Subgroups — C₅₃×#22544 #27230056

Order 13568.

1098 conjugacy classes of subgroups.

The subgroup lattice has 508 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₅₃×#556	C₂	1	1	yes	maximal, characteristic
2	C₂×C₅₃×#446	C₂	1	1	yes	maximal, characteristic, complemented
3	Q₈×C₄⋊C₄×C₅₃	C₂	1	1	yes	maximal, characteristic, complemented
4	C₅₃×#1983	C₂	1	2	yes	maximal
5	C₅₃×#1981	C₂	1	1	yes	maximal, characteristic, complemented
6	C₅₃×#1959	C₂	1	2	yes	maximal
7	C₅₃×#1956	C₂	1	1	yes	maximal, characteristic, complemented
8	C₅₃×#1803	C₂	1	2	yes	maximal, complemented
9	C₅₃×#1802	C₂	1	1	yes	maximal, characteristic
10	C₅₃×#1763	C₂	1	2	yes	maximal, complemented
11	C₄×#102×C₅₃	C₂	1	1	yes	maximal, characteristic
12	C₅₃×#556	—	2	4	no	—
13	C₅₃×#556	C₄	1	8	yes	complemented
14	C₂×C₄⋊Q₈×C₅₃	C₂²	1	1	yes	characteristic
15	C₂×#126×C₅₃	—	2	2	no	—
16	C₂×#124×C₅₃	C₂²	1	2	yes	—
17	C₂×C₂²⋊Q₈×C₅₃	—	2	1	no	—
18	C₂×C₄×Q₈×C₅₃	—	2	1	no	—
19	C₂×C₄×Q₈×C₅₃	C₂²	1	2	yes	—
20	C₂×C₄×Q₈×C₅₃	C₂²	1	1	yes	characteristic
21	C₂×C₄×D₄×C₅₃	C₂²	1	1	yes	characteristic
22	C₂×#117×C₅₃	—	2	1	no	—
23	C₅₃×#446	—	2	4	no	—
24	C₂×C₄⋊C₈×C₅₃	C₂²	1	1	yes	characteristic
25	C₂×C₄⋊C₈×C₅₃	C₂²	1	2	yes	—
26	C₂×#103×C₅₃	C₂²	1	2	yes	—
27	C₂×#103×C₅₃	C₂²	1	2	yes	—
28	C₂×#103×C₅₃	C₂²	1	2	yes	—
29	C₂×#103×C₅₃	C₂²	1	2	yes	—
30	C₂×#102×C₅₃	C₂²	1	1	yes	characteristic
31	C₂×#102×C₅₃	C₂²	1	1	yes	characteristic
32	C₂×#102×C₅₃	C₂²	1	2	yes	—
33	C₂×#97×C₅₃	C₂²	1	2	yes	—
34	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
35	C₅₃×#389	—	2	1	no	—
36	C₅₃×#389	—	2	2	no	—
37	C₅₃×#384	C₂²	1	1	yes	characteristic
38	C₅₃×#384	C₂²	1	2	yes	—
39	C₄×C₄⋊C₄×C₅₃	C₂²	1	1	yes	characteristic
40	C₄×C₄⋊C₄×C₅₃	C₂²	1	2	yes	—
41	C₅₃×#340	C₂²	1	1	yes	characteristic
42	C₅₃×#340	C₂²	1	1	yes	characteristic
43	C₅₃×#340	C₂²	1	2	yes	—
44	C₂²×Q₈×C₅₃	D₄	1	1	yes	characteristic
45	C₂²×Q₈×C₅₃	C₂³	1	2	yes	—
46	C₂²×D₄×C₅₃	C₂³	1	1	yes	characteristic
47	C₂³×C₄×C₅₃	—	2	1	no	—
