Subgroups — C₅₃×#7929 #27215441

Order 13568.

1058 conjugacy classes of subgroups.

The subgroup lattice has 514 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₅₃×#407	C₂	1	1	yes	maximal, characteristic
2	C₂×C₅₃×#407	C₂	1	1	yes	maximal, characteristic
3	C₂×C₄×C₂²⋊C₄×C₅₃	C₂	1	1	yes	maximal, characteristic
4	C₅₃×#1282	C₂	1	4	yes	maximal, complemented
5	C₂⁴×C₄×C₅₃	C₂²	1	1	yes	characteristic
6	C₂²×M₁₆×C₅₃	—	2	1	no	—
7	C₂²×M₁₆×C₅₃	C₄	1	2	yes	complemented
8	C₂²×C₂²⋊C₄×C₅₃	C₄	1	2	yes	—
9	C₂²×C₄²×C₅₃	—	2	1	no	—
10	C₅₃×#407	—	2	1	no	—
11	C₅₃×#407	—	2	1	no	—
12	C₅₃×#407	—	2	1	no	—
13	C₅₃×#407	—	2	1	no	—
14	C₅₃×#407	—	2	1	no	—
15	C₅₃×#407	—	2	1	no	—
16	C₅₃×#407	C₄	1	2	yes	complemented
17	C₅₃×#407	C₄	1	2	yes	complemented
18	C₂×C₂²⋊C₈×C₅₃	—	2	1	no	—
19	C₂×C₂²⋊C₈×C₅₃	C₄	1	2	yes	complemented
20	C₂×C₂²⋊C₈×C₅₃	C₂²	1	2	yes	—
21	C₂×C₂²⋊C₈×C₅₃	C₂²	1	2	yes	—
22	C₄×C₂²⋊C₄×C₅₃	—	2	1	no	—
23	C₄×C₂²⋊C₄×C₅₃	—	2	1	no	—
24	C₄×C₂²⋊C₄×C₅₃	C₄	1	2	yes	—
25	C₄×C₂²⋊C₄×C₅₃	C₂²	1	2	yes	—
26	C₂×#95×C₅₃	—	2	1	no	—
27	C₂⁵×C₅₃	C₂×C₄	1	1	yes	characteristic
28	C₂³×C₄×C₅₃	—	2	2	no	—
29	C₂³×C₄×C₅₃	—	4	4	no	—
30	C₂³×C₄×C₅₃	C₂×C₄	1	1	yes	characteristic
31	C₂³×C₄×C₅₃	C₂×C₄	1	1	yes	characteristic
32	C₂³×C₄×C₅₃	C₂×C₄	1	1	yes	characteristic
33	C₂³×C₄×C₅₃	C₂×C₄	1	1	yes	characteristic
34	C₂³×C₄×C₅₃	C₂×C₄	1	1	yes	characteristic
35	C₂³×C₄×C₅₃	D₄	1	1	yes	characteristic
36	C₂³×C₄×C₅₃	D₄	1	1	yes	characteristic
37	C₂³×C₄×C₅₃	D₄	1	1	yes	characteristic
38	C₂³×C₄×C₅₃	D₄	1	1	yes	characteristic
39	C₂³×C₄×C₅₃	D₄	1	1	yes	characteristic
40	C₂³×C₄×C₅₃	D₄	1	1	yes	characteristic
41	C₂³×C₄×C₅₃	Q₈	1	1	yes	characteristic
42	C₂³×C₄×C₅₃	Q₈	1	1	yes	characteristic
43	C₂³×C₄×C₅₃	C₂³	1	1	yes	characteristic
44	C₂×M₁₆×C₅₃	—	2	2	no	—
45	C₂×M₁₆×C₅₃	—	2	2	no	—
46	C₂×M₁₆×C₅₃	—	2	2	no	—
47	C₂×M₁₆×C₅₃	—	2	2	no	—
48	C₂×M₁₆×C₅₃	—	4	4	no	—
49	C₂×M₁₆×C₅₃	—	4	4	no	—
50	C₂²×C₈×C₅₃	—	2	2	no	—
51	C₂²×C₈×C₅₃	C₂×C₄	1	4	yes	complemented
52	C₂×C₂²⋊C₄×C₅₃	—	2	8	no	—
53	C₂×C₂²⋊C₄×C₅₃	—	2	2	no	—
54	C₂×C₂²⋊C₄×C₅₃	C₂×C₄	1	4	yes	—
55	C₂×C₄²×C₅₃	—	2	2	no	—
