Subgroups — C₅₃×#18073 #27225585

Order 13568.

714 conjugacy classes of subgroups.

The subgroup lattice has 308 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₅₃×#3169	C₂	1	1	yes	maximal, characteristic
2	C₅₃×#2935	C₂	1	2	yes	maximal, complemented
3	C₅₃×#1594	C₂	1	2	yes	maximal, complemented
4	C₅₃×#1586	C₂	1	4	yes	maximal, complemented
5	C₅₃×#1580	C₂	1	2	yes	maximal, complemented
6	C₂×C₅₃×#334	C₂	1	1	yes	maximal, characteristic, complemented
7	C₅₃×#1480	C₂	1	2	yes	maximal, complemented
8	C₅₃×#1451	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂×#131×C₅₃	—	2	2	no	—
10	C₅₃×#532	C₂²	1	2	yes	—
11	C₅₃×#500	C₄	1	1	yes	characteristic
12	C₅₃×#493	C₄	1	1	yes	characteristic
13	C₅₃×#486	C₄	1	2	yes	—
14	C₅₃×#466	C₄	1	2	yes	—
15	C₅₃×#466	C₄	1	2	yes	—
16	C₄×D₈×C₅₃	C₂²	1	2	yes	—
17	C₄×D₈×C₅₃	C₂²	1	2	yes	—
18	C₅₃×#436	C₂²	1	4	yes	—
19	C₅₃×#435	—	2	4	no	—
20	C₈×D₄×C₅₃	C₂²	1	4	yes	—
21	C₅₃×#431	C₂²	1	1	yes	characteristic
22	C₂×C₄⋊C₈×C₅₃	C₂²	1	2	yes	—
23	C₅₃×#408	—	2	2	no	—
24	C₄×M₁₆×C₅₃	C₂²	1	1	yes	characteristic
25	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
26	C₅₃×#334	C₂²	1	1	yes	characteristic
27	C₅₃×#334	C₂²	1	2	yes	—
28	C₅₃×#334	C₂²	1	1	yes	characteristic
29	C₅₃×D₄⋊C₈	C₂²	1	4	yes	—
30	C₅₃×D₄⋊C₈	C₂²	1	4	yes	—
31	C₅₃×#322	C₂²	1	2	yes	—
32	C₅₃×#322	C₂²	1	2	yes	—
33	C₂×#41×C₅₃	—	2	2	no	—
34	C₂×D₈×C₅₃	C₂×C₄	1	2	yes	—
35	C₂×D₈×C₅₃	C₂×C₄	1	2	yes	—
36	#131×C₅₃	—	4	8	no	—
37	C₂×M₁₆×C₅₃	—	2	4	no	—
38	C₂×M₁₆×C₅₃	D₄	1	2	yes	—
39	C₂²×C₈×C₅₃	—	2	4	no	—
40	C₂²×C₈×C₅₃	—	2	2	no	—
41	C₂²×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
42	C₂²×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
43	C₄⋊Q₈×C₅₃	C₂×C₄	1	2	yes	—
44	#127×C₅₃	C₂×C₄	1	2	yes	—
45	#124×C₅₃	—	2	2	no	—
46	#121×C₅₃	—	2	2	no	—
47	#121×C₅₃	C₂×C₄	1	4	yes	—
48	C₄×D₄×C₅₃	—	2	2	no	—
49	C₄×D₄×C₅₃	C₂³	1	4	yes	—
50	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
51	#107×C₅₃	C₂×C₄	1	2	yes	—
52	#107×C₅₃	C₂×C₄	1	2	yes	—
53	C₄⋊C₈×C₅₃	—	2	2	no	—
54	C₄⋊C₈×C₅₃	—	2	1	no	—
55	C₄⋊C₈×C₅₃	C₂³	1	2	yes	—
56	C₄⋊C₈×C₅₃	C₂³	1	2	yes	—
57	#102×C₅₃	C₂×C₄	1	4	yes	—
58	#102×C₅₃	C₂×C₄	1	4	yes	—
59	C₂²⋊C₈×C₅₃	—	2	4	no	—
