Group #14517675 — subgroups

Subgroups — C₂×#6899⋊C₄ #14517675

Order 1944.

308 conjugacy classes of subgroups; only the 106 normal subgroups are shown.

normal subgroups only

This group has 308 conjugacy classes of subgroups — too many to draw. Only its 106 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	C₂²×#6899	C₂	1	1	yes	maximal, characteristic
2	#6899⋊C₄	C₂	1	2	yes	maximal, direct product factor
3	C₂×C₃⋊(3₋¹⁺²⋊C₄)	—	3	1	no	maximal
4	C₂×C₉⋊Dic₉	C₃	1	1	yes	maximal, characteristic, complemented
5	C₂×#66316	—	3	3	no	maximal
6	C₂×#6899	C₄	1	2	yes	complemented
7	C₂×#6899	C₂²	1	1	yes	characteristic
8	C₂²×C₃×3₋¹⁺²	D₃	1	1	yes	characteristic
9	C₃⋊(3₋¹⁺²⋊C₄)	—	3	2	no	—
10	C₂²×C₉⋊C₉	—	2	1	no	—
11	C₂²×C₉⋊C₉	—	2	3	no	—
12	C₂²×C₃²⋊C₉	D₃	1	3	yes	—
13	C₂²×C₉²	C₂×C₃	1	1	yes	characteristic
14	C₉⋊Dic₉	C₂×C₃	1	2	yes	complemented
15	#66316	—	3	6	no	—
16	#6899	C₂×C₄	1	1	yes	characteristic, complemented, Sylow
17	C₂×C₃×C₃⋊Dic₃	—	9	1	no	—
18	C₂×C₃⋊Dic₉	—	3	1	no	—
19	C₂×C₃⋊Dic₉	—	3	3	no	—
20	C₂×3₋¹⁺²⋊C₄	—	9	3	no	—
21	C₂×C₃×3₋¹⁺²	Dic₃	1	2	yes	—
22	C₂×C₃×3₋¹⁺²	C₂×D₃	1	1	yes	characteristic
23	C₂×C₉⋊C₉	—	2	2	no	—
24	C₂×C₉⋊C₉	—	2	6	no	—
25	C₂×C₉⋊C₉	—	2	1	no	—
26	C₂×C₉⋊C₉	—	2	3	no	—
27	C₂×C₃²⋊C₉	Dic₃	1	6	yes	—
28	C₂×C₃²⋊C₉	C₂×D₃	1	3	yes	—
29	C₂×C₉²	C₃×C₄	1	2	yes	complemented
30	C₂×C₉²	C₂²×C₃	1	1	yes	characteristic
31	C₂²×C₃³	C₃⋊D₃	1	1	yes	characteristic
32	C₃×C₃⋊Dic₃	—	9	2	no	—
33	C₂²×3₋¹⁺²	—	3	3	no	—
34	C₂²×C₃×C₉	—	2	3	no	—
35	C₂²×C₃×C₉	—	2	1	no	—
36	C₂²×C₃×C₉	C₃×D₃	1	3	yes	—
37	C₂²×C₃×C₉	C₃×D₃	1	1	yes	characteristic
38	C₃⋊Dic₉	—	3	2	no	—
39	C₃⋊Dic₉	—	3	6	no	—
40	3₋¹⁺²⋊C₄	—	9	6	no	—
41	C₃×3₋¹⁺²	C₂×Dic₃	1	1	yes	characteristic
42	C₉⋊C₉	—	2	1	no	—
43	C₉⋊C₉	—	2	3	no	—
44	C₃²⋊C₉	C₂×Dic₃	1	3	yes	—
45	C₉²	C₂×C₃×C₄	1	1	yes	characteristic, complemented
46	C₂×C₃⋊Dic₃	—	9	1	no	—
47	C₂×C₃×Dic₃	—	27	1	no	—
48	C₂×C₃×Dic₃	—	27	3	no	—
49	C₂×Dic₉	—	9	3	no	—
50	C₂×Dic₉	—	27	3	no	—
51	C₂×C₃³	C₃⋊Dic₃	1	2	yes	—
52	C₂×C₃³	C₂×C₃⋊D₃	1	1	yes	characteristic
