Group #15686536 — subgroups

Subgroups — #15686536

Order 2944.

370 conjugacy classes of subgroups; only the 129 normal subgroups are shown.

normal subgroups only

This group has 370 conjugacy classes of subgroups — too many to draw. Only its 129 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	#13093222	C₂	1	1	yes	maximal, characteristic, complemented
2	#13092853	C₂	1	3	yes	maximal, complemented
3	#13092852	C₂	1	3	yes	maximal, complemented
4	#13092847	C₂	1	3	yes	maximal, complemented
5	C₂×#10625008	C₂	1	3	yes	maximal, complemented
6	#13092834	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂₃×#576	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂₃×H(5,𝔽₂)	—	2	1	no	—
9	#10625160	—	2	1	no	—
10	C₂₃×AGL(1,ℤ/8)	—	2	3	no	—
11	C₂₃×#135	C₂²	1	3	yes	complemented
12	C₂×D₈×C₂₃	C₂²	1	3	yes	complemented
13	C₂₃×#131	C₂²	1	1	yes	characteristic
14	#10625079	—	2	3	no	—
15	#10625075	C₂²	1	3	yes	complemented
16	C₂×D₁₈₄	C₂²	1	3	yes	complemented
17	#10625011	C₂²	1	1	yes	characteristic, complemented
18	#10625010	C₂²	1	3	yes	complemented
19	#10625009	C₂²	1	3	yes	complemented
20	#10625008	C₂²	1	3	yes	complemented
21	#10625008	C₂²	1	6	yes	complemented
22	#10624994	C₂²	1	3	yes	complemented
23	C₂×C₂₃⋊C₁₆	C₂²	1	3	yes	complemented
24	#41×C₂₃	—	2	3	no	—
25	#41×C₂₃	—	4	1	no	—
26	#41×C₂₃	D₄	1	1	yes	characteristic
27	C₂×D₄×C₂₃	—	2	3	no	—
28	C₂×D₄×C₂₃	—	4	3	no	—
29	#67292	—	4	1	no	—
30	D₄×D₂₃	—	4	3	no	—
31	#67288	—	2	3	no	—
32	C₂×D₉₂	—	2	3	no	—
33	Q₁₆×C₂₃	C₂³	1	1	yes	characteristic
34	SD₁₆×C₂₃	—	2	3	no	—
35	D₈×C₂₃	—	2	3	no	—
36	D₈×C₂₃	C₂³	1	3	yes	—
37	M₁₆×C₂₃	D₄	1	3	yes	—
38	C₂×C₈×C₂₃	C₂³	1	3	yes	—
39	Dic₉₂	C₂³	1	1	yes	characteristic
40	D₁₈₄	—	2	3	no	—
41	D₁₈₄	C₂³	1	3	yes	—
42	#67263	—	2	3	no	—
43	C₂₃⋊C₁₆	C₂³	1	1	yes	characteristic
44	C₂₃⋊C₁₆	C₂³	1	3	yes	—
45	C₂³×C₂₃	—	4	3	no	—
46	C₂²×D₂₃	—	4	3	no	—
47	Q₈×C₂₃	—	2	1	no	—
48	Q₈×C₂₃	D₈	1	1	yes	characteristic
49	D₄×C₂₃	—	2	3	no	—
50	D₄×C₂₃	—	4	3	no	—
51	D₄×C₂₃	—	4	3	no	—
52	D₄×C₂₃	D₈	1	3	yes	—
53	C₂×C₄×C₂₃	—	4	3	no	—
54	C₂×C₄×C₂₃	C₂×D₄	1	3	yes	—
55	C₂²⋊D₂₃	—	4	3	no	—
56	D₉₂	—	2	3	no	—
57	D₉₂	—	4	3	no	—
58	C₄×D₂₃	—	4	3	no	—
59	Dic₄₆	—	2	1	no	—
60	C₈×C₂₃	C₂×D₄	1	3	yes	—
61	C₈×C₂₃	C₂⁴	1	1	yes	characteristic
