Subgroups — C₅₃×#36728 #27244240

Order 13568.

1854 conjugacy classes of subgroups.

The subgroup lattice has 522 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₅₃×#2848	C₂	1	2	yes	maximal
2	C₅₃×#2747	C₂	1	4	yes	maximal, complemented
3	C₅₃×#2722	C₂	1	4	yes	maximal, complemented
4	C₅₃×#2704	C₂	1	4	yes	maximal, complemented
5	C₅₃×#2690	C₂	1	4	yes	maximal, complemented
6	C₅₃×#2659	C₂	1	1	yes	maximal, characteristic, complemented
7	C₅₃×#2646	C₂	1	1	yes	maximal, characteristic, complemented
8	C₅₃×#2480	C₂	1	4	yes	maximal
9	C₅₃×#2473	C₂	1	4	yes	maximal, complemented
10	C₅₃×#2329	C₂	1	2	yes	maximal, complemented
11	C₂×C₅₃×#380	C₂	1	1	yes	maximal, characteristic
12	C₂⁴×C₄×C₅₃	C₂²	1	1	yes	characteristic
13	C₂×#126×C₅₃	C₂²	1	4	yes	—
14	C₂×#125×C₅₃	C₂²	1	4	yes	complemented
15	C₂×C₂²⋊Q₈×C₅₃	C₂²	1	2	yes	—
16	C₂×C₂²⋊Q₈×C₅₃	C₂²	1	2	yes	—
17	C₂×C₂²⋊Q₈×C₅₃	C₂²	1	2	yes	—
18	C₂²×C₄⋊C₄×C₅₃	C₂²	1	2	yes	—
19	C₂²×C₂²⋊C₄×C₅₃	C₂²	1	2	yes	—
20	C₅₃×#400	C₂²	1	4	yes	complemented
21	C₅₃×#400	C₂²	1	4	yes	complemented
22	C₅₃×#398	C₂²	1	4	yes	complemented
23	C₅₃×#398	C₂²	1	4	yes	complemented
24	C₅₃×#398	C₂²	1	4	yes	complemented
25	C₅₃×#398	C₂²	1	4	yes	complemented
26	C₅₃×#398	C₂²	1	4	yes	complemented
27	C₅₃×#397	C₂²	1	4	yes	—
28	C₅₃×#397	C₂²	1	4	yes	—
29	C₅₃×#397	C₂²	1	4	yes	—
30	C₅₃×#396	C₂²	1	4	yes	—
31	C₅₃×#395	C₂²	1	4	yes	complemented
32	C₅₃×#391	C₂²	1	4	yes	complemented
33	C₅₃×#389	C₂²	1	4	yes	complemented
34	C₅₃×#389	C₂²	1	4	yes	complemented
35	C₅₃×#388	C₂²	1	4	yes	—
36	C₅₃×#388	C₂²	1	4	yes	—
37	C₅₃×#388	C₂²	1	4	yes	—
38	C₅₃×#387	C₂²	1	4	yes	complemented
39	C₅₃×#387	C₂²	1	4	yes	complemented
40	C₅₃×#387	C₂²	1	4	yes	complemented
41	C₅₃×#387	C₂²	1	4	yes	complemented
42	C₅₃×#385	C₂²	1	4	yes	—
43	C₅₃×#385	C₂²	1	4	yes	—
44	C₅₃×#385	C₂²	1	4	yes	—
45	C₅₃×#385	C₂²	1	4	yes	—
46	C₅₃×#385	C₂²	1	4	yes	—
47	C₅₃×#380	C₂²	1	4	yes	—
48	C₅₃×#380	C₂²	1	4	yes	—
49	C₅₃×#380	C₂²	1	2	yes	—
50	C₅₃×#380	C₂²	1	4	yes	—
51	C₅₃×#380	C₂²	1	2	yes	—
52	C₅₃×#380	C₂²	1	2	yes	—
53	C₅₃×#379	C₂²	1	4	yes	—
54	C₄×C₂²⋊C₄×C₅₃	C₂²	1	4	yes	—
55	C₂×#95×C₅₃	C₂²	1	1	yes	characteristic
