Subgroups — #27353186

Order 13568.

702 conjugacy classes of subgroups.

The subgroup lattice has 426 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20283347	C₂	1	1	yes	maximal, characteristic
2	#20283343	C₂	1	1	yes	maximal, characteristic
3	C₂×C₄⋊C₈×D₅₃	C₂	1	1	yes	maximal, characteristic
4	#20273497	C₂	1	4	yes	maximal, complemented
5	#20273490	C₂	1	1	yes	maximal, characteristic, complemented
6	#20273489	C₂	1	1	yes	maximal, characteristic, complemented
7	#20273487	C₂	1	1	yes	maximal, characteristic, complemented
8	#20273464	C₂	1	4	yes	maximal, complemented
9	C₅₃×#1570	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂²×C₈×D₅₃	—	2	1	no	—
11	C₂×C₄²×D₅₃	C₂²	1	1	yes	characteristic
12	#16810679	—	2	1	no	—
13	#16810679	C₄	1	2	yes	—
14	#16810679	C₂²	1	2	yes	—
15	#16810677	C₄	1	2	yes	—
16	#16810676	C₂²	1	1	yes	characteristic
17	#16810675	—	2	1	no	—
18	#16810675	C₂²	1	2	yes	—
19	#16810674	C₂²	1	2	yes	—
20	#16810647	C₂²	1	1	yes	characteristic
21	#16810645	C₄	1	2	yes	—
22	C₂×C₄⋊#88322	C₂²	1	1	yes	characteristic
23	C₂×#13133146	—	2	1	no	—
24	C₄⋊C₈×D₅₃	—	2	1	no	—
25	C₄⋊C₈×D₅₃	—	2	1	no	—
26	C₄⋊C₈×D₅₃	C₂²	1	2	yes	—
27	C₄⋊C₈×D₅₃	C₂²	1	2	yes	—
28	#16810586	—	2	1	no	—
29	#16810585	C₄	1	2	yes	—
30	M₁₆×Dic₅₃	—	2	1	no	—
31	C₄×#11719912	C₂²	1	2	yes	—
32	#16810376	C₂²	1	4	yes	complemented
33	#16810375	C₂²	1	4	yes	complemented
34	#16810373	C₂²	1	4	yes	complemented
35	C₅₃×#431	C₂²	1	1	yes	characteristic
36	C₅₃×#423	C₂²	1	1	yes	characteristic
37	C₂×C₄⋊C₈×C₅₃	C₂²	1	1	yes	characteristic
38	C₅₃×#334	C₂²	1	4	yes	complemented
39	C₂²×C₄×D₅₃	C₂×C₄	1	1	yes	characteristic
40	C₂²×C₄×D₅₃	D₄	1	1	yes	characteristic
41	C₂²×C₄×D₅₃	Q₈	1	1	yes	characteristic
42	C₂×#11719917	—	2	1	no	—
43	C₂×#11719912	C₂×C₄	1	4	yes	—
44	C₂²×#88322	—	2	1	no	—
45	C₂×C₈×D₅₃	—	2	2	no	—
46	C₂×C₈×D₅₃	—	2	2	no	—
47	C₂×C₈×D₅₃	—	4	4	no	—
48	C₂×C₄×Dic₅₃	C₂×C₄	1	1	yes	characteristic
49	C₂×C₄×Dic₅₃	D₄	1	1	yes	characteristic
50	C₂×C₄×Dic₅₃	Q₈	1	1	yes	characteristic
51	C₄²×D₅₃	C₂×C₄	1	1	yes	characteristic
52	C₄²×D₅₃	C₂×C₄	1	1	yes	characteristic
53	C₄²×D₅₃	D₄	1	1	yes	characteristic
54	C₄²×D₅₃	D₄	1	1	yes	characteristic
