Group #15671266 — subgroups

Subgroups — #15671266

Order 2944.

136 conjugacy classes of subgroups; only the 61 normal subgroups are shown.

normal subgroups only

This group has 136 conjugacy classes of subgroups — too many to draw. Only its 61 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	1	yes	maximal, characteristic
2	C₂×C₄⋊(C₂₃⋊C₈)	C₂	1	1	yes	maximal, characteristic
3	C₂₃×#432	C₂	1	1	yes	maximal, characteristic
4	C₂×C₄²×C₂₃	C₂²	1	1	yes	characteristic
5	C₂²×C₂₃⋊C₈	C₄	1	2	yes	—
6	C₂₃×C₄⋊C₈	—	2	1	no	—
7	C₂₃×C₂²⋊C₈	—	2	1	no	—
8	C₂₃×#97	C₄	1	2	yes	—
9	C₂²⋊(C₂₃⋊C₈)	—	2	1	no	—
10	C₄⋊(C₂₃⋊C₈)	—	2	1	no	—
11	C₄⋊(C₂₃⋊C₈)	—	2	1	no	—
12	C₄⋊(C₂₃⋊C₈)	—	2	1	no	—
13	#10624985	C₄	1	2	yes	—
14	C₂²×C₄×C₂₃	C₂×C₄	1	1	yes	characteristic
15	C₂²×C₄×C₂₃	D₄	1	1	yes	characteristic
16	C₂²×C₄×C₂₃	Q₈	1	1	yes	characteristic
17	C₂×C₈×C₂₃	—	2	2	no	—
18	C₄²×C₂₃	C₂×C₄	1	1	yes	characteristic
19	C₄²×C₂₃	C₂×C₄	1	1	yes	characteristic
20	C₄²×C₂₃	D₄	1	1	yes	characteristic
21	C₄²×C₂₃	D₄	1	1	yes	characteristic
22	C₂×C₂₃⋊C₈	—	2	2	no	—
23	C₂×C₂₃⋊C₈	—	2	2	no	—
24	C₂×C₂₃⋊C₈	—	4	2	no	—
25	C₂³×C₂₃	C₄⋊C₄	1	1	yes	characteristic
26	C₂×C₄×C₂₃	—	2	2	no	—
27	C₂×C₄×C₂₃	—	2	1	no	—
28	C₂×C₄×C₂₃	—	2	1	no	—
29	C₂×C₄×C₂₃	—	4	1	no	—
30	C₂×C₄×C₂₃	C₄²	1	1	yes	characteristic
31	C₂×C₄×C₂₃	C₂²⋊C₄	1	1	yes	characteristic
32	C₂×C₄×C₂₃	C₂²⋊C₄	1	1	yes	characteristic
33	C₂×C₄×C₂₃	C₂²⋊C₄	1	1	yes	characteristic
34	C₂×C₄×C₂₃	C₄⋊C₄	1	1	yes	characteristic
35	C₂×C₄×C₂₃	C₄⋊C₄	1	1	yes	characteristic
36	C₈×C₂₃	—	4	2	no	—
37	C₂₃⋊C₈	—	4	2	no	—
38	C₂₃⋊C₈	—	4	2	no	—
39	#1290	—	23	1	no	maximal, Sylow
40	C₂²×C₂₃	—	2	1	no	—
41	C₂²×C₂₃	—	2	1	no	—
42	C₂²×C₂₃	#95	1	1	yes	characteristic
43	C₂²×C₂₃	#108	1	2	yes	—
44	C₄×C₂₃	—	2	1	no	—
45	C₄×C₂₃	—	2	1	no	—
46	C₄×C₂₃	—	2	1	no	—
47	C₄×C₂₃	—	2	1	no	—
48	C₄×C₂₃	—	2	1	no	—
