Subgroups — #27354700

Order 13568.

662 conjugacy classes of subgroups.

The subgroup lattice has 464 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20283691	C₂	1	1	yes	maximal, characteristic, complemented
2	#20283427	C₂	1	1	yes	maximal, characteristic, complemented
3	#20273606	C₂	1	2	yes	maximal, complemented
4	#20273600	C₂	1	2	yes	maximal, complemented
5	#20273539	C₂	1	2	yes	maximal, complemented
6	SD₁₆⋊#88322	C₂	1	2	yes	maximal, complemented
7	#20273482	C₂	1	1	yes	maximal, characteristic, complemented
8	#20273184	C₂	1	1	yes	maximal, characteristic, complemented
9	#20273180	C₂	1	1	yes	maximal, characteristic, complemented
10	M₁₆×#88322	C₂	1	1	yes	maximal, characteristic, complemented
11	C₅₃×#3171	C₂	1	1	yes	maximal, characteristic
12	C₂×#13133224	—	2	1	no	—
13	#16810754	C₂²	1	1	yes	characteristic
14	Q₈×#88322	C₂²	1	1	yes	characteristic
15	#16810701	—	2	1	no	—
16	#16810701	C₂²	1	2	yes	—
17	#16810701	C₂²	1	1	yes	characteristic
18	#16810695	—	2	1	no	—
19	D₄×#88322	—	2	1	no	—
20	D₄×#88322	—	2	1	no	—
21	D₄×#88322	C₂²	1	2	yes	—
22	D₄×#88322	C₂²	1	1	yes	characteristic
23	#16810684	—	2	1	no	—
24	#16810680	C₂²	1	1	yes	characteristic
25	#16810676	C₂²	1	1	yes	characteristic
26	#16810647	C₂²	1	1	yes	characteristic
27	#16810590	—	2	1	no	—
28	#16810579	—	2	1	no	—
29	C₂×C₄×#88322	C₂²	1	1	yes	characteristic
30	#16810376	C₂²	1	2	yes	—
31	#16810374	C₂²	1	2	yes	—
32	Q₈⋊#88322	C₂²	1	2	yes	—
33	D₄⋊#88322	C₂²	1	2	yes	—
34	D₄⋊#88322	C₂²	1	2	yes	—
35	D₄⋊#88322	C₂²	1	2	yes	—
36	#16810344	C₂²	1	2	yes	—
37	C₈×#88322	C₂²	1	2	yes	—
38	C₅₃×#532	C₂²	1	1	yes	characteristic
39	C₅₃×#529	C₂²	1	1	yes	characteristic
40	C₅₃×#501	C₄	1	1	yes	characteristic
41	C₅₃×#496	C₄	1	1	yes	characteristic
42	C₅₃×#490	C₄	1	1	yes	characteristic
43	C₅₃×#489	C₄	1	1	yes	characteristic
44	C₅₃×#469	C₄	1	2	yes	—
45	C₅₃×#468	C₄	1	2	yes	—
46	C₄×SD₁₆×C₅₃	C₂²	1	2	yes	—
47	C₄×D₈×C₅₃	C₂²	1	2	yes	—
48	C₄×M₁₆×C₅₃	C₂²	1	1	yes	characteristic
49	#13133224	—	4	4	no	—
50	C₂×#11719917	—	2	1	no	—
51	C₂×#11719917	—	2	2	no	—
52	C₂×#11719917	—	2	1	no	—
53	C₂²×#88322	—	2	1	no	—
