Subgroups — C₅₃×#36590 #27244102

Order 13568.

1666 conjugacy classes of subgroups.

The subgroup lattice has 658 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×#2594	C₂	1	4	yes	maximal, complemented
4	C₅₃×#2582	C₂	1	1	yes	maximal, characteristic, complemented
5	C₅₃×#2538	C₂	1	1	yes	maximal, characteristic
6	C₅₃×#2525	C₂	1	4	yes	maximal, complemented
7	C₅₃×#2514	C₂	1	2	yes	maximal, complemented
8	C₅₃×#2513	C₂	1	2	yes	maximal
9	C₅₃×#2494	C₂	1	2	yes	maximal, complemented
10	C₅₃×#2421	C₂	1	2	yes	maximal, complemented
11	C₅₃×#2362	C₂	1	2	yes	maximal, complemented
12	C₅₃×#2355	C₂	1	1	yes	maximal, characteristic, complemented
13	C₅₃×#2354	C₂	1	1	yes	maximal, characteristic
14	C₅₃×#2347	C₂	1	2	yes	maximal, complemented
15	C₄×#125×C₅₃	C₂	1	2	yes	maximal, complemented
16	C₄×#124×C₅₃	C₂	1	1	yes	maximal, characteristic
17	C₄²×Q₈×C₅₃	C₂	1	1	yes	maximal, characteristic, complemented
18	C₂×#125×C₅₃	C₂²	1	2	yes	—
19	C₂×#124×C₅₃	C₂²	1	1	yes	characteristic
20	C₂×C₂²⋊Q₈×C₅₃	C₂²	1	2	yes	—
21	C₂×C₄×Q₈×C₅₃	C₂²	1	1	yes	characteristic
22	C₂×C₄×Q₈×C₅₃	C₂²	1	1	yes	characteristic
23	C₂×C₄×Q₈×C₅₃	C₂²	1	1	yes	characteristic
24	C₂×C₄×Q₈×C₅₃	C₂²	1	1	yes	characteristic
25	C₂×C₄×D₄×C₅₃	C₂²	1	1	yes	characteristic
26	C₂×#117×C₅₃	C₂²	1	2	yes	—
27	C₅₃×#401	C₂²	1	4	yes	—
28	C₅₃×#400	C₂²	1	4	yes	—
29	C₅₃×#398	C₂²	1	4	yes	—
30	C₅₃×#397	C₂²	1	4	yes	—
31	C₅₃×#397	C₂²	1	4	yes	—
32	C₅₃×#396	C₂²	1	4	yes	—
33	C₅₃×#391	C₂²	1	1	yes	characteristic
34	C₅₃×#391	C₂²	1	2	yes	—
35	C₅₃×#391	C₂²	1	2	yes	—
36	C₅₃×#391	C₂²	1	2	yes	—
37	C₅₃×#390	C₂²	1	1	yes	characteristic
38	C₅₃×#390	C₂²	1	2	yes	—
39	C₅₃×#390	C₂²	1	2	yes	—
40	C₅₃×#389	C₂²	1	2	yes	—
41	C₅₃×#389	C₂²	1	2	yes	—
42	C₅₃×#389	C₂²	1	2	yes	—
43	C₅₃×#389	C₂²	1	2	yes	—
44	C₅₃×#389	C₂²	1	2	yes	—
45	C₅₃×#389	C₂²	1	2	yes	—
46	C₅₃×#389	C₂²	1	2	yes	—
47	C₅₃×#388	C₂²	1	4	yes	—
48	C₅₃×#388	C₂²	1	4	yes	—
49	C₅₃×#388	C₂²	1	4	yes	—
50	C₅₃×#388	C₂²	1	2	yes	—
51	C₅₃×#388	C₂²	1	4	yes	—
52	C₅₃×#388	C₂²	1	4	yes	—
53	C₅₃×#388	C₂²	1	2	yes	—
54	C₅₃×#388	C₂²	1	4	yes	—
55	C₅₃×#388	C₂²	1	4	yes	—
56	C₅₃×#387	C₂²	1	4	yes	—
57	C₅₃×#387	C₂²	1	4	yes	—
58	C₅₃×#387	C₂²	1	4	yes	—
59	C₅₃×#387	C₂²	1	4	yes	—
60	C₅₃×#384	C₂²	1	1	yes	characteristic
61	C₅₃×#384	C₂²	1	1	yes	characteristic
62	C₅₃×#383	C₂²	1	4	yes	—
63	C₅₃×#382	C₂²	1	2	yes	—
64	C₅₃×#382	C₂²	1	2	yes	—
65	C₅₃×#382	C₂²	1	1	yes	characteristic
66	C₅₃×#380	C₂²	1	2	yes	—
67	C₅₃×#380	C₂²	1	2	yes	—
68	C₄×C₄⋊C₄×C₅₃	C₂²	1	1	yes	characteristic
69	C₄×C₄⋊C₄×C₅₃	C₂²	1	2	yes	—
70	C₄×C₄⋊C₄×C₅₃	C₂²	1	2	yes	—
71	C₄×C₄⋊C₄×C₅₃	C₂²	1	2	yes	—
72	C₄×C₄⋊C₄×C₅₃	C₂²	1	2	yes	—
