Group #13186010 — subgroups

Subgroups — C₇×#8794 #13186010

Order 1792.

270 conjugacy classes of subgroups; only the 126 normal subgroups are shown.

normal subgroups only

This group has 270 conjugacy classes of subgroups — too many to draw. Only its 126 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×#1848	C₂	1	1	yes	maximal, characteristic
2	C₇×#1835	C₂	1	1	yes	maximal, characteristic
3	C₇×#1452	C₂	1	1	yes	maximal, characteristic
4	C₇×#1286	C₂	1	4	yes	maximal
5	C₇×#431	C₂²	1	2	yes	—
6	C₂×C₇×#108	—	2	2	no	—
7	C₇×#423	C₂²	1	2	yes	—
8	C₂×C₄×C₇×C₈	C₂²	1	1	yes	characteristic
9	C₇×#382	C₂²	1	2	yes	—
10	C₇×#340	—	2	2	no	—
11	C₇×#322	C₄	1	4	yes	complemented
12	#8794	C₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
13	C₂×C₇×M₁₆	—	2	2	no	—
14	C₂²×C₇×C₈	D₄	1	2	yes	—
15	C₂×C₇×C₄⋊C₄	—	2	2	no	—
16	C₂×C₄²×C₇	C₂³	1	1	yes	characteristic
17	C₇×#108	—	4	4	no	—
18	C₇×C₄⋊C₈	—	2	2	no	—
19	C₇×C₄⋊C₈	—	2	4	no	—
20	C₇×C₂²⋊C₈	—	2	2	no	—
21	C₄×C₇×C₈	C₂×C₄	1	4	yes	—
22	C₄×C₇×C₈	C₂×C₄	1	2	yes	—
23	C₄×C₇×C₈	Q₈	1	2	yes	—
24	C₇×#95	—	2	4	no	—
25	#1848	C₂×C₇	1	1	yes	characteristic
26	#1835	C₂×C₇	1	1	yes	characteristic
27	#1452	C₂×C₇	1	1	yes	characteristic
28	#1286	C₂×C₇	1	4	yes	—
29	C₂²×C₄×C₇	—	4	2	no	—
30	C₂²×C₄×C₇	C₂×D₄	1	1	yes	characteristic
31	C₂²×C₄×C₇	#41	1	2	yes	—
32	C₇×M₁₆	—	4	4	no	—
33	C₂×C₇×C₈	—	2	2	no	—
34	C₂×C₇×C₈	—	2	2	no	—
35	C₂×C₇×C₈	—	2	4	no	—
36	C₂×C₇×C₈	—	4	2	no	—
37	C₂×C₇×C₈	C₄⋊C₄	1	4	yes	—
38	C₇×C₄⋊C₄	—	4	2	no	—
39	C₇×C₄⋊C₄	—	4	2	no	—
40	C₄²×C₇	C₂²×C₄	1	1	yes	characteristic
41	C₄²×C₇	C₂×Q₈	1	1	yes	characteristic
42	C₄²×C₇	#41	1	2	yes	—
43	#431	C₂²×C₇	1	2	yes	—
44	C₂×#108	—	2	2	no	—
45	#423	C₂²×C₇	1	2	yes	—
46	C₂×C₄×C₈	C₂²×C₇	1	1	yes	characteristic
47	#382	C₂²×C₇	1	2	yes	—
48	#340	—	2	2	no	—
49	#322	C₄×C₇	1	4	yes	complemented
50	C₂³×C₇	#124	1	1	yes	characteristic
51	C₂×C₄×C₇	—	4	2	no	—
52	C₂×C₄×C₇	—	8	4	no	—
53	C₂×C₄×C₇	—	8	2	no	—
54	C₂×C₄×C₇	C₂×C₄⋊C₄	1	1	yes	characteristic
55	C₂×C₄×C₇	#117	1	2	yes	—
