Subgroups — C₅₃×#7898 #27215410

Order 13568.

710 conjugacy classes of subgroups.

The subgroup lattice has 284 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	C₂²×C₄×C₈×C₅₃	C₂	1	1	yes	maximal, characteristic
3	C₅₃×#2283	C₂	1	1	yes	maximal, characteristic
4	C₅₃×#1279	C₂	1	4	yes	maximal, complemented
5	C₂³×C₈×C₅₃	—	2	1	no	—
6	C₂²×C₄²×C₅₃	C₂²	1	1	yes	characteristic
7	C₅₃×#432	—	2	2	no	—
8	C₅₃×#432	C₄	1	4	yes	complemented
9	C₂×C₄⋊C₈×C₅₃	C₂²	1	2	yes	—
10	C₂×C₂²⋊C₈×C₅₃	—	2	1	no	—
11	C₂×#97×C₅₃	—	2	1	no	—
12	C₂×C₄×C₈×C₅₃	—	2	1	no	—
13	C₂×C₄×C₈×C₅₃	—	2	2	no	—
14	C₂×C₄×C₈×C₅₃	C₄	1	4	yes	complemented
15	C₂×C₄×C₈×C₅₃	C₂²	1	2	yes	—
16	C₅₃×#383	—	2	1	no	—
17	C₅₃×#382	C₄	1	2	yes	—
18	C₅₃×#381	—	2	1	no	—
19	C₅₃×#380	C₄	1	2	yes	—
20	C₄×C₄⋊C₄×C₅₃	C₂²	1	2	yes	—
21	C₄×C₂²⋊C₄×C₅₃	—	2	1	no	—
22	C₅₃×#376	—	2	1	no	—
23	C₂³×C₄×C₅₃	C₂×C₄	1	1	yes	characteristic
24	C₂³×C₄×C₅₃	D₄	1	1	yes	characteristic
25	C₂³×C₄×C₅₃	D₄	1	1	yes	characteristic
26	C₂²×C₈×C₅₃	—	2	8	no	—
27	C₂²×C₈×C₅₃	—	2	4	no	—
28	C₂²×C₈×C₅₃	—	2	2	no	—
29	C₂²×C₈×C₅₃	—	2	2	no	—
30	C₂×C₄⋊C₄×C₅₃	C₂×C₄	1	4	yes	—
31	C₂×C₂²⋊C₄×C₅₃	—	2	2	no	—
32	C₂×C₄²×C₅₃	—	2	2	no	—
33	C₂×C₄²×C₅₃	C₂×C₄	1	2	yes	—
34	C₂×C₄²×C₅₃	C₂×C₄	1	2	yes	—
35	C₂×C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
36	C₂×C₄²×C₅₃	D₄	1	2	yes	—
37	C₂×C₄²×C₅₃	D₄	1	1	yes	characteristic
38	C₂×C₄²×C₅₃	D₄	1	1	yes	characteristic
39	C₂×C₄²×C₅₃	Q₈	1	2	yes	—
40	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
41	C₄⋊C₈×C₅₃	—	2	2	no	—
42	C₄⋊C₈×C₅₃	C₂×C₄	1	4	yes	—
43	C₂²⋊C₈×C₅₃	—	2	4	no	—
44	#97×C₅₃	—	2	4	no	—
45	C₄×C₈×C₅₃	—	2	4	no	—
46	C₄×C₈×C₅₃	—	2	2	no	—
47	C₄×C₈×C₅₃	C₂×C₄	1	4	yes	—
48	#95×C₅₃	—	2	2	no	—
49	#95×C₅₃	—	2	2	no	—
50	#95×C₅₃	—	2	2	no	—
51	#95×C₅₃	—	2	2	no	—
52	C₂⁴×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
53	C₂²×C₄×C₅₃	—	2	4	no	—
54	C₂²×C₄×C₅₃	—	2	4	no	—
55	C₂²×C₄×C₅₃	—	2	4	no	—
56	C₂²×C₄×C₅₃	—	2	4	no	—
57	C₂²×C₄×C₅₃	C₄²	1	2	yes	—
58	C₂²×C₄×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
59	C₂²×C₄×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
60	C₂²×C₄×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
61	C₂²×C₄×C₅₃	C₂²⋊C₄	1	2	yes	—
62	C₂²×C₄×C₅₃	C₂²⋊C₄	1	2	yes	—
63	C₂²×C₄×C₅₃	C₄⋊C₄	1	2	yes	—
64	C₂²×C₄×C₅₃	C₄⋊C₄	1	2	yes	—
65	C₂²×C₄×C₅₃	C₄⋊C₄	1	2	yes	—
66	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
