Subgroups — C₅₃×#26889 #27234401

Order 13568.

1012 conjugacy classes of subgroups.

The subgroup lattice has 738 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₅₃×#474	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×C₄×D₈×C₅₃	C₂	1	1	yes	maximal, characteristic, complemented
3	C₅₃×#2452	C₂	1	1	yes	maximal, characteristic, complemented
4	C₅₃×#2366	C₂	1	1	yes	maximal, characteristic, complemented
5	C₅₃×#2054	C₂	1	1	yes	maximal, characteristic, complemented
6	C₅₃×#2041	C₂	1	1	yes	maximal, characteristic, complemented
7	C₅₃×#2023	C₂	1	1	yes	maximal, characteristic, complemented
8	C₅₃×#2011	C₂	1	1	yes	maximal, characteristic
9	C₅₃×#1934	C₂	1	1	yes	maximal, characteristic, complemented
10	C₅₃×#1925	C₂	1	1	yes	maximal, characteristic, complemented
11	C₅₃×#1918	C₂	1	1	yes	maximal, characteristic, complemented
12	C₅₃×#1916	C₂	1	1	yes	maximal, characteristic, complemented
13	C₅₃×#1879	C₂	1	1	yes	maximal, characteristic, complemented
14	C₅₃×#1879	C₂	1	1	yes	maximal, characteristic, complemented
15	C₅₃×#1862	C₂	1	1	yes	maximal, characteristic, complemented
16	C₂²×D₈×C₅₃	C₂²	1	1	yes	characteristic
17	C₂²×D₈×C₅₃	C₂²	1	1	yes	characteristic
18	C₂×C₄⋊Q₈×C₅₃	C₂²	1	1	yes	characteristic
19	C₂×#127×C₅₃	C₂²	1	1	yes	characteristic
20	C₂×C₂²⋊Q₈×C₅₃	—	2	1	no	—
21	C₂×C₄×D₄×C₅₃	—	2	1	no	—
22	C₂×C₄×D₄×C₅₃	C₂²	1	1	yes	characteristic
23	C₂×C₄×D₄×C₅₃	C₂²	1	1	yes	characteristic
24	C₂×C₄×D₄×C₅₃	C₂²	1	1	yes	characteristic
25	C₅₃×#474	—	2	4	no	—
26	C₄×D₈×C₅₃	—	2	4	no	—
27	C₂×#107×C₅₃	C₂²	1	1	yes	characteristic
28	C₂×#107×C₅₃	C₂²	1	1	yes	characteristic
29	C₂×C₄⋊C₈×C₅₃	C₂²	1	1	yes	characteristic
30	C₂×#102×C₅₃	C₂²	1	1	yes	characteristic
31	C₂×#102×C₅₃	C₂²	1	1	yes	characteristic
32	C₂×#102×C₅₃	C₂²	1	1	yes	characteristic
33	C₂×#102×C₅₃	C₂²	1	1	yes	characteristic
34	C₂×#102×C₅₃	C₂²	1	1	yes	characteristic
35	C₂×#102×C₅₃	C₂²	1	1	yes	characteristic
36	C₂×#102×C₅₃	C₂²	1	1	yes	characteristic
37	C₂×#102×C₅₃	C₂²	1	1	yes	characteristic
38	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
39	C₅₃×#396	—	2	1	no	—
40	C₅₃×#393	—	2	1	no	—
41	C₅₃×#390	—	2	1	no	—
42	C₅₃×#390	C₂²	1	1	yes	characteristic
43	C₅₃×#390	C₂²	1	1	yes	characteristic
44	C₅₃×#390	C₂²	1	1	yes	characteristic
45	C₅₃×#390	C₂²	1	1	yes	characteristic
46	C₅₃×#389	C₂²	1	1	yes	characteristic
47	C₅₃×#389	C₂²	1	1	yes	characteristic
48	C₅₃×#386	—	2	1	no	—
49	C₅₃×#386	—	2	1	no	—
50	C₅₃×#385	—	2	1	no	—
51	C₄×C₄⋊C₄×C₅₃	C₂²	1	1	yes	characteristic
52	C₄×C₄⋊C₄×C₅₃	C₂²	1	1	yes	characteristic
53	C₅₃×#340	C₂²	1	1	yes	characteristic
54	C₅₃×#340	C₂²	1	1	yes	characteristic
55	C₅₃×#340	C₂²	1	1	yes	characteristic
56	C₅₃×#340	C₂²	1	1	yes	characteristic
57	C₅₃×#336	C₂²	1	1	yes	characteristic
58	C₅₃×#336	C₂²	1	1	yes	characteristic
59	C₅₃×#336	C₂²	1	1	yes	characteristic
60	C₅₃×#336	C₂²	1	1	yes	characteristic
61	C₂²×Q₈×C₅₃	—	2	1	no	—
62	C₂²×D₄×C₅₃	—	2	1	no	—
63	C₂²×D₄×C₅₃	—	2	1	no	—
