Subgroups — C₅₃×#8245 #27215757

Order 13568.

444 conjugacy classes of subgroups.

The subgroup lattice has 296 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₅₃×#2931	C₂	1	1	yes	maximal, characteristic
2	C₅₃×#1543	C₂	1	1	yes	maximal, characteristic
3	C₅₃×#1525	C₂	1	1	yes	maximal, characteristic
4	C₅₃×#1451	C₂	1	1	yes	maximal, characteristic, complemented
5	C₈×M₁₆×C₅₃	C₂	1	1	yes	maximal, characteristic, complemented
6	C₅₃×#1299	C₂	1	1	yes	maximal, characteristic, complemented
7	C₅₃×#1299	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×#131×C₅₃	C₄	1	2	yes	—
9	C₅₃×#532	C₂²	1	1	yes	characteristic
10	C₅₃×#436	—	2	1	no	—
11	C₅₃×#436	—	2	1	no	—
12	C₈×D₄×C₅₃	—	2	1	no	—
13	C₈×D₄×C₅₃	—	2	1	no	—
14	C₅₃×#431	C₂²	1	1	yes	characteristic
15	C₅₃×#431	C₂²	1	1	yes	characteristic
16	C₅₃×#423	C₂²	1	1	yes	characteristic
17	C₅₃×#408	C₄	1	2	yes	—
18	C₄×M₁₆×C₅₃	C₂²	1	1	yes	characteristic
19	C₄×M₁₆×C₅₃	C₂²	1	1	yes	characteristic
20	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
21	C₅₃×#333	C₄	1	2	yes	—
22	C₅₃×#331	C₄	1	2	yes	—
23	C₅₃×#329	C₄	1	4	yes	—
24	C₅₃×#322	—	2	1	no	—
25	C₅₃×#322	—	2	1	no	—
26	C₅₃×#322	—	2	1	no	—
27	C₈²×C₅₃	—	2	1	no	—
28	C₂×#41×C₅₃	C₂×C₄	1	2	yes	—
29	#131×C₅₃	—	4	4	no	—
30	C₂×M₁₆×C₅₃	—	2	2	no	—
31	C₂×M₁₆×C₅₃	—	2	1	no	—
32	C₂×M₁₆×C₅₃	—	2	1	no	—
33	C₂×M₁₆×C₅₃	C₂×C₄	1	2	yes	—
34	C₂²×C₈×C₅₃	—	2	2	no	—
35	C₂²×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
36	C₂²×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
37	C₄⋊Q₈×C₅₃	C₂×C₄	1	1	yes	characteristic
38	#127×C₅₃	C₂×C₄	1	1	yes	characteristic
39	#124×C₅₃	C₂×C₄	1	2	yes	—
40	#121×C₅₃	—	2	1	no	—
41	#121×C₅₃	—	2	1	no	—
42	C₄×D₄×C₅₃	—	2	1	no	—
43	C₄×D₄×C₅₃	—	2	1	no	—
44	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
45	C₄⋊C₈×C₅₃	—	2	1	no	—
46	C₄⋊C₈×C₅₃	—	2	1	no	—
47	C₄⋊C₈×C₅₃	C₂×C₄	1	2	yes	—
48	C₄⋊C₈×C₅₃	C₂×C₄	1	2	yes	—
49	C₂²⋊C₈×C₅₃	—	2	2	no	—
50	C₂²⋊C₈×C₅₃	—	2	2	no	—
51	C₂²⋊C₈×C₅₃	—	2	1	no	—
52	C₂²⋊C₈×C₅₃	—	2	1	no	—
53	#97×C₅₃	C₂×C₄	1	2	yes	—
54	#97×C₅₃	C₂×C₄	1	2	yes	—
