Subgroups — #27353717

Order 13568.

700 conjugacy classes of subgroups.

The subgroup lattice has 470 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20283738	C₂	1	1	yes	maximal, characteristic, complemented
2	C₄×#13133242	C₂	1	1	yes	maximal, characteristic
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	#20273465	C₂	1	1	yes	maximal, characteristic, complemented
8	#20273097	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂×#16810350	C₂	1	1	yes	maximal, characteristic, complemented
10	#20273005	C₂	1	1	yes	maximal, characteristic, complemented
11	C₅₃×#2989	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×#13133242	C₄	1	1	yes	characteristic
13	C₂×C₄×D₂₁₂	C₂²	1	1	yes	characteristic
14	#16810733	C₄	1	2	yes	—
15	#16810721	C₄	1	2	yes	—
16	C₄×#11719924	C₂²	1	2	yes	—
17	C₄×#11719922	C₂²	1	2	yes	—
18	C₈×D₂₁₂	C₂²	1	2	yes	—
19	C₈×D₂₁₂	C₂²	1	2	yes	—
20	#16810688	—	2	1	no	—
21	#16810688	—	2	1	no	—
22	#16810676	C₂²	1	1	yes	characteristic
23	#16810666	C₄	1	1	yes	characteristic
24	#16810627	C₄	1	1	yes	characteristic
25	#16810624	C₄	1	1	yes	characteristic
26	C₂²⋊C₈×D₅₃	—	2	2	no	—
27	C₄×#11719917	C₂²	1	1	yes	characteristic
28	C₂×C₄×#88322	C₂²	1	1	yes	characteristic
29	#16810375	C₂²	1	2	yes	complemented
30	#16810373	C₂²	1	2	yes	complemented
31	Q₈⋊#88322	C₂²	1	1	yes	characteristic, complemented
32	Q₈⋊#88322	C₂²	1	1	yes	characteristic, complemented
33	D₄⋊#88322	C₂²	1	1	yes	characteristic, complemented
34	D₄⋊#88322	C₂²	1	1	yes	characteristic, complemented
35	#16810350	C₂²	1	2	yes	—
36	#16810350	C₂²	1	2	yes	—
37	#16810345	C₂²	1	2	yes	complemented
38	#16810344	C₂²	1	2	yes	complemented
39	C₄×#41×C₅₃	C₂²	1	1	yes	characteristic
40	C₅₃×#446	C₂²	1	2	yes	complemented
41	C₅₃×#436	C₂²	1	2	yes	complemented
42	C₅₃×#435	—	2	2	no	—
43	C₅₃×#432	—	2	1	no	—
44	C₅₃×#431	C₂²	1	1	yes	characteristic
45	C₅₃×#423	—	2	1	no	—
46	C₂×C₄⋊C₈×C₅₃	C₂²	1	1	yes	characteristic
47	C₂²×D₂₁₂	C₂×C₄	1	1	yes	characteristic
48	C₂²×C₄×D₅₃	—	2	1	no	—
49	#13133242	—	2	4	no	—
50	C₂×#11719924	C₂×C₄	1	2	yes	—
51	C₂×#11719922	C₂×C₄	1	2	yes	—
52	C₂×#11719917	C₂×C₄	1	1	yes	characteristic
