Group #15673027 — subgroups

Subgroups — #15673027

Order 2944.

220 conjugacy classes of subgroups; only the 61 normal subgroups are shown.

normal subgroups only

This group has 220 conjugacy classes of subgroups — too many to draw. Only its 61 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	#13092813	C₂	1	1	yes	maximal, characteristic, complemented
2	#13092793	C₂	1	1	yes	maximal, characteristic
3	#13092791	C₂	1	1	yes	maximal, characteristic, complemented
4	#13092790	C₂	1	1	yes	maximal, characteristic, complemented
5	#13092763	C₂	1	1	yes	maximal, characteristic, complemented
6	#13092139	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂₃×#486	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂₃×C₄⋊Q₈	C₂²	1	1	yes	characteristic
9	C₂₃×#127	C₂²	1	1	yes	characteristic
10	C₂×#67274	—	2	2	no	—
11	C₂×#67272	—	2	2	no	—
12	C₂×#67263	—	2	2	no	—
13	#10625045	C₂²	1	1	yes	characteristic
14	#10625040	C₂²	1	1	yes	characteristic
15	C₂₃×#102	—	2	2	no	—
16	C₄×C₈×C₂₃	C₂²	1	1	yes	characteristic
17	#10625014	—	2	2	no	—
18	#10624990	—	2	2	no	—
19	C₄×C₂₃⋊C₈	C₂²	1	1	yes	characteristic
20	C₄×C₂₃⋊C₈	C₂²	1	1	yes	characteristic
21	C₂×Q₈×C₂₃	—	2	1	no	—
22	C₂×D₄×C₂₃	—	2	1	no	—
23	C₂×D₄×C₂₃	—	2	1	no	—
24	C₂×D₄×C₂₃	—	2	1	no	—
25	C₂×D₉₂	—	2	1	no	—
26	C₂×D₉₂	—	2	1	no	—
27	C₂×D₉₂	—	2	1	no	—
28	C₂×Dic₄₆	—	2	1	no	—
29	C₂×C₈×C₂₃	D₄	1	2	yes	—
30	C₄⋊C₄×C₂₃	—	2	1	no	—
31	C₄⋊C₄×C₂₃	—	2	1	no	—
32	C₄²×C₂₃	C₂³	1	1	yes	characteristic
33	#67274	—	4	4	no	—
34	#67272	—	4	4	no	—
35	C₄⋊Dic₂₃	—	2	1	no	—
36	C₄⋊Dic₂₃	—	2	1	no	—
37	C₂×C₂₃⋊C₈	D₄	1	2	yes	—
38	C₂×C₂₃⋊C₈	D₄	1	2	yes	—
39	#67263	—	4	4	no	—
40	C₂³×C₂₃	—	4	1	no	—
41	C₂²×D₂₃	—	4	1	no	—
42	Q₈×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	—	4	1	no	—
47	C₂×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
48	C₂×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
49	C₂×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
50	C₂×Dic₂₃	—	4	1	no	—
51	D₉₂	—	4	2	no	—
52	D₉₂	—	4	2	no	—
53	D₉₂	—	4	2	no	—
54	Dic₄₆	—	4	2	no	—
55	C₈×C₂₃	—	2	2	no	—
56	C₂₃⋊C₈	—	2	2	no	—
57	C₂₃⋊C₈	SD₁₆	1	4	yes	complemented
58	#1715	—	23	1	no	maximal, Sylow
59	C₂²×C₂₃	—	8	1	no	—
60	C₂²×C₂₃	—	8	1	no	—
61	C₂²×C₂₃	—	8	1	no	—
62	C₂²×C₂₃	#127	1	1	yes	characteristic
