Subgroups — #27354073

Order 13568.

632 conjugacy classes of subgroups.

The subgroup lattice has 444 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20284795	C₂	1	1	yes	maximal, characteristic
2	#20283690	C₂	1	1	yes	maximal, characteristic
3	#20273593	C₂	1	2	yes	maximal, complemented
4	#20273586	C₂	1	2	yes	maximal, complemented
5	C₈×#11719925	C₂	1	2	yes	maximal, complemented
6	C₈×#11719924	C₂	1	2	yes	maximal, complemented
7	#20273498	C₂	1	1	yes	maximal, characteristic, complemented
8	#20273162	C₂	1	1	yes	maximal, characteristic, complemented
9	#20273126	C₂	1	1	yes	maximal, characteristic
10	C₈×#11719917	C₂	1	1	yes	maximal, characteristic, complemented
11	C₂×C₈×Q₈×C₅₃	C₂	1	1	yes	maximal, characteristic, complemented
12	#16811157	C₂²	1	1	yes	characteristic
13	#16810953	C₄	1	1	yes	characteristic
14	#16810936	C₄	1	1	yes	characteristic
15	#16810930	C₄	1	2	yes	—
16	#16810845	C₄	1	2	yes	—
17	#16810779	C₄	1	1	yes	characteristic
18	#16810762	C₄	1	1	yes	characteristic
19	#16810753	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	#16810702	C₂²	1	1	yes	characteristic
24	#16810694	—	2	1	no	—
25	C₈×D₂₁₂	C₂²	1	1	yes	characteristic
26	C₈×C₂²⋊D₅₃	—	2	1	no	—
27	#16810682	—	2	1	no	—
28	#16810680	C₂²	1	1	yes	characteristic
29	#16810676	C₂²	1	1	yes	characteristic
30	#16810577	—	2	1	no	—
31	C₄×#11719917	C₂²	1	1	yes	characteristic
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	C₈×#88322	C₂²	1	2	yes	—
40	C₂×C₄×Q₈×C₅₃	C₂²	1	1	yes	characteristic
41	C₈×Q₈×C₅₃	—	2	1	no	—
42	C₈×Q₈×C₅₃	—	2	1	no	—
43	C₈×Q₈×C₅₃	C₂²	1	2	yes	—
44	C₈×Q₈×C₅₃	C₂²	1	2	yes	—
45	C₂×C₄⋊C₈×C₅₃	—	2	1	no	—
46	C₂×C₄⋊C₈×C₅₃	C₂²	1	1	yes	characteristic
47	C₂×C₄×C₈×C₅₃	—	2	1	no	—
48	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
49	C₂×#11719925	C₂×C₄	1	2	yes	—
50	C₂×#11719924	C₂×C₄	1	2	yes	—
51	C₂×#11719917	—	2	1	no	—
52	C₂×#11719912	—	2	1	no	—
53	C₂×C₈×D₅₃	—	2	1	no	—
54	#13133211	—	2	1	no	—
55	#13133209	C₂×C₄	1	1	yes	characteristic
56	#13133205	C₂×C₄	1	1	yes	characteristic
57	#13133200	—	2	1	no	—
58	#13133196	C₂×C₄	1	1	yes	characteristic
59	#13133191	C₂×C₄	1	1	yes	characteristic
60	C₄×Dic₁₀₆	C₂³	1	1	yes	characteristic
61	C₄×D₂₁₂	C₂³	1	1	yes	characteristic
