Subgroups — #27300526

Order 13568.

374 conjugacy classes of subgroups.

The subgroup lattice has 142 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#16810449	C₂	1	4	yes	maximal
2	#20274875	C₂	1	2	yes	maximal
3	C₂×C₄×C₁₆×C₅₃	C₂	1	1	yes	maximal, characteristic
4	C₂×#13133151	—	2	2	no	—
5	C₂×#13133151	C₂²	1	4	yes	—
6	#16810493	—	2	2	no	—
7	#16810449	—	2	8	no	—
8	C₂²×C₁₆×C₅₃	C₂²	1	2	yes	—
9	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
10	C₄×C₁₆×C₅₃	C₄	1	4	yes	complemented
11	C₂×C₄⋊Dic₅₃	—	2	4	no	—
12	C₂×C₄⋊Dic₅₃	—	4	4	no	—
13	#13133151	—	2	8	no	—
14	#13133151	—	4	4	no	—
15	C₂²×C₈×C₅₃	D₄	1	1	yes	characteristic
16	C₂²×C₈×C₅₃	C₂³	1	1	yes	characteristic
17	C₂×C₄²×C₅₃	D₄	1	1	yes	characteristic
18	C₂×C₁₆×C₅₃	C₂×C₄	1	4	yes	—
19	C₂×C₁₆×C₅₃	D₄	1	4	yes	—
20	C₂×C₁₆×C₅₃	Q₈	1	4	yes	—
21	C₄×C₈×C₅₃	C₂×C₄	1	2	yes	—
22	C₄×C₈×C₅₃	Q₈	1	2	yes	—
23	C₂²×C₄×C₅₃	D₈	1	2	yes	—
24	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
25	C₂²×Dic₅₃	—	4	4	no	—
26	C₂×C₈×C₅₃	C₄⋊C₄	1	2	yes	—
27	C₂×C₈×C₅₃	D₈	1	2	yes	—
28	C₂×C₈×C₅₃	Q₁₆	1	2	yes	—
29	C₂×C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
30	C₂×C₈×C₅₃	C₂×D₄	1	2	yes	—
31	C₂×C₈×C₅₃	C₂×Q₈	1	2	yes	—
32	C₂×C₈×C₅₃	C₂×Q₈	1	1	yes	characteristic
33	C₄²×C₅₃	C₄⋊C₄	1	2	yes	—
34	C₄²×C₅₃	Q₁₆	1	2	yes	—
35	C₄⋊Dic₅₃	—	4	4	no	—
36	C₄⋊Dic₅₃	—	4	4	no	—
37	C₄⋊Dic₅₃	—	8	8	no	—
38	C₁₆×C₅₃	C₄⋊C₄	1	8	yes	complemented
39	C₂³×C₅₃	C₂×D₈	1	1	yes	characteristic
40	C₂×C₄×C₅₃	#107	1	4	yes	—
41	C₂×C₄×C₅₃	D₁₆	1	4	yes	—
42	C₂×C₄×C₅₃	Q₃₂	1	4	yes	—
43	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
44	C₂×C₄×C₅₃	#127	1	1	yes	characteristic
45	C₂×C₄×C₅₃	C₄⋊Q₈	1	1	yes	characteristic
46	C₂×C₄×C₅₃	C₂×D₈	1	1	yes	characteristic
47	C₂×C₄×C₅₃	C₂×Q₁₆	1	1	yes	characteristic
48	C₂×C₄×C₅₃	C₂×Q₁₆	1	1	yes	characteristic
49	C₂×Dic₅₃	—	8	4	no	—
50	C₂×Dic₅₃	—	8	8	no	—
51	C₈×C₅₃	#107	1	4	yes	—
52	C₈×C₅₃	C₂×C₄⋊C₄	1	2	yes	—
53	C₈×C₅₃	C₄⋊Q₈	1	2	yes	—
54	#12905	—	53	1	no	maximal, Sylow
55	C₂²×C₅₃	C₂×#107	1	1	yes	characteristic
56	C₂²×C₅₃	#493	1	1	yes	characteristic
57	C₂²×C₅₃	#500	1	1	yes	characteristic
58	C₂²×C₅₃	C₂×D₁₆	1	2	yes	—
59	C₂²×C₅₃	C₂×Q₃₂	1	2	yes	—
60	C₄×C₅₃	#366	1	8	yes	complemented
61	C₄×C₅₃	#384	1	1	yes	characteristic
62	C₄×C₅₃	C₂×#107	1	1	yes	characteristic
63	C₄×C₅₃	#494	1	1	yes	characteristic
64	C₄×C₅₃	#500	1	1	yes	characteristic
65	Dic₅₃	—	16	4	no	—
66	C₂×#366	—	53	4	no	—
67	C₂×C₄×C₁₆	D₅₃	1	1	yes	characteristic
68	#1845	—	53	2	no	—
69	C₂×C₅₃	#1845	1	1	yes	characteristic
70	C₂×C₅₃	C₂×#366	1	2	yes	—
71	C₂×C₅₃	#2246	1	1	yes	characteristic
72	C₂×C₅₃	#2247	1	1	yes	characteristic
73	C₂×C₅₃	#2252	1	2	yes	—
74	C₂²×C₁₆	C₂×D₅₃	1	2	yes	—
75	C₂×#107	—	53	4	no	—
76	C₂×#107	—	106	2	no	—
77	C₂×C₄×C₈	C₂×D₅₃	1	1	yes	characteristic
78	#384	—	106	2	no	—
79	#366	—	106	8	no	—
