Subgroups — C₅₃×#7916 #27215428

Order 13568.

622 conjugacy classes of subgroups.

The subgroup lattice has 292 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₅₃×#432	C₂	1	1	yes	maximal, characteristic
2	C₂×C₅₃×#431	C₂	1	1	yes	maximal, characteristic
3	C₂×C₅₃×#383	C₂	1	1	yes	maximal, characteristic
4	C₅₃×#1280	C₂	1	4	yes	maximal
5	C₂²×C₄²×C₅₃	C₂²	1	1	yes	characteristic
6	C₅₃×#432	—	2	2	no	—
7	C₅₃×#432	C₄	1	4	yes	complemented
8	C₅₃×#431	—	2	2	no	—
9	C₅₃×#431	C₄	1	4	yes	complemented
10	C₂×C₄⋊C₈×C₅₃	—	2	1	no	—
11	C₂×C₄⋊C₈×C₅₃	—	2	1	no	—
12	C₂×C₂²⋊C₈×C₅₃	C₂²	1	2	yes	—
13	C₂×C₂²⋊C₈×C₅₃	C₂²	1	2	yes	—
14	C₂×#97×C₅₃	—	2	1	no	—
15	C₂×C₄×C₈×C₅₃	—	2	1	no	—
16	C₅₃×#383	—	2	1	no	—
17	C₅₃×#383	—	2	1	no	—
18	C₅₃×#383	—	2	1	no	—
19	C₅₃×#383	—	2	1	no	—
20	C₂×#95×C₅₃	C₄	1	2	yes	—
21	C₂×#95×C₅₃	C₄	1	2	yes	—
22	C₂×#95×C₅₃	C₂²	1	2	yes	—
23	C₂³×C₄×C₅₃	C₂×C₄	1	1	yes	characteristic
24	C₂³×C₄×C₅₃	C₂×C₄	1	4	yes	—
25	C₂³×C₄×C₅₃	C₂×C₄	1	1	yes	characteristic
26	C₂³×C₄×C₅₃	C₂³	1	1	yes	characteristic
27	C₂²×C₈×C₅₃	—	2	2	no	—
28	C₂²×C₈×C₅₃	—	2	2	no	—
29	C₂×C₄²×C₅₃	C₂×C₄	1	2	yes	—
30	C₂×C₄²×C₅₃	C₂×C₄	1	2	yes	—
31	C₂×C₄²×C₅₃	D₄	1	1	yes	characteristic
32	C₂×C₄²×C₅₃	D₄	1	1	yes	characteristic
33	C₂×C₄²×C₅₃	D₄	1	1	yes	characteristic
34	C₂×C₄²×C₅₃	D₄	1	1	yes	characteristic
35	C₂×C₄²×C₅₃	D₄	1	2	yes	—
36	C₂×C₄²×C₅₃	Q₈	1	2	yes	—
37	C₄⋊C₈×C₅₃	—	2	4	no	—
38	C₄⋊C₈×C₅₃	—	2	4	no	—
39	C₂²⋊C₈×C₅₃	—	2	2	no	—
40	C₂²⋊C₈×C₅₃	—	2	2	no	—
41	C₂²⋊C₈×C₅₃	C₂×C₄	1	4	yes	—
42	C₂²⋊C₈×C₅₃	C₂×C₄	1	4	yes	—
43	#97×C₅₃	—	2	4	no	—
44	C₄×C₈×C₅₃	—	2	4	no	—
45	#95×C₅₃	—	2	2	no	—
46	#95×C₅₃	—	2	2	no	—
47	#95×C₅₃	—	2	2	no	—
48	#95×C₅₃	—	2	2	no	—
49	#95×C₅₃	—	2	2	no	—
50	#95×C₅₃	—	2	2	no	—
51	C₂⁴×C₅₃	C₂²×C₄	1	1	yes	characteristic
52	C₂²×C₄×C₅₃	—	2	4	no	—
53	C₂²×C₄×C₅₃	—	2	4	no	—
54	C₂²×C₄×C₅₃	—	2	8	no	—
55	C₂²×C₄×C₅₃	—	4	2	no	—
56	C₂²×C₄×C₅₃	—	4	2	no	—
57	C₂²×C₄×C₅₃	—	4	4	no	—
58	C₂²×C₄×C₅₃	—	4	4	no	—
59	C₂²×C₄×C₅₃	C₄²	1	2	yes	—
60	C₂²×C₄×C₅₃	C₄²	1	2	yes	—
61	C₂²×C₄×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
62	C₂²×C₄×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
63	C₂²×C₄×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
64	C₂²×C₄×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
65	C₂²×C₄×C₅₃	C₄⋊C₄	1	2	yes	—
66	C₂²×C₄×C₅₃	C₄⋊C₄	1	2	yes	—
67	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