56	C₂×#95×C₅₃	C₂²	1	1	yes	characteristic
57	C₂⁵×C₅₃	C₂³	1	1	yes	characteristic
58	C₂²×Q₈×C₅₃	C₂³	1	2	yes	—
59	C₂³×C₄×C₅₃	—	2	4	no	—
60	C₂³×C₄×C₅₃	—	4	4	no	—
61	C₂³×C₄×C₅₃	C₂³	1	1	yes	characteristic
62	C₂³×C₄×C₅₃	C₂³	1	2	yes	—
63	C₂³×C₄×C₅₃	C₂³	1	2	yes	—
64	C₂³×C₄×C₅₃	C₂³	1	2	yes	—
65	C₂³×C₄×C₅₃	C₂³	1	2	yes	—
66	C₂³×C₄×C₅₃	C₂³	1	1	yes	characteristic
67	#126×C₅₃	—	4	8	no	—
68	#125×C₅₃	—	4	8	no	—
69	C₂²⋊Q₈×C₅₃	—	2	8	no	—
70	C₂²⋊Q₈×C₅₃	—	4	4	no	—
71	C₂²⋊Q₈×C₅₃	—	4	4	no	—
72	C₂×C₄⋊C₄×C₅₃	—	4	4	no	—
73	C₂×C₄⋊C₄×C₅₃	C₂³	1	4	yes	—
74	C₂×C₄⋊C₄×C₅₃	C₂³	1	4	yes	—
75	C₂×C₄⋊C₄×C₅₃	C₂³	1	4	yes	—
76	C₂×C₄⋊C₄×C₅₃	C₂³	1	2	yes	—
77	C₂×C₄⋊C₄×C₅₃	C₂³	1	4	yes	—
78	C₂×C₄⋊C₄×C₅₃	C₂³	1	4	yes	—
79	C₂×C₄⋊C₄×C₅₃	C₂³	1	2	yes	—
80	C₂×C₄⋊C₄×C₅₃	C₂³	1	2	yes	—
81	C₂×C₄⋊C₄×C₅₃	C₂³	1	4	yes	—
82	C₂×C₄⋊C₄×C₅₃	C₂³	1	4	yes	—
83	C₂×C₄⋊C₄×C₅₃	C₂³	1	4	yes	—
84	C₂×C₄⋊C₄×C₅₃	C₂³	1	4	yes	—
85	C₂×C₄⋊C₄×C₅₃	C₂³	1	4	yes	—
86	C₂×C₄⋊C₄×C₅₃	C₂³	1	4	yes	—
87	C₂×C₂²⋊C₄×C₅₃	—	4	4	no	—
88	C₂×C₂²⋊C₄×C₅₃	C₂³	1	4	yes	—
89	C₂×C₂²⋊C₄×C₅₃	C₂³	1	4	yes	—
90	C₂×C₂²⋊C₄×C₅₃	C₂³	1	4	yes	—
91	C₂×C₂²⋊C₄×C₅₃	C₂³	1	4	yes	—
92	C₂×C₂²⋊C₄×C₅₃	C₂³	1	4	yes	—
93	C₂×C₂²⋊C₄×C₅₃	C₂³	1	4	yes	—
94	C₂×C₂²⋊C₄×C₅₃	C₂³	1	4	yes	—
95	C₂×C₂²⋊C₄×C₅₃	C₂³	1	4	yes	—
96	C₂×C₄²×C₅₃	C₂³	1	4	yes	—
97	C₂×C₄²×C₅₃	C₂³	1	4	yes	—
98	#95×C₅₃	C₂³	1	4	yes	—
99	#95×C₅₃	C₂³	1	4	yes	—
100	#95×C₅₃	C₂³	1	4	yes	—
101	#95×C₅₃	C₂³	1	4	yes	—
102	#95×C₅₃	C₂³	1	4	yes	—
103	#95×C₅₃	C₂³	1	4	yes	—
104	#95×C₅₃	C₂³	1	4	yes	—
105	#95×C₅₃	C₂³	1	4	yes	—
106	#95×C₅₃	C₂³	1	4	yes	—
107	#95×C₅₃	C₂³	1	4	yes	—
108	#95×C₅₃	C₂³	1	4	yes	—
109	#95×C₅₃	C₂³	1	4	yes	—
110	#95×C₅₃	C₂³	1	4	yes	—
111	C₂⁴×C₅₃	—	2	2	no	—
112	C₂⁴×C₅₃	—	2	2	no	—
113	C₂⁴×C₅₃	—	2	2	no	—
114	C₂⁴×C₅₃	—	2	4	no	—
115	C₂⁴×C₅₃	—	4	2	no	—
116	C₂⁴×C₅₃	C₂⁴	1	1	yes	characteristic
117	C₂⁴×C₅₃	C₂⁴	1	1	yes	characteristic
118	C₂⁴×C₅₃	C₂⁴	1	1	yes	characteristic
119	C₂×Q₈×C₅₃	—	2	4	no	—
120	C₂×Q₈×C₅₃	—	4	2	no	—
121	C₂×Q₈×C₅₃	—	4	2	no	—
122	C₂²×C₄×C₅₃	—	2	2	no	—
123	C₂²×C₄×C₅₃	—	2	4	no	—
124	C₂²×C₄×C₅₃	—	2	4	no	—
125	C₂²×C₄×C₅₃	—	2	4	no	—
126	C₂²×C₄×C₅₃	—	2	4	no	—
127	C₂²×C₄×C₅₃	—	2	4	no	—
