Group #18069562 — subgroups

Subgroups — #18069562

Order 4864.

212 conjugacy classes of subgroups; only the 99 normal subgroups are shown.

normal subgroups only

This group has 212 conjugacy classes of subgroups — too many to draw. Only its 99 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	#14565434	C₂	1	1	yes	maximal, characteristic
2	#14565391	C₂	1	1	yes	maximal, characteristic
3	C₈×M₁₆×C₁₉	C₂	1	1	yes	maximal, characteristic
4	C₂×#10597269	C₄	1	2	yes	—
5	C₄⋊(C₁₉⋊C₁₆)	—	2	1	no	—
6	C₄⋊(C₁₉⋊C₁₆)	—	2	1	no	—
7	C₄⋊(C₁₉⋊C₁₆)	—	2	1	no	—
8	C₂²⋊(C₁₉⋊C₁₆)	C₄	1	2	yes	—
9	C₄×C₁₉⋊C₁₆	—	2	1	no	—
10	C₁₉×#431	C₄	1	1	yes	characteristic
11	C₄×M₁₆×C₁₉	C₄	1	1	yes	characteristic
12	C₂×C₄×C₈×C₁₉	C₂²	1	1	yes	characteristic
13	C₁₉×#322	—	2	1	no	—
14	C₈²×C₁₉	—	2	1	no	—
15	C₂×M₁₆×C₁₉	—	2	1	no	—
16	C₂²×C₈×C₁₉	C₂×C₄	1	1	yes	characteristic
17	C₂²×C₈×C₁₉	C₂×C₄	1	1	yes	characteristic
18	C₂×C₄²×C₁₉	C₂×C₄	1	1	yes	characteristic
19	C₁₉×C₄⋊C₈	C₈	1	2	yes	—
20	C₁₉×C₂²⋊C₈	—	2	1	no	—
21	C₁₉×#97	C₈	1	2	yes	—
22	C₄×C₈×C₁₉	—	2	1	no	—
23	C₄×C₈×C₁₉	—	2	1	no	—
24	C₄×C₈×C₁₉	D₄	1	1	yes	characteristic
25	C₄×C₈×C₁₉	D₄	1	1	yes	characteristic
26	C₄×C₈×C₁₉	D₄	1	1	yes	characteristic
27	C₄×C₈×C₁₉	Q₈	1	1	yes	characteristic
28	#10597269	—	4	2	no	—
29	C₂×C₁₉⋊C₁₆	—	2	2	no	—
30	C₂×C₁₉⋊C₁₆	—	2	2	no	—
31	C₂²×C₄×C₁₉	C₄²	1	1	yes	characteristic
32	C₂²×C₄×C₁₉	M₁₆	1	1	yes	characteristic
33	C₂²×C₄×C₁₉	M₁₆	1	1	yes	characteristic
34	M₁₆×C₁₉	—	2	4	no	—
35	C₂×C₈×C₁₉	—	2	2	no	—
36	C₂×C₈×C₁₉	—	2	1	no	—
37	C₂×C₈×C₁₉	—	2	1	no	—
38	C₂×C₈×C₁₉	—	2	2	no	—
39	C₂×C₈×C₁₉	—	4	1	no	—
40	C₂×C₈×C₁₉	C₂²⋊C₄	1	1	yes	characteristic
41	C₂×C₈×C₁₉	C₂²⋊C₄	1	1	yes	characteristic
42	C₂×C₈×C₁₉	C₄⋊C₄	1	1	yes	characteristic
43	C₂×C₈×C₁₉	C₄⋊C₄	1	1	yes	characteristic
44	C₄²×C₁₉	C₂²⋊C₄	1	1	yes	characteristic
45	C₄²×C₁₉	C₄⋊C₄	1	1	yes	characteristic
46	C₄²×C₁₉	C₂×C₈	1	1	yes	characteristic
