Group #15673498 — subgroups

Subgroups — #15673498

Order 2944.

262 conjugacy classes of subgroups; only the 81 normal subgroups are shown.

normal subgroups only

This group has 262 conjugacy classes of subgroups — too many to draw. Only its 81 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	#41×Dic₂₃	C₂	1	1	yes	maximal, characteristic
2	#13092401	C₂	1	1	yes	maximal, characteristic
3	#13092176	C₂	1	2	yes	maximal, complemented
4	#13092173	C₂	1	2	yes	maximal, complemented
5	C₂₃×#410	C₂	1	1	yes	maximal, characteristic
6	C₂×#41×C₂₃	C₂²	1	1	yes	characteristic
7	C₂₃×#117	C₂²	1	2	yes	—
8	Q₈×Dic₂₃	—	2	1	no	—
9	D₄×Dic₂₃	—	2	1	no	—
10	D₄×Dic₂₃	—	2	2	no	—
11	#10625097	C₄	1	2	yes	complemented
12	#10625097	C₂²	1	1	yes	characteristic
13	C₂×C₄×Dic₂₃	C₄	1	2	yes	complemented
14	C₂×C₄×Dic₂₃	C₂²	1	1	yes	characteristic
15	C₂×#67267	C₂²	1	2	yes	—
16	C₂₃×#99	—	2	2	no	—
17	#10625018	—	2	1	no	—
18	#10625015	—	2	1	no	—
19	#41×C₂₃	—	2	2	no	—
20	#41×C₂₃	—	2	2	no	—
21	C₂×Q₈×C₂₃	D₄	1	1	yes	characteristic
22	C₂×D₄×C₂₃	D₄	1	1	yes	characteristic
23	C₂×D₄×C₂₃	D₄	1	2	yes	—
24	C₂²×C₄×C₂₃	C₂×C₄	1	2	yes	—
25	C₂²×C₄×C₂₃	C₂³	1	1	yes	characteristic
26	C₂²×Dic₂₃	—	2	2	no	—
27	C₂²×Dic₂₃	—	2	1	no	—
28	C₄⋊C₄×C₂₃	—	2	2	no	—
29	C₂²⋊C₄×C₂₃	—	2	2	no	—
30	C₄²×C₂₃	—	2	2	no	—
31	C₂²⋊Dic₂₃	—	2	2	no	—
32	C₂²⋊Dic₂₃	—	2	1	no	—
33	C₄⋊Dic₂₃	—	2	2	no	—
34	C₄⋊Dic₂₃	—	2	1	no	—
35	C₄×Dic₂₃	—	2	2	no	—
36	C₄×Dic₂₃	—	2	1	no	—
37	C₄×Dic₂₃	C₂×C₄	1	1	yes	characteristic
38	C₄×Dic₂₃	C₂×C₄	1	2	yes	—
39	C₄×Dic₂₃	C₂×C₄	1	1	yes	characteristic
40	#67267	—	2	2	no	—
41	#67267	—	2	2	no	—
42	C₂×C₂₃⋊C₈	—	2	2	no	—
43	C₂³×C₂₃	C₂²⋊C₄	1	2	yes	—
44	C₂³×C₂₃	C₂×D₄	1	1	yes	characteristic
45	Q₈×C₂₃	—	2	2	no	—
46	D₄×C₂₃	—	2	4	no	—
47	D₄×C₂₃	—	2	2	no	—
48	C₂×C₄×C₂₃	—	2	2	no	—
49	C₂×C₄×C₂₃	—	2	2	no	—
50	C₂×C₄×C₂₃	—	4	2	no	—
51	C₂×C₄×C₂₃	C₂²⋊C₄	1	2	yes	—
52	C₂×C₄×C₂₃	C₂²×C₄	1	1	yes	characteristic
53	C₂×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
54	C₂×C₄×C₂₃	#41	1	2	yes	—
55	C₂×C₄×C₂₃	#41	1	2	yes	—
56	C₂×Dic₂₃	—	2	1	no	—
57	C₂×Dic₂₃	—	2	2	no	—
58	C₂×Dic₂₃	—	2	1	no	—
59	C₂×Dic₂₃	—	4	2	no	—
60	C₂×Dic₂₃	—	4	4	no	—
61	C₂₃⋊C₈	—	4	2	no	—
62	#1827	—	23	1	no	maximal, Sylow
63	C₂²×C₂₃	—	2	2	no	—
64	C₂²×C₂₃	—	4	1	no	—
65	C₂²×C₂₃	—	4	2	no	—
66	C₂²×C₂₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
67	C₂²×C₂₃	#123	1	2	yes	—
