Subgroups — C₅₃×#22523 #27230035

Order 13568.

1250 conjugacy classes of subgroups.

The subgroup lattice has 502 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×D₄×Q₈×C₅₃	C₂	1	1	yes	maximal, characteristic
2	C₂×C₅₃×#446	C₂	1	1	yes	maximal, characteristic, complemented
3	C₅₃×#2355	C₂	1	1	yes	maximal, characteristic, complemented
4	C₅₃×#1993	C₂	1	2	yes	maximal, complemented
5	C₅₃×#1990	C₂	1	1	yes	maximal, characteristic, complemented
6	C₅₃×#1988	C₂	1	1	yes	maximal, characteristic, complemented
7	C₅₃×#1987	C₂	1	2	yes	maximal, complemented
8	C₅₃×#1804	C₂	1	2	yes	maximal
9	C₅₃×#1802	C₂	1	1	yes	maximal, characteristic
10	C₅₃×#1762	C₂	1	2	yes	maximal
11	C₄×#102×C₅₃	C₂	1	1	yes	maximal, characteristic
12	C₂³×Q₈×C₅₃	—	2	1	no	—
13	D₄×Q₈×C₅₃	—	2	4	no	—
14	D₄×Q₈×C₅₃	C₄	1	8	yes	complemented
15	C₂×C₄⋊Q₈×C₅₃	C₂²	1	2	yes	—
16	C₂×C₄⋊Q₈×C₅₃	C₂²	1	1	yes	characteristic
17	C₂×C₂²⋊Q₈×C₅₃	—	2	1	no	—
18	C₂×C₂²⋊Q₈×C₅₃	—	2	2	no	—
19	C₂×C₄×Q₈×C₅₃	C₂²	1	1	yes	characteristic
20	C₂×C₄×D₄×C₅₃	C₂²	1	1	yes	characteristic
21	C₂×C₄×D₄×C₅₃	C₂²	1	2	yes	—
22	C₅₃×#446	—	2	4	no	—
23	C₂×C₄⋊C₈×C₅₃	C₂²	1	2	yes	—
24	C₂×C₄⋊C₈×C₅₃	C₂²	1	1	yes	characteristic
25	C₂×#102×C₅₃	C₂²	1	1	yes	characteristic
26	C₂×#102×C₅₃	C₂²	1	2	yes	—
27	C₂×#102×C₅₃	C₂²	1	1	yes	characteristic
28	C₂×#97×C₅₃	C₂²	1	2	yes	—
29	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
30	C₅₃×#391	—	2	1	no	—
31	C₅₃×#387	—	2	2	no	—
32	C₅₃×#384	C₂²	1	1	yes	characteristic
33	C₅₃×#382	C₂²	1	2	yes	—
34	C₄×C₄⋊C₄×C₅₃	—	2	1	no	—
35	C₄×C₄⋊C₄×C₅₃	C₂²	1	2	yes	—
36	C₄×C₄⋊C₄×C₅₃	C₂²	1	1	yes	characteristic
37	C₅₃×#340	C₂²	1	1	yes	characteristic
38	C₅₃×#340	C₂²	1	2	yes	—
39	C₅₃×#340	C₂²	1	2	yes	—
40	C₅₃×#340	C₂²	1	2	yes	—
41	C₅₃×#340	C₂²	1	2	yes	—
42	C₅₃×#340	C₂²	1	2	yes	—
43	C₅₃×#340	C₂²	1	1	yes	characteristic
44	C₂²×Q₈×C₅₃	—	2	8	no	—
45	C₂²×Q₈×C₅₃	—	2	2	no	—
46	C₂²×Q₈×C₅₃	—	2	1	no	—
47	C₂²×Q₈×C₅₃	—	4	4	no	—
48	C₂²×Q₈×C₅₃	—	4	4	no	—
49	C₂²×Q₈×C₅₃	D₄	1	1	yes	characteristic
50	C₂²×D₄×C₅₃	C₂³	1	1	yes	characteristic
51	C₂³×C₄×C₅₃	—	2	2	no	—
52	C₂³×C₄×C₅₃	—	2	1	no	—
