Group #15673509 — subgroups

Subgroups — #15673509

Order 2944.

236 conjugacy classes of subgroups; only the 81 normal subgroups are shown.

normal subgroups only

This group has 236 conjugacy classes of subgroups — too many to draw. Only its 81 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	#13092895	C₂	1	1	yes	maximal, characteristic
2	C₂×#10625018	C₂	1	1	yes	maximal, characteristic
3	#13092189	C₂	1	2	yes	maximal
4	#13092188	C₂	1	2	yes	maximal
5	C₂×C₂₃×#101	C₂	1	1	yes	maximal, characteristic
6	C₂²×Q₈×C₂₃	C₂²	1	1	yes	characteristic
7	C₂×M₁₆×C₂₃	C₂²	1	2	yes	—
8	Q₈×Dic₂₃	—	2	2	no	—
9	Q₈×Dic₂₃	—	2	2	no	—
10	#10625097	C₄	1	4	yes	—
11	#10625097	C₂²	1	1	yes	characteristic
12	#10625097	C₂²	1	1	yes	characteristic
13	C₂×#67267	C₂²	1	2	yes	—
14	C₂₃×#101	—	2	2	no	—
15	#10625018	—	2	2	no	—
16	C₂×Q₈×C₂₃	—	2	2	no	—
17	C₂×Q₈×C₂₃	—	2	2	no	—
18	C₂×Q₈×C₂₃	D₄	1	2	yes	—
19	C₂×Q₈×C₂₃	D₄	1	2	yes	—
20	C₂²×C₄×C₂₃	C₂×C₄	1	2	yes	—
21	C₂²×C₄×C₂₃	C₂³	1	1	yes	characteristic
22	M₁₆×C₂₃	—	2	2	no	—
23	M₁₆×C₂₃	—	2	2	no	—
24	C₂×C₈×C₂₃	—	2	2	no	—
25	C₂²⋊Dic₂₃	C₂×C₄	1	2	yes	—
26	C₂²⋊Dic₂₃	C₂×C₄	1	2	yes	—
27	C₄⋊Dic₂₃	—	2	4	no	—
28	C₄⋊Dic₂₃	—	2	1	no	—
29	C₄⋊Dic₂₃	—	2	1	no	—
30	C₄×Dic₂₃	—	2	4	no	—
31	C₄×Dic₂₃	—	2	1	no	—
32	C₄×Dic₂₃	—	2	1	no	—
33	#67267	—	2	2	no	—
34	#67267	—	2	2	no	—
35	C₂×C₂₃⋊C₈	—	2	2	no	—
36	C₂³×C₂₃	C₂²×C₄	1	1	yes	characteristic
37	Q₈×C₂₃	—	2	4	no	—
38	Q₈×C₂₃	—	2	4	no	—
39	C₂×C₄×C₂₃	—	2	2	no	—
40	C₂×C₄×C₂₃	—	2	2	no	—
41	C₂×C₄×C₂₃	C₂²⋊C₄	1	4	yes	—
42	C₂×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
43	C₂×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
44	C₂×C₄×C₂₃	#41	1	2	yes	—
45	C₂×C₄×C₂₃	#41	1	2	yes	—
46	C₂×Dic₂₃	—	2	2	no	—
47	C₂×Dic₂₃	—	2	2	no	—
48	C₈×C₂₃	—	4	2	no	—
49	C₂₃⋊C₈	—	4	2	no	—
50	#1830	—	23	1	no	maximal, Sylow
51	C₂²×C₂₃	—	4	1	no	—
52	C₂²×C₂₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
53	C₂²×C₂₃	#117	1	1	yes	characteristic
