Group #14573237 — subgroups

Subgroups — C₂×#11926802 #14573237

Order 2432.

492 conjugacy classes of subgroups; only the 165 normal subgroups are shown.

normal subgroups only

This group has 492 conjugacy classes of subgroups — too many to draw. Only its 165 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	C₂²×Dic₇₆	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×#10597370	C₂	1	2	yes	maximal
3	#11926802	C₂	1	8	yes	maximal, direct product factor
4	C₂×#10597274	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×#10597273	C₂	1	2	yes	maximal, complemented
6	C₂²×M₁₆×C₁₉	C₂	1	1	yes	maximal, characteristic
7	C₂³×C₄×C₁₉	C₂²	1	1	yes	characteristic
8	C₂²×Dic₃₈	C₂²	1	2	yes	—
9	C₂×M₁₆×C₁₉	—	2	4	no	—
10	C₂×M₁₆×C₁₉	C₂²	1	4	yes	—
11	C₂²×C₈×C₁₉	C₂²	1	1	yes	characteristic
12	C₂²×C₈×C₁₉	C₂²	1	1	yes	characteristic
13	C₂×C₂²⋊Dic₁₉	—	2	2	no	—
14	C₂×C₄⋊Dic₁₉	C₂²	1	2	yes	—
15	#10597370	—	2	4	no	—
16	#10597370	C₂²	1	8	yes	—
17	C₂×Dic₁₉⋊C₄	—	2	2	no	—
18	C₂×Dic₇₆	—	2	4	no	—
19	C₂×Dic₇₆	C₂²	1	4	yes	complemented
20	#10597274	C₂²	1	4	yes	complemented
21	#10597273	C₂²	1	8	yes	complemented
22	C₂⁴×C₁₉	D₄	1	1	yes	characteristic
23	C₂²×C₄×C₁₉	—	2	2	no	—
24	C₂²×C₄×C₁₉	—	2	2	no	—
25	C₂²×C₄×C₁₉	D₄	1	1	yes	characteristic
26	C₂²×C₄×C₁₉	D₄	1	2	yes	—
27	C₂²×C₄×C₁₉	C₂³	1	2	yes	—
28	C₂²×C₄×C₁₉	C₂³	1	1	yes	characteristic
29	C₂²×Dic₁₉	—	2	2	no	—
30	C₂²×Dic₁₉	—	2	2	no	—
31	C₂×Dic₃₈	—	2	4	no	—
32	C₂×Dic₃₈	—	2	4	no	—
33	C₂×Dic₃₈	—	2	2	no	—
34	C₂×Dic₃₈	C₂³	1	4	yes	—
35	M₁₆×C₁₉	—	2	8	no	—
36	C₂×C₈×C₁₉	—	2	2	no	—
37	C₂×C₈×C₁₉	D₄	1	4	yes	—
38	C₂×C₈×C₁₉	C₂³	1	2	yes	—
39	C₂×C₈×C₁₉	C₂³	1	2	yes	—
40	C₂²⋊Dic₁₉	—	2	8	no	—
41	C₄⋊Dic₁₉	—	2	2	no	—
42	C₄⋊Dic₁₉	C₂³	1	4	yes	—
