Subgroups — C₅₃×#7877 #27215389

Order 13568.

674 conjugacy classes of subgroups.

The subgroup lattice has 308 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₅₃×#431	C₂	1	1	yes	maximal, characteristic
2	C₂×C₅₃×#423	C₂	1	1	yes	maximal, characteristic
3	C₂²×C₄⋊C₈×C₅₃	C₂	1	1	yes	maximal, characteristic
4	C₅₃×#1277	C₂	1	4	yes	maximal, complemented
5	C₂²×M₁₆×C₅₃	—	2	1	no	—
6	C₂³×C₈×C₅₃	—	2	1	no	—
7	C₂²×C₄²×C₅₃	C₂²	1	1	yes	characteristic
8	C₅₃×#431	—	2	2	no	—
9	C₅₃×#431	—	2	2	no	—
10	C₅₃×#423	—	2	2	no	—
11	C₅₃×#423	C₄	1	4	yes	—
12	C₂×C₄⋊C₈×C₅₃	—	2	1	no	—
13	C₂×C₄⋊C₈×C₅₃	—	2	2	no	—
14	C₂×C₄⋊C₈×C₅₃	—	2	1	no	—
15	C₂×C₄⋊C₈×C₅₃	C₄	1	2	yes	—
16	C₂×C₄⋊C₈×C₅₃	C₄	1	4	yes	—
17	C₂×C₄⋊C₈×C₅₃	C₂²	1	2	yes	—
18	C₂×C₄⋊C₈×C₅₃	C₂²	1	2	yes	—
19	C₂×C₂²⋊C₈×C₅₃	C₄	1	2	yes	—
20	C₂×C₄×C₈×C₅₃	C₂²	1	2	yes	—
21	C₂³×C₄×C₅₃	C₂×C₄	1	1	yes	characteristic
22	C₂³×C₄×C₅₃	D₄	1	1	yes	characteristic
23	C₂³×C₄×C₅₃	Q₈	1	1	yes	characteristic
24	C₂×M₁₆×C₅₃	—	2	4	no	—
25	C₂×M₁₆×C₅₃	—	4	2	no	—
26	C₂×M₁₆×C₅₃	—	4	2	no	—
27	C₂²×C₈×C₅₃	—	2	4	no	—
28	C₂²×C₈×C₅₃	—	2	2	no	—
29	C₂²×C₈×C₅₃	—	2	2	no	—
30	C₂²×C₈×C₅₃	—	4	2	no	—
31	C₂²×C₈×C₅₃	—	4	2	no	—
32	C₂²×C₈×C₅₃	C₂×C₄	1	4	yes	—
33	C₂×C₄²×C₅₃	C₂×C₄	1	2	yes	—
34	C₂×C₄²×C₅₃	C₂×C₄	1	2	yes	—
35	C₂×C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
36	C₂×C₄²×C₅₃	D₄	1	2	yes	—
37	C₂×C₄²×C₅₃	D₄	1	2	yes	—
38	C₂×C₄²×C₅₃	D₄	1	1	yes	characteristic
39	C₂×C₄²×C₅₃	Q₈	1	1	yes	characteristic
40	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
41	C₄⋊C₈×C₅₃	—	2	4	no	—
42	C₄⋊C₈×C₅₃	—	2	4	no	—
43	C₄⋊C₈×C₅₃	—	2	4	no	—
44	C₄⋊C₈×C₅₃	—	2	2	no	—
45	C₄⋊C₈×C₅₃	—	2	2	no	—
46	C₄⋊C₈×C₅₃	C₂×C₄	1	4	yes	—
47	C₄⋊C₈×C₅₃	C₂×C₄	1	4	yes	—
48	C₂²⋊C₈×C₅₃	—	2	4	no	—
49	C₄×C₈×C₅₃	—	2	2	no	—
50	C₄×C₈×C₅₃	—	2	2	no	—
51	C₂⁴×C₅₃	C₄⋊C₄	1	1	yes	characteristic
52	C₂²×C₄×C₅₃	—	2	4	no	—
53	C₂²×C₄×C₅₃	—	4	2	no	—
54	C₂²×C₄×C₅₃	C₄²	1	2	yes	—
55	C₂²×C₄×C₅₃	C₂²⋊C₄	1	2	yes	—
56	C₂²×C₄×C₅₃	C₂²⋊C₄	1	2	yes	—
57	C₂²×C₄×C₅₃	C₂²⋊C₄	1	2	yes	—
58	C₂²×C₄×C₅₃	C₄⋊C₄	1	1	yes	characteristic
59	C₂²×C₄×C₅₃	C₄⋊C₄	1	1	yes	characteristic
60	C₂²×C₄×C₅₃	C₄⋊C₄	1	2	yes	—