56	C₂×C₄×C₇	#124	1	1	yes	characteristic
57	C₂×C₄×C₇	#125	1	2	yes	—
58	C₂×C₄×C₇	AGL(1,ℤ/8)	1	2	yes	—
59	C₂×C₄×C₇	#137	1	2	yes	—
60	C₇×C₈	—	2	4	no	—
61	C₇×C₈	—	4	4	no	—
62	C₇×C₈	—	8	2	no	—
63	C₂×M₁₆	—	2	2	no	—
64	C₂²×C₈	C₇×D₄	1	2	yes	—
65	C₂×C₄⋊C₄	—	2	2	no	—
66	C₂×C₄²	C₂³×C₇	1	1	yes	characteristic
67	#108	—	4	4	no	—
68	C₄⋊C₈	—	2	2	no	—
69	C₄⋊C₈	—	2	4	no	—
70	C₂²⋊C₈	—	2	2	no	—
71	C₄×C₈	C₂×C₄×C₇	1	4	yes	—
72	C₄×C₈	C₂×C₄×C₇	1	2	yes	—
73	C₄×C₈	C₇×Q₈	1	2	yes	—
74	#95	—	2	4	no	—
75	C₂²×C₇	—	2	2	no	—
76	C₂²×C₇	#382	1	1	yes	characteristic
77	C₂²×C₇	#489	1	2	yes	—
78	C₄×C₇	—	2	2	no	—
79	C₄×C₇	—	2	2	no	—
80	C₄×C₇	—	16	2	no	—
81	C₄×C₇	#428	1	2	yes	—
82	C₄×C₇	#490	1	1	yes	characteristic
83	C₄×C₇	#491	1	1	yes	characteristic
84	C₂²×C₄	—	4	2	no	—
85	C₂²×C₄	C₂×C₇×D₄	1	1	yes	characteristic
86	C₂²×C₄	C₇×#41	1	2	yes	—
87	M₁₆	—	4	4	no	—
88	C₂×C₈	—	2	2	no	—
89	C₂×C₈	—	2	2	no	—
90	C₂×C₈	—	2	4	no	—
91	C₂×C₈	—	4	2	no	—
92	C₂×C₈	C₇×C₄⋊C₄	1	4	yes	—
93	C₄⋊C₄	—	4	2	no	—
94	C₄⋊C₄	—	4	2	no	—
95	C₄²	C₂²×C₄×C₇	1	1	yes	characteristic
96	C₄²	C₂×C₇×Q₈	1	1	yes	characteristic
97	C₄²	C₇×#41	1	2	yes	—
98	C₂×C₇	—	2	2	no	—
99	C₂×C₇	#1833	1	2	yes	—
100	C₂×C₇	#1836	1	1	yes	characteristic
101	C₂³	C₇×#124	1	1	yes	characteristic
102	C₂×C₄	—	4	2	no	—
103	C₂×C₄	—	8	4	no	—
104	C₂×C₄	—	8	2	no	—
105	C₂×C₄	C₂×C₇×C₄⋊C₄	1	1	yes	characteristic
106	C₂×C₄	C₇×#117	1	2	yes	—
107	C₂×C₄	C₇×#124	1	1	yes	characteristic
108	C₂×C₄	C₇×#125	1	2	yes	—
109	C₂×C₄	C₇×AGL(1,ℤ/8)	1	2	yes	—
110	C₂×C₄	C₇×#137	1	2	yes	—
111	C₈	—	2	4	no	—
112	C₈	—	4	4	no	—
113	C₈	—	8	2	no	—
114	C₇	#8794	1	1	yes	characteristic, direct product factor, Sylow
115	C₂²	—	2	2	no	—
116	C₂²	C₇×#382	1	1	yes	characteristic
117	C₂²	C₇×#489	1	2	yes	—
118	C₄	—	2	2	no	—
119	C₄	—	2	2	no	—
120	C₄	—	16	2	no	—
121	C₄	C₇×#428	1	2	yes	—
122	C₄	C₇×#490	1	1	yes	characteristic
123	C₄	C₇×#491	1	1	yes	characteristic
124	C₂	—	2	2	no	—
125	C₂	C₇×#1833	1	2	yes	—
126	C₂	C₇×#1836	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₇×#1848