47	C₄²×C₁₉	C₂×C₈	1	1	yes	characteristic
48	C₁₉⋊C₁₆	—	4	2	no	—
49	C₁₉⋊C₁₆	—	4	2	no	—
50	#7212	—	19	1	no	maximal, Sylow
51	C₂³×C₁₉	#97	1	1	yes	characteristic
52	C₂×C₄×C₁₉	—	2	1	no	—
53	C₂×C₄×C₁₉	—	2	1	no	—
54	C₂×C₄×C₁₉	—	2	2	no	—
55	C₂×C₄×C₁₉	—	2	2	no	—
56	C₂×C₄×C₁₉	#95	1	1	yes	characteristic
57	C₂×C₄×C₁₉	C₄×C₈	1	1	yes	characteristic
58	C₂×C₄×C₁₉	C₂²⋊C₈	1	1	yes	characteristic
59	C₂×C₄×C₁₉	C₂²⋊C₈	1	1	yes	characteristic
60	C₂×C₄×C₁₉	C₄⋊C₈	1	1	yes	characteristic
61	C₂×C₄×C₁₉	C₄⋊C₈	1	1	yes	characteristic
62	C₈×C₁₉	—	2	4	no	—
63	C₈×C₁₉	—	2	1	no	—
64	C₈×C₁₉	—	2	1	no	—
65	C₈×C₁₉	—	2	1	no	—
66	C₈×C₁₉	—	4	2	no	—
67	C₈×C₁₉	#100	1	1	yes	characteristic
68	C₈×C₁₉	#101	1	1	yes	characteristic
69	#2162	—	19	1	no	—
70	#2150	—	19	1	no	—
71	C₈×M₁₆	D₁₉	1	1	yes	characteristic
72	C₂²×C₁₉	—	2	1	no	—
73	C₂²×C₁₉	—	2	1	no	—
74	C₂²×C₁₉	—	2	1	no	—
75	C₂²×C₁₉	#336	1	1	yes	characteristic
76	C₄×C₁₉	—	2	1	no	—
77	C₄×C₁₉	—	2	1	no	—
78	C₄×C₁₉	—	2	1	no	—
79	C₄×C₁₉	#343	1	1	yes	characteristic
80	C₄×C₁₉	#344	1	1	yes	characteristic
81	C₄×C₁₉	#350	1	2	yes	—
82	C₄×C₁₉	#364	1	2	yes	—
83	C₂×M₃₂	—	19	2	no	—
84	#431	Dic₁₉	1	1	yes	characteristic
85	C₄×M₁₆	Dic₁₉	1	1	yes	characteristic
86	C₂×C₄×C₈	C₂×D₁₉	1	1	yes	characteristic
87	C₄⋊C₁₆	—	38	1	no	—
88	C₄⋊C₁₆	—	38	1	no	—
89	C₄⋊C₁₆	—	38	1	no	—
90	C₂²⋊C₁₆	—	19	2	no	—
91	C₄×C₁₆	—	38	1	no	—
92	#322	—	2	1	no	—
93	C₈²	—	2	1	no	—
94	C₂×C₁₉	—	4	1	no	—
95	C₂×C₁₉	#1283	1	1	yes	characteristic
96	C₂×C₁₉	#1376	1	1	yes	characteristic
97	C₂×C₁₉	#1379	1	1	yes	characteristic
98	C₂×M₁₆	—	2	1	no	—
99	C₂²×C₈	C₄×D₁₉	1	1	yes	characteristic
100	C₂²×C₈	C₄×D₁₉	1	1	yes	characteristic
101	C₂×C₄²	C₂×Dic₁₉	1	1	yes	characteristic
102	M₃₂	—	76	2	no	—
103	C₂×C₁₆	—	38	2	no	—
104	C₂×C₁₆	—	38	2	no	—
105	C₄⋊C₈	C₁₉⋊C₈	1	2	yes	—
106	C₂²⋊C₈	—	2	1	no	—
107	#97	C₁₉⋊C₈	1	2	yes	—
108	C₄×C₈	—	2	1	no	—
109	C₄×C₈	—	2	1	no	—
110	C₄×C₈	Dic₃₈	1	1	yes	characteristic
