Subgroups — #27537597

Order 13568.

498 conjugacy classes of subgroups.

The subgroup lattice has 452 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20283644	C₂	1	1	yes	maximal, characteristic
2	#20276624	C₂	1	1	yes	maximal, characteristic, complemented
3	#20276622	C₂	1	1	yes	maximal, characteristic, complemented
4	#20276622	C₂	1	1	yes	maximal, characteristic, complemented
5	#20276621	C₂	1	1	yes	maximal, characteristic, complemented
6	#20276621	C₂	1	1	yes	maximal, characteristic, complemented
7	C₁₆×Dic₁₀₆	C₂	1	1	yes	maximal, characteristic, complemented
8	#20276617	C₂	1	1	yes	maximal, characteristic, complemented
9	#20276472	C₂	1	1	yes	maximal, characteristic
10	#20276472	C₂	1	1	yes	maximal, characteristic
11	C₄⋊C₁₆×D₅₃	C₂	1	1	yes	maximal, characteristic
12	#20276313	C₂	1	1	yes	maximal, characteristic
13	#20276310	C₂	1	1	yes	maximal, characteristic
14	#20276308	C₂	1	1	yes	maximal, characteristic
15	C₅₃×#2183	C₂	1	1	yes	maximal, characteristic, complemented
16	#16811177	C₄	1	1	yes	characteristic
17	C₂×#13133156	C₂²	1	1	yes	characteristic
18	C₂×#13133156	C₂²	1	1	yes	characteristic
19	C₂×C₁₆×D₅₃	C₂²	1	1	yes	characteristic
20	C₂×C₁₆×D₅₃	C₂²	1	1	yes	characteristic
21	#16810754	C₄	1	1	yes	characteristic
22	#16810752	C₄	1	1	yes	characteristic
23	#16810752	C₄	1	1	yes	characteristic
24	#16810750	C₄	1	1	yes	characteristic
25	C₈×Dic₁₀₆	C₂²	1	1	yes	characteristic
26	C₈×Dic₁₀₆	C₂²	1	1	yes	characteristic
27	C₈×Dic₁₀₆	C₂²	1	1	yes	characteristic
28	#16810648	C₄	1	1	yes	characteristic
29	#16810648	C₄	1	1	yes	characteristic
30	#16810645	C₄	1	1	yes	characteristic
31	#16810577	C₂²	1	1	yes	characteristic
32	#16810577	C₂²	1	1	yes	characteristic
33	C₄×C₈×D₅₃	C₂²	1	1	yes	characteristic
34	C₄⋊#11719908	C₂²	1	1	yes	characteristic
35	C₄⋊#11719908	C₂²	1	1	yes	characteristic
36	C₄⋊#11719908	C₂²	1	1	yes	characteristic
37	#16810441	C₂²	1	1	yes	characteristic
38	#16810441	C₂²	1	1	yes	characteristic
39	#16810441	C₂²	1	1	yes	characteristic
40	#16810441	C₂²	1	1	yes	characteristic
41	#16810441	C₂²	1	1	yes	characteristic
42	#16810441	C₂²	1	1	yes	characteristic
43	#16810399	C₂²	1	1	yes	characteristic
44	#16810399	C₂²	1	1	yes	characteristic
45	#16810398	C₂²	1	1	yes	characteristic
46	#16810398	C₂²	1	1	yes	characteristic
