Group #15674954 — subgroups

Subgroups — #15674954

Order 2944.

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

normal subgroups only

This group has 214 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	#13092315	C₂	1	1	yes	maximal, characteristic
2	#13092179	C₂	1	2	yes	maximal
3	#13092179	C₂	1	2	yes	maximal
4	#13092172	C₂	1	1	yes	maximal, characteristic
5	C₂₃×#384	C₂	1	1	yes	maximal, characteristic
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	C₂²	1	1	yes	characteristic
11	C₂×C₄⋊Dic₂₃	C₂²	1	1	yes	characteristic
12	C₂×Dic₂₃⋊C₄	—	2	1	no	—
13	C₂²×C₂₃⋊C₈	C₂²	1	2	yes	—
14	C₂₃⋊#95	—	2	1	no	—
15	C₂²×C₄×C₂₃	—	2	2	no	—
16	C₂²×C₄×C₂₃	D₄	1	1	yes	characteristic
17	C₂²×C₄×C₂₃	D₄	1	1	yes	characteristic
18	C₂²×C₄×C₂₃	C₂³	1	1	yes	characteristic
19	C₂²×Dic₂₃	—	2	1	no	—
20	C₂²×Dic₂₃	—	2	1	no	—
21	C₄⋊C₄×C₂₃	—	2	4	no	—
22	C₄⋊C₄×C₂₃	—	4	2	no	—
23	C₄⋊C₄×C₂₃	—	4	2	no	—
24	C₄²×C₂₃	—	2	2	no	—
25	C₄⋊Dic₂₃	—	2	1	no	—
26	C₄⋊Dic₂₃	—	2	2	no	—
27	C₄⋊Dic₂₃	Q₈	1	2	yes	—
28	Dic₂₃⋊C₄	—	4	2	no	—
29	C₂×C₂₃⋊C₈	—	2	2	no	—
30	C₂×C₂₃⋊C₈	—	2	2	no	—
31	C₂×C₂₃⋊C₈	—	2	2	no	—
32	C₂³×C₂₃	C₂×D₄	1	1	yes	characteristic
33	C₂×C₄×C₂₃	—	2	1	no	—
34	C₂×C₄×C₂₃	—	2	1	no	—
35	C₂×C₄×C₂₃	—	2	1	no	—
36	C₂×C₄×C₂₃	—	2	1	no	—
37	C₂×C₄×C₂₃	—	4	2	no	—
38	C₂×C₄×C₂₃	—	4	2	no	—
39	C₂×C₄×C₂₃	—	4	2	no	—
40	C₂×C₄×C₂₃	D₈	1	2	yes	—
41	C₂×C₄×C₂₃	SD₁₆	1	2	yes	—
42	C₂×C₄×C₂₃	C₂×Q₈	1	1	yes	characteristic
43	C₂×C₄×C₂₃	#41	1	2	yes	—
44	C₂×C₄×C₂₃	#41	1	2	yes	—
45	C₂×C₄×C₂₃	#41	1	1	yes	characteristic
46	C₂×Dic₂₃	—	2	2	no	—
47	C₂×Dic₂₃	—	4	1	no	—
48	C₂×Dic₂₃	—	4	1	no	—
49	C₂×Dic₂₃	—	4	1	no	—
50	C₂×Dic₂₃	—	4	1	no	—
51	C₂×Dic₂₃	—	4	1	no	—
52	C₂₃⋊C₈	—	4	2	no	—
53	#2099	—	23	1	no	maximal, Sylow
54	C₂²×C₂₃	C₂²⋊Q₈	1	1	yes	characteristic
55	C₂²×C₂₃	#123	1	1	yes	characteristic
56	C₂²×C₂₃	#124	1	1	yes	characteristic
57	C₂²×C₂₃	C₂×D₈	1	1	yes	characteristic
58	C₂²×C₂₃	C₂×SD₁₆	1	1	yes	characteristic
59	C₂²×C₂₃	#137	1	1	yes	characteristic
60	C₂²×C₂₃	#137	1	1	yes	characteristic
61	C₄×C₂₃	—	2	2	no	—
62	C₄×C₂₃	—	2	2	no	—
63	C₄×C₂₃	—	8	2	no	—
64	C₄×C₂₃	#125	1	2	yes	—
65	C₄×C₂₃	#126	1	2	yes	—
66	Dic₂₃	—	4	2	no	—
67	Dic₂₃	—	8	1	no	—
68	#389	—	23	1	no	—
69	#384	D₂₃	1	1	yes	characteristic
70	#340	—	23	2	no	—
71	#340	—	23	2	no	—
72	#336	—	23	1	no	—
73	C₂×C₂₃	#400	1	1	yes	characteristic
