Group #15674685 — subgroups

Subgroups — #15674685

Order 2944.

266 conjugacy classes of subgroups; only the 63 normal subgroups are shown.

normal subgroups only

This group has 266 conjugacy classes of subgroups — too many to draw. Only its 63 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	#13093217	C₂	1	1	yes	maximal, characteristic
2	#13092440	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×#10625007	C₂	1	2	yes	maximal, complemented
4	#13092399	C₂	1	2	yes	maximal, complemented
5	C₂₃×#338	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×#67288	C₂²	1	2	yes	—
7	C₂×M₁₆×C₂₃	C₂²	1	2	yes	—
8	C₂×M₁₆×C₂₃	C₂²	1	1	yes	characteristic
9	C₂×C₄×Dic₂₃	C₂²	1	1	yes	characteristic
10	C₂×#67267	C₂²	1	1	yes	characteristic
11	#10625080	—	2	2	no	—
12	#10625079	—	2	2	no	—
13	#10625075	—	2	2	no	—
14	#10625007	—	2	2	no	—
15	#10625007	—	2	2	no	—
16	#10625006	—	2	2	no	—
17	#10625005	—	2	2	no	—
18	Dic₂₃⋊C₈	—	2	2	no	—
19	C₂²×C₄×C₂₃	C₂³	1	1	yes	characteristic
20	C₂×C₂²⋊D₂₃	—	2	2	no	—
21	C₂²×Dic₂₃	—	2	1	no	—
22	#67288	—	2	2	no	—
23	#67288	—	2	2	no	—
24	#67288	—	2	2	no	—
25	#67288	—	2	2	no	—
26	C₂×D₉₂	D₄	1	2	yes	—
27	C₂×C₄×D₂₃	—	2	2	no	—
28	C₂×Dic₄₆	D₄	1	2	yes	—
29	M₁₆×C₂₃	—	2	2	no	—
30	M₁₆×C₂₃	—	2	2	no	—
31	M₁₆×C₂₃	—	2	2	no	—
32	C₂×C₈×C₂₃	—	2	2	no	—
33	C₂×C₈×C₂₃	D₄	1	2	yes	—
34	C₄×Dic₂₃	—	2	1	no	—
35	C₄×Dic₂₃	D₄	1	1	yes	characteristic
36	C₄×Dic₂₃	D₄	1	1	yes	characteristic
37	#67267	—	2	2	no	—
38	C₂×C₂₃⋊C₈	—	2	1	no	—
39	Dic₉₂	—	2	2	no	—
40	D₁₈₄	—	2	2	no	—
41	#67263	—	2	4	no	—
42	C₂³×C₂₃	#41	1	1	yes	characteristic
43	C₂²×D₂₃	—	2	2	no	—
44	C₂×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
45	C₂×C₄×C₂₃	C₂×D₄	1	2	yes	—
46	C₂×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
47	C₂×C₄×C₂₃	#41	1	2	yes	—
48	C₂²⋊D₂₃	—	4	2	no	—
49	C₂²⋊D₂₃	—	4	2	no	—
50	C₂×Dic₂₃	—	2	1	no	—
51	C₂×Dic₂₃	—	2	1	no	—
52	C₂×Dic₂₃	—	2	2	no	—
53	C₂×Dic₂₃	—	4	2	no	—
54	D₉₂	—	2	2	no	—
55	D₉₂	—	2	2	no	—
56	C₄×D₂₃	—	4	2	no	—
57	C₄×D₂₃	—	4	2	no	—
58	Dic₄₆	—	2	2	no	—
59	Dic₄₆	—	2	2	no	—
60	C₈×C₂₃	—	2	2	no	—
61	C₈×C₂₃	—	4	2	no	—
62	C₂₃⋊C₈	—	4	1	no	—
63	#2050	—	23	1	no	maximal, Sylow
64	C₂²×C₂₃	—	4	1	no	—
65	C₂²×C₂₃	#121	1	1	yes	characteristic
66	C₂²×C₂₃	#123	1	2	yes	—
67	C₂×D₂₃	—	4	2	no	—
