Subgroups — #27273774

Order 13568.

666 conjugacy classes of subgroups.

The subgroup lattice has 434 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#16810675	C₂	1	1	yes	maximal, characteristic
2	#431×D₅₃	C₂	1	1	yes	maximal, characteristic
3	C₄×M₁₆×D₅₃	C₂	1	1	yes	maximal, characteristic
4	#20272566	C₂	1	1	yes	maximal, characteristic, complemented
5	#20272565	C₂	1	1	yes	maximal, characteristic, complemented
6	#20272564	C₂	1	1	yes	maximal, characteristic, complemented
7	C₅₃×#1283	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×M₁₆×D₅₃	C₄	1	2	yes	—
9	C₂×C₄²×D₅₃	C₂²	1	1	yes	characteristic
10	#16810676	C₂²	1	1	yes	characteristic
11	#16810675	—	2	1	no	—
12	#16810675	—	2	1	no	—
13	#16810675	—	2	1	no	—
14	#16810675	—	2	1	no	—
15	#16810675	—	2	1	no	—
16	#16810674	—	2	1	no	—
17	#16810673	C₄	1	2	yes	—
18	C₂×C₄⋊#88322	C₂²	1	1	yes	characteristic
19	C₄⋊C₈×D₅₃	—	2	1	no	—
20	C₂×#13133129	C₄	1	2	yes	—
21	C₂²⋊C₈×D₅₃	C₄	1	2	yes	—
22	C₄×#11719917	C₂²	1	1	yes	characteristic
23	#16810575	—	2	1	no	—
24	M₁₆×Dic₅₃	C₄	1	2	yes	—
25	C₄×#11719912	—	2	1	no	—
26	#97×D₅₃	—	2	1	no	—
27	C₂×C₈×Dic₅₃	C₄	1	2	yes	—
28	C₄×C₈×D₅₃	—	2	1	no	—
29	C₄×C₈×D₅₃	—	2	1	no	—
30	C₅₃×#431	C₂²	1	1	yes	characteristic
31	C₂×C₄⋊C₈×C₅₃	C₂²	1	1	yes	characteristic
32	C₄×M₁₆×C₅₃	C₂²	1	1	yes	characteristic
33	C₂²×C₄×D₅₃	C₂×C₄	1	1	yes	characteristic
34	C₂²×C₄×D₅₃	C₂×C₄	1	1	yes	characteristic
35	C₂²×C₄×D₅₃	C₂×C₄	1	1	yes	characteristic
36	C₂×#11719917	C₂×C₄	1	2	yes	—
37	C₂×#11719912	—	2	2	no	—
38	M₁₆×D₅₃	—	2	4	no	—
39	M₁₆×D₅₃	C₈	1	8	yes	complemented
40	C₂²×#88322	C₂×C₄	1	2	yes	—
41	C₂×C₈×D₅₃	—	2	2	no	—
42	C₂×C₈×D₅₃	—	2	2	no	—
43	C₂×C₄×Dic₅₃	C₂×C₄	1	1	yes	characteristic
44	C₂×C₄×Dic₅₃	C₂×C₄	1	1	yes	characteristic
45	C₂×C₄×Dic₅₃	C₂×C₄	1	1	yes	characteristic
46	C₄²×D₅₃	D₄	1	1	yes	characteristic
47	C₄²×D₅₃	D₄	1	1	yes	characteristic
48	C₄²×D₅₃	D₄	1	1	yes	characteristic
49	C₄²×D₅₃	D₄	1	1	yes	characteristic
50	C₄²×D₅₃	D₄	1	1	yes	characteristic
