Subgroups — C₅₃×#18017 #27225529

Order 13568.

600 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	C₅₃×#3108	C₂	1	1	yes	maximal, characteristic
2	C₅₃×#2994	C₂	1	1	yes	maximal, characteristic
3	C₅₃×#2984	C₂	1	1	yes	maximal, characteristic, complemented
4	C₅₃×#1591	C₂	1	1	yes	maximal, characteristic, complemented
5	C₅₃×#1590	C₂	1	1	yes	maximal, characteristic, complemented
6	C₅₃×#1588	C₂	1	1	yes	maximal, characteristic, complemented
7	C₅₃×#1587	C₂	1	1	yes	maximal, characteristic, complemented
8	C₅₃×#1587	C₂	1	1	yes	maximal, characteristic, complemented
9	C₅₃×#1585	C₂	1	1	yes	maximal, characteristic, complemented
10	C₈×Q₁₆×C₅₃	C₂	1	1	yes	maximal, characteristic, complemented
11	C₈×SD₁₆×C₅₃	C₂	1	1	yes	maximal, characteristic, complemented
12	C₅₃×#1573	C₂	1	1	yes	maximal, characteristic, complemented
13	C₅₃×#1494	C₂	1	1	yes	maximal, characteristic
14	C₂×C₅₃×Q₈⋊C₈	C₂	1	1	yes	maximal, characteristic, complemented
15	C₅₃×#1450	C₂	1	1	yes	maximal, characteristic, complemented
16	C₅₃×#529	C₂²	1	1	yes	characteristic
17	C₂×C₄×Q₈×C₅₃	C₂²	1	1	yes	characteristic
18	C₅₃×#483	C₄	1	1	yes	characteristic
19	C₅₃×#480	C₄	1	1	yes	characteristic
20	C₅₃×#479	C₄	1	1	yes	characteristic
21	C₅₃×#479	C₄	1	1	yes	characteristic
22	C₅₃×#464	C₄	1	1	yes	characteristic
23	C₅₃×#464	C₄	1	1	yes	characteristic
24	C₅₃×#451	C₄	1	1	yes	characteristic
25	C₅₃×#448	C₄	1	1	yes	characteristic
26	C₅₃×#446	—	2	1	no	—
27	C₅₃×#446	—	2	1	no	—
28	C₅₃×#446	C₂²	1	1	yes	characteristic
29	C₅₃×#446	C₂²	1	1	yes	characteristic
30	C₅₃×#446	C₂²	1	1	yes	characteristic
31	C₈×Q₈×C₅₃	C₂²	1	1	yes	characteristic
32	C₈×Q₈×C₅₃	C₂²	1	1	yes	characteristic
33	C₈×Q₈×C₅₃	C₂²	1	1	yes	characteristic
34	C₄×Q₁₆×C₅₃	C₂²	1	1	yes	characteristic
35	C₄×Q₁₆×C₅₃	C₂²	1	1	yes	characteristic
36	C₄×SD₁₆×C₅₃	C₂²	1	1	yes	characteristic
37	C₄×SD₁₆×C₅₃	C₂²	1	1	yes	characteristic
38	C₅₃×#436	C₂²	1	1	yes	characteristic
39	C₅₃×#435	—	2	1	no	—
40	C₈×D₄×C₅₃	—	2	1	no	—
41	C₈×D₄×C₅₃	C₂²	1	1	yes	characteristic
42	C₅₃×#433	—	2	1	no	—
43	C₅₃×#432	—	2	1	no	—
44	C₅₃×#432	C₂²	1	1	yes	characteristic
45	C₅₃×#431	—	2	1	no	—
46	C₅₃×#431	C₂²	1	1	yes	characteristic
47	C₅₃×#431	C₂²	1	1	yes	characteristic
48	C₅₃×#424	—	2	1	no	—
49	C₂×C₄⋊C₈×C₅₃	C₂²	1	1	yes	characteristic
50	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
51	C₅₃×#335	C₂²	1	1	yes	characteristic
52	C₅₃×#335	C₂²	1	1	yes	characteristic
53	C₅₃×#334	C₂²	1	1	yes	characteristic
54	C₅₃×#334	C₂²	1	1	yes	characteristic
55	C₅₃×Q₈⋊C₈	C₂²	1	1	yes	characteristic
56	C₅₃×Q₈⋊C₈	C₂²	1	1	yes	characteristic
57	C₅₃×Q₈⋊C₈	C₂²	1	1	yes	characteristic
