Subgroups — #27402537

Order 13568.

858 conjugacy classes of subgroups.

The subgroup lattice has 576 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#16810743	C₂	1	2	yes	maximal
2	C₄×#13133237	C₂	1	1	yes	maximal, characteristic
3	#20283280	C₂	1	1	yes	maximal, characteristic, complemented
4	#20275465	C₂	1	2	yes	maximal, complemented
5	#20274990	C₂	1	2	yes	maximal, complemented
6	#20274989	C₂	1	2	yes	maximal, complemented
7	#20274886	C₂	1	1	yes	maximal, characteristic, complemented
8	#20274735	C₂	1	1	yes	maximal, characteristic, complemented
9	#20274732	C₂	1	1	yes	maximal, characteristic, complemented
10	#20274585	C₂	1	1	yes	maximal, characteristic, complemented
11	C₅₃×#2906	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×#13133237	C₂²	1	1	yes	characteristic
13	C₂×#13133223	C₂²	1	2	yes	—
14	C₄×#11719945	C₂²	1	1	yes	characteristic
15	#16810743	—	2	8	no	—
16	C₄×#11719925	C₄	1	1	yes	characteristic, complemented
17	C₄×#11719925	C₄	1	1	yes	characteristic, complemented
18	C₄×#11719924	C₄	1	1	yes	characteristic, complemented
19	C₄×#11719924	C₄	1	1	yes	characteristic, complemented
20	C₄×#11719923	C₄	1	1	yes	characteristic, complemented
21	C₄×#11719923	C₄	1	1	yes	characteristic, complemented
22	C₄×#11719922	C₄	1	1	yes	characteristic, complemented
23	C₄×#11719922	C₄	1	1	yes	characteristic, complemented
24	C₂×#13133152	C₂²	1	2	yes	—
25	C₂×#13133152	C₂²	1	2	yes	—
26	#16810659	C₂²	1	1	yes	characteristic
27	#16810649	C₂²	1	2	yes	—
28	#16810648	—	2	2	no	—
29	#16810647	C₂²	1	1	yes	characteristic
30	#16810645	—	2	1	no	—
31	C₄⋊C₈×D₅₃	—	2	1	no	—
32	C₂×#13133144	C₂²	1	2	yes	—
33	C₂×#13133143	C₂²	1	2	yes	—
34	#16810619	C₂²	1	1	yes	characteristic
35	#16810617	C₂²	1	1	yes	characteristic
36	C₂×#13133127	C₂²	1	2	yes	—
37	C₂×C₄×#88322	C₂²	1	1	yes	characteristic
38	C₄×C₄⋊Dic₅₃	C₂²	1	1	yes	characteristic
39	#16810384	C₂²	1	2	yes	—
40	#16810382	C₂²	1	2	yes	—
41	#16810377	C₂²	1	2	yes	—
42	C₂×#131×C₅₃	C₂²	1	2	yes	—
43	C₄×#41×C₅₃	C₂²	1	1	yes	characteristic
44	C₅₃×#424	—	2	2	no	—
45	C₅₃×#424	C₂²	1	2	yes	—
46	C₅₃×#423	—	2	1	no	—
47	C₅₃×#423	C₂²	1	1	yes	characteristic
48	C₂×C₄⋊C₈×C₅₃	—	2	1	no	—
49	C₂×C₄⋊C₈×C₅₃	C₂²	1	1	yes	characteristic
50	C₂×#11719945	C₂³	1	1	yes	characteristic
51	#13133237	D₄	1	4	yes	—
52	#13133237	Q₈	1	4	yes	—
53	C₂×#11719925	C₂×C₄	1	1	yes	characteristic
