Subgroups — #27353673

Order 13568.

748 conjugacy classes of subgroups.

The subgroup lattice has 494 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×#13133242	C₂	1	1	yes	maximal, characteristic
2	C₂×C₈×D₂₁₂	C₂	1	1	yes	maximal, characteristic, complemented
3	#20273577	C₂	1	2	yes	maximal, complemented
4	#20273570	C₂	1	2	yes	maximal, complemented
5	#20273549	C₂	1	2	yes	maximal, complemented
6	#20273542	C₂	1	2	yes	maximal, complemented
7	#20273468	C₂	1	1	yes	maximal, characteristic, complemented
8	#20273105	C₂	1	1	yes	maximal, characteristic, complemented
9	#20273102	C₂	1	1	yes	maximal, characteristic, complemented
10	#20273004	C₂	1	1	yes	maximal, characteristic, complemented
11	C₅₃×#2965	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×#13133242	C₄	1	1	yes	characteristic
13	C₂²×C₈×D₅₃	—	2	1	no	—
14	C₂×C₄×D₂₁₂	C₂²	1	1	yes	characteristic
15	#16810733	C₄	1	2	yes	—
16	#16810721	C₄	1	2	yes	—
17	C₄×#11719924	C₂²	1	2	yes	—
18	C₄×#11719922	C₂²	1	2	yes	—
19	C₈×D₂₁₂	—	2	1	no	—
20	C₈×D₂₁₂	—	2	1	no	—
21	C₈×D₂₁₂	C₂²	1	2	yes	—
22	C₈×D₂₁₂	C₂²	1	2	yes	—
23	#16810676	C₂²	1	1	yes	characteristic
24	#16810666	C₄	1	1	yes	characteristic
25	C₂×C₄⋊#88322	C₂²	1	1	yes	characteristic
26	#16810627	C₄	1	1	yes	characteristic
27	#16810624	C₄	1	1	yes	characteristic
28	C₂×#13133129	—	2	1	no	—
29	C₄×#11719917	C₂²	1	1	yes	characteristic
30	#16810375	C₂²	1	2	yes	complemented
31	#16810373	C₂²	1	2	yes	complemented
32	Q₈⋊#88322	C₂²	1	2	yes	complemented
33	D₄⋊#88322	C₂²	1	2	yes	complemented
34	#16810350	C₂²	1	2	yes	—
35	#16810350	C₂²	1	2	yes	—
36	#16810345	C₂²	1	2	yes	complemented
37	#16810344	C₂²	1	2	yes	complemented
38	C₄×#41×C₅₃	C₂²	1	1	yes	characteristic
39	C₅₃×#446	C₂²	1	1	yes	characteristic, complemented
40	C₅₃×#446	C₂²	1	1	yes	characteristic, complemented
41	C₅₃×#436	C₂²	1	1	yes	characteristic, complemented
42	C₅₃×#436	C₂²	1	1	yes	characteristic, complemented
43	C₅₃×#435	—	2	1	no	—
44	C₅₃×#435	—	2	1	no	—
45	C₅₃×#431	—	2	1	no	—
46	C₅₃×#431	C₂²	1	1	yes	characteristic
