Group #14572766 — subgroups

Subgroups — #14572766

Order 2432.

394 conjugacy classes of subgroups; only the 179 normal subgroups are shown.

normal subgroups only

This group has 394 conjugacy classes of subgroups — too many to draw. Only its 179 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	C₄×#64545	C₂	1	1	yes	maximal, characteristic
2	Q₈×C₁₉⋊C₈	C₂	1	1	yes	maximal, characteristic, complemented
3	#11926623	C₂	1	3	yes	maximal, complemented
4	#11926621	C₂	1	3	yes	maximal, complemented
5	#11926595	C₂	1	3	yes	maximal, complemented
6	#11926566	C₂	1	3	yes	maximal, complemented
7	C₈×Q₈×C₁₉	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×#64545	C₄	1	1	yes	characteristic
9	C₄×Q₈×C₁₉	C₂²	1	1	yes	characteristic
10	Q₈×Dic₁₉	C₄	1	1	yes	characteristic
11	C₂×#64515	C₂²	1	6	yes	—
12	#10597339	C₄	1	3	yes	—
13	#10597338	C₄	1	3	yes	—
14	C₄×D₇₆	C₂²	1	3	yes	—
15	C₄²×D₁₉	C₂²	1	3	yes	—
16	C₁₉×C₄⋊C₈	C₂²	1	3	yes	—
17	C₄×C₈×C₁₉	C₂²	1	3	yes	—
18	(C₂×D₁₉)⋊C₈	C₂²	1	3	yes	—
19	(C₂×D₁₉)⋊C₈	C₂²	1	3	yes	—
20	Dic₁₉⋊C₈	C₂²	1	3	yes	—
21	Dic₁₉⋊C₈	C₂²	1	1	yes	characteristic
22	C₄⋊(C₁₉⋊C₈)	C₂²	1	3	yes	—
23	C₄×C₁₉⋊C₈	C₂²	1	3	yes	—
24	C₂×Q₈×C₁₉	C₂×C₄	1	1	yes	characteristic
25	#64545	—	2	4	no	—
26	C₂×D₇₆	C₂×C₄	1	3	yes	—
27	C₂×C₄×D₁₉	C₂×C₄	1	3	yes	—
28	C₂×C₄×D₁₉	C₂×C₄	1	3	yes	—
29	C₂×C₄×D₁₉	C₂³	1	3	yes	—
30	C₂×C₈×C₁₉	C₂³	1	1	yes	characteristic
31	C₂×C₈×C₁₉	C₂³	1	3	yes	—
32	C₄⋊C₄×C₁₉	C₂×C₄	1	3	yes	—
33	C₄²×C₁₉	C₂³	1	3	yes	—
34	(C₂×D₁₉)⋊C₄	C₂×C₄	1	6	yes	—
35	C₄⋊Dic₁₉	C₂×C₄	1	3	yes	—
36	C₄×Dic₁₉	C₂×C₄	1	3	yes	—
37	C₄×Dic₁₉	C₂×C₄	1	3	yes	—
38	C₄×Dic₁₉	C₂³	1	1	yes	characteristic
39	C₂×C₁₉⋊C₈	C₂³	1	1	yes	characteristic
40	C₂×C₁₉⋊C₈	C₂³	1	3	yes	—
41	#64515	—	4	6	no	—
42	C₂²×D₁₉	C₂²×C₄	1	3	yes	—
43	Q₈×C₁₉	M₁₆	1	4	yes	complemented
44	C₂×C₄×C₁₉	C₂²×C₄	1	3	yes	—
45	C₂×C₄×C₁₉	C₂²×C₄	1	3	yes	—
46	C₂×C₄×C₁₉	C₂⁴	1	1	yes	characteristic
47	C₂×Dic₁₉	C₂²×C₄	1	1	yes	characteristic
48	C₂×Dic₁₉	C₂²×C₄	1	3	yes	—
49	C₂×Dic₁₉	C₂²×C₄	1	1	yes	characteristic
50	D₇₆	—	2	6	no	—
51	C₄×D₁₉	—	2	6	no	—
52	C₄×D₁₉	—	2	3	no	—
53	C₄×D₁₉	—	2	3	no	—
54	C₄×D₁₉	—	2	6	no	—
55	C₈×C₁₉	—	2	3	no	—
56	C₈×C₁₉	—	2	1	no	—
57	C₁₉⋊C₈	—	2	3	no	—
58	C₁₉⋊C₈	—	2	1	no	—
59	#2991	—	19	1	no	maximal, Sylow
60	C₂²×C₁₉	C₂³×C₄	1	1	yes	characteristic
61	C₂×D₁₉	—	2	3	no	—
62	C₂×D₁₉	—	2	3	no	—
