Subgroups — C₅₃×#22290 #27229802

Order 13568.

702 conjugacy classes of subgroups.

The subgroup lattice has 414 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₅₃×#2396	C₂	1	1	yes	maximal, characteristic
2	C₅₃×#1984	C₂	1	2	yes	maximal, complemented
3	C₅₃×#1940	C₂	1	2	yes	maximal, complemented
4	C₅₃×#1917	C₂	1	2	yes	maximal, complemented
5	C₅₃×#1916	C₂	1	2	yes	maximal, complemented
6	C₅₃×#1850	C₂	1	2	yes	maximal
7	C₅₃×#1798	C₂	1	1	yes	maximal, characteristic
8	C₅₃×#1753	C₂	1	2	yes	maximal
9	C₅₃×#1729	C₂	1	1	yes	maximal, characteristic
10	C₂²×M₁₆×C₅₃	C₂²	1	1	yes	characteristic
11	C₂³×C₈×C₅₃	C₂²	1	1	yes	characteristic
12	C₂×C₄×Q₈×C₅₃	C₂²	1	1	yes	characteristic
13	C₂×#117×C₅₃	C₂²	1	1	yes	characteristic
14	C₂×C₄⋊C₈×C₅₃	C₂²	1	1	yes	characteristic
15	C₂×C₄⋊C₈×C₅₃	C₂²	1	2	yes	—
16	C₂×C₄⋊C₈×C₅₃	C₂²	1	1	yes	characteristic
17	C₂×C₄⋊C₈×C₅₃	C₂²	1	2	yes	—
18	C₂×C₂²⋊C₈×C₅₃	C₂²	1	2	yes	—
19	C₂×#97×C₅₃	C₂²	1	2	yes	—
20	C₂×C₄×C₈×C₅₃	C₂²	1	2	yes	—
21	C₅₃×#400	C₄	1	1	yes	characteristic
22	C₅₃×#399	C₄	1	1	yes	characteristic
23	C₅₃×#398	C₄	1	2	yes	—
24	C₅₃×#396	C₄	1	2	yes	—
25	C₅₃×#395	C₄	1	1	yes	characteristic
26	C₅₃×#393	C₄	1	1	yes	characteristic
27	C₄×C₄⋊C₄×C₅₃	C₂²	1	2	yes	—
28	C₄×C₄⋊C₄×C₅₃	C₂²	1	1	yes	characteristic
29	C₄×C₂²⋊C₄×C₅₃	C₂²	1	2	yes	—
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	2	yes	—
36	C₅₃×#336	C₂²	1	2	yes	—
37	C₂²×Q₈×C₅₃	C₂×C₄	1	1	yes	characteristic
38	C₂³×C₄×C₅₃	C₂³	1	1	yes	characteristic
39	C₂×M₁₆×C₅₃	—	4	1	no	—
40	C₂×M₁₆×C₅₃	—	4	1	no	—
41	C₂×M₁₆×C₅₃	Q₈	1	4	yes	complemented
42	C₂²×C₈×C₅₃	—	2	2	no	—
43	C₂²×C₈×C₅₃	—	4	1	no	—
44	C₂²×C₈×C₅₃	—	4	1	no	—
45	C₂²×C₈×C₅₃	C₂³	1	2	yes	—
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	C₄×Q₈×C₅₃	—	4	1	no	—
50	C₄×Q₈×C₅₃	—	4	1	no	—
51	#117×C₅₃	—	4	1	no	—
52	#117×C₅₃	—	4	1	no	—
53	C₂×C₄⋊C₄×C₅₃	C₂×C₄	1	2	yes	—
54	C₂×C₄⋊C₄×C₅₃	C₂×C₄	1	1	yes	characteristic
55	C₂×C₄⋊C₄×C₅₃	C₂×C₄	1	1	yes	characteristic
56	C₂×C₄⋊C₄×C₅₃	C₂×C₄	1	2	yes	—
57	C₂×C₄⋊C₄×C₅₃	C₂×C₄	1	1	yes	characteristic
58	C₂×C₄⋊C₄×C₅₃	C₂×C₄	1	2	yes	—
