Group #18069188 — subgroups

Subgroups — #18069188

Order 4864.

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

normal subgroups only

This group has 162 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	#14561705	C₂	1	1	yes	maximal, characteristic, complemented
2	C₈×C₁₉⋊C₁₆	C₂	1	1	yes	maximal, characteristic, complemented
3	C₈×D₈×C₁₉	C₂	1	1	yes	maximal, characteristic
4	C₄⋊(C₁₉⋊C₁₆)	—	2	1	no	—
5	C₄×C₁₉⋊C₁₆	—	2	1	no	—
6	C₄×D₈×C₁₉	C₄	1	1	yes	characteristic
7	C₈×D₄×C₁₉	—	2	1	no	—
8	C₁₉×#335	C₄	1	1	yes	characteristic
9	C₁₉×D₄⋊C₈	—	2	1	no	—
10	C₈²×C₁₉	C₂²	1	1	yes	characteristic
11	C₂×D₈×C₁₉	C₈	1	1	yes	characteristic
12	C₂²×C₈×C₁₉	—	4	1	no	—
13	C₄×D₄×C₁₉	—	2	1	no	—
14	C₁₉×#107	C₈	1	1	yes	characteristic
15	C₁₉×C₄⋊C₈	—	2	1	no	—
16	C₁₉×#102	—	2	1	no	—
17	C₁₉×C₂²⋊C₈	—	4	1	no	—
18	C₄×C₈×C₁₉	C₂×C₄	1	1	yes	characteristic
19	C₄×C₈×C₁₉	D₄	1	1	yes	characteristic
20	C₄×C₈×C₁₉	D₄	1	1	yes	characteristic
21	C₂×C₁₉⋊C₁₆	—	2	2	no	—
22	C₂×C₁₉⋊C₁₆	—	4	1	no	—
23	C₂×D₄×C₁₉	—	2	1	no	—
24	C₂²×C₄×C₁₉	—	4	1	no	—
25	D₈×C₁₉	—	2	1	no	—
26	D₈×C₁₉	C₁₆	1	2	yes	complemented
27	C₂×C₈×C₁₉	—	2	1	no	—
28	C₂×C₈×C₁₉	—	4	1	no	—
29	C₂×C₈×C₁₉	—	8	1	no	—
30	C₂×C₈×C₁₉	—	8	1	no	—
31	C₂×C₈×C₁₉	C₂×C₈	1	1	yes	characteristic
32	C₂×C₈×C₁₉	M₁₆	1	1	yes	characteristic
33	C₂×C₈×C₁₉	D₈	1	1	yes	characteristic
34	C₂×C₈×C₁₉	SD₁₆	1	1	yes	characteristic
35	C₄⋊C₄×C₁₉	—	2	1	no	—
36	C₂²⋊C₄×C₁₉	—	4	1	no	—
37	C₄²×C₁₉	C₂²⋊C₄	1	1	yes	characteristic
38	C₁₉⋊C₁₆	—	2	4	no	—
39	C₁₉⋊C₁₆	—	8	1	no	—
40	#7068	—	19	1	no	maximal, Sylow
41	C₂³×C₁₉	—	4	1	no	—
42	D₄×C₁₉	—	2	2	no	—
43	D₄×C₁₉	—	4	1	no	—
44	C₂×C₄×C₁₉	—	4	1	no	—
45	C₂×C₄×C₁₉	—	8	1	no	—
46	C₂×C₄×C₁₉	—	8	1	no	—
47	C₂×C₄×C₁₉	C₂²⋊C₈	1	1	yes	characteristic
48	C₂×C₄×C₁₉	#102	1	1	yes	characteristic
49	C₂×C₄×C₁₉	#104	1	1	yes	characteristic
50	C₈×C₁₉	—	2	2	no	—
51	C₈×C₁₉	—	2	1	no	—
52	C₈×C₁₉	—	2	1	no	—
53	C₈×C₁₉	—	8	1	no	—
54	C₈×C₁₉	C₂×C₁₆	1	1	yes	characteristic
55	C₈×C₁₉	M₃₂	1	1	yes	characteristic
56	C₈×C₁₉	D₁₆	1	1	yes	characteristic
57	C₈×C₁₉	SD₃₂	1	1	yes	characteristic
58	C₈×D₈	D₁₉	1	1	yes	characteristic
59	#1370	—	19	1	no	—
60	C₈×C₁₆	—	19	1	no	—
61	C₂²×C₁₉	—	4	2	no	—
62	C₂²×C₁₉	—	8	1	no	—
63	C₂²×C₁₉	—	8	1	no	—
64	C₂²×C₁₉	D₄⋊C₈	1	1	yes	characteristic
65	C₄×C₁₉	—	2	1	no	—
66	C₄×C₁₉	—	8	1	no	—
67	C₄×C₁₉	C₂²⋊C₁₆	1	1	yes	characteristic