53	C₂×#11719917	C₂×C₄	1	1	yes	characteristic
54	C₂²×#88322	—	2	1	no	—
55	C₂×C₈×D₅₃	—	2	2	no	—
56	C₂×C₈×D₅₃	—	2	2	no	—
57	C₄×D₂₁₂	—	2	1	no	—
58	C₄×D₂₁₂	—	2	1	no	—
59	C₄×D₂₁₂	C₂×C₄	1	2	yes	—
60	C₄×D₂₁₂	C₂³	1	2	yes	—
61	C₂×C₄⋊Dic₅₃	C₂×C₄	1	1	yes	characteristic
62	C₂×#11719921	—	2	1	no	—
63	#13133150	C₂×C₄	1	2	yes	—
64	#13133148	C₂×C₄	1	2	yes	—
65	C₄⋊#88322	C₂³	1	1	yes	characteristic
66	C₄⋊#88322	C₂³	1	1	yes	characteristic
67	#13133141	C₂×C₄	1	2	yes	—
68	#13133138	C₂×C₄	1	2	yes	—
69	#13133137	C₂×C₄	1	2	yes	—
70	#13133137	C₂×C₄	1	2	yes	—
71	C₂²⋊#88322	—	2	1	no	—
72	#13133129	—	2	2	no	—
73	#13133129	—	2	2	no	—
74	#13133128	C₂×C₄	1	2	yes	—
75	C₄×#88322	—	2	1	no	—
76	C₄×#88322	—	2	1	no	—
77	C₄×#88322	C₂³	1	1	yes	characteristic
78	C₄×#88322	C₂³	1	1	yes	characteristic
79	C₄×#88322	C₂³	1	2	yes	—
80	C₂×#41×C₅₃	C₂×C₄	1	1	yes	characteristic
81	C₂×M₁₆×C₅₃	—	2	2	no	—
82	C₂²×C₈×C₅₃	—	2	1	no	—
83	C₄×Q₈×C₅₃	C₂×C₄	1	1	yes	characteristic
84	C₄×Q₈×C₅₃	C₂³	1	1	yes	characteristic
85	C₄×D₄×C₅₃	—	2	1	no	—
86	C₄×D₄×C₅₃	—	2	1	no	—
87	C₄×D₄×C₅₃	C₂×C₄	1	1	yes	characteristic
88	C₄×D₄×C₅₃	C₂³	1	1	yes	characteristic
89	#117×C₅₃	—	2	1	no	—
90	#117×C₅₃	C₂×C₄	1	1	yes	characteristic
91	C₂×C₄²×C₅₃	—	2	1	no	—
92	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
93	C₄⋊C₈×C₅₃	—	2	2	no	—
94	C₄⋊C₈×C₅₃	—	2	1	no	—
95	C₄⋊C₈×C₅₃	D₄	1	1	yes	characteristic
96	C₄⋊C₈×C₅₃	D₄	1	1	yes	characteristic
97	C₄⋊C₈×C₅₃	C₂³	1	2	yes	—
98	C₂²⋊C₈×C₅₃	—	2	2	no	—
99	C₂²⋊C₈×C₅₃	D₄	1	2	yes	—
100	#97×C₅₃	—	2	2	no	—
101	C₄×C₈×C₅₃	C₂³	1	2	yes	—
102	C₂³×D₅₃	—	2	1	no	—
103	#41×C₅₃	—	2	2	no	—
104	#41×C₅₃	—	2	2	no	—
105	C₂×Q₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
106	C₂×D₄×C₅₃	—	2	1	no	—
107	C₂×D₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
108	C₂²×C₄×C₅₃	—	2	1	no	—
109	C₂²×C₄×C₅₃	—	2	1	no	—
110	C₂²×C₄×C₅₃	—	2	1	no	—
111	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
112	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
113	C₂²×C₄×C₅₃	#41	1	1	yes	characteristic
114	C₂²×Dic₅₃	—	2	1	no	—
115	C₂×D₂₁₂	—	2	1	no	—
116	C₂×D₂₁₂	—	2	2	no	—
117	C₂×D₂₁₂	M₁₆	1	4	yes	complemented
118	C₂×D₂₁₂	C₂²×C₄	1	2	yes	—
119	C₂×C₄×D₅₃	—	2	2	no	—
