Subgroups — C₅₃×#22264 #27229776

Order 13568.

1018 conjugacy classes of subgroups.

The subgroup lattice has 274 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₅₃×#436	C₂	1	2	yes	maximal, complemented
2	C₄×#127×C₅₃	C₂	1	1	yes	maximal, characteristic
3	C₅₃×#1954	C₂	1	2	yes	maximal, complemented
4	C₅₃×#1870	C₂	1	8	yes	maximal, complemented
5	C₅₃×#1831	C₂	1	1	yes	maximal, characteristic, complemented
6	C₅₃×#1744	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂²×M₁₆×C₅₃	C₂²	1	4	yes	—
8	C₂×#127×C₅₃	C₄	1	1	yes	characteristic
9	C₂×C₄×D₄×C₅₃	C₂²	1	2	yes	—
10	C₂×C₄×D₄×C₅₃	C₂²	1	4	yes	—
11	C₅₃×#436	—	2	8	no	—
12	C₂×C₄⋊C₈×C₅₃	C₂²	1	2	yes	—
13	C₂×C₂²⋊C₈×C₅₃	C₂²	1	4	yes	—
14	C₂×C₂²⋊C₈×C₅₃	C₂²	1	8	yes	—
15	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
16	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
17	C₅₃×#390	C₄	1	2	yes	—
18	C₅₃×#390	C₄	1	4	yes	—
19	C₅₃×#384	C₄	1	1	yes	characteristic
20	C₄³×C₅₃	C₂²	1	1	yes	characteristic
21	C₅₃×#336	C₂²	1	4	yes	—
22	C₅₃×#336	C₂²	1	4	yes	—
23	C₂²×D₄×C₅₃	C₂×C₄	1	2	yes	—
24	C₂²×D₄×C₅₃	C₂×C₄	1	4	yes	—
25	C₂³×C₄×C₅₃	C₂³	1	4	yes	—
26	C₂×M₁₆×C₅₃	—	2	8	no	—
27	C₂×M₁₆×C₅₃	—	4	8	no	—
28	C₂²×C₈×C₅₃	C₂³	1	4	yes	—
29	C₂²×C₈×C₅₃	C₂³	1	2	yes	—
30	C₂²×C₈×C₅₃	C₂³	1	2	yes	—
31	#127×C₅₃	—	2	4	no	—
32	C₄×D₄×C₅₃	—	2	4	no	—
33	C₄×D₄×C₅₃	—	2	4	no	—
34	C₄×D₄×C₅₃	—	4	8	no	—
35	C₂×C₄⋊C₄×C₅₃	C₂×C₄	1	2	yes	—
36	C₂×C₄⋊C₄×C₅₃	C₂×C₄	1	4	yes	—
37	C₂×C₂²⋊C₄×C₅₃	C₂×C₄	1	4	yes	—
38	C₂×C₂²⋊C₄×C₅₃	C₂×C₄	1	8	yes	—
39	C₂×C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
40	C₂×C₄²×C₅₃	C₂×C₄	1	2	yes	—
41	C₂×C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
42	C₂×C₄²×C₅₃	C₂³	1	2	yes	—
43	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
44	C₄⋊C₈×C₅₃	—	2	4	no	—
45	C₂²⋊C₈×C₅₃	—	2	8	no	—
46	C₂²⋊C₈×C₅₃	—	4	8	no	—
47	C₄×C₈×C₅₃	—	2	2	no	—
48	C₄×C₈×C₅₃	D₄	1	4	yes	complemented
