Subgroups — #27274406

Order 13568.

742 conjugacy classes of subgroups.

The subgroup lattice has 474 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	C₄×C₄⋊C₄×D₅₃	C₂	1	1	yes	maximal, characteristic
4	#20272618	C₂	1	1	yes	maximal, characteristic, complemented
5	#20272617	C₂	1	1	yes	maximal, characteristic, complemented
6	#20272616	C₂	1	1	yes	maximal, characteristic, complemented
7	C₅₃×#1303	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×C₄⋊C₄×D₅₃	—	2	1	no	—
9	C₂×C₄²×D₅₃	—	2	1	no	—
10	C₂×C₄²×D₅₃	C₂²	1	1	yes	characteristic
11	#16810679	—	2	1	no	—
12	#16810679	C₄	1	2	yes	complemented
13	#16810679	C₄	1	2	yes	complemented
14	#16810677	—	2	1	no	—
15	#16810676	C₂²	1	1	yes	characteristic
16	#16810675	—	2	1	no	—
17	#16810675	C₄	1	2	yes	complemented
18	#16810674	—	2	1	no	—
19	#16810647	C₂²	1	1	yes	characteristic
20	#16810645	C₄	1	2	yes	complemented
21	#16810586	—	2	1	no	—
22	#16810585	—	2	1	no	—
23	M₁₆×Dic₅₃	—	2	1	no	—
24	C₄×#11719912	—	2	1	no	—
25	C₄⋊C₄×Dic₅₃	—	2	1	no	—
26	C₄×#11719919	—	2	1	no	—
27	C₄×C₄⋊Dic₅₃	C₂²	1	1	yes	characteristic
28	#16810474	C₄	1	2	yes	—
29	#95×D₅₃	C₄	1	2	yes	—
30	C₅₃×#431	C₂²	1	1	yes	characteristic
31	C₅₃×#423	C₂²	1	1	yes	characteristic
32	C₄×C₄⋊C₄×C₅₃	C₂²	1	1	yes	characteristic
33	C₂²×C₄×D₅₃	—	2	2	no	—
34	C₂²×C₄×D₅₃	C₂×C₄	1	1	yes	characteristic
35	C₂²×C₄×D₅₃	D₄	1	1	yes	characteristic
36	C₂²×C₄×D₅₃	D₄	1	1	yes	characteristic
37	C₂×#11719917	—	2	1	no	—
38	C₂×#11719912	—	2	2	no	—
39	C₂×C₄⋊Dic₅₃	—	2	1	no	—
40	C₂×#11719919	—	2	2	no	—
41	C₄⋊C₄×D₅₃	—	2	8	no	—
42	C₂×C₄×Dic₅₃	—	2	1	no	—
43	C₂×C₄×Dic₅₃	—	2	2	no	—
44	C₂×C₄×Dic₅₃	C₂×C₄	1	1	yes	characteristic
45	C₂×C₄×Dic₅₃	D₄	1	1	yes	characteristic
46	C₂×C₄×Dic₅₃	D₄	1	1	yes	characteristic
47	C₄²×D₅₃	—	4	2	no	—
48	C₄²×D₅₃	—	4	2	no	—
49	C₄²×D₅₃	C₂×C₄	1	1	yes	characteristic
50	C₄²×D₅₃	C₂×C₄	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	C₄⋊#88322	—	2	1	no	—
58	C₄⋊#88322	C₂×C₄	1	2	yes	complemented
59	C₄⋊#88322	C₂×C₄	1	2	yes	complemented
60	#13133146	—	2	2	no	—
61	C₂²⋊#88322	—	2	1	no	—
62	#13133129	—	2	2	no	—
63	#13133129	—	2	2	no	—
