Subgroups — C₅₃×#36601 #27244113

Order 13568.

1658 conjugacy classes of subgroups.

The subgroup lattice has 564 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₅₃×#2861	C₂	1	2	yes	maximal
2	C₅₃×#2859	C₂	1	1	yes	maximal, characteristic, complemented
3	C₅₃×#2596	C₂	1	4	yes	maximal, complemented
4	C₅₃×#2581	C₂	1	1	yes	maximal, characteristic, complemented
5	C₅₃×#2580	C₂	1	1	yes	maximal, characteristic
6	C₅₃×#2573	C₂	1	4	yes	maximal, complemented
7	C₅₃×#2512	C₂	1	2	yes	maximal
8	C₅₃×#2511	C₂	1	2	yes	maximal, complemented
9	C₅₃×#2469	C₂	1	4	yes	maximal, complemented
10	C₅₃×#2373	C₂	1	4	yes	maximal, complemented
11	C₅₃×#2357	C₂	1	1	yes	maximal, characteristic
12	C₅₃×#2356	C₂	1	1	yes	maximal, characteristic, complemented
13	C₄×#125×C₅₃	C₂	1	2	yes	maximal, complemented
14	C₄×#124×C₅₃	C₂	1	1	yes	maximal, characteristic
15	C₄²×Q₈×C₅₃	C₂	1	1	yes	maximal, characteristic, complemented
16	C₂×#126×C₅₃	C₂²	1	4	yes	—
17	C₂×#125×C₅₃	C₂²	1	2	yes	—
18	C₂×#124×C₅₃	C₂²	1	1	yes	characteristic
19	C₂×C₄×Q₈×C₅₃	C₂²	1	2	yes	—
20	C₂×C₄×Q₈×C₅₃	C₂²	1	1	yes	characteristic
21	C₂×C₄×Q₈×C₅₃	C₂²	1	1	yes	characteristic
22	C₂×C₄×D₄×C₅₃	C₂²	1	1	yes	characteristic
23	C₅₃×#400	C₂²	1	4	yes	—
24	C₅₃×#400	C₂²	1	4	yes	—
25	C₅₃×#400	C₂²	1	2	yes	—
26	C₅₃×#399	C₂²	1	2	yes	—
27	C₅₃×#397	C₂²	1	4	yes	—
28	C₅₃×#396	C₂²	1	4	yes	—
29	C₅₃×#395	C₂²	1	2	yes	—
30	C₅₃×#393	C₂²	1	2	yes	—
31	C₅₃×#391	C₂²	1	1	yes	characteristic
32	C₅₃×#391	C₂²	1	2	yes	—
33	C₅₃×#391	C₂²	1	2	yes	—
34	C₅₃×#390	C₂²	1	1	yes	characteristic
35	C₅₃×#390	C₂²	1	2	yes	—
36	C₅₃×#390	C₂²	1	2	yes	—
37	C₅₃×#389	C₂²	1	4	yes	—
38	C₅₃×#389	C₂²	1	2	yes	—
39	C₅₃×#389	C₂²	1	2	yes	—
40	C₅₃×#389	C₂²	1	4	yes	—
41	C₅₃×#388	C₂²	1	4	yes	—
42	C₅₃×#388	C₂²	1	4	yes	—
43	C₅₃×#388	C₂²	1	4	yes	—
44	C₅₃×#388	C₂²	1	4	yes	—
45	C₅₃×#388	C₂²	1	4	yes	—
46	C₅₃×#388	C₂²	1	4	yes	—
47	C₅₃×#388	C₂²	1	4	yes	—
48	C₅₃×#387	C₂²	1	4	yes	—
49	C₅₃×#387	C₂²	1	4	yes	—
50	C₅₃×#387	C₂²	1	4	yes	—
51	C₅₃×#387	C₂²	1	4	yes	—
52	C₅₃×#387	C₂²	1	4	yes	—
53	C₅₃×#387	C₂²	1	4	yes	—
54	C₅₃×#385	C₂²	1	4	yes	—
55	C₅₃×#384	C₂²	1	2	yes	—
56	C₅₃×#383	C₂²	1	2	yes	—
57	C₅₃×#382	C₂²	1	1	yes	characteristic