56	C₂×C₄²×C₅₃	—	2	2	no	—
57	C₂×C₄²×C₅₃	—	4	4	no	—
58	C₂²⋊C₈×C₅₃	—	2	2	no	—
59	C₂²⋊C₈×C₅₃	—	2	2	no	—
60	C₂²⋊C₈×C₅₃	—	2	2	no	—
61	C₂²⋊C₈×C₅₃	—	2	2	no	—
62	C₂²⋊C₈×C₅₃	—	2	2	no	—
63	C₂²⋊C₈×C₅₃	—	2	2	no	—
64	C₂²⋊C₈×C₅₃	—	2	2	no	—
65	C₂²⋊C₈×C₅₃	C₂×C₄	1	4	yes	complemented
66	#95×C₅₃	—	2	2	no	—
67	#95×C₅₃	—	2	2	no	—
68	C₂⁴×C₅₃	—	2	4	no	—
69	C₂⁴×C₅₃	—	4	2	no	—
70	C₂⁴×C₅₃	—	4	2	no	—
71	C₂⁴×C₅₃	C₄²	1	1	yes	characteristic
72	C₂⁴×C₅₃	C₄²	1	1	yes	characteristic
73	C₂⁴×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
74	C₂⁴×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
75	C₂⁴×C₅₃	C₄⋊C₄	1	1	yes	characteristic
76	C₂⁴×C₅₃	C₄⋊C₄	1	1	yes	characteristic
77	C₂⁴×C₅₃	C₂²×C₄	1	1	yes	characteristic
78	C₂²×C₄×C₅₃	—	2	4	no	—
79	C₂²×C₄×C₅₃	—	2	4	no	—
80	C₂²×C₄×C₅₃	—	2	4	no	—
81	C₂²×C₄×C₅₃	—	2	8	no	—
82	C₂²×C₄×C₅₃	—	2	4	no	—
83	C₂²×C₄×C₅₃	—	4	2	no	—
84	C₂²×C₄×C₅₃	—	4	2	no	—
85	C₂²×C₄×C₅₃	—	4	2	no	—
86	C₂²×C₄×C₅₃	—	4	2	no	—
87	C₂²×C₄×C₅₃	—	4	2	no	—
88	C₂²×C₄×C₅₃	—	4	2	no	—
89	C₂²×C₄×C₅₃	—	4	2	no	—
90	C₂²×C₄×C₅₃	—	4	2	no	—
91	C₂²×C₄×C₅₃	—	4	2	no	—
92	C₂²×C₄×C₅₃	—	4	2	no	—
93	C₂²×C₄×C₅₃	—	4	2	no	—
94	C₂²×C₄×C₅₃	—	4	4	no	—
95	C₂²×C₄×C₅₃	—	4	4	no	—
96	C₂²×C₄×C₅₃	C₄²	1	1	yes	characteristic
97	C₂²×C₄×C₅₃	C₄²	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	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	C₂²⋊C₄	1	1	yes	characteristic
106	C₂²×C₄×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
107	C₂²×C₄×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
108	C₂²×C₄×C₅₃	C₄⋊C₄	1	1	yes	characteristic
109	C₂²×C₄×C₅₃	C₄⋊C₄	1	1	yes	characteristic
110	C₂²×C₄×C₅₃	C₄⋊C₄	1	1	yes	characteristic
111	C₂²×C₄×C₅₃	C₄⋊C₄	1	1	yes	characteristic
112	C₂²×C₄×C₅₃	C₄⋊C₄	1	1	yes	characteristic
113	C₂²×C₄×C₅₃	C₄⋊C₄	1	1	yes	characteristic
114	C₂²×C₄×C₅₃	C₄⋊C₄	1	1	yes	characteristic
115	C₂²×C₄×C₅₃	C₄⋊C₄	1	1	yes	characteristic
116	C₂²×C₄×C₅₃	C₄⋊C₄	1	1	yes	characteristic
117	C₂²×C₄×C₅₃	C₄⋊C₄	1	1	yes	characteristic
118	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
119	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
