Subgroups — C₅₃×#22969 #27230481

Order 13568.

886 conjugacy classes of subgroups.

The subgroup lattice has 468 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₅₃×#476	C₂	1	1	yes	maximal, characteristic
2	C₅₃×#2369	C₂	1	1	yes	maximal, characteristic, complemented
3	D₄×C₄⋊C₄×C₅₃	C₂	1	1	yes	maximal, characteristic
4	C₅₃×#1990	C₂	1	1	yes	maximal, characteristic, complemented
5	C₅₃×#1947	C₂	1	1	yes	maximal, characteristic, complemented
6	C₅₃×#1944	C₂	1	2	yes	maximal, complemented
7	C₅₃×#1920	C₂	1	2	yes	maximal, complemented
8	C₅₃×#1862	C₂	1	2	yes	maximal
9	C₅₃×#1770	C₂	1	1	yes	maximal, characteristic, complemented
10	C₅₃×#1762	C₂	1	2	yes	maximal
11	C₄×#102×C₅₃	C₂	1	1	yes	maximal, characteristic
12	C₂×C₄⋊Q₈×C₅₃	C₂²	1	1	yes	characteristic
13	C₂×C₄×D₄×C₅₃	C₂²	1	2	yes	—
14	C₂×C₄×D₄×C₅₃	C₂²	1	1	yes	characteristic
15	C₂²×C₄⋊C₄×C₅₃	—	2	1	no	—
16	C₅₃×#476	C₄	1	8	yes	complemented
17	C₂×#107×C₅₃	C₂²	1	2	yes	—
18	C₂×#106×C₅₃	C₂²	1	2	yes	—
19	C₂×C₄⋊C₈×C₅₃	C₂²	1	1	yes	characteristic
20	C₂×#102×C₅₃	C₂²	1	1	yes	characteristic
21	C₂×#102×C₅₃	C₂²	1	2	yes	—
22	C₂×#102×C₅₃	C₂²	1	1	yes	characteristic
23	C₂×#97×C₅₃	C₂²	1	2	yes	—
24	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
25	C₅₃×#391	—	2	2	no	—
26	C₅₃×#389	—	2	1	no	—
27	C₅₃×#389	C₂²	1	2	yes	—
28	C₅₃×#389	C₂²	1	2	yes	—
29	C₅₃×#385	—	2	2	no	—
30	C₅₃×#384	C₂²	1	1	yes	characteristic
31	C₅₃×#380	—	2	1	no	—
32	C₄×C₄⋊C₄×C₅₃	—	2	1	no	—
33	C₄×C₄⋊C₄×C₅₃	C₂²	1	2	yes	—
34	C₄×C₄⋊C₄×C₅₃	C₂²	1	1	yes	characteristic
35	C₄×C₄⋊C₄×C₅₃	C₂²	1	1	yes	characteristic
36	C₅₃×#340	C₂²	1	1	yes	characteristic
37	C₅₃×#340	C₂²	1	2	yes	—
38	C₅₃×#340	C₂²	1	2	yes	—
39	C₅₃×#340	C₂²	1	1	yes	characteristic
40	C₅₃×#340	C₂²	1	2	yes	—
41	C₅₃×#336	C₂²	1	2	yes	—
42	C₂²×Q₈×C₅₃	—	2	1	no	—
43	C₂²×D₄×C₅₃	C₂³	1	1	yes	characteristic
44	C₂³×C₄×C₅₃	—	2	1	no	—
45	C₂³×C₄×C₅₃	—	2	2	no	—
46	C₂²×C₈×C₅₃	C₂³	1	1	yes	characteristic
47	C₂²×C₈×C₅₃	C₂³	1	2	yes	—
