Subgroups — #27537677

Order 13568.

618 conjugacy classes of subgroups.

The subgroup lattice has 570 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₈×D₄×D₅₃	C₂	1	1	yes	maximal, characteristic
2	#20276543	C₂	1	1	yes	maximal, characteristic, complemented
3	#20276542	C₂	1	1	yes	maximal, characteristic, complemented
4	#20276541	C₂	1	1	yes	maximal, characteristic, complemented
5	#20276540	C₂	1	1	yes	maximal, characteristic, complemented
6	#20276539	C₂	1	1	yes	maximal, characteristic, complemented
7	#20276538	C₂	1	1	yes	maximal, characteristic, complemented
8	#20276537	C₂	1	1	yes	maximal, characteristic, complemented
9	#20276517	C₂	1	1	yes	maximal, characteristic, complemented
10	#20276516	C₂	1	1	yes	maximal, characteristic, complemented
11	#20276325	C₂	1	1	yes	maximal, characteristic, complemented
12	#20276324	C₂	1	1	yes	maximal, characteristic, complemented
13	C₂×#16810402	C₂	1	1	yes	maximal, characteristic, complemented
14	C₂²⋊C₁₆×D₅₃	C₂	1	1	yes	maximal, characteristic, complemented
15	C₅₃×#2168	C₂	1	1	yes	maximal, characteristic, complemented
16	C₂²×C₈×D₅₃	C₂²	1	1	yes	characteristic
17	C₂²×C₈×D₅₃	C₂²	1	1	yes	characteristic
18	C₄×D₄×D₅₃	C₄	1	1	yes	characteristic
19	C₂×#13133157	C₂²	1	1	yes	characteristic, complemented
20	C₂×#13133156	C₂²	1	1	yes	characteristic, complemented
21	C₂×#13133156	C₂²	1	1	yes	characteristic, complemented
22	C₂²×#11719908	C₂²	1	1	yes	characteristic, complemented
23	C₂×C₁₆×D₅₃	C₂²	1	1	yes	characteristic, complemented
24	C₂×C₁₆×D₅₃	C₂²	1	1	yes	characteristic, complemented
25	C₈×D₂₁₂	C₂²	1	1	yes	characteristic
26	D₄×#88322	C₄	1	1	yes	characteristic
27	#16810688	C₄	1	1	yes	characteristic
28	C₈×C₂²⋊D₅₃	C₂²	1	1	yes	characteristic
29	C₈×C₂²⋊D₅₃	C₂²	1	1	yes	characteristic
30	#16810686	C₄	1	1	yes	characteristic
31	#16810686	C₄	1	1	yes	characteristic
32	C₄⋊C₈×D₅₃	C₄	1	1	yes	characteristic
33	C₂²⋊C₈×D₅₃	C₄	1	1	yes	characteristic
34	C₂²⋊C₈×D₅₃	C₄	1	1	yes	characteristic
35	C₄×C₈×D₅₃	C₂²	1	1	yes	characteristic
36	C₄⋊#11719908	C₂²	1	1	yes	characteristic, complemented
37	#16810441	C₂²	1	1	yes	characteristic, complemented
38	#16810441	C₂²	1	1	yes	characteristic, complemented
39	C₂²⋊#11719908	C₂²	1	1	yes	characteristic, complemented
40	C₂²⋊#11719908	C₂²	1	1	yes	characteristic, complemented
41	#16810402	C₂²	1	1	yes	characteristic
42	#16810402	C₂²	1	1	yes	characteristic
43	#16810402	C₂²	1	1	yes	characteristic
44	#16810402	C₂²	1	1	yes	characteristic
45	#16810402	C₂²	1	1	yes	characteristic, complemented
46	#16810402	C₂²	1	1	yes	characteristic, complemented
