Subgroups — C₅₃×#17453 #27224965

Order 13568.

1222 conjugacy classes of subgroups.

The subgroup lattice has 674 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×H(5,𝔽₂)×C₅₃	C₂	1	1	yes	maximal, characteristic
2	C₅₃×#2989	C₂	1	1	yes	maximal, characteristic, complemented
3	C₅₃×#2965	C₂	1	1	yes	maximal, characteristic, complemented
4	C₅₃×#1490	C₂	1	2	yes	maximal, complemented
5	C₅₃×#1489	C₂	1	2	yes	maximal, complemented
6	C₅₃×#1488	C₂	1	2	yes	maximal, complemented
7	C₅₃×#1486	C₂	1	2	yes	maximal, complemented
8	C₅₃×#1478	C₂	1	1	yes	maximal, characteristic, complemented
9	C₅₃×#1477	C₂	1	1	yes	maximal, characteristic, complemented
10	C₅₃×#1476	C₂	1	1	yes	maximal, characteristic, complemented
11	C₂×C₅₃×D₄⋊C₈	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×H(5,𝔽₂)×C₅₃	C₄	1	1	yes	characteristic
13	C₅₃×#520	—	2	1	no	—
14	C₅₃×#520	C₄	1	2	yes	—
15	C₅₃×#520	C₄	1	1	yes	characteristic
16	C₅₃×#520	C₄	1	1	yes	characteristic
17	C₅₃×#518	—	2	2	no	—
18	C₅₃×#518	—	2	1	no	—
19	C₅₃×#518	C₄	1	2	yes	—
20	C₅₃×#518	C₄	1	1	yes	characteristic
21	C₄×#41×C₅₃	—	2	1	no	—
22	C₄×#41×C₅₃	C₂²	1	1	yes	characteristic
23	C₄×#41×C₅₃	C₂²	1	2	yes	—
24	C₄×#41×C₅₃	C₂²	1	1	yes	characteristic
25	C₂×C₄×D₄×C₅₃	—	2	2	no	—
26	C₂×C₄×D₄×C₅₃	—	2	1	no	—
27	C₂×C₄×D₄×C₅₃	C₂²	1	1	yes	characteristic
28	C₂×C₄×D₄×C₅₃	C₂²	1	2	yes	—
29	C₅₃×#446	—	2	1	no	—
30	C₅₃×#446	—	2	1	no	—
31	C₅₃×#436	—	2	1	no	—
32	C₅₃×#436	—	2	1	no	—
33	C₅₃×#435	—	2	2	no	—
34	C₅₃×#435	—	2	2	no	—
35	C₅₃×#432	C₂²	1	2	yes	complemented
36	C₅₃×#431	C₂²	1	1	yes	characteristic, complemented
37	C₅₃×#431	C₂²	1	1	yes	characteristic, complemented
38	C₅₃×#431	C₂²	1	2	yes	complemented
39	C₅₃×#423	C₂²	1	2	yes	complemented
40	C₂×C₄⋊C₈×C₅₃	C₂²	1	1	yes	characteristic, complemented
41	C₄×M₁₆×C₅₃	C₂²	1	2	yes	complemented
42	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic, complemented
43	C₅₃×Q₈⋊C₈	C₂²	1	1	yes	characteristic, complemented
44	C₅₃×Q₈⋊C₈	C₂²	1	1	yes	characteristic, complemented
45	C₅₃×Q₈⋊C₈	C₂²	1	2	yes	complemented
46	C₅₃×D₄⋊C₈	C₂²	1	2	yes	complemented
47	C₅₃×D₄⋊C₈	C₂²	1	2	yes	complemented
48	C₅₃×D₄⋊C₈	C₂²	1	1	yes	characteristic, complemented
49	C₅₃×D₄⋊C₈	C₂²	1	1	yes	characteristic, complemented
50	C₅₃×D₄⋊C₈	C₂²	1	2	yes	complemented
51	C₅₃×D₄⋊C₈	C₂²	1	2	yes	complemented
52	C₅₃×D₄⋊C₈	C₂²	1	2	yes	complemented
53	H(5,𝔽₂)×C₅₃	—	2	4	no	—
54	H(5,𝔽₂)×C₅₃	—	2	4	no	—
55	C₂×#41×C₅₃	—	2	1	no	—
56	C₂×#41×C₅₃	C₂×C₄	1	2	yes	—
57	C₂×#41×C₅₃	C₂×C₄	1	1	yes	characteristic
58	C₂×#41×C₅₃	C₂×C₄	1	1	yes	characteristic
59	C₂²×D₄×C₅₃	—	2	2	no	—
60	C₂²×D₄×C₅₃	—	2	1	no	—
61	C₂²×D₄×C₅₃	C₂×C₄	1	2	yes	—
62	C₂²×D₄×C₅₃	C₂×C₄	1	1	yes	characteristic
63	C₂³×C₄×C₅₃	—	2	1	no	—
64	C₂³×C₄×C₅₃	—	2	2	no	—
65	C₂×M₁₆×C₅₃	—	2	2	no	—
66	C₂×M₁₆×C₅₃	—	2	2	no	—
67	C₂²×C₈×C₅₃	—	2	1	no	—
68	C₂²×C₈×C₅₃	—	2	1	no	—
69	C₄×Q₈×C₅₃	—	2	1	no	—
70	C₄×Q₈×C₅₃	C₂×C₄	1	1	yes	characteristic