53	C₂×3₋¹⁺²	—	3	6	no	—
54	C₂×3₋¹⁺²	—	3	3	no	—
55	C₂×C₃×C₉	—	2	6	no	—
56	C₂×C₃×C₉	—	2	2	no	—
57	C₂×C₃×C₉	—	2	3	no	—
58	C₂×C₃×C₉	—	2	1	no	—
59	C₂×C₃×C₉	C₃×Dic₃	1	6	yes	—
60	C₂×C₃×C₉	C₃×Dic₃	1	2	yes	—
61	C₂×C₃×C₉	C₂×C₃×D₃	1	3	yes	—
62	C₂×C₃×C₉	C₂×C₃×D₃	1	1	yes	characteristic
63	C₂²×C₃²	—	3	1	no	—
64	C₂²×C₃²	—	3	3	no	—
65	C₂²×C₃²	C₃×C₃⋊D₃	1	1	yes	characteristic
66	C₃⋊Dic₃	—	9	2	no	—
67	C₃×Dic₃	—	27	2	no	—
68	C₃×Dic₃	—	27	6	no	—
69	C₂²×C₉	—	3	3	no	—
70	C₂²×C₉	—	6	1	no	—
71	C₂²×C₉	—	6	3	no	—
72	C₂²×C₉	AGL(1,ℤ/9)	1	3	yes	—
73	Dic₉	—	9	6	no	—
74	Dic₉	—	27	6	no	—
75	C₃³	C₂×C₃⋊Dic₃	1	1	yes	characteristic
76	3₋¹⁺²	—	3	3	no	—
77	C₃×C₉	—	2	3	no	—
78	C₃×C₉	—	2	1	no	—
79	C₃×C₉	C₂×C₃×Dic₃	1	3	yes	—
80	C₃×C₉	C₂×C₃×Dic₃	1	1	yes	characteristic
81	C₂×C₃×C₄	—	81	1	no	—
82	C₂×Dic₃	—	27	1	no	—
83	C₂×Dic₃	—	27	3	no	—
84	C₂×C₃²	—	3	2	no	—
85	C₂×C₃²	—	3	6	no	—
86	C₂×C₃²	—	3	1	no	—
87	C₂×C₃²	—	3	3	no	—
88	C₂×C₃²	C₃×C₃⋊Dic₃	1	2	yes	—
89	C₂×C₃²	C₂×C₃×C₃⋊D₃	1	1	yes	characteristic
90	C₂×C₉	—	3	6	no	—
91	C₂×C₉	—	3	3	no	—
92	C₂×C₉	—	6	2	no	—
93	C₂×C₉	—	6	6	no	—
94	C₂×C₉	—	6	1	no	—
95	C₂×C₉	—	6	3	no	—
96	C₂×C₉	3₋¹⁺²⋊C₄	1	6	yes	—
97	C₂×C₉	C₂×AGL(1,ℤ/9)	1	3	yes	—
98	C₂²×C₃	—	9	1	no	—
99	C₂²×C₃	C₃⋊AGL(1,ℤ/9)	1	1	yes	characteristic
100	C₂²×C₃	#4207	1	3	yes	—
101	C₃×C₄	—	81	2	no	—
102	Dic₃	—	27	2	no	—
103	Dic₃	—	27	6	no	—
104	C₃²	—	3	1	no	—
105	C₃²	—	3	3	no	—
106	C₃²	C₂×C₃×C₃⋊Dic₃	1	1	yes	characteristic
107	C₉	—	3	3	no	—
108	C₉	—	6	1	no	—
109	C₉	—	6	3	no	—
110	C₉	C₂×3₋¹⁺²⋊C₄	1	3	yes	—
111	C₂×C₄	—	81	1	no	Sylow
112	C₂×C₃	—	9	2	no	—
113	C₂×C₃	—	9	1	no	—
114	C₂×C₃	C₃⋊(3₋¹⁺²⋊C₄)	1	2	yes	—
115	C₂×C₃	#66409	1	6	yes	—
116	C₂×C₃	C₂×C₃⋊AGL(1,ℤ/9)	1	1	yes	characteristic
117	C₂×C₃	C₂×#4207	1	3	yes	—
118	C₂²	#6899⋊C₂	1	1	yes	characteristic
119	C₄	—	81	2	no	—
120	C₃	—	9	1	no	—
121	C₃	C₂×C₃⋊(3₋¹⁺²⋊C₄)	1	1	yes	characteristic
122	C₃	C₂×#66409	1	3	yes	—
123	C₂	#6899⋊C₄	1	2	yes	direct product factor
124	C₂	C₂×#6899⋊C₂	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂²×#6899

