Subgroups — C₅₃×#10707 #27218219

Order 13568.

886 conjugacy classes of subgroups.

The subgroup lattice has 304 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₅₃×#405	C₂	1	2	yes	maximal
2	C₂²×#97×C₅₃	C₂	1	1	yes	maximal, characteristic
3	C₂×C₅₃×#341	C₂	1	1	yes	maximal, characteristic
4	C₅₃×#1735	C₂	1	1	yes	maximal, characteristic
5	C₂×C₅₃×#339	C₂	1	2	yes	maximal
6	C₂²×M₁₆×C₅₃	C₂²	1	2	yes	—
7	C₂³×C₈×C₅₃	C₂²	1	1	yes	characteristic
8	C₂³×C₈×C₅₃	C₂²	1	1	yes	characteristic
9	C₂×#117×C₅₃	C₂²	1	2	yes	—
10	C₂²×C₄²×C₅₃	C₂²	1	1	yes	characteristic
11	C₅₃×#405	—	2	4	no	—
12	C₅₃×#405	C₄	1	8	yes	complemented
13	C₂×#97×C₅₃	—	2	4	no	—
14	C₂×#97×C₅₃	—	2	2	no	—
15	C₂×#97×C₅₃	C₄	1	4	yes	complemented
16	C₂×C₄×C₈×C₅₃	—	2	2	no	—
17	C₅₃×#341	C₄	1	4	yes	complemented
18	C₅₃×#340	C₄	1	4	yes	complemented
19	C₅₃×#339	C₄	1	8	yes	complemented
20	C₂³×C₄×C₅₃	—	2	1	no	—
21	C₂³×C₄×C₅₃	C₂³	1	1	yes	characteristic
22	C₂×M₁₆×C₅₃	—	2	2	no	—
23	C₂×M₁₆×C₅₃	—	2	2	no	—
24	C₂×M₁₆×C₅₃	—	2	2	no	—
25	C₂×M₁₆×C₅₃	C₂×C₄	1	4	yes	—
26	C₂×M₁₆×C₅₃	C₂×C₄	1	4	yes	—
27	C₂×M₁₆×C₅₃	C₂×C₄	1	4	yes	—
28	C₂²×C₈×C₅₃	—	2	2	no	—
29	C₂²×C₈×C₅₃	—	2	2	no	—
30	C₂²×C₈×C₅₃	—	2	2	no	—
31	C₂²×C₈×C₅₃	C₂×C₄	1	2	yes	—
32	C₂²×C₈×C₅₃	C₂×C₄	1	2	yes	—
33	C₂²×C₈×C₅₃	C₂×C₄	1	2	yes	—
34	C₂²×C₈×C₅₃	C₂×C₄	1	2	yes	—
35	C₂²×C₈×C₅₃	C₂×C₄	1	4	yes	—
36	C₂²×C₈×C₅₃	D₄	1	4	yes	—
37	C₂²×C₈×C₅₃	D₄	1	2	yes	—
38	C₂²×C₈×C₅₃	Q₈	1	2	yes	—
39	#117×C₅₃	—	2	2	no	—
40	#117×C₅₃	—	2	2	no	—
41	#117×C₅₃	C₂×C₄	1	4	yes	—
42	#117×C₅₃	C₂×C₄	1	4	yes	—
43	C₂×C₄⋊C₄×C₅₃	C₂×C₄	1	4	yes	—
44	C₂×C₂²⋊C₄×C₅₃	—	2	2	no	—
45	C₂×C₄²×C₅₃	—	2	2	no	—
46	C₂×C₄²×C₅₃	—	2	2	no	—
47	C₂×C₄²×C₅₃	—	2	1	no	—
48	C₂×C₄²×C₅₃	C₂×C₄	1	4	yes	—
49	C₂×C₄²×C₅₃	C₂×C₄	1	2	yes	—
50	#97×C₅₃	—	2	8	no	—
51	#97×C₅₃	—	2	8	no	—
52	C₄×C₈×C₅₃	—	2	8	no	—
53	C₂⁴×C₅₃	#41	1	1	yes	characteristic
54	C₂²×C₄×C₅₃	—	2	2	no	—
55	C₂²×C₄×C₅₃	—	2	4	no	—
56	C₂²×C₄×C₅₃	—	2	4	no	—
57	C₂²×C₄×C₅₃	—	4	4	no	—
58	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
59	C₂²×C₄×C₅₃	C₂²×C₄	1	2	yes	—
60	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
61	C₂²×C₄×C₅₃	C₂²×C₄	1	2	yes	—
62	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
63	C₂²×C₄×C₅₃	C₂×D₄	1	2	yes	—
64	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
65	C₂²×C₄×C₅₃	C₂×Q₈	1	1	yes	characteristic
66	C₂²×C₄×C₅₃	#41	1	1	yes	characteristic
67	C₂²×C₄×C₅₃	#41	1	2	yes	—
68	M₁₆×C₅₃	—	2	4	no	—
69	M₁₆×C₅₃	—	2	4	no	—
