Subgroups — C₅₃×#22490 #27230002

Order 13568.

1142 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×C₈×D₄×C₅₃	C₂	1	1	yes	maximal, characteristic
3	C₅₃×#2354	C₂	1	1	yes	maximal, characteristic
4	C₅₃×#1990	C₂	1	1	yes	maximal, characteristic, complemented
5	C₅₃×#1988	C₂	1	1	yes	maximal, characteristic, complemented
6	C₅₃×#1892	C₂	1	2	yes	maximal, complemented
7	C₅₃×#1873	C₂	1	2	yes	maximal, complemented
8	C₅₃×#1825	C₂	1	2	yes	maximal, complemented
9	C₅₃×#1803	C₂	1	1	yes	maximal, characteristic, complemented
10	C₄×#103×C₅₃	C₂	1	1	yes	maximal, characteristic, complemented
11	C₅₃×#1735	C₂	1	2	yes	maximal, complemented
12	C₂³×C₈×C₅₃	C₂²	1	2	yes	—
13	C₅₃×#556	—	2	4	no	—
14	C₅₃×#556	C₄	1	8	yes	complemented
15	C₂×C₄⋊Q₈×C₅₃	C₂²	1	1	yes	characteristic
16	C₂×#126×C₅₃	—	2	2	no	—
17	C₂×#124×C₅₃	—	2	1	no	—
18	C₂×C₂²⋊Q₈×C₅₃	C₂²	1	2	yes	—
19	C₂×C₄×Q₈×C₅₃	—	2	1	no	—
20	C₂×C₄×Q₈×C₅₃	C₂²	1	1	yes	characteristic
21	C₂×C₄×D₄×C₅₃	C₂²	1	1	yes	characteristic
22	C₂×#117×C₅₃	C₂²	1	2	yes	—
23	C₂×#117×C₅₃	C₂²	1	2	yes	—
24	C₈×D₄×C₅₃	—	2	4	no	—
25	C₂×C₄⋊C₈×C₅₃	C₂²	1	1	yes	characteristic
26	C₂×#103×C₅₃	C₂²	1	1	yes	characteristic
27	C₂×#103×C₅₃	C₂²	1	1	yes	characteristic
28	C₂×#103×C₅₃	C₂²	1	2	yes	—
29	C₂×C₂²⋊C₈×C₅₃	C₂²	1	2	yes	—
30	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
31	C₅₃×#390	—	2	1	no	—
32	C₅₃×#388	—	2	2	no	—
33	C₅₃×#384	C₂²	1	1	yes	characteristic
34	C₅₃×#380	C₂²	1	2	yes	—
35	C₄×C₄⋊C₄×C₅₃	—	2	1	no	—
36	C₄×C₄⋊C₄×C₅₃	C₂²	1	1	yes	characteristic
37	C₅₃×#340	C₂²	1	2	yes	—
38	C₅₃×#340	C₂²	1	2	yes	—
39	C₅₃×#340	C₂²	1	2	yes	—
40	C₅₃×#340	C₂²	1	2	yes	—
41	C₅₃×#340	C₂²	1	1	yes	characteristic
42	C₅₃×#340	C₂²	1	2	yes	—
43	C₅₃×#340	C₂²	1	1	yes	characteristic
44	C₂²×Q₈×C₅₃	—	2	1	no	—
45	C₂²×Q₈×C₅₃	C₂³	1	1	yes	characteristic
46	C₂²×D₄×C₅₃	D₄	1	1	yes	characteristic
47	C₂³×C₄×C₅₃	C₂³	1	2	yes	—
48	C₂²×C₈×C₅₃	—	2	4	no	—
49	C₂²×C₈×C₅₃	—	2	4	no	—
