Subgroups — C₅₃×#22812 #27230324

Order 13568.

802 conjugacy classes of subgroups.

The subgroup lattice has 442 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄×SD₁₆×C₅₃	C₂	1	1	yes	maximal, characteristic
2	C₅₃×#2357	C₂	1	1	yes	maximal, characteristic
3	C₅₃×#2356	C₂	1	1	yes	maximal, characteristic, complemented
4	C₅₃×#1973	C₂	1	2	yes	maximal
5	C₅₃×#1927	C₂	1	2	yes	maximal, complemented
6	C₅₃×#1926	C₂	1	2	yes	maximal
7	C₅₃×#1846	C₂	1	1	yes	maximal, characteristic, complemented
8	C₅₃×#1805	C₂	1	1	yes	maximal, characteristic, complemented
9	C₅₃×#1804	C₂	1	1	yes	maximal, characteristic
10	C₅₃×#1776	C₂	1	2	yes	maximal, complemented
11	C₅₃×#1770	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂²×SD₁₆×C₅₃	C₂²	1	1	yes	characteristic
13	C₂×C₄×Q₈×C₅₃	C₂²	1	1	yes	characteristic
14	C₂×C₄×D₄×C₅₃	C₂²	1	1	yes	characteristic
15	C₄×SD₁₆×C₅₃	C₄	1	8	yes	complemented
16	C₂×#107×C₅₃	C₂²	1	2	yes	—
17	C₂×#106×C₅₃	C₂²	1	2	yes	—
18	C₂×#106×C₅₃	C₂²	1	1	yes	characteristic
19	C₂×C₄⋊C₈×C₅₃	C₂²	1	1	yes	characteristic
20	C₂×#103×C₅₃	C₂²	1	2	yes	—
21	C₂×#103×C₅₃	C₂²	1	1	yes	characteristic
22	C₂×#102×C₅₃	C₂²	1	2	yes	—
23	C₂×#102×C₅₃	C₂²	1	1	yes	characteristic
24	C₂×#97×C₅₃	C₂²	1	2	yes	—
25	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
26	C₅₃×#391	C₂²	1	2	yes	—
27	C₅₃×#390	C₂²	1	2	yes	—
28	C₅₃×#388	—	2	2	no	—
29	C₅₃×#387	—	2	2	no	—
30	C₅₃×#383	—	2	1	no	—
31	C₅₃×#382	C₂²	1	1	yes	characteristic
32	C₅₃×#382	C₂²	1	1	yes	characteristic
33	C₅₃×#381	—	2	1	no	—
34	C₄×C₄⋊C₄×C₅₃	C₂²	1	2	yes	—
35	C₄×C₄⋊C₄×C₅₃	C₂²	1	2	yes	—
36	C₄×C₄⋊C₄×C₅₃	C₂²	1	1	yes	characteristic
37	C₄×C₂²⋊C₄×C₅₃	—	2	1	no	—
38	C₅₃×#376	—	2	1	no	—
39	C₅₃×#340	C₂²	1	2	yes	—
40	C₅₃×#340	C₂²	1	2	yes	—
41	C₅₃×#336	C₂²	1	2	yes	—
42	C₂²×Q₈×C₅₃	C₂³	1	1	yes	characteristic
43	C₂²×D₄×C₅₃	C₂³	1	1	yes	characteristic
44	C₂³×C₄×C₅₃	—	2	1	no	—
45	C₂×SD₁₆×C₅₃	—	2	2	no	—
46	C₂×SD₁₆×C₅₃	—	2	2	no	—
47	C₂×SD₁₆×C₅₃	C₂×C₄	1	4	yes	—
48	C₂²×C₈×C₅₃	C₂³	1	2	yes	—
49	C₂²×C₈×C₅₃	C₂³	1	1	yes	characteristic
