Subgroups — #27388776

Order 13568.

814 conjugacy classes of subgroups.

The subgroup lattice has 428 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×Q₈×Dic₅₃	C₂	1	1	yes	maximal, characteristic
2	#20275652	C₂	1	1	yes	maximal, characteristic
3	#20275433	C₂	1	1	yes	maximal, characteristic
4	#20275431	C₂	1	1	yes	maximal, characteristic
5	#20275273	C₂	1	2	yes	maximal
6	#20275271	C₂	1	2	yes	maximal
7	C₄⋊C₈×Dic₅₃	C₂	1	1	yes	maximal, characteristic
8	#20274563	C₂	1	2	yes	maximal
9	#20274459	C₂	1	2	yes	maximal
10	C₄×C₈×Dic₅₃	C₂	1	1	yes	maximal, characteristic
11	C₅₃×#1984	C₂	1	1	yes	maximal, characteristic
12	C₂×Q₈×Dic₅₃	C₄	1	1	yes	characteristic
13	C₂×Q₈×Dic₅₃	C₄	1	1	yes	characteristic
14	C₂×C₄⋊#88322	C₂²	1	1	yes	characteristic
15	C₂×#13133146	C₂²	1	2	yes	—
16	C₂×#13133146	C₂²	1	2	yes	—
17	C₂×C₄×#88322	C₂²	1	1	yes	characteristic
18	C₂×C₈×Dic₅₃	C₂²	1	2	yes	—
19	C₂×C₈×Dic₅₃	C₂²	1	1	yes	characteristic
20	C₂×C₈×Dic₅₃	C₂²	1	1	yes	characteristic
21	C₂×C₈×Dic₅₃	C₂²	1	1	yes	characteristic
22	C₂×C₈×Dic₅₃	C₂²	1	1	yes	characteristic
23	C₄⋊C₄×Dic₅₃	C₄	1	2	yes	—
24	C₄⋊C₄×Dic₅₃	C₄	1	1	yes	characteristic
25	C₄⋊C₄×Dic₅₃	C₄	1	2	yes	—
26	C₄⋊C₄×Dic₅₃	C₄	1	1	yes	characteristic
27	C₄×C₄⋊Dic₅₃	C₂²	1	2	yes	—
28	C₄×C₄⋊Dic₅₃	C₂²	1	1	yes	characteristic
29	C₄²×Dic₅₃	C₂²	1	2	yes	—
30	C₄²×Dic₅₃	C₂²	1	1	yes	characteristic
31	#16810378	C₂²	1	2	yes	—
32	#16810378	C₂²	1	2	yes	—
33	#16810377	C₂²	1	2	yes	—
34	#16810377	C₂²	1	2	yes	—
35	#16810377	C₂²	1	2	yes	—
36	C₂×C₄×Q₈×C₅₃	C₂²	1	1	yes	characteristic
37	C₂×C₄⋊C₈×C₅₃	C₂²	1	1	yes	characteristic
38	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
39	C₅₃×#336	C₂²	1	2	yes	—
40	C₅₃×#336	C₂²	1	2	yes	—
41	C₂²×#88322	C₂³	1	2	yes	—
42	C₂²×#88322	C₂³	1	1	yes	characteristic
43	C₂²×#88322	C₂³	1	1	yes	characteristic
44	Q₈×Dic₅₃	—	2	4	no	—
45	Q₈×Dic₅₃	C₈	1	8	yes	complemented
46	C₂×C₄⋊Dic₅₃	C₂×C₄	1	1	yes	characteristic
47	C₂×C₄⋊Dic₅₃	C₂×C₄	1	2	yes	—
