Subgroups — C₅₃×#22836 #27230348

Order 13568.

752 conjugacy classes of subgroups.

The subgroup lattice has 468 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₅₃×#2293	C₂	1	1	yes	maximal, characteristic
2	C₅₃×#1954	C₂	1	1	yes	maximal, characteristic
3	C₅₃×#1954	C₂	1	1	yes	maximal, characteristic
4	C₅₃×#1924	C₂	1	2	yes	maximal, complemented
5	C₅₃×#1923	C₂	1	2	yes	maximal, complemented
6	C₅₃×#1831	C₂	1	1	yes	maximal, characteristic, complemented
7	C₅₃×#1818	C₂	1	2	yes	maximal, complemented
8	C₅₃×#1770	C₂	1	1	yes	maximal, characteristic, complemented
9	C₈×C₄⋊C₄×C₅₃	C₂	1	1	yes	maximal, characteristic, complemented
10	C₄×C₄⋊C₈×C₅₃	C₂	1	1	yes	maximal, characteristic, complemented
11	C₅₃×#1750	C₂	1	2	yes	maximal, complemented
12	C₂×C₄×D₄×C₅₃	C₂²	1	1	yes	characteristic
13	C₂×#117×C₅₃	C₂²	1	2	yes	—
14	C₂×C₄⋊C₈×C₅₃	C₂²	1	1	yes	characteristic
15	C₂×C₄⋊C₈×C₅₃	C₂²	1	1	yes	characteristic
16	C₂×C₄⋊C₈×C₅₃	C₂²	1	1	yes	characteristic
17	C₂×C₄⋊C₈×C₅₃	C₂²	1	1	yes	characteristic
18	C₂×C₂²⋊C₈×C₅₃	C₂²	1	2	yes	—
19	C₂×C₂²⋊C₈×C₅₃	C₂²	1	2	yes	—
20	C₂×C₂²⋊C₈×C₅₃	C₂²	1	2	yes	—
21	C₂×C₂²⋊C₈×C₅₃	C₂²	1	2	yes	—
22	C₂×#97×C₅₃	C₂²	1	2	yes	—
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	2	yes	—
26	C₅₃×#390	C₄	1	1	yes	characteristic
27	C₅₃×#390	C₄	1	1	yes	characteristic
28	C₅₃×#389	C₄	1	1	yes	characteristic
29	C₅₃×#389	C₄	1	1	yes	characteristic
30	C₅₃×#388	C₄	1	2	yes	—
31	C₅₃×#388	C₄	1	2	yes	—
32	C₄×C₄⋊C₄×C₅₃	C₂²	1	1	yes	characteristic
33	C₄×C₂²⋊C₄×C₅₃	C₂²	1	2	yes	—
34	C₄³×C₅₃	C₂²	1	1	yes	characteristic
35	C₅₃×#336	C₂²	1	2	yes	—
36	C₅₃×#336	C₂²	1	2	yes	—
37	C₅₃×#336	C₂²	1	2	yes	—
38	C₅₃×#336	C₂²	1	1	yes	characteristic
39	C₅₃×#336	C₂²	1	2	yes	—
40	C₅₃×#336	C₂²	1	1	yes	characteristic
41	C₂²×D₄×C₅₃	C₂×C₄	1	1	yes	characteristic
42	C₂³×C₄×C₅₃	C₂³	1	2	yes	—
43	C₂²×C₈×C₅₃	C₂³	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	2	yes	—
47	C₂²×C₈×C₅₃	C₂³	1	1	yes	characteristic