47	#16810402	C₂²	1	1	yes	characteristic, complemented
48	#16810402	C₂²	1	1	yes	characteristic, complemented
49	#16810399	C₂²	1	1	yes	characteristic, complemented
50	#16810398	C₂²	1	1	yes	characteristic, complemented
51	#16810398	C₂²	1	1	yes	characteristic, complemented
52	C₂×M₃₂×C₅₃	C₂²	1	1	yes	characteristic, complemented
53	C₂²×C₁₆×C₅₃	C₂²	1	1	yes	characteristic, complemented
54	C₈×D₄×C₅₃	C₂²	1	1	yes	characteristic
55	C₅₃×C₄⋊C₁₆	C₂²	1	1	yes	characteristic, complemented
56	C₅₃×C₂²⋊C₁₆	C₂²	1	1	yes	characteristic, complemented
57	C₅₃×C₂²⋊C₁₆	C₂²	1	1	yes	characteristic, complemented
58	C₅₃×#346	C₂²	1	1	yes	characteristic, complemented
59	C₂×D₄×D₅₃	C₈	1	1	yes	characteristic
60	C₂²×C₄×D₅₃	C₂×C₄	1	1	yes	characteristic
61	C₂²×C₄×D₅₃	C₂×C₄	1	1	yes	characteristic
62	C₂²×#88322	C₂×C₄	1	1	yes	characteristic
63	C₂²×#88322	C₂×C₄	1	1	yes	characteristic
64	C₂×C₈×D₅₃	—	4	1	no	—
65	C₂×C₈×D₅₃	—	4	1	no	—
66	C₂×C₈×D₅₃	—	4	1	no	—
67	C₂×C₈×D₅₃	—	4	1	no	—
68	C₂×C₈×D₅₃	C₂×C₄	1	1	yes	characteristic
69	C₂×C₈×D₅₃	C₂×C₄	1	1	yes	characteristic
70	C₂×C₈×D₅₃	C₂×C₄	1	1	yes	characteristic
71	C₂×C₈×D₅₃	C₂×C₄	1	1	yes	characteristic
72	C₂×C₈×D₅₃	C₂×C₄	1	1	yes	characteristic
73	C₂×C₈×D₅₃	C₂³	1	1	yes	characteristic
74	C₂×C₈×D₅₃	C₂³	1	1	yes	characteristic
75	C₂×C₈×D₅₃	C₂³	1	1	yes	characteristic
76	C₄×D₂₁₂	C₂×C₄	1	1	yes	characteristic
77	D₄×Dic₅₃	C₈	1	1	yes	characteristic
78	#13133179	C₈	1	1	yes	characteristic
79	C₄×C₂²⋊D₅₃	C₂×C₄	1	1	yes	characteristic
80	C₄×C₂²⋊D₅₃	C₂×C₄	1	1	yes	characteristic
81	#13133177	C₈	1	1	yes	characteristic
82	#13133177	C₈	1	1	yes	characteristic
83	C₄⋊C₄×D₅₃	C₈	1	1	yes	characteristic
84	C₂²⋊C₄×D₅₃	C₈	1	1	yes	characteristic
85	C₂²⋊C₄×D₅₃	C₈	1	1	yes	characteristic
86	C₄²×D₅₃	C₂×C₄	1	1	yes	characteristic
87	#13133157	—	4	1	no	—
88	#13133156	—	4	1	no	—
89	#13133156	—	4	1	no	—
90	C₂×#11719908	—	4	1	no	—
91	C₂×#11719908	C₂³	1	1	yes	characteristic
92	C₂×#11719908	C₂³	1	1	yes	characteristic
93	C₂×#11719908	C₂³	1	1	yes	characteristic
94	C₂×#11719908	C₂³	1	1	yes	characteristic
95	C₁₆×D₅₃	—	4	1	no	—
96	C₁₆×D₅₃	—	4	1	no	—
97	C₄⋊#88322	C₂×C₄	1	1	yes	characteristic
98	#13133146	C₂×C₄	1	1	yes	characteristic
99	#13133146	C₂×C₄	1	1	yes	characteristic
100	C₂²⋊#88322	C₂×C₄	1	1	yes	characteristic
101	C₂²⋊#88322	C₂×C₄	1	1	yes	characteristic
102	#13133129	C₂×C₄	1	1	yes	characteristic
103	#13133129	C₂×C₄	1	1	yes	characteristic
104	#13133129	C₂×C₄	1	1	yes	characteristic
105	#13133129	C₂×C₄	1	1	yes	characteristic
106	C₄×#88322	C₂×C₄	1	1	yes	characteristic
107	C₈×Dic₅₃	C₂×C₄	1	1	yes	characteristic
