Group #15672167 — subgroups

Subgroups — #15672167

Order 2944.

168 conjugacy classes of subgroups; only the 89 normal subgroups are shown.

normal subgroups only

This group has 168 conjugacy classes of subgroups — too many to draw. Only its 89 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	#13092441	C₂	1	1	yes	maximal, characteristic
3	#13092167	C₂	1	4	yes	maximal, complemented
4	C₂×C₂₃×C₄⋊C₈	C₂	1	1	yes	maximal, characteristic
5	C₂²×C₈×C₂₃	—	2	1	no	—
6	C₂×C₄²×C₂₃	C₂²	1	1	yes	characteristic
7	C₂×#67267	—	2	1	no	—
8	C₂₃×C₄⋊C₈	—	2	1	no	—
9	C₂₃×C₄⋊C₈	C₂²	1	2	yes	—
10	C₂²⋊(C₂₃⋊C₈)	C₄	1	2	yes	—
11	C₄⋊(C₂₃⋊C₈)	—	2	1	no	—
12	C₄⋊(C₂₃⋊C₈)	C₄	1	2	yes	—
13	C₄⋊(C₂₃⋊C₈)	C₂²	1	2	yes	—
14	C₄×C₂₃⋊C₈	C₂²	1	2	yes	—
15	C₂²×C₄×C₂₃	C₂×C₄	1	1	yes	characteristic
16	C₂²×C₄×C₂₃	D₄	1	1	yes	characteristic
17	C₂²×C₄×C₂₃	Q₈	1	1	yes	characteristic
18	C₂×C₈×C₂₃	—	2	2	no	—
19	C₂×C₈×C₂₃	—	4	2	no	—
20	C₄²×C₂₃	C₂×C₄	1	1	yes	characteristic
21	C₄²×C₂₃	D₄	1	1	yes	characteristic
22	C₄²×C₂₃	Q₈	1	1	yes	characteristic
23	C₄²×C₂₃	C₂³	1	1	yes	characteristic
24	#67267	—	4	2	no	—
25	C₂×C₂₃⋊C₈	—	2	2	no	—
26	C₂×C₂₃⋊C₈	C₂×C₄	1	4	yes	—
27	C₂³×C₂₃	C₄⋊C₄	1	1	yes	characteristic
28	C₂×C₄×C₂₃	—	2	2	no	—
29	C₂×C₄×C₂₃	—	4	1	no	—
30	C₂×C₄×C₂₃	C₄⋊C₄	1	1	yes	characteristic
31	C₂×C₄×C₂₃	C₄⋊C₄	1	1	yes	characteristic
32	C₂×C₄×C₂₃	C₄⋊C₄	1	1	yes	characteristic
33	C₂×C₄×C₂₃	SD₁₆	1	2	yes	—
34	C₂×C₄×C₂₃	SD₁₆	1	2	yes	—
35	C₂×C₄×C₂₃	C₂²×C₄	1	1	yes	characteristic
36	C₂×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
37	C₂×C₄×C₂₃	C₂×Q₈	1	1	yes	characteristic
38	C₈×C₂₃	—	4	2	no	—
39	C₂₃⋊C₈	—	2	4	no	—
40	C₂₃⋊C₈	—	4	2	no	—
41	#1570	—	23	1	no	maximal, Sylow
42	C₂²×C₂₃	—	2	1	no	—
43	C₂²×C₂₃	—	2	1	no	—
44	C₂²×C₂₃	#106	1	2	yes	—
45	C₂²×C₂₃	C₂×C₄⋊C₄	1	1	yes	characteristic
46	C₄×C₂₃	—	2	1	no	—
47	C₄×C₂₃	—	2	1	no	—
48	C₄×C₂₃	—	2	1	no	—
49	C₄×C₂₃	#106	1	2	yes	—
50	C₄×C₂₃	#131	1	2	yes	—
