Group #15683773 — subgroups

Subgroups — #15683773

Order 2944.

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

normal subgroups only

This group has 484 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	C₂×#10625045	C₂	1	1	yes	maximal, characteristic
2	#13092783	C₂	1	4	yes	maximal, complemented
3	#13092765	C₂	1	4	yes	maximal, complemented
4	#13092421	C₂	1	4	yes	maximal, complemented
5	C₄×#67267	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂₃×#533	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂²×D₉₂	—	2	2	no	—
8	C₂²×D₉₂	C₂²	1	2	yes	—
9	C₂₃×C₄⋊Q₈	C₂²	1	2	yes	complemented
10	C₂₃×#125	C₂²	1	2	yes	complemented
11	C₂₃×C₂²⋊Q₈	—	2	2	no	—
12	C₄×Q₈×C₂₃	—	2	2	no	—
13	C₂₃×#117	C₂²	1	2	yes	—
14	C₂×C₄²×C₂₃	C₂²	1	1	yes	characteristic
15	C₂×#67267	C₂²	1	2	yes	—
16	#10625045	—	2	2	no	—
17	#10625045	C₂²	1	2	yes	—
18	#10625045	C₂²	1	2	yes	—
19	#10624991	C₂²	1	16	yes	complemented
20	#10624985	C₂²	1	2	yes	complemented
21	C₄×C₂₃⋊C₈	C₂²	1	2	yes	complemented
22	C₂³×D₂₃	—	2	2	no	—
23	C₂×Q₈×C₂₃	—	2	2	no	—
24	C₂²×C₄×C₂₃	D₄	1	2	yes	—
25	C₂²×C₄×C₂₃	C₂³	1	1	yes	characteristic
26	C₂×D₉₂	—	2	8	no	—
27	C₂×D₉₂	—	2	2	no	—
28	C₂×D₉₂	—	2	4	no	—
29	C₂×D₉₂	—	2	4	no	—
30	C₂×D₉₂	—	4	8	no	—
31	C₂×D₉₂	C₂³	1	4	yes	—
32	C₄⋊C₄×C₂₃	—	2	4	no	—
33	C₄⋊C₄×C₂₃	—	2	2	no	—
34	C₄⋊C₄×C₂₃	C₂³	1	4	yes	—
35	C₂²⋊C₄×C₂₃	—	2	2	no	—
36	C₄²×C₂₃	—	2	2	no	—
37	C₄²×C₂₃	D₄	1	2	yes	—
38	C₄²×C₂₃	C₂³	1	1	yes	characteristic
39	C₄²×C₂₃	C₂³	1	1	yes	characteristic
40	#67267	—	2	4	no	—
41	C₂×C₂₃⋊C₈	C₂³	1	4	yes	—
42	C₂³×C₂₃	C₂×D₄	1	1	yes	characteristic
43	C₂²×D₂₃	—	2	4	no	—
44	C₂²×D₂₃	—	4	8	no	—
45	C₂²×D₂₃	—	4	4	no	—
46	Q₈×C₂₃	—	4	4	no	—
47	C₂×C₄×C₂₃	—	2	4	no	—
48	C₂×C₄×C₂₃	—	2	4	no	—
49	C₂×C₄×C₂₃	C₂×D₄	1	2	yes	—
50	C₂×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
51	C₂×C₄×C₂₃	C₂×D₄	1	2	yes	—
52	C₂×C₄×C₂₃	#41	1	4	yes	—
53	C₂×C₄×C₂₃	C₂⁴	1	1	yes	characteristic
54	D₉₂	—	2	8	no	—
55	D₉₂	—	4	8	no	—
56	D₉₂	—	4	4	no	—
57	D₉₂	—	4	8	no	—
58	C₂₃⋊C₈	—	2	4	no	—
59	#3138	—	23	1	no	maximal, Sylow
60	C₂²×C₂₃	—	2	2	no	—
61	C₂²×C₂₃	#124	1	2	yes	—
62	C₂²×C₂₃	C₂²×D₄	1	1	yes	characteristic
63	C₂×D₂₃	—	4	4	no	—
64	C₂×D₂₃	—	4	8	no	—
65	C₂×D₂₃	—	8	8	no	—
66	C₄×C₂₃	—	2	2	no	—
