Subgroups — C₅₃×#10783 #27218295

Order 13568.

1204 conjugacy classes of subgroups.

The subgroup lattice has 640 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₅₃×#3428	C₂	1	1	yes	maximal, characteristic
2	C₂×C₅₃×#421	C₂	1	1	yes	maximal, characteristic
3	C₂×C₅₃×#421	C₂	1	1	yes	maximal, characteristic
4	C₂×C₅₃×#344	C₂	1	1	yes	maximal, characteristic, complemented
5	C₅₃×#1740	C₂	1	1	yes	maximal, characteristic, complemented
6	C₅₃×#1733	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂×C₅₃×H(3,ℤ/4)	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂²×#41×C₅₃	C₂²	1	1	yes	characteristic
9	C₂²×M₁₆×C₅₃	C₂²	1	1	yes	characteristic
10	C₂²×M₁₆×C₅₃	C₂²	1	1	yes	characteristic
11	C₅₃×#520	—	2	2	no	—
12	C₅₃×#520	—	2	2	no	—
13	C₅₃×#519	C₄	1	2	yes	complemented
14	C₅₃×#518	—	2	2	no	—
15	C₅₃×#518	C₄	1	2	yes	complemented
16	C₄×#41×C₅₃	—	2	2	no	—
17	C₄×#41×C₅₃	—	2	2	no	—
18	C₂×#117×C₅₃	—	2	2	no	—
19	C₂×#117×C₅₃	C₂²	1	1	yes	characteristic
20	C₂×#117×C₅₃	C₂²	1	1	yes	characteristic
21	C₂×#117×C₅₃	C₂²	1	1	yes	characteristic
22	C₂×#117×C₅₃	C₂²	1	1	yes	characteristic
23	C₅₃×#421	—	2	2	no	—
24	C₅₃×#421	—	2	2	no	—
25	C₅₃×#421	—	2	2	no	—
26	C₅₃×#421	C₄	1	4	yes	complemented
27	C₂×#104×C₅₃	—	2	2	no	—
28	C₂×#104×C₅₃	C₄	1	4	yes	complemented
29	C₅₃×#344	—	2	2	no	—
30	C₅₃×#342	—	2	2	no	—
31	C₅₃×#339	—	2	2	no	—
32	C₅₃×H(3,ℤ/4)	—	2	2	no	—
33	C₂×#41×C₅₃	—	2	1	no	—
34	C₂×#41×C₅₃	—	2	1	no	—
35	C₂×#41×C₅₃	—	2	1	no	—
36	C₂×#41×C₅₃	—	2	1	no	—
37	C₂×#41×C₅₃	—	2	1	no	—
38	C₂×#41×C₅₃	—	2	1	no	—
39	C₂×#41×C₅₃	C₂×C₄	1	2	yes	—
40	C₂×#41×C₅₃	C₂×C₄	1	2	yes	—
41	C₂×#41×C₅₃	D₄	1	2	yes	—
42	C₂×#41×C₅₃	D₄	1	2	yes	—
43	C₂×#41×C₅₃	D₄	1	2	yes	—
44	C₂×#41×C₅₃	Q₈	1	2	yes	—
45	C₂²×Q₈×C₅₃	C₂×C₄	1	1	yes	characteristic
46	C₂²×D₄×C₅₃	—	2	1	no	—
47	C₂²×D₄×C₅₃	C₂×C₄	1	1	yes	characteristic
48	C₂³×C₄×C₅₃	—	2	1	no	—
49	C₂³×C₄×C₅₃	C₂³	1	1	yes	characteristic
50	C₂×M₁₆×C₅₃	—	2	1	no	—
51	C₂×M₁₆×C₅₃	—	2	1	no	—
52	C₂×M₁₆×C₅₃	—	2	1	no	—
53	C₂×M₁₆×C₅₃	—	2	1	no	—
54	C₂×M₁₆×C₅₃	—	2	1	no	—
55	C₂×M₁₆×C₅₃	—	2	1	no	—
56	C₂×M₁₆×C₅₃	—	2	1	no	—
57	C₂×M₁₆×C₅₃	—	2	1	no	—
58	C₂×M₁₆×C₅₃	—	2	1	no	—
59	C₂×M₁₆×C₅₃	—	2	1	no	—
60	C₂×M₁₆×C₅₃	C₂×C₄	1	2	yes	complemented
61	C₂×M₁₆×C₅₃	C₂×C₄	1	2	yes	complemented
62	C₂²×C₈×C₅₃	—	2	1	no	—
63	C₂²×C₈×C₅₃	—	2	1	no	—
64	C₄×Q₈×C₅₃	—	2	4	no	—
65	C₄×Q₈×C₅₃	—	2	4	no	—
66	C₄×D₄×C₅₃	—	2	8	no	—
67	C₄×D₄×C₅₃	—	2	8	no	—
68	C₄×D₄×C₅₃	—	2	4	no	—
69	C₄×D₄×C₅₃	—	2	4	no	—
70	#117×C₅₃	—	2	4	no	—
71	#117×C₅₃	—	2	4	no	—
72	#117×C₅₃	—	2	4	no	—
73	#117×C₅₃	—	2	1	no	—
74	#117×C₅₃	—	2	1	no	—
75	#117×C₅₃	—	2	1	no	—
76	#117×C₅₃	—	2	1	no	—