47	C₄×M₁₆×C₅₃	—	2	1	no	—
48	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
49	C₂²×D₂₁₂	C₂×C₄	1	1	yes	characteristic
50	C₂²×C₄×D₅₃	—	2	1	no	—
51	#13133242	—	2	4	no	—
52	C₂×#11719924	C₂×C₄	1	2	yes	—
53	C₂×#11719922	C₂×C₄	1	2	yes	—
54	C₂×#11719917	C₂×C₄	1	1	yes	characteristic
55	C₂×#11719917	C₂×C₄	1	1	yes	characteristic
56	C₂²×#88322	—	2	1	no	—
57	C₂×C₈×D₅₃	—	2	2	no	—
58	C₂×C₈×D₅₃	—	2	2	no	—
59	C₂×C₈×D₅₃	—	4	4	no	—
60	C₄×D₂₁₂	—	2	1	no	—
61	C₄×D₂₁₂	—	2	1	no	—
62	C₄×D₂₁₂	C₂×C₄	1	2	yes	—
63	C₄×D₂₁₂	C₂³	1	2	yes	—
64	C₂×C₄⋊Dic₅₃	C₂×C₄	1	1	yes	characteristic
65	C₂×#11719921	—	2	1	no	—
66	#13133150	C₂×C₄	1	2	yes	—
67	#13133148	C₂×C₄	1	2	yes	—
68	C₄⋊#88322	—	2	1	no	—
69	C₄⋊#88322	—	2	1	no	—
70	C₄⋊#88322	C₂³	1	2	yes	—
71	#13133141	C₂×C₄	1	2	yes	—
72	#13133138	C₂×C₄	1	2	yes	—
73	#13133137	C₂×C₄	1	2	yes	—
74	#13133137	C₂×C₄	1	2	yes	—
75	C₂²⋊#88322	—	2	1	no	—
76	#13133129	—	2	2	no	—
77	#13133129	—	2	2	no	—
78	#13133128	C₂×C₄	1	2	yes	—
79	C₄×#88322	C₂³	1	2	yes	—
80	C₄×#88322	C₂³	1	2	yes	—
81	C₂×#41×C₅₃	C₂×C₄	1	1	yes	characteristic
82	C₂×M₁₆×C₅₃	—	2	1	no	—
83	C₂×M₁₆×C₅₃	—	2	1	no	—
84	C₂²×C₈×C₅₃	D₄	1	1	yes	characteristic
85	C₂²×C₈×C₅₃	D₄	1	1	yes	characteristic
86	C₄×Q₈×C₅₃	C₂×C₄	1	1	yes	characteristic
87	C₄×Q₈×C₅₃	C₂³	1	1	yes	characteristic
88	C₄×D₄×C₅₃	—	2	1	no	—
89	C₄×D₄×C₅₃	—	2	1	no	—
90	C₄×D₄×C₅₃	C₂×C₄	1	1	yes	characteristic
91	C₄×D₄×C₅₃	C₂³	1	1	yes	characteristic
92	#117×C₅₃	—	2	1	no	—
93	#117×C₅₃	C₂×C₄	1	1	yes	characteristic
94	C₂×C₄²×C₅₃	—	2	1	no	—
95	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
96	C₄⋊C₈×C₅₃	—	2	1	no	—
97	C₄⋊C₈×C₅₃	—	2	1	no	—
98	C₄⋊C₈×C₅₃	C₂³	1	1	yes	characteristic
99	C₄⋊C₈×C₅₃	C₂³	1	1	yes	characteristic
100	C₂²⋊C₈×C₅₃	—	2	1	no	—
101	C₂²⋊C₈×C₅₃	—	2	1	no	—
102	C₂²⋊C₈×C₅₃	—	2	1	no	—
103	#97×C₅₃	—	2	1	no	—
104	#97×C₅₃	—	2	1	no	—
105	C₄×C₈×C₅₃	—	2	1	no	—
106	C₄×C₈×C₅₃	D₄	1	1	yes	characteristic
107	C₄×C₈×C₅₃	D₄	1	1	yes	characteristic
108	C₄×C₈×C₅₃	C₂³	1	1	yes	characteristic
109	C₄×C₈×C₅₃	C₂³	1	1	yes	characteristic
