Group #13057126 — subgroups

Subgroups — C₇×#5406 #13057126

Order 1344.

124 conjugacy classes of subgroups; only the 26 normal subgroups are shown.

normal subgroups only

This group has 124 conjugacy classes of subgroups — too many to draw. Only its 26 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	C₇×#986	C₂	1	1	yes	maximal, characteristic
2	C₇×#984	C₂	1	1	yes	maximal, characteristic, complemented
3	C₇×#131⋊C₃	C₂	1	1	yes	maximal, characteristic, complemented
4	C₇×#471	—	3	1	no	maximal, Hall
5	C₇×#41⋊C₃	C₂²	1	1	yes	characteristic
6	C₇×GL(2,𝔽₃)	—	2	1	no	—
7	C₇×#226	—	2	1	no	—
8	C₇×#204	—	4	1	no	maximal
9	C₇×#137	—	3	1	no	—
10	C₇×AGL(1,ℤ/8)	—	3	1	no	—
11	C₂×C₇×SD₁₆	—	3	1	no	—
12	C₇×#131	D₃	1	1	yes	characteristic, complemented
13	C₇×#108	—	3	1	no	—
14	C₇×#101	—	3	1	no	—
15	C₇×#100	—	3	1	no	—
16	#5406	C₇	1	1	yes	maximal, characteristic, direct product factor, Hall
17	C₇×D₁₂	—	4	1	no	—
18	C₇×Dic₆	—	4	1	no	—
19	C₇×SL(2,𝔽₃)	D₄	1	1	yes	characteristic
20	C₃×C₇×C₈	—	4	1	no	—
21	C₇×#41	C₂×D₃	1	1	yes	characteristic
22	C₂×C₇×Q₈	—	3	1	no	—
23	C₂×C₇×D₄	—	3	1	no	—
24	C₇×Q₁₆	—	6	1	no	—
25	C₇×SD₁₆	—	6	1	no	—
26	C₇×SD₁₆	—	6	1	no	—
27	C₇×SD₁₆	—	6	1	no	—
28	C₇×SD₁₆	—	6	1	no	—
29	C₇×D₈	—	6	1	no	—
30	C₇×M₁₆	—	3	1	no	—
31	C₇×M₁₆	—	3	1	no	—
32	C₇×M₁₆	—	3	1	no	—
33	C₂×C₇×C₈	—	3	1	no	—
34	#986	C₂×C₇	1	1	yes	characteristic
35	#984	C₂×C₇	1	1	yes	characteristic, complemented
36	#131⋊C₃	C₂×C₇	1	1	yes	characteristic, complemented
37	C₂×D₃×C₇	—	8	1	no	—
38	C₃×C₄×C₇	—	4	1	no	—
39	C₇×Dic₃	—	8	1	no	—
40	#471	—	3	1	no	Sylow
41	C₂³×C₇	—	6	1	no	—
42	C₇×Q₈	—	6	1	no	—
43	C₇×Q₈	—	6	1	no	—
44	C₇×Q₈	D₁₂	1	1	yes	characteristic
45	C₇×D₄	—	3	1	no	—
46	C₇×D₄	—	6	1	no	—
47	C₇×D₄	—	6	1	no	—
48	C₂×C₄×C₇	—	3	1	no	—
49	C₂×C₄×C₇	—	6	1	no	—
50	C₇×C₈	—	3	1	no	—
51	C₇×C₈	—	6	1	no	—
52	C₇×C₈	—	6	1	no	—
53	C₇×C₈	S₄	1	1	yes	characteristic
54	#41⋊C₃	C₂²×C₇	1	1	yes	characteristic
55	GL(2,𝔽₃)	—	2	1	no	—
56	#226	—	2	1	no	—
57	#204	—	4	1	no	—
58	C₂×C₃×C₇	—	4	1	no	—
59	D₃×C₇	—	16	1	no	—
60	#137	—	3	1	no	—
61	AGL(1,ℤ/8)	—	3	1	no	—
62	C₂×SD₁₆	—	3	1	no	—
63	#131	D₃×C₇	1	1	yes	characteristic, complemented
64	#108	—	3	1	no	—
65	#101	—	3	1	no	—
66	#100	—	3	1	no	—
67	C₂²×C₇	—	3	1	no	—
68	C₂²×C₇	—	12	1	no	—
69	C₂²×C₇	—	24	1	no	—
70	C₄×C₇	—	3	1	no	—
71	C₄×C₇	—	12	1	no	—
72	C₄×C₇	C₂×S₄	1	1	yes	characteristic
73	D₁₂	—	4	1	no	—
74	Dic₆	—	4	1	no	—
75	SL(2,𝔽₃)	C₇×D₄	1	1	yes	characteristic
76	C₃×C₈	—	4	1	no	—
77	C₃×C₇	—	4	1	no	Hall
78	#41	C₂×D₃×C₇	1	1	yes	characteristic
79	C₂×Q₈	—	3	1	no	—
80	C₂×D₄	—	3	1	no	—
81	Q₁₆	—	6	1	no	—
82	SD₁₆	—	6	1	no	—
83	SD₁₆	—	6	1	no	—
84	SD₁₆	—	6	1	no	—
85	SD₁₆	—	6	1	no	—
86	D₈	—	6	1	no	—
87	M₁₆	—	3	1	no	—
88	M₁₆	—	3	1	no	—
89	M₁₆	—	3	1	no	—
90	C₂×C₈	—	3	1	no	—
91	C₂×C₇	—	6	1	no	—
92	C₂×C₇	—	24	1	no	—
93	C₂×C₇	C₄⋊S₄	1	1	yes	characteristic
94	C₂×D₃	—	8	1	no	—
95	C₃×C₄	—	4	1	no	—
96	Dic₃	—	8	1	no	—
97	C₂³	—	6	1	no	—
98	Q₈	—	6	1	no	—
99	Q₈	—	6	1	no	—
100	Q₈	C₇×D₁₂	1	1	yes	characteristic
101	D₄	—	3	1	no	—
102	D₄	—	6	1	no	—
103	D₄	—	6	1	no	—
104	C₂×C₄	—	3	1	no	—
105	C₂×C₄	—	6	1	no	—
106	C₈	—	3	1	no	—
107	C₈	—	6	1	no	—
108	C₈	—	6	1	no	—
109	C₈	C₇×S₄	1	1	yes	characteristic
110	C₇	#5406	1	1	yes	characteristic, direct product factor, Sylow
111	C₂×C₃	—	4	1	no	—
112	D₃	—	16	1	no	—
113	C₂²	—	3	1	no	—
114	C₂²	—	12	1	no	—
115	C₂²	—	24	1	no	—
116	C₄	—	3	1	no	—
117	C₄	—	12	1	no	—
118	C₄	C₂×C₇×S₄	1	1	yes	characteristic
119	C₃	—	4	1	no	Sylow
120	C₂	—	6	1	no	—
121	C₂	—	24	1	no	—
122	C₂	C₇×C₄⋊S₄	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₇×#986

