Group #15673278 — subgroups

Subgroups — #15673278

Order 2944.

232 conjugacy classes of subgroups; only the 117 normal subgroups are shown.

normal subgroups only

This group has 232 conjugacy classes of subgroups — too many to draw. Only its 117 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	#13092373	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	C₄	1	4	yes	complemented
14	C₈×Dic₂₃	C₄	1	4	yes	complemented
15	#10624985	—	2	2	no	—
16	C₄×C₂₃⋊C₈	—	2	2	no	—
17	C₂²×C₄×C₂₃	C₂³	1	1	yes	characteristic
18	C₂²×Dic₂₃	—	2	1	no	—
19	C₂×C₈×C₂₃	C₂×C₄	1	2	yes	—
20	C₂×C₈×C₂₃	C₂×C₄	1	2	yes	—
21	C₂×C₈×C₂₃	C₂×C₄	1	1	yes	characteristic
22	C₂×C₈×C₂₃	C₂×C₄	1	1	yes	characteristic
23	C₄⋊C₄×C₂₃	C₂×C₄	1	4	yes	—
24	C₂²⋊C₄×C₂₃	—	2	2	no	—
25	C₄²×C₂₃	—	2	2	no	—
26	C₄×Dic₂₃	—	2	1	no	—
27	C₄×Dic₂₃	C₂×C₄	1	2	yes	—
28	#67267	—	2	2	no	—
29	#67267	C₂×C₄	1	4	yes	—
30	C₂×C₂₃⋊C₈	—	2	2	no	—
31	C₂×C₂₃⋊C₈	C₂×C₄	1	2	yes	—
32	C₂×C₂₃⋊C₈	D₄	1	1	yes	characteristic
33	C₂×C₂₃⋊C₈	D₄	1	1	yes	characteristic
34	C₂×C₂₃⋊C₈	Q₈	1	2	yes	—
35	C₂³×C₂₃	#41	1	1	yes	characteristic
36	C₂×C₄×C₂₃	—	2	4	no	—
37	C₂×C₄×C₂₃	C₂²×C₄	1	1	yes	characteristic
38	C₂×C₄×C₂₃	C₂²×C₄	1	1	yes	characteristic
39	C₂×C₄×C₂₃	C₂²×C₄	1	1	yes	characteristic
40	C₂×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
41	C₂×C₄×C₂₃	C₂×Q₈	1	1	yes	characteristic
42	C₂×C₄×C₂₃	#41	1	1	yes	characteristic
43	C₂×Dic₂₃	—	2	4	no	—
44	C₂×Dic₂₃	—	2	2	no	—
45	C₈×C₂₃	C₄²	1	2	yes	—
46	C₈×C₂₃	C₄²	1	2	yes	—
47	C₂₃⋊C₈	—	2	4	no	—
48	C₂₃⋊C₈	C₄⋊C₄	1	2	yes	—
49	C₂₃⋊C₈	C₄⋊C₄	1	2	yes	—
50	#1772	—	23	1	no	maximal, Sylow
51	C₂²×C₂₃	—	2	2	no	—
52	C₂²×C₂₃	#117	1	1	yes	characteristic
53	C₂²×C₂₃	C₄×D₄	1	1	yes	characteristic
54	C₂²×C₂₃	C₄×Q₈	1	1	yes	characteristic
55	C₄×C₂₃	—	4	4	no	—
56	C₄×C₂₃	C₂×C₄²	1	1	yes	characteristic
57	C₄×C₂₃	C₂×C₄⋊C₄	1	1	yes	characteristic
58	C₄×C₂₃	C₄×D₄	1	1	yes	characteristic
59	C₄×C₂₃	C₄×Q₈	1	1	yes	characteristic
60	Dic₂₃	—	2	4	no	—
61	C₂×#108	—	23	1	no	—
62	#427	D₂₃	1	1	yes	characteristic
