Group #13206 — subgroups

Subgroups — #13206

Order 256.

295 conjugacy classes of subgroups; only the 91 normal subgroups are shown.

normal subgroups only

This group has 295 conjugacy classes of subgroups — too many to draw. Only its 91 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×#493	C₂	1	1	yes	maximal, characteristic
2	C₂×#357	C₂	1	4	yes	maximal, complemented
3	C₂×#346	C₂	1	1	yes	maximal, characteristic, complemented
4	#1846	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂²×D₈	—	2	1	no	—
6	C₂²×D₈	C₂²	1	2	yes	—
7	C₂×#127	—	2	1	no	—
8	C₂²×C₁₆	C₂²	1	2	yes	—
9	#493	—	2	2	no	—
10	#493	C₄	1	4	yes	complemented
11	C₂×#107	C₂²	1	2	yes	—
12	C₂×#106	—	2	1	no	—
13	C₂×C₄×C₈	C₂²	1	1	yes	characteristic
14	#382	—	2	1	no	—
15	#357	—	2	8	no	—
16	#346	—	2	2	no	—
17	C₂²×D₄	—	2	2	no	—
18	C₂²×D₄	—	4	2	no	—
19	C₂×D₈	—	2	4	no	—
20	C₂×D₈	—	2	4	no	—
21	C₂×D₈	—	2	2	no	—
22	C₂×D₈	—	2	4	no	—
23	C₂×D₈	—	4	2	no	—
24	C₂×D₈	—	4	2	no	—
25	C₂×D₈	C₂×C₄	1	4	yes	—
26	C₂²×C₈	D₄	1	1	yes	characteristic
27	C₂²×C₈	C₂³	1	1	yes	characteristic
28	#127	—	2	4	no	—
29	#127	—	4	2	no	—
30	C₂×C₄⋊C₄	—	2	2	no	—
31	C₂×C₄²	D₄	1	1	yes	characteristic
32	C₂×C₁₆	—	2	2	no	—
33	C₂×C₁₆	—	2	2	no	—
34	C₂×C₁₆	D₄	1	4	yes	complemented
35	#107	—	2	4	no	—
36	#106	—	4	2	no	—
37	C₄×C₈	C₂×C₄	1	2	yes	—
38	C₄×C₈	D₄	1	2	yes	—
39	#95	—	4	2	no	—
40	C₂⁴	—	4	2	no	—
41	C₂×D₄	—	2	4	no	—
42	C₂×D₄	—	4	8	no	—
43	C₂×D₄	—	4	2	no	—
44	C₂×D₄	—	4	2	no	—
45	C₂×D₄	—	4	2	no	—
46	C₂×D₄	—	4	2	no	—
47	C₂×D₄	—	4	2	no	—
48	C₂×D₄	—	8	8	no	—
49	C₂²×C₄	—	4	2	no	—
50	C₂²×C₄	D₈	1	2	yes	—
51	C₂²×C₄	C₂×D₄	1	1	yes	characteristic
52	D₈	—	2	8	no	—
53	D₈	—	4	8	no	—
54	D₈	—	4	4	no	—
55	C₂×C₈	C₂²⋊C₄	1	2	yes	—
56	C₂×C₈	D₈	1	2	yes	—
57	C₂×C₈	SD₁₆	1	2	yes	—
58	C₂×C₈	C₂²×C₄	1	1	yes	characteristic
59	C₂×C₈	C₂×D₄	1	2	yes	—
60	C₂×C₈	C₂×D₄	1	1	yes	characteristic
61	C₂×C₈	#41	1	2	yes	—
62	C₄⋊C₄	—	4	2	no	—
63	C₄⋊C₄	—	4	2	no	—
64	C₄²	C₂²⋊C₄	1	2	yes	—
65	C₄²	SD₁₆	1	2	yes	—
66	C₁₆	—	2	4	no	—
67	C₂³	—	4	4	no	—
68	C₂³	—	8	2	no	—
69	C₂³	—	8	2	no	—
70	C₂³	—	8	8	no	—
71	C₂³	C₂×D₈	1	1	yes	characteristic
72	D₄	—	4	4	no	—
73	D₄	—	4	4	no	—
74	D₄	—	8	16	no	—
75	D₄	—	8	2	no	—
76	D₄	—	8	2	no	—
77	C₂×C₄	—	2	4	no	—
78	C₂×C₄	—	8	2	no	—
79	C₂×C₄	—	8	4	no	—
80	C₂×C₄	#102	1	4	yes	—
81	C₂×C₄	C₂×C₂²⋊C₄	1	1	yes	characteristic
82	C₂×C₄	#124	1	1	yes	characteristic
83	C₂×C₄	#127	1	1	yes	characteristic
84	C₂×C₄	C₂×D₈	1	1	yes	characteristic
85	C₂×C₄	C₂×SD₁₆	1	1	yes	characteristic
86	C₂×C₄	C₂×SD₁₆	1	1	yes	characteristic
87	C₈	#102	1	4	yes	—
88	C₈	C₄×D₄	1	2	yes	—
89	C₈	#121	1	2	yes	—
90	C₂²	—	8	8	no	—
91	C₂²	—	8	4	no	—
92	C₂²	—	16	4	no	—
93	C₂²	—	16	4	no	—
94	C₂²	C₂×#102	1	1	yes	characteristic
95	C₂²	#486	1	1	yes	characteristic
96	C₂²	#493	1	1	yes	characteristic
97	C₂²	#509	1	4	yes	—
98	C₄	—	2	4	no	—
99	C₄	—	16	2	no	—
100	C₄	#390	1	1	yes	characteristic
101	C₄	C₂×#102	1	1	yes	characteristic
102	C₄	#486	1	1	yes	characteristic
103	C₄	#492	1	1	yes	characteristic
104	C₂	—	16	4	no	—
105	C₂	#1957	1	1	yes	characteristic
106	C₂	#2140	1	2	yes	—
107	C₂	#2244	1	2	yes	—
108	C₂	#2250	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#493

