Subgroups — C₅₃×#22656 #27230168

Order 13568.

1010 conjugacy classes of subgroups.

The subgroup lattice has 512 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₈×D₄×C₅₃	C₂	1	1	yes	maximal, characteristic
2	C₂×C₅₃×#427	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂²×#107×C₅₃	C₂	1	1	yes	maximal, characteristic, complemented
4	C₅₃×#2352	C₂	1	2	yes	maximal
5	C₅₃×#1949	C₂	1	2	yes	maximal, complemented
6	C₅₃×#1854	C₂	1	2	yes	maximal, complemented
7	C₅₃×#1853	C₂	1	2	yes	maximal, complemented
8	C₅₃×#1846	C₂	1	1	yes	maximal, characteristic, complemented
9	C₅₃×#1808	C₂	1	1	yes	maximal, characteristic, complemented
10	C₅₃×#1808	C₂	1	1	yes	maximal, characteristic, complemented
11	C₄×#107×C₅₃	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂³×C₈×C₅₃	C₂²	1	1	yes	characteristic
13	C₂³×C₈×C₅₃	C₂²	1	1	yes	characteristic
14	C₂×C₄×D₄×C₅₃	C₂²	1	1	yes	characteristic
15	C₂×#117×C₅₃	C₂²	1	2	yes	—
16	C₂²×C₄⋊C₄×C₅₃	C₂²	1	2	yes	—
17	C₈×D₄×C₅₃	C₄	1	8	yes	complemented
18	C₅₃×#427	—	2	4	no	—
19	C₂×#107×C₅₃	—	2	4	no	—
20	C₂×#107×C₅₃	C₂²	1	2	yes	—
21	C₂×#107×C₅₃	C₂²	1	1	yes	characteristic
22	C₂×#107×C₅₃	C₂²	1	1	yes	characteristic
23	C₂×#107×C₅₃	C₂²	1	1	yes	characteristic
24	C₂×#107×C₅₃	C₂²	1	1	yes	characteristic
25	C₂×#106×C₅₃	C₂²	1	2	yes	—
26	C₂×#106×C₅₃	C₂²	1	2	yes	—
27	C₂×C₄⋊C₈×C₅₃	C₂²	1	1	yes	characteristic
28	C₂×C₂²⋊C₈×C₅₃	C₂²	1	1	yes	characteristic
29	C₂×C₂²⋊C₈×C₅₃	C₂²	1	1	yes	characteristic
30	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
31	C₅₃×#388	—	2	2	no	—
32	C₅₃×#387	—	2	2	no	—
33	C₅₃×#386	—	2	2	no	—
34	C₅₃×#382	C₂²	1	2	yes	—
35	C₅₃×#380	C₂²	1	2	yes	—
36	C₅₃×#380	C₂²	1	2	yes	—
37	C₄×C₄⋊C₄×C₅₃	C₂²	1	2	yes	—
38	C₄×C₂²⋊C₄×C₅₃	—	2	2	no	—
39	C₅₃×#340	C₂²	1	2	yes	—
40	C₅₃×#340	C₂²	1	2	yes	—
41	C₅₃×#340	C₂²	1	2	yes	—
42	C₂²×D₄×C₅₃	D₄	1	1	yes	characteristic
43	C₂³×C₄×C₅₃	—	2	2	no	—
44	C₂³×C₄×C₅₃	C₂³	1	1	yes	characteristic
45	C₂³×C₄×C₅₃	C₂³	1	1	yes	characteristic
46	C₂²×C₈×C₅₃	—	2	2	no	—
47	C₂²×C₈×C₅₃	—	2	2	no	—
48	C₂²×C₈×C₅₃	—	2	2	no	—
49	C₂²×C₈×C₅₃	—	2	2	no	—
50	C₂²×C₈×C₅₃	C₂×C₄	1	4	yes	—
51	C₂²×C₈×C₅₃	C₂×C₄	1	4	yes	complemented
52	C₂²×C₈×C₅₃	C₂³	1	1	yes	characteristic
