Subgroups — D₅₃×#1771 #27382024

Order 13568.

1310 conjugacy classes of subgroups.

The subgroup lattice has 562 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄⋊C₈×D₅₃	C₂	1	1	yes	maximal, characteristic
2	C₂×#97×D₅₃	C₂	1	1	yes	maximal, characteristic
3	C₂×#97×D₅₃	C₂	1	1	yes	maximal, characteristic
4	C₂×C₄×C₈×D₅₃	C₂	1	1	yes	maximal, characteristic
5	C₄×C₄⋊C₄×D₅₃	C₂	1	1	yes	maximal, characteristic
6	#20274592	C₂	1	1	yes	maximal, characteristic, complemented
7	#20274591	C₂	1	1	yes	maximal, characteristic, complemented
8	#20274590	C₂	1	1	yes	maximal, characteristic, complemented
9	#20274589	C₂	1	1	yes	maximal, characteristic, complemented
10	#20274588	C₂	1	1	yes	maximal, characteristic, complemented
11	#20274587	C₂	1	1	yes	maximal, characteristic, complemented
12	#20274586	C₂	1	1	yes	maximal, characteristic, complemented
13	#336×D₅₃	C₂	1	1	yes	maximal, characteristic
14	#336×D₅₃	C₂	1	1	yes	maximal, characteristic
15	C₅₃×#1771	C₂	1	1	yes	maximal, characteristic, complemented
16	C₂²×C₈×D₅₃	C₂²	1	1	yes	characteristic
17	C₂²×C₈×D₅₃	C₂²	1	1	yes	characteristic
18	C₂²×C₈×D₅₃	C₂²	1	1	yes	characteristic
19	C₂²×C₈×D₅₃	C₂²	1	1	yes	characteristic
20	C₂×C₄⋊C₄×D₅₃	C₄	1	2	yes	—
21	C₂×C₄²×D₅₃	C₂²	1	1	yes	characteristic
22	C₂×C₄²×D₅₃	C₂²	1	1	yes	characteristic
23	C₂×C₄²×D₅₃	C₂²	1	1	yes	characteristic
24	C₂×C₄⋊#88322	C₂²	1	1	yes	characteristic
25	C₂×#13133146	C₂²	1	1	yes	characteristic
26	C₂×#13133146	C₂²	1	1	yes	characteristic
27	C₄⋊C₈×D₅₃	C₄	1	8	yes	complemented
28	C₂×#13133128	C₂²	1	1	yes	characteristic
29	C₂×#13133128	C₂²	1	1	yes	characteristic
30	C₂×#13133127	C₂²	1	1	yes	characteristic
31	C₂×#13133127	C₂²	1	1	yes	characteristic
32	C₂×#13133127	C₂²	1	1	yes	characteristic
33	C₂×#13133127	C₂²	1	1	yes	characteristic
34	#97×D₅₃	—	2	4	no	—
35	#97×D₅₃	C₄	1	8	yes	complemented
36	C₂×C₄×#88322	C₂²	1	1	yes	characteristic
37	C₂×C₈×Dic₅₃	C₂²	1	1	yes	characteristic
38	C₂×C₈×Dic₅₃	C₂²	1	1	yes	characteristic
39	C₄×C₈×D₅₃	—	2	4	no	—
40	C₄⋊C₄×Dic₅₃	C₄	1	2	yes	—
41	C₄×#11719919	C₂²	1	1	yes	characteristic
42	C₄×#11719919	C₂²	1	1	yes	characteristic
43	C₄×C₄⋊Dic₅₃	C₂²	1	1	yes	characteristic
44	#16810474	C₄	1	2	yes	—
45	#95×D₅₃	C₄	1	2	yes	—
46	#16810378	C₂²	1	1	yes	characteristic
47	#16810378	C₂²	1	1	yes	characteristic
48	#16810377	C₂²	1	1	yes	characteristic