51	C₄²×D₅₃	D₄	1	1	yes	characteristic
52	C₄²×D₅₃	Q₈	1	1	yes	characteristic
53	C₄²×D₅₃	Q₈	1	1	yes	characteristic
54	C₄⋊#88322	—	2	1	no	—
55	C₄⋊#88322	—	2	1	no	—
56	C₄⋊#88322	—	2	1	no	—
57	#13133146	—	2	2	no	—
58	C₂²⋊#88322	C₂×C₄	1	2	yes	—
59	#13133129	—	2	2	no	—
60	#13133129	—	2	2	no	—
61	#13133129	—	2	2	no	—
62	#13133128	—	2	1	no	—
63	#13133127	—	2	2	no	—
64	C₄×#88322	—	2	1	no	—
65	C₄×#88322	—	2	1	no	—
66	C₈×Dic₅₃	—	2	2	no	—
67	C₈×Dic₅₃	—	2	2	no	—
68	C₈×Dic₅₃	—	2	2	no	—
69	C₈×Dic₅₃	—	2	2	no	—
70	C₂×M₁₆×C₅₃	C₂×C₄	1	2	yes	—
71	C₂²×C₈×C₅₃	C₂×C₄	1	2	yes	—
72	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
73	C₄⋊C₈×C₅₃	—	2	1	no	—
74	C₄⋊C₈×C₅₃	—	2	1	no	—
75	C₄⋊C₈×C₅₃	—	2	1	no	—
76	C₂²⋊C₈×C₅₃	C₂×C₄	1	2	yes	—
77	#97×C₅₃	—	2	1	no	—
78	C₄×C₈×C₅₃	—	2	1	no	—
79	C₄×C₈×C₅₃	—	2	1	no	—
80	C₂³×D₅₃	C₄²	1	1	yes	characteristic
81	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
82	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
83	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
84	C₂²×Dic₅₃	C₄²	1	1	yes	characteristic
85	C₂²×Dic₅₃	C₄²	1	1	yes	characteristic
86	C₂²×Dic₅₃	C₄²	1	1	yes	characteristic
87	C₂×C₄×D₅₃	—	4	2	no	—
88	C₂×C₄×D₅₃	—	4	1	no	—
89	C₂×C₄×D₅₃	—	4	1	no	—
90	C₂×C₄×D₅₃	C₂²⋊C₄	1	1	yes	characteristic
91	C₂×C₄×D₅₃	C₂²⋊C₄	1	1	yes	characteristic
92	C₂×C₄×D₅₃	C₂²⋊C₄	1	1	yes	characteristic
93	C₂×C₄×D₅₃	C₂²⋊C₄	1	1	yes	characteristic
94	C₂×C₄×D₅₃	C₂²⋊C₄	1	1	yes	characteristic
95	C₂×C₄×D₅₃	C₂²⋊C₄	1	1	yes	characteristic
96	C₂×C₄×D₅₃	C₄⋊C₄	1	1	yes	characteristic
97	C₂×C₄×D₅₃	C₄⋊C₄	1	1	yes	characteristic
98	C₂×C₄×D₅₃	C₄⋊C₄	1	1	yes	characteristic
99	C₂×C₄×D₅₃	C₄⋊C₄	1	1	yes	characteristic
100	C₂×C₄×D₅₃	C₄⋊C₄	1	1	yes	characteristic
101	C₂×C₄×D₅₃	C₄⋊C₄	1	1	yes	characteristic
102	C₂×C₄×D₅₃	C₂×C₈	1	4	yes	—
103	C₂×C₄×D₅₃	M₁₆	1	4	yes	—
104	M₁₆×C₅₃	—	2	2	no	—
105	M₁₆×C₅₃	C₂×C₈	1	4	yes	complemented
106	C₂×C₈×C₅₃	—	2	2	no	—
107	C₂×C₈×C₅₃	—	2	2	no	—
108	C₂×C₈×C₅₃	—	2	2	no	—
