Group #15674144 — subgroups

Subgroups — #15674144

Order 2944.

208 conjugacy classes of subgroups; only the 83 normal subgroups are shown.

normal subgroups only

This group has 208 conjugacy classes of subgroups — too many to draw. Only its 83 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×#10625004	C₂	1	1	yes	maximal, characteristic
2	#13092453	C₂	1	1	yes	maximal, characteristic
3	#13092445	C₂	1	1	yes	maximal, characteristic
4	#13092444	C₂	1	1	yes	maximal, characteristic
5	C₂×C₄×C₂₃⋊C₈	C₂	1	1	yes	maximal, characteristic
6	#13092177	C₂	1	1	yes	maximal, characteristic
7	C₂₃×#339	C₂	1	1	yes	maximal, characteristic
8	C₂×M₁₆×C₂₃	C₂²	1	1	yes	characteristic
9	C₂₃×#117	C₂²	1	1	yes	characteristic
10	C₂×C₄²×C₂₃	C₂²	1	1	yes	characteristic
11	#10625097	C₂²	1	1	yes	characteristic
12	C₂×#67267	C₂²	1	1	yes	characteristic
13	C₂²×C₂₃⋊C₈	C₂²	1	1	yes	characteristic
14	C₂²×C₂₃⋊C₈	C₂²	1	1	yes	characteristic
15	#10625004	—	2	2	no	—
16	Dic₂₃⋊C₈	—	2	2	no	—
17	#10624990	C₄	1	2	yes	complemented
18	#10624989	C₄	1	2	yes	complemented
19	C₄⋊(C₂₃⋊C₈)	—	2	2	no	—
20	C₄×C₂₃⋊C₈	—	2	2	no	—
21	C₂²×C₄×C₂₃	—	2	1	no	—
22	C₂²×C₄×C₂₃	C₂³	1	1	yes	characteristic
23	M₁₆×C₂₃	—	2	1	no	—
24	M₁₆×C₂₃	—	2	1	no	—
25	C₂×C₈×C₂₃	—	2	1	no	—
26	C₄⋊C₄×C₂₃	C₂×C₄	1	2	yes	—
27	C₂²⋊C₄×C₂₃	—	2	1	no	—
28	C₄²×C₂₃	—	2	1	no	—
29	C₄²×C₂₃	—	2	1	no	—
30	C₄²×C₂₃	Q₈	1	2	yes	—
31	C₂²⋊Dic₂₃	—	2	1	no	—
32	C₄⋊Dic₂₃	C₂×C₄	1	2	yes	—
33	C₄×Dic₂₃	—	2	1	no	—
34	#67267	—	2	1	no	—
35	#67267	—	2	1	no	—
36	C₂×C₂₃⋊C₈	—	2	1	no	—
37	C₂×C₂₃⋊C₈	—	2	1	no	—
38	C₂×C₂₃⋊C₈	—	2	1	no	—
39	C₂×C₂₃⋊C₈	C₂×C₄	1	1	yes	characteristic
40	C₂×C₂₃⋊C₈	C₂×C₄	1	1	yes	characteristic
41	C₂×C₂₃⋊C₈	D₄	1	2	yes	—
42	C₂×C₂₃⋊C₈	D₄	1	2	yes	—
43	C₂×C₂₃⋊C₈	Q₈	1	2	yes	—
44	C₂³×C₂₃	#41	1	1	yes	characteristic
45	C₂×C₄×C₂₃	—	2	4	no	—
46	C₂×C₄×C₂₃	—	2	2	no	—
47	C₂×C₄×C₂₃	—	2	2	no	—
48	C₂×C₄×C₂₃	C₂²×C₄	1	1	yes	characteristic
49	C₂×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
50	C₂×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
51	C₂×C₄×C₂₃	C₂×Q₈	1	1	yes	characteristic
52	C₂×C₄×C₂₃	C₂×Q₈	1	1	yes	characteristic
53	C₂×C₄×C₂₃	#41	1	1	yes	characteristic
54	C₂×Dic₂₃	—	2	2	no	—
55	C₈×C₂₃	—	4	1	no	—
56	C₂₃⋊C₈	—	2	2	no	—
57	C₂₃⋊C₈	—	4	1	no	—
58	C₂₃⋊C₈	C₄⋊C₄	1	2	yes	—
59	C₂₃⋊C₈	C₄⋊C₄	1	2	yes	—
60	#1945	—	23	1	no	maximal, Sylow
61	C₂²×C₂₃	—	2	2	no	—
62	C₂²×C₂₃	C₄×D₄	1	1	yes	characteristic
63	C₂²×C₂₃	C₂²⋊Q₈	1	1	yes	characteristic