50	C₂²×C₈×C₅₃	—	4	2	no	—
51	C₂²×C₈×C₅₃	C₂³	1	2	yes	—
52	C₂²×C₈×C₅₃	C₂³	1	1	yes	characteristic
53	C₂²×C₈×C₅₃	C₂³	1	1	yes	characteristic
54	C₄⋊Q₈×C₅₃	—	2	2	no	—
55	C₄⋊Q₈×C₅₃	C₂×C₄	1	4	yes	—
56	#126×C₅₃	—	2	16	no	—
57	#124×C₅₃	—	2	8	no	—
58	C₂²⋊Q₈×C₅₃	—	2	4	no	—
59	C₂²⋊Q₈×C₅₃	C₂×C₄	1	8	yes	—
60	C₄×Q₈×C₅₃	—	2	8	no	—
61	C₄×Q₈×C₅₃	—	2	2	no	—
62	C₄×Q₈×C₅₃	C₂×C₄	1	4	yes	—
63	C₄×D₄×C₅₃	C₂×C₄	1	4	yes	—
64	C₄×D₄×C₅₃	D₄	1	4	yes	—
65	#117×C₅₃	—	2	4	no	—
66	#117×C₅₃	—	2	4	no	—
67	#117×C₅₃	—	4	2	no	—
68	#117×C₅₃	C₂×C₄	1	8	yes	—
69	C₂×C₄⋊C₄×C₅₃	—	2	2	no	—
70	C₂×C₄⋊C₄×C₅₃	—	2	1	no	—
71	C₂×C₄⋊C₄×C₅₃	—	2	1	no	—
72	C₂×C₄⋊C₄×C₅₃	—	2	1	no	—
73	C₂×C₄⋊C₄×C₅₃	—	2	1	no	—
74	C₂×C₄⋊C₄×C₅₃	D₄	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×C₄⋊C₄×C₅₃	C₂³	1	2	yes	—
79	C₂×C₄⋊C₄×C₅₃	C₂³	1	1	yes	characteristic
80	C₂×C₂²⋊C₄×C₅₃	—	2	2	no	—
81	C₂×C₂²⋊C₄×C₅₃	—	2	2	no	—
82	C₂×C₂²⋊C₄×C₅₃	—	2	2	no	—
83	C₂×C₂²⋊C₄×C₅₃	—	2	2	no	—
84	C₂×C₂²⋊C₄×C₅₃	D₄	1	2	yes	—
85	C₂×C₄²×C₅₃	—	2	2	no	—
86	C₂×C₄²×C₅₃	—	2	1	no	—
87	C₂×C₄²×C₅₃	—	2	2	no	—
88	C₂×C₄²×C₅₃	—	2	1	no	—
89	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
90	C₄⋊C₈×C₅₃	—	2	2	no	—
91	#103×C₅₃	—	2	2	no	—
92	#103×C₅₃	—	2	2	no	—
93	#103×C₅₃	—	4	2	no	—
94	C₂²⋊C₈×C₅₃	—	2	4	no	—
95	C₄×C₈×C₅₃	—	2	2	no	—
96	#95×C₅₃	—	2	2	no	—
97	#95×C₅₃	—	2	1	no	—
98	C₂⁴×C₅₃	C₂×D₄	1	2	yes	—
99	C₂×Q₈×C₅₃	—	2	4	no	—
100	C₂×Q₈×C₅₃	—	2	1	no	—
101	C₂×Q₈×C₅₃	—	2	1	no	—
102	C₂×Q₈×C₅₃	—	2	1	no	—
103	C₂×Q₈×C₅₃	—	2	1	no	—
104	C₂×Q₈×C₅₃	—	2	1	no	—
105	C₂×Q₈×C₅₃	—	4	2	no	—
106	C₂×Q₈×C₅₃	—	4	2	no	—
107	C₂×Q₈×C₅₃	C₂²×C₄	1	2	yes	—
108	C₂×D₄×C₅₃	—	2	2	no	—
109	C₂×D₄×C₅₃	—	2	2	no	—
110	C₂×D₄×C₅₃	C₂²⋊C₄	1	4	yes	—
111	C₂²×C₄×C₅₃	—	2	2	no	—
112	C₂²×C₄×C₅₃	—	2	2	no	—
113	C₂²×C₄×C₅₃	—	2	2	no	—
114	C₂²×C₄×C₅₃	—	2	2	no	—
115	C₂²×C₄×C₅₃	—	2	1	no	—
116	C₂²×C₄×C₅₃	—	2	1	no	—
117	C₂²×C₄×C₅₃	—	2	1	no	—
118	C₂²×C₄×C₅₃	—	2	2	no	—
