Group #13132447 — subgroups

Subgroups — Dic₇⋊ΓL(1,𝔽₁₆) #13132447

Order 1680.

104 conjugacy classes of subgroups; only the 40 normal subgroups are shown.

normal subgroups only

This group has 104 conjugacy classes of subgroups — too many to draw. Only its 40 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×C₃⋊(C₇⋊AGL(1,𝔽₅))	C₂	1	1	yes	maximal, characteristic
2	C₂×C₇×ΓL(1,𝔽₁₆)	C₂	1	1	yes	maximal, characteristic
3	C₃×D₅×Dic₇	C₂	1	1	yes	maximal, characteristic
4	Dic₇⋊AGL(1,𝔽₅)	—	3	1	no	maximal, Hall
5	C₂×C₃×C₇×D₅	C₂²	1	1	yes	characteristic
6	C₃⋊(C₇⋊AGL(1,𝔽₅))	—	2	1	no	—
7	C₇×ΓL(1,𝔽₁₆)	—	2	1	no	—
8	C₃×Dic₃₅	C₄	1	1	yes	characteristic, complemented
9	C₃×C₅×Dic₇	C₄	1	1	yes	characteristic, complemented
10	Dic₇⋊Dic₃	—	5	1	no	maximal, Hall
11	C₂×C₇⋊AGL(1,𝔽₅)	—	3	1	no	—
12	C₂×C₇×AGL(1,𝔽₅)	—	3	1	no	—
13	D₅×Dic₇	D₃	1	1	yes	characteristic
14	C₄⋊ΓL(1,𝔽₁₆)	—	7	1	no	maximal, Hall
15	C₂×C₃×C₅×C₇	C₂×C₄	1	1	yes	characteristic
16	C₃×C₇×D₅	D₄	1	1	yes	characteristic
17	C₃×C₇×D₅	Q₈	1	1	yes	characteristic
18	C₂×Dic₂₁	—	5	1	no	—
19	C₂×C₇×Dic₃	—	5	1	no	—
20	C₂×C₃×Dic₇	—	5	1	no	—
21	C₂×C₇×D₅	C₂×D₃	1	1	yes	characteristic
22	C₇⋊AGL(1,𝔽₅)	—	6	1	no	—
23	C₇×AGL(1,𝔽₅)	—	6	1	no	—
24	Dic₃₅	Dic₃	1	1	yes	characteristic, complemented
25	C₅×Dic₇	Dic₃	1	1	yes	characteristic, complemented
26	C₂×ΓL(1,𝔽₁₆)	—	7	1	no	—
27	C₂×ΓL(1,𝔽₁₆)	D₇	1	1	yes	characteristic
28	C₃×C₄×D₅	—	7	1	no	—
29	Dic₇⋊C₄	—	15	1	no	Hall
30	C₃×C₅×C₇	C₄⋊C₄	1	1	yes	characteristic, complemented, Hall
31	C₂²×C₃×C₇	—	5	1	no	—
32	Dic₂₁	—	10	1	no	—
33	C₃×Dic₇	—	5	1	no	—
34	C₃×Dic₇	AGL(1,𝔽₅)	1	1	yes	characteristic, complemented
35	C₇×Dic₃	—	10	1	no	—
36	C₄⋊AGL(1,𝔽₅)	—	21	1	no	Hall
37	C₂×C₅×C₇	C₂×Dic₃	1	1	yes	characteristic
38	C₇×D₅	Dic₆	1	1	yes	characteristic
39	C₇×D₅	D₁₂	1	1	yes	characteristic
40	C₂×C₃×D₅	C₂×D₇	1	1	yes	characteristic
41	ΓL(1,𝔽₁₆)	—	2	1	no	—
42	ΓL(1,𝔽₁₆)	—	14	1	no	—
