Subgroups — #27354099

Order 13568.

632 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	#20284787	C₂	1	1	yes	maximal, characteristic
2	C₈×#11719945	C₂	1	1	yes	maximal, characteristic
3	#20283649	C₂	1	1	yes	maximal, characteristic, complemented
4	#20273588	C₂	1	2	yes	maximal, complemented
5	#20273574	C₂	1	2	yes	maximal, complemented
6	C₈×Dic₂₁₂	C₂	1	2	yes	maximal, complemented
7	C₈×#11719913	C₂	1	2	yes	maximal, complemented
8	#20273504	C₂	1	1	yes	maximal, characteristic, complemented
9	#20273155	C₂	1	1	yes	maximal, characteristic
10	#20273137	C₂	1	1	yes	maximal, characteristic, complemented
11	C₈×M₁₆×C₅₃	C₂	1	1	yes	maximal, characteristic, complemented
12	#16811180	C₂²	1	1	yes	characteristic
13	C₄×#11719945	C₂²	1	1	yes	characteristic
14	#16810965	C₄	1	2	yes	—
15	#16810909	C₄	1	1	yes	characteristic
16	#16810904	C₄	1	1	yes	characteristic
17	#16810834	C₄	1	1	yes	characteristic
18	#16810829	C₄	1	1	yes	characteristic
19	#16810787	C₄	1	2	yes	—
20	#16810753	—	2	1	no	—
21	#16810753	C₂²	1	2	yes	—
22	C₈×Dic₁₀₆	—	2	1	no	—
23	C₈×Dic₁₀₆	C₂²	1	2	yes	—
24	C₈×Dic₁₀₆	C₂²	1	1	yes	characteristic
25	C₈×Dic₁₀₆	C₂²	1	1	yes	characteristic
26	C₄×Dic₂₁₂	C₂²	1	2	yes	—
27	C₄×#11719913	C₂²	1	2	yes	—
28	C₈×D₂₁₂	C₂²	1	1	yes	characteristic
29	C₈×D₂₁₂	C₂²	1	1	yes	characteristic
30	C₈×C₂²⋊D₅₃	—	2	1	no	—
31	C₈×C₂²⋊D₅₃	—	2	1	no	—
32	#16810676	C₂²	1	1	yes	characteristic
33	#16810674	—	2	1	no	—
34	#16810649	—	2	1	no	—
35	#16810647	C₂²	1	1	yes	characteristic
36	#16810578	—	2	1	no	—
37	C₄×C₈×D₅₃	—	2	1	no	—
38	#16810376	C₂²	1	2	yes	—
39	#16810374	C₂²	1	2	yes	—
40	#16810355	C₂²	1	2	yes	—
41	#16810355	C₂²	1	2	yes	—
42	#16810355	C₂²	1	2	yes	—
43	#16810352	C₂²	1	2	yes	—
44	C₅₃×#431	C₂²	1	1	yes	characteristic
45	C₄×M₁₆×C₅₃	C₂²	1	1	yes	characteristic
46	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
47	C₅₃×#322	C₂²	1	2	yes	—
48	C₈²×C₅₃	C₂²	1	2	yes	—
49	C₂×#11719945	C₂×C₄	1	1	yes	characteristic
50	C₂×Dic₂₁₂	C₂×C₄	1	2	yes	—
51	C₂×#11719913	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	#13133217	C₂×C₄	1	1	yes	characteristic
56	#13133217	C₂×C₄	1	1	yes	characteristic
57	#13133209	—	2	1	no	—
58	#13133196	—	2	1	no	—
59	#13133196	C₂×C₄	1	2	yes	—
60	C₄×Dic₁₀₆	—	2	1	no	—
61	C₄×Dic₁₀₆	C₂×C₄	1	1	yes	characteristic
62	C₄×Dic₁₀₆	C₂³	1	2	yes	—
63	C₄×Dic₁₀₆	C₂³	1	1	yes	characteristic
64	C₄×D₂₁₂	C₂×C₄	1	1	yes	characteristic
65	C₄×D₂₁₂	C₂³	1	1	yes	characteristic
66	C₄×C₂²⋊D₅₃	—	2	1	no	—
67	C₄×C₂²⋊D₅₃	—	2	1	no	—
68	#13133176	—	2	1	no	—