80	C₄×C₁₆	Dic₅₃	1	4	yes	complemented
81	C₅₃	#12905	1	1	yes	characteristic, complemented, Sylow
82	C₂²×C₈	D₂₁₂	1	1	yes	characteristic
83	C₂²×C₈	C₂²×D₅₃	1	1	yes	characteristic
84	C₂×C₄⋊C₄	—	106	4	no	—
85	C₂×C₄⋊C₄	—	212	4	no	—
86	C₂×C₄²	D₂₁₂	1	1	yes	characteristic
87	C₂×C₁₆	Dic₁₀₆	1	4	yes	—
88	C₂×C₁₆	D₂₁₂	1	4	yes	—
89	C₂×C₁₆	C₂×Dic₅₃	1	4	yes	—
90	#107	—	106	8	no	—
91	#107	—	212	4	no	—
92	C₄×C₈	Dic₁₀₆	1	2	yes	—
93	C₄×C₈	C₂×Dic₅₃	1	2	yes	—
94	C₂²×C₄	—	212	4	no	—
95	C₂²×C₄	D₄₂₄	1	2	yes	—
96	C₂²×C₄	C₂×D₂₁₂	1	1	yes	characteristic
97	C₂×C₈	D₄₂₄	1	2	yes	—
98	C₂×C₈	Dic₂₁₂	1	2	yes	—
99	C₂×C₈	C₄⋊Dic₅₃	1	2	yes	—
100	C₂×C₈	C₂×Dic₁₀₆	1	2	yes	—
101	C₂×C₈	C₂×Dic₁₀₆	1	1	yes	characteristic
102	C₂×C₈	C₂×D₂₁₂	1	2	yes	—
103	C₂×C₈	C₂²×Dic₅₃	1	1	yes	characteristic
104	C₄⋊C₄	—	424	8	no	—
105	C₄⋊C₄	—	212	4	no	—
106	C₄⋊C₄	—	212	4	no	—
107	C₄²	Dic₂₁₂	1	2	yes	—
108	C₄²	C₄⋊Dic₅₃	1	2	yes	—
109	C₁₆	C₄⋊Dic₅₃	1	8	yes	complemented
110	C₂³	C₂×D₄₂₄	1	1	yes	characteristic
111	C₂×C₄	—	424	8	no	—
112	C₂×C₄	—	424	4	no	—
113	C₂×C₄	#13133151	1	4	yes	—
114	C₂×C₄	D₈₄₈	1	4	yes	—
115	C₂×C₄	Dic₄₂₄	1	4	yes	—
116	C₂×C₄	C₂×C₄⋊Dic₅₃	1	1	yes	characteristic
117	C₂×C₄	#13133214	1	1	yes	characteristic
118	C₂×C₄	#13133217	1	1	yes	characteristic
119	C₂×C₄	C₂×D₄₂₄	1	1	yes	characteristic
120	C₂×C₄	C₂×Dic₂₁₂	1	1	yes	characteristic
121	C₂×C₄	C₂×Dic₂₁₂	1	1	yes	characteristic
122	C₈	#13133151	1	4	yes	—
123	C₈	C₂×C₄⋊Dic₅₃	1	2	yes	—
124	C₈	#13133217	1	2	yes	—
125	C₂²	C₂×#13133151	1	1	yes	characteristic
126	C₂²	#16811000	1	1	yes	characteristic
127	C₂²	#16811028	1	1	yes	characteristic
128	C₂²	C₂×D₈₄₈	1	2	yes	—
129	C₂²	C₂×Dic₄₂₄	1	2	yes	—
130	C₄	—	848	4	no	—
131	C₄	#16810449	1	8	yes	complemented
132	C₄	#16810493	1	1	yes	characteristic
133	C₄	C₂×#13133151	1	1	yes	characteristic
134	C₄	#16811003	1	1	yes	characteristic
135	C₄	#16811028	1	1	yes	characteristic
136	C₂	#20274875	1	1	yes	characteristic
137	C₂	C₂×#16810449	1	2	yes	—
138	C₂	#20276965	1	1	yes	characteristic
139	C₂	#20276968	1	1	yes	characteristic
140	C₂	#20276990	1	2	yes	—

	subgroup	G/H
center	C₂³	C₂×D₄₂₄
commutator	C₂×C₈×C₅₃	C₂²×C₄
Frattini	C₂²×C₈	C₂²×D₅₃
Fitting	C₂×C₄×C₁₆×C₅₃	C₂
socle, prime radical	C₂³×C₅₃	C₂×D₈
cosocle	C₂²×C₈×C₅₃	C₂³
hypercenter	C₂×C₄×C₁₆	D₅₃
hypocenter	C₅₃	#12905
LCS	G ▷ C₂×C₈×C₅₃ ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ C₂×C₄×C₁₆

Complements

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

N	H ≅ G/N	type
10. C₄×C₁₆×C₅₃	C₄	semidirect
38. C₁₆×C₅₃	C₄⋊C₄	semidirect
60. C₄×C₅₃	#366	semidirect
80. C₄×C₁₆	Dic₅₃	semidirect
81. C₅₃	#12905	semidirect
109. C₁₆	C₄⋊Dic₅₃	semidirect
131. C₄	#16810449	semidirect

Maximal subgroups

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

subgroup	index
C₂×#16810449	2
#20274875	2
C₂×C₄×C₁₆×C₅₃	2
#12905	53