Subgroups — C₅₃×#8882 #27216394

Order 13568.

618 conjugacy classes of subgroups.

The subgroup lattice has 426 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	D₄×C₄⋊C₄×C₅₃	C₂	1	1	yes	maximal, characteristic
2	C₅₃×#1947	C₂	1	1	yes	maximal, characteristic, complemented
3	C₅₃×#1845	C₂	1	1	yes	maximal, characteristic, complemented
4	C₅₃×#1802	C₂	1	1	yes	maximal, characteristic
5	C₂×C₅₃×D₄⋊C₈	C₂	1	1	yes	maximal, characteristic
6	C₅₃×#1296	C₂	1	1	yes	maximal, characteristic, complemented
7	C₅₃×#1279	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×C₄×D₄×C₅₃	—	2	1	no	—
9	C₂×C₄×D₄×C₅₃	C₂²	1	1	yes	characteristic
10	C₂²×C₄⋊C₄×C₅₃	—	2	1	no	—
11	C₂×#107×C₅₃	—	2	1	no	—
12	C₂×#107×C₅₃	—	2	2	no	—
13	C₂×C₄⋊C₈×C₅₃	C₂²	1	1	yes	characteristic
14	C₂×C₄⋊C₈×C₅₃	C₂²	1	1	yes	characteristic
15	C₂×#102×C₅₃	—	2	1	no	—
16	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
17	C₅₃×#389	—	2	1	no	—
18	C₅₃×#385	—	2	2	no	—
19	C₅₃×#384	C₂²	1	1	yes	characteristic
20	C₅₃×#384	C₂²	1	1	yes	characteristic
21	C₅₃×#380	—	2	1	no	—
22	C₄×C₄⋊C₄×C₅₃	C₂²	1	1	yes	characteristic
23	C₅₃×#340	—	2	1	no	—
24	C₅₃×#340	—	2	1	no	—
25	C₅₃×D₄⋊C₈	C₄	1	4	yes	complemented
26	C₂²×D₄×C₅₃	D₄	1	1	yes	characteristic
27	C₂³×C₄×C₅₃	—	2	1	no	—
28	C₂³×C₄×C₅₃	—	2	2	no	—
29	C₂²×C₈×C₅₃	—	2	1	no	—
30	C₂²×C₈×C₅₃	—	2	1	no	—
31	C₂²×C₈×C₅₃	—	2	1	no	—
32	C₄×D₄×C₅₃	—	2	1	no	—
33	C₄×D₄×C₅₃	—	2	1	no	—
34	C₄×D₄×C₅₃	—	4	2	no	—
35	C₄×D₄×C₅₃	—	4	2	no	—
36	C₄×D₄×C₅₃	C₂×C₄	1	2	yes	—
37	C₄×D₄×C₅₃	Q₈	1	2	yes	—
38	C₂×C₄⋊C₄×C₅₃	—	2	2	no	—
39	C₂×C₄⋊C₄×C₅₃	—	2	1	no	—
40	C₂×C₄⋊C₄×C₅₃	—	2	1	no	—
41	C₂×C₄⋊C₄×C₅₃	—	2	1	no	—
42	C₂×C₄⋊C₄×C₅₃	—	2	1	no	—
43	C₂×C₄⋊C₄×C₅₃	—	2	1	no	—
44	C₂×C₄⋊C₄×C₅₃	—	4	2	no	—
45	C₂×C₄⋊C₄×C₅₃	—	4	2	no	—
46	C₂×C₄⋊C₄×C₅₃	D₄	1	1	yes	characteristic
47	C₂×C₄⋊C₄×C₅₃	D₄	1	1	yes	characteristic
48	C₂×C₄⋊C₄×C₅₃	D₄	1	1	yes	characteristic
49	C₂×C₄⋊C₄×C₅₃	D₄	1	2	yes	—
50	C₂×C₂²⋊C₄×C₅₃	—	2	2	no	—
51	C₂×C₂²⋊C₄×C₅₃	—	2	1	no	—
52	C₂×C₄²×C₅₃	—	2	1	no	—
53	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
