Group #15672152 — subgroups

Subgroups — #15672152

Order 2944.

184 conjugacy classes of subgroups; only the 97 normal subgroups are shown.

normal subgroups only

This group has 184 conjugacy classes of subgroups — too many to draw. Only its 97 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	1	yes	maximal, characteristic
2	C₂×C₄×C₂₃⋊C₈	C₂	1	1	yes	maximal, characteristic
3	#13092167	C₂	1	4	yes	maximal, complemented
4	C₂₃×#423	C₂	1	1	yes	maximal, characteristic
5	C₂×M₁₆×C₂₃	—	2	1	no	—
6	C₂×C₄²×C₂₃	C₂²	1	1	yes	characteristic
7	C₂×#67267	—	2	1	no	—
8	C₂²×C₂₃⋊C₈	C₄	1	2	yes	—
9	C₂₃×C₄⋊C₈	—	2	1	no	—
10	C₂₃×C₄⋊C₈	C₂²	1	2	yes	—
11	C₄⋊(C₂₃⋊C₈)	—	2	1	no	—
12	C₄⋊(C₂₃⋊C₈)	C₂²	1	2	yes	—
13	C₄×C₂₃⋊C₈	C₄	1	1	yes	characteristic
14	C₄×C₂₃⋊C₈	C₄	1	1	yes	characteristic
15	C₄×C₂₃⋊C₈	C₂²	1	2	yes	—
16	C₂²×C₄×C₂₃	C₂×C₄	1	1	yes	characteristic
17	C₂²×C₄×C₂₃	D₄	1	1	yes	characteristic
18	C₂²×C₄×C₂₃	Q₈	1	1	yes	characteristic
19	M₁₆×C₂₃	—	4	2	no	—
20	C₂×C₈×C₂₃	—	2	2	no	—
21	C₄²×C₂₃	C₂×C₄	1	1	yes	characteristic
22	C₄²×C₂₃	D₄	1	1	yes	characteristic
23	C₄²×C₂₃	Q₈	1	1	yes	characteristic
24	C₄²×C₂₃	C₂³	1	1	yes	characteristic
25	#67267	—	4	2	no	—
26	C₂×C₂₃⋊C₈	—	2	4	no	—
27	C₂×C₂₃⋊C₈	—	2	2	no	—
28	C₂×C₂₃⋊C₈	C₂×C₄	1	2	yes	—
29	C₂×C₂₃⋊C₈	C₂×C₄	1	2	yes	—
30	C₂³×C₂₃	C₄⋊C₄	1	1	yes	characteristic
31	C₂×C₄×C₂₃	—	2	2	no	—
32	C₂×C₄×C₂₃	—	2	2	no	—
33	C₂×C₄×C₂₃	C₄⋊C₄	1	1	yes	characteristic
34	C₂×C₄×C₂₃	C₄⋊C₄	1	1	yes	characteristic
35	C₂×C₄×C₂₃	C₄⋊C₄	1	1	yes	characteristic
36	C₂×C₄×C₂₃	SD₁₆	1	2	yes	—
37	C₂×C₄×C₂₃	SD₁₆	1	2	yes	—
38	C₂×C₄×C₂₃	C₂²×C₄	1	1	yes	characteristic
39	C₂×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
40	C₂×C₄×C₂₃	C₂×Q₈	1	1	yes	characteristic
41	C₈×C₂₃	—	4	2	no	—
42	C₂₃⋊C₈	—	2	2	no	—
43	C₂₃⋊C₈	—	4	2	no	—
44	C₂₃⋊C₈	M₁₆	1	4	yes	complemented
45	#1566	—	23	1	no	maximal, Sylow
46	C₂²×C₂₃	—	2	1	no	—
47	C₂²×C₂₃	—	2	1	no	—
48	C₂²×C₂₃	#106	1	2	yes	—
