Group #15683731 — subgroups

Subgroups — C₂×#13092781 #15683731

Order 2944.

488 conjugacy classes of subgroups; only the 149 normal subgroups are shown.

normal subgroups only

This group has 488 conjugacy classes of subgroups — too many to draw. Only its 149 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×#10625041	C₂	1	1	yes	maximal, characteristic, complemented
2	#13092781	C₂	1	8	yes	maximal, direct product factor
3	C₂×#10625014	C₂	1	4	yes	maximal, complemented
4	C₂×#10624985	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×C₂₃×#127	C₂	1	1	yes	maximal, characteristic
6	C₂²×D₄×C₂₃	—	2	2	no	—
7	C₂²×D₄×C₂₃	C₂²	1	2	yes	—
8	C₂₃×#127	—	2	2	no	—
9	C₂₃×#127	C₂²	1	4	yes	—
10	C₂×C₄²×C₂₃	C₂²	1	1	yes	characteristic
11	C₂×C₄⋊Dic₂₃	C₂²	1	2	yes	—
12	C₂×Dic₂₃⋊C₄	—	2	2	no	—
13	C₂²×C₂₃⋊C₈	C₂²	1	2	yes	—
14	#10625041	—	2	2	no	—
15	#10625041	C₂²	1	4	yes	complemented
16	#10625014	C₂²	1	16	yes	complemented
17	#10624985	C₂²	1	4	yes	complemented
18	C₂⁴×C₂₃	—	2	2	no	—
19	C₂×D₄×C₂₃	—	2	2	no	—
20	C₂×D₄×C₂₃	—	2	8	no	—
21	C₂×D₄×C₂₃	—	2	8	no	—
22	C₂×D₄×C₂₃	—	4	8	no	—
23	C₂×D₄×C₂₃	C₂³	1	4	yes	—
24	C₂²×C₄×C₂₃	D₄	1	2	yes	—
25	C₂²×C₄×C₂₃	C₂³	1	1	yes	characteristic
26	C₂²×Dic₂₃	—	2	2	no	—
27	C₄²×C₂₃	D₄	1	2	yes	—
28	C₄²×C₂₃	C₂³	1	2	yes	—
29	C₄⋊Dic₂₃	—	2	2	no	—
30	C₄⋊Dic₂₃	C₂³	1	4	yes	—
31	Dic₂₃⋊C₄	—	2	8	no	—
32	C₂×C₂₃⋊C₈	—	2	4	no	—
33	C₂×C₂₃⋊C₈	C₂³	1	4	yes	—
34	C₂³×C₂₃	—	2	4	no	—
35	C₂³×C₂₃	—	4	4	no	—
36	C₂³×C₂₃	—	4	8	no	—
37	C₂³×C₂₃	C₂×D₄	1	1	yes	characteristic
38	D₄×C₂₃	—	2	8	no	—
39	D₄×C₂₃	—	4	16	no	—
40	D₄×C₂₃	—	4	4	no	—
41	C₂×C₄×C₂₃	—	2	4	no	—
42	C₂×C₄×C₂₃	C₂×D₄	1	4	yes	—
43	C₂×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
44	C₂×C₄×C₂₃	#41	1	4	yes	—
45	C₂×C₄×C₂₃	C₂⁴	1	1	yes	characteristic
46	C₂×Dic₂₃	—	2	4	no	—
47	C₂×Dic₂₃	—	4	4	no	—
48	C₂₃⋊C₈	—	2	4	no	—
49	C₂×#490	—	23	1	no	maximal, Sylow
50	C₂²×C₂₃	—	4	4	no	—
51	C₂²×C₂₃	—	4	8	no	—
52	C₂²×C₂₃	—	8	8	no	—
53	C₂²×C₂₃	#124	1	2	yes	—
54	C₂²×C₂₃	AGL(1,ℤ/8)	1	4	yes	—