58	C₅₃×Q₈⋊C₈	C₂²	1	1	yes	characteristic
59	C₅₃×Q₈⋊C₈	C₂²	1	1	yes	characteristic
60	C₅₃×Q₈⋊C₈	C₂²	1	1	yes	characteristic
61	C₅₃×D₄⋊C₈	C₂²	1	1	yes	characteristic
62	C₅₃×D₄⋊C₈	C₂²	1	1	yes	characteristic
63	C₅₃×#322	C₂²	1	1	yes	characteristic
64	C₅₃×#322	C₂²	1	1	yes	characteristic
65	C₈²×C₅₃	C₂²	1	1	yes	characteristic
66	C₈²×C₅₃	C₂²	1	1	yes	characteristic
67	C₂²×Q₈×C₅₃	C₂×C₄	1	1	yes	characteristic
68	C₂×Q₁₆×C₅₃	C₂×C₄	1	1	yes	characteristic
69	C₂×Q₁₆×C₅₃	C₂×C₄	1	1	yes	characteristic
70	C₂×SD₁₆×C₅₃	C₂×C₄	1	1	yes	characteristic
71	C₂×SD₁₆×C₅₃	C₂×C₄	1	1	yes	characteristic
72	C₂×M₁₆×C₅₃	—	2	1	no	—
73	C₂²×C₈×C₅₃	—	2	1	no	—
74	C₂²×C₈×C₅₃	—	2	1	no	—
75	C₂²×C₈×C₅₃	—	2	1	no	—
76	#126×C₅₃	—	2	1	no	—
77	#125×C₅₃	C₂×C₄	1	1	yes	characteristic
78	#124×C₅₃	C₂×C₄	1	1	yes	characteristic
79	#123×C₅₃	—	2	1	no	—
80	C₂²⋊Q₈×C₅₃	C₂×C₄	1	1	yes	characteristic
81	#121×C₅₃	C₂×C₄	1	1	yes	characteristic
82	C₄×Q₈×C₅₃	—	2	1	no	—
83	C₄×Q₈×C₅₃	—	2	1	no	—
84	C₄×Q₈×C₅₃	C₂×C₄	1	1	yes	characteristic
85	C₄×Q₈×C₅₃	C₂×C₄	1	1	yes	characteristic
86	C₄×Q₈×C₅₃	C₂³	1	1	yes	characteristic
87	C₄×Q₈×C₅₃	C₂³	1	1	yes	characteristic
88	C₄×Q₈×C₅₃	C₂³	1	1	yes	characteristic
89	C₄×D₄×C₅₃	—	2	1	no	—
90	C₄×D₄×C₅₃	C₂³	1	1	yes	characteristic
91	#117×C₅₃	—	2	1	no	—
92	C₂×C₄⋊C₄×C₅₃	—	2	1	no	—
93	C₂×C₄⋊C₄×C₅₃	C₂×C₄	1	1	yes	characteristic
94	C₂×C₄²×C₅₃	—	2	1	no	—
95	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
96	#107×C₅₃	C₂×C₄	1	1	yes	characteristic
97	#107×C₅₃	C₂×C₄	1	1	yes	characteristic
98	#106×C₅₃	C₂×C₄	1	1	yes	characteristic
99	#106×C₅₃	C₂×C₄	1	1	yes	characteristic
100	C₄⋊C₈×C₅₃	—	2	1	no	—
101	C₄⋊C₈×C₅₃	—	2	1	no	—
102	C₄⋊C₈×C₅₃	—	2	1	no	—
103	C₄⋊C₈×C₅₃	—	2	1	no	—
104	C₄⋊C₈×C₅₃	—	2	1	no	—
105	C₄⋊C₈×C₅₃	—	2	1	no	—
106	C₄⋊C₈×C₅₃	—	2	1	no	—
107	C₄⋊C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
108	C₄⋊C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
109	C₄⋊C₈×C₅₃	D₄	1	1	yes	characteristic
110	C₄⋊C₈×C₅₃	D₄	1	1	yes	characteristic
111	C₄⋊C₈×C₅₃	C₂³	1	1	yes	characteristic
112	C₄⋊C₈×C₅₃	C₂³	1	1	yes	characteristic
113	C₄⋊C₈×C₅₃	C₂³	1	1	yes	characteristic
114	C₄⋊C₈×C₅₃	C₂³	1	1	yes	characteristic
115	#103×C₅₃	C₂×C₄	1	1	yes	characteristic
116	#103×C₅₃	C₂×C₄	1	1	yes	characteristic
117	#103×C₅₃	C₂×C₄	1	1	yes	characteristic
118	#103×C₅₃	C₂×C₄	1	1	yes	characteristic
119	#103×C₅₃	C₂×C₄	1	1	yes	characteristic
120	#103×C₅₃	C₂×C₄	1	1	yes	characteristic
121	#102×C₅₃	C₂×C₄	1	1	yes	characteristic
122	#102×C₅₃	C₂×C₄	1	1	yes	characteristic
123	C₂²⋊C₈×C₅₃	—	2	1	no	—
124	C₂²⋊C₈×C₅₃	—	2	1	no	—
125	C₂²⋊C₈×C₅₃	—	2	1	no	—
