Subgroups — #27537310

Order 13568.

682 conjugacy classes of subgroups.

The subgroup lattice has 568 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#16811041	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×#16811039	C₂	1	1	yes	maximal, characteristic, complemented
3	#20283422	C₂	1	1	yes	maximal, characteristic
4	#20276545	C₂	1	1	yes	maximal, characteristic, complemented
5	#20276544	C₂	1	1	yes	maximal, characteristic, complemented
6	#20276542	C₂	1	1	yes	maximal, characteristic, complemented
7	#20276541	C₂	1	1	yes	maximal, characteristic, complemented
8	#20276539	C₂	1	1	yes	maximal, characteristic, complemented
9	C₁₆×C₂²⋊D₅₃	C₂	1	1	yes	maximal, characteristic, complemented
10	#20276532	C₂	1	1	yes	maximal, characteristic, complemented
11	C₄⋊C₁₆×D₅₃	C₂	1	1	yes	maximal, characteristic, complemented
12	#20276327	C₂	1	1	yes	maximal, characteristic, complemented
13	#20276327	C₂	1	1	yes	maximal, characteristic, complemented
14	#20276308	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	#16811041	—	2	4	no	—
20	#16811039	—	2	4	no	—
21	C₂×#13133157	C₂²	1	1	yes	characteristic
22	C₂×#13133156	C₂²	1	1	yes	characteristic
23	C₂×#13133156	C₂²	1	1	yes	characteristic
24	C₂×#13133156	C₂²	1	1	yes	characteristic
25	C₂×#13133156	C₂²	1	1	yes	characteristic
26	C₂²×#11719908	C₂²	1	1	yes	characteristic
27	C₂×C₁₆×D₅₃	C₂²	1	1	yes	characteristic
28	C₂×C₁₆×D₅₃	C₂²	1	1	yes	characteristic
29	C₂×C₁₆×D₅₃	C₂²	1	1	yes	characteristic
30	C₂×C₁₆×D₅₃	C₂²	1	1	yes	characteristic
31	#16810701	C₄	1	1	yes	characteristic
32	#16810700	C₄	1	1	yes	characteristic
33	#16810698	C₄	1	1	yes	characteristic
34	#16810698	C₄	1	1	yes	characteristic
35	C₈×D₂₁₂	C₂²	1	1	yes	characteristic
36	C₈×C₂²⋊D₅₃	C₂²	1	1	yes	characteristic
37	C₈×C₂²⋊D₅₃	C₂²	1	1	yes	characteristic
38	#16810645	C₄	1	1	yes	characteristic
39	#16810588	C₄	1	1	yes	characteristic
40	#16810588	C₄	1	1	yes	characteristic
41	C₄×C₈×D₅₃	C₂²	1	1	yes	characteristic
42	C₄⋊#11719908	C₂²	1	1	yes	characteristic
43	#16810441	C₂²	1	1	yes	characteristic
44	#16810441	C₂²	1	1	yes	characteristic
45	C₂²⋊#11719908	C₂²	1	1	yes	characteristic
46	C₂²⋊#11719908	C₂²	1	1	yes	characteristic
47	#16810402	C₂²	1	1	yes	characteristic
