Group #15683899 — subgroups

Subgroups — C₂×#13092817 #15683899

Order 2944.

360 conjugacy classes of subgroups; only the 165 normal subgroups are shown.

normal subgroups only

This group has 360 conjugacy classes of subgroups — too many to draw. Only its 165 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×#10625061	C₂	1	2	yes	maximal
2	#13092817	C₂	1	8	yes	maximal, direct product factor
3	C₂×#10624989	C₂	1	2	yes	maximal
4	C₂×#10624999	C₂	1	1	yes	maximal, characteristic
5	C₂×C₈×Dic₂₃	C₂	1	1	yes	maximal, characteristic
6	C₂×C₂₃×#106	C₂	1	1	yes	maximal, characteristic
7	C₂²×C₈×C₂₃	C₂²	1	1	yes	characteristic
8	C₂×C₄⋊C₄×C₂₃	C₂²	1	2	yes	—
9	C₂×C₄⋊Dic₂₃	—	2	2	no	—
10	C₂×C₄⋊Dic₂₃	C₂²	1	2	yes	—
11	C₂×Dic₂₃⋊C₄	—	2	2	no	—
12	C₂×C₄×Dic₂₃	C₂²	1	1	yes	characteristic
13	C₂²×C₂₃⋊C₈	C₂²	1	1	yes	characteristic
14	#10625061	—	2	4	no	—
15	#10625061	C₂²	1	8	yes	—
16	C₂₃×#106	C₂²	1	4	yes	—
17	#10624999	C₂²	1	4	yes	—
18	C₈×Dic₂₃	C₂²	1	4	yes	—
19	#10624989	C₂²	1	8	yes	—
20	C₂²×C₄×C₂₃	—	2	2	no	—
21	C₂²×C₄×C₂₃	C₂³	1	1	yes	characteristic
22	C₂²×Dic₂₃	—	2	2	no	—
23	C₂²×Dic₂₃	D₄	1	2	yes	—
24	C₂×C₈×C₂₃	Q₈	1	4	yes	—
25	C₂×C₈×C₂₃	C₂³	1	2	yes	—
26	C₄⋊C₄×C₂₃	—	2	2	no	—
27	C₄⋊C₄×C₂₃	C₂³	1	4	yes	—
28	C₄⋊Dic₂₃	—	2	2	no	—
29	C₄⋊Dic₂₃	—	2	8	no	—
30	C₄⋊Dic₂₃	C₂³	1	4	yes	—
31	Dic₂₃⋊C₄	—	2	8	no	—
32	C₄×Dic₂₃	D₄	1	2	yes	—
33	C₄×Dic₂₃	C₂³	1	2	yes	—
34	C₂×C₂₃⋊C₈	Q₈	1	4	yes	—
35	C₂×C₂₃⋊C₈	C₂³	1	2	yes	—
36	C₂³×C₂₃	C₂×D₄	1	1	yes	characteristic
37	C₂×C₄×C₂₃	—	2	4	no	—
38	C₂×C₄×C₂₃	—	4	4	no	—
39	C₂×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
40	C₂×C₄×C₂₃	C₂×Q₈	1	2	yes	—
41	C₂×C₄×C₂₃	C₂×Q₈	1	2	yes	—
42	C₂×C₄×C₂₃	C₂⁴	1	1	yes	characteristic
43	C₂×Dic₂₃	—	2	4	no	—
44	C₂×Dic₂₃	—	2	4	no	—
45	C₂×Dic₂₃	—	4	4	no	—
46	C₂×Dic₂₃	C₂×D₄	1	4	yes	—
47	C₈×C₂₃	C₂×Q₈	1	4	yes	—
48	C₂₃⋊C₈	C₂×Q₈	1	4	yes	—
49	C₂×#499	—	23	1	no	maximal, Sylow
50	C₂²×C₂₃	C₄⋊Q₈	1	2	yes	—
51	C₂²×C₂₃	#135	1	4	yes	—
52	C₂²×C₂₃	C₂²×D₄	1	1	yes	characteristic
53	C₄×C₂₃	—	4	4	no	—