59	C₂×C₄⋊C₄×C₅₃	C₂×C₄	1	2	yes	—
60	C₂×C₂²⋊C₄×C₅₃	C₂×C₄	1	2	yes	—
61	C₂×C₂²⋊C₄×C₅₃	C₂×C₄	1	1	yes	characteristic
62	C₂×C₂²⋊C₄×C₅₃	C₂×C₄	1	1	yes	characteristic
63	C₂×C₂²⋊C₄×C₅₃	C₂×C₄	1	2	yes	—
64	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
65	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
66	C₂×C₄²×C₅₃	C₂³	1	2	yes	—
67	C₂×C₄²×C₅₃	C₂³	1	2	yes	—
68	C₄⋊C₈×C₅₃	—	4	2	no	—
69	C₄⋊C₈×C₅₃	—	4	1	no	—
70	C₄⋊C₈×C₅₃	—	4	1	no	—
71	C₄⋊C₈×C₅₃	—	4	2	no	—
72	C₂²⋊C₈×C₅₃	—	2	4	no	—
73	#97×C₅₃	—	4	2	no	—
74	C₄×C₈×C₅₃	—	4	2	no	—
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	#95×C₅₃	C₂×C₄	1	1	yes	characteristic
79	#95×C₅₃	C₂×C₄	1	1	yes	characteristic
80	#95×C₅₃	C₂×C₄	1	2	yes	—
81	C₂⁴×C₅₃	C₂²×C₄	1	1	yes	characteristic
82	C₂×Q₈×C₅₃	—	2	4	no	—
83	C₂×Q₈×C₅₃	—	4	1	no	—
84	C₂²×C₄×C₅₃	—	2	1	no	—
85	C₂²×C₄×C₅₃	—	2	1	no	—
86	C₂²×C₄×C₅₃	—	2	1	no	—
87	C₂²×C₄×C₅₃	—	2	1	no	—
88	C₂²×C₄×C₅₃	C₂²×C₄	1	2	yes	—
89	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
90	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
91	C₂²×C₄×C₅₃	C₂²×C₄	1	2	yes	—
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	1	yes	characteristic
95	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
96	C₂²×C₄×C₅₃	C₂²×C₄	1	2	yes	—
97	C₂²×C₄×C₅₃	C₂×Q₈	1	2	yes	—
98	C₂²×C₄×C₅₃	#41	1	2	yes	—
99	C₂²×C₄×C₅₃	C₂⁴	1	1	yes	characteristic
100	M₁₆×C₅₃	—	4	4	no	—
101	C₂×C₈×C₅₃	—	2	2	no	—
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	—	4	4	no	—
113	C₂×C₈×C₅₃	C₂×Q₈	1	4	yes	—
114	C₄⋊C₄×C₅₃	—	2	4	no	—
115	C₄⋊C₄×C₅₃	—	4	2	no	—
116	C₄⋊C₄×C₅₃	—	4	2	no	—
117	C₄⋊C₄×C₅₃	—	4	1	no	—
118	C₄⋊C₄×C₅₃	—	4	1	no	—
119	C₄⋊C₄×C₅₃	—	4	2	no	—
120	C₄⋊C₄×C₅₃	—	4	1	no	—
121	C₂²⋊C₄×C₅₃	—	2	4	no	—
122	C₂²⋊C₄×C₅₃	—	4	2	no	—
123	C₂²⋊C₄×C₅₃	—	4	1	no	—
124	C₂²⋊C₄×C₅₃	—	4	1	no	—
125	C₄²×C₅₃	—	2	2	no	—
126	C₄²×C₅₃	—	2	2	no	—
127	C₄²×C₅₃	—	2	1	no	—
128	C₄²×C₅₃	—	2	1	no	—
129	C₄²×C₅₃	—	2	2	no	—
130	C₄²×C₅₃	—	2	1	no	—
131	C₄²×C₅₃	—	2	1	no	—
132	C₄²×C₅₃	—	2	2	no	—
133	C₂³×C₅₃	—	2	1	no	—
134	C₂³×C₅₃	—	2	1	no	—
135	C₂³×C₅₃	—	2	2	no	—
