Subgroups — #27537720

Order 13568.

556 conjugacy classes of subgroups.

The subgroup lattice has 556 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20283422	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	#20276330	C₂	1	1	yes	maximal, characteristic, complemented
12	#20276328	C₂	1	1	yes	maximal, characteristic, complemented
13	#20276327	C₂	1	1	yes	maximal, characteristic, complemented
14	#20276327	C₂	1	1	yes	maximal, characteristic, complemented
15	C₅₃×#2168	C₂	1	1	yes	maximal, characteristic, complemented
16	C₂×#13133223	C₂²	1	1	yes	characteristic
17	C₂×#13133223	C₂²	1	1	yes	characteristic
18	#16811170	C₄	1	1	yes	characteristic
19	C₂×#13133157	C₂²	1	1	yes	characteristic
20	C₂×#13133156	C₂²	1	1	yes	characteristic
21	C₂×#13133156	C₂²	1	1	yes	characteristic
22	C₂²×#11719908	C₂²	1	1	yes	characteristic
23	C₂×C₁₆×D₅₃	C₂²	1	1	yes	characteristic
24	C₂×C₁₆×D₅₃	C₂²	1	1	yes	characteristic
25	#16810701	C₄	1	1	yes	characteristic
26	#16810700	C₄	1	1	yes	characteristic
27	#16810698	C₄	1	1	yes	characteristic
28	#16810698	C₄	1	1	yes	characteristic
29	C₈×D₂₁₂	C₂²	1	1	yes	characteristic
30	C₈×C₂²⋊D₅₃	C₂²	1	1	yes	characteristic
31	C₈×C₂²⋊D₅₃	C₂²	1	1	yes	characteristic
32	#16810645	C₄	1	1	yes	characteristic
33	#16810588	C₄	1	1	yes	characteristic
34	#16810588	C₄	1	1	yes	characteristic
35	C₄×C₈×D₅₃	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	C₂²⋊#11719908	C₂²	1	1	yes	characteristic
40	C₂²⋊#11719908	C₂²	1	1	yes	characteristic
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
46	#16810402	C₂²	1	1	yes	characteristic
47	#16810402	C₂²	1	1	yes	characteristic
48	#16810402	C₂²	1	1	yes	characteristic
49	#16810399	C₂²	1	1	yes	characteristic
50	#16810398	C₂²	1	1	yes	characteristic
51	#16810398	C₂²	1	1	yes	characteristic
52	C₂×M₃₂×C₅₃	C₂²	1	1	yes	characteristic
53	C₂²×C₁₆×C₅₃	C₂²	1	1	yes	characteristic
54	C₈×D₄×C₅₃	C₂²	1	1	yes	characteristic
55	C₅₃×C₄⋊C₁₆	C₂²	1	1	yes	characteristic
56	C₅₃×C₂²⋊C₁₆	C₂²	1	1	yes	characteristic
57	C₅₃×C₂²⋊C₁₆	C₂²	1	1	yes	characteristic
58	C₅₃×#346	C₂²	1	1	yes	characteristic
59	C₂×#11719945	C₂×C₄	1	1	yes	characteristic
60	C₂×#11719945	C₂×C₄	1	1	yes	characteristic
61	#13133223	—	4	1	no	—