C₂

yes

maximal, characteristic

#6899⋊C₄

C₂

yes

maximal, direct product factor

C₂×C₃⋊(3₋¹⁺²⋊C₄)

—

maximal

C₂×C₉⋊Dic₉

C₃

yes

maximal, characteristic, complemented

C₂×#66316

—

maximal

C₂×#6899

C₄

yes

complemented

C₂×#6899

C₂²

yes

characteristic

C₂²×C₃×3₋¹⁺²

D₃

yes

characteristic

C₃⋊(3₋¹⁺²⋊C₄)

—

C₂²×C₉⋊C₉

—

C₂²×C₉⋊C₉

—

C₂²×C₃²⋊C₉

D₃

yes

—

C₂²×C₉²

C₂×C₃

yes

characteristic

C₉⋊Dic₉

C₂×C₃

yes

complemented

#66316

—

#6899

C₂×C₄

yes

characteristic, complemented, Sylow

C₂×C₃×C₃⋊Dic₃

—

C₂×C₃⋊Dic₉

—

C₂×C₃⋊Dic₉

—

C₂×3₋¹⁺²⋊C₄

—

C₂×C₃×3₋¹⁺²

Dic₃

yes

—

C₂×C₃×3₋¹⁺²

C₂×D₃

yes

characteristic

C₂×C₉⋊C₉

—

C₂×C₉⋊C₉

—

C₂×C₉⋊C₉

—

C₂×C₉⋊C₉

—

C₂×C₃²⋊C₉

Dic₃

yes

—

C₂×C₃²⋊C₉

C₂×D₃

yes

—

C₂×C₉²

C₃×C₄

yes

complemented

C₂×C₉²

C₂²×C₃

yes

characteristic

C₂²×C₃³

C₃⋊D₃

yes

characteristic

C₃×C₃⋊Dic₃

—

C₂²×3₋¹⁺²

—

C₂²×C₃×C₉

—

C₂²×C₃×C₉

—

C₂²×C₃×C₉

C₃×D₃

yes

—

C₂²×C₃×C₉

C₃×D₃

yes

characteristic

C₃⋊Dic₉

—

C₃⋊Dic₉

—

3₋¹⁺²⋊C₄

—

C₃×3₋¹⁺²

C₂×Dic₃

yes

characteristic

C₉⋊C₉

—

C₉⋊C₉

—

C₃²⋊C₉

C₂×Dic₃

yes

—

C₉²

C₂×C₃×C₄

yes

characteristic, complemented

C₂×C₃⋊Dic₃

—

C₂×C₃×Dic₃

—

C₂×C₃×Dic₃

—

C₂×Dic₉

—

C₂×Dic₉

—

C₂×C₃³

C₃⋊Dic₃

yes

—

C₂×C₃³

C₂×C₃⋊D₃

yes

characteristic

C₂×3₋¹⁺²

—

C₂×3₋¹⁺²

—

C₂×C₃×C₉

—

C₂×C₃×C₉

—

C₂×C₃×C₉

—

C₂×C₃×C₉

—

C₂×C₃×C₉

C₃×Dic₃

yes

—

C₂×C₃×C₉

C₃×Dic₃

yes

—

C₂×C₃×C₉

C₂×C₃×D₃

yes

—

C₂×C₃×C₉

C₂×C₃×D₃

yes

characteristic

C₂²×C₃²

—

C₂²×C₃²

—

C₂²×C₃²

C₃×C₃⋊D₃

yes

characteristic

C₃⋊Dic₃

—

C₃×Dic₃

—

C₃×Dic₃

—

C₂²×C₉

—

C₂²×C₉

—

C₂²×C₉

—

C₂²×C₉

AGL(1,ℤ/9)

yes

—

Dic₉

—

Dic₉

—

C₃³

C₂×C₃⋊Dic₃

yes

characteristic

3₋¹⁺²

—

C₃×C₉

—

C₃×C₉

—

C₃×C₉

C₂×C₃×Dic₃

yes

—

C₃×C₉

C₂×C₃×Dic₃

yes

characteristic

C₂×C₃×C₄

—

C₂×Dic₃

—

C₂×Dic₃

—

C₂×C₃²

—

C₂×C₃²

—

C₂×C₃²

—

C₂×C₃²

—

C₂×C₃²

C₃×C₃⋊Dic₃

yes

—

C₂×C₃²

C₂×C₃×C₃⋊D₃

yes

characteristic

C₂×C₉

—

C₂×C₉

—

C₂×C₉

—

C₂×C₉

—

C₂×C₉

—

C₂×C₉

—

C₂×C₉

3₋¹⁺²⋊C₄

yes

—

C₂×C₉

C₂×AGL(1,ℤ/9)

