Group #15675149 — subgroups

Subgroups — C₂×Dic₂₃⋊C₁₆ #15675149

Order 2944.

184 conjugacy classes of subgroups; only the 117 normal subgroups are shown.

normal subgroups only

This group has 184 conjugacy classes of subgroups — too many to draw. Only its 117 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	C₂²×C₂₃⋊C₁₆	C₂	1	1	yes	maximal, characteristic
2	C₂×C₈×Dic₂₃	C₂	1	1	yes	maximal, characteristic
3	Dic₂₃⋊C₁₆	C₂	1	4	yes	maximal, direct product factor
4	C₂²×C₁₆×C₂₃	C₂	1	1	yes	maximal, characteristic
5	C₂²×C₈×C₂₃	C₂²	1	1	yes	characteristic
6	C₂×C₄×Dic₂₃	C₄	1	1	yes	characteristic
7	C₂²×C₂₃⋊C₈	C₄	1	1	yes	characteristic
8	C₂×C₁₆×C₂₃	—	2	2	no	—
9	C₂×C₁₆×C₂₃	C₂²	1	2	yes	—
10	C₈×Dic₂₃	C₄	1	2	yes	—
11	C₈×Dic₂₃	C₂²	1	2	yes	—
12	C₂×C₂₃⋊C₁₆	—	2	2	no	—
13	C₂×C₂₃⋊C₁₆	C₂²	1	2	yes	—
14	C₂²×C₄×C₂₃	C₂×C₄	1	1	yes	characteristic
15	C₂²×Dic₂₃	C₈	1	1	yes	characteristic
16	C₂²×Dic₂₃	C₈	1	1	yes	characteristic
17	C₂×C₈×C₂₃	C₂×C₄	1	1	yes	characteristic
18	C₂×C₈×C₂₃	D₄	1	2	yes	—
19	C₂×C₈×C₂₃	Q₈	1	2	yes	—
20	C₂×C₈×C₂₃	C₂³	1	1	yes	characteristic
21	C₄×Dic₂₃	C₈	1	2	yes	—
22	C₄×Dic₂₃	C₂×C₄	1	2	yes	—
23	C₂×C₂₃⋊C₈	—	2	2	no	—
24	C₂×C₂₃⋊C₈	C₂×C₄	1	2	yes	—
25	C₁₆×C₂₃	—	2	2	no	—
26	C₂₃⋊C₁₆	—	2	2	no	—
27	C₂³×C₂₃	C₂×C₈	1	1	yes	characteristic
28	C₂×C₄×C₂₃	C₄⋊C₄	1	2	yes	—
29	C₂×C₄×C₂₃	C₂×C₈	1	1	yes	characteristic
30	C₂×C₄×C₂₃	M₁₆	1	2	yes	—
31	C₂×C₄×C₂₃	C₂²×C₄	1	1	yes	characteristic
32	C₂×Dic₂₃	—	2	2	no	—
33	C₂×Dic₂₃	C₁₆	1	4	yes	complemented
34	C₂×Dic₂₃	C₂×C₈	1	2	yes	—
35	C₂×Dic₂₃	C₂×C₈	1	2	yes	—
36	C₈×C₂₃	C₄⋊C₄	1	2	yes	—
37	C₈×C₂₃	C₂×D₄	1	1	yes	characteristic
38	C₈×C₂₃	C₂×Q₈	1	1	yes	characteristic
39	C₂₃⋊C₈	—	2	2	no	—
40	C₂×C₄⋊C₁₆	—	23	1	no	maximal, Sylow
41	C₂²×C₂₃	C₄⋊C₈	1	2	yes	—
42	C₂²×C₂₃	C₂×C₁₆	1	2	yes	—
43	C₂²×C₂₃	M₃₂	1	2	yes	—
44	C₂²×C₂₃	C₂²×C₈	1	1	yes	characteristic
45	C₄×C₂₃	C₄⋊C₈	1	2	yes	—
46	C₄×C₂₃	C₂×C₄⋊C₄	1	1	yes	characteristic