48	C₂×C₄⋊Dic₅₃	C₂×C₄	1	2	yes	—
49	C₂×C₄⋊Dic₅₃	C₂×C₄	1	1	yes	characteristic
50	C₂×C₄×Dic₅₃	C₂×C₄	1	2	yes	—
51	C₂×C₄×Dic₅₃	C₂×C₄	1	2	yes	—
52	C₂×C₄×Dic₅₃	C₂×C₄	1	2	yes	—
53	C₂×C₄×Dic₅₃	C₂×C₄	1	1	yes	characteristic
54	C₂×C₄×Dic₅₃	C₂×C₄	1	1	yes	characteristic
55	C₂×C₄×Dic₅₃	C₂×C₄	1	2	yes	—
56	C₂×C₄×Dic₅₃	C₂×C₄	1	1	yes	characteristic
57	C₂×C₄×Dic₅₃	C₂×C₄	1	1	yes	characteristic
58	C₂×C₄×Dic₅₃	C₂³	1	1	yes	characteristic
59	C₂×C₄×Dic₅₃	C₂³	1	2	yes	—
60	C₂×C₄×Dic₅₃	C₂³	1	1	yes	characteristic
61	C₄⋊#88322	—	2	2	no	—
62	#13133146	—	2	4	no	—
63	#13133146	—	2	4	no	—
64	C₄×#88322	—	2	2	no	—
65	C₈×Dic₅₃	—	2	2	no	—
66	C₈×Dic₅₃	—	2	2	no	—
67	C₈×Dic₅₃	—	2	4	no	—
68	C₈×Dic₅₃	—	2	2	no	—
69	C₈×Dic₅₃	Q₈	1	4	yes	complemented
70	#13133124	C₂×C₄	1	2	yes	—
71	#13133124	C₂×C₄	1	2	yes	—
72	#13133124	C₂×C₄	1	2	yes	—
73	C₂²×Q₈×C₅₃	C₂×C₄	1	1	yes	characteristic
74	C₂²×C₈×C₅₃	C₂³	1	2	yes	—
75	C₂²×C₈×C₅₃	C₂³	1	1	yes	characteristic
76	C₂²×C₈×C₅₃	C₂³	1	1	yes	characteristic
77	C₄×Q₈×C₅₃	—	2	2	no	—
78	C₄×Q₈×C₅₃	—	2	2	no	—
79	C₂×C₄⋊C₄×C₅₃	C₂×C₄	1	1	yes	characteristic
80	C₂×C₄⋊C₄×C₅₃	C₂×C₄	1	2	yes	—
81	C₂×C₄²×C₅₃	C₂³	1	2	yes	—
82	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
83	C₄⋊C₈×C₅₃	—	2	2	no	—
84	C₄×C₈×C₅₃	—	2	2	no	—
85	C₂×Q₈×C₅₃	—	2	2	no	—
86	C₂×Q₈×C₅₃	—	2	2	no	—
87	C₂×Q₈×C₅₃	C₂×C₈	1	4	yes	—
88	C₂²×C₄×C₅₃	C₂²×C₄	1	2	yes	—
89	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
90	C₂²×C₄×C₅₃	C₂²×C₄	1	2	yes	—
91	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
92	C₂²×C₄×C₅₃	C₂⁴	1	1	yes	characteristic
93	C₂²×Dic₅₃	C₂²×C₄	1	2	yes	—
94	C₂²×Dic₅₃	C₂²×C₄	1	1	yes	characteristic
95	C₂²×Dic₅₃	C₂²×C₄	1	2	yes	—
96	C₂²×Dic₅₃	C₂²×C₄	1	1	yes	characteristic
97	C₂²×Dic₅₃	C₂²×C₄	1	1	yes	characteristic
98	C₂²×Dic₅₃	C₂²×C₄	1	1	yes	characteristic
99	C₂×C₈×C₅₃	—	2	1	no	—
100	C₂×C₈×C₅₃	—	2	1	no	—
101	C₂×C₈×C₅₃	—	2	1	no	—
102	C₂×C₈×C₅₃	—	2	1	no	—
103	C₂×C₈×C₅₃	—	2	2	no	—
104	C₂×C₈×C₅₃	—	2	2	no	—
