Group #14574221 — subgroups

Subgroups — #14574221

Order 2432.

368 conjugacy classes of subgroups; only the 149 normal subgroups are shown.

normal subgroups only

This group has 368 conjugacy classes of subgroups — too many to draw. Only its 149 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	2	yes	maximal
2	#11926949	C₂	1	1	yes	maximal, characteristic, complemented
3	#11926924	C₂	1	1	yes	maximal, characteristic, complemented
4	#11926895	C₂	1	2	yes	maximal, complemented
5	#11926785	C₂	1	4	yes	maximal
6	C₄×Dic₇₆	C₂	1	4	yes	maximal, complemented
7	C₂×C₄×C₈×C₁₉	C₂	1	1	yes	maximal, characteristic
8	C₂²×C₈×C₁₉	C₂²	1	2	yes	—
9	C₂×C₄²×C₁₉	C₂²	1	1	yes	characteristic
10	#10597372	—	2	2	no	—
11	#10597370	—	2	2	no	—
12	#10597370	C₂²	1	4	yes	—
13	C₂×Dic₇₆	C₂²	1	4	yes	—
14	#10597316	—	2	2	no	—
15	#10597315	C₂²	1	2	yes	—
16	#10597315	C₂²	1	2	yes	—
17	C₄×Dic₃₈	—	2	2	no	—
18	C₄×Dic₃₈	C₂²	1	4	yes	—
19	C₄×C₈×C₁₉	C₂²	1	1	yes	characteristic
20	C₄×C₈×C₁₉	C₂²	1	2	yes	—
21	C₄×C₈×C₁₉	C₂²	1	1	yes	characteristic
22	#10597275	C₂²	1	4	yes	—
23	#10597273	C₂²	1	8	yes	—
24	C₂²×C₄×C₁₉	D₄	1	1	yes	characteristic
25	C₂²×C₄×C₁₉	D₄	1	1	yes	characteristic
26	C₂²×C₄×C₁₉	C₂³	1	1	yes	characteristic
27	C₂×Dic₃₈	—	2	2	no	—
28	C₂×Dic₃₈	C₂³	1	4	yes	—
29	C₂×C₈×C₁₉	—	2	4	no	—
30	C₂×C₈×C₁₉	C₂³	1	2	yes	—
31	C₂×C₈×C₁₉	C₂³	1	2	yes	—
32	C₄²×C₁₉	D₄	1	1	yes	characteristic
33	C₄²×C₁₉	D₄	1	1	yes	characteristic
34	C₄²×C₁₉	C₂³	1	1	yes	characteristic
35	C₄²×C₁₉	C₂³	1	1	yes	characteristic
36	C₂²⋊Dic₁₉	—	2	4	no	—
37	C₄⋊Dic₁₉	—	2	2	no	—
38	C₄⋊Dic₁₉	—	2	2	no	—
39	C₄⋊Dic₁₉	—	2	2	no	—
40	C₄⋊Dic₁₉	C₂³	1	4	yes	—
41	Dic₁₉⋊C₄	—	2	4	no	—
42	Dic₁₉⋊C₄	—	2	4	no	—
43	C₄×Dic₁₉	—	2	4	no	—
44	Dic₇₆	—	2	8	no	—
45	C₂³×C₁₉	C₂×D₄	1	1	yes	characteristic
46	C₂×C₄×C₁₉	—	2	2	no	—
47	C₂×C₄×C₁₉	—	2	2	no	—
48	C₂×C₄×C₁₉	Q₁₆	1	4	yes	complemented
49	C₂×C₄×C₁₉	C₂×D₄	1	1	yes	characteristic
50	C₂×C₄×C₁₉	C₂×D₄	1	1	yes	characteristic
51	C₂×C₄×C₁₉	C₂×D₄	1	1	yes	characteristic
52	C₂×C₄×C₁₉	C₂×D₄	1	1	yes	characteristic
53	C₂×C₄×C₁₉	C₂×D₄	1	1	yes	characteristic
54	C₂×C₄×C₁₉	C₂⁴	1	1	yes	characteristic
55	C₂×Dic₁₉	—	2	4	no	—
56	C₂×Dic₁₉	—	2	4	no	—
57	Dic₃₈	—	2	8	no	—
58	Dic₃₈	—	4	4	no	—
59	C₈×C₁₉	—	2	2	no	—
60	C₈×C₁₉	#41	1	4	yes	—
61	#3170	—	19	1	no	maximal, Sylow
62	C₂²×C₁₉	—	2	1	no	—
63	C₂²×C₁₉	—	2	1	no	—
64	C₂²×C₁₉	C₂×Q₁₆	1	2	yes	—
65	C₂²×C₁₉	C₂²×D₄	1	1	yes	characteristic
66	C₄×C₁₉	—	2	1	no	—
67	C₄×C₁₉	—	2	1	no	—
68	C₄×C₁₉	C₂×Q₁₆	1	2	yes	—
