Group #15674357 — subgroups

Subgroups — #15674357

Order 2944.

300 conjugacy classes of subgroups; only the 137 normal subgroups are shown.

normal subgroups only

This group has 300 conjugacy classes of subgroups — too many to draw. Only its 137 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×#10625040	C₂	1	1	yes	maximal, characteristic
2	C₂×C₄×C₂₃⋊C₈	C₂	1	1	yes	maximal, characteristic
3	#13092179	C₂	1	4	yes	maximal
4	C₂₃×#384	C₂	1	1	yes	maximal, characteristic
5	C₂²×Dic₄₆	—	2	1	no	—
6	C₂×C₄⋊C₄×C₂₃	—	2	1	no	—
7	C₂×C₄⋊C₄×C₂₃	—	2	1	no	—
8	C₂×C₄⋊C₄×C₂₃	C₂²	1	2	yes	—
9	C₂×C₄²×C₂₃	C₂²	1	1	yes	characteristic
10	C₂×C₄⋊Dic₂₃	—	2	1	no	—
11	C₂×C₄⋊Dic₂₃	C₂²	1	2	yes	—
12	C₂²×C₂₃⋊C₈	C₂²	1	2	yes	—
13	#10625040	—	2	2	no	—
14	#10625040	C₄	1	4	yes	complemented
15	C₄×C₂₃⋊C₈	—	2	2	no	—
16	C₂²×C₄×C₂₃	—	2	2	no	—
17	C₂²×C₄×C₂₃	D₄	1	1	yes	characteristic
18	C₂²×C₄×C₂₃	D₄	1	1	yes	characteristic
19	C₂²×C₄×C₂₃	C₂³	1	1	yes	characteristic
20	C₂²×Dic₂₃	—	2	2	no	—
21	C₂×Dic₄₆	—	2	4	no	—
22	C₂×Dic₄₆	—	4	2	no	—
23	C₂×Dic₄₆	—	4	2	no	—
24	C₄⋊C₄×C₂₃	—	2	4	no	—
25	C₄⋊C₄×C₂₃	—	4	2	no	—
26	C₄⋊C₄×C₂₃	—	4	2	no	—
27	C₄²×C₂₃	C₂×C₄	1	2	yes	—
28	C₄²×C₂₃	D₄	1	2	yes	—
29	C₄⋊Dic₂₃	—	2	4	no	—
30	C₄⋊Dic₂₃	—	2	2	no	—
31	C₄⋊Dic₂₃	C₂×C₄	1	4	yes	—
32	C₂×C₂₃⋊C₈	—	2	2	no	—
33	C₂×C₂₃⋊C₈	—	2	2	no	—
34	C₂×C₂₃⋊C₈	Q₈	1	4	yes	complemented
35	C₂³×C₂₃	C₂×D₄	1	1	yes	characteristic
36	C₂×C₄×C₂₃	—	4	2	no	—
37	C₂×C₄×C₂₃	—	4	2	no	—
38	C₂×C₄×C₂₃	—	4	2	no	—
39	C₂×C₄×C₂₃	C₂²⋊C₄	1	2	yes	—
40	C₂×C₄×C₂₃	C₂²⋊C₄	1	2	yes	—
41	C₂×C₄×C₂₃	D₈	1	2	yes	—
42	C₂×C₄×C₂₃	SD₁₆	1	2	yes	—
43	C₂×C₄×C₂₃	SD₁₆	1	2	yes	—
44	C₂×C₄×C₂₃	Q₁₆	1	2	yes	—
45	C₂×C₄×C₂₃	C₂²×C₄	1	1	yes	characteristic
46	C₂×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
47	C₂×C₄×C₂₃	C₂×Q₈	1	2	yes	—
48	C₂×C₄×C₂₃	#41	1	2	yes	—
49	C₂×Dic₂₃	—	2	4	no	—
50	C₂×Dic₂₃	—	4	2	no	—
51	C₂×Dic₂₃	—	4	2	no	—
52	Dic₄₆	—	4	8	no	—
53	C₂₃⋊C₈	—	2	4	no	—
54	#1990	—	23	1	no	maximal, Sylow
55	C₂²×C₂₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
56	C₂²×C₂₃	#124	1	1	yes	characteristic
57	C₂²×C₂₃	C₄⋊Q₈	1	1	yes	characteristic
58	C₂²×C₂₃	C₂×D₈	1	1	yes	characteristic
59	C₂²×C₂₃	C₂×SD₁₆	1	1	yes	characteristic
60	C₂²×C₂₃	C₂×SD₁₆	1	1	yes	characteristic
61	C₂²×C₂₃	C₂×Q₁₆	1	1	yes	characteristic
62	C₄×C₂₃	—	8	2	no	—
63	C₄×C₂₃	#102	1	4	yes	—
64	C₄×C₂₃	#103	1	4	yes	—
65	C₄×C₂₃	C₄×Q₈	1	2	yes	—
66	C₄×C₂₃	C₂²⋊Q₈	1	2	yes	—
67	Dic₂₃	—	4	4	no	—
68	C₂×C₄⋊Q₈	—	23	1	no	—
69	C₂×C₄×C₈	—	23	1	no	—