51	C₄×C₂₃	C₂×SD₁₆	1	1	yes	characteristic
52	C₄×C₂₃	C₂×SD₁₆	1	1	yes	characteristic
53	#431	—	23	1	no	—
54	#423	—	23	1	no	—
55	C₂×C₄⋊C₈	D₂₃	1	1	yes	characteristic
56	#334	—	23	4	no	—
57	C₂×C₂₃	—	2	2	no	—
58	C₂×C₂₃	#424	1	1	yes	characteristic
59	C₂×C₂₃	C₂×#106	1	1	yes	characteristic
60	C₂×C₂₃	#430	1	1	yes	characteristic
61	C₂×M₁₆	—	46	1	no	—
62	C₂²×C₈	—	2	1	no	—
63	C₂×C₄²	C₂×D₂₃	1	1	yes	characteristic
64	C₄⋊C₈	—	2	1	no	—
65	C₄⋊C₈	—	23	2	no	—
66	C₄⋊C₈	—	23	2	no	—
67	C₄⋊C₈	—	46	1	no	—
68	C₄⋊C₈	C₂×D₂₃	1	2	yes	—
69	C₂²⋊C₈	—	23	2	no	—
70	C₄×C₈	—	23	2	no	—
71	C₂₃	#1570	1	1	yes	characteristic, complemented, Sylow
72	C₂²×C₄	Dic₄₆	1	1	yes	characteristic
73	C₂²×C₄	C₄×D₂₃	1	1	yes	characteristic
74	C₂²×C₄	C₂²⋊D₂₃	1	1	yes	characteristic
75	M₁₆	—	92	2	no	—
76	C₂×C₈	—	2	2	no	—
77	C₂×C₈	—	4	2	no	—
78	C₂×C₈	—	23	4	no	—
79	C₂×C₈	—	46	2	no	—
80	C₄²	Dic₄₆	1	1	yes	characteristic
81	C₄²	C₄×D₂₃	1	1	yes	characteristic
82	C₄²	C₂²⋊D₂₃	1	1	yes	characteristic
83	C₄²	C₂²×D₂₃	1	1	yes	characteristic
84	C₂³	Dic₂₃⋊C₄	1	1	yes	characteristic
85	C₂×C₄	—	2	2	no	—
86	C₂×C₄	—	4	1	no	—
87	C₂×C₄	Dic₂₃⋊C₄	1	1	yes	characteristic
88	C₂×C₄	Dic₂₃⋊C₄	1	1	yes	characteristic
89	C₂×C₄	Dic₂₃⋊C₄	1	1	yes	characteristic
90	C₂×C₄	#67273	1	2	yes	—
91	C₂×C₄	#67274	1	2	yes	—
92	C₂×C₄	C₂×Dic₄₆	1	1	yes	characteristic
93	C₂×C₄	C₂×C₄×D₂₃	1	1	yes	characteristic
94	C₂×C₄	C₂×C₂²⋊D₂₃	1	1	yes	characteristic
95	C₈	—	4	2	no	—
96	C₈	—	46	4	no	—
97	C₈	—	92	2	no	—
98	C₂²	—	2	1	no	—
99	C₂²	—	2	1	no	—
100	C₂²	#10624990	1	2	yes	—
101	C₂²	C₂×Dic₂₃⋊C₄	1	1	yes	characteristic
102	C₄	—	2	1	no	—
103	C₄	—	2	1	no	—
104	C₄	—	2	1	no	—
105	C₄	#10624990	1	2	yes	—
106	C₄	#10625072	1	2	yes	—
107	C₄	C₂×#67273	1	1	yes	characteristic
108	C₄	C₂×#67274	1	1	yes	characteristic
109	C₂	—	2	2	no	—
110	C₂	#13092443	1	1	yes	characteristic
111	C₂	C₂×#10624990	1	1	yes	characteristic
112	C₂	#13092465	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#13092470

