Group #14558320 — subgroups

Subgroups — C₂×#11924562 #14558320

Order 2368.

298 conjugacy classes of subgroups; only the 111 normal subgroups are shown.

normal subgroups only

This group has 298 conjugacy classes of subgroups — too many to draw. Only its 111 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×#10596962	C₂	1	1	yes	maximal, characteristic, complemented
2	#11924562	C₂	1	8	yes	maximal, direct product factor
3	C₂×#10596940	C₂	1	2	yes	maximal, complemented
4	C₂×#10596939	C₂	1	2	yes	maximal, complemented
5	C₂×#10596934	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂²×D₄×C₃₇	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂⁴×C₃₇	—	2	1	no	—
8	C₂×D₄×C₃₇	—	2	1	no	—
9	C₂×D₄×C₃₇	—	2	2	no	—
10	C₂×D₄×C₃₇	C₂²	1	2	yes	—
11	C₂×D₄×C₃₇	C₂²	1	4	yes	complemented
12	C₂²×C₄×C₃₇	C₂²	1	1	yes	characteristic
13	C₂×C₂²⋊D₃₇	—	2	1	no	—
14	#10596962	—	2	2	no	—
15	#10596962	C₂²	1	4	yes	complemented
16	C₂×D₁₄₈	C₂²	1	1	yes	characteristic, complemented
17	C₂×C₄×D₃₇	—	2	1	no	—
18	C₂×Dic₇₄	C₂²	1	1	yes	characteristic, complemented
19	#10596940	C₂²	1	8	yes	complemented
20	#10596939	C₂²	1	8	yes	complemented
21	#10596934	C₂²	1	4	yes	complemented
22	C₂×#64436	C₂²	1	2	yes	complemented
23	C₂³×C₃₇	—	2	2	no	—
24	C₂³×C₃₇	—	2	4	no	—
25	C₂³×C₃₇	—	4	4	no	—
26	C₂³×C₃₇	D₄	1	1	yes	characteristic
27	C₂²×D₃₇	—	2	1	no	—
28	D₄×C₃₇	—	2	2	no	—
29	D₄×C₃₇	—	2	4	no	—
30	D₄×C₃₇	C₂³	1	4	yes	—
31	C₂×C₄×C₃₇	D₄	1	1	yes	characteristic
32	C₂×C₄×C₃₇	D₄	1	2	yes	—
33	C₂×C₄×C₃₇	C₂³	1	1	yes	characteristic
34	C₂×C₄×C₃₇	C₂³	1	2	yes	—
35	C₂²⋊D₃₇	—	2	4	no	—
36	C₂×Dic₃₇	—	2	1	no	—
37	D₁₄₈	—	2	1	no	—
38	D₁₄₈	C₂³	1	2	yes	complemented
39	C₄×D₃₇	—	2	4	no	—
40	Dic₇₄	—	2	1	no	—
41	Dic₇₄	C₂³	1	2	yes	complemented
42	#64436	C₂³	1	4	yes	complemented
43	C₂²×C₃₇	—	2	2	no	—
44	C₂²×C₃₇	—	2	4	no	—
45	C₂²×C₃₇	—	4	8	no	—
46	C₂²×C₃₇	—	4	4	no	—
47	C₂²×C₃₇	C₂×D₄	1	2	yes	—
48	C₂²×C₃₇	C₂×D₄	1	1	yes	characteristic
49	C₂×D₃₇	—	2	2	no	—
