Group #15672132 — subgroups

Subgroups — #15672132

Order 2944.

254 conjugacy classes of subgroups; only the 85 normal subgroups are shown.

normal subgroups only

This group has 254 conjugacy classes of subgroups — too many to draw. Only its 85 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	#13092441	C₂	1	2	yes	maximal, complemented
2	#13092166	C₂	1	2	yes	maximal, complemented
3	#13092161	C₂	1	2	yes	maximal, complemented
4	C₂₃×#532	C₂	1	1	yes	maximal, characteristic
5	C₂×#41×C₂₃	C₄	1	2	yes	—
6	C₂₃×C₄⋊Q₈	C₂²	1	1	yes	characteristic
7	C₂₃×#127	C₂²	1	1	yes	characteristic
8	C₂₃×#124	C₄	1	2	yes	—
9	C₂₃×#121	—	2	2	no	—
10	C₄×D₄×C₂₃	—	2	2	no	—
11	C₂×C₄²×C₂₃	C₂²	1	1	yes	characteristic
12	C₂×#67267	—	2	2	no	—
13	C₄⋊(C₂₃⋊C₈)	—	2	2	no	—
14	C₄⋊(C₂₃⋊C₈)	C₂²	1	4	yes	—
15	#41×C₂₃	—	4	4	no	—
16	C₂×Q₈×C₂₃	C₂×C₄	1	2	yes	—
17	C₂×D₄×C₂₃	—	2	2	no	—
18	C₂×D₄×C₂₃	—	2	2	no	—
19	C₂×D₄×C₂₃	C₂×C₄	1	2	yes	—
20	C₂²×C₄×C₂₃	—	2	2	no	—
21	C₂²×C₄×C₂₃	C₂×C₄	1	1	yes	characteristic
22	C₂²×C₄×C₂₃	D₄	1	2	yes	—
23	C₄⋊C₄×C₂₃	—	2	2	no	—
24	C₂²⋊C₄×C₂₃	—	2	4	no	—
25	C₄²×C₂₃	C₂×C₄	1	1	yes	characteristic
26	C₄²×C₂₃	D₄	1	2	yes	—
27	C₄²×C₂₃	C₂³	1	1	yes	characteristic
28	#67267	—	4	4	no	—
29	C₂×C₂₃⋊C₈	—	2	4	no	—
30	C₂³×C₂₃	—	2	2	no	—
31	C₂³×C₂₃	C₂²⋊C₄	1	1	yes	characteristic
32	Q₈×C₂₃	—	4	2	no	—
33	D₄×C₂₃	—	4	2	no	—
34	D₄×C₂₃	—	4	4	no	—
35	D₄×C₂₃	—	4	2	no	—
36	D₄×C₂₃	—	4	2	no	—
37	C₂×C₄×C₂₃	—	2	2	no	—
38	C₂×C₄×C₂₃	—	2	2	no	—
39	C₂×C₄×C₂₃	—	2	2	no	—
40	C₂×C₄×C₂₃	—	2	2	no	—
41	C₂×C₄×C₂₃	—	4	4	no	—
42	C₂×C₄×C₂₃	C₂²⋊C₄	1	2	yes	—
43	C₂×C₄×C₂₃	C₂²⋊C₄	1	1	yes	characteristic
44	C₂×C₄×C₂₃	C₂²×C₄	1	1	yes	characteristic
45	C₂×C₄×C₂₃	C₂×D₄	1	2	yes	—
46	C₂₃⋊C₈	—	4	4	no	—
47	#1560	—	23	1	no	maximal, Sylow
48	C₂²×C₂₃	—	2	1	no	—
49	C₂²×C₂₃	—	2	2	no	—
50	C₂²×C₂₃	—	4	2	no	—
51	C₂²×C₂₃	—	4	4	no	—
52	C₂²×C₂₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
53	C₄×C₂₃	—	2	1	no	—
54	C₄×C₂₃	—	2	2	no	—
55	C₄×C₂₃	—	4	2	no	—
56	C₄×C₂₃	#100	1	2	yes	—
57	C₄×C₂₃	AGL(1,ℤ/8)	1	2	yes	—
58	C₄×C₂₃	#137	1	2	yes	—
59	#532	D₂₃	1	1	yes	characteristic
60	#423	—	23	2	no	—
61	#333	—	23	2	no	—
62	#331	—	23	2	no	—
63	C₂×C₂₃	—	4	1	no	—
64	C₂×C₂₃	—	8	2	no	—
65	C₂×C₂₃	C₂×#100	1	1	yes	characteristic
66	C₂×C₂₃	#417	1	2	yes	—
67	C₂×#41	Dic₂₃	1	2	yes	—
68	C₂×M₁₆	—	46	2	no	—
69	C₄⋊Q₈	C₂×D₂₃	1	1	yes	characteristic
70	#127	C₂×D₂₃	1	1	yes	characteristic
71	#124	Dic₂₃	1	2	yes	—
72	#121	—	2	2	no	—
73	C₄×D₄	—	2	2	no	—
74	C₂×C₄²	C₂×D₂₃	1	1	yes	characteristic
75	C₄⋊C₈	—	23	4	no	—
76	C₄⋊C₈	—	46	2	no	—
77	C₂₃	#1560	1	1	yes	characteristic, complemented, Sylow
78	#41	—	4	4	no	—
79	C₂×Q₈	C₂×Dic₂₃	1	2	yes	—
80	C₂×D₄	—	2	2	no	—
81	C₂×D₄	—	2	2	no	—
82	C₂×D₄	C₂×Dic₂₃	1	2	yes	—
83	C₂²×C₄	—	2	2	no	—
84	C₂²×C₄	C₂×Dic₂₃	1	1	yes	characteristic
85	C₂²×C₄	C₂²⋊D₂₃	1	2	yes	—
86	M₁₆	—	92	4	no	—
87	C₂×C₈	—	46	4	no	—
88	C₄⋊C₄	—	2	2	no	—
89	C₂²⋊C₄	—	2	4	no	—
90	C₄²	C₂×Dic₂₃	1	1	yes	characteristic
91	C₄²	C₂²⋊D₂₃	1	2	yes	—
92	C₄²	C₂²×D₂₃	1	1	yes	characteristic
93	C₂³	—	2	2	no	—
94	C₂³	C₂²⋊Dic₂₃	1	1	yes	characteristic
95	Q₈	—	4	2	no	—
96	D₄	—	4	2	no	—
97	D₄	—	4	4	no	—
98	D₄	—	4	2	no	—
99	D₄	—	4	2	no	—
100	C₂×C₄	—	2	2	no	—
101	C₂×C₄	—	2	2	no	—
102	C₂×C₄	—	2	2	no	—
103	C₂×C₄	—	2	2	no	—
104	C₂×C₄	—	4	4	no	—
105	C₂×C₄	C₂²⋊Dic₂₃	1	2	yes	—
106	C₂×C₄	C₂²⋊Dic₂₃	1	1	yes	characteristic
107	C₂×C₄	C₂²×Dic₂₃	1	1	yes	characteristic
108	C₂×C₄	C₂×C₂²⋊D₂₃	1	2	yes	—
109	C₈	—	92	4	no	—
110	C₂²	—	2	1	no	—
111	C₂²	—	2	2	no	—
112	C₂²	—	4	2	no	—
113	C₂²	—	4	4	no	—
114	C₂²	C₂×C₂²⋊Dic₂₃	1	1	yes	characteristic
115	C₄	—	2	1	no	—
116	C₄	—	2	2	no	—
117	C₄	—	4	2	no	—
118	C₄	#10625015	1	2	yes	—
119	C₄	#10625120	1	2	yes	—
120	C₄	#10625122	1	2	yes	—
121	C₂	—	4	1	no	—
122	C₂	—	8	2	no	—
123	C₂	C₂×#10625015	1	1	yes	characteristic
124	C₂	#13092418	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#13092441