119	C₂²×C₄×C₅₃	—	2	1	no	—
120	C₂²×C₄×C₅₃	—	2	2	no	—
121	C₂²×C₄×C₅₃	C₂²×C₄	1	4	yes	—
122	C₂²×C₄×C₅₃	C₂×D₄	1	2	yes	—
123	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
124	C₂²×C₄×C₅₃	C₂×D₄	1	4	yes	—
125	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
126	C₂²×C₄×C₅₃	C₂⁴	1	1	yes	characteristic
127	C₂×C₈×C₅₃	—	2	1	no	—
128	C₂×C₈×C₅₃	—	2	1	no	—
129	C₂×C₈×C₅₃	—	2	1	no	—
130	C₂×C₈×C₅₃	—	2	2	no	—
131	C₂×C₈×C₅₃	—	2	1	no	—
132	C₂×C₈×C₅₃	—	2	2	no	—
133	C₂×C₈×C₅₃	—	2	2	no	—
134	C₂×C₈×C₅₃	—	2	1	no	—
135	C₂×C₈×C₅₃	—	4	8	no	—
136	C₂×C₈×C₅₃	#41	1	2	yes	—
137	C₄⋊C₄×C₅₃	—	2	8	no	—
138	C₄⋊C₄×C₅₃	—	2	4	no	—
139	C₄⋊C₄×C₅₃	—	2	4	no	—
140	C₄⋊C₄×C₅₃	—	2	1	no	—
141	C₄⋊C₄×C₅₃	—	2	2	no	—
142	C₄⋊C₄×C₅₃	—	2	1	no	—
143	C₄⋊C₄×C₅₃	—	2	1	no	—
144	C₄⋊C₄×C₅₃	—	2	2	no	—
145	C₄⋊C₄×C₅₃	—	2	2	no	—
146	C₄⋊C₄×C₅₃	—	2	1	no	—
147	C₄⋊C₄×C₅₃	—	4	2	no	—
148	C₄⋊C₄×C₅₃	—	8	1	no	—
149	C₄⋊C₄×C₅₃	C₂²⋊C₄	1	4	yes	—
150	C₄⋊C₄×C₅₃	C₂²×C₄	1	2	yes	—
151	C₄⋊C₄×C₅₃	C₂²×C₄	1	4	yes	—
152	C₄⋊C₄×C₅₃	#41	1	2	yes	—
153	C₂²⋊C₄×C₅₃	—	2	8	no	—
154	C₂²⋊C₄×C₅₃	—	2	8	no	—
155	C₂²⋊C₄×C₅₃	—	4	4	no	—
156	C₂²⋊C₄×C₅₃	—	8	2	no	—
157	C₂²⋊C₄×C₅₃	C₂²⋊C₄	1	8	yes	—
158	C₄²×C₅₃	—	2	8	no	—
159	C₄²×C₅₃	—	2	4	no	—
160	C₄²×C₅₃	—	4	4	no	—
161	C₄²×C₅₃	—	4	2	no	—
162	C₄²×C₅₃	C₂²×C₄	1	2	yes	—
163	C₄²×C₅₃	C₂×D₄	1	2	yes	—
164	C₂³×C₅₃	—	2	2	no	—
165	C₂³×C₅₃	—	2	2	no	—
166	C₂³×C₅₃	—	2	2	no	—
167	C₂³×C₅₃	—	2	2	no	—
168	C₂³×C₅₃	—	2	2	no	—
169	C₂³×C₅₃	—	2	2	no	—
170	C₂³×C₅₃	C₂×C₂²⋊C₄	1	4	yes	—
171	C₂³×C₅₃	C₂²×D₄	1	1	yes	characteristic
172	Q₈×C₅₃	—	2	2	no	—
173	Q₈×C₅₃	—	2	2	no	—
174	Q₈×C₅₃	—	4	1	no	—
175	Q₈×C₅₃	—	4	1	no	—
176	Q₈×C₅₃	—	4	8	no	—
177	D₄×C₅₃	—	2	8	no	—
178	C₂×C₄×C₅₃	—	2	2	no	—
179	C₂×C₄×C₅₃	—	2	2	no	—
180	C₂×C₄×C₅₃	—	2	1	no	—
181	C₂×C₄×C₅₃	—	2	1	no	—
182	C₂×C₄×C₅₃	—	2	4	no	—
183	C₂×C₄×C₅₃	—	2	4	no	—
184	C₂×C₄×C₅₃	—	2	4	no	—
185	C₂×C₄×C₅₃	—	2	4	no	—
186	C₂×C₄×C₅₃	—	2	4	no	—
187	C₂×C₄×C₅₃	—	2	2	no	—