55	C₄²×D₅₃	Q₈	1	1	yes	characteristic
56	C₄²×D₅₃	Q₈	1	1	yes	characteristic
57	C₄²×D₅₃	C₂³	1	1	yes	characteristic
58	C₄²×D₅₃	C₂³	1	1	yes	characteristic
59	C₄⋊#88322	—	2	1	no	—
60	C₄⋊#88322	—	2	1	no	—
61	C₄⋊#88322	C₂×C₄	1	2	yes	—
62	C₄⋊#88322	C₂³	1	2	yes	—
63	C₄⋊#88322	C₂³	1	2	yes	—
64	#13133146	—	2	2	no	—
65	#13133146	—	2	2	no	—
66	#13133146	C₂×C₄	1	4	yes	—
67	C₂²⋊#88322	C₂×C₄	1	2	yes	—
68	#13133129	—	2	2	no	—
69	#13133129	—	2	2	no	—
70	#13133129	C₂×C₄	1	4	yes	—
71	#13133127	—	2	2	no	—
72	#13133127	C₂×C₄	1	4	yes	—
73	C₄×#88322	C₂³	1	2	yes	—
74	C₈×Dic₅₃	—	2	2	no	—
75	C₂×M₁₆×C₅₃	—	2	1	no	—
76	C₂²×C₈×C₅₃	—	2	1	no	—
77	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
78	C₄⋊C₈×C₅₃	—	2	1	no	—
79	C₄⋊C₈×C₅₃	—	2	1	no	—
80	C₄⋊C₈×C₅₃	C₂×C₄	1	2	yes	—
81	C₄⋊C₈×C₅₃	C₂³	1	2	yes	—
82	C₄⋊C₈×C₅₃	C₂³	1	2	yes	—
83	C₂²⋊C₈×C₅₃	C₂×C₄	1	2	yes	—
84	C₄×C₈×C₅₃	C₂³	1	2	yes	—
85	C₂³×D₅₃	C₄⋊C₄	1	1	yes	characteristic
86	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
87	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
88	C₂²×C₄×C₅₃	C₂×Q₈	1	1	yes	characteristic
89	C₂²×Dic₅₃	C₄⋊C₄	1	1	yes	characteristic
90	C₂²×Dic₅₃	C₄⋊C₄	1	1	yes	characteristic
91	C₂²×Dic₅₃	C₄⋊C₄	1	1	yes	characteristic
92	C₂×C₄×D₅₃	—	2	2	no	—
93	C₂×C₄×D₅₃	—	2	2	no	—
94	C₂×C₄×D₅₃	—	2	4	no	—
95	C₂×C₄×D₅₃	—	4	2	no	—
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	C₂²×C₄	1	1	yes	characteristic
103	C₂×C₄×D₅₃	C₂²×C₄	1	1	yes	characteristic
104	C₂×C₄×D₅₃	C₂×D₄	1	1	yes	characteristic
105	C₂×C₄×D₅₃	C₂×D₄	1	1	yes	characteristic
106	C₂×C₄×D₅₃	C₂×Q₈	1	1	yes	characteristic
107	C₂×C₄×D₅₃	C₂×Q₈	1	1	yes	characteristic
108	M₁₆×C₅₃	—	4	2	no	—
109	C₂×C₈×C₅₃	—	2	2	no	—
110	C₂×C₈×C₅₃	—	2	2	no	—
111	C₂×C₈×C₅₃	—	4	2	no	—
112	C₂×C₈×C₅₃	C₂²×C₄	1	4	yes	—
113	C₄²×C₅₃	C₂²×C₄	1	1	yes	characteristic
114	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
115	C₄²×C₅₃	C₂×Q₈	1	1	yes	characteristic
116	C₄²×C₅₃	C₂⁴	1	1	yes	characteristic
117	C₄×Dic₅₃	C₄⋊C₄	1	1	yes	characteristic
118	C₄×Dic₅₃	C₄⋊C₄	1	1	yes	characteristic
119	C₄×Dic₅₃	C₄⋊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	C₂²×C₄	1	1	yes	characteristic
124	C₄×Dic₅₃	C₂²×C₄	1	1	yes	characteristic
