Group #15673563 — subgroups

Subgroups — #15673563

Order 2944.

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

normal subgroups only

This group has 256 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	C₂×#10624989	C₂	1	2	yes	maximal
2	C₂×#10624990	C₂	1	2	yes	maximal
3	C₂×#10624985	C₂	1	1	yes	maximal, characteristic
4	#13092283	C₂	1	1	yes	maximal, characteristic
5	C₂₃×#384	C₂	1	1	yes	maximal, characteristic
6	C₂×C₄⋊C₄×C₂₃	—	2	2	no	—
7	C₂×C₄⋊C₄×C₂₃	C₂²	1	2	yes	—
8	C₂×C₄²×C₂₃	C₂²	1	1	yes	characteristic
9	C₂×C₄⋊Dic₂₃	C₂²	1	1	yes	characteristic
10	C₂×C₄⋊Dic₂₃	C₂²	1	1	yes	characteristic
11	C₂²×C₂₃⋊C₈	C₂²	1	2	yes	—
12	C₂₃⋊#95	—	2	2	no	—
13	#10624990	—	2	4	no	—
14	#10624989	—	2	4	no	—
15	#10624985	C₄	1	4	yes	complemented
16	C₂²×C₄×C₂₃	—	2	2	no	—
17	C₂²×C₄×C₂₃	D₄	1	2	yes	—
18	C₂²×C₄×C₂₃	C₂³	1	1	yes	characteristic
19	C₂²×Dic₂₃	—	2	1	no	—
20	C₂²×Dic₂₃	—	2	1	no	—
21	C₄⋊C₄×C₂₃	—	2	2	no	—
22	C₄⋊C₄×C₂₃	—	2	2	no	—
23	C₄⋊C₄×C₂₃	—	4	4	no	—
24	C₄²×C₂₃	C₂×C₄	1	2	yes	—
25	C₄²×C₂₃	Q₈	1	2	yes	—
26	C₄⋊Dic₂₃	—	2	2	no	—
27	C₄⋊Dic₂₃	—	2	2	no	—
28	C₂×C₂₃⋊C₈	C₂×C₄	1	4	yes	—
29	C₂×C₂₃⋊C₈	D₄	1	4	yes	—
30	C₂×C₂₃⋊C₈	Q₈	1	4	yes	—
31	C₂³×C₂₃	C₂×D₄	1	1	yes	characteristic
32	C₂×C₄×C₂₃	—	2	2	no	—
33	C₂×C₄×C₂₃	—	2	2	no	—
34	C₂×C₄×C₂₃	—	4	2	no	—
35	C₂×C₄×C₂₃	—	4	4	no	—
36	C₂×C₄×C₂₃	C₄⋊C₄	1	4	yes	—
37	C₂×C₄×C₂₃	C₂²×C₄	1	1	yes	characteristic
38	C₂×C₄×C₂₃	C₂×D₄	1	2	yes	—
39	C₂×C₄×C₂₃	C₂×Q₈	1	1	yes	characteristic
40	C₂×C₄×C₂₃	C₂×Q₈	1	2	yes	—
41	C₂×Dic₂₃	—	4	1	no	—
42	C₂×Dic₂₃	—	4	1	no	—
43	C₂×Dic₂₃	—	4	2	no	—
44	C₂×Dic₂₃	—	4	2	no	—
45	C₂₃⋊C₈	C₄⋊C₄	1	8	yes	complemented
46	#1847	—	23	1	no	maximal, Sylow
47	C₂²×C₂₃	C₂×C₄⋊C₄	1	1	yes	characteristic
48	C₂²×C₂₃	#127	1	1	yes	characteristic
49	C₂²×C₂₃	C₄⋊Q₈	1	1	yes	characteristic
50	C₂²×C₂₃	AGL(1,ℤ/8)	1	2	yes	—
51	C₂²×C₂₃	#137	1	2	yes	—
52	C₄×C₂₃	—	2	4	no	—
53	C₄×C₂₃	—	8	2	no	—
54	C₄×C₂₃	C₂×C₄⋊C₄	1	2	yes	—
55	C₄×C₂₃	C₄⋊Q₈	1	2	yes	—
56	Dic₂₃	—	8	1	no	—
