Subgroups — C₅₃×#21996 #27229508

Order 13568.

930 conjugacy classes of subgroups.

The subgroup lattice has 566 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×#1923	C₂	1	2	yes	maximal, complemented
3	C₅₃×#1916	C₂	1	2	yes	maximal, complemented
4	C₅₃×#1869	C₂	1	2	yes	maximal
5	C₅₃×#1850	C₂	1	2	yes	maximal, complemented
6	C₅₃×#1839	C₂	1	1	yes	maximal, characteristic, complemented
7	C₅₃×#1831	C₂	1	1	yes	maximal, characteristic, complemented
8	C₅₃×#1817	C₂	1	1	yes	maximal, characteristic, complemented
9	C₅₃×#1797	C₂	1	1	yes	maximal, characteristic, complemented
10	C₄×C₂²⋊C₈×C₅₃	C₂	1	1	yes	maximal, characteristic, complemented
11	C₂×C₅₃×#336	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂³×C₈×C₅₃	C₂²	1	2	yes	—
13	C₂×C₄×D₄×C₅₃	C₄	1	1	yes	characteristic
14	C₂×C₄×D₄×C₅₃	C₄	1	1	yes	characteristic
15	C₂×C₄×D₄×C₅₃	C₂²	1	1	yes	characteristic
16	C₂²×C₄²×C₅₃	C₂²	1	1	yes	characteristic
17	C₂²×C₄²×C₅₃	C₂²	1	1	yes	characteristic
18	C₂×C₄⋊C₈×C₅₃	C₂²	1	2	yes	—
19	C₂×C₄⋊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×C₂²⋊C₈×C₅₃	C₂²	1	2	yes	—
23	C₂×C₄×C₈×C₅₃	C₂²	1	2	yes	—
24	C₄×C₄⋊C₄×C₅₃	C₄	1	1	yes	characteristic
25	C₄×C₄⋊C₄×C₅₃	C₄	1	1	yes	characteristic
26	C₄×C₄⋊C₄×C₅₃	C₂²	1	1	yes	characteristic
27	C₄×C₂²⋊C₄×C₅₃	C₄	1	1	yes	characteristic
28	C₄×C₂²⋊C₄×C₅₃	C₄	1	1	yes	characteristic
29	C₄×C₂²⋊C₄×C₅₃	C₄	1	1	yes	characteristic
30	C₄×C₂²⋊C₄×C₅₃	C₄	1	1	yes	characteristic
31	C₄×C₂²⋊C₄×C₅₃	C₂²	1	1	yes	characteristic
32	C₄×C₂²⋊C₄×C₅₃	C₂²	1	1	yes	characteristic
33	C₄³×C₅₃	C₂²	1	1	yes	characteristic
34	C₅₃×#336	C₂²	1	2	yes	—
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	2	yes	—
39	C₅₃×#336	C₂²	1	1	yes	characteristic
40	C₅₃×#336	C₂²	1	1	yes	characteristic
41	C₅₃×#336	C₂²	1	1	yes	characteristic
42	C₅₃×#336	C₂²	1	1	yes	characteristic
43	C₂²×D₄×C₅₃	C₂×C₄	1	1	yes	characteristic
44	C₂³×C₄×C₅₃	C₂×C₄	1	1	yes	characteristic
45	C₂³×C₄×C₅₃	C₂×C₄	1	1	yes	characteristic
46	C₂³×C₄×C₅₃	C₂×C₄	1	1	yes	characteristic
47	C₂³×C₄×C₅₃	C₂×C₄	1	1	yes	characteristic
48	C₂³×C₄×C₅₃	C₂³	1	1	yes	characteristic
49	C₂³×C₄×C₅₃	C₂³	1	1	yes	characteristic
50	C₂²×C₈×C₅₃	—	2	4	no	—
51	C₂²×C₈×C₅₃	—	4	2	no	—
52	C₂²×C₈×C₅₃	—	4	2	no	—
53	C₂²×C₈×C₅₃	C₂³	1	2	yes	—
54	C₂²×C₈×C₅₃	C₂³	1	2	yes	—
55	C₂²×C₈×C₅₃	C₂³	1	2	yes	—
56	C₂²×C₈×C₅₃	C₂³	1	2	yes	—
57	C₄×D₄×C₅₃	—	2	4	no	—
58	C₄×D₄×C₅₃	—	2	4	no	—
59	C₄×D₄×C₅₃	C₈	1	8	yes	complemented
60	C₂×C₄⋊C₄×C₅₃	C₂×C₄	1	1	yes	characteristic