74	C₂×C₂₃	#474	1	1	yes	characteristic
75	C₂×C₂₃	#475	1	1	yes	characteristic
76	C₂×C₂₃	#480	1	1	yes	characteristic
77	C₂×C₂₃	#483	1	1	yes	characteristic
78	C₂×C₂₃	#486	1	1	yes	characteristic
79	C₂×C₂₃	#491	1	1	yes	characteristic
80	C₂²×C₈	—	23	2	no	—
81	C₂×C₄⋊C₄	—	2	1	no	—
82	C₂×C₄⋊C₄	—	2	1	no	—
83	C₂×C₄⋊C₄	—	23	1	no	—
84	C₂×C₄⋊C₄	—	23	1	no	—
85	C₂×C₄⋊C₄	—	46	1	no	—
86	C₂×C₄⋊C₄	C₂×D₂₃	1	2	yes	—
87	C₂×C₄²	C₂×D₂₃	1	1	yes	characteristic
88	#95	—	46	1	no	—
89	C₂₃	#2099	1	1	yes	characteristic, complemented, Sylow
90	C₂²×C₄	—	2	2	no	—
91	C₂²×C₄	—	46	1	no	—
92	C₂²×C₄	—	46	1	no	—
93	C₂²×C₄	C₂²⋊D₂₃	1	1	yes	characteristic
94	C₂²×C₄	C₂²⋊D₂₃	1	1	yes	characteristic
95	C₂²×C₄	C₂²×D₂₃	1	1	yes	characteristic
96	C₂×C₈	—	46	2	no	—
97	C₂×C₈	—	46	2	no	—
98	C₂×C₈	—	46	2	no	—
99	C₄⋊C₄	—	2	4	no	—
100	C₄⋊C₄	—	4	2	no	—
101	C₄⋊C₄	—	4	2	no	—
102	C₄⋊C₄	—	23	2	no	—
103	C₄⋊C₄	—	46	1	no	—
104	C₄⋊C₄	—	46	2	no	—
105	C₄⋊C₄	—	92	2	no	—
106	C₄²	—	2	2	no	—
107	C₂³	C₂×C₂²⋊D₂₃	1	1	yes	characteristic
108	C₂×C₄	—	2	1	no	—
109	C₂×C₄	—	2	1	no	—
110	C₂×C₄	—	2	1	no	—
111	C₂×C₄	—	2	1	no	—
112	C₂×C₄	—	4	2	no	—
113	C₂×C₄	—	4	2	no	—
114	C₂×C₄	—	4	2	no	—
115	C₂×C₄	—	46	2	no	—
116	C₂×C₄	—	92	1	no	—
117	C₂×C₄	—	92	1	no	—
118	C₂×C₄	—	92	1	no	—
119	C₂×C₄	—	92	1	no	—
120	C₂×C₄	—	92	1	no	—
121	C₂×C₄	#67272	1	2	yes	—
122	C₂×C₄	#67273	1	2	yes	—
123	C₂×C₄	#67288	1	2	yes	—
124	C₂×C₄	#67290	1	2	yes	—
125	C₂×C₄	Q₈×D₂₃	1	1	yes	characteristic
126	C₂×C₄	#67292	1	1	yes	characteristic
127	C₈	—	92	2	no	—
128	C₂²	#10625100	1	1	yes	characteristic
129	C₂²	C₂×#67272	1	1	yes	characteristic
130	C₂²	C₂×#67273	1	1	yes	characteristic
131	C₂²	#10625107	1	1	yes	characteristic
132	C₂²	#10625113	1	1	yes	characteristic
133	C₂²	#10625113	1	1	yes	characteristic
134	C₂²	#10625117	1	1	yes	characteristic
135	C₄	—	2	2	no	—
136	C₄	—	2	2	no	—
137	C₄	—	8	2	no	—
138	C₄	—	92	2	no	—
139	C₄	—	184	1	no	—
140	C₄	#10625060	1	2	yes	—
141	C₄	#10625069	1	2	yes	—
142	C₂	#13092355	1	1	yes	characteristic
143	C₂	#13092694	1	1	yes	characteristic
144	C₂	#13092699	1	1	yes	characteristic
145	C₂	#13092730	1	1	yes	characteristic
146	C₂	#13092747	1	1	yes	characteristic
147	C₂	#13092763	1	1	yes	characteristic
148	C₂	#13092785	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#13092315