120	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
121	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
122	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
123	C₂²×C₄×C₅₃	C₂×Q₈	1	1	yes	characteristic
124	M₁₆×C₅₃	—	4	4	no	—
125	M₁₆×C₅₃	—	4	4	no	—
126	M₁₆×C₅₃	—	4	4	no	—
127	M₁₆×C₅₃	—	4	4	no	—
128	C₂×C₈×C₅₃	—	2	4	no	—
129	C₂×C₈×C₅₃	—	2	4	no	—
130	C₂×C₈×C₅₃	—	4	4	no	—
131	C₂×C₈×C₅₃	—	4	4	no	—
132	C₂²⋊C₄×C₅₃	—	2	8	no	—
133	C₂²⋊C₄×C₅₃	—	2	4	no	—
134	C₂²⋊C₄×C₅₃	M₁₆	1	8	yes	complemented
135	C₄²×C₅₃	—	4	4	no	—
136	C₄²×C₅₃	—	4	4	no	—
137	C₂³×C₅₃	—	2	2	no	—
138	C₂³×C₅₃	—	2	2	no	—
139	C₂³×C₅₃	—	2	2	no	—
140	C₂³×C₅₃	—	2	2	no	—
141	C₂³×C₅₃	—	2	2	no	—
142	C₂³×C₅₃	—	2	2	no	—
143	C₂³×C₅₃	—	2	2	no	—
144	C₂³×C₅₃	—	2	2	no	—
145	C₂³×C₅₃	—	2	2	no	—
146	C₂³×C₅₃	—	2	2	no	—
147	C₂³×C₅₃	—	2	2	no	—
148	C₂³×C₅₃	—	2	2	no	—
149	C₂³×C₅₃	—	4	1	no	—
150	C₂³×C₅₃	—	4	1	no	—
151	C₂³×C₅₃	—	4	1	no	—
152	C₂³×C₅₃	—	4	1	no	—
153	C₂³×C₅₃	—	4	1	no	—
154	C₂³×C₅₃	—	4	1	no	—
155	C₂³×C₅₃	—	4	1	no	—
156	C₂³×C₅₃	—	4	1	no	—
157	C₂³×C₅₃	—	4	16	no	—
158	C₂³×C₅₃	#95	1	1	yes	characteristic
159	C₂³×C₅₃	#95	1	1	yes	characteristic
160	C₂³×C₅₃	#95	1	1	yes	characteristic
161	C₂³×C₅₃	#95	1	1	yes	characteristic
162	C₂³×C₅₃	C₂×C₄²	1	1	yes	characteristic
163	C₂³×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
164	C₂³×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
165	C₂³×C₅₃	C₂×M₁₆	1	4	yes	—
166	C₂×C₄×C₅₃	—	2	2	no	—
167	C₂×C₄×C₅₃	—	2	2	no	—
168	C₂×C₄×C₅₃	—	2	2	no	—
169	C₂×C₄×C₅₃	—	2	2	no	—
170	C₂×C₄×C₅₃	—	2	2	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	2	no	—
175	C₂×C₄×C₅₃	—	2	2	no	—
176	C₂×C₄×C₅₃	—	2	2	no	—
177	C₂×C₄×C₅₃	—	2	2	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	8	no	—
182	C₂×C₄×C₅₃	—	4	1	no	—
183	C₂×C₄×C₅₃	—	4	1	no	—
184	C₂×C₄×C₅₃	—	4	1	no	—
185	C₂×C₄×C₅₃	—	4	1	no	—
186	C₂×C₄×C₅₃	—	4	1	no	—
187	C₂×C₄×C₅₃	—	4	1	no	—
188	C₂×C₄×C₅₃	—	4	1	no	—
189	C₂×C₄×C₅₃	—	4	1	no	—
190	C₂×C₄×C₅₃	—	4	1	no	—
191	C₂×C₄×C₅₃	—	4	1	no	—
192	C₂×C₄×C₅₃	—	4	1	no	—
193	C₂×C₄×C₅₃	—	4	1	no	—