62	C₄×C₂²⋊D₅₃	—	2	1	no	—
63	#13133174	—	2	1	no	—
64	#13133150	C₂×C₄	1	2	yes	—
65	#13133148	C₂×C₄	1	2	yes	—
66	C₄⋊#88322	C₂×C₄	1	2	yes	—
67	C₄⋊#88322	C₂³	1	1	yes	characteristic
68	C₄⋊#88322	C₂³	1	1	yes	characteristic
69	#13133146	—	2	1	no	—
70	#13133146	—	2	1	no	—
71	#13133141	C₂×C₄	1	2	yes	—
72	#13133141	C₂×C₄	1	2	yes	—
73	#13133140	C₂×C₄	1	2	yes	—
74	#13133137	C₂×C₄	1	2	yes	—
75	C₂²⋊#88322	—	2	1	no	—
76	C₂²⋊#88322	C₂×C₄	1	1	yes	characteristic
77	C₂²⋊#88322	C₂×C₄	1	1	yes	characteristic
78	#13133129	—	2	1	no	—
79	#13133129	—	2	1	no	—
80	#13133128	—	2	1	no	—
81	#13133128	—	2	1	no	—
82	#13133127	—	2	1	no	—
83	C₄×#88322	C₂³	1	2	yes	—
84	C₄×#88322	C₂³	1	2	yes	—
85	C₈×Dic₅₃	—	2	1	no	—
86	C₂²×Q₈×C₅₃	C₂×C₄	1	1	yes	characteristic
87	C₂²×C₈×C₅₃	—	2	1	no	—
88	C₂²×C₈×C₅₃	D₄	1	1	yes	characteristic
89	C₂²×C₈×C₅₃	D₄	1	1	yes	characteristic
90	C₄×Q₈×C₅₃	—	2	1	no	—
91	C₄×Q₈×C₅₃	—	2	1	no	—
92	C₄×Q₈×C₅₃	C₂×C₄	1	2	yes	—
93	C₄×Q₈×C₅₃	C₂³	1	2	yes	—
94	C₂×C₄⋊C₄×C₅₃	—	2	1	no	—
95	C₂×C₄⋊C₄×C₅₃	C₂×C₄	1	1	yes	characteristic
96	C₂×C₄²×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	C₄×C₈×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	4	no	—
109	C₂×Q₈×C₅₃	—	2	1	no	—
110	C₂×Q₈×C₅₃	C₂²×C₄	1	2	yes	—
111	C₂²×C₄×C₅₃	—	2	1	no	—
112	C₂²×C₄×C₅₃	—	2	1	no	—
113	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
114	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
115	C₂²×C₄×C₅₃	#41	1	1	yes	characteristic
116	C₂×C₂²⋊D₅₃	—	2	1	no	—
117	C₂×D₂₁₂	C₂²×C₄	1	1	yes	characteristic
118	C₂×C₄×D₅₃	—	2	1	no	—
119	C₂×Dic₁₀₆	C₂²×C₄	1	1	yes	characteristic
120	C₂×C₈×C₅₃	—	2	2	no	—
121	C₂×C₈×C₅₃	—	2	2	no	—
122	C₂×C₈×C₅₃	—	4	1	no	—
123	C₂×C₈×C₅₃	—	4	2	no	—
124	C₂×C₈×C₅₃	C₂×D₄	1	1	yes	characteristic
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	C₄⋊C₄×C₅₃	—	2	2	no	—
129	C₄⋊C₄×C₅₃	—	2	1	no	—
130	C₄⋊C₄×C₅₃	—	2	2	no	—
131	C₄⋊C₄×C₅₃	C₂²×C₄	1	2	yes	—
132	C₄²×C₅₃	—	2	2	no	—
133	C₄²×C₅₃	—	2	2	no	—
134	C₄²×C₅₃	C₂²×C₄	1	1	yes	characteristic
135	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
136	C₄²×C₅₃	#41	1	1	yes	characteristic
137	C₄²×C₅₃	C₂⁴	1	1	yes	characteristic
138	C₂²⋊Dic₅₃	—	2	1	no	—
139	#11719925	—	2	4	no	—
140	#11719924	—	2	4	no	—
141	#11719921	—	2	1	no	—