54	C₂×#11719923	C₂×C₄	1	1	yes	characteristic
55	C₂×#11719924	C₂×C₄	1	1	yes	characteristic
56	C₂×#11719922	C₂×C₄	1	1	yes	characteristic
57	#13133223	—	2	4	no	—
58	#13133223	—	4	2	no	—
59	C₂×#11719917	C₂³	1	2	yes	—
60	C₂×#11719912	—	2	2	no	—
61	C₂²×#88322	C₂³	1	1	yes	characteristic
62	C₂²×#88322	C₂³	1	1	yes	characteristic
63	C₂×C₈×D₅₃	—	2	2	no	—
64	C₄×Dic₁₀₆	C₂×C₄	1	1	yes	characteristic
65	C₄×Dic₁₀₆	C₂×C₄	1	1	yes	characteristic
66	C₄×D₂₁₂	C₂×C₄	1	1	yes	characteristic
67	C₄×D₂₁₂	C₂×C₄	1	1	yes	characteristic
68	C₄×C₂²⋊D₅₃	—	2	1	no	—
69	C₄×C₂²⋊D₅₃	—	2	1	no	—
70	#13133176	C₂³	1	1	yes	characteristic
71	#13133174	—	2	1	no	—
72	C₂×C₄⋊Dic₅₃	—	2	1	no	—
73	C₂×C₄×Dic₅₃	C₂³	1	2	yes	—
74	C₄²×D₅₃	—	2	1	no	—
75	#13133152	—	2	4	no	—
76	#13133152	—	4	2	no	—
77	#13133150	C₂×C₄	1	1	yes	characteristic
78	#13133150	C₂×C₄	1	1	yes	characteristic
79	#13133148	C₂×C₄	1	1	yes	characteristic
80	#13133148	C₂×C₄	1	1	yes	characteristic
81	C₄⋊#88322	—	2	1	no	—
82	C₄⋊#88322	—	2	1	no	—
83	#13133146	—	2	2	no	—
84	#13133146	—	2	2	no	—
85	#13133144	—	2	4	no	—
86	#13133143	—	2	4	no	—
87	#13133141	C₂×C₄	1	1	yes	characteristic
88	#13133141	C₂×C₄	1	1	yes	characteristic
89	#13133140	C₂×C₄	1	1	yes	characteristic
90	#13133140	C₂×C₄	1	1	yes	characteristic
91	#13133138	C₂×C₄	1	1	yes	characteristic
92	#13133138	C₂×C₄	1	1	yes	characteristic
93	#13133137	C₂×C₄	1	1	yes	characteristic
94	#13133137	C₂×C₄	1	1	yes	characteristic
95	#13133127	—	2	4	no	—
96	C₄×#88322	C₂×C₄	1	1	yes	characteristic
97	C₄×#88322	C₂×C₄	1	1	yes	characteristic
98	C₄×#88322	C₂×C₄	1	1	yes	characteristic
99	C₄×#88322	C₂×C₄	1	1	yes	characteristic
100	#13133124	—	2	1	no	—
101	C₂×#41×C₅₃	C₂³	1	1	yes	characteristic
102	#131×C₅₃	—	2	4	no	—
103	#131×C₅₃	—	4	2	no	—
104	C₂×M₁₆×C₅₃	—	2	2	no	—
105	C₂×M₁₆×C₅₃	C₂³	1	2	yes	—
106	C₂²×C₈×C₅₃	—	2	2	no	—
107	C₂²×C₈×C₅₃	C₂³	1	2	yes	—
108	C₄×Q₈×C₅₃	C₂×C₄	1	1	yes	characteristic
109	C₄×Q₈×C₅₃	C₂×C₄	1	1	yes	characteristic
110	C₄×D₄×C₅₃	—	2	1	no	—
111	C₄×D₄×C₅₃	—	2	1	no	—
112	C₄×D₄×C₅₃	C₂×C₄	1	1	yes	characteristic
113	C₄×D₄×C₅₃	C₂×C₄	1	1	yes	characteristic
114	#117×C₅₃	—	2	1	no	—
115	#117×C₅₃	C₂³	1	1	yes	characteristic
116	C₂×C₄²×C₅₃	—	2	1	no	—
117	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
118	C₄⋊C₈×C₅₃	—	2	2	no	—
119	C₄⋊C₈×C₅₃	—	2	2	no	—
120	C₄⋊C₈×C₅₃	—	2	1	no	—