73	C₄×C₄⋊C₄×C₅₃	C₂²	1	4	yes	—
74	C₄×C₄⋊C₄×C₅₃	C₂²	1	2	yes	—
75	C₄×C₄⋊C₄×C₅₃	C₂²	1	1	yes	characteristic
76	C₄×C₄⋊C₄×C₅₃	C₂²	1	1	yes	characteristic
77	C₄×C₄⋊C₄×C₅₃	C₂²	1	4	yes	—
78	C₄×C₄⋊C₄×C₅₃	C₂²	1	2	yes	—
79	C₄×C₂²⋊C₄×C₅₃	C₂²	1	2	yes	—
80	C₄×C₂²⋊C₄×C₅₃	C₂²	1	2	yes	—
81	C₄³×C₅₃	C₂²	1	1	yes	characteristic
82	C₄³×C₅₃	C₂²	1	2	yes	—
83	C₂²×Q₈×C₅₃	C₂³	1	1	yes	characteristic
84	C₂²×Q₈×C₅₃	C₂³	1	1	yes	characteristic
85	C₂²×D₄×C₅₃	C₂³	1	1	yes	characteristic
86	C₂³×C₄×C₅₃	C₂³	1	2	yes	—
87	#125×C₅₃	—	2	8	no	—
88	#124×C₅₃	—	2	4	no	—
89	C₂²⋊Q₈×C₅₃	—	2	8	no	—
90	C₄×Q₈×C₅₃	—	2	4	no	—
91	C₄×Q₈×C₅₃	—	2	4	no	—
92	C₄×Q₈×C₅₃	—	2	4	no	—
93	C₄×Q₈×C₅₃	—	2	4	no	—
94	C₄×D₄×C₅₃	—	2	4	no	—
95	#117×C₅₃	—	2	8	no	—
96	C₂×C₄⋊C₄×C₅₃	C₂³	1	2	yes	—
97	C₂×C₄⋊C₄×C₅₃	C₂³	1	1	yes	characteristic
98	C₂×C₄⋊C₄×C₅₃	C₂³	1	1	yes	characteristic
99	C₂×C₄⋊C₄×C₅₃	C₂³	1	1	yes	characteristic
100	C₂×C₄⋊C₄×C₅₃	C₂³	1	1	yes	characteristic
101	C₂×C₄⋊C₄×C₅₃	C₂³	1	2	yes	—
102	C₂×C₄⋊C₄×C₅₃	C₂³	1	2	yes	—
103	C₂×C₄⋊C₄×C₅₃	C₂³	1	2	yes	—
104	C₂×C₄⋊C₄×C₅₃	C₂³	1	2	yes	—
105	C₂×C₄⋊C₄×C₅₃	C₂³	1	2	yes	—
106	C₂×C₄⋊C₄×C₅₃	C₂³	1	4	yes	—
107	C₂×C₄⋊C₄×C₅₃	C₂³	1	2	yes	—
108	C₂×C₄⋊C₄×C₅₃	C₂³	1	2	yes	—
109	C₂×C₄⋊C₄×C₅₃	C₂³	1	4	yes	—
110	C₂×C₄⋊C₄×C₅₃	C₂³	1	4	yes	—
111	C₂×C₄⋊C₄×C₅₃	C₂³	1	4	yes	—
112	C₂×C₄⋊C₄×C₅₃	C₂³	1	2	yes	—
113	C₂×C₄⋊C₄×C₅₃	C₂³	1	2	yes	—
114	C₂×C₄⋊C₄×C₅₃	C₂³	1	1	yes	characteristic
115	C₂×C₂²⋊C₄×C₅₃	C₂³	1	4	yes	—
116	C₂×C₂²⋊C₄×C₅₃	C₂³	1	2	yes	—
117	C₂×C₂²⋊C₄×C₅₃	C₂³	1	4	yes	—
118	C₂×C₂²⋊C₄×C₅₃	C₂³	1	2	yes	—
119	C₂×C₂²⋊C₄×C₅₃	C₂³	1	2	yes	—
120	C₂×C₂²⋊C₄×C₅₃	C₂³	1	2	yes	—
121	C₂×C₂²⋊C₄×C₅₃	C₂³	1	2	yes	—
122	C₂×C₂²⋊C₄×C₅₃	C₂³	1	4	yes	—
123	C₂×C₂²⋊C₄×C₅₃	C₂³	1	4	yes	—
124	C₂×C₄²×C₅₃	C₂³	1	4	yes	—
125	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
126	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
127	C₂×C₄²×C₅₃	C₂³	1	2	yes	—
128	C₂×C₄²×C₅₃	C₂³	1	2	yes	—
129	C₂×C₄²×C₅₃	C₂³	1	2	yes	—
130	C₂×C₄²×C₅₃	C₂³	1	2	yes	—
131	C₂×C₄²×C₅₃	C₂³	1	2	yes	—
132	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
133	C₂×C₄²×C₅₃	C₂³	1	2	yes	—
134	C₂×C₄²×C₅₃	C₂³	1	2	yes	—
135	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
136	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
137	C₂×C₄²×C₅₃	C₂³	1	4	yes	—
138	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
139	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
140	#95×C₅₃	C₂³	1	2	yes	—
141	#95×C₅₃	C₂³	1	2	yes	—
142	#95×C₅₃	C₂³	1	2	yes	—
143	#95×C₅₃	C₂³	1	2	yes	—
144	#95×C₅₃	C₂³	1	2	yes	—
145	#95×C₅₃	C₂³	1	2	yes	—
146	#95×C₅₃	C₂³	1	2	yes	—
147	#95×C₅₃	C₂³	1	4	yes	—
148	#95×C₅₃	C₂³	1	4	yes	—
149	#95×C₅₃	C₂³	1	4	yes	—