142	#11719921	—	2	1	no	—
143	#11719921	—	2	1	no	—
144	#11719921	—	2	1	no	—
145	C₄⋊Dic₅₃	C₂²×C₄	1	1	yes	characteristic
146	C₄⋊Dic₅₃	C₂²×C₄	1	1	yes	characteristic
147	#11719919	—	2	1	no	—
148	#11719919	—	2	1	no	—
149	#11719919	—	2	1	no	—
150	#11719919	—	2	1	no	—
151	C₄×Dic₅₃	—	2	1	no	—
152	#11719917	—	2	4	no	—
153	C₂×#88322	—	2	2	no	—
154	C₂×#88322	—	2	1	no	—
155	C₂×#88322	—	2	1	no	—
156	C₂×#88322	C₂²×C₄	1	2	yes	—
157	C₂×#88322	C₂²×C₄	1	2	yes	—
158	#11719912	—	4	2	no	—
159	C₈×D₅₃	—	4	2	no	—
160	C₂³×C₅₃	C₄×D₄	1	1	yes	characteristic
161	C₂²×D₅₃	—	2	1	no	—
162	Q₈×C₅₃	—	2	4	no	—
163	Q₈×C₅₃	—	2	2	no	—
164	Q₈×C₅₃	#131	1	4	yes	—
165	C₂×C₄×C₅₃	—	2	4	no	—
166	C₂×C₄×C₅₃	—	2	2	no	—
167	C₂×C₄×C₅₃	—	2	2	no	—
168	C₂×C₄×C₅₃	—	2	2	no	—
169	C₂×C₄×C₅₃	—	2	1	no	—
170	C₂×C₄×C₅₃	—	2	1	no	—
171	C₂×C₄×C₅₃	—	2	2	no	—
172	C₂×C₄×C₅₃	—	4	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	—	4	2	no	—
180	C₂×Dic₅₃	—	2	1	no	—
181	C₂×Dic₅₃	—	2	1	no	—
182	C₂×Dic₅₃	—	2	1	no	—
183	D₂₁₂	—	2	2	no	—
184	C₄×D₅₃	—	4	1	no	—
185	C₄×D₅₃	—	4	1	no	—
186	Dic₁₀₆	—	2	2	no	—
187	C₈×C₅₃	—	2	1	no	—
188	C₈×C₅₃	—	2	1	no	—
189	C₈×C₅₃	—	2	1	no	—
190	C₈×C₅₃	—	4	2	no	—
191	C₈×C₅₃	#137	1	2	yes	—
192	#88322	—	2	4	no	—
193	#88322	—	2	2	no	—
194	#88322	—	2	2	no	—
195	#88322	—	4	1	no	—
196	#88322	—	4	1	no	—
197	#17911	—	53	1	no	maximal, Sylow
198	C₂²×C₅₃	—	2	1	no	—
199	C₂²×C₅₃	—	2	1	no	—
200	C₂²×C₅₃	#443	1	2	yes	—
201	C₂²×C₅₃	C₂×C₄×D₄	1	1	yes	characteristic
202	C₂×D₅₃	—	4	1	no	—
203	C₂×D₅₃	—	4	1	no	—
204	C₂×D₅₃	—	4	1	no	—
205	C₄×C₅₃	—	2	4	no	—
206	C₄×C₅₃	—	2	1	no	—
207	C₄×C₅₃	—	2	1	no	—
208	C₄×C₅₃	—	2	1	no	—
209	C₄×C₅₃	—	2	1	no	—
210	C₄×C₅₃	—	4	2	no	—
211	C₄×C₅₃	C₂×#131	1	2	yes	—
212	C₄×C₅₃	C₂×#137	1	1	yes	characteristic
213	C₄×C₅₃	#575	1	1	yes	characteristic
214	Dic₅₃	—	4	1	no	—
215	Dic₅₃	—	4	1	no	—
216	Dic₅₃	—	4	1	no	—
217	#3108	—	53	1	no	—
218	#2966	—	53	1	no	—
219	C₂×C₈×Q₈	D₅₃	1	1	yes	characteristic, complemented
220	#1588	—	53	2	no	—
221	#1587	—	53	2	no	—
222	C₈×Q₁₆	—	53	2	no	—
223	C₈×SD₁₆	—	53	2	no	—
224	#1573	—	53	1	no	—
225	#1491	—	53	1	no	—
226	#1483	—	53	1	no	—
227	C₈×M₁₆	—	53	1	no	—
228	C₂×C₅₃	—	2	2	no	—
229	C₂×C₅₃	#2929	1	1	yes	characteristic
230	C₂×C₅₃	C₄×#137	1	1	yes	characteristic
