Subgroups — C₅₃×#22256 #27229768

Order 13568.

614 conjugacy classes of subgroups.

The subgroup lattice has 250 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₅₃×#3160	C₂	1	1	yes	maximal, characteristic
2	C₅₃×#2960	C₂	1	2	yes	maximal
3	C₅₃×#1994	C₂	1	2	yes	maximal
4	C₅₃×#1922	C₂	1	8	yes	maximal
5	C₅₃×#1837	C₂	1	1	yes	maximal, characteristic
6	C₅₃×#1744	C₂	1	1	yes	maximal, characteristic
7	C₅₃×#533	C₂²	1	2	yes	—
8	C₅₃×#500	C₄	1	1	yes	characteristic, complemented
9	C₅₃×#500	C₄	1	1	yes	characteristic, complemented
10	C₅₃×#499	C₄	1	4	yes	complemented
11	C₅₃×#498	C₄	1	2	yes	complemented
12	C₅₃×#446	—	2	2	no	—
13	C₅₃×#446	—	2	2	no	—
14	C₈×Q₈×C₅₃	—	2	2	no	—
15	C₈×Q₈×C₅₃	—	2	2	no	—
16	C₅₃×#427	C₂²	1	4	yes	—
17	C₅₃×#424	C₂²	1	4	yes	—
18	C₅₃×#423	C₂²	1	2	yes	—
19	C₅₃×#405	C₂²	1	4	yes	—
20	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
21	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
22	C₄³×C₅₃	C₂²	1	1	yes	characteristic
23	C₅₃×#341	C₂²	1	4	yes	—
24	C₅₃×#339	C₂²	1	8	yes	—
25	C₅₃×#336	C₂²	1	4	yes	—
26	C₂×M₁₆×C₅₃	C₂³	1	4	yes	—
27	C₂²×C₈×C₅₃	C₂³	1	2	yes	—
28	C₂²×C₈×C₅₃	C₂³	1	2	yes	—
29	C₄⋊Q₈×C₅₃	C₂×C₄	1	2	yes	—
30	C₄⋊Q₈×C₅₃	C₂×C₄	1	2	yes	—
31	#125×C₅₃	C₂×C₄	1	4	yes	—
32	C₂²⋊Q₈×C₅₃	—	2	2	no	—
33	C₂²⋊Q₈×C₅₃	—	2	2	no	—
34	C₄×Q₈×C₅₃	—	2	2	no	—
35	C₄×Q₈×C₅₃	—	2	2	no	—
36	#117×C₅₃	C₂³	1	4	yes	—
37	C₂×C₄²×C₅₃	—	2	1	no	—
38	C₂×C₄²×C₅₃	—	2	1	no	—
39	C₂×C₄²×C₅₃	C₂³	1	2	yes	—
40	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
41	#107×C₅₃	C₂×C₄	1	4	yes	—
42	#107×C₅₃	C₂×C₄	1	4	yes	—
43	#106×C₅₃	C₂×C₄	1	8	yes	—
44	C₄⋊C₈×C₅₃	—	2	4	no	—
45	C₄⋊C₈×C₅₃	—	2	4	no	—
46	C₄⋊C₈×C₅₃	—	2	2	no	—
47	C₄⋊C₈×C₅₃	—	2	2	no	—
48	#97×C₅₃	—	2	4	no	—
49	C₄×C₈×C₅₃	—	2	4	no	—
50	C₄×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
51	C₄×C₈×C₅₃	C₂×C₄	1	2	yes	—
52	C₄×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
53	C₄×C₈×C₅₃	D₄	1	1	yes	characteristic
54	C₄×C₈×C₅₃	D₄	1	2	yes	—
55	C₄×C₈×C₅₃	D₄	1	1	yes	characteristic
56	C₂×Q₈×C₅₃	—	2	2	no	—
57	C₂×Q₈×C₅₃	—	2	2	no	—
58	C₂²×C₄×C₅₃	—	2	2	no	—
59	C₂²×C₄×C₅₃	#41	1	1	yes	characteristic
60	C₂²×C₄×C₅₃	#41	1	1	yes	characteristic
