Group #15672449 — subgroups

Subgroups — #15672449

Order 2944.

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

normal subgroups only

This group has 336 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	#13093052	C₂	1	1	yes	maximal, characteristic, complemented
2	#13092609	C₂	1	2	yes	maximal, complemented
3	#13092161	C₂	1	1	yes	maximal, characteristic, complemented
4	#13092146	C₂	1	2	yes	maximal, complemented
5	C₂₃×#493	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×D₄×D₂₃	—	2	2	no	—
7	C₂×D₈×C₂₃	—	2	2	no	—
8	C₂₃×#127	C₂²	1	1	yes	characteristic
9	C₂₃×#127	C₂²	1	1	yes	characteristic, complemented
10	#10625109	—	2	2	no	—
11	#10625108	—	2	2	no	—
12	C₂×#67272	—	2	2	no	—
13	C₄×D₉₂	C₂²	1	2	yes	—
14	C₄×C₈×C₂₃	C₂²	1	1	yes	characteristic, complemented
15	#10625014	—	2	2	no	—
16	C₄⋊(C₂₃⋊C₈)	C₂²	1	2	yes	complemented
17	C₂³×D₂₃	—	2	2	no	—
18	C₂×D₄×C₂₃	—	2	2	no	—
19	C₂×D₄×C₂₃	—	2	2	no	—
20	C₂×D₄×C₂₃	—	2	1	no	—
21	C₂×D₄×C₂₃	D₄	1	2	yes	—
22	C₂×C₂²⋊D₂₃	—	2	4	no	—
23	D₄×D₂₃	—	4	4	no	—
24	D₄×D₂₃	—	4	4	no	—
25	C₂×D₉₂	D₄	1	2	yes	—
26	C₂×C₄×D₂₃	—	2	2	no	—
27	D₈×C₂₃	—	4	4	no	—
28	C₂×C₈×C₂₃	—	2	1	no	—
29	C₄²×C₂₃	C₂³	1	1	yes	characteristic
30	C₂²⋊Dic₂₃	—	2	2	no	—
31	#67272	—	4	4	no	—
32	(C₂×D₂₃)⋊C₄	—	2	2	no	—
33	C₄⋊Dic₂₃	D₄	1	2	yes	—
34	C₂×C₂₃⋊C₈	—	2	2	no	—
35	C₂³×C₂₃	—	2	2	no	—
36	C₂³×C₂₃	—	4	1	no	—
37	C₂²×D₂₃	—	2	2	no	—
38	C₂²×D₂₃	—	4	8	no	—
39	C₂²×D₂₃	—	8	2	no	—
40	D₄×C₂₃	—	4	2	no	—
41	D₄×C₂₃	—	4	2	no	—
42	D₄×C₂₃	—	4	2	no	—
43	D₄×C₂₃	—	4	2	no	—
44	D₄×C₂₃	—	4	2	no	—
45	D₄×C₂₃	—	4	2	no	—
46	C₂×C₄×C₂₃	C₂×D₄	1	2	yes	—
47	C₂×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
48	C₂²⋊D₂₃	—	4	8	no	—
49	C₂×Dic₂₃	—	2	2	no	—
50	D₉₂	—	2	2	no	—
51	D₉₂	D₈	1	4	yes	complemented
52	C₄×D₂₃	—	4	2	no	—
53	C₄×D₂₃	—	4	2	no	—
54	C₈×C₂₃	—	2	2	no	—
55	C₂₃⋊C₈	—	4	2	no	—
56	#1619	—	23	1	no	maximal, Sylow
57	C₂²×C₂₃	—	4	4	no	—
58	C₂²×C₂₃	—	4	2	no	—
59	C₂²×C₂₃	—	8	2	no	—
60	C₂²×C₂₃	—	8	1	no	—
61	C₂²×C₂₃	#120	1	1	yes	characteristic
62	C₂×D₂₃	—	2	4	no	—
63	C₂×D₂₃	—	4	2	no	—
64	C₂×D₂₃	—	4	2	no	—
65	C₂×D₂₃	—	8	8	no	—
