Subgroups — D₅₃×#1584 #27353608

Order 13568.

674 conjugacy classes of subgroups.

The subgroup lattice has 572 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#446×D₅₃	C₂	1	1	yes	maximal, characteristic
2	C₈×Q₈×D₅₃	C₂	1	1	yes	maximal, characteristic
3	C₄×Q₁₆×D₅₃	C₂	1	1	yes	maximal, characteristic
4	#20273568	C₂	1	1	yes	maximal, characteristic, complemented
5	#20273567	C₂	1	1	yes	maximal, characteristic, complemented
6	#20273566	C₂	1	1	yes	maximal, characteristic, complemented
7	#20273565	C₂	1	1	yes	maximal, characteristic, complemented
8	#20273564	C₂	1	1	yes	maximal, characteristic, complemented
9	#20273563	C₂	1	1	yes	maximal, characteristic, complemented
10	#20273562	C₂	1	1	yes	maximal, characteristic, complemented
11	#335×D₅₃	C₂	1	1	yes	maximal, characteristic
12	Q₈⋊C₈×D₅₃	C₂	1	1	yes	maximal, characteristic
13	Q₈⋊C₈×D₅₃	C₂	1	1	yes	maximal, characteristic
14	#322×D₅₃	C₂	1	1	yes	maximal, characteristic
15	C₅₃×#1584	C₂	1	1	yes	maximal, characteristic, complemented
16	C₂×Q₁₆×D₅₃	C₄	1	1	yes	characteristic
17	C₄×Q₈×D₅₃	C₂²	1	1	yes	characteristic
18	C₄×Q₈×D₅₃	C₂²	1	1	yes	characteristic
19	#16810754	C₂²	1	1	yes	characteristic
20	#16810753	C₂²	1	1	yes	characteristic
21	#16810752	—	2	1	no	—
22	#16810751	—	2	1	no	—
23	#16810750	C₂²	1	1	yes	characteristic
24	Q₈×#88322	C₂²	1	1	yes	characteristic
25	C₈×Dic₁₀₆	—	2	1	no	—
26	C₈×Dic₁₀₆	C₂²	1	1	yes	characteristic
27	#16810747	—	2	1	no	—
28	#16810747	C₂²	1	1	yes	characteristic
29	C₄×Dic₂₁₂	C₂²	1	1	yes	characteristic
30	Q₁₆×Dic₅₃	C₄	1	1	yes	characteristic
31	#16810717	C₄	1	1	yes	characteristic
32	C₄×#11719925	C₂²	1	1	yes	characteristic
33	C₄×#11719925	C₂²	1	1	yes	characteristic
34	#16810715	C₄	1	1	yes	characteristic
35	#16810715	C₄	1	1	yes	characteristic
36	#107×D₅₃	C₄	1	1	yes	characteristic
37	C₄⋊C₈×D₅₃	—	2	1	no	—
38	C₄⋊C₈×D₅₃	—	2	1	no	—
39	C₄⋊C₈×D₅₃	C₂²	1	1	yes	characteristic
40	C₄⋊C₈×D₅₃	C₂²	1	1	yes	characteristic
41	#103×D₅₃	C₄	1	1	yes	characteristic
42	#103×D₅₃	C₄	1	1	yes	characteristic
43	#97×D₅₃	—	2	1	no	—
44	C₄×C₈×D₅₃	—	2	1	no	—
45	C₄×C₈×D₅₃	C₂²	1	1	yes	characteristic
46	C₄×C₈×D₅₃	C₂²	1	1	yes	characteristic
47	C₄×C₈×D₅₃	C₂²	1	1	yes	characteristic
48	#16810376	C₂²	1	1	yes	characteristic
49	#16810375	C₂²	1	1	yes	characteristic
50	#16810375	C₂²	1	1	yes	characteristic
