Group #15673523 — subgroups

Subgroups — #15673523

Order 2944.

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

normal subgroups only

This group has 248 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×C₈×Dic₂₃	C₂	1	1	yes	maximal, characteristic
2	#13092286	C₂	1	2	yes	maximal
3	#13092180	C₂	1	1	yes	maximal, characteristic
4	#13092179	C₂	1	2	yes	maximal
5	C₂₃×#340	C₂	1	1	yes	maximal, characteristic
6	C₂²×C₈×C₂₃	C₂²	1	1	yes	characteristic
7	C₂×C₄⋊C₄×C₂₃	C₂²	1	2	yes	—
8	C₂×C₄⋊Dic₂₃	C₂²	1	2	yes	—
9	C₂×Dic₂₃⋊C₄	—	2	2	no	—
10	C₂×Dic₂₃⋊C₄	—	2	2	no	—
11	C₂×C₄×Dic₂₃	C₂²	1	1	yes	characteristic
12	C₂²×C₂₃⋊C₈	C₂²	1	1	yes	characteristic
13	C₈×Dic₂₃	C₄	1	4	yes	complemented
14	C₂²×C₄×C₂₃	—	2	2	no	—
15	C₂²×C₄×C₂₃	C₂³	1	1	yes	characteristic
16	C₂²×Dic₂₃	—	2	2	no	—
17	C₂²×Dic₂₃	D₄	1	1	yes	characteristic
18	C₂²×Dic₂₃	D₄	1	1	yes	characteristic
19	C₂×C₈×C₂₃	—	2	2	no	—
20	C₂×C₈×C₂₃	C₂×C₄	1	2	yes	—
21	C₄⋊C₄×C₂₃	—	2	2	no	—
22	C₄⋊C₄×C₂₃	—	2	2	no	—
23	C₄⋊Dic₂₃	—	2	2	no	—
24	C₄⋊Dic₂₃	—	2	2	no	—
25	Dic₂₃⋊C₄	—	4	4	no	—
26	Dic₂₃⋊C₄	—	4	4	no	—
27	C₄×Dic₂₃	C₂×C₄	1	2	yes	—
28	C₄×Dic₂₃	Q₈	1	2	yes	—
29	C₂×C₂₃⋊C₈	—	2	2	no	—
30	C₂×C₂₃⋊C₈	C₂×C₄	1	2	yes	—
31	C₂³×C₂₃	C₂×D₄	1	1	yes	characteristic
32	C₂×C₄×C₂₃	—	4	2	no	—
33	C₂×C₄×C₂₃	—	4	2	no	—
34	C₂×C₄×C₂₃	—	4	2	no	—
35	C₂×C₄×C₂₃	C₂²×C₄	1	1	yes	characteristic
36	C₂×C₄×C₂₃	C₂×Q₈	1	1	yes	characteristic
37	C₂×C₄×C₂₃	#41	1	2	yes	—
38	C₂×C₄×C₂₃	#41	1	2	yes	—
39	C₂×Dic₂₃	—	4	2	no	—
40	C₂×Dic₂₃	—	4	2	no	—
41	C₂×Dic₂₃	—	4	2	no	—
42	C₂×Dic₂₃	C₄⋊C₄	1	2	yes	—
43	C₂×Dic₂₃	C₄⋊C₄	1	2	yes	—
44	C₂×Dic₂₃	D₈	1	2	yes	—
45	C₂×Dic₂₃	SD₁₆	1	4	yes	—
46	C₂×Dic₂₃	Q₁₆	1	2	yes	—
47	C₈×C₂₃	—	2	2	no	—
48	C₂₃⋊C₈	—	2	2	no	—
49	#1834	—	23	1	no	maximal, Sylow
50	C₂²×C₂₃	C₂×C₄⋊C₄	1	1	yes	characteristic
51	C₂²×C₂₃	#124	1	2	yes	—
52	C₂²×C₂₃	C₂×D₈	1	1	yes	characteristic
53	C₂²×C₂₃	C₂×SD₁₆	1	2	yes	—
54	C₂²×C₂₃	C₂×Q₁₆	1	1	yes	characteristic
55	C₄×C₂₃	—	8	2	no	—
56	C₄×C₂₃	#117	1	1	yes	characteristic
57	C₄×C₂₃	#117	1	1	yes	characteristic
58	C₄×C₂₃	#125	1	2	yes	—
59	Dic₂₃	—	8	2	no	—