48	#16810402	C₂²	1	1	yes	characteristic
49	#16810402	C₂²	1	1	yes	characteristic
50	#16810402	C₂²	1	1	yes	characteristic
51	#16810399	C₂²	1	1	yes	characteristic
52	C₁₆×Dic₅₃	C₂²	1	1	yes	characteristic
53	C₁₆×Dic₅₃	C₂²	1	1	yes	characteristic
54	C₂×M₃₂×C₅₃	C₂²	1	1	yes	characteristic
55	C₂²×C₁₆×C₅₃	C₂²	1	1	yes	characteristic
56	C₈×D₄×C₅₃	C₂²	1	1	yes	characteristic
57	C₅₃×C₄⋊C₁₆	C₂²	1	1	yes	characteristic
58	C₅₃×C₂²⋊C₁₆	C₂²	1	1	yes	characteristic
59	C₅₃×C₂²⋊C₁₆	C₂²	1	1	yes	characteristic
60	C₅₃×#346	C₂²	1	1	yes	characteristic
61	C₂×#11719945	C₂×C₄	1	1	yes	characteristic
62	C₂×#11719945	C₂×C₄	1	1	yes	characteristic
63	#13133223	—	2	2	no	—
64	#13133223	—	2	2	no	—
65	#13133223	—	2	2	no	—
66	#13133223	—	2	2	no	—
67	C₂×#11719917	C₂×C₄	1	1	yes	characteristic
68	C₂×#11719917	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×#11719912	C₂×C₄	1	1	yes	characteristic
72	C₂×#11719912	C₂×C₄	1	1	yes	characteristic
73	C₂×C₈×D₅₃	C₂×C₄	1	1	yes	characteristic
74	C₂×C₈×D₅₃	C₂³	1	1	yes	characteristic
75	C₂×C₈×D₅₃	C₂³	1	1	yes	characteristic
76	C₂×C₈×D₅₃	C₂³	1	1	yes	characteristic
77	#13133216	C₈	1	1	yes	characteristic
78	#13133213	C₈	1	1	yes	characteristic
79	#13133203	C₈	1	1	yes	characteristic
80	#13133203	C₈	1	1	yes	characteristic
81	#13133192	C₈	1	1	yes	characteristic
82	#13133190	C₈	1	1	yes	characteristic
83	#13133190	C₈	1	1	yes	characteristic
84	#13133188	C₈	1	1	yes	characteristic
85	C₄×D₂₁₂	C₂×C₄	1	1	yes	characteristic
86	C₄×C₂²⋊D₅₃	C₂×C₄	1	1	yes	characteristic
87	C₄×C₂²⋊D₅₃	C₂×C₄	1	1	yes	characteristic
88	C₄²×D₅₃	C₂×C₄	1	1	yes	characteristic
89	#13133157	—	2	2	no	—
90	#13133156	—	2	2	no	—
91	#13133156	—	2	2	no	—
92	#13133156	—	2	2	no	—
93	#13133156	D₄	1	4	yes	complemented
94	C₂×#11719908	—	2	2	no	—
95	C₂×#11719908	C₂³	1	1	yes	characteristic
96	C₂×#11719908	C₂³	1	1	yes	characteristic
97	C₂×#11719908	C₂³	1	1	yes	characteristic
98	C₂×#11719908	C₂³	1	1	yes	characteristic
99	C₁₆×D₅₃	—	2	2	no	—
100	C₁₆×D₅₃	—	2	2	no	—
101	C₁₆×D₅₃	—	2	2	no	—
102	C₁₆×D₅₃	—	2	2	no	—
103	C₄⋊#88322	C₂×C₄	1	1	yes	characteristic
104	#13133146	C₂×C₄	1	1	yes	characteristic
105	#13133146	C₂×C₄	1	1	yes	characteristic
106	C₂²⋊#88322	C₂×C₄	1	1	yes	characteristic
