Group #15673570 — subgroups

Subgroups — #15673570

Order 2944.

212 conjugacy classes of subgroups; only the 101 normal subgroups are shown.

normal subgroups only

This group has 212 conjugacy classes of subgroups — too many to draw. Only its 101 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	1	yes	maximal, characteristic
2	C₂×#10625004	C₂	1	2	yes	maximal
3	#13092440	C₂	1	2	yes	maximal
4	C₂×#10624997	C₂	1	1	yes	maximal, characteristic
5	C₂×C₂₃×#108	C₂	1	1	yes	maximal, characteristic
6	C₂×M₁₆×C₂₃	C₂²	1	2	yes	—
7	C₂²×C₈×C₂₃	C₂²	1	1	yes	characteristic
8	C₂×C₄×Dic₂₃	C₂²	1	1	yes	characteristic
9	C₂×#67267	C₂²	1	2	yes	—
10	C₂²×C₂₃⋊C₈	C₂²	1	1	yes	characteristic
11	C₂₃×#108	—	2	2	no	—
12	#10625004	—	2	4	no	—
13	#10625001	—	2	2	no	—
14	#10624997	C₄	1	4	yes	—
15	Dic₂₃⋊C₈	—	2	4	no	—
16	C₂²×C₄×C₂₃	C₂³	1	1	yes	characteristic
17	C₂²×Dic₂₃	Q₈	1	2	yes	—
18	M₁₆×C₂₃	—	2	2	no	—
19	M₁₆×C₂₃	—	2	2	no	—
20	C₂×C₈×C₂₃	—	2	2	no	—
21	C₂×C₈×C₂₃	C₂×C₄	1	2	yes	—
22	C₂×C₈×C₂₃	D₄	1	2	yes	—
23	C₂×C₈×C₂₃	Q₈	1	2	yes	—
24	C₄×Dic₂₃	C₂×C₄	1	2	yes	—
25	C₄×Dic₂₃	D₄	1	2	yes	—
26	#67267	—	2	2	no	—
27	#67267	—	2	2	no	—
28	C₂×C₂₃⋊C₈	—	2	2	no	—
29	C₂×C₂₃⋊C₈	C₂×C₄	1	2	yes	—
30	C₂×C₂₃⋊C₈	D₄	1	2	yes	—
31	C₂×C₂₃⋊C₈	Q₈	1	2	yes	—
32	C₂³×C₂₃	C₂×Q₈	1	1	yes	characteristic
33	C₂×C₄×C₂₃	C₂²×C₄	1	1	yes	characteristic
34	C₂×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
35	C₂×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
36	C₂×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
37	C₂×C₄×C₂₃	C₂×Q₈	1	1	yes	characteristic
38	C₂×C₄×C₂₃	C₂×Q₈	1	1	yes	characteristic
39	C₂×Dic₂₃	—	4	2	no	—
40	C₂×Dic₂₃	C₄⋊C₄	1	4	yes	—
41	C₈×C₂₃	—	4	2	no	—
42	C₈×C₂₃	C₄⋊C₄	1	4	yes	—
43	C₂₃⋊C₈	—	4	2	no	—
44	C₂₃⋊C₈	C₄⋊C₄	1	4	yes	—
45	#1849	—	23	1	no	maximal, Sylow
46	C₂²×C₂₃	—	2	2	no	—
47	C₂²×C₂₃	C₂×C₄⋊C₄	1	1	yes	characteristic
48	C₂²×C₂₃	C₄⋊Q₈	1	1	yes	characteristic
49	C₂²×C₂₃	C₄⋊Q₈	1	1	yes	characteristic
50	C₄×C₂₃	C₂×C₄⋊C₄	1	1	yes	characteristic
