Group #15673609 — subgroups

Subgroups — #15673609

Order 2944.

324 conjugacy classes of subgroups; only the 97 normal subgroups are shown.

normal subgroups only

This group has 324 conjugacy classes of subgroups — too many to draw. Only its 97 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	#13092892	C₂	1	1	yes	maximal, characteristic
2	#13092459	C₂	1	2	yes	maximal, complemented
3	#13092320	C₂	1	1	yes	maximal, characteristic
4	#13092178	C₂	1	2	yes	maximal, complemented
5	C₂×C₂₃×#100	C₂	1	1	yes	maximal, characteristic
6	C₂²×D₄×C₂₃	C₂²	1	1	yes	characteristic
7	C₂×M₁₆×C₂₃	C₂²	1	2	yes	—
8	C₂×C₂²⋊Dic₂₃	—	2	2	no	—
9	C₂×C₂²⋊Dic₂₃	—	2	2	no	—
10	D₄×Dic₂₃	—	2	2	no	—
11	D₄×Dic₂₃	—	2	2	no	—
12	#10625097	C₂²	1	2	yes	—
13	C₂×C₄⋊Dic₂₃	C₂²	1	1	yes	characteristic
14	C₂×C₄×Dic₂₃	C₂²	1	1	yes	characteristic
15	C₂₃×#100	C₄	1	4	yes	complemented
16	#10625000	—	2	2	no	—
17	#10624999	—	2	2	no	—
18	C₂⁴×C₂₃	D₄	1	2	yes	—
19	C₂×D₄×C₂₃	—	2	2	no	—
20	C₂×D₄×C₂₃	—	2	2	no	—
21	C₂×D₄×C₂₃	C₂×C₄	1	2	yes	—
22	C₂×D₄×C₂₃	Q₈	1	2	yes	—
23	C₂²×C₄×C₂₃	C₂³	1	1	yes	characteristic
24	C₂²×Dic₂₃	—	2	1	no	—
25	C₂²×Dic₂₃	—	2	2	no	—
26	C₂²×Dic₂₃	—	2	2	no	—
27	C₂²×Dic₂₃	—	2	1	no	—
28	M₁₆×C₂₃	—	2	2	no	—
29	M₁₆×C₂₃	C₂×C₄	1	4	yes	complemented
30	C₂×C₈×C₂₃	—	2	2	no	—
31	C₂²⋊Dic₂₃	—	2	4	no	—
32	C₂²⋊Dic₂₃	—	2	2	no	—
33	C₂²⋊Dic₂₃	—	4	4	no	—
34	C₄⋊Dic₂₃	—	2	2	no	—
35	C₄⋊Dic₂₃	D₄	1	2	yes	—
36	C₄⋊Dic₂₃	D₄	1	2	yes	—
37	C₄×Dic₂₃	—	2	2	no	—
38	C₄×Dic₂₃	—	2	2	no	—
39	C₂³×C₂₃	—	2	2	no	—
40	C₂³×C₂₃	—	2	2	no	—
41	C₂³×C₂₃	—	4	2	no	—
42	C₂³×C₂₃	—	4	2	no	—
43	C₂³×C₂₃	C₄⋊C₄	1	4	yes	—
44	C₂³×C₂₃	C₂×D₄	1	1	yes	characteristic
45	D₄×C₂₃	—	2	4	no	—
46	D₄×C₂₃	—	2	4	no	—
47	C₂×C₄×C₂₃	C₂²×C₄	1	1	yes	characteristic
48	C₂×C₄×C₂₃	C₂×D₄	1	2	yes	—
49	C₂×C₄×C₂₃	C₂×Q₈	1	1	yes	characteristic
50	C₂×C₄×C₂₃	#41	1	2	yes	—
51	C₂×Dic₂₃	—	2	2	no	—
52	C₂×Dic₂₃	—	2	2	no	—
53	C₂×Dic₂₃	—	2	2	no	—
54	C₂×Dic₂₃	—	4	1	no	—
55	C₂×Dic₂₃	—	4	1	no	—
56	C₂×Dic₂₃	—	4	2	no	—
57	C₂×Dic₂₃	—	4	1	no	—
58	C₂×Dic₂₃	—	8	2	no	—
59	C₂×Dic₂₃	—	8	2	no	—
60	C₈×C₂₃	—	2	4	no	—
61	#1856	—	23	1	no	maximal, Sylow
62	C₂²×C₂₃	—	2	1	no	—
63	C₂²×C₂₃	—	2	1	no	—
64	C₂²×C₂₃	—	2	4	no	—
65	C₂²×C₂₃	—	4	4	no	—
66	C₂²×C₂₃	—	4	2	no	—
67	C₂²×C₂₃	—	4	2	no	—
68	C₂²×C₂₃	—	8	2	no	—
69	C₂²×C₂₃	C₂×C₄⋊C₄	1	1	yes	characteristic
70	C₂²×C₂₃	#120	1	1	yes	characteristic
71	C₂²×C₂₃	#123	1	1	yes	characteristic
72	C₄×C₂₃	C₄×D₄	1	2	yes	—
73	C₄×C₂₃	C₂²⋊Q₈	1	2	yes	—
