Subgroups — #27354126

Order 13568.

624 conjugacy classes of subgroups.

The subgroup lattice has 454 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20284780	C₂	1	1	yes	maximal, characteristic
2	C₈×#11719945	C₂	1	1	yes	maximal, characteristic
3	#20273563	C₂	1	2	yes	maximal, complemented
4	#20273549	C₂	1	2	yes	maximal, complemented
5	#20273532	C₂	1	2	yes	maximal, complemented
6	#20273525	C₂	1	2	yes	maximal, complemented
7	#20273506	C₂	1	1	yes	maximal, characteristic, complemented
8	#20273152	C₂	1	1	yes	maximal, characteristic
9	#20273136	C₂	1	1	yes	maximal, characteristic, complemented
10	#20273004	C₂	1	1	yes	maximal, characteristic, complemented
11	C₅₃×#2960	C₂	1	1	yes	maximal, characteristic, complemented
12	C₄×#11719945	C₂²	1	1	yes	characteristic
13	#16810961	C₄	1	2	yes	—
14	#16810905	C₄	1	1	yes	characteristic
15	#16810900	C₄	1	1	yes	characteristic
16	#16810830	C₄	1	1	yes	characteristic
17	#16810825	C₄	1	1	yes	characteristic
18	#16810783	C₄	1	2	yes	—
19	C₈×Dic₁₀₆	C₂²	1	1	yes	characteristic
20	C₈×Dic₁₀₆	C₂²	1	1	yes	characteristic
21	C₄×#11719925	C₂²	1	2	yes	—
22	C₄×#11719924	C₂²	1	2	yes	—
23	C₈×D₂₁₂	C₂²	1	1	yes	characteristic
24	C₈×D₂₁₂	C₂²	1	1	yes	characteristic
25	C₈×C₂²⋊D₅₃	—	2	1	no	—
26	C₈×C₂²⋊D₅₃	—	2	1	no	—
27	#16810676	C₂²	1	1	yes	characteristic
28	#16810676	C₂²	1	1	yes	characteristic
29	#16810674	—	2	1	no	—
30	C₄×#11719917	C₂²	1	1	yes	characteristic
31	C₄×C₈×D₅₃	—	2	1	no	—
32	#16810375	C₂²	1	2	yes	—
33	#16810373	C₂²	1	2	yes	—
34	Q₈⋊#88322	C₂²	1	2	yes	—
35	Q₈⋊#88322	C₂²	1	2	yes	—
36	#16810353	C₂²	1	2	yes	—
37	#16810350	C₂²	1	2	yes	—
38	#16810345	C₂²	1	2	yes	—
39	#16810344	C₂²	1	2	yes	—
40	C₅₃×#533	C₂²	1	1	yes	characteristic
41	C₅₃×#446	—	2	1	no	—
42	C₅₃×#446	C₂²	1	2	yes	—
43	C₈×Q₈×C₅₃	—	2	1	no	—
44	C₈×Q₈×C₅₃	C₂²	1	2	yes	—
45	C₅₃×#424	—	2	1	no	—
46	C₅₃×#423	C₂²	1	1	yes	characteristic
47	C₅₃×#405	—	2	1	no	—
48	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
49	C₂×#11719945	C₂×C₄	1	1	yes	characteristic
50	C₂×#11719925	C₂×C₄	1	2	yes	—
