Group #15673531 — subgroups

Subgroups — #15673531

Order 2944.

224 conjugacy classes of subgroups; only the 109 normal subgroups are shown.

normal subgroups only

This group has 224 conjugacy classes of subgroups — too many to draw. Only its 109 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	#13092287	C₂	1	1	yes	maximal, characteristic
2	#13092283	C₂	1	1	yes	maximal, characteristic
3	#13092180	C₂	1	4	yes	maximal
4	C₂×C₂₃×#97	C₂	1	1	yes	maximal, characteristic
5	C₂²×C₈×C₂₃	C₂²	1	2	yes	—
6	C₂×C₄²×C₂₃	C₂²	1	1	yes	characteristic
7	C₂×C₄⋊Dic₂₃	—	2	2	no	—
8	C₂×C₄⋊Dic₂₃	C₂²	1	2	yes	—
9	C₂×C₄⋊Dic₂₃	C₂²	1	2	yes	—
10	C₂₃×#97	C₄	1	4	yes	complemented
11	C₂₃⋊#95	—	2	2	no	—
12	C₂²×C₄×C₂₃	D₄	1	2	yes	—
13	C₂²×C₄×C₂₃	C₂³	1	1	yes	characteristic
14	C₂²×Dic₂₃	—	2	2	no	—
15	C₂²×Dic₂₃	—	2	2	no	—
16	C₂×C₈×C₂₃	—	2	2	no	—
17	C₂×C₈×C₂₃	—	2	2	no	—
18	C₂×C₈×C₂₃	C₂×C₄	1	4	yes	—
19	C₄²×C₂₃	C₂×C₄	1	2	yes	—
20	C₄²×C₂₃	Q₈	1	2	yes	—
21	C₄⋊Dic₂₃	—	2	4	no	—
22	C₄⋊Dic₂₃	—	2	4	no	—
23	C₄⋊Dic₂₃	—	4	4	no	—
24	C₂³×C₂₃	C₂×D₄	1	1	yes	characteristic
25	C₂×C₄×C₂₃	—	2	2	no	—
26	C₂×C₄×C₂₃	—	2	2	no	—
27	C₂×C₄×C₂₃	C₄⋊C₄	1	4	yes	—
28	C₂×C₄×C₂₃	C₂²×C₄	1	1	yes	characteristic
29	C₂×C₄×C₂₃	C₂×Q₈	1	1	yes	characteristic
30	C₂×C₄×C₂₃	#41	1	2	yes	—
31	C₂×C₄×C₂₃	#41	1	2	yes	—
32	C₂×Dic₂₃	—	4	2	no	—
33	C₂×Dic₂₃	—	4	2	no	—
34	C₂×Dic₂₃	—	4	4	no	—
35	C₂×Dic₂₃	—	4	2	no	—
36	C₂×Dic₂₃	—	4	2	no	—
37	C₈×C₂₃	—	2	4	no	—
38	#1836	—	23	1	no	maximal, Sylow
39	C₂²×C₂₃	C₂×C₄⋊C₄	1	1	yes	characteristic
40	C₂²×C₂₃	#124	1	1	yes	characteristic
41	C₂²×C₂₃	#124	1	1	yes	characteristic
42	C₂²×C₂₃	AGL(1,ℤ/8)	1	2	yes	—
43	C₂²×C₂₃	#137	1	2	yes	—
44	C₄×C₂₃	—	2	4	no	—
45	C₄×C₂₃	#117	1	2	yes	—
46	C₄×C₂₃	#125	1	2	yes	—
47	Dic₂₃	—	8	2	no	—
