Subgroups — C₅₃×#24331 #27231843

Order 13568.

1078 conjugacy classes of subgroups.

The subgroup lattice has 524 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₅₃×#554	C₂	1	1	yes	maximal, characteristic
2	C₂×C₅₃×#477	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×C₄×SD₁₆×C₅₃	C₂	1	1	yes	maximal, characteristic
4	C₅₃×#2582	C₂	1	1	yes	maximal, characteristic, complemented
5	C₅₃×#2059	C₂	1	2	yes	maximal, complemented
6	C₅₃×#2026	C₂	1	2	yes	maximal, complemented
7	C₅₃×#1990	C₂	1	1	yes	maximal, characteristic, complemented
8	C₅₃×#1984	C₂	1	1	yes	maximal, characteristic, complemented
9	C₅₃×#1877	C₂	1	2	yes	maximal
10	C₅₃×#1862	C₂	1	2	yes	maximal
11	C₅₃×#1802	C₂	1	1	yes	maximal, characteristic
12	C₂²×SD₁₆×C₅₃	C₂²	1	1	yes	characteristic
13	C₅₃×#554	—	2	4	no	—
14	C₅₃×#554	—	2	4	no	—
15	C₂×C₄⋊Q₈×C₅₃	C₂²	1	1	yes	characteristic
16	C₂×#125×C₅₃	C₂²	1	2	yes	—
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×Q₈×C₅₃	C₂²	1	1	yes	characteristic
21	C₂×C₄×D₄×C₅₃	C₂²	1	2	yes	—
22	C₂×C₄×D₄×C₅₃	C₂²	1	1	yes	characteristic
23	C₂²×C₄⋊C₄×C₅₃	—	2	1	no	—
24	C₅₃×#477	—	2	4	no	—
25	C₄×SD₁₆×C₅₃	—	2	4	no	—
26	C₂×#107×C₅₃	C₂²	1	2	yes	—
27	C₂×#106×C₅₃	C₂²	1	1	yes	characteristic
28	C₂×C₄⋊C₈×C₅₃	C₂²	1	1	yes	characteristic
29	C₂×#103×C₅₃	C₂²	1	1	yes	characteristic
30	C₂×#103×C₅₃	C₂²	1	2	yes	—
31	C₂×#102×C₅₃	C₂²	1	1	yes	characteristic
32	C₂×#102×C₅₃	C₂²	1	2	yes	—
33	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
34	C₅₃×#398	—	2	2	no	—
35	C₅₃×#391	—	2	1	no	—
36	C₅₃×#391	C₂²	1	2	yes	—
37	C₅₃×#389	—	2	1	no	—
38	C₅₃×#389	C₂²	1	2	yes	—
39	C₅₃×#384	C₂²	1	1	yes	characteristic
40	C₅₃×#340	C₂²	1	2	yes	—
41	C₅₃×#340	C₂²	1	2	yes	—
42	C₅₃×#340	C₂²	1	2	yes	—
43	C₅₃×#336	C₂²	1	2	yes	—
44	C₅₃×#336	C₂²	1	2	yes	—
45	C₂²×Q₈×C₅₃	—	2	1	no	—
46	C₂²×Q₈×C₅₃	D₄	1	1	yes	characteristic
47	C₂²×Q₈×C₅₃	C₂³	1	1	yes	characteristic
48	C₂²×D₄×C₅₃	C₂³	1	1	yes	characteristic
49	C₂³×C₄×C₅₃	—	2	1	no	—
50	C₂³×C₄×C₅₃	—	2	2	no	—
51	C₂×SD₁₆×C₅₃	—	2	2	no	—
52	C₂×SD₁₆×C₅₃	—	2	2	no	—