60	C₂²⋊C₈×C₅₃	—	2	4	no	—
61	C₂²⋊C₈×C₅₃	—	2	1	no	—
62	#97×C₅₃	D₄	1	2	yes	—
63	C₄×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
64	C₄×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
65	C₄×C₈×C₅₃	C₂³	1	2	yes	—
66	C₄×C₈×C₅₃	C₂³	1	1	yes	characteristic
67	C₄×C₈×C₅₃	C₂³	1	1	yes	characteristic
68	C₄×C₈×C₅₃	C₂³	1	2	yes	—
69	#41×C₅₃	—	4	4	no	—
70	#41×C₅₃	—	4	4	no	—
71	C₂×Q₈×C₅₃	—	2	2	no	—
72	C₂×D₄×C₅₃	—	2	4	no	—
73	C₂×D₄×C₅₃	—	2	2	no	—
74	C₂×D₄×C₅₃	—	2	2	no	—
75	C₂×D₄×C₅₃	C₂²×C₄	1	4	yes	—
76	C₂²×C₄×C₅₃	—	2	4	no	—
77	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
78	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
79	C₂²×C₄×C₅₃	#41	1	1	yes	characteristic
80	D₈×C₅₃	—	2	4	no	—
81	D₈×C₅₃	—	4	2	no	—
82	M₁₆×C₅₃	—	2	4	no	—
83	M₁₆×C₅₃	—	4	8	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	1	no	—
89	C₂×C₈×C₅₃	—	4	8	no	—
90	C₂×C₈×C₅₃	—	4	4	no	—
91	C₂×C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
92	C₂×C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
93	C₂×C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
94	C₂×C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
95	C₂×C₈×C₅₃	C₂×D₄	1	4	yes	—
96	C₄⋊C₄×C₅₃	—	2	2	no	—
97	C₄⋊C₄×C₅₃	C₂²×C₄	1	4	yes	—
98	C₂²⋊C₄×C₅₃	—	2	4	no	—
99	C₂²⋊C₄×C₅₃	—	2	4	no	—
100	C₄²×C₅₃	C₂²×C₄	1	1	yes	characteristic
101	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
102	C₄²×C₅₃	#41	1	1	yes	characteristic
103	C₄²×C₅₃	C₂⁴	1	1	yes	characteristic
104	C₂³×C₅₃	—	2	4	no	—
105	C₂³×C₅₃	C₄×D₄	1	1	yes	characteristic
106	Q₈×C₅₃	—	4	4	no	—
107	D₄×C₅₃	—	2	4	no	—
108	D₄×C₅₃	—	2	4	no	—
109	D₄×C₅₃	—	4	8	no	—
110	D₄×C₅₃	—	4	4	no	—
111	D₄×C₅₃	—	8	2	no	—
112	C₂×C₄×C₅₃	—	2	1	no	—
113	C₂×C₄×C₅₃	—	2	1	no	—
114	C₂×C₄×C₅₃	—	2	4	no	—
115	C₂×C₄×C₅₃	—	4	1	no	—
116	C₂×C₄×C₅₃	—	4	4	no	—
117	C₂×C₄×C₅₃	—	4	4	no	—
118	C₂×C₄×C₅₃	C₄×D₄	1	1	yes	characteristic
119	C₂×C₄×C₅₃	C₄×D₄	1	1	yes	characteristic
120	C₂×C₄×C₅₃	C₄×D₄	1	1	yes	characteristic
121	C₂×C₄×C₅₃	AGL(1,ℤ/8)	1	4	yes	—
122	C₂×C₄×C₅₃	C₂³×C₄	1	1	yes	characteristic
123	C₂×C₄×C₅₃	C₂²×D₄	1	1	yes	characteristic
124	C₂×C₄×C₅₃	C₂×#41	1	1	yes	characteristic
125	C₈×C₅₃	—	2	1	no	—
126	C₈×C₅₃	—	2	1	no	—