48	C₂²×C₈×C₅₃	C₂³	1	1	yes	characteristic
49	C₄×D₄×C₅₃	—	2	2	no	—
50	C₄×D₄×C₅₃	—	2	2	no	—
51	#117×C₅₃	—	4	2	no	—
52	#117×C₅₃	—	4	2	no	—
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	1	yes	characteristic
56	C₂×C₄⋊C₄×C₅₃	C₂×C₄	1	2	yes	—
57	C₂×C₄⋊C₄×C₅₃	C₂×C₄	1	2	yes	—
58	C₂×C₂²⋊C₄×C₅₃	C₂×C₄	1	2	yes	—
59	C₂×C₂²⋊C₄×C₅₃	C₂×C₄	1	2	yes	—
60	C₂×C₂²⋊C₄×C₅₃	C₂×C₄	1	2	yes	—
61	C₂×C₂²⋊C₄×C₅₃	C₂×C₄	1	2	yes	—
62	C₂×C₂²⋊C₄×C₅₃	C₂×C₄	1	2	yes	—
63	C₂×C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
64	C₂×C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
65	C₂×C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
66	C₂×C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
67	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
68	C₂×C₄²×C₅₃	C₂³	1	2	yes	—
69	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
70	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
71	C₄⋊C₈×C₅₃	—	2	2	no	—
72	C₄⋊C₈×C₅₃	—	2	2	no	—
73	C₄⋊C₈×C₅₃	—	2	2	no	—
74	C₄⋊C₈×C₅₃	—	2	2	no	—
75	C₂²⋊C₈×C₅₃	—	2	4	no	—
76	C₂²⋊C₈×C₅₃	—	2	4	no	—
77	C₂²⋊C₈×C₅₃	—	4	2	no	—
78	C₂²⋊C₈×C₅₃	—	4	2	no	—
79	#97×C₅₃	—	4	2	no	—
80	C₄×C₈×C₅₃	—	2	2	no	—
81	C₄×C₈×C₅₃	—	2	2	no	—
82	C₄×C₈×C₅₃	—	4	2	no	—
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	C₂⁴×C₅₃	C₂²×C₄	1	2	yes	—
87	C₂×D₄×C₅₃	—	2	2	no	—
88	C₂×D₄×C₅₃	—	2	2	no	—
89	C₂×D₄×C₅₃	—	2	2	no	—
90	C₂²×C₄×C₅₃	—	2	2	no	—
91	C₂²×C₄×C₅₃	—	2	2	no	—
92	C₂²×C₄×C₅₃	—	2	2	no	—
93	C₂²×C₄×C₅₃	—	2	2	no	—
94	C₂²×C₄×C₅₃	—	2	2	no	—
95	C₂²×C₄×C₅₃	—	2	2	no	—
96	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
97	C₂²×C₄×C₅₃	C₂²×C₄	1	2	yes	—
98	C₂²×C₄×C₅₃	C₂²×C₄	1	2	yes	—
99	C₂²×C₄×C₅₃	C₂²×C₄	1	2	yes	—
100	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
101	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
102	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
103	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