53	C₂×SD₁₆×C₅₃	Q₈	1	4	yes	complemented
54	C₂²×C₈×C₅₃	C₂³	1	1	yes	characteristic
55	C₂²×C₈×C₅₃	C₂³	1	2	yes	—
56	C₂²×C₈×C₅₃	C₂³	1	1	yes	characteristic
57	C₄⋊Q₈×C₅₃	—	2	2	no	—
58	C₄⋊Q₈×C₅₃	D₄	1	4	yes	—
59	#125×C₅₃	—	2	4	no	—
60	#125×C₅₃	—	2	4	no	—
61	C₂²⋊Q₈×C₅₃	—	2	16	no	—
62	C₂²⋊Q₈×C₅₃	—	2	8	no	—
63	C₄×Q₈×C₅₃	—	2	2	no	—
64	C₄×Q₈×C₅₃	—	2	2	no	—
65	C₄×Q₈×C₅₃	—	2	2	no	—
66	C₄×Q₈×C₅₃	—	2	2	no	—
67	C₄×D₄×C₅₃	—	2	4	no	—
68	C₄×D₄×C₅₃	—	2	2	no	—
69	C₄×D₄×C₅₃	—	2	4	no	—
70	C₄×D₄×C₅₃	D₄	1	4	yes	—
71	C₂×C₄⋊C₄×C₅₃	—	2	8	no	—
72	C₂×C₄⋊C₄×C₅₃	—	2	2	no	—
73	C₂×C₄⋊C₄×C₅₃	—	2	1	no	—
74	C₂×C₄⋊C₄×C₅₃	—	2	2	no	—
75	C₂×C₄⋊C₄×C₅₃	—	2	1	no	—
76	C₂×C₄⋊C₄×C₅₃	—	2	1	no	—
77	C₂×C₄⋊C₄×C₅₃	—	2	1	no	—
78	C₂×C₄⋊C₄×C₅₃	—	2	2	no	—
79	C₂×C₄⋊C₄×C₅₃	D₄	1	1	yes	characteristic
80	C₂×C₄⋊C₄×C₅₃	D₄	1	2	yes	—
81	C₂×C₄⋊C₄×C₅₃	C₂³	1	1	yes	characteristic
82	C₂×C₄⋊C₄×C₅₃	C₂³	1	2	yes	—
83	C₂×C₄⋊C₄×C₅₃	C₂³	1	1	yes	characteristic
84	C₂×C₄⋊C₄×C₅₃	C₂³	1	2	yes	—
85	C₂×C₂²⋊C₄×C₅₃	—	2	1	no	—
86	C₂×C₂²⋊C₄×C₅₃	—	2	2	no	—
87	C₂×C₄²×C₅₃	—	2	1	no	—
88	C₂×C₄²×C₅₃	C₂³	1	2	yes	—
89	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
90	#107×C₅₃	—	2	4	no	—
91	#106×C₅₃	—	2	2	no	—
92	C₄⋊C₈×C₅₃	—	2	2	no	—
93	#103×C₅₃	—	2	2	no	—
94	#103×C₅₃	—	2	4	no	—
95	#102×C₅₃	—	2	2	no	—
96	#102×C₅₃	—	2	4	no	—
97	C₄×C₈×C₅₃	—	2	2	no	—
98	#95×C₅₃	—	2	2	no	—
99	#95×C₅₃	—	2	2	no	—
100	#95×C₅₃	—	2	2	no	—
101	C₂⁴×C₅₃	—	2	1	no	—
102	C₂×Q₈×C₅₃	—	2	2	no	—
103	C₂×Q₈×C₅₃	—	2	4	no	—
104	C₂×Q₈×C₅₃	—	2	2	no	—
105	C₂×Q₈×C₅₃	—	2	2	no	—
106	C₂×Q₈×C₅₃	—	2	1	no	—
107	C₂×Q₈×C₅₃	—	2	1	no	—
108	C₂×Q₈×C₅₃	—	2	1	no	—
109	C₂×Q₈×C₅₃	—	2	1	no	—
110	C₂×Q₈×C₅₃	—	2	1	no	—
111	C₂×Q₈×C₅₃	—	8	1	no	—
112	C₂×Q₈×C₅₃	—	8	1	no	—
113	C₂×Q₈×C₅₃	C₂×Q₈	1	2	yes	—
114	C₂×D₄×C₅₃	—	2	1	no	—
115	C₂×D₄×C₅₃	—	2	1	no	—
116	C₂×D₄×C₅₃	—	2	1	no	—
117	C₂×D₄×C₅₃	C₂×D₄	1	2	yes	—
118	C₂×D₄×C₅₃	C₂×Q₈	1	2	yes	—
119	C₂×D₄×C₅₃	#41	1	2	yes	—
120	C₂²×C₄×C₅₃	—	2	1	no	—
121	C₂²×C₄×C₅₃	—	2	1	no	—
122	C₂²×C₄×C₅₃	—	2	2	no	—
123	C₂²×C₄×C₅₃	—	2	8	no	—
124	C₂²×C₄×C₅₃	—	2	4	no	—
125	C₂²×C₄×C₅₃	—	2	2	no	—