61	C₂×C₄⋊C₄×C₅₃	C₂×C₄	1	1	yes	characteristic
62	C₂×C₄⋊C₄×C₅₃	C₂×C₄	1	1	yes	characteristic
63	C₂×C₂²⋊C₄×C₅₃	C₂×C₄	1	1	yes	characteristic
64	C₂×C₂²⋊C₄×C₅₃	C₂×C₄	1	1	yes	characteristic
65	C₂×C₂²⋊C₄×C₅₃	C₂×C₄	1	1	yes	characteristic
66	C₂×C₂²⋊C₄×C₅₃	C₂×C₄	1	1	yes	characteristic
67	C₂×C₂²⋊C₄×C₅₃	C₂×C₄	1	1	yes	characteristic
68	C₂×C₂²⋊C₄×C₅₃	C₂×C₄	1	1	yes	characteristic
69	C₂×C₄²×C₅₃	—	4	2	no	—
70	C₂×C₄²×C₅₃	—	4	2	no	—
71	C₂×C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
72	C₂×C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
73	C₂×C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
74	C₂×C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
75	C₂×C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
76	C₂×C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
77	C₂×C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
78	C₂×C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
79	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
80	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
81	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
82	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
83	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
84	C₄⋊C₈×C₅₃	—	4	2	no	—
85	C₄⋊C₈×C₅₃	—	4	2	no	—
86	C₂²⋊C₈×C₅₃	—	2	4	no	—
87	C₂²⋊C₈×C₅₃	—	4	2	no	—
88	C₂²⋊C₈×C₅₃	—	4	2	no	—
89	C₄×C₈×C₅₃	—	4	2	no	—
90	#95×C₅₃	C₂×C₄	1	1	yes	characteristic
91	#95×C₅₃	C₂×C₄	1	1	yes	characteristic
92	#95×C₅₃	C₂×C₄	1	1	yes	characteristic
93	#95×C₅₃	C₂×C₄	1	1	yes	characteristic
94	#95×C₅₃	C₂×C₄	1	1	yes	characteristic
95	#95×C₅₃	C₂×C₄	1	1	yes	characteristic
96	C₂⁴×C₅₃	C₂²×C₄	1	1	yes	characteristic
97	C₂⁴×C₅₃	C₂²×C₄	1	1	yes	characteristic
98	C₂×D₄×C₅₃	—	2	2	no	—
99	C₂×D₄×C₅₃	—	2	2	no	—
100	C₂×D₄×C₅₃	C₂×C₈	1	4	yes	—
101	C₂²×C₄×C₅₃	—	2	2	no	—
102	C₂²×C₄×C₅₃	—	2	1	no	—
103	C₂²×C₄×C₅₃	—	2	1	no	—
104	C₂²×C₄×C₅₃	—	2	2	no	—
105	C₂²×C₄×C₅₃	—	2	2	no	—
106	C₂²×C₄×C₅₃	—	2	1	no	—
107	C₂²×C₄×C₅₃	—	2	2	no	—
108	C₂²×C₄×C₅₃	—	2	1	no	—
109	C₂²×C₄×C₅₃	—	4	1	no	—
110	C₂²×C₄×C₅₃	—	4	1	no	—
111	C₂²×C₄×C₅₃	—	4	1	no	—
112	C₂²×C₄×C₅₃	—	4	1	no	—
113	C₂²×C₄×C₅₃	—	4	1	no	—
114	C₂²×C₄×C₅₃	—	4	1	no	—
115	C₂²×C₄×C₅₃	—	4	1	no	—
116	C₂²×C₄×C₅₃	—	4	1	no	—
117	C₂²×C₄×C₅₃	C₂×C₈	1	4	yes	—
118	C₂²×C₄×C₅₃	C₂×C₈	1	4	yes	complemented
119	C₂²×C₄×C₅₃	M₁₆	1	4	yes	—
120	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
121	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
122	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