48	C₂²×C₈×C₅₃	C₂³	1	1	yes	characteristic
49	C₂²×C₈×C₅₃	C₂³	1	2	yes	—
50	C₂²×C₈×C₅₃	C₂³	1	1	yes	characteristic
51	C₄⋊Q₈×C₅₃	—	2	2	no	—
52	C₄⋊Q₈×C₅₃	C₂×C₄	1	4	yes	—
53	#126×C₅₃	—	2	16	no	—
54	#124×C₅₃	—	2	4	no	—
55	#124×C₅₃	C₂×C₄	1	8	yes	—
56	C₂²⋊Q₈×C₅₃	—	2	8	no	—
57	C₄×Q₈×C₅₃	—	2	4	no	—
58	C₄×Q₈×C₅₃	—	4	4	no	—
59	C₄×Q₈×C₅₃	C₂×C₄	1	4	yes	—
60	C₄×Q₈×C₅₃	C₂×C₄	1	8	yes	—
61	C₄×Q₈×C₅₃	D₄	1	4	yes	—
62	C₄×D₄×C₅₃	—	2	2	no	—
63	C₄×D₄×C₅₃	C₂×C₄	1	4	yes	—
64	#117×C₅₃	—	2	8	no	—
65	C₂×C₄⋊C₄×C₅₃	—	2	1	no	—
66	C₂×C₄⋊C₄×C₅₃	—	2	2	no	—
67	C₂×C₄⋊C₄×C₅₃	—	2	1	no	—
68	C₂×C₄⋊C₄×C₅₃	—	2	2	no	—
69	C₂×C₄⋊C₄×C₅₃	—	2	1	no	—
70	C₂×C₄⋊C₄×C₅₃	—	2	2	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	1	yes	characteristic
78	C₂×C₂²⋊C₄×C₅₃	—	2	2	no	—
79	C₂×C₂²⋊C₄×C₅₃	—	2	1	no	—
80	C₂×C₂²⋊C₄×C₅₃	—	2	2	no	—
81	C₂×C₄²×C₅₃	—	2	2	no	—
82	C₂×C₄²×C₅₃	—	2	2	no	—
83	C₂×C₄²×C₅₃	—	2	1	no	—
84	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
85	C₂×C₄²×C₅₃	C₂³	1	2	yes	—
86	C₄⋊C₈×C₅₃	—	2	2	no	—
87	C₄⋊C₈×C₅₃	—	2	4	no	—
88	#103×C₅₃	—	2	4	no	—
89	#103×C₅₃	—	2	4	no	—
90	#103×C₅₃	—	4	2	no	—
91	#103×C₅₃	—	4	2	no	—
92	#102×C₅₃	—	2	2	no	—
93	#102×C₅₃	—	2	2	no	—
94	#102×C₅₃	—	4	2	no	—
95	#97×C₅₃	—	2	4	no	—
96	C₄×C₈×C₅₃	—	2	2	no	—
97	#95×C₅₃	—	2	2	no	—
98	#95×C₅₃	—	2	1	no	—
99	C₂⁴×C₅₃	—	2	1	no	—
100	C₂×Q₈×C₅₃	—	2	2	no	—
101	C₂×Q₈×C₅₃	—	2	2	no	—
102	C₂×Q₈×C₅₃	—	2	2	no	—
103	C₂×Q₈×C₅₃	—	2	2	no	—
104	C₂×Q₈×C₅₃	—	2	2	no	—
105	C₂×Q₈×C₅₃	—	2	2	no	—
106	C₂×Q₈×C₅₃	—	2	2	no	—
107	C₂×Q₈×C₅₃	C₂²⋊C₄	1	4	yes	—
108	C₂×Q₈×C₅₃	C₂²×C₄	1	4	yes	—
109	C₂×D₄×C₅₃	—	2	1	no	—
110	C₂×D₄×C₅₃	—	2	1	no	—
111	C₂×D₄×C₅₃	—	2	1	no	—
112	C₂×D₄×C₅₃	—	2	1	no	—
113	C₂×D₄×C₅₃	—	2	1	no	—
114	C₂×D₄×C₅₃	C₂²×C₄	1	2	yes	—
115	C₂²×C₄×C₅₃	—	2	1	no	—
116	C₂²×C₄×C₅₃	—	2	1	no	—
117	C₂²×C₄×C₅₃	—	2	2	no	—
118	C₂²×C₄×C₅₃	—	2	4	no	—
119	C₂²×C₄×C₅₃	—	2	1	no	—
120	C₂²×C₄×C₅₃	—	2	2	no	—
121	C₂²×C₄×C₅₃	—	4	2	no	—
122	C₂²×C₄×C₅₃	—	4	2	no	—
123	C₂²×C₄×C₅₃	C₂×D₄	1	2	yes	—