54	C₂²×#88322	—	2	2	no	—
55	C₂²×#88322	D₄	1	1	yes	characteristic
56	C₂²×#88322	D₄	1	1	yes	characteristic
57	C₄⋊#88322	—	2	1	no	—
58	C₄⋊#88322	—	2	1	no	—
59	C₄⋊#88322	—	2	1	no	—
60	C₄⋊#88322	—	2	1	no	—
61	C₄⋊#88322	C₂³	1	1	yes	characteristic
62	C₄⋊#88322	C₂³	1	1	yes	characteristic
63	C₄⋊#88322	C₂³	1	2	yes	—
64	C₂²⋊#88322	—	2	2	no	—
65	C₂²⋊#88322	—	2	2	no	—
66	C₂²⋊#88322	—	2	1	no	—
67	C₂²⋊#88322	—	2	1	no	—
68	C₂²⋊#88322	—	2	1	no	—
69	C₂²⋊#88322	—	2	1	no	—
70	#13133128	—	2	1	no	—
71	#13133128	—	2	1	no	—
72	C₄×#88322	—	2	1	no	—
73	C₄×#88322	D₄	1	1	yes	characteristic
74	C₄×#88322	D₄	1	1	yes	characteristic
75	C₄×#88322	C₂³	1	2	yes	—
76	C₄×#88322	C₂³	1	1	yes	characteristic
77	C₄×#88322	C₂³	1	1	yes	characteristic
78	C₂×#41×C₅₃	—	2	1	no	—
79	C₂×SD₁₆×C₅₃	C₂×C₄	1	2	yes	—
80	C₂×D₈×C₅₃	C₂×C₄	1	2	yes	—
81	C₂×M₁₆×C₅₃	C₂×C₄	1	2	yes	—
82	C₄⋊Q₈×C₅₃	C₂×C₄	1	1	yes	characteristic
83	#127×C₅₃	C₂×C₄	1	1	yes	characteristic
84	#126×C₅₃	—	2	1	no	—
85	#125×C₅₃	C₂×C₄	1	1	yes	characteristic
86	#124×C₅₃	—	2	1	no	—
87	#124×C₅₃	C₂×C₄	1	1	yes	characteristic
88	#123×C₅₃	—	2	1	no	—
89	C₂²⋊Q₈×C₅₃	C₂×C₄	1	1	yes	characteristic
90	#121×C₅₃	—	2	1	no	—
91	#121×C₅₃	C₂×C₄	1	1	yes	characteristic
92	#121×C₅₃	C₂×C₄	1	2	yes	—
93	C₄×Q₈×C₅₃	C₂³	1	1	yes	characteristic
94	C₄×D₄×C₅₃	—	2	1	no	—
95	C₄×D₄×C₅₃	—	2	1	no	—
96	C₄×D₄×C₅₃	C₂³	1	1	yes	characteristic
97	C₄×D₄×C₅₃	C₂³	1	2	yes	—
98	#117×C₅₃	—	2	1	no	—
99	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
100	#107×C₅₃	C₂×C₄	1	2	yes	—
101	#106×C₅₃	C₂×C₄	1	2	yes	—
102	#103×C₅₃	C₂×C₄	1	2	yes	—
103	#102×C₅₃	C₂×C₄	1	2	yes	—
104	#102×C₅₃	C₂×C₄	1	2	yes	—
105	#102×C₅₃	C₂×C₄	1	2	yes	—
106	#97×C₅₃	C₂×C₄	1	1	yes	characteristic
107	#97×C₅₃	C₂×C₄	1	1	yes	characteristic
108	C₄×C₈×C₅₃	C₂³	1	2	yes	—
109	#41×C₅₃	—	4	2	no	—
110	#41×C₅₃	—	4	2	no	—
111	C₂×Q₈×C₅₃	—	2	1	no	—
112	C₂×Q₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
113	C₂×D₄×C₅₃	—	2	1	no	—
114	C₂×D₄×C₅₃	—	2	2	no	—
115	C₂×D₄×C₅₃	—	2	1	no	—
116	C₂×D₄×C₅₃	—	2	1	no	—
117	C₂×D₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
