Subgroups — C₂²×C₅₃×#346 #27240833

Order 13568.

942 conjugacy classes of subgroups.

The subgroup lattice has 132 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂³×C₁₆×C₅₃	C₂	1	2	yes	maximal
2	C₂²×C₄×C₈×C₅₃	C₂	1	1	yes	maximal, characteristic
3	C₂×C₅₃×#346	C₂	1	12	yes	maximal, direct product factor
4	C₂³×C₈×C₅₃	C₄	1	1	yes	characteristic
5	C₂³×C₈×C₅₃	C₂²	1	1	yes	characteristic
6	C₂²×C₄²×C₅₃	C₄	1	1	yes	characteristic
7	C₂²×C₁₆×C₅₃	C₄	1	16	yes	complemented
8	C₂²×C₁₆×C₅₃	C₂²	1	12	yes	—
9	C₂×C₄×C₈×C₅₃	C₄	1	6	yes	—
10	C₂×C₄×C₈×C₅₃	C₂²	1	6	yes	—
11	C₅₃×#346	C₂²	1	16	yes	direct product factor
12	C₂³×C₄×C₅₃	C₈	1	2	yes	—
13	C₂³×C₄×C₅₃	C₂×C₄	1	1	yes	characteristic
14	C₂²×C₈×C₅₃	C₈	1	8	yes	—
15	C₂²×C₈×C₅₃	C₂×C₄	1	8	yes	—
16	C₂²×C₈×C₅₃	C₂×C₄	1	3	yes	—
17	C₂²×C₈×C₅₃	C₂×C₄	1	6	yes	—
18	C₂²×C₈×C₅₃	C₂³	1	3	yes	—
19	C₂×C₄²×C₅₃	C₈	1	6	yes	—
20	C₂×C₄²×C₅₃	C₂×C₄	1	6	yes	—
21	C₂×C₁₆×C₅₃	C₂×C₄	1	48	yes	complemented
22	C₂×C₁₆×C₅₃	C₂³	1	8	yes	—
23	C₄×C₈×C₅₃	C₂×C₄	1	12	yes	—
24	C₄×C₈×C₅₃	C₂³	1	4	yes	—
25	C₂⁴×C₅₃	C₂×C₈	1	1	yes	characteristic
26	C₂²×C₄×C₅₃	—	2	8	no	—
27	C₂²×C₄×C₅₃	C₄²	1	4	yes	—
28	C₂²×C₄×C₅₃	C₂×C₈	1	12	yes	—
29	C₂²×C₄×C₅₃	C₂×C₈	1	3	yes	—
30	C₂²×C₄×C₅₃	C₂×C₈	1	4	yes	—
31	C₂²×C₄×C₅₃	C₂²×C₄	1	3	yes	—
32	C₂×C₈×C₅₃	C₄²	1	12	yes	—
33	C₂×C₈×C₅₃	C₂×C₈	1	24	yes	—
34	C₂×C₈×C₅₃	C₂²×C₄	1	3	yes	—
35	C₂×C₈×C₅₃	C₂²×C₄	1	12	yes	—
36	C₂×C₈×C₅₃	C₂²×C₄	1	4	yes	—
37	C₂×C₈×C₅₃	C₂⁴	1	1	yes	characteristic
38	C₄²×C₅₃	C₂×C₈	1	12	yes	—
39	C₄²×C₅₃	C₂²×C₄	1	4	yes	—
40	C₁₆×C₅₃	C₂²×C₄	1	16	yes	complemented
41	C₂³×C₅₃	C₄×C₈	1	4	yes	—
42	C₂³×C₅₃	M₃₂	1	8	yes	—
43	C₂³×C₅₃	C₂²×C₈	1	3	yes	—
44	C₂×C₄×C₅₃	—	2	24	no	—
45	C₂×C₄×C₅₃	C₄×C₈	1	12	yes	—
46	C₂×C₄×C₅₃	C₂×C₄²	1	6	yes	—
47	C₂×C₄×C₅₃	C₂²×C₈	1	8	yes	—
48	C₂×C₄×C₅₃	C₂²×C₈	1	6	yes	—
49	C₂×C₄×C₅₃	C₂²×C₈	1	3	yes	—
50	C₂×C₄×C₅₃	C₂³×C₄	1	1	yes	characteristic
51	C₈×C₅₃	C₂×C₄²	1	6	yes	—
52	C₈×C₅₃	C₂²×C₈	1	8	yes	—
53	C₈×C₅₃	C₂³×C₄	1	2	yes	—
54	C₂²×#346	C₅₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
55	C₂²×C₅₃	#346	1	16	yes	direct product factor
56	C₂²×C₅₃	C₂×C₄×C₈	1	6	yes	—
57	C₂²×C₅₃	C₂×M₃₂	1	12	yes	—
58	C₂²×C₅₃	C₂³×C₈	1	1	yes	characteristic
59	C₄×C₅₃	—	2	8	no	—
60	C₄×C₅₃	C₂×C₄×C₈	1	6	yes	—
61	C₄×C₅₃	C₂²×C₄²	1	1	yes	characteristic
62	C₄×C₅₃	C₂³×C₈	1	1	yes	characteristic
63	C₂³×C₁₆	C₂×C₅₃	1	2	yes	—