150	#95×C₅₃	C₂³	1	4	yes	—
151	#95×C₅₃	C₂³	1	4	yes	—
152	#95×C₅₃	C₂³	1	4	yes	—
153	#95×C₅₃	C₂³	1	4	yes	—
154	#95×C₅₃	C₂³	1	4	yes	—
155	#95×C₅₃	C₂³	1	4	yes	—
156	#95×C₅₃	C₂³	1	4	yes	—
157	C₂⁴×C₅₃	C₂⁴	1	2	yes	—
158	C₂×Q₈×C₅₃	—	2	2	no	—
159	C₂×Q₈×C₅₃	—	2	2	no	—
160	C₂×Q₈×C₅₃	—	2	2	no	—
161	C₂×Q₈×C₅₃	—	2	2	no	—
162	C₂×Q₈×C₅₃	—	2	2	no	—
163	C₂×Q₈×C₅₃	#41	1	4	yes	—
164	C₂×D₄×C₅₃	—	2	2	no	—
165	C₂×D₄×C₅₃	—	2	2	no	—
166	C₂×D₄×C₅₃	—	2	2	no	—
167	C₂²×C₄×C₅₃	—	2	4	no	—
168	C₂²×C₄×C₅₃	—	2	4	no	—
169	C₂²×C₄×C₅₃	—	2	4	no	—
170	C₂²×C₄×C₅₃	C₂⁴	1	2	yes	—
171	C₂²×C₄×C₅₃	C₂⁴	1	1	yes	characteristic
172	C₂²×C₄×C₅₃	C₂⁴	1	1	yes	characteristic
173	C₂²×C₄×C₅₃	C₂⁴	1	2	yes	—
174	C₂²×C₄×C₅₃	C₂⁴	1	2	yes	—
175	C₂²×C₄×C₅₃	C₂⁴	1	2	yes	—
176	C₂²×C₄×C₅₃	C₂⁴	1	2	yes	—
177	C₂²×C₄×C₅₃	C₂⁴	1	4	yes	—
178	C₂²×C₄×C₅₃	C₂⁴	1	4	yes	—
179	C₂²×C₄×C₅₃	C₂⁴	1	2	yes	—
180	C₂²×C₄×C₅₃	C₂⁴	1	2	yes	—
181	C₂²×C₄×C₅₃	C₂⁴	1	1	yes	characteristic
182	C₂²×C₄×C₅₃	C₂⁴	1	1	yes	characteristic
183	C₂²×C₄×C₅₃	C₂⁴	1	1	yes	characteristic
184	C₂²×C₄×C₅₃	C₂⁴	1	1	yes	characteristic
185	C₂²×C₄×C₅₃	C₂⁴	1	1	yes	characteristic
186	C₄⋊C₄×C₅₃	—	2	4	no	—
187	C₄⋊C₄×C₅₃	—	2	2	no	—
188	C₄⋊C₄×C₅₃	—	2	4	no	—
189	C₄⋊C₄×C₅₃	—	2	4	no	—
190	C₄⋊C₄×C₅₃	—	2	4	no	—
191	C₄⋊C₄×C₅₃	—	2	4	no	—
192	C₄⋊C₄×C₅₃	—	2	2	no	—
193	C₄⋊C₄×C₅₃	—	2	2	no	—
194	C₄⋊C₄×C₅₃	—	2	2	no	—
195	C₄⋊C₄×C₅₃	—	2	8	no	—
196	C₄⋊C₄×C₅₃	—	2	4	no	—
197	C₄⋊C₄×C₅₃	—	2	8	no	—
198	C₄⋊C₄×C₅₃	—	2	8	no	—
199	C₄⋊C₄×C₅₃	—	2	4	no	—
200	C₄⋊C₄×C₅₃	—	2	4	no	—
201	C₄⋊C₄×C₅₃	—	4	2	no	—
202	C₄⋊C₄×C₅₃	—	4	2	no	—
203	C₄⋊C₄×C₅₃	—	4	4	no	—
204	C₄⋊C₄×C₅₃	#41	1	4	yes	—
205	C₂²⋊C₄×C₅₃	—	2	8	no	—
206	C₂²⋊C₄×C₅₃	—	2	4	no	—
207	C₂²⋊C₄×C₅₃	—	2	8	no	—
208	C₂²⋊C₄×C₅₃	—	2	4	no	—
209	C₂²⋊C₄×C₅₃	—	2	4	no	—
210	C₂²⋊C₄×C₅₃	—	4	2	no	—
211	C₂²⋊C₄×C₅₃	—	4	4	no	—
212	C₂²⋊C₄×C₅₃	—	4	2	no	—
213	C₂²⋊C₄×C₅₃	—	4	4	no	—
214	C₄²×C₅₃	—	2	8	no	—
215	C₄²×C₅₃	—	2	4	no	—
216	C₄²×C₅₃	—	2	4	no	—
217	C₄²×C₅₃	—	2	4	no	—
218	C₄²×C₅₃	—	2	2	no	—
219	C₄²×C₅₃	—	2	2	no	—
220	C₄²×C₅₃	—	2	4	no	—
221	C₄²×C₅₃	—	2	4	no	—
222	C₄²×C₅₃	—	2	2	no	—
223	C₄²×C₅₃	—	4	2	no	—
224	C₄²×C₅₃	—	4	1	no	—
225	C₄²×C₅₃	—	4	1	no	—
226	C₄²×C₅₃	—	4	4	no	—
227	C₄²×C₅₃	—	4	2	no	—
228	C₄²×C₅₃	#41	1	4	yes	—
229	C₄²×C₅₃	#41	1	4	yes	—
230	C₂³×C₅₃	—	2	2	no	—
231	C₂³×C₅₃	—	2	2	no	—
232	C₂³×C₅₃	—	2	2	no	—
233	C₂³×C₅₃	—	2	2	no	—
234	C₂³×C₅₃	—	2	2	no	—
235	C₂³×C₅₃	—	2	2	no	—
236	C₂³×C₅₃	—	2	2	no	—
237	C₂³×C₅₃	C₂⁵	1	1	yes	characteristic
238	Q₈×C₅₃	—	2	8	no	—
239	Q₈×C₅₃	—	4	4	no	—
240	D₄×C₅₃	—	4	4	no	—
241	C₂×C₄×C₅₃	—	2	2	no	—
242	C₂×C₄×C₅₃	—	2	2	no	—
243	C₂×C₄×C₅₃	—	2	1	no	—
244	C₂×C₄×C₅₃	—	2	1	no	—
245	C₂×C₄×C₅₃	—	2	2	no	—