64	C₂²×D₄×C₅₃	C₂³	1	1	yes	characteristic
65	C₂²×D₄×C₅₃	C₂³	1	1	yes	characteristic
66	C₂²×D₄×C₅₃	C₂³	1	1	yes	characteristic
67	C₂³×C₄×C₅₃	—	2	1	no	—
68	C₂³×C₄×C₅₃	—	2	1	no	—
69	C₂³×C₄×C₅₃	—	2	1	no	—
70	C₂×D₈×C₅₃	—	2	2	no	—
71	C₂×D₈×C₅₃	—	2	2	no	—
72	C₂×D₈×C₅₃	—	2	2	no	—
73	C₂×D₈×C₅₃	—	2	2	no	—
74	C₂×D₈×C₅₃	—	2	2	no	—
75	C₂×D₈×C₅₃	—	4	1	no	—
76	C₂²×C₈×C₅₃	C₂³	1	1	yes	characteristic
77	C₂²×C₈×C₅₃	C₂³	1	1	yes	characteristic
78	C₂²×C₈×C₅₃	C₂³	1	1	yes	characteristic
79	C₂²×C₈×C₅₃	C₂³	1	1	yes	characteristic
80	C₄⋊Q₈×C₅₃	—	2	2	no	—
81	C₄⋊Q₈×C₅₃	—	4	1	no	—
82	#127×C₅₃	—	2	2	no	—
83	#127×C₅₃	—	2	2	no	—
84	C₂²⋊Q₈×C₅₃	—	4	4	no	—
85	C₄×D₄×C₅₃	—	2	2	no	—
86	C₄×D₄×C₅₃	—	2	2	no	—
87	C₄×D₄×C₅₃	—	2	2	no	—
88	C₄×D₄×C₅₃	—	4	1	no	—
89	C₄×D₄×C₅₃	—	4	1	no	—
90	C₄×D₄×C₅₃	—	4	1	no	—
91	C₄×D₄×C₅₃	—	4	4	no	—
92	C₂×C₄⋊C₄×C₅₃	—	2	1	no	—
93	C₂×C₄⋊C₄×C₅₃	—	2	1	no	—
94	C₂×C₄⋊C₄×C₅₃	—	2	1	no	—
95	C₂×C₄⋊C₄×C₅₃	D₄	1	1	yes	characteristic
96	C₂×C₄⋊C₄×C₅₃	D₄	1	1	yes	characteristic
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	1	yes	characteristic
102	C₂×C₂²⋊C₄×C₅₃	—	2	1	no	—
103	C₂×C₂²⋊C₄×C₅₃	—	2	1	no	—
104	C₂×C₂²⋊C₄×C₅₃	—	2	1	no	—
105	C₂×C₂²⋊C₄×C₅₃	—	2	1	no	—
106	C₂×C₂²⋊C₄×C₅₃	—	2	1	no	—
107	C₂×C₂²⋊C₄×C₅₃	—	2	1	no	—
108	C₂×C₂²⋊C₄×C₅₃	—	2	1	no	—
109	C₂×C₂²⋊C₄×C₅₃	—	2	1	no	—
110	C₂×C₂²⋊C₄×C₅₃	—	2	1	no	—
111	C₂×C₂²⋊C₄×C₅₃	—	2	1	no	—
112	C₂×C₂²⋊C₄×C₅₃	—	2	1	no	—
113	C₂×C₄²×C₅₃	—	2	1	no	—
114	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
115	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
116	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
117	#107×C₅₃	—	2	2	no	—
118	#107×C₅₃	D₄	1	4	yes	complemented
119	C₄⋊C₈×C₅₃	—	2	2	no	—
120	#102×C₅₃	—	2	2	no	—
121	#102×C₅₃	—	2	2	no	—
122	#102×C₅₃	—	2	2	no	—
123	#102×C₅₃	—	2	2	no	—
124	#102×C₅₃	—	2	2	no	—
125	#102×C₅₃	—	2	2	no	—
126	#102×C₅₃	—	2	2	no	—
127	#102×C₅₃	—	4	1	no	—
128	C₄×C₈×C₅₃	—	2	2	no	—
129	#95×C₅₃	—	2	1	no	—
130	#95×C₅₃	—	2	1	no	—
131	#95×C₅₃	—	2	1	no	—
132	#95×C₅₃	D₄	1	1	yes	characteristic
133	#95×C₅₃	D₄	1	1	yes	characteristic
134	C₂⁴×C₅₃	—	2	1	no	—
135	C₂⁴×C₅₃	—	2	1	no	—
136	C₂⁴×C₅₃	—	2	1	no	—
137	C₂×Q₈×C₅₃	—	4	2	no	—
138	C₂×Q₈×C₅₃	—	4	2	no	—
139	C₂×Q₈×C₅₃	—	8	1	no	—
140	C₂×D₄×C₅₃	—	2	4	no	—
141	C₂×D₄×C₅₃	—	2	4	no	—
142	C₂×D₄×C₅₃	—	2	1	no	—
143	C₂×D₄×C₅₃	—	2	1	no	—
144	C₂×D₄×C₅₃	—	2	1	no	—
145	C₂×D₄×C₅₃	—	2	1	no	—
146	C₂×D₄×C₅₃	—	2	1	no	—
147	C₂×D₄×C₅₃	—	2	1	no	—
148	C₂×D₄×C₅₃	—	2	1	no	—
149	C₂×D₄×C₅₃	—	2	1	no	—
150	C₂×D₄×C₅₃	—	2	1	no	—
151	C₂×D₄×C₅₃	—	2	1	no	—
152	C₂×D₄×C₅₃	—	2	1	no	—
153	C₂×D₄×C₅₃	—	2	1	no	—
154	C₂×D₄×C₅₃	—	2	1	no	—
155	C₂×D₄×C₅₃	—	2	1	no	—
156	C₂×D₄×C₅₃	—	2	1	no	—
157	C₂×D₄×C₅₃	—	4	2	no	—
158	C₂×D₄×C₅₃	—	4	2	no	—
159	C₂×D₄×C₅₃	—	8	1	no	—
160	C₂×D₄×C₅₃	—	8	1	no	—
161	C₂×D₄×C₅₃	#41	1	2	yes	—
162	C₂×D₄×C₅₃	#41	1	2	yes	—
163	C₂×D₄×C₅₃	#41	1	2	yes	—
164	C₂²×C₄×C₅₃	—	2	1	no	—
165	C₂²×C₄×C₅₃	—	2	1	no	—
166	C₂²×C₄×C₅₃	—	2	1	no	—
167	C₂²×C₄×C₅₃	—	2	1	no	—
