Subgroups — C₅₃×#8241 #27215753

Order 13568.

452 conjugacy classes of subgroups.

The subgroup lattice has 214 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₅₃×#2931	C₂	1	1	yes	maximal, characteristic
2	C₅₃×#1543	C₂	1	1	yes	maximal, characteristic
3	C₅₃×#1525	C₂	1	1	yes	maximal, characteristic
4	C₈×M₁₆×C₅₃	C₂	1	2	yes	maximal, complemented
5	C₅₃×#1299	C₂	1	2	yes	maximal, complemented
6	C₂×#131×C₅₃	C₄	1	2	yes	—
7	C₅₃×#532	C₂²	1	1	yes	characteristic
8	C₅₃×#436	—	2	2	no	—
9	C₈×D₄×C₅₃	—	2	2	no	—
10	C₅₃×#431	C₂²	1	2	yes	—
11	C₅₃×#423	C₂²	1	1	yes	characteristic
12	C₅₃×#408	C₄	1	2	yes	—
13	C₄×M₁₆×C₅₃	C₂²	1	2	yes	—
14	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
15	C₅₃×#333	C₄	1	2	yes	—
16	C₅₃×#331	C₄	1	2	yes	—
17	C₅₃×#329	C₄	1	4	yes	—
18	C₅₃×#322	—	2	2	no	—
19	C₈²×C₅₃	—	2	2	no	—
20	C₂×#41×C₅₃	C₂×C₄	1	2	yes	—
21	#131×C₅₃	—	4	4	no	—
22	C₂×M₁₆×C₅₃	—	2	2	no	—
23	C₂×M₁₆×C₅₃	—	2	2	no	—
24	C₂×M₁₆×C₅₃	C₂×C₄	1	2	yes	—
25	C₂²×C₈×C₅₃	—	2	2	no	—
26	C₂²×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
27	C₂²×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
28	C₄⋊Q₈×C₅₃	C₂×C₄	1	1	yes	characteristic
29	#127×C₅₃	C₂×C₄	1	1	yes	characteristic
30	#124×C₅₃	C₂×C₄	1	2	yes	—
31	#121×C₅₃	—	2	2	no	—
32	C₄×D₄×C₅₃	—	2	2	no	—
33	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
34	C₄⋊C₈×C₅₃	—	2	2	no	—
35	C₄⋊C₈×C₅₃	C₂×C₄	1	4	yes	—
36	C₂²⋊C₈×C₅₃	—	2	4	no	—
37	C₂²⋊C₈×C₅₃	—	2	2	no	—
38	#97×C₅₃	C₂×C₄	1	4	yes	—
39	C₄×C₈×C₅₃	—	2	2	no	—
40	C₄×C₈×C₅₃	—	2	2	no	—
41	C₄×C₈×C₅₃	D₄	1	2	yes	—
42	C₄×C₈×C₅₃	D₄	1	2	yes	—
43	#41×C₅₃	—	4	2	no	—
44	#41×C₅₃	—	4	2	no	—
45	C₂×Q₈×C₅₃	C₄²	1	2	yes	—
46	C₂×D₄×C₅₃	—	2	2	no	—
47	C₂×D₄×C₅₃	—	2	2	no	—
48	C₂×D₄×C₅₃	C₄²	1	2	yes	—
49	C₂²×C₄×C₅₃	—	2	2	no	—
50	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
51	C₂²×C₄×C₅₃	#41	1	2	yes	—
52	M₁₆×C₅₃	—	2	8	no	—
53	M₁₆×C₅₃	—	4	4	no	—
54	M₁₆×C₅₃	—	4	2	no	—
55	C₂×C₈×C₅₃	—	2	4	no	—
56	C₂×C₈×C₅₃	—	2	2	no	—
57	C₂×C₈×C₅₃	—	2	2	no	—
58	C₂×C₈×C₅₃	—	2	4	no	—
59	C₂×C₈×C₅₃	—	4	1	no	—
60	C₂×C₈×C₅₃	—	4	4	no	—
61	C₂×C₈×C₅₃	C₂²⋊C₄	1	2	yes	—
62	C₂×C₈×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
63	C₂×C₈×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
64	C₄⋊C₄×C₅₃	—	2	2	no	—
65	C₂²⋊C₄×C₅₃	—	2	4	no	—
66	C₄²×C₅₃	C₂²×C₄	1	1	yes	characteristic
67	C₄²×C₅₃	C₂²×C₄	1	1	yes	characteristic
68	C₄²×C₅₃	C₂×D₄	1	2	yes	—
69	C₂³×C₅₃	—	2	2	no	—