53	C₂²×C₈×C₅₃	C₂³	1	1	yes	characteristic
54	C₂²×C₈×C₅₃	C₂³	1	1	yes	characteristic
55	C₂²×C₈×C₅₃	C₂³	1	1	yes	characteristic
56	C₄×D₄×C₅₃	C₂×C₄	1	4	yes	—
57	C₄×D₄×C₅₃	Q₈	1	4	yes	—
58	#117×C₅₃	—	2	4	no	—
59	#117×C₅₃	—	4	2	no	—
60	C₂×C₄⋊C₄×C₅₃	—	2	2	no	—
61	C₂×C₄⋊C₄×C₅₃	—	2	4	no	—
62	C₂×C₄⋊C₄×C₅₃	—	4	2	no	—
63	C₂×C₄⋊C₄×C₅₃	D₄	1	1	yes	characteristic
64	C₂×C₄⋊C₄×C₅₃	C₂³	1	2	yes	—
65	C₂×C₄⋊C₄×C₅₃	C₂³	1	2	yes	—
66	C₂×C₄⋊C₄×C₅₃	C₂³	1	2	yes	—
67	C₂×C₄⋊C₄×C₅₃	C₂³	1	2	yes	—
68	C₂×C₂²⋊C₄×C₅₃	—	2	2	no	—
69	C₂×C₂²⋊C₄×C₅₃	—	2	2	no	—
70	C₂×C₂²⋊C₄×C₅₃	—	2	2	no	—
71	C₂×C₂²⋊C₄×C₅₃	D₄	1	1	yes	characteristic
72	C₂×C₂²⋊C₄×C₅₃	D₄	1	1	yes	characteristic
73	C₂×C₄²×C₅₃	—	2	2	no	—
74	C₂×C₄²×C₅₃	—	2	2	no	—
75	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
76	#107×C₅₃	—	2	2	no	—
77	#107×C₅₃	—	2	2	no	—
78	#107×C₅₃	—	2	2	no	—
79	#107×C₅₃	—	2	4	no	—
80	#107×C₅₃	D₄	1	4	yes	complemented
81	#106×C₅₃	—	2	4	no	—
82	#106×C₅₃	—	4	2	no	—
83	C₄⋊C₈×C₅₃	C₂×C₄	1	4	yes	complemented
84	C₂²⋊C₈×C₅₃	C₂×C₄	1	4	yes	—
85	C₂²⋊C₈×C₅₃	C₂×C₄	1	4	yes	complemented
86	C₄×C₈×C₅₃	C₂×C₄	1	4	yes	complemented
87	#95×C₅₃	—	2	2	no	—
88	#95×C₅₃	—	2	2	no	—
89	#95×C₅₃	—	2	2	no	—
90	#95×C₅₃	—	2	2	no	—
91	#95×C₅₃	—	2	2	no	—
92	C₂⁴×C₅₃	C₂×D₄	1	1	yes	characteristic
93	C₂⁴×C₅₃	C₂×D₄	1	1	yes	characteristic
94	C₂×D₄×C₅₃	—	2	2	no	—
95	C₂×D₄×C₅₃	—	2	2	no	—
96	C₂×D₄×C₅₃	C₄⋊C₄	1	4	yes	—
97	C₂²×C₄×C₅₃	—	2	1	no	—
98	C₂²×C₄×C₅₃	—	2	1	no	—
99	C₂²×C₄×C₅₃	—	2	1	no	—
100	C₂²×C₄×C₅₃	—	2	1	no	—
101	C₂²×C₄×C₅₃	—	2	1	no	—
102	C₂²×C₄×C₅₃	—	2	1	no	—
103	C₂²×C₄×C₅₃	—	2	1	no	—
104	C₂²×C₄×C₅₃	—	2	1	no	—
105	C₂²×C₄×C₅₃	—	2	2	no	—
106	C₂²×C₄×C₅₃	—	2	2	no	—
107	C₂²×C₄×C₅₃	—	2	2	no	—
108	C₂²×C₄×C₅₃	—	2	2	no	—
109	C₂²×C₄×C₅₃	—	4	4	no	—
110	C₂²×C₄×C₅₃	—	4	4	no	—
111	C₂²×C₄×C₅₃	—	8	2	no	—
112	C₂²×C₄×C₅₃	C₂²×C₄	1	2	yes	—
113	C₂²×C₄×C₅₃	C₂²×C₄	1	2	yes	—
114	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
115	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
116	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
117	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
118	C₂²×C₄×C₅₃	C₂×Q₈	1	2	yes	—
119	C₂²×C₄×C₅₃	C₂×Q₈	1	2	yes	—
