Subgroups — #27406174

Order 13568.

708 conjugacy classes of subgroups.

The subgroup lattice has 490 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20283664	C₂	1	1	yes	maximal, characteristic, complemented
2	#20283283	C₂	1	1	yes	maximal, characteristic
3	#20275693	C₂	1	1	yes	maximal, characteristic, complemented
4	#20275491	C₂	1	1	yes	maximal, characteristic, complemented
5	#20275469	C₂	1	2	yes	maximal, complemented
6	#20275316	C₂	1	2	yes	maximal, complemented
7	#20274992	C₂	1	2	yes	maximal
8	#20274570	C₂	1	1	yes	maximal, characteristic, complemented
9	#20274562	C₂	1	2	yes	maximal
10	C₄×#13133144	C₂	1	1	yes	maximal, characteristic
11	C₅₃×#3076	C₂	1	1	yes	maximal, characteristic
12	C₂×#13133224	C₂²	1	2	yes	—
13	#16810754	—	2	2	no	—
14	#16810754	—	2	1	no	—
15	Q₈×#88322	—	2	1	no	—
16	C₂×#13133153	C₂²	1	2	yes	—
17	#16810650	—	2	2	no	—
18	#16810650	C₂²	1	2	yes	—
19	#16810650	C₂²	1	2	yes	—
20	#16810647	—	2	1	no	—
21	#16810647	C₂²	1	1	yes	characteristic
22	C₂×C₄⋊#88322	—	2	1	no	—
23	C₂×C₄⋊#88322	C₂²	1	1	yes	characteristic
24	C₂×#13133144	C₂²	1	1	yes	characteristic
25	C₂×#13133144	C₂²	1	1	yes	characteristic
26	#16810579	C₂²	1	2	yes	—
27	C₄×#11719917	C₂²	1	1	yes	characteristic
28	C₂×#13133127	C₂²	1	2	yes	—
29	C₄²×Dic₅₃	C₂²	1	1	yes	characteristic
30	#16810388	C₂²	1	2	yes	—
31	#16810382	C₂²	1	1	yes	characteristic
32	#16810382	C₂²	1	2	yes	—
33	#16810382	C₂²	1	2	yes	—
34	#16810382	C₂²	1	1	yes	characteristic
35	#16810377	C₂²	1	2	yes	—
36	C₅₃×#533	C₂²	1	1	yes	characteristic
37	C₄×#41×C₅₃	C₂²	1	1	yes	characteristic
38	C₅₃×#479	C₄	1	2	yes	complemented
39	C₅₃×#478	C₄	1	2	yes	complemented
40	C₅₃×#477	C₄	1	1	yes	characteristic, complemented
41	C₅₃×#476	C₄	1	1	yes	characteristic, complemented
42	C₅₃×#475	C₄	1	1	yes	characteristic, complemented
43	C₅₃×#474	C₄	1	1	yes	characteristic, complemented
44	C₅₃×#427	C₂²	1	2	yes	—
45	C₂×C₄⋊C₈×C₅₃	C₂²	1	1	yes	characteristic
46	C₅₃×#416	C₂²	1	2	yes	—
47	#13133224	—	2	4	no	—
