Group #15671741 — subgroups

Subgroups — #15671741

Order 2944.

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

normal subgroups only

This group has 236 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	#13092474	C₂	1	2	yes	maximal
2	#13092143	C₂	1	4	yes	maximal
3	C₂×C₄×Q₈×C₂₃	C₂	1	1	yes	maximal, characteristic
4	C₂²×Q₈×C₂₃	C₄	1	1	yes	characteristic
5	C₄×Q₈×C₂₃	—	2	2	no	—
6	C₄×Q₈×C₂₃	—	2	2	no	—
7	C₂×C₄⋊C₄×C₂₃	C₄	1	1	yes	characteristic
8	C₂×C₄⋊C₄×C₂₃	C₂²	1	2	yes	—
9	C₂×C₄²×C₂₃	C₄	1	2	yes	—
10	C₂×C₄²×C₂₃	C₂²	1	1	yes	characteristic
11	C₂²⋊(C₂₃⋊C₈)	C₂²	1	4	yes	—
12	C₄⋊(C₂₃⋊C₈)	—	2	2	no	—
13	#10624985	—	2	2	no	—
14	C₂×Q₈×C₂₃	—	2	2	no	—
15	C₂×Q₈×C₂₃	—	2	4	no	—
16	C₂²×C₄×C₂₃	C₂×C₄	1	1	yes	characteristic
17	C₂²×C₄×C₂₃	C₂×C₄	1	1	yes	characteristic
18	C₂²×C₄×C₂₃	C₂×C₄	1	2	yes	—
19	C₂²×C₄×C₂₃	C₂×C₄	1	2	yes	—
20	C₂²×C₄×C₂₃	C₂³	1	1	yes	characteristic
21	C₄⋊C₄×C₂₃	—	2	2	no	—
22	C₄⋊C₄×C₂₃	—	2	4	no	—
23	C₄²×C₂₃	—	2	2	no	—
24	C₄²×C₂₃	—	2	2	no	—
25	C₄²×C₂₃	D₄	1	2	yes	—
26	C₄²×C₂₃	D₄	1	2	yes	—
27	C₂×C₂₃⋊C₈	—	2	4	no	—
28	C₂³×C₂₃	C₂²×C₄	1	1	yes	characteristic
29	Q₈×C₂₃	—	2	8	no	—
30	C₂×C₄×C₂₃	—	2	2	no	—
31	C₂×C₄×C₂₃	—	2	2	no	—
32	C₂×C₄×C₂₃	—	2	2	no	—
33	C₂×C₄×C₂₃	—	2	2	no	—
34	C₂×C₄×C₂₃	—	2	2	no	—
35	C₂×C₄×C₂₃	—	4	1	no	—
36	C₂×C₄×C₂₃	—	4	2	no	—
37	C₂×C₄×C₂₃	C₂²⋊C₄	1	1	yes	characteristic
38	C₂×C₄×C₂₃	C₂²⋊C₄	1	2	yes	—
39	C₂×C₄×C₂₃	C₂²⋊C₄	1	1	yes	characteristic
40	C₂×C₄×C₂₃	M₁₆	1	4	yes	—
41	C₂×C₄×C₂₃	C₂×D₄	1	2	yes	—
42	C₂₃⋊C₈	—	4	4	no	—
43	#1467	—	23	1	no	maximal, Sylow
44	C₂²×C₂₃	—	2	1	no	—
45	C₂²×C₂₃	—	2	1	no	—
46	C₂²×C₂₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
47	C₂²×C₂₃	C₂×M₁₆	1	2	yes	—
48	C₄×C₂₃	—	2	2	no	—
49	C₄×C₂₃	—	2	4	no	—
50	C₄×C₂₃	—	2	2	no	—
51	C₄×C₂₃	—	2	1	no	—
52	C₄×C₂₃	—	4	2	no	—
