Subgroups — C₅₃×#62653 #27270165

Order 13568.

10930 conjugacy classes of subgroups.

The subgroup lattice has 140 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×2₋¹⁺⁴×C₅₃	C₂	1	1	yes	maximal, characteristic
2	C₄×2₋¹⁺⁴×C₅₃	C₂	1	32	yes	maximal, complemented
3	C₅₃×#3428	C₂	1	20	yes	maximal, complemented
4	C₂×C₅₃×#519	C₂	1	10	yes	maximal, complemented
5	C₂×2₋¹⁺⁴×C₅₃	C₄	1	32	yes	complemented
6	C₂×2₋¹⁺⁴×C₅₃	C₂²	1	16	yes	—
7	C₂²×#41×C₅₃	C₂²	1	10	yes	—
8	C₂³×Q₈×C₅₃	C₂²	1	5	yes	—
9	C₅₃×#520	C₂²	1	160	yes	complemented
10	C₅₃×#519	C₂²	1	40	yes	—
11	C₅₃×#519	C₂²	1	80	yes	complemented
12	C₅₃×#518	C₂²	1	40	yes	—
13	C₄×#41×C₅₃	C₂²	1	160	yes	complemented
14	C₂×C₄×Q₈×C₅₃	C₂²	1	80	yes	complemented
15	C₂×#117×C₅₃	C₂²	1	60	yes	—
16	2₋¹⁺⁴×C₅₃	C₂×C₄	1	256	yes	complemented
17	C₂×#41×C₅₃	C₂×C₄	1	160	yes	complemented
18	C₂×#41×C₅₃	C₂³	1	80	yes	—
19	C₂²×Q₈×C₅₃	C₂×C₄	1	80	yes	complemented
20	C₂²×Q₈×C₅₃	C₂³	1	40	yes	—
21	C₂²×Q₈×C₅₃	C₂³	1	10	yes	—
22	C₂²×D₄×C₅₃	C₂³	1	10	yes	—
23	C₂³×C₄×C₅₃	C₂³	1	15	yes	—
24	C₄×Q₈×C₅₃	C₂³	1	320	yes	—
25	C₄×D₄×C₅₃	C₂³	1	320	yes	—
26	#117×C₅₃	C₂³	1	80	yes	—
27	#117×C₅₃	C₂³	1	240	yes	—
28	C₂×C₄⋊C₄×C₅₃	C₂³	1	120	yes	—
29	C₂×C₂²⋊C₄×C₅₃	C₂³	1	40	yes	—
30	C₂×C₄²×C₅₃	C₂³	1	120	yes	—
31	C₂⁴×C₅₃	C₂⁴	1	5	yes	—
32	#41×C₅₃	C₂²×C₄	1	640	yes	—
33	C₂×Q₈×C₅₃	C₂²×C₄	1	80	yes	—
34	C₂×Q₈×C₅₃	C₂²×C₄	1	320	yes	—
35	C₂×Q₈×C₅₃	C₂⁴	1	40	yes	—
36	C₂×D₄×C₅₃	C₂²×C₄	1	80	yes	—
37	C₂×D₄×C₅₃	C₂⁴	1	40	yes	—
38	C₂²×C₄×C₅₃	C₂²×C₄	1	120	yes	—
39	C₂²×C₄×C₅₃	C₂⁴	1	80	yes	—
40	C₂²×C₄×C₅₃	C₂⁴	1	60	yes	—
41	C₂²×C₄×C₅₃	C₂⁴	1	10	yes	—
42	C₄⋊C₄×C₅₃	C₂⁴	1	160	yes	—
43	C₂²⋊C₄×C₅₃	C₂⁴	1	16	yes	—
44	C₂²⋊C₄×C₅₃	C₂⁴	1	80	yes	—
45	C₄²×C₅₃	C₂⁴	1	160	yes	—
46	C₂³×C₅₃	—	4	10	no	—
47	C₂³×C₅₃	C₂³×C₄	1	20	yes	—
48	C₂³×C₅₃	C₂⁵	1	10	yes	—
49	C₂³×C₅₃	C₂⁵	1	1	yes	characteristic
50	Q₈×C₅₃	C₂³×C₄	1	160	yes	—
51	D₄×C₅₃	C₂³×C₄	1	160	yes	—
52	C₂×C₄×C₅₃	—	4	80	no	—
53	C₂×C₄×C₅₃	C₂³×C₄	1	40	yes	—
54	C₂×C₄×C₅₃	C₂³×C₄	1	240	yes	—
55	C₂×C₄×C₅₃	C₂⁵	1	32	yes	—
56	C₂×C₄×C₅₃	C₂⁵	1	20	yes	—
57	#62653	C₅₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
58	C₂²×C₅₃	—	2	20	no	—
59	C₂²×C₅₃	—	2	2	no	—
60	C₂²×C₅₃	—	4	20	no	—
61	C₂²×C₅₃	C₂⁴×C₄	1	2	yes	—
62	C₂²×C₅₃	C₂⁴×C₄	1	20	yes	—
63	C₂²×C₅₃	C₂⁶	1	1	yes	characteristic
64	C₄×C₅₃	—	2	32	no	—
65	C₄×C₅₃	C₂⁴×C₄	1	40	yes	—