71	C₄×Q₈×C₅₃	C₂×C₄	1	2	yes	—
72	C₄×Q₈×C₅₃	D₄	1	1	yes	characteristic
73	C₄×Q₈×C₅₃	D₄	1	1	yes	characteristic
74	C₄×Q₈×C₅₃	C₂³	1	1	yes	characteristic
75	C₄×D₄×C₅₃	—	2	2	no	—
76	C₄×D₄×C₅₃	—	2	1	no	—
77	C₄×D₄×C₅₃	—	2	2	no	—
78	C₄×D₄×C₅₃	—	2	1	no	—
79	C₄×D₄×C₅₃	—	2	2	no	—
80	C₄×D₄×C₅₃	—	2	2	no	—
81	C₄×D₄×C₅₃	—	2	2	no	—
82	C₄×D₄×C₅₃	—	2	1	no	—
83	C₄×D₄×C₅₃	—	2	2	no	—
84	C₄×D₄×C₅₃	—	2	1	no	—
85	C₄×D₄×C₅₃	—	2	2	no	—
86	C₄×D₄×C₅₃	—	2	1	no	—
87	C₄×D₄×C₅₃	—	2	2	no	—
88	C₄×D₄×C₅₃	—	2	1	no	—
89	C₄×D₄×C₅₃	—	2	2	no	—
90	C₄×D₄×C₅₃	—	2	1	no	—
91	C₄×D₄×C₅₃	—	2	2	no	—
92	C₄×D₄×C₅₃	C₂×C₄	1	2	yes	—
93	C₄×D₄×C₅₃	C₂×C₄	1	2	yes	—
94	C₄×D₄×C₅₃	C₂×C₄	1	2	yes	—
95	C₄×D₄×C₅₃	C₂×C₄	1	1	yes	characteristic
96	C₄×D₄×C₅₃	C₂×C₄	1	2	yes	—
97	C₄×D₄×C₅₃	D₄	1	2	yes	—
98	C₄×D₄×C₅₃	D₄	1	1	yes	characteristic
99	C₄×D₄×C₅₃	D₄	1	2	yes	—
100	C₄×D₄×C₅₃	D₄	1	1	yes	characteristic
101	C₄×D₄×C₅₃	C₂³	1	2	yes	—
102	C₄×D₄×C₅₃	C₂³	1	1	yes	characteristic
103	#117×C₅₃	—	2	2	no	—
104	#117×C₅₃	—	2	1	no	—
105	#117×C₅₃	—	2	1	no	—
106	#117×C₅₃	—	2	2	no	—
107	#117×C₅₃	C₂×C₄	1	2	yes	—
108	#117×C₅₃	C₂×C₄	1	1	yes	characteristic
109	#117×C₅₃	C₂×C₄	1	2	yes	—
110	#117×C₅₃	C₂×C₄	1	1	yes	characteristic
111	C₂×C₄⋊C₄×C₅₃	—	2	2	no	—
112	C₂×C₄⋊C₄×C₅₃	—	2	1	no	—
113	C₂×C₄⋊C₄×C₅₃	C₂×C₄	1	2	yes	—
114	C₂×C₄⋊C₄×C₅₃	C₂×C₄	1	1	yes	characteristic
115	C₂×C₂²⋊C₄×C₅₃	—	2	2	no	—
116	C₂×C₂²⋊C₄×C₅₃	—	2	2	no	—
117	C₂×C₂²⋊C₄×C₅₃	—	2	2	no	—
118	C₂×C₂²⋊C₄×C₅₃	—	2	1	no	—
119	C₂×C₂²⋊C₄×C₅₃	—	2	2	no	—
120	C₂×C₄²×C₅₃	—	2	1	no	—
121	C₂×C₄²×C₅₃	—	2	2	no	—
122	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
123	C₂×C₄²×C₅₃	C₂³	1	2	yes	—
124	C₄⋊C₈×C₅₃	—	2	1	no	—
125	C₄⋊C₈×C₅₃	—	2	2	no	—
126	C₄⋊C₈×C₅₃	—	2	2	no	—
127	C₄⋊C₈×C₅₃	—	2	1	no	—
128	C₄⋊C₈×C₅₃	C₂³	1	2	yes	—
129	C₄⋊C₈×C₅₃	C₂³	1	1	yes	characteristic
130	C₄⋊C₈×C₅₃	C₂³	1	1	yes	characteristic
131	C₂²⋊C₈×C₅₃	—	2	1	no	—
132	C₂²⋊C₈×C₅₃	—	2	1	no	—
133	C₂²⋊C₈×C₅₃	—	2	2	no	—
134	C₂²⋊C₈×C₅₃	—	2	2	no	—
135	#97×C₅₃	—	2	2	no	—
136	#97×C₅₃	—	2	2	no	—
137	C₄×C₈×C₅₃	—	2	1	no	—
138	C₄×C₈×C₅₃	—	2	1	no	—
139	C₄×C₈×C₅₃	C₂³	1	1	yes	characteristic
140	C₄×C₈×C₅₃	C₂³	1	1	yes	characteristic
141	C₄×C₈×C₅₃	C₂³	1	2	yes	—
142	C₂⁴×C₅₃	—	2	2	no	—
143	C₂⁴×C₅₃	—	2	1	no	—
144	#41×C₅₃	—	2	8	no	—
145	#41×C₅₃	—	2	4	no	—
146	#41×C₅₃	—	2	4	no	—
147	#41×C₅₃	—	2	2	no	—
148	#41×C₅₃	—	2	2	no	—
149	#41×C₅₃	—	2	2	no	—
150	#41×C₅₃	M₁₆	1	4	yes	complemented
151	C₂×Q₈×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
152	C₂×Q₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
153	C₂×D₄×C₅₃	—	2	16	no	—
154	C₂×D₄×C₅₃	—	2	8	no	—
155	C₂×D₄×C₅₃	—	2	2	no	—
156	C₂×D₄×C₅₃	—	2	1	no	—
157	C₂×D₄×C₅₃	—	2	4	no	—
158	C₂×D₄×C₅₃	—	2	4	no	—
159	C₂×D₄×C₅₃	—	2	2	no	—
160	C₂×D₄×C₅₃	—	2	1	no	—
161	C₂×D₄×C₅₃	—	2	2	no	—
162	C₂×D₄×C₅₃	C₂²⋊C₄	1	2	yes	—
163	C₂×D₄×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
164	C₂×D₄×C₅₃	M₁₆	1	4	yes	complemented