123	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
124	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
125	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
126	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
127	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
128	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
129	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
130	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
131	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
132	C₂²×C₄×C₅₃	#41	1	4	yes	—
133	C₂²×C₄×C₅₃	C₂⁴	1	1	yes	characteristic
134	C₂×C₈×C₅₃	—	2	2	no	—
135	C₂×C₈×C₅₃	—	2	2	no	—
136	C₂×C₈×C₅₃	—	2	2	no	—
137	C₂×C₈×C₅₃	—	2	2	no	—
138	C₂×C₈×C₅₃	—	2	2	no	—
139	C₂×C₈×C₅₃	—	2	2	no	—
140	C₂×C₈×C₅₃	—	2	2	no	—
141	C₂×C₈×C₅₃	—	2	2	no	—
142	C₂×C₈×C₅₃	—	2	2	no	—
143	C₂×C₈×C₅₃	—	2	2	no	—
144	C₂×C₈×C₅₃	—	2	2	no	—
145	C₂×C₈×C₅₃	—	2	2	no	—
146	C₂×C₈×C₅₃	—	4	8	no	—
147	C₄⋊C₄×C₅₃	—	2	2	no	—
148	C₄⋊C₄×C₅₃	—	2	2	no	—
149	C₄⋊C₄×C₅₃	C₂×C₈	1	4	yes	complemented
150	C₂²⋊C₄×C₅₃	—	2	2	no	—
151	C₂²⋊C₄×C₅₃	—	2	2	no	—
152	C₂²⋊C₄×C₅₃	—	2	2	no	—
153	C₂²⋊C₄×C₅₃	—	2	2	no	—
154	C₂²⋊C₄×C₅₃	C₂×C₈	1	4	yes	—
155	C₂²⋊C₄×C₅₃	C₂×C₈	1	4	yes	complemented
156	C₄²×C₅₃	—	2	2	no	—
157	C₄²×C₅₃	—	2	2	no	—
158	C₄²×C₅₃	—	2	2	no	—
159	C₄²×C₅₃	—	2	1	no	—
160	C₄²×C₅₃	—	2	2	no	—
161	C₄²×C₅₃	—	2	2	no	—
162	C₄²×C₅₃	—	2	1	no	—
163	C₄²×C₅₃	—	4	1	no	—
164	C₄²×C₅₃	—	4	1	no	—
165	C₄²×C₅₃	—	4	1	no	—
166	C₄²×C₅₃	—	4	1	no	—
167	C₄²×C₅₃	—	4	1	no	—
168	C₄²×C₅₃	—	4	1	no	—
169	C₄²×C₅₃	C₂×C₈	1	4	yes	complemented
170	C₂³×C₅₃	—	2	1	no	—
171	C₂³×C₅₃	—	2	1	no	—
172	C₂³×C₅₃	—	2	1	no	—
173	C₂³×C₅₃	—	2	1	no	—
174	C₂³×C₅₃	—	2	1	no	—
175	C₂³×C₅₃	—	2	1	no	—
176	C₂³×C₅₃	—	2	1	no	—
177	C₂³×C₅₃	—	2	1	no	—
178	C₂³×C₅₃	—	2	1	no	—
179	C₂³×C₅₃	—	2	1	no	—
180	C₂³×C₅₃	#117	1	2	yes	—
181	C₂³×C₅₃	C₂²×C₈	1	2	yes	—
182	C₂³×C₅₃	C₂²×C₈	1	2	yes	—
183	C₂³×C₅₃	C₂×M₁₆	1	2	yes	—
184	C₂³×C₅₃	C₂³×C₄	1	1	yes	characteristic
185	D₄×C₅₃	—	2	8	no	—
186	C₂×C₄×C₅₃	—	2	1	no	—
187	C₂×C₄×C₅₃	—	2	1	no	—
188	C₂×C₄×C₅₃	—	2	1	no	—
189	C₂×C₄×C₅₃	—	2	1	no	—
190	C₂×C₄×C₅₃	—	2	1	no	—
191	C₂×C₄×C₅₃	—	2	1	no	—
192	C₂×C₄×C₅₃	—	2	1	no	—
193	C₂×C₄×C₅₃	—	2	1	no	—
194	C₂×C₄×C₅₃	—	2	1	no	—
195	C₂×C₄×C₅₃	—	2	1	no	—
196	C₂×C₄×C₅₃	—	2	1	no	—
197	C₂×C₄×C₅₃	—	2	1	no	—
198	C₂×C₄×C₅₃	—	2	1	no	—
199	C₂×C₄×C₅₃	—	2	1	no	—
200	C₂×C₄×C₅₃	—	2	1	no	—
201	C₂×C₄×C₅₃	—	2	1	no	—
202	C₂×C₄×C₅₃	—	2	4	no	—