51	C₂×#11719924	C₂×C₄	1	2	yes	—
52	C₂×#11719917	—	2	1	no	—
53	C₂×C₈×D₅₃	—	2	1	no	—
54	C₂×C₈×D₅₃	—	2	1	no	—
55	C₄×Dic₁₀₆	C₂×C₄	1	1	yes	characteristic
56	C₄×Dic₁₀₆	C₂³	1	1	yes	characteristic
57	C₄×D₂₁₂	C₂×C₄	1	1	yes	characteristic
58	C₄×D₂₁₂	C₂³	1	1	yes	characteristic
59	C₄×C₂²⋊D₅₃	—	2	1	no	—
60	C₄×C₂²⋊D₅₃	—	2	1	no	—
61	#13133176	C₂×C₄	1	1	yes	characteristic
62	#13133174	—	2	1	no	—
63	C₄²×D₅₃	—	2	1	no	—
64	#13133150	C₂×C₄	1	2	yes	—
65	#13133148	C₂×C₄	1	2	yes	—
66	C₄⋊#88322	C₂×C₄	1	1	yes	characteristic
67	C₄⋊#88322	C₂×C₄	1	1	yes	characteristic
68	C₄⋊#88322	C₂³	1	1	yes	characteristic
69	C₄⋊#88322	C₂³	1	1	yes	characteristic
70	#13133146	—	2	1	no	—
71	#13133146	—	2	1	no	—
72	#13133141	C₂×C₄	1	2	yes	—
73	#13133141	C₂×C₄	1	2	yes	—
74	#13133140	C₂×C₄	1	2	yes	—
75	#13133137	C₂×C₄	1	2	yes	—
76	C₂²⋊#88322	—	2	1	no	—
77	C₂²⋊#88322	C₂×C₄	1	2	yes	—
78	#13133129	—	2	1	no	—
79	#13133129	—	2	1	no	—
80	#13133128	—	2	1	no	—
81	C₄×#88322	—	2	1	no	—
82	C₄×#88322	C₂³	1	2	yes	—
83	C₄×#88322	C₂³	1	2	yes	—
84	C₈×Dic₅₃	—	2	1	no	—
85	C₈×Dic₅₃	—	2	1	no	—
86	C₂×M₁₆×C₅₃	—	2	1	no	—
87	C₂²×C₈×C₅₃	D₄	1	1	yes	characteristic
88	C₂²×C₈×C₅₃	D₄	1	1	yes	characteristic
89	C₄⋊Q₈×C₅₃	C₂×C₄	1	1	yes	characteristic
90	C₄⋊Q₈×C₅₃	C₂×C₄	1	1	yes	characteristic
91	#125×C₅₃	—	2	1	no	—
92	C₂²⋊Q₈×C₅₃	—	2	1	no	—
93	C₂²⋊Q₈×C₅₃	C₂×C₄	1	2	yes	—
94	C₄×Q₈×C₅₃	—	2	1	no	—
95	C₄×Q₈×C₅₃	C₂³	1	2	yes	—
96	#117×C₅₃	—	2	1	no	—
97	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
98	C₄⋊C₈×C₅₃	—	2	2	no	—
99	C₄⋊C₈×C₅₃	—	2	2	no	—
100	C₄⋊C₈×C₅₃	—	2	1	no	—
101	C₄⋊C₈×C₅₃	C₂³	1	2	yes	—
102	#97×C₅₃	—	2	2	no	—
103	C₄×C₈×C₅₃	—	2	2	no	—
104	C₄×C₈×C₅₃	D₄	1	1	yes	characteristic
105	C₄×C₈×C₅₃	D₄	1	1	yes	characteristic
106	C₄×C₈×C₅₃	C₂³	1	1	yes	characteristic
107	C₄×C₈×C₅₃	C₂³	1	1	yes	characteristic
108	C₂×Q₈×C₅₃	—	2	1	no	—
109	C₂×Q₈×C₅₃	C₂²×C₄	1	2	yes	—
110	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
111	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
112	C₂²×C₄×C₅₃	#41	1	1	yes	characteristic
113	C₂×C₂²⋊D₅₃	—	2	1	no	—
114	#11719945	—	2	2	no	—