194	C₂×C₄×C₅₃	—	4	16	no	—
195	C₂×C₄×C₅₃	—	4	4	no	—
196	C₂×C₄×C₅₃	—	4	4	no	—
197	C₂×C₄×C₅₃	#95	1	1	yes	characteristic
198	C₂×C₄×C₅₃	#95	1	1	yes	characteristic
199	C₂×C₄×C₅₃	#95	1	1	yes	characteristic
200	C₂×C₄×C₅₃	#95	1	1	yes	characteristic
201	C₂×C₄×C₅₃	#99	1	4	yes	—
202	C₂×C₄×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
203	C₂×C₄×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
204	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
205	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
206	C₈×C₅₃	—	4	4	no	—
207	C₈×C₅₃	—	4	4	no	—
208	#7929	C₅₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
209	C₂²×C₅₃	—	2	8	no	—
210	C₂²×C₅₃	—	2	1	no	—
211	C₂²×C₅₃	—	2	1	no	—
212	C₂²×C₅₃	—	2	1	no	—
213	C₂²×C₅₃	—	2	1	no	—
214	C₂²×C₅₃	—	2	1	no	—
215	C₂²×C₅₃	—	2	1	no	—
216	C₂²×C₅₃	—	2	1	no	—
217	C₂²×C₅₃	—	2	1	no	—
218	C₂²×C₅₃	—	2	1	no	—
219	C₂²×C₅₃	—	2	1	no	—
220	C₂²×C₅₃	—	2	1	no	—
221	C₂²×C₅₃	—	2	1	no	—
222	C₂²×C₅₃	—	2	1	no	—
223	C₂²×C₅₃	—	2	1	no	—
224	C₂²×C₅₃	—	2	1	no	—
225	C₂²×C₅₃	—	2	1	no	—
226	C₂²×C₅₃	—	4	4	no	—
227	C₂²×C₅₃	—	4	4	no	—
228	C₂²×C₅₃	—	4	4	no	—
229	C₂²×C₅₃	—	4	4	no	—
230	C₂²×C₅₃	—	4	4	no	—
231	C₂²×C₅₃	—	4	4	no	—
232	C₂²×C₅₃	#342	1	2	yes	—
233	C₂²×C₅₃	#344	1	2	yes	—
234	C₂²×C₅₃	C₂×#95	1	1	yes	characteristic
235	C₂²×C₅₃	C₄×M₁₆	1	2	yes	—
236	C₂²×C₅₃	#407	1	2	yes	—
237	C₂²×C₅₃	#423	1	2	yes	—
238	C₄×C₅₃	—	2	1	no	—
239	C₄×C₅₃	—	2	1	no	—
240	C₄×C₅₃	—	2	1	no	—
241	C₄×C₅₃	—	2	1	no	—
242	C₄×C₅₃	—	2	1	no	—
243	C₄×C₅₃	—	2	1	no	—
244	C₄×C₅₃	—	4	8	no	—
245	C₄×C₅₃	#342	1	2	yes	—
246	C₄×C₅₃	C₂×#99	1	2	yes	—
247	C₂×#407	C₂×C₅₃	1	1	yes	characteristic
248	C₂×#407	C₂×C₅₃	1	1	yes	characteristic
249	C₂×C₄×C₂²⋊C₄	C₂×C₅₃	1	1	yes	characteristic
250	#1282	C₂×C₅₃	1	4	yes	complemented
251	C₂×C₅₃	—	2	2	no	—
252	C₂×C₅₃	—	2	2	no	—
253	C₂×C₅₃	—	2	2	no	—
254	C₂×C₅₃	—	2	2	no	—
255	C₂×C₅₃	—	2	2	no	—
256	C₂×C₅₃	—	4	1	no	—
257	C₂×C₅₃	—	4	1	no	—
258	C₂×C₅₃	#1728	1	1	yes	characteristic
259	C₂×C₅₃	C₂×#342	1	1	yes	characteristic
260	C₂×C₅₃	C₂×#344	1	1	yes	characteristic