54	#107×C₅₃	—	4	4	no	—
55	#107×C₅₃	—	4	2	no	—
56	C₄⋊C₈×C₅₃	—	2	1	no	—
57	C₄⋊C₈×C₅₃	—	2	1	no	—
58	C₄⋊C₈×C₅₃	—	2	1	no	—
59	C₄⋊C₈×C₅₃	C₂×C₄	1	2	yes	complemented
60	#102×C₅₃	—	4	2	no	—
61	C₄×C₈×C₅₃	—	2	1	no	—
62	C₄×C₈×C₅₃	C₂×C₄	1	2	yes	complemented
63	#95×C₅₃	—	2	2	no	—
64	#95×C₅₃	—	2	1	no	—
65	C₂⁴×C₅₃	—	2	1	no	—
66	C₂×D₄×C₅₃	—	2	1	no	—
67	C₂×D₄×C₅₃	—	2	1	no	—
68	C₂×D₄×C₅₃	—	2	1	no	—
69	C₂×D₄×C₅₃	C₄⋊C₄	1	2	yes	—
70	C₂×D₄×C₅₃	D₈	1	2	yes	—
71	C₂×D₄×C₅₃	Q₁₆	1	2	yes	—
72	C₂²×C₄×C₅₃	—	2	1	no	—
73	C₂²×C₄×C₅₃	—	2	1	no	—
74	C₂²×C₄×C₅₃	—	2	2	no	—
75	C₂²×C₄×C₅₃	—	2	2	no	—
76	C₂²×C₄×C₅₃	—	2	1	no	—
77	C₂²×C₄×C₅₃	—	2	2	no	—
78	C₂²×C₄×C₅₃	—	4	4	no	—
79	C₂²×C₄×C₅₃	—	4	1	no	—
80	C₂²×C₄×C₅₃	—	4	1	no	—
81	C₂²×C₄×C₅₃	—	4	1	no	—
82	C₂²×C₄×C₅₃	—	4	1	no	—
83	C₂²×C₄×C₅₃	—	4	4	no	—
84	C₂²×C₄×C₅₃	—	4	1	no	—
85	C₂²×C₄×C₅₃	—	8	2	no	—
86	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
87	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
88	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
89	C₂×C₈×C₅₃	—	2	2	no	—
90	C₂×C₈×C₅₃	—	2	2	no	—
91	C₂×C₈×C₅₃	—	2	2	no	—
92	C₂×C₈×C₅₃	—	2	2	no	—
93	C₂×C₈×C₅₃	—	4	1	no	—
94	C₂×C₈×C₅₃	—	4	1	no	—
95	C₂×C₈×C₅₃	—	4	1	no	—
96	C₂×C₈×C₅₃	—	4	1	no	—
97	C₂×C₈×C₅₃	—	4	1	no	—
98	C₄⋊C₄×C₅₃	—	2	1	no	—
99	C₄⋊C₄×C₅₃	—	2	1	no	—
100	C₄⋊C₄×C₅₃	—	2	1	no	—
101	C₄⋊C₄×C₅₃	—	2	1	no	—
102	C₄⋊C₄×C₅₃	—	4	4	no	—
103	C₄⋊C₄×C₅₃	—	4	2	no	—
104	C₄⋊C₄×C₅₃	—	4	2	no	—
105	C₄⋊C₄×C₅₃	—	4	1	no	—
106	C₄⋊C₄×C₅₃	—	4	1	no	—
107	C₄⋊C₄×C₅₃	—	4	1	no	—
108	C₄⋊C₄×C₅₃	—	4	1	no	—
109	C₄⋊C₄×C₅₃	—	4	1	no	—
110	C₄⋊C₄×C₅₃	—	4	1	no	—
111	C₄⋊C₄×C₅₃	—	4	1	no	—
112	C₄⋊C₄×C₅₃	—	4	1	no	—
113	C₄⋊C₄×C₅₃	C₄⋊C₄	1	2	yes	—
114	C₄⋊C₄×C₅₃	D₈	1	2	yes	—
115	C₂²⋊C₄×C₅₃	—	2	2	no	—
116	C₂²⋊C₄×C₅₃	—	2	2	no	—
117	C₂²⋊C₄×C₅₃	—	4	4	no	—
118	C₄²×C₅₃	—	4	1	no	—
119	C₄²×C₅₃	—	4	1	no	—
120	C₄²×C₅₃	C₂²×C₄	1	1	yes	characteristic
121	C₄²×C₅₃	C₂×Q₈	1	1	yes	characteristic
122	C₄²×C₅₃	#41	1	1	yes	characteristic
123	C₄²×C₅₃	#41	1	1	yes	characteristic
124	C₂³×C₅₃	—	2	2	no	—
125	C₂³×C₅₃	—	2	2	no	—