47	C₄×#11719908	C₂²	1	1	yes	characteristic
48	C₁₆×Dic₅₃	C₂²	1	1	yes	characteristic
49	C₁₆×Dic₅₃	C₂²	1	1	yes	characteristic
50	C₁₆×Dic₅₃	C₂²	1	1	yes	characteristic
51	C₁₆×Dic₅₃	C₂²	1	1	yes	characteristic
52	C₈×Q₈×C₅₃	C₂²	1	1	yes	characteristic
53	C₅₃×C₄⋊C₁₆	C₂²	1	1	yes	characteristic
54	C₅₃×C₄⋊C₁₆	C₂²	1	1	yes	characteristic
55	C₅₃×C₄⋊C₁₆	C₂²	1	1	yes	characteristic
56	C₅₃×#346	C₂²	1	1	yes	characteristic
57	C₅₃×#346	C₂²	1	1	yes	characteristic
58	C₄×C₁₆×C₅₃	C₂²	1	1	yes	characteristic
59	C₂×#11719912	C₂×C₄	1	1	yes	characteristic
60	C₂×#11719912	C₂×C₄	1	1	yes	characteristic
61	C₂×C₈×D₅₃	C₂×C₄	1	1	yes	characteristic
62	C₂×C₈×D₅₃	C₂³	1	1	yes	characteristic
63	#13133216	C₈	1	1	yes	characteristic
64	#13133215	C₈	1	1	yes	characteristic
65	#13133207	C₈	1	1	yes	characteristic
66	#13133207	C₈	1	1	yes	characteristic
67	#13133197	C₈	1	1	yes	characteristic
68	#13133195	C₈	1	1	yes	characteristic
69	#13133195	C₈	1	1	yes	characteristic
70	#13133193	C₈	1	1	yes	characteristic
71	C₄×Dic₁₀₆	C₂×C₄	1	1	yes	characteristic
72	C₄×Dic₁₀₆	C₂×C₄	1	1	yes	characteristic
73	C₄×Dic₁₀₆	C₂×C₄	1	1	yes	characteristic
74	#13133174	C₂×C₄	1	1	yes	characteristic
75	#13133174	C₂×C₄	1	1	yes	characteristic
76	C₄²×D₅₃	C₂×C₄	1	1	yes	characteristic
77	#13133156	—	2	2	no	—
78	#13133156	Q₈	1	4	yes	complemented
79	C₂×#11719908	C₂³	1	1	yes	characteristic
80	C₂×#11719908	C₂³	1	1	yes	characteristic
81	C₂×#11719908	C₂³	1	1	yes	characteristic
82	C₂×#11719908	C₂³	1	1	yes	characteristic
83	C₁₆×D₅₃	—	2	2	no	—
84	C₁₆×D₅₃	—	2	2	no	—
85	C₄⋊#88322	C₂×C₄	1	1	yes	characteristic
86	C₄⋊#88322	C₂×C₄	1	1	yes	characteristic
87	C₄⋊#88322	C₂×C₄	1	1	yes	characteristic
88	#13133146	C₂×C₄	1	1	yes	characteristic
89	#13133146	C₂×C₄	1	1	yes	characteristic
90	#13133146	C₂×C₄	1	1	yes	characteristic
91	#13133146	C₂×C₄	1	1	yes	characteristic
92	#13133146	C₂×C₄	1	1	yes	characteristic
93	#13133146	C₂×C₄	1	1	yes	characteristic
94	#13133128	C₂×C₄	1	1	yes	characteristic
95	#13133128	C₂×C₄	1	1	yes	characteristic
96	#13133127	C₂×C₄	1	1	yes	characteristic
97	#13133127	C₂×C₄	1	1	yes	characteristic
98	C₄×#88322	C₂×C₄	1	1	yes	characteristic
99	C₈×Dic₅₃	C₂×C₄	1	1	yes	characteristic
100	C₈×Dic₅₃	C₂³	1	1	yes	characteristic
101	C₈×Dic₅₃	C₂³	1	1	yes	characteristic
102	C₈×Dic₅₃	C₂³	1	1	yes	characteristic