68	C₂×D₂₃	—	8	2	no	—
69	C₄×C₂₃	#120	1	1	yes	characteristic
70	C₄×C₂₃	#121	1	2	yes	—
71	C₄×C₂₃	#124	1	1	yes	characteristic
72	Dic₂₃	—	2	2	no	—
73	Dic₂₃	—	4	1	no	—
74	Dic₂₃	—	4	2	no	—
75	#575	—	23	1	no	—
76	#423	—	23	1	no	—
77	C₂×#104	—	23	2	no	—
78	#413	—	23	2	no	—
79	#338	D₂₃	1	1	yes	characteristic, complemented
80	C₂×C₂₃	—	2	1	no	—
81	C₂×C₂₃	—	2	2	no	—
82	C₂×C₂₃	#394	1	1	yes	characteristic
83	D₂₃	—	8	2	no	—
84	C₂×#41	—	23	2	no	—
85	#137	—	46	2	no	—
86	AGL(1,ℤ/8)	—	46	2	no	—
87	#135	—	46	2	no	—
88	C₂×M₁₆	—	23	1	no	—
89	C₂×M₁₆	C₂×D₂₃	1	2	yes	—
90	C₂×M₁₆	C₂×D₂₃	1	1	yes	characteristic
91	C₂×C₄²	—	23	1	no	—
92	C₄⋊C₈	—	46	2	no	—
93	#104	—	46	2	no	—
94	#104	—	46	2	no	—
95	#101	—	46	2	no	—
96	#100	—	46	2	no	—
97	C₂₃	#2050	1	1	yes	characteristic, complemented, Sylow
98	#41	—	46	2	no	—
99	#41	—	46	2	no	—
100	#41	—	46	2	no	—
101	#41	—	46	2	no	—
102	C₂×Q₈	—	23	2	no	—
103	C₂×D₄	—	23	2	no	—
104	C₂×D₄	—	46	2	no	—
105	C₂²×C₄	—	46	1	no	—
106	C₂²×C₄	—	46	2	no	—
107	C₂²×C₄	C₂²×D₂₃	1	1	yes	characteristic
108	Q₁₆	—	46	2	no	—
109	SD₁₆	—	46	4	no	—
110	D₈	—	46	2	no	—
111	M₁₆	—	2	2	no	—
112	M₁₆	—	2	2	no	—
113	M₁₆	—	2	2	no	—
114	M₁₆	—	46	2	no	—
115	C₂×C₈	—	2	2	no	—
116	C₂×C₈	—	46	1	no	—
117	C₂×C₈	D₉₂	1	2	yes	—
118	C₄²	—	23	1	no	—
119	C₄²	—	23	1	no	—
120	C₄²	—	46	1	no	—
121	C₂³	—	46	2	no	—
122	C₂³	#67292	1	1	yes	characteristic
123	Q₈	—	46	2	no	—
124	Q₈	—	46	2	no	—
125	D₄	—	46	2	no	—
126	D₄	—	46	2	no	—
127	D₄	—	92	2	no	—
128	D₄	—	92	2	no	—
129	C₂×C₄	—	46	1	no	—
130	C₂×C₄	—	46	2	no	—
131	C₂×C₄	—	46	1	no	—
132	C₂×C₄	—	92	2	no	—
133	C₂×C₄	—	92	2	no	—
134	C₂×C₄	—	92	2	no	—
135	C₂×C₄	C₂×D₉₂	1	1	yes	characteristic
136	C₂×C₄	#67288	1	2	yes	—
137	C₂×C₄	D₄×D₂₃	1	2	yes	—
138	C₂×C₄	D₄×D₂₃	1	1	yes	characteristic
139	C₈	—	2	2	no	—
140	C₈	—	4	2	no	—
141	C₈	—	92	1	no	—
142	C₂²	—	4	1	no	—
143	C₂²	—	92	2	no	—
144	C₂²	—	184	2	no	—
145	C₂²	#10625065	1	2	yes	—
146	C₂²	#10625066	1	1	yes	characteristic
147	C₄	—	46	2	no	—
148	C₄	—	92	1	no	—
149	C₄	—	92	2	no	—
150	C₄	#10625046	1	1	yes	characteristic
151	C₄	#10625053	1	1	yes	characteristic
152	C₄	#10625055	1	2	yes	—
153	C₂	—	2	1	no	—
154	C₂	—	2	2	no	—
155	C₂	—	184	2	no	—
156	C₂	#13092335	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#13093217