124	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
125	C₂²×C₄×C₅₃	C₂×D₄	1	2	yes	—
126	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
127	C₂²×C₄×C₅₃	C₂⁴	1	1	yes	characteristic
128	C₂×C₈×C₅₃	—	2	1	no	—
129	C₂×C₈×C₅₃	—	2	1	no	—
130	C₂×C₈×C₅₃	—	2	1	no	—
131	C₂×C₈×C₅₃	—	2	2	no	—
132	C₂×C₈×C₅₃	—	2	1	no	—
133	C₂×C₈×C₅₃	—	2	2	no	—
134	C₂×C₈×C₅₃	—	2	2	no	—
135	C₂×C₈×C₅₃	—	2	1	no	—
136	C₂×C₈×C₅₃	#41	1	2	yes	—
137	C₄⋊C₄×C₅₃	—	2	4	no	—
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	1	no	—
142	C₄⋊C₄×C₅₃	—	2	1	no	—
143	C₄⋊C₄×C₅₃	—	2	2	no	—
144	C₄⋊C₄×C₅₃	—	2	2	no	—
145	C₄⋊C₄×C₅₃	—	2	1	no	—
146	C₄⋊C₄×C₅₃	—	4	4	no	—
147	C₄⋊C₄×C₅₃	—	4	2	no	—
148	C₄⋊C₄×C₅₃	—	8	2	no	—
149	C₄⋊C₄×C₅₃	—	8	1	no	—
150	C₄⋊C₄×C₅₃	C₂²⋊C₄	1	4	yes	—
151	C₄⋊C₄×C₅₃	C₂²⋊C₄	1	8	yes	—
152	C₄⋊C₄×C₅₃	C₂²×C₄	1	2	yes	—
153	C₄⋊C₄×C₅₃	#41	1	2	yes	—
154	C₂²⋊C₄×C₅₃	—	2	8	no	—
155	C₂²⋊C₄×C₅₃	—	2	4	no	—
156	C₂²⋊C₄×C₅₃	—	2	8	no	—
157	C₄²×C₅₃	—	2	8	no	—
158	C₄²×C₅₃	—	4	4	no	—
159	C₄²×C₅₃	—	4	2	no	—
160	C₄²×C₅₃	C₂²×C₄	1	2	yes	—
161	C₄²×C₅₃	C₂²×C₄	1	4	yes	—
162	C₄²×C₅₃	C₂×D₄	1	2	yes	—
163	C₄²×C₅₃	C₂×D₄	1	4	yes	—
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	—	4	1	no	—
170	C₂³×C₅₃	—	4	1	no	—
171	C₂³×C₅₃	C₂²×D₄	1	1	yes	characteristic
172	Q₈×C₅₃	—	2	8	no	—
173	Q₈×C₅₃	—	2	4	no	—
174	Q₈×C₅₃	—	2	4	no	—
175	Q₈×C₅₃	—	4	2	no	—
176	Q₈×C₅₃	—	4	2	no	—
177	D₄×C₅₃	—	2	2	no	—
178	D₄×C₅₃	—	2	2	no	—
179	D₄×C₅₃	—	4	1	no	—
180	D₄×C₅₃	—	4	1	no	—
181	C₂×C₄×C₅₃	—	2	2	no	—
182	C₂×C₄×C₅₃	—	2	2	no	—
183	C₂×C₄×C₅₃	—	2	2	no	—
184	C₂×C₄×C₅₃	—	2	1	no	—
185	C₂×C₄×C₅₃	—	2	2	no	—
186	C₂×C₄×C₅₃	—	2	1	no	—
187	C₂×C₄×C₅₃	—	2	2	no	—
188	C₂×C₄×C₅₃	—	2	4	no	—
189	C₂×C₄×C₅₃	—	2	2	no	—
190	C₂×C₄×C₅₃	—	4	2	no	—
191	C₂×C₄×C₅₃	—	4	2	no	—
192	C₂×C₄×C₅₃	—	4	1	no	—
193	C₂×C₄×C₅₃	—	4	1	no	—
194	C₂×C₄×C₅₃	—	4	1	no	—
195	C₂×C₄×C₅₃	—	4	1	no	—
196	C₂×C₄×C₅₃	—	4	8	no	—
197	C₂×C₄×C₅₃	—	4	1	no	—
198	C₂×C₄×C₅₃	—	4	2	no	—
199	C₂×C₄×C₅₃	—	4	2	no	—
200	C₂×C₄×C₅₃	—	4	1	no	—