77	#117×C₅₃	—	2	1	no	—
78	#117×C₅₃	—	2	1	no	—
79	#117×C₅₃	—	2	1	no	—
80	#117×C₅₃	—	2	1	no	—
81	#117×C₅₃	—	2	1	no	—
82	#117×C₅₃	—	2	1	no	—
83	#117×C₅₃	—	2	1	no	—
84	#117×C₅₃	C₂×C₄	1	2	yes	—
85	#117×C₅₃	C₂×C₄	1	2	yes	—
86	#117×C₅₃	C₂×C₄	1	2	yes	complemented
87	#117×C₅₃	D₄	1	2	yes	—
88	#117×C₅₃	D₄	1	2	yes	—
89	C₂×C₄⋊C₄×C₅₃	—	2	4	no	—
90	C₂×C₄⋊C₄×C₅₃	—	2	1	no	—
91	C₂×C₄⋊C₄×C₅₃	—	2	1	no	—
92	C₂×C₄⋊C₄×C₅₃	—	2	1	no	—
93	C₂×C₄⋊C₄×C₅₃	—	2	1	no	—
94	C₂×C₂²⋊C₄×C₅₃	—	2	4	no	—
95	C₂×C₂²⋊C₄×C₅₃	—	2	1	no	—
96	C₂×C₂²⋊C₄×C₅₃	—	2	1	no	—
97	C₂×C₂²⋊C₄×C₅₃	—	2	1	no	—
98	C₂×C₂²⋊C₄×C₅₃	D₄	1	2	yes	—
99	C₂×C₄²×C₅₃	—	2	4	no	—
100	C₂×C₄²×C₅₃	—	2	1	no	—
101	C₂×C₄²×C₅₃	—	2	1	no	—
102	C₂×C₄²×C₅₃	C₂×C₄	1	2	yes	complemented
103	C₂×C₄²×C₅₃	D₄	1	2	yes	—
104	#104×C₅₃	—	2	8	no	—
105	#104×C₅₃	—	2	8	no	—
106	C₂⁴×C₅₃	—	2	1	no	—
107	C₂⁴×C₅₃	C₂×D₄	1	1	yes	characteristic
108	#41×C₅₃	—	2	2	no	—
109	#41×C₅₃	—	2	2	no	—
110	#41×C₅₃	—	2	2	no	—
111	#41×C₅₃	—	2	2	no	—
112	#41×C₅₃	—	2	2	no	—
113	#41×C₅₃	—	2	2	no	—
114	#41×C₅₃	—	2	2	no	—
115	#41×C₅₃	—	2	2	no	—
116	#41×C₅₃	—	2	2	no	—
117	#41×C₅₃	—	2	2	no	—
118	#41×C₅₃	—	2	2	no	—
119	#41×C₅₃	—	2	2	no	—
120	#41×C₅₃	C₂²⋊C₄	1	4	yes	—
121	#41×C₅₃	C₂²⋊C₄	1	4	yes	—
122	#41×C₅₃	C₄⋊C₄	1	4	yes	—
123	#41×C₅₃	C₄⋊C₄	1	4	yes	—
124	C₂×Q₈×C₅₃	—	2	1	no	—
125	C₂×Q₈×C₅₃	—	2	1	no	—
126	C₂×Q₈×C₅₃	—	2	1	no	—
127	C₂×Q₈×C₅₃	C₄²	1	2	yes	—
128	C₂×Q₈×C₅₃	C₂²⋊C₄	1	2	yes	—
129	C₂×Q₈×C₅₃	C₄⋊C₄	1	2	yes	—
130	C₂×D₄×C₅₃	—	2	2	no	—
131	C₂×D₄×C₅₃	—	2	2	no	—
132	C₂×D₄×C₅₃	—	2	2	no	—
133	C₂×D₄×C₅₃	—	2	2	no	—
134	C₂×D₄×C₅₃	—	2	2	no	—
135	C₂×D₄×C₅₃	—	2	2	no	—
136	C₂×D₄×C₅₃	—	2	1	no	—
137	C₂×D₄×C₅₃	—	2	1	no	—
138	C₂×D₄×C₅₃	—	2	1	no	—
139	C₂×D₄×C₅₃	C₄²	1	2	yes	—
140	C₂×D₄×C₅₃	C₂²⋊C₄	1	2	yes	—
141	C₂×D₄×C₅₃	C₄⋊C₄	1	2	yes	—
142	C₂²×C₄×C₅₃	—	2	1	no	—
143	C₂²×C₄×C₅₃	—	2	2	no	—
144	C₂²×C₄×C₅₃	—	2	2	no	—
145	C₂²×C₄×C₅₃	—	2	2	no	—
146	C₂²×C₄×C₅₃	—	2	2	no	—
147	C₂²×C₄×C₅₃	—	2	2	no	—
148	C₂²×C₄×C₅₃	—	2	2	no	—
149	C₂²×C₄×C₅₃	—	2	8	no	—
150	C₂²×C₄×C₅₃	—	2	2	no	—
151	C₂²×C₄×C₅₃	—	2	2	no	—
152	C₂²×C₄×C₅₃	—	2	2	no	—
153	C₂²×C₄×C₅₃	—	4	1	no	—
154	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
155	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
156	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
157	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
158	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
159	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
160	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
161	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
162	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
