Group #15673524 — subgroups

Subgroups — #15673524

Order 2944.

248 conjugacy classes of subgroups; only the 133 normal subgroups are shown.

normal subgroups only

This group has 248 conjugacy classes of subgroups — too many to draw. Only its 133 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×C₄×C₂₃⋊C₈	C₂	1	1	yes	maximal, characteristic
2	#13092287	C₂	1	1	yes	maximal, characteristic
3	#13092179	C₂	1	4	yes	maximal
4	C₂₃×#384	C₂	1	1	yes	maximal, characteristic
5	C₂×C₄⋊C₄×C₂₃	—	2	1	no	—
6	C₂×C₄⋊C₄×C₂₃	—	2	1	no	—
7	C₂×C₄⋊C₄×C₂₃	C₂²	1	2	yes	—
8	C₂×C₄²×C₂₃	C₂²	1	1	yes	characteristic
9	C₂×C₄⋊Dic₂₃	—	2	1	no	—
10	C₂×C₄⋊Dic₂₃	—	2	1	no	—
11	C₂×C₄⋊Dic₂₃	C₂²	1	2	yes	—
12	C₂²×C₂₃⋊C₈	C₂²	1	2	yes	—
13	C₄×C₂₃⋊C₈	C₄	1	4	yes	complemented
14	C₂²×C₄×C₂₃	—	2	2	no	—
15	C₂²×C₄×C₂₃	D₄	1	1	yes	characteristic
16	C₂²×C₄×C₂₃	D₄	1	1	yes	characteristic
17	C₂²×C₄×C₂₃	C₂³	1	1	yes	characteristic
18	C₂²×Dic₂₃	—	2	2	no	—
19	C₄⋊C₄×C₂₃	—	2	4	no	—
20	C₄⋊C₄×C₂₃	—	4	2	no	—
21	C₄⋊C₄×C₂₃	—	4	2	no	—
22	C₄²×C₂₃	C₂×C₄	1	2	yes	—
23	C₄²×C₂₃	Q₈	1	2	yes	—
24	C₄⋊Dic₂₃	—	2	4	no	—
25	C₄⋊Dic₂₃	—	4	2	no	—
26	C₄⋊Dic₂₃	—	4	2	no	—
27	C₂×C₂₃⋊C₈	—	2	2	no	—
28	C₂×C₂₃⋊C₈	—	2	2	no	—
29	C₂×C₂₃⋊C₈	C₂×C₄	1	4	yes	—
30	C₂³×C₂₃	C₂×D₄	1	1	yes	characteristic
31	C₂×C₄×C₂₃	—	4	2	no	—
32	C₂×C₄×C₂₃	—	4	2	no	—
33	C₂×C₄×C₂₃	—	4	2	no	—
34	C₂×C₄×C₂₃	C₄⋊C₄	1	2	yes	—
35	C₂×C₄×C₂₃	C₄⋊C₄	1	2	yes	—
36	C₂×C₄×C₂₃	D₈	1	2	yes	—
37	C₂×C₄×C₂₃	SD₁₆	1	2	yes	—
38	C₂×C₄×C₂₃	SD₁₆	1	2	yes	—
39	C₂×C₄×C₂₃	Q₁₆	1	2	yes	—
40	C₂×C₄×C₂₃	C₂²×C₄	1	1	yes	characteristic
41	C₂×C₄×C₂₃	C₂×Q₈	1	1	yes	characteristic
42	C₂×C₄×C₂₃	#41	1	2	yes	—
43	C₂×C₄×C₂₃	#41	1	2	yes	—
44	C₂×Dic₂₃	—	4	2	no	—
45	C₂×Dic₂₃	—	4	2	no	—
46	C₂×Dic₂₃	—	4	2	no	—
47	C₂₃⋊C₈	—	2	4	no	—
48	#1834	—	23	1	no	maximal, Sylow
49	C₂²×C₂₃	C₂×C₄⋊C₄	1	1	yes	characteristic
50	C₂²×C₂₃	#124	1	1	yes	characteristic
51	C₂²×C₂₃	#124	1	1	yes	characteristic
52	C₂²×C₂₃	C₂×D₈	1	1	yes	characteristic
53	C₂²×C₂₃	C₂×SD₁₆	1	1	yes	characteristic
54	C₂²×C₂₃	C₂×SD₁₆	1	1	yes	characteristic
55	C₂²×C₂₃	C₂×Q₁₆	1	1	yes	characteristic
56	C₄×C₂₃	—	8	2	no	—
57	C₄×C₂₃	#106	1	4	yes	—
58	C₄×C₂₃	#107	1	4	yes	—
59	C₄×C₂₃	#117	1	2	yes	—
60	C₄×C₂₃	#125	1	2	yes	—
61	Dic₂₃	—	8	2	no	—
62	C₂×C₄×C₈	—	23	1	no	—
63	#384	—	23	1	no	—
64	#384	D₂₃	1	1	yes	characteristic