58	C₅₃×#382	C₂²	1	1	yes	characteristic
59	C₅₃×#382	C₂²	1	1	yes	characteristic
60	C₅₃×#381	C₂²	1	2	yes	—
61	C₄×C₄⋊C₄×C₅₃	C₂²	1	4	yes	—
62	C₄×C₄⋊C₄×C₅₃	C₂²	1	2	yes	—
63	C₄×C₄⋊C₄×C₅₃	C₂²	1	2	yes	—
64	C₄×C₄⋊C₄×C₅₃	C₂²	1	2	yes	—
65	C₄×C₄⋊C₄×C₅₃	C₂²	1	2	yes	—
66	C₄×C₄⋊C₄×C₅₃	C₂²	1	2	yes	—
67	C₄×C₄⋊C₄×C₅₃	C₂²	1	1	yes	characteristic
68	C₄×C₄⋊C₄×C₅₃	C₂²	1	2	yes	—
69	C₄×C₄⋊C₄×C₅₃	C₂²	1	4	yes	—
70	C₄×C₂²⋊C₄×C₅₃	C₂²	1	4	yes	—
71	C₄×C₂²⋊C₄×C₅₃	C₂²	1	2	yes	—
72	C₅₃×#376	C₂²	1	2	yes	—
73	C₄³×C₅₃	C₂²	1	1	yes	characteristic
74	C₄³×C₅₃	C₂²	1	2	yes	—
75	C₂²×Q₈×C₅₃	C₂³	1	1	yes	characteristic
76	C₂²×Q₈×C₅₃	C₂³	1	1	yes	characteristic
77	C₂²×D₄×C₅₃	C₂³	1	1	yes	characteristic
78	C₂³×C₄×C₅₃	C₂³	1	2	yes	—
79	#126×C₅₃	—	2	16	no	—
80	#125×C₅₃	—	2	8	no	—
81	#124×C₅₃	—	2	4	no	—
82	C₄×Q₈×C₅₃	—	2	8	no	—
83	C₄×Q₈×C₅₃	—	2	4	no	—
84	C₄×Q₈×C₅₃	—	2	4	no	—
85	C₄×D₄×C₅₃	—	2	4	no	—
86	C₂×C₄⋊C₄×C₅₃	C₂³	1	1	yes	characteristic
87	C₂×C₄⋊C₄×C₅₃	C₂³	1	1	yes	characteristic
88	C₂×C₄⋊C₄×C₅₃	C₂³	1	2	yes	—
89	C₂×C₄⋊C₄×C₅₃	C₂³	1	2	yes	—
90	C₂×C₄⋊C₄×C₅₃	C₂³	1	2	yes	—
91	C₂×C₄⋊C₄×C₅₃	C₂³	1	1	yes	characteristic
92	C₂×C₄⋊C₄×C₅₃	C₂³	1	2	yes	—
93	C₂×C₄⋊C₄×C₅₃	C₂³	1	2	yes	—
94	C₂×C₄⋊C₄×C₅₃	C₂³	1	2	yes	—
95	C₂×C₄⋊C₄×C₅₃	C₂³	1	4	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×C₄⋊C₄×C₅₃	C₂³	1	4	yes	—
101	C₂×C₄⋊C₄×C₅₃	C₂³	1	2	yes	—
102	C₂×C₂²⋊C₄×C₅₃	C₂³	1	2	yes	—
103	C₂×C₂²⋊C₄×C₅₃	C₂³	1	4	yes	—
104	C₂×C₂²⋊C₄×C₅₃	C₂³	1	4	yes	—
105	C₂×C₂²⋊C₄×C₅₃	C₂³	1	4	yes	—
106	C₂×C₂²⋊C₄×C₅₃	C₂³	1	4	yes	—
107	C₂×C₂²⋊C₄×C₅₃	C₂³	1	4	yes	—
108	C₂×C₂²⋊C₄×C₅₃	C₂³	1	4	yes	—
109	C₂×C₄²×C₅₃	C₂³	1	2	yes	—
110	C₂×C₄²×C₅₃	C₂³	1	2	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	4	yes	—
115	C₂×C₄²×C₅₃	C₂³	1	2	yes	—
116	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
117	C₂×C₄²×C₅₃	C₂³	1	2	yes	—
118	C₂×C₄²×C₅₃	C₂³	1	2	yes	—
119	C₂×C₄²×C₅₃	C₂³	1	2	yes	—
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	1	yes	characteristic
124	#95×C₅₃	C₂³	1	2	yes	—
125	#95×C₅₃	C₂³	1	4	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	1	yes	characteristic
130	#95×C₅₃	C₂³	1	2	yes	—
131	#95×C₅₃	C₂³	1	2	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	2	yes	—