68	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
69	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
70	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
71	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
72	C₂²×C₄×C₅₃	C₂×Q₈	1	1	yes	characteristic
73	C₂×C₈×C₅₃	—	2	4	no	—
74	C₂×C₈×C₅₃	—	2	4	no	—
75	C₂×C₈×C₅₃	—	4	4	no	—
76	C₂×C₈×C₅₃	—	4	4	no	—
77	C₄²×C₅₃	C₂²⋊C₄	1	2	yes	—
78	C₄²×C₅₃	C₂²⋊C₄	1	2	yes	—
79	C₄²×C₅₃	C₂²⋊C₄	1	2	yes	—
80	C₄²×C₅₃	C₂²⋊C₄	1	2	yes	—
81	C₄²×C₅₃	C₄⋊C₄	1	2	yes	—
82	C₄²×C₅₃	C₄⋊C₄	1	2	yes	—
83	C₄²×C₅₃	C₄⋊C₄	1	2	yes	—
84	C₄²×C₅₃	C₄⋊C₄	1	2	yes	—
85	C₂³×C₅₃	—	2	2	no	—
86	C₂³×C₅₃	—	2	2	no	—
87	C₂³×C₅₃	C₂×C₄²	1	1	yes	characteristic
88	C₂³×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
89	C₂³×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
90	C₂³×C₅₃	#131	1	4	yes	—
91	C₂×C₄×C₅₃	—	2	4	no	—
92	C₂×C₄×C₅₃	—	2	2	no	—
93	C₂×C₄×C₅₃	—	2	2	no	—
94	C₂×C₄×C₅₃	—	2	2	no	—
95	C₂×C₄×C₅₃	—	2	2	no	—
96	C₂×C₄×C₅₃	—	2	2	no	—
97	C₂×C₄×C₅₃	—	2	2	no	—
98	C₂×C₄×C₅₃	—	2	4	no	—
99	C₂×C₄×C₅₃	—	4	2	no	—
100	C₂×C₄×C₅₃	—	4	2	no	—
101	C₂×C₄×C₅₃	—	4	1	no	—
102	C₂×C₄×C₅₃	—	4	1	no	—
103	C₂×C₄×C₅₃	—	4	1	no	—
104	C₂×C₄×C₅₃	—	4	1	no	—
105	C₂×C₄×C₅₃	—	4	4	no	—
106	C₂×C₄×C₅₃	—	4	4	no	—
107	C₂×C₄×C₅₃	—	4	4	no	—
108	C₂×C₄×C₅₃	—	8	8	no	—
109	C₂×C₄×C₅₃	#95	1	2	yes	—
110	C₂×C₄×C₅₃	#95	1	2	yes	—
111	C₂×C₄×C₅₃	#95	1	2	yes	—
112	C₂×C₄×C₅₃	#95	1	2	yes	—
113	C₂×C₄×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
114	C₂×C₄×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
115	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
116	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
117	C₈×C₅₃	—	4	4	no	—
118	C₈×C₅₃	—	4	4	no	—
119	#7916	C₅₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
120	C₂²×C₅₃	—	2	8	no	—
121	C₂²×C₅₃	—	2	1	no	—
122	C₂²×C₅₃	—	2	1	no	—
123	C₂²×C₅₃	—	2	1	no	—
124	C₂²×C₅₃	—	2	1	no	—
125	C₂²×C₅₃	C₂×#95	1	1	yes	characteristic
126	C₂²×C₅₃	#405	1	2	yes	—
127	C₂²×C₅₃	#408	1	2	yes	—
128	C₂²×C₅₃	#410	1	2	yes	—
129	C₂²×C₅₃	#413	1	2	yes	—
130	C₂²×C₅₃	#424	1	2	yes	—
131	C₄×C₅₃	—	2	2	no	—
132	C₄×C₅₃	—	2	2	no	—
133	C₄×C₅₃	—	2	1	no	—
134	C₄×C₅₃	—	2	1	no	—
135	C₄×C₅₃	—	2	1	no	—
136	C₄×C₅₃	—	2	1	no	—
137	C₄×C₅₃	—	2	2	no	—
138	C₄×C₅₃	—	2	2	no	—
139	C₄×C₅₃	—	8	4	no	—
140	C₂×#432	C₂×C₅₃	1	1	yes	characteristic
141	C₂×#431	C₂×C₅₃	1	1	yes	characteristic