67	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
68	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
69	C₂×C₈×C₅₃	—	2	16	no	—
70	C₂×C₈×C₅₃	—	2	8	no	—
71	C₂×C₈×C₅₃	—	2	4	no	—
72	C₂×C₈×C₅₃	—	2	4	no	—
73	C₂×C₈×C₅₃	—	4	4	no	—
74	C₄⋊C₄×C₅₃	—	2	8	no	—
75	C₂²⋊C₄×C₅₃	—	4	4	no	—
76	C₄²×C₅₃	—	4	4	no	—
77	C₄²×C₅₃	C₄²	1	2	yes	—
78	C₄²×C₅₃	C₂²⋊C₄	1	2	yes	—
79	C₄²×C₅₃	C₂²⋊C₄	1	2	yes	—
80	C₄²×C₅₃	C₄⋊C₄	1	2	yes	—
81	C₄²×C₅₃	C₄⋊C₄	1	2	yes	—
82	C₄²×C₅₃	C₄⋊C₄	1	2	yes	—
83	C₄²×C₅₃	C₂²×C₄	1	1	yes	characteristic
84	C₄²×C₅₃	C₂²×C₄	1	1	yes	characteristic
85	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
86	C₄²×C₅₃	C₂×Q₈	1	1	yes	characteristic
87	C₂³×C₅₃	—	2	2	no	—
88	C₂³×C₅₃	—	2	2	no	—
89	C₂³×C₅₃	#95	1	2	yes	—
90	C₂³×C₅₃	#104	1	4	yes	—
91	C₂³×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
92	C₂×C₄×C₅₃	—	2	4	no	—
93	C₂×C₄×C₅₃	—	2	4	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	—	2	4	no	—
100	C₂×C₄×C₅₃	—	2	4	no	—
101	C₂×C₄×C₅₃	—	2	4	no	—
102	C₂×C₄×C₅₃	—	4	4	no	—
103	C₂×C₄×C₅₃	—	4	4	no	—
104	C₂×C₄×C₅₃	—	4	4	no	—
105	C₂×C₄×C₅₃	#95	1	2	yes	—
106	C₂×C₄×C₅₃	#95	1	2	yes	—
107	C₂×C₄×C₅₃	#95	1	2	yes	—
108	C₂×C₄×C₅₃	C₂×C₄²	1	1	yes	characteristic
109	C₂×C₄×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
110	C₂×C₄×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
111	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
112	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
113	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
114	C₂×C₄×C₅₃	#131	1	4	yes	—
115	C₂×C₄×C₅₃	#135	1	4	yes	—
116	C₈×C₅₃	—	2	8	no	—
117	C₈×C₅₃	—	4	4	no	—
118	#7898	C₅₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
119	C₂²×C₅₃	—	2	2	no	—
120	C₂²×C₅₃	—	2	2	no	—
121	C₂²×C₅₃	—	2	8	no	—
122	C₂²×C₅₃	#339	1	4	yes	—
123	C₂²×C₅₃	C₂×#95	1	1	yes	characteristic
124	C₂²×C₅₃	#408	1	2	yes	—
125	C₂²×C₅₃	#416	1	2	yes	—
126	C₂²×C₅₃	C₂×#104	1	2	yes	—
127	C₄×C₅₃	—	2	2	no	—
128	C₄×C₅₃	—	2	1	no	—
129	C₄×C₅₃	—	2	2	no	—
130	C₄×C₅₃	—	2	1	no	—
131	C₄×C₅₃	—	2	2	no	—
132	C₄×C₅₃	—	8	4	no	—
133	C₄×C₅₃	#405	1	2	yes	—
134	C₄×C₅₃	#416	1	2	yes	—
135	C₄×C₅₃	#424	1	2	yes	—
136	C₄×C₅₃	#427	1	2	yes	—
137	C₂×#432	C₂×C₅₃	1	1	yes	characteristic
138	C₂²×C₄×C₈	C₂×C₅₃	1	1	yes	characteristic
139	#2283	C₂×C₅₃	1	1	yes	characteristic
140	#1279	C₂×C₅₃	1	4	yes	complemented
141	C₂×C₅₃	—	2	4	no	—
142	C₂×C₅₃	#1494	1	4	yes	—
