Group #18082306 — subgroups

Subgroups — #18082306

Order 4864.

558 conjugacy classes of subgroups; only the 211 normal subgroups are shown.

normal subgroups only

This group has 558 conjugacy classes of subgroups — too many to draw. Only its 211 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×#11926924	C₂	1	1	yes	maximal, characteristic
2	#14564596	C₂	1	4	yes	maximal
3	#14563795	C₂	1	1	yes	maximal, characteristic
4	C₂×C₈²×C₁₉	C₂	1	1	yes	maximal, characteristic
5	C₂²×Dic₇₆	—	2	2	no	—
6	C₂×#10597315	C₂²	1	2	yes	—
7	#11926924	—	2	2	no	—
8	#11926924	C₄	1	4	yes	complemented
9	C₂×#10597275	—	2	2	no	—
10	#11926414	C₂²	1	2	yes	—
11	#11926307	—	2	8	no	—
12	C₂×C₄×C₈×C₁₉	C₂²	1	2	yes	—
13	C₂×C₄×C₈×C₁₉	C₂²	1	1	yes	characteristic
14	C₈²×C₁₉	—	2	2	no	—
15	C₂²×Dic₃₈	—	2	2	no	—
16	C₂²×C₈×C₁₉	D₄	1	4	yes	—
17	C₂²×C₈×C₁₉	D₄	1	2	yes	—
18	C₂×C₄²×C₁₉	C₂³	1	1	yes	characteristic
19	C₂×C₄⋊Dic₁₉	—	2	4	no	—
20	C₂×C₄⋊Dic₁₉	—	2	4	no	—
21	C₂×C₄⋊Dic₁₉	—	2	2	no	—
22	C₂×Dic₇₆	—	2	8	no	—
23	C₂×Dic₇₆	—	4	4	no	—
24	C₂×Dic₇₆	—	4	4	no	—
25	#10597315	—	2	4	no	—
26	#10597315	—	2	2	no	—
27	#10597315	C₂×C₄	1	4	yes	—
28	C₄×C₈×C₁₉	—	2	2	no	—
29	C₄×C₈×C₁₉	—	2	2	no	—
30	C₄×C₈×C₁₉	C₂×C₄	1	2	yes	—
31	C₄×C₈×C₁₉	D₄	1	2	yes	—
32	#10597275	—	4	4	no	—
33	C₂²×C₄×C₁₉	C₂×D₄	1	1	yes	characteristic
34	C₂²×C₄×C₁₉	C₂×D₄	1	2	yes	—
35	C₂²×Dic₁₉	—	4	2	no	—
36	C₂²×Dic₁₉	—	4	2	no	—
37	C₂×Dic₃₈	—	2	4	no	—
38	C₂×Dic₃₈	—	4	4	no	—
39	C₂×Dic₃₈	—	4	4	no	—
40	C₂×Dic₃₈	—	4	2	no	—
41	C₂×C₈×C₁₉	—	2	4	no	—
42	C₂×C₈×C₁₉	—	2	4	no	—
43	C₂×C₈×C₁₉	C₂²⋊C₄	1	4	yes	—
44	C₂×C₈×C₁₉	D₈	1	4	yes	—
45	C₂×C₈×C₁₉	SD₁₆	1	4	yes	—
46	C₂×C₈×C₁₉	Q₁₆	1	8	yes	complemented
47	C₄²×C₁₉	C₂²×C₄	1	1	yes	characteristic
48	C₄²×C₁₉	C₂×D₄	1	1	yes	characteristic
49	C₄²×C₁₉	#41	1	2	yes	—
50	C₄⋊Dic₁₉	—	2	8	no	—
51	C₄⋊Dic₁₉	—	4	4	no	—
52	C₄⋊Dic₁₉	—	4	4	no	—