50	C₂×D₃₇	—	4	2	no	—
51	C₄×C₃₇	C₂×D₄	1	1	yes	characteristic
52	C₄×C₃₇	C₂×D₄	1	2	yes	—
53	C₄×C₃₇	C₂⁴	1	1	yes	characteristic
54	Dic₃₇	—	2	2	no	—
55	C₂×C₃₇	—	2	2	no	—
56	C₂×C₃₇	—	4	4	no	—
57	C₂×C₃₇	AGL(1,ℤ/8)	1	2	yes	complemented
58	C₂×C₃₇	C₂²×D₄	1	1	yes	characteristic
59	D₃₇	—	4	2	no	—
60	C₂×AGL(1,ℤ/8)	—	37	1	no	maximal, Sylow
61	C₃₇	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic, complemented, Sylow
62	C₂×#41	—	37	1	no	—
63	C₂²×D₄	D₃₇	1	1	yes	characteristic, complemented
64	AGL(1,ℤ/8)	—	37	8	no	—
65	C₂×SD₁₆	—	37	2	no	—
66	C₂×D₈	—	37	2	no	—
67	C₂×M₁₆	—	37	1	no	—
68	C₂⁴	—	2	1	no	—
69	#41	—	37	4	no	—
70	#41	—	74	2	no	—
71	C₂×Q₈	—	37	1	no	—
72	C₂×D₄	—	2	2	no	—
73	C₂×D₄	—	2	1	no	—
74	C₂×D₄	—	37	1	no	—
75	C₂×D₄	—	74	1	no	—
76	C₂×D₄	C₂×D₃₇	1	2	yes	—
77	C₂×D₄	C₂×D₃₇	1	4	yes	complemented
78	C₂²×C₄	—	74	1	no	—
79	C₂²×C₄	C₂×D₃₇	1	1	yes	characteristic
80	SD₁₆	—	37	8	no	—
81	D₈	—	37	8	no	—
82	M₁₆	—	37	4	no	—
83	C₂×C₈	—	37	2	no	—
84	C₂³	—	2	4	no	—
85	C₂³	—	2	2	no	—
86	C₂³	—	4	4	no	—
87	C₂³	—	74	1	no	—
88	C₂³	C₂²⋊D₃₇	1	1	yes	characteristic
89	Q₈	—	37	2	no	—
90	Q₈	—	74	1	no	—
91	D₄	—	2	2	no	—
92	D₄	—	2	4	no	—
93	D₄	—	37	2	no	—
94	D₄	—	74	4	no	—
95	D₄	—	74	1	no	—
96	D₄	C₂²×D₃₇	1	4	yes	—
97	C₂×C₄	—	74	1	no	—
98	C₂×C₄	—	74	4	no	—
99	C₂×C₄	C₂²⋊D₃₇	1	2	yes	—
100	C₂×C₄	C₂²⋊D₃₇	1	1	yes	characteristic
101	C₂×C₄	C₂²×D₃₇	1	2	yes	—
102	C₂×C₄	C₂²×D₃₇	1	1	yes	characteristic
103	C₈	—	37	4	no	—
104	C₂²	—	2	2	no	—
105	C₂²	—	2	4	no	—
106	C₂²	—	4	8	no	—
107	C₂²	—	4	4	no	—
108	C₂²	—	74	2	no	—
109	C₂²	—	148	2	no	—
110	C₂²	C₂×C₂²⋊D₃₇	1	2	yes	—
111	C₂²	C₂×C₂²⋊D₃₇	1	1	yes	characteristic
112	C₄	—	74	2	no	—
113	C₄	C₂×C₂²⋊D₃₇	1	2	yes	—
114	C₄	C₂×C₂²⋊D₃₇	1	1	yes	characteristic
115	C₄	C₂³×D₃₇	1	1	yes	characteristic
116	C₂	—	2	2	no	—
117	C₂	—	4	4	no	—
118	C₂	—	148	2	no	—
119	C₂	#11924562	1	2	yes	direct product factor
120	C₂	C₂²×C₂²⋊D₃₇	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#10596962

