Group #13057150 — subgroups

Subgroups — C₇×#5430 #13057150

Order 1344.

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

normal subgroups only

This group has 138 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	C₇×2₋¹⁺⁴⋊C₃	C₂	1	1	yes	maximal, characteristic
2	C₇×#984	C₂	1	1	yes	maximal, characteristic, complemented
3	C₇×GU(2,𝔽₃)	C₂	1	1	yes	maximal, characteristic, complemented
4	C₇×#456	—	3	1	no	maximal, Hall
5	C₇×#41⋊C₃	C₂²	1	1	yes	characteristic
6	C₂×C₇×SL(2,𝔽₃)	—	2	1	no	—
7	C₇×#226	—	2	1	no	—
8	C₇×#214	—	4	1	no	maximal
9	C₇×2₋¹⁺⁴	D₃	1	1	yes	characteristic
10	C₇×#137	—	3	1	no	—
11	C₇×#137	—	3	1	no	—
12	C₇×C₄⋊Q₈	—	3	1	no	—
13	C₇×#104	—	3	1	no	—
14	C₇×#104	—	3	1	no	—
15	C₇×#101	—	3	1	no	—
16	#5430	C₇	1	1	yes	maximal, characteristic, direct product factor, Hall
17	C₃×C₇×D₄	—	4	1	no	—
18	C₇×Dic₆	—	4	1	no	—
19	C₇×SL(2,𝔽₃)	D₄	1	1	yes	characteristic
20	C₇×C₃⋊C₈	—	4	1	no	—
21	C₇×#41	—	3	1	no	—
22	C₇×#41	—	6	1	no	—
23	C₇×#41	C₂×D₃	1	1	yes	characteristic
24	C₂×C₇×Q₈	—	2	1	no	—
25	C₂×C₇×Q₈	—	3	1	no	—
26	C₂×C₇×Q₈	—	3	1	no	—
27	C₇×Q₁₆	—	6	1	no	—
28	C₇×Q₁₆	—	6	1	no	—
29	C₇×SD₁₆	—	6	1	no	—
30	C₇×SD₁₆	—	6	1	no	—
31	C₇×M₁₆	—	3	1	no	—
32	C₇×M₁₆	—	3	1	no	—
33	C₇×C₄⋊C₄	—	6	2	no	—
34	C₄²×C₇	—	3	1	no	—
35	2₋¹⁺⁴⋊C₃	C₂×C₇	1	1	yes	characteristic
36	#984	C₂×C₇	1	1	yes	characteristic, complemented
37	GU(2,𝔽₃)	C₂×C₇	1	1	yes	characteristic, complemented
38	C₂²×C₃×C₇	—	8	1	no	—
39	C₃×C₄×C₇	—	4	1	no	—
40	C₇×Dic₃	—	8	1	no	—
41	#456	—	3	1	no	Sylow
42	C₇×Q₈	—	3	1	no	—
43	C₇×Q₈	—	6	1	no	—
44	C₇×Q₈	—	6	1	no	—
45	C₇×Q₈	—	6	1	no	—
46	C₇×Q₈	#67	1	1	yes	characteristic
47	C₇×D₄	—	3	1	no	—
48	C₇×D₄	—	6	1	no	—
49	C₇×D₄	S₄	1	1	yes	characteristic
50	C₂×C₄×C₇	—	3	1	no	—
51	C₂×C₄×C₇	—	3	2	no	—
52	C₂×C₄×C₇	—	6	1	no	—
53	C₂×C₄×C₇	—	6	1	no	—
54	C₂×C₄×C₇	—	6	1	no	—
55	C₇×C₈	—	6	1	no	—
56	C₇×C₈	—	6	1	no	—
57	#41⋊C₃	C₂²×C₇	1	1	yes	characteristic
58	C₂×SL(2,𝔽₃)	—	2	1	no	—
59	#226	—	2	1	no	—
60	#214	—	4	1	no	—
61	C₂×C₃×C₇	—	4	1	no	—
62	C₂×C₃×C₇	—	16	1	no	—
63	2₋¹⁺⁴	D₃×C₇	1	1	yes	characteristic
64	#137	—	3	1	no	—
65	#137	—	3	1	no	—
66	C₄⋊Q₈	—	3	1	no	—
67	#104	—	3	1	no	—
68	#104	—	3	1	no	—
69	#101	—	3	1	no	—
70	C₂²×C₇	—	2	1	no	—
71	C₂²×C₇	—	3	1	no	—
72	C₄×C₇	—	3	1	no	—
73	C₄×C₇	—	6	2	no	—
74	C₄×C₇	—	6	1	no	—
75	C₄×C₇	—	12	1	no	—
76	C₄×C₇	C₂×S₄	1	1	yes	characteristic
77	C₃×D₄	—	4	1	no	—
78	Dic₆	—	4	1	no	—
79	SL(2,𝔽₃)	C₇×D₄	1	1	yes	characteristic
80	C₃⋊C₈	—	4	1	no	—
81	C₃×C₇	—	4	1	no	Hall
82	#41	—	3	1	no	—
83	#41	—	6	1	no	—
84	#41	C₂×D₃×C₇	1	1	yes	characteristic
85	C₂×Q₈	—	2	1	no	—
86	C₂×Q₈	—	3	1	no	—
87	C₂×Q₈	—	3	1	no	—
88	Q₁₆	—	6	1	no	—
89	Q₁₆	—	6	1	no	—
90	SD₁₆	—	6	1	no	—
91	SD₁₆	—	6	1	no	—
92	M₁₆	—	3	1	no	—
93	M₁₆	—	3	1	no	—
94	C₄⋊C₄	—	6	2	no	—
95	C₄²	—	3	1	no	—
96	C₂×C₇	—	4	1	no	—
97	C₂×C₇	—	6	1	no	—
98	C₂×C₇	GL(2,ℤ/4)	1	1	yes	characteristic
99	C₂²×C₃	—	8	1	no	—
100	C₃×C₄	—	4	1	no	—
101	Dic₃	—	8	1	no	—
102	Q₈	—	3	1	no	—
103	Q₈	—	6	1	no	—
104	Q₈	—	6	1	no	—
105	Q₈	—	6	1	no	—
106	Q₈	C₇×#67	1	1	yes	characteristic
107	D₄	—	3	1	no	—
108	D₄	—	6	1	no	—
109	D₄	C₇×S₄	1	1	yes	characteristic
110	C₂×C₄	—	3	1	no	—
111	C₂×C₄	—	3	2	no	—
112	C₂×C₄	—	6	1	no	—
113	C₂×C₄	—	6	1	no	—
114	C₂×C₄	—	6	1	no	—
115	C₈	—	6	1	no	—
116	C₈	—	6	1	no	—
117	C₇	#5430	1	1	yes	characteristic, direct product factor, Sylow
118	C₂×C₃	—	4	1	no	—
119	C₂×C₃	—	16	1	no	—
120	C₂²	—	2	1	no	—
121	C₂²	—	3	1	no	—
122	C₄	—	3	1	no	—
123	C₄	—	6	2	no	—
124	C₄	—	6	1	no	—
125	C₄	—	12	1	no	—
126	C₄	C₂×C₇×S₄	1	1	yes	characteristic
127	C₃	—	4	1	no	Sylow
128	C₂	—	4	1	no	—
129	C₂	—	6	1	no	—
130	C₂	C₇×GL(2,ℤ/4)	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₇×2₋¹⁺⁴⋊C₃

