Group #15674313 — subgroups

Subgroups — #15674313

Order 2944.

276 conjugacy classes of subgroups; only the 97 normal subgroups are shown.

normal subgroups only

This group has 276 conjugacy classes of subgroups — too many to draw. Only its 97 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×#10625001	C₂	1	1	yes	maximal, characteristic
2	#13092426	C₂	1	2	yes	maximal
3	C₂×#10625018	C₂	1	2	yes	maximal
4	#13092372	C₂	1	1	yes	maximal, characteristic
5	C₂²×Q₁₆×C₂₃	C₂	1	1	yes	maximal, characteristic
6	C₂²×Q₈×C₂₃	C₂²	1	2	yes	—
7	C₂×Q₁₆×C₂₃	—	2	1	no	—
8	C₂×Q₁₆×C₂₃	—	2	1	no	—
9	C₂×Q₁₆×C₂₃	—	2	2	no	—
10	C₂×Q₁₆×C₂₃	C₄	1	4	yes	complemented
11	C₂²×C₈×C₂₃	C₂²	1	1	yes	characteristic
12	#10625097	C₂²	1	1	yes	characteristic
13	C₂×#67267	C₂²	1	2	yes	—
14	C₂×#67267	C₂²	1	1	yes	characteristic
15	#10625018	—	2	4	no	—
16	#10625017	—	2	4	no	—
17	#10625001	—	2	2	no	—
18	#10624997	—	2	1	no	—
19	C₈×Dic₂₃	—	2	1	no	—
20	C₂×Q₈×C₂₃	—	2	2	no	—
21	C₂×Q₈×C₂₃	—	2	2	no	—
22	C₂×Q₈×C₂₃	—	2	2	no	—
23	C₂×Q₈×C₂₃	—	2	2	no	—
24	C₂×Q₈×C₂₃	—	2	2	no	—
25	C₂×Q₈×C₂₃	C₂×C₄	1	4	yes	—
26	C₂²×C₄×C₂₃	—	2	2	no	—
27	C₂²×C₄×C₂₃	C₂³	1	1	yes	characteristic
28	Q₁₆×C₂₃	—	2	4	no	—
29	Q₁₆×C₂₃	—	2	4	no	—
30	C₂×C₈×C₂₃	C₂×C₄	1	2	yes	—
31	C₂×C₈×C₂₃	D₄	1	1	yes	characteristic
32	C₂×C₈×C₂₃	D₄	1	1	yes	characteristic
33	C₂×C₈×C₂₃	D₄	1	2	yes	—
34	C₂²⋊Dic₂₃	—	2	1	no	—
35	C₄⋊Dic₂₃	D₄	1	2	yes	—
36	C₄×Dic₂₃	—	2	1	no	—
37	#67267	—	2	1	no	—
38	#67267	—	2	2	no	—
39	#67267	—	2	2	no	—
40	#67267	D₄	1	2	yes	—
41	C₂×C₂₃⋊C₈	—	2	1	no	—
42	C₂×C₂₃⋊C₈	—	2	2	no	—
43	C₂³×C₂₃	#41	1	1	yes	characteristic
44	Q₈×C₂₃	—	2	4	no	—
45	Q₈×C₂₃	—	2	4	no	—
46	Q₈×C₂₃	—	4	2	no	—
47	Q₈×C₂₃	—	4	2	no	—
48	C₂×C₄×C₂₃	—	2	4	no	—
49	C₂×C₄×C₂₃	—	4	2	no	—
50	C₂×C₄×C₂₃	—	4	2	no	—
51	C₂×C₄×C₂₃	C₂²×C₄	1	1	yes	characteristic
52	C₂×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
53	C₂×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
54	C₂×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
55	C₂×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
56	C₂×C₄×C₂₃	#41	1	1	yes	characteristic
57	C₂×Dic₂₃	—	2	2	no	—
58	C₈×C₂₃	C₂²⋊C₄	1	2	yes	—
59	C₈×C₂₃	C₂²⋊C₄	1	2	yes	—
60	C₂₃⋊C₈	—	2	2	no	—
61	C₂₃⋊C₈	—	4	2	no	—
62	#1979	—	23	1	no	maximal, Sylow
63	C₂²×C₂₃	—	2	2	no	—
64	C₂²×C₂₃	C₄×D₄	1	1	yes	characteristic
65	C₂²×C₂₃	#121	1	1	yes	characteristic
66	C₂²×C₂₃	#124	1	1	yes	characteristic
67	C₄×C₂₃	—	4	4	no	—
68	C₄×C₂₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