49	C₂⁴×C₅₃	C₂²×C₄	1	4	yes	—
50	C₂×D₄×C₅₃	—	2	4	no	—
51	C₂×D₄×C₅₃	—	2	8	no	—
52	C₂×D₄×C₅₃	—	2	4	no	—
53	C₂×D₄×C₅₃	—	2	4	no	—
54	C₂×D₄×C₅₃	—	4	4	no	—
55	C₂×D₄×C₅₃	—	4	4	no	—
56	C₂²×C₄×C₅₃	—	2	4	no	—
57	C₂²×C₄×C₅₃	—	2	4	no	—
58	C₂²×C₄×C₅₃	—	2	8	no	—
59	C₂²×C₄×C₅₃	—	2	8	no	—
60	C₂²×C₄×C₅₃	C₂²×C₄	1	2	yes	—
61	C₂²×C₄×C₅₃	C₂²×C₄	1	4	yes	—
62	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
63	C₂²×C₄×C₅₃	C₂²×C₄	1	2	yes	—
64	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
65	C₂²×C₄×C₅₃	C₂⁴	1	1	yes	characteristic
66	M₁₆×C₅₃	—	4	16	no	—
67	C₂×C₈×C₅₃	—	2	4	no	—
68	C₂×C₈×C₅₃	—	2	4	no	—
69	C₂×C₈×C₅₃	—	2	2	no	—
70	C₂×C₈×C₅₃	—	2	4	no	—
71	C₂×C₈×C₅₃	—	2	8	no	—
72	C₂×C₈×C₅₃	C₂×D₄	1	4	yes	—
73	C₄⋊C₄×C₅₃	—	2	4	no	—
74	C₄⋊C₄×C₅₃	—	4	4	no	—
75	C₂²⋊C₄×C₅₃	—	2	8	no	—
76	C₂²⋊C₄×C₅₃	—	4	8	no	—
77	C₄²×C₅₃	—	2	2	no	—
78	C₄²×C₅₃	—	2	4	no	—
79	C₄²×C₅₃	—	4	2	no	—
80	C₄²×C₅₃	M₁₆	1	4	yes	complemented
81	C₄²×C₅₃	C₂×D₄	1	2	yes	—
82	C₄²×C₅₃	#41	1	2	yes	—
83	C₂³×C₅₃	—	2	4	no	—
84	C₂³×C₅₃	—	2	8	no	—
85	C₂³×C₅₃	—	2	4	no	—
86	C₂³×C₅₃	—	2	4	no	—
87	C₂³×C₅₃	—	2	8	no	—
88	C₂³×C₅₃	C₂³×C₄	1	1	yes	characteristic
89	D₄×C₅₃	—	4	16	no	—
90	D₄×C₅₃	—	4	8	no	—
91	C₂×C₄×C₅₃	—	2	2	no	—
92	C₂×C₄×C₅₃	—	2	2	no	—
93	C₂×C₄×C₅₃	—	2	2	no	—
94	C₂×C₄×C₅₃	—	2	4	no	—
95	C₂×C₄×C₅₃	—	2	8	no	—
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	—	4	4	no	—
100	C₂×C₄×C₅₃	—	4	4	no	—
101	C₂×C₄×C₅₃	—	4	8	no	—
102	C₂×C₄×C₅₃	C₄×D₄	1	2	yes	—
103	C₂×C₄×C₅₃	C₄×D₄	1	2	yes	—
104	C₂×C₄×C₅₃	C₂×M₁₆	1	4	yes	—
105	C₂×C₄×C₅₃	C₂×M₁₆	1	2	yes	—
106	C₂×C₄×C₅₃	#131	1	2	yes	—
107	C₂×C₄×C₅₃	C₂²×D₄	1	1	yes	characteristic
108	C₂×C₄×C₅₃	C₂×#41	1	1	yes	characteristic
109	C₂×C₄×C₅₃	H(5,𝔽₂)	1	4	yes	—
110	C₈×C₅₃	—	2	4	no	—
111	C₈×C₅₃	—	4	4	no	—
112	C₈×C₅₃	—	4	2	no	—
113	#22264	C₅₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