68	C₄×C₁₉	#350	1	1	yes	characteristic
69	C₄×C₁₉	#357	1	1	yes	characteristic
70	C₄×C₁₉	#359	1	1	yes	characteristic
71	C₄×D₈	Dic₁₉	1	1	yes	characteristic
72	C₈×D₄	—	2	1	no	—
73	C₄⋊C₁₆	—	38	1	no	—
74	C₄×C₁₆	—	38	1	no	—
75	#335	Dic₁₉	1	1	yes	characteristic
76	D₄⋊C₈	—	2	1	no	—
77	C₈²	C₂×D₁₉	1	1	yes	characteristic
78	C₂×C₁₉	—	8	2	no	—
79	C₂×C₁₉	D₄⋊C₁₆	1	1	yes	characteristic
80	C₂×C₁₉	#1331	1	1	yes	characteristic
81	C₂×C₁₉	#1335	1	1	yes	characteristic
82	C₂×D₈	C₁₉⋊C₈	1	1	yes	characteristic
83	C₂²×C₈	—	4	1	no	—
84	C₄×D₄	—	2	1	no	—
85	C₂×C₁₆	—	38	2	no	—
86	C₂×C₁₆	—	76	1	no	—
87	#107	C₁₉⋊C₈	1	1	yes	characteristic
88	C₄⋊C₈	—	2	1	no	—
89	#102	—	2	1	no	—
90	C₂²⋊C₈	—	4	1	no	—
91	C₄×C₈	C₂×Dic₁₉	1	1	yes	characteristic
92	C₄×C₈	C₂²⋊D₁₉	1	1	yes	characteristic
93	C₄×C₈	C₂²⋊D₁₉	1	1	yes	characteristic
94	C₁₉	#7068	1	1	yes	characteristic, complemented, Sylow
95	C₂×D₄	—	2	1	no	—
96	C₂²×C₄	—	4	1	no	—
97	D₈	—	2	1	no	—
98	D₈	C₁₉⋊C₁₆	1	2	yes	complemented
99	C₂×C₈	—	2	1	no	—
100	C₂×C₈	—	4	1	no	—
101	C₂×C₈	—	8	1	no	—
102	C₂×C₈	—	8	1	no	—
103	C₂×C₈	C₂×C₁₉⋊C₈	1	1	yes	characteristic
104	C₂×C₈	#64520	1	1	yes	characteristic
105	C₂×C₈	#64525	1	1	yes	characteristic
106	C₂×C₈	#64526	1	1	yes	characteristic
107	C₄⋊C₄	—	2	1	no	—
108	C₂²⋊C₄	—	4	1	no	—
109	C₄²	C₂²⋊Dic₁₉	1	1	yes	characteristic
110	C₁₆	—	38	4	no	—
111	C₁₆	—	152	1	no	—
112	C₂³	—	4	1	no	—
113	D₄	—	2	2	no	—
114	D₄	—	4	1	no	—
115	C₂×C₄	—	4	1	no	—
116	C₂×C₄	—	8	1	no	—
117	C₂×C₄	—	8	1	no	—
118	C₂×C₄	C₂²⋊(C₁₉⋊C₈)	1	1	yes	characteristic
119	C₂×C₄	#10597289	1	1	yes	characteristic
120	C₂×C₄	#10597294	1	1	yes	characteristic
121	C₈	—	2	2	no	—
122	C₈	—	2	1	no	—
123	C₈	—	2	1	no	—
124	C₈	—	8	1	no	—
125	C₈	C₂×C₁₉⋊C₁₆	1	1	yes	characteristic
126	C₈	#10597269	1	1	yes	characteristic
127	C₈	#10597283	1	1	yes	characteristic
128	C₈	#10597284	1	1	yes	characteristic
129	C₂²	—	4	2	no	—
130	C₂²	—	8	1	no	—
131	C₂²	—	8	1	no	—
132	C₂²	D₄⋊(C₁₉⋊C₈)	1	1	yes	characteristic
133	C₄	—	2	1	no	—
134	C₄	—	8	1	no	—
135	C₄	C₂²⋊(C₁₉⋊C₁₆)	1	1	yes	characteristic
136	C₄	#11926328	1	1	yes	characteristic
137	C₄	#11926345	1	1	yes	characteristic
138	C₄	#11926351	1	1	yes	characteristic
139	C₂	—	8	2	no	—
140	C₂	D₄⋊(C₁₉⋊C₁₆)	1	1	yes	characteristic
141	C₂	#14561596	1	1	yes	characteristic
142	C₂	#14561608	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#14561705

