Group #13261 — subgroups

Subgroups — #13261

Order 256.

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

normal subgroups only

This group has 283 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×#493	C₂	1	1	yes	maximal, characteristic
2	C₂×#364	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×#361	C₂	1	2	yes	maximal, complemented
4	#2140	C₂	1	2	yes	maximal, complemented
5	#1772	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂²×D₈	—	2	1	no	—
7	C₂²×D₈	C₂²	1	2	yes	—
8	C₂×#127	—	2	1	no	—
9	C₂×M₃₂	C₂²	1	2	yes	—
10	#493	—	2	2	no	—
11	#493	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	#361	—	2	4	no	—
18	#357	—	2	4	no	—
19	#339	—	2	1	no	—
20	C₂²×D₄	—	2	2	no	—
21	C₂²×D₄	—	4	2	no	—
22	C₂×D₈	—	2	4	no	—
23	C₂×D₈	—	2	2	no	—
24	C₂×D₈	—	2	2	no	—
25	C₂×D₈	—	2	2	no	—
26	C₂×D₈	—	2	2	no	—
27	C₂×D₈	—	2	2	no	—
28	C₂×D₈	—	4	1	no	—
29	C₂×D₈	—	4	1	no	—
30	C₂×D₈	—	4	2	no	—
31	C₂×D₈	C₂×C₄	1	4	yes	—
32	C₂×M₁₆	—	2	1	no	—
33	C₂²×C₈	D₄	1	1	yes	characteristic
34	C₂²×C₈	C₂³	1	1	yes	characteristic
35	#127	—	2	4	no	—
36	#127	—	4	2	no	—
37	#117	—	2	1	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	complemented
44	#107	D₄	1	2	yes	complemented
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	C₂⁴	—	4	2	no	—
51	C₂×D₄	—	2	4	no	—
52	C₂×D₄	—	4	8	no	—
53	C₂×D₄	—	4	2	no	—
54	C₂×D₄	—	4	2	no	—
55	C₂×D₄	—	4	2	no	—
56	C₂×D₄	—	4	2	no	—
57	C₂×D₄	—	4	2	no	—
58	C₂×D₄	—	8	8	no	—
59	C₂²×C₄	—	2	1	no	—
60	C₂²×C₄	C₂×D₄	1	1	yes	characteristic
61	D₈	—	2	4	no	—
62	D₈	—	2	4	no	—
63	D₈	—	4	4	no	—
64	D₈	—	4	4	no	—
65	D₈	—	4	2	no	—
66	D₈	—	4	2	no	—
67	M₁₆	—	2	2	no	—
68	M₁₆	—	4	1	no	—
69	C₂×C₈	—	4	1	no	—
70	C₂×C₈	C₂²⋊C₄	1	2	yes	—
71	C₂×C₈	D₈	1	1	yes	characteristic
72	C₂×C₈	D₈	1	1	yes	characteristic
73	C₂×C₈	SD₁₆	1	2	yes	—
74	C₂×C₈	C₂²×C₄	1	1	yes	characteristic
75	C₂×C₈	C₂×D₄	1	1	yes	characteristic
76	C₂×C₈	C₂×D₄	1	1	yes	characteristic
77	C₂×C₈	C₂×D₄	1	1	yes	characteristic
78	C₂×C₈	#41	1	1	yes	characteristic
79	C₂×C₈	#41	1	1	yes	characteristic
80	C₄⋊C₄	—	2	2	no	—
81	C₂²⋊C₄	—	4	1	no	—
82	C₄²	—	2	1	no	—
83	C₄²	—	4	1	no	—
84	C₄²	C₂²⋊C₄	1	2	yes	—
85	C₁₆	—	4	2	no	—
86	C₂³	—	4	4	no	—
87	C₂³	—	8	8	no	—
88	C₂³	—	8	2	no	—
89	C₂³	—	8	2	no	—
90	C₂³	AGL(1,ℤ/8)	1	1	yes	characteristic
91	D₄	—	4	4	no	—
92	D₄	—	4	4	no	—
93	D₄	—	8	16	no	—
94	D₄	—	8	2	no	—
95	D₄	—	8	2	no	—
96	C₂×C₄	—	2	4	no	—
97	C₂×C₄	—	2	2	no	—
98	C₂×C₄	—	4	2	no	—
99	C₂×C₄	C₂×C₂²⋊C₄	1	1	yes	characteristic
100	C₂×C₄	#121	1	1	yes	characteristic
101	C₂×C₄	#121	1	1	yes	characteristic
102	C₂×C₄	C₂×D₈	1	1	yes	characteristic
103	C₂×C₄	C₂×SD₁₆	1	1	yes	characteristic
104	C₂×C₄	AGL(1,ℤ/8)	1	1	yes	characteristic
105	C₈	—	4	2	no	—
106	C₈	#102	1	2	yes	—
107	C₈	#102	1	2	yes	—
108	C₈	C₄×D₄	1	1	yes	characteristic
109	C₈	C₄×D₄	1	1	yes	characteristic
110	C₈	#124	1	1	yes	characteristic
111	C₈	#127	1	1	yes	characteristic
112	C₂²	—	2	2	no	—
113	C₂²	—	8	8	no	—
114	C₂²	—	8	4	no	—
115	C₂²	—	16	4	no	—
116	C₂²	—	16	4	no	—
117	C₂²	#418	1	1	yes	characteristic
118	C₂²	#459	1	1	yes	characteristic
119	C₂²	#460	1	1	yes	characteristic
120	C₄	—	2	4	no	—
121	C₄	—	8	2	no	—
122	C₄	#390	1	1	yes	characteristic
123	C₄	C₂×#102	1	1	yes	characteristic
124	C₄	#459	1	1	yes	characteristic
125	C₄	#460	1	1	yes	characteristic
126	C₂	—	2	1	no	—
127	C₂	—	2	1	no	—
128	C₂	—	16	4	no	—
129	C₂	#1982	1	1	yes	characteristic
130	C₂	#2213	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#493

