Group #21426002 — subgroups

Subgroups — #21426002

Order 7552.

126 conjugacy classes of subgroups; only the 71 normal subgroups are shown.

normal subgroups only

This group has 126 conjugacy classes of subgroups — too many to draw. Only its 71 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₈×Dic₁₁₈	C₂	1	1	yes	maximal, characteristic
2	C₄⋊(C₅₉⋊C₁₆)	C₂	1	1	yes	maximal, characteristic
3	Dic₅₉⋊C₁₆	C₂	1	1	yes	maximal, characteristic
4	Dic₅₉⋊C₁₆	C₂	1	1	yes	maximal, characteristic
5	#16837177	C₂	1	1	yes	maximal, characteristic
6	C₁₆×Dic₅₉	C₂	1	1	yes	maximal, characteristic
7	C₅₉×#346	C₂	1	1	yes	maximal, characteristic
8	C₄×Dic₁₁₈	C₄	1	1	yes	characteristic
9	C₂×C₅₉⋊C₁₆	C₂²	1	1	yes	characteristic
10	C₂×C₅₉⋊C₁₆	C₂²	1	1	yes	characteristic
11	C₄⋊(C₅₉⋊C₈)	C₄	1	1	yes	characteristic
12	Dic₅₉⋊C₈	C₄	1	1	yes	characteristic
13	Dic₅₉⋊C₈	C₄	1	1	yes	characteristic
14	C₈×Dic₅₉	C₂²	1	1	yes	characteristic
15	C₈×Dic₅₉	C₂²	1	1	yes	characteristic
16	C₂×C₁₆×C₅₉	C₂²	1	1	yes	characteristic
17	C₂×C₁₆×C₅₉	C₂²	1	1	yes	characteristic
18	C₄×C₈×C₅₉	C₂²	1	1	yes	characteristic
19	C₂×Dic₁₁₈	C₈	1	1	yes	characteristic
20	C₂×C₈×C₅₉	C₂×C₄	1	1	yes	characteristic
21	C₂×C₈×C₅₉	C₂³	1	1	yes	characteristic
22	C₄²×C₅₉	C₂×C₄	1	1	yes	characteristic
23	C₄⋊Dic₅₉	C₈	1	1	yes	characteristic
24	Dic₅₉⋊C₄	C₈	1	1	yes	characteristic
25	Dic₅₉⋊C₄	C₈	1	1	yes	characteristic
26	C₄×Dic₅₉	C₂×C₄	1	1	yes	characteristic
27	C₄×Dic₅₉	C₂×C₄	1	1	yes	characteristic
28	C₂×C₅₉⋊C₈	C₂×C₄	1	1	yes	characteristic
29	C₂×C₅₉⋊C₈	C₂×C₄	1	1	yes	characteristic
30	C₁₆×C₅₉	—	2	1	no	—
31	C₁₆×C₅₉	Q₈	1	2	yes	complemented
32	C₅₉⋊C₁₆	—	2	1	no	—
33	C₅₉⋊C₁₆	—	2	1	no	—
34	C₂×C₄×C₅₉	C₂×C₈	1	1	yes	characteristic
35	C₂×C₄×C₅₉	C₂×C₈	1	1	yes	characteristic
36	C₂×C₄×C₅₉	C₂²×C₄	1	1	yes	characteristic
37	C₂×Dic₅₉	C₂×C₈	1	1	yes	characteristic
38	C₂×Dic₅₉	C₂×C₈	1	1	yes	characteristic
39	C₂×Dic₅₉	C₂×C₈	1	1	yes	characteristic
40	C₂×Dic₅₉	C₂×C₈	1	1	yes	characteristic
41	Dic₁₁₈	—	2	2	no	—
42	C₈×C₅₉	—	2	1	no	—
43	C₈×C₅₉	C₂×Q₈	1	1	yes	characteristic
44	C₈×C₅₉	#41	1	1	yes	characteristic
45	C₅₉⋊C₈	—	2	1	no	—