188	C₂×C₄×C₅₃	—	2	2	no	—
189	C₂×C₄×C₅₃	—	2	2	no	—
190	C₂×C₄×C₅₃	—	4	1	no	—
191	C₂×C₄×C₅₃	—	4	1	no	—
192	C₂×C₄×C₅₃	—	4	2	no	—
193	C₂×C₄×C₅₃	—	4	2	no	—
194	C₂×C₄×C₅₃	—	4	1	no	—
195	C₂×C₄×C₅₃	—	4	1	no	—
196	C₂×C₄×C₅₃	—	4	1	no	—
197	C₂×C₄×C₅₃	—	4	1	no	—
198	C₂×C₄×C₅₃	—	4	2	no	—
199	C₂×C₄×C₅₃	—	4	1	no	—
200	C₂×C₄×C₅₃	—	4	1	no	—
201	C₂×C₄×C₅₃	—	4	2	no	—
202	C₂×C₄×C₅₃	—	4	1	no	—
203	C₂×C₄×C₅₃	—	4	2	no	—
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	#135	1	2	yes	—
208	C₂×C₄×C₅₃	#135	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	2	no	—
217	C₈×C₅₃	—	4	1	no	—
218	#22490	C₅₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
219	C₂²×C₅₃	—	2	4	no	—
220	C₂²×C₅₃	—	2	4	no	—
221	C₂²×C₅₃	—	2	4	no	—
222	C₂²×C₅₃	—	4	2	no	—
223	C₂²×C₅₃	—	4	2	no	—
224	C₂²×C₅₃	—	4	2	no	—
225	C₂²×C₅₃	—	4	2	no	—
226	C₂²×C₅₃	C₂²×C₂²⋊C₄	1	1	yes	characteristic
227	C₂²×C₅₃	#546	1	1	yes	characteristic
228	C₂²×C₅₃	#547	1	1	yes	characteristic
229	C₂²×C₅₃	C₂×#135	1	1	yes	characteristic
230	C₂²×C₅₃	C₂×#135	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	2	no	—
235	C₄×C₅₃	—	4	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	#416	1	4	yes	—
242	C₄×C₅₃	C₄×#41	1	1	yes	characteristic
243	C₄×C₅₃	#520	1	1	yes	characteristic
244	C₄×C₅₃	#546	1	1	yes	characteristic
245	C₄×C₅₃	#547	1	1	yes	characteristic
246	C₂×#556	C₂×C₅₃	1	1	yes	characteristic
247	C₂×C₈×D₄	C₂×C₅₃	1	1	yes	characteristic
248	#2354	C₂×C₅₃	1	1	yes	characteristic
249	#1990	C₂×C₅₃	1	1	yes	characteristic, complemented
250	#1988	C₂×C₅₃	1	1	yes	characteristic, complemented
251	#1892	C₂×C₅₃	1	2	yes	complemented
252	#1873	C₂×C₅₃	1	2	yes	complemented
253	#1825	C₂×C₅₃	1	2	yes	complemented
254	#1803	C₂×C₅₃	1	1	yes	characteristic, complemented
255	C₄×#103	C₂×C₅₃	1	1	yes	characteristic, complemented
256	#1735	C₂×C₅₃	1	2	yes	complemented
257	C₂×C₅₃	—	4	4	no	—
258	C₂×C₅₃	#2339	1	1	yes	characteristic
