Subgroups — C₂×C₅₃×#1580 #27225360

Order 13568.

894 conjugacy classes of subgroups.

The subgroup lattice has 272 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄×D₈×C₅₃	C₂	1	1	yes	maximal, characteristic
2	C₂×C₈×D₄×C₅₃	C₂	1	2	yes	maximal, complemented
3	C₅₃×#1580	C₂	1	8	yes	maximal, direct product factor
4	C₂×C₅₃×#334	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×C₅₃×D₄⋊C₈	C₂	1	2	yes	maximal, complemented
6	C₂×C₅₃×#322	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂²×D₈×C₅₃	C₄	1	1	yes	characteristic
8	C₂³×C₈×C₅₃	—	2	2	no	—
9	C₂×C₄×D₄×C₅₃	C₂²	1	2	yes	—
10	C₄×D₈×C₅₃	C₄	1	4	yes	—
11	C₄×D₈×C₅₃	C₂²	1	4	yes	—
12	C₈×D₄×C₅₃	—	2	4	no	—
13	C₈×D₄×C₅₃	C₂²	1	8	yes	complemented
14	C₂×#107×C₅₃	C₄	1	1	yes	characteristic
15	C₂×C₄⋊C₈×C₅₃	C₂²	1	2	yes	—
16	C₂×#102×C₅₃	C₄	1	2	yes	—
17	C₂×C₂²⋊C₈×C₅₃	—	2	2	no	—
18	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
19	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
20	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
21	C₅₃×#334	C₂²	1	4	yes	complemented
22	C₅₃×D₄⋊C₈	C₂²	1	8	yes	complemented
23	C₅₃×#322	C₂²	1	4	yes	complemented
24	C₂²×D₄×C₅₃	C₂×C₄	1	2	yes	—
25	C₂³×C₄×C₅₃	—	2	2	no	—
26	C₂×D₈×C₅₃	C₈	1	8	yes	complemented
27	C₂×D₈×C₅₃	C₂×C₄	1	4	yes	—
28	C₂²×C₈×C₅₃	—	2	8	no	—
29	C₂²×C₈×C₅₃	—	2	2	no	—
30	C₂²×C₈×C₅₃	—	2	1	no	—
31	C₂²×C₈×C₅₃	—	4	8	no	—
32	C₂²×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
33	C₂²×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
34	C₂²×C₈×C₅₃	D₄	1	2	yes	—
35	C₄×D₄×C₅₃	—	2	2	no	—
36	C₄×D₄×C₅₃	—	2	2	no	—
37	C₄×D₄×C₅₃	C₂×C₄	1	4	yes	—
38	C₄×D₄×C₅₃	C₂³	1	4	yes	—
39	C₂×C₄⋊C₄×C₅₃	C₂×C₄	1	2	yes	—
40	C₂×C₂²⋊C₄×C₅₃	—	2	2	no	—
41	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
42	#107×C₅₃	C₂×C₄	1	4	yes	—
43	C₄⋊C₈×C₅₃	—	2	2	no	—
44	C₄⋊C₈×C₅₃	C₂³	1	4	yes	—
45	#102×C₅₃	C₂×C₄	1	8	yes	—
46	C₂²⋊C₈×C₅₃	—	2	8	no	—
47	C₄×C₈×C₅₃	—	2	1	no	—
48	C₄×C₈×C₅₃	C₂×C₄	1	2	yes	—
49	C₄×C₈×C₅₃	D₄	1	2	yes	—
50	C₄×C₈×C₅₃	C₂³	1	2	yes	—
51	C₄×C₈×C₅₃	C₂³	1	2	yes	—
52	C₄×C₈×C₅₃	C₂³	1	2	yes	—
53	C₂⁴×C₅₃	—	2	2	no	—
54	C₂×D₄×C₅₃	—	2	2	no	—