104	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
105	C₂²×C₄×C₅₃	C₂⁴	1	1	yes	characteristic
106	C₂×C₈×C₅₃	—	2	1	no	—
107	C₂×C₈×C₅₃	—	2	1	no	—
108	C₂×C₈×C₅₃	—	2	1	no	—
109	C₂×C₈×C₅₃	—	2	1	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	1	no	—
116	C₂×C₈×C₅₃	—	2	2	no	—
117	C₂×C₈×C₅₃	—	2	2	no	—
118	C₂×C₈×C₅₃	—	2	2	no	—
119	C₂×C₈×C₅₃	—	2	2	no	—
120	C₂×C₈×C₅₃	—	2	1	no	—
121	C₂×C₈×C₅₃	—	2	1	no	—
122	C₂×C₈×C₅₃	#41	1	2	yes	—
123	C₂×C₈×C₅₃	#41	1	2	yes	—
124	C₄⋊C₄×C₅₃	—	2	2	no	—
125	C₄⋊C₄×C₅₃	—	2	2	no	—
126	C₄⋊C₄×C₅₃	—	2	2	no	—
127	C₄⋊C₄×C₅₃	—	4	2	no	—
128	C₄⋊C₄×C₅₃	—	4	2	no	—
129	C₂²⋊C₄×C₅₃	—	2	4	no	—
130	C₂²⋊C₄×C₅₃	—	2	4	no	—
131	C₂²⋊C₄×C₅₃	—	4	2	no	—
132	C₂²⋊C₄×C₅₃	—	4	2	no	—
133	C₂²⋊C₄×C₅₃	—	4	2	no	—
134	C₄²×C₅₃	—	2	2	no	—
135	C₄²×C₅₃	—	2	2	no	—
136	C₄²×C₅₃	—	2	2	no	—
137	C₄²×C₅₃	—	2	1	no	—
138	C₄²×C₅₃	—	2	1	no	—
139	C₄²×C₅₃	—	2	2	no	—
140	C₄²×C₅₃	—	2	2	no	—
141	C₄²×C₅₃	—	2	1	no	—
142	C₄²×C₅₃	—	2	1	no	—
143	C₄²×C₅₃	—	4	1	no	—
144	C₄²×C₅₃	#41	1	2	yes	—
145	C₄²×C₅₃	#41	1	2	yes	—
146	C₂³×C₅₃	—	2	2	no	—
147	C₂³×C₅₃	—	2	2	no	—
148	C₂³×C₅₃	—	2	2	no	—
149	C₂³×C₅₃	—	2	2	no	—
150	C₂³×C₅₃	—	2	2	no	—
151	C₂³×C₅₃	—	2	2	no	—
152	C₂³×C₅₃	—	2	2	no	—
153	C₂³×C₅₃	C₂³×C₄	1	1	yes	characteristic
154	D₄×C₅₃	—	4	4	no	—
155	C₂×C₄×C₅₃	—	2	1	no	—
156	C₂×C₄×C₅₃	—	2	1	no	—
157	C₂×C₄×C₅₃	—	2	1	no	—
158	C₂×C₄×C₅₃	—	2	1	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	—	2	1	no	—
164	C₂×C₄×C₅₃	—	2	1	no	—
165	C₂×C₄×C₅₃	—	2	2	no	—
166	C₂×C₄×C₅₃	—	2	2	no	—
167	C₂×C₄×C₅₃	—	2	1	no	—
168	C₂×C₄×C₅₃	—	2	1	no	—
169	C₂×C₄×C₅₃	—	2	1	no	—
170	C₂×C₄×C₅₃	—	2	1	no	—
171	C₂×C₄×C₅₃	—	2	2	no	—
172	C₂×C₄×C₅₃	—	2	2	no	—
173	C₂×C₄×C₅₃	—	2	2	no	—
174	C₂×C₄×C₅₃	—	2	1	no	—
175	C₂×C₄×C₅₃	—	2	1	no	—
176	C₂×C₄×C₅₃	—	2	2	no	—
177	C₂×C₄×C₅₃	—	4	2	no	—
178	C₂×C₄×C₅₃	—	4	2	no	—
179	C₂×C₄×C₅₃	—	4	2	no	—
180	C₂×C₄×C₅₃	—	4	2	no	—
181	C₂×C₄×C₅₃	#117	1	2	yes	—
182	C₂×C₄×C₅₃	#117	1	2	yes	—
183	C₂×C₄×C₅₃	#131	1	2	yes	—