68	C₄×C₂₃	—	2	2	no	—
69	C₄×C₂₃	—	8	2	no	—
70	C₄×C₂₃	#117	1	1	yes	characteristic
71	C₄×C₂₃	#117	1	1	yes	characteristic
72	C₄×C₂₃	#123	1	2	yes	—
73	Dic₂₃	—	2	2	no	—
74	Dic₂₃	—	4	2	no	—
75	Dic₂₃	—	4	1	no	—
76	C₄×#41	—	23	1	no	—
77	#413	—	23	1	no	—
78	#410	D₂₃	1	1	yes	characteristic
79	#339	—	23	2	no	—
80	H(3,ℤ/4)	—	23	2	no	—
81	C₂×C₂₃	—	2	1	no	—
82	C₂×C₂₃	—	2	2	no	—
83	C₂×C₂₃	—	4	2	no	—
84	C₂×C₂₃	#381	1	1	yes	characteristic
85	C₂×#41	C₂×D₂₃	1	1	yes	characteristic
86	C₂×M₁₆	—	23	2	no	—
87	C₄×Q₈	—	46	1	no	—
88	C₄×D₄	—	46	1	no	—
89	C₄×D₄	—	46	2	no	—
90	#117	—	23	2	no	—
91	#117	—	23	1	no	—
92	#117	C₂×D₂₃	1	2	yes	—
93	C₂×C₄²	—	23	2	no	—
94	C₂×C₄²	—	23	1	no	—
95	#101	—	46	1	no	—
96	#100	—	46	1	no	—
97	#99	—	2	2	no	—
98	C₂₃	#1827	1	1	yes	characteristic, complemented, Sylow
99	#41	—	2	2	no	—
100	#41	—	2	2	no	—
101	C₂×Q₈	D₉₂	1	1	yes	characteristic
102	C₂×D₄	D₉₂	1	1	yes	characteristic
103	C₂×D₄	C₂²⋊D₂₃	1	2	yes	—
104	C₂²×C₄	—	46	2	no	—
105	C₂²×C₄	—	46	1	no	—
106	C₂²×C₄	C₄×D₂₃	1	2	yes	—
107	C₂²×C₄	C₂²×D₂₃	1	1	yes	characteristic
108	M₁₆	—	46	2	no	—
109	M₁₆	—	46	2	no	—
110	C₂×C₈	—	46	2	no	—
111	C₄⋊C₄	—	2	2	no	—
112	C₄⋊C₄	—	46	2	no	—
113	C₄⋊C₄	—	46	1	no	—
114	C₂²⋊C₄	—	2	2	no	—
115	C₂²⋊C₄	—	46	2	no	—
116	C₂²⋊C₄	—	46	1	no	—
117	C₄²	—	2	2	no	—
118	C₄²	—	23	1	no	—
119	C₄²	—	23	2	no	—
120	C₄²	—	23	1	no	—
121	C₄²	—	46	2	no	—
122	C₄²	—	46	1	no	—
123	C₂³	(C₂×D₂₃)⋊C₄	1	2	yes	—
124	C₂³	C₂×C₂²⋊D₂₃	1	1	yes	characteristic
125	Q₈	—	2	2	no	—
126	D₄	—	2	4	no	—
127	D₄	—	2	2	no	—
128	C₂×C₄	—	2	2	no	—
129	C₂×C₄	—	2	2	no	—
130	C₂×C₄	—	4	2	no	—
131	C₂×C₄	—	46	1	no	—
132	C₂×C₄	—	46	2	no	—
133	C₂×C₄	—	46	1	no	—
134	C₂×C₄	—	92	2	no	—
135	C₂×C₄	—	92	4	no	—
136	C₂×C₄	(C₂×D₂₃)⋊C₄	1	2	yes	—
137	C₂×C₄	C₂×C₄×D₂₃	1	1	yes	characteristic
138	C₂×C₄	C₂×D₉₂	1	1	yes	characteristic
139	C₂×C₄	#67290	1	2	yes	—
140	C₂×C₄	#67290	1	2	yes	—
141	C₈	—	92	2	no	—
142	C₂²	—	2	2	no	—
143	C₂²	—	4	1	no	—
144	C₂²	—	4	2	no	—
145	C₂²	C₂×(C₂×D₂₃)⋊C₄	1	1	yes	characteristic
146	C₂²	#10625106	1	2	yes	—
147	C₄	—	2	2	no	—
148	C₄	—	8	2	no	—
149	C₄	—	46	2	no	—
150	C₄	—	92	2	no	—
151	C₄	—	92	1	no	—
152	C₄	#10625048	1	1	yes	characteristic
153	C₄	#10625048	1	1	yes	characteristic
154	C₄	#10625057	1	2	yes	—
155	C₂	—	2	2	no	—
156	C₂	—	2	1	no	—
157	C₂	—	4	2	no	—
158	C₂	#13092278	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#41×Dic₂₃