74	Dic₂₃	—	4	2	no	—
75	Dic₂₃	—	4	2	no	—
76	Dic₂₃	—	4	2	no	—
77	Dic₂₃	—	8	1	no	—
78	#518	—	23	1	no	—
79	#428	—	23	2	no	—
80	C₂×#100	D₂₃	1	1	yes	characteristic
81	#390	—	23	1	no	—
82	#339	—	23	2	no	—
83	C₂×C₂₃	—	2	1	no	—
84	C₂×C₂₃	—	2	1	no	—
85	C₂×C₂₃	—	4	4	no	—
86	C₂×C₂₃	#385	1	1	yes	characteristic
87	C₂×C₂₃	#453	1	1	yes	characteristic
88	C₂×C₂₃	#455	1	1	yes	characteristic
89	C₂²×D₄	C₂×D₂₃	1	1	yes	characteristic
90	C₂×M₁₆	C₂×D₂₃	1	2	yes	—
91	C₄×D₄	—	46	2	no	—
92	C₄×D₄	—	46	2	no	—
93	#117	—	23	2	no	—
94	C₂×C₄⋊C₄	—	23	1	no	—
95	C₂×C₂²⋊C₄	—	46	2	no	—
96	C₂×C₂²⋊C₄	—	46	2	no	—
97	C₂×C₄²	—	23	1	no	—
98	#107	—	46	2	no	—
99	#106	—	46	2	no	—
100	#100	Dic₂₃	1	4	yes	complemented
101	C₂₃	#1856	1	1	yes	characteristic, complemented, Sylow
102	C₂⁴	D₉₂	1	2	yes	—
103	C₂×D₄	—	2	2	no	—
104	C₂×D₄	—	2	2	no	—
105	C₂×D₄	Dic₄₆	1	2	yes	—
106	C₂×D₄	C₂×Dic₂₃	1	2	yes	—
107	C₂²×C₄	—	46	1	no	—
108	C₂²×C₄	—	46	2	no	—
109	C₂²×C₄	—	46	2	no	—
110	C₂²×C₄	—	46	1	no	—
111	C₂²×C₄	C₂²×D₂₃	1	1	yes	characteristic
112	M₁₆	—	2	2	no	—
113	M₁₆	C₂×Dic₂₃	1	4	yes	complemented
114	C₂×C₈	—	2	2	no	—
115	C₄⋊C₄	—	23	2	no	—
116	C₄⋊C₄	—	23	2	no	—
117	C₄⋊C₄	—	46	2	no	—
118	C₂²⋊C₄	—	46	4	no	—
119	C₂²⋊C₄	—	46	2	no	—
120	C₂²⋊C₄	—	92	4	no	—
121	C₄²	—	46	2	no	—
122	C₄²	—	46	2	no	—
123	C₂³	—	2	2	no	—
124	C₂³	—	2	2	no	—
125	C₂³	—	4	2	no	—
126	C₂³	—	4	2	no	—
127	C₂³	C₄⋊Dic₂₃	1	4	yes	—
128	C₂³	C₂×D₉₂	1	1	yes	characteristic
129	D₄	—	2	4	no	—
130	D₄	—	2	4	no	—
131	C₂×C₄	—	46	2	no	—
132	C₂×C₄	—	46	2	no	—
133	C₂×C₄	—	46	2	no	—
134	C₂×C₄	—	92	1	no	—
135	C₂×C₄	—	92	1	no	—
136	C₂×C₄	—	92	2	no	—
137	C₂×C₄	—	92	1	no	—
138	C₂×C₄	—	184	2	no	—
139	C₂×C₄	—	184	2	no	—
140	C₂×C₄	C₂×Dic₄₆	1	1	yes	characteristic
141	C₂×C₄	D₄×D₂₃	1	2	yes	—
142	C₂×C₄	#67290	1	2	yes	—
143	C₂×C₄	C₂²×Dic₂₃	1	1	yes	characteristic
144	C₈	—	2	4	no	—
145	C₂²	—	2	1	no	—
146	C₂²	—	2	1	no	—
147	C₂²	—	2	4	no	—
148	C₂²	—	4	4	no	—
149	C₂²	—	4	2	no	—
150	C₂²	—	4	2	no	—
151	C₂²	—	8	2	no	—
152	C₂²	#10625053	1	1	yes	characteristic
153	C₂²	#10625057	1	1	yes	characteristic
154	C₂²	C₂×C₄⋊Dic₂₃	1	1	yes	characteristic
155	C₄	—	92	2	no	—
156	C₄	—	92	2	no	—
157	C₄	—	92	2	no	—
158	C₄	—	184	1	no	—
159	C₄	#10625049	1	2	yes	—
160	C₄	D₄×Dic₂₃	1	2	yes	—
161	C₂	—	2	1	no	—
162	C₂	—	2	1	no	—
163	C₂	—	4	4	no	—
164	C₂	#13092292	1	1	yes	characteristic
165	C₂	#13092587	1	1	yes	characteristic
166	C₂	#13092597	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#13092892