120	C₂²×C₄×C₅₃	C₂⁴	1	1	yes	characteristic
121	C₂×C₈×C₅₃	—	2	1	no	—
122	C₂×C₈×C₅₃	—	2	1	no	—
123	C₂×C₈×C₅₃	—	2	1	no	—
124	C₂×C₈×C₅₃	—	2	1	no	—
125	C₂×C₈×C₅₃	—	2	1	no	—
126	C₂×C₈×C₅₃	—	2	1	no	—
127	C₂×C₈×C₅₃	—	2	8	no	—
128	C₂×C₈×C₅₃	—	2	8	no	—
129	C₂×C₈×C₅₃	C₂²×C₄	1	2	yes	—
130	C₂×C₈×C₅₃	C₂²×C₄	1	2	yes	—
131	C₂×C₈×C₅₃	C₂²×C₄	1	2	yes	—
132	C₂×C₈×C₅₃	C₂²×C₄	1	2	yes	—
133	C₂×C₈×C₅₃	C₂×D₄	1	2	yes	—
134	C₂×C₈×C₅₃	#41	1	2	yes	—
135	C₄⋊C₄×C₅₃	—	2	2	no	—
136	C₄⋊C₄×C₅₃	—	2	2	no	—
137	C₄⋊C₄×C₅₃	—	2	2	no	—
138	C₄⋊C₄×C₅₃	—	2	2	no	—
139	C₄⋊C₄×C₅₃	—	2	2	no	—
140	C₄⋊C₄×C₅₃	—	2	2	no	—
141	C₄⋊C₄×C₅₃	—	2	2	no	—
142	C₄⋊C₄×C₅₃	—	4	4	no	—
143	C₄⋊C₄×C₅₃	C₄⋊C₄	1	4	yes	—
144	C₄⋊C₄×C₅₃	C₂×D₄	1	4	yes	—
145	C₂²⋊C₄×C₅₃	—	4	4	no	—
146	C₂²⋊C₄×C₅₃	—	4	4	no	—
147	C₂²⋊C₄×C₅₃	—	8	2	no	—
148	C₂²⋊C₄×C₅₃	C₄⋊C₄	1	4	yes	—
149	C₂²⋊C₄×C₅₃	C₄⋊C₄	1	4	yes	—
150	C₄²×C₅₃	—	4	4	no	—
151	C₄²×C₅₃	—	8	2	no	—
152	C₄²×C₅₃	C₂²×C₄	1	2	yes	—
153	C₄²×C₅₃	C₂×Q₈	1	2	yes	—
154	C₂³×C₅₃	—	2	1	no	—
155	C₂³×C₅₃	—	2	1	no	—
156	C₂³×C₅₃	—	2	1	no	—
157	C₂³×C₅₃	—	2	1	no	—
158	C₂³×C₅₃	—	2	1	no	—
159	C₂³×C₅₃	—	2	1	no	—
160	C₂³×C₅₃	—	2	1	no	—
161	C₂³×C₅₃	—	2	1	no	—
162	C₂³×C₅₃	—	2	1	no	—
163	C₂³×C₅₃	—	2	1	no	—
164	C₂³×C₅₃	C₂×C₄⋊C₄	1	2	yes	—
165	C₂³×C₅₃	C₂×C₄⋊C₄	1	2	yes	—
166	C₂³×C₅₃	#135	1	2	yes	—
167	C₂³×C₅₃	#135	1	2	yes	—
168	C₂³×C₅₃	C₂²×D₄	1	1	yes	characteristic
169	D₄×C₅₃	—	2	8	no	—
170	C₂×C₄×C₅₃	—	2	1	no	—
171	C₂×C₄×C₅₃	—	2	1	no	—
172	C₂×C₄×C₅₃	—	2	1	no	—
173	C₂×C₄×C₅₃	—	2	1	no	—
174	C₂×C₄×C₅₃	—	2	1	no	—
175	C₂×C₄×C₅₃	—	2	1	no	—
176	C₂×C₄×C₅₃	—	2	1	no	—
177	C₂×C₄×C₅₃	—	2	1	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	—	2	2	no	—
183	C₂×C₄×C₅₃	—	2	2	no	—
184	C₂×C₄×C₅₃	—	2	2	no	—
185	C₂×C₄×C₅₃	—	2	2	no	—
186	C₂×C₄×C₅₃	—	2	4	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	—	4	2	no	—
194	C₂×C₄×C₅₃	—	4	2	no	—
195	C₂×C₄×C₅₃	—	4	2	no	—
196	C₂×C₄×C₅₃	—	4	2	no	—
197	C₂×C₄×C₅₃	—	4	2	no	—
198	C₂×C₄×C₅₃	—	8	8	no	—
199	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	2	yes	—
200	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	2	yes	—
201	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	2	yes	—
202	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	2	yes	—
