Subgroups — C₅₃×#21280 #27228792

Order 13568.

814 conjugacy classes of subgroups.

The subgroup lattice has 422 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄×SD₁₆×C₅₃	C₂	1	1	yes	maximal, characteristic
2	C₅₃×#2320	C₂	1	1	yes	maximal, characteristic, complemented
3	C₅₃×#2319	C₂	1	1	yes	maximal, characteristic
4	C₅₃×#1968	C₂	1	2	yes	maximal
5	C₅₃×#1941	C₂	1	2	yes	maximal, complemented
6	C₅₃×#1936	C₂	1	2	yes	maximal
7	C₅₃×#1935	C₂	1	2	yes	maximal, complemented
8	C₄×#106×C₅₃	C₂	1	1	yes	maximal, characteristic, complemented
9	C₅₃×#1763	C₂	1	1	yes	maximal, characteristic, complemented
10	C₅₃×#1762	C₂	1	1	yes	maximal, characteristic
11	C₅₃×#1726	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂²×SD₁₆×C₅₃	C₂²	1	1	yes	characteristic
13	C₂×C₄×Q₈×C₅₃	C₂²	1	1	yes	characteristic
14	C₂×C₄×D₄×C₅₃	C₂²	1	1	yes	characteristic
15	C₄×SD₁₆×C₅₃	C₄	1	8	yes	complemented
16	C₂×#106×C₅₃	C₂²	1	2	yes	—
17	C₂×#106×C₅₃	C₂²	1	1	yes	characteristic
18	C₂×#103×C₅₃	C₂²	1	2	yes	—
19	C₂×#103×C₅₃	C₂²	1	1	yes	characteristic
20	C₂×#102×C₅₃	C₂²	1	2	yes	—
21	C₂×#102×C₅₃	C₂²	1	1	yes	characteristic
22	C₂×#97×C₅₃	C₂²	1	2	yes	—
23	C₂×#97×C₅₃	C₂²	1	1	yes	characteristic
24	C₂×C₄×C₈×C₅₃	C₂²	1	2	yes	—
25	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
26	C₅₃×#391	C₂²	1	2	yes	—
27	C₅₃×#390	C₂²	1	2	yes	—
28	C₅₃×#389	C₂²	1	2	yes	—
29	C₅₃×#389	C₂²	1	2	yes	—
30	C₅₃×#388	—	2	2	no	—
31	C₅₃×#387	—	2	2	no	—
32	C₅₃×#383	—	2	1	no	—
33	C₅₃×#382	—	2	1	no	—
34	C₅₃×#381	—	2	1	no	—
35	C₅₃×#380	—	2	1	no	—
36	C₄×C₄⋊C₄×C₅₃	C₂²	1	1	yes	characteristic
37	C₄×C₄⋊C₄×C₅₃	C₂²	1	1	yes	characteristic
38	C₅₃×#376	C₂²	1	1	yes	characteristic
39	C₅₃×#340	C₂²	1	2	yes	—
40	C₅₃×#340	C₂²	1	2	yes	—
41	C₅₃×#340	C₂²	1	2	yes	—
42	C₂²×Q₈×C₅₃	C₂³	1	1	yes	characteristic
43	C₂²×D₄×C₅₃	C₂³	1	1	yes	characteristic
44	C₂³×C₄×C₅₃	—	2	1	no	—
45	C₂×SD₁₆×C₅₃	—	2	4	no	—
46	C₂×SD₁₆×C₅₃	C₂×C₄	1	4	yes	—
47	C₂²×C₈×C₅₃	C₂³	1	1	yes	characteristic