62	#13133223	—	4	1	no	—
63	#13133223	—	4	1	no	—
64	#13133223	—	4	1	no	—
65	C₂×#11719917	C₂×C₄	1	1	yes	characteristic
66	C₂×#11719917	C₂×C₄	1	1	yes	characteristic
67	C₂×#11719912	C₂×C₄	1	1	yes	characteristic
68	C₂×#11719912	C₂×C₄	1	1	yes	characteristic
69	C₂×#11719912	C₂×C₄	1	1	yes	characteristic
70	C₂×#11719912	C₂×C₄	1	1	yes	characteristic
71	C₂×C₈×D₅₃	C₂×C₄	1	1	yes	characteristic
72	C₂×C₈×D₅₃	C₂³	1	1	yes	characteristic
73	C₂×C₈×D₅₃	C₂³	1	1	yes	characteristic
74	C₂×C₈×D₅₃	C₂³	1	1	yes	characteristic
75	#13133216	C₈	1	1	yes	characteristic
76	#13133213	C₈	1	1	yes	characteristic
77	#13133203	C₈	1	1	yes	characteristic
78	#13133203	C₈	1	1	yes	characteristic
79	#13133192	C₈	1	1	yes	characteristic
80	#13133190	C₈	1	1	yes	characteristic
81	#13133190	C₈	1	1	yes	characteristic
82	#13133188	C₈	1	1	yes	characteristic
83	C₄×D₂₁₂	C₂×C₄	1	1	yes	characteristic
84	C₄×C₂²⋊D₅₃	C₂×C₄	1	1	yes	characteristic
85	C₄×C₂²⋊D₅₃	C₂×C₄	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	C₂×D₄×C₅₃	C₂×C₈	1	1	yes	characteristic
123	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
124	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
125	C₂×C₂²⋊D₅₃	C₂×C₈	1	1	yes	characteristic
126	C₂×C₂²⋊D₅₃	C₂×C₈	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	C₂×C₂²⋊D₅₃	C₂×C₈	1	1	yes	characteristic
130	C₂×C₂²⋊D₅₃	C₂×C₈	1	1	yes	characteristic
131	#11719945	—	4	1	no	—
132	#11719945	—	4	1	no	—
133	#11719945	—	4	1	no	—
134	#11719945	—	4	1	no	—
135	C₂×D₂₁₂	C₂×C₈	1	1	yes	characteristic
136	C₂×D₂₁₂	C₂×C₈	1	1	yes	characteristic
137	C₂×D₂₁₂	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	C₂²×C₄	1	1	yes	characteristic
141	C₂×C₄×D₅₃	C₂²×C₄	1	1	yes	characteristic
142	C₂×C₄×D₅₃	C₂²×C₄	1	1	yes	characteristic
143	C₂×Dic₁₀₆	C₂×C₈	1	1	yes	characteristic
144	C₂×Dic₁₀₆	C₂×C₈	1	1	yes	characteristic
145	C₂×C₈×C₅₃	—	2	1	no	—
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	C₂²×C₄	1	1	yes	characteristic
150	C₂×C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
151	C₂×C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
152	C₂×C₈×C₅₃	C₂⁴	1	1	yes	characteristic
153	C₄⋊C₄×C₅₃	C₂×C₈	1	1	yes	characteristic
154	C₂²⋊C₄×C₅₃	C₂×C₈	1	1	yes	characteristic