48	C₂²×C₈×C₅₃	C₂³	1	1	yes	characteristic
49	C₄⋊Q₈×C₅₃	—	2	2	no	—
50	C₄⋊Q₈×C₅₃	C₂×C₄	1	4	yes	complemented
51	C₄×D₄×C₅₃	—	2	2	no	—
52	C₄×D₄×C₅₃	—	2	4	no	—
53	C₄×D₄×C₅₃	—	4	2	no	—
54	C₄×D₄×C₅₃	C₂×C₄	1	4	yes	—
55	C₂×C₄⋊C₄×C₅₃	—	2	1	no	—
56	C₂×C₄⋊C₄×C₅₃	—	2	1	no	—
57	C₂×C₄⋊C₄×C₅₃	—	2	2	no	—
58	C₂×C₄⋊C₄×C₅₃	—	2	1	no	—
59	C₂×C₄⋊C₄×C₅₃	—	2	2	no	—
60	C₂×C₄⋊C₄×C₅₃	—	4	4	no	—
61	C₂×C₄⋊C₄×C₅₃	D₄	1	1	yes	characteristic
62	C₂×C₄⋊C₄×C₅₃	D₄	1	1	yes	characteristic
63	C₂×C₄⋊C₄×C₅₃	C₂³	1	2	yes	—
64	C₂×C₄⋊C₄×C₅₃	C₂³	1	1	yes	characteristic
65	C₂×C₄⋊C₄×C₅₃	C₂³	1	1	yes	characteristic
66	C₂×C₄⋊C₄×C₅₃	C₂³	1	2	yes	—
67	C₂×C₄⋊C₄×C₅₃	C₂³	1	1	yes	characteristic
68	C₂×C₂²⋊C₄×C₅₃	—	2	1	no	—
69	C₂×C₂²⋊C₄×C₅₃	—	2	2	no	—
70	C₂×C₄²×C₅₃	—	2	1	no	—
71	C₂×C₄²×C₅₃	—	2	2	no	—
72	C₂×C₄²×C₅₃	C₂³	1	2	yes	—
73	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
74	#107×C₅₃	—	4	2	no	—
75	#106×C₅₃	C₂×C₄	1	8	yes	complemented
76	C₄⋊C₈×C₅₃	C₂×C₄	1	4	yes	complemented
77	#102×C₅₃	—	2	2	no	—
78	#102×C₅₃	C₂×C₄	1	8	yes	—
79	#102×C₅₃	Q₈	1	4	yes	complemented
80	#97×C₅₃	—	4	2	no	—
81	C₄×C₈×C₅₃	—	2	2	no	—
82	#95×C₅₃	—	2	2	no	—
83	#95×C₅₃	—	2	2	no	—
84	#95×C₅₃	—	2	1	no	—
85	#95×C₅₃	—	2	2	no	—
86	#95×C₅₃	D₄	1	2	yes	—
87	C₂⁴×C₅₃	—	2	1	no	—
88	C₂×Q₈×C₅₃	—	2	4	no	—
89	C₂×Q₈×C₅₃	—	8	1	no	—
90	C₂×Q₈×C₅₃	—	8	1	no	—
91	C₂×D₄×C₅₃	—	2	1	no	—
92	C₂×D₄×C₅₃	—	2	1	no	—
93	C₂×D₄×C₅₃	—	2	1	no	—
94	C₂×D₄×C₅₃	C₂²×C₄	1	2	yes	—
95	C₂×D₄×C₅₃	C₂×Q₈	1	2	yes	—
96	C₂×D₄×C₅₃	#41	1	2	yes	—
97	C₂²×C₄×C₅₃	—	2	1	no	—
98	C₂²×C₄×C₅₃	—	2	2	no	—
99	C₂²×C₄×C₅₃	—	2	1	no	—
100	C₂²×C₄×C₅₃	—	2	4	no	—
101	C₂²×C₄×C₅₃	—	2	1	no	—
102	C₂²×C₄×C₅₃	—	2	2	no	—
103	C₂²×C₄×C₅₃	—	4	4	no	—
104	C₂²×C₄×C₅₃	—	4	4	no	—
105	C₂²×C₄×C₅₃	—	4	2	no	—
106	C₂²×C₄×C₅₃	—	4	2	no	—
107	C₂²×C₄×C₅₃	—	8	2	no	—
108	C₂²×C₄×C₅₃	C₂×D₄	1	2	yes	—
109	C₂²×C₄×C₅₃	C₂×D₄	1	2	yes	—
110	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
