Subgroups — C₅₃×#22830 #27230342

Order 13568.

716 conjugacy classes of subgroups.

The subgroup lattice has 470 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄²×Q₈×C₅₃	C₂	1	1	yes	maximal, characteristic
2	C₅₃×#1985	C₂	1	2	yes	maximal
3	C₅₃×#1917	C₂	1	2	yes	maximal
4	C₅₃×#1916	C₂	1	2	yes	maximal
5	C₅₃×#1839	C₂	1	1	yes	maximal, characteristic
6	C₅₃×#1839	C₂	1	1	yes	maximal, characteristic
7	C₅₃×#1831	C₂	1	1	yes	maximal, characteristic
8	C₅₃×#1771	C₂	1	2	yes	maximal
9	C₅₃×#1767	C₂	1	1	yes	maximal, characteristic
10	C₅₃×#1745	C₂	1	1	yes	maximal, characteristic
11	C₅₃×#1744	C₂	1	1	yes	maximal, characteristic
12	C₂×C₄×Q₈×C₅₃	C₄	1	1	yes	characteristic
13	C₂×C₄×Q₈×C₅₃	C₄	1	1	yes	characteristic
14	C₂×C₄×Q₈×C₅₃	C₂²	1	1	yes	characteristic
15	C₂×C₄⋊C₈×C₅₃	C₂²	1	2	yes	—
16	C₂×C₄⋊C₈×C₅₃	C₂²	1	2	yes	—
17	C₂×#97×C₅₃	C₂²	1	2	yes	—
18	C₂×#97×C₅₃	C₂²	1	2	yes	—
19	C₂×C₄×C₈×C₅₃	C₂²	1	2	yes	—
20	C₄×C₄⋊C₄×C₅₃	C₄	1	1	yes	characteristic
21	C₄×C₄⋊C₄×C₅₃	C₄	1	1	yes	characteristic
22	C₄×C₄⋊C₄×C₅₃	C₄	1	1	yes	characteristic
23	C₄×C₄⋊C₄×C₅₃	C₄	1	1	yes	characteristic
24	C₄×C₄⋊C₄×C₅₃	C₄	1	1	yes	characteristic
25	C₄×C₄⋊C₄×C₅₃	C₄	1	1	yes	characteristic
26	C₄×C₄⋊C₄×C₅₃	C₂²	1	1	yes	characteristic
27	C₄×C₄⋊C₄×C₅₃	C₂²	1	1	yes	characteristic
28	C₄×C₄⋊C₄×C₅₃	C₂²	1	1	yes	characteristic
29	C₄³×C₅₃	C₂²	1	1	yes	characteristic
30	C₄³×C₅₃	C₂²	1	1	yes	characteristic
31	C₄³×C₅₃	C₂²	1	1	yes	characteristic
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	C₅₃×#336	C₂²	1	2	yes	—
38	C₅₃×#336	C₂²	1	2	yes	—
39	C₅₃×#336	C₂²	1	1	yes	characteristic
40	C₅₃×#336	C₂²	1	1	yes	characteristic
41	C₅₃×#336	C₂²	1	1	yes	characteristic
42	C₅₃×#336	C₂²	1	1	yes	characteristic
43	C₂²×Q₈×C₅₃	C₂×C₄	1	1	yes	characteristic
44	C₂²×C₈×C₅₃	C₂³	1	2	yes	—
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	C₄×Q₈×C₅₃	—	2	4	no	—
49	C₄×Q₈×C₅₃	—	2	4	no	—
50	C₄×Q₈×C₅₃	—	2	2	no	—
51	C₄×Q₈×C₅₃	—	2	2	no	—
52	C₂×C₄⋊C₄×C₅₃	C₂×C₄	1	1	yes	characteristic
53	C₂×C₄⋊C₄×C₅₃	C₂×C₄	1	1	yes	characteristic
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	1	yes	characteristic
57	C₂×C₄⋊C₄×C₅₃	C₂×C₄	1	1	yes	characteristic
58	C₂×C₄⋊C₄×C₅₃	C₂×C₄	1	1	yes	characteristic
59	C₂×C₄⋊C₄×C₅₃	C₂×C₄	1	1	yes	characteristic
60	C₂×C₄⋊C₄×C₅₃	C₂×C₄	1	1	yes	characteristic