46	C₅₉⋊C₈	—	2	1	no	—
47	C₂²×C₅₉	C₂²×C₈	1	1	yes	characteristic
48	C₄×C₅₉	—	2	1	no	—
49	C₄×C₅₉	—	2	1	no	—
50	C₄×C₅₉	C₄×Q₈	1	1	yes	characteristic
51	C₄×C₅₉	#131	1	1	yes	characteristic
52	Dic₅₉	—	2	1	no	—
53	Dic₅₉	—	2	1	no	—
54	Dic₅₉	—	2	1	no	—
55	Dic₅₉	M₃₂	1	2	yes	—
56	#2183	—	59	1	no	maximal, Sylow
57	C₂×C₅₉	C₈×Q₈	1	1	yes	characteristic
58	C₂×C₅₉	C₂×M₃₂	1	1	yes	characteristic
59	C₂×C₅₉	#504	1	1	yes	characteristic
60	C₈×Q₈	—	59	1	no	—
61	C₄⋊C₁₆	—	59	1	no	—
62	C₄⋊C₁₆	—	59	1	no	—
63	C₄⋊C₁₆	—	59	1	no	—
64	#346	—	59	1	no	—
65	#346	D₅₉	1	1	yes	characteristic
66	C₄×C₁₆	—	59	1	no	—
67	C₅₉	#2183	1	1	yes	characteristic, complemented, Sylow
68	C₄×Q₈	—	59	1	no	—
69	C₂×C₁₆	—	59	1	no	—
70	C₂×C₁₆	—	59	1	no	—
71	C₂×C₁₆	C₂×D₅₉	1	1	yes	characteristic
72	C₂×C₁₆	C₂×D₅₉	1	1	yes	characteristic
73	C₄⋊C₈	—	59	1	no	—
74	C₄⋊C₈	—	59	1	no	—
75	C₄⋊C₈	—	59	1	no	—
76	C₄×C₈	—	59	1	no	—
77	C₄×C₈	—	59	1	no	—
78	C₄×C₈	C₂×D₅₉	1	1	yes	characteristic
79	C₂×Q₈	—	59	1	no	—
80	C₂×C₈	—	59	1	no	—
81	C₂×C₈	—	59	1	no	—
82	C₂×C₈	C₄×D₅₉	1	1	yes	characteristic
83	C₂×C₈	C₂²×D₅₉	1	1	yes	characteristic
84	C₄⋊C₄	—	59	1	no	—
85	C₄⋊C₄	—	59	1	no	—
86	C₄⋊C₄	—	59	1	no	—
87	C₄²	—	59	1	no	—
88	C₄²	—	59	1	no	—
89	C₄²	C₄×D₅₉	1	1	yes	characteristic
90	C₁₆	—	2	1	no	—
91	C₁₆	—	118	1	no	—
92	C₁₆	—	118	1	no	—
93	C₁₆	Dic₁₁₈	1	2	yes	complemented
94	Q₈	—	118	2	no	—
95	C₂×C₄	—	59	1	no	—
96	C₂×C₄	—	59	1	no	—
97	C₂×C₄	—	59	1	no	—
98	C₂×C₄	—	59	1	no	—
99	C₂×C₄	C₈×D₅₉	1	1	yes	characteristic
100	C₂×C₄	C₈×D₅₉	1	1	yes	characteristic
101	C₂×C₄	C₂×C₄×D₅₉	1	1	yes	characteristic
102	C₈	—	2	1	no	—
103	C₈	—	118	1	no	—
104	C₈	—	118	1	no	—
105	C₈	C₂×Dic₁₁₈	1	1	yes	characteristic
106	C₈	#11745685	1	1	yes	characteristic
107	C₂²	C₂×C₈×D₅₉	1	1	yes	characteristic
108	C₄	—	2	1	no	—
109	C₄	—	2	1	no	—
110	C₄	—	59	2	no	—
111	C₄	—	118	1	no	—
112	C₄	—	118	1	no	—
113	C₄	—	118	1	no	—
114	C₄	C₄×Dic₁₁₈	1	1	yes	characteristic
115	C₄	#14273733	1	1	yes	characteristic
116	C₂	C₈×Dic₁₁₈	1	1	yes	characteristic
117	C₂	M₃₂×D₅₉	1	1	yes	characteristic
118	C₂	#16837820	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₈×Dic₁₁₈

