Group #14572705 — subgroups

Subgroups — #14572705

Order 2432.

394 conjugacy classes of subgroups; only the 157 normal subgroups are shown.

normal subgroups only

This group has 394 conjugacy classes of subgroups — too many to draw. Only its 157 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	#11927094	C₂	1	1	yes	maximal, characteristic
2	#11926617	C₂	1	4	yes	maximal, complemented
3	#11926614	C₂	1	4	yes	maximal, complemented
4	#11926601	C₂	1	1	yes	maximal, characteristic, complemented
5	#11926509	C₂	1	4	yes	maximal, complemented
6	C₁₉×#432	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂×#64541	C₂²	1	2	yes	—
8	C₂×C₄²×C₁₉	C₂²	1	1	yes	characteristic
9	#10597376	C₄	1	2	yes	—
10	#10597376	C₄	1	2	yes	—
11	C₂×#64515	C₂²	1	4	yes	—
12	C₂×C₈×D₁₉	C₂²	1	4	yes	—
13	#10597321	C₄	1	2	yes	—
14	#10597320	C₄	1	1	yes	characteristic
15	C₄×D₇₆	C₂²	1	2	yes	—
16	C₄×D₇₆	C₂²	1	2	yes	—
17	#10597315	C₄	1	1	yes	characteristic
18	C₁₉×C₄⋊C₈	C₂²	1	2	yes	—
19	C₁₉×C₂²⋊C₈	C₂²	1	2	yes	—
20	C₁₉×#97	C₂²	1	2	yes	—
21	C₂²⋊(C₁₉⋊C₈)	C₂²	1	2	yes	—
22	(C₂×D₁₉)⋊C₈	C₂²	1	8	yes	—
23	C₄⋊(C₁₉⋊C₈)	C₂²	1	2	yes	—
24	#10597260	C₂²	1	2	yes	—
25	C₂²×C₄×C₁₉	C₂×C₄	1	1	yes	characteristic
26	C₂²×C₄×C₁₉	C₂×C₄	1	1	yes	characteristic
27	C₂²×C₄×C₁₉	C₂³	1	1	yes	characteristic
28	C₂×C₂²⋊D₁₉	C₂×C₄	1	4	yes	—
29	#64541	—	4	2	no	—
30	#64541	—	4	2	no	—
31	C₂×D₇₆	C₂×C₄	1	2	yes	—
32	C₂×D₇₆	C₂×C₄	1	2	yes	—
33	C₂×D₇₆	C₂×C₄	1	2	yes	—
34	C₂×C₄×D₁₉	C₂³	1	4	yes	—
35	C₂×Dic₃₈	C₂×C₄	1	2	yes	—
36	C₂×C₈×C₁₉	C₂³	1	4	yes	—
37	C₄²×C₁₉	C₂×C₄	1	1	yes	characteristic
38	C₄²×C₁₉	C₂×C₄	1	1	yes	characteristic
39	C₄²×C₁₉	C₂³	1	1	yes	characteristic
40	C₄²×C₁₉	C₂³	1	1	yes	characteristic
41	(C₂×D₁₉)⋊C₄	C₂×C₄	1	4	yes	—
42	(C₂×D₁₉)⋊C₄	C₂×C₄	1	4	yes	—
43	C₄⋊Dic₁₉	C₂×C₄	1	2	yes	—
44	C₄⋊Dic₁₉	C₂×C₄	1	2	yes	—