61	C₂×C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
62	C₂×C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
63	C₂×C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
64	C₂×C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
65	C₂×C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
66	C₂×C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
67	C₂×C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
68	C₂×C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
69	C₂×C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
70	C₂×C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
71	C₂×C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
72	C₂×C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
73	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
74	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
75	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
76	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
77	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
78	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
79	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
80	C₄⋊C₈×C₅₃	—	2	4	no	—
81	C₄⋊C₈×C₅₃	—	2	4	no	—
82	#97×C₅₃	—	2	4	no	—
83	#97×C₅₃	—	2	4	no	—
84	C₄×C₈×C₅₃	—	2	4	no	—
85	#95×C₅₃	C₂×C₄	1	1	yes	characteristic
86	#95×C₅₃	C₂×C₄	1	1	yes	characteristic
87	#95×C₅₃	C₂×C₄	1	1	yes	characteristic
88	#95×C₅₃	C₂×C₄	1	1	yes	characteristic
89	#95×C₅₃	C₂×C₄	1	1	yes	characteristic
90	#95×C₅₃	C₂×C₄	1	1	yes	characteristic
91	C₂×Q₈×C₅₃	—	2	2	no	—
92	C₂×Q₈×C₅₃	—	2	2	no	—
93	C₂×Q₈×C₅₃	—	2	2	no	—
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	1	yes	characteristic
97	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
98	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
99	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
100	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
101	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
102	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
103	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
104	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
105	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
106	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
107	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