64	C₂²×C₂₃	#125	1	1	yes	characteristic
65	C₄×C₂₃	—	2	4	no	—
66	C₄×C₂₃	—	4	2	no	—
67	C₄×C₂₃	C₂×C₄⋊C₄	1	1	yes	characteristic
68	C₄×C₂₃	C₄×Q₈	1	1	yes	characteristic
69	C₄×C₂₃	#121	1	1	yes	characteristic
70	C₄×C₂₃	C₄⋊Q₈	1	1	yes	characteristic
71	Dic₂₃	—	4	2	no	—
72	C₂×#108	—	23	1	no	—
73	#427	—	23	1	no	—
74	#424	—	23	1	no	—
75	#424	—	23	1	no	—
76	C₂×C₄×C₈	—	23	1	no	—
77	#339	—	23	1	no	—
78	#339	D₂₃	1	1	yes	characteristic
79	C₂×C₂₃	—	2	1	no	—
80	C₂×C₂₃	—	2	1	no	—
81	C₂×C₂₃	#389	1	1	yes	characteristic
82	C₂×C₂₃	#443	1	2	yes	—
83	C₂×M₁₆	—	23	1	no	—
84	C₂×M₁₆	C₂×D₂₃	1	1	yes	characteristic
85	C₂²×C₈	—	23	1	no	—
86	C₂²×C₈	—	23	1	no	—
87	#117	—	23	1	no	—
88	#117	C₂×D₂₃	1	1	yes	characteristic
89	C₂×C₄²	C₂×D₂₃	1	1	yes	characteristic
90	#108	—	46	2	no	—
91	#107	—	23	2	no	—
92	#106	—	23	2	no	—
93	C₄⋊C₈	—	46	2	no	—
94	C₄⋊C₈	—	46	2	no	—
95	C₄×C₈	—	46	2	no	—
96	C₂₃	#1945	1	1	yes	characteristic, complemented, Sylow
97	C₂²×C₄	—	2	1	no	—
98	C₂²×C₄	C₂²×D₂₃	1	1	yes	characteristic
99	M₁₆	—	2	1	no	—
100	M₁₆	—	2	1	no	—
101	M₁₆	—	46	1	no	—
102	M₁₆	—	46	1	no	—
103	C₂×C₈	—	2	1	no	—
104	C₂×C₈	—	23	1	no	—
105	C₂×C₈	—	23	1	no	—
106	C₂×C₈	—	23	2	no	—
107	C₂×C₈	—	23	2	no	—
108	C₂×C₈	—	23	2	no	—
109	C₂×C₈	—	46	1	no	—
110	C₂×C₈	—	46	1	no	—
111	C₂×C₈	—	46	1	no	—
112	C₄⋊C₄	—	23	2	no	—
113	C₄⋊C₄	C₄×D₂₃	1	2	yes	—
114	C₂²⋊C₄	—	2	1	no	—
115	C₂²⋊C₄	—	46	1	no	—
116	C₄²	—	2	1	no	—
117	C₄²	—	2	1	no	—
118	C₄²	—	46	1	no	—
119	C₄²	Dic₄₆	1	2	yes	—
120	C₂³	#67290	1	1	yes	characteristic
121	C₂×C₄	—	2	4	no	—
122	C₂×C₄	—	2	2	no	—
123	C₂×C₄	—	2	2	no	—
124	C₂×C₄	—	46	2	no	—
125	C₂×C₄	C₂×Dic₄₆	1	1	yes	characteristic
126	C₂×C₄	C₂×C₄×D₂₃	1	1	yes	characteristic
127	C₂×C₄	#67288	1	1	yes	characteristic
128	C₂×C₄	D₄×D₂₃	1	1	yes	characteristic
129	C₂×C₄	D₄×D₂₃	1	1	yes	characteristic
130	C₂×C₄	Q₈×D₂₃	1	1	yes	characteristic
131	C₈	—	4	1	no	—
132	C₈	—	23	2	no	—
133	C₈	—	23	2	no	—
134	C₈	—	46	2	no	—
135	C₈	—	92	1	no	—
136	C₂²	—	2	2	no	—
137	C₂²	#10625049	1	1	yes	characteristic
138	C₂²	(C₂²⋊D₂₃)⋊C₄	1	1	yes	characteristic
139	C₂²	#10625060	1	1	yes	characteristic
140	C₄	—	2	4	no	—
141	C₄	—	4	2	no	—
142	C₄	—	92	2	no	—
143	C₄	C₄×Dic₄₆	1	1	yes	characteristic
144	C₄	#10625055	1	1	yes	characteristic
145	C₄	#10625059	1	1	yes	characteristic
146	C₄	C₄⋊C₄×D₂₃	1	1	yes	characteristic
147	C₂	—	2	1	no	—
148	C₂	—	2	1	no	—
149	C₂	#13092314	1	1	yes	characteristic
150	C₂	#13092531	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#10625004