165	C₂×D₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
166	C₂×D₄×C₅₃	C₂²×C₄	1	2	yes	—
167	C₂²×C₄×C₅₃	—	2	2	no	—
168	C₂²×C₄×C₅₃	—	2	1	no	—
169	C₂²×C₄×C₅₃	—	2	2	no	—
170	C₂²×C₄×C₅₃	—	2	1	no	—
171	C₂²×C₄×C₅₃	—	2	1	no	—
172	C₂²×C₄×C₅₃	—	2	2	no	—
173	C₂²×C₄×C₅₃	—	2	1	no	—
174	C₂²×C₄×C₅₃	—	2	2	no	—
175	C₂²×C₄×C₅₃	—	2	2	no	—
176	C₂²×C₄×C₅₃	—	2	2	no	—
177	C₂²×C₄×C₅₃	—	2	2	no	—
178	C₂²×C₄×C₅₃	—	8	1	no	—
179	C₂²×C₄×C₅₃	—	8	2	no	—
180	C₂²×C₄×C₅₃	—	8	1	no	—
181	C₂²×C₄×C₅₃	—	8	2	no	—
182	C₂²×C₄×C₅₃	C₂²×C₄	1	2	yes	—
183	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
184	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
185	C₂²×C₄×C₅₃	C₂×D₄	1	2	yes	—
186	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
187	C₂²×C₄×C₅₃	C₂×D₄	1	2	yes	—
188	M₁₆×C₅₃	—	4	4	no	—
189	M₁₆×C₅₃	—	8	2	no	—
190	C₂×C₈×C₅₃	—	2	1	no	—
191	C₂×C₈×C₅₃	—	2	1	no	—
192	C₂×C₈×C₅₃	—	2	2	no	—
193	C₂×C₈×C₅₃	—	2	1	no	—
194	C₂×C₈×C₅₃	—	2	1	no	—
195	C₂×C₈×C₅₃	—	2	2	no	—
196	C₂×C₈×C₅₃	—	4	2	no	—
197	C₂×C₈×C₅₃	—	8	1	no	—
198	C₄⋊C₄×C₅₃	—	2	1	no	—
199	C₄⋊C₄×C₅₃	—	2	2	no	—
200	C₄⋊C₄×C₅₃	—	2	2	no	—
201	C₄⋊C₄×C₅₃	—	2	1	no	—
202	C₄⋊C₄×C₅₃	—	2	1	no	—
203	C₄⋊C₄×C₅₃	C₂²⋊C₄	1	2	yes	—
204	C₄⋊C₄×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
205	C₄⋊C₄×C₅₃	C₂²⋊C₄	1	2	yes	—
206	C₄⋊C₄×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
207	C₄⋊C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
208	C₄⋊C₄×C₅₃	C₂²×C₄	1	2	yes	—
209	C₄⋊C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
210	C₂²⋊C₄×C₅₃	—	2	2	no	—
211	C₂²⋊C₄×C₅₃	—	2	1	no	—
212	C₂²⋊C₄×C₅₃	—	2	2	no	—
213	C₂²⋊C₄×C₅₃	—	2	2	no	—
214	C₂²⋊C₄×C₅₃	—	2	2	no	—
215	C₂²⋊C₄×C₅₃	—	2	1	no	—
216	C₂²⋊C₄×C₅₃	—	2	2	no	—
217	C₂²⋊C₄×C₅₃	—	2	2	no	—
218	C₂²⋊C₄×C₅₃	—	2	1	no	—
219	C₂²⋊C₄×C₅₃	C₂²⋊C₄	1	2	yes	—
220	C₂²⋊C₄×C₅₃	C₂²⋊C₄	1	2	yes	—
221	C₂²⋊C₄×C₅₃	C₂²⋊C₄	1	2	yes	—
222	C₄²×C₅₃	—	2	1	no	—
223	C₄²×C₅₃	—	2	2	no	—
224	C₄²×C₅₃	—	2	2	no	—
225	C₄²×C₅₃	—	2	1	no	—
226	C₄²×C₅₃	—	2	1	no	—
227	C₄²×C₅₃	C₂²×C₄	1	1	yes	characteristic
228	C₄²×C₅₃	C₂²×C₄	1	2	yes	—
229	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
230	C₄²×C₅₃	C₂×D₄	1	2	yes	—
231	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
232	C₄²×C₅₃	C₂×D₄	1	2	yes	—
233	C₄²×C₅₃	C₂⁴	1	1	yes	characteristic
234	C₂³×C₅₃	—	2	8	no	—
235	C₂³×C₅₃	—	2	2	no	—
236	C₂³×C₅₃	—	2	1	no	—
237	C₂³×C₅₃	—	2	4	no	—
238	C₂³×C₅₃	—	8	1	no	—
239	C₂³×C₅₃	—	8	2	no	—
240	C₂³×C₅₃	—	8	1	no	—
241	C₂³×C₅₃	—	8	2	no	—
242	C₂³×C₅₃	C₂×C₂²⋊C₄	1	2	yes	—
243	C₂³×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
244	Q₈×C₅₃	—	2	1	no	—
245	Q₈×C₅₃	—	2	2	no	—
246	Q₈×C₅₃	C₂×M₁₆	1	2	yes	—
247	D₄×C₅₃	—	2	8	no	—
248	D₄×C₅₃	—	2	8	no	—
249	D₄×C₅₃	—	2	4	no	—
250	D₄×C₅₃	—	2	4	no	—
251	D₄×C₅₃	—	2	2	no	—
252	D₄×C₅₃	—	2	1	no	—
253	D₄×C₅₃	—	2	2	no	—
254	D₄×C₅₃	—	2	4	no	—
255	D₄×C₅₃	C₂×M₁₆	1	2	yes	—
256	D₄×C₅₃	C₂×M₁₆	1	4	yes	—
257	C₂×C₄×C₅₃	—	2	8	no	—
258	C₂×C₄×C₅₃	—	2	1	no	—