43	C₃×C₄×C₅	—	7	1	no	—
44	C₃×Dic₅	—	7	1	no	—
45	C₂×C₄×C₇	—	15	1	no	—
46	C₂×Dic₇	—	5	1	no	—
47	C₂×Dic₇	—	15	1	no	—
48	C₄⋊Dic₃	—	35	1	no	Hall
49	C₂×C₃×C₇	—	5	1	no	—
50	C₂×C₃×C₇	—	5	1	no	—
51	C₂×C₃×C₇	C₂×AGL(1,𝔽₅)	1	1	yes	characteristic
52	C₂×AGL(1,𝔽₅)	—	3	1	no	—
53	C₂×AGL(1,𝔽₅)	—	21	1	no	—
54	C₄×D₅	—	7	1	no	—
55	C₅×C₇	C₄⋊Dic₃	1	1	yes	characteristic, complemented, Hall
56	C₂×C₃×C₅	C₄×D₇	1	1	yes	characteristic
57	C₃×D₅	Dic₁₄	1	1	yes	characteristic
58	C₃×D₅	C₂²⋊D₇	1	1	yes	characteristic
59	C₂²×C₇	—	5	1	no	—
60	C₄×C₇	—	30	1	no	—
61	Dic₇	—	5	1	no	—
62	Dic₇	—	30	1	no	—
63	Dic₇	ΓL(1,𝔽₁₆)	1	1	yes	characteristic, complemented
64	C₂×C₃×C₄	—	35	1	no	—
65	C₂×Dic₃	—	5	1	no	—
66	C₂×Dic₃	—	35	1	no	—
67	C₃×C₇	C₄⋊AGL(1,𝔽₅)	1	1	yes	characteristic, complemented, Hall
68	C₂×D₅	D₃×D₇	1	1	yes	characteristic
69	AGL(1,𝔽₅)	—	6	1	no	—
70	AGL(1,𝔽₅)	—	42	1	no	—
71	C₄×C₅	—	7	1	no	—
72	Dic₅	—	7	1	no	—
73	C₄⋊C₄	—	105	1	no	Sylow
74	C₃×C₅	Dic₇⋊C₄	1	1	yes	characteristic, complemented, Hall
75	C₂×C₇	—	5	1	no	—
76	C₂×C₇	—	5	1	no	—
77	C₂×C₇	C₂×ΓL(1,𝔽₁₆)	1	1	yes	characteristic
78	C₂²×C₃	—	5	1	no	—
79	C₃×C₄	—	7	1	no	—
80	C₃×C₄	—	35	1	no	—
81	Dic₃	—	10	1	no	—
82	Dic₃	—	70	1	no	—
83	C₂×C₅	Dic₃×D₇	1	1	yes	characteristic
84	D₅	#4247	1	1	yes	characteristic
85	D₅	#4248	1	1	yes	characteristic
86	C₂×C₄	—	15	1	no	—
87	C₂×C₄	—	35	1	no	—
88	C₂×C₄	—	105	1	no	—
89	C₇	C₄⋊ΓL(1,𝔽₁₆)	1	1	yes	characteristic, complemented, Sylow
90	C₂×C₃	—	5	1	no	—
91	C₂×C₃	—	5	1	no	—
92	C₂×C₃	D₇×AGL(1,𝔽₅)	1	1	yes	characteristic
93	C₅	Dic₇⋊Dic₃	1	1	yes	characteristic, complemented, Sylow
94	C₂²	—	5	1	no	—
95	C₄	—	7	1	no	—
96	C₄	—	30	1	no	—
97	C₄	—	35	1	no	—
98	C₄	—	210	1	no	—
99	C₃	Dic₇⋊AGL(1,𝔽₅)	1	1	yes	characteristic, complemented, Sylow
100	C₂	—	5	1	no	—
101	C₂	—	5	1	no	—
102	C₂	D₇×ΓL(1,𝔽₁₆)	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×C₃⋊(C₇⋊AGL(1,𝔽₅))