C₂

yes

maximal, characteristic, complemented

C₈×C₁₉⋊C₁₆

C₂

yes

maximal, characteristic, complemented

C₈×D₈×C₁₉

C₂

yes

maximal, characteristic

C₄⋊(C₁₉⋊C₁₆)

—

C₄×C₁₉⋊C₁₆

—

C₄×D₈×C₁₉

C₄

yes

characteristic

C₈×D₄×C₁₉

—

C₁₉×#335

C₄

yes

characteristic

C₁₉×D₄⋊C₈

—

C₈²×C₁₉

C₂²

yes

characteristic

C₂×D₈×C₁₉

C₈

yes

characteristic

C₂²×C₈×C₁₉

—

C₄×D₄×C₁₉

—

C₁₉×#107

C₈

yes

characteristic

C₁₉×C₄⋊C₈

—

C₁₉×#102

—

C₁₉×C₂²⋊C₈

—

C₄×C₈×C₁₉

C₂×C₄

yes

characteristic

C₄×C₈×C₁₉

D₄

yes

characteristic

C₄×C₈×C₁₉

D₄

yes

characteristic

C₂×C₁₉⋊C₁₆

—

C₂×C₁₉⋊C₁₆

—

C₂×D₄×C₁₉

—

C₂²×C₄×C₁₉

—

D₈×C₁₉

—

D₈×C₁₉

C₁₆

yes

complemented

C₂×C₈×C₁₉

—

C₂×C₈×C₁₉

—

C₂×C₈×C₁₉

—

C₂×C₈×C₁₉

—

C₂×C₈×C₁₉

C₂×C₈

yes

characteristic

C₂×C₈×C₁₉

M₁₆

yes

characteristic

C₂×C₈×C₁₉

D₈

yes

characteristic

C₂×C₈×C₁₉

SD₁₆

yes

characteristic

C₄⋊C₄×C₁₉

—

C₂²⋊C₄×C₁₉

—

C₄²×C₁₉

C₂²⋊C₄

yes

characteristic

C₁₉⋊C₁₆

—

C₁₉⋊C₁₆

—

#7068

—

maximal, Sylow

C₂³×C₁₉

—

D₄×C₁₉

—

D₄×C₁₉

—

C₂×C₄×C₁₉

—

C₂×C₄×C₁₉

—

C₂×C₄×C₁₉

—

C₂×C₄×C₁₉

C₂²⋊C₈

yes

C₂×C₄×C₁₉

yes

C₂×C₄×C₁₉

yes

C₈×C₁₉

—

C₈×C₁₉

—

C₈×C₁₉

—

C₈×C₁₉

—

C₈×C₁₉

C₂×C₁₆

yes

characteristic

C₈×C₁₉

M₃₂

yes

characteristic

C₈×C₁₉

D₁₆

yes

characteristic

C₈×C₁₉

SD₃₂

yes

characteristic

C₈×D₈

D₁₉

yes

characteristic

#1370

—

C₈×C₁₆

—

C₂²×C₁₉

—

C₂²×C₁₉

—

C₂²×C₁₉

—

C₂²×C₁₉

D₄⋊C₈

yes

characteristic

C₄×C₁₉

—

C₄×C₁₉

—

C₄×C₁₉

C₂²⋊C₁₆

yes

C₄×C₁₉

yes

C₄×C₁₉

yes

C₄×C₁₉

yes

C₄×D₈

Dic₁₉

yes

characteristic

C₈×D₄

—

C₄⋊C₁₆

—

C₄×C₁₆

—

#335

Dic₁₉

yes

characteristic

D₄⋊C₈

—

C₈²

C₂×D₁₉

yes

characteristic

C₂×C₁₉

—

C₂×C₁₉

D₄⋊C₁₆

yes

C₂×C₁₉

yes

C₂×C₁₉

yes

C₂×D₈

C₁₉⋊C₈

yes

characteristic

C₂²×C₈

—

C₄×D₄

—

C₂×C₁₆

—

C₂×C₁₆

—

#107

C₁₉⋊C₈

yes

characteristic

C₄⋊C₈

—

#102

—

C₂²⋊C₈

—

C₄×C₈

C₂×Dic₁₉

yes

characteristic

C₄×C₈

C₂²⋊D₁₉

yes

characteristic

C₄×C₈

C₂²⋊D₁₉

yes

characteristic

C₁₉

#7068

yes

characteristic, complemented, Sylow

C₂×D₄

—

C₂²×C₄

—

D₈

—

D₈

C₁₉⋊C₁₆

yes

complemented

C₂×C₈

—

100

C₂×C₈

—

101

C₂×C₈

—

102

C₂×C₈

—

103

C₂×C₈

C₂×C₁₉⋊C₈

yes

characteristic

104

C₂×C₈

#64520

yes

