Group #15673442 — subgroups

Subgroups — #15673442

Order 2944.

260 conjugacy classes of subgroups; only the 81 normal subgroups are shown.

normal subgroups only

This group has 260 conjugacy classes of subgroups — too many to draw. Only its 81 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×#10625078	C₂	1	1	yes	maximal, characteristic
2	#13092453	C₂	1	2	yes	maximal, complemented
3	#13092390	C₂	1	2	yes	maximal
4	#13092174	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂₃×H(3,ℤ/4)	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×#67288	C₂²	1	1	yes	characteristic
7	C₂×M₁₆×C₂₃	C₂²	1	1	yes	characteristic
8	C₂₃×#117	C₂²	1	2	yes	—
9	#10625097	C₂²	1	2	yes	—
10	C₂²×C₂₃⋊C₈	C₂²	1	1	yes	characteristic
11	#10625078	—	2	2	no	—
12	#10625078	—	2	2	no	—
13	C₂×#67262	C₄	1	2	yes	complemented
14	C₂×C₈×D₂₃	C₄	1	2	yes	complemented
15	#10624990	—	2	2	no	—
16	#10624989	—	2	2	no	—
17	#10624988	—	2	4	no	—
18	C₂²×C₄×C₂₃	C₂³	1	1	yes	characteristic
19	C₂×C₂²⋊D₂₃	D₄	1	2	yes	—
20	#67288	—	2	2	no	—
21	#67288	—	2	2	no	—
22	C₂×D₉₂	Q₈	1	1	yes	characteristic
23	C₂×C₄×D₂₃	C₂×C₄	1	2	yes	—
24	C₂×Dic₄₆	Q₈	1	1	yes	characteristic
25	M₁₆×C₂₃	—	2	2	no	—
26	C₂×C₈×C₂₃	C₂×C₄	1	2	yes	—
27	C₄⋊C₄×C₂₃	—	2	2	no	—
28	C₂²⋊C₄×C₂₃	—	2	2	no	—
29	C₄²×C₂₃	—	2	2	no	—
30	C₂²⋊Dic₂₃	—	2	2	no	—
31	C₄⋊Dic₂₃	—	2	2	no	—
32	C₄×Dic₂₃	—	2	2	no	—
33	C₂×C₂₃⋊C₈	C₂×C₄	1	1	yes	characteristic
34	C₂×C₂₃⋊C₈	C₂×C₄	1	1	yes	characteristic
35	C₂×C₂₃⋊C₈	D₄	1	2	yes	—
36	C₂×C₂₃⋊C₈	D₄	1	2	yes	—
37	#67262	—	2	4	no	—
38	C₈×D₂₃	—	2	2	no	—
39	C₈×D₂₃	—	2	2	no	—
40	C₂³×C₂₃	C₂×D₄	1	1	yes	characteristic
41	C₂²×D₂₃	C₄⋊C₄	1	2	yes	—
42	C₂×C₄×C₂₃	—	4	2	no	—
43	C₂×C₄×C₂₃	C₂²×C₄	1	1	yes	characteristic
44	C₂×C₄×C₂₃	C₂×D₄	1	2	yes	—
45	C₂×C₄×C₂₃	C₂×Q₈	1	1	yes	characteristic
46	C₂×C₄×C₂₃	#41	1	2	yes	—
47	C₂²⋊D₂₃	—	2	4	no	—
48	C₂×Dic₂₃	—	4	2	no	—
49	C₂×Dic₂₃	C₄⋊C₄	1	2	yes	—
50	D₉₂	—	2	2	no	—
51	C₄×D₂₃	—	2	2	no	—
52	C₄×D₂₃	—	2	2	no	—
53	Dic₄₆	—	2	2	no	—
54	C₈×C₂₃	—	2	2	no	—
55	C₂₃⋊C₈	C₂²⋊C₄	1	1	yes	characteristic
56	C₂₃⋊C₈	C₂²⋊C₄	1	1	yes	characteristic
57	C₂₃⋊C₈	C₂²⋊C₄	1	2	yes	—
58	#1812	—	23	1	no	maximal, Sylow
59	C₂²×C₂₃	—	4	1	no	—
60	C₂²×C₂₃	C₂×C₄⋊C₄	1	1	yes	characteristic
61	C₂²×C₂₃	#121	1	2	yes	—