53	C₂²×C₈×C₅₃	C₂³	1	1	yes	characteristic
54	C₂²×C₈×C₅₃	C₂³	1	2	yes	—
55	C₂²×C₈×C₅₃	C₂³	1	1	yes	characteristic
56	C₄⋊Q₈×C₅₃	—	2	2	no	—
57	C₄⋊Q₈×C₅₃	—	2	4	no	—
58	C₄⋊Q₈×C₅₃	C₂×C₄	1	8	yes	—
59	C₄⋊Q₈×C₅₃	C₂×C₄	1	4	yes	—
60	C₂²⋊Q₈×C₅₃	—	2	16	no	—
61	C₂²⋊Q₈×C₅₃	—	2	8	no	—
62	C₄×Q₈×C₅₃	C₂×C₄	1	4	yes	—
63	C₄×Q₈×C₅₃	D₄	1	4	yes	—
64	C₄×D₄×C₅₃	—	2	2	no	—
65	C₄×D₄×C₅₃	—	2	4	no	—
66	C₄×D₄×C₅₃	C₂×C₄	1	4	yes	—
67	C₄×D₄×C₅₃	C₂×C₄	1	8	yes	—
68	C₂×C₄⋊C₄×C₅₃	—	2	2	no	—
69	C₂×C₄⋊C₄×C₅₃	—	2	1	no	—
70	C₂×C₄⋊C₄×C₅₃	—	2	1	no	—
71	C₂×C₄⋊C₄×C₅₃	—	2	1	no	—
72	C₂×C₄⋊C₄×C₅₃	D₄	1	1	yes	characteristic
73	C₂×C₄⋊C₄×C₅₃	D₄	1	2	yes	—
74	C₂×C₄⋊C₄×C₅₃	C₂³	1	1	yes	characteristic
75	C₂×C₄⋊C₄×C₅₃	C₂³	1	2	yes	—
76	C₂×C₄⋊C₄×C₅₃	C₂³	1	1	yes	characteristic
77	C₂×C₄⋊C₄×C₅₃	C₂³	1	2	yes	—
78	C₂×C₄⋊C₄×C₅₃	C₂³	1	1	yes	characteristic
79	C₂×C₂²⋊C₄×C₅₃	—	2	2	no	—
80	C₂×C₂²⋊C₄×C₅₃	—	2	1	no	—
81	C₂×C₄²×C₅₃	—	2	1	no	—
82	C₂×C₄²×C₅₃	—	2	2	no	—
83	C₂×C₄²×C₅₃	C₂³	1	2	yes	—
84	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
85	C₄⋊C₈×C₅₃	—	2	2	no	—
86	C₄⋊C₈×C₅₃	—	2	4	no	—
87	#102×C₅₃	—	2	2	no	—
88	#102×C₅₃	—	2	4	no	—
89	#102×C₅₃	Q₈	1	4	yes	complemented
90	#97×C₅₃	—	2	4	no	—
91	C₄×C₈×C₅₃	—	2	2	no	—
92	#95×C₅₃	—	2	2	no	—
93	#95×C₅₃	—	2	1	no	—
94	#95×C₅₃	—	2	2	no	—
95	#95×C₅₃	—	2	2	no	—
96	C₂⁴×C₅₃	—	2	1	no	—
97	C₂×Q₈×C₅₃	—	2	8	no	—
98	C₂×Q₈×C₅₃	—	2	4	no	—
99	C₂×Q₈×C₅₃	—	2	2	no	—
100	C₂×Q₈×C₅₃	—	2	2	no	—
101	C₂×Q₈×C₅₃	—	4	4	no	—
102	C₂×Q₈×C₅₃	—	4	4	no	—
103	C₂×Q₈×C₅₃	—	4	32	no	—
104	C₂×Q₈×C₅₃	—	4	2	no	—
105	C₂×Q₈×C₅₃	—	4	2	no	—
106	C₂×Q₈×C₅₃	C₂²⋊C₄	1	4	yes	—
107	C₂×D₄×C₅₃	—	2	1	no	—
108	C₂×D₄×C₅₃	—	2	1	no	—
109	C₂×D₄×C₅₃	—	2	1	no	—
110	C₂×D₄×C₅₃	C₂²×C₄	1	2	yes	—
111	C₂×D₄×C₅₃	C₂×Q₈	1	2	yes	—
112	C₂×D₄×C₅₃	#41	1	2	yes	—
113	C₂²×C₄×C₅₃	—	2	1	no	—
114	C₂²×C₄×C₅₃	—	2	2	no	—
115	C₂²×C₄×C₅₃	—	2	1	no	—
116	C₂²×C₄×C₅₃	—	2	8	no	—
117	C₂²×C₄×C₅₃	—	2	4	no	—
118	C₂²×C₄×C₅₃	—	2	1	no	—
119	C₂²×C₄×C₅₃	—	2	2	no	—
120	C₂²×C₄×C₅₃	—	4	4	no	—
121	C₂²×C₄×C₅₃	—	4	4	no	—
122	C₂²×C₄×C₅₃	—	4	2	no	—
123	C₂²×C₄×C₅₃	—	4	2	no	—
124	C₂²×C₄×C₅₃	C₂×D₄	1	2	yes	—