C₂

yes

maximal, characteristic, complemented

#11924562

C₂

yes

maximal, direct product factor

C₂×#10596940

C₂

yes

maximal, complemented

C₂×#10596939

C₂

yes

maximal, complemented

C₂×#10596934

C₂

yes

maximal, characteristic, complemented

C₂²×D₄×C₃₇

C₂

yes

maximal, characteristic, complemented

C₂⁴×C₃₇

—

C₂×D₄×C₃₇

—

C₂×D₄×C₃₇

—

C₂×D₄×C₃₇

C₂²

yes

—

C₂×D₄×C₃₇

C₂²

yes

complemented

C₂²×C₄×C₃₇

C₂²

yes

characteristic

C₂×C₂²⋊D₃₇

—

#10596962

—

#10596962

C₂²

yes

complemented

C₂×D₁₄₈

C₂²

yes

characteristic, complemented

C₂×C₄×D₃₇

—

C₂×Dic₇₄

C₂²

yes

characteristic, complemented

#10596940

C₂²

yes

complemented

#10596939

C₂²

yes

complemented

#10596934

C₂²

yes

complemented

C₂×#64436

C₂²

yes

complemented

C₂³×C₃₇

—

C₂³×C₃₇

—

C₂³×C₃₇

—

C₂³×C₃₇

D₄

yes

characteristic

C₂²×D₃₇

—

D₄×C₃₇

—

D₄×C₃₇

—

D₄×C₃₇

C₂³

yes

—

C₂×C₄×C₃₇

D₄

yes

characteristic

C₂×C₄×C₃₇

D₄

yes

—

C₂×C₄×C₃₇

C₂³

yes

characteristic

C₂×C₄×C₃₇

C₂³

yes

—

C₂²⋊D₃₇

—

C₂×Dic₃₇

—

D₁₄₈

—

D₁₄₈

C₂³

yes

complemented

C₄×D₃₇

—

Dic₇₄

—

Dic₇₄

C₂³

yes

complemented

#64436

C₂³

yes

complemented

C₂²×C₃₇

—

C₂²×C₃₇

—

C₂²×C₃₇

—

C₂²×C₃₇

—

C₂²×C₃₇

C₂×D₄

yes

—

C₂²×C₃₇

C₂×D₄

yes

characteristic

C₂×D₃₇

—

C₂×D₃₇

—

C₄×C₃₇

C₂×D₄

yes

characteristic

C₄×C₃₇

C₂×D₄

yes

—

C₄×C₃₇

C₂⁴

yes

characteristic

Dic₃₇

—

C₂×C₃₇

—

C₂×C₃₇

—

C₂×C₃₇

AGL(1,ℤ/8)

yes

complemented

C₂×C₃₇

C₂²×D₄

yes

characteristic

D₃₇

—

C₂×AGL(1,ℤ/8)

—

maximal, Sylow

C₃₇

C₂×AGL(1,ℤ/8)

yes

characteristic, complemented, Sylow

C₂×#41

—

C₂²×D₄

D₃₇

yes

characteristic, complemented

AGL(1,ℤ/8)