C₂

yes

maximal, characteristic

#13092459

C₂

yes

maximal, complemented

#13092320

C₂

yes

maximal, characteristic

#13092178

C₂

yes

maximal, complemented

C₂×C₂₃×#100

C₂

yes

maximal, characteristic

C₂²×D₄×C₂₃

C₂²

yes

characteristic

C₂×M₁₆×C₂₃

C₂²

yes

—

C₂×C₂²⋊Dic₂₃

—

C₂×C₂²⋊Dic₂₃

—

D₄×Dic₂₃

—

D₄×Dic₂₃

—

#10625097

C₂²

yes

—

C₂×C₄⋊Dic₂₃

C₂²

yes

characteristic

C₂×C₄×Dic₂₃

C₂²

yes

characteristic

C₂₃×#100

C₄

yes

complemented

#10625000

—

#10624999

—

C₂⁴×C₂₃

D₄

yes

—

C₂×D₄×C₂₃

—

C₂×D₄×C₂₃

—

C₂×D₄×C₂₃

C₂×C₄

yes

—

C₂×D₄×C₂₃

Q₈

yes

—

C₂²×C₄×C₂₃

C₂³

yes

characteristic

C₂²×Dic₂₃

—

C₂²×Dic₂₃

—

C₂²×Dic₂₃

—

C₂²×Dic₂₃

—

M₁₆×C₂₃

—

M₁₆×C₂₃

C₂×C₄

yes

complemented

C₂×C₈×C₂₃

—

C₂²⋊Dic₂₃

—

C₂²⋊Dic₂₃

—

C₂²⋊Dic₂₃

—

C₄⋊Dic₂₃

—

C₄⋊Dic₂₃

D₄

yes

—

C₄⋊Dic₂₃

D₄

yes

—

C₄×Dic₂₃

—

C₄×Dic₂₃

—

C₂³×C₂₃

—

C₂³×C₂₃

—

C₂³×C₂₃

—

C₂³×C₂₃

—

C₂³×C₂₃

C₄⋊C₄

yes

—

C₂³×C₂₃

C₂×D₄

yes

characteristic

D₄×C₂₃

—

D₄×C₂₃

—

C₂×C₄×C₂₃

C₂²×C₄

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

—

C₂×Dic₂₃

—

C₂×Dic₂₃

—

C₂×Dic₂₃

—

C₂×Dic₂₃

—

C₈×C₂₃

—

#1856

—

maximal, Sylow

C₂²×C₂₃

—

C₂²×C₂₃

—

C₂²×C₂₃

—

C₂²×C₂₃

—

C₂²×C₂₃

—

C₂²×C₂₃

—

C₂²×C₂₃

—

C₂²×C₂₃

C₂×C₄⋊C₄

yes

C₂²×C₂₃

yes

C₂²×C₂₃

yes

C₄×C₂₃

C₄×D₄

yes

—

C₄×C₂₃

C₂²⋊Q₈

yes

—

Dic₂₃

—

Dic₂₃

—

Dic₂₃

—

Dic₂₃

—

#518

—

#428

—

C₂×#100

D₂₃

yes

characteristic

#390

—

#339

—

C₂×C₂₃

—

C₂×C₂₃

—

C₂×C₂₃

—

C₂×C₂₃

yes

C₂×C₂₃

yes

C₂×C₂₃

yes

C₂²×D₄

C₂×D₂₃

yes

characteristic

C₂×M₁₆

C₂×D₂₃

yes

—

C₄×D₄

—

C₄×D₄

—

#117

—

C₂×C₄⋊C₄

—

C₂×C₂²⋊C₄

—

C₂×C₂²⋊C₄

—

C₂×C₄²

—

#107

—

#106

—

100

#100

Dic₂₃

yes

complemented

101

C₂₃

#1856

yes

characteristic, complemented, Sylow

102

C₂⁴

D₉₂

yes

—

103

C₂×D₄

—

104

C₂×D₄

—

105

C₂×D₄

Dic₄₆

yes

—

106

C₂×D₄

C₂×Dic₂₃

yes

—

107

C₂²×C₄

—

108

C₂²×C₄

—

109

C₂²×C₄

—

110

C₂²×C₄

—

111

C₂²×C₄

C₂²×D₂₃

yes

characteristic

112

M₁₆

—

113

M₁₆

C₂×Dic₂₃

yes

complemented

114

C₂×C₈

—

115

C₄⋊C₄

—

116

C₄⋊C₄

—

117

C₄⋊C₄

—

118

C₂²⋊C₄

—

119

C₂²⋊C₄

—

120

C₂²⋊C₄

—

121

C₄²

—

122

C₄²

—

123

C₂³

—

124

C₂³

—

125

C₂³

—

126