57	Dic₂₃	—	8	1	no	—
58	C₂×#107	—	23	2	no	—
59	C₂×#106	—	23	2	no	—
60	C₂×#97	—	23	1	no	—
61	#384	D₂₃	1	1	yes	characteristic
62	#382	—	23	1	no	—
63	C₂×C₂₃	#384	1	1	yes	characteristic
64	C₂×C₂₃	#428	1	2	yes	—
65	C₂×C₂₃	#496	1	1	yes	characteristic
66	C₂×C₂₃	#497	1	1	yes	characteristic
67	C₂×C₂₃	#501	1	2	yes	—
68	C₂²×C₈	—	23	2	no	—
69	C₂×C₄⋊C₄	—	2	2	no	—
70	C₂×C₄⋊C₄	—	23	1	no	—
71	C₂×C₄⋊C₄	—	23	1	no	—
72	C₂×C₄⋊C₄	C₂×D₂₃	1	2	yes	—
73	C₂×C₄²	C₂×D₂₃	1	1	yes	characteristic
74	#107	—	46	4	no	—
75	#106	—	46	4	no	—
76	#97	—	23	4	no	—
77	#95	—	46	2	no	—
78	C₂₃	#1847	1	1	yes	characteristic, complemented, Sylow
79	C₂²×C₄	—	2	2	no	—
80	C₂²×C₄	—	46	1	no	—
81	C₂²×C₄	—	46	1	no	—
82	C₂²×C₄	C₂²⋊D₂₃	1	2	yes	—
83	C₂²×C₄	C₂²×D₂₃	1	1	yes	characteristic
84	C₂×C₈	—	23	4	no	—
85	C₂×C₈	—	23	4	no	—
86	C₂×C₈	—	23	4	no	—
87	C₄⋊C₄	—	2	2	no	—
88	C₄⋊C₄	—	2	2	no	—
89	C₄⋊C₄	—	4	4	no	—
90	C₄⋊C₄	—	46	2	no	—
91	C₄⋊C₄	—	46	2	no	—
92	C₄²	Dic₄₆	1	2	yes	—
93	C₄²	C₄×D₂₃	1	2	yes	—
94	C₂³	C₂×C₂²⋊D₂₃	1	1	yes	characteristic
95	C₂×C₄	—	2	2	no	—
96	C₂×C₄	—	2	2	no	—
97	C₂×C₄	—	4	4	no	—
98	C₂×C₄	—	4	2	no	—
99	C₂×C₄	—	92	2	no	—
100	C₂×C₄	—	92	1	no	—
101	C₂×C₄	—	92	1	no	—
102	C₂×C₄	—	92	2	no	—
103	C₂×C₄	Dic₂₃⋊C₄	1	4	yes	—
104	C₂×C₄	C₂×Dic₄₆	1	1	yes	characteristic
105	C₂×C₄	C₂×C₄×D₂₃	1	1	yes	characteristic
106	C₂×C₄	D₄×D₂₃	1	2	yes	—
107	C₂×C₄	Q₈×D₂₃	1	2	yes	—
108	C₈	—	23	8	no	—
109	C₂²	C₂×Dic₂₃⋊C₄	1	1	yes	characteristic
110	C₂²	#10625103	1	2	yes	—
111	C₂²	#10625111	1	1	yes	characteristic
112	C₂²	#10625113	1	2	yes	—
113	C₂²	#10625115	1	1	yes	characteristic
114	C₄	—	2	4	no	—
115	C₄	—	8	2	no	—
116	C₄	—	184	1	no	—
117	C₄	—	184	1	no	—
118	C₄	#10625059	1	2	yes	—
119	C₄	C₄⋊C₄×D₂₃	1	2	yes	—
120	C₂	#13092286	1	1	yes	characteristic
121	C₂	#13092457	1	2	yes	—
122	C₂	#13092804	1	1	yes	characteristic
123	C₂	#13092809	1	1	yes	characteristic
124	C₂	#13092823	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#10624989