70	#384	D₂₃	1	1	yes	characteristic
71	#340	—	23	4	no	—
72	C₂×C₂₃	#391	1	1	yes	characteristic
73	C₂×C₂₃	C₂×#102	1	1	yes	characteristic
74	C₂×C₂₃	C₂×#103	1	1	yes	characteristic
75	C₂×C₂₃	#486	1	1	yes	characteristic
76	C₂×C₂₃	#487	1	1	yes	characteristic
77	C₂×C₂₃	#498	1	1	yes	characteristic
78	C₂×C₂₃	#500	1	1	yes	characteristic
79	C₂²×Q₈	—	46	1	no	—
80	C₂²×C₈	—	23	2	no	—
81	C₄⋊Q₈	—	23	4	no	—
82	C₄⋊Q₈	—	46	2	no	—
83	C₂×C₄⋊C₄	—	2	1	no	—
84	C₂×C₄⋊C₄	—	2	1	no	—
85	C₂×C₄⋊C₄	—	23	2	no	—
86	C₂×C₄⋊C₄	—	46	1	no	—
87	C₂×C₄⋊C₄	C₂×D₂₃	1	2	yes	—
88	C₂×C₄²	C₂×D₂₃	1	1	yes	characteristic
89	C₄×C₈	—	46	2	no	—
90	C₂₃	#1990	1	1	yes	characteristic, complemented, Sylow
91	C₂×Q₈	—	46	4	no	—
92	C₂×Q₈	—	92	2	no	—
93	C₂×Q₈	—	92	2	no	—
94	C₂²×C₄	—	2	2	no	—
95	C₂²×C₄	—	46	2	no	—
96	C₂²×C₄	C₂²⋊D₂₃	1	1	yes	characteristic
97	C₂²×C₄	C₂²⋊D₂₃	1	1	yes	characteristic
98	C₂²×C₄	C₂²×D₂₃	1	1	yes	characteristic
99	C₂×C₈	—	23	4	no	—
100	C₂×C₈	—	46	2	no	—
101	C₂×C₈	—	46	2	no	—
102	C₄⋊C₄	—	2	4	no	—
103	C₄⋊C₄	—	4	2	no	—
104	C₄⋊C₄	—	4	2	no	—
105	C₄⋊C₄	—	23	4	no	—
106	C₄⋊C₄	—	46	4	no	—
107	C₄⋊C₄	—	46	2	no	—
108	C₄²	C₄×D₂₃	1	2	yes	—
109	C₄²	D₉₂	1	2	yes	—
110	C₂³	C₂×C₂²⋊D₂₃	1	1	yes	characteristic
111	Q₈	—	92	8	no	—
112	C₂×C₄	—	4	2	no	—
113	C₂×C₄	—	4	2	no	—
114	C₂×C₄	—	4	2	no	—
115	C₂×C₄	—	46	4	no	—
116	C₂×C₄	—	92	2	no	—
117	C₂×C₄	—	92	2	no	—
118	C₂×C₄	(C₂×D₂₃)⋊C₄	1	2	yes	—
119	C₂×C₄	(C₂×D₂₃)⋊C₄	1	2	yes	—
120	C₂×C₄	#67272	1	2	yes	—
121	C₂×C₄	#67273	1	2	yes	—
122	C₂×C₄	#67274	1	2	yes	—
123	C₂×C₄	#67275	1	2	yes	—
124	C₂×C₄	C₂×C₄×D₂₃	1	1	yes	characteristic
125	C₂×C₄	C₂×D₉₂	1	1	yes	characteristic
126	C₂×C₄	#67290	1	2	yes	—
127	C₂×C₄	Q₈×D₂₃	1	2	yes	—
128	C₈	—	46	4	no	—
129	C₂²	C₂×(C₂×D₂₃)⋊C₄	1	1	yes	characteristic
130	C₂²	C₂×#67272	1	1	yes	characteristic
131	C₂²	C₂×#67273	1	1	yes	characteristic
132	C₂²	#10625107	1	1	yes	characteristic
133	C₂²	C₂×#67274	1	1	yes	characteristic
134	C₂²	C₂×#67275	1	1	yes	characteristic
135	C₂²	#10625115	1	1	yes	characteristic
136	C₄	—	8	2	no	—
137	C₄	—	92	4	no	—
138	C₄	#10624991	1	4	yes	—
139	C₄	#10624992	1	4	yes	—
140	C₄	#10625058	1	2	yes	—
141	C₄	#10625068	1	2	yes	—
142	C₂	#13092322	1	1	yes	characteristic
143	C₂	C₂×#10624991	1	1	yes	characteristic
144	C₂	C₂×#10624992	1	1	yes	characteristic
145	C₂	#13092763	1	1	yes	characteristic
146	C₂	#13092767	1	1	yes	characteristic
147	C₂	#13092813	1	1	yes	characteristic
148	C₂	#13092820	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#10625040