C₂

yes

maximal, characteristic

C₇×#984

C₂

yes

maximal, characteristic, complemented

C₇×GU(2,𝔽₃)

C₂

yes

maximal, characteristic, complemented

C₇×#456

—

maximal, Hall

C₇×#41⋊C₃

C₂²

yes

characteristic

C₂×C₇×SL(2,𝔽₃)

—

C₇×#226

—

C₇×#214

—

maximal

C₇×2₋¹⁺⁴

D₃

yes

characteristic

C₇×#137

—

C₇×#137

—

C₇×C₄⋊Q₈

—

—

—

—

C₇

yes

maximal, characteristic, direct product factor, Hall

C₃×C₇×D₄

—

C₇×Dic₆

—

C₇×SL(2,𝔽₃)

D₄

yes

characteristic

C₇×C₃⋊C₈

—

C₇×#41

—

C₇×#41

—

C₇×#41

C₂×D₃

yes

characteristic

C₂×C₇×Q₈

—

C₂×C₇×Q₈

—

C₂×C₇×Q₈

—

C₇×Q₁₆

—

C₇×Q₁₆

—

C₇×SD₁₆

—

C₇×SD₁₆

—

C₇×M₁₆

—

C₇×M₁₆

—

C₇×C₄⋊C₄

—

C₄²×C₇

—

2₋¹⁺⁴⋊C₃

C₂×C₇

yes

characteristic

#984

C₂×C₇

yes

characteristic, complemented

GU(2,𝔽₃)

C₂×C₇

yes

characteristic, complemented

C₂²×C₃×C₇

—

C₃×C₄×C₇

—

C₇×Dic₃

—

#456

—

Sylow

C₇×Q₈

—

C₇×Q₈

—

C₇×Q₈

—

C₇×Q₈

—

C₇×Q₈

#67

yes

characteristic

C₇×D₄

—

C₇×D₄

—

C₇×D₄

S₄

yes

characteristic

C₂×C₄×C₇

—

C₂×C₄×C₇

—

C₂×C₄×C₇

—

C₂×C₄×C₇

—

C₂×C₄×C₇

—

C₇×C₈

—

C₇×C₈

—

#41⋊C₃

C₂²×C₇

yes

characteristic

C₂×SL(2,𝔽₃)

