Subgroups — #27353085

Order 13568.

734 conjugacy classes of subgroups.

The subgroup lattice has 418 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20283824	C₂	1	1	yes	maximal, characteristic
2	#20283823	C₂	1	1	yes	maximal, characteristic
3	C₄×#13133225	C₂	1	1	yes	maximal, characteristic
4	#20273467	C₂	1	1	yes	maximal, characteristic, complemented
5	#20273466	C₂	1	2	yes	maximal, complemented
6	#20273464	C₂	1	2	yes	maximal, complemented
7	#20273461	C₂	1	1	yes	maximal, characteristic, complemented
8	#20273460	C₂	1	2	yes	maximal, complemented
9	C₂×#16810375	C₂	1	1	yes	maximal, characteristic, complemented
10	#335×D₅₃	C₂	1	2	yes	maximal, complemented
11	C₅₃×#1565	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×#13133225	C₄	1	1	yes	characteristic
13	C₂×#13133225	C₄	1	1	yes	characteristic
14	C₄×#11719945	C₂²	1	1	yes	characteristic
15	C₈×Dic₁₀₆	—	2	1	no	—
16	#16810747	—	2	1	no	—
17	C₈×D₂₁₂	—	2	1	no	—
18	#16810688	—	2	1	no	—
19	C₈×C₂²⋊D₅₃	—	2	2	no	—
20	#16810686	—	2	2	no	—
21	#16810676	C₂²	1	1	yes	characteristic
22	#16810675	C₂²	1	2	yes	—
23	#16810675	C₂²	1	2	yes	—
24	C₂×C₄⋊#88322	C₂²	1	1	yes	characteristic
25	C₄⋊C₈×D₅₃	C₂²	1	2	yes	—
26	C₄⋊C₈×D₅₃	C₂²	1	2	yes	—
27	#16810581	C₄	1	1	yes	characteristic
28	#16810581	C₄	1	1	yes	characteristic
29	#16810580	C₄	1	2	yes	—
30	#16810580	C₄	1	2	yes	—
31	C₄×#11719912	C₂²	1	2	yes	—
32	C₂×C₄×#88322	C₂²	1	1	yes	characteristic
33	C₄×C₈×D₅₃	C₂²	1	2	yes	—
34	#16810376	C₂²	1	2	yes	—
35	#16810375	C₂²	1	1	yes	characteristic
36	#16810375	C₂²	1	2	yes	—
37	#16810375	C₂²	1	2	yes	—
38	#16810375	C₂²	1	1	yes	characteristic
39	#16810374	C₂²	1	2	yes	—
40	#16810373	C₂²	1	1	yes	characteristic
41	#16810373	C₂²	1	1	yes	characteristic
42	C₅₃×#431	C₂²	1	1	yes	characteristic
43	C₂×C₄⋊C₈×C₅₃	C₂²	1	1	yes	characteristic
44	C₄×M₁₆×C₅₃	C₂²	1	1	yes	characteristic
45	C₅₃×#335	C₂²	1	2	yes	—
46	C₅₃×#334	C₂²	1	2	yes	—
47	C₂×#11719945	C₂×C₄	1	1	yes	characteristic
48	#13133225	—	2	4	no	—
49	#13133225	C₈	1	8	yes	complemented
50	C₂×#11719912	C₂×C₄	1	2	yes	—
51	C₂×#11719912	C₂×C₄	1	2	yes	—
52	C₂²×#88322	—	2	1	no	—
53	C₂²×#88322	C₂×C₄	1	1	yes	characteristic
54	C₂²×#88322	C₂×C₄	1	1	yes	characteristic
55	C₂×C₈×D₅₃	—	2	2	no	—
56	C₂×C₈×D₅₃	—	2	2	no	—
