Group #13115 — subgroups

Subgroups — #13115

Order 256.

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

normal subgroups only

This group has 126 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	#2155	C₂	1	1	yes	maximal, characteristic
2	#1946	C₂	1	1	yes	maximal, characteristic
3	#1773	C₂	1	1	yes	maximal, characteristic
4	#1389	C₂	1	1	yes	maximal, characteristic
5	#1388	C₂	1	1	yes	maximal, characteristic
6	#1384	C₂	1	1	yes	maximal, characteristic
7	#1377	C₂	1	1	yes	maximal, characteristic
8	C₂×M₃₂	C₂²	1	1	yes	characteristic
9	C₂×M₃₂	C₂²	1	1	yes	characteristic
10	C₂×#108	C₂²	1	1	yes	characteristic
11	C₂×#108	C₂²	1	1	yes	characteristic
12	#427	C₂²	1	1	yes	characteristic
13	#427	C₂²	1	1	yes	characteristic
14	#424	—	2	1	no	—
15	#405	—	2	1	no	—
16	C₂×#97	C₂²	1	1	yes	characteristic
17	#365	C₄	1	4	yes	complemented
18	#339	—	2	1	no	—
19	#339	—	2	1	no	—
20	C₂×M₁₆	—	2	1	no	—
21	C₂×M₁₆	—	2	1	no	—
22	C₂²×C₈	D₄	1	1	yes	characteristic
23	C₂²×C₈	C₂³	1	1	yes	characteristic
24	#117	—	2	1	no	—
25	#117	—	2	1	no	—
26	C₂×C₄²	D₄	1	1	yes	characteristic
27	M₃₂	—	2	1	no	—
28	M₃₂	—	2	1	no	—
29	M₃₂	—	2	1	no	—
30	M₃₂	C₂×C₄	1	2	yes	—
31	C₂×C₁₆	—	2	1	no	—
32	C₂×C₁₆	—	2	1	no	—
33	#108	—	2	1	no	—
34	#108	—	2	1	no	—
35	#108	—	2	1	no	—
36	#108	C₂×C₄	1	2	yes	—
37	#107	—	2	1	no	—
38	#107	—	2	1	no	—
39	#106	C₂×C₄	1	2	yes	—
40	#106	Q₈	1	2	yes	—
41	C₄⋊C₈	—	4	2	no	—
42	#97	—	2	1	no	—
43	#97	—	2	1	no	—
44	#97	—	4	1	no	—
45	C₄×C₈	—	4	1	no	—
46	C₂²×C₄	—	2	1	no	—
47	C₂²×C₄	C₂×D₄	1	1	yes	characteristic
48	M₁₆	—	2	2	no	—
49	M₁₆	—	4	1	no	—
50	M₁₆	—	4	1	no	—
51	M₁₆	—	4	1	no	—
52	C₂×C₈	—	2	1	no	—
53	C₂×C₈	—	2	1	no	—
54	C₂×C₈	—	4	1	no	—
55	C₂×C₈	—	4	1	no	—
56	C₂×C₈	SD₁₆	1	2	yes	—
57	C₂×C₈	C₂²×C₄	1	1	yes	characteristic
58	C₂×C₈	C₂×Q₈	1	1	yes	characteristic
59	C₂×C₈	#41	1	1	yes	characteristic
60	C₂×C₈	#41	1	1	yes	characteristic
61	C₂×C₈	#41	1	1	yes	characteristic
62	C₂×C₈	#41	1	1	yes	characteristic
63	C₄⋊C₄	—	2	2	no	—
64	C₄⋊C₄	—	2	2	no	—
65	C₂²⋊C₄	—	4	1	no	—
66	C₂²⋊C₄	—	4	1	no	—
67	C₄²	—	2	1	no	—
68	C₄²	—	2	1	no	—
69	C₄²	—	4	1	no	—
70	C₄²	—	4	1	no	—
71	C₁₆	—	2	2	no	—
72	C₁₆	—	4	1	no	—
73	C₂³	#135	1	1	yes	characteristic
74	C₂×C₄	—	4	2	no	—
75	C₂×C₄	—	4	2	no	—
76	C₂×C₄	—	4	1	no	—
77	C₂×C₄	—	8	1	no	—
78	C₂×C₄	C₄×D₄	1	1	yes	characteristic
79	C₂×C₄	C₂²⋊Q₈	1	1	yes	characteristic
80	C₂×C₄	#123	1	1	yes	characteristic
81	C₂×C₄	C₂×SD₁₆	1	1	yes	characteristic
82	C₂×C₄	AGL(1,ℤ/8)	1	1	yes	characteristic
83	C₂×C₄	#137	1	1	yes	characteristic
84	C₈	—	2	2	no	—
85	C₈	—	4	2	no	—
86	C₈	—	8	1	no	—
87	C₈	#117	1	1	yes	characteristic
88	C₈	C₄×Q₈	1	1	yes	characteristic
89	C₈	#125	1	1	yes	characteristic
90	C₈	#126	1	1	yes	characteristic
91	C₂²	—	4	1	no	—
92	C₂²	C₄×SD₁₆	1	1	yes	characteristic
93	C₂²	#478	1	1	yes	characteristic
94	C₂²	#485	1	1	yes	characteristic
95	C₄	—	8	2	no	—
96	C₄	—	8	2	no	—
97	C₄	—	8	1	no	—
98	C₄	#387	1	1	yes	characteristic
99	C₄	#440	1	1	yes	characteristic
100	C₄	#475	1	1	yes	characteristic
101	C₄	#481	1	1	yes	characteristic
102	C₂	—	2	1	no	—
103	C₂	—	2	1	no	—
104	C₂	—	2	1	no	—
105	C₂	#1919	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#2155

