Group #15674146 — subgroups

Subgroups — #15674146

Order 2944.

208 conjugacy classes of subgroups; only the 97 normal subgroups are shown.

normal subgroups only

This group has 208 conjugacy classes of subgroups — too many to draw. Only its 97 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×#10625001	C₂	1	1	yes	maximal, characteristic
2	#13092444	C₂	1	2	yes	maximal
3	C₂×C₈×Dic₂₃	C₂	1	1	yes	maximal, characteristic
4	#13092176	C₂	1	2	yes	maximal
5	C₂₃×#427	C₂	1	1	yes	maximal, characteristic
6	C₂²×C₈×C₂₃	C₂²	1	1	yes	characteristic
7	C₂₃×#117	C₂²	1	2	yes	—
8	C₂×C₄×Dic₂₃	C₂²	1	1	yes	characteristic
9	C₂×#67267	C₂²	1	2	yes	—
10	C₂²×C₂₃⋊C₈	C₂²	1	1	yes	characteristic
11	C₂₃×#107	C₄	1	2	yes	complemented
12	C₂₃×#106	C₄	1	2	yes	complemented
13	#10625001	—	2	2	no	—
14	C₈×Dic₂₃	—	2	2	no	—
15	C₄⋊(C₂₃⋊C₈)	—	2	4	no	—
16	C₂²×C₄×C₂₃	C₂³	1	1	yes	characteristic
17	C₂²×Dic₂₃	—	2	1	no	—
18	C₂×C₈×C₂₃	C₂×C₄	1	1	yes	characteristic
19	C₂×C₈×C₂₃	C₂×C₄	1	1	yes	characteristic
20	C₂×C₈×C₂₃	D₄	1	2	yes	—
21	C₂×C₈×C₂₃	Q₈	1	2	yes	—
22	C₄⋊C₄×C₂₃	C₂×C₄	1	4	yes	—
23	C₂²⋊C₄×C₂₃	—	2	2	no	—
24	C₄²×C₂₃	—	2	2	no	—
25	C₄×Dic₂₃	—	2	1	no	—
26	C₄×Dic₂₃	Q₈	1	2	yes	—
27	#67267	—	2	2	no	—
28	#67267	—	2	2	no	—
29	C₂×C₂₃⋊C₈	—	2	1	no	—
30	C₂×C₂₃⋊C₈	—	2	1	no	—
31	C₂×C₂₃⋊C₈	—	2	2	no	—
32	C₂×C₂₃⋊C₈	D₄	1	2	yes	—
33	C₂³×C₂₃	#41	1	1	yes	characteristic
34	C₂×C₄×C₂₃	—	2	4	no	—
35	C₂×C₄×C₂₃	C₂²×C₄	1	1	yes	characteristic
36	C₂×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
37	C₂×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
38	C₂×C₄×C₂₃	C₂×Q₈	1	1	yes	characteristic
39	C₂×C₄×C₂₃	C₂×Q₈	1	1	yes	characteristic
40	C₂×C₄×C₂₃	#41	1	1	yes	characteristic
41	C₂×Dic₂₃	—	2	4	no	—
42	C₂×Dic₂₃	—	2	2	no	—
43	C₈×C₂₃	C₄⋊C₄	1	2	yes	—
44	C₈×C₂₃	C₄⋊C₄	1	2	yes	—
45	C₂₃⋊C₈	—	2	2	no	—
46	C₂₃⋊C₈	—	4	2	no	—
47	#1945	—	23	1	no	maximal, Sylow
48	C₂²×C₂₃	—	2	2	no	—
49	C₂²×C₂₃	C₄×D₄	1	1	yes	characteristic
50	C₂²×C₂₃	C₂²⋊Q₈	1	1	yes	characteristic
51	C₂²×C₂₃	#125	1	1	yes	characteristic
52	C₄×C₂₃	—	4	4	no	—
53	C₄×C₂₃	C₂×C₄⋊C₄	1	1	yes	characteristic
