Subgroups — C₅₃×#21130 #27228642

Order 13568.

890 conjugacy classes of subgroups.

The subgroup lattice has 428 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×#2275	C₂	1	1	yes	maximal, characteristic
4	C₅₃×#1928	C₂	1	2	yes	maximal, complemented
5	C₅₃×#1920	C₂	1	2	yes	maximal, complemented
6	C₅₃×#1877	C₂	1	2	yes	maximal
7	C₅₃×#1862	C₂	1	2	yes	maximal
8	C₄×#107×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×#1746	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	2	yes	—
15	C₂×C₄×D₄×C₅₃	C₂²	1	1	yes	characteristic
16	C₄×SD₁₆×C₅₃	C₄	1	8	yes	complemented
17	C₂×#107×C₅₃	C₂²	1	2	yes	—
18	C₂×#106×C₅₃	C₂²	1	1	yes	characteristic
19	C₂×#103×C₅₃	C₂²	1	2	yes	—
20	C₂×#103×C₅₃	C₂²	1	1	yes	characteristic
21	C₂×#102×C₅₃	C₂²	1	2	yes	—
22	C₂×#102×C₅₃	C₂²	1	1	yes	characteristic
23	C₂×#97×C₅₃	C₂²	1	1	yes	characteristic
24	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
25	C₅₃×#391	C₂²	1	2	yes	—
26	C₅₃×#389	C₂²	1	2	yes	—
27	C₅₃×#387	—	2	2	no	—
28	C₅₃×#383	—	2	1	no	—
29	C₅₃×#382	—	2	1	no	—
30	C₄×C₄⋊C₄×C₅₃	C₂²	1	2	yes	—
31	C₄×C₄⋊C₄×C₅₃	C₂²	1	1	yes	characteristic
32	C₄×C₄⋊C₄×C₅₃	C₂²	1	1	yes	characteristic
33	C₄×C₂²⋊C₄×C₅₃	—	2	1	no	—
34	C₄×C₂²⋊C₄×C₅₃	—	2	2	no	—
35	C₅₃×#376	C₂²	1	1	yes	characteristic
36	C₂×#95×C₅₃	—	2	1	no	—
37	C₅₃×#340	C₂²	1	2	yes	—
38	C₅₃×#340	C₂²	1	2	yes	—
39	C₅₃×#340	C₂²	1	2	yes	—
40	C₅₃×#336	C₂²	1	2	yes	—
41	C₅₃×#336	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	2	no	—
45	C₂³×C₄×C₅₃	—	2	1	no	—
46	C₂×SD₁₆×C₅₃	—	2	4	no	—
47	C₂×SD₁₆×C₅₃	C₂×C₄	1	4	yes	—
48	C₂²×C₈×C₅₃	C₂³	1	2	yes	—
49	C₂²×C₈×C₅₃	C₂³	1	1	yes	characteristic
50	C₂²×C₈×C₅₃	C₂³	1	1	yes	characteristic
51	C₄×Q₈×C₅₃	—	2	2	no	—
52	C₄×Q₈×C₅₃	C₂×C₄	1	4	yes	complemented
53	C₄×D₄×C₅₃	—	2	4	no	—
54	C₄×D₄×C₅₃	—	2	2	no	—
55	C₄×D₄×C₅₃	—	2	4	no	—
56	C₄×D₄×C₅₃	C₂×C₄	1	4	yes	—
57	C₂×C₄⋊C₄×C₅₃	—	2	1	no	—
