Group #14569018 — subgroups

Subgroups — #14569018

Order 2432.

468 conjugacy classes of subgroups; only the 185 normal subgroups are shown.

normal subgroups only

This group has 468 conjugacy classes of subgroups — too many to draw. Only its 185 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	C₂×#10597316	C₂	1	1	yes	maximal, characteristic
2	C₂×C₄×Dic₃₈	C₂	1	1	yes	maximal, characteristic
3	#11926476	C₂	1	2	yes	maximal
4	#11926465	C₂	1	2	yes	maximal
5	#11926443	C₂	1	2	yes	maximal
6	#11926443	C₂	1	2	yes	maximal
7	#11926442	C₂	1	4	yes	maximal
8	C₄×C₄⋊C₄×C₁₉	C₂	1	1	yes	maximal, characteristic
9	C₂²×Dic₃₈	C₂²	1	1	yes	characteristic
10	C₂×C₄⋊C₄×C₁₉	C₂²	1	2	yes	—
11	C₂×C₄²×C₁₉	C₂²	1	2	yes	—
12	C₂×C₄²×C₁₉	C₂²	1	1	yes	characteristic
13	C₂×C₄⋊Dic₁₉	C₂²	1	1	yes	characteristic
14	C₂×C₄⋊Dic₁₉	C₂²	1	1	yes	characteristic
15	C₂×C₄⋊Dic₁₉	C₂²	1	4	yes	—
16	C₂×C₄⋊Dic₁₉	C₂²	1	1	yes	characteristic
17	C₂×C₄⋊Dic₁₉	C₂²	1	4	yes	—
18	C₂×Dic₁₉⋊C₄	C₂²	1	2	yes	—
19	C₂×Dic₁₉⋊C₄	C₂²	1	2	yes	—
20	C₂×Dic₁₉⋊C₄	C₂²	1	2	yes	—
21	C₂×C₄×Dic₁₉	C₂²	1	2	yes	—
22	#10597316	—	2	4	no	—
23	C₄×Dic₃₈	—	2	4	no	—
24	C₁₉×#95	C₂²	1	2	yes	—
25	C₁₉⋊#95	C₂²	1	4	yes	—
26	C₁₉⋊#95	C₂²	1	4	yes	—
27	C₂²×C₄×C₁₉	C₂³	1	1	yes	characteristic
28	C₂²×C₄×C₁₉	C₂³	1	1	yes	characteristic
29	C₂²×C₄×C₁₉	C₂³	1	4	yes	—
30	C₂²×C₄×C₁₉	C₂³	1	1	yes	characteristic
31	C₂²×Dic₁₉	C₂³	1	2	yes	—
32	C₂²×Dic₁₉	C₂³	1	2	yes	—
33	C₂²×Dic₁₉	C₂³	1	2	yes	—
34	C₂²×Dic₁₉	C₂³	1	2	yes	—
35	C₂×Dic₃₈	—	2	2	no	—
36	C₂×Dic₃₈	—	2	4	no	—
37	C₄⋊C₄×C₁₉	Q₈	1	8	yes	complemented
38	C₄²×C₁₉	—	4	2	no	—
39	C₄²×C₁₉	D₄	1	4	yes	—
40	C₄⋊Dic₁₉	—	2	2	no	—
41	C₄⋊Dic₁₉	—	2	2	no	—
42	C₄⋊Dic₁₉	—	2	2	no	—
43	C₄⋊Dic₁₉	—	2	8	no	—
44	C₄⋊Dic₁₉	—	4	4	no	—
45	Dic₁₉⋊C₄	—	2	4	no	—
46	Dic₁₉⋊C₄	—	2	4	no	—
47	Dic₁₉⋊C₄	—	2	4	no	—
48	C₄×Dic₁₉	—	2	4	no	—