64	#13133129	—	2	2	no	—
65	#13133127	—	2	2	no	—
66	#13133127	—	2	2	no	—
67	C₄×#88322	—	2	1	no	—
68	C₈×Dic₅₃	—	2	2	no	—
69	#13133124	—	2	2	no	—
70	#13133124	C₂×C₄	1	2	yes	—
71	C₂×M₁₆×C₅₃	—	2	1	no	—
72	C₂×C₄⋊C₄×C₅₃	—	2	1	no	—
73	C₂×C₄²×C₅₃	—	2	1	no	—
74	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
75	C₄⋊C₈×C₅₃	—	2	1	no	—
76	C₄⋊C₈×C₅₃	C₂×C₄	1	2	yes	complemented
77	C₄⋊C₈×C₅₃	C₂×C₄	1	2	yes	complemented
78	C₂²⋊C₈×C₅₃	—	2	1	no	—
79	C₄×C₈×C₅₃	—	2	1	no	—
80	#95×C₅₃	C₂×C₄	1	2	yes	—
81	C₂³×D₅₃	C₂²⋊C₄	1	1	yes	characteristic
82	C₂²×C₄×C₅₃	—	2	2	no	—
83	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
84	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
85	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
86	C₂²×Dic₅₃	—	2	2	no	—
87	C₂²×Dic₅₃	C₂²⋊C₄	1	1	yes	characteristic
88	C₂²×Dic₅₃	C₂²⋊C₄	1	1	yes	characteristic
89	C₂²×Dic₅₃	C₂²⋊C₄	1	1	yes	characteristic
90	C₂×C₄×D₅₃	—	2	1	no	—
91	C₂×C₄×D₅₃	—	2	1	no	—
92	C₂×C₄×D₅₃	—	2	1	no	—
93	C₂×C₄×D₅₃	—	2	1	no	—
94	C₂×C₄×D₅₃	—	2	4	no	—
95	C₂×C₄×D₅₃	—	2	8	no	—
96	C₂×C₄×D₅₃	—	4	2	no	—
97	C₂×C₄×D₅₃	—	4	4	no	—
98	C₂×C₄×D₅₃	—	4	4	no	—
99	C₂×C₄×D₅₃	C₄²	1	1	yes	characteristic
100	C₂×C₄×D₅₃	C₄²	1	1	yes	characteristic
101	C₂×C₄×D₅₃	C₂²⋊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	C₂²⋊C₄	1	1	yes	characteristic
105	C₂×C₄×D₅₃	C₄⋊C₄	1	1	yes	characteristic
106	C₂×C₄×D₅₃	C₄⋊C₄	1	1	yes	characteristic
107	C₂×C₄×D₅₃	C₄⋊C₄	1	1	yes	characteristic
108	C₂×C₄×D₅₃	C₄⋊C₄	1	1	yes	characteristic
109	C₂×C₄×D₅₃	C₄⋊C₄	1	1	yes	characteristic
110	C₂×C₄×D₅₃	C₄⋊C₄	1	1	yes	characteristic
111	M₁₆×C₅₃	—	4	1	no	—
112	M₁₆×C₅₃	—	4	1	no	—
113	C₂×C₈×C₅₃	—	2	2	no	—
114	C₂×C₈×C₅₃	—	2	2	no	—
115	C₄⋊C₄×C₅₃	—	2	4	no	—
116	C₄²×C₅₃	—	4	1	no	—
117	C₄²×C₅₃	—	4	1	no	—
118	C₄²×C₅₃	C₂²×C₄	1	1	yes	characteristic
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	C₄⋊Dic₅₃	—	2	4	no	—
123	#11719919	—	2	8	no	—
124	C₄×Dic₅₃	—	4	4	no	—
125	C₄×Dic₅₃	—	4	1	no	—
126	C₄×Dic₅₃	—	4	1	no	—
127	C₄×Dic₅₃	C₄²	1	1	yes	characteristic
128	C₄×Dic₅₃	C₄²	1	1	yes	characteristic
129	C₄×Dic₅₃	C₂²⋊C₄	1	1	yes	characteristic