55	C₄×C₈×C₅₃	—	2	1	no	—
56	C₄×C₈×C₅₃	—	2	1	no	—
57	C₄×C₈×C₅₃	—	2	1	no	—
58	C₄×C₈×C₅₃	—	2	1	no	—
59	C₄×C₈×C₅₃	D₄	1	1	yes	characteristic
60	C₄×C₈×C₅₃	D₄	1	1	yes	characteristic
61	C₄×C₈×C₅₃	D₄	1	1	yes	characteristic
62	C₄×C₈×C₅₃	D₄	1	1	yes	characteristic
63	#41×C₅₃	—	4	2	no	—
64	#41×C₅₃	—	4	2	no	—
65	C₂×Q₈×C₅₃	C₄²	1	2	yes	—
66	C₂×D₄×C₅₃	—	2	2	no	—
67	C₂×D₄×C₅₃	—	2	1	no	—
68	C₂×D₄×C₅₃	—	2	1	no	—
69	C₂×D₄×C₅₃	C₄²	1	2	yes	—
70	C₂²×C₄×C₅₃	—	2	2	no	—
71	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
72	C₂²×C₄×C₅₃	#41	1	1	yes	characteristic
73	C₂²×C₄×C₅₃	#41	1	1	yes	characteristic
74	M₁₆×C₅₃	—	2	4	no	—
75	M₁₆×C₅₃	—	4	2	no	—
76	M₁₆×C₅₃	—	4	4	no	—
77	M₁₆×C₅₃	—	4	2	no	—
78	C₂×C₈×C₅₃	—	2	2	no	—
79	C₂×C₈×C₅₃	—	2	2	no	—
80	C₂×C₈×C₅₃	—	2	2	no	—
81	C₂×C₈×C₅₃	—	2	2	no	—
82	C₂×C₈×C₅₃	—	2	2	no	—
83	C₂×C₈×C₅₃	—	2	2	no	—
84	C₂×C₈×C₅₃	—	4	1	no	—
85	C₂×C₈×C₅₃	—	4	4	no	—
86	C₂×C₈×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
87	C₂×C₈×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
88	C₂×C₈×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
89	C₂×C₈×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
90	C₄⋊C₄×C₅₃	—	2	1	no	—
91	C₄⋊C₄×C₅₃	—	2	1	no	—
92	C₂²⋊C₄×C₅₃	—	2	2	no	—
93	C₂²⋊C₄×C₅₃	—	2	2	no	—
94	C₄²×C₅₃	C₂²×C₄	1	1	yes	characteristic
95	C₄²×C₅₃	C₂²×C₄	1	1	yes	characteristic
96	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
97	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
98	C₂³×C₅₃	—	2	2	no	—
99	C₂³×C₅₃	#117	1	1	yes	characteristic
100	Q₈×C₅₃	—	4	2	no	—
101	D₄×C₅₃	—	4	2	no	—
102	D₄×C₅₃	—	4	4	no	—
103	D₄×C₅₃	—	4	2	no	—
104	D₄×C₅₃	—	4	2	no	—
105	C₂×C₄×C₅₃	—	2	1	no	—
106	C₂×C₄×C₅₃	—	2	1	no	—
107	C₂×C₄×C₅₃	—	2	2	no	—
108	C₂×C₄×C₅₃	—	2	2	no	—
109	C₂×C₄×C₅₃	—	2	2	no	—
110	C₂×C₄×C₅₃	—	4	2	no	—
111	C₂×C₄×C₅₃	—	4	2	no	—
112	C₂×C₄×C₅₃	C₂×C₄²	1	1	yes	characteristic
113	C₂×C₄×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
114	C₂×C₄×C₅₃	C₄×D₄	1	1	yes	characteristic
115	C₂×C₄×C₅₃	C₄×D₄	1	1	yes	characteristic
116	C₂×C₄×C₅₃	C₄×D₄	1	1	yes	characteristic