C₂

yes

maximal, characteristic

C₂×#357

C₂

yes

maximal, complemented

C₂×#346

C₂

yes

maximal, characteristic, complemented

#1846

C₂

yes

maximal, characteristic, complemented

C₂²×D₈

—

C₂²×D₈

C₂²

yes

—

C₂×#127

—

C₂²×C₁₆

C₂²

yes

—

#493

—

#493

C₄

yes

complemented

C₂×#107

C₂²

yes

—

C₂×#106

—

C₂×C₄×C₈

C₂²

yes

—

—

—

C₂²×D₄

—

C₂²×D₄

—

C₂×D₈

—

C₂×D₈

—

C₂×D₈

—

C₂×D₈

—

C₂×D₈

—

C₂×D₈

—

C₂×D₈

C₂×C₄

yes

—

C₂²×C₈

D₄

yes

characteristic

C₂²×C₈

C₂³

yes

characteristic

#127

—

#127

—

C₂×C₄⋊C₄

—

C₂×C₄²

D₄

yes

characteristic

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

D₄

yes

complemented

#107

—

#106

—

C₄×C₈

C₂×C₄

yes

—

C₄×C₈

D₄

yes

—

#95

—

C₂⁴

—

C₂×D₄

—

C₂×D₄

—

C₂×D₄

—

C₂×D₄

—

C₂×D₄

—

C₂×D₄

—

C₂×D₄

—

C₂×D₄

—

C₂²×C₄

—

C₂²×C₄

D₈

yes

—

C₂²×C₄

C₂×D₄

yes

characteristic

D₈

—

D₈

—

D₈

—

C₂×C₈

C₂²⋊C₄

yes

—

C₂×C₈

D₈

yes

—

C₂×C₈

SD₁₆

yes

—

C₂×C₈

C₂²×C₄

yes

characteristic

C₂×C₈

C₂×D₄

yes

—

C₂×C₈

C₂×D₄

yes

characteristic

C₂×C₈

#41

yes

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄²

C₂²⋊C₄

yes

—

C₄²

SD₁₆

yes

—

C₁₆

—

C₂³

—

C₂³

—

C₂³

—

C₂³

—

C₂³

C₂×D₈

yes

characteristic

D₄

—

D₄

—

D₄

—

D₄

—

D₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

#102

yes

—

C₂×C₄

C₂×C₂²⋊C₄

yes

C₂×C₄

yes

C₂×C₄

yes

C₂×C₄

C₂×D₈

yes

characteristic

C₂×C₄

C₂×SD₁₆

yes

characteristic

C₂×C₄

C₂×SD₁₆

yes

characteristic

C₈

#102

yes

—

C₈

C₄×D₄

yes

—

C₈

#121

yes

—

C₂²

—

C₂²

—

C₂²

—

C₂²

—

C₂²

yes

C₂²

yes

C₂²

yes

C₂²

yes

—

C₄

—

C₄

—

100

C₄

#390

yes

characteristic

101

C₄

C₂×#102

yes

characteristic

102

C₄

#486

yes

characteristic

103

C₄

#492

yes

characteristic

104

C₂

—

105

C₂

#1957

yes

characteristic

106

C₂

#2140

yes

—

107

C₂

#2244

yes

—

108

C₂

#2250

yes

—

	subgroup	G/H
center, socle	C₂³	C₂×D₈
commutator	C₂×C₈	C₂²×C₄
Frattini, cosocle	C₂²×C₈	C₂³
prime radical	C₂×#493	C₂
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

C₂³

C₂×D₈

commutator

C₂×C₈

C₂²×C₄

Frattini, cosocle

C₂²×C₈

C₂³

prime radical

C₂×#493

C₂

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
2. C₂×#357	C₂	semidirect
3. C₂×#346	C₂	semidirect
4. #1846	C₂	semidirect
10. #493	C₄	semidirect
34. C₂×C₁₆	D₄	semidirect

H ≅ G/N

type

2. C₂×#357

C₂

semidirect

3. C₂×#346

C₂

semidirect

4. #1846

C₂

semidirect

10. #493

C₄

semidirect

34. C₂×C₁₆

D₄

semidirect

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

subgroup	index
C₂×#493	2
C₂×#357	2
C₂×#346	2
#1846	2

subgroup

index

Subgroups — #13206

Conjugacy classes of subgroups

Complements

Maximal subgroups