Group #13119 — subgroups

Subgroups — #13119

Order 256.

138 conjugacy classes of subgroups; only the 61 normal subgroups are shown.

normal subgroups only

This group has 138 conjugacy classes of subgroups — too many to draw. Only its 61 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	#2159	C₂	1	1	yes	maximal, characteristic
2	#1946	C₂	1	1	yes	maximal, characteristic
3	#1776	C₂	1	1	yes	maximal, characteristic
4	#1387	C₂	1	1	yes	maximal, characteristic
5	#1386	C₂	1	1	yes	maximal, characteristic
6	#1383	C₂	1	1	yes	maximal, characteristic
7	#1375	C₂	1	1	yes	maximal, characteristic
8	C₂×M₃₂	C₂²	1	1	yes	characteristic
9	C₂×M₃₂	C₂²	1	1	yes	characteristic
10	C₂×#108	C₂²	1	1	yes	characteristic
11	#427	C₂²	1	1	yes	characteristic
12	C₂×#107	C₂²	1	1	yes	characteristic
13	C₂×#106	C₂²	1	1	yes	characteristic
14	#424	—	2	1	no	—
15	C₂×#97	C₂²	1	1	yes	characteristic
16	C₄×C₄⋊C₄	—	2	1	no	—
17	#367	C₄	1	2	yes	complemented
18	#366	C₄	1	2	yes	complemented
19	#340	—	2	1	no	—
20	#339	—	2	1	no	—
21	C₂×M₁₆	—	2	1	no	—
22	C₂²×C₈	D₄	1	1	yes	characteristic
23	C₂²×C₈	C₂³	1	1	yes	characteristic
24	#117	—	2	1	no	—
25	C₂×C₄⋊C₄	—	2	1	no	—
26	C₂×C₄⋊C₄	—	2	1	no	—
27	C₂×C₄²	—	4	1	no	—
28	C₂×C₄²	D₄	1	1	yes	characteristic
29	M₃₂	—	2	1	no	—
30	M₃₂	—	2	1	no	—
31	M₃₂	—	2	1	no	—
32	M₃₂	—	2	1	no	—
33	C₂×C₁₆	—	2	1	no	—
34	C₂×C₁₆	C₂×C₄	1	2	yes	—
35	#108	—	2	1	no	—
36	#108	—	2	1	no	—
37	#107	—	2	1	no	—
38	#107	C₂×C₄	1	1	yes	characteristic
39	#107	C₂×C₄	1	2	yes	—
40	#107	C₂×C₄	1	1	yes	characteristic
41	#106	—	2	1	no	—
42	#106	—	2	1	no	—
43	#106	Q₈	1	2	yes	—
44	C₄⋊C₈	—	4	2	no	—
45	#97	—	2	1	no	—
46	#97	—	2	1	no	—
47	#95	—	4	1	no	—
48	C₂²×C₄	—	2	1	no	—
49	C₂²×C₄	—	4	1	no	—
50	C₂²×C₄	—	4	1	no	—
51	C₂²×C₄	C₂×D₄	1	1	yes	characteristic
52	M₁₆	—	4	1	no	—
53	M₁₆	—	4	1	no	—
54	C₂×C₈	—	2	1	no	—
55	C₂×C₈	—	2	1	no	—
56	C₂×C₈	—	2	1	no	—
57	C₂×C₈	—	4	1	no	—
58	C₂×C₈	C₂²×C₄	1	1	yes	characteristic
59	C₂×C₈	C₂×Q₈	1	1	yes	characteristic
60	C₂×C₈	#41	1	1	yes	characteristic
61	C₂×C₈	#41	1	1	yes	characteristic
62	C₂×C₈	#41	1	1	yes	characteristic
63	C₂×C₈	#41	1	1	yes	characteristic
64	C₄⋊C₄	—	2	1	no	—
65	C₄⋊C₄	—	2	1	no	—
66	C₄⋊C₄	—	2	2	no	—
67	C₄⋊C₄	—	2	2	no	—
68	C₄⋊C₄	—	4	1	no	—
69	C₄⋊C₄	—	4	1	no	—
70	C₂²⋊C₄	—	4	1	no	—
71	C₄²	—	2	1	no	—
72	C₄²	—	4	1	no	—
73	C₄²	—	8	2	no	—
74	C₄²	Q₁₆	1	2	yes	—
75	C₁₆	—	2	2	no	—
76	C₁₆	—	4	1	no	—
77	C₂³	AGL(1,ℤ/8)	1	1	yes	characteristic
78	C₂×C₄	—	2	2	no	—
79	C₂×C₄	—	4	1	no	—
80	C₂×C₄	—	4	1	no	—
81	C₂×C₄	—	4	1	no	—
82	C₂×C₄	—	4	1	no	—
83	C₂×C₄	—	4	2	no	—
84	C₂×C₄	—	4	2	no	—
85	C₂×C₄	—	8	1	no	—
86	C₂×C₄	—	8	1	no	—
87	C₂×C₄	—	8	1	no	—
88	C₂×C₄	—	8	1	no	—
89	C₂×C₄	C₄×D₄	1	1	yes	characteristic
90	C₂×C₄	C₂²⋊Q₈	1	1	yes	characteristic
91	C₂×C₄	#123	1	1	yes	characteristic
92	C₂×C₄	C₂×Q₁₆	1	1	yes	characteristic
93	C₂×C₄	#135	1	1	yes	characteristic
94	C₂×C₄	AGL(1,ℤ/8)	1	1	yes	characteristic
95	C₈	—	4	1	no	—
96	C₈	—	8	1	no	—
97	C₈	#117	1	1	yes	characteristic
98	C₈	C₄×Q₈	1	1	yes	characteristic
99	C₈	#125	1	1	yes	characteristic
100	C₈	#126	1	1	yes	characteristic
101	C₂²	—	2	1	no	—
102	C₂²	—	2	1	no	—
103	C₂²	#442	1	1	yes	characteristic
104	C₂²	#478	1	1	yes	characteristic
105	C₂²	#484	1	1	yes	characteristic
106	C₄	—	4	2	no	—
107	C₄	—	8	2	no	—
108	C₄	—	8	1	no	—
109	C₄	—	8	1	no	—
110	C₄	—	8	2	no	—
111	C₄	#387	1	1	yes	characteristic
112	C₄	C₄×Q₁₆	1	1	yes	characteristic
113	C₄	#477	1	1	yes	characteristic
114	C₄	#482	1	1	yes	characteristic
115	C₂	—	2	1	no	—
116	C₂	—	2	1	no	—
117	C₂	#1921	1	1	yes	characteristic
118	C₂	#2180	1	1	yes	characteristic
119	C₂	AGL(1,ℤ/16)	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#2159