126	C₂³×C₅₃	—	2	2	no	—
127	C₂³×C₅₃	—	4	1	no	—
128	C₂³×C₅₃	—	4	1	no	—
129	C₂³×C₅₃	—	4	1	no	—
130	C₂³×C₅₃	—	4	1	no	—
131	C₂³×C₅₃	#120	1	1	yes	characteristic
132	D₄×C₅₃	—	2	2	no	—
133	D₄×C₅₃	—	2	2	no	—
134	D₄×C₅₃	—	2	2	no	—
135	D₄×C₅₃	#107	1	4	yes	complemented
136	C₂×C₄×C₅₃	—	2	1	no	—
137	C₂×C₄×C₅₃	—	2	1	no	—
138	C₂×C₄×C₅₃	—	2	2	no	—
139	C₂×C₄×C₅₃	—	2	2	no	—
140	C₂×C₄×C₅₃	—	4	1	no	—
141	C₂×C₄×C₅₃	—	4	1	no	—
142	C₂×C₄×C₅₃	—	4	1	no	—
143	C₂×C₄×C₅₃	—	4	1	no	—
144	C₂×C₄×C₅₃	—	4	2	no	—
145	C₂×C₄×C₅₃	—	4	2	no	—
146	C₂×C₄×C₅₃	—	4	2	no	—
147	C₂×C₄×C₅₃	—	4	2	no	—
148	C₂×C₄×C₅₃	—	4	2	no	—
149	C₂×C₄×C₅₃	—	4	2	no	—
150	C₂×C₄×C₅₃	—	4	1	no	—
151	C₂×C₄×C₅₃	—	4	1	no	—
152	C₂×C₄×C₅₃	—	4	2	no	—
153	C₂×C₄×C₅₃	—	4	1	no	—
154	C₂×C₄×C₅₃	—	8	8	no	—
155	C₂×C₄×C₅₃	—	8	1	no	—
156	C₂×C₄×C₅₃	—	8	1	no	—
157	C₂×C₄×C₅₃	—	8	1	no	—
158	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
159	C₂×C₄×C₅₃	C₄×D₄	1	1	yes	characteristic
160	C₂×C₄×C₅₃	C₄×D₄	1	1	yes	characteristic
161	C₂×C₄×C₅₃	C₂²⋊Q₈	1	1	yes	characteristic
162	C₂×C₄×C₅₃	C₂²⋊Q₈	1	1	yes	characteristic
163	C₂×C₄×C₅₃	#123	1	1	yes	characteristic
164	C₂×C₄×C₅₃	C₂×D₈	1	1	yes	characteristic
165	C₂×C₄×C₅₃	C₂×D₈	1	1	yes	characteristic
166	C₂×C₄×C₅₃	C₂×Q₁₆	1	1	yes	characteristic
167	C₂×C₄×C₅₃	#135	1	1	yes	characteristic
168	C₂×C₄×C₅₃	AGL(1,ℤ/8)	1	1	yes	characteristic
169	C₂×C₄×C₅₃	AGL(1,ℤ/8)	1	1	yes	characteristic
170	C₂×C₄×C₅₃	#137	1	1	yes	characteristic
171	C₂×C₄×C₅₃	#137	1	1	yes	characteristic
172	C₈×C₅₃	—	2	4	no	—
173	C₈×C₅₃	—	4	2	no	—
174	C₈×C₅₃	—	8	1	no	—
175	#8882	C₅₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
176	C₂²×C₅₃	—	2	4	no	—
177	C₂²×C₅₃	—	4	2	no	—
178	C₂²×C₅₃	—	4	2	no	—
179	C₂²×C₅₃	—	4	2	no	—
180	C₂²×C₅₃	—	4	2	no	—
181	C₂²×C₅₃	—	4	2	no	—
182	C₂²×C₅₃	—	4	2	no	—
183	C₂²×C₅₃	#385	1	1	yes	characteristic
184	C₂²×C₅₃	#447	1	1	yes	characteristic
185	C₂²×C₅₃	#447	1	1	yes	characteristic
186	C₂²×C₅₃	#451	1	1	yes	characteristic
187	C₂²×C₅₃	#452	1	1	yes	characteristic
188	C₂²×C₅₃	#453	1	1	yes	characteristic