64	C₂²×C₄×C₈	C₂×C₅₃	1	1	yes	characteristic
65	C₂×#346	C₂×C₅₃	1	12	yes	direct product factor
66	C₂×C₅₃	C₂×#346	1	12	yes	direct product factor
67	C₂×C₅₃	C₂²×C₄×C₈	1	1	yes	characteristic
68	C₂×C₅₃	C₂²×M₃₂	1	2	yes	—
69	C₂³×C₈	C₄×C₅₃	1	1	yes	characteristic
70	C₂³×C₈	C₂²×C₅₃	1	1	yes	characteristic
71	C₂²×C₄²	C₄×C₅₃	1	1	yes	characteristic
72	C₂²×C₁₆	C₄×C₅₃	1	16	yes	complemented
73	C₂²×C₁₆	C₂²×C₅₃	1	12	yes	—
74	C₂×C₄×C₈	C₄×C₅₃	1	6	yes	—
75	C₂×C₄×C₈	C₂²×C₅₃	1	6	yes	—
76	#346	C₂²×C₅₃	1	16	yes	direct product factor
77	C₅₃	C₂²×#346	1	1	yes	characteristic, direct product factor, Sylow
78	C₂³×C₄	C₈×C₅₃	1	2	yes	—
79	C₂³×C₄	C₂×C₄×C₅₃	1	1	yes	characteristic
80	C₂²×C₈	C₈×C₅₃	1	8	yes	—
81	C₂²×C₈	C₂×C₄×C₅₃	1	8	yes	—
82	C₂²×C₈	C₂×C₄×C₅₃	1	3	yes	—
83	C₂²×C₈	C₂×C₄×C₅₃	1	6	yes	—
84	C₂²×C₈	C₂³×C₅₃	1	3	yes	—
85	C₂×C₄²	C₈×C₅₃	1	6	yes	—
86	C₂×C₄²	C₂×C₄×C₅₃	1	6	yes	—
87	C₂×C₁₆	C₂×C₄×C₅₃	1	48	yes	complemented
88	C₂×C₁₆	C₂³×C₅₃	1	8	yes	—
89	C₄×C₈	C₂×C₄×C₅₃	1	12	yes	—
90	C₄×C₈	C₂³×C₅₃	1	4	yes	—
91	C₂⁴	C₂×C₈×C₅₃	1	1	yes	characteristic
92	C₂²×C₄	—	2	8	no	—
93	C₂²×C₄	C₄²×C₅₃	1	4	yes	—
94	C₂²×C₄	C₂×C₈×C₅₃	1	12	yes	—
95	C₂²×C₄	C₂×C₈×C₅₃	1	3	yes	—
96	C₂²×C₄	C₂×C₈×C₅₃	1	4	yes	—
97	C₂²×C₄	C₂²×C₄×C₅₃	1	3	yes	—
98	C₂×C₈	C₄²×C₅₃	1	12	yes	—
99	C₂×C₈	C₂×C₈×C₅₃	1	24	yes	—
100	C₂×C₈	C₂²×C₄×C₅₃	1	3	yes	—
101	C₂×C₈	C₂²×C₄×C₅₃	1	12	yes	—
102	C₂×C₈	C₂²×C₄×C₅₃	1	4	yes	—
103	C₂×C₈	C₂⁴×C₅₃	1	1	yes	characteristic
104	C₄²	C₂×C₈×C₅₃	1	12	yes	—
105	C₄²	C₂²×C₄×C₅₃	1	4	yes	—
106	C₁₆	C₂²×C₄×C₅₃	1	16	yes	complemented
107	C₂³	C₄×C₈×C₅₃	1	4	yes	—
108	C₂³	M₃₂×C₅₃	1	8	yes	—
109	C₂³	C₂²×C₈×C₅₃	1	3	yes	—
110	C₂×C₄	—	2	24	no	—
111	C₂×C₄	C₄×C₈×C₅₃	1	12	yes	—
112	C₂×C₄	C₂×C₄²×C₅₃	1	6	yes	—
113	C₂×C₄	C₂²×C₈×C₅₃	1	8	yes	—
114	C₂×C₄	C₂²×C₈×C₅₃	1	6	yes	—
115	C₂×C₄	C₂²×C₈×C₅₃	1	3	yes	—
116	C₂×C₄	C₂³×C₄×C₅₃	1	1	yes	characteristic
117	C₈	C₂×C₄²×C₅₃	1	6	yes	—
118	C₈	C₂²×C₈×C₅₃	1	8	yes	—
119	C₈	C₂³×C₄×C₅₃	1	2	yes	—
120	C₂²	C₅₃×#346	1	16	yes	direct product factor
121	C₂²	C₂×C₄×C₈×C₅₃	1	6	yes	—
122	C₂²	C₂×M₃₂×C₅₃	1	12	yes	—
123	C₂²	C₂³×C₈×C₅₃	1	1	yes	characteristic
124	C₄	—	2	8	no	—
125	C₄	C₂×C₄×C₈×C₅₃	1	6	yes	—
126	C₄	C₂²×C₄²×C₅₃	1	1	yes	characteristic
127	C₄	C₂³×C₈×C₅₃	1	1	yes	characteristic
128	C₂	C₂×C₅₃×#346	1	12	yes	direct product factor
129	C₂	C₂²×C₄×C₈×C₅₃	1	1	yes	characteristic
130	C₂	C₂²×M₃₂×C₅₃	1	2	yes	—