203	C₂×C₄×C₅₃	C₂³×C₄	1	1	yes	characteristic
204	C₂×C₄×C₅₃	C₂²×D₄	1	1	yes	characteristic
205	C₂×C₄×C₅₃	C₂²×Q₈	1	1	yes	characteristic
206	C₂×C₄×C₅₃	C₂×#41	1	1	yes	characteristic
207	C₂×C₄×C₅₃	H(5,𝔽₂)	1	1	yes	characteristic
208	C₂×C₄×C₅₃	2₋¹⁺⁴	1	1	yes	characteristic
209	C₈×C₅₃	—	2	2	no	—
210	C₈×C₅₃	—	2	2	no	—
211	C₈×C₅₃	—	2	2	no	—
212	C₈×C₅₃	C₄×D₄	1	4	yes	—
213	#22656	C₅₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
214	C₂²×C₅₃	—	2	2	no	—
215	C₂²×C₅₃	—	2	2	no	—
216	C₂²×C₅₃	—	2	2	no	—
217	C₂²×C₅₃	—	2	2	no	—
218	C₂²×C₅₃	—	2	2	no	—
219	C₂²×C₅₃	—	2	2	no	—
220	C₂²×C₅₃	—	2	2	no	—
221	C₂²×C₅₃	—	2	2	no	—
222	C₂²×C₅₃	—	2	2	no	—
223	C₂²×C₅₃	—	4	1	no	—
224	C₂²×C₅₃	—	4	1	no	—
225	C₂²×C₅₃	—	4	1	no	—
226	C₂²×C₅₃	—	4	1	no	—
227	C₂²×C₅₃	#427	1	4	yes	complemented
228	C₂²×C₅₃	C₂²×C₄⋊C₄	1	1	yes	characteristic
229	C₂²×C₅₃	D₄²	1	1	yes	characteristic
230	C₂²×C₅₃	#547	1	1	yes	characteristic
231	C₂²×C₅₃	C₂×#135	1	1	yes	characteristic
232	C₂²×C₅₃	C₂×#135	1	1	yes	characteristic
233	C₂²×C₅₃	#576	1	1	yes	characteristic
234	C₂²×C₅₃	#578	1	1	yes	characteristic
235	C₄×C₅₃	—	2	2	no	—
236	C₄×C₅₃	—	2	2	no	—
237	C₄×C₅₃	—	2	2	no	—
238	C₄×C₅₃	—	2	2	no	—
239	C₄×C₅₃	—	4	4	no	—
240	C₄×C₅₃	—	8	2	no	—
241	C₄×C₅₃	—	8	2	no	—
242	C₄×C₅₃	—	8	2	no	—
243	C₄×C₅₃	C₂×C₄×D₄	1	1	yes	characteristic
244	C₄×C₅₃	#520	1	1	yes	characteristic
245	C₄×C₅₃	D₄×Q₈	1	1	yes	characteristic
246	C₄×C₅₃	#554	1	1	yes	characteristic
247	C₂×C₈×D₄	C₂×C₅₃	1	1	yes	characteristic
248	C₂×#427	C₂×C₅₃	1	1	yes	characteristic, complemented
249	C₂²×#107	C₂×C₅₃	1	1	yes	characteristic, complemented
250	#2352	C₂×C₅₃	1	2	yes	—
251	#1949	C₂×C₅₃	1	2	yes	complemented
252	#1854	C₂×C₅₃	1	2	yes	complemented
253	#1853	C₂×C₅₃	1	2	yes	complemented
254	#1846	C₂×C₅₃	1	1	yes	characteristic, complemented
255	#1808	C₂×C₅₃	1	1	yes	characteristic, complemented
256	#1808	C₂×C₅₃	1	1	yes	characteristic, complemented
257	C₄×#107	C₂×C₅₃	1	1	yes	characteristic, complemented
258	C₂×C₅₃	—	2	4	no	—
259	C₂×C₅₃	—	4	2	no	—
260	C₂×C₅₃	D₄×C₄⋊C₄	1	1	yes	characteristic
261	C₂×C₅₃	C₂×#427	1	1	yes	characteristic
262	C₂×C₅₃	#2913	1	1	yes	characteristic