C₂

yes

maximal, characteristic

C₄⋊(C₅₉⋊C₁₆)

C₂

yes

maximal, characteristic

Dic₅₉⋊C₁₆

C₂

yes

maximal, characteristic

Dic₅₉⋊C₁₆

C₂

yes

maximal, characteristic

#16837177

C₂

yes

maximal, characteristic

C₁₆×Dic₅₉

C₂

yes

maximal, characteristic

C₅₉×#346

C₂

yes

maximal, characteristic

C₄×Dic₁₁₈

C₄

yes

characteristic

C₂×C₅₉⋊C₁₆

C₂²

yes

characteristic

C₂×C₅₉⋊C₁₆

C₂²

yes

characteristic

C₄⋊(C₅₉⋊C₈)

C₄

yes

characteristic

Dic₅₉⋊C₈

C₄

yes

characteristic

Dic₅₉⋊C₈

C₄

yes

characteristic

C₈×Dic₅₉

C₂²

yes

characteristic

C₈×Dic₅₉

C₂²

yes

characteristic

C₂×C₁₆×C₅₉

C₂²

yes

characteristic

C₂×C₁₆×C₅₉

C₂²

yes

characteristic

C₄×C₈×C₅₉

C₂²

yes

characteristic

C₂×Dic₁₁₈

C₈

yes

characteristic

C₂×C₈×C₅₉

C₂×C₄

yes

characteristic

C₂×C₈×C₅₉

C₂³

yes

characteristic

C₄²×C₅₉

C₂×C₄

yes

characteristic

C₄⋊Dic₅₉

C₈

yes

characteristic

Dic₅₉⋊C₄

C₈

yes

characteristic

Dic₅₉⋊C₄

C₈

yes

characteristic

C₄×Dic₅₉

C₂×C₄

yes

characteristic

C₄×Dic₅₉

C₂×C₄

yes

characteristic

C₂×C₅₉⋊C₈

C₂×C₄

yes

characteristic

C₂×C₅₉⋊C₈

C₂×C₄

yes

characteristic

C₁₆×C₅₉

—

C₁₆×C₅₉

Q₈

yes

complemented

C₅₉⋊C₁₆

—

C₅₉⋊C₁₆

—

C₂×C₄×C₅₉

C₂×C₈

yes

characteristic

C₂×C₄×C₅₉

C₂×C₈

yes

characteristic

C₂×C₄×C₅₉

C₂²×C₄

yes

characteristic

C₂×Dic₅₉

C₂×C₈

yes

characteristic

C₂×Dic₅₉

C₂×C₈

yes

characteristic

C₂×Dic₅₉

C₂×C₈

yes

characteristic

C₂×Dic₅₉

C₂×C₈

yes

characteristic

Dic₁₁₈

—

C₈×C₅₉

—

C₈×C₅₉

C₂×Q₈

yes

characteristic

C₈×C₅₉

#41

yes

characteristic

C₅₉⋊C₈

—

C₅₉⋊C₈

—

C₂²×C₅₉

C₂²×C₈

yes

characteristic

C₄×C₅₉

—

C₄×C₅₉

—

C₄×C₅₉

C₄×Q₈

yes

characteristic

C₄×C₅₉

#131

yes

characteristic

Dic₅₉

—

Dic₅₉

—

Dic₅₉

—

Dic₅₉

M₃₂

yes

—

#2183

—

maximal, Sylow

C₂×C₅₉

C₈×Q₈

yes

characteristic

C₂×C₅₉

C₂×M₃₂

yes

characteristic

C₂×C₅₉

#504

yes

characteristic

C₈×Q₈

—

C₄⋊C₁₆

—

C₄⋊C₁₆

—

C₄⋊C₁₆

—

#346

—

#346

D₅₉

yes

characteristic

C₄×C₁₆

—

C₅₉

#2183

yes

characteristic, complemented, Sylow

C₄×Q₈

—

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

C₂×D₅₉

yes

characteristic

C₂×C₁₆

C₂×D₅₉

yes

characteristic

C₄⋊C₈

—

C₄⋊C₈

—

C₄⋊C₈

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

C₂×D₅₉

yes

characteristic

C₂×Q₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

C₄×D₅₉

yes

characteristic

C₂×C₈

C₂²×D₅₉

yes

characteristic

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄²

—

C₄²

—

C₄²

C₄×D₅₉

yes

characteristic

C₁₆

—

C₁₆

—

118

—

C₁₆

—

118

—

C₁₆

Dic₁₁₈

yes

complemented

Q₈

—

118

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

C₈×D₅₉

yes

characteristic

100

C₂×C₄

C₈×D₅₉

yes

characteristic

101

C₂×C₄

C₂×C₄×D₅₉

yes

characteristic

102

C₈

—

103

C₈

—

118

—

104

C₈

—

118

—

105

C₈

C₂×Dic₁₁₈

yes

characteristic

106

C₈

#11745685

yes

characteristic

107

C₂²

C₂×C₈×D₅₉

yes

characteristic

108

C₄

—

109

C₄

—

110

C₄

—

111

C₄

—

118

—

112

C₄

—

118

—

113

C₄

—

118

—

114

C₄

C₄×Dic₁₁₈

yes

characteristic

115

C₄

#14273733

yes

characteristic

116

C₂

C₈×Dic₁₁₈

yes

characteristic

117

C₂

M₃₂×D₅₉

yes

characteristic

118

C₂

#16837820

yes

characteristic

	subgroup	G/H
center, Frattini	C₂×C₈	C₂²×D₅₉
commutator, socle	C₂²×C₅₉	C₂²×C₈
Fitting	C₅₉×#346	C₂
hypercenter	#346	D₅₉
hypocenter	C₅₉	#2183
LCS	G ▷ C₂²×C₅₉ ▷ C₅₉
UCS	C₁ ◁ C₂×C₈ ◁ #346

subgroup

G/H

center, Frattini

C₂×C₈

C₂²×D₅₉

commutator, socle

C₂²×C₅₉

C₂²×C₈

Fitting

C₅₉×#346

C₂

D₅₉

C₅₉

G ▷ C₂²×C₅₉ ▷ C₅₉

C₁ ◁ C₂×C₈ ◁ #346

N	H ≅ G/N	type
31. C₁₆×C₅₉	Q₈	semidirect
67. C₅₉	#2183	semidirect
93. C₁₆	Dic₁₁₈	semidirect

H ≅ G/N

type

31. C₁₆×C₅₉

Q₈

semidirect

67. C₅₉

#2183

semidirect

93. C₁₆

Dic₁₁₈

semidirect

Subgroups — #21426002

Conjugacy classes of subgroups

Complements