Group #14568121 — subgroups

Subgroups — #14568121

Order 2432.

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

normal subgroups only

This group has 512 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×C₄⋊Dic₁₉	C₂	1	1	yes	maximal, characteristic
2	#11926483	C₂	1	4	yes	maximal, complemented
3	#11926451	C₂	1	2	yes	maximal, complemented
4	#11926409	C₂	1	2	yes	maximal, complemented
5	#11926405	C₂	1	4	yes	maximal
6	C₄×C₂²⋊Dic₁₉	C₂	1	1	yes	maximal, characteristic
7	C₂×C₁₉×C₂²⋊Q₈	C₂	1	1	yes	maximal, characteristic
8	C₂²×Q₈×C₁₉	C₂²	1	1	yes	characteristic
9	C₂³×C₄×C₁₉	C₂²	1	1	yes	characteristic
10	C₂³×Dic₁₉	C₂²	1	2	yes	—
11	C₁₉×C₂²⋊Q₈	—	2	4	no	—
12	C₂×C₄⋊C₄×C₁₉	C₂²	1	1	yes	characteristic
13	C₂×C₄⋊C₄×C₁₉	C₂²	1	2	yes	—
14	C₂×C₂²⋊C₄×C₁₉	C₂²	1	2	yes	—
15	C₂×C₂²⋊Dic₁₉	C₂²	1	2	yes	—
16	C₂×C₄⋊Dic₁₉	—	2	4	no	—
17	C₂×C₄⋊Dic₁₉	C₂²	1	2	yes	—
18	C₂×C₄⋊Dic₁₉	C₂²	1	2	yes	—
19	C₂×C₄×Dic₁₉	C₂²	1	2	yes	—
20	C₁₉⋊#95	C₂²	1	2	yes	—
21	C₁₉⋊#95	C₂²	1	8	yes	—
22	C₁₉⋊#95	C₂²	1	8	yes	—
23	C₂⁴×C₁₉	C₂³	1	1	yes	characteristic
24	C₂×Q₈×C₁₉	—	2	2	no	—
25	C₂×Q₈×C₁₉	—	2	4	no	—
26	C₂²×C₄×C₁₉	—	2	4	no	—
27	C₂²×C₄×C₁₉	D₄	1	4	yes	—
28	C₂²×C₄×C₁₉	C₂³	1	2	yes	—
29	C₂²×C₄×C₁₉	C₂³	1	2	yes	—
30	C₂²×C₄×C₁₉	C₂³	1	1	yes	characteristic
31	C₂²×C₄×C₁₉	C₂³	1	1	yes	characteristic
32	C₂²×Dic₁₉	—	2	4	no	—
33	C₂²×Dic₁₉	—	2	8	no	—
34	C₂²×Dic₁₉	C₂³	1	4	yes	—
35	C₂²×Dic₁₉	C₂³	1	4	yes	—
36	C₄⋊C₄×C₁₉	—	2	2	no	—
37	C₄⋊C₄×C₁₉	—	2	4	no	—
38	C₂²⋊C₄×C₁₉	—	2	4	no	—
39	C₂²⋊Dic₁₉	—	2	4	no	—
40	C₄⋊Dic₁₉	—	2	4	no	—
41	C₄⋊Dic₁₉	—	2	4	no	—
42	C₄×Dic₁₉	—	2	4	no	—
43	C₂³×C₁₉	—	2	2	no	—
44	C₂³×C₁₉	—	2	2	no	—
45	C₂³×C₁₉	C₂×D₄	1	2	yes	—
