Group #15671801 — subgroups

Subgroups — #15671801

Order 2944.

216 conjugacy classes of subgroups; only the 89 normal subgroups are shown.

normal subgroups only

This group has 216 conjugacy classes of subgroups — too many to draw. Only its 89 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	#13092974	C₂	1	1	yes	maximal, characteristic
2	C₂×C₄⋊(C₂₃⋊C₈)	C₂	1	1	yes	maximal, characteristic
3	#13092149	C₂	1	4	yes	maximal, complemented
4	C₂₃×#431	C₂	1	1	yes	maximal, characteristic
5	C₂×C₄²×C₂₃	C₂²	1	1	yes	characteristic
6	#10625097	—	2	1	no	—
7	#10625095	—	2	1	no	—
8	#10625095	C₄	1	2	yes	—
9	C₂²×C₂₃⋊C₈	—	2	1	no	—
10	#10625041	—	2	1	no	—
11	#10625040	C₄	1	1	yes	characteristic
12	#10625040	C₄	1	1	yes	characteristic
13	C₄×Dic₄₆	—	2	1	no	—
14	C₄×Dic₄₆	C₂²	1	2	yes	—
15	C₂₃×C₄⋊C₈	—	2	1	no	—
16	C₂₃×C₂²⋊C₈	—	2	1	no	—
17	C₄×C₈×C₂₃	C₂²	1	2	yes	—
18	C₄⋊(C₂₃⋊C₈)	—	2	1	no	—
19	C₄⋊(C₂₃⋊C₈)	C₂²	1	2	yes	—
20	C₂²×C₄×C₂₃	C₂×C₄	1	1	yes	characteristic
21	C₂²×C₄×C₂₃	D₄	1	1	yes	characteristic
22	C₂²×C₄×C₂₃	D₄	1	1	yes	characteristic
23	C₂×Dic₄₆	—	2	1	no	—
24	C₂×Dic₄₆	C₂×C₄	1	2	yes	—
25	C₂×C₈×C₂₃	—	2	2	no	—
26	C₄²×C₂₃	C₂×C₄	1	1	yes	characteristic
27	C₄²×C₂₃	D₄	1	1	yes	characteristic
28	C₄²×C₂₃	D₄	1	1	yes	characteristic
29	C₄²×C₂₃	C₂³	1	1	yes	characteristic
30	C₂²⋊Dic₂₃	—	2	2	no	—
31	C₄⋊Dic₂₃	—	2	1	no	—
32	C₄⋊Dic₂₃	—	2	1	no	—
33	C₄⋊Dic₂₃	—	2	1	no	—
34	C₄⋊Dic₂₃	C₂×C₄	1	2	yes	—
35	Dic₂₃⋊C₄	—	2	2	no	—
36	Dic₂₃⋊C₄	—	2	2	no	—
37	C₄×Dic₂₃	—	2	2	no	—
38	C₂×C₂₃⋊C₈	—	2	2	no	—
39	C₂×C₂₃⋊C₈	—	4	2	no	—
40	C₂³×C₂₃	C₂²⋊C₄	1	1	yes	characteristic
41	C₂×C₄×C₂₃	—	2	2	no	—
42	C₂×C₄×C₂₃	—	4	1	no	—
43	C₂×C₄×C₂₃	C₂²⋊C₄	1	1	yes	characteristic
44	C₂×C₄×C₂₃	C₂²⋊C₄	1	1	yes	characteristic
45	C₂×C₄×C₂₃	C₂²⋊C₄	1	1	yes	characteristic
46	C₂×C₄×C₂₃	SD₁₆	1	2	yes	—
47	C₂×C₄×C₂₃	Q₁₆	1	2	yes	—
48	C₂×C₄×C₂₃	C₂²×C₄	1	1	yes	characteristic
49	C₂×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
50	C₂×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
51	C₂×Dic₂₃	—	2	2	no	—
52	C₂×Dic₂₃	—	2	2	no	—
53	Dic₄₆	—	2	4	no	—
54	Dic₄₆	—	4	2	no	—
55	C₈×C₂₃	—	4	2	no	—
56	C₂₃⋊C₈	—	4	2	no	—
57	#1481	—	23	1	no	maximal, Sylow
58	C₂²×C₂₃	—	2	1	no	—
59	C₂²×C₂₃	—	2	1	no	—
60	C₂²×C₂₃	#103	1	2	yes	—
61	C₂²×C₂₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
62	C₄×C₂₃	—	2	1	no	—
63	C₄×C₂₃	—	2	1	no	—
64	C₄×C₂₃	—	2	1	no	—
65	C₄×C₂₃	#103	1	2	yes	—
66	C₄×C₂₃	#131	1	2	yes	—
