Group #15671292 — subgroups

Subgroups — #15671292

Order 2944.

148 conjugacy classes of subgroups; only the 77 normal subgroups are shown.

normal subgroups only

This group has 148 conjugacy classes of subgroups — too many to draw. Only its 77 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	#13092470	C₂	1	1	yes	maximal, characteristic
2	C₂×C₄⋊(C₂₃⋊C₈)	C₂	1	1	yes	maximal, characteristic
3	C₂₃×#423	C₂	1	1	yes	maximal, characteristic
4	C₂×M₁₆×C₂₃	—	2	1	no	—
5	C₂×C₄²×C₂₃	C₂²	1	1	yes	characteristic
6	C₂²×C₂₃⋊C₈	—	2	1	no	—
7	C₂₃×C₄⋊C₈	—	2	1	no	—
8	C₂₃×C₄⋊C₈	C₄	1	2	yes	—
9	C₂²⋊(C₂₃⋊C₈)	C₄	1	2	yes	—
10	C₄⋊(C₂₃⋊C₈)	—	2	1	no	—
11	C₄⋊(C₂₃⋊C₈)	—	2	1	no	—
12	C₄⋊(C₂₃⋊C₈)	C₄	1	2	yes	—
13	C₄×C₂₃⋊C₈	—	2	1	no	—
14	C₂²×C₄×C₂₃	C₂×C₄	1	1	yes	characteristic
15	C₂²×C₄×C₂₃	D₄	1	1	yes	characteristic
16	C₂²×C₄×C₂₃	Q₈	1	1	yes	characteristic
17	M₁₆×C₂₃	—	4	1	no	—
18	M₁₆×C₂₃	—	4	1	no	—
19	C₂×C₈×C₂₃	—	2	2	no	—
20	C₄²×C₂₃	C₂×C₄	1	1	yes	characteristic
21	C₄²×C₂₃	C₂×C₄	1	1	yes	characteristic
22	C₄²×C₂₃	D₄	1	1	yes	characteristic
23	C₄²×C₂₃	D₄	1	1	yes	characteristic
24	C₂×C₂₃⋊C₈	—	2	2	no	—
25	C₂×C₂₃⋊C₈	—	2	2	no	—
26	C₂×C₂₃⋊C₈	—	4	1	no	—
27	C₂×C₂₃⋊C₈	—	4	1	no	—
28	C₂³×C₂₃	C₄⋊C₄	1	1	yes	characteristic
29	C₂×C₄×C₂₃	—	2	2	no	—
30	C₂×C₄×C₂₃	—	4	1	no	—
31	C₂×C₄×C₂₃	C₄²	1	1	yes	characteristic
32	C₂×C₄×C₂₃	C₂²⋊C₄	1	1	yes	characteristic
33	C₂×C₄×C₂₃	C₂²⋊C₄	1	1	yes	characteristic
34	C₂×C₄×C₂₃	C₂²⋊C₄	1	1	yes	characteristic
35	C₂×C₄×C₂₃	C₄⋊C₄	1	1	yes	characteristic
36	C₂×C₄×C₂₃	C₄⋊C₄	1	1	yes	characteristic
37	C₂×C₄×C₂₃	D₈	1	1	yes	characteristic
38	C₂×C₄×C₂₃	SD₁₆	1	1	yes	characteristic
39	C₂×C₄×C₂₃	SD₁₆	1	1	yes	characteristic
40	C₂×C₄×C₂₃	Q₁₆	1	1	yes	characteristic
41	C₈×C₂₃	—	4	2	no	—
42	C₂₃⋊C₈	—	4	2	no	—
43	C₂₃⋊C₈	—	4	2	no	—
44	#1301	—	23	1	no	maximal, Sylow
45	C₂²×C₂₃	—	2	1	no	—
46	C₂²×C₂₃	—	2	1	no	—
47	C₂²×C₂₃	#95	1	1	yes	characteristic
48	C₂²×C₂₃	#106	1	1	yes	characteristic
49	C₂²×C₂₃	#107	1	1	yes	characteristic
50	C₄×C₂₃	—	2	1	no	—
51	C₄×C₂₃	—	2	1	no	—
52	C₄×C₂₃	—	2	1	no	—
53	C₄×C₂₃	#102	1	1	yes	characteristic
54	C₄×C₂₃	#102	1	1	yes	characteristic
55	C₄×C₂₃	#103	1	1	yes	characteristic
56	C₄×C₂₃	#103	1	1	yes	characteristic
