Group #15672513 — subgroups

Subgroups — #15672513

Order 2944.

314 conjugacy classes of subgroups; only the 65 normal subgroups are shown.

normal subgroups only

This group has 314 conjugacy classes of subgroups — too many to draw. Only its 65 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	#13092789	C₂	1	1	yes	maximal, characteristic, complemented
2	#13092707	C₂	1	2	yes	maximal, complemented
3	#13092161	C₂	1	1	yes	maximal, characteristic, complemented
4	D₄⋊(C₂₃⋊C₈)	C₂	1	2	yes	maximal, complemented
5	D₄²×C₂₃	C₂	1	1	yes	maximal, characteristic
6	C₂²×D₄×C₂₃	—	2	2	no	—
7	C₂₃×#127	C₂²	1	1	yes	characteristic
8	C₂₃×#121	—	2	2	no	—
9	C₂₃×#120	—	2	2	no	—
10	C₄×D₄×C₂₃	C₂²	1	2	yes	—
11	C₂×#67273	—	2	2	no	—
12	#10625040	C₂²	1	1	yes	characteristic, complemented
13	#10625014	—	2	2	no	—
14	#10624990	—	2	2	no	—
15	C₄⋊(C₂₃⋊C₈)	C₂²	1	2	yes	complemented
16	C₄×C₂₃⋊C₈	C₂²	1	1	yes	characteristic, complemented
17	C₂⁴×C₂₃	—	2	2	no	—
18	C₂×D₄×C₂₃	—	2	4	no	—
19	C₂×D₄×C₂₃	—	2	2	no	—
20	C₂×D₄×C₂₃	—	4	4	no	—
21	C₂×D₄×C₂₃	—	4	4	no	—
22	C₂×D₄×C₂₃	D₄	1	2	yes	—
23	C₂×D₄×C₂₃	D₄	1	2	yes	—
24	C₂²×C₄×C₂₃	—	2	2	no	—
25	C₂×Dic₄₆	—	2	1	no	—
26	C₄⋊C₄×C₂₃	D₄	1	2	yes	—
27	C₂²⋊C₄×C₂₃	—	2	2	no	—
28	C₂²⋊C₄×C₂₃	—	2	2	no	—
29	C₄²×C₂₃	C₂³	1	1	yes	characteristic
30	#67273	—	4	4	no	—
31	C₄⋊Dic₂₃	—	2	2	no	—
32	C₂×C₂₃⋊C₈	—	2	1	no	—
33	C₂×C₂₃⋊C₈	—	2	2	no	—
34	C₂³×C₂₃	—	2	2	no	—
35	C₂³×C₂₃	—	2	2	no	—
36	C₂³×C₂₃	—	4	8	no	—
37	C₂³×C₂₃	—	8	2	no	—
38	D₄×C₂₃	—	2	2	no	—
39	D₄×C₂₃	—	4	2	no	—
40	D₄×C₂₃	—	4	8	no	—
41	D₄×C₂₃	—	4	2	no	—
42	D₄×C₂₃	—	4	2	no	—
43	D₄×C₂₃	SD₁₆	1	4	yes	complemented
44	C₂×C₄×C₂₃	—	2	2	no	—
45	C₂×C₄×C₂₃	—	4	2	no	—
46	C₂×C₄×C₂₃	—	4	2	no	—
47	C₂×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
48	C₂×C₄×C₂₃	C₂×D₄	1	2	yes	—
49	C₂×Dic₂₃	—	4	1	no	—
50	Dic₄₆	—	4	2	no	—
51	C₂₃⋊C₈	—	2	2	no	—
52	C₂₃⋊C₈	—	4	2	no	—
53	#1629	—	23	1	no	maximal, Sylow
54	C₂²×C₂₃	—	2	4	no	—
55	C₂²×C₂₃	—	4	2	no	—
56	C₂²×C₂₃	—	4	2	no	—
57	C₂²×C₂₃	—	4	4	no	—