163	C₂²×C₄×C₅₃	C₂×Q₈	1	1	yes	characteristic
164	C₂²×C₄×C₅₃	#41	1	1	yes	characteristic
165	C₂²×C₄×C₅₃	#41	1	1	yes	characteristic
166	C₂²×C₄×C₅₃	#41	1	1	yes	characteristic
167	C₂²×C₄×C₅₃	#41	1	1	yes	characteristic
168	M₁₆×C₅₃	—	2	2	no	—
169	M₁₆×C₅₃	—	2	2	no	—
170	M₁₆×C₅₃	—	2	2	no	—
171	M₁₆×C₅₃	—	2	2	no	—
172	M₁₆×C₅₃	—	4	1	no	—
173	M₁₆×C₅₃	—	4	1	no	—
174	M₁₆×C₅₃	—	4	1	no	—
175	M₁₆×C₅₃	—	4	1	no	—
176	C₂×C₈×C₅₃	—	2	2	no	—
177	C₂×C₈×C₅₃	—	2	2	no	—
178	C₂×C₈×C₅₃	—	4	1	no	—
179	C₂×C₈×C₅₃	—	4	1	no	—
180	C₂×C₈×C₅₃	—	4	1	no	—
181	C₂×C₈×C₅₃	—	4	1	no	—
182	C₄⋊C₄×C₅₃	—	2	8	no	—
183	C₄⋊C₄×C₅₃	—	2	2	no	—
184	C₄⋊C₄×C₅₃	—	2	2	no	—
185	C₄⋊C₄×C₅₃	—	2	2	no	—
186	C₄⋊C₄×C₅₃	—	2	2	no	—
187	C₄⋊C₄×C₅₃	—	2	2	no	—
188	C₄⋊C₄×C₅₃	—	4	1	no	—
189	C₄⋊C₄×C₅₃	—	4	1	no	—
190	C₄⋊C₄×C₅₃	—	4	1	no	—
191	C₂²⋊C₄×C₅₃	—	2	8	no	—
192	C₂²⋊C₄×C₅₃	—	2	2	no	—
193	C₂²⋊C₄×C₅₃	—	2	2	no	—
194	C₂²⋊C₄×C₅₃	—	2	2	no	—
195	C₂²⋊C₄×C₅₃	—	2	2	no	—
196	C₂²⋊C₄×C₅₃	—	4	1	no	—
197	C₂²⋊C₄×C₅₃	—	4	1	no	—
198	C₂²⋊C₄×C₅₃	—	4	1	no	—
199	C₂²⋊C₄×C₅₃	C₂²⋊C₄	1	4	yes	—
200	C₄²×C₅₃	—	2	8	no	—
201	C₄²×C₅₃	—	2	2	no	—
202	C₄²×C₅₃	—	2	2	no	—
203	C₄²×C₅₃	—	2	2	no	—
204	C₄²×C₅₃	—	2	2	no	—
205	C₄²×C₅₃	—	2	2	no	—
206	C₄²×C₅₃	—	2	2	no	—
207	C₄²×C₅₃	—	4	1	no	—
208	C₄²×C₅₃	C₂²⋊C₄	1	4	yes	—
209	C₂³×C₅₃	—	2	2	no	—
210	C₂³×C₅₃	—	2	2	no	—
211	C₂³×C₅₃	—	2	2	no	—
212	C₂³×C₅₃	—	8	1	no	—
213	C₂³×C₅₃	—	8	2	no	—
214	C₂³×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
215	C₂³×C₅₃	C₄×D₄	1	1	yes	characteristic
216	C₂³×C₅₃	C₄×D₄	1	1	yes	characteristic
217	C₂³×C₅₃	#120	1	1	yes	characteristic
218	C₂³×C₅₃	#120	1	1	yes	characteristic
219	C₂³×C₅₃	#121	1	1	yes	characteristic
220	C₂³×C₅₃	C₂²⋊Q₈	1	1	yes	characteristic
221	Q₈×C₅₃	—	2	2	no	—
222	Q₈×C₅₃	—	2	2	no	—
223	Q₈×C₅₃	—	2	2	no	—
224	Q₈×C₅₃	#95	1	4	yes	—
225	D₄×C₅₃	—	2	4	no	—
226	D₄×C₅₃	—	2	4	no	—
227	D₄×C₅₃	—	2	4	no	—
228	D₄×C₅₃	—	2	4	no	—
229	D₄×C₅₃	—	2	2	no	—
230	D₄×C₅₃	—	2	2	no	—
231	D₄×C₅₃	—	2	2	no	—
232	D₄×C₅₃	#95	1	4	yes	—
233	C₂×C₄×C₅₃	—	2	2	no	—
234	C₂×C₄×C₅₃	—	2	2	no	—
235	C₂×C₄×C₅₃	—	2	2	no	—
236	C₂×C₄×C₅₃	—	2	4	no	—
237	C₂×C₄×C₅₃	—	2	4	no	—
238	C₂×C₄×C₅₃	—	2	4	no	—
239	C₂×C₄×C₅₃	—	2	4	no	—
240	C₂×C₄×C₅₃	—	2	4	no	—
241	C₂×C₄×C₅₃	—	2	4	no	—
242	C₂×C₄×C₅₃	—	4	2	no	—
243	C₂×C₄×C₅₃	—	4	2	no	—
244	C₂×C₄×C₅₃	—	8	2	no	—
245	C₂×C₄×C₅₃	—	8	2	no	—
246	C₂×C₄×C₅₃	—	8	8	no	—
247	C₂×C₄×C₅₃	—	8	2	no	—
248	C₂×C₄×C₅₃	—	8	2	no	—
249	C₂×C₄×C₅₃	C₂×C₄²	1	1	yes	characteristic
250	C₂×C₄×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
251	C₂×C₄×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