126	C₂²⋊C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
127	C₂²⋊C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
128	C₂²⋊C₈×C₅₃	D₄	1	1	yes	characteristic
129	C₂²⋊C₈×C₅₃	D₄	1	1	yes	characteristic
130	#97×C₅₃	—	2	1	no	—
131	#97×C₅₃	—	2	1	no	—
132	#97×C₅₃	—	2	1	no	—
133	#97×C₅₃	—	2	1	no	—
134	C₄×C₈×C₅₃	—	2	1	no	—
135	C₄×C₈×C₅₃	—	2	1	no	—
136	C₄×C₈×C₅₃	—	2	1	no	—
137	C₄×C₈×C₅₃	—	2	1	no	—
138	C₄×C₈×C₅₃	C₂³	1	1	yes	characteristic
139	C₄×C₈×C₅₃	C₂³	1	1	yes	characteristic
140	C₄×C₈×C₅₃	C₂³	1	1	yes	characteristic
141	C₄×C₈×C₅₃	C₂³	1	1	yes	characteristic
142	C₄×C₈×C₅₃	C₂³	1	1	yes	characteristic
143	C₄×C₈×C₅₃	C₂³	1	1	yes	characteristic
144	C₂×Q₈×C₅₃	—	2	1	no	—
145	C₂×Q₈×C₅₃	—	2	2	no	—
146	C₂×Q₈×C₅₃	—	4	1	no	—
147	C₂×Q₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
148	C₂×Q₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
149	C₂×Q₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
150	C₂×D₄×C₅₃	—	2	1	no	—
151	C₂×D₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
152	C₂²×C₄×C₅₃	—	2	1	no	—
153	C₂²×C₄×C₅₃	—	2	1	no	—
154	C₂²×C₄×C₅₃	—	2	1	no	—
155	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
156	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
157	C₂²×C₄×C₅₃	#41	1	1	yes	characteristic
158	Q₁₆×C₅₃	—	2	2	no	—
159	Q₁₆×C₅₃	—	4	1	no	—
160	SD₁₆×C₅₃	—	2	2	no	—
161	SD₁₆×C₅₃	—	4	1	no	—
162	M₁₆×C₅₃	—	8	1	no	—
163	C₂×C₈×C₅₃	—	2	4	no	—
164	C₂×C₈×C₅₃	—	2	1	no	—
165	C₂×C₈×C₅₃	—	2	1	no	—
166	C₂×C₈×C₅₃	—	2	1	no	—
167	C₂×C₈×C₅₃	—	2	1	no	—
168	C₂×C₈×C₅₃	—	2	1	no	—
169	C₂×C₈×C₅₃	—	2	1	no	—
170	C₂×C₈×C₅₃	—	8	1	no	—
171	C₂×C₈×C₅₃	—	8	1	no	—
172	C₂×C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
173	C₂×C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
174	C₂×C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
175	C₂×C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
176	C₂×C₈×C₅₃	C₂×D₄	1	1	yes	characteristic
177	C₂×C₈×C₅₃	C₂×D₄	1	1	yes	characteristic
178	C₂×C₈×C₅₃	C₂×D₄	1	1	yes	characteristic
179	C₂×C₈×C₅₃	C₂×D₄	1	1	yes	characteristic
180	C₄⋊C₄×C₅₃	—	2	1	no	—
181	C₄⋊C₄×C₅₃	—	2	1	no	—
182	C₄⋊C₄×C₅₃	—	2	1	no	—
183	C₄⋊C₄×C₅₃	—	2	1	no	—
184	C₄⋊C₄×C₅₃	—	2	1	no	—
185	C₄⋊C₄×C₅₃	—	2	1	no	—
186	C₄⋊C₄×C₅₃	—	2	1	no	—
187	C₄⋊C₄×C₅₃	—	2	1	no	—
188	C₄⋊C₄×C₅₃	—	2	1	no	—
189	C₄⋊C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
190	C₄⋊C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
191	C₄⋊C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
192	C₄⋊C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
193	C₂²⋊C₄×C₅₃	—	2	1	no	—
194	C₂²⋊C₄×C₅₃	—	2	1	no	—
195	C₂²⋊C₄×C₅₃	—	2	1	no	—
196	C₂²⋊C₄×C₅₃	—	2	1	no	—
