Subgroups — C₅₃×#20552 #27228064

Order 13568.

890 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×#433	C₂	1	1	yes	maximal, characteristic
2	C₂×C₅₃×#424	C₂	1	1	yes	maximal, characteristic
3	C₅₃×#2298	C₂	1	1	yes	maximal, characteristic
4	C₅₃×#1771	C₂	1	2	yes	maximal, complemented
5	C₈×C₄⋊C₄×C₅₃	C₂	1	2	yes	maximal, complemented
6	C₅₃×#1768	C₂	1	2	yes	maximal, complemented
7	C₄×C₄⋊C₈×C₅₃	C₂	1	2	yes	maximal, complemented
8	C₅₃×#1753	C₂	1	2	yes	maximal
9	C₈×C₂²⋊C₄×C₅₃	C₂	1	2	yes	maximal
10	C₂²×M₁₆×C₅₃	C₂²	1	1	yes	characteristic
11	C₂³×C₈×C₅₃	C₂²	1	1	yes	characteristic
12	C₂×#117×C₅₃	C₂²	1	1	yes	characteristic
13	C₅₃×#433	C₄	1	8	yes	complemented
14	C₅₃×#424	C₄	1	8	yes	complemented
15	C₂×C₄⋊C₈×C₅₃	C₂²	1	2	yes	—
16	C₂×C₄⋊C₈×C₅₃	C₂²	1	2	yes	—
17	C₂×C₄⋊C₈×C₅₃	C₂²	1	2	yes	—
18	C₂×C₂²⋊C₈×C₅₃	C₂²	1	2	yes	—
19	C₂×#97×C₅₃	C₂²	1	2	yes	—
20	C₂×#97×C₅₃	C₂²	1	1	yes	characteristic
21	C₂×C₄×C₈×C₅₃	C₂²	1	2	yes	—
22	C₂×C₄×C₈×C₅₃	C₂²	1	2	yes	—
23	C₂×C₄×C₈×C₅₃	C₂²	1	2	yes	—
24	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
25	C₅₃×#388	C₄	1	4	yes	—
26	C₅₃×#387	C₄	1	4	yes	—
27	C₄×C₄⋊C₄×C₅₃	C₂²	1	2	yes	—
28	C₄×C₂²⋊C₄×C₅₃	C₂²	1	2	yes	—
29	C₅₃×#376	C₂²	1	1	yes	characteristic
30	C₄³×C₅₃	C₂²	1	1	yes	characteristic
31	C₅₃×#336	C₂²	1	2	yes	—
32	C₅₃×#336	C₂²	1	2	yes	—
33	C₅₃×#336	C₂²	1	2	yes	—
34	C₅₃×#336	C₂²	1	2	yes	—
35	C₂³×C₄×C₅₃	C₂³	1	1	yes	characteristic
36	C₂×M₁₆×C₅₃	—	4	2	no	—
37	C₂×M₁₆×C₅₃	C₂×C₄	1	4	yes	—
38	C₂²×C₈×C₅₃	—	4	2	no	—
39	C₂²×C₈×C₅₃	C₂×C₄	1	4	yes	—
40	C₂²×C₈×C₅₃	C₂³	1	2	yes	—
41	C₂²×C₈×C₅₃	C₂³	1	2	yes	—
42	C₂²×C₈×C₅₃	C₂³	1	2	yes	—
43	C₂²×C₈×C₅₃	C₂³	1	2	yes	—
44	#117×C₅₃	C₂×C₄	1	4	yes	—
45	#117×C₅₃	C₂×C₄	1	4	yes	—
46	C₂×C₄⋊C₄×C₅₃	C₂×C₄	1	4	yes	—
47	C₂×C₄⋊C₄×C₅₃	C₂×C₄	1	2	yes	—
48	C₂×C₂²⋊C₄×C₅₃	C₂×C₄	1	4	yes	—
49	C₂×C₂²⋊C₄×C₅₃	C₂×C₄	1	2	yes	—
50	C₂×C₄²×C₅₃	C₂×C₄	1	4	yes	—
51	C₂×C₄²×C₅₃	C₂³	1	2	yes	—
52	C₂×C₄²×C₅₃	C₂³	1	2	yes	—
53	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
54	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
