Subgroups — D₅₃⋊#1565 #28316084

Order 13568.

834 conjugacy classes of subgroups.

The subgroup lattice has 462 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	D₅₃⋊#431	C₂	1	1	yes	maximal, characteristic
2	C₂×D₅₃⋊(C₄⋊C₈)	C₂	1	1	yes	maximal, characteristic
3	#20290268	C₂	1	2	yes	maximal, complemented
4	#20290267	C₂	1	4	yes	maximal, complemented
5	#20290266	C₂	1	2	yes	maximal, complemented
6	D₅₃⋊#335	C₂	1	2	yes	maximal, complemented
7	D₅₃⋊#334	C₂	1	2	yes	maximal, complemented
8	C₄×M₁₆×D₅₃	C₂	1	1	yes	maximal, characteristic
9	C₂²×D₅₃⋊C₈	—	2	1	no	—
10	#16811642	C₂²	1	2	yes	—
11	#16811641	—	2	1	no	—
12	#16811640	—	2	1	no	—
13	#16811639	C₂²	1	2	yes	—
14	C₂×#13133284	—	2	1	no	—
15	D₅₃⋊(C₄⋊C₈)	—	2	1	no	—
16	D₅₃⋊(C₄⋊C₈)	—	2	1	no	—
17	D₅₃⋊(C₄⋊C₈)	C₂²	1	2	yes	—
18	D₅₃⋊(C₄⋊C₈)	C₂²	1	2	yes	—
19	D₅₃⋊(C₄⋊C₈)	C₂²	1	1	yes	characteristic
20	D₅₃⋊(C₄⋊C₈)	C₂²	1	1	yes	characteristic
21	C₂×C₄⋊#88324	C₂²	1	2	yes	—
22	D₅₃⋊(C₂²⋊C₈)	—	2	1	no	—
23	C₄×D₅₃⋊C₈	—	2	1	no	—
24	#16811487	C₂²	1	4	yes	complemented
25	#16811486	C₂²	1	4	yes	complemented
26	#16811485	C₂²	1	4	yes	complemented
27	#16811484	C₂²	1	4	yes	complemented
28	C₂×M₁₆×D₅₃	C₄	1	1	yes	characteristic
29	C₂×M₁₆×D₅₃	C₄	1	1	yes	characteristic
30	C₂×C₄²×D₅₃	C₂²	1	1	yes	characteristic
31	C₄×#11719917	C₄	1	1	yes	characteristic
32	#16810575	C₂²	1	2	yes	—
33	M₁₆×Dic₅₃	C₂²	1	1	yes	characteristic
34	M₁₆×Dic₅₃	C₂²	1	1	yes	characteristic
35	C₄×#11719912	C₄	1	2	yes	—
36	#97×D₅₃	C₄	1	2	yes	—
37	C₄×C₈×D₅₃	C₂²	1	2	yes	—
38	C₄×M₁₆×C₅₃	C₄	1	1	yes	characteristic
39	C₂×D₅₃⋊C₈	—	2	1	no	—
40	C₂×D₅₃⋊C₈	—	2	1	no	—
41	C₂×D₅₃⋊C₈	—	2	2	no	—
42	C₂×D₅₃⋊C₈	—	2	2	no	—
43	C₂×D₅₃⋊C₈	—	4	4	no	—
44	C₂²×#88324	—	2	2	no	—
45	#13133284	—	2	2	no	—
46	#13133284	—	2	2	no	—
47	#13133284	—	2	1	no	—
48	#13133284	—	2	1	no	—
49	C₄⋊#88324	—	2	2	no	—
50	C₄⋊#88324	C₂³	1	4	yes	—
51	C₄⋊#88324	C₂³	1	2	yes	—
52	C₄⋊#88324	C₂³	1	2	yes	—
53	#13133274	—	2	2	no	—
54	C₂²⋊#88324	—	2	2	no	—
55	C₄×#88324	—	2	2	no	—
56	C₄×#88324	—	2	2	no	—
57	C₂²×C₄×D₅₃	C₂×C₄	1	1	yes	characteristic
58	C₂²×C₄×D₅₃	D₄	1	1	yes	characteristic
59	C₂²×C₄×D₅₃	Q₈	1	1	yes	characteristic
60	C₂×#11719917	C₂×C₄	1	1	yes	characteristic
61	C₂×#11719917	C₂×C₄	1	1	yes	characteristic