C₂

yes

maximal

C₂×#10624990

C₂

yes

maximal

C₂×#10624985

C₂

yes

maximal, characteristic

#13092283

C₂

yes

maximal, characteristic

C₂₃×#384

C₂

yes

maximal, characteristic

C₂×C₄⋊C₄×C₂₃

—

C₂×C₄⋊C₄×C₂₃

C₂²

yes

—

C₂×C₄²×C₂₃

C₂²

yes

characteristic

C₂×C₄⋊Dic₂₃

C₂²

yes

characteristic

C₂×C₄⋊Dic₂₃

C₂²

yes

characteristic

C₂²×C₂₃⋊C₈

C₂²

yes

—

C₂₃⋊#95

—

#10624990

—

#10624989

—

#10624985

C₄

yes

complemented

C₂²×C₄×C₂₃

—

C₂²×C₄×C₂₃

D₄

yes

—

C₂²×C₄×C₂₃

C₂³

yes

characteristic

C₂²×Dic₂₃

—

C₂²×Dic₂₃

—

C₄⋊C₄×C₂₃

—

C₄⋊C₄×C₂₃

—

C₄⋊C₄×C₂₃

—

C₄²×C₂₃

C₂×C₄

yes

—

C₄²×C₂₃

Q₈

yes

—

C₄⋊Dic₂₃

—

C₄⋊Dic₂₃

—

C₂×C₂₃⋊C₈

C₂×C₄

yes

—

C₂×C₂₃⋊C₈

D₄

yes

—

C₂×C₂₃⋊C₈

Q₈

yes

—

C₂³×C₂₃

C₂×D₄

yes

characteristic

C₂×C₄×C₂₃

—

C₂×C₄×C₂₃

—

C₂×C₄×C₂₃

—

C₂×C₄×C₂₃

—

C₂×C₄×C₂₃

C₄⋊C₄

yes

—

C₂×C₄×C₂₃

C₂²×C₄

yes

characteristic

C₂×C₄×C₂₃

C₂×D₄

yes

—

C₂×C₄×C₂₃

C₂×Q₈

yes

characteristic

C₂×C₄×C₂₃

C₂×Q₈

yes

—

C₂×Dic₂₃

—

C₂×Dic₂₃

—

C₂×Dic₂₃

—

C₂×Dic₂₃

—

C₂₃⋊C₈

C₄⋊C₄

yes

complemented

#1847

—

maximal, Sylow

C₂²×C₂₃

C₂×C₄⋊C₄

yes

characteristic

C₂²×C₂₃

#127

yes

characteristic

C₂²×C₂₃

C₄⋊Q₈

yes

characteristic

C₂²×C₂₃

AGL(1,ℤ/8)

yes

—

C₂²×C₂₃

#137

yes

—

C₄×C₂₃

—

C₄×C₂₃

—

C₄×C₂₃

C₂×C₄⋊C₄

yes

—

C₄×C₂₃

C₄⋊Q₈

yes

—

Dic₂₃

—

Dic₂₃

—

C₂×#107

—

C₂×#106

—

C₂×#97

—

#384

D₂₃

yes

characteristic

#382

—

C₂×C₂₃

#384

yes

characteristic

C₂×C₂₃

#428

yes

—

C₂×C₂₃

yes

C₂×C₂₃

yes

C₂×C₂₃

yes

—

C₂²×C₈

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

C₂×D₂₃

yes

—

C₂×C₄²

C₂×D₂₃

yes

—

—

—

—

C₂₃

#1847

yes

characteristic, complemented, Sylow

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

C₂²⋊D₂₃

yes

—

C₂²×C₄

C₂²×D₂₃

yes

characteristic

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄²

Dic₄₆

yes

—

C₄²

C₄×D₂₃

yes

—

C₂³

C₂×C₂²⋊D₂₃

yes

characteristic

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

100

C₂×C₄

—

101

C₂×C₄

—

102

C₂×C₄

—

103

C₂×C₄

Dic₂₃⋊C₄

yes

—

104

C₂×C₄

C₂×Dic₄₆

yes

characteristic

105

C₂×C₄

C₂×C₄×D₂₃

yes

characteristic

106

C₂×C₄

D₄×D₂₃

yes

—

107

C₂×C₄

Q₈×D₂₃

yes

—

108

C₈

—

109

C₂²

C₂×Dic₂₃⋊C₄

yes

characteristic

110

C₂²

#10625103

yes

—

111

C₂²

#10625111

yes

characteristic

112

C₂²

#10625113

yes

—

113

C₂²

#10625115

yes

characteristic

114

C₄

—

115

C₄

—

116

C₄

—

184

—

117

C₄

—

184

—

118

C₄

#10625059

yes

—

119

C₄

C₄⋊C₄×D₂₃

yes

—

120

C₂

#13092286

yes

characteristic

121

C₂

#13092457

yes

—

122

C₂

#13092804

yes

characteristic

123

C₂

#13092809

yes

characteristic

124

C₂

#13092823

yes

—

	subgroup	G/H
center	C₂³	C₂×C₂²⋊D₂₃
commutator	C₂×C₄×C₂₃	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₂₃
Fitting	C₂₃×#384	C₂
socle, prime radical	C₂³×C₂₃	C₂×D₄
cosocle	C₂²×C₄×C₂₃	C₂³
hypercenter	#384	D₂₃
hypocenter	C₂₃	#1847
LCS	G ▷ C₂×C₄×C₂₃ ▷ C₂×C₂₃ ▷ C₂₃
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #384

subgroup

G/H

center

C₂³

C₂×C₂²⋊D₂₃

commutator

C₂×C₄×C₂₃

C₂²×C₄

Frattini

C₂²×C₄

C₂²×D₂₃

Fitting

C₂₃×#384

C₂

socle, prime radical

C₂³×C₂₃

C₂×D₄

cosocle

C₂²×C₄×C₂₃

C₂³

D₂₃

C₂₃

G ▷ C₂×C₄×C₂₃ ▷ C₂×C₂₃ ▷ C₂₃

UCS

C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #384

N	H ≅ G/N	type
15. #10624985	C₄	semidirect
45. C₂₃⋊C₈	C₄⋊C₄	semidirect
78. C₂₃	#1847	semidirect

H ≅ G/N

type

15. #10624985

C₄

semidirect

45. C₂₃⋊C₈

C₄⋊C₄

semidirect

78. C₂₃

#1847

semidirect

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

subgroup	index
C₂×#10624989	2
C₂×#10624990	2
C₂×#10624985	2
#13092283	2
C₂₃×#384	2
#1847	23

subgroup

index

C₂₃×#384

Subgroups — #15673563

Conjugacy classes of subgroups

Complements

Maximal subgroups