Subgroups — C₅₃×#34281 #27241793

Order 13568.

2070 conjugacy classes of subgroups.

The subgroup lattice has 554 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₅₃×#541	C₂	1	1	yes	maximal, characteristic
2	C₅₃×#2373	C₂	1	4	yes	maximal
3	C₅₃×#2373	C₂	1	4	yes	maximal
4	C₅₃×#2371	C₂	1	4	yes	maximal, complemented
5	C₅₃×#2370	C₂	1	4	yes	maximal, complemented
6	C₅₃×#2330	C₂	1	1	yes	maximal, characteristic, complemented
7	C₅₃×#2327	C₂	1	1	yes	maximal, characteristic, complemented
8	C₅₃×#2323	C₂	1	2	yes	maximal
9	C₅₃×#2314	C₂	1	1	yes	maximal, characteristic
10	C₄×C₄⋊Q₈×C₅₃	C₂	1	1	yes	maximal, characteristic, complemented
11	C₄×#125×C₅₃	C₂	1	1	yes	maximal, characteristic, complemented
12	C₄×C₂²⋊Q₈×C₅₃	C₂	1	2	yes	maximal
13	C₅₃×#2295	C₂	1	2	yes	maximal, complemented
14	C₅₃×#2276	C₂	1	2	yes	maximal, complemented
15	C₄×#117×C₅₃	C₂	1	1	yes	maximal, characteristic
16	C₅₃×#541	C₄	1	16	yes	complemented
17	C₂×C₄⋊Q₈×C₅₃	C₂²	1	1	yes	characteristic
18	C₂×#126×C₅₃	C₂²	1	4	yes	—
19	C₂×#125×C₅₃	C₂²	1	1	yes	characteristic
20	C₂×#125×C₅₃	C₂²	1	4	yes	—
21	C₂×C₂²⋊Q₈×C₅₃	C₂²	1	2	yes	—
22	C₂×C₄×Q₈×C₅₃	C₂²	1	2	yes	—
23	C₂×C₄×Q₈×C₅₃	C₂²	1	2	yes	—
24	C₂×C₄×Q₈×C₅₃	C₂²	1	2	yes	—
25	C₂×#117×C₅₃	C₂²	1	1	yes	characteristic
26	C₂×#117×C₅₃	C₂²	1	1	yes	characteristic
27	C₂²×C₄²×C₅₃	C₂²	1	1	yes	characteristic
28	C₅₃×#391	C₂²	1	2	yes	—
29	C₅₃×#391	C₂²	1	2	yes	—
30	C₅₃×#389	C₂²	1	2	yes	—
31	C₅₃×#389	C₂²	1	2	yes	—
32	C₅₃×#389	C₂²	1	4	yes	—
33	C₅₃×#389	C₂²	1	4	yes	—
34	C₅₃×#389	C₂²	1	4	yes	—
35	C₅₃×#388	C₂²	1	4	yes	—
36	C₅₃×#388	C₂²	1	4	yes	—
37	C₅₃×#388	C₂²	1	4	yes	—
38	C₅₃×#387	C₂²	1	4	yes	—
39	C₅₃×#387	C₂²	1	4	yes	—
40	C₅₃×#387	C₂²	1	4	yes	—
41	C₅₃×#387	C₂²	1	4	yes	—
42	C₅₃×#387	C₂²	1	4	yes	—
43	C₅₃×#387	C₂²	1	4	yes	—
44	C₅₃×#387	C₂²	1	4	yes	—
45	C₅₃×#387	C₂²	1	4	yes	—
46	C₅₃×#387	C₂²	1	4	yes	—
47	C₅₃×#385	C₂²	1	4	yes	—
48	C₅₃×#384	C₂²	1	1	yes	characteristic
49	C₅₃×#382	C₂²	1	1	yes	characteristic
50	C₅₃×#380	C₂²	1	2	yes	—
51	C₄×C₄⋊C₄×C₅₃	C₂²	1	2	yes	—
52	C₄×C₄⋊C₄×C₅₃	C₂²	1	2	yes	—
53	C₄×C₄⋊C₄×C₅₃	C₂²	1	2	yes	—
54	C₄×C₄⋊C₄×C₅₃	C₂²	1	2	yes	—
55	C₄×C₄⋊C₄×C₅₃	C₂²	1	2	yes	—
56	C₄×C₄⋊C₄×C₅₃	C₂²	1	4	yes	—
57	C₄×C₄⋊C₄×C₅₃	C₂²	1	4	yes	—
58	C₄×C₄⋊C₄×C₅₃	C₂²	1	4	yes	—
59	C₄×C₄⋊C₄×C₅₃	C₂²	1	4	yes	—
60	C₄×C₄⋊C₄×C₅₃	C₂²	1	4	yes	—
61	C₄×C₄⋊C₄×C₅₃	C₂²	1	4	yes	—