246	C₂×C₄×C₅₃	—	2	1	no	—
247	C₂×C₄×C₅₃	—	2	1	no	—
248	C₂×C₄×C₅₃	—	2	2	no	—
249	C₂×C₄×C₅₃	—	2	1	no	—
250	C₂×C₄×C₅₃	—	2	1	no	—
251	C₂×C₄×C₅₃	—	2	2	no	—
252	C₂×C₄×C₅₃	—	2	2	no	—
253	C₂×C₄×C₅₃	—	2	2	no	—
254	C₂×C₄×C₅₃	—	2	2	no	—
255	C₂×C₄×C₅₃	—	2	2	no	—
256	C₂×C₄×C₅₃	—	2	4	no	—
257	C₂×C₄×C₅₃	—	2	2	no	—
258	C₂×C₄×C₅₃	—	2	2	no	—
259	C₂×C₄×C₅₃	—	2	1	no	—
260	C₂×C₄×C₅₃	—	2	1	no	—
261	C₂×C₄×C₅₃	—	2	2	no	—
262	C₂×C₄×C₅₃	—	2	4	no	—
263	C₂×C₄×C₅₃	—	2	4	no	—
264	C₂×C₄×C₅₃	—	2	4	no	—
265	C₂×C₄×C₅₃	—	2	2	no	—
266	C₂×C₄×C₅₃	—	2	1	no	—
267	C₂×C₄×C₅₃	—	2	1	no	—
268	C₂×C₄×C₅₃	—	2	2	no	—
269	C₂×C₄×C₅₃	—	2	2	no	—
270	C₂×C₄×C₅₃	—	2	2	no	—
271	C₂×C₄×C₅₃	—	2	1	no	—
272	C₂×C₄×C₅₃	—	2	1	no	—
273	C₂×C₄×C₅₃	—	2	2	no	—
274	C₂×C₄×C₅₃	—	2	4	no	—
275	C₂×C₄×C₅₃	—	2	2	no	—
276	C₂×C₄×C₅₃	—	2	4	no	—
277	C₂×C₄×C₅₃	—	2	2	no	—
278	C₂×C₄×C₅₃	—	4	8	no	—
279	C₂×C₄×C₅₃	C₂×#41	1	4	yes	—
280	C₂×C₄×C₅₃	C₂×#41	1	4	yes	—
281	C₂×C₄×C₅₃	C₂×#41	1	2	yes	—
282	C₂×C₄×C₅₃	C₂×#41	1	2	yes	—
283	C₂×C₄×C₅₃	C₂×#41	1	2	yes	—
284	C₂×C₄×C₅₃	C₂×#41	1	2	yes	—
285	C₂×C₄×C₅₃	C₂×#41	1	2	yes	—
286	C₂×C₄×C₅₃	C₂×#41	1	2	yes	—
287	C₂×C₄×C₅₃	C₂×#41	1	2	yes	—
288	C₂×C₄×C₅₃	C₂×#41	1	2	yes	—
289	C₂×C₄×C₅₃	2₋¹⁺⁴	1	2	yes	—
290	#36590	C₅₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
291	C₂²×C₅₃	—	4	2	no	—
292	C₂²×C₅₃	—	4	2	no	—
293	C₂²×C₅₃	—	4	2	no	—
294	C₂²×C₅₃	—	4	2	no	—
295	C₂²×C₅₃	—	4	2	no	—
296	C₂²×C₅₃	—	4	2	no	—
297	C₂²×C₅₃	—	4	2	no	—
298	C₂²×C₅₃	C₂²×#41	1	1	yes	characteristic
299	C₂²×C₅₃	C₂²×#41	1	1	yes	characteristic
300	C₂²×C₅₃	C₂²×#41	1	1	yes	characteristic
301	C₂²×C₅₃	C₂²×#41	1	1	yes	characteristic
302	C₂²×C₅₃	C₂×2₋¹⁺⁴	1	1	yes	characteristic
303	C₂²×C₅₃	#585	1	1	yes	characteristic
304	C₂²×C₅₃	#585	1	1	yes	characteristic
305	C₄×C₅₃	—	2	4	no	—
306	C₄×C₅₃	—	2	2	no	—
307	C₄×C₅₃	—	2	2	no	—
308	C₄×C₅₃	—	2	2	no	—
309	C₄×C₅₃	—	2	4	no	—
310	C₄×C₅₃	—	4	2	no	—
311	C₄×C₅₃	—	4	2	no	—
312	C₄×C₅₃	—	4	4	no	—
313	C₄×C₅₃	—	4	4	no	—
314	C₄×C₅₃	—	4	2	no	—
315	C₄×C₅₃	—	4	2	no	—
316	C₄×C₅₃	—	4	1	no	—
317	C₄×C₅₃	—	4	1	no	—
318	C₄×C₅₃	—	4	2	no	—
319	C₄×C₅₃	#529	1	4	yes	—
320	C₄×C₅₃	#556	1	4	yes	—
321	#2861	C₂×C₅₃	1	2	yes	—
322	#2859	C₂×C₅₃	1	1	yes	characteristic, complemented
323	#2594	C₂×C₅₃	1	4	yes	complemented
324	#2582	C₂×C₅₃	1	1	yes	characteristic, complemented
325	#2538	C₂×C₅₃	1	1	yes	characteristic
326	#2525	C₂×C₅₃	1	4	yes	complemented
327	#2514	C₂×C₅₃	1	2	yes	complemented
328	#2513	C₂×C₅₃	1	2	yes	—
329	#2494	C₂×C₅₃	1	2	yes	complemented
330	#2421	C₂×C₅₃	1	2	yes	complemented
331	#2362	C₂×C₅₃	1	2	yes	complemented
332	#2355	C₂×C₅₃	1	1	yes	characteristic, complemented