108	C₂²×C₄×C₅₃	C₂⁴	1	1	yes	characteristic
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	—	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	—	2	2	no	—
119	C₂×C₈×C₅₃	—	2	2	no	—
120	C₂×C₈×C₅₃	#41	1	4	yes	—
121	C₄⋊C₄×C₅₃	—	2	2	no	—
122	C₄⋊C₄×C₅₃	—	2	2	no	—
123	C₄⋊C₄×C₅₃	—	2	2	no	—
124	C₄⋊C₄×C₅₃	—	2	2	no	—
125	C₄⋊C₄×C₅₃	—	2	2	no	—
126	C₄⋊C₄×C₅₃	—	2	2	no	—
127	C₄⋊C₄×C₅₃	—	2	2	no	—
128	C₄⋊C₄×C₅₃	—	2	2	no	—
129	C₄⋊C₄×C₅₃	M₁₆	1	4	yes	—
130	C₄²×C₅₃	—	2	2	no	—
131	C₄²×C₅₃	—	2	2	no	—
132	C₄²×C₅₃	—	2	2	no	—
133	C₄²×C₅₃	—	2	2	no	—
134	C₄²×C₅₃	—	2	2	no	—
135	C₄²×C₅₃	—	2	2	no	—
136	C₄²×C₅₃	—	2	2	no	—
137	C₄²×C₅₃	—	2	2	no	—
138	C₄²×C₅₃	—	2	2	no	—
139	C₄²×C₅₃	—	2	2	no	—
140	C₄²×C₅₃	—	2	2	no	—
141	C₄²×C₅₃	—	2	1	no	—
142	C₄²×C₅₃	—	2	1	no	—
143	C₄²×C₅₃	—	2	1	no	—
144	C₄²×C₅₃	—	2	1	no	—
145	C₄²×C₅₃	—	2	2	no	—
146	C₄²×C₅₃	—	4	1	no	—
147	C₄²×C₅₃	—	4	1	no	—
148	C₄²×C₅₃	—	4	1	no	—
149	C₄²×C₅₃	M₁₆	1	4	yes	—
150	C₄²×C₅₃	#41	1	2	yes	—
151	C₄²×C₅₃	#41	1	2	yes	—
152	C₂³×C₅₃	C₂³×C₄	1	1	yes	characteristic
153	Q₈×C₅₃	—	2	8	no	—
154	C₂×C₄×C₅₃	—	2	1	no	—
155	C₂×C₄×C₅₃	—	2	1	no	—
156	C₂×C₄×C₅₃	—	2	1	no	—
157	C₂×C₄×C₅₃	—	2	1	no	—
158	C₂×C₄×C₅₃	—	2	1	no	—
159	C₂×C₄×C₅₃	—	2	1	no	—
160	C₂×C₄×C₅₃	—	2	1	no	—
161	C₂×C₄×C₅₃	—	2	1	no	—
162	C₂×C₄×C₅₃	—	2	1	no	—
163	C₂×C₄×C₅₃	—	2	1	no	—
164	C₂×C₄×C₅₃	—	2	1	no	—
165	C₂×C₄×C₅₃	—	2	1	no	—
166	C₂×C₄×C₅₃	—	2	1	no	—
167	C₂×C₄×C₅₃	—	2	1	no	—
168	C₂×C₄×C₅₃	—	2	1	no	—
169	C₂×C₄×C₅₃	—	2	1	no	—
170	C₂×C₄×C₅₃	—	2	1	no	—
171	C₂×C₄×C₅₃	—	2	1	no	—
172	C₂×C₄×C₅₃	—	2	1	no	—
173	C₂×C₄×C₅₃	—	2	1	no	—
174	C₂×C₄×C₅₃	—	2	1	no	—
175	C₂×C₄×C₅₃	—	2	1	no	—
176	C₂×C₄×C₅₃	—	2	1	no	—
177	C₂×C₄×C₅₃	—	2	1	no	—
178	C₂×C₄×C₅₃	—	2	1	no	—
179	C₂×C₄×C₅₃	—	2	1	no	—
180	C₂×C₄×C₅₃	—	2	1	no	—
181	C₂×C₄×C₅₃	—	2	1	no	—
182	C₂×C₄×C₅₃	—	2	1	no	—
183	C₂×C₄×C₅₃	—	2	1	no	—
184	C₂×C₄×C₅₃	—	2	1	no	—
185	C₂×C₄×C₅₃	—	2	1	no	—
186	C₂×C₄×C₅₃	—	2	1	no	—
187	C₂×C₄×C₅₃	#117	1	2	yes	—
188	C₂×C₄×C₅₃	#117	1	2	yes	—
189	C₂×C₄×C₅₃	C₂×M₁₆	1	2	yes	—
190	C₂×C₄×C₅₃	C₂×M₁₆	1	2	yes	—
191	C₂×C₄×C₅₃	C₂×M₁₆	1	2	yes	—
192	C₂×C₄×C₅₃	C₂×M₁₆	1	2	yes	—
193	C₂×C₄×C₅₃	C₂×M₁₆	1	2	yes	—
194	C₂×C₄×C₅₃	C₂×M₁₆	1	2	yes	—
195	C₂×C₄×C₅₃	#131	1	2	yes	—
196	C₂×C₄×C₅₃	C₂×#41	1	1	yes	characteristic
197	C₂×C₄×C₅₃	C₂×#41	1	2	yes	—