105	C₂×C₈×C₅₃	—	2	2	no	—
106	C₂×C₈×C₅₃	C₂×Q₈	1	2	yes	—
107	C₂×C₈×C₅₃	#41	1	2	yes	—
108	C₄⋊C₄×C₅₃	—	2	4	no	—
109	C₄⋊C₄×C₅₃	—	2	2	no	—
110	C₄²×C₅₃	—	2	2	no	—
111	C₄²×C₅₃	—	2	2	no	—
112	C₄²×C₅₃	#41	1	2	yes	—
113	C₄²×C₅₃	#41	1	2	yes	—
114	C₄⋊Dic₅₃	—	2	2	no	—
115	C₄⋊Dic₅₃	—	2	4	no	—
116	C₄⋊Dic₅₃	C₂×C₈	1	8	yes	—
117	C₄⋊Dic₅₃	C₂×C₈	1	4	yes	—
118	C₄×Dic₅₃	—	2	1	no	—
119	C₄×Dic₅₃	—	2	1	no	—
120	C₄×Dic₅₃	—	2	2	no	—
121	C₄×Dic₅₃	—	2	2	no	—
122	C₄×Dic₅₃	—	2	4	no	—
123	C₄×Dic₅₃	—	2	2	no	—
124	C₄×Dic₅₃	—	2	2	no	—
125	C₄×Dic₅₃	—	2	4	no	—
126	C₄×Dic₅₃	—	2	4	no	—
127	C₄×Dic₅₃	C₂×C₈	1	8	yes	—
128	C₄×Dic₅₃	C₂×C₈	1	4	yes	—
129	C₄×Dic₅₃	M₁₆	1	4	yes	—
130	C₄×Dic₅₃	C₂×Q₈	1	2	yes	—
131	C₄×Dic₅₃	#41	1	2	yes	—
132	C₂×#88322	—	2	1	no	—
133	C₂×#88322	—	2	1	no	—
134	C₂×#88322	—	2	1	no	—
135	C₂×#88322	—	2	1	no	—
136	C₂×#88322	—	2	2	no	—
137	C₂×#88322	—	2	2	no	—
138	C₂×#88322	—	2	2	no	—
139	C₂×#88322	C₂×Q₈	1	2	yes	—
140	C₂×#88322	#41	1	2	yes	—
141	C₂³×C₅₃	C₂³×C₄	1	1	yes	characteristic
142	Q₈×C₅₃	—	2	8	no	—
143	C₂×C₄×C₅₃	—	2	2	no	—
144	C₂×C₄×C₅₃	—	2	2	no	—
145	C₂×C₄×C₅₃	—	2	1	no	—
146	C₂×C₄×C₅₃	—	2	1	no	—
147	C₂×C₄×C₅₃	—	2	2	no	—
148	C₂×C₄×C₅₃	—	2	2	no	—
149	C₂×C₄×C₅₃	—	2	2	no	—
150	C₂×C₄×C₅₃	#117	1	2	yes	—
151	C₂×C₄×C₅₃	#117	1	2	yes	—
152	C₂×C₄×C₅₃	C₂²×C₈	1	4	yes	—
153	C₂×C₄×C₅₃	C₂²×C₈	1	2	yes	—
154	C₂×C₄×C₅₃	C₂×M₁₆	1	2	yes	—
155	C₂×C₄×C₅₃	C₂²×Q₈	1	1	yes	characteristic
156	C₂×C₄×C₅₃	C₂×#41	1	1	yes	characteristic
157	C₂×C₄×C₅₃	C₂×#41	1	1	yes	characteristic
158	C₂×C₄×C₅₃	C₂×#41	1	1	yes	characteristic
159	C₂×C₄×C₅₃	C₂×#41	1	1	yes	characteristic
160	C₂×C₄×C₅₃	C₂×#41	1	1	yes	characteristic
161	C₂×Dic₅₃	—	2	1	no	—
162	C₂×Dic₅₃	—	2	2	no	—
163	C₂×Dic₅₃	—	2	1	no	—
164	C₂×Dic₅₃	—	2	1	no	—
165	C₂×Dic₅₃	—	2	2	no	—
166	C₂×Dic₅₃	—	2	1	no	—
167	C₂×Dic₅₃	—	2	2	no	—
168	C₂×Dic₅₃	—	2	2	no	—
169	C₂×Dic₅₃	—	2	2	no	—
170	C₂×Dic₅₃	C₄×Q₈	1	2	yes	—
171	C₂×Dic₅₃	C₄×Q₈	1	2	yes	—
172	C₂×Dic₅₃	C₂²×C₈	1	2	yes	—