126	C₂²×C₄×C₅₃	—	2	1	no	—
127	C₂²×C₄×C₅₃	—	4	2	no	—
128	C₂²×C₄×C₅₃	—	4	2	no	—
129	C₂²×C₄×C₅₃	—	8	2	no	—
130	C₂²×C₄×C₅₃	—	8	2	no	—
131	C₂²×C₄×C₅₃	C₂×D₄	1	2	yes	—
132	C₂²×C₄×C₅₃	C₂×D₄	1	2	yes	—
133	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
134	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
135	C₂²×C₄×C₅₃	C₂⁴	1	1	yes	characteristic
136	SD₁₆×C₅₃	—	2	8	no	—
137	C₂×C₈×C₅₃	—	2	1	no	—
138	C₂×C₈×C₅₃	—	2	1	no	—
139	C₂×C₈×C₅₃	—	2	1	no	—
140	C₂×C₈×C₅₃	—	2	2	no	—
141	C₂×C₈×C₅₃	—	2	2	no	—
142	C₂×C₈×C₅₃	—	2	1	no	—
143	C₂×C₈×C₅₃	—	2	1	no	—
144	C₂×C₈×C₅₃	—	2	2	no	—
145	C₂×C₈×C₅₃	C₂×Q₈	1	2	yes	—
146	C₄⋊C₄×C₅₃	—	2	8	no	—
147	C₄⋊C₄×C₅₃	—	2	8	no	—
148	C₄⋊C₄×C₅₃	—	2	2	no	—
149	C₄⋊C₄×C₅₃	—	2	4	no	—
150	C₄⋊C₄×C₅₃	—	2	4	no	—
151	C₄⋊C₄×C₅₃	—	2	1	no	—
152	C₄⋊C₄×C₅₃	—	2	2	no	—
153	C₄⋊C₄×C₅₃	—	2	1	no	—
154	C₄⋊C₄×C₅₃	—	2	1	no	—
155	C₄⋊C₄×C₅₃	—	2	2	no	—
156	C₄⋊C₄×C₅₃	—	2	2	no	—
157	C₄⋊C₄×C₅₃	—	4	2	no	—
158	C₄⋊C₄×C₅₃	—	4	2	no	—
159	C₄⋊C₄×C₅₃	—	8	1	no	—
160	C₄⋊C₄×C₅₃	—	8	2	no	—
161	C₄⋊C₄×C₅₃	C₂×D₄	1	2	yes	—
162	C₄⋊C₄×C₅₃	C₂×D₄	1	4	yes	—
163	C₂²⋊C₄×C₅₃	—	2	8	no	—
164	C₂²⋊C₄×C₅₃	—	2	4	no	—
165	C₄²×C₅₃	—	2	2	no	—
166	C₄²×C₅₃	—	2	2	no	—
167	C₄²×C₅₃	—	4	2	no	—
168	C₄²×C₅₃	C₂×D₄	1	2	yes	—
169	C₄²×C₅₃	C₂×D₄	1	2	yes	—
170	C₂³×C₅₃	—	2	2	no	—
171	C₂³×C₅₃	—	2	2	no	—
172	C₂³×C₅₃	—	2	2	no	—
173	C₂³×C₅₃	—	4	1	no	—
174	C₂³×C₅₃	—	4	1	no	—
175	C₂³×C₅₃	—	4	1	no	—
176	C₂³×C₅₃	—	4	1	no	—
177	C₂³×C₅₃	C₂²×D₄	1	1	yes	characteristic
178	Q₈×C₅₃	—	2	2	no	—
179	Q₈×C₅₃	—	2	2	no	—
180	Q₈×C₅₃	—	4	1	no	—
181	Q₈×C₅₃	—	4	1	no	—
182	Q₈×C₅₃	—	4	4	no	—
183	Q₈×C₅₃	—	8	4	no	—
184	D₄×C₅₃	—	2	2	no	—
185	D₄×C₅₃	—	2	2	no	—
186	D₄×C₅₃	—	2	2	no	—
187	D₄×C₅₃	C₂²⋊Q₈	1	4	yes	—
188	C₂×C₄×C₅₃	—	2	1	no	—
189	C₂×C₄×C₅₃	—	2	1	no	—
190	C₂×C₄×C₅₃	—	2	2	no	—
191	C₂×C₄×C₅₃	—	2	4	no	—
192	C₂×C₄×C₅₃	—	2	2	no	—
193	C₂×C₄×C₅₃	—	2	2	no	—
194	C₂×C₄×C₅₃	—	2	2	no	—
195	C₂×C₄×C₅₃	—	2	2	no	—
196	C₂×C₄×C₅₃	—	2	2	no	—
197	C₂×C₄×C₅₃	—	2	2	no	—
198	C₂×C₄×C₅₃	—	2	2	no	—
199	C₂×C₄×C₅₃	—	4	1	no	—
200	C₂×C₄×C₅₃	—	4	1	no	—
201	C₂×C₄×C₅₃	—	4	1	no	—