203	C₂×C₄×C₅₃	—	2	4	no	—
204	C₂×C₄×C₅₃	—	2	1	no	—
205	C₂×C₄×C₅₃	—	2	1	no	—
206	C₂×C₄×C₅₃	—	2	1	no	—
207	C₂×C₄×C₅₃	—	2	1	no	—
208	C₂×C₄×C₅₃	—	2	1	no	—
209	C₂×C₄×C₅₃	—	2	1	no	—
210	C₂×C₄×C₅₃	—	2	1	no	—
211	C₂×C₄×C₅₃	—	2	1	no	—
212	C₂×C₄×C₅₃	—	2	1	no	—
213	C₂×C₄×C₅₃	—	2	1	no	—
214	C₂×C₄×C₅₃	—	2	1	no	—
215	C₂×C₄×C₅₃	—	2	1	no	—
216	C₂×C₄×C₅₃	—	4	4	no	—
217	C₂×C₄×C₅₃	—	4	4	no	—
218	C₂×C₄×C₅₃	—	4	4	no	—
219	C₂×C₄×C₅₃	—	4	4	no	—
220	C₂×C₄×C₅₃	—	4	2	no	—
221	C₂×C₄×C₅₃	—	4	2	no	—
222	C₂×C₄×C₅₃	#117	1	2	yes	—
223	C₂×C₄×C₅₃	C₂²×C₈	1	2	yes	—
224	C₂×C₄×C₅₃	C₂²×C₈	1	2	yes	—
225	C₂×C₄×C₅₃	C₂²×C₈	1	2	yes	—
226	C₂×C₄×C₅₃	C₂²×C₈	1	2	yes	—
227	C₂×C₄×C₅₃	C₂²×C₈	1	2	yes	—
228	C₂×C₄×C₅₃	C₂×M₁₆	1	2	yes	—
229	C₂×C₄×C₅₃	C₂×M₁₆	1	2	yes	—
230	C₂×C₄×C₅₃	C₂×#41	1	2	yes	—
231	C₂×C₄×C₅₃	H(5,𝔽₂)	1	1	yes	characteristic
232	C₂×C₄×C₅₃	H(5,𝔽₂)	1	2	yes	—
233	C₂×C₄×C₅₃	2₋¹⁺⁴	1	1	yes	characteristic
234	C₈×C₅₃	—	4	2	no	—
235	C₈×C₅₃	—	4	2	no	—
236	C₈×C₅₃	—	4	2	no	—
237	C₈×C₅₃	—	4	2	no	—
238	#21996	C₅₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
239	C₂²×C₅₃	—	2	2	no	—
240	C₂²×C₅₃	—	2	2	no	—
241	C₂²×C₅₃	—	2	2	no	—
242	C₂²×C₅₃	—	2	2	no	—
243	C₂²×C₅₃	—	2	2	no	—
244	C₂²×C₅₃	—	2	2	no	—
245	C₂²×C₅₃	—	2	2	no	—
246	C₂²×C₅₃	—	2	2	no	—
247	C₂²×C₅₃	—	2	2	no	—
248	C₂²×C₅₃	—	4	1	no	—
249	C₂²×C₅₃	—	4	1	no	—
250	C₂²×C₅₃	—	4	1	no	—
251	C₂²×C₅₃	—	4	1	no	—
252	C₂²×C₅₃	#431	1	4	yes	—
253	C₂²×C₅₃	C₂×#117	1	1	yes	characteristic
254	C₂²×C₅₃	#518	1	1	yes	characteristic
255	C₂²×C₅₃	#520	1	1	yes	characteristic
256	C₂²×C₅₃	C₂³×C₈	1	1	yes	characteristic
257	C₂²×C₅₃	C₂²×M₁₆	1	1	yes	characteristic
258	C₂²×C₅₃	#568	1	1	yes	characteristic
259	C₂²×C₅₃	#568	1	1	yes	characteristic
260	C₄×C₅₃	—	2	2	no	—
261	C₄×C₅₃	—	2	2	no	—
262	C₄×C₅₃	—	2	2	no	—
263	C₄×C₅₃	—	2	2	no	—
264	C₄×C₅₃	—	2	2	no	—
265	C₄×C₅₃	—	2	2	no	—
266	C₄×C₅₃	—	2	2	no	—
267	C₄×C₅₃	—	2	2	no	—
268	C₄×C₅₃	—	4	1	no	—
269	C₄×C₅₃	—	4	1	no	—
270	C₄×C₅₃	—	4	1	no	—
271	C₄×C₅₃	—	4	1	no	—
272	C₄×C₅₃	#538	1	2	yes	—
273	C₄×C₅₃	#539	1	2	yes	—
274	C₄²×D₄	C₂×C₅₃	1	1	yes	characteristic
275	#1923	C₂×C₅₃	1	2	yes	complemented
276	#1916	C₂×C₅₃	1	2	yes	complemented
277	#1869	C₂×C₅₃	1	2	yes	—
278	#1850	C₂×C₅₃	1	2	yes	complemented
279	#1839	C₂×C₅₃	1	1	yes	characteristic, complemented
280	#1831	C₂×C₅₃	1	1	yes	characteristic, complemented
281	#1817	C₂×C₅₃	1	1	yes	characteristic, complemented
282	#1797	C₂×C₅₃	1	1	yes	characteristic, complemented
283	C₄×C₂²⋊C₈	C₂×C₅₃	1	1	yes	characteristic, complemented
