Subgroups — C₅₃×#7897 #27215409

Order 13568.

798 conjugacy classes of subgroups.

The subgroup lattice has 332 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₅₃×#423	C₂	1	1	yes	maximal, characteristic
2	C₂²×C₄×C₈×C₅₃	C₂	1	1	yes	maximal, characteristic
3	C₅₃×#2285	C₂	1	1	yes	maximal, characteristic
4	C₅₃×#1279	C₂	1	4	yes	maximal, complemented
5	C₂²×M₁₆×C₅₃	—	2	1	no	—
6	C₂³×C₈×C₅₃	—	2	1	no	—
7	C₂×#117×C₅₃	—	2	1	no	—
8	C₂²×C₄²×C₅₃	C₂²	1	1	yes	characteristic
9	C₅₃×#423	—	2	2	no	—
10	C₅₃×#423	C₄	1	4	yes	complemented
11	C₂×C₄⋊C₈×C₅₃	—	2	1	no	—
12	C₂×C₄⋊C₈×C₅₃	C₂²	1	2	yes	—
13	C₂×C₄×C₈×C₅₃	—	2	1	no	—
14	C₂×C₄×C₈×C₅₃	—	2	2	no	—
15	C₂×C₄×C₈×C₅₃	C₄	1	4	yes	complemented
16	C₂×C₄×C₈×C₅₃	C₂²	1	2	yes	—
17	C₅₃×#384	C₄	1	2	yes	—
18	C₅₃×#382	—	2	1	no	—
19	C₅₃×#380	—	2	1	no	—
20	C₅₃×#380	C₄	1	2	yes	—
21	C₄×C₄⋊C₄×C₅₃	—	2	1	no	—
22	C₄×C₄⋊C₄×C₅₃	C₂²	1	2	yes	—
23	C₂³×C₄×C₅₃	C₂×C₄	1	1	yes	characteristic
24	C₂³×C₄×C₅₃	D₄	1	1	yes	characteristic
25	C₂³×C₄×C₅₃	D₄	1	1	yes	characteristic
26	C₂×M₁₆×C₅₃	—	2	4	no	—
27	C₂×M₁₆×C₅₃	—	4	2	no	—
28	C₂×M₁₆×C₅₃	—	4	2	no	—
29	C₂²×C₈×C₅₃	—	2	8	no	—
30	C₂²×C₈×C₅₃	—	2	4	no	—
31	C₂²×C₈×C₅₃	—	2	2	no	—
32	C₂²×C₈×C₅₃	—	2	2	no	—
33	#117×C₅₃	—	4	2	no	—
34	#117×C₅₃	—	4	2	no	—
35	C₂×C₄⋊C₄×C₅₃	—	2	2	no	—
36	C₂×C₄⋊C₄×C₅₃	—	2	2	no	—
37	C₂×C₄⋊C₄×C₅₃	C₂×C₄	1	4	yes	—
38	C₂×C₂²⋊C₄×C₅₃	—	2	2	no	—
39	C₂×C₄²×C₅₃	—	2	2	no	—
40	C₂×C₄²×C₅₃	C₂×C₄	1	2	yes	—
41	C₂×C₄²×C₅₃	C₂×C₄	1	2	yes	—
42	C₂×C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
43	C₂×C₄²×C₅₃	D₄	1	2	yes	—
44	C₂×C₄²×C₅₃	D₄	1	1	yes	characteristic
45	C₂×C₄²×C₅₃	D₄	1	1	yes	characteristic
46	C₂×C₄²×C₅₃	Q₈	1	2	yes	—
47	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
48	C₄⋊C₈×C₅₃	—	2	4	no	—
49	C₄⋊C₈×C₅₃	—	2	2	no	—
50	C₄⋊C₈×C₅₃	C₂×C₄	1	4	yes	—
51	C₄×C₈×C₅₃	—	2	4	no	—
52	C₄×C₈×C₅₃	—	2	2	no	—
53	C₄×C₈×C₅₃	C₂×C₄	1	4	yes	—
54	#95×C₅₃	—	2	2	no	—
55	#95×C₅₃	—	2	2	no	—