55	C₂²×C₂₃	C₂²×D₄	1	1	yes	characteristic
56	C₄×C₂₃	—	2	4	no	—
57	C₄×C₂₃	#124	1	2	yes	—
58	C₄×C₂₃	C₂×#41	1	2	yes	—
59	Dic₂₃	—	4	4	no	—
60	C₂×#127	D₂₃	1	1	yes	characteristic
61	C₂×#125	—	23	1	no	—
62	#490	—	23	8	no	—
63	C₂×#102	—	23	4	no	—
64	C₂×#97	—	23	1	no	—
65	C₂×C₂₃	—	8	4	no	—
66	C₂×C₂₃	#490	1	4	yes	complemented
67	C₂×C₂₃	C₂×#124	1	1	yes	characteristic
68	C₂×C₂₃	C₂×AGL(1,ℤ/8)	1	2	yes	—
69	C₂²×D₄	—	2	2	no	—
70	C₂²×D₄	C₂×D₂₃	1	2	yes	—
71	C₂²×C₈	—	23	2	no	—
72	#127	—	2	2	no	—
73	#127	C₂×D₂₃	1	4	yes	—
74	#125	—	23	4	no	—
75	#125	—	46	2	no	—
76	C₂×C₄⋊C₄	—	23	2	no	—
77	C₂×C₄⋊C₄	—	46	2	no	—
78	C₂×C₄²	C₂×D₂₃	1	1	yes	characteristic
79	#102	—	23	16	no	—
80	#97	—	23	4	no	—
81	C₂₃	C₂×#490	1	1	yes	characteristic, complemented, Sylow
82	C₂⁴	—	2	2	no	—
83	C₂×D₄	—	2	8	no	—
84	C₂×D₄	—	2	2	no	—
85	C₂×D₄	—	2	8	no	—
86	C₂×D₄	—	4	8	no	—
87	C₂×D₄	C₂²×D₂₃	1	4	yes	—
88	C₂²×C₄	—	46	2	no	—
89	C₂²×C₄	C₂²⋊D₂₃	1	2	yes	—
90	C₂²×C₄	C₂²×D₂₃	1	1	yes	characteristic
91	C₂×C₈	—	23	4	no	—
92	C₂×C₈	—	46	4	no	—
93	C₄⋊C₄	—	23	4	no	—
94	C₄⋊C₄	—	46	2	no	—
95	C₄⋊C₄	—	46	8	no	—
96	C₄²	C₂²⋊D₂₃	1	2	yes	—
97	C₄²	C₂²×D₂₃	1	2	yes	—
98	C₂³	—	2	4	no	—
99	C₂³	—	4	8	no	—
100	C₂³	—	4	4	no	—
101	C₂³	C₂×C₂²⋊D₂₃	1	1	yes	characteristic
102	D₄	—	2	8	no	—
103	D₄	—	4	16	no	—
104	D₄	—	4	4	no	—
105	C₂×C₄	—	2	4	no	—
106	C₂×C₄	—	46	4	no	—
107	C₂×C₄	—	92	4	no	—
108	C₂×C₄	#67290	1	4	yes	—
109	C₂×C₄	C₂×C₂²⋊D₂₃	1	4	yes	—
110	C₂×C₄	C₂×C₂²⋊D₂₃	1	1	yes	characteristic
111	C₂×C₄	C₂³×D₂₃	1	1	yes	characteristic
112	C₈	—	46	4	no	—
113	C₂²	—	4	4	no	—
114	C₂²	—	4	8	no	—
115	C₂²	—	8	8	no	—
116	C₂²	#10625103	1	4	yes	—
117	C₂²	#10625107	1	2	yes	—
118	C₂²	C₂²×C₂²⋊D₂₃	1	1	yes	characteristic
119	C₄	—	2	4	no	—
120	C₄	—	92	4	no	—
121	C₄	#10625107	1	2	yes	—
122	C₄	C₂×#67290	1	2	yes	—
123	C₂	—	8	4	no	—
124	C₂	#13092781	1	4	yes	direct product factor
125	C₂	C₂×#10625107	1	1	yes	characteristic
126	C₂	C₂×#10625103	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#10625041