characteristic

105

C₂×C₈

#64525

yes

characteristic

106

C₂×C₈

#64526

yes

characteristic

107

C₄⋊C₄

—

108

C₂²⋊C₄

—

109

C₄²

C₂²⋊Dic₁₉

yes

characteristic

110

C₁₆

—

111

C₁₆

—

152

—

112

C₂³

—

113

D₄

—

114

D₄

—

115

C₂×C₄

—

116

C₂×C₄

—

117

C₂×C₄

—

118

C₂×C₄

C₂²⋊(C₁₉⋊C₈)

yes

characteristic

119

C₂×C₄

#10597289

yes

characteristic

120

C₂×C₄

#10597294

yes

characteristic

121

C₈

—

122

C₈

—

123

C₈

—

124

C₈

—

125

C₈

C₂×C₁₉⋊C₁₆

yes

characteristic

126

C₈

#10597269

yes

characteristic

127

C₈

#10597283

yes

characteristic

128

C₈

#10597284

yes

characteristic

129

C₂²

—

130

C₂²

—

131

C₂²

—

132

C₂²

D₄⋊(C₁₉⋊C₈)

yes

characteristic

133

C₄

—

134

C₄

—

135

C₄

C₂²⋊(C₁₉⋊C₁₆)

yes

characteristic

136

C₄

#11926328

yes

characteristic

137

C₄

#11926345

yes

characteristic

138

C₄

#11926351

yes

characteristic

139

C₂

—

140

C₂

D₄⋊(C₁₉⋊C₁₆)

yes

characteristic

141

C₂

#14561596

yes

characteristic

142

C₂

#14561608

yes

characteristic

	subgroup	G/H
center	C₂×C₈	#64525
commutator	C₈×C₁₉	C₂×C₁₆
Frattini	C₈²	C₂×D₁₉
Fitting	C₈×D₈×C₁₉	C₂
socle	C₂²×C₁₉	D₄⋊C₈
prime radical	C₂×D₈×C₁₉	C₈
cosocle	C₈²×C₁₉	C₂²
hypercenter	C₈×D₈	D₁₉
hypocenter	C₁₉	#7068
LCS	G ▷ C₈×C₁₉ ▷ C₄×C₁₉ ▷ C₂×C₁₉ ▷ C₁₉
UCS	C₁ ◁ C₂×C₈ ◁ C₄×C₈ ◁ C₈² ◁ C₈×D₈

subgroup

G/H

center

C₂×C₈

#64525

commutator

C₈×C₁₉

C₂×C₁₆

Frattini

C₈²

C₂×D₁₉

Fitting

C₈×D₈×C₁₉

C₂

socle

C₂²×C₁₉

D₄⋊C₈

prime radical

C₂×D₈×C₁₉

C₈

cosocle

C₈²×C₁₉

C₂²

hypercenter

C₈×D₈

D₁₉

hypocenter

C₁₉

#7068

LCS

G ▷ C₈×C₁₉ ▷ C₄×C₁₉ ▷ C₂×C₁₉ ▷ C₁₉

UCS

C₁ ◁ C₂×C₈ ◁ C₄×C₈ ◁ C₈² ◁ C₈×D₈

N	H ≅ G/N	type
1. #14561705	C₂	semidirect
2. C₈×C₁₉⋊C₁₆	C₂	semidirect
26. D₈×C₁₉	C₁₆	semidirect
94. C₁₉	#7068	semidirect
98. D₈	C₁₉⋊C₁₆	semidirect

H ≅ G/N

type

1. #14561705

C₂

semidirect

2. C₈×C₁₉⋊C₁₆

C₂

semidirect

26. D₈×C₁₉

C₁₆

semidirect

94. C₁₉

#7068

semidirect

98. D₈

C₁₉⋊C₁₆

semidirect

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

subgroup	index
#14561705	2
C₈×C₁₉⋊C₁₆	2
C₈×D₈×C₁₉	2
#7068	19

subgroup

index

#14561705

C₈×C₁₉⋊C₁₆

C₈×D₈×C₁₉

#7068

Subgroups — #18069188

Conjugacy classes of subgroups

Complements

Maximal subgroups