107	C₂²⋊#88322	C₂×C₄	1	1	yes	characteristic
108	#13133129	C₂×C₄	1	1	yes	characteristic
109	#13133129	C₂×C₄	1	1	yes	characteristic
110	#13133129	C₂×C₄	1	1	yes	characteristic
111	#13133129	C₂×C₄	1	1	yes	characteristic
112	C₄×#88322	C₂×C₄	1	1	yes	characteristic
113	C₈×Dic₅₃	C₂×C₄	1	1	yes	characteristic
114	C₈×Dic₅₃	C₂³	1	1	yes	characteristic
115	C₂²×C₈×C₅₃	C₂³	1	1	yes	characteristic
116	C₂²×C₈×C₅₃	C₂³	1	1	yes	characteristic
117	C₄×D₄×C₅₃	C₂×C₄	1	1	yes	characteristic
118	M₃₂×C₅₃	—	2	2	no	—
119	C₂×C₁₆×C₅₃	—	2	2	no	—
120	C₂×C₁₆×C₅₃	C₂³	1	1	yes	characteristic
121	C₂×C₁₆×C₅₃	C₂³	1	1	yes	characteristic
122	C₂×C₁₆×C₅₃	C₂³	1	1	yes	characteristic
123	C₂×C₁₆×C₅₃	C₂³	1	1	yes	characteristic
124	C₄⋊C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
125	C₂²⋊C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
126	C₂²⋊C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
127	C₄×C₈×C₅₃	C₂³	1	1	yes	characteristic
128	C₂×D₄×C₅₃	C₂×C₈	1	1	yes	characteristic
129	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
130	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
131	C₂×C₂²⋊D₅₃	C₂×C₈	1	1	yes	characteristic
132	C₂×C₂²⋊D₅₃	C₂×C₈	1	1	yes	characteristic
133	C₂×C₂²⋊D₅₃	C₂×C₈	1	1	yes	characteristic
134	C₂×C₂²⋊D₅₃	C₂×C₈	1	1	yes	characteristic
135	C₂×C₂²⋊D₅₃	C₂×C₈	1	1	yes	characteristic
136	C₂×C₂²⋊D₅₃	C₂×C₈	1	1	yes	characteristic
137	#11719945	—	2	2	no	—
138	#11719945	—	2	2	no	—
139	#11719945	—	2	2	no	—
140	#11719945	—	2	2	no	—
141	C₂×D₂₁₂	C₂×C₈	1	1	yes	characteristic
142	C₂×D₂₁₂	C₂×C₈	1	1	yes	characteristic
143	C₂×D₂₁₂	C₂×C₈	1	1	yes	characteristic
144	C₂×C₄×D₅₃	C₂×C₈	1	1	yes	characteristic
145	C₂×C₄×D₅₃	C₂×C₈	1	1	yes	characteristic
146	C₂×C₄×D₅₃	C₂²×C₄	1	1	yes	characteristic
147	C₂×C₄×D₅₃	C₂²×C₄	1	1	yes	characteristic
148	C₂×C₄×D₅₃	C₂²×C₄	1	1	yes	characteristic
149	C₂×Dic₁₀₆	C₂×C₈	1	1	yes	characteristic
150	C₂×Dic₁₀₆	C₂×C₈	1	1	yes	characteristic
151	C₂×C₈×C₅₃	—	2	1	no	—
152	C₂×C₈×C₅₃	—	2	1	no	—
153	C₂×C₈×C₅₃	—	2	1	no	—
154	C₂×C₈×C₅₃	—	2	1	no	—
155	C₂×C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
156	C₂×C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
157	C₂×C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