168	C₂²×C₄×C₅₃	—	2	1	no	—
169	C₂²×C₄×C₅₃	—	4	2	no	—
170	C₂²×C₄×C₅₃	—	4	2	no	—
171	C₂²×C₄×C₅₃	—	4	2	no	—
172	C₂²×C₄×C₅₃	—	4	2	no	—
173	C₂²×C₄×C₅₃	—	4	2	no	—
174	C₂²×C₄×C₅₃	—	4	2	no	—
175	C₂²×C₄×C₅₃	—	8	1	no	—
176	C₂²×C₄×C₅₃	—	8	1	no	—
177	C₂²×C₄×C₅₃	—	8	1	no	—
178	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
179	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
180	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
181	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
182	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
183	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
184	C₂²×C₄×C₅₃	C₂⁴	1	1	yes	characteristic
185	D₈×C₅₃	—	2	8	no	—
186	D₈×C₅₃	—	4	4	no	—
187	C₂×C₈×C₅₃	—	2	1	no	—
188	C₂×C₈×C₅₃	—	2	1	no	—
189	C₂×C₈×C₅₃	—	2	1	no	—
190	C₂×C₈×C₅₃	—	2	1	no	—
191	C₂×C₈×C₅₃	—	2	1	no	—
192	C₂×C₈×C₅₃	—	2	1	no	—
193	C₂×C₈×C₅₃	—	2	1	no	—
194	C₂×C₈×C₅₃	—	2	1	no	—
195	C₂×C₈×C₅₃	—	2	1	no	—
196	C₂×C₈×C₅₃	—	2	1	no	—
197	C₂×C₈×C₅₃	—	2	1	no	—
198	C₂×C₈×C₅₃	C₂×D₄	1	2	yes	—
199	C₄⋊C₄×C₅₃	—	2	1	no	—
200	C₄⋊C₄×C₅₃	—	2	1	no	—
201	C₄⋊C₄×C₅₃	—	2	1	no	—
202	C₄⋊C₄×C₅₃	—	2	1	no	—
203	C₄⋊C₄×C₅₃	—	2	1	no	—
204	C₄⋊C₄×C₅₃	—	2	1	no	—
205	C₄⋊C₄×C₅₃	—	2	1	no	—
206	C₄⋊C₄×C₅₃	—	2	1	no	—
207	C₄⋊C₄×C₅₃	—	4	1	no	—
208	C₄⋊C₄×C₅₃	—	4	2	no	—
209	C₄⋊C₄×C₅₃	—	4	2	no	—
210	C₄⋊C₄×C₅₃	—	4	2	no	—
211	C₄⋊C₄×C₅₃	D₈	1	4	yes	complemented
212	C₄⋊C₄×C₅₃	C₂×D₄	1	2	yes	—
213	C₄⋊C₄×C₅₃	C₂×D₄	1	2	yes	—
214	C₂²⋊C₄×C₅₃	—	4	2	no	—
215	C₂²⋊C₄×C₅₃	—	4	2	no	—
216	C₂²⋊C₄×C₅₃	—	4	2	no	—
217	C₂²⋊C₄×C₅₃	—	4	2	no	—
218	C₂²⋊C₄×C₅₃	—	4	2	no	—
219	C₂²⋊C₄×C₅₃	—	4	2	no	—
220	C₂²⋊C₄×C₅₃	—	4	2	no	—
221	C₂²⋊C₄×C₅₃	—	4	2	no	—
222	C₂²⋊C₄×C₅₃	—	8	1	no	—
223	C₂²⋊C₄×C₅₃	—	8	1	no	—
224	C₂²⋊C₄×C₅₃	—	8	1	no	—
225	C₄²×C₅₃	—	2	1	no	—
226	C₄²×C₅₃	—	2	1	no	—
227	C₄²×C₅₃	—	2	1	no	—
228	C₄²×C₅₃	—	2	1	no	—
229	C₄²×C₅₃	—	2	1	no	—
230	C₄²×C₅₃	—	2	1	no	—
231	C₄²×C₅₃	—	4	2	no	—
232	C₂³×C₅₃	—	2	2	no	—
233	C₂³×C₅₃	—	2	2	no	—
234	C₂³×C₅₃	—	2	2	no	—
235	C₂³×C₅₃	—	4	1	no	—
236	C₂³×C₅₃	—	4	1	no	—
237	C₂³×C₅₃	—	4	1	no	—
238	C₂³×C₅₃	—	4	1	no	—
239	C₂³×C₅₃	—	4	1	no	—
240	C₂³×C₅₃	—	4	1	no	—
241	C₂³×C₅₃	—	4	1	no	—
242	C₂³×C₅₃	—	4	1	no	—
243	C₂³×C₅₃	—	4	1	no	—
244	C₂³×C₅₃	—	4	1	no	—
245	C₂³×C₅₃	—	4	1	no	—
246	C₂³×C₅₃	—	4	1	no	—
247	C₂³×C₅₃	—	4	1	no	—
248	C₂³×C₅₃	—	4	1	no	—
249	C₂³×C₅₃	—	4	1	no	—
250	C₂³×C₅₃	—	4	1	no	—
251	C₂³×C₅₃	—	4	1	no	—
252	C₂³×C₅₃	—	4	1	no	—
253	C₂³×C₅₃	C₂²×D₄	1	1	yes	characteristic
254	Q₈×C₅₃	—	8	4	no	—
255	D₄×C₅₃	—	2	2	no	—
256	D₄×C₅₃	—	2	2	no	—
257	D₄×C₅₃	—	2	2	no	—
258	D₄×C₅₃	—	2	2	no	—
259	D₄×C₅₃	—	2	2	no	—
260	D₄×C₅₃	—	2	2	no	—
261	D₄×C₅₃	—	4	8	no	—
262	D₄×C₅₃	—	4	1	no	—
263	D₄×C₅₃	—	4	1	no	—
264	D₄×C₅₃	—	4	1	no	—
265	D₄×C₅₃	—	4	1	no	—
266	D₄×C₅₃	—	4	1	no	—
267	D₄×C₅₃	—	4	1	no	—
268	D₄×C₅₃	—	8	4	no	—
269	C₂×C₄×C₅₃	—	2	1	no	—
270	C₂×C₄×C₅₃	—	2	1	no	—
271	C₂×C₄×C₅₃	—	2	1	no	—
272	C₂×C₄×C₅₃	—	2	1	no	—
273	C₂×C₄×C₅₃	—	2	1	no	—
274	C₂×C₄×C₅₃	—	2	1	no	—