108	C₈×Dic₅₃	C₂³	1	1	yes	characteristic
109	C₂²×C₈×C₅₃	C₂³	1	1	yes	characteristic
110	C₂²×C₈×C₅₃	C₂³	1	1	yes	characteristic
111	C₄×D₄×C₅₃	C₂×C₄	1	1	yes	characteristic
112	M₃₂×C₅₃	—	4	1	no	—
113	C₂×C₁₆×C₅₃	—	4	1	no	—
114	C₂×C₁₆×C₅₃	C₂³	1	1	yes	characteristic
115	C₂×C₁₆×C₅₃	C₂³	1	1	yes	characteristic
116	C₂×C₁₆×C₅₃	C₂³	1	1	yes	characteristic
117	C₂×C₁₆×C₅₃	C₂³	1	1	yes	characteristic
118	C₄⋊C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
119	C₂²⋊C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
120	C₂²⋊C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
121	C₄×C₈×C₅₃	C₂³	1	1	yes	characteristic
122	C₂³×D₅₃	C₂×C₈	1	1	yes	characteristic
123	C₂³×D₅₃	C₂×C₈	1	1	yes	characteristic
124	C₂×D₄×C₅₃	C₂×C₈	1	1	yes	characteristic
125	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
126	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
127	C₂×C₂²⋊D₅₃	C₂×C₈	1	1	yes	characteristic
128	C₂×C₂²⋊D₅₃	C₂×C₈	1	1	yes	characteristic
129	C₂²×Dic₅₃	C₂×C₈	1	1	yes	characteristic
130	C₂²×Dic₅₃	C₂×C₈	1	1	yes	characteristic
131	D₄×D₅₃	—	2	4	no	—
132	C₂×D₂₁₂	C₂×C₈	1	1	yes	characteristic
133	C₂×C₄×D₅₃	—	4	1	no	—
134	C₂×C₄×D₅₃	—	4	1	no	—
135	C₂×C₄×D₅₃	—	4	1	no	—
136	C₂×C₄×D₅₃	—	4	1	no	—
137	C₂×C₄×D₅₃	C₂×C₈	1	1	yes	characteristic
138	C₂×C₄×D₅₃	C₂×C₈	1	1	yes	characteristic
139	C₂×C₄×D₅₃	C₂×C₈	1	1	yes	characteristic
140	C₂×C₄×D₅₃	C₂×C₈	1	1	yes	characteristic
141	C₂×C₄×D₅₃	C₂×C₈	1	1	yes	characteristic
142	C₂×C₄×D₅₃	C₂×C₈	1	1	yes	characteristic
143	C₂×C₄×D₅₃	C₂²×C₄	1	1	yes	characteristic
144	C₂×C₄×D₅₃	C₂²×C₄	1	1	yes	characteristic
145	C₂×C₄×D₅₃	C₂²×C₄	1	1	yes	characteristic
146	C₂×C₈×C₅₃	—	2	1	no	—
147	C₂×C₈×C₅₃	—	2	1	no	—
148	C₂×C₈×C₅₃	—	2	1	no	—
149	C₂×C₈×C₅₃	—	2	1	no	—
150	C₂×C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
151	C₂×C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
152	C₂×C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
153	C₂×C₈×C₅₃	C₂⁴	1	1	yes	characteristic
154	C₄⋊C₄×C₅₃	C₂×C₈	1	1	yes	characteristic
155	C₂²⋊C₄×C₅₃	C₂×C₈	1	1	yes	characteristic
156	C₂²⋊C₄×C₅₃	C₂×C₈	1	1	yes	characteristic
157	C₄²×C₅₃	C₂²×C₄	1	1	yes	characteristic
158	C₂²⋊Dic₅₃	C₂×C₈	1	1	yes	characteristic
159	C₂²⋊Dic₅₃	C₂×C₈	1	1	yes	characteristic
160	#11719921	C₂×C₈	1	1	yes	characteristic
161	#11719921	C₂×C₈	1	1	yes	characteristic
162	#11719921	C₂×C₈	1	1	yes	characteristic
163	#11719921	C₂×C₈	1	1	yes	characteristic
164	C₄⋊Dic₅₃	C₂×C₈	1	1	yes	characteristic
165	#11719919	C₂×C₈	1	1	yes	characteristic
166	#11719919	C₂×C₈	1	1	yes	characteristic
167	C₄×Dic₅₃	C₂×C₈	1	1	yes	characteristic