231	C₂×C₅₃	C₂×#443	1	1	yes	characteristic
232	D₅₃	—	8	1	no	—
233	#529	—	53	1	no	—
234	C₂×C₄×Q₈	C₂×D₅₃	1	1	yes	characteristic
235	#483	—	53	1	no	—
236	#480	—	53	1	no	—
237	#479	—	53	2	no	—
238	#464	—	53	2	no	—
239	#451	—	53	1	no	—
240	#448	—	53	1	no	—
241	#446	—	53	1	no	—
242	C₈×Q₈	—	2	1	no	—
243	C₈×Q₈	—	2	1	no	—
244	C₈×Q₈	—	53	1	no	—
245	C₈×Q₈	C₂×D₅₃	1	2	yes	—
246	C₈×Q₈	C₂×D₅₃	1	2	yes	—
247	C₄×Q₁₆	—	53	2	no	—
248	C₄×SD₁₆	—	53	2	no	—
249	#436	—	53	1	no	—
250	#435	—	106	1	no	—
251	C₈×D₄	—	53	1	no	—
252	C₈×D₄	—	106	1	no	—
253	#433	—	106	1	no	—
254	#432	—	53	1	no	—
255	#431	—	53	1	no	—
256	C₂×C₄⋊C₈	—	2	1	no	—
257	C₂×C₄⋊C₈	C₂×D₅₃	1	1	yes	characteristic
258	#405	—	106	1	no	—
259	C₄×M₁₆	—	53	1	no	—
260	C₂×C₄×C₈	—	2	1	no	—
261	C₂×C₄×C₈	C₂×D₅₃	1	1	yes	characteristic
262	#335	—	53	2	no	—
263	#334	—	53	2	no	—
264	Q₈⋊C₈	—	53	2	no	—
265	Q₈⋊C₈	—	53	2	no	—
266	Q₈⋊C₈	—	53	2	no	—
267	D₄⋊C₈	—	53	2	no	—
268	#322	—	53	2	no	—
269	C₈²	—	53	2	no	—
270	C₅₃	#17911	1	1	yes	characteristic, complemented, Sylow
271	C₂²×Q₈	C₄×D₅₃	1	1	yes	characteristic
272	C₂×Q₁₆	—	53	2	no	—
273	C₂×SD₁₆	—	53	2	no	—
274	C₂×M₁₆	—	106	1	no	—
275	C₂×M₁₆	—	106	1	no	—
276	C₂²×C₈	—	2	1	no	—
277	C₂²×C₈	—	106	1	no	—
278	C₂²×C₈	C₂²⋊D₅₃	1	1	yes	characteristic
279	C₂²×C₈	C₂²⋊D₅₃	1	1	yes	characteristic
280	#126	—	106	1	no	—
281	#125	—	53	1	no	—
282	#124	—	53	1	no	—
283	#123	—	106	1	no	—
284	C₂²⋊Q₈	—	53	1	no	—
285	#121	—	53	1	no	—
286	C₄×Q₈	—	2	1	no	—
287	C₄×Q₈	—	2	1	no	—
288	C₄×Q₈	—	53	1	no	—
289	C₄×Q₈	C₄×D₅₃	1	2	yes	—
290	C₄×Q₈	C₂²×D₅₃	1	2	yes	—
291	C₄×D₄	—	53	1	no	—
292	C₄×D₄	—	106	1	no	—
293	#117	—	106	1	no	—
294	C₂×C₄⋊C₄	—	2	1	no	—
295	C₂×C₄⋊C₄	C₄×D₅₃	1	1	yes	characteristic
296	C₂×C₄²	—	2	1	no	—
297	C₂×C₄²	C₂²×D₅₃	1	1	yes	characteristic
298	#107	—	53	2	no	—
299	#106	—	53	2	no	—
300	C₄⋊C₈	—	2	2	no	—
301	C₄⋊C₈	—	2	2	no	—
302	C₄⋊C₈	—	2	1	no	—
303	C₄⋊C₈	—	53	2	no	—
304	C₄⋊C₈	—	53	1	no	—
305	C₄⋊C₈	—	53	1	no	—
306	C₄⋊C₈	—	106	1	no	—
307	C₄⋊C₈	—	106	1	no	—
308	C₄⋊C₈	C₂²×D₅₃	1	2	yes	—
309	#103	—	53	2	no	—
310	#103	—	53	2	no	—
311	#103	—	53	2	no	—
312	#102	—	53	2	no	—
313	C₂²⋊C₈	—	53	1	no	—
314	C₂²⋊C₈	—	53	1	no	—
315	C₂²⋊C₈	—	106	1	no	—
316	C₂²⋊C₈	—	106	1	no	—
317	C₂²⋊C₈	—	106	1	no	—
