Group #15674543 — subgroups

Subgroups — #15674543

Order 2944.

294 conjugacy classes of subgroups; only the 85 normal subgroups are shown.

normal subgroups only

This group has 294 conjugacy classes of subgroups — too many to draw. Only its 85 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×#10625041	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×#10625014	C₂	1	2	yes	maximal
3	#13092179	C₂	1	2	yes	maximal, complemented
4	#13092172	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂₃×#390	C₂	1	1	yes	maximal, characteristic
6	C₂²×D₄×C₂₃	C₂²	1	1	yes	characteristic
7	C₂×C₄⋊C₄×C₂₃	C₂²	1	1	yes	characteristic
8	C₂×C₂²⋊C₄×C₂₃	—	2	2	no	—
9	C₂×C₄²×C₂₃	C₂²	1	1	yes	characteristic
10	C₂×C₄⋊Dic₂₃	C₂²	1	2	yes	—
11	C₂×Dic₂₃⋊C₄	—	2	2	no	—
12	C₂²×C₂₃⋊C₈	C₂²	1	2	yes	—
13	#10625041	—	2	4	no	—
14	#10625014	—	2	4	no	—
15	C₂⁴×C₂₃	—	2	1	no	—
16	C₂×D₄×C₂₃	—	2	2	no	—
17	C₂×D₄×C₂₃	—	2	2	no	—
18	C₂×D₄×C₂₃	—	2	1	no	—
19	C₂×D₄×C₂₃	Q₈	1	2	yes	—
20	C₂²×C₄×C₂₃	—	2	1	no	—
21	C₂²×C₄×C₂₃	D₄	1	2	yes	—
22	C₂²×C₄×C₂₃	C₂³	1	1	yes	characteristic
23	C₂²×Dic₂₃	—	2	2	no	—
24	C₄⋊C₄×C₂₃	—	2	2	no	—
25	C₂²⋊C₄×C₂₃	—	4	4	no	—
26	C₄²×C₂₃	—	2	2	no	—
27	C₄⋊Dic₂₃	—	2	2	no	—
28	C₄⋊Dic₂₃	D₄	1	4	yes	—
29	Dic₂₃⋊C₄	—	2	8	no	—
30	C₂×C₂₃⋊C₈	—	2	2	no	—
31	C₂×C₂₃⋊C₈	—	2	2	no	—
32	C₂×C₂₃⋊C₈	—	2	2	no	—
33	C₂³×C₂₃	—	2	2	no	—
34	C₂³×C₂₃	—	4	1	no	—
35	C₂³×C₂₃	—	4	1	no	—
36	C₂³×C₂₃	—	4	2	no	—
37	C₂³×C₂₃	—	4	2	no	—
38	C₂³×C₂₃	C₂×D₄	1	1	yes	characteristic
39	D₄×C₂₃	—	2	4	no	—
40	D₄×C₂₃	—	4	2	no	—
41	C₂×C₄×C₂₃	—	2	2	no	—
42	C₂×C₄×C₂₃	—	2	2	no	—
43	C₂×C₄×C₂₃	—	2	2	no	—
44	C₂×C₄×C₂₃	—	4	1	no	—
45	C₂×C₄×C₂₃	—	4	2	no	—
46	C₂×C₄×C₂₃	C₂×D₄	1	2	yes	—
47	C₂×C₄×C₂₃	C₂×Q₈	1	1	yes	characteristic
48	C₂×C₄×C₂₃	#41	1	1	yes	characteristic
49	C₂×C₄×C₂₃	#41	1	2	yes	—
50	C₂×Dic₂₃	—	2	4	no	—
51	C₂×Dic₂₃	—	4	2	no	—
52	C₂×Dic₂₃	—	4	2	no	—
53	C₂₃⋊C₈	—	4	2	no	—
54	#2025	—	23	1	no	maximal, Sylow
55	C₂²×C₂₃	—	4	2	no	—
56	C₂²×C₂₃	—	4	4	no	—
57	C₂²×C₂₃	—	8	2	no	—
58	C₂²×C₂₃	—	8	1	no	—
59	C₂²×C₂₃	—	8	1	no	—
60	C₂²×C₂₃	#120	1	1	yes	characteristic
61	C₂²×C₂₃	C₂²⋊Q₈	1	1	yes	characteristic
62	C₂²×C₂₃	#124	1	1	yes	characteristic
63	C₂²×C₂₃	#135	1	2	yes	—
64	C₂²×C₂₃	AGL(1,ℤ/8)	1	2	yes	—
65	C₄×C₂₃	—	4	2	no	—
66	C₄×C₂₃	—	8	1	no	—
67	C₄×C₂₃	C₂²⋊Q₈	1	2	yes	—