202	C₂×C₄×C₅₃	—	4	1	no	—
203	C₂×C₄×C₅₃	—	4	2	no	—
204	C₂×C₄×C₅₃	—	4	2	no	—
205	C₂×C₄×C₅₃	—	4	8	no	—
206	C₂×C₄×C₅₃	—	4	1	no	—
207	C₂×C₄×C₅₃	—	4	2	no	—
208	C₂×C₄×C₅₃	—	4	2	no	—
209	C₂×C₄×C₅₃	—	4	2	no	—
210	C₂×C₄×C₅₃	—	4	1	no	—
211	C₂×C₄×C₅₃	—	4	1	no	—
212	C₂×C₄×C₅₃	—	8	8	no	—
213	C₂×C₄×C₅₃	#120	1	2	yes	—
214	C₂×C₄×C₅₃	#120	1	2	yes	—
215	C₂×C₄×C₅₃	#135	1	2	yes	—
216	C₂×C₄×C₅₃	#137	1	2	yes	—
217	C₂×C₄×C₅₃	C₂²×D₄	1	1	yes	characteristic
218	C₂×C₄×C₅₃	C₂²×D₄	1	1	yes	characteristic
219	C₂×C₄×C₅₃	C₂²×Q₈	1	1	yes	characteristic
220	C₂×C₄×C₅₃	C₂×#41	1	1	yes	characteristic
221	C₂×C₄×C₅₃	2₋¹⁺⁴	1	1	yes	characteristic
222	C₂×C₄×C₅₃	2₋¹⁺⁴	1	1	yes	characteristic
223	C₈×C₅₃	—	2	2	no	—
224	C₈×C₅₃	—	4	1	no	—
225	C₈×C₅₃	—	4	2	no	—
226	#24331	C₅₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
227	C₂²×C₅₃	—	2	4	no	—
228	C₂²×C₅₃	—	4	2	no	—
229	C₂²×C₅₃	—	4	2	no	—
230	C₂²×C₅₃	—	4	2	no	—
231	C₂²×C₅₃	—	4	2	no	—
232	C₂²×C₅₃	—	4	2	no	—
233	C₂²×C₅₃	—	4	2	no	—
234	C₂²×C₅₃	C₂×#120	1	1	yes	characteristic
235	C₂²×C₅₃	#548	1	1	yes	characteristic
236	C₂²×C₅₃	D₄×Q₈	1	1	yes	characteristic
237	C₂²×C₅₃	C₂×#135	1	1	yes	characteristic
238	C₂²×C₅₃	C₂×#137	1	1	yes	characteristic
239	C₂²×C₅₃	#576	1	1	yes	characteristic
240	C₂²×C₅₃	#577	1	1	yes	characteristic
241	C₄×C₅₃	—	2	2	no	—
242	C₄×C₅₃	—	4	2	no	—
243	C₄×C₅₃	—	4	2	no	—
244	C₄×C₅₃	—	4	2	no	—
245	C₄×C₅₃	—	4	2	no	—
246	C₄×C₅₃	—	4	4	no	—
247	C₄×C₅₃	—	8	2	no	—
248	C₄×C₅₃	—	8	1	no	—
249	C₄×C₅₃	#452	1	4	yes	—
250	C₄×C₅₃	C₂×C₂²⋊Q₈	1	1	yes	characteristic
251	C₄×C₅₃	#536	1	1	yes	characteristic
252	C₄×C₅₃	#548	1	1	yes	characteristic
253	C₄×C₅₃	D₄×Q₈	1	1	yes	characteristic
254	C₂×#554	C₂×C₅₃	1	1	yes	characteristic
255	C₂×#477	C₂×C₅₃	1	1	yes	characteristic, complemented
256	C₂×C₄×SD₁₆	C₂×C₅₃	1	1	yes	characteristic
257	#2582	C₂×C₅₃	1	1	yes	characteristic, complemented
258	#2059	C₂×C₅₃	1	2	yes	complemented
259	#2026	C₂×C₅₃	1	2	yes	complemented
260	#1990	C₂×C₅₃	1	1	yes	characteristic, complemented
261	#1984	C₂×C₅₃	1	1	yes	characteristic, complemented