103	C₄×Q₈×C₅₃	C₂×C₄	1	1	yes	characteristic
104	C₂×C₁₆×C₅₃	C₂³	1	1	yes	characteristic
105	C₂×C₁₆×C₅₃	C₂³	1	1	yes	characteristic
106	C₂×C₁₆×C₅₃	C₂³	1	1	yes	characteristic
107	C₂×C₁₆×C₅₃	C₂³	1	1	yes	characteristic
108	C₄⋊C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
109	C₄⋊C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
110	C₄⋊C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
111	C₄×C₈×C₅₃	C₂³	1	1	yes	characteristic
112	C₄×C₈×C₅₃	C₂³	1	1	yes	characteristic
113	C₄×C₈×C₅₃	C₂³	1	1	yes	characteristic
114	C₂×Q₈×C₅₃	C₂×C₈	1	1	yes	characteristic
115	C₂×C₄×D₅₃	C₂×C₈	1	1	yes	characteristic
116	C₂×C₄×D₅₃	C₂×C₈	1	1	yes	characteristic
117	C₂×C₄×D₅₃	C₂²×C₄	1	1	yes	characteristic
118	C₂×Dic₁₀₆	C₂×C₈	1	1	yes	characteristic
119	C₂×Dic₁₀₆	C₂×C₈	1	1	yes	characteristic
120	C₂×Dic₁₀₆	C₂×C₈	1	1	yes	characteristic
121	C₂×C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
122	C₂×C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
123	C₂×C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
124	C₂×C₈×C₅₃	C₂⁴	1	1	yes	characteristic
125	C₄⋊C₄×C₅₃	C₂×C₈	1	1	yes	characteristic
126	C₄⋊C₄×C₅₃	C₂×C₈	1	1	yes	characteristic
127	C₄⋊C₄×C₅₃	C₂×C₈	1	1	yes	characteristic
128	C₄²×C₅₃	C₂²×C₄	1	1	yes	characteristic
129	C₄²×C₅₃	C₂²×C₄	1	1	yes	characteristic
130	C₄²×C₅₃	C₂²×C₄	1	1	yes	characteristic
131	#11719921	C₂×C₈	1	1	yes	characteristic
132	#11719921	C₂×C₈	1	1	yes	characteristic
133	#11719921	C₂×C₈	1	1	yes	characteristic
134	#11719921	C₂×C₈	1	1	yes	characteristic
135	C₄⋊Dic₅₃	C₂×C₈	1	1	yes	characteristic
136	C₄⋊Dic₅₃	C₂×C₈	1	1	yes	characteristic
137	C₄⋊Dic₅₃	C₂×C₈	1	1	yes	characteristic
138	#11719919	C₂×C₈	1	1	yes	characteristic
139	#11719919	C₂×C₈	1	1	yes	characteristic
140	#11719919	C₂×C₈	1	1	yes	characteristic
141	#11719919	C₂×C₈	1	1	yes	characteristic
142	#11719919	C₂×C₈	1	1	yes	characteristic
143	#11719919	C₂×C₈	1	1	yes	characteristic
144	#11719919	C₂×C₈	1	1	yes	characteristic
145	#11719919	C₂×C₈	1	1	yes	characteristic
146	#11719919	C₂×C₈	1	1	yes	characteristic
147	#11719919	C₂×C₈	1	1	yes	characteristic
148	C₄×Dic₅₃	C₂×C₈	1	1	yes	characteristic
149	C₄×Dic₅₃	C₂×C₈	1	1	yes	characteristic
150	C₄×Dic₅₃	C₂²×C₄	1	1	yes	characteristic
151	C₄×Dic₅₃	C₂²×C₄	1	1	yes	characteristic
152	C₄×Dic₅₃	C₂²×C₄	1	1	yes	characteristic
