Subgroups — #27353545

Order 13568.

782 conjugacy classes of subgroups.

The subgroup lattice has 428 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20283736	C₂	1	2	yes	maximal, complemented
2	C₄×D₈×D₅₃	C₂	1	1	yes	maximal, characteristic
3	#20273537	C₂	1	2	yes	maximal, complemented
4	#20273536	C₂	1	2	yes	maximal, complemented
5	C₈×D₄₂₄	C₂	1	1	yes	maximal, characteristic, complemented
6	D₈×#88322	C₂	1	1	yes	maximal, characteristic, complemented
7	#20273509	C₂	1	1	yes	maximal, characteristic, complemented
8	#20273462	C₂	1	1	yes	maximal, characteristic, complemented
9	#20273104	C₂	1	2	yes	maximal, complemented
10	#20272997	C₂	1	1	yes	maximal, characteristic, complemented
11	C₈×D₈×C₅₃	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×D₈×D₅₃	C₄	1	1	yes	characteristic
13	C₄×D₄×D₅₃	C₂²	1	2	yes	—
14	C₄×D₄₂₄	C₂²	1	1	yes	characteristic
15	D₈×Dic₅₃	C₄	1	1	yes	characteristic
16	#16810705	C₄	1	1	yes	characteristic
17	C₄×#11719922	C₂²	1	2	yes	—
18	#16810703	C₄	1	2	yes	—
19	C₈×D₂₁₂	C₂²	1	2	yes	complemented
20	D₄×#88322	C₂²	1	2	yes	complemented
21	#16810688	C₂²	1	2	yes	complemented
22	C₈×C₂²⋊D₅₃	—	2	2	no	—
23	#16810686	—	2	2	no	—
24	#16810675	C₂²	1	2	yes	—
25	#16810674	C₂²	1	1	yes	characteristic
26	#16810674	C₂²	1	1	yes	characteristic
27	#107×D₅₃	C₄	1	1	yes	characteristic
28	#102×D₅₃	C₄	1	2	yes	—
29	C₂×#13133129	—	2	2	no	—
30	C₂²⋊C₈×D₅₃	—	2	2	no	—
31	C₄×C₈×D₅₃	C₂²	1	1	yes	characteristic
32	#16810376	C₂²	1	1	yes	characteristic, complemented
33	#16810373	C₂²	1	2	yes	complemented
34	#16810352	C₂²	1	2	yes	complemented
35	D₄⋊#88322	C₂²	1	2	yes	complemented
36	#16810350	C₂²	1	2	yes	complemented
37	#16810345	C₂²	1	1	yes	characteristic, complemented
38	#16810345	C₂²	1	1	yes	characteristic, complemented
39	C₈×#88322	C₂²	1	1	yes	characteristic, complemented
40	C₄×D₈×C₅₃	C₂²	1	1	yes	characteristic
41	C₈×D₄×C₅₃	C₂²	1	2	yes	complemented
42	C₅₃×#335	C₂²	1	1	yes	characteristic, complemented
43	C₅₃×D₄⋊C₈	C₂²	1	2	yes	complemented
44	C₈²×C₅₃	C₂²	1	1	yes	characteristic, complemented