262	#1877	C₂×C₅₃	1	2	yes	—
263	#1862	C₂×C₅₃	1	2	yes	—
264	#1802	C₂×C₅₃	1	1	yes	characteristic
265	C₂×C₅₃	—	4	4	no	—
266	C₂×C₅₃	#2409	1	1	yes	characteristic
267	C₂×C₅₃	C₂×#452	1	1	yes	characteristic
268	C₂×C₅₃	#3012	1	1	yes	characteristic
269	C₂×C₅₃	#3361	1	1	yes	characteristic
270	C₂×C₅₃	#3372	1	1	yes	characteristic
271	C₂×C₅₃	#3381	1	1	yes	characteristic
272	C₂×C₅₃	#3385	1	1	yes	characteristic
273	C₂²×SD₁₆	C₂²×C₅₃	1	1	yes	characteristic
274	#554	—	2	4	no	—
275	#554	—	2	4	no	—
276	C₂×C₄⋊Q₈	C₂²×C₅₃	1	1	yes	characteristic
277	C₂×#125	C₂²×C₅₃	1	2	yes	—
278	C₂×C₂²⋊Q₈	—	2	1	no	—
279	C₂×C₂²⋊Q₈	—	2	2	no	—
280	C₂×C₄×Q₈	C₂²×C₅₃	1	1	yes	characteristic
281	C₂×C₄×Q₈	C₂²×C₅₃	1	1	yes	characteristic
282	C₂×C₄×D₄	C₂²×C₅₃	1	2	yes	—
283	C₂×C₄×D₄	C₂²×C₅₃	1	1	yes	characteristic
284	C₂²×C₄⋊C₄	—	2	1	no	—
285	#477	—	2	4	no	—
286	C₄×SD₁₆	—	2	4	no	—
287	C₂×#107	C₂²×C₅₃	1	2	yes	—
288	C₂×#106	C₂²×C₅₃	1	1	yes	characteristic
289	C₂×C₄⋊C₈	C₂²×C₅₃	1	1	yes	characteristic
290	C₂×#103	C₂²×C₅₃	1	2	yes	—
291	C₂×#103	C₂²×C₅₃	1	1	yes	characteristic
292	C₂×#102	C₂²×C₅₃	1	2	yes	—
293	C₂×#102	C₂²×C₅₃	1	1	yes	characteristic
294	C₂×C₄×C₈	C₂²×C₅₃	1	1	yes	characteristic
295	#398	—	2	2	no	—
296	#391	—	2	1	no	—
297	#391	C₂²×C₅₃	1	2	yes	—
298	#389	—	2	1	no	—
299	#389	C₂²×C₅₃	1	2	yes	—
300	#384	C₂²×C₅₃	1	1	yes	characteristic
301	#340	C₂²×C₅₃	1	2	yes	—
302	#340	C₂²×C₅₃	1	2	yes	—
303	#340	C₂²×C₅₃	1	2	yes	—
304	#336	C₂²×C₅₃	1	2	yes	—
305	#336	C₂²×C₅₃	1	2	yes	—
306	C₅₃	#24331	1	1	yes	characteristic, direct product factor, Sylow
307	C₂²×Q₈	—	2	1	no	—
308	C₂²×Q₈	D₄×C₅₃	1	1	yes	characteristic
309	C₂²×Q₈	C₂³×C₅₃	1	1	yes	characteristic
310	C₂²×D₄	C₂³×C₅₃	1	1	yes	characteristic
311	C₂³×C₄	—	2	1	no	—
312	C₂³×C₄	—	2	2	no	—
313	C₂×SD₁₆	—	2	2	no	—
314	C₂×SD₁₆	—	2	2	no	—
315	C₂×SD₁₆	Q₈×C₅₃	1	4	yes	complemented
316	C₂²×C₈	C₂³×C₅₃	1	1	yes	characteristic
317	C₂²×C₈	C₂³×C₅₃	1	2	yes	—
318	C₂²×C₈	C₂³×C₅₃	1	1	yes	characteristic
319	C₄⋊Q₈	—	2	2	no	—
320	C₄⋊Q₈	D₄×C₅₃	1	4	yes	—
321	#125	—	2	4	no	—