155	C₂²⋊C₄×C₅₃	C₂×C₈	1	1	yes	characteristic
156	C₄²×C₅₃	C₂²×C₄	1	1	yes	characteristic
157	C₂²⋊Dic₅₃	C₂×C₈	1	1	yes	characteristic
158	C₂²⋊Dic₅₃	C₂×C₈	1	1	yes	characteristic
159	#11719921	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	C₄⋊Dic₅₃	C₂×C₈	1	1	yes	characteristic
164	#11719919	C₂×C₈	1	1	yes	characteristic
165	#11719919	C₂×C₈	1	1	yes	characteristic
166	C₄×Dic₅₃	C₂×C₈	1	1	yes	characteristic
167	C₄×Dic₅₃	C₂×C₈	1	1	yes	characteristic
168	C₄×Dic₅₃	C₂²×C₄	1	1	yes	characteristic
169	#11719917	—	4	1	no	—
170	#11719917	—	4	1	no	—
171	C₂×#88322	C₂²×C₄	1	1	yes	characteristic
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	#11719912	—	4	1	no	—
176	#11719912	—	4	1	no	—
177	#11719912	—	4	1	no	—
178	#11719912	—	4	1	no	—
179	C₈×D₅₃	—	2	1	no	—
180	C₈×D₅₃	—	2	1	no	—
181	C₈×D₅₃	—	2	1	no	—
182	C₈×D₅₃	—	2	1	no	—
183	C₈×D₅₃	—	2	1	no	—
184	C₈×D₅₃	—	2	1	no	—
185	C₈×D₅₃	—	4	1	no	—
186	C₁₆×C₅₃	—	2	1	no	—
187	C₁₆×C₅₃	—	2	1	no	—
188	C₁₆×C₅₃	—	2	1	no	—
189	C₁₆×C₅₃	—	2	1	no	—
190	#11719908	—	2	1	no	—
191	#11719908	—	2	1	no	—
192	#11719908	—	2	1	no	—
193	#11719908	—	2	1	no	—
194	C₂³×C₅₃	C₂²×C₈	1	1	yes	characteristic
195	C₂³×C₅₃	C₂²×C₈	1	1	yes	characteristic
196	C₂²×D₅₃	C₂²×C₈	1	1	yes	characteristic
197	C₂²×D₅₃	C₂²×C₈	1	1	yes	characteristic
198	C₂²×D₅₃	C₂²×C₈	1	1	yes	characteristic
199	D₄×C₅₃	—	4	1	no	—
200	C₂×C₄×C₅₃	—	2	1	no	—
201	C₂×C₄×C₅₃	—	2	1	no	—
202	C₂×C₄×C₅₃	—	2	1	no	—
203	C₂×C₄×C₅₃	—	2	1	no	—
204	C₂×C₄×C₅₃	C₂²×C₈	1	1	yes	characteristic
205	C₂×C₄×C₅₃	C₂²×C₈	1	1	yes	characteristic
206	C₂×C₄×C₅₃	C₂²×C₈	1	1	yes	characteristic
207	C₂×C₄×C₅₃	C₂²×C₈	1	1	yes	characteristic
208	C₂×C₄×C₅₃	C₂³×C₄	1	1	yes	characteristic
209	C₂²⋊D₅₃	—	4	1	no	—
210	C₂²⋊D₅₃	—	4	1	no	—
211	C₂²⋊D₅₃	—	4	1	no	—
212	C₂²⋊D₅₃	—	4	1	no	—
213	C₂²⋊D₅₃	—	4	1	no	—
214	C₂²⋊D₅₃	—	4	1	no	—
215	C₂×Dic₅₃	C₂²×C₈	1	1	yes	characteristic
216	C₂×Dic₅₃	C₂²×C₈	1	1	yes	characteristic
217	C₂×Dic₅₃	C₂²×C₈	1	1	yes	characteristic
218	C₂×Dic₅₃	C₂²×C₈	1	1	yes	characteristic
219	C₂×Dic₅₃	C₂²×C₈	1	1	yes	characteristic
220	D₂₁₂	—	4	1	no	—
221	D₂₁₂	—	4	1	no	—
222	D₂₁₂	—	4	1	no	—
223	C₄×D₅₃	—	2	1	no	—
224	C₄×D₅₃	—	2	1	no	—
225	C₄×D₅₃	—	2	1	no	—
226	C₄×D₅₃	—	2	1	no	—
227	C₄×D₅₃	—	2	1	no	—