189	C₂²×C₅₃	#456	1	1	yes	characteristic
190	C₄×C₅₃	—	2	1	no	—
191	C₄×C₅₃	—	2	1	no	—
192	C₄×C₅₃	—	4	2	no	—
193	C₄×C₅₃	—	4	2	no	—
194	C₄×C₅₃	—	8	2	no	—
195	C₄×C₅₃	—	8	1	no	—
196	C₄×C₅₃	—	16	1	no	—
197	C₄×C₅₃	C₂×#107	1	1	yes	characteristic
198	C₄×C₅₃	#428	1	1	yes	characteristic
199	C₄×C₅₃	C₄×D₈	1	1	yes	characteristic
200	C₄×C₅₃	#440	1	1	yes	characteristic
201	C₄×C₅₃	#474	1	1	yes	characteristic
202	C₄×C₅₃	#478	1	1	yes	characteristic
203	C₄×C₅₃	#480	1	1	yes	characteristic
204	C₄×C₅₃	#484	1	1	yes	characteristic
205	D₄×C₄⋊C₄	C₂×C₅₃	1	1	yes	characteristic
206	#1947	C₂×C₅₃	1	1	yes	characteristic, complemented
207	#1845	C₂×C₅₃	1	1	yes	characteristic, complemented
208	#1802	C₂×C₅₃	1	1	yes	characteristic
209	C₂×D₄⋊C₈	C₂×C₅₃	1	1	yes	characteristic
210	#1296	C₂×C₅₃	1	1	yes	characteristic, complemented
211	#1279	C₂×C₅₃	1	1	yes	characteristic, complemented
212	C₂×C₅₃	—	4	4	no	—
213	C₂×C₅₃	#1619	1	1	yes	characteristic
214	C₂×C₅₃	#1632	1	1	yes	characteristic
215	C₂×C₅₃	#1647	1	1	yes	characteristic
216	C₂×C₅₃	#1649	1	1	yes	characteristic
217	C₂×C₅₃	#1852	1	1	yes	characteristic
218	C₂×C₅₃	#1859	1	1	yes	characteristic
219	C₂×C₅₃	#1862	1	1	yes	characteristic
220	C₂×C₄×D₄	—	2	1	no	—
221	C₂×C₄×D₄	C₂²×C₅₃	1	1	yes	characteristic
222	C₂²×C₄⋊C₄	—	2	1	no	—
223	C₂×#107	—	2	2	no	—
224	C₂×#107	—	2	1	no	—
225	C₂×C₄⋊C₈	C₂²×C₅₃	1	1	yes	characteristic
226	C₂×C₄⋊C₈	C₂²×C₅₃	1	1	yes	characteristic
227	C₂×#102	—	2	1	no	—
228	C₂×C₄×C₈	C₂²×C₅₃	1	1	yes	characteristic
229	#389	—	2	1	no	—
230	#385	—	2	2	no	—
231	#384	C₂²×C₅₃	1	1	yes	characteristic
232	#384	C₂²×C₅₃	1	1	yes	characteristic
233	#380	—	2	1	no	—
234	C₄×C₄⋊C₄	C₂²×C₅₃	1	1	yes	characteristic
235	#340	—	2	1	no	—
236	#340	—	2	1	no	—
237	D₄⋊C₈	C₄×C₅₃	1	4	yes	complemented
238	C₅₃	#8882	1	1	yes	characteristic, direct product factor, Sylow
239	C₂²×D₄	D₄×C₅₃	1	1	yes	characteristic
240	C₂³×C₄	—	2	1	no	—
241	C₂³×C₄	—	2	2	no	—
242	C₂²×C₈	—	2	1	no	—
243	C₂²×C₈	—	2	1	no	—
244	C₂²×C₈	—	2	1	no	—
245	C₄×D₄	—	2	1	no	—