45	C₂×D₄×D₅₃	C₂×C₄	1	2	yes	—
46	C₂²×C₄×D₅₃	—	2	2	no	—
47	C₂×D₄₂₄	C₂×C₄	1	1	yes	characteristic
48	C₂×#11719922	C₂×C₄	1	2	yes	—
49	D₈×D₅₃	C₈	1	8	yes	complemented
50	C₂²×#88322	—	2	2	no	—
51	C₂×C₈×D₅₃	—	2	2	no	—
52	C₂×C₈×D₅₃	—	2	2	no	—
53	C₂×C₈×D₅₃	C₂×C₄	1	1	yes	characteristic
54	C₂×C₈×D₅₃	C₂×C₄	1	1	yes	characteristic
55	C₄×D₂₁₂	C₂³	1	2	yes	—
56	D₄×Dic₅₃	C₂×C₄	1	2	yes	—
57	#13133179	C₂×C₄	1	2	yes	—
58	C₄×C₂²⋊D₅₃	—	2	2	no	—
59	#13133177	—	2	2	no	—
60	C₄⋊C₄×D₅₃	C₂×C₄	1	2	yes	—
61	C₂²⋊C₄×D₅₃	—	2	2	no	—
62	C₄²×D₅₃	C₂³	1	1	yes	characteristic
63	#13133151	C₂×C₄	1	1	yes	characteristic
64	#13133150	C₂×C₄	1	2	yes	—
65	C₄⋊#88322	C₂³	1	2	yes	—
66	#13133146	—	2	1	no	—
67	#13133146	D₄	1	1	yes	characteristic
68	#13133146	D₄	1	1	yes	characteristic
69	#13133139	C₂×C₄	1	2	yes	—
70	#13133138	C₂×C₄	1	2	yes	—
71	#13133137	C₂×C₄	1	2	yes	—
72	C₂²⋊#88322	—	2	2	no	—
73	#13133129	—	2	2	no	—
74	#13133129	—	2	2	no	—
75	#13133129	—	2	2	no	—
76	#13133129	—	2	1	no	—
77	#13133129	D₄	1	1	yes	characteristic
78	#13133129	D₄	1	1	yes	characteristic
79	C₄×#88322	C₂³	1	1	yes	characteristic
80	C₄×#88322	C₂³	1	1	yes	characteristic
81	C₄×#88322	C₂³	1	1	yes	characteristic
82	C₈×Dic₅₃	—	2	2	no	—
83	C₈×Dic₅₃	C₂×C₄	1	1	yes	characteristic
84	C₈×Dic₅₃	C₂×C₄	1	1	yes	characteristic
85	C₂×D₈×C₅₃	C₂×C₄	1	1	yes	characteristic
86	C₂²×C₈×C₅₃	—	2	2	no	—
87	C₄×D₄×C₅₃	C₂³	1	2	yes	—
88	#107×C₅₃	C₂×C₄	1	1	yes	characteristic
89	C₄⋊C₈×C₅₃	C₂³	1	2	yes	—
90	#102×C₅₃	C₂×C₄	1	2	yes	—
91	C₂²⋊C₈×C₅₃	—	2	2	no	—
92	C₄×C₈×C₅₃	C₂³	1	1	yes	characteristic
93	C₄×C₈×C₅₃	C₂³	1	1	yes	characteristic
94	C₄×C₈×C₅₃	C₂³	1	1	yes	characteristic
95	C₂³×D₅₃	—	2	2	no	—
96	C₂×D₄×C₅₃	C₂²×C₄	1	2	yes	—
97	C₂²×C₄×C₅₃	—	2	2	no	—
98	C₂×C₂²⋊D₅₃	—	2	2	no	—
99	C₂²×Dic₅₃	—	2	2	no	—
100	D₄×D₅₃	—	2	4	no	—
101	D₄×D₅₃	C₂×C₈	1	8	yes	complemented
102	C₂×D₂₁₂	C₂²×C₄	1	2	yes	—
103	C₂×C₄×D₅₃	—	2	2	no	—
104	C₂×C₄×D₅₃	—	2	2	no	—
105	C₂×C₄×D₅₃	—	2	2	no	—
106	C₂×C₄×D₅₃	—	8	2	no	—
107	C₂×C₄×D₅₃	—	8	2	no	—
108	C₂×C₄×D₅₃	C₂²×C₄	1	1	yes	characteristic
109	C₂×C₄×D₅₃	C₂×D₄	1	1	yes	characteristic
