Group #13481 — subgroups

Subgroups — #13481

Order 256.

171 conjugacy classes of subgroups; only the 65 normal subgroups are shown.

normal subgroups only

This group has 171 conjugacy classes of subgroups — too many to draw. Only its 65 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×#358	C₂	1	2	yes	maximal
2	#1969	C₂	1	1	yes	maximal, characteristic
3	#1845	C₂	1	1	yes	maximal, characteristic
4	#1380	C₂	1	2	yes	maximal
5	#1374	C₂	1	1	yes	maximal, characteristic
6	C₂²×Q₁₆	C₂²	1	1	yes	characteristic
7	C₂²×C₁₆	C₂²	1	2	yes	—
8	C₂×#107	—	2	1	no	—
9	C₂×#107	C₂²	1	2	yes	—
10	C₂×#107	C₂²	1	1	yes	characteristic
11	C₂×#103	—	2	1	no	—
12	C₂×C₄×C₈	C₂²	1	1	yes	characteristic
13	#391	—	2	1	no	—
14	#384	—	2	1	no	—
15	#358	—	2	4	no	—
16	C₂²×Q₈	—	2	1	no	—
17	C₂×Q₁₆	—	2	2	no	—
18	C₂×Q₁₆	—	2	2	no	—
19	C₂×Q₁₆	—	2	1	no	—
20	C₂×Q₁₆	D₄	1	2	yes	—
21	C₂²×C₈	D₄	1	1	yes	characteristic
22	C₂²×C₈	C₂³	1	1	yes	characteristic
23	C₂×C₄⋊C₄	—	2	2	no	—
24	C₂×C₄⋊C₄	—	2	1	no	—
25	C₂×C₄⋊C₄	—	4	1	no	—
26	C₂×C₄⋊C₄	—	4	1	no	—
27	C₂×C₄²	D₄	1	1	yes	characteristic
28	C₂×C₁₆	—	2	2	no	—
29	C₂×C₁₆	—	2	2	no	—
30	C₂×C₁₆	—	2	2	no	—
31	#107	—	2	2	no	—
32	#107	—	2	2	no	—
33	#107	—	2	2	no	—
34	#107	—	4	2	no	—
35	#103	—	4	2	no	—
36	C₄×C₈	—	2	2	no	—
37	#95	—	4	1	no	—
38	#95	—	4	1	no	—
39	C₂×Q₈	—	2	2	no	—
40	C₂×Q₈	—	4	1	no	—
41	C₂×Q₈	—	4	1	no	—
42	C₂×Q₈	—	4	1	no	—
43	C₂×Q₈	—	4	1	no	—
44	C₂×Q₈	—	4	1	no	—
45	C₂²×C₄	—	4	1	no	—
46	C₂²×C₄	—	4	1	no	—
47	C₂²×C₄	—	4	2	no	—
48	C₂²×C₄	D₈	1	1	yes	characteristic
49	C₂²×C₄	D₈	1	1	yes	characteristic
50	C₂²×C₄	C₂×D₄	1	1	yes	characteristic
51	Q₁₆	—	2	4	no	—
52	Q₁₆	—	4	2	no	—
53	C₂×C₈	—	2	1	no	—
54	C₂×C₈	—	2	1	no	—
55	C₂×C₈	SD₁₆	1	2	yes	—
56	C₂×C₈	C₂×D₄	1	1	yes	characteristic
57	C₂×C₈	#41	1	2	yes	—
58	C₂×C₈	#41	1	2	yes	—
59	C₂×C₈	#41	1	1	yes	characteristic
60	C₄⋊C₄	—	4	2	no	—
61	C₄⋊C₄	—	4	1	no	—
62	C₄⋊C₄	—	4	1	no	—
63	C₄⋊C₄	—	4	2	no	—
64	C₄⋊C₄	—	8	2	no	—
65	C₄⋊C₄	—	8	2	no	—
66	C₄²	—	2	1	no	—
67	C₄²	—	2	1	no	—
68	C₁₆	—	4	2	no	—
69	C₂³	C₂×D₈	1	1	yes	characteristic
70	Q₈	—	4	2	no	—
71	Q₈	—	4	2	no	—
72	Q₈	—	8	1	no	—
73	Q₈	—	8	1	no	—
74	C₂×C₄	—	2	1	no	—
75	C₂×C₄	—	2	1	no	—
76	C₂×C₄	—	2	1	no	—
77	C₂×C₄	—	2	1	no	—
78	C₂×C₄	—	4	2	no	—
79	C₂×C₄	—	8	1	no	—
80	C₂×C₄	—	8	1	no	—
81	C₂×C₄	—	8	2	no	—
82	C₂×C₄	—	8	1	no	—
83	C₂×C₄	—	8	2	no	—
84	C₂×C₄	—	8	2	no	—
85	C₂×C₄	—	8	1	no	—
86	C₂×C₄	—	8	1	no	—
87	C₂×C₄	SD₃₂	1	2	yes	—
88	C₂×C₄	Q₃₂	1	2	yes	—
89	C₂×C₄	#121	1	1	yes	characteristic
90	C₂×C₄	#123	1	1	yes	characteristic
91	C₂×C₄	#124	1	1	yes	characteristic
92	C₂×C₄	C₂×SD₁₆	1	1	yes	characteristic
93	C₂×C₄	AGL(1,ℤ/8)	1	1	yes	characteristic
94	C₂×C₄	#137	1	1	yes	characteristic
95	C₈	—	2	2	no	—
96	C₈	#123	1	2	yes	—
97	C₈	#126	1	2	yes	—
98	C₂²	#459	1	1	yes	characteristic
99	C₂²	#480	1	1	yes	characteristic
100	C₂²	#486	1	1	yes	characteristic
101	C₂²	C₂×SD₃₂	1	1	yes	characteristic
102	C₂²	C₂×Q₃₂	1	1	yes	characteristic
103	C₂²	#509	1	1	yes	characteristic
104	C₂²	#510	1	1	yes	characteristic
105	C₄	—	2	2	no	—
106	C₄	—	2	2	no	—
107	C₄	—	8	2	no	—
108	C₄	—	16	2	no	—
109	C₄	—	16	1	no	—
110	C₄	#397	1	1	yes	characteristic
111	C₄	#461	1	1	yes	characteristic
112	C₄	#481	1	1	yes	characteristic
113	C₄	#489	1	1	yes	characteristic
114	C₂	#2071	1	1	yes	characteristic
115	C₂	#2207	1	1	yes	characteristic
116	C₂	#2209	1	1	yes	characteristic
117	C₂	#2235	1	1	yes	characteristic
118	C₂	#2236	1	1	yes	characteristic
119	C₂	#2241	1	1	yes	characteristic
120	C₂	#2243	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#358