115	#11719945	—	2	2	no	—
116	C₂×D₂₁₂	C₂²×C₄	1	1	yes	characteristic
117	C₂×C₄×D₅₃	—	2	1	no	—
118	C₂×C₄×D₅₃	—	2	1	no	—
119	C₂×C₄×D₅₃	—	2	1	no	—
120	C₂×Dic₁₀₆	C₂²×C₄	1	1	yes	characteristic
121	M₁₆×C₅₃	—	4	2	no	—
122	C₂×C₈×C₅₃	—	2	2	no	—
123	C₂×C₈×C₅₃	—	2	2	no	—
124	C₂×C₈×C₅₃	—	4	1	no	—
125	C₂×C₈×C₅₃	C₂×D₄	1	1	yes	characteristic
126	C₂×C₈×C₅₃	C₂×D₄	1	1	yes	characteristic
127	C₂×C₈×C₅₃	C₂×D₄	1	1	yes	characteristic
128	C₂×C₈×C₅₃	C₂×D₄	1	1	yes	characteristic
129	C₄⋊C₄×C₅₃	—	2	1	no	—
130	C₄⋊C₄×C₅₃	—	2	1	no	—
131	C₄⋊C₄×C₅₃	—	2	1	no	—
132	C₄⋊C₄×C₅₃	—	2	2	no	—
133	C₄⋊C₄×C₅₃	—	2	2	no	—
134	C₄⋊C₄×C₅₃	C₂²×C₄	1	2	yes	—
135	C₂²⋊C₄×C₅₃	—	2	2	no	—
136	C₄²×C₅₃	—	2	2	no	—
137	C₄²×C₅₃	C₂²×C₄	1	1	yes	characteristic
138	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
139	C₄²×C₅₃	#41	1	1	yes	characteristic
140	C₄²×C₅₃	C₂⁴	1	1	yes	characteristic
141	C₂²⋊Dic₅₃	—	2	1	no	—
142	#11719925	—	2	4	no	—
143	#11719924	—	2	4	no	—
144	#11719921	—	2	1	no	—
145	#11719921	—	2	1	no	—
146	C₄⋊Dic₅₃	C₂²×C₄	1	1	yes	characteristic
147	C₄⋊Dic₅₃	C₂²×C₄	1	1	yes	characteristic
148	#11719919	—	2	1	no	—
149	#11719919	—	2	1	no	—
150	C₄×Dic₅₃	—	2	1	no	—
151	C₄×Dic₅₃	—	2	1	no	—
152	C₄×Dic₅₃	—	2	1	no	—
153	#11719917	—	4	2	no	—
154	C₂×#88322	—	2	1	no	—
155	C₂×#88322	—	2	1	no	—
156	C₂×#88322	—	2	2	no	—
157	C₂×#88322	C₂²×C₄	1	2	yes	—
158	C₂×#88322	C₂²×C₄	1	2	yes	—
159	C₈×D₅₃	—	4	2	no	—
160	C₈×D₅₃	—	4	2	no	—
161	C₂³×C₅₃	C₄×D₄	1	1	yes	characteristic
162	C₂²×D₅₃	—	2	1	no	—
163	Q₈×C₅₃	—	2	2	no	—
164	Q₈×C₅₃	—	2	2	no	—
165	Q₈×C₅₃	—	4	2	no	—
166	C₂×C₄×C₅₃	—	2	1	no	—
167	C₂×C₄×C₅₃	—	2	1	no	—
168	C₂×C₄×C₅₃	—	2	1	no	—
169	C₂×C₄×C₅₃	—	2	1	no	—
170	C₂×C₄×C₅₃	—	2	2	no	—
171	C₂×C₄×C₅₃	—	2	2	no	—
172	C₂×C₄×C₅₃	—	2	2	no	—
173	C₂×C₄×C₅₃	C₄×D₄	1	1	yes	characteristic
174	C₂×C₄×C₅₃	C₄×D₄	1	1	yes	characteristic
175	C₂×C₄×C₅₃	C₄×D₄	1	1	yes	characteristic
176	C₂×C₄×C₅₃	C₂³×C₄	1	1	yes	characteristic
177	C₂×C₄×C₅₃	C₂²×D₄	1	1	yes	characteristic
178	C₂×C₄×C₅₃	C₂×#41	1	1	yes	characteristic
179	C₂²⋊D₅₃	—	2	4	no	—
180	C₂×Dic₅₃	—	2	1	no	—