110	C₂×C₄×D₅₃	#41	1	1	yes	characteristic
111	D₈×C₅₃	C₂×C₈	1	4	yes	complemented
112	C₂×C₈×C₅₃	—	2	2	no	—
113	C₂×C₈×C₅₃	—	2	1	no	—
114	C₂×C₈×C₅₃	—	8	2	no	—
115	C₂×C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
116	C₂×C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
117	C₂×C₈×C₅₃	C₂×D₄	1	1	yes	characteristic
118	C₂×C₈×C₅₃	C₂×D₄	1	1	yes	characteristic
119	C₄⋊C₄×C₅₃	C₂²×C₄	1	2	yes	—
120	C₂²⋊C₄×C₅₃	—	2	2	no	—
121	C₄²×C₅₃	C₂⁴	1	1	yes	characteristic
122	C₂²⋊Dic₅₃	—	2	2	no	—
123	#11719922	C₂×C₈	1	8	yes	complemented
124	#11719921	—	2	2	no	—
125	#11719921	—	2	2	no	—
126	C₄⋊Dic₅₃	C₂²×C₄	1	2	yes	—
127	#11719919	—	2	2	no	—
128	C₄×Dic₅₃	C₂²×C₄	1	1	yes	characteristic
129	C₄×Dic₅₃	C₂×D₄	1	1	yes	characteristic
130	C₄×Dic₅₃	#41	1	1	yes	characteristic
131	C₂×#88322	—	2	2	no	—
132	C₂×#88322	—	2	1	no	—
133	C₂×#88322	—	8	2	no	—
134	C₂×#88322	C₂²×C₄	1	1	yes	characteristic
135	C₂×#88322	C₂²×C₄	1	1	yes	characteristic
136	C₂×#88322	C₂×D₄	1	1	yes	characteristic
137	C₂×#88322	C₂×D₄	1	1	yes	characteristic
138	D₄₂₄	C₂×C₈	1	4	yes	complemented
139	C₈×D₅₃	—	2	2	no	—
140	C₈×D₅₃	—	8	2	no	—
141	C₈×D₅₃	—	8	2	no	—
142	C₈×D₅₃	C₂×C₈	1	4	yes	complemented
143	C₂³×C₅₃	—	2	2	no	—
144	C₂²×D₅₃	—	2	2	no	—
145	C₂²×D₅₃	—	2	8	no	—
146	C₂²×D₅₃	—	8	2	no	—
147	C₂²×D₅₃	—	8	2	no	—
148	C₂²×D₅₃	C₄×D₄	1	1	yes	characteristic
149	D₄×C₅₃	—	2	2	no	—
150	D₄×C₅₃	C₂²×C₈	1	4	yes	—
151	C₂×C₄×C₅₃	—	2	2	no	—
152	C₂×C₄×C₅₃	—	4	2	no	—
153	C₂×C₄×C₅₃	—	4	2	no	—
154	C₂×C₄×C₅₃	C₂³×C₄	1	1	yes	characteristic
155	C₂×C₄×C₅₃	C₂²×D₄	1	1	yes	characteristic
156	C₂×C₄×C₅₃	C₂×#41	1	1	yes	characteristic
157	C₂²⋊D₅₃	—	2	8	no	—
158	C₂×Dic₅₃	—	2	2	no	—
159	C₂×Dic₅₃	—	8	2	no	—
160	C₂×Dic₅₃	C₄×D₄	1	1	yes	characteristic
161	C₂×Dic₅₃	C₄×D₄	1	1	yes	characteristic
162	C₂×Dic₅₃	C₄×D₄	1	1	yes	characteristic
163	D₂₁₂	—	2	2	no	—
164	D₂₁₂	C₂²×C₈	1	4	yes	—
165	C₄×D₅₃	—	2	1	no	—
166	C₄×D₅₃	—	2	1	no	—
167	C₄×D₅₃	—	4	1	no	—
168	C₄×D₅₃	—	4	2	no	—
169	C₄×D₅₃	—	4	2	no	—
170	C₄×D₅₃	—	8	2	no	—
171	C₄×D₅₃	—	8	2	no	—
172	C₄×D₅₃	C₂²×C₈	1	2	yes	—
173	C₄×D₅₃	#131	1	2	yes	—