168	C₄×Dic₅₃	C₂×C₈	1	1	yes	characteristic
169	C₄×Dic₅₃	C₂²×C₄	1	1	yes	characteristic
170	C₂×#88322	—	4	1	no	—
171	C₂×#88322	—	4	1	no	—
172	C₂×#88322	C₂²×C₄	1	1	yes	characteristic
173	C₂×#88322	C₂²×C₄	1	1	yes	characteristic
174	C₂×#88322	C₂²×C₄	1	1	yes	characteristic
175	C₂×#88322	C₂²×C₄	1	1	yes	characteristic
176	C₈×D₅₃	—	2	1	no	—
177	C₈×D₅₃	—	2	1	no	—
178	C₈×D₅₃	—	2	1	no	—
179	C₈×D₅₃	—	2	1	no	—
180	C₈×D₅₃	—	2	1	no	—
181	C₈×D₅₃	—	2	1	no	—
182	C₈×D₅₃	—	4	1	no	—
183	C₈×D₅₃	—	4	1	no	—
184	C₈×D₅₃	—	4	1	no	—
185	C₈×D₅₃	—	4	1	no	—
186	C₈×D₅₃	—	4	1	no	—
187	C₁₆×C₅₃	—	2	1	no	—
188	C₁₆×C₅₃	—	2	1	no	—
189	C₁₆×C₅₃	—	2	1	no	—
190	C₁₆×C₅₃	—	2	1	no	—
191	#11719908	—	2	1	no	—
192	#11719908	—	2	1	no	—
193	#11719908	—	2	1	no	—
194	#11719908	—	2	1	no	—
195	C₂³×C₅₃	C₂²×C₈	1	1	yes	characteristic
196	C₂³×C₅₃	C₂²×C₈	1	1	yes	characteristic
197	C₂²×D₅₃	—	2	2	no	—
198	C₂²×D₅₃	—	4	1	no	—
199	C₂²×D₅₃	—	4	1	no	—
200	C₂²×D₅₃	—	4	1	no	—
201	C₂²×D₅₃	—	4	1	no	—
202	C₂²×D₅₃	M₃₂	1	4	yes	complemented
203	C₂²×D₅₃	C₂²×C₈	1	1	yes	characteristic
204	C₂²×D₅₃	C₂²×C₈	1	1	yes	characteristic
205	C₂²×D₅₃	C₂²×C₈	1	1	yes	characteristic
206	D₄×C₅₃	—	2	2	no	—
207	C₂×C₄×C₅₃	—	2	1	no	—
208	C₂×C₄×C₅₃	—	2	1	no	—
209	C₂×C₄×C₅₃	—	2	1	no	—
210	C₂×C₄×C₅₃	—	2	1	no	—
211	C₂×C₄×C₅₃	C₂²×C₈	1	1	yes	characteristic
212	C₂×C₄×C₅₃	C₂²×C₈	1	1	yes	characteristic
213	C₂×C₄×C₅₃	C₂²×C₈	1	1	yes	characteristic
214	C₂×C₄×C₅₃	C₂²×C₈	1	1	yes	characteristic
215	C₂×C₄×C₅₃	C₂³×C₄	1	1	yes	characteristic
216	C₂²⋊D₅₃	—	2	2	no	—
217	C₂²⋊D₅₃	—	2	2	no	—
218	C₂×Dic₅₃	—	4	1	no	—
219	C₂×Dic₅₃	—	4	1	no	—
220	C₂×Dic₅₃	C₂²×C₈	1	1	yes	characteristic
221	C₂×Dic₅₃	C₂²×C₈	1	1	yes	characteristic
222	C₂×Dic₅₃	C₂²×C₈	1	1	yes	characteristic
223	C₂×Dic₅₃	C₂²×C₈	1	1	yes	characteristic
224	C₂×Dic₅₃	C₂²×C₈	1	1	yes	characteristic
225	D₂₁₂	—	2	2	no	—
226	C₄×D₅₃	—	2	2	no	—
227	C₄×D₅₃	—	2	1	no	—
228	C₄×D₅₃	—	2	1	no	—
229	C₄×D₅₃	—	2	1	no	—
230	C₄×D₅₃	—	2	1	no	—
231	C₄×D₅₃	—	2	1	no	—
232	C₄×D₅₃	—	2	1	no	—
233	C₄×D₅₃	—	4	1	no	—
234	C₄×D₅₃	—	4	1	no	—
235	C₄×D₅₃	—	4	1	no	—
236	C₄×D₅₃	—	4	1	no	—
237	C₄×D₅₃	—	4	1	no	—
238	C₈×C₅₃	—	2	1	no	—
239	C₈×C₅₃	—	2	1	no	—
240	C₈×C₅₃	—	2	1	no	—
241	C₈×C₅₃	H(5,𝔽₂)	1	1	yes	characteristic
242	C₈×C₅₃	H(5,𝔽₂)	1	1	yes	characteristic
243	#88322	—	2	1	no	—
244	#88322	—	2	1	no	—
245	#88322	—	2	1	no	—
246	#88322	—	2	1	no	—
247	#33487	—	53	1	no	maximal, Sylow
248	C₂²×C₅₃	—	2	1	no	—
249	C₂²×C₅₃	—	2	1	no	—
250	C₂²×C₅₃	—	2	1	no	—
251	C₂²×C₅₃	—	2	1	no	—
252	C₂²×C₅₃	—	2	1	no	—
253	C₂²×C₅₃	C₂×M₃₂	1	2	yes	—
254	C₂²×C₅₃	C₂³×C₈	1	1	yes	characteristic