43	Dic₁₉⋊C₄	—	2	8	no	—
44	Dic₇₆	—	2	8	no	—
45	C₂³×C₁₉	—	2	2	no	—
46	C₂³×C₁₉	—	2	4	no	—
47	C₂³×C₁₉	C₂×D₄	1	2	yes	—
48	C₂³×C₁₉	C₂×D₄	1	1	yes	characteristic
49	C₂×C₄×C₁₉	—	2	2	no	—
50	C₂×C₄×C₁₉	—	2	4	no	—
51	C₂×C₄×C₁₉	—	2	2	no	—
52	C₂×C₄×C₁₉	—	2	2	no	—
53	C₂×C₄×C₁₉	C₂×D₄	1	2	yes	—
54	C₂×C₄×C₁₉	C₂×D₄	1	1	yes	characteristic
55	C₂×C₄×C₁₉	C₂×D₄	1	2	yes	—
56	C₂×C₄×C₁₉	#41	1	2	yes	—
57	C₂×C₄×C₁₉	C₂⁴	1	1	yes	characteristic
58	C₂×Dic₁₉	—	2	4	no	—
59	C₂×Dic₁₉	—	2	4	no	—
60	C₂×Dic₁₉	—	4	4	no	—
61	C₂×Dic₁₉	—	4	4	no	—
62	Dic₃₈	—	2	4	no	—
63	Dic₃₈	—	2	4	no	—
64	Dic₃₈	—	4	4	no	—
65	C₈×C₁₉	—	2	2	no	—
66	C₈×C₁₉	C₂×D₄	1	4	yes	—
67	C₂×#470	—	19	1	no	maximal, Sylow
68	C₂²×C₁₉	—	2	2	no	—
69	C₂²×C₁₉	—	2	4	no	—
70	C₂²×C₁₉	—	4	4	no	—
71	C₂²×C₁₉	#121	1	2	yes	—
72	C₂²×C₁₉	#137	1	4	yes	—
73	C₂²×C₁₉	C₂²×D₄	1	1	yes	characteristic
74	C₄×C₁₉	—	2	2	no	—
75	C₄×C₁₉	#121	1	2	yes	—
76	C₄×C₁₉	C₂²×D₄	1	1	yes	characteristic
77	C₄×C₁₉	C₂×#41	1	1	yes	characteristic
78	Dic₁₉	—	4	4	no	—
79	Dic₁₉	—	4	4	no	—
80	C₂²×Q₁₆	—	19	1	no	—
81	C₂²×M₁₆	D₁₉	1	1	yes	characteristic
82	C₂×C₂²⋊Q₈	—	19	2	no	—
83	#470	—	19	8	no	—
84	C₂×#106	—	19	1	no	—
85	C₂×#103	—	19	2	no	—
86	C₂×C₁₉	—	4	2	no	—
87	C₂×C₁₉	#470	1	4	yes	complemented
88	C₂×C₁₉	C₂×#121	1	1	yes	characteristic
89	C₂×C₁₉	C₂×#137	1	2	yes	—
90	C₂²×Q₈	—	19	2	no	—
91	C₂³×C₄	C₂×D₁₉	1	1	yes	characteristic
92	C₂×Q₁₆	—	19	4	no	—
93	C₂×Q₁₆	—	38	4	no	—
94	C₂×M₁₆	—	2	4	no	—
95	C₂×M₁₆	C₂×D₁₉	1	4	yes	—
96	C₂²×C₈	C₂×D₁₉	1	1	yes	characteristic
97	C₂²×C₈	C₂×D₁₉	1	1	yes	characteristic
98	C₂²⋊Q₈	—	19	8	no	—
99	C₂²⋊Q₈	—	38	4	no	—
100	C₂×C₄⋊C₄	—	19	2	no	—
101	C₂×C₄⋊C₄	—	38	2	no	—
102	C₂×C₂²⋊C₄	—	38	2	no	—
103	#106	—	19	4	no	—
104	#103	—	19	8	no	—