51	C₄×C₂₃	C₂×C₄⋊C₄	1	1	yes	characteristic
52	C₄×C₂₃	#127	1	1	yes	characteristic
53	C₄×C₂₃	C₄⋊Q₈	1	1	yes	characteristic
54	Dic₂₃	—	2	4	no	—
55	C₂×#108	—	23	2	no	—
56	C₂×#108	—	23	1	no	—
57	C₂×#108	D₂₃	1	1	yes	characteristic
58	#423	—	23	2	no	—
59	C₂×#97	—	23	1	no	—
60	C₂×C₂₃	—	2	1	no	—
61	C₂×C₂₃	—	2	1	no	—
62	C₂×C₂₃	#384	1	1	yes	characteristic
63	C₂×C₂₃	#430	1	2	yes	—
64	C₂×M₁₆	—	23	2	no	—
65	C₂×M₁₆	C₂×D₂₃	1	2	yes	—
66	C₂²×C₈	—	23	1	no	—
67	C₂²×C₈	C₂×D₂₃	1	1	yes	characteristic
68	C₂×C₄²	—	23	1	no	—
69	#108	—	2	2	no	—
70	#108	—	46	4	no	—
71	#108	—	46	2	no	—
72	C₄⋊C₈	—	46	4	no	—
73	#97	—	23	4	no	—
74	C₂₃	#1849	1	1	yes	characteristic, complemented, Sylow
75	C₂²×C₄	—	23	2	no	—
76	C₂²×C₄	C₂²×D₂₃	1	1	yes	characteristic
77	M₁₆	—	2	2	no	—
78	M₁₆	—	2	2	no	—
79	M₁₆	—	46	2	no	—
80	M₁₆	—	46	2	no	—
81	C₂×C₈	—	2	2	no	—
82	C₂×C₈	—	23	2	no	—
83	C₂×C₈	—	23	2	no	—
84	C₂×C₈	—	23	2	no	—
85	C₂×C₈	—	46	2	no	—
86	C₂×C₈	Dic₄₆	1	2	yes	—
87	C₂×C₈	C₄×D₂₃	1	2	yes	—
88	C₂×C₈	C₂²⋊D₂₃	1	2	yes	—
89	C₄²	—	23	2	no	—
90	C₄²	—	23	2	no	—
91	C₂³	Q₈×D₂₃	1	1	yes	characteristic
92	C₂×C₄	—	23	4	no	—
93	C₂×C₄	—	92	2	no	—
94	C₂×C₄	C₂×Dic₄₆	1	1	yes	characteristic
95	C₂×C₄	C₂×C₄×D₂₃	1	1	yes	characteristic
96	C₂×C₄	D₄×D₂₃	1	1	yes	characteristic
97	C₂×C₄	D₄×D₂₃	1	1	yes	characteristic
98	C₂×C₄	Q₈×D₂₃	1	1	yes	characteristic
99	C₂×C₄	C₂×C₂²⋊D₂₃	1	1	yes	characteristic
100	C₈	—	4	2	no	—
101	C₈	—	23	4	no	—
102	C₈	—	92	2	no	—
103	C₈	Dic₂₃⋊C₄	1	4	yes	—
104	C₂²	—	2	2	no	—
105	C₂²	#10625059	1	1	yes	characteristic
106	C₂²	C₄⋊C₄×D₂₃	1	1	yes	characteristic
107	C₂²	#10625115	1	1	yes	characteristic
108	C₄	—	46	4	no	—
109	C₄	#10625059	1	1	yes	characteristic
110	C₄	C₄⋊C₄×D₂₃	1	1	yes	characteristic
111	C₄	C₂×Dic₂₃⋊C₄	1	1	yes	characteristic
112	C₄	#10625111	1	1	yes	characteristic
113	C₂	—	2	1	no	—
114	C₂	—	2	1	no	—
115	C₂	#13092286	1	1	yes	characteristic
116	C₂	#13092464	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#10625001

