Subgroups — C₅₃×#22241 #27229753

Order 13568.

694 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	C₂×C₅₃×#446	C₂	1	1	yes	maximal, characteristic
2	C₂×C₈×Q₈×C₅₃	C₂	1	1	yes	maximal, characteristic
3	C₅₃×#2399	C₂	1	1	yes	maximal, characteristic
4	C₅₃×#1985	C₂	1	1	yes	maximal, characteristic
5	C₅₃×#1985	C₂	1	1	yes	maximal, characteristic
6	C₅₃×#1917	C₂	1	4	yes	maximal
7	C₅₃×#1916	C₂	1	4	yes	maximal
8	C₅₃×#1832	C₂	1	1	yes	maximal, characteristic
9	C₅₃×#1746	C₂	1	1	yes	maximal, characteristic
10	C₂×C₄×Q₈×C₅₃	C₂²	1	1	yes	characteristic
11	C₂×C₄×Q₈×C₅₃	C₂²	1	1	yes	characteristic
12	C₅₃×#446	—	2	4	no	—
13	C₈×Q₈×C₅₃	—	2	4	no	—
14	C₂×C₄⋊C₈×C₅₃	C₂²	1	2	yes	—
15	C₂×C₄⋊C₈×C₅₃	C₂²	1	2	yes	—
16	C₂×C₄⋊C₈×C₅₃	C₂²	1	1	yes	characteristic
17	C₂×C₄⋊C₈×C₅₃	C₂²	1	1	yes	characteristic
18	C₂×#97×C₅₃	C₂²	1	2	yes	—
19	C₂×#97×C₅₃	C₂²	1	1	yes	characteristic
20	C₂×C₄×C₈×C₅₃	C₂²	1	2	yes	—
21	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
22	C₅₃×#398	C₄	1	2	yes	—
23	C₅₃×#398	C₄	1	2	yes	—
24	C₅₃×#395	C₄	1	2	yes	—
25	C₅₃×#395	C₄	1	2	yes	—
26	C₄×C₄⋊C₄×C₅₃	C₂²	1	4	yes	—
27	C₅₃×#376	C₂²	1	1	yes	characteristic
28	C₅₃×#336	C₂²	1	2	yes	—
29	C₅₃×#336	C₂²	1	4	yes	—
30	C₅₃×#336	C₂²	1	4	yes	—
31	C₅₃×#336	C₂²	1	4	yes	—
32	C₅₃×#336	C₂²	1	2	yes	—
33	C₂²×Q₈×C₅₃	C₂×C₄	1	1	yes	characteristic
34	C₂²×Q₈×C₅₃	C₂×C₄	1	1	yes	characteristic
35	C₂²×C₈×C₅₃	C₂³	1	2	yes	—
36	C₂²×C₈×C₅₃	C₂³	1	2	yes	—
37	C₂²×C₈×C₅₃	C₂³	1	2	yes	—
38	C₂²×C₈×C₅₃	C₂³	1	1	yes	characteristic
39	C₂²×C₈×C₅₃	C₂³	1	1	yes	characteristic
40	C₄×Q₈×C₅₃	—	2	2	no	—
41	C₄×Q₈×C₅₃	—	2	2	no	—
42	C₄×Q₈×C₅₃	—	2	2	no	—
43	C₄×Q₈×C₅₃	—	2	2	no	—
44	C₂×C₄⋊C₄×C₅₃	C₂×C₄	1	2	yes	—
45	C₂×C₄⋊C₄×C₅₃	C₂×C₄	1	2	yes	—
46	C₂×C₄⋊C₄×C₅₃	C₂×C₄	1	1	yes	characteristic
47	C₂×C₄⋊C₄×C₅₃	C₂×C₄	1	1	yes	characteristic
48	C₂×C₄⋊C₄×C₅₃	C₂×C₄	1	4	yes	—
49	C₂×C₄⋊C₄×C₅₃	C₂×C₄	1	4	yes	—
50	C₂×C₄²×C₅₃	C₂³	1	2	yes	—
