Group #15674170 — subgroups

Subgroups — #15674170

Order 2944.

200 conjugacy classes of subgroups; only the 93 normal subgroups are shown.

normal subgroups only

This group has 200 conjugacy classes of subgroups — too many to draw. Only its 93 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	2	yes	maximal
2	C₂×C₄⋊(C₂₃⋊C₈)	C₂	1	1	yes	maximal, characteristic
3	C₄×#67267	C₂	1	1	yes	maximal, characteristic
4	#13092181	C₂	1	2	yes	maximal
5	C₂₃×#423	C₂	1	1	yes	maximal, characteristic
6	C₂×M₁₆×C₂₃	C₂²	1	2	yes	—
7	C₂×C₄²×C₂₃	C₂²	1	1	yes	characteristic
8	C₂×#67267	C₂²	1	1	yes	characteristic
9	C₂×#67267	C₂²	1	1	yes	characteristic
10	C₂²×C₂₃⋊C₈	C₂²	1	2	yes	—
11	C₂₃×C₄⋊C₈	—	2	1	no	—
12	C₂₃×C₄⋊C₈	—	2	1	no	—
13	#10625004	—	2	4	no	—
14	C₄⋊(C₂₃⋊C₈)	C₄	1	2	yes	—
15	C₄⋊(C₂₃⋊C₈)	C₄	1	2	yes	—
16	#10624985	—	2	1	no	—
17	C₄×C₂₃⋊C₈	—	2	1	no	—
18	C₂²×C₄×C₂₃	Q₈	1	1	yes	characteristic
19	C₂²×C₄×C₂₃	Q₈	1	1	yes	characteristic
20	C₂²×C₄×C₂₃	C₂³	1	1	yes	characteristic
21	M₁₆×C₂₃	—	2	2	no	—
22	M₁₆×C₂₃	—	2	2	no	—
23	C₂×C₈×C₂₃	—	2	2	no	—
24	C₄²×C₂₃	C₂×C₄	1	1	yes	characteristic
25	C₄²×C₂₃	C₂×C₄	1	1	yes	characteristic
26	C₄²×C₂₃	D₄	1	1	yes	characteristic
27	C₄²×C₂₃	D₄	1	1	yes	characteristic
28	#67267	—	2	1	no	—
29	#67267	—	2	1	no	—
30	#67267	D₄	1	2	yes	—
31	#67267	Q₈	1	2	yes	—
32	C₂×C₂₃⋊C₈	—	2	2	no	—
33	C₂×C₂₃⋊C₈	—	2	1	no	—
34	C₂×C₂₃⋊C₈	—	2	1	no	—
35	C₂×C₂₃⋊C₈	—	2	2	no	—
36	C₂×C₂₃⋊C₈	C₂×C₄	1	4	yes	—
37	C₂³×C₂₃	C₂×Q₈	1	1	yes	characteristic
38	C₂×C₄×C₂₃	—	2	2	no	—
39	C₂×C₄×C₂₃	—	4	1	no	—
40	C₂×C₄×C₂₃	C₄⋊C₄	1	1	yes	characteristic
41	C₂×C₄×C₂₃	C₄⋊C₄	1	1	yes	characteristic
42	C₂×C₄×C₂₃	C₄⋊C₄	1	1	yes	characteristic
43	C₂×C₄×C₂₃	C₄⋊C₄	1	1	yes	characteristic
44	C₂×C₄×C₂₃	C₂²×C₄	1	1	yes	characteristic
45	C₂×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
46	C₂×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
47	C₂×C₄×C₂₃	C₂×Q₈	1	1	yes	characteristic
48	C₂×C₄×C₂₃	#41	1	1	yes	characteristic
49	C₂×C₄×C₂₃	#41	1	1	yes	characteristic
50	C₈×C₂₃	—	4	2	no	—
51	C₂₃⋊C₈	—	2	4	no	—
52	C₂₃⋊C₈	—	2	2	no	—
53	C₂₃⋊C₈	—	2	2	no	—
54	#1950	—	23	1	no	maximal, Sylow
55	C₂²×C₂₃	—	2	1	no	—
56	C₂²×C₂₃	—	2	1	no	—
57	C₂²×C₂₃	C₂×C₄⋊C₄	1	1	yes	characteristic
58	C₂²×C₂₃	#125	1	1	yes	characteristic
59	C₂²×C₂₃	C₄⋊Q₈	1	1	yes	characteristic
60	C₄×C₂₃	—	2	1	no	—
61	C₄×C₂₃	—	2	1	no	—
62	C₄×C₂₃	—	2	1	no	—
63	C₄×C₂₃	#108	1	2	yes	complemented
64	C₄×C₂₃	C₄×D₄	1	1	yes	characteristic
65	C₄×C₂₃	C₄×Q₈	1	1	yes	characteristic