62	C₄×C₄⋊C₄×C₅₃	C₂²	1	4	yes	—
63	C₄×C₂²⋊C₄×C₅₃	C₂²	1	2	yes	—
64	C₄×C₂²⋊C₄×C₅₃	C₂²	1	2	yes	—
65	C₄×C₂²⋊C₄×C₅₃	C₂²	1	4	yes	—
66	C₅₃×#376	C₂²	1	1	yes	characteristic
67	C₅₃×#376	C₂²	1	1	yes	characteristic
68	C₅₃×#376	C₂²	1	4	yes	—
69	C₅₃×#376	C₂²	1	4	yes	—
70	C₄³×C₅₃	C₂²	1	1	yes	characteristic
71	C₄³×C₅₃	C₂²	1	1	yes	characteristic
72	C₂²×Q₈×C₅₃	C₂³	1	2	yes	—
73	C₂³×C₄×C₅₃	C₂³	1	2	yes	—
74	C₂³×C₄×C₅₃	C₂³	1	1	yes	characteristic
75	C₄⋊Q₈×C₅₃	C₂×C₄	1	8	yes	complemented
76	#126×C₅₃	C₂×C₄	1	32	yes	—
77	#125×C₅₃	C₂×C₄	1	8	yes	complemented
78	#125×C₅₃	C₂×C₄	1	32	yes	complemented
79	C₂²⋊Q₈×C₅₃	C₂×C₄	1	16	yes	—
80	C₄×Q₈×C₅₃	—	2	8	no	—
81	C₄×Q₈×C₅₃	—	2	8	no	—
82	C₄×Q₈×C₅₃	C₂×C₄	1	16	yes	complemented
83	#117×C₅₃	—	2	4	no	—
84	#117×C₅₃	C₂×C₄	1	8	yes	—
85	C₂×C₄⋊C₄×C₅₃	C₂³	1	1	yes	characteristic
86	C₂×C₄⋊C₄×C₅₃	C₂³	1	1	yes	characteristic
87	C₂×C₄⋊C₄×C₅₃	C₂³	1	2	yes	—
88	C₂×C₄⋊C₄×C₅₃	C₂³	1	2	yes	—
89	C₂×C₄⋊C₄×C₅₃	C₂³	1	4	yes	—
90	C₂×C₄⋊C₄×C₅₃	C₂³	1	4	yes	—
91	C₂×C₄⋊C₄×C₅₃	C₂³	1	4	yes	—
92	C₂×C₄⋊C₄×C₅₃	C₂³	1	4	yes	—
93	C₂×C₄⋊C₄×C₅₃	C₂³	1	4	yes	—
94	C₂×C₄⋊C₄×C₅₃	C₂³	1	2	yes	—
95	C₂×C₄⋊C₄×C₅₃	C₂³	1	2	yes	—
96	C₂×C₄⋊C₄×C₅₃	C₂³	1	4	yes	—
97	C₂×C₄⋊C₄×C₅₃	C₂³	1	4	yes	—
98	C₂×C₄⋊C₄×C₅₃	C₂³	1	4	yes	—
99	C₂×C₄⋊C₄×C₅₃	C₂³	1	4	yes	—
100	C₂×C₂²⋊C₄×C₅₃	C₂³	1	1	yes	characteristic
101	C₂×C₂²⋊C₄×C₅₃	C₂³	1	1	yes	characteristic
102	C₂×C₂²⋊C₄×C₅₃	C₂³	1	4	yes	—
103	C₂×C₂²⋊C₄×C₅₃	C₂³	1	2	yes	—
104	C₂×C₂²⋊C₄×C₅₃	C₂³	1	4	yes	—
105	C₂×C₄²×C₅₃	—	2	4	no	—
106	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
107	C₂×C₄²×C₅₃	C₂³	1	2	yes	—
108	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
109	C₂×C₄²×C₅₃	C₂³	1	4	yes	—
110	C₂×C₄²×C₅₃	C₂³	1	4	yes	—
111	C₂×C₄²×C₅₃	C₂³	1	2	yes	—
112	C₂×C₄²×C₅₃	C₂³	1	2	yes	—
113	C₂×C₄²×C₅₃	C₂³	1	2	yes	—
114	C₂×C₄²×C₅₃	C₂³	1	2	yes	—
115	C₂×C₄²×C₅₃	C₂³	1	4	yes	—
116	C₂×C₄²×C₅₃	C₂³	1	4	yes	—
117	C₂×C₄²×C₅₃	C₂³	1	4	yes	—
118	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
119	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
120	C₂×C₄²×C₅₃	C₂³	1	4	yes	—
121	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
122	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
123	#95×C₅₃	C₂³	1	2	yes	—
124	#95×C₅₃	C₂³	1	2	yes	—
125	#95×C₅₃	C₂³	1	2	yes	—
126	#95×C₅₃	C₂³	1	4	yes	—
127	#95×C₅₃	C₂³	1	2	yes	—
128	#95×C₅₃	C₂³	1	4	yes	—
