Subgroups — C₅₃×#18053 #27225565

Order 13568.

624 conjugacy classes of subgroups.

The subgroup lattice has 578 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₅₃×#2991	C₂	1	1	yes	maximal, characteristic
2	C₅₃×#2984	C₂	1	1	yes	maximal, characteristic, complemented
3	C₄×#137×C₅₃	C₂	1	1	yes	maximal, characteristic
4	C₅₃×#1592	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×#1584	C₂	1	1	yes	maximal, characteristic, complemented
9	C₅₃×#1583	C₂	1	1	yes	maximal, characteristic, complemented
10	C₅₃×#1579	C₂	1	1	yes	maximal, characteristic, complemented
11	C₈×Q₁₆×C₅₃	C₂	1	1	yes	maximal, characteristic, complemented
12	C₅₃×#1569	C₂	1	1	yes	maximal, characteristic, complemented
13	C₅₃×#1488	C₂	1	1	yes	maximal, characteristic
14	C₅₃×#1487	C₂	1	1	yes	maximal, characteristic, complemented
15	C₅₃×#1453	C₂	1	1	yes	maximal, characteristic, complemented
16	C₂×#137×C₅₃	C₄	1	1	yes	characteristic
17	C₄×#41×C₅₃	C₂²	1	1	yes	characteristic
18	C₂×C₄×Q₈×C₅₃	C₂²	1	1	yes	characteristic
19	C₅₃×#446	—	2	1	no	—
20	C₅₃×#446	—	2	1	no	—
21	C₅₃×#446	C₂²	1	1	yes	characteristic
22	C₅₃×#446	C₂²	1	1	yes	characteristic
23	C₈×Q₈×C₅₃	C₂²	1	1	yes	characteristic
24	C₈×Q₈×C₅₃	C₂²	1	1	yes	characteristic
25	C₈×Q₈×C₅₃	C₂²	1	1	yes	characteristic
26	C₈×Q₈×C₅₃	C₂²	1	1	yes	characteristic
27	C₅₃×#441	C₄	1	1	yes	characteristic
28	C₅₃×#441	C₄	1	1	yes	characteristic
29	C₅₃×#440	C₄	1	1	yes	characteristic
30	C₅₃×#440	C₄	1	1	yes	characteristic
31	C₄×Q₁₆×C₅₃	C₂²	1	1	yes	characteristic
32	C₄×Q₁₆×C₅₃	C₂²	1	1	yes	characteristic
33	C₄×SD₁₆×C₅₃	C₂²	1	1	yes	characteristic
34	C₄×SD₁₆×C₅₃	C₂²	1	1	yes	characteristic
35	C₅₃×#436	—	2	1	no	—
36	C₅₃×#436	—	2	1	no	—
37	C₅₃×#436	C₂²	1	1	yes	characteristic
38	C₅₃×#436	C₂²	1	1	yes	characteristic
39	C₅₃×#432	—	2	1	no	—
40	C₅₃×#432	C₂²	1	1	yes	characteristic
41	C₅₃×#431	—	2	1	no	—
42	C₅₃×#431	—	2	1	no	—
43	C₅₃×#431	C₂²	1	1	yes	characteristic
44	C₅₃×#431	C₂²	1	1	yes	characteristic
45	C₅₃×#428	C₄	1	1	yes	characteristic
46	C₅₃×#423	—	2	1	no	—
47	C₅₃×#423	C₂²	1	1	yes	characteristic
48	C₅₃×#419	C₄	1	1	yes	characteristic
49	C₅₃×#417	C₄	1	1	yes	characteristic
50	C₄×M₁₆×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	C₅₃×#322	C₂²	1	1	yes	characteristic
66	C₈²×C₅₃	C₂²	1	1	yes	characteristic
67	C₂×#41×C₅₃	C₂×C₄	1	1	yes	characteristic
68	C₂²×Q₈×C₅₃	C₂×C₄	1	1	yes	characteristic
69	#137×C₅₃	—	2	4	no	—
70	C₂×Q₁₆×C₅₃	C₂×C₄	1	1	yes	characteristic