47	C₄×C₂₃	C₂×M₁₆	1	1	yes	characteristic
48	Dic₂₃	—	2	2	no	—
49	Dic₂₃	C₂×C₁₆	1	4	yes	complemented
50	C₂²×C₁₆	—	23	1	no	—
51	C₂²×C₁₆	D₂₃	1	1	yes	characteristic
52	C₂×C₄×C₈	—	23	1	no	—
53	C₄⋊C₁₆	—	23	4	no	—
54	C₂×C₂₃	C₄⋊C₁₆	1	4	yes	complemented
55	C₂×C₂₃	C₂×C₄⋊C₈	1	1	yes	characteristic
56	C₂×C₂₃	C₂²×C₁₆	1	1	yes	characteristic
57	C₂×C₂₃	C₂×M₃₂	1	1	yes	characteristic
58	C₂²×C₈	—	23	1	no	—
59	C₂²×C₈	C₂×D₂₃	1	1	yes	characteristic
60	C₂×C₄²	—	23	1	no	—
61	C₂×C₁₆	—	2	2	no	—
62	C₂×C₁₆	—	23	2	no	—
63	C₂×C₁₆	—	46	2	no	—
64	C₂×C₁₆	C₂×D₂₃	1	2	yes	—
65	C₄×C₈	—	23	2	no	—
66	C₄×C₈	—	23	2	no	—
67	C₂₃	C₂×C₄⋊C₁₆	1	1	yes	characteristic, complemented, Sylow
68	C₂²×C₄	—	23	1	no	—
69	C₂²×C₄	—	23	1	no	—
70	C₂²×C₄	C₄×D₂₃	1	1	yes	characteristic
71	C₂×C₈	—	23	2	no	—
72	C₂×C₈	—	46	2	no	—
73	C₂×C₈	Dic₄₆	1	2	yes	—
74	C₂×C₈	C₄×D₂₃	1	1	yes	characteristic
75	C₂×C₈	C₂²⋊D₂₃	1	2	yes	—
76	C₂×C₈	C₂²×D₂₃	1	1	yes	characteristic
77	C₄²	—	23	2	no	—
78	C₄²	—	23	2	no	—
79	C₁₆	—	2	2	no	—
80	C₁₆	—	46	2	no	—
81	C₂³	C₈×D₂₃	1	1	yes	characteristic
82	C₂×C₄	—	23	2	no	—
83	C₂×C₄	—	23	4	no	—
84	C₂×C₄	—	23	2	no	—
85	C₂×C₄	—	46	2	no	—
86	C₂×C₄	C₈×D₂₃	1	1	yes	characteristic
87	C₂×C₄	#67262	1	2	yes	—
88	C₂×C₄	Dic₂₃⋊C₄	1	2	yes	—
89	C₂×C₄	C₂×C₄×D₂₃	1	1	yes	characteristic
90	C₈	—	46	2	no	—
91	C₈	Dic₂₃⋊C₄	1	2	yes	—
92	C₈	C₂×Dic₄₆	1	1	yes	characteristic
93	C₈	C₂×C₂²⋊D₂₃	1	1	yes	characteristic
94	C₂²	C₁₆×D₂₃	1	2	yes	—
95	C₂²	#10624980	1	2	yes	—
96	C₂²	Dic₂₃⋊C₈	1	2	yes	—
97	C₂²	C₂×C₈×D₂₃	1	1	yes	characteristic
98	C₄	—	23	4	no	—
99	C₄	—	46	2	no	—
100	C₄	Dic₂₃⋊C₈	1	2	yes	—
101	C₄	C₂×#67262	1	1	yes	characteristic
102	C₄	C₂×Dic₂₃⋊C₄	1	1	yes	characteristic
103	C₂	Dic₂₃⋊C₁₆	1	4	yes	direct product factor
104	C₂	C₂×Dic₂₃⋊C₈	1	1	yes	characteristic
105	C₂	C₂×C₁₆×D₂₃	1	1	yes	characteristic
106	C₂	C₂×#10624980	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂²×C₂₃⋊C₁₆