56	C₂⁴×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
57	C₂²×C₄×C₅₃	—	2	4	no	—
58	C₂²×C₄×C₅₃	—	2	4	no	—
59	C₂²×C₄×C₅₃	—	2	4	no	—
60	C₂²×C₄×C₅₃	C₄²	1	2	yes	—
61	C₂²×C₄×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
62	C₂²×C₄×C₅₃	C₂²⋊C₄	1	2	yes	—
63	C₂²×C₄×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
64	C₂²×C₄×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
65	C₂²×C₄×C₅₃	C₂²⋊C₄	1	2	yes	—
66	C₂²×C₄×C₅₃	C₄⋊C₄	1	2	yes	—
67	C₂²×C₄×C₅₃	C₄⋊C₄	1	2	yes	—
68	C₂²×C₄×C₅₃	C₄⋊C₄	1	2	yes	—
69	C₂²×C₄×C₅₃	D₈	1	2	yes	—
70	C₂²×C₄×C₅₃	SD₁₆	1	2	yes	—
71	C₂²×C₄×C₅₃	SD₁₆	1	2	yes	—
72	C₂²×C₄×C₅₃	Q₁₆	1	2	yes	—
73	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
74	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
75	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
76	M₁₆×C₅₃	—	4	8	no	—
77	C₂×C₈×C₅₃	—	2	8	no	—
78	C₂×C₈×C₅₃	—	2	16	no	—
79	C₂×C₈×C₅₃	—	2	4	no	—
80	C₂×C₈×C₅₃	—	2	4	no	—
81	C₂×C₈×C₅₃	—	4	2	no	—
82	C₂×C₈×C₅₃	—	4	2	no	—
83	C₄⋊C₄×C₅₃	—	2	8	no	—
84	C₄⋊C₄×C₅₃	—	4	4	no	—
85	C₄⋊C₄×C₅₃	—	4	4	no	—
86	C₂²⋊C₄×C₅₃	—	4	4	no	—
87	C₄²×C₅₃	—	4	2	no	—
88	C₄²×C₅₃	—	4	2	no	—
89	C₄²×C₅₃	C₄²	1	2	yes	—
90	C₄²×C₅₃	C₂²⋊C₄	1	2	yes	—
91	C₄²×C₅₃	C₂²⋊C₄	1	2	yes	—
92	C₄²×C₅₃	C₄⋊C₄	1	2	yes	—
93	C₄²×C₅₃	C₄⋊C₄	1	2	yes	—
94	C₄²×C₅₃	C₄⋊C₄	1	2	yes	—
95	C₄²×C₅₃	C₂²×C₄	1	1	yes	characteristic
96	C₄²×C₅₃	C₂²×C₄	1	1	yes	characteristic
97	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
98	C₄²×C₅₃	C₂×Q₈	1	1	yes	characteristic
99	C₂³×C₅₃	—	2	2	no	—
100	C₂³×C₅₃	—	2	2	no	—
101	C₂³×C₅₃	#95	1	2	yes	—
102	C₂³×C₅₃	#102	1	2	yes	—
103	C₂³×C₅₃	#103	1	2	yes	—
104	C₂³×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
105	C₂×C₄×C₅₃	—	2	2	no	—
106	C₂×C₄×C₅₃	—	2	2	no	—
107	C₂×C₄×C₅₃	—	2	4	no	—
108	C₂×C₄×C₅₃	—	2	4	no	—
109	C₂×C₄×C₅₃	—	2	4	no	—
110	C₂×C₄×C₅₃	—	2	4	no	—
111	C₂×C₄×C₅₃	—	4	4	no	—
112	C₂×C₄×C₅₃	—	4	4	no	—
113	C₂×C₄×C₅₃	—	4	4	no	—
114	C₂×C₄×C₅₃	#95	1	2	yes	—
