Group #15673443 — subgroups

Subgroups — #15673443

Order 2944.

260 conjugacy classes of subgroups; only the 105 normal subgroups are shown.

normal subgroups only

This group has 260 conjugacy classes of subgroups — too many to draw. Only its 105 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	#13092455	C₂	1	2	yes	maximal, complemented
2	#13092391	C₂	1	2	yes	maximal
3	#13092174	C₂	1	2	yes	maximal, complemented
4	C₂×C₂₃×#131	C₂	1	1	yes	maximal, characteristic
5	C₂×#41×C₂₃	C₂²	1	1	yes	characteristic
6	C₂₃×#131	—	2	2	no	—
7	C₂₃×#131	—	2	2	no	—
8	C₂×M₁₆×C₂₃	C₄	1	2	yes	complemented
9	C₂×M₁₆×C₂₃	C₂²	1	1	yes	characteristic
10	C₂²×C₈×C₂₃	C₄	1	2	yes	complemented
11	C₂²×C₈×C₂₃	C₂²	1	1	yes	characteristic
12	#10625097	C₂²	1	4	yes	—
13	#10625016	—	2	4	no	—
14	#10625000	—	2	2	no	—
15	#10624999	—	2	2	no	—
16	#41×C₂₃	—	2	2	no	—
17	#41×C₂₃	—	2	2	no	—
18	C₂×Q₈×C₂₃	Q₈	1	1	yes	characteristic
19	C₂×D₄×C₂₃	D₄	1	2	yes	—
20	C₂×D₄×C₂₃	Q₈	1	1	yes	characteristic
21	C₂²×C₄×C₂₃	C₂×C₄	1	2	yes	—
22	C₂²×C₄×C₂₃	C₂³	1	1	yes	characteristic
23	M₁₆×C₂₃	—	2	2	no	—
24	M₁₆×C₂₃	—	2	4	no	—
25	C₂×C₈×C₂₃	—	2	2	no	—
26	C₂×C₈×C₂₃	—	2	2	no	—
27	C₂×C₈×C₂₃	C₂×C₄	1	2	yes	—
28	C₂×C₈×C₂₃	C₂×C₄	1	1	yes	characteristic
29	C₂×C₈×C₂₃	C₂×C₄	1	1	yes	characteristic
30	C₂×C₈×C₂₃	D₄	1	2	yes	—
31	C₂×C₈×C₂₃	D₄	1	2	yes	—
32	C₂²⋊Dic₂₃	—	2	4	no	—
33	C₄⋊Dic₂₃	—	2	4	no	—
34	C₄×Dic₂₃	—	2	4	no	—
35	C₂³×C₂₃	C₄⋊C₄	1	2	yes	—
36	C₂³×C₂₃	C₂×D₄	1	1	yes	characteristic
37	Q₈×C₂₃	—	2	2	no	—
38	D₄×C₂₃	—	2	4	no	—
39	D₄×C₂₃	—	2	2	no	—
40	C₂×C₄×C₂₃	—	2	2	no	—
41	C₂×C₄×C₂₃	—	2	2	no	—
42	C₂×C₄×C₂₃	C₄⋊C₄	1	2	yes	—
43	C₂×C₄×C₂₃	C₂²×C₄	1	1	yes	characteristic
44	C₂×C₄×C₂₃	C₂×D₄	1	2	yes	—
45	C₂×C₄×C₂₃	C₂×Q₈	1	1	yes	characteristic
46	C₂×C₄×C₂₃	#41	1	2	yes	—
47	C₂×Dic₂₃	—	4	4	no	—
48	C₈×C₂₃	—	2	2	no	—
49	C₈×C₂₃	C₂²⋊C₄	1	1	yes	characteristic
50	C₈×C₂₃	C₂²⋊C₄	1	1	yes	characteristic
51	C₈×C₂₃	C₂²⋊C₄	1	2	yes	—
52	#1812	—	23	1	no	maximal, Sylow
53	C₂²×C₂₃	—	2	2	no	—
54	C₂²×C₂₃	—	4	1	no	—
55	C₂²×C₂₃	—	4	2	no	—
56	C₂²×C₂₃	C₂×C₄⋊C₄	1	1	yes	characteristic
57	C₂²×C₂₃	#121	1	2	yes	—
58	C₄×C₂₃	—	2	2	no	—
59	C₄×C₂₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
