Subgroups — C₅₃×#10721 #27218233

Order 13568.

1646 conjugacy classes of subgroups.

The subgroup lattice has 310 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂⁴×C₈×C₅₃	C₂	1	1	yes	maximal, characteristic
2	C₂×C₅₃×#380	C₂	1	2	yes	maximal
3	C₂×C₅₃×#340	C₂	1	4	yes	maximal, complemented
4	C₂⁴×C₄×C₅₃	C₂²	1	1	yes	characteristic
5	C₂³×C₈×C₅₃	—	2	4	no	—
6	C₂³×C₈×C₅₃	—	2	8	no	—
7	C₂³×C₈×C₅₃	C₄	1	4	yes	complemented
8	C₂³×C₈×C₅₃	C₂²	1	2	yes	—
9	C₂²×C₄⋊C₄×C₅₃	C₂²	1	4	yes	—
10	C₂²×C₂²⋊C₄×C₅₃	—	2	2	no	—
11	C₅₃×#380	—	2	4	no	—
12	C₅₃×#380	C₄	1	8	yes	complemented
13	C₂×#95×C₅₃	—	2	2	no	—
14	C₅₃×#340	—	2	8	no	—
15	C₂⁵×C₅₃	D₄	1	1	yes	characteristic
16	C₂³×C₄×C₅₃	—	2	4	no	—
17	C₂³×C₄×C₅₃	—	2	4	no	—
18	C₂³×C₄×C₅₃	—	2	4	no	—
19	C₂³×C₄×C₅₃	C₂×C₄	1	2	yes	—
20	C₂³×C₄×C₅₃	C₂×C₄	1	4	yes	—
21	C₂³×C₄×C₅₃	D₄	1	1	yes	characteristic
22	C₂³×C₄×C₅₃	D₄	1	4	yes	—
23	C₂³×C₄×C₅₃	Q₈	1	2	yes	—
24	C₂³×C₄×C₅₃	C₂³	1	1	yes	characteristic
25	C₂²×C₈×C₅₃	—	2	32	no	—
26	C₂²×C₈×C₅₃	—	2	16	no	—
27	C₂²×C₈×C₅₃	—	2	8	no	—
28	C₂²×C₈×C₅₃	—	2	4	no	—
29	C₂²×C₈×C₅₃	—	2	4	no	—
30	C₂²×C₈×C₅₃	C₂×C₄	1	4	yes	—
31	C₂²×C₈×C₅₃	D₄	1	8	yes	—
32	C₂×C₄⋊C₄×C₅₃	—	2	4	no	—
33	C₂×C₄⋊C₄×C₅₃	—	2	8	no	—
34	C₂×C₄⋊C₄×C₅₃	—	2	8	no	—
35	C₂×C₄⋊C₄×C₅₃	C₂×C₄	1	8	yes	—
36	C₂×C₂²⋊C₄×C₅₃	—	2	8	no	—
37	C₂×C₂²⋊C₄×C₅₃	—	4	4	no	—
38	C₂×C₂²⋊C₄×C₅₃	—	4	4	no	—
39	#95×C₅₃	—	2	8	no	—
40	C₂⁴×C₅₃	—	2	4	no	—
41	C₂⁴×C₅₃	—	2	4	no	—
42	C₂⁴×C₅₃	C₂²⋊C₄	1	4	yes	—
43	C₂⁴×C₅₃	C₄⋊C₄	1	2	yes	—
44	C₂⁴×C₅₃	D₈	1	2	yes	—
45	C₂⁴×C₅₃	SD₁₆	1	4	yes	—
46	C₂⁴×C₅₃	Q₁₆	1	2	yes	—
47	C₂⁴×C₅₃	C₂×D₄	1	1	yes	characteristic
48	C₂²×C₄×C₅₃	—	2	4	no	—
49	C₂²×C₄×C₅₃	—	2	4	no	—
50	C₂²×C₄×C₅₃	—	2	4	no	—
51	C₂²×C₄×C₅₃	—	2	16	no	—
52	C₂²×C₄×C₅₃	—	2	8	no	—
53	C₂²×C₄×C₅₃	—	2	8	no	—
54	C₂²×C₄×C₅₃	—	2	8	no	—
55	C₂²×C₄×C₅₃	—	2	8	no	—
56	C₂²×C₄×C₅₃	—	4	8	no	—
57	C₂²×C₄×C₅₃	—	4	4	no	—
58	C₂²×C₄×C₅₃	—	4	4	no	—
59	C₂²×C₄×C₅₃	—	4	8	no	—
60	C₂²×C₄×C₅₃	C₄²	1	4	yes	—
61	C₂²×C₄×C₅₃	C₂²⋊C₄	1	4	yes	—
62	C₂²×C₄×C₅₃	C₂²⋊C₄	1	4	yes	—
63	C₂²×C₄×C₅₃	C₄⋊C₄	1	2	yes	—
64	C₂²×C₄×C₅₃	C₄⋊C₄	1	8	yes	—
65	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
66	C₂²×C₄×C₅₃	C₂²×C₄	1	2	yes	—
67	C₂²×C₄×C₅₃	C₂×D₄	1	2	yes	—
68	C₂²×C₄×C₅₃	C₂×D₄	1	4	yes	—
69	C₂²×C₄×C₅₃	C₂×Q₈	1	1	yes	characteristic
70	C₂²×C₄×C₅₃	#41	1	4	yes	—
71	C₂×C₈×C₅₃	—	2	32	no	—
72	C₂×C₈×C₅₃	—	2	4	no	—