110	C₂³×D₅₃	—	2	1	no	—
111	#41×C₅₃	—	2	2	no	—
112	#41×C₅₃	—	2	2	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	C₂²×C₄	1	1	yes	characteristic
120	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
121	C₂²×C₄×C₅₃	#41	1	1	yes	characteristic
122	C₂²×Dic₅₃	—	2	1	no	—
123	C₂×D₂₁₂	—	2	1	no	—
124	C₂×D₂₁₂	—	2	2	no	—
125	C₂×D₂₁₂	M₁₆	1	4	yes	complemented
126	C₂×D₂₁₂	C₂²×C₄	1	2	yes	—
127	C₂×C₄×D₅₃	—	2	2	no	—
128	C₂×C₄×D₅₃	—	2	2	no	—
129	C₂×C₄×D₅₃	—	4	2	no	—
130	C₂×C₄×D₅₃	—	4	2	no	—
131	M₁₆×C₅₃	—	4	2	no	—
132	M₁₆×C₅₃	—	4	2	no	—
133	C₂×C₈×C₅₃	—	2	2	no	—
134	C₂×C₈×C₅₃	—	2	1	no	—
135	C₂×C₈×C₅₃	—	2	1	no	—
136	C₂×C₈×C₅₃	—	4	1	no	—
137	C₂×C₈×C₅₃	C₂×D₄	1	1	yes	characteristic
138	C₂×C₈×C₅₃	C₂×D₄	1	1	yes	characteristic
139	C₂×C₈×C₅₃	C₂×D₄	1	1	yes	characteristic
140	C₂×C₈×C₅₃	C₂×D₄	1	1	yes	characteristic
141	C₄⋊C₄×C₅₃	—	2	1	no	—
142	C₄⋊C₄×C₅₃	—	2	1	no	—
143	C₄⋊C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
144	C₄⋊C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
145	C₂²⋊C₄×C₅₃	—	2	1	no	—
146	C₂²⋊C₄×C₅₃	—	2	1	no	—
147	C₂²⋊C₄×C₅₃	—	2	1	no	—
148	C₄²×C₅₃	—	2	1	no	—
149	C₄²×C₅₃	—	2	1	no	—
150	C₄²×C₅₃	—	2	1	no	—
151	C₄²×C₅₃	C₂²×C₄	1	1	yes	characteristic
152	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
153	C₄²×C₅₃	#41	1	1	yes	characteristic
154	C₄²×C₅₃	C₂⁴	1	1	yes	characteristic
155	#11719924	—	2	4	no	—
156	#11719922	—	2	4	no	—
157	#11719921	—	2	2	no	—
158	#11719921	—	2	2	no	—
159	C₄⋊Dic₅₃	—	2	1	no	—
160	C₄⋊Dic₅₃	C₂²×C₄	1	2	yes	—
161	#11719917	—	2	2	no	—
162	#11719917	—	2	2	no	—
163	C₂×#88322	—	2	2	no	—
164	C₂×#88322	—	2	2	no	—
165	C₂×#88322	—	4	2	no	—
166	C₂×#88322	C₂²×C₄	1	2	yes	—
167	C₂×#88322	C₂²×C₄	1	2	yes	—
168	C₈×D₅₃	—	4	4	no	—
169	C₈×D₅₃	—	4	4	no	—
170	C₂³×C₅₃	—	2	1	no	—
171	C₂³×C₅₃	C₄×D₄	1	1	yes	characteristic
172	C₂²×D₅₃	—	2	2	no	—
173	C₂²×D₅₃	—	2	4	no	—
174	C₂²×D₅₃	—	4	2	no	—
175	C₂²×D₅₃	—	4	2	no	—
176	Q₈×C₅₃	—	2	1	no	—
177	Q₈×C₅₃	—	2	1	no	—
178	D₄×C₅₃	—	2	1	no	—
179	D₄×C₅₃	—	2	1	no	—
180	D₄×C₅₃	—	4	2	no	—
181	C₂×C₄×C₅₃	—	2	1	no	—