130	C₄×Dic₅₃	C₂²⋊C₄	1	1	yes	characteristic
131	C₄×Dic₅₃	C₂²⋊C₄	1	1	yes	characteristic
132	C₄×Dic₅₃	C₂²⋊C₄	1	1	yes	characteristic
133	C₄×Dic₅₃	C₄⋊C₄	1	1	yes	characteristic
134	C₄×Dic₅₃	C₄⋊C₄	1	1	yes	characteristic
135	C₄×Dic₅₃	C₄⋊C₄	1	1	yes	characteristic
136	C₄×Dic₅₃	C₄⋊C₄	1	1	yes	characteristic
137	C₄×Dic₅₃	C₄⋊C₄	1	1	yes	characteristic
138	C₄×Dic₅₃	C₄⋊C₄	1	1	yes	characteristic
139	#11719917	—	4	1	no	—
140	#11719917	—	4	1	no	—
141	C₂×#88322	—	2	2	no	—
142	C₂×#88322	—	2	2	no	—
143	#11719912	—	4	4	no	—
144	C₂³×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
145	C₂²×D₅₃	—	2	1	no	—
146	C₂²×D₅₃	—	2	1	no	—
147	C₂²×D₅₃	—	2	2	no	—
148	C₂²×D₅₃	—	2	2	no	—
149	C₂²×D₅₃	#95	1	1	yes	characteristic
150	C₂²×D₅₃	#95	1	1	yes	characteristic
151	C₂×C₄×C₅₃	—	2	2	no	—
152	C₂×C₄×C₅₃	—	2	4	no	—
153	C₂×C₄×C₅₃	—	4	1	no	—
154	C₂×C₄×C₅₃	—	4	2	no	—
155	C₂×C₄×C₅₃	—	4	2	no	—
156	C₂×C₄×C₅₃	C₂×C₄²	1	1	yes	characteristic
157	C₂×C₄×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
158	C₂×C₄×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
159	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
160	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
161	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
162	C₂×C₄×C₅₃	AGL(1,ℤ/8)	1	1	yes	characteristic
163	C₂×C₄×C₅₃	AGL(1,ℤ/8)	1	1	yes	characteristic
164	C₂×C₄×C₅₃	#137	1	1	yes	characteristic
165	C₂×C₄×C₅₃	#137	1	1	yes	characteristic
166	C₂×Dic₅₃	—	2	1	no	—
167	C₂×Dic₅₃	—	2	1	no	—
168	C₂×Dic₅₃	—	2	2	no	—
169	C₂×Dic₅₃	—	2	4	no	—
170	C₂×Dic₅₃	—	4	1	no	—
171	C₂×Dic₅₃	—	4	2	no	—
172	C₂×Dic₅₃	—	4	2	no	—
173	C₂×Dic₅₃	#95	1	1	yes	characteristic
174	C₂×Dic₅₃	#95	1	1	yes	characteristic
175	C₂×Dic₅₃	#95	1	1	yes	characteristic
176	C₂×Dic₅₃	#95	1	1	yes	characteristic
177	C₂×Dic₅₃	#95	1	1	yes	characteristic
178	C₂×Dic₅₃	#95	1	1	yes	characteristic
179	C₄×D₅₃	—	2	2	no	—
180	C₄×D₅₃	—	2	2	no	—
181	C₄×D₅₃	—	2	2	no	—
182	C₄×D₅₃	—	2	1	no	—
183	C₄×D₅₃	—	2	1	no	—
184	C₄×D₅₃	—	2	1	no	—
185	C₄×D₅₃	—	2	1	no	—
186	C₄×D₅₃	—	2	2	no	—
187	C₄×D₅₃	—	2	1	no	—
188	C₄×D₅₃	—	2	1	no	—