66	C₄×C₂₃	#121	1	1	yes	characteristic
67	C₄×C₂₃	C₂²⋊Q₈	1	1	yes	characteristic
68	C₂×#108	—	23	2	no	—
69	#423	D₂₃	1	1	yes	characteristic
70	C₂×C₄⋊C₈	—	23	1	no	—
71	C₄×M₁₆	—	23	1	no	—
72	#341	—	23	2	no	—
73	C₂×C₂₃	—	2	1	no	—
74	C₂×C₂₃	—	2	1	no	—
75	C₂×C₂₃	#389	1	1	yes	characteristic
76	C₂×C₂₃	C₂×#108	1	1	yes	characteristic
77	C₂×C₂₃	#430	1	1	yes	characteristic
78	C₂×M₁₆	—	23	1	no	—
79	C₂×M₁₆	—	23	1	no	—
80	C₂×M₁₆	C₂×D₂₃	1	2	yes	—
81	C₂²×C₈	—	23	2	no	—
82	C₂×C₄²	C₂×D₂₃	1	1	yes	characteristic
83	#108	—	46	4	no	—
84	C₄⋊C₈	—	2	1	no	—
85	C₄⋊C₈	—	2	1	no	—
86	C₄⋊C₈	—	23	2	no	—
87	C₄⋊C₈	—	23	2	no	—
88	#97	—	46	1	no	—
89	C₄×C₈	—	46	1	no	—
90	C₂₃	#1950	1	1	yes	characteristic, complemented, Sylow
91	C₂²×C₄	Dic₄₆	1	1	yes	characteristic
92	C₂²×C₄	Dic₄₆	1	1	yes	characteristic
93	C₂²×C₄	C₂²×D₂₃	1	1	yes	characteristic
94	M₁₆	—	2	2	no	—
95	M₁₆	—	2	2	no	—
96	M₁₆	—	23	2	no	—
97	M₁₆	—	23	2	no	—
98	M₁₆	—	46	1	no	—
99	M₁₆	—	46	1	no	—
100	C₂×C₈	—	2	2	no	—
101	C₂×C₈	—	23	4	no	—
102	C₂×C₈	—	46	2	no	—
103	C₂×C₈	—	46	2	no	—
104	C₂×C₈	—	46	1	no	—
105	C₂×C₈	—	46	1	no	—
106	C₄²	C₄×D₂₃	1	1	yes	characteristic
107	C₄²	C₄×D₂₃	1	1	yes	characteristic
108	C₄²	C₂²⋊D₂₃	1	1	yes	characteristic
109	C₄²	C₂²⋊D₂₃	1	1	yes	characteristic
110	C₂³	C₂×Dic₄₆	1	1	yes	characteristic
111	C₂×C₄	—	2	2	no	—
112	C₂×C₄	—	4	1	no	—
113	C₂×C₄	Dic₂₃⋊C₄	1	1	yes	characteristic
114	C₂×C₄	Dic₂₃⋊C₄	1	1	yes	characteristic
115	C₂×C₄	Dic₂₃⋊C₄	1	1	yes	characteristic
116	C₂×C₄	Dic₂₃⋊C₄	1	1	yes	characteristic
117	C₂×C₄	C₂×C₄×D₂₃	1	1	yes	characteristic
118	C₂×C₄	D₄×D₂₃	1	1	yes	characteristic
119	C₂×C₄	#67290	1	1	yes	characteristic
120	C₂×C₄	Q₈×D₂₃	1	1	yes	characteristic
121	C₂×C₄	#67292	1	1	yes	characteristic
122	C₂×C₄	C₂×C₂²⋊D₂₃	1	1	yes	characteristic
123	C₈	—	4	2	no	—
124	C₈	—	46	4	no	—
125	C₈	—	46	2	no	—
126	C₈	—	46	2	no	—
127	C₂²	—	2	1	no	—
128	C₂²	—	2	1	no	—
129	C₂²	#10625059	1	1	yes	characteristic
130	C₂²	#10625061	1	1	yes	characteristic
131	C₂²	C₂×Dic₂₃⋊C₄	1	1	yes	characteristic
132	C₄	—	2	1	no	—
133	C₄	—	2	1	no	—
134	C₄	—	2	1	no	—
135	C₄	#10625004	1	2	yes	complemented
136	C₄	#10625058	1	1	yes	characteristic
137	C₄	#10625064	1	1	yes	characteristic
138	C₄	#10625109	1	1	yes	characteristic
139	C₄	#10625117	1	1	yes	characteristic
140	C₂	—	2	1	no	—
141	C₂	—	2	1	no	—
142	C₂	#13092313	1	1	yes	characteristic
143	C₂	C₂×#10625004	1	1	yes	characteristic
144	C₂	#13092465	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#10625004

