Subgroups — C₅₃×#36599 #27244111

Order 13568.

1776 conjugacy classes of subgroups.

The subgroup lattice has 676 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₅₃×#2857	C₂	1	1	yes	maximal, characteristic
2	C₅₃×#2857	C₂	1	2	yes	maximal
3	C₅₃×#2585	C₂	1	4	yes	maximal, complemented
4	C₅₃×#2569	C₂	1	4	yes	maximal
5	C₅₃×#2552	C₂	1	1	yes	maximal, characteristic, complemented
6	C₅₃×#2546	C₂	1	1	yes	maximal, characteristic, complemented
7	C₅₃×#2511	C₂	1	2	yes	maximal, complemented
8	C₅₃×#2509	C₂	1	2	yes	maximal, complemented
9	C₅₃×#2446	C₂	1	2	yes	maximal, complemented
10	C₅₃×#2442	C₂	1	2	yes	maximal
11	C₅₃×#2370	C₂	1	2	yes	maximal, complemented
12	C₅₃×#2365	C₂	1	1	yes	maximal, characteristic, complemented
13	C₅₃×#2355	C₂	1	1	yes	maximal, characteristic, complemented
14	C₅₃×#2345	C₂	1	2	yes	maximal
15	C₄×C₂²⋊Q₈×C₅₃	C₂	1	1	yes	maximal, characteristic
16	C₄×C₂²⋊Q₈×C₅₃	C₂	1	2	yes	maximal
17	C₂×C₄×C₄⋊C₄×C₅₃	C₂	1	1	yes	maximal, characteristic
18	C₂×C₄⋊Q₈×C₅₃	C₂²	1	1	yes	characteristic
19	C₂×C₄⋊Q₈×C₅₃	C₂²	1	2	yes	—
20	C₂×#125×C₅₃	C₂²	1	2	yes	—
21	C₂×C₂²⋊Q₈×C₅₃	C₂²	1	1	yes	characteristic
22	C₂×C₂²⋊Q₈×C₅₃	C₂²	1	2	yes	—
23	C₂×C₄×Q₈×C₅₃	C₂²	1	1	yes	characteristic
24	C₂×C₄×Q₈×C₅₃	C₂²	1	2	yes	—
25	C₂²×C₄⋊C₄×C₅₃	C₂²	1	1	yes	characteristic
26	C₂²×C₄⋊C₄×C₅₃	C₂²	1	1	yes	characteristic
27	C₂²×C₄²×C₅₃	C₂²	1	1	yes	characteristic
28	C₂²×C₄²×C₅₃	C₂²	1	2	yes	—
29	C₅₃×#400	C₂²	1	4	yes	—
30	C₅₃×#400	C₂²	1	4	yes	—
31	C₅₃×#400	C₂²	1	4	yes	—
32	C₅₃×#398	C₂²	1	4	yes	—
33	C₅₃×#398	C₂²	1	4	yes	—
34	C₅₃×#398	C₂²	1	4	yes	—
35	C₅₃×#391	C₂²	1	1	yes	characteristic
36	C₅₃×#391	C₂²	1	2	yes	—
37	C₅₃×#391	C₂²	1	2	yes	—
38	C₅₃×#391	C₂²	1	2	yes	—
39	C₅₃×#391	C₂²	1	2	yes	—
40	C₅₃×#389	C₂²	1	2	yes	—
41	C₅₃×#389	C₂²	1	2	yes	—
42	C₅₃×#389	C₂²	1	2	yes	—
43	C₅₃×#389	C₂²	1	2	yes	—
44	C₅₃×#389	C₂²	1	2	yes	—
45	C₅₃×#389	C₂²	1	4	yes	—
46	C₅₃×#389	C₂²	1	2	yes	—
47	C₅₃×#389	C₂²	1	1	yes	characteristic
48	C₅₃×#389	C₂²	1	2	yes	—
49	C₅₃×#389	C₂²	1	4	yes	—
50	C₅₃×#387	C₂²	1	4	yes	—
51	C₅₃×#387	C₂²	1	4	yes	—
52	C₅₃×#387	C₂²	1	4	yes	—
53	C₅₃×#387	C₂²	1	4	yes	—
54	C₅₃×#387	C₂²	1	4	yes	—
55	C₅₃×#387	C₂²	1	2	yes	—
56	C₅₃×#387	C₂²	1	4	yes	—
57	C₅₃×#387	C₂²	1	4	yes	—
58	C₅₃×#385	C₂²	1	4	yes	—
59	C₅₃×#385	C₂²	1	2	yes	—
60	C₅₃×#384	C₂²	1	2	yes	—
61	C₅₃×#384	C₂²	1	1	yes	characteristic
62	C₅₃×#382	C₂²	1	2	yes	—
63	C₅₃×#380	C₂²	1	4	yes	—
64	C₅₃×#380	C₂²	1	2	yes	—
65	C₅₃×#380	C₂²	1	1	yes	characteristic
66	C₅₃×#380	C₂²	1	2	yes	—
67	C₅₃×#380	C₂²	1	4	yes	—
68	C₄×C₄⋊C₄×C₅₃	C₂²	1	2	yes	—
69	C₄×C₄⋊C₄×C₅₃	C₂²	1	1	yes	characteristic
70	C₄×C₄⋊C₄×C₅₃	C₂²	1	2	yes	—
71	C₄×C₄⋊C₄×C₅₃	C₂²	1	2	yes	—
72	C₄×C₄⋊C₄×C₅₃	C₂²	1	1	yes	characteristic
73	C₄×C₄⋊C₄×C₅₃	C₂²	1	2	yes	—
