	2	2	no	—
307	C₄²	—	2	2	no	—
308	C₄²	—	2	2	no	—
309	C₄²	—	2	4	no	—
310	C₄²	—	2	2	no	—
311	C₄²	—	4	2	no	—
312	C₄²	—	4	2	no	—
313	C₄²	—	4	1	no	—
314	C₄²	C₂×C₈×C₅₃	1	4	yes	—
315	C₄²	#41×C₅₃	1	2	yes	—
316	C₄²	#41×C₅₃	1	2	yes	—
317	C₂³	—	2	2	no	—
318	C₂³	—	2	2	no	—
319	C₂³	—	2	2	no	—
320	C₂³	—	2	2	no	—
321	C₂³	—	2	2	no	—
322	C₂³	—	2	2	no	—
323	C₂³	C₂²×C₈×C₅₃	1	4	yes	—
324	C₂³	C₂³×C₄×C₅₃	1	1	yes	characteristic
325	D₄	—	2	8	no	—
326	C₂×C₄	—	2	1	no	—
327	C₂×C₄	—	2	1	no	—
328	C₂×C₄	—	2	1	no	—
329	C₂×C₄	—	2	2	no	—
330	C₂×C₄	—	2	2	no	—
331	C₂×C₄	—	2	1	no	—
332	C₂×C₄	—	2	1	no	—
333	C₂×C₄	—	2	1	no	—
334	C₂×C₄	—	2	2	no	—
335	C₂×C₄	—	2	1	no	—
336	C₂×C₄	—	2	1	no	—
337	C₂×C₄	—	2	2	no	—
338	C₂×C₄	—	2	2	no	—
339	C₂×C₄	—	2	1	no	—
340	C₂×C₄	—	2	2	no	—
341	C₂×C₄	—	2	2	no	—
342	C₂×C₄	—	2	1	no	—
343	C₂×C₄	—	2	1	no	—
344	C₂×C₄	—	4	8	no	—
345	C₂×C₄	—	4	8	no	—
346	C₂×C₄	—	4	4	no	—
347	C₂×C₄	—	4	4	no	—
348	C₂×C₄	#117×C₅₃	1	2	yes	—
349	C₂×C₄	#117×C₅₃	1	2	yes	—
350	C₂×C₄	C₂²×C₈×C₅₃	1	2	yes	—
351	C₂×C₄	C₂²×C₈×C₅₃	1	2	yes	—
352	C₂×C₄	C₂²×C₈×C₅₃	1	4	yes	—
353	C₂×C₄	C₂²×C₈×C₅₃	1	2	yes	—
354	C₂×C₄	C₂×M₁₆×C₅₃	1	4	yes	—
355	C₂×C₄	C₂×M₁₆×C₅₃	1	2	yes	—
356	C₂×C₄	#131×C₅₃	1	2	yes	—
357	C₂×C₄	C₂×#41×C₅₃	1	1	yes	characteristic
358	C₂×C₄	C₂×#41×C₅₃	1	2	yes	—
359	C₂×C₄	C₂×#41×C₅₃	1	1	yes	characteristic
360	C₂×C₄	H(5,𝔽₂)×C₅₃	1	2	yes	—
361	C₈	—	2	4	no	—
362	C₈	—	4	2	no	—
363	C₈	—	4	4	no	—
364	C₂²	—	2	4	no	—
365	C₂²	—	2	4	no	—
366	C₂²	—	2	4	no	—
367	C₂²	—	4	2	no	—
368	C₂²	—	4	2	no	—
369	C₂²	—	4	2	no	—
370	C₂²	—	4	2	no	—
371	C₂²	C₂×#117×C₅₃	1	1	yes	characteristic
372	C₂²	C₄×#41×C₅₃	1	1	yes	characteristic
373	C₂²	C₅₃×#518	1	1	yes	characteristic
374	C₂²	C₂³×C₈×C₅₃	1	1	yes	characteristic
375	C₂²	C₂²×M₁₆×C₅₃	1	1	yes	characteristic
376	C₂²	C₂×#131×C₅₃	1	1	yes	characteristic
377	C₂²	C₅₃×#568	1	1	yes	characteristic
378	C₄	—	2	2	no	—
379	C₄	—	2	2	no	—
380	C₄	—	2	4	no	—
381	C₄	—	2	2	no	—
382	C₄	—	2	4	no	—
383	C₄	—	2	2	no	—
384	C₄	—	4	1	no	—
385	C₄	—	4	2	no	—
386	C₄	C₅₃×#431	1	4	yes	—
387	C₄	C₅₃×#553	1	2	yes	—
388	C₄	C₅₃×#559	1	2	yes	—
389	C₂	—	4	4	no	—
390	C₂	C₅₃×#2357	1	1	yes	characteristic
391	C₂	C₂×C₅₃×#431	1	1	yes	characteristic
392	C₂	C₅₃×#2925	1	1	yes	characteristic
393	C₂	C₈×#41×C₅₃	1	1	yes	characteristic
394	C₂	C₅₃×#2971	1	1	yes	characteristic
395	C₂	C₅₃×#2972	1	1	yes	characteristic
396	C₂	C₅₃×#2974	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄×C₅₃	C₂⁴
commutator	C₂²	C₂³×C₈×C₅₃
Frattini, cosocle	C₂²×C₄	C₂⁴×C₅₃
socle	C₂³×C₅₃	C₂³×C₄
prime radical	C₂²×D₄×C₅₃	C₂×C₄
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₄×C₅₃ ◁ G

Complements

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

N	H ≅ G/N	type
3. C₅₃×#1923	C₂	semidirect
4. C₅₃×#1831	C₂	semidirect
5. C₅₃×#1817	C₂	semidirect
6. C₈×C₄⋊C₄×C₅₃	C₂	semidirect
7. C₄×C₄⋊C₈×C₅₃	C₂	semidirect
8. C₄×C₂²⋊C₈×C₅₃	C₂	semidirect
41. C₄×D₄×C₅₃	C₈	semidirect
165. #22835	C₅₃	direct (N × H)
193. #1923	C₂×C₅₃	semidirect
194. #1831	C₂×C₅₃	semidirect
195. #1817	C₂×C₅₃	semidirect
196. C₈×C₄⋊C₄	C₂×C₅₃	semidirect
197. C₄×C₄⋊C₈	C₂×C₅₃	semidirect
198. C₄×C₂²⋊C₈	C₂×C₅₃	semidirect
229. C₅₃	#22835	direct (N × H)
240. C₄×D₄	C₈×C₅₃	semidirect

Maximal subgroups

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

subgroup	index
C₄²×D₄×C₅₃	2
C₅₃×#1954	2
C₅₃×#1923	2
C₅₃×#1831	2
C₅₃×#1817	2
C₈×C₄⋊C₄×C₅₃	2
C₄×C₄⋊C₈×C₅₃	2
C₄×C₂²⋊C₈×C₅₃	2
#22835	53

Subgroups — C53×#22835 #27230347

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅₃×#22835 #27230347