58	C₂²×C₂₃	—	4	2	no	—
59	C₂²×C₂₃	—	8	8	no	—
60	C₂²×C₂₃	#120	1	1	yes	characteristic
61	C₄×C₂₃	—	2	1	no	—
62	C₄×C₂₃	—	4	2	no	—
63	C₄×C₂₃	C₂×SD₁₆	1	2	yes	—
64	C₄×C₂₃	AGL(1,ℤ/8)	1	2	yes	—
65	Dic₂₃	—	8	1	no	—
66	D₄²	D₂₃	1	1	yes	characteristic
67	#492	—	23	1	no	—
68	#476	—	23	2	no	—
69	#331	—	23	1	no	—
70	D₄⋊C₈	—	23	2	no	—
71	C₂×C₂₃	—	4	4	no	—
72	C₂×C₂₃	—	8	2	no	—
73	C₂×C₂₃	#450	1	2	yes	—
74	C₂×C₂₃	#453	1	1	yes	characteristic
75	C₂²×D₄	—	2	2	no	—
76	C₂×SD₁₆	—	46	2	no	—
77	C₄⋊Q₈	—	23	1	no	—
78	#127	C₂×D₂₃	1	1	yes	characteristic
79	#121	—	2	2	no	—
80	#120	—	2	2	no	—
81	C₄×D₄	C₂×D₂₃	1	2	yes	—
82	#106	—	46	2	no	—
83	C₄⋊C₈	—	23	2	no	—
84	#102	—	46	2	no	—
85	C₄×C₈	—	23	1	no	—
86	C₂₃	#1629	1	1	yes	characteristic, complemented, Sylow
87	C₂⁴	—	2	2	no	—
88	C₂×Q₈	—	46	1	no	—
89	C₂×D₄	—	2	4	no	—
90	C₂×D₄	—	2	2	no	—
91	C₂×D₄	—	4	4	no	—
92	C₂×D₄	—	4	4	no	—
93	C₂×D₄	C₂²⋊D₂₃	1	2	yes	—
94	C₂×D₄	C₂²⋊D₂₃	1	2	yes	—
95	C₂²×C₄	—	2	2	no	—
96	SD₁₆	—	92	4	no	—
97	C₂×C₈	—	46	1	no	—
98	C₂×C₈	—	46	2	no	—
99	C₄⋊C₄	—	46	2	no	—
100	C₄⋊C₄	C₂²⋊D₂₃	1	2	yes	—
101	C₂²⋊C₄	—	2	2	no	—
102	C₂²⋊C₄	—	2	2	no	—
103	C₄²	C₂²×D₂₃	1	1	yes	characteristic
104	C₂³	—	2	2	no	—
105	C₂³	—	2	2	no	—
106	C₂³	—	4	8	no	—
107	C₂³	—	8	2	no	—
108	Q₈	—	92	2	no	—
109	D₄	—	2	2	no	—
110	D₄	—	4	2	no	—
111	D₄	—	4	8	no	—
112	D₄	—	4	2	no	—
113	D₄	—	4	2	no	—
114	D₄	#67273	1	4	yes	complemented
115	C₂×C₄	—	2	2	no	—
116	C₂×C₄	—	4	2	no	—
117	C₂×C₄	—	4	2	no	—
118	C₂×C₄	—	92	1	no	—
119	C₂×C₄	C₂×C₂²⋊D₂₃	1	1	yes	characteristic
120	C₂×C₄	C₂×C₂²⋊D₂₃	1	2	yes	—
121	C₈	—	46	2	no	—
122	C₈	—	92	2	no	—
123	C₂²	—	2	4	no	—
124	C₂²	—	4	2	no	—
125	C₂²	—	4	2	no	—
126	C₂²	—	4	4	no	—
127	C₂²	—	4	2	no	—
128	C₂²	—	8	8	no	—
129	C₂²	#10625124	1	1	yes	characteristic
130	C₄	—	2	1	no	—
131	C₄	—	4	2	no	—
132	C₄	—	184	1	no	—
133	C₄	#10625103	1	2	yes	—
134	C₄	C₂×#67273	1	2	yes	—
135	C₂	—	4	4	no	—
136	C₂	—	8	2	no	—
137	C₂	#13092568	1	2	yes	—
138	C₂	#13092585	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#13092789