105	C₁₉	C₂×#470	1	1	yes	characteristic, complemented, Sylow
106	C₂⁴	D₇₆	1	1	yes	characteristic
107	C₂×Q₈	—	19	4	no	—
108	C₂×Q₈	—	38	2	no	—
109	C₂×Q₈	—	38	4	no	—
110	C₂×Q₈	—	38	4	no	—
111	C₂²×C₄	—	2	2	no	—
112	C₂²×C₄	—	2	2	no	—
113	C₂²×C₄	—	38	2	no	—
114	C₂²×C₄	—	38	2	no	—
115	C₂²×C₄	D₇₆	1	2	yes	—
116	C₂²×C₄	D₇₆	1	1	yes	characteristic
117	C₂²×C₄	C₂²×D₁₉	1	2	yes	—
118	C₂²×C₄	C₂²×D₁₉	1	1	yes	characteristic
119	Q₁₆	—	38	8	no	—
120	M₁₆	—	2	8	no	—
121	C₂×C₈	—	2	2	no	—
122	C₂×C₈	C₂²⋊D₁₉	1	4	yes	—
123	C₂×C₈	C₂²×D₁₉	1	2	yes	—
124	C₂×C₈	C₂²×D₁₉	1	2	yes	—
125	C₄⋊C₄	—	19	4	no	—
126	C₄⋊C₄	—	38	8	no	—
127	C₄⋊C₄	—	38	2	no	—
128	C₂²⋊C₄	—	38	8	no	—
129	C₂³	—	2	2	no	—
130	C₂³	—	2	4	no	—
131	C₂³	C₂×D₇₆	1	2	yes	—
132	C₂³	C₂×D₇₆	1	1	yes	characteristic
133	Q₈	—	38	4	no	—
134	Q₈	—	38	4	no	—
135	Q₈	—	76	4	no	—
136	C₂×C₄	—	2	2	no	—
137	C₂×C₄	—	2	4	no	—
138	C₂×C₄	—	2	2	no	—
139	C₂×C₄	—	2	2	no	—
140	C₂×C₄	—	38	4	no	—
141	C₂×C₄	—	38	4	no	—
142	C₂×C₄	—	76	4	no	—
143	C₂×C₄	—	76	4	no	—
144	C₂×C₄	C₂×D₇₆	1	2	yes	—
145	C₂×C₄	C₂×D₇₆	1	1	yes	characteristic
146	C₂×C₄	#64541	1	2	yes	—
147	C₂×C₄	C₂×C₂²⋊D₁₉	1	2	yes	—
148	C₂×C₄	C₂³×D₁₉	1	1	yes	characteristic
149	C₈	—	2	2	no	—
150	C₈	C₂×C₂²⋊D₁₉	1	4	yes	—
151	C₂²	—	2	4	no	—
152	C₂²	—	2	2	no	—
153	C₂²	—	4	4	no	—
154	C₂²	#10597355	1	4	yes	—
155	C₂²	#10597376	1	2	yes	—
156	C₂²	C₂²×D₇₆	1	1	yes	characteristic
157	C₄	—	2	2	no	—
158	C₄	—	76	4	no	—
159	C₄	—	76	4	no	—
160	C₄	#10597376	1	2	yes	—
161	C₄	C₂×#64541	1	1	yes	characteristic
162	C₄	C₂²×C₂²⋊D₁₉	1	1	yes	characteristic
163	C₂	—	4	2	no	—
164	C₂	#11926802	1	4	yes	direct product factor
165	C₂	C₂×#10597376	1	1	yes	characteristic
166	C₂	C₂×#10597355	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂²×Dic₇₆