252	C₂×C₄×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
253	C₂×C₄×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
254	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
255	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
256	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
257	C₂×C₄×C₅₃	#117	1	1	yes	characteristic
258	C₂×C₄×C₅₃	#117	1	1	yes	characteristic
259	C₂×C₄×C₅₃	C₄×D₄	1	1	yes	characteristic
260	C₂×C₄×C₅₃	C₄×D₄	1	1	yes	characteristic
261	C₂×C₄×C₅₃	C₄×D₄	1	1	yes	characteristic
262	C₂×C₄×C₅₃	C₄×D₄	1	1	yes	characteristic
263	C₂×C₄×C₅₃	C₄×D₄	1	1	yes	characteristic
264	C₂×C₄×C₅₃	C₄×D₄	1	1	yes	characteristic
265	C₂×C₄×C₅₃	#120	1	1	yes	characteristic
266	C₂×C₄×C₅₃	#120	1	1	yes	characteristic
267	C₂×C₄×C₅₃	#121	1	1	yes	characteristic
268	C₂×C₄×C₅₃	#121	1	1	yes	characteristic
269	C₂×C₄×C₅₃	#121	1	1	yes	characteristic
270	C₂×C₄×C₅₃	C₂²⋊Q₈	1	1	yes	characteristic
271	C₂×C₄×C₅₃	C₂²⋊Q₈	1	1	yes	characteristic
272	C₂×C₄×C₅₃	C₂²⋊Q₈	1	1	yes	characteristic
273	C₂×C₄×C₅₃	#123	1	1	yes	characteristic
274	C₂×C₄×C₅₃	#123	1	1	yes	characteristic
275	C₂×C₄×C₅₃	#123	1	1	yes	characteristic
276	C₂×C₄×C₅₃	#123	1	1	yes	characteristic
277	C₈×C₅₃	—	4	2	no	—
278	C₈×C₅₃	—	4	2	no	—
279	#10783	C₅₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
280	C₂²×C₅₃	—	2	4	no	—
281	C₂²×C₅₃	—	4	1	no	—
282	C₂²×C₅₃	—	4	1	no	—
283	C₂²×C₅₃	—	4	1	no	—
284	C₂²×C₅₃	—	4	1	no	—
285	C₂²×C₅₃	—	4	1	no	—
286	C₂²×C₅₃	—	4	1	no	—
287	C₂²×C₅₃	—	4	1	no	—
288	C₂²×C₅₃	—	4	4	no	—
289	C₂²×C₅₃	—	8	2	no	—
290	C₂²×C₅₃	—	8	2	no	—
291	C₂²×C₅₃	C₄×C₂²⋊C₄	1	1	yes	characteristic
292	C₂²×C₅₃	#379	1	1	yes	characteristic
293	C₂²×C₅₃	#380	1	1	yes	characteristic
294	C₂²×C₅₃	#385	1	1	yes	characteristic
295	C₂²×C₅₃	#385	1	1	yes	characteristic
296	C₂²×C₅₃	#386	1	1	yes	characteristic
297	C₂²×C₅₃	#386	1	1	yes	characteristic
298	C₄×C₅₃	—	2	4	no	—
299	C₄×C₅₃	—	4	4	no	—
300	C₄×C₅₃	—	4	4	no	—
301	C₄×C₅₃	—	8	2	no	—
302	C₄×C₅₃	—	8	2	no	—
303	C₄×C₅₃	C₂×#95	1	1	yes	characteristic
304	C₄×C₅₃	C₄×C₂²⋊C₄	1	1	yes	characteristic
305	C₄×C₅₃	#380	1	1	yes	characteristic
306	C₄×C₅₃	#381	1	1	yes	characteristic
307	C₄×C₅₃	#385	1	1	yes	characteristic
308	C₄×C₅₃	#385	1	1	yes	characteristic
309	C₄×C₅₃	#386	1	1	yes	characteristic
310	C₄×C₅₃	#386	1	1	yes	characteristic
311	#3428	C₂×C₅₃	1	1	yes	characteristic
312	C₂×#421	C₂×C₅₃	1	1	yes	characteristic
313	C₂×#421	C₂×C₅₃	1	1	yes	characteristic
314	C₂×#344	C₂×C₅₃	1	1	yes	characteristic, complemented
315	#1740	C₂×C₅₃	1	1	yes	characteristic, complemented
316	#1733	C₂×C₅₃	1	1	yes	characteristic, complemented
317	C₂×H(3,ℤ/4)	C₂×C₅₃	1	1	yes	characteristic, complemented