284	C₂×#336	C₂×C₅₃	1	1	yes	characteristic, complemented
285	C₂×C₅₃	—	2	4	no	—
286	C₂×C₅₃	—	4	2	no	—
287	C₂×C₅₃	#2318	1	1	yes	characteristic
288	C₂×C₅₃	C₂×#431	1	1	yes	characteristic
289	C₂×C₅₃	#2923	1	1	yes	characteristic
290	C₂×C₅₃	#2972	1	1	yes	characteristic
291	C₂×C₅₃	#2973	1	1	yes	characteristic
292	C₂×C₅₃	#2988	1	1	yes	characteristic
293	C₂×C₅₃	#2989	1	1	yes	characteristic
294	C₂³×C₈	C₂²×C₅₃	1	2	yes	—
295	C₂×C₄×D₄	C₄×C₅₃	1	1	yes	characteristic
296	C₂×C₄×D₄	C₄×C₅₃	1	1	yes	characteristic
297	C₂×C₄×D₄	C₂²×C₅₃	1	1	yes	characteristic
298	C₂²×C₄²	C₂²×C₅₃	1	1	yes	characteristic
299	C₂²×C₄²	C₂²×C₅₃	1	1	yes	characteristic
300	C₂×C₄⋊C₈	C₂²×C₅₃	1	2	yes	—
301	C₂×C₄⋊C₈	C₂²×C₅₃	1	2	yes	—
302	C₂×C₂²⋊C₈	C₂²×C₅₃	1	2	yes	—
303	C₂×C₂²⋊C₈	C₂²×C₅₃	1	2	yes	—
304	C₂×C₂²⋊C₈	C₂²×C₅₃	1	2	yes	—
305	C₂×C₄×C₈	C₂²×C₅₃	1	2	yes	—
306	C₄×C₄⋊C₄	C₄×C₅₃	1	1	yes	characteristic
307	C₄×C₄⋊C₄	C₄×C₅₃	1	1	yes	characteristic
308	C₄×C₄⋊C₄	C₂²×C₅₃	1	1	yes	characteristic
309	C₄×C₂²⋊C₄	C₄×C₅₃	1	1	yes	characteristic
310	C₄×C₂²⋊C₄	C₄×C₅₃	1	1	yes	characteristic
311	C₄×C₂²⋊C₄	C₄×C₅₃	1	1	yes	characteristic
312	C₄×C₂²⋊C₄	C₄×C₅₃	1	1	yes	characteristic
313	C₄×C₂²⋊C₄	C₂²×C₅₃	1	1	yes	characteristic
314	C₄×C₂²⋊C₄	C₂²×C₅₃	1	1	yes	characteristic
315	C₄³	C₂²×C₅₃	1	1	yes	characteristic
316	#336	C₂²×C₅₃	1	1	yes	characteristic
317	#336	C₂²×C₅₃	1	1	yes	characteristic
318	#336	C₂²×C₅₃	1	1	yes	characteristic
319	#336	C₂²×C₅₃	1	1	yes	characteristic
320	#336	C₂²×C₅₃	1	2	yes	—
321	#336	C₂²×C₅₃	1	2	yes	—
322	#336	C₂²×C₅₃	1	2	yes	—
323	#336	C₂²×C₅₃	1	2	yes	—
324	#336	C₂²×C₅₃	1	2	yes	—
325	C₅₃	#21996	1	1	yes	characteristic, direct product factor, Sylow
326	C₂²×D₄	C₂×C₄×C₅₃	1	1	yes	characteristic
327	C₂³×C₄	C₂×C₄×C₅₃	1	1	yes	characteristic
328	C₂³×C₄	C₂×C₄×C₅₃	1	1	yes	characteristic
329	C₂³×C₄	C₂×C₄×C₅₃	1	1	yes	characteristic
330	C₂³×C₄	C₂×C₄×C₅₃	1	1	yes	characteristic
331	C₂³×C₄	C₂³×C₅₃	1	1	yes	characteristic
332	C₂³×C₄	C₂³×C₅₃	1	1	yes	characteristic
333	C₂²×C₈	—	2	4	no	—
334	C₂²×C₈	—	4	2	no	—
335	C₂²×C₈	—	4	2	no	—
336	C₂²×C₈	C₂³×C₅₃	1	2	yes	—
337	C₂²×C₈	C₂³×C₅₃	1	2	yes	—
338	C₂²×C₈	C₂³×C₅₃	1	2	yes	—
339	C₂²×C₈	C₂³×C₅₃	1	2	yes	—
340	C₄×D₄	—	2	4	no	—
341	C₄×D₄	—	2	4	no	—
342	C₄×D₄	C₈×C₅₃	1	8	yes	complemented
343	C₂×C₄⋊C₄	C₂×C₄×C₅₃	1	1	yes	characteristic
344	C₂×C₄⋊C₄	C₂×C₄×C₅₃	1	1	yes	characteristic
345	C₂×C₄⋊C₄	C₂×C₄×C₅₃	1	1	yes	characteristic
346	C₂×C₂²⋊C₄	C₂×C₄×C₅₃	1	1	yes	characteristic