137	#95×C₅₃	C₂³	1	2	yes	—
138	#95×C₅₃	C₂³	1	4	yes	—
139	#95×C₅₃	C₂³	1	4	yes	—
140	#95×C₅₃	C₂³	1	4	yes	—
141	C₂⁴×C₅₃	C₂⁴	1	2	yes	—
142	C₂×Q₈×C₅₃	—	2	4	no	—
143	C₂×Q₈×C₅₃	—	2	2	no	—
144	C₂×Q₈×C₅₃	—	2	2	no	—
145	C₂×Q₈×C₅₃	—	2	2	no	—
146	C₂×Q₈×C₅₃	—	2	2	no	—
147	C₂×D₄×C₅₃	—	2	2	no	—
148	C₂×D₄×C₅₃	—	2	2	no	—
149	C₂×D₄×C₅₃	—	2	2	no	—
150	C₂²×C₄×C₅₃	—	2	4	no	—
151	C₂²×C₄×C₅₃	—	4	4	no	—
152	C₂²×C₄×C₅₃	C₂⁴	1	2	yes	—
153	C₂²×C₄×C₅₃	C₂⁴	1	2	yes	—
154	C₂²×C₄×C₅₃	C₂⁴	1	2	yes	—
155	C₂²×C₄×C₅₃	C₂⁴	1	2	yes	—
156	C₂²×C₄×C₅₃	C₂⁴	1	2	yes	—
157	C₂²×C₄×C₅₃	C₂⁴	1	4	yes	—
158	C₂²×C₄×C₅₃	C₂⁴	1	4	yes	—
159	C₂²×C₄×C₅₃	C₂⁴	1	1	yes	characteristic
160	C₂²×C₄×C₅₃	C₂⁴	1	2	yes	—
161	C₂²×C₄×C₅₃	C₂⁴	1	2	yes	—
162	C₂²×C₄×C₅₃	C₂⁴	1	2	yes	—
163	C₂²×C₄×C₅₃	C₂⁴	1	1	yes	characteristic
164	C₂²×C₄×C₅₃	C₂⁴	1	2	yes	—
165	C₂²×C₄×C₅₃	C₂⁴	1	1	yes	characteristic
166	C₄⋊C₄×C₅₃	—	2	2	no	—
167	C₄⋊C₄×C₅₃	—	2	4	no	—
168	C₄⋊C₄×C₅₃	—	2	4	no	—
169	C₄⋊C₄×C₅₃	—	2	4	no	—
170	C₄⋊C₄×C₅₃	—	2	4	no	—
171	C₄⋊C₄×C₅₃	—	2	2	no	—
172	C₄⋊C₄×C₅₃	—	2	2	no	—
173	C₄⋊C₄×C₅₃	—	2	8	no	—
174	C₄⋊C₄×C₅₃	—	2	8	no	—
175	C₄⋊C₄×C₅₃	—	2	4	no	—
176	C₄⋊C₄×C₅₃	—	2	4	no	—
177	C₄⋊C₄×C₅₃	—	2	8	no	—
178	C₄⋊C₄×C₅₃	—	2	8	no	—
179	C₄⋊C₄×C₅₃	—	4	4	no	—
180	C₄⋊C₄×C₅₃	—	4	4	no	—
181	C₄⋊C₄×C₅₃	#41	1	8	yes	—
182	C₂²⋊C₄×C₅₃	—	2	4	no	—
183	C₂²⋊C₄×C₅₃	—	2	8	no	—
184	C₂²⋊C₄×C₅₃	—	2	8	no	—
185	C₂²⋊C₄×C₅₃	—	2	8	no	—
186	C₂²⋊C₄×C₅₃	—	2	8	no	—
187	C₂²⋊C₄×C₅₃	—	4	4	no	—
188	C₂²⋊C₄×C₅₃	—	4	4	no	—
189	C₄²×C₅₃	—	2	4	no	—
190	C₄²×C₅₃	—	2	4	no	—
191	C₄²×C₅₃	—	2	4	no	—
192	C₄²×C₅₃	—	2	4	no	—
193	C₄²×C₅₃	—	2	4	no	—
194	C₄²×C₅₃	—	2	8	no	—
195	C₄²×C₅₃	—	2	2	no	—
196	C₄²×C₅₃	—	2	4	no	—
197	C₄²×C₅₃	—	4	2	no	—
198	C₄²×C₅₃	—	4	2	no	—
199	C₄²×C₅₃	—	4	4	no	—
200	C₄²×C₅₃	—	4	2	no	—
201	C₄²×C₅₃	#41	1	4	yes	—
202	C₄²×C₅₃	#41	1	4	yes	—
203	C₂³×C₅₃	—	2	2	no	—
204	C₂³×C₅₃	—	2	2	no	—
205	C₂³×C₅₃	—	2	4	no	—
206	C₂³×C₅₃	—	2	2	no	—
207	C₂³×C₅₃	—	2	4	no	—
208	C₂³×C₅₃	C₂⁵	1	1	yes	characteristic
209	Q₈×C₅₃	—	2	8	no	—
210	Q₈×C₅₃	—	4	4	no	—
211	D₄×C₅₃	—	4	4	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	2	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	2	no	—
221	C₂×C₄×C₅₃	—	2	2	no	—