C₂

yes

maximal, characteristic

C₂×C₇×ΓL(1,𝔽₁₆)

C₂

yes

maximal, characteristic

C₃×D₅×Dic₇

C₂

yes

maximal, characteristic

Dic₇⋊AGL(1,𝔽₅)

—

maximal, Hall

C₂×C₃×C₇×D₅

C₂²

yes

characteristic

C₃⋊(C₇⋊AGL(1,𝔽₅))

—

C₇×ΓL(1,𝔽₁₆)

—

C₃×Dic₃₅

C₄

yes

characteristic, complemented

C₃×C₅×Dic₇

C₄

yes

characteristic, complemented

Dic₇⋊Dic₃

—

maximal, Hall

C₂×C₇⋊AGL(1,𝔽₅)

—

C₂×C₇×AGL(1,𝔽₅)

—

D₅×Dic₇

D₃

yes

characteristic

C₄⋊ΓL(1,𝔽₁₆)

—

maximal, Hall

C₂×C₃×C₅×C₇

C₂×C₄

yes

characteristic

C₃×C₇×D₅

D₄

yes

characteristic

C₃×C₇×D₅

Q₈

yes

characteristic

C₂×Dic₂₁

—

C₂×C₇×Dic₃

—

C₂×C₃×Dic₇

—

C₂×C₇×D₅

C₂×D₃

yes

characteristic

C₇⋊AGL(1,𝔽₅)

—

C₇×AGL(1,𝔽₅)

—

Dic₃₅

Dic₃

yes

characteristic, complemented

C₅×Dic₇

Dic₃

yes

characteristic, complemented

C₂×ΓL(1,𝔽₁₆)

—

C₂×ΓL(1,𝔽₁₆)

D₇

yes

characteristic

C₃×C₄×D₅

—

Dic₇⋊C₄

—

Hall

C₃×C₅×C₇

C₄⋊C₄

yes

characteristic, complemented, Hall

C₂²×C₃×C₇

—

Dic₂₁

—

C₃×Dic₇

—

C₃×Dic₇

AGL(1,𝔽₅)

yes

characteristic, complemented

C₇×Dic₃

—

C₄⋊AGL(1,𝔽₅)

—

Hall

C₂×C₅×C₇

C₂×Dic₃

yes

characteristic

C₇×D₅

Dic₆

yes

characteristic

C₇×D₅

D₁₂

yes

characteristic

C₂×C₃×D₅

C₂×D₇

yes

characteristic

ΓL(1,𝔽₁₆)

—

ΓL(1,𝔽₁₆)

—

C₃×C₄×C₅

—

C₃×Dic₅

—

C₂×C₄×C₇

—

C₂×Dic₇

—

C₂×Dic₇

—

C₄⋊Dic₃

—

Hall

C₂×C₃×C₇

—

C₂×C₃×C₇

—

C₂×C₃×C₇

yes

—

—

C₄×D₅

—

C₅×C₇

C₄⋊Dic₃

yes

characteristic, complemented, Hall

C₂×C₃×C₅

C₄×D₇

yes

characteristic

C₃×D₅

Dic₁₄

yes

characteristic

C₃×D₅

C₂²⋊D₇

yes

characteristic

C₂²×C₇

—

C₄×C₇

—

Dic₇

—

Dic₇

—

Dic₇

ΓL(1,𝔽₁₆)

yes

characteristic, complemented

C₂×C₃×C₄

—

C₂×Dic₃

—

C₂×Dic₃

—

C₃×C₇

C₄⋊AGL(1,𝔽₅)

yes

characteristic, complemented, Hall

C₂×D₅

D₃×D₇

yes

characteristic

AGL(1,𝔽₅)

—

AGL(1,𝔽₅)

—

C₄×C₅

—

Dic₅

—

C₄⋊C₄

—

105

Sylow

C₃×C₅

Dic₇⋊C₄

yes

characteristic, complemented, Hall

C₂×C₇

—

C₂×C₇

—

C₂×C₇

C₂×ΓL(1,𝔽₁₆)

yes

characteristic

C₂²×C₃

—

C₃×C₄

—

C₃×C₄

—

Dic₃

—

Dic₃

—

C₂×C₅

Dic₃×D₇

yes

D₅

yes

D₅

yes

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

105

—

C₇

C₄⋊ΓL(1,𝔽₁₆)

yes

characteristic, complemented, Sylow

C₂×C₃

—

C₂×C₃

—

C₂×C₃

D₇×AGL(1,𝔽₅)

yes

characteristic

C₅

Dic₇⋊Dic₃

yes

characteristic, complemented, Sylow

C₂²

—

C₄

—

C₄

—

C₄

—

C₄

—

210

—

C₃

Dic₇⋊AGL(1,𝔽₅)