67	C₄×C₂₃	C₂×SD₁₆	1	1	yes	characteristic
68	C₄×C₂₃	C₂×Q₁₆	1	1	yes	characteristic
69	Dic₂₃	—	4	2	no	—
70	Dic₂₃	—	4	2	no	—
71	#533	—	23	1	no	—
72	#431	D₂₃	1	1	yes	characteristic
73	C₂×C₄⋊C₈	—	23	1	no	—
74	Q₈⋊C₈	—	23	4	no	—
75	C₂×C₂₃	—	2	2	no	—
76	C₂×C₂₃	#408	1	1	yes	characteristic
77	C₂×C₂₃	C₂×#103	1	1	yes	characteristic
78	C₂×C₂₃	#421	1	1	yes	characteristic
79	C₂²×C₈	—	46	1	no	—
80	C₄⋊Q₈	—	23	1	no	—
81	C₄⋊Q₈	—	23	1	no	—
82	#125	—	46	1	no	—
83	C₂²⋊Q₈	—	23	2	no	—
84	C₂²⋊Q₈	—	46	1	no	—
85	C₄×Q₈	—	23	2	no	—
86	C₄×Q₈	—	46	1	no	—
87	#117	—	46	1	no	—
88	C₂×C₄²	C₂×D₂₃	1	1	yes	characteristic
89	C₄⋊C₈	—	2	1	no	—
90	C₄⋊C₈	—	23	2	no	—
91	C₄⋊C₈	—	46	1	no	—
92	C₂²⋊C₈	—	2	1	no	—
93	C₄×C₈	C₂×D₂₃	1	2	yes	—
94	C₂₃	#1481	1	1	yes	characteristic, complemented, Sylow
95	C₂×Q₈	—	23	2	no	—
96	C₂×Q₈	—	46	1	no	—
97	C₂²×C₄	C₄×D₂₃	1	1	yes	characteristic
98	C₂²×C₄	D₉₂	1	1	yes	characteristic
99	C₂²×C₄	C₂²⋊D₂₃	1	1	yes	characteristic
100	C₂×C₈	—	2	2	no	—
101	C₂×C₈	—	46	2	no	—
102	C₂×C₈	—	92	2	no	—
103	C₄⋊C₄	—	23	2	no	—
104	C₄⋊C₄	—	46	1	no	—
105	C₄⋊C₄	—	46	1	no	—
106	C₄⋊C₄	—	46	2	no	—
107	C₄⋊C₄	—	46	1	no	—
108	C₄⋊C₄	—	46	2	no	—
109	C₂²⋊C₄	—	46	2	no	—
110	C₄²	—	46	2	no	—
111	C₄²	C₄×D₂₃	1	1	yes	characteristic
112	C₄²	D₉₂	1	1	yes	characteristic
113	C₄²	C₂²⋊D₂₃	1	1	yes	characteristic
114	C₄²	C₂²×D₂₃	1	1	yes	characteristic
115	C₂³	(C₂×D₂₃)⋊C₄	1	1	yes	characteristic
116	Q₈	—	46	4	no	—
117	Q₈	—	92	2	no	—
118	C₂×C₄	—	2	2	no	—
119	C₂×C₄	—	4	1	no	—
120	C₂×C₄	—	46	2	no	—
121	C₂×C₄	—	46	2	no	—
122	C₂×C₄	#67263	1	2	yes	—
123	C₂×C₄	Dic₉₂	1	2	yes	—
124	C₂×C₄	(C₂×D₂₃)⋊C₄	1	1	yes	characteristic
125	C₂×C₄	(C₂×D₂₃)⋊C₄	1	1	yes	characteristic
126	C₂×C₄	(C₂×D₂₃)⋊C₄	1	1	yes	characteristic
127	C₂×C₄	C₂×C₄×D₂₃	1	1	yes	characteristic
128	C₂×C₄	C₂×D₉₂	1	1	yes	characteristic
129	C₂×C₄	C₂×C₂²⋊D₂₃	1	1	yes	characteristic
130	C₈	—	4	2	no	—
131	C₈	—	92	2	no	—
132	C₂²	—	2	1	no	—
133	C₂²	—	2	1	no	—
134	C₂²	#10624998	1	2	yes	—
135	C₂²	C₂×(C₂×D₂₃)⋊C₄	1	1	yes	characteristic
136	C₄	—	2	1	no	—
137	C₄	—	2	1	no	—
138	C₄	—	2	1	no	—
139	C₄	—	92	2	no	—
140	C₄	—	92	2	no	—
141	C₄	#10624998	1	2	yes	—
142	C₄	C₂×#67263	1	1	yes	characteristic
143	C₄	C₂×Dic₉₂	1	1	yes	characteristic
144	C₄	#10625078	1	2	yes	—
145	C₂	—	2	2	no	—
146	C₂	#13092385	1	1	yes	characteristic
147	C₂	C₂×#10624998	1	1	yes	characteristic
148	C₂	#13092435	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#13092974