189	C₄×D₅₃	—	2	1	no	—
190	C₄×D₅₃	—	2	1	no	—
191	C₄×D₅₃	—	2	2	no	—
192	C₄×D₅₃	—	4	1	no	—
193	C₄×D₅₃	—	4	1	no	—
194	C₄×D₅₃	—	4	16	no	—
195	C₄×D₅₃	#104	1	4	yes	—
196	C₈×C₅₃	—	4	2	no	—
197	C₈×C₅₃	—	4	2	no	—
198	#88322	—	4	2	no	—
199	#88322	—	4	2	no	—
200	#8098	—	53	1	no	maximal, Sylow
201	C₂²×C₅₃	—	2	1	no	—
202	C₂²×C₅₃	—	2	1	no	—
203	C₂²×C₅₃	C₂×#95	1	1	yes	characteristic
204	C₂²×C₅₃	#418	1	1	yes	characteristic
205	C₂²×C₅₃	#419	1	1	yes	characteristic
206	C₂×D₅₃	—	2	4	no	—
207	C₂×D₅₃	—	2	4	no	—
208	C₂×D₅₃	—	2	1	no	—
209	C₂×D₅₃	—	2	1	no	—
210	C₂×D₅₃	—	2	1	no	—
211	C₂×D₅₃	—	2	1	no	—
212	C₂×D₅₃	#339	1	2	yes	—
213	C₂×D₅₃	#344	1	2	yes	—
214	C₄×C₅₃	—	2	1	no	—
215	C₄×C₅₃	—	2	1	no	—
216	C₄×C₅₃	—	2	1	no	—
217	C₄×C₅₃	—	4	4	no	—
218	C₄×C₅₃	#417	1	1	yes	characteristic
219	C₄×C₅₃	#417	1	1	yes	characteristic
220	C₄×C₅₃	C₂×#104	1	2	yes	—
221	C₄×C₅₃	#428	1	1	yes	characteristic
222	C₄×C₅₃	#428	1	1	yes	characteristic
223	Dic₅₃	—	2	1	no	—
224	Dic₅₃	—	2	1	no	—
225	Dic₅₃	—	2	1	no	—
226	Dic₅₃	—	2	1	no	—
227	Dic₅₃	—	2	1	no	—
228	Dic₅₃	—	4	4	no	—
229	Dic₅₃	#339	1	2	yes	—
230	#2920	—	53	1	no	—
231	#2920	—	53	1	no	—
232	C₂×C₄×C₄⋊C₄	—	53	1	no	—
233	#1303	—	53	1	no	—
234	#1303	—	53	1	no	—
235	#1303	—	53	1	no	—
236	#1303	D₅₃	1	1	yes	characteristic, complemented
237	C₂×C₅₃	—	2	1	no	—
238	C₂×C₅₃	—	2	1	no	—
239	C₂×C₅₃	C₂×#339	1	1	yes	characteristic
240	C₂×C₅₃	#1736	1	1	yes	characteristic
241	C₂×C₅₃	C₂×#344	1	1	yes	characteristic
242	D₅₃	—	2	2	no	—
243	D₅₃	—	2	2	no	—
244	C₂²×C₄⋊C₄	—	106	1	no	—
245	C₂²×C₄²	—	53	1	no	—
246	C₂²×C₄²	—	106	1	no	—
247	#432	—	53	2	no	—
248	#432	—	53	2	no	—
249	#432	—	106	1	no	—
250	#432	—	106	1	no	—
251	#431	—	53	1	no	—
252	#431	—	53	2	no	—
253	#431	—	106	1	no	—
254	#431	—	106	1	no	—
255	#431	C₂×D₅₃	1	1	yes	characteristic
256	#423	—	53	1	no	—
257	#423	—	53	2	no	—
258	#423	C₂×D₅₃	1	1	yes	characteristic
259	#407	—	106	1	no	—
260	#407	—	106	1	no	—
261	C₄×M₁₆	—	106	1	no	—
262	C₄×M₁₆	—	106	1	no	—
263	C₄×C₄⋊C₄	—	53	1	no	—
264	C₄×C₄⋊C₄	—	53	2	no	—
265	C₄×C₄⋊C₄	—	106	1	no	—
266	C₄×C₄⋊C₄	—	106	1	no	—
267	C₄×C₄⋊C₄	C₂×D₅₃	1	1	yes	characteristic
