Subgroups — C₅₃×#22384 #27229896

Order 13568.

578 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×#2402	C₂	1	1	yes	maximal, characteristic
2	C₅₃×#1940	C₂	1	2	yes	maximal
3	C₅₃×#1917	C₂	1	2	yes	maximal
4	C₅₃×#1916	C₂	1	2	yes	maximal
5	C₅₃×#1916	C₂	1	2	yes	maximal
6	C₅₃×#1916	C₂	1	2	yes	maximal
7	C₅₃×#1840	C₂	1	1	yes	maximal, characteristic
8	C₅₃×#1770	C₂	1	2	yes	maximal
9	C₅₃×#1768	C₂	1	1	yes	maximal, characteristic
10	C₂×C₄⋊C₈×C₅₃	C₂²	1	2	yes	—
11	C₂×C₄⋊C₈×C₅₃	C₂²	1	2	yes	—
12	C₂×C₄⋊C₈×C₅₃	C₂²	1	2	yes	—
13	C₂×#97×C₅₃	C₂²	1	2	yes	—
14	C₂×#97×C₅₃	C₂²	1	1	yes	characteristic
15	C₂×C₄×C₈×C₅₃	C₂²	1	2	yes	—
16	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
17	C₅₃×#400	C₄	1	2	yes	—
18	C₅₃×#400	C₄	1	2	yes	—
19	C₅₃×#398	C₄	1	2	yes	—
20	C₅₃×#398	C₄	1	2	yes	—
21	C₄×C₄⋊C₄×C₅₃	C₂²	1	1	yes	characteristic
22	C₄×C₄⋊C₄×C₅₃	C₂²	1	2	yes	—
23	C₄×C₄⋊C₄×C₅₃	C₂²	1	2	yes	—
24	C₄×C₄⋊C₄×C₅₃	C₂²	1	1	yes	characteristic
25	C₅₃×#376	C₂²	1	1	yes	characteristic
26	C₅₃×#336	C₂²	1	2	yes	—
27	C₅₃×#336	C₂²	1	2	yes	—
28	C₅₃×#336	C₂²	1	2	yes	—
29	C₅₃×#336	C₂²	1	1	yes	characteristic
30	C₅₃×#336	C₂²	1	2	yes	—
31	C₅₃×#336	C₂²	1	2	yes	—
32	C₅₃×#336	C₂²	1	2	yes	—
33	C₅₃×#336	C₂²	1	2	yes	—
34	C₅₃×#336	C₂²	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	2	yes	—
39	C₂×C₄⋊C₄×C₅₃	C₂×C₄	1	2	yes	—
40	C₂×C₄⋊C₄×C₅₃	C₂×C₄	1	2	yes	—
41	C₂×C₄⋊C₄×C₅₃	C₂×C₄	1	2	yes	—
42	C₂×C₄⋊C₄×C₅₃	C₂×C₄	1	2	yes	—
43	C₂×C₄⋊C₄×C₅₃	C₂×C₄	1	2	yes	—
44	C₂×C₄⋊C₄×C₅₃	C₂×C₄	1	2	yes	—
45	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
46	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
47	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
48	C₂×C₄²×C₅₃	C₂³	1	2	yes	—
49	C₂×C₄²×C₅₃	C₂³	1	2	yes	—
50	C₄⋊C₈×C₅₃	—	2	4	no	—
51	C₄⋊C₈×C₅₃	—	2	4	no	—
52	C₄⋊C₈×C₅₃	—	4	2	no	—
53	#97×C₅₃	—	4	1	no	—
54	#97×C₅₃	—	4	2	no	—
55	C₄×C₈×C₅₃	—	2	4	no	—
56	C₄×C₈×C₅₃	—	4	1	no	—
57	#95×C₅₃	C₂×C₄	1	2	yes	—
58	#95×C₅₃	C₂×C₄	1	2	yes	—
59	#95×C₅₃	C₂×C₄	1	1	yes	characteristic
60	#95×C₅₃	C₂×C₄	1	1	yes	characteristic
61	#95×C₅₃	C₂×C₄	1	2	yes	—
62	#95×C₅₃	C₂×C₄	1	2	yes	—
63	#95×C₅₃	C₂×C₄	1	1	yes	characteristic
64	#95×C₅₃	C₂×C₄	1	2	yes	—
65	#95×C₅₃	C₂×C₄	1	2	yes	—
66	#95×C₅₃	C₂×C₄	1	1	yes	characteristic
67	C₂²×C₄×C₅₃	C₂²×C₄	1	2	yes	—