48	C₂²×C₈×C₅₃	C₂³	1	2	yes	—
49	C₂²×C₈×C₅₃	C₂³	1	1	yes	characteristic
50	C₄×Q₈×C₅₃	—	2	2	no	—
51	C₄×Q₈×C₅₃	C₂×C₄	1	4	yes	—
52	C₄×D₄×C₅₃	—	2	2	no	—
53	C₄×D₄×C₅₃	C₂×C₄	1	4	yes	—
54	C₂×C₄⋊C₄×C₅₃	—	2	1	no	—
55	C₂×C₄⋊C₄×C₅₃	—	2	2	no	—
56	C₂×C₄⋊C₄×C₅₃	—	2	2	no	—
57	C₂×C₄⋊C₄×C₅₃	C₂³	1	2	yes	—
58	C₂×C₄⋊C₄×C₅₃	C₂³	1	2	yes	—
59	C₂×C₄⋊C₄×C₅₃	C₂³	1	1	yes	characteristic
60	C₂×C₄⋊C₄×C₅₃	C₂³	1	1	yes	characteristic
61	C₂×C₂²⋊C₄×C₅₃	—	2	1	no	—
62	C₂×C₂²⋊C₄×C₅₃	—	2	2	no	—
63	C₂×C₂²⋊C₄×C₅₃	—	2	2	no	—
64	C₂×C₄²×C₅₃	—	2	1	no	—
65	C₂×C₄²×C₅₃	—	2	1	no	—
66	C₂×C₄²×C₅₃	—	2	1	no	—
67	C₂×C₄²×C₅₃	C₂³	1	2	yes	—
68	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
69	#106×C₅₃	—	2	4	no	—
70	#106×C₅₃	C₂×C₄	1	4	yes	—
71	#103×C₅₃	—	2	4	no	—
72	#103×C₅₃	C₂×C₄	1	4	yes	—
73	#102×C₅₃	—	2	4	no	—
74	#102×C₅₃	C₂×C₄	1	4	yes	—
75	#97×C₅₃	—	2	4	no	—
76	#97×C₅₃	—	2	2	no	—
77	C₄×C₈×C₅₃	—	2	4	no	—
78	C₄×C₈×C₅₃	C₂×C₄	1	4	yes	—
79	#95×C₅₃	—	2	1	no	—
80	#95×C₅₃	—	2	1	no	—
81	#95×C₅₃	—	2	2	no	—
82	#95×C₅₃	—	2	2	no	—
83	#95×C₅₃	—	2	2	no	—
84	#95×C₅₃	—	2	2	no	—
85	#95×C₅₃	D₄	1	1	yes	characteristic
86	#95×C₅₃	D₄	1	1	yes	characteristic
87	#95×C₅₃	D₄	1	2	yes	—
88	C₂⁴×C₅₃	—	2	1	no	—
89	C₂×Q₈×C₅₃	—	2	1	no	—
90	C₂×Q₈×C₅₃	—	2	2	no	—
91	C₂×Q₈×C₅₃	—	2	2	no	—
92	C₂×Q₈×C₅₃	C₂²×C₄	1	2	yes	—
93	C₂×D₄×C₅₃	—	2	2	no	—
94	C₂×D₄×C₅₃	—	2	2	no	—
95	C₂×D₄×C₅₃	—	2	1	no	—
96	C₂×D₄×C₅₃	C₂²×C₄	1	2	yes	—
97	C₂²×C₄×C₅₃	—	2	1	no	—
98	C₂²×C₄×C₅₃	—	2	1	no	—
99	C₂²×C₄×C₅₃	—	2	1	no	—
100	C₂²×C₄×C₅₃	—	2	4	no	—
101	C₂²×C₄×C₅₃	—	2	2	no	—
102	C₂²×C₄×C₅₃	—	2	2	no	—
103	C₂²×C₄×C₅₃	—	8	2	no	—
104	C₂²×C₄×C₅₃	C₂×D₄	1	2	yes	—
105	C₂²×C₄×C₅₃	C₂×D₄	1	2	yes	—
106	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
107	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
108	C₂²×C₄×C₅₃	C₂⁴	1	1	yes	characteristic
109	SD₁₆×C₅₃	—	2	8	no	—
110	C₂×C₈×C₅₃	—	2	2	no	—
111	C₂×C₈×C₅₃	—	2	2	no	—
112	C₂×C₈×C₅₃	—	2	2	no	—
113	C₂×C₈×C₅₃	C₂²×C₄	1	2	yes	—