120	C₂×C₄×D₅₃	—	2	2	no	—
121	C₂×C₄×D₅₃	—	8	1	no	—
122	C₂×C₄×D₅₃	—	8	1	no	—
123	M₁₆×C₅₃	—	8	2	no	—
124	C₂×C₈×C₅₃	—	2	2	no	—
125	C₂×C₈×C₅₃	—	8	1	no	—
126	C₂×C₈×C₅₃	C₂×D₄	1	2	yes	—
127	C₂×C₈×C₅₃	C₂×D₄	1	2	yes	—
128	C₄⋊C₄×C₅₃	—	2	1	no	—
129	C₄⋊C₄×C₅₃	—	2	1	no	—
130	C₄⋊C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
131	C₄⋊C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
132	C₂²⋊C₄×C₅₃	—	2	1	no	—
133	C₂²⋊C₄×C₅₃	—	2	1	no	—
134	C₂²⋊C₄×C₅₃	—	2	1	no	—
135	C₄²×C₅₃	—	2	1	no	—
136	C₄²×C₅₃	—	2	1	no	—
137	C₄²×C₅₃	—	2	1	no	—
138	C₄²×C₅₃	C₂²×C₄	1	1	yes	characteristic
139	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
140	C₄²×C₅₃	#41	1	1	yes	characteristic
141	C₄²×C₅₃	C₂⁴	1	1	yes	characteristic
142	#11719924	—	2	4	no	—
143	#11719922	—	2	4	no	—
144	#11719921	—	2	2	no	—
145	#11719921	—	2	2	no	—
146	C₄⋊Dic₅₃	—	2	1	no	—
147	C₄⋊Dic₅₃	C₂²×C₄	1	2	yes	—
148	#11719917	—	2	2	no	—
149	#11719917	—	2	2	no	—
150	C₂×#88322	—	2	1	no	—
151	C₂×#88322	—	2	1	no	—
152	C₂×#88322	—	2	1	no	—
153	C₂×#88322	—	2	1	no	—
154	C₂×#88322	—	4	2	no	—
155	C₂×#88322	C₂²×C₄	1	2	yes	—
156	C₂×#88322	C₂²×C₄	1	2	yes	—
157	C₈×D₅₃	—	8	2	no	—
158	C₈×D₅₃	—	8	2	no	—
159	C₂³×C₅₃	—	2	1	no	—
160	C₂³×C₅₃	C₄×D₄	1	1	yes	characteristic
161	C₂²×D₅₃	—	2	2	no	—
162	C₂²×D₅₃	—	2	4	no	—
163	C₂²×D₅₃	—	8	1	no	—
164	C₂²×D₅₃	—	8	1	no	—
165	Q₈×C₅₃	—	2	1	no	—
166	Q₈×C₅₃	—	2	1	no	—
167	D₄×C₅₃	—	2	1	no	—
168	D₄×C₅₃	—	2	1	no	—
169	D₄×C₅₃	—	4	2	no	—
170	C₂×C₄×C₅₃	—	2	1	no	—
171	C₂×C₄×C₅₃	—	2	1	no	—
172	C₂×C₄×C₅₃	—	2	1	no	—
173	C₂×C₄×C₅₃	—	2	1	no	—
174	C₂×C₄×C₅₃	—	2	1	no	—
175	C₂×C₄×C₅₃	—	4	1	no	—
176	C₂×C₄×C₅₃	—	4	1	no	—
177	C₂×C₄×C₅₃	—	4	1	no	—
178	C₂×C₄×C₅₃	—	4	2	no	—
179	C₂×C₄×C₅₃	—	8	1	no	—
180	C₂×C₄×C₅₃	C₄×D₄	1	1	yes	characteristic
181	C₂×C₄×C₅₃	C₄×D₄	1	1	yes	characteristic
182	C₂×C₄×C₅₃	C₄×D₄	1	1	yes	characteristic
183	C₂×C₄×C₅₃	C₂×M₁₆	1	2	yes	—
184	C₂×C₄×C₅₃	#131	1	2	yes	—
185	C₂×C₄×C₅₃	C₂³×C₄	1	1	yes	characteristic
186	C₂×C₄×C₅₃	C₂²×D₄	1	1	yes	characteristic
187	C₂×C₄×C₅₃	C₂×#41	1	1	yes	characteristic
188	C₂×Dic₅₃	—	2	2	no	—
189	C₂×Dic₅₃	—	8	1	no	—
190	D₂₁₂	—	2	2	no	—
191	D₂₁₂	—	2	4	no	—
192	D₂₁₂	C₂×M₁₆	1	4	yes	—