68	C₂²×C₄×C₅₃	C₂²×C₄	1	2	yes	—
69	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
70	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
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	2	yes	—
75	C₂²×C₄×C₅₃	C₂⁴	1	1	yes	characteristic
76	C₂×C₈×C₅₃	—	2	2	no	—
77	C₂×C₈×C₅₃	—	2	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	—	2	2	no	—
85	C₂×C₈×C₅₃	—	2	2	no	—
86	C₂×C₈×C₅₃	—	2	2	no	—
87	C₂×C₈×C₅₃	#41	1	4	yes	—
88	C₄⋊C₄×C₅₃	—	2	4	no	—
89	C₄⋊C₄×C₅₃	—	2	4	no	—
90	C₄⋊C₄×C₅₃	—	2	4	no	—
91	C₄⋊C₄×C₅₃	—	4	2	no	—
92	C₄⋊C₄×C₅₃	—	4	2	no	—
93	C₄⋊C₄×C₅₃	—	4	2	no	—
94	C₄²×C₅₃	—	2	2	no	—
95	C₄²×C₅₃	—	2	2	no	—
96	C₄²×C₅₃	—	2	2	no	—
97	C₄²×C₅₃	—	2	1	no	—
98	C₄²×C₅₃	—	2	2	no	—
99	C₄²×C₅₃	—	2	2	no	—
100	C₄²×C₅₃	—	2	1	no	—
101	C₄²×C₅₃	—	2	2	no	—
102	C₂³×C₅₃	C₂³×C₄	1	1	yes	characteristic
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	2	no	—
107	C₂×C₄×C₅₃	—	2	1	no	—
108	C₂×C₄×C₅₃	—	2	1	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	2	no	—
114	C₂×C₄×C₅₃	—	2	2	no	—
115	C₂×C₄×C₅₃	—	2	2	no	—
116	C₂×C₄×C₅₃	—	2	2	no	—
117	C₂×C₄×C₅₃	—	2	2	no	—
118	C₂×C₄×C₅₃	—	2	2	no	—
119	C₂×C₄×C₅₃	—	2	2	no	—
120	C₂×C₄×C₅₃	—	2	2	no	—
121	C₂×C₄×C₅₃	—	2	2	no	—
122	C₂×C₄×C₅₃	—	2	2	no	—
123	C₂×C₄×C₅₃	—	2	2	no	—
124	C₂×C₄×C₅₃	#131	1	4	yes	—
125	C₂×C₄×C₅₃	C₂×#41	1	2	yes	—
126	C₂×C₄×C₅₃	H(5,𝔽₂)	1	1	yes	characteristic
127	C₂×C₄×C₅₃	2₋¹⁺⁴	1	2	yes	—
128	C₂×C₄×C₅₃	2₋¹⁺⁴	1	1	yes	characteristic
129	C₈×C₅₃	—	2	4	no	—
130	C₈×C₅₃	—	4	2	no	—
131	C₈×C₅₃	—	4	2	no	—
132	C₈×C₅₃	—	4	2	no	—
133	#22384	C₅₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
134	C₂²×C₅₃	C₄×#41	1	1	yes	characteristic
135	C₂²×C₅₃	#519	1	1	yes	characteristic
136	C₂²×C₅₃	#520	1	1	yes	characteristic
137	C₂²×C₅₃	C₂×#131	1	2	yes	—
138	C₂²×C₅₃	#568	1	2	yes	—
139	C₄×C₅₃	—	2	4	no	—
140	C₄×C₅₃	—	4	2	no	—
141	C₄×C₅₃	—	4	2	no	—
142	C₄×C₅₃	—	4	1	no	—
143	C₄×C₅₃	—	4	1	no	—
144	C₄×C₅₃	—	4	2	no	—
145	C₄×C₅₃	—	4	2	no	—
146	C₄×C₅₃	—	4	2	no	—
147	C₄×C₅₃	#539	1	2	yes	—
148	C₄×C₅₃	#541	1	2	yes	—
149	#2402	C₂×C₅₃	1	1	yes	characteristic
150	#1940	C₂×C₅₃	1	2	yes	—
151	#1917	C₂×C₅₃	1	2	yes	—
152	#1916	C₂×C₅₃	1	2	yes	—
153	#1916	C₂×C₅₃	1	2	yes	—
154	#1916	C₂×C₅₃	1	2	yes	—
155	#1840	C₂×C₅₃	1	1	yes	characteristic
156	#1770	C₂×C₅₃	1	2	yes	—