318	C₂×C₅₃	—	2	1	no	—
319	C₂×C₅₃	—	2	1	no	—
320	C₂×C₅₃	—	2	1	no	—
321	C₂×C₅₃	—	2	1	no	—
322	C₂×C₅₃	—	2	1	no	—
323	C₂×C₅₃	—	2	1	no	—
324	C₂×C₅₃	—	2	1	no	—
325	C₂×C₅₃	—	4	4	no	—
326	C₂×C₅₃	C₂²⋊#95	1	1	yes	characteristic
327	C₂²×#41	C₂²×C₅₃	1	1	yes	characteristic
328	C₂²×M₁₆	C₂²×C₅₃	1	1	yes	characteristic
329	C₂²×M₁₆	C₂²×C₅₃	1	1	yes	characteristic
330	#520	—	2	2	no	—
331	#520	—	2	2	no	—
332	#519	C₄×C₅₃	1	2	yes	complemented
333	#518	—	2	2	no	—
334	#518	C₄×C₅₃	1	2	yes	complemented
335	C₄×#41	—	2	2	no	—
336	C₄×#41	—	2	2	no	—
337	C₂×#117	—	2	2	no	—
338	C₂×#117	C₂²×C₅₃	1	1	yes	characteristic
339	C₂×#117	C₂²×C₅₃	1	1	yes	characteristic
340	C₂×#117	C₂²×C₅₃	1	1	yes	characteristic
341	C₂×#117	C₂²×C₅₃	1	1	yes	characteristic
342	#421	—	2	2	no	—
343	#421	—	2	2	no	—
344	#421	—	2	2	no	—
345	#421	C₄×C₅₃	1	4	yes	complemented
346	C₂×#104	—	2	2	no	—
347	C₂×#104	C₄×C₅₃	1	4	yes	complemented
348	#344	—	2	2	no	—
349	#342	—	2	2	no	—
350	#339	—	2	2	no	—
351	H(3,ℤ/4)	—	2	2	no	—
352	C₅₃	#10783	1	1	yes	characteristic, direct product factor, Sylow
353	C₂×#41	—	2	1	no	—
354	C₂×#41	—	2	1	no	—
355	C₂×#41	—	2	1	no	—
356	C₂×#41	—	2	1	no	—
357	C₂×#41	—	2	1	no	—
358	C₂×#41	—	2	1	no	—
359	C₂×#41	C₂×C₄×C₅₃	1	2	yes	—
360	C₂×#41	C₂×C₄×C₅₃	1	2	yes	—
361	C₂×#41	D₄×C₅₃	1	2	yes	—
362	C₂×#41	D₄×C₅₃	1	2	yes	—
363	C₂×#41	D₄×C₅₃	1	2	yes	—
364	C₂×#41	Q₈×C₅₃	1	2	yes	—
365	C₂²×Q₈	C₂×C₄×C₅₃	1	1	yes	characteristic
366	C₂²×D₄	—	2	1	no	—
367	C₂²×D₄	C₂×C₄×C₅₃	1	1	yes	characteristic
368	C₂³×C₄	—	2	1	no	—
369	C₂³×C₄	C₂³×C₅₃	1	1	yes	characteristic
370	C₂×M₁₆	—	2	1	no	—
371	C₂×M₁₆	—	2	1	no	—
372	C₂×M₁₆	—	2	1	no	—
373	C₂×M₁₆	—	2	1	no	—
374	C₂×M₁₆	—	2	1	no	—
375	C₂×M₁₆	—	2	1	no	—
376	C₂×M₁₆	—	2	1	no	—
377	C₂×M₁₆	—	2	1	no	—
378	C₂×M₁₆	—	2	1	no	—
379	C₂×M₁₆	—	2	1	no	—
380	C₂×M₁₆	C₂×C₄×C₅₃	1	2	yes	complemented
381	C₂×M₁₆	C₂×C₄×C₅₃	1	2	yes	complemented
382	C₂²×C₈	—	2	1	no	—
383	C₂²×C₈	—	2	1	no	—
384	C₄×Q₈	—	2	4	no	—
385	C₄×Q₈	—	2	4	no	—
386	C₄×D₄	—	2	8	no	—
387	C₄×D₄	—	2	8	no	—
388	C₄×D₄	—	2	4	no	—
389	C₄×D₄	—	2	4	no	—
390	#117	—	2	4	no	—
391	#117	—	2	4	no	—
392	#117	—	2	1	no	—
393	#117	—	2	1	no	—
394	#117	—	2	1	no	—
395	#117	—	2	1	no	—
396	#117	—	2	4	no	—
397	#117	—	2	1	no	—
398	#117	—	2	1	no	—
399	#117	—	2	1	no	—
400	#117	—	2	1	no	—
401	#117	—	2	1	no	—
402	#117	—	2	1	no	—
403	#117	—	2	1	no	—
404	#117	C₂×C₄×C₅₃	1	2	yes	—
405	#117	C₂×C₄×C₅₃	1	2	yes	—
406	#117	C₂×C₄×C₅₃	1	2	yes	complemented
407	#117	D₄×C₅₃	1	2	yes	—
408	#117	D₄×C₅₃	1	2	yes	—
409	C₂×C₄⋊C₄	—	2	4	no	—
410	C₂×C₄⋊C₄	—	2	1	no	—
411	C₂×C₄⋊C₄	—	2	1	no	—
412	C₂×C₄⋊C₄	—	2	1	no	—