	subgroup	G/H
center	C₂³×C₈×C₅₃	C₂²
commutator	C₂	C₂²×C₄×C₈×C₅₃
Frattini, cosocle	C₂×C₈	C₂⁴×C₅₃
socle, prime radical	C₂⁴×C₅₃	C₂×C₈
LCS	G ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₈×C₅₃ ◁ G

Complements

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

N	H ≅ G/N	type
3. C₂×C₅₃×#346	C₂	direct (N × H)
7. C₂²×C₁₆×C₅₃	C₄	semidirect
11. C₅₃×#346	C₂²	direct (N × H)
21. C₂×C₁₆×C₅₃	C₂×C₄	semidirect
40. C₁₆×C₅₃	C₂²×C₄	semidirect
54. C₂²×#346	C₅₃	direct (N × H)
55. C₂²×C₅₃	#346	direct (N × H)
65. C₂×#346	C₂×C₅₃	direct (N × H)
66. C₂×C₅₃	C₂×#346	direct (N × H)
72. C₂²×C₁₆	C₄×C₅₃	semidirect
76. #346	C₂²×C₅₃	direct (N × H)
77. C₅₃	C₂²×#346	direct (N × H)
87. C₂×C₁₆	C₂×C₄×C₅₃	semidirect
106. C₁₆	C₂²×C₄×C₅₃	semidirect
120. C₂²	C₅₃×#346	direct (N × H)
128. C₂	C₂×C₅₃×#346	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂³×C₁₆×C₅₃	2
C₂²×C₄×C₈×C₅₃	2
C₂×C₅₃×#346	2
C₂²×#346	53

Subgroups — C22×C53×#346 #27240833

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×C₅₃×#346 #27240833