268	C₂×#95	—	53	2	no	—
269	C₅₃	#8098	1	1	yes	characteristic, complemented, Sylow
270	C₂³×C₄	—	53	1	no	—
271	C₂³×C₄	—	53	1	no	—
272	C₂³×C₄	—	53	1	no	—
273	C₂³×C₄	—	106	2	no	—
274	C₂×M₁₆	—	2	1	no	—
275	C₂×M₁₆	—	106	2	no	—
276	C₂×M₁₆	—	106	1	no	—
277	C₂×C₄⋊C₄	—	2	1	no	—
278	C₂×C₄⋊C₄	—	106	8	no	—
279	C₂×C₄⋊C₄	—	106	2	no	—
280	C₂×C₄⋊C₄	—	106	1	no	—
281	C₂×C₄²	—	2	1	no	—
282	C₂×C₄²	—	53	1	no	—
283	C₂×C₄²	—	53	1	no	—
284	C₂×C₄²	—	53	1	no	—
285	C₂×C₄²	—	53	1	no	—
286	C₂×C₄²	—	53	1	no	—
287	C₂×C₄²	—	53	1	no	—
288	C₂×C₄²	—	53	1	no	—
289	C₂×C₄²	—	53	1	no	—
290	C₂×C₄²	—	53	1	no	—
291	C₂×C₄²	—	53	1	no	—
292	C₂×C₄²	—	53	1	no	—
293	C₂×C₄²	—	106	2	no	—
294	C₂×C₄²	—	106	1	no	—
295	C₂×C₄²	—	212	2	no	—
296	C₂×C₄²	—	212	2	no	—
297	C₂×C₄²	C₂²×D₅₃	1	1	yes	characteristic
298	C₄⋊C₈	—	2	1	no	—
299	C₄⋊C₈	—	53	2	no	—
300	C₄⋊C₈	—	53	2	no	—
301	C₄⋊C₈	—	106	2	no	—
302	C₄⋊C₈	—	106	1	no	—
303	C₄⋊C₈	C₄×D₅₃	1	2	yes	complemented
304	C₄⋊C₈	C₄×D₅₃	1	2	yes	complemented
305	C₂²⋊C₈	—	2	1	no	—
306	C₂²⋊C₈	—	106	2	no	—
307	C₂²⋊C₈	—	106	2	no	—
308	C₂²⋊C₈	—	106	2	no	—
309	C₂²⋊C₈	—	106	1	no	—
310	#97	—	106	2	no	—
311	#97	—	106	2	no	—
312	C₄×C₈	—	2	1	no	—
313	C₄×C₈	—	106	2	no	—
314	C₄×C₈	—	106	1	no	—
315	#95	—	53	2	no	—
316	#95	—	106	2	no	—
317	#95	C₄×D₅₃	1	2	yes	—
318	C₂⁴	—	53	1	no	—
319	C₂²×C₄	—	2	2	no	—
320	C₂²×C₄	—	53	1	no	—
321	C₂²×C₄	—	53	1	no	—
322	C₂²×C₄	—	53	1	no	—
323	C₂²×C₄	—	53	1	no	—
324	C₂²×C₄	—	53	1	no	—
325	C₂²×C₄	—	53	1	no	—
326	C₂²×C₄	—	53	1	no	—
327	C₂²×C₄	—	53	1	no	—
328	C₂²×C₄	—	53	1	no	—
329	C₂²×C₄	—	53	1	no	—
330	C₂²×C₄	—	53	1	no	—
331	C₂²×C₄	—	53	1	no	—
332	C₂²×C₄	—	53	1	no	—
333	C₂²×C₄	—	53	1	no	—
334	C₂²×C₄	—	53	1	no	—
335	C₂²×C₄	—	106	4	no	—
336	C₂²×C₄	—	106	8	no	—
337	C₂²×C₄	—	106	1	no	—
338	C₂²×C₄	—	106	1	no	—
339	C₂²×C₄	—	106	1	no	—
340	C₂²×C₄	—	106	1	no	—
341	C₂²×C₄	—	106	2	no	—
342	C₂²×C₄	—	212	2	no	—
343	C₂²×C₄	—	212	4	no	—
344	C₂²×C₄	—	212	4	no	—
345	C₂²×C₄	C₂×C₄×D₅₃	1	1	yes	characteristic
346	C₂²×C₄	D₄×D₅₃	1	1	yes	characteristic
347	C₂²×C₄	D₄×D₅₃	1	1	yes	characteristic
348	M₁₆	—	4	1	no	—