413	C₂×C₄⋊C₄	—	2	1	no	—
414	C₂×C₂²⋊C₄	—	2	1	no	—
415	C₂×C₂²⋊C₄	—	2	4	no	—
416	C₂×C₂²⋊C₄	—	2	1	no	—
417	C₂×C₂²⋊C₄	—	2	1	no	—
418	C₂×C₂²⋊C₄	D₄×C₅₃	1	2	yes	—
419	C₂×C₄²	—	2	4	no	—
420	C₂×C₄²	—	2	1	no	—
421	C₂×C₄²	—	2	1	no	—
422	C₂×C₄²	C₂×C₄×C₅₃	1	2	yes	complemented
423	C₂×C₄²	D₄×C₅₃	1	2	yes	—
424	#104	—	2	8	no	—
425	#104	—	2	8	no	—
426	C₂⁴	—	2	1	no	—
427	C₂⁴	C₂×D₄×C₅₃	1	1	yes	characteristic
428	#41	—	2	2	no	—
429	#41	—	2	2	no	—
430	#41	—	2	2	no	—
431	#41	—	2	2	no	—
432	#41	—	2	2	no	—
433	#41	—	2	2	no	—
434	#41	—	2	2	no	—
435	#41	—	2	2	no	—
436	#41	—	2	2	no	—
437	#41	—	2	2	no	—
438	#41	—	2	2	no	—
439	#41	—	2	2	no	—
440	#41	C₂²⋊C₄×C₅₃	1	4	yes	—
441	#41	C₂²⋊C₄×C₅₃	1	4	yes	—
442	#41	C₄⋊C₄×C₅₃	1	4	yes	—
443	#41	C₄⋊C₄×C₅₃	1	4	yes	—
444	C₂×Q₈	—	2	1	no	—
445	C₂×Q₈	—	2	1	no	—
446	C₂×Q₈	—	2	1	no	—
447	C₂×Q₈	C₄²×C₅₃	1	2	yes	—
448	C₂×Q₈	C₂²⋊C₄×C₅₃	1	2	yes	—
449	C₂×Q₈	C₄⋊C₄×C₅₃	1	2	yes	—
450	C₂×D₄	—	2	2	no	—
451	C₂×D₄	—	2	2	no	—
452	C₂×D₄	—	2	2	no	—
453	C₂×D₄	—	2	2	no	—
454	C₂×D₄	—	2	2	no	—
455	C₂×D₄	—	2	2	no	—
456	C₂×D₄	—	2	1	no	—
457	C₂×D₄	—	2	1	no	—
458	C₂×D₄	—	2	1	no	—
459	C₂×D₄	C₄²×C₅₃	1	2	yes	—
460	C₂×D₄	C₂²⋊C₄×C₅₃	1	2	yes	—
461	C₂×D₄	C₄⋊C₄×C₅₃	1	2	yes	—
462	C₂²×C₄	—	2	1	no	—
463	C₂²×C₄	—	2	2	no	—
464	C₂²×C₄	—	2	2	no	—
465	C₂²×C₄	—	2	2	no	—
466	C₂²×C₄	—	2	2	no	—
467	C₂²×C₄	—	2	2	no	—
468	C₂²×C₄	—	2	2	no	—
469	C₂²×C₄	—	2	8	no	—
470	C₂²×C₄	—	2	2	no	—
471	C₂²×C₄	—	2	2	no	—
472	C₂²×C₄	—	2	2	no	—
473	C₂²×C₄	—	4	1	no	—
474	C₂²×C₄	C₂²×C₄×C₅₃	1	1	yes	characteristic
475	C₂²×C₄	C₂²×C₄×C₅₃	1	1	yes	characteristic
476	C₂²×C₄	C₂²×C₄×C₅₃	1	1	yes	characteristic
477	C₂²×C₄	C₂×D₄×C₅₃	1	1	yes	characteristic
478	C₂²×C₄	C₂×D₄×C₅₃	1	1	yes	characteristic
479	C₂²×C₄	C₂×D₄×C₅₃	1	1	yes	characteristic
480	C₂²×C₄	C₂×D₄×C₅₃	1	1	yes	characteristic
481	C₂²×C₄	C₂×D₄×C₅₃	1	1	yes	characteristic
482	C₂²×C₄	C₂×D₄×C₅₃	1	1	yes	characteristic
483	C₂²×C₄	C₂×Q₈×C₅₃	1	1	yes	characteristic
484	C₂²×C₄	#41×C₅₃	1	1	yes	characteristic
485	C₂²×C₄	#41×C₅₃	1	1	yes	characteristic
486	C₂²×C₄	#41×C₅₃	1	1	yes	characteristic
487	C₂²×C₄	#41×C₅₃	1	1	yes	characteristic
488	M₁₆	—	2	2	no	—
489	M₁₆	—	2	2	no	—
490	M₁₆	—	2	2	no	—
491	M₁₆	—	2	2	no	—
492	M₁₆	—	4	1	no	—
493	M₁₆	—	4	1	no	—
494	M₁₆	—	4	1	no	—
495	M₁₆	—	4	1	no	—
496	C₂×C₈	—	2	2	no	—
497	C₂×C₈	—	2	2	no	—
498	C₂×C₈	—	4	1	no	—
499	C₂×C₈	—	4	1	no	—
500	C₂×C₈	—	4	1	no	—
501	C₂×C₈	—	4	1	no	—
502	C₄⋊C₄	—	2	8	no	—
503	C₄⋊C₄	—	2	2	no	—
504	C₄⋊C₄	—	2	2	no	—
505	C₄⋊C₄	—	2	2	no	—
506	C₄⋊C₄	—	2	2	no	—
507	C₄⋊C₄	—	2	2	no	—
508	C₄⋊C₄	—	4	1	no	—
509	C₄⋊C₄	—	4	1	no	—
510	C₄⋊C₄	—	4	1	no	—
511	C₂²⋊C₄	—	2	8	no	—
