Subgroups — C₅₃×#12731 #27220243

Order 13568.

1190 conjugacy classes of subgroups.

The subgroup lattice has 294 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂³×D₈×C₅₃	C₂	1	1	yes	maximal, characteristic
2	C₂×C₅₃×#357	C₂	1	4	yes	maximal, complemented
3	C₂×C₅₃×C₂²⋊C₁₆	C₂	1	1	yes	maximal, characteristic, complemented
4	C₅₃×#1808	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂³×D₄×C₅₃	—	2	1	no	—
6	C₂²×D₈×C₅₃	—	2	8	no	—
7	C₂²×D₈×C₅₃	—	2	2	no	—
8	C₂²×D₈×C₅₃	—	2	1	no	—
9	C₂²×D₈×C₅₃	C₄	1	4	yes	complemented
10	C₂²×D₈×C₅₃	C₂²	1	2	yes	—
11	C₂³×C₈×C₅₃	C₂²	1	1	yes	characteristic
12	C₂²×C₁₆×C₅₃	C₂²	1	2	yes	complemented
13	C₂×#107×C₅₃	C₂²	1	2	yes	complemented
14	C₅₃×#380	—	2	1	no	—
15	C₅₃×#357	—	2	8	no	—
16	C₅₃×C₂²⋊C₁₆	—	2	2	no	—
17	C₅₃×#340	—	2	1	no	—
18	C₂⁵×C₅₃	—	4	1	no	—
19	C₂²×D₄×C₅₃	—	2	8	no	—
20	C₂²×D₄×C₅₃	—	2	4	no	—
21	C₂²×D₄×C₅₃	—	2	2	no	—
22	C₂²×D₄×C₅₃	—	4	4	no	—
23	C₂²×D₄×C₅₃	—	4	1	no	—
24	C₂²×D₄×C₅₃	—	4	2	no	—
25	C₂³×C₄×C₅₃	D₄	1	1	yes	characteristic
26	C₂×D₈×C₅₃	—	2	16	no	—
27	C₂×D₈×C₅₃	—	2	8	no	—
28	C₂×D₈×C₅₃	—	2	16	no	—
29	C₂×D₈×C₅₃	—	2	4	no	—
30	C₂×D₈×C₅₃	—	2	2	no	—
31	C₂×D₈×C₅₃	—	2	4	no	—
32	C₂×D₈×C₅₃	C₂×C₄	1	4	yes	—
33	C₂×D₈×C₅₃	D₄	1	8	yes	—
34	C₂²×C₈×C₅₃	—	2	4	no	—
35	C₂²×C₈×C₅₃	C₂×C₄	1	2	yes	—
36	C₂²×C₈×C₅₃	D₄	1	2	yes	—
37	C₂²×C₈×C₅₃	D₄	1	1	yes	characteristic
38	C₂²×C₈×C₅₃	C₂³	1	1	yes	characteristic
39	C₂×C₄⋊C₄×C₅₃	—	2	2	no	—
40	C₂×C₂²⋊C₄×C₅₃	—	4	1	no	—
41	C₂×C₁₆×C₅₃	—	2	2	no	—
42	C₂×C₁₆×C₅₃	—	2	4	no	—
43	#107×C₅₃	—	2	4	no	—
44	#95×C₅₃	—	4	1	no	—
45	C₂⁴×C₅₃	—	4	8	no	—
46	C₂⁴×C₅₃	—	4	4	no	—
47	C₂⁴×C₅₃	—	4	2	no	—
48	C₂⁴×C₅₃	—	8	4	no	—
49	C₂⁴×C₅₃	—	8	2	no	—
50	C₂⁴×C₅₃	—	8	2	no	—
51	C₂⁴×C₅₃	D₈	1	1	yes	characteristic
52	C₂×D₄×C₅₃	—	2	16	no	—
53	C₂×D₄×C₅₃	—	2	4	no	—
54	C₂×D₄×C₅₃	—	2	8	no	—
55	C₂×D₄×C₅₃	—	4	8	no	—
56	C₂×D₄×C₅₃	—	4	8	no	—
57	C₂×D₄×C₅₃	—	4	16	no	—
58	C₂×D₄×C₅₃	—	4	4	no	—
59	C₂×D₄×C₅₃	—	4	2	no	—
60	C₂×D₄×C₅₃	—	4	4	no	—
61	C₂²×C₄×C₅₃	—	2	2	no	—
62	C₂²×C₄×C₅₃	—	2	2	no	—
63	C₂²×C₄×C₅₃	—	4	2	no	—
64	C₂²×C₄×C₅₃	C₂²⋊C₄	1	2	yes	—
65	C₂²×C₄×C₅₃	D₈	1	1	yes	characteristic
66	C₂²×C₄×C₅₃	SD₁₆	1	2	yes	—