222	C₂×C₄×C₅₃	—	2	2	no	—
223	C₂×C₄×C₅₃	—	2	2	no	—
224	C₂×C₄×C₅₃	—	2	2	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	2	no	—
229	C₂×C₄×C₅₃	—	2	2	no	—
230	C₂×C₄×C₅₃	—	2	4	no	—
231	C₂×C₄×C₅₃	—	2	4	no	—
232	C₂×C₄×C₅₃	—	2	4	no	—
233	C₂×C₄×C₅₃	—	2	2	no	—
234	C₂×C₄×C₅₃	—	2	2	no	—
235	C₂×C₄×C₅₃	—	2	4	no	—
236	C₂×C₄×C₅₃	—	2	2	no	—
237	C₂×C₄×C₅₃	—	2	2	no	—
238	C₂×C₄×C₅₃	—	2	4	no	—
239	C₂×C₄×C₅₃	—	2	2	no	—
240	C₂×C₄×C₅₃	—	2	4	no	—
241	C₂×C₄×C₅₃	—	8	4	no	—
242	C₂×C₄×C₅₃	C₂×#41	1	2	yes	—
243	C₂×C₄×C₅₃	C₂×#41	1	4	yes	—
244	C₂×C₄×C₅₃	C₂×#41	1	2	yes	—
245	C₂×C₄×C₅₃	C₂×#41	1	4	yes	—
246	C₂×C₄×C₅₃	C₂×#41	1	4	yes	—
247	C₂×C₄×C₅₃	C₂×#41	1	4	yes	—
248	C₂×C₄×C₅₃	C₂×#41	1	2	yes	—
249	C₂×C₄×C₅₃	C₂×#41	1	2	yes	—
250	C₂×C₄×C₅₃	2₋¹⁺⁴	1	2	yes	—
251	#36601	C₅₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
252	C₂²×C₅₃	—	4	2	no	—
253	C₂²×C₅₃	—	4	2	no	—
254	C₂²×C₅₃	—	4	4	no	—
255	C₂²×C₅₃	—	4	2	no	—
256	C₂²×C₅₃	—	4	4	no	—
257	C₂²×C₅₃	C₂²×#41	1	1	yes	characteristic
258	C₂²×C₅₃	C₂²×#41	1	2	yes	—
259	C₂²×C₅₃	C₂²×#41	1	1	yes	characteristic
260	C₂²×C₅₃	C₂×2₋¹⁺⁴	1	1	yes	characteristic
261	C₂²×C₅₃	#585	1	2	yes	—
262	C₄×C₅₃	—	2	4	no	—
263	C₄×C₅₃	—	2	2	no	—
264	C₄×C₅₃	—	2	4	no	—
265	C₄×C₅₃	—	2	2	no	—
266	C₄×C₅₃	—	2	4	no	—
267	C₄×C₅₃	—	2	4	no	—
268	C₄×C₅₃	—	4	2	no	—
269	C₄×C₅₃	—	4	2	no	—
270	C₄×C₅₃	—	4	4	no	—
271	C₄×C₅₃	—	4	4	no	—
272	C₄×C₅₃	—	4	2	no	—
273	C₄×C₅₃	—	4	2	no	—
274	C₄×C₅₃	—	4	2	no	—
275	C₄×C₅₃	#556	1	4	yes	—
276	#2861	C₂×C₅₃	1	2	yes	—
277	#2859	C₂×C₅₃	1	1	yes	characteristic, complemented
278	#2596	C₂×C₅₃	1	4	yes	complemented
279	#2581	C₂×C₅₃	1	1	yes	characteristic, complemented
280	#2580	C₂×C₅₃	1	1	yes	characteristic
281	#2573	C₂×C₅₃	1	4	yes	complemented
282	#2512	C₂×C₅₃	1	2	yes	—
283	#2511	C₂×C₅₃	1	2	yes	complemented
284	#2469	C₂×C₅₃	1	4	yes	complemented
285	#2373	C₂×C₅₃	1	4	yes	complemented
286	#2357	C₂×C₅₃	1	1	yes	characteristic
287	#2356	C₂×C₅₃	1	1	yes	characteristic, complemented
288	C₄×#125	C₂×C₅₃	1	2	yes	complemented
289	C₄×#124	C₂×C₅₃	1	1	yes	characteristic
290	C₄²×Q₈	C₂×C₅₃	1	1	yes	characteristic, complemented
291	C₂×C₅₃	—	8	2	no	—
292	C₂×C₅₃	#3444	1	1	yes	characteristic