125	C₂²×C₄×C₅₃	C₂×D₄	1	2	yes	—
126	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
127	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
128	C₂²×C₄×C₅₃	C₂⁴	1	1	yes	characteristic
129	C₂×C₈×C₅₃	—	2	1	no	—
130	C₂×C₈×C₅₃	—	2	1	no	—
131	C₂×C₈×C₅₃	—	2	1	no	—
132	C₂×C₈×C₅₃	—	2	2	no	—
133	C₂×C₈×C₅₃	—	2	1	no	—
134	C₂×C₈×C₅₃	—	2	2	no	—
135	C₂×C₈×C₅₃	—	2	1	no	—
136	C₂×C₈×C₅₃	—	2	2	no	—
137	C₂×C₈×C₅₃	C₂×Q₈	1	2	yes	complemented
138	C₄⋊C₄×C₅₃	—	2	8	no	—
139	C₄⋊C₄×C₅₃	—	2	4	no	—
140	C₄⋊C₄×C₅₃	—	2	1	no	—
141	C₄⋊C₄×C₅₃	—	2	2	no	—
142	C₄⋊C₄×C₅₃	—	2	1	no	—
143	C₄⋊C₄×C₅₃	—	2	1	no	—
144	C₄⋊C₄×C₅₃	—	2	2	no	—
145	C₄⋊C₄×C₅₃	—	2	1	no	—
146	C₄⋊C₄×C₅₃	—	2	2	no	—
147	C₄⋊C₄×C₅₃	—	4	2	no	—
148	C₄⋊C₄×C₅₃	—	8	1	no	—
149	C₄⋊C₄×C₅₃	C₂²⋊C₄	1	8	yes	—
150	C₄⋊C₄×C₅₃	C₂²⋊C₄	1	4	yes	—
151	C₄⋊C₄×C₅₃	C₂²×C₄	1	2	yes	—
152	C₄⋊C₄×C₅₃	C₂²×C₄	1	4	yes	—
153	C₄⋊C₄×C₅₃	C₂×Q₈	1	2	yes	complemented
154	C₂²⋊C₄×C₅₃	—	2	8	no	—
155	C₂²⋊C₄×C₅₃	—	2	4	no	—
156	C₄²×C₅₃	—	4	2	no	—
157	C₄²×C₅₃	—	8	2	no	—
158	C₄²×C₅₃	C₂²×C₄	1	4	yes	—
159	C₄²×C₅₃	C₂²×C₄	1	2	yes	—
160	C₄²×C₅₃	C₂×D₄	1	2	yes	—
161	C₄²×C₅₃	C₂×D₄	1	4	yes	—
162	C₂³×C₅₃	—	2	2	no	—
163	C₂³×C₅₃	—	2	2	no	—
164	C₂³×C₅₃	—	2	2	no	—
165	C₂³×C₅₃	—	4	1	no	—
166	C₂³×C₅₃	—	4	1	no	—
167	C₂³×C₅₃	—	4	1	no	—
168	C₂³×C₅₃	—	4	1	no	—
169	C₂³×C₅₃	C₂²×D₄	1	1	yes	characteristic
170	Q₈×C₅₃	—	2	8	no	—
171	Q₈×C₅₃	—	4	16	no	—
172	Q₈×C₅₃	—	4	8	no	—
173	D₄×C₅₃	—	2	2	no	—
174	D₄×C₅₃	—	2	2	no	—
175	D₄×C₅₃	—	2	2	no	—
176	D₄×C₅₃	C₄×Q₈	1	4	yes	—
177	C₂×C₄×C₅₃	—	2	2	no	—
178	C₂×C₄×C₅₃	—	2	2	no	—
179	C₂×C₄×C₅₃	—	2	2	no	—
180	C₂×C₄×C₅₃	—	2	1	no	—
181	C₂×C₄×C₅₃	—	2	2	no	—
182	C₂×C₄×C₅₃	—	2	1	no	—
183	C₂×C₄×C₅₃	—	2	4	no	—
184	C₂×C₄×C₅₃	—	2	2	no	—
185	C₂×C₄×C₅₃	—	2	2	no	—
186	C₂×C₄×C₅₃	—	4	16	no	—
187	C₂×C₄×C₅₃	—	4	2	no	—
188	C₂×C₄×C₅₃	—	4	1	no	—
189	C₂×C₄×C₅₃	—	4	2	no	—
190	C₂×C₄×C₅₃	—	4	1	no	—
191	C₂×C₄×C₅₃	—	4	1	no	—
192	C₂×C₄×C₅₃	—	4	1	no	—
193	C₂×C₄×C₅₃	—	4	8	no	—
194	C₂×C₄×C₅₃	—	4	1	no	—
195	C₂×C₄×C₅₃	—	4	2	no	—
196	C₂×C₄×C₅₃	—	4	1	no	—
197	C₂×C₄×C₅₃	—	4	2	no	—
198	C₂×C₄×C₅₃	—	4	2	no	—