58	C₂×C₄⋊C₄×C₅₃	—	2	2	no	—
59	C₂×C₄⋊C₄×C₅₃	C₂³	1	2	yes	—
60	C₂×C₄⋊C₄×C₅₃	C₂³	1	1	yes	characteristic
61	C₂×C₄⋊C₄×C₅₃	C₂³	1	1	yes	characteristic
62	C₂×C₄⋊C₄×C₅₃	C₂³	1	2	yes	—
63	C₂×C₂²⋊C₄×C₅₃	—	2	2	no	—
64	C₂×C₂²⋊C₄×C₅₃	—	2	1	no	—
65	C₂×C₄²×C₅₃	—	2	1	no	—
66	C₂×C₄²×C₅₃	—	2	2	no	—
67	C₂×C₄²×C₅₃	—	2	1	no	—
68	C₂×C₄²×C₅₃	—	2	1	no	—
69	C₂×C₄²×C₅₃	C₂³	1	2	yes	—
70	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
71	#107×C₅₃	—	2	4	no	—
72	#106×C₅₃	C₂×C₄	1	4	yes	complemented
73	#103×C₅₃	—	2	4	no	—
74	#103×C₅₃	C₂×C₄	1	4	yes	complemented
75	#102×C₅₃	—	2	4	no	—
76	#102×C₅₃	C₂×C₄	1	4	yes	—
77	#97×C₅₃	—	2	2	no	—
78	C₄×C₈×C₅₃	C₂×C₄	1	4	yes	complemented
79	#95×C₅₃	—	2	2	no	—
80	#95×C₅₃	—	2	1	no	—
81	#95×C₅₃	—	2	1	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	1	no	—
95	C₂×D₄×C₅₃	C₂²×C₄	1	2	yes	—
96	C₂×D₄×C₅₃	#41	1	4	yes	—
97	C₂²×C₄×C₅₃	—	2	1	no	—
98	C₂²×C₄×C₅₃	—	2	1	no	—
99	C₂²×C₄×C₅₃	—	2	2	no	—
100	C₂²×C₄×C₅₃	—	2	1	no	—
101	C₂²×C₄×C₅₃	—	2	8	no	—
102	C₂²×C₄×C₅₃	—	2	4	no	—
103	C₂²×C₄×C₅₃	—	2	2	no	—
104	C₂²×C₄×C₅₃	—	8	2	no	—
105	C₂²×C₄×C₅₃	—	8	2	no	—
106	C₂²×C₄×C₅₃	—	8	2	no	—
107	C₂²×C₄×C₅₃	C₂×D₄	1	2	yes	—
108	C₂²×C₄×C₅₃	C₂×D₄	1	2	yes	—
109	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
110	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
111	C₂²×C₄×C₅₃	C₂⁴	1	1	yes	characteristic
112	SD₁₆×C₅₃	—	2	8	no	—
113	C₂×C₈×C₅₃	—	2	2	no	—
114	C₂×C₈×C₅₃	—	2	2	no	—
115	C₂×C₈×C₅₃	—	2	2	no	—
116	C₂×C₈×C₅₃	—	2	2	no	—
117	C₂×C₈×C₅₃	—	2	2	no	—
118	C₂×C₈×C₅₃	C₂²×C₄	1	2	yes	—
119	C₂×C₈×C₅₃	C₂²×C₄	1	2	yes	—
120	C₄⋊C₄×C₅₃	—	2	4	no	—
121	C₄⋊C₄×C₅₃	—	2	1	no	—
122	C₄⋊C₄×C₅₃	—	2	2	no	—
123	C₄⋊C₄×C₅₃	—	2	1	no	—
124	C₄⋊C₄×C₅₃	—	2	2	no	—
125	C₄⋊C₄×C₅₃	—	2	2	no	—
126	C₄⋊C₄×C₅₃	—	8	2	no	—
127	C₄⋊C₄×C₅₃	C₂²×C₄	1	2	yes	—
128	C₄⋊C₄×C₅₃	C₂²×C₄	1	2	yes	—
129	C₄⋊C₄×C₅₃	#41	1	4	yes	—
130	C₂²⋊C₄×C₅₃	—	2	8	no	—