333	#2354	C₂×C₅₃	1	1	yes	characteristic
334	#2347	C₂×C₅₃	1	2	yes	complemented
335	C₄×#125	C₂×C₅₃	1	2	yes	complemented
336	C₄×#124	C₂×C₅₃	1	1	yes	characteristic
337	C₄²×Q₈	C₂×C₅₃	1	1	yes	characteristic, complemented
338	C₂×C₅₃	—	8	2	no	—
339	C₂×C₅₃	C₂×#529	1	1	yes	characteristic
340	C₂×C₅₃	C₂×#556	1	1	yes	characteristic
341	C₂×C₅₃	#3480	1	1	yes	characteristic
342	C₂×C₅₃	#3480	1	1	yes	characteristic
343	C₂×C₅₃	#3512	1	1	yes	characteristic
344	C₂×C₅₃	#3515	1	1	yes	characteristic
345	C₂×C₅₃	#3518	1	1	yes	characteristic
346	C₂×#125	C₂²×C₅₃	1	2	yes	—
347	C₂×#124	C₂²×C₅₃	1	1	yes	characteristic
348	C₂×C₂²⋊Q₈	C₂²×C₅₃	1	2	yes	—
349	C₂×C₄×Q₈	C₂²×C₅₃	1	1	yes	characteristic
350	C₂×C₄×Q₈	C₂²×C₅₃	1	1	yes	characteristic
351	C₂×C₄×Q₈	C₂²×C₅₃	1	1	yes	characteristic
352	C₂×C₄×Q₈	C₂²×C₅₃	1	1	yes	characteristic
353	C₂×C₄×D₄	C₂²×C₅₃	1	1	yes	characteristic
354	C₂×#117	C₂²×C₅₃	1	2	yes	—
355	#401	C₂²×C₅₃	1	4	yes	—
356	#400	C₂²×C₅₃	1	4	yes	—
357	#398	C₂²×C₅₃	1	4	yes	—
358	#397	C₂²×C₅₃	1	4	yes	—
359	#397	C₂²×C₅₃	1	4	yes	—
360	#396	C₂²×C₅₃	1	4	yes	—
361	#391	C₂²×C₅₃	1	2	yes	—
362	#391	C₂²×C₅₃	1	2	yes	—
363	#391	C₂²×C₅₃	1	2	yes	—
364	#391	C₂²×C₅₃	1	1	yes	characteristic
365	#390	C₂²×C₅₃	1	2	yes	—
366	#390	C₂²×C₅₃	1	2	yes	—
367	#390	C₂²×C₅₃	1	1	yes	characteristic
368	#389	C₂²×C₅₃	1	2	yes	—
369	#389	C₂²×C₅₃	1	2	yes	—
370	#389	C₂²×C₅₃	1	2	yes	—
371	#389	C₂²×C₅₃	1	2	yes	—
372	#389	C₂²×C₅₃	1	2	yes	—
373	#389	C₂²×C₅₃	1	2	yes	—
374	#389	C₂²×C₅₃	1	2	yes	—
375	#388	C₂²×C₅₃	1	2	yes	—
376	#388	C₂²×C₅₃	1	4	yes	—
377	#388	C₂²×C₅₃	1	4	yes	—
378	#388	C₂²×C₅₃	1	4	yes	—
379	#388	C₂²×C₅₃	1	4	yes	—
380	#388	C₂²×C₅₃	1	4	yes	—
381	#388	C₂²×C₅₃	1	4	yes	—
382	#388	C₂²×C₅₃	1	4	yes	—
383	#388	C₂²×C₅₃	1	2	yes	—
384	#387	C₂²×C₅₃	1	4	yes	—
385	#387	C₂²×C₅₃	1	4	yes	—
386	#387	C₂²×C₅₃	1	4	yes	—
387	#387	C₂²×C₅₃	1	4	yes	—
388	#384	C₂²×C₅₃	1	1	yes	characteristic
389	#384	C₂²×C₅₃	1	1	yes	characteristic
390	#383	C₂²×C₅₃	1	4	yes	—
391	#382	C₂²×C₅₃	1	1	yes	characteristic
392	#382	C₂²×C₅₃	1	2	yes	—
393	#382	C₂²×C₅₃	1	2	yes	—
394	#380	C₂²×C₅₃	1	2	yes	—
395	#380	C₂²×C₅₃	1	2	yes	—
396	C₄×C₄⋊C₄	C₂²×C₅₃	1	4	yes	—
397	C₄×C₄⋊C₄	C₂²×C₅₃	1	2	yes	—
398	C₄×C₄⋊C₄	C₂²×C₅₃	1	2	yes	—
399	C₄×C₄⋊C₄	C₂²×C₅₃	1	4	yes	—
400	C₄×C₄⋊C₄	C₂²×C₅₃	1	2	yes	—
401	C₄×C₄⋊C₄	C₂²×C₅₃	1	1	yes	characteristic
402	C₄×C₄⋊C₄	C₂²×C₅₃	1	2	yes	—
403	C₄×C₄⋊C₄	C₂²×C₅₃	1	1	yes	characteristic
404	C₄×C₄⋊C₄	C₂²×C₅₃	1	1	yes	characteristic
405	C₄×C₄⋊C₄	C₂²×C₅₃	1	2	yes	—
406	C₄×C₄⋊C₄	C₂²×C₅₃	1	2	yes	—
407	C₄×C₂²⋊C₄	C₂²×C₅₃	1	2	yes	—
408	C₄×C₂²⋊C₄	C₂²×C₅₃	1	2	yes	—
409	C₄³	C₂²×C₅₃	1	2	yes	—
410	C₄³	C₂²×C₅₃	1	1	yes	characteristic