48	#13133224	—	4	2	no	—
49	C₂×#11719917	—	2	2	no	—
50	C₂×#11719917	C₂³	1	1	yes	characteristic
51	C₂×#11719917	C₂³	1	2	yes	—
52	C₂×#11719917	C₂³	1	1	yes	characteristic
53	C₂²×#88322	—	2	2	no	—
54	C₂²×#88322	C₂³	1	2	yes	—
55	C₂×C₄×Dic₅₃	—	2	1	no	—
56	C₂×C₄×Dic₅₃	—	2	1	no	—
57	C₂×C₄×Dic₅₃	C₂³	1	1	yes	characteristic
58	C₂×C₄×Dic₅₃	C₂³	1	1	yes	characteristic
59	#13133153	—	4	2	no	—
60	C₄⋊#88322	—	2	1	no	—
61	C₄⋊#88322	—	2	1	no	—
62	C₄⋊#88322	—	2	2	no	—
63	C₄⋊#88322	—	2	2	no	—
64	C₄⋊#88322	—	2	2	no	—
65	C₄⋊#88322	—	2	2	no	—
66	C₄⋊#88322	D₄	1	1	yes	characteristic
67	C₄⋊#88322	D₄	1	2	yes	—
68	C₄⋊#88322	D₄	1	1	yes	characteristic
69	#13133144	—	2	2	no	—
70	#13133144	Q₈	1	4	yes	complemented
71	#13133128	—	2	1	no	—
72	#13133128	—	2	2	no	—
73	#13133127	—	4	2	no	—
74	C₄×#88322	—	2	1	no	—
75	C₄×#88322	—	2	2	no	—
76	C₂×#41×C₅₃	C₂³	1	1	yes	characteristic
77	C₂²×C₈×C₅₃	C₂³	1	2	yes	—
78	C₄⋊Q₈×C₅₃	C₂×C₄	1	1	yes	characteristic
79	C₄⋊Q₈×C₅₃	C₂×C₄	1	1	yes	characteristic
80	#125×C₅₃	C₂×C₄	1	2	yes	—
81	C₂²⋊Q₈×C₅₃	—	2	1	no	—
82	C₂²⋊Q₈×C₅₃	—	2	1	no	—
83	C₄×Q₈×C₅₃	—	2	1	no	—
84	C₄×Q₈×C₅₃	—	2	1	no	—
85	C₄×Q₈×C₅₃	C₂×C₄	1	1	yes	characteristic
86	C₄×Q₈×C₅₃	C₂×C₄	1	1	yes	characteristic
87	C₄×D₄×C₅₃	—	2	1	no	—
88	C₄×D₄×C₅₃	—	2	1	no	—
89	C₄×D₄×C₅₃	C₂×C₄	1	1	yes	characteristic
90	C₄×D₄×C₅₃	C₂×C₄	1	1	yes	characteristic
91	#117×C₅₃	—	2	1	no	—
92	#117×C₅₃	C₂³	1	1	yes	characteristic
93	#117×C₅₃	C₂³	1	2	yes	—
94	C₂×C₄²×C₅₃	—	2	1	no	—
95	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
96	#107×C₅₃	C₂×C₄	1	2	yes	—
97	#107×C₅₃	C₂×C₄	1	2	yes	—
98	#106×C₅₃	C₂×C₄	1	2	yes	—
99	#106×C₅₃	C₂×C₄	1	2	yes	—
100	C₄⋊C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
101	C₄⋊C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
102	C₄⋊C₈×C₅₃	C₂×C₄	1	2	yes	—
103	#103×C₅₃	C₂×C₄	1	2	yes	—
104	#103×C₅₃	C₂×C₄	1	2	yes	—
105	#102×C₅₃	C₂×C₄	1	2	yes	—
106	#102×C₅₃	C₂×C₄	1	2	yes	—
107	#41×C₅₃	—	2	1	no	—
108	#41×C₅₃	—	2	1	no	—
109	#41×C₅₃	C₂×Q₈	1	2	yes	—
110	#41×C₅₃	#41	1	2	yes	—
