Group #14564578 — subgroups

Subgroups — #14564578

Order 2432.

260 conjugacy classes of subgroups; only the 93 normal subgroups are shown.

normal subgroups only

This group has 260 conjugacy classes of subgroups — too many to draw. Only its 93 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	#11927101	C₂	1	1	yes	maximal, characteristic
2	C₄×#64520	C₂	1	1	yes	maximal, characteristic
3	#11926304	C₂	1	4	yes	maximal
4	C₄×C₄⋊C₄×C₁₉	C₂	1	1	yes	maximal, characteristic
5	C₂×C₄⋊C₄×C₁₉	—	2	1	no	—
6	C₂×C₄²×C₁₉	C₂²	1	2	yes	—
7	C₂×C₄²×C₁₉	C₂²	1	1	yes	characteristic
8	#10597372	C₂²	1	2	yes	—
9	#10597370	—	2	1	no	—
10	#10597370	—	2	1	no	—
11	C₂×#64520	C₂²	1	2	yes	—
12	#10597316	C₄	1	2	yes	complemented
13	#10597315	C₄	1	1	yes	characteristic, complemented
14	#10597315	C₄	1	1	yes	characteristic, complemented
15	C₄×Dic₃₈	—	2	1	no	—
16	C₄×Dic₃₈	—	2	1	no	—
17	C₁₉×#95	—	2	1	no	—
18	#10597260	—	2	1	no	—
19	C₄×C₁₉⋊C₈	—	2	1	no	—
20	C₂²×C₄×C₁₉	—	2	2	no	—
21	C₂²×C₄×C₁₉	D₄	1	1	yes	characteristic
22	C₂²×C₄×C₁₉	D₄	1	1	yes	characteristic
23	C₂²×C₄×C₁₉	C₂³	1	1	yes	characteristic
24	C₂×Dic₃₈	—	2	1	no	—
25	C₂×Dic₃₈	—	2	1	no	—
26	C₄⋊C₄×C₁₉	—	2	4	no	—
27	C₄²×C₁₉	—	2	4	no	—
28	C₄²×C₁₉	C₂×C₄	1	1	yes	characteristic
29	C₄²×C₁₉	C₂×C₄	1	1	yes	characteristic
30	C₄²×C₁₉	D₄	1	1	yes	characteristic
31	C₄²×C₁₉	D₄	1	1	yes	characteristic
32	C₂²⋊Dic₁₉	—	2	2	no	—
33	C₄⋊Dic₁₉	—	2	1	no	—
34	C₄⋊Dic₁₉	—	2	1	no	—
35	C₄⋊Dic₁₉	C₂×C₄	1	2	yes	—
36	C₄⋊Dic₁₉	C₂×C₄	1	2	yes	—
37	Dic₁₉⋊C₄	—	2	2	no	—
38	Dic₁₉⋊C₄	—	2	2	no	—
39	C₄×Dic₁₉	—	2	2	no	—
40	#64520	—	2	2	no	—
41	#64520	Q₈	1	4	yes	complemented
42	C₂×C₁₉⋊C₈	—	2	2	no	—
43	C₂³×C₁₉	C₂×D₄	1	1	yes	characteristic
44	C₂×C₄×C₁₉	—	2	2	no	—