322	#125	—	2	4	no	—
323	C₂²⋊Q₈	—	2	8	no	—
324	C₂²⋊Q₈	—	2	16	no	—
325	C₄×Q₈	—	2	2	no	—
326	C₄×Q₈	—	2	2	no	—
327	C₄×Q₈	—	2	2	no	—
328	C₄×Q₈	—	2	2	no	—
329	C₄×D₄	—	2	4	no	—
330	C₄×D₄	—	2	4	no	—
331	C₄×D₄	—	2	2	no	—
332	C₄×D₄	D₄×C₅₃	1	4	yes	—
333	C₂×C₄⋊C₄	—	2	8	no	—
334	C₂×C₄⋊C₄	—	2	2	no	—
335	C₂×C₄⋊C₄	—	2	1	no	—
336	C₂×C₄⋊C₄	—	2	2	no	—
337	C₂×C₄⋊C₄	—	2	1	no	—
338	C₂×C₄⋊C₄	—	2	1	no	—
339	C₂×C₄⋊C₄	—	2	1	no	—
340	C₂×C₄⋊C₄	—	2	2	no	—
341	C₂×C₄⋊C₄	D₄×C₅₃	1	1	yes	characteristic
342	C₂×C₄⋊C₄	D₄×C₅₃	1	2	yes	—
343	C₂×C₄⋊C₄	C₂³×C₅₃	1	1	yes	characteristic
344	C₂×C₄⋊C₄	C₂³×C₅₃	1	2	yes	—
345	C₂×C₄⋊C₄	C₂³×C₅₃	1	1	yes	characteristic
346	C₂×C₄⋊C₄	C₂³×C₅₃	1	2	yes	—
347	C₂×C₂²⋊C₄	—	2	2	no	—
348	C₂×C₂²⋊C₄	—	2	1	no	—
349	C₂×C₄²	—	2	1	no	—
350	C₂×C₄²	C₂³×C₅₃	1	2	yes	—
351	C₂×C₄²	C₂³×C₅₃	1	1	yes	characteristic
352	#107	—	2	4	no	—
353	#106	—	2	2	no	—
354	C₄⋊C₈	—	2	2	no	—
355	#103	—	2	2	no	—
356	#103	—	2	4	no	—
357	#102	—	2	2	no	—
358	#102	—	2	4	no	—
359	C₄×C₈	—	2	2	no	—
360	#95	—	2	2	no	—
361	#95	—	2	2	no	—
362	#95	—	2	2	no	—
363	C₂⁴	—	2	1	no	—
364	C₂×Q₈	—	2	2	no	—
365	C₂×Q₈	—	2	4	no	—
366	C₂×Q₈	—	2	2	no	—
367	C₂×Q₈	—	2	2	no	—
368	C₂×Q₈	—	2	1	no	—
369	C₂×Q₈	—	2	1	no	—
370	C₂×Q₈	—	2	1	no	—
371	C₂×Q₈	—	2	1	no	—
372	C₂×Q₈	—	2	1	no	—
373	C₂×Q₈	—	8	1	no	—
374	C₂×Q₈	—	8	1	no	—
375	C₂×Q₈	C₂×Q₈×C₅₃	1	2	yes	—
376	C₂×D₄	—	2	1	no	—
377	C₂×D₄	—	2	1	no	—
378	C₂×D₄	—	2	1	no	—
379	C₂×D₄	C₂×D₄×C₅₃	1	2	yes	—
380	C₂×D₄	C₂×Q₈×C₅₃	1	2	yes	—
381	C₂×D₄	#41×C₅₃	1	2	yes	—
382	C₂²×C₄	—	2	1	no	—
383	C₂²×C₄	—	2	1	no	—
384	C₂²×C₄	—	2	2	no	—
385	C₂²×C₄	—	2	8	no	—
386	C₂²×C₄	—	2	4	no	—
387	C₂²×C₄	—	2	2	no	—
388	C₂²×C₄	—	2	1	no	—
389	C₂²×C₄	—	4	2	no	—
390	C₂²×C₄	—	4	2	no	—
391	C₂²×C₄	—	8	2	no	—
392	C₂²×C₄	—	8	2	no	—
393	C₂²×C₄	C₂×D₄×C₅₃	1	2	yes	—
394	C₂²×C₄	C₂×D₄×C₅₃	1	2	yes	—
395	C₂²×C₄	C₂×D₄×C₅₃	1	1	yes	characteristic
396	C₂²×C₄	C₂×D₄×C₅₃	1	1	yes	characteristic
397	C₂²×C₄	C₂⁴×C₅₃	1	1	yes	characteristic
398	SD₁₆	—	2	8	no	—
399	C₂×C₈	—	2	1	no	—
400	C₂×C₈	—	2	1	no	—
401	C₂×C₈	—	2	1	no	—
402	C₂×C₈	—	2	2	no	—