142	C₂×#383	C₂×C₅₃	1	1	yes	characteristic
143	#1280	C₂×C₅₃	1	4	yes	—
144	C₂×C₅₃	—	2	2	no	—
145	C₂×C₅₃	—	2	2	no	—
146	C₂×C₅₃	#1473	1	4	yes	—
147	C₂×C₅₃	#1729	1	1	yes	characteristic
148	C₂×C₅₃	#1740	1	1	yes	characteristic
149	C₂×C₅₃	#1742	1	1	yes	characteristic
150	C₂²×C₄²	C₂²×C₅₃	1	1	yes	characteristic
151	#432	—	2	2	no	—
152	#432	C₄×C₅₃	1	4	yes	complemented
153	#431	—	2	2	no	—
154	#431	C₄×C₅₃	1	4	yes	complemented
155	C₂×C₄⋊C₈	—	2	1	no	—
156	C₂×C₄⋊C₈	—	2	1	no	—
157	C₂×C₂²⋊C₈	C₂²×C₅₃	1	2	yes	—
158	C₂×C₂²⋊C₈	C₂²×C₅₃	1	2	yes	—
159	C₂×#97	—	2	1	no	—
160	C₂×C₄×C₈	—	2	1	no	—
161	#383	—	2	1	no	—
162	#383	—	2	1	no	—
163	#383	—	2	1	no	—
164	#383	—	2	1	no	—
165	C₂×#95	C₄×C₅₃	1	2	yes	—
166	C₂×#95	C₄×C₅₃	1	2	yes	—
167	C₂×#95	C₂²×C₅₃	1	2	yes	—
168	C₅₃	#7916	1	1	yes	characteristic, direct product factor, Sylow
169	C₂³×C₄	C₂×C₄×C₅₃	1	1	yes	characteristic
170	C₂³×C₄	C₂×C₄×C₅₃	1	4	yes	—
171	C₂³×C₄	C₂×C₄×C₅₃	1	1	yes	characteristic
172	C₂³×C₄	C₂³×C₅₃	1	1	yes	characteristic
173	C₂²×C₈	—	2	2	no	—
174	C₂²×C₈	—	2	2	no	—
175	C₂×C₄²	C₂×C₄×C₅₃	1	2	yes	—
176	C₂×C₄²	C₂×C₄×C₅₃	1	2	yes	—
177	C₂×C₄²	D₄×C₅₃	1	2	yes	—
178	C₂×C₄²	D₄×C₅₃	1	1	yes	characteristic
179	C₂×C₄²	D₄×C₅₃	1	1	yes	characteristic
180	C₂×C₄²	D₄×C₅₃	1	1	yes	characteristic
181	C₂×C₄²	D₄×C₅₃	1	1	yes	characteristic
182	C₂×C₄²	Q₈×C₅₃	1	2	yes	—
183	C₄⋊C₈	—	2	4	no	—
184	C₄⋊C₈	—	2	4	no	—
185	C₂²⋊C₈	—	2	2	no	—
186	C₂²⋊C₈	—	2	2	no	—
187	C₂²⋊C₈	C₂×C₄×C₅₃	1	4	yes	—
188	C₂²⋊C₈	C₂×C₄×C₅₃	1	4	yes	—
189	#97	—	2	4	no	—
190	C₄×C₈	—	2	4	no	—
191	#95	—	2	2	no	—
192	#95	—	2	2	no	—
193	#95	—	2	2	no	—
194	#95	—	2	2	no	—
195	#95	—	2	2	no	—
196	#95	—	2	2	no	—
197	C₂⁴	C₂²×C₄×C₅₃	1	1	yes	characteristic
198	C₂²×C₄	—	2	4	no	—
199	C₂²×C₄	—	2	4	no	—
200	C₂²×C₄	—	2	8	no	—
201	C₂²×C₄	—	4	2	no	—
202	C₂²×C₄	—	4	2	no	—
203	C₂²×C₄	—	4	4	no	—
204	C₂²×C₄	—	4	4	no	—
205	C₂²×C₄	C₄²×C₅₃	1	2	yes	—
206	C₂²×C₄	C₄²×C₅₃	1	2	yes	—
207	C₂²×C₄	C₂²⋊C₄×C₅₃	1	1	yes	characteristic
208	C₂²×C₄	C₂²⋊C₄×C₅₃	1	1	yes	characteristic
209	C₂²×C₄	C₂²⋊C₄×C₅₃	1	1	yes	characteristic
210	C₂²×C₄	C₂²⋊C₄×C₅₃	1	1	yes	characteristic
211	C₂²×C₄	C₄⋊C₄×C₅₃	1	2	yes	—
212	C₂²×C₄	C₄⋊C₄×C₅₃	1	2	yes	—
213	C₂²×C₄	C₂²×C₄×C₅₃	1	1	yes	characteristic
214	C₂²×C₄	C₂²×C₄×C₅₃	1	1	yes	characteristic
215	C₂²×C₄	C₂×D₄×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	C₂²×C₄	C₂×Q₈×C₅₃	1	1	yes	characteristic
219	C₂×C₈	—	2	4	no	—
220	C₂×C₈	—	2	4	no	—