128	C₂²×C₄×C₅₃	—	2	4	no	—
129	C₂²×C₄×C₅₃	—	2	4	no	—
130	C₂²×C₄×C₅₃	—	4	2	no	—
131	C₂²×C₄×C₅₃	—	4	2	no	—
132	C₂²×C₄×C₅₃	—	4	2	no	—
133	C₂²×C₄×C₅₃	—	4	2	no	—
134	C₂²×C₄×C₅₃	—	4	2	no	—
135	C₂²×C₄×C₅₃	—	4	2	no	—
136	C₂²×C₄×C₅₃	—	4	2	no	—
137	C₂²×C₄×C₅₃	—	4	16	no	—
138	C₂²×C₄×C₅₃	#41	1	4	yes	—
139	C₂²×C₄×C₅₃	C₂⁴	1	4	yes	—
140	C₂²×C₄×C₅₃	C₂⁴	1	4	yes	—
141	C₂²×C₄×C₅₃	C₂⁴	1	4	yes	—
142	C₂²×C₄×C₅₃	C₂⁴	1	4	yes	—
143	C₂²×C₄×C₅₃	C₂⁴	1	4	yes	—
144	C₂²×C₄×C₅₃	C₂⁴	1	4	yes	—
145	C₂²×C₄×C₅₃	C₂⁴	1	2	yes	—
146	C₂²×C₄×C₅₃	C₂⁴	1	2	yes	—
147	C₄⋊C₄×C₅₃	—	2	8	no	—
148	C₄⋊C₄×C₅₃	—	2	4	no	—
149	C₄⋊C₄×C₅₃	—	2	8	no	—
150	C₄⋊C₄×C₅₃	—	2	8	no	—
151	C₄⋊C₄×C₅₃	—	2	8	no	—
152	C₄⋊C₄×C₅₃	—	4	4	no	—
153	C₄⋊C₄×C₅₃	—	4	2	no	—
154	C₄⋊C₄×C₅₃	—	4	2	no	—
155	C₄⋊C₄×C₅₃	—	4	4	no	—
156	C₄⋊C₄×C₅₃	—	4	4	no	—
157	C₄⋊C₄×C₅₃	—	4	4	no	—
158	C₄⋊C₄×C₅₃	—	4	4	no	—
159	C₄⋊C₄×C₅₃	—	4	4	no	—
160	C₄⋊C₄×C₅₃	—	4	4	no	—
161	C₂²⋊C₄×C₅₃	—	2	8	no	—
162	C₂²⋊C₄×C₅₃	—	2	8	no	—
163	C₂²⋊C₄×C₅₃	—	2	8	no	—
164	C₂²⋊C₄×C₅₃	—	2	8	no	—
165	C₂²⋊C₄×C₅₃	—	2	8	no	—
166	C₂²⋊C₄×C₅₃	—	4	4	no	—
167	C₂²⋊C₄×C₅₃	—	4	4	no	—
168	C₂²⋊C₄×C₅₃	—	4	4	no	—
169	C₄²×C₅₃	—	4	4	no	—
170	C₄²×C₅₃	—	4	4	no	—
171	C₂³×C₅₃	—	2	2	no	—
172	C₂³×C₅₃	—	2	1	no	—
173	C₂³×C₅₃	—	2	1	no	—
174	C₂³×C₅₃	—	2	8	no	—
175	C₂³×C₅₃	—	2	1	no	—
176	C₂³×C₅₃	—	2	1	no	—
177	C₂³×C₅₃	—	2	2	no	—
178	C₂³×C₅₃	—	2	2	no	—
179	C₂³×C₅₃	—	2	1	no	—
180	C₂³×C₅₃	—	2	1	no	—
181	C₂³×C₅₃	—	2	2	no	—
182	C₂³×C₅₃	—	2	2	no	—
183	C₂³×C₅₃	—	4	8	no	—
184	C₂³×C₅₃	—	4	8	no	—
185	C₂³×C₅₃	—	4	4	no	—
186	C₂³×C₅₃	—	4	4	no	—
187	C₂³×C₅₃	C₂×#41	1	2	yes	—
188	C₂³×C₅₃	H(5,𝔽₂)	1	2	yes	—
189	C₂³×C₅₃	2₋¹⁺⁴	1	2	yes	—
190	C₂³×C₅₃	2₋¹⁺⁴	1	4	yes	—
191	C₂³×C₅₃	C₂⁵	1	1	yes	characteristic
192	Q₈×C₅₃	—	8	4	no	—
193	C₂×C₄×C₅₃	—	2	8	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	4	no	—
199	C₂×C₄×C₅₃	—	2	4	no	—
200	C₂×C₄×C₅₃	—	2	4	no	—
201	C₂×C₄×C₅₃	—	2	4	no	—
202	C₂×C₄×C₅₃	—	2	4	no	—
203	C₂×C₄×C₅₃	—	2	4	no	—
204	C₂×C₄×C₅₃	—	2	4	no	—
205	C₂×C₄×C₅₃	—	2	4	no	—
206	C₂×C₄×C₅₃	—	2	4	no	—
207	C₂×C₄×C₅₃	—	2	4	no	—
208	C₂×C₄×C₅₃	—	2	4	no	—
209	C₂×C₄×C₅₃	—	2	4	no	—
210	C₂×C₄×C₅₃	—	2	4	no	—