yes

characteristic, complemented, Sylow

100

C₂

—

101

C₂

—

102

C₂

D₇×ΓL(1,𝔽₁₆)

yes

characteristic

	subgroup	G/H
center, Frattini, hypercenter	C₂	D₇×ΓL(1,𝔽₁₆)
commutator, Fitting, socle	C₂×C₃×C₅×C₇	C₂×C₄
prime radical, cosocle	C₂×C₃×C₇×D₅	C₂²
hypocenter	C₃×C₅×C₇	C₄⋊C₄
LCS	G ▷ C₂×C₃×C₅×C₇ ▷ C₃×C₅×C₇
UCS	C₁ ◁ C₂

subgroup

G/H

center, Frattini, hypercenter

C₂

D₇×ΓL(1,𝔽₁₆)

commutator, Fitting, socle

C₂×C₃×C₅×C₇

C₂×C₄

prime radical, cosocle

C₂×C₃×C₇×D₅

C₂²

hypocenter

C₃×C₅×C₇

C₄⋊C₄

LCS

G ▷ C₂×C₃×C₅×C₇ ▷ C₃×C₅×C₇

UCS

C₁ ◁ C₂

N	H ≅ G/N	type
8. C₃×Dic₃₅	C₄	semidirect
9. C₃×C₅×Dic₇	C₄	semidirect
24. Dic₃₅	Dic₃	semidirect
25. C₅×Dic₇	Dic₃	semidirect
30. C₃×C₅×C₇	C₄⋊C₄	semidirect
34. C₃×Dic₇	AGL(1,𝔽₅)	semidirect
55. C₅×C₇	C₄⋊Dic₃	semidirect
63. Dic₇	ΓL(1,𝔽₁₆)	semidirect
67. C₃×C₇	C₄⋊AGL(1,𝔽₅)	semidirect
74. C₃×C₅	Dic₇⋊C₄	semidirect
89. C₇	C₄⋊ΓL(1,𝔽₁₆)	semidirect
93. C₅	Dic₇⋊Dic₃	semidirect
99. C₃	Dic₇⋊AGL(1,𝔽₅)	semidirect

H ≅ G/N

type

8. C₃×Dic₃₅

C₄

semidirect

9. C₃×C₅×Dic₇

C₄

semidirect

24. Dic₃₅

Dic₃

semidirect

25. C₅×Dic₇

Dic₃

semidirect

30. C₃×C₅×C₇

C₄⋊C₄

semidirect

34. C₃×Dic₇

AGL(1,𝔽₅)

semidirect

55. C₅×C₇

C₄⋊Dic₃

semidirect

63. Dic₇

ΓL(1,𝔽₁₆)

semidirect

67. C₃×C₇

C₄⋊AGL(1,𝔽₅)

semidirect

74. C₃×C₅

Dic₇⋊C₄

semidirect

89. C₇

C₄⋊ΓL(1,𝔽₁₆)

semidirect

93. C₅

Dic₇⋊Dic₃

semidirect

99. C₃

Dic₇⋊AGL(1,𝔽₅)

semidirect

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

subgroup	index
C₂×C₃⋊(C₇⋊AGL(1,𝔽₅))	2
C₂×C₇×ΓL(1,𝔽₁₆)	2
C₃×D₅×Dic₇	2
Dic₇⋊AGL(1,𝔽₅)	3
Dic₇⋊Dic₃	5
C₄⋊ΓL(1,𝔽₁₆)	7

subgroup

index

C₂×C₃⋊(C₇⋊AGL(1,𝔽₅))

C₂×C₇×ΓL(1,𝔽₁₆)

C₃×D₅×Dic₇

Dic₇⋊AGL(1,𝔽₅)

Dic₇⋊Dic₃

C₄⋊ΓL(1,𝔽₁₆)

Subgroups — Dic₇⋊ΓL(1,𝔽₁₆) #13132447

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — Dic7⋊ΓL(1,𝔽16) #13132447

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — Dic₇⋊ΓL(1,𝔽₁₆) #13132447