201	C₂×C₄×C₅₃	—	4	1	no	—
202	C₂×C₄×C₅₃	—	4	2	no	—
203	C₂×C₄×C₅₃	C₂×C₂²⋊C₄	1	4	yes	—
204	C₂×C₄×C₅₃	C₂×C₂²⋊C₄	1	4	yes	—
205	C₂×C₄×C₅₃	C₂×C₂²⋊C₄	1	2	yes	—
206	C₂×C₄×C₅₃	C₂×C₂²⋊C₄	1	2	yes	—
207	C₂×C₄×C₅₃	AGL(1,ℤ/8)	1	2	yes	—
208	C₂×C₄×C₅₃	#137	1	2	yes	—
209	C₂×C₄×C₅₃	C₂³×C₄	1	1	yes	characteristic
210	C₂×C₄×C₅₃	C₂²×D₄	1	1	yes	characteristic
211	C₂×C₄×C₅₃	C₂×#41	1	1	yes	characteristic
212	C₂×C₄×C₅₃	C₂×#41	1	1	yes	characteristic
213	C₂×C₄×C₅₃	H(5,𝔽₂)	1	1	yes	characteristic
214	C₂×C₄×C₅₃	2₋¹⁺⁴	1	1	yes	characteristic
215	C₈×C₅₃	—	2	2	no	—
216	C₈×C₅₃	—	4	1	no	—
217	C₈×C₅₃	—	4	2	no	—
218	#22544	C₅₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
219	C₂²×C₅₃	—	4	2	no	—
220	C₂²×C₅₃	—	4	2	no	—
221	C₂²×C₅₃	—	4	2	no	—
222	C₂²×C₅₃	—	8	1	no	—
223	C₂²×C₅₃	—	8	1	no	—
224	C₂²×C₅₃	—	8	1	no	—
225	C₂²×C₅₃	—	8	1	no	—
226	C₂²×C₅₃	C₂²×C₂²⋊C₄	1	1	yes	characteristic
227	C₂²×C₅₃	#548	1	1	yes	characteristic
228	C₂²×C₅₃	#550	1	1	yes	characteristic
229	C₂²×C₅₃	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
230	C₂²×C₅₃	C₂×#137	1	1	yes	characteristic
231	C₂²×C₅₃	#577	1	1	yes	characteristic
232	C₂²×C₅₃	#578	1	1	yes	characteristic
233	C₄×C₅₃	—	2	4	no	—
234	C₄×C₅₃	—	2	4	no	—
235	C₄×C₅₃	—	2	2	no	—
236	C₄×C₅₃	—	4	4	no	—
237	C₄×C₅₃	—	4	2	no	—
238	C₄×C₅₃	—	4	2	no	—
239	C₄×C₅₃	—	8	1	no	—
240	C₄×C₅₃	—	8	2	no	—
241	C₄×C₅₃	#417	1	4	yes	—
242	C₄×C₅₃	C₄×#41	1	1	yes	characteristic
243	C₄×C₅₃	#520	1	1	yes	characteristic
244	C₄×C₅₃	#548	1	1	yes	characteristic
245	C₄×C₅₃	#550	1	1	yes	characteristic
246	C₂×#556	C₂×C₅₃	1	1	yes	characteristic
247	C₂×#446	C₂×C₅₃	1	1	yes	characteristic, complemented
248	Q₈×C₄⋊C₄	C₂×C₅₃	1	1	yes	characteristic, complemented
249	#1983	C₂×C₅₃	1	2	yes	—
250	#1981	C₂×C₅₃	1	1	yes	characteristic, complemented
251	#1959	C₂×C₅₃	1	2	yes	—
252	#1956	C₂×C₅₃	1	1	yes	characteristic, complemented
253	#1803	C₂×C₅₃	1	2	yes	complemented
254	#1802	C₂×C₅₃	1	1	yes	characteristic