211	C₂×C₄×C₅₃	—	2	4	no	—
212	C₂×C₄×C₅₃	—	2	4	no	—
213	C₂×C₄×C₅₃	—	2	4	no	—
214	C₂×C₄×C₅₃	—	2	4	no	—
215	C₂×C₄×C₅₃	—	2	4	no	—
216	C₂×C₄×C₅₃	—	4	4	no	—
217	C₂×C₄×C₅₃	—	4	8	no	—
218	C₂×C₄×C₅₃	—	4	8	no	—
219	C₂×C₄×C₅₃	—	8	4	no	—
220	C₂×C₄×C₅₃	—	8	4	no	—
221	C₂×C₄×C₅₃	C₂×#41	1	4	yes	—
222	C₂×C₄×C₅₃	H(5,𝔽₂)	1	4	yes	—
223	#36728	C₅₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
224	C₂²×C₅₃	—	2	2	no	—
225	C₂²×C₅₃	—	2	2	no	—
226	C₂²×C₅₃	—	2	4	no	—
227	C₂²×C₅₃	—	2	4	no	—
228	C₂²×C₅₃	—	2	2	no	—
229	C₂²×C₅₃	—	2	2	no	—
230	C₂²×C₅₃	—	2	4	no	—
231	C₂²×C₅₃	—	2	4	no	—
232	C₂²×C₅₃	—	4	16	no	—
233	C₂²×C₅₃	—	4	1	no	—
234	C₂²×C₅₃	—	4	1	no	—
235	C₂²×C₅₃	—	4	1	no	—
236	C₂²×C₅₃	—	4	1	no	—
237	C₂²×C₅₃	—	4	2	no	—
238	C₂²×C₅₃	—	4	2	no	—
239	C₂²×C₅₃	#541	1	4	yes	—
240	C₂²×C₅₃	C₂²×#41	1	1	yes	characteristic
241	C₂²×C₅₃	C₂×H(5,𝔽₂)	1	2	yes	—
242	C₂²×C₅₃	C₂×H(5,𝔽₂)	1	1	yes	characteristic
243	C₂²×C₅₃	C₂×2₋¹⁺⁴	1	2	yes	—
244	C₂²×C₅₃	C₂×2₋¹⁺⁴	1	1	yes	characteristic
245	C₄×C₅₃	—	2	4	no	—
246	C₄×C₅₃	—	2	4	no	—
247	C₄×C₅₃	—	4	4	no	—
248	C₄×C₅₃	—	4	4	no	—
249	C₄×C₅₃	—	4	4	no	—
250	C₄×C₅₃	—	4	4	no	—
251	C₄×C₅₃	—	4	4	no	—
252	C₄×C₅₃	—	4	4	no	—
253	#2848	C₂×C₅₃	1	2	yes	—
254	#2747	C₂×C₅₃	1	4	yes	complemented
255	#2722	C₂×C₅₃	1	4	yes	complemented
256	#2704	C₂×C₅₃	1	4	yes	complemented
257	#2690	C₂×C₅₃	1	4	yes	complemented
258	#2659	C₂×C₅₃	1	1	yes	characteristic, complemented
259	#2646	C₂×C₅₃	1	1	yes	characteristic, complemented
260	#2480	C₂×C₅₃	1	4	yes	—
261	#2473	C₂×C₅₃	1	4	yes	complemented
262	#2329	C₂×C₅₃	1	2	yes	complemented
263	C₂×#380	C₂×C₅₃	1	1	yes	characteristic
264	C₂×C₅₃	—	2	4	no	—
265	C₂×C₅₃	—	4	2	no	—
266	C₂×C₅₃	—	4	2	no	—
267	C₂×C₅₃	C₂×#541	1	1	yes	characteristic
268	C₂×C₅₃	#3490	1	1	yes	characteristic
269	C₂×C₅₃	#3492	1	1	yes	characteristic
270	C₂×C₅₃	#3536	1	2	yes	—
271	C₂×C₅₃	#3538	1	2	yes	—
272	C₂⁴×C₄	C₂²×C₅₃	1	1	yes	characteristic
273	C₂×#126	C₂²×C₅₃	1	4	yes	—
274	C₂×#125	C₂²×C₅₃	1	4	yes	complemented
275	C₂×C₂²⋊Q₈	C₂²×C₅₃	1	2	yes	—
276	C₂×C₂²⋊Q₈	C₂²×C₅₃	1	2	yes	—
277	C₂×C₂²⋊Q₈	C₂²×C₅₃	1	2	yes	—
278	C₂²×C₄⋊C₄	C₂²×C₅₃	1	2	yes	—
279	C₂²×C₂²⋊C₄	C₂²×C₅₃	1	2	yes	—