55	C₂×D₄×C₅₃	—	2	4	no	—
56	C₂×D₄×C₅₃	C₂×C₈	1	8	yes	complemented
57	C₂×D₄×C₅₃	C₂²×C₄	1	4	yes	—
58	C₂²×C₄×C₅₃	—	2	2	no	—
59	C₂²×C₄×C₅₃	—	2	4	no	—
60	C₂²×C₄×C₅₃	—	2	4	no	—
61	C₂²×C₄×C₅₃	—	4	4	no	—
62	C₂²×C₄×C₅₃	—	4	4	no	—
63	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
64	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
65	C₂²×C₄×C₅₃	#41	1	1	yes	characteristic
66	D₈×C₅₃	C₂×C₈	1	16	yes	complemented
67	C₂×C₈×C₅₃	—	2	2	no	—
68	C₂×C₈×C₅₃	—	2	4	no	—
69	C₂×C₈×C₅₃	—	2	4	no	—
70	C₂×C₈×C₅₃	—	2	2	no	—
71	C₂×C₈×C₅₃	—	4	16	no	—
72	C₂×C₈×C₅₃	—	4	2	no	—
73	C₂×C₈×C₅₃	—	4	4	no	—
74	C₂×C₈×C₅₃	C₂×C₈	1	4	yes	complemented
75	C₂×C₈×C₅₃	C₂²×C₄	1	2	yes	—
76	C₂×C₈×C₅₃	C₂²×C₄	1	2	yes	—
77	C₂×C₈×C₅₃	C₂×D₄	1	4	yes	—
78	C₄⋊C₄×C₅₃	—	2	2	no	—
79	C₄⋊C₄×C₅₃	C₂²×C₄	1	4	yes	—
80	C₂²⋊C₄×C₅₃	—	2	4	no	—
81	C₂²⋊C₄×C₅₃	—	2	4	no	—
82	C₄²×C₅₃	C₂²×C₄	1	1	yes	characteristic
83	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
84	C₄²×C₅₃	#41	1	1	yes	characteristic
85	C₄²×C₅₃	C₂⁴	1	1	yes	characteristic
86	C₂³×C₅₃	—	2	4	no	—
87	C₂³×C₅₃	—	2	8	no	—
88	C₂³×C₅₃	—	4	4	no	—
89	C₂³×C₅₃	—	4	4	no	—
90	C₂³×C₅₃	C₄×D₄	1	1	yes	characteristic
91	D₄×C₅₃	—	2	4	no	—
92	D₄×C₅₃	—	2	8	no	—
93	D₄×C₅₃	C₂²×C₈	1	8	yes	complemented
94	C₂×C₄×C₅₃	—	2	2	no	—
95	C₂×C₄×C₅₃	—	2	4	no	—
96	C₂×C₄×C₅₃	—	4	4	no	—
97	C₂×C₄×C₅₃	—	4	4	no	—
98	C₂×C₄×C₅₃	—	4	4	no	—
99	C₂×C₄×C₅₃	—	4	8	no	—
100	C₂×C₄×C₅₃	C₄×D₄	1	1	yes	characteristic
101	C₂×C₄×C₅₃	C₄×D₄	1	1	yes	characteristic
102	C₂×C₄×C₅₃	C₄×D₄	1	1	yes	characteristic
103	C₂×C₄×C₅₃	C₂²×C₈	1	2	yes	—
104	C₂×C₄×C₅₃	#131	1	2	yes	—
105	C₂×C₄×C₅₃	AGL(1,ℤ/8)	1	4	yes	—
106	C₂×C₄×C₅₃	C₂³×C₄	1	1	yes	characteristic
107	C₂×C₄×C₅₃	C₂²×D₄	1	1	yes	characteristic
108	C₂×C₄×C₅₃	C₂×#41	1	1	yes	characteristic
109	C₈×C₅₃	—	2	2	no	—
110	C₈×C₅₃	—	2	4	no	—
111	C₈×C₅₃	—	4	2	no	—
112	C₈×C₅₃	—	4	4	no	—
113	C₈×C₅₃	C₂²×C₈	1	4	yes	complemented
114	C₂×#1580	C₅₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
115	C₂²×C₅₃	—	2	8	no	—
116	C₂²×C₅₃	—	4	16	no	—
117	C₂²×C₅₃	—	4	4	no	—
118	C₂²×C₅₃	—	4	4	no	—
119	C₂²×C₅₃	C₈×D₄	1	2	yes	—
120	C₂²×C₅₃	#442	1	2	yes	—
121	C₂²×C₅₃	#444	1	2	yes	—