111	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
112	C₂²×C₄×C₅₃	C₂⁴	1	1	yes	characteristic
113	C₂×C₈×C₅₃	—	2	1	no	—
114	C₂×C₈×C₅₃	—	2	1	no	—
115	C₂×C₈×C₅₃	—	2	2	no	—
116	C₂×C₈×C₅₃	—	2	2	no	—
117	C₂×C₈×C₅₃	—	2	1	no	—
118	C₂×C₈×C₅₃	—	2	1	no	—
119	C₂×C₈×C₅₃	—	2	1	no	—
120	C₂×C₈×C₅₃	C₂²×C₄	1	4	yes	—
121	C₂×C₈×C₅₃	C₂×Q₈	1	2	yes	—
122	C₄⋊C₄×C₅₃	—	2	4	no	—
123	C₄⋊C₄×C₅₃	—	2	2	no	—
124	C₄⋊C₄×C₅₃	—	2	2	no	—
125	C₄⋊C₄×C₅₃	—	2	1	no	—
126	C₄⋊C₄×C₅₃	—	2	2	no	—
127	C₄⋊C₄×C₅₃	—	2	1	no	—
128	C₄⋊C₄×C₅₃	—	2	1	no	—
129	C₄⋊C₄×C₅₃	—	2	2	no	—
130	C₄⋊C₄×C₅₃	—	2	1	no	—
131	C₄⋊C₄×C₅₃	—	2	2	no	—
132	C₄⋊C₄×C₅₃	—	4	4	no	—
133	C₄⋊C₄×C₅₃	—	4	2	no	—
134	C₄⋊C₄×C₅₃	—	8	1	no	—
135	C₄⋊C₄×C₅₃	—	8	2	no	—
136	C₄⋊C₄×C₅₃	C₂²×C₄	1	4	yes	—
137	C₄⋊C₄×C₅₃	C₂²×C₄	1	2	yes	—
138	C₄⋊C₄×C₅₃	C₂×Q₈	1	2	yes	—
139	C₂²⋊C₄×C₅₃	—	2	4	no	—
140	C₂²⋊C₄×C₅₃	—	4	4	no	—
141	C₄²×C₅₃	—	2	2	no	—
142	C₄²×C₅₃	—	2	2	no	—
143	C₄²×C₅₃	—	4	2	no	—
144	C₄²×C₅₃	—	8	2	no	—
145	C₄²×C₅₃	C₂²×C₄	1	2	yes	—
146	C₄²×C₅₃	#41	1	2	yes	—
147	C₂³×C₅₃	—	2	2	no	—
148	C₂³×C₅₃	—	2	2	no	—
149	C₂³×C₅₃	—	2	2	no	—
150	C₂³×C₅₃	—	4	1	no	—
151	C₂³×C₅₃	—	4	1	no	—
152	C₂³×C₅₃	—	4	1	no	—
153	C₂³×C₅₃	—	4	1	no	—
154	C₂³×C₅₃	C₂²×D₄	1	1	yes	characteristic
155	Q₈×C₅₃	—	8	4	no	—
156	D₄×C₅₃	—	2	2	no	—
157	D₄×C₅₃	—	2	2	no	—
158	D₄×C₅₃	—	2	2	no	—
159	D₄×C₅₃	C₄×Q₈	1	4	yes	—
160	C₂×C₄×C₅₃	—	2	1	no	—
161	C₂×C₄×C₅₃	—	2	1	no	—
162	C₂×C₄×C₅₃	—	2	2	no	—
163	C₂×C₄×C₅₃	—	2	4	no	—
164	C₂×C₄×C₅₃	—	2	2	no	—
165	C₂×C₄×C₅₃	—	2	2	no	—
166	C₂×C₄×C₅₃	—	2	2	no	—
167	C₂×C₄×C₅₃	—	2	2	no	—
168	C₂×C₄×C₅₃	—	2	2	no	—
169	C₂×C₄×C₅₃	—	2	2	no	—
170	C₂×C₄×C₅₃	—	2	2	no	—
171	C₂×C₄×C₅₃	—	4	1	no	—
172	C₂×C₄×C₅₃	—	4	1	no	—
173	C₂×C₄×C₅₃	—	4	1	no	—
174	C₂×C₄×C₅₃	—	4	2	no	—
175	C₂×C₄×C₅₃	—	4	2	no	—
176	C₂×C₄×C₅₃	—	4	1	no	—
177	C₂×C₄×C₅₃	—	4	8	no	—
178	C₂×C₄×C₅₃	—	4	2	no	—
179	C₂×C₄×C₅₃	—	4	1	no	—
180	C₂×C₄×C₅₃	—	4	2	no	—
181	C₂×C₄×C₅₃	—	4	1	no	—
182	C₂×C₄×C₅₃	—	4	2	no	—
183	C₂×C₄×C₅₃	—	4	1	no	—