129	#95×C₅₃	C₂³	1	4	yes	—
130	#95×C₅₃	C₂³	1	4	yes	—
131	#95×C₅₃	C₂³	1	4	yes	—
132	#95×C₅₃	C₂³	1	4	yes	—
133	#95×C₅₃	C₂³	1	4	yes	—
134	#95×C₅₃	C₂³	1	4	yes	—
135	#95×C₅₃	C₂³	1	4	yes	—
136	#95×C₅₃	C₂³	1	4	yes	—
137	#95×C₅₃	C₂³	1	4	yes	—
138	C₂⁴×C₅₃	C₂⁴	1	1	yes	characteristic
139	C₂×Q₈×C₅₃	—	2	4	no	—
140	C₂×Q₈×C₅₃	—	2	4	no	—
141	C₂×Q₈×C₅₃	C₂²×C₄	1	8	yes	—
142	C₂²×C₄×C₅₃	—	2	2	no	—
143	C₂²×C₄×C₅₃	—	2	2	no	—
144	C₂²×C₄×C₅₃	—	2	4	no	—
145	C₂²×C₄×C₅₃	—	2	4	no	—
146	C₂²×C₄×C₅₃	—	4	2	no	—
147	C₂²×C₄×C₅₃	C₂²×C₄	1	4	yes	—
148	C₂²×C₄×C₅₃	C₂⁴	1	1	yes	characteristic
149	C₂²×C₄×C₅₃	C₂⁴	1	4	yes	—
150	C₂²×C₄×C₅₃	C₂⁴	1	1	yes	characteristic
151	C₂²×C₄×C₅₃	C₂⁴	1	4	yes	—
152	C₂²×C₄×C₅₃	C₂⁴	1	2	yes	—
153	C₂²×C₄×C₅₃	C₂⁴	1	2	yes	—
154	C₂²×C₄×C₅₃	C₂⁴	1	4	yes	—
155	C₂²×C₄×C₅₃	C₂⁴	1	4	yes	—
156	C₂²×C₄×C₅₃	C₂⁴	1	1	yes	characteristic
157	C₂²×C₄×C₅₃	C₂⁴	1	1	yes	characteristic
158	C₂²×C₄×C₅₃	C₂⁴	1	2	yes	—
159	C₂²×C₄×C₅₃	C₂⁴	1	1	yes	characteristic
160	C₂²×C₄×C₅₃	C₂⁴	1	2	yes	—
161	C₂²×C₄×C₅₃	C₂⁴	1	1	yes	characteristic
162	C₄⋊C₄×C₅₃	—	2	2	no	—
163	C₄⋊C₄×C₅₃	—	2	2	no	—
164	C₄⋊C₄×C₅₃	—	2	8	no	—
165	C₄⋊C₄×C₅₃	—	2	8	no	—
166	C₄⋊C₄×C₅₃	—	2	4	no	—
167	C₄⋊C₄×C₅₃	—	2	4	no	—
168	C₄⋊C₄×C₅₃	—	2	8	no	—
169	C₄⋊C₄×C₅₃	—	4	4	no	—
170	C₄⋊C₄×C₅₃	—	4	4	no	—
171	C₄⋊C₄×C₅₃	C₂²×C₄	1	8	yes	—
172	C₄⋊C₄×C₅₃	C₂²×C₄	1	8	yes	—
173	C₄⋊C₄×C₅₃	C₂²×C₄	1	16	yes	—
174	C₄⋊C₄×C₅₃	C₂²×C₄	1	16	yes	—
175	C₄⋊C₄×C₅₃	C₂²×C₄	1	16	yes	—
176	C₄⋊C₄×C₅₃	C₂²×C₄	1	16	yes	—
177	C₂²⋊C₄×C₅₃	—	2	2	no	—
178	C₂²⋊C₄×C₅₃	—	2	2	no	—
179	C₂²⋊C₄×C₅₃	—	4	4	no	—
180	C₂²⋊C₄×C₅₃	C₂²×C₄	1	8	yes	—
181	C₂²⋊C₄×C₅₃	C₂²×C₄	1	16	yes	—
182	C₄²×C₅₃	—	2	4	no	—
183	C₄²×C₅₃	—	2	2	no	—
184	C₄²×C₅₃	—	2	2	no	—
185	C₄²×C₅₃	—	2	8	no	—
186	C₄²×C₅₃	—	2	8	no	—
187	C₄²×C₅₃	—	2	4	no	—
188	C₄²×C₅₃	—	2	4	no	—
189	C₄²×C₅₃	—	4	2	no	—
190	C₄²×C₅₃	—	4	2	no	—
191	C₄²×C₅₃	—	4	4	no	—
192	C₄²×C₅₃	—	4	4	no	—
193	C₄²×C₅₃	—	4	4	no	—
194	C₄²×C₅₃	C₂²×C₄	1	4	yes	—
195	C₄²×C₅₃	C₂²×C₄	1	4	yes	—
196	C₄²×C₅₃	C₂²×C₄	1	16	yes	—
197	C₄²×C₅₃	#41	1	4	yes	—
198	C₄²×C₅₃	#41	1	4	yes	—
199	C₂³×C₅₃	—	2	1	no	—
200	C₂³×C₅₃	—	2	1	no	—
201	C₂³×C₅₃	—	2	2	no	—
202	C₂³×C₅₃	—	2	2	no	—
203	C₂³×C₅₃	C₂³×C₄	1	2	yes	—
204	C₂³×C₅₃	C₂⁵	1	1	yes	characteristic
205	Q₈×C₅₃	—	2	16	no	—
206	C₂×C₄×C₅₃	—	2	1	no	—
207	C₂×C₄×C₅₃	—	2	2	no	—
208	C₂×C₄×C₅₃	—	2	1	no	—
209	C₂×C₄×C₅₃	—	2	4	no	—