49	C₂²×C₂₃	C₂×C₄⋊C₄	1	1	yes	characteristic
50	C₄×C₂₃	—	2	1	no	—
51	C₄×C₂₃	—	2	1	no	—
52	C₄×C₂₃	#106	1	2	yes	—
53	C₄×C₂₃	#108	1	2	yes	—
54	C₄×C₂₃	C₂×M₁₆	1	2	yes	—
55	C₄×C₂₃	C₂×SD₁₆	1	1	yes	characteristic
56	C₄×C₂₃	C₂×SD₁₆	1	1	yes	characteristic
57	#423	—	23	1	no	—
58	#423	D₂₃	1	1	yes	characteristic
59	C₂×C₄×C₈	—	23	1	no	—
60	#334	—	23	4	no	—
61	C₂×C₂₃	—	2	2	no	—
62	C₂×C₂₃	#423	1	1	yes	characteristic
63	C₂×C₂₃	C₂×#106	1	1	yes	characteristic
64	C₂×C₂₃	C₂×#108	1	1	yes	characteristic
65	C₂×M₁₆	—	2	1	no	—
66	C₂×M₁₆	—	46	1	no	—
67	C₂²×C₈	—	23	2	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	C₂×D₂₃	1	2	yes	—
73	C₄×C₈	—	23	1	no	—
74	C₄×C₈	—	23	1	no	—
75	C₄×C₈	—	23	2	no	—
76	C₂₃	#1566	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	2	no	—
81	M₁₆	—	92	2	no	—
82	C₂×C₈	—	2	2	no	—
83	C₂×C₈	—	23	2	no	—
84	C₂×C₈	—	23	2	no	—
85	C₂×C₈	—	46	4	no	—
86	C₂×C₈	—	46	2	no	—
87	C₄²	Dic₄₆	1	1	yes	characteristic
88	C₄²	C₄×D₂₃	1	1	yes	characteristic
89	C₄²	C₂²⋊D₂₃	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	—	2	2	no	—
94	C₂×C₄	Dic₂₃⋊C₄	1	1	yes	characteristic
95	C₂×C₄	Dic₂₃⋊C₄	1	1	yes	characteristic
96	C₂×C₄	Dic₂₃⋊C₄	1	1	yes	characteristic
97	C₂×C₄	#67273	1	2	yes	—
98	C₂×C₄	#67274	1	2	yes	—
99	C₂×C₄	C₂×Dic₄₆	1	1	yes	characteristic
100	C₂×C₄	C₂×C₄×D₂₃	1	1	yes	characteristic
101	C₂×C₄	C₂×C₂²⋊D₂₃	1	1	yes	characteristic
102	C₈	—	4	2	no	—
103	C₈	—	23	4	no	—
104	C₈	—	46	2	no	—
105	C₈	—	92	2	no	—
106	C₂²	—	2	1	no	—
107	C₂²	—	2	1	no	—
108	C₂²	#10624990	1	2	yes	—
109	C₂²	C₂×Dic₂₃⋊C₄	1	1	yes	characteristic
110	C₄	—	2	1	no	—
111	C₄	—	2	1	no	—
112	C₄	#10624990	1	2	yes	—
113	C₄	#10625004	1	2	yes	—
114	C₄	M₁₆×D₂₃	1	2	yes	—
115	C₄	C₂×#67273	1	1	yes	characteristic
116	C₄	C₂×#67274	1	1	yes	characteristic
117	C₂	—	2	2	no	—
118	C₂	#13092440	1	1	yes	characteristic
119	C₂	C₂×#10624990	1	1	yes	characteristic
120	C₂	C₂×#10625004	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#13092441