121	C₄⋊C₈×C₅₃	—	2	1	no	—
122	C₄⋊C₈×C₅₃	D₄	1	1	yes	characteristic
123	C₄⋊C₈×C₅₃	D₄	1	1	yes	characteristic
124	C₄⋊C₈×C₅₃	D₄	1	2	yes	—
125	#41×C₅₃	—	2	1	no	—
126	#41×C₅₃	—	2	1	no	—
127	#41×C₅₃	C₂×D₄	1	2	yes	—
128	#41×C₅₃	C₂×Q₈	1	2	yes	—
129	C₂×Q₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
130	C₂×D₄×C₅₃	—	2	1	no	—
131	C₂×D₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
132	C₂²×C₄×C₅₃	—	2	1	no	—
133	C₂²×C₄×C₅₃	—	2	1	no	—
134	C₂²×C₄×C₅₃	—	2	1	no	—
135	C₂²×C₄×C₅₃	#41	1	1	yes	characteristic
136	C₂²×C₄×C₅₃	#41	1	1	yes	characteristic
137	C₂²×C₄×C₅₃	C₂⁴	1	1	yes	characteristic
138	C₂×C₂²⋊D₅₃	—	2	1	no	—
139	C₂²×Dic₅₃	—	2	2	no	—
140	#11719945	—	2	1	no	—
141	#11719945	—	2	1	no	—
142	#11719945	C₂×D₄	1	2	yes	—
143	#11719945	C₂×Q₈	1	2	yes	—
144	C₂×D₂₁₂	C₂²×C₄	1	1	yes	characteristic
145	C₂×C₄×D₅₃	—	2	1	no	—
146	C₂×C₄×D₅₃	—	2	1	no	—
147	C₂×C₄×D₅₃	—	2	1	no	—
148	C₂×Dic₁₀₆	C₂²×C₄	1	1	yes	characteristic
149	M₁₆×C₅₃	—	2	2	no	—
150	M₁₆×C₅₃	—	2	2	no	—
151	M₁₆×C₅₃	—	4	4	no	—
152	C₂×C₈×C₅₃	—	2	2	no	—
153	C₂×C₈×C₅₃	—	2	2	no	—
154	C₂×C₈×C₅₃	—	2	2	no	—
155	C₂×C₈×C₅₃	—	4	4	no	—
156	C₂×C₈×C₅₃	C₂×D₄	1	2	yes	—
157	C₂×C₈×C₅₃	C₂×D₄	1	2	yes	—
158	C₄⋊C₄×C₅₃	—	2	1	no	—
159	C₄⋊C₄×C₅₃	—	2	1	no	—
160	C₄⋊C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
161	C₄⋊C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
162	C₂²⋊C₄×C₅₃	—	2	1	no	—
163	C₂²⋊C₄×C₅₃	—	2	1	no	—
164	C₂²⋊C₄×C₅₃	—	2	1	no	—
165	C₄²×C₅₃	—	2	1	no	—
166	C₄²×C₅₃	—	2	1	no	—
167	C₄²×C₅₃	—	2	1	no	—
168	C₄²×C₅₃	C₂²×C₄	1	1	yes	characteristic
169	C₄²×C₅₃	C₂²×C₄	1	1	yes	characteristic
170	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
171	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
172	C₂²⋊Dic₅₃	—	2	1	no	—
173	#11719925	C₄⋊C₄	1	4	yes	complemented
174	#11719924	C₄⋊C₄	1	4	yes	complemented
175	#11719923	C₄⋊C₄	1	4	yes	complemented
176	#11719922	C₄⋊C₄	1	4	yes	complemented
177	#11719921	—	2	1	no	—
178	#11719921	—	2	1	no	—
179	C₄⋊Dic₅₃	—	4	2	no	—
180	C₄⋊Dic₅₃	C₂²×C₄	1	1	yes	characteristic
181	C₄⋊Dic₅₃	C₂²×C₄	1	1	yes	characteristic
182	#11719919	—	2	1	no	—
183	#11719919	—	2	1	no	—
184	C₄×Dic₅₃	—	2	1	no	—
185	C₄×Dic₅₃	—	2	1	no	—
186	C₄×Dic₅₃	—	2	1	no	—
187	C₄×Dic₅₃	—	2	2	no	—
188	C₄×Dic₅₃	—	2	2	no	—
189	#11719917	—	2	2	no	—
190	#11719917	—	2	2	no	—
191	C₂×#88322	—	2	1	no	—
192	C₂×#88322	—	2	1	no	—