C₂

yes

maximal, characteristic

#13092179

C₂

yes

maximal

#13092179

C₂

yes

maximal

#13092172

C₂

yes

maximal, characteristic

C₂₃×#384

C₂

yes

maximal, characteristic

C₂×C₄⋊C₄×C₂₃

—

C₂×C₄⋊C₄×C₂₃

—

C₂×C₄⋊C₄×C₂₃

C₂²

yes

—

C₂×C₄²×C₂₃

C₂²

yes

characteristic

C₂×C₄⋊Dic₂₃

C₂²

yes

characteristic

C₂×C₄⋊Dic₂₃

C₂²

yes

characteristic

C₂×Dic₂₃⋊C₄

—

C₂²×C₂₃⋊C₈

C₂²

yes

—

C₂₃⋊#95

—

C₂²×C₄×C₂₃

—

C₂²×C₄×C₂₃

D₄

yes

characteristic

C₂²×C₄×C₂₃

D₄

yes

characteristic

C₂²×C₄×C₂₃

C₂³

yes

characteristic

C₂²×Dic₂₃

—

C₂²×Dic₂₃

—

C₄⋊C₄×C₂₃

—

C₄⋊C₄×C₂₃

—

C₄⋊C₄×C₂₃

—

C₄²×C₂₃

—

C₄⋊Dic₂₃

—

C₄⋊Dic₂₃

—

C₄⋊Dic₂₃

Q₈

yes

—

Dic₂₃⋊C₄

—

C₂×C₂₃⋊C₈

—

C₂×C₂₃⋊C₈

—

C₂×C₂₃⋊C₈

—

C₂³×C₂₃

C₂×D₄

yes

characteristic

C₂×C₄×C₂₃

—

C₂×C₄×C₂₃

—

C₂×C₄×C₂₃

—

C₂×C₄×C₂₃

—

C₂×C₄×C₂₃

—

C₂×C₄×C₂₃

—

C₂×C₄×C₂₃

—

C₂×C₄×C₂₃

D₈

yes

—

C₂×C₄×C₂₃

SD₁₆

yes

—

C₂×C₄×C₂₃

C₂×Q₈

yes

characteristic

C₂×C₄×C₂₃

#41

yes

—

C₂×C₄×C₂₃

#41

yes

—

C₂×C₄×C₂₃

#41

yes

characteristic

C₂×Dic₂₃

—

C₂×Dic₂₃

—

C₂×Dic₂₃

—

C₂×Dic₂₃

—

C₂×Dic₂₃

—

C₂×Dic₂₃

—

C₂₃⋊C₈

—

#2099

—

maximal, Sylow

C₂²×C₂₃

C₂²⋊Q₈

yes

C₂²×C₂₃

yes

C₂²×C₂₃

yes

C₂²×C₂₃

C₂×D₈

yes

characteristic

C₂²×C₂₃

C₂×SD₁₆

yes

C₂²×C₂₃

yes

C₂²×C₂₃

yes

C₄×C₂₃

—

C₄×C₂₃

—

C₄×C₂₃

—

C₄×C₂₃

#125

yes

—

C₄×C₂₃

#126

yes

—

Dic₂₃

—

Dic₂₃

—

#389

—

#384

D₂₃

yes

—

—

—

C₂×C₂₃

yes

C₂×C₂₃

yes

C₂×C₂₃

yes

C₂×C₂₃

yes

C₂×C₂₃

yes

C₂×C₂₃

yes

C₂×C₂₃

yes

C₂²×C₈

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

C₂×D₂₃

yes

—

C₂×C₄²

C₂×D₂₃

yes

characteristic

#95

—

C₂₃

#2099

yes

characteristic, complemented, Sylow

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

C₂²⋊D₂₃

yes

characteristic

C₂²×C₄

C₂²⋊D₂₃

yes

characteristic

C₂²×C₄

C₂²×D₂₃

yes

characteristic

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₄⋊C₄

—

100

C₄⋊C₄

—

101

C₄⋊C₄

—

102

C₄⋊C₄

—

103

C₄⋊C₄

—

104

C₄⋊C₄

—

105

C₄⋊C₄

—

106

C₄²

—

107

C₂³

