Group #15683910 — subgroups

Subgroups — #15683910

Order 2944.

356 conjugacy classes of subgroups; only the 141 normal subgroups are shown.

normal subgroups only

This group has 356 conjugacy classes of subgroups — too many to draw. Only its 141 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	#13092973	C₂	1	2	yes	maximal
2	#13092821	C₂	1	2	yes	maximal, complemented
3	#13092819	C₂	1	1	yes	maximal, characteristic, complemented
4	#13092816	C₂	1	2	yes	maximal, complemented
5	#13092814	C₂	1	2	yes	maximal, complemented
6	#13092454	C₂	1	2	yes	maximal
7	#13092452	C₂	1	2	yes	maximal
8	C₄×C₈×D₂₃	C₂	1	1	yes	maximal, characteristic
9	C₂₃×#499	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂₃×#125	C₂²	1	2	yes	—
11	C₂×C₈×D₂₃	C₂²	1	1	yes	characteristic
12	C₂×C₈×D₂₃	C₂²	1	1	yes	characteristic
13	#10625067	—	2	4	no	—
14	#10625063	C₂²	1	4	yes	—
15	#10625059	C₂²	1	4	yes	—
16	#10625058	—	2	4	no	—
17	C₄²×D₂₃	C₂²	1	1	yes	characteristic
18	#10625041	C₂²	1	2	yes	—
19	C₂₃×#107	C₂²	1	2	yes	—
20	C₂₃×#106	C₂²	1	2	yes	—
21	C₄×C₈×C₂₃	C₂²	1	1	yes	characteristic
22	#10625000	C₂²	1	2	yes	—
23	#10624999	C₂²	1	2	yes	—
24	C₈×Dic₂₃	C₂²	1	1	yes	characteristic
25	C₈×Dic₂₃	C₂²	1	1	yes	characteristic
26	#10624990	C₂²	1	4	yes	—
27	#10624989	C₂²	1	4	yes	—
28	C₄×C₂₃⋊C₈	C₂²	1	1	yes	characteristic
29	C₂×C₄×D₂₃	D₄	1	2	yes	—
30	C₂×C₄×D₂₃	C₂³	1	1	yes	characteristic
31	C₂×Dic₄₆	—	2	4	no	—
32	C₂×C₈×C₂₃	C₂³	1	1	yes	characteristic
33	C₂×C₈×C₂₃	C₂³	1	1	yes	characteristic
34	C₄⋊C₄×C₂₃	—	2	4	no	—
35	C₄⋊C₄×C₂₃	C₂³	1	4	yes	—
36	C₄²×C₂₃	C₂³	1	1	yes	characteristic
37	(C₂×D₂₃)⋊C₄	—	2	4	no	—
38	C₄⋊Dic₂₃	C₂³	1	4	yes	—
39	Dic₂₃⋊C₄	—	2	4	no	—
40	Dic₂₃⋊C₄	—	2	4	no	—
41	C₄×Dic₂₃	—	2	4	no	—
42	C₄×Dic₂₃	D₄	1	2	yes	—
43	C₄×Dic₂₃	C₂³	1	1	yes	characteristic
44	C₂×C₂₃⋊C₈	C₂³	1	1	yes	characteristic
45	C₂×C₂₃⋊C₈	C₂³	1	1	yes	characteristic
46	C₈×D₂₃	Q₈	1	4	yes	complemented
47	C₈×D₂₃	Q₈	1	4	yes	complemented
48	C₂²×D₂₃	C₂×D₄	1	1	yes	characteristic
49	C₂×C₄×C₂₃	—	2	4	no	—
50	C₂×C₄×C₂₃	C₂×D₄	1	2	yes	—
51	C₂×C₄×C₂₃	C₂⁴	1	1	yes	characteristic
52	C₂×Dic₂₃	—	2	4	no	—
53	C₂×Dic₂₃	C₂×D₄	1	1	yes	characteristic
54	C₂×Dic₂₃	C₂×D₄	1	2	yes	—
55	C₄×D₂₃	—	2	4	no	—
56	C₄×D₂₃	C₂×Q₈	1	2	yes	—
57	C₄×D₂₃	C₂×Q₈	1	2	yes	—
58	Dic₄₆	—	4	8	no	—
59	C₈×C₂₃	C₂×Q₈	1	2	yes	—
60	C₈×C₂₃	C₂×Q₈	1	2	yes	—
61	C₂₃⋊C₈	C₂×Q₈	1	2	yes	—
62	C₂₃⋊C₈	C₂×Q₈	1	2	yes	—