280	#400	C₂²×C₅₃	1	4	yes	complemented
281	#400	C₂²×C₅₃	1	4	yes	complemented
282	#398	C₂²×C₅₃	1	4	yes	complemented
283	#398	C₂²×C₅₃	1	4	yes	complemented
284	#398	C₂²×C₅₃	1	4	yes	complemented
285	#398	C₂²×C₅₃	1	4	yes	complemented
286	#398	C₂²×C₅₃	1	4	yes	complemented
287	#397	C₂²×C₅₃	1	4	yes	—
288	#397	C₂²×C₅₃	1	4	yes	—
289	#397	C₂²×C₅₃	1	4	yes	—
290	#396	C₂²×C₅₃	1	4	yes	—
291	#395	C₂²×C₅₃	1	4	yes	complemented
292	#391	C₂²×C₅₃	1	4	yes	complemented
293	#389	C₂²×C₅₃	1	4	yes	complemented
294	#389	C₂²×C₅₃	1	4	yes	complemented
295	#388	C₂²×C₅₃	1	4	yes	—
296	#388	C₂²×C₅₃	1	4	yes	—
297	#388	C₂²×C₅₃	1	4	yes	—
298	#387	C₂²×C₅₃	1	4	yes	complemented
299	#387	C₂²×C₅₃	1	4	yes	complemented
300	#387	C₂²×C₅₃	1	4	yes	complemented
301	#387	C₂²×C₅₃	1	4	yes	complemented
302	#385	C₂²×C₅₃	1	4	yes	—
303	#385	C₂²×C₅₃	1	4	yes	—
304	#385	C₂²×C₅₃	1	4	yes	—
305	#385	C₂²×C₅₃	1	4	yes	—
306	#385	C₂²×C₅₃	1	4	yes	—
307	#380	C₂²×C₅₃	1	2	yes	—
308	#380	C₂²×C₅₃	1	2	yes	—
309	#380	C₂²×C₅₃	1	2	yes	—
310	#380	C₂²×C₅₃	1	4	yes	—
311	#380	C₂²×C₅₃	1	4	yes	—
312	#380	C₂²×C₅₃	1	4	yes	—
313	#379	C₂²×C₅₃	1	4	yes	—
314	C₄×C₂²⋊C₄	C₂²×C₅₃	1	4	yes	—
315	C₂×#95	C₂²×C₅₃	1	1	yes	characteristic
316	C₂×#95	C₂²×C₅₃	1	1	yes	characteristic
317	C₅₃	#36728	1	1	yes	characteristic, direct product factor, Sylow
318	C₂⁵	C₂³×C₅₃	1	1	yes	characteristic
319	C₂²×Q₈	C₂³×C₅₃	1	2	yes	—
320	C₂³×C₄	—	2	4	no	—
321	C₂³×C₄	—	4	4	no	—
322	C₂³×C₄	C₂³×C₅₃	1	1	yes	characteristic
323	C₂³×C₄	C₂³×C₅₃	1	2	yes	—
324	C₂³×C₄	C₂³×C₅₃	1	2	yes	—
325	C₂³×C₄	C₂³×C₅₃	1	2	yes	—
326	C₂³×C₄	C₂³×C₅₃	1	2	yes	—
327	C₂³×C₄	C₂³×C₅₃	1	1	yes	characteristic
328	#126	—	4	8	no	—
329	#125	—	4	8	no	—
330	C₂²⋊Q₈	—	2	8	no	—
331	C₂²⋊Q₈	—	4	4	no	—
332	C₂²⋊Q₈	—	4	4	no	—
333	C₂×C₄⋊C₄	—	4	4	no	—
334	C₂×C₄⋊C₄	C₂³×C₅₃	1	4	yes	—
335	C₂×C₄⋊C₄	C₂³×C₅₃	1	4	yes	—
336	C₂×C₄⋊C₄	C₂³×C₅₃	1	4	yes	—
337	C₂×C₄⋊C₄	C₂³×C₅₃	1	2	yes	—
338	C₂×C₄⋊C₄	C₂³×C₅₃	1	4	yes	—
339	C₂×C₄⋊C₄	C₂³×C₅₃	1	4	yes	—
340	C₂×C₄⋊C₄	C₂³×C₅₃	1	2	yes	—
341	C₂×C₄⋊C₄	C₂³×C₅₃	1	2	yes	—
342	C₂×C₄⋊C₄	C₂³×C₅₃	1	4	yes	—
343	C₂×C₄⋊C₄	C₂³×C₅₃	1	4	yes	—
344	C₂×C₄⋊C₄	C₂³×C₅₃	1	4	yes	—