349	M₁₆	—	4	1	no	—
350	M₁₆	—	212	4	no	—
351	M₁₆	—	212	1	no	—
352	M₁₆	—	212	1	no	—
353	C₂×C₈	—	2	2	no	—
354	C₂×C₈	—	2	2	no	—
355	C₂×C₈	—	106	2	no	—
356	C₂×C₈	—	106	2	no	—
357	C₄⋊C₄	—	2	4	no	—
358	C₄⋊C₄	—	106	8	no	—
359	C₄⋊C₄	—	106	4	no	—
360	C₄²	—	4	1	no	—
361	C₄²	—	4	1	no	—
362	C₄²	—	53	1	no	—
363	C₄²	—	53	1	no	—
364	C₄²	—	53	1	no	—
365	C₄²	—	53	1	no	—
366	C₄²	—	53	1	no	—
367	C₄²	—	53	1	no	—
368	C₄²	—	53	1	no	—
369	C₄²	—	53	1	no	—
370	C₄²	—	53	1	no	—
371	C₄²	—	53	1	no	—
372	C₄²	—	53	1	no	—
373	C₄²	—	53	1	no	—
374	C₄²	—	212	4	no	—
375	C₄²	—	212	1	no	—
376	C₄²	—	212	1	no	—
377	C₄²	C₂×C₄×D₅₃	1	1	yes	characteristic
378	C₄²	C₂×C₄×D₅₃	1	1	yes	characteristic
379	C₄²	D₄×D₅₃	1	1	yes	characteristic
380	C₄²	Q₈×D₅₃	1	1	yes	characteristic
381	C₂³	—	53	1	no	—
382	C₂³	—	53	1	no	—
383	C₂³	—	106	2	no	—
384	C₂³	—	106	2	no	—
385	C₂³	—	106	1	no	—
386	C₂³	—	106	1	no	—
387	C₂³	C₂²⋊C₄×D₅₃	1	1	yes	characteristic
388	C₂×C₄	—	2	2	no	—
389	C₂×C₄	—	2	4	no	—
390	C₂×C₄	—	4	1	no	—
391	C₂×C₄	—	4	2	no	—
392	C₂×C₄	—	4	2	no	—
393	C₂×C₄	—	53	1	no	—
394	C₂×C₄	—	53	4	no	—
395	C₂×C₄	—	53	1	no	—
396	C₂×C₄	—	53	1	no	—
397	C₂×C₄	—	53	1	no	—
398	C₂×C₄	—	53	1	no	—
399	C₂×C₄	—	53	1	no	—
400	C₂×C₄	—	106	2	no	—
401	C₂×C₄	—	106	2	no	—
402	C₂×C₄	—	106	2	no	—
403	C₂×C₄	—	106	2	no	—
404	C₂×C₄	—	106	2	no	—
405	C₂×C₄	—	106	1	no	—
406	C₂×C₄	—	106	1	no	—
407	C₂×C₄	—	106	1	no	—
408	C₂×C₄	—	106	1	no	—
409	C₂×C₄	—	106	1	no	—
410	C₂×C₄	—	106	1	no	—
411	C₂×C₄	—	106	2	no	—
412	C₂×C₄	—	106	1	no	—
413	C₂×C₄	—	106	1	no	—
414	C₂×C₄	—	106	1	no	—
415	C₂×C₄	—	106	1	no	—
416	C₂×C₄	—	106	4	no	—
417	C₂×C₄	—	212	1	no	—
418	C₂×C₄	—	212	1	no	—
419	C₂×C₄	—	212	16	no	—
420	C₂×C₄	—	212	1	no	—
421	C₂×C₄	—	212	2	no	—
422	C₂×C₄	—	212	2	no	—
423	C₂×C₄	C₄²×D₅₃	1	1	yes	characteristic
424	C₂×C₄	C₂²⋊C₄×D₅₃	1	1	yes	characteristic
425	C₂×C₄	C₂²⋊C₄×D₅₃	1	1	yes	characteristic
426	C₂×C₄	C₄⋊C₄×D₅₃	1	1	yes	characteristic
427	C₂×C₄	C₄⋊C₄×D₅₃	1	1	yes	characteristic
428	C₂×C₄	C₄⋊C₄×D₅₃	1	1	yes	characteristic
429	C₂×C₄	#13133241	1	1	yes	characteristic
430	C₂×C₄	#13133244	1	1	yes	characteristic