70	M₁₆×C₅₃	—	2	4	no	—
71	M₁₆×C₅₃	C₄²	1	8	yes	—
72	C₂×C₈×C₅₃	—	2	4	no	—
73	C₂×C₈×C₅₃	—	2	2	no	—
74	C₂×C₈×C₅₃	—	2	2	no	—
75	C₂×C₈×C₅₃	—	2	4	no	—
76	C₂×C₈×C₅₃	—	2	4	no	—
77	C₂×C₈×C₅₃	—	2	4	no	—
78	C₂×C₈×C₅₃	—	2	4	no	—
79	C₂×C₈×C₅₃	C₄²	1	4	yes	—
80	C₂×C₈×C₅₃	C₄²	1	4	yes	—
81	C₂×C₈×C₅₃	C₂²⋊C₄	1	4	yes	—
82	C₂×C₈×C₅₃	C₂²⋊C₄	1	4	yes	—
83	C₂×C₈×C₅₃	C₂²⋊C₄	1	4	yes	—
84	C₂×C₈×C₅₃	C₄⋊C₄	1	8	yes	—
85	C₂×C₈×C₅₃	C₄⋊C₄	1	4	yes	—
86	C₄⋊C₄×C₅₃	—	2	4	no	—
87	C₄⋊C₄×C₅₃	C₄²	1	8	yes	—
88	C₂²⋊C₄×C₅₃	—	2	4	no	—
89	C₂²⋊C₄×C₅₃	—	2	4	no	—
90	C₄²×C₅₃	—	2	4	no	—
91	C₄²×C₅₃	—	2	2	no	—
92	C₄²×C₅₃	—	2	4	no	—
93	C₄²×C₅₃	—	2	4	no	—
94	C₄²×C₅₃	C₄²	1	4	yes	—
95	C₂³×C₅₃	—	2	4	no	—
96	C₂³×C₅₃	#117	1	1	yes	characteristic
97	C₂³×C₅₃	#117	1	2	yes	—
98	C₂³×C₅₃	C₄×D₄	1	2	yes	—
99	C₂³×C₅₃	C₄×D₄	1	1	yes	characteristic
100	C₂³×C₅₃	C₄×Q₈	1	1	yes	characteristic
101	C₂×C₄×C₅₃	—	2	4	no	—
102	C₂×C₄×C₅₃	—	2	8	no	—
103	C₂×C₄×C₅₃	—	4	8	no	—
104	C₂×C₄×C₅₃	—	4	4	no	—
105	C₂×C₄×C₅₃	—	4	8	no	—
106	C₂×C₄×C₅₃	C₂×C₄²	1	1	yes	characteristic
107	C₂×C₄×C₅₃	C₂×C₄²	1	1	yes	characteristic
108	C₂×C₄×C₅₃	C₂×C₄²	1	2	yes	—
109	C₂×C₄×C₅₃	C₂×C₄²	1	1	yes	characteristic
110	C₂×C₄×C₅₃	C₂×C₄²	1	2	yes	—
111	C₂×C₄×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
112	C₂×C₄×C₅₃	C₂×C₂²⋊C₄	1	2	yes	—
113	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	2	yes	—
114	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
115	C₂×C₄×C₅₃	#117	1	1	yes	characteristic
116	C₂×C₄×C₅₃	#117	1	2	yes	—
117	C₂×C₄×C₅₃	C₄×D₄	1	2	yes	—
118	C₂×C₄×C₅₃	C₄×D₄	1	1	yes	characteristic
119	C₂×C₄×C₅₃	C₄×D₄	1	2	yes	—
120	C₂×C₄×C₅₃	C₄×D₄	1	2	yes	—
121	C₂×C₄×C₅₃	C₄×D₄	1	2	yes	—
122	C₂×C₄×C₅₃	C₄×Q₈	1	2	yes	—
123	C₂×C₄×C₅₃	C₄×Q₈	1	1	yes	characteristic
124	C₈×C₅₃	—	2	4	no	—
125	C₈×C₅₃	—	2	8	no	—
126	C₈×C₅₃	#95	1	8	yes	—
127	#10707	C₅₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
128	C₂²×C₅₃	—	2	4	no	—
129	C₂²×C₅₃	—	2	4	no	—
130	C₂²×C₅₃	—	2	2	no	—
131	C₂²×C₅₃	—	2	2	no	—
132	C₂²×C₅₃	#376	1	1	yes	characteristic
133	C₂²×C₅₃	C₄×C₂²⋊C₄	1	1	yes	characteristic
134	C₂²×C₅₃	C₄×C₂²⋊C₄	1	2	yes	—
135	C₂²×C₅₃	C₄×C₄⋊C₄	1	2	yes	—
136	C₂²×C₅₃	C₄×C₄⋊C₄	1	1	yes	characteristic
137	C₂²×C₅₃	#444	1	4	yes	—
138	C₄×C₅₃	—	4	4	no	—
139	C₄×C₅₃	—	4	8	no	—
140	C₄×C₅₃	C₄³	1	1	yes	characteristic
141	C₄×C₅₃	C₂×#95	1	1	yes	characteristic
142	C₄×C₅₃	C₄×C₂²⋊C₄	1	1	yes	characteristic
143	C₄×C₅₃	C₄×C₂²⋊C₄	1	2	yes	—