111	C₂×Q₈×C₅₃	—	2	1	no	—
112	C₂×Q₈×C₅₃	—	2	1	no	—
113	C₂×Q₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
114	C₂×D₄×C₅₃	—	2	1	no	—
115	C₂×D₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
116	C₂²×C₄×C₅₃	—	2	1	no	—
117	C₂²×C₄×C₅₃	—	2	1	no	—
118	C₂²×C₄×C₅₃	—	2	1	no	—
119	C₂²×C₄×C₅₃	#41	1	1	yes	characteristic
120	C₂²×C₄×C₅₃	#41	1	1	yes	characteristic
121	C₂²×C₄×C₅₃	C₂⁴	1	1	yes	characteristic
122	C₂²×Dic₅₃	—	2	1	no	—
123	C₂²×Dic₅₃	—	2	1	no	—
124	C₂×C₈×C₅₃	—	2	2	no	—
125	C₂×C₈×C₅₃	—	2	2	no	—
126	C₂×C₈×C₅₃	C₂²×C₄	1	2	yes	—
127	C₂×C₈×C₅₃	C₂²×C₄	1	2	yes	—
128	C₄⋊C₄×C₅₃	—	2	1	no	—
129	C₄⋊C₄×C₅₃	—	2	1	no	—
130	C₄⋊C₄×C₅₃	—	2	2	no	—
131	C₄⋊C₄×C₅₃	—	2	2	no	—
132	C₄⋊C₄×C₅₃	—	2	1	no	—
133	C₄⋊C₄×C₅₃	—	2	1	no	—
134	C₄⋊C₄×C₅₃	C₂²×C₄	1	2	yes	—
135	C₄⋊C₄×C₅₃	C₂²×C₄	1	2	yes	—
136	C₄⋊C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
137	C₄⋊C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
138	C₂²⋊C₄×C₅₃	—	2	1	no	—
139	C₂²⋊C₄×C₅₃	—	2	2	no	—
140	C₂²⋊C₄×C₅₃	—	2	1	no	—
141	C₂²⋊C₄×C₅₃	—	2	1	no	—
142	C₄²×C₅₃	—	2	1	no	—
143	C₄²×C₅₃	—	2	1	no	—
144	C₄²×C₅₃	—	2	1	no	—
145	C₄²×C₅₃	—	2	2	no	—
146	C₄²×C₅₃	C₂²×C₄	1	1	yes	characteristic
147	C₄²×C₅₃	C₂²×C₄	1	1	yes	characteristic
148	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
149	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
150	C₄×Dic₅₃	—	2	1	no	—
151	C₄×Dic₅₃	—	2	1	no	—
152	C₄×Dic₅₃	—	2	1	no	—
153	C₄×Dic₅₃	—	4	2	no	—
154	C₄×Dic₅₃	—	4	2	no	—
155	C₄×Dic₅₃	C₂×Q₈	1	2	yes	—
156	#11719917	—	2	1	no	—
157	#11719917	—	2	1	no	—
158	#11719917	—	2	2	no	—
159	#11719917	—	2	1	no	—
160	#11719917	—	2	2	no	—
161	#11719917	—	4	4	no	—
162	#11719917	C₂×Q₈	1	2	yes	—
163	C₂×#88322	—	2	1	no	—
164	C₂×#88322	—	2	2	no	—
165	C₂×#88322	—	2	2	no	—
166	C₂×#88322	—	2	1	no	—
167	C₂×#88322	—	2	2	no	—
168	C₂×#88322	—	4	4	no	—
169	C₂×#88322	C₂×D₄	1	2	yes	—
170	C₂×#88322	C₂×D₄	1	2	yes	—
171	C₂³×C₅₃	—	2	1	no	—
172	C₂³×C₅₃	C₂×#41	1	1	yes	characteristic
173	Q₈×C₅₃	—	2	1	no	—