48	Dic₂₃	—	8	2	no	—
49	C₂×#97	D₂₃	1	1	yes	characteristic
50	#384	—	23	1	no	—
51	#382	—	23	1	no	—
52	#340	—	23	4	no	—
53	C₂×C₂₃	#382	1	1	yes	characteristic
54	C₂×C₂₃	#428	1	2	yes	—
55	C₂×C₂₃	#489	1	2	yes	—
56	C₂×C₂₃	#490	1	1	yes	characteristic
57	C₂×C₂₃	#491	1	1	yes	characteristic
58	C₂²×C₈	C₂×D₂₃	1	2	yes	—
59	C₂×C₄⋊C₄	—	23	2	no	—
60	C₂×C₄⋊C₄	—	23	2	no	—
61	C₂×C₄⋊C₄	—	46	2	no	—
62	C₂×C₄²	C₂×D₂₃	1	1	yes	characteristic
63	#97	Dic₂₃	1	4	yes	complemented
64	#95	—	46	2	no	—
65	C₂₃	#1836	1	1	yes	characteristic, complemented, Sylow
66	C₂²×C₄	—	46	2	no	—
67	C₂²×C₄	—	46	2	no	—
68	C₂²×C₄	D₉₂	1	2	yes	—
69	C₂²×C₄	C₂²×D₂₃	1	1	yes	characteristic
70	C₂×C₈	—	2	2	no	—
71	C₂×C₈	—	2	2	no	—
72	C₂×C₈	C₂×Dic₂₃	1	4	yes	—
73	C₄⋊C₄	—	46	4	no	—
74	C₄⋊C₄	—	46	4	no	—
75	C₄⋊C₄	—	92	4	no	—
76	C₄²	Dic₄₆	1	2	yes	—
77	C₄²	C₂×Dic₂₃	1	2	yes	—
78	C₂³	C₂×D₉₂	1	1	yes	characteristic
79	C₂×C₄	—	2	2	no	—
80	C₂×C₄	—	2	2	no	—
81	C₂×C₄	—	92	2	no	—
82	C₂×C₄	—	92	2	no	—
83	C₂×C₄	—	92	4	no	—
84	C₂×C₄	—	92	2	no	—
85	C₂×C₄	—	92	2	no	—
86	C₂×C₄	C₄⋊Dic₂₃	1	4	yes	—
87	C₂×C₄	C₂×Dic₄₆	1	1	yes	characteristic
88	C₂×C₄	#67288	1	2	yes	—
89	C₂×C₄	#67288	1	2	yes	—
90	C₂×C₄	C₂²×Dic₂₃	1	1	yes	characteristic
91	C₈	—	2	4	no	—
92	C₂²	#10625046	1	1	yes	characteristic
93	C₂²	#10625046	1	1	yes	characteristic
94	C₂²	#10625079	1	2	yes	—
95	C₂²	#10625080	1	2	yes	—
96	C₂²	C₂×C₄⋊Dic₂₃	1	1	yes	characteristic
97	C₄	—	2	4	no	—
98	C₄	—	184	2	no	—
99	C₄	—	184	2	no	—
100	C₄	#10625041	1	2	yes	—
101	C₄	#10625097	1	2	yes	—
102	C₂	#13092283	1	1	yes	characteristic
103	C₂	#13092459	1	2	yes	—
104	C₂	#13092777	1	2	yes	—
105	C₂	#13092782	1	1	yes	characteristic
106	C₂	#13092786	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#13092287