73	C₂×C₈×C₅₃	—	2	4	no	—
74	C₂×C₈×C₅₃	—	2	16	no	—
75	C₂×C₈×C₅₃	C₂²⋊C₄	1	8	yes	complemented
76	C₄⋊C₄×C₅₃	—	2	16	no	—
77	C₄⋊C₄×C₅₃	—	4	4	no	—
78	C₄⋊C₄×C₅₃	—	4	4	no	—
79	C₂²⋊C₄×C₅₃	—	4	16	no	—
80	C₂³×C₅₃	—	2	4	no	—
81	C₂³×C₅₃	—	2	8	no	—
82	C₂³×C₅₃	—	2	8	no	—
83	C₂³×C₅₃	—	2	4	no	—
84	C₂³×C₅₃	—	2	16	no	—
85	C₂³×C₅₃	—	2	8	no	—
86	C₂³×C₅₃	—	2	8	no	—
87	C₂³×C₅₃	#95	1	4	yes	—
88	C₂³×C₅₃	#102	1	8	yes	—
89	C₂³×C₅₃	#103	1	8	yes	—
90	C₂³×C₅₃	#106	1	4	yes	—
91	C₂³×C₅₃	#107	1	4	yes	—
92	C₂³×C₅₃	C₂×C₂²⋊C₄	1	2	yes	—
93	C₂³×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
94	C₂³×C₅₃	#121	1	4	yes	—
95	C₂³×C₅₃	C₂×D₈	1	1	yes	characteristic
96	C₂³×C₅₃	C₂×SD₁₆	1	2	yes	—
97	C₂³×C₅₃	C₂×Q₁₆	1	1	yes	characteristic
98	C₂³×C₅₃	#135	1	4	yes	—
99	C₂×C₄×C₅₃	—	2	4	no	—
100	C₂×C₄×C₅₃	—	2	8	no	—
101	C₂×C₄×C₅₃	—	2	8	no	—
102	C₂×C₄×C₅₃	—	2	4	no	—
103	C₂×C₄×C₅₃	—	2	4	no	—
104	C₂×C₄×C₅₃	—	2	16	no	—
105	C₂×C₄×C₅₃	—	4	8	no	—
106	C₂×C₄×C₅₃	—	4	8	no	—
107	C₂×C₄×C₅₃	—	4	8	no	—
108	C₂×C₄×C₅₃	—	8	8	no	—
109	C₂×C₄×C₅₃	—	8	8	no	—
110	C₂×C₄×C₅₃	#95	1	4	yes	—
111	C₂×C₄×C₅₃	C₂×C₄²	1	1	yes	characteristic
112	C₂×C₄×C₅₃	C₂×C₂²⋊C₄	1	2	yes	—
113	C₂×C₄×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
114	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	2	yes	—
115	C₂×C₄×C₅₃	#117	1	2	yes	—
116	C₂×C₄×C₅₃	C₄×D₄	1	8	yes	—
117	C₂×C₄×C₅₃	#120	1	4	yes	—
118	C₂×C₄×C₅₃	C₂²⋊Q₈	1	4	yes	—
119	C₂×C₄×C₅₃	#123	1	4	yes	—
120	C₈×C₅₃	—	2	8	no	—
121	#10721	C₅₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
122	C₂²×C₅₃	—	2	8	no	—
123	C₂²×C₅₃	—	2	8	no	—
124	C₂²×C₅₃	—	2	16	no	—
125	C₂²×C₅₃	—	2	16	no	—
126	C₂²×C₅₃	—	2	4	no	—
127	C₂²×C₅₃	—	2	4	no	—
128	C₂²×C₅₃	#340	1	8	yes	complemented
129	C₂²×C₅₃	C₂×#95	1	1	yes	characteristic
130	C₂²×C₅₃	#380	1	2	yes	—
131	C₂²×C₅₃	#386	1	4	yes	—
132	C₂²×C₅₃	C₂×#102	1	2	yes	—
133	C₂²×C₅₃	C₂×#103	1	2	yes	—
134	C₂²×C₅₃	#416	1	4	yes	—
135	C₂²×C₅₃	C₂×#106	1	1	yes	characteristic
136	C₂²×C₅₃	C₂×#107	1	1	yes	characteristic
137	C₂²×C₅₃	#427	1	2	yes	—
138	C₂²×C₅₃	#465	1	8	yes	—
139	C₂²×C₅₃	#466	1	4	yes	—
140	C₂²×C₅₃	#467	1	4	yes	—
141	C₄×C₅₃	—	2	4	no	—
142	C₄×C₅₃	—	8	8	no	—
143	C₄×C₅₃	C₄×C₂²⋊C₄	1	2	yes	—
144	C₄×C₅₃	#379	1	1	yes	characteristic
145	C₄×C₅₃	#381	1	1	yes	characteristic
146	C₄×C₅₃	#385	1	4	yes	—
147	C₂⁴×C₈	C₂×C₅₃	1	1	yes	characteristic
