Subgroups — C₅₃×#34526 #27242038

Order 13568.

1800 conjugacy classes of subgroups.

The subgroup lattice has 170 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₅₃×#564	C₂	1	1	yes	maximal, characteristic
2	C₅₃×#2382	C₂	1	3	yes	maximal
3	C₅₃×#2370	C₂	1	12	yes	maximal
4	C₅₃×#2336	C₂	1	12	yes	maximal
5	C₄×#125×C₅₃	C₂	1	3	yes	maximal
6	C₅₃×#564	C₄	1	16	yes	complemented
7	C₂×#125×C₅₃	C₂²	1	3	yes	—
8	C₂×#125×C₅₃	C₂²	1	12	yes	—
9	C₅₃×#389	C₂²	1	6	yes	—
10	C₅₃×#389	C₂²	1	12	yes	—
11	C₅₃×#389	C₂²	1	12	yes	—
12	C₅₃×#387	C₂²	1	12	yes	—
13	C₅₃×#387	C₂²	1	12	yes	—
14	C₅₃×#387	C₂²	1	12	yes	—
15	C₅₃×#387	C₂²	1	12	yes	—
16	C₅₃×#387	C₂²	1	12	yes	—
17	C₅₃×#383	C₂²	1	4	yes	—
18	C₅₃×#382	C₂²	1	3	yes	—
19	C₄×C₄⋊C₄×C₅₃	C₂²	1	6	yes	—
20	C₄×C₄⋊C₄×C₅₃	C₂²	1	12	yes	—
21	C₄×C₄⋊C₄×C₅₃	C₂²	1	12	yes	—
22	C₄×C₄⋊C₄×C₅₃	C₂²	1	12	yes	—
23	C₄³×C₅₃	C₂²	1	1	yes	characteristic
24	#125×C₅₃	C₂×C₄	1	24	yes	—
25	#125×C₅₃	C₂×C₄	1	96	yes	—
26	C₂×C₄⋊C₄×C₅₃	C₂³	1	12	yes	—
27	C₂×C₄⋊C₄×C₅₃	C₂³	1	6	yes	—
28	C₂×C₄⋊C₄×C₅₃	C₂³	1	12	yes	—
29	C₂×C₄⋊C₄×C₅₃	C₂³	1	6	yes	—
30	C₂×C₄⋊C₄×C₅₃	C₂³	1	12	yes	—
31	C₂×C₄⋊C₄×C₅₃	C₂³	1	12	yes	—
32	C₂×C₄²×C₅₃	C₂³	1	3	yes	—
33	C₂×C₄²×C₅₃	C₂³	1	3	yes	—
34	C₂×C₄²×C₅₃	C₂³	1	12	yes	—
35	C₂×C₄²×C₅₃	C₂³	1	12	yes	—
36	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
37	C₂×C₄²×C₅₃	C₂³	1	4	yes	—
38	#95×C₅₃	C₂³	1	12	yes	—
39	#95×C₅₃	C₂³	1	12	yes	—
40	#95×C₅₃	C₂³	1	12	yes	—
41	#95×C₅₃	C₂³	1	12	yes	—
42	#95×C₅₃	C₂³	1	12	yes	—
43	C₂²×C₄×C₅₃	C₂⁴	1	1	yes	characteristic
44	C₂²×C₄×C₅₃	C₂⁴	1	12	yes	—
45	C₂²×C₄×C₅₃	C₂⁴	1	3	yes	—
46	C₂²×C₄×C₅₃	C₂⁴	1	12	yes	—
47	C₂²×C₄×C₅₃	C₂⁴	1	3	yes	—
48	C₄⋊C₄×C₅₃	—	2	24	no	—
49	C₄⋊C₄×C₅₃	—	4	12	no	—
50	C₄⋊C₄×C₅₃	—	4	6	no	—
51	C₄⋊C₄×C₅₃	C₂²×C₄	1	24	yes	—
52	C₄⋊C₄×C₅₃	C₂²×C₄	1	48	yes	—
53	C₄⋊C₄×C₅₃	C₂²×C₄	1	48	yes	—
54	C₄²×C₅₃	—	2	6	no	—
55	C₄²×C₅₃	—	4	3	no	—
56	C₄²×C₅₃	—	4	12	no	—
57	C₄²×C₅₃	—	4	12	no	—
58	C₄²×C₅₃	C₂²×C₄	1	4	yes	—