C₂

yes

maximal, characteristic

#13092283

C₂

yes

maximal, characteristic

#13092180

C₂

yes

maximal

C₂×C₂₃×#97

C₂

yes

maximal, characteristic

C₂²×C₈×C₂₃

C₂²

yes

—

C₂×C₄²×C₂₃

C₂²

yes

characteristic

C₂×C₄⋊Dic₂₃

—

C₂×C₄⋊Dic₂₃

C₂²

yes

—

C₂×C₄⋊Dic₂₃

C₂²

yes

—

C₂₃×#97

C₄

yes

complemented

C₂₃⋊#95

—

C₂²×C₄×C₂₃

D₄

yes

—

C₂²×C₄×C₂₃

C₂³

yes

characteristic

C₂²×Dic₂₃

—

C₂²×Dic₂₃

—

C₂×C₈×C₂₃

—

C₂×C₈×C₂₃

—

C₂×C₈×C₂₃

C₂×C₄

yes

—

C₄²×C₂₃

C₂×C₄

yes

—

C₄²×C₂₃

Q₈

yes

—

C₄⋊Dic₂₃

—

C₄⋊Dic₂₃

—

C₄⋊Dic₂₃

—

C₂³×C₂₃

C₂×D₄

yes

characteristic

C₂×C₄×C₂₃

—

C₂×C₄×C₂₃

—

C₂×C₄×C₂₃

C₄⋊C₄

yes

—

C₂×C₄×C₂₃

C₂²×C₄

yes

characteristic

C₂×C₄×C₂₃

C₂×Q₈

yes

characteristic

C₂×C₄×C₂₃

#41

yes

—

C₂×C₄×C₂₃

#41

yes

—

C₂×Dic₂₃

—

C₂×Dic₂₃

—

C₂×Dic₂₃

—

C₂×Dic₂₃

—

C₂×Dic₂₃

—

C₈×C₂₃

—

#1836

—

maximal, Sylow

C₂²×C₂₃

C₂×C₄⋊C₄

yes

C₂²×C₂₃

yes

C₂²×C₂₃

yes

C₂²×C₂₃

yes

—

C₂²×C₂₃

#137

yes

—

C₄×C₂₃

—

C₄×C₂₃

#117

yes

—

C₄×C₂₃

#125

yes

—

Dic₂₃

—

Dic₂₃

—

C₂×#97

D₂₃

yes

—

—

—

C₂×C₂₃

#382

yes

characteristic

C₂×C₂₃

#428

yes

—

C₂×C₂₃

#489

yes

—

C₂×C₂₃

yes

C₂×C₂₃

yes

C₂²×C₈

C₂×D₂₃

yes

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄²

C₂×D₂₃

yes

characteristic

#97

Dic₂₃

yes

complemented

#95

—

C₂₃

#1836

yes

characteristic, complemented, Sylow

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

D₉₂

yes

—

C₂²×C₄

C₂²×D₂₃

yes

characteristic

C₂×C₈

—

C₂×C₈

—

C₂×C₈

C₂×Dic₂₃

yes

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄²

Dic₄₆

yes

—

C₄²

C₂×Dic₂₃

yes

—

C₂³

C₂×D₉₂

yes

characteristic

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

C₄⋊Dic₂₃

yes

—

C₂×C₄

C₂×Dic₄₆

yes

characteristic

C₂×C₄

#67288

yes

—

C₂×C₄

#67288

yes

—

C₂×C₄

C₂²×Dic₂₃

yes

characteristic

C₈

—

C₂²

yes

C₂²

yes

C₂²

yes

—

C₂²

#10625080

yes

—

C₂²

C₂×C₄⋊Dic₂₃

yes

characteristic

C₄

—

C₄

—

184

—

C₄

—

184

—

100

C₄

#10625041

yes

—

101

C₄

#10625097

yes

—

102

C₂

#13092283

yes

characteristic

103

C₂

#13092459

yes

—

104

C₂

#13092777

yes

—

105

C₂

#13092782

yes

characteristic

106

C₂

#13092786

yes

characteristic

	subgroup	G/H
center	C₂³	C₂×D₉₂
commutator	C₂×C₄×C₂₃	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₂₃
Fitting	C₂×C₂₃×#97	C₂
socle, prime radical	C₂³×C₂₃	C₂×D₄
cosocle	C₂²×C₄×C₂₃	C₂³
hypercenter	C₂×#97	D₂₃
hypocenter	C₂₃	#1836
LCS	G ▷ C₂×C₄×C₂₃ ▷ C₂×C₂₃ ▷ C₂₃
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×#97

subgroup

G/H

center

C₂³

C₂×D₉₂

commutator

C₂×C₄×C₂₃

C₂²×C₄

Frattini

C₂²×C₄

C₂²×D₂₃

Fitting

C₂×C₂₃×#97

C₂

socle, prime radical

C₂³×C₂₃

C₂×D₄

cosocle

C₂²×C₄×C₂₃

C₂³

hypercenter

C₂×#97

D₂₃

hypocenter

C₂₃

#1836

LCS

G ▷ C₂×C₄×C₂₃ ▷ C₂×C₂₃ ▷ C₂₃

UCS

C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×#97

N	H ≅ G/N	type
10. C₂₃×#97	C₄	semidirect
63. #97	Dic₂₃	semidirect
65. C₂₃	#1836	semidirect

H ≅ G/N

type

10. C₂₃×#97

C₄

semidirect

63. #97

Dic₂₃

semidirect

65. C₂₃

#1836

semidirect

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

subgroup	index
#13092287	2
#13092283	2
#13092180	2
C₂×C₂₃×#97	2
#1836	23

subgroup

index

C₂×C₂₃×#97

Subgroups — #15673531

Conjugacy classes of subgroups

Complements

Maximal subgroups