210	C₂×C₄×C₅₃	—	2	4	no	—
211	C₂×C₄×C₅₃	—	2	2	no	—
212	C₂×C₄×C₅₃	—	2	2	no	—
213	C₂×C₄×C₅₃	—	2	2	no	—
214	C₂×C₄×C₅₃	—	2	2	no	—
215	C₂×C₄×C₅₃	—	2	4	no	—
216	C₂×C₄×C₅₃	—	2	2	no	—
217	C₂×C₄×C₅₃	—	2	2	no	—
218	C₂×C₄×C₅₃	—	2	2	no	—
219	C₂×C₄×C₅₃	—	2	2	no	—
220	C₂×C₄×C₅₃	—	2	4	no	—
221	C₂×C₄×C₅₃	—	2	2	no	—
222	C₂×C₄×C₅₃	—	2	8	no	—
223	C₂×C₄×C₅₃	—	2	1	no	—
224	C₂×C₄×C₅₃	—	2	1	no	—
225	C₂×C₄×C₅₃	—	2	4	no	—
226	C₂×C₄×C₅₃	—	2	4	no	—
227	C₂×C₄×C₅₃	—	2	2	no	—
228	C₂×C₄×C₅₃	—	2	4	no	—
229	C₂×C₄×C₅₃	—	2	4	no	—
230	C₂×C₄×C₅₃	—	2	4	no	—
231	C₂×C₄×C₅₃	—	2	2	no	—
232	C₂×C₄×C₅₃	—	2	4	no	—
233	C₂×C₄×C₅₃	—	4	8	no	—
234	C₂×C₄×C₅₃	C₂³×C₄	1	2	yes	—
235	C₂×C₄×C₅₃	C₂³×C₄	1	8	yes	—
236	C₂×C₄×C₅₃	C₂³×C₄	1	4	yes	—
237	C₂×C₄×C₅₃	C₂³×C₄	1	4	yes	—
238	C₂×C₄×C₅₃	C₂³×C₄	1	8	yes	—
239	C₂×C₄×C₅₃	C₂³×C₄	1	2	yes	—
240	C₂×C₄×C₅₃	C₂×#41	1	4	yes	—
241	C₂×C₄×C₅₃	C₂×#41	1	2	yes	—
242	C₂×C₄×C₅₃	C₂×#41	1	2	yes	—
243	C₂×C₄×C₅₃	C₂×#41	1	2	yes	—
244	C₂×C₄×C₅₃	C₂×#41	1	2	yes	—
245	C₂×C₄×C₅₃	2₋¹⁺⁴	1	4	yes	—
246	#34281	C₅₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
247	C₂²×C₅₃	—	2	2	no	—
248	C₂²×C₅₃	—	2	4	no	—
249	C₂²×C₅₃	—	4	2	no	—
250	C₂²×C₅₃	—	4	1	no	—
251	C₂²×C₅₃	—	4	1	no	—
252	C₂²×C₅₃	C₂⁴×C₄	1	1	yes	characteristic
253	C₂²×C₅₃	C₂²×#41	1	1	yes	characteristic
254	C₂²×C₅₃	C₂²×#41	1	1	yes	characteristic
255	C₂²×C₅₃	C₂×2₋¹⁺⁴	1	2	yes	—
256	C₂²×C₅₃	#585	1	2	yes	—
257	C₄×C₅₃	—	2	2	no	—
258	C₄×C₅₃	—	2	8	no	—
259	C₄×C₅₃	—	2	4	no	—
260	C₄×C₅₃	—	2	4	no	—
261	C₄×C₅₃	—	2	8	no	—
262	C₄×C₅₃	—	2	4	no	—
263	C₄×C₅₃	—	2	4	no	—
264	C₄×C₅₃	—	2	2	no	—
265	C₄×C₅₃	—	2	2	no	—
266	C₄×C₅₃	—	4	4	no	—
267	C₄×C₅₃	—	4	1	no	—
268	C₄×C₅₃	—	4	1	no	—
269	C₄×C₅₃	—	4	4	no	—
270	C₄×C₅₃	C₄×#41	1	4	yes	—
271	C₂×#541	C₂×C₅₃	1	1	yes	characteristic
272	#2373	C₂×C₅₃	1	4	yes	—
273	#2373	C₂×C₅₃	1	4	yes	—
274	#2371	C₂×C₅₃	1	4	yes	complemented
275	#2370	C₂×C₅₃	1	4	yes	complemented
276	#2330	C₂×C₅₃	1	1	yes	characteristic, complemented
277	#2327	C₂×C₅₃	1	1	yes	characteristic, complemented
278	#2323	C₂×C₅₃	1	2	yes	—
279	#2314	C₂×C₅₃	1	1	yes	characteristic
280	C₄×C₄⋊Q₈	C₂×C₅₃	1	1	yes	characteristic, complemented
281	C₄×#125	C₂×C₅₃	1	1	yes	characteristic, complemented
282	C₄×C₂²⋊Q₈	C₂×C₅₃	1	2	yes	—
283	#2295	C₂×C₅₃	1	2	yes	complemented
284	#2276	C₂×C₅₃	1	2	yes	complemented
285	C₄×#117	C₂×C₅₃	1	1	yes	characteristic