261	C₂⁴×C₄	C₂²×C₅₃	1	1	yes	characteristic
262	C₂²×M₁₆	—	2	1	no	—
263	C₂²×M₁₆	C₄×C₅₃	1	2	yes	complemented
264	C₂²×C₂²⋊C₄	C₄×C₅₃	1	2	yes	—
265	C₂²×C₄²	—	2	1	no	—
266	#407	—	2	1	no	—
267	#407	—	2	1	no	—
268	#407	—	2	1	no	—
269	#407	—	2	1	no	—
270	#407	—	2	1	no	—
271	#407	—	2	1	no	—
272	#407	C₄×C₅₃	1	2	yes	complemented
273	#407	C₄×C₅₃	1	2	yes	complemented
274	C₂×C₂²⋊C₈	—	2	1	no	—
275	C₂×C₂²⋊C₈	C₄×C₅₃	1	2	yes	complemented
276	C₂×C₂²⋊C₈	C₂²×C₅₃	1	2	yes	—
277	C₂×C₂²⋊C₈	C₂²×C₅₃	1	2	yes	—
278	C₄×C₂²⋊C₄	—	2	1	no	—
279	C₄×C₂²⋊C₄	—	2	1	no	—
280	C₄×C₂²⋊C₄	C₄×C₅₃	1	2	yes	—
281	C₄×C₂²⋊C₄	C₂²×C₅₃	1	2	yes	—
282	C₂×#95	—	2	1	no	—
283	C₅₃	#7929	1	1	yes	characteristic, direct product factor, Sylow
284	C₂⁵	C₂×C₄×C₅₃	1	1	yes	characteristic
285	C₂³×C₄	—	2	2	no	—
286	C₂³×C₄	—	4	4	no	—
287	C₂³×C₄	C₂×C₄×C₅₃	1	1	yes	characteristic
288	C₂³×C₄	C₂×C₄×C₅₃	1	1	yes	characteristic
289	C₂³×C₄	C₂×C₄×C₅₃	1	1	yes	characteristic
290	C₂³×C₄	C₂×C₄×C₅₃	1	1	yes	characteristic
291	C₂³×C₄	C₂×C₄×C₅₃	1	1	yes	characteristic
292	C₂³×C₄	D₄×C₅₃	1	1	yes	characteristic
293	C₂³×C₄	D₄×C₅₃	1	1	yes	characteristic
294	C₂³×C₄	D₄×C₅₃	1	1	yes	characteristic
295	C₂³×C₄	D₄×C₅₃	1	1	yes	characteristic
296	C₂³×C₄	D₄×C₅₃	1	1	yes	characteristic
297	C₂³×C₄	D₄×C₅₃	1	1	yes	characteristic
298	C₂³×C₄	Q₈×C₅₃	1	1	yes	characteristic
299	C₂³×C₄	Q₈×C₅₃	1	1	yes	characteristic
300	C₂³×C₄	C₂³×C₅₃	1	1	yes	characteristic
301	C₂×M₁₆	—	2	2	no	—
302	C₂×M₁₆	—	2	2	no	—
303	C₂×M₁₆	—	2	2	no	—
304	C₂×M₁₆	—	2	2	no	—
305	C₂×M₁₆	—	4	4	no	—
306	C₂×M₁₆	—	4	4	no	—
307	C₂²×C₈	—	2	2	no	—
308	C₂²×C₈	C₂×C₄×C₅₃	1	4	yes	complemented
309	C₂×C₂²⋊C₄	—	2	2	no	—
310	C₂×C₂²⋊C₄	—	2	8	no	—
311	C₂×C₂²⋊C₄	C₂×C₄×C₅₃	1	4	yes	—
312	C₂×C₄²	—	2	2	no	—
313	C₂×C₄²	—	2	2	no	—
314	C₂×C₄²	—	4	4	no	—
315	C₂²⋊C₈	—	2	2	no	—
316	C₂²⋊C₈	—	2	2	no	—
317	C₂²⋊C₈	—	2	2	no	—
318	C₂²⋊C₈	—	2	2	no	—
319	C₂²⋊C₈	—	2	2	no	—
320	C₂²⋊C₈	—	2	2	no	—
321	C₂²⋊C₈	—	2	2	no	—
322	C₂²⋊C₈	C₂×C₄×C₅₃	1	4	yes	complemented
323	#95	—	2	2	no	—
324	#95	—	2	2	no	—
325	C₂⁴	—	2	4	no	—
326	C₂⁴	—	4	2	no	—
327	C₂⁴	—	4	2	no	—
328	C₂⁴	C₄²×C₅₃	1	1	yes	characteristic