67	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
68	D₈×C₅₃	—	2	16	no	—
69	D₈×C₅₃	—	2	4	no	—
70	D₈×C₅₃	—	2	8	no	—
71	D₈×C₅₃	C₂²⋊C₄	1	8	yes	complemented
72	C₂×C₈×C₅₃	—	2	2	no	—
73	C₂×C₈×C₅₃	—	2	4	no	—
74	C₂×C₈×C₅₃	—	2	4	no	—
75	C₂×C₈×C₅₃	C₂²⋊C₄	1	2	yes	—
76	C₂×C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
77	C₂×C₈×C₅₃	C₂×D₄	1	4	yes	—
78	C₂×C₈×C₅₃	C₂×D₄	1	1	yes	characteristic
79	C₄⋊C₄×C₅₃	—	4	2	no	—
80	C₄⋊C₄×C₅₃	—	4	2	no	—
81	C₂²⋊C₄×C₅₃	—	8	2	no	—
82	C₁₆×C₅₃	—	4	2	no	—
83	C₂³×C₅₃	—	2	2	no	—
84	C₂³×C₅₃	—	2	2	no	—
85	C₂³×C₅₃	—	4	16	no	—
86	C₂³×C₅₃	—	4	4	no	—
87	C₂³×C₅₃	—	4	8	no	—
88	C₂³×C₅₃	—	8	8	no	—
89	C₂³×C₅₃	—	8	16	no	—
90	C₂³×C₅₃	—	8	16	no	—
91	C₂³×C₅₃	—	8	8	no	—
92	C₂³×C₅₃	—	8	2	no	—
93	C₂³×C₅₃	—	8	2	no	—
94	C₂³×C₅₃	—	8	4	no	—
95	C₂³×C₅₃	#102	1	2	yes	—
96	C₂³×C₅₃	D₁₆	1	2	yes	—
97	C₂³×C₅₃	SD₃₂	1	2	yes	—
98	C₂³×C₅₃	C₂×D₈	1	1	yes	characteristic
99	D₄×C₅₃	—	2	8	no	—
100	D₄×C₅₃	—	4	16	no	—
101	D₄×C₅₃	—	4	4	no	—
102	D₄×C₅₃	—	4	8	no	—
103	C₂×C₄×C₅₃	—	2	2	no	—
104	C₂×C₄×C₅₃	—	2	2	no	—
105	C₂×C₄×C₅₃	—	2	2	no	—
106	C₂×C₄×C₅₃	—	2	4	no	—
107	C₂×C₄×C₅₃	—	8	2	no	—
108	C₂×C₄×C₅₃	—	8	2	no	—
109	C₂×C₄×C₅₃	—	8	2	no	—
110	C₂×C₄×C₅₃	#102	1	2	yes	—
111	C₂×C₄×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
112	C₂×C₄×C₅₃	#120	1	2	yes	—
113	C₂×C₄×C₅₃	C₂×SD₁₆	1	1	yes	characteristic
114	C₂×C₄×C₅₃	AGL(1,ℤ/8)	1	2	yes	—
115	C₈×C₅₃	—	2	2	no	—
116	C₈×C₅₃	C₂×C₂²⋊C₄	1	2	yes	—
117	C₈×C₅₃	#120	1	2	yes	—
118	#12731	C₅₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
119	C₂²×C₅₃	—	2	2	no	—
120	C₂²×C₅₃	—	2	4	no	—
121	C₂²×C₅₃	—	2	4	no	—
122	C₂²×C₅₃	—	2	2	no	—
123	C₂²×C₅₃	—	4	8	no	—
124	C₂²×C₅₃	—	8	32	no	—
125	C₂²×C₅₃	—	8	4	no	—
126	C₂²×C₅₃	—	8	4	no	—
127	C₂²×C₅₃	—	8	8	no	—
128	C₂²×C₅₃	—	8	8	no	—
129	C₂²×C₅₃	#357	1	4	yes	complemented
130	C₂²×C₅₃	C₂×#102	1	1	yes	characteristic
131	C₂²×C₅₃	#447	1	2	yes	—
132	C₂²×C₅₃	C₂×D₁₆	1	1	yes	characteristic
133	C₂²×C₅₃	C₂×SD₃₂	1	1	yes	characteristic
134	C₂²×C₅₃	#509	1	2	yes	—
135	C₄×C₅₃	—	2	2	no	—
136	C₄×C₅₃	—	16	2	no	—
137	C₄×C₅₃	#379	1	1	yes	characteristic
138	C₄×C₅₃	#418	1	1	yes	characteristic