109	C₂×C₈×C₅₃	—	4	2	no	—
110	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
111	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
112	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
113	C₄²×C₅₃	C₂×Q₈	1	1	yes	characteristic
114	C₄×Dic₅₃	C₂²⋊C₄	1	1	yes	characteristic
115	C₄×Dic₅₃	C₂²⋊C₄	1	1	yes	characteristic
116	C₄×Dic₅₃	C₂²⋊C₄	1	1	yes	characteristic
117	C₄×Dic₅₃	C₂²⋊C₄	1	1	yes	characteristic
118	C₄×Dic₅₃	C₂²⋊C₄	1	1	yes	characteristic
119	C₄×Dic₅₃	C₂²⋊C₄	1	1	yes	characteristic
120	C₄×Dic₅₃	C₄⋊C₄	1	1	yes	characteristic
121	C₄×Dic₅₃	C₄⋊C₄	1	1	yes	characteristic
122	C₄×Dic₅₃	C₄⋊C₄	1	1	yes	characteristic
123	C₄×Dic₅₃	C₄⋊C₄	1	1	yes	characteristic
124	C₄×Dic₅₃	C₄⋊C₄	1	1	yes	characteristic
125	C₄×Dic₅₃	C₄⋊C₄	1	1	yes	characteristic
126	#11719917	—	2	2	no	—
127	#11719917	C₂×C₈	1	4	yes	complemented
128	C₂×#88322	—	2	2	no	—
129	C₂×#88322	—	2	2	no	—
130	C₂×#88322	—	2	2	no	—
131	C₂×#88322	—	4	2	no	—
132	#11719912	—	2	8	no	—
133	C₈×D₅₃	—	2	8	no	—
134	C₈×D₅₃	—	4	4	no	—
135	C₂³×C₅₃	C₂×C₄²	1	1	yes	characteristic
136	C₂²×D₅₃	—	4	1	no	—
137	C₂²×D₅₃	—	4	1	no	—
138	C₂²×D₅₃	#95	1	1	yes	characteristic
139	C₂²×D₅₃	#95	1	1	yes	characteristic
140	C₂²×D₅₃	C₄×C₈	1	2	yes	—
141	C₂²×D₅₃	#97	1	2	yes	—
142	C₂×C₄×C₅₃	—	2	1	no	—
143	C₂×C₄×C₅₃	—	2	1	no	—
144	C₂×C₄×C₅₃	—	4	1	no	—
145	C₂×C₄×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
146	C₂×C₄×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
147	C₂×C₄×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
148	C₂×C₄×C₅₃	C₂×C₄⋊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	C₂²×C₈	1	2	yes	—
152	C₂×C₄×C₅₃	C₂×M₁₆	1	2	yes	—
153	C₂×Dic₅₃	—	2	2	no	—
154	C₂×Dic₅₃	—	4	1	no	—
155	C₂×Dic₅₃	#95	1	1	yes	characteristic
156	C₂×Dic₅₃	#95	1	1	yes	characteristic
157	C₂×Dic₅₃	#95	1	1	yes	characteristic
158	C₂×Dic₅₃	#95	1	1	yes	characteristic
159	C₂×Dic₅₃	#95	1	1	yes	characteristic
160	C₂×Dic₅₃	#95	1	1	yes	characteristic
161	C₂×Dic₅₃	C₄×C₈	1	2	yes	—