193	C₂×#88322	—	2	2	no	—
194	C₂×#88322	C₂²×C₄	1	1	yes	characteristic
195	C₂×#88322	C₂²×C₄	1	1	yes	characteristic
196	C₂×#88322	C₂²×C₄	1	1	yes	characteristic
197	C₂×#88322	C₂²×C₄	1	1	yes	characteristic
198	C₂×#88322	C₂×D₄	1	2	yes	—
199	C₂×#88322	C₂×Q₈	1	2	yes	—
200	#11719912	—	4	4	no	—
201	C₈×D₅₃	—	4	4	no	—
202	C₂³×C₅₃	—	2	1	no	—
203	C₂³×C₅₃	C₂×#41	1	1	yes	characteristic
204	C₂²×D₅₃	—	2	1	no	—
205	Q₈×C₅₃	—	2	1	no	—
206	Q₈×C₅₃	C₂×C₄⋊C₄	1	2	yes	—
207	D₄×C₅₃	—	2	2	no	—
208	D₄×C₅₃	—	2	2	no	—
209	D₄×C₅₃	—	2	1	no	—
210	D₄×C₅₃	C₂×C₄⋊C₄	1	2	yes	—
211	C₂×C₄×C₅₃	—	2	1	no	—
212	C₂×C₄×C₅₃	—	2	2	no	—
213	C₂×C₄×C₅₃	—	2	2	no	—
214	C₂×C₄×C₅₃	—	2	2	no	—
215	C₂×C₄×C₅₃	—	2	1	no	—
216	C₂×C₄×C₅₃	—	2	1	no	—
217	C₂×C₄×C₅₃	—	4	1	no	—
218	C₂×C₄×C₅₃	—	4	2	no	—
219	C₂×C₄×C₅₃	—	8	1	no	—
220	C₂×C₄×C₅₃	C₄×D₄	1	1	yes	characteristic
221	C₂×C₄×C₅₃	C₄×D₄	1	1	yes	characteristic
222	C₂×C₄×C₅₃	C₄×D₄	1	1	yes	characteristic
223	C₂×C₄×C₅₃	C₄×D₄	1	1	yes	characteristic
224	C₂×C₄×C₅₃	C₂³×C₄	1	1	yes	characteristic
225	C₂×C₄×C₅₃	C₂²×D₄	1	1	yes	characteristic
226	C₂×C₄×C₅₃	C₂²×D₄	1	1	yes	characteristic
227	C₂×C₄×C₅₃	C₂²×Q₈	1	1	yes	characteristic
228	C₂×C₄×C₅₃	H(5,𝔽₂)	1	1	yes	characteristic
229	C₂×C₄×C₅₃	2₋¹⁺⁴	1	1	yes	characteristic
230	C₂²⋊D₅₃	—	2	2	no	—
231	C₂²⋊D₅₃	—	2	2	no	—
232	C₂×Dic₅₃	—	2	1	no	—
233	C₂×Dic₅₃	—	2	1	no	—
234	C₂×Dic₅₃	—	2	1	no	—
235	C₂×Dic₅₃	—	4	2	no	—
236	C₂×Dic₅₃	—	4	2	no	—
237	C₂×Dic₅₃	—	4	2	no	—
238	D₂₁₂	—	2	1	no	—
239	D₂₁₂	C₂×C₄⋊C₄	1	2	yes	—
240	C₄×D₅₃	—	2	2	no	—
241	C₄×D₅₃	—	2	2	no	—
242	C₄×D₅₃	—	4	2	no	—
243	C₄×D₅₃	—	8	1	no	—
244	Dic₁₀₆	—	2	1	no	—
245	Dic₁₀₆	C₂×C₄⋊C₄	1	2	yes	—
246	C₈×C₅₃	—	2	2	no	—
247	C₈×C₅₃	—	2	2	no	—
248	C₈×C₅₃	—	4	2	no	—
249	#88322	—	2	1	no	—
250	#88322	—	2	1	no	—
251	#88322	—	4	2	no	—
252	#88322	C₂×C₄⋊C₄	1	2	yes	—
253	#88322	C₂×C₄⋊C₄	1	2	yes	—
254	#22686	—	53	1	no	maximal, Sylow
255	C₂²×C₅₃	—	2	1	no	—
256	C₂²×C₅₃	—	2	1	no	—
257	C₂²×C₅₃	—	2	2	no	—
258	C₂²×C₅₃	—	4	1	no	—
259	C₂²×C₅₃	—	4	1	no	—
260	C₂²×C₅₃	C₂×C₄×D₄	1	1	yes	characteristic
261	C₂²×C₅₃	#547	1	1	yes	characteristic
262	C₂²×C₅₃	#554	1	1	yes	characteristic
263	C₂×D₅₃	—	2	2	no	—
264	C₂×D₅₃	—	4	1	no	—
265	C₂×D₅₃	—	4	1	no	—
266	C₄×C₅₃	—	2	2	no	—
267	C₄×C₅₃	—	2	1	no	—
268	C₄×C₅₃	—	2	1	no	—
269	C₄×C₅₃	—	2	1	no	—
270	C₄×C₅₃	—	4	1	no	—
271	C₄×C₅₃	—	4	1	no	—
272	C₄×C₅₃	#444	1	2	yes	complemented
