Subgroups — #27274202

Order 13568.

730 conjugacy classes of subgroups.

The subgroup lattice has 448 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#16810645	C₂	1	1	yes	maximal, characteristic
2	C₂×C₄⋊C₈×D₅₃	C₂	1	1	yes	maximal, characteristic
3	#20277120	C₂	1	1	yes	maximal, characteristic
4	#20272599	C₂	1	1	yes	maximal, characteristic, complemented
5	#20272599	C₂	1	1	yes	maximal, characteristic, complemented
6	#20272598	C₂	1	1	yes	maximal, characteristic, complemented
7	C₅₃×#1296	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂²×#11719912	—	2	1	no	—
9	C₂²×C₈×D₅₃	—	2	1	no	—
10	C₂×#13133175	—	2	1	no	—
11	C₂×C₄²×D₅₃	C₂²	1	1	yes	characteristic
12	#16810645	—	2	2	no	—
13	#16810645	C₄	1	4	yes	complemented
14	C₂×C₄⋊#88322	C₂²	1	1	yes	characteristic
15	C₂×C₄⋊#88322	C₂²	1	1	yes	characteristic
16	C₂×#13133146	—	2	1	no	—
17	C₂×#13133146	—	2	1	no	—
18	C₄⋊C₈×D₅₃	—	2	2	no	—
19	C₄⋊C₈×D₅₃	C₄	1	4	yes	complemented
20	#16810493	C₂²	1	1	yes	characteristic
21	#16810488	—	2	1	no	—
22	#16810480	—	2	1	no	—
23	#16810480	C₄	1	2	yes	—
24	C₄⋊C₄×Dic₅₃	—	2	1	no	—
25	C₄⋊C₄×Dic₅₃	C₄	1	2	yes	—
26	C₂×C₄⋊C₈×C₅₃	C₂²	1	1	yes	characteristic
27	C₂×C₄⋊C₈×C₅₃	C₂²	1	1	yes	characteristic
28	C₅₃×#384	C₂²	1	1	yes	characteristic
29	C₂²×C₄×D₅₃	C₂×C₄	1	1	yes	characteristic
30	C₂²×C₄×D₅₃	D₄	1	1	yes	characteristic
31	C₂²×C₄×D₅₃	D₄	1	1	yes	characteristic
32	C₂×#11719912	—	2	4	no	—
33	C₂×#11719912	—	4	2	no	—
34	C₂×#11719912	—	4	2	no	—
35	C₂²×#88322	—	2	1	no	—
36	C₂²×#88322	—	2	1	no	—
37	C₂×C₈×D₅₃	—	2	4	no	—
38	C₂×C₈×D₅₃	—	4	2	no	—
39	C₂×C₈×D₅₃	—	4	2	no	—
40	#13133175	—	4	2	no	—
41	#13133175	—	4	2	no	—
42	C₂×C₄⋊Dic₅₃	—	2	1	no	—
43	C₂×C₄⋊Dic₅₃	—	2	1	no	—
44	C₂×C₄⋊Dic₅₃	C₂×C₄	1	2	yes	—
45	C₂×#11719921	—	2	2	no	—
46	C₂×C₄×Dic₅₃	—	2	2	no	—
47	C₂×C₄×Dic₅₃	C₂×C₄	1	1	yes	characteristic
48	C₂×C₄×Dic₅₃	D₄	1	1	yes	characteristic
49	C₂×C₄×Dic₅₃	D₄	1	1	yes	characteristic
50	C₄²×D₅₃	C₂×C₄	1	2	yes	—
51	C₄²×D₅₃	C₂×C₄	1	2	yes	—
52	C₄²×D₅₃	D₄	1	2	yes	—
53	C₄²×D₅₃	Q₈	1	2	yes	—
54	C₄⋊#88322	—	2	1	no	—
55	C₄⋊#88322	—	2	1	no	—