63	C₂×D₂₃	—	8	1	no	—
64	C₂×D₂₃	—	8	1	no	—
65	C₂×D₂₃	—	8	1	no	—
66	C₄×C₂₃	—	8	1	no	—
67	C₄×C₂₃	C₂×SD₁₆	1	2	yes	—
68	C₄×C₂₃	AGL(1,ℤ/8)	1	2	yes	—
69	C₄×C₂₃	AGL(1,ℤ/8)	1	2	yes	—
70	Dic₂₃	—	8	1	no	—
71	#498	—	23	1	no	—
72	#493	—	23	1	no	—
73	#492	—	23	1	no	—
74	#492	—	23	1	no	—
75	#486	—	23	1	no	—
76	#486	D₂₃	1	1	yes	characteristic, complemented
77	#322	—	23	1	no	—
78	C₂×C₂₃	—	16	1	no	—
79	C₂×C₂₃	#492	1	1	yes	characteristic
80	C₂×C₂₃	#496	1	1	yes	characteristic
81	C₂×C₂₃	#496	1	1	yes	characteristic
82	D₂₃	—	16	1	no	—
83	C₂×SD₁₆	—	46	2	no	—
84	C₂×SD₁₆	—	46	2	no	—
85	C₂×D₈	—	46	2	no	—
86	C₄⋊Q₈	—	23	1	no	—
87	C₄⋊Q₈	C₂×D₂₃	1	1	yes	characteristic
88	#127	—	23	1	no	—
89	#127	C₂×D₂₃	1	1	yes	characteristic
90	#106	—	46	2	no	—
91	#102	—	2	2	no	—
92	#102	—	46	2	no	—
93	C₄×C₈	—	23	1	no	—
94	C₄×C₈	—	23	1	no	—
95	C₄×C₈	C₂×D₂₃	1	1	yes	characteristic
96	C₂₃	#1715	1	1	yes	characteristic, complemented, Sylow
97	C₂×Q₈	—	2	1	no	—
98	C₂×Q₈	—	46	1	no	—
99	C₂×D₄	—	2	1	no	—
100	C₂×D₄	—	2	1	no	—
101	C₂×D₄	—	2	1	no	—
102	C₂×D₄	—	46	1	no	—
103	C₂×D₄	—	46	1	no	—
104	C₂×D₄	—	46	1	no	—
105	SD₁₆	—	92	4	no	—
106	SD₁₆	—	92	4	no	—
107	D₈	—	92	4	no	—
108	C₂×C₈	—	23	2	no	—
109	C₂×C₈	—	23	2	no	—
110	C₂×C₈	C₂²⋊D₂₃	1	2	yes	—
111	C₄⋊C₄	—	2	1	no	—
112	C₄⋊C₄	—	2	1	no	—
113	C₄⋊C₄	—	46	1	no	—
114	C₄⋊C₄	—	46	1	no	—
115	C₄²	C₂²×D₂₃	1	1	yes	characteristic
116	C₂³	—	4	1	no	—
117	C₂³	—	92	1	no	—
118	Q₈	—	4	2	no	—
119	Q₈	—	92	2	no	—
120	D₄	—	4	2	no	—
121	D₄	—	4	2	no	—
122	D₄	—	4	2	no	—
123	D₄	—	92	2	no	—
124	D₄	—	92	2	no	—
125	D₄	—	92	2	no	—
126	C₂×C₄	—	4	1	no	—
127	C₂×C₄	—	92	1	no	—
128	C₂×C₄	D₄×D₂₃	1	1	yes	characteristic
129	C₂×C₄	D₄×D₂₃	1	1	yes	characteristic
130	C₂×C₄	C₂×C₂²⋊D₂₃	1	1	yes	characteristic
131	C₈	—	2	2	no	—
132	C₈	—	23	4	no	—
133	C₈	—	46	2	no	—
134	C₂²	—	8	1	no	—
135	C₂²	—	8	1	no	—
136	C₂²	—	8	1	no	—
137	C₂²	—	184	1	no	—
138	C₂²	—	184	1	no	—
139	C₂²	—	184	1	no	—
140	C₂²	#10625111	1	1	yes	characteristic
141	C₄	—	8	1	no	—
142	C₄	—	184	1	no	—
143	C₄	#10625082	1	2	yes	—
144	C₄	SD₁₆×D₂₃	1	2	yes	—
145	C₄	#10625120	1	2	yes	—
146	C₂	—	16	1	no	—
147	C₂	—	368	1	no	—
148	C₂	#13092788	1	1	yes	characteristic
149	C₂	#13092805	1	1	yes	characteristic
150	C₂	#13092807	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#13092813