139	C₄×C₅₃	#450	1	2	yes	—
140	C₂³×D₈	C₂×C₅₃	1	1	yes	characteristic
141	C₂×#357	C₂×C₅₃	1	4	yes	complemented
142	C₂×C₂²⋊C₁₆	C₂×C₅₃	1	1	yes	characteristic, complemented
143	#1808	C₂×C₅₃	1	1	yes	characteristic, complemented
144	C₂×C₅₃	—	2	4	no	—
145	C₂×C₅₃	—	8	8	no	—
146	C₂×C₅₃	#1786	1	1	yes	characteristic
147	C₂×C₅₃	C₂×#357	1	1	yes	characteristic
148	C₂×C₅₃	#2140	1	1	yes	characteristic
149	C₂×C₅₃	#2184	1	2	yes	—
150	C₂×C₅₃	#2187	1	2	yes	—
151	C₂³×D₄	—	2	1	no	—
152	C₂²×D₈	—	2	8	no	—
153	C₂²×D₈	—	2	2	no	—
154	C₂²×D₈	—	2	1	no	—
155	C₂²×D₈	C₄×C₅₃	1	4	yes	complemented
156	C₂²×D₈	C₂²×C₅₃	1	2	yes	—
157	C₂³×C₈	C₂²×C₅₃	1	1	yes	characteristic
158	C₂²×C₁₆	C₂²×C₅₃	1	2	yes	complemented
159	C₂×#107	C₂²×C₅₃	1	2	yes	complemented
160	#380	—	2	1	no	—
161	#357	—	2	8	no	—
162	C₂²⋊C₁₆	—	2	2	no	—
163	#340	—	2	1	no	—
164	C₅₃	#12731	1	1	yes	characteristic, direct product factor, Sylow
165	C₂⁵	—	4	1	no	—
166	C₂²×D₄	—	2	8	no	—
167	C₂²×D₄	—	2	2	no	—
168	C₂²×D₄	—	2	4	no	—
169	C₂²×D₄	—	4	4	no	—
170	C₂²×D₄	—	4	1	no	—
171	C₂²×D₄	—	4	2	no	—
172	C₂³×C₄	D₄×C₅₃	1	1	yes	characteristic
173	C₂×D₈	—	2	8	no	—
174	C₂×D₈	—	2	16	no	—
175	C₂×D₈	—	2	16	no	—
176	C₂×D₈	—	2	4	no	—
177	C₂×D₈	—	2	2	no	—
178	C₂×D₈	—	2	4	no	—
179	C₂×D₈	C₂×C₄×C₅₃	1	4	yes	—
180	C₂×D₈	D₄×C₅₃	1	8	yes	—
181	C₂²×C₈	—	2	4	no	—
182	C₂²×C₈	C₂×C₄×C₅₃	1	2	yes	—
183	C₂²×C₈	D₄×C₅₃	1	2	yes	—
184	C₂²×C₈	D₄×C₅₃	1	1	yes	characteristic
185	C₂²×C₈	C₂³×C₅₃	1	1	yes	characteristic
186	C₂×C₄⋊C₄	—	2	2	no	—
187	C₂×C₂²⋊C₄	—	4	1	no	—
188	C₂×C₁₆	—	2	2	no	—
189	C₂×C₁₆	—	2	4	no	—
190	#107	—	2	4	no	—
191	#95	—	4	1	no	—
192	C₂⁴	—	4	8	no	—
193	C₂⁴	—	4	4	no	—
194	C₂⁴	—	4	2	no	—
195	C₂⁴	—	8	4	no	—
196	C₂⁴	—	8	2	no	—
197	C₂⁴	—	8	2	no	—
198	C₂⁴	D₈×C₅₃	1	1	yes	characteristic
199	C₂×D₄	—	2	16	no	—
200	C₂×D₄	—	2	4	no	—
201	C₂×D₄	—	2	8	no	—
202	C₂×D₄	—	4	8	no	—
203	C₂×D₄	—	4	8	no	—
204	C₂×D₄	—	4	16	no	—
205	C₂×D₄	—	4	4	no	—
206	C₂×D₄	—	4	2	no	—
207	C₂×D₄	—	4	4	no	—
208	C₂²×C₄	—	2	2	no	—
209	C₂²×C₄	—	2	2	no	—
210	C₂²×C₄	—	4	2	no	—
211	C₂²×C₄	C₂²⋊C₄×C₅₃	1	2	yes	—
212	C₂²×C₄	D₈×C₅₃	1	1	yes	characteristic
