Group #13160 — subgroups

Subgroups — #13160

Order 256.

169 conjugacy classes of subgroups; only the 95 normal subgroups are shown.

normal subgroups only

This group has 169 conjugacy classes of subgroups — too many to draw. Only its 95 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×C₄⋊C₁₆	C₂	1	2	yes	maximal
2	C₂×C₄⋊C₁₆	C₂	1	1	yes	maximal, characteristic
3	C₂×#346	C₂	1	1	yes	maximal, characteristic
4	C₈×C₄⋊C₄	C₂	1	1	yes	maximal, characteristic
5	#1374	C₂	1	2	yes	maximal
6	C₂²×C₁₆	C₂²	1	2	yes	—
7	C₂²×C₁₆	C₂²	1	1	yes	characteristic
8	C₂²×C₁₆	C₂²	1	1	yes	characteristic
9	C₂×C₄⋊C₈	C₄	1	1	yes	characteristic
10	C₂×C₄×C₈	C₂²	1	2	yes	—
11	C₂×C₄×C₈	C₂²	1	1	yes	characteristic
12	C₄×C₄⋊C₄	C₄	1	1	yes	characteristic
13	C₄⋊C₁₆	—	2	4	no	—
14	C₄⋊C₁₆	—	2	2	no	—
15	#346	—	2	2	no	—
16	#336	C₄	1	2	yes	—
17	C₂²×C₈	C₂×C₄	1	2	yes	—
18	C₂²×C₈	C₂×C₄	1	1	yes	characteristic
19	C₂²×C₈	C₂³	1	1	yes	characteristic
20	C₂×C₄⋊C₄	C₈	1	2	yes	—
21	C₂×C₄²	C₂×C₄	1	2	yes	—
22	C₂×C₄²	C₂×C₄	1	1	yes	characteristic
23	C₂×C₁₆	—	2	1	no	—
24	C₂×C₁₆	—	2	1	no	—
25	C₂×C₁₆	—	2	1	no	—
26	C₂×C₁₆	—	2	1	no	—
27	C₂×C₁₆	—	2	2	no	—
28	C₂×C₁₆	—	2	2	no	—
29	C₂×C₁₆	—	2	2	no	—
30	C₂×C₁₆	D₄	1	2	yes	—
31	C₂×C₁₆	Q₈	1	2	yes	—
32	C₄⋊C₈	—	2	2	no	—
33	C₄×C₈	—	2	2	no	—
34	C₄×C₈	—	2	2	no	—
35	C₄×C₈	D₄	1	2	yes	—
36	C₄×C₈	Q₈	1	2	yes	—
37	#95	C₈	1	1	yes	characteristic
38	#95	C₈	1	1	yes	characteristic
39	C₂²×C₄	C₂×C₈	1	2	yes	—
40	C₂²×C₄	C₂×C₈	1	1	yes	characteristic
41	C₂²×C₄	C₂×C₈	1	1	yes	characteristic
42	C₂²×C₄	C₂×C₈	1	2	yes	—
43	C₂²×C₄	C₂²×C₄	1	1	yes	characteristic
44	C₂×C₈	—	2	1	no	—
45	C₂×C₈	—	2	1	no	—
46	C₂×C₈	—	2	2	no	—
47	C₂×C₈	—	2	2	no	—
48	C₂×C₈	—	2	2	no	—
49	C₂×C₈	C₄⋊C₄	1	2	yes	—
50	C₂×C₈	C₂×D₄	1	1	yes	characteristic
51	C₂×C₈	C₂×D₄	1	1	yes	characteristic
52	C₂×C₈	C₂×Q₈	1	1	yes	characteristic
53	C₂×C₈	C₂×Q₈	1	1	yes	characteristic
54	C₂×C₈	#41	1	1	yes	characteristic
55	C₂×C₈	#41	1	1	yes	characteristic
56	C₄⋊C₄	—	2	4	no	—
57	C₄²	—	2	2	no	—
58	C₄²	—	2	2	no	—
59	C₄²	C₄⋊C₄	1	2	yes	—
60	C₄²	M₁₆	1	2	yes	—
61	C₁₆	—	2	2	no	—
62	C₁₆	—	2	2	no	—
63	C₁₆	—	4	2	no	—
64	C₂³	C₂²×C₈	1	1	yes	characteristic
65	C₂×C₄	—	2	2	no	—
66	C₂×C₄	—	2	2	no	—
67	C₂×C₄	—	2	2	no	—
68	C₂×C₄	—	2	2	no	—
69	C₂×C₄	—	2	2	no	—
70	C₂×C₄	—	2	2	no	—
71	C₂×C₄	—	2	2	no	—
72	C₂×C₄	C₄⋊C₈	1	2	yes	—
73	C₂×C₄	C₄⋊C₈	1	2	yes	—
74	C₂×C₄	M₃₂	1	4	yes	—
75	C₂×C₄	C₂×C₄⋊C₄	1	1	yes	characteristic
76	C₂×C₄	C₄×D₄	1	1	yes	characteristic
77	C₂×C₄	C₄×Q₈	1	1	yes	characteristic
78	C₂×C₄	C₂×M₁₆	1	1	yes	characteristic
79	C₂×C₄	#131	1	1	yes	characteristic
80	C₂×C₄	#131	1	1	yes	characteristic
81	C₈	—	2	2	no	—
82	C₈	—	4	2	no	—
83	C₈	#121	1	1	yes	characteristic
84	C₈	C₂²⋊Q₈	1	1	yes	characteristic
85	C₈	#125	1	1	yes	characteristic
86	C₈	C₄⋊Q₈	1	1	yes	characteristic
87	C₂²	C₂×C₄⋊C₈	1	1	yes	characteristic
88	C₂²	C₈×D₄	1	1	yes	characteristic
89	C₂²	C₈×Q₈	1	1	yes	characteristic
90	C₂²	C₂×M₃₂	1	2	yes	—
91	C₂²	#504	1	2	yes	—
92	C₄	—	2	2	no	—
93	C₄	—	2	2	no	—
94	C₄	—	2	4	no	—
95	C₄	—	4	2	no	—
96	C₄	#389	1	1	yes	characteristic
97	C₄	#424	1	1	yes	characteristic
98	C₄	#436	1	1	yes	characteristic
99	C₄	#446	1	1	yes	characteristic
100	C₂	#1939	1	1	yes	characteristic
101	C₂	#2150	1	1	yes	characteristic
102	C₂	#2151	1	1	yes	characteristic
103	C₂	#2168	1	2	yes	—
104	C₂	#2183	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×C₄⋊C₁₆