51	C₂×C₄²×C₅₃	C₂³	1	2	yes	—
52	C₂×C₄²×C₅₃	C₂³	1	2	yes	—
53	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
54	C₄⋊C₈×C₅₃	—	2	4	no	—
55	C₄⋊C₈×C₅₃	—	2	4	no	—
56	C₄⋊C₈×C₅₃	—	2	2	no	—
57	C₄⋊C₈×C₅₃	—	2	2	no	—
58	#97×C₅₃	—	2	4	no	—
59	#97×C₅₃	—	2	2	no	—
60	C₄×C₈×C₅₃	—	2	4	no	—
61	C₄×C₈×C₅₃	D₄	1	4	yes	—
62	#95×C₅₃	C₂×C₄	1	2	yes	—
63	#95×C₅₃	C₂×C₄	1	2	yes	—
64	#95×C₅₃	C₂×C₄	1	4	yes	—
65	#95×C₅₃	C₂×C₄	1	4	yes	—
66	C₂×Q₈×C₅₃	—	2	2	no	—
67	C₂×Q₈×C₅₃	—	2	2	no	—
68	C₂×Q₈×C₅₃	—	2	4	no	—
69	C₂×Q₈×C₅₃	—	4	2	no	—
70	C₂²×C₄×C₅₃	C₂²×C₄	1	4	yes	—
71	C₂²×C₄×C₅₃	C₂²×C₄	1	2	yes	—
72	C₂²×C₄×C₅₃	C₂²×C₄	1	2	yes	—
73	C₂²×C₄×C₅₃	C₂²×C₄	1	2	yes	—
74	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
75	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
76	C₂²×C₄×C₅₃	C₂²×C₄	1	2	yes	—
77	C₂²×C₄×C₅₃	C₂⁴	1	1	yes	characteristic
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	4	no	—
84	C₂×C₈×C₅₃	—	2	4	no	—
85	C₂×C₈×C₅₃	—	2	2	no	—
86	C₂×C₈×C₅₃	—	2	2	no	—
87	C₂×C₈×C₅₃	C₂×D₄	1	2	yes	—
88	C₂×C₈×C₅₃	C₂×D₄	1	2	yes	—
89	C₄⋊C₄×C₅₃	—	2	4	no	—
90	C₄⋊C₄×C₅₃	—	2	4	no	—
91	C₄⋊C₄×C₅₃	—	2	2	no	—
92	C₄⋊C₄×C₅₃	—	2	2	no	—
93	C₄⋊C₄×C₅₃	—	2	8	no	—
94	C₄⋊C₄×C₅₃	—	4	4	no	—
95	C₄²×C₅₃	—	2	2	no	—
96	C₄²×C₅₃	—	2	2	no	—
97	C₄²×C₅₃	—	2	2	no	—
98	C₄²×C₅₃	—	2	2	no	—
99	C₄²×C₅₃	—	2	2	no	—
100	C₄²×C₅₃	—	2	2	no	—
101	C₄²×C₅₃	C₂×D₄	1	2	yes	—
102	C₄²×C₅₃	#41	1	2	yes	—
103	C₂³×C₅₃	C₂³×C₄	1	1	yes	characteristic
104	Q₈×C₅₃	—	2	8	no	—
105	Q₈×C₅₃	—	4	4	no	—
106	C₂×C₄×C₅₃	—	2	2	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	—	2	2	no	—
111	C₂×C₄×C₅₃	—	2	2	no	—
112	C₂×C₄×C₅₃	—	2	2	no	—
113	C₂×C₄×C₅₃	—	2	4	no	—
114	C₂×C₄×C₅₃	—	2	4	no	—
115	C₂×C₄×C₅₃	—	2	4	no	—
116	C₂×C₄×C₅₃	—	2	2	no	—
117	C₂×C₄×C₅₃	—	2	2	no	—
118	C₂×C₄×C₅₃	—	2	4	no	—
119	C₂×C₄×C₅₃	—	2	4	no	—
120	C₂×C₄×C₅₃	C₄×D₄	1	2	yes	—
121	C₂×C₄×C₅₃	C₄×D₄	1	2	yes	—
122	C₂×C₄×C₅₃	#131	1	4	yes	—
123	C₂×C₄×C₅₃	C₂²×D₄	1	1	yes	characteristic