66	C₄×C₂₃	—	2	1	no	—
67	C₄×C₂₃	C₂×D₈	1	2	yes	—
68	C₄×C₂₃	AGL(1,ℤ/8)	1	2	yes	—
69	Dic₂₃	—	4	2	no	—
70	D₄²	—	23	1	no	—
71	#493	D₂₃	1	1	yes	characteristic, complemented
72	#459	—	23	2	no	—
73	#331	—	23	1	no	—
74	D₄⋊C₈	—	23	2	no	—
75	C₂×C₂₃	—	8	2	no	—
76	C₂×C₂₃	—	16	1	no	—
77	C₂×C₂₃	#447	1	2	yes	—
78	C₂×C₂₃	#453	1	1	yes	characteristic
79	D₂₃	—	4	4	no	—
80	C₂²×D₄	—	46	2	no	—
81	C₂×D₈	—	2	2	no	—
82	C₂×D₈	—	46	2	no	—
83	#127	C₂×D₂₃	1	1	yes	characteristic
84	#127	C₂×D₂₃	1	1	yes	characteristic, complemented
85	#121	—	46	2	no	—
86	#120	—	46	2	no	—
87	C₄×D₄	—	23	2	no	—
88	C₄⋊C₈	—	23	2	no	—
89	#102	—	46	2	no	—
90	C₄×C₈	C₂×D₂₃	1	1	yes	characteristic, complemented
91	C₂₃	#1619	1	1	yes	characteristic, complemented, Sylow
92	C₂⁴	—	46	2	no	—
93	C₂×D₄	—	2	2	no	—
94	C₂×D₄	—	2	2	no	—
95	C₂×D₄	—	2	1	no	—
96	C₂×D₄	—	23	2	no	—
97	C₂×D₄	—	46	4	no	—
98	C₂×D₄	—	92	4	no	—
99	C₂×D₄	—	92	4	no	—
100	C₂×D₄	C₂²⋊D₂₃	1	2	yes	—
101	C₂²×C₄	—	46	2	no	—
102	D₈	—	4	4	no	—
103	D₈	—	92	4	no	—
104	C₂×C₈	—	2	1	no	—
105	C₂×C₈	—	46	2	no	—
106	C₄⋊C₄	—	23	2	no	—
107	C₂²⋊C₄	—	46	2	no	—
108	C₂²⋊C₄	—	46	2	no	—
109	C₄²	C₂²×D₂₃	1	1	yes	characteristic
110	C₂³	—	2	2	no	—
111	C₂³	—	4	1	no	—
112	C₂³	—	46	2	no	—
113	C₂³	—	92	8	no	—
114	C₂³	—	184	2	no	—
115	D₄	—	4	2	no	—
116	D₄	—	4	2	no	—
117	D₄	—	4	2	no	—
118	D₄	—	4	2	no	—
119	D₄	—	4	2	no	—
120	D₄	—	4	2	no	—
121	D₄	—	23	4	no	—
122	D₄	—	46	2	no	—
123	D₄	—	92	8	no	—
124	C₂×C₄	—	46	2	no	—
125	C₂×C₄	—	92	2	no	—
126	C₂×C₄	—	92	2	no	—
127	C₂×C₄	D₄×D₂₃	1	2	yes	—
128	C₂×C₄	C₂×C₂²⋊D₂₃	1	1	yes	characteristic
129	C₈	—	2	2	no	—
130	C₈	—	92	2	no	—
131	C₂²	—	4	4	no	—
132	C₂²	—	4	2	no	—
133	C₂²	—	8	2	no	—
134	C₂²	—	8	1	no	—
135	C₂²	—	46	4	no	—
136	C₂²	—	92	2	no	—
137	C₂²	—	92	2	no	—
138	C₂²	—	184	8	no	—
139	C₂²	#10625108	1	1	yes	characteristic
140	C₄	—	2	1	no	—
141	C₄	—	92	2	no	—
142	C₄	D₈×D₂₃	1	2	yes	—
143	C₄	#10625082	1	2	yes	—
144	C₂	—	8	2	no	—
145	C₂	—	16	1	no	—
146	C₂	—	92	4	no	—
147	C₂	#13092549	1	2	yes	—
148	C₂	#13092584	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#13093052