62	C₂×D₂₃	—	2	2	no	—
63	C₂×D₂₃	—	4	2	no	—
64	C₄×C₂₃	—	8	2	no	—
65	C₄×C₂₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
66	C₄×C₂₃	#117	1	1	yes	characteristic
67	C₄×C₂₃	C₂²⋊Q₈	1	2	yes	—
68	Dic₂₃	—	2	2	no	—
69	Dic₂₃	—	8	2	no	—
70	C₂×#131	—	23	1	no	—
71	#427	—	23	2	no	—
72	#410	—	23	2	no	—
73	H(3,ℤ/4)	—	23	1	no	—
74	H(3,ℤ/4)	D₂₃	1	1	yes	characteristic, complemented
75	C₂×C₂₃	—	2	2	no	—
76	C₂×C₂₃	—	2	1	no	—
77	C₂×C₂₃	#380	1	1	yes	characteristic
78	D₂₃	—	4	2	no	—
79	C₂×#41	—	23	1	no	—
80	#131	—	46	2	no	—
81	#131	—	46	2	no	—
82	C₂×M₁₆	—	23	2	no	—
83	C₂×M₁₆	C₂×D₂₃	1	1	yes	characteristic
84	C₂²×C₈	—	23	1	no	—
85	C₂²×C₈	—	23	2	no	—
86	#117	—	23	2	no	—
87	#117	C₂×D₂₃	1	2	yes	—
88	#107	—	46	2	no	—
89	#106	—	46	2	no	—
90	#99	—	46	4	no	—
91	C₂₃	#1812	1	1	yes	characteristic, complemented, Sylow
92	#41	—	46	2	no	—
93	#41	—	46	2	no	—
94	C₂×Q₈	—	23	1	no	—
95	C₂×D₄	—	23	2	no	—
96	C₂×D₄	—	23	1	no	—
97	C₂²×C₄	—	23	2	no	—
98	C₂²×C₄	C₂²×D₂₃	1	1	yes	characteristic
99	M₁₆	—	2	2	no	—
100	M₁₆	—	46	4	no	—
101	C₂×C₈	—	23	1	no	—
102	C₂×C₈	—	23	1	no	—
103	C₂×C₈	—	23	2	no	—
104	C₂×C₈	—	23	2	no	—
105	C₂×C₈	—	46	2	no	—
106	C₂×C₈	—	46	2	no	—
107	C₂×C₈	C₄×D₂₃	1	2	yes	—
108	C₄⋊C₄	—	2	2	no	—
109	C₄⋊C₄	—	46	2	no	—
110	C₂²⋊C₄	—	2	2	no	—
111	C₂²⋊C₄	—	46	2	no	—
112	C₄²	—	2	2	no	—
113	C₄²	—	46	2	no	—
114	C₂³	—	23	2	no	—
115	C₂³	D₄×D₂₃	1	1	yes	characteristic
116	Q₈	—	46	2	no	—
117	D₄	—	46	4	no	—
118	D₄	—	46	2	no	—
119	C₂×C₄	—	4	2	no	—
120	C₂×C₄	—	23	2	no	—
121	C₂×C₄	—	46	2	no	—
122	C₂×C₄	—	46	2	no	—
123	C₂×C₄	—	92	2	no	—
124	C₂×C₄	C₂×C₄×D₂₃	1	1	yes	characteristic
125	C₂×C₄	#67288	1	2	yes	—
126	C₂×C₄	D₄×D₂₃	1	2	yes	—
127	C₂×C₄	Q₈×D₂₃	1	1	yes	characteristic
128	C₈	—	2	2	no	—
129	C₈	—	23	1	no	—
130	C₈	—	23	1	no	—
131	C₈	—	23	2	no	—
132	C₂²	—	4	1	no	—
133	C₂²	—	46	2	no	—
134	C₂²	—	92	2	no	—
135	C₂²	#10625055	1	2	yes	—
136	C₂²	C₄⋊C₄×D₂₃	1	1	yes	characteristic
137	C₄	—	8	2	no	—
138	C₄	—	46	2	no	—
139	C₄	—	184	2	no	—
140	C₄	#10625043	1	1	yes	characteristic
141	C₄	C₂²⋊C₄×D₂₃	1	1	yes	characteristic
142	C₄	#10625067	1	2	yes	—
143	C₂	—	2	2	no	—
144	C₂	—	2	1	no	—
145	C₂	—	92	2	no	—
146	C₂	#13092276	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#10625078