69	#13133176	C₂×C₄	1	1	yes	characteristic
70	#13133174	—	2	1	no	—
71	C₄²×D₅₃	—	2	1	no	—
72	#13133151	C₂×C₄	1	2	yes	—
73	#13133149	C₂×C₄	1	2	yes	—
74	C₄⋊#88322	—	2	1	no	—
75	C₄⋊#88322	C₂³	1	2	yes	—
76	C₄⋊#88322	C₂³	1	1	yes	characteristic
77	C₄⋊#88322	C₂³	1	1	yes	characteristic
78	#13133146	—	2	2	no	—
79	#13133146	—	2	2	no	—
80	#13133146	—	2	1	no	—
81	#13133146	—	2	1	no	—
82	#13133142	C₂×C₄	1	2	yes	—
83	#13133142	C₂×C₄	1	2	yes	—
84	#13133142	C₂×C₄	1	2	yes	—
85	#13133139	C₂×C₄	1	2	yes	—
86	C₂²⋊#88322	—	2	1	no	—
87	#13133129	—	2	1	no	—
88	#13133129	—	2	1	no	—
89	#13133127	—	2	2	no	—
90	C₄×#88322	—	2	1	no	—
91	C₈×Dic₅₃	—	2	1	no	—
92	C₈×Dic₅₃	—	2	1	no	—
93	C₈×Dic₅₃	—	2	2	no	—
94	C₂×M₁₆×C₅₃	—	2	1	no	—
95	C₂²×C₈×C₅₃	D₄	1	1	yes	characteristic
96	C₂²×C₈×C₅₃	D₄	1	1	yes	characteristic
97	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
98	C₄⋊C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
99	C₄⋊C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
100	C₂²⋊C₈×C₅₃	C₂×C₄	1	2	yes	—
101	#97×C₅₃	—	2	1	no	—
102	C₄×C₈×C₅₃	D₄	1	1	yes	characteristic
103	C₄×C₈×C₅₃	D₄	1	1	yes	characteristic
104	C₄×C₈×C₅₃	C₂³	1	2	yes	—
105	C₄×C₈×C₅₃	C₂³	1	2	yes	—
106	C₄×C₈×C₅₃	C₂³	1	1	yes	characteristic
107	C₄×C₈×C₅₃	C₂³	1	1	yes	characteristic
108	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
109	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
110	C₂²×C₄×C₅₃	#41	1	1	yes	characteristic
111	C₂×C₂²⋊D₅₃	—	2	1	no	—
112	#11719945	—	2	2	no	—
113	#11719945	—	2	2	no	—
114	C₂×D₂₁₂	C₂²×C₄	1	1	yes	characteristic
115	C₂×C₄×D₅₃	—	2	1	no	—
116	C₂×C₄×D₅₃	—	2	1	no	—
117	C₂×C₄×D₅₃	—	2	1	no	—
118	C₂×Dic₁₀₆	—	2	1	no	—
119	C₂×Dic₁₀₆	C₂²×C₄	1	2	yes	—
120	C₂×Dic₁₀₆	C₂²×C₄	1	1	yes	characteristic
121	M₁₆×C₅₃	—	2	4	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	C₂²×C₄	1	2	yes	—
126	C₂×C₈×C₅₃	C₂²×C₄	1	2	yes	—
127	C₂×C₈×C₅₃	C₂×D₄	1	1	yes	characteristic
128	C₂×C₈×C₅₃	C₂×D₄	1	1	yes	characteristic
129	C₂×C₈×C₅₃	C₂×D₄	1	1	yes	characteristic
130	C₂×C₈×C₅₃	C₂×D₄	1	1	yes	characteristic
131	C₄²×C₅₃	C₂²×C₄	1	1	yes	characteristic
132	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
133	C₄²×C₅₃	#41	1	1	yes	characteristic
134	C₄²×C₅₃	C₂⁴	1	1	yes	characteristic
135	C₂²⋊Dic₅₃	—	2	2	no	—
136	C₂²⋊Dic₅₃	—	2	1	no	—
137	#11719921	—	2	1	no	—
138	#11719921	—	2	1	no	—
139	C₄⋊Dic₅₃	—	2	1	no	—
140	C₄⋊Dic₅₃	—	2	1	no	—
141	C₄⋊Dic₅₃	—	2	1	no	—
142	C₄⋊Dic₅₃	C₂²×C₄	1	2	yes	—
143	C₄⋊Dic₅₃	C₂²×C₄	1	1	yes	characteristic
144	C₄⋊Dic₅₃	C₂²×C₄	1	1	yes	characteristic
