Group #13439 — subgroups

Subgroups — #13439

Order 256.

209 conjugacy classes of subgroups; only the 55 normal subgroups are shown.

normal subgroups only

This group has 209 conjugacy classes of subgroups — too many to draw. Only its 55 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×D₁₆	C₂	1	1	yes	maximal, characteristic
2	#2151	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×#361	C₂	1	2	yes	maximal, complemented
4	#1978	C₂	1	2	yes	maximal, complemented
5	#1381	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂²×D₈	C₂²	1	2	yes	—
7	C₂×D₁₆	—	2	4	no	—
8	C₂×D₁₆	—	2	2	no	—
9	C₂×M₃₂	C₂²	1	1	yes	characteristic
10	C₂²×C₁₆	C₂²	1	1	yes	characteristic
11	C₂×#108	C₂²	1	2	yes	—
12	#418	—	2	2	no	—
13	C₂×#100	—	2	2	no	—
14	#405	C₂²	1	1	yes	characteristic
15	C₄⋊C₁₆	—	2	2	no	—
16	#361	—	2	4	no	—
17	C₂²×D₄	—	2	2	no	—
18	C₂×D₈	—	2	2	no	—
19	C₂×D₈	—	2	2	no	—
20	C₂×D₈	—	2	2	no	—
21	C₂×D₈	—	2	2	no	—
22	C₂×D₈	—	2	2	no	—
23	C₂×D₈	D₄	1	4	yes	—
24	C₂×M₁₆	—	2	2	no	—
25	C₂×M₁₆	D₄	1	1	yes	characteristic
26	C₂²×C₈	C₂³	1	1	yes	characteristic
27	#117	D₄	1	1	yes	characteristic
28	D₁₆	—	2	8	no	—
29	M₃₂	—	2	2	no	—
30	C₂×C₁₆	—	2	2	no	—
31	C₂×C₁₆	—	2	1	no	—
32	C₂×C₁₆	D₄	1	2	yes	—
33	#108	—	2	2	no	—
34	#108	—	2	2	no	—
35	#102	—	4	4	no	—
36	#100	—	4	4	no	—
37	#97	—	2	1	no	—
38	C₄×C₈	—	2	1	no	—
39	C₂⁴	—	4	2	no	—
40	C₂×D₄	—	2	4	no	—
41	C₂×D₄	—	4	2	no	—
42	C₂×D₄	—	4	2	no	—
43	C₂×D₄	—	4	2	no	—
44	C₂×D₄	—	4	2	no	—
45	C₂×D₄	—	4	2	no	—
46	C₂²×C₄	C₂×D₄	1	1	yes	characteristic
47	D₈	—	2	4	no	—
48	D₈	—	2	4	no	—
49	D₈	—	4	2	no	—
50	D₈	—	4	2	no	—
51	M₁₆	—	2	2	no	—
52	M₁₆	—	4	2	no	—
53	M₁₆	—	4	2	no	—
54	C₂×C₈	—	2	1	no	—
55	C₂×C₈	—	4	2	no	—
56	C₂×C₈	C₂×D₄	1	2	yes	—
57	C₂×C₈	C₂×D₄	1	1	yes	characteristic
58	C₂×C₈	#41	1	1	yes	characteristic
59	C₂×C₈	#41	1	2	yes	—
60	C₄⋊C₄	D₈	1	2	yes	—
61	C₂²⋊C₄	—	2	1	no	—
62	C₄²	—	2	1	no	—
63	C₁₆	—	2	2	no	—
64	C₁₆	—	4	1	no	—
65	C₂³	—	4	4	no	—
66	C₂³	—	8	2	no	—
67	C₂³	—	8	2	no	—
68	C₂³	—	16	4	no	—
69	C₂³	#135	1	1	yes	characteristic
70	D₄	—	4	4	no	—
71	D₄	—	4	4	no	—
72	D₄	—	8	2	no	—
73	D₄	—	8	2	no	—
74	C₂×C₄	—	2	2	no	—
75	C₂×C₄	#121	1	2	yes	—
76	C₂×C₄	#121	1	1	yes	characteristic
77	C₂×C₄	C₂×D₈	1	1	yes	characteristic
78	C₂×C₄	AGL(1,ℤ/8)	1	2	yes	—
79	C₈	—	4	1	no	—
80	C₈	—	8	2	no	—
81	C₈	#120	1	1	yes	characteristic
82	C₈	#123	1	2	yes	—
83	C₈	#124	1	1	yes	characteristic
84	C₂²	—	2	2	no	—
85	C₂²	—	8	4	no	—
86	C₂²	—	16	4	no	—
87	C₂²	—	16	4	no	—
88	C₂²	—	16	4	no	—
89	C₂²	#463	1	2	yes	—
90	C₂²	#466	1	1	yes	characteristic
91	C₄	—	4	2	no	—
92	C₄	#394	1	1	yes	characteristic
93	C₄	#459	1	2	yes	—
94	C₄	#469	1	1	yes	characteristic
95	C₂	—	2	2	no	—
96	C₂	—	16	4	no	—
97	C₂	#2054	1	1	yes	characteristic
98	C₂	#2221	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂²×D₁₆