182	C₂×C₄×C₅₃	—	2	1	no	—
183	C₂×C₄×C₅₃	—	2	1	no	—
184	C₂×C₄×C₅₃	—	2	1	no	—
185	C₂×C₄×C₅₃	—	2	1	no	—
186	C₂×C₄×C₅₃	—	4	1	no	—
187	C₂×C₄×C₅₃	—	4	2	no	—
188	C₂×C₄×C₅₃	—	4	1	no	—
189	C₂×C₄×C₅₃	—	4	1	no	—
190	C₂×C₄×C₅₃	—	4	2	no	—
191	C₂×C₄×C₅₃	C₄×D₄	1	1	yes	characteristic
192	C₂×C₄×C₅₃	C₄×D₄	1	1	yes	characteristic
193	C₂×C₄×C₅₃	C₄×D₄	1	1	yes	characteristic
194	C₂×C₄×C₅₃	C₂×M₁₆	1	2	yes	—
195	C₂×C₄×C₅₃	#131	1	2	yes	—
196	C₂×C₄×C₅₃	C₂³×C₄	1	1	yes	characteristic
197	C₂×C₄×C₅₃	C₂²×D₄	1	1	yes	characteristic
198	C₂×C₄×C₅₃	C₂×#41	1	1	yes	characteristic
199	C₂×Dic₅₃	—	2	2	no	—
200	C₂×Dic₅₃	—	4	2	no	—
201	D₂₁₂	—	2	2	no	—
202	D₂₁₂	—	2	4	no	—
203	D₂₁₂	C₂×M₁₆	1	4	yes	—
204	C₄×D₅₃	—	4	4	no	—
205	C₄×D₅₃	—	4	2	no	—
206	C₄×D₅₃	—	4	2	no	—
207	C₈×C₅₃	—	2	1	no	—
208	C₈×C₅₃	—	2	1	no	—
209	C₈×C₅₃	—	2	1	no	—
210	C₈×C₅₃	—	4	1	no	—
211	C₈×C₅₃	—	4	1	no	—
212	C₈×C₅₃	AGL(1,ℤ/8)	1	2	yes	complemented
213	#88322	—	2	2	no	—
214	#88322	—	2	2	no	—
215	#88322	—	4	2	no	—
216	#88322	—	4	2	no	—
217	#17869	—	53	1	no	maximal, Sylow
218	C₂²×C₅₃	—	2	1	no	—
219	C₂²×C₅₃	—	2	1	no	—
220	C₂²×C₅₃	—	4	1	no	—
221	C₂²×C₅₃	—	4	1	no	—
222	C₂²×C₅₃	—	4	1	no	—
223	C₂²×C₅₃	#435	1	2	yes	—
224	C₂²×C₅₃	C₂×C₄×D₄	1	1	yes	characteristic
225	C₂×D₅₃	—	2	4	no	—
226	C₂×D₅₃	—	4	8	no	—
227	C₂×D₅₃	—	4	2	no	—
228	C₂×D₅₃	—	4	2	no	—
229	C₄×C₅₃	—	2	1	no	—
230	C₄×C₅₃	—	2	1	no	—
231	C₄×C₅₃	—	2	1	no	—
232	C₄×C₅₃	—	4	1	no	—
233	C₄×C₅₃	—	4	1	no	—
234	C₄×C₅₃	—	4	1	no	—
235	C₄×C₅₃	#435	1	2	yes	—
236	C₄×C₅₃	C₂²×M₁₆	1	1	yes	characteristic
237	C₄×C₅₃	C₂×#131	1	1	yes	characteristic
238	C₄×C₅₃	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
239	C₄×C₅₃	#575	1	1	yes	characteristic
240	Dic₅₃	—	4	2	no	—
241	#2965	D₅₃	1	1	yes	characteristic, complemented
242	C₄×AGL(1,ℤ/8)	—	53	1	no	—
243	C₂×C₈×D₄	—	53	1	no	—
244	#1586	—	53	2	no	—
245	#1585	—	53	2	no	—
246	#1582	—	53	2	no	—
247	#1581	—	53	2	no	—
248	#1565	—	53	1	no	—
249	#1478	—	53	1	no	—
250	#1477	—	53	1	no	—
251	#1451	—	53	1	no	—
252	C₂×C₅₃	—	2	2	no	—
253	C₂×C₅₃	—	8	1	no	—
