Group #15673439 — subgroups

Subgroups — #15673439

Order 2944.

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

normal subgroups only

This group has 272 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	C₂×#10625001	C₂	1	2	yes	maximal, complemented
2	#13092381	C₂	1	2	yes	maximal
3	#13092182	C₂	1	2	yes	maximal, complemented
4	C₂²×M₁₆×C₂₃	C₂	1	1	yes	maximal, characteristic
5	C₂³×C₄×C₂₃	C₂²	1	1	yes	characteristic
6	C₂×M₁₆×C₂₃	—	2	4	no	—
7	C₂×M₁₆×C₂₃	C₄	1	4	yes	—
8	C₂²×C₈×C₂₃	C₂²	1	1	yes	characteristic
9	C₂²×C₈×C₂₃	C₂²	1	1	yes	characteristic
10	C₂×#67267	C₂²	1	4	yes	—
11	C₂²⋊(C₂₃⋊C₈)	—	2	4	no	—
12	#10625001	—	2	4	no	—
13	C₂⁴×C₂₃	Q₈	1	1	yes	characteristic
14	C₂²×C₄×C₂₃	—	2	2	no	—
15	C₂²×C₄×C₂₃	—	2	2	no	—
16	C₂²×C₄×C₂₃	C₂×C₄	1	2	yes	—
17	C₂²×C₄×C₂₃	D₄	1	2	yes	—
18	C₂²×C₄×C₂₃	Q₈	1	1	yes	characteristic
19	C₂²×C₄×C₂₃	C₂³	1	1	yes	characteristic
20	M₁₆×C₂₃	—	2	8	no	—
21	C₂×C₈×C₂₃	—	2	2	no	—
22	C₂×C₈×C₂₃	C₂×C₄	1	2	yes	—
23	C₂×C₈×C₂₃	C₂×C₄	1	2	yes	—
24	C₂×C₈×C₂₃	D₄	1	4	yes	—
25	#67267	—	2	4	no	—
26	#67267	—	2	4	no	—
27	C₂×C₂₃⋊C₈	—	2	4	no	—
28	C₂³×C₂₃	—	2	2	no	—
29	C₂³×C₂₃	—	4	2	no	—
30	C₂³×C₂₃	C₄⋊C₄	1	2	yes	—
31	C₂³×C₂₃	C₂×Q₈	1	1	yes	characteristic
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	—	2	4	no	—
36	C₂×C₄×C₂₃	C₄⋊C₄	1	2	yes	—
37	C₂×C₄×C₂₃	C₂²×C₄	1	1	yes	characteristic
38	C₂×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
39	C₂×C₄×C₂₃	C₂×D₄	1	2	yes	—
40	C₂×C₄×C₂₃	#41	1	2	yes	—
41	C₈×C₂₃	—	2	2	no	—
42	C₈×C₂₃	C₂²⋊C₄	1	4	yes	—
43	C₂₃⋊C₈	—	4	4	no	—
44	#1811	—	23	1	no	maximal, Sylow
45	C₂²×C₂₃	—	2	2	no	—
46	C₂²×C₂₃	—	2	2	no	—
47	C₂²×C₂₃	—	2	4	no	—
48	C₂²×C₂₃	—	4	4	no	—
49	C₂²×C₂₃	C₂×C₄⋊C₄	1	1	yes	characteristic
50	C₂²×C₂₃	C₂²⋊Q₈	1	2	yes	—
51	C₄×C₂₃	—	2	2	no	—
52	C₄×C₂₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
