Group #15675303 — subgroups

Subgroups — #15675303

Order 2944.

126 conjugacy classes of subgroups; only the 71 normal subgroups are shown.

normal subgroups only

This group has 126 conjugacy classes of subgroups — too many to draw. Only its 71 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	C₈×Dic₄₆	C₂	1	1	yes	maximal, characteristic
2	C₄⋊(C₂₃⋊C₁₆)	C₂	1	1	yes	maximal, characteristic
3	Dic₂₃⋊C₁₆	C₂	1	1	yes	maximal, characteristic
4	Dic₂₃⋊C₁₆	C₂	1	1	yes	maximal, characteristic
5	#13092192	C₂	1	1	yes	maximal, characteristic
6	C₁₆×Dic₂₃	C₂	1	1	yes	maximal, characteristic
7	C₂₃×#346	C₂	1	1	yes	maximal, characteristic
8	C₄×Dic₄₆	C₄	1	1	yes	characteristic
9	C₂×C₁₆×C₂₃	C₂²	1	1	yes	characteristic
10	C₂×C₁₆×C₂₃	C₂²	1	1	yes	characteristic
11	C₄×C₈×C₂₃	C₂²	1	1	yes	characteristic
12	Dic₂₃⋊C₈	C₄	1	1	yes	characteristic
13	Dic₂₃⋊C₈	C₄	1	1	yes	characteristic
14	C₈×Dic₂₃	C₂²	1	1	yes	characteristic
15	C₈×Dic₂₃	C₂²	1	1	yes	characteristic
16	C₂×C₂₃⋊C₁₆	C₂²	1	1	yes	characteristic
17	C₂×C₂₃⋊C₁₆	C₂²	1	1	yes	characteristic
18	C₄⋊(C₂₃⋊C₈)	C₄	1	1	yes	characteristic
19	C₂×Dic₄₆	C₈	1	1	yes	characteristic
20	C₂×C₈×C₂₃	C₂×C₄	1	1	yes	characteristic
21	C₂×C₈×C₂₃	C₂³	1	1	yes	characteristic
22	C₄²×C₂₃	C₂×C₄	1	1	yes	characteristic
23	C₄⋊Dic₂₃	C₈	1	1	yes	characteristic
24	Dic₂₃⋊C₄	C₈	1	1	yes	characteristic
25	Dic₂₃⋊C₄	C₈	1	1	yes	characteristic
26	C₄×Dic₂₃	C₂×C₄	1	1	yes	characteristic
27	C₄×Dic₂₃	C₂×C₄	1	1	yes	characteristic
28	C₂×C₂₃⋊C₈	C₂×C₄	1	1	yes	characteristic
29	C₂×C₂₃⋊C₈	C₂×C₄	1	1	yes	characteristic
30	C₁₆×C₂₃	—	2	1	no	—
31	C₁₆×C₂₃	Q₈	1	2	yes	complemented
32	C₂₃⋊C₁₆	—	2	1	no	—
33	C₂₃⋊C₁₆	—	2	1	no	—
34	C₂×C₄×C₂₃	C₂×C₈	1	1	yes	characteristic
35	C₂×C₄×C₂₃	C₂×C₈	1	1	yes	characteristic
36	C₂×C₄×C₂₃	C₂²×C₄	1	1	yes	characteristic
37	C₂×Dic₂₃	C₂×C₈	1	1	yes	characteristic
38	C₂×Dic₂₃	C₂×C₈	1	1	yes	characteristic
39	C₂×Dic₂₃	C₂×C₈	1	1	yes	characteristic
40	C₂×Dic₂₃	C₂×C₈	1	1	yes	characteristic
41	Dic₄₆	—	2	2	no	—
42	C₈×C₂₃	—	2	1	no	—
43	C₈×C₂₃	C₂×Q₈	1	1	yes	characteristic
44	C₈×C₂₃	#41	1	1	yes	characteristic
45	C₂₃⋊C₈	—	2	1	no	—