63	#405	—	23	2	no	—
64	C₂×C₄×C₈	—	23	1	no	—
65	#339	—	23	2	no	—
66	C₂×C₂₃	—	2	1	no	—
67	C₂×C₂₃	—	2	1	no	—
68	C₂×C₂₃	C₄×C₄⋊C₄	1	1	yes	characteristic
69	C₂×C₂₃	#443	1	2	yes	—
70	C₂×M₁₆	—	23	2	no	—
71	C₂²×C₈	—	23	1	no	—
72	C₂²×C₈	C₂×D₂₃	1	1	yes	characteristic
73	#117	C₂×D₂₃	1	2	yes	—
74	C₂×C₄²	—	23	1	no	—
75	#108	—	23	4	no	—
76	#107	Dic₂₃	1	2	yes	complemented
77	#106	Dic₂₃	1	2	yes	complemented
78	#97	—	46	2	no	—
79	C₄×C₈	—	23	4	no	—
80	C₄×C₈	—	46	2	no	—
81	C₂₃	#1772	1	1	yes	characteristic, complemented, Sylow
82	C₂²×C₄	—	46	1	no	—
83	C₂²×C₄	C₂²×D₂₃	1	1	yes	characteristic
84	M₁₆	—	23	4	no	—
85	M₁₆	—	46	2	no	—
86	C₂×C₈	—	23	2	no	—
87	C₂×C₈	—	23	1	no	—
88	C₂×C₈	—	23	1	no	—
89	C₂×C₈	—	23	2	no	—
90	C₂×C₈	—	46	2	no	—
91	C₂×C₈	C₄×D₂₃	1	2	yes	—
92	C₂×C₈	C₄×D₂₃	1	2	yes	—
93	C₂×C₈	C₂×Dic₂₃	1	1	yes	characteristic
94	C₂×C₈	C₂×Dic₂₃	1	1	yes	characteristic
95	C₄⋊C₄	C₂×Dic₂₃	1	4	yes	—
96	C₂²⋊C₄	—	2	2	no	—
97	C₄²	—	2	2	no	—
98	C₄²	—	23	2	no	—
99	C₄²	—	46	1	no	—
100	C₂³	#67290	1	1	yes	characteristic
101	C₂×C₄	—	2	4	no	—
102	C₂×C₄	—	46	4	no	—
103	C₂×C₄	—	46	2	no	—
104	C₂×C₄	C₂×C₄×D₂₃	1	1	yes	characteristic
105	C₂×C₄	C₂×C₄×D₂₃	1	1	yes	characteristic
106	C₂×C₄	D₄×D₂₃	1	1	yes	characteristic
107	C₂×C₄	Q₈×D₂₃	1	1	yes	characteristic
108	C₂×C₄	#67292	1	1	yes	characteristic
109	C₂×C₄	C₂²×Dic₂₃	1	1	yes	characteristic
110	C₈	—	23	2	no	—
111	C₈	—	23	2	no	—
112	C₈	—	46	4	no	—
113	C₈	C₄×Dic₂₃	1	2	yes	—
114	C₈	C₄×Dic₂₃	1	2	yes	—
115	C₂²	—	2	2	no	—
116	C₂²	#10625058	1	1	yes	characteristic
117	C₂²	#10625063	1	1	yes	characteristic
118	C₂²	D₄×Dic₂₃	1	1	yes	characteristic
119	C₄	—	4	4	no	—
120	C₄	—	46	4	no	—
121	C₄	C₄⋊C₄×D₂₃	1	1	yes	characteristic
122	C₄	#10625064	1	1	yes	characteristic
123	C₄	C₂×C₄×Dic₂₃	1	1	yes	characteristic
124	C₄	Q₈×Dic₂₃	1	1	yes	characteristic
125	C₂	—	2	1	no	—
126	C₂	—	2	1	no	—
127	C₂	C₄⋊C₄×Dic₂₃	1	1	yes	characteristic
128	C₂	#13092534	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#10625001