68	C₄×C₂₃	#124	1	2	yes	—
69	Dic₂₃	—	4	4	no	—
70	C₂×#125	—	23	1	no	—
71	C₂×#102	—	23	2	no	—
72	#390	D₂₃	1	1	yes	characteristic
73	#340	—	23	2	no	—
74	#336	—	23	1	no	—
75	C₂×C₂₃	—	8	2	no	—
76	C₂×C₂₃	#393	1	1	yes	characteristic
77	C₂×C₂₃	#449	1	2	yes	—
78	C₂×C₂₃	#478	1	2	yes	—
79	C₂×C₂₃	#488	1	1	yes	characteristic
80	C₂×C₂₃	#490	1	1	yes	characteristic
81	C₂²×D₄	C₂×D₂₃	1	1	yes	characteristic
82	C₂²×C₈	—	23	2	no	—
83	#125	—	46	4	no	—
84	C₂×C₄⋊C₄	—	23	2	no	—
85	C₂×C₄⋊C₄	—	46	2	no	—
86	C₂×C₄⋊C₄	C₂×D₂₃	1	1	yes	characteristic
87	C₂×C₂²⋊C₄	—	2	2	no	—
88	C₂×C₄²	C₂×D₂₃	1	1	yes	characteristic
89	#102	—	46	4	no	—
90	C₂₃	#2025	1	1	yes	characteristic, complemented, Sylow
91	C₂⁴	—	2	1	no	—
92	C₂×D₄	—	2	1	no	—
93	C₂×D₄	—	2	2	no	—
94	C₂×D₄	—	2	2	no	—
95	C₂×D₄	Dic₄₆	1	2	yes	—
96	C₂²×C₄	—	2	1	no	—
97	C₂²×C₄	—	46	2	no	—
98	C₂²×C₄	C₂²⋊D₂₃	1	2	yes	—
99	C₂²×C₄	C₂²×D₂₃	1	1	yes	characteristic
100	C₂×C₈	—	46	2	no	—
101	C₂×C₈	—	46	2	no	—
102	C₂×C₈	—	46	2	no	—
103	C₄⋊C₄	—	2	2	no	—
104	C₄⋊C₄	—	23	4	no	—
105	C₄⋊C₄	—	46	8	no	—
106	C₄⋊C₄	—	46	2	no	—
107	C₂²⋊C₄	—	4	4	no	—
108	C₄²	—	2	2	no	—
109	C₂³	—	2	2	no	—
110	C₂³	—	4	2	no	—
111	C₂³	—	4	1	no	—
112	C₂³	—	4	1	no	—
113	C₂³	—	4	2	no	—
114	C₂³	C₂×C₂²⋊D₂₃	1	1	yes	characteristic
115	D₄	—	2	4	no	—
116	D₄	—	4	2	no	—
117	C₂×C₄	—	2	2	no	—
118	C₂×C₄	—	2	2	no	—
119	C₂×C₄	—	2	2	no	—
120	C₂×C₄	—	4	2	no	—
121	C₂×C₄	—	4	1	no	—
122	C₂×C₄	—	46	4	no	—
123	C₂×C₄	—	92	2	no	—
124	C₂×C₄	—	92	2	no	—
125	C₂×C₄	C₂×Dic₄₆	1	1	yes	characteristic
126	C₂×C₄	#67288	1	1	yes	characteristic
127	C₂×C₄	D₄×D₂₃	1	2	yes	—
128	C₂×C₄	#67290	1	2	yes	—
129	C₈	—	92	2	no	—
130	C₂²	—	4	4	no	—
131	C₂²	—	4	2	no	—
132	C₂²	—	8	1	no	—
133	C₂²	—	8	1	no	—
134	C₂²	—	8	2	no	—
135	C₂²	#10625095	1	1	yes	characteristic
136	C₂²	#10625103	1	2	yes	—
137	C₂²	#10625107	1	1	yes	characteristic
138	C₂²	#10625108	1	1	yes	characteristic
139	C₂²	#10625121	1	2	yes	—
140	C₄	—	4	2	no	—
141	C₄	—	8	1	no	—
142	C₄	—	92	4	no	—
143	C₄	#10625049	1	2	yes	—
144	C₄	#10625056	1	2	yes	—
145	C₂	—	8	2	no	—
146	C₂	#13092329	1	1	yes	characteristic
147	C₂	#13092560	1	2	yes	—
148	C₂	#13092717	1	2	yes	—
149	C₂	#13092771	1	1	yes	characteristic
150	C₂	#13092781	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#10625041