144	C₄×C₅₃	C₄×C₄⋊C₄	1	1	yes	characteristic
145	C₄×C₅₃	C₄×C₄⋊C₄	1	2	yes	—
146	C₂×#405	C₂×C₅₃	1	2	yes	—
147	C₂²×#97	C₂×C₅₃	1	1	yes	characteristic
148	C₂×#341	C₂×C₅₃	1	1	yes	characteristic
149	#1735	C₂×C₅₃	1	1	yes	characteristic
150	C₂×#339	C₂×C₅₃	1	2	yes	—
151	C₂×C₅₃	—	2	1	no	—
152	C₂×C₅₃	—	2	1	no	—
153	C₂×C₅₃	—	2	2	no	—
154	C₂×C₅₃	C₄×#95	1	1	yes	characteristic
155	C₂×C₅₃	#1765	1	4	yes	—
156	C₂×C₅₃	#1773	1	2	yes	—
157	C₂²×M₁₆	C₂²×C₅₃	1	2	yes	—
158	C₂³×C₈	C₂²×C₅₃	1	1	yes	characteristic
159	C₂³×C₈	C₂²×C₅₃	1	1	yes	characteristic
160	C₂×#117	C₂²×C₅₃	1	2	yes	—
161	C₂²×C₄²	C₂²×C₅₃	1	1	yes	characteristic
162	#405	—	2	4	no	—
163	#405	C₄×C₅₃	1	8	yes	complemented
164	C₂×#97	—	2	4	no	—
165	C₂×#97	—	2	2	no	—
166	C₂×#97	C₄×C₅₃	1	4	yes	complemented
167	C₂×C₄×C₈	—	2	2	no	—
168	#341	C₄×C₅₃	1	4	yes	complemented
169	#340	C₄×C₅₃	1	4	yes	complemented
170	#339	C₄×C₅₃	1	8	yes	complemented
171	C₅₃	#10707	1	1	yes	characteristic, direct product factor, Sylow
172	C₂³×C₄	—	2	1	no	—
173	C₂³×C₄	C₂³×C₅₃	1	1	yes	characteristic
174	C₂×M₁₆	—	2	2	no	—
175	C₂×M₁₆	—	2	2	no	—
176	C₂×M₁₆	—	2	2	no	—
177	C₂×M₁₆	C₂×C₄×C₅₃	1	4	yes	—
178	C₂×M₁₆	C₂×C₄×C₅₃	1	4	yes	—
179	C₂×M₁₆	C₂×C₄×C₅₃	1	4	yes	—
180	C₂²×C₈	—	2	2	no	—
181	C₂²×C₈	—	2	2	no	—
182	C₂²×C₈	—	2	2	no	—
183	C₂²×C₈	C₂×C₄×C₅₃	1	2	yes	—
184	C₂²×C₈	C₂×C₄×C₅₃	1	2	yes	—
185	C₂²×C₈	C₂×C₄×C₅₃	1	2	yes	—
186	C₂²×C₈	C₂×C₄×C₅₃	1	2	yes	—
187	C₂²×C₈	C₂×C₄×C₅₃	1	4	yes	—
188	C₂²×C₈	D₄×C₅₃	1	2	yes	—
189	C₂²×C₈	D₄×C₅₃	1	4	yes	—
190	C₂²×C₈	Q₈×C₅₃	1	2	yes	—
191	#117	—	2	2	no	—
192	#117	—	2	2	no	—
193	#117	C₂×C₄×C₅₃	1	4	yes	—
194	#117	C₂×C₄×C₅₃	1	4	yes	—
195	C₂×C₄⋊C₄	C₂×C₄×C₅₃	1	4	yes	—
196	C₂×C₂²⋊C₄	—	2	2	no	—
197	C₂×C₄²	—	2	2	no	—
198	C₂×C₄²	—	2	2	no	—
199	C₂×C₄²	—	2	1	no	—
200	C₂×C₄²	C₂×C₄×C₅₃	1	4	yes	—
201	C₂×C₄²	C₂×C₄×C₅₃	1	2	yes	—
202	#97	—	2	8	no	—
203	#97	—	2	8	no	—
204	C₄×C₈	—	2	8	no	—
205	C₂⁴	#41×C₅₃	1	1	yes	characteristic
206	C₂²×C₄	—	2	2	no	—
207	C₂²×C₄	—	2	4	no	—
208	C₂²×C₄	—	2	4	no	—
209	C₂²×C₄	—	4	4	no	—
210	C₂²×C₄	C₂²×C₄×C₅₃	1	1	yes	characteristic
211	C₂²×C₄	C₂²×C₄×C₅₃	1	2	yes	—
212	C₂²×C₄	C₂²×C₄×C₅₃	1	1	yes	characteristic
213	C₂²×C₄	C₂²×C₄×C₅₃	1	2	yes	—
214	C₂²×C₄	C₂²×C₄×C₅₃	1	1	yes	characteristic
215	C₂²×C₄	C₂×D₄×C₅₃	1	2	yes	—
216	C₂²×C₄	C₂×D₄×C₅₃	1	1	yes	characteristic
217	C₂²×C₄	C₂×Q₈×C₅₃	1	1	yes	characteristic
218	C₂²×C₄	#41×C₅₃	1	1	yes	characteristic
219	C₂²×C₄	#41×C₅₃	1	2	yes	—
220	M₁₆	—	2	4	no	—
