Group #15673392 — subgroups

Subgroups — #15673392

Order 2944.

272 conjugacy classes of subgroups; only the 87 normal subgroups are shown.

normal subgroups only

This group has 272 conjugacy classes of subgroups — too many to draw. Only its 87 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×#10625058	C₂	1	1	yes	maximal, characteristic
2	#13092440	C₂	1	2	yes	maximal
3	C₂×#10625006	C₂	1	2	yes	maximal
4	#13092186	C₂	1	1	yes	maximal, characteristic
5	C₂₃×#343	C₂	1	1	yes	maximal, characteristic
6	C₂²×Dic₄₆	C₂²	1	1	yes	characteristic
7	C₂×M₁₆×C₂₃	C₂²	1	2	yes	—
8	C₂×C₄⋊C₄×C₂₃	C₂²	1	1	yes	characteristic
9	C₂×Dic₂₃⋊C₄	C₄	1	2	yes	—
10	C₂×C₄×Dic₂₃	C₄	1	2	yes	—
11	C₂×C₄×Dic₂₃	C₂²	1	1	yes	characteristic
12	C₂×#67267	C₂²	1	2	yes	—
13	#10625058	—	2	2	no	—
14	#10625058	—	2	2	no	—
15	#10625006	—	2	4	no	—
16	Dic₂₃⋊C₈	—	2	2	no	—
17	Dic₂₃⋊C₈	—	2	2	no	—
18	C₂²×C₄×C₂₃	C₂×C₄	1	2	yes	—
19	C₂²×C₄×C₂₃	C₂³	1	1	yes	characteristic
20	C₂²×Dic₂₃	C₂×C₄	1	1	yes	characteristic
21	C₂²×Dic₂₃	C₂×C₄	1	2	yes	—
22	C₂²×Dic₂₃	C₂×C₄	1	1	yes	characteristic
23	C₂×Dic₄₆	—	2	2	no	—
24	C₂×Dic₄₆	—	2	2	no	—
25	C₂×Dic₄₆	D₄	1	2	yes	—
26	C₂×Dic₄₆	Q₈	1	2	yes	—
27	M₁₆×C₂₃	—	2	2	no	—
28	M₁₆×C₂₃	—	2	2	no	—
29	C₂×C₈×C₂₃	—	2	2	no	—
30	C₄⋊C₄×C₂₃	—	2	2	no	—
31	Dic₂₃⋊C₄	—	2	4	no	—
32	C₄×Dic₂₃	—	2	2	no	—
33	C₄×Dic₂₃	—	2	2	no	—
34	C₄×Dic₂₃	D₄	1	2	yes	—
35	C₄×Dic₂₃	D₄	1	2	yes	—
36	#67267	—	2	2	no	—
37	#67267	—	2	2	no	—
38	C₂×C₂₃⋊C₈	—	2	2	no	—
39	C₂³×C₂₃	C₂²×C₄	1	1	yes	characteristic
40	C₂×C₄×C₂₃	—	2	2	no	—
41	C₂×C₄×C₂₃	—	4	2	no	—
42	C₂×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
43	C₂×C₄×C₂₃	C₂×D₄	1	2	yes	—
44	C₂×C₄×C₂₃	C₂×Q₈	1	1	yes	characteristic
45	C₂×C₄×C₂₃	#41	1	2	yes	—
46	C₂×Dic₂₃	—	2	2	no	—
47	C₂×Dic₂₃	—	2	2	no	—
48	C₂×Dic₂₃	—	2	2	no	—
49	C₂×Dic₂₃	—	4	1	no	—
50	C₂×Dic₂₃	C₂²⋊C₄	1	1	yes	characteristic
51	C₂×Dic₂₃	C₂²⋊C₄	1	2	yes	—
52	C₂×Dic₂₃	C₂²⋊C₄	1	1	yes	characteristic
53	C₂×Dic₂₃	C₄⋊C₄	1	4	yes	—
54	Dic₄₆	—	2	4	no	—
55	Dic₄₆	—	2	4	no	—
56	C₈×C₂₃	—	4	2	no	—
57	C₂₃⋊C₈	—	4	2	no	—
58	#1801	—	23	1	no	maximal, Sylow
59	C₂²×C₂₃	—	2	1	no	—
60	C₂²×C₂₃	—	2	1	no	—
61	C₂²×C₂₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
62	C₂²×C₂₃	C₂×C₄⋊C₄	1	1	yes	characteristic
63	C₂²×C₂₃	#117	1	1	yes	characteristic
64	C₄×C₂₃	—	4	2	no	—