54	C₄×C₂₃	C₄⋊Q₈	1	2	yes	—
55	C₄×C₂₃	C₂²×Q₈	1	1	yes	characteristic
56	C₄×C₂₃	C₂²×Q₈	1	1	yes	characteristic
57	Dic₂₃	—	2	4	no	—
58	Dic₂₃	—	4	4	no	—
59	C₂×#125	—	23	2	no	—
60	#499	—	23	8	no	—
61	C₂×#107	—	23	2	no	—
62	C₂×#106	—	23	1	no	—
63	C₂×#106	D₂₃	1	1	yes	characteristic
64	C₂×C₄×C₈	—	23	1	no	—
65	C₂×C₂₃	#499	1	4	yes	complemented
66	C₂×C₂₃	C₂×C₄⋊Q₈	1	1	yes	characteristic
67	C₂×C₂₃	C₂×#135	1	2	yes	—
68	C₂²×C₈	—	23	1	no	—
69	C₂²×C₈	C₂×D₂₃	1	1	yes	characteristic
70	#125	—	23	8	no	—
71	#125	—	46	4	no	—
72	C₂×C₄⋊C₄	—	23	2	no	—
73	C₂×C₄⋊C₄	—	46	2	no	—
74	C₂×C₄⋊C₄	—	46	2	no	—
75	C₂×C₄⋊C₄	C₂×D₂₃	1	2	yes	—
76	C₂×C₄²	—	23	1	no	—
77	#107	—	23	8	no	—
78	#106	—	23	4	no	—
79	#106	C₂×D₂₃	1	4	yes	—
80	C₄×C₈	—	23	4	no	—
81	C₂₃	C₂×#499	1	1	yes	characteristic, complemented, Sylow
82	C₂²×C₄	—	2	2	no	—
83	C₂²×C₄	—	23	2	no	—
84	C₂²×C₄	—	46	2	no	—
85	C₂²×C₄	C₂²×D₂₃	1	1	yes	characteristic
86	C₂×C₈	—	23	4	no	—
87	C₂×C₈	—	23	2	no	—
88	C₂×C₈	Dic₄₆	1	4	yes	—
89	C₂×C₈	C₂²×D₂₃	1	2	yes	—
90	C₄⋊C₄	—	2	2	no	—
91	C₄⋊C₄	—	23	4	no	—
92	C₄⋊C₄	—	46	8	no	—
93	C₄⋊C₄	—	46	8	no	—
94	C₄⋊C₄	—	46	2	no	—
95	C₄⋊C₄	C₂²×D₂₃	1	4	yes	—
96	C₄²	—	23	2	no	—
97	C₄²	—	23	2	no	—
98	C₂³	D₄×D₂₃	1	1	yes	characteristic
99	C₂×C₄	—	2	4	no	—
100	C₂×C₄	—	4	4	no	—
101	C₂×C₄	—	23	4	no	—
102	C₂×C₄	—	46	4	no	—
103	C₂×C₄	—	46	4	no	—
104	C₂×C₄	—	92	4	no	—
105	C₂×C₄	C₂×Dic₄₆	1	2	yes	—
106	C₂×C₄	D₄×D₂₃	1	1	yes	characteristic
107	C₂×C₄	Q₈×D₂₃	1	2	yes	—
108	C₂×C₄	C₂³×D₂₃	1	1	yes	characteristic
109	C₈	—	23	4	no	—
110	C₈	C₂×Dic₄₆	1	4	yes	—
111	C₂²	#10625059	1	2	yes	—
112	C₂²	#10625087	1	4	yes	—
113	C₂²	C₂×D₄×D₂₃	1	1	yes	characteristic
114	C₄	—	4	4	no	—
115	C₄	—	46	4	no	—
116	C₄	—	92	4	no	—
117	C₄	#10625059	1	2	yes	—
118	C₄	C₂²×Dic₄₆	1	1	yes	characteristic
119	C₄	C₂×Q₈×D₂₃	1	1	yes	characteristic
120	C₂	#13092817	1	4	yes	direct product factor
121	C₂	C₂×#10625059	1	1	yes	characteristic
122	C₂	C₂×#10625087	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#10625061