213	C₂²×C₄	SD₁₆×C₅₃	1	2	yes	—
214	C₂²×C₄	C₂×D₄×C₅₃	1	1	yes	characteristic
215	D₈	—	2	16	no	—
216	D₈	—	2	4	no	—
217	D₈	—	2	8	no	—
218	D₈	C₂²⋊C₄×C₅₃	1	8	yes	complemented
219	C₂×C₈	—	2	2	no	—
220	C₂×C₈	—	2	4	no	—
221	C₂×C₈	—	2	4	no	—
222	C₂×C₈	C₂²⋊C₄×C₅₃	1	2	yes	—
223	C₂×C₈	C₂²×C₄×C₅₃	1	1	yes	characteristic
224	C₂×C₈	C₂×D₄×C₅₃	1	4	yes	—
225	C₂×C₈	C₂×D₄×C₅₃	1	1	yes	characteristic
226	C₄⋊C₄	—	4	2	no	—
227	C₄⋊C₄	—	4	2	no	—
228	C₂²⋊C₄	—	8	2	no	—
229	C₁₆	—	4	2	no	—
230	C₂³	—	2	2	no	—
231	C₂³	—	2	2	no	—
232	C₂³	—	4	16	no	—
233	C₂³	—	4	4	no	—
234	C₂³	—	4	8	no	—
235	C₂³	—	8	16	no	—
236	C₂³	—	8	16	no	—
237	C₂³	—	8	8	no	—
238	C₂³	—	8	2	no	—
239	C₂³	—	8	2	no	—
240	C₂³	—	8	4	no	—
241	C₂³	—	8	8	no	—
242	C₂³	#102×C₅₃	1	2	yes	—
243	C₂³	D₁₆×C₅₃	1	2	yes	—
244	C₂³	SD₃₂×C₅₃	1	2	yes	—
245	C₂³	C₂×D₈×C₅₃	1	1	yes	characteristic
246	D₄	—	2	8	no	—
247	D₄	—	4	16	no	—
248	D₄	—	4	4	no	—
249	D₄	—	4	8	no	—
250	C₂×C₄	—	2	2	no	—
251	C₂×C₄	—	2	2	no	—
252	C₂×C₄	—	2	2	no	—
253	C₂×C₄	—	2	4	no	—
254	C₂×C₄	—	8	2	no	—
255	C₂×C₄	—	8	2	no	—
256	C₂×C₄	—	8	2	no	—
257	C₂×C₄	#102×C₅₃	1	2	yes	—
258	C₂×C₄	C₂×C₂²⋊C₄×C₅₃	1	1	yes	characteristic
259	C₂×C₄	#120×C₅₃	1	2	yes	—
260	C₂×C₄	C₂×SD₁₆×C₅₃	1	1	yes	characteristic
261	C₂×C₄	AGL(1,ℤ/8)×C₅₃	1	2	yes	—
262	C₈	—	2	2	no	—
263	C₈	C₂×C₂²⋊C₄×C₅₃	1	2	yes	—
264	C₈	#120×C₅₃	1	2	yes	—
265	C₂²	—	2	4	no	—
266	C₂²	—	2	4	no	—
267	C₂²	—	2	2	no	—
268	C₂²	—	2	2	no	—
269	C₂²	—	4	8	no	—
270	C₂²	—	8	32	no	—
271	C₂²	—	8	8	no	—
272	C₂²	—	8	8	no	—
273	C₂²	—	8	4	no	—
274	C₂²	—	8	4	no	—
275	C₂²	C₅₃×#357	1	4	yes	complemented
276	C₂²	C₂×#102×C₅₃	1	1	yes	characteristic
277	C₂²	C₅₃×#447	1	2	yes	—
278	C₂²	C₂×D₁₆×C₅₃	1	1	yes	characteristic
279	C₂²	C₂×SD₃₂×C₅₃	1	1	yes	characteristic
280	C₂²	C₅₃×#509	1	2	yes	—
281	C₄	—	2	2	no	—
282	C₄	—	16	2	no	—
283	C₄	C₅₃×#379	1	1	yes	characteristic
284	C₄	C₅₃×#418	1	1	yes	characteristic
285	C₄	C₅₃×#450	1	2	yes	—
286	C₂	—	2	4	no	—
287	C₂	—	8	8	no	—
288	C₂	C₅₃×#1786	1	1	yes	characteristic
289	C₂	C₂×C₅₃×#357	1	1	yes	characteristic
290	C₂	C₅₃×#2140	1	1	yes	characteristic
291	C₂	C₅₃×#2184	1	2	yes	—
292	C₂	C₅₃×#2187	1	2	yes	—