259	C₂×C₄×C₅₃	—	2	2	no	—
260	C₂×C₄×C₅₃	—	2	2	no	—
261	C₂×C₄×C₅₃	—	2	1	no	—
262	C₂×C₄×C₅₃	—	2	2	no	—
263	C₂×C₄×C₅₃	—	2	1	no	—
264	C₂×C₄×C₅₃	—	2	2	no	—
265	C₂×C₄×C₅₃	—	2	4	no	—
266	C₂×C₄×C₅₃	—	2	1	no	—
267	C₂×C₄×C₅₃	—	2	2	no	—
268	C₂×C₄×C₅₃	—	2	1	no	—
269	C₂×C₄×C₅₃	—	4	1	no	—
270	C₂×C₄×C₅₃	—	4	2	no	—
271	C₂×C₄×C₅₃	—	4	2	no	—
272	C₂×C₄×C₅₃	—	4	1	no	—
273	C₂×C₄×C₅₃	—	4	2	no	—
274	C₂×C₄×C₅₃	—	4	1	no	—
275	C₂×C₄×C₅₃	—	8	1	no	—
276	C₂×C₄×C₅₃	—	8	2	no	—
277	C₂×C₄×C₅₃	—	8	2	no	—
278	C₂×C₄×C₅₃	—	8	2	no	—
279	C₂×C₄×C₅₃	—	8	2	no	—
280	C₂×C₄×C₅₃	—	8	1	no	—
281	C₂×C₄×C₅₃	—	8	2	no	—
282	C₂×C₄×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
283	C₂×C₄×C₅₃	C₂×C₂²⋊C₄	1	2	yes	—
284	C₂×C₄×C₅₃	C₂×C₂²⋊C₄	1	2	yes	—
285	C₂×C₄×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
286	C₂×C₄×C₅₃	C₂×C₂²⋊C₄	1	2	yes	—
287	C₂×C₄×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
288	C₂×C₄×C₅₃	C₂×M₁₆	1	2	yes	—
289	C₂×C₄×C₅₃	C₂×M₁₆	1	2	yes	—
290	C₂×C₄×C₅₃	C₂³×C₄	1	1	yes	characteristic
291	C₂×C₄×C₅₃	C₂²×D₄	1	1	yes	characteristic
292	C₂×C₄×C₅₃	C₂²×D₄	1	1	yes	characteristic
293	C₈×C₅₃	—	2	2	no	—
294	C₈×C₅₃	—	4	1	no	—
295	C₈×C₅₃	—	4	2	no	—
296	C₈×C₅₃	—	4	2	no	—
297	C₈×C₅₃	—	4	1	no	—
298	C₈×C₅₃	—	4	1	no	—
299	#17453	C₅₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
300	C₂²×C₅₃	—	2	1	no	—
301	C₂²×C₅₃	—	2	1	no	—
302	C₂²×C₅₃	—	2	2	no	—
303	C₂²×C₅₃	—	2	4	no	—
304	C₂²×C₅₃	—	2	2	no	—
305	C₂²×C₅₃	—	2	2	no	—
306	C₂²×C₅₃	—	2	2	no	—
307	C₂²×C₅₃	—	4	2	no	—
308	C₂²×C₅₃	—	4	1	no	—
309	C₂²×C₅₃	—	4	2	no	—
310	C₂²×C₅₃	—	4	1	no	—
311	C₂²×C₅₃	—	8	4	no	—
312	C₂²×C₅₃	—	8	8	no	—
313	C₂²×C₅₃	#407	1	2	yes	—
314	C₂²×C₅₃	C₂²×C₂²⋊C₄	1	1	yes	characteristic
315	C₄×C₅₃	—	2	2	no	—
316	C₄×C₅₃	—	2	2	no	—
317	C₄×C₅₃	—	2	1	no	—
318	C₄×C₅₃	—	2	2	no	—
319	C₄×C₅₃	—	2	1	no	—
320	C₄×C₅₃	—	2	1	no	—
321	C₄×C₅₃	—	2	2	no	—
322	C₄×C₅₃	—	4	2	no	—
323	C₄×C₅₃	—	4	1	no	—
324	C₄×C₅₃	—	4	1	no	—
325	C₄×C₅₃	#407	1	2	yes	—
326	C₄×C₅₃	C₂²×M₁₆	1	1	yes	characteristic
327	C₄×C₅₃	C₂²×M₁₆	1	1	yes	characteristic
328	C₄×C₅₃	#576	1	1	yes	characteristic
329	C₄×C₅₃	#577	1	1	yes	characteristic
330	C₄×H(5,𝔽₂)	C₂×C₅₃	1	1	yes	characteristic
331	#2989	C₂×C₅₃	1	1	yes	characteristic, complemented
332	#2965	C₂×C₅₃	1	1	yes	characteristic, complemented
333	#1490	C₂×C₅₃	1	2	yes	complemented
334	#1489	C₂×C₅₃	1	2	yes	complemented
335	#1488	C₂×C₅₃	1	2	yes	complemented
336	#1486	C₂×C₅₃	1	2	yes	complemented
337	#1478	C₂×C₅₃	1	1	yes	characteristic, complemented
338	#1477	C₂×C₅₃	1	1	yes	characteristic, complemented
339	#1476	C₂×C₅₃	1	1	yes	characteristic, complemented
340	C₂×D₄⋊C₈	C₂×C₅₃	1	1	yes	characteristic, complemented
341	C₂×C₅₃	—	2	2	no	—
342	C₂×C₅₃	—	4	2	no	—
343	C₂×C₅₃	—	4	4	no	—
344	C₂×C₅₃	—	4	2	no	—
345	C₂×C₅₃	C₂×#407	1	1	yes	characteristic
346	C₂×C₅₃	#2897	1	1	yes	characteristic
347	C₂×C₅₃	#2901	1	1	yes	characteristic
348	C₂×H(5,𝔽₂)	C₄×C₅₃	1	1	yes	characteristic