46	C₂³×C₁₉	#41	1	4	yes	—
47	C₂³×C₁₉	C₂⁴	1	1	yes	characteristic
48	Q₈×C₁₉	—	4	4	no	—
49	C₂×C₄×C₁₉	—	2	2	no	—
50	C₂×C₄×C₁₉	—	2	8	no	—
51	C₂×C₄×C₁₉	—	2	2	no	—
52	C₂×C₄×C₁₉	—	2	2	no	—
53	C₂×C₄×C₁₉	—	2	4	no	—
54	C₂×C₄×C₁₉	—	2	4	no	—
55	C₂×C₄×C₁₉	C₂×D₄	1	2	yes	—
56	C₂×C₄×C₁₉	C₂×D₄	1	2	yes	—
57	C₂×C₄×C₁₉	#41	1	4	yes	—
58	C₂×Dic₁₉	—	2	4	no	—
59	C₂×Dic₁₉	—	2	4	no	—
60	C₂×Dic₁₉	—	2	4	no	—
61	C₂×Dic₁₉	—	2	8	no	—
62	C₂×Dic₁₉	—	2	4	no	—
63	C₂×Dic₁₉	—	4	8	no	—
64	#2503	—	19	1	no	maximal, Sylow
65	C₂²×C₁₉	—	2	2	no	—
66	C₂²×C₁₉	—	2	2	no	—
67	C₂²×C₁₉	—	2	8	no	—
68	C₂²×C₁₉	#124	1	4	yes	—
69	C₂²×C₁₉	C₂²×D₄	1	1	yes	characteristic
70	C₂²×C₁₉	C₂×#41	1	2	yes	—
71	C₂²×C₁₉	C₂×#41	1	2	yes	—
72	C₂²×C₁₉	2₋¹⁺⁴	1	2	yes	—
73	C₄×C₁₉	—	2	2	no	—
74	C₄×C₁₉	—	4	2	no	—
75	C₄×C₁₉	—	4	2	no	—
76	C₄×C₁₉	#123	1	4	yes	—
77	Dic₁₉	—	4	4	no	—
78	Dic₁₉	—	4	4	no	—
79	C₂×C₂²⋊Q₈	D₁₉	1	1	yes	characteristic
80	C₂²×C₄⋊C₄	—	19	1	no	—
81	#400	—	19	4	no	—
82	#391	—	19	2	no	—
83	#382	—	19	2	no	—
84	#381	—	19	4	no	—
85	C₄×C₂²⋊C₄	—	19	1	no	—
86	C₂×C₁₉	—	2	4	no	—
87	C₂×C₁₉	C₂×#123	1	1	yes	characteristic
88	C₂×C₁₉	C₂×#124	1	1	yes	characteristic
89	C₂×C₁₉	#536	1	1	yes	characteristic
90	C₂×C₁₉	#552	1	4	yes	—
91	C₂²×Q₈	C₂×D₁₉	1	1	yes	characteristic
92	C₂³×C₄	—	19	2	no	—
93	C₂³×C₄	C₂×D₁₉	1	1	yes	characteristic
94	C₂²⋊Q₈	—	2	4	no	—
95	C₂×C₄⋊C₄	—	19	2	no	—
96	C₂×C₄⋊C₄	—	19	2	no	—
97	C₂×C₄⋊C₄	—	38	4	no	—
98	C₂×C₄⋊C₄	C₂×D₁₉	1	1	yes	characteristic
99	C₂×C₄⋊C₄	C₂×D₁₉	1	2	yes	—
100	C₂×C₂²⋊C₄	—	19	2	no	—
101	C₂×C₂²⋊C₄	C₂×D₁₉	1	2	yes	—
102	C₂×C₄²	—	19	2	no	—
103	#95	—	19	2	no	—
104	#95	—	19	8	no	—
105	#95	—	19	8	no	—
106	C₁₉	#2503	1	1	yes	characteristic, complemented, Sylow