411	C₅₃	#36590	1	1	yes	characteristic, direct product factor, Sylow
412	C₂²×Q₈	C₂³×C₅₃	1	1	yes	characteristic
413	C₂²×Q₈	C₂³×C₅₃	1	1	yes	characteristic
414	C₂²×D₄	C₂³×C₅₃	1	1	yes	characteristic
415	C₂³×C₄	C₂³×C₅₃	1	2	yes	—
416	#125	—	2	8	no	—
417	#124	—	2	4	no	—
418	C₂²⋊Q₈	—	2	8	no	—
419	C₄×Q₈	—	2	4	no	—
420	C₄×Q₈	—	2	4	no	—
421	C₄×Q₈	—	2	4	no	—
422	C₄×Q₈	—	2	4	no	—
423	C₄×D₄	—	2	4	no	—
424	#117	—	2	8	no	—
425	C₂×C₄⋊C₄	C₂³×C₅₃	1	2	yes	—
426	C₂×C₄⋊C₄	C₂³×C₅₃	1	1	yes	characteristic
427	C₂×C₄⋊C₄	C₂³×C₅₃	1	1	yes	characteristic
428	C₂×C₄⋊C₄	C₂³×C₅₃	1	1	yes	characteristic
429	C₂×C₄⋊C₄	C₂³×C₅₃	1	1	yes	characteristic
430	C₂×C₄⋊C₄	C₂³×C₅₃	1	2	yes	—
431	C₂×C₄⋊C₄	C₂³×C₅₃	1	2	yes	—
432	C₂×C₄⋊C₄	C₂³×C₅₃	1	2	yes	—
433	C₂×C₄⋊C₄	C₂³×C₅₃	1	2	yes	—
434	C₂×C₄⋊C₄	C₂³×C₅₃	1	2	yes	—
435	C₂×C₄⋊C₄	C₂³×C₅₃	1	4	yes	—
436	C₂×C₄⋊C₄	C₂³×C₅₃	1	2	yes	—
437	C₂×C₄⋊C₄	C₂³×C₅₃	1	2	yes	—
438	C₂×C₄⋊C₄	C₂³×C₅₃	1	4	yes	—
439	C₂×C₄⋊C₄	C₂³×C₅₃	1	4	yes	—
440	C₂×C₄⋊C₄	C₂³×C₅₃	1	4	yes	—
441	C₂×C₄⋊C₄	C₂³×C₅₃	1	2	yes	—
442	C₂×C₄⋊C₄	C₂³×C₅₃	1	2	yes	—
443	C₂×C₄⋊C₄	C₂³×C₅₃	1	1	yes	characteristic
444	C₂×C₂²⋊C₄	C₂³×C₅₃	1	4	yes	—
445	C₂×C₂²⋊C₄	C₂³×C₅₃	1	2	yes	—
446	C₂×C₂²⋊C₄	C₂³×C₅₃	1	4	yes	—
447	C₂×C₂²⋊C₄	C₂³×C₅₃	1	2	yes	—
448	C₂×C₂²⋊C₄	C₂³×C₅₃	1	2	yes	—
449	C₂×C₂²⋊C₄	C₂³×C₅₃	1	2	yes	—
450	C₂×C₂²⋊C₄	C₂³×C₅₃	1	2	yes	—
451	C₂×C₂²⋊C₄	C₂³×C₅₃	1	4	yes	—
452	C₂×C₂²⋊C₄	C₂³×C₅₃	1	4	yes	—
453	C₂×C₄²	C₂³×C₅₃	1	4	yes	—
454	C₂×C₄²	C₂³×C₅₃	1	1	yes	characteristic
455	C₂×C₄²	C₂³×C₅₃	1	1	yes	characteristic
456	C₂×C₄²	C₂³×C₅₃	1	2	yes	—
457	C₂×C₄²	C₂³×C₅₃	1	2	yes	—
458	C₂×C₄²	C₂³×C₅₃	1	2	yes	—
459	C₂×C₄²	C₂³×C₅₃	1	2	yes	—
460	C₂×C₄²	C₂³×C₅₃	1	2	yes	—
461	C₂×C₄²	C₂³×C₅₃	1	1	yes	characteristic
462	C₂×C₄²	C₂³×C₅₃	1	2	yes	—
463	C₂×C₄²	C₂³×C₅₃	1	2	yes	—
464	C₂×C₄²	C₂³×C₅₃	1	1	yes	characteristic
465	C₂×C₄²	C₂³×C₅₃	1	1	yes	characteristic
466	C₂×C₄²	C₂³×C₅₃	1	4	yes	—
467	C₂×C₄²	C₂³×C₅₃	1	1	yes	characteristic
468	C₂×C₄²	C₂³×C₅₃	1	1	yes	characteristic
469	#95	C₂³×C₅₃	1	2	yes	—
470	#95	C₂³×C₅₃	1	2	yes	—
471	#95	C₂³×C₅₃	1	2	yes	—
472	#95	C₂³×C₅₃	1	2	yes	—
473	#95	C₂³×C₅₃	1	2	yes	—
474	#95	C₂³×C₅₃	1	2	yes	—
475	#95	C₂³×C₅₃	1	2	yes	—
476	#95	C₂³×C₅₃	1	4	yes	—
477	#95	C₂³×C₅₃	1	4	yes	—
478	#95	C₂³×C₅₃	1	4	yes	—
479	#95	C₂³×C₅₃	1	4	yes	—
480	#95	C₂³×C₅₃	1	4	yes	—
481	#95	C₂³×C₅₃	1	4	yes	—
482	#95	C₂³×C₅₃	1	4	yes	—
483	#95	C₂³×C₅₃	1	4	yes	—
484	#95	C₂³×C₅₃	1	4	yes	—
485	#95	C₂³×C₅₃	1	4	yes	—
486	C₂⁴	C₂⁴×C₅₃	1	2	yes	—
487	C₂×Q₈	—	2	2	no	—
488	C₂×Q₈	—	2	2	no	—
489	C₂×Q₈	—	2	2	no	—
490	C₂×Q₈	—	2	2	no	—
491	C₂×Q₈	—	2	2	no	—
492	C₂×Q₈	#41×C₅₃	1	4	yes	—
493	C₂×D₄	—	2	2	no	—