349	#520	—	2	1	no	—
350	#520	C₄×C₅₃	1	2	yes	—
351	#520	C₄×C₅₃	1	1	yes	characteristic
352	#520	C₄×C₅₃	1	1	yes	characteristic
353	#518	—	2	2	no	—
354	#518	—	2	1	no	—
355	#518	C₄×C₅₃	1	2	yes	—
356	#518	C₄×C₅₃	1	1	yes	characteristic
357	C₄×#41	—	2	1	no	—
358	C₄×#41	C₂²×C₅₃	1	1	yes	characteristic
359	C₄×#41	C₂²×C₅₃	1	2	yes	—
360	C₄×#41	C₂²×C₅₃	1	1	yes	characteristic
361	C₂×C₄×D₄	—	2	2	no	—
362	C₂×C₄×D₄	—	2	1	no	—
363	C₂×C₄×D₄	C₂²×C₅₃	1	1	yes	characteristic
364	C₂×C₄×D₄	C₂²×C₅₃	1	2	yes	—
365	#446	—	2	1	no	—
366	#446	—	2	1	no	—
367	#436	—	2	1	no	—
368	#436	—	2	1	no	—
369	#435	—	2	2	no	—
370	#435	—	2	2	no	—
371	#432	C₂²×C₅₃	1	2	yes	complemented
372	#431	C₂²×C₅₃	1	1	yes	characteristic, complemented
373	#431	C₂²×C₅₃	1	1	yes	characteristic, complemented
374	#431	C₂²×C₅₃	1	2	yes	complemented
375	#423	C₂²×C₅₃	1	2	yes	complemented
376	C₂×C₄⋊C₈	C₂²×C₅₃	1	1	yes	characteristic, complemented
377	C₄×M₁₆	C₂²×C₅₃	1	2	yes	complemented
378	C₂×C₄×C₈	C₂²×C₅₃	1	1	yes	characteristic, complemented
379	Q₈⋊C₈	C₂²×C₅₃	1	1	yes	characteristic, complemented
380	Q₈⋊C₈	C₂²×C₅₃	1	1	yes	characteristic, complemented
381	Q₈⋊C₈	C₂²×C₅₃	1	2	yes	complemented
382	D₄⋊C₈	C₂²×C₅₃	1	2	yes	complemented
383	D₄⋊C₈	C₂²×C₅₃	1	2	yes	complemented
384	D₄⋊C₈	C₂²×C₅₃	1	1	yes	characteristic, complemented
385	D₄⋊C₈	C₂²×C₅₃	1	1	yes	characteristic, complemented
386	D₄⋊C₈	C₂²×C₅₃	1	2	yes	complemented
387	D₄⋊C₈	C₂²×C₅₃	1	2	yes	complemented
388	D₄⋊C₈	C₂²×C₅₃	1	2	yes	complemented
389	C₅₃	#17453	1	1	yes	characteristic, direct product factor, Sylow
390	H(5,𝔽₂)	—	2	4	no	—
391	H(5,𝔽₂)	—	2	4	no	—
392	C₂×#41	—	2	1	no	—
393	C₂×#41	C₂×C₄×C₅₃	1	2	yes	—
394	C₂×#41	C₂×C₄×C₅₃	1	1	yes	characteristic
395	C₂×#41	C₂×C₄×C₅₃	1	1	yes	characteristic
396	C₂²×D₄	—	2	2	no	—
397	C₂²×D₄	—	2	1	no	—
398	C₂²×D₄	C₂×C₄×C₅₃	1	2	yes	—
399	C₂²×D₄	C₂×C₄×C₅₃	1	1	yes	characteristic
400	C₂³×C₄	—	2	1	no	—
401	C₂³×C₄	—	2	2	no	—
402	C₂×M₁₆	—	2	2	no	—
403	C₂×M₁₆	—	2	2	no	—
404	C₂²×C₈	—	2	1	no	—
405	C₂²×C₈	—	2	1	no	—
406	C₄×Q₈	—	2	1	no	—
407	C₄×Q₈	C₂×C₄×C₅₃	1	1	yes	characteristic
408	C₄×Q₈	C₂×C₄×C₅₃	1	2	yes	—
409	C₄×Q₈	D₄×C₅₃	1	1	yes	characteristic
410	C₄×Q₈	D₄×C₅₃	1	1	yes	characteristic
411	C₄×Q₈	C₂³×C₅₃	1	1	yes	characteristic
412	C₄×D₄	—	2	2	no	—
413	C₄×D₄	—	2	1	no	—
414	C₄×D₄	—	2	1	no	—
415	C₄×D₄	—	2	2	no	—
416	C₄×D₄	—	2	1	no	—
417	C₄×D₄	—	2	2	no	—
418	C₄×D₄	—	2	2	no	—
419	C₄×D₄	—	2	1	no	—
420	C₄×D₄	—	2	2	no	—
421	C₄×D₄	—	2	1	no	—
422	C₄×D₄	—	2	2	no	—
423	C₄×D₄	—	2	2	no	—
424	C₄×D₄	—	2	2	no	—
425	C₄×D₄	—	2	1	no	—
426	C₄×D₄	—	2	2	no	—
427	C₄×D₄	—	2	1	no	—
428	C₄×D₄	—	2	2	no	—
429	C₄×D₄	C₂×C₄×C₅₃	1	2	yes	—
430	C₄×D₄	C₂×C₄×C₅₃	1	2	yes	—
431	C₄×D₄	C₂×C₄×C₅₃	1	2	yes	—
432	C₄×D₄	C₂×C₄×C₅₃	1	1	yes	characteristic
433	C₄×D₄	C₂×C₄×C₅₃	1	2	yes	—
434	C₄×D₄	D₄×C₅₃	1	2	yes	—
435	C₄×D₄	D₄×C₅₃	1	1	yes	characteristic
436	C₄×D₄	D₄×C₅₃	1	2	yes	—
437	C₄×D₄	D₄×C₅₃	1	1	yes	characteristic