148	C₂×#380	C₂×C₅₃	1	2	yes	—
149	C₂×#340	C₂×C₅₃	1	4	yes	complemented
150	C₂×C₅₃	—	2	8	no	—
151	C₂×C₅₃	C₂²⋊#95	1	1	yes	characteristic
152	C₂×C₅₃	C₂×#340	1	1	yes	characteristic
153	C₂×C₅₃	#1735	1	1	yes	characteristic
154	C₂×C₅₃	#1807	1	2	yes	—
155	C₂×C₅₃	#1808	1	2	yes	—
156	C₂×C₅₃	#1892	1	4	yes	—
157	C₂×C₅₃	#1893	1	4	yes	—
158	C₂⁴×C₄	C₂²×C₅₃	1	1	yes	characteristic
159	C₂³×C₈	—	2	4	no	—
160	C₂³×C₈	—	2	8	no	—
161	C₂³×C₈	C₄×C₅₃	1	4	yes	complemented
162	C₂³×C₈	C₂²×C₅₃	1	2	yes	—
163	C₂²×C₄⋊C₄	C₂²×C₅₃	1	4	yes	—
164	C₂²×C₂²⋊C₄	—	2	2	no	—
165	#380	—	2	4	no	—
166	#380	C₄×C₅₃	1	8	yes	complemented
167	C₂×#95	—	2	2	no	—
168	#340	—	2	8	no	—
169	C₅₃	#10721	1	1	yes	characteristic, direct product factor, Sylow
170	C₂⁵	D₄×C₅₃	1	1	yes	characteristic
171	C₂³×C₄	—	2	4	no	—
172	C₂³×C₄	—	2	4	no	—
173	C₂³×C₄	—	2	4	no	—
174	C₂³×C₄	C₂×C₄×C₅₃	1	2	yes	—
175	C₂³×C₄	C₂×C₄×C₅₃	1	4	yes	—
176	C₂³×C₄	D₄×C₅₃	1	4	yes	—
177	C₂³×C₄	D₄×C₅₃	1	1	yes	characteristic
178	C₂³×C₄	Q₈×C₅₃	1	2	yes	—
179	C₂³×C₄	C₂³×C₅₃	1	1	yes	characteristic
180	C₂²×C₈	—	2	32	no	—
181	C₂²×C₈	—	2	4	no	—
182	C₂²×C₈	—	2	4	no	—
183	C₂²×C₈	—	2	8	no	—
184	C₂²×C₈	—	2	16	no	—
185	C₂²×C₈	C₂×C₄×C₅₃	1	4	yes	—
186	C₂²×C₈	D₄×C₅₃	1	8	yes	—
187	C₂×C₄⋊C₄	—	2	8	no	—
188	C₂×C₄⋊C₄	—	2	8	no	—
189	C₂×C₄⋊C₄	—	2	4	no	—
190	C₂×C₄⋊C₄	C₂×C₄×C₅₃	1	8	yes	—
191	C₂×C₂²⋊C₄	—	2	8	no	—
192	C₂×C₂²⋊C₄	—	4	4	no	—
193	C₂×C₂²⋊C₄	—	4	4	no	—
194	#95	—	2	8	no	—
195	C₂⁴	—	2	4	no	—
196	C₂⁴	—	2	4	no	—
197	C₂⁴	C₂²⋊C₄×C₅₃	1	4	yes	—
198	C₂⁴	C₄⋊C₄×C₅₃	1	2	yes	—
199	C₂⁴	D₈×C₅₃	1	2	yes	—
200	C₂⁴	SD₁₆×C₅₃	1	4	yes	—
201	C₂⁴	Q₁₆×C₅₃	1	2	yes	—
202	C₂⁴	C₂×D₄×C₅₃	1	1	yes	characteristic
203	C₂²×C₄	—	2	4	no	—
204	C₂²×C₄	—	2	4	no	—
205	C₂²×C₄	—	2	4	no	—
206	C₂²×C₄	—	2	16	no	—
207	C₂²×C₄	—	2	8	no	—
208	C₂²×C₄	—	2	8	no	—
209	C₂²×C₄	—	2	8	no	—
210	C₂²×C₄	—	2	8	no	—
211	C₂²×C₄	—	4	8	no	—
212	C₂²×C₄	—	4	4	no	—
213	C₂²×C₄	—	4	4	no	—
214	C₂²×C₄	—	4	8	no	—
215	C₂²×C₄	C₄²×C₅₃	1	4	yes	—
216	C₂²×C₄	C₂²⋊C₄×C₅₃	1	4	yes	—
217	C₂²×C₄	C₂²⋊C₄×C₅₃	1	4	yes	—
218	C₂²×C₄	C₄⋊C₄×C₅₃	1	2	yes	—
219	C₂²×C₄	C₄⋊C₄×C₅₃	1	8	yes	—
220	C₂²×C₄	C₂²×C₄×C₅₃	1	1	yes	characteristic
221	C₂²×C₄	C₂²×C₄×C₅₃	1	2	yes	—
222	C₂²×C₄	C₂×D₄×C₅₃	1	2	yes	—
223	C₂²×C₄	C₂×D₄×C₅₃	1	4	yes	—
224	C₂²×C₄	C₂×Q₈×C₅₃	1	1	yes	characteristic
225	C₂²×C₄	#41×C₅₃	1	4	yes	—
226	C₂×C₈	—	2	32	no	—