56	C₄⋊#88322	C₂×C₄	1	2	yes	—
57	C₄⋊#88322	C₂×C₄	1	2	yes	—
58	#13133146	—	2	4	no	—
59	#13133146	—	2	4	no	—
60	#13133124	—	2	2	no	—
61	#13133124	—	2	2	no	—
62	C₂²×C₈×C₅₃	—	2	1	no	—
63	C₂²×C₈×C₅₃	—	2	1	no	—
64	C₂×C₄⋊C₄×C₅₃	—	2	1	no	—
65	C₂×C₄⋊C₄×C₅₃	—	2	1	no	—
66	C₂×C₄⋊C₄×C₅₃	C₂×C₄	1	2	yes	—
67	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
68	C₄⋊C₈×C₅₃	—	2	1	no	—
69	C₄⋊C₈×C₅₃	—	2	1	no	—
70	C₄⋊C₈×C₅₃	C₂×C₄	1	2	yes	—
71	C₄⋊C₈×C₅₃	C₂×C₄	1	2	yes	—
72	C₂³×D₅₃	C₂²⋊C₄	1	1	yes	characteristic
73	C₂²×C₄×C₅₃	—	2	2	no	—
74	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
75	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
76	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
77	C₂²×Dic₅₃	—	2	2	no	—
78	C₂²×Dic₅₃	C₂²⋊C₄	1	1	yes	characteristic
79	C₂²×Dic₅₃	C₂²⋊C₄	1	1	yes	characteristic
80	C₂²×Dic₅₃	C₂²⋊C₄	1	1	yes	characteristic
81	C₂×C₄×D₅₃	—	2	1	no	—
82	C₂×C₄×D₅₃	—	2	1	no	—
83	C₂×C₄×D₅₃	—	2	1	no	—
84	C₂×C₄×D₅₃	—	2	1	no	—
85	C₂×C₄×D₅₃	—	4	1	no	—
86	C₂×C₄×D₅₃	—	4	1	no	—
87	C₂×C₄×D₅₃	C₄²	1	2	yes	—
88	C₂×C₄×D₅₃	C₂²⋊C₄	1	2	yes	—
89	C₂×C₄×D₅₃	C₂²⋊C₄	1	2	yes	—
90	C₂×C₄×D₅₃	C₄⋊C₄	1	2	yes	—
91	C₂×C₄×D₅₃	C₄⋊C₄	1	2	yes	—
92	C₂×C₄×D₅₃	C₄⋊C₄	1	2	yes	—
93	C₂×C₄×D₅₃	D₈	1	2	yes	—
94	C₂×C₄×D₅₃	SD₁₆	1	2	yes	—
95	C₂×C₄×D₅₃	SD₁₆	1	2	yes	—
96	C₂×C₄×D₅₃	Q₁₆	1	2	yes	—
97	C₂×C₈×C₅₃	—	2	2	no	—
98	C₂×C₈×C₅₃	—	2	2	no	—
99	C₂×C₈×C₅₃	—	4	1	no	—
100	C₂×C₈×C₅₃	—	4	1	no	—
101	C₂×C₈×C₅₃	—	4	1	no	—
102	C₂×C₈×C₅₃	—	4	1	no	—
103	C₄⋊C₄×C₅₃	—	4	2	no	—
104	C₄⋊C₄×C₅₃	—	4	1	no	—
105	C₄⋊C₄×C₅₃	—	4	1	no	—
106	C₄⋊C₄×C₅₃	—	4	1	no	—
107	C₄⋊C₄×C₅₃	—	4	1	no	—
108	C₄²×C₅₃	C₂²×C₄	1	1	yes	characteristic
109	C₄²×C₅₃	C₂²×C₄	1	1	yes	characteristic
110	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
111	C₄²×C₅₃	C₂×Q₈	1	1	yes	characteristic
112	#11719921	—	4	4	no	—
113	C₄⋊Dic₅₃	—	4	1	no	—
114	C₄⋊Dic₅₃	—	4	1	no	—
115	C₄⋊Dic₅₃	—	4	1	no	—
116	C₄⋊Dic₅₃	—	4	1	no	—
117	C₄⋊Dic₅₃	—	4	2	no	—
118	C₄×Dic₅₃	—	4	4	no	—
119	C₄×Dic₅₃	C₄²	1	2	yes	—
120	C₄×Dic₅₃	C₂²⋊C₄	1	2	yes	—
121	C₄×Dic₅₃	C₂²⋊C₄	1	2	yes	—