199	C₂×C₄×C₅₃	—	4	1	no	—
200	C₂×C₄×C₅₃	C₂×C₂²⋊C₄	1	4	yes	—
201	C₂×C₄×C₅₃	C₂×C₂²⋊C₄	1	4	yes	—
202	C₂×C₄×C₅₃	C₂×C₂²⋊C₄	1	2	yes	—
203	C₂×C₄×C₅₃	C₂×C₂²⋊C₄	1	2	yes	—
204	C₂×C₄×C₅₃	AGL(1,ℤ/8)	1	2	yes	—
205	C₂×C₄×C₅₃	AGL(1,ℤ/8)	1	2	yes	—
206	C₂×C₄×C₅₃	C₂³×C₄	1	1	yes	characteristic
207	C₂×C₄×C₅₃	C₂²×D₄	1	1	yes	characteristic
208	C₂×C₄×C₅₃	C₂²×Q₈	1	1	yes	characteristic
209	C₂×C₄×C₅₃	C₂×#41	1	1	yes	characteristic
210	C₂×C₄×C₅₃	2₋¹⁺⁴	1	1	yes	characteristic
211	C₂×C₄×C₅₃	2₋¹⁺⁴	1	1	yes	characteristic
212	C₈×C₅₃	—	2	2	no	—
213	C₈×C₅₃	—	4	1	no	—
214	C₈×C₅₃	—	4	2	no	—
215	#22523	C₅₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
216	C₂²×C₅₃	—	2	4	no	—
217	C₂²×C₅₃	—	4	2	no	—
218	C₂²×C₅₃	—	4	2	no	—
219	C₂²×C₅₃	—	4	2	no	—
220	C₂²×C₅₃	—	4	2	no	—
221	C₂²×C₅₃	—	4	2	no	—
222	C₂²×C₅₃	—	4	2	no	—
223	C₂²×C₅₃	C₂²×C₂²⋊C₄	1	1	yes	characteristic
224	C₂²×C₅₃	#548	1	1	yes	characteristic
225	C₂²×C₅₃	D₄×Q₈	1	1	yes	characteristic
226	C₂²×C₅₃	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
227	C₂²×C₅₃	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
228	C₂²×C₅₃	#577	1	1	yes	characteristic
229	C₂²×C₅₃	#578	1	1	yes	characteristic
230	C₄×C₅₃	—	2	4	no	—
231	C₄×C₅₃	—	2	4	no	—
232	C₄×C₅₃	—	2	2	no	—
233	C₄×C₅₃	—	4	2	no	—
234	C₄×C₅₃	—	4	4	no	—
235	C₄×C₅₃	—	4	2	no	—
236	C₄×C₅₃	—	8	1	no	—
237	C₄×C₅₃	—	8	2	no	—
238	C₄×C₅₃	#418	1	4	yes	—
239	C₄×C₅₃	C₂×C₄×Q₈	1	1	yes	characteristic
240	C₄×C₅₃	#519	1	1	yes	characteristic
241	C₄×C₅₃	#548	1	1	yes	characteristic
242	C₄×C₅₃	D₄×Q₈	1	1	yes	characteristic
243	C₂×D₄×Q₈	C₂×C₅₃	1	1	yes	characteristic
244	C₂×#446	C₂×C₅₃	1	1	yes	characteristic, complemented
245	#2355	C₂×C₅₃	1	1	yes	characteristic, complemented
246	#1993	C₂×C₅₃	1	2	yes	complemented
247	#1990	C₂×C₅₃	1	1	yes	characteristic, complemented
248	#1988	C₂×C₅₃	1	1	yes	characteristic, complemented
249	#1987	C₂×C₅₃	1	2	yes	complemented
250	#1804	C₂×C₅₃	1	2	yes	—
251	#1802	C₂×C₅₃	1	1	yes	characteristic
252	#1762	C₂×C₅₃	1	2	yes	—
253	C₄×#102	C₂×C₅₃	1	1	yes	characteristic
254	C₂×C₅₃	—	4	4	no	—
255	C₂×C₅₃	Q₈×C₂²⋊C₄	1	1	yes	characteristic