347	C₂×C₂²⋊C₄	C₂×C₄×C₅₃	1	1	yes	characteristic
348	C₂×C₂²⋊C₄	C₂×C₄×C₅₃	1	1	yes	characteristic
349	C₂×C₂²⋊C₄	C₂×C₄×C₅₃	1	1	yes	characteristic
350	C₂×C₂²⋊C₄	C₂×C₄×C₅₃	1	1	yes	characteristic
351	C₂×C₂²⋊C₄	C₂×C₄×C₅₃	1	1	yes	characteristic
352	C₂×C₄²	—	4	2	no	—
353	C₂×C₄²	—	4	2	no	—
354	C₂×C₄²	C₂×C₄×C₅₃	1	1	yes	characteristic
355	C₂×C₄²	C₂×C₄×C₅₃	1	1	yes	characteristic
356	C₂×C₄²	C₂×C₄×C₅₃	1	1	yes	characteristic
357	C₂×C₄²	C₂×C₄×C₅₃	1	1	yes	characteristic
358	C₂×C₄²	C₂×C₄×C₅₃	1	1	yes	characteristic
359	C₂×C₄²	C₂×C₄×C₅₃	1	1	yes	characteristic
360	C₂×C₄²	C₂×C₄×C₅₃	1	1	yes	characteristic
361	C₂×C₄²	C₂×C₄×C₅₃	1	1	yes	characteristic
362	C₂×C₄²	C₂³×C₅₃	1	1	yes	characteristic
363	C₂×C₄²	C₂³×C₅₃	1	1	yes	characteristic
364	C₂×C₄²	C₂³×C₅₃	1	1	yes	characteristic
365	C₂×C₄²	C₂³×C₅₃	1	1	yes	characteristic
366	C₂×C₄²	C₂³×C₅₃	1	1	yes	characteristic
367	C₄⋊C₈	—	4	2	no	—
368	C₄⋊C₈	—	4	2	no	—
369	C₂²⋊C₈	—	2	4	no	—
370	C₂²⋊C₈	—	4	2	no	—
371	C₂²⋊C₈	—	4	2	no	—
372	C₄×C₈	—	4	2	no	—
373	#95	C₂×C₄×C₅₃	1	1	yes	characteristic
374	#95	C₂×C₄×C₅₃	1	1	yes	characteristic
375	#95	C₂×C₄×C₅₃	1	1	yes	characteristic
376	#95	C₂×C₄×C₅₃	1	1	yes	characteristic
377	#95	C₂×C₄×C₅₃	1	1	yes	characteristic
378	#95	C₂×C₄×C₅₃	1	1	yes	characteristic
379	C₂⁴	C₂²×C₄×C₅₃	1	1	yes	characteristic
380	C₂⁴	C₂²×C₄×C₅₃	1	1	yes	characteristic
381	C₂×D₄	—	2	2	no	—
382	C₂×D₄	—	2	2	no	—
383	C₂×D₄	C₂×C₈×C₅₃	1	4	yes	—
384	C₂²×C₄	—	2	2	no	—
385	C₂²×C₄	—	2	1	no	—
386	C₂²×C₄	—	2	1	no	—
387	C₂²×C₄	—	2	2	no	—
388	C₂²×C₄	—	2	2	no	—
389	C₂²×C₄	—	2	1	no	—
390	C₂²×C₄	—	2	2	no	—
391	C₂²×C₄	—	2	1	no	—
392	C₂²×C₄	—	4	1	no	—
393	C₂²×C₄	—	4	1	no	—
394	C₂²×C₄	—	4	1	no	—
395	C₂²×C₄	—	4	1	no	—
396	C₂²×C₄	—	4	1	no	—
397	C₂²×C₄	—	4	1	no	—
398	C₂²×C₄	—	4	1	no	—
399	C₂²×C₄	—	4	1	no	—
400	C₂²×C₄	C₂×C₈×C₅₃	1	4	yes	—
401	C₂²×C₄	C₂×C₈×C₅₃	1	4	yes	complemented
402	C₂²×C₄	M₁₆×C₅₃	1	4	yes	—
403	C₂²×C₄	C₂²×C₄×C₅₃	1	1	yes	characteristic
404	C₂²×C₄	C₂²×C₄×C₅₃	1	1	yes	characteristic
405	C₂²×C₄	C₂²×C₄×C₅₃	1	1	yes	characteristic
406	C₂²×C₄	C₂²×C₄×C₅₃	1	1	yes	characteristic
407	C₂²×C₄	C₂²×C₄×C₅₃	1	1	yes	characteristic
408	C₂²×C₄	C₂²×C₄×C₅₃	1	1	yes	characteristic
409	C₂²×C₄	C₂²×C₄×C₅₃	1	1	yes	characteristic
410	C₂²×C₄	C₂²×C₄×C₅₃	1	1	yes	characteristic
411	C₂²×C₄	C₂²×C₄×C₅₃	1	1	yes	characteristic
412	C₂²×C₄	C₂²×C₄×C₅₃	1	1	yes	characteristic
413	C₂²×C₄	C₂²×C₄×C₅₃	1	1	yes	characteristic
414	C₂²×C₄	C₂²×C₄×C₅₃	1	1	yes	characteristic
415	C₂²×C₄	#41×C₅₃	1	4	yes	—