61	C₂²×C₄×C₅₃	C₂⁴	1	1	yes	characteristic
62	M₁₆×C₅₃	—	2	8	no	—
63	C₂×C₈×C₅₃	—	2	4	no	—
64	C₂×C₈×C₅₃	—	2	4	no	—
65	C₂×C₈×C₅₃	—	2	4	no	—
66	C₂×C₈×C₅₃	C₂²×C₄	1	2	yes	—
67	C₂×C₈×C₅₃	C₂²×C₄	1	2	yes	—
68	C₂×C₈×C₅₃	C₂×D₄	1	2	yes	—
69	C₂×C₈×C₅₃	C₂×D₄	1	2	yes	—
70	C₄⋊C₄×C₅₃	—	2	4	no	—
71	C₄⋊C₄×C₅₃	—	2	4	no	—
72	C₄⋊C₄×C₅₃	—	2	2	no	—
73	C₄⋊C₄×C₅₃	—	2	2	no	—
74	C₄⋊C₄×C₅₃	C₂²×C₄	1	4	yes	—
75	C₄⋊C₄×C₅₃	C₂²×C₄	1	4	yes	—
76	C₂²⋊C₄×C₅₃	—	2	4	no	—
77	C₄²×C₅₃	—	2	4	no	—
78	C₄²×C₅₃	—	2	4	no	—
79	C₄²×C₅₃	—	2	4	no	—
80	C₄²×C₅₃	—	4	2	no	—
81	C₄²×C₅₃	C₂²×C₄	1	1	yes	characteristic
82	C₄²×C₅₃	C₂²×C₄	1	1	yes	characteristic
83	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
84	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
85	C₂³×C₅₃	C₂×#41	1	1	yes	characteristic
86	Q₈×C₅₃	—	4	4	no	—
87	Q₈×C₅₃	—	4	4	no	—
88	C₂×C₄×C₅₃	—	2	2	no	—
89	C₂×C₄×C₅₃	—	2	2	no	—
90	C₂×C₄×C₅₃	—	2	4	no	—
91	C₂×C₄×C₅₃	—	2	4	no	—
92	C₂×C₄×C₅₃	—	2	4	no	—
93	C₂×C₄×C₅₃	—	4	2	no	—
94	C₂×C₄×C₅₃	—	4	2	no	—
95	C₂×C₄×C₅₃	C₄×D₄	1	1	yes	characteristic
96	C₂×C₄×C₅₃	C₄×D₄	1	1	yes	characteristic
97	C₂×C₄×C₅₃	C₄×D₄	1	1	yes	characteristic
98	C₂×C₄×C₅₃	C₄×D₄	1	1	yes	characteristic
99	C₂×C₄×C₅₃	C₂³×C₄	1	1	yes	characteristic
100	C₂×C₄×C₅₃	C₂²×D₄	1	1	yes	characteristic
101	C₂×C₄×C₅₃	2₋¹⁺⁴	1	4	yes	—
102	C₈×C₅₃	—	2	2	no	—
103	C₈×C₅₃	—	2	2	no	—
104	C₈×C₅₃	—	2	2	no	—
105	C₈×C₅₃	—	2	2	no	—
106	C₈×C₅₃	—	4	4	no	—
107	#22256	C₅₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
108	C₂²×C₅₃	—	2	1	no	—
109	C₂²×C₅₃	—	2	1	no	—
110	C₂²×C₅₃	C₂×C₄×D₄	1	1	yes	characteristic
111	C₂²×C₅₃	#541	1	2	yes	—
112	C₄×C₅₃	—	2	1	no	—
113	C₄×C₅₃	—	2	1	no	—
114	C₄×C₅₃	—	4	4	no	—
115	C₄×C₅₃	—	4	4	no	—
116	C₄×C₅₃	—	4	4	no	—
117	C₄×C₅₃	#443	1	2	yes	—
118	C₄×C₅₃	#444	1	2	yes	—
119	C₄×C₅₃	#519	1	2	yes	—
120	C₄×C₅₃	#536	1	2	yes	—
121	#3160	C₂×C₅₃	1	1	yes	characteristic
122	#2960	C₂×C₅₃	1	2	yes	—
123	#1994	C₂×C₅₃	1	2	yes	—
124	#1922	C₂×C₅₃	1	8	yes	—
125	#1837	C₂×C₅₃	1	1	yes	characteristic
126	#1744	C₂×C₅₃	1	1	yes	characteristic
127	C₂×C₅₃	—	2	2	no	—
128	C₂×C₅₃	#2327	1	1	yes	characteristic
129	C₂×C₅₃	C₂×#443	1	1	yes	characteristic