51	#16810355	C₂²	1	1	yes	characteristic
52	#16810355	C₂²	1	1	yes	characteristic
53	Q₈⋊#88322	C₂²	1	1	yes	characteristic
54	Q₈⋊#88322	C₂²	1	1	yes	characteristic
55	#16810353	C₂²	1	1	yes	characteristic
56	#16810353	C₂²	1	1	yes	characteristic
57	#16810345	C₂²	1	1	yes	characteristic
58	#16810345	C₂²	1	1	yes	characteristic
59	#16810344	C₂²	1	1	yes	characteristic
60	C₅₃×#446	C₂²	1	1	yes	characteristic
61	C₈×Q₈×C₅₃	C₂²	1	1	yes	characteristic
62	C₄×Q₁₆×C₅₃	C₂²	1	1	yes	characteristic
63	C₅₃×#335	C₂²	1	1	yes	characteristic
64	C₅₃×Q₈⋊C₈	C₂²	1	1	yes	characteristic
65	C₅₃×Q₈⋊C₈	C₂²	1	1	yes	characteristic
66	C₅₃×#322	C₂²	1	1	yes	characteristic
67	C₂×Q₈×D₅₃	C₂×C₄	1	1	yes	characteristic
68	C₂×Q₈×D₅₃	C₂×C₄	1	1	yes	characteristic
69	C₂×Dic₂₁₂	C₂×C₄	1	1	yes	characteristic
70	C₂×#11719925	C₂×C₄	1	1	yes	characteristic
71	C₂×#11719925	C₂×C₄	1	1	yes	characteristic
72	Q₁₆×D₅₃	—	2	4	no	—
73	C₂×C₈×D₅₃	—	2	1	no	—
74	C₂×C₈×D₅₃	—	2	1	no	—
75	C₂×C₈×D₅₃	—	2	1	no	—
76	C₂×C₈×D₅₃	C₂×C₄	1	1	yes	characteristic
77	C₂×C₈×D₅₃	C₂×C₄	1	1	yes	characteristic
78	C₂×C₈×D₅₃	D₄	1	1	yes	characteristic
79	C₂×C₈×D₅₃	D₄	1	1	yes	characteristic
80	Q₈×Dic₅₃	C₂×C₄	1	1	yes	characteristic
81	Q₈×Dic₅₃	C₂×C₄	1	1	yes	characteristic
82	C₄×Dic₁₀₆	—	2	1	no	—
83	C₄×Dic₁₀₆	—	2	1	no	—
84	C₄×Dic₁₀₆	C₂³	1	1	yes	characteristic
85	C₄×Dic₁₀₆	C₂³	1	1	yes	characteristic
86	#13133182	—	2	1	no	—
87	#13133182	—	2	1	no	—
88	#13133182	C₂×C₄	1	1	yes	characteristic
89	#13133182	C₂×C₄	1	1	yes	characteristic
90	C₄⋊C₄×D₅₃	—	2	1	no	—
91	C₄⋊C₄×D₅₃	—	2	1	no	—
92	C₄⋊C₄×D₅₃	C₂×C₄	1	1	yes	characteristic
93	C₄⋊C₄×D₅₃	C₂×C₄	1	1	yes	characteristic
94	C₄²×D₅₃	—	2	1	no	—
95	C₄²×D₅₃	—	2	1	no	—
96	C₄²×D₅₃	C₂³	1	1	yes	characteristic
97	#13133151	C₂×C₄	1	1	yes	characteristic
98	#13133150	C₂×C₄	1	1	yes	characteristic
99	#13133150	C₂×C₄	1	1	yes	characteristic
100	C₄⋊#88322	—	2	1	no	—
101	C₄⋊#88322	—	2	1	no	—
102	C₄⋊#88322	C₂³	1	1	yes	characteristic
103	C₄⋊#88322	C₂³	1	1	yes	characteristic
104	#13133146	—	2	1	no	—
105	#13133146	—	2	1	no	—
106	#13133146	—	2	1	no	—
107	#13133146	—	2	1	no	—
108	#13133146	—	2	1	no	—
109	#13133146	—	2	1	no	—
110	#13133142	C₂×C₄	1	1	yes	characteristic
111	#13133142	C₂×C₄	1	1	yes	characteristic
112	#13133141	C₂×C₄	1	1	yes	characteristic
113	#13133141	C₂×C₄	1	1	yes	characteristic
114	#13133140	C₂×C₄	1	1	yes	characteristic
