Subgroups — C₅₃×#22001 #27229513

Order 13568.

780 conjugacy classes of subgroups.

The subgroup lattice has 420 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₅₃×#2290	C₂	1	1	yes	maximal, characteristic
2	C₅₃×#1924	C₂	1	2	yes	maximal, complemented
3	C₅₃×#1917	C₂	1	2	yes	maximal, complemented
4	C₅₃×#1869	C₂	1	2	yes	maximal
5	C₅₃×#1850	C₂	1	2	yes	maximal, complemented
6	C₅₃×#1840	C₂	1	1	yes	maximal, characteristic, complemented
7	C₅₃×#1832	C₂	1	1	yes	maximal, characteristic, complemented
8	C₅₃×#1818	C₂	1	1	yes	maximal, characteristic, complemented
9	C₅₃×#1798	C₂	1	1	yes	maximal, characteristic, complemented
10	C₅₃×#1749	C₂	1	1	yes	maximal, characteristic, complemented
11	C₂×C₅₃×#336	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂³×C₈×C₅₃	C₂²	1	2	yes	—
13	C₂×C₄×D₄×C₅₃	C₂²	1	1	yes	characteristic
14	C₂×#117×C₅₃	C₂²	1	1	yes	characteristic
15	C₂²×C₄²×C₅₃	C₂²	1	1	yes	characteristic
16	C₂×C₄⋊C₈×C₅₃	C₂²	1	2	yes	—
17	C₂×C₄⋊C₈×C₅₃	C₂²	1	2	yes	—
18	C₂×C₂²⋊C₈×C₅₃	C₂²	1	2	yes	—
19	C₂×C₂²⋊C₈×C₅₃	C₂²	1	2	yes	—
20	C₂×C₂²⋊C₈×C₅₃	C₂²	1	2	yes	—
21	C₂×#97×C₅₃	C₂²	1	2	yes	—
22	C₅₃×#388	C₄	1	2	yes	—
23	C₅₃×#387	C₄	1	2	yes	—
24	C₅₃×#386	C₄	1	2	yes	—
25	C₅₃×#385	C₄	1	2	yes	—
26	C₄×C₄⋊C₄×C₅₃	C₂²	1	1	yes	characteristic
27	C₄×C₂²⋊C₄×C₅₃	C₂²	1	1	yes	characteristic
28	C₄×C₂²⋊C₄×C₅₃	C₂²	1	1	yes	characteristic
29	C₅₃×#376	C₂²	1	1	yes	characteristic
30	C₅₃×#336	C₂²	1	2	yes	—
31	C₅₃×#336	C₂²	1	2	yes	—
32	C₅₃×#336	C₂²	1	2	yes	—
33	C₅₃×#336	C₂²	1	2	yes	—
34	C₅₃×#336	C₂²	1	2	yes	—
35	C₅₃×#336	C₂²	1	1	yes	characteristic
36	C₅₃×#336	C₂²	1	1	yes	characteristic
37	C₅₃×#336	C₂²	1	1	yes	characteristic
38	C₅₃×#336	C₂²	1	1	yes	characteristic
39	C₂²×D₄×C₅₃	C₂×C₄	1	1	yes	characteristic
40	C₂³×C₄×C₅₃	C₂×C₄	1	2	yes	—
41	C₂³×C₄×C₅₃	C₂³	1	1	yes	characteristic
42	C₂³×C₄×C₅₃	C₂³	1	1	yes	characteristic
43	C₂²×C₈×C₅₃	—	2	4	no	—
44	C₂²×C₈×C₅₃	—	4	2	no	—
45	C₂²×C₈×C₅₃	—	4	2	no	—