C₂

yes

maximal, characteristic

#13092440

C₂

yes

maximal, characteristic, complemented

C₂×#10625007

C₂

yes

maximal, complemented

#13092399

C₂

yes

maximal, complemented

C₂₃×#338

C₂

yes

maximal, characteristic, complemented

C₂×#67288

C₂²

yes

—

C₂×M₁₆×C₂₃

C₂²

yes

—

C₂×M₁₆×C₂₃

C₂²

yes

characteristic

C₂×C₄×Dic₂₃

C₂²

yes

characteristic

C₂×#67267

C₂²

yes

—

—

—

—

—

—

—

Dic₂₃⋊C₈

—

C₂²×C₄×C₂₃

C₂³

yes

characteristic

C₂×C₂²⋊D₂₃

—

C₂²×Dic₂₃

—

—

—

—

—

C₂×D₉₂

D₄

yes

—

C₂×C₄×D₂₃

—

C₂×Dic₄₆

D₄

yes

—

M₁₆×C₂₃

—

M₁₆×C₂₃

—

M₁₆×C₂₃

—

C₂×C₈×C₂₃

—

C₂×C₈×C₂₃

D₄

yes

—

C₄×Dic₂₃

—

C₄×Dic₂₃

D₄

yes

characteristic

C₄×Dic₂₃

D₄

yes

characteristic

#67267

—

C₂×C₂₃⋊C₈

—

Dic₉₂

—

D₁₈₄

—

#67263

—

C₂³×C₂₃

#41

yes

characteristic

C₂²×D₂₃

—

C₂×C₄×C₂₃

C₂×D₄

yes

characteristic

C₂×C₄×C₂₃

C₂×D₄

yes

—

C₂×C₄×C₂₃

C₂×D₄

yes

characteristic

C₂×C₄×C₂₃

#41

yes

—

C₂²⋊D₂₃

—

C₂²⋊D₂₃

—

C₂×Dic₂₃

—

C₂×Dic₂₃

—

C₂×Dic₂₃

—

C₂×Dic₂₃

—

D₉₂

—

D₉₂

—

C₄×D₂₃

—

C₄×D₂₃

—

Dic₄₆

—

Dic₄₆

—

C₈×C₂₃

—

C₈×C₂₃

—

C₂₃⋊C₈

—

#2050

—

maximal, Sylow

C₂²×C₂₃

—

C₂²×C₂₃

#121

yes

characteristic

C₂²×C₂₃

#123

yes

—

C₂×D₂₃

—

C₂×D₂₃

—

C₄×C₂₃

#120

yes

characteristic

C₄×C₂₃

#121

yes

—

C₄×C₂₃

#124

yes

characteristic

Dic₂₃

—

Dic₂₃

—

Dic₂₃

—

—

—

—

—

D₂₃

yes

characteristic, complemented

C₂×C₂₃

—

C₂×C₂₃

—

C₂×C₂₃

#394

yes

characteristic

D₂₃

—

C₂×#41

—

#137

—

AGL(1,ℤ/8)