63	C₂×D₁₉	—	2	3	no	—
64	C₄×C₁₉	—	2	3	no	—
65	C₄×C₁₉	C₂×M₁₆	1	6	yes	—
66	C₄×C₁₉	H(5,𝔽₂)	1	1	yes	characteristic
67	C₄×C₁₉	2₋¹⁺⁴	1	1	yes	characteristic
68	Dic₁₉	—	2	3	no	—
69	Dic₁₉	—	2	1	no	—
70	Dic₁₉	#131	1	2	yes	—
71	C₄×#41	—	19	1	no	—
72	C₈×Q₈	—	19	1	no	—
73	C₈×Q₈	D₁₉	1	1	yes	characteristic, complemented
74	#436	—	19	3	no	—
75	#436	—	19	3	no	—
76	#431	—	19	3	no	—
77	#423	—	19	3	no	—
78	C₂×C₁₉	#520	1	1	yes	characteristic
79	C₂×C₁₉	C₂²×M₁₆	1	1	yes	characteristic
80	C₂×C₁₉	C₂×#131	1	1	yes	characteristic
81	D₁₉	—	4	3	no	—
82	C₂×#41	—	19	1	no	—
83	C₂×M₁₆	—	19	6	no	—
84	C₄×Q₈	—	19	1	no	—
85	C₄×Q₈	C₂×D₁₉	1	1	yes	characteristic
86	C₄×D₄	—	19	3	no	—
87	C₄×D₄	—	19	3	no	—
88	#117	—	19	3	no	—
89	C₂×C₄²	—	19	3	no	—
90	C₄⋊C₈	—	19	3	no	—
91	C₄⋊C₈	—	19	1	no	—
92	C₄⋊C₈	—	19	3	no	—
93	C₄⋊C₈	C₂×D₁₉	1	3	yes	—
94	C₂²⋊C₈	—	19	3	no	—
95	C₂²⋊C₈	—	19	3	no	—
96	C₄×C₈	—	19	3	no	—
97	C₄×C₈	C₂×D₁₉	1	3	yes	—
98	C₁₉	#2991	1	1	yes	characteristic, complemented, Sylow
99	#41	—	38	4	no	—
100	C₂×Q₈	C₄×D₁₉	1	1	yes	characteristic
101	C₂×D₄	—	19	3	no	—
102	C₂²×C₄	—	19	3	no	—
103	C₂²×C₄	—	19	3	no	—
104	C₂²×C₄	—	19	3	no	—
105	M₁₆	—	76	6	no	—
106	C₂×C₈	—	19	1	no	—
107	C₂×C₈	—	19	3	no	—
108	C₂×C₈	C₂²×D₁₉	1	1	yes	characteristic
109	C₂×C₈	C₂²×D₁₉	1	3	yes	—
110	C₄⋊C₄	—	19	3	no	—
111	C₄⋊C₄	C₄×D₁₉	1	3	yes	—
112	C₂²⋊C₄	—	19	6	no	—
113	C₄²	—	19	3	no	—
114	C₄²	—	19	3	no	—
115	C₄²	—	19	1	no	—
116	C₄²	C₂²×D₁₉	1	3	yes	—
117	C₂³	—	19	3	no	—
118	Q₈	#64515	1	4	yes	complemented
119	D₄	—	38	6	no	—
120	C₂×C₄	—	19	1	no	—
121	C₂×C₄	—	19	1	no	—
122	C₂×C₄	—	19	3	no	—
123	C₂×C₄	—	38	6	no	—
124	C₂×C₄	—	38	3	no	—
125	C₂×C₄	—	38	3	no	—
126	C₂×C₄	—	38	6	no	—
127	C₂×C₄	C₂×C₄×D₁₉	1	3	yes	—
128	C₂×C₄	C₂×C₄×D₁₉	1	3	yes	—
129	C₂×C₄	C₂³×D₁₉	1	1	yes	characteristic
130	C₈	—	2	1	no	—
131	C₈	—	2	3	no	—
132	C₈	—	38	1	no	—
133	C₈	—	38	3	no	—
134	C₂²	—	38	3	no	—
135	C₂²	—	38	3	no	—
136	C₂²	—	38	3	no	—
137	C₂²	C₂²×C₄×D₁₉	1	1	yes	characteristic
138	C₄	—	2	3	no	—
139	C₄	—	19	2	no	—
140	C₄	—	38	1	no	—
141	C₄	—	38	3	no	—
142	C₄	C₂×#64515	1	6	yes	—
143	C₄	#10597433	1	1	yes	characteristic
144	C₄	#10597435	1	1	yes	characteristic
145	C₂	—	76	3	no	—
146	C₂	#11927026	1	1	yes	characteristic
147	C₂	C₂²×#64515	1	1	yes	characteristic
148	C₂	#131×D₁₉	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₄×#64545