C₂

yes

maximal, characteristic

#13092453

C₂

yes

maximal, complemented

#13092390

C₂

yes

maximal

#13092174

C₂

yes

maximal, characteristic, complemented

C₂₃×H(3,ℤ/4)

C₂

yes

maximal, characteristic, complemented

C₂×#67288

C₂²

yes

characteristic

C₂×M₁₆×C₂₃

C₂²

yes

characteristic

C₂₃×#117

C₂²

yes

—

#10625097

C₂²

yes

—

C₂²×C₂₃⋊C₈

C₂²

yes

—

—

C₄

yes

complemented

C₂×C₈×D₂₃

C₄

yes

—

—

—

C₂²×C₄×C₂₃

C₂³

yes

characteristic

C₂×C₂²⋊D₂₃

D₄

yes

—

#67288

—

#67288

—

C₂×D₉₂

Q₈

yes

characteristic

C₂×C₄×D₂₃

C₂×C₄

yes

—

C₂×Dic₄₆

Q₈

yes

characteristic

M₁₆×C₂₃

—

C₂×C₈×C₂₃

C₂×C₄

yes

—

C₄⋊C₄×C₂₃

—

C₂²⋊C₄×C₂₃

—

C₄²×C₂₃

—

C₂²⋊Dic₂₃

—

C₄⋊Dic₂₃

—

C₄×Dic₂₃

—

C₂×C₂₃⋊C₈

C₂×C₄

yes

characteristic

C₂×C₂₃⋊C₈

C₂×C₄

yes

characteristic

C₂×C₂₃⋊C₈

D₄

yes

—

C₂×C₂₃⋊C₈

D₄

yes

—

#67262

—

C₈×D₂₃

—

C₈×D₂₃

—

C₂³×C₂₃

C₂×D₄

yes

characteristic

C₂²×D₂₃

C₄⋊C₄

yes

—

C₂×C₄×C₂₃

—

C₂×C₄×C₂₃

C₂²×C₄

yes

characteristic

C₂×C₄×C₂₃

C₂×D₄

yes

—

C₂×C₄×C₂₃

C₂×Q₈

yes

characteristic

C₂×C₄×C₂₃

#41

yes

—

C₂²⋊D₂₃

—

C₂×Dic₂₃

—

C₂×Dic₂₃

C₄⋊C₄

yes

—

D₉₂

—

C₄×D₂₃

—

C₄×D₂₃

—

Dic₄₆

—

C₈×C₂₃

—

C₂₃⋊C₈

C₂²⋊C₄

yes

characteristic

C₂₃⋊C₈

C₂²⋊C₄

yes

characteristic

C₂₃⋊C₈

C₂²⋊C₄

yes

—

#1812

—

maximal, Sylow

C₂²×C₂₃

—

C₂²×C₂₃

C₂×C₄⋊C₄

yes

characteristic

C₂²×C₂₃

#121

yes

—

C₂×D₂₃

—

C₂×D₂₃

—

C₄×C₂₃

—

C₄×C₂₃

C₂×C₂²⋊C₄

yes

characteristic

C₄×C₂₃

#117

yes

characteristic

C₄×C₂₃

C₂²⋊Q₈

yes

—

Dic₂₃

—

Dic₂₃

—

—

—

—

—

D₂₃

yes

characteristic, complemented

C₂×C₂₃

—

C₂×C₂₃

—

C₂×C₂₃

#380

yes

characteristic

D₂₃

—

C₂×#41

—

#131

—

#131

—

C₂×M₁₆

—

C₂×M₁₆

C₂×D₂₃

yes

characteristic

C₂²×C₈

—

C₂²×C₈

—

#117

—

#117

C₂×D₂₃

yes

—

#107

—

#106

—

#99

—

C₂₃

#1812

yes

characteristic, complemented, Sylow

#41

—

#41

—

C₂×Q₈

—

C₂×D₄

—

C₂×D₄

—

C₂²×C₄

—

C₂²×C₄

C₂²×D₂₃

yes

characteristic

M₁₆

—

100

M₁₆

—

101

C₂×C₈

—

102

C₂×C₈

—

103

C₂×C₈

—

104

C₂×C₈

—

105

C₂×C₈

—

106