329	C₂⁴	C₄²×C₅₃	1	1	yes	characteristic
330	C₂⁴	C₂²⋊C₄×C₅₃	1	1	yes	characteristic
331	C₂⁴	C₂²⋊C₄×C₅₃	1	1	yes	characteristic
332	C₂⁴	C₄⋊C₄×C₅₃	1	1	yes	characteristic
333	C₂⁴	C₄⋊C₄×C₅₃	1	1	yes	characteristic
334	C₂⁴	C₂²×C₄×C₅₃	1	1	yes	characteristic
335	C₂²×C₄	—	2	4	no	—
336	C₂²×C₄	—	2	4	no	—
337	C₂²×C₄	—	2	4	no	—
338	C₂²×C₄	—	2	8	no	—
339	C₂²×C₄	—	2	4	no	—
340	C₂²×C₄	—	4	2	no	—
341	C₂²×C₄	—	4	2	no	—
342	C₂²×C₄	—	4	2	no	—
343	C₂²×C₄	—	4	2	no	—
344	C₂²×C₄	—	4	2	no	—
345	C₂²×C₄	—	4	2	no	—
346	C₂²×C₄	—	4	2	no	—
347	C₂²×C₄	—	4	2	no	—
348	C₂²×C₄	—	4	2	no	—
349	C₂²×C₄	—	4	2	no	—
350	C₂²×C₄	—	4	2	no	—
351	C₂²×C₄	—	4	4	no	—
352	C₂²×C₄	—	4	4	no	—
353	C₂²×C₄	C₄²×C₅₃	1	1	yes	characteristic
354	C₂²×C₄	C₄²×C₅₃	1	1	yes	characteristic
355	C₂²×C₄	C₂²⋊C₄×C₅₃	1	1	yes	characteristic
356	C₂²×C₄	C₂²⋊C₄×C₅₃	1	1	yes	characteristic
357	C₂²×C₄	C₂²⋊C₄×C₅₃	1	1	yes	characteristic
358	C₂²×C₄	C₂²⋊C₄×C₅₃	1	1	yes	characteristic
359	C₂²×C₄	C₂²⋊C₄×C₅₃	1	1	yes	characteristic
360	C₂²×C₄	C₂²⋊C₄×C₅₃	1	1	yes	characteristic
361	C₂²×C₄	C₂²⋊C₄×C₅₃	1	1	yes	characteristic
362	C₂²×C₄	C₂²⋊C₄×C₅₃	1	1	yes	characteristic
363	C₂²×C₄	C₂²⋊C₄×C₅₃	1	1	yes	characteristic
364	C₂²×C₄	C₂²⋊C₄×C₅₃	1	1	yes	characteristic
365	C₂²×C₄	C₄⋊C₄×C₅₃	1	1	yes	characteristic
366	C₂²×C₄	C₄⋊C₄×C₅₃	1	1	yes	characteristic
367	C₂²×C₄	C₄⋊C₄×C₅₃	1	1	yes	characteristic
368	C₂²×C₄	C₄⋊C₄×C₅₃	1	1	yes	characteristic
369	C₂²×C₄	C₄⋊C₄×C₅₃	1	1	yes	characteristic
370	C₂²×C₄	C₄⋊C₄×C₅₃	1	1	yes	characteristic
371	C₂²×C₄	C₄⋊C₄×C₅₃	1	1	yes	characteristic
372	C₂²×C₄	C₄⋊C₄×C₅₃	1	1	yes	characteristic
373	C₂²×C₄	C₄⋊C₄×C₅₃	1	1	yes	characteristic
374	C₂²×C₄	C₄⋊C₄×C₅₃	1	1	yes	characteristic
375	C₂²×C₄	C₂²×C₄×C₅₃	1	1	yes	characteristic
376	C₂²×C₄	C₂²×C₄×C₅₃	1	1	yes	characteristic
377	C₂²×C₄	C₂×D₄×C₅₃	1	1	yes	characteristic
378	C₂²×C₄	C₂×D₄×C₅₃	1	1	yes	characteristic
379	C₂²×C₄	C₂×D₄×C₅₃	1	1	yes	characteristic
380	C₂²×C₄	C₂×Q₈×C₅₃	1	1	yes	characteristic
381	M₁₆	—	4	4	no	—
382	M₁₆	—	4	4	no	—
383	M₁₆	—	4	4	no	—
384	M₁₆	—	4	4	no	—
385	C₂×C₈	—	2	4	no	—
