Group #15673048 — subgroups

Subgroups — #15673048

Order 2944.

186 conjugacy classes of subgroups; only the 57 normal subgroups are shown.

normal subgroups only

This group has 186 conjugacy classes of subgroups — too many to draw. Only its 57 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	#13092820	C₂	1	1	yes	maximal, characteristic, complemented
2	#13092769	C₂	1	1	yes	maximal, characteristic, complemented
3	#13092768	C₂	1	1	yes	maximal, characteristic, complemented
4	#13092763	C₂	1	1	yes	maximal, characteristic
5	#13092763	C₂	1	1	yes	maximal, characteristic
6	#13092139	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂₃×#492	C₂	1	1	yes	maximal, characteristic
8	C₂×SD₁₆×C₂₃	—	2	2	no	—
9	C₂₃×C₄⋊Q₈	C₂²	1	1	yes	characteristic
10	C₂₃×#127	C₂²	1	1	yes	characteristic
11	#10625040	C₂²	1	1	yes	characteristic
12	#10625040	C₂²	1	1	yes	characteristic
13	C₄×C₈×C₂₃	C₂²	1	1	yes	characteristic
14	#10625014	—	2	2	no	—
15	#10625014	—	2	2	no	—
16	#10624998	—	2	2	no	—
17	#10624992	—	2	2	no	—
18	#10624989	—	2	2	no	—
19	C₄×C₂₃⋊C₈	C₂²	1	1	yes	characteristic
20	C₄×C₂₃⋊C₈	C₂²	1	1	yes	characteristic
21	C₂×Q₈×C₂₃	—	2	1	no	—
22	C₂×D₄×C₂₃	—	2	1	no	—
23	C₂×D₄×C₂₃	—	2	1	no	—
24	C₂×D₄×C₂₃	—	2	1	no	—
25	C₂×Dic₄₆	—	2	1	no	—
26	C₂×Dic₄₆	—	2	1	no	—
27	SD₁₆×C₂₃	—	4	4	no	—
28	C₂×C₈×C₂₃	D₄	1	2	yes	—
29	C₄⋊C₄×C₂₃	—	2	1	no	—
30	C₄⋊C₄×C₂₃	—	2	1	no	—
31	C₄²×C₂₃	C₂³	1	1	yes	characteristic
32	C₄⋊Dic₂₃	—	2	1	no	—
33	C₄⋊Dic₂₃	—	2	1	no	—
34	C₄⋊Dic₂₃	—	2	1	no	—
35	C₄⋊Dic₂₃	—	2	1	no	—
36	C₂×C₂₃⋊C₈	D₄	1	2	yes	—
37	C₂×C₂₃⋊C₈	D₄	1	2	yes	—
38	C₂³×C₂₃	—	4	1	no	—
39	Q₈×C₂₃	—	4	2	no	—
40	D₄×C₂₃	—	4	2	no	—
41	D₄×C₂₃	—	4	2	no	—
42	D₄×C₂₃	—	4	2	no	—
43	C₂×C₄×C₂₃	—	4	1	no	—
44	C₂×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
45	C₂×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
46	C₂×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
47	C₂×Dic₂₃	—	4	1	no	—
48	C₂×Dic₂₃	—	4	1	no	—
49	Dic₄₆	—	4	2	no	—
50	Dic₄₆	—	4	2	no	—
51	C₈×C₂₃	—	2	2	no	—
52	C₂₃⋊C₈	—	2	2	no	—
53	C₂₃⋊C₈	—	2	2	no	—
54	#1719	—	23	1	no	maximal, Sylow
55	C₂²×C₂₃	—	8	1	no	—