67	C₄×C₂₃	—	4	4	no	—
68	C₄×C₂₃	#124	1	2	yes	—
69	C₄×C₂₃	AGL(1,ℤ/8)	1	4	yes	—
70	C₄×C₂₃	C₂×#41	1	2	yes	—
71	#533	D₂₃	1	1	yes	characteristic, complemented
72	C₂×#127	—	23	1	no	—
73	#490	—	23	4	no	—
74	#486	—	23	4	no	—
75	#418	—	23	4	no	—
76	C₄×M₁₆	—	23	1	no	—
77	C₂×C₂₃	—	2	2	no	—
78	C₂×C₂₃	C₂×#124	1	1	yes	characteristic
79	C₂×C₂₃	C₂×AGL(1,ℤ/8)	1	2	yes	—
80	D₂₃	—	8	4	no	—
81	C₂²×D₄	—	23	2	no	—
82	C₂²×D₄	—	46	2	no	—
83	C₂×M₁₆	—	23	2	no	—
84	C₄⋊Q₈	C₂×D₂₃	1	2	yes	complemented
85	#127	—	23	2	no	—
86	#127	—	23	2	no	—
87	#127	—	46	2	no	—
88	#125	C₂×D₂₃	1	2	yes	complemented
89	C₂²⋊Q₈	—	2	2	no	—
90	C₄×Q₈	—	2	2	no	—
91	#117	C₂×D₂₃	1	2	yes	—
92	C₂×C₄²	C₂×D₂₃	1	1	yes	characteristic
93	#102	—	23	16	no	—
94	#97	—	23	2	no	—
95	C₄×C₈	—	23	2	no	—
96	C₂₃	#3138	1	1	yes	characteristic, complemented, Sylow
97	C₂⁴	—	46	2	no	—
98	C₂×Q₈	—	2	2	no	—
99	C₂×D₄	—	23	4	no	—
100	C₂×D₄	—	46	8	no	—
101	C₂×D₄	—	46	4	no	—
102	C₂×D₄	—	46	4	no	—
103	C₂×D₄	—	46	2	no	—
104	C₂×D₄	—	92	8	no	—
105	C₂²×C₄	C₂²⋊D₂₃	1	2	yes	—
106	C₂²×C₄	C₂²×D₂₃	1	1	yes	characteristic
107	M₁₆	—	46	4	no	—
108	C₂×C₈	—	23	4	no	—
109	C₄⋊C₄	—	2	4	no	—
110	C₄⋊C₄	—	2	2	no	—
111	C₄⋊C₄	C₂²×D₂₃	1	4	yes	—
112	C₂²⋊C₄	—	2	2	no	—
113	C₄²	—	2	2	no	—
114	C₄²	C₂²⋊D₂₃	1	2	yes	—
115	C₄²	C₂²×D₂₃	1	1	yes	characteristic
116	C₄²	C₂²×D₂₃	1	1	yes	characteristic
117	C₂³	—	46	4	no	—
118	C₂³	—	92	8	no	—
119	C₂³	—	92	4	no	—
120	C₂³	C₂×C₂²⋊D₂₃	1	1	yes	characteristic
121	Q₈	—	4	4	no	—
122	D₄	—	46	8	no	—
123	D₄	—	92	8	no	—
124	D₄	—	92	4	no	—
125	D₄	—	92	8	no	—
126	C₂×C₄	—	2	4	no	—
127	C₂×C₄	—	2	4	no	—
128	C₂×C₄	#67292	1	4	yes	—
129	C₂×C₄	C₂×C₂²⋊D₂₃	1	2	yes	—
130	C₂×C₄	C₂×C₂²⋊D₂₃	1	1	yes	characteristic
131	C₂×C₄	C₂×C₂²⋊D₂₃	1	2	yes	—
132	C₂×C₄	C₂³×D₂₃	1	1	yes	characteristic
133	C₈	—	46	4	no	—
134	C₂²	—	2	2	no	—
135	C₂²	—	92	4	no	—
136	C₂²	—	92	8	no	—
137	C₂²	—	184	8	no	—
138	C₂²	#10625118	1	2	yes	—
139	C₂²	C₂²×C₂²⋊D₂₃	1	1	yes	characteristic
140	C₄	—	2	2	no	—
141	C₄	—	4	4	no	—
142	C₄	#10625118	1	2	yes	—
143	C₄	#10625120	1	4	yes	—
144	C₄	C₂×#67292	1	2	yes	—
145	C₂	—	2	2	no	—
146	C₂	—	184	4	no	—
147	C₂	C₂×#10625118	1	1	yes	characteristic
148	C₂	C₂×#10625120	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#10625045