71	C₂×Q₁₆×C₅₃	C₂×C₄	1	1	yes	characteristic
72	C₂×SD₁₆×C₅₃	C₂×C₄	1	1	yes	characteristic
73	C₂×SD₁₆×C₅₃	C₂×C₄	1	1	yes	characteristic
74	C₂×M₁₆×C₅₃	—	2	1	no	—
75	C₂×M₁₆×C₅₃	—	2	1	no	—
76	C₂×M₁₆×C₅₃	—	2	1	no	—
77	C₂×M₁₆×C₅₃	C₂×C₄	1	1	yes	characteristic
78	C₂×M₁₆×C₅₃	C₂×C₄	1	1	yes	characteristic
79	C₄×Q₈×C₅₃	—	2	1	no	—
80	C₄×Q₈×C₅₃	—	2	1	no	—
81	C₄×Q₈×C₅₃	C₂×C₄	1	1	yes	characteristic
82	C₄×Q₈×C₅₃	C₂×C₄	1	1	yes	characteristic
83	C₄×Q₈×C₅₃	C₂×C₄	1	1	yes	characteristic
84	C₄×Q₈×C₅₃	C₂³	1	1	yes	characteristic
85	C₄×Q₈×C₅₃	C₂³	1	1	yes	characteristic
86	C₄×Q₈×C₅₃	C₂³	1	1	yes	characteristic
87	C₄×D₄×C₅₃	—	2	1	no	—
88	C₄×D₄×C₅₃	—	2	1	no	—
89	C₄×D₄×C₅₃	C₂×C₄	1	1	yes	characteristic
90	C₄×D₄×C₅₃	C₂³	1	1	yes	characteristic
91	#117×C₅₃	—	2	1	no	—
92	#117×C₅₃	C₂×C₄	1	1	yes	characteristic
93	C₂×C₄⋊C₄×C₅₃	—	2	1	no	—
94	C₂×C₄⋊C₄×C₅₃	C₂×C₄	1	1	yes	characteristic
95	C₂×C₄²×C₅₃	—	2	1	no	—
96	C₂×C₄²×C₅₃	—	2	1	no	—
97	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
98	#107×C₅₃	C₂×C₄	1	1	yes	characteristic
99	#107×C₅₃	C₂×C₄	1	1	yes	characteristic
100	#106×C₅₃	C₂×C₄	1	1	yes	characteristic
101	#106×C₅₃	C₂×C₄	1	1	yes	characteristic
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	—	2	1	no	—
108	C₄⋊C₈×C₅₃	—	2	1	no	—
109	C₄⋊C₈×C₅₃	—	2	1	no	—
110	C₄⋊C₈×C₅₃	D₄	1	1	yes	characteristic
111	C₄⋊C₈×C₅₃	D₄	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	C₄⋊C₈×C₅₃	C₂³	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	#103×C₅₃	C₂×C₄	1	1	yes	characteristic
122	#102×C₅₃	C₂×C₄	1	1	yes	characteristic
123	#102×C₅₃	C₂×C₄	1	1	yes	characteristic
124	C₂²⋊C₈×C₅₃	—	2	1	no	—
125	C₂²⋊C₈×C₅₃	—	2	1	no	—
126	C₂²⋊C₈×C₅₃	—	2	1	no	—
127	C₂²⋊C₈×C₅₃	—	2	1	no	—
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	C₂×C₄	1	1	yes	characteristic
134	#97×C₅₃	C₂×C₄	1	1	yes	characteristic
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	—	2	1	no	—
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	C₄×C₈×C₅₃	C₂³	1	1	yes	characteristic
145	#41×C₅₃	—	2	2	no	—
146	#41×C₅₃	—	4	1	no	—
147	C₂×Q₈×C₅₃	—	2	1	no	—
148	C₂×Q₈×C₅₃	—	2	2	no	—
149	C₂×Q₈×C₅₃	—	4	1	no	—
150	C₂×Q₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
151	C₂×Q₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
152	C₂×Q₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
153	C₂×D₄×C₅₃	—	2	1	no	—
154	C₂×D₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