438	C₄×D₄	C₂³×C₅₃	1	2	yes	—
439	C₄×D₄	C₂³×C₅₃	1	1	yes	characteristic
440	#117	—	2	2	no	—
441	#117	—	2	1	no	—
442	#117	—	2	1	no	—
443	#117	—	2	2	no	—
444	#117	C₂×C₄×C₅₃	1	2	yes	—
445	#117	C₂×C₄×C₅₃	1	1	yes	characteristic
446	#117	C₂×C₄×C₅₃	1	2	yes	—
447	#117	C₂×C₄×C₅₃	1	1	yes	characteristic
448	C₂×C₄⋊C₄	—	2	2	no	—
449	C₂×C₄⋊C₄	—	2	1	no	—
450	C₂×C₄⋊C₄	C₂×C₄×C₅₃	1	2	yes	—
451	C₂×C₄⋊C₄	C₂×C₄×C₅₃	1	1	yes	characteristic
452	C₂×C₂²⋊C₄	—	2	1	no	—
453	C₂×C₂²⋊C₄	—	2	2	no	—
454	C₂×C₂²⋊C₄	—	2	2	no	—
455	C₂×C₂²⋊C₄	—	2	2	no	—
456	C₂×C₂²⋊C₄	—	2	2	no	—
457	C₂×C₄²	—	2	1	no	—
458	C₂×C₄²	—	2	2	no	—
459	C₂×C₄²	C₂³×C₅₃	1	1	yes	characteristic
460	C₂×C₄²	C₂³×C₅₃	1	2	yes	—
461	C₄⋊C₈	—	2	1	no	—
462	C₄⋊C₈	—	2	2	no	—
463	C₄⋊C₈	—	2	2	no	—
464	C₄⋊C₈	—	2	1	no	—
465	C₄⋊C₈	C₂³×C₅₃	1	2	yes	—
466	C₄⋊C₈	C₂³×C₅₃	1	1	yes	characteristic
467	C₄⋊C₈	C₂³×C₅₃	1	1	yes	characteristic
468	C₂²⋊C₈	—	2	1	no	—
469	C₂²⋊C₈	—	2	1	no	—
470	C₂²⋊C₈	—	2	2	no	—
471	C₂²⋊C₈	—	2	2	no	—
472	#97	—	2	2	no	—
473	#97	—	2	2	no	—
474	C₄×C₈	—	2	1	no	—
475	C₄×C₈	—	2	1	no	—
476	C₄×C₈	C₂³×C₅₃	1	1	yes	characteristic
477	C₄×C₈	C₂³×C₅₃	1	1	yes	characteristic
478	C₄×C₈	C₂³×C₅₃	1	2	yes	—
479	C₂⁴	—	2	2	no	—
480	C₂⁴	—	2	1	no	—
481	#41	—	2	8	no	—
482	#41	—	2	4	no	—
483	#41	—	2	4	no	—
484	#41	—	2	2	no	—
485	#41	—	2	2	no	—
486	#41	—	2	2	no	—
487	#41	M₁₆×C₅₃	1	4	yes	complemented
488	C₂×Q₈	C₂²⋊C₄×C₅₃	1	1	yes	characteristic
489	C₂×Q₈	C₂²×C₄×C₅₃	1	1	yes	characteristic
490	C₂×D₄	—	2	16	no	—
491	C₂×D₄	—	2	8	no	—
492	C₂×D₄	—	2	2	no	—
493	C₂×D₄	—	2	1	no	—
494	C₂×D₄	—	2	4	no	—
495	C₂×D₄	—	2	4	no	—
496	C₂×D₄	—	2	2	no	—
497	C₂×D₄	—	2	1	no	—
498	C₂×D₄	—	2	2	no	—
499	C₂×D₄	C₂²⋊C₄×C₅₃	1	2	yes	—
500	C₂×D₄	C₂²⋊C₄×C₅₃	1	1	yes	characteristic
501	C₂×D₄	M₁₆×C₅₃	1	4	yes	complemented
502	C₂×D₄	C₂²×C₄×C₅₃	1	1	yes	characteristic
503	C₂×D₄	C₂²×C₄×C₅₃	1	2	yes	—
504	C₂²×C₄	—	2	2	no	—
505	C₂²×C₄	—	2	1	no	—
506	C₂²×C₄	—	2	2	no	—
507	C₂²×C₄	—	2	1	no	—
508	C₂²×C₄	—	2	1	no	—
509	C₂²×C₄	—	2	2	no	—
510	C₂²×C₄	—	2	1	no	—
511	C₂²×C₄	—	2	2	no	—
512	C₂²×C₄	—	2	2	no	—
513	C₂²×C₄	—	2	2	no	—
514	C₂²×C₄	—	2	2	no	—
515	C₂²×C₄	—	8	1	no	—
516	C₂²×C₄	—	8	2	no	—
517	C₂²×C₄	—	8	1	no	—
518	C₂²×C₄	—	8	2	no	—
519	C₂²×C₄	C₂²×C₄×C₅₃	1	2	yes	—
520	C₂²×C₄	C₂²×C₄×C₅₃	1	1	yes	characteristic
521	C₂²×C₄	C₂×D₄×C₅₃	1	1	yes	characteristic
522	C₂²×C₄	C₂×D₄×C₅₃	1	2	yes	—
523	C₂²×C₄	C₂×D₄×C₅₃	1	1	yes	characteristic
524	C₂²×C₄	C₂×D₄×C₅₃	1	2	yes	—
525	M₁₆	—	4	4	no	—
526	M₁₆	—	8	2	no	—
527	C₂×C₈	—	2	1	no	—
528	C₂×C₈	—	2	1	no	—
529	C₂×C₈	—	2	1	no	—
530	C₂×C₈	—	2	1	no	—
531	C₂×C₈	—	2	2	no	—
532	C₂×C₈	—	2	2	no	—
533	C₂×C₈	—	4	2	no	—
534	C₂×C₈	—	8	1	no	—
535	C₄⋊C₄	—	2	1	no	—
536	C₄⋊C₄	—	2	2	no	—
537	C₄⋊C₄	—	2	2	no	—
538	C₄⋊C₄	—	2	1	no	—
539	C₄⋊C₄	—	2	1	no	—
540	C₄⋊C₄	C₂²⋊C₄×C₅₃	1	2	yes	—
541	C₄⋊C₄	C₂²⋊C₄×C₅₃	1	1	yes	characteristic
542	C₄⋊C₄	C₂²⋊C₄×C₅₃	1	2	yes	—
543	C₄⋊C₄	C₂²⋊C₄×C₅₃	1	1	yes	characteristic