158	C₂×C₈×C₅₃	C₂⁴	1	1	yes	characteristic
159	C₄⋊C₄×C₅₃	C₂×C₈	1	1	yes	characteristic
160	C₂²⋊C₄×C₅₃	C₂×C₈	1	1	yes	characteristic
161	C₂²⋊C₄×C₅₃	C₂×C₈	1	1	yes	characteristic
162	C₄²×C₅₃	C₂²×C₄	1	1	yes	characteristic
163	C₂²⋊Dic₅₃	C₂×C₈	1	1	yes	characteristic
164	C₂²⋊Dic₅₃	C₂×C₈	1	1	yes	characteristic
165	#11719921	C₂×C₈	1	1	yes	characteristic
166	#11719921	C₂×C₈	1	1	yes	characteristic
167	#11719921	C₂×C₈	1	1	yes	characteristic
168	#11719921	C₂×C₈	1	1	yes	characteristic
169	C₄⋊Dic₅₃	C₂×C₈	1	1	yes	characteristic
170	#11719919	C₂×C₈	1	1	yes	characteristic
171	#11719919	C₂×C₈	1	1	yes	characteristic
172	C₄×Dic₅₃	C₂×C₈	1	1	yes	characteristic
173	C₄×Dic₅₃	C₂×C₈	1	1	yes	characteristic
174	C₄×Dic₅₃	C₂²×C₄	1	1	yes	characteristic
175	#11719917	—	2	2	no	—
176	#11719917	—	2	2	no	—
177	C₂×#88322	C₂²×C₄	1	1	yes	characteristic
178	C₂×#88322	C₂²×C₄	1	1	yes	characteristic
179	C₂×#88322	C₂²×C₄	1	1	yes	characteristic
180	C₂×#88322	C₂²×C₄	1	1	yes	characteristic
181	#11719912	—	2	2	no	—
182	#11719912	—	2	2	no	—
183	#11719912	—	2	2	no	—
184	#11719912	—	2	2	no	—
185	C₈×D₅₃	—	2	2	no	—
186	C₈×D₅₃	—	2	1	no	—
187	C₈×D₅₃	—	2	1	no	—
188	C₈×D₅₃	—	2	1	no	—
189	C₈×D₅₃	—	2	1	no	—
190	C₈×D₅₃	C₂×D₄	1	2	yes	—
191	C₈×D₅₃	#41	1	2	yes	—
192	C₁₆×C₅₃	—	2	1	no	—
193	C₁₆×C₅₃	—	2	1	no	—
194	C₁₆×C₅₃	—	2	1	no	—
195	C₁₆×C₅₃	C₂×D₄	1	2	yes	—
196	#11719908	—	2	1	no	—
197	#11719908	—	2	1	no	—
198	#11719908	—	2	1	no	—
199	#11719908	C₂×D₄	1	2	yes	—
200	C₂³×C₅₃	C₂²×C₈	1	1	yes	characteristic
201	C₂³×C₅₃	C₂²×C₈	1	1	yes	characteristic
202	C₂²×D₅₃	C₂²×C₈	1	1	yes	characteristic
203	C₂²×D₅₃	C₂²×C₈	1	1	yes	characteristic
204	C₂²×D₅₃	C₂²×C₈	1	1	yes	characteristic
205	D₄×C₅₃	—	4	1	no	—
206	C₂×C₄×C₅₃	—	2	1	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	C₂²×C₈	1	1	yes	characteristic
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	C₂³×C₄	1	1	yes	characteristic
215	C₂²⋊D₅₃	—	2	2	no	—
216	C₂²⋊D₅₃	—	2	2	no	—
217	C₂²⋊D₅₃	—	2	2	no	—
218	C₂²⋊D₅₃	—	2	2	no	—
219	C₂²⋊D₅₃	—	2	2	no	—
220	C₂²⋊D₅₃	—	2	2	no	—
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	C₂×Dic₅₃	C₂²×C₈	1	1	yes	characteristic
226	D₂₁₂	—	2	2	no	—
227	D₂₁₂	—	2	2	no	—