C₂

yes

maximal, characteristic

C₂×C₄⋊(C₂₃⋊C₈)

C₂

yes

maximal, characteristic

#13092149

C₂

yes

maximal, complemented

C₂₃×#431

C₂

yes

maximal, characteristic

C₂×C₄²×C₂₃

C₂²

yes

—

—

C₄

yes

—

C₂²×C₂₃⋊C₈

—

#10625041

—

#10625040

C₄

yes

characteristic

#10625040

C₄

yes

characteristic

C₄×Dic₄₆

—

C₄×Dic₄₆

C₂²

yes

—

C₂₃×C₄⋊C₈

—

C₂₃×C₂²⋊C₈

—

C₄×C₈×C₂₃

C₂²

yes

—

C₄⋊(C₂₃⋊C₈)

—

C₄⋊(C₂₃⋊C₈)

C₂²

yes

—

C₂²×C₄×C₂₃

C₂×C₄

yes

characteristic

C₂²×C₄×C₂₃

D₄

yes

characteristic

C₂²×C₄×C₂₃

D₄

yes

characteristic

C₂×Dic₄₆

—

C₂×Dic₄₆

C₂×C₄

yes

—

C₂×C₈×C₂₃

—

C₄²×C₂₃

C₂×C₄

yes

characteristic

C₄²×C₂₃

D₄

yes

characteristic

C₄²×C₂₃

D₄

yes

characteristic

C₄²×C₂₃

C₂³

yes

characteristic

C₂²⋊Dic₂₃

—

C₄⋊Dic₂₃

—

C₄⋊Dic₂₃

—

C₄⋊Dic₂₃

—

C₄⋊Dic₂₃

C₂×C₄

yes

—

Dic₂₃⋊C₄

—

Dic₂₃⋊C₄

—

C₄×Dic₂₃

—

C₂×C₂₃⋊C₈

—

C₂×C₂₃⋊C₈

—

C₂³×C₂₃

C₂²⋊C₄

yes

characteristic

C₂×C₄×C₂₃

—

C₂×C₄×C₂₃

—

C₂×C₄×C₂₃

C₂²⋊C₄

yes

characteristic

C₂×C₄×C₂₃

C₂²⋊C₄

yes

characteristic

C₂×C₄×C₂₃

C₂²⋊C₄

yes

characteristic

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×D₄

yes

characteristic

C₂×Dic₂₃

—

C₂×Dic₂₃

—

Dic₄₆

—

Dic₄₆

—

C₈×C₂₃

—

C₂₃⋊C₈

—

#1481

—

maximal, Sylow

C₂²×C₂₃

—

C₂²×C₂₃

—

C₂²×C₂₃

#103

yes

—

C₂²×C₂₃

C₂×C₂²⋊C₄

yes

characteristic

C₄×C₂₃

—

C₄×C₂₃

—

C₄×C₂₃

—

C₄×C₂₃

#103

yes

—

C₄×C₂₃

#131

yes

—

C₄×C₂₃

C₂×SD₁₆

yes

characteristic

C₄×C₂₃

C₂×Q₁₆

yes

characteristic

Dic₂₃

—

Dic₂₃

—

#533

—

#431

D₂₃

yes

characteristic

C₂×C₄⋊C₈

—

Q₈⋊C₈

—

C₂×C₂₃

—

C₂×C₂₃

yes

C₂×C₂₃

yes

C₂×C₂₃

yes

C₂²×C₈

—

C₄⋊Q₈

—

C₄⋊Q₈

—

#125

—

C₂²⋊Q₈

—

C₂²⋊Q₈

—

C₄×Q₈

—

C₄×Q₈

—

#117

—

C₂×C₄²

C₂×D₂₃

yes

characteristic

C₄⋊C₈

—

C₄⋊C₈

—

C₄⋊C₈

—

C₂²⋊C₈

—

C₄×C₈

C₂×D₂₃

yes

—

C₂₃

#1481

yes

characteristic, complemented, Sylow

C₂×Q₈

—

C₂×Q₈

—

C₂²×C₄

C₄×D₂₃

yes

characteristic

C₂²×C₄

D₉₂

yes

characteristic

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

C₄⋊C₄

—

108

C₄⋊C₄

—

109

C₂²⋊C₄

—

110

C₄²

—

111

C₄²

C₄×D₂₃