255	C₂×D₅₃	—	2	1	no	—
256	C₂×D₅₃	—	2	1	no	—
257	C₂×D₅₃	—	2	1	no	—
258	C₂×D₅₃	—	2	1	no	—
259	C₂×D₅₃	—	2	1	no	—
260	C₂×D₅₃	—	2	1	no	—
261	C₂×D₅₃	—	2	1	no	—
262	C₂×D₅₃	—	4	4	no	—
263	C₂×D₅₃	—	4	4	no	—
264	C₂×D₅₃	C₂×M₃₂	1	2	yes	—
265	C₂×D₅₃	C₂×M₃₂	1	2	yes	—
266	C₄×C₅₃	—	2	1	no	—
267	C₄×C₅₃	—	2	1	no	—
268	C₄×C₅₃	—	2	1	no	—
269	C₄×C₅₃	—	2	1	no	—
270	C₄×C₅₃	#518	1	1	yes	characteristic
271	C₄×C₅₃	#568	1	1	yes	characteristic
272	Dic₅₃	—	2	1	no	—
273	Dic₅₃	—	2	1	no	—
274	Dic₅₃	—	2	1	no	—
275	Dic₅₃	—	2	1	no	—
276	Dic₅₃	—	2	1	no	—
277	C₂×C₈×D₄	—	53	1	no	—
278	#2168	—	53	1	no	—
279	#2168	—	53	1	no	—
280	#2168	—	53	1	no	—
281	#2168	—	53	1	no	—
282	#2168	—	53	1	no	—
283	#2168	—	53	1	no	—
284	#2168	—	53	1	no	—
285	#2168	D₅₃	1	1	yes	characteristic, complemented
286	#2163	—	53	1	no	—
287	#2163	—	53	1	no	—
288	#2112	—	53	1	no	—
289	#2112	—	53	1	no	—
290	C₂×C₂²⋊C₁₆	—	53	1	no	—
291	C₂×C₂²⋊C₁₆	—	53	1	no	—
292	C₂×C₅₃	—	2	2	no	—
293	C₂×C₅₃	—	4	1	no	—
294	C₂×C₅₃	#2972	1	1	yes	characteristic
295	C₂×C₅₃	C₂²×M₃₂	1	1	yes	characteristic
296	C₂×C₅₃	#3407	1	1	yes	characteristic
297	D₅₃	—	2	2	no	—
298	D₅₃	—	2	2	no	—
299	D₅₃	—	4	1	no	—
300	C₂³×C₈	—	53	1	no	—
301	C₂³×C₈	—	53	1	no	—
302	C₂×C₄×D₄	—	53	1	no	—
303	C₂×M₃₂	—	53	1	no	—
304	C₂×M₃₂	—	53	1	no	—
305	C₂×M₃₂	—	53	1	no	—
306	C₂×M₃₂	C₂×D₅₃	1	1	yes	characteristic, complemented
307	C₂²×C₁₆	—	53	1	no	—
308	C₂²×C₁₆	—	53	1	no	—
309	C₂²×C₁₆	—	53	1	no	—
310	C₂²×C₁₆	C₂×D₅₃	1	1	yes	characteristic, complemented
311	C₈×D₄	—	53	1	no	—
312	C₈×D₄	—	53	1	no	—
313	C₈×D₄	—	53	1	no	—
314	C₈×D₄	—	53	1	no	—
315	C₈×D₄	—	53	1	no	—
316	C₈×D₄	—	53	1	no	—
317	C₈×D₄	—	53	1	no	—
318	C₈×D₄	C₂×D₅₃	1	1	yes	characteristic
319	C₂×C₄⋊C₈	—	53	1	no	—
320	C₂×C₂²⋊C₈	—	53	1	no	—
321	C₂×C₂²⋊C₈	—	53	1	no	—
322	C₂×C₄×C₈	—	53	1	no	—
323	C₄⋊C₁₆	—	53	1	no	—
324	C₄⋊C₁₆	—	53	1	no	—
325	C₄⋊C₁₆	—	53	1	no	—
326	C₄⋊C₁₆	C₂×D₅₃	1	1	yes	characteristic, complemented
327	C₂²⋊C₁₆	—	53	1	no	—
328	C₂²⋊C₁₆	—	53	1	no	—
329	C₂²⋊C₁₆	—	53	1	no	—
330	C₂²⋊C₁₆	—	53	1	no	—
331	C₂²⋊C₁₆	—	53	1	no	—
332	C₂²⋊C₁₆	—	53	1	no	—
333	C₂²⋊C₁₆	—	53	1	no	—
334	C₂²⋊C₁₆	—	53	1	no	—
335	C₂²⋊C₁₆	—	53	1	no	—
336	C₂²⋊C₁₆	—	53	1	no	—
337	C₂²⋊C₁₆	C₂×D₅₃	1	1	yes	characteristic, complemented
338	C₂²⋊C₁₆	C₂×D₅₃	1	1	yes	characteristic, complemented
339	#346	—	53	1	no	—
340	#346	—	53	1	no	—
341	#346	—	53	1	no	—
342	#346	C₂×D₅₃	1	1	yes	characteristic, complemented
343	C₅₃	#33487	1	1	yes	characteristic, complemented, Sylow
344	C₂²×D₄	—	53	1	no	—
345	C₂³×C₄	—	53	1	no	—
346	C₂³×C₄	—	53	1	no	—
347	C₂²×C₈	—	53	1	no	—
348	C₂²×C₈	—	53	1	no	—