124	C₂×C₄×C₅₃	C₂×#41	1	1	yes	characteristic
125	C₂×C₄×C₅₃	2₋¹⁺⁴	1	2	yes	—
126	C₂×C₄×C₅₃	2₋¹⁺⁴	1	2	yes	—
127	C₈×C₅₃	—	2	2	no	—
128	C₈×C₅₃	—	2	2	no	—
129	C₈×C₅₃	—	4	2	no	—
130	C₈×C₅₃	—	4	2	no	—
131	C₈×C₅₃	—	4	2	no	—
132	#22241	C₅₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
133	C₂²×C₅₃	C₂×C₄×D₄	1	1	yes	characteristic
134	C₂²×C₅₃	#519	1	2	yes	—
135	C₂²×C₅₃	C₂×#131	1	2	yes	—
136	C₂²×C₅₃	#568	1	2	yes	—
137	C₄×C₅₃	—	2	2	no	—
138	C₄×C₅₃	—	2	4	no	—
139	C₄×C₅₃	—	2	2	no	—
140	C₄×C₅₃	—	4	2	no	—
141	C₄×C₅₃	—	4	2	no	—
142	C₄×C₅₃	—	4	2	no	—
143	C₄×C₅₃	—	4	4	no	—
144	C₄×C₅₃	#536	1	1	yes	characteristic
145	C₄×C₅₃	#536	1	1	yes	characteristic
146	C₄×C₅₃	#541	1	2	yes	—
147	C₂×#446	C₂×C₅₃	1	1	yes	characteristic
148	C₂×C₈×Q₈	C₂×C₅₃	1	1	yes	characteristic
149	#2399	C₂×C₅₃	1	1	yes	characteristic
150	#1985	C₂×C₅₃	1	1	yes	characteristic
151	#1985	C₂×C₅₃	1	1	yes	characteristic
152	#1917	C₂×C₅₃	1	4	yes	—
153	#1916	C₂×C₅₃	1	4	yes	—
154	#1832	C₂×C₅₃	1	1	yes	characteristic
155	#1746	C₂×C₅₃	1	1	yes	characteristic
156	C₂×C₅₃	#2327	1	1	yes	characteristic
157	C₂×C₅₃	#2929	1	1	yes	characteristic
158	C₂×C₅₃	#2930	1	1	yes	characteristic
159	C₂×C₅₃	#2985	1	4	yes	—
160	C₂×C₄×Q₈	C₂²×C₅₃	1	1	yes	characteristic
161	C₂×C₄×Q₈	C₂²×C₅₃	1	1	yes	characteristic
162	#446	—	2	4	no	—
163	C₈×Q₈	—	2	4	no	—
164	C₂×C₄⋊C₈	C₂²×C₅₃	1	1	yes	characteristic
165	C₂×C₄⋊C₈	C₂²×C₅₃	1	1	yes	characteristic
166	C₂×C₄⋊C₈	C₂²×C₅₃	1	2	yes	—
167	C₂×C₄⋊C₈	C₂²×C₅₃	1	2	yes	—
168	C₂×#97	C₂²×C₅₃	1	2	yes	—
169	C₂×#97	C₂²×C₅₃	1	1	yes	characteristic
170	C₂×C₄×C₈	C₂²×C₅₃	1	2	yes	—
171	C₂×C₄×C₈	C₂²×C₅₃	1	1	yes	characteristic
172	#398	C₄×C₅₃	1	2	yes	—
173	#398	C₄×C₅₃	1	2	yes	—
174	#395	C₄×C₅₃	1	2	yes	—
175	#395	C₄×C₅₃	1	2	yes	—
176	C₄×C₄⋊C₄	C₂²×C₅₃	1	4	yes	—
177	#376	C₂²×C₅₃	1	1	yes	characteristic
178	#336	C₂²×C₅₃	1	4	yes	—
179	#336	C₂²×C₅₃	1	2	yes	—
180	#336	C₂²×C₅₃	1	4	yes	—