C₂

yes

maximal

C₂×C₄⋊(C₂₃⋊C₈)

C₂

yes

maximal, characteristic

C₄×#67267

C₂

yes

maximal, characteristic

#13092181

C₂

yes

maximal

C₂₃×#423

C₂

yes

maximal, characteristic

C₂×M₁₆×C₂₃

C₂²

yes

—

C₂×C₄²×C₂₃

C₂²

yes

characteristic

C₂×#67267

C₂²

yes

characteristic

C₂×#67267

C₂²

yes

characteristic

C₂²×C₂₃⋊C₈

C₂²

yes

—

C₂₃×C₄⋊C₈

—

C₂₃×C₄⋊C₈

—

#10625004

—

C₄⋊(C₂₃⋊C₈)

C₄

yes

—

C₄⋊(C₂₃⋊C₈)

C₄

yes

—

#10624985

—

C₄×C₂₃⋊C₈

—

C₂²×C₄×C₂₃

Q₈

yes

characteristic

C₂²×C₄×C₂₃

Q₈

yes

characteristic

C₂²×C₄×C₂₃

C₂³

yes

characteristic

M₁₆×C₂₃

—

M₁₆×C₂₃

—

C₂×C₈×C₂₃

—

C₄²×C₂₃

C₂×C₄

yes

characteristic

C₄²×C₂₃

C₂×C₄

yes

characteristic

C₄²×C₂₃

D₄

yes

characteristic

C₄²×C₂₃

D₄

yes

—

—

D₄

yes

—

#67267

Q₈

yes

—

C₂×C₂₃⋊C₈

—

C₂×C₂₃⋊C₈

—

C₂×C₂₃⋊C₈

—

C₂×C₂₃⋊C₈

—

C₂×C₂₃⋊C₈

C₂×C₄

yes

—

C₂³×C₂₃

C₂×Q₈

yes

characteristic

C₂×C₄×C₂₃

—

C₂×C₄×C₂₃

—

C₂×C₄×C₂₃

C₄⋊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×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