49	#16810377	C₂²	1	1	yes	characteristic
50	#16810377	C₂²	1	1	yes	characteristic
51	#16810377	C₂²	1	1	yes	characteristic
52	C₂×C₄⋊C₈×C₅₃	C₂²	1	1	yes	characteristic
53	C₂×#97×C₅₃	C₂²	1	1	yes	characteristic
54	C₂×#97×C₅₃	C₂²	1	1	yes	characteristic
55	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
56	C₄×C₄⋊C₄×C₅₃	C₂²	1	1	yes	characteristic
57	C₅₃×#336	C₂²	1	1	yes	characteristic
58	C₅₃×#336	C₂²	1	1	yes	characteristic
59	C₂²×C₄×D₅₃	C₂×C₄	1	2	yes	—
60	C₂²×C₄×D₅₃	C₂×C₄	1	1	yes	characteristic
61	C₂²×C₄×D₅₃	C₂×C₄	1	1	yes	characteristic
62	C₂²×C₄×D₅₃	C₂×C₄	1	2	yes	—
63	C₂²×C₄×D₅₃	C₂³	1	1	yes	characteristic
64	C₂²×#88322	C₂³	1	1	yes	characteristic
65	C₂²×#88322	C₂³	1	1	yes	characteristic
66	C₂²×#88322	C₂³	1	1	yes	characteristic
67	C₂²×#88322	C₂³	1	1	yes	characteristic
68	C₂×C₈×D₅₃	—	2	2	no	—
69	C₂×C₈×D₅₃	—	2	2	no	—
70	C₂×C₈×D₅₃	—	2	2	no	—
71	C₂×C₈×D₅₃	—	2	2	no	—
72	C₂×C₈×D₅₃	—	2	2	no	—
73	C₂×C₈×D₅₃	—	2	2	no	—
74	C₂×C₈×D₅₃	C₂×C₄	1	4	yes	—
75	C₂×C₈×D₅₃	C₂×C₄	1	4	yes	—
76	C₂×C₈×D₅₃	C₂×C₄	1	4	yes	—
77	C₂×C₈×D₅₃	C₂×C₄	1	4	yes	—
78	C₂×C₈×D₅₃	D₄	1	4	yes	—
79	C₂×C₈×D₅₃	Q₈	1	4	yes	—
80	C₂×C₄⋊Dic₅₃	C₂×C₄	1	2	yes	—
81	C₂×#11719919	C₂×C₄	1	2	yes	—
82	C₂×#11719919	C₂×C₄	1	2	yes	—
83	C₄⋊C₄×D₅₃	—	2	8	no	—
84	C₂×C₄×Dic₅₃	C₂×C₄	1	2	yes	—
85	C₂×C₄×Dic₅₃	C₂×C₄	1	2	yes	—
86	C₂×C₄×Dic₅₃	C₂×C₄	1	1	yes	characteristic
87	C₂×C₄×Dic₅₃	C₂×C₄	1	1	yes	characteristic
88	C₂×C₄×Dic₅₃	C₂³	1	1	yes	characteristic
89	C₂×C₄×Dic₅₃	C₂³	1	1	yes	characteristic
90	C₂×C₄×Dic₅₃	C₂³	1	1	yes	characteristic
91	C₄²×D₅₃	—	2	2	no	—
92	C₄²×D₅₃	—	2	2	no	—
93	C₄²×D₅₃	—	2	2	no	—
94	C₄²×D₅₃	—	2	2	no	—
95	C₄²×D₅₃	C₂×C₄	1	4	yes	—
96	C₄²×D₅₃	C₂×C₄	1	4	yes	—
97	C₄⋊#88322	C₂×C₄	1	4	yes	complemented
98	#13133146	C₂×C₄	1	4	yes	complemented
99	#13133146	C₂×C₄	1	4	yes	complemented
100	#13133128	—	2	2	no	—
101	#13133128	C₂×C₄	1	4	yes	complemented
102	#13133127	—	2	2	no	—
103	#13133127	—	2	2	no	—
104	#13133127	C₂×C₄	1	4	yes	complemented
105	#13133127	C₂×C₄	1	4	yes	complemented
106	C₄×#88322	—	2	2	no	—
107	C₈×Dic₅₃	—	2	2	no	—
108	C₈×Dic₅₃	—	2	2	no	—
109	#13133124	C₂×C₄	1	2	yes	—
110	#13133124	C₂×C₄	1	2	yes	—
111	#13133124	C₂×C₄	1	2	yes	—
112	C₂²×C₈×C₅₃	C₂³	1	1	yes	characteristic
113	C₂²×C₈×C₅₃	C₂³	1	1	yes	characteristic
114	C₂²×C₈×C₅₃	C₂³	1	1	yes	characteristic
115	C₂²×C₈×C₅₃	C₂³	1	1	yes	characteristic