125	C₄×Dic₅₃	C₂×D₄	1	1	yes	characteristic
126	C₄×Dic₅₃	C₂×D₄	1	1	yes	characteristic
127	C₄×Dic₅₃	C₂×Q₈	1	1	yes	characteristic
128	C₄×Dic₅₃	C₂×Q₈	1	1	yes	characteristic
129	#11719917	—	4	2	no	—
130	C₂×#88322	—	2	2	no	—
131	C₂×#88322	—	2	2	no	—
132	C₂×#88322	—	4	2	no	—
133	C₂×#88322	C₂²×C₄	1	4	yes	—
134	#11719912	—	2	8	no	—
135	C₈×D₅₃	—	4	4	no	—
136	C₈×D₅₃	—	4	4	no	—
137	C₂³×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
138	C₂²×D₅₃	—	2	2	no	—
139	C₂²×D₅₃	—	2	2	no	—
140	C₂²×D₅₃	—	2	2	no	—
141	C₂²×D₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
142	C₂²×D₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
143	C₂×C₄×C₅₃	—	2	2	no	—
144	C₂×C₄×C₅₃	—	4	1	no	—
145	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
146	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
147	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
148	C₂×C₄×C₅₃	AGL(1,ℤ/8)	1	2	yes	—
149	C₂×C₄×C₅₃	#137	1	2	yes	—
150	C₂×C₄×C₅₃	C₂³×C₄	1	1	yes	characteristic
151	C₂×C₄×C₅₃	C₂²×D₄	1	1	yes	characteristic
152	C₂×C₄×C₅₃	C₂²×Q₈	1	1	yes	characteristic
153	C₂×Dic₅₃	—	2	2	no	—
154	C₂×Dic₅₃	—	2	2	no	—
155	C₂×Dic₅₃	—	4	1	no	—
156	C₂×Dic₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
157	C₂×Dic₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
158	C₂×Dic₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
159	C₂×Dic₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
160	C₂×Dic₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
161	C₂×Dic₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
162	C₄×D₅₃	—	2	2	no	—
163	C₄×D₅₃	—	2	2	no	—
164	C₄×D₅₃	—	2	2	no	—
165	C₄×D₅₃	—	2	2	no	—
166	C₄×D₅₃	—	2	1	no	—
167	C₄×D₅₃	—	2	1	no	—
168	C₄×D₅₃	—	2	1	no	—
169	C₄×D₅₃	—	2	1	no	—
170	C₄×D₅₃	—	2	2	no	—
171	C₄×D₅₃	—	2	2	no	—
172	C₄×D₅₃	—	2	2	no	—
173	C₄×D₅₃	—	4	1	no	—
174	C₄×D₅₃	—	4	1	no	—
175	C₄×D₅₃	#131	1	4	yes	—
176	C₈×C₅₃	—	2	4	no	—
177	C₈×C₅₃	—	4	2	no	—
178	C₈×C₅₃	—	4	2	no	—
179	#88322	—	2	4	no	—
180	#88322	—	4	2	no	—
181	#88322	—	4	2	no	—
182	#17807	—	53	1	no	maximal, Sylow
183	C₂²×C₅₃	—	2	1	no	—
184	C₂²×C₅₃	—	2	1	no	—
185	C₂²×C₅₃	#428	1	2	yes	—
186	C₂²×C₅₃	C₂²×C₄⋊C₄	1	1	yes	characteristic