228	D₂₁₂	—	4	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	—	2	2	no	—
234	C₄×D₅₃	—	4	1	no	—
235	C₄×D₅₃	C₄×D₄	1	2	yes	—
236	C₄×D₅₃	#131	1	2	yes	—
237	Dic₁₀₆	—	2	2	no	—
238	Dic₁₀₆	—	2	2	no	—
239	C₈×C₅₃	—	2	1	no	—
240	C₈×C₅₃	—	2	1	no	—
241	C₈×C₅₃	—	2	1	no	—
242	C₈×C₅₃	C₂²×D₄	1	1	yes	characteristic
243	C₈×C₅₃	C₂×#41	1	1	yes	characteristic
244	#88322	—	2	1	no	—
245	#88322	—	2	1	no	—
246	#88322	—	2	1	no	—
247	#88322	C₄×D₄	1	2	yes	—
248	#33438	—	53	1	no	maximal, Sylow
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	—	2	1	no	—
254	C₂²×C₅₃	—	2	1	no	—
255	C₂²×C₅₃	C₂³×C₈	1	1	yes	characteristic
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	—	2	1	no	—
263	C₂×D₅₃	—	2	1	no	—
264	C₂×D₅₃	C₈×D₄	1	2	yes	—
265	C₄×C₅₃	—	2	1	no	—
266	C₄×C₅₃	—	2	1	no	—
267	C₄×C₅₃	—	2	1	no	—
268	C₄×C₅₃	—	2	1	no	—
269	C₄×C₅₃	C₂×C₄×D₄	1	1	yes	characteristic
270	C₄×C₅₃	C₂×#131	1	1	yes	characteristic
271	Dic₅₃	—	2	1	no	—
272	Dic₅₃	—	2	1	no	—
273	Dic₅₃	—	2	1	no	—
274	Dic₅₃	C₈×D₄	1	2	yes	—
275	Dic₅₃	#504	1	2	yes	complemented
276	C₂×#504	—	53	1	no	—
277	C₂×#504	—	53	1	no	—
278	#2931	—	53	1	no	—
279	#2169	—	53	1	no	—
280	#2169	—	53	1	no	—
281	#2168	—	53	1	no	—
282	#2168	—	53	1	no	—
283	#2168	—	53	1	no	—
284	#2168	D₅₃	1	1	yes	characteristic, complemented
285	D₄×C₁₆	—	53	1	no	—
286	D₄×C₁₆	—	53	1	no	—
287	C₂×C₄⋊C₁₆	—	53	1	no	—
288	#2113	—	53	1	no	—
289	#2113	—	53	1	no	—
290	C₄×M₃₂	—	53	1	no	—
291	C₂×C₅₃	—	4	1	no	—
292	C₂×C₅₃	—	4	1	no	—
293	C₂×C₅₃	C₂×C₈×D₄	1	1	yes	characteristic
294	C₂×C₅₃	C₂×#504	1	1	yes	characteristic
295	C₂×C₅₃	#3407	1	1	yes	characteristic
296	D₅₃	—	2	2	no	—
297	D₅₃	—	4	1	no	—
298	D₅₃	—	4	1	no	—
299	C₂×#131	—	53	1	no	—
300	C₂×#131	—	53	1	no	—
301	#532	—	53	1	no	—
302	#504	—	106	4	no	—
303	#504	—	106	4	no	—
304	C₂×M₃₂	—	53	1	no	—
305	C₂×M₃₂	—	53	1	no	—
306	C₂×M₃₂	—	53	1	no	—
307	C₂×M₃₂	—	53	1	no	—
308	C₂×M₃₂	—	53	1	no	—
309	C₂×M₃₂	C₂×D₅₃	1	1	yes	characteristic
310	C₂²×C₁₆	—	53	1	no	—
311	C₂²×C₁₆	—	53	1	no	—
312	C₂²×C₁₆	—	53	1	no	—
313	C₂²×C₁₆	—	53	1	no	—
314	C₂²×C₁₆	—	53	1	no	—
315	C₂²×C₁₆	C₂×D₅₃	1	1	yes	characteristic
316	#436	—	53	1	no	—
317	#436	—	53	1	no	—
318	#436	—	53	1	no	—
319	#436	—	53	1	no	—
320	C₈×D₄	—	53	1	no	—