C₂

yes

maximal, characteristic, complemented

C₂×#10625014

C₂

yes

maximal

#13092179

C₂

yes

maximal, complemented

#13092172

C₂

yes

maximal, characteristic, complemented

C₂₃×#390

C₂

yes

maximal, characteristic

C₂²×D₄×C₂₃

C₂²

yes

characteristic

C₂×C₄⋊C₄×C₂₃

C₂²

yes

characteristic

C₂×C₂²⋊C₄×C₂₃

—

C₂×C₄²×C₂₃

C₂²

yes

characteristic

C₂×C₄⋊Dic₂₃

C₂²

yes

—

C₂×Dic₂₃⋊C₄

—

C₂²×C₂₃⋊C₈

C₂²

yes

—

#10625041

—

#10625014

—

C₂⁴×C₂₃

—

C₂×D₄×C₂₃

—

C₂×D₄×C₂₃

—

C₂×D₄×C₂₃

—

C₂×D₄×C₂₃

Q₈

yes

—

C₂²×C₄×C₂₃

—

C₂²×C₄×C₂₃

D₄

yes

—

C₂²×C₄×C₂₃

C₂³

yes

characteristic

C₂²×Dic₂₃

—

C₄⋊C₄×C₂₃

—

C₂²⋊C₄×C₂₃

—

C₄²×C₂₃

—

C₄⋊Dic₂₃

—

C₄⋊Dic₂₃

D₄

yes

—

Dic₂₃⋊C₄

—

C₂×C₂₃⋊C₈

—

C₂×C₂₃⋊C₈

—

C₂×C₂₃⋊C₈

—

C₂³×C₂₃

—

C₂³×C₂₃

—

C₂³×C₂₃

—

C₂³×C₂₃

—

C₂³×C₂₃

—

C₂³×C₂₃

C₂×D₄

yes

characteristic

D₄×C₂₃

—

D₄×C₂₃

—

C₂×C₄×C₂₃

—

C₂×C₄×C₂₃

—

C₂×C₄×C₂₃

—

C₂×C₄×C₂₃

—

C₂×C₄×C₂₃

—

C₂×C₄×C₂₃

C₂×D₄

yes

—

C₂×C₄×C₂₃

C₂×Q₈

yes

C₂×C₄×C₂₃

yes

C₂×C₄×C₂₃

yes

—

C₂×Dic₂₃

—

C₂×Dic₂₃

—

C₂×Dic₂₃

—

C₂₃⋊C₈

—

#2025

—

maximal, Sylow

C₂²×C₂₃

—

C₂²×C₂₃

—

C₂²×C₂₃

—

C₂²×C₂₃

—

C₂²×C₂₃

—

C₂²×C₂₃

#120

yes

characteristic

C₂²×C₂₃

C₂²⋊Q₈

yes

C₂²×C₂₃

yes

C₂²×C₂₃

yes

—

C₂²×C₂₃

AGL(1,ℤ/8)