275	C₂×C₄×C₅₃	—	2	1	no	—
276	C₂×C₄×C₅₃	—	2	2	no	—
277	C₂×C₄×C₅₃	—	2	2	no	—
278	C₂×C₄×C₅₃	—	2	1	no	—
279	C₂×C₄×C₅₃	—	2	1	no	—
280	C₂×C₄×C₅₃	—	2	1	no	—
281	C₂×C₄×C₅₃	—	2	1	no	—
282	C₂×C₄×C₅₃	—	2	1	no	—
283	C₂×C₄×C₅₃	—	2	1	no	—
284	C₂×C₄×C₅₃	—	2	1	no	—
285	C₂×C₄×C₅₃	—	2	1	no	—
286	C₂×C₄×C₅₃	—	4	1	no	—
287	C₂×C₄×C₅₃	—	4	1	no	—
288	C₂×C₄×C₅₃	—	4	1	no	—
289	C₂×C₄×C₅₃	—	4	1	no	—
290	C₂×C₄×C₅₃	—	4	1	no	—
291	C₂×C₄×C₅₃	—	4	1	no	—
292	C₂×C₄×C₅₃	—	4	1	no	—
293	C₂×C₄×C₅₃	—	4	1	no	—
294	C₂×C₄×C₅₃	—	4	1	no	—
295	C₂×C₄×C₅₃	—	4	1	no	—
296	C₂×C₄×C₅₃	—	4	1	no	—
297	C₂×C₄×C₅₃	—	4	1	no	—
298	C₂×C₄×C₅₃	—	4	1	no	—
299	C₂×C₄×C₅₃	—	8	4	no	—
300	C₂×C₄×C₅₃	—	8	4	no	—
301	C₂×C₄×C₅₃	—	8	4	no	—
302	C₂×C₄×C₅₃	C₂×D₈	1	2	yes	—
303	C₂×C₄×C₅₃	C₂×D₈	1	2	yes	—
304	C₂×C₄×C₅₃	C₂×D₈	1	2	yes	—
305	C₂×C₄×C₅₃	C₂²×D₄	1	1	yes	characteristic
306	C₂×C₄×C₅₃	C₂×#41	1	1	yes	characteristic
307	C₂×C₄×C₅₃	C₂×#41	1	1	yes	characteristic
308	C₂×C₄×C₅₃	C₂×#41	1	1	yes	characteristic
309	C₂×C₄×C₅₃	H(5,𝔽₂)	1	1	yes	characteristic
310	C₂×C₄×C₅₃	2₋¹⁺⁴	1	1	yes	characteristic
311	C₈×C₅₃	—	2	2	no	—
312	C₈×C₅₃	—	4	1	no	—
313	C₈×C₅₃	—	4	1	no	—
314	C₈×C₅₃	—	4	1	no	—
315	#26889	C₅₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
316	C₂²×C₅₃	—	4	2	no	—
317	C₂²×C₅₃	—	4	2	no	—
318	C₂²×C₅₃	—	4	2	no	—
319	C₂²×C₅₃	—	4	2	no	—
320	C₂²×C₅₃	—	4	2	no	—
321	C₂²×C₅₃	—	4	2	no	—
322	C₂²×C₅₃	—	4	2	no	—
323	C₂²×C₅₃	—	4	2	no	—
324	C₂²×C₅₃	—	4	2	no	—
325	C₂²×C₅₃	—	8	1	no	—
326	C₂²×C₅₃	—	8	1	no	—
327	C₂²×C₅₃	—	8	1	no	—
328	C₂²×C₅₃	—	8	1	no	—
329	C₂²×C₅₃	—	8	1	no	—
330	C₂²×C₅₃	—	8	1	no	—
331	C₂²×C₅₃	—	8	1	no	—
332	C₂²×C₅₃	—	8	1	no	—
333	C₂²×C₅₃	—	8	1	no	—
334	C₂²×C₅₃	—	8	1	no	—
335	C₂²×C₅₃	—	8	1	no	—
336	C₂²×C₅₃	—	8	1	no	—
337	C₂²×C₅₃	#525	1	1	yes	characteristic
338	C₂²×C₅₃	D₄²	1	1	yes	characteristic
339	C₂²×C₅₃	#548	1	1	yes	characteristic
340	C₂²×C₅₃	C₂²×D₈	1	1	yes	characteristic
341	C₂²×C₅₃	#575	1	1	yes	characteristic
342	C₂²×C₅₃	#576	1	1	yes	characteristic
343	C₂²×C₅₃	#577	1	1	yes	characteristic
344	C₄×C₅₃	—	2	2	no	—
345	C₄×C₅₃	—	2	2	no	—
346	C₄×C₅₃	—	2	2	no	—
347	C₄×C₅₃	—	4	1	no	—
348	C₄×C₅₃	—	4	1	no	—
349	C₄×C₅₃	—	4	1	no	—
350	C₄×C₅₃	—	4	2	no	—
351	C₄×C₅₃	—	4	2	no	—
352	C₄×C₅₃	—	8	1	no	—
353	C₄×C₅₃	—	8	1	no	—
354	C₄×C₅₃	—	8	1	no	—
355	C₄×C₅₃	#525	1	1	yes	characteristic
356	C₄×C₅₃	#539	1	1	yes	characteristic
357	C₄×C₅₃	#548	1	1	yes	characteristic
358	C₄×C₅₃	#551	1	1	yes	characteristic
359	C₂×#474	C₂×C₅₃	1	1	yes	characteristic, complemented
360	C₂×C₄×D₈	C₂×C₅₃	1	1	yes	characteristic, complemented
361	#2452	C₂×C₅₃	1	1	yes	characteristic, complemented
362	#2366	C₂×C₅₃	1	1	yes	characteristic, complemented
363	#2054	C₂×C₅₃	1	1	yes	characteristic, complemented
364	#2041	C₂×C₅₃	1	1	yes	characteristic, complemented
365	#2023	C₂×C₅₃	1	1	yes	characteristic, complemented
366	#2011	C₂×C₅₃	1	1	yes	characteristic
367	#1934	C₂×C₅₃	1	1	yes	characteristic, complemented
368	#1925	C₂×C₅₃	1	1	yes	characteristic, complemented