114	C₂²×C₅₃	—	4	4	no	—
115	C₂²×C₅₃	—	4	4	no	—
116	C₂²×C₅₃	—	4	4	no	—
117	C₂²×C₅₃	—	4	8	no	—
118	C₂²×C₅₃	—	4	8	no	—
119	C₂²×C₅₃	C₂×C₄×D₄	1	1	yes	characteristic
120	C₂²×C₅₃	#518	1	2	yes	—
121	C₂²×C₅₃	C₂²×M₁₆	1	1	yes	characteristic
122	C₂²×C₅₃	C₂×#131	1	1	yes	characteristic
123	C₂²×C₅₃	#568	1	2	yes	—
124	C₄×C₅₃	—	2	2	no	—
125	C₄×C₅₃	—	2	4	no	—
126	C₄×C₅₃	—	4	4	no	—
127	C₄×C₅₃	—	4	2	no	—
128	C₄×C₅₃	#436	1	4	yes	—
129	C₄×C₅₃	#535	1	2	yes	—
130	C₄×C₅₃	#540	1	2	yes	—
131	C₂×#436	C₂×C₅₃	1	2	yes	complemented
132	C₄×#127	C₂×C₅₃	1	1	yes	characteristic
133	#1954	C₂×C₅₃	1	2	yes	complemented
134	#1870	C₂×C₅₃	1	8	yes	complemented
135	#1831	C₂×C₅₃	1	1	yes	characteristic, complemented
136	#1744	C₂×C₅₃	1	1	yes	characteristic, complemented
137	C₂×C₅₃	—	8	4	no	—
138	C₂×C₅₃	#2328	1	1	yes	characteristic
139	C₂×C₅₃	C₂×#436	1	1	yes	characteristic
140	C₂×C₅₃	#2932	1	1	yes	characteristic
141	C₂×C₅₃	#2975	1	2	yes	—
142	C₂×C₅₃	#2981	1	2	yes	—
143	C₂²×M₁₆	C₂²×C₅₃	1	4	yes	—
144	C₂×#127	C₄×C₅₃	1	1	yes	characteristic
145	C₂×C₄×D₄	C₂²×C₅₃	1	2	yes	—
146	C₂×C₄×D₄	C₂²×C₅₃	1	4	yes	—
147	#436	—	2	8	no	—
148	C₂×C₄⋊C₈	C₂²×C₅₃	1	2	yes	—
149	C₂×C₂²⋊C₈	C₂²×C₅₃	1	4	yes	—
150	C₂×C₂²⋊C₈	C₂²×C₅₃	1	8	yes	—
151	C₂×C₄×C₈	C₂²×C₅₃	1	1	yes	characteristic
152	C₂×C₄×C₈	C₂²×C₅₃	1	1	yes	characteristic
153	#390	C₄×C₅₃	1	2	yes	—
154	#390	C₄×C₅₃	1	4	yes	—
155	#384	C₄×C₅₃	1	1	yes	characteristic
156	C₄³	C₂²×C₅₃	1	1	yes	characteristic
157	#336	C₂²×C₅₃	1	4	yes	—
158	#336	C₂²×C₅₃	1	4	yes	—
159	C₅₃	#22264	1	1	yes	characteristic, direct product factor, Sylow
160	C₂²×D₄	C₂×C₄×C₅₃	1	2	yes	—
161	C₂²×D₄	C₂×C₄×C₅₃	1	4	yes	—
162	C₂³×C₄	C₂³×C₅₃	1	4	yes	—
163	C₂×M₁₆	—	2	8	no	—
164	C₂×M₁₆	—	4	8	no	—
165	C₂²×C₈	C₂³×C₅₃	1	4	yes	—
166	C₂²×C₈	C₂³×C₅₃	1	2	yes	—
167	C₂²×C₈	C₂³×C₅₃	1	2	yes	—
168	#127	—	2	4	no	—
169	C₄×D₄	—	2	4	no	—
170	C₄×D₄	—	2	4	no	—
171	C₄×D₄	—	4	8	no	—