115	#13133140	C₂×C₄	1	1	yes	characteristic
116	#13133128	—	2	1	no	—
117	#13133127	—	2	1	no	—
118	#13133127	—	2	1	no	—
119	C₄×#88322	—	2	1	no	—
120	C₄×#88322	C₂³	1	1	yes	characteristic
121	C₄×#88322	C₂³	1	1	yes	characteristic
122	C₄×#88322	C₂³	1	1	yes	characteristic
123	C₈×Dic₅₃	—	2	1	no	—
124	C₈×Dic₅₃	—	2	1	no	—
125	C₈×Dic₅₃	—	2	1	no	—
126	C₈×Dic₅₃	C₂×C₄	1	1	yes	characteristic
127	C₈×Dic₅₃	C₂×C₄	1	1	yes	characteristic
128	C₈×Dic₅₃	D₄	1	1	yes	characteristic
129	C₈×Dic₅₃	D₄	1	1	yes	characteristic
130	C₂×Q₁₆×C₅₃	C₂×C₄	1	1	yes	characteristic
131	C₄×Q₈×C₅₃	C₂³	1	1	yes	characteristic
132	C₄×Q₈×C₅₃	C₂³	1	1	yes	characteristic
133	#107×C₅₃	C₂×C₄	1	1	yes	characteristic
134	C₄⋊C₈×C₅₃	—	2	1	no	—
135	C₄⋊C₈×C₅₃	—	2	1	no	—
136	C₄⋊C₈×C₅₃	C₂³	1	1	yes	characteristic
137	C₄⋊C₈×C₅₃	C₂³	1	1	yes	characteristic
138	#103×C₅₃	C₂×C₄	1	1	yes	characteristic
139	#103×C₅₃	C₂×C₄	1	1	yes	characteristic
140	#97×C₅₃	—	2	1	no	—
141	C₄×C₈×C₅₃	—	2	1	no	—
142	C₄×C₈×C₅₃	C₂³	1	1	yes	characteristic
143	C₄×C₈×C₅₃	C₂³	1	1	yes	characteristic
144	C₄×C₈×C₅₃	C₂³	1	1	yes	characteristic
145	C₂×Q₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
146	C₂×Q₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
147	Q₈×D₅₃	—	2	2	no	—
148	Q₈×D₅₃	—	2	2	no	—
149	Q₈×D₅₃	—	2	2	no	—
150	Q₈×D₅₃	M₁₆	1	4	yes	—
151	C₂×C₄×D₅₃	—	2	1	no	—
152	C₂×C₄×D₅₃	—	2	1	no	—
153	C₂×C₄×D₅₃	—	2	1	no	—
154	C₂×C₄×D₅₃	—	2	1	no	—
155	C₂×C₄×D₅₃	C₂²×C₄	1	1	yes	characteristic
156	C₂×C₄×D₅₃	C₂×D₄	1	1	yes	characteristic
157	C₂×C₄×D₅₃	#41	1	1	yes	characteristic
158	C₂×Dic₁₀₆	—	2	1	no	—
159	C₂×Dic₁₀₆	—	2	1	no	—
160	C₂×Dic₁₀₆	C₂²×C₄	1	1	yes	characteristic
161	C₂×Dic₁₀₆	C₂²×C₄	1	1	yes	characteristic
162	Q₁₆×C₅₃	—	2	2	no	—
163	C₂×C₈×C₅₃	—	2	1	no	—
164	C₂×C₈×C₅₃	—	2	1	no	—
165	C₂×C₈×C₅₃	—	2	1	no	—
166	C₂×C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
167	C₂×C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
168	C₂×C₈×C₅₃	C₂×D₄	1	1	yes	characteristic
169	C₂×C₈×C₅₃	C₂×D₄	1	1	yes	characteristic
170	C₄⋊C₄×C₅₃	—	2	1	no	—
171	C₄⋊C₄×C₅₃	—	2	1	no	—
172	C₄⋊C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
173	C₄⋊C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
174	C₄²×C₅₃	—	2	1	no	—
175	C₄²×C₅₃	—	2	1	no	—
176	C₄²×C₅₃	C₂⁴	1	1	yes	characteristic
177	#11719925	—	2	2	no	—
178	#11719925	—	2	2	no	—