228	C₄×D₅₃	—	2	1	no	—
229	C₄×D₅₃	—	4	1	no	—
230	C₄×D₅₃	—	4	1	no	—
231	Dic₁₀₆	—	4	1	no	—
232	Dic₁₀₆	—	4	1	no	—
233	C₈×C₅₃	—	2	1	no	—
234	C₈×C₅₃	—	2	1	no	—
235	C₈×C₅₃	—	2	1	no	—
236	C₈×C₅₃	H(5,𝔽₂)	1	1	yes	characteristic
237	C₈×C₅₃	H(5,𝔽₂)	1	1	yes	characteristic
238	#88322	—	2	1	no	—
239	#88322	—	2	1	no	—
240	#88322	—	2	1	no	—
241	#88322	—	2	1	no	—
242	#33494	—	53	1	no	maximal, Sylow
243	C₂²×C₅₃	—	2	1	no	—
244	C₂²×C₅₃	—	2	1	no	—
245	C₂²×C₅₃	—	2	1	no	—
246	C₂²×C₅₃	—	2	1	no	—
247	C₂²×C₅₃	—	2	1	no	—
248	C₂²×C₅₃	—	2	1	no	—
249	C₂²×C₅₃	C₂³×C₈	1	1	yes	characteristic
250	C₂×D₅₃	—	2	1	no	—
251	C₂×D₅₃	—	2	1	no	—
252	C₂×D₅₃	—	2	1	no	—
253	C₂×D₅₃	—	2	1	no	—
254	C₂×D₅₃	—	2	1	no	—
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	C₄×C₅₃	—	2	1	no	—
260	C₄×C₅₃	—	2	1	no	—
261	C₄×C₅₃	—	2	1	no	—
262	C₄×C₅₃	—	2	1	no	—
263	C₄×C₅₃	#518	1	1	yes	characteristic
264	C₄×C₅₃	#568	1	1	yes	characteristic
265	Dic₅₃	—	2	1	no	—
266	Dic₅₃	—	2	1	no	—
267	Dic₅₃	—	2	1	no	—
268	Dic₅₃	—	2	1	no	—
269	Dic₅₃	—	2	1	no	—
270	#2931	—	53	1	no	—
271	#2168	—	53	1	no	—
272	#2168	—	53	1	no	—
273	#2168	—	53	1	no	—
274	#2168	—	53	1	no	—
275	#2168	—	53	1	no	—
276	#2168	—	53	1	no	—
277	#2168	—	53	1	no	—
278	#2168	D₅₃	1	1	yes	characteristic, complemented
279	#2163	—	53	1	no	—
280	#2163	—	53	1	no	—
281	#2113	—	53	1	no	—
282	#2113	—	53	1	no	—
283	#2113	—	53	1	no	—
284	#2113	—	53	1	no	—
285	C₂×C₅₃	—	4	1	no	—
286	C₂×C₅₃	—	4	1	no	—
287	C₂×C₅₃	#2972	1	1	yes	characteristic
288	C₂×C₅₃	#3407	1	1	yes	characteristic
289	C₂×C₅₃	#3407	1	1	yes	characteristic
290	D₅₃	—	4	1	no	—
291	D₅₃	—	4	1	no	—
292	D₅₃	—	4	1	no	—
293	C₂×#131	—	53	1	no	—
294	C₂×#131	—	53	1	no	—
295	#532	—	53	1	no	—
296	C₂×M₃₂	—	53	1	no	—
297	C₂×M₃₂	—	53	1	no	—
298	C₂×M₃₂	—	53	1	no	—
299	C₂×M₃₂	C₂×D₅₃	1	1	yes	characteristic
300	C₂²×C₁₆	—	53	1	no	—
301	C₂²×C₁₆	—	53	1	no	—
302	C₂²×C₁₆	—	53	1	no	—
303	C₂²×C₁₆	C₂×D₅₃	1	1	yes	characteristic
304	#436	—	53	1	no	—
305	#436	—	53	1	no	—
306	#436	—	53	1	no	—
307	#436	—	53	1	no	—
308	C₈×D₄	—	53	1	no	—
309	C₈×D₄	—	53	1	no	—
310	C₈×D₄	—	53	1	no	—
311	C₈×D₄	C₂×D₅₃	1	1	yes	characteristic
312	#423	—	53	1	no	—
313	#408	—	53	1	no	—
314	#408	—	53	1	no	—
315	C₂×C₄×C₈	—	53	1	no	—