263	C₂×C₅₃	#3280	1	1	yes	characteristic
264	C₂×C₅₃	#3285	1	1	yes	characteristic
265	C₂×C₅₃	#3336	1	1	yes	characteristic
266	C₂×C₅₃	#3339	1	1	yes	characteristic
267	C₂³×C₈	C₂²×C₅₃	1	1	yes	characteristic
268	C₂³×C₈	C₂²×C₅₃	1	1	yes	characteristic
269	C₂×C₄×D₄	C₂²×C₅₃	1	1	yes	characteristic
270	C₂×#117	C₂²×C₅₃	1	2	yes	—
271	C₂²×C₄⋊C₄	C₂²×C₅₃	1	2	yes	—
272	C₈×D₄	C₄×C₅₃	1	8	yes	complemented
273	#427	—	2	4	no	—
274	C₂×#107	—	2	4	no	—
275	C₂×#107	C₂²×C₅₃	1	1	yes	characteristic
276	C₂×#107	C₂²×C₅₃	1	1	yes	characteristic
277	C₂×#107	C₂²×C₅₃	1	1	yes	characteristic
278	C₂×#107	C₂²×C₅₃	1	2	yes	—
279	C₂×#107	C₂²×C₅₃	1	1	yes	characteristic
280	C₂×#106	C₂²×C₅₃	1	2	yes	—
281	C₂×#106	C₂²×C₅₃	1	2	yes	—
282	C₂×C₄⋊C₈	C₂²×C₅₃	1	1	yes	characteristic
283	C₂×C₂²⋊C₈	C₂²×C₅₃	1	1	yes	characteristic
284	C₂×C₂²⋊C₈	C₂²×C₅₃	1	1	yes	characteristic
285	C₂×C₄×C₈	C₂²×C₅₃	1	1	yes	characteristic
286	#388	—	2	2	no	—
287	#387	—	2	2	no	—
288	#386	—	2	2	no	—
289	#382	C₂²×C₅₃	1	2	yes	—
290	#380	C₂²×C₅₃	1	2	yes	—
291	#380	C₂²×C₅₃	1	2	yes	—
292	C₄×C₄⋊C₄	C₂²×C₅₃	1	2	yes	—
293	C₄×C₂²⋊C₄	—	2	2	no	—
294	#340	C₂²×C₅₃	1	2	yes	—
295	#340	C₂²×C₅₃	1	2	yes	—
296	#340	C₂²×C₅₃	1	2	yes	—
297	C₅₃	#22656	1	1	yes	characteristic, direct product factor, Sylow
298	C₂²×D₄	D₄×C₅₃	1	1	yes	characteristic
299	C₂³×C₄	—	2	2	no	—
300	C₂³×C₄	C₂³×C₅₃	1	1	yes	characteristic
301	C₂³×C₄	C₂³×C₅₃	1	1	yes	characteristic
302	C₂²×C₈	—	2	2	no	—
303	C₂²×C₈	—	2	2	no	—
304	C₂²×C₈	—	2	2	no	—
305	C₂²×C₈	—	2	2	no	—
306	C₂²×C₈	C₂×C₄×C₅₃	1	4	yes	—
307	C₂²×C₈	C₂×C₄×C₅₃	1	4	yes	complemented
308	C₂²×C₈	C₂³×C₅₃	1	1	yes	characteristic
309	C₂²×C₈	C₂³×C₅₃	1	1	yes	characteristic
310	C₂²×C₈	C₂³×C₅₃	1	1	yes	characteristic
311	C₂²×C₈	C₂³×C₅₃	1	1	yes	characteristic
312	C₄×D₄	C₂×C₄×C₅₃	1	4	yes	—
313	C₄×D₄	Q₈×C₅₃	1	4	yes	—
314	#117	—	2	4	no	—
315	#117	—	4	2	no	—
316	C₂×C₄⋊C₄	—	2	4	no	—
317	C₂×C₄⋊C₄	—	2	2	no	—
318	C₂×C₄⋊C₄	—	4	2	no	—
319	C₂×C₄⋊C₄	D₄×C₅₃	1	1	yes	characteristic
320	C₂×C₄⋊C₄	C₂³×C₅₃	1	2	yes	—
321	C₂×C₄⋊C₄	C₂³×C₅₃	1	2	yes	—
322	C₂×C₄⋊C₄	C₂³×C₅₃	1	2	yes	—
323	C₂×C₄⋊C₄	C₂³×C₅₃	1	2	yes	—
324	C₂×C₂²⋊C₄	—	2	2	no	—
325	C₂×C₂²⋊C₄	—	2	2	no	—
326	C₂×C₂²⋊C₄	—	2	2	no	—
