Group #14574151 — subgroups

Subgroups — C₂×#11926950 #14574151

Order 2432.

384 conjugacy classes of subgroups; only the 165 normal subgroups are shown.

normal subgroups only

This group has 384 conjugacy classes of subgroups — too many to draw. Only its 165 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	#11926950	C₂	1	8	yes	maximal, direct product factor
3	C₂×#10597264	C₂	1	2	yes	maximal
4	C₂×#10597265	C₂	1	2	yes	maximal
5	C₂×#10597260	C₂	1	1	yes	maximal, characteristic
6	C₂×C₁₉×C₄⋊Q₈	C₂	1	1	yes	maximal, characteristic
7	C₂²×Q₈×C₁₉	—	2	1	no	—
8	C₁₉×C₄⋊Q₈	—	2	2	no	—
9	C₁₉×C₄⋊Q₈	C₂²	1	4	yes	—
10	C₂×C₄⋊C₄×C₁₉	—	2	1	no	—
11	C₂×C₄⋊C₄×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×Dic₁₉⋊C₄	—	2	1	no	—
16	C₂×Dic₁₉⋊C₄	—	2	1	no	—
17	C₂²×C₁₉⋊C₈	C₂²	1	2	yes	—
18	#10597316	—	2	2	no	—
19	#10597316	C₂²	1	4	yes	—
20	#10597265	C₂²	1	8	yes	—
21	#10597264	C₂²	1	8	yes	—
22	#10597260	C₂²	1	4	yes	—
23	C₂×Q₈×C₁₉	—	2	4	no	—
24	C₂×Q₈×C₁₉	—	4	4	no	—
25	C₂²×C₄×C₁₉	—	2	2	no	—
26	C₂²×C₄×C₁₉	D₄	1	1	yes	characteristic
27	C₂²×C₄×C₁₉	D₄	1	1	yes	characteristic
28	C₂²×C₄×C₁₉	C₂³	1	1	yes	characteristic
29	C₂²×Dic₁₉	—	2	1	no	—
30	C₂²×Dic₁₉	—	2	1	no	—
31	C₄⋊C₄×C₁₉	—	2	4	no	—
32	C₄⋊C₄×C₁₉	—	2	2	no	—
33	C₄⋊C₄×C₁₉	C₂³	1	4	yes	—
34	C₄²×C₁₉	D₄	1	2	yes	—
35	C₄²×C₁₉	C₂³	1	2	yes	—
36	C₄⋊Dic₁₉	—	2	1	no	—
37	C₄⋊Dic₁₉	—	2	1	no	—
38	C₄⋊Dic₁₉	C₂³	1	2	yes	—
39	C₄⋊Dic₁₉	C₂³	1	2	yes	—
40	Dic₁₉⋊C₄	—	2	4	no	—
41	Dic₁₉⋊C₄	—	2	4	no	—
42	C₂×C₁₉⋊C₈	Q₈	1	8	yes	complemented
43	C₂×C₁₉⋊C₈	C₂³	1	4	yes	—
44	C₂³×C₁₉	C₂×D₄	1	1	yes	characteristic
45	Q₈×C₁₉	—	4	8	no	—
46	C₂×C₄×C₁₉	—	2	2	no	—
47	C₂×C₄×C₁₉	—	2	2	no	—
48	C₂×C₄×C₁₉	—	2	4	no	—
49	C₂×C₄×C₁₉	—	4	4	no	—
50	C₂×C₄×C₁₉	C₂×D₄	1	2	yes	—
51	C₂×C₄×C₁₉	C₂×D₄	1	2	yes	—
52	C₂×C₄×C₁₉	C₂×D₄	1	1	yes	characteristic
53	C₂×C₄×C₁₉	C₂×Q₈	1	4	yes	—
54	C₂×C₄×C₁₉	C₂⁴	1	1	yes	characteristic
55	C₂×Dic₁₉	—	2	2	no	—
56	C₂×Dic₁₉	—	2	2	no	—
57	C₂×Dic₁₉	—	4	2	no	—
58	C₂×Dic₁₉	—	4	2	no	—
59	C₁₉⋊C₈	C₂×Q₈	1	8	yes	complemented
60	C₂×#501	—	19	1	no	maximal, Sylow
61	C₂²×C₁₉	C₄⋊Q₈	1	2	yes	—
62	C₂²×C₁₉	AGL(1,ℤ/8)	1	2	yes	—
63	C₂²×C₁₉	#137	1	2	yes	—
64	C₂²×C₁₉	C₂²×D₄	1	1	yes	characteristic
65	C₄×C₁₉	—	2	2	no	—
66	C₄×C₁₉	—	2	2	no	—
67	C₄×C₁₉	—	4	4	no	—
68	C₄×C₁₉	C₄⋊Q₈	1	2	yes	—
69	C₄×C₁₉	C₂²×Q₈	1	2	yes	—
70	Dic₁₉	—	4	2	no	—