145	#11719919	—	2	1	no	—
146	#11719919	—	2	2	no	—
147	#11719919	—	2	1	no	—
148	#11719919	—	2	2	no	—
149	C₄×Dic₅₃	—	2	1	no	—
150	C₄×Dic₅₃	—	2	1	no	—
151	C₄×Dic₅₃	—	2	1	no	—
152	C₄×Dic₅₃	—	2	2	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	Dic₂₁₂	—	2	4	no	—
158	#11719913	—	2	4	no	—
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	C₂×C₄×C₅₃	—	2	1	no	—
164	C₂×C₄×C₅₃	—	2	1	no	—
165	C₂×C₄×C₅₃	—	2	1	no	—
166	C₂×C₄×C₅₃	—	2	1	no	—
167	C₂×C₄×C₅₃	—	2	2	no	—
168	C₂×C₄×C₅₃	C₄×D₄	1	1	yes	characteristic
169	C₂×C₄×C₅₃	C₄×D₄	1	1	yes	characteristic
170	C₂×C₄×C₅₃	C₄×D₄	1	1	yes	characteristic
171	C₂×C₄×C₅₃	C₂³×C₄	1	1	yes	characteristic
172	C₂×C₄×C₅₃	C₂²×D₄	1	1	yes	characteristic
173	C₂×C₄×C₅₃	C₂×#41	1	1	yes	characteristic
174	C₂²⋊D₅₃	—	2	4	no	—
175	C₂×Dic₅₃	—	2	1	no	—
176	C₂×Dic₅₃	—	2	2	no	—
177	C₂×Dic₅₃	—	2	2	no	—
178	C₂×Dic₅₃	—	2	1	no	—
179	C₂×Dic₅₃	—	2	1	no	—
180	D₂₁₂	—	2	1	no	—
181	D₂₁₂	#131	1	2	yes	—
182	C₄×D₅₃	—	2	4	no	—
183	C₄×D₅₃	—	4	1	no	—
184	C₄×D₅₃	—	4	1	no	—
185	C₄×D₅₃	—	4	2	no	—
186	Dic₁₀₆	—	2	1	no	—
187	Dic₁₀₆	—	2	2	no	—
188	Dic₁₀₆	—	2	2	no	—
189	Dic₁₀₆	—	4	2	no	—
190	Dic₁₀₆	#131	1	2	yes	—
191	C₈×C₅₃	—	2	4	no	—
192	C₈×C₅₃	—	2	2	no	—
193	C₈×C₅₃	—	2	2	no	—
194	C₈×C₅₃	—	2	1	no	—
195	C₈×C₅₃	—	2	1	no	—
196	C₈×C₅₃	—	2	1	no	—
197	C₈×C₅₃	#137	1	2	yes	—
198	#88322	—	4	1	no	—
199	#88322	—	4	1	no	—
200	#88322	—	4	2	no	—
201	#17913	—	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	#443	1	2	yes	—
213	C₄×C₅₃	C₂×#131	1	1	yes	characteristic
214	C₄×C₅₃	C₂×#131	1	1	yes	characteristic
215	C₄×C₅₃	C₂×#137	1	1	yes	characteristic
216	C₄×C₅₃	#575	1	1	yes	characteristic
217	Dic₅₃	—	2	2	no	—
218	Dic₅₃	—	4	2	no	—
219	Dic₅₃	—	4	1	no	—
220	Dic₅₃	—	4	2	no	—
221	Dic₅₃	—	4	1	no	—
222	#3107	—	53	1	no	—
223	C₈×#41	—	53	1	no	—
224	#2960	—	53	1	no	—
225	#1587	—	53	2	no	—
226	#1585	—	53	2	no	—
227	C₈×Q₁₆	—	53	2	no	—
228	C₈×SD₁₆	—	53	2	no	—
229	#1574	—	53	1	no	—
230	#1489	—	53	1	no	—
231	#1485	—	53	1	no	—
232	C₈×M₁₆	D₅₃	1	1	yes	characteristic, complemented
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×#443	1	1	yes	characteristic
237	D₅₃	—	4	2	no	—
238	#533	—	53	1	no	—
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	—	53	2	no	—
247	#446	—	106	1	no	—
248	C₈×Q₈	—	53	2	no	—
249	C₈×Q₈	—	53	1	no	—
250	C₈×Q₈	—	53	1	no	—
251	C₈×Q₈	—	106	1	no	—
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	—	106	1	no	—
260	#431	C₂×D₅₃	1	1	yes	characteristic
261	#424	—	106	1	no	—
262	#423	—	53	1	no	—
263	#405	—	106	1	no	—