327	C₂×C₂²⋊C₄	D₄×C₅₃	1	1	yes	characteristic
328	C₂×C₂²⋊C₄	D₄×C₅₃	1	1	yes	characteristic
329	C₂×C₄²	—	2	2	no	—
330	C₂×C₄²	—	2	2	no	—
331	C₂×C₄²	C₂³×C₅₃	1	1	yes	characteristic
332	#107	—	2	2	no	—
333	#107	—	2	2	no	—
334	#107	—	2	2	no	—
335	#107	—	2	4	no	—
336	#107	D₄×C₅₃	1	4	yes	complemented
337	#106	—	2	4	no	—
338	#106	—	4	2	no	—
339	C₄⋊C₈	C₂×C₄×C₅₃	1	4	yes	complemented
340	C₂²⋊C₈	C₂×C₄×C₅₃	1	4	yes	—
341	C₂²⋊C₈	C₂×C₄×C₅₃	1	4	yes	complemented
342	C₄×C₈	C₂×C₄×C₅₃	1	4	yes	complemented
343	#95	—	2	2	no	—
344	#95	—	2	2	no	—
345	#95	—	2	2	no	—
346	#95	—	2	2	no	—
347	#95	—	2	2	no	—
348	C₂⁴	C₂×D₄×C₅₃	1	1	yes	characteristic
349	C₂⁴	C₂×D₄×C₅₃	1	1	yes	characteristic
350	C₂×D₄	—	2	2	no	—
351	C₂×D₄	—	2	2	no	—
352	C₂×D₄	C₄⋊C₄×C₅₃	1	4	yes	—
353	C₂²×C₄	—	2	1	no	—
354	C₂²×C₄	—	2	1	no	—
355	C₂²×C₄	—	2	1	no	—
356	C₂²×C₄	—	2	1	no	—
357	C₂²×C₄	—	2	1	no	—
358	C₂²×C₄	—	2	1	no	—
359	C₂²×C₄	—	2	1	no	—
360	C₂²×C₄	—	2	1	no	—
361	C₂²×C₄	—	2	2	no	—
362	C₂²×C₄	—	2	2	no	—
363	C₂²×C₄	—	2	2	no	—
364	C₂²×C₄	—	2	2	no	—
365	C₂²×C₄	—	4	4	no	—
366	C₂²×C₄	—	4	4	no	—
367	C₂²×C₄	—	8	2	no	—
368	C₂²×C₄	C₂²×C₄×C₅₃	1	2	yes	—
369	C₂²×C₄	C₂²×C₄×C₅₃	1	2	yes	—
370	C₂²×C₄	C₂×D₄×C₅₃	1	1	yes	characteristic
371	C₂²×C₄	C₂×D₄×C₅₃	1	1	yes	characteristic
372	C₂²×C₄	C₂×D₄×C₅₃	1	1	yes	characteristic
373	C₂²×C₄	C₂×D₄×C₅₃	1	1	yes	characteristic
374	C₂²×C₄	C₂×Q₈×C₅₃	1	2	yes	—
375	C₂²×C₄	C₂×Q₈×C₅₃	1	2	yes	—
376	C₂²×C₄	C₂⁴×C₅₃	1	1	yes	characteristic
377	C₂×C₈	—	2	1	no	—
378	C₂×C₈	—	2	1	no	—
379	C₂×C₈	—	2	1	no	—
380	C₂×C₈	—	2	1	no	—
381	C₂×C₈	—	2	1	no	—
382	C₂×C₈	—	2	1	no	—
383	C₂×C₈	—	2	8	no	—
384	C₂×C₈	—	2	8	no	—
385	C₂×C₈	C₂²×C₄×C₅₃	1	2	yes	—
386	C₂×C₈	C₂²×C₄×C₅₃	1	2	yes	—
387	C₂×C₈	C₂²×C₄×C₅₃	1	2	yes	—
388	C₂×C₈	C₂²×C₄×C₅₃	1	2	yes	—
389	C₂×C₈	C₂×D₄×C₅₃	1	2	yes	—
390	C₂×C₈	#41×C₅₃	1	2	yes	—
391	C₄⋊C₄	—	2	2	no	—
392	C₄⋊C₄	—	2	2	no	—
393	C₄⋊C₄	—	2	2	no	—
394	C₄⋊C₄	—	2	2	no	—
395	C₄⋊C₄	—	2	2	no	—
396	C₄⋊C₄	—	2	2	no	—
397	C₄⋊C₄	—	2	2	no	—
398	C₄⋊C₄	—	4	4	no	—
399	C₄⋊C₄	C₄⋊C₄×C₅₃	1	4	yes	—
400	C₄⋊C₄	C₂×D₄×C₅₃	1	4	yes	—
401	C₂²⋊C₄	—	4	4	no	—
402	C₂²⋊C₄	—	4	4	no	—
403	C₂²⋊C₄	—	8	2	no	—
404	C₂²⋊C₄	C₄⋊C₄×C₅₃	1	4	yes	—
405	C₂²⋊C₄	C₄⋊C₄×C₅₃	1	4	yes	—