63	#3160	—	23	1	no	maximal, Sylow
64	C₂²×C₂₃	C₂²×D₄	1	1	yes	characteristic
65	C₂×D₂₃	—	2	2	no	—
66	C₂×D₂₃	C₄⋊Q₈	1	2	yes	—
67	C₄×C₂₃	—	2	2	no	—
68	C₄×C₂₃	—	4	4	no	—
69	C₄×C₂₃	C₂²×Q₈	1	1	yes	characteristic
70	C₄×C₂₃	C₂²×Q₈	1	1	yes	characteristic
71	Dic₂₃	—	4	4	no	—
72	Dic₂₃	C₄⋊Q₈	1	2	yes	—
73	Dic₂₃	#135	1	4	yes	—
74	#533	—	23	2	no	—
75	#500	—	23	2	no	—
76	#499	—	23	2	no	—
77	#499	—	23	1	no	—
78	#499	D₂₃	1	1	yes	characteristic, complemented
79	#498	—	23	2	no	—
80	#427	—	23	2	no	—
81	#427	—	23	2	no	—
82	C₂×C₄×C₈	—	23	1	no	—
83	C₂×C₂₃	C₂×C₄⋊Q₈	1	1	yes	characteristic
84	C₂×C₂₃	C₂×#135	1	2	yes	—
85	D₂₃	—	2	2	no	—
86	C₂²×C₈	—	23	1	no	—
87	C₂²×C₈	—	23	1	no	—
88	C₄⋊Q₈	—	23	4	no	—
89	#125	—	23	2	no	—
90	#125	C₂×D₂₃	1	2	yes	—
91	C₂²⋊Q₈	—	46	4	no	—
92	C₄×Q₈	—	46	4	no	—
93	#117	—	23	4	no	—
94	C₂×C₄²	—	23	1	no	—
95	#107	—	23	2	no	—
96	#107	—	23	4	no	—
97	#107	C₂×D₂₃	1	2	yes	—
98	#106	—	23	2	no	—
99	#106	—	23	4	no	—
100	#106	C₂×D₂₃	1	2	yes	—
101	C₄×C₈	—	23	1	no	—
102	C₄×C₈	—	23	1	no	—
103	C₄×C₈	—	23	1	no	—
104	C₄×C₈	C₂×D₂₃	1	1	yes	characteristic
105	C₂₃	#3160	1	1	yes	characteristic, complemented, Sylow
106	C₂×Q₈	—	46	4	no	—
107	C₂²×C₄	—	23	1	no	—
108	C₂²×C₄	—	23	2	no	—
109	C₂×C₈	—	23	1	no	—
110	C₂×C₈	—	23	1	no	—
111	C₂×C₈	—	23	4	no	—
112	C₂×C₈	—	23	4	no	—
113	C₂×C₈	C₂²×D₂₃	1	1	yes	characteristic
114	C₂×C₈	C₂²×D₂₃	1	1	yes	characteristic
115	C₄⋊C₄	—	2	4	no	—
116	C₄⋊C₄	—	23	4	no	—
117	C₄⋊C₄	—	46	4	no	—
118	C₄⋊C₄	—	46	4	no	—
119	C₄⋊C₄	C₂²×D₂₃	1	4	yes	—
120	C₂²⋊C₄	—	46	4	no	—
121	C₄²	—	23	2	no	—
122	C₄²	—	23	1	no	—
123	C₄²	—	46	4	no	—
124	C₄²	C₂²×D₂₃	1	1	yes	characteristic
125	C₂³	—	23	1	no	—
126	Q₈	—	92	8	no	—
127	C₂×C₄	—	2	4	no	—
128	C₂×C₄	—	23	1	no	—
129	C₂×C₄	—	23	2	no	—
130	C₂×C₄	—	23	2	no	—
131	C₂×C₄	—	23	2	no	—
132	C₂×C₄	—	46	4	no	—
133	C₂×C₄	—	46	4	no	—
134	C₂×C₄	D₄×D₂₃	1	2	yes	—
135	C₂×C₄	C₂³×D₂₃	1	1	yes	characteristic
136	C₈	—	23	2	no	—
137	C₈	—	23	2	no	—
138	C₈	Q₈×D₂₃	1	2	yes	—
139	C₈	Q₈×D₂₃	1	2	yes	—
140	C₂²	—	23	2	no	—
141	C₂²	—	46	2	no	—
142	C₂²	C₂×D₄×D₂₃	1	1	yes	characteristic
143	C₄	—	2	2	no	—
144	C₄	—	4	4	no	—
145	C₄	—	23	4	no	—
146	C₄	—	23	2	no	—
147	C₄	—	92	4	no	—
148	C₄	C₂×Q₈×D₂₃	1	1	yes	characteristic
149	C₄	C₂×Q₈×D₂₃	1	1	yes	characteristic
150	C₂	—	46	2	no	—
151	C₂	C₄⋊Q₈×D₂₃	1	1	yes	characteristic
152	C₂	#135×D₂₃	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#13092973