61	C₂²×C₄×C₅₃	C₄⋊C₄	1	2	yes	—
62	C₂²×C₄×C₅₃	C₄⋊C₄	1	1	yes	characteristic
63	C₂²×C₄×C₅₃	D₈	1	2	yes	—
64	C₂²×C₄×C₅₃	SD₁₆	1	2	yes	—
65	C₂²×C₄×C₅₃	SD₁₆	1	2	yes	—
66	C₂²×C₄×C₅₃	Q₁₆	1	2	yes	—
67	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
68	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
69	C₂²×C₄×C₅₃	C₂×Q₈	1	1	yes	characteristic
70	M₁₆×C₅₃	—	4	8	no	—
71	C₂×C₈×C₅₃	—	2	4	no	—
72	C₂×C₈×C₅₃	—	2	4	no	—
73	C₂×C₈×C₅₃	—	2	4	no	—
74	C₂×C₈×C₅₃	—	2	8	no	—
75	C₂×C₈×C₅₃	—	4	8	no	—
76	C₂×C₈×C₅₃	—	4	2	no	—
77	C₂×C₈×C₅₃	—	4	2	no	—
78	C₂×C₈×C₅₃	—	4	2	no	—
79	C₂×C₈×C₅₃	—	4	2	no	—
80	C₄²×C₅₃	C₄²	1	2	yes	—
81	C₄²×C₅₃	C₂²⋊C₄	1	2	yes	—
82	C₄²×C₅₃	C₂²⋊C₄	1	2	yes	—
83	C₄²×C₅₃	C₂²⋊C₄	1	2	yes	—
84	C₄²×C₅₃	C₄⋊C₄	1	2	yes	—
85	C₄²×C₅₃	C₄⋊C₄	1	2	yes	—
86	C₄²×C₅₃	C₂²×C₄	1	1	yes	characteristic
87	C₄²×C₅₃	C₂²×C₄	1	1	yes	characteristic
88	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
89	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
90	C₂³×C₅₃	—	2	2	no	—
91	C₂³×C₅₃	—	2	2	no	—
92	C₂³×C₅₃	#95	1	2	yes	—
93	C₂³×C₅₃	#106	1	2	yes	—
94	C₂³×C₅₃	#107	1	2	yes	—
95	C₂³×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
96	C₂×C₄×C₅₃	—	2	2	no	—
97	C₂×C₄×C₅₃	—	2	2	no	—
98	C₂×C₄×C₅₃	—	2	2	no	—
99	C₂×C₄×C₅₃	—	2	4	no	—
100	C₂×C₄×C₅₃	—	2	4	no	—
101	C₂×C₄×C₅₃	—	4	2	no	—
102	C₂×C₄×C₅₃	—	4	2	no	—
103	C₂×C₄×C₅₃	#95	1	2	yes	—
104	C₂×C₄×C₅₃	#95	1	2	yes	—
105	C₂×C₄×C₅₃	#95	1	2	yes	—
106	C₂×C₄×C₅₃	#102	1	4	yes	—
107	C₂×C₄×C₅₃	#102	1	4	yes	—
108	C₂×C₄×C₅₃	#103	1	4	yes	—
109	C₂×C₄×C₅₃	#103	1	4	yes	—
110	C₂×C₄×C₅₃	#106	1	2	yes	—
111	C₂×C₄×C₅₃	#107	1	2	yes	—
112	C₂×C₄×C₅₃	C₂×C₄²	1	1	yes	characteristic
113	C₂×C₄×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
114	C₂×C₄×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
115	C₂×C₄×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
116	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
117	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
118	C₂×C₄×C₅₃	#131	1	4	yes	—
119	C₂×C₄×C₅₃	C₂×D₈	1	1	yes	characteristic
120	C₂×C₄×C₅₃	C₂×SD₁₆	1	1	yes	characteristic