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

Complements

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

N	H ≅ G/N	type
2. C₂×C₅₃×#357	C₂	semidirect
3. C₂×C₅₃×C₂²⋊C₁₆	C₂	semidirect
4. C₅₃×#1808	C₂	semidirect
9. C₂²×D₈×C₅₃	C₄	semidirect
12. C₂²×C₁₆×C₅₃	C₂²	semidirect
13. C₂×#107×C₅₃	C₂²	semidirect
71. D₈×C₅₃	C₂²⋊C₄	semidirect
118. #12731	C₅₃	direct (N × H)
129. C₂²×C₅₃	#357	semidirect
141. C₂×#357	C₂×C₅₃	semidirect
142. C₂×C₂²⋊C₁₆	C₂×C₅₃	semidirect
143. #1808	C₂×C₅₃	semidirect
155. C₂²×D₈	C₄×C₅₃	semidirect
158. C₂²×C₁₆	C₂²×C₅₃	semidirect
159. C₂×#107	C₂²×C₅₃	semidirect
164. C₅₃	#12731	direct (N × H)
218. D₈	C₂²⋊C₄×C₅₃	semidirect
275. C₂²	C₅₃×#357	semidirect

Maximal subgroups

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

subgroup	index
C₂³×D₈×C₅₃	2
C₂×C₅₃×#357	2
C₂×C₅₃×C₂²⋊C₁₆	2
C₅₃×#1808	2
#12731	53

Subgroups — C53×#12731 #27220243

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅₃×#12731 #27220243