116	C₂×C₄⋊C₄×C₅₃	C₂×C₄	1	2	yes	—
117	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
118	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
119	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
120	C₄⋊C₈×C₅₃	C₂×C₄	1	4	yes	complemented
121	#97×C₅₃	—	2	2	no	—
122	#97×C₅₃	C₂×C₄	1	4	yes	complemented
123	C₄×C₈×C₅₃	—	2	2	no	—
124	#95×C₅₃	C₂×C₄	1	2	yes	—
125	C₂³×D₅₃	C₂²×C₄	1	1	yes	characteristic
126	C₂²×C₄×C₅₃	C₂²×C₄	1	2	yes	—
127	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
128	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
129	C₂²×C₄×C₅₃	C₂²×C₄	1	2	yes	—
130	C₂²×C₄×C₅₃	C₂⁴	1	1	yes	characteristic
131	C₂²×Dic₅₃	C₂²×C₄	1	2	yes	—
132	C₂²×Dic₅₃	C₂²×C₄	1	1	yes	characteristic
133	C₂²×Dic₅₃	C₂²×C₄	1	1	yes	characteristic
134	C₂²×Dic₅₃	C₂²×C₄	1	2	yes	—
135	C₂²×Dic₅₃	C₂²×C₄	1	1	yes	characteristic
136	C₂×C₄×D₅₃	—	2	4	no	—
137	C₂×C₄×D₅₃	—	2	4	no	—
138	C₂×C₄×D₅₃	—	2	4	no	—
139	C₂×C₄×D₅₃	—	2	4	no	—
140	C₂×C₄×D₅₃	—	2	4	no	—
141	C₂×C₄×D₅₃	—	2	4	no	—
142	C₂×C₄×D₅₃	—	2	4	no	—
143	C₂×C₄×D₅₃	C₄²	1	4	yes	—
144	C₂×C₄×D₅₃	C₄²	1	4	yes	—
145	C₂×C₄×D₅₃	M₁₆	1	8	yes	—
146	C₂×C₄×D₅₃	C₂²×C₄	1	2	yes	—
147	C₂×C₄×D₅₃	C₂²×C₄	1	2	yes	—
148	C₂×C₄×D₅₃	C₂×D₄	1	2	yes	—
149	C₂×C₄×D₅₃	C₂×Q₈	1	2	yes	—
150	C₂×C₄×D₅₃	#41	1	2	yes	—
151	C₂×C₄×D₅₃	#41	1	2	yes	—
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	—	2	1	no	—
156	C₂×C₈×C₅₃	—	2	1	no	—
157	C₂×C₈×C₅₃	—	2	1	no	—
158	C₂×C₈×C₅₃	C₂²×C₄	1	2	yes	—
159	C₂×C₈×C₅₃	C₂²×C₄	1	2	yes	—
160	C₂×C₈×C₅₃	C₂²×C₄	1	2	yes	—
161	C₂×C₈×C₅₃	C₂²×C₄	1	2	yes	—
162	C₂×C₈×C₅₃	C₂×D₄	1	2	yes	—
163	C₂×C₈×C₅₃	C₂×Q₈	1	2	yes	—
164	C₄⋊C₄×C₅₃	—	2	4	no	—
165	C₄²×C₅₃	—	2	1	no	—
166	C₄²×C₅₃	—	2	1	no	—
167	C₄²×C₅₃	—	2	1	no	—
168	C₄²×C₅₃	—	2	1	no	—
169	C₄²×C₅₃	C₂²×C₄	1	2	yes	—
170	C₄²×C₅₃	C₂²×C₄	1	2	yes	—
171	C₄⋊Dic₅₃	—	2	4	no	—
172	#11719919	—	2	4	no	—
173	#11719919	—	2	4	no	—
174	C₄×Dic₅₃	—	2	1	no	—
175	C₄×Dic₅₃	—	2	1	no	—
176	C₄×Dic₅₃	—	2	1	no	—
177	C₄×Dic₅₃	—	2	1	no	—
178	C₄×Dic₅₃	—	2	4	no	—
179	C₄×Dic₅₃	—	2	4	no	—
180	C₄×Dic₅₃	C₄²	1	4	yes	—
181	C₄×Dic₅₃	C₄²	1	4	yes	—
182	C₄×Dic₅₃	C₂²×C₄	1	2	yes	—
183	C₄×Dic₅₃	C₂²×C₄	1	2	yes	—
184	C₂×#88322	—	2	1	no	—
185	C₂×#88322	—	2	1	no	—
186	C₂×#88322	—	2	1	no	—
187	C₂×#88322	—	2	1	no	—
188	C₂×#88322	—	2	1	no	—
189	C₂×#88322	—	2	1	no	—
190	C₂×#88322	C₂²×C₄	1	2	yes	—