255	#1763	C₂×C₅₃	1	2	yes	complemented
256	C₄×#102	C₂×C₅₃	1	1	yes	characteristic
257	C₂×C₅₃	—	8	2	no	—
258	C₂×C₅₃	#2341	1	1	yes	characteristic
259	C₂×C₅₃	C₂×#417	1	1	yes	characteristic
260	C₂×C₅₃	#2898	1	1	yes	characteristic
261	C₂×C₅₃	#3367	1	1	yes	characteristic
262	C₂×C₅₃	#3369	1	1	yes	characteristic
263	C₂×C₅₃	#3397	1	1	yes	characteristic
264	C₂×C₅₃	#3398	1	1	yes	characteristic
265	#556	—	2	4	no	—
266	#556	C₄×C₅₃	1	8	yes	complemented
267	C₂×C₄⋊Q₈	C₂²×C₅₃	1	1	yes	characteristic
268	C₂×#126	—	2	2	no	—
269	C₂×#124	C₂²×C₅₃	1	2	yes	—
270	C₂×C₂²⋊Q₈	—	2	1	no	—
271	C₂×C₄×Q₈	—	2	1	no	—
272	C₂×C₄×Q₈	C₂²×C₅₃	1	2	yes	—
273	C₂×C₄×Q₈	C₂²×C₅₃	1	1	yes	characteristic
274	C₂×C₄×D₄	C₂²×C₅₃	1	1	yes	characteristic
275	C₂×#117	—	2	1	no	—
276	#446	—	2	4	no	—
277	C₂×C₄⋊C₈	C₂²×C₅₃	1	1	yes	characteristic
278	C₂×C₄⋊C₈	C₂²×C₅₃	1	2	yes	—
279	C₂×#103	C₂²×C₅₃	1	2	yes	—
280	C₂×#103	C₂²×C₅₃	1	2	yes	—
281	C₂×#103	C₂²×C₅₃	1	2	yes	—
282	C₂×#103	C₂²×C₅₃	1	2	yes	—
283	C₂×#102	C₂²×C₅₃	1	1	yes	characteristic
284	C₂×#102	C₂²×C₅₃	1	1	yes	characteristic
285	C₂×#102	C₂²×C₅₃	1	2	yes	—
286	C₂×#97	C₂²×C₅₃	1	2	yes	—
287	C₂×C₄×C₈	C₂²×C₅₃	1	1	yes	characteristic
288	#389	—	2	1	no	—
289	#389	—	2	2	no	—
290	#384	C₂²×C₅₃	1	1	yes	characteristic
291	#384	C₂²×C₅₃	1	2	yes	—
292	C₄×C₄⋊C₄	C₂²×C₅₃	1	1	yes	characteristic
293	C₄×C₄⋊C₄	C₂²×C₅₃	1	2	yes	—
294	#340	C₂²×C₅₃	1	2	yes	—
295	#340	C₂²×C₅₃	1	1	yes	characteristic
296	#340	C₂²×C₅₃	1	1	yes	characteristic
297	C₅₃	#22544	1	1	yes	characteristic, direct product factor, Sylow
298	C₂²×Q₈	D₄×C₅₃	1	1	yes	characteristic
299	C₂²×Q₈	C₂³×C₅₃	1	2	yes	—
300	C₂²×D₄	C₂³×C₅₃	1	1	yes	characteristic
301	C₂³×C₄	—	2	1	no	—
302	C₂²×C₈	C₂³×C₅₃	1	1	yes	characteristic
303	C₂²×C₈	C₂³×C₅₃	1	2	yes	—
304	C₂²×C₈	C₂³×C₅₃	1	1	yes	characteristic
305	C₄⋊Q₈	—	2	2	no	—
306	C₄⋊Q₈	C₂×C₄×C₅₃	1	4	yes	—
307	#126	—	2	16	no	—
308	#124	—	2	4	no	—
309	#124	C₂×C₄×C₅₃	1	8	yes	—
310	C₂²⋊Q₈	—	2	8	no	—
311	C₄×Q₈	—	2	4	no	—