273	C₄×C₅₃	C₂²×C₄⋊C₄	1	1	yes	characteristic
274	C₄×C₅₃	#520	1	1	yes	characteristic
275	C₄×C₅₃	D₄²	1	1	yes	characteristic
276	C₄×C₅₃	D₄×Q₈	1	1	yes	characteristic
277	Dic₅₃	—	2	2	no	—
278	Dic₅₃	—	4	1	no	—
279	Dic₅₃	—	4	1	no	—
280	Dic₅₃	—	8	2	no	—
281	C₂×#444	—	53	2	no	—
282	C₄×#135	—	53	1	no	—
283	#2906	—	53	1	no	—
284	#2906	D₅₃	1	1	yes	characteristic, complemented
285	#1946	—	53	2	no	—
286	#1866	—	53	2	no	—
287	#1866	—	53	2	no	—
288	#1848	—	53	1	no	—
289	#1806	—	53	1	no	—
290	#1806	—	53	1	no	—
291	#1770	—	53	1	no	—
292	C₂×C₅₃	—	2	1	no	—
293	C₂×C₅₃	—	2	1	no	—
294	C₂×C₅₃	—	4	2	no	—
295	C₂×C₅₃	D₄×C₄⋊C₄	1	1	yes	characteristic
296	C₂×C₅₃	C₂×#444	1	1	yes	characteristic
297	C₂×C₅₃	#2957	1	1	yes	characteristic
298	D₅₃	—	4	2	no	—
299	C₂×#135	—	53	1	no	—
300	C₂×#131	—	53	2	no	—
301	C₂×#131	C₂×D₅₃	1	2	yes	—
302	C₄×#41	—	53	1	no	—
303	C₄×#41	C₂×D₅₃	1	1	yes	characteristic
304	#444	—	106	8	no	—
305	C₄×Q₁₆	—	53	1	no	—
306	C₄×Q₁₆	—	53	1	no	—
307	C₄×SD₁₆	—	53	1	no	—
308	C₄×SD₁₆	—	53	1	no	—
309	C₄×SD₁₆	—	53	1	no	—
310	C₄×SD₁₆	—	53	1	no	—
311	C₄×D₈	—	53	1	no	—
312	C₄×D₈	—	53	1	no	—
313	C₂×#108	—	53	2	no	—
314	C₂×#108	—	53	2	no	—
315	#427	—	53	1	no	—
316	#424	—	2	2	no	—
317	#424	—	53	2	no	—
318	#424	—	106	2	no	—
319	#424	C₂×D₅₃	1	2	yes	—
320	#423	—	2	1	no	—
321	#423	—	53	1	no	—
322	#423	—	106	1	no	—
323	#423	C₂×D₅₃	1	1	yes	characteristic
324	C₂×C₄⋊C₈	—	2	1	no	—
325	C₂×C₄⋊C₈	—	106	1	no	—
326	C₂×C₄⋊C₈	C₂×D₅₃	1	1	yes	characteristic
327	C₂×#104	—	53	2	no	—
328	C₂×#104	—	53	2	no	—
329	#416	—	53	1	no	—
330	#416	—	53	1	no	—
331	C₂×#97	—	53	2	no	—
332	C₂×C₄×C₈	—	53	1	no	—
333	C₄×C₄⋊C₄	—	53	1	no	—
334	#339	—	53	2	no	—
335	#339	—	53	2	no	—
336	#336	—	53	2	no	—
337	C₅₃	#22686	1	1	yes	characteristic, complemented, Sylow
338	C₂×#41	—	53	1	no	—
339	C₂×#41	C₂²×D₅₃	1	1	yes	characteristic
340	#135	—	53	4	no	—
341	#135	—	53	4	no	—
342	C₂×Q₁₆	—	53	1	no	—
343	C₂×SD₁₆	—	53	1	no	—
344	C₂×SD₁₆	—	53	1	no	—
345	C₂×D₈	—	53	1	no	—
346	#131	—	2	4	no	—
347	#131	—	4	2	no	—
348	#131	—	106	4	no	—
349	#131	—	212	2	no	—
350	C₂×M₁₆	—	2	2	no	—
351	C₂×M₁₆	—	53	2	no	—
352	C₂×M₁₆	—	106	2	no	—
353	C₂×M₁₆	C₂²×D₅₃	1	2	yes	—
354	C₂²×C₈	—	2	2	no	—
355	C₂²×C₈	—	53	1	no	—
356	C₂²×C₈	—	53	1	no	—
357	C₂²×C₈	—	106	2	no	—
358	C₂²×C₈	C₂²×D₅₃	1	2	yes	—
359	C₄×Q₈	—	53	1	no	—
360	C₄×Q₈	—	53	1	no	—