74	C₄×C₄⋊C₄×C₅₃	C₂²	1	1	yes	characteristic
75	C₄×C₄⋊C₄×C₅₃	C₂²	1	1	yes	characteristic
76	C₄×C₄⋊C₄×C₅₃	C₂²	1	1	yes	characteristic
77	C₄×C₄⋊C₄×C₅₃	C₂²	1	4	yes	—
78	C₄×C₄⋊C₄×C₅₃	C₂²	1	2	yes	—
79	C₄×C₄⋊C₄×C₅₃	C₂²	1	2	yes	—
80	C₄×C₂²⋊C₄×C₅₃	C₂²	1	2	yes	—
81	C₄×C₂²⋊C₄×C₅₃	C₂²	1	4	yes	—
82	C₂×#95×C₅₃	C₂²	1	2	yes	—
83	C₂²×Q₈×C₅₃	C₂³	1	1	yes	characteristic
84	C₂²×Q₈×C₅₃	C₂³	1	2	yes	—
85	C₂³×C₄×C₅₃	C₂³	1	2	yes	—
86	C₂³×C₄×C₅₃	C₂³	1	1	yes	characteristic
87	C₂³×C₄×C₅₃	C₂³	1	1	yes	characteristic
88	C₂³×C₄×C₅₃	C₂³	1	1	yes	characteristic
89	C₂³×C₄×C₅₃	C₂³	1	2	yes	—
90	C₄⋊Q₈×C₅₃	—	2	4	no	—
91	C₄⋊Q₈×C₅₃	—	4	4	no	—
92	#125×C₅₃	—	2	8	no	—
93	C₂²⋊Q₈×C₅₃	—	2	4	no	—
94	C₂²⋊Q₈×C₅₃	—	2	8	no	—
95	C₄×Q₈×C₅₃	—	2	4	no	—
96	C₄×Q₈×C₅₃	—	4	4	no	—
97	C₂×C₄⋊C₄×C₅₃	—	2	4	no	—
98	C₂×C₄⋊C₄×C₅₃	Q₈	1	8	yes	complemented
99	C₂×C₄⋊C₄×C₅₃	C₂³	1	1	yes	characteristic
100	C₂×C₄⋊C₄×C₅₃	C₂³	1	2	yes	—
101	C₂×C₄⋊C₄×C₅₃	C₂³	1	1	yes	characteristic
102	C₂×C₄⋊C₄×C₅₃	C₂³	1	1	yes	characteristic
103	C₂×C₄⋊C₄×C₅₃	C₂³	1	2	yes	—
104	C₂×C₄⋊C₄×C₅₃	C₂³	1	2	yes	—
105	C₂×C₄⋊C₄×C₅₃	C₂³	1	1	yes	characteristic
106	C₂×C₄⋊C₄×C₅₃	C₂³	1	2	yes	—
107	C₂×C₄⋊C₄×C₅₃	C₂³	1	4	yes	—
108	C₂×C₄⋊C₄×C₅₃	C₂³	1	2	yes	—
109	C₂×C₄⋊C₄×C₅₃	C₂³	1	4	yes	—
110	C₂×C₄⋊C₄×C₅₃	C₂³	1	4	yes	—
111	C₂×C₄⋊C₄×C₅₃	C₂³	1	2	yes	—
112	C₂×C₄⋊C₄×C₅₃	C₂³	1	2	yes	—
113	C₂×C₄⋊C₄×C₅₃	C₂³	1	4	yes	—
114	C₂×C₄⋊C₄×C₅₃	C₂³	1	4	yes	—
115	C₂×C₄⋊C₄×C₅₃	C₂³	1	4	yes	—
116	C₂×C₄⋊C₄×C₅₃	C₂³	1	2	yes	—
117	C₂×C₄⋊C₄×C₅₃	C₂³	1	2	yes	—
118	C₂×C₄⋊C₄×C₅₃	C₂³	1	4	yes	—
119	C₂×C₄⋊C₄×C₅₃	C₂³	1	1	yes	characteristic
120	C₂×C₄⋊C₄×C₅₃	C₂³	1	2	yes	—
121	C₂×C₄⋊C₄×C₅₃	C₂³	1	1	yes	characteristic
122	C₂×C₄⋊C₄×C₅₃	C₂³	1	1	yes	characteristic
123	C₂×C₂²⋊C₄×C₅₃	C₂³	1	2	yes	—
124	C₂×C₂²⋊C₄×C₅₃	C₂³	1	4	yes	—
125	C₂×C₂²⋊C₄×C₅₃	C₂³	1	2	yes	—
126	C₂×C₄²×C₅₃	—	2	4	no	—
127	C₂×C₄²×C₅₃	—	4	4	no	—
128	C₂×C₄²×C₅₃	C₂³	1	2	yes	—
129	C₂×C₄²×C₅₃	C₂³	1	2	yes	—
130	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
131	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
132	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
133	C₂×C₄²×C₅₃	C₂³	1	2	yes	—
134	C₂×C₄²×C₅₃	C₂³	1	2	yes	—
135	C₂×C₄²×C₅₃	C₂³	1	2	yes	—
136	C₂×C₄²×C₅₃	C₂³	1	2	yes	—
137	C₂×C₄²×C₅₃	C₂³	1	4	yes	—
138	C₂×C₄²×C₅₃	C₂³	1	4	yes	—
139	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
140	#95×C₅₃	C₂³	1	2	yes	—
141	#95×C₅₃	C₂³	1	2	yes	—
142	#95×C₅₃	C₂³	1	2	yes	—
143	#95×C₅₃	C₂³	1	2	yes	—
144	#95×C₅₃	C₂³	1	2	yes	—
145	#95×C₅₃	C₂³	1	2	yes	—
146	#95×C₅₃	C₂³	1	4	yes	—
147	#95×C₅₃	C₂³	1	4	yes	—
148	#95×C₅₃	C₂³	1	2	yes	—
149	#95×C₅₃	C₂³	1	4	yes	—
150	#95×C₅₃	C₂³	1	4	yes	—
151	#95×C₅₃	C₂³	1	4	yes	—
152	#95×C₅₃	C₂³	1	2	yes	—
153	#95×C₅₃	C₂³	1	2	yes	—
154	#95×C₅₃	C₂³	1	4	yes	—
155	#95×C₅₃	C₂³	1	4	yes	—