431	C₂×C₄	#13133246	1	1	yes	characteristic
432	C₂×C₄	#13133249	1	1	yes	characteristic
433	C₈	—	4	2	no	—
434	C₈	—	4	2	no	—
435	C₈	—	212	2	no	—
436	C₈	—	212	2	no	—
437	C₂²	—	2	1	no	—
438	C₂²	—	2	1	no	—
439	C₂²	—	53	2	no	—
440	C₂²	—	53	2	no	—
441	C₂²	—	106	4	no	—
442	C₂²	—	106	4	no	—
443	C₂²	—	106	1	no	—
444	C₂²	—	106	1	no	—
445	C₂²	—	106	1	no	—
446	C₂²	—	106	1	no	—
447	C₂²	#95×D₅₃	1	1	yes	characteristic
448	C₂²	#16810626	1	1	yes	characteristic
449	C₂²	#16810630	1	1	yes	characteristic
450	C₄	—	2	1	no	—
451	C₄	—	2	1	no	—
452	C₄	—	2	1	no	—
453	C₄	—	4	4	no	—
454	C₄	—	53	2	no	—
455	C₄	—	106	1	no	—
456	C₄	—	106	1	no	—
457	C₄	—	106	1	no	—
458	C₄	—	106	1	no	—
459	C₄	—	106	1	no	—
460	C₄	—	212	4	no	—
461	C₄	#16810621	1	1	yes	characteristic
462	C₄	#16810623	1	1	yes	characteristic
463	C₄	#104×D₅₃	1	2	yes	—
464	C₄	#16810662	1	1	yes	characteristic
465	C₄	#16810664	1	1	yes	characteristic
466	C₂	—	2	1	no	—
467	C₂	—	2	1	no	—
468	C₂	—	106	2	no	—
469	C₂	—	106	2	no	—
470	C₂	#339×D₅₃	1	1	yes	characteristic
471	C₂	#20274433	1	1	yes	characteristic
472	C₂	#344×D₅₃	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂²⋊C₄×D₅₃
commutator	C₂×C₄×C₅₃	C₂×C₄²
Frattini	C₂×C₄²	C₂²×D₅₃
Fitting	C₅₃×#1303	C₂
socle	C₂²×C₅₃	C₂×#95
prime radical	C₂³×D₅₃	C₂²⋊C₄
cosocle	C₂×C₄²×C₅₃	C₂³
hypercenter	#1303	D₅₃
hypocenter	C₅₃	#8098
LCS	G ▷ C₂×C₄×C₅₃ ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #1303

Complements

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

N	H ≅ G/N	type
4. #20272618	C₂	semidirect
5. #20272617	C₂	semidirect
6. #20272616	C₂	semidirect
7. C₅₃×#1303	C₂	semidirect
12. #16810679	C₄	semidirect
13. #16810679	C₄	semidirect
17. #16810675	C₄	semidirect
20. #16810645	C₄	semidirect
58. C₄⋊#88322	C₂×C₄	semidirect
59. C₄⋊#88322	C₂×C₄	semidirect
76. C₄⋊C₈×C₅₃	C₂×C₄	semidirect
77. C₄⋊C₈×C₅₃	C₂×C₄	semidirect
236. #1303	D₅₃	semidirect
269. C₅₃	#8098	semidirect
303. C₄⋊C₈	C₄×D₅₃	semidirect
304. C₄⋊C₈	C₄×D₅₃	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
C₄×C₄⋊C₄×D₅₃	2
#20272618	2
#20272617	2
#20272616	2
C₅₃×#1303	2
#8098	53