107	C₂⁴	C₂²×D₁₉	1	1	yes	characteristic
108	C₂×Q₈	—	2	4	no	—
109	C₂×Q₈	—	2	2	no	—
110	C₂²×C₄	—	2	4	no	—
111	C₂²×C₄	—	19	4	no	—
112	C₂²×C₄	—	19	4	no	—
113	C₂²×C₄	—	38	8	no	—
114	C₂²×C₄	—	38	4	no	—
115	C₂²×C₄	C₂²⋊D₁₉	1	4	yes	—
116	C₂²×C₄	C₂²×D₁₉	1	2	yes	—
117	C₂²×C₄	C₂²×D₁₉	1	2	yes	—
118	C₂²×C₄	C₂²×D₁₉	1	1	yes	characteristic
119	C₂²×C₄	C₂²×D₁₉	1	1	yes	characteristic
120	C₄⋊C₄	—	2	2	no	—
121	C₄⋊C₄	—	2	4	no	—
122	C₄⋊C₄	—	38	4	no	—
123	C₄⋊C₄	—	38	4	no	—
124	C₂²⋊C₄	—	2	4	no	—
125	C₂²⋊C₄	—	38	4	no	—
126	C₄²	—	38	4	no	—
127	C₂³	—	2	2	no	—
128	C₂³	—	2	2	no	—
129	C₂³	#64545	1	4	yes	—
130	C₂³	C₂×C₂²⋊D₁₉	1	2	yes	—
131	C₂³	C₂³×D₁₉	1	1	yes	characteristic
132	Q₈	—	4	4	no	—
133	C₂×C₄	—	2	2	no	—
134	C₂×C₄	—	2	8	no	—
135	C₂×C₄	—	2	4	no	—
136	C₂×C₄	—	2	4	no	—
137	C₂×C₄	—	2	2	no	—
138	C₂×C₄	—	2	2	no	—
139	C₂×C₄	—	38	4	no	—
140	C₂×C₄	—	38	4	no	—
141	C₂×C₄	—	38	8	no	—
142	C₂×C₄	—	38	4	no	—
143	C₂×C₄	—	38	4	no	—
144	C₂×C₄	—	76	8	no	—
145	C₂×C₄	#64543	1	4	yes	—
146	C₂×C₄	C₂×C₂²⋊D₁₉	1	2	yes	—
147	C₂×C₄	C₂×C₂²⋊D₁₉	1	2	yes	—
148	C₂²	—	2	2	no	—
149	C₂²	—	2	2	no	—
150	C₂²	—	2	8	no	—
151	C₂²	#10597393	1	4	yes	—
152	C₂²	C₂×#64543	1	2	yes	—
153	C₂²	C₂×#64545	1	2	yes	—
154	C₂²	#10597436	1	2	yes	—
155	C₂²	C₂²×C₂²⋊D₁₉	1	1	yes	characteristic
156	C₄	—	2	2	no	—
157	C₄	—	4	2	no	—
158	C₄	—	4	2	no	—
159	C₄	—	76	4	no	—
160	C₄	—	76	4	no	—
161	C₄	#10597381	1	4	yes	—
162	C₂	—	2	4	no	—
163	C₂	C₂×#10597381	1	1	yes	characteristic
164	C₂	C₂×#10597393	1	1	yes	characteristic
165	C₂	#11927115	1	1	yes	characteristic
166	C₂	#11927236	1	4	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂²×C₄⋊Dic₁₉

