Group #13057075 — subgroups

Subgroups — D₇×#859 #13057075

Order 1344.

194 conjugacy classes of subgroups; only the 39 normal subgroups are shown.

normal subgroups only

This group has 194 conjugacy classes of subgroups — too many to draw. Only its 39 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×D₇×SL(2,𝔽₃)	C₂	1	1	yes	maximal, characteristic
2	C₇⋊#859	C₂	1	1	yes	maximal, characteristic, complemented
3	C₇×#859	C₂	1	1	yes	maximal, characteristic, complemented
4	D₇×#103	—	3	1	no	maximal, Hall
5	C₂×C₇×SL(2,𝔽₃)	C₂²	1	1	yes	characteristic
6	C₂×Dic₃×D₇	—	4	1	no	maximal
7	D₇×SL(2,𝔽₃)	C₄	1	2	yes	complemented
8	C₂×Q₈×D₇	D₃	1	1	yes	characteristic
9	C₂×C₈×D₇	—	3	1	no	—
10	D₇×C₄⋊C₄	—	3	1	no	—
11	C₇×#103	—	3	1	no	—
12	#6409	—	3	1	no	—
13	#6390	—	3	1	no	—
14	#6384	—	3	1	no	—
15	C₂×#859	—	7	1	no	maximal, Hall
16	C₂²×C₃×D₇	—	4	1	no	—
17	C₂×Dic₂₁	—	4	1	no	—
18	C₂×C₇×Dic₃	—	4	1	no	—
19	C₇×SL(2,𝔽₃)	C₂×C₄	1	1	yes	characteristic, complemented
20	Dic₃×D₇	—	4	4	no	—
21	C₂×C₇×Q₈	C₂×D₃	1	1	yes	characteristic
22	Q₈×D₇	—	3	2	no	—
23	Q₈×D₇	Dic₃	1	2	yes	complemented
24	C₂×C₄×D₇	—	3	1	no	—
25	C₂×C₄×D₇	—	6	1	no	—
26	C₂×Dic₁₄	—	3	1	no	—
27	C₂×C₇×C₈	—	3	1	no	—
28	C₇×C₄⋊C₄	—	3	1	no	—
29	C₄⋊Dic₇	—	3	1	no	—
30	Dic₇⋊C₄	—	6	1	no	—
31	C₂×C₇⋊C₈	—	3	1	no	—
32	C₈×D₇	—	6	2	no	—
33	C₂²×SL(2,𝔽₃)	—	7	1	no	—
34	#859	—	7	1	no	—
35	#859	D₇	1	1	yes	characteristic, direct product factor
36	C₂²×C₃×C₇	—	4	1	no	—
37	C₂×C₃×D₇	—	4	2	no	—
38	C₂×C₃×D₇	—	4	2	no	—
39	C₂×C₃×D₇	—	4	2	no	—
40	Dic₂₁	—	4	2	no	—
41	C₇×Dic₃	—	4	2	no	—
42	C₂×#103	—	21	1	no	Sylow
43	C₂²×D₇	S₄	1	1	yes	characteristic
44	C₇×Q₈	—	3	1	no	—
45	C₇×Q₈	C₂×Dic₃	1	1	yes	characteristic, complemented
46	C₂×C₄×C₇	—	3	1	no	—
47	C₂×C₄×C₇	—	6	1	no	—
48	C₂×Dic₇	—	3	1	no	—
49	C₂×Dic₇	—	6	1	no	—
50	C₄×D₇	—	3	2	no	—
51	C₄×D₇	—	3	2	no	—
52	C₄×D₇	—	12	2	no	—
53	Dic₁₄	—	3	2	no	—
54	Dic₁₄	—	6	1	no	—
55	C₇×C₈	—	6	1	no	—
56	C₇⋊C₈	—	6	1	no	—
57	C₂²×Dic₃	—	28	1	no	—
58	C₂×SL(2,𝔽₃)	—	7	2	no	—
59	C₂×SL(2,𝔽₃)	C₂×D₇	1	1	yes	characteristic
60	C₂×C₃×C₇	—	4	1	no	—
61	C₂×C₃×C₇	—	4	1	no	—