111	C₄×C₈	D₇₆	1	1	yes	characteristic
112	C₄×C₈	C₂²⋊D₁₉	1	1	yes	characteristic
113	C₄×C₈	C₂²⋊D₁₉	1	1	yes	characteristic
114	C₁₉	#7212	1	1	yes	characteristic, complemented, Sylow
115	C₂²×C₄	#64520	1	1	yes	characteristic
116	C₂²×C₄	#64520	1	1	yes	characteristic
117	C₂²×C₄	C₄×Dic₁₉	1	1	yes	characteristic
118	M₁₆	—	2	4	no	—
119	C₂×C₈	—	2	2	no	—
120	C₂×C₈	—	2	1	no	—
121	C₂×C₈	—	2	1	no	—
122	C₂×C₈	—	2	2	no	—
123	C₂×C₈	—	4	1	no	—
124	C₂×C₈	Dic₁₉⋊C₄	1	1	yes	characteristic
125	C₂×C₈	Dic₁₉⋊C₄	1	1	yes	characteristic
126	C₂×C₈	(C₂×D₁₉)⋊C₄	1	1	yes	characteristic
127	C₂×C₈	(C₂×D₁₉)⋊C₄	1	1	yes	characteristic
128	C₄²	C₂×C₁₉⋊C₈	1	1	yes	characteristic
129	C₄²	C₂×C₁₉⋊C₈	1	1	yes	characteristic
130	C₄²	C₄⋊Dic₁₉	1	1	yes	characteristic
131	C₄²	C₂²⋊Dic₁₉	1	1	yes	characteristic
132	C₁₆	—	76	2	no	—
133	C₁₆	—	76	2	no	—
134	C₂³	#10597260	1	1	yes	characteristic
135	C₂×C₄	—	2	1	no	—
136	C₂×C₄	—	2	1	no	—
137	C₂×C₄	—	2	2	no	—
138	C₂×C₄	—	2	2	no	—
139	C₂×C₄	C₄×C₁₉⋊C₈	1	1	yes	characteristic
140	C₂×C₄	C₄⋊(C₁₉⋊C₈)	1	1	yes	characteristic
141	C₂×C₄	C₄⋊(C₁₉⋊C₈)	1	1	yes	characteristic
142	C₂×C₄	C₂²⋊(C₁₉⋊C₈)	1	1	yes	characteristic
143	C₂×C₄	C₂²⋊(C₁₉⋊C₈)	1	1	yes	characteristic
144	C₂×C₄	C₁₉⋊#95	1	1	yes	characteristic
145	C₈	—	2	4	no	—
146	C₈	—	2	1	no	—
147	C₈	—	2	1	no	—
148	C₈	—	2	1	no	—
149	C₈	—	4	2	no	—
150	C₈	#10597280	1	1	yes	characteristic
151	C₈	#10597281	1	1	yes	characteristic
152	C₂²	—	2	1	no	—
153	C₂²	—	2	1	no	—
154	C₂²	—	2	1	no	—
155	C₂²	#11926299	1	1	yes	characteristic
156	C₄	—	2	1	no	—
157	C₄	—	2	1	no	—
158	C₄	—	2	1	no	—
159	C₄	#11926312	1	1	yes	characteristic
160	C₄	#11926316	1	1	yes	characteristic
161	C₄	#11926328	1	2	yes	—
162	C₄	#11926365	1	2	yes	—
163	C₂	—	4	1	no	—
164	C₂	#14561486	1	1	yes	characteristic
165	C₂	#14561717	1	1	yes	characteristic
166	C₂	#14561724	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#14565434