494	C₂×D₄	—	2	2	no	—
495	C₂×D₄	—	2	2	no	—
496	C₂²×C₄	—	2	4	no	—
497	C₂²×C₄	—	2	4	no	—
498	C₂²×C₄	—	2	4	no	—
499	C₂²×C₄	C₂⁴×C₅₃	1	2	yes	—
500	C₂²×C₄	C₂⁴×C₅₃	1	1	yes	characteristic
501	C₂²×C₄	C₂⁴×C₅₃	1	1	yes	characteristic
502	C₂²×C₄	C₂⁴×C₅₃	1	2	yes	—
503	C₂²×C₄	C₂⁴×C₅₃	1	2	yes	—
504	C₂²×C₄	C₂⁴×C₅₃	1	2	yes	—
505	C₂²×C₄	C₂⁴×C₅₃	1	2	yes	—
506	C₂²×C₄	C₂⁴×C₅₃	1	4	yes	—
507	C₂²×C₄	C₂⁴×C₅₃	1	4	yes	—
508	C₂²×C₄	C₂⁴×C₅₃	1	2	yes	—
509	C₂²×C₄	C₂⁴×C₅₃	1	2	yes	—
510	C₂²×C₄	C₂⁴×C₅₃	1	1	yes	characteristic
511	C₂²×C₄	C₂⁴×C₅₃	1	1	yes	characteristic
512	C₂²×C₄	C₂⁴×C₅₃	1	1	yes	characteristic
513	C₂²×C₄	C₂⁴×C₅₃	1	1	yes	characteristic
514	C₂²×C₄	C₂⁴×C₅₃	1	1	yes	characteristic
515	C₄⋊C₄	—	2	4	no	—
516	C₄⋊C₄	—	2	2	no	—
517	C₄⋊C₄	—	2	4	no	—
518	C₄⋊C₄	—	2	4	no	—
519	C₄⋊C₄	—	2	4	no	—
520	C₄⋊C₄	—	2	2	no	—
521	C₄⋊C₄	—	2	2	no	—
522	C₄⋊C₄	—	2	2	no	—
523	C₄⋊C₄	—	2	4	no	—
524	C₄⋊C₄	—	2	4	no	—
525	C₄⋊C₄	—	2	8	no	—
526	C₄⋊C₄	—	2	8	no	—
527	C₄⋊C₄	—	2	8	no	—
528	C₄⋊C₄	—	2	4	no	—
529	C₄⋊C₄	—	2	4	no	—
530	C₄⋊C₄	—	4	2	no	—
531	C₄⋊C₄	—	4	2	no	—
532	C₄⋊C₄	—	4	4	no	—
533	C₄⋊C₄	#41×C₅₃	1	4	yes	—
534	C₂²⋊C₄	—	2	8	no	—
535	C₂²⋊C₄	—	2	4	no	—
536	C₂²⋊C₄	—	2	8	no	—
537	C₂²⋊C₄	—	2	4	no	—
538	C₂²⋊C₄	—	2	4	no	—
539	C₂²⋊C₄	—	4	2	no	—
540	C₂²⋊C₄	—	4	4	no	—
541	C₂²⋊C₄	—	4	2	no	—
542	C₂²⋊C₄	—	4	4	no	—
543	C₄²	—	2	8	no	—
544	C₄²	—	2	4	no	—
545	C₄²	—	2	4	no	—
546	C₄²	—	2	2	no	—
547	C₄²	—	2	2	no	—
548	C₄²	—	2	4	no	—
549	C₄²	—	2	4	no	—
550	C₄²	—	2	4	no	—
551	C₄²	—	2	2	no	—
552	C₄²	—	4	2	no	—
553	C₄²	—	4	1	no	—
554	C₄²	—	4	1	no	—
555	C₄²	—	4	4	no	—
556	C₄²	—	4	2	no	—
557	C₄²	#41×C₅₃	1	4	yes	—
558	C₄²	#41×C₅₃	1	4	yes	—
559	C₂³	—	2	2	no	—
560	C₂³	—	2	2	no	—
561	C₂³	—	2	2	no	—
562	C₂³	—	2	2	no	—
563	C₂³	—	2	2	no	—
564	C₂³	—	2	2	no	—
565	C₂³	—	2	2	no	—
566	C₂³	C₂⁵×C₅₃	1	1	yes	characteristic
567	Q₈	—	2	8	no	—
568	Q₈	—	4	4	no	—
569	D₄	—	4	4	no	—
570	C₂×C₄	—	2	1	no	—
571	C₂×C₄	—	2	1	no	—
572	C₂×C₄	—	2	2	no	—
573	C₂×C₄	—	2	2	no	—
574	C₂×C₄	—	2	2	no	—
575	C₂×C₄	—	2	1	no	—
576	C₂×C₄	—	2	1	no	—
577	C₂×C₄	—	2	2	no	—
578	C₂×C₄	—	2	1	no	—
579	C₂×C₄	—	2	1	no	—
580	C₂×C₄	—	2	2	no	—
581	C₂×C₄	—	2	4	no	—
582	C₂×C₄	—	2	2	no	—
583	C₂×C₄	—	2	2	no	—
584	C₂×C₄	—	2	1	no	—
585	C₂×C₄	—	2	2	no	—
586	C₂×C₄	—	2	1	no	—
587	C₂×C₄	—	2	2	no	—
588	C₂×C₄	—	2	2	no	—
589	C₂×C₄	—	2	4	no	—
590	C₂×C₄	—	2	2	no	—
591	C₂×C₄	—	2	4	no	—
592	C₂×C₄	—	2	4	no	—
593	C₂×C₄	—	2	2	no	—
594	C₂×C₄	—	2	1	no	—
595	C₂×C₄	—	2	2	no	—
596	C₂×C₄	—	2	1	no	—
597	C₂×C₄	—	2	2	no	—
598	C₂×C₄	—	2	2	no	—
599	C₂×C₄	—	2	2	no	—
600	C₂×C₄	—	2	1	no	—
601	C₂×C₄	—	2	1	no	—
602	C₂×C₄	—	2	2	no	—
603	C₂×C₄	—	2	4	no	—
604	C₂×C₄	—	2	2	no	—