321	C₈×D₄	—	53	1	no	—
322	C₈×D₄	—	53	1	no	—
323	C₈×D₄	C₂×D₅₃	1	1	yes	characteristic
324	#423	—	53	1	no	—
325	#408	—	53	1	no	—
326	#408	—	53	1	no	—
327	C₂×C₄×C₈	—	53	1	no	—
328	C₄⋊C₁₆	—	53	1	no	—
329	C₄⋊C₁₆	—	53	1	no	—
330	C₄⋊C₁₆	—	53	1	no	—
331	C₄⋊C₁₆	C₂×D₅₃	1	1	yes	characteristic
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	—	53	1	no	—
338	C₂²⋊C₁₆	C₂×D₅₃	1	1	yes	characteristic
339	C₂²⋊C₁₆	C₂×D₅₃	1	1	yes	characteristic
340	#346	—	53	1	no	—
341	#346	C₂×D₅₃	1	1	yes	characteristic
342	C₄×C₁₆	—	53	1	no	—
343	C₄×C₁₆	—	53	1	no	—
344	C₅₃	#33438	1	1	yes	characteristic, complemented, Sylow
345	C₂×#41	—	53	1	no	—
346	C₂×#41	—	53	1	no	—
347	#131	—	106	2	no	—
348	#131	—	106	2	no	—
349	#131	—	106	2	no	—
350	#131	—	106	2	no	—
351	C₂×M₁₆	—	53	1	no	—
352	C₂×M₁₆	—	53	1	no	—
353	C₂×M₁₆	—	53	1	no	—
354	C₂×M₁₆	—	53	1	no	—
355	C₂×M₁₆	—	53	1	no	—
356	C₂×M₁₆	—	53	1	no	—
357	C₂²×C₈	—	53	1	no	—
358	C₂²×C₈	—	53	1	no	—
359	C₂²×C₈	—	53	1	no	—
360	C₂²×C₈	—	53	1	no	—
361	C₂²×C₈	C₂²×D₅₃	1	1	yes	characteristic
362	C₂²×C₈	C₂²×D₅₃	1	1	yes	characteristic
363	C₄⋊Q₈	—	53	1	no	—
364	#127	—	53	1	no	—
365	#124	—	53	1	no	—
366	#124	—	53	1	no	—
367	#121	—	53	1	no	—
368	#121	—	53	1	no	—
369	#121	—	53	1	no	—
370	#121	—	53	1	no	—
371	C₄×D₄	—	53	1	no	—
372	C₄×D₄	—	53	1	no	—
373	C₄×D₄	—	53	1	no	—
374	C₄×D₄	C₄×D₅₃	1	1	yes	characteristic
375	C₂×C₄²	—	53	1	no	—
376	M₃₂	—	2	2	no	—
377	M₃₂	—	53	4	no	—
378	M₃₂	—	106	2	no	—
379	M₃₂	—	106	2	no	—
380	M₃₂	—	106	2	no	—
381	M₃₂	—	106	2	no	—
382	C₂×C₁₆	—	2	2	no	—
383	C₂×C₁₆	—	53	1	no	—
384	C₂×C₁₆	—	53	1	no	—
385	C₂×C₁₆	—	53	1	no	—
386	C₂×C₁₆	—	53	1	no	—
387	C₂×C₁₆	—	106	2	no	—
388	C₂×C₁₆	—	106	2	no	—
389	C₂×C₁₆	—	106	2	no	—
390	C₂×C₁₆	—	106	2	no	—
391	C₂×C₁₆	—	106	2	no	—
392	C₂×C₁₆	C₂²×D₅₃	1	1	yes	characteristic
393	C₂×C₁₆	C₂²×D₅₃	1	1	yes	characteristic
394	C₂×C₁₆	C₂²×D₅₃	1	1	yes	characteristic
395	C₂×C₁₆	C₂²×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	C₄×D₅₃	1	1	yes	characteristic
400	C₂²⋊C₈	—	53	1	no	—
401	C₂²⋊C₈	—	53	1	no	—
402	C₂²⋊C₈	—	53	1	no	—
403	C₂²⋊C₈	—	53	1	no	—
404	C₂²⋊C₈	—	53	1	no	—
405	C₂²⋊C₈	—	53	1	no	—
406	C₂²⋊C₈	C₄×D₅₃	1	1	yes	characteristic
407	C₂²⋊C₈	C₄×D₅₃	1	1	yes	characteristic