143	C₂×C₅₃	#1729	1	1	yes	characteristic
144	C₂×C₅₃	C₂×#339	1	1	yes	characteristic
145	C₂×C₅₃	#1735	1	1	yes	characteristic
146	C₂³×C₈	—	2	1	no	—
147	C₂²×C₄²	C₂²×C₅₃	1	1	yes	characteristic
148	#432	—	2	2	no	—
149	#432	C₄×C₅₃	1	4	yes	complemented
150	C₂×C₄⋊C₈	C₂²×C₅₃	1	2	yes	—
151	C₂×C₂²⋊C₈	—	2	1	no	—
152	C₂×#97	—	2	1	no	—
153	C₂×C₄×C₈	—	2	2	no	—
154	C₂×C₄×C₈	—	2	1	no	—
155	C₂×C₄×C₈	C₄×C₅₃	1	4	yes	complemented
156	C₂×C₄×C₈	C₂²×C₅₃	1	2	yes	—
157	#383	—	2	1	no	—
158	#382	C₄×C₅₃	1	2	yes	—
159	#381	—	2	1	no	—
160	#380	C₄×C₅₃	1	2	yes	—
161	C₄×C₄⋊C₄	C₂²×C₅₃	1	2	yes	—
162	C₄×C₂²⋊C₄	—	2	1	no	—
163	#376	—	2	1	no	—
164	C₅₃	#7898	1	1	yes	characteristic, direct product factor, Sylow
165	C₂³×C₄	C₂×C₄×C₅₃	1	1	yes	characteristic
166	C₂³×C₄	D₄×C₅₃	1	1	yes	characteristic
167	C₂³×C₄	D₄×C₅₃	1	1	yes	characteristic
168	C₂²×C₈	—	2	8	no	—
169	C₂²×C₈	—	2	2	no	—
170	C₂²×C₈	—	2	4	no	—
171	C₂²×C₈	—	2	2	no	—
172	C₂×C₄⋊C₄	C₂×C₄×C₅₃	1	4	yes	—
173	C₂×C₂²⋊C₄	—	2	2	no	—
174	C₂×C₄²	—	2	2	no	—
175	C₂×C₄²	C₂×C₄×C₅₃	1	2	yes	—
176	C₂×C₄²	C₂×C₄×C₅₃	1	2	yes	—
177	C₂×C₄²	C₂×C₄×C₅₃	1	1	yes	characteristic
178	C₂×C₄²	D₄×C₅₃	1	2	yes	—
179	C₂×C₄²	D₄×C₅₃	1	1	yes	characteristic
180	C₂×C₄²	D₄×C₅₃	1	1	yes	characteristic
181	C₂×C₄²	Q₈×C₅₃	1	2	yes	—
182	C₂×C₄²	C₂³×C₅₃	1	1	yes	characteristic
183	C₄⋊C₈	—	2	2	no	—
184	C₄⋊C₈	C₂×C₄×C₅₃	1	4	yes	—
185	C₂²⋊C₈	—	2	4	no	—
186	#97	—	2	4	no	—
187	C₄×C₈	—	2	2	no	—
188	C₄×C₈	—	2	4	no	—
189	C₄×C₈	C₂×C₄×C₅₃	1	4	yes	—
190	#95	—	2	2	no	—
191	#95	—	2	2	no	—
192	#95	—	2	2	no	—
193	#95	—	2	2	no	—
194	C₂⁴	C₂²⋊C₄×C₅₃	1	1	yes	characteristic
195	C₂²×C₄	—	2	4	no	—
196	C₂²×C₄	—	2	4	no	—
197	C₂²×C₄	—	2	4	no	—
198	C₂²×C₄	—	2	4	no	—
199	C₂²×C₄	C₄²×C₅₃	1	2	yes	—
200	C₂²×C₄	C₂²⋊C₄×C₅₃	1	1	yes	characteristic
201	C₂²×C₄	C₂²⋊C₄×C₅₃	1	1	yes	characteristic
202	C₂²×C₄	C₂²⋊C₄×C₅₃	1	1	yes	characteristic
203	C₂²×C₄	C₂²⋊C₄×C₅₃	1	2	yes	—
204	C₂²×C₄	C₂²⋊C₄×C₅₃	1	2	yes	—
205	C₂²×C₄	C₄⋊C₄×C₅₃	1	2	yes	—
206	C₂²×C₄	C₄⋊C₄×C₅₃	1	2	yes	—
207	C₂²×C₄	C₄⋊C₄×C₅₃	1	2	yes	—
208	C₂²×C₄	C₂²×C₄×C₅₃	1	1	yes	characteristic
209	C₂²×C₄	C₂×D₄×C₅₃	1	1	yes	characteristic
210	C₂²×C₄	C₂×D₄×C₅₃	1	1	yes	characteristic
211	C₂×C₈	—	2	16	no	—
212	C₂×C₈	—	2	8	no	—
213	C₂×C₈	—	2	4	no	—
214	C₂×C₈	—	2	4	no	—
215	C₂×C₈	—	4	4	no	—
216	C₄⋊C₄	—	2	8	no	—
217	C₂²⋊C₄	—	4	4	no	—
218	C₄²	—	4	4	no	—
219	C₄²	C₄²×C₅₃	1	2	yes	—
220	C₄²	C₂²⋊C₄×C₅₃	1	2	yes	—