369	#1918	C₂×C₅₃	1	1	yes	characteristic, complemented
370	#1916	C₂×C₅₃	1	1	yes	characteristic, complemented
371	#1879	C₂×C₅₃	1	1	yes	characteristic, complemented
372	#1879	C₂×C₅₃	1	1	yes	characteristic, complemented
373	#1862	C₂×C₅₃	1	1	yes	characteristic, complemented
374	C₂×C₅₃	—	8	2	no	—
375	C₂×C₅₃	—	8	2	no	—
376	C₂×C₅₃	—	8	2	no	—
377	C₂×C₅₃	#2463	1	1	yes	characteristic
378	C₂×C₅₃	#3100	1	1	yes	characteristic
379	C₂×C₅₃	#3118	1	1	yes	characteristic
380	C₂×C₅₃	#3271	1	1	yes	characteristic
381	C₂×C₅₃	D₄×D₈	1	1	yes	characteristic
382	C₂×C₅₃	#3358	1	1	yes	characteristic
383	C₂×C₅₃	#3370	1	1	yes	characteristic
384	C₂²×D₈	C₂²×C₅₃	1	1	yes	characteristic
385	C₂²×D₈	C₂²×C₅₃	1	1	yes	characteristic
386	C₂×C₄⋊Q₈	C₂²×C₅₃	1	1	yes	characteristic
387	C₂×#127	C₂²×C₅₃	1	1	yes	characteristic
388	C₂×C₂²⋊Q₈	—	2	1	no	—
389	C₂×C₄×D₄	—	2	1	no	—
390	C₂×C₄×D₄	C₂²×C₅₃	1	1	yes	characteristic
391	C₂×C₄×D₄	C₂²×C₅₃	1	1	yes	characteristic
392	C₂×C₄×D₄	C₂²×C₅₃	1	1	yes	characteristic
393	#474	—	2	4	no	—
394	C₄×D₈	—	2	4	no	—
395	C₂×#107	C₂²×C₅₃	1	1	yes	characteristic
396	C₂×#107	C₂²×C₅₃	1	1	yes	characteristic
397	C₂×C₄⋊C₈	C₂²×C₅₃	1	1	yes	characteristic
398	C₂×#102	C₂²×C₅₃	1	1	yes	characteristic
399	C₂×#102	C₂²×C₅₃	1	1	yes	characteristic
400	C₂×#102	C₂²×C₅₃	1	1	yes	characteristic
401	C₂×#102	C₂²×C₅₃	1	1	yes	characteristic
402	C₂×#102	C₂²×C₅₃	1	1	yes	characteristic
403	C₂×#102	C₂²×C₅₃	1	1	yes	characteristic
404	C₂×#102	C₂²×C₅₃	1	1	yes	characteristic
405	C₂×#102	C₂²×C₅₃	1	1	yes	characteristic
406	C₂×C₄×C₈	C₂²×C₅₃	1	1	yes	characteristic
407	#396	—	2	1	no	—
408	#393	—	2	1	no	—
409	#390	—	2	1	no	—
410	#390	C₂²×C₅₃	1	1	yes	characteristic
411	#390	C₂²×C₅₃	1	1	yes	characteristic
412	#390	C₂²×C₅₃	1	1	yes	characteristic
413	#390	C₂²×C₅₃	1	1	yes	characteristic
414	#389	C₂²×C₅₃	1	1	yes	characteristic
415	#389	C₂²×C₅₃	1	1	yes	characteristic
416	#386	—	2	1	no	—
417	#386	—	2	1	no	—
418	#385	—	2	1	no	—
419	C₄×C₄⋊C₄	C₂²×C₅₃	1	1	yes	characteristic
420	C₄×C₄⋊C₄	C₂²×C₅₃	1	1	yes	characteristic
421	#340	C₂²×C₅₃	1	1	yes	characteristic
422	#340	C₂²×C₅₃	1	1	yes	characteristic
423	#340	C₂²×C₅₃	1	1	yes	characteristic
424	#340	C₂²×C₅₃	1	1	yes	characteristic
425	#336	C₂²×C₅₃	1	1	yes	characteristic
426	#336	C₂²×C₅₃	1	1	yes	characteristic
427	#336	C₂²×C₅₃	1	1	yes	characteristic
428	#336	C₂²×C₅₃	1	1	yes	characteristic
429	C₅₃	#26889	1	1	yes	characteristic, direct product factor, Sylow
430	C₂²×Q₈	—	2	1	no	—
431	C₂²×D₄	—	2	1	no	—
432	C₂²×D₄	—	2	1	no	—
433	C₂²×D₄	C₂³×C₅₃	1	1	yes	characteristic
434	C₂²×D₄	C₂³×C₅₃	1	1	yes	characteristic
435	C₂²×D₄	C₂³×C₅₃	1	1	yes	characteristic
436	C₂³×C₄	—	2	1	no	—
437	C₂³×C₄	—	2	1	no	—
438	C₂³×C₄	—	2	1	no	—
439	C₂×D₈	—	2	2	no	—
440	C₂×D₈	—	2	2	no	—
441	C₂×D₈	—	2	2	no	—
442	C₂×D₈	—	2	2	no	—
443	C₂×D₈	—	2	2	no	—
444	C₂×D₈	—	4	1	no	—
445	C₂²×C₈	C₂³×C₅₃	1	1	yes	characteristic
446	C₂²×C₈	C₂³×C₅₃	1	1	yes	characteristic