156	#95×C₅₃	C₂³	1	4	yes	—
157	#95×C₅₃	C₂³	1	4	yes	—
158	#95×C₅₃	C₂³	1	4	yes	—
159	C₂⁴×C₅₃	C₂⁴	1	1	yes	characteristic
160	C₂×Q₈×C₅₃	—	2	2	no	—
161	C₂×Q₈×C₅₃	—	2	2	no	—
162	C₂×Q₈×C₅₃	—	2	4	no	—
163	C₂×Q₈×C₅₃	—	2	4	no	—
164	C₂×Q₈×C₅₃	—	2	4	no	—
165	C₂×Q₈×C₅₃	—	2	2	no	—
166	C₂²×C₄×C₅₃	—	2	2	no	—
167	C₂²×C₄×C₅₃	—	2	2	no	—
168	C₂²×C₄×C₅₃	—	2	2	no	—
169	C₂²×C₄×C₅₃	—	2	2	no	—
170	C₂²×C₄×C₅₃	—	2	4	no	—
171	C₂²×C₄×C₅₃	—	2	2	no	—
172	C₂²×C₄×C₅₃	—	2	2	no	—
173	C₂²×C₄×C₅₃	—	2	4	no	—
174	C₂²×C₄×C₅₃	—	4	2	no	—
175	C₂²×C₄×C₅₃	—	4	2	no	—
176	C₂²×C₄×C₅₃	—	4	2	no	—
177	C₂²×C₄×C₅₃	—	4	1	no	—
178	C₂²×C₄×C₅₃	C₂×Q₈	1	4	yes	—
179	C₂²×C₄×C₅₃	C₂×Q₈	1	8	yes	—
180	C₂²×C₄×C₅₃	#41	1	4	yes	—
181	C₂²×C₄×C₅₃	C₂⁴	1	1	yes	characteristic
182	C₂²×C₄×C₅₃	C₂⁴	1	2	yes	—
183	C₂²×C₄×C₅₃	C₂⁴	1	2	yes	—
184	C₂²×C₄×C₅₃	C₂⁴	1	2	yes	—
185	C₂²×C₄×C₅₃	C₂⁴	1	2	yes	—
186	C₂²×C₄×C₅₃	C₂⁴	1	4	yes	—
187	C₂²×C₄×C₅₃	C₂⁴	1	2	yes	—
188	C₂²×C₄×C₅₃	C₂⁴	1	4	yes	—
189	C₂²×C₄×C₅₃	C₂⁴	1	1	yes	characteristic
190	C₂²×C₄×C₅₃	C₂⁴	1	2	yes	—
191	C₂²×C₄×C₅₃	C₂⁴	1	2	yes	—
192	C₂²×C₄×C₅₃	C₂⁴	1	1	yes	characteristic
193	C₂²×C₄×C₅₃	C₂⁴	1	2	yes	—
194	C₂²×C₄×C₅₃	C₂⁴	1	1	yes	characteristic
195	C₂²×C₄×C₅₃	C₂⁴	1	1	yes	characteristic
196	C₂²×C₄×C₅₃	C₂⁴	1	1	yes	characteristic
197	C₄⋊C₄×C₅₃	—	2	4	no	—
198	C₄⋊C₄×C₅₃	—	2	2	no	—
199	C₄⋊C₄×C₅₃	—	2	4	no	—
200	C₄⋊C₄×C₅₃	—	2	2	no	—
201	C₄⋊C₄×C₅₃	—	2	4	no	—
202	C₄⋊C₄×C₅₃	—	2	8	no	—
203	C₄⋊C₄×C₅₃	—	2	4	no	—
204	C₄⋊C₄×C₅₃	—	2	2	no	—
205	C₄⋊C₄×C₅₃	—	2	8	no	—
206	C₄⋊C₄×C₅₃	—	2	8	no	—
207	C₄⋊C₄×C₅₃	—	2	4	no	—
208	C₄⋊C₄×C₅₃	—	2	4	no	—
209	C₄⋊C₄×C₅₃	—	2	4	no	—
210	C₄⋊C₄×C₅₃	—	2	2	no	—
211	C₄⋊C₄×C₅₃	—	4	2	no	—
212	C₄⋊C₄×C₅₃	—	4	4	no	—
213	C₄⋊C₄×C₅₃	—	4	2	no	—
214	C₄⋊C₄×C₅₃	—	4	4	no	—
215	C₄⋊C₄×C₅₃	—	4	4	no	—
216	C₄⋊C₄×C₅₃	—	4	4	no	—
217	C₄⋊C₄×C₅₃	C₂×Q₈	1	8	yes	—
218	C₄⋊C₄×C₅₃	C₂×Q₈	1	4	yes	—
219	C₄⋊C₄×C₅₃	C₂×Q₈	1	4	yes	—
220	C₄⋊C₄×C₅₃	#41	1	4	yes	—
221	C₂²⋊C₄×C₅₃	—	2	4	no	—
222	C₂²⋊C₄×C₅₃	—	2	8	no	—
223	C₂²⋊C₄×C₅₃	—	2	4	no	—
224	C₄²×C₅₃	—	2	2	no	—
225	C₄²×C₅₃	—	2	2	no	—
226	C₄²×C₅₃	—	2	2	no	—
227	C₄²×C₅₃	—	2	4	no	—
228	C₄²×C₅₃	—	2	4	no	—
229	C₄²×C₅₃	—	2	4	no	—
230	C₄²×C₅₃	—	2	4	no	—
231	C₄²×C₅₃	—	4	2	no	—
232	C₄²×C₅₃	—	4	2	no	—
233	C₄²×C₅₃	—	4	4	no	—
234	C₄²×C₅₃	—	4	4	no	—
235	C₄²×C₅₃	#41	1	4	yes	—
236	C₂³×C₅₃	—	2	1	no	—
237	C₂³×C₅₃	—	2	1	no	—
238	C₂³×C₅₃	—	2	1	no	—
239	C₂³×C₅₃	—	2	1	no	—
240	C₂³×C₅₃	C₂²×Q₈	1	2	yes	—
241	C₂³×C₅₃	C₂×#41	1	2	yes	—
242	C₂³×C₅₃	2₋¹⁺⁴	1	2	yes	—
243	C₂³×C₅₃	C₂⁵	1	1	yes	characteristic
244	Q₈×C₅₃	—	4	4	no	—
245	Q₈×C₅₃	—	4	8	no	—
246	C₂×C₄×C₅₃	—	2	1	no	—
247	C₂×C₄×C₅₃	—	2	1	no	—
248	C₂×C₄×C₅₃	—	2	1	no	—
249	C₂×C₄×C₅₃	—	2	1	no	—
250	C₂×C₄×C₅₃	—	2	2	no	—
251	C₂×C₄×C₅₃	—	2	2	no	—
252	C₂×C₄×C₅₃	—	2	2	no	—
253	C₂×C₄×C₅₃	—	2	1	no	—