C₂

yes

maximal

#1969

C₂

yes

maximal, characteristic

#1845

C₂

yes

maximal, characteristic

#1380

C₂

yes

maximal

#1374

C₂

yes

maximal, characteristic

C₂²×Q₁₆

C₂²

yes

characteristic

C₂²×C₁₆

C₂²

yes

—

C₂×#107

—

C₂×#107

C₂²

yes

—

C₂×#107

C₂²

yes

characteristic

C₂×#103

—

C₂×C₄×C₈

C₂²

yes

—

—

—

C₂²×Q₈

—

C₂×Q₁₆

—

C₂×Q₁₆

—

C₂×Q₁₆

—

C₂×Q₁₆

D₄

yes

—

C₂²×C₈

D₄

yes

characteristic

C₂²×C₈

C₂³

yes

characteristic

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄²

D₄

yes

characteristic

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

—

—

—

—

—

—

C₄×C₈

—

#95

—

#95

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

D₈

yes

characteristic

C₂²×C₄

D₈

yes

characteristic

C₂²×C₄

C₂×D₄

yes

characteristic

Q₁₆

—

Q₁₆

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

SD₁₆

yes

—

C₂×C₈

C₂×D₄

yes

characteristic

C₂×C₈

#41

yes

—

C₂×C₈

#41

yes

—

C₂×C₈

#41

yes

characteristic

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄²

—

C₄²

—

C₁₆

—

C₂³

C₂×D₈

yes

characteristic

Q₈

—

Q₈

—

Q₈

—

Q₈

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

SD₃₂

yes

—

C₂×C₄

Q₃₂

yes

—

C₂×C₄

yes

C₂×C₄

yes

C₂×C₄

yes

C₂×C₄

C₂×SD₁₆

yes

C₂×C₄

yes

C₂×C₄

yes

C₈

—

C₈

#123

yes

—

C₈

#126

yes

—

C₂²

yes

C₂²

yes

100

C₂²

#486

yes

characteristic

101

C₂²

C₂×SD₃₂

yes

characteristic

102

C₂²

C₂×Q₃₂

yes

characteristic

103

C₂²

#509

yes

characteristic

104

C₂²

#510

yes

characteristic

105

C₄

—

106

C₄

—

107

C₄

—

108

C₄

—

109

C₄

—

110

C₄

#397

yes

characteristic

111

C₄

#461

yes

characteristic

112

C₄

#481

yes

characteristic

113

C₄

#489

yes

characteristic

114

C₂

#2071

yes

characteristic

115

C₂

#2207

yes

characteristic

116

C₂

#2209

yes

characteristic

117

C₂

#2235

yes

characteristic

118

C₂

#2236

yes

characteristic

119

C₂

#2241

yes

characteristic

120

C₂

#2243

yes

characteristic

	subgroup	G/H
center, socle, prime radical	C₂³	C₂×D₈
commutator, Frattini, cosocle	C₂²×C₈	C₂³
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

subgroup

G/H

center, socle, prime radical

C₂³

C₂×D₈

commutator, Frattini, cosocle

C₂²×C₈

C₂³

LCS

G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

UCS

C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ G

LECS

G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

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

subgroup	index
C₂×#358	2
#1969	2
#1845	2
#1380	2
#1374	2

subgroup

index

Subgroups — #13481

Conjugacy classes of subgroups

Maximal subgroups