C₂

yes

maximal, characteristic

C₂×#364

C₂

yes

maximal, characteristic, complemented

C₂×#361

C₂

yes

maximal, complemented

#2140

C₂

yes

maximal, complemented

#1772

C₂

yes

maximal, characteristic, complemented

C₂²×D₈

—

C₂²×D₈

C₂²

yes

—

C₂×#127

—

C₂×M₃₂

C₂²

yes

—

#493

—

#493

C₄

yes

complemented

C₂×#108

C₂²

yes

characteristic

#427

C₂²

yes

characteristic

#405

—

C₂×C₄×C₈

C₂²

yes

—

—

—

—

C₂²×D₄

—

C₂²×D₄

—

C₂×D₈

—

C₂×D₈

—

C₂×D₈

—

C₂×D₈

—

C₂×D₈

—

C₂×D₈

—

C₂×D₈

—

C₂×D₈

—

C₂×D₈

—

C₂×D₈

C₂×C₄

yes

—

C₂×M₁₆

—

C₂²×C₈

D₄

yes

characteristic

C₂²×C₈

C₂³

yes

—

—

—

C₂×C₄²

D₄

yes

characteristic

M₃₂

—

M₃₂

—

C₂×C₁₆

—

#108

—

#108

D₄

yes

complemented

#107

D₄

yes

complemented

#106

—

#97

—

C₄×C₈

—

C₄×C₈

C₂×C₄

yes

—

C₄×C₈

D₄

yes

—

C₂⁴

—

C₂×D₄

—

C₂×D₄

—

C₂×D₄

—

C₂×D₄

—

C₂×D₄

—

C₂×D₄

—

C₂×D₄

—

C₂×D₄

—

C₂²×C₄

—

C₂²×C₄

C₂×D₄

yes

characteristic

D₈

—

D₈

—

D₈

—

D₈

—

D₈

—

D₈

—

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

—

C₂²⋊C₄

—

C₄²

—

C₄²

—

C₄²

C₂²⋊C₄

yes

—

C₁₆

—

C₂³

—

C₂³

—

C₂³

—

C₂³

—

C₂³

AGL(1,ℤ/8)

yes

characteristic

D₄

—

D₄

—

D₄

—

D₄

—

D₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

C₂×C₂²⋊C₄

yes

characteristic

100

C₂×C₄

#121

yes

characteristic

101

C₂×C₄

#121

yes

characteristic

102

C₂×C₄

C₂×D₈

yes

characteristic

103

C₂×C₄

C₂×SD₁₆

yes

characteristic

104

C₂×C₄

AGL(1,ℤ/8)

yes

characteristic

105

C₈

—

106

C₈

#102

yes

—

107

C₈

#102

yes

—

108

C₈

C₄×D₄

yes

characteristic

109

C₈

C₄×D₄

yes

characteristic

110

C₈

#124

yes

characteristic

111

C₈

#127

yes

characteristic

112

C₂²

—

113

C₂²

—

114

C₂²

—

115

C₂²

—

116

C₂²

—

117

C₂²

#418

yes

characteristic

118

C₂²

#459

yes

characteristic

119

C₂²

#460

yes

characteristic

120

C₄

—

121

C₄

—

122

C₄

#390

yes

characteristic

123

C₄

C₂×#102

yes

characteristic

124

C₄

#459

yes

characteristic

125

C₄

#460

yes

characteristic

126

C₂

—

127

C₂

—

128

C₂

—

129

C₂

#1982

yes

characteristic

130

C₂

#2213

yes

—

	subgroup	G/H
center, socle	C₂²	#459
commutator	C₂×C₈	C₂²×C₄
Frattini, cosocle	C₂²×C₈	C₂³
prime radical	C₂×#493	C₂
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

C₂×#493

C₂

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
2. C₂×#364	C₂	semidirect
3. C₂×#361	C₂	semidirect
4. #2140	C₂	semidirect
5. #1772	C₂	semidirect
11. #493	C₄	semidirect
43. #108	D₄	semidirect
44. #107	D₄	semidirect

H ≅ G/N

type

2. C₂×#364

C₂

semidirect

3. C₂×#361

C₂

semidirect

4. #2140

C₂

semidirect

5. #1772

C₂

semidirect

11. #493

C₄

semidirect

43. #108

D₄

semidirect

44. #107

D₄

semidirect

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

subgroup	index
C₂×#493	2
C₂×#364	2
C₂×#361	2
#2140	2
#1772	2

subgroup

index

Subgroups — #13261

Conjugacy classes of subgroups

Complements

Maximal subgroups