Group #15672162 — subgroups

Subgroups — #15672162

Order 2944.

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

normal subgroups only

This group has 166 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	#13092474	C₂	1	1	yes	maximal, characteristic
2	C₄×#67267	C₂	1	1	yes	maximal, characteristic
3	#13092169	C₂	1	2	yes	maximal, complemented
4	#13092167	C₂	1	2	yes	maximal, complemented
5	C₂₃×#423	C₂	1	1	yes	maximal, characteristic
6	C₂×M₁₆×C₂₃	—	2	1	no	—
7	C₂×C₄²×C₂₃	C₂²	1	1	yes	characteristic
8	C₂×#67267	C₄	1	2	yes	—
9	C₂₃×C₄⋊C₈	—	2	1	no	—
10	C₂₃×C₄⋊C₈	C₂²	1	2	yes	—
11	C₂²⋊(C₂₃⋊C₈)	—	2	1	no	—
12	C₄⋊(C₂₃⋊C₈)	C₂²	1	1	yes	characteristic
13	C₄⋊(C₂₃⋊C₈)	C₂²	1	1	yes	characteristic
14	#10624985	—	2	1	no	—
15	#10624985	C₄	1	1	yes	characteristic
16	#10624985	C₄	1	1	yes	characteristic
17	C₄×C₂₃⋊C₈	C₂²	1	2	yes	—
18	C₂²×C₄×C₂₃	C₂×C₄	1	1	yes	characteristic
19	C₂²×C₄×C₂₃	D₄	1	1	yes	characteristic
20	C₂²×C₄×C₂₃	Q₈	1	1	yes	characteristic
21	M₁₆×C₂₃	—	4	2	no	—
22	C₂×C₈×C₂₃	—	2	2	no	—
23	C₄²×C₂₃	C₂×C₄	1	1	yes	characteristic
24	C₄²×C₂₃	D₄	1	1	yes	characteristic
25	C₄²×C₂₃	Q₈	1	1	yes	characteristic
26	C₄²×C₂₃	C₂³	1	1	yes	characteristic
27	#67267	—	2	4	no	—
28	C₂×C₂₃⋊C₈	—	2	1	no	—
29	C₂×C₂₃⋊C₈	—	2	1	no	—
30	C₂×C₂₃⋊C₈	C₂×C₄	1	2	yes	—
31	C₂×C₂₃⋊C₈	C₂×C₄	1	2	yes	—
32	C₂³×C₂₃	C₄⋊C₄	1	1	yes	characteristic
33	C₂×C₄×C₂₃	—	2	2	no	—
34	C₂×C₄×C₂₃	—	2	1	no	—
35	C₂×C₄×C₂₃	—	2	1	no	—
36	C₂×C₄×C₂₃	—	4	1	no	—
37	C₂×C₄×C₂₃	C₄⋊C₄	1	1	yes	characteristic
38	C₂×C₄×C₂₃	C₄⋊C₄	1	1	yes	characteristic
39	C₂×C₄×C₂₃	C₄⋊C₄	1	1	yes	characteristic
40	C₂×C₄×C₂₃	C₂²×C₄	1	1	yes	characteristic
41	C₂×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
42	C₂×C₄×C₂₃	C₂×Q₈	1	1	yes	characteristic
43	C₈×C₂₃	—	4	2	no	—
44	C₂₃⋊C₈	—	2	2	no	—
45	C₂₃⋊C₈	—	4	1	no	—
46	C₂₃⋊C₈	—	4	1	no	—
47	C₂₃⋊C₈	M₁₆	1	4	yes	complemented
48	#1569	—	23	1	no	maximal, Sylow
49	C₂²×C₂₃	—	2	1	no	—
50	C₂²×C₂₃	—	2	1	no	—
51	C₂²×C₂₃	—	2	1	no	—
52	C₂²×C₂₃	C₂×C₄⋊C₄	1	1	yes	characteristic
53	C₄×C₂₃	—	2	1	no	—
54	C₄×C₂₃	—	2	1	no	—
55	C₄×C₂₃	—	2	1	no	—
56	C₄×C₂₃	—	2	1	no	—