C₂

yes

maximal, characteristic, complemented

#13092609

C₂

yes

maximal, complemented

#13092161

C₂

yes

maximal, characteristic, complemented

#13092146

C₂

yes

maximal, complemented

C₂₃×#493

C₂

yes

maximal, characteristic, complemented

C₂×D₄×D₂₃

—

C₂×D₈×C₂₃

—

C₂₃×#127

C₂²

yes

characteristic

C₂₃×#127

C₂²

yes

characteristic, complemented

#10625109

—

#10625108

—

C₂×#67272

—

C₄×D₉₂

C₂²

yes

—

C₄×C₈×C₂₃

C₂²

yes

characteristic, complemented

#10625014

—

C₄⋊(C₂₃⋊C₈)

C₂²

yes

complemented

C₂³×D₂₃

—

C₂×D₄×C₂₃

—

C₂×D₄×C₂₃

—

C₂×D₄×C₂₃

—

C₂×D₄×C₂₃

D₄

yes

—

C₂×C₂²⋊D₂₃

—

D₄×D₂₃

—

D₄×D₂₃

—

C₂×D₉₂

D₄

yes

—

C₂×C₄×D₂₃

—

D₈×C₂₃

—

C₂×C₈×C₂₃

—

C₄²×C₂₃

C₂³

yes

characteristic

C₂²⋊Dic₂₃

—

#67272

—

(C₂×D₂₃)⋊C₄

—

C₄⋊Dic₂₃

D₄

yes

—

C₂×C₂₃⋊C₈

—

C₂³×C₂₃

—

C₂³×C₂₃

—

C₂²×D₂₃

—

C₂²×D₂₃

—

C₂²×D₂₃

—

D₄×C₂₃

—

D₄×C₂₃

—

D₄×C₂₃

—

D₄×C₂₃

—

D₄×C₂₃

—

D₄×C₂₃

—

C₂×C₄×C₂₃

C₂×D₄

yes

—

C₂×C₄×C₂₃

C₂×D₄

yes

characteristic

C₂²⋊D₂₃

—

C₂×Dic₂₃

—

D₉₂

—

D₉₂

D₈

yes

complemented

C₄×D₂₃

—

C₄×D₂₃

—

C₈×C₂₃

—

C₂₃⋊C₈

—

#1619

—

maximal, Sylow

C₂²×C₂₃

—

C₂²×C₂₃

—

C₂²×C₂₃

—

C₂²×C₂₃

—

C₂²×C₂₃

#120

yes

characteristic

C₂×D₂₃

—

C₂×D₂₃

—

C₂×D₂₃

—

C₂×D₂₃

—

C₄×C₂₃

—

C₄×C₂₃

C₂×D₈

yes

—

C₄×C₂₃

AGL(1,ℤ/8)