49	C₄×C₂₃	#100	1	1	yes	characteristic
50	C₄×C₂₃	#101	1	1	yes	characteristic
51	#432	—	23	1	no	—
52	#432	D₂₃	1	1	yes	characteristic
53	C₂×C₄⋊C₈	—	23	1	no	—
54	C₂×C₂₃	—	2	2	no	—
55	C₂×C₂₃	H(3,ℤ/4)	1	1	yes	characteristic
56	C₂×C₂₃	#341	1	1	yes	characteristic
57	C₂×C₂₃	#343	1	1	yes	characteristic
58	C₂²×C₈	—	23	2	no	—
59	C₂×C₄²	C₂×D₂₃	1	1	yes	characteristic
60	C₄⋊C₈	—	2	1	no	—
61	C₄⋊C₈	—	46	1	no	—
62	C₄⋊C₈	—	46	1	no	—
63	C₄⋊C₈	—	46	1	no	—
64	C₂²⋊C₈	—	2	1	no	—
65	C₂²⋊C₈	—	46	1	no	—
66	#97	—	23	2	no	—
67	#97	Dic₂₃	1	2	yes	—
68	C₂₃	#1290	1	1	yes	characteristic, complemented, Sylow
69	C₂²×C₄	Dic₄₆	1	1	yes	characteristic
70	C₂²×C₄	C₄×D₂₃	1	1	yes	characteristic
71	C₂²×C₄	C₂²⋊D₂₃	1	1	yes	characteristic
72	C₂×C₈	—	2	2	no	—
73	C₂×C₈	—	46	2	no	—
74	C₂×C₈	—	46	2	no	—
75	C₂×C₈	—	92	2	no	—
76	C₄²	C₄×D₂₃	1	1	yes	characteristic
77	C₄²	D₉₂	1	1	yes	characteristic
78	C₄²	C₂×Dic₂₃	1	1	yes	characteristic
79	C₄²	C₂²⋊D₂₃	1	1	yes	characteristic
80	C₂³	Dic₂₃⋊C₄	1	1	yes	characteristic
81	C₂×C₄	—	2	2	no	—
82	C₂×C₄	—	2	1	no	—
83	C₂×C₄	—	2	1	no	—
84	C₂×C₄	—	4	1	no	—
85	C₂×C₄	C₄×Dic₂₃	1	1	yes	characteristic
86	C₂×C₄	Dic₂₃⋊C₄	1	1	yes	characteristic
87	C₂×C₄	C₄⋊Dic₂₃	1	1	yes	characteristic
88	C₂×C₄	(C₂×D₂₃)⋊C₄	1	1	yes	characteristic
89	C₂×C₄	(C₂×D₂₃)⋊C₄	1	1	yes	characteristic
90	C₂×C₄	C₂²⋊Dic₂₃	1	1	yes	characteristic
91	C₈	—	4	2	no	—
92	C₈	—	92	2	no	—
93	C₈	—	92	2	no	—
94	C₂²	—	2	1	no	—
95	C₂²	—	2	1	no	—
96	C₂²	#10625004	1	2	yes	—
97	C₂²	C₂₃⋊#95	1	1	yes	characteristic
98	C₄	—	2	1	no	—
99	C₄	—	2	1	no	—
100	C₄	—	2	1	no	—
101	C₄	—	2	1	no	—
102	C₄	—	2	1	no	—
103	C₄	#10625005	1	1	yes	characteristic
104	C₄	#10625006	1	1	yes	characteristic
105	C₂	—	2	2	no	—
106	C₂	#13092173	1	1	yes	characteristic
107	C₂	#13092181	1	1	yes	characteristic
108	C₂	#13092185	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#13092474