193	C₄×D₅₃	—	4	2	no	—
194	C₄×D₅₃	—	4	2	no	—
195	C₄×D₅₃	—	8	2	no	—
196	C₈×C₅₃	—	2	2	no	—
197	C₈×C₅₃	—	2	2	no	—
198	C₈×C₅₃	—	4	2	no	—
199	#88322	—	2	2	no	—
200	#88322	—	2	2	no	—
201	#88322	—	2	2	no	—
202	#88322	—	4	1	no	—
203	#88322	—	4	1	no	—
204	#88322	—	4	1	no	—
205	#17873	—	53	1	no	maximal, Sylow
206	C₂²×C₅₃	—	2	1	no	—
207	C₂²×C₅₃	—	2	1	no	—
208	C₂²×C₅₃	—	4	1	no	—
209	C₂²×C₅₃	—	4	1	no	—
210	C₂²×C₅₃	—	4	1	no	—
211	C₂²×C₅₃	#435	1	2	yes	—
212	C₂²×C₅₃	C₂×C₄×D₄	1	1	yes	characteristic
213	C₂×D₅₃	—	2	4	no	—
214	C₂×D₅₃	—	4	2	no	—
215	C₂×D₅₃	—	4	2	no	—
216	C₂×D₅₃	—	8	4	no	—
217	C₄×C₅₃	—	2	1	no	—
218	C₄×C₅₃	—	2	1	no	—
219	C₄×C₅₃	—	2	1	no	—
220	C₄×C₅₃	—	4	1	no	—
221	C₄×C₅₃	—	4	1	no	—
222	C₄×C₅₃	—	4	1	no	—
223	C₄×C₅₃	#435	1	2	yes	—
224	C₄×C₅₃	C₂²×M₁₆	1	1	yes	characteristic
225	C₄×C₅₃	C₂×#131	1	1	yes	characteristic
226	C₄×C₅₃	#576	1	1	yes	characteristic
227	C₄×C₅₃	#577	1	1	yes	characteristic
228	Dic₅₃	—	4	2	no	—
229	#2989	D₅₃	1	1	yes	characteristic, complemented
230	#2972	—	53	1	no	—
231	C₄×AGL(1,ℤ/8)	—	53	1	no	—
232	#1586	—	53	2	no	—
233	#1585	—	53	2	no	—
234	#1582	—	53	2	no	—
235	#1581	—	53	2	no	—
236	#1565	—	53	1	no	—
237	#1476	—	53	1	no	—
238	C₂×D₄⋊C₈	—	53	1	no	—
239	#1451	—	53	1	no	—
240	C₂×C₅₃	—	2	2	no	—
241	C₂×C₅₃	—	8	1	no	—
242	C₂×C₅₃	C₂×#435	1	1	yes	characteristic
243	C₂×C₅₃	#2946	1	1	yes	characteristic
244	C₂×C₅₃	#2956	1	1	yes	characteristic
245	D₅₃	—	4	4	no	—
246	C₂×AGL(1,ℤ/8)	—	53	1	no	—
247	C₄×#41	C₂×D₅₃	1	1	yes	characteristic
248	C₂×C₄×D₄	—	53	1	no	—
249	#446	C₂×D₅₃	1	2	yes	complemented
250	#442	—	53	2	no	—
251	#440	—	53	2	no	—
252	C₄×SD₁₆	—	53	2	no	—
253	C₄×D₈	—	53	2	no	—
254	#436	C₂×D₅₃	1	2	yes	complemented
255	#435	—	2	2	no	—
256	C₈×D₄	—	53	2	no	—
257	C₈×D₄	—	53	2	no	—
258	C₈×D₄	—	106	1	no	—
259	C₈×D₄	—	106	1	no	—
260	#432	—	2	1	no	—
261	#431	—	53	1	no	—
262	#431	C₂×D₅₃	1	1	yes	characteristic
263	#428	—	53	1	no	—
264	#423	—	2	1	no	—
265	C₂×C₄⋊C₈	C₂×D₅₃	1	1	yes	characteristic
266	#418	—	53	1	no	—
267	#417	—	53	1	no	—
268	C₂×C₂²⋊C₈	—	106	2	no	—
269	C₄×M₁₆	—	53	1	no	—
270	C₂×C₄×C₈	—	53	1	no	—
271	#335	—	53	2	no	—
272	#334	—	53	2	no	—
273	Q₈⋊C₈	—	53	1	no	—