184	C₂×C₄×C₅₃	—	8	8	no	—
185	C₂×C₄×C₅₃	C₄×D₄	1	2	yes	—
186	C₂×C₄×C₅₃	C₄×D₄	1	2	yes	—
187	C₂×C₄×C₅₃	AGL(1,ℤ/8)	1	2	yes	—
188	C₂×C₄×C₅₃	#137	1	2	yes	—
189	C₂×C₄×C₅₃	C₂³×C₄	1	1	yes	characteristic
190	C₂×C₄×C₅₃	C₂²×Q₈	1	1	yes	characteristic
191	C₂×C₄×C₅₃	C₂×#41	1	1	yes	characteristic
192	C₂×C₄×C₅₃	C₂×#41	1	1	yes	characteristic
193	C₂×C₄×C₅₃	H(5,𝔽₂)	1	1	yes	characteristic
194	C₂×C₄×C₅₃	2₋¹⁺⁴	1	1	yes	characteristic
195	C₈×C₅₃	—	2	4	no	—
196	C₈×C₅₃	—	2	2	no	—
197	C₈×C₅₃	—	4	1	no	—
198	#22969	C₅₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
199	C₂²×C₅₃	—	2	4	no	—
200	C₂²×C₅₃	—	4	2	no	—
201	C₂²×C₅₃	—	4	2	no	—
202	C₂²×C₅₃	—	4	2	no	—
203	C₂²×C₅₃	—	4	2	no	—
204	C₂²×C₅₃	—	4	2	no	—
205	C₂²×C₅₃	—	4	2	no	—
206	C₂²×C₅₃	C₂×C₄×D₄	1	1	yes	characteristic
207	C₂²×C₅₃	#546	1	1	yes	characteristic
208	C₂²×C₅₃	D₄×Q₈	1	1	yes	characteristic
209	C₂²×C₅₃	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
210	C₂²×C₅₃	C₂×#137	1	1	yes	characteristic
211	C₂²×C₅₃	#576	1	1	yes	characteristic
212	C₂²×C₅₃	#577	1	1	yes	characteristic
213	C₄×C₅₃	—	2	2	no	—
214	C₄×C₅₃	—	4	2	no	—
215	C₄×C₅₃	—	4	2	no	—
216	C₄×C₅₃	—	4	2	no	—
217	C₄×C₅₃	—	4	4	no	—
218	C₄×C₅₃	—	4	2	no	—
219	C₄×C₅₃	—	8	1	no	—
220	C₄×C₅₃	—	8	2	no	—
221	C₄×C₅₃	#440	1	4	yes	—
222	C₄×C₅₃	C₂×C₄×Q₈	1	1	yes	characteristic
223	C₄×C₅₃	#520	1	1	yes	characteristic
224	C₄×C₅₃	#552	1	1	yes	characteristic
225	C₄×C₅₃	#554	1	1	yes	characteristic
226	C₂×#476	C₂×C₅₃	1	1	yes	characteristic
227	#2369	C₂×C₅₃	1	1	yes	characteristic, complemented
228	D₄×C₄⋊C₄	C₂×C₅₃	1	1	yes	characteristic
229	#1990	C₂×C₅₃	1	1	yes	characteristic, complemented
230	#1947	C₂×C₅₃	1	1	yes	characteristic, complemented
231	#1944	C₂×C₅₃	1	2	yes	complemented
232	#1920	C₂×C₅₃	1	2	yes	complemented
233	#1862	C₂×C₅₃	1	2	yes	—
234	#1770	C₂×C₅₃	1	1	yes	characteristic, complemented
235	#1762	C₂×C₅₃	1	2	yes	—
236	C₄×#102	C₂×C₅₃	1	1	yes	characteristic