198	C₂×C₄×C₅₃	C₂×#41	1	1	yes	characteristic
199	C₂×C₄×C₅₃	2₋¹⁺⁴	1	2	yes	—
200	C₈×C₅₃	—	2	4	no	—
201	C₈×C₅₃	—	4	2	no	—
202	C₈×C₅₃	—	4	2	no	—
203	C₈×C₅₃	—	4	2	no	—
204	#22830	C₅₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
205	C₂²×C₅₃	C₂×#117	1	1	yes	characteristic
206	C₂²×C₅₃	C₄×#41	1	1	yes	characteristic
207	C₂²×C₅₃	#519	1	1	yes	characteristic
208	C₂²×C₅₃	C₂²×M₁₆	1	1	yes	characteristic
209	C₂²×C₅₃	C₂²×M₁₆	1	1	yes	characteristic
210	C₂²×C₅₃	C₂×#131	1	1	yes	characteristic
211	C₂²×C₅₃	#568	1	1	yes	characteristic
212	C₄×C₅₃	—	2	2	no	—
213	C₄×C₅₃	—	2	2	no	—
214	C₄×C₅₃	—	2	2	no	—
215	C₄×C₅₃	—	2	2	no	—
216	C₄×C₅₃	—	2	2	no	—
217	C₄×C₅₃	—	2	2	no	—
218	C₄×C₅₃	—	2	2	no	—
219	C₄×C₅₃	—	2	2	no	—
220	C₄×C₅₃	—	2	2	no	—
221	C₄×C₅₃	—	4	1	no	—
222	C₄×C₅₃	—	4	1	no	—
223	C₄×C₅₃	—	4	1	no	—
224	C₄×C₅₃	—	4	1	no	—
225	C₄×C₅₃	—	4	1	no	—
226	C₄×C₅₃	#552	1	2	yes	—
227	C₄×C₅₃	#556	1	2	yes	—
228	C₄²×Q₈	C₂×C₅₃	1	1	yes	characteristic
229	#1985	C₂×C₅₃	1	2	yes	—
230	#1917	C₂×C₅₃	1	2	yes	—
231	#1916	C₂×C₅₃	1	2	yes	—
232	#1839	C₂×C₅₃	1	1	yes	characteristic
233	#1839	C₂×C₅₃	1	1	yes	characteristic
234	#1831	C₂×C₅₃	1	1	yes	characteristic
235	#1771	C₂×C₅₃	1	2	yes	—
236	#1767	C₂×C₅₃	1	1	yes	characteristic
237	#1745	C₂×C₅₃	1	1	yes	characteristic
238	#1744	C₂×C₅₃	1	1	yes	characteristic
239	C₂×C₅₃	#2356	1	1	yes	characteristic
240	C₂×C₅₃	#2920	1	1	yes	characteristic
241	C₂×C₅₃	#2925	1	1	yes	characteristic
242	C₂×C₅₃	#2965	1	1	yes	characteristic
243	C₂×C₅₃	#2970	1	1	yes	characteristic
244	C₂×C₅₃	#2983	1	1	yes	characteristic
245	C₂×C₅₃	#2984	1	1	yes	characteristic
246	C₂×C₄×Q₈	C₄×C₅₃	1	1	yes	characteristic
247	C₂×C₄×Q₈	C₄×C₅₃	1	1	yes	characteristic
248	C₂×C₄×Q₈	C₂²×C₅₃	1	1	yes	characteristic
249	C₂×C₄⋊C₈	C₂²×C₅₃	1	2	yes	—
250	C₂×C₄⋊C₈	C₂²×C₅₃	1	2	yes	—
251	C₂×#97	C₂²×C₅₃	1	2	yes	—
252	C₂×#97	C₂²×C₅₃	1	2	yes	—
253	C₂×C₄×C₈	C₂²×C₅₃	1	2	yes	—
254	C₄×C₄⋊C₄	C₄×C₅₃	1	1	yes	characteristic
255	C₄×C₄⋊C₄	C₄×C₅₃	1	1	yes	characteristic
256	C₄×C₄⋊C₄	C₄×C₅₃	1	1	yes	characteristic
257	C₄×C₄⋊C₄	C₄×C₅₃	1	1	yes	characteristic
258	C₄×C₄⋊C₄	C₄×C₅₃	1	1	yes	characteristic
259	C₄×C₄⋊C₄	C₄×C₅₃	1	1	yes	characteristic
260	C₄×C₄⋊C₄	C₂²×C₅₃	1	1	yes	characteristic
261	C₄×C₄⋊C₄	C₂²×C₅₃	1	1	yes	characteristic
262	C₄×C₄⋊C₄	C₂²×C₅₃	1	1	yes	characteristic