55	C₄⋊C₈×C₅₃	C₂×C₄	1	8	yes	—
56	C₄⋊C₈×C₅₃	C₂×C₄	1	8	yes	—
57	C₄⋊C₈×C₅₃	C₂×C₄	1	8	yes	—
58	C₂²⋊C₈×C₅₃	C₂×C₄	1	8	yes	—
59	#97×C₅₃	—	4	2	no	—
60	#97×C₅₃	C₂×C₄	1	4	yes	—
61	C₄×C₈×C₅₃	—	4	2	no	—
62	C₄×C₈×C₅₃	—	4	2	no	—
63	C₄×C₈×C₅₃	—	4	2	no	—
64	C₄×C₈×C₅₃	C₂×C₄	1	4	yes	—
65	#95×C₅₃	C₂×C₄	1	4	yes	—
66	#95×C₅₃	C₂×C₄	1	4	yes	—
67	#95×C₅₃	C₂×C₄	1	4	yes	—
68	C₂⁴×C₅₃	C₂²×C₄	1	1	yes	characteristic
69	C₂²×C₄×C₅₃	—	2	2	no	—
70	C₂²×C₄×C₅₃	—	2	2	no	—
71	C₂²×C₄×C₅₃	C₂²×C₄	1	4	yes	—
72	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
73	C₂²×C₄×C₅₃	C₂²×C₄	1	2	yes	—
74	C₂²×C₄×C₅₃	C₂²×C₄	1	2	yes	—
75	C₂²×C₄×C₅₃	C₂²×C₄	1	2	yes	—
76	C₂²×C₄×C₅₃	C₂²×C₄	1	2	yes	—
77	C₂²×C₄×C₅₃	C₂²×C₄	1	4	yes	—
78	C₂²×C₄×C₅₃	C₂⁴	1	1	yes	characteristic
79	M₁₆×C₅₃	—	4	4	no	—
80	C₂×C₈×C₅₃	—	2	2	no	—
81	C₂×C₈×C₅₃	—	2	2	no	—
82	C₂×C₈×C₅₃	—	2	2	no	—
83	C₂×C₈×C₅₃	—	2	2	no	—
84	C₂×C₈×C₅₃	—	2	2	no	—
85	C₂×C₈×C₅₃	—	2	2	no	—
86	C₂×C₈×C₅₃	—	2	2	no	—
87	C₂×C₈×C₅₃	—	2	2	no	—
88	C₂×C₈×C₅₃	—	4	4	no	—
89	C₂×C₈×C₅₃	C₂²×C₄	1	4	yes	—
90	C₂×C₈×C₅₃	C₂²×C₄	1	4	yes	—
91	C₂×C₈×C₅₃	C₂²×C₄	1	4	yes	—
92	C₂×C₈×C₅₃	C₂²×C₄	1	4	yes	—
93	C₄⋊C₄×C₅₃	—	2	8	no	—
94	C₄⋊C₄×C₅₃	C₄²	1	8	yes	—
95	C₂²⋊C₄×C₅₃	—	2	8	no	—
96	C₂²⋊C₄×C₅₃	C₄²	1	8	yes	—
97	C₄²×C₅₃	—	2	8	no	—
98	C₄²×C₅₃	—	2	2	no	—
99	C₄²×C₅₃	—	2	2	no	—
100	C₄²×C₅₃	—	2	2	no	—
101	C₄²×C₅₃	—	2	2	no	—
102	C₄²×C₅₃	C₂²×C₄	1	2	yes	—
103	C₄²×C₅₃	C₂²×C₄	1	2	yes	—
104	C₄²×C₅₃	C₂²×C₄	1	2	yes	—
105	C₄²×C₅₃	C₂²×C₄	1	2	yes	—
106	C₂³×C₅₃	—	2	2	no	—
107	C₂³×C₅₃	—	2	4	no	—
108	C₂³×C₅₃	C₂×C₄²	1	2	yes	—
109	C₂³×C₅₃	C₂³×C₄	1	1	yes	characteristic
110	C₂×C₄×C₅₃	—	2	2	no	—
111	C₂×C₄×C₅₃	—	2	4	no	—
112	C₂×C₄×C₅₃	—	2	4	no	—
113	C₂×C₄×C₅₃	—	2	4	no	—
114	C₂×C₄×C₅₃	—	2	4	no	—
115	C₂×C₄×C₅₃	—	2	2	no	—
116	C₂×C₄×C₅₃	—	2	2	no	—
117	C₂×C₄×C₅₃	—	2	4	no	—
118	C₂×C₄×C₅₃	—	2	4	no	—
119	C₂×C₄×C₅₃	—	2	4	no	—
120	C₂×C₄×C₅₃	—	2	4	no	—
121	C₂×C₄×C₅₃	C₂×C₄²	1	2	yes	—
122	C₂×C₄×C₅₃	C₂×C₄²	1	4	yes	—
123	C₂×C₄×C₅₃	C₂×C₄²	1	4	yes	—
124	C₂×C₄×C₅₃	#131	1	8	yes	—
125	C₂×C₄×C₅₃	C₂³×C₄	1	1	yes	characteristic