361	C₄×Q₈	C₄×D₅₃	1	1	yes	characteristic
362	C₄×Q₈	C₄×D₅₃	1	1	yes	characteristic
363	C₄×D₄	—	2	1	no	—
364	C₄×D₄	—	2	1	no	—
365	C₄×D₄	—	53	1	no	—
366	C₄×D₄	—	53	1	no	—
367	C₄×D₄	—	106	1	no	—
368	C₄×D₄	—	106	1	no	—
369	C₄×D₄	C₄×D₅₃	1	1	yes	characteristic
370	C₄×D₄	C₄×D₅₃	1	1	yes	characteristic
371	#117	—	2	1	no	—
372	#117	—	53	1	no	—
373	#117	—	106	1	no	—
374	#117	C₂²×D₅₃	1	1	yes	characteristic
375	C₂×C₄⋊C₄	—	106	1	no	—
376	C₂×C₄²	—	2	1	no	—
377	C₂×C₄²	—	53	2	no	—
378	C₂×C₄²	—	106	1	no	—
379	C₂×C₄²	C₂²×D₅₃	1	1	yes	characteristic
380	#108	—	106	4	no	—
381	#108	—	212	2	no	—
382	#107	—	53	1	no	—
383	#107	—	53	1	no	—
384	#106	—	53	1	no	—
385	#106	—	53	1	no	—
386	C₄⋊C₈	—	2	2	no	—
387	C₄⋊C₈	—	2	2	no	—
388	C₄⋊C₈	—	2	1	no	—
389	C₄⋊C₈	—	2	1	no	—
390	C₄⋊C₈	—	106	2	no	—
391	C₄⋊C₈	—	106	2	no	—
392	C₄⋊C₈	—	106	1	no	—
393	C₄⋊C₈	—	106	1	no	—
394	C₄⋊C₈	C₂²⋊D₅₃	1	1	yes	characteristic
395	C₄⋊C₈	C₂²⋊D₅₃	1	1	yes	characteristic
396	C₄⋊C₈	C₂²⋊D₅₃	1	2	yes	—
397	#104	—	106	4	no	—
398	#104	—	106	4	no	—
399	#103	—	53	1	no	—
400	#103	—	53	1	no	—
401	#103	—	53	1	no	—
402	#103	—	53	1	no	—
403	#102	—	53	1	no	—
404	#102	—	53	1	no	—
405	#102	—	53	1	no	—
406	#102	—	53	1	no	—
407	#97	—	106	4	no	—
408	C₄×C₈	—	53	1	no	—
409	C₄×C₈	—	53	1	no	—
410	C₄×C₈	—	53	1	no	—
411	C₄×C₈	—	53	1	no	—
412	#95	—	106	1	no	—
413	#41	—	2	1	no	—
414	#41	—	2	1	no	—
415	#41	—	53	2	no	—
416	#41	—	53	2	no	—
417	#41	—	106	1	no	—
418	#41	—	106	1	no	—
419	#41	D₄×D₅₃	1	2	yes	—
420	#41	Q₈×D₅₃	1	2	yes	—
421	C₂×Q₈	—	53	1	no	—
422	C₂×Q₈	C₂×C₄×D₅₃	1	1	yes	characteristic
423	C₂×D₄	—	2	1	no	—
424	C₂×D₄	—	53	1	no	—
425	C₂×D₄	—	106	1	no	—
426	C₂×D₄	C₂×C₄×D₅₃	1	1	yes	characteristic
427	C₂²×C₄	—	2	1	no	—
428	C₂²×C₄	—	2	1	no	—
429	C₂²×C₄	—	2	1	no	—
430	C₂²×C₄	—	106	1	no	—
431	C₂²×C₄	—	106	1	no	—
432	C₂²×C₄	—	106	1	no	—
433	C₂²×C₄	—	106	2	no	—
434	C₂²×C₄	#11719945	1	1	yes	characteristic
435	C₂²×C₄	#11719945	1	1	yes	characteristic
436	C₂²×C₄	C₂³×D₅₃	1	1	yes	characteristic
437	Q₁₆	—	53	4	no	—
438	SD₁₆	—	53	4	no	—
439	SD₁₆	—	53	4	no	—
440	D₈	—	53	4	no	—
441	M₁₆	—	2	2	no	—
442	M₁₆	—	2	2	no	—
443	M₁₆	—	4	4	no	—
444	M₁₆	—	106	2	no	—
445	M₁₆	—	106	2	no	—
446	M₁₆	—	212	4	no	—
447	C₂×C₈	—	2	2	no	—
448	C₂×C₈	—	2	2	no	—
449	C₂×C₈	—	2	2	no	—
450	C₂×C₈	—	4	4	no	—
451	C₂×C₈	—	53	1	no	—
452	C₂×C₈	—	53	1	no	—
453	C₂×C₈	—	53	2	no	—