157	#1768	C₂×C₅₃	1	1	yes	characteristic
158	C₂×C₅₃	#2336	1	1	yes	characteristic
159	C₂×C₅₃	#2968	1	2	yes	—
160	C₂×C₅₃	#2986	1	1	yes	characteristic
161	C₂×C₅₃	#2987	1	1	yes	characteristic
162	C₂×C₅₃	#2993	1	2	yes	—
163	C₂×C₄⋊C₈	C₂²×C₅₃	1	2	yes	—
164	C₂×C₄⋊C₈	C₂²×C₅₃	1	2	yes	—
165	C₂×C₄⋊C₈	C₂²×C₅₃	1	2	yes	—
166	C₂×#97	C₂²×C₅₃	1	2	yes	—
167	C₂×#97	C₂²×C₅₃	1	1	yes	characteristic
168	C₂×C₄×C₈	C₂²×C₅₃	1	2	yes	—
169	C₂×C₄×C₈	C₂²×C₅₃	1	1	yes	characteristic
170	#400	C₄×C₅₃	1	2	yes	—
171	#400	C₄×C₅₃	1	2	yes	—
172	#398	C₄×C₅₃	1	2	yes	—
173	#398	C₄×C₅₃	1	2	yes	—
174	C₄×C₄⋊C₄	C₂²×C₅₃	1	1	yes	characteristic
175	C₄×C₄⋊C₄	C₂²×C₅₃	1	2	yes	—
176	C₄×C₄⋊C₄	C₂²×C₅₃	1	2	yes	—
177	C₄×C₄⋊C₄	C₂²×C₅₃	1	1	yes	characteristic
178	#376	C₂²×C₅₃	1	1	yes	characteristic
179	#336	C₂²×C₅₃	1	1	yes	characteristic
180	#336	C₂²×C₅₃	1	2	yes	—
181	#336	C₂²×C₅₃	1	2	yes	—
182	#336	C₂²×C₅₃	1	1	yes	characteristic
183	#336	C₂²×C₅₃	1	2	yes	—
184	#336	C₂²×C₅₃	1	2	yes	—
185	#336	C₂²×C₅₃	1	2	yes	—
186	#336	C₂²×C₅₃	1	2	yes	—
187	#336	C₂²×C₅₃	1	2	yes	—
188	C₅₃	#22384	1	1	yes	characteristic, direct product factor, Sylow
189	C₂²×C₈	C₂³×C₅₃	1	2	yes	—
190	C₂²×C₈	C₂³×C₅₃	1	2	yes	—
191	C₂²×C₈	C₂³×C₅₃	1	2	yes	—
192	C₂²×C₈	C₂³×C₅₃	1	2	yes	—
193	C₂×C₄⋊C₄	C₂×C₄×C₅₃	1	2	yes	—
194	C₂×C₄⋊C₄	C₂×C₄×C₅₃	1	2	yes	—
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	2	yes	—
198	C₂×C₄⋊C₄	C₂×C₄×C₅₃	1	2	yes	—
199	C₂×C₄²	C₂³×C₅₃	1	1	yes	characteristic
200	C₂×C₄²	C₂³×C₅₃	1	1	yes	characteristic
201	C₂×C₄²	C₂³×C₅₃	1	1	yes	characteristic
202	C₂×C₄²	C₂³×C₅₃	1	2	yes	—
203	C₂×C₄²	C₂³×C₅₃	1	2	yes	—
204	C₄⋊C₈	—	2	4	no	—
205	C₄⋊C₈	—	2	4	no	—
206	C₄⋊C₈	—	4	2	no	—
207	#97	—	4	1	no	—
208	#97	—	4	2	no	—
209	C₄×C₈	—	2	4	no	—
210	C₄×C₈	—	4	1	no	—
211	#95	C₂×C₄×C₅₃	1	2	yes	—
212	#95	C₂×C₄×C₅₃	1	2	yes	—
213	#95	C₂×C₄×C₅₃	1	1	yes	characteristic
214	#95	C₂×C₄×C₅₃	1	1	yes	characteristic
215	#95	C₂×C₄×C₅₃	1	2	yes	—
216	#95	C₂×C₄×C₅₃	1	2	yes	—
217	#95	C₂×C₄×C₅₃	1	1	yes	characteristic
218	#95	C₂×C₄×C₅₃	1	2	yes	—
219	#95	C₂×C₄×C₅₃	1	2	yes	—
220	#95	C₂×C₄×C₅₃	1	1	yes	characteristic
221	C₂²×C₄	C₂²×C₄×C₅₃	1	2	yes	—
222	C₂²×C₄	C₂²×C₄×C₅₃	1	2	yes	—
223	C₂²×C₄	C₂²×C₄×C₅₃	1	1	yes	characteristic
224	C₂²×C₄	C₂²×C₄×C₅₃	1	1	yes	characteristic
225	C₂²×C₄	C₂²×C₄×C₅₃	1	2	yes	—