155	C₂²×C₄×C₅₃	—	2	1	no	—
156	C₂²×C₄×C₅₃	—	2	1	no	—
157	C₂²×C₄×C₅₃	—	2	1	no	—
158	C₂²×C₄×C₅₃	—	2	1	no	—
159	C₂²×C₄×C₅₃	—	2	1	no	—
160	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
161	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
162	C₂²×C₄×C₅₃	#41	1	1	yes	characteristic
163	Q₁₆×C₅₃	—	2	2	no	—
164	Q₁₆×C₅₃	M₁₆	1	4	yes	complemented
165	SD₁₆×C₅₃	—	2	2	no	—
166	SD₁₆×C₅₃	—	2	2	no	—
167	M₁₆×C₅₃	—	2	2	no	—
168	M₁₆×C₅₃	—	4	1	no	—
169	M₁₆×C₅₃	—	8	1	no	—
170	M₁₆×C₅₃	—	8	1	no	—
171	M₁₆×C₅₃	—	8	1	no	—
172	C₂×C₈×C₅₃	—	2	1	no	—
173	C₂×C₈×C₅₃	—	2	1	no	—
174	C₂×C₈×C₅₃	—	2	1	no	—
175	C₂×C₈×C₅₃	—	2	1	no	—
176	C₂×C₈×C₅₃	—	2	1	no	—
177	C₂×C₈×C₅₃	—	2	1	no	—
178	C₂×C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
179	C₂×C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
180	C₂×C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
181	C₂×C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
182	C₂×C₈×C₅₃	C₂×D₄	1	1	yes	characteristic
183	C₂×C₈×C₅₃	C₂×D₄	1	1	yes	characteristic
184	C₂×C₈×C₅₃	C₂×D₄	1	1	yes	characteristic
185	C₂×C₈×C₅₃	C₂×D₄	1	1	yes	characteristic
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	—	2	1	no	—
190	C₄⋊C₄×C₅₃	—	2	1	no	—
191	C₄⋊C₄×C₅₃	—	2	1	no	—
192	C₄⋊C₄×C₅₃	—	2	1	no	—
193	C₄⋊C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
194	C₄⋊C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
195	C₄⋊C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
196	C₄⋊C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
197	C₂²⋊C₄×C₅₃	—	2	1	no	—
198	C₂²⋊C₄×C₅₃	—	2	1	no	—
199	C₂²⋊C₄×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	—	2	1	no	—
204	C₄²×C₅₃	—	2	1	no	—
205	C₄²×C₅₃	—	2	1	no	—
206	C₄²×C₅₃	—	2	1	no	—
207	C₄²×C₅₃	C₂²×C₄	1	1	yes	characteristic
208	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
209	C₄²×C₅₃	#41	1	1	yes	characteristic
210	C₄²×C₅₃	C₂⁴	1	1	yes	characteristic
211	C₂³×C₅₃	—	2	1	no	—
212	C₂³×C₅₃	C₄×D₄	1	1	yes	characteristic
213	Q₈×C₅₃	—	2	1	no	—
214	Q₈×C₅₃	—	2	1	no	—
215	Q₈×C₅₃	—	2	1	no	—
216	Q₈×C₅₃	—	2	1	no	—
217	Q₈×C₅₃	—	4	2	no	—
218	Q₈×C₅₃	C₂×M₁₆	1	2	yes	—
219	Q₈×C₅₃	C₂×M₁₆	1	2	yes	—
220	D₄×C₅₃	—	2	1	no	—
221	D₄×C₅₃	—	2	1	no	—
222	D₄×C₅₃	—	4	2	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	—	2	1	no	—
232	C₂×C₄×C₅₃	—	2	1	no	—
233	C₂×C₄×C₅₃	—	2	1	no	—
234	C₂×C₄×C₅₃	—	4	1	no	—
235	C₂×C₄×C₅₃	—	4	2	no	—
236	C₂×C₄×C₅₃	—	4	2	no	—
237	C₂×C₄×C₅₃	—	4	2	no	—