187	C₂×D₅₃	—	2	8	no	—
188	C₂×D₅₃	—	2	1	no	—
189	C₂×D₅₃	—	2	1	no	—
190	C₂×D₅₃	—	2	1	no	—
191	C₂×D₅₃	—	2	1	no	—
192	C₂×D₅₃	#424	1	2	yes	—
193	C₂×D₅₃	#430	1	2	yes	—
194	C₄×C₅₃	—	2	1	no	—
195	C₄×C₅₃	—	2	1	no	—
196	C₄×C₅₃	—	2	1	no	—
197	C₄×C₅₃	#428	1	2	yes	—
198	C₄×C₅₃	C₂×#131	1	2	yes	—
199	C₄×C₅₃	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
200	C₄×C₅₃	C₂×#137	1	1	yes	characteristic
201	Dic₅₃	—	2	1	no	—
202	Dic₅₃	—	2	1	no	—
203	Dic₅₃	—	2	1	no	—
204	Dic₅₃	—	2	1	no	—
205	Dic₅₃	—	2	1	no	—
206	Dic₅₃	#424	1	2	yes	—
207	#2920	—	53	1	no	—
208	#2920	—	53	1	no	—
209	C₂²×C₄⋊C₈	—	53	1	no	—
210	#1573	—	53	4	no	—
211	#1571	—	53	1	no	—
212	#1571	—	53	1	no	—
213	#1570	—	53	1	no	—
214	#1570	D₅₃	1	1	yes	characteristic, complemented
215	#1565	—	53	4	no	—
216	C₂×C₅₃	—	2	2	no	—
217	C₂×C₅₃	C₂×#424	1	1	yes	characteristic
218	C₂×C₅₃	C₂×#428	1	1	yes	characteristic
219	C₂×C₅₃	C₂×#430	1	1	yes	characteristic
220	D₅₃	—	2	2	no	—
221	D₅₃	—	2	2	no	—
222	C₂³×C₈	—	106	1	no	—
223	C₂²×C₄²	—	53	1	no	—
224	#432	—	53	2	no	—
225	#432	—	53	2	no	—
226	#432	—	53	2	no	—
227	#432	—	106	1	no	—
228	#431	—	53	1	no	—
229	#431	—	53	2	no	—
230	#431	—	53	2	no	—
231	#431	—	106	1	no	—
232	#431	C₂×D₅₃	1	1	yes	characteristic
233	#423	—	53	2	no	—
234	#423	—	53	1	no	—
235	#423	C₂×D₅₃	1	1	yes	characteristic
236	C₂×C₄⋊C₈	—	53	1	no	—
237	C₂×C₄⋊C₈	—	53	2	no	—
238	C₂×C₄⋊C₈	—	53	2	no	—
239	C₂×C₄⋊C₈	—	106	1	no	—
240	C₂×C₄⋊C₈	—	106	1	no	—
241	C₂×C₄⋊C₈	—	106	1	no	—
242	C₂×C₄⋊C₈	C₂×D₅₃	1	1	yes	characteristic
243	#407	—	53	2	no	—
244	#407	—	106	1	no	—
245	C₄×M₁₆	—	53	2	no	—
246	C₄×M₁₆	—	106	1	no	—
247	#335	—	53	4	no	—
248	#335	—	53	4	no	—
249	#334	—	53	4	no	—
250	#334	C₂×D₅₃	1	4	yes	complemented
251	C₅₃	#17807	1	1	yes	characteristic, complemented, Sylow
252	C₂³×C₄	—	53	1	no	—
253	C₂³×C₄	—	53	1	no	—
254	C₂³×C₄	—	53	1	no	—
255	C₂×M₁₆	—	2	1	no	—
256	C₂×M₁₆	—	53	4	no	—
257	C₂×M₁₆	—	106	1	no	—
258	C₂²×C₈	—	2	1	no	—
259	C₂²×C₈	—	106	2	no	—
260	C₂²×C₈	—	106	2	no	—
261	C₂²×C₈	—	106	1	no	—
262	C₂²×C₈	—	212	4	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	—	53	1	no	—
274	C₂×C₄²	C₂²×D₅₃	1	1	yes	characteristic
275	C₄⋊C₈	—	2	1	no	—
276	C₄⋊C₈	—	2	1	no	—