56	C₂²×C₂₃	—	8	1	no	—
57	C₂²×C₂₃	—	8	1	no	—
58	C₂²×C₂₃	#127	1	1	yes	characteristic
59	C₄×C₂₃	—	8	1	no	—
60	C₄×C₂₃	#135	1	2	yes	—
61	C₄×C₂₃	AGL(1,ℤ/8)	1	2	yes	—
62	C₄×C₂₃	#137	1	2	yes	—
63	Dic₂₃	—	8	1	no	—
64	Dic₂₃	—	8	1	no	—
65	#500	—	23	1	no	—
66	#492	D₂₃	1	1	yes	characteristic
67	#487	—	23	1	no	—
68	#487	—	23	1	no	—
69	#486	—	23	1	no	—
70	#486	—	23	1	no	—
71	#322	—	23	1	no	—
72	C₂×C₂₃	—	16	1	no	—
73	C₂×C₂₃	#495	1	1	yes	characteristic
74	C₂×C₂₃	#496	1	1	yes	characteristic
75	C₂×C₂₃	#497	1	1	yes	characteristic
76	C₂×SD₁₆	—	2	2	no	—
77	C₄⋊Q₈	—	23	1	no	—
78	C₄⋊Q₈	—	23	1	no	—
79	C₄⋊Q₈	C₂×D₂₃	1	1	yes	characteristic
80	#127	C₂×D₂₃	1	1	yes	characteristic
81	#107	—	46	2	no	—
82	#103	—	46	2	no	—
83	#103	—	46	2	no	—
84	#102	—	46	2	no	—
85	#102	—	46	2	no	—
86	C₄×C₈	—	23	1	no	—
87	C₄×C₈	—	23	1	no	—
88	C₄×C₈	C₂×D₂₃	1	1	yes	characteristic
89	C₂₃	#1719	1	1	yes	characteristic, complemented, Sylow
90	C₂×Q₈	—	2	1	no	—
91	C₂×Q₈	—	46	1	no	—
92	C₂×Q₈	—	46	1	no	—
93	C₂×D₄	—	2	1	no	—
94	C₂×D₄	—	2	1	no	—
95	C₂×D₄	—	2	1	no	—
96	SD₁₆	—	4	4	no	—
97	C₂×C₈	—	23	2	no	—
98	C₂×C₈	—	23	2	no	—
99	C₂×C₈	C₂²⋊D₂₃	1	2	yes	—
100	C₄⋊C₄	—	2	1	no	—
101	C₄⋊C₄	—	2	1	no	—
102	C₄⋊C₄	—	46	1	no	—
103	C₄⋊C₄	—	46	1	no	—
104	C₄⋊C₄	—	46	1	no	—
105	C₄⋊C₄	—	46	1	no	—
106	C₄²	C₂²×D₂₃	1	1	yes	characteristic
107	C₂³	—	4	1	no	—
108	Q₈	—	4	2	no	—
109	Q₈	—	92	2	no	—
110	Q₈	—	92	2	no	—
111	D₄	—	4	2	no	—
112	D₄	—	4	2	no	—
113	D₄	—	4	2	no	—
114	C₂×C₄	—	4	1	no	—
115	C₂×C₄	—	92	1	no	—
116	C₂×C₄	—	92	1	no	—
117	C₂×C₄	D₄×D₂₃	1	1	yes	characteristic
118	C₂×C₄	D₄×D₂₃	1	1	yes	characteristic
119	C₂×C₄	C₂×C₂²⋊D₂₃	1	1	yes	characteristic
120	C₈	—	2	2	no	—
121	C₈	—	46	2	no	—
122	C₈	—	46	2	no	—
123	C₂²	—	8	1	no	—
124	C₂²	—	8	1	no	—
125	C₂²	—	8	1	no	—
126	C₂²	#10625111	1	1	yes	characteristic
127	C₄	—	8	1	no	—
128	C₄	—	184	1	no	—
129	C₄	—	184	1	no	—
130	C₄	#10625086	1	2	yes	—
131	C₄	#10625087	1	2	yes	—
132	C₄	#10625103	1	2	yes	—
133	C₂	—	16	1	no	—
134	C₂	#13092798	1	1	yes	characteristic
135	C₂	#13092804	1	1	yes	characteristic
136	C₂	#13092808	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#13092820