512	C₂²⋊C₄	—	2	2	no	—
513	C₂²⋊C₄	—	2	2	no	—
514	C₂²⋊C₄	—	2	2	no	—
515	C₂²⋊C₄	—	2	2	no	—
516	C₂²⋊C₄	—	4	1	no	—
517	C₂²⋊C₄	—	4	1	no	—
518	C₂²⋊C₄	—	4	1	no	—
519	C₂²⋊C₄	C₂²⋊C₄×C₅₃	1	4	yes	—
520	C₄²	—	2	8	no	—
521	C₄²	—	2	2	no	—
522	C₄²	—	2	2	no	—
523	C₄²	—	2	2	no	—
524	C₄²	—	2	2	no	—
525	C₄²	—	2	2	no	—
526	C₄²	—	2	2	no	—
527	C₄²	—	4	1	no	—
528	C₄²	C₂²⋊C₄×C₅₃	1	4	yes	—
529	C₂³	—	2	2	no	—
530	C₂³	—	2	2	no	—
531	C₂³	—	2	2	no	—
532	C₂³	—	8	1	no	—
533	C₂³	—	8	2	no	—
534	C₂³	C₂×C₂²⋊C₄×C₅₃	1	1	yes	characteristic
535	C₂³	C₄×D₄×C₅₃	1	1	yes	characteristic
536	C₂³	C₄×D₄×C₅₃	1	1	yes	characteristic
537	C₂³	#120×C₅₃	1	1	yes	characteristic
538	C₂³	#120×C₅₃	1	1	yes	characteristic
539	C₂³	#121×C₅₃	1	1	yes	characteristic
540	C₂³	C₂²⋊Q₈×C₅₃	1	1	yes	characteristic
541	Q₈	—	2	2	no	—
542	Q₈	—	2	2	no	—
543	Q₈	—	2	2	no	—
544	Q₈	#95×C₅₃	1	4	yes	—
545	D₄	—	2	4	no	—
546	D₄	—	2	4	no	—
547	D₄	—	2	4	no	—
548	D₄	—	2	4	no	—
549	D₄	—	2	2	no	—
550	D₄	—	2	2	no	—
551	D₄	—	2	2	no	—
552	D₄	#95×C₅₃	1	4	yes	—
553	C₂×C₄	—	2	2	no	—
554	C₂×C₄	—	2	2	no	—
555	C₂×C₄	—	2	2	no	—
556	C₂×C₄	—	2	4	no	—
557	C₂×C₄	—	2	4	no	—
558	C₂×C₄	—	2	4	no	—
559	C₂×C₄	—	2	4	no	—
560	C₂×C₄	—	2	4	no	—
561	C₂×C₄	—	2	4	no	—
562	C₂×C₄	—	4	2	no	—
563	C₂×C₄	—	4	2	no	—
564	C₂×C₄	—	8	2	no	—
565	C₂×C₄	—	8	2	no	—
566	C₂×C₄	—	8	8	no	—
567	C₂×C₄	—	8	2	no	—
568	C₂×C₄	—	8	2	no	—
569	C₂×C₄	C₂×C₄²×C₅₃	1	1	yes	characteristic
570	C₂×C₄	C₂×C₂²⋊C₄×C₅₃	1	1	yes	characteristic
571	C₂×C₄	C₂×C₂²⋊C₄×C₅₃	1	1	yes	characteristic
572	C₂×C₄	C₂×C₂²⋊C₄×C₅₃	1	1	yes	characteristic
573	C₂×C₄	C₂×C₂²⋊C₄×C₅₃	1	1	yes	characteristic
574	C₂×C₄	C₂×C₄⋊C₄×C₅₃	1	1	yes	characteristic
575	C₂×C₄	C₂×C₄⋊C₄×C₅₃	1	1	yes	characteristic
576	C₂×C₄	C₂×C₄⋊C₄×C₅₃	1	1	yes	characteristic
577	C₂×C₄	#117×C₅₃	1	1	yes	characteristic
578	C₂×C₄	#117×C₅₃	1	1	yes	characteristic
579	C₂×C₄	C₄×D₄×C₅₃	1	1	yes	characteristic
580	C₂×C₄	C₄×D₄×C₅₃	1	1	yes	characteristic
581	C₂×C₄	C₄×D₄×C₅₃	1	1	yes	characteristic
582	C₂×C₄	C₄×D₄×C₅₃	1	1	yes	characteristic
583	C₂×C₄	C₄×D₄×C₅₃	1	1	yes	characteristic
584	C₂×C₄	C₄×D₄×C₅₃	1	1	yes	characteristic
585	C₂×C₄	#120×C₅₃	1	1	yes	characteristic
586	C₂×C₄	#120×C₅₃	1	1	yes	characteristic
587	C₂×C₄	#121×C₅₃	1	1	yes	characteristic
588	C₂×C₄	#121×C₅₃	1	1	yes	characteristic
589	C₂×C₄	#121×C₅₃	1	1	yes	characteristic
590	C₂×C₄	C₂²⋊Q₈×C₅₃	1	1	yes	characteristic
591	C₂×C₄	C₂²⋊Q₈×C₅₃	1	1	yes	characteristic
592	C₂×C₄	C₂²⋊Q₈×C₅₃	1	1	yes	characteristic
593	C₂×C₄	#123×C₅₃	1	1	yes	characteristic
594	C₂×C₄	#123×C₅₃	1	1	yes	characteristic
595	C₂×C₄	#123×C₅₃	1	1	yes	characteristic