274	Q₈⋊C₈	—	53	1	no	—
275	D₄⋊C₈	—	53	1	no	—
276	D₄⋊C₈	—	53	1	no	—
277	D₄⋊C₈	—	53	2	no	—
278	D₄⋊C₈	—	53	2	no	—
279	#322	—	53	2	no	—
280	#322	—	53	2	no	—
281	C₅₃	#17873	1	1	yes	characteristic, complemented, Sylow
282	C₂×#41	C₄×D₅₃	1	1	yes	characteristic
283	C₂²×D₄	—	53	1	no	—
284	C₂³×C₄	—	106	1	no	—
285	AGL(1,ℤ/8)	—	106	4	no	—
286	C₂×SD₁₆	—	53	2	no	—
287	C₂×D₈	—	53	2	no	—
288	C₂×M₁₆	—	2	2	no	—
289	C₂×M₁₆	—	53	1	no	—
290	C₂×M₁₆	—	53	1	no	—
291	C₂²×C₈	—	2	1	no	—
292	C₂²×C₈	—	106	1	no	—
293	C₂²×C₈	—	106	2	no	—
294	C₂²×C₈	—	106	2	no	—
295	C₄×Q₈	C₄×D₅₃	1	1	yes	characteristic
296	C₄×Q₈	C₂²×D₅₃	1	1	yes	characteristic
297	C₄×D₄	—	2	1	no	—
298	C₄×D₄	—	2	1	no	—
299	C₄×D₄	—	53	2	no	—
300	C₄×D₄	—	53	2	no	—
301	C₄×D₄	—	106	1	no	—
302	C₄×D₄	—	106	1	no	—
303	C₄×D₄	C₄×D₅₃	1	1	yes	characteristic
304	C₄×D₄	C₂²×D₅₃	1	1	yes	characteristic
305	#117	—	2	1	no	—
306	#117	C₄×D₅₃	1	1	yes	characteristic
307	C₂×C₄⋊C₄	—	53	1	no	—
308	C₂×C₂²⋊C₄	—	106	1	no	—
309	C₂×C₄²	—	2	1	no	—
310	C₂×C₄²	C₂²×D₅₃	1	1	yes	characteristic
311	#107	—	53	2	no	—
312	#106	—	53	2	no	—
313	C₄⋊C₈	—	2	2	no	—
314	C₄⋊C₈	—	2	1	no	—
315	C₄⋊C₈	—	53	1	no	—
316	C₄⋊C₈	—	53	1	no	—
317	C₄⋊C₈	C₂²⋊D₅₃	1	1	yes	characteristic
318	C₄⋊C₈	C₂²⋊D₅₃	1	1	yes	characteristic
319	C₄⋊C₈	C₂²×D₅₃	1	2	yes	—
320	#103	—	53	2	no	—
321	#102	—	53	2	no	—
322	#102	—	53	2	no	—
323	#102	—	53	2	no	—
324	C₂²⋊C₈	—	2	2	no	—
325	C₂²⋊C₈	—	106	1	no	—
326	C₂²⋊C₈	—	106	2	no	—
327	C₂²⋊C₈	—	106	2	no	—
328	C₂²⋊C₈	C₂²⋊D₅₃	1	2	yes	—
329	#97	—	2	2	no	—
330	#97	—	53	2	no	—
331	C₄×C₈	—	53	2	no	—
332	C₄×C₈	—	53	1	no	—
333	C₄×C₈	—	53	1	no	—
334	C₄×C₈	—	106	1	no	—
335	C₄×C₈	—	106	1	no	—
336	C₄×C₈	C₂²×D₅₃	1	2	yes	—
337	C₂⁴	—	106	1	no	—
338	#41	—	2	2	no	—
339	#41	—	2	2	no	—
340	C₂×Q₈	C₂×C₄×D₅₃	1	1	yes	characteristic
341	C₂×D₄	—	2	1	no	—
342	C₂×D₄	—	53	4	no	—
343	C₂×D₄	—	53	2	no	—
344	C₂×D₄	—	106	1	no	—
345	C₂×D₄	—	106	2	no	—
346	C₂×D₄	C₂×C₄×D₅₃	1	1	yes	characteristic
347	C₂²×C₄	—	2	1	no	—
348	C₂²×C₄	—	2	1	no	—
349	C₂²×C₄	—	2	1	no	—
350	C₂²×C₄	—	106	1	no	—
351	C₂²×C₄	—	106	2	no	—
352	C₂²×C₄	—	106	2	no	—
353	C₂²×C₄	—	424	1	no	—
354	C₂²×C₄	—	424	1	no	—
355	C₂²×C₄	C₂×C₄×D₅₃	1	1	yes	characteristic