62	#3415	—	23	1	no	maximal, Sylow
63	C₂²×C₂₃	—	4	3	no	—
64	C₂²×C₂₃	—	16	3	no	—
65	C₂²×C₂₃	C₂×D₈	1	3	yes	—
66	C₂×D₂₃	—	4	3	no	—
67	C₂×D₂₃	—	16	3	no	—
68	C₄×C₂₃	—	4	1	no	—
69	C₄×C₂₃	C₂×D₈	1	3	yes	—
70	C₄×C₂₃	C₂²×D₄	1	1	yes	characteristic
71	Dic₂₃	—	4	1	no	—
72	#576	—	23	1	no	—
73	#576	D₂₃	1	1	yes	characteristic, complemented
74	#509	—	23	3	no	—
75	#509	—	23	3	no	—
76	#508	—	23	3	no	—
77	C₂×D₁₆	—	23	3	no	—
78	#504	—	23	1	no	—
79	C₂×C₂₃	—	2	3	no	—
80	C₂×C₂₃	—	8	3	no	—
81	C₂×C₂₃	C₂²×D₈	1	1	yes	characteristic
82	D₂₃	—	8	3	no	—
83	H(5,𝔽₂)	—	2	1	no	—
84	H(5,𝔽₂)	—	46	1	no	—
85	AGL(1,ℤ/8)	—	2	3	no	—
86	AGL(1,ℤ/8)	—	46	3	no	—
87	#135	—	23	3	no	—
88	#135	C₂×D₂₃	1	3	yes	complemented
89	C₂×D₈	—	23	3	no	—
90	C₂×D₈	C₂×D₂₃	1	3	yes	complemented
91	#131	C₂×D₂₃	1	1	yes	characteristic
92	Q₃₂	—	23	1	no	—
93	SD₃₂	—	23	3	no	—
94	SD₃₂	—	23	3	no	—
95	D₁₆	—	23	3	no	—
96	D₁₆	—	23	6	no	—
97	M₃₂	—	23	3	no	—
98	C₂×C₁₆	—	23	3	no	—
99	C₂₃	#3415	1	1	yes	characteristic, complemented, Sylow
100	#41	—	2	3	no	—
101	#41	—	4	1	no	—
102	#41	—	46	3	no	—
103	#41	—	92	1	no	—
104	#41	C₂²⋊D₂₃	1	1	yes	characteristic
105	C₂×D₄	—	2	3	no	—
106	C₂×D₄	—	4	3	no	—
107	C₂×D₄	—	46	3	no	—
108	C₂×D₄	—	92	3	no	—
109	Q₁₆	—	23	1	no	—
110	Q₁₆	C₂²×D₂₃	1	1	yes	characteristic
111	SD₁₆	—	2	3	no	—
112	SD₁₆	—	46	3	no	—
113	D₈	—	2	3	no	—
114	D₈	—	23	3	no	—
115	D₈	—	46	3	no	—
116	D₈	C₂²×D₂₃	1	3	yes	—
117	M₁₆	C₂²⋊D₂₃	1	3	yes	—
118	C₂×C₈	C₂²×D₂₃	1	3	yes	—
119	C₁₆	—	23	1	no	—
120	C₁₆	—	23	3	no	—
121	C₂³	—	4	3	no	—
122	C₂³	—	92	3	no	—
123	Q₈	—	2	1	no	—
124	Q₈	—	46	1	no	—
125	Q₈	#67272	1	1	yes	characteristic
126	D₄	—	2	3	no	—
127	D₄	—	4	3	no	—
128	D₄	—	4	3	no	—
129	D₄	—	46	3	no	—
130	D₄	—	92	3	no	—
131	D₄	—	92	3	no	—
132	D₄	#67272	1	3	yes	—
133	C₂×C₄	—	4	3	no	—
134	C₂×C₄	—	92	3	no	—
135	C₂×C₄	C₂×C₂²⋊D₂₃	1	3	yes	—
136	C₈	C₂×C₂²⋊D₂₃	1	3	yes	—
137	C₈	C₂³×D₂₃	1	1	yes	characteristic
138	C₂²	—	4	3	no	—
139	C₂²	—	16	3	no	—
140	C₂²	—	92	3	no	—
141	C₂²	—	368	3	no	—
142	C₂²	C₂×#67272	1	3	yes	—
143	C₄	—	4	1	no	—
144	C₄	—	92	1	no	—
145	C₄	C₂×#67272	1	3	yes	—
146	C₄	C₂²×C₂²⋊D₂₃	1	1	yes	characteristic
147	C₂	—	2	3	no	—
148	C₂	—	8	3	no	—
149	C₂	—	184	3	no	—
150	C₂	C₂²×#67272	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#13093222