C₂

yes

maximal

#13092821

C₂

yes

maximal, complemented

#13092819

C₂

yes

maximal, characteristic, complemented

#13092816

C₂

yes

maximal, complemented

#13092814

C₂

yes

maximal, complemented

#13092454

C₂

yes

maximal

#13092452

C₂

yes

maximal

C₄×C₈×D₂₃

C₂

yes

maximal, characteristic

C₂₃×#499

C₂

yes

maximal, characteristic, complemented

C₂₃×#125

C₂²

yes

—

C₂×C₈×D₂₃

C₂²

yes

characteristic

C₂×C₈×D₂₃

C₂²

yes

characteristic

#10625067

—

#10625063

C₂²

yes

—

#10625059

C₂²

yes

—

#10625058

—

C₄²×D₂₃

C₂²

yes

characteristic

#10625041

C₂²

yes

—

C₂₃×#107

C₂²

yes

—

C₂₃×#106

C₂²

yes

—

C₄×C₈×C₂₃

C₂²

yes

characteristic

#10625000

C₂²

yes

—

#10624999

C₂²

yes

—

C₈×Dic₂₃

C₂²

yes

characteristic

C₈×Dic₂₃

C₂²

yes

characteristic

#10624990

C₂²

yes

—

#10624989

C₂²

yes

—

C₄×C₂₃⋊C₈

C₂²

yes

characteristic

C₂×C₄×D₂₃

D₄

yes

—

C₂×C₄×D₂₃

C₂³

yes

characteristic

C₂×Dic₄₆

—

C₂×C₈×C₂₃

C₂³

yes

characteristic

C₂×C₈×C₂₃

C₂³

yes

characteristic

C₄⋊C₄×C₂₃

—

C₄⋊C₄×C₂₃

C₂³

yes

—

C₄²×C₂₃

C₂³

yes

characteristic

(C₂×D₂₃)⋊C₄

—

C₄⋊Dic₂₃

C₂³

yes

—

Dic₂₃⋊C₄

—

Dic₂₃⋊C₄

—

C₄×Dic₂₃

—

C₄×Dic₂₃

D₄

yes

—

C₄×Dic₂₃

C₂³

yes

characteristic

C₂×C₂₃⋊C₈

C₂³

yes

characteristic

C₂×C₂₃⋊C₈

C₂³

yes

characteristic

C₈×D₂₃

Q₈

yes

complemented

C₈×D₂₃

Q₈

yes

complemented

C₂²×D₂₃

C₂×D₄

yes

characteristic

C₂×C₄×C₂₃

—

C₂×C₄×C₂₃

C₂×D₄

yes

—

C₂×C₄×C₂₃

C₂⁴

yes

characteristic

C₂×Dic₂₃

—

C₂×Dic₂₃

C₂×D₄

yes

characteristic

C₂×Dic₂₃

C₂×D₄

yes

—

C₄×D₂₃

—

C₄×D₂₃

C₂×Q₈

yes

—

C₄×D₂₃

C₂×Q₈

yes

—

Dic₄₆

—

C₈×C₂₃

C₂×Q₈

yes

—

C₈×C₂₃

C₂×Q₈

yes

—

C₂₃⋊C₈

C₂×Q₈

yes

—

C₂₃⋊C₈

C₂×Q₈

yes

—

#3160

—

maximal, Sylow

C₂²×C₂₃

C₂²×D₄

yes

characteristic

C₂×D₂₃

—

C₂×D₂₃

C₄⋊Q₈

yes

—

C₄×C₂₃

—

C₄×C₂₃

—

C₄×C₂₃

C₂²×Q₈

yes

characteristic

C₄×C₂₃

C₂²×Q₈

yes

characteristic

Dic₂₃

—

Dic₂₃

C₄⋊Q₈

yes

—

Dic₂₃

#135

yes

—

—

—

—

—

D₂₃

yes

characteristic, complemented

#498

—

#427

—

#427

—

C₂×C₄×C₈

—

C₂×C₂₃

C₂×C₄⋊Q₈

yes

characteristic

C₂×C₂₃

C₂×#135

yes

—

D₂₃

—

C₂²×C₈

—

C₂²×C₈

—

C₄⋊Q₈

—

#125

—

#125

C₂×D₂₃

yes

—

C₂²⋊Q₈

—

C₄×Q₈

—

#117

—

C₂×C₄²

—

#107

—

#107

—

#107

C₂×D₂₃

yes

—

#106

—

#106

—

100

#106

C₂×D₂₃

yes

—

101

C₄×C₈

—

102

C₄×C₈

—

103

C₄×C₈

—

104

C₄×C₈

C₂×D₂₃

yes