312	C₄×Q₈	—	4	4	no	—
313	C₄×Q₈	C₂×C₄×C₅₃	1	8	yes	—
314	C₄×Q₈	C₂×C₄×C₅₃	1	4	yes	—
315	C₄×Q₈	D₄×C₅₃	1	4	yes	—
316	C₄×D₄	—	2	2	no	—
317	C₄×D₄	C₂×C₄×C₅₃	1	4	yes	—
318	#117	—	2	8	no	—
319	C₂×C₄⋊C₄	—	2	1	no	—
320	C₂×C₄⋊C₄	—	2	2	no	—
321	C₂×C₄⋊C₄	—	2	1	no	—
322	C₂×C₄⋊C₄	—	2	2	no	—
323	C₂×C₄⋊C₄	—	2	1	no	—
324	C₂×C₄⋊C₄	—	2	2	no	—
325	C₂×C₄⋊C₄	—	2	1	no	—
326	C₂×C₄⋊C₄	D₄×C₅₃	1	1	yes	characteristic
327	C₂×C₄⋊C₄	D₄×C₅₃	1	2	yes	—
328	C₂×C₄⋊C₄	C₂³×C₅₃	1	1	yes	characteristic
329	C₂×C₄⋊C₄	C₂³×C₅₃	1	2	yes	—
330	C₂×C₄⋊C₄	C₂³×C₅₃	1	1	yes	characteristic
331	C₂×C₄⋊C₄	C₂³×C₅₃	1	1	yes	characteristic
332	C₂×C₂²⋊C₄	—	2	2	no	—
333	C₂×C₂²⋊C₄	—	2	1	no	—
334	C₂×C₂²⋊C₄	—	2	2	no	—
335	C₂×C₄²	—	2	2	no	—
336	C₂×C₄²	—	2	2	no	—
337	C₂×C₄²	—	2	1	no	—
338	C₂×C₄²	C₂³×C₅₃	1	1	yes	characteristic
339	C₂×C₄²	C₂³×C₅₃	1	2	yes	—
340	C₄⋊C₈	—	2	2	no	—
341	C₄⋊C₈	—	2	4	no	—
342	#103	—	2	4	no	—
343	#103	—	2	4	no	—
344	#103	—	4	2	no	—
345	#103	—	4	2	no	—
346	#102	—	2	2	no	—
347	#102	—	2	2	no	—
348	#102	—	4	2	no	—
349	#97	—	2	4	no	—
350	C₄×C₈	—	2	2	no	—
351	#95	—	2	2	no	—
352	#95	—	2	1	no	—
353	C₂⁴	—	2	1	no	—
354	C₂×Q₈	—	2	2	no	—
355	C₂×Q₈	—	2	2	no	—
356	C₂×Q₈	—	2	2	no	—
357	C₂×Q₈	—	2	2	no	—
358	C₂×Q₈	—	2	2	no	—
359	C₂×Q₈	—	2	2	no	—
360	C₂×Q₈	—	2	2	no	—
361	C₂×Q₈	C₂²⋊C₄×C₅₃	1	4	yes	—
362	C₂×Q₈	C₂²×C₄×C₅₃	1	4	yes	—
363	C₂×D₄	—	2	1	no	—
364	C₂×D₄	—	2	1	no	—
365	C₂×D₄	—	2	1	no	—
366	C₂×D₄	—	2	1	no	—
367	C₂×D₄	—	2	1	no	—
368	C₂×D₄	C₂²×C₄×C₅₃	1	2	yes	—
369	C₂²×C₄	—	2	1	no	—
370	C₂²×C₄	—	2	1	no	—
371	C₂²×C₄	—	2	2	no	—
372	C₂²×C₄	—	2	4	no	—
373	C₂²×C₄	—	2	1	no	—
374	C₂²×C₄	—	2	2	no	—
375	C₂²×C₄	—	4	2	no	—
376	C₂²×C₄	—	4	2	no	—
377	C₂²×C₄	C₂×D₄×C₅₃	1	2	yes	—
378	C₂²×C₄	C₂×D₄×C₅₃	1	1	yes	characteristic
379	C₂²×C₄	C₂×D₄×C₅₃	1	2	yes	—
380	C₂²×C₄	C₂×D₄×C₅₃	1	1	yes	characteristic
381	C₂²×C₄	C₂⁴×C₅₃	1	1	yes	characteristic
382	C₂×C₈	—	2	1	no	—
383	C₂×C₈	—	2	1	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	—	2	2	no	—
389	C₂×C₈	—	2	1	no	—
