Subgroups — C₅₃×#22232 #27229744

Order 13568.

838 conjugacy classes of subgroups.

The subgroup lattice has 416 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₅₃×#490	C₂	1	1	yes	maximal, characteristic
2	C₅₃×#2330	C₂	1	1	yes	maximal, characteristic, complemented
3	C₅₃×#2328	C₂	1	1	yes	maximal, characteristic
4	C₅₃×#1991	C₂	1	1	yes	maximal, characteristic, complemented
5	C₅₃×#1957	C₂	1	1	yes	maximal, characteristic
6	C₅₃×#1926	C₂	1	2	yes	maximal, complemented
7	C₅₃×#1925	C₂	1	2	yes	maximal, complemented
8	C₅₃×#1864	C₂	1	2	yes	maximal, complemented
9	C₅₃×#1862	C₂	1	2	yes	maximal, complemented
10	C₅₃×#1847	C₂	1	1	yes	maximal, characteristic, complemented
11	C₅₃×#1746	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×#127×C₅₃	C₂²	1	1	yes	characteristic
13	C₂×#125×C₅₃	C₂²	1	1	yes	characteristic
14	C₂×C₄×D₄×C₅₃	C₂²	1	1	yes	characteristic
15	C₂×C₄×D₄×C₅₃	C₂²	1	1	yes	characteristic
16	C₅₃×#490	C₄	1	8	yes	complemented
17	C₂×#107×C₅₃	C₂²	1	2	yes	—
18	C₂×#106×C₅₃	C₂²	1	2	yes	—
19	C₂×#102×C₅₃	C₂²	1	2	yes	—
20	C₂×#102×C₅₃	C₂²	1	2	yes	—
21	C₂×#102×C₅₃	C₂²	1	2	yes	—
22	C₂×#102×C₅₃	C₂²	1	2	yes	—
23	C₂×#97×C₅₃	C₂²	1	1	yes	characteristic
24	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
25	C₅₃×#390	C₂²	1	1	yes	characteristic
26	C₅₃×#390	C₂²	1	1	yes	characteristic
27	C₅₃×#389	—	2	2	no	—
28	C₅₃×#389	C₂²	1	2	yes	—
29	C₅₃×#387	—	2	2	no	—
30	C₅₃×#386	—	2	2	no	—
31	C₅₃×#386	—	2	2	no	—
32	C₅₃×#384	C₂²	1	1	yes	characteristic
33	C₅₃×#382	C₂²	1	1	yes	characteristic
34	C₄×C₄⋊C₄×C₅₃	C₂²	1	2	yes	—
35	C₅₃×#376	C₂²	1	1	yes	characteristic
36	C₅₃×#340	C₂²	1	2	yes	—
37	C₅₃×#340	C₂²	1	2	yes	—
38	C₅₃×#336	C₂²	1	2	yes	—
39	C₅₃×#336	C₂²	1	2	yes	—
40	C₂²×D₄×C₅₃	—	2	2	no	—
41	C₂²×D₄×C₅₃	C₂³	1	1	yes	characteristic
42	C₂²×D₄×C₅₃	C₂³	1	1	yes	characteristic
43	C₂³×C₄×C₅₃	—	2	1	no	—
44	C₂³×C₄×C₅₃	—	2	1	no	—
45	C₂²×C₈×C₅₃	C₂³	1	2	yes	—
46	C₂²×C₈×C₅₃	C₂³	1	2	yes	—