53	C₄⋊Dic₁₉	—	4	8	no	—
54	Dic₇₆	—	4	16	no	—
55	#9825	—	19	1	no	maximal, Sylow
56	C₂³×C₁₉	#127	1	1	yes	characteristic
57	C₂×C₄×C₁₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
58	C₂×C₄×C₁₉	C₄×D₄	1	2	yes	—
59	C₂×C₄×C₁₉	#121	1	2	yes	—
60	C₂×C₄×C₁₉	#124	1	1	yes	characteristic
61	C₂×C₄×C₁₉	C₂×D₈	1	2	yes	—
62	C₂×C₄×C₁₉	C₂×SD₁₆	1	2	yes	—
63	C₂×C₄×C₁₉	C₂×Q₁₆	1	4	yes	—
64	C₂×C₄×C₁₉	#135	1	4	yes	—
65	C₂×Dic₁₉	—	4	4	no	—
66	C₂×Dic₁₉	—	8	2	no	—
67	C₂×Dic₁₉	—	8	2	no	—
68	C₂×Dic₁₉	—	8	2	no	—
69	C₂×Dic₁₉	—	8	4	no	—
70	Dic₃₈	—	4	8	no	—
71	Dic₃₈	—	8	4	no	—
72	C₈×C₁₉	—	2	8	no	—
73	C₈×C₁₉	#102	1	8	yes	—
74	C₂×#494	—	19	1	no	—
75	#1990	—	19	4	no	—
76	#1845	—	19	1	no	—
77	C₂×C₈²	D₁₉	1	1	yes	characteristic
78	C₂²×C₁₉	#390	1	1	yes	characteristic
79	C₂²×C₁₉	#492	1	1	yes	characteristic
80	C₂²×C₁₉	#493	1	1	yes	characteristic
81	C₂²×C₁₉	#494	1	2	yes	—
82	C₂²×C₁₉	#495	1	2	yes	—
83	C₄×C₁₉	C₂×#102	1	2	yes	—
84	C₄×C₁₉	C₄×Q₁₆	1	4	yes	—
85	C₄×C₁₉	#467	1	4	yes	—
86	C₄×C₁₉	#486	1	2	yes	—
87	Dic₁₉	—	8	4	no	—
88	Dic₁₉	—	16	2	no	—
89	C₂²×Q₁₆	—	38	2	no	—
90	C₂×C₄⋊Q₈	—	19	2	no	—
91	#494	—	19	4	no	—
92	#494	—	38	2	no	—
93	C₂×#107	—	38	2	no	—
94	C₂×C₄×C₈	C₂×D₁₉	1	2	yes	—
95	C₂×C₄×C₈	C₂×D₁₉	1	1	yes	characteristic
96	#384	—	19	2	no	—
97	#340	—	38	8	no	—
98	C₈²	—	2	2	no	—
99	C₂×C₁₉	#1707	1	2	yes	—
100	C₂×C₁₉	#1709	1	1	yes	characteristic
101	C₂×C₁₉	#1713	1	1	yes	characteristic
102	C₂×C₁₉	#1957	1	1	yes	characteristic
103	C₂×C₁₉	#1969	1	2	yes	—
104	C₂²×Q₈	—	38	2	no	—
105	C₂×Q₁₆	—	38	8	no	—
106	C₂×Q₁₆	—	76	4	no	—
107	C₂×Q₁₆	—	76	4	no	—
108	C₂²×C₈	D₇₆	1	4	yes	—
109	C₂²×C₈	D₇₆	1	2	yes	—
110	C₄⋊Q₈	—	19	4	no	—
111	C₄⋊Q₈	—	38	4	no	—
112	C₄⋊Q₈	—	38	2	no	—
113	C₂×C₄⋊C₄	—	38	4	no	—
114	C₂×C₄⋊C₄	—	38	4	no	—
115	C₂×C₄⋊C₄	—	38	2	no	—
116	C₂×C₄²	C₂²×D₁₉	1	1	yes	characteristic
117	#107	—	76	4	no	—
118	C₄×C₈	—	2	2	no	—