121	C₂×C₄×C₅₃	C₂×SD₁₆	1	1	yes	characteristic
122	C₂×C₄×C₅₃	C₂×Q₁₆	1	1	yes	characteristic
123	C₈×C₅₃	—	4	4	no	—
124	C₈×C₅₃	—	4	4	no	—
125	C₈×C₅₃	—	4	4	no	—
126	#7877	C₅₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
127	C₂²×C₅₃	—	2	4	no	—
128	C₂²×C₅₃	—	2	4	no	—
129	C₂²×C₅₃	—	2	2	no	—
130	C₂²×C₅₃	—	2	2	no	—
131	C₂²×C₅₃	#340	1	4	yes	—
132	C₂²×C₅₃	C₂×#95	1	1	yes	characteristic
133	C₂²×C₅₃	#424	1	2	yes	—
134	C₂²×C₅₃	C₂×#106	1	1	yes	characteristic
135	C₂²×C₅₃	C₂×#107	1	1	yes	characteristic
136	C₂²×C₅₃	#430	1	2	yes	—
137	C₄×C₅₃	—	2	2	no	—
138	C₄×C₅₃	—	2	2	no	—
139	C₄×C₅₃	—	2	1	no	—
140	C₄×C₅₃	—	2	1	no	—
141	C₄×C₅₃	#340	1	4	yes	—
142	C₄×C₅₃	#405	1	2	yes	—
143	C₄×C₅₃	#408	1	2	yes	—
144	C₄×C₅₃	C₂×#102	1	1	yes	characteristic
145	C₄×C₅₃	C₂×#102	1	1	yes	characteristic
146	C₄×C₅₃	C₂×#103	1	1	yes	characteristic
147	C₄×C₅₃	C₂×#103	1	1	yes	characteristic
148	C₂×#431	C₂×C₅₃	1	1	yes	characteristic
149	C₂×#423	C₂×C₅₃	1	1	yes	characteristic
150	C₂²×C₄⋊C₈	C₂×C₅₃	1	1	yes	characteristic
151	#1277	C₂×C₅₃	1	4	yes	complemented
152	C₂×C₅₃	—	2	4	no	—
153	C₂×C₅₃	#1570	1	2	yes	—
154	C₂×C₅₃	#1571	1	2	yes	—
155	C₂×C₅₃	#1729	1	1	yes	characteristic
156	C₂×C₅₃	C₂×#340	1	1	yes	characteristic
157	C₂×C₅₃	#1738	1	1	yes	characteristic
158	C₂²×M₁₆	—	2	1	no	—
159	C₂³×C₈	—	2	1	no	—
160	C₂²×C₄²	C₂²×C₅₃	1	1	yes	characteristic
161	#431	—	2	2	no	—
162	#431	—	2	2	no	—
163	#423	—	2	2	no	—
164	#423	C₄×C₅₃	1	4	yes	—
165	C₂×C₄⋊C₈	—	2	2	no	—
166	C₂×C₄⋊C₈	—	2	1	no	—
167	C₂×C₄⋊C₈	—	2	1	no	—
168	C₂×C₄⋊C₈	C₄×C₅₃	1	2	yes	—
169	C₂×C₄⋊C₈	C₄×C₅₃	1	4	yes	—
170	C₂×C₄⋊C₈	C₂²×C₅₃	1	2	yes	—
171	C₂×C₄⋊C₈	C₂²×C₅₃	1	2	yes	—
172	C₂×C₂²⋊C₈	C₄×C₅₃	1	2	yes	—
173	C₂×C₄×C₈	C₂²×C₅₃	1	2	yes	—
174	C₅₃	#7877	1	1	yes	characteristic, direct product factor, Sylow
175	C₂³×C₄	C₂×C₄×C₅₃	1	1	yes	characteristic
176	C₂³×C₄	D₄×C₅₃	1	1	yes	characteristic
177	C₂³×C₄	Q₈×C₅₃	1	1	yes	characteristic
178	C₂×M₁₆	—	2	4	no	—
179	C₂×M₁₆	—	4	2	no	—
180	C₂×M₁₆	—	4	2	no	—
181	C₂²×C₈	—	2	2	no	—
182	C₂²×C₈	—	2	2	no	—
183	C₂²×C₈	—	2	4	no	—
184	C₂²×C₈	—	4	2	no	—
185	C₂²×C₈	—	4	2	no	—