65	C₄×C₂₃	#121	1	2	yes	—
66	C₄×C₂₃	C₂²⋊Q₈	1	2	yes	—
67	Dic₂₃	—	2	4	no	—
68	Dic₂₃	—	2	1	no	—
69	Dic₂₃	—	2	2	no	—
70	Dic₂₃	#101	1	2	yes	complemented
71	C₂×C₄×Q₈	—	23	1	no	—
72	#423	—	23	2	no	—
73	C₂×#101	—	23	2	no	—
74	#343	—	23	1	no	—
75	#343	D₂₃	1	1	yes	characteristic
76	C₂×C₂₃	—	2	1	no	—
77	C₂×C₂₃	—	2	1	no	—
78	C₂×C₂₃	#380	1	1	yes	characteristic
79	C₂×C₂₃	C₂×#101	1	1	yes	characteristic
80	C₂×C₂₃	#413	1	1	yes	characteristic
81	C₂²×Q₈	—	23	1	no	—
82	C₂×M₁₆	—	23	2	no	—
83	C₂×M₁₆	C₂×D₂₃	1	2	yes	—
84	C₄×Q₈	—	46	2	no	—
85	C₄×Q₈	—	46	2	no	—
86	C₂×C₄⋊C₄	—	23	2	no	—
87	C₂×C₄⋊C₄	C₂×D₂₃	1	1	yes	characteristic
88	C₂×C₄²	—	23	1	no	—
89	C₂×C₄²	—	23	2	no	—
90	C₄⋊C₈	—	46	2	no	—
91	C₄⋊C₈	—	46	2	no	—
92	#101	—	46	4	no	—
93	C₂₃	#1801	1	1	yes	characteristic, complemented, Sylow
94	C₂×Q₈	—	23	2	no	—
95	C₂×Q₈	—	23	2	no	—
96	C₂×Q₈	—	46	2	no	—
97	C₂×Q₈	—	46	2	no	—
98	C₂²×C₄	—	23	1	no	—
99	C₂²×C₄	—	23	2	no	—
100	C₂²×C₄	—	23	1	no	—
101	C₂²×C₄	C₄×D₂₃	1	2	yes	—
102	C₂²×C₄	C₂²×D₂₃	1	1	yes	characteristic
103	M₁₆	—	2	2	no	—
104	M₁₆	—	2	2	no	—
105	M₁₆	—	46	2	no	—
106	M₁₆	—	46	2	no	—
107	C₂×C₈	—	2	2	no	—
108	C₂×C₈	—	46	2	no	—
109	C₄⋊C₄	—	2	2	no	—
110	C₄⋊C₄	—	46	4	no	—
111	C₄²	—	23	2	no	—
112	C₄²	—	23	2	no	—
113	C₄²	—	46	2	no	—
114	C₄²	—	46	2	no	—
115	C₂³	C₂×C₄×D₂₃	1	1	yes	characteristic
116	Q₈	—	46	4	no	—
117	Q₈	—	46	4	no	—
118	C₂×C₄	—	2	2	no	—
119	C₂×C₄	—	4	2	no	—
120	C₂×C₄	—	23	1	no	—
121	C₂×C₄	—	23	1	no	—
122	C₂×C₄	—	23	2	no	—
123	C₂×C₄	—	23	4	no	—
124	C₂×C₄	—	46	2	no	—
125	C₂×C₄	—	46	2	no	—
126	C₂×C₄	—	46	2	no	—
127	C₂×C₄	—	92	1	no	—
128	C₂×C₄	#67288	1	2	yes	—
129	C₂×C₄	D₄×D₂₃	1	1	yes	characteristic
130	C₂×C₄	D₄×D₂₃	1	2	yes	—
131	C₂×C₄	Q₈×D₂₃	1	1	yes	characteristic
132	C₈	—	4	2	no	—
133	C₈	—	92	2	no	—
134	C₂²	—	2	1	no	—
135	C₂²	—	2	1	no	—
136	C₂²	#10625043	1	1	yes	characteristic
137	C₂²	C₂²⋊C₄×D₂₃	1	1	yes	characteristic
138	C₂²	C₄⋊C₄×D₂₃	1	1	yes	characteristic
139	C₄	—	4	2	no	—
140	C₄	—	23	2	no	—
141	C₄	—	46	1	no	—
142	C₄	—	46	2	no	—
143	C₄	—	46	4	no	—
144	C₄	#10625055	1	2	yes	—
145	C₄	#10625067	1	2	yes	—
146	C₂	—	2	1	no	—
147	C₂	—	2	1	no	—
148	C₂	#13092276	1	1	yes	characteristic
149	C₂	#101×D₂₃	1	1	yes	characteristic
150	C₂	#13092398	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#10625058