238	C₂×C₄×C₅₃	—	4	2	no	—
239	C₂×C₄×C₅₃	—	4	1	no	—
240	C₂×C₄×C₅₃	—	4	1	no	—
241	C₂×C₄×C₅₃	C₄×D₄	1	1	yes	characteristic
242	C₂×C₄×C₅₃	C₄×D₄	1	1	yes	characteristic
243	C₂×C₄×C₅₃	C₄×D₄	1	1	yes	characteristic
244	C₂×C₄×C₅₃	C₂³×C₄	1	1	yes	characteristic
245	C₂×C₄×C₅₃	C₂²×D₄	1	1	yes	characteristic
246	C₂×C₄×C₅₃	C₂×#41	1	1	yes	characteristic
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	—	2	1	no	—
252	C₈×C₅₃	—	2	1	no	—
253	C₈×C₅₃	—	2	1	no	—
254	C₈×C₅₃	—	4	1	no	—
255	C₈×C₅₃	—	4	1	no	—
256	C₈×C₅₃	—	4	1	no	—
257	C₈×C₅₃	—	4	1	no	—
258	C₈×C₅₃	—	4	1	no	—
259	C₈×C₅₃	—	4	1	no	—
260	C₈×C₅₃	C₂×M₁₆	1	2	yes	—
261	#18053	C₅₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
262	C₂²×C₅₃	—	2	1	no	—
263	C₂²×C₅₃	—	2	1	no	—
264	C₂²×C₅₃	—	2	1	no	—
265	C₂²×C₅₃	—	4	1	no	—
266	C₂²×C₅₃	—	4	1	no	—
267	C₂²×C₅₃	—	4	1	no	—
268	C₂²×C₅₃	C₂×C₄×D₄	1	1	yes	characteristic
269	C₄×C₅₃	—	2	1	no	—
270	C₄×C₅₃	—	2	2	no	—
271	C₄×C₅₃	—	2	2	no	—
272	C₄×C₅₃	—	2	1	no	—
273	C₄×C₅₃	—	2	1	no	—
274	C₄×C₅₃	—	2	1	no	—
275	C₄×C₅₃	—	4	1	no	—
276	C₄×C₅₃	—	4	1	no	—
277	C₄×C₅₃	—	4	1	no	—
278	C₄×C₅₃	—	4	1	no	—
279	C₄×C₅₃	—	4	1	no	—
280	C₄×C₅₃	C₂²×M₁₆	1	1	yes	characteristic
281	C₄×C₅₃	#568	1	1	yes	characteristic
282	C₄×C₅₃	#577	1	1	yes	characteristic
283	C₄×C₅₃	#578	1	1	yes	characteristic
284	#2991	C₂×C₅₃	1	1	yes	characteristic
285	#2984	C₂×C₅₃	1	1	yes	characteristic, complemented
286	C₄×#137	C₂×C₅₃	1	1	yes	characteristic
287	#1592	C₂×C₅₃	1	1	yes	characteristic, complemented
288	#1590	C₂×C₅₃	1	1	yes	characteristic, complemented
289	#1588	C₂×C₅₃	1	1	yes	characteristic, complemented
290	#1587	C₂×C₅₃	1	1	yes	characteristic, complemented
291	#1584	C₂×C₅₃	1	1	yes	characteristic, complemented
292	#1583	C₂×C₅₃	1	1	yes	characteristic, complemented
293	#1579	C₂×C₅₃	1	1	yes	characteristic, complemented
294	C₈×Q₁₆	C₂×C₅₃	1	1	yes	characteristic, complemented
295	#1569	C₂×C₅₃	1	1	yes	characteristic, complemented
296	#1488	C₂×C₅₃	1	1	yes	characteristic
297	#1487	C₂×C₅₃	1	1	yes	characteristic, complemented
298	#1453	C₂×C₅₃	1	1	yes	characteristic, complemented
299	C₂×C₅₃	—	4	1	no	—
300	C₂×C₅₃	—	8	1	no	—
301	C₂×C₅₃	D₄×M₁₆	1	1	yes	characteristic
302	C₂×C₅₃	#2947	1	1	yes	characteristic
303	C₂×C₅₃	#2955	1	1	yes	characteristic
304	C₂×#137	C₄×C₅₃	1	1	yes	characteristic
305	C₄×#41	C₂²×C₅₃	1	1	yes	characteristic
306	C₂×C₄×Q₈	C₂²×C₅₃	1	1	yes	characteristic
307	#446	—	2	1	no	—