54	C₂²×C₂₃	#117	1	1	yes	characteristic
55	C₄×C₂₃	—	2	4	no	—
56	C₄×C₂₃	#123	1	2	yes	—
57	C₄×C₂₃	#123	1	2	yes	—
58	Dic₂₃	—	4	2	no	—
59	Dic₂₃	—	4	2	no	—
60	#519	—	23	1	no	—
61	C₂×#101	—	23	1	no	—
62	C₂×#101	D₂₃	1	1	yes	characteristic
63	#344	—	23	2	no	—
64	#344	—	23	2	no	—
65	C₂×C₂₃	—	2	1	no	—
66	C₂×C₂₃	—	2	1	no	—
67	C₂×C₂₃	—	2	1	no	—
68	C₂×C₂₃	#381	1	1	yes	characteristic
69	C₂²×Q₈	C₂×D₂₃	1	1	yes	characteristic
70	C₂×M₁₆	—	23	2	no	—
71	C₂×M₁₆	C₂×D₂₃	1	2	yes	—
72	C₄×Q₈	—	46	2	no	—
73	C₄×Q₈	—	46	2	no	—
74	#117	—	23	4	no	—
75	#117	—	23	1	no	—
76	#117	—	23	1	no	—
77	#101	—	2	2	no	—
78	#101	—	46	2	no	—
79	C₂₃	#1830	1	1	yes	characteristic, complemented, Sylow
80	C₂×Q₈	—	2	2	no	—
81	C₂×Q₈	—	2	2	no	—
82	C₂×Q₈	D₉₂	1	2	yes	—
83	C₂×Q₈	C₂²⋊D₂₃	1	2	yes	—
84	C₂²×C₄	C₄×D₂₃	1	2	yes	—
85	C₂²×C₄	C₂²×D₂₃	1	1	yes	characteristic
86	M₁₆	—	2	2	no	—
87	M₁₆	—	2	2	no	—
88	M₁₆	—	46	2	no	—
89	M₁₆	—	46	2	no	—
90	C₂×C₈	—	2	2	no	—
91	C₂×C₈	—	46	2	no	—
92	C₄⋊C₄	—	46	4	no	—
93	C₄⋊C₄	—	46	1	no	—
94	C₄⋊C₄	—	46	1	no	—
95	C₂²⋊C₄	—	23	2	no	—
96	C₂²⋊C₄	—	23	2	no	—
97	C₄²	—	46	4	no	—
98	C₄²	—	46	1	no	—
99	C₄²	—	46	1	no	—
100	C₂³	C₂×C₄×D₂₃	1	1	yes	characteristic
101	Q₈	—	2	4	no	—
102	Q₈	—	2	4	no	—
103	C₂×C₄	—	2	2	no	—
104	C₂×C₄	—	2	2	no	—
105	C₂×C₄	—	46	2	no	—
106	C₂×C₄	—	46	2	no	—
107	C₂×C₄	(C₂×D₂₃)⋊C₄	1	4	yes	—
108	C₂×C₄	C₂×D₉₂	1	1	yes	characteristic
109	C₂×C₄	#67290	1	2	yes	—
110	C₂×C₄	#67290	1	2	yes	—
111	C₂×C₄	C₂×C₂²⋊D₂₃	1	1	yes	characteristic
112	C₈	—	4	2	no	—
113	C₈	—	92	2	no	—
114	C₂²	—	4	1	no	—
115	C₂²	#10625048	1	1	yes	characteristic
116	C₂²	#10625048	1	1	yes	characteristic
117	C₂²	C₂×(C₂×D₂₃)⋊C₄	1	1	yes	characteristic
118	C₄	—	2	4	no	—
119	C₄	—	92	2	no	—
120	C₄	—	92	2	no	—
121	C₄	#10625057	1	2	yes	—
122	C₄	#10625106	1	2	yes	—
123	C₂	—	2	1	no	—
124	C₂	—	2	1	no	—
125	C₂	—	2	1	no	—
126	C₂	#13092278	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#13092895