46	C₂²×C₈×C₅₃	C₂³	1	2	yes	—
47	C₂²×C₈×C₅₃	C₂³	1	2	yes	—
48	C₂²×C₈×C₅₃	C₂³	1	2	yes	—
49	C₂²×C₈×C₅₃	C₂³	1	2	yes	—
50	C₄×D₄×C₅₃	—	2	4	no	—
51	#117×C₅₃	—	4	2	no	—
52	C₂×C₄⋊C₄×C₅₃	C₂×C₄	1	1	yes	characteristic
53	C₂×C₄⋊C₄×C₅₃	C₂×C₄	1	2	yes	—
54	C₂×C₄⋊C₄×C₅₃	C₂×C₄	1	2	yes	—
55	C₂×C₂²⋊C₄×C₅₃	C₂×C₄	1	1	yes	characteristic
56	C₂×C₂²⋊C₄×C₅₃	C₂×C₄	1	1	yes	characteristic
57	C₂×C₂²⋊C₄×C₅₃	C₂×C₄	1	2	yes	—
58	C₂×C₂²⋊C₄×C₅₃	C₂×C₄	1	2	yes	—
59	C₂×C₂²⋊C₄×C₅₃	C₂×C₄	1	2	yes	—
60	C₂×C₄²×C₅₃	—	4	2	no	—
61	C₂×C₄²×C₅₃	C₂×C₄	1	2	yes	—
62	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
63	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
64	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
65	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
66	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
67	C₄⋊C₈×C₅₃	—	4	2	no	—
68	C₄⋊C₈×C₅₃	—	4	2	no	—
69	C₂²⋊C₈×C₅₃	—	2	4	no	—
70	C₂²⋊C₈×C₅₃	—	4	2	no	—
71	C₂²⋊C₈×C₅₃	—	4	2	no	—
72	#97×C₅₃	—	4	2	no	—
73	#95×C₅₃	C₂×C₄	1	2	yes	—
74	#95×C₅₃	C₂×C₄	1	2	yes	—
75	#95×C₅₃	C₂×C₄	1	2	yes	—
76	#95×C₅₃	C₂×C₄	1	2	yes	—
77	#95×C₅₃	C₂×C₄	1	2	yes	—
78	C₂⁴×C₅₃	C₂²×C₄	1	1	yes	characteristic
79	C₂⁴×C₅₃	C₂²×C₄	1	1	yes	characteristic
80	C₂×D₄×C₅₃	—	2	2	no	—
81	C₂×D₄×C₅₃	—	2	4	no	—
82	C₂²×C₄×C₅₃	—	2	2	no	—
83	C₂²×C₄×C₅₃	—	2	1	no	—
84	C₂²×C₄×C₅₃	—	2	1	no	—
85	C₂²×C₄×C₅₃	—	2	4	no	—
86	C₂²×C₄×C₅₃	—	2	2	no	—
87	C₂²×C₄×C₅₃	—	2	1	no	—
88	C₂²×C₄×C₅₃	—	2	2	no	—
89	C₂²×C₄×C₅₃	—	2	1	no	—
90	C₂²×C₄×C₅₃	—	4	2	no	—
91	C₂²×C₄×C₅₃	—	4	2	no	—
92	C₂²×C₄×C₅₃	C₂²×C₄	1	2	yes	—
93	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
94	C₂²×C₄×C₅₃	C₂²×C₄	1	2	yes	—
95	C₂²×C₄×C₅₃	C₂²×C₄	1	2	yes	—
96	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
97	C₂²×C₄×C₅₃	C₂²×C₄	1	2	yes	—
98	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
99	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
100	C₂²×C₄×C₅₃	#41	1	4	yes	—