122	C₂²×C₅₃	C₂×C₄×D₄	1	1	yes	characteristic
123	C₄×C₅₃	—	2	1	no	—
124	C₄×C₅₃	—	2	1	no	—
125	C₄×C₅₃	—	4	4	no	—
126	C₄×C₅₃	C₈×D₄	1	2	yes	—
127	C₄×C₅₃	#442	1	2	yes	—
128	C₄×C₅₃	C₂³×C₈	1	1	yes	characteristic
129	C₄×C₅₃	C₂×#131	1	1	yes	characteristic
130	C₄×C₅₃	C₂×AGL(1,ℤ/8)	1	2	yes	—
131	C₂×C₄×D₈	C₂×C₅₃	1	1	yes	characteristic
132	C₂×C₈×D₄	C₂×C₅₃	1	2	yes	complemented
133	#1580	C₂×C₅₃	1	8	yes	direct product factor
134	C₂×#334	C₂×C₅₃	1	1	yes	characteristic, complemented
135	C₂×D₄⋊C₈	C₂×C₅₃	1	2	yes	complemented
136	C₂×#322	C₂×C₅₃	1	1	yes	characteristic, complemented
137	C₂×C₅₃	—	4	8	no	—
138	C₂×C₅₃	#1580	1	4	yes	direct product factor
139	C₂×C₅₃	C₂×C₈×D₄	1	1	yes	characteristic
140	C₂×C₅₃	C₂×#442	1	1	yes	characteristic
141	C₂×C₅₃	C₂×#444	1	1	yes	characteristic
142	C₂²×D₈	C₄×C₅₃	1	1	yes	characteristic
143	C₂³×C₈	—	2	2	no	—
144	C₂×C₄×D₄	C₂²×C₅₃	1	2	yes	—
145	C₄×D₈	C₄×C₅₃	1	4	yes	—
146	C₄×D₈	C₂²×C₅₃	1	4	yes	—
147	C₈×D₄	—	2	4	no	—
148	C₈×D₄	C₂²×C₅₃	1	8	yes	complemented
149	C₂×#107	C₄×C₅₃	1	1	yes	characteristic
150	C₂×C₄⋊C₈	C₂²×C₅₃	1	2	yes	—
151	C₂×#102	C₄×C₅₃	1	2	yes	—
152	C₂×C₂²⋊C₈	—	2	2	no	—
153	C₂×C₄×C₈	C₂²×C₅₃	1	1	yes	characteristic
154	C₂×C₄×C₈	C₂²×C₅₃	1	1	yes	characteristic
155	C₂×C₄×C₈	C₂²×C₅₃	1	1	yes	characteristic
156	#334	C₂²×C₅₃	1	4	yes	complemented
157	D₄⋊C₈	C₂²×C₅₃	1	8	yes	complemented
158	#322	C₂²×C₅₃	1	4	yes	complemented
159	C₅₃	C₂×#1580	1	1	yes	characteristic, direct product factor, Sylow
160	C₂²×D₄	C₂×C₄×C₅₃	1	2	yes	—
161	C₂³×C₄	—	2	2	no	—
162	C₂×D₈	C₈×C₅₃	1	8	yes	complemented
163	C₂×D₈	C₂×C₄×C₅₃	1	4	yes	—
164	C₂²×C₈	—	2	2	no	—
165	C₂²×C₈	—	2	1	no	—
166	C₂²×C₈	—	2	8	no	—
167	C₂²×C₈	—	4	8	no	—
168	C₂²×C₈	C₂×C₄×C₅₃	1	1	yes	characteristic
169	C₂²×C₈	C₂×C₄×C₅₃	1	1	yes	characteristic
170	C₂²×C₈	D₄×C₅₃	1	2	yes	—
171	C₄×D₄	—	2	2	no	—
172	C₄×D₄	—	2	2	no	—
173	C₄×D₄	C₂×C₄×C₅₃	1	4	yes	—
174	C₄×D₄	C₂³×C₅₃	1	4	yes	—
175	C₂×C₄⋊C₄	C₂×C₄×C₅₃	1	2	yes	—
176	C₂×C₂²⋊C₄	—	2	2	no	—
177	C₂×C₄²	C₂³×C₅₃	1	1	yes	characteristic
178	#107	C₂×C₄×C₅₃	1	4	yes	—
179	C₄⋊C₈	—	2	2	no	—
180	C₄⋊C₈	C₂³×C₅₃	1	4	yes	—
181	#102	C₂×C₄×C₅₃	1	8	yes	—