46	C₂₃⋊C₈	—	2	1	no	—
47	#2183	—	23	1	no	maximal, Sylow
48	C₂²×C₂₃	C₂²×C₈	1	1	yes	characteristic
49	C₄×C₂₃	—	2	1	no	—
50	C₄×C₂₃	—	2	1	no	—
51	C₄×C₂₃	C₄×Q₈	1	1	yes	characteristic
52	C₄×C₂₃	#131	1	1	yes	characteristic
53	Dic₂₃	—	2	1	no	—
54	Dic₂₃	—	2	1	no	—
55	Dic₂₃	—	2	1	no	—
56	Dic₂₃	M₃₂	1	2	yes	—
57	C₈×Q₈	—	23	1	no	—
58	C₄⋊C₁₆	—	23	1	no	—
59	C₄⋊C₁₆	—	23	1	no	—
60	C₄⋊C₁₆	—	23	1	no	—
61	#346	—	23	1	no	—
62	#346	D₂₃	1	1	yes	characteristic
63	C₄×C₁₆	—	23	1	no	—
64	C₂×C₂₃	C₈×Q₈	1	1	yes	characteristic
65	C₂×C₂₃	C₂×M₃₂	1	1	yes	characteristic
66	C₂×C₂₃	#504	1	1	yes	characteristic
67	C₄×Q₈	—	23	1	no	—
68	C₂×C₁₆	—	23	1	no	—
69	C₂×C₁₆	—	23	1	no	—
70	C₂×C₁₆	C₂×D₂₃	1	1	yes	characteristic
71	C₂×C₁₆	C₂×D₂₃	1	1	yes	characteristic
72	C₄⋊C₈	—	23	1	no	—
73	C₄⋊C₈	—	23	1	no	—
74	C₄⋊C₈	—	23	1	no	—
75	C₄×C₈	—	23	1	no	—
76	C₄×C₈	—	23	1	no	—
77	C₄×C₈	C₂×D₂₃	1	1	yes	characteristic
78	C₂₃	#2183	1	1	yes	characteristic, complemented, Sylow
79	C₂×Q₈	—	23	1	no	—
80	C₂×C₈	—	23	1	no	—
81	C₂×C₈	—	23	1	no	—
82	C₂×C₈	C₄×D₂₃	1	1	yes	characteristic
83	C₂×C₈	C₂²×D₂₃	1	1	yes	characteristic
84	C₄⋊C₄	—	23	1	no	—
85	C₄⋊C₄	—	23	1	no	—
86	C₄⋊C₄	—	23	1	no	—
87	C₄²	—	23	1	no	—
88	C₄²	—	23	1	no	—
89	C₄²	C₄×D₂₃	1	1	yes	characteristic
90	C₁₆	—	2	1	no	—
91	C₁₆	—	46	1	no	—
92	C₁₆	—	46	1	no	—
93	C₁₆	Dic₄₆	1	2	yes	complemented
94	Q₈	—	46	2	no	—
95	C₂×C₄	—	23	1	no	—
96	C₂×C₄	—	23	1	no	—
97	C₂×C₄	—	23	1	no	—
98	C₂×C₄	—	23	1	no	—
99	C₂×C₄	C₈×D₂₃	1	1	yes	characteristic
100	C₂×C₄	C₈×D₂₃	1	1	yes	characteristic
101	C₂×C₄	C₂×C₄×D₂₃	1	1	yes	characteristic
102	C₈	—	2	1	no	—
103	C₈	—	46	1	no	—
104	C₈	—	46	1	no	—
105	C₈	C₂×Dic₄₆	1	1	yes	characteristic
106	C₈	#67288	1	1	yes	characteristic
107	C₂²	C₂×C₈×D₂₃	1	1	yes	characteristic
108	C₄	—	2	1	no	—
109	C₄	—	2	1	no	—
110	C₄	—	23	2	no	—
111	C₄	—	46	1	no	—
112	C₄	—	46	1	no	—
113	C₄	—	46	1	no	—
114	C₄	C₄×Dic₄₆	1	1	yes	characteristic
115	C₄	#10625072	1	1	yes	characteristic
116	C₂	C₈×Dic₄₆	1	1	yes	characteristic
117	C₂	M₃₂×D₂₃	1	1	yes	characteristic
118	C₂	#13092835	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₈×Dic₄₆