191	C₂×#88322	C₂²×C₄	1	2	yes	—
192	C₂×#88322	C₂²×C₄	1	2	yes	—
193	C₂×#88322	C₂²×C₄	1	2	yes	—
194	C₂×#88322	C₂×D₄	1	2	yes	—
195	C₂×#88322	C₂×Q₈	1	2	yes	—
196	C₈×D₅₃	—	2	8	no	—
197	C₈×D₅₃	—	2	8	no	—
198	C₈×D₅₃	—	2	8	no	—
199	C₈×D₅₃	C₄⋊C₄	1	16	yes	complemented
200	C₂³×C₅₃	C₂³×C₄	1	1	yes	characteristic
201	C₂²×D₅₃	C₂×C₄²	1	2	yes	—
202	C₂²×D₅₃	C₄×D₄	1	2	yes	—
203	C₂²×D₅₃	C₄×Q₈	1	2	yes	—
204	C₂²×D₅₃	C₂×M₁₆	1	4	yes	—
205	C₂²×D₅₃	#131	1	4	yes	—
206	C₂×C₄×C₅₃	—	2	2	no	—
207	C₂×C₄×C₅₃	—	2	2	no	—
208	C₂×C₄×C₅₃	—	2	2	no	—
209	C₂×C₄×C₅₃	—	2	2	no	—
210	C₂×C₄×C₅₃	—	2	2	no	—
211	C₂×C₄×C₅₃	—	2	2	no	—
212	C₂×C₄×C₅₃	—	2	2	no	—
213	C₂×C₄×C₅₃	C₂×C₄²	1	2	yes	—
214	C₂×C₄×C₅₃	C₂×C₄²	1	2	yes	—
215	C₂×C₄×C₅₃	C₂×M₁₆	1	4	yes	—
216	C₂×C₄×C₅₃	C₂³×C₄	1	1	yes	characteristic
217	C₂×C₄×C₅₃	C₂³×C₄	1	1	yes	characteristic
218	C₂×C₄×C₅₃	C₂²×D₄	1	1	yes	characteristic
219	C₂×C₄×C₅₃	C₂²×Q₈	1	1	yes	characteristic
220	C₂×C₄×C₅₃	C₂×#41	1	1	yes	characteristic
221	C₂×C₄×C₅₃	C₂×#41	1	1	yes	characteristic
222	C₂×Dic₅₃	—	2	2	no	—
223	C₂×Dic₅₃	—	2	2	no	—
224	C₂×Dic₅₃	—	2	2	no	—
225	C₂×Dic₅₃	—	2	2	no	—
226	C₂×Dic₅₃	—	2	2	no	—
227	C₂×Dic₅₃	—	2	2	no	—
228	C₂×Dic₅₃	—	2	2	no	—
229	C₂×Dic₅₃	C₂×C₄²	1	2	yes	—
230	C₂×Dic₅₃	C₂×C₄²	1	2	yes	—
231	C₂×Dic₅₃	C₂×C₄²	1	2	yes	—
232	C₂×Dic₅₃	C₄×D₄	1	2	yes	—
233	C₂×Dic₅₃	C₄×Q₈	1	2	yes	—
234	C₂×Dic₅₃	C₂×M₁₆	1	4	yes	—
235	C₄×D₅₃	—	2	8	no	—
236	C₄×D₅₃	—	2	8	no	—
237	C₄×D₅₃	—	2	16	no	—
238	C₄×D₅₃	—	4	8	no	—
239	C₄×D₅₃	C₂×C₄⋊C₄	1	4	yes	—
240	C₄×D₅₃	#117	1	4	yes	—
241	C₄×D₅₃	C₄×D₄	1	4	yes	—
242	C₄×D₅₃	C₄×Q₈	1	4	yes	—
243	C₈×C₅₃	—	2	2	no	—
244	C₈×C₅₃	—	2	2	no	—
245	C₈×C₅₃	—	2	2	no	—
246	C₈×C₅₃	C₂×C₄⋊C₄	1	4	yes	complemented
247	#88322	—	2	2	no	—
248	#88322	—	2	2	no	—
249	#88322	—	2	2	no	—
250	#88322	C₂×C₄⋊C₄	1	4	yes	complemented
251	C₂×#1771	—	53	1	no	maximal, Sylow
252	C₂²×C₅₃	C₂²×C₄²	1	1	yes	characteristic
253	C₂²×C₅₃	C₂×C₄×D₄	1	1	yes	characteristic
254	C₂²×C₅₃	C₂×C₄×Q₈	1	1	yes	characteristic
255	C₂²×C₅₃	C₂²×M₁₆	1	2	yes	—
256	C₂²×C₅₃	C₂×#131	1	2	yes	—
257	C₂×D₅₃	C₄×C₄⋊C₄	1	4	yes	—
258	C₂×D₅₃	C₄×M₁₆	1	4	yes	—
259	C₂×D₅₃	#405	1	4	yes	—
260	C₂×D₅₃	#435	1	8	yes	—
261	C₂×D₅₃	#446	1	8	yes	—
262	C₄×C₅₃	—	2	2	no	—
263	C₄×C₅₃	—	2	2	no	—
264	C₄×C₅₃	—	2	4	no	—
265	C₄×C₅₃	—	4	2	no	—
266	C₄×C₅₃	C₂²×C₄⋊C₄	1	1	yes	characteristic
267	C₄×C₅₃	C₂×#117	1	1	yes	characteristic