349	C₂²×C₈	—	53	1	no	—
350	C₂²×C₈	—	53	1	no	—
351	C₂²×C₈	—	53	1	no	—
352	C₂²×C₈	—	53	1	no	—
353	C₂²×C₈	—	53	1	no	—
354	C₂²×C₈	—	53	1	no	—
355	C₂²×C₈	—	53	1	no	—
356	C₂²×C₈	—	53	1	no	—
357	C₂²×C₈	—	212	1	no	—
358	C₂²×C₈	—	212	1	no	—
359	C₂²×C₈	—	212	1	no	—
360	C₂²×C₈	—	212	1	no	—
361	C₂²×C₈	C₂²×D₅₃	1	1	yes	characteristic
362	C₂²×C₈	C₂²×D₅₃	1	1	yes	characteristic
363	C₄×D₄	—	53	1	no	—
364	C₄×D₄	—	53	1	no	—
365	C₄×D₄	—	53	1	no	—
366	C₄×D₄	—	53	1	no	—
367	C₄×D₄	—	53	1	no	—
368	C₄×D₄	—	53	1	no	—
369	C₄×D₄	—	53	1	no	—
370	C₄×D₄	C₄×D₅₃	1	1	yes	characteristic
371	C₂×C₄⋊C₄	—	53	1	no	—
372	C₂×C₂²⋊C₄	—	53	1	no	—
373	C₂×C₂²⋊C₄	—	53	1	no	—
374	C₂×C₄²	—	53	1	no	—
375	M₃₂	—	4	1	no	—
376	M₃₂	—	212	1	no	—
377	M₃₂	—	212	1	no	—
378	M₃₂	—	212	1	no	—
379	C₂×C₁₆	—	4	1	no	—
380	C₂×C₁₆	—	53	1	no	—
381	C₂×C₁₆	—	53	1	no	—
382	C₂×C₁₆	—	53	1	no	—
383	C₂×C₁₆	—	53	1	no	—
384	C₂×C₁₆	—	212	1	no	—
385	C₂×C₁₆	—	212	1	no	—
386	C₂×C₁₆	—	212	1	no	—
387	C₂×C₁₆	C₂²×D₅₃	1	1	yes	characteristic
388	C₂×C₁₆	C₂²×D₅₃	1	1	yes	characteristic
389	C₂×C₁₆	C₂²×D₅₃	1	1	yes	characteristic
390	C₂×C₁₆	C₂²×D₅₃	1	1	yes	characteristic
391	C₄⋊C₈	—	53	1	no	—
392	C₄⋊C₈	—	53	1	no	—
393	C₄⋊C₈	—	53	1	no	—
394	C₄⋊C₈	C₄×D₅₃	1	1	yes	characteristic
395	C₂²⋊C₈	—	53	1	no	—
396	C₂²⋊C₈	—	53	1	no	—
397	C₂²⋊C₈	—	53	1	no	—
398	C₂²⋊C₈	—	53	1	no	—
399	C₂²⋊C₈	—	53	1	no	—
400	C₂²⋊C₈	—	53	1	no	—
401	C₂²⋊C₈	C₄×D₅₃	1	1	yes	characteristic
402	C₂²⋊C₈	C₄×D₅₃	1	1	yes	characteristic
403	C₄×C₈	—	53	1	no	—
404	C₄×C₈	—	53	1	no	—
405	C₄×C₈	—	53	1	no	—
406	C₄×C₈	C₂²×D₅₃	1	1	yes	characteristic
407	C₂⁴	—	53	1	no	—
408	C₂⁴	—	53	1	no	—
409	C₂×D₄	—	53	1	no	—
410	C₂×D₄	—	53	1	no	—
411	C₂×D₄	—	53	1	no	—
412	C₂×D₄	—	106	4	no	—
413	C₂×D₄	C₈×D₅₃	1	1	yes	characteristic
414	C₂²×C₄	—	53	1	no	—
415	C₂²×C₄	—	53	1	no	—
416	C₂²×C₄	—	53	1	no	—
417	C₂²×C₄	—	53	1	no	—
418	C₂²×C₄	—	53	1	no	—
419	C₂²×C₄	—	53	1	no	—
420	C₂²×C₄	—	53	1	no	—
421	C₂²×C₄	—	53	1	no	—
422	C₂²×C₄	—	53	1	no	—
423	C₂²×C₄	—	53	1	no	—
424	C₂²×C₄	—	53	1	no	—
425	C₂²×C₄	—	212	1	no	—
426	C₂²×C₄	—	212	1	no	—
427	C₂²×C₄	—	212	1	no	—
428	C₂²×C₄	—	212	1	no	—
429	C₂²×C₄	C₂×C₄×D₅₃	1	1	yes	characteristic
430	C₂²×C₄	C₂×C₄×D₅₃	1	1	yes	characteristic
431	C₂×C₈	—	2	1	no	—
432	C₂×C₈	—	2	1	no	—
433	C₂×C₈	—	2	1	no	—
434	C₂×C₈	—	2	1	no	—
435	C₂×C₈	—	53	1	no	—
436	C₂×C₈	—	53	1	no	—
437	C₂×C₈	—	53	1	no	—
438	C₂×C₈	—	53	1	no	—
439	C₂×C₈	—	106	1	no	—
440	C₂×C₈	—	106	1	no	—
441	C₂×C₈	—	106	1	no	—
442	C₂×C₈	—	106	1	no	—
443	C₂×C₈	—	106	1	no	—
444	C₂×C₈	—	106	1	no	—
445	C₂×C₈	—	212	1	no	—
446	C₂×C₈	—	212	1	no	—
447	C₂×C₈	—	212	1	no	—