C₂

yes

maximal, characteristic

#13092783

C₂

yes

maximal, complemented

#13092765

C₂

yes

maximal, complemented

#13092421

C₂

yes

maximal, complemented

C₄×#67267

C₂

yes

maximal, characteristic, complemented

C₂₃×#533

C₂

yes

maximal, characteristic, complemented

C₂²×D₉₂

—

C₂²×D₉₂

C₂²

yes

—

C₂₃×C₄⋊Q₈

C₂²

yes

complemented

C₂₃×#125

C₂²

yes

complemented

C₂₃×C₂²⋊Q₈

—

C₄×Q₈×C₂₃

—

C₂₃×#117

C₂²

yes

—

C₂×C₄²×C₂₃

C₂²

yes

characteristic

C₂×#67267

C₂²

yes

—

#10625045

—

#10625045

C₂²

yes

—

#10625045

C₂²

yes

—

#10624991

C₂²

yes

complemented

#10624985

C₂²

yes

complemented

C₄×C₂₃⋊C₈

C₂²

yes

complemented

C₂³×D₂₃

—

C₂×Q₈×C₂₃

—

C₂²×C₄×C₂₃

D₄

yes

—

C₂²×C₄×C₂₃

C₂³

yes

characteristic

C₂×D₉₂

—

C₂×D₉₂

—

C₂×D₉₂

—

C₂×D₉₂

—

C₂×D₉₂

—

C₂×D₉₂

C₂³

yes

—

C₄⋊C₄×C₂₃

—

C₄⋊C₄×C₂₃

—

C₄⋊C₄×C₂₃

C₂³

yes

—

C₂²⋊C₄×C₂₃

—

C₄²×C₂₃

—

C₄²×C₂₃

D₄

yes

—

C₄²×C₂₃

C₂³

yes

characteristic

C₄²×C₂₃

C₂³

yes

characteristic

#67267

—

C₂×C₂₃⋊C₈

C₂³

yes

—

C₂³×C₂₃

C₂×D₄

yes

characteristic

C₂²×D₂₃

—

C₂²×D₂₃

—

C₂²×D₂₃

—

Q₈×C₂₃

—

C₂×C₄×C₂₃

—

C₂×C₄×C₂₃

—

C₂×C₄×C₂₃

C₂×D₄

yes

—

C₂×C₄×C₂₃

C₂×D₄

yes

characteristic

C₂×C₄×C₂₃

C₂×D₄

yes

—

C₂×C₄×C₂₃

#41

yes

—

C₂×C₄×C₂₃

C₂⁴

yes

characteristic

D₉₂

—

D₉₂

—

D₉₂

—

D₉₂

—

C₂₃⋊C₈

—

#3138

—

maximal, Sylow

C₂²×C₂₃

—

C₂²×C₂₃

#124

yes

—

C₂²×C₂₃

C₂²×D₄

yes

characteristic

C₂×D₂₃

—

C₂×D₂₃

—

C₂×D₂₃

—

C₄×C₂₃

—

C₄×C₂₃

—

C₄×C₂₃

#124

yes

—

C₄×C₂₃

AGL(1,ℤ/8)

yes

—

C₄×C₂₃

C₂×#41

yes

—

#533

D₂₃

yes

characteristic, complemented

—

—

—

—

C₄×M₁₆

—

C₂×C₂₃

—

C₂×C₂₃

C₂×#124

yes

characteristic

C₂×C₂₃

C₂×AGL(1,ℤ/8)