390	C₂×C₈	#41×C₅₃	1	2	yes	—
391	C₄⋊C₄	—	2	4	no	—
392	C₄⋊C₄	—	2	8	no	—
393	C₄⋊C₄	—	2	4	no	—
394	C₄⋊C₄	—	2	1	no	—
395	C₄⋊C₄	—	2	1	no	—
396	C₄⋊C₄	—	2	1	no	—
397	C₄⋊C₄	—	2	2	no	—
398	C₄⋊C₄	—	2	2	no	—
399	C₄⋊C₄	—	2	1	no	—
400	C₄⋊C₄	—	4	4	no	—
401	C₄⋊C₄	—	4	2	no	—
402	C₄⋊C₄	—	8	2	no	—
403	C₄⋊C₄	—	8	1	no	—
404	C₄⋊C₄	C₂²⋊C₄×C₅₃	1	4	yes	—
405	C₄⋊C₄	C₂²⋊C₄×C₅₃	1	8	yes	—
406	C₄⋊C₄	C₂²×C₄×C₅₃	1	2	yes	—
407	C₄⋊C₄	#41×C₅₃	1	2	yes	—
408	C₂²⋊C₄	—	2	8	no	—
409	C₂²⋊C₄	—	2	4	no	—
410	C₂²⋊C₄	—	2	8	no	—
411	C₄²	—	2	8	no	—
412	C₄²	—	4	4	no	—
413	C₄²	—	4	2	no	—
414	C₄²	C₂²×C₄×C₅₃	1	2	yes	—
415	C₄²	C₂²×C₄×C₅₃	1	4	yes	—
416	C₄²	C₂×D₄×C₅₃	1	2	yes	—
417	C₄²	C₂×D₄×C₅₃	1	4	yes	—
418	C₂³	—	2	2	no	—
419	C₂³	—	4	1	no	—
420	C₂³	—	4	1	no	—
421	C₂³	—	4	1	no	—
422	C₂³	—	4	1	no	—
423	C₂³	—	4	1	no	—
424	C₂³	—	4	1	no	—
425	C₂³	C₂²×D₄×C₅₃	1	1	yes	characteristic
426	Q₈	—	2	8	no	—
427	Q₈	—	2	4	no	—
428	Q₈	—	2	4	no	—
429	Q₈	—	4	2	no	—
430	Q₈	—	4	2	no	—
431	D₄	—	2	2	no	—
432	D₄	—	2	2	no	—
433	D₄	—	4	1	no	—
434	D₄	—	4	1	no	—
435	C₂×C₄	—	2	2	no	—
436	C₂×C₄	—	2	2	no	—
437	C₂×C₄	—	2	2	no	—
438	C₂×C₄	—	2	1	no	—
439	C₂×C₄	—	2	2	no	—
440	C₂×C₄	—	2	1	no	—
441	C₂×C₄	—	2	2	no	—
442	C₂×C₄	—	2	4	no	—
443	C₂×C₄	—	2	2	no	—
444	C₂×C₄	—	4	2	no	—
445	C₂×C₄	—	4	2	no	—
446	C₂×C₄	—	4	1	no	—
447	C₂×C₄	—	4	1	no	—
448	C₂×C₄	—	4	1	no	—
449	C₂×C₄	—	4	1	no	—
450	C₂×C₄	—	4	8	no	—
451	C₂×C₄	—	4	1	no	—
452	C₂×C₄	—	4	2	no	—
453	C₂×C₄	—	4	2	no	—
454	C₂×C₄	—	4	1	no	—
455	C₂×C₄	—	4	1	no	—
456	C₂×C₄	—	4	2	no	—
457	C₂×C₄	C₂×C₂²⋊C₄×C₅₃	1	4	yes	—
458	C₂×C₄	C₂×C₂²⋊C₄×C₅₃	1	2	yes	—
459	C₂×C₄	C₂×C₂²⋊C₄×C₅₃	1	4	yes	—
460	C₂×C₄	C₂×C₂²⋊C₄×C₅₃	1	2	yes	—
461	C₂×C₄	AGL(1,ℤ/8)×C₅₃	1	2	yes	—
462	C₂×C₄	#137×C₅₃	1	2	yes	—
463	C₂×C₄	C₂³×C₄×C₅₃	1	1	yes	characteristic
464	C₂×C₄	C₂²×D₄×C₅₃	1	1	yes	characteristic
465	C₂×C₄	C₂×#41×C₅₃	1	1	yes	characteristic
466	C₂×C₄	C₂×#41×C₅₃	1	1	yes	characteristic
467	C₂×C₄	H(5,𝔽₂)×C₅₃	1	1	yes	characteristic