268	C₄×C₅₃	C₂×C₄×D₄	1	1	yes	characteristic
269	C₄×C₅₃	C₂×C₄×Q₈	1	1	yes	characteristic
270	Dic₅₃	—	2	2	no	—
271	Dic₅₃	—	2	2	no	—
272	Dic₅₃	—	2	4	no	—
273	Dic₅₃	—	4	2	no	—
274	Dic₅₃	C₄×C₄⋊C₄	1	4	yes	—
275	C₂²×C₄⋊C₈	—	53	1	no	—
276	C₂²×#97	—	53	1	no	—
277	C₂²×#97	—	53	1	no	—
278	C₂²×C₄×C₈	—	53	1	no	—
279	C₂×C₄×C₄⋊C₄	—	53	1	no	—
280	#1771	—	53	1	no	—
281	#1771	—	53	1	no	—
282	#1771	—	53	1	no	—
283	#1771	—	53	1	no	—
284	#1771	—	53	1	no	—
285	#1771	—	53	1	no	—
286	#1771	—	53	1	no	—
287	#1771	D₅₃	1	1	yes	characteristic, direct product factor
288	C₂×#336	—	53	1	no	—
289	C₂×#336	—	53	1	no	—
290	C₂×C₅₃	C₂×C₄×C₄⋊C₄	1	1	yes	characteristic
291	C₂×C₅₃	C₂×C₄×M₁₆	1	1	yes	characteristic
292	C₂×C₅₃	C₂×#405	1	1	yes	characteristic
293	C₂×C₅₃	C₂×#435	1	2	yes	—
294	C₂×C₅₃	C₂×#446	1	2	yes	—
295	D₅₃	#1771	1	8	yes	direct product factor
296	C₂³×C₈	—	53	1	no	—
297	C₂³×C₈	—	53	1	no	—
298	C₂³×C₈	—	53	1	no	—
299	C₂³×C₈	—	53	1	no	—
300	C₂²×C₄⋊C₄	—	53	2	no	—
301	C₂²×C₄²	—	53	1	no	—
302	C₂²×C₄²	—	53	1	no	—
303	C₂²×C₄²	—	53	1	no	—
304	C₂×C₄⋊C₈	—	53	1	no	—
305	C₂×C₄⋊C₈	—	53	1	no	—
306	C₂×C₄⋊C₈	—	53	1	no	—
307	C₂×C₄⋊C₈	—	53	8	no	—
308	C₂×C₄⋊C₈	C₂×D₅₃	1	1	yes	characteristic
309	C₂×#97	—	53	1	no	—
310	C₂×#97	—	53	1	no	—
311	C₂×#97	—	53	1	no	—
312	C₂×#97	—	53	8	no	—
313	C₂×#97	—	53	1	no	—
314	C₂×#97	—	53	1	no	—
315	C₂×#97	—	53	1	no	—
316	C₂×#97	—	106	4	no	—
317	C₂×#97	C₂×D₅₃	1	1	yes	characteristic
318	C₂×#97	C₂×D₅₃	1	1	yes	characteristic
319	C₂×C₄×C₈	—	53	1	no	—
320	C₂×C₄×C₈	—	53	1	no	—
321	C₂×C₄×C₈	—	53	1	no	—
322	C₂×C₄×C₈	—	106	4	no	—
323	C₂×C₄×C₈	C₂×D₅₃	1	1	yes	characteristic
324	C₄×C₄⋊C₄	—	53	2	no	—
325	C₄×C₄⋊C₄	—	53	1	no	—
326	C₄×C₄⋊C₄	—	53	1	no	—
327	C₄×C₄⋊C₄	—	53	1	no	—
328	C₄×C₄⋊C₄	—	53	2	no	—
329	C₄×C₄⋊C₄	C₂×D₅₃	1	1	yes	characteristic
330	C₂×#95	—	53	2	no	—
331	#336	—	53	1	no	—
332	#336	—	53	1	no	—
333	#336	—	53	1	no	—
334	#336	—	53	1	no	—
335	#336	—	53	1	no	—
336	#336	—	53	1	no	—
337	#336	C₂×D₅₃	1	1	yes	characteristic
338	#336	C₂×D₅₃	1	1	yes	characteristic
339	C₅₃	C₂×#1771	1	1	yes	characteristic, complemented, Sylow
340	C₂³×C₄	—	53	1	no	—
341	C₂³×C₄	—	53	1	no	—
342	C₂³×C₄	—	53	2	no	—
343	C₂³×C₄	—	53	1	no	—
344	C₂³×C₄	—	53	2	no	—
345	C₂²×C₈	—	53	1	no	—
346	C₂²×C₈	—	53	4	no	—
347	C₂²×C₈	—	53	4	no	—
348	C₂²×C₈	—	53	4	no	—
349	C₂²×C₈	—	53	1	no	—
350	C₂²×C₈	—	53	1	no	—
