Group #13154 — subgroups

Subgroups — #13154

Order 256.

157 conjugacy classes of subgroups; only the 59 normal subgroups are shown.

normal subgroups only

This group has 157 conjugacy classes of subgroups — too many to draw. Only its 59 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	C₂×#362	C₂	1	1	yes	maximal, characteristic
2	#2141	C₂	1	1	yes	maximal, characteristic
3	C₂×#347	C₂	1	1	yes	maximal, characteristic
4	#1971	C₂	1	1	yes	maximal, characteristic
5	#1946	C₂	1	1	yes	maximal, characteristic
6	#1386	C₂	1	1	yes	maximal, characteristic
7	#1383	C₂	1	1	yes	maximal, characteristic
8	C₂²×Q₁₆	C₂²	1	1	yes	characteristic
9	C₂×M₃₂	C₂²	1	1	yes	characteristic
10	C₂×M₃₂	C₂²	1	1	yes	characteristic
11	C₂×#108	C₂²	1	1	yes	characteristic
12	#427	C₂²	1	1	yes	characteristic
13	C₂×#106	C₂²	1	1	yes	characteristic
14	#424	—	2	1	no	—
15	C₂×#103	—	2	1	no	—
16	C₂×#97	C₂²	1	1	yes	characteristic
17	#391	—	2	1	no	—
18	#362	—	2	2	no	—
19	#358	C₄	1	4	yes	complemented
20	#347	—	2	2	no	—
21	#339	—	2	1	no	—
22	C₂²×Q₈	—	2	1	no	—
23	C₂×Q₁₆	—	2	1	no	—
24	C₂×Q₁₆	—	2	1	no	—
25	C₂×Q₁₆	—	2	1	no	—
26	C₂×Q₁₆	—	2	1	no	—
27	C₂×Q₁₆	—	2	1	no	—
28	C₂×Q₁₆	C₂×C₄	1	2	yes	—
29	C₂×M₁₆	—	2	1	no	—
30	C₂²×C₈	D₄	1	1	yes	characteristic
31	C₂²×C₈	C₂³	1	1	yes	characteristic
32	#117	—	2	1	no	—
33	C₂×C₄⋊C₄	—	2	1	no	—
34	C₂×C₄²	D₄	1	1	yes	characteristic
35	M₃₂	—	2	1	no	—
36	M₃₂	—	2	1	no	—
37	M₃₂	—	2	1	no	—
38	M₃₂	D₄	1	2	yes	—
39	C₂×C₁₆	—	2	1	no	—
40	C₂×C₁₆	C₂×C₄	1	2	yes	—
41	#108	—	2	1	no	—
42	#108	—	2	1	no	—
43	#107	C₂×C₄	1	2	yes	—
44	#106	—	2	1	no	—
45	#106	—	2	1	no	—
46	#106	—	2	1	no	—
47	C₄⋊C₈	—	4	2	no	—
48	#103	—	4	2	no	—
49	#97	—	2	1	no	—
50	#97	—	2	1	no	—
51	#95	—	4	2	no	—
52	C₂×Q₈	—	2	2	no	—
53	C₂×Q₈	—	4	1	no	—
54	C₂×Q₈	—	4	1	no	—
55	C₂×Q₈	—	4	1	no	—
56	C₂×Q₈	—	4	1	no	—
57	C₂×Q₈	—	4	1	no	—
58	C₂²×C₄	—	2	1	no	—
59	C₂²×C₄	—	4	1	no	—
60	C₂²×C₄	—	4	1	no	—
61	C₂²×C₄	C₂×D₄	1	1	yes	characteristic
62	Q₁₆	—	2	2	no	—
63	Q₁₆	—	2	2	no	—
64	Q₁₆	—	4	1	no	—
65	Q₁₆	—	4	1	no	—
66	M₁₆	—	4	1	no	—
67	M₁₆	—	4	1	no	—
68	C₂×C₈	—	2	1	no	—
69	C₂×C₈	—	2	1	no	—
70	C₂×C₈	—	2	1	no	—
71	C₂×C₈	—	4	1	no	—
72	C₂×C₈	C₂²×C₄	1	1	yes	characteristic
73	C₂×C₈	C₂×D₄	1	1	yes	characteristic
74	C₂×C₈	#41	1	1	yes	characteristic
75	C₂×C₈	#41	1	1	yes	characteristic
76	C₂×C₈	#41	1	1	yes	characteristic
77	C₂×C₈	#41	1	1	yes	characteristic
78	C₄⋊C₄	—	2	2	no	—
79	C₄⋊C₄	—	4	1	no	—
80	C₄⋊C₄	—	4	1	no	—
81	C₂²⋊C₄	—	4	1	no	—
82	C₄²	—	2	1	no	—
83	C₄²	—	2	1	no	—
84	C₄²	—	4	1	no	—
85	C₁₆	—	2	2	no	—
86	C₁₆	—	2	2	no	—
87	C₂³	AGL(1,ℤ/8)	1	1	yes	characteristic
88	Q₈	—	4	2	no	—
89	Q₈	—	4	2	no	—
90	Q₈	—	8	1	no	—
91	Q₈	—	8	1	no	—
92	C₂×C₄	—	4	2	no	—
93	C₂×C₄	—	4	2	no	—
94	C₂×C₄	—	4	2	no	—
95	C₂×C₄	—	4	1	no	—
96	C₂×C₄	—	8	1	no	—
97	C₂×C₄	—	8	1	no	—
98	C₂×C₄	—	8	2	no	—
99	C₂×C₄	—	8	1	no	—
100	C₂×C₄	C₄×D₄	1	1	yes	characteristic
101	C₂×C₄	#121	1	1	yes	characteristic
102	C₂×C₄	#124	1	1	yes	characteristic
103	C₂×C₄	AGL(1,ℤ/8)	1	1	yes	characteristic
104	C₂×C₄	AGL(1,ℤ/8)	1	1	yes	characteristic
105	C₂×C₄	#137	1	1	yes	characteristic
106	C₈	—	4	1	no	—
107	C₈	—	8	1	no	—
108	C₈	#117	1	1	yes	characteristic
109	C₈	C₄×D₄	1	1	yes	characteristic
110	C₈	#123	1	1	yes	characteristic
111	C₈	#126	1	1	yes	characteristic
112	C₂²	—	2	2	no	—
113	C₂²	#442	1	1	yes	characteristic
114	C₂²	#468	1	1	yes	characteristic
115	C₂²	#490	1	1	yes	characteristic
116	C₄	—	8	1	no	—
117	C₄	—	8	2	no	—
118	C₄	—	8	2	no	—
119	C₄	—	16	1	no	—
120	C₄	#388	1	1	yes	characteristic
121	C₄	#440	1	1	yes	characteristic
122	C₄	#469	1	1	yes	characteristic
123	C₄	#489	1	1	yes	characteristic
124	C₂	—	2	1	no	—
125	C₂	—	2	1	no	—
126	C₂	#1936	1	1	yes	characteristic
127	C₂	#2180	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#362