184	C₂×C₄×C₅₃	#131	1	2	yes	—
185	C₂×C₄×C₅₃	#131	1	2	yes	—
186	C₂×C₄×C₅₃	C₂×#41	1	1	yes	characteristic
187	C₂×C₄×C₅₃	C₂×#41	1	1	yes	characteristic
188	C₂×C₄×C₅₃	C₂×#41	1	1	yes	characteristic
189	C₂×C₄×C₅₃	C₂×#41	1	1	yes	characteristic
190	C₂×C₄×C₅₃	H(5,𝔽₂)	1	1	yes	characteristic
191	C₂×C₄×C₅₃	H(5,𝔽₂)	1	1	yes	characteristic
192	C₈×C₅₃	—	2	2	no	—
193	C₈×C₅₃	—	2	2	no	—
194	C₈×C₅₃	—	4	1	no	—
195	C₈×C₅₃	—	4	1	no	—
196	C₈×C₅₃	—	4	2	no	—
197	C₈×C₅₃	—	4	2	no	—
198	#22836	C₅₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
199	C₂²×C₅₃	—	4	2	no	—
200	C₂²×C₅₃	—	4	2	no	—
201	C₂²×C₅₃	—	4	2	no	—
202	C₂²×C₅₃	—	4	2	no	—
203	C₂²×C₅₃	—	4	2	no	—
204	C₂²×C₅₃	—	4	2	no	—
205	C₂²×C₅₃	—	4	2	no	—
206	C₂²×C₅₃	C₂×#117	1	1	yes	characteristic
207	C₂²×C₅₃	C₄×#41	1	1	yes	characteristic
208	C₂²×C₅₃	#518	1	1	yes	characteristic
209	C₂²×C₅₃	C₂×#131	1	1	yes	characteristic
210	C₂²×C₅₃	C₂×#131	1	1	yes	characteristic
211	C₂²×C₅₃	C₂×#131	1	1	yes	characteristic
212	C₂²×C₅₃	#568	1	1	yes	characteristic
213	C₄×C₅₃	—	2	2	no	—
214	C₄×C₅₃	—	2	2	no	—
215	C₄×C₅₃	—	4	2	no	—
216	C₄×C₅₃	—	4	1	no	—
217	C₄×C₅₃	—	4	1	no	—
218	C₄×C₅₃	—	4	2	no	—
219	C₄×C₅₃	—	4	1	no	—
220	C₄×C₅₃	—	4	2	no	—
221	C₄×C₅₃	#433	1	4	yes	—
222	C₄×C₅₃	#553	1	1	yes	characteristic
223	C₄×C₅₃	#553	1	1	yes	characteristic
224	C₄×C₅₃	#559	1	1	yes	characteristic
225	C₄×C₅₃	#559	1	1	yes	characteristic
226	#2293	C₂×C₅₃	1	1	yes	characteristic
227	#1954	C₂×C₅₃	1	1	yes	characteristic
228	#1954	C₂×C₅₃	1	1	yes	characteristic
229	#1924	C₂×C₅₃	1	2	yes	complemented
230	#1923	C₂×C₅₃	1	2	yes	complemented
231	#1831	C₂×C₅₃	1	1	yes	characteristic, complemented
232	#1818	C₂×C₅₃	1	2	yes	complemented
233	#1770	C₂×C₅₃	1	1	yes	characteristic, complemented
234	C₈×C₄⋊C₄	C₂×C₅₃	1	1	yes	characteristic, complemented
235	C₄×C₄⋊C₈	C₂×C₅₃	1	1	yes	characteristic, complemented
236	#1750	C₂×C₅₃	1	2	yes	complemented
237	C₂×C₅₃	—	8	2	no	—
238	C₂×C₅₃	#2357	1	1	yes	characteristic