286	C₂×C₅₃	—	4	2	no	—
287	C₂×C₅₃	C₂×C₄×#41	1	1	yes	characteristic
288	C₂×C₅₃	C₄×2₋¹⁺⁴	1	2	yes	—
289	C₂×C₅₃	#3495	1	2	yes	—
290	C₂×C₅₃	#3509	1	1	yes	characteristic
291	C₂×C₅₃	#3512	1	1	yes	characteristic
292	#541	C₄×C₅₃	1	16	yes	complemented
293	C₂×C₄⋊Q₈	C₂²×C₅₃	1	1	yes	characteristic
294	C₂×#126	C₂²×C₅₃	1	4	yes	—
295	C₂×#125	C₂²×C₅₃	1	1	yes	characteristic
296	C₂×#125	C₂²×C₅₃	1	4	yes	—
297	C₂×C₂²⋊Q₈	C₂²×C₅₃	1	2	yes	—
298	C₂×C₄×Q₈	C₂²×C₅₃	1	2	yes	—
299	C₂×C₄×Q₈	C₂²×C₅₃	1	2	yes	—
300	C₂×C₄×Q₈	C₂²×C₅₃	1	2	yes	—
301	C₂×#117	C₂²×C₅₃	1	1	yes	characteristic
302	C₂×#117	C₂²×C₅₃	1	1	yes	characteristic
303	C₂²×C₄²	C₂²×C₅₃	1	1	yes	characteristic
304	#391	C₂²×C₅₃	1	2	yes	—
305	#391	C₂²×C₅₃	1	2	yes	—
306	#389	C₂²×C₅₃	1	2	yes	—
307	#389	C₂²×C₅₃	1	4	yes	—
308	#389	C₂²×C₅₃	1	2	yes	—
309	#389	C₂²×C₅₃	1	4	yes	—
310	#389	C₂²×C₅₃	1	4	yes	—
311	#388	C₂²×C₅₃	1	4	yes	—
312	#388	C₂²×C₅₃	1	4	yes	—
313	#388	C₂²×C₅₃	1	4	yes	—
314	#387	C₂²×C₅₃	1	4	yes	—
315	#387	C₂²×C₅₃	1	4	yes	—
316	#387	C₂²×C₅₃	1	4	yes	—
317	#387	C₂²×C₅₃	1	4	yes	—
318	#387	C₂²×C₅₃	1	4	yes	—
319	#387	C₂²×C₅₃	1	4	yes	—
320	#387	C₂²×C₅₃	1	4	yes	—
321	#387	C₂²×C₅₃	1	4	yes	—
322	#387	C₂²×C₅₃	1	4	yes	—
323	#385	C₂²×C₅₃	1	4	yes	—
324	#384	C₂²×C₅₃	1	1	yes	characteristic
325	#382	C₂²×C₅₃	1	1	yes	characteristic
326	#380	C₂²×C₅₃	1	2	yes	—
327	C₄×C₄⋊C₄	C₂²×C₅₃	1	2	yes	—
328	C₄×C₄⋊C₄	C₂²×C₅₃	1	4	yes	—
329	C₄×C₄⋊C₄	C₂²×C₅₃	1	4	yes	—
330	C₄×C₄⋊C₄	C₂²×C₅₃	1	2	yes	—
331	C₄×C₄⋊C₄	C₂²×C₅₃	1	2	yes	—
332	C₄×C₄⋊C₄	C₂²×C₅₃	1	2	yes	—
333	C₄×C₄⋊C₄	C₂²×C₅₃	1	4	yes	—
334	C₄×C₄⋊C₄	C₂²×C₅₃	1	4	yes	—
335	C₄×C₄⋊C₄	C₂²×C₅₃	1	4	yes	—
336	C₄×C₄⋊C₄	C₂²×C₅₃	1	4	yes	—
337	C₄×C₄⋊C₄	C₂²×C₅₃	1	2	yes	—
338	C₄×C₄⋊C₄	C₂²×C₅₃	1	4	yes	—
339	C₄×C₂²⋊C₄	C₂²×C₅₃	1	2	yes	—
340	C₄×C₂²⋊C₄	C₂²×C₅₃	1	2	yes	—
341	C₄×C₂²⋊C₄	C₂²×C₅₃	1	4	yes	—
342	#376	C₂²×C₅₃	1	1	yes	characteristic
343	#376	C₂²×C₅₃	1	4	yes	—
344	#376	C₂²×C₅₃	1	4	yes	—
345	#376	C₂²×C₅₃	1	1	yes	characteristic
346	C₄³	C₂²×C₅₃	1	1	yes	characteristic
347	C₄³	C₂²×C₅₃	1	1	yes	characteristic
348	C₅₃	#34281	1	1	yes	characteristic, direct product factor, Sylow
349	C₂²×Q₈	C₂³×C₅₃	1	2	yes	—
350	C₂³×C₄	C₂³×C₅₃	1	2	yes	—
351	C₂³×C₄	C₂³×C₅₃	1	1	yes	characteristic
352	C₄⋊Q₈	C₂×C₄×C₅₃	1	8	yes	complemented
353	#126	C₂×C₄×C₅₃	1	32	yes	—
354	#125	C₂×C₄×C₅₃	1	8	yes	complemented