172	C₂×C₄⋊C₄	C₂×C₄×C₅₃	1	2	yes	—
173	C₂×C₄⋊C₄	C₂×C₄×C₅₃	1	4	yes	—
174	C₂×C₂²⋊C₄	C₂×C₄×C₅₃	1	4	yes	—
175	C₂×C₂²⋊C₄	C₂×C₄×C₅₃	1	8	yes	—
176	C₂×C₄²	C₂×C₄×C₅₃	1	1	yes	characteristic
177	C₂×C₄²	C₂×C₄×C₅₃	1	2	yes	—
178	C₂×C₄²	C₂×C₄×C₅₃	1	1	yes	characteristic
179	C₂×C₄²	C₂³×C₅₃	1	2	yes	—
180	C₂×C₄²	C₂³×C₅₃	1	1	yes	characteristic
181	C₄⋊C₈	—	2	4	no	—
182	C₂²⋊C₈	—	2	8	no	—
183	C₂²⋊C₈	—	4	8	no	—
184	C₄×C₈	—	2	2	no	—
185	C₄×C₈	D₄×C₅₃	1	4	yes	complemented
186	C₂⁴	C₂²×C₄×C₅₃	1	4	yes	—
187	C₂×D₄	—	2	4	no	—
188	C₂×D₄	—	2	8	no	—
189	C₂×D₄	—	2	4	no	—
190	C₂×D₄	—	2	4	no	—
191	C₂×D₄	—	4	4	no	—
192	C₂×D₄	—	4	4	no	—
193	C₂²×C₄	—	2	4	no	—
194	C₂²×C₄	—	2	4	no	—
195	C₂²×C₄	—	2	8	no	—
196	C₂²×C₄	—	2	8	no	—
197	C₂²×C₄	C₂²×C₄×C₅₃	1	2	yes	—
198	C₂²×C₄	C₂²×C₄×C₅₃	1	4	yes	—
199	C₂²×C₄	C₂²×C₄×C₅₃	1	1	yes	characteristic
200	C₂²×C₄	C₂²×C₄×C₅₃	1	2	yes	—
201	C₂²×C₄	C₂²×C₄×C₅₃	1	1	yes	characteristic
202	C₂²×C₄	C₂⁴×C₅₃	1	1	yes	characteristic
203	M₁₆	—	4	16	no	—
204	C₂×C₈	—	2	4	no	—
205	C₂×C₈	—	2	4	no	—
206	C₂×C₈	—	2	2	no	—
207	C₂×C₈	—	2	4	no	—
208	C₂×C₈	—	2	8	no	—
209	C₂×C₈	C₂×D₄×C₅₃	1	4	yes	—
210	C₄⋊C₄	—	2	4	no	—
211	C₄⋊C₄	—	4	4	no	—
212	C₂²⋊C₄	—	2	8	no	—
213	C₂²⋊C₄	—	4	8	no	—
214	C₄²	—	2	2	no	—
215	C₄²	—	2	4	no	—
216	C₄²	—	4	2	no	—
217	C₄²	M₁₆×C₅₃	1	4	yes	complemented
218	C₄²	C₂×D₄×C₅₃	1	2	yes	—
219	C₄²	#41×C₅₃	1	2	yes	—
220	C₂³	—	2	4	no	—
221	C₂³	—	2	8	no	—
222	C₂³	—	2	4	no	—
223	C₂³	—	2	4	no	—
224	C₂³	—	2	8	no	—
225	C₂³	C₂³×C₄×C₅₃	1	1	yes	characteristic
226	D₄	—	4	16	no	—
227	D₄	—	4	8	no	—
228	C₂×C₄	—	2	2	no	—
229	C₂×C₄	—	2	2	no	—
230	C₂×C₄	—	2	2	no	—
231	C₂×C₄	—	2	4	no	—
232	C₂×C₄	—	2	2	no	—
233	C₂×C₄	—	2	8	no	—
234	C₂×C₄	—	2	2	no	—
235	C₂×C₄	—	2	2	no	—
236	C₂×C₄	—	4	4	no	—
237	C₂×C₄	—	4	4	no	—
238	C₂×C₄	—	4	8	no	—
239	C₂×C₄	C₄×D₄×C₅₃	1	2	yes	—
240	C₂×C₄	C₄×D₄×C₅₃	1	2	yes	—
241	C₂×C₄	C₂×M₁₆×C₅₃	1	4	yes	—