C₂

yes

maximal, characteristic

#13092401

C₂

yes

maximal, characteristic

#13092176

C₂

yes

maximal, complemented

#13092173

C₂

yes

maximal, complemented

C₂₃×#410

C₂

yes

maximal, characteristic

C₂×#41×C₂₃

C₂²

yes

characteristic

C₂₃×#117

C₂²

yes

—

Q₈×Dic₂₃

—

D₄×Dic₂₃

—

D₄×Dic₂₃

—

#10625097

C₄

yes

complemented

#10625097

C₂²

yes

characteristic

C₂×C₄×Dic₂₃

C₄

yes

complemented

C₂×C₄×Dic₂₃

C₂²

yes

characteristic

C₂×#67267

C₂²

yes

—

C₂₃×#99

—

#10625018

—

#10625015

—

#41×C₂₃

—

#41×C₂₃

—

C₂×Q₈×C₂₃

D₄

yes

characteristic

C₂×D₄×C₂₃

D₄

yes

characteristic

C₂×D₄×C₂₃

D₄

yes

—

C₂²×C₄×C₂₃

C₂×C₄

yes

—

C₂²×C₄×C₂₃

C₂³

yes

characteristic

C₂²×Dic₂₃

—

C₂²×Dic₂₃

—

C₄⋊C₄×C₂₃

—

C₂²⋊C₄×C₂₃

—

C₄²×C₂₃

—

C₂²⋊Dic₂₃

—

C₂²⋊Dic₂₃

—

C₄⋊Dic₂₃

—

C₄⋊Dic₂₃

—

C₄×Dic₂₃

—

C₄×Dic₂₃

—

C₄×Dic₂₃

C₂×C₄

yes

characteristic

C₄×Dic₂₃

C₂×C₄

yes

—

C₄×Dic₂₃

C₂×C₄

yes

characteristic

#67267

—

#67267

—

C₂×C₂₃⋊C₈

—

C₂³×C₂₃

C₂²⋊C₄

yes

—

C₂³×C₂₃

C₂×D₄

yes

characteristic

Q₈×C₂₃

—

D₄×C₂₃

—

D₄×C₂₃

—

C₂×C₄×C₂₃

—

C₂×C₄×C₂₃

—

C₂×C₄×C₂₃

—

C₂×C₄×C₂₃

C₂²⋊C₄

yes

—

C₂×C₄×C₂₃

C₂²×C₄

yes

characteristic

C₂×C₄×C₂₃

C₂×D₄

yes

characteristic

C₂×C₄×C₂₃

#41

yes

—

C₂×C₄×C₂₃

#41

yes

—

C₂×Dic₂₃

—

C₂×Dic₂₃

—

C₂×Dic₂₃

—

C₂×Dic₂₃

—

C₂×Dic₂₃

—

C₂₃⋊C₈

—

#1827

—

maximal, Sylow

C₂²×C₂₃

—

C₂²×C₂₃

—

C₂²×C₂₃

—

C₂²×C₂₃

C₂×C₂²⋊C₄

yes

characteristic

C₂²×C₂₃

#123

yes

—

C₄×C₂₃

—

C₄×C₂₃

—

C₄×C₂₃

yes

C₄×C₂₃

yes

C₄×C₂₃

yes

—

Dic₂₃

—

Dic₂₃

—

Dic₂₃

—

C₄×#41

—

#413

—

#410

D₂₃

yes

characteristic

#339

—

H(3,ℤ/4)