70	C₂³×C₅₃	#117	1	1	yes	characteristic
71	Q₈×C₅₃	—	4	2	no	—
72	D₄×C₅₃	—	4	2	no	—
73	D₄×C₅₃	—	4	4	no	—
74	D₄×C₅₃	—	4	4	no	—
75	C₂×C₄×C₅₃	—	2	1	no	—
76	C₂×C₄×C₅₃	—	2	1	no	—
77	C₂×C₄×C₅₃	—	2	4	no	—
78	C₂×C₄×C₅₃	—	2	2	no	—
79	C₂×C₄×C₅₃	—	4	2	no	—
80	C₂×C₄×C₅₃	—	4	2	no	—
81	C₂×C₄×C₅₃	C₂×C₄²	1	1	yes	characteristic
82	C₂×C₄×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
83	C₂×C₄×C₅₃	C₄×D₄	1	2	yes	—
84	C₂×C₄×C₅₃	C₄×D₄	1	2	yes	—
85	C₈×C₅₃	—	2	8	no	—
86	C₈×C₅₃	—	2	1	no	—
87	C₈×C₅₃	—	2	2	no	—
88	C₈×C₅₃	—	4	2	no	—
89	C₈×C₅₃	—	4	4	no	—
90	C₈×C₅₃	#100	1	2	yes	—
91	#8241	C₅₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
92	C₂²×C₅₃	—	2	1	no	—
93	C₂²×C₅₃	—	2	2	no	—
94	C₂²×C₅₃	—	4	2	no	—
95	C₂²×C₅₃	—	4	4	no	—
96	C₂²×C₅₃	C₄×C₂²⋊C₄	1	1	yes	characteristic
97	C₄×C₅₃	—	2	1	no	—
98	C₄×C₅₃	—	2	2	no	—
99	C₄×C₅₃	—	4	2	no	—
100	C₄×C₅₃	C₂×#100	1	1	yes	characteristic
101	C₄×C₅₃	#413	1	1	yes	characteristic
102	C₄×C₅₃	#443	1	4	yes	—
103	#2931	C₂×C₅₃	1	1	yes	characteristic
104	#1543	C₂×C₅₃	1	1	yes	characteristic
105	#1525	C₂×C₅₃	1	1	yes	characteristic
106	C₈×M₁₆	C₂×C₅₃	1	2	yes	complemented
107	#1299	C₂×C₅₃	1	2	yes	complemented
108	C₂×C₅₃	—	4	1	no	—
109	C₂×C₅₃	—	8	2	no	—
110	C₂×C₅₃	C₄×#100	1	1	yes	characteristic
111	C₂×C₅₃	#1764	1	2	yes	—
112	C₂×#131	C₄×C₅₃	1	2	yes	—
113	#532	C₂²×C₅₃	1	1	yes	characteristic
114	#436	—	2	2	no	—
115	C₈×D₄	—	2	2	no	—
116	#431	C₂²×C₅₃	1	2	yes	—
117	#423	C₂²×C₅₃	1	1	yes	characteristic
118	#408	C₄×C₅₃	1	2	yes	—
119	C₄×M₁₆	C₂²×C₅₃	1	2	yes	—
120	C₂×C₄×C₈	C₂²×C₅₃	1	1	yes	characteristic
121	#333	C₄×C₅₃	1	2	yes	—
122	#331	C₄×C₅₃	1	2	yes	—
123	#329	C₄×C₅₃	1	4	yes	—
124	#322	—	2	2	no	—
125	C₈²	—	2	2	no	—
126	C₅₃	#8241	1	1	yes	characteristic, direct product factor, Sylow
127	C₂×#41	C₂×C₄×C₅₃	1	2	yes	—
128	#131	—	4	4	no	—
129	C₂×M₁₆	—	2	2	no	—
130	C₂×M₁₆	—	2	2	no	—
131	C₂×M₁₆	C₂×C₄×C₅₃	1	2	yes	—
132	C₂²×C₈	—	2	2	no	—
133	C₂²×C₈	C₂×C₄×C₅₃	1	1	yes	characteristic
134	C₂²×C₈	C₂×C₄×C₅₃	1	1	yes	characteristic
135	C₄⋊Q₈	C₂×C₄×C₅₃	1	1	yes	characteristic
136	#127	C₂×C₄×C₅₃	1	1	yes	characteristic
137	#124	C₂×C₄×C₅₃	1	2	yes	—
138	#121	—	2	2	no	—
139	C₄×D₄	—	2	2	no	—
140	C₂×C₄²	C₂³×C₅₃	1	1	yes	characteristic
141	C₄⋊C₈	—	2	2	no	—
142	C₄⋊C₈	C₂×C₄×C₅₃	1	4	yes	—
143	C₂²⋊C₈	—	2	2	no	—
144	C₂²⋊C₈	—	2	4	no	—
145	#97	C₂×C₄×C₅₃	1	4	yes	—
146	C₄×C₈	—	2	2	no	—
147	C₄×C₈	—	2	2	no	—
148	C₄×C₈	D₄×C₅₃	1	2	yes	—