C₂

yes

maximal, characteristic, complemented

#13092793

C₂

yes

maximal, characteristic

#13092791

C₂

yes

maximal, characteristic, complemented

#13092790

C₂

yes

maximal, characteristic, complemented

#13092763

C₂

yes

maximal, characteristic, complemented

#13092139

C₂

yes

maximal, characteristic, complemented

C₂₃×#486

C₂

yes

maximal, characteristic, complemented

C₂₃×C₄⋊Q₈

C₂²

yes

characteristic

C₂₃×#127

C₂²

yes

—

—

—

C₂²

yes

characteristic

#10625040

C₂²

yes

characteristic

C₂₃×#102

—

C₄×C₈×C₂₃

C₂²

yes

characteristic

#10625014

—

#10624990

—

C₄×C₂₃⋊C₈

C₂²

yes

characteristic

C₄×C₂₃⋊C₈

C₂²

yes

characteristic

C₂×Q₈×C₂₃

—

C₂×D₄×C₂₃

—

C₂×D₄×C₂₃

—

C₂×D₄×C₂₃

—

C₂×D₉₂

—

C₂×D₉₂

—

C₂×D₉₂

—

C₂×Dic₄₆

—

C₂×C₈×C₂₃

D₄

yes

—

C₄⋊C₄×C₂₃

—

C₄⋊C₄×C₂₃

—

C₄²×C₂₃

C₂³

yes

characteristic

#67274

—

#67272

—

C₄⋊Dic₂₃

—

C₄⋊Dic₂₃

—

C₂×C₂₃⋊C₈

D₄

yes

—

C₂×C₂₃⋊C₈

D₄

yes

—

#67263

—

C₂³×C₂₃

—

C₂²×D₂₃

—

Q₈×C₂₃

—

D₄×C₂₃

—

D₄×C₂₃

—

D₄×C₂₃

—

C₂×C₄×C₂₃

—

C₂×C₄×C₂₃

C₂×D₄

yes

characteristic

C₂×C₄×C₂₃

C₂×D₄

yes

characteristic

C₂×C₄×C₂₃

C₂×D₄

yes

characteristic

C₂×Dic₂₃

—

D₉₂

—

D₉₂

—

D₉₂

—

Dic₄₆

—

C₈×C₂₃

—

C₂₃⋊C₈

—

C₂₃⋊C₈

SD₁₆

yes

complemented

#1715

—

maximal, Sylow

C₂²×C₂₃

—

C₂²×C₂₃

—

C₂²×C₂₃

—

C₂²×C₂₃

#127

yes

characteristic

C₂×D₂₃

—

C₂×D₂₃

—

C₂×D₂₃

—

C₄×C₂₃

—

C₄×C₂₃

C₂×SD₁₆

yes

—

C₄×C₂₃

AGL(1,ℤ/8)

yes

—

C₄×C₂₃

AGL(1,ℤ/8)

