Group #15674974 — subgroups

Subgroups — #15674974

Order 2944.

158 conjugacy classes of subgroups; only the 57 normal subgroups are shown.

normal subgroups only

This group has 158 conjugacy classes of subgroups — too many to draw. Only its 57 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	#13092465	C₂	1	1	yes	maximal, characteristic
2	#13092440	C₂	1	1	yes	maximal, characteristic
3	#13092189	C₂	1	1	yes	maximal, characteristic
4	#13092178	C₂	1	1	yes	maximal, characteristic
5	#13092176	C₂	1	1	yes	maximal, characteristic
6	#13092173	C₂	1	1	yes	maximal, characteristic
7	C₂₃×#344	C₂	1	1	yes	maximal, characteristic
8	C₂×M₁₆×C₂₃	C₂²	1	1	yes	characteristic
9	C₂×M₁₆×C₂₃	C₂²	1	1	yes	characteristic
10	C₂₃×#117	C₂²	1	1	yes	characteristic
11	#10625097	C₂²	1	1	yes	characteristic
12	C₂×C₄×Dic₂₃	C₂²	1	1	yes	characteristic
13	C₂×#67267	C₂²	1	1	yes	characteristic
14	C₂×#67267	C₂²	1	1	yes	characteristic
15	#10625004	—	2	2	no	—
16	Dic₂₃⋊C₈	—	2	2	no	—
17	C₂²×C₄×C₂₃	C₂³	1	1	yes	characteristic
18	C₂²×Dic₂₃	—	2	1	no	—
19	M₁₆×C₂₃	—	2	1	no	—
20	M₁₆×C₂₃	—	2	1	no	—
21	M₁₆×C₂₃	—	2	1	no	—
22	M₁₆×C₂₃	—	2	1	no	—
23	C₂×C₈×C₂₃	—	2	1	no	—
24	C₂×C₈×C₂₃	—	2	1	no	—
25	C₄⋊C₄×C₂₃	—	2	1	no	—
26	C₂²⋊C₄×C₂₃	—	2	1	no	—
27	C₄²×C₂₃	—	2	1	no	—
28	C₂²⋊Dic₂₃	—	2	1	no	—
29	C₄⋊Dic₂₃	—	2	1	no	—
30	C₄×Dic₂₃	—	2	1	no	—
31	C₄×Dic₂₃	—	2	1	no	—
32	C₄×Dic₂₃	Q₈	1	1	yes	characteristic
33	C₄×Dic₂₃	Q₈	1	1	yes	characteristic
34	#67267	—	2	1	no	—
35	#67267	—	2	1	no	—
36	#67267	—	2	1	no	—
37	#67267	—	2	1	no	—
38	C₂×C₂₃⋊C₈	—	2	1	no	—
39	C₂×C₂₃⋊C₈	D₄	1	2	yes	—
40	C₂³×C₂₃	#41	1	1	yes	characteristic
41	C₂×C₄×C₂₃	—	4	1	no	—
42	C₂×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
43	C₂×C₄×C₂₃	C₂×Q₈	1	1	yes	characteristic
44	C₂×C₄×C₂₃	#41	1	1	yes	characteristic
45	C₂×C₄×C₂₃	#41	1	1	yes	characteristic
46	C₂×C₄×C₂₃	#41	1	1	yes	characteristic
47	C₂×C₄×C₂₃	#41	1	1	yes	characteristic
48	C₂×Dic₂₃	—	2	1	no	—
49	C₂×Dic₂₃	—	2	1	no	—
50	C₂×Dic₂₃	—	4	2	no	—
51	C₂×Dic₂₃	—	4	1	no	—
52	C₈×C₂₃	—	4	1	no	—
53	C₈×C₂₃	—	4	1	no	—
54	C₂₃⋊C₈	—	2	2	no	—
55	C₂₃⋊C₈	—	4	1	no	—
56	#2102	—	23	1	no	maximal, Sylow
57	C₂²×C₂₃	—	4	1	no	—
58	C₂²×C₂₃	C₂²⋊Q₈	1	1	yes	characteristic
59	C₂²×C₂₃	#126	1	1	yes	characteristic
60	C₂²×C₂₃	#126	1	1	yes	characteristic
61	C₄×C₂₃	—	8	1	no	—