62	C₂×C₃×C₇	—	4	1	no	—
63	C₃×D₇	—	4	4	no	—
64	C₂²×Q₈	—	7	1	no	—
65	C₂²×C₈	—	21	1	no	—
66	C₂×C₄⋊C₄	—	21	1	no	—
67	#103	—	3	1	no	—
68	#103	—	21	1	no	—
69	#103	—	21	1	no	—
70	#103	—	21	1	no	—
71	C₂²×C₇	C₂×S₄	1	1	yes	characteristic
72	C₂×D₇	#226	1	2	yes	—
73	C₂×D₇	GL(2,𝔽₃)	1	2	yes	—
74	C₂×D₇	SL(2,ℤ/4)	1	2	yes	—
75	C₄×C₇	—	3	1	no	—
76	C₄×C₇	—	3	1	no	—
77	C₄×C₇	—	12	1	no	—
78	Dic₇	—	3	2	no	—
79	Dic₇	—	12	1	no	—
80	C₂³×C₃	—	28	1	no	—
81	C₂×Dic₃	—	4	1	no	—
82	C₂×Dic₃	—	28	1	no	—
83	C₂×Dic₃	—	28	4	no	—
84	SL(2,𝔽₃)	C₄×D₇	1	1	yes	characteristic, complemented
85	C₃×C₇	—	4	1	no	Hall
86	C₂×Q₈	—	7	2	no	—
87	C₂×Q₈	—	21	1	no	—
88	C₂×Q₈	—	21	2	no	—
89	C₂×Q₈	D₃×D₇	1	1	yes	characteristic
90	C₂²×C₄	—	21	1	no	—
91	C₂²×C₄	—	42	1	no	—
92	C₂×C₈	—	3	1	no	—
93	C₂×C₈	—	21	1	no	—
94	C₂×C₈	—	42	2	no	—
95	C₄⋊C₄	—	3	1	no	—
96	C₄⋊C₄	—	21	1	no	—
97	C₄⋊C₄	—	42	1	no	—
98	C₂×C₇	C₂×#226	1	1	yes	characteristic
99	C₂×C₇	C₂×GL(2,𝔽₃)	1	1	yes	characteristic
100	C₂×C₇	C₂×SL(2,ℤ/4)	1	1	yes	characteristic
101	D₇	#859	1	4	yes	direct product factor
102	C₂²×C₃	—	4	1	no	—
103	C₂²×C₃	—	28	2	no	—
104	C₂²×C₃	—	28	2	no	—
105	C₂²×C₃	—	28	2	no	—
106	Dic₃	—	4	2	no	—
107	Dic₃	—	28	2	no	—
108	C₂³	—	7	1	no	—
109	Q₈	—	3	1	no	—
110	Q₈	—	21	2	no	—
111	Q₈	—	42	1	no	—
112	Q₈	Dic₃×D₇	1	1	yes	characteristic, complemented
113	C₂×C₄	—	3	1	no	—
114	C₂×C₄	—	6	1	no	—
115	C₂×C₄	—	21	2	no	—
116	C₂×C₄	—	21	2	no	—
117	C₂×C₄	—	21	1	no	—
118	C₂×C₄	—	42	1	no	—
119	C₂×C₄	—	84	2	no	—
120	C₈	—	6	1	no	—
121	C₈	—	42	1	no	—
122	C₇	C₂×#859	1	1	yes	characteristic, complemented, Sylow
123	C₂×C₃	—	4	1	no	—
124	C₂×C₃	—	4	1	no	—
125	C₂×C₃	—	4	1	no	—
126	C₂×C₃	—	28	4	no	—
127	C₂²	—	7	2	no	—
128	C₂²	—	7	2	no	—
129	C₂²	—	7	2	no	—
130	C₂²	D₇×S₄	1	1	yes	characteristic
131	C₄	—	3	1	no	—
132	C₄	—	3	1	no	—
133	C₄	—	12	1	no	—
134	C₄	—	21	2	no	—
135	C₄	—	84	1	no	—
136	C₃	—	4	1	no	Sylow
137	C₂	—	7	4	no	—
138	C₂	D₇×#226	1	1	yes	characteristic
139	C₂	D₇×GL(2,𝔽₃)	1	1	yes	characteristic
140	C₂	D₇×SL(2,ℤ/4)	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×D₇×SL(2,𝔽₃)