153	C₂×#88322	C₂²×C₄	1	1	yes	characteristic
154	C₂×#88322	C₂²×C₄	1	1	yes	characteristic
155	C₂×#88322	C₂²×C₄	1	1	yes	characteristic
156	C₂×#88322	C₂²×C₄	1	1	yes	characteristic
157	#11719912	—	2	2	no	—
158	#11719912	—	2	2	no	—
159	C₈×D₅₃	—	2	2	no	—
160	C₈×D₅₃	C₂×Q₈	1	2	yes	—
161	C₈×D₅₃	#41	1	2	yes	—
162	C₁₆×C₅₃	—	2	1	no	—
163	C₁₆×C₅₃	—	2	1	no	—
164	C₁₆×C₅₃	—	2	1	no	—
165	C₁₆×C₅₃	C₂×Q₈	1	2	yes	—
166	#11719908	—	2	1	no	—
167	#11719908	—	2	1	no	—
168	#11719908	—	2	1	no	—
169	#11719908	C₂×Q₈	1	2	yes	—
170	C₂²×D₅₃	C₂²×C₈	1	1	yes	characteristic
171	Q₈×C₅₃	—	4	1	no	—
172	C₂×C₄×C₅₃	C₂²×C₈	1	1	yes	characteristic
173	C₂×C₄×C₅₃	C₂²×C₈	1	1	yes	characteristic
174	C₂×C₄×C₅₃	C₂²×C₈	1	1	yes	characteristic
175	C₂×C₄×C₅₃	C₂²×C₈	1	1	yes	characteristic
176	C₂×C₄×C₅₃	C₂²×C₈	1	1	yes	characteristic
177	C₂×C₄×C₅₃	C₂²×C₈	1	1	yes	characteristic
178	C₂×C₄×C₅₃	C₂³×C₄	1	1	yes	characteristic
179	C₂×Dic₅₃	C₂²×C₈	1	1	yes	characteristic
180	C₂×Dic₅₃	C₂²×C₈	1	1	yes	characteristic
181	C₂×Dic₅₃	C₂²×C₈	1	1	yes	characteristic
182	C₂×Dic₅₃	C₂²×C₈	1	1	yes	characteristic
183	C₂×Dic₅₃	C₂²×C₈	1	1	yes	characteristic
184	C₂×Dic₅₃	C₂²×C₈	1	1	yes	characteristic
185	C₂×Dic₅₃	C₂²×C₈	1	1	yes	characteristic
186	C₄×D₅₃	—	2	2	no	—
187	C₄×D₅₃	—	4	1	no	—
188	C₄×D₅₃	C₄×Q₈	1	2	yes	—
189	C₄×D₅₃	#131	1	2	yes	—
190	Dic₁₀₆	—	2	2	no	—
191	Dic₁₀₆	—	2	2	no	—
192	Dic₁₀₆	—	4	1	no	—
193	C₈×C₅₃	—	2	1	no	—
194	C₈×C₅₃	—	2	1	no	—
195	C₈×C₅₃	—	2	1	no	—
196	C₈×C₅₃	C₂²×Q₈	1	1	yes	characteristic
197	C₈×C₅₃	C₂×#41	1	1	yes	characteristic
198	#88322	—	2	1	no	—
199	#88322	—	2	1	no	—
200	#88322	—	2	1	no	—
201	#88322	C₄×Q₈	1	2	yes	—
202	#33476	—	53	1	no	maximal, Sylow
203	C₂²×C₅₃	C₂³×C₈	1	1	yes	characteristic
204	C₂×D₅₃	—	2	1	no	—
205	C₂×D₅₃	—	2	1	no	—
206	C₂×D₅₃	C₈×Q₈	1	2	yes	—
207	C₄×C₅₃	—	2	1	no	—
208	C₄×C₅₃	—	2	1	no	—
209	C₄×C₅₃	—	2	1	no	—
210	C₄×C₅₃	—	2	1	no	—
211	C₄×C₅₃	—	2	1	no	—
212	C₄×C₅₃	—	2	1	no	—
213	C₄×C₅₃	C₂×C₄×Q₈	1	1	yes	characteristic
214	C₄×C₅₃	C₂×#131	1	1	yes	characteristic
215	Dic₅₃	—	2	1	no	—
216	Dic₅₃	—	2	1	no	—
217	Dic₅₃	—	2	1	no	—
218	Dic₅₃	—	2	1	no	—
219	Dic₅₃	—	2	1	no	—
220	Dic₅₃	C₈×Q₈	1	2	yes	—
221	Dic₅₃	#504	1	2	yes	—
222	#2960	—	53	1	no	—
223	#2183	—	53	1	no	—
224	#2183	—	53	1	no	—
225	#2183	—	53	1	no	—
226	#2183	—	53	1	no	—