221	M₁₆	—	2	4	no	—
222	M₁₆	—	2	4	no	—
223	M₁₆	C₄²×C₅₃	1	8	yes	—
224	C₂×C₈	—	2	4	no	—
225	C₂×C₈	—	2	2	no	—
226	C₂×C₈	—	2	2	no	—
227	C₂×C₈	—	2	4	no	—
228	C₂×C₈	—	2	4	no	—
229	C₂×C₈	—	2	4	no	—
230	C₂×C₈	—	2	4	no	—
231	C₂×C₈	C₄²×C₅₃	1	4	yes	—
232	C₂×C₈	C₄²×C₅₃	1	4	yes	—
233	C₂×C₈	C₂²⋊C₄×C₅₃	1	4	yes	—
234	C₂×C₈	C₂²⋊C₄×C₅₃	1	4	yes	—
235	C₂×C₈	C₂²⋊C₄×C₅₃	1	4	yes	—
236	C₂×C₈	C₄⋊C₄×C₅₃	1	4	yes	—
237	C₂×C₈	C₄⋊C₄×C₅₃	1	8	yes	—
238	C₄⋊C₄	—	2	4	no	—
239	C₄⋊C₄	C₄²×C₅₃	1	8	yes	—
240	C₂²⋊C₄	—	2	4	no	—
241	C₂²⋊C₄	—	2	4	no	—
242	C₄²	—	2	4	no	—
243	C₄²	—	2	2	no	—
244	C₄²	—	2	4	no	—
245	C₄²	—	2	4	no	—
246	C₄²	C₄²×C₅₃	1	4	yes	—
247	C₂³	—	2	4	no	—
248	C₂³	#117×C₅₃	1	2	yes	—
249	C₂³	#117×C₅₃	1	1	yes	characteristic
250	C₂³	C₄×D₄×C₅₃	1	2	yes	—
251	C₂³	C₄×D₄×C₅₃	1	1	yes	characteristic
252	C₂³	C₄×Q₈×C₅₃	1	1	yes	characteristic
253	C₂×C₄	—	2	4	no	—
254	C₂×C₄	—	2	8	no	—
255	C₂×C₄	—	4	8	no	—
256	C₂×C₄	—	4	4	no	—
257	C₂×C₄	—	4	8	no	—
258	C₂×C₄	C₂×C₄²×C₅₃	1	1	yes	characteristic
259	C₂×C₄	C₂×C₄²×C₅₃	1	1	yes	characteristic
260	C₂×C₄	C₂×C₄²×C₅₃	1	2	yes	—
261	C₂×C₄	C₂×C₄²×C₅₃	1	1	yes	characteristic
262	C₂×C₄	C₂×C₄²×C₅₃	1	2	yes	—
263	C₂×C₄	C₂×C₂²⋊C₄×C₅₃	1	1	yes	characteristic
264	C₂×C₄	C₂×C₂²⋊C₄×C₅₃	1	2	yes	—
265	C₂×C₄	C₂×C₄⋊C₄×C₅₃	1	2	yes	—
266	C₂×C₄	C₂×C₄⋊C₄×C₅₃	1	1	yes	characteristic
267	C₂×C₄	#117×C₅₃	1	1	yes	characteristic
268	C₂×C₄	#117×C₅₃	1	2	yes	—
269	C₂×C₄	C₄×D₄×C₅₃	1	1	yes	characteristic
270	C₂×C₄	C₄×D₄×C₅₃	1	2	yes	—
271	C₂×C₄	C₄×D₄×C₅₃	1	2	yes	—
272	C₂×C₄	C₄×D₄×C₅₃	1	2	yes	—
273	C₂×C₄	C₄×D₄×C₅₃	1	2	yes	—
274	C₂×C₄	C₄×Q₈×C₅₃	1	2	yes	—
275	C₂×C₄	C₄×Q₈×C₅₃	1	1	yes	characteristic
276	C₈	—	2	4	no	—
277	C₈	—	2	8	no	—
278	C₈	#95×C₅₃	1	8	yes	—
279	C₂²	—	2	4	no	—
280	C₂²	—	2	4	no	—
281	C₂²	—	2	2	no	—
282	C₂²	—	2	2	no	—
283	C₂²	C₅₃×#376	1	1	yes	characteristic
284	C₂²	C₄×C₂²⋊C₄×C₅₃	1	1	yes	characteristic
285	C₂²	C₄×C₂²⋊C₄×C₅₃	1	2	yes	—
286	C₂²	C₄×C₄⋊C₄×C₅₃	1	2	yes	—
287	C₂²	C₄×C₄⋊C₄×C₅₃	1	1	yes	characteristic
288	C₂²	C₅₃×#444	1	4	yes	—
289	C₄	—	4	4	no	—
290	C₄	—	4	8	no	—
291	C₄	C₄³×C₅₃	1	1	yes	characteristic
292	C₄	C₂×#95×C₅₃	1	1	yes	characteristic
293	C₄	C₄×C₂²⋊C₄×C₅₃	1	1	yes	characteristic
294	C₄	C₄×C₂²⋊C₄×C₅₃	1	2	yes	—
295	C₄	C₄×C₄⋊C₄×C₅₃	1	1	yes	characteristic
296	C₄	C₄×C₄⋊C₄×C₅₃	1	2	yes	—
297	C₂	—	2	1	no	—
298	C₂	—	2	1	no	—
299	C₂	—	2	2	no	—
300	C₂	C₄×#95×C₅₃	1	1	yes	characteristic
301	C₂	C₅₃×#1765	1	4	yes	—
302	C₂	C₅₃×#1773	1	2	yes	—