174	Q₈×C₅₃	—	4	2	no	—
175	Q₈×C₅₃	—	8	1	no	—
176	Q₈×C₅₃	C₄×Q₈	1	2	yes	—
177	D₄×C₅₃	—	2	2	no	—
178	D₄×C₅₃	—	2	2	no	—
179	D₄×C₅₃	—	2	1	no	—
180	D₄×C₅₃	C₄×Q₈	1	2	yes	—
181	C₂×C₄×C₅₃	—	2	1	no	—
182	C₂×C₄×C₅₃	—	2	2	no	—
183	C₂×C₄×C₅₃	—	2	2	no	—
184	C₂×C₄×C₅₃	—	2	2	no	—
185	C₂×C₄×C₅₃	—	2	1	no	—
186	C₂×C₄×C₅₃	—	2	1	no	—
187	C₂×C₄×C₅₃	—	2	2	no	—
188	C₂×C₄×C₅₃	—	2	2	no	—
189	C₂×C₄×C₅₃	—	4	1	no	—
190	C₂×C₄×C₅₃	—	4	2	no	—
191	C₂×C₄×C₅₃	—	8	1	no	—
192	C₂×C₄×C₅₃	C₄×D₄	1	1	yes	characteristic
193	C₂×C₄×C₅₃	C₄×D₄	1	1	yes	characteristic
194	C₂×C₄×C₅₃	C₄×D₄	1	1	yes	characteristic
195	C₂×C₄×C₅₃	C₄×D₄	1	1	yes	characteristic
196	C₂×C₄×C₅₃	C₂³×C₄	1	1	yes	characteristic
197	C₂×C₄×C₅₃	C₂²×D₄	1	1	yes	characteristic
198	C₂×C₄×C₅₃	C₂²×Q₈	1	1	yes	characteristic
199	C₂×C₄×C₅₃	C₂×#41	1	1	yes	characteristic
200	C₂×C₄×C₅₃	H(5,𝔽₂)	1	1	yes	characteristic
201	C₂×C₄×C₅₃	2₋¹⁺⁴	1	1	yes	characteristic
202	C₂×Dic₅₃	—	2	2	no	—
203	C₂×Dic₅₃	—	2	2	no	—
204	C₂×Dic₅₃	—	4	1	no	—
205	C₂×Dic₅₃	—	4	1	no	—
206	C₂×Dic₅₃	—	4	1	no	—
207	C₂×Dic₅₃	—	4	1	no	—
208	C₈×C₅₃	—	2	2	no	—
209	C₈×C₅₃	—	2	2	no	—
210	#88322	—	2	2	no	—
211	#88322	—	2	2	no	—
212	#88322	—	2	2	no	—
213	#88322	—	4	2	no	—
214	#88322	—	4	1	no	—
215	#22983	—	53	1	no	maximal, Sylow
216	C₂²×C₅₃	—	2	1	no	—
217	C₂²×C₅₃	—	2	1	no	—
218	C₂²×C₅₃	—	2	2	no	—
219	C₂²×C₅₃	—	4	1	no	—
220	C₂²×C₅₃	—	4	1	no	—
221	C₂²×C₅₃	C₂×C₄×D₄	1	1	yes	characteristic
222	C₂²×C₅₃	#552	1	1	yes	characteristic
223	C₂²×C₅₃	#554	1	1	yes	characteristic
224	C₄×C₅₃	—	2	2	no	—
225	C₄×C₅₃	—	2	1	no	—
226	C₄×C₅₃	—	2	1	no	—
227	C₄×C₅₃	—	2	1	no	—
228	C₄×C₅₃	—	4	1	no	—
229	C₄×C₅₃	—	4	2	no	—
230	C₄×C₅₃	—	4	1	no	—
231	C₄×C₅₃	—	4	2	no	—
232	C₄×C₅₃	#444	1	2	yes	—
233	C₄×C₅₃	C₂×C₄×Q₈	1	1	yes	characteristic
234	C₄×C₅₃	#520	1	1	yes	characteristic
235	C₄×C₅₃	#546	1	1	yes	characteristic
236	C₄×C₅₃	D₄×Q₈	1	1	yes	characteristic
237	Dic₅₃	—	4	2	no	—
238	Dic₅₃	—	4	2	no	—
239	#3076	D₅₃	1	1	yes	characteristic
240	#2962	—	53	1	no	—
241	#2906	—	53	1	no	—
