Group #13262 — subgroups

Subgroups — #13262

Order 256.

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

normal subgroups only

This group has 231 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	C₂×#494	C₂	1	1	yes	maximal, characteristic
2	C₂×#364	C₂	1	1	yes	maximal, characteristic
3	C₂×#362	C₂	1	2	yes	maximal
4	#2141	C₂	1	2	yes	maximal
5	#1772	C₂	1	1	yes	maximal, characteristic
6	C₂²×Q₁₆	—	2	1	no	—
7	C₂²×Q₁₆	C₂²	1	2	yes	—
8	C₂×C₄⋊Q₈	—	2	1	no	—
9	C₂×M₃₂	C₂²	1	2	yes	—
10	#494	—	2	2	no	—
11	#494	C₄	1	4	yes	complemented
12	C₂×#108	C₂²	1	1	yes	characteristic
13	#427	C₂²	1	1	yes	characteristic
14	#405	—	2	1	no	—
15	C₂×C₄×C₈	C₂²	1	1	yes	characteristic
16	#364	—	2	2	no	—
17	#362	—	2	4	no	—
18	#358	—	2	4	no	—
19	#339	—	2	1	no	—
20	C₂²×Q₈	—	2	2	no	—
21	C₂×Q₁₆	—	2	4	no	—
22	C₂×Q₁₆	—	2	2	no	—
23	C₂×Q₁₆	—	2	2	no	—
24	C₂×Q₁₆	—	2	2	no	—
25	C₂×Q₁₆	—	2	2	no	—
26	C₂×Q₁₆	—	2	2	no	—
27	C₂×Q₁₆	—	4	1	no	—
28	C₂×Q₁₆	—	4	1	no	—
29	C₂×Q₁₆	—	4	2	no	—
30	C₂×Q₁₆	C₂×C₄	1	4	yes	—
31	C₂×M₁₆	—	2	1	no	—
32	C₂²×C₈	D₄	1	1	yes	characteristic
33	C₂²×C₈	C₂³	1	1	yes	characteristic
34	C₄⋊Q₈	—	2	4	no	—
35	C₄⋊Q₈	—	4	2	no	—
36	#117	—	2	1	no	—
37	C₂×C₄⋊C₄	—	4	2	no	—
38	C₂×C₄²	D₄	1	1	yes	characteristic
39	M₃₂	—	2	2	no	—
40	M₃₂	—	2	2	no	—
41	C₂×C₁₆	—	2	2	no	—
42	#108	—	2	1	no	—
43	#108	D₄	1	2	yes	—
44	#107	D₄	1	2	yes	—
45	#106	—	2	1	no	—
46	#97	—	4	1	no	—
47	C₄×C₈	—	4	1	no	—
48	C₄×C₈	C₂×C₄	1	2	yes	—
49	C₄×C₈	D₄	1	2	yes	—
50	C₂×Q₈	—	2	4	no	—
51	C₂×Q₈	—	4	2	no	—
52	C₂×Q₈	—	4	2	no	—
53	C₂×Q₈	—	4	2	no	—
54	C₂×Q₈	—	4	2	no	—
55	C₂×Q₈	—	4	2	no	—
56	C₂²×C₄	—	2	1	no	—
57	C₂²×C₄	—	4	2	no	—
58	C₂²×C₄	C₂×D₄	1	1	yes	characteristic
59	Q₁₆	—	2	4	no	—
60	Q₁₆	—	2	4	no	—
61	Q₁₆	—	4	4	no	—
62	Q₁₆	—	4	4	no	—
63	Q₁₆	—	4	2	no	—
64	Q₁₆	—	4	2	no	—
65	M₁₆	—	2	2	no	—
66	M₁₆	—	4	1	no	—
67	C₂×C₈	—	4	1	no	—
68	C₂×C₈	C₂²⋊C₄	1	2	yes	—
69	C₂×C₈	D₈	1	1	yes	characteristic
70	C₂×C₈	D₈	1	1	yes	characteristic
71	C₂×C₈	SD₁₆	1	2	yes	—
72	C₂×C₈	C₂²×C₄	1	1	yes	characteristic
73	C₂×C₈	C₂×D₄	1	1	yes	characteristic
74	C₂×C₈	C₂×D₄	1	1	yes	characteristic
75	C₂×C₈	C₂×D₄	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	8	no	—
80	C₂²⋊C₄	—	4	1	no	—
81	C₄²	—	2	1	no	—
82	C₄²	—	4	1	no	—
83	C₄²	C₂²⋊C₄	1	2	yes	—
84	C₁₆	—	4	2	no	—
85	C₂³	AGL(1,ℤ/8)	1	1	yes	characteristic
86	Q₈	—	4	4	no	—
87	Q₈	—	4	4	no	—
88	Q₈	—	8	2	no	—
89	Q₈	—	8	2	no	—
90	C₂×C₄	—	2	4	no	—
91	C₂×C₄	—	2	2	no	—
92	C₂×C₄	—	4	4	no	—
93	C₂×C₄	—	4	2	no	—
94	C₂×C₄	—	8	2	no	—
95	C₂×C₄	—	8	2	no	—
96	C₂×C₄	C₂×C₂²⋊C₄	1	1	yes	characteristic
97	C₂×C₄	#121	1	1	yes	characteristic
98	C₂×C₄	#121	1	1	yes	characteristic
99	C₂×C₄	C₂×D₈	1	1	yes	characteristic
100	C₂×C₄	C₂×SD₁₆	1	1	yes	characteristic
101	C₂×C₄	AGL(1,ℤ/8)	1	1	yes	characteristic
102	C₈	—	4	2	no	—
103	C₈	#102	1	2	yes	—
104	C₈	#102	1	2	yes	—
105	C₈	C₄×D₄	1	1	yes	characteristic
106	C₈	C₄×D₄	1	1	yes	characteristic
107	C₈	#124	1	1	yes	characteristic
108	C₈	#127	1	1	yes	characteristic
109	C₂²	—	2	2	no	—
110	C₂²	#418	1	1	yes	characteristic
111	C₂²	#459	1	1	yes	characteristic
112	C₂²	#460	1	1	yes	characteristic
113	C₄	—	2	4	no	—
114	C₄	—	8	2	no	—
115	C₄	—	8	4	no	—
116	C₄	#390	1	1	yes	characteristic
117	C₄	C₂×#102	1	1	yes	characteristic
118	C₄	#459	1	1	yes	characteristic
119	C₄	#460	1	1	yes	characteristic
120	C₂	—	2	1	no	—
121	C₂	—	2	1	no	—
122	C₂	#1982	1	1	yes	characteristic
123	C₂	#2214	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#494