181	C₂×Dic₅₃	—	2	1	no	—
182	C₂×Dic₅₃	—	2	1	no	—
183	D₂₁₂	—	2	1	no	—
184	D₂₁₂	#131	1	2	yes	—
185	C₄×D₅₃	—	2	4	no	—
186	C₄×D₅₃	—	4	1	no	—
187	C₄×D₅₃	—	4	1	no	—
188	C₄×D₅₃	—	4	2	no	—
189	Dic₁₀₆	—	2	1	no	—
190	Dic₁₀₆	#131	1	2	yes	—
191	C₈×C₅₃	—	2	1	no	—
192	C₈×C₅₃	—	2	1	no	—
193	C₈×C₅₃	—	2	1	no	—
194	C₈×C₅₃	—	4	2	no	—
195	C₈×C₅₃	#137	1	2	yes	—
196	#88322	—	2	2	no	—
197	#88322	—	2	2	no	—
198	#88322	—	4	2	no	—
199	#88322	—	4	1	no	—
200	#88322	—	4	1	no	—
201	#17916	—	53	1	no	maximal, Sylow
202	C₂²×C₅₃	—	2	1	no	—
203	C₂²×C₅₃	—	2	1	no	—
204	C₂²×C₅₃	—	2	1	no	—
205	C₂²×C₅₃	C₂×C₄×D₄	1	1	yes	characteristic
206	C₂×D₅₃	—	2	2	no	—
207	C₂×D₅₃	—	4	1	no	—
208	C₂×D₅₃	—	4	1	no	—
209	C₄×C₅₃	—	2	1	no	—
210	C₄×C₅₃	—	2	1	no	—
211	C₄×C₅₃	—	2	1	no	—
212	C₄×C₅₃	—	4	2	no	—
213	C₄×C₅₃	—	4	2	no	—
214	C₄×C₅₃	#444	1	2	yes	—
215	C₄×C₅₃	C₂×#131	1	1	yes	characteristic
216	C₄×C₅₃	C₂×#131	1	1	yes	characteristic
217	C₄×C₅₃	C₂×#137	1	1	yes	characteristic
218	C₄×C₅₃	#575	1	1	yes	characteristic
219	Dic₅₃	—	2	2	no	—
220	Dic₅₃	—	4	1	no	—
221	Dic₅₃	—	4	1	no	—
222	#3107	—	53	1	no	—
223	C₈×#41	—	53	1	no	—
224	#2960	D₅₃	1	1	yes	characteristic, complemented
225	#1584	—	53	2	no	—
226	#1582	—	53	2	no	—
227	#1579	—	53	2	no	—
228	SD₁₆⋊C₈	—	53	2	no	—
229	#1574	—	53	1	no	—
230	#1489	—	53	1	no	—
231	#1485	—	53	1	no	—
232	#1451	—	53	1	no	—
233	C₂×C₅₃	—	4	1	no	—
234	C₂×C₅₃	#2929	1	1	yes	characteristic
235	C₂×C₅₃	C₄×#137	1	1	yes	characteristic
236	C₂×C₅₃	C₂×#444	1	1	yes	characteristic
237	D₅₃	—	4	2	no	—
238	#533	C₂×D₅₃	1	1	yes	characteristic
239	C₄×#41	—	53	1	no	—
240	#485	—	53	2	no	—
241	#475	—	53	1	no	—
242	#474	—	53	1	no	—
243	#462	—	53	1	no	—
244	#461	—	53	1	no	—
245	#452	—	53	2	no	—
246	#446	—	2	1	no	—
247	#446	C₂×D₅₃	1	2	yes	—
248	C₈×Q₈	—	2	1	no	—
249	C₈×Q₈	—	53	1	no	—
250	C₈×Q₈	—	53	1	no	—
251	C₈×Q₈	C₂×D₅₃	1	2	yes	—
252	C₄×Q₁₆	—	53	2	no	—
253	C₄×SD₁₆	—	53	2	no	—
254	C₈×D₄	—	53	1	no	—
255	C₈×D₄	—	53	1	no	—
256	C₈×D₄	—	106	1	no	—
257	C₈×D₄	—	106	1	no	—
258	#431	—	53	1	no	—
259	#431	—	53	1	no	—
260	#431	—	106	1	no	—