122	C₄×Dic₅₃	C₄⋊C₄	1	2	yes	—
123	C₄×Dic₅₃	C₄⋊C₄	1	2	yes	—
124	C₄×Dic₅₃	C₄⋊C₄	1	2	yes	—
125	C₂×#88322	—	2	2	no	—
126	C₂×#88322	—	2	2	no	—
127	C₂×#88322	—	4	1	no	—
128	C₂×#88322	—	4	1	no	—
129	C₂×#88322	—	4	1	no	—
130	C₂×#88322	—	4	1	no	—
131	#11719912	—	4	8	no	—
132	C₈×D₅₃	—	4	8	no	—
133	C₂³×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
134	C₂²×D₅₃	—	2	1	no	—
135	C₂²×D₅₃	—	2	1	no	—
136	C₂²×D₅₃	—	2	1	no	—
137	C₂²×D₅₃	—	2	1	no	—
138	C₂²×D₅₃	#95	1	2	yes	—
139	C₂²×D₅₃	#102	1	2	yes	—
140	C₂²×D₅₃	#103	1	2	yes	—
141	C₂×C₄×C₅₃	—	2	1	no	—
142	C₂×C₄×C₅₃	—	2	1	no	—
143	C₂×C₄×C₅₃	—	4	2	no	—
144	C₂×C₄×C₅₃	—	4	2	no	—
145	C₂×C₄×C₅₃	—	4	2	no	—
146	C₂×C₄×C₅₃	C₂×C₄²	1	1	yes	characteristic
147	C₂×C₄×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
148	C₂×C₄×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
149	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
150	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
151	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
152	C₂×C₄×C₅₃	C₂×D₈	1	1	yes	characteristic
153	C₂×C₄×C₅₃	C₂×SD₁₆	1	1	yes	characteristic
154	C₂×C₄×C₅₃	C₂×SD₁₆	1	1	yes	characteristic
155	C₂×C₄×C₅₃	C₂×Q₁₆	1	1	yes	characteristic
156	C₂×C₄×C₅₃	AGL(1,ℤ/8)	1	1	yes	characteristic
157	C₂×C₄×C₅₃	AGL(1,ℤ/8)	1	1	yes	characteristic
158	C₂×C₄×C₅₃	#137	1	1	yes	characteristic
159	C₂×C₄×C₅₃	#137	1	1	yes	characteristic
160	C₂×Dic₅₃	—	2	1	no	—
161	C₂×Dic₅₃	—	2	1	no	—
162	C₂×Dic₅₃	—	2	1	no	—
163	C₂×Dic₅₃	—	2	1	no	—
164	C₂×Dic₅₃	—	4	2	no	—
165	C₂×Dic₅₃	—	4	2	no	—
166	C₂×Dic₅₃	—	4	2	no	—
167	C₂×Dic₅₃	#95	1	2	yes	—
168	C₂×Dic₅₃	#95	1	2	yes	—
169	C₂×Dic₅₃	#95	1	2	yes	—
170	C₂×Dic₅₃	#102	1	2	yes	—
171	C₂×Dic₅₃	#103	1	2	yes	—
172	C₄×D₅₃	—	2	2	no	—
173	C₄×D₅₃	—	2	2	no	—
174	C₄×D₅₃	—	2	2	no	—
175	C₄×D₅₃	—	2	2	no	—
176	C₄×D₅₃	—	4	2	no	—
177	C₄×D₅₃	—	4	2	no	—
178	C₄×D₅₃	#102	1	4	yes	—
179	C₄×D₅₃	#103	1	4	yes	—
180	C₄×D₅₃	#106	1	4	yes	—
181	C₄×D₅₃	#107	1	4	yes	—
182	C₈×C₅₃	—	4	2	no	—
183	C₈×C₅₃	—	4	2	no	—
184	#88322	—	4	2	no	—
185	#88322	—	4	2	no	—
186	#8045	—	53	1	no	maximal, Sylow