69	C₄×C₂₃	C₄×D₄	1	1	yes	characteristic
70	C₄×C₂₃	#121	1	1	yes	characteristic
71	C₄×C₂₃	#127	1	1	yes	characteristic
72	Dic₂₃	—	4	2	no	—
73	C₂²×Q₁₆	D₂₃	1	1	yes	characteristic
74	C₂×#108	—	23	1	no	—
75	#419	—	23	2	no	—
76	C₂×#101	—	23	2	no	—
77	#405	—	23	1	no	—
78	C₂×C₂₃	—	2	1	no	—
79	C₂×C₂₃	—	2	1	no	—
80	C₂×C₂₃	#390	1	1	yes	characteristic
81	C₂×C₂₃	#473	1	2	yes	—
82	C₂²×Q₈	C₂×D₂₃	1	2	yes	—
83	C₂×Q₁₆	—	2	1	no	—
84	C₂×Q₁₆	—	2	1	no	—
85	C₂×Q₁₆	—	2	2	no	—
86	C₂×Q₁₆	Dic₂₃	1	4	yes	complemented
87	C₂×M₁₆	—	23	1	no	—
88	C₂×M₁₆	—	23	2	no	—
89	C₂²×C₈	C₂×D₂₃	1	1	yes	characteristic
90	#117	—	23	1	no	—
91	#108	—	46	2	no	—
92	#103	—	46	4	no	—
93	#101	—	46	4	no	—
94	#97	—	46	1	no	—
95	C₄×C₈	—	46	1	no	—
96	C₂₃	#1979	1	1	yes	characteristic, complemented, Sylow
97	C₂×Q₈	—	2	2	no	—
98	C₂×Q₈	—	2	2	no	—
99	C₂×Q₈	—	2	2	no	—
100	C₂×Q₈	—	2	2	no	—
101	C₂×Q₈	—	2	2	no	—
102	C₂×Q₈	C₂×Dic₂₃	1	4	yes	—
103	C₂²×C₄	—	2	2	no	—
104	C₂²×C₄	C₂²×D₂₃	1	1	yes	characteristic
105	Q₁₆	—	2	4	no	—
106	Q₁₆	—	2	4	no	—
107	M₁₆	—	23	2	no	—
108	M₁₆	—	46	1	no	—
109	M₁₆	—	46	2	no	—
110	M₁₆	—	46	2	no	—
111	C₂×C₈	—	46	1	no	—
112	C₂×C₈	—	46	2	no	—
113	C₂×C₈	C₂×Dic₂₃	1	2	yes	—
114	C₂×C₈	C₂²⋊D₂₃	1	1	yes	characteristic
115	C₂×C₈	C₂²⋊D₂₃	1	1	yes	characteristic
116	C₂×C₈	C₂²⋊D₂₃	1	2	yes	—
117	C₄⋊C₄	—	23	2	no	—
118	C₂²⋊C₄	—	46	1	no	—
119	C₄²	—	46	1	no	—
120	C₂³	#67290	1	1	yes	characteristic
121	Q₈	—	2	4	no	—
122	Q₈	—	2	4	no	—
123	Q₈	—	4	2	no	—
124	Q₈	—	4	2	no	—
125	C₂×C₄	—	2	4	no	—
126	C₂×C₄	—	4	2	no	—
127	C₂×C₄	—	4	2	no	—
128	C₂×C₄	—	46	2	no	—
129	C₂×C₄	D₄×D₂₃	1	1	yes	characteristic
130	C₂×C₄	D₄×D₂₃	1	1	yes	characteristic
131	C₂×C₄	#67290	1	1	yes	characteristic
132	C₂×C₄	C₂²×Dic₂₃	1	1	yes	characteristic
133	C₂×C₄	C₂×C₂²⋊D₂₃	1	1	yes	characteristic
134	C₂×C₄	C₂×C₂²⋊D₂₃	1	1	yes	characteristic
135	C₈	—	46	2	no	—
136	C₈	—	92	2	no	—
137	C₈	C₂²⋊Dic₂₃	1	2	yes	—
138	C₈	C₂²⋊Dic₂₃	1	2	yes	—
139	C₂²	—	2	2	no	—
140	C₂²	D₄×Dic₂₃	1	1	yes	characteristic
141	C₂²	#10625107	1	1	yes	characteristic
142	C₂²	#10625109	1	1	yes	characteristic
143	C₄	—	4	4	no	—
144	C₄	—	92	2	no	—
145	C₄	D₄×Dic₂₃	1	1	yes	characteristic
146	C₄	#10625109	1	1	yes	characteristic
147	C₄	#10625111	1	1	yes	characteristic
148	C₄	C₂×C₂²⋊Dic₂₃	1	1	yes	characteristic
149	C₂	—	2	1	no	—
150	C₂	—	2	1	no	—
151	C₂	#13092320	1	1	yes	characteristic
152	C₂	#13092689	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#10625001