261	#424	—	2	1	no	—
262	#423	C₂×D₅₃	1	1	yes	characteristic
263	#405	—	2	1	no	—
264	C₄×M₁₆	—	53	1	no	—
265	C₂×C₄×C₈	—	106	1	no	—
266	C₂×C₄×C₈	C₂×D₅₃	1	1	yes	characteristic
267	#335	—	53	2	no	—
268	#334	—	53	2	no	—
269	Q₈⋊C₈	—	53	2	no	—
270	Q₈⋊C₈	—	53	2	no	—
271	Q₈⋊C₈	—	53	2	no	—
272	D₄⋊C₈	—	53	2	no	—
273	#322	—	53	2	no	—
274	#322	—	53	2	no	—
275	C₅₃	#17916	1	1	yes	characteristic, complemented, Sylow
276	C₂×#41	—	53	1	no	—
277	C₂×Q₁₆	—	53	2	no	—
278	C₂×SD₁₆	—	53	2	no	—
279	C₂×M₁₆	—	2	1	no	—
280	C₂×M₁₆	—	106	1	no	—
281	C₂²×C₈	—	106	1	no	—
282	C₂²×C₈	—	106	1	no	—
283	C₂²×C₈	C₂²⋊D₅₃	1	1	yes	characteristic
284	C₂²×C₈	C₂²⋊D₅₃	1	1	yes	characteristic
285	C₄⋊Q₈	C₄×D₅₃	1	1	yes	characteristic
286	C₄⋊Q₈	C₄×D₅₃	1	1	yes	characteristic
287	#125	—	2	1	no	—
288	C₂²⋊Q₈	—	2	1	no	—
289	C₂²⋊Q₈	C₄×D₅₃	1	2	yes	—
290	C₄×Q₈	—	2	1	no	—
291	C₄×Q₈	—	53	1	no	—
292	C₄×Q₈	—	53	1	no	—
293	C₄×Q₈	C₂²×D₅₃	1	2	yes	—
294	C₄×D₄	—	53	1	no	—
295	C₄×D₄	—	53	1	no	—
296	C₄×D₄	—	106	1	no	—
297	C₄×D₄	—	106	1	no	—
298	#117	—	2	1	no	—
299	#117	—	53	1	no	—
300	#117	—	106	1	no	—
301	C₂×C₄²	—	106	1	no	—
302	C₂×C₄²	C₂²×D₅₃	1	1	yes	characteristic
303	#107	—	53	2	no	—
304	#106	—	53	2	no	—
305	C₄⋊C₈	—	2	2	no	—
306	C₄⋊C₈	—	2	2	no	—
307	C₄⋊C₈	—	2	1	no	—
308	C₄⋊C₈	—	53	1	no	—
309	C₄⋊C₈	—	53	1	no	—
310	C₄⋊C₈	—	53	1	no	—
311	C₄⋊C₈	—	53	1	no	—
312	C₄⋊C₈	—	106	1	no	—
313	C₄⋊C₈	—	106	1	no	—
314	C₄⋊C₈	C₂²×D₅₃	1	2	yes	—
315	#103	—	53	2	no	—
316	#103	—	53	2	no	—
317	#103	—	53	2	no	—
318	#102	—	53	2	no	—
319	C₂²⋊C₈	—	53	2	no	—
320	C₂²⋊C₈	—	106	1	no	—
321	C₂²⋊C₈	—	106	1	no	—
322	C₂²⋊C₈	—	106	1	no	—
323	#97	—	2	2	no	—
324	#97	—	106	1	no	—
325	C₄×C₈	—	2	2	no	—
326	C₄×C₈	—	53	2	no	—
327	C₄×C₈	—	53	2	no	—
328	C₄×C₈	—	106	1	no	—
329	C₄×C₈	—	106	1	no	—
330	C₄×C₈	—	106	1	no	—
331	C₄×C₈	C₂²⋊D₅₃	1	1	yes	characteristic
332	C₄×C₈	C₂²⋊D₅₃	1	1	yes	characteristic
333	C₄×C₈	C₂²×D₅₃	1	1	yes	characteristic
334	C₄×C₈	C₂²×D₅₃	1	1	yes	characteristic
335	#41	—	106	2	no	—
336	#41	—	106	2	no	—
337	C₂×Q₈	—	2	1	no	—
338	C₂×Q₈	—	53	1	no	—