yes

characteristic

112

C₄²

D₉₂

yes

characteristic

113

C₄²

C₂²⋊D₂₃

yes

characteristic

114

C₄²

C₂²×D₂₃

yes

characteristic

115

C₂³

(C₂×D₂₃)⋊C₄

yes

characteristic

116

Q₈

—

117

Q₈

—

118

C₂×C₄

—

119

C₂×C₄

—

120

C₂×C₄

—

121

C₂×C₄

—

122

C₂×C₄

#67263

yes

—

123

C₂×C₄

Dic₉₂

yes

—

124

C₂×C₄

(C₂×D₂₃)⋊C₄

yes

characteristic

125

C₂×C₄

(C₂×D₂₃)⋊C₄

yes

characteristic

126

C₂×C₄

(C₂×D₂₃)⋊C₄

yes

characteristic

127

C₂×C₄

C₂×C₄×D₂₃

yes

characteristic

128

C₂×C₄

C₂×D₉₂

yes

characteristic

129

C₂×C₄

C₂×C₂²⋊D₂₃

yes

characteristic

130

C₈

—

131

C₈

—

132

C₂²

—

133

C₂²

—

134

C₂²

#10624998

yes

—

135

C₂²

C₂×(C₂×D₂₃)⋊C₄

yes

characteristic

136

C₄

—

137

C₄

—

138

C₄

—

139

C₄

—

140

C₄

—

141

C₄

#10624998

yes

—

142

C₄

C₂×#67263

yes

characteristic

143

C₄

C₂×Dic₉₂

yes

characteristic

144

C₄

#10625078

yes

—

145

C₂

—

146

C₂

#13092385

yes

characteristic

147

C₂

C₂×#10624998

yes

characteristic

148

C₂

#13092435

yes

characteristic

	subgroup	G/H
center	C₂×C₄	C₂×D₉₂
commutator	C₂×C₄×C₂₃	C₂²×C₄
Frattini	C₄²	C₂²×D₂₃
Fitting	C₂₃×#431	C₂
socle	C₂²×C₂₃	C₂×C₂²⋊C₄
prime radical	C₂³×C₂₃	C₂²⋊C₄
cosocle	C₄²×C₂₃	C₂³
hypercenter	#431	D₂₃
hypocenter	C₂₃	#1481
LCS	G ▷ C₂×C₄×C₂₃ ▷ C₂×C₂₃ ▷ C₂₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #431

subgroup

G/H

center

C₂×C₄

C₂×D₉₂

commutator

C₂×C₄×C₂₃

C₂²×C₄

Frattini

C₄²

C₂²×D₂₃

Fitting

C₂₃×#431

C₂

socle

C₂²×C₂₃

C₂×C₂²⋊C₄

prime radical

C₂³×C₂₃

C₂²⋊C₄

cosocle

C₄²×C₂₃

C₂³

D₂₃

C₂₃

G ▷ C₂×C₄×C₂₃ ▷ C₂×C₂₃ ▷ C₂₃

UCS

C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #431

N	H ≅ G/N	type
3. #13092149	C₂	semidirect
94. C₂₃	#1481	semidirect

H ≅ G/N

type

3. #13092149

C₂

semidirect

94. C₂₃

#1481

semidirect

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

subgroup	index
#13092974	2
C₂×C₄⋊(C₂₃⋊C₈)	2
#13092149	2
C₂₃×#431	2
#1481	23

subgroup

index

#13092974

C₂×C₄⋊(C₂₃⋊C₈)

#13092149

C₂₃×#431

#1481

Subgroups — #15671801

Conjugacy classes of subgroups

Complements

Maximal subgroups