173	C₂×Dic₅₃	C₂²×C₈	1	4	yes	—
174	C₂×Dic₅₃	C₂²×C₈	1	2	yes	—
175	C₂×Dic₅₃	C₂×M₁₆	1	2	yes	—
176	C₂×Dic₅₃	C₂×M₁₆	1	2	yes	—
177	C₂×Dic₅₃	#131	1	2	yes	—
178	C₂×Dic₅₃	#131	1	2	yes	—
179	C₈×C₅₃	—	2	2	no	—
180	C₈×C₅₃	—	2	2	no	—
181	C₈×C₅₃	—	4	2	no	—
182	#88322	—	2	2	no	—
183	#88322	—	2	2	no	—
184	#88322	—	4	2	no	—
185	#21339	—	53	1	no	maximal, Sylow
186	C₂²×C₅₃	C₂×#117	1	1	yes	characteristic
187	C₂²×C₅₃	C₂×C₄×Q₈	1	1	yes	characteristic
188	C₂²×C₅₃	C₄×#41	1	1	yes	characteristic
189	C₂²×C₅₃	C₂³×C₈	1	1	yes	characteristic
190	C₂²×C₅₃	C₂²×M₁₆	1	1	yes	characteristic
191	C₂²×C₅₃	C₂×#131	1	1	yes	characteristic
192	C₂²×C₅₃	C₂×#131	1	1	yes	characteristic
193	C₄×C₅₃	—	2	4	no	—
194	C₄×C₅₃	—	2	2	no	—
195	C₄×C₅₃	—	4	2	no	—
196	C₄×C₅₃	#431	1	4	yes	—
197	C₄×C₅₃	#529	1	1	yes	characteristic
198	C₄×C₅₃	#529	1	1	yes	characteristic
199	C₄×C₅₃	#533	1	1	yes	characteristic
200	C₄×C₅₃	#533	1	1	yes	characteristic
201	Dic₅₃	—	2	2	no	—
202	Dic₅₃	—	2	4	no	—
203	Dic₅₃	—	2	2	no	—
204	Dic₅₃	—	4	2	no	—
205	Dic₅₃	C₈×Q₈	1	4	yes	—
206	Dic₅₃	#446	1	4	yes	—
207	C₄²×Q₈	—	53	1	no	—
208	#1984	—	53	1	no	—
209	#1984	D₅₃	1	1	yes	characteristic
210	#1939	—	53	1	no	—
211	#1939	—	53	1	no	—
212	#1916	—	53	2	no	—
213	#1916	—	53	2	no	—
214	C₄×C₄⋊C₈	—	53	1	no	—
215	C₄×C₄⋊C₈	—	53	2	no	—
216	#1744	—	53	2	no	—
217	C₄²×C₈	—	53	1	no	—
218	C₂×C₅₃	#2295	1	1	yes	characteristic
219	C₂×C₅₃	C₂×#431	1	1	yes	characteristic
220	C₂×C₅₃	#2922	1	1	yes	characteristic
221	C₂×C₅₃	C₂×C₈×Q₈	1	1	yes	characteristic
222	C₂×C₅₃	C₂×#446	1	1	yes	characteristic
223	C₂×C₅₃	C₈×#41	1	1	yes	characteristic
224	C₂×C₅₃	#2965	1	1	yes	characteristic
225	C₂×C₄×Q₈	—	53	1	no	—
226	C₂×C₄×Q₈	—	53	1	no	—
227	C₂×C₄×Q₈	C₂×D₅₃	1	1	yes	characteristic
228	C₂×C₄⋊C₈	—	53	2	no	—
229	C₂×C₄⋊C₈	—	53	2	no	—
230	C₂×C₄⋊C₈	—	53	1	no	—
231	C₂×C₄⋊C₈	C₂×D₅₃	1	1	yes	characteristic
232	C₂×C₄×C₈	—	53	1	no	—
233	C₂×C₄×C₈	—	53	1	no	—
234	C₂×C₄×C₈	—	53	1	no	—
235	C₂×C₄×C₈	—	53	1	no	—