293	C₂×C₅₃	C₂×#556	1	1	yes	characteristic
294	C₂×C₅₃	#3480	1	2	yes	—
295	C₂×C₅₃	#3510	1	1	yes	characteristic
296	C₂×C₅₃	#3515	1	2	yes	—
297	C₂×#126	C₂²×C₅₃	1	4	yes	—
298	C₂×#125	C₂²×C₅₃	1	2	yes	—
299	C₂×#124	C₂²×C₅₃	1	1	yes	characteristic
300	C₂×C₄×Q₈	C₂²×C₅₃	1	1	yes	characteristic
301	C₂×C₄×Q₈	C₂²×C₅₃	1	1	yes	characteristic
302	C₂×C₄×Q₈	C₂²×C₅₃	1	2	yes	—
303	C₂×C₄×D₄	C₂²×C₅₃	1	1	yes	characteristic
304	#400	C₂²×C₅₃	1	4	yes	—
305	#400	C₂²×C₅₃	1	4	yes	—
306	#400	C₂²×C₅₃	1	2	yes	—
307	#399	C₂²×C₅₃	1	2	yes	—
308	#397	C₂²×C₅₃	1	4	yes	—
309	#396	C₂²×C₅₃	1	4	yes	—
310	#395	C₂²×C₅₃	1	2	yes	—
311	#393	C₂²×C₅₃	1	2	yes	—
312	#391	C₂²×C₅₃	1	2	yes	—
313	#391	C₂²×C₅₃	1	2	yes	—
314	#391	C₂²×C₅₃	1	1	yes	characteristic
315	#390	C₂²×C₅₃	1	2	yes	—
316	#390	C₂²×C₅₃	1	2	yes	—
317	#390	C₂²×C₅₃	1	1	yes	characteristic
318	#389	C₂²×C₅₃	1	2	yes	—
319	#389	C₂²×C₅₃	1	2	yes	—
320	#389	C₂²×C₅₃	1	4	yes	—
321	#389	C₂²×C₅₃	1	4	yes	—
322	#388	C₂²×C₅₃	1	4	yes	—
323	#388	C₂²×C₅₃	1	4	yes	—
324	#388	C₂²×C₅₃	1	4	yes	—
325	#388	C₂²×C₅₃	1	4	yes	—
326	#388	C₂²×C₅₃	1	4	yes	—
327	#388	C₂²×C₅₃	1	4	yes	—
328	#388	C₂²×C₅₃	1	4	yes	—
329	#387	C₂²×C₅₃	1	4	yes	—
330	#387	C₂²×C₅₃	1	4	yes	—
331	#387	C₂²×C₅₃	1	4	yes	—
332	#387	C₂²×C₅₃	1	4	yes	—
333	#387	C₂²×C₅₃	1	4	yes	—
334	#387	C₂²×C₅₃	1	4	yes	—
335	#385	C₂²×C₅₃	1	4	yes	—
336	#384	C₂²×C₅₃	1	2	yes	—
337	#383	C₂²×C₅₃	1	2	yes	—
338	#382	C₂²×C₅₃	1	1	yes	characteristic
339	#382	C₂²×C₅₃	1	1	yes	characteristic
340	#382	C₂²×C₅₃	1	1	yes	characteristic
341	#381	C₂²×C₅₃	1	2	yes	—
342	C₄×C₄⋊C₄	C₂²×C₅₃	1	4	yes	—
343	C₄×C₄⋊C₄	C₂²×C₅₃	1	2	yes	—
344	C₄×C₄⋊C₄	C₂²×C₅₃	1	2	yes	—
345	C₄×C₄⋊C₄	C₂²×C₅₃	1	4	yes	—
346	C₄×C₄⋊C₄	C₂²×C₅₃	1	2	yes	—
347	C₄×C₄⋊C₄	C₂²×C₅₃	1	2	yes	—
348	C₄×C₄⋊C₄	C₂²×C₅₃	1	1	yes	characteristic
349	C₄×C₄⋊C₄	C₂²×C₅₃	1	2	yes	—
350	C₄×C₄⋊C₄	C₂²×C₅₃	1	2	yes	—
351	C₄×C₂²⋊C₄	C₂²×C₅₃	1	4	yes	—
352	C₄×C₂²⋊C₄	C₂²×C₅₃	1	2	yes	—
353	#376	C₂²×C₅₃	1	2	yes	—
354	C₄³	C₂²×C₅₃	1	2	yes	—
355	C₄³	C₂²×C₅₃	1	1	yes	characteristic
356	C₅₃	#36601	1	1	yes	characteristic, direct product factor, Sylow
357	C₂²×Q₈	C₂³×C₅₃	1	1	yes	characteristic
358	C₂²×Q₈	C₂³×C₅₃	1	1	yes	characteristic
359	C₂²×D₄	C₂³×C₅₃	1	1	yes	characteristic