65	#340	—	23	4	no	—
66	C₂×C₂₃	#382	1	1	yes	characteristic
67	C₂×C₂₃	C₂×#106	1	1	yes	characteristic
68	C₂×C₂₃	C₂×#107	1	1	yes	characteristic
69	C₂×C₂₃	#486	1	1	yes	characteristic
70	C₂×C₂₃	#486	1	1	yes	characteristic
71	C₂×C₂₃	#487	1	1	yes	characteristic
72	C₂×C₂₃	#487	1	1	yes	characteristic
73	C₂²×C₈	—	23	2	no	—
74	C₂×C₄⋊C₄	—	2	1	no	—
75	C₂×C₄⋊C₄	—	2	1	no	—
76	C₂×C₄⋊C₄	—	23	2	no	—
77	C₂×C₄⋊C₄	—	46	1	no	—
78	C₂×C₄⋊C₄	—	46	1	no	—
79	C₂×C₄⋊C₄	C₂×D₂₃	1	2	yes	—
80	C₂×C₄²	C₂×D₂₃	1	1	yes	characteristic
81	C₄×C₈	—	23	4	no	—
82	C₂₃	#1834	1	1	yes	characteristic, complemented, Sylow
83	C₂²×C₄	—	2	2	no	—
84	C₂²×C₄	—	46	2	no	—
85	C₂²×C₄	C₂²⋊D₂₃	1	1	yes	characteristic
86	C₂²×C₄	C₂²⋊D₂₃	1	1	yes	characteristic
87	C₂²×C₄	C₂²×D₂₃	1	1	yes	characteristic
88	C₂×C₈	—	23	4	no	—
89	C₂×C₈	—	46	2	no	—
90	C₂×C₈	—	46	2	no	—
91	C₄⋊C₄	—	2	4	no	—
92	C₄⋊C₄	—	4	2	no	—
93	C₄⋊C₄	—	4	2	no	—
94	C₄⋊C₄	—	46	4	no	—
95	C₄⋊C₄	—	92	2	no	—
96	C₄⋊C₄	—	92	2	no	—
97	C₄²	Dic₄₆	1	2	yes	—
98	C₄²	C₄×D₂₃	1	2	yes	—
99	C₂³	C₂×C₂²⋊D₂₃	1	1	yes	characteristic
100	C₂×C₄	—	4	2	no	—
101	C₂×C₄	—	4	2	no	—
102	C₂×C₄	—	4	2	no	—
103	C₂×C₄	—	92	2	no	—
104	C₂×C₄	—	92	2	no	—
105	C₂×C₄	—	92	2	no	—
106	C₂×C₄	Dic₂₃⋊C₄	1	2	yes	—
107	C₂×C₄	Dic₂₃⋊C₄	1	2	yes	—
108	C₂×C₄	#67272	1	2	yes	—
109	C₂×C₄	#67273	1	2	yes	—
110	C₂×C₄	#67274	1	2	yes	—
111	C₂×C₄	#67275	1	2	yes	—
112	C₂×C₄	C₂×Dic₄₆	1	1	yes	characteristic
113	C₂×C₄	C₂×C₄×D₂₃	1	1	yes	characteristic
114	C₂×C₄	#67290	1	2	yes	—
115	C₂×C₄	#67292	1	2	yes	—
116	C₈	—	46	4	no	—
117	C₂²	C₂×Dic₂₃⋊C₄	1	1	yes	characteristic
118	C₂²	C₂×#67272	1	1	yes	characteristic
119	C₂²	C₂×#67273	1	1	yes	characteristic
120	C₂²	#10625107	1	1	yes	characteristic
121	C₂²	C₂×#67274	1	1	yes	characteristic
122	C₂²	C₂×#67275	1	1	yes	characteristic
123	C₂²	#10625118	1	1	yes	characteristic
124	C₄	—	8	2	no	—
125	C₄	—	184	2	no	—
126	C₄	#10624989	1	4	yes	—
127	C₄	#10624990	1	4	yes	—
128	C₄	#10625061	1	2	yes	—
129	C₄	#10625063	1	2	yes	—
130	C₂	#13092282	1	1	yes	characteristic
131	C₂	C₂×#10624990	1	1	yes	characteristic
132	C₂	C₂×#10624989	1	1	yes	characteristic
133	C₂	#13092763	1	1	yes	characteristic
134	C₂	#13092765	1	1	yes	characteristic
135	C₂	#13092767	1	1	yes	characteristic
136	C₂	#13092769	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×C₄×C₂₃⋊C₈

