Group #13240 — subgroups

Subgroups — #13240

Order 256.

215 conjugacy classes of subgroups; only the 79 normal subgroups are shown.

normal subgroups only

This group has 215 conjugacy classes of subgroups — too many to draw. Only its 79 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	C₂²×Q₃₂	C₂	1	1	yes	maximal, characteristic
2	C₂×#367	C₂	1	1	yes	maximal, characteristic
3	C₂×#358	C₂	1	2	yes	maximal
4	C₂×#346	C₂	1	1	yes	maximal, characteristic
5	#1969	C₂	1	2	yes	maximal
6	C₂²×Q₁₆	C₂²	1	2	yes	—
7	C₂×Q₃₂	—	2	2	no	—
8	C₂×Q₃₂	—	2	2	no	—
9	C₂×Q₃₂	C₄	1	4	yes	complemented
10	C₂²×C₁₆	C₂²	1	1	yes	characteristic
11	C₂²×C₁₆	C₂²	1	1	yes	characteristic
12	C₂×#107	C₂²	1	2	yes	—
13	C₂×#103	—	2	2	no	—
14	C₂×C₄×C₈	C₂²	1	1	yes	characteristic
15	#391	—	2	2	no	—
16	#367	—	2	2	no	—
17	#358	—	2	4	no	—
18	#346	—	2	2	no	—
19	C₂²×Q₈	—	2	2	no	—
20	C₂×Q₁₆	—	2	2	no	—
21	C₂×Q₁₆	—	2	2	no	—
22	C₂×Q₁₆	—	2	2	no	—
23	C₂×Q₁₆	—	2	2	no	—
24	C₂×Q₁₆	—	2	2	no	—
25	C₂×Q₁₆	C₂×C₄	1	4	yes	—
26	C₂²×C₈	D₄	1	1	yes	characteristic
27	C₂²×C₈	C₂³	1	1	yes	characteristic
28	C₂×C₄⋊C₄	—	2	2	no	—
29	C₂×C₄²	D₄	1	1	yes	characteristic
30	Q₃₂	—	2	8	no	—
31	C₂×C₁₆	—	2	1	no	—
32	C₂×C₁₆	—	2	1	no	—
33	C₂×C₁₆	C₂×C₄	1	2	yes	—
34	C₂×C₁₆	D₄	1	2	yes	—
35	C₂×C₁₆	D₄	1	2	yes	—
36	C₂×C₁₆	D₄	1	2	yes	—
37	#107	—	2	2	no	—
38	#107	—	2	2	no	—
39	#103	—	4	4	no	—
40	C₄×C₈	—	2	1	no	—
41	C₄×C₈	—	2	1	no	—
42	#95	—	4	4	no	—
43	C₂×Q₈	—	2	4	no	—
44	C₂×Q₈	—	4	2	no	—
45	C₂×Q₈	—	4	2	no	—
46	C₂×Q₈	—	4	2	no	—
47	C₂×Q₈	—	4	2	no	—
48	C₂×Q₈	—	4	2	no	—
49	C₂²×C₄	—	4	2	no	—
50	C₂²×C₄	—	4	2	no	—
51	C₂²×C₄	D₈	1	2	yes	—
52	C₂²×C₄	C₂×D₄	1	1	yes	characteristic
53	Q₁₆	—	2	4	no	—
54	Q₁₆	—	2	4	no	—
55	Q₁₆	—	4	2	no	—
56	Q₁₆	—	4	2	no	—
57	C₂×C₈	—	2	1	no	—
58	C₂×C₈	—	2	1	no	—
59	C₂×C₈	D₈	1	2	yes	—
60	C₂×C₈	C₂²×C₄	1	1	yes	characteristic
61	C₂×C₈	C₂×D₄	1	1	yes	characteristic
62	C₂×C₈	C₂×D₄	1	1	yes	characteristic
63	C₂×C₈	C₂×D₄	1	1	yes	characteristic
64	C₂×C₈	#41	1	1	yes	characteristic
65	C₂×C₈	#41	1	1	yes	characteristic
66	C₄⋊C₄	—	4	2	no	—
67	C₄⋊C₄	—	4	2	no	—
68	C₄²	—	2	1	no	—
69	C₄²	—	2	1	no	—
70	C₁₆	—	2	2	no	—
71	C₁₆	C₂²⋊C₄	1	4	yes	—
72	C₂³	C₂×D₈	1	1	yes	characteristic
73	Q₈	—	4	4	no	—
74	Q₈	—	4	4	no	—
75	Q₈	—	8	2	no	—
76	Q₈	—	8	2	no	—
77	C₂×C₄	—	2	2	no	—
78	C₂×C₄	—	2	4	no	—
79	C₂×C₄	—	4	4	no	—
80	C₂×C₄	—	8	2	no	—
81	C₂×C₄	—	8	2	no	—
82	C₂×C₄	—	8	2	no	—
83	C₂×C₄	—	8	4	no	—
84	C₂×C₄	C₄×D₄	1	1	yes	characteristic
85	C₂×C₄	#121	1	1	yes	characteristic
86	C₂×C₄	#127	1	1	yes	characteristic
87	C₂×C₄	C₂×D₈	1	1	yes	characteristic
88	C₂×C₄	#135	1	1	yes	characteristic
89	C₂×C₄	#135	1	1	yes	characteristic
90	C₈	—	2	2	no	—
91	C₈	C₂×C₂²⋊C₄	1	1	yes	characteristic
92	C₈	C₄×D₄	1	1	yes	characteristic
93	C₈	#121	1	1	yes	characteristic
94	C₈	#124	1	1	yes	characteristic
95	C₂²	C₄×D₈	1	1	yes	characteristic
96	C₂²	#466	1	1	yes	characteristic
97	C₂²	#493	1	1	yes	characteristic
98	C₂²	#510	1	4	yes	—
99	C₄	—	4	2	no	—
100	C₄	—	8	4	no	—
101	C₄	—	16	2	no	—
102	C₄	#390	1	1	yes	characteristic
103	C₄	C₄×D₈	1	1	yes	characteristic
104	C₄	#466	1	1	yes	characteristic
105	C₄	#495	1	1	yes	characteristic
106	C₂	#1970	1	1	yes	characteristic
107	C₂	#2176	1	2	yes	—
108	C₂	#2219	1	2	yes	—
109	C₂	#2251	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂²×Q₃₂

