Group #18069573 — subgroups

Subgroups — #18069573

Order 4864.

276 conjugacy classes of subgroups; only the 67 normal subgroups are shown.

normal subgroups only

This group has 276 conjugacy classes of subgroups — too many to draw. Only its 67 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	#14563718	C₂	1	1	yes	maximal, characteristic
2	#14561969	C₂	1	1	yes	maximal, characteristic
3	C₁₉×#1750	C₂	1	1	yes	maximal, characteristic
4	#11926601	—	2	1	no	—
5	#11926508	—	2	1	no	—
6	#11926313	—	2	1	no	—
7	#11926313	C₄	1	2	yes	—
8	#11926270	—	2	1	no	—
9	#11926268	C₄	1	2	yes	—
10	C₂×C₁₉×#117	C₂²	1	1	yes	characteristic
11	C₂×C₁₉×C₂²⋊C₈	—	2	1	no	—
12	C₂×C₁₉×#97	C₄	1	1	yes	characteristic
13	C₂×C₄×C₈×C₁₉	C₄	1	1	yes	characteristic
14	C₁₉×#336	—	2	1	no	—
15	C₂³×C₄×C₁₉	D₄	1	1	yes	characteristic
16	C₂²×C₈×C₁₉	—	2	1	no	—
17	C₂²×C₈×C₁₉	—	2	1	no	—
18	C₁₉×#117	—	4	1	no	—
19	C₁₉×#117	—	4	1	no	—
20	C₂×C₄⋊C₄×C₁₉	C₂×C₄	1	1	yes	characteristic
21	C₂×C₄⋊C₄×C₁₉	D₄	1	1	yes	characteristic
22	C₂×C₂²⋊C₄×C₁₉	C₂×C₄	1	1	yes	characteristic
23	C₂×C₂²⋊C₄×C₁₉	Q₈	1	1	yes	characteristic
24	C₂×C₄²×C₁₉	C₂×C₄	1	1	yes	characteristic
25	C₂×C₄²×C₁₉	D₄	1	1	yes	characteristic
26	C₂×#64520	—	2	2	no	—
27	C₁₉×C₂²⋊C₈	—	2	4	no	—
28	C₁₉×#97	—	4	1	no	—
29	C₄×C₈×C₁₉	—	4	1	no	—
30	C₂²⋊(C₁₉⋊C₈)	—	2	2	no	—
31	C₂²⋊(C₁₉⋊C₈)	—	4	1	no	—
32	C₂²⋊(C₁₉⋊C₈)	—	4	1	no	—
33	C₄⋊(C₁₉⋊C₈)	—	4	1	no	—
34	#10597260	—	4	1	no	—
35	C₂⁴×C₁₉	C₄⋊C₄	1	1	yes	characteristic
36	C₂²×C₄×C₁₉	—	2	1	no	—
37	C₂²×C₄×C₁₉	—	2	1	no	—
38	C₂²×C₄×C₁₉	—	2	2	no	—
39	C₂²×C₄×C₁₉	—	4	1	no	—
40	C₂²×C₄×C₁₉	C₄²	1	1	yes	characteristic
41	C₂²×C₄×C₁₉	C₂²⋊C₄	1	1	yes	characteristic
42	C₂²×C₄×C₁₉	C₂²⋊C₄	1	1	yes	characteristic
43	C₂²×C₄×C₁₉	C₂²⋊C₄	1	1	yes	characteristic
44	C₂²×C₄×C₁₉	C₄⋊C₄	1	1	yes	characteristic
45	C₂²×C₄×C₁₉	C₄⋊C₄	1	1	yes	characteristic
46	C₂×C₈×C₁₉	—	2	2	no	—
47	C₂×C₈×C₁₉	—	2	2	no	—
48	C₂×C₈×C₁₉	—	4	1	no	—
49	C₂×C₈×C₁₉	—	4	1	no	—
50	C₂×C₈×C₁₉	—	4	1	no	—
51	C₂×C₈×C₁₉	—	4	1	no	—
52	C₄⋊C₄×C₁₉	—	4	1	no	—
53	C₄⋊C₄×C₁₉	—	4	1	no	—
54	C₂²⋊C₄×C₁₉	—	4	1	no	—
55	C₂²⋊C₄×C₁₉	—	4	1	no	—
56	C₄²×C₁₉	—	2	1	no	—
57	C₄²×C₁₉	—	2	1	no	—
58	C₄²×C₁₉	—	2	1	no	—
59	C₄²×C₁₉	—	2	1	no	—
60	#64520	—	4	2	no	—
61	#64520	—	4	2	no	—
62	C₂×C₁₉⋊C₈	—	4	2	no	—