256	C₂×C₅₃	C₂×#418	1	1	yes	characteristic
257	C₂×C₅₃	#2898	1	1	yes	characteristic
258	C₂×C₅₃	#3367	1	1	yes	characteristic
259	C₂×C₅₃	#3373	1	1	yes	characteristic
260	C₂×C₅₃	#3384	1	1	yes	characteristic
261	C₂×C₅₃	#3386	1	1	yes	characteristic
262	C₂³×Q₈	—	2	1	no	—
263	D₄×Q₈	—	2	4	no	—
264	D₄×Q₈	C₄×C₅₃	1	8	yes	complemented
265	C₂×C₄⋊Q₈	C₂²×C₅₃	1	2	yes	—
266	C₂×C₄⋊Q₈	C₂²×C₅₃	1	1	yes	characteristic
267	C₂×C₂²⋊Q₈	—	2	1	no	—
268	C₂×C₂²⋊Q₈	—	2	2	no	—
269	C₂×C₄×Q₈	C₂²×C₅₃	1	1	yes	characteristic
270	C₂×C₄×D₄	C₂²×C₅₃	1	1	yes	characteristic
271	C₂×C₄×D₄	C₂²×C₅₃	1	2	yes	—
272	#446	—	2	4	no	—
273	C₂×C₄⋊C₈	C₂²×C₅₃	1	1	yes	characteristic
274	C₂×C₄⋊C₈	C₂²×C₅₃	1	2	yes	—
275	C₂×#102	C₂²×C₅₃	1	2	yes	—
276	C₂×#102	C₂²×C₅₃	1	1	yes	characteristic
277	C₂×#102	C₂²×C₅₃	1	1	yes	characteristic
278	C₂×#97	C₂²×C₅₃	1	2	yes	—
279	C₂×C₄×C₈	C₂²×C₅₃	1	1	yes	characteristic
280	#391	—	2	1	no	—
281	#387	—	2	2	no	—
282	#384	C₂²×C₅₃	1	1	yes	characteristic
283	#382	C₂²×C₅₃	1	2	yes	—
284	C₄×C₄⋊C₄	—	2	1	no	—
285	C₄×C₄⋊C₄	C₂²×C₅₃	1	2	yes	—
286	C₄×C₄⋊C₄	C₂²×C₅₃	1	1	yes	characteristic
287	#340	C₂²×C₅₃	1	2	yes	—
288	#340	C₂²×C₅₃	1	2	yes	—
289	#340	C₂²×C₅₃	1	2	yes	—
290	#340	C₂²×C₅₃	1	1	yes	characteristic
291	#340	C₂²×C₅₃	1	2	yes	—
292	#340	C₂²×C₅₃	1	2	yes	—
293	#340	C₂²×C₅₃	1	1	yes	characteristic
294	C₅₃	#22523	1	1	yes	characteristic, direct product factor, Sylow
295	C₂²×Q₈	—	2	8	no	—
296	C₂²×Q₈	—	2	2	no	—
297	C₂²×Q₈	—	2	1	no	—
298	C₂²×Q₈	—	4	4	no	—
299	C₂²×Q₈	—	4	4	no	—
300	C₂²×Q₈	D₄×C₅₃	1	1	yes	characteristic
301	C₂²×D₄	C₂³×C₅₃	1	1	yes	characteristic
302	C₂³×C₄	—	2	2	no	—
303	C₂³×C₄	—	2	1	no	—
304	C₂²×C₈	C₂³×C₅₃	1	1	yes	characteristic
305	C₂²×C₈	C₂³×C₅₃	1	2	yes	—
306	C₂²×C₈	C₂³×C₅₃	1	1	yes	characteristic
307	C₄⋊Q₈	—	2	2	no	—
308	C₄⋊Q₈	—	2	4	no	—
309	C₄⋊Q₈	C₂×C₄×C₅₃	1	8	yes	—
310	C₄⋊Q₈	C₂×C₄×C₅₃	1	4	yes	—
311	C₂²⋊Q₈	—	2	16	no	—
312	C₂²⋊Q₈	—	2	8	no	—
313	C₄×Q₈	C₂×C₄×C₅₃	1	4	yes	—
314	C₄×Q₈	D₄×C₅₃	1	4	yes	—
315	C₄×D₄	—	2	2	no	—
316	C₄×D₄	—	2	4	no	—
317	C₄×D₄	C₂×C₄×C₅₃	1	4	yes	—
318	C₄×D₄	C₂×C₄×C₅₃	1	8	yes	—