197	C₂²⋊C₄×C₅₃	—	2	1	no	—
198	C₄²×C₅₃	—	2	1	no	—
199	C₄²×C₅₃	—	2	1	no	—
200	C₄²×C₅₃	—	2	1	no	—
201	C₄²×C₅₃	—	2	1	no	—
202	C₄²×C₅₃	—	2	1	no	—
203	C₄²×C₅₃	C₂²×C₄	1	1	yes	characteristic
204	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
205	C₄²×C₅₃	#41	1	1	yes	characteristic
206	C₄²×C₅₃	C₂⁴	1	1	yes	characteristic
207	C₂³×C₅₃	—	2	1	no	—
208	C₂³×C₅₃	C₄×D₄	1	1	yes	characteristic
209	Q₈×C₅₃	—	2	1	no	—
210	Q₈×C₅₃	—	2	1	no	—
211	Q₈×C₅₃	—	2	1	no	—
212	Q₈×C₅₃	—	2	1	no	—
213	Q₈×C₅₃	—	2	1	no	—
214	Q₈×C₅₃	—	4	2	no	—
215	Q₈×C₅₃	#131	1	2	yes	—
216	D₄×C₅₃	—	2	1	no	—
217	D₄×C₅₃	—	2	1	no	—
218	D₄×C₅₃	—	4	2	no	—
219	C₂×C₄×C₅₃	—	2	1	no	—
220	C₂×C₄×C₅₃	—	2	2	no	—
221	C₂×C₄×C₅₃	—	2	1	no	—
222	C₂×C₄×C₅₃	—	2	1	no	—
223	C₂×C₄×C₅₃	—	2	1	no	—
224	C₂×C₄×C₅₃	—	2	1	no	—
225	C₂×C₄×C₅₃	—	2	1	no	—
226	C₂×C₄×C₅₃	—	2	1	no	—
227	C₂×C₄×C₅₃	—	2	1	no	—
228	C₂×C₄×C₅₃	—	2	1	no	—
229	C₂×C₄×C₅₃	—	2	1	no	—
230	C₂×C₄×C₅₃	—	2	1	no	—
231	C₂×C₄×C₅₃	—	4	2	no	—
232	C₂×C₄×C₅₃	—	4	2	no	—
233	C₂×C₄×C₅₃	—	4	1	no	—
234	C₂×C₄×C₅₃	—	4	1	no	—
235	C₂×C₄×C₅₃	C₄×D₄	1	1	yes	characteristic
236	C₂×C₄×C₅₃	C₄×D₄	1	1	yes	characteristic
237	C₂×C₄×C₅₃	C₄×D₄	1	1	yes	characteristic
238	C₂×C₄×C₅₃	C₂³×C₄	1	1	yes	characteristic
239	C₂×C₄×C₅₃	C₂²×D₄	1	1	yes	characteristic
240	C₂×C₄×C₅₃	C₂×#41	1	1	yes	characteristic
241	C₈×C₅₃	—	2	2	no	—
242	C₈×C₅₃	—	2	2	no	—
243	C₈×C₅₃	—	2	1	no	—
244	C₈×C₅₃	—	2	1	no	—
245	C₈×C₅₃	—	2	1	no	—
246	C₈×C₅₃	—	2	1	no	—
247	C₈×C₅₃	—	2	1	no	—
248	C₈×C₅₃	—	2	1	no	—
249	C₈×C₅₃	—	2	1	no	—
250	C₈×C₅₃	—	2	1	no	—
251	C₈×C₅₃	—	4	1	no	—
252	C₈×C₅₃	—	4	1	no	—
253	C₈×C₅₃	—	4	1	no	—
254	C₈×C₅₃	—	4	1	no	—
255	#18017	C₅₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
256	C₂²×C₅₃	—	2	1	no	—
257	C₂²×C₅₃	—	2	1	no	—
258	C₂²×C₅₃	—	4	1	no	—
259	C₂²×C₅₃	—	4	1	no	—
260	C₂²×C₅₃	—	4	1	no	—
261	C₂²×C₅₃	#443	1	2	yes	—
262	C₂²×C₅₃	C₂×C₄×D₄	1	1	yes	characteristic
263	C₄×C₅₃	—	2	1	no	—
264	C₄×C₅₃	—	2	2	no	—
265	C₄×C₅₃	—	2	1	no	—
266	C₄×C₅₃	—	2	1	no	—
267	C₄×C₅₃	—	2	1	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	—	4	1	no	—
273	C₄×C₅₃	—	4	1	no	—
274	C₄×C₅₃	C₂×#131	1	1	yes	characteristic
275	C₄×C₅₃	#568	1	1	yes	characteristic
276	C₄×C₅₃	#577	1	1	yes	characteristic
277	C₄×C₅₃	#578	1	1	yes	characteristic
278	#3108	C₂×C₅₃	1	1	yes	characteristic
279	#2994	C₂×C₅₃	1	1	yes	characteristic
280	#2984	C₂×C₅₃	1	1	yes	characteristic, complemented
281	#1591	C₂×C₅₃	1	1	yes	characteristic, complemented
282	#1590	C₂×C₅₃	1	1	yes	characteristic, complemented
283	#1588	C₂×C₅₃	1	1	yes	characteristic, complemented
284	#1587	C₂×C₅₃	1	1	yes	characteristic, complemented