596	C₂×C₄	#123×C₅₃	1	1	yes	characteristic
597	C₈	—	4	2	no	—
598	C₈	—	4	2	no	—
599	C₂²	—	2	4	no	—
600	C₂²	—	4	1	no	—
601	C₂²	—	4	1	no	—
602	C₂²	—	4	1	no	—
603	C₂²	—	4	1	no	—
604	C₂²	—	4	1	no	—
605	C₂²	—	4	1	no	—
606	C₂²	—	4	1	no	—
607	C₂²	—	4	4	no	—
608	C₂²	—	8	2	no	—
609	C₂²	—	8	2	no	—
610	C₂²	C₄×C₂²⋊C₄×C₅₃	1	1	yes	characteristic
611	C₂²	C₅₃×#379	1	1	yes	characteristic
612	C₂²	C₅₃×#380	1	1	yes	characteristic
613	C₂²	C₅₃×#385	1	1	yes	characteristic
614	C₂²	C₅₃×#385	1	1	yes	characteristic
615	C₂²	C₅₃×#386	1	1	yes	characteristic
616	C₂²	C₅₃×#386	1	1	yes	characteristic
617	C₄	—	2	4	no	—
618	C₄	—	4	4	no	—
619	C₄	—	4	4	no	—
620	C₄	—	8	2	no	—
621	C₄	—	8	2	no	—
622	C₄	C₂×#95×C₅₃	1	1	yes	characteristic
623	C₄	C₄×C₂²⋊C₄×C₅₃	1	1	yes	characteristic
624	C₄	C₅₃×#380	1	1	yes	characteristic
625	C₄	C₅₃×#381	1	1	yes	characteristic
626	C₄	C₅₃×#385	1	1	yes	characteristic
627	C₄	C₅₃×#385	1	1	yes	characteristic
628	C₄	C₅₃×#386	1	1	yes	characteristic
629	C₄	C₅₃×#386	1	1	yes	characteristic
630	C₂	—	2	1	no	—
631	C₂	—	2	1	no	—
632	C₂	—	2	1	no	—
633	C₂	—	2	1	no	—
634	C₂	—	2	1	no	—
635	C₂	—	2	1	no	—
636	C₂	—	2	1	no	—
637	C₂	—	4	4	no	—
638	C₂	C₅₃×C₂²⋊#95	1	1	yes	characteristic

	subgroup	G/H
center	C₄×C₅₃	#386
commutator	C₂×C₄	C₂×C₄²×C₅₃
Frattini, cosocle	C₂³×C₄	C₂³×C₅₃
socle	C₂×C₅₃	C₂²⋊#95
prime radical	C₂²×#41×C₅₃	C₂²
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄×C₅₃ ◁ C₂³×C₄×C₅₃ ◁ G

Complements

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

N	H ≅ G/N	type
4. C₂×C₅₃×#344	C₂	semidirect
5. C₅₃×#1740	C₂	semidirect
6. C₅₃×#1733	C₂	semidirect
7. C₂×C₅₃×H(3,ℤ/4)	C₂	semidirect
13. C₅₃×#519	C₄	semidirect
15. C₅₃×#518	C₄	semidirect
26. C₅₃×#421	C₄	semidirect
28. C₂×#104×C₅₃	C₄	semidirect
60. C₂×M₁₆×C₅₃	C₂×C₄	semidirect
61. C₂×M₁₆×C₅₃	C₂×C₄	semidirect
86. #117×C₅₃	C₂×C₄	semidirect
102. C₂×C₄²×C₅₃	C₂×C₄	semidirect
279. #10783	C₅₃	direct (N × H)
314. C₂×#344	C₂×C₅₃	semidirect
315. #1740	C₂×C₅₃	semidirect
316. #1733	C₂×C₅₃	semidirect
317. C₂×H(3,ℤ/4)	C₂×C₅₃	semidirect
332. #519	C₄×C₅₃	semidirect
334. #518	C₄×C₅₃	semidirect
345. #421	C₄×C₅₃	semidirect
347. C₂×#104	C₄×C₅₃	semidirect
352. C₅₃	#10783	direct (N × H)
380. C₂×M₁₆	C₂×C₄×C₅₃	semidirect
381. C₂×M₁₆	C₂×C₄×C₅₃	semidirect
406. #117	C₂×C₄×C₅₃	semidirect
422. C₂×C₄²	C₂×C₄×C₅₃	semidirect

Maximal subgroups

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

subgroup	index
C₅₃×#3428	2
C₂×C₅₃×#421	2
C₂×C₅₃×#344	2
C₅₃×#1740	2
C₅₃×#1733	2
C₂×C₅₃×H(3,ℤ/4)	2
#10783	53

Subgroups — C53×#10783 #27218295

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅₃×#10783 #27218295