246	C₄×D₄	—	2	1	no	—
247	C₄×D₄	—	4	2	no	—
248	C₄×D₄	—	4	2	no	—
249	C₄×D₄	C₂×C₄×C₅₃	1	2	yes	—
250	C₄×D₄	Q₈×C₅₃	1	2	yes	—
251	C₂×C₄⋊C₄	—	2	2	no	—
252	C₂×C₄⋊C₄	—	2	1	no	—
253	C₂×C₄⋊C₄	—	2	1	no	—
254	C₂×C₄⋊C₄	—	2	1	no	—
255	C₂×C₄⋊C₄	—	2	1	no	—
256	C₂×C₄⋊C₄	—	2	1	no	—
257	C₂×C₄⋊C₄	—	4	2	no	—
258	C₂×C₄⋊C₄	—	4	2	no	—
259	C₂×C₄⋊C₄	D₄×C₅₃	1	1	yes	characteristic
260	C₂×C₄⋊C₄	D₄×C₅₃	1	1	yes	characteristic
261	C₂×C₄⋊C₄	D₄×C₅₃	1	1	yes	characteristic
262	C₂×C₄⋊C₄	D₄×C₅₃	1	2	yes	—
263	C₂×C₂²⋊C₄	—	2	1	no	—
264	C₂×C₂²⋊C₄	—	2	2	no	—
265	C₂×C₄²	—	2	1	no	—
266	C₂×C₄²	C₂³×C₅₃	1	1	yes	characteristic
267	#107	—	4	4	no	—
268	#107	—	4	2	no	—
269	C₄⋊C₈	—	2	1	no	—
270	C₄⋊C₈	—	2	1	no	—
271	C₄⋊C₈	—	2	1	no	—
272	C₄⋊C₈	C₂×C₄×C₅₃	1	2	yes	complemented
273	#102	—	4	2	no	—
274	C₄×C₈	—	2	1	no	—
275	C₄×C₈	C₂×C₄×C₅₃	1	2	yes	complemented
276	#95	—	2	2	no	—
277	#95	—	2	1	no	—
278	C₂⁴	—	2	1	no	—
279	C₂×D₄	—	2	1	no	—
280	C₂×D₄	—	2	1	no	—
281	C₂×D₄	—	2	1	no	—
282	C₂×D₄	C₄⋊C₄×C₅₃	1	2	yes	—
283	C₂×D₄	D₈×C₅₃	1	2	yes	—
284	C₂×D₄	Q₁₆×C₅₃	1	2	yes	—
285	C₂²×C₄	—	2	1	no	—
286	C₂²×C₄	—	2	1	no	—
287	C₂²×C₄	—	2	2	no	—
288	C₂²×C₄	—	2	2	no	—
289	C₂²×C₄	—	2	1	no	—
290	C₂²×C₄	—	2	2	no	—
291	C₂²×C₄	—	4	4	no	—
292	C₂²×C₄	—	4	1	no	—
293	C₂²×C₄	—	4	1	no	—
294	C₂²×C₄	—	4	1	no	—
295	C₂²×C₄	—	4	1	no	—
296	C₂²×C₄	—	4	4	no	—
297	C₂²×C₄	—	4	1	no	—
298	C₂²×C₄	—	8	2	no	—
299	C₂²×C₄	C₂×D₄×C₅₃	1	1	yes	characteristic
300	C₂²×C₄	C₂×D₄×C₅₃	1	1	yes	characteristic
301	C₂²×C₄	C₂×D₄×C₅₃	1	1	yes	characteristic
302	C₂×C₈	—	2	2	no	—
303	C₂×C₈	—	2	2	no	—
304	C₂×C₈	—	2	2	no	—
305	C₂×C₈	—	2	2	no	—
306	C₂×C₈	—	4	1	no	—
307	C₂×C₈	—	4	1	no	—
308	C₂×C₈	—	4	1	no	—
309	C₂×C₈	—	4	1	no	—
310	C₂×C₈	—	4	1	no	—
311	C₄⋊C₄	—	2	1	no	—
312	C₄⋊C₄	—	2	1	no	—
313	C₄⋊C₄	—	2	1	no	—
314	C₄⋊C₄	—	2	1	no	—
315	C₄⋊C₄	—	4	2	no	—
316	C₄⋊C₄	—	4	4	no	—
317	C₄⋊C₄	—	4	2	no	—
318	C₄⋊C₄	—	4	1	no	—
319	C₄⋊C₄	—	4	1	no	—
320	C₄⋊C₄	—	4	1	no	—
321	C₄⋊C₄	—	4	1	no	—
322	C₄⋊C₄	—	4	1	no	—
323	C₄⋊C₄	—	4	1	no	—
324	C₄⋊C₄	—	4	1	no	—
325	C₄⋊C₄	—	4	1	no	—