544	C₄⋊C₄	C₂²×C₄×C₅₃	1	1	yes	characteristic
545	C₄⋊C₄	C₂²×C₄×C₅₃	1	2	yes	—
546	C₄⋊C₄	C₂²×C₄×C₅₃	1	1	yes	characteristic
547	C₂²⋊C₄	—	2	2	no	—
548	C₂²⋊C₄	—	2	1	no	—
549	C₂²⋊C₄	—	2	1	no	—
550	C₂²⋊C₄	—	2	2	no	—
551	C₂²⋊C₄	—	2	2	no	—
552	C₂²⋊C₄	—	2	2	no	—
553	C₂²⋊C₄	—	2	2	no	—
554	C₂²⋊C₄	—	2	2	no	—
555	C₂²⋊C₄	—	2	1	no	—
556	C₂²⋊C₄	C₂²⋊C₄×C₅₃	1	2	yes	—
557	C₂²⋊C₄	C₂²⋊C₄×C₅₃	1	2	yes	—
558	C₂²⋊C₄	C₂²⋊C₄×C₅₃	1	2	yes	—
559	C₄²	—	2	1	no	—
560	C₄²	—	2	2	no	—
561	C₄²	—	2	2	no	—
562	C₄²	—	2	1	no	—
563	C₄²	—	2	1	no	—
564	C₄²	C₂²×C₄×C₅₃	1	1	yes	characteristic
565	C₄²	C₂²×C₄×C₅₃	1	2	yes	—
566	C₄²	C₂×D₄×C₅₃	1	1	yes	characteristic
567	C₄²	C₂×D₄×C₅₃	1	1	yes	characteristic
568	C₄²	C₂×D₄×C₅₃	1	2	yes	—
569	C₄²	C₂×D₄×C₅₃	1	2	yes	—
570	C₄²	C₂⁴×C₅₃	1	1	yes	characteristic
571	C₂³	—	2	8	no	—
572	C₂³	—	2	2	no	—
573	C₂³	—	2	1	no	—
574	C₂³	—	2	4	no	—
575	C₂³	—	8	1	no	—
576	C₂³	—	8	2	no	—
577	C₂³	—	8	1	no	—
578	C₂³	—	8	2	no	—
579	C₂³	C₂×C₂²⋊C₄×C₅₃	1	2	yes	—
580	C₂³	C₂×C₂²⋊C₄×C₅₃	1	1	yes	characteristic
581	Q₈	—	2	1	no	—
582	Q₈	—	2	2	no	—
583	Q₈	C₂×M₁₆×C₅₃	1	2	yes	—
584	D₄	—	2	8	no	—
585	D₄	—	2	8	no	—
586	D₄	—	2	4	no	—
587	D₄	—	2	4	no	—
588	D₄	—	2	2	no	—
589	D₄	—	2	1	no	—
590	D₄	—	2	2	no	—
591	D₄	—	2	4	no	—
592	D₄	C₂×M₁₆×C₅₃	1	2	yes	—
593	D₄	C₂×M₁₆×C₅₃	1	4	yes	—
594	C₂×C₄	—	2	8	no	—
595	C₂×C₄	—	2	1	no	—
596	C₂×C₄	—	2	2	no	—
597	C₂×C₄	—	2	2	no	—
598	C₂×C₄	—	2	1	no	—
599	C₂×C₄	—	2	2	no	—
600	C₂×C₄	—	2	1	no	—
601	C₂×C₄	—	2	2	no	—
602	C₂×C₄	—	2	4	no	—
603	C₂×C₄	—	2	1	no	—
604	C₂×C₄	—	2	2	no	—
605	C₂×C₄	—	2	1	no	—
606	C₂×C₄	—	4	1	no	—
607	C₂×C₄	—	4	2	no	—
608	C₂×C₄	—	4	2	no	—
609	C₂×C₄	—	4	1	no	—
610	C₂×C₄	—	4	2	no	—
611	C₂×C₄	—	4	1	no	—
612	C₂×C₄	—	8	1	no	—
613	C₂×C₄	—	8	2	no	—
614	C₂×C₄	—	8	2	no	—
615	C₂×C₄	—	8	2	no	—
616	C₂×C₄	—	8	2	no	—
617	C₂×C₄	—	8	1	no	—
618	C₂×C₄	—	8	2	no	—
619	C₂×C₄	C₂×C₂²⋊C₄×C₅₃	1	1	yes	characteristic
620	C₂×C₄	C₂×C₂²⋊C₄×C₅₃	1	2	yes	—
621	C₂×C₄	C₂×C₂²⋊C₄×C₅₃	1	2	yes	—
622	C₂×C₄	C₂×C₂²⋊C₄×C₅₃	1	1	yes	characteristic
623	C₂×C₄	C₂×C₂²⋊C₄×C₅₃	1	2	yes	—
624	C₂×C₄	C₂×C₂²⋊C₄×C₅₃	1	1	yes	characteristic
625	C₂×C₄	C₂×M₁₆×C₅₃	1	2	yes	—
626	C₂×C₄	C₂×M₁₆×C₅₃	1	2	yes	—
627	C₂×C₄	C₂³×C₄×C₅₃	1	1	yes	characteristic
628	C₂×C₄	C₂²×D₄×C₅₃	1	1	yes	characteristic
629	C₂×C₄	C₂²×D₄×C₅₃	1	1	yes	characteristic
630	C₈	—	2	2	no	—
631	C₈	—	4	1	no	—
632	C₈	—	4	2	no	—
633	C₈	—	4	2	no	—
634	C₈	—	4	1	no	—
635	C₈	—	4	1	no	—
636	C₂²	—	2	1	no	—
637	C₂²	—	2	1	no	—
638	C₂²	—	2	2	no	—
639	C₂²	—	2	4	no	—
640	C₂²	—	2	2	no	—
641	C₂²	—	2	2	no	—
642	C₂²	—	2	2	no	—
643	C₂²	—	4	2	no	—
644	C₂²	—	4	1	no	—
645	C₂²	—	4	2	no	—
646	C₂²	—	4	1	no	—
647	C₂²	—	8	4	no	—
648	C₂²	—	8	8	no	—
649	C₂²	C₅₃×#407	1	2	yes	—
650	C₂²	C₂²×C₂²⋊C₄×C₅₃	1	1	yes	characteristic
651	C₄	—	2	2	no	—
652	C₄	—	2	2	no	—
653	C₄	—	2	1	no	—
654	C₄	—	2	2	no	—
655	C₄	—	2	1	no	—
656	C₄	—	2	1	no	—
657	C₄	—	2	2	no	—
658	C₄	—	4	2	no	—
659	C₄	—	4	1	no	—
660	C₄	—	4	1	no	—