351	C₂²×C₈	—	53	1	no	—
352	C₂²×C₈	—	53	4	no	—
353	C₂²×C₈	—	53	4	no	—
354	C₂²×C₈	—	53	4	no	—
355	C₂²×C₈	—	106	2	no	—
356	C₂²×C₈	—	106	2	no	—
357	C₂²×C₈	—	106	2	no	—
358	C₂²×C₈	—	106	2	no	—
359	C₂²×C₈	—	106	2	no	—
360	C₂²×C₈	—	106	2	no	—
361	C₂²×C₈	C₂²×D₅₃	1	1	yes	characteristic
362	C₂²×C₈	C₂²×D₅₃	1	1	yes	characteristic
363	C₂²×C₈	C₂²×D₅₃	1	1	yes	characteristic
364	C₂²×C₈	C₂²×D₅₃	1	1	yes	characteristic
365	C₂×C₄⋊C₄	—	53	2	no	—
366	C₂×C₄⋊C₄	—	53	2	no	—
367	C₂×C₄⋊C₄	—	53	2	no	—
368	C₂×C₄⋊C₄	—	106	8	no	—
369	C₂×C₄⋊C₄	C₄×D₅₃	1	2	yes	—
370	C₂×C₄²	—	53	1	no	—
371	C₂×C₄²	—	53	1	no	—
372	C₂×C₄²	—	53	1	no	—
373	C₂×C₄²	—	53	4	no	—
374	C₂×C₄²	—	53	4	no	—
375	C₂×C₄²	—	53	2	no	—
376	C₂×C₄²	—	53	2	no	—
377	C₂×C₄²	—	53	1	no	—
378	C₂×C₄²	—	53	1	no	—
379	C₂×C₄²	—	106	2	no	—
380	C₂×C₄²	—	106	2	no	—
381	C₂×C₄²	—	106	2	no	—
382	C₂×C₄²	—	106	2	no	—
383	C₂×C₄²	C₂²×D₅₃	1	1	yes	characteristic
384	C₂×C₄²	C₂²×D₅₃	1	1	yes	characteristic
385	C₂×C₄²	C₂²×D₅₃	1	1	yes	characteristic
386	C₄⋊C₈	—	53	4	no	—
387	C₄⋊C₈	—	53	4	no	—
388	C₄⋊C₈	—	53	4	no	—
389	C₄⋊C₈	C₄×D₅₃	1	4	yes	complemented
390	#97	—	2	2	no	—
391	#97	—	53	4	no	—
392	#97	—	53	4	no	—
393	#97	—	53	4	no	—
394	#97	—	106	2	no	—
395	#97	—	106	2	no	—
396	#97	—	106	2	no	—
397	#97	C₄×D₅₃	1	4	yes	complemented
398	C₄×C₈	—	2	2	no	—
399	C₄×C₈	—	106	2	no	—
400	C₄×C₈	—	106	2	no	—
401	C₄×C₈	—	106	2	no	—
402	#95	—	53	2	no	—
403	#95	—	53	2	no	—
404	#95	—	53	2	no	—
405	#95	C₄×D₅₃	1	2	yes	—
406	C₂⁴	—	53	1	no	—
407	C₂²×C₄	—	53	1	no	—
408	C₂²×C₄	—	53	2	no	—
409	C₂²×C₄	—	53	2	no	—
410	C₂²×C₄	—	53	2	no	—
411	C₂²×C₄	—	53	2	no	—
412	C₂²×C₄	—	53	2	no	—
413	C₂²×C₄	—	53	2	no	—
414	C₂²×C₄	—	53	1	no	—
415	C₂²×C₄	—	53	2	no	—
416	C₂²×C₄	—	53	1	no	—
417	C₂²×C₄	—	53	4	no	—
418	C₂²×C₄	—	53	4	no	—
419	C₂²×C₄	—	53	8	no	—
420	C₂²×C₄	—	53	2	no	—
421	C₂²×C₄	—	106	4	no	—
422	C₂²×C₄	—	106	4	no	—
423	C₂²×C₄	—	106	4	no	—
424	C₂²×C₄	—	106	4	no	—
425	C₂²×C₄	—	106	4	no	—
426	C₂²×C₄	—	106	4	no	—
427	C₂²×C₄	—	106	4	no	—
428	C₂²×C₄	C₂×C₄×D₅₃	1	2	yes	—
429	C₂²×C₄	C₂×C₄×D₅₃	1	1	yes	characteristic
430	C₂²×C₄	C₂×C₄×D₅₃	1	1	yes	characteristic
431	C₂²×C₄	C₂×C₄×D₅₃	1	2	yes	—
432	C₂²×C₄	C₂³×D₅₃	1	1	yes	characteristic
433	C₂×C₈	—	2	1	no	—
434	C₂×C₈	—	2	1	no	—
435	C₂×C₈	—	2	1	no	—
436	C₂×C₈	—	2	1	no	—
437	C₂×C₈	—	2	1	no	—
438	C₂×C₈	—	2	1	no	—