54	C₄×C₂₃	C₄×Q₈	1	1	yes	characteristic
55	C₄×C₂₃	#121	1	1	yes	characteristic
56	C₄×C₂₃	C₄⋊Q₈	1	1	yes	characteristic
57	Dic₂₃	—	2	4	no	—
58	C₂×#108	—	23	1	no	—
59	#427	D₂₃	1	1	yes	characteristic
60	#424	—	23	2	no	—
61	C₂×C₄×C₈	—	23	1	no	—
62	#339	—	23	2	no	—
63	C₂×C₂₃	—	2	1	no	—
64	C₂×C₂₃	—	2	1	no	—
65	C₂×C₂₃	#389	1	1	yes	characteristic
66	C₂×C₂₃	#443	1	2	yes	—
67	C₂×M₁₆	—	23	2	no	—
68	C₂²×C₈	—	23	1	no	—
69	C₂²×C₈	C₂×D₂₃	1	1	yes	characteristic
70	#117	C₂×D₂₃	1	2	yes	—
71	C₂×C₄²	—	23	1	no	—
72	#108	—	46	2	no	—
73	#107	Dic₂₃	1	2	yes	complemented
74	#106	Dic₂₃	1	2	yes	complemented
75	C₄⋊C₈	—	46	4	no	—
76	C₄×C₈	—	46	2	no	—
77	C₂₃	#1945	1	1	yes	characteristic, complemented, Sylow
78	C₂²×C₄	—	46	1	no	—
79	C₂²×C₄	C₂²×D₂₃	1	1	yes	characteristic
80	M₁₆	—	46	2	no	—
81	M₁₆	—	46	2	no	—
82	C₂×C₈	—	23	2	no	—
83	C₂×C₈	—	46	1	no	—
84	C₂×C₈	—	46	1	no	—
85	C₂×C₈	—	46	2	no	—
86	C₂×C₈	Dic₄₆	1	2	yes	—
87	C₂×C₈	D₉₂	1	2	yes	—
88	C₂×C₈	C₂×Dic₂₃	1	1	yes	characteristic
89	C₂×C₈	C₂×Dic₂₃	1	1	yes	characteristic
90	C₄⋊C₄	C₂×Dic₂₃	1	4	yes	—
91	C₂²⋊C₄	—	2	2	no	—
92	C₄²	—	2	2	no	—
93	C₄²	—	23	2	no	—
94	C₄²	—	46	1	no	—
95	C₂³	#67290	1	1	yes	characteristic
96	C₂×C₄	—	2	4	no	—
97	C₂×C₄	—	46	4	no	—
98	C₂×C₄	—	46	2	no	—
99	C₂×C₄	C₂×Dic₄₆	1	1	yes	characteristic
100	C₂×C₄	C₂×D₉₂	1	1	yes	characteristic
101	C₂×C₄	D₄×D₂₃	1	1	yes	characteristic
102	C₂×C₄	Q₈×D₂₃	1	1	yes	characteristic
103	C₂×C₄	#67292	1	1	yes	characteristic
104	C₂×C₄	C₂²×Dic₂₃	1	1	yes	characteristic
105	C₈	—	46	2	no	—
106	C₈	—	92	2	no	—
107	C₈	C₄⋊Dic₂₃	1	2	yes	—
108	C₈	C₄⋊Dic₂₃	1	2	yes	—
109	C₂²	—	2	2	no	—
110	C₂²	#10625061	1	1	yes	characteristic
111	C₂²	#10625068	1	1	yes	characteristic
112	C₂²	D₄×Dic₂₃	1	1	yes	characteristic
113	C₄	—	4	4	no	—
114	C₄	—	46	4	no	—
115	C₄	#10625059	1	1	yes	characteristic
116	C₄	#10625066	1	1	yes	characteristic
117	C₄	C₂×C₄⋊Dic₂₃	1	1	yes	characteristic
118	C₄	Q₈×Dic₂₃	1	1	yes	characteristic
119	C₂	—	2	1	no	—
120	C₂	—	2	1	no	—
121	C₂	#13092316	1	1	yes	characteristic
122	C₂	#13092534	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#10625001