C₂

yes

maximal, characteristic

#13092287

C₂

yes

maximal, characteristic

#13092179

C₂

yes

maximal

C₂₃×#384

C₂

yes

maximal, characteristic

C₂×C₄⋊C₄×C₂₃

—

C₂×C₄⋊C₄×C₂₃

—

C₂×C₄⋊C₄×C₂₃

C₂²

yes

—

C₂×C₄²×C₂₃

C₂²

yes

characteristic

C₂×C₄⋊Dic₂₃

—

C₂×C₄⋊Dic₂₃

—

C₂×C₄⋊Dic₂₃

C₂²

yes

—

C₂²×C₂₃⋊C₈

C₂²

yes

—

C₄×C₂₃⋊C₈

C₄

yes

complemented

C₂²×C₄×C₂₃

—

C₂²×C₄×C₂₃

D₄

yes

characteristic

C₂²×C₄×C₂₃

D₄

yes

characteristic

C₂²×C₄×C₂₃

C₂³

yes

characteristic

C₂²×Dic₂₃

—

C₄⋊C₄×C₂₃

—

C₄⋊C₄×C₂₃

—

C₄⋊C₄×C₂₃

—

C₄²×C₂₃

C₂×C₄

yes

—

C₄²×C₂₃

Q₈

yes

—

C₄⋊Dic₂₃

—

C₄⋊Dic₂₃

—

C₄⋊Dic₂₃

—

C₂×C₂₃⋊C₈

—

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

C₄⋊C₄

yes

—

C₂×C₄×C₂₃

C₄⋊C₄

yes

—

C₂×C₄×C₂₃

D₈

yes

—

C₂×C₄×C₂₃

SD₁₆

yes

—

C₂×C₄×C₂₃

SD₁₆

yes

—

C₂×C₄×C₂₃

Q₁₆

yes

—

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

—

C₂₃⋊C₈

—

#1834

—

maximal, Sylow

C₂²×C₂₃

C₂×C₄⋊C₄

yes

C₂²×C₂₃

yes

C₂²×C₂₃

yes

C₂²×C₂₃

C₂×D₈

yes

characteristic

C₂²×C₂₃

C₂×SD₁₆

yes

characteristic

C₂²×C₂₃

C₂×SD₁₆

yes

characteristic

C₂²×C₂₃

C₂×Q₁₆

yes

characteristic

C₄×C₂₃

—

C₄×C₂₃

#106

yes

—

C₄×C₂₃

#107

yes

—

C₄×C₂₃

#117

yes

—

C₄×C₂₃

#125

yes

—

Dic₂₃

—

C₂×C₄×C₈

—

#384

—

#384

D₂₃

yes

characteristic

#340

—

C₂×C₂₃

yes

C₂×C₂₃

yes

C₂×C₂₃

yes

C₂×C₂₃

yes

C₂×C₂₃

yes

C₂×C₂₃

yes

C₂×C₂₃

yes

C₂²×C₈

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

C₂×D₂₃

yes

—

C₂×C₄²

C₂×D₂₃

yes

characteristic

C₄×C₈

—

C₂₃

#1834

yes

characteristic, complemented, Sylow

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

C₂²⋊D₂₃

yes

characteristic

C₂²×C₄

C₂²⋊D₂₃

yes

characteristic

C₂²×C₄

C₂²×D₂₃

yes

characteristic

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄²

Dic₄₆

yes

—

C₄²

C₄×D₂₃

yes

—

C₂³

C₂×C₂²⋊D₂₃

yes

characteristic

100

C₂×C₄

—

101

C₂×C₄

—

102

C₂×C₄

—

103

C₂×C₄

—

104

C₂×C₄

—

105

C₂×C₄

—

106

C₂×C₄