416	C₂²×C₄	C₂⁴×C₅₃	1	1	yes	characteristic
417	C₂×C₈	—	2	2	no	—
418	C₂×C₈	—	2	2	no	—
419	C₂×C₈	—	2	2	no	—
420	C₂×C₈	—	2	2	no	—
421	C₂×C₈	—	2	2	no	—
422	C₂×C₈	—	2	2	no	—
423	C₂×C₈	—	2	2	no	—
424	C₂×C₈	—	2	2	no	—
425	C₂×C₈	—	2	2	no	—
426	C₂×C₈	—	2	2	no	—
427	C₂×C₈	—	2	2	no	—
428	C₂×C₈	—	2	2	no	—
429	C₂×C₈	—	4	8	no	—
430	C₄⋊C₄	—	2	2	no	—
431	C₄⋊C₄	—	2	2	no	—
432	C₄⋊C₄	C₂×C₈×C₅₃	1	4	yes	complemented
433	C₂²⋊C₄	—	2	2	no	—
434	C₂²⋊C₄	—	2	2	no	—
435	C₂²⋊C₄	—	2	2	no	—
436	C₂²⋊C₄	—	2	2	no	—
437	C₂²⋊C₄	C₂×C₈×C₅₃	1	4	yes	—
438	C₂²⋊C₄	C₂×C₈×C₅₃	1	4	yes	complemented
439	C₄²	—	2	2	no	—
440	C₄²	—	2	2	no	—
441	C₄²	—	2	2	no	—
442	C₄²	—	2	1	no	—
443	C₄²	—	2	2	no	—
444	C₄²	—	2	2	no	—
445	C₄²	—	2	1	no	—
446	C₄²	—	4	1	no	—
447	C₄²	—	4	1	no	—
448	C₄²	—	4	1	no	—
449	C₄²	—	4	1	no	—
450	C₄²	—	4	1	no	—
451	C₄²	—	4	1	no	—
452	C₄²	C₂×C₈×C₅₃	1	4	yes	complemented
453	C₂³	—	2	1	no	—
454	C₂³	—	2	1	no	—
455	C₂³	—	2	1	no	—
456	C₂³	—	2	1	no	—
457	C₂³	—	2	1	no	—
458	C₂³	—	2	1	no	—
459	C₂³	—	2	1	no	—
460	C₂³	—	2	1	no	—
461	C₂³	—	2	1	no	—
462	C₂³	—	2	1	no	—
463	C₂³	#117×C₅₃	1	2	yes	—
464	C₂³	C₂²×C₈×C₅₃	1	2	yes	—
465	C₂³	C₂²×C₈×C₅₃	1	2	yes	—
466	C₂³	C₂×M₁₆×C₅₃	1	2	yes	—
467	C₂³	C₂³×C₄×C₅₃	1	1	yes	characteristic
468	D₄	—	2	8	no	—
469	C₂×C₄	—	2	1	no	—
470	C₂×C₄	—	2	1	no	—
471	C₂×C₄	—	2	1	no	—
472	C₂×C₄	—	2	1	no	—
473	C₂×C₄	—	2	1	no	—
474	C₂×C₄	—	2	1	no	—
475	C₂×C₄	—	2	1	no	—
476	C₂×C₄	—	2	1	no	—
477	C₂×C₄	—	2	1	no	—
478	C₂×C₄	—	2	1	no	—
479	C₂×C₄	—	2	1	no	—
480	C₂×C₄	—	2	1	no	—
481	C₂×C₄	—	2	1	no	—
482	C₂×C₄	—	2	1	no	—
483	C₂×C₄	—	2	1	no	—
484	C₂×C₄	—	2	1	no	—
485	C₂×C₄	—	2	4	no	—
486	C₂×C₄	—	2	4	no	—
487	C₂×C₄	—	2	1	no	—
488	C₂×C₄	—	2	1	no	—
489	C₂×C₄	—	2	1	no	—
490	C₂×C₄	—	2	1	no	—
491	C₂×C₄	—	2	1	no	—
492	C₂×C₄	—	2	1	no	—
493	C₂×C₄	—	2	1	no	—
494	C₂×C₄	—	2	1	no	—
495	C₂×C₄	—	2	1	no	—
496	C₂×C₄	—	2	1	no	—
497	C₂×C₄	—	2	1	no	—
498	C₂×C₄	—	2	1	no	—
499	C₂×C₄	—	4	4	no	—
500	C₂×C₄	—	4	4	no	—
501	C₂×C₄	—	4	4	no	—
502	C₂×C₄	—	4	4	no	—
503	C₂×C₄	—	4	2	no	—
504	C₂×C₄	—	4	2	no	—
505	C₂×C₄	#117×C₅₃	1	2	yes	—
506	C₂×C₄	C₂²×C₈×C₅₃	1	2	yes	—
507	C₂×C₄	C₂²×C₈×C₅₃	1	2	yes	—
508	C₂×C₄	C₂²×C₈×C₅₃	1	2	yes	—
509	C₂×C₄	C₂²×C₈×C₅₃	1	2	yes	—
510	C₂×C₄	C₂²×C₈×C₅₃	1	2	yes	—
511	C₂×C₄	C₂×M₁₆×C₅₃	1	2	yes	—
512	C₂×C₄	C₂×M₁₆×C₅₃	1	2	yes	—
513	C₂×C₄	C₂×#41×C₅₃	1	2	yes	—
514	C₂×C₄	H(5,𝔽₂)×C₅₃	1	1	yes	characteristic
515	C₂×C₄	H(5,𝔽₂)×C₅₃	1	2	yes	—
516	C₂×C₄	2₋¹⁺⁴×C₅₃	1	1	yes	characteristic
517	C₈	—	4	2	no	—
518	C₈	—	4	2	no	—