C₂

yes

maximal, characteristic

#14565391

C₂

yes

maximal, characteristic

C₈×M₁₆×C₁₉

C₂

yes

maximal, characteristic

C₂×#10597269

C₄

yes

—

C₄⋊(C₁₉⋊C₁₆)

—

C₄⋊(C₁₉⋊C₁₆)

—

C₄⋊(C₁₉⋊C₁₆)

—

C₂²⋊(C₁₉⋊C₁₆)

C₄

yes

—

C₄×C₁₉⋊C₁₆

—

C₁₉×#431

C₄

yes

characteristic

C₄×M₁₆×C₁₉

C₄

yes

characteristic

C₂×C₄×C₈×C₁₉

C₂²

yes

characteristic

C₁₉×#322

—

C₈²×C₁₉

—

C₂×M₁₆×C₁₉

—

C₂²×C₈×C₁₉

C₂×C₄

yes

characteristic

C₂²×C₈×C₁₉

C₂×C₄

yes

characteristic

C₂×C₄²×C₁₉

C₂×C₄

yes

characteristic

C₁₉×C₄⋊C₈

C₈

yes

—

C₁₉×C₂²⋊C₈

—

C₁₉×#97

C₈

yes

—

C₄×C₈×C₁₉

—

C₄×C₈×C₁₉

—

C₄×C₈×C₁₉

D₄

yes

characteristic

C₄×C₈×C₁₉

D₄

yes

characteristic

C₄×C₈×C₁₉

D₄

yes

characteristic

C₄×C₈×C₁₉

Q₈

yes

characteristic

#10597269

—

C₂×C₁₉⋊C₁₆

—

C₂×C₁₉⋊C₁₆

—

C₂²×C₄×C₁₉

C₄²

yes

characteristic

C₂²×C₄×C₁₉

M₁₆

yes

characteristic

C₂²×C₄×C₁₉

M₁₆

yes

characteristic

M₁₆×C₁₉

—

C₂×C₈×C₁₉

—

C₂×C₈×C₁₉

—

C₂×C₈×C₁₉

—

C₂×C₈×C₁₉

—

C₂×C₈×C₁₉

—

C₂×C₈×C₁₉

C₂²⋊C₄

yes

characteristic

C₂×C₈×C₁₉

C₂²⋊C₄

yes

characteristic

C₂×C₈×C₁₉

C₄⋊C₄

yes

characteristic

C₂×C₈×C₁₉

C₄⋊C₄

yes

characteristic

C₄²×C₁₉

C₂²⋊C₄

yes

characteristic

C₄²×C₁₉

C₄⋊C₄

yes

characteristic

C₄²×C₁₉

C₂×C₈

yes

characteristic

C₄²×C₁₉

C₂×C₈

yes

characteristic

C₁₉⋊C₁₆

—

C₁₉⋊C₁₆

—

—

C₂³×C₁₉

yes

C₂×C₄×C₁₉

—

C₂×C₄×C₁₉

—

C₂×C₄×C₁₉

—

C₂×C₄×C₁₉

—

C₂×C₄×C₁₉

#95

yes

characteristic

C₂×C₄×C₁₉

C₄×C₈

yes

characteristic

C₂×C₄×C₁₉

C₂²⋊C₈

yes

characteristic

C₂×C₄×C₁₉

C₂²⋊C₈

yes

characteristic

C₂×C₄×C₁₉

C₄⋊C₈

yes

characteristic

C₂×C₄×C₁₉

C₄⋊C₈

yes

characteristic

C₈×C₁₉

—

C₈×C₁₉

—

C₈×C₁₉

—

C₈×C₁₉

—

C₈×C₁₉

—

C₈×C₁₉

yes

C₈×C₁₉

yes

—

—

C₈×M₁₆

D₁₉

yes

characteristic

C₂²×C₁₉

—

C₂²×C₁₉

—

C₂²×C₁₉

—

C₂²×C₁₉

#336

yes

characteristic

C₄×C₁₉

—

C₄×C₁₉

—

C₄×C₁₉

—

C₄×C₁₉

yes

C₄×C₁₉

yes

C₄×C₁₉

yes

—

C₄×C₁₉

#364

yes

—

C₂×M₃₂

—

#431

Dic₁₉

yes

characteristic

C₄×M₁₆

Dic₁₉

yes

characteristic

C₂×C₄×C₈

C₂×D₁₉

yes

characteristic

C₄⋊C₁₆

—

C₄⋊C₁₆

—

C₄⋊C₁₆

—

C₂²⋊C₁₆

—

C₄×C₁₆

—

#322

—

C₈²

—

C₂×C₁₉

—

C₂×C₁₉

yes

C₂×C₁₉

yes

C₂×C₁₉

yes

C₂×M₁₆

—

C₂²×C₈

C₄×D₁₉

yes

characteristic

100

C₂²×C₈

C₄×D₁₉

yes

characteristic

101

C₂×C₄²

C₂×Dic₁₉

yes

characteristic

102

M₃₂

—

103

C₂×C₁₆

—

104

C₂×C₁₆

—

105

C₄⋊C₈

C₁₉⋊C₈

yes

—

106

C₂²⋊C₈

—

107

#97

C₁₉⋊C₈

yes

—

108

C₄×C₈

—

109

C₄×C₈

—

110

C₄×C₈

Dic₃₈

yes

characteristic

111

C₄×C₈

D₇₆

yes

characteristic

112

C₄×C₈

C₂²⋊D₁₉

yes