101	C₂²×C₄×C₅₃	C₂⁴	1	1	yes	characteristic
102	C₂×C₈×C₅₃	—	2	2	no	—
103	C₂×C₈×C₅₃	—	2	2	no	—
104	C₂×C₈×C₅₃	—	2	2	no	—
105	C₂×C₈×C₅₃	—	2	2	no	—
106	C₂×C₈×C₅₃	—	2	2	no	—
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	—	2	2	no	—
113	C₂×C₈×C₅₃	—	2	2	no	—
114	C₂×C₈×C₅₃	—	4	8	no	—
115	C₄⋊C₄×C₅₃	—	2	2	no	—
116	C₄⋊C₄×C₅₃	—	4	2	no	—
117	C₄⋊C₄×C₅₃	—	4	2	no	—
118	C₂²⋊C₄×C₅₃	—	2	2	no	—
119	C₂²⋊C₄×C₅₃	—	2	4	no	—
120	C₂²⋊C₄×C₅₃	—	2	2	no	—
121	C₂²⋊C₄×C₅₃	—	4	2	no	—
122	C₂²⋊C₄×C₅₃	—	4	2	no	—
123	C₄²×C₅₃	—	2	2	no	—
124	C₄²×C₅₃	—	2	2	no	—
125	C₄²×C₅₃	—	2	1	no	—
126	C₄²×C₅₃	—	2	2	no	—
127	C₄²×C₅₃	—	2	2	no	—
128	C₄²×C₅₃	—	2	1	no	—
129	C₄²×C₅₃	—	4	2	no	—
130	C₂³×C₅₃	—	2	1	no	—
131	C₂³×C₅₃	—	2	1	no	—
132	C₂³×C₅₃	—	2	2	no	—
133	C₂³×C₅₃	—	2	1	no	—
134	C₂³×C₅₃	—	2	1	no	—
135	C₂³×C₅₃	—	2	1	no	—
136	C₂³×C₅₃	—	2	2	no	—
137	C₂³×C₅₃	—	2	2	no	—
138	C₂³×C₅₃	#117	1	2	yes	—
139	C₂³×C₅₃	#131	1	4	yes	—
140	C₂³×C₅₃	C₂³×C₄	1	1	yes	characteristic
141	D₄×C₅₃	—	4	4	no	—
142	C₂×C₄×C₅₃	—	2	1	no	—
143	C₂×C₄×C₅₃	—	2	1	no	—
144	C₂×C₄×C₅₃	—	2	4	no	—
145	C₂×C₄×C₅₃	—	2	4	no	—
146	C₂×C₄×C₅₃	—	2	1	no	—
147	C₂×C₄×C₅₃	—	2	1	no	—
148	C₂×C₄×C₅₃	—	2	1	no	—
149	C₂×C₄×C₅₃	—	2	1	no	—
150	C₂×C₄×C₅₃	—	2	2	no	—
151	C₂×C₄×C₅₃	—	2	2	no	—
152	C₂×C₄×C₅₃	—	2	2	no	—
153	C₂×C₄×C₅₃	—	2	1	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	—	2	2	no	—
163	C₂×C₄×C₅₃	—	2	2	no	—
164	C₂×C₄×C₅₃	—	2	2	no	—
165	C₂×C₄×C₅₃	—	4	8	no	—
166	C₂×C₄×C₅₃	—	4	2	no	—
167	C₂×C₄×C₅₃	—	4	2	no	—
168	C₂×C₄×C₅₃	#117	1	2	yes	—
169	C₂×C₄×C₅₃	C₂×#41	1	2	yes	—
170	C₂×C₄×C₅₃	H(5,𝔽₂)	1	1	yes	characteristic
171	C₂×C₄×C₅₃	H(5,𝔽₂)	1	2	yes	—
172	C₂×C₄×C₅₃	2₋¹⁺⁴	1	1	yes	characteristic
173	C₈×C₅₃	—	4	2	no	—
174	C₈×C₅₃	—	4	2	no	—
175	C₈×C₅₃	—	4	2	no	—
176	C₈×C₅₃	—	4	2	no	—
177	#22001	C₅₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
178	C₂²×C₅₃	—	2	2	no	—
179	C₂²×C₅₃	—	2	2	no	—
180	C₂²×C₅₃	—	2	4	no	—
181	C₂²×C₅₃	—	2	4	no	—