454	C₂×C₈	—	53	2	no	—
455	C₂×C₈	—	53	1	no	—
456	C₂×C₈	—	53	1	no	—
457	C₂×C₈	—	106	1	no	—
458	C₂×C₈	—	106	1	no	—
459	C₂×C₈	—	106	2	no	—
460	C₂×C₈	—	212	4	no	—
461	C₂×C₈	C₂×C₂²⋊D₅₃	1	2	yes	—
462	C₂×C₈	C₂×C₂²⋊D₅₃	1	2	yes	—
463	C₄⋊C₄	—	2	1	no	—
464	C₄⋊C₄	—	2	1	no	—
465	C₄⋊C₄	—	53	1	no	—
466	C₄⋊C₄	—	53	1	no	—
467	C₄⋊C₄	—	106	1	no	—
468	C₄⋊C₄	—	106	1	no	—
469	C₄⋊C₄	—	212	2	no	—
470	C₄⋊C₄	C₂×C₄×D₅₃	1	1	yes	characteristic
471	C₄⋊C₄	C₂×C₄×D₅₃	1	1	yes	characteristic
472	C₂²⋊C₄	—	2	1	no	—
473	C₂²⋊C₄	—	2	1	no	—
474	C₂²⋊C₄	—	2	1	no	—
475	C₂²⋊C₄	—	106	1	no	—
476	C₂²⋊C₄	—	106	1	no	—
477	C₂²⋊C₄	—	106	1	no	—
478	C₄²	—	2	1	no	—
479	C₄²	—	2	1	no	—
480	C₄²	—	2	1	no	—
481	C₄²	—	106	1	no	—
482	C₄²	—	106	1	no	—
483	C₄²	—	106	1	no	—
484	C₄²	—	106	2	no	—
485	C₄²	—	106	2	no	—
486	C₄²	C₂×C₄×D₅₃	1	1	yes	characteristic
487	C₄²	C₂×C₄×D₅₃	1	1	yes	characteristic
488	C₄²	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
489	C₄²	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
490	C₂³	—	2	1	no	—
491	C₂³	—	106	1	no	—
492	C₂³	C₂×#11719945	1	1	yes	characteristic
493	Q₈	—	2	1	no	—
494	Q₈	—	53	2	no	—
495	Q₈	—	106	1	no	—
496	Q₈	C₄⋊C₄×D₅₃	1	2	yes	—
497	D₄	—	2	2	no	—
498	D₄	—	2	2	no	—
499	D₄	—	2	1	no	—
500	D₄	—	53	2	no	—
501	D₄	—	106	2	no	—
502	D₄	—	106	2	no	—
503	D₄	—	106	1	no	—
504	D₄	C₄⋊C₄×D₅₃	1	2	yes	—
505	C₂×C₄	—	2	1	no	—
506	C₂×C₄	—	2	2	no	—
507	C₂×C₄	—	2	2	no	—
508	C₂×C₄	—	2	2	no	—
509	C₂×C₄	—	2	1	no	—
510	C₂×C₄	—	2	1	no	—
511	C₂×C₄	—	4	1	no	—
512	C₂×C₄	—	4	2	no	—
513	C₂×C₄	—	8	1	no	—
514	C₂×C₄	—	106	1	no	—
515	C₂×C₄	—	106	2	no	—
516	C₂×C₄	—	106	2	no	—
517	C₂×C₄	—	106	1	no	—
518	C₂×C₄	—	106	1	no	—
519	C₂×C₄	—	424	1	no	—
520	C₂×C₄	—	212	2	no	—
521	C₂×C₄	—	212	2	no	—
522	C₂×C₄	—	212	2	no	—
523	C₂×C₄	—	212	2	no	—
524	C₂×C₄	C₄×C₂²⋊D₅₃	1	1	yes	characteristic
525	C₂×C₄	C₄×C₂²⋊D₅₃	1	1	yes	characteristic
526	C₂×C₄	C₄×C₂²⋊D₅₃	1	1	yes	characteristic
527	C₂×C₄	C₄×C₂²⋊D₅₃	1	1	yes	characteristic
528	C₂×C₄	C₂²×C₄×D₅₃	1	1	yes	characteristic
529	C₂×C₄	C₂×D₄×D₅₃	1	1	yes	characteristic
530	C₂×C₄	C₂²×C₂²⋊D₅₃	1	1	yes	characteristic
531	C₂×C₄	C₂×Q₈×D₅₃	1	1	yes	characteristic
532	C₂×C₄	#13133262	1	1	yes	characteristic
533	C₂×C₄	#13133265	1	1	yes	characteristic
534	C₈	—	2	2	no	—
535	C₈	—	2	2	no	—
536	C₈	—	4	2	no	—
537	C₈	—	53	2	no	—
538	C₈	—	53	2	no	—
539	C₈	—	106	1	no	—