254	C₂×C₅₃	C₂×#435	1	1	yes	characteristic
255	C₂×C₅₃	C₄×AGL(1,ℤ/8)	1	1	yes	characteristic
256	C₂×C₅₃	#2955	1	1	yes	characteristic
257	D₅₃	—	4	4	no	—
258	C₂×AGL(1,ℤ/8)	—	53	1	no	—
259	C₂³×C₈	—	106	1	no	—
260	C₄×#41	C₂×D₅₃	1	1	yes	characteristic
261	C₂×C₄×D₄	—	53	1	no	—
262	#446	C₂×D₅₃	1	1	yes	characteristic, complemented
263	#446	C₂×D₅₃	1	1	yes	characteristic, complemented
264	#442	—	53	2	no	—
265	#440	—	53	2	no	—
266	C₄×SD₁₆	—	53	2	no	—
267	C₄×D₈	—	53	2	no	—
268	#436	C₂×D₅₃	1	1	yes	characteristic, complemented
269	#436	C₂×D₅₃	1	1	yes	characteristic, complemented
270	#435	—	2	1	no	—
271	#435	—	2	1	no	—
272	C₈×D₄	—	53	2	no	—
273	C₈×D₄	—	53	2	no	—
274	C₈×D₄	—	106	1	no	—
275	C₈×D₄	—	106	1	no	—
276	#431	—	2	1	no	—
277	#431	—	53	1	no	—
278	#431	C₂×D₅₃	1	1	yes	characteristic
279	#428	—	53	1	no	—
280	C₂×C₄⋊C₈	—	53	1	no	—
281	#418	—	53	1	no	—
282	#417	—	53	1	no	—
283	C₂×C₂²⋊C₈	—	106	1	no	—
284	C₄×M₁₆	—	2	1	no	—
285	C₄×M₁₆	—	53	1	no	—
286	C₂×C₄×C₈	C₂×D₅₃	1	1	yes	characteristic
287	#335	—	53	2	no	—
288	#334	—	53	2	no	—
289	Q₈⋊C₈	—	53	2	no	—
290	D₄⋊C₈	—	53	2	no	—
291	D₄⋊C₈	—	53	2	no	—
292	D₄⋊C₈	—	53	2	no	—
293	#322	—	53	2	no	—
294	#322	—	53	2	no	—
295	C₅₃	#17869	1	1	yes	characteristic, complemented, Sylow
296	C₂×#41	C₄×D₅₃	1	1	yes	characteristic
297	C₂²×D₄	—	53	1	no	—
298	C₂³×C₄	—	106	1	no	—
299	AGL(1,ℤ/8)	—	106	4	no	—
300	C₂×SD₁₆	—	53	2	no	—
301	C₂×D₈	—	53	2	no	—
302	C₂×M₁₆	—	2	1	no	—
303	C₂×M₁₆	—	2	1	no	—
304	C₂×M₁₆	—	53	1	no	—
305	C₂×M₁₆	—	53	1	no	—
306	C₂²×C₈	—	106	1	no	—
307	C₂²×C₈	—	106	2	no	—
308	C₂²×C₈	—	106	2	no	—
309	C₂²×C₈	—	212	4	no	—
310	C₂²×C₈	C₂²⋊D₅₃	1	1	yes	characteristic
311	C₂²×C₈	C₂²⋊D₅₃	1	1	yes	characteristic
312	C₄×Q₈	C₄×D₅₃	1	1	yes	characteristic
313	C₄×Q₈	C₂²×D₅₃	1	1	yes	characteristic
314	C₄×D₄	—	2	1	no	—
315	C₄×D₄	—	2	1	no	—
316	C₄×D₄	—	53	2	no	—
317	C₄×D₄	—	53	2	no	—
318	C₄×D₄	—	106	1	no	—
319	C₄×D₄	—	106	1	no	—
320	C₄×D₄	C₄×D₅₃	1	1	yes	characteristic
321	C₄×D₄	C₂²×D₅₃	1	1	yes	characteristic
322	#117	—	2	1	no	—