45	C₂×C₄×C₁₉	—	2	4	no	—
46	C₂×C₄×C₁₉	—	4	1	no	—
47	C₂×C₄×C₁₉	—	4	2	no	—
48	C₂×C₄×C₁₉	—	4	2	no	—
49	C₂×C₄×C₁₉	C₂²⋊C₄	1	1	yes	characteristic
50	C₂×C₄×C₁₉	C₂²⋊C₄	1	1	yes	characteristic
51	C₂×C₄×C₁₉	C₂²⋊C₄	1	1	yes	characteristic
52	C₂×C₄×C₁₉	C₂²⋊C₄	1	1	yes	characteristic
53	C₂×C₄×C₁₉	C₂²×C₄	1	1	yes	characteristic
54	C₂×C₄×C₁₉	C₂×D₄	1	1	yes	characteristic
55	C₂×C₄×C₁₉	C₂×Q₈	1	2	yes	—
56	C₂×C₄×C₁₉	#41	1	2	yes	—
57	C₂×Dic₁₉	—	2	2	no	—
58	C₂×Dic₁₉	—	2	2	no	—
59	Dic₃₈	—	4	2	no	—
60	Dic₃₈	—	4	2	no	—
61	C₁₉⋊C₈	—	2	4	no	—
62	#1986	—	19	1	no	maximal, Sylow
63	C₂²×C₁₉	—	2	1	no	—
64	C₂²×C₁₉	—	2	1	no	—
65	C₂²×C₁₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
66	C₂²×C₁₉	#124	1	1	yes	characteristic
67	C₂²×C₁₉	C₄⋊Q₈	1	1	yes	characteristic
68	C₄×C₁₉	—	2	1	no	—
69	C₄×C₁₉	—	2	1	no	—
70	C₄×C₁₉	—	2	1	no	—
71	C₄×C₁₉	—	4	4	no	—
72	C₄×C₁₉	#104	1	2	yes	—
73	C₄×C₁₉	C₄×Q₈	1	2	yes	—
74	C₄×C₁₉	C₂²⋊Q₈	1	2	yes	—
75	Dic₁₉	—	4	2	no	—
76	Dic₁₉	—	4	2	no	—
77	#533	—	19	1	no	—
78	C₄×M₁₆	—	19	1	no	—
79	C₄×C₄⋊C₄	D₁₉	1	1	yes	characteristic
80	#339	—	19	4	no	—
81	C₂×C₁₉	—	2	1	no	—
82	C₂×C₁₉	—	2	1	no	—
83	C₂×C₁₉	#391	1	1	yes	characteristic
84	C₂×C₁₉	C₂×#104	1	1	yes	characteristic
85	C₂×C₁₉	#421	1	1	yes	characteristic
86	C₂×M₁₆	—	19	2	no	—
87	C₄⋊Q₈	—	19	1	no	—
88	C₄⋊Q₈	—	19	1	no	—
89	#125	—	19	2	no	—
90	C₂²⋊Q₈	—	38	1	no	—
91	C₂²⋊Q₈	—	38	1	no	—
92	C₄×Q₈	—	38	1	no	—
93	C₄×Q₈	—	38	1	no	—
94	#117	—	19	2	no	—
95	C₂×C₄⋊C₄	—	2	1	no	—
96	C₂×C₄²	C₂×D₁₉	1	2	yes	—
97	C₂×C₄²	C₂×D₁₉	1	1	yes	characteristic
98	#97	—	38	1	no	—
99	C₄×C₈	—	38	1	no	—
100	#95	—	2	1	no	—
101	C₁₉	#1986	1	1	yes	characteristic, complemented, Sylow
102	C₂×Q₈	—	38	1	no	—
103	C₂×Q₈	—	38	1	no	—
104	C₂²×C₄	—	2	2	no	—