C₂

yes

maximal, characteristic

C₄⋊(C₂₃⋊C₁₆)

C₂

yes

maximal, characteristic

Dic₂₃⋊C₁₆

C₂

yes

maximal, characteristic

Dic₂₃⋊C₁₆

C₂

yes

maximal, characteristic

#13092192

C₂

yes

maximal, characteristic

C₁₆×Dic₂₃

C₂

yes

maximal, characteristic

C₂₃×#346

C₂

yes

maximal, characteristic

C₄×Dic₄₆

C₄

yes

characteristic

C₂×C₁₆×C₂₃

C₂²

yes

characteristic

C₂×C₁₆×C₂₃

C₂²

yes

characteristic

C₄×C₈×C₂₃

C₂²

yes

characteristic

Dic₂₃⋊C₈

C₄

yes

characteristic

Dic₂₃⋊C₈

C₄

yes

characteristic

C₈×Dic₂₃

C₂²

yes

characteristic

C₈×Dic₂₃

C₂²

yes

characteristic

C₂×C₂₃⋊C₁₆

C₂²

yes

characteristic

C₂×C₂₃⋊C₁₆

C₂²

yes

characteristic

C₄⋊(C₂₃⋊C₈)

C₄

yes

characteristic

C₂×Dic₄₆

C₈

yes

characteristic

C₂×C₈×C₂₃

C₂×C₄

yes

characteristic

C₂×C₈×C₂₃

C₂³

yes

characteristic

C₄²×C₂₃

C₂×C₄

yes

characteristic

C₄⋊Dic₂₃

C₈

yes

characteristic

Dic₂₃⋊C₄

C₈

yes

characteristic

Dic₂₃⋊C₄

C₈

yes

characteristic

C₄×Dic₂₃

C₂×C₄

yes

characteristic

C₄×Dic₂₃

C₂×C₄

yes

characteristic

C₂×C₂₃⋊C₈

C₂×C₄

yes

characteristic

C₂×C₂₃⋊C₈

C₂×C₄

yes

characteristic

C₁₆×C₂₃

—

C₁₆×C₂₃

Q₈

yes

complemented

C₂₃⋊C₁₆

—

C₂₃⋊C₁₆

—

C₂×C₄×C₂₃

C₂×C₈

yes

characteristic

C₂×C₄×C₂₃

C₂×C₈

yes

characteristic

C₂×C₄×C₂₃

C₂²×C₄

yes

characteristic

C₂×Dic₂₃

C₂×C₈

yes

characteristic

C₂×Dic₂₃

C₂×C₈

yes

characteristic

C₂×Dic₂₃

C₂×C₈

yes

characteristic

C₂×Dic₂₃

C₂×C₈

yes

characteristic

Dic₄₆

—

C₈×C₂₃

—

C₈×C₂₃

C₂×Q₈

yes

characteristic

C₈×C₂₃

#41

yes

characteristic

C₂₃⋊C₈

—

C₂₃⋊C₈

—

#2183

—

maximal, Sylow

C₂²×C₂₃

C₂²×C₈

yes

characteristic

C₄×C₂₃

—

C₄×C₂₃

—

C₄×C₂₃

C₄×Q₈

yes

characteristic

C₄×C₂₃

#131

yes

characteristic

Dic₂₃

—

Dic₂₃

—

Dic₂₃

—

Dic₂₃

M₃₂

yes

—

C₈×Q₈

—

C₄⋊C₁₆

—

C₄⋊C₁₆

—

C₄⋊C₁₆

—

#346

—

#346

D₂₃

yes

characteristic

C₄×C₁₆

—

C₂×C₂₃

C₈×Q₈

yes

characteristic

C₂×C₂₃

C₂×M₃₂

yes

characteristic

C₂×C₂₃

#504

yes

characteristic

C₄×Q₈

—

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

C₂×D₂₃