242	#1994	—	53	1	no	—
243	#1950	—	53	1	no	—
244	#1946	—	53	2	no	—
245	#1922	—	53	2	no	—
246	#1866	—	53	2	no	—
247	#1767	—	53	1	no	—
248	#1765	—	53	2	no	—
249	C₄×#104	—	53	1	no	—
250	C₂×C₅₃	—	2	1	no	—
251	C₂×C₅₃	—	2	1	no	—
252	C₂×C₅₃	—	4	2	no	—
253	C₂×C₅₃	#2360	1	1	yes	characteristic
254	C₂×C₅₃	C₂×#444	1	1	yes	characteristic
255	C₂×C₅₃	#2956	1	1	yes	characteristic
256	C₂×#131	—	53	2	no	—
257	#533	C₂×D₅₃	1	1	yes	characteristic
258	C₄×#41	C₂×D₅₃	1	1	yes	characteristic
259	#479	Dic₅₃	1	2	yes	complemented
260	#478	Dic₅₃	1	2	yes	complemented
261	#477	Dic₅₃	1	1	yes	characteristic, complemented
262	#476	Dic₅₃	1	1	yes	characteristic, complemented
263	#475	Dic₅₃	1	1	yes	characteristic, complemented
264	#474	Dic₅₃	1	1	yes	characteristic, complemented
265	#446	—	106	2	no	—
266	#446	—	106	1	no	—
267	C₈×Q₈	—	106	1	no	—
268	C₂×#108	—	53	2	no	—
269	#427	C₂×D₅₃	1	2	yes	—
270	#424	—	53	2	no	—
271	#424	—	53	2	no	—
272	#424	—	106	2	no	—
273	#423	—	53	1	no	—
274	#423	—	106	1	no	—
275	C₂×C₄⋊C₈	—	53	1	no	—
276	C₂×C₄⋊C₈	—	106	1	no	—
277	C₂×C₄⋊C₈	C₂×D₅₃	1	1	yes	characteristic
278	C₂×#104	—	53	1	no	—
279	C₂×#104	—	53	1	no	—
280	#416	C₂×D₅₃	1	2	yes	—
281	#405	—	53	2	no	—
282	C₄×M₁₆	—	53	1	no	—
283	C₂×#97	—	53	2	no	—
284	C₄³	—	53	1	no	—
285	#341	—	53	2	no	—
286	#339	—	53	2	no	—
287	#339	—	53	1	no	—
288	#339	—	53	1	no	—
289	#339	—	53	2	no	—
290	#336	—	53	2	no	—
291	C₅₃	#22983	1	1	yes	characteristic, complemented, Sylow
292	C₂×#41	C₂²×D₅₃	1	1	yes	characteristic
293	#131	—	106	4	no	—
294	#131	—	212	2	no	—
295	C₂×M₁₆	—	53	1	no	—
296	C₂×M₁₆	—	53	2	no	—
297	C₂×M₁₆	—	53	1	no	—
298	C₂×M₁₆	—	106	2	no	—
299	C₂²×C₈	—	53	2	no	—
300	C₂²×C₈	—	106	2	no	—
301	C₂²×C₈	C₂²×D₅₃	1	2	yes	—
302	C₄⋊Q₈	C₂×Dic₅₃	1	1	yes	characteristic
303	C₄⋊Q₈	C₂×Dic₅₃	1	1	yes	characteristic
304	#125	C₂×Dic₅₃	1	2	yes	—
305	C₂²⋊Q₈	—	2	1	no	—
306	C₂²⋊Q₈	—	2	1	no	—
307	C₄×Q₈	—	2	1	no	—
308	C₄×Q₈	—	2	1	no	—
309	C₄×Q₈	C₂×Dic₅₃	1	1	yes	characteristic
310	C₄×Q₈	C₂×Dic₅₃	1	1	yes	characteristic