63	C₂×C₁₉⋊C₈	—	4	2	no	—
64	#7216	—	19	1	no	maximal, Sylow
65	C₂³×C₁₉	—	2	1	no	—
66	C₂³×C₁₉	—	2	1	no	—
67	C₂³×C₁₉	—	2	1	no	—
68	C₂³×C₁₉	—	4	1	no	—
69	C₂³×C₁₉	—	4	1	no	—
70	C₂³×C₁₉	#95	1	1	yes	characteristic
71	C₂×C₄×C₁₉	—	2	1	no	—
72	C₂×C₄×C₁₉	—	2	1	no	—
73	C₂×C₄×C₁₉	—	2	1	no	—
74	C₂×C₄×C₁₉	—	2	1	no	—
75	C₂×C₄×C₁₉	—	2	1	no	—
76	C₂×C₄×C₁₉	—	2	1	no	—
77	C₂×C₄×C₁₉	—	2	1	no	—
78	C₂×C₄×C₁₉	—	2	1	no	—
79	C₂×C₄×C₁₉	—	4	2	no	—
80	C₂×C₄×C₁₉	—	4	2	no	—
81	C₂×C₄×C₁₉	—	4	1	no	—
82	C₂×C₄×C₁₉	—	4	1	no	—
83	C₂×C₄×C₁₉	—	4	1	no	—
84	C₂×C₄×C₁₉	—	4	1	no	—
85	C₂×C₄×C₁₉	#99	1	1	yes	characteristic
86	C₂×C₄×C₁₉	#99	1	1	yes	characteristic
87	C₂×C₄×C₁₉	#104	1	2	yes	—
88	C₈×C₁₉	—	4	2	no	—
89	C₈×C₁₉	—	4	2	no	—
90	C₁₉⋊C₈	—	8	2	no	—
91	C₁₉⋊C₈	—	8	2	no	—
92	#1822	—	19	1	no	—
93	#1750	D₁₉	1	1	yes	characteristic
94	#1464	—	19	1	no	—
95	C₂²×C₁₉	—	2	1	no	—
96	C₂²×C₁₉	—	2	1	no	—
97	C₂²×C₁₉	—	4	2	no	—
98	C₂²×C₁₉	—	4	1	no	—
99	C₂²×C₁₉	—	4	1	no	—
100	C₂²×C₁₉	—	4	1	no	—
101	C₂²×C₁₉	—	8	1	no	—
102	C₂²×C₁₉	H(3,ℤ/4)	1	1	yes	characteristic
103	C₂²×C₁₉	#339	1	1	yes	characteristic
104	C₂²×C₁₉	#342	1	1	yes	characteristic
105	C₄×C₁₉	—	2	2	no	—
106	C₄×C₁₉	—	2	1	no	—
107	C₄×C₁₉	—	2	1	no	—
108	C₄×C₁₉	—	4	1	no	—
109	C₄×C₁₉	—	4	1	no	—
110	C₄×C₁₉	—	4	1	no	—
111	C₄×C₁₉	—	4	1	no	—
112	C₂×#117	C₂×D₁₉	1	1	yes	characteristic
113	#432	—	38	1	no	—
114	#407	—	38	1	no	—
115	C₂×C₂²⋊C₈	—	2	1	no	—
116	C₂×#97	Dic₁₉	1	1	yes	characteristic
117	C₂×C₄×C₈	Dic₁₉	1	1	yes	characteristic
118	#343	—	19	2	no	—
119	#343	—	38	1	no	—
120	#336	—	2	1	no	—
121	#324	—	38	1	no	—
122	#323	—	19	2	no	—
123	C₂×C₁₉	—	2	1	no	—
124	C₂×C₁₉	—	2	1	no	—
125	C₂×C₁₉	—	8	1	no	—
126	C₂×C₁₉	#1284	1	1	yes	characteristic
127	C₂×C₁₉	#1392	1	1	yes	characteristic
128	C₂×C₁₉	#1395	1	1	yes	characteristic
129	C₂³×C₄	C₂²⋊D₁₉	1	1	yes	characteristic
130	C₂×M₁₆	—	38	2	no	—
131	C₂²×C₈	—	2	1	no	—
132	C₂²×C₈	—	2	1	no	—
133	#117	—	4	1	no	—
134	#117	—	4	1	no	—
135	C₂×C₄⋊C₄	C₄×D₁₉	1	1	yes	characteristic
136	C₂×C₄⋊C₄	D₇₆	1	1	yes	characteristic
137	C₂×C₂²⋊C₄	Dic₃₈	1	1	yes	characteristic
138	C₂×C₂²⋊C₄	C₄×D₁₉	1	1	yes	characteristic
139	C₂×C₄²	C₂×Dic₁₉	1	1	yes	characteristic
140	C₂×C₄²	C₂²⋊D₁₉	1	1	yes	characteristic
141	C₄⋊C₈	—	76	1	no	—
142	C₂²⋊C₈	—	2	4	no	—