62	C₂×#11719912	C₂×C₄	1	2	yes	—
63	C₂×#11719912	C₂×C₄	1	2	yes	—
64	M₁₆×D₅₃	—	2	4	no	—
65	M₁₆×D₅₃	C₈	1	8	yes	complemented
66	C₂×C₈×D₅₃	C₂×C₄	1	2	yes	—
67	C₂×C₈×D₅₃	C₂×C₄	1	2	yes	—
68	C₂×C₄×Dic₅₃	D₄	1	1	yes	characteristic
69	C₂×C₄×Dic₅₃	Q₈	1	1	yes	characteristic
70	C₂×C₄×Dic₅₃	C₂³	1	1	yes	characteristic
71	C₄²×D₅₃	C₂×C₄	1	1	yes	characteristic
72	C₄²×D₅₃	C₂×C₄	1	1	yes	characteristic
73	C₄²×D₅₃	D₄	1	1	yes	characteristic
74	C₄²×D₅₃	D₄	1	1	yes	characteristic
75	C₄²×D₅₃	Q₈	1	1	yes	characteristic
76	C₄²×D₅₃	Q₈	1	1	yes	characteristic
77	C₄²×D₅₃	C₂³	1	1	yes	characteristic
78	C₄²×D₅₃	C₂³	1	1	yes	characteristic
79	#13133128	C₂×C₄	1	2	yes	—
80	#13133127	C₂×C₄	1	2	yes	—
81	#13133127	C₂×C₄	1	2	yes	—
82	C₄×#88322	C₂×C₄	1	2	yes	—
83	C₈×Dic₅₃	C₂³	1	2	yes	—
84	C₈×Dic₅₃	C₂³	1	2	yes	—
85	C₂×M₁₆×C₅₃	C₂×C₄	1	1	yes	characteristic
86	C₂×M₁₆×C₅₃	C₂×C₄	1	1	yes	characteristic
87	C₂×C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
88	#97×C₅₃	C₂×C₄	1	2	yes	—
89	C₄×C₈×C₅₃	C₂×C₄	1	2	yes	—
90	C₂³×D₅₃	C₄⋊C₄	1	1	yes	characteristic
91	C₂²×C₄×C₅₃	C₄⋊C₄	1	1	yes	characteristic
92	C₂²×C₄×C₅₃	C₄⋊C₄	1	1	yes	characteristic
93	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
94	C₂²×Dic₅₃	C₄⋊C₄	1	1	yes	characteristic
95	C₂²×Dic₅₃	C₂×D₄	1	1	yes	characteristic
96	C₂²×Dic₅₃	C₂×Q₈	1	1	yes	characteristic
97	C₂×C₄×D₅₃	—	2	2	no	—
98	C₂×C₄×D₅₃	—	2	2	no	—
99	C₂×C₄×D₅₃	—	4	2	no	—
100	C₂×C₄×D₅₃	C₄⋊C₄	1	1	yes	characteristic
101	C₂×C₄×D₅₃	C₄⋊C₄	1	1	yes	characteristic
102	C₂×C₄×D₅₃	C₄⋊C₄	1	1	yes	characteristic
103	C₂×C₄×D₅₃	C₄⋊C₄	1	1	yes	characteristic
104	C₂×C₄×D₅₃	C₄⋊C₄	1	1	yes	characteristic
105	C₂×C₄×D₅₃	C₄⋊C₄	1	1	yes	characteristic
106	C₂×C₄×D₅₃	C₂×C₈	1	4	yes	—
107	C₂×C₄×D₅₃	M₁₆	1	4	yes	—
108	C₂×C₄×D₅₃	C₂²×C₄	1	1	yes	characteristic
109	C₂×C₄×D₅₃	C₂²×C₄	1	1	yes	characteristic
110	C₂×C₄×D₅₃	C₂×D₄	1	1	yes	characteristic
111	C₂×C₄×D₅₃	C₂×D₄	1	1	yes	characteristic
112	C₂×C₄×D₅₃	C₂×Q₈	1	1	yes	characteristic
113	C₂×C₄×D₅₃	C₂×Q₈	1	1	yes	characteristic
114	C₂×#88324	—	2	2	no	—
115	C₂×#88324	—	2	2	no	—
116	C₂×#88324	—	2	4	no	—
117	C₂×#88324	—	4	4	no	—
118	D₅₃⋊C₈	—	4	4	no	—
119	D₅₃⋊C₈	—	4	2	no	—
120	D₅₃⋊C₈	—	4	4	no	—
121	D₅₃⋊C₈	—	4	2	no	—