57	C₄×C₂₃	#108	1	2	yes	—
58	#431	—	23	1	no	—
59	#423	D₂₃	1	1	yes	characteristic
60	C₂×C₄⋊C₈	—	23	1	no	—
61	C₂×C₂₃	—	2	1	no	—
62	C₂×C₂₃	—	2	1	no	—
63	C₂×C₂₃	#340	1	1	yes	characteristic
64	C₂×C₂₃	#341	1	1	yes	characteristic
65	C₂×C₂₃	#344	1	1	yes	characteristic
66	C₂×M₁₆	—	2	1	no	—
67	C₂²×C₈	—	46	1	no	—
68	C₂×C₄²	C₂×D₂₃	1	1	yes	characteristic
69	C₄⋊C₈	—	2	1	no	—
70	C₄⋊C₈	—	23	2	no	—
71	C₄⋊C₈	—	46	1	no	—
72	C₄⋊C₈	—	46	1	no	—
73	C₄⋊C₈	Dic₂₃	1	2	yes	—
74	C₂²⋊C₈	—	23	2	no	—
75	C₄×C₈	—	46	1	no	—
76	C₂₃	#1301	1	1	yes	characteristic, complemented, Sylow
77	C₂²×C₄	Dic₄₆	1	1	yes	characteristic
78	C₂²×C₄	C₄×D₂₃	1	1	yes	characteristic
79	C₂²×C₄	C₂²⋊D₂₃	1	1	yes	characteristic
80	M₁₆	—	4	1	no	—
81	M₁₆	—	4	1	no	—
82	C₂×C₈	—	2	2	no	—
83	C₂×C₈	—	46	2	no	—
84	C₂×C₈	—	46	2	no	—
85	C₂×C₈	—	92	1	no	—
86	C₂×C₈	—	92	1	no	—
87	C₄²	C₄×D₂₃	1	1	yes	characteristic
88	C₄²	D₉₂	1	1	yes	characteristic
89	C₄²	C₂×Dic₂₃	1	1	yes	characteristic
90	C₄²	C₂²⋊D₂₃	1	1	yes	characteristic
91	C₂³	Dic₂₃⋊C₄	1	1	yes	characteristic
92	C₂×C₄	—	2	2	no	—
93	C₂×C₄	—	4	1	no	—
94	C₂×C₄	C₄×Dic₂₃	1	1	yes	characteristic
95	C₂×C₄	Dic₂₃⋊C₄	1	1	yes	characteristic
96	C₂×C₄	C₄⋊Dic₂₃	1	1	yes	characteristic
97	C₂×C₄	(C₂×D₂₃)⋊C₄	1	1	yes	characteristic
98	C₂×C₄	(C₂×D₂₃)⋊C₄	1	1	yes	characteristic
99	C₂×C₄	#67272	1	1	yes	characteristic
100	C₂×C₄	#67273	1	1	yes	characteristic
101	C₂×C₄	#67274	1	1	yes	characteristic
102	C₂×C₄	#67275	1	1	yes	characteristic
103	C₂×C₄	C₂²⋊Dic₂₃	1	1	yes	characteristic
104	C₈	—	4	2	no	—
105	C₈	—	92	2	no	—
106	C₈	—	92	2	no	—
107	C₂²	—	2	1	no	—
108	C₂²	—	2	1	no	—
109	C₂²	#10624989	1	1	yes	characteristic
110	C₂²	#10624990	1	1	yes	characteristic
111	C₂²	C₂₃⋊#95	1	1	yes	characteristic
112	C₄	—	2	1	no	—
113	C₄	—	2	1	no	—
114	C₄	—	2	1	no	—
115	C₄	#10624991	1	1	yes	characteristic
116	C₄	#10624992	1	1	yes	characteristic
117	C₄	#10625004	1	2	yes	—
118	C₄	#10625014	1	1	yes	characteristic
119	C₄	#10625017	1	1	yes	characteristic
120	C₂	—	2	1	no	—
121	C₂	—	2	1	no	—
122	C₂	#13092179	1	1	yes	characteristic
123	C₂	#13092181	1	1	yes	characteristic
124	C₂	#13092188	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#13092470

