Group #26137251 — subgroups

Subgroups — #26137251

Order 12032.

182 conjugacy classes of subgroups; only the 75 normal subgroups are shown.

normal subgroups only

This group has 182 conjugacy classes of subgroups — too many to draw. Only its 75 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	#20245663	C₂	1	1	yes	maximal, characteristic
2	#20245641	C₂	1	1	yes	maximal, characteristic
3	#20241570	C₂	1	4	yes	maximal
4	C₄×C₃₂×C₄₇	C₂	1	1	yes	maximal, characteristic
5	C₂×Dic₃₇₆	—	2	1	no	—
6	C₂×Dic₃₇₆	C₂²	1	2	yes	—
7	#15689528	—	2	1	no	—
8	#15689503	—	2	1	no	—
9	#15688949	—	2	1	no	—
10	#15688949	C₂²	1	2	yes	—
11	C₂×C₃₂×C₄₇	C₂²	1	2	yes	—
12	C₄×C₁₆×C₄₇	C₂²	1	1	yes	characteristic
13	C₂×Dic₁₈₈	—	2	2	no	—
14	C₂×Dic₁₈₈	—	4	1	no	—
15	#13094024	—	4	1	no	—
16	#13094024	—	4	1	no	—
17	Dic₃₇₆	—	2	4	no	—
18	Dic₃₇₆	—	4	2	no	—
19	#13093958	—	2	2	no	—
20	#13093958	—	4	1	no	—
21	C₂×C₁₆×C₄₇	D₄	1	1	yes	characteristic
22	C₂×C₁₆×C₄₇	C₂³	1	1	yes	characteristic
23	C₄×C₈×C₄₇	D₄	1	1	yes	characteristic
24	C₃₂×C₄₇	—	2	2	no	—
25	C₂×Dic₉₄	—	4	2	no	—
26	C₂×Dic₉₄	—	8	1	no	—
27	C₂×C₈×C₄₇	D₈	1	1	yes	characteristic
28	C₂×C₈×C₄₇	C₂×D₄	1	1	yes	characteristic
29	C₄²×C₄₇	D₈	1	1	yes	characteristic
30	C₄⋊Dic₄₇	—	4	2	no	—
31	C₄⋊Dic₄₇	—	8	1	no	—
32	C₄⋊Dic₄₇	—	8	1	no	—
33	C₄⋊Dic₄₇	—	8	1	no	—
34	Dic₁₈₈	—	4	2	no	—
35	Dic₁₈₈	—	4	2	no	—
36	Dic₁₈₈	—	8	2	no	—
37	C₁₆×C₄₇	SD₁₆	1	2	yes	—
38	C₁₆×C₄₇	#41	1	2	yes	—
39	C₂×C₄×C₄₇	D₁₆	1	1	yes	characteristic
40	C₂×C₄×C₄₇	D₁₆	1	1	yes	characteristic
41	C₂×C₄×C₄₇	C₂×D₈	1	1	yes	characteristic
42	C₂×Dic₄₇	—	8	2	no	—
43	C₂×Dic₄₇	—	8	2	no	—
44	Dic₉₄	—	8	2	no	—
45	Dic₉₄	—	8	2	no	—
46	Dic₉₄	—	16	2	no	—
47	C₈×C₄₇	SD₃₂	1	2	yes	—
48	C₈×C₄₇	#124	1	1	yes	characteristic
49	C₈×C₄₇	C₂×SD₁₆	1	1	yes	characteristic
50	#13720	—	47	1	no	maximal, Sylow
51	C₂²×C₄₇	C₂×D₁₆	1	1	yes	characteristic
52	C₄×C₄₇	SD₆₄	1	2	yes	—
53	C₄×C₄₇	Q₆₄	1	2	yes	—
54	C₄×C₄₇	#486	1	1	yes	characteristic
55	C₄×C₄₇	C₂×SD₃₂	1	1	yes	characteristic