122	M₁₆×C₅₃	—	2	2	no	—
123	M₁₆×C₅₃	C₂×C₈	1	4	yes	complemented
124	C₂×C₈×C₅₃	C₂²×C₄	1	2	yes	—
125	C₂×C₈×C₅₃	C₂²×C₄	1	2	yes	—
126	C₄²×C₅₃	C₄⋊C₄	1	1	yes	characteristic
127	C₄²×C₅₃	C₄⋊C₄	1	1	yes	characteristic
128	C₄²×C₅₃	C₂²×C₄	1	1	yes	characteristic
129	C₄²×C₅₃	C₂²×C₄	1	1	yes	characteristic
130	C₄×Dic₅₃	C₄⋊C₄	1	1	yes	characteristic
131	C₄×Dic₅₃	C₄⋊C₄	1	1	yes	characteristic
132	C₄×Dic₅₃	C₄⋊C₄	1	1	yes	characteristic
133	C₄×Dic₅₃	C₄⋊C₄	1	1	yes	characteristic
134	C₄×Dic₅₃	C₂²×C₄	1	1	yes	characteristic
135	C₄×Dic₅₃	C₂×D₄	1	1	yes	characteristic
136	C₄×Dic₅₃	C₂×D₄	1	1	yes	characteristic
137	C₄×Dic₅₃	C₂×D₄	1	1	yes	characteristic
138	C₄×Dic₅₃	C₂×Q₈	1	1	yes	characteristic
139	C₄×Dic₅₃	C₂×Q₈	1	1	yes	characteristic
140	C₄×Dic₅₃	C₂×Q₈	1	1	yes	characteristic
141	C₄×Dic₅₃	C₂⁴	1	1	yes	characteristic
142	#11719917	—	2	2	no	—
143	#11719917	C₂×C₈	1	4	yes	complemented
144	C₂×#88322	C₂²×C₄	1	2	yes	—
145	C₂×#88322	C₂²×C₄	1	2	yes	—
146	#11719912	—	2	4	no	—
147	#11719912	C₂×C₈	1	8	yes	complemented
148	C₈×D₅₃	—	2	4	no	—
149	C₈×D₅₃	C₂×C₈	1	8	yes	complemented
150	C₂³×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
151	C₂²×D₅₃	—	2	2	no	—
152	C₂²×D₅₃	—	2	2	no	—
153	C₂²×D₅₃	C₄⋊C₈	1	4	yes	—
154	C₂²×D₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
155	C₂²×D₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
156	C₂×C₄×C₅₃	—	2	2	no	—
157	C₂×C₄×C₅₃	—	4	1	no	—
158	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
159	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
160	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
161	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
162	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
163	C₂×C₄×C₅₃	C₂²×C₈	1	2	yes	—
164	C₂×C₄×C₅₃	C₂×M₁₆	1	2	yes	—
165	C₂×C₄×C₅₃	C₂³×C₄	1	1	yes	characteristic
166	C₂×Dic₅₃	—	2	1	no	—
167	C₂×Dic₅₃	—	2	1	no	—
168	C₂×Dic₅₃	—	4	1	no	—
169	C₂×Dic₅₃	C₄⋊C₈	1	4	yes	—
170	C₂×Dic₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
171	C₂×Dic₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
172	C₂×Dic₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
173	C₂×Dic₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
174	C₂×Dic₅₃	C₂²×D₄	1	1	yes	characteristic