254	C₂×C₄×C₅₃	—	2	2	no	—
255	C₂×C₄×C₅₃	—	2	2	no	—
256	C₂×C₄×C₅₃	—	2	1	no	—
257	C₂×C₄×C₅₃	—	2	2	no	—
258	C₂×C₄×C₅₃	—	2	8	no	—
259	C₂×C₄×C₅₃	—	2	8	no	—
260	C₂×C₄×C₅₃	—	2	1	no	—
261	C₂×C₄×C₅₃	—	2	2	no	—
262	C₂×C₄×C₅₃	—	2	2	no	—
263	C₂×C₄×C₅₃	—	2	2	no	—
264	C₂×C₄×C₅₃	—	2	4	no	—
265	C₂×C₄×C₅₃	—	2	4	no	—
266	C₂×C₄×C₅₃	—	2	2	no	—
267	C₂×C₄×C₅₃	—	2	2	no	—
268	C₂×C₄×C₅₃	—	2	2	no	—
269	C₂×C₄×C₅₃	—	2	2	no	—
270	C₂×C₄×C₅₃	—	2	4	no	—
271	C₂×C₄×C₅₃	—	2	4	no	—
272	C₂×C₄×C₅₃	—	2	2	no	—
273	C₂×C₄×C₅₃	—	2	4	no	—
274	C₂×C₄×C₅₃	—	2	2	no	—
275	C₂×C₄×C₅₃	—	2	2	no	—
276	C₂×C₄×C₅₃	—	2	1	no	—
277	C₂×C₄×C₅₃	—	2	2	no	—
278	C₂×C₄×C₅₃	—	2	2	no	—
279	C₂×C₄×C₅₃	—	2	2	no	—
280	C₂×C₄×C₅₃	—	2	1	no	—
281	C₂×C₄×C₅₃	—	2	2	no	—
282	C₂×C₄×C₅₃	—	2	4	no	—
283	C₂×C₄×C₅₃	—	2	2	no	—
284	C₂×C₄×C₅₃	—	4	8	no	—
285	C₂×C₄×C₅₃	—	4	4	no	—
286	C₂×C₄×C₅₃	—	4	8	no	—
287	C₂×C₄×C₅₃	C₂²×Q₈	1	2	yes	—
288	C₂×C₄×C₅₃	C₂²×Q₈	1	4	yes	—
289	C₂×C₄×C₅₃	C₂²×Q₈	1	4	yes	—
290	C₂×C₄×C₅₃	C₂²×Q₈	1	2	yes	—
291	C₂×C₄×C₅₃	C₂×#41	1	2	yes	—
292	C₂×C₄×C₅₃	C₂×#41	1	4	yes	—
293	C₂×C₄×C₅₃	C₂×#41	1	2	yes	—
294	C₂×C₄×C₅₃	C₂×#41	1	2	yes	—
295	C₂×C₄×C₅₃	C₂×#41	1	2	yes	—
296	C₂×C₄×C₅₃	C₂×#41	1	2	yes	—
297	C₂×C₄×C₅₃	C₂×#41	1	2	yes	—
298	C₂×C₄×C₅₃	H(5,𝔽₂)	1	2	yes	—
299	#36599	C₅₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
300	C₂²×C₅₃	—	2	2	no	—
301	C₂²×C₅₃	—	2	2	no	—
302	C₂²×C₅₃	—	2	2	no	—
303	C₂²×C₅₃	—	2	2	no	—
304	C₂²×C₅₃	—	2	2	no	—
305	C₂²×C₅₃	—	2	2	no	—
306	C₂²×C₅₃	#557	1	4	yes	—
307	C₂²×C₅₃	C₂³×Q₈	1	1	yes	characteristic
308	C₂²×C₅₃	C₂²×#41	1	1	yes	characteristic
309	C₂²×C₅₃	C₂²×#41	1	1	yes	characteristic
310	C₂²×C₅₃	C₂²×#41	1	1	yes	characteristic
311	C₂²×C₅₃	C₂×H(5,𝔽₂)	1	1	yes	characteristic
312	C₂²×C₅₃	C₂×2₋¹⁺⁴	1	1	yes	characteristic
313	C₂²×C₅₃	#585	1	1	yes	characteristic
314	C₄×C₅₃	—	2	2	no	—
315	C₄×C₅₃	—	2	2	no	—
316	C₄×C₅₃	—	2	2	no	—
317	C₄×C₅₃	—	2	4	no	—
318	C₄×C₅₃	—	2	4	no	—
319	C₄×C₅₃	—	2	4	no	—
320	C₄×C₅₃	—	4	2	no	—
321	C₄×C₅₃	—	4	4	no	—
322	C₄×C₅₃	—	4	2	no	—
323	C₄×C₅₃	—	4	4	no	—
324	C₄×C₅₃	—	4	2	no	—
325	C₄×C₅₃	—	4	1	no	—
326	C₄×C₅₃	—	4	2	no	—
327	C₄×C₅₃	—	4	2	no	—
328	C₄×C₅₃	#533	1	4	yes	—
329	C₄×C₅₃	#553	1	4	yes	—
330	#2857	C₂×C₅₃	1	1	yes	characteristic
331	#2857	C₂×C₅₃	1	2	yes	—
332	#2585	C₂×C₅₃	1	4	yes	complemented
333	#2569	C₂×C₅₃	1	4	yes	—
334	#2552	C₂×C₅₃	1	1	yes	characteristic, complemented
335	#2546	C₂×C₅₃	1	1	yes	characteristic, complemented
336	#2511	C₂×C₅₃	1	2	yes	complemented
337	#2509	C₂×C₅₃	1	2	yes	complemented
338	#2446	C₂×C₅₃	1	2	yes	complemented
339	#2442	C₂×C₅₃	1	2	yes	—
340	#2370	C₂×C₅₃	1	2	yes	complemented
341	#2365	C₂×C₅₃	1	1	yes	characteristic, complemented
342	#2355	C₂×C₅₃	1	1	yes	characteristic, complemented
343	#2345	C₂×C₅₃	1	2	yes	—