119	C₄×C₈	—	2	2	no	—
120	C₄×C₈	C₄×D₁₉	1	2	yes	—
121	C₄×C₈	C₂²⋊D₁₉	1	2	yes	—
122	C₁₉	#9825	1	1	yes	characteristic, complemented, Sylow
123	C₂×Q₈	—	38	4	no	—
124	C₂×Q₈	—	76	2	no	—
125	C₂×Q₈	—	76	4	no	—
126	C₂×Q₈	—	76	4	no	—
127	C₂²×C₄	—	76	2	no	—
128	C₂²×C₄	—	76	2	no	—
129	C₂²×C₄	C₂×D₇₆	1	2	yes	—
130	C₂²×C₄	C₂×D₇₆	1	1	yes	characteristic
131	Q₁₆	—	76	16	no	—
132	C₂×C₈	—	2	4	no	—
133	C₂×C₈	—	2	4	no	—
134	C₂×C₈	#64516	1	4	yes	—
135	C₂×C₈	D₁₅₂	1	4	yes	—
136	C₂×C₈	Dic₇₆	1	8	yes	complemented
137	C₂×C₈	(C₂×D₁₉)⋊C₄	1	4	yes	—
138	C₄⋊C₄	—	38	8	no	—
139	C₄⋊C₄	—	76	4	no	—
140	C₄⋊C₄	—	76	4	no	—
141	C₄⋊C₄	—	76	8	no	—
142	C₄²	C₂×C₄×D₁₉	1	1	yes	characteristic
143	C₄²	#64541	1	2	yes	—
144	C₄²	C₂×C₂²⋊D₁₉	1	1	yes	characteristic
145	C₂³	#10597320	1	1	yes	characteristic
146	Q₈	—	76	8	no	—
147	Q₈	—	152	4	no	—
148	C₂×C₄	—	76	4	no	—
149	C₂×C₄	—	152	2	no	—
150	C₂×C₄	—	152	4	no	—
151	C₂×C₄	—	152	2	no	—
152	C₂×C₄	—	152	2	no	—
153	C₂×C₄	C₄×D₇₆	1	2	yes	—
154	C₂×C₄	#10597321	1	1	yes	characteristic
155	C₂×C₄	C₂×#64516	1	2	yes	—
156	C₂×C₄	C₂×D₁₅₂	1	2	yes	—
157	C₂×C₄	#10597350	1	4	yes	—
158	C₂×C₄	C₂×Dic₇₆	1	4	yes	—
159	C₂×C₄	C₂×(C₂×D₁₉)⋊C₄	1	1	yes	characteristic
160	C₂×C₄	#10597376	1	2	yes	—
161	C₈	—	2	8	no	—
162	C₈	#10597278	1	8	yes	—
163	C₂²	#11926446	1	1	yes	characteristic
164	C₂²	#11926917	1	1	yes	characteristic
165	C₂²	#11926921	1	1	yes	characteristic
166	C₂²	#11926924	1	2	yes	—
167	C₂²	#11926928	1	2	yes	—
168	C₄	—	304	2	no	—
169	C₄	—	152	4	no	—
170	C₄	C₂×#10597278	1	2	yes	—
171	C₄	C₄×Dic₇₆	1	4	yes	—
172	C₄	#11926785	1	4	yes	—
173	C₄	#11926891	1	2	yes	—
174	C₂	#14563232	1	2	yes	—
175	C₂	#14563241	1	1	yes	characteristic
176	C₂	#14563253	1	1	yes	characteristic
177	C₂	#14564441	1	1	yes	characteristic
178	C₂	#14564496	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#11926924