175	C₂×Dic₅₃	C₂²×Q₈	1	1	yes	characteristic
176	C₄×D₅₃	—	2	2	no	—
177	C₄×D₅₃	—	2	2	no	—
178	C₄×D₅₃	—	2	2	no	—
179	C₄×D₅₃	—	2	2	no	—
180	C₄×D₅₃	—	2	1	no	—
181	C₄×D₅₃	—	2	1	no	—
182	C₄×D₅₃	—	4	1	no	—
183	C₄×D₅₃	—	4	1	no	—
184	C₄×D₅₃	C₄⋊C₈	1	4	yes	—
185	C₄×D₅₃	C₄⋊C₈	1	4	yes	—
186	C₄×D₅₃	C₂²×C₈	1	2	yes	—
187	C₄×D₅₃	C₂²×C₈	1	2	yes	—
188	C₄×D₅₃	C₂×M₁₆	1	2	yes	—
189	C₄×D₅₃	C₂×M₁₆	1	2	yes	—
190	C₄×D₅₃	AGL(1,ℤ/8)	1	2	yes	—
191	C₄×D₅₃	#137	1	2	yes	—
192	#88324	—	4	4	no	—
193	#88324	—	4	2	no	—
194	#88324	—	4	2	no	—
195	C₈×C₅₃	—	2	2	no	—
196	C₈×C₅₃	C₂²×C₈	1	4	yes	complemented
197	#88322	—	2	2	no	—
198	#88322	C₂²×C₈	1	4	yes	complemented
199	C₂×#1565	—	53	1	no	maximal, Sylow
200	C₂²×C₅₃	—	2	1	no	—
201	C₂²×C₅₃	—	2	1	no	—
202	C₂²×C₅₃	C₂×C₄⋊C₈	1	2	yes	—
203	C₂²×C₅₃	C₂²×C₄⋊C₄	1	1	yes	characteristic
204	C₂×D₅₃	—	2	1	no	—
205	C₂×D₅₃	—	2	1	no	—
206	C₂×D₅₃	—	2	8	no	—
207	C₂×D₅₃	C₂×C₄⋊C₈	1	2	yes	—
208	C₂×D₅₃	C₂×C₄⋊C₈	1	2	yes	—
209	C₂×D₅₃	#428	1	2	yes	—
210	C₂×D₅₃	#430	1	2	yes	—
211	C₄×C₅₃	—	2	1	no	—
212	C₄×C₅₃	—	2	1	no	—
213	C₄×C₅₃	—	2	1	no	—
214	C₄×C₅₃	C₂×C₄⋊C₈	1	2	yes	—
215	C₄×C₅₃	#428	1	2	yes	—
216	C₄×C₅₃	C₂³×C₈	1	1	yes	characteristic
217	C₄×C₅₃	C₂²×M₁₆	1	1	yes	characteristic
218	Dic₅₃	—	2	1	no	—
219	Dic₅₃	—	2	1	no	—
220	Dic₅₃	—	2	1	no	—
221	Dic₅₃	C₂×C₄⋊C₈	1	2	yes	—
222	Dic₅₃	C₂×C₄⋊C₈	1	2	yes	—
223	Dic₅₃	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
224	Dic₅₃	C₂×#137	1	1	yes	characteristic
225	C₂×#431	—	53	1	no	—
226	C₂²×C₄⋊C₈	—	53	1	no	—
227	C₂×C₄×M₁₆	—	53	1	no	—
228	#1565	—	53	2	no	—
229	#1565	—	53	4	no	—
230	#1565	—	53	2	no	—
231	C₂×#335	—	53	2	no	—
232	C₂×#334	—	53	2	no	—
233	C₂×C₅₃	—	2	2	no	—
234	C₂×C₅₃	C₂²×C₄⋊C₈	1	1	yes	characteristic
235	C₂×C₅₃	C₂×#428	1	1	yes	characteristic
236	C₂×C₅₃	C₂×#430	1	1	yes	characteristic
237	D₅₃	—	2	2	no	—
238	D₅₃	#1565	1	4	yes	complemented
239	C₂²×M₁₆	—	53	1	no	—
240	C₂²×M₁₆	—	53	1	no	—
241	C₂³×C₈	—	106	1	no	—
242	C₂²×C₄²	—	53	1	no	—
243	#431	—	53	2	no	—
244	#431	—	53	2	no	—
245	#431	—	106	1	no	—
246	#431	—	106	1	no	—
247	C₂×C₄⋊C₈	—	53	1	no	—
248	C₂×C₄⋊C₈	—	53	2	no	—
249	C₂×C₄⋊C₈	—	53	1	no	—
250	C₂×C₄⋊C₈	—	53	2	no	—
251	C₂×C₄⋊C₈	—	53	2	no	—
252	C₂×C₄⋊C₈	—	106	1	no	—
253	C₂×C₄⋊C₈	—	106	1	no	—
