Group #13181 — subgroups

Subgroups — #13181

Order 256.

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

normal subgroups only

This group has 155 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	#2159	C₂	1	2	yes	maximal
2	C₂×#364	C₂	1	1	yes	maximal, characteristic
3	#1845	C₂	1	1	yes	maximal, characteristic
4	#1772	C₂	1	1	yes	maximal, characteristic
5	#1387	C₂	1	2	yes	maximal
6	C₂×M₃₂	C₂²	1	2	yes	—
7	C₂×#108	C₂²	1	1	yes	characteristic
8	#427	C₂²	1	1	yes	characteristic
9	C₂×#107	—	2	1	no	—
10	C₂×#107	C₂²	1	2	yes	—
11	#405	—	2	1	no	—
12	C₂×C₄×C₈	C₂²	1	1	yes	characteristic
13	#384	—	2	1	no	—
14	#367	—	2	2	no	—
15	#366	—	2	2	no	—
16	#364	C₄	1	4	yes	complemented
17	#339	—	2	1	no	—
18	C₂×M₁₆	—	2	1	no	—
19	C₂²×C₈	D₄	1	1	yes	characteristic
20	C₂²×C₈	C₂³	1	1	yes	characteristic
21	#117	—	2	1	no	—
22	C₂×C₄⋊C₄	—	2	2	no	—
23	C₂×C₄⋊C₄	—	4	2	no	—
24	C₂×C₄²	D₄	1	1	yes	characteristic
25	M₃₂	—	2	2	no	—
26	M₃₂	C₂×C₄	1	4	yes	—
27	C₂×C₁₆	—	2	2	no	—
28	#108	—	2	1	no	—
29	#108	Q₈	1	2	yes	—
30	#107	—	2	2	no	—
31	#107	—	2	2	no	—
32	#107	—	4	1	no	—
33	#107	—	4	1	no	—
34	#107	D₄	1	1	yes	characteristic
35	#107	D₄	1	1	yes	characteristic
36	#106	—	2	1	no	—
37	#97	—	4	1	no	—
38	C₄×C₈	—	4	1	no	—
39	C₄×C₈	C₂×C₄	1	2	yes	—
40	C₄×C₈	Q₈	1	2	yes	—
41	C₂²×C₄	—	2	1	no	—
42	C₂²×C₄	—	4	2	no	—
43	C₂²×C₄	C₂×D₄	1	1	yes	characteristic
44	M₁₆	—	2	2	no	—
45	M₁₆	—	4	1	no	—
46	C₂×C₈	—	4	1	no	—
47	C₂×C₈	C₄⋊C₄	1	2	yes	—
48	C₂×C₈	D₈	1	1	yes	characteristic
49	C₂×C₈	D₈	1	1	yes	characteristic
50	C₂×C₈	Q₁₆	1	2	yes	—
51	C₂×C₈	C₂²×C₄	1	1	yes	characteristic
52	C₂×C₈	C₂×D₄	1	1	yes	characteristic
53	C₂×C₈	C₂×Q₈	1	1	yes	characteristic
54	C₂×C₈	C₂×Q₈	1	1	yes	characteristic
55	C₂×C₈	#41	1	1	yes	characteristic
56	C₂×C₈	#41	1	1	yes	characteristic
57	C₄⋊C₄	—	2	1	no	—
58	C₄⋊C₄	—	2	1	no	—
59	C₄⋊C₄	—	4	2	no	—
60	C₄⋊C₄	—	4	2	no	—
61	C₄⋊C₄	—	8	4	no	—
62	C₂²⋊C₄	—	4	1	no	—
63	C₄²	—	2	1	no	—
64	C₄²	—	4	1	no	—
65	C₄²	C₄⋊C₄	1	2	yes	—
66	C₁₆	—	2	4	no	—
67	C₂³	AGL(1,ℤ/8)	1	1	yes	characteristic
68	C₂×C₄	—	2	2	no	—
69	C₂×C₄	—	2	2	no	—
70	C₂×C₄	—	2	2	no	—
71	C₂×C₄	—	4	1	no	—
72	C₂×C₄	—	4	1	no	—
73	C₂×C₄	—	8	4	no	—
74	C₂×C₄	—	8	2	no	—
75	C₂×C₄	C₂×C₄⋊C₄	1	1	yes	characteristic
76	C₂×C₄	#121	1	1	yes	characteristic
77	C₂×C₄	C₂²⋊Q₈	1	1	yes	characteristic
78	C₂×C₄	C₂×D₈	1	1	yes	characteristic
79	C₂×C₄	C₂×Q₁₆	1	1	yes	characteristic
80	C₂×C₄	#137	1	1	yes	characteristic
81	C₈	—	4	2	no	—
82	C₈	#107	1	2	yes	—
83	C₈	#107	1	2	yes	—
84	C₈	C₄×D₄	1	1	yes	characteristic
85	C₈	C₄×Q₈	1	1	yes	characteristic
86	C₈	#125	1	1	yes	characteristic
87	C₈	C₄⋊Q₈	1	1	yes	characteristic
88	C₂²	—	2	1	no	—
89	C₂²	—	2	1	no	—
90	C₂²	#428	1	1	yes	characteristic
91	C₂²	#459	1	1	yes	characteristic
92	C₂²	#477	1	1	yes	characteristic
93	C₄	—	2	4	no	—
94	C₄	—	8	1	no	—
95	C₄	—	8	1	no	—
96	C₄	—	16	2	no	—
97	C₄	#389	1	1	yes	characteristic
98	C₄	C₂×#107	1	1	yes	characteristic
99	C₄	#462	1	1	yes	characteristic
100	C₄	#474	1	1	yes	characteristic
101	C₂	—	2	1	no	—
102	C₂	—	2	1	no	—
103	C₂	#1947	1	1	yes	characteristic
104	C₂	#2213	1	1	yes	characteristic
105	C₂	#2214	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#2159