605	C₂×C₄	—	2	4	no	—
606	C₂×C₄	—	2	2	no	—
607	C₂×C₄	—	4	8	no	—
608	C₂×C₄	C₂×#41×C₅₃	1	4	yes	—
609	C₂×C₄	C₂×#41×C₅₃	1	4	yes	—
610	C₂×C₄	C₂×#41×C₅₃	1	2	yes	—
611	C₂×C₄	C₂×#41×C₅₃	1	2	yes	—
612	C₂×C₄	C₂×#41×C₅₃	1	2	yes	—
613	C₂×C₄	C₂×#41×C₅₃	1	2	yes	—
614	C₂×C₄	C₂×#41×C₅₃	1	2	yes	—
615	C₂×C₄	C₂×#41×C₅₃	1	2	yes	—
616	C₂×C₄	C₂×#41×C₅₃	1	2	yes	—
617	C₂×C₄	C₂×#41×C₅₃	1	2	yes	—
618	C₂×C₄	2₋¹⁺⁴×C₅₃	1	2	yes	—
619	C₂²	—	4	2	no	—
620	C₂²	—	4	2	no	—
621	C₂²	—	4	2	no	—
622	C₂²	—	4	2	no	—
623	C₂²	—	4	2	no	—
624	C₂²	—	4	2	no	—
625	C₂²	—	4	2	no	—
626	C₂²	C₂²×#41×C₅₃	1	1	yes	characteristic
627	C₂²	C₂²×#41×C₅₃	1	1	yes	characteristic
628	C₂²	C₂²×#41×C₅₃	1	1	yes	characteristic
629	C₂²	C₂²×#41×C₅₃	1	1	yes	characteristic
630	C₂²	C₂×2₋¹⁺⁴×C₅₃	1	1	yes	characteristic
631	C₂²	C₅₃×#585	1	1	yes	characteristic
632	C₂²	C₅₃×#585	1	1	yes	characteristic
633	C₄	—	2	2	no	—
634	C₄	—	2	2	no	—
635	C₄	—	2	2	no	—
636	C₄	—	2	4	no	—
637	C₄	—	2	4	no	—
638	C₄	—	4	2	no	—
639	C₄	—	4	4	no	—
640	C₄	—	4	4	no	—
641	C₄	—	4	2	no	—
642	C₄	—	4	2	no	—
643	C₄	—	4	1	no	—
644	C₄	—	4	2	no	—
645	C₄	—	4	1	no	—
646	C₄	—	4	2	no	—
647	C₄	C₅₃×#529	1	4	yes	—
648	C₄	C₅₃×#556	1	4	yes	—
649	C₂	—	8	2	no	—
650	C₂	C₂×C₅₃×#529	1	1	yes	characteristic
651	C₂	C₂×C₅₃×#556	1	1	yes	characteristic
652	C₂	C₅₃×#3480	1	1	yes	characteristic
653	C₂	C₅₃×#3480	1	1	yes	characteristic
654	C₂	C₅₃×#3512	1	1	yes	characteristic
655	C₂	C₅₃×#3515	1	1	yes	characteristic
656	C₂	C₅₃×#3518	1	1	yes	characteristic

	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×#2594	C₂	semidirect
4. C₅₃×#2582	C₂	semidirect
6. C₅₃×#2525	C₂	semidirect
7. C₅₃×#2514	C₂	semidirect
9. C₅₃×#2494	C₂	semidirect
10. C₅₃×#2421	C₂	semidirect
11. C₅₃×#2362	C₂	semidirect
12. C₅₃×#2355	C₂	semidirect
14. C₅₃×#2347	C₂	semidirect
15. C₄×#125×C₅₃	C₂	semidirect
17. C₄²×Q₈×C₅₃	C₂	semidirect
290. #36590	C₅₃	direct (N × H)
322. #2859	C₂×C₅₃	semidirect
323. #2594	C₂×C₅₃	semidirect
324. #2582	C₂×C₅₃	semidirect
326. #2525	C₂×C₅₃	semidirect
327. #2514	C₂×C₅₃	semidirect
329. #2494	C₂×C₅₃	semidirect
330. #2421	C₂×C₅₃	semidirect
331. #2362	C₂×C₅₃	semidirect
332. #2355	C₂×C₅₃	semidirect
334. #2347	C₂×C₅₃	semidirect
335. C₄×#125	C₂×C₅₃	semidirect
337. C₄²×Q₈	C₂×C₅₃	semidirect
411. C₅₃	#36590	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×#2594	2
C₅₃×#2582	2
C₅₃×#2538	2
C₅₃×#2525	2
C₅₃×#2514	2
C₅₃×#2513	2
C₅₃×#2494	2
C₅₃×#2421	2
C₅₃×#2362	2
C₅₃×#2355	2
C₅₃×#2354	2
C₅₃×#2347	2
C₄×#125×C₅₃	2
C₄×#124×C₅₃	2
C₄²×Q₈×C₅₃	2
#36590	53

Subgroups — C53×#36590 #27244102

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅₃×#36590 #27244102