57	C₂×C₈×D₅₃	C₂×C₄	1	2	yes	—
58	C₂×C₈×D₅₃	C₂×C₄	1	2	yes	—
59	C₄×Dic₁₀₆	D₄	1	1	yes	characteristic
60	C₄×Dic₁₀₆	Q₈	1	1	yes	characteristic
61	C₄×D₂₁₂	D₄	1	1	yes	characteristic
62	C₄×D₂₁₂	Q₈	1	1	yes	characteristic
63	C₄×C₂²⋊D₅₃	D₄	1	2	yes	—
64	C₄×C₂²⋊D₅₃	Q₈	1	2	yes	—
65	#13133176	C₂×C₄	1	1	yes	characteristic
66	#13133174	C₂×C₄	1	2	yes	—
67	C₄²×D₅₃	C₂³	1	2	yes	—
68	C₄⋊#88322	—	2	1	no	—
69	C₄⋊#88322	C₂³	1	1	yes	characteristic
70	C₄⋊#88322	C₂³	1	1	yes	characteristic
71	C₄⋊#88322	C₂³	1	2	yes	—
72	#13133146	—	2	2	no	—
73	#13133146	—	2	2	no	—
74	C₂²⋊#88322	—	2	1	no	—
75	#13133129	—	2	2	no	—
76	#13133129	—	2	2	no	—
77	#13133127	C₂×C₄	1	2	yes	—
78	#13133127	C₂×C₄	1	2	yes	—
79	C₄×#88322	—	2	1	no	—
80	C₄×#88322	C₂×C₄	1	1	yes	characteristic
81	C₄×#88322	C₂×C₄	1	1	yes	characteristic
82	C₄×#88322	C₂³	1	1	yes	characteristic
83	C₄×#88322	C₂³	1	1	yes	characteristic
84	C₈×Dic₅₃	—	2	2	no	—
85	C₈×Dic₅₃	—	2	2	no	—
86	C₈×Dic₅₃	C₂×C₄	1	2	yes	—
87	C₈×Dic₅₃	C₂×C₄	1	2	yes	—
88	C₂×M₁₆×C₅₃	C₂×C₄	1	1	yes	characteristic
89	C₂×M₁₆×C₅₃	C₂×C₄	1	1	yes	characteristic
90	C₂²×C₈×C₅₃	—	2	1	no	—
91	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
92	C₄⋊C₈×C₅₃	—	2	1	no	—
93	C₄⋊C₈×C₅₃	C₂³	1	1	yes	characteristic
94	C₄⋊C₈×C₅₃	C₂³	1	2	yes	—
95	C₄⋊C₈×C₅₃	C₂³	1	1	yes	characteristic
96	C₂²⋊C₈×C₅₃	—	2	1	no	—
97	#97×C₅₃	C₂×C₄	1	2	yes	—
98	C₄×C₈×C₅₃	—	2	1	no	—
99	C₄×C₈×C₅₃	C₂³	1	2	yes	—
100	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
101	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
102	C₂²×C₄×C₅₃	C₂×Q₈	1	1	yes	characteristic
103	C₂×C₂²⋊D₅₃	C₄⋊C₄	1	2	yes	—
104	#11719945	C₂×C₈	1	4	yes	—
105	#11719945	M₁₆	1	4	yes	—
106	C₂×D₂₁₂	C₄⋊C₄	1	1	yes	characteristic
107	C₂×C₄×D₅₃	C₂²×C₄	1	2	yes	—
108	C₂×C₄×D₅₃	C₂×D₄	1	2	yes	—
109	C₂×C₄×D₅₃	C₂×Q₈	1	2	yes	—
110	C₂×Dic₁₀₆	C₄⋊C₄	1	1	yes	characteristic
111	M₁₆×C₅₃	—	2	2	no	—
112	M₁₆×C₅₃	C₂×C₈	1	4	yes	complemented
113	C₂×C₈×C₅₃	—	2	1	no	—
114	C₂×C₈×C₅₃	—	2	1	no	—
115	C₂×C₈×C₅₃	—	2	2	no	—
116	C₂×C₈×C₅₃	—	8	1	no	—
117	C₂×C₈×C₅₃	C₂²×C₄	1	2	yes	—
118	C₂×C₈×C₅₃	C₂²×C₄	1	2	yes	—
119	C₄²×C₅₃	C₂²×C₄	1	1	yes	characteristic
120	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
121	C₄²×C₅₃	C₂×Q₈	1	1	yes	characteristic
122	C₄²×C₅₃	C₂⁴	1	1	yes	characteristic