115	C₂×C₄×C₅₃	#95	1	2	yes	—
116	C₂×C₄×C₅₃	#95	1	2	yes	—
117	C₂×C₄×C₅₃	#102	1	2	yes	—
118	C₂×C₄×C₅₃	#102	1	4	yes	—
119	C₂×C₄×C₅₃	#103	1	2	yes	—
120	C₂×C₄×C₅₃	#103	1	4	yes	—
121	C₂×C₄×C₅₃	#104	1	4	yes	—
122	C₂×C₄×C₅₃	#106	1	4	yes	—
123	C₂×C₄×C₅₃	#107	1	4	yes	—
124	C₂×C₄×C₅₃	C₂×C₄²	1	1	yes	characteristic
125	C₂×C₄×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
126	C₂×C₄×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
127	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
128	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
129	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
130	C₂×C₄×C₅₃	#131	1	4	yes	—
131	C₂×C₄×C₅₃	C₂×D₈	1	1	yes	characteristic
132	C₂×C₄×C₅₃	C₂×SD₁₆	1	1	yes	characteristic
133	C₂×C₄×C₅₃	C₂×SD₁₆	1	1	yes	characteristic
134	C₂×C₄×C₅₃	C₂×Q₁₆	1	1	yes	characteristic
135	C₈×C₅₃	—	2	8	no	—
136	C₈×C₅₃	—	4	4	no	—
137	#7897	C₅₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
138	C₂²×C₅₃	—	2	2	no	—
139	C₂²×C₅₃	—	2	2	no	—
140	C₂²×C₅₃	—	2	4	no	—
141	C₂²×C₅₃	—	2	4	no	—
142	C₂²×C₅₃	#340	1	4	yes	—
143	C₂²×C₅₃	C₂×#95	1	1	yes	characteristic
144	C₂²×C₅₃	#408	1	2	yes	—
145	C₂²×C₅₃	C₂×#102	1	1	yes	characteristic
146	C₂²×C₅₃	C₂×#103	1	1	yes	characteristic
147	C₂²×C₅₃	C₂×#104	1	2	yes	—
148	C₄×C₅₃	—	2	2	no	—
149	C₄×C₅₃	—	2	1	no	—
150	C₄×C₅₃	—	2	1	no	—
151	C₄×C₅₃	—	8	4	no	—
152	C₄×C₅₃	#339	1	4	yes	—
153	C₄×C₅₃	#340	1	4	yes	—
154	C₄×C₅₃	#405	1	2	yes	—
155	C₄×C₅₃	C₂×#102	1	1	yes	characteristic
156	C₄×C₅₃	C₂×#103	1	1	yes	characteristic
157	C₄×C₅₃	#424	1	2	yes	—
158	C₄×C₅₃	C₂×#106	1	1	yes	characteristic
159	C₄×C₅₃	C₂×#107	1	1	yes	characteristic
160	C₂×#423	C₂×C₅₃	1	1	yes	characteristic
161	C₂²×C₄×C₈	C₂×C₅₃	1	1	yes	characteristic
162	#2285	C₂×C₅₃	1	1	yes	characteristic
163	#1279	C₂×C₅₃	1	4	yes	complemented
164	C₂×C₅₃	—	2	4	no	—
165	C₂×C₅₃	#1492	1	2	yes	—
166	C₂×C₅₃	#1493	1	2	yes	—
167	C₂×C₅₃	#1729	1	1	yes	characteristic