114	C₂×C₈×C₅₃	C₂²×C₄	1	2	yes	—
115	C₂×C₈×C₅₃	#41	1	4	yes	—
116	C₂×C₈×C₅₃	#41	1	4	yes	—
117	C₄⋊C₄×C₅₃	—	2	4	no	—
118	C₄⋊C₄×C₅₃	—	2	1	no	—
119	C₄⋊C₄×C₅₃	—	2	1	no	—
120	C₄⋊C₄×C₅₃	—	2	2	no	—
121	C₄⋊C₄×C₅₃	—	2	2	no	—
122	C₄⋊C₄×C₅₃	—	2	2	no	—
123	C₄⋊C₄×C₅₃	—	2	2	no	—
124	C₄⋊C₄×C₅₃	—	8	2	no	—
125	C₄⋊C₄×C₅₃	—	8	2	no	—
126	C₄⋊C₄×C₅₃	C₂²×C₄	1	2	yes	—
127	C₄⋊C₄×C₅₃	C₂²×C₄	1	2	yes	—
128	C₂²⋊C₄×C₅₃	—	2	4	no	—
129	C₂²⋊C₄×C₅₃	—	8	2	no	—
130	C₂²⋊C₄×C₅₃	—	8	2	no	—
131	C₄²×C₅₃	—	2	4	no	—
132	C₄²×C₅₃	—	2	2	no	—
133	C₄²×C₅₃	—	2	2	no	—
134	C₄²×C₅₃	—	8	1	no	—
135	C₄²×C₅₃	—	8	1	no	—
136	C₄²×C₅₃	C₂²×C₄	1	2	yes	—
137	C₄²×C₅₃	#41	1	2	yes	—
138	C₂³×C₅₃	—	2	2	no	—
139	C₂³×C₅₃	—	4	2	no	—
140	C₂³×C₅₃	—	4	2	no	—
141	C₂³×C₅₃	—	4	1	no	—
142	C₂³×C₅₃	—	4	1	no	—
143	C₂³×C₅₃	C₂²×D₄	1	1	yes	characteristic
144	Q₈×C₅₃	—	2	2	no	—
145	Q₈×C₅₃	—	2	2	no	—
146	Q₈×C₅₃	—	4	2	no	—
147	D₄×C₅₃	—	2	2	no	—
148	D₄×C₅₃	—	2	2	no	—
149	D₄×C₅₃	—	4	2	no	—
150	C₂×C₄×C₅₃	—	2	2	no	—
151	C₂×C₄×C₅₃	—	2	1	no	—
152	C₂×C₄×C₅₃	—	2	1	no	—
153	C₂×C₄×C₅₃	—	2	2	no	—
154	C₂×C₄×C₅₃	—	2	2	no	—
155	C₂×C₄×C₅₃	—	2	2	no	—
156	C₂×C₄×C₅₃	—	2	2	no	—
157	C₂×C₄×C₅₃	—	2	2	no	—
158	C₂×C₄×C₅₃	—	2	2	no	—
159	C₂×C₄×C₅₃	—	2	2	no	—
160	C₂×C₄×C₅₃	—	2	2	no	—
161	C₂×C₄×C₅₃	—	2	2	no	—
162	C₂×C₄×C₅₃	—	4	2	no	—
163	C₂×C₄×C₅₃	—	4	1	no	—
164	C₂×C₄×C₅₃	—	4	2	no	—
165	C₂×C₄×C₅₃	—	4	1	no	—
166	C₂×C₄×C₅₃	—	4	2	no	—
167	C₂×C₄×C₅₃	—	4	2	no	—
168	C₂×C₄×C₅₃	—	4	2	no	—
169	C₂×C₄×C₅₃	—	4	2	no	—
170	C₂×C₄×C₅₃	—	4	2	no	—
171	C₂×C₄×C₅₃	—	4	2	no	—
172	C₂×C₄×C₅₃	—	8	4	no	—
173	C₂×C₄×C₅₃	C₄×D₄	1	2	yes	—
174	C₂×C₄×C₅₃	C₄×D₄	1	2	yes	—
175	C₂×C₄×C₅₃	C₂³×C₄	1	1	yes	characteristic
176	C₂×C₄×C₅₃	C₂×#41	1	2	yes	—
177	C₂×C₄×C₅₃	C₂×#41	1	2	yes	—
178	C₂×C₄×C₅₃	C₂×#41	1	1	yes	characteristic
179	C₈×C₅₃	—	2	2	no	—
180	C₈×C₅₃	—	2	4	no	—
181	C₈×C₅₃	#117	1	4	yes	—
182	#21280	C₅₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
183	C₂²×C₅₃	—	4	2	no	—
184	C₂²×C₅₃	—	4	2	no	—