403	C₂×C₈	—	2	2	no	—
404	C₂×C₈	—	2	1	no	—
405	C₂×C₈	—	2	1	no	—
406	C₂×C₈	—	2	2	no	—
407	C₂×C₈	C₂×Q₈×C₅₃	1	2	yes	—
408	C₄⋊C₄	—	2	8	no	—
409	C₄⋊C₄	—	2	8	no	—
410	C₄⋊C₄	—	2	2	no	—
411	C₄⋊C₄	—	2	4	no	—
412	C₄⋊C₄	—	2	4	no	—
413	C₄⋊C₄	—	2	1	no	—
414	C₄⋊C₄	—	2	2	no	—
415	C₄⋊C₄	—	2	1	no	—
416	C₄⋊C₄	—	2	1	no	—
417	C₄⋊C₄	—	2	2	no	—
418	C₄⋊C₄	—	2	2	no	—
419	C₄⋊C₄	—	4	2	no	—
420	C₄⋊C₄	—	4	2	no	—
421	C₄⋊C₄	—	8	1	no	—
422	C₄⋊C₄	—	8	2	no	—
423	C₄⋊C₄	C₂×D₄×C₅₃	1	2	yes	—
424	C₄⋊C₄	C₂×D₄×C₅₃	1	4	yes	—
425	C₂²⋊C₄	—	2	8	no	—
426	C₂²⋊C₄	—	2	4	no	—
427	C₄²	—	2	2	no	—
428	C₄²	—	2	2	no	—
429	C₄²	—	4	2	no	—
430	C₄²	C₂×D₄×C₅₃	1	2	yes	—
431	C₄²	C₂×D₄×C₅₃	1	2	yes	—
432	C₂³	—	2	2	no	—
433	C₂³	—	2	2	no	—
434	C₂³	—	2	2	no	—
435	C₂³	—	4	1	no	—
436	C₂³	—	4	1	no	—
437	C₂³	—	4	1	no	—
438	C₂³	—	4	1	no	—
439	C₂³	C₂²×D₄×C₅₃	1	1	yes	characteristic
440	Q₈	—	2	2	no	—
441	Q₈	—	2	2	no	—
442	Q₈	—	4	1	no	—
443	Q₈	—	4	1	no	—
444	Q₈	—	4	4	no	—
445	Q₈	—	8	4	no	—
446	D₄	—	2	2	no	—
447	D₄	—	2	2	no	—
448	D₄	—	2	2	no	—
449	D₄	C₂²⋊Q₈×C₅₃	1	4	yes	—
450	C₂×C₄	—	2	1	no	—
451	C₂×C₄	—	2	1	no	—
452	C₂×C₄	—	2	2	no	—
453	C₂×C₄	—	2	2	no	—
454	C₂×C₄	—	2	2	no	—
455	C₂×C₄	—	2	4	no	—
456	C₂×C₄	—	2	2	no	—
457	C₂×C₄	—	2	2	no	—
458	C₂×C₄	—	2	2	no	—
459	C₂×C₄	—	2	2	no	—
460	C₂×C₄	—	2	2	no	—
461	C₂×C₄	—	4	1	no	—
462	C₂×C₄	—	4	1	no	—
463	C₂×C₄	—	4	1	no	—
464	C₂×C₄	—	4	1	no	—
465	C₂×C₄	—	4	2	no	—
466	C₂×C₄	—	4	2	no	—
467	C₂×C₄	—	4	8	no	—
468	C₂×C₄	—	4	1	no	—
469	C₂×C₄	—	4	2	no	—
470	C₂×C₄	—	4	2	no	—
471	C₂×C₄	—	4	1	no	—
472	C₂×C₄	—	4	2	no	—
473	C₂×C₄	—	4	1	no	—
474	C₂×C₄	—	8	8	no	—
475	C₂×C₄	#120×C₅₃	1	2	yes	—
476	C₂×C₄	#120×C₅₃	1	2	yes	—
477	C₂×C₄	#135×C₅₃	1	2	yes	—
478	C₂×C₄	#137×C₅₃	1	2	yes	—
479	C₂×C₄	C₂²×D₄×C₅₃	1	1	yes	characteristic
480	C₂×C₄	C₂²×D₄×C₅₃	1	1	yes	characteristic
481	C₂×C₄	C₂²×Q₈×C₅₃	1	1	yes	characteristic
482	C₂×C₄	C₂×#41×C₅₃	1	1	yes	characteristic
483	C₂×C₄	2₋¹⁺⁴×C₅₃	1	1	yes	characteristic
484	C₂×C₄	2₋¹⁺⁴×C₅₃	1	1	yes	characteristic
485	C₈	—	2	2	no	—
486	C₈	—	4	1	no	—
487	C₈	—	4	2	no	—
488	C₂²	—	2	4	no	—