168	C₂×C₅₃	C₂×#339	1	1	yes	characteristic
169	C₂×C₅₃	C₂×#340	1	1	yes	characteristic
170	C₂²×M₁₆	—	2	1	no	—
171	C₂³×C₈	—	2	1	no	—
172	C₂×#117	—	2	1	no	—
173	C₂²×C₄²	C₂²×C₅₃	1	1	yes	characteristic
174	#423	—	2	2	no	—
175	#423	C₄×C₅₃	1	4	yes	complemented
176	C₂×C₄⋊C₈	—	2	1	no	—
177	C₂×C₄⋊C₈	C₂²×C₅₃	1	2	yes	—
178	C₂×C₄×C₈	—	2	2	no	—
179	C₂×C₄×C₈	—	2	1	no	—
180	C₂×C₄×C₈	C₄×C₅₃	1	4	yes	complemented
181	C₂×C₄×C₈	C₂²×C₅₃	1	2	yes	—
182	#384	C₄×C₅₃	1	2	yes	—
183	#382	—	2	1	no	—
184	#380	—	2	1	no	—
185	#380	C₄×C₅₃	1	2	yes	—
186	C₄×C₄⋊C₄	—	2	1	no	—
187	C₄×C₄⋊C₄	C₂²×C₅₃	1	2	yes	—
188	C₅₃	#7897	1	1	yes	characteristic, direct product factor, Sylow
189	C₂³×C₄	C₂×C₄×C₅₃	1	1	yes	characteristic
190	C₂³×C₄	D₄×C₅₃	1	1	yes	characteristic
191	C₂³×C₄	D₄×C₅₃	1	1	yes	characteristic
192	C₂×M₁₆	—	2	4	no	—
193	C₂×M₁₆	—	4	2	no	—
194	C₂×M₁₆	—	4	2	no	—
195	C₂²×C₈	—	2	8	no	—
196	C₂²×C₈	—	2	2	no	—
197	C₂²×C₈	—	2	4	no	—
198	C₂²×C₈	—	2	2	no	—
199	#117	—	4	2	no	—
200	#117	—	4	2	no	—
201	C₂×C₄⋊C₄	—	2	2	no	—
202	C₂×C₄⋊C₄	—	2	2	no	—
203	C₂×C₄⋊C₄	C₂×C₄×C₅₃	1	4	yes	—
204	C₂×C₂²⋊C₄	—	2	2	no	—
205	C₂×C₄²	—	2	2	no	—
206	C₂×C₄²	C₂×C₄×C₅₃	1	2	yes	—
207	C₂×C₄²	C₂×C₄×C₅₃	1	2	yes	—
208	C₂×C₄²	C₂×C₄×C₅₃	1	1	yes	characteristic
209	C₂×C₄²	D₄×C₅₃	1	2	yes	—
210	C₂×C₄²	D₄×C₅₃	1	1	yes	characteristic
211	C₂×C₄²	D₄×C₅₃	1	1	yes	characteristic
212	C₂×C₄²	Q₈×C₅₃	1	2	yes	—
213	C₂×C₄²	C₂³×C₅₃	1	1	yes	characteristic
214	C₄⋊C₈	—	2	2	no	—
215	C₄⋊C₈	—	2	4	no	—
216	C₄⋊C₈	C₂×C₄×C₅₃	1	4	yes	—
217	C₄×C₈	—	2	2	no	—
218	C₄×C₈	—	2	4	no	—
219	C₄×C₈	C₂×C₄×C₅₃	1	4	yes	—
220	#95	—	2	2	no	—
221	#95	—	2	2	no	—
222	C₂⁴	C₂²⋊C₄×C₅₃	1	1	yes	characteristic
223	C₂²×C₄	—	2	4	no	—
224	C₂²×C₄	—	2	4	no	—
225	C₂²×C₄	—	2	4	no	—
226	C₂²×C₄	C₄²×C₅₃	1	2	yes	—
227	C₂²×C₄	C₂²⋊C₄×C₅₃	1	1	yes	characteristic
228	C₂²×C₄	C₂²⋊C₄×C₅₃	1	2	yes	—
229	C₂²×C₄	C₂²⋊C₄×C₅₃	1	1	yes	characteristic
230	C₂²×C₄	C₂²⋊C₄×C₅₃	1	1	yes	characteristic