406	C₄²	—	4	4	no	—
407	C₄²	—	8	2	no	—
408	C₄²	C₂²×C₄×C₅₃	1	2	yes	—
409	C₄²	C₂×Q₈×C₅₃	1	2	yes	—
410	C₂³	—	2	1	no	—
411	C₂³	—	2	1	no	—
412	C₂³	—	2	1	no	—
413	C₂³	—	2	1	no	—
414	C₂³	—	2	1	no	—
415	C₂³	—	2	1	no	—
416	C₂³	—	2	1	no	—
417	C₂³	—	2	1	no	—
418	C₂³	—	2	1	no	—
419	C₂³	—	2	1	no	—
420	C₂³	C₂×C₄⋊C₄×C₅₃	1	2	yes	—
421	C₂³	C₂×C₄⋊C₄×C₅₃	1	2	yes	—
422	C₂³	#135×C₅₃	1	2	yes	—
423	C₂³	#135×C₅₃	1	2	yes	—
424	C₂³	C₂²×D₄×C₅₃	1	1	yes	characteristic
425	D₄	—	2	8	no	—
426	C₂×C₄	—	2	1	no	—
427	C₂×C₄	—	2	1	no	—
428	C₂×C₄	—	2	1	no	—
429	C₂×C₄	—	2	1	no	—
430	C₂×C₄	—	2	1	no	—
431	C₂×C₄	—	2	1	no	—
432	C₂×C₄	—	2	1	no	—
433	C₂×C₄	—	2	1	no	—
434	C₂×C₄	—	2	2	no	—
435	C₂×C₄	—	2	2	no	—
436	C₂×C₄	—	2	2	no	—
437	C₂×C₄	—	2	2	no	—
438	C₂×C₄	—	2	2	no	—
439	C₂×C₄	—	2	2	no	—
440	C₂×C₄	—	2	2	no	—
441	C₂×C₄	—	2	2	no	—
442	C₂×C₄	—	2	4	no	—
443	C₂×C₄	—	4	2	no	—
444	C₂×C₄	—	4	2	no	—
445	C₂×C₄	—	4	2	no	—
446	C₂×C₄	—	4	2	no	—
447	C₂×C₄	—	4	2	no	—
448	C₂×C₄	—	4	2	no	—
449	C₂×C₄	—	4	2	no	—
450	C₂×C₄	—	4	2	no	—
451	C₂×C₄	—	4	2	no	—
452	C₂×C₄	—	4	2	no	—
453	C₂×C₄	—	4	2	no	—
454	C₂×C₄	—	8	8	no	—
455	C₂×C₄	C₂×C₄⋊C₄×C₅₃	1	2	yes	—
456	C₂×C₄	C₂×C₄⋊C₄×C₅₃	1	2	yes	—
457	C₂×C₄	C₂×C₄⋊C₄×C₅₃	1	2	yes	—
458	C₂×C₄	C₂×C₄⋊C₄×C₅₃	1	2	yes	—
459	C₂×C₄	C₂³×C₄×C₅₃	1	1	yes	characteristic
460	C₂×C₄	C₂²×D₄×C₅₃	1	1	yes	characteristic
461	C₂×C₄	C₂²×Q₈×C₅₃	1	1	yes	characteristic
462	C₂×C₄	C₂×#41×C₅₃	1	1	yes	characteristic
463	C₂×C₄	H(5,𝔽₂)×C₅₃	1	1	yes	characteristic
464	C₂×C₄	2₋¹⁺⁴×C₅₃	1	1	yes	characteristic
465	C₈	—	2	2	no	—
466	C₈	—	2	2	no	—
467	C₈	—	2	2	no	—
468	C₈	C₄×D₄×C₅₃	1	4	yes	—
469	C₂²	—	2	2	no	—
470	C₂²	—	2	2	no	—
471	C₂²	—	2	2	no	—
472	C₂²	—	2	2	no	—
473	C₂²	—	2	2	no	—
474	C₂²	—	2	2	no	—
475	C₂²	—	2	2	no	—
476	C₂²	—	2	2	no	—
477	C₂²	—	2	2	no	—
478	C₂²	—	4	1	no	—
479	C₂²	—	4	1	no	—
480	C₂²	—	4	1	no	—
481	C₂²	—	4	1	no	—
482	C₂²	C₅₃×#427	1	4	yes	complemented
483	C₂²	C₂²×C₄⋊C₄×C₅₃	1	1	yes	characteristic
484	C₂²	D₄²×C₅₃	1	1	yes	characteristic
485	C₂²	C₅₃×#547	1	1	yes	characteristic
486	C₂²	C₂×#135×C₅₃	1	1	yes	characteristic
487	C₂²	C₂×#135×C₅₃	1	1	yes	characteristic
488	C₂²	C₅₃×#576	1	1	yes	characteristic