174	C₈×C₅₃	—	2	1	no	—
175	C₈×C₅₃	—	2	1	no	—
176	C₈×C₅₃	—	2	1	no	—
177	C₈×C₅₃	—	4	1	no	—
178	C₈×C₅₃	—	4	2	no	—
179	C₈×C₅₃	C₂²×C₈	1	2	yes	—
180	#88322	—	2	1	no	—
181	#88322	—	2	1	no	—
182	#88322	—	2	1	no	—
183	#88322	—	4	1	no	—
184	#88322	—	4	2	no	—
185	#88322	C₂²×C₈	1	2	yes	—
186	#17854	—	53	1	no	maximal, Sylow
187	C₂²×C₅₃	—	2	4	no	—
188	C₂²×C₅₃	—	4	2	no	—
189	C₂²×C₅₃	—	4	2	no	—
190	C₂²×C₅₃	C₂×C₄×D₄	1	1	yes	characteristic
191	C₂×D₅₃	—	2	1	no	—
192	C₂×D₅₃	—	2	1	no	—
193	C₂×D₅₃	—	2	4	no	—
194	C₂×D₅₃	—	4	2	no	—
195	C₂×D₅₃	—	4	2	no	—
196	C₂×D₅₃	—	8	8	no	—
197	C₂×D₅₃	C₈×D₄	1	2	yes	—
198	C₄×C₅₃	—	2	1	no	—
199	C₄×C₅₃	—	4	2	no	—
200	C₄×C₅₃	C₂³×C₈	1	1	yes	characteristic
201	C₄×C₅₃	C₂×#131	1	1	yes	characteristic
202	C₄×C₅₃	#576	1	1	yes	characteristic
203	C₄×C₅₃	#576	1	1	yes	characteristic
204	Dic₅₃	—	2	1	no	—
205	Dic₅₃	—	2	1	no	—
206	Dic₅₃	—	4	2	no	—
207	Dic₅₃	C₈×D₄	1	2	yes	—
208	#2972	—	53	2	no	—
209	C₂×C₄×D₈	—	53	1	no	—
210	#1580	—	53	2	no	—
211	#1580	—	53	2	no	—
212	C₈×D₈	—	53	1	no	—
213	C₈×D₈	—	53	1	no	—
214	C₈×D₈	—	53	1	no	—
215	C₈×D₈	D₅₃	1	1	yes	characteristic, complemented
216	#1564	—	53	1	no	—
217	#1478	—	53	2	no	—
218	#1450	—	53	1	no	—
219	C₂×C₅₃	—	4	4	no	—
220	C₂×C₅₃	C₂×C₈×D₄	1	1	yes	characteristic
221	C₂×C₅₃	#2948	1	1	yes	characteristic
222	C₂×C₅₃	#2957	1	1	yes	characteristic
223	D₅₃	—	2	2	no	—
224	D₅₃	—	4	4	no	—
225	C₂²×D₈	—	53	1	no	—
226	C₂×C₄×D₄	—	53	2	no	—
227	C₄×D₈	—	53	1	no	—
228	C₄×D₈	—	53	2	no	—
229	C₄×D₈	—	53	1	no	—
230	C₄×D₈	—	53	2	no	—
231	C₄×D₈	—	53	1	no	—
232	C₄×D₈	C₂×D₅₃	1	1	yes	characteristic
233	C₈×D₄	—	53	2	no	—
234	C₈×D₄	—	53	2	no	—
235	C₈×D₄	—	53	2	no	—
236	C₈×D₄	—	106	2	no	—
237	C₈×D₄	—	106	2	no	—
238	C₈×D₄	C₂×D₅₃	1	2	yes	complemented
239	#431	—	53	1	no	—
240	#431	—	53	1	no	—
241	#431	—	53	2	no	—
242	C₂×#107	—	53	1	no	—
243	C₂×#102	—	53	2	no	—
244	C₂×C₂²⋊C₈	—	106	2	no	—
245	C₂×C₂²⋊C₈	—	106	2	no	—
246	C₂×C₄×C₈	—	53	1	no	—
247	#335	—	53	1	no	—
248	#335	C₂×D₅₃	1	1	yes	characteristic, complemented
249	#334	—	53	2	no	—
250	D₄⋊C₈	—	53	2	no	—
251	D₄⋊C₈	—	53	2	no	—