185	C₂²×C₅₃	—	4	2	no	—
186	C₂²×C₅₃	—	8	2	no	—
187	C₂²×C₅₃	—	8	2	no	—
188	C₂²×C₅₃	C₂×C₄×D₄	1	1	yes	characteristic
189	C₂²×C₅₃	#532	1	2	yes	—
190	C₂²×C₅₃	#577	1	1	yes	characteristic
191	C₂²×C₅₃	#577	1	2	yes	—
192	C₂²×C₅₃	#577	1	1	yes	characteristic
193	C₄×C₅₃	—	2	2	no	—
194	C₄×C₅₃	—	2	2	no	—
195	C₄×C₅₃	—	4	2	no	—
196	C₄×C₅₃	—	4	2	no	—
197	C₄×C₅₃	—	4	2	no	—
198	C₄×C₅₃	—	4	2	no	—
199	C₄×C₅₃	—	4	2	no	—
200	C₄×C₅₃	—	8	2	no	—
201	C₄×C₅₃	—	8	2	no	—
202	C₄×C₅₃	C₂×#117	1	1	yes	characteristic
203	C₄×C₅₃	C₄×#41	1	1	yes	characteristic
204	C₄×C₅₃	#529	1	2	yes	—
205	C₂×C₄×SD₁₆	C₂×C₅₃	1	1	yes	characteristic
206	#2320	C₂×C₅₃	1	1	yes	characteristic, complemented
207	#2319	C₂×C₅₃	1	1	yes	characteristic
208	#1968	C₂×C₅₃	1	2	yes	—
209	#1941	C₂×C₅₃	1	2	yes	complemented
210	#1936	C₂×C₅₃	1	2	yes	—
211	#1935	C₂×C₅₃	1	2	yes	complemented
212	C₄×#106	C₂×C₅₃	1	1	yes	characteristic, complemented
213	#1763	C₂×C₅₃	1	1	yes	characteristic, complemented
214	#1762	C₂×C₅₃	1	1	yes	characteristic
215	#1726	C₂×C₅₃	1	1	yes	characteristic, complemented
216	C₂×C₅₃	—	8	2	no	—
217	C₂×C₅₃	#2293	1	1	yes	characteristic
218	C₂×C₅₃	#2949	1	1	yes	characteristic
219	C₂×C₅₃	#2952	1	1	yes	characteristic
220	C₂×C₅₃	#3173	1	2	yes	—
221	C₂×C₅₃	#3174	1	2	yes	—
222	C₂²×SD₁₆	C₂²×C₅₃	1	1	yes	characteristic
223	C₂×C₄×Q₈	C₂²×C₅₃	1	1	yes	characteristic
224	C₂×C₄×D₄	C₂²×C₅₃	1	1	yes	characteristic
225	C₄×SD₁₆	C₄×C₅₃	1	8	yes	complemented
226	C₂×#106	C₂²×C₅₃	1	2	yes	—
227	C₂×#106	C₂²×C₅₃	1	1	yes	characteristic
228	C₂×#103	C₂²×C₅₃	1	2	yes	—
229	C₂×#103	C₂²×C₅₃	1	1	yes	characteristic
230	C₂×#102	C₂²×C₅₃	1	2	yes	—
231	C₂×#102	C₂²×C₅₃	1	1	yes	characteristic
232	C₂×#97	C₂²×C₅₃	1	2	yes	—
233	C₂×#97	C₂²×C₅₃	1	1	yes	characteristic
234	C₂×C₄×C₈	C₂²×C₅₃	1	2	yes	—
235	C₂×C₄×C₈	C₂²×C₅₃	1	1	yes	characteristic
236	#391	C₂²×C₅₃	1	2	yes	—
237	#390	C₂²×C₅₃	1	2	yes	—
238	#389	C₂²×C₅₃	1	2	yes	—
239	#389	C₂²×C₅₃	1	2	yes	—
240	#388	—	2	2	no	—