62	C₄×C₂₃	#123	1	1	yes	characteristic
63	C₄×C₂₃	#124	1	1	yes	characteristic
64	C₄×C₂₃	#125	1	1	yes	characteristic
65	C₄×C₂₃	#125	1	1	yes	characteristic
66	Dic₂₃	—	2	2	no	—
67	Dic₂₃	—	4	1	no	—
68	Dic₂₃	—	8	1	no	—
69	#430	—	23	1	no	—
70	#423	—	23	1	no	—
71	#344	—	23	1	no	—
72	#344	D₂₃	1	1	yes	characteristic
73	#339	—	23	1	no	—
74	#339	—	23	1	no	—
75	H(3,ℤ/4)	—	23	1	no	—
76	C₂×C₂₃	—	2	1	no	—
77	C₂×C₂₃	—	2	1	no	—
78	C₂×C₂₃	—	2	1	no	—
79	C₂×C₂₃	#400	1	1	yes	characteristic
80	C₂×M₁₆	—	23	1	no	—
81	C₂×M₁₆	—	23	1	no	—
82	C₂×M₁₆	C₂×D₂₃	1	1	yes	characteristic
83	C₂×M₁₆	C₂×D₂₃	1	1	yes	characteristic
84	#117	—	23	1	no	—
85	#117	C₂×D₂₃	1	1	yes	characteristic
86	C₂×C₄²	—	23	1	no	—
87	#108	—	46	2	no	—
88	C₄⋊C₈	—	46	2	no	—
89	C₂₃	#2102	1	1	yes	characteristic, complemented, Sylow
90	C₂²×C₄	—	46	1	no	—
91	C₂²×C₄	C₂²×D₂₃	1	1	yes	characteristic
92	M₁₆	—	2	1	no	—
93	M₁₆	—	2	1	no	—
94	M₁₆	—	2	1	no	—
95	M₁₆	—	2	1	no	—
96	M₁₆	—	46	1	no	—
97	M₁₆	—	46	1	no	—
98	M₁₆	—	46	1	no	—
99	M₁₆	—	46	1	no	—
100	C₂×C₈	—	2	1	no	—
101	C₂×C₈	—	2	1	no	—
102	C₂×C₈	—	23	2	no	—
103	C₂×C₈	—	46	1	no	—
104	C₄⋊C₄	—	2	1	no	—
105	C₄⋊C₄	—	46	1	no	—
106	C₂²⋊C₄	—	2	1	no	—
107	C₂²⋊C₄	—	46	1	no	—
108	C₄²	—	2	1	no	—
109	C₄²	—	23	1	no	—
110	C₄²	—	23	1	no	—
111	C₄²	—	46	1	no	—
112	C₄²	—	46	1	no	—
113	C₂³	#67288	1	1	yes	characteristic
114	C₂×C₄	—	4	1	no	—
115	C₂×C₄	—	46	1	no	—
116	C₂×C₄	—	46	1	no	—
117	C₂×C₄	—	92	2	no	—
118	C₂×C₄	—	92	1	no	—
119	C₂×C₄	#67288	1	1	yes	characteristic
120	C₂×C₄	#67288	1	1	yes	characteristic
121	C₂×C₄	D₄×D₂₃	1	1	yes	characteristic
122	C₂×C₄	#67290	1	1	yes	characteristic
123	C₂×C₄	#67290	1	1	yes	characteristic
124	C₂×C₄	Q₈×D₂₃	1	1	yes	characteristic
125	C₈	—	4	1	no	—
126	C₈	—	4	1	no	—
127	C₈	—	46	2	no	—
128	C₈	—	92	1	no	—
129	C₂²	—	4	1	no	—
130	C₂²	#10625047	1	1	yes	characteristic
131	C₂²	#10625050	1	1	yes	characteristic
132	C₂²	#10625067	1	1	yes	characteristic
133	C₄	—	8	1	no	—
134	C₄	—	46	2	no	—
135	C₄	—	92	1	no	—
136	C₄	—	184	1	no	—
137	C₄	#10625054	1	1	yes	characteristic
138	C₄	#10625056	1	1	yes	characteristic
139	C₄	#10625060	1	1	yes	characteristic
140	C₄	#10625060	1	1	yes	characteristic
141	C₂	—	2	1	no	—
142	C₂	—	2	1	no	—
143	C₂	—	2	1	no	—
144	C₂	#13092358	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#13092465