131	C₂²⋊C₄×C₅₃	—	2	4	no	—
132	C₄²×C₅₃	—	2	4	no	—
133	C₄²×C₅₃	—	2	2	no	—
134	C₄²×C₅₃	—	2	2	no	—
135	C₄²×C₅₃	—	8	1	no	—
136	C₄²×C₅₃	—	8	1	no	—
137	C₄²×C₅₃	—	8	2	no	—
138	C₄²×C₅₃	C₂²×C₄	1	2	yes	—
139	C₄²×C₅₃	#41	1	2	yes	—
140	C₂³×C₅₃	—	2	2	no	—
141	C₂³×C₅₃	—	2	4	no	—
142	C₂³×C₅₃	—	4	1	no	—
143	C₂³×C₅₃	—	4	2	no	—
144	C₂³×C₅₃	—	4	1	no	—
145	C₂³×C₅₃	C₂²×D₄	1	1	yes	characteristic
146	Q₈×C₅₃	—	2	2	no	—
147	Q₈×C₅₃	—	2	2	no	—
148	Q₈×C₅₃	—	4	2	no	—
149	D₄×C₅₃	—	2	2	no	—
150	D₄×C₅₃	—	2	4	no	—
151	D₄×C₅₃	#117	1	4	yes	—
152	C₂×C₄×C₅₃	—	2	1	no	—
153	C₂×C₄×C₅₃	—	2	2	no	—
154	C₂×C₄×C₅₃	—	2	1	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	4	no	—
160	C₂×C₄×C₅₃	—	2	2	no	—
161	C₂×C₄×C₅₃	—	2	2	no	—
162	C₂×C₄×C₅₃	—	2	2	no	—
163	C₂×C₄×C₅₃	—	2	2	no	—
164	C₂×C₄×C₅₃	—	2	2	no	—
165	C₂×C₄×C₅₃	—	4	2	no	—
166	C₂×C₄×C₅₃	—	4	2	no	—
167	C₂×C₄×C₅₃	—	4	1	no	—
168	C₂×C₄×C₅₃	—	4	2	no	—
169	C₂×C₄×C₅₃	—	4	2	no	—
170	C₂×C₄×C₅₃	—	4	1	no	—
171	C₂×C₄×C₅₃	—	4	2	no	—
172	C₂×C₄×C₅₃	—	4	2	no	—
173	C₂×C₄×C₅₃	—	4	2	no	—
174	C₂×C₄×C₅₃	—	8	4	no	—
175	C₂×C₄×C₅₃	—	8	8	no	—
176	C₂×C₄×C₅₃	C₄×D₄	1	2	yes	—
177	C₂×C₄×C₅₃	C₄×D₄	1	2	yes	—
178	C₂×C₄×C₅₃	C₂³×C₄	1	1	yes	characteristic
179	C₂×C₄×C₅₃	C₂×#41	1	2	yes	—
180	C₂×C₄×C₅₃	C₂×#41	1	2	yes	—
181	C₂×C₄×C₅₃	C₂×#41	1	1	yes	characteristic
182	C₈×C₅₃	—	2	2	no	—
183	C₈×C₅₃	—	2	2	no	—
184	C₈×C₅₃	—	4	2	no	—
185	#21130	C₅₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
186	C₂²×C₅₃	—	2	4	no	—
187	C₂²×C₅₃	—	4	4	no	—
188	C₂²×C₅₃	—	4	4	no	—
189	C₂²×C₅₃	—	4	2	no	—
190	C₂²×C₅₃	—	4	2	no	—
191	C₂²×C₅₃	C₂×C₄×D₄	1	1	yes	characteristic
192	C₂²×C₅₃	#525	1	2	yes	—
193	C₂²×C₅₃	#576	1	1	yes	characteristic
194	C₂²×C₅₃	#577	1	2	yes	—
195	C₂²×C₅₃	#578	1	1	yes	characteristic
196	C₄×C₅₃	—	2	2	no	—
197	C₄×C₅₃	—	2	2	no	—
198	C₄×C₅₃	—	4	2	no	—
199	C₄×C₅₃	—	4	4	no	—
200	C₄×C₅₃	—	4	2	no	—
