Group #15689910 — subgroups

Subgroups — C₂×#13094050 #15689910

Order 3008.

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×#10625350	C₂	1	1	yes	maximal, characteristic, complemented
2	#13094050	C₂	1	8	yes	maximal, direct product factor
3	C₂×#10625335	C₂	1	2	yes	maximal, complemented
4	C₂×#10625334	C₂	1	2	yes	maximal, complemented
5	C₂×#10625329	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂²×D₄×C₄₇	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂⁴×C₄₇	—	2	1	no	—
8	C₂×D₄×C₄₇	—	2	1	no	—
9	C₂×D₄×C₄₇	—	2	2	no	—
10	C₂×D₄×C₄₇	C₂²	1	2	yes	—
11	C₂×D₄×C₄₇	C₂²	1	4	yes	complemented
12	C₂²×C₄×C₄₇	C₂²	1	1	yes	characteristic
13	C₂×C₂²⋊D₄₇	—	2	1	no	—
14	#10625350	—	2	2	no	—
15	#10625350	C₂²	1	4	yes	complemented
16	C₂×D₁₈₈	C₂²	1	1	yes	characteristic, complemented
17	C₂×C₄×D₄₇	—	2	1	no	—
18	C₂×Dic₉₄	C₂²	1	1	yes	characteristic, complemented
19	#10625335	C₂²	1	8	yes	complemented
20	#10625334	C₂²	1	8	yes	complemented
21	#10625329	C₂²	1	4	yes	complemented
22	C₂×C₄₇⋊C₈	C₂²	1	2	yes	complemented
23	C₂³×C₄₇	—	2	2	no	—
24	C₂³×C₄₇	—	2	4	no	—
25	C₂³×C₄₇	—	4	4	no	—
26	C₂³×C₄₇	D₄	1	1	yes	characteristic
27	C₂²×D₄₇	—	2	1	no	—
28	D₄×C₄₇	—	2	2	no	—
29	D₄×C₄₇	—	2	4	no	—
30	D₄×C₄₇	C₂³	1	4	yes	—
31	C₂×C₄×C₄₇	D₄	1	1	yes	characteristic
32	C₂×C₄×C₄₇	D₄	1	2	yes	—
33	C₂×C₄×C₄₇	C₂³	1	1	yes	characteristic
34	C₂×C₄×C₄₇	C₂³	1	2	yes	—
35	C₂²⋊D₄₇	—	2	4	no	—
36	C₂×Dic₄₇	—	2	1	no	—
37	D₁₈₈	—	2	1	no	—
38	D₁₈₈	C₂³	1	2	yes	complemented
39	C₄×D₄₇	—	2	4	no	—
40	Dic₉₄	—	2	1	no	—
41	Dic₉₄	C₂³	1	2	yes	complemented
42	C₄₇⋊C₈	C₂³	1	4	yes	complemented
43	C₂²×C₄₇	—	2	2	no	—
44	C₂²×C₄₇	—	2	4	no	—
45	C₂²×C₄₇	—	4	8	no	—
46	C₂²×C₄₇	—	4	4	no	—
47	C₂²×C₄₇	C₂×D₄	1	2	yes	—
48	C₂²×C₄₇	C₂×D₄	1	1	yes	characteristic
49	C₂×D₄₇	—	2	2	no	—
50	C₂×D₄₇	—	4	2	no	—
51	C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
52	C₄×C₄₇	C₂×D₄	1	2	yes	—