252	D₄⋊C₈	—	53	2	no	—
253	D₄⋊C₈	C₂×D₅₃	1	2	yes	complemented
254	#322	—	53	1	no	—
255	#322	—	53	1	no	—
256	C₈²	—	53	1	no	—
257	C₈²	C₂×D₅₃	1	1	yes	characteristic, complemented
258	C₅₃	#17854	1	1	yes	characteristic, complemented, Sylow
259	C₂²×D₄	—	53	2	no	—
260	C₂³×C₄	—	106	2	no	—
261	C₂×D₈	—	53	2	no	—
262	C₂×D₈	—	53	8	no	—
263	C₂×D₈	—	53	1	no	—
264	C₂×D₈	C₄×D₅₃	1	1	yes	characteristic
265	C₂²×C₈	—	2	2	no	—
266	C₂²×C₈	—	53	1	no	—
267	C₂²×C₈	—	53	1	no	—
268	C₂²×C₈	—	106	2	no	—
269	C₂²×C₈	—	106	2	no	—
270	C₂²×C₈	—	106	2	no	—
271	C₄×D₄	—	53	2	no	—
272	C₄×D₄	—	53	2	no	—
273	C₄×D₄	—	53	2	no	—
274	C₄×D₄	—	106	2	no	—
275	C₄×D₄	—	106	2	no	—
276	C₄×D₄	C₂²×D₅₃	1	2	yes	—
277	C₂×C₄⋊C₄	—	53	2	no	—
278	C₂×C₂²⋊C₄	—	106	2	no	—
279	C₂×C₄²	—	53	1	no	—
280	#107	—	53	1	no	—
281	#107	—	53	2	no	—
282	#107	C₄×D₅₃	1	1	yes	characteristic
283	C₄⋊C₈	—	53	1	no	—
284	C₄⋊C₈	—	53	1	no	—
285	C₄⋊C₈	—	53	2	no	—
286	C₄⋊C₈	—	106	1	no	—
287	C₄⋊C₈	C₂²×D₅₃	1	2	yes	—
288	#102	—	53	2	no	—
289	#102	—	53	2	no	—
290	#102	—	53	2	no	—
291	#102	C₄×D₅₃	1	2	yes	—
292	C₂²⋊C₈	—	2	2	no	—
293	C₂²⋊C₈	—	53	1	no	—
294	C₂²⋊C₈	—	53	1	no	—
295	C₂²⋊C₈	—	106	2	no	—
296	C₂²⋊C₈	—	106	2	no	—
297	C₂²⋊C₈	—	106	1	no	—
298	C₂²⋊C₈	—	106	2	no	—
299	C₂²⋊C₈	—	106	2	no	—
300	C₄×C₈	—	53	1	no	—
301	C₄×C₈	—	53	1	no	—
302	C₄×C₈	—	53	1	no	—
303	C₄×C₈	—	53	1	no	—
304	C₄×C₈	—	53	1	no	—
305	C₄×C₈	—	106	2	no	—
306	C₄×C₈	C₂²×D₅₃	1	1	yes	characteristic
307	C₄×C₈	C₂²×D₅₃	1	1	yes	characteristic
308	C₄×C₈	C₂²×D₅₃	1	1	yes	characteristic
309	C₂⁴	—	106	2	no	—
310	C₂×D₄	—	53	8	no	—
311	C₂×D₄	—	53	2	no	—
312	C₂×D₄	—	106	4	no	—
313	C₂×D₄	—	106	2	no	—
314	C₂×D₄	C₂×C₄×D₅₃	1	2	yes	—
315	C₂²×C₄	—	2	2	no	—
316	C₂²×C₄	—	53	1	no	—
317	C₂²×C₄	—	53	1	no	—
318	C₂²×C₄	—	53	1	no	—
319	C₂²×C₄	—	106	2	no	—
320	C₂²×C₄	—	106	2	no	—
321	C₂²×C₄	—	106	2	no	—
322	C₂²×C₄	—	106	2	no	—
323	C₂²×C₄	—	424	2	no	—
324	C₂²×C₄	—	424	2	no	—
325	D₈	—	53	8	no	—
326	D₈	—	53	4	no	—
327	D₈	C₈×D₅₃	1	4	yes	complemented
328	C₂×C₈	—	2	2	no	—
329	C₂×C₈	—	2	1	no	—
330	C₂×C₈	—	8	2	no	—
331	C₂×C₈	—	53	1	no	—