50	C₂²×C₈×C₅₃	C₂³	1	1	yes	characteristic
51	C₄×Q₈×C₅₃	—	2	2	no	—
52	C₄×Q₈×C₅₃	C₂×C₄	1	4	yes	—
53	C₄×D₄×C₅₃	—	2	2	no	—
54	C₄×D₄×C₅₃	C₂×C₄	1	4	yes	—
55	C₂×C₄⋊C₄×C₅₃	—	2	1	no	—
56	C₂×C₄⋊C₄×C₅₃	D₄	1	2	yes	—
57	C₂×C₄⋊C₄×C₅₃	C₂³	1	2	yes	—
58	C₂×C₄⋊C₄×C₅₃	C₂³	1	2	yes	—
59	C₂×C₄⋊C₄×C₅₃	C₂³	1	1	yes	characteristic
60	C₂×C₄⋊C₄×C₅₃	C₂³	1	1	yes	characteristic
61	C₂×C₂²⋊C₄×C₅₃	—	2	1	no	—
62	C₂×C₂²⋊C₄×C₅₃	—	2	2	no	—
63	C₂×C₂²⋊C₄×C₅₃	—	2	2	no	—
64	C₂×C₄²×C₅₃	—	2	1	no	—
65	C₂×C₄²×C₅₃	—	2	2	no	—
66	C₂×C₄²×C₅₃	—	2	2	no	—
67	C₂×C₄²×C₅₃	C₂³	1	2	yes	—
68	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
69	#107×C₅₃	—	2	4	no	—
70	#106×C₅₃	—	4	2	no	—
71	#106×C₅₃	C₂×C₄	1	4	yes	—
72	C₄⋊C₈×C₅₃	—	2	2	no	—
73	#103×C₅₃	—	4	2	no	—
74	#103×C₅₃	C₂×C₄	1	4	yes	—
75	#102×C₅₃	—	4	2	no	—
76	#102×C₅₃	C₂×C₄	1	4	yes	—
77	#97×C₅₃	—	2	4	no	—
78	C₄×C₈×C₅₃	C₂×C₄	1	4	yes	—
79	#95×C₅₃	—	2	1	no	—
80	#95×C₅₃	—	2	2	no	—
81	#95×C₅₃	—	2	1	no	—
82	#95×C₅₃	—	2	2	no	—
83	#95×C₅₃	—	2	2	no	—
84	#95×C₅₃	—	2	1	no	—
85	#95×C₅₃	—	2	1	no	—
86	#95×C₅₃	—	2	2	no	—
87	#95×C₅₃	D₄	1	1	yes	characteristic
88	#95×C₅₃	D₄	1	1	yes	characteristic
89	C₂⁴×C₅₃	—	2	1	no	—
90	C₂×Q₈×C₅₃	—	2	1	no	—
91	C₂×Q₈×C₅₃	—	2	1	no	—
92	C₂×Q₈×C₅₃	—	2	1	no	—
93	C₂×Q₈×C₅₃	—	2	1	no	—
94	C₂×Q₈×C₅₃	—	2	1	no	—
95	C₂×Q₈×C₅₃	C₂²×C₄	1	2	yes	—
96	C₂×D₄×C₅₃	—	2	1	no	—
97	C₂×D₄×C₅₃	—	2	1	no	—
98	C₂×D₄×C₅₃	—	2	1	no	—
99	C₂×D₄×C₅₃	—	2	1	no	—
100	C₂×D₄×C₅₃	—	2	1	no	—
101	C₂×D₄×C₅₃	C₂²×C₄	1	2	yes	—
102	C₂²×C₄×C₅₃	—	2	1	no	—
103	C₂²×C₄×C₅₃	—	2	1	no	—
104	C₂²×C₄×C₅₃	—	2	1	no	—
105	C₂²×C₄×C₅₃	—	2	4	no	—
106	C₂²×C₄×C₅₃	—	2	2	no	—
107	C₂²×C₄×C₅₃	—	2	2	no	—
108	C₂²×C₄×C₅₃	—	8	2	no	—
109	C₂²×C₄×C₅₃	C₂×D₄	1	2	yes	—
110	C₂²×C₄×C₅₃	C₂×D₄	1	2	yes	—
111	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
112	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
113	C₂²×C₄×C₅₃	C₂⁴	1	1	yes	characteristic
114	SD₁₆×C₅₃	—	2	8	no	—
115	C₂×C₈×C₅₃	—	2	1	no	—
116	C₂×C₈×C₅₃	—	2	1	no	—
117	C₂×C₈×C₅₃	—	2	2	no	—
118	C₂×C₈×C₅₃	—	2	2	no	—
119	C₂×C₈×C₅₃	—	2	2	no	—