49	C₂³×C₁₉	C₂⁴	1	1	yes	characteristic
50	C₂×C₄×C₁₉	—	2	4	no	—
51	C₂×C₄×C₁₉	—	2	2	no	—
52	C₂×C₄×C₁₉	—	2	4	no	—
53	C₂×C₄×C₁₉	—	2	1	no	—
54	C₂×C₄×C₁₉	—	2	1	no	—
55	C₂×C₄×C₁₉	C₂×D₄	1	2	yes	—
56	C₂×C₄×C₁₉	C₂×D₄	1	2	yes	—
57	C₂×C₄×C₁₉	C₂×D₄	1	2	yes	—
58	C₂×C₄×C₁₉	C₂×Q₈	1	8	yes	—
59	C₂×C₄×C₁₉	C₂×Q₈	1	4	yes	—
60	C₂×Dic₁₉	—	2	2	no	—
61	C₂×Dic₁₉	—	2	2	no	—
62	C₂×Dic₁₉	—	2	2	no	—
63	C₂×Dic₁₉	—	2	2	no	—
64	C₂×Dic₁₉	—	2	2	no	—
65	C₂×Dic₁₉	—	2	4	no	—
66	C₂×Dic₁₉	—	2	2	no	—
67	C₂×Dic₁₉	—	2	2	no	—
68	C₂×Dic₁₉	—	2	2	no	—
69	C₂×Dic₁₉	—	2	2	no	—
70	C₂×Dic₁₉	#41	1	4	yes	—
71	Dic₃₈	—	2	8	no	—
72	#2584	—	19	1	no	maximal, Sylow
73	C₂²×C₁₉	C₂²×D₄	1	1	yes	characteristic
74	C₂²×C₁₉	C₂²×Q₈	1	2	yes	—
75	C₂²×C₁₉	C₂×#41	1	2	yes	—
76	C₂²×C₁₉	H(5,𝔽₂)	1	1	yes	characteristic
77	C₂²×C₁₉	2₋¹⁺⁴	1	1	yes	characteristic
78	C₄×C₁₉	—	2	2	no	—
79	C₄×C₁₉	—	2	2	no	—
80	C₄×C₁₉	—	2	8	no	—
81	C₄×C₁₉	C₄⋊Q₈	1	4	yes	—
82	Dic₁₉	—	2	4	no	—
83	Dic₁₉	—	4	2	no	—
84	Dic₁₉	—	4	2	no	—
85	Dic₁₉	—	4	2	no	—
86	C₂×#125	—	19	1	no	—
87	C₂×C₄×Q₈	—	19	1	no	—
88	#398	—	19	2	no	—
89	#395	—	19	2	no	—
90	#389	—	19	2	no	—
91	#389	—	19	2	no	—
92	#389	—	19	4	no	—
93	C₄×C₄⋊C₄	D₁₉	1	1	yes	characteristic
94	C₂×C₁₉	#525	1	1	yes	characteristic
95	C₂×C₁₉	C₂×C₄⋊Q₈	1	1	yes	characteristic
96	C₂×C₁₉	#537	1	1	yes	characteristic
97	C₂×C₁₉	#554	1	2	yes	—
98	C₂×C₁₉	#557	1	2	yes	—
99	C₂²×Q₈	—	19	1	no	—
100	#125	—	38	4	no	—
101	C₄×Q₈	—	38	4	no	—
102	C₂×C₄⋊C₄	—	19	2	no	—
103	C₂×C₄⋊C₄	—	19	1	no	—
104	C₂×C₄⋊C₄	—	19	2	no	—
105	C₂×C₄⋊C₄	—	19	1	no	—
106	C₂×C₄⋊C₄	—	19	1	no	—
107	C₂×C₄⋊C₄	—	19	2	no	—
108	C₂×C₄⋊C₄	—	19	4	no	—
109	C₂×C₄⋊C₄	—	19	4	no	—
110	C₂×C₄⋊C₄	C₂×D₁₉	1	2	yes	—
111	C₂×C₄²	—	19	2	no	—
112	C₂×C₄²	C₂×D₁₉	1	2	yes	—
113	C₂×C₄²	C₂×D₁₉	1	1	yes	characteristic
114	#95	—	19	4	no	—
115	#95	—	19	4	no	—