186	C₂²×C₈	C₂×C₄×C₅₃	1	4	yes	—
187	C₂×C₄²	C₂×C₄×C₅₃	1	2	yes	—
188	C₂×C₄²	C₂×C₄×C₅₃	1	2	yes	—
189	C₂×C₄²	C₂×C₄×C₅₃	1	1	yes	characteristic
190	C₂×C₄²	D₄×C₅₃	1	2	yes	—
191	C₂×C₄²	D₄×C₅₃	1	2	yes	—
192	C₂×C₄²	D₄×C₅₃	1	1	yes	characteristic
193	C₂×C₄²	Q₈×C₅₃	1	1	yes	characteristic
194	C₂×C₄²	C₂³×C₅₃	1	1	yes	characteristic
195	C₄⋊C₈	—	2	2	no	—
196	C₄⋊C₈	—	2	2	no	—
197	C₄⋊C₈	—	2	4	no	—
198	C₄⋊C₈	—	2	4	no	—
199	C₄⋊C₈	—	2	4	no	—
200	C₄⋊C₈	C₂×C₄×C₅₃	1	4	yes	—
201	C₄⋊C₈	C₂×C₄×C₅₃	1	4	yes	—
202	C₂²⋊C₈	—	2	4	no	—
203	C₄×C₈	—	2	2	no	—
204	C₄×C₈	—	2	2	no	—
205	C₂⁴	C₄⋊C₄×C₅₃	1	1	yes	characteristic
206	C₂²×C₄	—	2	4	no	—
207	C₂²×C₄	—	4	2	no	—
208	C₂²×C₄	C₄²×C₅₃	1	2	yes	—
209	C₂²×C₄	C₂²⋊C₄×C₅₃	1	2	yes	—
210	C₂²×C₄	C₂²⋊C₄×C₅₃	1	2	yes	—
211	C₂²×C₄	C₂²⋊C₄×C₅₃	1	2	yes	—
212	C₂²×C₄	C₄⋊C₄×C₅₃	1	1	yes	characteristic
213	C₂²×C₄	C₄⋊C₄×C₅₃	1	1	yes	characteristic
214	C₂²×C₄	C₄⋊C₄×C₅₃	1	2	yes	—
215	C₂²×C₄	C₄⋊C₄×C₅₃	1	2	yes	—
216	C₂²×C₄	C₄⋊C₄×C₅₃	1	1	yes	characteristic
217	C₂²×C₄	D₈×C₅₃	1	2	yes	—
218	C₂²×C₄	SD₁₆×C₅₃	1	2	yes	—
219	C₂²×C₄	SD₁₆×C₅₃	1	2	yes	—
220	C₂²×C₄	Q₁₆×C₅₃	1	2	yes	—
221	C₂²×C₄	C₂²×C₄×C₅₃	1	1	yes	characteristic
222	C₂²×C₄	C₂×D₄×C₅₃	1	1	yes	characteristic
223	C₂²×C₄	C₂×Q₈×C₅₃	1	1	yes	characteristic
224	M₁₆	—	4	8	no	—
225	C₂×C₈	—	2	4	no	—
226	C₂×C₈	—	2	4	no	—
227	C₂×C₈	—	2	4	no	—
228	C₂×C₈	—	2	8	no	—
229	C₂×C₈	—	4	8	no	—
230	C₂×C₈	—	4	2	no	—
231	C₂×C₈	—	4	2	no	—
232	C₂×C₈	—	4	2	no	—
233	C₂×C₈	—	4	2	no	—
234	C₄²	C₄²×C₅₃	1	2	yes	—
235	C₄²	C₂²⋊C₄×C₅₃	1	2	yes	—
236	C₄²	C₂²⋊C₄×C₅₃	1	2	yes	—
237	C₄²	C₂²⋊C₄×C₅₃	1	2	yes	—
238	C₄²	C₄⋊C₄×C₅₃	1	2	yes	—
239	C₄²	C₄⋊C₄×C₅₃	1	2	yes	—
240	C₄²	C₂²×C₄×C₅₃	1	1	yes	characteristic
241	C₄²	C₂²×C₄×C₅₃	1	1	yes	characteristic
242	C₄²	C₂×D₄×C₅₃	1	1	yes	characteristic
243	C₄²	C₂×D₄×C₅₃	1	1	yes	characteristic
244	C₂³	—	2	2	no	—
245	C₂³	—	2	2	no	—
246	C₂³	#95×C₅₃	1	2	yes	—
247	C₂³	#106×C₅₃	1	2	yes	—
248	C₂³	#107×C₅₃	1	2	yes	—
249	C₂³	C₂×C₄⋊C₄×C₅₃	1	1	yes	characteristic
250	C₂×C₄	—	2	2	no	—
251	C₂×C₄	—	2	2	no	—
252	C₂×C₄	—	2	2	no	—
253	C₂×C₄	—	2	4	no	—
254	C₂×C₄	—	2	4	no	—
255	C₂×C₄	—	4	2	no	—