236	C₂×C₄×C₈	—	53	2	no	—
237	C₂×C₄×C₈	—	53	1	no	—
238	C₂×C₄×C₈	C₂×D₅₃	1	1	yes	characteristic
239	C₄×C₄⋊C₄	—	53	1	no	—
240	C₄×C₄⋊C₄	—	53	2	no	—
241	C₄×C₄⋊C₄	—	53	1	no	—
242	C₄×C₄⋊C₄	—	53	2	no	—
243	C₄×C₄⋊C₄	—	53	2	no	—
244	C₄×C₄⋊C₄	—	53	1	no	—
245	C₄³	—	53	1	no	—
246	C₄³	—	53	2	no	—
247	#336	—	53	2	no	—
248	#336	—	53	2	no	—
249	#336	—	53	2	no	—
250	#336	—	53	2	no	—
251	#336	—	53	2	no	—
252	#336	C₂×D₅₃	1	2	yes	—
253	#336	C₂×D₅₃	1	2	yes	—
254	C₅₃	#21339	1	1	yes	characteristic, complemented, Sylow
255	C₂²×Q₈	C₄×D₅₃	1	1	yes	characteristic
256	C₂²×C₈	—	53	1	no	—
257	C₂²×C₈	—	53	1	no	—
258	C₂²×C₈	—	53	2	no	—
259	C₂²×C₈	C₂²×D₅₃	1	2	yes	—
260	C₂²×C₈	C₂²×D₅₃	1	1	yes	characteristic
261	C₂²×C₈	C₂²×D₅₃	1	1	yes	characteristic
262	C₄×Q₈	—	2	2	no	—
263	C₄×Q₈	—	2	2	no	—
264	C₄×Q₈	—	53	8	no	—
265	C₄×Q₈	—	106	4	no	—
266	C₂×C₄⋊C₄	—	53	2	no	—
267	C₂×C₄⋊C₄	—	53	1	no	—
268	C₂×C₄⋊C₄	—	53	2	no	—
269	C₂×C₄⋊C₄	—	53	1	no	—
270	C₂×C₄⋊C₄	C₄×D₅₃	1	1	yes	characteristic
271	C₂×C₄⋊C₄	C₄×D₅₃	1	2	yes	—
272	C₂×C₄²	—	53	2	no	—
273	C₂×C₄²	—	53	1	no	—
274	C₂×C₄²	—	53	1	no	—
275	C₂×C₄²	—	53	1	no	—
276	C₂×C₄²	—	53	2	no	—
277	C₂×C₄²	—	53	2	no	—
278	C₂×C₄²	—	53	1	no	—
279	C₂×C₄²	—	53	1	no	—
280	C₂×C₄²	—	53	2	no	—
281	C₂×C₄²	—	53	1	no	—
282	C₂×C₄²	—	53	2	no	—
283	C₂×C₄²	C₂²×D₅₃	1	2	yes	—
284	C₂×C₄²	C₂²×D₅₃	1	1	yes	characteristic
285	C₄⋊C₈	—	2	2	no	—
286	C₄⋊C₈	—	106	4	no	—
287	C₄⋊C₈	—	106	4	no	—
288	C₄⋊C₈	—	106	2	no	—
289	C₄×C₈	—	2	2	no	—
290	C₄×C₈	—	53	4	no	—
291	C₄×C₈	—	106	2	no	—
292	C₄×C₈	—	106	4	no	—
293	C₄×C₈	—	106	2	no	—
294	C₄×C₈	—	106	2	no	—
295	C₄×C₈	—	106	2	no	—
296	#95	—	53	2	no	—
297	#95	—	53	2	no	—
298	#95	—	53	2	no	—
299	C₂×Q₈	—	2	2	no	—
300	C₂×Q₈	—	2	2	no	—
301	C₂×Q₈	C₈×D₅₃	1	4	yes	—
302	C₂²×C₄	—	53	1	no	—
303	C₂²×C₄	—	53	1	no	—
304	C₂²×C₄	—	53	1	no	—
305	C₂²×C₄	—	53	2	no	—
306	C₂²×C₄	—	53	1	no	—
307	C₂²×C₄	—	53	2	no	—
308	C₂²×C₄	C₂×C₄×D₅₃	1	2	yes	—
309	C₂²×C₄	C₂×C₄×D₅₃	1	1	yes	characteristic