127	C₈×C₅₃	—	2	1	no	—
128	C₈×C₅₃	—	2	4	no	—
129	C₈×C₅₃	—	4	2	no	—
130	C₈×C₅₃	—	4	2	no	—
131	C₈×C₅₃	—	4	2	no	—
132	C₈×C₅₃	#131	1	2	yes	complemented
133	#18073	C₅₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
134	C₂²×C₅₃	—	2	1	no	—
135	C₂²×C₅₃	—	2	1	no	—
136	C₂²×C₅₃	—	4	4	no	—
137	C₂²×C₅₃	—	4	4	no	—
138	C₂²×C₅₃	—	4	4	no	—
139	C₂²×C₅₃	#442	1	2	yes	—
140	C₂²×C₅₃	C₂×C₄×D₄	1	1	yes	characteristic
141	C₄×C₅₃	—	2	1	no	—
142	C₄×C₅₃	—	2	1	no	—
143	C₄×C₅₃	—	2	1	no	—
144	C₄×C₅₃	—	4	4	no	—
145	C₄×C₅₃	#442	1	2	yes	—
146	C₄×C₅₃	C₂×#131	1	1	yes	characteristic
147	C₄×C₅₃	#568	1	1	yes	characteristic
148	C₄×C₅₃	C₂×AGL(1,ℤ/8)	1	2	yes	—
149	#3169	C₂×C₅₃	1	1	yes	characteristic
150	#2935	C₂×C₅₃	1	2	yes	complemented
151	#1594	C₂×C₅₃	1	2	yes	complemented
152	#1586	C₂×C₅₃	1	4	yes	complemented
153	#1580	C₂×C₅₃	1	2	yes	complemented
154	C₂×#334	C₂×C₅₃	1	1	yes	characteristic, complemented
155	#1480	C₂×C₅₃	1	2	yes	complemented
156	#1451	C₂×C₅₃	1	1	yes	characteristic, complemented
157	C₂×C₅₃	—	2	2	no	—
158	C₂×C₅₃	—	8	4	no	—
159	C₂×C₅₃	#2935	1	1	yes	characteristic
160	C₂×C₅₃	C₂×#442	1	1	yes	characteristic
161	C₂×C₅₃	#2957	1	1	yes	characteristic
162	C₂×#131	—	2	2	no	—
163	#532	C₂²×C₅₃	1	2	yes	—
164	#500	C₄×C₅₃	1	1	yes	characteristic
165	#493	C₄×C₅₃	1	1	yes	characteristic
166	#486	C₄×C₅₃	1	2	yes	—
167	#466	C₄×C₅₃	1	2	yes	—
168	#466	C₄×C₅₃	1	2	yes	—
169	C₄×D₈	C₂²×C₅₃	1	2	yes	—
170	C₄×D₈	C₂²×C₅₃	1	2	yes	—
171	#436	C₂²×C₅₃	1	4	yes	—
172	#435	—	2	4	no	—
173	C₈×D₄	C₂²×C₅₃	1	4	yes	—
174	#431	C₂²×C₅₃	1	1	yes	characteristic
175	C₂×C₄⋊C₈	C₂²×C₅₃	1	2	yes	—
176	#408	—	2	2	no	—
177	C₄×M₁₆	C₂²×C₅₃	1	1	yes	characteristic
178	C₂×C₄×C₈	C₂²×C₅₃	1	1	yes	characteristic
179	#334	C₂²×C₅₃	1	1	yes	characteristic
180	#334	C₂²×C₅₃	1	2	yes	—
181	#334	C₂²×C₅₃	1	1	yes	characteristic
182	D₄⋊C₈	C₂²×C₅₃	1	4	yes	—
183	D₄⋊C₈	C₂²×C₅₃	1	4	yes	—
184	#322	C₂²×C₅₃	1	2	yes	—
185	#322	C₂²×C₅₃	1	2	yes	—
186	C₅₃	#18073	1	1	yes	characteristic, direct product factor, Sylow
187	C₂×#41	—	2	2	no	—
188	C₂×D₈	C₂×C₄×C₅₃	1	2	yes	—
189	C₂×D₈	C₂×C₄×C₅₃	1	2	yes	—
190	#131	—	4	8	no	—
191	C₂×M₁₆	—	2	4	no	—
192	C₂×M₁₆	D₄×C₅₃	1	2	yes	—
193	C₂²×C₈	—	2	2	no	—