285	#1587	C₂×C₅₃	1	1	yes	characteristic, complemented
286	#1585	C₂×C₅₃	1	1	yes	characteristic, complemented
287	C₈×Q₁₆	C₂×C₅₃	1	1	yes	characteristic, complemented
288	C₈×SD₁₆	C₂×C₅₃	1	1	yes	characteristic, complemented
289	#1573	C₂×C₅₃	1	1	yes	characteristic, complemented
290	#1494	C₂×C₅₃	1	1	yes	characteristic
291	C₂×Q₈⋊C₈	C₂×C₅₃	1	1	yes	characteristic, complemented
292	#1450	C₂×C₅₃	1	1	yes	characteristic, complemented
293	C₂×C₅₃	—	2	2	no	—
294	C₂×C₅₃	—	8	1	no	—
295	C₂×C₅₃	#2933	1	1	yes	characteristic
296	C₂×C₅₃	#2947	1	1	yes	characteristic
297	C₂×C₅₃	C₂×#443	1	1	yes	characteristic
298	#529	C₂²×C₅₃	1	1	yes	characteristic
299	C₂×C₄×Q₈	C₂²×C₅₃	1	1	yes	characteristic
300	#483	C₄×C₅₃	1	1	yes	characteristic
301	#480	C₄×C₅₃	1	1	yes	characteristic
302	#479	C₄×C₅₃	1	1	yes	characteristic
303	#479	C₄×C₅₃	1	1	yes	characteristic
304	#464	C₄×C₅₃	1	1	yes	characteristic
305	#464	C₄×C₅₃	1	1	yes	characteristic
306	#451	C₄×C₅₃	1	1	yes	characteristic
307	#448	C₄×C₅₃	1	1	yes	characteristic
308	#446	—	2	1	no	—
309	#446	—	2	1	no	—
310	#446	C₂²×C₅₃	1	1	yes	characteristic
311	#446	C₂²×C₅₃	1	1	yes	characteristic
312	#446	C₂²×C₅₃	1	1	yes	characteristic
313	C₈×Q₈	C₂²×C₅₃	1	1	yes	characteristic
314	C₈×Q₈	C₂²×C₅₃	1	1	yes	characteristic
315	C₈×Q₈	C₂²×C₅₃	1	1	yes	characteristic
316	C₄×Q₁₆	C₂²×C₅₃	1	1	yes	characteristic
317	C₄×Q₁₆	C₂²×C₅₃	1	1	yes	characteristic
318	C₄×SD₁₆	C₂²×C₅₃	1	1	yes	characteristic
319	C₄×SD₁₆	C₂²×C₅₃	1	1	yes	characteristic
320	#436	C₂²×C₅₃	1	1	yes	characteristic
321	#435	—	2	1	no	—
322	C₈×D₄	—	2	1	no	—
323	C₈×D₄	C₂²×C₅₃	1	1	yes	characteristic
324	#433	—	2	1	no	—
325	#432	—	2	1	no	—
326	#432	C₂²×C₅₃	1	1	yes	characteristic
327	#431	—	2	1	no	—
328	#431	C₂²×C₅₃	1	1	yes	characteristic
329	#431	C₂²×C₅₃	1	1	yes	characteristic
330	#424	—	2	1	no	—
331	C₂×C₄⋊C₈	C₂²×C₅₃	1	1	yes	characteristic
332	C₂×C₄×C₈	C₂²×C₅₃	1	1	yes	characteristic
333	#335	C₂²×C₅₃	1	1	yes	characteristic
334	#335	C₂²×C₅₃	1	1	yes	characteristic
335	#334	C₂²×C₅₃	1	1	yes	characteristic
336	#334	C₂²×C₅₃	1	1	yes	characteristic
337	Q₈⋊C₈	C₂²×C₅₃	1	1	yes	characteristic
338	Q₈⋊C₈	C₂²×C₅₃	1	1	yes	characteristic
339	Q₈⋊C₈	C₂²×C₅₃	1	1	yes	characteristic
340	Q₈⋊C₈	C₂²×C₅₃	1	1	yes	characteristic
341	Q₈⋊C₈	C₂²×C₅₃	1	1	yes	characteristic
342	Q₈⋊C₈	C₂²×C₅₃	1	1	yes	characteristic
343	D₄⋊C₈	C₂²×C₅₃	1	1	yes	characteristic
344	D₄⋊C₈	C₂²×C₅₃	1	1	yes	characteristic
345	#322	C₂²×C₅₃	1	1	yes	characteristic
346	#322	C₂²×C₅₃	1	1	yes	characteristic
347	C₈²	C₂²×C₅₃	1	1	yes	characteristic
348	C₈²	C₂²×C₅₃	1	1	yes	characteristic