386	C₂×C₈	—	2	4	no	—
387	C₂×C₈	—	4	4	no	—
388	C₂×C₈	—	4	4	no	—
389	C₂²⋊C₄	—	2	8	no	—
390	C₂²⋊C₄	—	2	4	no	—
391	C₂²⋊C₄	M₁₆×C₅₃	1	8	yes	complemented
392	C₄²	—	4	4	no	—
393	C₄²	—	4	4	no	—
394	C₂³	—	2	2	no	—
395	C₂³	—	2	2	no	—
396	C₂³	—	2	2	no	—
397	C₂³	—	2	2	no	—
398	C₂³	—	2	2	no	—
399	C₂³	—	2	2	no	—
400	C₂³	—	2	2	no	—
401	C₂³	—	2	2	no	—
402	C₂³	—	2	2	no	—
403	C₂³	—	2	2	no	—
404	C₂³	—	2	2	no	—
405	C₂³	—	2	2	no	—
406	C₂³	—	4	1	no	—
407	C₂³	—	4	1	no	—
408	C₂³	—	4	1	no	—
409	C₂³	—	4	1	no	—
410	C₂³	—	4	1	no	—
411	C₂³	—	4	1	no	—
412	C₂³	—	4	1	no	—
413	C₂³	—	4	1	no	—
414	C₂³	—	4	16	no	—
415	C₂³	#95×C₅₃	1	1	yes	characteristic
416	C₂³	#95×C₅₃	1	1	yes	characteristic
417	C₂³	#95×C₅₃	1	1	yes	characteristic
418	C₂³	#95×C₅₃	1	1	yes	characteristic
419	C₂³	C₂×C₄²×C₅₃	1	1	yes	characteristic
420	C₂³	C₂×C₂²⋊C₄×C₅₃	1	1	yes	characteristic
421	C₂³	C₂×C₄⋊C₄×C₅₃	1	1	yes	characteristic
422	C₂³	C₂×M₁₆×C₅₃	1	4	yes	—
423	C₂×C₄	—	2	2	no	—
424	C₂×C₄	—	2	2	no	—
425	C₂×C₄	—	2	2	no	—
426	C₂×C₄	—	2	2	no	—
427	C₂×C₄	—	2	2	no	—
428	C₂×C₄	—	2	2	no	—
429	C₂×C₄	—	2	2	no	—
430	C₂×C₄	—	2	2	no	—
431	C₂×C₄	—	2	2	no	—
432	C₂×C₄	—	2	2	no	—
433	C₂×C₄	—	2	2	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	8	no	—
439	C₂×C₄	—	4	1	no	—
440	C₂×C₄	—	4	1	no	—
441	C₂×C₄	—	4	1	no	—
442	C₂×C₄	—	4	1	no	—
443	C₂×C₄	—	4	1	no	—
444	C₂×C₄	—	4	1	no	—
445	C₂×C₄	—	4	1	no	—
446	C₂×C₄	—	4	1	no	—
447	C₂×C₄	—	4	1	no	—
448	C₂×C₄	—	4	1	no	—
449	C₂×C₄	—	4	1	no	—
450	C₂×C₄	—	4	1	no	—
451	C₂×C₄	—	4	4	no	—
452	C₂×C₄	—	4	16	no	—
453	C₂×C₄	—	4	4	no	—
454	C₂×C₄	#95×C₅₃	1	1	yes	characteristic
455	C₂×C₄	#95×C₅₃	1	1	yes	characteristic
456	C₂×C₄	#95×C₅₃	1	1	yes	characteristic
457	C₂×C₄	#95×C₅₃	1	1	yes	characteristic
458	C₂×C₄	#99×C₅₃	1	4	yes	—
459	C₂×C₄	C₂×C₂²⋊C₄×C₅₃	1	1	yes	characteristic
460	C₂×C₄	C₂×C₂²⋊C₄×C₅₃	1	1	yes	characteristic
461	C₂×C₄	C₂×C₄⋊C₄×C₅₃	1	1	yes	characteristic
462	C₂×C₄	C₂×C₄⋊C₄×C₅₃	1	1	yes	characteristic
463	C₈	—	4	4	no	—
464	C₈	—	4	4	no	—
465	C₂²	—	2	8	no	—
466	C₂²	—	2	1	no	—
467	C₂²	—	2	1	no	—
468	C₂²	—	2	1	no	—
469	C₂²	—	2	1	no	—