308	#446	—	2	1	no	—
309	#446	C₂²×C₅₃	1	1	yes	characteristic
310	#446	C₂²×C₅₃	1	1	yes	characteristic
311	C₈×Q₈	C₂²×C₅₃	1	1	yes	characteristic
312	C₈×Q₈	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	#441	C₄×C₅₃	1	1	yes	characteristic
316	#441	C₄×C₅₃	1	1	yes	characteristic
317	#440	C₄×C₅₃	1	1	yes	characteristic
318	#440	C₄×C₅₃	1	1	yes	characteristic
319	C₄×Q₁₆	C₂²×C₅₃	1	1	yes	characteristic
320	C₄×Q₁₆	C₂²×C₅₃	1	1	yes	characteristic
321	C₄×SD₁₆	C₂²×C₅₃	1	1	yes	characteristic
322	C₄×SD₁₆	C₂²×C₅₃	1	1	yes	characteristic
323	#436	—	2	1	no	—
324	#436	—	2	1	no	—
325	#436	C₂²×C₅₃	1	1	yes	characteristic
326	#436	C₂²×C₅₃	1	1	yes	characteristic
327	#432	—	2	1	no	—
328	#432	C₂²×C₅₃	1	1	yes	characteristic
329	#431	—	2	1	no	—
330	#431	—	2	1	no	—
331	#431	C₂²×C₅₃	1	1	yes	characteristic
332	#431	C₂²×C₅₃	1	1	yes	characteristic
333	#428	C₄×C₅₃	1	1	yes	characteristic
334	#423	—	2	1	no	—
335	#423	C₂²×C₅₃	1	1	yes	characteristic
336	#419	C₄×C₅₃	1	1	yes	characteristic
337	#417	C₄×C₅₃	1	1	yes	characteristic
338	C₄×M₁₆	C₂²×C₅₃	1	1	yes	characteristic
339	#335	C₂²×C₅₃	1	1	yes	characteristic
340	#335	C₂²×C₅₃	1	1	yes	characteristic
341	#334	C₂²×C₅₃	1	1	yes	characteristic
342	#334	C₂²×C₅₃	1	1	yes	characteristic
343	Q₈⋊C₈	C₂²×C₅₃	1	1	yes	characteristic
344	Q₈⋊C₈	C₂²×C₅₃	1	1	yes	characteristic
345	Q₈⋊C₈	C₂²×C₅₃	1	1	yes	characteristic
346	Q₈⋊C₈	C₂²×C₅₃	1	1	yes	characteristic
347	Q₈⋊C₈	C₂²×C₅₃	1	1	yes	characteristic
348	Q₈⋊C₈	C₂²×C₅₃	1	1	yes	characteristic
349	D₄⋊C₈	C₂²×C₅₃	1	1	yes	characteristic
350	D₄⋊C₈	C₂²×C₅₃	1	1	yes	characteristic
351	#322	C₂²×C₅₃	1	1	yes	characteristic
352	#322	C₂²×C₅₃	1	1	yes	characteristic
353	#322	C₂²×C₅₃	1	1	yes	characteristic
354	C₈²	C₂²×C₅₃	1	1	yes	characteristic
355	C₅₃	#18053	1	1	yes	characteristic, direct product factor, Sylow
356	C₂×#41	C₂×C₄×C₅₃	1	1	yes	characteristic
357	C₂²×Q₈	C₂×C₄×C₅₃	1	1	yes	characteristic
358	#137	—	2	4	no	—
359	C₂×Q₁₆	C₂×C₄×C₅₃	1	1	yes	characteristic
360	C₂×Q₁₆	C₂×C₄×C₅₃	1	1	yes	characteristic
361	C₂×SD₁₆	C₂×C₄×C₅₃	1	1	yes	characteristic
362	C₂×SD₁₆	C₂×C₄×C₅₃	1	1	yes	characteristic
363	C₂×M₁₆	—	2	1	no	—
364	C₂×M₁₆	—	2	1	no	—
365	C₂×M₁₆	—	2	1	no	—
366	C₂×M₁₆	C₂×C₄×C₅₃	1	1	yes	characteristic
367	C₂×M₁₆	C₂×C₄×C₅₃	1	1	yes	characteristic
368	C₄×Q₈	—	2	1	no	—
369	C₄×Q₈	—	2	1	no	—
370	C₄×Q₈	C₂×C₄×C₅₃	1	1	yes	characteristic
371	C₄×Q₈	C₂×C₄×C₅₃	1	1	yes	characteristic
372	C₄×Q₈	C₂×C₄×C₅₃	1	1	yes	characteristic