—

C₂×SD₁₆

—

C₂×D₈

—

C₂×M₁₆

—

C₂⁴

—

#41

—

#41

—

C₂×Q₈

—

C₂×D₄

—

C₂×D₄

—

C₂×D₄

—

C₂×D₄

—

C₂×D₄

C₂×D₃₇

yes

—

C₂×D₄

C₂×D₃₇

yes

complemented

C₂²×C₄

—

C₂²×C₄

C₂×D₃₇

yes

characteristic

SD₁₆

—

D₈

—

M₁₆

—

C₂×C₈

—

C₂³

—

C₂³

—

C₂³

—

C₂³

—

C₂³

C₂²⋊D₃₇

yes

characteristic

Q₈

—

Q₈

—

D₄

—

D₄

—

D₄

—

D₄

—

D₄

—

D₄

C₂²×D₃₇

yes

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

C₂²⋊D₃₇

yes

—

100

C₂×C₄

C₂²⋊D₃₇

yes

characteristic

101

C₂×C₄

C₂²×D₃₇

yes

—

102

C₂×C₄

C₂²×D₃₇

yes

characteristic

103

C₈

—

104

C₂²

—

105

C₂²

—

106

C₂²

—

107

C₂²

—

108

C₂²

—

109

C₂²

—

148

—

110

C₂²

C₂×C₂²⋊D₃₇

yes

—

111

C₂²

C₂×C₂²⋊D₃₇

yes

characteristic

112

C₄

—

113

C₄

C₂×C₂²⋊D₃₇

yes

—

114

C₄

C₂×C₂²⋊D₃₇

yes

characteristic

115

C₄

C₂³×D₃₇

yes

characteristic

116

C₂

—

117

C₂

—

118

C₂

—

148

—

119

C₂

#11924562

yes

direct product factor

120

C₂

C₂²×C₂²⋊D₃₇

yes

characteristic

	subgroup	G/H
center	C₂²	C₂×C₂²⋊D₃₇
commutator, cosocle	C₄×C₃₇	C₂⁴
Frattini	C₄	C₂³×D₃₇
Fitting	C₂²×D₄×C₃₇	C₂
socle	C₂²×C₃₇	C₂×D₄
hypercenter	C₂²×D₄	D₃₇
hypocenter	C₃₇	C₂×AGL(1,ℤ/8)
LCS	G ▷ C₄×C₃₇ ▷ C₂×C₃₇ ▷ C₃₇
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂²×D₄

subgroup

G/H

center

C₂²

C₂×C₂²⋊D₃₇

commutator, cosocle

C₄×C₃₇

C₂⁴

Frattini

C₄

C₂³×D₃₇

Fitting

C₂²×D₄×C₃₇

C₂

socle

C₂²×C₃₇

C₂×D₄

hypercenter

C₂²×D₄

D₃₇

hypocenter

C₃₇

C₂×AGL(1,ℤ/8)

LCS

G ▷ C₄×C₃₇ ▷ C₂×C₃₇ ▷ C₃₇

UCS

C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂²×D₄

N	H ≅ G/N	type
1. C₂×#10596962	C₂	semidirect
2. #11924562	C₂	direct (N × H)
3. C₂×#10596940	C₂	semidirect
4. C₂×#10596939	C₂	semidirect
5. C₂×#10596934	C₂	semidirect
6. C₂²×D₄×C₃₇	C₂	semidirect
11. C₂×D₄×C₃₇	C₂²	semidirect
15. #10596962	C₂²	semidirect
16. C₂×D₁₄₈	C₂²	semidirect
18. C₂×Dic₇₄	C₂²	semidirect
19. #10596940	C₂²	semidirect
20. #10596939	C₂²	semidirect
21. #10596934	C₂²	semidirect
22. C₂×#64436	C₂²	semidirect
38. D₁₄₈	C₂³	semidirect
41. Dic₇₄	C₂³	semidirect
42. #64436	C₂³	semidirect
57. C₂×C₃₇	AGL(1,ℤ/8)	semidirect
61. C₃₇	C₂×AGL(1,ℤ/8)	semidirect
63. C₂²×D₄	D₃₇	semidirect
77. C₂×D₄	C₂×D₃₇	semidirect
119. C₂	#11924562	direct (N × H)

H ≅ G/N

type

1. C₂×#10596962

C₂

semidirect

2. #11924562

C₂

direct (N × H)

3. C₂×#10596940

C₂

semidirect

4. C₂×#10596939

C₂

semidirect

5. C₂×#10596934

C₂

semidirect

6. C₂²×D₄×C₃₇

C₂

semidirect

11. C₂×D₄×C₃₇

C₂²

semidirect

15. #10596962

C₂²

semidirect

16. C₂×D₁₄₈

C₂²

semidirect

18. C₂×Dic₇₄

C₂²

semidirect

19. #10596940

C₂²

semidirect

20. #10596939

C₂²

semidirect

21. #10596934

C₂²

semidirect

22. C₂×#64436

C₂²

semidirect

38. D₁₄₈

C₂³

semidirect

41. Dic₇₄

C₂³

semidirect

42. #64436

C₂³

semidirect

57. C₂×C₃₇

AGL(1,ℤ/8)

semidirect

61. C₃₇

C₂×AGL(1,ℤ/8)

semidirect

63. C₂²×D₄

D₃₇

semidirect

77. C₂×D₄

C₂×D₃₇

semidirect

119. C₂

#11924562

direct (N × H)

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

subgroup	index
C₂×#10596962	2
#11924562	2
C₂×#10596940	2
C₂×#10596939	2
C₂×#10596934	2
C₂²×D₄×C₃₇	2
C₂×AGL(1,ℤ/8)	37

subgroup

index

C₂²×D₄×C₃₇

Subgroups — C₂×#11924562 #14558320

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C2×#11924562 #14558320

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#11924562 #14558320