263	C₄³	C₂²×C₅₃	1	1	yes	characteristic
264	C₄³	C₂²×C₅₃	1	1	yes	characteristic
265	C₄³	C₂²×C₅₃	1	1	yes	characteristic
266	#336	C₂²×C₅₃	1	1	yes	characteristic
267	#336	C₂²×C₅₃	1	1	yes	characteristic
268	#336	C₂²×C₅₃	1	2	yes	—
269	#336	C₂²×C₅₃	1	1	yes	characteristic
270	#336	C₂²×C₅₃	1	2	yes	—
271	#336	C₂²×C₅₃	1	2	yes	—
272	#336	C₂²×C₅₃	1	2	yes	—
273	#336	C₂²×C₅₃	1	2	yes	—
274	#336	C₂²×C₅₃	1	2	yes	—
275	#336	C₂²×C₅₃	1	2	yes	—
276	#336	C₂²×C₅₃	1	1	yes	characteristic
277	C₅₃	#22830	1	1	yes	characteristic, direct product factor, Sylow
278	C₂²×Q₈	C₂×C₄×C₅₃	1	1	yes	characteristic
279	C₂²×C₈	C₂³×C₅₃	1	2	yes	—
280	C₂²×C₈	C₂³×C₅₃	1	2	yes	—
281	C₂²×C₈	C₂³×C₅₃	1	2	yes	—
282	C₂²×C₈	C₂³×C₅₃	1	2	yes	—
283	C₄×Q₈	—	2	4	no	—
284	C₄×Q₈	—	2	2	no	—
285	C₄×Q₈	—	2	2	no	—
286	C₄×Q₈	—	2	4	no	—
287	C₂×C₄⋊C₄	C₂×C₄×C₅₃	1	1	yes	characteristic
288	C₂×C₄⋊C₄	C₂×C₄×C₅₃	1	1	yes	characteristic
289	C₂×C₄⋊C₄	C₂×C₄×C₅₃	1	1	yes	characteristic
290	C₂×C₄⋊C₄	C₂×C₄×C₅₃	1	1	yes	characteristic
291	C₂×C₄⋊C₄	C₂×C₄×C₅₃	1	1	yes	characteristic
292	C₂×C₄⋊C₄	C₂×C₄×C₅₃	1	1	yes	characteristic
293	C₂×C₄⋊C₄	C₂×C₄×C₅₃	1	1	yes	characteristic
294	C₂×C₄⋊C₄	C₂×C₄×C₅₃	1	1	yes	characteristic
295	C₂×C₄⋊C₄	C₂×C₄×C₅₃	1	1	yes	characteristic
296	C₂×C₄²	C₂×C₄×C₅₃	1	1	yes	characteristic
297	C₂×C₄²	C₂×C₄×C₅₃	1	1	yes	characteristic
298	C₂×C₄²	C₂×C₄×C₅₃	1	1	yes	characteristic
299	C₂×C₄²	C₂×C₄×C₅₃	1	1	yes	characteristic
300	C₂×C₄²	C₂×C₄×C₅₃	1	1	yes	characteristic
301	C₂×C₄²	C₂×C₄×C₅₃	1	1	yes	characteristic
302	C₂×C₄²	C₂×C₄×C₅₃	1	1	yes	characteristic
303	C₂×C₄²	C₂×C₄×C₅₃	1	1	yes	characteristic
304	C₂×C₄²	C₂×C₄×C₅₃	1	1	yes	characteristic
305	C₂×C₄²	C₂×C₄×C₅₃	1	1	yes	characteristic
306	C₂×C₄²	C₂×C₄×C₅₃	1	1	yes	characteristic
307	C₂×C₄²	C₂×C₄×C₅₃	1	1	yes	characteristic
308	C₂×C₄²	C₂³×C₅₃	1	1	yes	characteristic
309	C₂×C₄²	C₂³×C₅₃	1	1	yes	characteristic
310	C₂×C₄²	C₂³×C₅₃	1	1	yes	characteristic
311	C₂×C₄²	C₂³×C₅₃	1	1	yes	characteristic
312	C₂×C₄²	C₂³×C₅₃	1	1	yes	characteristic
313	C₂×C₄²	C₂³×C₅₃	1	1	yes	characteristic
314	C₂×C₄²	C₂³×C₅₃	1	1	yes	characteristic
315	C₄⋊C₈	—	2	4	no	—
316	C₄⋊C₈	—	2	4	no	—
317	#97	—	2	4	no	—
318	#97	—	2	4	no	—
319	C₄×C₈	—	2	4	no	—
320	#95	C₂×C₄×C₅₃	1	1	yes	characteristic
321	#95	C₂×C₄×C₅₃	1	1	yes	characteristic
322	#95	C₂×C₄×C₅₃	1	1	yes	characteristic
