Group #15675623 — subgroups

Subgroups — #15675623

Order 2944.

140 conjugacy classes of subgroups; only the 51 normal subgroups are shown.

normal subgroups only

This group has 140 conjugacy classes of subgroups — too many to draw. Only its 51 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	#13092821	C₂	1	1	yes	maximal, characteristic
2	#13092795	C₂	1	1	yes	maximal, characteristic
3	#13092222	C₂	1	1	yes	maximal, characteristic
4	#13092221	C₂	1	1	yes	maximal, characteristic
5	#13092220	C₂	1	1	yes	maximal, characteristic
6	C₁₆×Dic₂₃	C₂	1	1	yes	maximal, characteristic
7	C₂₃×#358	C₂	1	1	yes	maximal, characteristic
8	C₂×Q₁₆×C₂₃	C₂²	1	1	yes	characteristic
9	#10625115	—	2	1	no	—
10	C₂×#67275	—	2	1	no	—
11	C₂×Dic₉₂	C₂²	1	1	yes	characteristic
12	#10625059	—	2	1	no	—
13	C₂×C₁₆×C₂₃	C₂²	1	1	yes	characteristic
14	C₂₃×#107	C₂²	1	1	yes	characteristic
15	#10625000	C₂²	1	1	yes	characteristic
16	C₈×Dic₂₃	C₂²	1	1	yes	characteristic
17	C₂×C₂₃⋊C₁₆	C₂²	1	1	yes	characteristic
18	#10624989	—	2	1	no	—
19	C₂×Q₈×C₂₃	—	2	1	no	—
20	C₂×Dic₄₆	—	2	1	no	—
21	C₂×Dic₄₆	—	4	1	no	—
22	Q₁₆×C₂₃	—	2	1	no	—
23	Q₁₆×C₂₃	—	2	1	no	—
24	C₂×C₈×C₂₃	C₂³	1	1	yes	characteristic
25	C₄⋊C₄×C₂₃	—	2	1	no	—
26	#67275	—	4	2	no	—
27	C₄⋊Dic₂₃	—	2	1	no	—
28	Dic₂₃⋊C₄	—	4	1	no	—
29	Dic₂₃⋊C₄	—	4	1	no	—
30	Dic₂₃⋊C₄	—	4	1	no	—
31	C₄×Dic₂₃	D₄	1	1	yes	characteristic
32	C₂×C₂₃⋊C₈	D₄	1	1	yes	characteristic
33	Dic₉₂	—	2	1	no	—
34	Dic₉₂	—	2	1	no	—
35	C₁₆×C₂₃	—	2	1	no	—
36	C₂₃⋊C₁₆	—	2	1	no	—
37	Q₈×C₂₃	—	4	1	no	—
38	Q₈×C₂₃	—	4	1	no	—
39	C₂×C₄×C₂₃	—	4	1	no	—
40	C₂×C₄×C₂₃	—	4	1	no	—
41	C₂×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
42	C₂×Dic₂₃	—	4	1	no	—
43	C₂×Dic₂₃	—	4	1	no	—
44	C₂×Dic₂₃	D₈	1	1	yes	characteristic
45	C₂×Dic₂₃	D₈	1	1	yes	characteristic
46	Dic₄₆	—	4	1	no	—
47	Dic₄₆	—	4	1	no	—
48	Dic₄₆	—	8	2	no	—
49	C₈×C₂₃	#41	1	1	yes	characteristic
50	C₈×C₂₃	#41	1	1	yes	characteristic
51	C₂₃⋊C₈	SD₁₆	1	2	yes	—
52	#2241	—	23	1	no	maximal, Sylow
53	C₂²×C₂₃	C₂×D₈	1	1	yes	characteristic