69	C₄×C₁₉	C₂×Q₁₆	1	2	yes	—
70	C₄×C₁₉	#135	1	2	yes	—
71	C₄×C₁₉	C₂×#41	1	2	yes	—
72	Dic₁₉	—	4	4	no	—
73	Dic₁₉	—	4	4	no	—
74	#533	—	19	2	no	—
75	#500	—	19	1	no	—
76	#494	—	19	1	no	—
77	#487	—	19	2	no	—
78	#467	—	19	4	no	—
79	C₄×Q₁₆	—	19	4	no	—
80	C₂×C₄×C₈	D₁₉	1	1	yes	characteristic
81	C₂×C₁₉	—	2	2	no	—
82	C₂×C₁₉	#532	1	1	yes	characteristic
83	C₂×C₁₉	C₂²×Q₁₆	1	1	yes	characteristic
84	C₂×C₁₉	C₂×#135	1	1	yes	characteristic
85	C₂×Q₁₆	—	19	4	no	—
86	C₂²×C₈	C₂×D₁₉	1	2	yes	—
87	C₄⋊Q₈	—	19	2	no	—
88	C₄⋊Q₈	—	19	2	no	—
89	#125	—	38	2	no	—
90	C₂²⋊Q₈	—	19	4	no	—
91	C₂²⋊Q₈	—	38	2	no	—
92	C₄×Q₈	—	19	4	no	—
93	C₄×Q₈	—	38	2	no	—
94	#117	—	38	2	no	—
95	C₂×C₄²	C₂×D₁₉	1	1	yes	characteristic
96	#107	—	19	4	no	—
97	#103	—	19	8	no	—
98	C₄×C₈	C₂×D₁₉	1	2	yes	—
99	C₄×C₈	C₂×D₁₉	1	1	yes	characteristic
100	C₄×C₈	C₂×D₁₉	1	1	yes	characteristic
101	C₁₉	#3170	1	1	yes	characteristic, complemented, Sylow
102	C₂×Q₈	—	19	4	no	—
103	C₂×Q₈	—	38	2	no	—
104	C₂²×C₄	D₇₆	1	1	yes	characteristic
105	C₂²×C₄	D₇₆	1	1	yes	characteristic
106	C₂²×C₄	C₂²×D₁₉	1	1	yes	characteristic
107	Q₁₆	—	38	8	no	—
108	C₂×C₈	—	2	4	no	—
109	C₂×C₈	C₂²×D₁₉	1	2	yes	—
110	C₂×C₈	C₂²×D₁₉	1	2	yes	—
111	C₄⋊C₄	—	19	4	no	—
112	C₄⋊C₄	—	38	4	no	—
113	C₄⋊C₄	—	38	4	no	—
114	C₄⋊C₄	—	38	2	no	—
115	C₄⋊C₄	—	38	2	no	—
116	C₄⋊C₄	—	38	2	no	—
117	C₂²⋊C₄	—	38	4	no	—
118	C₄²	—	38	4	no	—
119	C₄²	D₇₆	1	1	yes	characteristic
120	C₄²	D₇₆	1	1	yes	characteristic
121	C₄²	C₂²×D₁₉	1	1	yes	characteristic
122	C₄²	C₂²×D₁₉	1	1	yes	characteristic
123	C₂³	C₂×D₇₆	1	1	yes	characteristic
124	Q₈	—	38	8	no	—
125	Q₈	—	76	4	no	—
126	C₂×C₄	—	2	2	no	—
127	C₂×C₄	—	2	2	no	—
128	C₂×C₄	—	38	4	no	—
129	C₂×C₄	—	38	4	no	—
130	C₂×C₄	Dic₇₆	1	4	yes	complemented
131	C₂×C₄	C₂×D₇₆	1	1	yes	characteristic
132	C₂×C₄	C₂×D₇₆	1	1	yes	characteristic
133	C₂×C₄	C₂×D₇₆	1	1	yes	characteristic
134	C₂×C₄	C₂×D₇₆	1	1	yes	characteristic
135	C₂×C₄	C₂×D₇₆	1	1	yes	characteristic
136	C₂×C₄	C₂³×D₁₉	1	1	yes	characteristic
137	C₈	—	2	2	no	—
138	C₈	#64541	1	4	yes	—
139	C₂²	—	2	1	no	—
140	C₂²	—	2	1	no	—
141	C₂²	C₂×Dic₇₆	1	2	yes	—
142	C₂²	C₂²×D₇₆	1	1	yes	characteristic
143	C₄	—	2	1	no	—
144	C₄	—	2	1	no	—
145	C₄	—	76	4	no	—
146	C₄	—	76	4	no	—
147	C₄	#10597350	1	2	yes	—
148	C₄	C₂×Dic₇₆	1	2	yes	—
149	C₄	C₂×Dic₇₆	1	2	yes	—
150	C₄	C₂×#64541	1	2	yes	—
151	C₂	—	2	2	no	—
152	C₂	#11927094	1	1	yes	characteristic
153	C₂	C₂²×Dic₇₆	1	1	yes	characteristic
154	C₂	C₂×#10597350	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#11927101