C₂

yes

maximal, characteristic

C₇×#984

C₂

yes

maximal, characteristic, complemented

C₇×#131⋊C₃

C₂

yes

maximal, characteristic, complemented

C₇×#471

—

maximal, Hall

C₇×#41⋊C₃

C₂²

yes

characteristic

C₇×GL(2,𝔽₃)

—

—

—

—

—

C₂×C₇×SD₁₆

—

C₇×#131

D₃

yes

characteristic, complemented

—

—

—

C₇

yes

maximal, characteristic, direct product factor, Hall

C₇×D₁₂

—

C₇×Dic₆

—

C₇×SL(2,𝔽₃)

D₄

yes

characteristic

C₃×C₇×C₈

—

C₇×#41

C₂×D₃

yes

characteristic

C₂×C₇×Q₈

—

C₂×C₇×D₄

—

C₇×Q₁₆

—

C₇×SD₁₆

—

C₇×SD₁₆

—

C₇×SD₁₆

—

C₇×SD₁₆

—

C₇×D₈

—

C₇×M₁₆

—

C₇×M₁₆

—

C₇×M₁₆

—

C₂×C₇×C₈

—

#986

C₂×C₇

yes

characteristic

#984

C₂×C₇

yes

characteristic, complemented

#131⋊C₃

C₂×C₇

yes

characteristic, complemented

C₂×D₃×C₇

—

C₃×C₄×C₇

—

C₇×Dic₃

—

#471

—

Sylow

C₂³×C₇

—

C₇×Q₈

—

C₇×Q₈

—

C₇×Q₈

D₁₂

yes

characteristic

C₇×D₄

—

C₇×D₄

—

C₇×D₄

—

C₂×C₄×C₇

—

C₂×C₄×C₇

—

C₇×C₈

—

C₇×C₈

—

C₇×C₈

—

C₇×C₈

S₄

yes

characteristic

#41⋊C₃

C₂²×C₇

yes

—

—

—

C₂×C₃×C₇

—

D₃×C₇

—

#137

—

AGL(1,ℤ/8)