311	C₄×D₄	—	2	1	no	—
312	C₄×D₄	—	2	1	no	—
313	C₄×D₄	C₂×Dic₅₃	1	1	yes	characteristic
314	C₄×D₄	C₂×Dic₅₃	1	1	yes	characteristic
315	#117	—	2	1	no	—
316	#117	C₂²×D₅₃	1	1	yes	characteristic
317	#117	C₂²×D₅₃	1	2	yes	—
318	C₂×C₄²	—	2	1	no	—
319	C₂×C₄²	—	53	1	no	—
320	C₂×C₄²	—	53	1	no	—
321	C₂×C₄²	—	106	1	no	—
322	C₂×C₄²	—	106	1	no	—
323	C₂×C₄²	C₂²×D₅₃	1	1	yes	characteristic
324	#108	—	212	2	no	—
325	#107	C₂×Dic₅₃	1	2	yes	—
326	#107	C₂×Dic₅₃	1	2	yes	—
327	#106	C₂×Dic₅₃	1	2	yes	—
328	#106	C₂×Dic₅₃	1	2	yes	—
329	C₄⋊C₈	—	53	1	no	—
330	C₄⋊C₈	—	53	1	no	—
331	C₄⋊C₈	—	53	2	no	—
332	C₄⋊C₈	—	106	2	no	—
333	C₄⋊C₈	—	106	2	no	—
334	C₄⋊C₈	—	106	2	no	—
335	C₄⋊C₈	—	106	2	no	—
336	C₄⋊C₈	—	106	1	no	—
337	C₄⋊C₈	—	106	1	no	—
338	C₄⋊C₈	C₂×Dic₅₃	1	1	yes	characteristic
339	C₄⋊C₈	C₂×Dic₅₃	1	1	yes	characteristic
340	C₄⋊C₈	C₂×Dic₅₃	1	2	yes	—
341	#104	—	53	4	no	—
342	#104	—	106	2	no	—
343	#103	C₂×Dic₅₃	1	2	yes	—
344	#103	C₂×Dic₅₃	1	2	yes	—
345	#102	C₂×Dic₅₃	1	2	yes	—
346	#102	C₂×Dic₅₃	1	2	yes	—
347	#97	—	106	2	no	—
348	#97	—	106	1	no	—
349	#97	—	212	2	no	—
350	C₄×C₈	—	106	2	no	—
351	C₄×C₈	—	106	1	no	—
352	#41	—	2	1	no	—
353	#41	—	2	1	no	—
354	#41	Q₈×D₅₃	1	2	yes	—
355	#41	#11719949	1	2	yes	—
356	C₂×Q₈	—	2	1	no	—
357	C₂×Q₈	—	2	1	no	—
358	C₂×Q₈	C₂²×Dic₅₃	1	1	yes	characteristic
359	C₂×D₄	—	2	1	no	—
360	C₂×D₄	C₂²×Dic₅₃	1	1	yes	characteristic
361	C₂²×C₄	—	2	1	no	—
362	C₂²×C₄	—	2	1	no	—
363	C₂²×C₄	—	2	1	no	—
364	C₂²×C₄	—	106	1	no	—
365	C₂²×C₄	—	106	1	no	—
366	C₂²×C₄	#11719947	1	1	yes	characteristic
367	C₂²×C₄	#11719947	1	1	yes	characteristic
368	C₂²×C₄	C₂³×D₅₃	1	1	yes	characteristic
369	M₁₆	—	53	2	no	—
370	M₁₆	—	106	1	no	—
371	M₁₆	—	106	1	no	—
372	M₁₆	—	106	2	no	—
373	M₁₆	—	106	1	no	—
374	M₁₆	—	106	2	no	—
375	M₁₆	—	212	4	no	—
376	C₂×C₈	—	2	2	no	—
377	C₂×C₈	—	2	2	no	—
378	C₂×C₈	—	53	2	no	—
379	C₂×C₈	—	53	2	no	—
380	C₂×C₈	—	106	1	no	—
381	C₂×C₈	—	106	2	no	—
382	C₂×C₈	—	106	2	no	—
383	C₂×C₈	—	106	1	no	—
384	C₂×C₈	—	106	2	no	—
385	C₂×C₈	—	212	4	no	—