182	C₂²×C₅₃	—	4	1	no	—
183	C₂²×C₅₃	—	4	1	no	—
184	C₂²×C₅₃	—	4	1	no	—
185	C₂²×C₅₃	—	4	2	no	—
186	C₂²×C₅₃	—	4	2	no	—
187	C₂²×C₅₃	#433	1	4	yes	—
188	C₂²×C₅₃	C₂×#117	1	1	yes	characteristic
189	C₂²×C₅₃	#518	1	1	yes	characteristic
190	C₂²×C₅₃	#520	1	1	yes	characteristic
191	C₂²×C₅₃	C₂×#131	1	2	yes	—
192	C₂²×C₅₃	#568	1	2	yes	—
193	C₄×C₅₃	—	2	2	no	—
194	C₄×C₅₃	—	4	1	no	—
195	C₄×C₅₃	—	4	1	no	—
196	C₄×C₅₃	—	4	1	no	—
197	C₄×C₅₃	—	4	2	no	—
198	C₄×C₅₃	—	4	2	no	—
199	C₄×C₅₃	—	4	2	no	—
200	C₄×C₅₃	—	4	2	no	—
201	C₄×C₅₃	#538	1	2	yes	—
202	C₄×C₅₃	#539	1	2	yes	—
203	#2290	C₂×C₅₃	1	1	yes	characteristic
204	#1924	C₂×C₅₃	1	2	yes	complemented
205	#1917	C₂×C₅₃	1	2	yes	complemented
206	#1869	C₂×C₅₃	1	2	yes	—
207	#1850	C₂×C₅₃	1	2	yes	complemented
208	#1840	C₂×C₅₃	1	1	yes	characteristic, complemented
209	#1832	C₂×C₅₃	1	1	yes	characteristic, complemented
210	#1818	C₂×C₅₃	1	1	yes	characteristic, complemented
211	#1798	C₂×C₅₃	1	1	yes	characteristic, complemented
212	#1749	C₂×C₅₃	1	1	yes	characteristic, complemented
213	C₂×#336	C₂×C₅₃	1	1	yes	characteristic, complemented
214	C₂×C₅₃	—	2	4	no	—
215	C₂×C₅₃	—	8	1	no	—
216	C₂×C₅₃	#2318	1	1	yes	characteristic
217	C₂×C₅₃	C₂×#433	1	1	yes	characteristic
218	C₂×C₅₃	#2923	1	1	yes	characteristic
219	C₂×C₅₃	#2976	1	2	yes	—
220	C₂×C₅₃	#2992	1	2	yes	—
221	C₂³×C₈	C₂²×C₅₃	1	2	yes	—
222	C₂×C₄×D₄	C₂²×C₅₃	1	1	yes	characteristic
223	C₂×#117	C₂²×C₅₃	1	1	yes	characteristic
224	C₂²×C₄²	C₂²×C₅₃	1	1	yes	characteristic
225	C₂×C₄⋊C₈	C₂²×C₅₃	1	2	yes	—
226	C₂×C₄⋊C₈	C₂²×C₅₃	1	2	yes	—
227	C₂×C₂²⋊C₈	C₂²×C₅₃	1	2	yes	—
228	C₂×C₂²⋊C₈	C₂²×C₅₃	1	2	yes	—
229	C₂×C₂²⋊C₈	C₂²×C₅₃	1	2	yes	—
230	C₂×#97	C₂²×C₅₃	1	2	yes	—
231	#388	C₄×C₅₃	1	2	yes	—
232	#387	C₄×C₅₃	1	2	yes	—
233	#386	C₄×C₅₃	1	2	yes	—
234	#385	C₄×C₅₃	1	2	yes	—
235	C₄×C₄⋊C₄	C₂²×C₅₃	1	1	yes	characteristic
236	C₄×C₂²⋊C₄	C₂²×C₅₃	1	1	yes	characteristic
237	C₄×C₂²⋊C₄	C₂²×C₅₃	1	1	yes	characteristic