66	C₂²×2₋¹⁺⁴	C₂×C₅₃	1	1	yes	characteristic
67	C₄×2₋¹⁺⁴	C₂×C₅₃	1	32	yes	complemented
68	#3428	C₂×C₅₃	1	20	yes	complemented
69	C₂×#519	C₂×C₅₃	1	10	yes	complemented
70	C₂×C₅₃	—	2	2	no	—
71	C₂×C₅₃	—	2	20	no	—
72	C₂×C₅₃	C₂⁵×C₄	1	1	yes	characteristic
73	C₂×C₅₃	2₋¹⁺⁶	1	2	yes	—
74	C₂×2₋¹⁺⁴	C₄×C₅₃	1	32	yes	complemented
75	C₂×2₋¹⁺⁴	C₂²×C₅₃	1	16	yes	—
76	C₂²×#41	C₂²×C₅₃	1	10	yes	—
77	C₂³×Q₈	C₂²×C₅₃	1	5	yes	—
78	#520	C₂²×C₅₃	1	160	yes	complemented
79	#519	C₂²×C₅₃	1	40	yes	—
80	#519	C₂²×C₅₃	1	80	yes	complemented
81	#518	C₂²×C₅₃	1	40	yes	—
82	C₄×#41	C₂²×C₅₃	1	160	yes	complemented
83	C₂×C₄×Q₈	C₂²×C₅₃	1	80	yes	complemented
84	C₂×#117	C₂²×C₅₃	1	60	yes	—
85	C₅₃	#62653	1	1	yes	characteristic, direct product factor, Sylow
86	2₋¹⁺⁴	C₂×C₄×C₅₃	1	256	yes	complemented
87	C₂×#41	C₂×C₄×C₅₃	1	160	yes	complemented
88	C₂×#41	C₂³×C₅₃	1	80	yes	—
89	C₂²×Q₈	C₂×C₄×C₅₃	1	80	yes	complemented
90	C₂²×Q₈	C₂³×C₅₃	1	40	yes	—
91	C₂²×Q₈	C₂³×C₅₃	1	10	yes	—
92	C₂²×D₄	C₂³×C₅₃	1	10	yes	—
93	C₂³×C₄	C₂³×C₅₃	1	15	yes	—
94	C₄×Q₈	C₂³×C₅₃	1	320	yes	—
95	C₄×D₄	C₂³×C₅₃	1	320	yes	—
96	#117	C₂³×C₅₃	1	80	yes	—
97	#117	C₂³×C₅₃	1	240	yes	—
98	C₂×C₄⋊C₄	C₂³×C₅₃	1	120	yes	—
99	C₂×C₂²⋊C₄	C₂³×C₅₃	1	40	yes	—
100	C₂×C₄²	C₂³×C₅₃	1	120	yes	—
101	C₂⁴	C₂⁴×C₅₃	1	5	yes	—
102	#41	C₂²×C₄×C₅₃	1	640	yes	—
103	C₂×Q₈	C₂²×C₄×C₅₃	1	80	yes	—
104	C₂×Q₈	C₂²×C₄×C₅₃	1	320	yes	—
105	C₂×Q₈	C₂⁴×C₅₃	1	40	yes	—
106	C₂×D₄	C₂²×C₄×C₅₃	1	80	yes	—
107	C₂×D₄	C₂⁴×C₅₃	1	40	yes	—
108	C₂²×C₄	C₂²×C₄×C₅₃	1	120	yes	—
109	C₂²×C₄	C₂⁴×C₅₃	1	80	yes	—
110	C₂²×C₄	C₂⁴×C₅₃	1	60	yes	—
111	C₂²×C₄	C₂⁴×C₅₃	1	10	yes	—
112	C₄⋊C₄	C₂⁴×C₅₃	1	160	yes	—
113	C₂²⋊C₄	C₂⁴×C₅₃	1	16	yes	—
114	C₂²⋊C₄	C₂⁴×C₅₃	1	80	yes	—
115	C₄²	C₂⁴×C₅₃	1	160	yes	—
116	C₂³	—	4	10	no	—
117	C₂³	C₂³×C₄×C₅₃	1	20	yes	—
118	C₂³	C₂⁵×C₅₃	1	1	yes	characteristic
119	C₂³	C₂⁵×C₅₃	1	10	yes	—
120	Q₈	C₂³×C₄×C₅₃	1	160	yes	—
121	D₄	C₂³×C₄×C₅₃	1	160	yes	—
122	C₂×C₄	—	4	80	no	—
123	C₂×C₄	C₂³×C₄×C₅₃	1	40	yes	—
124	C₂×C₄	C₂³×C₄×C₅₃	1	240	yes	—
125	C₂×C₄	C₂⁵×C₅₃	1	32	yes	—
126	C₂×C₄	C₂⁵×C₅₃	1	20	yes	—
127	C₂²	—	2	20	no	—
128	C₂²	—	2	2	no	—
129	C₂²	—	4	20	no	—
130	C₂²	C₂⁴×C₄×C₅₃	1	2	yes	—
131	C₂²	C₂⁴×C₄×C₅₃	1	20	yes	—
132	C₂²	C₂⁶×C₅₃	1	1	yes	characteristic
133	C₄	—	2	32	no	—
134	C₄	C₂⁴×C₄×C₅₃	1	40	yes	—
135	C₂	—	2	2	no	—
136	C₂	—	2	20	no	—
137	C₂	C₂⁵×C₄×C₅₃	1	1	yes	characteristic
138	C₂	C₅₃×2₋¹⁺⁶	1	2	yes	—