323	#117	C₄×D₅₃	1	1	yes	characteristic
324	C₂×C₄⋊C₄	—	53	1	no	—
325	C₂×C₂²⋊C₄	—	106	1	no	—
326	C₂×C₄²	—	2	1	no	—
327	C₂×C₄²	C₂²×D₅₃	1	1	yes	characteristic
328	#107	—	53	2	no	—
329	#106	—	53	2	no	—
330	C₄⋊C₈	—	2	1	no	—
331	C₄⋊C₈	—	2	1	no	—
332	C₄⋊C₈	—	53	2	no	—
333	C₄⋊C₈	—	106	1	no	—
334	C₄⋊C₈	—	106	1	no	—
335	C₄⋊C₈	C₂²×D₅₃	1	1	yes	characteristic
336	C₄⋊C₈	C₂²×D₅₃	1	1	yes	characteristic
337	#103	—	53	2	no	—
338	#102	—	53	2	no	—
339	#102	—	53	2	no	—
340	#102	—	53	2	no	—
341	C₂²⋊C₈	—	2	1	no	—
342	C₂²⋊C₈	—	2	1	no	—
343	C₂²⋊C₈	—	2	1	no	—
344	C₂²⋊C₈	—	106	1	no	—
345	C₂²⋊C₈	—	106	2	no	—
346	C₂²⋊C₈	—	106	2	no	—
347	#97	—	2	1	no	—
348	#97	—	2	1	no	—
349	#97	—	53	2	no	—
350	C₄×C₈	—	2	1	no	—
351	C₄×C₈	—	53	2	no	—
352	C₄×C₈	—	53	2	no	—
353	C₄×C₈	C₂²⋊D₅₃	1	1	yes	characteristic
354	C₄×C₈	C₂²⋊D₅₃	1	1	yes	characteristic
355	C₄×C₈	C₂²×D₅₃	1	1	yes	characteristic
356	C₄×C₈	C₂²×D₅₃	1	1	yes	characteristic
357	C₂⁴	—	106	1	no	—
358	#41	—	2	2	no	—
359	#41	—	2	2	no	—
360	C₂×Q₈	C₂×C₄×D₅₃	1	1	yes	characteristic
361	C₂×D₄	—	2	1	no	—
362	C₂×D₄	—	53	4	no	—
363	C₂×D₄	—	53	2	no	—
364	C₂×D₄	—	106	1	no	—
365	C₂×D₄	—	106	2	no	—
366	C₂×D₄	C₂×C₄×D₅₃	1	1	yes	characteristic
367	C₂²×C₄	—	2	1	no	—
368	C₂²×C₄	—	2	1	no	—
369	C₂²×C₄	—	2	1	no	—
370	C₂²×C₄	—	106	1	no	—
371	C₂²×C₄	—	106	2	no	—
372	C₂²×C₄	—	106	2	no	—
373	C₂²×C₄	—	212	2	no	—
374	C₂²×C₄	—	212	2	no	—
375	C₂²×C₄	C₂×C₄×D₅₃	1	1	yes	characteristic
376	C₂²×C₄	#11719945	1	1	yes	characteristic
377	C₂²×C₄	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
378	SD₁₆	—	106	4	no	—
379	D₈	—	106	4	no	—
380	M₁₆	—	4	2	no	—
381	M₁₆	—	4	2	no	—
382	M₁₆	—	106	2	no	—
383	M₁₆	—	106	2	no	—
384	C₂×C₈	—	2	2	no	—
385	C₂×C₈	—	2	1	no	—
386	C₂×C₈	—	2	1	no	—
387	C₂×C₈	—	4	1	no	—
388	C₂×C₈	—	53	2	no	—
389	C₂×C₈	—	53	2	no	—
390	C₂×C₈	—	106	2	no	—
391	C₂×C₈	—	106	2	no	—
392	C₂×C₈	—	212	4	no	—
393	C₂×C₈	—	212	2	no	—
394	C₂×C₈	—	212	4	no	—
395	C₂×C₈	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
396	C₂×C₈	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