C₂

yes

maximal, characteristic

#11926483

C₂

yes

maximal, complemented

#11926451

C₂

yes

maximal, complemented

#11926409

C₂

yes

maximal, complemented

#11926405

C₂

yes

maximal

C₄×C₂²⋊Dic₁₉

C₂

yes

maximal, characteristic

C₂×C₁₉×C₂²⋊Q₈

C₂

yes

maximal, characteristic

C₂²×Q₈×C₁₉

C₂²

yes

characteristic

C₂³×C₄×C₁₉

C₂²

yes

characteristic

C₂³×Dic₁₉

C₂²

yes

—

C₁₉×C₂²⋊Q₈

—

C₂×C₄⋊C₄×C₁₉

C₂²

yes

characteristic

C₂×C₄⋊C₄×C₁₉

C₂²

yes

—

C₂×C₂²⋊C₄×C₁₉

C₂²

yes

—

C₂×C₂²⋊Dic₁₉

C₂²

yes

—

C₂×C₄⋊Dic₁₉

—

C₂×C₄⋊Dic₁₉

C₂²

yes

—

C₂×C₄⋊Dic₁₉

C₂²

yes

—

C₂×C₄×Dic₁₉

C₂²

yes

—

C₁₉⋊#95

C₂²

yes

—

C₁₉⋊#95

C₂²

yes

—

C₁₉⋊#95

C₂²

yes

—

C₂⁴×C₁₉

C₂³

yes

characteristic

C₂×Q₈×C₁₉

—

C₂×Q₈×C₁₉

—

C₂²×C₄×C₁₉

—

C₂²×C₄×C₁₉

D₄

yes

—

C₂²×C₄×C₁₉

C₂³

yes

—

C₂²×C₄×C₁₉

C₂³

yes

—

C₂²×C₄×C₁₉

C₂³

yes

characteristic

C₂²×C₄×C₁₉

C₂³

yes

characteristic

C₂²×Dic₁₉

—

C₂²×Dic₁₉

—

C₂²×Dic₁₉

C₂³

yes

—

C₂²×Dic₁₉

C₂³

yes

—

C₄⋊C₄×C₁₉

—

C₄⋊C₄×C₁₉

—

C₂²⋊C₄×C₁₉

—

C₂²⋊Dic₁₉

—

C₄⋊Dic₁₉

—

C₄⋊Dic₁₉

—

C₄×Dic₁₉

—

C₂³×C₁₉

—

C₂³×C₁₉

—

C₂³×C₁₉

C₂×D₄

yes

—

C₂³×C₁₉

#41

yes

—

C₂³×C₁₉

C₂⁴

yes

characteristic

Q₈×C₁₉

—

C₂×C₄×C₁₉

—

C₂×C₄×C₁₉

—

C₂×C₄×C₁₉

—

C₂×C₄×C₁₉

—

C₂×C₄×C₁₉

—

C₂×C₄×C₁₉

—

C₂×C₄×C₁₉

C₂×D₄

yes

—

C₂×C₄×C₁₉

C₂×D₄

yes

—

C₂×C₄×C₁₉

#41

yes

—

C₂×Dic₁₉

—

C₂×Dic₁₉

—

C₂×Dic₁₉

—

C₂×Dic₁₉

—

C₂×Dic₁₉

—

C₂×Dic₁₉

—

#2503

—

maximal, Sylow

C₂²×C₁₉

—

C₂²×C₁₉

—

C₂²×C₁₉

—

C₂²×C₁₉

#124

yes

—

C₂²×C₁₉

C₂²×D₄

yes

characteristic

C₂²×C₁₉

C₂×#41

yes

—

C₂²×C₁₉

C₂×#41

yes

—

C₂²×C₁₉

2₋¹⁺⁴

yes

—

C₄×C₁₉

—

C₄×C₁₉

—

C₄×C₁₉

—

C₄×C₁₉

#123

yes

—

Dic₁₉

—

Dic₁₉

—

C₂×C₂²⋊Q₈

D₁₉

yes

characteristic

C₂²×C₄⋊C₄

—

—

—

—

—

C₄×C₂²⋊C₄

—

C₂×C₁₉

—

C₂×C₁₉

yes

C₂×C₁₉

yes

C₂×C₁₉

yes

C₂×C₁₉

yes

—

C₂²×Q₈

C₂×D₁₉

yes

characteristic

C₂³×C₄

—

C₂³×C₄

C₂×D₁₉

yes

characteristic

C₂²⋊Q₈

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

C₂×D₁₉

yes

characteristic

C₂×C₄⋊C₄

C₂×D₁₉

yes

—

100

C₂×C₂²⋊C₄

—

101

C₂×C₂²⋊C₄

C₂×D₁₉

yes

—

102

C₂×C₄²

—

103

#95

—

104

#95

—

105

#95

—

106

C₁₉

#2503

yes

characteristic, complemented, Sylow

107

C₂⁴