—

C₂×C₂₃

—

C₂×C₂₃

—

C₂×C₂₃

—

C₂×C₂₃

#381

yes

characteristic

C₂×#41

C₂×D₂₃

yes

characteristic

C₂×M₁₆

—

C₄×Q₈

—

C₄×D₄

—

C₄×D₄

—

#117

—

#117

—

#117

C₂×D₂₃

yes

—

C₂×C₄²

—

C₂×C₄²

—

#101

—

#100

—

#99

—

C₂₃

#1827

yes

characteristic, complemented, Sylow

#41

—

100

#41

—

101

C₂×Q₈

D₉₂

yes

characteristic

102

C₂×D₄

D₉₂

yes

characteristic

103

C₂×D₄

C₂²⋊D₂₃

yes

—

104

C₂²×C₄

—

105

C₂²×C₄

—

106

C₂²×C₄

C₄×D₂₃

yes

—

107

C₂²×C₄

C₂²×D₂₃

yes

characteristic

108

M₁₆

—

109

M₁₆

—

110

C₂×C₈

—

111

C₄⋊C₄

—

112

C₄⋊C₄

—

113

C₄⋊C₄

—

114

C₂²⋊C₄

—

115

C₂²⋊C₄

—

116

C₂²⋊C₄

—

117

C₄²

—

118

C₄²

—

119

C₄²

—

120

C₄²

—

121

C₄²

—

122

C₄²

—

123

C₂³

(C₂×D₂₃)⋊C₄

yes

—

124

C₂³

C₂×C₂²⋊D₂₃

yes

characteristic

125

Q₈

—

126

D₄

—

127

D₄

—

128

C₂×C₄

—

129

C₂×C₄

—

130

C₂×C₄

—

131

C₂×C₄

—

132

C₂×C₄

—

133

C₂×C₄

—

134

C₂×C₄

—

135

C₂×C₄

—

136

C₂×C₄

(C₂×D₂₃)⋊C₄

yes

—

137

C₂×C₄

C₂×C₄×D₂₃

yes

characteristic

138

C₂×C₄

C₂×D₉₂

yes

characteristic

139

C₂×C₄

#67290

yes

—

140

C₂×C₄

#67290

yes

—

141

C₈

—

142

C₂²

—

143

C₂²

—

144

C₂²

—

145

C₂²

C₂×(C₂×D₂₃)⋊C₄

yes

characteristic

146

C₂²

#10625106

yes

—

147

C₄

—

148

C₄

—

149

C₄

—

150

C₄

—

151

C₄

—

152

C₄

#10625048

yes

characteristic

153

C₄

#10625048

yes

characteristic

154

C₄

#10625057

yes

—

155

C₂

—

156

C₂

—

157

C₂

—

158

C₂

#13092278

yes

characteristic

	subgroup	G/H
center	C₄	#10625048
commutator	C₂×C₄×C₂₃	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₂₃
Fitting	C₂₃×#410	C₂
socle	C₂×C₂₃	#381
prime radical	C₂×#41×C₂₃	C₂²
cosocle	C₂²×C₄×C₂₃	C₂³
hypercenter	#410	D₂₃
hypocenter	C₂₃	#1827
LCS	G ▷ C₂×C₄×C₂₃ ▷ C₂×C₂₃ ▷ C₂₃
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ #410

subgroup

G/H

center

C₄

#10625048

commutator

C₂×C₄×C₂₃

C₂²×C₄

Frattini

C₂²×C₄

C₂²×D₂₃

Fitting

C₂₃×#410

C₂

socle

C₂×C₂₃

#381

prime radical

C₂×#41×C₂₃

C₂²

cosocle

C₂²×C₄×C₂₃

C₂³

D₂₃

C₂₃

G ▷ C₂×C₄×C₂₃ ▷ C₂×C₂₃ ▷ C₂₃

UCS

C₁ ◁ C₄ ◁ C₂²×C₄ ◁ #410

N	H ≅ G/N	type
3. #13092176	C₂	semidirect
4. #13092173	C₂	semidirect
11. #10625097	C₄	semidirect
13. C₂×C₄×Dic₂₃	C₄	semidirect
98. C₂₃	#1827	semidirect

H ≅ G/N

type

3. #13092176

C₂

semidirect

4. #13092173

C₂

semidirect

11. #10625097

C₄

semidirect

13. C₂×C₄×Dic₂₃

C₄

semidirect

98. C₂₃

#1827

semidirect

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

subgroup	index
#41×Dic₂₃	2
#13092401	2
#13092176	2
#13092173	2
C₂₃×#410	2
#1827	23

subgroup

index

#41×Dic₂₃

C₂₃×#410

Subgroups — #15673498

Conjugacy classes of subgroups

Complements

Maximal subgroups