C₂

yes

maximal, characteristic

#2151

C₂

yes

maximal, characteristic, complemented

C₂×#361

C₂

yes

maximal, complemented

#1978

C₂

yes

maximal, complemented

#1381

C₂

yes

maximal, characteristic, complemented

C₂²×D₈

C₂²

yes

—

C₂×D₁₆

—

C₂×D₁₆

—

C₂×M₃₂

C₂²

yes

characteristic

C₂²×C₁₆

C₂²

yes

characteristic

C₂×#108

C₂²

yes

—

#418

—

C₂×#100

—

#405

C₂²

yes

characteristic

C₄⋊C₁₆

—

#361

—

C₂²×D₄

—

C₂×D₈

—

C₂×D₈

—

C₂×D₈

—

C₂×D₈

—

C₂×D₈

—

C₂×D₈

D₄

yes

—

C₂×M₁₆

—

C₂×M₁₆

D₄

yes

characteristic

C₂²×C₈

C₂³

yes

characteristic

#117

D₄

yes

characteristic

D₁₆

—

M₃₂

—

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

D₄

yes

—

—

—

—

—

—

C₄×C₈

—

C₂⁴

—

C₂×D₄

—

C₂×D₄

—

C₂×D₄

—

C₂×D₄

—

C₂×D₄

—

C₂×D₄

—

C₂²×C₄

C₂×D₄

yes

characteristic

D₈

—

D₈

—

D₈

—

D₈

—

M₁₆

—

M₁₆

—

M₁₆

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

C₂×D₄

yes

—

C₂×C₈

C₂×D₄

yes

C₂×C₈

yes

C₂×C₈

yes

—

C₄⋊C₄

D₈

yes

—

C₂²⋊C₄

—

C₄²

—

C₁₆

—

C₁₆

—

C₂³

—

C₂³

—

C₂³

—

C₂³

—

C₂³

#135

yes

characteristic

D₄

—

D₄

—

D₄

—

D₄

—

C₂×C₄

—

C₂×C₄

#121

yes

—

C₂×C₄

#121

yes

characteristic

C₂×C₄

C₂×D₈

yes

characteristic

C₂×C₄

AGL(1,ℤ/8)

yes

—

C₈

—

C₈

—

C₈

#120

yes

characteristic

C₈

#123

yes

—

C₈

#124

yes

characteristic

C₂²

—

C₂²

—

C₂²

—

C₂²

—

C₂²

—

C₂²

#463

yes

—

C₂²

#466

yes

characteristic

C₄

—

C₄

#394

yes

characteristic

C₄

#459

yes

—

C₄

#469

yes

characteristic

C₂

—

C₂

—

C₂

#2054

yes

characteristic

C₂

#2221

yes

—

	subgroup	G/H
center, socle	C₂²	#466
commutator, Frattini, cosocle	C₂²×C₈	C₂³
prime radical	C₂²×D₁₆	C₂
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #405 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

subgroup

G/H

center, socle

C₂²

#466

commutator, Frattini, cosocle

C₂²×C₈

C₂³

prime radical

C₂²×D₁₆

C₂

LCS

G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

UCS

C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #405 ◁ G

LECS

G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

N	H ≅ G/N	type
2. #2151	C₂	semidirect
3. C₂×#361	C₂	semidirect
4. #1978	C₂	semidirect
5. #1381	C₂	semidirect

H ≅ G/N

type

2. #2151

C₂

semidirect

3. C₂×#361

C₂

semidirect

4. #1978

C₂

semidirect

5. #1381

C₂

semidirect

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

subgroup	index
C₂²×D₁₆	2
#2151	2
C₂×#361	2
#1978	2
#1381	2

subgroup

index

C₂²×D₁₆

Subgroups — #13439

Conjugacy classes of subgroups

Complements

Maximal subgroups