yes

—

Dic₂₃

—

D₄²

—

#493

D₂₃

yes

characteristic, complemented

#459

—

#331

—

D₄⋊C₈

—

C₂×C₂₃

—

C₂×C₂₃

—

C₂×C₂₃

#447

yes

—

C₂×C₂₃

#453

yes

characteristic

D₂₃

—

C₂²×D₄

—

C₂×D₈

—

C₂×D₈

—

#127

C₂×D₂₃

yes

characteristic

#127

C₂×D₂₃

yes

characteristic, complemented

#121

—

#120

—

C₄×D₄

—

C₄⋊C₈

—

#102

—

C₄×C₈

C₂×D₂₃

yes

characteristic, complemented

C₂₃

#1619

yes

characteristic, complemented, Sylow

C₂⁴

—

C₂×D₄

—

C₂×D₄

—

C₂×D₄

—

C₂×D₄

—

C₂×D₄

—

C₂×D₄

—

C₂×D₄

—

100

C₂×D₄

C₂²⋊D₂₃

yes

—

101

C₂²×C₄

—

102

D₈

—

103

D₈

—

104

C₂×C₈

—

105

C₂×C₈

—

106

C₄⋊C₄

—

107

C₂²⋊C₄

—

108

C₂²⋊C₄

—

109

C₄²

C₂²×D₂₃

yes

characteristic

110

C₂³

—

111

C₂³

—

112

C₂³

—

113

C₂³

—

114

C₂³

—

184

—

115

D₄

—

116

D₄

—

117

D₄

—

118

D₄

—

119

D₄

—

120

D₄

—

121

D₄

—

122

D₄

—

123

D₄

—

124

C₂×C₄

—

125

C₂×C₄

—

126

C₂×C₄

—

127

C₂×C₄

D₄×D₂₃

yes

—

128

C₂×C₄

C₂×C₂²⋊D₂₃

yes

characteristic

129

C₈

—

130

C₈

—

131

C₂²

—

132

C₂²

—

133

C₂²

—

134

C₂²

—

135

C₂²

—

136

C₂²

—

137

C₂²

—

138

C₂²

—

184

—

139

C₂²

#10625108

yes

characteristic

140

C₄

—

141

C₄

—

142

C₄

D₈×D₂₃

yes

—

143

C₄

#10625082

yes

—

144

C₂

—

145

C₂

—

146

C₂

—

147

C₂

#13092549

yes

—

148

C₂

#13092584

yes

characteristic

	subgroup	G/H
center	C₂²	#10625108
commutator, cosocle	C₄²×C₂₃	C₂³
Frattini	C₄²	C₂²×D₂₃
Fitting	C₂₃×#493	C₂
socle	C₂²×C₂₃	#120
hypercenter	#493	D₂₃
hypocenter	C₂₃	#1619
LCS	G ▷ C₄²×C₂₃ ▷ C₂²×C₂₃ ▷ C₂₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #493

subgroup

G/H

center

C₂²

#10625108

commutator, cosocle

C₄²×C₂₃

C₂³

Frattini

C₄²

C₂²×D₂₃

Fitting

C₂₃×#493

C₂

C₂²×C₂₃

D₂₃

C₂₃

G ▷ C₄²×C₂₃ ▷ C₂²×C₂₃ ▷ C₂₃

UCS

C₁ ◁ C₂² ◁ C₄² ◁ #493

N	H ≅ G/N	type
1. #13093052	C₂	semidirect
2. #13092609	C₂	semidirect
3. #13092161	C₂	semidirect
4. #13092146	C₂	semidirect
5. C₂₃×#493	C₂	semidirect
9. C₂₃×#127	C₂²	semidirect
14. C₄×C₈×C₂₃	C₂²	semidirect
16. C₄⋊(C₂₃⋊C₈)	C₂²	semidirect
51. D₉₂	D₈	semidirect
71. #493	D₂₃	semidirect
84. #127	C₂×D₂₃	semidirect
90. C₄×C₈	C₂×D₂₃	semidirect
91. C₂₃	#1619	semidirect

H ≅ G/N

type

1. #13093052

C₂

semidirect

2. #13092609

C₂

semidirect

3. #13092161

C₂

semidirect

4. #13092146

C₂

semidirect

5. C₂₃×#493

C₂

semidirect

9. C₂₃×#127

C₂²

semidirect

14. C₄×C₈×C₂₃

C₂²

semidirect

16. C₄⋊(C₂₃⋊C₈)

C₂²

semidirect

51. D₉₂

D₈

semidirect

71. #493

D₂₃

semidirect

84. #127

C₂×D₂₃

semidirect

90. C₄×C₈

C₂×D₂₃

semidirect

91. C₂₃

#1619

semidirect

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

subgroup	index
#13093052	2
#13092609	2
#13092161	2
#13092146	2
C₂₃×#493	2
#1619	23

subgroup

index

C₂₃×#493

Subgroups — #15672449

Conjugacy classes of subgroups

Complements

Maximal subgroups