—

#226

—

#214

—

C₂×C₃×C₇

—

C₂×C₃×C₇

—

2₋¹⁺⁴

D₃×C₇

yes

characteristic

#137

—

#137

—

C₄⋊Q₈

—

#104

—

#104

—

#101

—

C₂²×C₇

—

C₂²×C₇

—

C₄×C₇

—

C₄×C₇

—

C₄×C₇

—

C₄×C₇

—

C₄×C₇

C₂×S₄

yes

characteristic

C₃×D₄

—

Dic₆

—

SL(2,𝔽₃)

C₇×D₄

yes

characteristic

C₃⋊C₈

—

C₃×C₇

—

—

—

C₂×D₃×C₇

yes

characteristic

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

Q₁₆

—

Q₁₆

—

SD₁₆

—

SD₁₆

—

M₁₆

—

M₁₆

—

C₄⋊C₄

—

C₄²

—

C₂×C₇

—

C₂×C₇

—

C₂×C₇

GL(2,ℤ/4)

yes

characteristic

C₂²×C₃

—

100

C₃×C₄

—

101

Dic₃

—

102

Q₈

—

103

Q₈

—

104

Q₈

—

105

Q₈

—

106

Q₈

C₇×#67

yes

characteristic

107

D₄

—

108

D₄

—

109

D₄

C₇×S₄

yes

characteristic

110

C₂×C₄

—

111

C₂×C₄

—

112

C₂×C₄

—

113

C₂×C₄

—

114

C₂×C₄

—

115

C₈

—

116

C₈

—

117

C₇

#5430

yes

characteristic, direct product factor, Sylow

118

C₂×C₃

—

119

C₂×C₃

—

120

C₂²

—

121

C₂²

—

122

C₄

—

123

C₄

—

124

C₄

—

125

C₄

—

126

C₄

C₂×C₇×S₄

yes

characteristic

127

C₃

—

Sylow

128

C₂

—

129

C₂

—

130

C₂

C₇×GL(2,ℤ/4)

yes

characteristic

	subgroup	G/H
center, socle	C₂×C₇	GL(2,ℤ/4)
commutator, cosocle	#41⋊C₃	C₂²×C₇
Frattini	C₄	C₂×C₇×S₄
Fitting	C₇×2₋¹⁺⁴	D₃
prime radical	C₇×2₋¹⁺⁴⋊C₃	C₂
hypercenter	C₇×D₄	S₄
hypocenter	SL(2,𝔽₃)	C₇×D₄
LCS	G ▷ #41⋊C₃ ▷ SL(2,𝔽₃)
UCS	C₁ ◁ C₂×C₇ ◁ C₄×C₇ ◁ C₇×D₄

subgroup

G/H

C₂×C₇

C₂²×C₇

C₄

C₂×C₇×S₄

Fitting

C₇×2₋¹⁺⁴

D₃

prime radical

C₇×2₋¹⁺⁴⋊C₃

C₂

hypercenter

C₇×D₄

S₄

hypocenter

SL(2,𝔽₃)

C₇×D₄

LCS

G ▷ #41⋊C₃ ▷ SL(2,𝔽₃)

UCS

C₁ ◁ C₂×C₇ ◁ C₄×C₇ ◁ C₇×D₄

N	H ≅ G/N	type
2. C₇×#984	C₂	semidirect
3. C₇×GU(2,𝔽₃)	C₂	semidirect
16. #5430	C₇	direct (N × H)
36. #984	C₂×C₇	semidirect
37. GU(2,𝔽₃)	C₂×C₇	semidirect
117. C₇	#5430	direct (N × H)

H ≅ G/N

type

2. C₇×#984

C₂

semidirect

3. C₇×GU(2,𝔽₃)

C₂

semidirect

16. #5430

C₇

direct (N × H)

36. #984

C₂×C₇

semidirect

37. GU(2,𝔽₃)

C₂×C₇

semidirect

117. C₇

#5430

direct (N × H)

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

subgroup	index
C₇×2₋¹⁺⁴⋊C₃	2
C₇×#984	2
C₇×GU(2,𝔽₃)	2
C₇×#456	3
C₇×#214	4
#5430	7

subgroup

index

C₇×2₋¹⁺⁴⋊C₃

C₇×GU(2,𝔽₃)

Subgroups — C₇×#5430 #13057150

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C7×#5430 #13057150

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₇×#5430 #13057150