53	C₄×C₂₃	#101	1	2	yes	—
54	C₂×C₄×Q₈	D₂₃	1	1	yes	characteristic
55	#432	—	23	2	no	—
56	#324	—	23	4	no	—
57	C₂×C₂₃	—	2	2	no	—
58	C₂×C₂₃	#407	1	1	yes	characteristic
59	C₂×C₂₃	#410	1	1	yes	characteristic
60	C₂×C₂₃	C₂×#101	1	1	yes	characteristic
61	C₂²×Q₈	Dic₂₃	1	1	yes	characteristic
62	C₄×Q₈	—	2	2	no	—
63	C₄×Q₈	—	2	2	no	—
64	C₂×C₄⋊C₄	Dic₂₃	1	1	yes	characteristic
65	C₂×C₄⋊C₄	C₂×D₂₃	1	2	yes	—
66	C₂×C₄²	Dic₂₃	1	2	yes	—
67	C₂×C₄²	C₂×D₂₃	1	1	yes	characteristic
68	C₄⋊C₈	—	46	2	no	—
69	C₂²⋊C₈	—	23	4	no	—
70	#97	—	46	2	no	—
71	C₂₃	#1467	1	1	yes	characteristic, complemented, Sylow
72	C₂×Q₈	—	2	4	no	—
73	C₂×Q₈	—	2	2	no	—
74	C₂²×C₄	C₂×Dic₂₃	1	2	yes	—
75	C₂²×C₄	C₂×Dic₂₃	1	1	yes	characteristic
76	C₂²×C₄	C₂×Dic₂₃	1	1	yes	characteristic
77	C₂²×C₄	C₂×Dic₂₃	1	2	yes	—
78	C₂²×C₄	C₂²×D₂₃	1	1	yes	characteristic
79	C₂×C₈	—	46	4	no	—
80	C₄⋊C₄	—	2	2	no	—
81	C₄⋊C₄	—	2	4	no	—
82	C₄²	—	2	2	no	—
83	C₄²	—	2	2	no	—
84	C₄²	C₂²⋊D₂₃	1	2	yes	—
85	C₄²	C₂²⋊D₂₃	1	2	yes	—
86	C₂³	C₂²×Dic₂₃	1	1	yes	characteristic
87	Q₈	—	2	8	no	—
88	C₂×C₄	—	2	2	no	—
89	C₂×C₄	—	2	2	no	—
90	C₂×C₄	—	2	2	no	—
91	C₂×C₄	—	2	2	no	—
92	C₂×C₄	—	2	2	no	—
93	C₂×C₄	—	4	1	no	—
94	C₂×C₄	—	4	2	no	—
95	C₂×C₄	#67267	1	4	yes	—
96	C₂×C₄	C₂²⋊Dic₂₃	1	1	yes	characteristic
97	C₂×C₄	C₂²⋊Dic₂₃	1	2	yes	—
98	C₂×C₄	C₂²⋊Dic₂₃	1	1	yes	characteristic
99	C₂×C₄	C₂×C₂²⋊D₂₃	1	2	yes	—
100	C₈	—	92	4	no	—
101	C₂²	—	2	1	no	—
102	C₂²	—	2	1	no	—
103	C₂²	C₂×#67267	1	2	yes	—
104	C₂²	C₂×C₂²⋊Dic₂₃	1	1	yes	characteristic
105	C₄	—	2	4	no	—
106	C₄	—	2	2	no	—
107	C₄	—	2	2	no	—
108	C₄	—	2	1	no	—
109	C₄	—	4	2	no	—
110	C₄	#10625018	1	2	yes	—
111	C₂	—	2	2	no	—
112	C₂	#13092381	1	1	yes	characteristic
113	C₂	#13092391	1	1	yes	characteristic
114	C₂	C₂×#10625018	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#13092474