	subgroup	G/H
center, socle	C₂²×C₅₃	C₂⁶
commutator	C₂	C₂⁵×C₄×C₅₃
Frattini, cosocle	C₂²	C₂⁶×C₅₃
prime radical	C₂²×2₋¹⁺⁴×C₅₃	C₂
LCS	G ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₅₃ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₄×2₋¹⁺⁴×C₅₃	C₂	semidirect
3. C₅₃×#3428	C₂	semidirect
4. C₂×C₅₃×#519	C₂	semidirect
5. C₂×2₋¹⁺⁴×C₅₃	C₄	semidirect
9. C₅₃×#520	C₂²	semidirect
11. C₅₃×#519	C₂²	semidirect
13. C₄×#41×C₅₃	C₂²	semidirect
14. C₂×C₄×Q₈×C₅₃	C₂²	semidirect
16. 2₋¹⁺⁴×C₅₃	C₂×C₄	semidirect
17. C₂×#41×C₅₃	C₂×C₄	semidirect
19. C₂²×Q₈×C₅₃	C₂×C₄	semidirect
57. #62653	C₅₃	direct (N × H)
67. C₄×2₋¹⁺⁴	C₂×C₅₃	semidirect
68. #3428	C₂×C₅₃	semidirect
69. C₂×#519	C₂×C₅₃	semidirect
74. C₂×2₋¹⁺⁴	C₄×C₅₃	semidirect
78. #520	C₂²×C₅₃	semidirect
80. #519	C₂²×C₅₃	semidirect
82. C₄×#41	C₂²×C₅₃	semidirect
83. C₂×C₄×Q₈	C₂²×C₅₃	semidirect
85. C₅₃	#62653	direct (N × H)
86. 2₋¹⁺⁴	C₂×C₄×C₅₃	semidirect
87. C₂×#41	C₂×C₄×C₅₃	semidirect
89. C₂²×Q₈	C₂×C₄×C₅₃	semidirect

Maximal subgroups

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

subgroup	index
C₂²×2₋¹⁺⁴×C₅₃	2
C₄×2₋¹⁺⁴×C₅₃	2
C₅₃×#3428	2
C₂×C₅₃×#519	2
#62653	53

Subgroups — C53×#62653 #27270165

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅₃×#62653 #27270165