231	C₂²×C₄	C₂²⋊C₄×C₅₃	1	2	yes	—
232	C₂²×C₄	C₄⋊C₄×C₅₃	1	2	yes	—
233	C₂²×C₄	C₄⋊C₄×C₅₃	1	2	yes	—
234	C₂²×C₄	C₄⋊C₄×C₅₃	1	2	yes	—
235	C₂²×C₄	D₈×C₅₃	1	2	yes	—
236	C₂²×C₄	SD₁₆×C₅₃	1	2	yes	—
237	C₂²×C₄	SD₁₆×C₅₃	1	2	yes	—
238	C₂²×C₄	Q₁₆×C₅₃	1	2	yes	—
239	C₂²×C₄	C₂²×C₄×C₅₃	1	1	yes	characteristic
240	C₂²×C₄	C₂×D₄×C₅₃	1	1	yes	characteristic
241	C₂²×C₄	C₂×D₄×C₅₃	1	1	yes	characteristic
242	M₁₆	—	4	8	no	—
243	C₂×C₈	—	2	8	no	—
244	C₂×C₈	—	2	16	no	—
245	C₂×C₈	—	2	4	no	—
246	C₂×C₈	—	2	4	no	—
247	C₂×C₈	—	4	2	no	—
248	C₂×C₈	—	4	2	no	—
249	C₄⋊C₄	—	2	8	no	—
250	C₄⋊C₄	—	4	4	no	—
251	C₄⋊C₄	—	4	4	no	—
252	C₂²⋊C₄	—	4	4	no	—
253	C₄²	—	4	2	no	—
254	C₄²	—	4	2	no	—
255	C₄²	C₄²×C₅₃	1	2	yes	—
256	C₄²	C₂²⋊C₄×C₅₃	1	2	yes	—
257	C₄²	C₂²⋊C₄×C₅₃	1	2	yes	—
258	C₄²	C₄⋊C₄×C₅₃	1	2	yes	—
259	C₄²	C₄⋊C₄×C₅₃	1	2	yes	—
260	C₄²	C₄⋊C₄×C₅₃	1	2	yes	—
261	C₄²	C₂²×C₄×C₅₃	1	1	yes	characteristic
262	C₄²	C₂²×C₄×C₅₃	1	1	yes	characteristic
263	C₄²	C₂×D₄×C₅₃	1	1	yes	characteristic
264	C₄²	C₂×Q₈×C₅₃	1	1	yes	characteristic
265	C₂³	—	2	2	no	—
266	C₂³	—	2	2	no	—
267	C₂³	#95×C₅₃	1	2	yes	—
268	C₂³	#102×C₅₃	1	2	yes	—
269	C₂³	#103×C₅₃	1	2	yes	—
270	C₂³	C₂×C₂²⋊C₄×C₅₃	1	1	yes	characteristic
271	C₂×C₄	—	2	2	no	—
272	C₂×C₄	—	2	2	no	—
273	C₂×C₄	—	2	4	no	—
274	C₂×C₄	—	2	4	no	—
275	C₂×C₄	—	2	4	no	—
276	C₂×C₄	—	2	4	no	—
277	C₂×C₄	—	4	4	no	—
278	C₂×C₄	—	4	4	no	—
279	C₂×C₄	—	4	4	no	—
280	C₂×C₄	#95×C₅₃	1	2	yes	—
281	C₂×C₄	#95×C₅₃	1	2	yes	—
282	C₂×C₄	#95×C₅₃	1	2	yes	—
283	C₂×C₄	#102×C₅₃	1	2	yes	—
284	C₂×C₄	#102×C₅₃	1	4	yes	—
285	C₂×C₄	#103×C₅₃	1	2	yes	—
286	C₂×C₄	#103×C₅₃	1	4	yes	—
287	C₂×C₄	#104×C₅₃	1	4	yes	—
288	C₂×C₄	#106×C₅₃	1	4	yes	—
289	C₂×C₄	#107×C₅₃	1	4	yes	—
290	C₂×C₄	C₂×C₄²×C₅₃	1	1	yes	characteristic
291	C₂×C₄	C₂×C₂²⋊C₄×C₅₃	1	1	yes	characteristic
292	C₂×C₄	C₂×C₂²⋊C₄×C₅₃	1	1	yes	characteristic
293	C₂×C₄	C₂×C₄⋊C₄×C₅₃	1	1	yes	characteristic
294	C₂×C₄	C₂×C₄⋊C₄×C₅₃	1	1	yes	characteristic