56	Dic₄₇	—	16	2	no	—
57	Dic₄₇	—	16	2	no	—
58	#2252	—	47	1	no	—
59	#2247	—	47	1	no	—
60	#1416	—	47	4	no	—
61	C₄×C₃₂	D₄₇	1	1	yes	characteristic
62	C₂×C₄₇	#2240	1	1	yes	characteristic
63	C₂×C₄₇	C₂×SD₆₄	1	1	yes	characteristic
64	C₂×C₄₇	C₂×Q₆₄	1	1	yes	characteristic
65	C₂×Q₃₂	—	47	2	no	—
66	C₂×Q₃₂	—	94	1	no	—
67	#500	—	94	1	no	—
68	#494	—	94	1	no	—
69	C₂×C₃₂	C₂×D₄₇	1	2	yes	—
70	#366	—	47	2	no	—
71	#366	—	94	1	no	—
72	C₄×C₁₆	C₂×D₄₇	1	1	yes	characteristic
73	C₄₇	#13720	1	1	yes	characteristic, complemented, Sylow
74	C₂×Q₁₆	—	94	2	no	—
75	C₂×Q₁₆	—	188	1	no	—
76	C₄⋊Q₈	—	188	1	no	—
77	C₄⋊Q₈	—	188	1	no	—
78	Q₃₂	—	94	4	no	—
79	Q₃₂	—	188	2	no	—
80	C₂×C₁₆	D₁₈₈	1	1	yes	characteristic
81	C₂×C₁₆	C₂²×D₄₇	1	1	yes	characteristic
82	#107	—	94	2	no	—
83	#107	—	188	1	no	—
84	C₄×C₈	D₁₈₈	1	1	yes	characteristic
85	C₃₂	—	2	2	no	—
86	C₂×Q₈	—	376	1	no	—
87	C₂×Q₈	—	188	2	no	—
88	Q₁₆	—	376	2	no	—
89	Q₁₆	—	188	2	no	—
90	Q₁₆	—	188	2	no	—
91	C₂×C₈	D₃₇₆	1	1	yes	characteristic
92	C₂×C₈	C₂×D₁₈₈	1	1	yes	characteristic
93	C₄⋊C₄	—	376	1	no	—
94	C₄⋊C₄	—	376	1	no	—
95	C₄⋊C₄	—	376	1	no	—
96	C₄⋊C₄	—	188	2	no	—
97	C₄²	D₃₇₆	1	1	yes	characteristic
98	C₁₆	#10625325	1	2	yes	—
99	C₁₆	#10625350	1	2	yes	—
100	Q₈	—	376	2	no	—
101	Q₈	—	376	2	no	—
102	Q₈	—	752	2	no	—
103	C₂×C₄	—	376	2	no	—
104	C₂×C₄	—	376	2	no	—
105	C₂×C₄	D₇₅₂	1	1	yes	characteristic
106	C₂×C₄	D₇₅₂	1	1	yes	characteristic
107	C₂×C₄	C₂×D₃₇₆	1	1	yes	characteristic
108	C₈	#13093969	1	2	yes	—
109	C₈	#13094012	1	1	yes	characteristic
110	C₈	C₂×#10625325	1	1	yes	characteristic
111	C₂²	C₂×D₇₅₂	1	1	yes	characteristic
112	C₄	—	752	2	no	—
113	C₄	—	752	2	no	—
114	C₄	#15688962	1	2	yes	—
115	C₄	Dic₇₅₂	1	2	yes	—
116	C₄	#15689470	1	1	yes	characteristic
117	C₄	C₂×#13093969	1	1	yes	characteristic
118	C₂	#20245612	1	1	yes	characteristic
119	C₂	C₂×#15688962	1	1	yes	characteristic
120	C₂	C₂×Dic₇₅₂	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#20245663