143	C₂²⋊C₈	—	38	2	no	—
144	C₂²⋊C₈	—	76	1	no	—
145	C₂²⋊C₈	—	76	1	no	—
146	#97	—	4	1	no	—
147	#97	—	76	1	no	—
148	C₄×C₈	—	4	1	no	—
149	C₁₉	#7216	1	1	yes	characteristic, complemented, Sylow
150	C₂⁴	Dic₁₉⋊C₄	1	1	yes	characteristic
151	C₂²×C₄	—	2	2	no	—
152	C₂²×C₄	—	2	1	no	—
153	C₂²×C₄	—	2	1	no	—
154	C₂²×C₄	—	4	1	no	—
155	C₂²×C₄	C₄×Dic₁₉	1	1	yes	characteristic
156	C₂²×C₄	Dic₁₉⋊C₄	1	1	yes	characteristic
157	C₂²×C₄	C₄⋊Dic₁₉	1	1	yes	characteristic
158	C₂²×C₄	(C₂×D₁₉)⋊C₄	1	1	yes	characteristic
159	C₂²×C₄	(C₂×D₁₉)⋊C₄	1	1	yes	characteristic
160	C₂²×C₄	C₂²⋊Dic₁₉	1	1	yes	characteristic
161	M₁₆	—	76	2	no	—
162	M₁₆	—	76	2	no	—
163	C₂×C₈	—	2	2	no	—
164	C₂×C₈	—	2	2	no	—
165	C₂×C₈	—	4	1	no	—
166	C₂×C₈	—	4	1	no	—
167	C₂×C₈	—	4	1	no	—
168	C₂×C₈	—	4	1	no	—
169	C₂×C₈	—	76	2	no	—
170	C₂×C₈	—	76	2	no	—
171	C₄⋊C₄	—	4	1	no	—
172	C₄⋊C₄	—	4	1	no	—
173	C₂²⋊C₄	—	4	1	no	—
174	C₂²⋊C₄	—	4	1	no	—
175	C₄²	—	2	1	no	—
176	C₄²	—	2	1	no	—
177	C₄²	—	2	1	no	—
178	C₄²	—	2	1	no	—
179	C₂³	—	2	1	no	—
180	C₂³	—	2	1	no	—
181	C₂³	—	2	1	no	—
182	C₂³	—	4	1	no	—
183	C₂³	—	4	1	no	—
184	C₂³	C₁₉⋊#95	1	1	yes	characteristic
185	C₂×C₄	—	2	1	no	—
186	C₂×C₄	—	2	1	no	—
187	C₂×C₄	—	2	1	no	—
188	C₂×C₄	—	2	1	no	—
189	C₂×C₄	—	2	1	no	—
190	C₂×C₄	—	2	1	no	—
191	C₂×C₄	—	2	1	no	—
192	C₂×C₄	—	2	1	no	—
193	C₂×C₄	—	4	2	no	—
194	C₂×C₄	—	4	2	no	—
195	C₂×C₄	—	4	1	no	—
196	C₂×C₄	—	4	1	no	—
197	C₂×C₄	—	4	1	no	—
198	C₂×C₄	—	4	1	no	—
199	C₂×C₄	#10597262	1	2	yes	—
200	C₂×C₄	#10597291	1	1	yes	characteristic
201	C₂×C₄	#10597291	1	1	yes	characteristic
202	C₈	—	4	2	no	—
203	C₈	—	4	2	no	—
204	C₈	—	152	2	no	—
205	C₈	—	152	2	no	—
206	C₂²	—	2	1	no	—
207	C₂²	—	2	1	no	—
208	C₂²	—	4	2	no	—
209	C₂²	—	4	1	no	—
210	C₂²	—	4	1	no	—
211	C₂²	—	4	1	no	—
212	C₂²	—	8	1	no	—
213	C₂²	#11926300	1	1	yes	characteristic
214	C₂²	#11926304	1	1	yes	characteristic
215	C₂²	#11926311	1	1	yes	characteristic
216	C₄	—	2	1	no	—
217	C₄	—	2	1	no	—
218	C₄	—	2	2	no	—
219	C₄	—	4	1	no	—
220	C₄	—	4	1	no	—
221	C₄	—	4	1	no	—
222	C₄	—	4	1	no	—
223	C₂	—	2	1	no	—
224	C₂	—	2	1	no	—
225	C₂	—	8	1	no	—
226	C₂	#14561488	1	1	yes	characteristic
227	C₂	#14561758	1	1	yes	characteristic
228	C₂	#14561764	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#14563718