C₂

yes

maximal, characteristic

#14564596

C₂

yes

maximal

#14563795

C₂

yes

maximal, characteristic

C₂×C₈²×C₁₉

C₂

yes

maximal, characteristic

C₂²×Dic₇₆

—

C₂×#10597315

C₂²

yes

—

#11926924

—

#11926924

C₄

yes

complemented

C₂×#10597275

—

#11926414

C₂²

yes

—

#11926307

—

C₂×C₄×C₈×C₁₉

C₂²

yes

—

C₂×C₄×C₈×C₁₉

C₂²

yes

characteristic

C₈²×C₁₉

—

C₂²×Dic₃₈

—

C₂²×C₈×C₁₉

D₄

yes

—

C₂²×C₈×C₁₉

D₄

yes

—

C₂×C₄²×C₁₉

C₂³

yes

characteristic

C₂×C₄⋊Dic₁₉

—

C₂×C₄⋊Dic₁₉

—

C₂×C₄⋊Dic₁₉

—

C₂×Dic₇₆

—

C₂×Dic₇₆

—

C₂×Dic₇₆

—

#10597315

—

#10597315

—

#10597315

C₂×C₄

yes

—

C₄×C₈×C₁₉

—

C₄×C₈×C₁₉

—

C₄×C₈×C₁₉

C₂×C₄

yes

—

C₄×C₈×C₁₉

D₄

yes

—

#10597275

—

C₂²×C₄×C₁₉

C₂×D₄

yes

characteristic

C₂²×C₄×C₁₉

C₂×D₄

yes

—

C₂²×Dic₁₉

—

C₂²×Dic₁₉

—

C₂×Dic₃₈

—

C₂×Dic₃₈

—

C₂×Dic₃₈

—

C₂×Dic₃₈

—

C₂×C₈×C₁₉

—

C₂×C₈×C₁₉

—

C₂×C₈×C₁₉

C₂²⋊C₄

yes

—

C₂×C₈×C₁₉

D₈

yes

—

C₂×C₈×C₁₉

SD₁₆

yes

—

C₂×C₈×C₁₉

Q₁₆

yes

complemented

C₄²×C₁₉

C₂²×C₄

yes

characteristic

C₄²×C₁₉

C₂×D₄

yes

characteristic

C₄²×C₁₉

#41

yes

—

C₄⋊Dic₁₉

—

C₄⋊Dic₁₉

—

C₄⋊Dic₁₉

—

C₄⋊Dic₁₉

—

Dic₇₆

—

—

C₂³×C₁₉

yes

C₂×C₄×C₁₉

C₂×C₂²⋊C₄

yes

characteristic

C₂×C₄×C₁₉

C₄×D₄

yes

—

C₂×C₄×C₁₉

#121

yes

—

C₂×C₄×C₁₉

#124

yes

characteristic

C₂×C₄×C₁₉

C₂×D₈

yes

—

C₂×C₄×C₁₉

C₂×SD₁₆

yes

—

C₂×C₄×C₁₉

C₂×Q₁₆

yes

—

C₂×C₄×C₁₉

#135

yes

—

C₂×Dic₁₉

—

C₂×Dic₁₉

—

C₂×Dic₁₉

—

C₂×Dic₁₉

—

C₂×Dic₁₉

—

Dic₃₈

—

Dic₃₈

—

C₈×C₁₉

—

C₈×C₁₉

#102

yes

—

C₂×#494

—

#1990

—

#1845

—

C₂×C₈²

D₁₉

yes

C₂²×C₁₉

yes

C₂²×C₁₉

yes

C₂²×C₁₉

yes

C₂²×C₁₉

yes

—

C₂²×C₁₉

#495

yes

—

C₄×C₁₉

C₂×#102

yes

—

C₄×C₁₉

C₄×Q₁₆

yes

—

C₄×C₁₉

#467

yes

—

C₄×C₁₉

#486

yes

—

Dic₁₉

—

Dic₁₉

—

C₂²×Q₁₆

—

C₂×C₄⋊Q₈

—

#494

—

#494

—

C₂×#107

—

C₂×C₄×C₈

C₂×D₁₉

yes

—

C₂×C₄×C₈

C₂×D₁₉

yes

characteristic

#384

—

#340

—

C₈²

—

C₂×C₁₉

#1707

yes

—

100

C₂×C₁₉

#1709

yes

characteristic

101

C₂×C₁₉

#1713

yes

characteristic

102

C₂×C₁₉

#1957

yes

characteristic

103

C₂×C₁₉

#1969

yes

—

104

C₂²×Q₈

—

105

C₂×Q₁₆

—

106

C₂×Q₁₆

—

107

C₂×Q₁₆

—

108

C₂²×C₈

D₇₆

yes

—

109

C₂²×C₈

D₇₆

yes

—

110

C₄⋊Q₈

—

111

C₄⋊Q₈

—

112

C₄⋊Q₈

—

113

C₂×C₄⋊C₄

—

114

C₂×C₄⋊C₄

—

115

C₂×C₄⋊C₄

—

116

C₂×C₄²

C₂²×D₁₉

yes

characteristic

117

#107

—

118

C₄×C₈

—

119

C₄×C₈

—

120

C₄×C₈

C₄×D₁₉

yes

—

121

C₄×C₈

C₂²⋊D₁₉

yes

—

122

C₁₉

#9825

yes

characteristic, complemented, Sylow

123

C₂×Q₈

—

124

C₂×Q₈

—

125

C₂×Q₈

—

126

C₂×Q₈

—

127

C₂²×C₄

—

128