360	C₂³×C₄	C₂³×C₅₃	1	2	yes	—
361	#126	—	2	16	no	—
362	#125	—	2	8	no	—
363	#124	—	2	4	no	—
364	C₄×Q₈	—	2	8	no	—
365	C₄×Q₈	—	2	4	no	—
366	C₄×Q₈	—	2	4	no	—
367	C₄×D₄	—	2	4	no	—
368	C₂×C₄⋊C₄	C₂³×C₅₃	1	1	yes	characteristic
369	C₂×C₄⋊C₄	C₂³×C₅₃	1	1	yes	characteristic
370	C₂×C₄⋊C₄	C₂³×C₅₃	1	2	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	1	yes	characteristic
374	C₂×C₄⋊C₄	C₂³×C₅₃	1	2	yes	—
375	C₂×C₄⋊C₄	C₂³×C₅₃	1	2	yes	—
376	C₂×C₄⋊C₄	C₂³×C₅₃	1	2	yes	—
377	C₂×C₄⋊C₄	C₂³×C₅₃	1	4	yes	—
378	C₂×C₄⋊C₄	C₂³×C₅₃	1	4	yes	—
379	C₂×C₄⋊C₄	C₂³×C₅₃	1	4	yes	—
380	C₂×C₄⋊C₄	C₂³×C₅₃	1	4	yes	—
381	C₂×C₄⋊C₄	C₂³×C₅₃	1	4	yes	—
382	C₂×C₄⋊C₄	C₂³×C₅₃	1	4	yes	—
383	C₂×C₄⋊C₄	C₂³×C₅₃	1	2	yes	—
384	C₂×C₂²⋊C₄	C₂³×C₅₃	1	2	yes	—
385	C₂×C₂²⋊C₄	C₂³×C₅₃	1	4	yes	—
386	C₂×C₂²⋊C₄	C₂³×C₅₃	1	4	yes	—
387	C₂×C₂²⋊C₄	C₂³×C₅₃	1	4	yes	—
388	C₂×C₂²⋊C₄	C₂³×C₅₃	1	4	yes	—
389	C₂×C₂²⋊C₄	C₂³×C₅₃	1	4	yes	—
390	C₂×C₂²⋊C₄	C₂³×C₅₃	1	4	yes	—
391	C₂×C₄²	C₂³×C₅₃	1	2	yes	—
392	C₂×C₄²	C₂³×C₅₃	1	2	yes	—
393	C₂×C₄²	C₂³×C₅₃	1	2	yes	—
394	C₂×C₄²	C₂³×C₅₃	1	2	yes	—
395	C₂×C₄²	C₂³×C₅₃	1	2	yes	—
396	C₂×C₄²	C₂³×C₅₃	1	4	yes	—
397	C₂×C₄²	C₂³×C₅₃	1	2	yes	—
398	C₂×C₄²	C₂³×C₅₃	1	1	yes	characteristic
399	C₂×C₄²	C₂³×C₅₃	1	2	yes	—
400	C₂×C₄²	C₂³×C₅₃	1	2	yes	—
401	C₂×C₄²	C₂³×C₅₃	1	2	yes	—
402	C₂×C₄²	C₂³×C₅₃	1	4	yes	—
403	C₂×C₄²	C₂³×C₅₃	1	1	yes	characteristic
404	C₂×C₄²	C₂³×C₅₃	1	1	yes	characteristic
405	#95	C₂³×C₅₃	1	1	yes	characteristic
406	#95	C₂³×C₅₃	1	2	yes	—
407	#95	C₂³×C₅₃	1	4	yes	—
408	#95	C₂³×C₅₃	1	4	yes	—
409	#95	C₂³×C₅₃	1	2	yes	—
410	#95	C₂³×C₅₃	1	4	yes	—
411	#95	C₂³×C₅₃	1	1	yes	characteristic
412	#95	C₂³×C₅₃	1	2	yes	—
413	#95	C₂³×C₅₃	1	2	yes	—
414	#95	C₂³×C₅₃	1	4	yes	—
415	#95	C₂³×C₅₃	1	4	yes	—
416	#95	C₂³×C₅₃	1	4	yes	—
417	#95	C₂³×C₅₃	1	4	yes	—
418	#95	C₂³×C₅₃	1	2	yes	—
419	#95	C₂³×C₅₃	1	2	yes	—
420	#95	C₂³×C₅₃	1	4	yes	—
421	#95	C₂³×C₅₃	1	4	yes	—
422	#95	C₂³×C₅₃	1	4	yes	—
423	C₂⁴	C₂⁴×C₅₃	1	2	yes	—
424	C₂×Q₈	—	2	4	no	—
425	C₂×Q₈	—	2	2	no	—
426	C₂×Q₈	—	2	2	no	—
427	C₂×Q₈	—	2	2	no	—
428	C₂×Q₈	—	2	2	no	—
429	C₂×D₄	—	2	2	no	—
430	C₂×D₄	—	2	2	no	—
431	C₂×D₄	—	2	2	no	—
432	C₂²×C₄	—	2	4	no	—
433	C₂²×C₄	—	4	4	no	—
434	C₂²×C₄	C₂⁴×C₅₃	1	2	yes	—