C₂

yes

maximal

#13092817

C₂

yes

maximal, direct product factor

C₂×#10624989

C₂

yes

maximal

C₂×#10624999

C₂

yes

maximal, characteristic

C₂×C₈×Dic₂₃

C₂

yes

maximal, characteristic

C₂×C₂₃×#106

C₂

yes

maximal, characteristic

C₂²×C₈×C₂₃

C₂²

yes

characteristic

C₂×C₄⋊C₄×C₂₃

C₂²

yes

—

C₂×C₄⋊Dic₂₃

—

C₂×C₄⋊Dic₂₃

C₂²

yes

—

C₂×Dic₂₃⋊C₄

—

C₂×C₄×Dic₂₃

C₂²

yes

characteristic

C₂²×C₂₃⋊C₈

C₂²

yes

characteristic

#10625061

—

#10625061

C₂²

yes

—

C₂₃×#106

C₂²

yes

—

#10624999

C₂²

yes

—

C₈×Dic₂₃

C₂²

yes

—

#10624989

C₂²

yes

—

C₂²×C₄×C₂₃

—

C₂²×C₄×C₂₃

C₂³

yes

characteristic

C₂²×Dic₂₃

—

C₂²×Dic₂₃

D₄

yes

—

C₂×C₈×C₂₃

Q₈

yes

—

C₂×C₈×C₂₃

C₂³

yes

—

C₄⋊C₄×C₂₃

—

C₄⋊C₄×C₂₃

C₂³

yes

—

C₄⋊Dic₂₃

—

C₄⋊Dic₂₃

—

C₄⋊Dic₂₃

C₂³

yes

—

Dic₂₃⋊C₄

—

C₄×Dic₂₃

D₄

yes

—

C₄×Dic₂₃

C₂³

yes

—

C₂×C₂₃⋊C₈

Q₈

yes

—

C₂×C₂₃⋊C₈

C₂³

yes

—

C₂³×C₂₃

C₂×D₄

yes

characteristic

C₂×C₄×C₂₃

—

C₂×C₄×C₂₃

—

C₂×C₄×C₂₃

C₂×D₄

yes

characteristic

C₂×C₄×C₂₃

C₂×Q₈

yes

—

C₂×C₄×C₂₃

C₂×Q₈

yes

—

C₂×C₄×C₂₃

C₂⁴

yes

characteristic

C₂×Dic₂₃

—

C₂×Dic₂₃

—

C₂×Dic₂₃

—

C₂×Dic₂₃

C₂×D₄

yes

—

C₈×C₂₃

C₂×Q₈

yes

—

C₂₃⋊C₈

C₂×Q₈

yes

—

C₂×#499

—

maximal, Sylow

C₂²×C₂₃

C₄⋊Q₈

yes

—

C₂²×C₂₃

#135

yes

—

C₂²×C₂₃

C₂²×D₄

yes

characteristic

C₄×C₂₃

—

C₄×C₂₃

C₄⋊Q₈

yes

—

C₄×C₂₃

C₂²×Q₈

yes

characteristic

C₄×C₂₃

C₂²×Q₈

yes

characteristic

Dic₂₃

—

Dic₂₃

—

—

—

—

—

D₂₃

yes

characteristic

C₂×C₄×C₈

—

C₂×C₂₃

#499

yes

complemented

C₂×C₂₃

C₂×C₄⋊Q₈

yes

characteristic

C₂×C₂₃

C₂×#135

yes

—

C₂²×C₈

—

C₂²×C₈

C₂×D₂₃

yes

characteristic

#125

—

#125

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

C₂×D₂₃

yes

—

C₂×C₄²

—

#107

—

#106

—

#106

C₂×D₂₃

yes

—

C₄×C₈

—

C₂₃

C₂×#499

yes

characteristic, complemented, Sylow

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

C₂²×D₂₃

yes

characteristic

C₂×C₈

—

C₂×C₈

—

C₂×C₈

Dic₄₆

yes

—

C₂×C₈