181	#336	C₂²×C₅₃	1	4	yes	—
182	#336	C₂²×C₅₃	1	2	yes	—
183	C₅₃	#22241	1	1	yes	characteristic, direct product factor, Sylow
184	C₂²×Q₈	C₂×C₄×C₅₃	1	1	yes	characteristic
185	C₂²×Q₈	C₂×C₄×C₅₃	1	1	yes	characteristic
186	C₂²×C₈	C₂³×C₅₃	1	2	yes	—
187	C₂²×C₈	C₂³×C₅₃	1	2	yes	—
188	C₂²×C₈	C₂³×C₅₃	1	2	yes	—
189	C₂²×C₈	C₂³×C₅₃	1	1	yes	characteristic
190	C₂²×C₈	C₂³×C₅₃	1	1	yes	characteristic
191	C₄×Q₈	—	2	2	no	—
192	C₄×Q₈	—	2	2	no	—
193	C₄×Q₈	—	2	2	no	—
194	C₄×Q₈	—	2	2	no	—
195	C₂×C₄⋊C₄	C₂×C₄×C₅₃	1	2	yes	—
196	C₂×C₄⋊C₄	C₂×C₄×C₅₃	1	2	yes	—
197	C₂×C₄⋊C₄	C₂×C₄×C₅₃	1	1	yes	characteristic
198	C₂×C₄⋊C₄	C₂×C₄×C₅₃	1	1	yes	characteristic
199	C₂×C₄⋊C₄	C₂×C₄×C₅₃	1	4	yes	—
200	C₂×C₄⋊C₄	C₂×C₄×C₅₃	1	4	yes	—
201	C₂×C₄²	C₂³×C₅₃	1	2	yes	—
202	C₂×C₄²	C₂³×C₅₃	1	2	yes	—
203	C₂×C₄²	C₂³×C₅₃	1	2	yes	—
204	C₂×C₄²	C₂³×C₅₃	1	1	yes	characteristic
205	C₄⋊C₈	—	2	4	no	—
206	C₄⋊C₈	—	2	4	no	—
207	C₄⋊C₈	—	2	2	no	—
208	C₄⋊C₈	—	2	2	no	—
209	#97	—	2	4	no	—
210	#97	—	2	2	no	—
211	C₄×C₈	—	2	4	no	—
212	C₄×C₈	D₄×C₅₃	1	4	yes	—
213	#95	C₂×C₄×C₅₃	1	2	yes	—
214	#95	C₂×C₄×C₅₃	1	2	yes	—
215	#95	C₂×C₄×C₅₃	1	4	yes	—
216	#95	C₂×C₄×C₅₃	1	4	yes	—
217	C₂×Q₈	—	2	2	no	—
218	C₂×Q₈	—	2	2	no	—
219	C₂×Q₈	—	2	4	no	—
220	C₂×Q₈	—	4	2	no	—
221	C₂²×C₄	C₂²×C₄×C₅₃	1	4	yes	—
222	C₂²×C₄	C₂²×C₄×C₅₃	1	2	yes	—
223	C₂²×C₄	C₂²×C₄×C₅₃	1	2	yes	—
224	C₂²×C₄	C₂²×C₄×C₅₃	1	2	yes	—
225	C₂²×C₄	C₂²×C₄×C₅₃	1	1	yes	characteristic
226	C₂²×C₄	C₂²×C₄×C₅₃	1	1	yes	characteristic
227	C₂²×C₄	C₂²×C₄×C₅₃	1	2	yes	—
228	C₂²×C₄	C₂⁴×C₅₃	1	1	yes	characteristic
229	C₂×C₈	—	2	2	no	—
230	C₂×C₈	—	2	2	no	—
231	C₂×C₈	—	2	2	no	—
232	C₂×C₈	—	2	2	no	—
233	C₂×C₈	—	2	2	no	—
234	C₂×C₈	—	2	4	no	—
235	C₂×C₈	—	2	4	no	—
236	C₂×C₈	—	2	2	no	—
237	C₂×C₈	—	2	2	no	—
238	C₂×C₈	C₂×D₄×C₅₃	1	2	yes	—
239	C₂×C₈	C₂×D₄×C₅₃	1	2	yes	—
240	C₄⋊C₄	—	2	4	no	—
241	C₄⋊C₄	—	2	2	no	—
242	C₄⋊C₄	—	2	2	no	—
243	C₄⋊C₄	—	2	4	no	—
244	C₄⋊C₄	—	2	8	no	—