344	C₄×C₂²⋊Q₈	C₂×C₅₃	1	1	yes	characteristic
345	C₄×C₂²⋊Q₈	C₂×C₅₃	1	2	yes	—
346	C₂×C₄×C₄⋊C₄	C₂×C₅₃	1	1	yes	characteristic
347	C₂×C₅₃	—	2	4	no	—
348	C₂×C₅₃	C₂×#533	1	1	yes	characteristic
349	C₂×C₅₃	C₂×#553	1	1	yes	characteristic
350	C₂×C₅₃	C₂×#557	1	1	yes	characteristic
351	C₂×C₅₃	#3480	1	1	yes	characteristic
352	C₂×C₅₃	#3495	1	1	yes	characteristic
353	C₂×C₅₃	#3501	1	1	yes	characteristic
354	C₂×C₅₃	#3520	1	1	yes	characteristic
355	C₂×C₄⋊Q₈	C₂²×C₅₃	1	1	yes	characteristic
356	C₂×C₄⋊Q₈	C₂²×C₅₃	1	2	yes	—
357	C₂×#125	C₂²×C₅₃	1	2	yes	—
358	C₂×C₂²⋊Q₈	C₂²×C₅₃	1	2	yes	—
359	C₂×C₂²⋊Q₈	C₂²×C₅₃	1	1	yes	characteristic
360	C₂×C₄×Q₈	C₂²×C₅₃	1	1	yes	characteristic
361	C₂×C₄×Q₈	C₂²×C₅₃	1	2	yes	—
362	C₂²×C₄⋊C₄	C₂²×C₅₃	1	1	yes	characteristic
363	C₂²×C₄⋊C₄	C₂²×C₅₃	1	1	yes	characteristic
364	C₂²×C₄²	C₂²×C₅₃	1	1	yes	characteristic
365	C₂²×C₄²	C₂²×C₅₃	1	2	yes	—
366	#400	C₂²×C₅₃	1	4	yes	—
367	#400	C₂²×C₅₃	1	4	yes	—
368	#400	C₂²×C₅₃	1	4	yes	—
369	#398	C₂²×C₅₃	1	4	yes	—
370	#398	C₂²×C₅₃	1	4	yes	—
371	#398	C₂²×C₅₃	1	4	yes	—
372	#391	C₂²×C₅₃	1	2	yes	—
373	#391	C₂²×C₅₃	1	2	yes	—
374	#391	C₂²×C₅₃	1	2	yes	—
375	#391	C₂²×C₅₃	1	1	yes	characteristic
376	#391	C₂²×C₅₃	1	2	yes	—
377	#389	C₂²×C₅₃	1	2	yes	—
378	#389	C₂²×C₅₃	1	4	yes	—
379	#389	C₂²×C₅₃	1	4	yes	—
380	#389	C₂²×C₅₃	1	2	yes	—
381	#389	C₂²×C₅₃	1	2	yes	—
382	#389	C₂²×C₅₃	1	2	yes	—
383	#389	C₂²×C₅₃	1	2	yes	—
384	#389	C₂²×C₅₃	1	2	yes	—
385	#389	C₂²×C₅₃	1	2	yes	—
386	#389	C₂²×C₅₃	1	1	yes	characteristic
387	#387	C₂²×C₅₃	1	4	yes	—
388	#387	C₂²×C₅₃	1	4	yes	—
389	#387	C₂²×C₅₃	1	4	yes	—
390	#387	C₂²×C₅₃	1	4	yes	—
391	#387	C₂²×C₅₃	1	2	yes	—
392	#387	C₂²×C₅₃	1	4	yes	—
393	#387	C₂²×C₅₃	1	4	yes	—
394	#387	C₂²×C₅₃	1	4	yes	—
395	#385	C₂²×C₅₃	1	2	yes	—
396	#385	C₂²×C₅₃	1	4	yes	—
397	#384	C₂²×C₅₃	1	1	yes	characteristic
398	#384	C₂²×C₅₃	1	2	yes	—
399	#382	C₂²×C₅₃	1	2	yes	—
400	#380	C₂²×C₅₃	1	2	yes	—
401	#380	C₂²×C₅₃	1	4	yes	—
402	#380	C₂²×C₅₃	1	4	yes	—
403	#380	C₂²×C₅₃	1	2	yes	—
404	#380	C₂²×C₅₃	1	1	yes	characteristic
405	C₄×C₄⋊C₄	C₂²×C₅₃	1	2	yes	—
406	C₄×C₄⋊C₄	C₂²×C₅₃	1	4	yes	—
407	C₄×C₄⋊C₄	C₂²×C₅₃	1	2	yes	—
408	C₄×C₄⋊C₄	C₂²×C₅₃	1	2	yes	—
409	C₄×C₄⋊C₄	C₂²×C₅₃	1	1	yes	characteristic
410	C₄×C₄⋊C₄	C₂²×C₅₃	1	2	yes	—
411	C₄×C₄⋊C₄	C₂²×C₅₃	1	1	yes	characteristic
412	C₄×C₄⋊C₄	C₂²×C₅₃	1	1	yes	characteristic
413	C₄×C₄⋊C₄	C₂²×C₅₃	1	2	yes	—
414	C₄×C₄⋊C₄	C₂²×C₅₃	1	2	yes	—
415	C₄×C₄⋊C₄	C₂²×C₅₃	1	1	yes	characteristic
416	C₄×C₄⋊C₄	C₂²×C₅₃	1	1	yes	characteristic
417	C₄×C₂²⋊C₄	C₂²×C₅₃	1	4	yes	—
418	C₄×C₂²⋊C₄	C₂²×C₅₃	1	2	yes	—
419	C₂×#95	C₂²×C₅₃	1	2	yes	—
420	C₅₃	#36599	1	1	yes	characteristic, direct product factor, Sylow
421	C₂²×Q₈	C₂³×C₅₃	1	1	yes	characteristic
422	C₂²×Q₈	C₂³×C₅₃	1	2	yes	—
423	C₂³×C₄	C₂³×C₅₃	1	2	yes	—