136	C₂³×C₅₃	—	2	2	no	—
137	C₂³×C₅₃	C₄×Q₈	1	2	yes	—
138	C₂³×C₅₃	C₂³×C₄	1	1	yes	characteristic
139	Q₈×C₅₃	—	4	4	no	—
140	C₂×C₄×C₅₃	—	2	1	no	—
141	C₂×C₄×C₅₃	—	2	2	no	—
142	C₂×C₄×C₅₃	—	2	1	no	—
143	C₂×C₄×C₅₃	—	2	2	no	—
144	C₂×C₄×C₅₃	—	2	2	no	—
145	C₂×C₄×C₅₃	—	2	2	no	—
146	C₂×C₄×C₅₃	—	2	2	no	—
147	C₂×C₄×C₅₃	—	2	2	no	—
148	C₂×C₄×C₅₃	—	2	1	no	—
149	C₂×C₄×C₅₃	—	2	1	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	—	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	C₄×Q₈	1	2	yes	—
166	C₂×C₄×C₅₃	#131	1	4	yes	—
167	C₂×C₄×C₅₃	C₂²×Q₈	1	1	yes	characteristic
168	C₂×C₄×C₅₃	C₂×#41	1	1	yes	characteristic
169	C₂×C₄×C₅₃	H(5,𝔽₂)	1	1	yes	characteristic
170	C₂×C₄×C₅₃	H(5,𝔽₂)	1	1	yes	characteristic
171	C₂×C₄×C₅₃	H(5,𝔽₂)	1	1	yes	characteristic
172	C₂×C₄×C₅₃	2₋¹⁺⁴	1	1	yes	characteristic
173	C₈×C₅₃	—	2	4	no	—
174	C₈×C₅₃	—	4	2	no	—
175	C₈×C₅₃	—	4	2	no	—
176	C₈×C₅₃	—	4	2	no	—
177	#22290	C₅₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
178	C₂²×C₅₃	—	2	2	no	—
179	C₂²×C₅₃	—	2	4	no	—
180	C₂²×C₅₃	—	4	1	no	—
181	C₂²×C₅₃	—	4	1	no	—
182	C₂²×C₅₃	—	4	2	no	—
183	C₂²×C₅₃	C₂×C₄×Q₈	1	1	yes	characteristic
184	C₂²×C₅₃	#518	1	1	yes	characteristic
185	C₂²×C₅₃	#520	1	1	yes	characteristic
186	C₂²×C₅₃	C₂×#131	1	2	yes	—
187	C₂²×C₅₃	#568	1	2	yes	—
188	C₄×C₅₃	—	2	2	no	—
189	C₄×C₅₃	—	2	4	no	—
190	C₄×C₅₃	—	4	1	no	—
191	C₄×C₅₃	—	4	1	no	—
192	C₄×C₅₃	—	4	1	no	—
193	C₄×C₅₃	—	4	1	no	—
194	C₄×C₅₃	—	4	2	no	—
195	C₄×C₅₃	—	4	2	no	—
196	C₄×C₅₃	—	4	2	no	—
197	C₄×C₅₃	#538	1	1	yes	characteristic
198	C₄×C₅₃	#539	1	1	yes	characteristic
199	C₄×C₅₃	#543	1	1	yes	characteristic
200	C₄×C₅₃	#544	1	1	yes	characteristic
201	#2396	C₂×C₅₃	1	1	yes	characteristic
202	#1984	C₂×C₅₃	1	2	yes	complemented
203	#1940	C₂×C₅₃	1	2	yes	complemented
204	#1917	C₂×C₅₃	1	2	yes	complemented
205	#1916	C₂×C₅₃	1	2	yes	complemented
206	#1850	C₂×C₅₃	1	2	yes	—
207	#1798	C₂×C₅₃	1	1	yes	characteristic
208	#1753	C₂×C₅₃	1	2	yes	—
209	#1729	C₂×C₅₃	1	1	yes	characteristic
210	C₂×C₅₃	—	4	2	no	—
211	C₂×C₅₃	#2329	1	1	yes	characteristic
212	C₂×C₅₃	#2962	1	2	yes	—