318	#97	—	106	1	no	—
319	#97	—	106	1	no	—
320	#97	—	106	1	no	—
321	C₄×C₈	—	2	2	no	—
322	C₄×C₈	—	2	2	no	—
323	C₄×C₈	—	53	2	no	—
324	C₄×C₈	—	53	2	no	—
325	C₄×C₈	—	106	1	no	—
326	C₄×C₈	C₂²⋊D₅₃	1	1	yes	characteristic
327	C₄×C₈	C₂²⋊D₅₃	1	1	yes	characteristic
328	C₄×C₈	C₂²×D₅₃	1	1	yes	characteristic
329	C₄×C₈	C₂²×D₅₃	1	1	yes	characteristic
330	C₂×Q₈	—	2	4	no	—
331	C₂×Q₈	—	2	1	no	—
332	C₂×Q₈	—	53	1	no	—
333	C₂×Q₈	C₂×C₄×D₅₃	1	2	yes	—
334	C₂×D₄	—	53	1	no	—
335	C₂×D₄	—	106	1	no	—
336	C₂²×C₄	—	2	1	no	—
337	C₂²×C₄	—	2	1	no	—
338	C₂²×C₄	—	106	1	no	—
339	C₂²×C₄	C₂×C₄×D₅₃	1	1	yes	characteristic
340	C₂²×C₄	#11719945	1	1	yes	characteristic
341	C₂²×C₄	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
342	Q₁₆	—	106	4	no	—
343	SD₁₆	—	106	4	no	—
344	M₁₆	—	106	4	no	—
345	M₁₆	—	212	2	no	—
346	C₂×C₈	—	2	2	no	—
347	C₂×C₈	—	2	2	no	—
348	C₂×C₈	—	4	1	no	—
349	C₂×C₈	—	4	2	no	—
350	C₂×C₈	—	53	2	no	—
351	C₂×C₈	—	53	2	no	—
352	C₂×C₈	—	106	2	no	—
353	C₂×C₈	—	106	1	no	—
354	C₂×C₈	—	106	1	no	—
355	C₂×C₈	—	212	2	no	—
356	C₂×C₈	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
357	C₂×C₈	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
358	C₂×C₈	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
359	C₂×C₈	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
360	C₄⋊C₄	—	2	2	no	—
361	C₄⋊C₄	—	2	1	no	—
362	C₄⋊C₄	—	2	2	no	—
363	C₄⋊C₄	—	53	1	no	—
364	C₄⋊C₄	—	53	1	no	—
365	C₄⋊C₄	—	106	1	no	—
366	C₄⋊C₄	—	106	1	no	—
367	C₄⋊C₄	—	106	1	no	—
368	C₄⋊C₄	—	106	1	no	—
369	C₄⋊C₄	C₂×C₄×D₅₃	1	2	yes	—
370	C₂²⋊C₄	—	106	1	no	—
371	C₂²⋊C₄	—	106	1	no	—
372	C₂²⋊C₄	—	106	1	no	—
373	C₂²⋊C₄	—	106	1	no	—
374	C₂²⋊C₄	—	106	1	no	—
375	C₄²	—	2	2	no	—
376	C₄²	—	2	2	no	—
377	C₄²	—	106	1	no	—
378	C₄²	C₂×C₄×D₅₃	1	1	yes	characteristic
379	C₄²	#11719945	1	1	yes	characteristic
380	C₄²	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
381	C₄²	C₂³×D₅₃	1	1	yes	characteristic
382	C₂³	—	106	1	no	—
383	C₂³	C₄×C₂²⋊D₅₃	1	1	yes	characteristic
384	Q₈	—	2	4	no	—
385	Q₈	—	2	2	no	—
386	Q₈	—	106	2	no	—
387	Q₈	#13133223	1	4	yes	—
388	D₄	—	106	2	no	—
389	D₄	—	212	2	no	—
390	C₂×C₄	—	2	4	no	—
391	C₂×C₄	—	2	2	no	—
392	C₂×C₄	—	2	2	no	—
393	C₂×C₄	—	2	2	no	—
394	C₂×C₄	—	2	1	no	—
395	C₂×C₄	—	2	1	no	—
396	C₂×C₄	—	2	2	no	—