264	C₄×M₁₆	C₂×D₅₃	1	1	yes	characteristic
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	C₂×D₅₃	1	2	yes	—
274	C₈²	C₂×D₅₃	1	2	yes	—
275	C₅₃	#17913	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	D₂₁₂	1	1	yes	characteristic
284	C₂²×C₈	D₂₁₂	1	1	yes	characteristic
285	C₄⋊Q₈	—	53	1	no	—
286	C₄⋊Q₈	—	53	1	no	—
287	#125	—	106	1	no	—
288	C₂²⋊Q₈	—	53	2	no	—
289	C₂²⋊Q₈	—	106	1	no	—
290	C₄×Q₈	—	53	1	no	—
291	C₄×Q₈	—	53	1	no	—
292	C₄×Q₈	—	53	2	no	—
293	C₄×Q₈	—	106	1	no	—
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	—	53	1	no	—
299	#117	—	106	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	—	53	2	no	—
306	C₄⋊C₈	—	53	1	no	—
307	C₄⋊C₈	—	53	1	no	—
308	C₄⋊C₈	—	106	2	no	—
309	C₄⋊C₈	—	106	2	no	—
310	C₄⋊C₈	—	106	1	no	—
311	C₄⋊C₈	—	106	1	no	—
312	C₄⋊C₈	—	106	1	no	—
313	C₄⋊C₈	C₄×D₅₃	1	1	yes	characteristic
314	C₄⋊C₈	C₄×D₅₃	1	1	yes	characteristic
315	#103	—	53	2	no	—
316	#103	—	53	2	no	—
317	#103	—	53	2	no	—
318	#102	—	53	2	no	—
319	C₂²⋊C₈	—	106	1	no	—
320	C₂²⋊C₈	—	106	1	no	—
321	C₂²⋊C₈	—	106	1	no	—
322	C₂²⋊C₈	C₄×D₅₃	1	2	yes	—
323	#97	—	2	1	no	—
324	#97	—	106	2	no	—
325	C₄×C₈	—	106	1	no	—
326	C₄×C₈	—	106	1	no	—
327	C₄×C₈	—	106	2	no	—
328	C₄×C₈	—	106	1	no	—
329	C₄×C₈	D₂₁₂	1	1	yes	characteristic
330	C₄×C₈	D₂₁₂	1	1	yes	characteristic
331	C₄×C₈	C₂²×D₅₃	1	2	yes	—
332	C₄×C₈	C₂²×D₅₃	1	2	yes	—
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	—	53	1	no	—
338	C₂×Q₈	—	53	2	no	—
339	C₂×Q₈	—	106	1	no	—
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	C₂×D₂₁₂	1	1	yes	characteristic
347	C₂²×C₄	#11719945	1	1	yes	characteristic
348	Q₁₆	—	106	4	no	—
349	SD₁₆	—	106	4	no	—
350	M₁₆	—	2	4	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	—	106	1	no	—
356	C₂×C₈	—	106	1	no	—
357	C₂×C₈	—	106	2	no	—
358	C₂×C₈	—	212	2	no	—
359	C₂×C₈	—	212	2	no	—
360	C₂×C₈	C₂×C₄×D₅₃	1	2	yes	—
361	C₂×C₈	C₂×C₄×D₅₃	1	2	yes	—
362	C₂×C₈	C₂×D₂₁₂	1	1	yes	characteristic
363	C₂×C₈	C₂×D₂₁₂	1	1	yes	characteristic
364	C₂×C₈	C₂×D₂₁₂	1	1	yes	characteristic
365	C₂×C₈	C₂×D₂₁₂	1	1	yes	characteristic
366	C₄⋊C₄	—	53	2	no	—
367	C₄⋊C₄	—	53	1	no	—
368	C₄⋊C₄	—	53	1	no	—
369	C₄⋊C₄	—	106	1	no	—
370	C₄⋊C₄	—	106	1	no	—
371	C₄⋊C₄	—	106	1	no	—
372	C₄⋊C₄	—	106	2	no	—
373	C₄⋊C₄	—	106	1	no	—
374	C₄⋊C₄	—	106	1	no	—
375	C₄⋊C₄	—	106	2	no	—
376	C₂²⋊C₄	—	106	1	no	—
377	C₂²⋊C₄	—	106	2	no	—
378	C₂²⋊C₄	—	106	1	no	—
379	C₂²⋊C₄	—	106	1	no	—
380	C₄²	—	106	1	no	—
381	C₄²	—	106	1	no	—
382	C₄²	—	106	1	no	—
383	C₄²	—	106	2	no	—
384	C₄²	C₂×C₄×D₅₃	1	1	yes	characteristic
385	C₄²	C₂×D₂₁₂	1	1	yes	characteristic