yes

—

C₂²×C₃

—

C₂²×C₃

C₃⋊AGL(1,ℤ/9)

yes

characteristic

100

C₂²×C₃

#4207

yes

—

101

C₃×C₄

—

102

Dic₃

—

103

Dic₃

—

104

C₃²

—

105

C₃²

—

106

C₃²

C₂×C₃×C₃⋊Dic₃

yes

characteristic

107

C₉

—

108

C₉

—

109

C₉

—

110

C₉

C₂×3₋¹⁺²⋊C₄

yes

—

111

C₂×C₄

—

Sylow

112

C₂×C₃

—

113

C₂×C₃

—

114

C₂×C₃

C₃⋊(3₋¹⁺²⋊C₄)

yes

—

115

C₂×C₃

#66409

yes

—

116

C₂×C₃

C₂×C₃⋊AGL(1,ℤ/9)

yes

characteristic

117

C₂×C₃

C₂×#4207

yes

—

118

C₂²

#6899⋊C₂

yes

characteristic

119

C₄

—

120

C₃

—

121

C₃

C₂×C₃⋊(3₋¹⁺²⋊C₄)

yes

characteristic

122

C₃

C₂×#66409

yes

—

123

C₂

#6899⋊C₄

yes

direct product factor

124

C₂

C₂×#6899⋊C₂

yes

characteristic

	subgroup	G/H
center, hypercenter	C₂²	#6899⋊C₂
commutator, hypocenter	C₉²	C₂×C₃×C₄
Frattini	C₂×C₃²	C₂×C₃×C₃⋊D₃
Fitting	C₂²×#6899	C₂
socle	C₂²×C₃²	C₃×C₃⋊D₃
prime radical	C₂²×C₃³	C₃⋊D₃
cosocle	C₂×C₉²	C₂²×C₃
LCS	G ▷ C₉²
UCS	C₁ ◁ C₂²

subgroup

G/H

center, hypercenter

C₂²

#6899⋊C₂

commutator, hypocenter

C₉²

C₂×C₃×C₄

Frattini

C₂×C₃²

C₂×C₃×C₃⋊D₃

Fitting

C₂²×#6899

C₂

socle

C₂²×C₃²

C₃×C₃⋊D₃

prime radical

C₂²×C₃³

C₃⋊D₃

cosocle

C₂×C₉²

C₂²×C₃

LCS

G ▷ C₉²

UCS

C₁ ◁ C₂²

N	H ≅ G/N	type
2. #6899⋊C₄	C₂	direct (N × H)
4. C₂×C₉⋊Dic₉	C₃	semidirect
6. C₂×#6899	C₄	semidirect
14. C₉⋊Dic₉	C₂×C₃	semidirect
16. #6899	C₂×C₄	semidirect
29. C₂×C₉²	C₃×C₄	semidirect
45. C₉²	C₂×C₃×C₄	semidirect
123. C₂	#6899⋊C₄	direct (N × H)

H ≅ G/N

type

2. #6899⋊C₄

C₂

direct (N × H)

4. C₂×C₉⋊Dic₉

C₃

semidirect

6. C₂×#6899

C₄

semidirect

14. C₉⋊Dic₉

C₂×C₃

semidirect

16. #6899

C₂×C₄

semidirect

29. C₂×C₉²

C₃×C₄

semidirect

45. C₉²

C₂×C₃×C₄

semidirect

123. C₂

#6899⋊C₄

direct (N × H)

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

subgroup	index
C₂²×#6899	2
#6899⋊C₄	2
C₂×C₃⋊(3₋¹⁺²⋊C₄)	3
C₂×C₉⋊Dic₉	3
C₂×#66316	3

subgroup

index

C₂²×#6899

#6899⋊C₄

C₂×C₃⋊(3₋¹⁺²⋊C₄)

C₂×C₉⋊Dic₉

C₂×#66316

Subgroups — C₂×#6899⋊C₄ #14517675

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C2×#6899⋊C4 #14517675

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#6899⋊C₄ #14517675