126	C₂×C₄×C₅₃	C₂³×C₄	1	1	yes	characteristic
127	C₂×C₄×C₅₃	H(5,𝔽₂)	1	2	yes	—
128	C₂×C₄×C₅₃	2₋¹⁺⁴	1	2	yes	—
129	C₈×C₅₃	—	2	4	no	—
130	C₈×C₅₃	—	2	4	no	—
131	C₈×C₅₃	—	2	4	no	—
132	C₈×C₅₃	—	2	4	no	—
133	#20552	C₅₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
134	C₂²×C₅₃	—	2	4	no	—
135	C₂²×C₅₃	—	2	2	no	—
136	C₂²×C₅₃	—	4	4	no	—
137	C₂²×C₅₃	C₂²×C₄²	1	1	yes	characteristic
138	C₂²×C₅₃	#520	1	2	yes	—
139	C₂²×C₅₃	C₂×#131	1	4	yes	—
140	C₄×C₅₃	—	2	4	no	—
141	C₄×C₅₃	—	2	4	no	—
142	C₄×C₅₃	—	2	8	no	—
143	C₄×C₅₃	—	2	2	no	—
144	C₄×C₅₃	—	4	4	no	—
145	C₄×C₅₃	#518	1	1	yes	characteristic
146	C₄×C₅₃	#519	1	1	yes	characteristic
147	C₄×C₅₃	#520	1	2	yes	—
148	C₂×#433	C₂×C₅₃	1	1	yes	characteristic
149	C₂×#424	C₂×C₅₃	1	1	yes	characteristic
150	#2298	C₂×C₅₃	1	1	yes	characteristic
151	#1771	C₂×C₅₃	1	2	yes	complemented
152	C₈×C₄⋊C₄	C₂×C₅₃	1	2	yes	complemented
153	#1768	C₂×C₅₃	1	2	yes	complemented
154	C₄×C₄⋊C₈	C₂×C₅₃	1	2	yes	complemented
155	#1753	C₂×C₅₃	1	2	yes	—
156	C₈×C₂²⋊C₄	C₂×C₅₃	1	2	yes	—
157	C₂×C₅₃	—	4	2	no	—
158	C₂×C₅₃	#2274	1	1	yes	characteristic
159	C₂×C₅₃	C₄×#131	1	2	yes	—
160	C₂×C₅₃	#2994	1	4	yes	—
161	C₂²×M₁₆	C₂²×C₅₃	1	1	yes	characteristic
162	C₂³×C₈	C₂²×C₅₃	1	1	yes	characteristic
163	C₂×#117	C₂²×C₅₃	1	1	yes	characteristic
164	#433	C₄×C₅₃	1	8	yes	complemented
165	#424	C₄×C₅₃	1	8	yes	complemented
166	C₂×C₄⋊C₈	C₂²×C₅₃	1	2	yes	—
167	C₂×C₄⋊C₈	C₂²×C₅₃	1	2	yes	—
168	C₂×C₄⋊C₈	C₂²×C₅₃	1	2	yes	—
169	C₂×C₂²⋊C₈	C₂²×C₅₃	1	2	yes	—
170	C₂×#97	C₂²×C₅₃	1	2	yes	—
171	C₂×#97	C₂²×C₅₃	1	1	yes	characteristic
172	C₂×C₄×C₈	C₂²×C₅₃	1	2	yes	—
173	C₂×C₄×C₈	C₂²×C₅₃	1	2	yes	—
174	C₂×C₄×C₈	C₂²×C₅₃	1	2	yes	—
175	C₂×C₄×C₈	C₂²×C₅₃	1	1	yes	characteristic
176	#388	C₄×C₅₃	1	4	yes	—
177	#387	C₄×C₅₃	1	4	yes	—
178	C₄×C₄⋊C₄	C₂²×C₅₃	1	2	yes	—
179	C₄×C₂²⋊C₄	C₂²×C₅₃	1	2	yes	—
180	#376	C₂²×C₅₃	1	1	yes	characteristic
181	C₄³	C₂²×C₅₃	1	1	yes	characteristic
182	#336	C₂²×C₅₃	1	2	yes	—
183	#336	C₂²×C₅₃	1	2	yes	—
184	#336	C₂²×C₅₃	1	2	yes	—
185	#336	C₂²×C₅₃	1	2	yes	—
186	C₅₃	#20552	1	1	yes	characteristic, direct product factor, Sylow