C₂

yes

maximal, characteristic

#14561969

C₂

yes

maximal, characteristic

C₁₉×#1750

C₂

yes

maximal, characteristic

—

—

—

C₄

yes

—

#11926270

—

#11926268

C₄

yes

—

C₂×C₁₉×#117

C₂²

yes

characteristic

C₂×C₁₉×C₂²⋊C₈

—

C₂×C₁₉×#97

C₄

yes

characteristic

C₂×C₄×C₈×C₁₉

C₄

yes

characteristic

C₁₉×#336

—

C₂³×C₄×C₁₉

D₄

yes

characteristic

C₂²×C₈×C₁₉

—

C₂²×C₈×C₁₉

—

C₁₉×#117

—

C₁₉×#117

—

C₂×C₄⋊C₄×C₁₉

C₂×C₄

yes

characteristic

C₂×C₄⋊C₄×C₁₉

D₄

yes

characteristic

C₂×C₂²⋊C₄×C₁₉

C₂×C₄

yes

characteristic

C₂×C₂²⋊C₄×C₁₉

Q₈

yes

characteristic

C₂×C₄²×C₁₉

C₂×C₄

yes

characteristic

C₂×C₄²×C₁₉

D₄

yes

characteristic

C₂×#64520

—

C₁₉×C₂²⋊C₈

—

C₁₉×#97

—

C₄×C₈×C₁₉

—

C₂²⋊(C₁₉⋊C₈)

—

C₂²⋊(C₁₉⋊C₈)

—

C₂²⋊(C₁₉⋊C₈)

—

C₄⋊(C₁₉⋊C₈)