489	C₂²	—	4	2	no	—
490	C₂²	—	4	2	no	—
491	C₂²	—	4	2	no	—
492	C₂²	—	4	2	no	—
493	C₂²	—	4	2	no	—
494	C₂²	—	4	2	no	—
495	C₂²	C₂×#120×C₅₃	1	1	yes	characteristic
496	C₂²	C₅₃×#548	1	1	yes	characteristic
497	C₂²	D₄×Q₈×C₅₃	1	1	yes	characteristic
498	C₂²	C₂×#135×C₅₃	1	1	yes	characteristic
499	C₂²	C₂×#137×C₅₃	1	1	yes	characteristic
500	C₂²	C₅₃×#576	1	1	yes	characteristic
501	C₂²	C₅₃×#577	1	1	yes	characteristic
502	C₄	—	2	2	no	—
503	C₄	—	4	2	no	—
504	C₄	—	4	2	no	—
505	C₄	—	4	2	no	—
506	C₄	—	4	4	no	—
507	C₄	—	4	2	no	—
508	C₄	—	8	1	no	—
509	C₄	—	8	2	no	—
510	C₄	C₅₃×#452	1	4	yes	—
511	C₄	C₂×C₂²⋊Q₈×C₅₃	1	1	yes	characteristic
512	C₄	C₅₃×#536	1	1	yes	characteristic
513	C₄	C₅₃×#548	1	1	yes	characteristic
514	C₄	D₄×Q₈×C₅₃	1	1	yes	characteristic
515	C₂	—	4	4	no	—
516	C₂	C₅₃×#2409	1	1	yes	characteristic
517	C₂	C₂×C₅₃×#452	1	1	yes	characteristic
518	C₂	C₅₃×#3012	1	1	yes	characteristic
519	C₂	C₅₃×#3361	1	1	yes	characteristic
520	C₂	C₅₃×#3372	1	1	yes	characteristic
521	C₂	C₅₃×#3381	1	1	yes	characteristic
522	C₂	C₅₃×#3385	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂³×C₅₃	C₂²×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁴×C₅₃
prime radical	C₂²×D₄×C₅₃	C₂³
LCS	G ▷ C₂²×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×#477	C₂	semidirect
4. C₅₃×#2582	C₂	semidirect
5. C₅₃×#2059	C₂	semidirect
6. C₅₃×#2026	C₂	semidirect
7. C₅₃×#1990	C₂	semidirect
8. C₅₃×#1984	C₂	semidirect
53. C₂×SD₁₆×C₅₃	Q₈	semidirect
226. #24331	C₅₃	direct (N × H)
255. C₂×#477	C₂×C₅₃	semidirect
257. #2582	C₂×C₅₃	semidirect
258. #2059	C₂×C₅₃	semidirect
259. #2026	C₂×C₅₃	semidirect
260. #1990	C₂×C₅₃	semidirect
261. #1984	C₂×C₅₃	semidirect
306. C₅₃	#24331	direct (N × H)
315. C₂×SD₁₆	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×#554	2
C₂×C₅₃×#477	2
C₂×C₄×SD₁₆×C₅₃	2
C₅₃×#2582	2
C₅₃×#2059	2
C₅₃×#2026	2
C₅₃×#1990	2
C₅₃×#1984	2
C₅₃×#1877	2
C₅₃×#1862	2
C₅₃×#1802	2
#24331	53

Subgroups — C53×#24331 #27231843

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅₃×#24331 #27231843