71	Dic₁₉	—	4	2	no	—
72	C₂×C₄⋊Q₈	D₁₉	1	1	yes	characteristic
73	C₂×#125	—	19	1	no	—
74	#501	—	19	8	no	—
75	C₂×#107	—	19	2	no	—
76	C₂×#106	—	19	2	no	—
77	C₂×#97	—	19	1	no	—
78	C₂×C₁₉	#501	1	4	yes	complemented
79	C₂×C₁₉	C₂×C₄⋊Q₈	1	1	yes	characteristic
80	C₂×C₁₉	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
81	C₂×C₁₉	C₂×#137	1	1	yes	characteristic
82	C₂²×Q₈	—	2	1	no	—
83	C₂²×C₈	—	19	2	no	—
84	C₄⋊Q₈	—	2	2	no	—
85	C₄⋊Q₈	C₂×D₁₉	1	4	yes	—
86	#125	—	19	4	no	—
87	#125	—	38	2	no	—
88	C₂×C₄⋊C₄	—	2	1	no	—
89	C₂×C₄⋊C₄	—	19	1	no	—
90	C₂×C₄⋊C₄	—	19	1	no	—
91	C₂×C₄⋊C₄	—	38	1	no	—
92	C₂×C₄⋊C₄	—	38	1	no	—
93	C₂×C₄⋊C₄	C₂×D₁₉	1	2	yes	—
94	C₂×C₄²	C₂×D₁₉	1	1	yes	characteristic
95	#107	—	19	8	no	—
96	#106	—	19	8	no	—
97	#97	—	19	4	no	—
98	C₁₉	C₂×#501	1	1	yes	characteristic, complemented, Sylow
99	C₂×Q₈	—	2	4	no	—
100	C₂×Q₈	—	4	4	no	—
101	C₂²×C₄	—	2	2	no	—
102	C₂²×C₄	—	38	1	no	—
103	C₂²×C₄	—	38	1	no	—
104	C₂²×C₄	C₂²⋊D₁₉	1	1	yes	characteristic
105	C₂²×C₄	C₂²⋊D₁₉	1	1	yes	characteristic
106	C₂²×C₄	C₂²×D₁₉	1	1	yes	characteristic
107	C₂×C₈	—	19	8	no	—
108	C₂×C₈	—	19	4	no	—
109	C₄⋊C₄	—	2	4	no	—
110	C₄⋊C₄	—	2	2	no	—
111	C₄⋊C₄	—	19	2	no	—
112	C₄⋊C₄	—	19	2	no	—
113	C₄⋊C₄	—	38	4	no	—
114	C₄⋊C₄	—	38	4	no	—
115	C₄⋊C₄	—	38	1	no	—
116	C₄⋊C₄	—	38	1	no	—
117	C₄⋊C₄	C₂²×D₁₉	1	4	yes	—
118	C₄²	C₂²⋊D₁₉	1	2	yes	—
119	C₄²	C₂²×D₁₉	1	2	yes	—
120	C₂³	C₂×C₂²⋊D₁₉	1	1	yes	characteristic
121	Q₈	—	4	8	no	—
122	C₂×C₄	—	2	2	no	—
123	C₂×C₄	—	2	2	no	—
124	C₂×C₄	—	2	4	no	—
125	C₂×C₄	—	4	4	no	—
126	C₂×C₄	—	38	2	no	—
127	C₂×C₄	—	38	2	no	—
128	C₂×C₄	—	76	2	no	—
129	C₂×C₄	—	76	2	no	—
130	C₂×C₄	Q₈×D₁₉	1	4	yes	—
131	C₂×C₄	C₂×C₂²⋊D₁₉	1	2	yes	—
132	C₂×C₄	C₂×C₂²⋊D₁₉	1	2	yes	—
133	C₂×C₄	C₂×C₂²⋊D₁₉	1	1	yes	characteristic
134	C₂×C₄	C₂³×D₁₉	1	1	yes	characteristic
135	C₈	—	19	8	no	—
136	C₂²	#10597378	1	2	yes	—
137	C₂²	#10597388	1	2	yes	—
138	C₂²	#10597390	1	2	yes	—
139	C₂²	C₂²×C₂²⋊D₁₉	1	1	yes	characteristic
140	C₄	—	2	2	no	—
141	C₄	—	2	2	no	—
142	C₄	—	4	4	no	—
143	C₄	—	76	2	no	—
144	C₄	—	76	2	no	—
145	C₄	#10597390	1	2	yes	—
146	C₄	C₂×Q₈×D₁₉	1	2	yes	—
147	C₂	#11926950	1	4	yes	direct product factor
148	C₂	C₂×#10597390	1	1	yes	characteristic
149	C₂	C₂×#10597378	1	1	yes	characteristic
150	C₂	C₂×#10597388	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#10597316