316	C₄⋊C₁₆	—	53	1	no	—
317	C₄⋊C₁₆	—	53	1	no	—
318	C₄⋊C₁₆	—	53	1	no	—
319	C₄⋊C₁₆	C₂×D₅₃	1	1	yes	characteristic
320	C₂²⋊C₁₆	—	53	1	no	—
321	C₂²⋊C₁₆	—	53	1	no	—
322	C₂²⋊C₁₆	—	53	1	no	—
323	C₂²⋊C₁₆	—	53	1	no	—
324	C₂²⋊C₁₆	—	53	1	no	—
325	C₂²⋊C₁₆	—	53	1	no	—
326	C₂²⋊C₁₆	—	53	1	no	—
327	C₂²⋊C₁₆	—	53	1	no	—
328	C₂²⋊C₁₆	—	53	1	no	—
329	C₂²⋊C₁₆	—	53	1	no	—
330	C₂²⋊C₁₆	C₂×D₅₃	1	1	yes	characteristic
331	C₂²⋊C₁₆	C₂×D₅₃	1	1	yes	characteristic
332	#346	—	53	1	no	—
333	#346	—	53	1	no	—
334	#346	—	53	1	no	—
335	#346	C₂×D₅₃	1	1	yes	characteristic
336	C₅₃	#33494	1	1	yes	characteristic, complemented, Sylow
337	C₂×#41	—	53	1	no	—
338	C₂×#41	—	53	1	no	—
339	#131	—	212	1	no	—
340	#131	—	212	1	no	—
341	#131	—	212	1	no	—
342	#131	—	212	1	no	—
343	C₂×M₁₆	—	53	1	no	—
344	C₂×M₁₆	—	53	1	no	—
345	C₂×M₁₆	—	53	1	no	—
346	C₂×M₁₆	—	53	1	no	—
347	C₂×M₁₆	—	53	1	no	—
348	C₂×M₁₆	—	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	C₂²×D₅₃	1	1	yes	characteristic
354	C₂²×C₈	C₂²×D₅₃	1	1	yes	characteristic
355	C₄⋊Q₈	—	53	1	no	—
356	#127	—	53	1	no	—
357	#124	—	53	1	no	—
358	#124	—	53	1	no	—
359	#121	—	53	1	no	—
360	#121	—	53	1	no	—
361	#121	—	53	1	no	—
362	#121	—	53	1	no	—
363	C₄×D₄	—	53	1	no	—
364	C₄×D₄	—	53	1	no	—
365	C₄×D₄	—	53	1	no	—
366	C₄×D₄	C₄×D₅₃	1	1	yes	characteristic
367	C₂×C₄²	—	53	1	no	—
368	M₃₂	—	4	1	no	—
369	M₃₂	—	212	1	no	—
370	M₃₂	—	212	1	no	—
371	M₃₂	—	212	1	no	—
372	C₂×C₁₆	—	4	1	no	—
373	C₂×C₁₆	—	53	1	no	—
374	C₂×C₁₆	—	53	1	no	—
375	C₂×C₁₆	—	53	1	no	—
376	C₂×C₁₆	—	53	1	no	—
377	C₂×C₁₆	—	212	1	no	—
378	C₂×C₁₆	—	212	1	no	—
379	C₂×C₁₆	—	212	1	no	—
380	C₂×C₁₆	C₂²×D₅₃	1	1	yes	characteristic
381	C₂×C₁₆	C₂²×D₅₃	1	1	yes	characteristic
382	C₂×C₁₆	C₂²×D₅₃	1	1	yes	characteristic
383	C₂×C₁₆	C₂²×D₅₃	1	1	yes	characteristic
384	C₄⋊C₈	—	53	1	no	—
385	C₄⋊C₈	—	53	1	no	—
386	C₄⋊C₈	—	53	1	no	—
387	C₄⋊C₈	C₄×D₅₃	1	1	yes	characteristic
388	C₂²⋊C₈	—	53	1	no	—
389	C₂²⋊C₈	—	53	1	no	—
390	C₂²⋊C₈	—	53	1	no	—
391	C₂²⋊C₈	—	53	1	no	—
392	C₂²⋊C₈	—	53	1	no	—
393	C₂²⋊C₈	—	53	1	no	—
394	C₂²⋊C₈	C₄×D₅₃	1	1	yes	characteristic
395	C₂²⋊C₈	C₄×D₅₃	1	1	yes	characteristic
396	C₄×C₈	—	53	1	no	—
397	C₄×C₈	—	53	1	no	—
398	C₄×C₈	—	53	1	no	—
399	C₄×C₈	C₂²×D₅₃	1	1	yes	characteristic