241	#387	—	2	2	no	—
242	#383	—	2	1	no	—
243	#382	—	2	1	no	—
244	#381	—	2	1	no	—
245	#380	—	2	1	no	—
246	C₄×C₄⋊C₄	C₂²×C₅₃	1	1	yes	characteristic
247	C₄×C₄⋊C₄	C₂²×C₅₃	1	1	yes	characteristic
248	#376	C₂²×C₅₃	1	1	yes	characteristic
249	#340	C₂²×C₅₃	1	2	yes	—
250	#340	C₂²×C₅₃	1	2	yes	—
251	#340	C₂²×C₅₃	1	2	yes	—
252	C₅₃	#21280	1	1	yes	characteristic, direct product factor, Sylow
253	C₂²×Q₈	C₂³×C₅₃	1	1	yes	characteristic
254	C₂²×D₄	C₂³×C₅₃	1	1	yes	characteristic
255	C₂³×C₄	—	2	1	no	—
256	C₂×SD₁₆	—	2	4	no	—
257	C₂×SD₁₆	C₂×C₄×C₅₃	1	4	yes	—
258	C₂²×C₈	C₂³×C₅₃	1	1	yes	characteristic
259	C₂²×C₈	C₂³×C₅₃	1	2	yes	—
260	C₂²×C₈	C₂³×C₅₃	1	1	yes	characteristic
261	C₄×Q₈	—	2	2	no	—
262	C₄×Q₈	C₂×C₄×C₅₃	1	4	yes	—
263	C₄×D₄	—	2	2	no	—
264	C₄×D₄	C₂×C₄×C₅₃	1	4	yes	—
265	C₂×C₄⋊C₄	—	2	1	no	—
266	C₂×C₄⋊C₄	—	2	2	no	—
267	C₂×C₄⋊C₄	—	2	2	no	—
268	C₂×C₄⋊C₄	C₂³×C₅₃	1	2	yes	—
269	C₂×C₄⋊C₄	C₂³×C₅₃	1	2	yes	—
270	C₂×C₄⋊C₄	C₂³×C₅₃	1	1	yes	characteristic
271	C₂×C₄⋊C₄	C₂³×C₅₃	1	1	yes	characteristic
272	C₂×C₂²⋊C₄	—	2	1	no	—
273	C₂×C₂²⋊C₄	—	2	2	no	—
274	C₂×C₂²⋊C₄	—	2	2	no	—
275	C₂×C₄²	—	2	1	no	—
276	C₂×C₄²	—	2	1	no	—
277	C₂×C₄²	—	2	1	no	—
278	C₂×C₄²	C₂³×C₅₃	1	2	yes	—
279	C₂×C₄²	C₂³×C₅₃	1	1	yes	characteristic
280	#106	—	2	4	no	—
281	#106	C₂×C₄×C₅₃	1	4	yes	—
282	#103	—	2	4	no	—
283	#103	C₂×C₄×C₅₃	1	4	yes	—
284	#102	—	2	4	no	—
285	#102	C₂×C₄×C₅₃	1	4	yes	—
286	#97	—	2	4	no	—
287	#97	—	2	2	no	—
288	C₄×C₈	—	2	4	no	—
289	C₄×C₈	C₂×C₄×C₅₃	1	4	yes	—
290	#95	—	2	1	no	—
291	#95	—	2	1	no	—
292	#95	—	2	2	no	—
293	#95	—	2	2	no	—
294	#95	—	2	2	no	—
295	#95	—	2	2	no	—
296	#95	D₄×C₅₃	1	1	yes	characteristic
297	#95	D₄×C₅₃	1	1	yes	characteristic
298	#95	D₄×C₅₃	1	2	yes	—
299	C₂⁴	—	2	1	no	—
300	C₂×Q₈	—	2	1	no	—
301	C₂×Q₈	—	2	2	no	—
302	C₂×Q₈	—	2	2	no	—
303	C₂×Q₈	C₂²×C₄×C₅₃	1	2	yes	—
304	C₂×D₄	—	2	2	no	—
305	C₂×D₄	—	2	2	no	—
306	C₂×D₄	—	2	1	no	—
307	C₂×D₄	C₂²×C₄×C₅₃	1	2	yes	—
308	C₂²×C₄	—	2	1	no	—