117	C₂×C₄×C₅₃	C₄×D₄	1	1	yes	characteristic
118	C₈×C₅₃	—	2	4	no	—
119	C₈×C₅₃	—	2	1	no	—
120	C₈×C₅₃	—	2	1	no	—
121	C₈×C₅₃	—	2	1	no	—
122	C₈×C₅₃	—	4	2	no	—
123	C₈×C₅₃	—	4	2	no	—
124	C₈×C₅₃	—	4	2	no	—
125	C₈×C₅₃	—	4	2	no	—
126	C₈×C₅₃	#100	1	2	yes	—
127	#8245	C₅₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
128	C₂²×C₅₃	—	2	1	no	—
129	C₂²×C₅₃	—	2	1	no	—
130	C₂²×C₅₃	—	2	1	no	—
131	C₂²×C₅₃	—	4	2	no	—
132	C₂²×C₅₃	—	4	2	no	—
133	C₂²×C₅₃	—	4	2	no	—
134	C₂²×C₅₃	C₄×C₂²⋊C₄	1	1	yes	characteristic
135	C₄×C₅₃	—	2	1	no	—
136	C₄×C₅₃	—	2	1	no	—
137	C₄×C₅₃	—	2	1	no	—
138	C₄×C₅₃	—	4	2	no	—
139	C₄×C₅₃	C₂×#100	1	1	yes	characteristic
140	C₄×C₅₃	#413	1	1	yes	characteristic
141	C₄×C₅₃	#443	1	2	yes	—
142	C₄×C₅₃	#444	1	2	yes	—
143	#2931	C₂×C₅₃	1	1	yes	characteristic
144	#1543	C₂×C₅₃	1	1	yes	characteristic
145	#1525	C₂×C₅₃	1	1	yes	characteristic
146	#1451	C₂×C₅₃	1	1	yes	characteristic, complemented
147	C₈×M₁₆	C₂×C₅₃	1	1	yes	characteristic, complemented
148	#1299	C₂×C₅₃	1	1	yes	characteristic, complemented
149	#1299	C₂×C₅₃	1	1	yes	characteristic, complemented
150	C₂×C₅₃	—	4	1	no	—
151	C₂×C₅₃	—	8	2	no	—
152	C₂×C₅₃	C₄×#100	1	1	yes	characteristic
153	C₂×C₅₃	#1764	1	1	yes	characteristic
154	C₂×C₅₃	#1765	1	1	yes	characteristic
155	C₂×#131	C₄×C₅₃	1	2	yes	—
156	#532	C₂²×C₅₃	1	1	yes	characteristic
157	#436	—	2	1	no	—
158	#436	—	2	1	no	—
159	C₈×D₄	—	2	1	no	—
160	C₈×D₄	—	2	1	no	—
161	#431	C₂²×C₅₃	1	1	yes	characteristic
162	#431	C₂²×C₅₃	1	1	yes	characteristic
163	#423	C₂²×C₅₃	1	1	yes	characteristic
164	#408	C₄×C₅₃	1	2	yes	—
165	C₄×M₁₆	C₂²×C₅₃	1	1	yes	characteristic
166	C₄×M₁₆	C₂²×C₅₃	1	1	yes	characteristic
167	C₂×C₄×C₈	C₂²×C₅₃	1	1	yes	characteristic
168	#333	C₄×C₅₃	1	2	yes	—
169	#331	C₄×C₅₃	1	2	yes	—
170	#329	C₄×C₅₃	1	4	yes	—
171	#322	—	2	1	no	—
172	#322	—	2	1	no	—
173	#322	—	2	1	no	—
174	C₈²	—	2	1	no	—
175	C₅₃	#8245	1	1	yes	characteristic, direct product factor, Sylow
176	C₂×#41	C₂×C₄×C₅₃	1	2	yes	—
177	#131	—	4	4	no	—
178	C₂×M₁₆	—	2	1	no	—
179	C₂×M₁₆	—	2	1	no	—
180	C₂×M₁₆	—	2	2	no	—