397	C₂×C₈	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
398	C₂×C₈	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
399	C₄⋊C₄	—	2	1	no	—
400	C₄⋊C₄	—	2	1	no	—
401	C₄⋊C₄	—	53	2	no	—
402	C₄⋊C₄	—	106	1	no	—
403	C₄⋊C₄	C₂×C₄×D₅₃	1	1	yes	characteristic
404	C₄⋊C₄	C₂×C₄×D₅₃	1	1	yes	characteristic
405	C₂²⋊C₄	—	2	1	no	—
406	C₂²⋊C₄	—	2	1	no	—
407	C₂²⋊C₄	—	2	1	no	—
408	C₂²⋊C₄	—	106	2	no	—
409	C₂²⋊C₄	—	106	2	no	—
410	C₄²	—	2	1	no	—
411	C₄²	—	2	1	no	—
412	C₄²	—	2	1	no	—
413	C₄²	C₂×C₄×D₅₃	1	1	yes	characteristic
414	C₄²	#11719945	1	1	yes	characteristic
415	C₄²	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
416	C₄²	C₂³×D₅₃	1	1	yes	characteristic
417	C₂³	—	2	1	no	—
418	C₂³	—	106	2	no	—
419	C₂³	—	106	4	no	—
420	C₂³	—	212	2	no	—
421	C₂³	—	212	2	no	—
422	C₂³	C₄×C₂²⋊D₅₃	1	1	yes	characteristic
423	Q₈	—	2	1	no	—
424	Q₈	—	2	1	no	—
425	D₄	—	2	1	no	—
426	D₄	—	2	1	no	—
427	D₄	—	4	2	no	—
428	D₄	—	53	4	no	—
429	D₄	—	106	2	no	—
430	D₄	—	106	4	no	—
431	C₂×C₄	—	2	1	no	—
432	C₂×C₄	—	2	1	no	—
433	C₂×C₄	—	2	1	no	—
434	C₂×C₄	—	2	1	no	—
435	C₂×C₄	—	2	1	no	—
436	C₂×C₄	—	4	1	no	—
437	C₂×C₄	—	4	2	no	—
438	C₂×C₄	—	4	1	no	—
439	C₂×C₄	—	4	1	no	—
440	C₂×C₄	—	4	2	no	—
441	C₂×C₄	—	106	2	no	—
442	C₂×C₄	—	212	4	no	—
443	C₂×C₄	—	212	2	no	—
444	C₂×C₄	—	212	2	no	—
445	C₂×C₄	—	212	2	no	—
446	C₂×C₄	C₄×C₂²⋊D₅₃	1	1	yes	characteristic
447	C₂×C₄	C₄×C₂²⋊D₅₃	1	1	yes	characteristic
448	C₂×C₄	C₄×C₂²⋊D₅₃	1	1	yes	characteristic
449	C₂×C₄	M₁₆×D₅₃	1	2	yes	—
450	C₂×C₄	#13133225	1	2	yes	—
451	C₂×C₄	C₂²×C₄×D₅₃	1	1	yes	characteristic
452	C₂×C₄	C₂²×C₂²⋊D₅₃	1	1	yes	characteristic
453	C₂×C₄	C₂×#11719945	1	1	yes	characteristic
454	C₈	—	2	1	no	—
455	C₈	—	2	1	no	—
456	C₈	—	2	1	no	—
457	C₈	—	4	1	no	—
458	C₈	—	4	1	no	—
459	C₈	—	106	2	no	—
460	C₈	—	106	2	no	—
461	C₈	—	212	2	no	—
462	C₈	—	212	2	no	—
463	C₈	#13133242	1	2	yes	complemented
464	C₂²	—	2	1	no	—
465	C₂²	—	2	1	no	—
466	C₂²	—	4	1	no	—
467	C₂²	—	4	1	no	—
468	C₂²	—	4	1	no	—
469	C₂²	—	106	4	no	—
470	C₂²	—	212	8	no	—
471	C₂²	—	212	2	no	—