439	C₂×C₈	—	53	2	no	—
440	C₂×C₈	—	53	2	no	—
441	C₂×C₈	—	53	2	no	—
442	C₂×C₈	—	53	16	no	—
443	C₂×C₈	—	53	2	no	—
444	C₂×C₈	—	53	2	no	—
445	C₂×C₈	—	53	2	no	—
446	C₂×C₈	—	106	8	no	—
447	C₂×C₈	—	106	8	no	—
448	C₂×C₈	—	106	8	no	—
449	C₂×C₈	—	106	1	no	—
450	C₂×C₈	—	106	1	no	—
451	C₂×C₈	—	106	1	no	—
452	C₂×C₈	—	106	1	no	—
453	C₂×C₈	—	106	1	no	—
454	C₂×C₈	—	106	1	no	—
455	C₂×C₈	C₂×C₄×D₅₃	1	2	yes	—
456	C₂×C₈	C₂×C₄×D₅₃	1	2	yes	—
457	C₂×C₈	C₂×C₄×D₅₃	1	2	yes	—
458	C₂×C₈	C₂×C₄×D₅₃	1	2	yes	—
459	C₂×C₈	D₄×D₅₃	1	2	yes	—
460	C₂×C₈	Q₈×D₅₃	1	2	yes	—
461	C₄⋊C₄	—	2	4	no	—
462	C₄⋊C₄	—	106	4	no	—
463	C₄⋊C₄	—	106	4	no	—
464	C₄⋊C₄	—	106	4	no	—
465	C₄²	—	2	1	no	—
466	C₄²	—	2	1	no	—
467	C₄²	—	2	1	no	—
468	C₄²	—	2	1	no	—
469	C₄²	—	53	4	no	—
470	C₄²	—	53	4	no	—
471	C₄²	—	53	2	no	—
472	C₄²	—	53	2	no	—
473	C₄²	—	106	4	no	—
474	C₄²	—	106	4	no	—
475	C₄²	—	106	1	no	—
476	C₄²	—	106	1	no	—
477	C₄²	—	106	1	no	—
478	C₄²	—	106	1	no	—
479	C₄²	C₂×C₄×D₅₃	1	2	yes	—
480	C₄²	C₂×C₄×D₅₃	1	2	yes	—
481	C₂³	—	53	2	no	—
482	C₂³	—	53	2	no	—
483	C₂³	—	53	2	no	—
484	C₂³	—	53	4	no	—
485	C₂³	—	53	4	no	—
486	C₂³	C₂²×C₄×D₅₃	1	1	yes	characteristic
487	C₂×C₄	—	2	2	no	—
488	C₂×C₄	—	2	2	no	—
489	C₂×C₄	—	2	2	no	—
490	C₂×C₄	—	2	2	no	—
491	C₂×C₄	—	2	2	no	—
492	C₂×C₄	—	2	2	no	—
493	C₂×C₄	—	2	2	no	—
494	C₂×C₄	—	53	2	no	—
495	C₂×C₄	—	53	2	no	—
496	C₂×C₄	—	53	2	no	—
497	C₂×C₄	—	53	4	no	—
498	C₂×C₄	—	53	4	no	—
499	C₂×C₄	—	53	4	no	—
500	C₂×C₄	—	53	4	no	—
501	C₂×C₄	—	53	2	no	—
502	C₂×C₄	—	53	2	no	—
503	C₂×C₄	—	53	4	no	—
504	C₂×C₄	—	106	8	no	—
505	C₂×C₄	—	106	8	no	—
506	C₂×C₄	—	106	16	no	—
507	C₂×C₄	—	106	2	no	—
508	C₂×C₄	—	106	2	no	—
509	C₂×C₄	—	106	2	no	—
510	C₂×C₄	—	106	2	no	—
511	C₂×C₄	—	106	2	no	—
512	C₂×C₄	—	106	2	no	—
513	C₂×C₄	—	106	2	no	—
514	C₂×C₄	—	212	8	no	—
515	C₂×C₄	C₄²×D₅₃	1	2	yes	—
516	C₂×C₄	C₄²×D₅₃	1	2	yes	—
517	C₂×C₄	M₁₆×D₅₃	1	4	yes	—
518	C₂×C₄	C₂²×C₄×D₅₃	1	1	yes	characteristic
519	C₂×C₄	C₂²×C₄×D₅₃	1	1	yes	characteristic
520	C₂×C₄	C₂×D₄×D₅₃	1	1	yes	characteristic
521	C₂×C₄	C₂×Q₈×D₅₃	1	1	yes	characteristic
522	C₂×C₄	#41×D₅₃	1	1	yes	characteristic
523	C₂×C₄	#41×D₅₃	1	1	yes	characteristic
524	C₈	—	2	2	no	—
525	C₈	—	2	2	no	—
526	C₈	—	2	2	no	—
527	C₈	—	53	4	no	—
528	C₈	—	106	2	no	—
529	C₈	—	106	2	no	—
530	C₈	—	106	2	no	—
531	C₈	C₄⋊C₄×D₅₃	1	4	yes	complemented
532	C₂²	—	53	4	no	—
533	C₂²	—	53	4	no	—