60	C₄×C₂₃	#117	1	1	yes	characteristic
61	C₄×C₂₃	C₂²⋊Q₈	1	2	yes	—
62	Dic₂₃	—	8	4	no	—
63	C₂×#131	D₂₃	1	1	yes	characteristic
64	#427	—	23	2	no	—
65	#410	—	23	2	no	—
66	H(3,ℤ/4)	—	23	2	no	—
67	C₂×C₂₃	—	2	2	no	—
68	C₂×C₂₃	—	2	1	no	—
69	C₂×C₂₃	—	4	2	no	—
70	C₂×C₂₃	#380	1	1	yes	characteristic
71	C₂×#41	C₂×D₂₃	1	1	yes	characteristic
72	#131	—	2	2	no	—
73	#131	—	2	2	no	—
74	C₂×M₁₆	Dic₂₃	1	2	yes	complemented
75	C₂×M₁₆	C₂×D₂₃	1	1	yes	characteristic
76	C₂²×C₈	Dic₂₃	1	2	yes	complemented
77	C₂²×C₈	C₂×D₂₃	1	1	yes	characteristic
78	#117	—	23	4	no	—
79	#107	—	46	2	no	—
80	#106	—	46	2	no	—
81	#99	—	46	4	no	—
82	C₂₃	#1812	1	1	yes	characteristic, complemented, Sylow
83	#41	—	2	2	no	—
84	#41	—	2	2	no	—
85	C₂×Q₈	Dic₄₆	1	1	yes	characteristic
86	C₂×D₄	Dic₄₆	1	1	yes	characteristic
87	C₂×D₄	D₉₂	1	2	yes	—
88	C₂²×C₄	C₂×Dic₂₃	1	2	yes	—
89	C₂²×C₄	C₂²×D₂₃	1	1	yes	characteristic
90	M₁₆	—	2	2	no	—
91	M₁₆	—	2	4	no	—
92	C₂×C₈	—	2	2	no	—
93	C₂×C₈	—	2	2	no	—
94	C₂×C₈	C₂×Dic₂₃	1	2	yes	—
95	C₂×C₈	C₂×Dic₂₃	1	1	yes	characteristic
96	C₂×C₈	C₂×Dic₂₃	1	1	yes	characteristic
97	C₂×C₈	C₂²⋊D₂₃	1	2	yes	—
98	C₂×C₈	C₂²⋊D₂₃	1	2	yes	—
99	C₄⋊C₄	—	46	4	no	—
100	C₂²⋊C₄	—	46	4	no	—
101	C₄²	—	46	4	no	—
102	C₂³	C₄⋊Dic₂₃	1	2	yes	—
103	C₂³	C₂×D₉₂	1	1	yes	characteristic
104	Q₈	—	2	2	no	—
105	D₄	—	2	4	no	—
106	D₄	—	2	2	no	—
107	C₂×C₄	—	2	2	no	—
108	C₂×C₄	—	2	2	no	—
109	C₂×C₄	—	92	4	no	—
110	C₂×C₄	C₄⋊Dic₂₃	1	2	yes	—
111	C₂×C₄	C₂×Dic₄₆	1	1	yes	characteristic
112	C₂×C₄	#67288	1	2	yes	—
113	C₂×C₄	C₂²×Dic₂₃	1	1	yes	characteristic
114	C₂×C₄	C₂×C₂²⋊D₂₃	1	2	yes	—
115	C₈	—	2	2	no	—
116	C₈	C₂²⋊Dic₂₃	1	1	yes	characteristic
117	C₈	C₂²⋊Dic₂₃	1	1	yes	characteristic
118	C₈	C₂²⋊Dic₂₃	1	2	yes	—
119	C₂²	—	2	2	no	—
120	C₂²	—	4	1	no	—
121	C₂²	—	4	2	no	—
122	C₂²	C₂×C₄⋊Dic₂₃	1	1	yes	characteristic
123	C₂²	#10625101	1	2	yes	—
124	C₄	—	2	2	no	—
125	C₄	—	184	4	no	—
126	C₄	#10625095	1	2	yes	—
127	C₄	#10625097	1	1	yes	characteristic
128	C₄	C₂×C₂²⋊Dic₂₃	1	1	yes	characteristic
129	C₂	—	2	2	no	—
130	C₂	—	2	1	no	—
131	C₂	—	4	2	no	—
132	C₂	#13092277	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#13092455