239	C₂×C₅₃	C₂×#433	1	1	yes	characteristic
240	C₂×C₅₃	#2925	1	1	yes	characteristic
241	C₂×C₅₃	#2966	1	1	yes	characteristic
242	C₂×C₅₃	#2969	1	1	yes	characteristic
243	C₂×C₅₃	#2979	1	1	yes	characteristic
244	C₂×C₅₃	#2981	1	1	yes	characteristic
245	C₂×C₄×D₄	C₂²×C₅₃	1	1	yes	characteristic
246	C₂×#117	C₂²×C₅₃	1	2	yes	—
247	C₂×C₄⋊C₈	C₂²×C₅₃	1	1	yes	characteristic
248	C₂×C₄⋊C₈	C₂²×C₅₃	1	1	yes	characteristic
249	C₂×C₄⋊C₈	C₂²×C₅₃	1	1	yes	characteristic
250	C₂×C₄⋊C₈	C₂²×C₅₃	1	1	yes	characteristic
251	C₂×C₂²⋊C₈	C₂²×C₅₃	1	2	yes	—
252	C₂×C₂²⋊C₈	C₂²×C₅₃	1	2	yes	—
253	C₂×C₂²⋊C₈	C₂²×C₅₃	1	2	yes	—
254	C₂×C₂²⋊C₈	C₂²×C₅₃	1	2	yes	—
255	C₂×#97	C₂²×C₅₃	1	2	yes	—
256	C₂×C₄×C₈	C₂²×C₅₃	1	1	yes	characteristic
257	C₂×C₄×C₈	C₂²×C₅₃	1	1	yes	characteristic
258	C₂×C₄×C₈	C₂²×C₅₃	1	2	yes	—
259	#390	C₄×C₅₃	1	1	yes	characteristic
260	#390	C₄×C₅₃	1	1	yes	characteristic
261	#389	C₄×C₅₃	1	1	yes	characteristic
262	#389	C₄×C₅₃	1	1	yes	characteristic
263	#388	C₄×C₅₃	1	2	yes	—
264	#388	C₄×C₅₃	1	2	yes	—
265	C₄×C₄⋊C₄	C₂²×C₅₃	1	1	yes	characteristic
266	C₄×C₂²⋊C₄	C₂²×C₅₃	1	2	yes	—
267	C₄³	C₂²×C₅₃	1	1	yes	characteristic
268	#336	C₂²×C₅₃	1	1	yes	characteristic
269	#336	C₂²×C₅₃	1	2	yes	—
270	#336	C₂²×C₅₃	1	2	yes	—
271	#336	C₂²×C₅₃	1	2	yes	—
272	#336	C₂²×C₅₃	1	2	yes	—
273	#336	C₂²×C₅₃	1	1	yes	characteristic
274	C₅₃	#22836	1	1	yes	characteristic, direct product factor, Sylow
275	C₂²×D₄	C₂×C₄×C₅₃	1	1	yes	characteristic
276	C₂³×C₄	C₂³×C₅₃	1	2	yes	—
277	C₂²×C₈	C₂³×C₅₃	1	1	yes	characteristic
278	C₂²×C₈	C₂³×C₅₃	1	1	yes	characteristic
279	C₂²×C₈	C₂³×C₅₃	1	2	yes	—
280	C₂²×C₈	C₂³×C₅₃	1	2	yes	—
281	C₂²×C₈	C₂³×C₅₃	1	1	yes	characteristic
282	C₂²×C₈	C₂³×C₅₃	1	1	yes	characteristic
283	C₄×D₄	—	2	2	no	—
284	C₄×D₄	—	2	2	no	—
285	#117	—	4	2	no	—
286	#117	—	4	2	no	—
287	C₂×C₄⋊C₄	C₂×C₄×C₅₃	1	1	yes	characteristic
288	C₂×C₄⋊C₄	C₂×C₄×C₅₃	1	1	yes	characteristic
289	C₂×C₄⋊C₄	C₂×C₄×C₅₃	1	1	yes	characteristic