C₂²×D₁₉

yes

characteristic

108

C₂×Q₈

—

109

C₂×Q₈

—

110

C₂²×C₄

—

111

C₂²×C₄

—

112

C₂²×C₄

—

113

C₂²×C₄

—

114

C₂²×C₄

—

115

C₂²×C₄

C₂²⋊D₁₉

yes

—

116

C₂²×C₄

C₂²×D₁₉

yes

—

117

C₂²×C₄

C₂²×D₁₉

yes

—

118

C₂²×C₄

C₂²×D₁₉

yes

characteristic

119

C₂²×C₄

C₂²×D₁₉

yes

characteristic

120

C₄⋊C₄

—

121

C₄⋊C₄

—

122

C₄⋊C₄

—

123

C₄⋊C₄

—

124

C₂²⋊C₄

—

125

C₂²⋊C₄

—

126

C₄²

—

127

C₂³

—

128

C₂³

—

129

C₂³

#64545

yes

—

130

C₂³

C₂×C₂²⋊D₁₉

yes

—

131

C₂³

C₂³×D₁₉

yes

characteristic

132

Q₈

—

133

C₂×C₄

—

134

C₂×C₄

—

135

C₂×C₄

—

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

—

145

C₂×C₄

#64543

yes

—

146

C₂×C₄

C₂×C₂²⋊D₁₉

yes

—

147

C₂×C₄

C₂×C₂²⋊D₁₉

yes

—

148

C₂²

—

149

C₂²

—

150

C₂²

—

151

C₂²

#10597393

yes

—

152

C₂²

C₂×#64543

yes

—

153

C₂²

C₂×#64545

yes

—

154

C₂²

#10597436

yes

—

155

C₂²

C₂²×C₂²⋊D₁₉

yes

characteristic

156

C₄

—

157

C₄

—

158

C₄

—

159

C₄

—

160

C₄

—

161

C₄

#10597381

yes

—

162

C₂

—

163

C₂

C₂×#10597381

yes

characteristic

164

C₂

C₂×#10597393

yes

characteristic

165

C₂

#11927115

yes

characteristic

166

C₂

#11927236

yes

—

	subgroup	G/H
center, Frattini	C₂³	C₂³×D₁₉
commutator, socle, cosocle	C₂³×C₁₉	C₂⁴
Fitting	C₂×C₁₉×C₂²⋊Q₈	C₂
prime radical	C₂⁴×C₁₉	C₂³
hypercenter	C₂×C₂²⋊Q₈	D₁₉
hypocenter	C₁₉	#2503
LCS	G ▷ C₂³×C₁₉ ▷ C₁₉
UCS	C₁ ◁ C₂³ ◁ C₂×C₂²⋊Q₈

subgroup

G/H

center, Frattini

C₂³

C₂³×D₁₉

commutator, socle, cosocle

C₂³×C₁₉

C₂⁴

Fitting

C₂×C₁₉×C₂²⋊Q₈

C₂

prime radical

C₂⁴×C₁₉

C₂³

hypercenter

C₂×C₂²⋊Q₈

D₁₉

C₁₉

G ▷ C₂³×C₁₉ ▷ C₁₉

C₁ ◁ C₂³ ◁ C₂×C₂²⋊Q₈

N	H ≅ G/N	type
2. #11926483	C₂	semidirect
3. #11926451	C₂	semidirect
4. #11926409	C₂	semidirect
106. C₁₉	#2503	semidirect

H ≅ G/N

type

2. #11926483

C₂

semidirect

3. #11926451

C₂

semidirect

4. #11926409

C₂

semidirect

106. C₁₉

#2503

semidirect

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

subgroup	index
C₂²×C₄⋊Dic₁₉	2
#11926483	2
#11926451	2
#11926409	2
#11926405	2
C₄×C₂²⋊Dic₁₉	2
C₂×C₁₉×C₂²⋊Q₈	2
#2503	19

subgroup

index

C₂²×C₄⋊Dic₁₉

C₄×C₂²⋊Dic₁₉

C₂×C₁₉×C₂²⋊Q₈

#2503

Subgroups — #14568121

Conjugacy classes of subgroups

Complements

Maximal subgroups