47	#127×C₅₃	—	2	2	no	—
48	#127×C₅₃	C₂×C₄	1	4	yes	—
49	#125×C₅₃	—	2	2	no	—
50	#125×C₅₃	C₂×C₄	1	4	yes	complemented
51	C₄×D₄×C₅₃	—	4	2	no	—
52	C₄×D₄×C₅₃	—	4	2	no	—
53	C₂×C₄⋊C₄×C₅₃	—	2	2	no	—
54	C₂×C₄⋊C₄×C₅₃	—	2	2	no	—
55	C₂×C₄⋊C₄×C₅₃	—	2	2	no	—
56	C₂×C₄⋊C₄×C₅₃	C₂³	1	1	yes	characteristic
57	C₂×C₄⋊C₄×C₅₃	C₂³	1	1	yes	characteristic
58	C₂×C₄⋊C₄×C₅₃	C₂³	1	2	yes	—
59	C₂×C₄⋊C₄×C₅₃	C₂³	1	2	yes	—
60	C₂×C₂²⋊C₄×C₅₃	—	2	1	no	—
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	2	no	—
65	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
66	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
67	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
68	#107×C₅₃	—	4	2	no	—
69	#106×C₅₃	—	4	2	no	—
70	#102×C₅₃	—	4	2	no	—
71	#102×C₅₃	—	4	2	no	—
72	#102×C₅₃	C₂×C₄	1	8	yes	complemented
73	#102×C₅₃	C₂×C₄	1	8	yes	complemented
74	#97×C₅₃	C₂×C₄	1	4	yes	complemented
75	C₄×C₈×C₅₃	—	2	2	no	—
76	#95×C₅₃	—	2	2	no	—
77	#95×C₅₃	—	2	2	no	—
78	#95×C₅₃	—	2	2	no	—
79	#95×C₅₃	D₄	1	2	yes	—
80	#95×C₅₃	D₄	1	2	yes	—
81	C₂⁴×C₅₃	—	2	1	no	—
82	C₂⁴×C₅₃	—	2	1	no	—
83	C₂×D₄×C₅₃	—	2	8	no	—
84	C₂×D₄×C₅₃	—	2	1	no	—
85	C₂×D₄×C₅₃	—	2	1	no	—
86	C₂×D₄×C₅₃	—	2	2	no	—
87	C₂×D₄×C₅₃	—	2	2	no	—
88	C₂×D₄×C₅₃	—	2	2	no	—
89	C₂×D₄×C₅₃	—	2	2	no	—
90	C₂×D₄×C₅₃	—	8	2	no	—
91	C₂×D₄×C₅₃	—	8	2	no	—
92	C₂×D₄×C₅₃	C₂²×C₄	1	2	yes	—
93	C₂×D₄×C₅₃	C₂²×C₄	1	2	yes	—
94	C₂²×C₄×C₅₃	—	2	2	no	—
95	C₂²×C₄×C₅₃	—	2	2	no	—
96	C₂²×C₄×C₅₃	—	2	1	no	—
97	C₂²×C₄×C₅₃	—	2	1	no	—
98	C₂²×C₄×C₅₃	—	4	2	no	—
99	C₂²×C₄×C₅₃	—	4	2	no	—
100	C₂²×C₄×C₅₃	—	8	2	no	—
101	C₂²×C₄×C₅₃	—	8	2	no	—
102	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
103	C₂²×C₄×C₅₃	C₂×D₄	1	2	yes	—
104	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
105	C₂²×C₄×C₅₃	C₂×D₄	1	2	yes	—
106	C₂²×C₄×C₅₃	C₂⁴	1	1	yes	characteristic
107	C₂×C₈×C₅₃	—	2	2	no	—
108	C₂×C₈×C₅₃	—	2	2	no	—
109	C₂×C₈×C₅₃	—	2	2	no	—
110	C₂×C₈×C₅₃	—	2	2	no	—