397	C₂×C₄	—	4	2	no	—
398	C₂×C₄	—	106	1	no	—
399	C₂×C₄	—	106	1	no	—
400	C₂×C₄	—	106	1	no	—
401	C₂×C₄	—	212	1	no	—
402	C₂×C₄	—	212	1	no	—
403	C₂×C₄	C₄×C₂²⋊D₅₃	1	1	yes	characteristic
404	C₂×C₄	C₄×C₂²⋊D₅₃	1	1	yes	characteristic
405	C₂×C₄	C₄×C₂²⋊D₅₃	1	1	yes	characteristic
406	C₂×C₄	C₂²×C₄×D₅₃	1	1	yes	characteristic
407	C₂×C₄	C₂²×C₂²⋊D₅₃	1	1	yes	characteristic
408	C₂×C₄	C₂×#11719945	1	1	yes	characteristic
409	C₈	—	2	1	no	—
410	C₈	—	2	1	no	—
411	C₈	—	2	1	no	—
412	C₈	—	4	2	no	—
413	C₈	—	106	4	no	—
414	C₈	—	106	2	no	—
415	C₈	—	106	2	no	—
416	C₈	—	212	1	no	—
417	C₈	—	212	1	no	—
418	C₈	#13133248	1	2	yes	—
419	C₂²	—	2	1	no	—
420	C₂²	—	2	1	no	—
421	C₂²	—	212	1	no	—
422	C₂²	—	212	1	no	—
423	C₂²	—	212	1	no	—
424	C₂²	#16810738	1	2	yes	—
425	C₂²	C₂×C₄×C₂²⋊D₅₃	1	1	yes	characteristic
426	C₄	—	2	4	no	—
427	C₄	—	2	1	no	—
428	C₄	—	2	1	no	—
429	C₄	—	2	1	no	—
430	C₄	—	2	1	no	—
431	C₄	—	4	2	no	—
432	C₄	—	212	1	no	—
433	C₄	—	212	1	no	—
434	C₄	—	212	1	no	—
435	C₄	C₂×#13133223	1	2	yes	—
436	C₄	C₂×#13133248	1	1	yes	characteristic
437	C₄	#16811421	1	1	yes	characteristic
438	C₂	—	2	2	no	—
439	C₂	—	424	1	no	—
440	C₂	#20283409	1	1	yes	characteristic
441	C₂	C₄×#13133248	1	1	yes	characteristic
442	C₂	C₂×#16810738	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₈	C₂×C₂²⋊D₅₃
commutator	C₂×C₄×C₅₃	C₂³×C₄
Frattini	C₄²	C₂³×D₅₃
Fitting	C₂×C₈×Q₈×C₅₃	C₂
socle	C₂²×C₅₃	C₂×C₄×D₄
prime radical	#13133191	C₂×C₄
cosocle	C₄²×C₅₃	C₂⁴
hypercenter	C₂×C₈×Q₈	D₅₃
hypocenter	C₅₃	#17911
LCS	G ▷ C₂×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂×C₈ ◁ C₂×C₄×C₈ ◁ C₂×C₈×Q₈

Complements

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

N	H ≅ G/N	type
3. #20273593	C₂	semidirect
4. #20273586	C₂	semidirect
5. C₈×#11719925	C₂	semidirect
6. C₈×#11719924	C₂	semidirect
7. #20273498	C₂	semidirect
8. #20273162	C₂	semidirect
10. C₈×#11719917	C₂	semidirect
11. C₂×C₈×Q₈×C₅₃	C₂	semidirect
219. C₂×C₈×Q₈	D₅₃	semidirect
270. C₅₃	#17911	semidirect

Maximal subgroups

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

subgroup	index
#20284795	2
#20283690	2
#20273593	2
#20273586	2
C₈×#11719925	2
C₈×#11719924	2
#20273498	2
#20273162	2
#20273126	2
C₈×#11719917	2
C₂×C₈×Q₈×C₅₃	2
#17911	53