45	C₂×C₁₉⋊C₈	C₂³	1	4	yes	—
46	#64515	—	4	4	no	—
47	C₈×D₁₉	—	4	4	no	—
48	C₂³×C₁₉	C₂²×C₄	1	1	yes	characteristic
49	C₂²×D₁₉	C₂²×C₄	1	4	yes	—
50	C₂×C₄×C₁₉	—	2	2	no	—
51	C₂×C₄×C₁₉	—	4	1	no	—
52	C₂×C₄×C₁₉	—	4	1	no	—
53	C₂×C₄×C₁₉	C₂²×C₄	1	1	yes	characteristic
54	C₂×C₄×C₁₉	C₂²×C₄	1	1	yes	characteristic
55	C₂×C₄×C₁₉	C₂²×C₄	1	1	yes	characteristic
56	C₂×C₄×C₁₉	C₂²×C₄	1	1	yes	characteristic
57	C₂×C₄×C₁₉	C₂²×C₄	1	1	yes	characteristic
58	C₂×C₄×C₁₉	C₂⁴	1	1	yes	characteristic
59	C₂²⋊D₁₉	—	4	4	no	—
60	C₂×Dic₁₉	C₂²×C₄	1	4	yes	—
61	D₇₆	—	4	2	no	—
62	D₇₆	—	4	2	no	—
63	D₇₆	—	4	2	no	—
64	C₄×D₁₉	—	2	4	no	—
65	C₄×D₁₉	—	2	4	no	—
66	Dic₃₈	—	4	2	no	—
67	C₈×C₁₉	—	2	4	no	—
68	C₁₉⋊C₈	—	2	4	no	—
69	#2980	—	19	1	no	maximal, Sylow
70	C₂²×C₁₉	—	2	1	no	—
71	C₂²×C₁₉	—	2	1	no	—
72	C₂²×C₁₉	—	2	1	no	—
73	C₂²×C₁₉	C₂³×C₄	1	1	yes	characteristic
74	C₂×D₁₉	—	2	4	no	—
75	C₂×D₁₉	—	2	4	no	—
76	C₂×D₁₉	—	2	4	no	—
77	C₄×C₁₉	—	2	1	no	—
78	C₄×C₁₉	—	2	1	no	—
79	C₄×C₁₉	—	2	1	no	—
80	C₄×C₁₉	—	2	1	no	—
81	C₄×C₁₉	—	2	1	no	—
82	C₄×C₁₉	H(5,𝔽₂)	1	2	yes	—
83	Dic₁₉	—	2	4	no	—
84	#532	—	19	1	no	—
85	#435	—	19	4	no	—
86	#435	—	19	4	no	—
87	#432	—	19	1	no	—
88	#432	D₁₉	1	1	yes	characteristic, complemented
89	#408	—	19	4	no	—
90	C₂×C₁₉	—	4	1	no	—
91	C₂×C₁₉	#518	1	1	yes	characteristic
92	C₂×C₁₉	#568	1	1	yes	characteristic
93	C₂×C₁₉	#568	1	1	yes	characteristic
94	D₁₉	—	4	4	no	—
95	C₂×#41	—	19	2	no	—
96	C₂×M₁₆	—	19	4	no	—
97	C₂²×C₈	—	19	4	no	—
98	C₄⋊Q₈	—	19	1	no	—
99	#127	—	19	1	no	—
100	#124	—	19	2	no	—
101	#121	—	19	2	no	—
102	#121	—	19	2	no	—
103	C₄×D₄	—	19	2	no	—
104	C₄×D₄	—	19	2	no	—
105	C₂×C₄²	C₂×D₁₉	1	1	yes	characteristic
106	C₄⋊C₈	—	19	2	no	—
107	C₄⋊C₈	C₂×D₁₉	1	2	yes	—
108	C₂²⋊C₈	—	19	2	no	—
109	C₂²⋊C₈	—	19	8	no	—
110	C₂²⋊C₈	C₂×D₁₉	1	2	yes	—
111	#97	—	19	2	no	—
112	#97	C₂×D₁₉	1	2	yes	—