201	C₄×C₅₃	—	4	2	no	—
202	C₄×C₅₃	—	4	2	no	—
203	C₄×C₅₃	—	4	2	no	—
204	C₄×C₅₃	—	8	2	no	—
205	C₄×C₅₃	C₂×#117	1	1	yes	characteristic
206	C₄×C₅₃	C₄×#41	1	1	yes	characteristic
207	C₄×C₅₃	#529	1	2	yes	—
208	C₂×C₄×SD₁₆	C₂×C₅₃	1	1	yes	characteristic
209	#2320	C₂×C₅₃	1	1	yes	characteristic, complemented
210	#2275	C₂×C₅₃	1	1	yes	characteristic
211	#1928	C₂×C₅₃	1	2	yes	complemented
212	#1920	C₂×C₅₃	1	2	yes	complemented
213	#1877	C₂×C₅₃	1	2	yes	—
214	#1862	C₂×C₅₃	1	2	yes	—
215	C₄×#107	C₂×C₅₃	1	1	yes	characteristic, complemented
216	#1763	C₂×C₅₃	1	1	yes	characteristic, complemented
217	#1762	C₂×C₅₃	1	1	yes	characteristic
218	#1746	C₂×C₅₃	1	1	yes	characteristic, complemented
219	C₂×C₅₃	—	4	4	no	—
220	C₂×C₅₃	#2290	1	1	yes	characteristic
221	C₂×C₅₃	#2949	1	1	yes	characteristic
222	C₂×C₅₃	#2951	1	1	yes	characteristic
223	C₂×C₅₃	#3118	1	2	yes	—
224	C₂×C₅₃	#3120	1	2	yes	—
225	C₂²×SD₁₆	C₂²×C₅₃	1	1	yes	characteristic
226	C₂×C₄×Q₈	C₂²×C₅₃	1	1	yes	characteristic
227	C₂×C₄×D₄	C₂²×C₅₃	1	2	yes	—
228	C₂×C₄×D₄	C₂²×C₅₃	1	1	yes	characteristic
229	C₄×SD₁₆	C₄×C₅₃	1	8	yes	complemented
230	C₂×#107	C₂²×C₅₃	1	2	yes	—
231	C₂×#106	C₂²×C₅₃	1	1	yes	characteristic
232	C₂×#103	C₂²×C₅₃	1	2	yes	—
233	C₂×#103	C₂²×C₅₃	1	1	yes	characteristic
234	C₂×#102	C₂²×C₅₃	1	2	yes	—
235	C₂×#102	C₂²×C₅₃	1	1	yes	characteristic
236	C₂×#97	C₂²×C₅₃	1	1	yes	characteristic
237	C₂×C₄×C₈	C₂²×C₅₃	1	1	yes	characteristic
238	#391	C₂²×C₅₃	1	2	yes	—
239	#389	C₂²×C₅₃	1	2	yes	—
240	#387	—	2	2	no	—
241	#383	—	2	1	no	—
242	#382	—	2	1	no	—
243	C₄×C₄⋊C₄	C₂²×C₅₃	1	1	yes	characteristic
244	C₄×C₄⋊C₄	C₂²×C₅₃	1	2	yes	—
245	C₄×C₄⋊C₄	C₂²×C₅₃	1	1	yes	characteristic
246	C₄×C₂²⋊C₄	—	2	1	no	—
247	C₄×C₂²⋊C₄	—	2	2	no	—
248	#376	C₂²×C₅₃	1	1	yes	characteristic
249	C₂×#95	—	2	1	no	—
250	#340	C₂²×C₅₃	1	2	yes	—
251	#340	C₂²×C₅₃	1	2	yes	—
252	#340	C₂²×C₅₃	1	2	yes	—