323	#95	C₂×C₄×C₅₃	1	1	yes	characteristic
324	#95	C₂×C₄×C₅₃	1	1	yes	characteristic
325	#95	C₂×C₄×C₅₃	1	1	yes	characteristic
326	C₂×Q₈	—	2	2	no	—
327	C₂×Q₈	—	2	2	no	—
328	C₂×Q₈	—	2	2	no	—
329	C₂²×C₄	C₂²×C₄×C₅₃	1	1	yes	characteristic
330	C₂²×C₄	C₂²×C₄×C₅₃	1	1	yes	characteristic
331	C₂²×C₄	C₂²×C₄×C₅₃	1	1	yes	characteristic
332	C₂²×C₄	C₂²×C₄×C₅₃	1	1	yes	characteristic
333	C₂²×C₄	C₂²×C₄×C₅₃	1	1	yes	characteristic
334	C₂²×C₄	C₂²×C₄×C₅₃	1	1	yes	characteristic
335	C₂²×C₄	C₂²×C₄×C₅₃	1	1	yes	characteristic
336	C₂²×C₄	C₂²×C₄×C₅₃	1	1	yes	characteristic
337	C₂²×C₄	C₂²×C₄×C₅₃	1	1	yes	characteristic
338	C₂²×C₄	C₂²×C₄×C₅₃	1	1	yes	characteristic
339	C₂²×C₄	C₂²×C₄×C₅₃	1	1	yes	characteristic
340	C₂²×C₄	C₂²×C₄×C₅₃	1	1	yes	characteristic
341	C₂²×C₄	C₂²×C₄×C₅₃	1	1	yes	characteristic
342	C₂²×C₄	C₂²×C₄×C₅₃	1	1	yes	characteristic
343	C₂²×C₄	C₂⁴×C₅₃	1	1	yes	characteristic
344	C₂×C₈	—	2	2	no	—
345	C₂×C₈	—	2	2	no	—
346	C₂×C₈	—	2	2	no	—
347	C₂×C₈	—	2	2	no	—
348	C₂×C₈	—	2	2	no	—
349	C₂×C₈	—	2	2	no	—
350	C₂×C₈	—	2	2	no	—
351	C₂×C₈	—	2	2	no	—
352	C₂×C₈	—	2	2	no	—
353	C₂×C₈	—	2	2	no	—
354	C₂×C₈	—	2	2	no	—
355	C₂×C₈	#41×C₅₃	1	4	yes	—
356	C₄⋊C₄	—	2	2	no	—
357	C₄⋊C₄	—	2	2	no	—
358	C₄⋊C₄	—	2	2	no	—
359	C₄⋊C₄	—	2	2	no	—
360	C₄⋊C₄	—	2	2	no	—
361	C₄⋊C₄	—	2	2	no	—
362	C₄⋊C₄	—	2	2	no	—
363	C₄⋊C₄	—	2	2	no	—
364	C₄⋊C₄	M₁₆×C₅₃	1	4	yes	—
365	C₄²	—	2	2	no	—
366	C₄²	—	2	2	no	—
367	C₄²	—	2	2	no	—
368	C₄²	—	2	2	no	—
369	C₄²	—	2	2	no	—
370	C₄²	—	2	2	no	—
371	C₄²	—	2	2	no	—
372	C₄²	—	2	2	no	—
373	C₄²	—	2	2	no	—
374	C₄²	—	2	2	no	—
375	C₄²	—	2	2	no	—
376	C₄²	—	2	1	no	—
377	C₄²	—	2	1	no	—
378	C₄²	—	2	1	no	—
379	C₄²	—	2	1	no	—
380	C₄²	—	2	2	no	—
381	C₄²	—	4	1	no	—
382	C₄²	—	4	1	no	—
383	C₄²	—	4	1	no	—
384	C₄²	M₁₆×C₅₃	1	4	yes	—
385	C₄²	#41×C₅₃	1	2	yes	—
386	C₄²	#41×C₅₃	1	2	yes	—
387	C₂³	C₂³×C₄×C₅₃	1	1	yes	characteristic
388	Q₈	—	2	8	no	—
389	C₂×C₄	—	2	1	no	—
390	C₂×C₄	—	2	1	no	—
391	C₂×C₄	—	2	1	no	—
392	C₂×C₄	—	2	1	no	—
393	C₂×C₄	—	2	1	no	—
394	C₂×C₄	—	2	1	no	—
395	C₂×C₄	—	2	1	no	—
396	C₂×C₄	—	2	1	no	—
397	C₂×C₄	—	2	1	no	—
398	C₂×C₄	—	2	1	no	—
399	C₂×C₄	—	2	1	no	—
400	C₂×C₄	—	2	1	no	—
401	C₂×C₄	—	2	1	no	—
402	C₂×C₄	—	2	1	no	—
403	C₂×C₄	—	2	1	no	—
404	C₂×C₄	—	2	1	no	—
405	C₂×C₄	—	2	1	no	—
406	C₂×C₄	—	2	1	no	—
407	C₂×C₄	—	2	1	no	—
408	C₂×C₄	—	2	1	no	—