373	C₄×Q₈	C₂³×C₅₃	1	1	yes	characteristic
374	C₄×Q₈	C₂³×C₅₃	1	1	yes	characteristic
375	C₄×Q₈	C₂³×C₅₃	1	1	yes	characteristic
376	C₄×D₄	—	2	1	no	—
377	C₄×D₄	—	2	1	no	—
378	C₄×D₄	C₂×C₄×C₅₃	1	1	yes	characteristic
379	C₄×D₄	C₂³×C₅₃	1	1	yes	characteristic
380	#117	—	2	1	no	—
381	#117	C₂×C₄×C₅₃	1	1	yes	characteristic
382	C₂×C₄⋊C₄	—	2	1	no	—
383	C₂×C₄⋊C₄	C₂×C₄×C₅₃	1	1	yes	characteristic
384	C₂×C₄²	—	2	1	no	—
385	C₂×C₄²	—	2	1	no	—
386	C₂×C₄²	C₂³×C₅₃	1	1	yes	characteristic
387	#107	C₂×C₄×C₅₃	1	1	yes	characteristic
388	#107	C₂×C₄×C₅₃	1	1	yes	characteristic
389	#106	C₂×C₄×C₅₃	1	1	yes	characteristic
390	#106	C₂×C₄×C₅₃	1	1	yes	characteristic
391	C₄⋊C₈	—	2	1	no	—
392	C₄⋊C₈	—	2	1	no	—
393	C₄⋊C₈	—	2	1	no	—
394	C₄⋊C₈	—	2	1	no	—
395	C₄⋊C₈	—	2	1	no	—
396	C₄⋊C₈	—	2	1	no	—
397	C₄⋊C₈	—	2	1	no	—
398	C₄⋊C₈	—	2	1	no	—
399	C₄⋊C₈	D₄×C₅₃	1	1	yes	characteristic
400	C₄⋊C₈	D₄×C₅₃	1	1	yes	characteristic
401	C₄⋊C₈	C₂³×C₅₃	1	1	yes	characteristic
402	C₄⋊C₈	C₂³×C₅₃	1	1	yes	characteristic
403	C₄⋊C₈	C₂³×C₅₃	1	1	yes	characteristic
404	C₄⋊C₈	C₂³×C₅₃	1	1	yes	characteristic
405	#103	C₂×C₄×C₅₃	1	1	yes	characteristic
406	#103	C₂×C₄×C₅₃	1	1	yes	characteristic
407	#103	C₂×C₄×C₅₃	1	1	yes	characteristic
408	#103	C₂×C₄×C₅₃	1	1	yes	characteristic
409	#103	C₂×C₄×C₅₃	1	1	yes	characteristic
410	#103	C₂×C₄×C₅₃	1	1	yes	characteristic
411	#102	C₂×C₄×C₅₃	1	1	yes	characteristic
412	#102	C₂×C₄×C₅₃	1	1	yes	characteristic
413	C₂²⋊C₈	—	2	1	no	—
414	C₂²⋊C₈	—	2	1	no	—
415	C₂²⋊C₈	—	2	1	no	—
416	C₂²⋊C₈	—	2	1	no	—
417	C₂²⋊C₈	D₄×C₅₃	1	1	yes	characteristic
418	C₂²⋊C₈	D₄×C₅₃	1	1	yes	characteristic
419	#97	—	2	1	no	—
420	#97	—	2	1	no	—
421	#97	—	2	1	no	—
422	#97	C₂×C₄×C₅₃	1	1	yes	characteristic
423	#97	C₂×C₄×C₅₃	1	1	yes	characteristic
424	C₄×C₈	—	2	1	no	—
425	C₄×C₈	—	2	1	no	—
426	C₄×C₈	—	2	1	no	—
427	C₄×C₈	—	2	1	no	—
428	C₄×C₈	C₂³×C₅₃	1	1	yes	characteristic
429	C₄×C₈	C₂³×C₅₃	1	1	yes	characteristic
430	C₄×C₈	C₂³×C₅₃	1	1	yes	characteristic
431	C₄×C₈	C₂³×C₅₃	1	1	yes	characteristic
432	C₄×C₈	C₂³×C₅₃	1	1	yes	characteristic
433	C₄×C₈	C₂³×C₅₃	1	1	yes	characteristic
434	#41	—	2	2	no	—
435	#41	—	4	1	no	—
436	C₂×Q₈	—	2	1	no	—
437	C₂×Q₈	—	2	2	no	—
438	C₂×Q₈	—	4	1	no	—
439	C₂×Q₈	C₂²×C₄×C₅₃	1	1	yes	characteristic
440	C₂×Q₈	C₂²×C₄×C₅₃	1	1	yes	characteristic
441	C₂×Q₈	C₂²×C₄×C₅₃	1	1	yes	characteristic
442	C₂×D₄	—	2	1	no	—
443	C₂×D₄	C₂²×C₄×C₅₃	1	1	yes	characteristic