C₂

yes

maximal, characteristic, complemented

#13092853

C₂

yes

maximal, complemented

#13092852

C₂

yes

maximal, complemented

#13092847

C₂

yes

maximal, complemented

C₂×#10625008

C₂

yes

maximal, complemented

#13092834

C₂

yes

maximal, characteristic, complemented

C₂₃×#576

C₂

yes

maximal, characteristic, complemented

C₂₃×H(5,𝔽₂)

—

#10625160

—

C₂₃×AGL(1,ℤ/8)

—

C₂₃×#135

C₂²

yes

complemented

C₂×D₈×C₂₃

C₂²

yes

complemented

C₂₃×#131

C₂²

yes

characteristic

#10625079

—

#10625075

C₂²

yes

complemented

C₂×D₁₈₄

C₂²

yes

complemented

#10625011

C₂²

yes

characteristic, complemented

#10625010

C₂²

yes

complemented

#10625009

C₂²

yes

complemented

#10625008

C₂²

yes

complemented

#10625008

C₂²

yes

complemented

#10624994

C₂²

yes

complemented

C₂×C₂₃⋊C₁₆

C₂²

yes

complemented

#41×C₂₃

—

#41×C₂₃

—

#41×C₂₃

D₄

yes

characteristic

C₂×D₄×C₂₃

—

C₂×D₄×C₂₃

—

#67292

—

D₄×D₂₃

—

#67288

—

C₂×D₉₂

—

Q₁₆×C₂₃

C₂³

yes

characteristic

SD₁₆×C₂₃

—

D₈×C₂₃

—

D₈×C₂₃

C₂³

yes

—

M₁₆×C₂₃

D₄

yes

—

C₂×C₈×C₂₃

C₂³

yes

—

Dic₉₂

C₂³

yes

characteristic

D₁₈₄

—

D₁₈₄

C₂³

yes

—

#67263

—

C₂₃⋊C₁₆

C₂³

yes

characteristic

C₂₃⋊C₁₆

C₂³

yes

—

C₂³×C₂₃

—

C₂²×D₂₃

—

Q₈×C₂₃

—

Q₈×C₂₃

D₈

yes

characteristic

D₄×C₂₃

—

D₄×C₂₃

—

D₄×C₂₃

—

D₄×C₂₃

D₈

yes

—

C₂×C₄×C₂₃

—

C₂×C₄×C₂₃

C₂×D₄

yes

—

C₂²⋊D₂₃

—

D₉₂

—

D₉₂

—

C₄×D₂₃

—

Dic₄₆

—

C₈×C₂₃

C₂×D₄

yes

—

C₈×C₂₃

C₂⁴

yes

characteristic

#3415

—

maximal, Sylow

C₂²×C₂₃

—

C₂²×C₂₃

—

C₂²×C₂₃

C₂×D₈

yes

—

C₂×D₂₃

—

C₂×D₂₃

—

C₄×C₂₃

—

C₄×C₂₃

C₂×D₈

yes

—

C₄×C₂₃

C₂²×D₄

yes

characteristic

Dic₂₃

—

#576

—

#576

D₂₃

yes

characteristic, complemented

#509

—

#509

—

#508

—

C₂×D₁₆

—

#504

—

C₂×C₂₃

—

C₂×C₂₃

—

C₂×C₂₃

C₂²×D₈

yes

characteristic

D₂₃

—

—

—

—

—

—

C₂×D₂₃

yes

complemented

C₂×D₈

—

C₂×D₈

C₂×D₂₃

yes

complemented

#131

C₂×D₂₃

yes

characteristic

Q₃₂

—

SD₃₂

—

SD₃₂

—

D₁₆

—

D₁₆

—

M₃₂

—

C₂×C₁₆

—

C₂₃

#3415

yes

characteristic, complemented, Sylow

100

#41

—

101

#41

—

102

#41

—

103

#41

—

104

#41

C₂²⋊D₂₃

yes

characteristic

105

C₂×D₄

—

106

C₂×D₄

—

107

C₂×D₄

—

108

C₂×D₄

—

109

Q₁₆

—

110

Q₁₆

C₂²×D₂₃

yes

characteristic

111

SD₁₆

—

112

SD₁₆

—

113

D₈

—

114

D₈

—

115

D₈

—

116

D₈

C₂²×D₂₃

yes

—

117

M₁₆

C₂²⋊D₂₃

yes

—

118

C₂×C₈

C₂²×D₂₃

yes

—

119

C₁₆

—

120

C₁₆

—

121

C₂³

—

122

C₂³

—

123

Q₈

—

124

Q₈

—

125

Q₈

#67272

yes

characteristic

126

D₄

—

