Subgroups — C₅₃×#8170 #27215682

Order 13568.

718 conjugacy classes of subgroups.

The subgroup lattice has 302 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄²×D₄×C₅₃	C₂	1	1	yes	maximal, characteristic
2	C₄×C₄⋊C₈×C₅₃	C₂	1	1	yes	maximal, characteristic, complemented
3	C₅₃×#1744	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×C₅₃×D₄⋊C₈	C₂	1	2	yes	maximal
5	C₅₃×#1279	C₂	1	2	yes	maximal, complemented
6	C₂×C₄×D₄×C₅₃	C₄	1	2	yes	—
7	C₂×C₄×D₄×C₅₃	C₂²	1	1	yes	characteristic
8	C₂²×C₄²×C₅₃	—	2	1	no	—
9	C₂×C₄⋊C₈×C₅₃	C₂²	1	2	yes	—
10	C₂×C₄×C₈×C₅₃	—	2	1	no	—
11	C₂×C₄×C₈×C₅₃	C₂²	1	2	yes	—
12	C₄×C₄⋊C₄×C₅₃	C₄	1	2	yes	—
13	C₄×C₄⋊C₄×C₅₃	C₂²	1	1	yes	characteristic
14	C₄×C₂²⋊C₄×C₅₃	—	2	1	no	—
15	C₄×C₂²⋊C₄×C₅₃	—	2	2	no	—
16	C₄³×C₅₃	C₂²	1	1	yes	characteristic
17	C₅₃×#336	—	2	2	no	—
18	C₅₃×#336	—	2	1	no	—
19	C₅₃×D₄⋊C₈	C₄	1	8	yes	complemented
20	C₂²×D₄×C₅₃	C₂×C₄	1	1	yes	characteristic
21	C₂³×C₄×C₅₃	—	2	2	no	—
22	C₂³×C₄×C₅₃	—	2	1	no	—
23	C₂²×C₈×C₅₃	—	2	2	no	—
24	C₂²×C₈×C₅₃	—	2	2	no	—
25	C₄×D₄×C₅₃	—	2	4	no	—
26	C₄×D₄×C₅₃	—	2	2	no	—
27	C₄×D₄×C₅₃	C₈	1	8	yes	complemented
28	C₄×D₄×C₅₃	C₂×C₄	1	4	yes	—
29	C₂×C₄⋊C₄×C₅₃	C₂×C₄	1	1	yes	characteristic
30	C₂×C₄⋊C₄×C₅₃	C₂×C₄	1	2	yes	—
31	C₂×C₂²⋊C₄×C₅₃	—	2	2	no	—
32	C₂×C₂²⋊C₄×C₅₃	—	2	1	no	—
33	C₂×C₄²×C₅₃	—	2	2	no	—
34	C₂×C₄²×C₅₃	—	2	1	no	—
35	C₂×C₄²×C₅₃	—	4	4	no	—
36	C₂×C₄²×C₅₃	C₂×C₄	1	2	yes	—
37	C₂×C₄²×C₅₃	D₄	1	2	yes	—
38	C₂×C₄²×C₅₃	D₄	1	1	yes	characteristic
39	C₂×C₄²×C₅₃	D₄	1	1	yes	characteristic
40	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
41	C₄⋊C₈×C₅₃	—	2	2	no	—
42	C₄⋊C₈×C₅₃	C₂×C₄	1	4	yes	complemented
43	C₄×C₈×C₅₃	—	2	2	no	—
44	C₄×C₈×C₅₃	—	4	2	no	—
45	C₄×C₈×C₅₃	C₂×C₄	1	4	yes	complemented
46	#95×C₅₃	—	2	2	no	—
47	#95×C₅₃	—	2	1	no	—
48	C₂⁴×C₅₃	—	2	1	no	—
49	C₂×D₄×C₅₃	—	2	1	no	—
50	C₂×D₄×C₅₃	—	2	2	no	—
51	C₂×D₄×C₅₃	C₄²	1	2	yes	—
52	C₂×D₄×C₅₃	C₂×C₈	1	4	yes	—
53	C₂²×C₄×C₅₃	—	2	1	no	—
54	C₂²×C₄×C₅₃	—	2	2	no	—
55	C₂²×C₄×C₅₃	—	2	8	no	—
56	C₂²×C₄×C₅₃	—	2	4	no	—
57	C₂²×C₄×C₅₃	—	4	4	no	—
58	C₂²×C₄×C₅₃	—	4	4	no	—