C₂

yes

maximal

#13092143

C₂

yes

maximal

C₂×C₄×Q₈×C₂₃

C₂

yes

maximal, characteristic

C₂²×Q₈×C₂₃

C₄

yes

characteristic

C₄×Q₈×C₂₃

—

C₄×Q₈×C₂₃

—

C₂×C₄⋊C₄×C₂₃

C₄

yes

characteristic

C₂×C₄⋊C₄×C₂₃

C₂²

yes

—

C₂×C₄²×C₂₃

C₄

yes

—

C₂×C₄²×C₂₃

C₂²

yes

characteristic

C₂²⋊(C₂₃⋊C₈)

C₂²

yes

—

C₄⋊(C₂₃⋊C₈)

—

#10624985

—

C₂×Q₈×C₂₃

—

C₂×Q₈×C₂₃

—

C₂²×C₄×C₂₃

C₂×C₄

yes

characteristic

C₂²×C₄×C₂₃

C₂×C₄

yes

characteristic

C₂²×C₄×C₂₃

C₂×C₄

yes

—

C₂²×C₄×C₂₃

C₂×C₄

yes

—

C₂²×C₄×C₂₃

C₂³

yes

characteristic

C₄⋊C₄×C₂₃

—

C₄⋊C₄×C₂₃

—

C₄²×C₂₃

—

C₄²×C₂₃

—

C₄²×C₂₃

D₄

yes

—

C₄²×C₂₃

D₄

yes

—

C₂×C₂₃⋊C₈

—

C₂³×C₂₃

C₂²×C₄

yes

characteristic

Q₈×C₂₃

—

C₂×C₄×C₂₃

—

C₂×C₄×C₂₃

—

C₂×C₄×C₂₃

—

C₂×C₄×C₂₃

—

C₂×C₄×C₂₃

—

C₂×C₄×C₂₃

—

C₂×C₄×C₂₃

—

C₂×C₄×C₂₃

C₂²⋊C₄

yes

characteristic

C₂×C₄×C₂₃

C₂²⋊C₄

yes

—

C₂×C₄×C₂₃

C₂²⋊C₄

yes

characteristic

C₂×C₄×C₂₃

M₁₆

yes

—

C₂×C₄×C₂₃

C₂×D₄

yes

—

C₂₃⋊C₈

—

#1467

—

maximal, Sylow

C₂²×C₂₃

—

C₂²×C₂₃

—

C₂²×C₂₃

C₂×C₂²⋊C₄

yes

characteristic

C₂²×C₂₃

C₂×M₁₆

yes

—

C₄×C₂₃

—

C₄×C₂₃

—

C₄×C₂₃

—

C₄×C₂₃

—

C₄×C₂₃

—

C₄×C₂₃

#101

yes

—

C₂×C₄×Q₈

D₂₃

yes

characteristic

#432

—

#324

—

C₂×C₂₃

—

C₂×C₂₃

yes

C₂×C₂₃

yes

C₂×C₂₃

yes

C₂²×Q₈

Dic₂₃

yes

characteristic

C₄×Q₈

—

C₄×Q₈

—

C₂×C₄⋊C₄

Dic₂₃

yes

characteristic

C₂×C₄⋊C₄

C₂×D₂₃

yes

—

C₂×C₄²

Dic₂₃

yes

—

C₂×C₄²

C₂×D₂₃

yes

characteristic

C₄⋊C₈

—

C₂²⋊C₈

—

#97

—

C₂₃

#1467

yes

characteristic, complemented, Sylow

C₂×Q₈

—

C₂×Q₈

—

C₂²×C₄

C₂×Dic₂₃

yes

—

C₂²×C₄

C₂×Dic₂₃

yes

characteristic

C₂²×C₄

C₂×Dic₂₃

yes

characteristic

C₂²×C₄

C₂×Dic₂₃

yes

—

C₂²×C₄

C₂²×D₂₃

yes

characteristic

C₂×C₈

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄²

—

C₄²

—

C₄²

C₂²⋊D₂₃

yes

—

C₄²

C₂²⋊D₂₃

yes

—

C₂³

C₂²×Dic₂₃

yes

characteristic

Q₈

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

#67267

yes

—

C₂×C₄

C₂²⋊Dic₂₃

yes

characteristic

C₂×C₄

C₂²⋊Dic₂₃

yes

—

C₂×C₄

C₂²⋊Dic₂₃

yes

characteristic

C₂×C₄

C₂×C₂²⋊D₂₃

yes

—

100

C₈

—

101

C₂²

—

102

C₂²

—

103

C₂²

C₂×#67267

yes

—

104

C₂²

C₂×C₂²⋊Dic₂₃

yes

characteristic

105

C₄

—

106

C₄

—

107

C₄

—

108

C₄

—

109

C₄

—

110

C₄

#10625018

yes

—

111

C₂

—

112

C₂

#13092381

yes

characteristic

113

C₂

#13092391

yes

characteristic

114

C₂

C₂×#10625018

yes

characteristic

	subgroup	G/H
center	C₂²	C₂×C₂²⋊Dic₂₃
commutator, prime radical	C₂³×C₂₃	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₂₃
Fitting	C₂×C₄×Q₈×C₂₃	C₂
socle	C₂²×C₂₃	C₂×C₂²⋊C₄
cosocle	C₂²×C₄×C₂₃	C₂³
hypercenter	C₂×C₄×Q₈	D₂₃
hypocenter	C₂₃	#1467
LCS	G ▷ C₂³×C₂₃ ▷ C₂×C₂₃ ▷ C₂₃
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×C₄×Q₈

subgroup

G/H

center

C₂²

C₂×C₂²⋊Dic₂₃

commutator, prime radical

C₂³×C₂₃

C₂²×C₄

Frattini

C₂²×C₄

C₂²×D₂₃

Fitting

C₂×C₄×Q₈×C₂₃

C₂

socle

C₂²×C₂₃

C₂×C₂²⋊C₄

cosocle

C₂²×C₄×C₂₃

C₂³

hypercenter

C₂×C₄×Q₈

D₂₃

hypocenter

C₂₃

#1467

LCS

G ▷ C₂³×C₂₃ ▷ C₂×C₂₃ ▷ C₂₃

UCS

C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×C₄×Q₈

N	H ≅ G/N	type
71. C₂₃	#1467	semidirect

H ≅ G/N

type

71. C₂₃

#1467

semidirect

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

subgroup	index
#13092474	2
#13092143	2
C₂×C₄×Q₈×C₂₃	2
#1467	23

subgroup

index

#13092474

#13092143

C₂×C₄×Q₈×C₂₃

#1467

Subgroups — #15671741

Conjugacy classes of subgroups

Complements

Maximal subgroups