519	C₈	—	4	2	no	—
520	C₈	—	4	2	no	—
521	C₂²	—	2	2	no	—
522	C₂²	—	2	2	no	—
523	C₂²	—	2	2	no	—
524	C₂²	—	2	2	no	—
525	C₂²	—	2	2	no	—
526	C₂²	—	2	2	no	—
527	C₂²	—	2	2	no	—
528	C₂²	—	2	2	no	—
529	C₂²	—	2	2	no	—
530	C₂²	—	4	1	no	—
531	C₂²	—	4	1	no	—
532	C₂²	—	4	1	no	—
533	C₂²	—	4	1	no	—
534	C₂²	C₅₃×#431	1	4	yes	—
535	C₂²	C₂×#117×C₅₃	1	1	yes	characteristic
536	C₂²	C₅₃×#518	1	1	yes	characteristic
537	C₂²	C₅₃×#520	1	1	yes	characteristic
538	C₂²	C₂³×C₈×C₅₃	1	1	yes	characteristic
539	C₂²	C₂²×M₁₆×C₅₃	1	1	yes	characteristic
540	C₂²	C₅₃×#568	1	1	yes	characteristic
541	C₂²	C₅₃×#568	1	1	yes	characteristic
542	C₄	—	2	2	no	—
543	C₄	—	2	2	no	—
544	C₄	—	2	2	no	—
545	C₄	—	2	2	no	—
546	C₄	—	2	2	no	—
547	C₄	—	2	2	no	—
548	C₄	—	2	2	no	—
549	C₄	—	2	2	no	—
550	C₄	—	4	1	no	—
551	C₄	—	4	1	no	—
552	C₄	—	4	1	no	—
553	C₄	—	4	1	no	—
554	C₄	C₅₃×#538	1	2	yes	—
555	C₄	C₅₃×#539	1	2	yes	—
556	C₂	—	2	4	no	—
557	C₂	—	4	2	no	—
558	C₂	C₅₃×#2318	1	1	yes	characteristic
559	C₂	C₂×C₅₃×#431	1	1	yes	characteristic
560	C₂	C₅₃×#2923	1	1	yes	characteristic
561	C₂	C₅₃×#2972	1	1	yes	characteristic
562	C₂	C₅₃×#2973	1	1	yes	characteristic
563	C₂	C₅₃×#2988	1	1	yes	characteristic
564	C₂	C₅₃×#2989	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
2. C₅₃×#1923	C₂	semidirect
3. C₅₃×#1916	C₂	semidirect
5. C₅₃×#1850	C₂	semidirect
6. C₅₃×#1839	C₂	semidirect
7. C₅₃×#1831	C₂	semidirect
8. C₅₃×#1817	C₂	semidirect
9. C₅₃×#1797	C₂	semidirect
10. C₄×C₂²⋊C₈×C₅₃	C₂	semidirect
11. C₂×C₅₃×#336	C₂	semidirect
59. C₄×D₄×C₅₃	C₈	semidirect
118. C₂²×C₄×C₅₃	C₂×C₈	semidirect
149. C₄⋊C₄×C₅₃	C₂×C₈	semidirect
155. C₂²⋊C₄×C₅₃	C₂×C₈	semidirect
169. C₄²×C₅₃	C₂×C₈	semidirect
238. #21996	C₅₃	direct (N × H)
275. #1923	C₂×C₅₃	semidirect
276. #1916	C₂×C₅₃	semidirect
278. #1850	C₂×C₅₃	semidirect
279. #1839	C₂×C₅₃	semidirect
280. #1831	C₂×C₅₃	semidirect
281. #1817	C₂×C₅₃	semidirect
282. #1797	C₂×C₅₃	semidirect
283. C₄×C₂²⋊C₈	C₂×C₅₃	semidirect
284. C₂×#336	C₂×C₅₃	semidirect
325. C₅₃	#21996	direct (N × H)
342. C₄×D₄	C₈×C₅₃	semidirect
401. C₂²×C₄	C₂×C₈×C₅₃	semidirect
432. C₄⋊C₄	C₂×C₈×C₅₃	semidirect
438. C₂²⋊C₄	C₂×C₈×C₅₃	semidirect
452. C₄²	C₂×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×#1923	2
C₅₃×#1916	2
C₅₃×#1869	2
C₅₃×#1850	2
C₅₃×#1839	2
C₅₃×#1831	2
C₅₃×#1817	2
C₅₃×#1797	2
C₄×C₂²⋊C₈×C₅₃	2
C₂×C₅₃×#336	2
#21996	53

Subgroups — C53×#21996 #27229508

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅₃×#21996 #27229508