120	C₂×C₈×C₅₃	C₂²×C₄	1	2	yes	—
121	C₂×C₈×C₅₃	C₂²×C₄	1	2	yes	—
122	C₂×C₈×C₅₃	#41	1	2	yes	—
123	C₂×C₈×C₅₃	#41	1	2	yes	—
124	C₄⋊C₄×C₅₃	—	2	4	no	—
125	C₄⋊C₄×C₅₃	—	2	1	no	—
126	C₄⋊C₄×C₅₃	—	2	1	no	—
127	C₄⋊C₄×C₅₃	—	2	2	no	—
128	C₄⋊C₄×C₅₃	—	2	2	no	—
129	C₄⋊C₄×C₅₃	—	2	2	no	—
130	C₄⋊C₄×C₅₃	—	2	2	no	—
131	C₄⋊C₄×C₅₃	—	4	2	no	—
132	C₄⋊C₄×C₅₃	C₂²×C₄	1	2	yes	—
133	C₄⋊C₄×C₅₃	C₂²×C₄	1	2	yes	—
134	C₂²⋊C₄×C₅₃	—	2	4	no	—
135	C₂²⋊C₄×C₅₃	—	4	4	no	—
136	C₂²⋊C₄×C₅₃	—	8	2	no	—
137	C₄²×C₅₃	—	2	4	no	—
138	C₄²×C₅₃	—	2	2	no	—
139	C₄²×C₅₃	—	2	2	no	—
140	C₄²×C₅₃	—	4	4	no	—
141	C₄²×C₅₃	—	8	2	no	—
142	C₄²×C₅₃	C₂²×C₄	1	2	yes	—
143	C₄²×C₅₃	#41	1	2	yes	—
144	C₂³×C₅₃	—	2	2	no	—
145	C₂³×C₅₃	—	4	1	no	—
146	C₂³×C₅₃	—	4	1	no	—
147	C₂³×C₅₃	—	4	2	no	—
148	C₂³×C₅₃	—	4	2	no	—
149	C₂³×C₅₃	C₂²×D₄	1	1	yes	characteristic
150	Q₈×C₅₃	—	2	2	no	—
151	Q₈×C₅₃	—	2	2	no	—
152	Q₈×C₅₃	—	4	1	no	—
153	Q₈×C₅₃	—	4	1	no	—
154	D₄×C₅₃	—	2	2	no	—
155	D₄×C₅₃	—	2	2	no	—
156	D₄×C₅₃	—	4	1	no	—
157	D₄×C₅₃	—	4	1	no	—
158	C₂×C₄×C₅₃	—	2	2	no	—
159	C₂×C₄×C₅₃	—	2	2	no	—
160	C₂×C₄×C₅₃	—	2	2	no	—
161	C₂×C₄×C₅₃	—	2	2	no	—
162	C₂×C₄×C₅₃	—	2	2	no	—
163	C₂×C₄×C₅₃	—	2	2	no	—
164	C₂×C₄×C₅₃	—	2	2	no	—
165	C₂×C₄×C₅₃	—	2	2	no	—
166	C₂×C₄×C₅₃	—	2	2	no	—
167	C₂×C₄×C₅₃	—	2	2	no	—
168	C₂×C₄×C₅₃	—	2	2	no	—
169	C₂×C₄×C₅₃	—	4	1	no	—
170	C₂×C₄×C₅₃	—	4	1	no	—
171	C₂×C₄×C₅₃	—	4	2	no	—
172	C₂×C₄×C₅₃	—	4	2	no	—
173	C₂×C₄×C₅₃	—	4	2	no	—
174	C₂×C₄×C₅₃	—	4	2	no	—
175	C₂×C₄×C₅₃	—	4	2	no	—
176	C₂×C₄×C₅₃	—	4	2	no	—
177	C₂×C₄×C₅₃	—	4	2	no	—
178	C₂×C₄×C₅₃	—	4	2	no	—
179	C₂×C₄×C₅₃	—	8	4	no	—
180	C₂×C₄×C₅₃	C₄×D₄	1	2	yes	—
181	C₂×C₄×C₅₃	C₄×D₄	1	2	yes	—
182	C₂×C₄×C₅₃	C₂³×C₄	1	1	yes	characteristic
183	C₂×C₄×C₅₃	C₂×#41	1	1	yes	characteristic
184	C₂×C₄×C₅₃	C₂×#41	1	1	yes	characteristic
185	C₂×C₄×C₅₃	C₂×#41	1	1	yes	characteristic
186	C₂×C₄×C₅₃	H(5,𝔽₂)	1	1	yes	characteristic
187	C₂×C₄×C₅₃	2₋¹⁺⁴	1	1	yes	characteristic
188	C₈×C₅₃	—	2	2	no	—
189	C₈×C₅₃	—	4	2	no	—
190	C₈×C₅₃	#117	1	4	yes	—
191	#22812	C₅₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