149	C₄×C₈	D₄×C₅₃	1	2	yes	—
150	#41	—	4	2	no	—
151	#41	—	4	2	no	—
152	C₂×Q₈	C₄²×C₅₃	1	2	yes	—
153	C₂×D₄	—	2	2	no	—
154	C₂×D₄	—	2	2	no	—
155	C₂×D₄	C₄²×C₅₃	1	2	yes	—
156	C₂²×C₄	—	2	2	no	—
157	C₂²×C₄	C₂²×C₄×C₅₃	1	1	yes	characteristic
158	C₂²×C₄	#41×C₅₃	1	2	yes	—
159	M₁₆	—	2	8	no	—
160	M₁₆	—	4	4	no	—
161	M₁₆	—	4	2	no	—
162	C₂×C₈	—	2	4	no	—
163	C₂×C₈	—	2	2	no	—
164	C₂×C₈	—	2	2	no	—
165	C₂×C₈	—	2	4	no	—
166	C₂×C₈	—	4	1	no	—
167	C₂×C₈	—	4	4	no	—
168	C₂×C₈	C₂²⋊C₄×C₅₃	1	2	yes	—
169	C₂×C₈	C₂²⋊C₄×C₅₃	1	1	yes	characteristic
170	C₂×C₈	C₂²⋊C₄×C₅₃	1	1	yes	characteristic
171	C₄⋊C₄	—	2	2	no	—
172	C₂²⋊C₄	—	2	4	no	—
173	C₄²	C₂²×C₄×C₅₃	1	1	yes	characteristic
174	C₄²	C₂²×C₄×C₅₃	1	1	yes	characteristic
175	C₄²	C₂×D₄×C₅₃	1	2	yes	—
176	C₂³	—	2	2	no	—
177	C₂³	#117×C₅₃	1	1	yes	characteristic
178	Q₈	—	4	2	no	—
179	D₄	—	4	2	no	—
180	D₄	—	4	4	no	—
181	D₄	—	4	4	no	—
182	C₂×C₄	—	2	1	no	—
183	C₂×C₄	—	2	1	no	—
184	C₂×C₄	—	2	4	no	—
185	C₂×C₄	—	2	2	no	—
186	C₂×C₄	—	4	2	no	—
187	C₂×C₄	—	4	2	no	—
188	C₂×C₄	C₂×C₄²×C₅₃	1	1	yes	characteristic
189	C₂×C₄	C₂×C₂²⋊C₄×C₅₃	1	1	yes	characteristic
190	C₂×C₄	C₄×D₄×C₅₃	1	2	yes	—
191	C₂×C₄	C₄×D₄×C₅₃	1	2	yes	—
192	C₈	—	2	8	no	—
193	C₈	—	2	1	no	—
194	C₈	—	2	2	no	—
195	C₈	—	4	2	no	—
196	C₈	—	4	4	no	—
197	C₈	#100×C₅₃	1	2	yes	—
198	C₂²	—	2	1	no	—
199	C₂²	—	2	2	no	—
200	C₂²	—	4	2	no	—
201	C₂²	—	4	4	no	—
202	C₂²	C₄×C₂²⋊C₄×C₅₃	1	1	yes	characteristic
203	C₄	—	2	1	no	—
204	C₄	—	2	2	no	—
205	C₄	—	4	2	no	—
206	C₄	C₂×#100×C₅₃	1	1	yes	characteristic
207	C₄	C₅₃×#413	1	1	yes	characteristic
208	C₄	C₅₃×#443	1	4	yes	—
209	C₂	—	4	1	no	—
210	C₂	—	8	2	no	—
211	C₂	C₄×#100×C₅₃	1	1	yes	characteristic
212	C₂	C₅₃×#1764	1	2	yes	—

	subgroup	G/H
center	C₂×C₈×C₅₃	C₂²⋊C₄
commutator	C₂×C₄	C₂×C₄²×C₅₃
Frattini, cosocle	C₂×C₄²	C₂³×C₅₃
socle	C₂²×C₅₃	C₄×C₂²⋊C₄
prime radical	C₅₃×#532	C₂²
LCS	G ▷ C₂×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₈×C₅₃ ◁ C₂×C₄×C₈×C₅₃ ◁ G

Complements

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

N	H ≅ G/N	type
4. C₈×M₁₆×C₅₃	C₂	semidirect
5. C₅₃×#1299	C₂	semidirect
91. #8241	C₅₃	direct (N × H)
106. C₈×M₁₆	C₂×C₅₃	semidirect
107. #1299	C₂×C₅₃	semidirect
126. C₅₃	#8241	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×#2931	2
C₅₃×#1543	2
C₅₃×#1525	2
C₈×M₁₆×C₅₃	2
C₅₃×#1299	2
#8241	53

Subgroups — C53×#8241 #27215753

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅₃×#8241 #27215753