C₂

yes

maximal, characteristic

#13092426

C₂

yes

maximal

C₂×#10625018

C₂

yes

maximal

#13092372

C₂

yes

maximal, characteristic

C₂²×Q₁₆×C₂₃

C₂

yes

maximal, characteristic

C₂²×Q₈×C₂₃

C₂²

yes

—

C₂×Q₁₆×C₂₃

—

C₂×Q₁₆×C₂₃

—

C₂×Q₁₆×C₂₃

—

C₂×Q₁₆×C₂₃

C₄

yes

complemented

C₂²×C₈×C₂₃

C₂²

yes

characteristic

#10625097

C₂²

yes

characteristic

C₂×#67267

C₂²

yes

—

C₂×#67267

C₂²

yes

—

—

—

—

C₈×Dic₂₃

—

C₂×Q₈×C₂₃

—

C₂×Q₈×C₂₃

—

C₂×Q₈×C₂₃

—

C₂×Q₈×C₂₃

—

C₂×Q₈×C₂₃

—

C₂×Q₈×C₂₃

C₂×C₄

yes

—

C₂²×C₄×C₂₃

—

C₂²×C₄×C₂₃

C₂³

yes

characteristic

Q₁₆×C₂₃

—

Q₁₆×C₂₃

—

C₂×C₈×C₂₃

C₂×C₄

yes

—

C₂×C₈×C₂₃

D₄

yes

characteristic

C₂×C₈×C₂₃

D₄

yes

characteristic

C₂×C₈×C₂₃

D₄

yes

—

C₂²⋊Dic₂₃

—

C₄⋊Dic₂₃

D₄

yes

—

C₄×Dic₂₃

—

—

—

—

D₄

yes

—

C₂×C₂₃⋊C₈

—

C₂×C₂₃⋊C₈

—

C₂³×C₂₃

#41

yes

characteristic

Q₈×C₂₃

—

Q₈×C₂₃

—

Q₈×C₂₃

—

Q₈×C₂₃

—

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×D₄

yes

characteristic

C₂×C₄×C₂₃

C₂×D₄

yes

characteristic

C₂×C₄×C₂₃

#41

yes

characteristic

C₂×Dic₂₃

—

C₈×C₂₃

C₂²⋊C₄

yes

—

C₈×C₂₃

C₂²⋊C₄

yes

—

C₂₃⋊C₈

—

C₂₃⋊C₈

—

#1979

—

maximal, Sylow

C₂²×C₂₃

—

C₂²×C₂₃

C₄×D₄

yes

C₂²×C₂₃

yes

C₂²×C₂₃

yes

C₄×C₂₃

—

C₄×C₂₃

C₂×C₂²⋊C₄

yes

characteristic

C₄×C₂₃

C₄×D₄

yes

C₄×C₂₃

yes

C₄×C₂₃

yes

Dic₂₃

—

C₂²×Q₁₆

D₂₃

yes

—

—

—

—

C₂×C₂₃

—

C₂×C₂₃

—

C₂×C₂₃

#390

yes

characteristic

C₂×C₂₃

#473

yes

—

C₂²×Q₈

C₂×D₂₃

yes

—

C₂×Q₁₆

—

C₂×Q₁₆

—

C₂×Q₁₆

—

C₂×Q₁₆

Dic₂₃

yes

complemented

C₂×M₁₆

—

C₂×M₁₆

—

C₂²×C₈

C₂×D₂₃

yes

—

—

—

—

—

C₄×C₈

—

C₂₃

#1979

yes

characteristic, complemented, Sylow

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

100

C₂×Q₈

—

101

C₂×Q₈

—

102

C₂×Q₈

C₂×Dic₂₃

yes

—

103

C₂²×C₄

—

104

C₂²×C₄

C₂²×D₂₃

yes

characteristic

105

Q₁₆

—

106

Q₁₆

—

107

M₁₆

—

108

M₁₆

—

109

M₁₆

—

110

M₁₆

—

111

C₂×C₈

—

112

C₂×C₈

—

113

C₂×C₈

C₂×Dic₂₃

yes

—

114

C₂×C₈

C₂²⋊D₂₃

yes

characteristic

115

C₂×C₈

C₂²⋊D₂₃

yes

characteristic

116

C₂×C₈

C₂²⋊D₂₃