192	C₂²×C₅₃	—	4	2	no	—
193	C₂²×C₅₃	—	4	2	no	—
194	C₂²×C₅₃	—	4	2	no	—
195	C₂²×C₅₃	—	8	2	no	—
196	C₂²×C₅₃	—	8	2	no	—
197	C₂²×C₅₃	C₂×C₄×D₄	1	1	yes	characteristic
198	C₂²×C₅₃	#547	1	1	yes	characteristic
199	C₂²×C₅₃	#550	1	1	yes	characteristic
200	C₂²×C₅₃	#575	1	2	yes	—
201	C₂²×C₅₃	#577	1	2	yes	—
202	C₄×C₅₃	—	2	2	no	—
203	C₄×C₅₃	—	2	2	no	—
204	C₄×C₅₃	—	4	2	no	—
205	C₄×C₅₃	—	4	2	no	—
206	C₄×C₅₃	—	4	2	no	—
207	C₄×C₅₃	—	4	2	no	—
208	C₄×C₅₃	—	4	2	no	—
209	C₄×C₅₃	—	8	2	no	—
210	C₄×C₅₃	—	8	2	no	—
211	C₄×C₅₃	C₂×#117	1	1	yes	characteristic
212	C₄×C₅₃	#520	1	1	yes	characteristic
213	C₄×C₅₃	#555	1	1	yes	characteristic
214	C₄×C₅₃	#556	1	1	yes	characteristic
215	C₂×C₄×SD₁₆	C₂×C₅₃	1	1	yes	characteristic
216	#2357	C₂×C₅₃	1	1	yes	characteristic
217	#2356	C₂×C₅₃	1	1	yes	characteristic, complemented
218	#1973	C₂×C₅₃	1	2	yes	—
219	#1927	C₂×C₅₃	1	2	yes	complemented
220	#1926	C₂×C₅₃	1	2	yes	—
221	#1846	C₂×C₅₃	1	1	yes	characteristic, complemented
222	#1805	C₂×C₅₃	1	1	yes	characteristic, complemented
223	#1804	C₂×C₅₃	1	1	yes	characteristic
224	#1776	C₂×C₅₃	1	2	yes	complemented
225	#1770	C₂×C₅₃	1	1	yes	characteristic, complemented
226	C₂×C₅₃	—	8	2	no	—
227	C₂×C₅₃	#2354	1	1	yes	characteristic
228	C₂×C₅₃	#2943	1	1	yes	characteristic
229	C₂×C₅₃	#2949	1	1	yes	characteristic
230	C₂×C₅₃	#3351	1	2	yes	—
231	C₂×C₅₃	#3401	1	2	yes	—
232	C₂²×SD₁₆	C₂²×C₅₃	1	1	yes	characteristic
233	C₂×C₄×Q₈	C₂²×C₅₃	1	1	yes	characteristic
234	C₂×C₄×D₄	C₂²×C₅₃	1	1	yes	characteristic
235	C₄×SD₁₆	C₄×C₅₃	1	8	yes	complemented
236	C₂×#107	C₂²×C₅₃	1	2	yes	—
237	C₂×#106	C₂²×C₅₃	1	2	yes	—
238	C₂×#106	C₂²×C₅₃	1	1	yes	characteristic
239	C₂×C₄⋊C₈	C₂²×C₅₃	1	1	yes	characteristic
240	C₂×#103	C₂²×C₅₃	1	2	yes	—
241	C₂×#103	C₂²×C₅₃	1	1	yes	characteristic
242	C₂×#102	C₂²×C₅₃	1	2	yes	—
243	C₂×#102	C₂²×C₅₃	1	1	yes	characteristic
244	C₂×#97	C₂²×C₅₃	1	2	yes	—
245	C₂×C₄×C₈	C₂²×C₅₃	1	1	yes	characteristic
246	#391	C₂²×C₅₃	1	2	yes	—
247	#390	C₂²×C₅₃	1	2	yes	—
248	#388	—	2	2	no	—
249	#387	—	2	2	no	—
250	#383	—	2	1	no	—