187	C₂³×C₄	C₂³×C₅₃	1	1	yes	characteristic
188	C₂×M₁₆	—	4	2	no	—
189	C₂×M₁₆	C₂×C₄×C₅₃	1	4	yes	—
190	C₂²×C₈	—	4	2	no	—
191	C₂²×C₈	C₂×C₄×C₅₃	1	4	yes	—
192	C₂²×C₈	C₂³×C₅₃	1	2	yes	—
193	C₂²×C₈	C₂³×C₅₃	1	2	yes	—
194	C₂²×C₈	C₂³×C₅₃	1	2	yes	—
195	C₂²×C₈	C₂³×C₅₃	1	2	yes	—
196	#117	C₂×C₄×C₅₃	1	4	yes	—
197	#117	C₂×C₄×C₅₃	1	4	yes	—
198	C₂×C₄⋊C₄	C₂×C₄×C₅₃	1	4	yes	—
199	C₂×C₄⋊C₄	C₂×C₄×C₅₃	1	2	yes	—
200	C₂×C₂²⋊C₄	C₂×C₄×C₅₃	1	4	yes	—
201	C₂×C₂²⋊C₄	C₂×C₄×C₅₃	1	2	yes	—
202	C₂×C₄²	C₂×C₄×C₅₃	1	4	yes	—
203	C₂×C₄²	C₂³×C₅₃	1	2	yes	—
204	C₂×C₄²	C₂³×C₅₃	1	2	yes	—
205	C₂×C₄²	C₂³×C₅₃	1	1	yes	characteristic
206	C₂×C₄²	C₂³×C₅₃	1	1	yes	characteristic
207	C₄⋊C₈	C₂×C₄×C₅₃	1	8	yes	—
208	C₄⋊C₈	C₂×C₄×C₅₃	1	8	yes	—
209	C₄⋊C₈	C₂×C₄×C₅₃	1	8	yes	—
210	C₂²⋊C₈	C₂×C₄×C₅₃	1	8	yes	—
211	#97	—	4	2	no	—
212	#97	C₂×C₄×C₅₃	1	4	yes	—
213	C₄×C₈	—	4	2	no	—
214	C₄×C₈	—	4	2	no	—
215	C₄×C₈	—	4	2	no	—
216	C₄×C₈	C₂×C₄×C₅₃	1	4	yes	—
217	#95	C₂×C₄×C₅₃	1	4	yes	—
218	#95	C₂×C₄×C₅₃	1	4	yes	—
219	#95	C₂×C₄×C₅₃	1	4	yes	—
220	C₂⁴	C₂²×C₄×C₅₃	1	1	yes	characteristic
221	C₂²×C₄	—	2	2	no	—
222	C₂²×C₄	—	2	2	no	—
223	C₂²×C₄	C₂²×C₄×C₅₃	1	4	yes	—
224	C₂²×C₄	C₂²×C₄×C₅₃	1	1	yes	characteristic
225	C₂²×C₄	C₂²×C₄×C₅₃	1	2	yes	—
226	C₂²×C₄	C₂²×C₄×C₅₃	1	2	yes	—
227	C₂²×C₄	C₂²×C₄×C₅₃	1	2	yes	—
228	C₂²×C₄	C₂²×C₄×C₅₃	1	2	yes	—
229	C₂²×C₄	C₂²×C₄×C₅₃	1	4	yes	—
230	C₂²×C₄	C₂⁴×C₅₃	1	1	yes	characteristic
231	M₁₆	—	4	4	no	—
232	C₂×C₈	—	2	2	no	—
233	C₂×C₈	—	2	2	no	—
234	C₂×C₈	—	2	2	no	—
235	C₂×C₈	—	2	2	no	—
236	C₂×C₈	—	2	2	no	—
237	C₂×C₈	—	2	2	no	—
238	C₂×C₈	—	2	2	no	—
239	C₂×C₈	—	2	2	no	—
240	C₂×C₈	—	4	4	no	—
241	C₂×C₈	C₂²×C₄×C₅₃	1	4	yes	—
242	C₂×C₈	C₂²×C₄×C₅₃	1	4	yes	—
243	C₂×C₈	C₂²×C₄×C₅₃	1	4	yes	—
244	C₂×C₈	C₂²×C₄×C₅₃	1	4	yes	—
245	C₄⋊C₄	—	2	8	no	—
246	C₄⋊C₄	C₄²×C₅₃	1	8	yes	—
247	C₂²⋊C₄	—	2	8	no	—
248	C₂²⋊C₄	C₄²×C₅₃	1	8	yes	—
249	C₄²	—	2	8	no	—
250	C₄²	—	2	2	no	—
251	C₄²	—	2	2	no	—
252	C₄²	—	2	2	no	—
253	C₄²	—	2	2	no	—
254	C₄²	C₂²×C₄×C₅₃	1	2	yes	—
255	C₄²	C₂²×C₄×C₅₃	1	2	yes	—
256	C₄²	C₂²×C₄×C₅₃	1	2	yes	—
257	C₄²	C₂²×C₄×C₅₃	1	2	yes	—