226	C₂²×C₄	C₂²×C₄×C₅₃	1	2	yes	—
227	C₂²×C₄	C₂²×C₄×C₅₃	1	2	yes	—
228	C₂²×C₄	C₂²×C₄×C₅₃	1	2	yes	—
229	C₂²×C₄	C₂⁴×C₅₃	1	1	yes	characteristic
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	2	no	—
235	C₂×C₈	—	2	2	no	—
236	C₂×C₈	—	2	2	no	—
237	C₂×C₈	—	2	2	no	—
238	C₂×C₈	—	2	2	no	—
239	C₂×C₈	—	2	2	no	—
240	C₂×C₈	—	2	2	no	—
241	C₂×C₈	#41×C₅₃	1	4	yes	—
242	C₄⋊C₄	—	2	4	no	—
243	C₄⋊C₄	—	2	4	no	—
244	C₄⋊C₄	—	2	4	no	—
245	C₄⋊C₄	—	4	2	no	—
246	C₄⋊C₄	—	4	2	no	—
247	C₄⋊C₄	—	4	2	no	—
248	C₄²	—	2	2	no	—
249	C₄²	—	2	2	no	—
250	C₄²	—	2	2	no	—
251	C₄²	—	2	1	no	—
252	C₄²	—	2	2	no	—
253	C₄²	—	2	2	no	—
254	C₄²	—	2	1	no	—
255	C₄²	—	2	2	no	—
256	C₂³	C₂³×C₄×C₅₃	1	1	yes	characteristic
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	1	no	—
262	C₂×C₄	—	2	1	no	—
263	C₂×C₄	—	2	2	no	—
264	C₂×C₄	—	2	2	no	—
265	C₂×C₄	—	2	2	no	—
266	C₂×C₄	—	2	2	no	—
267	C₂×C₄	—	2	2	no	—
268	C₂×C₄	—	2	2	no	—
269	C₂×C₄	—	2	2	no	—
270	C₂×C₄	—	2	2	no	—
271	C₂×C₄	—	2	2	no	—
272	C₂×C₄	—	2	2	no	—
273	C₂×C₄	—	2	2	no	—
274	C₂×C₄	—	2	2	no	—
275	C₂×C₄	—	2	2	no	—
276	C₂×C₄	—	2	2	no	—
277	C₂×C₄	—	2	2	no	—
278	C₂×C₄	#131×C₅₃	1	4	yes	—
279	C₂×C₄	C₂×#41×C₅₃	1	2	yes	—
280	C₂×C₄	H(5,𝔽₂)×C₅₃	1	1	yes	characteristic
281	C₂×C₄	2₋¹⁺⁴×C₅₃	1	2	yes	—
282	C₂×C₄	2₋¹⁺⁴×C₅₃	1	1	yes	characteristic
283	C₈	—	2	4	no	—
284	C₈	—	4	2	no	—
285	C₈	—	4	2	no	—
286	C₈	—	4	2	no	—
287	C₂²	C₄×#41×C₅₃	1	1	yes	characteristic
288	C₂²	C₅₃×#519	1	1	yes	characteristic
289	C₂²	C₅₃×#520	1	1	yes	characteristic
290	C₂²	C₂×#131×C₅₃	1	2	yes	—
291	C₂²	C₅₃×#568	1	2	yes	—
292	C₄	—	2	4	no	—
293	C₄	—	4	2	no	—
294	C₄	—	4	2	no	—
295	C₄	—	4	1	no	—
296	C₄	—	4	1	no	—
297	C₄	—	4	2	no	—
298	C₄	—	4	2	no	—
299	C₄	—	4	2	no	—
300	C₄	C₅₃×#539	1	2	yes	—
301	C₄	C₅₃×#541	1	2	yes	—
302	C₂	C₅₃×#2336	1	1	yes	characteristic
303	C₂	C₅₃×#2968	1	2	yes	—
304	C₂	C₅₃×#2986	1	1	yes	characteristic
305	C₂	C₅₃×#2987	1	1	yes	characteristic
306	C₂	C₅₃×#2993	1	2	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
133. #22384	C₅₃	direct (N × H)
188. C₅₃	#22384	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×#2402	2
C₅₃×#1940	2
C₅₃×#1917	2
C₅₃×#1916	2
C₅₃×#1840	2
C₅₃×#1770	2
C₅₃×#1768	2
#22384	53

Subgroups — C53×#22384 #27229896

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅₃×#22384 #27229896