424	C₂³×C₄	C₂³×C₅₃	1	1	yes	characteristic
425	C₂³×C₄	C₂³×C₅₃	1	1	yes	characteristic
426	C₂³×C₄	C₂³×C₅₃	1	1	yes	characteristic
427	C₂³×C₄	C₂³×C₅₃	1	2	yes	—
428	C₄⋊Q₈	—	2	4	no	—
429	C₄⋊Q₈	—	4	4	no	—
430	#125	—	2	8	no	—
431	C₂²⋊Q₈	—	2	8	no	—
432	C₂²⋊Q₈	—	2	4	no	—
433	C₄×Q₈	—	2	4	no	—
434	C₄×Q₈	—	4	4	no	—
435	C₂×C₄⋊C₄	—	2	4	no	—
436	C₂×C₄⋊C₄	Q₈×C₅₃	1	8	yes	complemented
437	C₂×C₄⋊C₄	C₂³×C₅₃	1	1	yes	characteristic
438	C₂×C₄⋊C₄	C₂³×C₅₃	1	2	yes	—
439	C₂×C₄⋊C₄	C₂³×C₅₃	1	1	yes	characteristic
440	C₂×C₄⋊C₄	C₂³×C₅₃	1	1	yes	characteristic
441	C₂×C₄⋊C₄	C₂³×C₅₃	1	2	yes	—
442	C₂×C₄⋊C₄	C₂³×C₅₃	1	2	yes	—
443	C₂×C₄⋊C₄	C₂³×C₅₃	1	1	yes	characteristic
444	C₂×C₄⋊C₄	C₂³×C₅₃	1	4	yes	—
445	C₂×C₄⋊C₄	C₂³×C₅₃	1	2	yes	—
446	C₂×C₄⋊C₄	C₂³×C₅₃	1	2	yes	—
447	C₂×C₄⋊C₄	C₂³×C₅₃	1	4	yes	—
448	C₂×C₄⋊C₄	C₂³×C₅₃	1	4	yes	—
449	C₂×C₄⋊C₄	C₂³×C₅₃	1	2	yes	—
450	C₂×C₄⋊C₄	C₂³×C₅₃	1	2	yes	—
451	C₂×C₄⋊C₄	C₂³×C₅₃	1	4	yes	—
452	C₂×C₄⋊C₄	C₂³×C₅₃	1	4	yes	—
453	C₂×C₄⋊C₄	C₂³×C₅₃	1	4	yes	—
454	C₂×C₄⋊C₄	C₂³×C₅₃	1	2	yes	—
455	C₂×C₄⋊C₄	C₂³×C₅₃	1	2	yes	—
456	C₂×C₄⋊C₄	C₂³×C₅₃	1	4	yes	—
457	C₂×C₄⋊C₄	C₂³×C₅₃	1	1	yes	characteristic
458	C₂×C₄⋊C₄	C₂³×C₅₃	1	2	yes	—
459	C₂×C₄⋊C₄	C₂³×C₅₃	1	1	yes	characteristic
460	C₂×C₄⋊C₄	C₂³×C₅₃	1	1	yes	characteristic
461	C₂×C₂²⋊C₄	C₂³×C₅₃	1	4	yes	—
462	C₂×C₂²⋊C₄	C₂³×C₅₃	1	2	yes	—
463	C₂×C₂²⋊C₄	C₂³×C₅₃	1	2	yes	—
464	C₂×C₄²	—	2	4	no	—
465	C₂×C₄²	—	4	4	no	—
466	C₂×C₄²	C₂³×C₅₃	1	2	yes	—
467	C₂×C₄²	C₂³×C₅₃	1	2	yes	—
468	C₂×C₄²	C₂³×C₅₃	1	2	yes	—
469	C₂×C₄²	C₂³×C₅₃	1	2	yes	—
470	C₂×C₄²	C₂³×C₅₃	1	1	yes	characteristic
471	C₂×C₄²	C₂³×C₅₃	1	1	yes	characteristic
472	C₂×C₄²	C₂³×C₅₃	1	1	yes	characteristic
473	C₂×C₄²	C₂³×C₅₃	1	4	yes	—
474	C₂×C₄²	C₂³×C₅₃	1	4	yes	—
475	C₂×C₄²	C₂³×C₅₃	1	2	yes	—
476	C₂×C₄²	C₂³×C₅₃	1	2	yes	—
477	C₂×C₄²	C₂³×C₅₃	1	1	yes	characteristic
478	#95	C₂³×C₅₃	1	2	yes	—
479	#95	C₂³×C₅₃	1	2	yes	—
480	#95	C₂³×C₅₃	1	2	yes	—
481	#95	C₂³×C₅₃	1	2	yes	—
482	#95	C₂³×C₅₃	1	2	yes	—
483	#95	C₂³×C₅₃	1	2	yes	—
484	#95	C₂³×C₅₃	1	4	yes	—
485	#95	C₂³×C₅₃	1	4	yes	—
486	#95	C₂³×C₅₃	1	2	yes	—
487	#95	C₂³×C₅₃	1	4	yes	—
488	#95	C₂³×C₅₃	1	4	yes	—
489	#95	C₂³×C₅₃	1	4	yes	—
490	#95	C₂³×C₅₃	1	2	yes	—
491	#95	C₂³×C₅₃	1	2	yes	—
492	#95	C₂³×C₅₃	1	4	yes	—
493	#95	C₂³×C₅₃	1	4	yes	—
494	#95	C₂³×C₅₃	1	4	yes	—
495	#95	C₂³×C₅₃	1	4	yes	—
496	#95	C₂³×C₅₃	1	4	yes	—
497	C₂⁴	C₂⁴×C₅₃	1	1	yes	characteristic
498	C₂×Q₈	—	2	2	no	—
499	C₂×Q₈	—	2	2	no	—
500	C₂×Q₈	—	2	4	no	—
501	C₂×Q₈	—	2	4	no	—
502	C₂×Q₈	—	2	4	no	—
503	C₂×Q₈	—	2	2	no	—
504	C₂²×C₄	—	2	2	no	—
505	C₂²×C₄	—	2	2	no	—
506	C₂²×C₄	—	2	2	no	—
507	C₂²×C₄	—	2	2	no	—
508	C₂²×C₄	—	2	4	no	—
509	C₂²×C₄	—	2	2	no	—
510	C₂²×C₄	—	2	2	no	—
511	C₂²×C₄	—	2	4	no	—
512	C₂²×C₄	—	4	2	no	—
513	C₂²×C₄	—	4	2	no	—
514	C₂²×C₄	—	4	2	no	—