57	C₄×C₂₃	C₂×M₁₆	1	2	yes	—
58	C₄×C₂₃	AGL(1,ℤ/8)	1	1	yes	characteristic
59	C₄×C₂₃	#137	1	1	yes	characteristic
60	#432	—	23	1	no	—
61	#423	D₂₃	1	1	yes	characteristic
62	C₄×M₁₆	—	23	1	no	—
63	#335	—	23	2	no	—
64	#334	—	23	2	no	—
65	C₂×C₂₃	—	4	1	no	—
66	C₂×C₂₃	#423	1	1	yes	characteristic
67	C₂×C₂₃	#428	1	1	yes	characteristic
68	C₂×C₂₃	#430	1	1	yes	characteristic
69	C₂×M₁₆	—	2	1	no	—
70	C₂×M₁₆	—	23	2	no	—
71	C₂×C₄²	C₂×D₂₃	1	1	yes	characteristic
72	C₄⋊C₈	—	2	1	no	—
73	C₄⋊C₈	—	23	1	no	—
74	C₄⋊C₈	—	23	1	no	—
75	C₄⋊C₈	C₂×D₂₃	1	2	yes	—
76	C₂²⋊C₈	—	46	1	no	—
77	#97	—	23	1	no	—
78	#97	—	23	1	no	—
79	#97	—	46	1	no	—
80	C₄×C₈	—	23	2	no	—
81	C₂₃	#1569	1	1	yes	characteristic, complemented, Sylow
82	C₂²×C₄	Dic₄₆	1	1	yes	characteristic
83	C₂²×C₄	C₄×D₂₃	1	1	yes	characteristic
84	C₂²×C₄	C₂²⋊D₂₃	1	1	yes	characteristic
85	M₁₆	—	4	2	no	—
86	M₁₆	—	46	4	no	—
87	C₂×C₈	—	2	2	no	—
88	C₂×C₈	—	23	2	no	—
89	C₂×C₈	—	23	2	no	—
90	C₂×C₈	—	46	1	no	—
91	C₂×C₈	—	46	1	no	—
92	C₄²	Dic₄₆	1	1	yes	characteristic
93	C₄²	C₄×D₂₃	1	1	yes	characteristic
94	C₄²	C₂²⋊D₂₃	1	1	yes	characteristic
95	C₄²	C₂²×D₂₃	1	1	yes	characteristic
96	C₂³	Dic₂₃⋊C₄	1	1	yes	characteristic
97	C₂×C₄	—	2	2	no	—
98	C₂×C₄	—	2	1	no	—
99	C₂×C₄	—	2	1	no	—
100	C₂×C₄	—	4	1	no	—
101	C₂×C₄	Dic₂₃⋊C₄	1	1	yes	characteristic
102	C₂×C₄	Dic₂₃⋊C₄	1	1	yes	characteristic
103	C₂×C₄	Dic₂₃⋊C₄	1	1	yes	characteristic
104	C₂×C₄	C₂×Dic₄₆	1	1	yes	characteristic
105	C₂×C₄	C₂×C₄×D₂₃	1	1	yes	characteristic
106	C₂×C₄	C₂×C₂²⋊D₂₃	1	1	yes	characteristic
107	C₈	—	4	2	no	—
108	C₈	—	23	4	no	—
109	C₈	—	46	2	no	—
110	C₈	—	92	1	no	—
111	C₈	—	92	1	no	—
112	C₂²	—	2	1	no	—
113	C₂²	—	2	1	no	—
114	C₂²	—	2	1	no	—
115	C₂²	C₂×Dic₂₃⋊C₄	1	1	yes	characteristic
116	C₄	—	2	1	no	—
117	C₄	—	2	1	no	—
118	C₄	—	2	1	no	—
119	C₄	—	2	1	no	—
120	C₄	M₁₆×D₂₃	1	2	yes	—
121	C₄	#10625103	1	1	yes	characteristic
122	C₄	#10625113	1	1	yes	characteristic
123	C₂	—	4	1	no	—
124	C₂	#13092440	1	1	yes	characteristic
125	C₂	#13092457	1	1	yes	characteristic
126	C₂	#13092465	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#13092474