yes

—

C₄×C₂₃

—

C₄×C₂₃

—

C₄×C₂₃

C₂²⋊Q₈

yes

—

C₄×C₂₃

#124

yes

—

Dic₂₃

—

C₂×#125

—

C₂×#102

—

#390

D₂₃

yes

characteristic

#340

—

#336

—

C₂×C₂₃

—

C₂×C₂₃

#393

yes

characteristic

C₂×C₂₃

#449

yes

—

C₂×C₂₃

#478

yes

—

C₂×C₂₃

yes

C₂×C₂₃

yes

C₂²×D₄

C₂×D₂₃

yes

characteristic

C₂²×C₈

—

#125

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

C₂×D₂₃

yes

characteristic

C₂×C₂²⋊C₄

—

C₂×C₄²

C₂×D₂₃

yes

characteristic

#102

—

C₂₃

#2025

yes

characteristic, complemented, Sylow

C₂⁴

—

C₂×D₄

—

C₂×D₄

—

C₂×D₄

—

C₂×D₄

Dic₄₆

yes

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

C₂²⋊D₂₃

yes

—

C₂²×C₄

C₂²×D₂₃

yes

characteristic

100

C₂×C₈

—

101

C₂×C₈

—

102

C₂×C₈

—

103

C₄⋊C₄

—

104

C₄⋊C₄

—

105

C₄⋊C₄

—

106

C₄⋊C₄

—

107

C₂²⋊C₄

—

108

C₄²

—

109

C₂³

—

110

C₂³

—

111

C₂³

—

112

C₂³

—

113

C₂³

—

114

C₂³

C₂×C₂²⋊D₂₃

yes

characteristic

115

D₄

—

116

D₄

—

117

C₂×C₄

—

118

C₂×C₄

—

119

C₂×C₄

—

120

C₂×C₄

—

121

C₂×C₄

—

122

C₂×C₄

—

123

C₂×C₄

—

124

C₂×C₄

—

125

C₂×C₄

C₂×Dic₄₆

yes

characteristic

126

C₂×C₄

#67288

yes

characteristic

127

C₂×C₄

D₄×D₂₃

yes

—

128

C₂×C₄

#67290

yes

—

129

C₈

—

130

C₂²

—

131

C₂²

—

132

C₂²

—

133

C₂²

—

134

C₂²

—

135

C₂²

#10625095

yes

characteristic

136

C₂²

#10625103

yes

—

137

C₂²

#10625107

yes

characteristic

138

C₂²

#10625108

yes

characteristic

139

C₂²

#10625121

yes

—

140

C₄

—

141

C₄

—

142

C₄

—

143

C₄

#10625049

yes

—

144

C₄

#10625056

yes

—

145

C₂

—

146

C₂

#13092329

yes

characteristic

147

C₂

#13092560

yes

—

148

C₂

#13092717

yes

—

149

C₂

#13092771

yes

characteristic

150

C₂

#13092781

yes

characteristic

	subgroup	G/H
center	C₂³	C₂×C₂²⋊D₂₃
commutator, cosocle	C₂²×C₄×C₂₃	C₂³
Frattini	C₂²×C₄	C₂²×D₂₃
Fitting	C₂₃×#390	C₂
socle	C₂³×C₂₃	C₂×D₄
prime radical	C₂²×D₄×C₂₃	C₂²
hypercenter	#390	D₂₃
hypocenter	C₂₃	#2025
LCS	G ▷ C₂²×C₄×C₂₃ ▷ C₂×C₂₃ ▷ C₂₃
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #390

subgroup

G/H

center

C₂³

C₂×C₂²⋊D₂₃

commutator, cosocle

C₂²×C₄×C₂₃

C₂³

Frattini

C₂²×C₄

C₂²×D₂₃

Fitting

C₂₃×#390

C₂

socle

C₂³×C₂₃

C₂×D₄

prime radical

C₂²×D₄×C₂₃

C₂²

D₂₃

C₂₃

G ▷ C₂²×C₄×C₂₃ ▷ C₂×C₂₃ ▷ C₂₃

UCS

C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #390

N	H ≅ G/N	type
1. C₂×#10625041	C₂	semidirect
3. #13092179	C₂	semidirect
4. #13092172	C₂	semidirect
90. C₂₃	#2025	semidirect

H ≅ G/N

type

1. C₂×#10625041

C₂

semidirect

3. #13092179

C₂

semidirect

4. #13092172

C₂

semidirect

90. C₂₃

#2025

semidirect

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

subgroup	index
C₂×#10625041	2
C₂×#10625014	2
#13092179	2
#13092172	2
C₂₃×#390	2
#2025	23

subgroup

index

C₂₃×#390

Subgroups — #15674543

Conjugacy classes of subgroups

Complements

Maximal subgroups