C₂

yes

maximal, characteristic

C₇×#1835

C₂

yes

maximal, characteristic

C₇×#1452

C₂

yes

maximal, characteristic

C₇×#1286

C₂

yes

maximal

C₇×#431

C₂²

yes

—

C₂×C₇×#108

—

C₇×#423

C₂²

yes

—

C₂×C₄×C₇×C₈

C₂²

yes

characteristic

C₇×#382

C₂²

yes

—

C₇×#340

—

C₇×#322

C₄

yes

complemented

#8794

C₇

yes

maximal, characteristic, direct product factor, Sylow

C₂×C₇×M₁₆

—

C₂²×C₇×C₈

D₄

yes

—

C₂×C₇×C₄⋊C₄

—

C₂×C₄²×C₇

C₂³

yes

characteristic

C₇×#108

—

C₇×C₄⋊C₈

—

C₇×C₄⋊C₈

—

C₇×C₂²⋊C₈

—

C₄×C₇×C₈

C₂×C₄

yes

—

C₄×C₇×C₈

C₂×C₄

yes

—

C₄×C₇×C₈

Q₈

yes

—

C₇×#95

—

#1848

C₂×C₇

yes

characteristic

#1835

C₂×C₇

yes

characteristic

#1452

C₂×C₇

yes

characteristic

#1286

C₂×C₇

yes

—

C₂²×C₄×C₇

—

C₂²×C₄×C₇

C₂×D₄

yes

characteristic

C₂²×C₄×C₇

#41

yes

—

C₇×M₁₆

—

C₂×C₇×C₈

—

C₂×C₇×C₈

—

C₂×C₇×C₈

—

C₂×C₇×C₈

—

C₂×C₇×C₈

C₄⋊C₄

yes

—

C₇×C₄⋊C₄

—

C₇×C₄⋊C₄

—

C₄²×C₇

C₂²×C₄

yes

characteristic

C₄²×C₇

C₂×Q₈

yes

characteristic

C₄²×C₇

#41

yes

—

#431

C₂²×C₇

yes

—

C₂×#108

—

#423

C₂²×C₇

yes

—

C₂×C₄×C₈

C₂²×C₇

yes

characteristic

#382

C₂²×C₇

yes

—

#340

—

#322

C₄×C₇

yes

complemented

C₂³×C₇

#124

yes

characteristic

C₂×C₄×C₇

—

C₂×C₄×C₇

—

C₂×C₄×C₇

—

C₂×C₄×C₇

C₂×C₄⋊C₄

yes

characteristic

C₂×C₄×C₇

#117

yes

—

C₂×C₄×C₇

#124

yes

characteristic

C₂×C₄×C₇

#125

yes

—

C₂×C₄×C₇

AGL(1,ℤ/8)

yes

—

C₂×C₄×C₇

#137

yes

—

C₇×C₈

—

C₇×C₈

—

C₇×C₈

—

C₂×M₁₆

—

C₂²×C₈

C₇×D₄

yes

—

C₂×C₄⋊C₄

—

C₂×C₄²

C₂³×C₇

yes

characteristic

#108

—

C₄⋊C₈

—

C₄⋊C₈

—

C₂²⋊C₈

—

C₄×C₈

C₂×C₄×C₇

yes

—

C₄×C₈

C₂×C₄×C₇

yes

—

C₄×C₈

C₇×Q₈

yes

—

#95

—

C₂²×C₇

—

C₂²×C₇

#382

yes

characteristic

C₂²×C₇

#489

yes

—

C₄×C₇

—

C₄×C₇

—

C₄×C₇

—

C₄×C₇

#428

yes

—

C₄×C₇

yes

C₄×C₇

yes

C₂²×C₄

—

C₂²×C₄

C₂×C₇×D₄

yes

characteristic

C₂²×C₄

C₇×#41

yes

—

M₁₆

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

C₇×C₄⋊C₄

yes

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄²

C₂²×C₄×C₇

yes

characteristic

C₄²

C₂×C₇×Q₈

yes

characteristic

C₄²

C₇×#41

yes

—

C₂×C₇

—

C₂×C₇

#1833

yes

—

100

C₂×C₇

#1836

yes

characteristic

101

C₂³

C₇×#124

yes

characteristic

102

C₂×C₄

—

103

C₂×C₄

—

104

C₂×C₄

—

105

C₂×C₄

C₂×C₇×C₄⋊C₄

yes

characteristic

106

C₂×C₄

C₇×#117

yes

—

107

C₂×C₄

C₇×#124

yes

characteristic

108

C₂×C₄

C₇×#125

yes

—

109

C₂×C₄

C₇×AGL(1,ℤ/8)

yes

—

110

C₂×C₄

C₇×#137

yes

—

111

C₈

—

112

C₈

—

113

C₈

—

114

C₇

#8794

yes

characteristic, direct product factor, Sylow

115

C₂²

—

116

C₂²

C₇×#382

yes

characteristic

117

C₂²

C₇×#489

yes

—

118

C₄

—

119

C₄

—

120

C₄

—

121

C₄

C₇×#428

yes

—

122

C₄

C₇×#490

yes

characteristic

123

C₄

C₇×#491

yes

characteristic

124

C₂

—

125

C₂

C₇×#1833

yes

—

126

C₂

C₇×#1836

yes

characteristic

	subgroup	G/H
center, socle	C₂²×C₇	#382
commutator	C₄²	C₂²×C₄×C₇
Frattini, cosocle	C₂×C₄²	C₂³×C₇
prime radical	C₂³×C₇	#124
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₇ ◁ C₂×C₄²×C₇ ◁ G

subgroup

G/H

center, socle

C₂²×C₇

#382

commutator

C₄²

C₂²×C₄×C₇

Frattini, cosocle

C₂×C₄²

C₂³×C₇

C₂³×C₇

G ▷ C₄² ▷ C₂² ▷ C₁

C₁ ◁ C₂²×C₇ ◁ C₂×C₄²×C₇ ◁ G

N	H ≅ G/N	type
11. C₇×#322	C₄	semidirect
12. #8794	C₇	direct (N × H)
49. #322	C₄×C₇	semidirect
114. C₇	#8794	direct (N × H)

H ≅ G/N

type

11. C₇×#322

C₄

semidirect

12. #8794

C₇

direct (N × H)

49. #322

C₄×C₇

semidirect

114. C₇

#8794

direct (N × H)

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

subgroup	index
C₇×#1848	2
C₇×#1835	2
C₇×#1452	2
C₇×#1286	2
#8794	7

subgroup

index

Subgroups — C₇×#8794 #13186010

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C7×#8794 #13186010

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₇×#8794 #13186010