661	C₄	C₅₃×#407	1	2	yes	—
662	C₄	C₂²×M₁₆×C₅₃	1	1	yes	characteristic
663	C₄	C₂²×M₁₆×C₅₃	1	1	yes	characteristic
664	C₄	C₅₃×#576	1	1	yes	characteristic
665	C₄	C₅₃×#577	1	1	yes	characteristic
666	C₂	—	2	2	no	—
667	C₂	—	4	2	no	—
668	C₂	—	4	4	no	—
669	C₂	—	4	2	no	—
670	C₂	C₂×C₅₃×#407	1	1	yes	characteristic
671	C₂	C₅₃×#2897	1	1	yes	characteristic
672	C₂	C₅₃×#2901	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄×C₅₃	C₂²×D₄
commutator	C₂×C₄	C₂³×C₄×C₅₃
Frattini, cosocle	C₄²	C₂⁴×C₅₃
socle	C₂²×C₅₃	C₂²×C₂²⋊C₄
prime radical	C₂×H(5,𝔽₂)×C₅₃	C₄
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₅₃ ◁ C₄×#41×C₅₃ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₅₃×#2989	C₂	semidirect
3. C₅₃×#2965	C₂	semidirect
4. C₅₃×#1490	C₂	semidirect
5. C₅₃×#1489	C₂	semidirect
6. C₅₃×#1488	C₂	semidirect
7. C₅₃×#1486	C₂	semidirect
8. C₅₃×#1478	C₂	semidirect
9. C₅₃×#1477	C₂	semidirect
10. C₅₃×#1476	C₂	semidirect
11. C₂×C₅₃×D₄⋊C₈	C₂	semidirect
35. C₅₃×#432	C₂²	semidirect
36. C₅₃×#431	C₂²	semidirect
37. C₅₃×#431	C₂²	semidirect
38. C₅₃×#431	C₂²	semidirect
39. C₅₃×#423	C₂²	semidirect
40. C₂×C₄⋊C₈×C₅₃	C₂²	semidirect
41. C₄×M₁₆×C₅₃	C₂²	semidirect
42. C₂×C₄×C₈×C₅₃	C₂²	semidirect
43. C₅₃×Q₈⋊C₈	C₂²	semidirect
44. C₅₃×Q₈⋊C₈	C₂²	semidirect
45. C₅₃×Q₈⋊C₈	C₂²	semidirect
46. C₅₃×D₄⋊C₈	C₂²	semidirect
47. C₅₃×D₄⋊C₈	C₂²	semidirect
48. C₅₃×D₄⋊C₈	C₂²	semidirect
49. C₅₃×D₄⋊C₈	C₂²	semidirect
50. C₅₃×D₄⋊C₈	C₂²	semidirect
51. C₅₃×D₄⋊C₈	C₂²	semidirect
52. C₅₃×D₄⋊C₈	C₂²	semidirect
150. #41×C₅₃	M₁₆	semidirect
164. C₂×D₄×C₅₃	M₁₆	semidirect
299. #17453	C₅₃	direct (N × H)
331. #2989	C₂×C₅₃	semidirect
332. #2965	C₂×C₅₃	semidirect
333. #1490	C₂×C₅₃	semidirect
334. #1489	C₂×C₅₃	semidirect
335. #1488	C₂×C₅₃	semidirect
336. #1486	C₂×C₅₃	semidirect
337. #1478	C₂×C₅₃	semidirect
338. #1477	C₂×C₅₃	semidirect
339. #1476	C₂×C₅₃	semidirect
340. C₂×D₄⋊C₈	C₂×C₅₃	semidirect
371. #432	C₂²×C₅₃	semidirect
372. #431	C₂²×C₅₃	semidirect
373. #431	C₂²×C₅₃	semidirect
374. #431	C₂²×C₅₃	semidirect
375. #423	C₂²×C₅₃	semidirect
376. C₂×C₄⋊C₈	C₂²×C₅₃	semidirect
377. C₄×M₁₆	C₂²×C₅₃	semidirect
378. C₂×C₄×C₈	C₂²×C₅₃	semidirect
379. Q₈⋊C₈	C₂²×C₅₃	semidirect
380. Q₈⋊C₈	C₂²×C₅₃	semidirect
381. Q₈⋊C₈	C₂²×C₅₃	semidirect
382. D₄⋊C₈	C₂²×C₅₃	semidirect
383. D₄⋊C₈	C₂²×C₅₃	semidirect
384. D₄⋊C₈	C₂²×C₅₃	semidirect
385. D₄⋊C₈	C₂²×C₅₃	semidirect
386. D₄⋊C₈	C₂²×C₅₃	semidirect
387. D₄⋊C₈	C₂²×C₅₃	semidirect
388. D₄⋊C₈	C₂²×C₅₃	semidirect
389. C₅₃	#17453	direct (N × H)
487. #41	M₁₆×C₅₃	semidirect
501. C₂×D₄	M₁₆×C₅₃	semidirect

Maximal subgroups

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

subgroup	index
C₄×H(5,𝔽₂)×C₅₃	2
C₅₃×#2989	2
C₅₃×#2965	2
C₅₃×#1490	2
C₅₃×#1489	2
C₅₃×#1488	2
C₅₃×#1486	2
C₅₃×#1478	2
C₅₃×#1477	2
C₅₃×#1476	2
C₂×C₅₃×D₄⋊C₈	2
#17453	53

Subgroups — C53×#17453 #27224965

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅₃×#17453 #27224965