53	C₄×C₂₃	#117	1	1	yes	characteristic
54	C₄×C₂₃	#121	1	2	yes	—
55	C₂²×M₁₆	D₂₃	1	1	yes	characteristic
56	C₂×#108	—	23	2	no	—
57	#407	—	23	2	no	—
58	#341	—	23	2	no	—
59	C₂×C₂₃	—	2	2	no	—
60	C₂×C₂₃	—	4	2	no	—
61	C₂×C₂₃	#380	1	1	yes	characteristic
62	C₂×C₂₃	#430	1	2	yes	—
63	C₂³×C₄	C₂×D₂₃	1	1	yes	characteristic
64	C₂×M₁₆	—	2	4	no	—
65	C₂×M₁₆	—	23	4	no	—
66	C₂×M₁₆	Dic₂₃	1	4	yes	—
67	C₂²×C₈	C₂×D₂₃	1	1	yes	characteristic
68	C₂²×C₈	C₂×D₂₃	1	1	yes	characteristic
69	#108	—	46	4	no	—
70	C₂²⋊C₈	—	46	4	no	—
71	C₂₃	#1811	1	1	yes	characteristic, complemented, Sylow
72	C₂⁴	Dic₄₆	1	1	yes	characteristic
73	C₂²×C₄	—	2	2	no	—
74	C₂²×C₄	—	2	2	no	—
75	C₂²×C₄	Dic₄₆	1	1	yes	characteristic
76	C₂²×C₄	D₉₂	1	2	yes	—
77	C₂²×C₄	C₂×Dic₂₃	1	2	yes	—
78	C₂²×C₄	C₂²×D₂₃	1	1	yes	characteristic
79	M₁₆	—	2	8	no	—
80	M₁₆	—	46	4	no	—
81	M₁₆	—	46	4	no	—
82	C₂×C₈	—	2	2	no	—
83	C₂×C₈	—	46	4	no	—
84	C₂×C₈	C₂×Dic₂₃	1	2	yes	—
85	C₂×C₈	C₂×Dic₂₃	1	2	yes	—
86	C₂×C₈	C₂²⋊D₂₃	1	4	yes	—
87	C₂³	—	2	2	no	—
88	C₂³	—	4	2	no	—
89	C₂³	C₄⋊Dic₂₃	1	2	yes	—
90	C₂³	C₂×Dic₄₆	1	1	yes	characteristic
91	C₂×C₄	—	2	2	no	—
92	C₂×C₄	—	2	2	no	—
93	C₂×C₄	—	2	4	no	—
94	C₂×C₄	—	2	2	no	—
95	C₂×C₄	C₄⋊Dic₂₃	1	2	yes	—
96	C₂×C₄	C₂×D₉₂	1	1	yes	characteristic
97	C₂×C₄	#67288	1	2	yes	—
98	C₂×C₄	C₂²×Dic₂₃	1	1	yes	characteristic
99	C₂×C₄	C₂×C₂²⋊D₂₃	1	2	yes	—
100	C₈	—	2	2	no	—
101	C₈	—	92	4	no	—
102	C₈	C₂²⋊Dic₂₃	1	4	yes	—
103	C₂²	—	2	2	no	—
104	C₂²	—	2	4	no	—
105	C₂²	—	2	2	no	—
106	C₂²	—	4	4	no	—
107	C₂²	#10625095	1	2	yes	—
108	C₂²	C₂×C₄⋊Dic₂₃	1	1	yes	characteristic
109	C₄	—	2	2	no	—
110	C₄	#10625097	1	1	yes	characteristic
111	C₄	#10625101	1	2	yes	—
112	C₄	C₂×C₂²⋊Dic₂₃	1	1	yes	characteristic
113	C₂	—	2	2	no	—
114	C₂	—	4	2	no	—
115	C₂	#13092277	1	1	yes	characteristic
116	C₂	#13092466	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#10625001