59	C₂²×C₄×C₅₃	—	4	2	no	—
60	C₂²×C₄×C₅₃	—	4	1	no	—
61	C₂²×C₄×C₅₃	—	4	1	no	—
62	C₂²×C₄×C₅₃	C₂²⋊C₄	1	2	yes	—
63	C₂²×C₄×C₅₃	C₂²⋊C₄	1	2	yes	—
64	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
65	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
66	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
67	C₂×C₈×C₅₃	—	2	4	no	—
68	C₂×C₈×C₅₃	—	2	4	no	—
69	C₂×C₈×C₅₃	—	4	4	no	—
70	C₂×C₈×C₅₃	—	4	2	no	—
71	C₂×C₈×C₅₃	—	4	2	no	—
72	C₄⋊C₄×C₅₃	—	2	1	no	—
73	C₄⋊C₄×C₅₃	—	2	2	no	—
74	C₄⋊C₄×C₅₃	C₄²	1	2	yes	—
75	C₄⋊C₄×C₅₃	C₂×C₈	1	4	yes	—
76	C₂²⋊C₄×C₅₃	—	2	8	no	—
77	C₂²⋊C₄×C₅₃	—	2	4	no	—
78	C₄²×C₅₃	—	2	1	no	—
79	C₄²×C₅₃	—	2	2	no	—
80	C₄²×C₅₃	—	2	1	no	—
81	C₄²×C₅₃	—	4	2	no	—
82	C₄²×C₅₃	—	4	4	no	—
83	C₄²×C₅₃	—	4	2	no	—
84	C₄²×C₅₃	C₂×C₈	1	4	yes	—
85	C₄²×C₅₃	M₁₆	1	4	yes	—
86	C₄²×C₅₃	C₂²×C₄	1	2	yes	—
87	C₄²×C₅₃	#41	1	2	yes	—
88	C₂³×C₅₃	—	2	2	no	—
89	C₂³×C₅₃	—	2	4	no	—
90	C₂³×C₅₃	—	4	1	no	—
91	C₂³×C₅₃	—	4	1	no	—
92	C₂³×C₅₃	—	4	2	no	—
93	C₂³×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
94	D₄×C₅₃	—	2	2	no	—
95	D₄×C₅₃	—	2	4	no	—
96	D₄×C₅₃	C₄×C₈	1	4	yes	complemented
97	C₂×C₄×C₅₃	—	2	2	no	—
98	C₂×C₄×C₅₃	—	2	2	no	—
99	C₂×C₄×C₅₃	—	2	2	no	—
100	C₂×C₄×C₅₃	—	2	2	no	—
101	C₂×C₄×C₅₃	—	2	2	no	—
102	C₂×C₄×C₅₃	—	2	2	no	—
103	C₂×C₄×C₅₃	—	2	4	no	—
104	C₂×C₄×C₅₃	—	4	16	no	—
105	C₂×C₄×C₅₃	—	4	1	no	—
106	C₂×C₄×C₅₃	—	4	2	no	—
107	C₂×C₄×C₅₃	—	4	1	no	—
108	C₂×C₄×C₅₃	—	4	2	no	—
109	C₂×C₄×C₅₃	—	4	4	no	—
110	C₂×C₄×C₅₃	—	4	4	no	—
111	C₂×C₄×C₅₃	C₂²⋊C₈	1	4	yes	—
112	C₂×C₄×C₅₃	C₂²⋊C₈	1	4	yes	—
113	C₂×C₄×C₅₃	C₂×C₄²	1	1	yes	characteristic
114	C₂×C₄×C₅₃	#117	1	1	yes	characteristic
115	C₂×C₄×C₅₃	C₄×D₄	1	2	yes	—
116	C₂×C₄×C₅₃	C₄×D₄	1	2	yes	—
117	C₂×C₄×C₅₃	C₂²×C₈	1	2	yes	—
118	C₂×C₄×C₅₃	C₂×M₁₆	1	2	yes	—
119	C₂×C₄×C₅₃	AGL(1,ℤ/8)	1	2	yes	—
120	C₂×C₄×C₅₃	AGL(1,ℤ/8)	1	1	yes	characteristic
121	C₂×C₄×C₅₃	#137	1	1	yes	characteristic
122	C₈×C₅₃	—	4	4	no	—
123	C₈×C₅₃	—	4	4	no	—
124	#8170	C₅₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
125	C₂²×C₅₃	—	2	4	no	—
126	C₂²×C₅₃	—	4	4	no	—
127	C₂²×C₅₃	—	4	4	no	—