227	#2183	—	53	1	no	—
228	#2183	D₅₃	1	1	yes	characteristic, complemented
229	Q₈×C₁₆	—	53	1	no	—
230	Q₈×C₁₆	—	53	1	no	—
231	#2151	—	53	1	no	—
232	#2151	—	53	1	no	—
233	C₂×C₄⋊C₁₆	—	53	1	no	—
234	#2108	—	53	1	no	—
235	#2108	—	53	1	no	—
236	C₄×M₃₂	—	53	1	no	—
237	C₂×C₅₃	C₂×C₈×Q₈	1	1	yes	characteristic
238	C₂×C₅₃	C₂×#504	1	1	yes	characteristic
239	C₂×C₅₃	#3407	1	1	yes	characteristic
240	D₅₃	—	2	2	no	—
241	#533	—	53	1	no	—
242	C₂×M₃₂	—	53	1	no	—
243	C₂×M₃₂	—	53	1	no	—
244	C₂²×C₁₆	—	53	1	no	—
245	C₂²×C₁₆	—	53	1	no	—
246	#446	—	53	1	no	—
247	#446	—	53	1	no	—
248	#446	—	53	1	no	—
249	#446	—	53	1	no	—
250	C₈×Q₈	—	53	1	no	—
251	C₈×Q₈	—	53	1	no	—
252	C₈×Q₈	—	53	1	no	—
253	C₈×Q₈	C₂×D₅₃	1	1	yes	characteristic
254	#424	—	53	1	no	—
255	#424	—	53	1	no	—
256	#423	—	53	1	no	—
257	#405	—	53	1	no	—
258	#405	—	53	1	no	—
259	C₂×C₄×C₈	—	53	1	no	—
260	C₄⋊C₁₆	—	53	1	no	—
261	C₄⋊C₁₆	—	53	1	no	—
262	C₄⋊C₁₆	—	53	1	no	—
263	C₄⋊C₁₆	—	53	1	no	—
264	C₄⋊C₁₆	—	53	1	no	—
265	C₄⋊C₁₆	—	53	1	no	—
266	C₄⋊C₁₆	—	53	1	no	—
267	C₄⋊C₁₆	—	53	1	no	—
268	C₄⋊C₁₆	—	53	1	no	—
269	C₄⋊C₁₆	C₂×D₅₃	1	1	yes	characteristic
270	C₄⋊C₁₆	C₂×D₅₃	1	1	yes	characteristic
271	C₄⋊C₁₆	C₂×D₅₃	1	1	yes	characteristic
272	#346	—	53	1	no	—
273	#346	—	53	1	no	—
274	#346	—	53	1	no	—
275	#346	—	53	1	no	—
276	#346	C₂×D₅₃	1	1	yes	characteristic
277	#346	C₂×D₅₃	1	1	yes	characteristic
278	C₄×C₁₆	—	53	1	no	—
279	C₄×C₁₆	—	53	1	no	—
280	C₄×C₁₆	—	53	1	no	—
281	C₄×C₁₆	—	53	1	no	—
282	C₄×C₁₆	—	53	1	no	—
283	C₄×C₁₆	C₂×D₅₃	1	1	yes	characteristic
284	C₅₃	#33476	1	1	yes	characteristic, complemented, Sylow
285	C₂×M₁₆	—	53	1	no	—
286	C₂×M₁₆	—	53	1	no	—
287	C₂²×C₈	—	53	1	no	—
288	C₂²×C₈	—	53	1	no	—
289	C₄⋊Q₈	—	53	1	no	—
290	C₄⋊Q₈	—	53	1	no	—
291	#125	—	53	1	no	—
292	#125	—	53	1	no	—
293	C₂²⋊Q₈	—	53	1	no	—
294	C₂²⋊Q₈	—	53	1	no	—
295	C₂²⋊Q₈	—	53	1	no	—
296	C₂²⋊Q₈	—	53	1	no	—
297	C₄×Q₈	—	53	1	no	—
298	C₄×Q₈	—	53	1	no	—
299	C₄×Q₈	—	53	1	no	—
300	C₄×Q₈	C₄×D₅₃	1	1	yes	characteristic
301	#117	—	53	1	no	—
302	#117	—	53	1	no	—
303	C₂×C₄²	—	53	1	no	—
304	M₃₂	—	53	4	no	—
305	M₃₂	—	106	2	no	—
306	C₂×C₁₆	—	53	1	no	—
307	C₂×C₁₆	—	53	1	no	—
308	C₂×C₁₆	—	53	1	no	—
309	C₂×C₁₆	—	53	1	no	—
310	C₂×C₁₆	—	106	2	no	—
311	C₂×C₁₆	—	106	2	no	—
312	C₂×C₁₆	C₂²×D₅₃	1	1	yes	characteristic
313	C₂×C₁₆	C₂²×D₅₃	1	1	yes	characteristic