290	C₂×C₄⋊C₄	C₂×C₄×C₅₃	1	2	yes	—
291	C₂×C₄⋊C₄	C₂×C₄×C₅₃	1	2	yes	—
292	C₂×C₂²⋊C₄	C₂×C₄×C₅₃	1	2	yes	—
293	C₂×C₂²⋊C₄	C₂×C₄×C₅₃	1	2	yes	—
294	C₂×C₂²⋊C₄	C₂×C₄×C₅₃	1	2	yes	—
295	C₂×C₂²⋊C₄	C₂×C₄×C₅₃	1	2	yes	—
296	C₂×C₂²⋊C₄	C₂×C₄×C₅₃	1	2	yes	—
297	C₂×C₄²	C₂×C₄×C₅₃	1	1	yes	characteristic
298	C₂×C₄²	C₂×C₄×C₅₃	1	1	yes	characteristic
299	C₂×C₄²	C₂×C₄×C₅₃	1	1	yes	characteristic
300	C₂×C₄²	C₂×C₄×C₅₃	1	1	yes	characteristic
301	C₂×C₄²	C₂³×C₅₃	1	1	yes	characteristic
302	C₂×C₄²	C₂³×C₅₃	1	2	yes	—
303	C₂×C₄²	C₂³×C₅₃	1	1	yes	characteristic
304	C₂×C₄²	C₂³×C₅₃	1	1	yes	characteristic
305	C₄⋊C₈	—	2	2	no	—
306	C₄⋊C₈	—	2	2	no	—
307	C₄⋊C₈	—	2	2	no	—
308	C₄⋊C₈	—	2	2	no	—
309	C₂²⋊C₈	—	2	4	no	—
310	C₂²⋊C₈	—	2	4	no	—
311	C₂²⋊C₈	—	4	2	no	—
312	C₂²⋊C₈	—	4	2	no	—
313	#97	—	4	2	no	—
314	C₄×C₈	—	2	2	no	—
315	C₄×C₈	—	2	2	no	—
316	C₄×C₈	—	4	2	no	—
317	#95	C₂×C₄×C₅₃	1	2	yes	—
318	#95	C₂×C₄×C₅₃	1	2	yes	—
319	#95	C₂×C₄×C₅₃	1	2	yes	—
320	C₂⁴	C₂²×C₄×C₅₃	1	2	yes	—
321	C₂×D₄	—	2	2	no	—
322	C₂×D₄	—	2	2	no	—
323	C₂×D₄	—	2	2	no	—
324	C₂²×C₄	—	2	2	no	—
325	C₂²×C₄	—	2	2	no	—
326	C₂²×C₄	—	2	2	no	—
327	C₂²×C₄	—	2	2	no	—
328	C₂²×C₄	—	2	2	no	—
329	C₂²×C₄	—	2	2	no	—
330	C₂²×C₄	C₂²×C₄×C₅₃	1	1	yes	characteristic
331	C₂²×C₄	C₂²×C₄×C₅₃	1	2	yes	—
332	C₂²×C₄	C₂²×C₄×C₅₃	1	2	yes	—
333	C₂²×C₄	C₂²×C₄×C₅₃	1	2	yes	—
334	C₂²×C₄	C₂²×C₄×C₅₃	1	1	yes	characteristic
335	C₂²×C₄	C₂²×C₄×C₅₃	1	1	yes	characteristic
336	C₂²×C₄	C₂²×C₄×C₅₃	1	1	yes	characteristic
337	C₂²×C₄	C₂²×C₄×C₅₃	1	1	yes	characteristic
338	C₂²×C₄	C₂²×C₄×C₅₃	1	1	yes	characteristic
339	C₂²×C₄	C₂⁴×C₅₃	1	1	yes	characteristic
340	C₂×C₈	—	2	1	no	—
341	C₂×C₈	—	2	1	no	—
342	C₂×C₈	—	2	1	no	—
343	C₂×C₈	—	2	1	no	—
344	C₂×C₈	—	2	1	no	—
345	C₂×C₈	—	2	1	no	—
346	C₂×C₈	—	2	2	no	—
347	C₂×C₈	—	2	2	no	—
348	C₂×C₈	—	2	1	no	—
349	C₂×C₈	—	2	1	no	—
350	C₂×C₈	—	2	2	no	—
351	C₂×C₈	—	2	2	no	—
352	C₂×C₈	—	2	2	no	—
353	C₂×C₈	—	2	2	no	—
354	C₂×C₈	—	2	1	no	—
355	C₂×C₈	—	2	1	no	—
356	C₂×C₈	#41×C₅₃	1	2	yes	—