179	C₄⋊Dic₅₃	—	2	1	no	—
180	C₄⋊Dic₅₃	—	2	1	no	—
181	C₄⋊Dic₅₃	C₂²×C₄	1	1	yes	characteristic
182	C₄⋊Dic₅₃	C₂²×C₄	1	1	yes	characteristic
183	#11719919	—	2	1	no	—
184	#11719919	—	2	1	no	—
185	#11719919	—	2	1	no	—
186	#11719919	—	2	1	no	—
187	#11719919	—	2	1	no	—
188	#11719919	—	2	1	no	—
189	C₄×Dic₅₃	—	2	1	no	—
190	C₄×Dic₅₃	—	2	1	no	—
191	C₄×Dic₅₃	—	2	1	no	—
192	C₄×Dic₅₃	—	2	1	no	—
193	C₄×Dic₅₃	—	2	1	no	—
194	C₄×Dic₅₃	—	2	1	no	—
195	C₄×Dic₅₃	C₂²×C₄	1	1	yes	characteristic
196	C₄×Dic₅₃	C₂×D₄	1	1	yes	characteristic
197	C₄×Dic₅₃	#41	1	1	yes	characteristic
198	C₂×#88322	—	2	1	no	—
199	C₂×#88322	—	2	1	no	—
200	C₂×#88322	—	2	1	no	—
201	C₂×#88322	C₂²×C₄	1	1	yes	characteristic
202	C₂×#88322	C₂²×C₄	1	1	yes	characteristic
203	C₂×#88322	C₂×D₄	1	1	yes	characteristic
204	C₂×#88322	C₂×D₄	1	1	yes	characteristic
205	Dic₂₁₂	—	2	2	no	—
206	C₈×D₅₃	—	2	2	no	—
207	C₈×D₅₃	—	2	2	no	—
208	C₈×D₅₃	—	2	2	no	—
209	C₈×D₅₃	—	4	2	no	—
210	C₈×D₅₃	—	4	2	no	—
211	C₈×D₅₃	—	4	2	no	—
212	C₈×D₅₃	Q₁₆	1	4	yes	complemented
213	C₂²×D₅₃	C₄×D₄	1	1	yes	characteristic
214	Q₈×C₅₃	—	2	1	no	—
215	Q₈×C₅₃	—	2	1	no	—
216	Q₈×C₅₃	—	2	1	no	—
217	Q₈×C₅₃	C₂×M₁₆	1	2	yes	—
218	C₂×C₄×C₅₃	—	2	1	no	—
219	C₂×C₄×C₅₃	—	2	1	no	—
220	C₂×C₄×C₅₃	—	2	1	no	—
221	C₂×C₄×C₅₃	—	2	1	no	—
222	C₂×C₄×C₅₃	C₂³×C₄	1	1	yes	characteristic
223	C₂×C₄×C₅₃	C₂²×D₄	1	1	yes	characteristic
224	C₂×C₄×C₅₃	C₂×#41	1	1	yes	characteristic
225	C₂×Dic₅₃	—	2	1	no	—
226	C₂×Dic₅₃	—	2	1	no	—
227	C₂×Dic₅₃	—	2	1	no	—
228	C₂×Dic₅₃	—	2	1	no	—
229	C₂×Dic₅₃	C₄×D₄	1	1	yes	characteristic
230	C₂×Dic₅₃	C₄×D₄	1	1	yes	characteristic
231	C₂×Dic₅₃	C₄×D₄	1	1	yes	characteristic
232	C₄×D₅₃	—	2	4	no	—
233	C₄×D₅₃	—	2	2	no	—
234	C₄×D₅₃	—	4	2	no	—
235	C₄×D₅₃	—	4	2	no	—
236	C₄×D₅₃	—	4	2	no	—
237	C₄×D₅₃	C₂×M₁₆	1	2	yes	—
238	C₄×D₅₃	#131	1	2	yes	—
239	C₄×D₅₃	C₂×Q₁₆	1	2	yes	—
240	C₄×D₅₃	#135	1	2	yes	—
241	Dic₁₀₆	—	2	1	no	—
242	Dic₁₀₆	—	2	4	no	—
243	Dic₁₀₆	—	2	1	no	—
244	Dic₁₀₆	—	2	1	no	—
245	Dic₁₀₆	—	4	2	no	—
246	Dic₁₀₆	C₂×M₁₆	1	2	yes	—
247	C₈×C₅₃	—	2	1	no	—
248	C₈×C₅₃	—	2	1	no	—
249	C₈×C₅₃	—	2	1	no	—
250	C₈×C₅₃	—	4	1	no	—
251	C₈×C₅₃	—	4	1	no	—
252	C₈×C₅₃	—	4	1	no	—
253	C₈×C₅₃	C₂×Q₁₆	1	2	yes	complemented
254	#88322	—	2	1	no	—
255	#88322	—	2	1	no	—
256	#88322	—	2	1	no	—
257	#88322	—	4	1	no	—
258	#88322	—	4	1	no	—
259	#88322	—	4	1	no	—