254	C₂×C₄⋊C₈	—	106	1	no	—
255	C₂×C₂²⋊C₈	—	106	1	no	—
256	C₄×M₁₆	—	53	1	no	—
257	C₄×M₁₆	—	53	1	no	—
258	C₄×M₁₆	—	53	1	no	—
259	C₄×M₁₆	—	53	2	no	—
260	C₄×M₁₆	—	53	2	no	—
261	C₄×M₁₆	#6157	1	1	yes	characteristic
262	C₂×#97	—	53	2	no	—
263	C₂×C₄×C₈	—	53	2	no	—
264	C₂×C₄×C₈	—	106	1	no	—
265	#335	—	53	4	no	—
266	#335	—	53	4	no	—
267	#334	—	53	4	no	—
268	#334	—	53	4	no	—
269	C₅₃	C₂×#1565	1	1	yes	characteristic, complemented, Sylow
270	C₂³×C₄	—	53	1	no	—
271	C₂³×C₄	—	53	1	no	—
272	C₂³×C₄	—	53	1	no	—
273	C₂×M₁₆	—	53	1	no	—
274	C₂×M₁₆	—	53	8	no	—
275	C₂×M₁₆	—	53	1	no	—
276	C₂×M₁₆	—	53	2	no	—
277	C₂×M₁₆	—	53	2	no	—
278	C₂×M₁₆	—	106	4	no	—
279	C₂×M₁₆	C₂×#6157	1	1	yes	characteristic
280	C₂×M₁₆	C₂×#6157	1	1	yes	characteristic
281	C₂²×C₈	—	53	2	no	—
282	C₂²×C₈	—	53	2	no	—
283	C₂²×C₈	—	106	2	no	—
284	C₂²×C₈	—	106	2	no	—
285	C₂²×C₈	—	106	2	no	—
286	C₂²×C₈	—	106	1	no	—
287	C₂²×C₈	—	106	1	no	—
288	C₂²×C₈	—	212	4	no	—
289	C₂×C₄²	—	53	1	no	—
290	C₂×C₄²	—	53	1	no	—
291	C₂×C₄²	—	53	1	no	—
292	C₂×C₄²	—	53	1	no	—
293	C₂×C₄²	—	53	1	no	—
294	C₂×C₄²	—	53	1	no	—
295	C₂×C₄²	—	53	1	no	—
296	C₂×C₄²	—	53	1	no	—
297	C₂×C₄²	—	53	1	no	—
298	C₂×C₄²	—	53	1	no	—
299	C₂×C₄²	—	53	1	no	—
300	C₂×C₄²	C₂×#6157	1	1	yes	characteristic
301	C₄⋊C₈	—	53	2	no	—
302	C₄⋊C₈	—	53	4	no	—
303	C₄⋊C₈	—	53	2	no	—
304	C₄⋊C₈	—	106	2	no	—
305	C₄⋊C₈	—	106	2	no	—
306	C₄⋊C₈	—	106	2	no	—
307	C₄⋊C₈	—	106	1	no	—
308	C₄⋊C₈	—	106	1	no	—
309	C₂²⋊C₈	—	106	2	no	—
310	C₂²⋊C₈	—	106	2	no	—
311	#97	—	53	2	no	—
312	#97	—	53	2	no	—
313	#97	—	53	2	no	—
314	#97	C₂×#6157	1	2	yes	—
315	C₄×C₈	—	53	2	no	—
316	C₄×C₈	—	53	2	no	—
317	C₄×C₈	—	53	2	no	—
318	C₄×C₈	—	106	2	no	—
319	C₄×C₈	—	106	2	no	—
320	C₄×C₈	C₂×#6157	1	2	yes	—
321	C₂⁴	—	53	1	no	—
322	C₂²×C₄	—	53	1	no	—
323	C₂²×C₄	—	53	1	no	—
324	C₂²×C₄	—	53	1	no	—
325	C₂²×C₄	—	53	1	no	—
326	C₂²×C₄	—	53	4	no	—
327	C₂²×C₄	—	53	4	no	—
328	C₂²×C₄	—	53	1	no	—
329	C₂²×C₄	—	53	1	no	—
330	C₂²×C₄	—	53	1	no	—
331	C₂²×C₄	—	53	1	no	—
332	C₂²×C₄	—	53	1	no	—
333	C₂²×C₄	—	53	1	no	—
334	C₂²×C₄	—	53	1	no	—
335	C₂²×C₄	—	53	1	no	—
336	C₂²×C₄	—	53	1	no	—
337	C₂²×C₄	—	53	1	no	—
338	C₂²×C₄	—	53	1	no	—
339	C₂²×C₄	—	106	2	no	—
340	C₂²×C₄	—	106	2	no	—
341	C₂²×C₄	—	212	2	no	—