113	C₁₉	#2980	1	1	yes	characteristic, complemented, Sylow
114	#41	—	76	2	no	—
115	#41	—	76	2	no	—
116	C₂×Q₈	—	19	2	no	—
117	C₂×D₄	—	19	4	no	—
118	C₂×D₄	—	19	2	no	—
119	C₂×D₄	—	19	2	no	—
120	C₂×D₄	—	19	2	no	—
121	C₂²×C₄	—	19	4	no	—
122	C₂²×C₄	C₄×D₁₉	1	1	yes	characteristic
123	C₂²×C₄	C₄×D₁₉	1	1	yes	characteristic
124	C₂²×C₄	C₂²×D₁₉	1	1	yes	characteristic
125	M₁₆	—	76	4	no	—
126	C₂×C₈	—	19	4	no	—
127	C₂×C₈	—	76	4	no	—
128	C₂×C₈	C₂²×D₁₉	1	4	yes	—
129	C₄⋊C₄	—	19	2	no	—
130	C₄⋊C₄	—	19	2	no	—
131	C₂²⋊C₄	—	19	4	no	—
132	C₂²⋊C₄	—	19	4	no	—
133	C₄²	C₄×D₁₉	1	1	yes	characteristic
134	C₄²	C₄×D₁₉	1	1	yes	characteristic
135	C₄²	C₂²×D₁₉	1	1	yes	characteristic
136	C₄²	C₂²×D₁₉	1	1	yes	characteristic
137	C₂³	—	19	4	no	—
138	C₂³	C₂×C₄×D₁₉	1	1	yes	characteristic
139	Q₈	—	76	2	no	—
140	D₄	—	76	2	no	—
141	D₄	—	76	2	no	—
142	D₄	—	76	2	no	—
143	D₄	—	76	4	no	—
144	C₂×C₄	—	2	2	no	—
145	C₂×C₄	—	4	1	no	—
146	C₂×C₄	—	4	1	no	—
147	C₂×C₄	—	19	4	no	—
148	C₂×C₄	—	38	4	no	—
149	C₂×C₄	—	38	4	no	—
150	C₂×C₄	C₂×C₄×D₁₉	1	1	yes	characteristic
151	C₂×C₄	C₂×C₄×D₁₉	1	1	yes	characteristic
152	C₂×C₄	C₂×C₄×D₁₉	1	1	yes	characteristic
153	C₂×C₄	C₂×C₄×D₁₉	1	1	yes	characteristic
154	C₂×C₄	C₂×C₄×D₁₉	1	1	yes	characteristic
155	C₂×C₄	C₂³×D₁₉	1	1	yes	characteristic
156	C₈	—	2	4	no	—
157	C₈	—	38	4	no	—
158	C₂²	—	2	1	no	—
159	C₂²	—	2	1	no	—
160	C₂²	—	2	1	no	—
161	C₂²	—	38	4	no	—
162	C₂²	—	38	4	no	—
163	C₂²	—	38	4	no	—
164	C₂²	C₂²×C₄×D₁₉	1	1	yes	characteristic
165	C₄	—	2	1	no	—
166	C₄	—	2	1	no	—
167	C₄	—	2	1	no	—
168	C₄	—	2	1	no	—
169	C₄	—	2	1	no	—
170	C₄	—	38	4	no	—
171	C₄	#10597435	1	2	yes	—
172	C₂	—	4	1	no	—
173	C₂	—	76	4	no	—
174	C₂	#11927018	1	1	yes	characteristic
175	C₂	#11927310	1	1	yes	characteristic
176	C₂	#11927310	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#11927094

