Subgroups — C₅₃×#8284 #27215796

Order 13568.

430 conjugacy classes of subgroups.

The subgroup lattice has 198 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×#107×C₅₃	C₂	1	1	yes	maximal, characteristic
2	C₈×C₄⋊C₄×C₅₃	C₂	1	2	yes	maximal
3	C₂×C₅₃×#334	C₂	1	1	yes	maximal, characteristic
4	C₂×C₅₃×#322	C₂	1	1	yes	maximal, characteristic
5	C₅₃×#1279	C₂	1	2	yes	maximal
6	C₂×#107×C₅₃	C₄	1	2	yes	—
7	C₂×C₄⋊C₈×C₅₃	C₂²	1	2	yes	—
8	C₂×C₄×C₈×C₅₃	—	2	2	no	—
9	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
10	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
11	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
12	C₄×C₄⋊C₄×C₅₃	C₂²	1	2	yes	—
13	C₅₃×#340	C₄	1	2	yes	—
14	C₅₃×#336	—	2	2	no	—
15	C₅₃×#334	C₄	1	4	yes	complemented
16	C₅₃×#322	C₄	1	4	yes	complemented
17	C₂²×C₈×C₅₃	—	2	2	no	—
18	C₂²×C₈×C₅₃	—	2	1	no	—
19	C₂²×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
20	C₂²×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
21	C₂²×C₈×C₅₃	D₄	1	2	yes	—
22	C₂×C₄⋊C₄×C₅₃	C₂×C₄	1	4	yes	—
23	C₂×C₄²×C₅₃	—	2	2	no	—
24	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
25	#107×C₅₃	C₈	1	8	yes	complemented
26	C₄⋊C₈×C₅₃	—	2	2	no	—
27	C₄⋊C₈×C₅₃	C₂×C₄	1	4	yes	—
28	C₄×C₈×C₅₃	—	2	1	no	—
29	C₄×C₈×C₅₃	—	4	4	no	—
30	C₄×C₈×C₅₃	C₂×C₄	1	2	yes	—
31	C₄×C₈×C₅₃	C₂×C₄	1	2	yes	—
32	C₄×C₈×C₅₃	C₂×C₄	1	2	yes	—
33	C₄×C₈×C₅₃	C₂×C₄	1	2	yes	—
34	C₄×C₈×C₅₃	Q₈	1	2	yes	—
35	#95×C₅₃	—	2	2	no	—
36	C₂²×C₄×C₅₃	—	2	4	no	—
37	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
38	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
39	C₂²×C₄×C₅₃	#41	1	1	yes	characteristic
40	C₂×C₈×C₅₃	—	2	2	no	—
41	C₂×C₈×C₅₃	—	2	2	no	—
42	C₂×C₈×C₅₃	—	2	2	no	—
43	C₂×C₈×C₅₃	—	2	2	no	—
44	C₂×C₈×C₅₃	—	2	4	no	—
45	C₂×C₈×C₅₃	—	4	2	no	—
46	C₂×C₈×C₅₃	—	4	2	no	—
47	C₂×C₈×C₅₃	—	4	2	no	—
48	C₂×C₈×C₅₃	C₄²	1	2	yes	—
49	C₂×C₈×C₅₃	C₄²	1	2	yes	—
50	C₂×C₈×C₅₃	C₄⋊C₄	1	4	yes	—
51	C₂×C₈×C₅₃	C₂×C₈	1	4	yes	—
52	C₄⋊C₄×C₅₃	—	2	4	no	—
53	C₄⋊C₄×C₅₃	C₂×C₈	1	8	yes	—
54	C₄²×C₅₃	—	4	2	no	—
55	C₄²×C₅₃	—	4	2	no	—
56	C₄²×C₅₃	C₂²×C₄	1	1	yes	characteristic
57	C₄²×C₅₃	C₂²×C₄	1	1	yes	characteristic
58	C₄²×C₅₃	C₂×Q₈	1	1	yes	characteristic
59	C₄²×C₅₃	#41	1	1	yes	characteristic
60	C₂³×C₅₃	C₄×D₄	1	1	yes	characteristic
61	C₂×C₄×C₅₃	—	2	2	no	—
62	C₂×C₄×C₅₃	—	2	8	no	—
63	C₂×C₄×C₅₃	—	4	4	no	—
64	C₂×C₄×C₅₃	—	4	4	no	—
65	C₂×C₄×C₅₃	C₂×C₄²	1	1	yes	characteristic
66	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
67	C₂×C₄×C₅₃	#117	1	1	yes	characteristic