118	C₂×D₄×C₅₃	C₂²×C₄	1	2	yes	—
119	C₂²×C₄×C₅₃	—	2	1	no	—
120	C₂²×C₄×C₅₃	—	2	2	no	—
121	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
122	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
123	C₂²×C₄×C₅₃	#41	1	1	yes	characteristic
124	SD₁₆×C₅₃	—	2	4	no	—
125	D₈×C₅₃	—	2	4	no	—
126	M₁₆×C₅₃	—	2	4	no	—
127	C₂×C₈×C₅₃	C₂²×C₄	1	2	yes	—
128	C₂×C₈×C₅₃	C₂²×C₄	1	2	yes	—
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	1	no	—
133	C₄⋊C₄×C₅₃	—	2	1	no	—
134	C₄⋊C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
135	C₄⋊C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
136	C₄⋊C₄×C₅₃	C₂²×C₄	1	2	yes	—
137	C₂²⋊C₄×C₅₃	—	2	1	no	—
138	C₂²⋊C₄×C₅₃	—	2	1	no	—
139	C₂²⋊C₄×C₅₃	—	2	1	no	—
140	C₂²⋊C₄×C₅₃	—	2	2	no	—
141	C₂²⋊C₄×C₅₃	—	2	1	no	—
142	C₂²⋊C₄×C₅₃	—	2	2	no	—
143	C₂²⋊C₄×C₅₃	—	2	1	no	—
144	C₄²×C₅₃	—	2	1	no	—
145	C₄²×C₅₃	C₂²×C₄	1	1	yes	characteristic
146	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
147	C₄²×C₅₃	#41	1	1	yes	characteristic
148	C₄²×C₅₃	C₂⁴	1	1	yes	characteristic
149	#11719917	—	4	2	no	—
150	#11719917	—	4	2	no	—
151	#11719917	—	4	4	no	—
152	C₂×#88322	—	2	2	no	—
153	C₂×#88322	—	2	1	no	—
154	C₂×#88322	—	2	1	no	—
155	C₂×#88322	—	2	2	no	—
156	C₂×#88322	—	2	2	no	—
157	C₂×#88322	—	4	1	no	—
158	C₂×#88322	—	4	2	no	—
159	C₂×#88322	—	4	4	no	—
160	C₂×#88322	C₂×D₄	1	1	yes	characteristic
161	C₂×#88322	C₂×D₄	1	1	yes	characteristic
162	C₂×#88322	C₂×D₄	1	1	yes	characteristic
163	C₂×#88322	C₂×D₄	1	1	yes	characteristic
164	C₂³×C₅₃	—	2	2	no	—
165	C₂³×C₅₃	—	2	1	no	—
166	C₂³×C₅₃	C₄×D₄	1	1	yes	characteristic
167	Q₈×C₅₃	—	2	2	no	—
168	Q₈×C₅₃	—	4	2	no	—
169	D₄×C₅₃	—	2	2	no	—
170	D₄×C₅₃	—	2	2	no	—
171	D₄×C₅₃	—	2	2	no	—
172	D₄×C₅₃	—	4	2	no	—
173	D₄×C₅₃	—	4	2	no	—
174	D₄×C₅₃	—	4	2	no	—
175	D₄×C₅₃	—	4	4	no	—
176	C₂×C₄×C₅₃	—	2	2	no	—
177	C₂×C₄×C₅₃	—	2	1	no	—
178	C₂×C₄×C₅₃	—	2	1	no	—
179	C₂×C₄×C₅₃	—	2	2	no	—
180	C₂×C₄×C₅₃	—	2	1	no	—
181	C₂×C₄×C₅₃	—	2	1	no	—
182	C₂×C₄×C₅₃	—	2	1	no	—
183	C₂×C₄×C₅₃	—	4	1	no	—
184	C₂×C₄×C₅₃	—	4	2	no	—
185	C₂×C₄×C₅₃	—	4	1	no	—
186	C₂×C₄×C₅₃	—	4	2	no	—
187	C₂×C₄×C₅₃	—	4	1	no	—