C₂

yes

maximal, characteristic

C₄×#67267

C₂

yes

maximal, characteristic

#13092169

C₂

yes

maximal, complemented

#13092167

C₂

yes

maximal, complemented

C₂₃×#423

C₂

yes

maximal, characteristic

C₂×M₁₆×C₂₃

—

C₂×C₄²×C₂₃

C₂²

yes

characteristic

C₂×#67267

C₄

yes

—

C₂₃×C₄⋊C₈

—

C₂₃×C₄⋊C₈

C₂²

yes

—

C₂²⋊(C₂₃⋊C₈)

—

C₄⋊(C₂₃⋊C₈)

C₂²

yes

characteristic

C₄⋊(C₂₃⋊C₈)

C₂²

yes

characteristic

#10624985

—

#10624985

C₄

yes

characteristic

#10624985

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

—

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

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

—

C₂₃⋊C₈

M₁₆

yes

complemented

#1569

—

maximal, Sylow

C₂²×C₂₃

—

C₂²×C₂₃

—

C₂²×C₂₃

—

C₂²×C₂₃

C₂×C₄⋊C₄

yes

characteristic

C₄×C₂₃

—

C₄×C₂₃

—

C₄×C₂₃

—

C₄×C₂₃

—

C₄×C₂₃

C₂×M₁₆

yes

—

C₄×C₂₃

yes

C₄×C₂₃

yes

—

D₂₃

yes

characteristic

C₄×M₁₆

—

#335

—

#334

—

C₂×C₂₃

—

C₂×C₂₃

yes

C₂×C₂₃

yes

C₂×C₂₃

yes

C₂×M₁₆

—

C₂×M₁₆

—

C₂×C₄²

C₂×D₂₃

yes

characteristic

C₄⋊C₈

—

C₄⋊C₈

—

C₄⋊C₈

—

C₄⋊C₈

C₂×D₂₃

yes

—

C₂²⋊C₈

—

#97

—

#97

—

#97

—

C₄×C₈

—

C₂₃

#1569

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

—

100

C₂×C₄

—

101

C₂×C₄

Dic₂₃⋊C₄

yes

characteristic

102

C₂×C₄

Dic₂₃⋊C₄

yes

characteristic

103

C₂×C₄

Dic₂₃⋊C₄

yes

characteristic

104

C₂×C₄

C₂×Dic₄₆

yes

characteristic

105

C₂×C₄

C₂×C₄×D₂₃

yes

characteristic

106

C₂×C₄

C₂×C₂²⋊D₂₃

yes

characteristic

107

C₈

—

108

C₈

—

109

C₈

—

110

C₈

—

111

C₈

—

112

C₂²

—

113

C₂²

—

114

C₂²

—

115

C₂²

C₂×Dic₂₃⋊C₄

yes

characteristic

116

C₄

—

117

C₄

—

118

C₄

—

119

C₄

—

120

C₄

M₁₆×D₂₃

yes

—

121

C₄

#10625103

yes

characteristic

122

C₄

#10625113

yes

characteristic

123

C₂

—

124

C₂

#13092440

yes

characteristic

125

C₂

#13092457

yes

characteristic

126

C₂

#13092465

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	#1569
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. #13092169	C₂	semidirect
4. #13092167	C₂	semidirect
47. C₂₃⋊C₈	M₁₆	semidirect
81. C₂₃	#1569	semidirect

H ≅ G/N

type

3. #13092169

C₂

semidirect

4. #13092167

C₂

semidirect

47. C₂₃⋊C₈

M₁₆

semidirect

81. C₂₃

#1569

semidirect

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

subgroup	index
#13092474	2
C₄×#67267	2
#13092169	2
#13092167	2
C₂₃×#423	2
#1569	23

subgroup

index

C₂₃×#423

Subgroups — #15672162

Conjugacy classes of subgroups

Complements

Maximal subgroups