C₂

yes

maximal, characteristic

C₂×C₄×C₂₃⋊C₈

C₂

yes

maximal, characteristic

#13092167

C₂

yes

maximal, complemented

C₂₃×#423

C₂

yes

maximal, characteristic

C₂×M₁₆×C₂₃

—

C₂×C₄²×C₂₃

C₂²

yes

characteristic

C₂×#67267

—

C₂²×C₂₃⋊C₈

C₄

yes

—

C₂₃×C₄⋊C₈

—

C₂₃×C₄⋊C₈

C₂²

yes

—

C₄⋊(C₂₃⋊C₈)

—

C₄⋊(C₂₃⋊C₈)

C₂²

yes

—

C₄×C₂₃⋊C₈

C₄

yes

characteristic

C₄×C₂₃⋊C₈

C₄

yes

characteristic

C₄×C₂₃⋊C₈

C₂²

yes

—

C₂²×C₄×C₂₃

C₂×C₄

yes

characteristic

C₂²×C₄×C₂₃

D₄

yes

characteristic

C₂²×C₄×C₂₃

Q₈

yes

characteristic

M₁₆×C₂₃

—

C₂×C₈×C₂₃

—

C₄²×C₂₃

C₂×C₄

yes

characteristic

C₄²×C₂₃

D₄

yes

characteristic

C₄²×C₂₃

Q₈

yes

characteristic

C₄²×C₂₃

C₂³

yes

characteristic

#67267

—

C₂×C₂₃⋊C₈

—

C₂×C₂₃⋊C₈

—

C₂×C₂₃⋊C₈

C₂×C₄

yes

—

C₂×C₂₃⋊C₈

C₂×C₄

yes

—

C₂³×C₂₃

C₄⋊C₄

yes

characteristic

C₂×C₄×C₂₃

—

C₂×C₄×C₂₃

—

C₂×C₄×C₂₃

C₄⋊C₄

yes

characteristic

C₂×C₄×C₂₃

C₄⋊C₄

yes

characteristic

C₂×C₄×C₂₃

C₄⋊C₄

yes

characteristic

C₂×C₄×C₂₃

SD₁₆

yes

—

C₂×C₄×C₂₃

SD₁₆

yes

—

C₂×C₄×C₂₃

C₂²×C₄

yes

characteristic

C₂×C₄×C₂₃

C₂×D₄

yes

characteristic

C₂×C₄×C₂₃

C₂×Q₈

yes

characteristic

C₈×C₂₃

—

C₂₃⋊C₈

—

C₂₃⋊C₈

—

C₂₃⋊C₈

M₁₆

yes

complemented

#1566

—

maximal, Sylow

C₂²×C₂₃

—

C₂²×C₂₃

—

C₂²×C₂₃

#106

yes

—

C₂²×C₂₃

C₂×C₄⋊C₄

yes

characteristic

C₄×C₂₃

—

C₄×C₂₃

—

C₄×C₂₃

#106

yes

—

C₄×C₂₃

#108

yes

—

C₄×C₂₃

C₂×M₁₆

yes

—

C₄×C₂₃

C₂×SD₁₆

yes

characteristic

C₄×C₂₃

C₂×SD₁₆

yes

characteristic

#423

—

#423

D₂₃

yes

characteristic

C₂×C₄×C₈

—

#334

—

C₂×C₂₃

—

C₂×C₂₃

yes

C₂×C₂₃

yes

C₂×C₂₃

yes

C₂×M₁₆

—

C₂×M₁₆

—

C₂²×C₈

—

C₂×C₄²

C₂×D₂₃

yes

characteristic

C₄⋊C₈

—

C₄⋊C₈

—

C₄⋊C₈

—

C₄⋊C₈

C₂×D₂₃

yes

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

—

C₂₃

#1566

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

Dic₄₆

yes

characteristic

C₄²

C₄×D₂₃

yes

characteristic

C₄²

C₂²⋊D₂₃

yes

characteristic

C₄²

C₂²×D₂₃

yes

characteristic

C₂³

Dic₂₃⋊C₄

yes

characteristic

C₂×C₄

—

C₂×C₄

—

C₂×C₄

Dic₂₃⋊C₄

yes

characteristic

C₂×C₄

Dic₂₃⋊C₄

yes

characteristic

C₂×C₄

Dic₂₃⋊C₄

yes

characteristic

C₂×C₄

#67273

yes

—

C₂×C₄

#67274

yes

—

C₂×C₄

C₂×Dic₄₆

yes

characteristic

100

C₂×C₄

C₂×C₄×D₂₃

yes

characteristic

101

C₂×C₄

C₂×C₂²⋊D₂₃

yes

characteristic

102

C₈

—

103

C₈

—

104

C₈

—

105

C₈

—

106

C₂²

—

107

C₂²

—

108

C₂²

#10624990

yes

—

109

C₂²

C₂×Dic₂₃⋊C₄

yes

characteristic

110

C₄

—

111

C₄

—

112

C₄

#10624990

yes

—

113

C₄

#10625004

yes

—

114

C₄

M₁₆×D₂₃

yes

—

115

C₄

C₂×#67273

yes

characteristic

116

C₄

C₂×#67274

yes

characteristic

117

C₂

—

118

C₂

#13092440

yes

characteristic

119

C₂

C₂×#10624990

yes

characteristic

120

C₂

C₂×#10625004

yes

characteristic

	subgroup	G/H
center	C₂×C₄	C₂×C₂²⋊D₂₃
commutator	C₂×C₄×C₂₃	C₂²×C₄
Frattini	C₄²	C₂²×D₂₃
Fitting	C₂₃×#423	C₂
socle	C₂²×C₂₃	C₂×C₄⋊C₄
prime radical	C₂³×C₂₃	C₄⋊C₄
cosocle	C₄²×C₂₃	C₂³
hypercenter	#423	D₂₃
hypocenter	C₂₃	#1566
LCS	G ▷ C₂×C₄×C₂₃ ▷ C₂×C₂₃ ▷ C₂₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #423

subgroup

G/H

center

C₂×C₄

C₂×C₂²⋊D₂₃

commutator

C₂×C₄×C₂₃

C₂²×C₄

Frattini

C₄²

C₂²×D₂₃

Fitting

C₂₃×#423

C₂

socle

C₂²×C₂₃

C₂×C₄⋊C₄

prime radical

C₂³×C₂₃

C₄⋊C₄

cosocle

C₄²×C₂₃

C₂³

D₂₃

C₂₃

G ▷ C₂×C₄×C₂₃ ▷ C₂×C₂₃ ▷ C₂₃

UCS

C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #423

N	H ≅ G/N	type
3. #13092167	C₂	semidirect
44. C₂₃⋊C₈	M₁₆	semidirect
76. C₂₃	#1566	semidirect

H ≅ G/N

type

3. #13092167

C₂

semidirect

44. C₂₃⋊C₈

M₁₆

semidirect

76. C₂₃

#1566

semidirect

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

subgroup	index
#13092441	2
C₂×C₄×C₂₃⋊C₈	2
#13092167	2
C₂₃×#423	2
#1566	23

subgroup

index

#13092441

C₂×C₄×C₂₃⋊C₈

#13092167

C₂₃×#423

#1566

Subgroups — #15672152

Conjugacy classes of subgroups

Complements

Maximal subgroups