68	C₂×C₄×C₅₃	C₄×D₄	1	1	yes	characteristic
69	C₂×C₄×C₅₃	C₄×Q₈	1	1	yes	characteristic
70	C₂×C₄×C₅₃	C₄×Q₈	1	1	yes	characteristic
71	C₂×C₄×C₅₃	C₂²×C₈	1	2	yes	—
72	C₂×C₄×C₅₃	#131	1	2	yes	—
73	C₂×C₄×C₅₃	AGL(1,ℤ/8)	1	2	yes	—
74	C₂×C₄×C₅₃	#137	1	2	yes	—
75	C₈×C₅₃	—	2	2	no	—
76	C₈×C₅₃	—	2	4	no	—
77	C₈×C₅₃	—	4	2	no	—
78	C₈×C₅₃	—	4	4	no	—
79	C₈×C₅₃	C₄×C₈	1	4	yes	complemented
80	#8284	C₅₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
81	C₂²×C₅₃	C₄×C₄⋊C₄	1	1	yes	characteristic
82	C₂²×C₅₃	C₈×D₄	1	2	yes	—
83	C₂²×C₅₃	#441	1	1	yes	characteristic
84	C₂²×C₅₃	#442	1	1	yes	characteristic
85	C₂²×C₅₃	#444	1	2	yes	—
86	C₄×C₅₃	—	2	1	no	—
87	C₄×C₅₃	—	2	1	no	—
88	C₄×C₅₃	—	4	8	no	—
89	C₄×C₅₃	C₂×C₄×C₈	1	1	yes	characteristic
90	C₄×C₅₃	#405	1	1	yes	characteristic
91	C₄×C₅₃	#428	1	2	yes	—
92	C₄×C₅₃	#440	1	2	yes	—
93	C₄×C₅₃	C₈×Q₈	1	2	yes	—
94	C₄×#107	C₂×C₅₃	1	1	yes	characteristic
95	C₈×C₄⋊C₄	C₂×C₅₃	1	2	yes	—
96	C₂×#334	C₂×C₅₃	1	1	yes	characteristic
97	C₂×#322	C₂×C₅₃	1	1	yes	characteristic
98	#1279	C₂×C₅₃	1	2	yes	—
99	C₂×C₅₃	#1579	1	2	yes	—
100	C₂×C₅₃	#1580	1	2	yes	—
101	C₂×C₅₃	C₈×C₄⋊C₄	1	1	yes	characteristic
102	C₂×C₅₃	#1773	1	1	yes	characteristic
103	C₂×C₅₃	#1776	1	1	yes	characteristic
104	C₂×#107	C₄×C₅₃	1	2	yes	—
105	C₂×C₄⋊C₈	C₂²×C₅₃	1	2	yes	—
106	C₂×C₄×C₈	—	2	2	no	—
107	C₂×C₄×C₈	C₂²×C₅₃	1	1	yes	characteristic
108	C₂×C₄×C₈	C₂²×C₅₃	1	1	yes	characteristic
109	C₂×C₄×C₈	C₂²×C₅₃	1	1	yes	characteristic
110	C₄×C₄⋊C₄	C₂²×C₅₃	1	2	yes	—
111	#340	C₄×C₅₃	1	2	yes	—
112	#336	—	2	2	no	—
113	#334	C₄×C₅₃	1	4	yes	complemented
114	#322	C₄×C₅₃	1	4	yes	complemented
115	C₅₃	#8284	1	1	yes	characteristic, direct product factor, Sylow
116	C₂²×C₈	—	2	2	no	—
117	C₂²×C₈	—	2	1	no	—
118	C₂²×C₈	C₂×C₄×C₅₃	1	1	yes	characteristic
119	C₂²×C₈	C₂×C₄×C₅₃	1	1	yes	characteristic
120	C₂²×C₈	D₄×C₅₃	1	2	yes	—
121	C₂×C₄⋊C₄	C₂×C₄×C₅₃	1	4	yes	—
122	C₂×C₄²	—	2	2	no	—
123	C₂×C₄²	C₂³×C₅₃	1	1	yes	characteristic
124	#107	C₈×C₅₃	1	8	yes	complemented
125	C₄⋊C₈	—	2	2	no	—
126	C₄⋊C₈	C₂×C₄×C₅₃	1	4	yes	—
127	C₄×C₈	—	2	1	no	—
128	C₄×C₈	—	4	4	no	—
129	C₄×C₈	C₂×C₄×C₅₃	1	2	yes	—
130	C₄×C₈	C₂×C₄×C₅₃	1	2	yes	—
131	C₄×C₈	C₂×C₄×C₅₃	1	2	yes	—
132	C₄×C₈	C₂×C₄×C₅₃	1	2	yes	—
133	C₄×C₈	Q₈×C₅₃	1	2	yes	—
134	#95	—	2	2	no	—
135	C₂²×C₄	—	2	4	no	—
136	C₂²×C₄	C₂²×C₄×C₅₃	1	1	yes	characteristic
137	C₂²×C₄	C₂×D₄×C₅₃	1	1	yes	characteristic
138	C₂²×C₄	#41×C₅₃	1	1	yes	characteristic
139	C₂×C₈	—	2	2	no	—
140	C₂×C₈	—	2	2	no	—
141	C₂×C₈	—	2	2	no	—
142	C₂×C₈	—	2	2	no	—
143	C₂×C₈	—	2	4	no	—
144	C₂×C₈	—	4	2	no	—