C₂×C₂²⋊D₂₃

yes

characteristic

108

C₂×C₄

—

109

C₂×C₄

—

110

C₂×C₄

—

111

C₂×C₄

—

112

C₂×C₄

—

113

C₂×C₄

—

114

C₂×C₄

—

115

C₂×C₄

—

116

C₂×C₄

—

117

C₂×C₄

—

118

C₂×C₄

—

119

C₂×C₄

—

120

C₂×C₄

—

121

C₂×C₄

#67272

yes

—

122

C₂×C₄

#67273

yes

—

123

C₂×C₄

#67288

yes

—

124

C₂×C₄

#67290

yes

—

125

C₂×C₄

Q₈×D₂₃

yes

characteristic

126

C₂×C₄

#67292

yes

characteristic

127

C₈

—

128

C₂²

#10625100

yes

characteristic

129

C₂²

C₂×#67272

yes

characteristic

130

C₂²

C₂×#67273

yes

characteristic

131

C₂²

#10625107

yes

characteristic

132

C₂²

#10625113

yes

characteristic

133

C₂²

#10625113

yes

characteristic

134

C₂²

#10625117

yes

characteristic

135

C₄

—

136

C₄

—

137

C₄

—

138

C₄

—

139

C₄

—

184

—

140

C₄

#10625060

yes

—

141

C₄

#10625069

yes

—

142

C₂

#13092355

yes

characteristic

143

C₂

#13092694

yes

characteristic

144

C₂

#13092699

yes

characteristic

145

C₂

#13092730

yes

characteristic

146

C₂

#13092747

yes

characteristic

147

C₂

#13092763

yes

characteristic

148

C₂

#13092785

yes

characteristic

	subgroup	G/H
center	C₂³	C₂×C₂²⋊D₂₃
commutator, cosocle	C₂²×C₄×C₂₃	C₂³
Frattini	C₂²×C₄	C₂²×D₂₃
Fitting	C₂₃×#384	C₂
socle, prime radical	C₂³×C₂₃	C₂×D₄
hypercenter	#384	D₂₃
hypocenter	C₂₃	#2099
LCS	G ▷ C₂²×C₄×C₂₃ ▷ C₂×C₂₃ ▷ C₂₃
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #384

subgroup

G/H

center

C₂³

C₂×C₂²⋊D₂₃

commutator, cosocle

C₂²×C₄×C₂₃

C₂³

Frattini

C₂²×C₄

C₂²×D₂₃

Fitting

C₂₃×#384

C₂

socle, prime radical

C₂³×C₂₃

C₂×D₄

D₂₃

C₂₃

G ▷ C₂²×C₄×C₂₃ ▷ C₂×C₂₃ ▷ C₂₃

UCS

C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #384

N	H ≅ G/N	type
89. C₂₃	#2099	semidirect

H ≅ G/N

type

89. C₂₃

#2099

semidirect

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

subgroup	index
#13092315	2
#13092179	2
#13092172	2
C₂₃×#384	2
#2099	23

subgroup

index

C₂₃×#384

Subgroups — #15674954

Conjugacy classes of subgroups

Complements

Maximal subgroups