123	C₂²⋊Dic₅₃	C₄⋊C₄	1	2	yes	—
124	#11719921	C₄⋊C₄	1	2	yes	—
125	#11719921	C₄⋊C₄	1	2	yes	—
126	C₄⋊Dic₅₃	C₄⋊C₄	1	1	yes	characteristic
127	C₄⋊Dic₅₃	C₄⋊C₄	1	1	yes	characteristic
128	#11719919	C₄⋊C₄	1	2	yes	—
129	#11719919	C₄⋊C₄	1	2	yes	—
130	C₄×Dic₅₃	C₂²×C₄	1	2	yes	—
131	C₄×Dic₅₃	C₂×D₄	1	2	yes	—
132	C₄×Dic₅₃	C₂×Q₈	1	2	yes	—
133	C₂×#88322	—	2	2	no	—
134	C₂×#88322	—	2	1	no	—
135	C₂×#88322	—	2	2	no	—
136	C₂×#88322	—	2	1	no	—
137	C₂×#88322	—	8	1	no	—
138	C₂×#88322	C₂×C₈	1	4	yes	complemented
139	C₂×#88322	C₂²×C₄	1	1	yes	characteristic
140	C₂×#88322	C₂²×C₄	1	1	yes	characteristic
141	C₂×#88322	C₂²×C₄	1	1	yes	characteristic
142	C₂×#88322	C₂²×C₄	1	1	yes	characteristic
143	#11719912	—	2	4	no	—
144	#11719912	C₂×C₈	1	8	yes	complemented
145	C₈×D₅₃	—	2	4	no	—
146	C₈×D₅₃	—	8	2	no	—
147	C₈×D₅₃	—	8	2	no	—
148	C₈×D₅₃	C₂×C₈	1	8	yes	complemented
149	C₂³×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
150	C₂²×D₅₃	C₂×C₄⋊C₄	1	2	yes	—
151	C₂×C₄×C₅₃	—	2	1	no	—
152	C₂×C₄×C₅₃	—	2	1	no	—
153	C₂×C₄×C₅₃	—	4	1	no	—
154	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
155	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
156	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
157	C₂×C₄×C₅₃	C₂²×C₈	1	2	yes	—
158	C₂×C₄×C₅₃	C₂×M₁₆	1	2	yes	—
159	C₂×C₄×C₅₃	C₂³×C₄	1	1	yes	characteristic
160	C₂×C₄×C₅₃	C₂²×D₄	1	1	yes	characteristic
161	C₂×C₄×C₅₃	C₂²×Q₈	1	1	yes	characteristic
162	C₂²⋊D₅₃	C₄⋊C₈	1	8	yes	—
163	C₂×Dic₅₃	C₂×C₄⋊C₄	1	2	yes	—
164	C₂×Dic₅₃	C₂×C₄⋊C₄	1	2	yes	—
165	C₂×Dic₅₃	C₂×C₄⋊C₄	1	2	yes	—
166	D₂₁₂	C₄⋊C₈	1	4	yes	—
167	C₄×D₅₃	—	2	2	no	—
168	C₄×D₅₃	—	2	2	no	—
169	C₄×D₅₃	—	4	2	no	—
170	C₄×D₅₃	C₂²×C₈	1	4	yes	—
171	C₄×D₅₃	C₂×M₁₆	1	4	yes	—
172	Dic₁₀₆	C₄⋊C₈	1	4	yes	—
173	C₈×C₅₃	—	2	2	no	—
174	C₈×C₅₃	—	4	1	no	—
175	C₈×C₅₃	—	4	2	no	—
176	C₈×C₅₃	—	4	1	no	—
177	C₈×C₅₃	C₂²×C₈	1	4	yes	—
178	#88322	—	2	1	no	—
179	#88322	—	2	1	no	—
180	#88322	—	4	1	no	—
181	#88322	—	4	2	no	—
182	#88322	—	4	1	no	—
183	#88322	C₂²×C₈	1	2	yes	—
184	#88322	C₂²×C₈	1	2	yes	—
185	#17789	—	53	1	no	maximal, Sylow
186	C₂²×C₅₃	—	2	1	no	—
187	C₂²×C₅₃	—	2	1	no	—
188	C₂²×C₅₃	C₂×C₄⋊C₈	1	2	yes	—
189	C₂²×C₅₃	C₂²×C₄⋊C₄	1	1	yes	characteristic
190	C₂×D₅₃	—	2	2	no	—
191	C₂×D₅₃	—	2	2	no	—
192	C₂×D₅₃	C₂×C₄⋊C₈	1	4	yes	—