C₂

yes

maximal, characteristic

#11926617

C₂

yes

maximal, complemented

#11926614

C₂

yes

maximal, complemented

#11926601

C₂

yes

maximal, characteristic, complemented

#11926509

C₂

yes

maximal, complemented

C₁₉×#432

C₂

yes

maximal, characteristic, complemented

C₂×#64541

C₂²

yes

—

C₂×C₄²×C₁₉

C₂²

yes

characteristic

#10597376

C₄

yes

—

#10597376

C₄

yes

—

C₂×#64515

C₂²

yes

—

C₂×C₈×D₁₉

C₂²

yes

—

#10597321

C₄

yes

—

#10597320

C₄

yes

characteristic

C₄×D₇₆

C₂²

yes

—

C₄×D₇₆

C₂²

yes

—

#10597315

C₄

yes

characteristic

C₁₉×C₄⋊C₈

C₂²

yes

—

C₁₉×C₂²⋊C₈

C₂²

yes

—

C₁₉×#97

C₂²

yes

—

C₂²⋊(C₁₉⋊C₈)

C₂²

yes

—

(C₂×D₁₉)⋊C₈

C₂²

yes

—

C₄⋊(C₁₉⋊C₈)

C₂²

yes

—

#10597260

C₂²

yes

—

C₂²×C₄×C₁₉

C₂×C₄

yes

characteristic

C₂²×C₄×C₁₉

C₂×C₄

yes

characteristic

C₂²×C₄×C₁₉

C₂³

yes

characteristic

C₂×C₂²⋊D₁₉

C₂×C₄

yes

—

#64541

—

#64541

—

C₂×D₇₆

C₂×C₄

yes

—

C₂×D₇₆

C₂×C₄

yes

—

C₂×D₇₆

C₂×C₄

yes

—

C₂×C₄×D₁₉

C₂³

yes

—

C₂×Dic₃₈

C₂×C₄

yes

—

C₂×C₈×C₁₉

C₂³

yes

—

C₄²×C₁₉

C₂×C₄

yes

characteristic

C₄²×C₁₉

C₂×C₄

yes

characteristic

C₄²×C₁₉

C₂³

yes

characteristic

C₄²×C₁₉

C₂³

yes

characteristic

(C₂×D₁₉)⋊C₄

C₂×C₄

yes

—

(C₂×D₁₉)⋊C₄

C₂×C₄

yes

—

C₄⋊Dic₁₉

C₂×C₄

yes

—

C₄⋊Dic₁₉

C₂×C₄

yes

—

C₂×C₁₉⋊C₈

C₂³

yes

—

#64515

—

C₈×D₁₉

—

C₂³×C₁₉

C₂²×C₄

yes

characteristic

C₂²×D₁₉

C₂²×C₄

yes

—

C₂×C₄×C₁₉

—

C₂×C₄×C₁₉

—

C₂×C₄×C₁₉

—

C₂×C₄×C₁₉

C₂²×C₄

yes

characteristic

C₂×C₄×C₁₉

C₂²×C₄

yes

characteristic

C₂×C₄×C₁₉

C₂²×C₄

yes

characteristic

C₂×C₄×C₁₉

C₂²×C₄

yes

characteristic

C₂×C₄×C₁₉

C₂²×C₄

yes

characteristic

C₂×C₄×C₁₉

C₂⁴

yes

characteristic

C₂²⋊D₁₉

—

C₂×Dic₁₉

C₂²×C₄

yes

—

D₇₆

—

D₇₆

—

D₇₆

—

C₄×D₁₉

—

C₄×D₁₉

—

Dic₃₈

—

C₈×C₁₉

—

C₁₉⋊C₈

—

#2980

—

maximal, Sylow

C₂²×C₁₉

—

C₂²×C₁₉

—

C₂²×C₁₉

—

C₂²×C₁₉

C₂³×C₄

yes

characteristic

C₂×D₁₉

—

C₂×D₁₉

—

C₂×D₁₉

—

C₄×C₁₉

—

C₄×C₁₉

—

C₄×C₁₉

—

C₄×C₁₉

—

C₄×C₁₉

—

C₄×C₁₉

H(5,𝔽₂)