C₂

yes

maximal, characteristic, complemented

C₂×#10597370

C₂

yes

maximal

#11926802

C₂

yes

maximal, direct product factor

C₂×#10597274

C₂

yes

maximal, characteristic, complemented

C₂×#10597273

C₂

yes

maximal, complemented

C₂²×M₁₆×C₁₉

C₂

yes

maximal, characteristic

C₂³×C₄×C₁₉

C₂²

yes

characteristic

C₂²×Dic₃₈

C₂²

yes

—

C₂×M₁₆×C₁₉

—

C₂×M₁₆×C₁₉

C₂²

yes

—

C₂²×C₈×C₁₉

C₂²

yes

characteristic

C₂²×C₈×C₁₉

C₂²

yes

characteristic

C₂×C₂²⋊Dic₁₉

—

C₂×C₄⋊Dic₁₉

C₂²

yes

—

#10597370

—

#10597370

C₂²

yes

—

C₂×Dic₁₉⋊C₄

—

C₂×Dic₇₆

—

C₂×Dic₇₆

C₂²

yes

complemented

#10597274

C₂²

yes

complemented

#10597273

C₂²

yes

complemented

C₂⁴×C₁₉

D₄

yes

characteristic

C₂²×C₄×C₁₉

—

C₂²×C₄×C₁₉

—

C₂²×C₄×C₁₉

D₄

yes

characteristic

C₂²×C₄×C₁₉

D₄

yes

—

C₂²×C₄×C₁₉

C₂³

yes

—

C₂²×C₄×C₁₉

C₂³

yes

characteristic

C₂²×Dic₁₉

—

C₂²×Dic₁₉

—

C₂×Dic₃₈

—

C₂×Dic₃₈

—

C₂×Dic₃₈

—

C₂×Dic₃₈

C₂³

yes

—

M₁₆×C₁₉

—

C₂×C₈×C₁₉

—

C₂×C₈×C₁₉

D₄

yes

—

C₂×C₈×C₁₉

C₂³

yes

—

C₂×C₈×C₁₉

C₂³

yes

—

C₂²⋊Dic₁₉

—

C₄⋊Dic₁₉

—

C₄⋊Dic₁₉

C₂³

yes

—

Dic₁₉⋊C₄

—

Dic₇₆

—

C₂³×C₁₉

—

C₂³×C₁₉

—

C₂³×C₁₉

C₂×D₄

yes

—

C₂³×C₁₉

C₂×D₄

yes

characteristic

C₂×C₄×C₁₉

—

C₂×C₄×C₁₉

—

C₂×C₄×C₁₉

—

C₂×C₄×C₁₉

—

C₂×C₄×C₁₉

C₂×D₄

yes

—

C₂×C₄×C₁₉

C₂×D₄

yes

characteristic

C₂×C₄×C₁₉

C₂×D₄

yes

—

C₂×C₄×C₁₉

#41

yes

—

C₂×C₄×C₁₉

C₂⁴

yes

characteristic

C₂×Dic₁₉

—

C₂×Dic₁₉

—

C₂×Dic₁₉

—

C₂×Dic₁₉

—

Dic₃₈

—

Dic₃₈

—

Dic₃₈

—

C₈×C₁₉

—

C₈×C₁₉

C₂×D₄

yes

—

C₂×#470

—

maximal, Sylow

C₂²×C₁₉

—

C₂²×C₁₉

—

C₂²×C₁₉

—

C₂²×C₁₉

#121

yes

—

C₂²×C₁₉

#137

yes

—

C₂²×C₁₉

C₂²×D₄

yes

characteristic

C₄×C₁₉

—

C₄×C₁₉

#121

yes

—

C₄×C₁₉

C₂²×D₄

yes

characteristic

C₄×C₁₉

C₂×#41

yes

characteristic

Dic₁₉

—

Dic₁₉

—

C₂²×Q₁₆

—

C₂²×M₁₆

D₁₉

yes

characteristic

C₂×C₂²⋊Q₈

—

#470

—

C₂×#106

—

C₂×#103

—

C₂×C₁₉

—

C₂×C₁₉

yes

C₂×C₁₉

yes

C₂×C₁₉

yes

—

C₂²×Q₈

—

C₂³×C₄

C₂×D₁₉

yes

characteristic

C₂×Q₁₆

—

C₂×Q₁₆

—

C₂×M₁₆

—

C₂×M₁₆

C₂×D₁₉

yes

—

C₂²×C₈

C₂×D₁₉

yes

characteristic

C₂²×C₈

C₂×D₁₉

yes

characteristic

C₂²⋊Q₈

—

C₂²⋊Q₈

—

100

C₂×C₄⋊C₄

—

101

C₂×C₄⋊C₄

—

102

C₂×C₂²⋊C₄

—

103

#106

—

104

#103

—

105

C₁₉

C₂×#470

yes

characteristic, complemented, Sylow

106

C₂⁴

D₇₆

yes

characteristic

107

C₂×Q₈

—

108

C₂×Q₈

—

109

C₂×Q₈

—

110

C₂×Q₈

—

111

C₂²×C₄

—

112

C₂²×C₄

—

113

C₂²×C₄

—

114

C₂²×C₄

—

115

C₂²×C₄

D₇₆

yes

—

116

C₂²×C₄

D₇₆

yes

characteristic

117

C₂²×C₄

C₂²×D₁₉

yes

—

118

C₂²×C₄

C₂²×D₁₉

yes

characteristic

119

Q₁₆

—

120

M₁₆

—

121

C₂×C₈