54	C₄×C₂₃	—	8	1	no	—
55	C₄×C₂₃	—	8	1	no	—
56	C₄×C₂₃	#124	1	1	yes	characteristic
57	C₄×C₂₃	C₂×SD₁₆	1	1	yes	characteristic
58	Dic₂₃	—	8	1	no	—
59	Dic₂₃	—	8	1	no	—
60	Dic₂₃	SD₃₂	1	2	yes	—
61	Dic₂₃	Q₃₂	1	2	yes	—
62	#500	—	23	1	no	—
63	#494	—	23	1	no	—
64	#358	—	23	1	no	—
65	#358	—	23	1	no	—
66	#358	—	23	1	no	—
67	#358	D₂₃	1	1	yes	characteristic
68	C₄×C₁₆	—	23	1	no	—
69	C₂×C₂₃	#486	1	1	yes	characteristic
70	C₂×C₂₃	C₂×SD₃₂	1	1	yes	characteristic
71	C₂×C₂₃	C₂×Q₃₂	1	1	yes	characteristic
72	C₂×Q₁₆	—	23	1	no	—
73	C₂×Q₁₆	—	46	1	no	—
74	C₂×Q₁₆	C₂×D₂₃	1	1	yes	characteristic
75	C₄⋊Q₈	—	46	1	no	—
76	C₄⋊Q₈	—	46	1	no	—
77	C₂×C₁₆	—	23	1	no	—
78	C₂×C₁₆	C₂×D₂₃	1	1	yes	characteristic
79	#107	—	23	1	no	—
80	#107	—	46	1	no	—
81	#107	C₂×D₂₃	1	1	yes	characteristic
82	C₄×C₈	—	23	1	no	—
83	C₂₃	#2241	1	1	yes	characteristic, complemented, Sylow
84	C₂×Q₈	—	2	1	no	—
85	C₂×Q₈	—	46	1	no	—
86	C₂×Q₈	—	92	1	no	—
87	Q₁₆	—	2	1	no	—
88	Q₁₆	—	2	1	no	—
89	Q₁₆	—	46	1	no	—
90	Q₁₆	—	46	1	no	—
91	Q₁₆	—	92	2	no	—
92	C₂×C₈	—	23	1	no	—
93	C₂×C₈	C₂²×D₂₃	1	1	yes	characteristic
94	C₄⋊C₄	—	2	1	no	—
95	C₄⋊C₄	—	46	1	no	—
96	C₄⋊C₄	—	92	1	no	—
97	C₄⋊C₄	—	92	1	no	—
98	C₄⋊C₄	—	92	1	no	—
99	C₄²	—	23	1	no	—
100	C₁₆	—	2	1	no	—
101	C₁₆	—	46	1	no	—
102	Q₈	—	4	1	no	—
103	Q₈	—	4	1	no	—
104	Q₈	—	92	1	no	—
105	Q₈	—	92	1	no	—
106	Q₈	—	184	2	no	—
107	C₂×C₄	—	4	1	no	—
108	C₂×C₄	—	4	1	no	—
109	C₂×C₄	—	23	1	no	—
110	C₂×C₄	—	23	1	no	—
111	C₂×C₄	—	92	1	no	—
112	C₂×C₄	—	92	1	no	—
113	C₂×C₄	D₄×D₂₃	1	1	yes	characteristic
114	C₈	—	23	2	no	—
115	C₈	#67288	1	1	yes	characteristic
116	C₈	#67290	1	1	yes	characteristic
117	C₂²	D₈×D₂₃	1	1	yes	characteristic
118	C₄	—	8	1	no	—
119	C₄	—	8	1	no	—
120	C₄	—	23	2	no	—
121	C₄	—	23	2	no	—
122	C₄	—	184	1	no	—
123	C₄	—	184	1	no	—
124	C₄	#10625056	1	1	yes	characteristic
125	C₄	SD₁₆×D₂₃	1	1	yes	characteristic
126	C₂	#13092762	1	1	yes	characteristic
127	C₂	SD₃₂×D₂₃	1	1	yes	characteristic
128	C₂	Q₃₂×D₂₃	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#13092821