105	C₂²×C₄	C₂²⋊D₁₉	1	1	yes	characteristic
106	C₂²×C₄	C₂²⋊D₁₉	1	1	yes	characteristic
107	C₂²×C₄	C₂²×D₁₉	1	1	yes	characteristic
108	M₁₆	—	19	4	no	—
109	M₁₆	—	38	2	no	—
110	C₂×C₈	—	38	2	no	—
111	C₄⋊C₄	—	2	4	no	—
112	C₄⋊C₄	—	19	2	no	—
113	C₄⋊C₄	—	19	2	no	—
114	C₄⋊C₄	—	38	2	no	—
115	C₄⋊C₄	—	38	2	no	—
116	C₄⋊C₄	—	38	1	no	—
117	C₄⋊C₄	—	38	1	no	—
118	C₂²⋊C₄	—	38	2	no	—
119	C₄²	—	2	4	no	—
120	C₄²	—	38	2	no	—
121	C₄²	C₄×D₁₉	1	1	yes	characteristic
122	C₄²	C₄×D₁₉	1	1	yes	characteristic
123	C₄²	D₇₆	1	1	yes	characteristic
124	C₄²	D₇₆	1	1	yes	characteristic
125	C₂³	C₂×C₂²⋊D₁₉	1	1	yes	characteristic
126	Q₈	—	76	2	no	—
127	Q₈	—	76	2	no	—
128	C₂×C₄	—	2	2	no	—
129	C₂×C₄	—	2	4	no	—
130	C₂×C₄	—	4	1	no	—
131	C₂×C₄	—	4	2	no	—
132	C₂×C₄	—	4	2	no	—
133	C₂×C₄	—	38	2	no	—
134	C₂×C₄	—	38	2	no	—
135	C₂×C₄	(C₂×D₁₉)⋊C₄	1	1	yes	characteristic
136	C₂×C₄	(C₂×D₁₉)⋊C₄	1	1	yes	characteristic
137	C₂×C₄	(C₂×D₁₉)⋊C₄	1	1	yes	characteristic
138	C₂×C₄	(C₂×D₁₉)⋊C₄	1	1	yes	characteristic
139	C₂×C₄	C₂×C₄×D₁₉	1	1	yes	characteristic
140	C₂×C₄	C₂×D₇₆	1	1	yes	characteristic
141	C₂×C₄	#64543	1	2	yes	—
142	C₂×C₄	Q₈×D₁₉	1	2	yes	—
143	C₈	—	38	4	no	—
144	C₂²	—	2	1	no	—
145	C₂²	—	2	1	no	—
146	C₂²	C₂×(C₂×D₁₉)⋊C₄	1	1	yes	characteristic
147	C₂²	#10597382	1	1	yes	characteristic
148	C₂²	#10597390	1	1	yes	characteristic
149	C₄	—	2	1	no	—
150	C₄	—	2	1	no	—
151	C₄	—	2	1	no	—
152	C₄	—	4	4	no	—
153	C₄	—	76	2	no	—
154	C₄	—	76	2	no	—
155	C₄	#10597262	1	2	yes	—
156	C₄	#10597333	1	2	yes	—
157	C₄	#10597343	1	2	yes	—
158	C₂	—	2	1	no	—
159	C₂	—	2	1	no	—
160	C₂	#11926449	1	1	yes	characteristic
161	C₂	C₂×#10597262	1	1	yes	characteristic
162	C₂	#11926562	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#11927101