194	C₂²×C₈	—	2	4	no	—
195	C₂²×C₈	C₂×C₄×C₅₃	1	1	yes	characteristic
196	C₂²×C₈	C₂×C₄×C₅₃	1	1	yes	characteristic
197	C₄⋊Q₈	C₂×C₄×C₅₃	1	2	yes	—
198	#127	C₂×C₄×C₅₃	1	2	yes	—
199	#124	—	2	2	no	—
200	#121	—	2	2	no	—
201	#121	C₂×C₄×C₅₃	1	4	yes	—
202	C₄×D₄	—	2	2	no	—
203	C₄×D₄	C₂³×C₅₃	1	4	yes	—
204	C₂×C₄²	C₂³×C₅₃	1	1	yes	characteristic
205	#107	C₂×C₄×C₅₃	1	2	yes	—
206	#107	C₂×C₄×C₅₃	1	2	yes	—
207	C₄⋊C₈	—	2	2	no	—
208	C₄⋊C₈	—	2	1	no	—
209	C₄⋊C₈	C₂³×C₅₃	1	2	yes	—
210	C₄⋊C₈	C₂³×C₅₃	1	2	yes	—
211	#102	C₂×C₄×C₅₃	1	4	yes	—
212	#102	C₂×C₄×C₅₃	1	4	yes	—
213	C₂²⋊C₈	—	2	1	no	—
214	C₂²⋊C₈	—	2	4	no	—
215	C₂²⋊C₈	—	2	4	no	—
216	#97	D₄×C₅₃	1	2	yes	—
217	C₄×C₈	C₂×C₄×C₅₃	1	1	yes	characteristic
218	C₄×C₈	C₂×C₄×C₅₃	1	1	yes	characteristic
219	C₄×C₈	C₂³×C₅₃	1	2	yes	—
220	C₄×C₈	C₂³×C₅₃	1	1	yes	characteristic
221	C₄×C₈	C₂³×C₅₃	1	1	yes	characteristic
222	C₄×C₈	C₂³×C₅₃	1	2	yes	—
223	#41	—	4	4	no	—
224	#41	—	4	4	no	—
225	C₂×Q₈	—	2	2	no	—
226	C₂×D₄	—	2	4	no	—
227	C₂×D₄	—	2	2	no	—
228	C₂×D₄	—	2	2	no	—
229	C₂×D₄	C₂²×C₄×C₅₃	1	4	yes	—
230	C₂²×C₄	—	2	4	no	—
231	C₂²×C₄	C₂²×C₄×C₅₃	1	1	yes	characteristic
232	C₂²×C₄	C₂×D₄×C₅₃	1	1	yes	characteristic
233	C₂²×C₄	#41×C₅₃	1	1	yes	characteristic
234	D₈	—	2	4	no	—
235	D₈	—	4	2	no	—
236	M₁₆	—	2	4	no	—
237	M₁₆	—	4	8	no	—
238	C₂×C₈	—	2	2	no	—
239	C₂×C₈	—	2	2	no	—
240	C₂×C₈	—	2	2	no	—
241	C₂×C₈	—	2	2	no	—
242	C₂×C₈	—	4	1	no	—
243	C₂×C₈	—	4	8	no	—
244	C₂×C₈	—	4	4	no	—
245	C₂×C₈	C₂²×C₄×C₅₃	1	1	yes	characteristic
246	C₂×C₈	C₂²×C₄×C₅₃	1	1	yes	characteristic
247	C₂×C₈	C₂²×C₄×C₅₃	1	1	yes	characteristic
248	C₂×C₈	C₂²×C₄×C₅₃	1	1	yes	characteristic
249	C₂×C₈	C₂×D₄×C₅₃	1	4	yes	—
250	C₄⋊C₄	—	2	2	no	—
251	C₄⋊C₄	C₂²×C₄×C₅₃	1	4	yes	—
252	C₂²⋊C₄	—	2	4	no	—
253	C₂²⋊C₄	—	2	4	no	—
254	C₄²	C₂²×C₄×C₅₃	1	1	yes	characteristic
255	C₄²	C₂×D₄×C₅₃	1	1	yes	characteristic
256	C₄²	#41×C₅₃	1	1	yes	characteristic
257	C₄²	C₂⁴×C₅₃	1	1	yes	characteristic
258	C₂³	—	2	4	no	—
259	C₂³	C₄×D₄×C₅₃	1	1	yes	characteristic
260	Q₈	—	4	4	no	—
261	D₄	—	2	4	no	—
262	D₄	—	2	4	no	—
263	D₄	—	4	8	no	—
264	D₄	—	4	4	no	—
265	D₄	—	8	2	no	—
266	C₂×C₄	—	2	1	no	—
267	C₂×C₄	—	2	1	no	—
268	C₂×C₄	—	2	4	no	—
269	C₂×C₄	—	4	1	no	—
270	C₂×C₄	—	4	4	no	—