C₂

yes

maximal, complemented

#13092166

C₂

yes

maximal, complemented

#13092161

C₂

yes

maximal, complemented

C₂₃×#532

C₂

yes

maximal, characteristic

C₂×#41×C₂₃

C₄

yes

—

C₂₃×C₄⋊Q₈

C₂²

yes

characteristic

C₂₃×#127

C₂²

yes

characteristic

C₂₃×#124

C₄

yes

—

C₂₃×#121

—

C₄×D₄×C₂₃

—

C₂×C₄²×C₂₃

C₂²

yes

characteristic

C₂×#67267

—

C₄⋊(C₂₃⋊C₈)

—

C₄⋊(C₂₃⋊C₈)

C₂²

yes

—

#41×C₂₃

—

C₂×Q₈×C₂₃

C₂×C₄

yes

—

C₂×D₄×C₂₃

—

C₂×D₄×C₂₃

—

C₂×D₄×C₂₃

C₂×C₄

yes

—

C₂²×C₄×C₂₃

—

C₂²×C₄×C₂₃

C₂×C₄

yes

characteristic

C₂²×C₄×C₂₃

D₄

yes

—

C₄⋊C₄×C₂₃

—

C₂²⋊C₄×C₂₃

—

C₄²×C₂₃

C₂×C₄

yes

characteristic

C₄²×C₂₃

D₄

yes

—

C₄²×C₂₃

C₂³

yes

characteristic

#67267

—

C₂×C₂₃⋊C₈

—

C₂³×C₂₃

—

C₂³×C₂₃

C₂²⋊C₄

yes

characteristic

Q₈×C₂₃

—

D₄×C₂₃

—

D₄×C₂₃

—

D₄×C₂₃

—

D₄×C₂₃

—

C₂×C₄×C₂₃

—

C₂×C₄×C₂₃

—

C₂×C₄×C₂₃

—

C₂×C₄×C₂₃

—

C₂×C₄×C₂₃

—

C₂×C₄×C₂₃

C₂²⋊C₄

yes

—

C₂×C₄×C₂₃

C₂²⋊C₄

yes

characteristic

C₂×C₄×C₂₃

C₂²×C₄

yes

characteristic

C₂×C₄×C₂₃

C₂×D₄

yes

—

C₂₃⋊C₈

—

#1560

—

maximal, Sylow

C₂²×C₂₃

—

C₂²×C₂₃

—

C₂²×C₂₃

—

C₂²×C₂₃

—

C₂²×C₂₃

C₂×C₂²⋊C₄

yes

characteristic

C₄×C₂₃

—

C₄×C₂₃

—

C₄×C₂₃

—

C₄×C₂₃

#100

yes

—

C₄×C₂₃

AGL(1,ℤ/8)