C₂

yes

maximal, characteristic

#1946

C₂

yes

maximal, characteristic

#1776

C₂

yes

maximal, characteristic

#1387

C₂

yes

maximal, characteristic

#1386

C₂

yes

maximal, characteristic

#1383

C₂

yes

maximal, characteristic

#1375

C₂

yes

maximal, characteristic

C₂×M₃₂

C₂²

yes

characteristic

C₂×M₃₂

C₂²

yes

characteristic

C₂×#108

C₂²

yes

characteristic

#427

C₂²

yes

characteristic

C₂×#107

C₂²

yes

characteristic

C₂×#106

C₂²

yes

characteristic

#424

—

C₂×#97

C₂²

yes

characteristic

C₄×C₄⋊C₄

—

#367

C₄

yes

complemented

#366

C₄

yes

complemented

#340

—

#339

—

C₂×M₁₆

—

C₂²×C₈

D₄

yes

characteristic

C₂²×C₈

C₂³

yes

characteristic

#117

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄²

—

C₂×C₄²

D₄

yes

characteristic

M₃₂

—

M₃₂

—

M₃₂

—

M₃₂

—

C₂×C₁₆

—

C₂×C₁₆

C₂×C₄

yes

—

—

—

—

C₂×C₄

yes

characteristic

#107

C₂×C₄

yes

—

#107

C₂×C₄

yes

—

—

Q₈

yes

—

C₄⋊C₈

—

#97

—

#97

—

#95

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

C₂×D₄

yes

characteristic

M₁₆

—

M₁₆

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

C₂²×C₄

yes

characteristic

C₂×C₈

C₂×Q₈

yes

C₂×C₈

yes

C₂×C₈

yes

C₂×C₈

yes

C₂×C₈

yes

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₂²⋊C₄

—

C₄²

—

C₄²

—

C₄²

—

C₄²

Q₁₆

yes

—

C₁₆

—

C₁₆

—

C₂³

AGL(1,ℤ/8)

yes

characteristic

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

C₄×D₄

yes

characteristic

C₂×C₄

C₂²⋊Q₈

yes

characteristic

C₂×C₄

#123

yes

characteristic

C₂×C₄

C₂×Q₁₆

yes

C₂×C₄

yes

C₂×C₄

yes

C₈

—

C₈

—

C₈

#117

yes

characteristic

C₈

C₄×Q₈

yes

characteristic

C₈

#125

yes

characteristic

100

C₈

#126

yes

characteristic

101

C₂²

—

102

C₂²

—

103

C₂²

#442

yes

characteristic

104

C₂²

#478

yes

characteristic

105

C₂²

#484

yes

characteristic

106

C₄

—

107

C₄

—

108

C₄

—

109

C₄

—

110

C₄

—

111

C₄

#387

yes

characteristic

112

C₄

C₄×Q₁₆

yes

characteristic

113

C₄

#477

yes

characteristic

114

C₄

#482

yes

characteristic

115

C₂

—

116

C₂

—

117

C₂

#1921

yes

characteristic

118

C₂

#2180

yes

characteristic

119

C₂

AGL(1,ℤ/16)

yes

characteristic

	subgroup	G/H
center, socle	C₂²	#442
commutator	C₂×C₈	C₂²×C₄
Frattini, cosocle	C₂²×C₈	C₂³
prime radical	C₂³	AGL(1,ℤ/8)
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×#97 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

subgroup

G/H

center, socle

C₂²

#442

commutator

C₂×C₈

C₂²×C₄

Frattini, cosocle

C₂²×C₈

C₂³

prime radical

C₂³

AGL(1,ℤ/8)

LCS

G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁

UCS

C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×#97 ◁ G

LECS

G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

N	H ≅ G/N	type
17. #367	C₄	semidirect
18. #366	C₄	semidirect

H ≅ G/N

type

17. #367

C₄

semidirect

18. #366

C₄

semidirect

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

subgroup	index
#2159	2
#1946	2
#1776	2
#1387	2
#1386	2
#1383	2
#1375	2

subgroup

index

Subgroups — #13119

Conjugacy classes of subgroups

Complements

Maximal subgroups