C₂

yes

maximal, characteristic

C₂×#367

C₂

yes

maximal, characteristic

C₂×#358

C₂

yes

maximal

C₂×#346

C₂

yes

maximal, characteristic

#1969

C₂

yes

maximal

C₂²×Q₁₆

C₂²

yes

—

C₂×Q₃₂

—

C₂×Q₃₂

—

C₂×Q₃₂

C₄

yes

complemented

C₂²×C₁₆

C₂²

yes

characteristic

C₂²×C₁₆

C₂²

yes

characteristic

C₂×#107

C₂²

yes

—

C₂×#103

—

C₂×C₄×C₈

C₂²

yes

—

—

—

—

C₂²×Q₈

—

C₂×Q₁₆

—

C₂×Q₁₆

—

C₂×Q₁₆

—

C₂×Q₁₆

—

C₂×Q₁₆

—

C₂×Q₁₆

C₂×C₄

yes

—

C₂²×C₈

D₄

yes

characteristic

C₂²×C₈

C₂³

yes

characteristic

C₂×C₄⋊C₄

—

C₂×C₄²

D₄

yes

characteristic

Q₃₂

—

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

C₂×C₄

yes

—

C₂×C₁₆

D₄

yes

—

C₂×C₁₆

D₄

yes

—

C₂×C₁₆

D₄

yes

—

#107

—

#107

—

#103

—

C₄×C₈

—

C₄×C₈

—

#95

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

D₈

yes

—

C₂²×C₄

C₂×D₄

yes

characteristic

Q₁₆

—

Q₁₆

—

Q₁₆

—

Q₁₆

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

D₈

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

—

C₄⋊C₄

—

C₄²

—

C₄²

—

C₁₆

—

C₁₆

C₂²⋊C₄

yes

—

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

C₄×D₄

yes

C₂×C₄

yes

C₂×C₄

yes

C₂×C₄

C₂×D₈

yes

C₂×C₄

yes

C₂×C₄

yes

C₈

—

C₈

C₂×C₂²⋊C₄

yes

characteristic

C₈

C₄×D₄

yes

C₈

yes

C₈

yes

C₂²

C₄×D₈

yes

C₂²

yes

C₂²

yes

C₂²

yes

—

C₄

—

100

C₄

—

101

C₄

—

102

C₄

#390

yes

characteristic

103

C₄

C₄×D₈

yes

characteristic

104

C₄

#466

yes

characteristic

105

C₄

#495

yes

characteristic

106

C₂

#1970

yes

characteristic

107

C₂

#2176

yes

—

108

C₂

#2219

yes

—

109

C₂

#2251

yes

—

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

C₂×C₈

C₂²×C₄

Frattini, cosocle

C₂²×C₈

C₂³

LCS

G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁

UCS

C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ G

LECS

G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

N	H ≅ G/N	type
9. C₂×Q₃₂	C₄	semidirect

H ≅ G/N

type

9. C₂×Q₃₂

C₄

semidirect

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

subgroup	index
C₂²×Q₃₂	2
C₂×#367	2
C₂×#358	2
C₂×#346	2
#1969	2

subgroup

index

C₂²×Q₃₂

Subgroups — #13240

Conjugacy classes of subgroups

Complements

Maximal subgroups