470	C₂²	—	2	1	no	—
471	C₂²	—	2	1	no	—
472	C₂²	—	2	1	no	—
473	C₂²	—	2	1	no	—
474	C₂²	—	2	1	no	—
475	C₂²	—	2	1	no	—
476	C₂²	—	2	1	no	—
477	C₂²	—	2	1	no	—
478	C₂²	—	2	1	no	—
479	C₂²	—	2	1	no	—
480	C₂²	—	2	1	no	—
481	C₂²	—	2	1	no	—
482	C₂²	—	4	4	no	—
483	C₂²	—	4	4	no	—
484	C₂²	—	4	4	no	—
485	C₂²	—	4	4	no	—
486	C₂²	—	4	4	no	—
487	C₂²	—	4	4	no	—
488	C₂²	C₅₃×#342	1	2	yes	—
489	C₂²	C₅₃×#344	1	2	yes	—
490	C₂²	C₂×#95×C₅₃	1	1	yes	characteristic
491	C₂²	C₄×M₁₆×C₅₃	1	2	yes	—
492	C₂²	C₅₃×#407	1	2	yes	—
493	C₂²	C₅₃×#423	1	2	yes	—
494	C₄	—	2	1	no	—
495	C₄	—	2	1	no	—
496	C₄	—	2	1	no	—
497	C₄	—	2	1	no	—
498	C₄	—	2	1	no	—
499	C₄	—	2	1	no	—
500	C₄	—	4	8	no	—
501	C₄	C₅₃×#342	1	2	yes	—
502	C₄	C₂×#99×C₅₃	1	2	yes	—
503	C₂	—	2	2	no	—
504	C₂	—	2	2	no	—
505	C₂	—	2	2	no	—
506	C₂	—	2	2	no	—
507	C₂	—	2	2	no	—
508	C₂	—	4	1	no	—
509	C₂	—	4	1	no	—
510	C₂	C₅₃×#1728	1	1	yes	characteristic
511	C₂	C₂×C₅₃×#342	1	1	yes	characteristic
512	C₂	C₂×C₅₃×#344	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄×C₅₃	C₂×C₂²⋊C₄
commutator	C₂³	C₂×C₄²×C₅₃
Frattini, cosocle	C₂³×C₄	C₂³×C₅₃
socle	C₂²×C₅₃	C₂×#95
prime radical	C₂⁵×C₅₃	C₂×C₄
LCS	G ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₅₃ ◁ C₂⁴×C₄×C₅₃ ◁ G

Complements

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

N	H ≅ G/N	type
4. C₅₃×#1282	C₂	semidirect
7. C₂²×M₁₆×C₅₃	C₄	semidirect
16. C₅₃×#407	C₄	semidirect
17. C₅₃×#407	C₄	semidirect
19. C₂×C₂²⋊C₈×C₅₃	C₄	semidirect
51. C₂²×C₈×C₅₃	C₂×C₄	semidirect
65. C₂²⋊C₈×C₅₃	C₂×C₄	semidirect
134. C₂²⋊C₄×C₅₃	M₁₆	semidirect
208. #7929	C₅₃	direct (N × H)
250. #1282	C₂×C₅₃	semidirect
263. C₂²×M₁₆	C₄×C₅₃	semidirect
272. #407	C₄×C₅₃	semidirect
273. #407	C₄×C₅₃	semidirect
275. C₂×C₂²⋊C₈	C₄×C₅₃	semidirect
283. C₅₃	#7929	direct (N × H)
308. C₂²×C₈	C₂×C₄×C₅₃	semidirect
322. C₂²⋊C₈	C₂×C₄×C₅₃	semidirect
391. C₂²⋊C₄	M₁₆×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×#407	2
C₂×C₄×C₂²⋊C₄×C₅₃	2
C₅₃×#1282	2
#7929	53

Subgroups — C53×#7929 #27215441

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅₃×#7929 #27215441