386	C₄²	#11719945	1	1	yes	characteristic
387	C₄²	C₂³×D₅₃	1	1	yes	characteristic
388	C₂³	—	106	1	no	—
389	C₂³	C₄×D₂₁₂	1	1	yes	characteristic
390	Q₈	—	53	2	no	—
391	Q₈	—	106	1	no	—
392	Q₈	—	106	2	no	—
393	Q₈	—	106	2	no	—
394	Q₈	—	212	2	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	—	106	4	no	—
404	C₂×C₄	—	106	1	no	—
405	C₂×C₄	—	106	1	no	—
406	C₂×C₄	—	106	1	no	—
407	C₂×C₄	—	106	2	no	—
408	C₂×C₄	—	106	2	no	—
409	C₂×C₄	—	212	1	no	—
410	C₂×C₄	—	212	1	no	—
411	C₂×C₄	—	212	2	no	—
412	C₂×C₄	C₄×D₂₁₂	1	1	yes	characteristic
413	C₂×C₄	C₄×D₂₁₂	1	1	yes	characteristic
414	C₂×C₄	C₄×D₂₁₂	1	1	yes	characteristic
415	C₂×C₄	C₂²×C₄×D₅₃	1	1	yes	characteristic
416	C₂×C₄	C₂²×D₂₁₂	1	1	yes	characteristic
417	C₂×C₄	C₂×#11719945	1	1	yes	characteristic
418	C₈	—	2	4	no	—
419	C₈	—	2	2	no	—
420	C₈	—	2	2	no	—
421	C₈	—	2	1	no	—
422	C₈	—	2	1	no	—
423	C₈	—	2	1	no	—
424	C₈	—	212	1	no	—
425	C₈	—	212	1	no	—
426	C₈	—	212	2	no	—
427	C₈	#13133250	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×D₂₁₂	1	1	yes	characteristic
435	C₄	—	2	1	no	—
436	C₄	—	2	1	no	—
437	C₄	—	2	1	no	—
438	C₄	—	106	2	no	—
439	C₄	—	212	2	no	—
440	C₄	—	212	1	no	—
441	C₄	—	212	1	no	—
442	C₄	—	212	2	no	—
443	C₄	#16810741	1	2	yes	—
444	C₄	#131×D₅₃	1	1	yes	characteristic
445	C₄	#131×D₅₃	1	1	yes	characteristic
446	C₄	C₂×#13133250	1	1	yes	characteristic
447	C₄	#16811423	1	1	yes	characteristic
448	C₂	—	4	1	no	—
449	C₂	—	212	2	no	—
450	C₂	#20283411	1	1	yes	characteristic
451	C₂	C₄×#13133250	1	1	yes	characteristic
452	C₂	C₂×#16810741	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₈	C₂×D₂₁₂
commutator	C₂×C₄×C₅₃	C₂³×C₄
Frattini	C₄²	C₂³×D₅₃
Fitting	C₈×M₁₆×C₅₃	C₂
socle	C₂²×C₅₃	C₂×C₄×D₄
prime radical	C₂×#11719945	C₂×C₄
cosocle	C₄²×C₅₃	C₂⁴
hypercenter	C₈×M₁₆	D₅₃
hypocenter	C₅₃	#17913
LCS	G ▷ C₂×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂×C₈ ◁ C₂×C₄×C₈ ◁ C₈×M₁₆

Complements

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

N	H ≅ G/N	type
3. #20283649	C₂	semidirect
4. #20273588	C₂	semidirect
5. #20273574	C₂	semidirect
6. C₈×Dic₂₁₂	C₂	semidirect
7. C₈×#11719913	C₂	semidirect
8. #20273504	C₂	semidirect
10. #20273137	C₂	semidirect
11. C₈×M₁₆×C₅₃	C₂	semidirect
232. C₈×M₁₆	D₅₃	semidirect
275. C₅₃	#17913	semidirect

Maximal subgroups

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

subgroup	index
#20284787	2
C₈×#11719945	2
#20283649	2
#20273588	2
#20273574	2
C₈×Dic₂₁₂	2
C₈×#11719913	2
#20273504	2
#20273155	2
#20273137	2
C₈×M₁₆×C₅₃	2
#17913	53