342	C₂²×C₄	C₄⋊#6157	1	1	yes	characteristic
343	C₂²×C₄	C₄⋊#6157	1	1	yes	characteristic
344	C₂²×C₄	C₂²×#6157	1	1	yes	characteristic
345	M₁₆	—	2	2	no	—
346	M₁₆	—	53	4	no	—
347	M₁₆	—	53	8	no	—
348	M₁₆	—	106	4	no	—
349	M₁₆	—	106	2	no	—
350	M₁₆	D₅₃⋊C₈	1	4	yes	complemented
351	C₂×C₈	—	53	2	no	—
352	C₂×C₈	—	53	8	no	—
353	C₂×C₈	—	53	2	no	—
354	C₂×C₈	—	106	4	no	—
355	C₂×C₈	—	106	2	no	—
356	C₂×C₈	—	106	2	no	—
357	C₂×C₈	—	106	4	no	—
358	C₂×C₈	—	212	4	no	—
359	C₂×C₈	—	212	2	no	—
360	C₂×C₈	—	212	2	no	—
361	C₂×C₈	—	212	4	no	—
362	C₂×C₈	—	212	4	no	—
363	C₂×C₈	C₂²×#6157	1	2	yes	—
364	C₂×C₈	C₂²×#6157	1	2	yes	—
365	C₄²	—	53	1	no	—
366	C₄²	—	53	1	no	—
367	C₄²	—	53	1	no	—
368	C₄²	—	53	1	no	—
369	C₄²	—	53	1	no	—
370	C₄²	—	53	1	no	—
371	C₄²	—	53	1	no	—
372	C₄²	—	53	1	no	—
373	C₄²	—	53	1	no	—
374	C₄²	—	53	1	no	—
375	C₄²	—	53	1	no	—
376	C₄²	—	53	1	no	—
377	C₄²	C₄⋊#6157	1	1	yes	characteristic
378	C₄²	C₄⋊#6157	1	1	yes	characteristic
379	C₄²	C₂²×#6157	1	1	yes	characteristic
380	C₄²	C₂²×#6157	1	1	yes	characteristic
381	C₂³	—	53	1	no	—
382	C₂³	—	53	4	no	—
383	C₂³	—	53	1	no	—
384	C₂³	—	106	2	no	—
385	C₂³	—	106	2	no	—
386	C₂³	C₂×C₄⋊#6157	1	1	yes	characteristic
387	C₂×C₄	—	2	2	no	—
388	C₂×C₄	—	4	1	no	—
389	C₂×C₄	—	53	4	no	—
390	C₂×C₄	—	53	1	no	—
391	C₂×C₄	—	53	1	no	—
392	C₂×C₄	—	53	1	no	—
393	C₂×C₄	—	53	4	no	—
394	C₂×C₄	—	53	4	no	—
395	C₂×C₄	—	53	2	no	—
396	C₂×C₄	—	53	2	no	—
397	C₂×C₄	—	53	2	no	—
398	C₂×C₄	—	53	2	no	—
399	C₂×C₄	—	53	2	no	—
400	C₂×C₄	—	53	2	no	—
401	C₂×C₄	—	53	1	no	—
402	C₂×C₄	—	53	1	no	—
403	C₂×C₄	—	53	1	no	—
404	C₂×C₄	—	106	1	no	—
405	C₂×C₄	—	106	1	no	—
406	C₂×C₄	—	106	2	no	—
407	C₂×C₄	—	106	2	no	—
408	C₂×C₄	—	106	2	no	—
409	C₂×C₄	—	106	1	no	—
410	C₂×C₄	—	106	1	no	—
411	C₂×C₄	—	106	2	no	—
412	C₂×C₄	—	212	1	no	—
413	C₂×C₄	—	212	1	no	—
414	C₂×C₄	—	212	1	no	—
415	C₂×C₄	C₂×C₄⋊#6157	1	1	yes	characteristic
416	C₂×C₄	C₂×C₄⋊#6157	1	1	yes	characteristic
417	C₂×C₄	C₂×C₄⋊#6157	1	1	yes	characteristic
418	C₂×C₄	C₂×C₄⋊#6157	1	1	yes	characteristic
419	C₂×C₄	C₂×C₄⋊#6157	1	1	yes	characteristic
420	C₂×C₄	C₂×D₅₃⋊C₈	1	2	yes	—
421	C₂×C₄	D₅₃⋊M₁₆	1	2	yes	—
422	C₂×C₄	C₂³×#6157	1	1	yes	characteristic
423	C₈	—	2	2	no	—
424	C₈	—	53	4	no	—