C₂

yes

maximal, characteristic

C₇⋊#859

C₂

yes

maximal, characteristic, complemented

C₇×#859

C₂

yes

maximal, characteristic, complemented

D₇×#103

—

maximal, Hall

C₂×C₇×SL(2,𝔽₃)

C₂²

yes

characteristic

C₂×Dic₃×D₇

—

maximal

D₇×SL(2,𝔽₃)

C₄

yes

complemented

C₂×Q₈×D₇

D₃

yes

characteristic

C₂×C₈×D₇

—

D₇×C₄⋊C₄

—

—

—

—

—

—

C₂²×C₃×D₇

—

C₂×Dic₂₁

—

C₂×C₇×Dic₃

—

C₇×SL(2,𝔽₃)

C₂×C₄

yes

characteristic, complemented

Dic₃×D₇

—

C₂×C₇×Q₈

C₂×D₃

yes

characteristic

Q₈×D₇

—

Q₈×D₇

Dic₃

yes

complemented

C₂×C₄×D₇

—

C₂×C₄×D₇

—

C₂×Dic₁₄

—

C₂×C₇×C₈

—

C₇×C₄⋊C₄

—

C₄⋊Dic₇

—

Dic₇⋊C₄

—

C₂×C₇⋊C₈

—

C₈×D₇

—

C₂²×SL(2,𝔽₃)

—

#859

—

#859

D₇

yes

characteristic, direct product factor

C₂²×C₃×C₇

—

C₂×C₃×D₇

—

C₂×C₃×D₇

—

C₂×C₃×D₇

—

Dic₂₁

—

C₇×Dic₃

—

C₂×#103

—

Sylow

C₂²×D₇

S₄

yes

characteristic

C₇×Q₈

—

C₇×Q₈

C₂×Dic₃

yes

characteristic, complemented

C₂×C₄×C₇

—

C₂×C₄×C₇

—

C₂×Dic₇

—

C₂×Dic₇

—

C₄×D₇

—

C₄×D₇

—

C₄×D₇

—

Dic₁₄

—

Dic₁₄

—

C₇×C₈

—

C₇⋊C₈

—

C₂²×Dic₃

—

C₂×SL(2,𝔽₃)

—

C₂×SL(2,𝔽₃)

C₂×D₇

yes

characteristic

C₂×C₃×C₇

—

C₂×C₃×C₇

—

C₂×C₃×C₇

—

C₃×D₇

—

C₂²×Q₈

—

C₂²×C₈

—

C₂×C₄⋊C₄

—

—

—

—

—

C₂²×C₇

C₂×S₄

yes

characteristic

C₂×D₇

#226

yes

—

C₂×D₇

GL(2,𝔽₃)

yes

—

C₂×D₇

SL(2,ℤ/4)

yes

—

C₄×C₇

—

C₄×C₇

—

C₄×C₇

—

Dic₇

—

Dic₇

—

C₂³×C₃

—

C₂×Dic₃

—

C₂×Dic₃

—

C₂×Dic₃

—

SL(2,𝔽₃)