C₂×C₈

—

107

C₂×C₈

C₄×D₂₃

yes

—

108

C₄⋊C₄

—

109

C₄⋊C₄

—

110

C₂²⋊C₄

—

111

C₂²⋊C₄

—

112

C₄²

—

113

C₄²

—

114

C₂³

—

115

C₂³

D₄×D₂₃

yes

characteristic

116

Q₈

—

117

D₄

—

118

D₄

—

119

C₂×C₄

—

120

C₂×C₄

—

121

C₂×C₄

—

122

C₂×C₄

—

123

C₂×C₄

—

124

C₂×C₄

C₂×C₄×D₂₃

yes

characteristic

125

C₂×C₄

#67288

yes

—

126

C₂×C₄

D₄×D₂₃

yes

—

127

C₂×C₄

Q₈×D₂₃

yes

characteristic

128

C₈

—

129

C₈

—

130

C₈

—

131

C₈

—

132

C₂²

—

133

C₂²

—

134

C₂²

—

135

C₂²

#10625055

yes

—

136

C₂²

C₄⋊C₄×D₂₃

yes

characteristic

137

C₄

—

138

C₄

—

139

C₄

—

184

—

140

C₄

#10625043

yes

characteristic

141

C₄

C₂²⋊C₄×D₂₃

yes

characteristic

142

C₄

#10625067

yes

—

143

C₂

—

144

C₂

—

145

C₂

—

146

C₂

#13092276

yes

characteristic

	subgroup	G/H
center	C₄	#10625043
commutator	C₂×C₄×C₂₃	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₂₃
Fitting	C₂₃×H(3,ℤ/4)	C₂
socle	C₂×C₂₃	#380
prime radical	C₂×#67288	C₂²
cosocle	C₂²×C₄×C₂₃	C₂³
hypercenter	H(3,ℤ/4)	D₂₃
hypocenter	C₂₃	#1812
LCS	G ▷ C₂×C₄×C₂₃ ▷ C₂×C₂₃ ▷ C₂₃
UCS	C₁ ◁ C₄ ◁ C₂×M₁₆ ◁ H(3,ℤ/4)

subgroup

G/H

center

C₄

#10625043

commutator

C₂×C₄×C₂₃

C₂²×C₄

Frattini

C₂²×C₄

C₂²×D₂₃

Fitting

C₂₃×H(3,ℤ/4)

C₂

C₂×C₂₃

C₂²

C₂²×C₄×C₂₃

C₂³

D₂₃

C₂₃

G ▷ C₂×C₄×C₂₃ ▷ C₂×C₂₃ ▷ C₂₃

UCS

C₁ ◁ C₄ ◁ C₂×M₁₆ ◁ H(3,ℤ/4)

N	H ≅ G/N	type
2. #13092453	C₂	semidirect
4. #13092174	C₂	semidirect
5. C₂₃×H(3,ℤ/4)	C₂	semidirect
13. C₂×#67262	C₄	semidirect
14. C₂×C₈×D₂₃	C₄	semidirect
74. H(3,ℤ/4)	D₂₃	semidirect
91. C₂₃	#1812	semidirect

H ≅ G/N

type

2. #13092453

C₂

semidirect

4. #13092174

C₂

semidirect

5. C₂₃×H(3,ℤ/4)

C₂

semidirect

13. C₂×#67262

C₄

semidirect

14. C₂×C₈×D₂₃

C₄

semidirect

74. H(3,ℤ/4)

D₂₃

semidirect

91. C₂₃

#1812

semidirect

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

subgroup	index
C₂×#10625078	2
#13092453	2
#13092390	2
#13092174	2
C₂₃×H(3,ℤ/4)	2
#1812	23

subgroup

index

C₂₃×H(3,ℤ/4)

Subgroups — #15673442

Conjugacy classes of subgroups

Complements

Maximal subgroups