C₂

yes

maximal

C₂×C₄⋊C₁₆

C₂

yes

maximal, characteristic

C₂×#346

C₂

yes

maximal, characteristic

C₈×C₄⋊C₄

C₂

yes

maximal, characteristic

#1374

C₂

yes

maximal

C₂²×C₁₆

C₂²

yes

—

C₂²×C₁₆

C₂²

yes

characteristic

C₂²×C₁₆

C₂²

yes

characteristic

C₂×C₄⋊C₈

C₄

yes

characteristic

C₂×C₄×C₈

C₂²

yes

—

C₂×C₄×C₈

C₂²

yes

characteristic

C₄×C₄⋊C₄

C₄

yes

characteristic

C₄⋊C₁₆

—

C₄⋊C₁₆

—

#346

—

#336

C₄

yes

—

C₂²×C₈

C₂×C₄

yes

—

C₂²×C₈

C₂×C₄

yes

characteristic

C₂²×C₈

C₂³

yes

characteristic

C₂×C₄⋊C₄

C₈

yes

—

C₂×C₄²

C₂×C₄

yes

—

C₂×C₄²

C₂×C₄

yes

characteristic

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

D₄

yes

—

C₂×C₁₆

Q₈

yes

—

C₄⋊C₈

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

D₄

yes

—

C₄×C₈

Q₈

yes

—

#95

C₈

yes

characteristic

#95

C₈

yes

characteristic

C₂²×C₄

C₂×C₈

yes

—

C₂²×C₄

C₂×C₈

yes

characteristic

C₂²×C₄

C₂×C₈

yes

characteristic

C₂²×C₄

C₂×C₈

yes

—

C₂²×C₄

yes

characteristic

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

C₄⋊C₄

yes

—

C₂×C₈

C₂×D₄

yes

characteristic

C₂×C₈

C₂×D₄

yes

characteristic

C₂×C₈

C₂×Q₈

yes

characteristic

C₂×C₈

C₂×Q₈

yes

C₂×C₈

yes

C₂×C₈

yes

C₄⋊C₄

—

C₄²

—

C₄²

—

C₄²

C₄⋊C₄

yes

—

C₄²

M₁₆

yes

—

C₁₆

—

C₁₆

—

C₁₆

—

C₂³

C₂²×C₈

yes

characteristic

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

C₄⋊C₈

yes

—

C₂×C₄

C₄⋊C₈

yes

—

C₂×C₄

M₃₂

yes

—

C₂×C₄

C₂×C₄⋊C₄

yes

characteristic

C₂×C₄

C₄×D₄

yes

characteristic

C₂×C₄

C₄×Q₈

yes

characteristic

C₂×C₄

C₂×M₁₆

yes

C₂×C₄

yes

C₂×C₄

yes

C₈

—

C₈

—

C₈

#121

yes

characteristic

C₈

C₂²⋊Q₈

yes

characteristic

C₈

#125

yes

characteristic

C₈

C₄⋊Q₈

yes

characteristic

C₂²

C₂×C₄⋊C₈

yes

characteristic

C₂²

C₈×D₄

yes

characteristic

C₂²

C₈×Q₈

yes

characteristic

C₂²

C₂×M₃₂

yes

—

C₂²

#504

yes

—

C₄

—

C₄

—

C₄

—

C₄

—

C₄

yes

C₄

yes

C₄

yes

C₄

yes

100

C₂

#1939

yes

characteristic

101

C₂

#2150

yes

characteristic

102

C₂

#2151

yes

characteristic

103

C₂

#2168

yes

—

104

C₂

#2183

yes

—

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

subgroup

G/H

center, Frattini, cosocle

C₂²×C₈

C₂³

commutator, socle, prime radical

C₂³

C₂²×C₈

LCS

G ▷ C₂³ ▷ C₁

UCS

C₁ ◁ C₂²×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×C₄⋊C₁₆	2
C₂×#346	2
C₈×C₄⋊C₄	2
#1374	2

subgroup

index

C₂×C₄⋊C₁₆

C₂×#346

C₈×C₄⋊C₄

#1374

Subgroups — #13160

Conjugacy classes of subgroups

Maximal subgroups