259	C₂×C₅₃	C₂×#416	1	1	yes	characteristic
260	C₂×C₅₃	#2898	1	1	yes	characteristic
261	C₂×C₅₃	#3302	1	1	yes	characteristic
262	C₂×C₅₃	#3304	1	1	yes	characteristic
263	C₂×C₅₃	#3340	1	1	yes	characteristic
264	C₂×C₅₃	#3342	1	1	yes	characteristic
265	C₂³×C₈	C₂²×C₅₃	1	2	yes	—
266	#556	—	2	4	no	—
267	#556	C₄×C₅₃	1	8	yes	complemented
268	C₂×C₄⋊Q₈	C₂²×C₅₃	1	1	yes	characteristic
269	C₂×#126	—	2	2	no	—
270	C₂×#124	—	2	1	no	—
271	C₂×C₂²⋊Q₈	C₂²×C₅₃	1	2	yes	—
272	C₂×C₄×Q₈	—	2	1	no	—
273	C₂×C₄×Q₈	C₂²×C₅₃	1	1	yes	characteristic
274	C₂×C₄×D₄	C₂²×C₅₃	1	1	yes	characteristic
275	C₂×#117	C₂²×C₅₃	1	2	yes	—
276	C₂×#117	C₂²×C₅₃	1	2	yes	—
277	C₈×D₄	—	2	4	no	—
278	C₂×C₄⋊C₈	C₂²×C₅₃	1	1	yes	characteristic
279	C₂×#103	C₂²×C₅₃	1	1	yes	characteristic
280	C₂×#103	C₂²×C₅₃	1	1	yes	characteristic
281	C₂×#103	C₂²×C₅₃	1	2	yes	—
282	C₂×C₂²⋊C₈	C₂²×C₅₃	1	2	yes	—
283	C₂×C₄×C₈	C₂²×C₅₃	1	1	yes	characteristic
284	#390	—	2	1	no	—
285	#388	—	2	2	no	—
286	#384	C₂²×C₅₃	1	1	yes	characteristic
287	#380	C₂²×C₅₃	1	2	yes	—
288	C₄×C₄⋊C₄	—	2	1	no	—
289	C₄×C₄⋊C₄	C₂²×C₅₃	1	1	yes	characteristic
290	#340	C₂²×C₅₃	1	2	yes	—
291	#340	C₂²×C₅₃	1	1	yes	characteristic
292	#340	C₂²×C₅₃	1	2	yes	—
293	#340	C₂²×C₅₃	1	2	yes	—
294	#340	C₂²×C₅₃	1	2	yes	—
295	#340	C₂²×C₅₃	1	2	yes	—
296	#340	C₂²×C₅₃	1	1	yes	characteristic
297	C₅₃	#22490	1	1	yes	characteristic, direct product factor, Sylow
298	C₂²×Q₈	—	2	1	no	—
299	C₂²×Q₈	C₂³×C₅₃	1	1	yes	characteristic
300	C₂²×D₄	D₄×C₅₃	1	1	yes	characteristic
301	C₂³×C₄	C₂³×C₅₃	1	2	yes	—
302	C₂²×C₈	—	2	4	no	—
303	C₂²×C₈	—	2	4	no	—
304	C₂²×C₈	—	4	2	no	—
305	C₂²×C₈	C₂³×C₅₃	1	2	yes	—
306	C₂²×C₈	C₂³×C₅₃	1	1	yes	characteristic
307	C₂²×C₈	C₂³×C₅₃	1	1	yes	characteristic
308	C₄⋊Q₈	—	2	2	no	—
309	C₄⋊Q₈	C₂×C₄×C₅₃	1	4	yes	—
310	#126	—	2	16	no	—
311	#124	—	2	8	no	—
312	C₂²⋊Q₈	—	2	4	no	—
313	C₂²⋊Q₈	C₂×C₄×C₅₃	1	8	yes	—
314	C₄×Q₈	—	2	8	no	—
315	C₄×Q₈	—	2	2	no	—