309	C₂²×C₄	—	2	1	no	—
310	C₂²×C₄	—	2	1	no	—
311	C₂²×C₄	—	2	4	no	—
312	C₂²×C₄	—	2	2	no	—
313	C₂²×C₄	—	2	2	no	—
314	C₂²×C₄	—	8	2	no	—
315	C₂²×C₄	C₂×D₄×C₅₃	1	2	yes	—
316	C₂²×C₄	C₂×D₄×C₅₃	1	2	yes	—
317	C₂²×C₄	C₂×D₄×C₅₃	1	1	yes	characteristic
318	C₂²×C₄	C₂×D₄×C₅₃	1	1	yes	characteristic
319	C₂²×C₄	C₂⁴×C₅₃	1	1	yes	characteristic
320	SD₁₆	—	2	8	no	—
321	C₂×C₈	—	2	2	no	—
322	C₂×C₈	—	2	2	no	—
323	C₂×C₈	—	2	2	no	—
324	C₂×C₈	C₂²×C₄×C₅₃	1	2	yes	—
325	C₂×C₈	C₂²×C₄×C₅₃	1	2	yes	—
326	C₂×C₈	#41×C₅₃	1	4	yes	—
327	C₂×C₈	#41×C₅₃	1	4	yes	—
328	C₄⋊C₄	—	2	4	no	—
329	C₄⋊C₄	—	2	1	no	—
330	C₄⋊C₄	—	2	1	no	—
331	C₄⋊C₄	—	2	2	no	—
332	C₄⋊C₄	—	2	2	no	—
333	C₄⋊C₄	—	2	2	no	—
334	C₄⋊C₄	—	2	2	no	—
335	C₄⋊C₄	—	8	2	no	—
336	C₄⋊C₄	—	8	2	no	—
337	C₄⋊C₄	C₂²×C₄×C₅₃	1	2	yes	—
338	C₄⋊C₄	C₂²×C₄×C₅₃	1	2	yes	—
339	C₂²⋊C₄	—	2	4	no	—
340	C₂²⋊C₄	—	8	2	no	—
341	C₂²⋊C₄	—	8	2	no	—
342	C₄²	—	2	4	no	—
343	C₄²	—	2	2	no	—
344	C₄²	—	2	2	no	—
345	C₄²	—	8	1	no	—
346	C₄²	—	8	1	no	—
347	C₄²	C₂²×C₄×C₅₃	1	2	yes	—
348	C₄²	#41×C₅₃	1	2	yes	—
349	C₂³	—	2	2	no	—
350	C₂³	—	4	2	no	—
351	C₂³	—	4	1	no	—
352	C₂³	—	4	2	no	—
353	C₂³	—	4	1	no	—
354	C₂³	C₂²×D₄×C₅₃	1	1	yes	characteristic
355	Q₈	—	2	2	no	—
356	Q₈	—	2	2	no	—
357	Q₈	—	4	2	no	—
358	D₄	—	2	2	no	—
359	D₄	—	2	2	no	—
360	D₄	—	4	2	no	—
361	C₂×C₄	—	2	1	no	—
362	C₂×C₄	—	2	1	no	—
363	C₂×C₄	—	2	2	no	—
364	C₂×C₄	—	2	2	no	—
365	C₂×C₄	—	2	2	no	—
366	C₂×C₄	—	2	2	no	—
367	C₂×C₄	—	2	2	no	—
368	C₂×C₄	—	2	2	no	—
369	C₂×C₄	—	2	2	no	—
370	C₂×C₄	—	2	2	no	—
371	C₂×C₄	—	2	2	no	—
372	C₂×C₄	—	2	2	no	—
373	C₂×C₄	—	4	2	no	—
374	C₂×C₄	—	4	1	no	—
375	C₂×C₄	—	4	2	no	—
376	C₂×C₄	—	4	1	no	—
377	C₂×C₄	—	4	2	no	—
378	C₂×C₄	—	4	2	no	—
379	C₂×C₄	—	4	2	no	—
380	C₂×C₄	—	4	2	no	—
381	C₂×C₄	—	4	2	no	—
382	C₂×C₄	—	4	2	no	—
383	C₂×C₄	—	8	4	no	—
384	C₂×C₄	C₄×D₄×C₅₃	1	2	yes	—
385	C₂×C₄	C₄×D₄×C₅₃	1	2	yes	—
386	C₂×C₄	C₂³×C₄×C₅₃	1	1	yes	characteristic
387	C₂×C₄	C₂×#41×C₅₃	1	2	yes	—
388	C₂×C₄	C₂×#41×C₅₃	1	2	yes	—