310	C₂²×C₄	C₂×C₄×D₅₃	1	2	yes	—
311	C₂²×C₄	C₂×C₄×D₅₃	1	1	yes	characteristic
312	C₂²×C₄	C₂³×D₅₃	1	1	yes	characteristic
313	C₂×C₈	—	2	1	no	—
314	C₂×C₈	—	2	1	no	—
315	C₂×C₈	—	2	1	no	—
316	C₂×C₈	—	2	1	no	—
317	C₂×C₈	—	2	2	no	—
318	C₂×C₈	—	2	2	no	—
319	C₂×C₈	—	2	2	no	—
320	C₂×C₈	—	53	2	no	—
321	C₂×C₈	—	53	2	no	—
322	C₂×C₈	—	106	1	no	—
323	C₂×C₈	—	106	1	no	—
324	C₂×C₈	—	106	1	no	—
325	C₂×C₈	—	106	1	no	—
326	C₂×C₈	—	106	2	no	—
327	C₂×C₈	—	106	2	no	—
328	C₂×C₈	—	106	2	no	—
329	C₂×C₈	Q₈×D₅₃	1	2	yes	—
330	C₂×C₈	#11719949	1	2	yes	—
331	C₄⋊C₄	—	2	4	no	—
332	C₄⋊C₄	—	2	2	no	—
333	C₄⋊C₄	—	53	8	no	—
334	C₄⋊C₄	—	53	4	no	—
335	C₄⋊C₄	—	106	4	no	—
336	C₄⋊C₄	—	106	2	no	—
337	C₄²	—	2	2	no	—
338	C₄²	—	2	2	no	—
339	C₄²	—	53	8	no	—
340	C₄²	—	53	4	no	—
341	C₄²	—	53	4	no	—
342	C₄²	—	53	2	no	—
343	C₄²	—	53	2	no	—
344	C₄²	—	106	4	no	—
345	C₄²	—	106	2	no	—
346	C₄²	—	106	2	no	—
347	C₄²	—	106	4	no	—
348	C₄²	—	106	4	no	—
349	C₄²	—	106	1	no	—
350	C₄²	—	106	2	no	—
351	C₄²	—	106	1	no	—
352	C₄²	—	106	2	no	—
353	C₄²	#11719947	1	2	yes	—
354	C₄²	#11719947	1	2	yes	—
355	C₂³	C₂²×C₄×D₅₃	1	1	yes	characteristic
356	Q₈	—	2	8	no	—
357	C₂×C₄	—	2	2	no	—
358	C₂×C₄	—	2	2	no	—
359	C₂×C₄	—	2	1	no	—
360	C₂×C₄	—	2	1	no	—
361	C₂×C₄	—	2	2	no	—
362	C₂×C₄	—	2	2	no	—
363	C₂×C₄	—	2	2	no	—
364	C₂×C₄	—	53	2	no	—
365	C₂×C₄	—	53	2	no	—
366	C₂×C₄	—	53	2	no	—
367	C₂×C₄	—	53	2	no	—
368	C₂×C₄	—	53	2	no	—
369	C₂×C₄	—	53	2	no	—
370	C₂×C₄	—	53	4	no	—
371	C₂×C₄	—	53	2	no	—
372	C₂×C₄	—	53	2	no	—
373	C₂×C₄	—	106	1	no	—
374	C₂×C₄	—	106	2	no	—
375	C₂×C₄	—	106	1	no	—
376	C₂×C₄	—	106	1	no	—
377	C₂×C₄	—	106	2	no	—
378	C₂×C₄	—	106	1	no	—
379	C₂×C₄	—	106	2	no	—
380	C₂×C₄	—	106	2	no	—
381	C₂×C₄	—	106	2	no	—
382	C₂×C₄	#13133173	1	2	yes	—
383	C₂×C₄	#13133173	1	2	yes	—
384	C₂×C₄	C₂×C₈×D₅₃	1	4	yes	—
385	C₂×C₄	C₂×C₈×D₅₃	1	2	yes	—
386	C₂×C₄	M₁₆×D₅₃	1	2	yes	—
387	C₂×C₄	C₂×Q₈×D₅₃	1	1	yes	characteristic
388	C₂×C₄	#41×D₅₃	1	1	yes	characteristic