145	C₂×C₈	—	4	2	no	—
146	C₂×C₈	—	4	2	no	—
147	C₂×C₈	C₄²×C₅₃	1	2	yes	—
148	C₂×C₈	C₄²×C₅₃	1	2	yes	—
149	C₂×C₈	C₄⋊C₄×C₅₃	1	4	yes	—
150	C₂×C₈	C₂×C₈×C₅₃	1	4	yes	—
151	C₄⋊C₄	—	2	4	no	—
152	C₄⋊C₄	C₂×C₈×C₅₃	1	8	yes	—
153	C₄²	—	4	2	no	—
154	C₄²	—	4	2	no	—
155	C₄²	C₂²×C₄×C₅₃	1	1	yes	characteristic
156	C₄²	C₂²×C₄×C₅₃	1	1	yes	characteristic
157	C₄²	C₂×Q₈×C₅₃	1	1	yes	characteristic
158	C₄²	#41×C₅₃	1	1	yes	characteristic
159	C₂³	C₄×D₄×C₅₃	1	1	yes	characteristic
160	C₂×C₄	—	2	2	no	—
161	C₂×C₄	—	2	8	no	—
162	C₂×C₄	—	4	4	no	—
163	C₂×C₄	—	4	4	no	—
164	C₂×C₄	C₂×C₄²×C₅₃	1	1	yes	characteristic
165	C₂×C₄	C₂×C₄⋊C₄×C₅₃	1	1	yes	characteristic
166	C₂×C₄	#117×C₅₃	1	1	yes	characteristic
167	C₂×C₄	C₄×D₄×C₅₃	1	1	yes	characteristic
168	C₂×C₄	C₄×Q₈×C₅₃	1	1	yes	characteristic
169	C₂×C₄	C₄×Q₈×C₅₃	1	1	yes	characteristic
170	C₂×C₄	C₂²×C₈×C₅₃	1	2	yes	—
171	C₂×C₄	#131×C₅₃	1	2	yes	—
172	C₂×C₄	AGL(1,ℤ/8)×C₅₃	1	2	yes	—
173	C₂×C₄	#137×C₅₃	1	2	yes	—
174	C₈	—	2	2	no	—
175	C₈	—	2	4	no	—
176	C₈	—	4	2	no	—
177	C₈	—	4	4	no	—
178	C₈	C₄×C₈×C₅₃	1	4	yes	complemented
179	C₂²	C₄×C₄⋊C₄×C₅₃	1	1	yes	characteristic
180	C₂²	C₈×D₄×C₅₃	1	2	yes	—
181	C₂²	C₅₃×#441	1	1	yes	characteristic
182	C₂²	C₅₃×#442	1	1	yes	characteristic
183	C₂²	C₅₃×#444	1	2	yes	—
184	C₄	—	2	1	no	—
185	C₄	—	2	1	no	—
186	C₄	—	4	8	no	—
187	C₄	C₂×C₄×C₈×C₅₃	1	1	yes	characteristic
188	C₄	C₅₃×#405	1	1	yes	characteristic
189	C₄	C₅₃×#428	1	2	yes	—
190	C₄	C₅₃×#440	1	2	yes	—
191	C₄	C₈×Q₈×C₅₃	1	2	yes	—
192	C₂	C₅₃×#1579	1	2	yes	—
193	C₂	C₅₃×#1580	1	2	yes	—
194	C₂	C₈×C₄⋊C₄×C₅₃	1	1	yes	characteristic
195	C₂	C₅₃×#1773	1	1	yes	characteristic
196	C₂	C₅₃×#1776	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄×C₅₃	C₂×D₄
commutator	C₄	C₂×C₄×C₈×C₅₃
Frattini, cosocle	C₂×C₄²	C₂³×C₅₃
socle, prime radical	C₂³×C₅₃	C₄×D₄
LCS	G ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄×C₅₃ ◁ C₂×C₄×C₈×C₅₃ ◁ G

Complements

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

N	H ≅ G/N	type
15. C₅₃×#334	C₄	semidirect
16. C₅₃×#322	C₄	semidirect
25. #107×C₅₃	C₈	semidirect
79. C₈×C₅₃	C₄×C₈	semidirect
80. #8284	C₅₃	direct (N × H)
113. #334	C₄×C₅₃	semidirect
114. #322	C₄×C₅₃	semidirect
115. C₅₃	#8284	direct (N × H)
124. #107	C₈×C₅₃	semidirect
178. C₈	C₄×C₈×C₅₃	semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
C₄×#107×C₅₃	2
C₈×C₄⋊C₄×C₅₃	2
C₂×C₅₃×#334	2
C₂×C₅₃×#322	2
C₅₃×#1279	2
#8284	53

Subgroups — C53×#8284 #27215796

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅₃×#8284 #27215796