237	C₂×C₅₃	—	4	4	no	—
238	C₂×C₅₃	#2360	1	1	yes	characteristic
239	C₂×C₅₃	C₂×#440	1	1	yes	characteristic
240	C₂×C₅₃	#2946	1	1	yes	characteristic
241	C₂×C₅₃	#3325	1	1	yes	characteristic
242	C₂×C₅₃	#3331	1	1	yes	characteristic
243	C₂×C₅₃	#3384	1	1	yes	characteristic
244	C₂×C₅₃	#3385	1	1	yes	characteristic
245	C₂×C₄⋊Q₈	C₂²×C₅₃	1	1	yes	characteristic
246	C₂×C₄×D₄	C₂²×C₅₃	1	2	yes	—
247	C₂×C₄×D₄	C₂²×C₅₃	1	1	yes	characteristic
248	C₂²×C₄⋊C₄	—	2	1	no	—
249	#476	C₄×C₅₃	1	8	yes	complemented
250	C₂×#107	C₂²×C₅₃	1	2	yes	—
251	C₂×#106	C₂²×C₅₃	1	2	yes	—
252	C₂×C₄⋊C₈	C₂²×C₅₃	1	1	yes	characteristic
253	C₂×#102	C₂²×C₅₃	1	1	yes	characteristic
254	C₂×#102	C₂²×C₅₃	1	2	yes	—
255	C₂×#102	C₂²×C₅₃	1	1	yes	characteristic
256	C₂×#97	C₂²×C₅₃	1	2	yes	—
257	C₂×C₄×C₈	C₂²×C₅₃	1	1	yes	characteristic
258	#391	—	2	2	no	—
259	#389	—	2	1	no	—
260	#389	C₂²×C₅₃	1	2	yes	—
261	#389	C₂²×C₅₃	1	2	yes	—
262	#385	—	2	2	no	—
263	#384	C₂²×C₅₃	1	1	yes	characteristic
264	#380	—	2	1	no	—
265	C₄×C₄⋊C₄	—	2	1	no	—
266	C₄×C₄⋊C₄	C₂²×C₅₃	1	2	yes	—
267	C₄×C₄⋊C₄	C₂²×C₅₃	1	1	yes	characteristic
268	C₄×C₄⋊C₄	C₂²×C₅₃	1	1	yes	characteristic
269	#340	C₂²×C₅₃	1	2	yes	—
270	#340	C₂²×C₅₃	1	1	yes	characteristic
271	#340	C₂²×C₅₃	1	2	yes	—
272	#340	C₂²×C₅₃	1	1	yes	characteristic
273	#340	C₂²×C₅₃	1	2	yes	—
274	#336	C₂²×C₅₃	1	2	yes	—
275	C₅₃	#22969	1	1	yes	characteristic, direct product factor, Sylow
276	C₂²×Q₈	—	2	1	no	—
277	C₂²×D₄	C₂³×C₅₃	1	1	yes	characteristic
278	C₂³×C₄	—	2	1	no	—
279	C₂³×C₄	—	2	2	no	—
280	C₂²×C₈	C₂³×C₅₃	1	1	yes	characteristic
281	C₂²×C₈	C₂³×C₅₃	1	2	yes	—
282	C₂²×C₈	C₂³×C₅₃	1	1	yes	characteristic
283	C₄⋊Q₈	—	2	2	no	—
284	C₄⋊Q₈	C₂×C₄×C₅₃	1	4	yes	complemented
285	C₄×D₄	—	2	4	no	—
286	C₄×D₄	—	2	2	no	—
287	C₄×D₄	—	4	2	no	—
288	C₄×D₄	C₂×C₄×C₅₃	1	4	yes	—
289	C₂×C₄⋊C₄	—	2	1	no	—
290	C₂×C₄⋊C₄	—	2	1	no	—
291	C₂×C₄⋊C₄	—	2	2	no	—
292	C₂×C₄⋊C₄	—	2	1	no	—
293	C₂×C₄⋊C₄	—	2	2	no	—
294	C₂×C₄⋊C₄	—	4	4	no	—