181	C₂×M₁₆	C₂×C₄×C₅₃	1	2	yes	—
182	C₂²×C₈	—	2	2	no	—
183	C₂²×C₈	C₂×C₄×C₅₃	1	1	yes	characteristic
184	C₂²×C₈	C₂×C₄×C₅₃	1	1	yes	characteristic
185	C₄⋊Q₈	C₂×C₄×C₅₃	1	1	yes	characteristic
186	#127	C₂×C₄×C₅₃	1	1	yes	characteristic
187	#124	C₂×C₄×C₅₃	1	2	yes	—
188	#121	—	2	1	no	—
189	#121	—	2	1	no	—
190	C₄×D₄	—	2	1	no	—
191	C₄×D₄	—	2	1	no	—
192	C₂×C₄²	C₂³×C₅₃	1	1	yes	characteristic
193	C₄⋊C₈	—	2	1	no	—
194	C₄⋊C₈	—	2	1	no	—
195	C₄⋊C₈	C₂×C₄×C₅₃	1	2	yes	—
196	C₄⋊C₈	C₂×C₄×C₅₃	1	2	yes	—
197	C₂²⋊C₈	—	2	1	no	—
198	C₂²⋊C₈	—	2	1	no	—
199	C₂²⋊C₈	—	2	2	no	—
200	C₂²⋊C₈	—	2	2	no	—
201	#97	C₂×C₄×C₅₃	1	2	yes	—
202	#97	C₂×C₄×C₅₃	1	2	yes	—
203	C₄×C₈	—	2	1	no	—
204	C₄×C₈	—	2	1	no	—
205	C₄×C₈	—	2	1	no	—
206	C₄×C₈	—	2	1	no	—
207	C₄×C₈	D₄×C₅₃	1	1	yes	characteristic
208	C₄×C₈	D₄×C₅₃	1	1	yes	characteristic
209	C₄×C₈	D₄×C₅₃	1	1	yes	characteristic
210	C₄×C₈	D₄×C₅₃	1	1	yes	characteristic
211	#41	—	4	2	no	—
212	#41	—	4	2	no	—
213	C₂×Q₈	C₄²×C₅₃	1	2	yes	—
214	C₂×D₄	—	2	2	no	—
215	C₂×D₄	—	2	1	no	—
216	C₂×D₄	—	2	1	no	—
217	C₂×D₄	C₄²×C₅₃	1	2	yes	—
218	C₂²×C₄	—	2	2	no	—
219	C₂²×C₄	C₂²×C₄×C₅₃	1	1	yes	characteristic
220	C₂²×C₄	#41×C₅₃	1	1	yes	characteristic
221	C₂²×C₄	#41×C₅₃	1	1	yes	characteristic
222	M₁₆	—	2	4	no	—
223	M₁₆	—	4	2	no	—
224	M₁₆	—	4	4	no	—
225	M₁₆	—	4	2	no	—
226	C₂×C₈	—	2	2	no	—
227	C₂×C₈	—	2	2	no	—
228	C₂×C₈	—	2	2	no	—
229	C₂×C₈	—	2	2	no	—
230	C₂×C₈	—	2	2	no	—
231	C₂×C₈	—	2	2	no	—
232	C₂×C₈	—	4	1	no	—
233	C₂×C₈	—	4	4	no	—
234	C₂×C₈	C₂²⋊C₄×C₅₃	1	1	yes	characteristic
235	C₂×C₈	C₂²⋊C₄×C₅₃	1	1	yes	characteristic
236	C₂×C₈	C₂²⋊C₄×C₅₃	1	1	yes	characteristic
237	C₂×C₈	C₂²⋊C₄×C₅₃	1	1	yes	characteristic
238	C₄⋊C₄	—	2	1	no	—
239	C₄⋊C₄	—	2	1	no	—
240	C₂²⋊C₄	—	2	2	no	—
241	C₂²⋊C₄	—	2	2	no	—
242	C₄²	C₂²×C₄×C₅₃	1	1	yes	characteristic
243	C₄²	C₂²×C₄×C₅₃	1	1	yes	characteristic
244	C₄²	C₂×D₄×C₅₃	1	1	yes	characteristic
245	C₄²	C₂×D₄×C₅₃	1	1	yes	characteristic
246	C₂³	—	2	2	no	—
247	C₂³	#117×C₅₃	1	1	yes	characteristic
248	Q₈	—	4	2	no	—
249	D₄	—	4	2	no	—
250	D₄	—	4	4	no	—
251	D₄	—	4	2	no	—
252	D₄	—	4	2	no	—