355	#125	C₂×C₄×C₅₃	1	32	yes	complemented
356	C₂²⋊Q₈	C₂×C₄×C₅₃	1	16	yes	—
357	C₄×Q₈	—	2	8	no	—
358	C₄×Q₈	—	2	8	no	—
359	C₄×Q₈	C₂×C₄×C₅₃	1	16	yes	complemented
360	#117	—	2	4	no	—
361	#117	C₂×C₄×C₅₃	1	8	yes	—
362	C₂×C₄⋊C₄	C₂³×C₅₃	1	1	yes	characteristic
363	C₂×C₄⋊C₄	C₂³×C₅₃	1	1	yes	characteristic
364	C₂×C₄⋊C₄	C₂³×C₅₃	1	2	yes	—
365	C₂×C₄⋊C₄	C₂³×C₅₃	1	2	yes	—
366	C₂×C₄⋊C₄	C₂³×C₅₃	1	4	yes	—
367	C₂×C₄⋊C₄	C₂³×C₅₃	1	4	yes	—
368	C₂×C₄⋊C₄	C₂³×C₅₃	1	4	yes	—
369	C₂×C₄⋊C₄	C₂³×C₅₃	1	4	yes	—
370	C₂×C₄⋊C₄	C₂³×C₅₃	1	4	yes	—
371	C₂×C₄⋊C₄	C₂³×C₅₃	1	2	yes	—
372	C₂×C₄⋊C₄	C₂³×C₅₃	1	2	yes	—
373	C₂×C₄⋊C₄	C₂³×C₅₃	1	4	yes	—
374	C₂×C₄⋊C₄	C₂³×C₅₃	1	4	yes	—
375	C₂×C₄⋊C₄	C₂³×C₅₃	1	4	yes	—
376	C₂×C₄⋊C₄	C₂³×C₅₃	1	4	yes	—
377	C₂×C₂²⋊C₄	C₂³×C₅₃	1	1	yes	characteristic
378	C₂×C₂²⋊C₄	C₂³×C₅₃	1	1	yes	characteristic
379	C₂×C₂²⋊C₄	C₂³×C₅₃	1	4	yes	—
380	C₂×C₂²⋊C₄	C₂³×C₅₃	1	2	yes	—
381	C₂×C₂²⋊C₄	C₂³×C₅₃	1	4	yes	—
382	C₂×C₄²	—	2	4	no	—
383	C₂×C₄²	C₂³×C₅₃	1	1	yes	characteristic
384	C₂×C₄²	C₂³×C₅₃	1	4	yes	—
385	C₂×C₄²	C₂³×C₅₃	1	4	yes	—
386	C₂×C₄²	C₂³×C₅₃	1	1	yes	characteristic
387	C₂×C₄²	C₂³×C₅₃	1	2	yes	—
388	C₂×C₄²	C₂³×C₅₃	1	2	yes	—
389	C₂×C₄²	C₂³×C₅₃	1	2	yes	—
390	C₂×C₄²	C₂³×C₅₃	1	2	yes	—
391	C₂×C₄²	C₂³×C₅₃	1	2	yes	—
392	C₂×C₄²	C₂³×C₅₃	1	4	yes	—
393	C₂×C₄²	C₂³×C₅₃	1	4	yes	—
394	C₂×C₄²	C₂³×C₅₃	1	4	yes	—
395	C₂×C₄²	C₂³×C₅₃	1	1	yes	characteristic
396	C₂×C₄²	C₂³×C₅₃	1	1	yes	characteristic
397	C₂×C₄²	C₂³×C₅₃	1	4	yes	—
398	C₂×C₄²	C₂³×C₅₃	1	1	yes	characteristic
399	C₂×C₄²	C₂³×C₅₃	1	1	yes	characteristic
400	#95	C₂³×C₅₃	1	2	yes	—
401	#95	C₂³×C₅₃	1	2	yes	—
402	#95	C₂³×C₅₃	1	2	yes	—
403	#95	C₂³×C₅₃	1	4	yes	—
404	#95	C₂³×C₅₃	1	2	yes	—
405	#95	C₂³×C₅₃	1	4	yes	—
406	#95	C₂³×C₅₃	1	4	yes	—
407	#95	C₂³×C₅₃	1	4	yes	—
408	#95	C₂³×C₅₃	1	4	yes	—
409	#95	C₂³×C₅₃	1	4	yes	—
410	#95	C₂³×C₅₃	1	4	yes	—
411	#95	C₂³×C₅₃	1	4	yes	—
412	#95	C₂³×C₅₃	1	4	yes	—
413	#95	C₂³×C₅₃	1	4	yes	—
414	#95	C₂³×C₅₃	1	4	yes	—
415	C₂⁴	C₂⁴×C₅₃	1	1	yes	characteristic
416	C₂×Q₈	—	2	4	no	—
417	C₂×Q₈	—	2	4	no	—
418	C₂×Q₈	C₂²×C₄×C₅₃	1	8	yes	—
419	C₂²×C₄	—	2	2	no	—
420	C₂²×C₄	—	2	2	no	—
421	C₂²×C₄	—	2	4	no	—
422	C₂²×C₄	—	2	4	no	—
423	C₂²×C₄	—	4	2	no	—
424	C₂²×C₄	C₂²×C₄×C₅₃	1	4	yes	—
425	C₂²×C₄	C₂⁴×C₅₃	1	1	yes	characteristic
426	C₂²×C₄	C₂⁴×C₅₃	1	4	yes	—
427	C₂²×C₄	C₂⁴×C₅₃	1	1	yes	characteristic
428	C₂²×C₄	C₂⁴×C₅₃	1	4	yes	—