188	C₂×C₄×C₅₃	C₄×D₄	1	1	yes	characteristic
189	C₂×C₄×C₅₃	C₄×D₄	1	1	yes	characteristic
190	C₂×C₄×C₅₃	C₄×D₄	1	1	yes	characteristic
191	C₂×C₄×C₅₃	C₂³×C₄	1	1	yes	characteristic
192	C₂×C₄×C₅₃	C₂²×D₄	1	1	yes	characteristic
193	C₂×C₄×C₅₃	C₂×#41	1	1	yes	characteristic
194	C₈×C₅₃	—	2	2	no	—
195	C₈×C₅₃	#131	1	4	yes	complemented
196	#88322	—	2	1	no	—
197	#88322	—	2	1	no	—
198	#88322	—	2	1	no	—
199	#88322	—	4	2	no	—
200	#88322	—	4	2	no	—
201	#88322	—	4	1	no	—
202	#88322	—	4	1	no	—
203	#88322	AGL(1,ℤ/8)	1	2	yes	—
204	#17966	—	53	1	no	maximal, Sylow
205	C₂²×C₅₃	—	2	1	no	—
206	C₂²×C₅₃	—	2	1	no	—
207	C₂²×C₅₃	—	2	1	no	—
208	C₂²×C₅₃	—	4	1	no	—
209	C₂²×C₅₃	—	4	2	no	—
210	C₂²×C₅₃	—	4	2	no	—
211	C₂²×C₅₃	—	4	1	no	—
212	C₂²×C₅₃	—	4	2	no	—
213	C₂²×C₅₃	—	4	1	no	—
214	C₂²×C₅₃	C₂×C₄×D₄	1	1	yes	characteristic
215	C₄×C₅₃	—	2	1	no	—
216	C₄×C₅₃	—	2	1	no	—
217	C₄×C₅₃	—	2	1	no	—
218	C₄×C₅₃	—	2	1	no	—
219	C₄×C₅₃	—	4	1	no	—
220	C₄×C₅₃	—	4	1	no	—
221	C₄×C₅₃	—	4	2	no	—
222	C₄×C₅₃	—	4	1	no	—
223	C₄×C₅₃	C₂×#131	1	2	yes	—
224	C₄×C₅₃	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
225	C₄×C₅₃	#575	1	1	yes	characteristic
226	#3171	D₅₃	1	1	yes	characteristic
227	#2966	—	53	1	no	—
228	#2931	—	53	1	no	—
229	#1590	—	53	2	no	—
230	#1589	—	53	2	no	—
231	#1580	—	53	2	no	—
232	SD₁₆⋊C₈	—	53	2	no	—
233	#1569	—	53	1	no	—
234	#1496	—	53	1	no	—
235	#1495	—	53	1	no	—
236	C₈×M₁₆	—	53	1	no	—
237	C₂×C₅₃	—	4	1	no	—
238	C₂×C₅₃	—	8	2	no	—
239	C₂×C₅₃	—	8	1	no	—
240	C₂×C₅₃	#2931	1	1	yes	characteristic
241	C₂×C₅₃	C₄×AGL(1,ℤ/8)	1	1	yes	characteristic
242	C₂×C₅₃	#2955	1	1	yes	characteristic
243	C₂×#131	—	106	1	no	—
244	#532	C₂×D₅₃	1	1	yes	characteristic
245	#529	C₂×D₅₃	1	1	yes	characteristic
246	#501	Dic₅₃	1	1	yes	characteristic
247	#496	Dic₅₃	1	1	yes	characteristic
248	#490	Dic₅₃	1	1	yes	characteristic
249	#489	Dic₅₃	1	1	yes	characteristic
250	#469	Dic₅₃	1	2	yes	—
251	#468	Dic₅₃	1	2	yes	—
252	#446	—	53	1	no	—
253	C₈×Q₈	—	53	1	no	—
254	C₄×SD₁₆	C₂×D₅₃	1	2	yes	—
255	C₄×D₈	C₂×D₅₃	1	2	yes	—
256	#436	—	53	1	no	—
257	#436	—	53	2	no	—
258	#436	—	106	1	no	—
259	#435	—	106	1	no	—