162	C₂×Dic₅₃	#97	1	2	yes	—
163	C₄×D₅₃	—	2	1	no	—
164	C₄×D₅₃	—	2	1	no	—
165	C₄×D₅₃	—	2	1	no	—
166	C₄×D₅₃	—	2	1	no	—
167	C₄×D₅₃	—	2	2	no	—
168	C₄×D₅₃	—	2	2	no	—
169	C₄×D₅₃	—	4	1	no	—
170	C₄×D₅₃	—	4	1	no	—
171	C₄×D₅₃	—	4	1	no	—
172	C₄×D₅₃	—	4	1	no	—
173	C₄×D₅₃	C₂²⋊C₈	1	4	yes	—
174	C₄×D₅₃	C₂²⋊C₈	1	4	yes	—
175	C₄×D₅₃	C₄⋊C₈	1	4	yes	—
176	C₄×D₅₃	C₄⋊C₈	1	4	yes	—
177	C₈×C₅₃	—	2	4	no	—
178	C₈×C₅₃	—	4	2	no	—
179	C₈×C₅₃	—	4	2	no	—
180	#88322	—	2	4	no	—
181	#88322	—	4	2	no	—
182	#88322	—	4	2	no	—
183	#7932	—	53	1	no	maximal, Sylow
184	C₂²×C₅₃	—	2	1	no	—
185	C₂²×C₅₃	—	2	1	no	—
186	C₂²×C₅₃	C₂×#95	1	1	yes	characteristic
187	C₂²×C₅₃	C₂×C₄×C₈	1	1	yes	characteristic
188	C₂²×C₅₃	C₂×#97	1	1	yes	characteristic
189	C₂×D₅₃	—	2	1	no	—
190	C₂×D₅₃	—	2	1	no	—
191	C₂×D₅₃	—	2	1	no	—
192	C₂×D₅₃	—	2	1	no	—
193	C₂×D₅₃	—	4	2	no	—
194	C₂×D₅₃	—	4	2	no	—
195	C₂×D₅₃	#336	1	2	yes	—
196	C₂×D₅₃	#336	1	2	yes	—
197	C₄×C₅₃	—	2	1	no	—
198	C₄×C₅₃	—	2	1	no	—
199	C₄×C₅₃	—	2	1	no	—
200	C₄×C₅₃	C₂×C₂²⋊C₈	1	2	yes	—
201	C₄×C₅₃	#413	1	1	yes	characteristic
202	C₄×C₅₃	#413	1	1	yes	characteristic
203	C₄×C₅₃	C₂×C₄⋊C₈	1	2	yes	—
204	Dic₅₃	—	2	1	no	—
205	Dic₅₃	—	2	1	no	—
206	Dic₅₃	—	2	1	no	—
207	Dic₅₃	#336	1	2	yes	—
208	Dic₅₃	#336	1	2	yes	—
209	Dic₅₃	#344	1	2	yes	—
210	C₂×#431	—	53	1	no	—
211	C₂×#431	—	53	1	no	—
212	C₂×C₄×M₁₆	—	53	1	no	—
213	#1283	—	53	1	no	—
214	#1283	—	53	1	no	—
215	#1283	—	53	1	no	—
216	#1283	D₅₃	1	1	yes	characteristic, complemented
217	C₂×C₅₃	—	2	1	no	—
218	C₂×C₅₃	—	2	1	no	—
219	C₂×C₅₃	C₂×#336	1	1	yes	characteristic
220	C₂×C₅₃	#1742	1	1	yes	characteristic
221	C₂×C₅₃	C₂×#344	1	1	yes	characteristic
222	D₅₃	—	2	2	no	—
223	D₅₃	—	2	2	no	—
224	C₂²×M₁₆	—	53	2	no	—
225	C₂²×C₄²	—	53	1	no	—
226	#431	—	53	1	no	—
227	#431	—	53	2	no	—
228	#431	—	106	1	no	—
229	#431	—	106	1	no	—
230	#431	—	106	1	no	—
231	#431	—	106	1	no	—
232	#431	—	106	1	no	—
233	#431	—	106	1	no	—
234	#431	C₂×D₅₃	1	1	yes	characteristic
235	C₂×C₄⋊C₈	—	53	1	no	—