yes

—

Dic₂₃

—

—

—

—

—

—

D₂₃

yes

characteristic, complemented

#322

—

C₂×C₂₃

—

C₂×C₂₃

yes

C₂×C₂₃

yes

C₂×C₂₃

yes

D₂₃

—

C₂×SD₁₆

—

C₂×SD₁₆

—

C₂×D₈

—

C₄⋊Q₈

—

C₄⋊Q₈

C₂×D₂₃

yes

characteristic

#127

—

#127

C₂×D₂₃

yes

—

—

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

C₂×D₂₃

yes

characteristic

C₂₃

#1715

yes

characteristic, complemented, Sylow

C₂×Q₈

—

C₂×Q₈

—

C₂×D₄

—

100

C₂×D₄

—

101

C₂×D₄

—

102

C₂×D₄

—

103

C₂×D₄

—

104

C₂×D₄

—

105

SD₁₆

—

106

SD₁₆

—

107

D₈

—

108

C₂×C₈

—

109

C₂×C₈

—

110

C₂×C₈

C₂²⋊D₂₃

yes

—

111

C₄⋊C₄

—

112

C₄⋊C₄

—

113

C₄⋊C₄

—

114

C₄⋊C₄

—

115

C₄²

C₂²×D₂₃

yes

characteristic

116

C₂³

—

117

C₂³

—

118

Q₈

—

119

Q₈

—

120

D₄

—

121

D₄

—

122

D₄

—

123

D₄

—

124

D₄

—

125

D₄

—

126

C₂×C₄

—

127

C₂×C₄

—

128

C₂×C₄

D₄×D₂₃

yes

characteristic

129

C₂×C₄

D₄×D₂₃

yes

characteristic

130

C₂×C₄

C₂×C₂²⋊D₂₃

yes

characteristic

131

C₈

—

132

C₈

—

133

C₈

—

134

C₂²

—

135

C₂²

—

136

C₂²

—

137

C₂²

—

184

—

138

C₂²

—

184

—

139

C₂²

—

184

—

140

C₂²

#10625111

yes

characteristic

141

C₄

—

142

C₄

—

184

—

143

C₄

#10625082

yes

—

144

C₄

SD₁₆×D₂₃

yes

—

145

C₄

#10625120

yes

—

146

C₂

—

147

C₂

—

368

—

148

C₂

#13092788

yes

characteristic

149

C₂

#13092805

yes

characteristic

150

C₂

#13092807

yes

characteristic

	subgroup	G/H
center	C₂²	#10625111
commutator, cosocle	C₄²×C₂₃	C₂³
Frattini	C₄²	C₂²×D₂₃
Fitting	C₂₃×#486	C₂
socle	C₂²×C₂₃	#127
prime radical	#13092793	C₂
hypercenter	#486	D₂₃
hypocenter	C₂₃	#1715
LCS	G ▷ C₄²×C₂₃ ▷ C₂²×C₂₃ ▷ C₂₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #486

subgroup

G/H

center

C₂²

#10625111

commutator, cosocle

C₄²×C₂₃

C₂³

Frattini

C₄²

C₂²×D₂₃

Fitting

C₂₃×#486

C₂

C₂²×C₂₃

C₂

D₂₃

C₂₃

G ▷ C₄²×C₂₃ ▷ C₂²×C₂₃ ▷ C₂₃

UCS

C₁ ◁ C₂² ◁ C₄² ◁ #486

N	H ≅ G/N	type
1. #13092813	C₂	semidirect
3. #13092791	C₂	semidirect
4. #13092790	C₂	semidirect
5. #13092763	C₂	semidirect
6. #13092139	C₂	semidirect
7. C₂₃×#486	C₂	semidirect
57. C₂₃⋊C₈	SD₁₆	semidirect
76. #486	D₂₃	semidirect
96. C₂₃	#1715	semidirect

H ≅ G/N

type

1. #13092813

C₂

semidirect

3. #13092791

C₂

semidirect

4. #13092790

C₂

semidirect

5. #13092763

C₂

semidirect

6. #13092139

C₂

semidirect

7. C₂₃×#486

C₂

semidirect

57. C₂₃⋊C₈

SD₁₆

semidirect

76. #486

D₂₃

semidirect

96. C₂₃

#1715

semidirect

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

subgroup	index
#13092813	2
#13092793	2
#13092791	2
#13092790	2
#13092763	2
#13092139	2
C₂₃×#486	2
#1715	23

subgroup

index

C₂₃×#486

Subgroups — #15673027

Conjugacy classes of subgroups

Complements

Maximal subgroups