227	C₂×C₈	—	2	4	no	—
228	C₂×C₈	—	2	4	no	—
229	C₂×C₈	—	2	16	no	—
230	C₂×C₈	C₂²⋊C₄×C₅₃	1	8	yes	complemented
231	C₄⋊C₄	—	2	16	no	—
232	C₄⋊C₄	—	4	4	no	—
233	C₄⋊C₄	—	4	4	no	—
234	C₂²⋊C₄	—	4	16	no	—
235	C₂³	—	2	4	no	—
236	C₂³	—	2	8	no	—
237	C₂³	—	2	8	no	—
238	C₂³	—	2	4	no	—
239	C₂³	—	2	16	no	—
240	C₂³	—	2	8	no	—
241	C₂³	—	2	8	no	—
242	C₂³	#95×C₅₃	1	4	yes	—
243	C₂³	#102×C₅₃	1	8	yes	—
244	C₂³	#103×C₅₃	1	8	yes	—
245	C₂³	#106×C₅₃	1	4	yes	—
246	C₂³	#107×C₅₃	1	4	yes	—
247	C₂³	C₂×C₂²⋊C₄×C₅₃	1	2	yes	—
248	C₂³	C₂×C₄⋊C₄×C₅₃	1	1	yes	characteristic
249	C₂³	#121×C₅₃	1	4	yes	—
250	C₂³	C₂×D₈×C₅₃	1	1	yes	characteristic
251	C₂³	C₂×SD₁₆×C₅₃	1	2	yes	—
252	C₂³	C₂×Q₁₆×C₅₃	1	1	yes	characteristic
253	C₂³	#135×C₅₃	1	4	yes	—
254	C₂×C₄	—	2	4	no	—
255	C₂×C₄	—	2	8	no	—
256	C₂×C₄	—	2	8	no	—
257	C₂×C₄	—	2	4	no	—
258	C₂×C₄	—	2	4	no	—
259	C₂×C₄	—	2	16	no	—
260	C₂×C₄	—	4	8	no	—
261	C₂×C₄	—	4	8	no	—
262	C₂×C₄	—	4	8	no	—
263	C₂×C₄	—	8	8	no	—
264	C₂×C₄	—	8	8	no	—
265	C₂×C₄	#95×C₅₃	1	4	yes	—
266	C₂×C₄	C₂×C₄²×C₅₃	1	1	yes	characteristic
267	C₂×C₄	C₂×C₂²⋊C₄×C₅₃	1	2	yes	—
268	C₂×C₄	C₂×C₂²⋊C₄×C₅₃	1	1	yes	characteristic
269	C₂×C₄	C₂×C₄⋊C₄×C₅₃	1	2	yes	—
270	C₂×C₄	#117×C₅₃	1	2	yes	—
271	C₂×C₄	C₄×D₄×C₅₃	1	8	yes	—
272	C₂×C₄	#120×C₅₃	1	4	yes	—
273	C₂×C₄	C₂²⋊Q₈×C₅₃	1	4	yes	—
274	C₂×C₄	#123×C₅₃	1	4	yes	—
275	C₈	—	2	8	no	—
276	C₂²	—	2	4	no	—
277	C₂²	—	2	4	no	—
278	C₂²	—	2	8	no	—
279	C₂²	—	2	8	no	—
280	C₂²	—	2	16	no	—
281	C₂²	—	2	16	no	—
282	C₂²	C₅₃×#340	1	8	yes	complemented
283	C₂²	C₂×#95×C₅₃	1	1	yes	characteristic
284	C₂²	C₅₃×#380	1	2	yes	—
285	C₂²	C₅₃×#386	1	4	yes	—
286	C₂²	C₂×#102×C₅₃	1	2	yes	—
287	C₂²	C₂×#103×C₅₃	1	2	yes	—
288	C₂²	C₅₃×#416	1	4	yes	—
289	C₂²	C₂×#106×C₅₃	1	1	yes	characteristic
290	C₂²	C₂×#107×C₅₃	1	1	yes	characteristic
291	C₂²	C₅₃×#427	1	2	yes	—
292	C₂²	C₅₃×#465	1	8	yes	—
293	C₂²	C₅₃×#466	1	4	yes	—
294	C₂²	C₅₃×#467	1	4	yes	—
295	C₄	—	2	4	no	—
296	C₄	—	8	8	no	—
297	C₄	C₄×C₂²⋊C₄×C₅₃	1	2	yes	—
298	C₄	C₅₃×#379	1	1	yes	characteristic
299	C₄	C₅₃×#381	1	1	yes	characteristic
300	C₄	C₅₃×#385	1	4	yes	—
301	C₂	—	2	8	no	—
302	C₂	C₅₃×C₂²⋊#95	1	1	yes	characteristic
303	C₂	C₂×C₅₃×#340	1	1	yes	characteristic
304	C₂	C₅₃×#1735	1	1	yes	characteristic
305	C₂	C₅₃×#1807	1	2	yes	—
306	C₂	C₅₃×#1808	1	2	yes	—
307	C₂	C₅₃×#1892	1	4	yes	—
308	C₂	C₅₃×#1893	1	4	yes	—