116	#95	C₂×D₁₉	1	2	yes	—
117	C₁₉	#2584	1	1	yes	characteristic, complemented, Sylow
118	C₂×Q₈	—	38	4	no	—
119	C₂×Q₈	—	38	2	no	—
120	C₂²×C₄	—	19	2	no	—
121	C₂²×C₄	—	19	2	no	—
122	C₂²×C₄	—	19	2	no	—
123	C₂²×C₄	—	19	2	no	—
124	C₂²×C₄	C₂²×D₁₉	1	4	yes	—
125	C₂²×C₄	C₂²×D₁₉	1	1	yes	characteristic
126	C₂²×C₄	C₂²×D₁₉	1	1	yes	characteristic
127	C₂²×C₄	C₂²×D₁₉	1	1	yes	characteristic
128	C₄⋊C₄	—	38	8	no	—
129	C₄⋊C₄	—	38	4	no	—
130	C₄⋊C₄	—	38	4	no	—
131	C₄⋊C₄	—	38	4	no	—
132	C₄⋊C₄	—	38	2	no	—
133	C₄⋊C₄	—	38	2	no	—
134	C₄⋊C₄	—	38	2	no	—
135	C₄⋊C₄	—	76	4	no	—
136	C₄⋊C₄	Dic₃₈	1	8	yes	complemented
137	C₄²	—	4	2	no	—
138	C₄²	—	38	4	no	—
139	C₄²	D₇₆	1	4	yes	—
140	C₂³	C₂³×D₁₉	1	1	yes	characteristic
141	Q₈	—	38	8	no	—
142	C₂×C₄	—	2	1	no	—
143	C₂×C₄	—	2	4	no	—
144	C₂×C₄	—	2	1	no	—
145	C₂×C₄	—	2	4	no	—
146	C₂×C₄	—	2	2	no	—
147	C₂×C₄	—	19	4	no	—
148	C₂×C₄	—	38	2	no	—
149	C₂×C₄	—	38	2	no	—
150	C₂×C₄	—	38	2	no	—
151	C₂×C₄	—	38	2	no	—
152	C₂×C₄	—	38	2	no	—
153	C₂×C₄	—	38	2	no	—
154	C₂×C₄	—	38	2	no	—
155	C₂×C₄	—	38	2	no	—
156	C₂×C₄	—	38	2	no	—
157	C₂×C₄	—	38	4	no	—
158	C₂×C₄	C₂×Dic₃₈	1	8	yes	—
159	C₂×C₄	C₂×Dic₃₈	1	4	yes	—
160	C₂×C₄	C₂×D₇₆	1	2	yes	—
161	C₂×C₄	C₂×D₇₆	1	2	yes	—
162	C₂×C₄	C₂×D₇₆	1	2	yes	—
163	C₂²	C₂²×Dic₃₈	1	2	yes	—
164	C₂²	C₂²×D₇₆	1	1	yes	characteristic
165	C₂²	#10597430	1	1	yes	characteristic
166	C₂²	#10597433	1	1	yes	characteristic
167	C₂²	#41×D₁₉	1	2	yes	—
168	C₄	—	2	2	no	—
169	C₄	—	2	2	no	—
170	C₄	—	2	8	no	—
171	C₄	—	38	4	no	—
172	C₄	—	76	2	no	—
173	C₄	—	76	2	no	—
174	C₄	—	76	2	no	—
175	C₄	#10597315	1	4	yes	—
176	C₂	#11927056	1	1	yes	characteristic
177	C₂	C₂×#10597315	1	1	yes	characteristic
178	C₂	#11927125	1	1	yes	characteristic
179	C₂	#11927253	1	2	yes	—
180	C₂	#11927270	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#10597316