C₂

yes

maximal, characteristic

#20245641

C₂

yes

maximal, characteristic

#20241570

C₂

yes

maximal

C₄×C₃₂×C₄₇

C₂

yes

maximal, characteristic

C₂×Dic₃₇₆

—

C₂×Dic₃₇₆

C₂²

yes

—

—

—

—

C₂²

yes

—

C₂×C₃₂×C₄₇

C₂²

yes

—

C₄×C₁₆×C₄₇

C₂²

yes

characteristic

C₂×Dic₁₈₈

—

C₂×Dic₁₈₈

—

#13094024

—

#13094024

—

Dic₃₇₆

—

Dic₃₇₆

—

#13093958

—

#13093958

—

C₂×C₁₆×C₄₇

D₄

yes

characteristic

C₂×C₁₆×C₄₇

C₂³

yes

characteristic

C₄×C₈×C₄₇

D₄

yes

characteristic

C₃₂×C₄₇

—

C₂×Dic₉₄

—

C₂×Dic₉₄

—

C₂×C₈×C₄₇

D₈

yes

characteristic

C₂×C₈×C₄₇

C₂×D₄

yes

characteristic

C₄²×C₄₇

D₈

yes

characteristic

C₄⋊Dic₄₇

—

C₄⋊Dic₄₇

—

C₄⋊Dic₄₇

—

C₄⋊Dic₄₇

—

Dic₁₈₈

—

Dic₁₈₈

—

Dic₁₈₈

—

C₁₆×C₄₇

SD₁₆

yes

—

C₁₆×C₄₇

#41

yes

—

C₂×C₄×C₄₇

D₁₆

yes

characteristic

C₂×C₄×C₄₇

D₁₆

yes

characteristic

C₂×C₄×C₄₇

C₂×D₈

yes

characteristic

C₂×Dic₄₇

—

C₂×Dic₄₇

—

Dic₉₄

—

Dic₉₄

—

Dic₉₄

—

C₈×C₄₇

SD₃₂

yes

—

C₈×C₄₇

#124

yes

characteristic

C₈×C₄₇

C₂×SD₁₆

yes

characteristic

#13720

—

maximal, Sylow

C₂²×C₄₇

C₂×D₁₆

yes

characteristic

C₄×C₄₇

SD₆₄

yes

—

C₄×C₄₇

Q₆₄

yes

—

C₄×C₄₇

#486

yes

characteristic

C₄×C₄₇

C₂×SD₃₂

yes

characteristic

Dic₄₇

—

Dic₄₇

—

#2252

—

#2247

—

#1416

—

C₄×C₃₂

D₄₇

yes

characteristic

C₂×C₄₇

#2240

yes

characteristic

C₂×C₄₇

C₂×SD₆₄

yes

characteristic

C₂×C₄₇

C₂×Q₆₄

yes

characteristic

C₂×Q₃₂

—

C₂×Q₃₂

—

#500

—

#494

—

C₂×C₃₂

C₂×D₄₇

yes

—

#366

—

#366

—

C₄×C₁₆

C₂×D₄₇

yes

characteristic

C₄₇

#13720

yes

characteristic, complemented, Sylow

C₂×Q₁₆

—

C₂×Q₁₆

—

188

—

C₄⋊Q₈

—

188

—

C₄⋊Q₈

—

188

—

Q₃₂

—

Q₃₂

—

188

—

C₂×C₁₆

D₁₈₈

yes

characteristic

C₂×C₁₆

C₂²×D₄₇

yes

characteristic

#107

—

#107

—

188

—

C₄×C₈

D₁₈₈

yes

characteristic

C₃₂

—

C₂×Q₈

—

376

—

C₂×Q₈

—

188

—

Q₁₆

—

376

—

Q₁₆

—

188

—

Q₁₆

—

188

—

C₂×C₈

D₃₇₆

yes

characteristic

C₂×C₈

C₂×D₁₈₈

yes

characteristic

C₄⋊C₄

—

376

—

C₄⋊C₄

—

376

—

C₄⋊C₄

—

376

—

C₄⋊C₄

—

188

—

C₄²

D₃₇₆

yes

characteristic

C₁₆

#10625325

yes

—

C₁₆

#10625350

yes

—

100

Q₈

—

376

—

101

Q₈

—

376

—

102

Q₈

—

752

—

103

C₂×C₄

—

376

—

104

C₂×C₄

—

376

—

105

C₂×C₄

D₇₅₂

yes

characteristic

106

C₂×C₄

D₇₅₂

yes

characteristic

107

C₂×C₄

C₂×D₃₇₆

yes

characteristic

108

C₈

#13093969

yes

—

109

C₈

#13094012

yes

characteristic

110

C₈

C₂×#10625325

yes

characteristic

111

C₂²

C₂×D₇₅₂

yes

characteristic

112

C₄

—

752

—

113

C₄

—

752

—

114

C₄

#15688962

yes

—

115

C₄

Dic₇₅₂

yes

—

116

C₄

#15689470

yes

characteristic

117

C₄

C₂×#13093969

yes

characteristic

118

C₂

#20245612

yes

characteristic

119

C₂

C₂×#15688962

yes

characteristic

120

C₂

C₂×Dic₇₅₂

yes

characteristic

	subgroup	G/H
center	C₂²	C₂×D₇₅₂
commutator, cosocle	C₂×C₁₆×C₄₇	C₂³
Frattini	C₂×C₁₆	C₂²×D₄₇
Fitting	C₄×C₃₂×C₄₇	C₂
socle, prime radical	C₂²×C₄₇	C₂×D₁₆
hypercenter	C₄×C₃₂	D₄₇
hypocenter	C₄₇	#13720
LCS	G ▷ C₂×C₁₆×C₄₇ ▷ C₈×C₄₇ ▷ C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ C₄×C₁₆ ◁ C₄×C₃₂

subgroup

G/H

center

C₂²

C₂×D₇₅₂

commutator, cosocle

C₂×C₁₆×C₄₇

C₂³

Frattini

C₂×C₁₆

C₂²×D₄₇

Fitting

C₄×C₃₂×C₄₇

C₂

socle, prime radical

C₂²×C₄₇

C₂×D₁₆

hypercenter

C₄×C₃₂

D₄₇

hypocenter

C₄₇

#13720

LCS

G ▷ C₂×C₁₆×C₄₇ ▷ C₈×C₄₇ ▷ C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇

UCS

C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ C₄×C₁₆ ◁ C₄×C₃₂

N	H ≅ G/N	type
73. C₄₇	#13720	semidirect

H ≅ G/N

type

73. C₄₇

#13720

semidirect

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

subgroup	index
#20245663	2
#20245641	2
#20241570	2
C₄×C₃₂×C₄₇	2
#13720	47

subgroup

index

C₄×C₃₂×C₄₇

Subgroups — #26137251

Conjugacy classes of subgroups

Complements

Maximal subgroups