447	C₂²×C₈	C₂³×C₅₃	1	1	yes	characteristic
448	C₂²×C₈	C₂³×C₅₃	1	1	yes	characteristic
449	C₄⋊Q₈	—	2	2	no	—
450	C₄⋊Q₈	—	4	1	no	—
451	#127	—	2	2	no	—
452	#127	—	2	2	no	—
453	C₂²⋊Q₈	—	4	4	no	—
454	C₄×D₄	—	2	2	no	—
455	C₄×D₄	—	2	2	no	—
456	C₄×D₄	—	2	2	no	—
457	C₄×D₄	—	4	1	no	—
458	C₄×D₄	—	4	1	no	—
459	C₄×D₄	—	4	1	no	—
460	C₄×D₄	—	4	4	no	—
461	C₂×C₄⋊C₄	—	2	1	no	—
462	C₂×C₄⋊C₄	—	2	1	no	—
463	C₂×C₄⋊C₄	—	2	1	no	—
464	C₂×C₄⋊C₄	D₄×C₅₃	1	1	yes	characteristic
465	C₂×C₄⋊C₄	D₄×C₅₃	1	1	yes	characteristic
466	C₂×C₄⋊C₄	C₂³×C₅₃	1	1	yes	characteristic
467	C₂×C₄⋊C₄	C₂³×C₅₃	1	1	yes	characteristic
468	C₂×C₄⋊C₄	C₂³×C₅₃	1	1	yes	characteristic
469	C₂×C₄⋊C₄	C₂³×C₅₃	1	1	yes	characteristic
470	C₂×C₄⋊C₄	C₂³×C₅₃	1	1	yes	characteristic
471	C₂×C₂²⋊C₄	—	2	1	no	—
472	C₂×C₂²⋊C₄	—	2	1	no	—
473	C₂×C₂²⋊C₄	—	2	1	no	—
474	C₂×C₂²⋊C₄	—	2	1	no	—
475	C₂×C₂²⋊C₄	—	2	1	no	—
476	C₂×C₂²⋊C₄	—	2	1	no	—
477	C₂×C₂²⋊C₄	—	2	1	no	—
478	C₂×C₂²⋊C₄	—	2	1	no	—
479	C₂×C₂²⋊C₄	—	2	1	no	—
480	C₂×C₂²⋊C₄	—	2	1	no	—
481	C₂×C₂²⋊C₄	—	2	1	no	—
482	C₂×C₄²	—	2	1	no	—
483	C₂×C₄²	C₂³×C₅₃	1	1	yes	characteristic
484	C₂×C₄²	C₂³×C₅₃	1	1	yes	characteristic
485	C₂×C₄²	C₂³×C₅₃	1	1	yes	characteristic
486	#107	—	2	2	no	—
487	#107	D₄×C₅₃	1	4	yes	complemented
488	C₄⋊C₈	—	2	2	no	—
489	#102	—	2	2	no	—
490	#102	—	2	2	no	—
491	#102	—	2	2	no	—
492	#102	—	2	2	no	—
493	#102	—	2	2	no	—
494	#102	—	2	2	no	—
495	#102	—	2	2	no	—
496	#102	—	4	1	no	—
497	C₄×C₈	—	2	2	no	—
498	#95	—	2	1	no	—
499	#95	—	2	1	no	—
500	#95	—	2	1	no	—
501	#95	D₄×C₅₃	1	1	yes	characteristic
502	#95	D₄×C₅₃	1	1	yes	characteristic
503	C₂⁴	—	2	1	no	—
504	C₂⁴	—	2	1	no	—
505	C₂⁴	—	2	1	no	—
506	C₂×Q₈	—	4	2	no	—
507	C₂×Q₈	—	4	2	no	—
508	C₂×Q₈	—	8	1	no	—
509	C₂×D₄	—	2	4	no	—
510	C₂×D₄	—	2	4	no	—
511	C₂×D₄	—	2	1	no	—
512	C₂×D₄	—	2	1	no	—
513	C₂×D₄	—	2	1	no	—
514	C₂×D₄	—	2	1	no	—
515	C₂×D₄	—	2	1	no	—
516	C₂×D₄	—	2	1	no	—
517	C₂×D₄	—	2	1	no	—
518	C₂×D₄	—	2	1	no	—
519	C₂×D₄	—	2	1	no	—
520	C₂×D₄	—	2	1	no	—
521	C₂×D₄	—	2	1	no	—
522	C₂×D₄	—	2	1	no	—
523	C₂×D₄	—	2	1	no	—
524	C₂×D₄	—	2	1	no	—
525	C₂×D₄	—	2	1	no	—
526	C₂×D₄	—	4	2	no	—
527	C₂×D₄	—	4	2	no	—
528	C₂×D₄	—	8	1	no	—
529	C₂×D₄	—	8	1	no	—
530	C₂×D₄	#41×C₅₃	1	2	yes	—
531	C₂×D₄	#41×C₅₃	1	2	yes	—
532	C₂×D₄	#41×C₅₃	1	2	yes	—
533	C₂²×C₄	—	2	1	no	—
534	C₂²×C₄	—	2	1	no	—
535	C₂²×C₄	—	2	1	no	—
536	C₂²×C₄	—	2	1	no	—
537	C₂²×C₄	—	2	1	no	—
538	C₂²×C₄	—	4	2	no	—
539	C₂²×C₄	—	4	2	no	—
540	C₂²×C₄	—	4	2	no	—
541	C₂²×C₄	—	4	2	no	—
542	C₂²×C₄	—	4	2	no	—
543	C₂²×C₄	—	4	2	no	—
544	C₂²×C₄	—	8	1	no	—
545	C₂²×C₄	—	8	1	no	—
546	C₂²×C₄	—	8	1	no	—
547	C₂²×C₄	C₂×D₄×C₅₃	1	1	yes	characteristic
548	C₂²×C₄	C₂×D₄×C₅₃	1	1	yes	characteristic
549	C₂²×C₄	C₂×D₄×C₅₃	1	1	yes	characteristic
550	C₂²×C₄	C₂×D₄×C₅₃	1	1	yes	characteristic
551	C₂²×C₄	C₂×D₄×C₅₃	1	1	yes	characteristic
552	C₂²×C₄	C₂×D₄×C₅₃	1	1	yes	characteristic
553	C₂²×C₄	C₂⁴×C₅₃	1	1	yes	characteristic
554	D₈	—	2	8	no	—
555	D₈	—	4	4	no	—
556	C₂×C₈	—	2	1	no	—