295	C₂×C₄⋊C₄	D₄×C₅₃	1	1	yes	characteristic
296	C₂×C₄⋊C₄	D₄×C₅₃	1	1	yes	characteristic
297	C₂×C₄⋊C₄	C₂³×C₅₃	1	2	yes	—
298	C₂×C₄⋊C₄	C₂³×C₅₃	1	1	yes	characteristic
299	C₂×C₄⋊C₄	C₂³×C₅₃	1	1	yes	characteristic
300	C₂×C₄⋊C₄	C₂³×C₅₃	1	2	yes	—
301	C₂×C₄⋊C₄	C₂³×C₅₃	1	1	yes	characteristic
302	C₂×C₂²⋊C₄	—	2	1	no	—
303	C₂×C₂²⋊C₄	—	2	2	no	—
304	C₂×C₄²	—	2	1	no	—
305	C₂×C₄²	—	2	2	no	—
306	C₂×C₄²	C₂³×C₅₃	1	2	yes	—
307	C₂×C₄²	C₂³×C₅₃	1	1	yes	characteristic
308	#107	—	4	2	no	—
309	#106	C₂×C₄×C₅₃	1	8	yes	complemented
310	C₄⋊C₈	C₂×C₄×C₅₃	1	4	yes	complemented
311	#102	—	2	2	no	—
312	#102	C₂×C₄×C₅₃	1	8	yes	—
313	#102	Q₈×C₅₃	1	4	yes	complemented
314	#97	—	4	2	no	—
315	C₄×C₈	—	2	2	no	—
316	#95	—	2	2	no	—
317	#95	—	2	2	no	—
318	#95	—	2	1	no	—
319	#95	—	2	2	no	—
320	#95	D₄×C₅₃	1	2	yes	—
321	C₂⁴	—	2	1	no	—
322	C₂×Q₈	—	2	4	no	—
323	C₂×Q₈	—	8	1	no	—
324	C₂×Q₈	—	8	1	no	—
325	C₂×D₄	—	2	1	no	—
326	C₂×D₄	—	2	1	no	—
327	C₂×D₄	—	2	1	no	—
328	C₂×D₄	C₂²×C₄×C₅₃	1	2	yes	—
329	C₂×D₄	C₂×Q₈×C₅₃	1	2	yes	—
330	C₂×D₄	#41×C₅₃	1	2	yes	—
331	C₂²×C₄	—	2	1	no	—
332	C₂²×C₄	—	2	2	no	—
333	C₂²×C₄	—	2	1	no	—
334	C₂²×C₄	—	2	4	no	—
335	C₂²×C₄	—	2	1	no	—
336	C₂²×C₄	—	2	2	no	—
337	C₂²×C₄	—	4	4	no	—
338	C₂²×C₄	—	4	4	no	—
339	C₂²×C₄	—	4	2	no	—
340	C₂²×C₄	—	4	2	no	—
341	C₂²×C₄	—	8	2	no	—
342	C₂²×C₄	C₂×D₄×C₅₃	1	2	yes	—
343	C₂²×C₄	C₂×D₄×C₅₃	1	2	yes	—
344	C₂²×C₄	C₂×D₄×C₅₃	1	1	yes	characteristic
345	C₂²×C₄	C₂×D₄×C₅₃	1	1	yes	characteristic
346	C₂²×C₄	C₂⁴×C₅₃	1	1	yes	characteristic
347	C₂×C₈	—	2	1	no	—
348	C₂×C₈	—	2	1	no	—
349	C₂×C₈	—	2	2	no	—
350	C₂×C₈	—	2	2	no	—
351	C₂×C₈	—	2	1	no	—
352	C₂×C₈	—	2	1	no	—
353	C₂×C₈	—	2	1	no	—
354	C₂×C₈	C₂²×C₄×C₅₃	1	4	yes	—
355	C₂×C₈	C₂×Q₈×C₅₃	1	2	yes	—
356	C₄⋊C₄	—	2	4	no	—
357	C₄⋊C₄	—	2	2	no	—
358	C₄⋊C₄	—	2	2	no	—
359	C₄⋊C₄	—	2	1	no	—
360	C₄⋊C₄	—	2	2	no	—
361	C₄⋊C₄	—	2	1	no	—
362	C₄⋊C₄	—	2	1	no	—
363	C₄⋊C₄	—	2	2	no	—
364	C₄⋊C₄	—	2	1	no	—
365	C₄⋊C₄	—	2	2	no	—