C₂

yes

maximal, characteristic, complemented

#13092769

C₂

yes

maximal, characteristic, complemented

#13092768

C₂

yes

maximal, characteristic, complemented

#13092763

C₂

yes

maximal, characteristic

#13092763

C₂

yes

maximal, characteristic

#13092139

C₂

yes

maximal, characteristic, complemented

C₂₃×#492

C₂

yes

maximal, characteristic

C₂×SD₁₆×C₂₃

—

C₂₃×C₄⋊Q₈

C₂²

yes

characteristic

C₂₃×#127

C₂²

yes

characteristic

#10625040

C₂²

yes

characteristic

#10625040

C₂²

yes

characteristic

C₄×C₈×C₂₃

C₂²

yes

—

—

—

—

—

C₄×C₂₃⋊C₈

C₂²

yes

characteristic

C₄×C₂₃⋊C₈

C₂²

yes

characteristic

C₂×Q₈×C₂₃

—

C₂×D₄×C₂₃

—

C₂×D₄×C₂₃

—

C₂×D₄×C₂₃

—

C₂×Dic₄₆

—

C₂×Dic₄₆

—

SD₁₆×C₂₃

—

C₂×C₈×C₂₃

D₄

yes

—

C₄⋊C₄×C₂₃

—

C₄⋊C₄×C₂₃

—

C₄²×C₂₃

C₂³

yes

characteristic

C₄⋊Dic₂₃

—

C₄⋊Dic₂₃

—

C₄⋊Dic₂₃

—

C₄⋊Dic₂₃

—

C₂×C₂₃⋊C₈

D₄

yes

—

C₂×C₂₃⋊C₈

D₄

yes

—

C₂³×C₂₃

—

Q₈×C₂₃

—

D₄×C₂₃

—

D₄×C₂₃

—

D₄×C₂₃

—

C₂×C₄×C₂₃

—

C₂×C₄×C₂₃

C₂×D₄

yes

characteristic

C₂×C₄×C₂₃

C₂×D₄

yes

characteristic

C₂×C₄×C₂₃

C₂×D₄

yes

characteristic

C₂×Dic₂₃

—

C₂×Dic₂₃

—

Dic₄₆

—

Dic₄₆

—

C₈×C₂₃

—

C₂₃⋊C₈

—

C₂₃⋊C₈

—

#1719

—

maximal, Sylow

C₂²×C₂₃

—

C₂²×C₂₃

—

C₂²×C₂₃

—

C₂²×C₂₃

#127

yes

characteristic

C₄×C₂₃

—

C₄×C₂₃

#135

yes

—

C₄×C₂₃

AGL(1,ℤ/8)

yes

—

C₄×C₂₃

#137

yes

—

Dic₂₃

—

Dic₂₃

—

#500

—

#492

D₂₃

yes

—

—

—

—

—

C₂×C₂₃

—

C₂×C₂₃

yes

C₂×C₂₃

yes

C₂×C₂₃

yes

C₂×SD₁₆

—

C₄⋊Q₈

—

C₄⋊Q₈

—

C₄⋊Q₈

C₂×D₂₃

yes

characteristic

#127

C₂×D₂₃

yes

—

—

—

—

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

C₂×D₂₃

yes

characteristic

C₂₃

#1719

yes

characteristic, complemented, Sylow

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

C₂×D₄

—

C₂×D₄

—

C₂×D₄

—

SD₁₆

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

C₂²⋊D₂₃

yes

—

100

C₄⋊C₄

—

101

C₄⋊C₄

—

102

C₄⋊C₄

—

103

C₄⋊C₄

—

104

C₄⋊C₄

—

105

C₄⋊C₄

—

106

C₄²

C₂²×D₂₃

yes

characteristic

107

C₂³

—

108

Q₈

—

109

Q₈

—

110

Q₈

—

111

D₄

—

112

D₄

—

113

D₄

—

114

C₂×C₄

—

115

C₂×C₄

—

116

C₂×C₄

—

117

C₂×C₄

D₄×D₂₃

yes

characteristic

118

C₂×C₄

D₄×D₂₃

yes

characteristic

119

C₂×C₄

C₂×C₂²⋊D₂₃

yes

characteristic

120

C₈

—

121

C₈

—

122

C₈

—

123

C₂²

—

124

C₂²

—

125

C₂²

—

126

C₂²

#10625111

yes

characteristic

127

C₄

—

128

C₄

—

184

—

129

C₄

—

184

—

130

C₄

#10625086

yes

—

131

C₄

#10625087

yes

—

132

C₄

#10625103

yes

—

133

C₂

—

134

C₂

#13092798

yes

characteristic

135

C₂

#13092804

yes

characteristic

136

C₂

#13092808

yes

characteristic

	subgroup	G/H
center	C₂²	#10625111
commutator, cosocle	C₄²×C₂₃	C₂³
Frattini	C₄²	C₂²×D₂₃
Fitting	C₂₃×#492	C₂
socle	C₂²×C₂₃	#127
prime radical	C₂₃×#127	C₂²
hypercenter	#492	D₂₃
hypocenter	C₂₃	#1719
LCS	G ▷ C₄²×C₂₃ ▷ C₂²×C₂₃ ▷ C₂₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #492

subgroup

G/H

center

C₂²

#10625111

commutator, cosocle

C₄²×C₂₃

C₂³

Frattini

C₄²

C₂²×D₂₃

Fitting

C₂₃×#492

C₂

socle

C₂²×C₂₃

#127

prime radical

C₂₃×#127

C₂²

D₂₃

C₂₃

G ▷ C₄²×C₂₃ ▷ C₂²×C₂₃ ▷ C₂₃

UCS

C₁ ◁ C₂² ◁ C₄² ◁ #492

N	H ≅ G/N	type
1. #13092820	C₂	semidirect
2. #13092769	C₂	semidirect
3. #13092768	C₂	semidirect
6. #13092139	C₂	semidirect
89. C₂₃	#1719	semidirect

H ≅ G/N

type

1. #13092820

C₂

semidirect

2. #13092769

C₂

semidirect

3. #13092768

C₂

semidirect

6. #13092139

C₂

semidirect

89. C₂₃

#1719

semidirect

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

subgroup	index
#13092820	2
#13092769	2
#13092768	2
#13092763	2
#13092139	2
C₂₃×#492	2
#1719	23

subgroup

index

C₂₃×#492

Subgroups — #15673048

Conjugacy classes of subgroups

Complements

Maximal subgroups