389	C₂×C₄	C₂×#41×C₅₃	1	1	yes	characteristic
390	C₈	—	2	2	no	—
391	C₈	—	2	4	no	—
392	C₈	#117×C₅₃	1	4	yes	—
393	C₂²	—	4	2	no	—
394	C₂²	—	4	2	no	—
395	C₂²	—	4	2	no	—
396	C₂²	—	8	2	no	—
397	C₂²	—	8	2	no	—
398	C₂²	C₂×C₄×D₄×C₅₃	1	1	yes	characteristic
399	C₂²	C₅₃×#532	1	2	yes	—
400	C₂²	C₅₃×#577	1	2	yes	—
401	C₂²	C₅₃×#577	1	1	yes	characteristic
402	C₂²	C₅₃×#577	1	1	yes	characteristic
403	C₄	—	2	2	no	—
404	C₄	—	2	2	no	—
405	C₄	—	4	2	no	—
406	C₄	—	4	2	no	—
407	C₄	—	4	2	no	—
408	C₄	—	4	2	no	—
409	C₄	—	4	2	no	—
410	C₄	—	8	2	no	—
411	C₄	—	8	2	no	—
412	C₄	C₂×#117×C₅₃	1	1	yes	characteristic
413	C₄	C₄×#41×C₅₃	1	1	yes	characteristic
414	C₄	C₅₃×#529	1	2	yes	—
415	C₂	—	8	2	no	—
416	C₂	C₅₃×#2293	1	1	yes	characteristic
417	C₂	C₅₃×#2949	1	1	yes	characteristic
418	C₂	C₅₃×#2952	1	1	yes	characteristic
419	C₂	C₅₃×#3173	1	2	yes	—
420	C₂	C₅₃×#3174	1	2	yes	—

	subgroup	G/H
center, socle	C₂³×C₅₃	C₂²×D₄
commutator	C₂×C₄	C₂³×C₄×C₅₃
Frattini, cosocle	C₂²×C₄	C₂⁴×C₅₃
prime radical	C₂²×D₄×C₅₃	C₂³
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₅₃ ◁ C₅₃×#376 ◁ G

Complements

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

N	H ≅ G/N	type
2. C₅₃×#2320	C₂	semidirect
5. C₅₃×#1941	C₂	semidirect
7. C₅₃×#1935	C₂	semidirect
8. C₄×#106×C₅₃	C₂	semidirect
9. C₅₃×#1763	C₂	semidirect
11. C₅₃×#1726	C₂	semidirect
15. C₄×SD₁₆×C₅₃	C₄	semidirect
182. #21280	C₅₃	direct (N × H)
206. #2320	C₂×C₅₃	semidirect
209. #1941	C₂×C₅₃	semidirect
211. #1935	C₂×C₅₃	semidirect
212. C₄×#106	C₂×C₅₃	semidirect
213. #1763	C₂×C₅₃	semidirect
215. #1726	C₂×C₅₃	semidirect
225. C₄×SD₁₆	C₄×C₅₃	semidirect
252. C₅₃	#21280	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×C₄×SD₁₆×C₅₃	2
C₅₃×#2320	2
C₅₃×#2319	2
C₅₃×#1968	2
C₅₃×#1941	2
C₅₃×#1936	2
C₅₃×#1935	2
C₄×#106×C₅₃	2
C₅₃×#1763	2
C₅₃×#1762	2
C₅₃×#1726	2
#21280	53

Subgroups — C53×#21280 #27228792

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅₃×#21280 #27228792