448	C₂×C₈	—	212	1	no	—
449	C₂×C₈	—	212	1	no	—
450	C₂×C₈	—	212	1	no	—
451	C₂×C₈	—	212	1	no	—
452	C₂×C₈	C₂×C₄×D₅₃	1	1	yes	characteristic
453	C₂×C₈	C₂×C₄×D₅₃	1	1	yes	characteristic
454	C₂×C₈	C₂×C₄×D₅₃	1	1	yes	characteristic
455	C₂×C₈	C₂³×D₅₃	1	1	yes	characteristic
456	C₄⋊C₄	—	53	1	no	—
457	C₄⋊C₄	—	53	1	no	—
458	C₄⋊C₄	—	53	1	no	—
459	C₄⋊C₄	C₈×D₅₃	1	1	yes	characteristic
460	C₂²⋊C₄	—	53	1	no	—
461	C₂²⋊C₄	—	53	1	no	—
462	C₂²⋊C₄	—	53	1	no	—
463	C₂²⋊C₄	—	53	1	no	—
464	C₂²⋊C₄	—	53	1	no	—
465	C₂²⋊C₄	—	53	1	no	—
466	C₂²⋊C₄	C₈×D₅₃	1	1	yes	characteristic
467	C₂²⋊C₄	C₈×D₅₃	1	1	yes	characteristic
468	C₄²	—	53	1	no	—
469	C₄²	—	53	1	no	—
470	C₄²	—	53	1	no	—
471	C₄²	C₂×C₄×D₅₃	1	1	yes	characteristic
472	C₁₆	—	2	1	no	—
473	C₁₆	—	2	1	no	—
474	C₁₆	—	2	1	no	—
475	C₁₆	—	2	1	no	—
476	C₁₆	—	106	1	no	—
477	C₁₆	—	106	1	no	—
478	C₁₆	—	106	1	no	—
479	C₁₆	—	106	1	no	—
480	C₂³	—	53	1	no	—
481	C₂³	—	53	1	no	—
482	C₂³	—	53	4	no	—
483	C₂³	—	53	1	no	—
484	C₂³	—	106	2	no	—
485	C₂³	—	212	1	no	—
486	C₂³	—	212	1	no	—
487	C₂³	—	212	1	no	—
488	C₂³	—	212	1	no	—
489	C₂³	C₂×C₈×D₅₃	1	1	yes	characteristic
490	C₂³	C₂×C₈×D₅₃	1	1	yes	characteristic
491	D₄	—	2	2	no	—
492	D₄	—	106	2	no	—
493	D₄	—	106	2	no	—
494	D₄	—	106	2	no	—
495	C₂×C₄	—	2	1	no	—
496	C₂×C₄	—	2	1	no	—
497	C₂×C₄	—	2	1	no	—
498	C₂×C₄	—	2	1	no	—
499	C₂×C₄	—	53	1	no	—
500	C₂×C₄	—	53	1	no	—
501	C₂×C₄	—	53	1	no	—
502	C₂×C₄	—	53	1	no	—
503	C₂×C₄	—	53	1	no	—
504	C₂×C₄	—	106	2	no	—
505	C₂×C₄	—	106	1	no	—
506	C₂×C₄	—	106	1	no	—
507	C₂×C₄	—	106	1	no	—
508	C₂×C₄	—	106	1	no	—
509	C₂×C₄	—	106	1	no	—
510	C₂×C₄	—	106	1	no	—
511	C₂×C₄	—	212	1	no	—
512	C₂×C₄	—	212	1	no	—
513	C₂×C₄	—	212	1	no	—
514	C₂×C₄	—	212	1	no	—
515	C₂×C₄	—	212	1	no	—
516	C₂×C₄	—	212	1	no	—
517	C₂×C₄	—	212	1	no	—
518	C₂×C₄	C₂×C₈×D₅₃	1	1	yes	characteristic
519	C₂×C₄	C₂×C₈×D₅₃	1	1	yes	characteristic
520	C₂×C₄	C₂×C₈×D₅₃	1	1	yes	characteristic
521	C₂×C₄	C₂×C₈×D₅₃	1	1	yes	characteristic
522	C₂×C₄	C₂²×C₄×D₅₃	1	1	yes	characteristic
523	C₈	—	2	1	no	—
524	C₈	—	2	1	no	—
525	C₈	—	2	1	no	—
526	C₈	—	106	1	no	—
527	C₈	—	106	1	no	—
528	C₈	—	106	1	no	—
529	C₈	—	106	1	no	—
530	C₈	#13133262	1	1	yes	characteristic
531	C₈	#13133263	1	1	yes	characteristic
532	C₂²	—	2	1	no	—
533	C₂²	—	2	1	no	—
534	C₂²	—	2	1	no	—
535	C₂²	—	2	1	no	—
536	C₂²	—	2	1	no	—
537	C₂²	—	53	2	no	—
538	C₂²	—	53	2	no	—
539	C₂²	—	106	1	no	—
540	C₂²	—	106	1	no	—
541	C₂²	—	106	1	no	—
542	C₂²	—	106	1	no	—
543	C₂²	—	106	1	no	—
544	C₂²	—	106	1	no	—
545	C₂²	—	106	1	no	—
546	C₂²	—	212	4	no	—
547	C₂²	—	212	4	no	—
548	C₂²	M₃₂×D₅₃	1	2	yes	—