332	C₂×C₈	—	53	4	no	—
333	C₂×C₈	—	53	1	no	—
334	C₂×C₈	—	53	1	no	—
335	C₂×C₈	—	53	1	no	—
336	C₂×C₈	—	106	2	no	—
337	C₂×C₈	—	106	2	no	—
338	C₂×C₈	—	106	1	no	—
339	C₂×C₈	—	424	2	no	—
340	C₂×C₈	—	424	2	no	—
341	C₂×C₈	—	424	2	no	—
342	C₂×C₈	C₂×C₄×D₅₃	1	1	yes	characteristic
343	C₂×C₈	C₂×C₄×D₅₃	1	1	yes	characteristic
344	C₂×C₈	D₄×D₅₃	1	1	yes	characteristic
345	C₂×C₈	D₄×D₅₃	1	1	yes	characteristic
346	C₄⋊C₄	—	53	2	no	—
347	C₄⋊C₄	—	106	2	no	—
348	C₄⋊C₄	C₂×C₄×D₅₃	1	2	yes	—
349	C₂²⋊C₄	—	2	2	no	—
350	C₂²⋊C₄	—	106	2	no	—
351	C₂²⋊C₄	—	106	2	no	—
352	C₂²⋊C₄	—	106	2	no	—
353	C₄²	—	53	1	no	—
354	C₄²	—	53	1	no	—
355	C₄²	—	53	1	no	—
356	C₄²	C₂³×D₅₃	1	1	yes	characteristic
357	C₂³	—	2	2	no	—
358	C₂³	—	53	1	no	—
359	C₂³	—	106	8	no	—
360	C₂³	—	106	2	no	—
361	C₂³	—	424	2	no	—
362	C₂³	—	424	2	no	—
363	D₄	—	2	2	no	—
364	D₄	—	53	4	no	—
365	D₄	—	106	8	no	—
366	D₄	—	106	2	no	—
367	D₄	C₂×C₈×D₅₃	1	4	yes	—
368	C₂×C₄	—	2	2	no	—
369	C₂×C₄	—	4	2	no	—
370	C₂×C₄	—	4	2	no	—
371	C₂×C₄	—	53	1	no	—
372	C₂×C₄	—	53	2	no	—
373	C₂×C₄	—	53	2	no	—
374	C₂×C₄	—	53	1	no	—
375	C₂×C₄	—	53	1	no	—
376	C₂×C₄	—	106	1	no	—
377	C₂×C₄	—	106	1	no	—
378	C₂×C₄	—	106	2	no	—
379	C₂×C₄	—	424	2	no	—
380	C₂×C₄	—	424	2	no	—
381	C₂×C₄	—	424	2	no	—
382	C₂×C₄	—	212	1	no	—
383	C₂×C₄	—	212	2	no	—
384	C₂×C₄	—	212	2	no	—
385	C₂×C₄	C₂²×C₄×D₅₃	1	1	yes	characteristic
386	C₂×C₄	C₂×D₄×D₅₃	1	1	yes	characteristic
387	C₂×C₄	#41×D₅₃	1	1	yes	characteristic
388	C₈	—	2	1	no	—
389	C₈	—	2	1	no	—
390	C₈	—	2	1	no	—
391	C₈	—	4	1	no	—
392	C₈	—	4	2	no	—
393	C₈	—	53	2	no	—
394	C₈	—	106	1	no	—
395	C₈	—	106	1	no	—
396	C₈	—	106	1	no	—
397	C₈	—	212	1	no	—
398	C₈	—	212	2	no	—
399	C₈	C₂×C₈×D₅₃	1	2	yes	—
400	C₂²	—	2	4	no	—
401	C₂²	—	4	2	no	—
402	C₂²	—	4	2	no	—
403	C₂²	—	53	2	no	—
404	C₂²	—	106	1	no	—
405	C₂²	—	106	1	no	—
406	C₂²	—	106	4	no	—
407	C₂²	—	424	8	no	—
408	C₂²	—	212	2	no	—
409	C₂²	—	212	2	no	—
410	C₂²	C₄×D₄×D₅₃	1	1	yes	characteristic
411	C₄	—	2	1	no	—
412	C₄	—	4	2	no	—
413	C₄	—	53	2	no	—
414	C₄	—	106	1	no	—
415	C₄	—	106	1	no	—
416	C₄	—	212	2	no	—