408	C₄×C₈	—	53	1	no	—
409	C₄×C₈	—	53	1	no	—
410	C₄×C₈	—	53	1	no	—
411	C₄×C₈	C₂²×D₅₃	1	1	yes	characteristic
412	#41	—	106	2	no	—
413	#41	—	106	2	no	—
414	#41	—	106	2	no	—
415	#41	—	106	2	no	—
416	C₂×Q₈	—	53	1	no	—
417	C₂×Q₈	—	53	1	no	—
418	C₂×D₄	—	53	1	no	—
419	C₂×D₄	—	53	1	no	—
420	C₂×D₄	—	53	1	no	—
421	C₂×D₄	—	53	1	no	—
422	C₂×D₄	—	53	1	no	—
423	C₂×D₄	—	53	1	no	—
424	C₂×D₄	—	53	1	no	—
425	C₂×D₄	—	53	1	no	—
426	C₂×D₄	—	53	1	no	—
427	C₂×D₄	C₈×D₅₃	1	1	yes	characteristic
428	C₂²×C₄	—	53	1	no	—
429	C₂²×C₄	—	53	1	no	—
430	C₂²×C₄	—	53	1	no	—
431	C₂²×C₄	—	53	1	no	—
432	C₂²×C₄	—	53	1	no	—
433	C₂²×C₄	C₂×C₄×D₅₃	1	1	yes	characteristic
434	C₂²×C₄	C₂×C₄×D₅₃	1	1	yes	characteristic
435	M₁₆	—	106	2	no	—
436	M₁₆	—	106	2	no	—
437	M₁₆	—	106	2	no	—
438	M₁₆	—	106	2	no	—
439	M₁₆	—	106	2	no	—
440	M₁₆	—	106	2	no	—
441	C₂×C₈	—	2	1	no	—
442	C₂×C₈	—	2	1	no	—
443	C₂×C₈	—	2	1	no	—
444	C₂×C₈	—	2	1	no	—
445	C₂×C₈	—	53	1	no	—
446	C₂×C₈	—	53	2	no	—
447	C₂×C₈	—	53	2	no	—
448	C₂×C₈	—	53	1	no	—
449	C₂×C₈	—	53	1	no	—
450	C₂×C₈	—	53	1	no	—
451	C₂×C₈	—	106	2	no	—
452	C₂×C₈	—	106	1	no	—
453	C₂×C₈	—	106	1	no	—
454	C₂×C₈	—	106	1	no	—
455	C₂×C₈	—	106	1	no	—
456	C₂×C₈	C₂×C₄×D₅₃	1	1	yes	characteristic
457	C₂×C₈	C₂×C₄×D₅₃	1	1	yes	characteristic
458	C₂×C₈	C₂×C₄×D₅₃	1	1	yes	characteristic
459	C₂×C₈	C₂³×D₅₃	1	1	yes	characteristic
460	C₄⋊C₄	—	53	1	no	—
461	C₄⋊C₄	—	53	1	no	—
462	C₄⋊C₄	—	53	1	no	—
463	C₄⋊C₄	C₈×D₅₃	1	1	yes	characteristic
464	C₂²⋊C₄	—	53	1	no	—
465	C₂²⋊C₄	—	53	1	no	—
466	C₂²⋊C₄	—	53	1	no	—
467	C₂²⋊C₄	—	53	1	no	—
468	C₂²⋊C₄	—	53	1	no	—
469	C₂²⋊C₄	—	53	1	no	—
470	C₂²⋊C₄	C₈×D₅₃	1	1	yes	characteristic
471	C₂²⋊C₄	C₈×D₅₃	1	1	yes	characteristic
472	C₄²	—	53	1	no	—
473	C₄²	—	53	1	no	—
474	C₄²	—	53	1	no	—
475	C₄²	C₂×C₄×D₅₃	1	1	yes	characteristic
476	C₁₆	—	2	1	no	—
477	C₁₆	—	2	1	no	—
478	C₁₆	—	2	1	no	—
479	C₁₆	—	53	2	no	—
480	C₁₆	—	106	1	no	—
481	C₁₆	—	106	1	no	—
482	C₁₆	—	106	1	no	—
483	C₁₆	D₄×D₅₃	1	2	yes	—
484	C₂³	—	53	1	no	—
485	C₂³	—	53	1	no	—
486	C₂³	—	53	1	no	—
487	C₂³	C₂×C₈×D₅₃	1	1	yes	characteristic
488	C₂³	C₂×C₈×D₅₃	1	1	yes	characteristic
489	Q₈	—	106	2	no	—
490	Q₈	—	106	2	no	—
491	D₄	—	4	1	no	—
492	D₄	—	106	2	no	—
493	D₄	—	106	2	no	—
494	D₄	—	106	2	no	—