316	C₄×Q₈	C₂×C₄×C₅₃	1	4	yes	—
317	C₄×D₄	C₂×C₄×C₅₃	1	4	yes	—
318	C₄×D₄	D₄×C₅₃	1	4	yes	—
319	#117	—	2	4	no	—
320	#117	—	2	4	no	—
321	#117	—	4	2	no	—
322	#117	C₂×C₄×C₅₃	1	8	yes	—
323	C₂×C₄⋊C₄	—	2	2	no	—
324	C₂×C₄⋊C₄	—	2	1	no	—
325	C₂×C₄⋊C₄	—	2	1	no	—
326	C₂×C₄⋊C₄	—	2	1	no	—
327	C₂×C₄⋊C₄	—	2	1	no	—
328	C₂×C₄⋊C₄	D₄×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×C₄⋊C₄	C₂³×C₅₃	1	2	yes	—
333	C₂×C₄⋊C₄	C₂³×C₅₃	1	1	yes	characteristic
334	C₂×C₂²⋊C₄	—	2	2	no	—
335	C₂×C₂²⋊C₄	—	2	2	no	—
336	C₂×C₂²⋊C₄	—	2	2	no	—
337	C₂×C₂²⋊C₄	—	2	2	no	—
338	C₂×C₂²⋊C₄	D₄×C₅₃	1	2	yes	—
339	C₂×C₄²	—	2	2	no	—
340	C₂×C₄²	—	2	1	no	—
341	C₂×C₄²	—	2	2	no	—
342	C₂×C₄²	—	2	1	no	—
343	C₂×C₄²	C₂³×C₅₃	1	1	yes	characteristic
344	C₄⋊C₈	—	2	2	no	—
345	#103	—	2	2	no	—
346	#103	—	2	2	no	—
347	#103	—	4	2	no	—
348	C₂²⋊C₈	—	2	4	no	—
349	C₄×C₈	—	2	2	no	—
350	#95	—	2	2	no	—
351	#95	—	2	1	no	—
352	C₂⁴	C₂×D₄×C₅₃	1	2	yes	—
353	C₂×Q₈	—	2	4	no	—
354	C₂×Q₈	—	2	1	no	—
355	C₂×Q₈	—	2	1	no	—
356	C₂×Q₈	—	2	1	no	—
357	C₂×Q₈	—	2	1	no	—
358	C₂×Q₈	—	2	1	no	—
359	C₂×Q₈	—	4	2	no	—
360	C₂×Q₈	—	4	2	no	—
361	C₂×Q₈	C₂²×C₄×C₅₃	1	2	yes	—
362	C₂×D₄	—	2	2	no	—
363	C₂×D₄	—	2	2	no	—
364	C₂×D₄	C₂²⋊C₄×C₅₃	1	4	yes	—
365	C₂²×C₄	—	2	2	no	—
366	C₂²×C₄	—	2	2	no	—
367	C₂²×C₄	—	2	2	no	—
368	C₂²×C₄	—	2	2	no	—
369	C₂²×C₄	—	2	1	no	—
370	C₂²×C₄	—	2	1	no	—
371	C₂²×C₄	—	2	1	no	—
372	C₂²×C₄	—	2	2	no	—
373	C₂²×C₄	—	2	1	no	—
374	C₂²×C₄	—	2	2	no	—
375	C₂²×C₄	C₂²×C₄×C₅₃	1	4	yes	—
376	C₂²×C₄	C₂×D₄×C₅₃	1	2	yes	—
377	C₂²×C₄	C₂×D₄×C₅₃	1	1	yes	characteristic
378	C₂²×C₄	C₂×D₄×C₅₃	1	4	yes	—
379	C₂²×C₄	C₂×D₄×C₅₃	1	1	yes	characteristic
380	C₂²×C₄	C₂⁴×C₅₃	1	1	yes	characteristic
381	C₂×C₈	—	2	1	no	—
382	C₂×C₈	—	2	1	no	—
383	C₂×C₈	—	2	1	no	—
384	C₂×C₈	—	2	2	no	—
385	C₂×C₈	—	2	1	no	—
386	C₂×C₈	—	2	2	no	—
387	C₂×C₈	—	2	2	no	—
388	C₂×C₈	—	2	1	no	—
389	C₂×C₈	—	4	8	no	—
390	C₂×C₈	#41×C₅₃	1	2	yes	—
391	C₄⋊C₄	—	2	8	no	—