386	C₂×C₈	C₂²×Dic₅₃	1	2	yes	—
387	C₂×C₈	C₂²×Dic₅₃	1	2	yes	—
388	C₄⋊C₄	—	2	1	no	—
389	C₄⋊C₄	—	2	1	no	—
390	C₄⋊C₄	—	2	2	no	—
391	C₄⋊C₄	—	2	2	no	—
392	C₄⋊C₄	—	2	1	no	—
393	C₄⋊C₄	—	2	1	no	—
394	C₄⋊C₄	C₂²×Dic₅₃	1	2	yes	—
395	C₄⋊C₄	C₂²×Dic₅₃	1	2	yes	—
396	C₄⋊C₄	C₂²×Dic₅₃	1	1	yes	characteristic
397	C₄⋊C₄	C₂²×Dic₅₃	1	1	yes	characteristic
398	C₂²⋊C₄	—	2	1	no	—
399	C₂²⋊C₄	—	2	2	no	—
400	C₂²⋊C₄	—	2	1	no	—
401	C₂²⋊C₄	—	2	1	no	—
402	C₄²	—	2	1	no	—
403	C₄²	—	2	1	no	—
404	C₄²	—	2	2	no	—
405	C₄²	—	2	1	no	—
406	C₄²	—	53	2	no	—
407	C₄²	—	106	1	no	—
408	C₄²	—	106	1	no	—
409	C₄²	—	106	1	no	—
410	C₄²	—	212	2	no	—
411	C₄²	—	212	2	no	—
412	C₄²	D₄×D₅₃	1	1	yes	characteristic
413	C₄²	D₄×D₅₃	1	1	yes	characteristic
414	C₄²	C₂²×Dic₅₃	1	1	yes	characteristic
415	C₄²	C₂²×Dic₅₃	1	1	yes	characteristic
416	C₂³	—	2	1	no	—
417	C₂³	C₂×#11719947	1	1	yes	characteristic
418	Q₈	—	2	1	no	—
419	Q₈	—	4	2	no	—
420	Q₈	—	8	1	no	—
421	Q₈	Q₈×Dic₅₃	1	2	yes	—
422	D₄	—	2	2	no	—
423	D₄	—	2	2	no	—
424	D₄	—	2	1	no	—
425	D₄	Q₈×Dic₅₃	1	2	yes	—
426	C₂×C₄	—	2	1	no	—
427	C₂×C₄	—	2	2	no	—
428	C₂×C₄	—	2	2	no	—
429	C₂×C₄	—	2	2	no	—
430	C₂×C₄	—	2	1	no	—
431	C₂×C₄	—	2	1	no	—
432	C₂×C₄	—	2	2	no	—
433	C₂×C₄	—	2	2	no	—
434	C₂×C₄	—	4	1	no	—
435	C₂×C₄	—	4	2	no	—
436	C₂×C₄	—	8	1	no	—
437	C₂×C₄	—	106	2	no	—
438	C₂×C₄	—	106	2	no	—
439	C₂×C₄	—	212	1	no	—
440	C₂×C₄	—	212	1	no	—
441	C₂×C₄	—	212	1	no	—
442	C₂×C₄	—	212	1	no	—
443	C₂×C₄	D₄×Dic₅₃	1	1	yes	characteristic
444	C₂×C₄	D₄×Dic₅₃	1	1	yes	characteristic
445	C₂×C₄	D₄×Dic₅₃	1	1	yes	characteristic
446	C₂×C₄	D₄×Dic₅₃	1	1	yes	characteristic
447	C₂×C₄	C₂³×Dic₅₃	1	1	yes	characteristic
448	C₂×C₄	C₂×D₄×D₅₃	1	1	yes	characteristic
449	C₂×C₄	C₂×Q₈×D₅₃	1	1	yes	characteristic
450	C₂×C₄	C₂×#11719949	1	1	yes	characteristic
451	C₂×C₄	#13133263	1	1	yes	characteristic
452	C₂×C₄	#13133264	1	1	yes	characteristic
453	C₈	—	2	2	no	—
454	C₈	—	2	2	no	—
455	C₈	—	106	2	no	—
456	C₈	—	106	2	no	—
457	C₈	—	106	2	no	—
458	C₈	—	212	2	no	—
459	C₈	—	212	1	no	—