C₂²×D₂₃

yes

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

C₂²×D₂₃

yes

—

C₄²

—

C₄²

—

C₂³

D₄×D₂₃

yes

characteristic

C₂×C₄

—

100

C₂×C₄

—

101

C₂×C₄

—

102

C₂×C₄

—

103

C₂×C₄

—

104

C₂×C₄

—

105

C₂×C₄

C₂×Dic₄₆

yes

—

106

C₂×C₄

D₄×D₂₃

yes

characteristic

107

C₂×C₄

Q₈×D₂₃

yes

—

108

C₂×C₄

C₂³×D₂₃

yes

characteristic

109

C₈

—

110

C₈

C₂×Dic₄₆

yes

—

111

C₂²

#10625059

yes

—

112

C₂²

#10625087

yes

—

113

C₂²

C₂×D₄×D₂₃

yes

characteristic

114

C₄

—

115

C₄

—

116

C₄

—

117

C₄

#10625059

yes

—

118

C₄

C₂²×Dic₄₆

yes

characteristic

119

C₄

C₂×Q₈×D₂₃

yes

characteristic

120

C₂

#13092817

yes

direct product factor

121

C₂

C₂×#10625059

yes

characteristic

122

C₂

C₂×#10625087

yes

—

	subgroup	G/H
center	C₂³	D₄×D₂₃
commutator, cosocle	C₂×C₄×C₂₃	C₂⁴
Frattini	C₂×C₄	C₂³×D₂₃
Fitting	C₂×C₂₃×#106	C₂
socle, prime radical	C₂³×C₂₃	C₂×D₄
hypercenter	C₂×#106	D₂₃
hypocenter	C₂₃	C₂×#499
LCS	G ▷ C₂×C₄×C₂₃ ▷ C₂×C₂₃ ▷ C₂₃
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄ ◁ C₂×#106

subgroup

G/H

center

C₂³

D₄×D₂₃

commutator, cosocle

C₂×C₄×C₂₃

C₂⁴

Frattini

C₂×C₄

C₂³×D₂₃

Fitting

C₂×C₂₃×#106

C₂

socle, prime radical

C₂³×C₂₃

C₂×D₄

D₂₃

C₂₃

G ▷ C₂×C₄×C₂₃ ▷ C₂×C₂₃ ▷ C₂₃

UCS

C₁ ◁ C₂³ ◁ C₂²×C₄ ◁ C₂×#106

N	H ≅ G/N	type
2. #13092817	C₂	direct (N × H)
65. C₂×C₂₃	#499	semidirect
81. C₂₃	C₂×#499	semidirect
120. C₂	#13092817	direct (N × H)

H ≅ G/N

type

2. #13092817

C₂

direct (N × H)

65. C₂×C₂₃

#499

semidirect

81. C₂₃

C₂×#499

semidirect

120. C₂

#13092817

direct (N × H)

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

subgroup	index
C₂×#10625061	2
#13092817	2
C₂×#10624989	2
C₂×#10624999	2
C₂×C₈×Dic₂₃	2
C₂×C₂₃×#106	2
C₂×#499	23

subgroup

index

C₂×C₈×Dic₂₃

C₂×C₂₃×#106

C₂×#499

Subgroups — C₂×#13092817 #15683899

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C2×#13092817 #15683899

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#13092817 #15683899