295	C₂×C₄	C₂×C₄⋊C₄×C₅₃	1	1	yes	characteristic
296	C₂×C₄	#131×C₅₃	1	4	yes	—
297	C₂×C₄	C₂×D₈×C₅₃	1	1	yes	characteristic
298	C₂×C₄	C₂×SD₁₆×C₅₃	1	1	yes	characteristic
299	C₂×C₄	C₂×SD₁₆×C₅₃	1	1	yes	characteristic
300	C₂×C₄	C₂×Q₁₆×C₅₃	1	1	yes	characteristic
301	C₈	—	2	8	no	—
302	C₈	—	4	4	no	—
303	C₂²	—	2	2	no	—
304	C₂²	—	2	2	no	—
305	C₂²	—	2	4	no	—
306	C₂²	—	2	4	no	—
307	C₂²	C₅₃×#340	1	4	yes	—
308	C₂²	C₂×#95×C₅₃	1	1	yes	characteristic
309	C₂²	C₅₃×#408	1	2	yes	—
310	C₂²	C₂×#102×C₅₃	1	1	yes	characteristic
311	C₂²	C₂×#103×C₅₃	1	1	yes	characteristic
312	C₂²	C₂×#104×C₅₃	1	2	yes	—
313	C₄	—	2	2	no	—
314	C₄	—	2	1	no	—
315	C₄	—	2	1	no	—
316	C₄	—	8	4	no	—
317	C₄	C₅₃×#339	1	4	yes	—
318	C₄	C₅₃×#340	1	4	yes	—
319	C₄	C₅₃×#405	1	2	yes	—
320	C₄	C₂×#102×C₅₃	1	1	yes	characteristic
321	C₄	C₂×#103×C₅₃	1	1	yes	characteristic
322	C₄	C₅₃×#424	1	2	yes	—
323	C₄	C₂×#106×C₅₃	1	1	yes	characteristic
324	C₄	C₂×#107×C₅₃	1	1	yes	characteristic
325	C₂	—	2	4	no	—
326	C₂	C₅₃×#1492	1	2	yes	—
327	C₂	C₅₃×#1493	1	2	yes	—
328	C₂	C₅₃×#1729	1	1	yes	characteristic
329	C₂	C₂×C₅₃×#339	1	1	yes	characteristic
330	C₂	C₂×C₅₃×#340	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄×C₅₃	C₂×D₄
commutator	C₂×C₄	C₂×C₄²×C₅₃
Frattini, cosocle	C₂×C₄²	C₂³×C₅₃
socle	C₂³×C₅₃	C₂×C₂²⋊C₄
prime radical	C₂⁴×C₅₃	C₂²⋊C₄
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄×C₅₃ ◁ C₂²×C₄²×C₅₃ ◁ G

Complements

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

N	H ≅ G/N	type
4. C₅₃×#1279	C₂	semidirect
10. C₅₃×#423	C₄	semidirect
15. C₂×C₄×C₈×C₅₃	C₄	semidirect
137. #7897	C₅₃	direct (N × H)
163. #1279	C₂×C₅₃	semidirect
175. #423	C₄×C₅₃	semidirect
180. C₂×C₄×C₈	C₄×C₅₃	semidirect
188. C₅₃	#7897	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×C₅₃×#423	2
C₂²×C₄×C₈×C₅₃	2
C₅₃×#2285	2
C₅₃×#1279	2
#7897	53

Subgroups — C53×#7897 #27215409

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅₃×#7897 #27215409