409	C₂×C₄	—	2	1	no	—
410	C₂×C₄	—	2	1	no	—
411	C₂×C₄	—	2	1	no	—
412	C₂×C₄	—	2	1	no	—
413	C₂×C₄	—	2	1	no	—
414	C₂×C₄	—	2	1	no	—
415	C₂×C₄	—	2	1	no	—
416	C₂×C₄	—	2	1	no	—
417	C₂×C₄	—	2	1	no	—
418	C₂×C₄	—	2	1	no	—
419	C₂×C₄	—	2	1	no	—
420	C₂×C₄	—	2	1	no	—
421	C₂×C₄	—	2	1	no	—
422	C₂×C₄	#117×C₅₃	1	2	yes	—
423	C₂×C₄	#117×C₅₃	1	2	yes	—
424	C₂×C₄	C₂×M₁₆×C₅₃	1	2	yes	—
425	C₂×C₄	C₂×M₁₆×C₅₃	1	2	yes	—
426	C₂×C₄	C₂×M₁₆×C₅₃	1	2	yes	—
427	C₂×C₄	C₂×M₁₆×C₅₃	1	2	yes	—
428	C₂×C₄	C₂×M₁₆×C₅₃	1	2	yes	—
429	C₂×C₄	C₂×M₁₆×C₅₃	1	2	yes	—
430	C₂×C₄	#131×C₅₃	1	2	yes	—
431	C₂×C₄	C₂×#41×C₅₃	1	1	yes	characteristic
432	C₂×C₄	C₂×#41×C₅₃	1	2	yes	—
433	C₂×C₄	C₂×#41×C₅₃	1	1	yes	characteristic
434	C₂×C₄	2₋¹⁺⁴×C₅₃	1	2	yes	—
435	C₈	—	2	4	no	—
436	C₈	—	4	2	no	—
437	C₈	—	4	2	no	—
438	C₈	—	4	2	no	—
439	C₂²	C₂×#117×C₅₃	1	1	yes	characteristic
440	C₂²	C₄×#41×C₅₃	1	1	yes	characteristic
441	C₂²	C₅₃×#519	1	1	yes	characteristic
442	C₂²	C₂²×M₁₆×C₅₃	1	1	yes	characteristic
443	C₂²	C₂²×M₁₆×C₅₃	1	1	yes	characteristic
444	C₂²	C₂×#131×C₅₃	1	1	yes	characteristic
445	C₂²	C₅₃×#568	1	1	yes	characteristic
446	C₄	—	2	2	no	—
447	C₄	—	2	2	no	—
448	C₄	—	2	2	no	—
449	C₄	—	2	2	no	—
450	C₄	—	2	2	no	—
451	C₄	—	2	2	no	—
452	C₄	—	2	2	no	—
453	C₄	—	2	2	no	—
454	C₄	—	2	2	no	—
455	C₄	—	4	1	no	—
456	C₄	—	4	1	no	—
457	C₄	—	4	1	no	—
458	C₄	—	4	1	no	—
459	C₄	—	4	1	no	—
460	C₄	C₅₃×#552	1	2	yes	—
461	C₄	C₅₃×#556	1	2	yes	—
462	C₂	C₅₃×#2356	1	1	yes	characteristic
463	C₂	C₅₃×#2920	1	1	yes	characteristic
464	C₂	C₅₃×#2925	1	1	yes	characteristic
465	C₂	C₅₃×#2965	1	1	yes	characteristic
466	C₂	C₅₃×#2970	1	1	yes	characteristic
467	C₂	C₅₃×#2983	1	1	yes	characteristic
468	C₂	C₅₃×#2984	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄×C₅₃	C₂⁴
commutator	C₂³	C₂³×C₄×C₅₃
Frattini, cosocle	C₂²×C₄	C₂⁴×C₅₃
socle, prime radical	C₂³×C₅₃	C₂³×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
204. #22830	C₅₃	direct (N × H)
277. C₅₃	#22830	direct (N × H)

Maximal subgroups

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

subgroup	index
C₄²×Q₈×C₅₃	2
C₅₃×#1985	2
C₅₃×#1917	2
C₅₃×#1916	2
C₅₃×#1839	2
C₅₃×#1831	2
C₅₃×#1771	2
C₅₃×#1767	2
C₅₃×#1745	2
C₅₃×#1744	2
#22830	53

Subgroups — C53×#22830 #27230342

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅₃×#22830 #27230342