213	C₂×C₅₃	#2977	1	1	yes	characteristic
214	C₂×C₅₃	#2980	1	1	yes	characteristic
215	C₂×C₅₃	#2993	1	2	yes	—
216	C₂²×M₁₆	C₂²×C₅₃	1	1	yes	characteristic
217	C₂³×C₈	C₂²×C₅₃	1	1	yes	characteristic
218	C₂×C₄×Q₈	C₂²×C₅₃	1	1	yes	characteristic
219	C₂×#117	C₂²×C₅₃	1	1	yes	characteristic
220	C₂×C₄⋊C₈	C₂²×C₅₃	1	1	yes	characteristic
221	C₂×C₄⋊C₈	C₂²×C₅₃	1	1	yes	characteristic
222	C₂×C₄⋊C₈	C₂²×C₅₃	1	2	yes	—
223	C₂×C₄⋊C₈	C₂²×C₅₃	1	2	yes	—
224	C₂×C₂²⋊C₈	C₂²×C₅₃	1	2	yes	—
225	C₂×#97	C₂²×C₅₃	1	2	yes	—
226	C₂×C₄×C₈	C₂²×C₅₃	1	2	yes	—
227	#400	C₄×C₅₃	1	1	yes	characteristic
228	#399	C₄×C₅₃	1	1	yes	characteristic
229	#398	C₄×C₅₃	1	2	yes	—
230	#396	C₄×C₅₃	1	2	yes	—
231	#395	C₄×C₅₃	1	1	yes	characteristic
232	#393	C₄×C₅₃	1	1	yes	characteristic
233	C₄×C₄⋊C₄	C₂²×C₅₃	1	2	yes	—
234	C₄×C₄⋊C₄	C₂²×C₅₃	1	1	yes	characteristic
235	C₄×C₂²⋊C₄	C₂²×C₅₃	1	2	yes	—
236	#336	C₂²×C₅₃	1	2	yes	—
237	#336	C₂²×C₅₃	1	2	yes	—
238	#336	C₂²×C₅₃	1	2	yes	—
239	#336	C₂²×C₅₃	1	2	yes	—
240	#336	C₂²×C₅₃	1	2	yes	—
241	#336	C₂²×C₅₃	1	2	yes	—
242	#336	C₂²×C₅₃	1	2	yes	—
243	C₅₃	#22290	1	1	yes	characteristic, direct product factor, Sylow
244	C₂²×Q₈	C₂×C₄×C₅₃	1	1	yes	characteristic
245	C₂³×C₄	C₂³×C₅₃	1	1	yes	characteristic
246	C₂×M₁₆	—	4	1	no	—
247	C₂×M₁₆	—	4	1	no	—
248	C₂×M₁₆	Q₈×C₅₃	1	4	yes	complemented
249	C₂²×C₈	—	2	2	no	—
250	C₂²×C₈	—	4	1	no	—
251	C₂²×C₈	—	4	1	no	—
252	C₂²×C₈	C₂³×C₅₃	1	2	yes	—
253	C₂²×C₈	C₂³×C₅₃	1	2	yes	—
254	C₂²×C₈	C₂³×C₅₃	1	2	yes	—
255	C₂²×C₈	C₂³×C₅₃	1	2	yes	—
256	C₄×Q₈	—	4	1	no	—
257	C₄×Q₈	—	4	1	no	—
258	#117	—	4	1	no	—
259	#117	—	4	1	no	—
260	C₂×C₄⋊C₄	C₂×C₄×C₅₃	1	2	yes	—
261	C₂×C₄⋊C₄	C₂×C₄×C₅₃	1	1	yes	characteristic
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	1	yes	characteristic
265	C₂×C₄⋊C₄	C₂×C₄×C₅₃	1	2	yes	—
266	C₂×C₄⋊C₄	C₂×C₄×C₅₃	1	2	yes	—
267	C₂×C₂²⋊C₄	C₂×C₄×C₅₃	1	2	yes	—
268	C₂×C₂²⋊C₄	C₂×C₄×C₅₃	1	1	yes	characteristic
269	C₂×C₂²⋊C₄	C₂×C₄×C₅₃	1	1	yes	characteristic
270	C₂×C₂²⋊C₄	C₂×C₄×C₅₃	1	2	yes	—
271	C₂×C₄²	C₂³×C₅₃	1	1	yes	characteristic
272	C₂×C₄²	C₂³×C₅₃	1	1	yes	characteristic