460	C₂²	—	2	1	no	—
461	C₂²	—	2	1	no	—
462	C₂²	—	2	2	no	—
463	C₂²	—	4	1	no	—
464	C₂²	—	4	1	no	—
465	C₂²	C₂×D₄×Dic₅₃	1	1	yes	characteristic
466	C₂²	#16811315	1	1	yes	characteristic
467	C₂²	#16811331	1	1	yes	characteristic
468	C₄	—	2	1	no	—
469	C₄	—	2	2	no	—
470	C₄	—	2	1	no	—
471	C₄	—	2	1	no	—
472	C₄	—	4	1	no	—
473	C₄	—	4	2	no	—
474	C₄	—	4	1	no	—
475	C₄	—	4	2	no	—
476	C₄	—	212	2	no	—
477	C₄	—	212	2	no	—
478	C₄	#16810745	1	2	yes	—
479	C₄	C₂×Q₈×Dic₅₃	1	1	yes	characteristic
480	C₄	#16811106	1	1	yes	characteristic
481	C₄	#16811264	1	1	yes	characteristic
482	C₄	#16811286	1	1	yes	characteristic
483	C₂	—	2	1	no	—
484	C₂	—	2	1	no	—
485	C₂	—	4	2	no	—
486	C₂	#20277724	1	1	yes	characteristic
487	C₂	C₂×#16810745	1	1	yes	characteristic
488	C₂	#20283625	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂×D₄×D₅₃
commutator	C₂×C₄×C₅₃	C₂³×C₄
Frattini	C₂²×C₄	C₂³×D₅₃
Fitting	C₅₃×#3076	C₂
socle	C₂²×C₅₃	C₂×C₄×D₄
prime radical	C₂×#41×C₅₃	C₂³
cosocle	C₂²×C₄×C₅₃	C₂⁴
hypercenter	#3076	D₅₃
hypocenter	C₅₃	#22983
LCS	G ▷ C₂×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #3076

Complements

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

N	H ≅ G/N	type
1. #20283664	C₂	semidirect
3. #20275693	C₂	semidirect
4. #20275491	C₂	semidirect
5. #20275469	C₂	semidirect
6. #20275316	C₂	semidirect
8. #20274570	C₂	semidirect
38. C₅₃×#479	C₄	semidirect
39. C₅₃×#478	C₄	semidirect
40. C₅₃×#477	C₄	semidirect
41. C₅₃×#476	C₄	semidirect
42. C₅₃×#475	C₄	semidirect
43. C₅₃×#474	C₄	semidirect
70. #13133144	Q₈	semidirect
259. #479	Dic₅₃	semidirect
260. #478	Dic₅₃	semidirect
261. #477	Dic₅₃	semidirect
262. #476	Dic₅₃	semidirect
263. #475	Dic₅₃	semidirect
264. #474	Dic₅₃	semidirect
291. C₅₃	#22983	semidirect

Maximal subgroups

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

subgroup	index
#20283664	2
#20283283	2
#20275693	2
#20275491	2
#20275469	2
#20275316	2
#20274992	2
#20274570	2
#20274562	2
C₄×#13133144	2
C₅₃×#3076	2
#22983	53