C₂

yes

maximal, characteristic, complemented

#13092781

C₂

yes

maximal, direct product factor

C₂×#10625014

C₂

yes

maximal, complemented

C₂×#10624985

C₂

yes

maximal, characteristic, complemented

C₂×C₂₃×#127

C₂

yes

maximal, characteristic

C₂²×D₄×C₂₃

—

C₂²×D₄×C₂₃

C₂²

yes

—

C₂₃×#127

—

C₂₃×#127

C₂²

yes

—

C₂×C₄²×C₂₃

C₂²

yes

characteristic

C₂×C₄⋊Dic₂₃

C₂²

yes

—

C₂×Dic₂₃⋊C₄

—

C₂²×C₂₃⋊C₈

C₂²

yes

—

#10625041

—

#10625041

C₂²

yes

complemented

#10625014

C₂²

yes

complemented

#10624985

C₂²

yes

complemented

C₂⁴×C₂₃

—

C₂×D₄×C₂₃

—

C₂×D₄×C₂₃

—

C₂×D₄×C₂₃

—

C₂×D₄×C₂₃

—

C₂×D₄×C₂₃

C₂³

yes

—

C₂²×C₄×C₂₃

D₄

yes

—

C₂²×C₄×C₂₃

C₂³

yes

characteristic

C₂²×Dic₂₃

—

C₄²×C₂₃

D₄

yes

—

C₄²×C₂₃

C₂³

yes

—

C₄⋊Dic₂₃

—

C₄⋊Dic₂₃

C₂³

yes

—

Dic₂₃⋊C₄

—

C₂×C₂₃⋊C₈

—

C₂×C₂₃⋊C₈

C₂³

yes

—

C₂³×C₂₃

—

C₂³×C₂₃

—

C₂³×C₂₃

—

C₂³×C₂₃

C₂×D₄

yes

characteristic

D₄×C₂₃

—

D₄×C₂₃

—

D₄×C₂₃

—

C₂×C₄×C₂₃

—

C₂×C₄×C₂₃

C₂×D₄

yes

—

C₂×C₄×C₂₃

C₂×D₄

yes

characteristic

C₂×C₄×C₂₃

#41

yes

—

C₂×C₄×C₂₃

C₂⁴

yes

characteristic

C₂×Dic₂₃

—

C₂×Dic₂₃

—

C₂₃⋊C₈

—

C₂×#490

—

maximal, Sylow

C₂²×C₂₃

—

C₂²×C₂₃

—

C₂²×C₂₃

—

C₂²×C₂₃

#124

yes

—

C₂²×C₂₃

AGL(1,ℤ/8)