221	C₂×C₈	—	4	4	no	—
222	C₂×C₈	—	4	4	no	—
223	C₄²	C₂²⋊C₄×C₅₃	1	2	yes	—
224	C₄²	C₂²⋊C₄×C₅₃	1	2	yes	—
225	C₄²	C₂²⋊C₄×C₅₃	1	2	yes	—
226	C₄²	C₂²⋊C₄×C₅₃	1	2	yes	—
227	C₄²	C₄⋊C₄×C₅₃	1	2	yes	—
228	C₄²	C₄⋊C₄×C₅₃	1	2	yes	—
229	C₄²	C₄⋊C₄×C₅₃	1	2	yes	—
230	C₄²	C₄⋊C₄×C₅₃	1	2	yes	—
231	C₂³	—	2	2	no	—
232	C₂³	—	2	2	no	—
233	C₂³	C₂×C₄²×C₅₃	1	1	yes	characteristic
234	C₂³	C₂×C₂²⋊C₄×C₅₃	1	1	yes	characteristic
235	C₂³	C₂×C₄⋊C₄×C₅₃	1	1	yes	characteristic
236	C₂³	#131×C₅₃	1	4	yes	—
237	C₂×C₄	—	2	4	no	—
238	C₂×C₄	—	2	2	no	—
239	C₂×C₄	—	2	2	no	—
240	C₂×C₄	—	2	2	no	—
241	C₂×C₄	—	2	2	no	—
242	C₂×C₄	—	2	2	no	—
243	C₂×C₄	—	2	2	no	—
244	C₂×C₄	—	2	4	no	—
245	C₂×C₄	—	4	1	no	—
246	C₂×C₄	—	4	1	no	—
247	C₂×C₄	—	4	1	no	—
248	C₂×C₄	—	4	1	no	—
249	C₂×C₄	—	4	2	no	—
250	C₂×C₄	—	4	2	no	—
251	C₂×C₄	—	4	4	no	—
252	C₂×C₄	—	4	4	no	—
253	C₂×C₄	—	4	4	no	—
254	C₂×C₄	—	8	8	no	—
255	C₂×C₄	#95×C₅₃	1	2	yes	—
256	C₂×C₄	#95×C₅₃	1	2	yes	—
257	C₂×C₄	#95×C₅₃	1	2	yes	—
258	C₂×C₄	#95×C₅₃	1	2	yes	—
259	C₂×C₄	C₂×C₂²⋊C₄×C₅₃	1	1	yes	characteristic
260	C₂×C₄	C₂×C₂²⋊C₄×C₅₃	1	1	yes	characteristic
261	C₂×C₄	C₂×C₄⋊C₄×C₅₃	1	1	yes	characteristic
262	C₂×C₄	C₂×C₄⋊C₄×C₅₃	1	1	yes	characteristic
263	C₈	—	4	4	no	—
264	C₈	—	4	4	no	—
265	C₂²	—	2	8	no	—
266	C₂²	—	2	1	no	—
267	C₂²	—	2	1	no	—
268	C₂²	—	2	1	no	—
269	C₂²	—	2	1	no	—
270	C₂²	C₂×#95×C₅₃	1	1	yes	characteristic
271	C₂²	C₅₃×#405	1	2	yes	—
272	C₂²	C₅₃×#408	1	2	yes	—
273	C₂²	C₅₃×#410	1	2	yes	—
274	C₂²	C₅₃×#413	1	2	yes	—
275	C₂²	C₅₃×#424	1	2	yes	—
276	C₄	—	2	2	no	—
277	C₄	—	2	2	no	—
278	C₄	—	2	1	no	—
279	C₄	—	2	1	no	—
280	C₄	—	2	1	no	—
281	C₄	—	2	1	no	—
282	C₄	—	2	2	no	—
283	C₄	—	2	2	no	—
284	C₄	—	8	4	no	—
285	C₂	—	2	2	no	—
286	C₂	—	2	2	no	—
287	C₂	C₅₃×#1473	1	4	yes	—
288	C₂	C₅₃×#1729	1	1	yes	characteristic
289	C₂	C₅₃×#1740	1	1	yes	characteristic
290	C₂	C₅₃×#1742	1	1	yes	characteristic

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

Complements

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

N	H ≅ G/N	type
7. C₅₃×#432	C₄	semidirect
9. C₅₃×#431	C₄	semidirect
119. #7916	C₅₃	direct (N × H)
152. #432	C₄×C₅₃	semidirect
154. #431	C₄×C₅₃	semidirect
168. C₅₃	#7916	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×#432	2
C₂×C₅₃×#431	2
C₂×C₅₃×#383	2
C₅₃×#1280	2
#7916	53

Subgroups — C53×#7916 #27215428

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅₃×#7916 #27215428