53	C₄×C₄₇	C₂⁴	1	1	yes	characteristic
54	Dic₄₇	—	2	2	no	—
55	C₂×C₄₇	—	2	2	no	—
56	C₂×C₄₇	—	4	4	no	—
57	C₂×C₄₇	AGL(1,ℤ/8)	1	2	yes	complemented
58	C₂×C₄₇	C₂²×D₄	1	1	yes	characteristic
59	D₄₇	—	4	2	no	—
60	C₂×AGL(1,ℤ/8)	—	47	1	no	maximal, Sylow
61	C₄₇	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic, complemented, Sylow
62	C₂×#41	—	47	1	no	—
63	C₂²×D₄	D₄₇	1	1	yes	characteristic, complemented
64	AGL(1,ℤ/8)	—	47	8	no	—
65	C₂×SD₁₆	—	47	2	no	—
66	C₂×D₈	—	47	2	no	—
67	C₂×M₁₆	—	47	1	no	—
68	C₂⁴	—	2	1	no	—
69	#41	—	47	4	no	—
70	#41	—	94	2	no	—
71	C₂×Q₈	—	47	1	no	—
72	C₂×D₄	—	2	2	no	—
73	C₂×D₄	—	2	1	no	—
74	C₂×D₄	—	47	1	no	—
75	C₂×D₄	—	94	1	no	—
76	C₂×D₄	C₂×D₄₇	1	2	yes	—
77	C₂×D₄	C₂×D₄₇	1	4	yes	complemented
78	C₂²×C₄	—	94	1	no	—
79	C₂²×C₄	C₂×D₄₇	1	1	yes	characteristic
80	SD₁₆	—	47	8	no	—
81	D₈	—	47	8	no	—
82	M₁₆	—	47	4	no	—
83	C₂×C₈	—	47	2	no	—
84	C₂³	—	2	4	no	—
85	C₂³	—	2	2	no	—
86	C₂³	—	4	4	no	—
87	C₂³	—	94	1	no	—
88	C₂³	C₂²⋊D₄₇	1	1	yes	characteristic
89	Q₈	—	47	2	no	—
90	Q₈	—	94	1	no	—
91	D₄	—	2	2	no	—
92	D₄	—	2	4	no	—
93	D₄	—	47	2	no	—
94	D₄	—	94	4	no	—
95	D₄	—	94	1	no	—
96	D₄	C₂²×D₄₇	1	4	yes	—
97	C₂×C₄	—	94	1	no	—
98	C₂×C₄	—	94	4	no	—
99	C₂×C₄	C₂²⋊D₄₇	1	2	yes	—
100	C₂×C₄	C₂²⋊D₄₇	1	1	yes	characteristic
101	C₂×C₄	C₂²×D₄₇	1	2	yes	—
102	C₂×C₄	C₂²×D₄₇	1	1	yes	characteristic
103	C₈	—	47	4	no	—
104	C₂²	—	2	2	no	—
105	C₂²	—	2	4	no	—
106	C₂²	—	4	8	no	—
107	C₂²	—	4	4	no	—
108	C₂²	—	94	2	no	—
109	C₂²	—	188	2	no	—
110	C₂²	C₂×C₂²⋊D₄₇	1	2	yes	—
111	C₂²	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
112	C₄	—	94	2	no	—
113	C₄	C₂×C₂²⋊D₄₇	1	2	yes	—
114	C₄	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
115	C₄	C₂³×D₄₇	1	1	yes	characteristic
116	C₂	—	2	2	no	—
117	C₂	—	4	4	no	—
118	C₂	—	188	2	no	—
119	C₂	#13094050	1	2	yes	direct product factor
120	C₂	C₂²×C₂²⋊D₄₇	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#10625350