characteristic

113

C₄×C₈

C₂²⋊D₁₉

yes

characteristic

114

C₁₉

#7212

yes

characteristic, complemented, Sylow

115

C₂²×C₄

#64520

yes

characteristic

116

C₂²×C₄

#64520

yes

characteristic

117

C₂²×C₄

C₄×Dic₁₉

yes

characteristic

118

M₁₆

—

119

C₂×C₈

—

120

C₂×C₈

—

121

C₂×C₈

—

122

C₂×C₈

—

123

C₂×C₈

—

124

C₂×C₈

Dic₁₉⋊C₄

yes

characteristic

125

C₂×C₈

Dic₁₉⋊C₄

yes

characteristic

126

C₂×C₈

(C₂×D₁₉)⋊C₄

yes

characteristic

127

C₂×C₈

(C₂×D₁₉)⋊C₄

yes

characteristic

128

C₄²

C₂×C₁₉⋊C₈

yes

characteristic

129

C₄²

C₂×C₁₉⋊C₈

yes

characteristic

130

C₄²

C₄⋊Dic₁₉

yes

characteristic

131

C₄²

C₂²⋊Dic₁₉

yes

characteristic

132

C₁₆

—

133

C₁₆

—

134

C₂³

#10597260

yes

characteristic

135

C₂×C₄

—

136

C₂×C₄

—

137

C₂×C₄

—

138

C₂×C₄

—

139

C₂×C₄

C₄×C₁₉⋊C₈

yes

characteristic

140

C₂×C₄

C₄⋊(C₁₉⋊C₈)

yes

characteristic

141

C₂×C₄

C₄⋊(C₁₉⋊C₈)

yes

characteristic

142

C₂×C₄

C₂²⋊(C₁₉⋊C₈)

yes

characteristic

143

C₂×C₄

C₂²⋊(C₁₉⋊C₈)

yes

characteristic

144

C₂×C₄

C₁₉⋊#95

yes

characteristic

145

C₈

—

146

C₈

—

147

C₈

—

148

C₈

—

149

C₈

—

150

C₈

#10597280

yes

characteristic

151

C₈

#10597281

yes

characteristic

152

C₂²

—

153

C₂²

—

154

C₂²

—

155

C₂²

#11926299

yes

characteristic

156

C₄

—

157

C₄

—

158

C₄

—

159

C₄

#11926312

yes

characteristic

160

C₄

#11926316

yes

characteristic

161

C₄

#11926328

yes

—

162

C₄

#11926365

yes

—

163

C₂

—

164

C₂

#14561486

yes

characteristic

165

C₂

#14561717

yes

characteristic

166

C₂

#14561724

yes

characteristic

	subgroup	G/H
center	C₂×C₈	(C₂×D₁₉)⋊C₄
commutator	C₂×C₄×C₁₉	C₄×C₈
Frattini	C₂×C₄×C₈	C₂×D₁₉
Fitting	C₈×M₁₆×C₁₉	C₂
socle	C₂²×C₁₉	#336
prime radical	C₂³×C₁₉	#97
cosocle	C₂×C₄×C₈×C₁₉	C₂²
hypercenter	C₈×M₁₆	D₁₉
hypocenter	C₁₉	#7212
LCS	G ▷ C₂×C₄×C₁₉ ▷ C₂²×C₁₉ ▷ C₁₉
UCS	C₁ ◁ C₂×C₈ ◁ C₂×C₄×C₈ ◁ C₈×M₁₆

subgroup

G/H

center

C₂×C₈

(C₂×D₁₉)⋊C₄

commutator

C₂×C₄×C₁₉

C₄×C₈

Frattini

C₂×C₄×C₈

C₂×D₁₉

Fitting

C₈×M₁₆×C₁₉

C₂

C₂²×C₁₉

C₂³×C₁₉

C₂×C₄×C₈×C₁₉

C₂²

hypercenter

C₈×M₁₆

D₁₉

hypocenter

C₁₉

#7212

LCS

G ▷ C₂×C₄×C₁₉ ▷ C₂²×C₁₉ ▷ C₁₉

UCS

C₁ ◁ C₂×C₈ ◁ C₂×C₄×C₈ ◁ C₈×M₁₆

N	H ≅ G/N	type
114. C₁₉	#7212	semidirect

H ≅ G/N

type

114. C₁₉

#7212

semidirect

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

subgroup	index
#14565434	2
#14565391	2
C₈×M₁₆×C₁₉	2
#7212	19

subgroup

index

#14565434

#14565391

C₈×M₁₆×C₁₉

#7212

Subgroups — #18069562

Conjugacy classes of subgroups

Complements

Maximal subgroups