314	C₂×C₁₆	C₂²×D₅₃	1	1	yes	characteristic
315	C₂×C₁₆	C₂²×D₅₃	1	1	yes	characteristic
316	C₄⋊C₈	—	53	1	no	—
317	C₄⋊C₈	—	53	1	no	—
318	C₄⋊C₈	—	53	1	no	—
319	C₄⋊C₈	—	53	1	no	—
320	C₄⋊C₈	—	53	1	no	—
321	C₄⋊C₈	—	53	1	no	—
322	C₄⋊C₈	—	53	1	no	—
323	C₄⋊C₈	—	53	1	no	—
324	C₄⋊C₈	—	53	1	no	—
325	C₄⋊C₈	C₄×D₅₃	1	1	yes	characteristic
326	C₄⋊C₈	C₄×D₅₃	1	1	yes	characteristic
327	C₄⋊C₈	C₄×D₅₃	1	1	yes	characteristic
328	#97	—	53	1	no	—
329	#97	—	53	1	no	—
330	#97	—	53	1	no	—
331	#97	—	53	1	no	—
332	C₄×C₈	—	53	1	no	—
333	C₄×C₈	—	53	1	no	—
334	C₄×C₈	—	53	1	no	—
335	C₄×C₈	—	53	1	no	—
336	C₄×C₈	—	53	1	no	—
337	C₄×C₈	C₂²×D₅₃	1	1	yes	characteristic
338	C₄×C₈	C₂²×D₅₃	1	1	yes	characteristic
339	C₄×C₈	C₂²×D₅₃	1	1	yes	characteristic
340	C₂×Q₈	—	53	1	no	—
341	C₂×Q₈	—	53	1	no	—
342	C₂×Q₈	—	53	1	no	—
343	C₂×Q₈	C₈×D₅₃	1	1	yes	characteristic
344	C₂²×C₄	—	53	1	no	—
345	C₂²×C₄	—	53	1	no	—
346	C₂²×C₄	—	53	1	no	—
347	M₁₆	—	106	2	no	—
348	M₁₆	—	106	2	no	—
349	C₂×C₈	—	53	1	no	—
350	C₂×C₈	—	53	2	no	—
351	C₂×C₈	—	53	2	no	—
352	C₂×C₈	—	53	1	no	—
353	C₂×C₈	—	53	1	no	—
354	C₂×C₈	—	53	1	no	—
355	C₂×C₈	—	106	2	no	—
356	C₂×C₈	C₂×C₄×D₅₃	1	1	yes	characteristic
357	C₂×C₈	C₂×C₄×D₅₃	1	1	yes	characteristic
358	C₂×C₈	C₂×C₄×D₅₃	1	1	yes	characteristic
359	C₂×C₈	C₂³×D₅₃	1	1	yes	characteristic
360	C₄⋊C₄	—	53	1	no	—
361	C₄⋊C₄	—	53	1	no	—
362	C₄⋊C₄	—	53	1	no	—
363	C₄⋊C₄	—	53	1	no	—
364	C₄⋊C₄	—	53	1	no	—
365	C₄⋊C₄	—	53	1	no	—
366	C₄⋊C₄	—	53	1	no	—
367	C₄⋊C₄	—	53	1	no	—
368	C₄⋊C₄	—	53	1	no	—
369	C₄⋊C₄	—	53	1	no	—
370	C₄⋊C₄	—	53	1	no	—
371	C₄⋊C₄	—	53	1	no	—
372	C₄⋊C₄	—	53	1	no	—
373	C₄⋊C₄	C₈×D₅₃	1	1	yes	characteristic
374	C₄⋊C₄	C₈×D₅₃	1	1	yes	characteristic
375	C₄⋊C₄	C₈×D₅₃	1	1	yes	characteristic
376	C₂²⋊C₄	—	53	1	no	—
377	C₂²⋊C₄	—	53	1	no	—
378	C₂²⋊C₄	—	53	1	no	—
379	C₂²⋊C₄	—	53	1	no	—
380	C₄²	—	53	1	no	—
381	C₄²	—	53	1	no	—
382	C₄²	—	53	1	no	—
383	C₄²	—	53	1	no	—
384	C₄²	—	53	1	no	—
385	C₄²	C₂×C₄×D₅₃	1	1	yes	characteristic
386	C₄²	C₂×C₄×D₅₃	1	1	yes	characteristic
387	C₄²	C₂×C₄×D₅₃	1	1	yes	characteristic
388	C₁₆	—	2	1	no	—
389	C₁₆	—	2	1	no	—
390	C₁₆	—	2	1	no	—
391	C₁₆	—	53	2	no	—
392	C₁₆	—	106	1	no	—
393	C₁₆	—	106	1	no	—
394	C₁₆	—	106	1	no	—
395	C₁₆	Q₈×D₅₃	1	2	yes	—
396	C₂³	—	53	1	no	—
397	Q₈	—	4	1	no	—
398	Q₈	—	106	2	no	—
399	Q₈	—	106	2	no	—