417	C₄	C₂²×C₈×D₅₃	1	1	yes	characteristic
418	C₄	#131×D₅₃	1	1	yes	characteristic
419	C₄	#16811424	1	1	yes	characteristic
420	C₄	#16811425	1	1	yes	characteristic
421	C₂	—	4	4	no	—
422	C₂	—	106	2	no	—
423	C₂	—	212	4	no	—
424	C₂	C₈×D₄×D₅₃	1	1	yes	characteristic
425	C₂	#20283562	1	1	yes	characteristic
426	C₂	#20283627	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂×D₄×D₅₃
commutator	C₄×C₅₃	C₂³×C₈
Frattini	C₄²	C₂³×D₅₃
Fitting	C₈×D₈×C₅₃	C₂
socle	C₂²×C₅₃	C₂×C₄×D₄
prime radical	C₂×D₈×D₅₃	C₄
cosocle	C₄²×C₅₃	C₂⁴
hypercenter	C₈×D₈	D₅₃
hypocenter	C₅₃	#17854
LCS	G ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂×C₄ ◁ C₄×C₈ ◁ C₈×D₈

Complements

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

N	H ≅ G/N	type
1. #20283736	C₂	semidirect
3. #20273537	C₂	semidirect
4. #20273536	C₂	semidirect
5. C₈×D₄₂₄	C₂	semidirect
6. D₈×#88322	C₂	semidirect
7. #20273509	C₂	semidirect
8. #20273462	C₂	semidirect
9. #20273104	C₂	semidirect
10. #20272997	C₂	semidirect
11. C₈×D₈×C₅₃	C₂	semidirect
19. C₈×D₂₁₂	C₂²	semidirect
20. D₄×#88322	C₂²	semidirect
21. #16810688	C₂²	semidirect
32. #16810376	C₂²	semidirect
33. #16810373	C₂²	semidirect
34. #16810352	C₂²	semidirect
35. D₄⋊#88322	C₂²	semidirect
36. #16810350	C₂²	semidirect
37. #16810345	C₂²	semidirect
38. #16810345	C₂²	semidirect
39. C₈×#88322	C₂²	semidirect
41. C₈×D₄×C₅₃	C₂²	semidirect
42. C₅₃×#335	C₂²	semidirect
43. C₅₃×D₄⋊C₈	C₂²	semidirect
44. C₈²×C₅₃	C₂²	semidirect
49. D₈×D₅₃	C₈	semidirect
101. D₄×D₅₃	C₂×C₈	semidirect
111. D₈×C₅₃	C₂×C₈	semidirect
123. #11719922	C₂×C₈	semidirect
138. D₄₂₄	C₂×C₈	semidirect
142. C₈×D₅₃	C₂×C₈	semidirect
215. C₈×D₈	D₅₃	semidirect
238. C₈×D₄	C₂×D₅₃	semidirect
248. #335	C₂×D₅₃	semidirect
253. D₄⋊C₈	C₂×D₅₃	semidirect
257. C₈²	C₂×D₅₃	semidirect
258. C₅₃	#17854	semidirect
327. D₈	C₈×D₅₃	semidirect

Maximal subgroups

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

subgroup	index
#20283736	2
C₄×D₈×D₅₃	2
#20273537	2
#20273536	2
C₈×D₄₂₄	2
D₈×#88322	2
#20273509	2
#20273462	2
#20273104	2
#20272997	2
C₈×D₈×C₅₃	2
#17854	53