193	C₄×C₅₃	—	2	1	no	—
194	C₄×C₅₃	—	2	1	no	—
195	C₄×C₅₃	—	2	1	no	—
196	C₄×C₅₃	C₂×C₄⋊C₈	1	2	yes	—
197	C₄×C₅₃	C₂³×C₈	1	1	yes	characteristic
198	C₄×C₅₃	C₂²×M₁₆	1	1	yes	characteristic
199	C₄×C₅₃	#576	1	1	yes	characteristic
200	C₄×C₅₃	#578	1	1	yes	characteristic
201	Dic₅₃	—	2	2	no	—
202	Dic₅₃	—	2	2	no	—
203	Dic₅₃	C₂×C₄⋊C₈	1	4	yes	—
204	#2988	—	53	1	no	—
205	#2988	—	53	1	no	—
206	C₄×#131	—	53	1	no	—
207	#1565	—	53	1	no	—
208	#1565	—	53	2	no	—
209	#1565	—	53	2	no	—
210	#1565	D₅₃	1	1	yes	characteristic, complemented
211	#1564	—	53	1	no	—
212	#1564	—	53	2	no	—
213	C₂×#335	—	53	1	no	—
214	C₂×#335	—	53	2	no	—
215	C₂×C₅₃	—	2	2	no	—
216	C₂×C₅₃	C₂²×C₄⋊C₈	1	1	yes	characteristic
217	C₂×C₅₃	#2913	1	1	yes	characteristic
218	C₂×C₅₃	#2916	1	1	yes	characteristic
219	D₅₃	—	2	4	no	—
220	C₂×#131	—	53	1	no	—
221	C₂×#131	—	53	1	no	—
222	C₄×#41	—	53	1	no	—
223	C₈×Q₈	—	106	1	no	—
224	C₈×Q₈	—	106	1	no	—
225	C₈×D₄	—	106	1	no	—
226	C₈×D₄	—	106	2	no	—
227	C₈×D₄	—	106	1	no	—
228	C₈×D₄	—	106	2	no	—
229	#431	—	53	1	no	—
230	#431	—	53	2	no	—
231	#431	—	53	2	no	—
232	#431	C₂×D₅₃	1	1	yes	characteristic
233	C₂×C₄⋊C₈	—	53	1	no	—
234	C₂×C₄⋊C₈	—	53	2	no	—
235	C₂×C₄⋊C₈	—	53	2	no	—
236	C₂×C₄⋊C₈	C₂×D₅₃	1	1	yes	characteristic
237	#405	—	53	1	no	—
238	#405	—	53	1	no	—
239	#405	—	53	2	no	—
240	#405	—	53	2	no	—
241	C₄×M₁₆	—	53	2	no	—
242	C₄×M₁₆	C₂×D₅₃	1	1	yes	characteristic
243	C₂×C₄×C₈	—	53	1	no	—
244	C₂×C₄×C₈	—	53	2	no	—
245	#335	—	53	1	no	—
246	#335	—	53	2	no	—
247	#335	—	53	2	no	—
248	#335	—	53	1	no	—
249	#335	—	53	2	no	—
250	#335	C₂×D₅₃	1	2	yes	—
251	#334	—	53	1	no	—
252	#334	—	53	1	no	—
253	#334	—	53	2	no	—
254	#334	C₂×D₅₃	1	2	yes	—
255	C₅₃	#17789	1	1	yes	characteristic, complemented, Sylow
256	C₂×#41	—	53	1	no	—
257	#131	—	53	8	no	—
258	#131	—	106	4	no	—
259	C₂×M₁₆	—	53	2	no	—
260	C₂×M₁₆	—	53	2	no	—
261	C₂×M₁₆	C₄×D₅₃	1	1	yes	characteristic
262	C₂×M₁₆	C₄×D₅₃	1	1	yes	characteristic
263	C₂²×C₈	—	2	1	no	—
264	C₂²×C₈	—	53	1	no	—
265	C₂²×C₈	—	53	2	no	—
266	C₂²×C₈	—	53	1	no	—
267	C₂²×C₈	—	53	2	no	—
268	C₂²×C₈	—	106	2	no	—
269	C₂²×C₈	—	106	2	no	—
270	C₂²×C₈	—	106	1	no	—
271	C₄×Q₈	—	53	1	no	—
272	C₄×Q₈	—	53	1	no	—
273	C₄×D₄	—	53	2	no	—
274	C₄×D₄	—	53	2	no	—
275	C₄×D₄	—	53	1	no	—
276	C₄×D₄	—	53	1	no	—