111	C₂×C₈×C₅₃	—	2	2	no	—
112	C₂×C₈×C₅₃	C₂²×C₄	1	4	yes	—
113	C₄⋊C₄×C₅₃	—	2	8	no	—
114	C₄⋊C₄×C₅₃	—	2	2	no	—
115	C₄⋊C₄×C₅₃	—	2	2	no	—
116	C₄⋊C₄×C₅₃	—	2	2	no	—
117	C₄⋊C₄×C₅₃	—	2	2	no	—
118	C₄⋊C₄×C₅₃	—	2	2	no	—
119	C₄⋊C₄×C₅₃	—	8	2	no	—
120	C₄⋊C₄×C₅₃	—	8	2	no	—
121	C₄⋊C₄×C₅₃	C₂²×C₄	1	4	yes	—
122	C₂²⋊C₄×C₅₃	—	4	2	no	—
123	C₂²⋊C₄×C₅₃	—	4	2	no	—
124	C₂²⋊C₄×C₅₃	—	8	2	no	—
125	C₂²⋊C₄×C₅₃	—	8	2	no	—
126	C₄²×C₅₃	—	2	2	no	—
127	C₄²×C₅₃	—	2	2	no	—
128	C₄²×C₅₃	—	8	2	no	—
129	C₄²×C₅₃	C₂²×C₄	1	2	yes	—
130	C₄²×C₅₃	#41	1	2	yes	—
131	C₂³×C₅₃	—	2	2	no	—
132	C₂³×C₅₃	—	2	2	no	—
133	C₂³×C₅₃	—	4	1	no	—
134	C₂³×C₅₃	—	4	1	no	—
135	C₂³×C₅₃	—	4	1	no	—
136	C₂³×C₅₃	—	4	1	no	—
137	C₂³×C₅₃	—	4	2	no	—
138	C₂³×C₅₃	—	4	2	no	—
139	C₂³×C₅₃	—	4	2	no	—
140	C₂³×C₅₃	—	4	2	no	—
141	C₂³×C₅₃	C₂²×D₄	1	1	yes	characteristic
142	D₄×C₅₃	—	2	4	no	—
143	D₄×C₅₃	—	2	4	no	—
144	D₄×C₅₃	—	4	2	no	—
145	D₄×C₅₃	—	4	2	no	—
146	D₄×C₅₃	—	8	8	no	—
147	C₂×C₄×C₅₃	—	2	2	no	—
148	C₂×C₄×C₅₃	—	2	2	no	—
149	C₂×C₄×C₅₃	—	2	4	no	—
150	C₂×C₄×C₅₃	—	2	1	no	—
151	C₂×C₄×C₅₃	—	2	1	no	—
152	C₂×C₄×C₅₃	—	2	2	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	—	4	2	no	—
158	C₂×C₄×C₅₃	—	4	2	no	—
159	C₂×C₄×C₅₃	—	4	1	no	—
160	C₂×C₄×C₅₃	—	4	1	no	—
161	C₂×C₄×C₅₃	—	4	2	no	—
162	C₂×C₄×C₅₃	—	4	2	no	—
163	C₂×C₄×C₅₃	—	4	2	no	—
164	C₂×C₄×C₅₃	—	4	2	no	—
165	C₂×C₄×C₅₃	—	4	2	no	—
166	C₂×C₄×C₅₃	—	8	4	no	—
167	C₂×C₄×C₅₃	—	8	4	no	—
168	C₂×C₄×C₅₃	C₄×D₄	1	4	yes	—
169	C₂×C₄×C₅₃	C₂³×C₄	1	1	yes	characteristic
170	C₂×C₄×C₅₃	C₂×#41	1	1	yes	characteristic
171	C₂×C₄×C₅₃	H(5,𝔽₂)	1	2	yes	—
172	C₂×C₄×C₅₃	2₋¹⁺⁴	1	2	yes	—
173	C₈×C₅₃	—	2	4	no	—
174	C₈×C₅₃	—	4	2	no	—
175	#22232	C₅₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
176	C₂²×C₅₃	—	4	2	no	—
177	C₂²×C₅₃	—	4	2	no	—
178	C₂²×C₅₃	—	4	4	no	—
179	C₂²×C₅₃	—	4	4	no	—
180	C₂²×C₅₃	—	8	1	no	—
181	C₂²×C₅₃	—	8	1	no	—
182	C₂²×C₅₃	—	8	1	no	—
183	C₂²×C₅₃	—	8	1	no	—
184	C₂²×C₅₃	—	8	2	no	—