435	C₂²×C₄	C₂⁴×C₅₃	1	2	yes	—
436	C₂²×C₄	C₂⁴×C₅₃	1	2	yes	—
437	C₂²×C₄	C₂⁴×C₅₃	1	2	yes	—
438	C₂²×C₄	C₂⁴×C₅₃	1	2	yes	—
439	C₂²×C₄	C₂⁴×C₅₃	1	4	yes	—
440	C₂²×C₄	C₂⁴×C₅₃	1	4	yes	—
441	C₂²×C₄	C₂⁴×C₅₃	1	1	yes	characteristic
442	C₂²×C₄	C₂⁴×C₅₃	1	2	yes	—
443	C₂²×C₄	C₂⁴×C₅₃	1	2	yes	—
444	C₂²×C₄	C₂⁴×C₅₃	1	2	yes	—
445	C₂²×C₄	C₂⁴×C₅₃	1	1	yes	characteristic
446	C₂²×C₄	C₂⁴×C₅₃	1	2	yes	—
447	C₂²×C₄	C₂⁴×C₅₃	1	1	yes	characteristic
448	C₄⋊C₄	—	2	2	no	—
449	C₄⋊C₄	—	2	4	no	—
450	C₄⋊C₄	—	2	4	no	—
451	C₄⋊C₄	—	2	4	no	—
452	C₄⋊C₄	—	2	2	no	—
453	C₄⋊C₄	—	2	2	no	—
454	C₄⋊C₄	—	2	4	no	—
455	C₄⋊C₄	—	2	8	no	—
456	C₄⋊C₄	—	2	8	no	—
457	C₄⋊C₄	—	2	4	no	—
458	C₄⋊C₄	—	2	4	no	—
459	C₄⋊C₄	—	2	8	no	—
460	C₄⋊C₄	—	2	8	no	—
461	C₄⋊C₄	—	4	4	no	—
462	C₄⋊C₄	—	4	4	no	—
463	C₄⋊C₄	#41×C₅₃	1	8	yes	—
464	C₂²⋊C₄	—	2	4	no	—
465	C₂²⋊C₄	—	2	8	no	—
466	C₂²⋊C₄	—	2	8	no	—
467	C₂²⋊C₄	—	2	8	no	—
468	C₂²⋊C₄	—	2	8	no	—
469	C₂²⋊C₄	—	4	4	no	—
470	C₂²⋊C₄	—	4	4	no	—
471	C₄²	—	2	4	no	—
472	C₄²	—	2	4	no	—
473	C₄²	—	2	4	no	—
474	C₄²	—	2	4	no	—
475	C₄²	—	2	4	no	—
476	C₄²	—	2	8	no	—
477	C₄²	—	2	2	no	—
478	C₄²	—	2	4	no	—
479	C₄²	—	4	2	no	—
480	C₄²	—	4	2	no	—
481	C₄²	—	4	2	no	—
482	C₄²	—	4	4	no	—
483	C₄²	#41×C₅₃	1	4	yes	—
484	C₄²	#41×C₅₃	1	4	yes	—
485	C₂³	—	2	4	no	—
486	C₂³	—	2	2	no	—
487	C₂³	—	2	2	no	—
488	C₂³	—	2	2	no	—
489	C₂³	—	2	4	no	—
490	C₂³	C₂⁵×C₅₃	1	1	yes	characteristic
491	Q₈	—	2	8	no	—
492	Q₈	—	4	4	no	—
493	D₄	—	4	4	no	—
494	C₂×C₄	—	2	2	no	—
495	C₂×C₄	—	2	2	no	—
496	C₂×C₄	—	2	2	no	—
497	C₂×C₄	—	2	2	no	—
498	C₂×C₄	—	2	2	no	—
499	C₂×C₄	—	2	2	no	—
500	C₂×C₄	—	2	2	no	—
501	C₂×C₄	—	2	2	no	—
502	C₂×C₄	—	2	2	no	—
503	C₂×C₄	—	2	2	no	—
504	C₂×C₄	—	2	4	no	—
505	C₂×C₄	—	2	4	no	—
506	C₂×C₄	—	2	2	no	—
507	C₂×C₄	—	2	2	no	—
508	C₂×C₄	—	2	2	no	—
509	C₂×C₄	—	2	2	no	—
510	C₂×C₄	—	2	2	no	—
511	C₂×C₄	—	2	4	no	—
512	C₂×C₄	—	2	2	no	—
513	C₂×C₄	—	2	4	no	—
514	C₂×C₄	—	2	4	no	—
515	C₂×C₄	—	2	2	no	—
516	C₂×C₄	—	2	2	no	—
517	C₂×C₄	—	2	2	no	—
518	C₂×C₄	—	2	2	no	—
519	C₂×C₄	—	2	4	no	—
520	C₂×C₄	—	2	2	no	—
521	C₂×C₄	—	2	4	no	—
522	C₂×C₄	—	2	4	no	—
523	C₂×C₄	—	8	4	no	—
524	C₂×C₄	C₂×#41×C₅₃	1	4	yes	—
525	C₂×C₄	C₂×#41×C₅₃	1	2	yes	—
526	C₂×C₄	C₂×#41×C₅₃	1	4	yes	—