C₂

yes

maximal, characteristic

#11926950

C₂

yes

maximal, direct product factor

C₂×#10597264

C₂

yes

maximal

C₂×#10597265

C₂

yes

maximal

C₂×#10597260

C₂

yes

maximal, characteristic

C₂×C₁₉×C₄⋊Q₈

C₂

yes

maximal, characteristic

C₂²×Q₈×C₁₉

—

C₁₉×C₄⋊Q₈

—

C₁₉×C₄⋊Q₈

C₂²

yes

—

C₂×C₄⋊C₄×C₁₉

—

C₂×C₄⋊C₄×C₁₉

C₂²

yes

—

C₂×C₄²×C₁₉

C₂²

yes

characteristic

C₂×C₄⋊Dic₁₉

C₂²

yes

characteristic

C₂×C₄⋊Dic₁₉

C₂²

yes

characteristic

C₂×Dic₁₉⋊C₄

—

C₂×Dic₁₉⋊C₄

—

C₂²×C₁₉⋊C₈

C₂²

yes

—

#10597316

—

#10597316

C₂²

yes

—

#10597265

C₂²

yes

—

#10597264

C₂²

yes

—

#10597260

C₂²

yes

—

C₂×Q₈×C₁₉

—

C₂×Q₈×C₁₉

—

C₂²×C₄×C₁₉

—

C₂²×C₄×C₁₉

D₄

yes

characteristic

C₂²×C₄×C₁₉

D₄

yes

characteristic

C₂²×C₄×C₁₉

C₂³

yes

characteristic

C₂²×Dic₁₉

—

C₂²×Dic₁₉

—

C₄⋊C₄×C₁₉

—

C₄⋊C₄×C₁₉

—

C₄⋊C₄×C₁₉

C₂³

yes

—

C₄²×C₁₉

D₄

yes

—

C₄²×C₁₉

C₂³

yes

—

C₄⋊Dic₁₉

—

C₄⋊Dic₁₉

—

C₄⋊Dic₁₉

C₂³

yes

—

C₄⋊Dic₁₉

C₂³

yes

—

Dic₁₉⋊C₄

—

Dic₁₉⋊C₄

—

C₂×C₁₉⋊C₈

Q₈

yes

complemented

C₂×C₁₉⋊C₈

C₂³

yes

—

C₂³×C₁₉

C₂×D₄

yes

characteristic

Q₈×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

characteristic

C₂×C₄×C₁₉

C₂×Q₈

yes

—

C₂×C₄×C₁₉

C₂⁴

yes

characteristic

C₂×Dic₁₉

—

C₂×Dic₁₉

—

C₂×Dic₁₉

—

C₂×Dic₁₉

—

C₁₉⋊C₈

C₂×Q₈

yes

complemented

C₂×#501

—

maximal, Sylow

C₂²×C₁₉

C₄⋊Q₈

yes

—

C₂²×C₁₉

AGL(1,ℤ/8)