256	C₂×C₄	—	4	2	no	—
257	C₂×C₄	#95×C₅₃	1	2	yes	—
258	C₂×C₄	#95×C₅₃	1	2	yes	—
259	C₂×C₄	#95×C₅₃	1	2	yes	—
260	C₂×C₄	#102×C₅₃	1	4	yes	—
261	C₂×C₄	#102×C₅₃	1	4	yes	—
262	C₂×C₄	#103×C₅₃	1	4	yes	—
263	C₂×C₄	#103×C₅₃	1	4	yes	—
264	C₂×C₄	#106×C₅₃	1	2	yes	—
265	C₂×C₄	#107×C₅₃	1	2	yes	—
266	C₂×C₄	C₂×C₄²×C₅₃	1	1	yes	characteristic
267	C₂×C₄	C₂×C₂²⋊C₄×C₅₃	1	1	yes	characteristic
268	C₂×C₄	C₂×C₂²⋊C₄×C₅₃	1	1	yes	characteristic
269	C₂×C₄	C₂×C₂²⋊C₄×C₅₃	1	1	yes	characteristic
270	C₂×C₄	C₂×C₄⋊C₄×C₅₃	1	1	yes	characteristic
271	C₂×C₄	C₂×C₄⋊C₄×C₅₃	1	1	yes	characteristic
272	C₂×C₄	#131×C₅₃	1	4	yes	—
273	C₂×C₄	C₂×D₈×C₅₃	1	1	yes	characteristic
274	C₂×C₄	C₂×SD₁₆×C₅₃	1	1	yes	characteristic
275	C₂×C₄	C₂×SD₁₆×C₅₃	1	1	yes	characteristic
276	C₂×C₄	C₂×Q₁₆×C₅₃	1	1	yes	characteristic
277	C₈	—	4	4	no	—
278	C₈	—	4	4	no	—
279	C₈	—	4	4	no	—
280	C₂²	—	2	2	no	—
281	C₂²	—	2	2	no	—
282	C₂²	—	2	4	no	—
283	C₂²	—	2	4	no	—
284	C₂²	C₅₃×#340	1	4	yes	—
285	C₂²	C₂×#95×C₅₃	1	1	yes	characteristic
286	C₂²	C₅₃×#424	1	2	yes	—
287	C₂²	C₂×#106×C₅₃	1	1	yes	characteristic
288	C₂²	C₂×#107×C₅₃	1	1	yes	characteristic
289	C₂²	C₅₃×#430	1	2	yes	—
290	C₄	—	2	2	no	—
291	C₄	—	2	2	no	—
292	C₄	—	2	1	no	—
293	C₄	—	2	1	no	—
294	C₄	C₅₃×#340	1	4	yes	—
295	C₄	C₅₃×#405	1	2	yes	—
296	C₄	C₅₃×#408	1	2	yes	—
297	C₄	C₂×#102×C₅₃	1	1	yes	characteristic
298	C₄	C₂×#102×C₅₃	1	1	yes	characteristic
299	C₄	C₂×#103×C₅₃	1	1	yes	characteristic
300	C₄	C₂×#103×C₅₃	1	1	yes	characteristic
301	C₂	—	2	4	no	—
302	C₂	C₅₃×#1570	1	2	yes	—
303	C₂	C₅₃×#1571	1	2	yes	—
304	C₂	C₅₃×#1729	1	1	yes	characteristic
305	C₂	C₂×C₅₃×#340	1	1	yes	characteristic
306	C₂	C₅₃×#1738	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×C₄⋊C₄
prime radical	C₂⁴×C₅₃	C₄⋊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×#1277	C₂	semidirect
126. #7877	C₅₃	direct (N × H)
151. #1277	C₂×C₅₃	semidirect
174. C₅₃	#7877	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×#431	2
C₂×C₅₃×#423	2
C₂²×C₄⋊C₈×C₅₃	2
C₅₃×#1277	2
#7877	53

Subgroups — C53×#7877 #27215389

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅₃×#7877 #27215389