238	#376	C₂²×C₅₃	1	1	yes	characteristic
239	#336	C₂²×C₅₃	1	1	yes	characteristic
240	#336	C₂²×C₅₃	1	1	yes	characteristic
241	#336	C₂²×C₅₃	1	1	yes	characteristic
242	#336	C₂²×C₅₃	1	1	yes	characteristic
243	#336	C₂²×C₅₃	1	2	yes	—
244	#336	C₂²×C₅₃	1	2	yes	—
245	#336	C₂²×C₅₃	1	2	yes	—
246	#336	C₂²×C₅₃	1	2	yes	—
247	#336	C₂²×C₅₃	1	2	yes	—
248	C₅₃	#22001	1	1	yes	characteristic, direct product factor, Sylow
249	C₂²×D₄	C₂×C₄×C₅₃	1	1	yes	characteristic
250	C₂³×C₄	C₂×C₄×C₅₃	1	2	yes	—
251	C₂³×C₄	C₂³×C₅₃	1	1	yes	characteristic
252	C₂³×C₄	C₂³×C₅₃	1	1	yes	characteristic
253	C₂²×C₈	—	2	4	no	—
254	C₂²×C₈	—	4	2	no	—
255	C₂²×C₈	—	4	2	no	—
256	C₂²×C₈	C₂³×C₅₃	1	2	yes	—
257	C₂²×C₈	C₂³×C₅₃	1	2	yes	—
258	C₂²×C₈	C₂³×C₅₃	1	2	yes	—
259	C₂²×C₈	C₂³×C₅₃	1	2	yes	—
260	C₄×D₄	—	2	4	no	—
261	#117	—	4	2	no	—
262	C₂×C₄⋊C₄	C₂×C₄×C₅₃	1	1	yes	characteristic
263	C₂×C₄⋊C₄	C₂×C₄×C₅₃	1	2	yes	—
264	C₂×C₄⋊C₄	C₂×C₄×C₅₃	1	2	yes	—
265	C₂×C₂²⋊C₄	C₂×C₄×C₅₃	1	1	yes	characteristic
266	C₂×C₂²⋊C₄	C₂×C₄×C₅₃	1	1	yes	characteristic
267	C₂×C₂²⋊C₄	C₂×C₄×C₅₃	1	2	yes	—
268	C₂×C₂²⋊C₄	C₂×C₄×C₅₃	1	2	yes	—
269	C₂×C₂²⋊C₄	C₂×C₄×C₅₃	1	2	yes	—
270	C₂×C₄²	—	4	2	no	—
271	C₂×C₄²	C₂×C₄×C₅₃	1	2	yes	—
272	C₂×C₄²	C₂³×C₅₃	1	1	yes	characteristic
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	C₂×C₄²	C₂³×C₅₃	1	1	yes	characteristic
277	C₄⋊C₈	—	4	2	no	—
278	C₄⋊C₈	—	4	2	no	—
279	C₂²⋊C₈	—	2	4	no	—
280	C₂²⋊C₈	—	4	2	no	—
281	C₂²⋊C₈	—	4	2	no	—
282	#97	—	4	2	no	—
283	#95	C₂×C₄×C₅₃	1	2	yes	—
284	#95	C₂×C₄×C₅₃	1	2	yes	—
285	#95	C₂×C₄×C₅₃	1	2	yes	—
286	#95	C₂×C₄×C₅₃	1	2	yes	—
287	#95	C₂×C₄×C₅₃	1	2	yes	—
288	C₂⁴	C₂²×C₄×C₅₃	1	1	yes	characteristic
289	C₂⁴	C₂²×C₄×C₅₃	1	1	yes	characteristic
290	C₂×D₄	—	2	2	no	—
291	C₂×D₄	—	2	4	no	—
292	C₂²×C₄	—	2	2	no	—
293	C₂²×C₄	—	2	1	no	—
294	C₂²×C₄	—	2	1	no	—
295	C₂²×C₄	—	2	4	no	—
296	C₂²×C₄	—	2	2	no	—
297	C₂²×C₄	—	2	1	no	—
298	C₂²×C₄	—	2	2	no	—
299	C₂²×C₄	—	2	1	no	—