C₂

yes

maximal, characteristic

#13092440

C₂

yes

maximal, characteristic

#13092189

C₂

yes

maximal, characteristic

#13092178

C₂

yes

maximal, characteristic

#13092176

C₂

yes

maximal, characteristic

#13092173

C₂

yes

maximal, characteristic

C₂₃×#344

C₂

yes

maximal, characteristic

C₂×M₁₆×C₂₃

C₂²

yes

characteristic

C₂×M₁₆×C₂₃

C₂²

yes

characteristic

C₂₃×#117

C₂²

yes

characteristic

#10625097

C₂²

yes

characteristic

C₂×C₄×Dic₂₃

C₂²

yes

characteristic

C₂×#67267

C₂²

yes

characteristic

C₂×#67267

C₂²

yes

characteristic

#10625004

—

Dic₂₃⋊C₈

—

C₂²×C₄×C₂₃

C₂³

yes

characteristic

C₂²×Dic₂₃

—

M₁₆×C₂₃

—

M₁₆×C₂₃

—

M₁₆×C₂₃

—

M₁₆×C₂₃

—

C₂×C₈×C₂₃

—

C₂×C₈×C₂₃

—

C₄⋊C₄×C₂₃

—

C₂²⋊C₄×C₂₃

—

C₄²×C₂₃

—

C₂²⋊Dic₂₃

—

C₄⋊Dic₂₃

—

C₄×Dic₂₃

—

C₄×Dic₂₃

—

C₄×Dic₂₃

Q₈

yes

characteristic

C₄×Dic₂₃

Q₈

yes

—

—

—

—

C₂×C₂₃⋊C₈

—

C₂×C₂₃⋊C₈

D₄

yes

—

C₂³×C₂₃

#41

yes

characteristic

C₂×C₄×C₂₃

—

C₂×C₄×C₂₃

C₂×D₄

yes

characteristic

C₂×C₄×C₂₃

C₂×Q₈

yes

C₂×C₄×C₂₃

yes

C₂×C₄×C₂₃

yes

C₂×C₄×C₂₃

yes

C₂×C₄×C₂₃

yes

C₂×Dic₂₃

—

C₂×Dic₂₃

—

C₂×Dic₂₃

—

C₂×Dic₂₃

—

C₈×C₂₃

—

C₈×C₂₃

—

C₂₃⋊C₈

—

C₂₃⋊C₈

—

#2102

—

maximal, Sylow

C₂²×C₂₃

—

C₂²×C₂₃

C₂²⋊Q₈

yes

C₂²×C₂₃

yes

C₂²×C₂₃

yes

C₄×C₂₃

—

C₄×C₂₃

yes

C₄×C₂₃

yes

C₄×C₂₃

yes

C₄×C₂₃

yes

Dic₂₃

—

Dic₂₃

—

Dic₂₃

—

—

—

—

D₂₃

yes

—

—

—

C₂×C₂₃

—

C₂×C₂₃

—

C₂×C₂₃

—

C₂×C₂₃

#400

yes

characteristic

C₂×M₁₆

—

C₂×M₁₆

—

C₂×M₁₆

C₂×D₂₃

yes

characteristic

C₂×M₁₆

C₂×D₂₃

yes

characteristic

#117

—

#117

C₂×D₂₃

yes

characteristic

C₂×C₄²

—

#108

—

C₄⋊C₈

—

C₂₃

#2102

yes

characteristic, complemented, Sylow

C₂²×C₄

—

C₂²×C₄

C₂²×D₂₃

yes

characteristic

M₁₆

—

M₁₆

—

M₁₆

—

M₁₆

—

M₁₆

—

M₁₆

—

M₁₆

—

M₁₆

—

100

C₂×C₈

—

101

C₂×C₈

—

102

C₂×C₈

—

103

C₂×C₈

—

104

C₄⋊C₄

—

105

C₄⋊C₄

—

106

C₂²⋊C₄

—

107

C₂²⋊C₄

—

108

C₄²

—

109

C₄²

—

110

C₄²

—

111

C₄²

—

112

C₄²

—

113

C₂³

#67288

yes

characteristic

114

C₂×C₄

—

115

C₂×C₄

—

116

C₂×C₄

—

117

C₂×C₄

—

118

C₂×C₄

—

119

C₂×C₄

#67288

yes

characteristic

120

C₂×C₄

#67288

yes

characteristic

121

C₂×C₄

D₄×D₂₃

yes

characteristic

122

C₂×C₄

#67290

yes

characteristic

123

C₂×C₄

#67290

yes

characteristic

124

C₂×C₄

Q₈×D₂₃

yes

characteristic

125

C₈

—

126

C₈

—

127

C₈

—

128

C₈

—

129

C₂²

—

130

C₂²

#10625047

yes

characteristic

131

C₂²

#10625050

yes

characteristic

132

C₂²

#10625067

yes

characteristic

133

C₄

—

134

C₄

—

135

C₄

—

136

C₄

—

184

—

137

C₄

#10625054

yes

characteristic

138

C₄

#10625056

yes

characteristic

139

C₄

#10625060

yes

characteristic

140

C₄

#10625060

yes

characteristic

141

C₂

—

142

C₂

—

143

C₂

—

144

C₂

#13092358

yes

characteristic

	subgroup	G/H
center	C₄	#10625056
commutator, cosocle	C₂²×C₄×C₂₃	C₂³
Frattini	C₂²×C₄	C₂²×D₂₃
Fitting	C₂₃×#344	C₂
socle	C₂×C₂₃	#400
prime radical	C₂³×C₂₃	#41
hypercenter	#344	D₂₃
hypocenter	C₂₃	#2102
LCS	G ▷ C₂²×C₄×C₂₃ ▷ C₂×C₂₃ ▷ C₂₃
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ #344

subgroup

G/H

center

C₄

#10625056

commutator, cosocle

C₂²×C₄×C₂₃

C₂³

Frattini

C₂²×C₄

C₂²×D₂₃

Fitting

C₂₃×#344

C₂

C₂×C₂₃

C₂³×C₂₃

D₂₃

C₂₃

G ▷ C₂²×C₄×C₂₃ ▷ C₂×C₂₃ ▷ C₂₃

UCS

C₁ ◁ C₄ ◁ C₂²×C₄ ◁ #344

N	H ≅ G/N	type
89. C₂₃	#2102	semidirect

H ≅ G/N

type

89. C₂₃

#2102

semidirect

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

subgroup	index
#13092465	2
#13092440	2
#13092189	2
#13092178	2
#13092176	2
#13092173	2
C₂₃×#344	2
#2102	23

subgroup

index

C₂₃×#344

Subgroups — #15674974

Conjugacy classes of subgroups

Complements

Maximal subgroups