Group #15671279 — subgroups

Subgroups — #15671279

Order 2944.

180 conjugacy classes of subgroups; only the 73 normal subgroups are shown.

normal subgroups only

This group has 180 conjugacy classes of subgroups — too many to draw. Only its 73 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×#10624985	C₂	1	1	yes	maximal, characteristic
2	#13092283	C₂	1	1	yes	maximal, characteristic
3	C₂×C₂₃×#97	C₂	1	1	yes	maximal, characteristic
4	C₂²×C₈×C₂₃	—	2	1	no	—
5	C₂×C₄²×C₂₃	C₂²	1	1	yes	characteristic
6	C₂×C₄⋊Dic₂₃	C₄	1	2	yes	—
7	C₂²×C₂₃⋊C₈	—	2	1	no	—
8	C₂₃×#97	—	2	1	no	—
9	C₂₃×#97	C₄	1	2	yes	complemented
10	C₂₃⋊#95	—	2	1	no	—
11	C₂₃⋊#95	—	2	1	no	—
12	#10624985	—	2	1	no	—
13	#10624985	C₄	1	2	yes	complemented
14	C₂²×C₄×C₂₃	C₂×C₄	1	1	yes	characteristic
15	C₂²×C₄×C₂₃	D₄	1	1	yes	characteristic
16	C₂²×C₄×C₂₃	D₄	1	1	yes	characteristic
17	C₂²×Dic₂₃	—	2	2	no	—
18	C₂×C₈×C₂₃	—	2	4	no	—
19	C₂×C₈×C₂₃	—	2	2	no	—
20	C₄²×C₂₃	C₂×C₄	1	1	yes	characteristic
21	C₄²×C₂₃	C₂×C₄	1	1	yes	characteristic
22	C₄²×C₂₃	D₄	1	1	yes	characteristic
23	C₄²×C₂₃	Q₈	1	1	yes	characteristic
24	C₄⋊Dic₂₃	—	4	2	no	—
25	C₂×C₂₃⋊C₈	—	2	4	no	—
26	C₂×C₂₃⋊C₈	—	2	2	no	—
27	C₂³×C₂₃	C₂²⋊C₄	1	1	yes	characteristic
28	C₂×C₄×C₂₃	—	2	1	no	—
29	C₂×C₄×C₂₃	—	2	1	no	—
30	C₂×C₄×C₂₃	—	2	2	no	—
31	C₂×C₄×C₂₃	—	2	2	no	—
32	C₂×C₄×C₂₃	C₄²	1	1	yes	characteristic
33	C₂×C₄×C₂₃	C₂²⋊C₄	1	1	yes	characteristic
34	C₂×C₄×C₂₃	C₂²⋊C₄	1	1	yes	characteristic
35	C₂×C₄×C₂₃	C₄⋊C₄	1	1	yes	characteristic
36	C₂×C₄×C₂₃	C₄⋊C₄	1	1	yes	characteristic
37	C₂×C₄×C₂₃	C₄⋊C₄	1	1	yes	characteristic
38	C₂×Dic₂₃	—	4	2	no	—
39	C₂×Dic₂₃	—	4	2	no	—
40	C₂×Dic₂₃	—	4	2	no	—
41	C₈×C₂₃	—	2	4	no	—
42	C₂₃⋊C₈	—	2	4	no	—
43	#1295	—	23	1	no	maximal, Sylow
44	C₂²×C₂₃	#95	1	1	yes	characteristic
45	C₂²×C₂₃	#100	1	1	yes	characteristic
46	C₂²×C₂₃	#101	1	1	yes	characteristic
47	C₂²×C₂₃	#104	1	2	yes	—
48	C₂²×C₂₃	#104	1	2	yes	—
49	C₄×C₂₃	—	2	1	no	—
50	C₄×C₂₃	—	2	1	no	—
51	C₄×C₂₃	—	2	1	no	—
52	C₄×C₂₃	—	2	1	no	—
53	C₄×C₂₃	—	2	1	no	—