400	Q₈	—	212	1	no	—
401	C₂×C₄	—	53	1	no	—
402	C₂×C₄	—	53	1	no	—
403	C₂×C₄	—	53	2	no	—
404	C₂×C₄	—	53	2	no	—
405	C₂×C₄	—	53	1	no	—
406	C₂×C₄	—	53	1	no	—
407	C₂×C₄	—	53	1	no	—
408	C₂×C₄	—	53	1	no	—
409	C₂×C₄	—	53	1	no	—
410	C₂×C₄	—	106	2	no	—
411	C₂×C₄	—	212	1	no	—
412	C₂×C₄	C₂×C₈×D₅₃	1	1	yes	characteristic
413	C₂×C₄	C₂×C₈×D₅₃	1	1	yes	characteristic
414	C₂×C₄	C₂×C₈×D₅₃	1	1	yes	characteristic
415	C₂×C₄	C₂×C₈×D₅₃	1	1	yes	characteristic
416	C₂×C₄	C₂×C₈×D₅₃	1	1	yes	characteristic
417	C₂×C₄	C₂×C₈×D₅₃	1	1	yes	characteristic
418	C₂×C₄	C₂²×C₄×D₅₃	1	1	yes	characteristic
419	C₈	—	2	1	no	—
420	C₈	—	2	1	no	—
421	C₈	—	2	1	no	—
422	C₈	—	53	2	no	—
423	C₈	—	106	1	no	—
424	C₈	—	106	1	no	—
425	C₈	—	106	1	no	—
426	C₈	C₂×Q₈×D₅₃	1	1	yes	characteristic
427	C₈	#41×D₅₃	1	1	yes	characteristic
428	C₂²	—	53	2	no	—
429	C₂²	—	106	1	no	—
430	C₂²	—	106	1	no	—
431	C₂²	C₂²×C₈×D₅₃	1	1	yes	characteristic
432	C₄	—	2	1	no	—
433	C₄	—	2	1	no	—
434	C₄	—	2	1	no	—
435	C₄	—	2	1	no	—
436	C₄	—	2	1	no	—
437	C₄	—	2	1	no	—
438	C₄	—	53	2	no	—
439	C₄	—	53	2	no	—
440	C₄	—	106	1	no	—
441	C₄	—	106	1	no	—
442	C₄	—	106	1	no	—
443	C₄	—	106	1	no	—
444	C₄	—	106	1	no	—
445	C₄	C₄×Q₈×D₅₃	1	1	yes	characteristic
446	C₄	#131×D₅₃	1	1	yes	characteristic
447	C₂	—	106	2	no	—
448	C₂	C₈×Q₈×D₅₃	1	1	yes	characteristic
449	C₂	#504×D₅₃	1	1	yes	characteristic
450	C₂	#20287817	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂×C₈	C₂³×D₅₃
commutator, socle	C₂²×C₅₃	C₂³×C₈
Fitting	C₅₃×#2183	C₂
prime radical	C₂²×D₅₃	C₂²×C₈
cosocle	C₂×C₈×C₅₃	C₂⁴
hypercenter	#2183	D₅₃
hypocenter	C₅₃	#33476
LCS	G ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂×C₈ ◁ #2183

Complements

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

N	H ≅ G/N	type
2. #20276624	C₂	semidirect
3. #20276622	C₂	semidirect
4. #20276622	C₂	semidirect
5. #20276621	C₂	semidirect
6. #20276621	C₂	semidirect
7. C₁₆×Dic₁₀₆	C₂	semidirect
8. #20276617	C₂	semidirect
15. C₅₃×#2183	C₂	semidirect
78. #13133156	Q₈	semidirect
228. #2183	D₅₃	semidirect
284. C₅₃	#33476	semidirect

Maximal subgroups

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

subgroup	index
#20283644	2
#20276624	2
#20276622	2
#20276621	2
C₁₆×Dic₁₀₆	2
#20276617	2
#20276472	2
C₄⋊C₁₆×D₅₃	2
#20276313	2
#20276310	2
#20276308	2
C₅₃×#2183	2
#33476	53