C₂

yes

maximal, characteristic

#13092441

C₂

yes

maximal, characteristic

#13092167

C₂

yes

maximal, complemented

C₂×C₂₃×C₄⋊C₈

C₂

yes

maximal, characteristic

C₂²×C₈×C₂₃

—

C₂×C₄²×C₂₃

C₂²

yes

characteristic

C₂×#67267

—

C₂₃×C₄⋊C₈

—

C₂₃×C₄⋊C₈

C₂²

yes

—

C₂²⋊(C₂₃⋊C₈)

C₄

yes

—

C₄⋊(C₂₃⋊C₈)

—

C₄⋊(C₂₃⋊C₈)

C₄

yes

—

C₄⋊(C₂₃⋊C₈)

C₂²

yes

—

C₄×C₂₃⋊C₈

C₂²

yes

—

C₂²×C₄×C₂₃

C₂×C₄

yes

characteristic

C₂²×C₄×C₂₃

D₄

yes

characteristic

C₂²×C₄×C₂₃

Q₈

yes

characteristic

C₂×C₈×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×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

—

#1570

—

maximal, Sylow

C₂²×C₂₃

—

C₂²×C₂₃

—

C₂²×C₂₃

#106

yes

—

C₂²×C₂₃

C₂×C₄⋊C₄

yes

characteristic

C₄×C₂₃

—

C₄×C₂₃

—

C₄×C₂₃

—

C₄×C₂₃

#106

yes

—

C₄×C₂₃

#131

yes

—

C₄×C₂₃

C₂×SD₁₆

yes

characteristic

C₄×C₂₃

C₂×SD₁₆

yes

characteristic

#431

—

#423

—

C₂×C₄⋊C₈

D₂₃

yes

characteristic

#334

—

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

C₂×D₂₃

yes

—

C₂²⋊C₈

—

C₄×C₈

—

C₂₃

#1570

yes

characteristic, complemented, Sylow

C₂²×C₄

Dic₄₆

yes

characteristic

C₂²×C₄

C₄×D₂₃

yes

characteristic

C₂²×C₄

C₂²⋊D₂₃

yes

characteristic

M₁₆

—

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

C₂×C₄

C₂×C₄×D₂₃

yes

characteristic

C₂×C₄

C₂×C₂²⋊D₂₃

yes

characteristic

C₈

—

C₈

—

C₈

—

C₂²

—

C₂²

—

100

C₂²

#10624990

yes

—

101

C₂²

C₂×Dic₂₃⋊C₄

yes

characteristic

102

C₄

—

103

C₄

—

104

C₄

—

105

C₄

#10624990

yes

—

106

C₄

#10625072

yes

—

107

C₄

C₂×#67273

yes

characteristic

108

C₄

C₂×#67274

yes

characteristic

109

C₂

—

110

C₂

#13092443

yes

characteristic

111

C₂

C₂×#10624990

yes

characteristic

112

C₂

#13092465

yes

characteristic

	subgroup	G/H
center	C₂×C₄	C₂×C₂²⋊D₂₃
commutator	C₂×C₄×C₂₃	C₂²×C₄
Frattini	C₄²	C₂²×D₂₃
Fitting	C₂×C₂₃×C₄⋊C₈	C₂
socle	C₂²×C₂₃	C₂×C₄⋊C₄
prime radical	C₂³×C₂₃	C₄⋊C₄
cosocle	C₄²×C₂₃	C₂³
hypercenter	C₂×C₄⋊C₈	D₂₃
hypocenter	C₂₃	#1570
LCS	G ▷ C₂×C₄×C₂₃ ▷ C₂×C₂₃ ▷ C₂₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₂×C₄⋊C₈

subgroup

G/H

center

C₂×C₄

C₂×C₂²⋊D₂₃

commutator

C₂×C₄×C₂₃

C₂²×C₄

Frattini

C₄²

C₂²×D₂₃

Fitting

C₂×C₂₃×C₄⋊C₈

C₂

socle

C₂²×C₂₃

C₂×C₄⋊C₄

prime radical

C₂³×C₂₃

C₄⋊C₄

cosocle

C₄²×C₂₃

C₂³

hypercenter

C₂×C₄⋊C₈

D₂₃

hypocenter

C₂₃

#1570

LCS

G ▷ C₂×C₄×C₂₃ ▷ C₂×C₂₃ ▷ C₂₃

UCS

C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₂×C₄⋊C₈

N	H ≅ G/N	type
3. #13092167	C₂	semidirect
71. C₂₃	#1570	semidirect

H ≅ G/N

type

3. #13092167

C₂

semidirect

71. C₂₃

#1570

semidirect

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

subgroup	index
#13092470	2
#13092441	2
#13092167	2
C₂×C₂₃×C₄⋊C₈	2
#1570	23

subgroup

index

C₂×C₂₃×C₄⋊C₈

Subgroups — #15672167

Conjugacy classes of subgroups

Complements

Maximal subgroups