277	#117	—	53	1	no	—
278	#117	—	53	2	no	—
279	C₂×C₄²	—	53	2	no	—
280	C₂×C₄²	C₂²×D₅₃	1	1	yes	characteristic
281	C₄⋊C₈	—	2	1	no	—
282	C₄⋊C₈	—	53	1	no	—
283	C₄⋊C₈	—	53	1	no	—
284	C₄⋊C₈	—	53	2	no	—
285	C₄⋊C₈	—	106	2	no	—
286	C₄⋊C₈	—	106	2	no	—
287	C₄⋊C₈	—	106	1	no	—
288	C₄⋊C₈	C₂²×D₅₃	1	1	yes	characteristic
289	C₄⋊C₈	C₂²×D₅₃	1	1	yes	characteristic
290	C₄⋊C₈	C₂²×D₅₃	1	2	yes	—
291	C₂²⋊C₈	—	2	1	no	—
292	C₂²⋊C₈	—	106	2	no	—
293	C₂²⋊C₈	—	106	2	no	—
294	C₂²⋊C₈	—	106	1	no	—
295	#97	—	53	2	no	—
296	#97	—	53	2	no	—
297	#97	C₄×D₅₃	1	2	yes	—
298	C₄×C₈	—	2	1	no	—
299	C₄×C₈	—	53	2	no	—
300	C₄×C₈	—	53	2	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	—	106	2	no	—
307	C₄×C₈	—	106	1	no	—
308	C₄×C₈	C₂²×D₅₃	1	2	yes	—
309	#41	—	53	4	no	—
310	#41	—	53	4	no	—
311	C₂×Q₈	—	53	1	no	—
312	C₂×D₄	—	53	2	no	—
313	C₂×D₄	—	53	1	no	—
314	C₂²×C₄	—	53	2	no	—
315	C₂²×C₄	—	53	2	no	—
316	C₂²×C₄	—	53	2	no	—
317	C₂²×C₄	C₂×C₄×D₅₃	1	1	yes	characteristic
318	C₂²×C₄	D₄×D₅₃	1	1	yes	characteristic
319	C₂²×C₄	Q₈×D₅₃	1	1	yes	characteristic
320	M₁₆	—	2	2	no	—
321	M₁₆	—	53	8	no	—
322	M₁₆	—	106	4	no	—
323	M₁₆	C₈×D₅₃	1	4	yes	complemented
324	C₂×C₈	—	2	1	no	—
325	C₂×C₈	—	2	1	no	—
326	C₂×C₈	—	2	2	no	—
327	C₂×C₈	—	8	1	no	—
328	C₂×C₈	—	53	4	no	—
329	C₂×C₈	—	53	1	no	—
330	C₂×C₈	—	53	1	no	—
331	C₂×C₈	—	53	8	no	—
332	C₂×C₈	—	53	1	no	—
333	C₂×C₈	—	53	1	no	—
334	C₂×C₈	—	106	4	no	—
335	C₂×C₈	—	106	2	no	—
336	C₂×C₈	—	106	1	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	1	no	—
342	C₂×C₈	C₂×C₄×D₅₃	1	2	yes	—
343	C₂×C₈	C₂×C₄×D₅₃	1	2	yes	—
344	C₄⋊C₄	—	53	2	no	—
345	C₄⋊C₄	—	53	2	no	—
346	C₄⋊C₄	—	53	1	no	—
347	C₄⋊C₄	—	53	1	no	—
348	C₂²⋊C₄	—	53	2	no	—
349	C₂²⋊C₄	—	53	2	no	—
350	C₂²⋊C₄	—	53	2	no	—
351	C₄²	—	53	2	no	—
352	C₄²	—	53	2	no	—
353	C₄²	—	53	2	no	—
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	C₄²	C₂³×D₅₃	1	1	yes	characteristic
358	C₂³	—	53	2	no	—
359	C₂³	C₄⋊C₄×D₅₃	1	1	yes	characteristic
360	Q₈	—	53	4	no	—
361	D₄	—	53	8	no	—
362	D₄	—	53	4	no	—
363	C₂×C₄	—	2	1	no	—
364	C₂×C₄	—	2	1	no	—
365	C₂×C₄	—	4	1	no	—
366	C₂×C₄	—	53	2	no	—
367	C₂×C₄	—	53	4	no	—
368	C₂×C₄	—	53	4	no	—
369	C₂×C₄	—	53	2	no	—
370	C₂×C₄	—	53	2	no	—
371	C₂×C₄	—	106	2	no	—
372	C₂×C₄	—	106	2	no	—
373	C₂×C₄	—	212	2	no	—