	subgroup	G/H
center, socle	C₂⁴×C₅₃	C₂×D₄
commutator	C₂×C₄	C₂×C₄²×C₅₃
Frattini, cosocle	C₂³×C₄	C₂³×C₅₃
prime radical	C₂⁵×C₅₃	D₄
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂⁴×C₅₃ ◁ C₂⁴×C₄×C₅₃ ◁ G

Complements

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

N	H ≅ G/N	type
3. C₂×C₅₃×#340	C₂	semidirect
7. C₂³×C₈×C₅₃	C₄	semidirect
12. C₅₃×#380	C₄	semidirect
75. C₂×C₈×C₅₃	C₂²⋊C₄	semidirect
121. #10721	C₅₃	direct (N × H)
128. C₂²×C₅₃	#340	semidirect
149. C₂×#340	C₂×C₅₃	semidirect
161. C₂³×C₈	C₄×C₅₃	semidirect
166. #380	C₄×C₅₃	semidirect
169. C₅₃	#10721	direct (N × H)
230. C₂×C₈	C₂²⋊C₄×C₅₃	semidirect
282. C₂²	C₅₃×#340	semidirect

Maximal subgroups

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

subgroup	index
C₂⁴×C₈×C₅₃	2
C₂×C₅₃×#380	2
C₂×C₅₃×#340	2
#10721	53

Subgroups — C53×#10721 #27218233

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅₃×#10721 #27218233