C₂

yes

maximal, characteristic, complemented

#13092707

C₂

yes

maximal, complemented

#13092161

C₂

yes

maximal, characteristic, complemented

D₄⋊(C₂₃⋊C₈)

C₂

yes

maximal, complemented

D₄²×C₂₃

C₂

yes

maximal, characteristic

C₂²×D₄×C₂₃

—

C₂₃×#127

C₂²

yes

characteristic

C₂₃×#121

—

C₂₃×#120

—

C₄×D₄×C₂₃

C₂²

yes

—

C₂×#67273

—

#10625040

C₂²

yes

characteristic, complemented

#10625014

—

#10624990

—

C₄⋊(C₂₃⋊C₈)

C₂²

yes

complemented

C₄×C₂₃⋊C₈

C₂²

yes

characteristic, complemented

C₂⁴×C₂₃

—

C₂×D₄×C₂₃

—

C₂×D₄×C₂₃

—

C₂×D₄×C₂₃

—

C₂×D₄×C₂₃

—

C₂×D₄×C₂₃

D₄

yes

—

C₂×D₄×C₂₃

D₄

yes

—

C₂²×C₄×C₂₃

—

C₂×Dic₄₆

—

C₄⋊C₄×C₂₃

D₄

yes

—

C₂²⋊C₄×C₂₃

—

C₂²⋊C₄×C₂₃

—

C₄²×C₂₃

C₂³

yes

characteristic

#67273

—

C₄⋊Dic₂₃

—

C₂×C₂₃⋊C₈

—

C₂×C₂₃⋊C₈

—

C₂³×C₂₃

—

C₂³×C₂₃

—

C₂³×C₂₃

—

C₂³×C₂₃

—

D₄×C₂₃

—

D₄×C₂₃

—

D₄×C₂₃

—

D₄×C₂₃

—

D₄×C₂₃

—

D₄×C₂₃

SD₁₆

yes

complemented

C₂×C₄×C₂₃

—

C₂×C₄×C₂₃

—

C₂×C₄×C₂₃

—

C₂×C₄×C₂₃

C₂×D₄

yes

characteristic

C₂×C₄×C₂₃

C₂×D₄

yes

—

C₂×Dic₂₃

—

Dic₄₆

—

C₂₃⋊C₈

—

C₂₃⋊C₈

—

#1629

—

maximal, Sylow

C₂²×C₂₃

—

C₂²×C₂₃

—

C₂²×C₂₃

—

C₂²×C₂₃

—

C₂²×C₂₃

—

C₂²×C₂₃

—

C₂²×C₂₃

#120

yes

characteristic

C₄×C₂₃

—

C₄×C₂₃

—

C₄×C₂₃

C₂×SD₁₆

yes

—

C₄×C₂₃

AGL(1,ℤ/8)