273	C₂×C₄²	C₂³×C₅₃	1	2	yes	—
274	C₂×C₄²	C₂³×C₅₃	1	2	yes	—
275	C₄⋊C₈	—	4	2	no	—
276	C₄⋊C₈	—	4	1	no	—
277	C₄⋊C₈	—	4	1	no	—
278	C₄⋊C₈	—	4	2	no	—
279	C₂²⋊C₈	—	2	4	no	—
280	#97	—	4	2	no	—
281	C₄×C₈	—	4	2	no	—
282	#95	C₂×C₄×C₅₃	1	2	yes	—
283	#95	C₂×C₄×C₅₃	1	2	yes	—
284	#95	C₂×C₄×C₅₃	1	2	yes	—
285	#95	C₂×C₄×C₅₃	1	1	yes	characteristic
286	#95	C₂×C₄×C₅₃	1	1	yes	characteristic
287	#95	C₂×C₄×C₅₃	1	2	yes	—
288	C₂⁴	C₂²×C₄×C₅₃	1	1	yes	characteristic
289	C₂×Q₈	—	2	4	no	—
290	C₂×Q₈	—	4	1	no	—
291	C₂²×C₄	—	2	1	no	—
292	C₂²×C₄	—	2	1	no	—
293	C₂²×C₄	—	2	1	no	—
294	C₂²×C₄	—	2	1	no	—
295	C₂²×C₄	C₂²×C₄×C₅₃	1	2	yes	—
296	C₂²×C₄	C₂²×C₄×C₅₃	1	1	yes	characteristic
297	C₂²×C₄	C₂²×C₄×C₅₃	1	1	yes	characteristic
298	C₂²×C₄	C₂²×C₄×C₅₃	1	2	yes	—
299	C₂²×C₄	C₂²×C₄×C₅₃	1	2	yes	—
300	C₂²×C₄	C₂²×C₄×C₅₃	1	1	yes	characteristic
301	C₂²×C₄	C₂²×C₄×C₅₃	1	1	yes	characteristic
302	C₂²×C₄	C₂²×C₄×C₅₃	1	1	yes	characteristic
303	C₂²×C₄	C₂²×C₄×C₅₃	1	2	yes	—
304	C₂²×C₄	C₂×Q₈×C₅₃	1	2	yes	—
305	C₂²×C₄	#41×C₅₃	1	2	yes	—
306	C₂²×C₄	C₂⁴×C₅₃	1	1	yes	characteristic
307	M₁₆	—	4	4	no	—
308	C₂×C₈	—	2	2	no	—
309	C₂×C₈	—	2	2	no	—
310	C₂×C₈	—	2	2	no	—
311	C₂×C₈	—	2	2	no	—
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	—	4	4	no	—
320	C₂×C₈	C₂×Q₈×C₅₃	1	4	yes	—
321	C₄⋊C₄	—	2	4	no	—
322	C₄⋊C₄	—	4	1	no	—
323	C₄⋊C₄	—	4	1	no	—
324	C₄⋊C₄	—	4	2	no	—
325	C₄⋊C₄	—	4	1	no	—
326	C₄⋊C₄	—	4	2	no	—
327	C₄⋊C₄	—	4	2	no	—
328	C₂²⋊C₄	—	2	4	no	—
329	C₂²⋊C₄	—	4	1	no	—
330	C₂²⋊C₄	—	4	1	no	—
331	C₂²⋊C₄	—	4	2	no	—
332	C₄²	—	2	2	no	—
333	C₄²	—	2	2	no	—
334	C₄²	—	2	1	no	—
335	C₄²	—	2	1	no	—
336	C₄²	—	2	2	no	—
337	C₄²	—	2	1	no	—
338	C₄²	—	2	1	no	—
339	C₄²	—	2	2	no	—
340	C₂³	—	2	1	no	—
341	C₂³	—	2	1	no	—
342	C₂³	—	2	2	no	—
343	C₂³	—	2	2	no	—
344	C₂³	C₄×Q₈×C₅₃	1	2	yes	—
345	C₂³	C₂³×C₄×C₅₃	1	1	yes	characteristic
346	Q₈	—	4	4	no	—
347	C₂×C₄	—	2	2	no	—
348	C₂×C₄	—	2	1	no	—
349	C₂×C₄	—	2	1	no	—
350	C₂×C₄	—	2	2	no	—
351	C₂×C₄	—	2	2	no	—
352	C₂×C₄	—	2	2	no	—
353	C₂×C₄	—	2	2	no	—
354	C₂×C₄	—	2	2	no	—