C₄×D₇

yes

characteristic, complemented

C₃×C₇

—

Hall

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

D₃×D₇

yes

characteristic

C₂²×C₄

—

C₂²×C₄

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₂×C₇

C₂×#226

yes

characteristic

C₂×C₇

C₂×GL(2,𝔽₃)

yes

characteristic

100

C₂×C₇

C₂×SL(2,ℤ/4)

yes

characteristic

101

D₇

#859

yes

direct product factor

102

C₂²×C₃

—

103

C₂²×C₃

—

104

C₂²×C₃

—

105

C₂²×C₃

—

106

Dic₃

—

107

Dic₃

—

108

C₂³

—

109

Q₈

—

110

Q₈

—

111

Q₈

—

112

Q₈

Dic₃×D₇

yes

characteristic, complemented

113

C₂×C₄

—

114

C₂×C₄

—

115

C₂×C₄

—

116

C₂×C₄

—

117

C₂×C₄

—

118

C₂×C₄

—

119

C₂×C₄

—

120

C₈

—

121

C₈

—

122

C₇

C₂×#859

yes

characteristic, complemented, Sylow

123

C₂×C₃

—

124

C₂×C₃

—

125

C₂×C₃

—

126

C₂×C₃

—

127

C₂²

—

128

C₂²

—

129

C₂²

—

130

C₂²

D₇×S₄

yes

characteristic

131

C₄

—

132

C₄

—

133

C₄

—

134

C₄

—

135

C₄

—

136

C₃

—

Sylow

137

C₂

—

138

C₂

D₇×#226

yes

characteristic

139

C₂

D₇×GL(2,𝔽₃)

yes

characteristic

140

C₂

D₇×SL(2,ℤ/4)

yes

characteristic

	subgroup	G/H
center, Frattini, hypercenter	C₂²	D₇×S₄
commutator, hypocenter	C₇×SL(2,𝔽₃)	C₂×C₄
Fitting	C₂×C₇×Q₈	C₂×D₃
socle	C₂²×C₇	C₂×S₄
prime radical	C₂×D₇×SL(2,𝔽₃)	C₂
cosocle	C₂×C₇×SL(2,𝔽₃)	C₂²
LCS	G ▷ C₇×SL(2,𝔽₃)
UCS	C₁ ◁ C₂²

subgroup

G/H

center, Frattini, hypercenter

C₂²

D₇×S₄

commutator, hypocenter

C₇×SL(2,𝔽₃)

C₂×C₄

Fitting

C₂×C₇×Q₈

C₂×D₃

socle

C₂²×C₇

C₂×S₄

prime radical

C₂×D₇×SL(2,𝔽₃)

C₂

cosocle

C₂×C₇×SL(2,𝔽₃)

C₂²

LCS

G ▷ C₇×SL(2,𝔽₃)

UCS

C₁ ◁ C₂²

N	H ≅ G/N	type
2. C₇⋊#859	C₂	semidirect
3. C₇×#859	C₂	semidirect
7. D₇×SL(2,𝔽₃)	C₄	semidirect
19. C₇×SL(2,𝔽₃)	C₂×C₄	semidirect
23. Q₈×D₇	Dic₃	semidirect
35. #859	D₇	direct (N × H)
45. C₇×Q₈	C₂×Dic₃	semidirect
84. SL(2,𝔽₃)	C₄×D₇	semidirect
101. D₇	#859	direct (N × H)
112. Q₈	Dic₃×D₇	semidirect
122. C₇	C₂×#859	semidirect

H ≅ G/N

type

2. C₇⋊#859

C₂

semidirect

3. C₇×#859

C₂

semidirect

7. D₇×SL(2,𝔽₃)

C₄

semidirect

19. C₇×SL(2,𝔽₃)

C₂×C₄

semidirect

23. Q₈×D₇

Dic₃

semidirect

35. #859

D₇

direct (N × H)

45. C₇×Q₈

C₂×Dic₃

semidirect

84. SL(2,𝔽₃)

C₄×D₇

semidirect

101. D₇

#859

direct (N × H)

112. Q₈

Dic₃×D₇

semidirect

122. C₇

C₂×#859

semidirect

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

subgroup	index
C₂×D₇×SL(2,𝔽₃)	2
C₇⋊#859	2
C₇×#859	2
D₇×#103	3
C₂×Dic₃×D₇	4
C₂×#859	7

subgroup

index

C₂×D₇×SL(2,𝔽₃)

C₂×Dic₃×D₇

Subgroups — D₇×#859 #13057075

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — D7×#859 #13057075

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — D₇×#859 #13057075