C₂

yes

maximal, characteristic

C₂×C₄×Dic₃₈

C₂

yes

maximal, characteristic

#11926476

C₂

yes

maximal

#11926465

C₂

yes

maximal

#11926443

C₂

yes

maximal

#11926443

C₂

yes

maximal

#11926442

C₂

yes

maximal

C₄×C₄⋊C₄×C₁₉

C₂

yes

maximal, characteristic

C₂²×Dic₃₈

C₂²

yes

characteristic

C₂×C₄⋊C₄×C₁₉

C₂²

yes

—

C₂×C₄²×C₁₉

C₂²

yes

—

C₂×C₄²×C₁₉

C₂²

yes

characteristic

C₂×C₄⋊Dic₁₉

C₂²

yes

characteristic

C₂×C₄⋊Dic₁₉

C₂²

yes

characteristic

C₂×C₄⋊Dic₁₉

C₂²

yes

—

C₂×C₄⋊Dic₁₉

C₂²

yes

characteristic

C₂×C₄⋊Dic₁₉

C₂²

yes

—

C₂×Dic₁₉⋊C₄

C₂²

yes

—

C₂×Dic₁₉⋊C₄

C₂²

yes

—

C₂×Dic₁₉⋊C₄

C₂²

yes

—

C₂×C₄×Dic₁₉

C₂²

yes

—

#10597316

—

C₄×Dic₃₈

—

C₁₉×#95

C₂²

yes

—

C₁₉⋊#95

C₂²

yes

—

C₁₉⋊#95

C₂²

yes

—

C₂²×C₄×C₁₉

C₂³

yes

characteristic

C₂²×C₄×C₁₉

C₂³

yes

characteristic

C₂²×C₄×C₁₉

C₂³

yes

—

C₂²×C₄×C₁₉

C₂³

yes

characteristic

C₂²×Dic₁₉

C₂³

yes

—

C₂²×Dic₁₉

C₂³

yes

—

C₂²×Dic₁₉

C₂³

yes

—

C₂²×Dic₁₉

C₂³

yes

—

C₂×Dic₃₈

—

C₂×Dic₃₈

—

C₄⋊C₄×C₁₉

Q₈

yes

complemented

C₄²×C₁₉

—

C₄²×C₁₉

D₄

yes

—

C₄⋊Dic₁₉

—

C₄⋊Dic₁₉

—

C₄⋊Dic₁₉

—

C₄⋊Dic₁₉

—

C₄⋊Dic₁₉

—

Dic₁₉⋊C₄

—

Dic₁₉⋊C₄

—

Dic₁₉⋊C₄

—

C₄×Dic₁₉

—

C₂³×C₁₉

C₂⁴

yes

characteristic

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

C₂×D₄

yes

—

C₂×C₄×C₁₉

C₂×Q₈

yes

—

C₂×C₄×C₁₉

C₂×Q₈

yes

—

C₂×Dic₁₉

—

C₂×Dic₁₉

—

C₂×Dic₁₉

—

C₂×Dic₁₉

—

C₂×Dic₁₉

—

C₂×Dic₁₉

—

C₂×Dic₁₉

—

C₂×Dic₁₉

—

C₂×Dic₁₉

—

C₂×Dic₁₉

—

C₂×Dic₁₉

#41

yes

—

Dic₃₈

—

#2584

—

maximal, Sylow

C₂²×C₁₉

C₂²×D₄

yes

characteristic

C₂²×C₁₉

C₂²×Q₈

yes

—

C₂²×C₁₉

C₂×#41

yes

—

C₂²×C₁₉

H(5,𝔽₂)

