Subgroups — #28290715

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	2₋¹⁺⁴×Dic₅₃	C₂	1	32	yes	maximal, complemented
2	#20287918	C₂	1	20	yes	maximal, complemented
3	C₂×#16811101	C₂	1	10	yes	maximal, complemented
4	C₂²×2₋¹⁺⁴×C₅₃	C₂	1	1	yes	maximal, characteristic
5	#16811106	C₂²	1	160	yes	complemented
6	#16811101	C₂²	1	40	yes	—
7	#16811101	C₂²	1	80	yes	complemented
8	#16811098	C₂²	1	40	yes	—
9	#41×Dic₅₃	C₂²	1	160	yes	complemented
10	C₂×Q₈×Dic₅₃	C₂²	1	80	yes	complemented
11	C₂×#13133176	C₂²	1	60	yes	—
12	C₂×2₋¹⁺⁴×C₅₃	C₄	1	32	yes	complemented
13	C₂×2₋¹⁺⁴×C₅₃	C₂²	1	16	yes	—
14	C₂²×#41×C₅₃	C₂²	1	10	yes	—
15	C₂³×Q₈×C₅₃	C₂²	1	5	yes	—
16	Q₈×Dic₅₃	C₂³	1	320	yes	—
17	D₄×Dic₅₃	C₂³	1	320	yes	—
18	#13133176	C₂³	1	80	yes	—
19	#13133176	C₂³	1	240	yes	—
20	C₂×C₄⋊Dic₅₃	C₂³	1	120	yes	—
21	C₂×C₂²⋊Dic₅₃	C₂³	1	40	yes	—
22	C₂×C₄×Dic₅₃	C₂³	1	120	yes	—
23	2₋¹⁺⁴×C₅₃	C₂×C₄	1	256	yes	complemented
24	C₂×#41×C₅₃	C₂×C₄	1	160	yes	complemented
25	C₂×#41×C₅₃	C₂³	1	80	yes	—
26	C₂²×Q₈×C₅₃	C₂×C₄	1	80	yes	complemented
27	C₂²×Q₈×C₅₃	C₂³	1	40	yes	—
28	C₂²×Q₈×C₅₃	C₂³	1	10	yes	—
29	C₂²×D₄×C₅₃	C₂³	1	10	yes	—
30	C₂³×C₄×C₅₃	C₂³	1	15	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	10	yes	—
40	C₂²×C₄×C₅₃	C₂⁴	1	60	yes	—
41	C₂²×Dic₅₃	C₂⁴	1	80	yes	—
42	C₂²⋊Dic₅₃	C₂⁴	1	16	yes	—
43	C₂²⋊Dic₅₃	C₂⁴	1	80	yes	—
44	C₄⋊Dic₅₃	C₂⁴	1	160	yes	—
45	C₄×Dic₅₃	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	C₂³×C₄	1	40	yes	—
53	C₂×C₄×C₅₃	C₂³×C₄	1	240	yes	—
54	C₂×C₄×C₅₃	C₂⁵	1	20	yes	—
55	C₂×Dic₅₃	—	4	80	no	—
56	C₂×Dic₅₃	C₂⁵	1	32	yes	—
57	#62653	—	53	1	no	maximal, 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	C₂⁴×C₄	1	40	yes	—
65	Dic₅₃	—	2	32	no	—
66	C₂²×2₋¹⁺⁴	D₅₃	1	1	yes	characteristic
67	C₄×2₋¹⁺⁴	—	53	32	no	—
68	#3428	—	53	20	no	—
69	C₂×#519	—	53	10	no	—
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	Dic₅₃	1	32	yes	complemented
75	C₂×2₋¹⁺⁴	C₂×D₅₃	1	16	yes	—
76	C₂²×#41	C₂×D₅₃	1	10	yes	—
77	C₂³×Q₈	C₂×D₅₃	1	5	yes	—
78	#520	—	53	160	no	—
79	#519	—	53	40	no	—
80	#519	—	53	80	no	—
81	#518	—	53	40	no	—
82	C₄×#41	—	53	160	no	—
83	C₂×C₄×Q₈	—	53	80	no	—
84	C₂×#117	—	53	60	no	—
85	C₅₃	#62653	1	1	yes	characteristic, complemented, Sylow
86	2₋¹⁺⁴	C₂×Dic₅₃	1	256	yes	complemented
87	C₂×#41	C₂×Dic₅₃	1	160	yes	complemented
88	C₂×#41	C₂²×D₅₃	1	80	yes	—
89	C₂²×Q₈	C₂×Dic₅₃	1	80	yes	complemented
90	C₂²×Q₈	C₂²×D₅₃	1	40	yes	—
91	C₂²×Q₈	C₂²×D₅₃	1	10	yes	—
92	C₂²×D₄	C₂²×D₅₃	1	10	yes	—
93	C₂³×C₄	C₂²×D₅₃	1	15	yes	—
94	C₄×Q₈	—	53	320	no	—
95	C₄×D₄	—	53	320	no	—
96	#117	—	53	80	no	—
97	#117	—	53	240	no	—
98	C₂×C₄⋊C₄	—	53	120	no	—
99	C₂×C₂²⋊C₄	—	53	40	no	—
100	C₂×C₄²	—	53	120	no	—
101	C₂⁴	C₂³×D₅₃	1	5	yes	—
102	#41	C₂²×Dic₅₃	1	640	yes	—
103	C₂×Q₈	C₂²×Dic₅₃	1	80	yes	—
104	C₂×Q₈	C₂²×Dic₅₃	1	320	yes	—
105	C₂×Q₈	C₂³×D₅₃	1	40	yes	—
106	C₂×D₄	C₂²×Dic₅₃	1	80	yes	—
107	C₂×D₄	C₂³×D₅₃	1	40	yes	—
108	C₂²×C₄	—	53	80	no	—
109	C₂²×C₄	C₂²×Dic₅₃	1	120	yes	—
110	C₂²×C₄	C₂³×D₅₃	1	60	yes	—
111	C₂²×C₄	C₂³×D₅₃	1	10	yes	—
112	C₄⋊C₄	—	53	160	no	—
113	C₂²⋊C₄	—	53	16	no	—
114	C₂²⋊C₄	—	53	80	no	—
115	C₄²	—	53	160	no	—
116	C₂³	—	4	10	no	—
117	C₂³	C₂³×Dic₅₃	1	20	yes	—
118	C₂³	C₂⁴×D₅₃	1	1	yes	characteristic
119	C₂³	C₂⁴×D₅₃	1	10	yes	—
120	Q₈	C₂³×Dic₅₃	1	160	yes	—
121	D₄	C₂³×Dic₅₃	1	160	yes	—
122	C₂×C₄	—	53	32	no	—
123	C₂×C₄	—	212	80	no	—
124	C₂×C₄	C₂³×Dic₅₃	1	40	yes	—
125	C₂×C₄	C₂³×Dic₅₃	1	240	yes	—
126	C₂×C₄	C₂⁴×D₅₃	1	20	yes	—
127	C₂²	—	2	20	no	—
128	C₂²	—	2	2	no	—
129	C₂²	—	4	20	no	—
130	C₂²	C₂⁴×Dic₅₃	1	2	yes	—
131	C₂²	C₂⁴×Dic₅₃	1	20	yes	—
132	C₂²	C₂⁵×D₅₃	1	1	yes	characteristic
133	C₄	—	106	32	no	—
134	C₄	C₂⁴×Dic₅₃	1	40	yes	—
135	C₂	—	2	2	no	—
136	C₂	—	2	20	no	—
137	C₂	C₂⁵×Dic₅₃	1	1	yes	characteristic
138	C₂	#20289527	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂²	C₂⁵×D₅₃
commutator	C₂×C₅₃	C₂⁵×C₄
Fitting, prime radical	C₂²×2₋¹⁺⁴×C₅₃	C₂
socle, cosocle	C₂²×C₅₃	C₂⁶
hypercenter	C₂²×2₋¹⁺⁴	D₅₃
hypocenter	C₅₃	#62653
LCS	G ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂² ◁ C₂²×2₋¹⁺⁴