374	C₂×C₄	C₄⋊C₄×D₅₃	1	1	yes	characteristic
375	C₂×C₄	C₄⋊C₄×D₅₃	1	1	yes	characteristic
376	C₂×C₄	C₄⋊C₄×D₅₃	1	1	yes	characteristic
377	C₂×C₄	C₂×C₈×D₅₃	1	2	yes	—
378	C₂×C₄	M₁₆×D₅₃	1	2	yes	—
379	C₂×C₄	C₂²×C₄×D₅₃	1	1	yes	characteristic
380	C₂×C₄	C₂×D₄×D₅₃	1	1	yes	characteristic
381	C₂×C₄	C₂×Q₈×D₅₃	1	1	yes	characteristic
382	C₈	—	2	2	no	—
383	C₈	—	4	2	no	—
384	C₈	—	4	1	no	—
385	C₈	—	4	1	no	—
386	C₈	—	53	2	no	—
387	C₈	—	53	2	no	—
388	C₈	—	106	1	no	—
389	C₈	—	106	1	no	—
390	C₈	—	212	1	no	—
391	C₈	—	212	2	no	—
392	C₈	—	212	1	no	—
393	C₈	C₂×C₈×D₅₃	1	4	yes	—
394	C₂²	—	2	1	no	—
395	C₂²	—	2	1	no	—
396	C₂²	—	53	4	no	—
397	C₂²	—	106	2	no	—
398	C₂²	—	106	2	no	—
399	C₂²	C₄⋊C₈×D₅₃	1	2	yes	—
400	C₂²	C₂×C₄⋊C₄×D₅₃	1	1	yes	characteristic
401	C₄	—	2	1	no	—
402	C₄	—	2	1	no	—
403	C₄	—	2	1	no	—
404	C₄	—	53	4	no	—
405	C₄	—	106	2	no	—
406	C₄	—	106	2	no	—
407	C₄	C₄⋊C₈×D₅₃	1	2	yes	—
408	C₄	C₂²×C₈×D₅₃	1	1	yes	characteristic
409	C₄	C₂×M₁₆×D₅₃	1	1	yes	characteristic
410	C₄	#16811427	1	1	yes	characteristic
411	C₄	#16811439	1	1	yes	characteristic
412	C₂	—	2	2	no	—
413	C₂	—	106	4	no	—
414	C₂	C₂×C₄⋊C₈×D₅₃	1	1	yes	characteristic
415	C₂	#20283314	1	1	yes	characteristic
416	C₂	#20283325	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂×D₄×D₅₃
commutator	C₄×C₅₃	C₂³×C₈
Frattini	C₄²	C₂³×D₅₃
Fitting	C₅₃×#1565	C₂
socle	C₂²×C₅₃	C₂²×C₄⋊C₄
prime radical	C₂×#11719945	C₂×C₄
cosocle	C₄²×C₅₃	C₂⁴
hypercenter	#1565	D₅₃
hypocenter	C₅₃	#17789
LCS	G ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #1565

Complements

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

N	H ≅ G/N	type
4. #20273467	C₂	semidirect
5. #20273466	C₂	semidirect
6. #20273464	C₂	semidirect
7. #20273461	C₂	semidirect
8. #20273460	C₂	semidirect
9. C₂×#16810375	C₂	semidirect
10. #335×D₅₃	C₂	semidirect
11. C₅₃×#1565	C₂	semidirect
49. #13133225	C₈	semidirect
112. M₁₆×C₅₃	C₂×C₈	semidirect
138. C₂×#88322	C₂×C₈	semidirect
144. #11719912	C₂×C₈	semidirect
148. C₈×D₅₃	C₂×C₈	semidirect
210. #1565	D₅₃	semidirect
255. C₅₃	#17789	semidirect
323. 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
#20283824	2
#20283823	2
C₄×#13133225	2
#20273467	2
#20273466	2
#20273464	2
#20273461	2
#20273460	2
C₂×#16810375	2
#335×D₅₃	2
C₅₃×#1565	2
#17789	53