C₂

yes

maximal, characteristic

#13092453

C₂

yes

maximal, characteristic

#13092445

C₂

yes

maximal, characteristic

#13092444

C₂

yes

maximal, characteristic

C₂×C₄×C₂₃⋊C₈

C₂

yes

maximal, characteristic

#13092177

C₂

yes

maximal, characteristic

C₂₃×#339

C₂

yes

maximal, characteristic

C₂×M₁₆×C₂₃

C₂²

yes

characteristic

C₂₃×#117

C₂²

yes

characteristic

C₂×C₄²×C₂₃

C₂²

yes

characteristic

#10625097

C₂²

yes

characteristic

C₂×#67267

C₂²

yes

characteristic

C₂²×C₂₃⋊C₈

C₂²

yes

characteristic

C₂²×C₂₃⋊C₈

C₂²

yes

characteristic

#10625004

—

Dic₂₃⋊C₈

—

#10624990

C₄

yes

complemented

#10624989

C₄

yes

complemented

C₄⋊(C₂₃⋊C₈)

—

C₄×C₂₃⋊C₈

—

C₂²×C₄×C₂₃

—

C₂²×C₄×C₂₃

C₂³

yes

characteristic

M₁₆×C₂₃

—

M₁₆×C₂₃

—

C₂×C₈×C₂₃

—

C₄⋊C₄×C₂₃

C₂×C₄

yes

—

C₂²⋊C₄×C₂₃

—

C₄²×C₂₃

—

C₄²×C₂₃

—

C₄²×C₂₃

Q₈

yes

—

C₂²⋊Dic₂₃

—

C₄⋊Dic₂₃

C₂×C₄

yes

—

C₄×Dic₂₃

—

#67267

—

#67267

—

C₂×C₂₃⋊C₈

—

C₂×C₂₃⋊C₈

—

C₂×C₂₃⋊C₈

—

C₂×C₂₃⋊C₈

C₂×C₄

yes

characteristic

C₂×C₂₃⋊C₈

C₂×C₄

yes

characteristic

C₂×C₂₃⋊C₈

D₄

yes

—

C₂×C₂₃⋊C₈

D₄

yes

—

C₂×C₂₃⋊C₈

Q₈

yes

—

C₂³×C₂₃

#41

yes

characteristic

C₂×C₄×C₂₃

—

C₂×C₄×C₂₃

—

C₂×C₄×C₂₃

—

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×Q₈

yes

characteristic

C₂×C₄×C₂₃

C₂×Q₈

yes

characteristic

C₂×C₄×C₂₃

#41

yes

characteristic

C₂×Dic₂₃

—

C₈×C₂₃

—

C₂₃⋊C₈

—

C₂₃⋊C₈

—

C₂₃⋊C₈

C₄⋊C₄

yes

—

C₂₃⋊C₈

C₄⋊C₄

yes

—

#1945

—

maximal, Sylow

C₂²×C₂₃

—

C₂²×C₂₃

C₄×D₄

yes

characteristic

C₂²×C₂₃

C₂²⋊Q₈

yes

characteristic

C₂²×C₂₃

#125

yes

characteristic

C₄×C₂₃

—

C₄×C₂₃

—

C₄×C₂₃

C₂×C₄⋊C₄

yes

characteristic

C₄×C₂₃

C₄×Q₈

yes

characteristic

C₄×C₂₃

#121

yes

characteristic

C₄×C₂₃

C₄⋊Q₈

yes

characteristic

Dic₂₃

—

—

—

—

—

C₂×C₄×C₈

—

#339

—

#339

D₂₃

yes

characteristic

C₂×C₂₃

—

C₂×C₂₃

—

C₂×C₂₃

#389

yes

characteristic

C₂×C₂₃

#443

yes

—

C₂×M₁₆

—

C₂×M₁₆

C₂×D₂₃

yes

characteristic

C₂²×C₈

—

C₂²×C₈

—

#117

—

#117

C₂×D₂₃

yes

characteristic

C₂×C₄²

C₂×D₂₃

yes

—

—

—

C₄⋊C₈

—

C₄⋊C₈

—

C₄×C₈

—

C₂₃

#1945

yes

characteristic, complemented, Sylow

C₂²×C₄

—

C₂²×C₄

C₂²×D₂₃

yes

characteristic

M₁₆

—

100

M₁₆

—

101

M₁₆