Dic₂₃⋊C₄

yes

—

107

C₂×C₄

Dic₂₃⋊C₄

yes

—

108

C₂×C₄

#67272

yes

—

109

C₂×C₄

#67273

yes

—

110

C₂×C₄

#67274

yes

—

111

C₂×C₄

#67275

yes

—

112

C₂×C₄

C₂×Dic₄₆

yes

characteristic

113

C₂×C₄

C₂×C₄×D₂₃

yes

characteristic

114

C₂×C₄

#67290

yes

—

115

C₂×C₄

#67292

yes

—

116

C₈

—

117

C₂²

C₂×Dic₂₃⋊C₄

yes

characteristic

118

C₂²

C₂×#67272

yes

characteristic

119

C₂²

C₂×#67273

yes

characteristic

120

C₂²

#10625107

yes

characteristic

121

C₂²

C₂×#67274

yes

characteristic

122

C₂²

C₂×#67275

yes

characteristic

123

C₂²

#10625118

yes

characteristic

124

C₄

—

125

C₄

—

184

—

126

C₄

#10624989

yes

—

127

C₄

#10624990

yes

—

128

C₄

#10625061

yes

—

129

C₄

#10625063

yes

—

130

C₂

#13092282

yes

characteristic

131

C₂

C₂×#10624990

yes

characteristic

132

C₂

C₂×#10624989

yes

characteristic

133

C₂

#13092763

yes

characteristic

134

C₂

#13092765

yes

characteristic

135

C₂

#13092767

yes

characteristic

136

C₂

#13092769

yes

characteristic

	subgroup	G/H
center	C₂³	C₂×C₂²⋊D₂₃
commutator	C₂×C₄×C₂₃	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₂₃
Fitting	C₂₃×#384	C₂
socle, prime radical	C₂³×C₂₃	C₂×D₄
cosocle	C₂²×C₄×C₂₃	C₂³
hypercenter	#384	D₂₃
hypocenter	C₂₃	#1834
LCS	G ▷ C₂×C₄×C₂₃ ▷ C₂×C₂₃ ▷ C₂₃
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #384

subgroup

G/H

center

C₂³

C₂×C₂²⋊D₂₃

commutator

C₂×C₄×C₂₃

C₂²×C₄

Frattini

C₂²×C₄

C₂²×D₂₃

Fitting

C₂₃×#384

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 ◁ C₂×C₄² ◁ #384

N	H ≅ G/N	type
13. C₄×C₂₃⋊C₈	C₄	semidirect
82. C₂₃	#1834	semidirect

H ≅ G/N

type

13. C₄×C₂₃⋊C₈

C₄

semidirect

82. C₂₃

#1834

semidirect

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

subgroup	index
C₂×C₄×C₂₃⋊C₈	2
#13092287	2
#13092179	2
C₂₃×#384	2
#1834	23

subgroup

index

C₂×C₄×C₂₃⋊C₈

#13092287

#13092179

C₂₃×#384

#1834

Subgroups — #15673524

Conjugacy classes of subgroups

Complements

Maximal subgroups