182	C₂²⋊C₈	—	2	8	no	—
183	C₄×C₈	—	2	1	no	—
184	C₄×C₈	C₂×C₄×C₅₃	1	2	yes	—
185	C₄×C₈	D₄×C₅₃	1	2	yes	—
186	C₄×C₈	C₂³×C₅₃	1	2	yes	—
187	C₄×C₈	C₂³×C₅₃	1	2	yes	—
188	C₄×C₈	C₂³×C₅₃	1	2	yes	—
189	C₂⁴	—	2	2	no	—
190	C₂×D₄	—	2	2	no	—
191	C₂×D₄	—	2	4	no	—
192	C₂×D₄	C₂×C₈×C₅₃	1	8	yes	complemented
193	C₂×D₄	C₂²×C₄×C₅₃	1	4	yes	—
194	C₂²×C₄	—	2	2	no	—
195	C₂²×C₄	—	2	4	no	—
196	C₂²×C₄	—	2	4	no	—
197	C₂²×C₄	—	4	4	no	—
198	C₂²×C₄	—	4	4	no	—
199	C₂²×C₄	C₂²×C₄×C₅₃	1	1	yes	characteristic
200	C₂²×C₄	C₂×D₄×C₅₃	1	1	yes	characteristic
201	C₂²×C₄	#41×C₅₃	1	1	yes	characteristic
202	D₈	C₂×C₈×C₅₃	1	16	yes	complemented
203	C₂×C₈	—	2	2	no	—
204	C₂×C₈	—	2	4	no	—
205	C₂×C₈	—	2	4	no	—
206	C₂×C₈	—	2	2	no	—
207	C₂×C₈	—	4	16	no	—
208	C₂×C₈	—	4	2	no	—
209	C₂×C₈	—	4	4	no	—
210	C₂×C₈	C₂×C₈×C₅₃	1	4	yes	complemented
211	C₂×C₈	C₂²×C₄×C₅₃	1	2	yes	—
212	C₂×C₈	C₂²×C₄×C₅₃	1	2	yes	—
213	C₂×C₈	C₂×D₄×C₅₃	1	4	yes	—
214	C₄⋊C₄	—	2	2	no	—
215	C₄⋊C₄	C₂²×C₄×C₅₃	1	4	yes	—
216	C₂²⋊C₄	—	2	4	no	—
217	C₂²⋊C₄	—	2	4	no	—
218	C₄²	C₂²×C₄×C₅₃	1	1	yes	characteristic
219	C₄²	C₂×D₄×C₅₃	1	1	yes	characteristic
220	C₄²	#41×C₅₃	1	1	yes	characteristic
221	C₄²	C₂⁴×C₅₃	1	1	yes	characteristic
222	C₂³	—	2	4	no	—
223	C₂³	—	2	8	no	—
224	C₂³	—	4	4	no	—
225	C₂³	—	4	4	no	—
226	C₂³	C₄×D₄×C₅₃	1	1	yes	characteristic
227	D₄	—	2	4	no	—
228	D₄	—	2	8	no	—
229	D₄	C₂²×C₈×C₅₃	1	8	yes	complemented
230	C₂×C₄	—	2	2	no	—
231	C₂×C₄	—	2	4	no	—
232	C₂×C₄	—	4	4	no	—
233	C₂×C₄	—	4	4	no	—
234	C₂×C₄	—	4	4	no	—
235	C₂×C₄	—	4	8	no	—
236	C₂×C₄	C₄×D₄×C₅₃	1	1	yes	characteristic
237	C₂×C₄	C₄×D₄×C₅₃	1	1	yes	characteristic
238	C₂×C₄	C₄×D₄×C₅₃	1	1	yes	characteristic
239	C₂×C₄	C₂²×C₈×C₅₃	1	2	yes	—
240	C₂×C₄	#131×C₅₃	1	2	yes	—
241	C₂×C₄	AGL(1,ℤ/8)×C₅₃	1	4	yes	—
242	C₂×C₄	C₂³×C₄×C₅₃	1	1	yes	characteristic
243	C₂×C₄	C₂²×D₄×C₅₃	1	1	yes	characteristic
244	C₂×C₄	C₂×#41×C₅₃	1	1	yes	characteristic
245	C₈	—	2	2	no	—
246	C₈	—	2	4	no	—
247	C₈	—	4	2	no	—
248	C₈	—	4	4	no	—
249	C₈	C₂²×C₈×C₅₃	1	4	yes	complemented
250	C₂²	—	2	8	no	—
251	C₂²	—	4	4	no	—
252	C₂²	—	4	4	no	—
253	C₂²	—	4	16	no	—
254	C₂²	C₈×D₄×C₅₃	1	2	yes	—