C₂

yes

maximal, characteristic

Q₈×C₁₉⋊C₈

C₂

yes

maximal, characteristic, complemented

#11926623

C₂

yes

maximal, complemented

#11926621

C₂

yes

maximal, complemented

#11926595

C₂

yes

maximal, complemented

#11926566

C₂

yes

maximal, complemented

C₈×Q₈×C₁₉

C₂

yes

maximal, characteristic, complemented

C₂×#64545

C₄

yes

characteristic

C₄×Q₈×C₁₉

C₂²

yes

characteristic

Q₈×Dic₁₉

C₄

yes

characteristic

C₂×#64515

C₂²

yes

—

#10597339

C₄

yes

—

#10597338

C₄

yes

—

C₄×D₇₆

C₂²

yes

—

C₄²×D₁₉

C₂²

yes

—

C₁₉×C₄⋊C₈

C₂²

yes

—

C₄×C₈×C₁₉

C₂²

yes

—

(C₂×D₁₉)⋊C₈

C₂²

yes

—

(C₂×D₁₉)⋊C₈

C₂²

yes

—

Dic₁₉⋊C₈

C₂²

yes

—

Dic₁₉⋊C₈

C₂²

yes

characteristic

C₄⋊(C₁₉⋊C₈)

C₂²

yes

—

C₄×C₁₉⋊C₈

C₂²

yes

—

C₂×Q₈×C₁₉

C₂×C₄

yes

characteristic

#64545

—

C₂×D₇₆

C₂×C₄

yes

—

C₂×C₄×D₁₉

C₂×C₄

yes

—

C₂×C₄×D₁₉

C₂×C₄

yes

—

C₂×C₄×D₁₉

C₂³

yes

—

C₂×C₈×C₁₉

C₂³

yes

characteristic

C₂×C₈×C₁₉

C₂³

yes

—

C₄⋊C₄×C₁₉

C₂×C₄

yes

—

C₄²×C₁₉

C₂³

yes

—

(C₂×D₁₉)⋊C₄

C₂×C₄

yes

—

C₄⋊Dic₁₉

C₂×C₄

yes

—

C₄×Dic₁₉

C₂×C₄

yes

—

C₄×Dic₁₉

C₂×C₄

yes

—

C₄×Dic₁₉

C₂³

yes

characteristic

C₂×C₁₉⋊C₈

C₂³

yes

characteristic

C₂×C₁₉⋊C₈

C₂³

yes

—

#64515

—

C₂²×D₁₉

C₂²×C₄

yes

—

Q₈×C₁₉

M₁₆

yes

complemented

C₂×C₄×C₁₉

C₂²×C₄

yes

—

C₂×C₄×C₁₉

C₂²×C₄

yes

—

C₂×C₄×C₁₉

C₂⁴

yes

characteristic

C₂×Dic₁₉

C₂²×C₄

yes

characteristic

C₂×Dic₁₉

C₂²×C₄

yes

—

C₂×Dic₁₉

C₂²×C₄

yes

characteristic

D₇₆

—

C₄×D₁₉

—

C₄×D₁₉

—

C₄×D₁₉

—

C₄×D₁₉

—

C₈×C₁₉

—

C₈×C₁₉

—

C₁₉⋊C₈

—

C₁₉⋊C₈

—

#2991

—

maximal, Sylow

C₂²×C₁₉

C₂³×C₄

yes

characteristic

C₂×D₁₉

—

C₂×D₁₉

—

C₂×D₁₉

—

C₄×C₁₉

—

C₄×C₁₉

C₂×M₁₆

yes

—

C₄×C₁₉

H(5,𝔽₂)