260	#88322	C₂×Q₁₆	1	2	yes	complemented
261	C₂×#1584	—	53	1	no	maximal, Sylow
262	C₂²×C₅₃	C₂×C₄×D₄	1	1	yes	characteristic
263	C₂×D₅₃	#435	1	2	yes	—
264	C₂×D₅₃	C₄×Q₁₆	1	2	yes	—
265	C₂×D₅₃	#444	1	2	yes	—
266	C₄×C₅₃	—	2	2	no	—
267	C₄×C₅₃	—	2	1	no	—
268	C₄×C₅₃	—	4	1	no	—
269	C₄×C₅₃	—	4	1	no	—
270	C₄×C₅₃	—	4	1	no	—
271	C₄×C₅₃	C₂²×M₁₆	1	1	yes	characteristic
272	C₄×C₅₃	C₂×#131	1	1	yes	characteristic
273	C₄×C₅₃	C₂²×Q₁₆	1	1	yes	characteristic
274	C₄×C₅₃	C₂×#135	1	1	yes	characteristic
275	Dic₅₃	—	2	2	no	—
276	Dic₅₃	—	2	1	no	—
277	Dic₅₃	—	4	1	no	—
278	Dic₅₃	—	4	1	no	—
279	Dic₅₃	—	4	1	no	—
280	Dic₅₃	#435	1	2	yes	—
281	Dic₅₃	C₄×Q₁₆	1	2	yes	—
282	C₂×#446	—	53	1	no	—
283	C₂×C₈×Q₈	—	53	1	no	—
284	C₂×C₄×Q₁₆	—	53	1	no	—
285	#1584	—	53	1	no	—
286	#1584	—	53	1	no	—
287	#1584	—	53	1	no	—
288	#1584	—	53	1	no	—
289	#1584	—	53	1	no	—
290	#1584	—	53	1	no	—
291	#1584	—	53	1	no	—
292	#1584	D₅₃	1	1	yes	characteristic, direct product factor
293	C₂×#335	—	53	1	no	—
294	C₂×Q₈⋊C₈	—	53	1	no	—
295	C₂×Q₈⋊C₈	—	53	1	no	—
296	C₂×#322	—	53	1	no	—
297	C₂×C₅₃	C₂×#435	1	1	yes	characteristic
298	C₂×C₅₃	C₂×C₄×Q₁₆	1	1	yes	characteristic
299	C₂×C₅₃	C₂×#444	1	1	yes	characteristic
300	D₅₃	#1584	1	4	yes	direct product factor
301	C₂²×Q₁₆	—	53	1	no	—
302	C₂×C₄×Q₈	—	53	1	no	—
303	C₂×C₄×Q₈	—	53	1	no	—
304	#446	—	53	1	no	—
305	#446	—	53	1	no	—
306	#446	—	53	1	no	—
307	#446	—	106	1	no	—
308	#446	—	106	1	no	—
309	#446	C₂×D₅₃	1	1	yes	characteristic
310	C₈×Q₈	—	53	1	no	—
311	C₈×Q₈	—	53	1	no	—
312	C₈×Q₈	—	53	1	no	—
313	C₈×Q₈	—	106	1	no	—
314	C₈×Q₈	—	106	1	no	—
315	C₈×Q₈	C₂×D₅₃	1	1	yes	characteristic
316	C₄×Q₁₆	—	53	1	no	—
317	C₄×Q₁₆	—	53	1	no	—
318	C₄×Q₁₆	—	53	1	no	—
319	C₄×Q₁₆	—	53	1	no	—
320	C₄×Q₁₆	—	53	1	no	—
321	C₄×Q₁₆	—	53	1	no	—
322	C₄×Q₁₆	—	53	1	no	—
323	C₄×Q₁₆	C₂×D₅₃	1	1	yes	characteristic
324	C₂×#107	—	53	1	no	—
325	C₂×C₄⋊C₈	—	53	1	no	—
326	C₂×C₄⋊C₈	—	53	1	no	—
327	C₂×C₄⋊C₈	—	106	1	no	—
328	C₂×C₄⋊C₈	—	106	1	no	—
329	C₂×#103	—	53	1	no	—
330	C₂×#103	—	53	1	no	—
331	C₂×#97	—	106	1	no	—
332	C₂×C₄×C₈	—	53	1	no	—
333	C₂×C₄×C₈	—	53	1	no	—
334	C₂×C₄×C₈	—	53	1	no	—
335	C₂×C₄×C₈	—	106	1	no	—
336	#335	—	53	1	no	—
337	#335	—	53	1	no	—
338	#335	—	53	1	no	—
339	#335	C₂×D₅₃	1	1	yes	characteristic
340	Q₈⋊C₈	—	53	1	no	—
341	Q₈⋊C₈	—	53	1	no	—
342	Q₈⋊C₈	—	53	1	no	—
343	Q₈⋊C₈	—	53	1	no	—
344	Q₈⋊C₈	—	53	1	no	—