300	C₂²×C₄	—	4	2	no	—
301	C₂²×C₄	—	4	2	no	—
302	C₂²×C₄	C₂²×C₄×C₅₃	1	2	yes	—
303	C₂²×C₄	C₂²×C₄×C₅₃	1	1	yes	characteristic
304	C₂²×C₄	C₂²×C₄×C₅₃	1	2	yes	—
305	C₂²×C₄	C₂²×C₄×C₅₃	1	2	yes	—
306	C₂²×C₄	C₂²×C₄×C₅₃	1	1	yes	characteristic
307	C₂²×C₄	C₂²×C₄×C₅₃	1	2	yes	—
308	C₂²×C₄	C₂²×C₄×C₅₃	1	1	yes	characteristic
309	C₂²×C₄	C₂²×C₄×C₅₃	1	1	yes	characteristic
310	C₂²×C₄	#41×C₅₃	1	4	yes	—
311	C₂²×C₄	C₂⁴×C₅₃	1	1	yes	characteristic
312	C₂×C₈	—	2	2	no	—
313	C₂×C₈	—	2	2	no	—
314	C₂×C₈	—	2	2	no	—
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	—	2	2	no	—
321	C₂×C₈	—	2	2	no	—
322	C₂×C₈	—	2	2	no	—
323	C₂×C₈	—	2	2	no	—
324	C₂×C₈	—	4	8	no	—
325	C₄⋊C₄	—	2	2	no	—
326	C₄⋊C₄	—	4	2	no	—
327	C₄⋊C₄	—	4	2	no	—
328	C₂²⋊C₄	—	2	2	no	—
329	C₂²⋊C₄	—	2	4	no	—
330	C₂²⋊C₄	—	2	2	no	—
331	C₂²⋊C₄	—	4	2	no	—
332	C₂²⋊C₄	—	4	2	no	—
333	C₄²	—	2	2	no	—
334	C₄²	—	2	2	no	—
335	C₄²	—	2	1	no	—
336	C₄²	—	2	2	no	—
337	C₄²	—	2	2	no	—
338	C₄²	—	2	1	no	—
339	C₄²	—	4	2	no	—
340	C₂³	—	2	1	no	—
341	C₂³	—	2	1	no	—
342	C₂³	—	2	2	no	—
343	C₂³	—	2	1	no	—
344	C₂³	—	2	1	no	—
345	C₂³	—	2	1	no	—
346	C₂³	—	2	2	no	—
347	C₂³	—	2	2	no	—
348	C₂³	#117×C₅₃	1	2	yes	—
349	C₂³	#131×C₅₃	1	4	yes	—
350	C₂³	C₂³×C₄×C₅₃	1	1	yes	characteristic
351	D₄	—	4	4	no	—
352	C₂×C₄	—	2	1	no	—
353	C₂×C₄	—	2	1	no	—
354	C₂×C₄	—	2	4	no	—
355	C₂×C₄	—	2	4	no	—
356	C₂×C₄	—	2	1	no	—
357	C₂×C₄	—	2	1	no	—
358	C₂×C₄	—	2	1	no	—
359	C₂×C₄	—	2	2	no	—
360	C₂×C₄	—	2	2	no	—
361	C₂×C₄	—	2	2	no	—
362	C₂×C₄	—	2	1	no	—
363	C₂×C₄	—	2	2	no	—
364	C₂×C₄	—	2	1	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	—	2	2	no	—
374	C₂×C₄	—	2	2	no	—
375	C₂×C₄	—	4	8	no	—
376	C₂×C₄	—	4	2	no	—
377	C₂×C₄	—	4	2	no	—
378	C₂×C₄	#117×C₅₃	1	2	yes	—
379	C₂×C₄	C₂×#41×C₅₃	1	2	yes	—
380	C₂×C₄	H(5,𝔽₂)×C₅₃	1	1	yes	characteristic
381	C₂×C₄	H(5,𝔽₂)×C₅₃	1	2	yes	—
382	C₂×C₄	2₋¹⁺⁴×C₅₃	1	1	yes	characteristic
383	C₈	—	4	2	no	—
384	C₈	—	4	2	no	—
385	C₈	—	4	2	no	—