319	C₂×C₄⋊C₄	—	2	2	no	—
320	C₂×C₄⋊C₄	—	2	1	no	—
321	C₂×C₄⋊C₄	—	2	1	no	—
322	C₂×C₄⋊C₄	—	2	1	no	—
323	C₂×C₄⋊C₄	D₄×C₅₃	1	1	yes	characteristic
324	C₂×C₄⋊C₄	D₄×C₅₃	1	2	yes	—
325	C₂×C₄⋊C₄	C₂³×C₅₃	1	1	yes	characteristic
326	C₂×C₄⋊C₄	C₂³×C₅₃	1	2	yes	—
327	C₂×C₄⋊C₄	C₂³×C₅₃	1	1	yes	characteristic
328	C₂×C₄⋊C₄	C₂³×C₅₃	1	2	yes	—
329	C₂×C₄⋊C₄	C₂³×C₅₃	1	1	yes	characteristic
330	C₂×C₂²⋊C₄	—	2	2	no	—
331	C₂×C₂²⋊C₄	—	2	1	no	—
332	C₂×C₄²	—	2	1	no	—
333	C₂×C₄²	—	2	2	no	—
334	C₂×C₄²	C₂³×C₅₃	1	2	yes	—
335	C₂×C₄²	C₂³×C₅₃	1	1	yes	characteristic
336	C₄⋊C₈	—	2	2	no	—
337	C₄⋊C₈	—	2	4	no	—
338	#102	—	2	4	no	—
339	#102	—	2	2	no	—
340	#102	Q₈×C₅₃	1	4	yes	complemented
341	#97	—	2	4	no	—
342	C₄×C₈	—	2	2	no	—
343	#95	—	2	2	no	—
344	#95	—	2	1	no	—
345	#95	—	2	2	no	—
346	#95	—	2	2	no	—
347	C₂⁴	—	2	1	no	—
348	C₂×Q₈	—	2	8	no	—
349	C₂×Q₈	—	2	4	no	—
350	C₂×Q₈	—	2	2	no	—
351	C₂×Q₈	—	2	2	no	—
352	C₂×Q₈	—	4	4	no	—
353	C₂×Q₈	—	4	4	no	—
354	C₂×Q₈	—	4	32	no	—
355	C₂×Q₈	—	4	2	no	—
356	C₂×Q₈	—	4	2	no	—
357	C₂×Q₈	C₂²⋊C₄×C₅₃	1	4	yes	—
358	C₂×D₄	—	2	1	no	—
359	C₂×D₄	—	2	1	no	—
360	C₂×D₄	—	2	1	no	—
361	C₂×D₄	C₂²×C₄×C₅₃	1	2	yes	—
362	C₂×D₄	C₂×Q₈×C₅₃	1	2	yes	—
363	C₂×D₄	#41×C₅₃	1	2	yes	—
364	C₂²×C₄	—	2	1	no	—
365	C₂²×C₄	—	2	2	no	—
366	C₂²×C₄	—	2	1	no	—
367	C₂²×C₄	—	2	8	no	—
368	C₂²×C₄	—	2	4	no	—
369	C₂²×C₄	—	2	1	no	—
370	C₂²×C₄	—	2	2	no	—
371	C₂²×C₄	—	4	4	no	—
372	C₂²×C₄	—	4	4	no	—
373	C₂²×C₄	—	4	2	no	—
374	C₂²×C₄	—	4	2	no	—
375	C₂²×C₄	C₂×D₄×C₅₃	1	2	yes	—
376	C₂²×C₄	C₂×D₄×C₅₃	1	2	yes	—
377	C₂²×C₄	C₂×D₄×C₅₃	1	1	yes	characteristic
378	C₂²×C₄	C₂×D₄×C₅₃	1	1	yes	characteristic
379	C₂²×C₄	C₂⁴×C₅₃	1	1	yes	characteristic
380	C₂×C₈	—	2	1	no	—
381	C₂×C₈	—	2	1	no	—
382	C₂×C₈	—	2	1	no	—
383	C₂×C₈	—	2	2	no	—
384	C₂×C₈	—	2	1	no	—
385	C₂×C₈	—	2	2	no	—
386	C₂×C₈	—	2	1	no	—
387	C₂×C₈	—	2	2	no	—
388	C₂×C₈	C₂×Q₈×C₅₃	1	2	yes	complemented
389	C₄⋊C₄	—	2	4	no	—
390	C₄⋊C₄	—	2	8	no	—
391	C₄⋊C₄	—	2	1	no	—
392	C₄⋊C₄	—	2	2	no	—
393	C₄⋊C₄	—	2	1	no	—
394	C₄⋊C₄	—	2	1	no	—
395	C₄⋊C₄	—	2	2	no	—
396	C₄⋊C₄	—	2	1	no	—
397	C₄⋊C₄	—	2	2	no	—
398	C₄⋊C₄	—	4	2	no	—