187	C₂²×C₅₃	C₂×#95	1	1	yes	characteristic
188	C₂²×C₅₃	#413	1	1	yes	characteristic
189	C₂²×C₅₃	#413	1	1	yes	characteristic
190	C₂²×C₅₃	C₂×#102	1	1	yes	characteristic
191	C₂²×C₅₃	C₂×#103	1	1	yes	characteristic
192	C₂²×C₅₃	#417	1	1	yes	characteristic
193	C₂²×C₅₃	#417	1	1	yes	characteristic
194	C₂×D₅₃	—	2	2	no	—
195	C₂×D₅₃	—	2	2	no	—
196	C₂×D₅₃	—	2	2	no	—
197	C₂×D₅₃	—	2	2	no	—
198	C₂×D₅₃	—	2	2	no	—
199	C₂×D₅₃	—	2	2	no	—
200	C₂×D₅₃	#340	1	4	yes	—
201	C₄×C₅₃	—	2	1	no	—
202	C₄×C₅₃	—	2	1	no	—
203	C₄×C₅₃	—	8	2	no	—
204	C₄×C₅₃	C₂×#102	1	1	yes	characteristic
205	C₄×C₅₃	C₂×#103	1	1	yes	characteristic
206	C₄×C₅₃	#418	1	1	yes	characteristic
207	C₄×C₅₃	#419	1	1	yes	characteristic
208	C₄×C₅₃	C₂×#106	1	1	yes	characteristic
209	C₄×C₅₃	C₂×#107	1	1	yes	characteristic
210	C₄×C₅₃	#428	1	1	yes	characteristic
211	C₄×C₅₃	#428	1	1	yes	characteristic
212	Dic₅₃	—	2	2	no	—
213	Dic₅₃	—	2	2	no	—
214	Dic₅₃	—	8	2	no	—
215	Dic₅₃	#340	1	4	yes	—
216	C₂×#423	—	53	1	no	—
217	C₂²×C₄⋊C₈	—	53	1	no	—
218	#2285	—	53	1	no	—
219	#1296	—	53	1	no	—
220	#1296	—	53	1	no	—
221	#1296	—	53	1	no	—
222	#1296	D₅₃	1	1	yes	characteristic, complemented
223	C₂×C₅₃	#1547	1	1	yes	characteristic
224	C₂×C₅₃	#1547	1	1	yes	characteristic
225	C₂×C₅₃	#1550	1	1	yes	characteristic
226	C₂×C₅₃	#1550	1	1	yes	characteristic
227	C₂×C₅₃	C₂×#340	1	1	yes	characteristic
228	C₂×C₅₃	#1736	1	1	yes	characteristic
229	C₂×C₅₃	#1742	1	1	yes	characteristic
230	D₅₃	—	2	4	no	—
231	C₂²×M₁₆	—	106	1	no	—
232	C₂³×C₈	—	106	1	no	—
233	C₂×#117	—	106	1	no	—
234	C₂²×C₄²	—	53	1	no	—
235	#423	—	53	4	no	—
236	#423	—	106	2	no	—
237	C₂×C₄⋊C₈	—	53	1	no	—
238	C₂×C₄⋊C₈	—	53	1	no	—
239	C₂×C₄⋊C₈	—	53	4	no	—
240	C₂×C₄⋊C₈	—	106	2	no	—
241	C₂×C₄⋊C₈	—	106	1	no	—
242	C₂×C₄⋊C₈	—	106	1	no	—
243	C₂×C₄⋊C₈	C₂×D₅₃	1	1	yes	characteristic
244	C₂×C₄⋊C₈	C₂×D₅₃	1	1	yes	characteristic
245	#384	—	53	1	no	—
246	#384	C₂×D₅₃	1	1	yes	characteristic
247	#382	—	106	1	no	—
248	#380	—	53	2	no	—
249	#380	—	106	1	no	—
250	C₄×C₄⋊C₄	—	53	2	no	—
251	C₄×C₄⋊C₄	—	106	1	no	—