345	C₂×C₄⋊C₄	C₂³×C₅₃	1	4	yes	—
346	C₂×C₄⋊C₄	C₂³×C₅₃	1	4	yes	—
347	C₂×C₄⋊C₄	C₂³×C₅₃	1	4	yes	—
348	C₂×C₂²⋊C₄	—	4	4	no	—
349	C₂×C₂²⋊C₄	C₂³×C₅₃	1	4	yes	—
350	C₂×C₂²⋊C₄	C₂³×C₅₃	1	4	yes	—
351	C₂×C₂²⋊C₄	C₂³×C₅₃	1	4	yes	—
352	C₂×C₂²⋊C₄	C₂³×C₅₃	1	4	yes	—
353	C₂×C₂²⋊C₄	C₂³×C₅₃	1	4	yes	—
354	C₂×C₂²⋊C₄	C₂³×C₅₃	1	4	yes	—
355	C₂×C₂²⋊C₄	C₂³×C₅₃	1	4	yes	—
356	C₂×C₂²⋊C₄	C₂³×C₅₃	1	4	yes	—
357	C₂×C₄²	C₂³×C₅₃	1	4	yes	—
358	C₂×C₄²	C₂³×C₅₃	1	4	yes	—
359	#95	C₂³×C₅₃	1	4	yes	—
360	#95	C₂³×C₅₃	1	4	yes	—
361	#95	C₂³×C₅₃	1	4	yes	—
362	#95	C₂³×C₅₃	1	4	yes	—
363	#95	C₂³×C₅₃	1	4	yes	—
364	#95	C₂³×C₅₃	1	4	yes	—
365	#95	C₂³×C₅₃	1	4	yes	—
366	#95	C₂³×C₅₃	1	4	yes	—
367	#95	C₂³×C₅₃	1	4	yes	—
368	#95	C₂³×C₅₃	1	4	yes	—
369	#95	C₂³×C₅₃	1	4	yes	—
370	#95	C₂³×C₅₃	1	4	yes	—
371	#95	C₂³×C₅₃	1	4	yes	—
372	C₂⁴	—	2	2	no	—
373	C₂⁴	—	2	2	no	—
374	C₂⁴	—	2	2	no	—
375	C₂⁴	—	2	4	no	—
376	C₂⁴	—	4	2	no	—
377	C₂⁴	C₂⁴×C₅₃	1	1	yes	characteristic
378	C₂⁴	C₂⁴×C₅₃	1	1	yes	characteristic
379	C₂⁴	C₂⁴×C₅₃	1	1	yes	characteristic
380	C₂×Q₈	—	2	4	no	—
381	C₂×Q₈	—	4	2	no	—
382	C₂×Q₈	—	4	2	no	—
383	C₂²×C₄	—	2	2	no	—
384	C₂²×C₄	—	2	4	no	—
385	C₂²×C₄	—	2	4	no	—
386	C₂²×C₄	—	2	4	no	—
387	C₂²×C₄	—	2	4	no	—
388	C₂²×C₄	—	2	4	no	—
389	C₂²×C₄	—	2	4	no	—
390	C₂²×C₄	—	2	4	no	—
391	C₂²×C₄	—	4	2	no	—
392	C₂²×C₄	—	4	2	no	—
393	C₂²×C₄	—	4	2	no	—
394	C₂²×C₄	—	4	2	no	—
395	C₂²×C₄	—	4	2	no	—
396	C₂²×C₄	—	4	2	no	—
397	C₂²×C₄	—	4	2	no	—
398	C₂²×C₄	—	4	16	no	—
399	C₂²×C₄	#41×C₅₃	1	4	yes	—
400	C₂²×C₄	C₂⁴×C₅₃	1	4	yes	—
401	C₂²×C₄	C₂⁴×C₅₃	1	4	yes	—
402	C₂²×C₄	C₂⁴×C₅₃	1	4	yes	—
403	C₂²×C₄	C₂⁴×C₅₃	1	4	yes	—
404	C₂²×C₄	C₂⁴×C₅₃	1	4	yes	—
405	C₂²×C₄	C₂⁴×C₅₃	1	4	yes	—
406	C₂²×C₄	C₂⁴×C₅₃	1	2	yes	—
407	C₂²×C₄	C₂⁴×C₅₃	1	2	yes	—
408	C₄⋊C₄	—	2	4	no	—
409	C₄⋊C₄	—	2	8	no	—
410	C₄⋊C₄	—	2	8	no	—
411	C₄⋊C₄	—	2	8	no	—
412	C₄⋊C₄	—	2	8	no	—
413	C₄⋊C₄	—	4	4	no	—
414	C₄⋊C₄	—	4	2	no	—
415	C₄⋊C₄	—	4	2	no	—
416	C₄⋊C₄	—	4	4	no	—
417	C₄⋊C₄	—	4	4	no	—
418	C₄⋊C₄	—	4	4	no	—
419	C₄⋊C₄	—	4	4	no	—
420	C₄⋊C₄	—	4	4	no	—
421	C₄⋊C₄	—	4	4	no	—
422	C₂²⋊C₄	—	2	8	no	—
423	C₂²⋊C₄	—	2	8	no	—
424	C₂²⋊C₄	—	2	8	no	—
425	C₂²⋊C₄	—	2	8	no	—
426	C₂²⋊C₄	—	2	8	no	—