—

102

M₁₆

—

103

C₂×C₈

—

104

C₂×C₈

—

105

C₂×C₈

—

106

C₂×C₈

—

107

C₂×C₈

—

108

C₂×C₈

—

109

C₂×C₈

—

110

C₂×C₈

—

111

C₂×C₈

—

112

C₄⋊C₄

—

113

C₄⋊C₄

C₄×D₂₃

yes

—

114

C₂²⋊C₄

—

115

C₂²⋊C₄

—

116

C₄²

—

117

C₄²

—

118

C₄²

—

119

C₄²

Dic₄₆

yes

—

120

C₂³

#67290

yes

characteristic

121

C₂×C₄

—

122

C₂×C₄

—

123

C₂×C₄

—

124

C₂×C₄

—

125

C₂×C₄

C₂×Dic₄₆

yes

characteristic

126

C₂×C₄

C₂×C₄×D₂₃

yes

characteristic

127

C₂×C₄

#67288

yes

characteristic

128

C₂×C₄

D₄×D₂₃

yes

characteristic

129

C₂×C₄

D₄×D₂₃

yes

characteristic

130

C₂×C₄

Q₈×D₂₃

yes

characteristic

131

C₈

—

132

C₈

—

133

C₈

—

134

C₈

—

135

C₈

—

136

C₂²

—

137

C₂²

#10625049

yes

characteristic

138

C₂²

(C₂²⋊D₂₃)⋊C₄

yes

characteristic

139

C₂²

#10625060

yes

characteristic

140

C₄

—

141

C₄

—

142

C₄

—

143

C₄

C₄×Dic₄₆

yes

characteristic

144

C₄

#10625055

yes

characteristic

145

C₄

#10625059

yes

characteristic

146

C₄

C₄⋊C₄×D₂₃

yes

characteristic

147

C₂

—

148

C₂

—

149

C₂

#13092314

yes

characteristic

150

C₂

#13092531

yes

—

	subgroup	G/H
center	C₂×C₄	D₄×D₂₃
commutator	C₂×C₄×C₂₃	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₂₃
Fitting	C₂₃×#339	C₂
socle	C₂²×C₂₃	C₄×D₄
prime radical	C₂³×C₂₃	#41
cosocle	C₂²×C₄×C₂₃	C₂³
hypercenter	#339	D₂₃
hypocenter	C₂₃	#1945
LCS	G ▷ C₂×C₄×C₂₃ ▷ C₂×C₂₃ ▷ C₂₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄ ◁ #339

subgroup

G/H

center

C₂×C₄

D₄×D₂₃

commutator

C₂×C₄×C₂₃

C₂²×C₄

Frattini

C₂²×C₄

C₂²×D₂₃

Fitting

C₂₃×#339

C₂

socle

C₂²×C₂₃

C₄×D₄

prime radical

C₂³×C₂₃

#41

cosocle

C₂²×C₄×C₂₃

C₂³

D₂₃

C₂₃

G ▷ C₂×C₄×C₂₃ ▷ C₂×C₂₃ ▷ C₂₃

UCS

C₁ ◁ C₂×C₄ ◁ C₂²×C₄ ◁ #339

N	H ≅ G/N	type
17. #10624990	C₄	semidirect
18. #10624989	C₄	semidirect
96. C₂₃	#1945	semidirect

H ≅ G/N

type

17. #10624990

C₄

semidirect

18. #10624989

C₄

semidirect

96. C₂₃

#1945

semidirect

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

subgroup	index
C₂×#10625004	2
#13092453	2
#13092445	2
#13092444	2
C₂×C₄×C₂₃⋊C₈	2
#13092177	2
C₂₃×#339	2
#1945	23

subgroup

index

C₂×C₄×C₂₃⋊C₈

C₂₃×#339

Subgroups — #15674144

Conjugacy classes of subgroups

Complements

Maximal subgroups