252	C₅₃	#8045	1	1	yes	characteristic, complemented, Sylow
253	C₂³×C₄	—	53	1	no	—
254	C₂³×C₄	—	53	1	no	—
255	C₂³×C₄	—	53	1	no	—
256	C₂×M₁₆	—	106	4	no	—
257	C₂×M₁₆	—	212	2	no	—
258	C₂×M₁₆	—	212	2	no	—
259	C₂²×C₈	—	2	1	no	—
260	C₂²×C₈	—	2	1	no	—
261	C₂²×C₈	—	106	4	no	—
262	C₂²×C₈	—	106	1	no	—
263	C₂²×C₈	—	106	1	no	—
264	C₂²×C₈	—	212	2	no	—
265	C₂²×C₈	—	212	2	no	—
266	#117	—	212	2	no	—
267	#117	—	212	2	no	—
268	C₂×C₄⋊C₄	—	2	1	no	—
269	C₂×C₄⋊C₄	—	2	1	no	—
270	C₂×C₄⋊C₄	—	53	2	no	—
271	C₂×C₄⋊C₄	—	106	1	no	—
272	C₂×C₄⋊C₄	—	106	1	no	—
273	C₂×C₄⋊C₄	C₄×D₅₃	1	2	yes	—
274	C₂×C₂²⋊C₄	—	106	2	no	—
275	C₂×C₄²	—	53	1	no	—
276	C₂×C₄²	—	53	1	no	—
277	C₂×C₄²	—	53	1	no	—
278	C₂×C₄²	—	53	2	no	—
279	C₂×C₄²	—	53	2	no	—
280	C₂×C₄²	—	53	2	no	—
281	C₂×C₄²	—	53	2	no	—
282	C₂×C₄²	—	106	2	no	—
283	C₂×C₄²	C₂²×D₅₃	1	1	yes	characteristic
284	C₄⋊C₈	—	2	1	no	—
285	C₄⋊C₈	—	2	1	no	—
286	C₄⋊C₈	—	53	2	no	—
287	C₄⋊C₈	—	53	2	no	—
288	C₄⋊C₈	—	106	4	no	—
289	C₄⋊C₈	—	106	4	no	—
290	C₄⋊C₈	—	106	1	no	—
291	C₄⋊C₈	—	106	1	no	—
292	C₄⋊C₈	C₄×D₅₃	1	2	yes	—
293	C₄⋊C₈	C₄×D₅₃	1	2	yes	—
294	#95	—	106	2	no	—
295	#95	—	106	2	no	—
296	C₂⁴	—	53	1	no	—
297	C₂²×C₄	—	2	2	no	—
298	C₂²×C₄	—	53	1	no	—
299	C₂²×C₄	—	53	2	no	—
300	C₂²×C₄	—	53	2	no	—
301	C₂²×C₄	—	53	2	no	—
302	C₂²×C₄	—	53	2	no	—
303	C₂²×C₄	—	53	2	no	—
304	C₂²×C₄	—	53	2	no	—
305	C₂²×C₄	—	53	1	no	—
306	C₂²×C₄	—	53	1	no	—
307	C₂²×C₄	—	53	2	no	—
308	C₂²×C₄	—	53	2	no	—
309	C₂²×C₄	—	53	2	no	—
310	C₂²×C₄	—	53	2	no	—
311	C₂²×C₄	—	106	1	no	—
312	C₂²×C₄	—	106	1	no	—
313	C₂²×C₄	—	106	1	no	—
314	C₂²×C₄	—	106	1	no	—
315	C₂²×C₄	—	106	2	no	—
316	C₂²×C₄	—	212	1	no	—
317	C₂²×C₄	—	212	1	no	—
318	C₂²×C₄	C₂×C₄×D₅₃	1	1	yes	characteristic
319	C₂²×C₄	D₄×D₅₃	1	1	yes	characteristic
320	C₂²×C₄	D₄×D₅₃	1	1	yes	characteristic
321	M₁₆	—	212	8	no	—
322	C₂×C₈	—	2	2	no	—
323	C₂×C₈	—	2	2	no	—
324	C₂×C₈	—	4	1	no	—
325	C₂×C₈	—	4	1	no	—
326	C₂×C₈	—	4	1	no	—
327	C₂×C₈	—	4	1	no	—
328	C₂×C₈	—	106	2	no	—
329	C₂×C₈	—	106	2	no	—
330	C₂×C₈	—	212	8	no	—
331	C₂×C₈	—	212	1	no	—
332	C₂×C₈	—	212	1	no	—
333	C₂×C₈	—	212	1	no	—