429	C₂²×C₄	C₂⁴×C₅₃	1	2	yes	—
430	C₂²×C₄	C₂⁴×C₅₃	1	2	yes	—
431	C₂²×C₄	C₂⁴×C₅₃	1	4	yes	—
432	C₂²×C₄	C₂⁴×C₅₃	1	4	yes	—
433	C₂²×C₄	C₂⁴×C₅₃	1	1	yes	characteristic
434	C₂²×C₄	C₂⁴×C₅₃	1	1	yes	characteristic
435	C₂²×C₄	C₂⁴×C₅₃	1	2	yes	—
436	C₂²×C₄	C₂⁴×C₅₃	1	1	yes	characteristic
437	C₂²×C₄	C₂⁴×C₅₃	1	2	yes	—
438	C₂²×C₄	C₂⁴×C₅₃	1	1	yes	characteristic
439	C₄⋊C₄	—	2	2	no	—
440	C₄⋊C₄	—	2	2	no	—
441	C₄⋊C₄	—	2	8	no	—
442	C₄⋊C₄	—	2	8	no	—
443	C₄⋊C₄	—	2	4	no	—
444	C₄⋊C₄	—	2	4	no	—
445	C₄⋊C₄	—	2	8	no	—
446	C₄⋊C₄	—	4	4	no	—
447	C₄⋊C₄	—	4	4	no	—
448	C₄⋊C₄	C₂²×C₄×C₅₃	1	8	yes	—
449	C₄⋊C₄	C₂²×C₄×C₅₃	1	8	yes	—
450	C₄⋊C₄	C₂²×C₄×C₅₃	1	16	yes	—
451	C₄⋊C₄	C₂²×C₄×C₅₃	1	16	yes	—
452	C₄⋊C₄	C₂²×C₄×C₅₃	1	16	yes	—
453	C₄⋊C₄	C₂²×C₄×C₅₃	1	16	yes	—
454	C₂²⋊C₄	—	2	2	no	—
455	C₂²⋊C₄	—	2	2	no	—
456	C₂²⋊C₄	—	4	4	no	—
457	C₂²⋊C₄	C₂²×C₄×C₅₃	1	8	yes	—
458	C₂²⋊C₄	C₂²×C₄×C₅₃	1	16	yes	—
459	C₄²	—	2	4	no	—
460	C₄²	—	2	2	no	—
461	C₄²	—	2	2	no	—
462	C₄²	—	2	8	no	—
463	C₄²	—	2	8	no	—
464	C₄²	—	2	4	no	—
465	C₄²	—	2	4	no	—
466	C₄²	—	4	4	no	—
467	C₄²	—	4	2	no	—
468	C₄²	—	4	2	no	—
469	C₄²	—	4	4	no	—
470	C₄²	—	4	4	no	—
471	C₄²	C₂²×C₄×C₅₃	1	4	yes	—
472	C₄²	C₂²×C₄×C₅₃	1	4	yes	—
473	C₄²	C₂²×C₄×C₅₃	1	16	yes	—
474	C₄²	#41×C₅₃	1	4	yes	—
475	C₄²	#41×C₅₃	1	4	yes	—
476	C₂³	—	2	1	no	—
477	C₂³	—	2	1	no	—
478	C₂³	—	2	2	no	—
479	C₂³	—	2	2	no	—
480	C₂³	C₂³×C₄×C₅₃	1	2	yes	—
481	C₂³	C₂⁵×C₅₃	1	1	yes	characteristic
482	Q₈	—	2	16	no	—
483	C₂×C₄	—	2	2	no	—
484	C₂×C₄	—	2	1	no	—
485	C₂×C₄	—	2	4	no	—
486	C₂×C₄	—	2	4	no	—
487	C₂×C₄	—	2	1	no	—
488	C₂×C₄	—	2	2	no	—
489	C₂×C₄	—	2	2	no	—
490	C₂×C₄	—	2	2	no	—
491	C₂×C₄	—	2	4	no	—
492	C₂×C₄	—	2	2	no	—
493	C₂×C₄	—	2	2	no	—
494	C₂×C₄	—	2	2	no	—
495	C₂×C₄	—	2	4	no	—
496	C₂×C₄	—	2	2	no	—
497	C₂×C₄	—	2	2	no	—
498	C₂×C₄	—	2	2	no	—
499	C₂×C₄	—	2	8	no	—
500	C₂×C₄	—	2	1	no	—
501	C₂×C₄	—	2	1	no	—
502	C₂×C₄	—	2	4	no	—
503	C₂×C₄	—	2	4	no	—
504	C₂×C₄	—	2	2	no	—
505	C₂×C₄	—	2	4	no	—
506	C₂×C₄	—	2	4	no	—
507	C₂×C₄	—	2	4	no	—
508	C₂×C₄	—	2	2	no	—
509	C₂×C₄	—	2	4	no	—
510	C₂×C₄	—	4	8	no	—
511	C₂×C₄	C₂³×C₄×C₅₃	1	4	yes	—
512	C₂×C₄	C₂³×C₄×C₅₃	1	4	yes	—
513	C₂×C₄	C₂³×C₄×C₅₃	1	8	yes	—
514	C₂×C₄	C₂³×C₄×C₅₃	1	8	yes	—
515	C₂×C₄	C₂³×C₄×C₅₃	1	2	yes	—
516	C₂×C₄	C₂³×C₄×C₅₃	1	2	yes	—
517	C₂×C₄	C₂×#41×C₅₃	1	2	yes	—
518	C₂×C₄	C₂×#41×C₅₃	1	4	yes	—
519	C₂×C₄	C₂×#41×C₅₃	1	2	yes	—