185	C₂²×C₅₃	—	8	2	no	—
186	C₂²×C₅₃	C₂×C₄×D₄	1	1	yes	characteristic
187	C₂²×C₅₃	#535	1	1	yes	characteristic
188	C₂²×C₅₃	#536	1	1	yes	characteristic
189	C₂²×C₅₃	#576	1	2	yes	—
190	C₂²×C₅₃	#577	1	2	yes	—
191	C₄×C₅₃	—	2	4	no	—
192	C₄×C₅₃	—	4	4	no	—
193	C₄×C₅₃	—	4	1	no	—
194	C₄×C₅₃	—	4	1	no	—
195	C₄×C₅₃	—	4	2	no	—
196	C₄×C₅₃	—	8	2	no	—
197	C₄×C₅₃	—	8	1	no	—
198	C₄×C₅₃	—	8	1	no	—
199	C₄×C₅₃	#520	1	2	yes	—
200	C₄×C₅₃	#542	1	2	yes	—
201	C₂×#490	C₂×C₅₃	1	1	yes	characteristic
202	#2330	C₂×C₅₃	1	1	yes	characteristic, complemented
203	#2328	C₂×C₅₃	1	1	yes	characteristic
204	#1991	C₂×C₅₃	1	1	yes	characteristic, complemented
205	#1957	C₂×C₅₃	1	1	yes	characteristic
206	#1926	C₂×C₅₃	1	2	yes	complemented
207	#1925	C₂×C₅₃	1	2	yes	complemented
208	#1864	C₂×C₅₃	1	2	yes	complemented
209	#1862	C₂×C₅₃	1	2	yes	complemented
210	#1847	C₂×C₅₃	1	1	yes	characteristic, complemented
211	#1746	C₂×C₅₃	1	1	yes	characteristic, complemented
212	C₂×C₅₃	—	8	2	no	—
213	C₂×C₅₃	—	8	2	no	—
214	C₂×C₅₃	#2326	1	1	yes	characteristic
215	C₂×C₅₃	#2946	1	2	yes	—
216	C₂×C₅₃	#3220	1	1	yes	characteristic
217	C₂×C₅₃	#3222	1	1	yes	characteristic
218	C₂×C₅₃	#3241	1	2	yes	—
219	C₂×#127	C₂²×C₅₃	1	1	yes	characteristic
220	C₂×#125	C₂²×C₅₃	1	1	yes	characteristic
221	C₂×C₄×D₄	C₂²×C₅₃	1	1	yes	characteristic
222	C₂×C₄×D₄	C₂²×C₅₃	1	1	yes	characteristic
223	#490	C₄×C₅₃	1	8	yes	complemented
224	C₂×#107	C₂²×C₅₃	1	2	yes	—
225	C₂×#106	C₂²×C₅₃	1	2	yes	—
226	C₂×#102	C₂²×C₅₃	1	2	yes	—
227	C₂×#102	C₂²×C₅₃	1	2	yes	—
228	C₂×#102	C₂²×C₅₃	1	2	yes	—
229	C₂×#102	C₂²×C₅₃	1	2	yes	—
230	C₂×#97	C₂²×C₅₃	1	1	yes	characteristic
231	C₂×C₄×C₈	C₂²×C₅₃	1	1	yes	characteristic
232	#390	C₂²×C₅₃	1	1	yes	characteristic
233	#390	C₂²×C₅₃	1	1	yes	characteristic
234	#389	—	2	2	no	—
235	#389	C₂²×C₅₃	1	2	yes	—
236	#387	—	2	2	no	—
237	#386	—	2	2	no	—
238	#386	—	2	2	no	—
239	#384	C₂²×C₅₃	1	1	yes	characteristic