392	C₄⋊C₄	—	2	4	no	—
393	C₄⋊C₄	—	2	4	no	—
394	C₄⋊C₄	—	2	1	no	—
395	C₄⋊C₄	—	2	2	no	—
396	C₄⋊C₄	—	2	1	no	—
397	C₄⋊C₄	—	2	1	no	—
398	C₄⋊C₄	—	2	2	no	—
399	C₄⋊C₄	—	2	2	no	—
400	C₄⋊C₄	—	2	1	no	—
401	C₄⋊C₄	—	4	2	no	—
402	C₄⋊C₄	—	8	1	no	—
403	C₄⋊C₄	C₂²⋊C₄×C₅₃	1	4	yes	—
404	C₄⋊C₄	C₂²×C₄×C₅₃	1	2	yes	—
405	C₄⋊C₄	C₂²×C₄×C₅₃	1	4	yes	—
406	C₄⋊C₄	#41×C₅₃	1	2	yes	—
407	C₂²⋊C₄	—	2	8	no	—
408	C₂²⋊C₄	—	2	8	no	—
409	C₂²⋊C₄	—	4	4	no	—
410	C₂²⋊C₄	—	8	2	no	—
411	C₂²⋊C₄	C₂²⋊C₄×C₅₃	1	8	yes	—
412	C₄²	—	2	8	no	—
413	C₄²	—	2	4	no	—
414	C₄²	—	4	4	no	—
415	C₄²	—	4	2	no	—
416	C₄²	C₂²×C₄×C₅₃	1	2	yes	—
417	C₄²	C₂×D₄×C₅₃	1	2	yes	—
418	C₂³	—	2	2	no	—
419	C₂³	—	2	2	no	—
420	C₂³	—	2	2	no	—
421	C₂³	—	2	2	no	—
422	C₂³	—	2	2	no	—
423	C₂³	—	2	2	no	—
424	C₂³	C₂×C₂²⋊C₄×C₅₃	1	4	yes	—
425	C₂³	C₂²×D₄×C₅₃	1	1	yes	characteristic
426	Q₈	—	2	2	no	—
427	Q₈	—	2	2	no	—
428	Q₈	—	4	1	no	—
429	Q₈	—	4	1	no	—
430	Q₈	—	4	8	no	—
431	D₄	—	2	8	no	—
432	C₂×C₄	—	2	2	no	—
433	C₂×C₄	—	2	2	no	—
434	C₂×C₄	—	2	1	no	—
435	C₂×C₄	—	2	1	no	—
436	C₂×C₄	—	2	4	no	—
437	C₂×C₄	—	2	4	no	—
438	C₂×C₄	—	2	4	no	—
439	C₂×C₄	—	2	4	no	—
440	C₂×C₄	—	2	2	no	—
441	C₂×C₄	—	2	2	no	—
442	C₂×C₄	—	2	4	no	—
443	C₂×C₄	—	2	2	no	—
444	C₂×C₄	—	4	1	no	—
445	C₂×C₄	—	4	1	no	—
446	C₂×C₄	—	4	1	no	—
447	C₂×C₄	—	4	2	no	—
448	C₂×C₄	—	4	2	no	—
449	C₂×C₄	—	4	1	no	—
450	C₂×C₄	—	4	1	no	—
451	C₂×C₄	—	4	1	no	—
452	C₂×C₄	—	4	2	no	—
453	C₂×C₄	—	4	1	no	—
454	C₂×C₄	—	4	1	no	—
455	C₂×C₄	—	4	2	no	—
456	C₂×C₄	—	4	1	no	—
457	C₂×C₄	—	4	2	no	—
458	C₂×C₄	C₂×C₂²⋊C₄×C₅₃	1	4	yes	—
459	C₂×C₄	C₂×C₂²⋊C₄×C₅₃	1	2	yes	—
460	C₂×C₄	C₂×C₂²⋊C₄×C₅₃	1	2	yes	—
461	C₂×C₄	#135×C₅₃	1	2	yes	—
462	C₂×C₄	#135×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	2	no	—
471	C₈	—	4	1	no	—
472	C₂²	—	2	4	no	—
473	C₂²	—	2	4	no	—
474	C₂²	—	2	4	no	—
475	C₂²	—	4	2	no	—
476	C₂²	—	4	2	no	—
477	C₂²	—	4	2	no	—
478	C₂²	—	4	2	no	—
479	C₂²	C₂²×C₂²⋊C₄×C₅₃	1	1	yes	characteristic