245	C₄⋊C₄	—	4	4	no	—
246	C₄²	—	2	2	no	—
247	C₄²	—	2	2	no	—
248	C₄²	—	2	2	no	—
249	C₄²	—	2	2	no	—
250	C₄²	—	2	2	no	—
251	C₄²	—	2	2	no	—
252	C₄²	C₂×D₄×C₅₃	1	2	yes	—
253	C₄²	#41×C₅₃	1	2	yes	—
254	C₂³	C₂³×C₄×C₅₃	1	1	yes	characteristic
255	Q₈	—	2	8	no	—
256	Q₈	—	4	4	no	—
257	C₂×C₄	—	2	2	no	—
258	C₂×C₄	—	2	2	no	—
259	C₂×C₄	—	2	2	no	—
260	C₂×C₄	—	2	2	no	—
261	C₂×C₄	—	2	2	no	—
262	C₂×C₄	—	2	2	no	—
263	C₂×C₄	—	2	2	no	—
264	C₂×C₄	—	2	4	no	—
265	C₂×C₄	—	2	4	no	—
266	C₂×C₄	—	2	4	no	—
267	C₂×C₄	—	2	2	no	—
268	C₂×C₄	—	2	4	no	—
269	C₂×C₄	—	2	4	no	—
270	C₂×C₄	—	2	2	no	—
271	C₂×C₄	C₄×D₄×C₅₃	1	2	yes	—
272	C₂×C₄	C₄×D₄×C₅₃	1	2	yes	—
273	C₂×C₄	#131×C₅₃	1	4	yes	—
274	C₂×C₄	C₂²×D₄×C₅₃	1	1	yes	characteristic
275	C₂×C₄	C₂×#41×C₅₃	1	1	yes	characteristic
276	C₂×C₄	2₋¹⁺⁴×C₅₃	1	2	yes	—
277	C₂×C₄	2₋¹⁺⁴×C₅₃	1	2	yes	—
278	C₈	—	2	2	no	—
279	C₈	—	2	2	no	—
280	C₈	—	4	2	no	—
281	C₈	—	4	2	no	—
282	C₈	—	4	2	no	—
283	C₂²	C₂×C₄×D₄×C₅₃	1	1	yes	characteristic
284	C₂²	C₅₃×#519	1	2	yes	—
285	C₂²	C₂×#131×C₅₃	1	2	yes	—
286	C₂²	C₅₃×#568	1	2	yes	—
287	C₄	—	2	2	no	—
288	C₄	—	2	4	no	—
289	C₄	—	2	2	no	—
290	C₄	—	4	4	no	—
291	C₄	—	4	2	no	—
292	C₄	—	4	2	no	—
293	C₄	—	4	2	no	—
294	C₄	C₅₃×#536	1	1	yes	characteristic
295	C₄	C₅₃×#536	1	1	yes	characteristic
296	C₄	C₅₃×#541	1	2	yes	—
297	C₂	C₅₃×#2327	1	1	yes	characteristic
298	C₂	C₅₃×#2929	1	1	yes	characteristic
299	C₂	C₅₃×#2930	1	1	yes	characteristic
300	C₂	C₅₃×#2985	1	4	yes	—

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

Complements

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

N	H ≅ G/N	type
132. #22241	C₅₃	direct (N × H)
183. C₅₃	#22241	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×C₅₃×#446	2
C₂×C₈×Q₈×C₅₃	2
C₅₃×#2399	2
C₅₃×#1985	2
C₅₃×#1917	2
C₅₃×#1916	2
C₅₃×#1832	2
C₅₃×#1746	2
#22241	53

Subgroups — C53×#22241 #27229753

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅₃×#22241 #27229753