C₂

yes

maximal, complemented

#13092391

C₂

yes

maximal

#13092174

C₂

yes

maximal, complemented

C₂×C₂₃×#131

C₂

yes

maximal, characteristic

C₂×#41×C₂₃

C₂²

yes

characteristic

C₂₃×#131

—

C₂₃×#131

—

C₂×M₁₆×C₂₃

C₄

yes

complemented

C₂×M₁₆×C₂₃

C₂²

yes

characteristic

C₂²×C₈×C₂₃

C₄

yes

complemented

C₂²×C₈×C₂₃

C₂²

yes

characteristic

#10625097

C₂²

yes

—

#10625016

—

#10625000

—

#10624999

—

#41×C₂₃

—

#41×C₂₃

—

C₂×Q₈×C₂₃

Q₈

yes

characteristic

C₂×D₄×C₂₃

D₄

yes

—

C₂×D₄×C₂₃

Q₈

yes

characteristic

C₂²×C₄×C₂₃

C₂×C₄

yes

—

C₂²×C₄×C₂₃

C₂³

yes

characteristic

M₁₆×C₂₃

—

M₁₆×C₂₃

—

C₂×C₈×C₂₃

—

C₂×C₈×C₂₃

—

C₂×C₈×C₂₃

C₂×C₄

yes

—

C₂×C₈×C₂₃

C₂×C₄

yes

characteristic

C₂×C₈×C₂₃

C₂×C₄

yes

characteristic

C₂×C₈×C₂₃

D₄

yes

—

C₂×C₈×C₂₃

D₄

yes

—

C₂²⋊Dic₂₃

—

C₄⋊Dic₂₃

—

C₄×Dic₂₃

—

C₂³×C₂₃

C₄⋊C₄

yes

—

C₂³×C₂₃

C₂×D₄

yes

characteristic

Q₈×C₂₃

—

D₄×C₂₃

—

D₄×C₂₃

—

C₂×C₄×C₂₃

—

C₂×C₄×C₂₃

—

C₂×C₄×C₂₃

C₄⋊C₄

yes

—

C₂×C₄×C₂₃

C₂²×C₄

yes

characteristic

C₂×C₄×C₂₃

C₂×D₄

yes

—

C₂×C₄×C₂₃

C₂×Q₈

yes

characteristic

C₂×C₄×C₂₃

#41

yes

—

C₂×Dic₂₃

—

C₈×C₂₃

—

C₈×C₂₃

C₂²⋊C₄

yes

characteristic

C₈×C₂₃

C₂²⋊C₄

yes

characteristic

C₈×C₂₃

C₂²⋊C₄

yes

—

#1812

—

maximal, Sylow

C₂²×C₂₃

—

C₂²×C₂₃

—

C₂²×C₂₃

—

C₂²×C₂₃

C₂×C₄⋊C₄

yes

characteristic

C₂²×C₂₃

#121

yes

—

C₄×C₂₃

—

C₄×C₂₃

C₂×C₂²⋊C₄

yes

characteristic

C₄×C₂₃

#117

yes

characteristic

C₄×C₂₃

C₂²⋊Q₈

yes

—

Dic₂₃

—

C₂×#131

D₂₃

yes

—

—

—

C₂×C₂₃

—

C₂×C₂₃

—

C₂×C₂₃

—

C₂×C₂₃

#380

yes

characteristic

C₂×#41

C₂×D₂₃

yes

characteristic

#131

—

#131

—

C₂×M₁₆

Dic₂₃

yes

complemented

C₂×M₁₆

C₂×D₂₃

yes

characteristic

C₂²×C₈

Dic₂₃

yes

complemented

C₂²×C₈

C₂×D₂₃

yes

—

—

—

—

C₂₃

#1812

yes

characteristic, complemented, Sylow

#41

—

#41

—

C₂×Q₈

Dic₄₆

yes

characteristic

C₂×D₄

Dic₄₆

yes

characteristic

C₂×D₄

D₉₂

yes

—

C₂²×C₄

C₂×Dic₂₃

yes

—

C₂²×C₄

C₂²×D₂₃

yes

characteristic

M₁₆

—

M₁₆

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

C₂×Dic₂₃

yes

—

C₂×C₈

C₂×Dic₂₃

yes

characteristic

C₂×C₈

C₂×Dic₂₃

yes