253	#336	C₂²×C₅₃	1	2	yes	—
254	#336	C₂²×C₅₃	1	2	yes	—
255	C₅₃	#21130	1	1	yes	characteristic, direct product factor, Sylow
256	C₂²×Q₈	C₂³×C₅₃	1	1	yes	characteristic
257	C₂²×D₄	C₂³×C₅₃	1	1	yes	characteristic
258	C₂³×C₄	—	2	2	no	—
259	C₂³×C₄	—	2	1	no	—
260	C₂×SD₁₆	—	2	4	no	—
261	C₂×SD₁₆	C₂×C₄×C₅₃	1	4	yes	—
262	C₂²×C₈	C₂³×C₅₃	1	2	yes	—
263	C₂²×C₈	C₂³×C₅₃	1	1	yes	characteristic
264	C₂²×C₈	C₂³×C₅₃	1	1	yes	characteristic
265	C₄×Q₈	—	2	2	no	—
266	C₄×Q₈	C₂×C₄×C₅₃	1	4	yes	complemented
267	C₄×D₄	—	2	4	no	—
268	C₄×D₄	—	2	4	no	—
269	C₄×D₄	—	2	2	no	—
270	C₄×D₄	C₂×C₄×C₅₃	1	4	yes	—
271	C₂×C₄⋊C₄	—	2	1	no	—
272	C₂×C₄⋊C₄	—	2	2	no	—
273	C₂×C₄⋊C₄	C₂³×C₅₃	1	2	yes	—
274	C₂×C₄⋊C₄	C₂³×C₅₃	1	1	yes	characteristic
275	C₂×C₄⋊C₄	C₂³×C₅₃	1	1	yes	characteristic
276	C₂×C₄⋊C₄	C₂³×C₅₃	1	2	yes	—
277	C₂×C₂²⋊C₄	—	2	2	no	—
278	C₂×C₂²⋊C₄	—	2	1	no	—
279	C₂×C₄²	—	2	1	no	—
280	C₂×C₄²	—	2	2	no	—
281	C₂×C₄²	—	2	1	no	—
282	C₂×C₄²	—	2	1	no	—
283	C₂×C₄²	C₂³×C₅₃	1	2	yes	—
284	C₂×C₄²	C₂³×C₅₃	1	1	yes	characteristic
285	#107	—	2	4	no	—
286	#106	C₂×C₄×C₅₃	1	4	yes	complemented
287	#103	—	2	4	no	—
288	#103	C₂×C₄×C₅₃	1	4	yes	complemented
289	#102	—	2	4	no	—
290	#102	C₂×C₄×C₅₃	1	4	yes	—
291	#97	—	2	2	no	—
292	C₄×C₈	C₂×C₄×C₅₃	1	4	yes	complemented
293	#95	—	2	2	no	—
294	#95	—	2	1	no	—
295	#95	—	2	1	no	—
296	#95	—	2	2	no	—
297	#95	—	2	2	no	—
298	#95	—	2	2	no	—
299	#95	D₄×C₅₃	1	1	yes	characteristic
300	#95	D₄×C₅₃	1	1	yes	characteristic
301	#95	D₄×C₅₃	1	2	yes	—
302	C₂⁴	—	2	1	no	—
303	C₂×Q₈	—	2	1	no	—
304	C₂×Q₈	—	2	2	no	—
305	C₂×Q₈	—	2	2	no	—
306	C₂×Q₈	C₂²×C₄×C₅₃	1	2	yes	—
307	C₂×D₄	—	2	2	no	—
308	C₂×D₄	—	2	1	no	—
309	C₂×D₄	C₂²×C₄×C₅₃	1	2	yes	—
310	C₂×D₄	#41×C₅₃	1	4	yes	—
311	C₂²×C₄	—	2	1	no	—
312	C₂²×C₄	—	2	1	no	—
313	C₂²×C₄	—	2	2	no	—
314	C₂²×C₄	—	2	1	no	—
315	C₂²×C₄	—	2	8	no	—
316	C₂²×C₄	—	2	4	no	—
317	C₂²×C₄	—	2	2	no	—
318	C₂²×C₄	—	8	2	no	—