C₂

yes

maximal, characteristic

#13092444

C₂

yes

maximal

C₂×C₈×Dic₂₃

C₂

yes

maximal, characteristic

#13092176

C₂

yes

maximal

C₂₃×#427

C₂

yes

maximal, characteristic

C₂²×C₈×C₂₃

C₂²

yes

characteristic

C₂₃×#117

C₂²

yes

—

C₂×C₄×Dic₂₃

C₂²

yes

characteristic

C₂×#67267

C₂²

yes

—

C₂²×C₂₃⋊C₈

C₂²

yes

characteristic

C₂₃×#107

C₄

yes

complemented

C₂₃×#106

C₄

yes

complemented

#10625001

—

C₈×Dic₂₃

—

C₄⋊(C₂₃⋊C₈)

—

C₂²×C₄×C₂₃

C₂³

yes

characteristic

C₂²×Dic₂₃

—

C₂×C₈×C₂₃

C₂×C₄

yes

characteristic

C₂×C₈×C₂₃

C₂×C₄

yes

characteristic

C₂×C₈×C₂₃

D₄

yes

—

C₂×C₈×C₂₃

Q₈

yes

—

C₄⋊C₄×C₂₃

C₂×C₄

yes

—

C₂²⋊C₄×C₂₃

—

C₄²×C₂₃

—

C₄×Dic₂₃

—

C₄×Dic₂₃

Q₈

yes

—

#67267

—

#67267

—

C₂×C₂₃⋊C₈

—

C₂×C₂₃⋊C₈

—

C₂×C₂₃⋊C₈

—

C₂×C₂₃⋊C₈

D₄

yes

—

C₂³×C₂₃

#41

yes

characteristic

C₂×C₄×C₂₃

—

C₂×C₄×C₂₃

C₂²×C₄

yes

characteristic

C₂×C₄×C₂₃

C₂×D₄

yes

characteristic

C₂×C₄×C₂₃

C₂×D₄

yes

characteristic

C₂×C₄×C₂₃

C₂×Q₈

yes

characteristic

C₂×C₄×C₂₃

C₂×Q₈

yes

characteristic

C₂×C₄×C₂₃

#41

yes

characteristic

C₂×Dic₂₃

—

C₂×Dic₂₃

—

C₈×C₂₃

C₄⋊C₄

yes

—

C₈×C₂₃

C₄⋊C₄

yes

—

C₂₃⋊C₈

—

C₂₃⋊C₈

—

#1945

—

maximal, Sylow

C₂²×C₂₃

—

C₂²×C₂₃

C₄×D₄

yes

characteristic

C₂²×C₂₃

C₂²⋊Q₈

yes

characteristic

C₂²×C₂₃

#125

yes

characteristic

C₄×C₂₃

—

C₄×C₂₃

C₂×C₄⋊C₄

yes

characteristic

C₄×C₂₃

C₄×Q₈

yes

characteristic

C₄×C₂₃

#121

yes

characteristic

C₄×C₂₃

C₄⋊Q₈

yes

characteristic

Dic₂₃

—

C₂×#108

—

#427

D₂₃

yes

characteristic

#424

—

C₂×C₄×C₈

—

#339

—

C₂×C₂₃

—

C₂×C₂₃

—

C₂×C₂₃

#389

yes

characteristic

C₂×C₂₃

#443

yes

—

C₂×M₁₆

—

C₂²×C₈

—

C₂²×C₈

C₂×D₂₃

yes

characteristic

#117

C₂×D₂₃

yes

—

C₂×C₄²

—

#108

—

#107

Dic₂₃

yes

complemented

#106

Dic₂₃

yes

complemented

C₄⋊C₈

—

C₄×C₈

—

C₂₃

#1945

yes

characteristic, complemented, Sylow

C₂²×C₄

—

C₂²×C₄

C₂²×D₂₃

yes

characteristic

M₁₆

—

M₁₆

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

Dic₄₆

yes

—

C₂×C₈

D₉₂

yes

—

C₂×C₈

C₂×Dic₂₃

yes

characteristic

C₂×C₈

C₂×Dic₂₃