260	C₈×D₄	—	53	2	no	—
261	C₈×D₄	—	53	1	no	—
262	C₈×D₄	—	106	1	no	—
263	C₈×D₄	—	106	1	no	—
264	#433	—	106	1	no	—
265	#432	—	53	1	no	—
266	#431	—	53	1	no	—
267	#423	—	53	1	no	—
268	#408	—	106	1	no	—
269	#405	—	106	1	no	—
270	C₄×M₁₆	C₂×D₅₃	1	1	yes	characteristic
271	C₂×C₄×C₈	—	53	1	no	—
272	#335	—	53	2	no	—
273	#334	—	53	2	no	—
274	Q₈⋊C₈	—	53	2	no	—
275	D₄⋊C₈	—	53	2	no	—
276	D₄⋊C₈	—	53	2	no	—
277	D₄⋊C₈	—	53	2	no	—
278	#322	—	53	2	no	—
279	C₈²	—	53	2	no	—
280	C₅₃	#17966	1	1	yes	characteristic, complemented, Sylow
281	C₂×#41	—	2	1	no	—
282	C₂×SD₁₆	C₂×Dic₅₃	1	2	yes	—
283	C₂×D₈	C₂×Dic₅₃	1	2	yes	—
284	#131	—	212	4	no	—
285	C₂×M₁₆	—	106	2	no	—
286	C₂×M₁₆	—	106	1	no	—
287	C₂×M₁₆	—	106	1	no	—
288	C₂×M₁₆	C₂×Dic₅₃	1	2	yes	—
289	C₂²×C₈	—	53	1	no	—
290	C₂²×C₈	—	53	1	no	—
291	C₂²×C₈	—	106	2	no	—
292	C₂²×C₈	—	106	1	no	—
293	C₄⋊Q₈	C₂×Dic₅₃	1	1	yes	characteristic
294	#127	C₂×Dic₅₃	1	1	yes	characteristic
295	#126	—	2	1	no	—
296	#125	C₂×Dic₅₃	1	1	yes	characteristic
297	#124	—	2	1	no	—
298	#124	C₂×Dic₅₃	1	1	yes	characteristic
299	#123	—	2	1	no	—
300	C₂²⋊Q₈	C₂×Dic₅₃	1	1	yes	characteristic
301	#121	—	2	1	no	—
302	#121	C₂×Dic₅₃	1	1	yes	characteristic
303	#121	C₂×Dic₅₃	1	2	yes	—
304	C₄×Q₈	C₂²×D₅₃	1	1	yes	characteristic
305	C₄×D₄	—	2	1	no	—
306	C₄×D₄	—	2	1	no	—
307	C₄×D₄	C₂²×D₅₃	1	1	yes	characteristic
308	C₄×D₄	C₂²×D₅₃	1	2	yes	—
309	#117	—	2	1	no	—
310	C₂×C₄²	C₂²×D₅₃	1	1	yes	characteristic
311	#107	C₂×Dic₅₃	1	2	yes	—
312	#106	C₂×Dic₅₃	1	2	yes	—
313	C₄⋊C₈	—	53	1	no	—
314	C₄⋊C₈	—	53	1	no	—
315	C₄⋊C₈	—	53	2	no	—
316	C₄⋊C₈	—	106	1	no	—
317	C₄⋊C₈	—	106	1	no	—
318	C₄⋊C₈	—	106	1	no	—
319	C₄⋊C₈	—	106	1	no	—
320	#103	C₂×Dic₅₃	1	2	yes	—
321	#102	C₂×Dic₅₃	1	2	yes	—
322	#102	C₂×Dic₅₃	1	2	yes	—
323	#102	C₂×Dic₅₃	1	2	yes	—
324	C₂²⋊C₈	—	106	2	no	—
325	C₂²⋊C₈	—	106	2	no	—
326	C₂²⋊C₈	—	106	1	no	—
327	C₂²⋊C₈	—	106	1	no	—
328	C₂²⋊C₈	—	106	1	no	—
329	C₂²⋊C₈	—	106	1	no	—
330	#97	—	106	1	no	—
331	#97	—	106	1	no	—
332	#97	C₂×Dic₅₃	1	1	yes	characteristic
333	#97	C₂×Dic₅₃	1	1	yes	characteristic
334	C₄×C₈	—	53	1	no	—
335	C₄×C₈	—	53	1	no	—
336	C₄×C₈	—	53	1	no	—