C₂

yes

maximal, characteristic

C₂×C₈×Dic₂₃

C₂

yes

maximal, characteristic

Dic₂₃⋊C₁₆

C₂

yes

maximal, direct product factor

C₂²×C₁₆×C₂₃

C₂

yes

maximal, characteristic

C₂²×C₈×C₂₃

C₂²

yes

characteristic

C₂×C₄×Dic₂₃

C₄

yes

characteristic

C₂²×C₂₃⋊C₈

C₄

yes

characteristic

C₂×C₁₆×C₂₃

—

C₂×C₁₆×C₂₃

C₂²

yes

—

C₈×Dic₂₃

C₄

yes

—

C₈×Dic₂₃

C₂²

yes

—

C₂×C₂₃⋊C₁₆

—

C₂×C₂₃⋊C₁₆

C₂²

yes

—

C₂²×C₄×C₂₃

C₂×C₄

yes

characteristic

C₂²×Dic₂₃

C₈

yes

characteristic

C₂²×Dic₂₃

C₈

yes

characteristic

C₂×C₈×C₂₃

C₂×C₄

yes

characteristic

C₂×C₈×C₂₃

D₄

yes

—

C₂×C₈×C₂₃

Q₈

yes

—

C₂×C₈×C₂₃

C₂³

yes

characteristic

C₄×Dic₂₃

C₈

yes

—

C₄×Dic₂₃

C₂×C₄

yes

—

C₂×C₂₃⋊C₈

—

C₂×C₂₃⋊C₈

C₂×C₄

yes

—

C₁₆×C₂₃

—

C₂₃⋊C₁₆

—

C₂³×C₂₃

C₂×C₈

yes

characteristic

C₂×C₄×C₂₃

C₄⋊C₄

yes

—

C₂×C₄×C₂₃

C₂×C₈

yes

characteristic

C₂×C₄×C₂₃

M₁₆

yes

—

C₂×C₄×C₂₃

C₂²×C₄

yes

characteristic

C₂×Dic₂₃

—

C₂×Dic₂₃

C₁₆

yes

complemented

C₂×Dic₂₃

C₂×C₈

yes

—

C₂×Dic₂₃

C₂×C₈

yes

—

C₈×C₂₃

C₄⋊C₄

yes

—

C₈×C₂₃

C₂×D₄

yes

characteristic

C₈×C₂₃

C₂×Q₈

yes

characteristic

C₂₃⋊C₈

—

C₂×C₄⋊C₁₆

—

maximal, Sylow

C₂²×C₂₃

C₄⋊C₈

yes

—

C₂²×C₂₃

C₂×C₁₆

yes

—

C₂²×C₂₃

M₃₂

yes

—

C₂²×C₂₃

C₂²×C₈

yes

characteristic

C₄×C₂₃

C₄⋊C₈

yes

—

C₄×C₂₃

C₂×C₄⋊C₄

yes

characteristic

C₄×C₂₃

C₂×M₁₆

yes

characteristic

Dic₂₃

—

Dic₂₃

C₂×C₁₆

yes

complemented

C₂²×C₁₆

—

C₂²×C₁₆

D₂₃

yes

characteristic

C₂×C₄×C₈

—

C₄⋊C₁₆

—

C₂×C₂₃

C₄⋊C₁₆

yes

complemented

C₂×C₂₃

C₂×C₄⋊C₈

yes

characteristic

C₂×C₂₃

C₂²×C₁₆

yes

characteristic

C₂×C₂₃

C₂×M₃₂

yes

characteristic

C₂²×C₈

—

C₂²×C₈

C₂×D₂₃

yes

characteristic

C₂×C₄²

—

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

C₂×D₂₃

yes

—

C₄×C₈

—

C₄×C₈

—

C₂₃

C₂×C₄⋊C₁₆

yes

characteristic, complemented, Sylow

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

C₄×D₂₃

yes

characteristic

C₂×C₈

—

C₂×C₈

—

C₂×C₈

Dic₄₆

yes

—

C₂×C₈

C₄×D₂₃

yes

characteristic

C₂×C₈

C₂²⋊D₂₃

yes

—

C₂×C₈

C₂²×D₂₃

yes

characteristic

C₄²

—

C₄²

—

C₁₆