C₂

yes

maximal, characteristic

C₂×#364

C₂

yes

maximal, characteristic

C₂×#362

C₂

yes

maximal

#2141

C₂

yes

maximal

#1772

C₂

yes

maximal, characteristic

C₂²×Q₁₆

—

C₂²×Q₁₆

C₂²

yes

—

C₂×C₄⋊Q₈

—

C₂×M₃₂

C₂²

yes

—

#494

—

#494

C₄

yes

complemented

C₂×#108

C₂²

yes

characteristic

#427

C₂²

yes

characteristic

#405

—

C₂×C₄×C₈

C₂²

yes

—

—

—

—

C₂²×Q₈

—

C₂×Q₁₆

—

C₂×Q₁₆

—

C₂×Q₁₆

—

C₂×Q₁₆

—

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

C₄⋊Q₈

—

C₄⋊Q₈

—

#117

—

C₂×C₄⋊C₄

—

C₂×C₄²

D₄

yes

characteristic

M₃₂

—

M₃₂

—

C₂×C₁₆

—

#108

—

#108

D₄

yes

—

#107

D₄

yes

—

#106

—

#97

—

C₄×C₈

—

C₄×C₈

C₂×C₄

yes

—

C₄×C₈

D₄

yes

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

C₂×D₄

yes

characteristic

Q₁₆

—

Q₁₆

—

Q₁₆

—

Q₁₆

—

Q₁₆

—

Q₁₆

—

M₁₆

—

M₁₆

—

C₂×C₈

—

C₂×C₈

C₂²⋊C₄

yes

—

C₂×C₈

D₈

yes

characteristic

C₂×C₈

D₈

yes

characteristic

C₂×C₈

SD₁₆

yes

—

C₂×C₈

C₂²×C₄

yes

characteristic

C₂×C₈

C₂×D₄

yes

characteristic

C₂×C₈

C₂×D₄

yes

characteristic

C₂×C₈

C₂×D₄

yes

C₂×C₈

yes

C₂×C₈

yes

C₄⋊C₄

—

C₄⋊C₄

—

C₂²⋊C₄

—

C₄²

—

C₄²

—

C₄²

C₂²⋊C₄

yes

—

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×C₂²⋊C₄

yes

C₂×C₄

yes

C₂×C₄

yes

C₂×C₄

C₂×D₈

yes

characteristic

100

C₂×C₄

C₂×SD₁₆

yes

characteristic

101

C₂×C₄

AGL(1,ℤ/8)

yes

characteristic

102

C₈

—

103

C₈

#102

yes

—

104

C₈

#102

yes

—

105

C₈

C₄×D₄

yes

characteristic

106

C₈

C₄×D₄

yes

characteristic

107

C₈

#124

yes

characteristic

108

C₈

#127

yes

characteristic

109

C₂²

—

110

C₂²

#418

yes

characteristic

111

C₂²

#459

yes

characteristic

112

C₂²

#460

yes

characteristic

113

C₄

—

114

C₄

—

115

C₄

—

116

C₄

#390

yes

characteristic

117

C₄

C₂×#102

yes

characteristic

118

C₄

#459

yes

characteristic

119

C₄

#460

yes

characteristic

120

C₂

—

121

C₂

—

122

C₂

#1982

yes

characteristic

123

C₂

#2214

yes

—

	subgroup	G/H
center, socle	C₂²	#459
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×C₄×C₈ ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

subgroup

G/H

center, socle

C₂²

#459

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×C₄×C₈ ◁ G

LECS

G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

N	H ≅ G/N	type
11. #494	C₄	semidirect

H ≅ G/N

type

11. #494

C₄

semidirect

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

subgroup	index
C₂×#494	2
C₂×#364	2
C₂×#362	2
#2141	2
#1772	2

subgroup

index

Subgroups — #13262

Conjugacy classes of subgroups

Complements

Maximal subgroups