251	#382	C₂²×C₅₃	1	1	yes	characteristic
252	#382	C₂²×C₅₃	1	1	yes	characteristic
253	#381	—	2	1	no	—
254	C₄×C₄⋊C₄	C₂²×C₅₃	1	2	yes	—
255	C₄×C₄⋊C₄	C₂²×C₅₃	1	2	yes	—
256	C₄×C₄⋊C₄	C₂²×C₅₃	1	1	yes	characteristic
257	C₄×C₂²⋊C₄	—	2	1	no	—
258	#376	—	2	1	no	—
259	#340	C₂²×C₅₃	1	2	yes	—
260	#340	C₂²×C₅₃	1	2	yes	—
261	#336	C₂²×C₅₃	1	2	yes	—
262	C₅₃	#22812	1	1	yes	characteristic, direct product factor, Sylow
263	C₂²×Q₈	C₂³×C₅₃	1	1	yes	characteristic
264	C₂²×D₄	C₂³×C₅₃	1	1	yes	characteristic
265	C₂³×C₄	—	2	1	no	—
266	C₂×SD₁₆	—	2	2	no	—
267	C₂×SD₁₆	—	2	2	no	—
268	C₂×SD₁₆	C₂×C₄×C₅₃	1	4	yes	—
269	C₂²×C₈	C₂³×C₅₃	1	2	yes	—
270	C₂²×C₈	C₂³×C₅₃	1	1	yes	characteristic
271	C₂²×C₈	C₂³×C₅₃	1	1	yes	characteristic
272	C₄×Q₈	—	2	2	no	—
273	C₄×Q₈	C₂×C₄×C₅₃	1	4	yes	—
274	C₄×D₄	—	2	2	no	—
275	C₄×D₄	C₂×C₄×C₅₃	1	4	yes	—
276	C₂×C₄⋊C₄	—	2	1	no	—
277	C₂×C₄⋊C₄	D₄×C₅₃	1	2	yes	—
278	C₂×C₄⋊C₄	C₂³×C₅₃	1	2	yes	—
279	C₂×C₄⋊C₄	C₂³×C₅₃	1	2	yes	—
280	C₂×C₄⋊C₄	C₂³×C₅₃	1	1	yes	characteristic
281	C₂×C₄⋊C₄	C₂³×C₅₃	1	1	yes	characteristic
282	C₂×C₂²⋊C₄	—	2	1	no	—
283	C₂×C₂²⋊C₄	—	2	2	no	—
284	C₂×C₂²⋊C₄	—	2	2	no	—
285	C₂×C₄²	—	2	1	no	—
286	C₂×C₄²	—	2	2	no	—
287	C₂×C₄²	—	2	2	no	—
288	C₂×C₄²	C₂³×C₅₃	1	2	yes	—
289	C₂×C₄²	C₂³×C₅₃	1	1	yes	characteristic
290	#107	—	2	4	no	—
291	#106	—	4	2	no	—
292	#106	C₂×C₄×C₅₃	1	4	yes	—
293	C₄⋊C₈	—	2	2	no	—
294	#103	—	4	2	no	—
295	#103	C₂×C₄×C₅₃	1	4	yes	—
296	#102	—	4	2	no	—
297	#102	C₂×C₄×C₅₃	1	4	yes	—
298	#97	—	2	4	no	—
299	C₄×C₈	C₂×C₄×C₅₃	1	4	yes	—
300	#95	—	2	1	no	—
301	#95	—	2	2	no	—
302	#95	—	2	1	no	—
303	#95	—	2	2	no	—
304	#95	—	2	2	no	—
305	#95	—	2	1	no	—
306	#95	—	2	1	no	—
307	#95	—	2	2	no	—
308	#95	D₄×C₅₃	1	1	yes	characteristic
309	#95	D₄×C₅₃	1	1	yes	characteristic
310	C₂⁴	—	2	1	no	—
311	C₂×Q₈	—	2	1	no	—
312	C₂×Q₈	—	2	1	no	—
313	C₂×Q₈	—	2	1	no	—
314	C₂×Q₈	—	2	1	no	—
315	C₂×Q₈	—	2	1	no	—
316	C₂×Q₈	C₂²×C₄×C₅₃	1	2	yes	—
317	C₂×D₄	—	2	1	no	—
318	C₂×D₄	—	2	1	no	—
319	C₂×D₄	—	2	1	no	—
320	C₂×D₄	—	2	1	no	—
321	C₂×D₄	—	2	1	no	—
322	C₂×D₄	C₂²×C₄×C₅₃	1	2	yes	—