337	C₄×C₈	—	53	1	no	—
338	C₄×C₈	—	53	2	no	—
339	C₄×C₈	—	106	1	no	—
340	C₄×C₈	C₂²×D₅₃	1	2	yes	—
341	#41	—	4	2	no	—
342	#41	—	4	2	no	—
343	C₂×Q₈	—	2	1	no	—
344	C₂×Q₈	C₂²×Dic₅₃	1	1	yes	characteristic
345	C₂×D₄	—	2	1	no	—
346	C₂×D₄	—	2	2	no	—
347	C₂×D₄	—	2	1	no	—
348	C₂×D₄	—	2	1	no	—
349	C₂×D₄	C₂²×Dic₅₃	1	1	yes	characteristic
350	C₂×D₄	C₂²×Dic₅₃	1	2	yes	—
351	C₂²×C₄	—	2	1	no	—
352	C₂²×C₄	—	2	2	no	—
353	C₂²×C₄	D₄×D₅₃	1	1	yes	characteristic
354	C₂²×C₄	#11719947	1	1	yes	characteristic
355	C₂²×C₄	C₂²×Dic₅₃	1	1	yes	characteristic
356	SD₁₆	—	2	4	no	—
357	D₈	—	2	4	no	—
358	M₁₆	—	2	4	no	—
359	M₁₆	—	212	4	no	—
360	M₁₆	—	212	2	no	—
361	M₁₆	—	212	2	no	—
362	C₂×C₈	—	53	1	no	—
363	C₂×C₈	—	53	1	no	—
364	C₂×C₈	—	53	1	no	—
365	C₂×C₈	—	53	1	no	—
366	C₂×C₈	—	106	2	no	—
367	C₂×C₈	—	106	1	no	—
368	C₂×C₈	—	106	1	no	—
369	C₂×C₈	—	106	2	no	—
370	C₂×C₈	—	106	2	no	—
371	C₂×C₈	—	212	1	no	—
372	C₂×C₈	—	212	4	no	—
373	C₂×C₈	—	212	2	no	—
374	C₂×C₈	C₂²×Dic₅₃	1	2	yes	—
375	C₂×C₈	C₂²×Dic₅₃	1	2	yes	—
376	C₄⋊C₄	—	2	1	no	—
377	C₄⋊C₄	—	2	1	no	—
378	C₄⋊C₄	—	2	1	no	—
379	C₄⋊C₄	—	2	1	no	—
380	C₄⋊C₄	—	2	1	no	—
381	C₄⋊C₄	C₂²×Dic₅₃	1	1	yes	characteristic
382	C₄⋊C₄	C₂²×Dic₅₃	1	1	yes	characteristic
383	C₄⋊C₄	C₂²×Dic₅₃	1	2	yes	—
384	C₂²⋊C₄	—	2	1	no	—
385	C₂²⋊C₄	—	2	1	no	—
386	C₂²⋊C₄	—	2	1	no	—
387	C₂²⋊C₄	—	2	2	no	—
388	C₂²⋊C₄	—	2	1	no	—
389	C₂²⋊C₄	—	2	2	no	—
390	C₂²⋊C₄	—	2	1	no	—
391	C₄²	—	2	1	no	—
392	C₄²	D₄×D₅₃	1	1	yes	characteristic
393	C₄²	#11719947	1	1	yes	characteristic
394	C₄²	C₂²×Dic₅₃	1	1	yes	characteristic
395	C₄²	C₂³×D₅₃	1	1	yes	characteristic
396	C₂³	—	2	1	no	—
397	C₂³	—	2	2	no	—
398	C₂³	D₄×Dic₅₃	1	1	yes	characteristic
399	Q₈	—	2	2	no	—
400	Q₈	—	4	2	no	—
401	D₄	—	2	2	no	—
402	D₄	—	2	2	no	—
403	D₄	—	2	2	no	—
404	D₄	—	4	2	no	—
405	D₄	—	4	2	no	—
406	D₄	—	4	2	no	—
407	D₄	—	4	4	no	—
408	C₂×C₄	—	2	2	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	—	2	1	no	—
414	C₂×C₄	—	2	2	no	—
415	C₂×C₄	—	4	1	no	—
416	C₂×C₄	—	4	2	no	—
417	C₂×C₄	—	4	1	no	—
418	C₂×C₄	—	4	1	no	—
419	C₂×C₄	—	4	2	no	—
420	C₂×C₄	D₄×Dic₅₃	1	1	yes	characteristic