yes

characteristic

C₄×C₁₉

2₋¹⁺⁴

yes

characteristic

Dic₁₉

—

Dic₁₉

—

Dic₁₉

#131

yes

—

C₄×#41

—

C₈×Q₈

—

C₈×Q₈

D₁₉

yes

characteristic, complemented

—

—

—

—

C₂×C₁₉

#520

yes

characteristic

C₂×C₁₉

C₂²×M₁₆

yes

characteristic

C₂×C₁₉

C₂×#131

yes

characteristic

D₁₉

—

C₂×#41

—

C₂×M₁₆

—

C₄×Q₈

—

C₄×Q₈

C₂×D₁₉

yes

characteristic

C₄×D₄

—

C₄×D₄

—

#117

—

C₂×C₄²

—

C₄⋊C₈

—

C₄⋊C₈

—

C₄⋊C₈

—

C₄⋊C₈

C₂×D₁₉

yes

—

C₂²⋊C₈

—

C₂²⋊C₈

—

C₄×C₈

—

C₄×C₈

C₂×D₁₉

yes

—

C₁₉

#2991

yes

characteristic, complemented, Sylow

#41

—

100

C₂×Q₈

C₄×D₁₉

yes

characteristic

101

C₂×D₄

—

102

C₂²×C₄

—

103

C₂²×C₄

—

104

C₂²×C₄

—

105

M₁₆

—

106

C₂×C₈

—

107

C₂×C₈

—

108

C₂×C₈

C₂²×D₁₉

yes

characteristic

109

C₂×C₈

C₂²×D₁₉

yes

—

110

C₄⋊C₄

—

111

C₄⋊C₄

C₄×D₁₉

yes

—

112

C₂²⋊C₄

—

113

C₄²

—

114

C₄²

—

115

C₄²

—

116

C₄²

C₂²×D₁₉

yes

—

117

C₂³

—

118

Q₈

#64515

yes

complemented

119

D₄

—

120

C₂×C₄

—

121

C₂×C₄

—

122

C₂×C₄

—

123

C₂×C₄

—

124

C₂×C₄

—

125

C₂×C₄

—

126

C₂×C₄

—

127

C₂×C₄

C₂×C₄×D₁₉

yes

—

128

C₂×C₄

C₂×C₄×D₁₉

yes

—

129

C₂×C₄

C₂³×D₁₉

yes

characteristic

130

C₈

—

131

C₈

—

132

C₈

—

133

C₈

—

134

C₂²

—

135

C₂²

—

136

C₂²

—

137

C₂²

C₂²×C₄×D₁₉

yes

characteristic

138

C₄

—

139

C₄

—

140

C₄

—

141

C₄

—

142

C₄

C₂×#64515

yes

—

143

C₄

#10597433

yes

characteristic

144

C₄

#10597435

yes

characteristic

145

C₂

—

146

C₂

#11927026

yes

characteristic

147

C₂

C₂²×#64515

yes

characteristic

148

C₂

#131×D₁₉

yes

characteristic

	subgroup	G/H
center, Frattini	C₂×C₄	C₂³×D₁₉
commutator, socle	C₂²×C₁₉	C₂³×C₄
Fitting	C₈×Q₈×C₁₉	C₂
prime radical	C₂×#64545	C₄
cosocle	C₂×C₄×C₁₉	C₂⁴
hypercenter	C₈×Q₈	D₁₉
hypocenter	C₁₉	#2991
LCS	G ▷ C₂²×C₁₉ ▷ C₁₉
UCS	C₁ ◁ C₂×C₄ ◁ C₈×Q₈

subgroup

G/H

center, Frattini

C₂×C₄

C₂³×D₁₉

commutator, socle

C₂²×C₁₉

C₂³×C₄

Fitting

C₈×Q₈×C₁₉

C₂

prime radical

C₂×#64545

C₄

cosocle

C₂×C₄×C₁₉

C₂⁴

hypercenter

C₈×Q₈

D₁₉

C₁₉

G ▷ C₂²×C₁₉ ▷ C₁₉

C₁ ◁ C₂×C₄ ◁ C₈×Q₈

N	H ≅ G/N	type
2. Q₈×C₁₉⋊C₈	C₂	semidirect
3. #11926623	C₂	semidirect
4. #11926621	C₂	semidirect
5. #11926595	C₂	semidirect
6. #11926566	C₂	semidirect
7. C₈×Q₈×C₁₉	C₂	semidirect
43. Q₈×C₁₉	M₁₆	semidirect
73. C₈×Q₈	D₁₉	semidirect
98. C₁₉	#2991	semidirect
118. Q₈	#64515	semidirect

H ≅ G/N

type

2. Q₈×C₁₉⋊C₈

C₂

semidirect

3. #11926623

C₂

semidirect

4. #11926621

C₂

semidirect

5. #11926595

C₂

semidirect

6. #11926566

C₂

semidirect

7. C₈×Q₈×C₁₉

C₂

semidirect

43. Q₈×C₁₉

M₁₆

semidirect

73. C₈×Q₈

D₁₉

semidirect

98. C₁₉

#2991

semidirect

118. Q₈

#64515

semidirect

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

subgroup	index
C₄×#64545	2
Q₈×C₁₉⋊C₈	2
#11926623	2
#11926621	2
#11926595	2
#11926566	2
C₈×Q₈×C₁₉	2
#2991	19

subgroup

index

Q₈×C₁₉⋊C₈

C₈×Q₈×C₁₉

Subgroups — #14572766

Conjugacy classes of subgroups

Complements

Maximal subgroups