255	C₂²	C₅₃×#442	1	2	yes	—
256	C₂²	C₅₃×#444	1	2	yes	—
257	C₂²	C₂×C₄×D₄×C₅₃	1	1	yes	characteristic
258	C₄	—	2	1	no	—
259	C₄	—	2	1	no	—
260	C₄	—	4	4	no	—
261	C₄	C₈×D₄×C₅₃	1	2	yes	—
262	C₄	C₅₃×#442	1	2	yes	—
263	C₄	C₂³×C₈×C₅₃	1	1	yes	characteristic
264	C₄	C₂×#131×C₅₃	1	1	yes	characteristic
265	C₄	C₂×AGL(1,ℤ/8)×C₅₃	1	2	yes	—
266	C₂	—	4	8	no	—
267	C₂	C₅₃×#1580	1	4	yes	direct product factor
268	C₂	C₂×C₈×D₄×C₅₃	1	1	yes	characteristic
269	C₂	C₂×C₅₃×#442	1	1	yes	characteristic
270	C₂	C₂×C₅₃×#444	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄×C₅₃	C₂×D₄
commutator	C₄	C₂³×C₈×C₅₃
Frattini, cosocle	C₄²	C₂⁴×C₅₃
socle	C₂³×C₅₃	C₄×D₄
prime radical	C₂²×D₈×C₅₃	C₄
LCS	G ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄×C₅₃ ◁ C₂×C₄×C₈×C₅₃ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂×C₈×D₄×C₅₃	C₂	semidirect
3. C₅₃×#1580	C₂	direct (N × H)
4. C₂×C₅₃×#334	C₂	semidirect
5. C₂×C₅₃×D₄⋊C₈	C₂	semidirect
6. C₂×C₅₃×#322	C₂	semidirect
13. C₈×D₄×C₅₃	C₂²	semidirect
21. C₅₃×#334	C₂²	semidirect
22. C₅₃×D₄⋊C₈	C₂²	semidirect
23. C₅₃×#322	C₂²	semidirect
26. C₂×D₈×C₅₃	C₈	semidirect
56. C₂×D₄×C₅₃	C₂×C₈	semidirect
66. D₈×C₅₃	C₂×C₈	semidirect
74. C₂×C₈×C₅₃	C₂×C₈	semidirect
93. D₄×C₅₃	C₂²×C₈	semidirect
113. C₈×C₅₃	C₂²×C₈	semidirect
114. C₂×#1580	C₅₃	direct (N × H)
132. C₂×C₈×D₄	C₂×C₅₃	semidirect
133. #1580	C₂×C₅₃	direct (N × H)
134. C₂×#334	C₂×C₅₃	semidirect
135. C₂×D₄⋊C₈	C₂×C₅₃	semidirect
136. C₂×#322	C₂×C₅₃	semidirect
138. C₂×C₅₃	#1580	direct (N × H)
148. C₈×D₄	C₂²×C₅₃	semidirect
156. #334	C₂²×C₅₃	semidirect
157. D₄⋊C₈	C₂²×C₅₃	semidirect
158. #322	C₂²×C₅₃	semidirect
159. C₅₃	C₂×#1580	direct (N × H)
162. C₂×D₈	C₈×C₅₃	semidirect
192. C₂×D₄	C₂×C₈×C₅₃	semidirect
202. D₈	C₂×C₈×C₅₃	semidirect
210. C₂×C₈	C₂×C₈×C₅₃	semidirect
229. D₄	C₂²×C₈×C₅₃	semidirect
249. C₈	C₂²×C₈×C₅₃	semidirect
267. C₂	C₅₃×#1580	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×C₄×D₈×C₅₃	2
C₂×C₈×D₄×C₅₃	2
C₅₃×#1580	2
C₂×C₅₃×#334	2
C₂×C₅₃×D₄⋊C₈	2
C₂×C₅₃×#322	2
C₂×#1580	53

Subgroups — C2×C53×#1580 #27225360

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₅₃×#1580 #27225360