128	C₂²×C₅₃	—	4	2	no	—
129	C₂²×C₅₃	—	4	2	no	—
130	C₂²×C₅₃	C₄×C₂²⋊C₄	1	1	yes	characteristic
131	C₂²×C₅₃	C₂×C₂²⋊C₈	1	2	yes	—
132	C₂²×C₅₃	#417	1	1	yes	characteristic
133	C₂²×C₅₃	#418	1	1	yes	characteristic
134	C₂²×C₅₃	#421	1	2	yes	—
135	C₄×C₅₃	—	2	4	no	—
136	C₄×C₅₃	—	2	4	no	—
137	C₄×C₅₃	—	2	2	no	—
138	C₄×C₅₃	—	4	4	no	—
139	C₄×C₅₃	—	4	2	no	—
140	C₄×C₅₃	C₂×C₄×C₈	1	1	yes	characteristic
141	C₄×C₅₃	C₄×M₁₆	1	1	yes	characteristic
142	C₄×C₅₃	#431	1	2	yes	—
143	C₄×C₅₃	#440	1	2	yes	—
144	C₄×C₅₃	#442	1	2	yes	—
145	C₄²×D₄	C₂×C₅₃	1	1	yes	characteristic
146	C₄×C₄⋊C₈	C₂×C₅₃	1	1	yes	characteristic, complemented
147	#1744	C₂×C₅₃	1	1	yes	characteristic, complemented
148	C₂×D₄⋊C₈	C₂×C₅₃	1	2	yes	—
149	#1279	C₂×C₅₃	1	2	yes	complemented
150	C₂×C₅₃	—	4	4	no	—
151	C₂×C₅₃	#1477	1	2	yes	—
152	C₂×C₅₃	#1478	1	2	yes	—
153	C₂×C₅₃	C₄×C₂²⋊C₈	1	1	yes	characteristic
154	C₂×C₅₃	#1759	1	1	yes	characteristic
155	C₂×C₅₃	#1762	1	1	yes	characteristic
156	C₂×C₄×D₄	C₄×C₅₃	1	2	yes	—
157	C₂×C₄×D₄	C₂²×C₅₃	1	1	yes	characteristic
158	C₂²×C₄²	—	2	1	no	—
159	C₂×C₄⋊C₈	C₂²×C₅₃	1	2	yes	—
160	C₂×C₄×C₈	—	2	1	no	—
161	C₂×C₄×C₈	C₂²×C₅₃	1	2	yes	—
162	C₄×C₄⋊C₄	C₄×C₅₃	1	2	yes	—
163	C₄×C₄⋊C₄	C₂²×C₅₃	1	1	yes	characteristic
164	C₄×C₂²⋊C₄	—	2	2	no	—
165	C₄×C₂²⋊C₄	—	2	1	no	—
166	C₄³	C₂²×C₅₃	1	1	yes	characteristic
167	#336	—	2	2	no	—
168	#336	—	2	1	no	—
169	D₄⋊C₈	C₄×C₅₃	1	8	yes	complemented
170	C₅₃	#8170	1	1	yes	characteristic, direct product factor, Sylow
171	C₂²×D₄	C₂×C₄×C₅₃	1	1	yes	characteristic
172	C₂³×C₄	—	2	1	no	—
173	C₂³×C₄	—	2	2	no	—
174	C₂²×C₈	—	2	2	no	—
175	C₂²×C₈	—	2	2	no	—
176	C₄×D₄	—	2	2	no	—
177	C₄×D₄	—	2	4	no	—
178	C₄×D₄	C₈×C₅₃	1	8	yes	complemented
179	C₄×D₄	C₂×C₄×C₅₃	1	4	yes	—
180	C₂×C₄⋊C₄	C₂×C₄×C₅₃	1	1	yes	characteristic
181	C₂×C₄⋊C₄	C₂×C₄×C₅₃	1	2	yes	—
182	C₂×C₂²⋊C₄	—	2	1	no	—
183	C₂×C₂²⋊C₄	—	2	2	no	—
184	C₂×C₄²	—	2	1	no	—
185	C₂×C₄²	—	2	2	no	—
186	C₂×C₄²	—	4	4	no	—
187	C₂×C₄²	C₂×C₄×C₅₃	1	2	yes	—
188	C₂×C₄²	D₄×C₅₃	1	1	yes	characteristic
189	C₂×C₄²	D₄×C₅₃	1	1	yes	characteristic
190	C₂×C₄²	D₄×C₅₃	1	2	yes	—