C₂

yes

maximal, characteristic

C₂×C₄⋊(C₂₃⋊C₈)

C₂

yes

maximal, characteristic

C₂₃×#423

C₂

yes

maximal, characteristic

C₂×M₁₆×C₂₃

—

C₂×C₄²×C₂₃

C₂²

yes

characteristic

C₂²×C₂₃⋊C₈

—

C₂₃×C₄⋊C₈

—

C₂₃×C₄⋊C₈

C₄

yes

—

C₂²⋊(C₂₃⋊C₈)

C₄

yes

—

C₄⋊(C₂₃⋊C₈)

—

C₄⋊(C₂₃⋊C₈)

—

C₄⋊(C₂₃⋊C₈)

C₄

yes

—

C₄×C₂₃⋊C₈

—

C₂²×C₄×C₂₃

C₂×C₄

yes

characteristic

C₂²×C₄×C₂₃

D₄

yes

characteristic

C₂²×C₄×C₂₃

Q₈

yes

characteristic

M₁₆×C₂₃

—

M₁₆×C₂₃

—

C₂×C₈×C₂₃

—

C₄²×C₂₃

C₂×C₄

yes

characteristic

C₄²×C₂₃

C₂×C₄

yes

characteristic

C₄²×C₂₃

D₄

yes

characteristic

C₄²×C₂₃

D₄

yes

characteristic

C₂×C₂₃⋊C₈

—

C₂×C₂₃⋊C₈

—

C₂×C₂₃⋊C₈

—

C₂×C₂₃⋊C₈

—

C₂³×C₂₃

C₄⋊C₄

yes

characteristic

C₂×C₄×C₂₃

—

C₂×C₄×C₂₃

—

C₂×C₄×C₂₃

C₄²

yes

characteristic

C₂×C₄×C₂₃

C₂²⋊C₄

yes

characteristic

C₂×C₄×C₂₃

C₂²⋊C₄

yes

characteristic

C₂×C₄×C₂₃

C₂²⋊C₄

yes

characteristic

C₂×C₄×C₂₃

C₄⋊C₄

yes

characteristic

C₂×C₄×C₂₃

C₄⋊C₄

yes

characteristic

C₂×C₄×C₂₃

D₈

yes

characteristic

C₂×C₄×C₂₃

SD₁₆

yes

characteristic

C₂×C₄×C₂₃

SD₁₆

yes

characteristic

C₂×C₄×C₂₃

Q₁₆

yes

characteristic

C₈×C₂₃

—

C₂₃⋊C₈

—

C₂₃⋊C₈

—

#1301

—

maximal, Sylow

C₂²×C₂₃

—

C₂²×C₂₃

—

C₂²×C₂₃

yes

C₂²×C₂₃

yes

C₂²×C₂₃

yes

C₄×C₂₃

—

C₄×C₂₃

—

C₄×C₂₃

—

C₄×C₂₃

yes

C₄×C₂₃

yes

C₄×C₂₃

yes

C₄×C₂₃

yes

C₄×C₂₃

yes

—

#431

—

#423

D₂₃

yes

characteristic

C₂×C₄⋊C₈

—

C₂×C₂₃

—

C₂×C₂₃

—

C₂×C₂₃

yes

C₂×C₂₃

yes

C₂×C₂₃

yes

C₂×M₁₆

—

C₂²×C₈

—

C₂×C₄²

C₂×D₂₃

yes

characteristic

C₄⋊C₈

—

C₄⋊C₈

—

C₄⋊C₈

—

C₄⋊C₈

—

C₄⋊C₈

Dic₂₃

yes

—

C₂²⋊C₈

—

C₄×C₈

—

C₂₃

#1301

yes

characteristic, complemented, Sylow

C₂²×C₄

Dic₄₆

yes

characteristic

C₂²×C₄

C₄×D₂₃

yes

characteristic

C₂²×C₄

C₂²⋊D₂₃

yes

characteristic

M₁₆

—

M₁₆

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₄²

C₄×D₂₃

yes

characteristic

C₄²

D₉₂

yes

characteristic

C₄²

C₂×Dic₂₃

yes

characteristic

C₄²

C₂²⋊D₂₃

yes

characteristic