323	C₂²×C₄	—	2	1	no	—
324	C₂²×C₄	—	2	1	no	—
325	C₂²×C₄	—	2	1	no	—
326	C₂²×C₄	—	2	4	no	—
327	C₂²×C₄	—	2	2	no	—
328	C₂²×C₄	—	2	2	no	—
329	C₂²×C₄	—	8	2	no	—
330	C₂²×C₄	C₂×D₄×C₅₃	1	2	yes	—
331	C₂²×C₄	C₂×D₄×C₅₃	1	2	yes	—
332	C₂²×C₄	C₂×D₄×C₅₃	1	1	yes	characteristic
333	C₂²×C₄	C₂×D₄×C₅₃	1	1	yes	characteristic
334	C₂²×C₄	C₂⁴×C₅₃	1	1	yes	characteristic
335	SD₁₆	—	2	8	no	—
336	C₂×C₈	—	2	1	no	—
337	C₂×C₈	—	2	1	no	—
338	C₂×C₈	—	2	2	no	—
339	C₂×C₈	—	2	2	no	—
340	C₂×C₈	—	2	2	no	—
341	C₂×C₈	C₂²×C₄×C₅₃	1	2	yes	—
342	C₂×C₈	C₂²×C₄×C₅₃	1	2	yes	—
343	C₂×C₈	#41×C₅₃	1	2	yes	—
344	C₂×C₈	#41×C₅₃	1	2	yes	—
345	C₄⋊C₄	—	2	4	no	—
346	C₄⋊C₄	—	2	1	no	—
347	C₄⋊C₄	—	2	1	no	—
348	C₄⋊C₄	—	2	2	no	—
349	C₄⋊C₄	—	2	2	no	—
350	C₄⋊C₄	—	2	2	no	—
351	C₄⋊C₄	—	2	2	no	—
352	C₄⋊C₄	—	4	2	no	—
353	C₄⋊C₄	C₂²×C₄×C₅₃	1	2	yes	—
354	C₄⋊C₄	C₂²×C₄×C₅₃	1	2	yes	—
355	C₂²⋊C₄	—	2	4	no	—
356	C₂²⋊C₄	—	4	4	no	—
357	C₂²⋊C₄	—	8	2	no	—
358	C₄²	—	2	4	no	—
359	C₄²	—	2	2	no	—
360	C₄²	—	2	2	no	—
361	C₄²	—	4	4	no	—
362	C₄²	—	8	2	no	—
363	C₄²	C₂²×C₄×C₅₃	1	2	yes	—
364	C₄²	#41×C₅₃	1	2	yes	—
365	C₂³	—	2	2	no	—
366	C₂³	—	4	1	no	—
367	C₂³	—	4	1	no	—
368	C₂³	—	4	2	no	—
369	C₂³	—	4	2	no	—
370	C₂³	C₂²×D₄×C₅₃	1	1	yes	characteristic
371	Q₈	—	2	2	no	—
372	Q₈	—	2	2	no	—
373	Q₈	—	4	1	no	—
374	Q₈	—	4	1	no	—
375	D₄	—	2	2	no	—
376	D₄	—	2	2	no	—
377	D₄	—	4	1	no	—
378	D₄	—	4	1	no	—
379	C₂×C₄	—	2	2	no	—
380	C₂×C₄	—	2	2	no	—
381	C₂×C₄	—	2	2	no	—
382	C₂×C₄	—	2	2	no	—
383	C₂×C₄	—	2	2	no	—
384	C₂×C₄	—	2	2	no	—
385	C₂×C₄	—	2	2	no	—
386	C₂×C₄	—	2	2	no	—
387	C₂×C₄	—	2	2	no	—
388	C₂×C₄	—	2	2	no	—
389	C₂×C₄	—	2	2	no	—
390	C₂×C₄	—	4	1	no	—
391	C₂×C₄	—	4	1	no	—
392	C₂×C₄	—	4	2	no	—
393	C₂×C₄	—	4	2	no	—
394	C₂×C₄	—	4	2	no	—
395	C₂×C₄	—	4	2	no	—
396	C₂×C₄	—	4	2	no	—
397	C₂×C₄	—	4	2	no	—
398	C₂×C₄	—	4	2	no	—
399	C₂×C₄	—	4	2	no	—
400	C₂×C₄	—	8	4	no	—
401	C₂×C₄	C₄×D₄×C₅₃	1	2	yes	—
402	C₂×C₄	C₄×D₄×C₅₃	1	2	yes	—
403	C₂×C₄	C₂³×C₄×C₅₃	1	1	yes	characteristic
404	C₂×C₄	C₂×#41×C₅₃	1	1	yes	characteristic
405	C₂×C₄	C₂×#41×C₅₃	1	1	yes	characteristic