yes

characteristic

C₄⋊C₄

C₂×Dic₂₃

yes

—

C₂²⋊C₄

—

C₄²

—

C₄²

—

C₄²

—

C₂³

#67290

yes

characteristic

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

C₂×Dic₄₆

yes

characteristic

100

C₂×C₄

C₂×D₉₂

yes

characteristic

101

C₂×C₄

D₄×D₂₃

yes

characteristic

102

C₂×C₄

Q₈×D₂₃

yes

characteristic

103

C₂×C₄

#67292

yes

characteristic

104

C₂×C₄

C₂²×Dic₂₃

yes

characteristic

105

C₈

—

106

C₈

—

107

C₈

C₄⋊Dic₂₃

yes

—

108

C₈

C₄⋊Dic₂₃

yes

—

109

C₂²

—

110

C₂²

#10625061

yes

characteristic

111

C₂²

#10625068

yes

characteristic

112

C₂²

D₄×Dic₂₃

yes

characteristic

113

C₄

—

114

C₄

—

115

C₄

#10625059

yes

characteristic

116

C₄

#10625066

yes

characteristic

117

C₄

C₂×C₄⋊Dic₂₃

yes

characteristic

118

C₄

Q₈×Dic₂₃

yes

characteristic

119

C₂

—

120

C₂

—

121

C₂

#13092316

yes

characteristic

122

C₂

#13092534

yes

—

	subgroup	G/H
center	C₂×C₄	D₄×D₂₃
commutator	C₂×C₄×C₂₃	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₂₃
Fitting	C₂₃×#427	C₂
socle	C₂²×C₂₃	C₄×D₄
prime radical	C₂³×C₂₃	#41
cosocle	C₂²×C₄×C₂₃	C₂³
hypercenter	#427	D₂₃
hypocenter	C₂₃	#1945
LCS	G ▷ C₂×C₄×C₂₃ ▷ C₂×C₂₃ ▷ C₂₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄ ◁ #427

subgroup

G/H

center

C₂×C₄

D₄×D₂₃

commutator

C₂×C₄×C₂₃

C₂²×C₄

Frattini

C₂²×C₄

C₂²×D₂₃

Fitting

C₂₃×#427

C₂

socle

C₂²×C₂₃

C₄×D₄

prime radical

C₂³×C₂₃

#41

cosocle

C₂²×C₄×C₂₃

C₂³

D₂₃

C₂₃

G ▷ C₂×C₄×C₂₃ ▷ C₂×C₂₃ ▷ C₂₃

UCS

C₁ ◁ C₂×C₄ ◁ C₂²×C₄ ◁ #427

N	H ≅ G/N	type
11. C₂₃×#107	C₄	semidirect
12. C₂₃×#106	C₄	semidirect
73. #107	Dic₂₃	semidirect
74. #106	Dic₂₃	semidirect
77. C₂₃	#1945	semidirect

H ≅ G/N

type

11. C₂₃×#107

C₄

semidirect

12. C₂₃×#106

C₄

semidirect

73. #107

Dic₂₃

semidirect

74. #106

Dic₂₃

semidirect

77. C₂₃

#1945

semidirect

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

subgroup	index
C₂×#10625001	2
#13092444	2
C₂×C₈×Dic₂₃	2
#13092176	2
C₂₃×#427	2
#1945	23

subgroup

index

C₂×C₈×Dic₂₃

C₂₃×#427

Subgroups — #15674146

Conjugacy classes of subgroups

Complements

Maximal subgroups