357	C₂×C₈	#41×C₅₃	1	2	yes	—
358	C₄⋊C₄	—	2	2	no	—
359	C₄⋊C₄	—	2	2	no	—
360	C₄⋊C₄	—	2	2	no	—
361	C₄⋊C₄	—	4	2	no	—
362	C₄⋊C₄	—	4	2	no	—
363	C₂²⋊C₄	—	2	4	no	—
364	C₂²⋊C₄	—	2	4	no	—
365	C₂²⋊C₄	—	4	2	no	—
366	C₂²⋊C₄	—	4	2	no	—
367	C₂²⋊C₄	—	4	2	no	—
368	C₄²	—	2	2	no	—
369	C₄²	—	2	2	no	—
370	C₄²	—	2	2	no	—
371	C₄²	—	2	1	no	—
372	C₄²	—	2	1	no	—
373	C₄²	—	2	2	no	—
374	C₄²	—	2	2	no	—
375	C₄²	—	2	1	no	—
376	C₄²	—	2	1	no	—
377	C₄²	—	4	1	no	—
378	C₄²	#41×C₅₃	1	2	yes	—
379	C₄²	#41×C₅₃	1	2	yes	—
380	C₂³	—	2	2	no	—
381	C₂³	—	2	2	no	—
382	C₂³	—	2	2	no	—
383	C₂³	—	2	2	no	—
384	C₂³	—	2	2	no	—
385	C₂³	—	2	2	no	—
386	C₂³	—	2	2	no	—
387	C₂³	C₂³×C₄×C₅₃	1	1	yes	characteristic
388	D₄	—	4	4	no	—
389	C₂×C₄	—	2	1	no	—
390	C₂×C₄	—	2	1	no	—
391	C₂×C₄	—	2	1	no	—
392	C₂×C₄	—	2	1	no	—
393	C₂×C₄	—	2	2	no	—
394	C₂×C₄	—	2	1	no	—
395	C₂×C₄	—	2	2	no	—
396	C₂×C₄	—	2	1	no	—
397	C₂×C₄	—	2	1	no	—
398	C₂×C₄	—	2	2	no	—
399	C₂×C₄	—	2	2	no	—
400	C₂×C₄	—	2	2	no	—
401	C₂×C₄	—	2	1	no	—
402	C₂×C₄	—	2	1	no	—
403	C₂×C₄	—	2	1	no	—
404	C₂×C₄	—	2	1	no	—
405	C₂×C₄	—	2	2	no	—
406	C₂×C₄	—	2	2	no	—
407	C₂×C₄	—	2	2	no	—
408	C₂×C₄	—	2	1	no	—
409	C₂×C₄	—	2	1	no	—
410	C₂×C₄	—	2	2	no	—
411	C₂×C₄	—	4	2	no	—
412	C₂×C₄	—	4	2	no	—
413	C₂×C₄	—	4	2	no	—
414	C₂×C₄	—	4	2	no	—
415	C₂×C₄	#117×C₅₃	1	2	yes	—
416	C₂×C₄	#117×C₅₃	1	2	yes	—
417	C₂×C₄	#131×C₅₃	1	2	yes	—
418	C₂×C₄	#131×C₅₃	1	2	yes	—
419	C₂×C₄	#131×C₅₃	1	2	yes	—
420	C₂×C₄	C₂×#41×C₅₃	1	1	yes	characteristic
421	C₂×C₄	C₂×#41×C₅₃	1	1	yes	characteristic
422	C₂×C₄	C₂×#41×C₅₃	1	1	yes	characteristic
423	C₂×C₄	C₂×#41×C₅₃	1	1	yes	characteristic
424	C₂×C₄	H(5,𝔽₂)×C₅₃	1	1	yes	characteristic
425	C₂×C₄	H(5,𝔽₂)×C₅₃	1	1	yes	characteristic
426	C₈	—	2	2	no	—
427	C₈	—	2	2	no	—
428	C₈	—	4	1	no	—
429	C₈	—	4	1	no	—
430	C₈	—	4	2	no	—
431	C₈	—	4	2	no	—
432	C₂²	—	4	2	no	—
433	C₂²	—	4	2	no	—
434	C₂²	—	4	2	no	—
435	C₂²	—	4	2	no	—
436	C₂²	—	4	2	no	—
437	C₂²	—	4	2	no	—
438	C₂²	—	4	2	no	—