326	C₄⋊C₄	C₄⋊C₄×C₅₃	1	2	yes	—
327	C₄⋊C₄	D₈×C₅₃	1	2	yes	—
328	C₂²⋊C₄	—	2	2	no	—
329	C₂²⋊C₄	—	2	2	no	—
330	C₂²⋊C₄	—	4	4	no	—
331	C₄²	—	4	1	no	—
332	C₄²	—	4	1	no	—
333	C₄²	C₂²×C₄×C₅₃	1	1	yes	characteristic
334	C₄²	C₂×Q₈×C₅₃	1	1	yes	characteristic
335	C₄²	#41×C₅₃	1	1	yes	characteristic
336	C₄²	#41×C₅₃	1	1	yes	characteristic
337	C₂³	—	2	2	no	—
338	C₂³	—	2	2	no	—
339	C₂³	—	2	2	no	—
340	C₂³	—	4	1	no	—
341	C₂³	—	4	1	no	—
342	C₂³	—	4	1	no	—
343	C₂³	—	4	1	no	—
344	C₂³	#120×C₅₃	1	1	yes	characteristic
345	D₄	—	2	2	no	—
346	D₄	—	2	2	no	—
347	D₄	—	2	2	no	—
348	D₄	#107×C₅₃	1	4	yes	complemented
349	C₂×C₄	—	2	1	no	—
350	C₂×C₄	—	2	1	no	—
351	C₂×C₄	—	2	2	no	—
352	C₂×C₄	—	2	2	no	—
353	C₂×C₄	—	4	1	no	—
354	C₂×C₄	—	4	1	no	—
355	C₂×C₄	—	4	1	no	—
356	C₂×C₄	—	4	1	no	—
357	C₂×C₄	—	4	2	no	—
358	C₂×C₄	—	4	2	no	—
359	C₂×C₄	—	4	2	no	—
360	C₂×C₄	—	4	2	no	—
361	C₂×C₄	—	4	1	no	—
362	C₂×C₄	—	4	2	no	—
363	C₂×C₄	—	4	2	no	—
364	C₂×C₄	—	4	1	no	—
365	C₂×C₄	—	4	2	no	—
366	C₂×C₄	—	4	1	no	—
367	C₂×C₄	—	8	8	no	—
368	C₂×C₄	—	8	1	no	—
369	C₂×C₄	—	8	1	no	—
370	C₂×C₄	—	8	1	no	—
371	C₂×C₄	C₂×C₄⋊C₄×C₅₃	1	1	yes	characteristic
372	C₂×C₄	C₄×D₄×C₅₃	1	1	yes	characteristic
373	C₂×C₄	C₄×D₄×C₅₃	1	1	yes	characteristic
374	C₂×C₄	C₂²⋊Q₈×C₅₃	1	1	yes	characteristic
375	C₂×C₄	C₂²⋊Q₈×C₅₃	1	1	yes	characteristic
376	C₂×C₄	#123×C₅₃	1	1	yes	characteristic
377	C₂×C₄	C₂×D₈×C₅₃	1	1	yes	characteristic
378	C₂×C₄	C₂×D₈×C₅₃	1	1	yes	characteristic
379	C₂×C₄	C₂×Q₁₆×C₅₃	1	1	yes	characteristic
380	C₂×C₄	#135×C₅₃	1	1	yes	characteristic
381	C₂×C₄	AGL(1,ℤ/8)×C₅₃	1	1	yes	characteristic
382	C₂×C₄	AGL(1,ℤ/8)×C₅₃	1	1	yes	characteristic
383	C₂×C₄	#137×C₅₃	1	1	yes	characteristic
384	C₂×C₄	#137×C₅₃	1	1	yes	characteristic
385	C₈	—	2	4	no	—
386	C₈	—	4	2	no	—
387	C₈	—	8	1	no	—
388	C₂²	—	2	4	no	—
389	C₂²	—	4	2	no	—
390	C₂²	—	4	2	no	—
391	C₂²	—	4	2	no	—
392	C₂²	—	4	2	no	—
393	C₂²	—	4	2	no	—
394	C₂²	—	4	2	no	—
395	C₂²	C₅₃×#385	1	1	yes	characteristic
396	C₂²	C₅₃×#447	1	1	yes	characteristic
397	C₂²	C₅₃×#447	1	1	yes	characteristic