355	C₂×C₄	—	2	1	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	2	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	C₄×Q₈×C₅₃	1	2	yes	—
373	C₂×C₄	#131×C₅₃	1	4	yes	—
374	C₂×C₄	C₂²×Q₈×C₅₃	1	1	yes	characteristic
375	C₂×C₄	C₂×#41×C₅₃	1	1	yes	characteristic
376	C₂×C₄	H(5,𝔽₂)×C₅₃	1	1	yes	characteristic
377	C₂×C₄	H(5,𝔽₂)×C₅₃	1	1	yes	characteristic
378	C₂×C₄	H(5,𝔽₂)×C₅₃	1	1	yes	characteristic
379	C₂×C₄	2₋¹⁺⁴×C₅₃	1	1	yes	characteristic
380	C₈	—	2	4	no	—
381	C₈	—	4	2	no	—
382	C₈	—	4	2	no	—
383	C₈	—	4	2	no	—
384	C₂²	—	2	4	no	—
385	C₂²	—	2	2	no	—
386	C₂²	—	4	1	no	—
387	C₂²	—	4	1	no	—
388	C₂²	—	4	2	no	—
389	C₂²	C₂×C₄×Q₈×C₅₃	1	1	yes	characteristic
390	C₂²	C₅₃×#518	1	1	yes	characteristic
391	C₂²	C₅₃×#520	1	1	yes	characteristic
392	C₂²	C₂×#131×C₅₃	1	2	yes	—
393	C₂²	C₅₃×#568	1	2	yes	—
394	C₄	—	2	4	no	—
395	C₄	—	2	2	no	—
396	C₄	—	4	1	no	—
397	C₄	—	4	1	no	—
398	C₄	—	4	1	no	—
399	C₄	—	4	1	no	—
400	C₄	—	4	2	no	—
401	C₄	—	4	2	no	—
402	C₄	—	4	2	no	—
403	C₄	C₅₃×#538	1	1	yes	characteristic
404	C₄	C₅₃×#539	1	1	yes	characteristic
405	C₄	C₅₃×#543	1	1	yes	characteristic
406	C₄	C₅₃×#544	1	1	yes	characteristic
407	C₂	—	4	2	no	—
408	C₂	C₅₃×#2329	1	1	yes	characteristic
409	C₂	C₅₃×#2962	1	2	yes	—
410	C₂	C₅₃×#2977	1	1	yes	characteristic
411	C₂	C₅₃×#2980	1	1	yes	characteristic
412	C₂	C₅₃×#2993	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	C₂⁴×C₅₃	C₂²×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×#1984	C₂	semidirect
3. C₅₃×#1940	C₂	semidirect
4. C₅₃×#1917	C₂	semidirect
5. C₅₃×#1916	C₂	semidirect
41. C₂×M₁₆×C₅₃	Q₈	semidirect
177. #22290	C₅₃	direct (N × H)
202. #1984	C₂×C₅₃	semidirect
203. #1940	C₂×C₅₃	semidirect
204. #1917	C₂×C₅₃	semidirect
205. #1916	C₂×C₅₃	semidirect
243. C₅₃	#22290	direct (N × H)
248. C₂×M₁₆	Q₈×C₅₃	semidirect

Maximal subgroups

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

subgroup	index
C₅₃×#2396	2
C₅₃×#1984	2
C₅₃×#1940	2
C₅₃×#1917	2
C₅₃×#1916	2
C₅₃×#1850	2
C₅₃×#1798	2
C₅₃×#1753	2
C₅₃×#1729	2
#22290	53

Subgroups — C53×#22290 #27229802

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅₃×#22290 #27229802