yes

—

117

C₄⋊C₄

—

118

C₂²⋊C₄

—

119

C₄²

—

120

C₂³

#67290

yes

characteristic

121

Q₈

—

122

Q₈

—

123

Q₈

—

124

Q₈

—

125

C₂×C₄

—

126

C₂×C₄

—

127

C₂×C₄

—

128

C₂×C₄

—

129

C₂×C₄

D₄×D₂₃

yes

characteristic

130

C₂×C₄

D₄×D₂₃

yes

characteristic

131

C₂×C₄

#67290

yes

characteristic

132

C₂×C₄

C₂²×Dic₂₃

yes

characteristic

133

C₂×C₄

C₂×C₂²⋊D₂₃

yes

characteristic

134

C₂×C₄

C₂×C₂²⋊D₂₃

yes

characteristic

135

C₈

—

136

C₈

—

137

C₈

C₂²⋊Dic₂₃

yes

—

138

C₈

C₂²⋊Dic₂₃

yes

—

139

C₂²

—

140

C₂²

D₄×Dic₂₃

yes

characteristic

141

C₂²

#10625107

yes

characteristic

142

C₂²

#10625109

yes

characteristic

143

C₄

—

144

C₄

—

145

C₄

D₄×Dic₂₃

yes

characteristic

146

C₄

#10625109

yes

characteristic

147

C₄

#10625111

yes

characteristic

148

C₄

C₂×C₂²⋊Dic₂₃

yes

characteristic

149

C₂

—

150

C₂

—

151

C₂

#13092320

yes

characteristic

152

C₂

#13092689

yes

—

	subgroup	G/H
center	C₂²	#10625109
commutator	C₂×C₄×C₂₃	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₂₃
Fitting	C₂²×Q₁₆×C₂₃	C₂
socle	C₂²×C₂₃	#121
prime radical	C₂³×C₂₃	#41
cosocle	C₂²×C₄×C₂₃	C₂³
hypercenter	C₂²×Q₁₆	D₂₃
hypocenter	C₂₃	#1979
LCS	G ▷ C₂×C₄×C₂₃ ▷ C₂×C₂₃ ▷ C₂₃
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂²×Q₁₆

subgroup

G/H

center

C₂²

#10625109

commutator

C₂×C₄×C₂₃

C₂²×C₄

Frattini

C₂²×C₄

C₂²×D₂₃

Fitting

C₂²×Q₁₆×C₂₃

C₂

C₂²×C₂₃

C₂³×C₂₃

C₂²×C₄×C₂₃

C₂³

hypercenter

C₂²×Q₁₆

D₂₃

hypocenter

C₂₃

#1979

LCS

G ▷ C₂×C₄×C₂₃ ▷ C₂×C₂₃ ▷ C₂₃

UCS

C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂²×Q₁₆

N	H ≅ G/N	type
10. C₂×Q₁₆×C₂₃	C₄	semidirect
86. C₂×Q₁₆	Dic₂₃	semidirect
96. C₂₃	#1979	semidirect

H ≅ G/N

type

10. C₂×Q₁₆×C₂₃

C₄

semidirect

86. C₂×Q₁₆

Dic₂₃

semidirect

96. C₂₃

#1979

semidirect

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

subgroup	index
C₂×#10625001	2
#13092426	2
C₂×#10625018	2
#13092372	2
C₂²×Q₁₆×C₂₃	2
#1979	23

subgroup

index

C₂²×Q₁₆×C₂₃

Subgroups — #15674313

Conjugacy classes of subgroups

Complements

Maximal subgroups