59	C₄²×C₅₃	C₂²×C₄	1	16	yes	—
60	C₂³×C₅₃	C₂⁵	1	1	yes	characteristic
61	C₂×C₄×C₅₃	—	2	3	no	—
62	C₂×C₄×C₅₃	—	2	12	no	—
63	C₂×C₄×C₅₃	—	2	6	no	—
64	C₂×C₄×C₅₃	—	2	12	no	—
65	C₂×C₄×C₅₃	—	2	3	no	—
66	C₂×C₄×C₅₃	—	2	3	no	—
67	C₂×C₄×C₅₃	—	2	12	no	—
68	C₂×C₄×C₅₃	—	2	12	no	—
69	C₂×C₄×C₅₃	—	2	12	no	—
70	C₂×C₄×C₅₃	C₂³×C₄	1	6	yes	—
71	C₂×C₄×C₅₃	C₂³×C₄	1	24	yes	—
72	C₂×C₄×C₅₃	2₋¹⁺⁴	1	6	yes	—
73	#34526	C₅₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
74	C₂²×C₅₃	C₂⁴×C₄	1	1	yes	characteristic
75	C₂²×C₅₃	C₂×2₋¹⁺⁴	1	3	yes	—
76	C₂²×C₅₃	#585	1	3	yes	—
77	C₄×C₅₃	—	2	6	no	—
78	C₄×C₅₃	—	2	24	no	—
79	C₄×C₅₃	—	2	6	no	—
80	C₄×C₅₃	—	4	12	no	—
81	C₄×C₅₃	—	4	1	no	—
82	C₂×#564	C₂×C₅₃	1	1	yes	characteristic
83	#2382	C₂×C₅₃	1	3	yes	—
84	#2370	C₂×C₅₃	1	12	yes	—
85	#2336	C₂×C₅₃	1	12	yes	—
86	C₄×#125	C₂×C₅₃	1	3	yes	—
87	C₂×C₅₃	C₄×2₋¹⁺⁴	1	3	yes	—
88	C₂×C₅₃	#3553	1	3	yes	—
89	C₂×C₅₃	#3556	1	1	yes	characteristic
90	#564	C₄×C₅₃	1	16	yes	complemented
91	C₂×#125	C₂²×C₅₃	1	3	yes	—
92	C₂×#125	C₂²×C₅₃	1	12	yes	—
93	#389	C₂²×C₅₃	1	6	yes	—
94	#389	C₂²×C₅₃	1	12	yes	—
95	#389	C₂²×C₅₃	1	12	yes	—
96	#387	C₂²×C₅₃	1	12	yes	—
97	#387	C₂²×C₅₃	1	12	yes	—
98	#387	C₂²×C₅₃	1	12	yes	—
99	#387	C₂²×C₅₃	1	12	yes	—
100	#387	C₂²×C₅₃	1	12	yes	—
101	#383	C₂²×C₅₃	1	4	yes	—
102	#382	C₂²×C₅₃	1	3	yes	—
103	C₄×C₄⋊C₄	C₂²×C₅₃	1	6	yes	—
104	C₄×C₄⋊C₄	C₂²×C₅₃	1	12	yes	—
105	C₄×C₄⋊C₄	C₂²×C₅₃	1	12	yes	—
106	C₄×C₄⋊C₄	C₂²×C₅₃	1	12	yes	—
107	C₄³	C₂²×C₅₃	1	1	yes	characteristic
108	C₅₃	#34526	1	1	yes	characteristic, direct product factor, Sylow
109	#125	C₂×C₄×C₅₃	1	24	yes	—
110	#125	C₂×C₄×C₅₃	1	96	yes	—
111	C₂×C₄⋊C₄	C₂³×C₅₃	1	12	yes	—
112	C₂×C₄⋊C₄	C₂³×C₅₃	1	6	yes	—
113	C₂×C₄⋊C₄	C₂³×C₅₃	1	12	yes	—
114	C₂×C₄⋊C₄	C₂³×C₅₃	1	6	yes	—
115	C₂×C₄⋊C₄	C₂³×C₅₃	1	12	yes	—
116	C₂×C₄⋊C₄	C₂³×C₅₃	1	12	yes	—
117	C₂×C₄²	C₂³×C₅₃	1	3	yes	—
118	C₂×C₄²	C₂³×C₅₃	1	3	yes	—
119	C₂×C₄²	C₂³×C₅₃	1	12	yes	—
120	C₂×C₄²	C₂³×C₅₃	1	12	yes	—