520	C₂×C₄	C₂×#41×C₅₃	1	2	yes	—
521	C₂×C₄	C₂×#41×C₅₃	1	2	yes	—
522	C₂×C₄	2₋¹⁺⁴×C₅₃	1	4	yes	—
523	C₂²	—	2	2	no	—
524	C₂²	—	2	4	no	—
525	C₂²	—	4	2	no	—
526	C₂²	—	4	1	no	—
527	C₂²	—	4	1	no	—
528	C₂²	C₂⁴×C₄×C₅₃	1	1	yes	characteristic
529	C₂²	C₂²×#41×C₅₃	1	1	yes	characteristic
530	C₂²	C₂²×#41×C₅₃	1	1	yes	characteristic
531	C₂²	C₂×2₋¹⁺⁴×C₅₃	1	2	yes	—
532	C₂²	C₅₃×#585	1	2	yes	—
533	C₄	—	2	4	no	—
534	C₄	—	2	4	no	—
535	C₄	—	2	2	no	—
536	C₄	—	2	2	no	—
537	C₄	—	2	2	no	—
538	C₄	—	2	8	no	—
539	C₄	—	2	4	no	—
540	C₄	—	2	4	no	—
541	C₄	—	2	8	no	—
542	C₄	—	4	4	no	—
543	C₄	—	4	1	no	—
544	C₄	—	4	1	no	—
545	C₄	—	4	4	no	—
546	C₄	C₄×#41×C₅₃	1	4	yes	—
547	C₂	—	4	2	no	—
548	C₂	C₂×C₄×#41×C₅₃	1	1	yes	characteristic
549	C₂	C₄×2₋¹⁺⁴×C₅₃	1	2	yes	—
550	C₂	C₅₃×#3495	1	2	yes	—
551	C₂	C₅₃×#3509	1	1	yes	characteristic
552	C₂	C₅₃×#3512	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂³×C₅₃	C₂⁵
commutator	C₂²	C₂⁴×C₄×C₅₃
Frattini, cosocle	C₂³	C₂⁵×C₅₃
prime radical	C₂⁴×C₅₃	C₂⁴
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³×C₅₃ ◁ G

Complements

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

N	H ≅ G/N	type
4. C₅₃×#2371	C₂	semidirect
5. C₅₃×#2370	C₂	semidirect
6. C₅₃×#2330	C₂	semidirect
7. C₅₃×#2327	C₂	semidirect
10. C₄×C₄⋊Q₈×C₅₃	C₂	semidirect
11. C₄×#125×C₅₃	C₂	semidirect
13. C₅₃×#2295	C₂	semidirect
14. C₅₃×#2276	C₂	semidirect
16. C₅₃×#541	C₄	semidirect
75. C₄⋊Q₈×C₅₃	C₂×C₄	semidirect
77. #125×C₅₃	C₂×C₄	semidirect
78. #125×C₅₃	C₂×C₄	semidirect
82. C₄×Q₈×C₅₃	C₂×C₄	semidirect
246. #34281	C₅₃	direct (N × H)
274. #2371	C₂×C₅₃	semidirect
275. #2370	C₂×C₅₃	semidirect
276. #2330	C₂×C₅₃	semidirect
277. #2327	C₂×C₅₃	semidirect
280. C₄×C₄⋊Q₈	C₂×C₅₃	semidirect
281. C₄×#125	C₂×C₅₃	semidirect
283. #2295	C₂×C₅₃	semidirect
284. #2276	C₂×C₅₃	semidirect
292. #541	C₄×C₅₃	semidirect
348. C₅₃	#34281	direct (N × H)
352. C₄⋊Q₈	C₂×C₄×C₅₃	semidirect
354. #125	C₂×C₄×C₅₃	semidirect
355. #125	C₂×C₄×C₅₃	semidirect
359. C₄×Q₈	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×#541	2
C₅₃×#2373	2
C₅₃×#2371	2
C₅₃×#2370	2
C₅₃×#2330	2
C₅₃×#2327	2
C₅₃×#2323	2
C₅₃×#2314	2
C₄×C₄⋊Q₈×C₅₃	2
C₄×#125×C₅₃	2
C₄×C₂²⋊Q₈×C₅₃	2
C₅₃×#2295	2
C₅₃×#2276	2
C₄×#117×C₅₃	2
#34281	53

Subgroups — C53×#34281 #27241793

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅₃×#34281 #27241793