characteristic

105

C₂₃

#3160

yes

characteristic, complemented, Sylow

106

C₂×Q₈

—

107

C₂²×C₄

—

108

C₂²×C₄

—

109

C₂×C₈

—

110

C₂×C₈

—

111

C₂×C₈

—

112

C₂×C₈

—

113

C₂×C₈

C₂²×D₂₃

yes

characteristic

114

C₂×C₈

C₂²×D₂₃

yes

characteristic

115

C₄⋊C₄

—

116

C₄⋊C₄

—

117

C₄⋊C₄

—

118

C₄⋊C₄

—

119

C₄⋊C₄

C₂²×D₂₃

yes

—

120

C₂²⋊C₄

—

121

C₄²

—

122

C₄²

—

123

C₄²

—

124

C₄²

C₂²×D₂₃

yes

characteristic

125

C₂³

—

126

Q₈

—

127

C₂×C₄

—

128

C₂×C₄

—

129

C₂×C₄

—

130

C₂×C₄

—

131

C₂×C₄

—

132

C₂×C₄

—

133

C₂×C₄

—

134

C₂×C₄

D₄×D₂₃

yes

—

135

C₂×C₄

C₂³×D₂₃

yes

characteristic

136

C₈

—

137

C₈

—

138

C₈

Q₈×D₂₃

yes

—

139

C₈

Q₈×D₂₃

yes

—

140

C₂²

—

141

C₂²

—

142

C₂²

C₂×D₄×D₂₃

yes

characteristic

143

C₄

—

144

C₄

—

145

C₄

—

146

C₄

—

147

C₄

—

148

C₄

C₂×Q₈×D₂₃

yes

characteristic

149

C₄

C₂×Q₈×D₂₃

yes

characteristic

150

C₂

—

151

C₂

C₄⋊Q₈×D₂₃

yes

characteristic

152

C₂

#135×D₂₃

yes

—

	subgroup	G/H
center	C₂²	C₂×D₄×D₂₃
commutator, cosocle	C₂×C₄×C₂₃	C₂⁴
Frattini	C₂×C₄	C₂³×D₂₃
Fitting	C₂₃×#499	C₂
socle	C₂²×C₂₃	C₂²×D₄
prime radical	C₂²×D₂₃	C₂×D₄
hypercenter	#499	D₂₃
hypocenter	C₂₃	#3160
LCS	G ▷ C₂×C₄×C₂₃ ▷ C₂×C₂₃ ▷ C₂₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #499

subgroup

G/H

center

C₂²

C₂×D₄×D₂₃

commutator, cosocle

C₂×C₄×C₂₃

C₂⁴

Frattini

C₂×C₄

C₂³×D₂₃

Fitting

C₂₃×#499

C₂

socle

C₂²×C₂₃

C₂²×D₄

prime radical

C₂²×D₂₃

C₂×D₄

D₂₃

C₂₃

G ▷ C₂×C₄×C₂₃ ▷ C₂×C₂₃ ▷ C₂₃

UCS

C₁ ◁ C₂² ◁ C₄² ◁ #499

N	H ≅ G/N	type
2. #13092821	C₂	semidirect
3. #13092819	C₂	semidirect
4. #13092816	C₂	semidirect
5. #13092814	C₂	semidirect
9. C₂₃×#499	C₂	semidirect
46. C₈×D₂₃	Q₈	semidirect
47. C₈×D₂₃	Q₈	semidirect
78. #499	D₂₃	semidirect
105. C₂₃	#3160	semidirect

H ≅ G/N

type

2. #13092821

C₂

semidirect

3. #13092819

C₂

semidirect

4. #13092816

C₂

semidirect

5. #13092814

C₂

semidirect

9. C₂₃×#499

C₂

semidirect

46. C₈×D₂₃

Q₈

semidirect

47. C₈×D₂₃

Q₈

semidirect

78. #499

D₂₃

semidirect

105. C₂₃

#3160

semidirect

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

subgroup	index
#13092973	2
#13092821	2
#13092819	2
#13092816	2
#13092814	2
#13092454	2
#13092452	2
C₄×C₈×D₂₃	2
C₂₃×#499	2
#3160	23

subgroup

index

C₄×C₈×D₂₃

C₂₃×#499

#3160

Subgroups — #15683910

Conjugacy classes of subgroups

Complements

Maximal subgroups