349	C₅₃	#18017	1	1	yes	characteristic, direct product factor, Sylow
350	C₂²×Q₈	C₂×C₄×C₅₃	1	1	yes	characteristic
351	C₂×Q₁₆	C₂×C₄×C₅₃	1	1	yes	characteristic
352	C₂×Q₁₆	C₂×C₄×C₅₃	1	1	yes	characteristic
353	C₂×SD₁₆	C₂×C₄×C₅₃	1	1	yes	characteristic
354	C₂×SD₁₆	C₂×C₄×C₅₃	1	1	yes	characteristic
355	C₂×M₁₆	—	2	1	no	—
356	C₂²×C₈	—	2	1	no	—
357	C₂²×C₈	—	2	1	no	—
358	C₂²×C₈	—	2	1	no	—
359	#126	—	2	1	no	—
360	#125	C₂×C₄×C₅₃	1	1	yes	characteristic
361	#124	C₂×C₄×C₅₃	1	1	yes	characteristic
362	#123	—	2	1	no	—
363	C₂²⋊Q₈	C₂×C₄×C₅₃	1	1	yes	characteristic
364	#121	C₂×C₄×C₅₃	1	1	yes	characteristic
365	C₄×Q₈	—	2	1	no	—
366	C₄×Q₈	—	2	1	no	—
367	C₄×Q₈	C₂×C₄×C₅₃	1	1	yes	characteristic
368	C₄×Q₈	C₂×C₄×C₅₃	1	1	yes	characteristic
369	C₄×Q₈	C₂³×C₅₃	1	1	yes	characteristic
370	C₄×Q₈	C₂³×C₅₃	1	1	yes	characteristic
371	C₄×Q₈	C₂³×C₅₃	1	1	yes	characteristic
372	C₄×D₄	—	2	1	no	—
373	C₄×D₄	C₂³×C₅₃	1	1	yes	characteristic
374	#117	—	2	1	no	—
375	C₂×C₄⋊C₄	—	2	1	no	—
376	C₂×C₄⋊C₄	C₂×C₄×C₅₃	1	1	yes	characteristic
377	C₂×C₄²	—	2	1	no	—
378	C₂×C₄²	C₂³×C₅₃	1	1	yes	characteristic
379	#107	C₂×C₄×C₅₃	1	1	yes	characteristic
380	#107	C₂×C₄×C₅₃	1	1	yes	characteristic
381	#106	C₂×C₄×C₅₃	1	1	yes	characteristic
382	#106	C₂×C₄×C₅₃	1	1	yes	characteristic
383	C₄⋊C₈	—	2	1	no	—
384	C₄⋊C₈	—	2	1	no	—
385	C₄⋊C₈	—	2	1	no	—
386	C₄⋊C₈	—	2	1	no	—
387	C₄⋊C₈	—	2	1	no	—
388	C₄⋊C₈	—	2	1	no	—
389	C₄⋊C₈	—	2	1	no	—
390	C₄⋊C₈	C₂×C₄×C₅₃	1	1	yes	characteristic
391	C₄⋊C₈	C₂×C₄×C₅₃	1	1	yes	characteristic
392	C₄⋊C₈	D₄×C₅₃	1	1	yes	characteristic
393	C₄⋊C₈	D₄×C₅₃	1	1	yes	characteristic
394	C₄⋊C₈	C₂³×C₅₃	1	1	yes	characteristic
395	C₄⋊C₈	C₂³×C₅₃	1	1	yes	characteristic
396	C₄⋊C₈	C₂³×C₅₃	1	1	yes	characteristic
397	C₄⋊C₈	C₂³×C₅₃	1	1	yes	characteristic
398	#103	C₂×C₄×C₅₃	1	1	yes	characteristic
399	#103	C₂×C₄×C₅₃	1	1	yes	characteristic
400	#103	C₂×C₄×C₅₃	1	1	yes	characteristic
401	#103	C₂×C₄×C₅₃	1	1	yes	characteristic
402	#103	C₂×C₄×C₅₃	1	1	yes	characteristic
403	#103	C₂×C₄×C₅₃	1	1	yes	characteristic
404	#102	C₂×C₄×C₅₃	1	1	yes	characteristic
405	#102	C₂×C₄×C₅₃	1	1	yes	characteristic
406	C₂²⋊C₈	—	2	1	no	—
407	C₂²⋊C₈	—	2	1	no	—
408	C₂²⋊C₈	—	2	1	no	—
409	C₂²⋊C₈	C₂×C₄×C₅₃	1	1	yes	characteristic
410	C₂²⋊C₈	C₂×C₄×C₅₃	1	1	yes	characteristic
411	C₂²⋊C₈	D₄×C₅₃	1	1	yes	characteristic
412	C₂²⋊C₈	D₄×C₅₃	1	1	yes	characteristic
413	#97	—	2	1	no	—
414	#97	—	2	1	no	—
415	#97	—	2	1	no	—
416	#97	—	2	1	no	—
417	C₄×C₈	—	2	1	no	—
418	C₄×C₈	—	2	1	no	—
419	C₄×C₈	—	2	1	no	—
420	C₄×C₈	—	2	1	no	—
421	C₄×C₈	C₂³×C₅₃	1	1	yes	characteristic
422	C₄×C₈	C₂³×C₅₃	1	1	yes	characteristic
423	C₄×C₈	C₂³×C₅₃	1	1	yes	characteristic