400	#41	—	212	1	no	—
401	#41	—	212	1	no	—
402	#41	—	212	1	no	—
403	#41	—	212	1	no	—
404	C₂×Q₈	—	53	1	no	—
405	C₂×Q₈	—	53	1	no	—
406	C₂×D₄	—	53	1	no	—
407	C₂×D₄	—	53	1	no	—
408	C₂×D₄	—	53	1	no	—
409	C₂×D₄	—	53	1	no	—
410	C₂×D₄	—	53	1	no	—
411	C₂×D₄	—	53	1	no	—
412	C₂×D₄	—	53	1	no	—
413	C₂×D₄	—	53	1	no	—
414	C₂×D₄	—	53	1	no	—
415	C₂×D₄	C₈×D₅₃	1	1	yes	characteristic
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	C₂×C₄×D₅₃	1	1	yes	characteristic
422	C₂²×C₄	C₂×C₄×D₅₃	1	1	yes	characteristic
423	M₁₆	—	212	1	no	—
424	M₁₆	—	212	1	no	—
425	M₁₆	—	212	1	no	—
426	M₁₆	—	212	1	no	—
427	M₁₆	—	212	1	no	—
428	M₁₆	—	212	1	no	—
429	C₂×C₈	—	2	1	no	—
430	C₂×C₈	—	2	1	no	—
431	C₂×C₈	—	2	1	no	—
432	C₂×C₈	—	2	1	no	—
433	C₂×C₈	—	53	1	no	—
434	C₂×C₈	—	53	1	no	—
435	C₂×C₈	—	53	1	no	—
436	C₂×C₈	—	53	1	no	—
437	C₂×C₈	—	106	1	no	—
438	C₂×C₈	—	106	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	—	212	1	no	—
444	C₂×C₈	C₂×C₄×D₅₃	1	1	yes	characteristic
445	C₂×C₈	C₂×C₄×D₅₃	1	1	yes	characteristic
446	C₂×C₈	C₂×C₄×D₅₃	1	1	yes	characteristic
447	C₂×C₈	C₂³×D₅₃	1	1	yes	characteristic
448	C₄⋊C₄	—	53	1	no	—
449	C₄⋊C₄	—	53	1	no	—
450	C₄⋊C₄	—	53	1	no	—
451	C₄⋊C₄	C₈×D₅₃	1	1	yes	characteristic
452	C₂²⋊C₄	—	53	1	no	—
453	C₂²⋊C₄	—	53	1	no	—
454	C₂²⋊C₄	—	53	1	no	—
455	C₂²⋊C₄	—	53	1	no	—
456	C₂²⋊C₄	—	53	1	no	—
457	C₂²⋊C₄	—	53	1	no	—
458	C₂²⋊C₄	C₈×D₅₃	1	1	yes	characteristic
459	C₂²⋊C₄	C₈×D₅₃	1	1	yes	characteristic
460	C₄²	—	53	1	no	—
461	C₄²	—	53	1	no	—
462	C₄²	—	53	1	no	—
463	C₄²	C₂×C₄×D₅₃	1	1	yes	characteristic
464	C₁₆	—	2	1	no	—
465	C₁₆	—	2	1	no	—
466	C₁₆	—	2	1	no	—
467	C₁₆	—	2	1	no	—
468	C₁₆	—	106	1	no	—
469	C₁₆	—	106	1	no	—
470	C₁₆	—	106	1	no	—
471	C₁₆	—	106	1	no	—
472	C₂³	—	53	1	no	—
473	C₂³	—	53	1	no	—
474	C₂³	—	53	1	no	—
475	C₂³	C₂×C₈×D₅₃	1	1	yes	characteristic
476	C₂³	C₂×C₈×D₅₃	1	1	yes	characteristic
477	Q₈	—	212	1	no	—
478	Q₈	—	212	1	no	—
479	D₄	—	4	1	no	—
480	D₄	—	212	1	no	—
481	D₄	—	212	1	no	—
482	D₄	—	212	1	no	—
483	D₄	—	212	1	no	—
484	D₄	—	212	1	no	—
485	D₄	—	212	1	no	—
486	D₄	—	212	1	no	—
487	D₄	—	212	1	no	—
488	D₄	—	212	1	no	—
489	C₂×C₄	—	2	1	no	—
490	C₂×C₄	—	2	1	no	—
491	C₂×C₄	—	2	1	no	—
492	C₂×C₄	—	2	1	no	—
493	C₂×C₄	—	53	1	no	—