515	C₂²×C₄	—	4	1	no	—
516	C₂²×C₄	C₂×Q₈×C₅₃	1	4	yes	—
517	C₂²×C₄	C₂×Q₈×C₅₃	1	8	yes	—
518	C₂²×C₄	#41×C₅₃	1	4	yes	—
519	C₂²×C₄	C₂⁴×C₅₃	1	1	yes	characteristic
520	C₂²×C₄	C₂⁴×C₅₃	1	2	yes	—
521	C₂²×C₄	C₂⁴×C₅₃	1	2	yes	—
522	C₂²×C₄	C₂⁴×C₅₃	1	2	yes	—
523	C₂²×C₄	C₂⁴×C₅₃	1	2	yes	—
524	C₂²×C₄	C₂⁴×C₅₃	1	4	yes	—
525	C₂²×C₄	C₂⁴×C₅₃	1	2	yes	—
526	C₂²×C₄	C₂⁴×C₅₃	1	4	yes	—
527	C₂²×C₄	C₂⁴×C₅₃	1	1	yes	characteristic
528	C₂²×C₄	C₂⁴×C₅₃	1	2	yes	—
529	C₂²×C₄	C₂⁴×C₅₃	1	2	yes	—
530	C₂²×C₄	C₂⁴×C₅₃	1	1	yes	characteristic
531	C₂²×C₄	C₂⁴×C₅₃	1	2	yes	—
532	C₂²×C₄	C₂⁴×C₅₃	1	1	yes	characteristic
533	C₂²×C₄	C₂⁴×C₅₃	1	1	yes	characteristic
534	C₂²×C₄	C₂⁴×C₅₃	1	1	yes	characteristic
535	C₄⋊C₄	—	2	4	no	—
536	C₄⋊C₄	—	2	2	no	—
537	C₄⋊C₄	—	2	2	no	—
538	C₄⋊C₄	—	2	4	no	—
539	C₄⋊C₄	—	2	4	no	—
540	C₄⋊C₄	—	2	2	no	—
541	C₄⋊C₄	—	2	8	no	—
542	C₄⋊C₄	—	2	4	no	—
543	C₄⋊C₄	—	2	8	no	—
544	C₄⋊C₄	—	2	8	no	—
545	C₄⋊C₄	—	2	4	no	—
546	C₄⋊C₄	—	2	4	no	—
547	C₄⋊C₄	—	2	4	no	—
548	C₄⋊C₄	—	2	2	no	—
549	C₄⋊C₄	—	4	4	no	—
550	C₄⋊C₄	—	4	4	no	—
551	C₄⋊C₄	—	4	4	no	—
552	C₄⋊C₄	—	4	2	no	—
553	C₄⋊C₄	—	4	4	no	—
554	C₄⋊C₄	—	4	2	no	—
555	C₄⋊C₄	C₂×Q₈×C₅₃	1	8	yes	—
556	C₄⋊C₄	C₂×Q₈×C₅₃	1	4	yes	—
557	C₄⋊C₄	C₂×Q₈×C₅₃	1	4	yes	—
558	C₄⋊C₄	#41×C₅₃	1	4	yes	—
559	C₂²⋊C₄	—	2	8	no	—
560	C₂²⋊C₄	—	2	4	no	—
561	C₂²⋊C₄	—	2	4	no	—
562	C₄²	—	2	2	no	—
563	C₄²	—	2	2	no	—
564	C₄²	—	2	2	no	—
565	C₄²	—	2	4	no	—
566	C₄²	—	2	4	no	—
567	C₄²	—	2	4	no	—
568	C₄²	—	2	4	no	—
569	C₄²	—	4	2	no	—
570	C₄²	—	4	2	no	—
571	C₄²	—	4	4	no	—
572	C₄²	—	4	4	no	—
573	C₄²	#41×C₅₃	1	4	yes	—
574	C₂³	—	2	1	no	—
575	C₂³	—	2	1	no	—
576	C₂³	—	2	1	no	—
577	C₂³	—	2	1	no	—
578	C₂³	C₂²×Q₈×C₅₃	1	2	yes	—
579	C₂³	C₂×#41×C₅₃	1	2	yes	—
580	C₂³	2₋¹⁺⁴×C₅₃	1	2	yes	—
581	C₂³	C₂⁵×C₅₃	1	1	yes	characteristic
582	Q₈	—	4	4	no	—
583	Q₈	—	4	8	no	—
584	C₂×C₄	—	2	1	no	—
585	C₂×C₄	—	2	1	no	—
586	C₂×C₄	—	2	1	no	—
587	C₂×C₄	—	2	2	no	—
588	C₂×C₄	—	2	2	no	—
589	C₂×C₄	—	2	2	no	—
590	C₂×C₄	—	2	1	no	—
591	C₂×C₄	—	2	1	no	—
592	C₂×C₄	—	2	2	no	—
593	C₂×C₄	—	2	2	no	—
594	C₂×C₄	—	2	1	no	—
595	C₂×C₄	—	2	2	no	—
596	C₂×C₄	—	2	8	no	—
597	C₂×C₄	—	2	8	no	—
598	C₂×C₄	—	2	1	no	—
599	C₂×C₄	—	2	2	no	—
600	C₂×C₄	—	2	2	no	—
601	C₂×C₄	—	2	2	no	—
602	C₂×C₄	—	2	2	no	—
603	C₂×C₄	—	2	2	no	—
604	C₂×C₄	—	2	4	no	—
605	C₂×C₄	—	2	2	no	—
606	C₂×C₄	—	2	4	no	—
607	C₂×C₄	—	2	4	no	—
608	C₂×C₄	—	2	2	no	—
609	C₂×C₄	—	2	4	no	—
610	C₂×C₄	—	2	2	no	—
611	C₂×C₄	—	2	2	no	—
612	C₂×C₄	—	2	4	no	—
613	C₂×C₄	—	2	2	no	—
614	C₂×C₄	—	2	2	no	—
615	C₂×C₄	—	2	2	no	—
616	C₂×C₄	—	2	1	no	—
617	C₂×C₄	—	2	2	no	—
618	C₂×C₄	—	2	2	no	—
619	C₂×C₄	—	2	4	no	—
620	C₂×C₄	—	2	2	no	—
621	C₂×C₄	—	2	1	no	—
622	C₂×C₄	—	4	8	no	—
623	C₂×C₄	—	4	4	no	—
624	C₂×C₄	—	4	8	no	—
625	C₂×C₄	C₂²×Q₈×C₅₃	1	4	yes	—
626	C₂×C₄	C₂²×Q₈×C₅₃	1	4	yes	—