277	C₄⋊C₈	—	53	4	no	—
278	C₄⋊C₈	—	53	2	no	—
279	C₄⋊C₈	—	53	2	no	—
280	C₄⋊C₈	—	53	2	no	—
281	C₄⋊C₈	—	106	2	no	—
282	C₄⋊C₈	—	106	2	no	—
283	C₄⋊C₈	—	106	1	no	—
284	C₄⋊C₈	—	106	1	no	—
285	C₄⋊C₈	C₄×D₅₃	1	2	yes	—
286	C₄⋊C₈	C₂²×D₅₃	1	2	yes	—
287	C₄⋊C₈	C₂²×D₅₃	1	2	yes	—
288	C₂²⋊C₈	—	53	2	no	—
289	C₂²⋊C₈	—	53	4	no	—
290	C₂²⋊C₈	—	106	2	no	—
291	C₂²⋊C₈	—	106	2	no	—
292	C₂²⋊C₈	C₄×D₅₃	1	2	yes	—
293	#97	—	53	4	no	—
294	#97	—	106	2	no	—
295	C₄×C₈	—	53	2	no	—
296	C₄×C₈	—	106	2	no	—
297	C₄×C₈	C₂²×D₅₃	1	2	yes	—
298	C₂⁴	—	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	1	no	—
304	C₂²×C₄	—	53	1	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	—	106	4	no	—
315	C₂²×C₄	—	106	2	no	—
316	C₂²×C₄	—	106	2	no	—
317	C₂²×C₄	—	212	2	no	—
318	C₂²×C₄	C₂×C₄×D₅₃	1	1	yes	characteristic
319	C₂²×C₄	D₄×D₅₃	1	1	yes	characteristic
320	C₂²×C₄	Q₈×D₅₃	1	1	yes	characteristic
321	M₁₆	—	4	2	no	—
322	M₁₆	—	106	8	no	—
323	M₁₆	—	212	2	no	—
324	C₂×C₈	—	2	2	no	—
325	C₂×C₈	—	2	2	no	—
326	C₂×C₈	—	4	2	no	—
327	C₂×C₈	—	53	4	no	—
328	C₂×C₈	—	106	2	no	—
329	C₂×C₈	—	106	2	no	—
330	C₂×C₈	—	212	4	no	—
331	C₂×C₈	—	212	4	no	—
332	C₂×C₈	—	212	2	no	—
333	C₂×C₈	C₂×C₄×D₅₃	1	4	yes	—
334	C₄²	—	53	1	no	—
335	C₄²	—	53	1	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	C₂×C₄×D₅₃	1	1	yes	characteristic
347	C₄²	D₄×D₅₃	1	1	yes	characteristic
348	C₄²	Q₈×D₅₃	1	1	yes	characteristic
349	C₄²	C₂³×D₅₃	1	1	yes	characteristic
350	C₂³	—	53	1	no	—
351	C₂³	—	53	1	no	—
352	C₂³	—	106	2	no	—
353	C₂³	—	106	2	no	—
354	C₂³	—	106	2	no	—
355	C₂³	C₄⋊C₄×D₅₃	1	1	yes	characteristic
356	C₂×C₄	—	2	2	no	—
357	C₂×C₄	—	4	1	no	—
358	C₂×C₄	—	53	1	no	—
359	C₂×C₄	—	53	4	no	—
360	C₂×C₄	—	53	1	no	—
361	C₂×C₄	—	53	1	no	—
362	C₂×C₄	—	53	1	no	—
363	C₂×C₄	—	53	1	no	—
364	C₂×C₄	—	53	1	no	—
365	C₂×C₄	—	106	2	no	—
366	C₂×C₄	—	106	2	no	—
367	C₂×C₄	—	106	2	no	—
368	C₂×C₄	—	106	2	no	—
369	C₂×C₄	—	106	2	no	—
370	C₂×C₄	—	106	2	no	—
371	C₂×C₄	—	106	2	no	—
372	C₂×C₄	—	106	1	no	—
373	C₂×C₄	—	106	1	no	—
374	C₂×C₄	—	106	1	no	—
375	C₂×C₄	—	106	1	no	—
376	C₂×C₄	—	106	2	no	—
377	C₂×C₄	—	106	2	no	—
378	C₂×C₄	—	212	1	no	—
379	C₂×C₄	—	212	1	no	—
380	C₂×C₄	—	212	1	no	—