366	C₄⋊C₄	—	4	4	no	—
367	C₄⋊C₄	—	4	2	no	—
368	C₄⋊C₄	—	8	1	no	—
369	C₄⋊C₄	—	8	2	no	—
370	C₄⋊C₄	C₂²×C₄×C₅₃	1	4	yes	—
371	C₄⋊C₄	C₂²×C₄×C₅₃	1	2	yes	—
372	C₄⋊C₄	C₂×Q₈×C₅₃	1	2	yes	—
373	C₂²⋊C₄	—	2	4	no	—
374	C₂²⋊C₄	—	4	4	no	—
375	C₄²	—	2	2	no	—
376	C₄²	—	2	2	no	—
377	C₄²	—	4	2	no	—
378	C₄²	—	8	2	no	—
379	C₄²	C₂²×C₄×C₅₃	1	2	yes	—
380	C₄²	#41×C₅₃	1	2	yes	—
381	C₂³	—	2	2	no	—
382	C₂³	—	2	2	no	—
383	C₂³	—	2	2	no	—
384	C₂³	—	4	1	no	—
385	C₂³	—	4	1	no	—
386	C₂³	—	4	1	no	—
387	C₂³	—	4	1	no	—
388	C₂³	C₂²×D₄×C₅₃	1	1	yes	characteristic
389	Q₈	—	8	4	no	—
390	D₄	—	2	2	no	—
391	D₄	—	2	2	no	—
392	D₄	—	2	2	no	—
393	D₄	C₄×Q₈×C₅₃	1	4	yes	—
394	C₂×C₄	—	2	1	no	—
395	C₂×C₄	—	2	1	no	—
396	C₂×C₄	—	2	2	no	—
397	C₂×C₄	—	2	4	no	—
398	C₂×C₄	—	2	2	no	—
399	C₂×C₄	—	2	2	no	—
400	C₂×C₄	—	2	2	no	—
401	C₂×C₄	—	2	2	no	—
402	C₂×C₄	—	2	2	no	—
403	C₂×C₄	—	2	2	no	—
404	C₂×C₄	—	2	2	no	—
405	C₂×C₄	—	4	1	no	—
406	C₂×C₄	—	4	1	no	—
407	C₂×C₄	—	4	1	no	—
408	C₂×C₄	—	4	2	no	—
409	C₂×C₄	—	4	1	no	—
410	C₂×C₄	—	4	2	no	—
411	C₂×C₄	—	4	8	no	—
412	C₂×C₄	—	4	2	no	—
413	C₂×C₄	—	4	1	no	—
414	C₂×C₄	—	4	1	no	—
415	C₂×C₄	—	4	2	no	—
416	C₂×C₄	—	4	1	no	—
417	C₂×C₄	—	4	2	no	—
418	C₂×C₄	—	8	8	no	—
419	C₂×C₄	C₄×D₄×C₅₃	1	2	yes	—
420	C₂×C₄	C₄×D₄×C₅₃	1	2	yes	—
421	C₂×C₄	AGL(1,ℤ/8)×C₅₃	1	2	yes	—
422	C₂×C₄	#137×C₅₃	1	2	yes	—
423	C₂×C₄	C₂³×C₄×C₅₃	1	1	yes	characteristic
424	C₂×C₄	C₂²×Q₈×C₅₃	1	1	yes	characteristic
425	C₂×C₄	C₂×#41×C₅₃	1	1	yes	characteristic
426	C₂×C₄	C₂×#41×C₅₃	1	1	yes	characteristic
427	C₂×C₄	H(5,𝔽₂)×C₅₃	1	1	yes	characteristic
428	C₂×C₄	2₋¹⁺⁴×C₅₃	1	1	yes	characteristic
429	C₈	—	2	4	no	—
430	C₈	—	2	2	no	—
431	C₈	—	4	1	no	—
432	C₂²	—	2	4	no	—
433	C₂²	—	4	2	no	—
434	C₂²	—	4	2	no	—
435	C₂²	—	4	2	no	—
436	C₂²	—	4	2	no	—
437	C₂²	—	4	2	no	—
438	C₂²	—	4	2	no	—
439	C₂²	C₂×C₄×D₄×C₅₃	1	1	yes	characteristic
440	C₂²	C₅₃×#546	1	1	yes	characteristic
441	C₂²	D₄×Q₈×C₅₃	1	1	yes	characteristic