236	C₂×C₄⋊C₈	—	106	1	no	—
237	C₂×C₄⋊C₈	C₂×D₅₃	1	1	yes	characteristic
238	C₂×C₂²⋊C₈	—	53	2	no	—
239	C₂×C₂²⋊C₈	—	53	2	no	—
240	C₄×M₁₆	—	53	2	no	—
241	C₄×M₁₆	—	53	1	no	—
242	C₄×M₁₆	—	106	1	no	—
243	C₄×M₁₆	—	106	1	no	—
244	C₄×M₁₆	C₂×D₅₃	1	1	yes	characteristic
245	C₂×#97	—	106	1	no	—
246	C₂×C₄×C₈	—	53	2	no	—
247	C₂×C₄×C₈	—	106	1	no	—
248	C₂×C₄×C₈	—	106	1	no	—
249	C₅₃	#7932	1	1	yes	characteristic, complemented, Sylow
250	C₂³×C₄	—	53	1	no	—
251	C₂³×C₄	—	53	1	no	—
252	C₂³×C₄	—	53	1	no	—
253	C₂×M₁₆	—	53	2	no	—
254	C₂×M₁₆	—	53	8	no	—
255	C₂×M₁₆	—	106	2	no	—
256	C₂×M₁₆	—	106	4	no	—
257	C₂×M₁₆	C₄×D₅₃	1	2	yes	—
258	C₂²×C₈	—	53	2	no	—
259	C₂²×C₈	—	106	2	no	—
260	C₂²×C₈	—	106	2	no	—
261	C₂²×C₈	C₄×D₅₃	1	2	yes	—
262	C₂×C₄²	—	53	1	no	—
263	C₂×C₄²	—	53	1	no	—
264	C₂×C₄²	—	53	1	no	—
265	C₂×C₄²	—	53	1	no	—
266	C₂×C₄²	—	53	1	no	—
267	C₂×C₄²	—	53	1	no	—
268	C₂×C₄²	—	53	1	no	—
269	C₂×C₄²	—	53	1	no	—
270	C₂×C₄²	—	53	1	no	—
271	C₂×C₄²	—	53	1	no	—
272	C₂×C₄²	—	53	1	no	—
273	C₂×C₄²	C₂²×D₅₃	1	1	yes	characteristic
274	C₄⋊C₈	—	2	1	no	—
275	C₄⋊C₈	—	2	1	no	—
276	C₄⋊C₈	—	2	1	no	—
277	C₄⋊C₈	—	106	2	no	—
278	C₄⋊C₈	—	106	1	no	—
279	C₄⋊C₈	—	106	1	no	—
280	C₄⋊C₈	—	106	1	no	—
281	C₂²⋊C₈	—	53	2	no	—
282	C₂²⋊C₈	—	106	2	no	—
283	C₂²⋊C₈	—	106	2	no	—
284	C₂²⋊C₈	—	106	2	no	—
285	C₂²⋊C₈	C₄×D₅₃	1	2	yes	—
286	#97	—	2	1	no	—
287	#97	—	106	2	no	—
288	#97	—	106	1	no	—
289	C₄×C₈	—	2	1	no	—
290	C₄×C₈	—	2	1	no	—
291	C₄×C₈	—	106	2	no	—
292	C₄×C₈	—	106	2	no	—
293	C₄×C₈	—	106	2	no	—
294	C₄×C₈	—	106	2	no	—
295	C₄×C₈	—	106	1	no	—
296	C₄×C₈	—	106	1	no	—
297	C₂⁴	—	53	1	no	—
298	C₂²×C₄	—	53	1	no	—
299	C₂²×C₄	—	53	1	no	—
300	C₂²×C₄	—	53	1	no	—
301	C₂²×C₄	—	53	1	no	—
302	C₂²×C₄	—	53	1	no	—
303	C₂²×C₄	—	53	4	no	—
304	C₂²×C₄	—	53	4	no	—
305	C₂²×C₄	—	53	1	no	—
306	C₂²×C₄	—	53	1	no	—
307	C₂²×C₄	—	53	1	no	—
308	C₂²×C₄	—	53	1	no	—
309	C₂²×C₄	—	53	1	no	—
310	C₂²×C₄	—	53	1	no	—
311	C₂²×C₄	—	53	1	no	—
312	C₂²×C₄	—	53	1	no	—
313	C₂²×C₄	—	53	1	no	—