—

C₂×SD₁₆

—

#131

D₃×C₇

yes

characteristic, complemented

#108

—

#101

—

#100

—

C₂²×C₇

—

C₂²×C₇

—

C₂²×C₇

—

C₄×C₇

—

C₄×C₇

—

C₄×C₇

C₂×S₄

yes

characteristic

D₁₂

—

Dic₆

—

SL(2,𝔽₃)

C₇×D₄

yes

characteristic

C₃×C₈

—

C₃×C₇

—

Hall

#41

C₂×D₃×C₇

yes

characteristic

C₂×Q₈

—

C₂×D₄

—

Q₁₆

—

SD₁₆

—

SD₁₆

—

SD₁₆

—

SD₁₆

—

D₈

—

M₁₆

—

M₁₆

—

M₁₆

—

C₂×C₈

—

C₂×C₇

—

C₂×C₇

—

C₂×C₇

C₄⋊S₄

yes

characteristic

C₂×D₃

—

C₃×C₄

—

Dic₃

—

C₂³

—

Q₈

—

Q₈

—

100

Q₈

C₇×D₁₂

yes

characteristic

101

D₄

—

102

D₄

—

103

D₄

—

104

C₂×C₄

—

105

C₂×C₄

—

106

C₈

—

107

C₈

—

108

C₈

—

109

C₈

C₇×S₄

yes

characteristic

110

C₇

#5406

yes

characteristic, direct product factor, Sylow

111

C₂×C₃

—

112

D₃

—

113

C₂²

—

114

C₂²

—

115

C₂²

—

116

C₄

—

117

C₄

—

118

C₄

C₂×C₇×S₄

yes

characteristic

119

C₃

—

Sylow

120

C₂

—

121

C₂

—

122

C₂

C₇×C₄⋊S₄

yes

characteristic

	subgroup	G/H
center, socle	C₂×C₇	C₄⋊S₄
commutator, cosocle	#41⋊C₃	C₂²×C₇
Frattini	C₄	C₂×C₇×S₄
Fitting	C₇×#131	D₃
prime radical	C₇×#986	C₂
hypercenter	C₇×C₈	S₄
hypocenter	SL(2,𝔽₃)	C₇×D₄
LCS	G ▷ #41⋊C₃ ▷ SL(2,𝔽₃)
UCS	C₁ ◁ C₂×C₇ ◁ C₄×C₇ ◁ C₇×C₈

subgroup

G/H

center, socle

C₂×C₇

C₄⋊S₄

commutator, cosocle

#41⋊C₃

C₂²×C₇

Frattini

C₄

C₂×C₇×S₄

D₃

C₂

C₇×C₈

S₄

hypocenter

SL(2,𝔽₃)

C₇×D₄

LCS

G ▷ #41⋊C₃ ▷ SL(2,𝔽₃)

UCS

C₁ ◁ C₂×C₇ ◁ C₄×C₇ ◁ C₇×C₈

N	H ≅ G/N	type
2. C₇×#984	C₂	semidirect
3. C₇×#131⋊C₃	C₂	semidirect
12. C₇×#131	D₃	semidirect
16. #5406	C₇	direct (N × H)
35. #984	C₂×C₇	semidirect
36. #131⋊C₃	C₂×C₇	semidirect
63. #131	D₃×C₇	semidirect
110. C₇	#5406	direct (N × H)

H ≅ G/N

type

2. C₇×#984

C₂

semidirect

3. C₇×#131⋊C₃

C₂

semidirect

12. C₇×#131

D₃

semidirect

16. #5406

C₇

direct (N × H)

35. #984

C₂×C₇

semidirect

36. #131⋊C₃

C₂×C₇

semidirect

63. #131

D₃×C₇

semidirect

110. C₇

#5406

direct (N × H)

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

subgroup	index
C₇×#986	2
C₇×#984	2
C₇×#131⋊C₃	2
C₇×#471	3
C₇×#204	4
#5406	7

subgroup

index

C₇×#131⋊C₃

Subgroups — C₇×#5406 #13057126

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C7×#5406 #13057126

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₇×#5406 #13057126