yes

—

D₂₃

—

C₂²×D₄

—

C₂²×D₄

—

C₂×M₁₆

—

C₄⋊Q₈

C₂×D₂₃

yes

—

—

—

C₂×D₂₃

yes

complemented

C₂²⋊Q₈

—

C₄×Q₈

—

#117

C₂×D₂₃

yes

—

C₂×C₄²

C₂×D₂₃

yes

characteristic

#102

—

#97

—

C₄×C₈

—

C₂₃

#3138

yes

characteristic, complemented, Sylow

C₂⁴

—

C₂×Q₈

—

C₂×D₄

—

100

C₂×D₄

—

101

C₂×D₄

—

102

C₂×D₄

—

103

C₂×D₄

—

104

C₂×D₄

—

105

C₂²×C₄

C₂²⋊D₂₃

yes

—

106

C₂²×C₄

C₂²×D₂₃

yes

characteristic

107

M₁₆

—

108

C₂×C₈

—

109

C₄⋊C₄

—

110

C₄⋊C₄

—

111

C₄⋊C₄

C₂²×D₂₃

yes

—

112

C₂²⋊C₄

—

113

C₄²

—

114

C₄²

C₂²⋊D₂₃

yes

—

115

C₄²

C₂²×D₂₃

yes

characteristic

116

C₄²

C₂²×D₂₃

yes

characteristic

117

C₂³

—

118

C₂³

—

119

C₂³

—

120

C₂³

C₂×C₂²⋊D₂₃

yes

characteristic

121

Q₈

—

122

D₄

—

123

D₄

—

124

D₄

—

125

D₄

—

126

C₂×C₄

—

127

C₂×C₄

—

128

C₂×C₄

#67292

yes

—

129

C₂×C₄

C₂×C₂²⋊D₂₃

yes

—

130

C₂×C₄

C₂×C₂²⋊D₂₃

yes

characteristic

131

C₂×C₄

C₂×C₂²⋊D₂₃

yes

—

132

C₂×C₄

C₂³×D₂₃

yes

characteristic

133

C₈

—

134

C₂²

—

135

C₂²

—

136

C₂²

—

137

C₂²

—

184

—

138

C₂²

#10625118

yes

—

139

C₂²

C₂²×C₂²⋊D₂₃

yes

characteristic

140

C₄

—

141

C₄

—

142

C₄

#10625118

yes

—

143

C₄

#10625120

yes

—

144

C₄

C₂×#67292

yes

—

145

C₂

—

146

C₂

—

184

—

147

C₂

C₂×#10625118

yes

characteristic

148

C₂

C₂×#10625120

yes

—

	subgroup	G/H
center	C₂²	C₂²×C₂²⋊D₂₃
commutator, cosocle	C₂×C₄×C₂₃	C₂⁴
Frattini	C₂×C₄	C₂³×D₂₃
Fitting	C₂₃×#533	C₂
socle	C₂²×C₂₃	C₂²×D₄
prime radical	C₂×#10625045	C₂
hypercenter	#533	D₂₃
hypocenter	C₂₃	#3138
LCS	G ▷ C₂×C₄×C₂₃ ▷ C₂×C₂₃ ▷ C₂₃
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #533

subgroup

G/H

center

C₂²

C₂²×C₂²⋊D₂₃

commutator, cosocle

C₂×C₄×C₂₃

C₂⁴

Frattini

C₂×C₄

C₂³×D₂₃

Fitting

C₂₃×#533

C₂

socle

C₂²×C₂₃

C₂²×D₄

C₂

D₂₃

C₂₃

G ▷ C₂×C₄×C₂₃ ▷ C₂×C₂₃ ▷ C₂₃

UCS

C₁ ◁ C₂² ◁ C₂×C₄² ◁ #533

N	H ≅ G/N	type
2. #13092783	C₂	semidirect
3. #13092765	C₂	semidirect
4. #13092421	C₂	semidirect
5. C₄×#67267	C₂	semidirect
6. C₂₃×#533	C₂	semidirect
9. C₂₃×C₄⋊Q₈	C₂²	semidirect
10. C₂₃×#125	C₂²	semidirect
19. #10624991	C₂²	semidirect
20. #10624985	C₂²	semidirect
21. C₄×C₂₃⋊C₈	C₂²	semidirect
71. #533	D₂₃	semidirect
84. C₄⋊Q₈	C₂×D₂₃	semidirect
88. #125	C₂×D₂₃	semidirect
96. C₂₃	#3138	semidirect

H ≅ G/N

type

2. #13092783

C₂

semidirect

3. #13092765

C₂

semidirect

4. #13092421

C₂

semidirect

5. C₄×#67267

C₂

semidirect

6. C₂₃×#533

C₂

semidirect

9. C₂₃×C₄⋊Q₈

C₂²

semidirect

10. C₂₃×#125

C₂²

semidirect

19. #10624991

C₂²

semidirect

20. #10624985

C₂²

semidirect

21. C₄×C₂₃⋊C₈

C₂²

semidirect

71. #533

D₂₃

semidirect

84. C₄⋊Q₈

C₂×D₂₃

semidirect

88. #125

C₂×D₂₃

semidirect

96. C₂₃

#3138

semidirect

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

subgroup	index
C₂×#10625045	2
#13092783	2
#13092765	2
#13092421	2
C₄×#67267	2
C₂₃×#533	2
#3138	23

subgroup

index

C₂₃×#533

Subgroups — #15683773

Conjugacy classes of subgroups

Complements

Maximal subgroups