yes

—

C₂²×C₁₉

#137

yes

—

C₂²×C₁₉

C₂²×D₄

yes

characteristic

C₄×C₁₉

—

C₄×C₁₉

—

C₄×C₁₉

—

C₄×C₁₉

C₄⋊Q₈

yes

—

C₄×C₁₉

C₂²×Q₈

yes

—

Dic₁₉

—

Dic₁₉

—

C₂×C₄⋊Q₈

D₁₉

yes

—

—

—

—

C₂×#97

—

C₂×C₁₉

#501

yes

complemented

C₂×C₁₉

C₂×C₄⋊Q₈

yes

C₂×C₁₉

yes

C₂×C₁₉

yes

C₂²×Q₈

—

C₂²×C₈

—

C₄⋊Q₈

—

C₄⋊Q₈

C₂×D₁₉

yes

—

#125

—

#125

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

C₂×D₁₉

yes

—

C₂×C₄²

C₂×D₁₉

yes

—

—

—

C₁₉

C₂×#501

yes

characteristic, complemented, Sylow

C₂×Q₈

—

100

C₂×Q₈

—

101

C₂²×C₄

—

102

C₂²×C₄

—

103

C₂²×C₄

—

104

C₂²×C₄

C₂²⋊D₁₉

yes

characteristic

105

C₂²×C₄

C₂²⋊D₁₉

yes

characteristic

106

C₂²×C₄

C₂²×D₁₉

yes

characteristic

107

C₂×C₈

—

108

C₂×C₈

—

109

C₄⋊C₄

—

110

C₄⋊C₄

—

111

C₄⋊C₄

—

112

C₄⋊C₄

—

113

C₄⋊C₄

—

114

C₄⋊C₄

—

115

C₄⋊C₄

—

116

C₄⋊C₄

—

117

C₄⋊C₄

C₂²×D₁₉

yes

—

118

C₄²

C₂²⋊D₁₉

yes

—

119

C₄²

C₂²×D₁₉

yes

—

120

C₂³

C₂×C₂²⋊D₁₉

yes

characteristic

121

Q₈

—

122

C₂×C₄

—

123

C₂×C₄

—

124

C₂×C₄

—

125

C₂×C₄

—

126

C₂×C₄

—

127

C₂×C₄

—

128

C₂×C₄

—

129

C₂×C₄

—

130

C₂×C₄

Q₈×D₁₉

yes

—

131

C₂×C₄

C₂×C₂²⋊D₁₉

yes

—

132

C₂×C₄

C₂×C₂²⋊D₁₉

yes

—

133

C₂×C₄

C₂×C₂²⋊D₁₉

yes

characteristic

134

C₂×C₄

C₂³×D₁₉

yes

characteristic

135

C₈

—

136

C₂²

#10597378

yes

—

137

C₂²

#10597388

yes

—

138

C₂²

#10597390

yes

—

139

C₂²

C₂²×C₂²⋊D₁₉

yes

characteristic

140

C₄

—

141

C₄

—

142

C₄

—

143

C₄

—

144

C₄

—

145

C₄

#10597390

yes

—

146

C₄

C₂×Q₈×D₁₉

yes

—

147

C₂

#11926950

yes

direct product factor

148

C₂

C₂×#10597390

yes

characteristic

149

C₂

C₂×#10597378

yes

characteristic

150

C₂

C₂×#10597388

yes

characteristic

	subgroup	G/H
center	C₂³	C₂×C₂²⋊D₁₉
commutator, cosocle	C₂×C₄×C₁₉	C₂⁴
Frattini	C₂×C₄	C₂³×D₁₉
Fitting	C₂×C₁₉×C₄⋊Q₈	C₂
socle, prime radical	C₂³×C₁₉	C₂×D₄
hypercenter	C₂×C₄⋊Q₈	D₁₉
hypocenter	C₁₉	C₂×#501
LCS	G ▷ C₂×C₄×C₁₉ ▷ C₂×C₁₉ ▷ C₁₉
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄⋊Q₈

subgroup

G/H

center

C₂³

C₂×C₂²⋊D₁₉

commutator, cosocle

C₂×C₄×C₁₉

C₂⁴

Frattini

C₂×C₄

C₂³×D₁₉

Fitting

C₂×C₁₉×C₄⋊Q₈

C₂

socle, prime radical

C₂³×C₁₉

C₂×D₄

hypercenter

C₂×C₄⋊Q₈

D₁₉

hypocenter

C₁₉

C₂×#501

LCS

G ▷ C₂×C₄×C₁₉ ▷ C₂×C₁₉ ▷ C₁₉

UCS

C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄⋊Q₈

N	H ≅ G/N	type
2. #11926950	C₂	direct (N × H)
42. C₂×C₁₉⋊C₈	Q₈	semidirect
59. C₁₉⋊C₈	C₂×Q₈	semidirect
78. C₂×C₁₉	#501	semidirect
98. C₁₉	C₂×#501	semidirect
147. C₂	#11926950	direct (N × H)

H ≅ G/N

type

2. #11926950

C₂

direct (N × H)

42. C₂×C₁₉⋊C₈

Q₈

semidirect

59. C₁₉⋊C₈

C₂×Q₈

semidirect

78. C₂×C₁₉

#501

semidirect

98. C₁₉

C₂×#501

semidirect

147. C₂

#11926950

direct (N × H)

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

subgroup	index
C₂×#10597316	2
#11926950	2
C₂×#10597264	2
C₂×#10597265	2
C₂×#10597260	2
C₂×C₁₉×C₄⋊Q₈	2
C₂×#501	19

subgroup

index

C₂×C₁₉×C₄⋊Q₈

Subgroups — C₂×#11926950 #14574151

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C2×#11926950 #14574151

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#11926950 #14574151