253	C₂×C₄	—	2	1	no	—
254	C₂×C₄	—	2	1	no	—
255	C₂×C₄	—	2	2	no	—
256	C₂×C₄	—	2	2	no	—
257	C₂×C₄	—	2	2	no	—
258	C₂×C₄	—	4	2	no	—
259	C₂×C₄	—	4	2	no	—
260	C₂×C₄	C₂×C₄²×C₅₃	1	1	yes	characteristic
261	C₂×C₄	C₂×C₂²⋊C₄×C₅₃	1	1	yes	characteristic
262	C₂×C₄	C₄×D₄×C₅₃	1	1	yes	characteristic
263	C₂×C₄	C₄×D₄×C₅₃	1	1	yes	characteristic
264	C₂×C₄	C₄×D₄×C₅₃	1	1	yes	characteristic
265	C₂×C₄	C₄×D₄×C₅₃	1	1	yes	characteristic
266	C₈	—	2	4	no	—
267	C₈	—	2	1	no	—
268	C₈	—	2	1	no	—
269	C₈	—	2	1	no	—
270	C₈	—	4	2	no	—
271	C₈	—	4	2	no	—
272	C₈	—	4	2	no	—
273	C₈	—	4	2	no	—
274	C₈	#100×C₅₃	1	2	yes	—
275	C₂²	—	2	1	no	—
276	C₂²	—	2	1	no	—
277	C₂²	—	2	1	no	—
278	C₂²	—	4	2	no	—
279	C₂²	—	4	2	no	—
280	C₂²	—	4	2	no	—
281	C₂²	C₄×C₂²⋊C₄×C₅₃	1	1	yes	characteristic
282	C₄	—	2	1	no	—
283	C₄	—	2	1	no	—
284	C₄	—	2	1	no	—
285	C₄	—	4	2	no	—
286	C₄	C₂×#100×C₅₃	1	1	yes	characteristic
287	C₄	C₅₃×#413	1	1	yes	characteristic
288	C₄	C₅₃×#443	1	2	yes	—
289	C₄	C₅₃×#444	1	2	yes	—
290	C₂	—	4	1	no	—
291	C₂	—	8	2	no	—
292	C₂	C₄×#100×C₅₃	1	1	yes	characteristic
293	C₂	C₅₃×#1764	1	1	yes	characteristic
294	C₂	C₅₃×#1765	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄×C₅₃	C₂×C₂²⋊C₄
commutator	C₂×C₄	C₂×C₄²×C₅₃
Frattini, cosocle	C₂×C₄²	C₂³×C₅₃
socle	C₂²×C₅₃	C₄×C₂²⋊C₄
prime radical	C₅₃×#532	C₂²
LCS	G ▷ C₂×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₅₃ ◁ C₂×C₄×C₈×C₅₃ ◁ G

Complements

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

N	H ≅ G/N	type
4. C₅₃×#1451	C₂	semidirect
5. C₈×M₁₆×C₅₃	C₂	semidirect
6. C₅₃×#1299	C₂	semidirect
7. C₅₃×#1299	C₂	semidirect
127. #8245	C₅₃	direct (N × H)
146. #1451	C₂×C₅₃	semidirect
147. C₈×M₁₆	C₂×C₅₃	semidirect
148. #1299	C₂×C₅₃	semidirect
149. #1299	C₂×C₅₃	semidirect
175. C₅₃	#8245	direct (N × H)

Maximal subgroups

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

subgroup	index
C₅₃×#2931	2
C₅₃×#1543	2
C₅₃×#1525	2
C₅₃×#1451	2
C₈×M₁₆×C₅₃	2
C₅₃×#1299	2
#8245	53

Subgroups — C53×#8245 #27215757

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅₃×#8245 #27215757