339	C₂×Q₈	C₂×C₄×D₅₃	1	2	yes	—
340	C₂×D₄	—	53	1	no	—
341	C₂×D₄	—	106	1	no	—
342	C₂²×C₄	—	106	1	no	—
343	C₂²×C₄	—	106	1	no	—
344	C₂²×C₄	—	106	1	no	—
345	C₂²×C₄	C₂×C₄×D₅₃	1	1	yes	characteristic
346	C₂²×C₄	#11719945	1	1	yes	characteristic
347	C₂²×C₄	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
348	Q₁₆	—	106	4	no	—
349	SD₁₆	—	106	4	no	—
350	M₁₆	—	4	2	no	—
351	M₁₆	—	212	2	no	—
352	C₂×C₈	—	2	2	no	—
353	C₂×C₈	—	2	2	no	—
354	C₂×C₈	—	4	1	no	—
355	C₂×C₈	—	53	2	no	—
356	C₂×C₈	—	53	2	no	—
357	C₂×C₈	—	106	1	no	—
358	C₂×C₈	—	106	1	no	—
359	C₂×C₈	—	106	2	no	—
360	C₂×C₈	—	212	2	no	—
361	C₂×C₈	—	212	2	no	—
362	C₂×C₈	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
363	C₂×C₈	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
364	C₂×C₈	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
365	C₂×C₈	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
366	C₄⋊C₄	—	2	1	no	—
367	C₄⋊C₄	—	2	1	no	—
368	C₄⋊C₄	—	2	1	no	—
369	C₄⋊C₄	—	2	2	no	—
370	C₄⋊C₄	—	2	2	no	—
371	C₄⋊C₄	—	53	1	no	—
372	C₄⋊C₄	—	53	1	no	—
373	C₄⋊C₄	—	106	1	no	—
374	C₄⋊C₄	—	106	1	no	—
375	C₄⋊C₄	C₂×C₄×D₅₃	1	2	yes	—
376	C₂²⋊C₄	—	2	2	no	—
377	C₂²⋊C₄	—	106	1	no	—
378	C₂²⋊C₄	—	106	1	no	—
379	C₂²⋊C₄	—	106	1	no	—
380	C₄²	—	2	2	no	—
381	C₄²	—	106	1	no	—
382	C₄²	—	106	1	no	—
383	C₄²	—	106	1	no	—
384	C₄²	C₂×C₄×D₅₃	1	1	yes	characteristic
385	C₄²	#11719945	1	1	yes	characteristic
386	C₄²	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
387	C₄²	C₂³×D₅₃	1	1	yes	characteristic
388	C₂³	—	106	1	no	—
389	C₂³	C₄×C₂²⋊D₅₃	1	1	yes	characteristic
390	Q₈	—	2	2	no	—
391	Q₈	—	2	2	no	—
392	Q₈	—	4	2	no	—
393	Q₈	—	53	2	no	—
394	Q₈	—	106	1	no	—
395	D₄	—	53	2	no	—
396	D₄	—	106	4	no	—
397	D₄	—	106	1	no	—
398	C₂×C₄	—	2	1	no	—
399	C₂×C₄	—	2	1	no	—
400	C₂×C₄	—	2	1	no	—
401	C₂×C₄	—	2	1	no	—
402	C₂×C₄	—	2	2	no	—
403	C₂×C₄	—	2	2	no	—
404	C₂×C₄	—	2	2	no	—
405	C₂×C₄	—	106	4	no	—
406	C₂×C₄	—	106	1	no	—
407	C₂×C₄	—	106	1	no	—
408	C₂×C₄	—	106	1	no	—
409	C₂×C₄	—	212	1	no	—
410	C₂×C₄	—	212	1	no	—
411	C₂×C₄	—	212	2	no	—
412	C₂×C₄	C₄×C₂²⋊D₅₃	1	1	yes	characteristic