C₂

yes

maximal, characteristic

C₂×#10625018

C₂

yes

maximal, characteristic

#13092189

C₂

yes

maximal

#13092188

C₂

yes

maximal

C₂×C₂₃×#101

C₂

yes

maximal, characteristic

C₂²×Q₈×C₂₃

C₂²

yes

characteristic

C₂×M₁₆×C₂₃

C₂²

yes

—

Q₈×Dic₂₃

—

Q₈×Dic₂₃

—

#10625097

C₄

yes

—

#10625097

C₂²

yes

characteristic

#10625097

C₂²

yes

characteristic

C₂×#67267

C₂²

yes

—

C₂₃×#101

—

#10625018

—

C₂×Q₈×C₂₃

—

C₂×Q₈×C₂₃

—

C₂×Q₈×C₂₃

D₄

yes

—

C₂×Q₈×C₂₃

D₄

yes

—

C₂²×C₄×C₂₃

C₂×C₄

yes

—

C₂²×C₄×C₂₃

C₂³

yes

characteristic

M₁₆×C₂₃

—

M₁₆×C₂₃

—

C₂×C₈×C₂₃

—

C₂²⋊Dic₂₃

C₂×C₄

yes

—

C₂²⋊Dic₂₃

C₂×C₄

yes

—

C₄⋊Dic₂₃

—

C₄⋊Dic₂₃

—

C₄⋊Dic₂₃

—

C₄×Dic₂₃

—

C₄×Dic₂₃

—

C₄×Dic₂₃

—

#67267

—

#67267

—

C₂×C₂₃⋊C₈

—

C₂³×C₂₃

C₂²×C₄

yes

characteristic

Q₈×C₂₃

—

Q₈×C₂₃

—

C₂×C₄×C₂₃

—

C₂×C₄×C₂₃

—

C₂×C₄×C₂₃

C₂²⋊C₄

yes

—

C₂×C₄×C₂₃

C₂×D₄

yes

characteristic

C₂×C₄×C₂₃

C₂×D₄

yes

characteristic

C₂×C₄×C₂₃

#41

yes

—

C₂×C₄×C₂₃

#41

yes

—

C₂×Dic₂₃

—

C₂×Dic₂₃

—

C₈×C₂₃

—

C₂₃⋊C₈

—

#1830

—

maximal, Sylow

C₂²×C₂₃

—

C₂²×C₂₃

C₂×C₂²⋊C₄

yes

C₂²×C₂₃

yes

C₂²×C₂₃

yes

C₄×C₂₃

—

C₄×C₂₃

#123

yes

—

C₄×C₂₃

#123

yes

—

Dic₂₃

—

Dic₂₃

—

#519

—

C₂×#101

—

C₂×#101

D₂₃

yes

characteristic

#344

—

#344

—

C₂×C₂₃

—

C₂×C₂₃

—

C₂×C₂₃

—

C₂×C₂₃

#381

yes

characteristic

C₂²×Q₈

C₂×D₂₃

yes

characteristic

C₂×M₁₆

—

C₂×M₁₆

C₂×D₂₃

yes

—

C₄×Q₈

—

C₄×Q₈

—

—

—

—

—

—

C₂₃

#1830

yes

characteristic, complemented, Sylow

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

D₉₂

yes

—

C₂×Q₈

C₂²⋊D₂₃

yes

—

C₂²×C₄

C₄×D₂₃

yes

—

C₂²×C₄

C₂²×D₂₃

yes

characteristic

M₁₆

—

M₁₆

—

M₁₆

—

M₁₆

—

C₂×C₈

—

C₂×C₈

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₂²⋊C₄

—

C₂²⋊C₄

—

C₄²

—

C₄²

—

C₄²

—

100

C₂³

C₂×C₄×D₂₃

yes

characteristic

101

Q₈

—

102

Q₈

—

103

C₂×C₄

—

104

C₂×C₄

—

105

C₂×C₄

—

106

C₂×C₄

—

107

C₂×C₄

(C₂×D₂₃)⋊C₄

yes

—

108

C₂×C₄

C₂×D₉₂

yes

characteristic

109

C₂×C₄

#67290

yes

—

110

C₂×C₄

#67290

yes

—

111

C₂×C₄

C₂×C₂²⋊D₂₃

yes

characteristic

112

C₈

—

113

C₈

—

114

C₂²

—

115

C₂²

#10625048

yes

characteristic

116

C₂²

#10625048

yes

characteristic

117

C₂²

C₂×(C₂×D₂₃)⋊C₄

yes

characteristic

118

C₄

—

119

C₄

—

120

C₄

—

121

C₄

#10625057

yes

—

122

C₄

#10625106

yes

—

123

C₂

—

124

C₂

—

125

C₂

—

126

C₂

#13092278

yes

characteristic

	subgroup	G/H
center	C₂	#13092278
commutator, prime radical	C₂³×C₂₃	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₂₃
Fitting	C₂×C₂₃×#101	C₂
socle	C₂×C₂₃	#381
cosocle	C₂²×C₄×C₂₃	C₂³
hypercenter	C₂×#101	D₂₃
hypocenter	C₂₃	#1830
LCS	G ▷ C₂³×C₂₃ ▷ C₂×C₂₃ ▷ C₂₃
UCS	C₁ ◁ C₂ ◁ C₂²×C₄ ◁ C₂×#101

subgroup

G/H

center

C₂

#13092278

commutator, prime radical

C₂³×C₂₃

C₂²×C₄

Frattini

C₂²×C₄

C₂²×D₂₃

Fitting

C₂×C₂₃×#101

C₂

socle

C₂×C₂₃

#381

cosocle

C₂²×C₄×C₂₃

C₂³

D₂₃

C₂₃

G ▷ C₂³×C₂₃ ▷ C₂×C₂₃ ▷ C₂₃

UCS

C₁ ◁ C₂ ◁ C₂²×C₄ ◁ C₂×#101

N	H ≅ G/N	type
79. C₂₃	#1830	semidirect

H ≅ G/N

type

79. C₂₃

#1830

semidirect

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

subgroup	index
#13092895	2
C₂×#10625018	2
#13092189	2
#13092188	2
C₂×C₂₃×#101	2
#1830	23

subgroup

index

C₂×C₂₃×#101

Subgroups — #15673509

Conjugacy classes of subgroups

Complements

Maximal subgroups