C₂

yes

maximal, characteristic

C₂×C₄×C₂₃⋊C₈

C₂

yes

maximal, characteristic

#13092179

C₂

yes

maximal

C₂₃×#384

C₂

yes

maximal, characteristic

C₂²×Dic₄₆

—

C₂×C₄⋊C₄×C₂₃

—

C₂×C₄⋊C₄×C₂₃

—

C₂×C₄⋊C₄×C₂₃

C₂²

yes

—

C₂×C₄²×C₂₃

C₂²

yes

characteristic

C₂×C₄⋊Dic₂₃

—

C₂×C₄⋊Dic₂₃

C₂²

yes

—

C₂²×C₂₃⋊C₈

C₂²

yes

—

#10625040

—

#10625040

C₄

yes

complemented

C₄×C₂₃⋊C₈

—

C₂²×C₄×C₂₃

—

C₂²×C₄×C₂₃

D₄

yes

characteristic

C₂²×C₄×C₂₃

D₄

yes

characteristic

C₂²×C₄×C₂₃

C₂³

yes

characteristic

C₂²×Dic₂₃

—

C₂×Dic₄₆

—

C₂×Dic₄₆

—

C₂×Dic₄₆

—

C₄⋊C₄×C₂₃

—

C₄⋊C₄×C₂₃

—

C₄⋊C₄×C₂₃

—

C₄²×C₂₃

C₂×C₄

yes

—

C₄²×C₂₃

D₄

yes

—

C₄⋊Dic₂₃

—

C₄⋊Dic₂₃

—

C₄⋊Dic₂₃

C₂×C₄

yes

—

C₂×C₂₃⋊C₈

—

C₂×C₂₃⋊C₈

—

C₂×C₂₃⋊C₈

Q₈

yes

complemented

C₂³×C₂₃

C₂×D₄

yes

characteristic

C₂×C₄×C₂₃

—

C₂×C₄×C₂₃

—

C₂×C₄×C₂₃

—

C₂×C₄×C₂₃

C₂²⋊C₄

yes

—

C₂×C₄×C₂₃

C₂²⋊C₄

yes

—

C₂×C₄×C₂₃

D₈

yes

—

C₂×C₄×C₂₃

SD₁₆

yes

—

C₂×C₄×C₂₃

SD₁₆

yes

—

C₂×C₄×C₂₃

Q₁₆

yes

—

C₂×C₄×C₂₃

C₂²×C₄

yes

characteristic

C₂×C₄×C₂₃

C₂×D₄

yes

characteristic

C₂×C₄×C₂₃

C₂×Q₈

yes

—

C₂×C₄×C₂₃

#41

yes

—

C₂×Dic₂₃

—

C₂×Dic₂₃

—

C₂×Dic₂₃

—

Dic₄₆

—

C₂₃⋊C₈

—

#1990

—

maximal, Sylow

C₂²×C₂₃

C₂×C₂²⋊C₄

yes

characteristic

C₂²×C₂₃

#124

yes

characteristic

C₂²×C₂₃

C₄⋊Q₈

yes

characteristic

C₂²×C₂₃

C₂×D₈

yes

characteristic

C₂²×C₂₃

C₂×SD₁₆

yes

characteristic

C₂²×C₂₃

C₂×SD₁₆

yes

characteristic

C₂²×C₂₃

C₂×Q₁₆

yes

characteristic

C₄×C₂₃

—

C₄×C₂₃

#102

yes

—

C₄×C₂₃

#103

yes

—

C₄×C₂₃

C₄×Q₈

yes

—

C₄×C₂₃

C₂²⋊Q₈

yes

—

Dic₂₃

—

C₂×C₄⋊Q₈

—

C₂×C₄×C₈

—

#384

D₂₃

yes

characteristic

#340

—

C₂×C₂₃

yes

C₂×C₂₃

yes

C₂×C₂₃

yes

C₂×C₂₃

yes

C₂×C₂₃

yes

C₂×C₂₃

yes

C₂×C₂₃

yes

C₂²×Q₈

—

C₂²×C₈

—

C₄⋊Q₈

—

C₄⋊Q₈

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

C₂×D₂₃

yes

—

C₂×C₄²

C₂×D₂₃

yes

characteristic

C₄×C₈

—

C₂₃

#1990

yes

characteristic, complemented, Sylow

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

C₂²⋊D₂₃

yes

characteristic

C₂²×C₄

C₂²⋊D₂₃

yes

characteristic

C₂²×C₄

C₂²×D₂₃

yes

characteristic

C₂×C₈

—

100

C₂×C₈

—

101

C₂×C₈

—

102

C₄⋊C₄

—

103

C₄⋊C₄

—

104

C₄⋊C₄

—

105

C₄⋊C₄

—

106

C₄⋊C₄

—

107

C₄⋊C₄

—

108

C₄²

C₄×D₂₃

yes

—

109

C₄²

D₉₂

yes

—