130	C₂×C₅₃	C₂×#444	1	1	yes	characteristic
131	#533	C₂²×C₅₃	1	2	yes	—
132	#500	C₄×C₅₃	1	1	yes	characteristic, complemented
133	#500	C₄×C₅₃	1	1	yes	characteristic, complemented
134	#499	C₄×C₅₃	1	4	yes	complemented
135	#498	C₄×C₅₃	1	2	yes	complemented
136	#446	—	2	2	no	—
137	#446	—	2	2	no	—
138	C₈×Q₈	—	2	2	no	—
139	C₈×Q₈	—	2	2	no	—
140	#427	C₂²×C₅₃	1	4	yes	—
141	#424	C₂²×C₅₃	1	4	yes	—
142	#423	C₂²×C₅₃	1	2	yes	—
143	#405	C₂²×C₅₃	1	4	yes	—
144	C₂×C₄×C₈	C₂²×C₅₃	1	1	yes	characteristic
145	C₂×C₄×C₈	C₂²×C₅₃	1	1	yes	characteristic
146	C₄³	C₂²×C₅₃	1	1	yes	characteristic
147	#341	C₂²×C₅₃	1	4	yes	—
148	#339	C₂²×C₅₃	1	8	yes	—
149	#336	C₂²×C₅₃	1	4	yes	—
150	C₅₃	#22256	1	1	yes	characteristic, direct product factor, Sylow
151	C₂×M₁₆	C₂³×C₅₃	1	4	yes	—
152	C₂²×C₈	C₂³×C₅₃	1	2	yes	—
153	C₂²×C₈	C₂³×C₅₃	1	2	yes	—
154	C₄⋊Q₈	C₂×C₄×C₅₃	1	2	yes	—
155	C₄⋊Q₈	C₂×C₄×C₅₃	1	2	yes	—
156	#125	C₂×C₄×C₅₃	1	4	yes	—
157	C₂²⋊Q₈	—	2	2	no	—
158	C₂²⋊Q₈	—	2	2	no	—
159	C₄×Q₈	—	2	2	no	—
160	C₄×Q₈	—	2	2	no	—
161	#117	C₂³×C₅₃	1	4	yes	—
162	C₂×C₄²	—	2	1	no	—
163	C₂×C₄²	—	2	1	no	—
164	C₂×C₄²	C₂³×C₅₃	1	2	yes	—
165	C₂×C₄²	C₂³×C₅₃	1	1	yes	characteristic
166	#107	C₂×C₄×C₅₃	1	4	yes	—
167	#107	C₂×C₄×C₅₃	1	4	yes	—
168	#106	C₂×C₄×C₅₃	1	8	yes	—
169	C₄⋊C₈	—	2	4	no	—
170	C₄⋊C₈	—	2	4	no	—
171	C₄⋊C₈	—	2	2	no	—
172	C₄⋊C₈	—	2	2	no	—
173	#97	—	2	4	no	—
174	C₄×C₈	—	2	4	no	—
175	C₄×C₈	C₂×C₄×C₅₃	1	1	yes	characteristic
176	C₄×C₈	C₂×C₄×C₅₃	1	2	yes	—
177	C₄×C₈	C₂×C₄×C₅₃	1	1	yes	characteristic
178	C₄×C₈	D₄×C₅₃	1	1	yes	characteristic
179	C₄×C₈	D₄×C₅₃	1	2	yes	—
180	C₄×C₈	D₄×C₅₃	1	1	yes	characteristic
181	C₂×Q₈	—	2	2	no	—
182	C₂×Q₈	—	2	2	no	—
183	C₂²×C₄	—	2	2	no	—
184	C₂²×C₄	#41×C₅₃	1	1	yes	characteristic
185	C₂²×C₄	#41×C₅₃	1	1	yes	characteristic
186	C₂²×C₄	C₂⁴×C₅₃	1	1	yes	characteristic
187	M₁₆	—	2	8	no	—
188	C₂×C₈	—	2	4	no	—
189	C₂×C₈	—	2	4	no	—
190	C₂×C₈	—	2	4	no	—
191	C₂×C₈	C₂²×C₄×C₅₃	1	2	yes	—
192	C₂×C₈	C₂²×C₄×C₅₃	1	2	yes	—
193	C₂×C₈	C₂×D₄×C₅₃	1	2	yes	—
194	C₂×C₈	C₂×D₄×C₅₃	1	2	yes	—
195	C₄⋊C₄	—	2	4	no	—
196	C₄⋊C₄	—	2	4	no	—
197	C₄⋊C₄	—	2	2	no	—
198	C₄⋊C₄	—	2	2	no	—
199	C₄⋊C₄	C₂²×C₄×C₅₃	1	4	yes	—
200	C₄⋊C₄	C₂²×C₄×C₅₃	1	4	yes	—