489	C₂²	C₅₃×#578	1	1	yes	characteristic
490	C₄	—	2	2	no	—
491	C₄	—	2	2	no	—
492	C₄	—	2	2	no	—
493	C₄	—	2	2	no	—
494	C₄	—	4	4	no	—
495	C₄	—	8	2	no	—
496	C₄	—	8	2	no	—
497	C₄	—	8	2	no	—
498	C₄	C₂×C₄×D₄×C₅₃	1	1	yes	characteristic
499	C₄	C₅₃×#520	1	1	yes	characteristic
500	C₄	D₄×Q₈×C₅₃	1	1	yes	characteristic
501	C₄	C₅₃×#554	1	1	yes	characteristic
502	C₂	—	2	4	no	—
503	C₂	—	4	2	no	—
504	C₂	D₄×C₄⋊C₄×C₅₃	1	1	yes	characteristic
505	C₂	C₂×C₅₃×#427	1	1	yes	characteristic
506	C₂	C₅₃×#2913	1	1	yes	characteristic
507	C₂	C₅₃×#3280	1	1	yes	characteristic
508	C₂	C₅₃×#3285	1	1	yes	characteristic
509	C₂	C₅₃×#3336	1	1	yes	characteristic
510	C₂	C₅₃×#3339	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	D₄
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₅₃ ◁ C₂×C₄×D₄×C₅₃ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂×C₅₃×#427	C₂	semidirect
3. C₂²×#107×C₅₃	C₂	semidirect
5. C₅₃×#1949	C₂	semidirect
6. C₅₃×#1854	C₂	semidirect
7. C₅₃×#1853	C₂	semidirect
8. C₅₃×#1846	C₂	semidirect
9. C₅₃×#1808	C₂	semidirect
10. C₅₃×#1808	C₂	semidirect
11. C₄×#107×C₅₃	C₂	semidirect
17. C₈×D₄×C₅₃	C₄	semidirect
51. C₂²×C₈×C₅₃	C₂×C₄	semidirect
80. #107×C₅₃	D₄	semidirect
83. C₄⋊C₈×C₅₃	C₂×C₄	semidirect
85. C₂²⋊C₈×C₅₃	C₂×C₄	semidirect
86. C₄×C₈×C₅₃	C₂×C₄	semidirect
213. #22656	C₅₃	direct (N × H)
227. C₂²×C₅₃	#427	semidirect
248. C₂×#427	C₂×C₅₃	semidirect
249. C₂²×#107	C₂×C₅₃	semidirect
251. #1949	C₂×C₅₃	semidirect
252. #1854	C₂×C₅₃	semidirect
253. #1853	C₂×C₅₃	semidirect
254. #1846	C₂×C₅₃	semidirect
255. #1808	C₂×C₅₃	semidirect
256. #1808	C₂×C₅₃	semidirect
257. C₄×#107	C₂×C₅₃	semidirect
272. C₈×D₄	C₄×C₅₃	semidirect
297. C₅₃	#22656	direct (N × H)
307. C₂²×C₈	C₂×C₄×C₅₃	semidirect
336. #107	D₄×C₅₃	semidirect
339. C₄⋊C₈	C₂×C₄×C₅₃	semidirect
341. C₂²⋊C₈	C₂×C₄×C₅₃	semidirect
342. C₄×C₈	C₂×C₄×C₅₃	semidirect
482. C₂²	C₅₃×#427	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₈×D₄×C₅₃	2
C₂×C₅₃×#427	2
C₂²×#107×C₅₃	2
C₅₃×#2352	2
C₅₃×#1949	2
C₅₃×#1854	2
C₅₃×#1853	2
C₅₃×#1846	2
C₅₃×#1808	2
C₄×#107×C₅₃	2
#22656	53

Subgroups — C53×#22656 #27230168

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅₃×#22656 #27230168