472	C₂²	—	212	2	no	—
473	C₂²	#16810692	1	2	yes	—
474	C₂²	C₂×C₄×C₂²⋊D₅₃	1	1	yes	characteristic
475	C₄	—	2	1	no	—
476	C₄	—	2	1	no	—
477	C₄	—	2	1	no	—
478	C₄	—	4	1	no	—
479	C₄	—	4	1	no	—
480	C₄	—	4	1	no	—
481	C₄	—	212	2	no	—
482	C₄	#16810692	1	2	yes	—
483	C₄	C₂×M₁₆×D₅₃	1	1	yes	characteristic
484	C₄	C₂×#13133225	1	1	yes	characteristic
485	C₄	C₂×#13133242	1	1	yes	characteristic
486	C₄	#16811421	1	1	yes	characteristic
487	C₂	—	2	2	no	—
488	C₂	—	8	1	no	—
489	C₂	—	212	4	no	—
490	C₂	C₂×#16810692	1	1	yes	characteristic
491	C₂	C₄×#13133242	1	1	yes	characteristic
492	C₂	#20283616	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×#2965	C₂
socle	C₂²×C₅₃	C₂×C₄×D₄
prime radical	C₂×#13133242	C₄
cosocle	C₄²×C₅₃	C₂⁴
hypercenter	#2965	D₅₃
hypocenter	C₅₃	#17869
LCS	G ▷ C₂×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄×C₈ ◁ #2965

Complements

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

N	H ≅ G/N	type
2. C₂×C₈×D₂₁₂	C₂	semidirect
3. #20273577	C₂	semidirect
4. #20273570	C₂	semidirect
5. #20273549	C₂	semidirect
6. #20273542	C₂	semidirect
7. #20273468	C₂	semidirect
8. #20273105	C₂	semidirect
9. #20273102	C₂	semidirect
10. #20273004	C₂	semidirect
11. C₅₃×#2965	C₂	semidirect
30. #16810375	C₂²	semidirect
31. #16810373	C₂²	semidirect
32. Q₈⋊#88322	C₂²	semidirect
33. D₄⋊#88322	C₂²	semidirect
36. #16810345	C₂²	semidirect
37. #16810344	C₂²	semidirect
39. C₅₃×#446	C₂²	semidirect
40. C₅₃×#446	C₂²	semidirect
41. C₅₃×#436	C₂²	semidirect
42. C₅₃×#436	C₂²	semidirect
125. C₂×D₂₁₂	M₁₆	semidirect
212. C₈×C₅₃	AGL(1,ℤ/8)	semidirect
241. #2965	D₅₃	semidirect
262. #446	C₂×D₅₃	semidirect
263. #446	C₂×D₅₃	semidirect
268. #436	C₂×D₅₃	semidirect
269. #436	C₂×D₅₃	semidirect
295. C₅₃	#17869	semidirect
463. C₈	#13133242	semidirect

Maximal subgroups

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

subgroup	index
C₄×#13133242	2
C₂×C₈×D₂₁₂	2
#20273577	2
#20273570	2
#20273549	2
#20273542	2
#20273468	2
#20273105	2
#20273102	2
#20273004	2
C₅₃×#2965	2
#17869	53