C₂

yes

maximal, characteristic

#2141

C₂

yes

maximal, characteristic

C₂×#347

C₂

yes

maximal, characteristic

#1971

C₂

yes

maximal, characteristic

#1946

C₂

yes

maximal, characteristic

#1386

C₂

yes

maximal, characteristic

#1383

C₂

yes

maximal, characteristic

C₂²×Q₁₆

C₂²

yes

characteristic

C₂×M₃₂

C₂²

yes

characteristic

C₂×M₃₂

C₂²

yes

characteristic

C₂×#108

C₂²

yes

characteristic

#427

C₂²

yes

characteristic

C₂×#106

C₂²

yes

characteristic

#424

—

C₂×#103

—

C₂×#97

C₂²

yes

—

—

C₄

yes

complemented

#347

—

#339

—

C₂²×Q₈

—

C₂×Q₁₆

—

C₂×Q₁₆

—

C₂×Q₁₆

—

C₂×Q₁₆

—

C₂×Q₁₆

—

C₂×Q₁₆

C₂×C₄

yes

—

C₂×M₁₆

—

C₂²×C₈

D₄

yes

characteristic

C₂²×C₈

C₂³

yes

characteristic

#117

—

C₂×C₄⋊C₄

—

C₂×C₄²

D₄

yes

characteristic

M₃₂

—

M₃₂

—

M₃₂

—

M₃₂

D₄

yes

—

C₂×C₁₆

—

C₂×C₁₆

C₂×C₄

yes

—

#108

—

#108

—

#107

C₂×C₄

yes

—

#106

—

#106

—

#106

—

C₄⋊C₈

—

—

—

—

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

C₂×D₄

yes

characteristic

Q₁₆

—

Q₁₆

—

Q₁₆

—

Q₁₆

—

M₁₆

—

M₁₆

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

C₂²×C₄

yes

characteristic

C₂×C₈

C₂×D₄

yes

C₂×C₈

yes

C₂×C₈

yes

C₂×C₈

yes

C₂×C₈

yes

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₂²⋊C₄

—

C₄²

—

C₄²

—

C₄²

—

C₁₆

—

C₁₆

—

C₂³

AGL(1,ℤ/8)

yes

characteristic

Q₈

—

Q₈

—

Q₈

—

Q₈

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

100

C₂×C₄

C₄×D₄

yes

characteristic

101

C₂×C₄

#121

yes

characteristic

102

C₂×C₄

#124

yes

characteristic

103

C₂×C₄

AGL(1,ℤ/8)

yes

characteristic

104

C₂×C₄

AGL(1,ℤ/8)

yes

characteristic

105

C₂×C₄

#137

yes

characteristic

106

C₈

—

107

C₈

—

108

C₈

#117

yes

characteristic

109

C₈

C₄×D₄

yes

characteristic

110

C₈

#123

yes

characteristic

111

C₈

#126

yes

characteristic

112

C₂²

—

113

C₂²

#442

yes

characteristic

114

C₂²

#468

yes

characteristic

115

C₂²

#490

yes

characteristic

116

C₄

—

117

C₄

—

118

C₄

—

119

C₄

—

120

C₄

#388

yes

characteristic

121

C₄

#440

yes

characteristic

122

C₄

#469

yes

characteristic

123

C₄

#489

yes

characteristic

124

C₂

—

125

C₂

—

126

C₂

#1936

yes

characteristic

127

C₂

#2180

yes

—

	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
19. #358	C₄	semidirect

H ≅ G/N

type

19. #358

C₄

semidirect

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

subgroup	index
C₂×#362	2
#2141	2
C₂×#347	2
#1971	2
#1946	2
#1386	2
#1383	2

subgroup

index

Subgroups — #13154

Conjugacy classes of subgroups

Complements

Maximal subgroups