345	Q₈⋊C₈	—	53	1	no	—
346	Q₈⋊C₈	C₂×D₅₃	1	1	yes	characteristic
347	Q₈⋊C₈	C₂×D₅₃	1	1	yes	characteristic
348	#322	—	53	1	no	—
349	#322	—	53	1	no	—
350	#322	—	53	1	no	—
351	#322	C₂×D₅₃	1	1	yes	characteristic
352	C₅₃	C₂×#1584	1	1	yes	characteristic, complemented, Sylow
353	C₂²×Q₈	—	53	1	no	—
354	C₂²×Q₈	—	53	1	no	—
355	C₂×Q₁₆	—	53	1	no	—
356	C₂×Q₁₆	—	53	1	no	—
357	C₂×Q₁₆	—	53	1	no	—
358	C₂×Q₁₆	—	106	4	no	—
359	C₂×Q₁₆	C₄×D₅₃	1	1	yes	characteristic
360	C₂²×C₈	—	53	1	no	—
361	C₂²×C₈	—	53	1	no	—
362	C₂²×C₈	—	53	1	no	—
363	C₂²×C₈	—	53	1	no	—
364	C₂²×C₈	—	106	1	no	—
365	C₂²×C₈	—	106	1	no	—
366	C₂²×C₈	—	106	1	no	—
367	C₄×Q₈	—	53	1	no	—
368	C₄×Q₈	—	53	1	no	—
369	C₄×Q₈	—	53	1	no	—
370	C₄×Q₈	—	53	1	no	—
371	C₄×Q₈	—	53	1	no	—
372	C₄×Q₈	—	53	1	no	—
373	C₄×Q₈	—	106	1	no	—
374	C₄×Q₈	—	106	1	no	—
375	C₄×Q₈	—	106	1	no	—
376	C₄×Q₈	—	106	1	no	—
377	C₄×Q₈	C₂²×D₅₃	1	1	yes	characteristic
378	C₄×Q₈	C₂²×D₅₃	1	1	yes	characteristic
379	C₂×C₄⋊C₄	—	53	1	no	—
380	C₂×C₄⋊C₄	—	53	1	no	—
381	C₂×C₄⋊C₄	—	106	1	no	—
382	C₂×C₄⋊C₄	—	106	1	no	—
383	C₂×C₄²	—	53	1	no	—
384	C₂×C₄²	—	106	1	no	—
385	C₂×C₄²	—	106	1	no	—
386	#107	—	53	1	no	—
387	#107	—	53	1	no	—
388	#107	—	53	1	no	—
389	#107	C₄×D₅₃	1	1	yes	characteristic
390	C₄⋊C₈	—	2	1	no	—
391	C₄⋊C₈	—	2	1	no	—
392	C₄⋊C₈	—	53	1	no	—
393	C₄⋊C₈	—	53	1	no	—
394	C₄⋊C₈	—	106	1	no	—
395	C₄⋊C₈	—	106	1	no	—
396	C₄⋊C₈	—	106	1	no	—
397	C₄⋊C₈	—	106	1	no	—
398	C₄⋊C₈	—	106	1	no	—
399	C₄⋊C₈	—	106	1	no	—
400	C₄⋊C₈	—	106	1	no	—
401	C₄⋊C₈	—	106	1	no	—
402	C₄⋊C₈	C₂²×D₅₃	1	1	yes	characteristic
403	C₄⋊C₈	C₂²×D₅₃	1	1	yes	characteristic
404	#103	—	53	1	no	—
405	#103	—	53	1	no	—
406	#103	—	53	1	no	—
407	#103	—	53	1	no	—
408	#103	—	53	1	no	—
409	#103	—	53	1	no	—
410	#103	C₄×D₅₃	1	1	yes	characteristic
411	#103	C₄×D₅₃	1	1	yes	characteristic
412	#97	—	2	1	no	—
413	#97	—	106	1	no	—
414	#97	—	106	1	no	—
415	#97	—	106	1	no	—
416	C₄×C₈	—	2	1	no	—
417	C₄×C₈	—	53	1	no	—
418	C₄×C₈	—	53	1	no	—
419	C₄×C₈	—	53	1	no	—
420	C₄×C₈	—	53	1	no	—
421	C₄×C₈	—	53	1	no	—
422	C₄×C₈	—	53	1	no	—
423	C₄×C₈	—	53	1	no	—
424	C₄×C₈	—	106	1	no	—
425	C₄×C₈	—	106	1	no	—
426	C₄×C₈	—	106	1	no	—
427	C₄×C₈	—	106	1	no	—
428	C₄×C₈	C₂²×D₅₃	1	1	yes	characteristic
429	C₄×C₈	C₂²×D₅₃	1	1	yes	characteristic
430	C₄×C₈	C₂²×D₅₃	1	1	yes	characteristic
431	C₂×Q₈	—	53	4	no	—
432	C₂×Q₈	—	53	1	no	—