480	C₂²	C₅₃×#546	1	1	yes	characteristic
481	C₂²	C₅₃×#547	1	1	yes	characteristic
482	C₂²	C₂×#135×C₅₃	1	1	yes	characteristic
483	C₂²	C₂×#135×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	2	no	—
488	C₄	—	4	2	no	—
489	C₄	—	4	2	no	—
490	C₄	—	4	4	no	—
491	C₄	—	4	2	no	—
492	C₄	—	8	2	no	—
493	C₄	—	8	1	no	—
494	C₄	C₅₃×#416	1	4	yes	—
495	C₄	C₄×#41×C₅₃	1	1	yes	characteristic
496	C₄	C₅₃×#520	1	1	yes	characteristic
497	C₄	C₅₃×#546	1	1	yes	characteristic
498	C₄	C₅₃×#547	1	1	yes	characteristic
499	C₂	—	4	4	no	—
500	C₂	C₅₃×#2339	1	1	yes	characteristic
501	C₂	C₂×C₅₃×#416	1	1	yes	characteristic
502	C₂	C₅₃×#2898	1	1	yes	characteristic
503	C₂	C₅₃×#3302	1	1	yes	characteristic
504	C₂	C₅₃×#3304	1	1	yes	characteristic
505	C₂	C₅₃×#3340	1	1	yes	characteristic
506	C₂	C₅₃×#3342	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	D₄
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₅₃ ◁ C₂×C₄×D₄×C₅₃ ◁ G

Complements

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

N	H ≅ G/N	type
4. C₅₃×#1990	C₂	semidirect
5. C₅₃×#1988	C₂	semidirect
6. C₅₃×#1892	C₂	semidirect
7. C₅₃×#1873	C₂	semidirect
8. C₅₃×#1825	C₂	semidirect
9. C₅₃×#1803	C₂	semidirect
10. C₄×#103×C₅₃	C₂	semidirect
11. C₅₃×#1735	C₂	semidirect
14. C₅₃×#556	C₄	semidirect
218. #22490	C₅₃	direct (N × H)
249. #1990	C₂×C₅₃	semidirect
250. #1988	C₂×C₅₃	semidirect
251. #1892	C₂×C₅₃	semidirect
252. #1873	C₂×C₅₃	semidirect
253. #1825	C₂×C₅₃	semidirect
254. #1803	C₂×C₅₃	semidirect
255. C₄×#103	C₂×C₅₃	semidirect
256. #1735	C₂×C₅₃	semidirect
267. #556	C₄×C₅₃	semidirect
297. C₅₃	#22490	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×C₈×D₄×C₅₃	2
C₅₃×#2354	2
C₅₃×#1990	2
C₅₃×#1988	2
C₅₃×#1892	2
C₅₃×#1873	2
C₅₃×#1825	2
C₅₃×#1803	2
C₄×#103×C₅₃	2
C₅₃×#1735	2
#22490	53

Subgroups — C53×#22490 #27230002

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅₃×#22490 #27230002