240	#382	C₂²×C₅₃	1	1	yes	characteristic
241	C₄×C₄⋊C₄	C₂²×C₅₃	1	2	yes	—
242	#376	C₂²×C₅₃	1	1	yes	characteristic
243	#340	C₂²×C₅₃	1	2	yes	—
244	#340	C₂²×C₅₃	1	2	yes	—
245	#336	C₂²×C₅₃	1	2	yes	—
246	#336	C₂²×C₅₃	1	2	yes	—
247	C₅₃	#22232	1	1	yes	characteristic, direct product factor, Sylow
248	C₂²×D₄	—	2	2	no	—
249	C₂²×D₄	C₂³×C₅₃	1	1	yes	characteristic
250	C₂²×D₄	C₂³×C₅₃	1	1	yes	characteristic
251	C₂³×C₄	—	2	1	no	—
252	C₂³×C₄	—	2	1	no	—
253	C₂²×C₈	C₂³×C₅₃	1	2	yes	—
254	C₂²×C₈	C₂³×C₅₃	1	2	yes	—
255	#127	—	2	2	no	—
256	#127	C₂×C₄×C₅₃	1	4	yes	—
257	#125	—	2	2	no	—
258	#125	C₂×C₄×C₅₃	1	4	yes	complemented
259	C₄×D₄	—	4	2	no	—
260	C₄×D₄	—	4	2	no	—
261	C₂×C₄⋊C₄	—	2	2	no	—
262	C₂×C₄⋊C₄	—	2	2	no	—
263	C₂×C₄⋊C₄	—	2	2	no	—
264	C₂×C₄⋊C₄	C₂³×C₅₃	1	1	yes	characteristic
265	C₂×C₄⋊C₄	C₂³×C₅₃	1	1	yes	characteristic
266	C₂×C₄⋊C₄	C₂³×C₅₃	1	2	yes	—
267	C₂×C₄⋊C₄	C₂³×C₅₃	1	2	yes	—
268	C₂×C₂²⋊C₄	—	2	1	no	—
269	C₂×C₂²⋊C₄	—	2	1	no	—
270	C₂×C₂²⋊C₄	—	2	2	no	—
271	C₂×C₂²⋊C₄	—	2	2	no	—
272	C₂×C₄²	—	2	2	no	—
273	C₂×C₄²	C₂³×C₅₃	1	1	yes	characteristic
274	C₂×C₄²	C₂³×C₅₃	1	1	yes	characteristic
275	C₂×C₄²	C₂³×C₅₃	1	1	yes	characteristic
276	#107	—	4	2	no	—
277	#106	—	4	2	no	—
278	#102	—	4	2	no	—
279	#102	—	4	2	no	—
280	#102	C₂×C₄×C₅₃	1	8	yes	complemented
281	#102	C₂×C₄×C₅₃	1	8	yes	complemented
282	#97	C₂×C₄×C₅₃	1	4	yes	complemented
283	C₄×C₈	—	2	2	no	—
284	#95	—	2	2	no	—
285	#95	—	2	2	no	—
286	#95	—	2	2	no	—
287	#95	D₄×C₅₃	1	2	yes	—
288	#95	D₄×C₅₃	1	2	yes	—
289	C₂⁴	—	2	1	no	—
290	C₂⁴	—	2	1	no	—
291	C₂×D₄	—	2	8	no	—
292	C₂×D₄	—	2	1	no	—
293	C₂×D₄	—	2	1	no	—
294	C₂×D₄	—	2	2	no	—
295	C₂×D₄	—	2	2	no	—
296	C₂×D₄	—	2	2	no	—
297	C₂×D₄	—	2	2	no	—
298	C₂×D₄	—	8	2	no	—
299	C₂×D₄	—	8	2	no	—
300	C₂×D₄	C₂²×C₄×C₅₃	1	2	yes	—
301	C₂×D₄	C₂²×C₄×C₅₃	1	2	yes	—
302	C₂²×C₄	—	2	2	no	—
303	C₂²×C₄	—	2	2	no	—
304	C₂²×C₄	—	2	1	no	—
305	C₂²×C₄	—	2	1	no	—
306	C₂²×C₄	—	4	2	no	—
307	C₂²×C₄	—	4	2	no	—