C₂

yes

maximal, characteristic

#13092373

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

C₄

yes

complemented

C₈×Dic₂₃

C₄

yes

complemented

#10624985

—

C₄×C₂₃⋊C₈

—

C₂²×C₄×C₂₃

C₂³

yes

characteristic

C₂²×Dic₂₃

—

C₂×C₈×C₂₃

C₂×C₄

yes

—

C₂×C₈×C₂₃

C₂×C₄

yes

—

C₂×C₈×C₂₃

C₂×C₄

yes

characteristic

C₂×C₈×C₂₃

C₂×C₄

yes

characteristic

C₄⋊C₄×C₂₃

C₂×C₄

yes

—

C₂²⋊C₄×C₂₃

—

C₄²×C₂₃

—

C₄×Dic₂₃

—

C₄×Dic₂₃

C₂×C₄

yes

—

#67267

—

#67267

C₂×C₄

yes

—

C₂×C₂₃⋊C₈

—

C₂×C₂₃⋊C₈

C₂×C₄

yes

—

C₂×C₂₃⋊C₈

D₄

yes

characteristic

C₂×C₂₃⋊C₈

D₄

yes

characteristic

C₂×C₂₃⋊C₈

Q₈

yes

—

C₂³×C₂₃

#41

yes

characteristic

C₂×C₄×C₂₃

—

C₂×C₄×C₂₃

C₂²×C₄

yes

characteristic

C₂×C₄×C₂₃

C₂²×C₄

yes

characteristic

C₂×C₄×C₂₃

C₂²×C₄

yes

characteristic

C₂×C₄×C₂₃

C₂×D₄

yes

characteristic

C₂×C₄×C₂₃

C₂×Q₈

yes

characteristic

C₂×C₄×C₂₃

#41

yes

characteristic

C₂×Dic₂₃

—

C₂×Dic₂₃

—

C₈×C₂₃

C₄²

yes

—

C₈×C₂₃

C₄²

yes

—

C₂₃⋊C₈

—

C₂₃⋊C₈

C₄⋊C₄

yes

—

C₂₃⋊C₈

C₄⋊C₄

yes

—

#1772

—

maximal, Sylow

C₂²×C₂₃

—

C₂²×C₂₃

#117

yes

characteristic

C₂²×C₂₃

C₄×D₄

yes

characteristic

C₂²×C₂₃

C₄×Q₈

yes

characteristic

C₄×C₂₃

—

C₄×C₂₃

C₂×C₄²

yes

characteristic

C₄×C₂₃

C₂×C₄⋊C₄

yes

characteristic

C₄×C₂₃

C₄×D₄

yes

characteristic

C₄×C₂₃

C₄×Q₈

yes

characteristic

Dic₂₃

—

C₂×#108

—

#427

D₂₃

yes

characteristic

#405

—

C₂×C₄×C₈

—

#339

—

C₂×C₂₃

—

C₂×C₂₃

—

C₂×C₂₃

C₄×C₄⋊C₄

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

#97

—

C₄×C₈

—

C₄×C₈

—

C₂₃

#1772

yes

characteristic, complemented, Sylow

C₂²×C₄

—

C₂²×C₄

C₂²×D₂₃

yes

characteristic

M₁₆

—

M₁₆

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

C₄×D₂₃

yes

—

C₂×C₈

C₄×D₂₃

yes

—

C₂×C₈

C₂×Dic₂₃

yes

characteristic

C₂×C₈

C₂×Dic₂₃

yes

characteristic

C₄⋊C₄

C₂×Dic₂₃

yes

—

C₂²⋊C₄

—

C₄²

—

C₄²

—

C₄²

—

100

C₂³

#67290

yes

characteristic

101

C₂×C₄

—

102

C₂×C₄

—

103

C₂×C₄

—

104

C₂×C₄

C₂×C₄×D₂₃

yes

characteristic

105

C₂×C₄

C₂×C₄×D₂₃

yes

characteristic

106

C₂×C₄

D₄×D₂₃

yes

characteristic

107

C₂×C₄

Q₈×D₂₃

yes

characteristic

108

C₂×C₄

#67292

yes

characteristic

109

C₂×C₄

C₂²×Dic₂₃

yes

characteristic

110

C₈

—

111

C₈

—

112

C₈

—

113

C₈

C₄×Dic₂₃

yes

—

114

C₈

C₄×Dic₂₃

yes

—

115

C₂²

—

116

C₂²

#10625058

yes

characteristic

117

C₂²

#10625063

yes

characteristic

118

C₂²

D₄×Dic₂₃

yes

characteristic

119

C₄

—

120

C₄

—

121

C₄

C₄⋊C₄×D₂₃

yes

characteristic

122

C₄

#10625064

yes

characteristic

123

C₄

C₂×C₄×Dic₂₃

yes

characteristic

124

C₄

Q₈×Dic₂₃

yes

characteristic

125

C₂

—

126

C₂

—

127

C₂

C₄⋊C₄×Dic₂₃

yes

characteristic

128

C₂

#13092534

yes

—

	subgroup	G/H
center	C₂×C₄	D₄×D₂₃
commutator	C₄×C₂₃	C₂×C₄²
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	#1772
LCS	G ▷ C₄×C₂₃ ▷ C₂×C₂₃ ▷ C₂₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄ ◁ #427

subgroup

G/H

center

C₂×C₄

D₄×D₂₃

commutator

C₄×C₂₃

C₂×C₄²

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 ▷ 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
13. #10625001	C₄	semidirect
14. C₈×Dic₂₃	C₄	semidirect
76. #107	Dic₂₃	semidirect
77. #106	Dic₂₃	semidirect
81. C₂₃	#1772	semidirect

H ≅ G/N

type

11. C₂₃×#107

C₄

semidirect

12. C₂₃×#106

C₄

semidirect

13. #10625001

C₄

semidirect

14. C₈×Dic₂₃

C₄

semidirect

76. #107

Dic₂₃

semidirect

77. #106

Dic₂₃

semidirect

81. C₂₃

#1772

semidirect

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

subgroup	index
C₂×#10625001	2
#13092373	2
C₂×C₈×Dic₂₃	2
#13092176	2
C₂₃×#427	2
#1772	23

subgroup

index

C₂×C₈×Dic₂₃

C₂₃×#427

Subgroups — #15673278

Conjugacy classes of subgroups

Complements

Maximal subgroups