433	C₂×Q₈	—	53	1	no	—
434	C₂×Q₈	—	106	1	no	—
435	C₂×Q₈	—	106	1	no	—
436	C₂×Q₈	—	106	2	no	—
437	C₂×Q₈	—	106	2	no	—
438	C₂×Q₈	—	106	2	no	—
439	C₂×Q₈	C₂×C₄×D₅₃	1	1	yes	characteristic
440	C₂×Q₈	C₂×C₄×D₅₃	1	1	yes	characteristic
441	C₂²×C₄	—	53	1	no	—
442	C₂²×C₄	—	53	1	no	—
443	C₂²×C₄	—	53	1	no	—
444	C₂²×C₄	—	106	1	no	—
445	C₂²×C₄	—	106	1	no	—
446	C₂²×C₄	—	106	1	no	—
447	C₂²×C₄	—	106	1	no	—
448	Q₁₆	—	2	2	no	—
449	Q₁₆	—	106	2	no	—
450	Q₁₆	—	106	2	no	—
451	Q₁₆	—	106	2	no	—
452	C₂×C₈	—	2	1	no	—
453	C₂×C₈	—	2	1	no	—
454	C₂×C₈	—	2	1	no	—
455	C₂×C₈	—	53	1	no	—
456	C₂×C₈	—	53	4	no	—
457	C₂×C₈	—	53	1	no	—
458	C₂×C₈	—	53	1	no	—
459	C₂×C₈	—	53	1	no	—
460	C₂×C₈	—	106	2	no	—
461	C₂×C₈	—	106	2	no	—
462	C₂×C₈	—	106	2	no	—
463	C₂×C₈	—	106	1	no	—
464	C₂×C₈	—	106	1	no	—
465	C₂×C₈	—	106	1	no	—
466	C₂×C₈	—	212	2	no	—
467	C₂×C₈	—	212	2	no	—
468	C₂×C₈	—	212	2	no	—
469	C₂×C₈	C₂×C₄×D₅₃	1	1	yes	characteristic
470	C₂×C₈	C₂×C₄×D₅₃	1	1	yes	characteristic
471	C₂×C₈	D₄×D₅₃	1	1	yes	characteristic
472	C₂×C₈	D₄×D₅₃	1	1	yes	characteristic
473	C₄⋊C₄	—	2	1	no	—
474	C₄⋊C₄	—	2	1	no	—
475	C₄⋊C₄	—	53	1	no	—
476	C₄⋊C₄	—	53	1	no	—
477	C₄⋊C₄	—	106	1	no	—
478	C₄⋊C₄	—	106	1	no	—
479	C₄⋊C₄	—	106	1	no	—
480	C₄⋊C₄	—	106	1	no	—
481	C₄⋊C₄	—	106	1	no	—
482	C₄⋊C₄	—	106	1	no	—
483	C₄⋊C₄	—	106	1	no	—
484	C₄⋊C₄	—	106	1	no	—
485	C₄⋊C₄	C₂×C₄×D₅₃	1	1	yes	characteristic
486	C₄⋊C₄	C₂×C₄×D₅₃	1	1	yes	characteristic
487	C₄²	—	2	1	no	—
488	C₄²	—	2	1	no	—
489	C₄²	—	53	1	no	—
490	C₄²	—	53	1	no	—
491	C₄²	—	53	1	no	—
492	C₄²	—	106	1	no	—
493	C₄²	—	106	1	no	—
494	C₄²	—	106	1	no	—
495	C₄²	—	106	1	no	—
496	C₄²	—	106	1	no	—
497	C₄²	—	106	1	no	—
498	C₄²	C₂³×D₅₃	1	1	yes	characteristic
499	C₂³	—	53	1	no	—
500	Q₈	—	2	1	no	—
501	Q₈	—	2	1	no	—
502	Q₈	—	2	1	no	—
503	Q₈	—	53	2	no	—
504	Q₈	—	106	4	no	—
505	Q₈	—	106	1	no	—
506	Q₈	—	106	1	no	—
507	Q₈	—	106	1	no	—
508	Q₈	—	212	2	no	—
509	Q₈	M₁₆×D₅₃	1	2	yes	—
510	C₂×C₄	—	2	1	no	—
511	C₂×C₄	—	2	1	no	—
512	C₂×C₄	—	2	1	no	—
513	C₂×C₄	—	2	1	no	—
514	C₂×C₄	—	53	1	no	—
515	C₂×C₄	—	53	1	no	—
516	C₂×C₄	—	53	2	no	—
517	C₂×C₄	—	53	2	no	—
518	C₂×C₄	—	53	2	no	—
519	C₂×C₄	—	53	2	no	—
520	C₂×C₄	—	53	1	no	—
521	C₂×C₄	—	106	2	no	—
522	C₂×C₄	—	106	4	no	—
523	C₂×C₄	—	106	1	no	—
524	C₂×C₄	—	106	1	no	—
525	C₂×C₄	—	106	1	no	—
526	C₂×C₄	—	106	1	no	—
527	C₂×C₄	—	212	2	no	—
528	C₂×C₄	—	212	2	no	—