444	C₂²×C₄	—	2	1	no	—
445	C₂²×C₄	—	2	1	no	—
446	C₂²×C₄	—	2	1	no	—
447	C₂²×C₄	—	2	1	no	—
448	C₂²×C₄	—	2	1	no	—
449	C₂²×C₄	C₂²×C₄×C₅₃	1	1	yes	characteristic
450	C₂²×C₄	C₂×D₄×C₅₃	1	1	yes	characteristic
451	C₂²×C₄	#41×C₅₃	1	1	yes	characteristic
452	Q₁₆	—	2	2	no	—
453	Q₁₆	M₁₆×C₅₃	1	4	yes	complemented
454	SD₁₆	—	2	2	no	—
455	SD₁₆	—	2	2	no	—
456	M₁₆	—	2	2	no	—
457	M₁₆	—	4	1	no	—
458	M₁₆	—	8	1	no	—
459	M₁₆	—	8	1	no	—
460	M₁₆	—	8	1	no	—
461	C₂×C₈	—	2	1	no	—
462	C₂×C₈	—	2	1	no	—
463	C₂×C₈	—	2	1	no	—
464	C₂×C₈	—	2	1	no	—
465	C₂×C₈	—	2	1	no	—
466	C₂×C₈	—	2	1	no	—
467	C₂×C₈	C₂²×C₄×C₅₃	1	1	yes	characteristic
468	C₂×C₈	C₂²×C₄×C₅₃	1	1	yes	characteristic
469	C₂×C₈	C₂²×C₄×C₅₃	1	1	yes	characteristic
470	C₂×C₈	C₂²×C₄×C₅₃	1	1	yes	characteristic
471	C₂×C₈	C₂×D₄×C₅₃	1	1	yes	characteristic
472	C₂×C₈	C₂×D₄×C₅₃	1	1	yes	characteristic
473	C₂×C₈	C₂×D₄×C₅₃	1	1	yes	characteristic
474	C₂×C₈	C₂×D₄×C₅₃	1	1	yes	characteristic
475	C₄⋊C₄	—	2	1	no	—
476	C₄⋊C₄	—	2	1	no	—
477	C₄⋊C₄	—	2	1	no	—
478	C₄⋊C₄	—	2	1	no	—
479	C₄⋊C₄	—	2	1	no	—
480	C₄⋊C₄	—	2	1	no	—
481	C₄⋊C₄	—	2	1	no	—
482	C₄⋊C₄	C₂²×C₄×C₅₃	1	1	yes	characteristic
483	C₄⋊C₄	C₂²×C₄×C₅₃	1	1	yes	characteristic
484	C₄⋊C₄	C₂²×C₄×C₅₃	1	1	yes	characteristic
485	C₄⋊C₄	C₂²×C₄×C₅₃	1	1	yes	characteristic
486	C₂²⋊C₄	—	2	1	no	—
487	C₂²⋊C₄	—	2	1	no	—
488	C₂²⋊C₄	—	2	1	no	—
489	C₄²	—	2	1	no	—
490	C₄²	—	2	1	no	—
491	C₄²	—	2	1	no	—
492	C₄²	—	2	1	no	—
493	C₄²	—	2	1	no	—
494	C₄²	—	2	1	no	—
495	C₄²	—	2	1	no	—
496	C₄²	C₂²×C₄×C₅₃	1	1	yes	characteristic
497	C₄²	C₂×D₄×C₅₃	1	1	yes	characteristic
498	C₄²	#41×C₅₃	1	1	yes	characteristic
499	C₄²	C₂⁴×C₅₃	1	1	yes	characteristic
500	C₂³	—	2	1	no	—
501	C₂³	C₄×D₄×C₅₃	1	1	yes	characteristic
502	Q₈	—	2	1	no	—
503	Q₈	—	2	1	no	—
504	Q₈	—	2	1	no	—
505	Q₈	—	2	1	no	—
506	Q₈	—	4	2	no	—
507	Q₈	C₂×M₁₆×C₅₃	1	2	yes	—
508	Q₈	C₂×M₁₆×C₅₃	1	2	yes	—
509	D₄	—	2	1	no	—
510	D₄	—	2	1	no	—
511	D₄	—	4	2	no	—
512	C₂×C₄	—	2	1	no	—
513	C₂×C₄	—	2	1	no	—
514	C₂×C₄	—	2	1	no	—
515	C₂×C₄	—	2	1	no	—
516	C₂×C₄	—	2	1	no	—
517	C₂×C₄	—	2	1	no	—
518	C₂×C₄	—	2	1	no	—
519	C₂×C₄	—	2	1	no	—
520	C₂×C₄	—	2	1	no	—
521	C₂×C₄	—	2	1	no	—
522	C₂×C₄	—	2	1	no	—
523	C₂×C₄	—	4	1	no	—
524	C₂×C₄	—	4	2	no	—
525	C₂×C₄	—	4	2	no	—
526	C₂×C₄	—	4	2	no	—
527	C₂×C₄	—	4	2	no	—
528	C₂×C₄	—	4	1	no	—
529	C₂×C₄	—	4	1	no	—
530	C₂×C₄	C₄×D₄×C₅₃	1	1	yes	characteristic
531	C₂×C₄	C₄×D₄×C₅₃	1	1	yes	characteristic
532	C₂×C₄	C₄×D₄×C₅₃	1	1	yes	characteristic