319	C₂²×C₄	—	8	2	no	—
320	C₂²×C₄	—	8	2	no	—
321	C₂²×C₄	C₂×D₄×C₅₃	1	2	yes	—
322	C₂²×C₄	C₂×D₄×C₅₃	1	2	yes	—
323	C₂²×C₄	C₂×D₄×C₅₃	1	1	yes	characteristic
324	C₂²×C₄	C₂×D₄×C₅₃	1	1	yes	characteristic
325	C₂²×C₄	C₂⁴×C₅₃	1	1	yes	characteristic
326	SD₁₆	—	2	8	no	—
327	C₂×C₈	—	2	2	no	—
328	C₂×C₈	—	2	2	no	—
329	C₂×C₈	—	2	2	no	—
330	C₂×C₈	—	2	2	no	—
331	C₂×C₈	—	2	2	no	—
332	C₂×C₈	C₂²×C₄×C₅₃	1	2	yes	—
333	C₂×C₈	C₂²×C₄×C₅₃	1	2	yes	—
334	C₄⋊C₄	—	2	4	no	—
335	C₄⋊C₄	—	2	1	no	—
336	C₄⋊C₄	—	2	2	no	—
337	C₄⋊C₄	—	2	1	no	—
338	C₄⋊C₄	—	2	2	no	—
339	C₄⋊C₄	—	2	2	no	—
340	C₄⋊C₄	—	8	2	no	—
341	C₄⋊C₄	C₂²×C₄×C₅₃	1	2	yes	—
342	C₄⋊C₄	C₂²×C₄×C₅₃	1	2	yes	—
343	C₄⋊C₄	#41×C₅₃	1	4	yes	—
344	C₂²⋊C₄	—	2	8	no	—
345	C₂²⋊C₄	—	2	4	no	—
346	C₄²	—	2	4	no	—
347	C₄²	—	2	2	no	—
348	C₄²	—	2	2	no	—
349	C₄²	—	8	1	no	—
350	C₄²	—	8	1	no	—
351	C₄²	—	8	2	no	—
352	C₄²	C₂²×C₄×C₅₃	1	2	yes	—
353	C₄²	#41×C₅₃	1	2	yes	—
354	C₂³	—	2	2	no	—
355	C₂³	—	2	4	no	—
356	C₂³	—	4	1	no	—
357	C₂³	—	4	2	no	—
358	C₂³	—	4	1	no	—
359	C₂³	C₂²×D₄×C₅₃	1	1	yes	characteristic
360	Q₈	—	2	2	no	—
361	Q₈	—	2	2	no	—
362	Q₈	—	4	2	no	—
363	D₄	—	2	2	no	—
364	D₄	—	2	4	no	—
365	D₄	#117×C₅₃	1	4	yes	—
366	C₂×C₄	—	2	1	no	—
367	C₂×C₄	—	2	2	no	—
368	C₂×C₄	—	2	1	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	—	2	2	no	—
374	C₂×C₄	—	2	4	no	—
375	C₂×C₄	—	2	2	no	—
376	C₂×C₄	—	2	2	no	—
377	C₂×C₄	—	2	2	no	—
378	C₂×C₄	—	2	2	no	—
379	C₂×C₄	—	4	2	no	—
380	C₂×C₄	—	4	2	no	—
381	C₂×C₄	—	4	1	no	—
382	C₂×C₄	—	4	2	no	—
383	C₂×C₄	—	4	2	no	—
384	C₂×C₄	—	4	1	no	—
385	C₂×C₄	—	4	2	no	—
386	C₂×C₄	—	4	2	no	—
387	C₂×C₄	—	4	2	no	—
388	C₂×C₄	—	8	4	no	—
389	C₂×C₄	—	8	8	no	—
390	C₂×C₄	C₄×D₄×C₅₃	1	2	yes	—
391	C₂×C₄	C₄×D₄×C₅₃	1	2	yes	—
392	C₂×C₄	C₂³×C₄×C₅₃	1	1	yes	characteristic
393	C₂×C₄	C₂×#41×C₅₃	1	2	yes	—
394	C₂×C₄	C₂×#41×C₅₃	1	2	yes	—
395	C₂×C₄	C₂×#41×C₅₃	1	1	yes	characteristic