Complements

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

N	H ≅ G/N	type
1. 2₋¹⁺⁴×Dic₅₃	C₂	semidirect
2. #20287918	C₂	semidirect
3. C₂×#16811101	C₂	semidirect
5. #16811106	C₂²	semidirect
7. #16811101	C₂²	semidirect
9. #41×Dic₅₃	C₂²	semidirect
10. C₂×Q₈×Dic₅₃	C₂²	semidirect
12. C₂×2₋¹⁺⁴×C₅₃	C₄	semidirect
23. 2₋¹⁺⁴×C₅₃	C₂×C₄	semidirect
24. C₂×#41×C₅₃	C₂×C₄	semidirect
26. C₂²×Q₈×C₅₃	C₂×C₄	semidirect
74. C₂×2₋¹⁺⁴	Dic₅₃	semidirect
85. C₅₃	#62653	semidirect
86. 2₋¹⁺⁴	C₂×Dic₅₃	semidirect
87. C₂×#41	C₂×Dic₅₃	semidirect
89. C₂²×Q₈	C₂×Dic₅₃	semidirect

Maximal subgroups

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

subgroup	index
2₋¹⁺⁴×Dic₅₃	2
#20287918	2
C₂×#16811101	2
C₂²×2₋¹⁺⁴×C₅₃	2
#62653	53