191	C₂×C₄²	C₂³×C₅₃	1	1	yes	characteristic
192	C₄⋊C₈	—	2	2	no	—
193	C₄⋊C₈	C₂×C₄×C₅₃	1	4	yes	complemented
194	C₄×C₈	—	2	2	no	—
195	C₄×C₈	—	4	2	no	—
196	C₄×C₈	C₂×C₄×C₅₃	1	4	yes	complemented
197	#95	—	2	1	no	—
198	#95	—	2	2	no	—
199	C₂⁴	—	2	1	no	—
200	C₂×D₄	—	2	1	no	—
201	C₂×D₄	—	2	2	no	—
202	C₂×D₄	C₄²×C₅₃	1	2	yes	—
203	C₂×D₄	C₂×C₈×C₅₃	1	4	yes	—
204	C₂²×C₄	—	2	1	no	—
205	C₂²×C₄	—	2	2	no	—
206	C₂²×C₄	—	2	8	no	—
207	C₂²×C₄	—	2	4	no	—
208	C₂²×C₄	—	4	4	no	—
209	C₂²×C₄	—	4	4	no	—
210	C₂²×C₄	—	4	2	no	—
211	C₂²×C₄	—	4	1	no	—
212	C₂²×C₄	—	4	1	no	—
213	C₂²×C₄	C₂²⋊C₄×C₅₃	1	2	yes	—
214	C₂²×C₄	C₂²⋊C₄×C₅₃	1	2	yes	—
215	C₂²×C₄	C₂²×C₄×C₅₃	1	1	yes	characteristic
216	C₂²×C₄	C₂×D₄×C₅₃	1	1	yes	characteristic
217	C₂²×C₄	C₂×D₄×C₅₃	1	1	yes	characteristic
218	C₂×C₈	—	2	4	no	—
219	C₂×C₈	—	2	4	no	—
220	C₂×C₈	—	4	4	no	—
221	C₂×C₈	—	4	2	no	—
222	C₂×C₈	—	4	2	no	—
223	C₄⋊C₄	—	2	1	no	—
224	C₄⋊C₄	—	2	2	no	—
225	C₄⋊C₄	C₄²×C₅₃	1	2	yes	—
226	C₄⋊C₄	C₂×C₈×C₅₃	1	4	yes	—
227	C₂²⋊C₄	—	2	8	no	—
228	C₂²⋊C₄	—	2	4	no	—
229	C₄²	—	2	1	no	—
230	C₄²	—	2	2	no	—
231	C₄²	—	2	1	no	—
232	C₄²	—	4	2	no	—
233	C₄²	—	4	4	no	—
234	C₄²	—	4	2	no	—
235	C₄²	C₂×C₈×C₅₃	1	4	yes	—
236	C₄²	M₁₆×C₅₃	1	4	yes	—
237	C₄²	C₂²×C₄×C₅₃	1	2	yes	—
238	C₄²	#41×C₅₃	1	2	yes	—
239	C₂³	—	2	2	no	—
240	C₂³	—	2	4	no	—
241	C₂³	—	4	1	no	—
242	C₂³	—	4	1	no	—
243	C₂³	—	4	2	no	—
244	C₂³	C₂×C₂²⋊C₄×C₅₃	1	1	yes	characteristic
245	D₄	—	2	2	no	—
246	D₄	—	2	4	no	—
247	D₄	C₄×C₈×C₅₃	1	4	yes	complemented
248	C₂×C₄	—	2	2	no	—
249	C₂×C₄	—	2	2	no	—
250	C₂×C₄	—	2	2	no	—
251	C₂×C₄	—	2	2	no	—
252	C₂×C₄	—	2	2	no	—
253	C₂×C₄	—	2	2	no	—
254	C₂×C₄	—	2	4	no	—
255	C₂×C₄	—	4	16	no	—
256	C₂×C₄	—	4	1	no	—
257	C₂×C₄	—	4	2	no	—
258	C₂×C₄	—	4	1	no	—
259	C₂×C₄	—	4	2	no	—
260	C₂×C₄	—	4	4	no	—
261	C₂×C₄	—	4	4	no	—
262	C₂×C₄	C₂²⋊C₈×C₅₃	1	4	yes	—
263	C₂×C₄	C₂²⋊C₈×C₅₃	1	4	yes	—
264	C₂×C₄	C₂×C₄²×C₅₃	1	1	yes	characteristic
265	C₂×C₄	#117×C₅₃	1	1	yes	characteristic
266	C₂×C₄	C₄×D₄×C₅₃	1	2	yes	—
267	C₂×C₄	C₄×D₄×C₅₃	1	2	yes	—
268	C₂×C₄	C₂²×C₈×C₅₃	1	2	yes	—
269	C₂×C₄	C₂×M₁₆×C₅₃	1	2	yes	—