yes

—

C₄×C₂₃

#137

yes

—

#532

D₂₃

yes

—

—

—

C₂×C₂₃

—

C₂×C₂₃

—

C₂×C₂₃

C₂×#100

yes

characteristic

C₂×C₂₃

#417

yes

—

C₂×#41

Dic₂₃

yes

—

C₂×M₁₆

—

C₄⋊Q₈

C₂×D₂₃

yes

characteristic

#127

C₂×D₂₃

yes

characteristic

#124

Dic₂₃

yes

—

#121

—

C₄×D₄

—

C₂×C₄²

C₂×D₂₃

yes

characteristic

C₄⋊C₈

—

C₄⋊C₈

—

C₂₃

#1560

yes

characteristic, complemented, Sylow

#41

—

C₂×Q₈

C₂×Dic₂₃

yes

—

C₂×D₄

—

C₂×D₄

—

C₂×D₄

C₂×Dic₂₃

yes

—

C₂²×C₄

—

C₂²×C₄

C₂×Dic₂₃

yes

characteristic

C₂²×C₄

C₂²⋊D₂₃

yes

—

M₁₆

—

C₂×C₈

—

C₄⋊C₄

—

C₂²⋊C₄

—

C₄²

C₂×Dic₂₃

yes

characteristic

C₄²

C₂²⋊D₂₃

yes

—

C₄²

C₂²×D₂₃

yes

characteristic

C₂³

—

C₂³

C₂²⋊Dic₂₃

yes

characteristic

Q₈

—

D₄

—

D₄

—

D₄

—

D₄

—

100

C₂×C₄

—

101

C₂×C₄

—

102

C₂×C₄

—

103

C₂×C₄

—

104

C₂×C₄

—

105

C₂×C₄

C₂²⋊Dic₂₃

yes

—

106

C₂×C₄

C₂²⋊Dic₂₃

yes

characteristic

107

C₂×C₄

C₂²×Dic₂₃

yes

characteristic

108

C₂×C₄

C₂×C₂²⋊D₂₃

yes

—

109

C₈

—

110

C₂²

—

111

C₂²

—

112

C₂²

—

113

C₂²

—

114

C₂²

C₂×C₂²⋊Dic₂₃

yes

characteristic

115

C₄

—

116

C₄

—

117

C₄

—

118

C₄

#10625015

yes

—

119

C₄

#10625120

yes

—

120

C₄

#10625122

yes

—

121

C₂

—

122

C₂

—

123

C₂

C₂×#10625015

yes

characteristic

124

C₂

#13092418

yes

—

	subgroup	G/H
center	C₂²	C₂×C₂²⋊Dic₂₃
commutator	C₂×C₄×C₂₃	C₂²×C₄
Frattini	C₄²	C₂²×D₂₃
Fitting, prime radical	C₂₃×#532	C₂
socle	C₂²×C₂₃	C₂×C₂²⋊C₄
cosocle	C₄²×C₂₃	C₂³
hypercenter	#532	D₂₃
hypocenter	C₂₃	#1560
LCS	G ▷ C₂×C₄×C₂₃ ▷ C₂²×C₂₃ ▷ C₂₃
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #532

subgroup

G/H

center

C₂²

C₂×C₂²⋊Dic₂₃

commutator

C₂×C₄×C₂₃

C₂²×C₄

Frattini

C₄²

C₂²×D₂₃

Fitting, prime radical

C₂₃×#532

C₂

socle

C₂²×C₂₃

C₂×C₂²⋊C₄

cosocle

C₄²×C₂₃

C₂³

D₂₃

C₂₃

G ▷ C₂×C₄×C₂₃ ▷ C₂²×C₂₃ ▷ C₂₃

UCS

C₁ ◁ C₂² ◁ C₂×C₄² ◁ #532

N	H ≅ G/N	type
1. #13092441	C₂	semidirect
2. #13092166	C₂	semidirect
3. #13092161	C₂	semidirect
77. C₂₃	#1560	semidirect

H ≅ G/N

type

1. #13092441

C₂

semidirect

2. #13092166

C₂

semidirect

3. #13092161

C₂

semidirect

77. C₂₃

#1560

semidirect

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

subgroup	index
#13092441	2
#13092166	2
#13092161	2
C₂₃×#532	2
#1560	23

subgroup

index

C₂₃×#532

Subgroups — #15672132

Conjugacy classes of subgroups

Complements

Maximal subgroups