549	C₂²	C₂²×C₈×D₅₃	1	1	yes	characteristic
550	C₄	—	2	1	no	—
551	C₄	—	2	1	no	—
552	C₄	—	2	1	no	—
553	C₄	—	2	1	no	—
554	C₄	—	106	1	no	—
555	C₄	—	106	1	no	—
556	C₄	—	106	1	no	—
557	C₄	—	106	1	no	—
558	C₄	—	106	1	no	—
559	C₄	#16811095	1	1	yes	characteristic
560	C₄	#16811388	1	1	yes	characteristic
561	C₂	—	2	2	no	—
562	C₂	—	4	1	no	—
563	C₂	—	106	2	no	—
564	C₂	—	106	2	no	—
565	C₂	—	212	1	no	—
566	C₂	#20283736	1	1	yes	characteristic
567	C₂	C₂×M₃₂×D₅₃	1	1	yes	characteristic
568	C₂	#20287816	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂×C₈	C₂³×D₅₃
commutator, socle	C₂²×C₅₃	C₂³×C₈
Fitting	C₅₃×#2168	C₂
prime radical	C₂×D₄×D₅₃	C₈
cosocle	C₂×C₈×C₅₃	C₂⁴
hypercenter	#2168	D₅₃
hypocenter	C₅₃	#33487
LCS	G ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂×C₈ ◁ #2168

Complements

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

N	H ≅ G/N	type
2. #20276543	C₂	semidirect
3. #20276542	C₂	semidirect
4. #20276541	C₂	semidirect
5. #20276540	C₂	semidirect
6. #20276539	C₂	semidirect
7. #20276538	C₂	semidirect
8. #20276537	C₂	semidirect
9. #20276517	C₂	semidirect
10. #20276516	C₂	semidirect
11. #20276325	C₂	semidirect
12. #20276324	C₂	semidirect
13. C₂×#16810402	C₂	semidirect
14. C₂²⋊C₁₆×D₅₃	C₂	semidirect
15. C₅₃×#2168	C₂	semidirect
19. C₂×#13133157	C₂²	semidirect
20. C₂×#13133156	C₂²	semidirect
21. C₂×#13133156	C₂²	semidirect
22. C₂²×#11719908	C₂²	semidirect
23. C₂×C₁₆×D₅₃	C₂²	semidirect
24. C₂×C₁₆×D₅₃	C₂²	semidirect
36. C₄⋊#11719908	C₂²	semidirect
37. #16810441	C₂²	semidirect
38. #16810441	C₂²	semidirect
39. C₂²⋊#11719908	C₂²	semidirect
40. C₂²⋊#11719908	C₂²	semidirect
45. #16810402	C₂²	semidirect
46. #16810402	C₂²	semidirect
47. #16810402	C₂²	semidirect
48. #16810402	C₂²	semidirect
49. #16810399	C₂²	semidirect
50. #16810398	C₂²	semidirect
51. #16810398	C₂²	semidirect
52. C₂×M₃₂×C₅₃	C₂²	semidirect
53. C₂²×C₁₆×C₅₃	C₂²	semidirect
55. C₅₃×C₄⋊C₁₆	C₂²	semidirect
56. C₅₃×C₂²⋊C₁₆	C₂²	semidirect
57. C₅₃×C₂²⋊C₁₆	C₂²	semidirect
58. C₅₃×#346	C₂²	semidirect
202. C₂²×D₅₃	M₃₂	semidirect
285. #2168	D₅₃	semidirect
306. C₂×M₃₂	C₂×D₅₃	semidirect
310. C₂²×C₁₆	C₂×D₅₃	semidirect
326. C₄⋊C₁₆	C₂×D₅₃	semidirect
337. C₂²⋊C₁₆	C₂×D₅₃	semidirect
338. C₂²⋊C₁₆	C₂×D₅₃	semidirect
342. #346	C₂×D₅₃	semidirect
343. C₅₃	#33487	semidirect

Maximal subgroups

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

subgroup	index
C₈×D₄×D₅₃	2
#20276543	2
#20276542	2
#20276541	2
#20276540	2
#20276539	2
#20276538	2
#20276537	2
#20276517	2
#20276516	2
#20276325	2
#20276324	2
C₂×#16810402	2
C₂²⋊C₁₆×D₅₃	2
C₅₃×#2168	2
#33487	53