468	C₂×C₄	2₋¹⁺⁴×C₅₃	1	1	yes	characteristic
469	C₈	—	2	2	no	—
470	C₈	—	4	1	no	—
471	C₈	—	4	2	no	—
472	C₂²	—	4	2	no	—
473	C₂²	—	4	2	no	—
474	C₂²	—	4	2	no	—
475	C₂²	—	8	1	no	—
476	C₂²	—	8	1	no	—
477	C₂²	—	8	1	no	—
478	C₂²	—	8	1	no	—
479	C₂²	C₂²×C₂²⋊C₄×C₅₃	1	1	yes	characteristic
480	C₂²	C₅₃×#548	1	1	yes	characteristic
481	C₂²	C₅₃×#550	1	1	yes	characteristic
482	C₂²	C₂×AGL(1,ℤ/8)×C₅₃	1	1	yes	characteristic
483	C₂²	C₂×#137×C₅₃	1	1	yes	characteristic
484	C₂²	C₅₃×#577	1	1	yes	characteristic
485	C₂²	C₅₃×#578	1	1	yes	characteristic
486	C₄	—	2	4	no	—
487	C₄	—	2	4	no	—
488	C₄	—	2	2	no	—
489	C₄	—	4	4	no	—
490	C₄	—	4	2	no	—
491	C₄	—	4	2	no	—
492	C₄	—	8	1	no	—
493	C₄	—	8	2	no	—
494	C₄	C₅₃×#417	1	4	yes	—
495	C₄	C₄×#41×C₅₃	1	1	yes	characteristic
496	C₄	C₅₃×#520	1	1	yes	characteristic
497	C₄	C₅₃×#548	1	1	yes	characteristic
498	C₄	C₅₃×#550	1	1	yes	characteristic
499	C₂	—	8	2	no	—
500	C₂	C₅₃×#2341	1	1	yes	characteristic
501	C₂	C₂×C₅₃×#417	1	1	yes	characteristic
502	C₂	C₅₃×#2898	1	1	yes	characteristic
503	C₂	C₅₃×#3367	1	1	yes	characteristic
504	C₂	C₅₃×#3369	1	1	yes	characteristic
505	C₂	C₅₃×#3397	1	1	yes	characteristic
506	C₂	C₅₃×#3398	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. Q₈×C₄⋊C₄×C₅₃	C₂	semidirect
5. C₅₃×#1981	C₂	semidirect
7. C₅₃×#1956	C₂	semidirect
8. C₅₃×#1803	C₂	semidirect
10. C₅₃×#1763	C₂	semidirect
13. C₅₃×#556	C₄	semidirect
218. #22544	C₅₃	direct (N × H)
247. C₂×#446	C₂×C₅₃	semidirect
248. Q₈×C₄⋊C₄	C₂×C₅₃	semidirect
250. #1981	C₂×C₅₃	semidirect
252. #1956	C₂×C₅₃	semidirect
253. #1803	C₂×C₅₃	semidirect
255. #1763	C₂×C₅₃	semidirect
266. #556	C₄×C₅₃	semidirect
297. C₅₃	#22544	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×C₅₃×#556	2
C₂×C₅₃×#446	2
Q₈×C₄⋊C₄×C₅₃	2
C₅₃×#1983	2
C₅₃×#1981	2
C₅₃×#1959	2
C₅₃×#1956	2
C₅₃×#1803	2
C₅₃×#1802	2
C₅₃×#1763	2
C₄×#102×C₅₃	2
#22544	53

Subgroups — C53×#22544 #27230056

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅₃×#22544 #27230056