yes

characteristic

C₂²×C₁₉

2₋¹⁺⁴

yes

characteristic

C₄×C₁₉

—

C₄×C₁₉

—

C₄×C₁₉

—

C₄×C₁₉

C₄⋊Q₈

yes

—

Dic₁₉

—

Dic₁₉

—

Dic₁₉

—

Dic₁₉

—

C₂×#125

—

C₂×C₄×Q₈

—

—

—

—

—

—

C₄×C₄⋊C₄

D₁₉

yes

characteristic

C₂×C₁₉

#525

yes

characteristic

C₂×C₁₉

C₂×C₄⋊Q₈

yes

C₂×C₁₉

yes

C₂×C₁₉

yes

—

C₂×C₁₉

#557

yes

—

C₂²×Q₈

—

100

#125

—

101

C₄×Q₈

—

102

C₂×C₄⋊C₄

—

103

C₂×C₄⋊C₄

—

104

C₂×C₄⋊C₄

—

105

C₂×C₄⋊C₄

—

106

C₂×C₄⋊C₄

—

107

C₂×C₄⋊C₄

—

108

C₂×C₄⋊C₄

—

109

C₂×C₄⋊C₄

—

110

C₂×C₄⋊C₄

C₂×D₁₉

yes

—

111

C₂×C₄²

—

112

C₂×C₄²

C₂×D₁₉

yes

—

113

C₂×C₄²

C₂×D₁₉

yes

characteristic

114

#95

—

115

#95

—

116

#95

C₂×D₁₉

yes

—

117

C₁₉

#2584

yes

characteristic, complemented, Sylow

118

C₂×Q₈

—

119

C₂×Q₈

—

120

C₂²×C₄

—

121

C₂²×C₄

—

122

C₂²×C₄

—

123

C₂²×C₄

—

124

C₂²×C₄

C₂²×D₁₉

yes

—

125

C₂²×C₄

C₂²×D₁₉

yes

characteristic

126

C₂²×C₄

C₂²×D₁₉

yes

characteristic

127

C₂²×C₄

C₂²×D₁₉

yes

characteristic

128

C₄⋊C₄

—

129

C₄⋊C₄

—

130

C₄⋊C₄

—

131

C₄⋊C₄

—

132

C₄⋊C₄

—

133

C₄⋊C₄

—

134

C₄⋊C₄

—

135

C₄⋊C₄

—

136

C₄⋊C₄

Dic₃₈

yes

complemented

137

C₄²

—

138

C₄²

—

139

C₄²

D₇₆

yes

—

140

C₂³

C₂³×D₁₉

yes

characteristic

141

Q₈

—

142

C₂×C₄

—

143

C₂×C₄

—

144

C₂×C₄

—

145

C₂×C₄

—

146

C₂×C₄

—

147

C₂×C₄

—

148

C₂×C₄

—

149

C₂×C₄

—

150

C₂×C₄

—

151

C₂×C₄

—

152

C₂×C₄

—

153

C₂×C₄

—

154

C₂×C₄

—

155

C₂×C₄

—

156

C₂×C₄

—

157

C₂×C₄

—

158

C₂×C₄

C₂×Dic₃₈

yes

—

159

C₂×C₄

C₂×Dic₃₈

yes

—

160

C₂×C₄

C₂×D₇₆

yes

—

161

C₂×C₄

C₂×D₇₆

yes

—

162

C₂×C₄

C₂×D₇₆

yes

—

163

C₂²

C₂²×Dic₃₈

yes

—

164

C₂²

C₂²×D₇₆

yes

characteristic

165

C₂²

#10597430

yes

characteristic

166

C₂²

#10597433

yes

characteristic

167

C₂²

#41×D₁₉

yes

—

168

C₄

—

169

C₄

—

170

C₄

—

171

C₄

—

172

C₄

—

173

C₄

—

174

C₄

—

175

C₄

#10597315

yes

—

176

C₂

#11927056

yes

characteristic

177

C₂

C₂×#10597315

yes

characteristic

178

C₂

#11927125

yes

characteristic

179

C₂

#11927253

yes

—

180

C₂

#11927270

yes

—

	subgroup	G/H
center, Frattini	C₂³	C₂³×D₁₉
commutator, socle, prime radical, cosocle	C₂³×C₁₉	C₂⁴
Fitting	C₄×C₄⋊C₄×C₁₉	C₂
hypercenter	C₄×C₄⋊C₄	D₁₉
hypocenter	C₁₉	#2584
LCS	G ▷ C₂³×C₁₉ ▷ C₁₉
UCS	C₁ ◁ C₂³ ◁ C₄×C₄⋊C₄

subgroup

G/H

center, Frattini

C₂³

C₂³×D₁₉

commutator, socle, prime radical, cosocle

C₂³×C₁₉

C₂⁴

Fitting

C₄×C₄⋊C₄×C₁₉

C₂

hypercenter

C₄×C₄⋊C₄

D₁₉

C₁₉

G ▷ C₂³×C₁₉ ▷ C₁₉

C₁ ◁ C₂³ ◁ C₄×C₄⋊C₄

N	H ≅ G/N	type
37. C₄⋊C₄×C₁₉	Q₈	semidirect
117. C₁₉	#2584	semidirect
136. C₄⋊C₄	Dic₃₈	semidirect

H ≅ G/N

type

37. C₄⋊C₄×C₁₉

Q₈

semidirect

117. C₁₉

#2584

semidirect

136. C₄⋊C₄

Dic₃₈

semidirect

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

subgroup	index
C₂×#10597316	2
C₂×C₄×Dic₃₈	2
#11926476	2
#11926465	2
#11926443	2
#11926442	2
C₄×C₄⋊C₄×C₁₉	2
#2584	19

subgroup

index

C₂×C₄×Dic₃₈

C₄×C₄⋊C₄×C₁₉

Subgroups — #14569018

Conjugacy classes of subgroups

Complements

Maximal subgroups