C₂

yes

maximal, characteristic

C₂×C₄⋊(C₂₃⋊C₈)

C₂

yes

maximal, characteristic

C₂₃×#432

C₂

yes

maximal, characteristic

C₂×C₄²×C₂₃

C₂²

yes

characteristic

C₂²×C₂₃⋊C₈

C₄

yes

—

C₂₃×C₄⋊C₈

—

C₂₃×C₂²⋊C₈

—

C₂₃×#97

C₄

yes

—

C₂²⋊(C₂₃⋊C₈)

—

C₄⋊(C₂₃⋊C₈)

—

C₄⋊(C₂₃⋊C₈)

—

C₄⋊(C₂₃⋊C₈)

—

#10624985

C₄

yes

—

C₂²×C₄×C₂₃

C₂×C₄

yes

characteristic

C₂²×C₄×C₂₃

D₄

yes

characteristic

C₂²×C₄×C₂₃

Q₈

yes

characteristic

C₂×C₈×C₂₃

—

C₄²×C₂₃

C₂×C₄

yes

characteristic

C₄²×C₂₃

C₂×C₄

yes

characteristic

C₄²×C₂₃

D₄

yes

characteristic

C₄²×C₂₃

D₄

yes

characteristic

C₂×C₂₃⋊C₈

—

C₂×C₂₃⋊C₈

—

C₂×C₂₃⋊C₈

—

C₂³×C₂₃

C₄⋊C₄

yes

characteristic

C₂×C₄×C₂₃

—

C₂×C₄×C₂₃

—

C₂×C₄×C₂₃

—

C₂×C₄×C₂₃

—

C₂×C₄×C₂₃

C₄²

yes

characteristic

C₂×C₄×C₂₃

C₂²⋊C₄

yes

characteristic

C₂×C₄×C₂₃

C₂²⋊C₄

yes

characteristic

C₂×C₄×C₂₃

C₂²⋊C₄

yes

characteristic

C₂×C₄×C₂₃

C₄⋊C₄

yes

characteristic

C₂×C₄×C₂₃

C₄⋊C₄

yes

characteristic

C₈×C₂₃

—

C₂₃⋊C₈

—

C₂₃⋊C₈

—

#1290

—

maximal, Sylow

C₂²×C₂₃

—

C₂²×C₂₃

—

C₂²×C₂₃

#95

yes

characteristic

C₂²×C₂₃

#108

yes

—

C₄×C₂₃

—

C₄×C₂₃

—

C₄×C₂₃

—

C₄×C₂₃

—

C₄×C₂₃

—

C₄×C₂₃

yes

C₄×C₂₃

yes

—

D₂₃

yes

characteristic

C₂×C₄⋊C₈

—

C₂×C₂₃

—

C₂×C₂₃

yes

C₂×C₂₃

yes

C₂×C₂₃

yes

C₂²×C₈

—

C₂×C₄²

C₂×D₂₃

yes

characteristic

C₄⋊C₈

—

C₄⋊C₈

—

C₄⋊C₈

—

C₄⋊C₈

—

C₂²⋊C₈

—

C₂²⋊C₈

—

#97

—

#97

Dic₂₃

yes

—

C₂₃

#1290

yes

characteristic, complemented, Sylow

C₂²×C₄

Dic₄₆

yes

characteristic

C₂²×C₄

C₄×D₂₃

yes

characteristic

C₂²×C₄

C₂²⋊D₂₃

yes

characteristic

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₄²

C₄×D₂₃

yes

characteristic

C₄²

D₉₂

yes

characteristic

C₄²

C₂×Dic₂₃

yes

characteristic

C₄²

C₂²⋊D₂₃

yes

characteristic

C₂³

Dic₂₃⋊C₄

yes

characteristic

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

C₄×Dic₂₃

yes

characteristic

C₂×C₄

Dic₂₃⋊C₄

yes

characteristic

C₂×C₄

C₄⋊Dic₂₃

yes

characteristic

C₂×C₄

(C₂×D₂₃)⋊C₄

yes

characteristic

C₂×C₄

(C₂×D₂₃)⋊C₄

yes

characteristic

C₂×C₄

C₂²⋊Dic₂₃

yes

characteristic

C₈

—

C₈

—

C₈

—

C₂²

—

C₂²

—

C₂²

#10625004

yes

—

C₂²

C₂₃⋊#95

yes

characteristic

C₄

—

C₄

—

100

C₄

—

101

C₄

—

102

C₄

—

103

C₄

#10625005

yes

characteristic

104

C₄

#10625006

yes

characteristic

105

C₂

—

106

C₂

#13092173

yes

characteristic

107

C₂

#13092181

yes

characteristic

108

C₂

#13092185

yes

characteristic

	subgroup	G/H
center	C₂²	C₂₃⋊#95
commutator	C₂×C₄×C₂₃	C₄²
Frattini	C₂×C₄²	C₂×D₂₃
Fitting	C₂₃×#432	C₂
socle	C₂²×C₂₃	#95
prime radical	C₂³×C₂₃	C₄⋊C₄
cosocle	C₂×C₄²×C₂₃	C₂²
hypercenter	#432	D₂₃
hypocenter	C₂₃	#1290
LCS	G ▷ C₂×C₄×C₂₃ ▷ C₂×C₂₃ ▷ C₂₃
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #432

subgroup

G/H

center

C₂²

C₂₃⋊#95

commutator

C₂×C₄×C₂₃

C₄²

Frattini

C₂×C₄²

C₂×D₂₃

Fitting

C₂₃×#432

C₂

socle

C₂²×C₂₃

#95

prime radical

C₂³×C₂₃

C₄⋊C₄

cosocle

C₂×C₄²×C₂₃

C₂²

D₂₃

C₂₃

G ▷ C₂×C₄×C₂₃ ▷ C₂×C₂₃ ▷ C₂₃

UCS

C₁ ◁ C₂² ◁ C₂×C₄² ◁ #432

N	H ≅ G/N	type
68. C₂₃	#1290	semidirect

H ≅ G/N

type

68. C₂₃

#1290

semidirect

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

subgroup	index
#13092474	2
C₂×C₄⋊(C₂₃⋊C₈)	2
C₂₃×#432	2
#1290	23

subgroup

index

#13092474

C₂×C₄⋊(C₂₃⋊C₈)

C₂₃×#432

#1290

Subgroups — #15671266

Conjugacy classes of subgroups

Complements

Maximal subgroups