314	C₂²×C₄	—	53	1	no	—
315	C₂²×C₄	—	212	1	no	—
316	C₂²×C₄	—	212	1	no	—
317	C₂²×C₄	—	212	2	no	—
318	C₂²×C₄	C₂×C₄×D₅₃	1	1	yes	characteristic
319	C₂²×C₄	C₂×C₄×D₅₃	1	1	yes	characteristic
320	C₂²×C₄	C₂×C₄×D₅₃	1	1	yes	characteristic
321	M₁₆	—	2	2	no	—
322	M₁₆	—	53	4	no	—
323	M₁₆	—	106	8	no	—
324	M₁₆	—	106	2	no	—
325	M₁₆	C₈×D₅₃	1	4	yes	complemented
326	C₂×C₈	—	2	2	no	—
327	C₂×C₈	—	2	2	no	—
328	C₂×C₈	—	2	2	no	—
329	C₂×C₈	—	4	2	no	—
330	C₂×C₈	—	106	8	no	—
331	C₂×C₈	—	106	2	no	—
332	C₂×C₈	—	106	2	no	—
333	C₂×C₈	—	106	2	no	—
334	C₂×C₈	—	212	4	no	—
335	C₂×C₈	—	212	2	no	—
336	C₄²	—	53	1	no	—
337	C₄²	—	53	1	no	—
338	C₄²	—	53	1	no	—
339	C₄²	—	53	1	no	—
340	C₄²	—	53	1	no	—
341	C₄²	—	53	1	no	—
342	C₄²	—	53	1	no	—
343	C₄²	—	53	1	no	—
344	C₄²	—	53	1	no	—
345	C₄²	—	53	1	no	—
346	C₄²	—	53	1	no	—
347	C₄²	—	53	1	no	—
348	C₄²	D₄×D₅₃	1	1	yes	characteristic
349	C₄²	D₄×D₅₃	1	1	yes	characteristic
350	C₄²	D₄×D₅₃	1	1	yes	characteristic
351	C₄²	Q₈×D₅₃	1	1	yes	characteristic
352	C₂³	—	53	1	no	—
353	C₂³	—	53	1	no	—
354	C₂³	—	53	2	no	—
355	C₂³	—	53	2	no	—
356	C₂³	—	212	1	no	—
357	C₂³	—	212	1	no	—
358	C₂³	C₄²×D₅₃	1	1	yes	characteristic
359	C₂×C₄	—	2	1	no	—
360	C₂×C₄	—	2	1	no	—
361	C₂×C₄	—	4	1	no	—
362	C₂×C₄	—	53	2	no	—
363	C₂×C₄	—	53	2	no	—
364	C₂×C₄	—	53	1	no	—
365	C₂×C₄	—	53	1	no	—
366	C₂×C₄	—	53	1	no	—
367	C₂×C₄	—	53	4	no	—
368	C₂×C₄	—	53	4	no	—
369	C₂×C₄	—	53	4	no	—
370	C₂×C₄	—	53	4	no	—
371	C₂×C₄	—	53	1	no	—
372	C₂×C₄	—	53	1	no	—
373	C₂×C₄	—	53	1	no	—
374	C₂×C₄	—	106	2	no	—
375	C₂×C₄	—	106	2	no	—
376	C₂×C₄	—	106	2	no	—
377	C₂×C₄	—	106	1	no	—
378	C₂×C₄	—	106	1	no	—
379	C₂×C₄	—	106	1	no	—
380	C₂×C₄	—	106	1	no	—
381	C₂×C₄	—	212	1	no	—
382	C₂×C₄	—	212	1	no	—
383	C₂×C₄	—	212	1	no	—
384	C₂×C₄	—	212	1	no	—
385	C₂×C₄	—	212	1	no	—
386	C₂×C₄	C₂²⋊C₄×D₅₃	1	1	yes	characteristic
387	C₂×C₄	C₂²⋊C₄×D₅₃	1	1	yes	characteristic
388	C₂×C₄	C₂²⋊C₄×D₅₃	1	1	yes	characteristic
389	C₂×C₄	C₄⋊C₄×D₅₃	1	1	yes	characteristic