421	C₂×C₄	D₄×Dic₅₃	1	1	yes	characteristic
422	C₂×C₄	D₄×Dic₅₃	1	1	yes	characteristic
423	C₂×C₄	C₂³×Dic₅₃	1	1	yes	characteristic
424	C₂×C₄	C₂×D₄×D₅₃	1	1	yes	characteristic
425	C₂×C₄	C₂×#11719947	1	1	yes	characteristic
426	C₈	—	2	2	no	—
427	C₈	—	53	2	no	—
428	C₈	—	106	1	no	—
429	C₈	—	106	1	no	—
430	C₈	—	106	1	no	—
431	C₈	—	212	2	no	—
432	C₈	—	212	2	no	—
433	C₈	—	212	1	no	—
434	C₈	—	212	1	no	—
435	C₈	#13133224	1	4	yes	complemented
436	C₂²	—	2	1	no	—
437	C₂²	—	2	1	no	—
438	C₂²	—	2	1	no	—
439	C₂²	—	4	1	no	—
440	C₂²	—	4	2	no	—
441	C₂²	—	4	2	no	—
442	C₂²	—	4	1	no	—
443	C₂²	—	4	2	no	—
444	C₂²	—	4	1	no	—
445	C₂²	C₂×D₄×Dic₅₃	1	1	yes	characteristic
446	C₄	—	2	1	no	—
447	C₄	—	2	1	no	—
448	C₄	—	2	1	no	—
449	C₄	—	2	1	no	—
450	C₄	—	4	1	no	—
451	C₄	—	4	2	no	—
452	C₄	—	4	1	no	—
453	C₄	—	4	1	no	—
454	C₄	C₂×#13133224	1	2	yes	—
455	C₄	AGL(1,ℤ/8)×D₅₃	1	1	yes	characteristic
456	C₄	#16811419	1	1	yes	characteristic
457	C₂	—	4	1	no	—
458	C₂	—	8	2	no	—
459	C₂	—	8	1	no	—
460	C₂	#20283427	1	1	yes	characteristic
461	C₂	AGL(1,ℤ/8)×Dic₅₃	1	1	yes	characteristic
462	C₂	#20283618	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂×D₄×D₅₃
commutator	C₂×C₄×C₅₃	C₂³×C₄
Frattini	C₄²	C₂³×D₅₃
Fitting, prime radical	C₅₃×#3171	C₂
socle	C₂²×C₅₃	C₂×C₄×D₄
cosocle	C₄²×C₅₃	C₂⁴
hypercenter	#3171	D₅₃
hypocenter	C₅₃	#17966
LCS	G ▷ C₂×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #3171

Complements

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

N	H ≅ G/N	type
1. #20283691	C₂	semidirect
2. #20283427	C₂	semidirect
3. #20273606	C₂	semidirect
4. #20273600	C₂	semidirect
5. #20273539	C₂	semidirect
6. SD₁₆⋊#88322	C₂	semidirect
7. #20273482	C₂	semidirect
8. #20273184	C₂	semidirect
9. #20273180	C₂	semidirect
10. M₁₆×#88322	C₂	semidirect
195. C₈×C₅₃	#131	semidirect
280. C₅₃	#17966	semidirect
435. C₈	#13133224	semidirect

Maximal subgroups

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

subgroup	index
#20283691	2
#20283427	2
#20273606	2
#20273600	2
#20273539	2
SD₁₆⋊#88322	2
#20273482	2
#20273184	2
#20273180	2
M₁₆×#88322	2
C₅₃×#3171	2
#17966	53