398	C₂²	C₅₃×#451	1	1	yes	characteristic
399	C₂²	C₅₃×#452	1	1	yes	characteristic
400	C₂²	C₅₃×#453	1	1	yes	characteristic
401	C₂²	C₅₃×#456	1	1	yes	characteristic
402	C₄	—	2	1	no	—
403	C₄	—	2	1	no	—
404	C₄	—	4	2	no	—
405	C₄	—	4	2	no	—
406	C₄	—	8	2	no	—
407	C₄	—	8	1	no	—
408	C₄	—	16	1	no	—
409	C₄	C₂×#107×C₅₃	1	1	yes	characteristic
410	C₄	C₅₃×#428	1	1	yes	characteristic
411	C₄	C₄×D₈×C₅₃	1	1	yes	characteristic
412	C₄	C₅₃×#440	1	1	yes	characteristic
413	C₄	C₅₃×#474	1	1	yes	characteristic
414	C₄	C₅₃×#478	1	1	yes	characteristic
415	C₄	C₅₃×#480	1	1	yes	characteristic
416	C₄	C₅₃×#484	1	1	yes	characteristic
417	C₂	—	4	4	no	—
418	C₂	C₅₃×#1619	1	1	yes	characteristic
419	C₂	C₅₃×#1632	1	1	yes	characteristic
420	C₂	C₅₃×#1647	1	1	yes	characteristic
421	C₂	C₅₃×#1649	1	1	yes	characteristic
422	C₂	C₅₃×#1852	1	1	yes	characteristic
423	C₂	C₅₃×#1859	1	1	yes	characteristic
424	C₂	C₅₃×#1862	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂³×C₅₃	#120
commutator	C₄²	C₂²×C₄×C₅₃
Frattini, cosocle	C₂×C₄²	C₂³×C₅₃
prime radical	C₂²×D₄×C₅₃	D₄
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³×C₅₃ ◁ C₂×C₄²×C₅₃ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₅₃×#1947	C₂	semidirect
3. C₅₃×#1845	C₂	semidirect
6. C₅₃×#1296	C₂	semidirect
7. C₅₃×#1279	C₂	semidirect
25. C₅₃×D₄⋊C₈	C₄	semidirect
59. C₄⋊C₈×C₅₃	C₂×C₄	semidirect
62. C₄×C₈×C₅₃	C₂×C₄	semidirect
135. D₄×C₅₃	#107	semidirect
175. #8882	C₅₃	direct (N × H)
206. #1947	C₂×C₅₃	semidirect
207. #1845	C₂×C₅₃	semidirect
210. #1296	C₂×C₅₃	semidirect
211. #1279	C₂×C₅₃	semidirect
237. D₄⋊C₈	C₄×C₅₃	semidirect
238. C₅₃	#8882	direct (N × H)
272. C₄⋊C₈	C₂×C₄×C₅₃	semidirect
275. C₄×C₈	C₂×C₄×C₅₃	semidirect
348. D₄	#107×C₅₃	semidirect

Maximal subgroups

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

subgroup	index
D₄×C₄⋊C₄×C₅₃	2
C₅₃×#1947	2
C₅₃×#1845	2
C₅₃×#1802	2
C₂×C₅₃×D₄⋊C₈	2
C₅₃×#1296	2
C₅₃×#1279	2
#8882	53

Subgroups — C53×#8882 #27216394

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅₃×#8882 #27216394