271	C₂×C₄	—	4	4	no	—
272	C₂×C₄	C₄×D₄×C₅₃	1	1	yes	characteristic
273	C₂×C₄	C₄×D₄×C₅₃	1	1	yes	characteristic
274	C₂×C₄	C₄×D₄×C₅₃	1	1	yes	characteristic
275	C₂×C₄	AGL(1,ℤ/8)×C₅₃	1	4	yes	—
276	C₂×C₄	C₂³×C₄×C₅₃	1	1	yes	characteristic
277	C₂×C₄	C₂²×D₄×C₅₃	1	1	yes	characteristic
278	C₂×C₄	C₂×#41×C₅₃	1	1	yes	characteristic
279	C₈	—	2	1	no	—
280	C₈	—	2	1	no	—
281	C₈	—	2	1	no	—
282	C₈	—	2	4	no	—
283	C₈	—	4	2	no	—
284	C₈	—	4	2	no	—
285	C₈	—	4	2	no	—
286	C₈	#131×C₅₃	1	2	yes	complemented
287	C₂²	—	2	1	no	—
288	C₂²	—	2	1	no	—
289	C₂²	—	4	4	no	—
290	C₂²	—	4	4	no	—
291	C₂²	—	4	4	no	—
292	C₂²	C₅₃×#442	1	2	yes	—
293	C₂²	C₂×C₄×D₄×C₅₃	1	1	yes	characteristic
294	C₄	—	2	1	no	—
295	C₄	—	2	1	no	—
296	C₄	—	2	1	no	—
297	C₄	—	4	4	no	—
298	C₄	C₅₃×#442	1	2	yes	—
299	C₄	C₂×#131×C₅₃	1	1	yes	characteristic
300	C₄	C₅₃×#568	1	1	yes	characteristic
301	C₄	C₂×AGL(1,ℤ/8)×C₅₃	1	2	yes	—
302	C₂	—	2	2	no	—
303	C₂	—	8	4	no	—
304	C₂	C₅₃×#2935	1	1	yes	characteristic
305	C₂	C₂×C₅₃×#442	1	1	yes	characteristic
306	C₂	C₅₃×#2957	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄×C₅₃	C₂²×D₄
commutator	C₂×C₄	C₂³×C₄×C₅₃
Frattini, cosocle	C₄²	C₂⁴×C₅₃
socle	C₂²×C₅₃	C₂×C₄×D₄
prime radical	C₅₃×#3169	C₂
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₅₃ ◁ C₄×M₁₆×C₅₃ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₅₃×#2935	C₂	semidirect
3. C₅₃×#1594	C₂	semidirect
4. C₅₃×#1586	C₂	semidirect
5. C₅₃×#1580	C₂	semidirect
6. C₂×C₅₃×#334	C₂	semidirect
7. C₅₃×#1480	C₂	semidirect
8. C₅₃×#1451	C₂	semidirect
132. C₈×C₅₃	#131	semidirect
133. #18073	C₅₃	direct (N × H)
150. #2935	C₂×C₅₃	semidirect
151. #1594	C₂×C₅₃	semidirect
152. #1586	C₂×C₅₃	semidirect
153. #1580	C₂×C₅₃	semidirect
154. C₂×#334	C₂×C₅₃	semidirect
155. #1480	C₂×C₅₃	semidirect
156. #1451	C₂×C₅₃	semidirect
186. C₅₃	#18073	direct (N × H)
286. C₈	#131×C₅₃	semidirect

Maximal subgroups

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

subgroup	index
C₅₃×#3169	2
C₅₃×#2935	2
C₅₃×#1594	2
C₅₃×#1586	2
C₅₃×#1580	2
C₂×C₅₃×#334	2
C₅₃×#1480	2
C₅₃×#1451	2
#18073	53

Subgroups — C53×#18073 #27225585

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅₃×#18073 #27225585