406	C₂×C₄	C₂×#41×C₅₃	1	1	yes	characteristic
407	C₂×C₄	H(5,𝔽₂)×C₅₃	1	1	yes	characteristic
408	C₂×C₄	2₋¹⁺⁴×C₅₃	1	1	yes	characteristic
409	C₈	—	2	2	no	—
410	C₈	—	4	2	no	—
411	C₈	#117×C₅₃	1	4	yes	—
412	C₂²	—	4	2	no	—
413	C₂²	—	4	2	no	—
414	C₂²	—	4	2	no	—
415	C₂²	—	8	2	no	—
416	C₂²	—	8	2	no	—
417	C₂²	C₂×C₄×D₄×C₅₃	1	1	yes	characteristic
418	C₂²	C₅₃×#547	1	1	yes	characteristic
419	C₂²	C₅₃×#550	1	1	yes	characteristic
420	C₂²	C₅₃×#575	1	2	yes	—
421	C₂²	C₅₃×#577	1	2	yes	—
422	C₄	—	2	2	no	—
423	C₄	—	2	2	no	—
424	C₄	—	4	2	no	—
425	C₄	—	4	2	no	—
426	C₄	—	4	2	no	—
427	C₄	—	4	2	no	—
428	C₄	—	4	2	no	—
429	C₄	—	8	2	no	—
430	C₄	—	8	2	no	—
431	C₄	C₂×#117×C₅₃	1	1	yes	characteristic
432	C₄	C₅₃×#520	1	1	yes	characteristic
433	C₄	C₅₃×#555	1	1	yes	characteristic
434	C₄	C₅₃×#556	1	1	yes	characteristic
435	C₂	—	8	2	no	—
436	C₂	C₅₃×#2354	1	1	yes	characteristic
437	C₂	C₅₃×#2943	1	1	yes	characteristic
438	C₂	C₅₃×#2949	1	1	yes	characteristic
439	C₂	C₅₃×#3351	1	2	yes	—
440	C₂	C₅₃×#3401	1	2	yes	—

	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	C₂³
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₅₃ ◁ C₄×C₄⋊C₄×C₅₃ ◁ G

Complements

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

N	H ≅ G/N	type
3. C₅₃×#2356	C₂	semidirect
5. C₅₃×#1927	C₂	semidirect
7. C₅₃×#1846	C₂	semidirect
8. C₅₃×#1805	C₂	semidirect
10. C₅₃×#1776	C₂	semidirect
11. C₅₃×#1770	C₂	semidirect
15. C₄×SD₁₆×C₅₃	C₄	semidirect
191. #22812	C₅₃	direct (N × H)
217. #2356	C₂×C₅₃	semidirect
219. #1927	C₂×C₅₃	semidirect
221. #1846	C₂×C₅₃	semidirect
222. #1805	C₂×C₅₃	semidirect
224. #1776	C₂×C₅₃	semidirect
225. #1770	C₂×C₅₃	semidirect
235. C₄×SD₁₆	C₄×C₅₃	semidirect
262. C₅₃	#22812	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×C₄×SD₁₆×C₅₃	2
C₅₃×#2357	2
C₅₃×#2356	2
C₅₃×#1973	2
C₅₃×#1927	2
C₅₃×#1926	2
C₅₃×#1846	2
C₅₃×#1805	2
C₅₃×#1804	2
C₅₃×#1776	2
C₅₃×#1770	2
#22812	53

Subgroups — C53×#22812 #27230324

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅₃×#22812 #27230324