425	C₈	—	106	2	no	—
426	C₈	—	212	2	no	—
427	C₈	—	212	2	no	—
428	C₈	—	212	4	no	—
429	C₈	C₂×D₅₃⋊C₈	1	4	yes	complemented
430	C₂²	—	2	1	no	—
431	C₂²	—	2	1	no	—
432	C₂²	—	53	2	no	—
433	C₂²	—	53	2	no	—
434	C₂²	—	53	2	no	—
435	C₂²	—	53	2	no	—
436	C₂²	—	106	8	no	—
437	C₂²	—	106	1	no	—
438	C₂²	—	106	1	no	—
439	C₂²	D₅₃⋊(C₄⋊C₈)	1	2	yes	—
440	C₂²	C₂²×C₄⋊#6157	1	1	yes	characteristic
441	C₄	—	2	1	no	—
442	C₄	—	2	1	no	—
443	C₄	—	2	1	no	—
444	C₄	—	53	2	no	—
445	C₄	—	53	2	no	—
446	C₄	—	53	1	no	—
447	C₄	—	53	1	no	—
448	C₄	—	106	1	no	—
449	C₄	—	106	1	no	—
450	C₄	—	106	1	no	—
451	C₄	D₅₃⋊(C₄⋊C₈)	1	2	yes	—
452	C₄	#16811634	1	2	yes	—
453	C₄	C₂²×D₅₃⋊C₈	1	1	yes	characteristic
454	C₄	C₂×D₅₃⋊M₁₆	1	1	yes	characteristic
455	C₂	—	2	2	no	—
456	C₂	—	53	4	no	—
457	C₂	—	106	2	no	—
458	C₂	C₂×D₅₃⋊(C₄⋊C₈)	1	1	yes	characteristic
459	C₂	C₂×#16811634	1	1	yes	characteristic
460	C₂	D₅₃⋊#430	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂×C₄⋊#6157
commutator	C₄×C₅₃	C₂³×C₈
Frattini	C₂×C₄	C₂³×#6157
Fitting	C₄×M₁₆×C₅₃	C₄
socle	C₂²×C₅₃	C₂²×C₄⋊C₄
prime radical	C₂³×D₅₃	C₄⋊C₄
cosocle	C₄×Dic₅₃	C₂⁴
hypercenter	C₄×M₁₆	#6157
hypocenter	C₅₃	C₂×#1565
LCS	G ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₄×M₁₆

Complements

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

N	H ≅ G/N	type
3. #20290268	C₂	semidirect
4. #20290267	C₂	semidirect
5. #20290266	C₂	semidirect
6. D₅₃⋊#335	C₂	semidirect
7. D₅₃⋊#334	C₂	semidirect
24. #16811487	C₂²	semidirect
25. #16811486	C₂²	semidirect
26. #16811485	C₂²	semidirect
27. #16811484	C₂²	semidirect
65. M₁₆×D₅₃	C₈	semidirect
123. M₁₆×C₅₃	C₂×C₈	semidirect
143. #11719917	C₂×C₈	semidirect
147. #11719912	C₂×C₈	semidirect
149. C₈×D₅₃	C₂×C₈	semidirect
196. C₈×C₅₃	C₂²×C₈	semidirect
198. #88322	C₂²×C₈	semidirect
238. D₅₃	#1565	semidirect
269. C₅₃	C₂×#1565	semidirect
350. M₁₆	D₅₃⋊C₈	semidirect
429. C₈	C₂×D₅₃⋊C₈	semidirect

Maximal subgroups

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

subgroup	index
D₅₃⋊#431	2
C₂×D₅₃⋊(C₄⋊C₈)	2
#20290268	2
#20290267	2
#20290266	2
D₅₃⋊#335	2
D₅₃⋊#334	2
C₄×M₁₆×D₅₃	2
C₂×#1565	53

Subgroups — D53⋊#1565 #28316084

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — D₅₃⋊#1565 #28316084