534	C₂²	—	53	4	no	—
535	C₂²	—	53	8	no	—
536	C₂²	—	53	8	no	—
537	C₂²	C₂×C₄²×D₅₃	1	1	yes	characteristic
538	C₂²	C₄×D₄×D₅₃	1	1	yes	characteristic
539	C₂²	C₄×Q₈×D₅₃	1	1	yes	characteristic
540	C₂²	C₂×M₁₆×D₅₃	1	2	yes	—
541	C₂²	#131×D₅₃	1	2	yes	—
542	C₄	—	2	2	no	—
543	C₄	—	2	2	no	—
544	C₄	—	2	4	no	—
545	C₄	—	4	2	no	—
546	C₄	—	53	4	no	—
547	C₄	—	106	2	no	—
548	C₄	—	106	2	no	—
549	C₄	—	106	4	no	—
550	C₄	—	212	2	no	—
551	C₄	C₂×C₄⋊C₄×D₅₃	1	1	yes	characteristic
552	C₄	#117×D₅₃	1	1	yes	characteristic
553	C₄	C₄×D₄×D₅₃	1	1	yes	characteristic
554	C₄	C₄×Q₈×D₅₃	1	1	yes	characteristic
555	C₂	—	53	8	no	—
556	C₂	C₄×C₄⋊C₄×D₅₃	1	1	yes	characteristic
557	C₂	C₄×M₁₆×D₅₃	1	1	yes	characteristic
558	C₂	#405×D₅₃	1	1	yes	characteristic
559	C₂	#435×D₅₃	1	2	yes	—
560	C₂	#446×D₅₃	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂²×C₄	C₂³×D₅₃
commutator	C₂²×C₅₃	C₂²×C₄²
Fitting	C₅₃×#1771	C₂
socle	C₂³×C₅₃	C₂³×C₄
prime radical	C₂³×D₅₃	C₂²×C₄
cosocle	C₂²×C₄×C₅₃	C₂⁴
hypercenter	#1771	D₅₃
hypocenter	C₅₃	C₂×#1771
LCS	G ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂²×C₄ ◁ #1771

Complements

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

N	H ≅ G/N	type
6. #20274592	C₂	semidirect
7. #20274591	C₂	semidirect
8. #20274590	C₂	semidirect
9. #20274589	C₂	semidirect
10. #20274588	C₂	semidirect
11. #20274587	C₂	semidirect
12. #20274586	C₂	semidirect
15. C₅₃×#1771	C₂	semidirect
27. C₄⋊C₈×D₅₃	C₄	semidirect
35. #97×D₅₃	C₄	semidirect
97. C₄⋊#88322	C₂×C₄	semidirect
98. #13133146	C₂×C₄	semidirect
99. #13133146	C₂×C₄	semidirect
101. #13133128	C₂×C₄	semidirect
104. #13133127	C₂×C₄	semidirect
105. #13133127	C₂×C₄	semidirect
120. C₄⋊C₈×C₅₃	C₂×C₄	semidirect
122. #97×C₅₃	C₂×C₄	semidirect
199. C₈×D₅₃	C₄⋊C₄	semidirect
246. C₈×C₅₃	C₂×C₄⋊C₄	semidirect
250. #88322	C₂×C₄⋊C₄	semidirect
287. #1771	D₅₃	direct (N × H)
295. D₅₃	#1771	direct (N × H)
339. C₅₃	C₂×#1771	semidirect
389. C₄⋊C₈	C₄×D₅₃	semidirect
397. #97	C₄×D₅₃	semidirect
531. C₈	C₄⋊C₄×D₅₃	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₄⋊C₈×D₅₃	2
C₂×#97×D₅₃	2
C₂×C₄×C₈×D₅₃	2
C₄×C₄⋊C₄×D₅₃	2
#20274592	2
#20274591	2
#20274590	2
#20274589	2
#20274588	2
#20274587	2
#20274586	2
#336×D₅₃	2
C₅₃×#1771	2
C₂×#1771	53

Subgroups — D53×#1771 #27382024

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — D₅₃×#1771 #27382024