427	C₂²⋊C₄	—	4	4	no	—
428	C₂²⋊C₄	—	4	4	no	—
429	C₂²⋊C₄	—	4	4	no	—
430	C₄²	—	4	4	no	—
431	C₄²	—	4	4	no	—
432	C₂³	—	2	2	no	—
433	C₂³	—	2	1	no	—
434	C₂³	—	2	1	no	—
435	C₂³	—	2	8	no	—
436	C₂³	—	2	1	no	—
437	C₂³	—	2	1	no	—
438	C₂³	—	2	1	no	—
439	C₂³	—	2	2	no	—
440	C₂³	—	2	2	no	—
441	C₂³	—	2	1	no	—
442	C₂³	—	2	2	no	—
443	C₂³	—	2	2	no	—
444	C₂³	—	4	8	no	—
445	C₂³	—	4	8	no	—
446	C₂³	—	4	4	no	—
447	C₂³	—	4	4	no	—
448	C₂³	C₂×#41×C₅₃	1	2	yes	—
449	C₂³	H(5,𝔽₂)×C₅₃	1	2	yes	—
450	C₂³	2₋¹⁺⁴×C₅₃	1	2	yes	—
451	C₂³	2₋¹⁺⁴×C₅₃	1	4	yes	—
452	C₂³	C₂⁵×C₅₃	1	1	yes	characteristic
453	Q₈	—	8	4	no	—
454	C₂×C₄	—	2	8	no	—
455	C₂×C₄	—	2	2	no	—
456	C₂×C₄	—	2	2	no	—
457	C₂×C₄	—	2	2	no	—
458	C₂×C₄	—	2	2	no	—
459	C₂×C₄	—	2	4	no	—
460	C₂×C₄	—	2	4	no	—
461	C₂×C₄	—	2	4	no	—
462	C₂×C₄	—	2	4	no	—
463	C₂×C₄	—	2	4	no	—
464	C₂×C₄	—	2	4	no	—
465	C₂×C₄	—	2	4	no	—
466	C₂×C₄	—	2	4	no	—
467	C₂×C₄	—	2	4	no	—
468	C₂×C₄	—	2	4	no	—
469	C₂×C₄	—	2	4	no	—
470	C₂×C₄	—	2	4	no	—
471	C₂×C₄	—	2	4	no	—
472	C₂×C₄	—	2	4	no	—
473	C₂×C₄	—	2	4	no	—
474	C₂×C₄	—	2	4	no	—
475	C₂×C₄	—	2	4	no	—
476	C₂×C₄	—	2	4	no	—
477	C₂×C₄	—	4	4	no	—
478	C₂×C₄	—	4	8	no	—
479	C₂×C₄	—	4	8	no	—
480	C₂×C₄	—	8	4	no	—
481	C₂×C₄	—	8	4	no	—
482	C₂×C₄	C₂×#41×C₅₃	1	4	yes	—
483	C₂×C₄	H(5,𝔽₂)×C₅₃	1	4	yes	—
484	C₂²	—	2	2	no	—
485	C₂²	—	2	2	no	—
486	C₂²	—	2	4	no	—
487	C₂²	—	2	4	no	—
488	C₂²	—	2	2	no	—
489	C₂²	—	2	2	no	—
490	C₂²	—	2	4	no	—
491	C₂²	—	2	4	no	—
492	C₂²	—	4	16	no	—
493	C₂²	—	4	1	no	—
494	C₂²	—	4	1	no	—
495	C₂²	—	4	1	no	—
496	C₂²	—	4	1	no	—
497	C₂²	—	4	2	no	—
498	C₂²	—	4	2	no	—
499	C₂²	C₅₃×#541	1	4	yes	—
500	C₂²	C₂²×#41×C₅₃	1	1	yes	characteristic
501	C₂²	C₂×H(5,𝔽₂)×C₅₃	1	2	yes	—
502	C₂²	C₂×H(5,𝔽₂)×C₅₃	1	1	yes	characteristic
503	C₂²	C₂×2₋¹⁺⁴×C₅₃	1	2	yes	—
504	C₂²	C₂×2₋¹⁺⁴×C₅₃	1	1	yes	characteristic
505	C₄	—	2	4	no	—
506	C₄	—	2	4	no	—
507	C₄	—	4	4	no	—
508	C₄	—	4	4	no	—
509	C₄	—	4	4	no	—
510	C₄	—	4	4	no	—
511	C₄	—	4	4	no	—
512	C₄	—	4	4	no	—
513	C₂	—	2	4	no	—
514	C₂	—	4	2	no	—
515	C₂	—	4	2	no	—
516	C₂	C₂×C₅₃×#541	1	1	yes	characteristic
517	C₂	C₅₃×#3490	1	1	yes	characteristic
518	C₂	C₅₃×#3492	1	1	yes	characteristic