557	C₂×C₈	—	2	1	no	—
558	C₂×C₈	—	2	1	no	—
559	C₂×C₈	—	2	1	no	—
560	C₂×C₈	—	2	1	no	—
561	C₂×C₈	—	2	1	no	—
562	C₂×C₈	—	2	1	no	—
563	C₂×C₈	—	2	1	no	—
564	C₂×C₈	—	2	1	no	—
565	C₂×C₈	—	2	1	no	—
566	C₂×C₈	—	2	1	no	—
567	C₂×C₈	C₂×D₄×C₅₃	1	2	yes	—
568	C₄⋊C₄	—	2	1	no	—
569	C₄⋊C₄	—	2	1	no	—
570	C₄⋊C₄	—	2	1	no	—
571	C₄⋊C₄	—	2	1	no	—
572	C₄⋊C₄	—	2	1	no	—
573	C₄⋊C₄	—	2	1	no	—
574	C₄⋊C₄	—	2	1	no	—
575	C₄⋊C₄	—	2	1	no	—
576	C₄⋊C₄	—	4	1	no	—
577	C₄⋊C₄	—	4	2	no	—
578	C₄⋊C₄	—	4	2	no	—
579	C₄⋊C₄	—	4	2	no	—
580	C₄⋊C₄	D₈×C₅₃	1	4	yes	complemented
581	C₄⋊C₄	C₂×D₄×C₅₃	1	2	yes	—
582	C₄⋊C₄	C₂×D₄×C₅₃	1	2	yes	—
583	C₂²⋊C₄	—	4	2	no	—
584	C₂²⋊C₄	—	4	2	no	—
585	C₂²⋊C₄	—	4	2	no	—
586	C₂²⋊C₄	—	4	2	no	—
587	C₂²⋊C₄	—	4	2	no	—
588	C₂²⋊C₄	—	4	2	no	—
589	C₂²⋊C₄	—	4	2	no	—
590	C₂²⋊C₄	—	4	2	no	—
591	C₂²⋊C₄	—	8	1	no	—
592	C₂²⋊C₄	—	8	1	no	—
593	C₂²⋊C₄	—	8	1	no	—
594	C₄²	—	2	1	no	—
595	C₄²	—	2	1	no	—
596	C₄²	—	2	1	no	—
597	C₄²	—	2	1	no	—
598	C₄²	—	2	1	no	—
599	C₄²	—	2	1	no	—
600	C₄²	—	4	2	no	—
601	C₂³	—	2	2	no	—
602	C₂³	—	2	2	no	—
603	C₂³	—	2	2	no	—
604	C₂³	—	4	1	no	—
605	C₂³	—	4	1	no	—
606	C₂³	—	4	1	no	—
607	C₂³	—	4	1	no	—
608	C₂³	—	4	1	no	—
609	C₂³	—	4	1	no	—
610	C₂³	—	4	1	no	—
611	C₂³	—	4	1	no	—
612	C₂³	—	4	1	no	—
613	C₂³	—	4	1	no	—
614	C₂³	—	4	1	no	—
615	C₂³	—	4	1	no	—
616	C₂³	—	4	1	no	—
617	C₂³	—	4	1	no	—
618	C₂³	—	4	1	no	—
619	C₂³	—	4	1	no	—
620	C₂³	—	4	1	no	—
621	C₂³	—	4	1	no	—
622	C₂³	C₂²×D₄×C₅₃	1	1	yes	characteristic
623	Q₈	—	8	4	no	—
624	D₄	—	2	2	no	—
625	D₄	—	2	2	no	—
626	D₄	—	2	2	no	—
627	D₄	—	2	2	no	—
628	D₄	—	2	2	no	—
629	D₄	—	2	2	no	—
630	D₄	—	4	8	no	—
631	D₄	—	4	1	no	—
632	D₄	—	4	1	no	—
633	D₄	—	4	1	no	—
634	D₄	—	4	1	no	—
635	D₄	—	4	1	no	—
636	D₄	—	4	1	no	—
637	D₄	—	8	4	no	—
638	C₂×C₄	—	2	1	no	—
639	C₂×C₄	—	2	1	no	—
640	C₂×C₄	—	2	1	no	—
641	C₂×C₄	—	2	1	no	—
642	C₂×C₄	—	2	1	no	—
643	C₂×C₄	—	2	1	no	—
644	C₂×C₄	—	2	1	no	—
645	C₂×C₄	—	2	1	no	—
646	C₂×C₄	—	2	1	no	—
647	C₂×C₄	—	2	2	no	—
648	C₂×C₄	—	2	2	no	—
649	C₂×C₄	—	2	1	no	—
650	C₂×C₄	—	2	1	no	—
651	C₂×C₄	—	2	1	no	—
652	C₂×C₄	—	2	1	no	—
653	C₂×C₄	—	2	1	no	—
654	C₂×C₄	—	2	1	no	—
655	C₂×C₄	—	4	1	no	—
656	C₂×C₄	—	4	1	no	—
657	C₂×C₄	—	4	1	no	—
658	C₂×C₄	—	4	1	no	—
659	C₂×C₄	—	4	1	no	—
660	C₂×C₄	—	4	1	no	—
661	C₂×C₄	—	4	1	no	—
662	C₂×C₄	—	4	1	no	—
663	C₂×C₄	—	4	1	no	—
664	C₂×C₄	—	4	1	no	—
665	C₂×C₄	—	4	1	no	—
666	C₂×C₄	—	4	1	no	—
667	C₂×C₄	—	4	1	no	—
668	C₂×C₄	—	8	4	no	—
669	C₂×C₄	—	8	4	no	—
670	C₂×C₄	—	8	4	no	—
671	C₂×C₄	C₂×D₈×C₅₃	1	2	yes	—
672	C₂×C₄	C₂×D₈×C₅₃	1	2	yes	—
673	C₂×C₄	C₂×D₈×C₅₃	1	2	yes	—
674	C₂×C₄	C₂²×D₄×C₅₃	1	1	yes	characteristic
675	C₂×C₄	C₂×#41×C₅₃	1	1	yes	characteristic
676	C₂×C₄	C₂×#41×C₅₃	1	1	yes	characteristic
677	C₂×C₄	C₂×#41×C₅₃	1	1	yes	characteristic
678	C₂×C₄	H(5,𝔽₂)×C₅₃	1	1	yes	characteristic
679	C₂×C₄	2₋¹⁺⁴×C₅₃	1	1	yes	characteristic
680	C₈	—	2	2	no	—
681	C₈	—	4	1	no	—
682	C₈	—	4	1	no	—
683	C₈	—	4	1	no	—
684	C₂²	—	4	2	no	—
685	C₂²	—	4	2	no	—
686	C₂²	—	4	2	no	—