540	C₈	—	106	1	no	—
541	C₈	—	212	2	no	—
542	C₂²	—	2	1	no	—
543	C₂²	—	2	1	no	—
544	C₂²	—	2	2	no	—
545	C₂²	—	4	1	no	—
546	C₂²	—	4	1	no	—
547	C₂²	—	106	2	no	—
548	C₂²	—	212	1	no	—
549	C₂²	—	212	1	no	—
550	C₂²	C₂×C₄×C₂²⋊D₅₃	1	1	yes	characteristic
551	C₂²	#16811274	1	1	yes	characteristic
552	C₂²	#16811328	1	1	yes	characteristic
553	C₄	—	2	2	no	—
554	C₄	—	2	1	no	—
555	C₄	—	2	1	no	—
556	C₄	—	2	1	no	—
557	C₄	—	4	1	no	—
558	C₄	—	4	1	no	—
559	C₄	—	106	2	no	—
560	C₄	—	424	2	no	—
561	C₄	—	212	1	no	—
562	C₄	—	212	1	no	—
563	C₄	#16810743	1	2	yes	complemented
564	C₄	C₂×C₄⋊C₄×D₅₃	1	1	yes	characteristic
565	C₄	#16811104	1	1	yes	characteristic
566	C₄	D₄×C₂²⋊D₅₃	1	1	yes	characteristic
567	C₄	Q₈×C₂²⋊D₅₃	1	1	yes	characteristic
568	C₂	—	2	1	no	—
569	C₂	—	2	1	no	—
570	C₂	—	4	2	no	—
571	C₂	—	212	2	no	—
572	C₂	C₄⋊C₄×C₂²⋊D₅₃	1	1	yes	characteristic
573	C₂	C₂×#16810743	1	1	yes	characteristic
574	C₂	#20283630	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂²×C₂²⋊D₅₃
commutator	C₂×C₄×C₅₃	C₂³×C₄
Frattini	C₂²×C₄	C₂³×D₅₃
Fitting	C₅₃×#2906	C₂
socle	C₂²×C₅₃	C₂×C₄×D₄
prime radical	C₂×#13133237	C₂²
cosocle	C₂²×C₄×C₅₃	C₂⁴
hypercenter	#2906	D₅₃
hypocenter	C₅₃	#22686
LCS	G ▷ C₂×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄⋊C₈ ◁ #2906

Complements

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

N	H ≅ G/N	type
3. #20283280	C₂	semidirect
4. #20275465	C₂	semidirect
5. #20274990	C₂	semidirect
6. #20274989	C₂	semidirect
7. #20274886	C₂	semidirect
8. #20274735	C₂	semidirect
9. #20274732	C₂	semidirect
10. #20274585	C₂	semidirect
11. C₅₃×#2906	C₂	semidirect
16. C₄×#11719925	C₄	semidirect
17. C₄×#11719925	C₄	semidirect
18. C₄×#11719924	C₄	semidirect
19. C₄×#11719924	C₄	semidirect
20. C₄×#11719923	C₄	semidirect
21. C₄×#11719923	C₄	semidirect
22. C₄×#11719922	C₄	semidirect
23. C₄×#11719922	C₄	semidirect
173. #11719925	C₄⋊C₄	semidirect
174. #11719924	C₄⋊C₄	semidirect
175. #11719923	C₄⋊C₄	semidirect
176. #11719922	C₄⋊C₄	semidirect
272. C₄×C₅₃	#444	semidirect
284. #2906	D₅₃	semidirect
337. C₅₃	#22686	semidirect
563. C₄	#16810743	semidirect

Maximal subgroups

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

subgroup	index
C₂×#16810743	2
C₄×#13133237	2
#20283280	2
#20275465	2
#20274990	2
#20274989	2
#20274886	2
#20274735	2
#20274732	2
#20274585	2
C₅₃×#2906	2
#22686	53