yes

—

C₂²×C₂₃

C₂²×D₄

yes

characteristic

C₄×C₂₃

—

C₄×C₂₃

#124

yes

—

C₄×C₂₃

C₂×#41

yes

—

Dic₂₃

—

C₂×#127

D₂₃

yes

—

—

—

C₂×#97

—

C₂×C₂₃

—

C₂×C₂₃

yes

C₂×C₂₃

yes

C₂×C₂₃

yes

—

C₂²×D₄

—

C₂²×D₄

C₂×D₂₃

yes

—

C₂²×C₈

—

#127

—

#127

C₂×D₂₃

yes

—

#125

—

#125

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄²

C₂×D₂₃

yes

characteristic

#102

—

#97

—

C₂₃

C₂×#490

yes

characteristic, complemented, Sylow

C₂⁴

—

C₂×D₄

—

C₂×D₄

—

C₂×D₄

—

C₂×D₄

—

C₂×D₄

C₂²×D₂₃

yes

—

C₂²×C₄

—

C₂²×C₄

C₂²⋊D₂₃

yes

—

C₂²×C₄

C₂²×D₂₃

yes

characteristic

C₂×C₈

—

C₂×C₈

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄²

C₂²⋊D₂₃

yes

—

C₄²

C₂²×D₂₃

yes

—

C₂³

—

C₂³

—

100

C₂³

—

101

C₂³

C₂×C₂²⋊D₂₃

yes

characteristic

102

D₄

—

103

D₄

—

104

D₄

—

105

C₂×C₄

—

106

C₂×C₄

—

107

C₂×C₄

—

108

C₂×C₄

#67290

yes

—

109

C₂×C₄

C₂×C₂²⋊D₂₃

yes

—

110

C₂×C₄

C₂×C₂²⋊D₂₃

yes

characteristic

111

C₂×C₄

C₂³×D₂₃

yes

characteristic

112

C₈

—

113

C₂²

—

114

C₂²

—

115

C₂²

—

116

C₂²

#10625103

yes

—

117

C₂²

#10625107

yes

—

118

C₂²

C₂²×C₂²⋊D₂₃

yes

characteristic

119

C₄

—

120

C₄

—

121

C₄

#10625107

yes

—

122

C₄

C₂×#67290

yes

—

123

C₂

—

124

C₂

#13092781

yes

direct product factor

125

C₂

C₂×#10625107

yes

characteristic

126

C₂

C₂×#10625103

yes

—

	subgroup	G/H
center	C₂³	C₂×C₂²⋊D₂₃
commutator, cosocle	C₂×C₄×C₂₃	C₂⁴
Frattini	C₂×C₄	C₂³×D₂₃
Fitting, prime radical	C₂×C₂₃×#127	C₂
socle	C₂³×C₂₃	C₂×D₄
hypercenter	C₂×#127	D₂₃
hypocenter	C₂₃	C₂×#490
LCS	G ▷ C₂×C₄×C₂₃ ▷ C₂×C₂₃ ▷ C₂₃
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×#127

subgroup

G/H

center

C₂³

C₂×C₂²⋊D₂₃

commutator, cosocle

C₂×C₄×C₂₃

C₂⁴

Frattini

C₂×C₄

C₂³×D₂₃

Fitting, prime radical

C₂×C₂₃×#127

C₂

socle

C₂³×C₂₃

C₂×D₄

D₂₃

C₂₃

G ▷ C₂×C₄×C₂₃ ▷ C₂×C₂₃ ▷ C₂₃

UCS

C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×#127

N	H ≅ G/N	type
1. C₂×#10625041	C₂	semidirect
2. #13092781	C₂	direct (N × H)
3. C₂×#10625014	C₂	semidirect
4. C₂×#10624985	C₂	semidirect
15. #10625041	C₂²	semidirect
16. #10625014	C₂²	semidirect
17. #10624985	C₂²	semidirect
66. C₂×C₂₃	#490	semidirect
81. C₂₃	C₂×#490	semidirect
124. C₂	#13092781	direct (N × H)

H ≅ G/N

type

1. C₂×#10625041

C₂

semidirect

2. #13092781

C₂

direct (N × H)

3. C₂×#10625014

C₂

semidirect

4. C₂×#10624985

C₂

semidirect

15. #10625041

C₂²

semidirect

16. #10625014

C₂²

semidirect

17. #10624985

C₂²

semidirect

66. C₂×C₂₃

#490

semidirect

81. C₂₃

C₂×#490

semidirect

124. C₂

#13092781

direct (N × H)

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

subgroup	index
C₂×#10625041	2
#13092781	2
C₂×#10625014	2
C₂×#10624985	2
C₂×C₂₃×#127	2
C₂×#490	23

subgroup

index

C₂×C₂₃×#127

Subgroups — C₂×#13092781 #15683731

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C2×#13092781 #15683731

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#13092781 #15683731