201	C₂²⋊C₄	—	2	4	no	—
202	C₄²	—	2	4	no	—
203	C₄²	—	2	4	no	—
204	C₄²	—	2	4	no	—
205	C₄²	—	4	2	no	—
206	C₄²	C₂²×C₄×C₅₃	1	1	yes	characteristic
207	C₄²	C₂²×C₄×C₅₃	1	1	yes	characteristic
208	C₄²	C₂×D₄×C₅₃	1	1	yes	characteristic
209	C₄²	C₂×D₄×C₅₃	1	1	yes	characteristic
210	C₂³	C₂×#41×C₅₃	1	1	yes	characteristic
211	Q₈	—	4	4	no	—
212	Q₈	—	4	4	no	—
213	C₂×C₄	—	2	2	no	—
214	C₂×C₄	—	2	2	no	—
215	C₂×C₄	—	2	4	no	—
216	C₂×C₄	—	2	4	no	—
217	C₂×C₄	—	2	4	no	—
218	C₂×C₄	—	4	2	no	—
219	C₂×C₄	—	4	2	no	—
220	C₂×C₄	C₄×D₄×C₅₃	1	1	yes	characteristic
221	C₂×C₄	C₄×D₄×C₅₃	1	1	yes	characteristic
222	C₂×C₄	C₄×D₄×C₅₃	1	1	yes	characteristic
223	C₂×C₄	C₄×D₄×C₅₃	1	1	yes	characteristic
224	C₂×C₄	C₂³×C₄×C₅₃	1	1	yes	characteristic
225	C₂×C₄	C₂²×D₄×C₅₃	1	1	yes	characteristic
226	C₂×C₄	2₋¹⁺⁴×C₅₃	1	4	yes	—
227	C₈	—	2	2	no	—
228	C₈	—	2	2	no	—
229	C₈	—	2	2	no	—
230	C₈	—	2	2	no	—
231	C₈	—	4	4	no	—
232	C₂²	—	2	1	no	—
233	C₂²	—	2	1	no	—
234	C₂²	C₂×C₄×D₄×C₅₃	1	1	yes	characteristic
235	C₂²	C₅₃×#541	1	2	yes	—
236	C₄	—	2	1	no	—
237	C₄	—	2	1	no	—
238	C₄	—	4	4	no	—
239	C₄	—	4	4	no	—
240	C₄	—	4	4	no	—
241	C₄	C₅₃×#443	1	2	yes	—
242	C₄	C₅₃×#444	1	2	yes	—
243	C₄	C₅₃×#519	1	2	yes	—
244	C₄	C₅₃×#536	1	2	yes	—
245	C₂	—	2	2	no	—
246	C₂	C₅₃×#2327	1	1	yes	characteristic
247	C₂	C₂×C₅₃×#443	1	1	yes	characteristic
248	C₂	C₂×C₅₃×#444	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄×C₅₃	C₂²×D₄
commutator	C₂×C₄	C₂³×C₄×C₅₃
Frattini, cosocle	C₂²×C₄	C₂⁴×C₅₃
socle	C₂²×C₅₃	C₂×C₄×D₄
prime radical	C₂³×C₅₃	C₂×#41
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ 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
8. C₅₃×#500	C₄	semidirect
9. C₅₃×#500	C₄	semidirect
10. C₅₃×#499	C₄	semidirect
11. C₅₃×#498	C₄	semidirect
107. #22256	C₅₃	direct (N × H)
132. #500	C₄×C₅₃	semidirect
133. #500	C₄×C₅₃	semidirect
134. #499	C₄×C₅₃	semidirect
135. #498	C₄×C₅₃	semidirect
150. C₅₃	#22256	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×#3160	2
C₅₃×#2960	2
C₅₃×#1994	2
C₅₃×#1922	2
C₅₃×#1837	2
C₅₃×#1744	2
#22256	53

Subgroups — C53×#22256 #27229768

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅₃×#22256 #27229768