yes

characteristic

C₂×C₁₆

C₂×D₂₃

yes

characteristic

C₄⋊C₈

—

C₄⋊C₈

—

C₄⋊C₈

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

C₂×D₂₃

yes

characteristic

C₂₃

#2183

yes

characteristic, complemented, Sylow

C₂×Q₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

C₄×D₂₃

yes

characteristic

C₂×C₈

C₂²×D₂₃

yes

characteristic

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄²

—

C₄²

—

C₄²

C₄×D₂₃

yes

characteristic

C₁₆

—

C₁₆

—

C₁₆

—

C₁₆

Dic₄₆

yes

complemented

Q₈

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

C₈×D₂₃

yes

characteristic

100

C₂×C₄

C₈×D₂₃

yes

characteristic

101

C₂×C₄

C₂×C₄×D₂₃

yes

characteristic

102

C₈

—

103

C₈

—

104

C₈

—

105

C₈

C₂×Dic₄₆

yes

characteristic

106

C₈

#67288

yes

characteristic

107

C₂²

C₂×C₈×D₂₃

yes

characteristic

108

C₄

—

109

C₄

—

110

C₄

—

111

C₄

—

112

C₄

—

113

C₄

—

114

C₄

C₄×Dic₄₆

yes

characteristic

115

C₄

#10625072

yes

characteristic

116

C₂

C₈×Dic₄₆

yes

characteristic

117

C₂

M₃₂×D₂₃

yes

characteristic

118

C₂

#13092835

yes

characteristic

	subgroup	G/H
center, Frattini	C₂×C₈	C₂²×D₂₃
commutator, socle, prime radical	C₂²×C₂₃	C₂²×C₈
Fitting	C₂₃×#346	C₂
cosocle	C₂×C₈×C₂₃	C₂³
hypercenter	#346	D₂₃
hypocenter	C₂₃	#2183
LCS	G ▷ C₂²×C₂₃ ▷ C₂₃
UCS	C₁ ◁ C₂×C₈ ◁ #346

subgroup

G/H

center, Frattini

C₂×C₈

C₂²×D₂₃

commutator, socle, prime radical

C₂²×C₂₃

C₂²×C₈

Fitting

C₂₃×#346

C₂

cosocle

C₂×C₈×C₂₃

C₂³

D₂₃

C₂₃

G ▷ C₂²×C₂₃ ▷ C₂₃

C₁ ◁ C₂×C₈ ◁ #346

N	H ≅ G/N	type
31. C₁₆×C₂₃	Q₈	semidirect
78. C₂₃	#2183	semidirect
93. C₁₆	Dic₄₆	semidirect

H ≅ G/N

type

31. C₁₆×C₂₃

Q₈

semidirect

78. C₂₃

#2183

semidirect

93. C₁₆

Dic₄₆

semidirect

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

subgroup	index
C₈×Dic₄₆	2
C₄⋊(C₂₃⋊C₁₆)	2
Dic₂₃⋊C₁₆	2
#13092192	2
C₁₆×Dic₂₃	2
C₂₃×#346	2
#2183	23

subgroup

index

C₈×Dic₄₆

C₄⋊(C₂₃⋊C₁₆)

Dic₂₃⋊C₁₆

#13092192

C₁₆×Dic₂₃

C₂₃×#346

#2183

Subgroups — #15675303

Conjugacy classes of subgroups

Complements

Maximal subgroups