54	C₄×C₂₃	—	2	1	no	—
55	Dic₂₃	—	8	2	no	—
56	C₂×#97	—	23	1	no	—
57	C₂×#97	D₂₃	1	1	yes	characteristic
58	#382	—	23	1	no	—
59	C₂×C₂₃	#329	1	2	yes	—
60	C₂×C₂₃	#330	1	2	yes	—
61	C₂×C₂₃	#339	1	1	yes	characteristic
62	C₂×C₂₃	#339	1	1	yes	characteristic
63	C₂×C₂₃	#343	1	1	yes	characteristic
64	C₂²×C₈	—	2	1	no	—
65	C₂²×C₈	—	46	1	no	—
66	C₂×C₄⋊C₄	—	23	2	no	—
67	C₂×C₄²	C₂×D₂₃	1	1	yes	characteristic
68	#97	—	2	1	no	—
69	#97	—	23	2	no	—
70	#97	—	46	1	no	—
71	#97	Dic₂₃	1	2	yes	complemented
72	#95	—	46	1	no	—
73	#95	—	46	1	no	—
74	C₂₃	#1295	1	1	yes	characteristic, complemented, Sylow
75	C₂²×C₄	—	46	2	no	—
76	C₂²×C₄	C₄×D₂₃	1	1	yes	characteristic
77	C₂²×C₄	D₉₂	1	1	yes	characteristic
78	C₂²×C₄	C₂²⋊D₂₃	1	1	yes	characteristic
79	C₂×C₈	—	2	4	no	—
80	C₂×C₈	—	2	2	no	—
81	C₂×C₈	—	46	4	no	—
82	C₂×C₈	—	46	2	no	—
83	C₄⋊C₄	—	92	2	no	—
84	C₄²	Dic₄₆	1	1	yes	characteristic
85	C₄²	C₄×D₂₃	1	1	yes	characteristic
86	C₄²	C₂×Dic₂₃	1	1	yes	characteristic
87	C₄²	C₂²⋊D₂₃	1	1	yes	characteristic
88	C₂³	(C₂×D₂₃)⋊C₄	1	1	yes	characteristic
89	C₂×C₄	—	2	2	no	—
90	C₂×C₄	—	2	1	no	—
91	C₂×C₄	—	2	1	no	—
92	C₂×C₄	—	2	2	no	—
93	C₂×C₄	—	92	2	no	—
94	C₂×C₄	—	92	2	no	—
95	C₂×C₄	—	92	2	no	—
96	C₂×C₄	C₄×Dic₂₃	1	1	yes	characteristic
97	C₂×C₄	Dic₂₃⋊C₄	1	1	yes	characteristic
98	C₂×C₄	Dic₂₃⋊C₄	1	1	yes	characteristic
99	C₂×C₄	C₄⋊Dic₂₃	1	1	yes	characteristic
100	C₂×C₄	(C₂×D₂₃)⋊C₄	1	1	yes	characteristic
101	C₂×C₄	C₂²⋊Dic₂₃	1	1	yes	characteristic
102	C₈	—	2	4	no	—
103	C₈	—	46	4	no	—
104	C₂²	#10624987	1	2	yes	—
105	C₂²	#10625005	1	1	yes	characteristic
106	C₂²	#10625006	1	1	yes	characteristic
107	C₂²	#10625007	1	2	yes	—
108	C₂²	C₂₃⋊#95	1	1	yes	characteristic
109	C₄	—	2	1	no	—
110	C₄	—	2	1	no	—
111	C₄	—	2	1	no	—
112	C₄	—	2	1	no	—
113	C₄	—	2	1	no	—
114	C₄	—	2	1	no	—
115	C₄	—	184	2	no	—
116	C₂	#13092156	1	2	yes	—
117	C₂	#13092158	1	2	yes	—
118	C₂	#13092177	1	1	yes	characteristic
119	C₂	#13092178	1	1	yes	characteristic
120	C₂	#13092185	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#10624985