C₂

yes

maximal, characteristic

#13092440

C₂

yes

maximal

C₂×#10625006

C₂

yes

maximal

#13092186

C₂

yes

maximal, characteristic

C₂₃×#343

C₂

yes

maximal, characteristic

C₂²×Dic₄₆

C₂²

yes

characteristic

C₂×M₁₆×C₂₃

C₂²

yes

—

C₂×C₄⋊C₄×C₂₃

C₂²

yes

characteristic

C₂×Dic₂₃⋊C₄

C₄

yes

—

C₂×C₄×Dic₂₃

C₄

yes

—

C₂×C₄×Dic₂₃

C₂²

yes

characteristic

C₂×#67267

C₂²

yes

—

#10625058

—

#10625058

—

#10625006

—

Dic₂₃⋊C₈

—

Dic₂₃⋊C₈

—

C₂²×C₄×C₂₃

C₂×C₄

yes

—

C₂²×C₄×C₂₃

C₂³

yes

characteristic

C₂²×Dic₂₃

C₂×C₄

yes

characteristic

C₂²×Dic₂₃

C₂×C₄

yes

—

C₂²×Dic₂₃

C₂×C₄

yes

characteristic

C₂×Dic₄₆

—

C₂×Dic₄₆

—

C₂×Dic₄₆

D₄

yes

—

C₂×Dic₄₆

Q₈

yes

—

M₁₆×C₂₃

—

M₁₆×C₂₃

—

C₂×C₈×C₂₃

—

C₄⋊C₄×C₂₃

—

Dic₂₃⋊C₄

—

C₄×Dic₂₃

—

C₄×Dic₂₃

—

C₄×Dic₂₃

D₄

yes

—

C₄×Dic₂₃

D₄

yes

—

#67267

—

#67267

—

C₂×C₂₃⋊C₈

—

C₂³×C₂₃

C₂²×C₄

yes

characteristic

C₂×C₄×C₂₃

—

C₂×C₄×C₂₃

—

C₂×C₄×C₂₃

C₂×D₄

yes

characteristic

C₂×C₄×C₂₃

C₂×D₄

yes

—

C₂×C₄×C₂₃

C₂×Q₈

yes

characteristic

C₂×C₄×C₂₃

#41

yes

—

C₂×Dic₂₃

—

C₂×Dic₂₃

—

C₂×Dic₂₃

—

C₂×Dic₂₃

—

C₂×Dic₂₃

C₂²⋊C₄

yes

characteristic

C₂×Dic₂₃

C₂²⋊C₄

yes

—

C₂×Dic₂₃

C₂²⋊C₄

yes

characteristic

C₂×Dic₂₃

C₄⋊C₄

yes

—

Dic₄₆

—

Dic₄₆

—

C₈×C₂₃

—

C₂₃⋊C₈

—

#1801

—

maximal, Sylow

C₂²×C₂₃

—

C₂²×C₂₃

—

C₂²×C₂₃

C₂×C₂²⋊C₄

yes

characteristic

C₂²×C₂₃

C₂×C₄⋊C₄

yes

characteristic

C₂²×C₂₃

#117

yes

characteristic

C₄×C₂₃

—

C₄×C₂₃

#121

yes

—

C₄×C₂₃

C₂²⋊Q₈

yes

—

Dic₂₃

—

Dic₂₃

—

Dic₂₃

—

Dic₂₃

#101

yes

complemented

C₂×C₄×Q₈

—

—

—

—

D₂₃

yes

characteristic

C₂×C₂₃

—

C₂×C₂₃

—

C₂×C₂₃

yes

C₂×C₂₃

yes

C₂×C₂₃

yes

C₂²×Q₈

—

C₂×M₁₆

—

C₂×M₁₆

C₂×D₂₃

yes

—

C₄×Q₈

—

C₄×Q₈

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

C₂×D₂₃

yes

characteristic

C₂×C₄²

—

C₂×C₄²

—

C₄⋊C₈

—

C₄⋊C₈

—

#101

—

C₂₃

#1801

yes

characteristic, complemented, Sylow

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

C₂²×C₄

—

C₂²×C₄

—

100

C₂²×C₄

—

101