533	C₂×C₄	C₂³×C₄×C₅₃	1	1	yes	characteristic
534	C₂×C₄	C₂²×D₄×C₅₃	1	1	yes	characteristic
535	C₂×C₄	C₂×#41×C₅₃	1	1	yes	characteristic
536	C₈	—	2	1	no	—
537	C₈	—	2	1	no	—
538	C₈	—	2	1	no	—
539	C₈	—	2	1	no	—
540	C₈	—	2	1	no	—
541	C₈	—	2	1	no	—
542	C₈	—	2	1	no	—
543	C₈	—	4	1	no	—
544	C₈	—	4	1	no	—
545	C₈	—	4	1	no	—
546	C₈	—	4	1	no	—
547	C₈	—	4	1	no	—
548	C₈	—	4	1	no	—
549	C₈	C₂×M₁₆×C₅₃	1	2	yes	—
550	C₂²	—	2	1	no	—
551	C₂²	—	2	1	no	—
552	C₂²	—	2	1	no	—
553	C₂²	—	4	1	no	—
554	C₂²	—	4	1	no	—
555	C₂²	—	4	1	no	—
556	C₂²	C₂×C₄×D₄×C₅₃	1	1	yes	characteristic
557	C₄	—	2	1	no	—
558	C₄	—	2	1	no	—
559	C₄	—	2	1	no	—
560	C₄	—	2	1	no	—
561	C₄	—	2	2	no	—
562	C₄	—	2	2	no	—
563	C₄	—	4	1	no	—
564	C₄	—	4	1	no	—
565	C₄	—	4	1	no	—
566	C₄	—	4	1	no	—
567	C₄	—	4	1	no	—
568	C₄	C₂²×M₁₆×C₅₃	1	1	yes	characteristic
569	C₄	C₅₃×#568	1	1	yes	characteristic
570	C₄	C₅₃×#577	1	1	yes	characteristic
571	C₄	C₅₃×#578	1	1	yes	characteristic
572	C₂	—	4	1	no	—
573	C₂	—	8	1	no	—
574	C₂	D₄×M₁₆×C₅₃	1	1	yes	characteristic
575	C₂	C₅₃×#2947	1	1	yes	characteristic
576	C₂	C₅₃×#2955	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	C₂×#41×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
2. C₅₃×#2984	C₂	semidirect
4. C₅₃×#1592	C₂	semidirect
5. C₅₃×#1590	C₂	semidirect
6. C₅₃×#1588	C₂	semidirect
7. C₅₃×#1587	C₂	semidirect
8. C₅₃×#1584	C₂	semidirect
9. C₅₃×#1583	C₂	semidirect
10. C₅₃×#1579	C₂	semidirect
11. C₈×Q₁₆×C₅₃	C₂	semidirect
12. C₅₃×#1569	C₂	semidirect
14. C₅₃×#1487	C₂	semidirect
15. C₅₃×#1453	C₂	semidirect
164. Q₁₆×C₅₃	M₁₆	semidirect
261. #18053	C₅₃	direct (N × H)
285. #2984	C₂×C₅₃	semidirect
287. #1592	C₂×C₅₃	semidirect
288. #1590	C₂×C₅₃	semidirect
289. #1588	C₂×C₅₃	semidirect
290. #1587	C₂×C₅₃	semidirect
291. #1584	C₂×C₅₃	semidirect
292. #1583	C₂×C₅₃	semidirect
293. #1579	C₂×C₅₃	semidirect
294. C₈×Q₁₆	C₂×C₅₃	semidirect
295. #1569	C₂×C₅₃	semidirect
297. #1487	C₂×C₅₃	semidirect
298. #1453	C₂×C₅₃	semidirect
355. C₅₃	#18053	direct (N × H)
453. Q₁₆	M₁₆×C₅₃	semidirect

Maximal subgroups

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

subgroup	index
C₅₃×#2991	2
C₅₃×#2984	2
C₄×#137×C₅₃	2
C₅₃×#1592	2
C₅₃×#1590	2
C₅₃×#1588	2
C₅₃×#1587	2
C₅₃×#1584	2
C₅₃×#1583	2
C₅₃×#1579	2
C₈×Q₁₆×C₅₃	2
C₅₃×#1569	2
C₅₃×#1488	2
C₅₃×#1487	2
C₅₃×#1453	2
#18053	53

Subgroups — C53×#18053 #27225565

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅₃×#18053 #27225565