242	C₂×C₄	C₂×M₁₆×C₅₃	1	2	yes	—
243	C₂×C₄	#131×C₅₃	1	2	yes	—
244	C₂×C₄	C₂²×D₄×C₅₃	1	1	yes	characteristic
245	C₂×C₄	C₂×#41×C₅₃	1	1	yes	characteristic
246	C₂×C₄	H(5,𝔽₂)×C₅₃	1	4	yes	—
247	C₈	—	2	4	no	—
248	C₈	—	4	4	no	—
249	C₈	—	4	2	no	—
250	C₂²	—	4	8	no	—
251	C₂²	—	4	8	no	—
252	C₂²	—	4	4	no	—
253	C₂²	—	4	4	no	—
254	C₂²	—	4	4	no	—
255	C₂²	C₂×C₄×D₄×C₅₃	1	1	yes	characteristic
256	C₂²	C₅₃×#518	1	2	yes	—
257	C₂²	C₂²×M₁₆×C₅₃	1	1	yes	characteristic
258	C₂²	C₂×#131×C₅₃	1	1	yes	characteristic
259	C₂²	C₅₃×#568	1	2	yes	—
260	C₄	—	2	4	no	—
261	C₄	—	2	2	no	—
262	C₄	—	4	2	no	—
263	C₄	—	4	4	no	—
264	C₄	C₅₃×#436	1	4	yes	—
265	C₄	C₅₃×#535	1	2	yes	—
266	C₄	C₅₃×#540	1	2	yes	—
267	C₂	—	8	4	no	—
268	C₂	C₅₃×#2328	1	1	yes	characteristic
269	C₂	C₂×C₅₃×#436	1	1	yes	characteristic
270	C₂	C₅₃×#2932	1	1	yes	characteristic
271	C₂	C₅₃×#2975	1	2	yes	—
272	C₂	C₅₃×#2981	1	2	yes	—

	subgroup	G/H
center	C₂²×C₄×C₅₃	C₂⁴
commutator	C₂³	C₂³×C₄×C₅₃
Frattini, cosocle	C₂²×C₄	C₂⁴×C₅₃
socle	C₂³×C₅₃	C₂³×C₄
prime radical	C₂×#127×C₅₃	C₄
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂²×C₄×C₅₃ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂×C₅₃×#436	C₂	semidirect
3. C₅₃×#1954	C₂	semidirect
4. C₅₃×#1870	C₂	semidirect
5. C₅₃×#1831	C₂	semidirect
6. C₅₃×#1744	C₂	semidirect
48. C₄×C₈×C₅₃	D₄	semidirect
80. C₄²×C₅₃	M₁₆	semidirect
113. #22264	C₅₃	direct (N × H)
131. C₂×#436	C₂×C₅₃	semidirect
133. #1954	C₂×C₅₃	semidirect
134. #1870	C₂×C₅₃	semidirect
135. #1831	C₂×C₅₃	semidirect
136. #1744	C₂×C₅₃	semidirect
159. C₅₃	#22264	direct (N × H)
185. C₄×C₈	D₄×C₅₃	semidirect
217. C₄²	M₁₆×C₅₃	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₅₃×#436	2
C₄×#127×C₅₃	2
C₅₃×#1954	2
C₅₃×#1870	2
C₅₃×#1831	2
C₅₃×#1744	2
#22264	53

Subgroups — C53×#22264 #27229776

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅₃×#22264 #27229776