413	C₂×C₄	C₄×C₂²⋊D₅₃	1	1	yes	characteristic
414	C₂×C₄	C₄×C₂²⋊D₅₃	1	1	yes	characteristic
415	C₂×C₄	C₂²×C₄×D₅₃	1	1	yes	characteristic
416	C₂×C₄	C₂²×C₂²⋊D₅₃	1	1	yes	characteristic
417	C₂×C₄	C₂×#11719945	1	1	yes	characteristic
418	C₈	—	2	1	no	—
419	C₈	—	2	1	no	—
420	C₈	—	2	1	no	—
421	C₈	—	4	2	no	—
422	C₈	—	106	2	no	—
423	C₈	—	106	2	no	—
424	C₈	—	212	2	no	—
425	C₈	—	212	1	no	—
426	C₈	—	212	1	no	—
427	C₈	#13133248	1	2	yes	—
428	C₂²	—	2	1	no	—
429	C₂²	—	2	1	no	—
430	C₂²	—	2	1	no	—
431	C₂²	—	106	2	no	—
432	C₂²	—	212	1	no	—
433	C₂²	—	212	1	no	—
434	C₂²	C₂×C₄×C₂²⋊D₅₃	1	1	yes	characteristic
435	C₄	—	2	1	no	—
436	C₄	—	2	1	no	—
437	C₄	—	2	1	no	—
438	C₄	—	4	2	no	—
439	C₄	—	4	2	no	—
440	C₄	—	106	2	no	—
441	C₄	—	212	1	no	—
442	C₄	—	212	1	no	—
443	C₄	#16810743	1	2	yes	—
444	C₄	#131×D₅₃	1	1	yes	characteristic
445	C₄	#131×D₅₃	1	1	yes	characteristic
446	C₄	C₂×#13133248	1	1	yes	characteristic
447	C₄	#16811421	1	1	yes	characteristic
448	C₂	—	4	1	no	—
449	C₂	—	212	2	no	—
450	C₂	#20283407	1	1	yes	characteristic
451	C₂	C₄×#13133248	1	1	yes	characteristic
452	C₂	C₂×#16810743	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂²×C₂²⋊D₅₃
commutator	C₂×C₄×C₅₃	C₂³×C₄
Frattini	C₄²	C₂³×D₅₃
Fitting	C₅₃×#2960	C₂
socle	C₂²×C₅₃	C₂×C₄×D₄
prime radical	C₂×#11719945	C₂×C₄
cosocle	C₄²×C₅₃	C₂⁴
hypercenter	#2960	D₅₃
hypocenter	C₅₃	#17916
LCS	G ▷ C₂×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄×C₈ ◁ #2960

Complements

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

N	H ≅ G/N	type
3. #20273563	C₂	semidirect
4. #20273549	C₂	semidirect
5. #20273532	C₂	semidirect
6. #20273525	C₂	semidirect
7. #20273506	C₂	semidirect
9. #20273136	C₂	semidirect
10. #20273004	C₂	semidirect
11. C₅₃×#2960	C₂	semidirect
224. #2960	D₅₃	semidirect
275. C₅₃	#17916	semidirect

Maximal subgroups

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

subgroup	index
#20284780	2
C₈×#11719945	2
#20273563	2
#20273549	2
#20273532	2
#20273525	2
#20273506	2
#20273152	2
#20273136	2
#20273004	2
C₅₃×#2960	2
#17916	53