270	C₂×C₄	AGL(1,ℤ/8)×C₅₃	1	2	yes	—
271	C₂×C₄	AGL(1,ℤ/8)×C₅₃	1	1	yes	characteristic
272	C₂×C₄	#137×C₅₃	1	1	yes	characteristic
273	C₈	—	4	4	no	—
274	C₈	—	4	4	no	—
275	C₂²	—	2	4	no	—
276	C₂²	—	4	4	no	—
277	C₂²	—	4	4	no	—
278	C₂²	—	4	2	no	—
279	C₂²	—	4	2	no	—
280	C₂²	C₄×C₂²⋊C₄×C₅₃	1	1	yes	characteristic
281	C₂²	C₂×C₂²⋊C₈×C₅₃	1	2	yes	—
282	C₂²	C₅₃×#417	1	1	yes	characteristic
283	C₂²	C₅₃×#418	1	1	yes	characteristic
284	C₂²	C₅₃×#421	1	2	yes	—
285	C₄	—	2	4	no	—
286	C₄	—	2	4	no	—
287	C₄	—	2	2	no	—
288	C₄	—	4	4	no	—
289	C₄	—	4	2	no	—
290	C₄	C₂×C₄×C₈×C₅₃	1	1	yes	characteristic
291	C₄	C₄×M₁₆×C₅₃	1	1	yes	characteristic
292	C₄	C₅₃×#431	1	2	yes	—
293	C₄	C₅₃×#440	1	2	yes	—
294	C₄	C₅₃×#442	1	2	yes	—
295	C₂	—	4	4	no	—
296	C₂	C₅₃×#1477	1	2	yes	—
297	C₂	C₅₃×#1478	1	2	yes	—
298	C₂	C₄×C₂²⋊C₈×C₅₃	1	1	yes	characteristic
299	C₂	C₅₃×#1759	1	1	yes	characteristic
300	C₂	C₅₃×#1762	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄×C₅₃	C₂×D₄
commutator	C₄	C₂×C₄×C₈×C₅₃
Frattini, cosocle	C₂×C₄²	C₂³×C₅₃
socle	C₂³×C₅₃	C₂×C₂²⋊C₄
prime radical	C₂²×D₄×C₅₃	C₂×C₄
LCS	G ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄×C₅₃ ◁ C₄³×C₅₃ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₄×C₄⋊C₈×C₅₃	C₂	semidirect
3. C₅₃×#1744	C₂	semidirect
5. C₅₃×#1279	C₂	semidirect
19. C₅₃×D₄⋊C₈	C₄	semidirect
27. C₄×D₄×C₅₃	C₈	semidirect
42. C₄⋊C₈×C₅₃	C₂×C₄	semidirect
45. C₄×C₈×C₅₃	C₂×C₄	semidirect
96. D₄×C₅₃	C₄×C₈	semidirect
124. #8170	C₅₃	direct (N × H)
146. C₄×C₄⋊C₈	C₂×C₅₃	semidirect
147. #1744	C₂×C₅₃	semidirect
149. #1279	C₂×C₅₃	semidirect
169. D₄⋊C₈	C₄×C₅₃	semidirect
170. C₅₃	#8170	direct (N × H)
178. C₄×D₄	C₈×C₅₃	semidirect
193. C₄⋊C₈	C₂×C₄×C₅₃	semidirect
196. C₄×C₈	C₂×C₄×C₅₃	semidirect
247. D₄	C₄×C₈×C₅₃	semidirect

Maximal subgroups

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

subgroup	index
C₄²×D₄×C₅₃	2
C₄×C₄⋊C₈×C₅₃	2
C₅₃×#1744	2
C₂×C₅₃×D₄⋊C₈	2
C₅₃×#1279	2
#8170	53

Subgroups — C53×#8170 #27215682

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅₃×#8170 #27215682