424	C₄×C₈	C₂³×C₅₃	1	1	yes	characteristic
425	C₄×C₈	C₂³×C₅₃	1	1	yes	characteristic
426	C₄×C₈	C₂³×C₅₃	1	1	yes	characteristic
427	C₂×Q₈	—	2	1	no	—
428	C₂×Q₈	—	2	2	no	—
429	C₂×Q₈	—	4	1	no	—
430	C₂×Q₈	C₂²×C₄×C₅₃	1	1	yes	characteristic
431	C₂×Q₈	C₂²×C₄×C₅₃	1	1	yes	characteristic
432	C₂×Q₈	C₂²×C₄×C₅₃	1	1	yes	characteristic
433	C₂×D₄	—	2	1	no	—
434	C₂×D₄	C₂²×C₄×C₅₃	1	1	yes	characteristic
435	C₂²×C₄	—	2	1	no	—
436	C₂²×C₄	—	2	1	no	—
437	C₂²×C₄	—	2	1	no	—
438	C₂²×C₄	C₂²×C₄×C₅₃	1	1	yes	characteristic
439	C₂²×C₄	C₂×D₄×C₅₃	1	1	yes	characteristic
440	C₂²×C₄	#41×C₅₃	1	1	yes	characteristic
441	Q₁₆	—	2	2	no	—
442	Q₁₆	—	4	1	no	—
443	SD₁₆	—	2	2	no	—
444	SD₁₆	—	4	1	no	—
445	M₁₆	—	8	1	no	—
446	C₂×C₈	—	2	4	no	—
447	C₂×C₈	—	2	1	no	—
448	C₂×C₈	—	2	1	no	—
449	C₂×C₈	—	2	1	no	—
450	C₂×C₈	—	2	1	no	—
451	C₂×C₈	—	2	1	no	—
452	C₂×C₈	—	2	1	no	—
453	C₂×C₈	—	8	1	no	—
454	C₂×C₈	—	8	1	no	—
455	C₂×C₈	C₂²×C₄×C₅₃	1	1	yes	characteristic
456	C₂×C₈	C₂²×C₄×C₅₃	1	1	yes	characteristic
457	C₂×C₈	C₂²×C₄×C₅₃	1	1	yes	characteristic
458	C₂×C₈	C₂²×C₄×C₅₃	1	1	yes	characteristic
459	C₂×C₈	C₂×D₄×C₅₃	1	1	yes	characteristic
460	C₂×C₈	C₂×D₄×C₅₃	1	1	yes	characteristic
461	C₂×C₈	C₂×D₄×C₅₃	1	1	yes	characteristic
462	C₂×C₈	C₂×D₄×C₅₃	1	1	yes	characteristic
463	C₄⋊C₄	—	2	1	no	—
464	C₄⋊C₄	—	2	1	no	—
465	C₄⋊C₄	—	2	1	no	—
466	C₄⋊C₄	—	2	1	no	—
467	C₄⋊C₄	—	2	1	no	—
468	C₄⋊C₄	—	2	1	no	—
469	C₄⋊C₄	—	2	1	no	—
470	C₄⋊C₄	—	2	1	no	—
471	C₄⋊C₄	—	2	1	no	—
472	C₄⋊C₄	C₂²×C₄×C₅₃	1	1	yes	characteristic
473	C₄⋊C₄	C₂²×C₄×C₅₃	1	1	yes	characteristic
474	C₄⋊C₄	C₂²×C₄×C₅₃	1	1	yes	characteristic
475	C₄⋊C₄	C₂²×C₄×C₅₃	1	1	yes	characteristic
476	C₂²⋊C₄	—	2	1	no	—
477	C₂²⋊C₄	—	2	1	no	—
478	C₂²⋊C₄	—	2	1	no	—
479	C₂²⋊C₄	—	2	1	no	—
480	C₂²⋊C₄	—	2	1	no	—
481	C₄²	—	2	1	no	—
482	C₄²	—	2	1	no	—
483	C₄²	—	2	1	no	—
484	C₄²	—	2	1	no	—
485	C₄²	—	2	1	no	—
486	C₄²	C₂²×C₄×C₅₃	1	1	yes	characteristic
487	C₄²	C₂×D₄×C₅₃	1	1	yes	characteristic
488	C₄²	#41×C₅₃	1	1	yes	characteristic
489	C₄²	C₂⁴×C₅₃	1	1	yes	characteristic
490	C₂³	—	2	1	no	—
491	C₂³	C₄×D₄×C₅₃	1	1	yes	characteristic
492	Q₈	—	2	1	no	—
493	Q₈	—	2	1	no	—
494	Q₈	—	2	1	no	—
495	Q₈	—	2	1	no	—
496	Q₈	—	2	1	no	—
497	Q₈	—	4	2	no	—
498	Q₈	#131×C₅₃	1	2	yes	—
499	D₄	—	2	1	no	—
500	D₄	—	2	1	no	—
501	D₄	—	4	2	no	—
502	C₂×C₄	—	2	1	no	—
503	C₂×C₄	—	2	2	no	—
504	C₂×C₄	—	2	1	no	—
505	C₂×C₄	—	2	1	no	—
506	C₂×C₄	—	2	1	no	—
507	C₂×C₄	—	2	1	no	—
508	C₂×C₄	—	2	1	no	—
509	C₂×C₄	—	2	1	no	—
510	C₂×C₄	—	2	1	no	—
511	C₂×C₄	—	2	1	no	—
512	C₂×C₄	—	2	1	no	—
513	C₂×C₄	—	2	1	no	—
514	C₂×C₄	—	4	2	no	—
515	C₂×C₄	—	4	2	no	—
516	C₂×C₄	—	4	1	no	—
517	C₂×C₄	—	4	1	no	—