308	C₂²×C₄	—	8	2	no	—
309	C₂²×C₄	—	8	2	no	—
310	C₂²×C₄	C₂×D₄×C₅₃	1	1	yes	characteristic
311	C₂²×C₄	C₂×D₄×C₅₃	1	2	yes	—
312	C₂²×C₄	C₂×D₄×C₅₃	1	1	yes	characteristic
313	C₂²×C₄	C₂×D₄×C₅₃	1	2	yes	—
314	C₂²×C₄	C₂⁴×C₅₃	1	1	yes	characteristic
315	C₂×C₈	—	2	2	no	—
316	C₂×C₈	—	2	2	no	—
317	C₂×C₈	—	2	2	no	—
318	C₂×C₈	—	2	2	no	—
319	C₂×C₈	—	2	2	no	—
320	C₂×C₈	C₂²×C₄×C₅₃	1	4	yes	—
321	C₄⋊C₄	—	2	8	no	—
322	C₄⋊C₄	—	2	2	no	—
323	C₄⋊C₄	—	2	2	no	—
324	C₄⋊C₄	—	2	2	no	—
325	C₄⋊C₄	—	2	2	no	—
326	C₄⋊C₄	—	2	2	no	—
327	C₄⋊C₄	—	8	2	no	—
328	C₄⋊C₄	—	8	2	no	—
329	C₄⋊C₄	C₂²×C₄×C₅₃	1	4	yes	—
330	C₂²⋊C₄	—	4	2	no	—
331	C₂²⋊C₄	—	4	2	no	—
332	C₂²⋊C₄	—	8	2	no	—
333	C₂²⋊C₄	—	8	2	no	—
334	C₄²	—	2	2	no	—
335	C₄²	—	2	2	no	—
336	C₄²	—	8	2	no	—
337	C₄²	C₂²×C₄×C₅₃	1	2	yes	—
338	C₄²	#41×C₅₃	1	2	yes	—
339	C₂³	—	2	2	no	—
340	C₂³	—	2	2	no	—
341	C₂³	—	4	1	no	—
342	C₂³	—	4	1	no	—
343	C₂³	—	4	1	no	—
344	C₂³	—	4	1	no	—
345	C₂³	—	4	2	no	—
346	C₂³	—	4	2	no	—
347	C₂³	—	4	2	no	—
348	C₂³	—	4	2	no	—
349	C₂³	C₂²×D₄×C₅₃	1	1	yes	characteristic
350	D₄	—	2	4	no	—
351	D₄	—	2	4	no	—
352	D₄	—	4	2	no	—
353	D₄	—	4	2	no	—
354	D₄	—	8	8	no	—
355	C₂×C₄	—	2	2	no	—
356	C₂×C₄	—	2	2	no	—
357	C₂×C₄	—	2	4	no	—
358	C₂×C₄	—	2	1	no	—
359	C₂×C₄	—	2	1	no	—
360	C₂×C₄	—	2	2	no	—
361	C₂×C₄	—	2	2	no	—
362	C₂×C₄	—	2	2	no	—
363	C₂×C₄	—	2	2	no	—
364	C₂×C₄	—	2	2	no	—
365	C₂×C₄	—	4	2	no	—
366	C₂×C₄	—	4	2	no	—
367	C₂×C₄	—	4	1	no	—
368	C₂×C₄	—	4	1	no	—
369	C₂×C₄	—	4	2	no	—
370	C₂×C₄	—	4	2	no	—
371	C₂×C₄	—	4	2	no	—
372	C₂×C₄	—	4	2	no	—
373	C₂×C₄	—	4	2	no	—
374	C₂×C₄	—	8	4	no	—
375	C₂×C₄	—	8	4	no	—
376	C₂×C₄	C₄×D₄×C₅₃	1	4	yes	—
377	C₂×C₄	C₂³×C₄×C₅₃	1	1	yes	characteristic
378	C₂×C₄	C₂×#41×C₅₃	1	1	yes	characteristic
379	C₂×C₄	H(5,𝔽₂)×C₅₃	1	2	yes	—
380	C₂×C₄	2₋¹⁺⁴×C₅₃	1	2	yes	—
381	C₈	—	2	4	no	—
382	C₈	—	4	2	no	—
383	C₂²	—	4	4	no	—
384	C₂²	—	4	4	no	—
385	C₂²	—	4	2	no	—
386	C₂²	—	4	2	no	—
387	C₂²	—	8	1	no	—
388	C₂²	—	8	1	no	—