C₂

yes

maximal, characteristic, complemented

#13094050

C₂

yes

maximal, direct product factor

C₂×#10625335

C₂

yes

maximal, complemented

C₂×#10625334

C₂

yes

maximal, complemented

C₂×#10625329

C₂

yes

maximal, characteristic, complemented

C₂²×D₄×C₄₇

C₂

yes

maximal, characteristic, complemented

C₂⁴×C₄₇

—

C₂×D₄×C₄₇

—

C₂×D₄×C₄₇

—

C₂×D₄×C₄₇

C₂²

yes

—

C₂×D₄×C₄₇

C₂²

yes

complemented

C₂²×C₄×C₄₇

C₂²

yes

characteristic

C₂×C₂²⋊D₄₇

—

#10625350

—

#10625350

C₂²

yes

complemented

C₂×D₁₈₈

C₂²

yes

characteristic, complemented

C₂×C₄×D₄₇

—

C₂×Dic₉₄

C₂²

yes

characteristic, complemented

#10625335

C₂²

yes

complemented

#10625334

C₂²

yes

complemented

#10625329

C₂²

yes

complemented

C₂×C₄₇⋊C₈

C₂²

yes

complemented

C₂³×C₄₇

—

C₂³×C₄₇

—

C₂³×C₄₇

—

C₂³×C₄₇

D₄

yes

characteristic

C₂²×D₄₇

—

D₄×C₄₇

—

D₄×C₄₇

—

D₄×C₄₇

C₂³

yes

—

C₂×C₄×C₄₇

D₄

yes

characteristic

C₂×C₄×C₄₇

D₄

yes

—

C₂×C₄×C₄₇

C₂³

yes

characteristic

C₂×C₄×C₄₇

C₂³

yes

—

C₂²⋊D₄₇

—

C₂×Dic₄₇

—

D₁₈₈

—

D₁₈₈

C₂³

yes

complemented

C₄×D₄₇

—

Dic₉₄

—

Dic₉₄

C₂³

yes

complemented

C₄₇⋊C₈

C₂³

yes

complemented

C₂²×C₄₇

—

C₂²×C₄₇

—

C₂²×C₄₇

—

C₂²×C₄₇

—

C₂²×C₄₇

C₂×D₄

yes

—

C₂²×C₄₇

C₂×D₄

yes

characteristic

C₂×D₄₇

—

C₂×D₄₇

—

C₄×C₄₇

C₂×D₄

yes

characteristic

C₄×C₄₇

C₂×D₄

yes

—

C₄×C₄₇

C₂⁴

yes

characteristic

Dic₄₇

—

C₂×C₄₇

—

C₂×C₄₇

—

C₂×C₄₇

AGL(1,ℤ/8)

yes

complemented

C₂×C₄₇

C₂²×D₄

yes

characteristic

D₄₇

—

C₂×AGL(1,ℤ/8)

—

maximal, Sylow

C₄₇

C₂×AGL(1,ℤ/8)

yes

characteristic, complemented, Sylow

C₂×#41

—

C₂²×D₄

D₄₇

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×D₄₇

yes

—

C₂×D₄

C₂×D₄₇

yes

complemented

C₂²×C₄

—

C₂²×C₄

C₂×D₄₇

yes

characteristic

SD₁₆

—

D₈

—

M₁₆

—

C₂×C₈

—

C₂³

—

C₂³

—

C₂³

—

C₂³

—

C₂³

C₂²⋊D₄₇

yes

characteristic

Q₈

—

Q₈

—

D₄

—

D₄

—

D₄

—

D₄

—

D₄

—

D₄

C₂²×D₄₇

yes

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

C₂²⋊D₄₇

yes

—

100

C₂×C₄

C₂²⋊D₄₇

yes

characteristic

101

C₂×C₄

C₂²×D₄₇

yes

—

102

C₂×C₄

C₂²×D₄₇

yes

characteristic

103

C₈

—

104

C₂²

—

105

C₂²

—

106

C₂²

—

107

C₂²

—

108

C₂²

—

109

C₂²

—

188

—

110

C₂²

C₂×C₂²⋊D₄₇

yes

—

111

C₂²

C₂×C₂²⋊D₄₇

yes

characteristic

112

C₄

—

113

C₄

C₂×C₂²⋊D₄₇

yes

—

114

C₄

C₂×C₂²⋊D₄₇

yes

characteristic

115

C₄

C₂³×D₄₇

yes

characteristic

116

C₂

—

117

C₂

—

118

C₂

—

188

—

119

C₂

#13094050

yes

direct product factor

120

C₂

C₂²×C₂²⋊D₄₇

yes

characteristic

	subgroup	G/H
center	C₂²	C₂×C₂²⋊D₄₇
commutator, cosocle	C₄×C₄₇	C₂⁴
Frattini	C₄	C₂³×D₄₇
Fitting	C₂²×D₄×C₄₇	C₂
socle	C₂²×C₄₇	C₂×D₄
hypercenter	C₂²×D₄	D₄₇
hypocenter	C₄₇	C₂×AGL(1,ℤ/8)
LCS	G ▷ C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂²×D₄