C₂³

Dic₂₃⋊C₄

yes

characteristic

C₂×C₄

—

C₂×C₄

—

C₂×C₄

C₄×Dic₂₃

yes

characteristic

C₂×C₄

Dic₂₃⋊C₄

yes

characteristic

C₂×C₄

C₄⋊Dic₂₃

yes

characteristic

C₂×C₄

(C₂×D₂₃)⋊C₄

yes

characteristic

C₂×C₄

(C₂×D₂₃)⋊C₄

yes

characteristic

C₂×C₄

#67272

yes

characteristic

100

C₂×C₄

#67273

yes

characteristic

101

C₂×C₄

#67274

yes

characteristic

102

C₂×C₄

#67275

yes

characteristic

103

C₂×C₄

C₂²⋊Dic₂₃

yes

characteristic

104

C₈

—

105

C₈

—

106

C₈

—

107

C₂²

—

108

C₂²

—

109

C₂²

#10624989

yes

characteristic

110

C₂²

#10624990

yes

characteristic

111

C₂²

C₂₃⋊#95

yes

characteristic

112

C₄

—

113

C₄

—

114

C₄

—

115

C₄

#10624991

yes

characteristic

116

C₄

#10624992

yes

characteristic

117

C₄

#10625004

yes

—

118

C₄

#10625014

yes

characteristic

119

C₄

#10625017

yes

characteristic

120

C₂

—

121

C₂

—

122

C₂

#13092179

yes

characteristic

123

C₂

#13092181

yes

characteristic

124

C₂

#13092188

yes

characteristic

	subgroup	G/H
center	C₂×C₄	C₂²⋊Dic₂₃
commutator	C₂×C₄×C₂₃	C₄²
Frattini	C₂×C₄²	C₂×D₂₃
Fitting	C₂₃×#423	C₂
socle	C₂²×C₂₃	#95
prime radical	C₂³×C₂₃	C₄⋊C₄
cosocle	C₂×C₄²×C₂₃	C₂²
hypercenter	#423	D₂₃
hypocenter	C₂₃	#1301
LCS	G ▷ C₂×C₄×C₂₃ ▷ C₂²×C₂₃ ▷ C₂₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #423

subgroup

G/H

center

C₂×C₄

C₂²⋊Dic₂₃

commutator

C₂×C₄×C₂₃

C₄²

Frattini

C₂×C₄²

C₂×D₂₃

Fitting

C₂₃×#423

C₂

socle

C₂²×C₂₃

#95

prime radical

C₂³×C₂₃

C₄⋊C₄

cosocle

C₂×C₄²×C₂₃

C₂²

D₂₃

C₂₃

G ▷ C₂×C₄×C₂₃ ▷ C₂²×C₂₃ ▷ C₂₃

UCS

C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #423

N	H ≅ G/N	type
76. C₂₃	#1301	semidirect

H ≅ G/N

type

76. C₂₃

#1301

semidirect

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

subgroup	index
#13092470	2
C₂×C₄⋊(C₂₃⋊C₈)	2
C₂₃×#423	2
#1301	23

subgroup

index

#13092470

C₂×C₄⋊(C₂₃⋊C₈)

C₂₃×#423

#1301

Subgroups — #15671292

Conjugacy classes of subgroups

Complements

Maximal subgroups