494	C₂×C₄	—	53	1	no	—
495	C₂×C₄	—	53	1	no	—
496	C₂×C₄	—	53	1	no	—
497	C₂×C₄	—	53	1	no	—
498	C₂×C₄	—	106	1	no	—
499	C₂×C₄	—	106	1	no	—
500	C₂×C₄	—	106	1	no	—
501	C₂×C₄	—	106	1	no	—
502	C₂×C₄	—	106	1	no	—
503	C₂×C₄	—	106	1	no	—
504	C₂×C₄	—	212	1	no	—
505	C₂×C₄	—	212	1	no	—
506	C₂×C₄	C₂×C₈×D₅₃	1	1	yes	characteristic
507	C₂×C₄	C₂×C₈×D₅₃	1	1	yes	characteristic
508	C₂×C₄	C₂×C₈×D₅₃	1	1	yes	characteristic
509	C₂×C₄	C₂×C₈×D₅₃	1	1	yes	characteristic
510	C₂×C₄	C₂²×C₄×D₅₃	1	1	yes	characteristic
511	C₈	—	2	1	no	—
512	C₈	—	2	1	no	—
513	C₈	—	2	1	no	—
514	C₈	—	106	1	no	—
515	C₈	—	106	1	no	—
516	C₈	—	106	1	no	—
517	C₈	—	106	1	no	—
518	C₈	#13133262	1	1	yes	characteristic
519	C₈	#13133263	1	1	yes	characteristic
520	C₂²	—	2	1	no	—
521	C₂²	—	2	1	no	—
522	C₂²	—	2	1	no	—
523	C₂²	—	2	1	no	—
524	C₂²	—	2	1	no	—
525	C₂²	—	2	1	no	—
526	C₂²	—	106	1	no	—
527	C₂²	—	106	1	no	—
528	C₂²	—	106	1	no	—
529	C₂²	—	106	1	no	—
530	C₂²	—	106	1	no	—
531	C₂²	—	106	1	no	—
532	C₂²	—	106	1	no	—
533	C₂²	—	106	1	no	—
534	C₂²	—	106	1	no	—
535	C₂²	C₂²×C₈×D₅₃	1	1	yes	characteristic
536	C₄	—	2	1	no	—
537	C₄	—	2	1	no	—
538	C₄	—	2	1	no	—
539	C₄	—	2	1	no	—
540	C₄	—	106	1	no	—
541	C₄	—	106	1	no	—
542	C₄	—	106	1	no	—
543	C₄	—	106	1	no	—
544	C₄	—	106	1	no	—
545	C₄	#16811095	1	1	yes	characteristic
546	C₄	#16811388	1	1	yes	characteristic
547	C₂	—	4	1	no	—
548	C₂	—	4	1	no	—
549	C₂	—	212	1	no	—
550	C₂	—	212	1	no	—
551	C₂	—	212	1	no	—
552	C₂	#20283736	1	1	yes	characteristic
553	C₂	#20287816	1	1	yes	characteristic
554	C₂	#20287817	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	#16811170	C₄
cosocle	C₂×C₈×C₅₃	C₂⁴
hypercenter	#2168	D₅₃
hypocenter	C₅₃	#33494
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. #20276330	C₂	semidirect
12. #20276328	C₂	semidirect
13. #20276327	C₂	semidirect
14. #20276327	C₂	semidirect
15. C₅₃×#2168	C₂	semidirect
278. #2168	D₅₃	semidirect
336. C₅₃	#33494	semidirect

Maximal subgroups

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

subgroup	index
#20283422	2
#20276543	2
#20276542	2
#20276541	2
#20276540	2
#20276539	2
#20276538	2
#20276537	2
#20276517	2
#20276516	2
#20276330	2
#20276328	2
#20276327	2
C₅₃×#2168	2
#33494	53