529	C₂×C₄	—	212	2	no	—
530	C₂×C₄	C₂²×C₄×D₅₃	1	1	yes	characteristic
531	C₂×C₄	C₂×D₄×D₅₃	1	1	yes	characteristic
532	C₂×C₄	#41×D₅₃	1	1	yes	characteristic
533	C₈	—	2	1	no	—
534	C₈	—	2	1	no	—
535	C₈	—	2	1	no	—
536	C₈	—	4	1	no	—
537	C₈	—	4	1	no	—
538	C₈	—	4	1	no	—
539	C₈	—	53	2	no	—
540	C₈	—	106	1	no	—
541	C₈	—	106	1	no	—
542	C₈	—	106	1	no	—
543	C₈	—	212	1	no	—
544	C₈	—	212	1	no	—
545	C₈	—	212	1	no	—
546	C₈	Q₁₆×D₅₃	1	2	yes	complemented
547	C₂²	—	53	2	no	—
548	C₂²	—	53	2	no	—
549	C₂²	—	53	2	no	—
550	C₂²	C₄×D₄×D₅₃	1	1	yes	characteristic
551	C₄	—	2	1	no	—
552	C₄	—	2	2	no	—
553	C₄	—	4	1	no	—
554	C₄	—	4	1	no	—
555	C₄	—	4	1	no	—
556	C₄	—	53	2	no	—
557	C₄	—	53	2	no	—
558	C₄	—	106	1	no	—
559	C₄	—	106	2	no	—
560	C₄	—	212	1	no	—
561	C₄	—	212	1	no	—
562	C₄	—	212	1	no	—
563	C₄	C₂×M₁₆×D₅₃	1	1	yes	characteristic
564	C₄	#131×D₅₃	1	1	yes	characteristic
565	C₄	C₂×Q₁₆×D₅₃	1	1	yes	characteristic
566	C₄	#135×D₅₃	1	1	yes	characteristic
567	C₂	—	53	4	no	—
568	C₂	#435×D₅₃	1	1	yes	characteristic
569	C₂	C₄×Q₁₆×D₅₃	1	1	yes	characteristic
570	C₂	#444×D₅₃	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂×D₄×D₅₃
commutator	C₂×C₄×C₅₃	C₂³×C₄
Frattini	C₄²	C₂³×D₅₃
Fitting	C₅₃×#1584	C₂
socle	C₂²×C₅₃	C₂×C₄×D₄
prime radical	C₂²×D₅₃	C₄×D₄
cosocle	C₄²×C₅₃	C₂⁴
hypercenter	#1584	D₅₃
hypocenter	C₅₃	C₂×#1584
LCS	G ▷ C₂×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂×C₄ ◁ C₄×C₈ ◁ #1584

Complements

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

N	H ≅ G/N	type
4. #20273568	C₂	semidirect
5. #20273567	C₂	semidirect
6. #20273566	C₂	semidirect
7. #20273565	C₂	semidirect
8. #20273564	C₂	semidirect
9. #20273563	C₂	semidirect
10. #20273562	C₂	semidirect
15. C₅₃×#1584	C₂	semidirect
212. C₈×D₅₃	Q₁₆	semidirect
253. C₈×C₅₃	C₂×Q₁₆	semidirect
260. #88322	C₂×Q₁₆	semidirect
292. #1584	D₅₃	direct (N × H)
300. D₅₃	#1584	direct (N × H)
352. C₅₃	C₂×#1584	semidirect
546. C₈	Q₁₆×D₅₃	semidirect

Maximal subgroups

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

subgroup	index
#446×D₅₃	2
C₈×Q₈×D₅₃	2
C₄×Q₁₆×D₅₃	2
#20273568	2
#20273567	2
#20273566	2
#20273565	2
#20273564	2
#20273563	2
#20273562	2
#335×D₅₃	2
Q₈⋊C₈×D₅₃	2
#322×D₅₃	2
C₅₃×#1584	2
C₂×#1584	53

Subgroups — D53×#1584 #27353608

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — D₅₃×#1584 #27353608