389	C₂×C₄	#41×D₅₃	1	1	yes	characteristic
390	C₂×C₄	C₂×#11719947	1	1	yes	characteristic
391	C₂×C₄	C₂×#11719947	1	1	yes	characteristic
392	C₂×C₄	C₂×#11719949	1	1	yes	characteristic
393	C₈	—	2	2	no	—
394	C₈	—	2	2	no	—
395	C₈	—	4	2	no	—
396	C₈	—	106	2	no	—
397	C₈	—	106	2	no	—
398	C₈	—	212	2	no	—
399	C₂²	C₂×#13133173	1	1	yes	characteristic
400	C₂²	C₄×Q₈×D₅₃	1	1	yes	characteristic
401	C₂²	C₄×#11719949	1	1	yes	characteristic
402	C₂²	C₂²×C₈×D₅₃	1	1	yes	characteristic
403	C₂²	C₂×M₁₆×D₅₃	1	1	yes	characteristic
404	C₂²	#131×D₅₃	1	1	yes	characteristic
405	C₂²	#131×D₅₃	1	1	yes	characteristic
406	C₄	—	2	4	no	—
407	C₄	—	2	2	no	—
408	C₄	—	4	2	no	—
409	C₄	—	53	4	no	—
410	C₄	—	53	4	no	—
411	C₄	—	106	2	no	—
412	C₄	—	106	4	no	—
413	C₄	—	106	2	no	—
414	C₄	—	212	2	no	—
415	C₄	#16810673	1	4	yes	—
416	C₄	#16811152	1	1	yes	characteristic
417	C₄	#16811159	1	1	yes	characteristic
418	C₄	#16811175	1	1	yes	characteristic
419	C₄	#16811178	1	1	yes	characteristic
420	C₂	#20277190	1	1	yes	characteristic
421	C₂	C₂×#16810673	1	1	yes	characteristic
422	C₂	#20283355	1	1	yes	characteristic
423	C₂	C₈×Q₈×D₅₃	1	1	yes	characteristic
424	C₂	#446×D₅₃	1	1	yes	characteristic
425	C₂	C₈×#11719949	1	1	yes	characteristic
426	C₂	#20283680	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂²×C₄	C₂³×D₅₃
commutator	C₂²×C₅₃	C₂³×C₈
Fitting	C₅₃×#1984	C₂
socle, prime radical	C₂³×C₅₃	C₂³×C₄
cosocle	C₂²×C₄×C₅₃	C₂⁴
hypercenter	#1984	D₅₃
hypocenter	C₅₃	#21339
LCS	G ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂²×C₄ ◁ #1984

Complements

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

N	H ≅ G/N	type
45. Q₈×Dic₅₃	C₈	semidirect
69. C₈×Dic₅₃	Q₈	semidirect
254. C₅₃	#21339	semidirect

Maximal subgroups

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

subgroup	index
C₄×Q₈×Dic₅₃	2
#20275652	2
#20275433	2
#20275431	2
#20275273	2
#20275271	2
C₄⋊C₈×Dic₅₃	2
#20274563	2
#20274459	2
C₄×C₈×Dic₅₃	2
C₅₃×#1984	2
#21339	53