258	C₂³	—	2	2	no	—
259	C₂³	—	2	4	no	—
260	C₂³	C₂×C₄²×C₅₃	1	2	yes	—
261	C₂³	C₂³×C₄×C₅₃	1	1	yes	characteristic
262	C₂×C₄	—	2	4	no	—
263	C₂×C₄	—	2	2	no	—
264	C₂×C₄	—	2	4	no	—
265	C₂×C₄	—	2	4	no	—
266	C₂×C₄	—	2	4	no	—
267	C₂×C₄	—	2	2	no	—
268	C₂×C₄	—	2	2	no	—
269	C₂×C₄	—	2	4	no	—
270	C₂×C₄	—	2	4	no	—
271	C₂×C₄	—	2	4	no	—
272	C₂×C₄	—	2	4	no	—
273	C₂×C₄	C₂×C₄²×C₅₃	1	4	yes	—
274	C₂×C₄	C₂×C₄²×C₅₃	1	4	yes	—
275	C₂×C₄	C₂×C₄²×C₅₃	1	2	yes	—
276	C₂×C₄	#131×C₅₃	1	8	yes	—
277	C₂×C₄	C₂³×C₄×C₅₃	1	1	yes	characteristic
278	C₂×C₄	C₂³×C₄×C₅₃	1	1	yes	characteristic
279	C₂×C₄	H(5,𝔽₂)×C₅₃	1	2	yes	—
280	C₂×C₄	2₋¹⁺⁴×C₅₃	1	2	yes	—
281	C₈	—	2	4	no	—
282	C₈	—	2	4	no	—
283	C₈	—	2	4	no	—
284	C₈	—	2	4	no	—
285	C₂²	—	2	2	no	—
286	C₂²	—	2	4	no	—
287	C₂²	—	4	4	no	—
288	C₂²	C₂²×C₄²×C₅₃	1	1	yes	characteristic
289	C₂²	C₅₃×#520	1	2	yes	—
290	C₂²	C₂×#131×C₅₃	1	4	yes	—
291	C₄	—	2	8	no	—
292	C₄	—	2	2	no	—
293	C₄	—	2	4	no	—
294	C₄	—	2	4	no	—
295	C₄	—	4	4	no	—
296	C₄	C₅₃×#518	1	1	yes	characteristic
297	C₄	C₅₃×#519	1	1	yes	characteristic
298	C₄	C₅₃×#520	1	2	yes	—
299	C₂	—	4	2	no	—
300	C₂	C₅₃×#2274	1	1	yes	characteristic
301	C₂	C₄×#131×C₅₃	1	2	yes	—
302	C₂	C₅₃×#2994	1	4	yes	—

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

Complements

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

N	H ≅ G/N	type
4. C₅₃×#1771	C₂	semidirect
5. C₈×C₄⋊C₄×C₅₃	C₂	semidirect
6. C₅₃×#1768	C₂	semidirect
7. C₄×C₄⋊C₈×C₅₃	C₂	semidirect
13. C₅₃×#433	C₄	semidirect
14. C₅₃×#424	C₄	semidirect
133. #20552	C₅₃	direct (N × H)
151. #1771	C₂×C₅₃	semidirect
152. C₈×C₄⋊C₄	C₂×C₅₃	semidirect
153. #1768	C₂×C₅₃	semidirect
154. C₄×C₄⋊C₈	C₂×C₅₃	semidirect
164. #433	C₄×C₅₃	semidirect
165. #424	C₄×C₅₃	semidirect
186. C₅₃	#20552	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×#433	2
C₂×C₅₃×#424	2
C₅₃×#2298	2
C₅₃×#1771	2
C₈×C₄⋊C₄×C₅₃	2
C₅₃×#1768	2
C₄×C₄⋊C₈×C₅₃	2
C₅₃×#1753	2
C₈×C₂²⋊C₄×C₅₃	2
#20552	53

Subgroups — C53×#20552 #27228064

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅₃×#20552 #27228064