C₂³

—

127

C₂³

C₄⋊Dic₂₃

yes

—

128

C₂³

C₂×D₉₂

yes

characteristic

129

D₄

—

130

D₄

—

131

C₂×C₄

—

132

C₂×C₄

—

133

C₂×C₄

—

134

C₂×C₄

—

135

C₂×C₄

—

136

C₂×C₄

—

137

C₂×C₄

—

138

C₂×C₄

—

184

—

139

C₂×C₄

—

184

—

140

C₂×C₄

C₂×Dic₄₆

yes

characteristic

141

C₂×C₄

D₄×D₂₃

yes

—

142

C₂×C₄

#67290

yes

—

143

C₂×C₄

C₂²×Dic₂₃

yes

characteristic

144

C₈

—

145

C₂²

—

146

C₂²

—

147

C₂²

—

148

C₂²

—

149

C₂²

—

150

C₂²

—

151

C₂²

—

152

C₂²

#10625053

yes

characteristic

153

C₂²

#10625057

yes

characteristic

154

C₂²

C₂×C₄⋊Dic₂₃

yes

characteristic

155

C₄

—

156

C₄

—

157

C₄

—

158

C₄

—

184

—

159

C₄

#10625049

yes

—

160

C₄

D₄×Dic₂₃

yes

—

161

C₂

—

162

C₂

—

163

C₂

—

164

C₂

#13092292

yes

characteristic

165

C₂

#13092587

yes

characteristic

166

C₂

#13092597

yes

characteristic

	subgroup	G/H
center	C₂²	#10625053
commutator	C₂×C₄×C₂₃	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₂₃
Fitting	C₂×C₂₃×#100	C₂
socle	C₂²×C₂₃	#120
prime radical	C₂²×D₄×C₂₃	C₂²
cosocle	C₂²×C₄×C₂₃	C₂³
hypercenter	C₂×#100	D₂₃
hypocenter	C₂₃	#1856
LCS	G ▷ C₂×C₄×C₂₃ ▷ C₂×C₂₃ ▷ C₂₃
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×#100

subgroup

G/H

center

C₂²

#10625053

commutator

C₂×C₄×C₂₃

C₂²×C₄

Frattini

C₂²×C₄

C₂²×D₂₃

Fitting

C₂×C₂₃×#100

C₂

socle

C₂²×C₂₃

#120

prime radical

C₂²×D₄×C₂₃

C₂²

cosocle

C₂²×C₄×C₂₃

C₂³

D₂₃

C₂₃

G ▷ C₂×C₄×C₂₃ ▷ C₂×C₂₃ ▷ C₂₃

UCS

C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×#100

N	H ≅ G/N	type
2. #13092459	C₂	semidirect
4. #13092178	C₂	semidirect
15. C₂₃×#100	C₄	semidirect
29. M₁₆×C₂₃	C₂×C₄	semidirect
100. #100	Dic₂₃	semidirect
101. C₂₃	#1856	semidirect
113. M₁₆	C₂×Dic₂₃	semidirect

H ≅ G/N

type

2. #13092459

C₂

semidirect

4. #13092178

C₂

semidirect

15. C₂₃×#100

C₄

semidirect

29. M₁₆×C₂₃

C₂×C₄

semidirect

100. #100

Dic₂₃

semidirect

101. C₂₃

#1856

semidirect

113. M₁₆

C₂×Dic₂₃

semidirect

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

subgroup	index
#13092892	2
#13092459	2
#13092320	2
#13092178	2
C₂×C₂₃×#100	2
#1856	23

subgroup

index

C₂×C₂₃×#100

Subgroups — #15673609

Conjugacy classes of subgroups

Complements

Maximal subgroups