C₂

yes

maximal, complemented

#13092381

C₂

yes

maximal

#13092182

C₂

yes

maximal, complemented

C₂²×M₁₆×C₂₃

C₂

yes

maximal, characteristic

C₂³×C₄×C₂₃

C₂²

yes

characteristic

C₂×M₁₆×C₂₃

—

C₂×M₁₆×C₂₃

C₄

yes

—

C₂²×C₈×C₂₃

C₂²

yes

characteristic

C₂²×C₈×C₂₃

C₂²

yes

characteristic

C₂×#67267

C₂²

yes

—

C₂²⋊(C₂₃⋊C₈)

—

#10625001

—

C₂⁴×C₂₃

Q₈

yes

characteristic

C₂²×C₄×C₂₃

—

C₂²×C₄×C₂₃

—

C₂²×C₄×C₂₃

C₂×C₄

yes

—

C₂²×C₄×C₂₃

D₄

yes

—

C₂²×C₄×C₂₃

Q₈

yes

characteristic

C₂²×C₄×C₂₃

C₂³

yes

characteristic

M₁₆×C₂₃

—

C₂×C₈×C₂₃

—

C₂×C₈×C₂₃

C₂×C₄

yes

—

C₂×C₈×C₂₃

C₂×C₄

yes

—

C₂×C₈×C₂₃

D₄

yes

—

#67267

—

#67267

—

C₂×C₂₃⋊C₈

—

C₂³×C₂₃

—

C₂³×C₂₃

—

C₂³×C₂₃

C₄⋊C₄

yes

—

C₂³×C₂₃

C₂×Q₈

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

characteristic

C₂×C₄×C₂₃

C₂×D₄

yes

—

C₂×C₄×C₂₃

#41

yes

—

C₈×C₂₃

—

C₈×C₂₃

C₂²⋊C₄

yes

—

C₂₃⋊C₈

—

#1811

—

maximal, Sylow

C₂²×C₂₃

—

C₂²×C₂₃

—

C₂²×C₂₃

—

C₂²×C₂₃

—

C₂²×C₂₃

C₂×C₄⋊C₄

yes

characteristic

C₂²×C₂₃

C₂²⋊Q₈

yes

—

C₄×C₂₃

—

C₄×C₂₃

C₂×C₂²⋊C₄

yes

C₄×C₂₃

yes

C₄×C₂₃

yes

—

C₂²×M₁₆

D₂₃

yes

—

—

—

C₂×C₂₃

—

C₂×C₂₃

—

C₂×C₂₃

#380

yes

characteristic

C₂×C₂₃

#430

yes

—

C₂³×C₄

C₂×D₂₃

yes

characteristic

C₂×M₁₆

—

C₂×M₁₆

—

C₂×M₁₆

Dic₂₃

yes

—

C₂²×C₈

C₂×D₂₃

yes

characteristic

C₂²×C₈

C₂×D₂₃

yes

characteristic

#108

—

C₂²⋊C₈

—

C₂₃

#1811

yes

characteristic, complemented, Sylow

C₂⁴

Dic₄₆

yes

characteristic

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

Dic₄₆

yes

characteristic

C₂²×C₄

D₉₂

yes

—

C₂²×C₄

C₂×Dic₂₃

yes

—

C₂²×C₄

C₂²×D₂₃

yes

characteristic

M₁₆

—

M₁₆

—

M₁₆

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

C₂×Dic₂₃

yes

—

C₂×C₈

C₂×Dic₂₃

yes

—

C₂×C₈

C₂²⋊D₂₃

yes

—

C₂³

—

C₂³

—

C₂³

C₄⋊Dic₂₃

yes

—

C₂³

C₂×Dic₄₆

yes

characteristic

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

C₄⋊Dic₂₃

yes

—

C₂×C₄

C₂×D₉₂

yes

characteristic

C₂×C₄

#67288

yes

—

C₂×C₄

C₂²×Dic₂₃

yes

characteristic

C₂×C₄

C₂×C₂²⋊D₂₃

yes

—

100

C₈

—

101

C₈

—

102

C₈

C₂²⋊Dic₂₃

yes

—

103

C₂²

—

104

C₂²

—

105

C₂²

—

106

C₂²

—

107

C₂²

#10625095

yes

—

108

C₂²

C₂×C₄⋊Dic₂₃

yes

characteristic

109

C₄

—

110

C₄

#10625097

yes

characteristic

111

C₄

#10625101

yes

—

112

C₄

C₂×C₂²⋊Dic₂₃

yes

characteristic

113

C₂

—

114

C₂

—

115

C₂

#13092277

yes

characteristic

116

C₂

#13092466

yes

—

	subgroup	G/H
center	C₂×C₄	C₂×D₉₂
commutator	C₂×C₄×C₂₃	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₂₃
Fitting	C₂²×M₁₆×C₂₃	C₂
socle	C₂²×C₂₃	C₂×C₄⋊C₄
prime radical	C₂⁴×C₂₃	Q₈
cosocle	C₂²×C₄×C₂₃	C₂³
hypercenter	C₂²×M₁₆	D₂₃
hypocenter	C₂₃	#1811
LCS	G ▷ C₂×C₄×C₂₃ ▷ C₂×C₂₃ ▷ C₂₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ C₂²×M₁₆

subgroup

G/H

center

C₂×C₄

C₂×D₉₂

commutator

C₂×C₄×C₂₃

C₂²×C₄

Frattini

C₂²×C₄

C₂²×D₂₃

Fitting

C₂²×M₁₆×C₂₃

C₂

socle

C₂²×C₂₃

C₂×C₄⋊C₄

prime radical

C₂⁴×C₂₃

Q₈

cosocle

C₂²×C₄×C₂₃

C₂³

hypercenter

C₂²×M₁₆

D₂₃

hypocenter

C₂₃

#1811

LCS

G ▷ C₂×C₄×C₂₃ ▷ C₂×C₂₃ ▷ C₂₃

UCS

C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ C₂²×M₁₆

N	H ≅ G/N	type
1. C₂×#10625001	C₂	semidirect
3. #13092182	C₂	semidirect
71. C₂₃	#1811	semidirect

H ≅ G/N

type

1. C₂×#10625001

C₂

semidirect

3. #13092182

C₂

semidirect

71. C₂₃

#1811

semidirect

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

subgroup	index
C₂×#10625001	2
#13092381	2
#13092182	2
C₂²×M₁₆×C₂₃	2
#1811	23

subgroup

index

C₂²×M₁₆×C₂₃

Subgroups — #15673439

Conjugacy classes of subgroups

Complements

Maximal subgroups