390	C₂×C₄	C₄⋊C₄×D₅₃	1	1	yes	characteristic
391	C₂×C₄	C₄⋊C₄×D₅₃	1	1	yes	characteristic
392	C₂×C₄	C₂×C₈×D₅₃	1	2	yes	—
393	C₂×C₄	M₁₆×D₅₃	1	2	yes	—
394	C₈	—	2	4	no	—
395	C₈	—	4	2	no	—
396	C₈	—	4	2	no	—
397	C₈	—	106	4	no	—
398	C₈	—	212	2	no	—
399	C₈	—	212	2	no	—
400	C₂²	—	2	1	no	—
401	C₂²	—	2	1	no	—
402	C₂²	—	53	2	no	—
403	C₂²	—	53	2	no	—
404	C₂²	—	106	1	no	—
405	C₂²	—	106	1	no	—
406	C₂²	—	106	1	no	—
407	C₂²	—	106	1	no	—
408	C₂²	—	212	2	no	—
409	C₂²	—	212	2	no	—
410	C₂²	#95×D₅₃	1	1	yes	characteristic
411	C₂²	C₄×C₈×D₅₃	1	1	yes	characteristic
412	C₂²	#97×D₅₃	1	1	yes	characteristic
413	C₄	—	2	1	no	—
414	C₄	—	2	1	no	—
415	C₄	—	2	1	no	—
416	C₄	—	53	2	no	—
417	C₄	—	53	2	no	—
418	C₄	—	53	2	no	—
419	C₄	—	106	1	no	—
420	C₄	—	106	1	no	—
421	C₄	—	106	1	no	—
422	C₄	C₂²⋊C₈×D₅₃	1	2	yes	—
423	C₄	#16810604	1	1	yes	characteristic
424	C₄	#16810606	1	1	yes	characteristic
425	C₄	C₄⋊C₈×D₅₃	1	2	yes	—
426	C₂	—	2	1	no	—
427	C₂	—	2	1	no	—
428	C₂	—	106	2	no	—
429	C₂	—	106	2	no	—
430	C₂	#336×D₅₃	1	1	yes	characteristic
431	C₂	#20274452	1	1	yes	characteristic
432	C₂	#344×D₅₃	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂²⋊C₄×D₅₃
commutator	C₂²×C₅₃	C₂×C₄×C₈
Frattini	C₂×C₄²	C₂²×D₅₃
Fitting	C₅₃×#1283	C₂
socle	C₂²×C₅₃	C₂×#95
prime radical	C₂³×D₅₃	C₄²
cosocle	C₂×C₄²×C₅₃	C₂³
hypercenter	#1283	D₅₃
hypocenter	C₅₃	#7932
LCS	G ▷ C₂²×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #1283

Complements

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

N	H ≅ G/N	type
4. #20272566	C₂	semidirect
5. #20272565	C₂	semidirect
6. #20272564	C₂	semidirect
7. C₅₃×#1283	C₂	semidirect
39. M₁₆×D₅₃	C₈	semidirect
105. M₁₆×C₅₃	C₂×C₈	semidirect
127. #11719917	C₂×C₈	semidirect
216. #1283	D₅₃	semidirect
249. C₅₃	#7932	semidirect
325. M₁₆	C₈×D₅₃	semidirect

Maximal subgroups

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

subgroup	index
C₂×#16810675	2
#431×D₅₃	2
C₄×M₁₆×D₅₃	2
#20272566	2
#20272565	2
#20272564	2
C₅₃×#1283	2
#7932	53