356	C₂²×C₄	#11719945	1	1	yes	characteristic
357	C₂²×C₄	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
358	SD₁₆	—	106	4	no	—
359	D₈	—	106	4	no	—
360	M₁₆	—	8	2	no	—
361	M₁₆	—	106	2	no	—
362	M₁₆	—	106	2	no	—
363	C₂×C₈	—	2	2	no	—
364	C₂×C₈	—	8	1	no	—
365	C₂×C₈	—	53	2	no	—
366	C₂×C₈	—	53	2	no	—
367	C₂×C₈	—	106	1	no	—
368	C₂×C₈	—	106	1	no	—
369	C₂×C₈	—	106	1	no	—
370	C₂×C₈	—	106	1	no	—
371	C₂×C₈	—	424	2	no	—
372	C₂×C₈	—	424	2	no	—
373	C₂×C₈	—	212	2	no	—
374	C₂×C₈	C₂×C₂²⋊D₅₃	1	2	yes	—
375	C₂×C₈	C₂×C₂²⋊D₅₃	1	2	yes	—
376	C₄⋊C₄	—	2	1	no	—
377	C₄⋊C₄	—	2	1	no	—
378	C₄⋊C₄	—	53	2	no	—
379	C₄⋊C₄	—	106	1	no	—
380	C₄⋊C₄	C₂×C₄×D₅₃	1	1	yes	characteristic
381	C₄⋊C₄	C₂×C₄×D₅₃	1	1	yes	characteristic
382	C₂²⋊C₄	—	2	1	no	—
383	C₂²⋊C₄	—	2	1	no	—
384	C₂²⋊C₄	—	2	1	no	—
385	C₂²⋊C₄	—	106	2	no	—
386	C₂²⋊C₄	—	106	2	no	—
387	C₄²	—	2	1	no	—
388	C₄²	—	2	1	no	—
389	C₄²	—	2	1	no	—
390	C₄²	C₂×C₄×D₅₃	1	1	yes	characteristic
391	C₄²	#11719945	1	1	yes	characteristic
392	C₄²	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
393	C₄²	C₂³×D₅₃	1	1	yes	characteristic
394	C₂³	—	2	1	no	—
395	C₂³	—	106	4	no	—
396	C₂³	—	106	2	no	—
397	C₂³	—	424	1	no	—
398	C₂³	—	424	1	no	—
399	C₂³	C₄×C₂²⋊D₅₃	1	1	yes	characteristic
400	Q₈	—	2	1	no	—
401	Q₈	—	2	1	no	—
402	D₄	—	2	1	no	—
403	D₄	—	2	1	no	—
404	D₄	—	4	2	no	—
405	D₄	—	53	4	no	—
406	D₄	—	106	2	no	—
407	D₄	—	106	4	no	—
408	C₂×C₄	—	2	1	no	—
409	C₂×C₄	—	2	1	no	—
410	C₂×C₄	—	2	1	no	—
411	C₂×C₄	—	2	1	no	—
412	C₂×C₄	—	2	1	no	—
413	C₂×C₄	—	4	1	no	—
414	C₂×C₄	—	4	1	no	—
415	C₂×C₄	—	4	1	no	—
416	C₂×C₄	—	4	2	no	—
417	C₂×C₄	—	8	1	no	—
418	C₂×C₄	—	106	2	no	—
419	C₂×C₄	—	424	2	no	—
420	C₂×C₄	—	424	1	no	—
421	C₂×C₄	—	212	2	no	—
422	C₂×C₄	—	212	2	no	—
423	C₂×C₄	C₄×C₂²⋊D₅₃	1	1	yes	characteristic
424	C₂×C₄	C₄×C₂²⋊D₅₃	1	1	yes	characteristic
425	C₂×C₄	C₄×C₂²⋊D₅₃	1	1	yes	characteristic
426	C₂×C₄	M₁₆×D₅₃	1	2	yes	—
427	C₂×C₄	#13133225	1	2	yes	—
428	C₂×C₄	C₂²×C₄×D₅₃	1	1	yes	characteristic
429	C₂×C₄	C₂²×C₂²⋊D₅₃	1	1	yes	characteristic
430	C₂×C₄	C₂×#11719945	1	1	yes	characteristic
431	C₈	—	2	2	no	—
432	C₈	—	2	2	no	—
433	C₈	—	4	2	no	—
434	C₈	—	106	2	no	—
435	C₈	—	106	2	no	—