687	C₂²	—	4	2	no	—
688	C₂²	—	4	2	no	—
689	C₂²	—	4	2	no	—
690	C₂²	—	4	2	no	—
691	C₂²	—	4	2	no	—
692	C₂²	—	4	2	no	—
693	C₂²	—	8	1	no	—
694	C₂²	—	8	1	no	—
695	C₂²	—	8	1	no	—
696	C₂²	—	8	1	no	—
697	C₂²	—	8	1	no	—
698	C₂²	—	8	1	no	—
699	C₂²	—	8	1	no	—
700	C₂²	—	8	1	no	—
701	C₂²	—	8	1	no	—
702	C₂²	—	8	1	no	—
703	C₂²	—	8	1	no	—
704	C₂²	—	8	1	no	—
705	C₂²	C₅₃×#525	1	1	yes	characteristic
706	C₂²	D₄²×C₅₃	1	1	yes	characteristic
707	C₂²	C₅₃×#548	1	1	yes	characteristic
708	C₂²	C₂²×D₈×C₅₃	1	1	yes	characteristic
709	C₂²	C₅₃×#575	1	1	yes	characteristic
710	C₂²	C₅₃×#576	1	1	yes	characteristic
711	C₂²	C₅₃×#577	1	1	yes	characteristic
712	C₄	—	2	2	no	—
713	C₄	—	2	2	no	—
714	C₄	—	2	2	no	—
715	C₄	—	4	1	no	—
716	C₄	—	4	1	no	—
717	C₄	—	4	2	no	—
718	C₄	—	4	2	no	—
719	C₄	—	4	1	no	—
720	C₄	—	8	1	no	—
721	C₄	—	8	1	no	—
722	C₄	—	8	1	no	—
723	C₄	C₅₃×#525	1	1	yes	characteristic
724	C₄	C₅₃×#539	1	1	yes	characteristic
725	C₄	C₅₃×#548	1	1	yes	characteristic
726	C₄	C₅₃×#551	1	1	yes	characteristic
727	C₂	—	8	2	no	—
728	C₂	—	8	2	no	—
729	C₂	—	8	2	no	—
730	C₂	C₅₃×#2463	1	1	yes	characteristic
731	C₂	C₅₃×#3100	1	1	yes	characteristic
732	C₂	C₅₃×#3118	1	1	yes	characteristic
733	C₂	C₅₃×#3271	1	1	yes	characteristic
734	C₂	D₄×D₈×C₅₃	1	1	yes	characteristic
735	C₂	C₅₃×#3358	1	1	yes	characteristic
736	C₂	C₅₃×#3370	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂³×C₅₃	C₂²×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁴×C₅₃
prime radical	C₅₃×#2011	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₅₃ ◁ C₄×C₄⋊C₄×C₅₃ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂×C₅₃×#474	C₂	semidirect
2. C₂×C₄×D₈×C₅₃	C₂	semidirect
3. C₅₃×#2452	C₂	semidirect
4. C₅₃×#2366	C₂	semidirect
5. C₅₃×#2054	C₂	semidirect
6. C₅₃×#2041	C₂	semidirect
7. C₅₃×#2023	C₂	semidirect
9. C₅₃×#1934	C₂	semidirect
10. C₅₃×#1925	C₂	semidirect
11. C₅₃×#1918	C₂	semidirect
12. C₅₃×#1916	C₂	semidirect
13. C₅₃×#1879	C₂	semidirect
14. C₅₃×#1879	C₂	semidirect
15. C₅₃×#1862	C₂	semidirect
118. #107×C₅₃	D₄	semidirect
211. C₄⋊C₄×C₅₃	D₈	semidirect
315. #26889	C₅₃	direct (N × H)
359. C₂×#474	C₂×C₅₃	semidirect
360. C₂×C₄×D₈	C₂×C₅₃	semidirect
361. #2452	C₂×C₅₃	semidirect
362. #2366	C₂×C₅₃	semidirect
363. #2054	C₂×C₅₃	semidirect
364. #2041	C₂×C₅₃	semidirect
365. #2023	C₂×C₅₃	semidirect
367. #1934	C₂×C₅₃	semidirect
368. #1925	C₂×C₅₃	semidirect
369. #1918	C₂×C₅₃	semidirect
370. #1916	C₂×C₅₃	semidirect
371. #1879	C₂×C₅₃	semidirect
372. #1879	C₂×C₅₃	semidirect
373. #1862	C₂×C₅₃	semidirect
429. C₅₃	#26889	direct (N × H)
487. #107	D₄×C₅₃	semidirect
580. C₄⋊C₄	D₈×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×#474	2
C₂×C₄×D₈×C₅₃	2
C₅₃×#2452	2
C₅₃×#2366	2
C₅₃×#2054	2
C₅₃×#2041	2
C₅₃×#2023	2
C₅₃×#2011	2
C₅₃×#1934	2
C₅₃×#1925	2
C₅₃×#1918	2
C₅₃×#1916	2
C₅₃×#1879	2
C₅₃×#1862	2
#26889	53

Subgroups — C53×#26889 #27234401

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅₃×#26889 #27234401