—

122

C₂×C₈

C₂²⋊D₁₉

yes

—

123

C₂×C₈

C₂²×D₁₉

yes

—

124

C₂×C₈

C₂²×D₁₉

yes

—

125

C₄⋊C₄

—

126

C₄⋊C₄

—

127

C₄⋊C₄

—

128

C₂²⋊C₄

—

129

C₂³

—

130

C₂³

—

131

C₂³

C₂×D₇₆

yes

—

132

C₂³

C₂×D₇₆

yes

characteristic

133

Q₈

—

134

Q₈

—

135

Q₈

—

136

C₂×C₄

—

137

C₂×C₄

—

138

C₂×C₄

—

139

C₂×C₄

—

140

C₂×C₄

—

141

C₂×C₄

—

142

C₂×C₄

—

143

C₂×C₄

—

144

C₂×C₄

C₂×D₇₆

yes

—

145

C₂×C₄

C₂×D₇₆

yes

characteristic

146

C₂×C₄

#64541

yes

—

147

C₂×C₄

C₂×C₂²⋊D₁₉

yes

—

148

C₂×C₄

C₂³×D₁₉

yes

characteristic

149

C₈

—

150

C₈

C₂×C₂²⋊D₁₉

yes

—

151

C₂²

—

152

C₂²

—

153

C₂²

—

154

C₂²

#10597355

yes

—

155

C₂²

#10597376

yes

—

156

C₂²

C₂²×D₇₆

yes

characteristic

157

C₄

—

158

C₄

—

159

C₄

—

160

C₄

#10597376

yes

—

161

C₄

C₂×#64541

yes

characteristic

162

C₄

C₂²×C₂²⋊D₁₉

yes

characteristic

163

C₂

—

164

C₂

#11926802

yes

direct product factor

165

C₂

C₂×#10597376

yes

characteristic

166

C₂

C₂×#10597355

yes

—

	subgroup	G/H
center	C₂³	C₂×D₇₆
commutator, cosocle	C₂×C₄×C₁₉	C₂⁴
Frattini	C₂×C₄	C₂³×D₁₉
Fitting	C₂²×M₁₆×C₁₉	C₂
socle	C₂³×C₁₉	C₂×D₄
prime radical	C₂⁴×C₁₉	D₄
hypercenter	C₂²×M₁₆	D₁₉
hypocenter	C₁₉	C₂×#470
LCS	G ▷ C₂×C₄×C₁₉ ▷ C₂×C₁₉ ▷ C₁₉
UCS	C₁ ◁ C₂³ ◁ C₂³×C₄ ◁ C₂²×M₁₆

subgroup

G/H

center

C₂³

C₂×D₇₆

commutator, cosocle

C₂×C₄×C₁₉

C₂⁴

Frattini

C₂×C₄

C₂³×D₁₉

Fitting

C₂²×M₁₆×C₁₉

C₂

socle

C₂³×C₁₉

C₂×D₄

prime radical

C₂⁴×C₁₉

D₄

hypercenter

C₂²×M₁₆

D₁₉

hypocenter

C₁₉

C₂×#470

LCS

G ▷ C₂×C₄×C₁₉ ▷ C₂×C₁₉ ▷ C₁₉

UCS

C₁ ◁ C₂³ ◁ C₂³×C₄ ◁ C₂²×M₁₆

N	H ≅ G/N	type
1. C₂²×Dic₇₆	C₂	semidirect
3. #11926802	C₂	direct (N × H)
4. C₂×#10597274	C₂	semidirect
5. C₂×#10597273	C₂	semidirect
19. C₂×Dic₇₆	C₂²	semidirect
20. #10597274	C₂²	semidirect
21. #10597273	C₂²	semidirect
87. C₂×C₁₉	#470	semidirect
105. C₁₉	C₂×#470	semidirect
164. C₂	#11926802	direct (N × H)

H ≅ G/N

type

1. C₂²×Dic₇₆

C₂

semidirect

3. #11926802

C₂

direct (N × H)

4. C₂×#10597274

C₂

semidirect

5. C₂×#10597273

C₂

semidirect

19. C₂×Dic₇₆

C₂²

semidirect

20. #10597274

C₂²

semidirect

21. #10597273

C₂²

semidirect

87. C₂×C₁₉

#470

semidirect

105. C₁₉

C₂×#470

semidirect

164. C₂

#11926802

direct (N × H)

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

subgroup	index
C₂²×Dic₇₆	2
C₂×#10597370	2
#11926802	2
C₂×#10597274	2
C₂×#10597273	2
C₂²×M₁₆×C₁₉	2
C₂×#470	19

subgroup

index

C₂²×Dic₇₆

C₂²×M₁₆×C₁₉

Subgroups — C₂×#11926802 #14573237

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C2×#11926802 #14573237

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#11926802 #14573237