519	C₂	C₅₃×#3536	1	2	yes	—
520	C₂	C₅₃×#3538	1	2	yes	—

	subgroup	G/H
center, socle	C₂³×C₅₃	C₂⁵
commutator, Frattini, cosocle	C₂³	C₂⁵×C₅₃
prime radical	C₂⁵×C₅₃	C₂³
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³×C₅₃ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₅₃×#2747	C₂	semidirect
3. C₅₃×#2722	C₂	semidirect
4. C₅₃×#2704	C₂	semidirect
5. C₅₃×#2690	C₂	semidirect
6. C₅₃×#2659	C₂	semidirect
7. C₅₃×#2646	C₂	semidirect
9. C₅₃×#2473	C₂	semidirect
10. C₅₃×#2329	C₂	semidirect
14. C₂×#125×C₅₃	C₂²	semidirect
20. C₅₃×#400	C₂²	semidirect
21. C₅₃×#400	C₂²	semidirect
22. C₅₃×#398	C₂²	semidirect
23. C₅₃×#398	C₂²	semidirect
24. C₅₃×#398	C₂²	semidirect
25. C₅₃×#398	C₂²	semidirect
26. C₅₃×#398	C₂²	semidirect
31. C₅₃×#395	C₂²	semidirect
32. C₅₃×#391	C₂²	semidirect
33. C₅₃×#389	C₂²	semidirect
34. C₅₃×#389	C₂²	semidirect
38. C₅₃×#387	C₂²	semidirect
39. C₅₃×#387	C₂²	semidirect
40. C₅₃×#387	C₂²	semidirect
41. C₅₃×#387	C₂²	semidirect
223. #36728	C₅₃	direct (N × H)
254. #2747	C₂×C₅₃	semidirect
255. #2722	C₂×C₅₃	semidirect
256. #2704	C₂×C₅₃	semidirect
257. #2690	C₂×C₅₃	semidirect
258. #2659	C₂×C₅₃	semidirect
259. #2646	C₂×C₅₃	semidirect
261. #2473	C₂×C₅₃	semidirect
262. #2329	C₂×C₅₃	semidirect
274. C₂×#125	C₂²×C₅₃	semidirect
280. #400	C₂²×C₅₃	semidirect
281. #400	C₂²×C₅₃	semidirect
282. #398	C₂²×C₅₃	semidirect
283. #398	C₂²×C₅₃	semidirect
284. #398	C₂²×C₅₃	semidirect
285. #398	C₂²×C₅₃	semidirect
286. #398	C₂²×C₅₃	semidirect
291. #395	C₂²×C₅₃	semidirect
292. #391	C₂²×C₅₃	semidirect
293. #389	C₂²×C₅₃	semidirect
294. #389	C₂²×C₅₃	semidirect
298. #387	C₂²×C₅₃	semidirect
299. #387	C₂²×C₅₃	semidirect
300. #387	C₂²×C₅₃	semidirect
301. #387	C₂²×C₅₃	semidirect
317. C₅₃	#36728	direct (N × H)

Maximal subgroups

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

subgroup	index
C₅₃×#2848	2
C₅₃×#2747	2
C₅₃×#2722	2
C₅₃×#2704	2
C₅₃×#2690	2
C₅₃×#2659	2
C₅₃×#2646	2
C₅₃×#2480	2
C₅₃×#2473	2
C₅₃×#2329	2
C₂×C₅₃×#380	2
#36728	53

Subgroups — C53×#36728 #27244240

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅₃×#36728 #27244240