386	C₈	—	4	2	no	—
387	C₂²	—	2	4	no	—
388	C₂²	—	2	4	no	—
389	C₂²	—	2	2	no	—
390	C₂²	—	2	2	no	—
391	C₂²	—	4	2	no	—
392	C₂²	—	4	2	no	—
393	C₂²	—	4	1	no	—
394	C₂²	—	4	1	no	—
395	C₂²	—	4	1	no	—
396	C₂²	C₅₃×#433	1	4	yes	—
397	C₂²	C₂×#117×C₅₃	1	1	yes	characteristic
398	C₂²	C₅₃×#518	1	1	yes	characteristic
399	C₂²	C₅₃×#520	1	1	yes	characteristic
400	C₂²	C₂×#131×C₅₃	1	2	yes	—
401	C₂²	C₅₃×#568	1	2	yes	—
402	C₄	—	2	2	no	—
403	C₄	—	4	2	no	—
404	C₄	—	4	2	no	—
405	C₄	—	4	2	no	—
406	C₄	—	4	2	no	—
407	C₄	—	4	1	no	—
408	C₄	—	4	1	no	—
409	C₄	—	4	1	no	—
410	C₄	C₅₃×#538	1	2	yes	—
411	C₄	C₅₃×#539	1	2	yes	—
412	C₂	—	2	4	no	—
413	C₂	—	8	1	no	—
414	C₂	C₅₃×#2318	1	1	yes	characteristic
415	C₂	C₂×C₅₃×#433	1	1	yes	characteristic
416	C₂	C₅₃×#2923	1	1	yes	characteristic
417	C₂	C₅₃×#2976	1	2	yes	—
418	C₂	C₅₃×#2992	1	2	yes	—

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

Complements

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

N	H ≅ G/N	type
2. C₅₃×#1924	C₂	semidirect
3. C₅₃×#1917	C₂	semidirect
5. C₅₃×#1850	C₂	semidirect
6. C₅₃×#1840	C₂	semidirect
7. C₅₃×#1832	C₂	semidirect
8. C₅₃×#1818	C₂	semidirect
9. C₅₃×#1798	C₂	semidirect
10. C₅₃×#1749	C₂	semidirect
11. C₂×C₅₃×#336	C₂	semidirect
177. #22001	C₅₃	direct (N × H)
204. #1924	C₂×C₅₃	semidirect
205. #1917	C₂×C₅₃	semidirect
207. #1850	C₂×C₅₃	semidirect
208. #1840	C₂×C₅₃	semidirect
209. #1832	C₂×C₅₃	semidirect
210. #1818	C₂×C₅₃	semidirect
211. #1798	C₂×C₅₃	semidirect
212. #1749	C₂×C₅₃	semidirect
213. C₂×#336	C₂×C₅₃	semidirect
248. C₅₃	#22001	direct (N × H)

Maximal subgroups

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

subgroup	index
C₅₃×#2290	2
C₅₃×#1924	2
C₅₃×#1917	2
C₅₃×#1869	2
C₅₃×#1850	2
C₅₃×#1840	2
C₅₃×#1832	2
C₅₃×#1818	2
C₅₃×#1798	2
C₅₃×#1749	2
C₂×C₅₃×#336	2
#22001	53

Subgroups — C53×#22001 #27229513

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅₃×#22001 #27229513