399	C₄⋊C₄	—	8	1	no	—
400	C₄⋊C₄	C₂²⋊C₄×C₅₃	1	8	yes	—
401	C₄⋊C₄	C₂²⋊C₄×C₅₃	1	4	yes	—
402	C₄⋊C₄	C₂²×C₄×C₅₃	1	2	yes	—
403	C₄⋊C₄	C₂²×C₄×C₅₃	1	4	yes	—
404	C₄⋊C₄	C₂×Q₈×C₅₃	1	2	yes	complemented
405	C₂²⋊C₄	—	2	4	no	—
406	C₂²⋊C₄	—	2	8	no	—
407	C₄²	—	4	2	no	—
408	C₄²	—	8	2	no	—
409	C₄²	C₂²×C₄×C₅₃	1	4	yes	—
410	C₄²	C₂²×C₄×C₅₃	1	2	yes	—
411	C₄²	C₂×D₄×C₅₃	1	2	yes	—
412	C₄²	C₂×D₄×C₅₃	1	4	yes	—
413	C₂³	—	2	2	no	—
414	C₂³	—	2	2	no	—
415	C₂³	—	2	2	no	—
416	C₂³	—	4	1	no	—
417	C₂³	—	4	1	no	—
418	C₂³	—	4	1	no	—
419	C₂³	—	4	1	no	—
420	C₂³	C₂²×D₄×C₅₃	1	1	yes	characteristic
421	Q₈	—	2	8	no	—
422	Q₈	—	4	16	no	—
423	Q₈	—	4	8	no	—
424	D₄	—	2	2	no	—
425	D₄	—	2	2	no	—
426	D₄	—	2	2	no	—
427	D₄	C₄×Q₈×C₅₃	1	4	yes	—
428	C₂×C₄	—	2	2	no	—
429	C₂×C₄	—	2	2	no	—
430	C₂×C₄	—	2	2	no	—
431	C₂×C₄	—	2	1	no	—
432	C₂×C₄	—	2	2	no	—
433	C₂×C₄	—	2	1	no	—
434	C₂×C₄	—	2	4	no	—
435	C₂×C₄	—	2	2	no	—
436	C₂×C₄	—	2	2	no	—
437	C₂×C₄	—	4	16	no	—
438	C₂×C₄	—	4	1	no	—
439	C₂×C₄	—	4	2	no	—
440	C₂×C₄	—	4	1	no	—
441	C₂×C₄	—	4	2	no	—
442	C₂×C₄	—	4	1	no	—
443	C₂×C₄	—	4	1	no	—
444	C₂×C₄	—	4	8	no	—
445	C₂×C₄	—	4	1	no	—
446	C₂×C₄	—	4	2	no	—
447	C₂×C₄	—	4	1	no	—
448	C₂×C₄	—	4	2	no	—
449	C₂×C₄	—	4	2	no	—
450	C₂×C₄	—	4	1	no	—
451	C₂×C₄	C₂×C₂²⋊C₄×C₅₃	1	2	yes	—
452	C₂×C₄	C₂×C₂²⋊C₄×C₅₃	1	4	yes	—
453	C₂×C₄	C₂×C₂²⋊C₄×C₅₃	1	4	yes	—
454	C₂×C₄	C₂×C₂²⋊C₄×C₅₃	1	2	yes	—
455	C₂×C₄	AGL(1,ℤ/8)×C₅₃	1	2	yes	—
456	C₂×C₄	AGL(1,ℤ/8)×C₅₃	1	2	yes	—
457	C₂×C₄	C₂³×C₄×C₅₃	1	1	yes	characteristic
458	C₂×C₄	C₂²×D₄×C₅₃	1	1	yes	characteristic
459	C₂×C₄	C₂²×Q₈×C₅₃	1	1	yes	characteristic
460	C₂×C₄	C₂×#41×C₅₃	1	1	yes	characteristic
461	C₂×C₄	2₋¹⁺⁴×C₅₃	1	1	yes	characteristic
462	C₂×C₄	2₋¹⁺⁴×C₅₃	1	1	yes	characteristic
463	C₈	—	2	2	no	—
464	C₈	—	4	1	no	—
465	C₈	—	4	2	no	—
466	C₂²	—	2	4	no	—
467	C₂²	—	4	2	no	—
468	C₂²	—	4	2	no	—
469	C₂²	—	4	2	no	—
470	C₂²	—	4	2	no	—
471	C₂²	—	4	2	no	—
472	C₂²	—	4	2	no	—
473	C₂²	C₂²×C₂²⋊C₄×C₅₃	1	1	yes	characteristic
474	C₂²	C₅₃×#548	1	1	yes	characteristic
475	C₂²	D₄×Q₈×C₅₃	1	1	yes	characteristic
476	C₂²	C₂×AGL(1,ℤ/8)×C₅₃	1	1	yes	characteristic