436	C₈	—	106	2	no	—
437	C₈	—	212	1	no	—
438	C₈	—	212	1	no	—
439	C₈	—	212	1	no	—
440	C₂²	—	2	1	no	—
441	C₂²	—	2	1	no	—
442	C₂²	—	4	1	no	—
443	C₂²	—	4	1	no	—
444	C₂²	—	4	1	no	—
445	C₂²	—	106	4	no	—
446	C₂²	—	424	4	no	—
447	C₂²	—	212	2	no	—
448	C₂²	—	212	2	no	—
449	C₂²	#16810692	1	2	yes	—
450	C₂²	C₂×C₄×C₂²⋊D₅₃	1	1	yes	characteristic
451	C₄	—	2	1	no	—
452	C₄	—	2	1	no	—
453	C₄	—	2	1	no	—
454	C₄	—	4	1	no	—
455	C₄	—	4	1	no	—
456	C₄	—	4	1	no	—
457	C₄	—	212	2	no	—
458	C₄	#16810692	1	2	yes	—
459	C₄	C₂×M₁₆×D₅₃	1	1	yes	characteristic
460	C₄	C₂×#13133225	1	1	yes	characteristic
461	C₄	#16811426	1	1	yes	characteristic
462	C₄	#16811431	1	1	yes	characteristic
463	C₂	—	2	2	no	—
464	C₂	—	8	1	no	—
465	C₂	—	212	4	no	—
466	C₂	C₂×#16810692	1	1	yes	characteristic
467	C₂	#20283543	1	1	yes	characteristic
468	C₂	#20283622	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×#2989	C₂
socle	C₂²×C₅₃	C₂×C₄×D₄
prime radical	C₂×#13133242	C₄
cosocle	C₄²×C₅₃	C₂⁴
hypercenter	#2989	D₅₃
hypocenter	C₅₃	#17873
LCS	G ▷ C₂×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂×C₄ ◁ #431 ◁ #2989

Complements

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

N	H ≅ G/N	type
1. #20283738	C₂	semidirect
3. #20273577	C₂	semidirect
4. #20273570	C₂	semidirect
5. #20273549	C₂	semidirect
6. #20273542	C₂	semidirect
7. #20273465	C₂	semidirect
8. #20273097	C₂	semidirect
9. C₂×#16810350	C₂	semidirect
10. #20273005	C₂	semidirect
11. C₅₃×#2989	C₂	semidirect
29. #16810375	C₂²	semidirect
30. #16810373	C₂²	semidirect
31. Q₈⋊#88322	C₂²	semidirect
32. Q₈⋊#88322	C₂²	semidirect
33. D₄⋊#88322	C₂²	semidirect
34. D₄⋊#88322	C₂²	semidirect
37. #16810345	C₂²	semidirect
38. #16810344	C₂²	semidirect
40. C₅₃×#446	C₂²	semidirect
41. C₅₃×#436	C₂²	semidirect
117. C₂×D₂₁₂	M₁₆	semidirect
229. #2989	D₅₃	semidirect
249. #446	C₂×D₅₃	semidirect
254. #436	C₂×D₅₃	semidirect
281. C₅₃	#17873	semidirect

Maximal subgroups

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

subgroup	index
#20283738	2
C₄×#13133242	2
#20273577	2
#20273570	2
#20273549	2
#20273542	2
#20273465	2
#20273097	2
C₂×#16810350	2
#20273005	2
C₅₃×#2989	2
#17873	53