439	C₂²	C₂×#117×C₅₃	1	1	yes	characteristic
440	C₂²	C₄×#41×C₅₃	1	1	yes	characteristic
441	C₂²	C₅₃×#518	1	1	yes	characteristic
442	C₂²	C₂×#131×C₅₃	1	1	yes	characteristic
443	C₂²	C₂×#131×C₅₃	1	1	yes	characteristic
444	C₂²	C₂×#131×C₅₃	1	1	yes	characteristic
445	C₂²	C₅₃×#568	1	1	yes	characteristic
446	C₄	—	2	2	no	—
447	C₄	—	2	2	no	—
448	C₄	—	4	1	no	—
449	C₄	—	4	1	no	—
450	C₄	—	4	2	no	—
451	C₄	—	4	1	no	—
452	C₄	—	4	2	no	—
453	C₄	—	4	2	no	—
454	C₄	C₅₃×#433	1	4	yes	—
455	C₄	C₅₃×#553	1	1	yes	characteristic
456	C₄	C₅₃×#553	1	1	yes	characteristic
457	C₄	C₅₃×#559	1	1	yes	characteristic
458	C₄	C₅₃×#559	1	1	yes	characteristic
459	C₂	—	8	2	no	—
460	C₂	C₅₃×#2357	1	1	yes	characteristic
461	C₂	C₂×C₅₃×#433	1	1	yes	characteristic
462	C₂	C₅₃×#2925	1	1	yes	characteristic
463	C₂	C₅₃×#2966	1	1	yes	characteristic
464	C₂	C₅₃×#2969	1	1	yes	characteristic
465	C₂	C₅₃×#2979	1	1	yes	characteristic
466	C₂	C₅₃×#2981	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	C₂²×D₄×C₅₃	C₂×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
4. C₅₃×#1924	C₂	semidirect
5. C₅₃×#1923	C₂	semidirect
6. C₅₃×#1831	C₂	semidirect
7. C₅₃×#1818	C₂	semidirect
8. C₅₃×#1770	C₂	semidirect
9. C₈×C₄⋊C₄×C₅₃	C₂	semidirect
10. C₄×C₄⋊C₈×C₅₃	C₂	semidirect
11. C₅₃×#1750	C₂	semidirect
198. #22836	C₅₃	direct (N × H)
229. #1924	C₂×C₅₃	semidirect
230. #1923	C₂×C₅₃	semidirect
231. #1831	C₂×C₅₃	semidirect
232. #1818	C₂×C₅₃	semidirect
233. #1770	C₂×C₅₃	semidirect
234. C₈×C₄⋊C₄	C₂×C₅₃	semidirect
235. C₄×C₄⋊C₈	C₂×C₅₃	semidirect
236. #1750	C₂×C₅₃	semidirect
274. C₅₃	#22836	direct (N × H)

Maximal subgroups

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

subgroup	index
C₅₃×#2293	2
C₅₃×#1954	2
C₅₃×#1924	2
C₅₃×#1923	2
C₅₃×#1831	2
C₅₃×#1818	2
C₅₃×#1770	2
C₈×C₄⋊C₄×C₅₃	2
C₄×C₄⋊C₈×C₅₃	2
C₅₃×#1750	2
#22836	53

Subgroups — C53×#22836 #27230348

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅₃×#22836 #27230348