60	Dic₂₃	#106	1	4	yes	—
61	Dic₂₃	#107	1	4	yes	—
62	C₂×C₄×C₈	—	23	1	no	—
63	#384	—	23	2	no	—
64	#340	—	23	1	no	—
65	#340	—	23	2	no	—
66	#340	D₂₃	1	1	yes	characteristic
67	C₂×C₂₃	#382	1	1	yes	characteristic
68	C₂×C₂₃	C₂×#106	1	1	yes	characteristic
69	C₂×C₂₃	C₂×#107	1	1	yes	characteristic
70	C₂×C₂₃	#486	1	2	yes	—
71	C₂×C₂₃	#487	1	2	yes	—
72	C₂²×C₈	—	23	1	no	—
73	C₂²×C₈	C₂×D₂₃	1	1	yes	characteristic
74	C₂×C₄⋊C₄	—	23	2	no	—
75	C₂×C₄⋊C₄	—	46	2	no	—
76	C₂×C₄⋊C₄	—	46	2	no	—
77	C₂×C₄⋊C₄	C₂×D₂₃	1	2	yes	—
78	C₂×C₄²	—	23	1	no	—
79	C₄×C₈	—	23	4	no	—
80	C₂₃	#1834	1	1	yes	characteristic, complemented, Sylow
81	C₂²×C₄	—	2	2	no	—
82	C₂²×C₄	—	23	1	no	—
83	C₂²×C₄	—	23	1	no	—
84	C₂²×C₄	—	46	2	no	—
85	C₂²×C₄	C₂²×D₂₃	1	1	yes	characteristic
86	C₂×C₈	—	2	2	no	—
87	C₂×C₈	—	23	2	no	—
88	C₂×C₈	—	46	2	no	—
89	C₂×C₈	C₄×D₂₃	1	2	yes	—
90	C₄⋊C₄	—	2	2	no	—
91	C₄⋊C₄	—	2	2	no	—
92	C₄⋊C₄	—	46	2	no	—
93	C₄⋊C₄	—	46	2	no	—
94	C₄⋊C₄	—	92	4	no	—
95	C₄⋊C₄	—	92	4	no	—
96	C₄²	—	23	2	no	—
97	C₄²	—	23	2	no	—
98	C₂³	D₄×D₂₃	1	1	yes	characteristic
99	C₂×C₄	—	4	2	no	—
100	C₂×C₄	—	4	2	no	—
101	C₂×C₄	—	4	2	no	—
102	C₂×C₄	—	23	2	no	—
103	C₂×C₄	—	23	2	no	—
104	C₂×C₄	—	23	2	no	—
105	C₂×C₄	—	23	4	no	—
106	C₂×C₄	—	23	2	no	—
107	C₂×C₄	—	92	2	no	—
108	C₂×C₄	—	92	2	no	—
109	C₂×C₄	—	92	2	no	—
110	C₂×C₄	C₂×C₄×D₂₃	1	1	yes	characteristic
111	C₂×C₄	#67288	1	2	yes	—
112	C₂×C₄	#67290	1	2	yes	—
113	C₂×C₄	Q₈×D₂₃	1	1	yes	characteristic
114	C₈	—	2	2	no	—
115	C₈	—	46	2	no	—
116	C₂²	#10625056	1	2	yes	—
117	C₂²	C₄⋊C₄×D₂₃	1	1	yes	characteristic
118	C₂²	D₈×D₂₃	1	1	yes	characteristic
119	C₂²	SD₁₆×D₂₃	1	2	yes	—
120	C₂²	Q₁₆×D₂₃	1	1	yes	characteristic
121	C₄	—	8	2	no	—
122	C₄	—	23	4	no	—
123	C₄	—	23	4	no	—
124	C₄	—	184	2	no	—
125	C₄	#10625043	1	1	yes	characteristic
126	C₄	#10625048	1	1	yes	characteristic
127	C₄	#10625060	1	2	yes	—
128	C₂	#13092281	1	1	yes	characteristic
129	C₂	#106×D₂₃	1	1	yes	characteristic
130	C₂	#107×D₂₃	1	1	yes	characteristic
131	C₂	#13092762	1	2	yes	—
132	C₂	#13092766	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×C₈×Dic₂₃