C₂

yes

maximal, characteristic

C₂×#10625004

C₂

yes

maximal

#13092440

C₂

yes

maximal

C₂×#10624997

C₂

yes

maximal, characteristic

C₂×C₂₃×#108

C₂

yes

maximal, characteristic

C₂×M₁₆×C₂₃

C₂²

yes

—

C₂²×C₈×C₂₃

C₂²

yes

characteristic

C₂×C₄×Dic₂₃

C₂²

yes

characteristic

C₂×#67267

C₂²

yes

—

C₂²×C₂₃⋊C₈

C₂²

yes

characteristic

C₂₃×#108

—

#10625004

—

#10625001

—

#10624997

C₄

yes

—

Dic₂₃⋊C₈

—

C₂²×C₄×C₂₃

C₂³

yes

characteristic

C₂²×Dic₂₃

Q₈

yes

—

M₁₆×C₂₃

—

M₁₆×C₂₃

—

C₂×C₈×C₂₃

—

C₂×C₈×C₂₃

C₂×C₄

yes

—

C₂×C₈×C₂₃

D₄

yes

—

C₂×C₈×C₂₃

Q₈

yes

—

C₄×Dic₂₃

C₂×C₄

yes

—

C₄×Dic₂₃

D₄

yes

—

#67267

—

#67267

—

C₂×C₂₃⋊C₈

—

C₂×C₂₃⋊C₈

C₂×C₄

yes

—

C₂×C₂₃⋊C₈

D₄

yes

—

C₂×C₂₃⋊C₈

Q₈

yes

—

C₂³×C₂₃

C₂×Q₈

yes

characteristic

C₂×C₄×C₂₃

C₂²×C₄

yes

characteristic

C₂×C₄×C₂₃

C₂×D₄

yes

characteristic

C₂×C₄×C₂₃

C₂×D₄

yes

characteristic

C₂×C₄×C₂₃

C₂×D₄

yes

characteristic

C₂×C₄×C₂₃

C₂×Q₈

yes

characteristic

C₂×C₄×C₂₃

C₂×Q₈

yes

characteristic

C₂×Dic₂₃

—

C₂×Dic₂₃

C₄⋊C₄

yes

—

C₈×C₂₃

—

C₈×C₂₃

C₄⋊C₄

yes

—

C₂₃⋊C₈

—

C₂₃⋊C₈

C₄⋊C₄

yes

—

#1849

—

maximal, Sylow

C₂²×C₂₃

—

C₂²×C₂₃

C₂×C₄⋊C₄

yes

characteristic

C₂²×C₂₃

C₄⋊Q₈

yes

characteristic

C₂²×C₂₃

C₄⋊Q₈

yes

characteristic

C₄×C₂₃

C₂×C₄⋊C₄

yes

characteristic

C₄×C₂₃

C₂×C₄⋊C₄

yes

characteristic

C₄×C₂₃

#127

yes

characteristic

C₄×C₂₃

C₄⋊Q₈

yes

characteristic

Dic₂₃

—

C₂×#108

—

C₂×#108

—

C₂×#108

D₂₃

yes

characteristic

#423

—

C₂×#97

—

C₂×C₂₃

—

C₂×C₂₃

—

C₂×C₂₃

#384

yes

characteristic

C₂×C₂₃

#430

yes

—

C₂×M₁₆

—

C₂×M₁₆

C₂×D₂₃

yes

—

C₂²×C₈

—

C₂²×C₈

C₂×D₂₃

yes

characteristic

C₂×C₄²

—

#108

—

#108

—

#108

—

C₄⋊C₈

—

#97

—

C₂₃

#1849

yes

characteristic, complemented, Sylow

C₂²×C₄

—

C₂²×C₄

C₂²×D₂₃

yes

characteristic

M₁₆

—

M₁₆

—

M₁₆

—

M₁₆

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

Dic₄₆

yes

—

C₂×C₈

C₄×D₂₃

yes

—

C₂×C₈

C₂²⋊D₂₃

yes

—

C₄²

—

C₄²

—

C₂³

Q₈×D₂₃

yes

characteristic

C₂×C₄

—

C₂×C₄

—

C₂×C₄

C₂×Dic₄₆

yes

characteristic

C₂×C₄

C₂×C₄×D₂₃

yes

characteristic

C₂×C₄

D₄×D₂₃

yes

characteristic

C₂×C₄

D₄×D₂₃

yes

characteristic

C₂×C₄

Q₈×D₂₃

yes

characteristic

C₂×C₄

C₂×C₂²⋊D₂₃

yes

characteristic

100

C₈

—

101

C₈

—

102

C₈

—

103

C₈

Dic₂₃⋊C₄

yes

—

104

C₂²

—

105

C₂²

#10625059

yes

characteristic

106

C₂²

C₄⋊C₄×D₂₃

yes

characteristic

107

C₂²

#10625115

yes

characteristic

108

C₄

—

109

C₄

#10625059

yes

characteristic

110

C₄

C₄⋊C₄×D₂₃

yes

characteristic

111

C₄

C₂×Dic₂₃⋊C₄

yes

characteristic

112

C₄

#10625111

yes

characteristic

113

C₂

—

114

C₂

—

115

C₂

#13092286

yes

characteristic

116

C₂

#13092464

yes

—

	subgroup	G/H
center	C₂×C₄	D₄×D₂₃
commutator	C₂×C₄×C₂₃	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₂₃
Fitting	C₂×C₂₃×#108	C₂
socle	C₂²×C₂₃	C₂×C₄⋊C₄
prime radical	C₂³×C₂₃	C₂×Q₈
cosocle	C₂²×C₄×C₂₃	C₂³
hypercenter	C₂×#108	D₂₃
hypocenter	C₂₃	#1849
LCS	G ▷ C₂×C₄×C₂₃ ▷ C₂×C₂₃ ▷ C₂₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄ ◁ C₂×#108

subgroup

G/H

center

C₂×C₄

D₄×D₂₃

commutator

C₂×C₄×C₂₃

C₂²×C₄

Frattini

C₂²×C₄

C₂²×D₂₃

Fitting

C₂×C₂₃×#108

C₂

socle

C₂²×C₂₃

C₂×C₄⋊C₄

prime radical

C₂³×C₂₃

C₂×Q₈

cosocle

C₂²×C₄×C₂₃

C₂³

D₂₃

C₂₃

G ▷ C₂×C₄×C₂₃ ▷ C₂×C₂₃ ▷ C₂₃

UCS

C₁ ◁ C₂×C₄ ◁ C₂²×C₄ ◁ C₂×#108

N	H ≅ G/N	type
74. C₂₃	#1849	semidirect

H ≅ G/N

type

74. C₂₃

#1849

semidirect

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

subgroup	index
C₂×#10625001	2
C₂×#10625004	2
#13092440	2
C₂×#10624997	2
C₂×C₂₃×#108	2
#1849	23

subgroup

index

C₂×C₂₃×#108

Subgroups — #15673570

Conjugacy classes of subgroups

Complements

Maximal subgroups