C₂

yes

maximal, characteristic

C₄×#64520

C₂

yes

maximal, characteristic

#11926304

C₂

yes

maximal

C₄×C₄⋊C₄×C₁₉

C₂

yes

maximal, characteristic

C₂×C₄⋊C₄×C₁₉

—

C₂×C₄²×C₁₉

C₂²

yes

—

C₂×C₄²×C₁₉

C₂²

yes

characteristic

#10597372

C₂²

yes

—

#10597370

—

#10597370

—

C₂×#64520

C₂²

yes

—

#10597316

C₄

yes

complemented

#10597315

C₄

yes

characteristic, complemented

#10597315

C₄

yes

characteristic, complemented

C₄×Dic₃₈

—

C₄×Dic₃₈

—

C₁₉×#95

—

#10597260

—

C₄×C₁₉⋊C₈

—

C₂²×C₄×C₁₉

—

C₂²×C₄×C₁₉

D₄

yes

characteristic

C₂²×C₄×C₁₉

D₄

yes

characteristic

C₂²×C₄×C₁₉

C₂³

yes

characteristic

C₂×Dic₃₈

—

C₂×Dic₃₈

—

C₄⋊C₄×C₁₉

—

C₄²×C₁₉

—

C₄²×C₁₉

C₂×C₄

yes

characteristic

C₄²×C₁₉

C₂×C₄

yes

characteristic

C₄²×C₁₉

D₄

yes

characteristic

C₄²×C₁₉

D₄

yes

characteristic

C₂²⋊Dic₁₉

—

C₄⋊Dic₁₉

—

C₄⋊Dic₁₉

—

C₄⋊Dic₁₉

C₂×C₄

yes

—

C₄⋊Dic₁₉

C₂×C₄

yes

—

Dic₁₉⋊C₄

—

Dic₁₉⋊C₄

—

C₄×Dic₁₉

—

#64520

—

#64520

Q₈

yes

complemented

C₂×C₁₉⋊C₈

—

C₂³×C₁₉

C₂×D₄

yes

characteristic

C₂×C₄×C₁₉

—

C₂×C₄×C₁₉

—

C₂×C₄×C₁₉

—

C₂×C₄×C₁₉

—

C₂×C₄×C₁₉

—

C₂×C₄×C₁₉

C₂²⋊C₄

yes

characteristic

C₂×C₄×C₁₉

C₂²⋊C₄

yes

characteristic

C₂×C₄×C₁₉

C₂²⋊C₄

yes

characteristic

C₂×C₄×C₁₉

C₂²⋊C₄

yes

characteristic

C₂×C₄×C₁₉

C₂²×C₄

yes

characteristic

C₂×C₄×C₁₉

C₂×D₄

yes

characteristic

C₂×C₄×C₁₉

C₂×Q₈

yes

—

C₂×C₄×C₁₉

#41

yes

—

C₂×Dic₁₉

—

C₂×Dic₁₉

—

Dic₃₈

—

Dic₃₈

—

C₁₉⋊C₈

—

#1986

—

maximal, Sylow

C₂²×C₁₉

—

C₂²×C₁₉

—

C₂²×C₁₉

C₂×C₂²⋊C₄

yes

characteristic

C₂²×C₁₉

#124

yes

characteristic

C₂²×C₁₉

C₄⋊Q₈

yes

characteristic

C₄×C₁₉

—

C₄×C₁₉

—

C₄×C₁₉

—

C₄×C₁₉

—

C₄×C₁₉

#104

yes

—

C₄×C₁₉

C₄×Q₈

yes

—

C₄×C₁₉

C₂²⋊Q₈

yes

—

Dic₁₉

—

Dic₁₉

—

#533

—

C₄×M₁₆

—

C₄×C₄⋊C₄

D₁₉

yes

characteristic

#339

—

C₂×C₁₉

—

C₂×C₁₉

—

C₂×C₁₉

yes

C₂×C₁₉

yes

C₂×C₁₉

yes

C₂×M₁₆

—

C₄⋊Q₈

—

C₄⋊Q₈

—

#125

—

C₂²⋊Q₈

—

C₂²⋊Q₈

—

C₄×Q₈

—

C₄×Q₈

—

#117

—

C₂×C₄⋊C₄

—

C₂×C₄²

C₂×D₁₉

yes

—

C₂×C₄²

C₂×D₁₉

yes

characteristic

#97

—

C₄×C₈

—

100

#95

—

101

C₁₉

#1986

yes

characteristic, complemented, Sylow

102

C₂×Q₈

—

103

C₂×Q₈

—

104

C₂²×C₄

—

105

C₂²×C₄

C₂²⋊D₁₉

yes

characteristic

106

C₂²×C₄

C₂²⋊D₁₉

yes

characteristic

107

C₂²×C₄

C₂²×D₁₉

yes

characteristic

108

M₁₆

—

109

M₁₆

—

110

C₂×C₈

—

111

C₄⋊C₄

—

112

C₄⋊C₄

—

113

C₄⋊C₄

—

114

C₄⋊C₄

—

115

C₄⋊C₄

—

116

C₄⋊C₄

—