C₂

yes

maximal

C₂×#364

C₂

yes

maximal, characteristic

#1845

C₂

yes

maximal, characteristic

#1772

C₂

yes

maximal, characteristic

#1387

C₂

yes

maximal

C₂×M₃₂

C₂²

yes

—

C₂×#108

C₂²

yes

characteristic

#427

C₂²

yes

characteristic

C₂×#107

—

C₂×#107

C₂²

yes

—

#405

—

C₂×C₄×C₈

C₂²

yes

—

—

—

C₄

yes

complemented

#339

—

C₂×M₁₆

—

C₂²×C₈

D₄

yes

characteristic

C₂²×C₈

C₂³

yes

characteristic

#117

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄²

D₄

yes

characteristic

M₃₂

—

M₃₂

C₂×C₄

yes

—

C₂×C₁₆

—

#108

—

#108

Q₈

yes

—

—

—

—

—

D₄

yes

characteristic

#107

D₄

yes

characteristic

#106

—

#97

—

C₄×C₈

—

C₄×C₈

C₂×C₄

yes

—

C₄×C₈

Q₈

yes

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

C₂×D₄

yes

characteristic

M₁₆

—

M₁₆

—

C₂×C₈

—

C₂×C₈

C₄⋊C₄

yes

—

C₂×C₈

D₈

yes

characteristic

C₂×C₈

D₈

yes

characteristic

C₂×C₈

Q₁₆

yes

—

C₂×C₈

C₂²×C₄

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

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₂²⋊C₄

—

C₄²

—

C₄²

—

C₄²

C₄⋊C₄

yes

—

C₁₆

—

C₂³

AGL(1,ℤ/8)

yes

characteristic

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

C₂×C₄⋊C₄

yes

characteristic

C₂×C₄

#121

yes

characteristic

C₂×C₄

C₂²⋊Q₈

yes

characteristic

C₂×C₄

C₂×D₈

yes

characteristic

C₂×C₄

C₂×Q₁₆

yes

characteristic

C₂×C₄

#137

yes

characteristic

C₈

—

C₈

#107

yes

—

C₈

#107

yes

—

C₈

C₄×D₄

yes

characteristic

C₈

C₄×Q₈

yes

characteristic

C₈

#125

yes

characteristic

C₈

C₄⋊Q₈

yes

characteristic

C₂²

—

C₂²

—

C₂²

yes

C₂²

yes

C₂²

yes

C₄

—

C₄

—

C₄

—

C₄

—

C₄

yes

C₄

yes

C₄

yes

100

C₄

#474

yes

characteristic

101

C₂

—

102

C₂

—

103

C₂

#1947

yes

characteristic

104

C₂

#2213

yes

characteristic

105

C₂

#2214

yes

characteristic

	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
16. #364	C₄	semidirect

H ≅ G/N

type

16. #364

C₄

semidirect

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

subgroup	index
#2159	2
C₂×#364	2
#1845	2
#1772	2
#1387	2

subgroup

index

Subgroups — #13181

Conjugacy classes of subgroups

Complements

Maximal subgroups