334	C₂×C₈	—	212	1	no	—
335	C₄⋊C₄	—	4	2	no	—
336	C₄⋊C₄	—	4	1	no	—
337	C₄⋊C₄	—	4	1	no	—
338	C₄⋊C₄	—	4	1	no	—
339	C₄⋊C₄	—	4	1	no	—
340	C₄⋊C₄	—	212	2	no	—
341	C₄⋊C₄	—	212	1	no	—
342	C₄⋊C₄	—	212	1	no	—
343	C₄⋊C₄	—	212	1	no	—
344	C₄⋊C₄	—	212	1	no	—
345	C₂²⋊C₄	—	212	4	no	—
346	C₄²	—	53	2	no	—
347	C₄²	—	53	2	no	—
348	C₄²	—	53	2	no	—
349	C₄²	—	53	2	no	—
350	C₄²	—	53	2	no	—
351	C₄²	—	53	2	no	—
352	C₄²	—	212	4	no	—
353	C₄²	C₂×C₄×D₅₃	1	1	yes	characteristic
354	C₄²	C₂×C₄×D₅₃	1	1	yes	characteristic
355	C₄²	D₄×D₅₃	1	1	yes	characteristic
356	C₄²	Q₈×D₅₃	1	1	yes	characteristic
357	C₂³	—	53	2	no	—
358	C₂³	—	53	2	no	—
359	C₂³	—	53	2	no	—
360	C₂³	—	106	1	no	—
361	C₂³	—	106	1	no	—
362	C₂³	—	106	1	no	—
363	C₂³	—	106	1	no	—
364	C₂³	C₂²⋊C₄×D₅₃	1	1	yes	characteristic
365	C₂×C₄	—	2	1	no	—
366	C₂×C₄	—	2	1	no	—
367	C₂×C₄	—	4	2	no	—
368	C₂×C₄	—	4	2	no	—
369	C₂×C₄	—	4	2	no	—
370	C₂×C₄	—	53	2	no	—
371	C₂×C₄	—	53	2	no	—
372	C₂×C₄	—	53	2	no	—
373	C₂×C₄	—	53	4	no	—
374	C₂×C₄	—	53	4	no	—
375	C₂×C₄	—	53	4	no	—
376	C₂×C₄	—	53	4	no	—
377	C₂×C₄	—	53	2	no	—
378	C₂×C₄	—	53	2	no	—
379	C₂×C₄	—	106	1	no	—
380	C₂×C₄	—	106	1	no	—
381	C₂×C₄	—	106	1	no	—
382	C₂×C₄	—	106	1	no	—
383	C₂×C₄	—	106	2	no	—
384	C₂×C₄	—	106	2	no	—
385	C₂×C₄	—	106	2	no	—
386	C₂×C₄	—	106	2	no	—
387	C₂×C₄	—	212	2	no	—
388	C₂×C₄	—	212	2	no	—
389	C₂×C₄	—	212	2	no	—
390	C₂×C₄	—	212	2	no	—
391	C₂×C₄	—	212	2	no	—
392	C₂×C₄	C₄²×D₅₃	1	1	yes	characteristic
393	C₂×C₄	C₂²⋊C₄×D₅₃	1	1	yes	characteristic
394	C₂×C₄	C₂²⋊C₄×D₅₃	1	1	yes	characteristic
395	C₂×C₄	C₄⋊C₄×D₅₃	1	1	yes	characteristic
396	C₂×C₄	C₄⋊C₄×D₅₃	1	1	yes	characteristic
397	C₂×C₄	C₄⋊C₄×D₅₃	1	1	yes	characteristic
398	C₂×C₄	D₈×D₅₃	1	1	yes	characteristic
399	C₂×C₄	SD₁₆×D₅₃	1	1	yes	characteristic
400	C₂×C₄	SD₁₆×D₅₃	1	1	yes	characteristic
401	C₂×C₄	Q₁₆×D₅₃	1	1	yes	characteristic
402	C₂×C₄	#13133241	1	1	yes	characteristic
403	C₂×C₄	#13133244	1	1	yes	characteristic
404	C₂×C₄	#13133246	1	1	yes	characteristic
405	C₂×C₄	#13133249	1	1	yes	characteristic
406	C₈	—	4	2	no	—
407	C₈	—	4	2	no	—