389	C₂²	—	8	1	no	—
390	C₂²	—	8	1	no	—
391	C₂²	—	8	2	no	—
392	C₂²	—	8	2	no	—
393	C₂²	C₂×C₄×D₄×C₅₃	1	1	yes	characteristic
394	C₂²	C₅₃×#535	1	1	yes	characteristic
395	C₂²	C₅₃×#536	1	1	yes	characteristic
396	C₂²	C₅₃×#576	1	2	yes	—
397	C₂²	C₅₃×#577	1	2	yes	—
398	C₄	—	2	4	no	—
399	C₄	—	4	4	no	—
400	C₄	—	4	1	no	—
401	C₄	—	4	1	no	—
402	C₄	—	4	2	no	—
403	C₄	—	8	1	no	—
404	C₄	—	8	2	no	—
405	C₄	—	8	1	no	—
406	C₄	C₅₃×#520	1	2	yes	—
407	C₄	C₅₃×#542	1	2	yes	—
408	C₂	—	8	2	no	—
409	C₂	—	8	2	no	—
410	C₂	C₅₃×#2326	1	1	yes	characteristic
411	C₂	C₅₃×#2946	1	2	yes	—
412	C₂	C₅₃×#3220	1	1	yes	characteristic
413	C₂	C₅₃×#3222	1	1	yes	characteristic
414	C₂	C₅₃×#3241	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	C₂×#127×C₅₃	C₂²
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×#2330	C₂	semidirect
4. C₅₃×#1991	C₂	semidirect
6. C₅₃×#1926	C₂	semidirect
7. C₅₃×#1925	C₂	semidirect
8. C₅₃×#1864	C₂	semidirect
9. C₅₃×#1862	C₂	semidirect
10. C₅₃×#1847	C₂	semidirect
11. C₅₃×#1746	C₂	semidirect
16. C₅₃×#490	C₄	semidirect
50. #125×C₅₃	C₂×C₄	semidirect
72. #102×C₅₃	C₂×C₄	semidirect
73. #102×C₅₃	C₂×C₄	semidirect
74. #97×C₅₃	C₂×C₄	semidirect
175. #22232	C₅₃	direct (N × H)
202. #2330	C₂×C₅₃	semidirect
204. #1991	C₂×C₅₃	semidirect
206. #1926	C₂×C₅₃	semidirect
207. #1925	C₂×C₅₃	semidirect
208. #1864	C₂×C₅₃	semidirect
209. #1862	C₂×C₅₃	semidirect
210. #1847	C₂×C₅₃	semidirect
211. #1746	C₂×C₅₃	semidirect
223. #490	C₄×C₅₃	semidirect
247. C₅₃	#22232	direct (N × H)
258. #125	C₂×C₄×C₅₃	semidirect
280. #102	C₂×C₄×C₅₃	semidirect
281. #102	C₂×C₄×C₅₃	semidirect
282. #97	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×C₅₃×#490	2
C₅₃×#2330	2
C₅₃×#2328	2
C₅₃×#1991	2
C₅₃×#1957	2
C₅₃×#1926	2
C₅₃×#1925	2
C₅₃×#1864	2
C₅₃×#1862	2
C₅₃×#1847	2
C₅₃×#1746	2
#22232	53

Subgroups — C53×#22232 #27229744

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅₃×#22232 #27229744