C₂×C₄×C₂₃

yes

C₂×C₄×C₂₃

yes

C₈×C₂₃

—

C₂₃⋊C₈

—

C₂₃⋊C₈

—

C₂₃⋊C₈

—

#1950

—

maximal, Sylow

C₂²×C₂₃

—

C₂²×C₂₃

—

C₂²×C₂₃

C₂×C₄⋊C₄

yes

characteristic

C₂²×C₂₃

#125

yes

characteristic

C₂²×C₂₃

C₄⋊Q₈

yes

characteristic

C₄×C₂₃

—

C₄×C₂₃

—

C₄×C₂₃

—

C₄×C₂₃

#108

yes

complemented

C₄×C₂₃

C₄×D₄

yes

characteristic

C₄×C₂₃

C₄×Q₈

yes

characteristic

C₄×C₂₃

#121

yes

characteristic

C₄×C₂₃

C₂²⋊Q₈

yes

characteristic

C₂×#108

—

#423

D₂₃

yes

characteristic

C₂×C₄⋊C₈

—

C₄×M₁₆

—

#341

—

C₂×C₂₃

—

C₂×C₂₃

—

C₂×C₂₃

yes

C₂×C₂₃

yes

C₂×C₂₃

yes

C₂×M₁₆

—

C₂×M₁₆

—

C₂×M₁₆

C₂×D₂₃

yes

—

C₂²×C₈

—

C₂×C₄²

C₂×D₂₃

yes

characteristic

#108

—

C₄⋊C₈

—

C₄⋊C₈

—

C₄⋊C₈

—

C₄⋊C₈

—

#97

—

C₄×C₈

—

C₂₃

#1950

yes

characteristic, complemented, Sylow

C₂²×C₄

Dic₄₆

yes

characteristic

C₂²×C₄

Dic₄₆

yes

characteristic

C₂²×C₄

C₂²×D₂₃

yes

characteristic

M₁₆

—

M₁₆

—

M₁₆

—

M₁₆

—

M₁₆

—

M₁₆

—

100

C₂×C₈

—

101

C₂×C₈

—

102

C₂×C₈

—

103

C₂×C₈

—

104

C₂×C₈

—

105

C₂×C₈

—

106

C₄²

C₄×D₂₃

yes

characteristic

107

C₄²

C₄×D₂₃

yes

characteristic

108

C₄²

C₂²⋊D₂₃

yes

characteristic

109

C₄²

C₂²⋊D₂₃

yes

characteristic

110

C₂³

C₂×Dic₄₆

yes

characteristic

111

C₂×C₄

—

112

C₂×C₄

—

113

C₂×C₄

Dic₂₃⋊C₄

yes

characteristic

114

C₂×C₄

Dic₂₃⋊C₄

yes

characteristic

115

C₂×C₄

Dic₂₃⋊C₄

yes

characteristic

116

C₂×C₄

Dic₂₃⋊C₄

yes

characteristic

117

C₂×C₄

C₂×C₄×D₂₃

yes

characteristic

118

C₂×C₄

D₄×D₂₃

yes

characteristic

119

C₂×C₄

#67290

yes

characteristic

120

C₂×C₄

Q₈×D₂₃

yes

characteristic

121

C₂×C₄

#67292

yes

characteristic

122

C₂×C₄

C₂×C₂²⋊D₂₃

yes

characteristic

123

C₈

—

124

C₈

—

125

C₈

—

126

C₈

—

127

C₂²

—

128

C₂²

—

129

C₂²

#10625059

yes

characteristic

130

C₂²

#10625061

yes

characteristic

131

C₂²

C₂×Dic₂₃⋊C₄

yes

characteristic

132

C₄

—

133

C₄

—

134

C₄

—

135

C₄

#10625004

yes

complemented

136

C₄

#10625058

yes

characteristic

137

C₄

#10625064

yes

characteristic

138

C₄

#10625109

yes

characteristic

139

C₄

#10625117

yes

characteristic

140

C₂

—

141

C₂

—

142

C₂

#13092313

yes

characteristic

143

C₂

C₂×#10625004

yes

characteristic

144

C₂

#13092465

yes

characteristic

	subgroup	G/H
center	C₂×C₄	C₂×C₂²⋊D₂₃
commutator	C₂×C₄×C₂₃	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₂₃
Fitting	C₂₃×#423	C₂
socle	C₂²×C₂₃	C₂×C₄⋊C₄
prime radical	C₂³×C₂₃	C₂×Q₈
cosocle	C₂²×C₄×C₂₃	C₂³
hypercenter	#423	D₂₃
hypocenter	C₂₃	#1950
LCS	G ▷ C₂×C₄×C₂₃ ▷ C₂×C₂₃ ▷ C₂₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #423

subgroup

G/H

center

C₂×C₄

C₂×C₂²⋊D₂₃

commutator

C₂×C₄×C₂₃

C₂²×C₄

Frattini

C₂²×C₄

C₂²×D₂₃

Fitting

C₂₃×#423

C₂

socle

C₂²×C₂₃

C₂×C₄⋊C₄

prime radical

C₂³×C₂₃

C₂×Q₈

cosocle

C₂²×C₄×C₂₃

C₂³

D₂₃

C₂₃

G ▷ C₂×C₄×C₂₃ ▷ C₂×C₂₃ ▷ C₂₃

UCS

C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #423

N	H ≅ G/N	type
63. C₄×C₂₃	#108	semidirect
90. C₂₃	#1950	semidirect
135. C₄	#10625004	semidirect

H ≅ G/N

type

63. C₄×C₂₃

#108

semidirect

90. C₂₃

#1950

semidirect

135. C₄

#10625004

semidirect

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

subgroup	index
C₂×#10625004	2
C₂×C₄⋊(C₂₃⋊C₈)	2
C₄×#67267	2
#13092181	2
C₂₃×#423	2
#1950	23

subgroup

index

C₂×C₄⋊(C₂₃⋊C₈)

C₂₃×#423

Subgroups — #15674170

Conjugacy classes of subgroups

Complements

Maximal subgroups