subgroup

G/H

center

C₂²

C₂×C₂²⋊D₄₇

commutator, cosocle

C₄×C₄₇

C₂⁴

Frattini

C₄

C₂³×D₄₇

Fitting

C₂²×D₄×C₄₇

C₂

socle

C₂²×C₄₇

C₂×D₄

hypercenter

C₂²×D₄

D₄₇

hypocenter

C₄₇

C₂×AGL(1,ℤ/8)

LCS

G ▷ C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇

UCS

C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂²×D₄

N	H ≅ G/N	type
1. C₂×#10625350	C₂	semidirect
2. #13094050	C₂	direct (N × H)
3. C₂×#10625335	C₂	semidirect
4. C₂×#10625334	C₂	semidirect
5. C₂×#10625329	C₂	semidirect
6. C₂²×D₄×C₄₇	C₂	semidirect
11. C₂×D₄×C₄₇	C₂²	semidirect
15. #10625350	C₂²	semidirect
16. C₂×D₁₈₈	C₂²	semidirect
18. C₂×Dic₉₄	C₂²	semidirect
19. #10625335	C₂²	semidirect
20. #10625334	C₂²	semidirect
21. #10625329	C₂²	semidirect
22. C₂×C₄₇⋊C₈	C₂²	semidirect
38. D₁₈₈	C₂³	semidirect
41. Dic₉₄	C₂³	semidirect
42. C₄₇⋊C₈	C₂³	semidirect
57. C₂×C₄₇	AGL(1,ℤ/8)	semidirect
61. C₄₇	C₂×AGL(1,ℤ/8)	semidirect
63. C₂²×D₄	D₄₇	semidirect
77. C₂×D₄	C₂×D₄₇	semidirect
119. C₂	#13094050	direct (N × H)

H ≅ G/N

type

1. C₂×#10625350

C₂

semidirect

2. #13094050

C₂

direct (N × H)

3. C₂×#10625335

C₂

semidirect

4. C₂×#10625334

C₂

semidirect

5. C₂×#10625329

C₂

semidirect

6. C₂²×D₄×C₄₇

C₂

semidirect

11. C₂×D₄×C₄₇

C₂²

semidirect

15. #10625350

C₂²

semidirect

16. C₂×D₁₈₈

C₂²

semidirect

18. C₂×Dic₉₄

C₂²

semidirect

19. #10625335

C₂²

semidirect

20. #10625334

C₂²

semidirect

21. #10625329

C₂²

semidirect

22. C₂×C₄₇⋊C₈

C₂²

semidirect

38. D₁₈₈

C₂³

semidirect

41. Dic₉₄

C₂³

semidirect

42. C₄₇⋊C₈

C₂³

semidirect

57. C₂×C₄₇

AGL(1,ℤ/8)

semidirect

61. C₄₇

C₂×AGL(1,ℤ/8)

semidirect

63. C₂²×D₄

D₄₇

semidirect

77. C₂×D₄

C₂×D₄₇

semidirect

119. C₂

#13094050

direct (N × H)

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

subgroup	index
C₂×#10625350	2
#13094050	2
C₂×#10625335	2
C₂×#10625334	2
C₂×#10625329	2
C₂²×D₄×C₄₇	2
C₂×AGL(1,ℤ/8)	47

subgroup

index

C₂²×D₄×C₄₇

Subgroups — C₂×#13094050 #15689910

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C2×#13094050 #15689910

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#13094050 #15689910