characteristic

C₂×C₈

C₂²⋊D₂₃

yes

—

C₂×C₈

C₂²⋊D₂₃

yes

—

C₄⋊C₄

—

100

C₂²⋊C₄

—

101

C₄²

—

102

C₂³

C₄⋊Dic₂₃

yes

—

103

C₂³

C₂×D₉₂

yes

characteristic

104

Q₈

—

105

D₄

—

106

D₄

—

107

C₂×C₄

—

108

C₂×C₄

—

109

C₂×C₄

—

110

C₂×C₄

C₄⋊Dic₂₃

yes

—

111

C₂×C₄

C₂×Dic₄₆

yes

characteristic

112

C₂×C₄

#67288

yes

—

113

C₂×C₄

C₂²×Dic₂₃

yes

characteristic

114

C₂×C₄

C₂×C₂²⋊D₂₃

yes

—

115

C₈

—

116

C₈

C₂²⋊Dic₂₃

yes

characteristic

117

C₈

C₂²⋊Dic₂₃

yes

characteristic

118

C₈

C₂²⋊Dic₂₃

yes

—

119

C₂²

—

120

C₂²

—

121

C₂²

—

122

C₂²

C₂×C₄⋊Dic₂₃

yes

characteristic

123

C₂²

#10625101

yes

—

124

C₄

—

125

C₄

—

184

—

126

C₄

#10625095

yes

—

127

C₄

#10625097

yes

characteristic

128

C₄

C₂×C₂²⋊Dic₂₃

yes

characteristic

129

C₂

—

130

C₂

—

131

C₂

—

132

C₂

#13092277

yes

characteristic

	subgroup	G/H
center	C₄	#10625097
commutator	C₂×C₄×C₂₃	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₂₃
Fitting	C₂×C₂₃×#131	C₂
socle	C₂×C₂₃	#380
prime radical	C₂×#41×C₂₃	C₂²
cosocle	C₂²×C₄×C₂₃	C₂³
hypercenter	C₂×#131	D₂₃
hypocenter	C₂₃	#1812
LCS	G ▷ C₂×C₄×C₂₃ ▷ C₂×C₂₃ ▷ C₂₃
UCS	C₁ ◁ C₄ ◁ C₂×M₁₆ ◁ C₂×#131

subgroup

G/H

center

C₄

#10625097

commutator

C₂×C₄×C₂₃

C₂²×C₄

Frattini

C₂²×C₄

C₂²×D₂₃

Fitting

C₂×C₂₃×#131

C₂

socle

C₂×C₂₃

#380

prime radical

C₂×#41×C₂₃

C₂²

cosocle

C₂²×C₄×C₂₃

C₂³

D₂₃

C₂₃

G ▷ C₂×C₄×C₂₃ ▷ C₂×C₂₃ ▷ C₂₃

UCS

C₁ ◁ C₄ ◁ C₂×M₁₆ ◁ C₂×#131

N	H ≅ G/N	type
1. #13092455	C₂	semidirect
3. #13092174	C₂	semidirect
8. C₂×M₁₆×C₂₃	C₄	semidirect
10. C₂²×C₈×C₂₃	C₄	semidirect
74. C₂×M₁₆	Dic₂₃	semidirect
76. C₂²×C₈	Dic₂₃	semidirect
82. C₂₃	#1812	semidirect

H ≅ G/N

type

1. #13092455

C₂

semidirect

3. #13092174

C₂

semidirect

8. C₂×M₁₆×C₂₃

C₄

semidirect

10. C₂²×C₈×C₂₃

C₄

semidirect

74. C₂×M₁₆

Dic₂₃

semidirect

76. C₂²×C₈

Dic₂₃

semidirect

82. C₂₃

#1812

semidirect

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

subgroup	index
#13092455	2
#13092391	2
#13092174	2
C₂×C₂₃×#131	2
#1812	23

subgroup

index

C₂×C₂₃×#131

Subgroups — #15673443

Conjugacy classes of subgroups

Complements

Maximal subgroups