—

C₁₆

—

C₂³

C₈×D₂₃

yes

characteristic

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

C₈×D₂₃

yes

characteristic

C₂×C₄

#67262

yes

—

C₂×C₄

Dic₂₃⋊C₄

yes

—

C₂×C₄

C₂×C₄×D₂₃

yes

characteristic

C₈

—

C₈

Dic₂₃⋊C₄

yes

—

C₈

C₂×Dic₄₆

yes

characteristic

C₈

C₂×C₂²⋊D₂₃

yes

characteristic

C₂²

C₁₆×D₂₃

yes

—

C₂²

#10624980

yes

—

C₂²

Dic₂₃⋊C₈

yes

—

C₂²

C₂×C₈×D₂₃

yes

characteristic

C₄

—

C₄

—

100

C₄

Dic₂₃⋊C₈

yes

—

101

C₄

C₂×#67262

yes

characteristic

102

C₄

C₂×Dic₂₃⋊C₄

yes

characteristic

103

C₂

Dic₂₃⋊C₁₆

yes

direct product factor

104

C₂

C₂×Dic₂₃⋊C₈

yes

characteristic

105

C₂

C₂×C₁₆×D₂₃

yes

characteristic

106

C₂

C₂×#10624980

yes

characteristic

	subgroup	G/H
center	C₂²×C₈	C₂×D₂₃
commutator	C₂×C₂₃	C₂²×C₁₆
Frattini	C₂×C₈	C₂²×D₂₃
Fitting	C₂²×C₁₆×C₂₃	C₂
socle, prime radical	C₂³×C₂₃	C₂×C₈
cosocle	C₂×C₈×C₂₃	C₂³
hypercenter	C₂²×C₁₆	D₂₃
hypocenter	C₂₃	C₂×C₄⋊C₁₆
LCS	G ▷ C₂×C₂₃ ▷ C₂₃
UCS	C₁ ◁ C₂²×C₈ ◁ C₂²×C₁₆

subgroup

G/H

center

C₂²×C₈

C₂×D₂₃

commutator

C₂×C₂₃

C₂²×C₁₆

Frattini

C₂×C₈

C₂²×D₂₃

Fitting

C₂²×C₁₆×C₂₃

C₂

socle, prime radical

C₂³×C₂₃

C₂×C₈

cosocle

C₂×C₈×C₂₃

C₂³

hypercenter

C₂²×C₁₆

D₂₃

hypocenter

C₂₃

C₂×C₄⋊C₁₆

LCS

G ▷ C₂×C₂₃ ▷ C₂₃

UCS

C₁ ◁ C₂²×C₈ ◁ C₂²×C₁₆

N	H ≅ G/N	type
3. Dic₂₃⋊C₁₆	C₂	direct (N × H)
33. C₂×Dic₂₃	C₁₆	semidirect
49. Dic₂₃	C₂×C₁₆	semidirect
54. C₂×C₂₃	C₄⋊C₁₆	semidirect
67. C₂₃	C₂×C₄⋊C₁₆	semidirect
103. C₂	Dic₂₃⋊C₁₆	direct (N × H)

H ≅ G/N

type

3. Dic₂₃⋊C₁₆

C₂

direct (N × H)

33. C₂×Dic₂₃

C₁₆

semidirect

49. Dic₂₃

C₂×C₁₆

semidirect

54. C₂×C₂₃

C₄⋊C₁₆

semidirect

67. C₂₃

C₂×C₄⋊C₁₆

semidirect

103. C₂

Dic₂₃⋊C₁₆

direct (N × H)

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

subgroup	index
C₂²×C₂₃⋊C₁₆	2
C₂×C₈×Dic₂₃	2
Dic₂₃⋊C₁₆	2
C₂²×C₁₆×C₂₃	2
C₂×C₄⋊C₁₆	23

subgroup

index

C₂²×C₂₃⋊C₁₆

C₂×C₈×Dic₂₃

Dic₂₃⋊C₁₆

C₂²×C₁₆×C₂₃

C₂×C₄⋊C₁₆

Subgroups — C₂×Dic₂₃⋊C₁₆ #15675149

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C2×Dic23⋊C16 #15675149

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×Dic₂₃⋊C₁₆ #15675149