—

#10597260

—

C₂⁴×C₁₉

C₄⋊C₄

yes

characteristic

C₂²×C₄×C₁₉

—

C₂²×C₄×C₁₉

—

C₂²×C₄×C₁₉

—

C₂²×C₄×C₁₉

—

C₂²×C₄×C₁₉

C₄²

yes

characteristic

C₂²×C₄×C₁₉

C₂²⋊C₄

yes

characteristic

C₂²×C₄×C₁₉

C₂²⋊C₄

yes

characteristic

C₂²×C₄×C₁₉

C₂²⋊C₄

yes

characteristic

C₂²×C₄×C₁₉

C₄⋊C₄

yes

characteristic

C₂²×C₄×C₁₉

C₄⋊C₄

yes

characteristic

C₂×C₈×C₁₉

—

C₂×C₈×C₁₉

—

C₂×C₈×C₁₉

—

C₂×C₈×C₁₉

—

C₂×C₈×C₁₉

—

C₂×C₈×C₁₉

—

C₄⋊C₄×C₁₉

—

C₄⋊C₄×C₁₉

—

C₂²⋊C₄×C₁₉

—

C₂²⋊C₄×C₁₉

—

C₄²×C₁₉

—

C₄²×C₁₉

—

C₄²×C₁₉

—

C₄²×C₁₉

—

#64520

—

#64520

—

C₂×C₁₉⋊C₈

—

C₂×C₁₉⋊C₈

—

#7216

—

maximal, Sylow

C₂³×C₁₉

—

C₂³×C₁₉

—

C₂³×C₁₉

—

C₂³×C₁₉

—

C₂³×C₁₉

—

C₂³×C₁₉

#95

yes

characteristic

C₂×C₄×C₁₉

—

C₂×C₄×C₁₉

—

C₂×C₄×C₁₉

—

C₂×C₄×C₁₉

—

C₂×C₄×C₁₉

—

C₂×C₄×C₁₉

—

C₂×C₄×C₁₉

—

C₂×C₄×C₁₉

—

C₂×C₄×C₁₉

—

C₂×C₄×C₁₉

—

C₂×C₄×C₁₉

—

C₂×C₄×C₁₉

—

C₂×C₄×C₁₉

—

C₂×C₄×C₁₉

—

C₂×C₄×C₁₉

yes

C₂×C₄×C₁₉

yes

C₂×C₄×C₁₉

yes

—

C₈×C₁₉

—

C₈×C₁₉

—

C₁₉⋊C₈

—

C₁₉⋊C₈

—

#1822

—

#1750

D₁₉

yes

characteristic

#1464

—

C₂²×C₁₉

—

C₂²×C₁₉

—

C₂²×C₁₉

—

C₂²×C₁₉

—

C₂²×C₁₉

—

100

C₂²×C₁₉

—

101

C₂²×C₁₉

—

102

C₂²×C₁₉

H(3,ℤ/4)