442	C₂²	C₂×AGL(1,ℤ/8)×C₅₃	1	1	yes	characteristic
443	C₂²	C₂×#137×C₅₃	1	1	yes	characteristic
444	C₂²	C₅₃×#576	1	1	yes	characteristic
445	C₂²	C₅₃×#577	1	1	yes	characteristic
446	C₄	—	2	2	no	—
447	C₄	—	4	2	no	—
448	C₄	—	4	4	no	—
449	C₄	—	4	2	no	—
450	C₄	—	4	2	no	—
451	C₄	—	4	2	no	—
452	C₄	—	8	2	no	—
453	C₄	—	8	1	no	—
454	C₄	C₅₃×#440	1	4	yes	—
455	C₄	C₂×C₄×Q₈×C₅₃	1	1	yes	characteristic
456	C₄	C₅₃×#520	1	1	yes	characteristic
457	C₄	C₅₃×#552	1	1	yes	characteristic
458	C₄	C₅₃×#554	1	1	yes	characteristic
459	C₂	—	4	4	no	—
460	C₂	C₅₃×#2360	1	1	yes	characteristic
461	C₂	C₂×C₅₃×#440	1	1	yes	characteristic
462	C₂	C₅₃×#2946	1	1	yes	characteristic
463	C₂	C₅₃×#3325	1	1	yes	characteristic
464	C₂	C₅₃×#3331	1	1	yes	characteristic
465	C₂	C₅₃×#3384	1	1	yes	characteristic
466	C₂	C₅₃×#3385	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 ◁ C₄×C₄⋊C₄×C₅₃ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₅₃×#2369	C₂	semidirect
4. C₅₃×#1990	C₂	semidirect
5. C₅₃×#1947	C₂	semidirect
6. C₅₃×#1944	C₂	semidirect
7. C₅₃×#1920	C₂	semidirect
9. C₅₃×#1770	C₂	semidirect
16. C₅₃×#476	C₄	semidirect
50. C₄⋊Q₈×C₅₃	C₂×C₄	semidirect
75. #106×C₅₃	C₂×C₄	semidirect
76. C₄⋊C₈×C₅₃	C₂×C₄	semidirect
79. #102×C₅₃	Q₈	semidirect
198. #22969	C₅₃	direct (N × H)
227. #2369	C₂×C₅₃	semidirect
229. #1990	C₂×C₅₃	semidirect
230. #1947	C₂×C₅₃	semidirect
231. #1944	C₂×C₅₃	semidirect
232. #1920	C₂×C₅₃	semidirect
234. #1770	C₂×C₅₃	semidirect
249. #476	C₄×C₅₃	semidirect
275. C₅₃	#22969	direct (N × H)
284. C₄⋊Q₈	C₂×C₄×C₅₃	semidirect
309. #106	C₂×C₄×C₅₃	semidirect
310. C₄⋊C₈	C₂×C₄×C₅₃	semidirect
313. #102	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×C₅₃×#476	2
C₅₃×#2369	2
D₄×C₄⋊C₄×C₅₃	2
C₅₃×#1990	2
C₅₃×#1947	2
C₅₃×#1944	2
C₅₃×#1920	2
C₅₃×#1862	2
C₅₃×#1770	2
C₅₃×#1762	2
C₄×#102×C₅₃	2
#22969	53

Subgroups — C53×#22969 #27230481

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅₃×#22969 #27230481