121	C₂×C₄²	C₂³×C₅₃	1	1	yes	characteristic
122	C₂×C₄²	C₂³×C₅₃	1	4	yes	—
123	#95	C₂³×C₅₃	1	12	yes	—
124	#95	C₂³×C₅₃	1	12	yes	—
125	#95	C₂³×C₅₃	1	12	yes	—
126	#95	C₂³×C₅₃	1	12	yes	—
127	#95	C₂³×C₅₃	1	12	yes	—
128	C₂²×C₄	C₂⁴×C₅₃	1	1	yes	characteristic
129	C₂²×C₄	C₂⁴×C₅₃	1	12	yes	—
130	C₂²×C₄	C₂⁴×C₅₃	1	3	yes	—
131	C₂²×C₄	C₂⁴×C₅₃	1	12	yes	—
132	C₂²×C₄	C₂⁴×C₅₃	1	3	yes	—
133	C₄⋊C₄	—	2	24	no	—
134	C₄⋊C₄	—	4	12	no	—
135	C₄⋊C₄	—	4	6	no	—
136	C₄⋊C₄	C₂²×C₄×C₅₃	1	24	yes	—
137	C₄⋊C₄	C₂²×C₄×C₅₃	1	48	yes	—
138	C₄⋊C₄	C₂²×C₄×C₅₃	1	48	yes	—
139	C₄²	—	2	6	no	—
140	C₄²	—	4	3	no	—
141	C₄²	—	4	12	no	—
142	C₄²	—	4	12	no	—
143	C₄²	C₂²×C₄×C₅₃	1	4	yes	—
144	C₄²	C₂²×C₄×C₅₃	1	16	yes	—
145	C₂³	C₂⁵×C₅₃	1	1	yes	characteristic
146	C₂×C₄	—	2	3	no	—
147	C₂×C₄	—	2	12	no	—
148	C₂×C₄	—	2	6	no	—
149	C₂×C₄	—	2	12	no	—
150	C₂×C₄	—	2	3	no	—
151	C₂×C₄	—	2	3	no	—
152	C₂×C₄	—	2	12	no	—
153	C₂×C₄	—	2	12	no	—
154	C₂×C₄	—	2	12	no	—
155	C₂×C₄	C₂³×C₄×C₅₃	1	6	yes	—
156	C₂×C₄	C₂³×C₄×C₅₃	1	24	yes	—
157	C₂×C₄	2₋¹⁺⁴×C₅₃	1	6	yes	—
158	C₂²	C₂⁴×C₄×C₅₃	1	1	yes	characteristic
159	C₂²	C₂×2₋¹⁺⁴×C₅₃	1	3	yes	—
160	C₂²	C₅₃×#585	1	3	yes	—
161	C₄	—	2	6	no	—
162	C₄	—	2	6	no	—
163	C₄	—	2	24	no	—
164	C₄	—	4	12	no	—
165	C₄	—	4	1	no	—
166	C₂	C₄×2₋¹⁺⁴×C₅₃	1	3	yes	—
167	C₂	C₅₃×#3553	1	3	yes	—
168	C₂	C₅₃×#3556	1	1	yes	characteristic

	subgroup	G/H
center, socle, prime radical	C₂³×C₅₃	C₂⁵
commutator	C₂²	C₂⁴×C₄×C₅₃
Frattini, cosocle	C₂³	C₂⁵×C₅₃
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³×C₅₃ ◁ G

Complements

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

N	H ≅ G/N	type
6. C₅₃×#564	C₄	semidirect
73. #34526	C₅₃	direct (N × H)
90. #564	C₄×C₅₃	semidirect
108. C₅₃	#34526	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×#564	2
C₅₃×#2382	2
C₅₃×#2370	2
C₅₃×#2336	2
C₄×#125×C₅₃	2
#34526	53

Subgroups — C53×#34526 #27242038

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅₃×#34526 #27242038