yes

—

Dic₂₃

—

D₄²

D₂₃

yes

—

—

—

D₄⋊C₈

—

C₂×C₂₃

—

C₂×C₂₃

—

C₂×C₂₃

#450

yes

—

C₂×C₂₃

#453

yes

characteristic

C₂²×D₄

—

C₂×SD₁₆

—

C₄⋊Q₈

—

#127

C₂×D₂₃

yes

characteristic

#121

—

#120

—

C₄×D₄

C₂×D₂₃

yes

—

#106

—

C₄⋊C₈

—

#102

—

C₄×C₈

—

C₂₃

#1629

yes

characteristic, complemented, Sylow

C₂⁴

—

C₂×Q₈

—

C₂×D₄

—

C₂×D₄

—

C₂×D₄

—

C₂×D₄

—

C₂×D₄

C₂²⋊D₂₃

yes

—

C₂×D₄

C₂²⋊D₂₃

yes

—

C₂²×C₄

—

SD₁₆

—

C₂×C₈

—

C₂×C₈

—

C₄⋊C₄

—

100

C₄⋊C₄

C₂²⋊D₂₃

yes

—

101

C₂²⋊C₄

—

102

C₂²⋊C₄

—

103

C₄²

C₂²×D₂₃

yes

characteristic

104

C₂³

—

105

C₂³

—

106

C₂³

—

107

C₂³

—

108

Q₈

—

109

D₄

—

110

D₄

—

111

D₄

—

112

D₄

—

113

D₄

—

114

D₄

#67273

yes

complemented

115

C₂×C₄

—

116

C₂×C₄

—

117

C₂×C₄

—

118

C₂×C₄

—

119

C₂×C₄

C₂×C₂²⋊D₂₃

yes

characteristic

120

C₂×C₄

C₂×C₂²⋊D₂₃

yes

—

121

C₈

—

122

C₈

—

123

C₂²

—

124

C₂²

—

125

C₂²

—

126

C₂²

—

127

C₂²

—

128

C₂²

—

129

C₂²

#10625124

yes

characteristic

130

C₄

—

131

C₄

—

132

C₄

—

184

—

133

C₄

#10625103

yes

—

134

C₄

C₂×#67273

yes

—

135

C₂

—

136

C₂

—

137

C₂

#13092568

yes

—

138

C₂

#13092585

yes

characteristic

	subgroup	G/H
center	C₂²	#10625124
commutator, cosocle	C₄²×C₂₃	C₂³
Frattini	C₄²	C₂²×D₂₃
Fitting, prime radical	D₄²×C₂₃	C₂
socle	C₂²×C₂₃	#120
hypercenter	D₄²	D₂₃
hypocenter	C₂₃	#1629
LCS	G ▷ C₄²×C₂₃ ▷ C₂²×C₂₃ ▷ C₂₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ D₄²

subgroup

G/H

center

C₂²

#10625124

commutator, cosocle

C₄²×C₂₃

C₂³

Frattini

C₄²

C₂²×D₂₃

Fitting, prime radical

D₄²×C₂₃

C₂

socle

C₂²×C₂₃

#120

hypercenter

D₄²

D₂₃

hypocenter

C₂₃

#1629

LCS

G ▷ C₄²×C₂₃ ▷ C₂²×C₂₃ ▷ C₂₃

UCS

C₁ ◁ C₂² ◁ C₄² ◁ D₄²

N	H ≅ G/N	type
1. #13092789	C₂	semidirect
2. #13092707	C₂	semidirect
3. #13092161	C₂	semidirect
4. D₄⋊(C₂₃⋊C₈)	C₂	semidirect
12. #10625040	C₂²	semidirect
15. C₄⋊(C₂₃⋊C₈)	C₂²	semidirect
16. C₄×C₂₃⋊C₈	C₂²	semidirect
43. D₄×C₂₃	SD₁₆	semidirect
86. C₂₃	#1629	semidirect
114. D₄	#67273	semidirect

H ≅ G/N

type

1. #13092789

C₂

semidirect

2. #13092707

C₂

semidirect

3. #13092161

C₂

semidirect

4. D₄⋊(C₂₃⋊C₈)

C₂

semidirect

12. #10625040

C₂²

semidirect

15. C₄⋊(C₂₃⋊C₈)

C₂²

semidirect

16. C₄×C₂₃⋊C₈

C₂²

semidirect

43. D₄×C₂₃

SD₁₆

semidirect

86. C₂₃

#1629

semidirect

114. D₄

#67273

semidirect

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

subgroup	index
#13092789	2
#13092707	2
#13092161	2
D₄⋊(C₂₃⋊C₈)	2
D₄²×C₂₃	2
#1629	23

subgroup

index

#13092789

#13092707

#13092161

D₄⋊(C₂₃⋊C₈)

D₄²×C₂₃

#1629

Subgroups — #15672513

Conjugacy classes of subgroups

Complements

Maximal subgroups