381	C₂×C₄	C₄⋊C₄×D₅₃	1	1	yes	characteristic
382	C₂×C₄	C₄⋊C₄×D₅₃	1	1	yes	characteristic
383	C₂×C₄	C₄⋊C₄×D₅₃	1	1	yes	characteristic
384	C₂×C₄	#13133244	1	2	yes	—
385	C₂×C₄	#13133246	1	2	yes	—
386	C₂×C₄	C₂²×C₄×D₅₃	1	1	yes	characteristic
387	C₂×C₄	C₂×D₄×D₅₃	1	1	yes	characteristic
388	C₂×C₄	C₂×Q₈×D₅₃	1	1	yes	characteristic
389	C₈	—	2	4	no	—
390	C₈	—	4	2	no	—
391	C₈	—	4	2	no	—
392	C₈	—	106	4	no	—
393	C₈	—	212	2	no	—
394	C₈	—	212	2	no	—
395	C₂²	—	2	1	no	—
396	C₂²	—	2	1	no	—
397	C₂²	—	53	2	no	—
398	C₂²	—	53	2	no	—
399	C₂²	—	106	8	no	—
400	C₂²	—	106	1	no	—
401	C₂²	—	106	1	no	—
402	C₂²	—	106	1	no	—
403	C₂²	—	106	1	no	—
404	C₂²	#16810662	1	2	yes	—
405	C₂²	C₂×C₄⋊C₄×D₅₃	1	1	yes	characteristic
406	C₄	—	2	1	no	—
407	C₄	—	2	1	no	—
408	C₄	—	2	1	no	—
409	C₄	—	53	2	no	—
410	C₄	—	106	1	no	—
411	C₄	—	106	1	no	—
412	C₄	—	106	1	no	—
413	C₄	—	106	1	no	—
414	C₄	—	106	1	no	—
415	C₄	#16810662	1	2	yes	—
416	C₄	#131×D₅₃	1	2	yes	—
417	C₄	C₂×#13133244	1	1	yes	characteristic
418	C₄	C₂×#13133246	1	1	yes	characteristic
419	C₂	—	2	2	no	—
420	C₂	—	106	2	no	—
421	C₂	—	106	2	no	—
422	C₂	#424×D₅₃	1	1	yes	characteristic
423	C₂	C₂×#16810662	1	1	yes	characteristic
424	C₂	#430×D₅₃	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂×D₄×D₅₃
commutator	C₂×C₄×C₅₃	C₂³×C₄
Frattini	C₄²	C₂³×D₅₃
Fitting	C₅₃×#1570	C₂
socle	C₂²×C₅₃	C₂²×C₄⋊C₄
prime radical	C₂³×D₅₃	C₄⋊C₄
cosocle	C₄²×C₅₃	C₂⁴
hypercenter	#1570	D₅₃
hypocenter	C₅₃	#17807
LCS	G ▷ C₂×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #1570

Complements

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

N	H ≅ G/N	type
4. #20273497	C₂	semidirect
5. #20273490	C₂	semidirect
6. #20273489	C₂	semidirect
7. #20273487	C₂	semidirect
8. #20273464	C₂	semidirect
9. C₅₃×#1570	C₂	semidirect
32. #16810376	C₂²	semidirect
33. #16810375	C₂²	semidirect
34. #16810373	C₂²	semidirect
38. C₅₃×#334	C₂²	semidirect
214. #1570	D₅₃	semidirect
250. #334	C₂×D₅₃	semidirect
251. C₅₃	#17807	semidirect

Maximal subgroups

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

subgroup	index
#20283347	2
#20283343	2
C₂×C₄⋊C₈×D₅₃	2
#20273497	2
#20273490	2
#20273489	2
#20273487	2
#20273464	2
C₅₃×#1570	2
#17807	53