C₂

yes

maximal, characteristic

#13092283

C₂

yes

maximal, characteristic

C₂×C₂₃×#97

C₂

yes

maximal, characteristic

C₂²×C₈×C₂₃

—

C₂×C₄²×C₂₃

C₂²

yes

characteristic

C₂×C₄⋊Dic₂₃

C₄

yes

—

C₂²×C₂₃⋊C₈

—

C₂₃×#97

—

C₂₃×#97

C₄

yes

complemented

C₂₃⋊#95

—

C₂₃⋊#95

—

#10624985

—

#10624985

C₄

yes

complemented

C₂²×C₄×C₂₃

C₂×C₄

yes

characteristic

C₂²×C₄×C₂₃

D₄

yes

characteristic

C₂²×C₄×C₂₃

D₄

yes

characteristic

C₂²×Dic₂₃

—

C₂×C₈×C₂₃

—

C₂×C₈×C₂₃

—

C₄²×C₂₃

C₂×C₄

yes

characteristic

C₄²×C₂₃

C₂×C₄

yes

characteristic

C₄²×C₂₃

D₄

yes

characteristic

C₄²×C₂₃

Q₈

yes

characteristic

C₄⋊Dic₂₃

—

C₂×C₂₃⋊C₈

—

C₂×C₂₃⋊C₈

—

C₂³×C₂₃

C₂²⋊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

C₄⋊C₄

yes

characteristic

C₂×C₄×C₂₃

C₄⋊C₄

yes

characteristic

C₂×C₄×C₂₃

C₄⋊C₄

yes

characteristic

C₂×Dic₂₃

—

C₂×Dic₂₃

—

C₂×Dic₂₃

—

C₈×C₂₃

—

C₂₃⋊C₈

—

—

C₂²×C₂₃

yes

C₂²×C₂₃

yes

C₂²×C₂₃

yes

C₂²×C₂₃

yes

—

C₂²×C₂₃

#104

yes

—

C₄×C₂₃

—

C₄×C₂₃

—

C₄×C₂₃

—

C₄×C₂₃

—

C₄×C₂₃

—

C₄×C₂₃

—

Dic₂₃

—

C₂×#97

—

C₂×#97

D₂₃

yes

characteristic

#382

—

C₂×C₂₃

#329

yes

—

C₂×C₂₃

#330

yes

—

C₂×C₂₃

yes

C₂×C₂₃

yes

C₂×C₂₃

yes

C₂²×C₈

—

C₂²×C₈

—

C₂×C₄⋊C₄

—

C₂×C₄²

C₂×D₂₃

yes

—

—

—

Dic₂₃

yes

complemented

#95

—

#95

—

C₂₃

#1295

yes

characteristic, complemented, Sylow

C₂²×C₄

—

C₂²×C₄

C₄×D₂₃

yes

characteristic

C₂²×C₄

D₉₂

yes

characteristic

C₂²×C₄

C₂²⋊D₂₃

yes

characteristic

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₄⋊C₄

—

C₄²

Dic₄₆

yes

characteristic

C₄²

C₄×D₂₃

yes

characteristic

C₄²

C₂×Dic₂₃

yes

characteristic

C₄²

C₂²⋊D₂₃

yes

characteristic

C₂³

(C₂×D₂₃)⋊C₄

yes

characteristic

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

C₄×Dic₂₃

yes

characteristic

C₂×C₄

Dic₂₃⋊C₄

yes

characteristic

C₂×C₄

Dic₂₃⋊C₄

yes

characteristic

C₂×C₄

C₄⋊Dic₂₃

yes

characteristic

100

C₂×C₄

(C₂×D₂₃)⋊C₄

yes

characteristic

101

C₂×C₄

C₂²⋊Dic₂₃

yes

characteristic

102

C₈

—

103

C₈

—

104

C₂²

#10624987

yes

—

105

C₂²

#10625005

yes

characteristic

106

C₂²

#10625006

yes

characteristic

107

C₂²

#10625007

yes

—

108

C₂²

C₂₃⋊#95

yes

characteristic

109

C₄

—

110

C₄

—

111

C₄

—

112

C₄

—

113

C₄

—

114

C₄

—

115

C₄

—

184

—

116

C₂

#13092156

yes

—

117

C₂

#13092158

yes

—

118

C₂

#13092177

yes

characteristic

119

C₂

#13092178

yes

characteristic

120

C₂

#13092185

yes

characteristic

	subgroup	G/H
center	C₂³	(C₂×D₂₃)⋊C₄
commutator	C₂×C₄×C₂₃	C₄²
Frattini	C₂×C₄²	C₂×D₂₃
Fitting	C₂×C₂₃×#97	C₂
socle, prime radical	C₂³×C₂₃	C₂²⋊C₄
cosocle	C₂×C₄²×C₂₃	C₂²
hypercenter	C₂×#97	D₂₃
hypocenter	C₂₃	#1295
LCS	G ▷ C₂×C₄×C₂₃ ▷ C₂²×C₂₃ ▷ C₂₃
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×#97

subgroup

G/H

center

C₂³

(C₂×D₂₃)⋊C₄

commutator

C₂×C₄×C₂₃

C₄²

Frattini

C₂×C₄²

C₂×D₂₃

Fitting

C₂×C₂₃×#97

C₂

socle, prime radical

C₂³×C₂₃

C₂²⋊C₄

cosocle

C₂×C₄²×C₂₃

C₂²

hypercenter

C₂×#97

D₂₃

hypocenter

C₂₃

#1295

LCS

G ▷ C₂×C₄×C₂₃ ▷ C₂²×C₂₃ ▷ C₂₃

UCS

C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×#97

N	H ≅ G/N	type
9. C₂₃×#97	C₄	semidirect
13. #10624985	C₄	semidirect
71. #97	Dic₂₃	semidirect
74. C₂₃	#1295	semidirect

H ≅ G/N

type

9. C₂₃×#97

C₄

semidirect

13. #10624985

C₄

semidirect

71. #97

Dic₂₃

semidirect

74. C₂₃

#1295

semidirect

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

subgroup	index
C₂×#10624985	2
#13092283	2
C₂×C₂₃×#97	2
#1295	23

subgroup

index

C₂×#10624985

#13092283

C₂×C₂₃×#97

#1295

Subgroups — #15671279

Conjugacy classes of subgroups

Complements

Maximal subgroups