527	C₂×C₄	C₂×#41×C₅₃	1	4	yes	—
528	C₂×C₄	C₂×#41×C₅₃	1	4	yes	—
529	C₂×C₄	C₂×#41×C₅₃	1	2	yes	—
530	C₂×C₄	C₂×#41×C₅₃	1	2	yes	—
531	C₂×C₄	C₂×#41×C₅₃	1	2	yes	—
532	C₂×C₄	2₋¹⁺⁴×C₅₃	1	2	yes	—
533	C₂²	—	4	2	no	—
534	C₂²	—	4	4	no	—
535	C₂²	—	4	2	no	—
536	C₂²	—	4	4	no	—
537	C₂²	—	4	2	no	—
538	C₂²	C₂²×#41×C₅₃	1	1	yes	characteristic
539	C₂²	C₂²×#41×C₅₃	1	1	yes	characteristic
540	C₂²	C₂²×#41×C₅₃	1	2	yes	—
541	C₂²	C₂×2₋¹⁺⁴×C₅₃	1	1	yes	characteristic
542	C₂²	C₅₃×#585	1	2	yes	—
543	C₄	—	2	4	no	—
544	C₄	—	2	2	no	—
545	C₄	—	2	4	no	—
546	C₄	—	2	4	no	—
547	C₄	—	2	4	no	—
548	C₄	—	2	2	no	—
549	C₄	—	4	2	no	—
550	C₄	—	4	4	no	—
551	C₄	—	4	4	no	—
552	C₄	—	4	2	no	—
553	C₄	—	4	2	no	—
554	C₄	—	4	2	no	—
555	C₄	—	4	2	no	—
556	C₄	C₅₃×#556	1	4	yes	—
557	C₂	—	8	2	no	—
558	C₂	C₅₃×#3444	1	1	yes	characteristic
559	C₂	C₂×C₅₃×#556	1	1	yes	characteristic
560	C₂	C₅₃×#3480	1	2	yes	—
561	C₂	C₅₃×#3510	1	1	yes	characteristic
562	C₂	C₅₃×#3515	1	2	yes	—

	subgroup	G/H
center, socle	C₂³×C₅₃	C₂⁵
commutator, Frattini, cosocle	C₂³	C₂⁵×C₅₃
prime radical	C₂²×D₄×C₅₃	C₂³
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³×C₅₃ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₅₃×#2859	C₂	semidirect
3. C₅₃×#2596	C₂	semidirect
4. C₅₃×#2581	C₂	semidirect
6. C₅₃×#2573	C₂	semidirect
8. C₅₃×#2511	C₂	semidirect
9. C₅₃×#2469	C₂	semidirect
10. C₅₃×#2373	C₂	semidirect
12. C₅₃×#2356	C₂	semidirect
13. C₄×#125×C₅₃	C₂	semidirect
15. C₄²×Q₈×C₅₃	C₂	semidirect
251. #36601	C₅₃	direct (N × H)
277. #2859	C₂×C₅₃	semidirect
278. #2596	C₂×C₅₃	semidirect
279. #2581	C₂×C₅₃	semidirect
281. #2573	C₂×C₅₃	semidirect
283. #2511	C₂×C₅₃	semidirect
284. #2469	C₂×C₅₃	semidirect
285. #2373	C₂×C₅₃	semidirect
287. #2356	C₂×C₅₃	semidirect
288. C₄×#125	C₂×C₅₃	semidirect
290. C₄²×Q₈	C₂×C₅₃	semidirect
356. C₅₃	#36601	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×#2861	2
C₅₃×#2859	2
C₅₃×#2596	2
C₅₃×#2581	2
C₅₃×#2580	2
C₅₃×#2573	2
C₅₃×#2512	2
C₅₃×#2511	2
C₅₃×#2469	2
C₅₃×#2373	2
C₅₃×#2357	2
C₅₃×#2356	2
C₄×#125×C₅₃	2
C₄×#124×C₅₃	2
C₄²×Q₈×C₅₃	2
#36601	53

Subgroups — C53×#36601 #27244113

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅₃×#36601 #27244113