396	C₈	—	2	2	no	—
397	C₈	—	2	2	no	—
398	C₈	—	4	2	no	—
399	C₂²	—	2	4	no	—
400	C₂²	—	4	2	no	—
401	C₂²	—	4	2	no	—
402	C₂²	—	4	4	no	—
403	C₂²	—	4	4	no	—
404	C₂²	C₂×C₄×D₄×C₅₃	1	1	yes	characteristic
405	C₂²	C₅₃×#525	1	2	yes	—
406	C₂²	C₅₃×#576	1	1	yes	characteristic
407	C₂²	C₅₃×#577	1	2	yes	—
408	C₂²	C₅₃×#578	1	1	yes	characteristic
409	C₄	—	2	2	no	—
410	C₄	—	2	2	no	—
411	C₄	—	4	2	no	—
412	C₄	—	4	2	no	—
413	C₄	—	4	2	no	—
414	C₄	—	4	2	no	—
415	C₄	—	4	4	no	—
416	C₄	—	4	2	no	—
417	C₄	—	8	2	no	—
418	C₄	C₂×#117×C₅₃	1	1	yes	characteristic
419	C₄	C₄×#41×C₅₃	1	1	yes	characteristic
420	C₄	C₅₃×#529	1	2	yes	—
421	C₂	—	4	4	no	—
422	C₂	C₅₃×#2290	1	1	yes	characteristic
423	C₂	C₅₃×#2949	1	1	yes	characteristic
424	C₂	C₅₃×#2951	1	1	yes	characteristic
425	C₂	C₅₃×#3118	1	2	yes	—
426	C₂	C₅₃×#3120	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
4. C₅₃×#1928	C₂	semidirect
5. C₅₃×#1920	C₂	semidirect
8. C₄×#107×C₅₃	C₂	semidirect
9. C₅₃×#1763	C₂	semidirect
11. C₅₃×#1746	C₂	semidirect
16. C₄×SD₁₆×C₅₃	C₄	semidirect
52. C₄×Q₈×C₅₃	C₂×C₄	semidirect
72. #106×C₅₃	C₂×C₄	semidirect
74. #103×C₅₃	C₂×C₄	semidirect
78. C₄×C₈×C₅₃	C₂×C₄	semidirect
185. #21130	C₅₃	direct (N × H)
209. #2320	C₂×C₅₃	semidirect
211. #1928	C₂×C₅₃	semidirect
212. #1920	C₂×C₅₃	semidirect
215. C₄×#107	C₂×C₅₃	semidirect
216. #1763	C₂×C₅₃	semidirect
218. #1746	C₂×C₅₃	semidirect
229. C₄×SD₁₆	C₄×C₅₃	semidirect
255. C₅₃	#21130	direct (N × H)
266. C₄×Q₈	C₂×C₄×C₅₃	semidirect
286. #106	C₂×C₄×C₅₃	semidirect
288. #103	C₂×C₄×C₅₃	semidirect
292. C₄×C₈	C₂×C₄×C₅₃	semidirect

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×#2275	2
C₅₃×#1928	2
C₅₃×#1920	2
C₅₃×#1877	2
C₅₃×#1862	2
C₄×#107×C₅₃	2
C₅₃×#1763	2
C₅₃×#1762	2
C₅₃×#1746	2
#21130	53

Subgroups — C53×#21130 #27228642

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅₃×#21130 #27228642