C₂

yes

maximal, characteristic

#13092795

C₂

yes

maximal, characteristic

#13092222

C₂

yes

maximal, characteristic

#13092221

C₂

yes

maximal, characteristic

#13092220

C₂

yes

maximal, characteristic

C₁₆×Dic₂₃

C₂

yes

maximal, characteristic

C₂₃×#358

C₂

yes

maximal, characteristic

C₂×Q₁₆×C₂₃

C₂²

yes

characteristic

#10625115

—

C₂×#67275

—

C₂×Dic₉₂

C₂²

yes

characteristic

#10625059

—

C₂×C₁₆×C₂₃

C₂²

yes

characteristic

C₂₃×#107

C₂²

yes

characteristic

#10625000

C₂²

yes

characteristic

C₈×Dic₂₃

C₂²

yes

characteristic

C₂×C₂₃⋊C₁₆

C₂²

yes

characteristic

#10624989

—

C₂×Q₈×C₂₃

—

C₂×Dic₄₆

—

C₂×Dic₄₆

—

Q₁₆×C₂₃

—

Q₁₆×C₂₃

—

C₂×C₈×C₂₃

C₂³

yes

characteristic

C₄⋊C₄×C₂₃

—

#67275

—

C₄⋊Dic₂₃

—

Dic₂₃⋊C₄

—

Dic₂₃⋊C₄

—

Dic₂₃⋊C₄

—

C₄×Dic₂₃

D₄

yes

characteristic

C₂×C₂₃⋊C₈

D₄

yes

characteristic

Dic₉₂

—

Dic₉₂

—

C₁₆×C₂₃

—

C₂₃⋊C₁₆

—

Q₈×C₂₃

—

Q₈×C₂₃

—

C₂×C₄×C₂₃

—

C₂×C₄×C₂₃

—

C₂×C₄×C₂₃

C₂×D₄

yes

characteristic

C₂×Dic₂₃

—

C₂×Dic₂₃

—

C₂×Dic₂₃

D₈

yes

characteristic

C₂×Dic₂₃

D₈

yes

characteristic

Dic₄₆

—

Dic₄₆

—

Dic₄₆

—

C₈×C₂₃

yes

C₈×C₂₃

yes

C₂₃⋊C₈

SD₁₆

yes

—

#2241

—

maximal, Sylow

C₂²×C₂₃

C₂×D₈

yes

characteristic

C₄×C₂₃

—

C₄×C₂₃

—

C₄×C₂₃

#124

yes

characteristic

C₄×C₂₃

C₂×SD₁₆

yes

characteristic

Dic₂₃

—

Dic₂₃

—

Dic₂₃

SD₃₂

yes

—

Dic₂₃

Q₃₂

yes

—

—

—

—

—

—

D₂₃

yes

characteristic

C₄×C₁₆

—

C₂×C₂₃

#486

yes

characteristic

C₂×C₂₃

C₂×SD₃₂

yes

characteristic

C₂×C₂₃

C₂×Q₃₂

yes

characteristic

C₂×Q₁₆

—

C₂×Q₁₆

—

C₂×Q₁₆

C₂×D₂₃

yes

characteristic

C₄⋊Q₈

—

C₄⋊Q₈

—

C₂×C₁₆

—

C₂×C₁₆

C₂×D₂₃

yes

—

—

C₂×D₂₃

yes

characteristic

C₄×C₈

—

C₂₃

#2241

yes

characteristic, complemented, Sylow

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

Q₁₆

—

Q₁₆

—

Q₁₆

—

Q₁₆

—

Q₁₆

—

C₂×C₈

—

C₂×C₈

C₂²×D₂₃

yes

characteristic

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄²

—

100

C₁₆

—

101

C₁₆

—

102

Q₈

—

103

Q₈

—

104

Q₈

—

105

Q₈

—

106

Q₈

—

184

—

107

C₂×C₄

—

108

C₂×C₄

—

109

C₂×C₄

—

110

C₂×C₄

—

111

C₂×C₄

—

112

C₂×C₄

—

113

C₂×C₄

D₄×D₂₃

yes

characteristic

114

C₈

—

115

C₈

#67288

yes

characteristic

116

C₈

#67290

yes

characteristic

117

C₂²

D₈×D₂₃

yes

characteristic

118

C₄

—

119

C₄

—

120

C₄

—

121

C₄

—

122

C₄

—

184

—

123

C₄

—

184

—

124

C₄

#10625056

yes

characteristic

125

C₄

SD₁₆×D₂₃

yes

characteristic

126

C₂

#13092762

yes

characteristic

127

C₂

SD₃₂×D₂₃

yes

characteristic

128

C₂

Q₃₂×D₂₃

yes

characteristic

	subgroup	G/H
center	C₂²	D₈×D₂₃
commutator, cosocle	C₂×C₈×C₂₃	C₂³
Frattini	C₂×C₈	C₂²×D₂₃
Fitting	C₂₃×#358	C₂
socle, prime radical	C₂²×C₂₃	C₂×D₈
hypercenter	#358	D₂₃
hypocenter	C₂₃	#2241
LCS	G ▷ C₂×C₈×C₂₃ ▷ C₄×C₂₃ ▷ C₂×C₂₃ ▷ C₂₃
UCS	C₁ ◁ C₂² ◁ C₂×C₄ ◁ C₂×C₈ ◁ #358

subgroup

G/H

center

C₂²

D₈×D₂₃

commutator, cosocle

C₂×C₈×C₂₃

C₂³

Frattini

C₂×C₈

C₂²×D₂₃

Fitting

C₂₃×#358

C₂

socle, prime radical

C₂²×C₂₃

C₂×D₈

D₂₃

C₂₃

G ▷ C₂×C₈×C₂₃ ▷ C₄×C₂₃ ▷ C₂×C₂₃ ▷ C₂₃

UCS

C₁ ◁ C₂² ◁ C₂×C₄ ◁ C₂×C₈ ◁ #358

N	H ≅ G/N	type
83. C₂₃	#2241	semidirect

H ≅ G/N

type

83. C₂₃

#2241

semidirect

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

subgroup	index
#13092821	2
#13092795	2
#13092222	2
#13092221	2
#13092220	2
C₁₆×Dic₂₃	2
C₂₃×#358	2
#2241	23

subgroup

index

C₁₆×Dic₂₃

C₂₃×#358

#2241

Subgroups — #15675623

Conjugacy classes of subgroups

Complements

Maximal subgroups