627	C₂×C₄	C₂²×Q₈×C₅₃	1	2	yes	—
628	C₂×C₄	C₂²×Q₈×C₅₃	1	2	yes	—
629	C₂×C₄	C₂×#41×C₅₃	1	2	yes	—
630	C₂×C₄	C₂×#41×C₅₃	1	2	yes	—
631	C₂×C₄	C₂×#41×C₅₃	1	4	yes	—
632	C₂×C₄	C₂×#41×C₅₃	1	2	yes	—
633	C₂×C₄	C₂×#41×C₅₃	1	2	yes	—
634	C₂×C₄	C₂×#41×C₅₃	1	2	yes	—
635	C₂×C₄	C₂×#41×C₅₃	1	2	yes	—
636	C₂×C₄	H(5,𝔽₂)×C₅₃	1	2	yes	—
637	C₂²	—	2	2	no	—
638	C₂²	—	2	2	no	—
639	C₂²	—	2	2	no	—
640	C₂²	—	2	2	no	—
641	C₂²	—	2	2	no	—
642	C₂²	—	2	2	no	—
643	C₂²	C₅₃×#557	1	4	yes	—
644	C₂²	C₂³×Q₈×C₅₃	1	1	yes	characteristic
645	C₂²	C₂²×#41×C₅₃	1	1	yes	characteristic
646	C₂²	C₂²×#41×C₅₃	1	1	yes	characteristic
647	C₂²	C₂²×#41×C₅₃	1	1	yes	characteristic
648	C₂²	C₂×H(5,𝔽₂)×C₅₃	1	1	yes	characteristic
649	C₂²	C₂×2₋¹⁺⁴×C₅₃	1	1	yes	characteristic
650	C₂²	C₅₃×#585	1	1	yes	characteristic
651	C₄	—	2	2	no	—
652	C₄	—	2	2	no	—
653	C₄	—	2	4	no	—
654	C₄	—	2	4	no	—
655	C₄	—	2	4	no	—
656	C₄	—	2	2	no	—
657	C₄	—	4	2	no	—
658	C₄	—	4	4	no	—
659	C₄	—	4	2	no	—
660	C₄	—	4	4	no	—
661	C₄	—	4	1	no	—
662	C₄	—	4	2	no	—
663	C₄	—	4	2	no	—
664	C₄	—	4	2	no	—
665	C₄	C₅₃×#533	1	4	yes	—
666	C₄	C₅₃×#553	1	4	yes	—
667	C₂	—	2	4	no	—
668	C₂	C₂×C₅₃×#533	1	1	yes	characteristic
669	C₂	C₂×C₅₃×#553	1	1	yes	characteristic
670	C₂	C₂×C₅₃×#557	1	1	yes	characteristic
671	C₂	C₅₃×#3480	1	1	yes	characteristic
672	C₂	C₅₃×#3495	1	1	yes	characteristic
673	C₂	C₅₃×#3501	1	1	yes	characteristic
674	C₂	C₅₃×#3520	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂³×C₅₃	C₂⁵
commutator, Frattini, cosocle	C₂³	C₂⁵×C₅₃
prime radical	C₂⁴×C₅₃	C₂⁴
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³×C₅₃ ◁ G

Complements

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

N	H ≅ G/N	type
3. C₅₃×#2585	C₂	semidirect
5. C₅₃×#2552	C₂	semidirect
6. C₅₃×#2546	C₂	semidirect
7. C₅₃×#2511	C₂	semidirect
8. C₅₃×#2509	C₂	semidirect
9. C₅₃×#2446	C₂	semidirect
11. C₅₃×#2370	C₂	semidirect
12. C₅₃×#2365	C₂	semidirect
13. C₅₃×#2355	C₂	semidirect
98. C₂×C₄⋊C₄×C₅₃	Q₈	semidirect
299. #36599	C₅₃	direct (N × H)
332. #2585	C₂×C₅₃	semidirect
334. #2552	C₂×C₅₃	semidirect
335. #2546	C₂×C₅₃	semidirect
336. #2511	C₂×C₅₃	semidirect
337. #2509	C₂×C₅₃	semidirect
338. #2446	C₂×C₅₃	semidirect
340. #2370	C₂×C₅₃	semidirect
341. #2365	C₂×C₅₃	semidirect
342. #2355	C₂×C₅₃	semidirect
420. C₅₃	#36599	direct (N × H)
436. C₂×C₄⋊C₄	Q₈×C₅₃	semidirect

Maximal subgroups

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

subgroup	index
C₅₃×#2857	2
C₅₃×#2585	2
C₅₃×#2569	2
C₅₃×#2552	2
C₅₃×#2546	2
C₅₃×#2511	2
C₅₃×#2509	2
C₅₃×#2446	2
C₅₃×#2442	2
C₅₃×#2370	2
C₅₃×#2365	2
C₅₃×#2355	2
C₅₃×#2345	2
C₄×C₂²⋊Q₈×C₅₃	2
C₂×C₄×C₄⋊C₄×C₅₃	2
#36599	53

Subgroups — C53×#36599 #27244111

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅₃×#36599 #27244111