	subgroup	G/H
center	C₂²×C₄×C₅₃	C₂×D₄
commutator	C₄	C₄³×C₅₃
Frattini, cosocle	C₂³×C₄	C₂³×C₅₃
socle	C₂³×C₅₃	C₄×D₄
prime radical	C₂⁴×C₅₃	#41
LCS	G ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄×C₅₃ ◁ C₂³×C₈×C₅₃ ◁ G

Complements

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

N	H ≅ G/N	type
12. C₅₃×#405	C₄	semidirect
15. C₂×#97×C₅₃	C₄	semidirect
17. C₅₃×#341	C₄	semidirect
18. C₅₃×#340	C₄	semidirect
19. C₅₃×#339	C₄	semidirect
127. #10707	C₅₃	direct (N × H)
163. #405	C₄×C₅₃	semidirect
166. C₂×#97	C₄×C₅₃	semidirect
168. #341	C₄×C₅₃	semidirect
169. #340	C₄×C₅₃	semidirect
170. #339	C₄×C₅₃	semidirect
171. C₅₃	#10707	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×#405	2
C₂²×#97×C₅₃	2
C₂×C₅₃×#341	2
C₅₃×#1735	2
C₂×C₅₃×#339	2
#10707	53

Subgroups — C53×#10707 #27218219

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅₃×#10707 #27218219