yes

characteristic

103

C₂²×C₁₉

#339

yes

characteristic

104

C₂²×C₁₉

#342

yes

characteristic

105

C₄×C₁₉

—

106

C₄×C₁₉

—

107

C₄×C₁₉

—

108

C₄×C₁₉

—

109

C₄×C₁₉

—

110

C₄×C₁₉

—

111

C₄×C₁₉

—

112

C₂×#117

C₂×D₁₉

yes

characteristic

113

#432

—

114

#407

—

115

C₂×C₂²⋊C₈

—

116

C₂×#97

Dic₁₉

yes

characteristic

117

C₂×C₄×C₈

Dic₁₉

yes

characteristic

118

#343

—

119

#343

—

120

#336

—

121

#324

—

122

#323

—

123

C₂×C₁₉

—

124

C₂×C₁₉

—

125

C₂×C₁₉

—

126

C₂×C₁₉

#1284

yes

characteristic

127

C₂×C₁₉

#1392

yes

characteristic

128

C₂×C₁₉

#1395

yes

characteristic

129

C₂³×C₄

C₂²⋊D₁₉

yes

characteristic

130

C₂×M₁₆

—

131

C₂²×C₈

—

132

C₂²×C₈

—

133

#117

—

134

#117

—

135

C₂×C₄⋊C₄

C₄×D₁₉

yes

characteristic

136

C₂×C₄⋊C₄

D₇₆

yes

characteristic

137

C₂×C₂²⋊C₄

Dic₃₈

yes

characteristic

138

C₂×C₂²⋊C₄

C₄×D₁₉

yes

characteristic

139

C₂×C₄²

C₂×Dic₁₉

yes

characteristic

140

C₂×C₄²

C₂²⋊D₁₉

yes

characteristic

141

C₄⋊C₈

—

142

C₂²⋊C₈

—

143

C₂²⋊C₈

—

144

C₂²⋊C₈

—

145

C₂²⋊C₈

—

146

#97

—

147

#97

—

148

C₄×C₈

—

149

C₁₉

#7216

yes

characteristic, complemented, Sylow

150

C₂⁴

Dic₁₉⋊C₄

yes

characteristic

151

C₂²×C₄

—

152

C₂²×C₄

—

153

C₂²×C₄

—

154

C₂²×C₄

—

155

C₂²×C₄

C₄×Dic₁₉

yes

characteristic

156

C₂²×C₄

Dic₁₉⋊C₄

yes

characteristic

157

C₂²×C₄

C₄⋊Dic₁₉

yes

characteristic

158

C₂²×C₄

(C₂×D₁₉)⋊C₄

yes

characteristic

159

C₂²×C₄

(C₂×D₁₉)⋊C₄

yes

characteristic

160

C₂²×C₄

C₂²⋊Dic₁₉

yes

characteristic

161

M₁₆

—

162

M₁₆

—

163

C₂×C₈

—

164

C₂×C₈

—

165

C₂×C₈

—

166

C₂×C₈

—

167

C₂×C₈

—

168

C₂×C₈

—

169

C₂×C₈

—

170

C₂×C₈

—

171

C₄⋊C₄

—

172

C₄⋊C₄

—

173

C₂²⋊C₄

—

174

C₂²⋊C₄

—

175

C₄²

—

176

C₄²

—

177

C₄²

—

178

C₄²

—

179

C₂³

—

180

C₂³

—

181

C₂³

—

182

C₂³

—

183

C₂³

—

184

C₂³

C₁₉⋊#95

yes

characteristic

185

C₂×C₄

—

186

C₂×C₄

—

187

C₂×C₄

—

188

C₂×C₄

—

189

C₂×C₄

—

190

C₂×C₄

—

191

C₂×C₄

—

192

C₂×C₄

—

193

C₂×C₄

—

194

C₂×C₄

—

195

C₂×C₄

—

196

C₂×C₄

—

197

C₂×C₄

—

198

C₂×C₄

—

199

C₂×C₄

#10597262

yes

—

200

C₂×C₄

#10597291

yes

characteristic

201

C₂×C₄

#10597291

yes

characteristic

202

C₈

—

203

C₈

—

204

C₈

—

152

—

205

C₈

—

152

—

206

C₂²

—

207

C₂²

—

208

C₂²

—

209

C₂²

—

210

C₂²

—

211

C₂²

—

212

C₂²

—

213

C₂²

#11926300

yes

characteristic

214

C₂²

#11926304

yes

characteristic

215

C₂²

#11926311

yes

characteristic

216

C₄

—

217

C₄

—

218

C₄

—

219

C₄

—

220

C₄

—

221

C₄

—

222

C₄

—

223

C₂

—

224

C₂

—

225

C₂

—

226

C₂

#14561488

yes

characteristic

227

C₂

#14561758

yes

characteristic

228

C₂

#14561764

yes

characteristic

	subgroup	G/H
center	C₂²	#11926311
commutator	C₂²×C₄×C₁₉	C₄²
Frattini	C₂×#117	C₂×D₁₉
Fitting	C₁₉×#1750	C₂
socle	C₂²×C₁₉	#342
prime radical	C₂⁴×C₁₉	C₄⋊C₄
cosocle	C₂×C₁₉×#117	C₂²
hypercenter	#1750	D₁₉
hypocenter	C₁₉	#7216
LCS	G ▷ C₂²×C₄×C₁₉ ▷ C₂²×C₁₉ ▷ C₂×C₁₉ ▷ C₁₉
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×#117 ◁ #1750

subgroup

G/H

center

C₂²

#11926311

commutator

C₂²×C₄×C₁₉

C₄²

Frattini

C₂×#117

C₂×D₁₉

Fitting

C₁₉×#1750

C₂

socle

C₂²×C₁₉

#342

prime radical

C₂⁴×C₁₉

C₄⋊C₄

cosocle

C₂×C₁₉×#117

C₂²

D₁₉

C₁₉

G ▷ C₂²×C₄×C₁₉ ▷ C₂²×C₁₉ ▷ C₂×C₁₉ ▷ C₁₉

UCS

C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×#117 ◁ #1750

N	H ≅ G/N	type
149. C₁₉	#7216	semidirect

H ≅ G/N

type

149. C₁₉

#7216

semidirect

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

subgroup	index
#14563718	2
#14561969	2
C₁₉×#1750	2
#7216	19

subgroup

index

#14563718

#14561969

C₁₉×#1750

#7216

Subgroups — #18069573

Conjugacy classes of subgroups

Complements

Maximal subgroups