221	C₄²	C₂²⋊C₄×C₅₃	1	2	yes	—
222	C₄²	C₄⋊C₄×C₅₃	1	2	yes	—
223	C₄²	C₄⋊C₄×C₅₃	1	2	yes	—
224	C₄²	C₄⋊C₄×C₅₃	1	2	yes	—
225	C₄²	C₂²×C₄×C₅₃	1	1	yes	characteristic
226	C₄²	C₂²×C₄×C₅₃	1	1	yes	characteristic
227	C₄²	C₂×D₄×C₅₃	1	1	yes	characteristic
228	C₄²	C₂×Q₈×C₅₃	1	1	yes	characteristic
229	C₂³	—	2	2	no	—
230	C₂³	—	2	2	no	—
231	C₂³	#95×C₅₃	1	2	yes	—
232	C₂³	#104×C₅₃	1	4	yes	—
233	C₂³	C₂×C₂²⋊C₄×C₅₃	1	1	yes	characteristic
234	C₂×C₄	—	2	4	no	—
235	C₂×C₄	—	2	4	no	—
236	C₂×C₄	—	2	2	no	—
237	C₂×C₄	—	2	2	no	—
238	C₂×C₄	—	2	2	no	—
239	C₂×C₄	—	2	2	no	—
240	C₂×C₄	—	2	4	no	—
241	C₂×C₄	—	2	4	no	—
242	C₂×C₄	—	2	4	no	—
243	C₂×C₄	—	2	4	no	—
244	C₂×C₄	—	4	4	no	—
245	C₂×C₄	—	4	4	no	—
246	C₂×C₄	—	4	4	no	—
247	C₂×C₄	#95×C₅₃	1	2	yes	—
248	C₂×C₄	#95×C₅₃	1	2	yes	—
249	C₂×C₄	#95×C₅₃	1	2	yes	—
250	C₂×C₄	C₂×C₄²×C₅₃	1	1	yes	characteristic
251	C₂×C₄	C₂×C₂²⋊C₄×C₅₃	1	1	yes	characteristic
252	C₂×C₄	C₂×C₂²⋊C₄×C₅₃	1	1	yes	characteristic
253	C₂×C₄	C₂×C₄⋊C₄×C₅₃	1	1	yes	characteristic
254	C₂×C₄	C₂×C₄⋊C₄×C₅₃	1	1	yes	characteristic
255	C₂×C₄	C₂×C₄⋊C₄×C₅₃	1	1	yes	characteristic
256	C₂×C₄	#131×C₅₃	1	4	yes	—
257	C₂×C₄	#135×C₅₃	1	4	yes	—
258	C₈	—	2	8	no	—
259	C₈	—	4	4	no	—
260	C₂²	—	2	2	no	—
261	C₂²	—	2	2	no	—
262	C₂²	—	2	8	no	—
263	C₂²	C₅₃×#339	1	4	yes	—
264	C₂²	C₂×#95×C₅₃	1	1	yes	characteristic
265	C₂²	C₅₃×#408	1	2	yes	—
266	C₂²	C₅₃×#416	1	2	yes	—
267	C₂²	C₂×#104×C₅₃	1	2	yes	—
268	C₄	—	2	2	no	—
269	C₄	—	2	1	no	—
270	C₄	—	2	2	no	—
271	C₄	—	2	1	no	—
272	C₄	—	2	2	no	—
273	C₄	—	8	4	no	—
274	C₄	C₅₃×#405	1	2	yes	—
275	C₄	C₅₃×#416	1	2	yes	—
276	C₄	C₅₃×#424	1	2	yes	—
277	C₄	C₅₃×#427	1	2	yes	—
278	C₂	—	2	4	no	—
279	C₂	C₅₃×#1494	1	4	yes	—
280	C₂	C₅₃×#1729	1	1	yes	characteristic
281	C₂	C₂×C₅₃×#339	1	1	yes	characteristic
282	C₂	C₅₃×#1735	1	1	yes	characteristic

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

Complements

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

N	H ≅ G/N	type
4. C₅₃×#1279	C₂	semidirect
8. C₅₃×#432	C₄	semidirect
14. C₂×C₄×C₈×C₅₃	C₄	semidirect
118. #7898	C₅₃	direct (N × H)
140. #1279	C₂×C₅₃	semidirect
149. #432	C₄×C₅₃	semidirect
155. C₂×C₄×C₈	C₄×C₅₃	semidirect
164. C₅₃	#7898	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
C₂²×C₄×C₈×C₅₃	2
C₅₃×#2283	2
C₅₃×#1279	2
#7898	53

Subgroups — C53×#7898 #27215410

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅₃×#7898 #27215410