—

#135

—

C₂×M₁₆

—

C₂×M₁₆

C₂×D₂₃

yes

—

C₂×M₁₆

C₂×D₂₃

yes

characteristic

C₂×C₄²

—

C₄⋊C₈

—

—

—

—

—

C₂₃

#2050

yes

characteristic, complemented, Sylow

#41

—

#41

—

100

#41

—

101

#41

—

102

C₂×Q₈

—

103

C₂×D₄

—

104

C₂×D₄

—

105

C₂²×C₄

—

106

C₂²×C₄

—

107

C₂²×C₄

C₂²×D₂₃

yes

characteristic

108

Q₁₆

—

109

SD₁₆

—

110

D₈

—

111

M₁₆

—

112

M₁₆

—

113

M₁₆

—

114

M₁₆

—

115

C₂×C₈

—

116

C₂×C₈

—

117

C₂×C₈

D₉₂

yes

—

118

C₄²

—

119

C₄²

—

120

C₄²

—

121

C₂³

—

122

C₂³

#67292

yes

characteristic

123

Q₈

—

124

Q₈

—

125

D₄

—

126

D₄

—

127

D₄

—

128

D₄

—

129

C₂×C₄

—

130

C₂×C₄

—

131

C₂×C₄

—

132

C₂×C₄

—

133

C₂×C₄

—

134

C₂×C₄

—

135

C₂×C₄

C₂×D₉₂

yes

characteristic

136

C₂×C₄

#67288

yes

—

137

C₂×C₄

D₄×D₂₃

yes

—

138

C₂×C₄

D₄×D₂₃

yes

characteristic

139

C₈

—

140

C₈

—

141

C₈

—

142

C₂²

—

143

C₂²

—

144

C₂²

—

184

—

145

C₂²

#10625065

yes

—

146

C₂²

#10625066

yes

characteristic

147

C₄

—

148

C₄

—

149

C₄

—

150

C₄

#10625046

yes

characteristic

151

C₄

#10625053

yes

characteristic

152

C₄

#10625055

yes

—

153

C₂

—

154

C₂

—

155

C₂

—

184

—

156

C₂

#13092335

yes

characteristic

	subgroup	G/H
center	C₄	#10625046
commutator, cosocle	C₂²×C₄×C₂₃	C₂³
Frattini	C₂²×C₄	C₂²×D₂₃
Fitting	C₂₃×#338	C₂
socle	C₂×C₂₃	#394
prime radical	#13093217	C₂
hypercenter	#338	D₂₃
hypocenter	C₂₃	#2050
LCS	G ▷ C₂²×C₄×C₂₃ ▷ C₂×C₂₃ ▷ C₂₃
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ #338

subgroup

G/H

center

C₄

#10625046

commutator, cosocle

C₂²×C₄×C₂₃

C₂³

Frattini

C₂²×C₄

C₂²×D₂₃

Fitting

C₂₃×#338

C₂

C₂×C₂₃

C₂

D₂₃

C₂₃

G ▷ C₂²×C₄×C₂₃ ▷ C₂×C₂₃ ▷ C₂₃

UCS

C₁ ◁ C₄ ◁ C₂²×C₄ ◁ #338

N	H ≅ G/N	type
2. #13092440	C₂	semidirect
3. C₂×#10625007	C₂	semidirect
4. #13092399	C₂	semidirect
5. C₂₃×#338	C₂	semidirect
79. #338	D₂₃	semidirect
97. C₂₃	#2050	semidirect

H ≅ G/N

type

2. #13092440

C₂

semidirect

3. C₂×#10625007

C₂

semidirect

4. #13092399

C₂

semidirect

5. C₂₃×#338

C₂

semidirect

79. #338

D₂₃

semidirect

97. C₂₃

#2050

semidirect

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

subgroup	index
#13093217	2
#13092440	2
C₂×#10625007	2
#13092399	2
C₂₃×#338	2
#2050	23

subgroup

index

C₂₃×#338

Subgroups — #15674685

Conjugacy classes of subgroups

Complements

Maximal subgroups