117

C₄⋊C₄

—

118

C₂²⋊C₄

—

119

C₄²

—

120

C₄²

—

121

C₄²

C₄×D₁₉

yes

characteristic

122

C₄²

C₄×D₁₉

yes

characteristic

123

C₄²

D₇₆

yes

characteristic

124

C₄²

D₇₆

yes

characteristic

125

C₂³

C₂×C₂²⋊D₁₉

yes

characteristic

126

Q₈

—

127

Q₈

—

128

C₂×C₄

—

129

C₂×C₄

—

130

C₂×C₄

—

131

C₂×C₄

—

132

C₂×C₄

—

133

C₂×C₄

—

134

C₂×C₄

—

135

C₂×C₄

(C₂×D₁₉)⋊C₄

yes

characteristic

136

C₂×C₄

(C₂×D₁₉)⋊C₄

yes

characteristic

137

C₂×C₄

(C₂×D₁₉)⋊C₄

yes

characteristic

138

C₂×C₄

(C₂×D₁₉)⋊C₄

yes

characteristic

139

C₂×C₄

C₂×C₄×D₁₉

yes

characteristic

140

C₂×C₄

C₂×D₇₆

yes

characteristic

141

C₂×C₄

#64543

yes

—

142

C₂×C₄

Q₈×D₁₉

yes

—

143

C₈

—

144

C₂²

—

145

C₂²

—

146

C₂²

C₂×(C₂×D₁₉)⋊C₄

yes

characteristic

147

C₂²

#10597382

yes

characteristic

148

C₂²

#10597390

yes

characteristic

149

C₄

—

150

C₄

—

151

C₄

—

152

C₄

—

153

C₄

—

154

C₄

—

155

C₄

#10597262

yes

—

156

C₄

#10597333

yes

—

157

C₄

#10597343

yes

—

158

C₂

—

159

C₂

—

160

C₂

#11926449

yes

characteristic

161

C₂

C₂×#10597262

yes

characteristic

162

C₂

#11926562

yes

characteristic

	subgroup	G/H
center	C₂×C₄	C₂×D₇₆
commutator	C₂×C₄×C₁₉	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₁₉
Fitting	C₄×C₄⋊C₄×C₁₉	C₂
socle	C₂²×C₁₉	C₂×C₂²⋊C₄
prime radical	C₂³×C₁₉	C₂×D₄
cosocle	C₂²×C₄×C₁₉	C₂³
hypercenter	C₄×C₄⋊C₄	D₁₉
hypocenter	C₁₉	#1986
LCS	G ▷ C₂×C₄×C₁₉ ▷ C₂×C₁₉ ▷ C₁₉
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₄×C₄⋊C₄

subgroup

G/H

center

C₂×C₄

C₂×D₇₆

commutator

C₂×C₄×C₁₉

C₂²×C₄

Frattini

C₂²×C₄

C₂²×D₁₉

Fitting

C₄×C₄⋊C₄×C₁₉

C₂

socle

C₂²×C₁₉

C₂×C₂²⋊C₄

prime radical

C₂³×C₁₉

C₂×D₄

cosocle

C₂²×C₄×C₁₉

C₂³

hypercenter

C₄×C₄⋊C₄

D₁₉

hypocenter

C₁₉

#1986

LCS

G ▷ C₂×C₄×C₁₉ ▷ C₂×C₁₉ ▷ C₁₉

UCS

C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₄×C₄⋊C₄

N	H ≅ G/N	type
12. #10597316	C₄	semidirect
13. #10597315	C₄	semidirect
14. #10597315	C₄	semidirect
41. #64520	Q₈	semidirect
101. C₁₉	#1986	semidirect

H ≅ G/N

type

12. #10597316

C₄

semidirect

13. #10597315

C₄

semidirect

14. #10597315

C₄

semidirect

41. #64520

Q₈

semidirect

101. C₁₉

#1986

semidirect

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

subgroup	index
#11927101	2
C₄×#64520	2
#11926304	2
C₄×C₄⋊C₄×C₁₉	2
#1986	19

subgroup

index

C₄×C₄⋊C₄×C₁₉

Subgroups — #14564578

Conjugacy classes of subgroups

Complements

Maximal subgroups