110

C₂³

C₂×C₂²⋊D₂₃

yes

characteristic

111

Q₈

—

112

C₂×C₄

—

113

C₂×C₄

—

114

C₂×C₄

—

115

C₂×C₄

—

116

C₂×C₄

—

117

C₂×C₄

—

118

C₂×C₄

(C₂×D₂₃)⋊C₄

yes

—

119

C₂×C₄

(C₂×D₂₃)⋊C₄

yes

—

120

C₂×C₄

#67272

yes

—

121

C₂×C₄

#67273

yes

—

122

C₂×C₄

#67274

yes

—

123

C₂×C₄

#67275

yes

—

124

C₂×C₄

C₂×C₄×D₂₃

yes

characteristic

125

C₂×C₄

C₂×D₉₂

yes

characteristic

126

C₂×C₄

#67290

yes

—

127

C₂×C₄

Q₈×D₂₃

yes

—

128

C₈

—

129

C₂²

C₂×(C₂×D₂₃)⋊C₄

yes

characteristic

130

C₂²

C₂×#67272

yes

characteristic

131

C₂²

C₂×#67273

yes

characteristic

132

C₂²

#10625107

yes

characteristic

133

C₂²

C₂×#67274

yes

characteristic

134

C₂²

C₂×#67275

yes

characteristic

135

C₂²

#10625115

yes

characteristic

136

C₄

—

137

C₄

—

138

C₄

#10624991

yes

—

139

C₄

#10624992

yes

—

140

C₄

#10625058

yes

—

141

C₄

#10625068

yes

—

142

C₂

#13092322

yes

characteristic

143

C₂

C₂×#10624991

yes

characteristic

144

C₂

C₂×#10624992

yes

characteristic

145

C₂

#13092763

yes

characteristic

146

C₂

#13092767

yes

characteristic

147

C₂

#13092813

yes

characteristic

148

C₂

#13092820

yes

characteristic

	subgroup	G/H
center	C₂³	C₂×C₂²⋊D₂₃
commutator	C₂×C₄×C₂₃	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₂₃
Fitting	C₂₃×#384	C₂
socle, prime radical	C₂³×C₂₃	C₂×D₄
cosocle	C₂²×C₄×C₂₃	C₂³
hypercenter	#384	D₂₃
hypocenter	C₂₃	#1990
LCS	G ▷ C₂×C₄×C₂₃ ▷ C₂×C₂₃ ▷ C₂₃
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #384

subgroup

G/H

center

C₂³

C₂×C₂²⋊D₂₃

commutator

C₂×C₄×C₂₃

C₂²×C₄

Frattini

C₂²×C₄

C₂²×D₂₃

Fitting

C₂₃×#384

C₂

socle, prime radical

C₂³×C₂₃

C₂×D₄

cosocle

C₂²×C₄×C₂₃

C₂³

D₂₃

C₂₃

G ▷ C₂×C₄×C₂₃ ▷ C₂×C₂₃ ▷ C₂₃

UCS

C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #384

N	H ≅ G/N	type
14. #10625040	C₄	semidirect
34. C₂×C₂₃⋊C₈	Q₈	semidirect
90. C₂₃	#1990	semidirect

H ≅ G/N

type

14. #10625040

C₄

semidirect

34. C₂×C₂₃⋊C₈

Q₈

semidirect

90. C₂₃

#1990

semidirect

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

subgroup	index
C₂×#10625040	2
C₂×C₄×C₂₃⋊C₈	2
#13092179	2
C₂₃×#384	2
#1990	23

subgroup

index

C₂×#10625040

C₂×C₄×C₂₃⋊C₈

#13092179

C₂₃×#384

#1990

Subgroups — #15674357

Conjugacy classes of subgroups

Complements

Maximal subgroups