518	C₂×C₄	C₄×D₄×C₅₃	1	1	yes	characteristic
519	C₂×C₄	C₄×D₄×C₅₃	1	1	yes	characteristic
520	C₂×C₄	C₄×D₄×C₅₃	1	1	yes	characteristic
521	C₂×C₄	C₂³×C₄×C₅₃	1	1	yes	characteristic
522	C₂×C₄	C₂²×D₄×C₅₃	1	1	yes	characteristic
523	C₂×C₄	C₂×#41×C₅₃	1	1	yes	characteristic
524	C₈	—	2	2	no	—
525	C₈	—	2	2	no	—
526	C₈	—	2	1	no	—
527	C₈	—	2	1	no	—
528	C₈	—	2	1	no	—
529	C₈	—	2	1	no	—
530	C₈	—	2	1	no	—
531	C₈	—	2	1	no	—
532	C₈	—	2	1	no	—
533	C₈	—	2	1	no	—
534	C₈	—	4	1	no	—
535	C₈	—	4	1	no	—
536	C₈	—	4	1	no	—
537	C₈	—	4	1	no	—
538	C₂²	—	2	1	no	—
539	C₂²	—	2	1	no	—
540	C₂²	—	4	1	no	—
541	C₂²	—	4	1	no	—
542	C₂²	—	4	1	no	—
543	C₂²	C₅₃×#443	1	2	yes	—
544	C₂²	C₂×C₄×D₄×C₅₃	1	1	yes	characteristic
545	C₄	—	2	1	no	—
546	C₄	—	2	1	no	—
547	C₄	—	2	1	no	—
548	C₄	—	2	1	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	—	4	1	no	—
555	C₄	—	4	1	no	—
556	C₄	C₂×#131×C₅₃	1	1	yes	characteristic
557	C₄	C₅₃×#568	1	1	yes	characteristic
558	C₄	C₅₃×#577	1	1	yes	characteristic
559	C₄	C₅₃×#578	1	1	yes	characteristic
560	C₂	—	2	2	no	—
561	C₂	—	8	1	no	—
562	C₂	C₅₃×#2933	1	1	yes	characteristic
563	C₂	C₅₃×#2947	1	1	yes	characteristic
564	C₂	C₂×C₅₃×#443	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄×C₅₃	C₂²×D₄
commutator	C₂×C₄	C₂³×C₄×C₅₃
Frattini, cosocle	C₄²	C₂⁴×C₅₃
socle	C₂²×C₅₃	C₂×C₄×D₄
prime radical	#121×C₅₃	C₂×C₄
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₅₃ ◁ C₅₃×#431 ◁ G

Complements

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

N	H ≅ G/N	type
3. C₅₃×#2984	C₂	semidirect
4. C₅₃×#1591	C₂	semidirect
5. C₅₃×#1590	C₂	semidirect
6. C₅₃×#1588	C₂	semidirect
7. C₅₃×#1587	C₂	semidirect
8. C₅₃×#1587	C₂	semidirect
9. C₅₃×#1585	C₂	semidirect
10. C₈×Q₁₆×C₅₃	C₂	semidirect
11. C₈×SD₁₆×C₅₃	C₂	semidirect
12. C₅₃×#1573	C₂	semidirect
14. C₂×C₅₃×Q₈⋊C₈	C₂	semidirect
15. C₅₃×#1450	C₂	semidirect
255. #18017	C₅₃	direct (N × H)
280. #2984	C₂×C₅₃	semidirect
281. #1591	C₂×C₅₃	semidirect
282. #1590	C₂×C₅₃	semidirect
283. #1588	C₂×C₅₃	semidirect
284. #1587	C₂×C₅₃	semidirect
285. #1587	C₂×C₅₃	semidirect
286. #1585	C₂×C₅₃	semidirect
287. C₈×Q₁₆	C₂×C₅₃	semidirect
288. C₈×SD₁₆	C₂×C₅₃	semidirect
289. #1573	C₂×C₅₃	semidirect
291. C₂×Q₈⋊C₈	C₂×C₅₃	semidirect
292. #1450	C₂×C₅₃	semidirect
349. C₅₃	#18017	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×#3108	2
C₅₃×#2994	2
C₅₃×#2984	2
C₅₃×#1591	2
C₅₃×#1590	2
C₅₃×#1588	2
C₅₃×#1587	2
C₅₃×#1585	2
C₈×Q₁₆×C₅₃	2
C₈×SD₁₆×C₅₃	2
C₅₃×#1573	2
C₅₃×#1494	2
C₂×C₅₃×Q₈⋊C₈	2
C₅₃×#1450	2
#18017	53

Subgroups — C53×#18017 #27225529

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅₃×#18017 #27225529