C₂²×C₄

C₄×D₂₃

yes

—

102

C₂²×C₄

C₂²×D₂₃

yes

characteristic

103

M₁₆

—

104

M₁₆

—

105

M₁₆

—

106

M₁₆

—

107

C₂×C₈

—

108

C₂×C₈

—

109

C₄⋊C₄

—

110

C₄⋊C₄

—

111

C₄²

—

112

C₄²

—

113

C₄²

—

114

C₄²

—

115

C₂³

C₂×C₄×D₂₃

yes

characteristic

116

Q₈

—

117

Q₈

—

118

C₂×C₄

—

119

C₂×C₄

—

120

C₂×C₄

—

121

C₂×C₄

—

122

C₂×C₄

—

123

C₂×C₄

—

124

C₂×C₄

—

125

C₂×C₄

—

126

C₂×C₄

—

127

C₂×C₄

—

128

C₂×C₄

#67288

yes

—

129

C₂×C₄

D₄×D₂₃

yes

characteristic

130

C₂×C₄

D₄×D₂₃

yes

—

131

C₂×C₄

Q₈×D₂₃

yes

characteristic

132

C₈

—

133

C₈

—

134

C₂²

—

135

C₂²

—

136

C₂²

#10625043

yes

characteristic

137

C₂²

C₂²⋊C₄×D₂₃

yes

characteristic

138

C₂²

C₄⋊C₄×D₂₃

yes

characteristic

139

C₄

—

140

C₄

—

141

C₄

—

142

C₄

—

143

C₄

—

144

C₄

#10625055

yes

—

145

C₄

#10625067

yes

—

146

C₂

—

147

C₂

—

148

C₂

#13092276

yes

characteristic

149

C₂

#101×D₂₃

yes

characteristic

150

C₂

#13092398

yes

characteristic

	subgroup	G/H
center	C₂²	C₂²⋊C₄×D₂₃
commutator, prime radical	C₂³×C₂₃	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₂₃
Fitting	C₂₃×#343	C₂
socle	C₂²×C₂₃	C₂×C₂²⋊C₄
cosocle	C₂²×C₄×C₂₃	C₂³
hypercenter	#343	D₂₃
hypocenter	C₂₃	#1801
LCS	G ▷ C₂³×C₂₃ ▷ C₂×C₂₃ ▷ C₂₃
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #343

subgroup

G/H

center

C₂²

C₂²⋊C₄×D₂₃

commutator, prime radical

C₂³×C₂₃

C₂²×C₄

Frattini

C₂²×C₄

C₂²×D₂₃

Fitting

C₂₃×#343

C₂

socle

C₂²×C₂₃

C₂×C₂²⋊C₄

cosocle

C₂²×C₄×C₂₃

C₂³

D₂₃

C₂₃

G ▷ C₂³×C₂₃ ▷ C₂×C₂₃ ▷ C₂₃

UCS

C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #343

N	H ≅ G/N	type
70. Dic₂₃	#101	semidirect
93. C₂₃	#1801	semidirect

H ≅ G/N

type

70. Dic₂₃

#101

semidirect

93. C₂₃

#1801

semidirect

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

subgroup	index
C₂×#10625058	2
#13092440	2
C₂×#10625006	2
#13092186	2
C₂₃×#343	2
#1801	23

subgroup

index

C₂₃×#343

Subgroups — #15673392

Conjugacy classes of subgroups

Complements

Maximal subgroups