408	C₈	—	212	2	no	—
409	C₈	—	212	2	no	—
410	C₂²	—	53	4	no	—
411	C₂²	—	106	2	no	—
412	C₂²	—	106	2	no	—
413	C₂²	—	106	2	no	—
414	C₂²	—	106	2	no	—
415	C₂²	—	106	2	no	—
416	C₂²	—	106	2	no	—
417	C₂²	#95×D₅₃	1	1	yes	characteristic
418	C₂²	#16810604	1	1	yes	characteristic
419	C₂²	#16810606	1	1	yes	characteristic
420	C₂²	#102×D₅₃	1	1	yes	characteristic
421	C₂²	#103×D₅₃	1	1	yes	characteristic
422	C₂²	#16810621	1	1	yes	characteristic
423	C₂²	#16810623	1	1	yes	characteristic
424	C₄	—	2	1	no	—
425	C₄	—	2	1	no	—
426	C₄	—	8	2	no	—
427	C₄	—	53	4	no	—
428	C₄	—	106	2	no	—
429	C₄	—	106	2	no	—
430	C₄	—	424	2	no	—
431	C₄	#102×D₅₃	1	1	yes	characteristic
432	C₄	#103×D₅₃	1	1	yes	characteristic
433	C₄	#16810626	1	1	yes	characteristic
434	C₄	#16810630	1	1	yes	characteristic
435	C₄	#106×D₅₃	1	1	yes	characteristic
436	C₄	#107×D₅₃	1	1	yes	characteristic
437	C₄	#16810662	1	1	yes	characteristic
438	C₄	#16810664	1	1	yes	characteristic
439	C₂	—	106	4	no	—
440	C₂	#20273389	1	1	yes	characteristic
441	C₂	#20273391	1	1	yes	characteristic
442	C₂	#20273402	1	1	yes	characteristic
443	C₂	#20273404	1	1	yes	characteristic
444	C₂	#340×D₅₃	1	1	yes	characteristic
445	C₂	#20274433	1	1	yes	characteristic
446	C₂	#20274452	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	C₂²⋊C₄×D₅₃
commutator	C₂×C₄×C₅₃	C₂×C₄²
Frattini	C₂×C₄²	C₂²×D₅₃
Fitting	C₅₃×#1296	C₂
socle	C₂³×C₅₃	C₂×C₂²⋊C₄
prime radical	C₂³×D₅₃	C₂²⋊C₄
cosocle	C₂×C₄²×C₅₃	C₂³
hypercenter	#1296	D₅₃
hypocenter	C₅₃	#8045
LCS	G ▷ C₂×C₄×C₅₃ ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #1296

Complements

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

N	H ≅ G/N	type
4. #20272599	C₂	semidirect
5. #20272599	C₂	semidirect
6. #20272598	C₂	semidirect
7. C₅₃×#1296	C₂	semidirect
13. #16810645	C₄	semidirect
19. C₄⋊C₈×D₅₃	C₄	semidirect
222. #1296	D₅₃	semidirect
252. C₅₃	#8045	semidirect

Maximal subgroups

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

subgroup	index
C₂×#16810645	2
C₂×C₄⋊C₈×D₅₃	2
#20277120	2
#20272599	2
#20272598	2
C₅₃×#1296	2
#8045	53