477	C₂²	C₂×AGL(1,ℤ/8)×C₅₃	1	1	yes	characteristic
478	C₂²	C₅₃×#577	1	1	yes	characteristic
479	C₂²	C₅₃×#578	1	1	yes	characteristic
480	C₄	—	2	4	no	—
481	C₄	—	2	4	no	—
482	C₄	—	2	2	no	—
483	C₄	—	4	2	no	—
484	C₄	—	4	4	no	—
485	C₄	—	4	2	no	—
486	C₄	—	8	1	no	—
487	C₄	—	8	2	no	—
488	C₄	C₅₃×#418	1	4	yes	—
489	C₄	C₂×C₄×Q₈×C₅₃	1	1	yes	characteristic
490	C₄	C₅₃×#519	1	1	yes	characteristic
491	C₄	C₅₃×#548	1	1	yes	characteristic
492	C₄	D₄×Q₈×C₅₃	1	1	yes	characteristic
493	C₂	—	4	4	no	—
494	C₂	Q₈×C₂²⋊C₄×C₅₃	1	1	yes	characteristic
495	C₂	C₂×C₅₃×#418	1	1	yes	characteristic
496	C₂	C₅₃×#2898	1	1	yes	characteristic
497	C₂	C₅₃×#3367	1	1	yes	characteristic
498	C₂	C₅₃×#3373	1	1	yes	characteristic
499	C₂	C₅₃×#3384	1	1	yes	characteristic
500	C₂	C₅₃×#3386	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂³×C₅₃	C₂²×D₄
commutator	C₂×C₄	C₂³×C₄×C₅₃
Frattini, cosocle	C₂²×C₄	C₂⁴×C₅₃
prime radical	C₂²×D₄×C₅₃	C₂³
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₅₃ ◁ C₂×C₄×Q₈×C₅₃ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂×C₅₃×#446	C₂	semidirect
3. C₅₃×#2355	C₂	semidirect
4. C₅₃×#1993	C₂	semidirect
5. C₅₃×#1990	C₂	semidirect
6. C₅₃×#1988	C₂	semidirect
7. C₅₃×#1987	C₂	semidirect
14. D₄×Q₈×C₅₃	C₄	semidirect
89. #102×C₅₃	Q₈	semidirect
137. C₂×C₈×C₅₃	C₂×Q₈	semidirect
153. C₄⋊C₄×C₅₃	C₂×Q₈	semidirect
215. #22523	C₅₃	direct (N × H)
244. C₂×#446	C₂×C₅₃	semidirect
245. #2355	C₂×C₅₃	semidirect
246. #1993	C₂×C₅₃	semidirect
247. #1990	C₂×C₅₃	semidirect
248. #1988	C₂×C₅₃	semidirect
249. #1987	C₂×C₅₃	semidirect
264. D₄×Q₈	C₄×C₅₃	semidirect
294. C₅₃	#22523	direct (N × H)
340. #102	Q₈×C₅₃	semidirect
388. C₂×C₈	C₂×Q₈×C₅₃	semidirect
404. C₄⋊C₄	C₂×Q₈×C₅₃	semidirect

Maximal subgroups

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

subgroup	index
C₂×D₄×Q₈×C₅₃	2
C₂×C₅₃×#446	2
C₅₃×#2355	2
C₅₃×#1993	2
C₅₃×#1990	2
C₅₃×#1988	2
C₅₃×#1987	2
C₅₃×#1804	2
C₅₃×#1802	2
C₅₃×#1762	2
C₄×#102×C₅₃	2
#22523	53

Subgroups — C53×#22523 #27230035

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅₃×#22523 #27230035