yes

—

Dic₁₉

—

—

—

—

—

D₁₉

yes

characteristic, complemented

#408

—

C₂×C₁₉

—

C₂×C₁₉

yes

C₂×C₁₉

yes

C₂×C₁₉

yes

D₁₉

—

C₂×#41

—

C₂×M₁₆

—

C₂²×C₈

—

C₄⋊Q₈

—

#127

—

100

#124

—

101

#121

—

102

#121

—

103

C₄×D₄

—

104

C₄×D₄

—

105

C₂×C₄²

C₂×D₁₉

yes

characteristic

106

C₄⋊C₈

—

107

C₄⋊C₈

C₂×D₁₉

yes

—

108

C₂²⋊C₈

—

109

C₂²⋊C₈

—

110

C₂²⋊C₈

C₂×D₁₉

yes

—

111

#97

—

112

#97

C₂×D₁₉

yes

—

113

C₁₉

#2980

yes

characteristic, complemented, Sylow

114

#41

—

115

#41

—

116

C₂×Q₈

—

117

C₂×D₄

—

118

C₂×D₄

—

119

C₂×D₄

—

120

C₂×D₄

—

121

C₂²×C₄

—

122

C₂²×C₄

C₄×D₁₉

yes

characteristic

123

C₂²×C₄

C₄×D₁₉

yes

characteristic

124

C₂²×C₄

C₂²×D₁₉

yes

characteristic

125

M₁₆

—

126

C₂×C₈

—

127

C₂×C₈

—

128

C₂×C₈

C₂²×D₁₉

yes

—

129

C₄⋊C₄

—

130

C₄⋊C₄

—

131

C₂²⋊C₄

—

132

C₂²⋊C₄

—

133

C₄²

C₄×D₁₉

yes

characteristic

134

C₄²

C₄×D₁₉

yes

characteristic

135

C₄²

C₂²×D₁₉

yes

characteristic

136

C₄²

C₂²×D₁₉

yes

characteristic

137

C₂³

—

138

C₂³

C₂×C₄×D₁₉

yes

characteristic

139

Q₈

—

140

D₄

—

141

D₄

—

142

D₄

—

143

D₄

—

144

C₂×C₄

—

145

C₂×C₄

—

146

C₂×C₄

—

147

C₂×C₄

—

148

C₂×C₄

—

149

C₂×C₄

—

150

C₂×C₄

C₂×C₄×D₁₉

yes

characteristic

151

C₂×C₄

C₂×C₄×D₁₉

yes

characteristic

152

C₂×C₄

C₂×C₄×D₁₉

yes

characteristic

153

C₂×C₄

C₂×C₄×D₁₉

yes

characteristic

154

C₂×C₄

C₂×C₄×D₁₉

yes

characteristic

155

C₂×C₄

C₂³×D₁₉

yes

characteristic

156

C₈

—

157

C₈

—

158

C₂²

—

159

C₂²

—

160

C₂²

—

161

C₂²

—

162

C₂²

—

163

C₂²

—

164

C₂²

C₂²×C₄×D₁₉

yes

characteristic

165

C₄

—

166

C₄

—

167

C₄

—

168

C₄

—

169

C₄

—

170

C₄

—

171

C₄

#10597435

yes

—

172

C₂

—

173

C₂

—

174

C₂

#11927018

yes

characteristic

175

C₂

#11927310

yes

characteristic

176

C₂

#11927310

yes

characteristic

	subgroup	G/H
center, Frattini	C₂×C₄	C₂³×D₁₉
commutator, socle	C₂²×C₁₉	C₂³×C₄
Fitting	C₁₉×#432	C₂
prime radical	#11927094	C₂
cosocle	C₂×C₄×C₁₉	C₂⁴
hypercenter	#432	D₁₉
hypocenter	C₁₉	#2980
LCS	G ▷ C₂²×C₁₉ ▷ C₁₉
UCS	C₁ ◁ C₂×C₄ ◁ #432

subgroup

G/H

center, Frattini

C₂×C₄

C₂³×D₁₉

commutator, socle

C₂²×C₁₉

C₂³×C₄

Fitting

C₁₉×#432

C₂

prime radical

#11927094

C₂

cosocle

C₂×C₄×C₁₉

C₂⁴

D₁₉

C₁₉

G ▷ C₂²×C₁₉ ▷ C₁₉

C₁ ◁ C₂×C₄ ◁ #432

N	H ≅ G/N	type
2. #11926617	C₂	semidirect
3. #11926614	C₂	semidirect
4. #11926601	C₂	semidirect
5. #11926509	C₂	semidirect
6. C₁₉×#432	C₂	semidirect
88. #432	D₁₉	semidirect
113. C₁₉	#2980	semidirect

H ≅ G/N

type

2. #11926617

C₂

semidirect

3. #11926614

C₂

semidirect

4. #11926601

C₂

semidirect

5. #11926509

C₂

semidirect

6. C₁₉×#432

C₂

semidirect

88. #432

D₁₉

semidirect

113. C₁₉

#2980

semidirect

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

subgroup	index
#11927094	2
#11926617	2
#11926614	2
#11926601	2
#11926509	2
C₁₉×#432	2
#2980	19

subgroup

index

C₁₉×#432

Subgroups — #14572705

Conjugacy classes of subgroups

Complements

Maximal subgroups