C₂

yes

maximal, characteristic

#1946

C₂

yes

maximal, characteristic

#1773

C₂

yes

maximal, characteristic

#1389

C₂

yes

maximal, characteristic

#1388

C₂

yes

maximal, characteristic

#1384

C₂

yes

maximal, characteristic

#1377

C₂

yes

maximal, characteristic

C₂×M₃₂

C₂²

yes

characteristic

C₂×M₃₂

C₂²

yes

characteristic

C₂×#108

C₂²

yes

characteristic

C₂×#108

C₂²

yes

characteristic

#427

C₂²

yes

characteristic

#427

C₂²

yes

characteristic

#424

—

#405

—

C₂×#97

C₂²

yes

characteristic

#365

C₄

yes

complemented

#339

—

#339

—

C₂×M₁₆

—

C₂×M₁₆

—

C₂²×C₈

D₄

yes

characteristic

C₂²×C₈

C₂³

yes

characteristic

#117

—

#117

—

C₂×C₄²

D₄

yes

characteristic

M₃₂

—

M₃₂

—

M₃₂

—

M₃₂

C₂×C₄

yes

—

C₂×C₁₆

—

C₂×C₁₆

—

—

—

—

C₂×C₄

yes

—

#107

—

#107

—

#106

C₂×C₄

yes

—

#106

Q₈

yes

—

C₄⋊C₈

—

#97

—

#97

—

#97

—

C₄×C₈

—

C₂²×C₄

—

C₂²×C₄

C₂×D₄

yes

characteristic

M₁₆

—

M₁₆

—

M₁₆

—

M₁₆

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

SD₁₆

yes

—

C₂×C₈

C₂²×C₄

yes

characteristic

C₂×C₈

C₂×Q₈

yes

C₂×C₈

yes

C₂×C₈

yes

C₂×C₈

yes

C₂×C₈

yes

C₄⋊C₄

—

C₄⋊C₄

—

C₂²⋊C₄

—

C₂²⋊C₄

—

C₄²

—

C₄²

—

C₄²

—

C₄²

—

C₁₆

—

C₁₆

—

C₂³

#135

yes

characteristic

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

C₄×D₄

yes

characteristic

C₂×C₄

C₂²⋊Q₈

yes

characteristic

C₂×C₄

#123

yes

characteristic

C₂×C₄

C₂×SD₁₆

yes

C₂×C₄

yes

C₂×C₄

yes

C₈

—

C₈

—

C₈

—

C₈

#117

yes

characteristic

C₈

C₄×Q₈

yes

C₈

yes

C₈

yes

C₂²

—

C₂²

C₄×SD₁₆

yes

C₂²

yes

C₂²

yes

C₄

—

C₄

—

C₄

—

C₄

yes

C₄

yes

100

C₄

#475

yes

characteristic

101

C₄

#481

yes

characteristic

102

C₂

—

103

C₂

—

104

C₂

—

105

C₂

#1919

yes

characteristic

	subgroup	G/H
center	C₄	#481
commutator	C₂×C₈	C₂²×C₄
Frattini, cosocle	C₂²×C₈	C₂³
socle	C₂	#1919
prime radical	C₂³	#135
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ C₂×#97 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

subgroup

G/H

center

C₄

#481

commutator

C₂×C₈

C₂²×C₄

Frattini, cosocle

C₂²×C₈

C₂³

C₂

C₂³

G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁

UCS

C₁ ◁ C₄ ◁ C₂²×C₄ ◁ C₂×#97 ◁ G

LECS

G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

N	H ≅ G/N	type
17. #365	C₄	semidirect

H ≅ G/N

type

17. #365

C₄

semidirect

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

subgroup	index
#2155	2
#1946	2
#1773	2
#1389	2
#1388	2
#1384	2
#1377	2

subgroup

index

Subgroups — #13115

Conjugacy classes of subgroups

Complements

Maximal subgroups