495	D₄	—	106	2	no	—
496	D₄	—	106	2	no	—
497	D₄	—	106	2	no	—
498	D₄	—	106	2	no	—
499	D₄	—	106	2	no	—
500	D₄	—	212	1	no	—
501	C₂×C₄	—	2	1	no	—
502	C₂×C₄	—	2	1	no	—
503	C₂×C₄	—	2	1	no	—
504	C₂×C₄	—	2	1	no	—
505	C₂×C₄	—	53	1	no	—
506	C₂×C₄	—	53	1	no	—
507	C₂×C₄	—	53	2	no	—
508	C₂×C₄	—	53	2	no	—
509	C₂×C₄	—	53	1	no	—
510	C₂×C₄	—	53	1	no	—
511	C₂×C₄	—	53	1	no	—
512	C₂×C₄	—	106	2	no	—
513	C₂×C₄	—	106	1	no	—
514	C₂×C₄	—	106	1	no	—
515	C₂×C₄	—	106	1	no	—
516	C₂×C₄	—	106	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	—	53	2	no	—
527	C₈	—	106	1	no	—
528	C₈	—	106	1	no	—
529	C₈	—	106	1	no	—
530	C₈	C₂×D₄×D₅₃	1	1	yes	characteristic
531	C₈	#41×D₅₃	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	—	2	1	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	—	106	1	no	—
547	C₂²	C₂²×C₈×D₅₃	1	1	yes	characteristic
548	C₄	—	2	1	no	—
549	C₄	—	2	1	no	—
550	C₄	—	2	1	no	—
551	C₄	—	2	1	no	—
552	C₄	—	53	2	no	—
553	C₄	—	53	2	no	—
554	C₄	—	106	1	no	—
555	C₄	—	106	1	no	—
556	C₄	—	106	1	no	—
557	C₄	C₄×D₄×D₅₃	1	1	yes	characteristic
558	C₄	#131×D₅₃	1	1	yes	characteristic
559	C₂	—	4	1	no	—
560	C₂	—	4	1	no	—
561	C₂	—	106	2	no	—
562	C₂	—	212	1	no	—
563	C₂	—	212	1	no	—
564	C₂	C₈×D₄×D₅₃	1	1	yes	characteristic
565	C₂	#504×D₅₃	1	1	yes	characteristic
566	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	#33438
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
1. C₂×#16811041	C₂	semidirect
2. C₂×#16811039	C₂	semidirect
4. #20276545	C₂	semidirect
5. #20276544	C₂	semidirect
6. #20276542	C₂	semidirect
7. #20276541	C₂	semidirect
8. #20276539	C₂	semidirect
9. C₁₆×C₂²⋊D₅₃	C₂	semidirect
10. #20276532	C₂	semidirect
11. C₄⋊C₁₆×D₅₃	C₂	semidirect
12. #20276327	C₂	semidirect
13. #20276327	C₂	semidirect
14. #20276308	C₂	semidirect
15. C₅₃×#2168	C₂	semidirect
93. #13133156	D₄	semidirect
275. Dic₅₃	#504	semidirect
284. #2168	D₅₃	semidirect
344. C₅₃	#33438	semidirect

Maximal subgroups

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

subgroup	index
C₂×#16811041	2
C₂×#16811039	2
#20283422	2
#20276545	2
#20276544	2
#20276542	2
#20276541	2
#20276539	2
C₁₆×C₂²⋊D₅₃	2
#20276532	2
C₄⋊C₁₆×D₅₃	2
#20276327	2
#20276308	2
C₅₃×#2168	2
#33438	53