C₂

yes

maximal, characteristic

#13092286

C₂

yes

maximal

#13092180

C₂

yes

maximal, characteristic

#13092179

C₂

yes

maximal

C₂₃×#340

C₂

yes

maximal, characteristic

C₂²×C₈×C₂₃

C₂²

yes

characteristic

C₂×C₄⋊C₄×C₂₃

C₂²

yes

—

C₂×C₄⋊Dic₂₃

C₂²

yes

—

C₂×Dic₂₃⋊C₄

—

C₂×Dic₂₃⋊C₄

—

C₂×C₄×Dic₂₃

C₂²

yes

characteristic

C₂²×C₂₃⋊C₈

C₂²

yes

characteristic

C₈×Dic₂₃

C₄

yes

complemented

C₂²×C₄×C₂₃

—

C₂²×C₄×C₂₃

C₂³

yes

characteristic

C₂²×Dic₂₃

—

C₂²×Dic₂₃

D₄

yes

characteristic

C₂²×Dic₂₃

D₄

yes

characteristic

C₂×C₈×C₂₃

—

C₂×C₈×C₂₃

C₂×C₄

yes

—

C₄⋊C₄×C₂₃

—

C₄⋊C₄×C₂₃

—

C₄⋊Dic₂₃

—

C₄⋊Dic₂₃

—

Dic₂₃⋊C₄

—

Dic₂₃⋊C₄

—

C₄×Dic₂₃

C₂×C₄

yes

—

C₄×Dic₂₃

Q₈

yes

—

C₂×C₂₃⋊C₈

—

C₂×C₂₃⋊C₈

C₂×C₄

yes

—

C₂³×C₂₃

C₂×D₄

yes

characteristic

C₂×C₄×C₂₃

—

C₂×C₄×C₂₃

—

C₂×C₄×C₂₃

—

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

C₄⋊C₄

yes

—

C₂×Dic₂₃

C₄⋊C₄

yes

—

C₂×Dic₂₃

D₈

yes

—

C₂×Dic₂₃

SD₁₆

yes

—

C₂×Dic₂₃

Q₁₆

yes

—

C₈×C₂₃

—

C₂₃⋊C₈

—

#1834

—

maximal, Sylow

C₂²×C₂₃

C₂×C₄⋊C₄

yes

characteristic

C₂²×C₂₃

#124

yes

—

C₂²×C₂₃

C₂×D₈

yes

characteristic

C₂²×C₂₃

C₂×SD₁₆

yes

—

C₂²×C₂₃

C₂×Q₁₆

yes

characteristic

C₄×C₂₃

—

C₄×C₂₃

yes

C₄×C₂₃

yes

C₄×C₂₃

yes

—

Dic₂₃

—

Dic₂₃

#106

yes

—

Dic₂₃

#107

yes

—

C₂×C₄×C₈

—

—

—

—

D₂₃

yes

C₂×C₂₃

yes

C₂×C₂₃

yes

C₂×C₂₃

yes

C₂×C₂₃

yes

—

C₂×C₂₃

#487

yes

—

C₂²×C₈

—

C₂²×C₈

C₂×D₂₃

yes

characteristic

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

C₂×D₂₃

yes

—

C₂×C₄²

—

C₄×C₈

—

C₂₃

#1834

yes

characteristic, complemented, Sylow

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

C₂²×D₂₃

yes

characteristic

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

C₄×D₂₃

yes

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄²

—

C₄²

—

C₂³

D₄×D₂₃

yes

characteristic

C₂×C₄

—

100

C₂×C₄

—

101

C₂×C₄

—

102

C₂×C₄

—

103

C₂×C₄

—

104

C₂×C₄

—

105

C₂×C₄

—

106

C₂×C₄

—

107

C₂×C₄

—

108

C₂×C₄

—

109

C₂×C₄

—

110

C₂×C₄

C₂×C₄×D₂₃

yes

characteristic

111

C₂×C₄

#67288

yes

—

112

C₂×C₄

#67290

yes

—

113

C₂×C₄

Q₈×D₂₃

yes

characteristic

114

C₈

—

115

C₈

—

116

C₂²

#10625056

yes

—

117

C₂²

C₄⋊C₄×D₂₃

yes

characteristic

118

C₂²

D₈×D₂₃

yes

characteristic

119

C₂²

SD₁₆×D₂₃

yes

—

120

C₂²

Q₁₆×D₂₃

yes

characteristic

121

C₄

—

122

C₄

—

123

C₄

—

124

C₄

—

184

—

125

C₄

#10625043

yes

characteristic

126

C₄

#10625048

yes

characteristic

127

C₄

#10625060

yes

—

128

C₂

#13092281

yes

characteristic

129

C₂

#106×D₂₃

yes

characteristic

130

C₂

#107×D₂₃

yes

characteristic

131

C₂

#13092762

yes

—

132

C₂

#13092766

yes

—

	subgroup	G/H
center	C₂³	D₄×D₂₃
commutator	C₂×C₄×C₂₃	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₂₃
Fitting	C₂₃×#340	C₂
socle, prime radical	C₂³×C₂₃	C₂×D₄
cosocle	C₂²×C₄×C₂₃	C₂³
hypercenter	#340	D₂₃
hypocenter	C₂₃	#1834
LCS	G ▷ C₂×C₄×C₂₃ ▷ C₂×C₂₃ ▷ C₂₃
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄ ◁ #340

subgroup

G/H

center

C₂³

D₄×D₂₃

commutator

C₂×C₄×C₂₃

C₂²×C₄

Frattini

C₂²×C₄

C₂²×D₂₃

Fitting

C₂₃×#340

C₂

socle, prime radical

C₂³×C₂₃

C₂×D₄

cosocle

C₂²×C₄×C₂₃

C₂³

D₂₃

C₂₃

G ▷ C₂×C₄×C₂₃ ▷ C₂×C₂₃ ▷ C₂₃

UCS

C₁ ◁ C₂³ ◁ C₂²×C₄ ◁ #340

N	H ≅ G/N	type
13. C₈×Dic₂₃	C₄	semidirect
80. C₂₃	#1834	semidirect

H ≅ G/N

type

13. C₈×Dic₂₃

C₄

semidirect

80. C₂₃

#1834

semidirect

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

subgroup	index
C₂×C₈×Dic₂₃	2
#13092286	2
#13092180	2
#13092179	2
C₂₃×#340	2
#1834	23

subgroup

index

C₂×C₈×Dic₂₃

C₂₃×#340

Subgroups — #15673523

Conjugacy classes of subgroups

Complements

Maximal subgroups