C₂²×C₄

—

129

C₂²×C₄

C₂×D₇₆

yes

—

130

C₂²×C₄

C₂×D₇₆

yes

characteristic

131

Q₁₆

—

132

C₂×C₈

—

133

C₂×C₈

—

134

C₂×C₈

#64516

yes

—

135

C₂×C₈

D₁₅₂

yes

—

136

C₂×C₈

Dic₇₆

yes

complemented

137

C₂×C₈

(C₂×D₁₉)⋊C₄

yes

—

138

C₄⋊C₄

—

139

C₄⋊C₄

—

140

C₄⋊C₄

—

141

C₄⋊C₄

—

142

C₄²

C₂×C₄×D₁₉

yes

characteristic

143

C₄²

#64541

yes

—

144

C₄²

C₂×C₂²⋊D₁₉

yes

characteristic

145

C₂³

#10597320

yes

characteristic

146

Q₈

—

147

Q₈

—

152

—

148

C₂×C₄

—

149

C₂×C₄

—

152

—

150

C₂×C₄

—

152

—

151

C₂×C₄

—

152

—

152

C₂×C₄

—

152

—

153

C₂×C₄

C₄×D₇₆

yes

—

154

C₂×C₄

#10597321

yes

characteristic

155

C₂×C₄

C₂×#64516

yes

—

156

C₂×C₄

C₂×D₁₅₂

yes

—

157

C₂×C₄

#10597350

yes

—

158

C₂×C₄

C₂×Dic₇₆

yes

—

159

C₂×C₄

C₂×(C₂×D₁₉)⋊C₄

yes

characteristic

160

C₂×C₄

#10597376

yes

—

161

C₈

—

162

C₈

#10597278

yes

—

163

C₂²

#11926446

yes

characteristic

164

C₂²

#11926917

yes

characteristic

165

C₂²

#11926921

yes

characteristic

166

C₂²

#11926924

yes

—

167

C₂²

#11926928

yes

—

168

C₄

—

304

—

169

C₄

—

152

—

170

C₄

C₂×#10597278

yes

—

171

C₄

C₄×Dic₇₆

yes

—

172

C₄

#11926785

yes

—

173

C₄

#11926891

yes

—

174

C₂

#14563232

yes

—

175

C₂

#14563241

yes

characteristic

176

C₂

#14563253

yes

characteristic

177

C₂

#14564441

yes

characteristic

178

C₂

#14564496

yes

—

	subgroup	G/H
center	C₂³	#10597320
commutator	C₄²×C₁₉	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₁₉
Fitting	C₂×C₈²×C₁₉	C₂
socle, prime radical	C₂³×C₁₉	#127
cosocle	C₂×C₄²×C₁₉	C₂³
hypercenter	C₂×C₈²	D₁₉
hypocenter	C₁₉	#9825
LCS	G ▷ C₄²×C₁₉ ▷ C₂²×C₁₉ ▷ C₁₉
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₈²

subgroup

G/H

center

C₂³

#10597320

commutator

C₄²×C₁₉

C₂²×C₄

Frattini

C₂×C₄²

C₂²×D₁₉

Fitting

C₂×C₈²×C₁₉

C₂

socle, prime radical

C₂³×C₁₉

#127

cosocle

C₂×C₄²×C₁₉

C₂³

hypercenter

C₂×C₈²

D₁₉

hypocenter

C₁₉

#9825

LCS

G ▷ C₄²×C₁₉ ▷ C₂²×C₁₉ ▷ C₁₉

UCS

C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₈²

N	H ≅ G/N	type
8. #11926924	C₄	semidirect
46. C₂×C₈×C₁₉	Q₁₆	semidirect
122. C₁₉	#9825	semidirect
136. C₂×C₈	Dic₇₆	semidirect

H ≅ G/N

type

8. #11926924

C₄

semidirect

46. C₂×C₈×C₁₉

Q₁₆

semidirect

122. C₁₉

#9825

semidirect

136. C₂×C₈

Dic₇₆

semidirect

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

subgroup	index
C₂×#11926924	2
#14564596	2
#14563795	2
C₂×C₈²×C₁₉	2
#9825	19

subgroup

index

C₂×C₈²×C₁₉

Subgroups — #18082306

Conjugacy classes of subgroups

Complements

Maximal subgroups