127

D₄

—

128

D₄

—

129

D₄

—

130

D₄

—

131

D₄

—

132

D₄

#67272

yes

—

133

C₂×C₄

—

134

C₂×C₄

—

135

C₂×C₄

C₂×C₂²⋊D₂₃

yes

—

136

C₈

C₂×C₂²⋊D₂₃

yes

—

137

C₈

C₂³×D₂₃

yes

characteristic

138

C₂²

—

139

C₂²

—

140

C₂²

—

141

C₂²

—

368

—

142

C₂²

C₂×#67272

yes

—

143

C₄

—

144

C₄

—

145

C₄

C₂×#67272

yes

—

146

C₄

C₂²×C₂²⋊D₂₃

yes

characteristic

147

C₂

—

148

C₂

—

149

C₂

—

184

—

150

C₂

C₂²×#67272

yes

characteristic

	subgroup	G/H
center	C₂	C₂²×#67272
commutator, cosocle	C₈×C₂₃	C₂⁴
Frattini	C₈	C₂³×D₂₃
Fitting	C₂₃×#576	C₂
socle	C₂×C₂₃	C₂²×D₈
hypercenter	#576	D₂₃
hypocenter	C₂₃	#3415
LCS	G ▷ C₈×C₂₃ ▷ C₄×C₂₃ ▷ C₂×C₂₃ ▷ C₂₃
UCS	C₁ ◁ C₂ ◁ #41 ◁ #131 ◁ #576

subgroup

G/H

center

C₂

C₂²×#67272

commutator, cosocle

C₈×C₂₃

C₂⁴

Frattini

C₈

C₂³×D₂₃

Fitting

C₂₃×#576

C₂

socle

C₂×C₂₃

C₂²×D₈

D₂₃

C₂₃

G ▷ C₈×C₂₃ ▷ C₄×C₂₃ ▷ C₂×C₂₃ ▷ C₂₃

UCS

C₁ ◁ C₂ ◁ #41 ◁ #131 ◁ #576

N	H ≅ G/N	type
1. #13093222	C₂	semidirect
2. #13092853	C₂	semidirect
3. #13092852	C₂	semidirect
4. #13092847	C₂	semidirect
5. C₂×#10625008	C₂	semidirect
6. #13092834	C₂	semidirect
7. C₂₃×#576	C₂	semidirect
11. C₂₃×#135	C₂²	semidirect
12. C₂×D₈×C₂₃	C₂²	semidirect
15. #10625075	C₂²	semidirect
16. C₂×D₁₈₄	C₂²	semidirect
17. #10625011	C₂²	semidirect
18. #10625010	C₂²	semidirect
19. #10625009	C₂²	semidirect
20. #10625008	C₂²	semidirect
21. #10625008	C₂²	semidirect
22. #10624994	C₂²	semidirect
23. C₂×C₂₃⋊C₁₆	C₂²	semidirect
73. #576	D₂₃	semidirect
88. #135	C₂×D₂₃	semidirect
90. C₂×D₈	C₂×D₂₃	semidirect
99. C₂₃	#3415	semidirect

H ≅ G/N

type

1. #13093222

C₂

semidirect

2. #13092853

C₂

semidirect

3. #13092852

C₂

semidirect

4. #13092847

C₂

semidirect

5. C₂×#10625008

C₂

semidirect

6. #13092834

C₂

semidirect

7. C₂₃×#576

C₂

semidirect

11. C₂₃×#135

C₂²

semidirect

12. C₂×D₈×C₂₃

C₂²

semidirect

15. #10625075

C₂²

semidirect

16. C₂×D₁₈₄

C₂²

semidirect

17. #10625011

C₂²

semidirect

18. #10625010

C₂²

semidirect

19. #10625009

C₂²

semidirect

20. #10625008

C₂²

semidirect

21. #10625008

C₂²

semidirect

22. #10624994

C₂²

semidirect

23. C₂×C₂₃⋊C₁₆

C₂²

semidirect

73. #576

D₂₃

semidirect

88. #135

C₂×D₂₃

semidirect

90. C₂×D₈

C₂×D₂₃

semidirect

99. C₂₃

#3415

semidirect

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

subgroup	index
#13093222	2
#13092853	2
#13092852	2
#13092847	2
C₂×#10625008	2
#13092834	2
C₂₃×#576	2
#3415	23

subgroup

index

C₂₃×#576

Subgroups — #15686536

Conjugacy classes of subgroups

Complements

Maximal subgroups