C₂

yes

maximal

#11926949

C₂

yes

maximal, characteristic, complemented

#11926924

C₂

yes

maximal, characteristic, complemented

#11926895

C₂

yes

maximal, complemented

#11926785

C₂

yes

maximal

C₄×Dic₇₆

C₂

yes

maximal, complemented

C₂×C₄×C₈×C₁₉

C₂

yes

maximal, characteristic

C₂²×C₈×C₁₉

C₂²

yes

—

C₂×C₄²×C₁₉

C₂²

yes

—

—

C₂²

yes

—

C₂×Dic₇₆

C₂²

yes

—

#10597316

—

#10597315

C₂²

yes

—

#10597315

C₂²

yes

—

C₄×Dic₃₈

—

C₄×Dic₃₈

C₂²

yes

—

C₄×C₈×C₁₉

C₂²

yes

characteristic

C₄×C₈×C₁₉

C₂²

yes

—

C₄×C₈×C₁₉

C₂²

yes

characteristic

#10597275

C₂²

yes

—

#10597273

C₂²

yes

—

C₂²×C₄×C₁₉

D₄

yes

characteristic

C₂²×C₄×C₁₉

D₄

yes

characteristic

C₂²×C₄×C₁₉

C₂³

yes

characteristic

C₂×Dic₃₈

—

C₂×Dic₃₈

C₂³

yes

—

C₂×C₈×C₁₉

—

C₂×C₈×C₁₉

C₂³

yes

—

C₂×C₈×C₁₉

C₂³

yes

—

C₄²×C₁₉

D₄

yes

characteristic

C₄²×C₁₉

D₄

yes

characteristic

C₄²×C₁₉

C₂³

yes

characteristic

C₄²×C₁₉

C₂³

yes

characteristic

C₂²⋊Dic₁₉

—

C₄⋊Dic₁₉

—

C₄⋊Dic₁₉

—

C₄⋊Dic₁₉

—

C₄⋊Dic₁₉

C₂³

yes

—

Dic₁₉⋊C₄

—

Dic₁₉⋊C₄

—

C₄×Dic₁₉

—

Dic₇₆

—

C₂³×C₁₉

C₂×D₄

yes

characteristic

C₂×C₄×C₁₉

—

C₂×C₄×C₁₉

—

C₂×C₄×C₁₉

Q₁₆

yes

complemented

C₂×C₄×C₁₉

C₂×D₄

yes

characteristic

C₂×C₄×C₁₉

C₂×D₄

yes

characteristic

C₂×C₄×C₁₉

C₂×D₄

yes

characteristic

C₂×C₄×C₁₉

C₂×D₄

yes

characteristic

C₂×C₄×C₁₉

C₂×D₄

yes

characteristic

C₂×C₄×C₁₉

C₂⁴

yes

characteristic

C₂×Dic₁₉

—

C₂×Dic₁₉

—

Dic₃₈

—

Dic₃₈

—

C₈×C₁₉

—

C₈×C₁₉

#41

yes

—

#3170

—

maximal, Sylow

C₂²×C₁₉

—

C₂²×C₁₉

—

C₂²×C₁₉

C₂×Q₁₆

yes

—

C₂²×C₁₉

C₂²×D₄

yes

characteristic

C₄×C₁₉

—

C₄×C₁₉

—

C₄×C₁₉

C₂×Q₁₆

yes

—

C₄×C₁₉

C₂×Q₁₆

yes

—

C₄×C₁₉

#135

yes

—

C₄×C₁₉

C₂×#41

yes

—

Dic₁₉

—

Dic₁₉

—

—

—

—

—

—

C₄×Q₁₆

—

C₂×C₄×C₈

D₁₉

yes

characteristic

C₂×C₁₉

—

C₂×C₁₉

#532

yes

characteristic

C₂×C₁₉

C₂²×Q₁₆

yes

characteristic

C₂×C₁₉

C₂×#135

yes

characteristic

C₂×Q₁₆

—

C₂²×C₈

C₂×D₁₉

yes

—

C₄⋊Q₈

—

C₄⋊Q₈

—

#125

—

C₂²⋊Q₈

—

C₂²⋊Q₈

—

C₄×Q₈

—

C₄×Q₈

—

#117

—

C₂×C₄²

C₂×D₁₉

yes

characteristic

#107

—

#103

—

C₄×C₈

C₂×D₁₉

yes

—

C₄×C₈

C₂×D₁₉

yes

characteristic

100

C₄×C₈

C₂×D₁₉

yes

characteristic

101

C₁₉

#3170

yes

characteristic, complemented, Sylow

102

C₂×Q₈

—

103

C₂×Q₈

—

104

C₂²×C₄

D₇₆

yes

characteristic

105

C₂²×C₄

D₇₆

yes

characteristic

106

C₂²×C₄

C₂²×D₁₉

yes

characteristic

107

Q₁₆

—

108

C₂×C₈

—

109

C₂×C₈

C₂²×D₁₉

yes

—

110

C₂×C₈

C₂²×D₁₉

yes

—

111

C₄⋊C₄

—

112

C₄⋊C₄

—

113

C₄⋊C₄

—

114

C₄⋊C₄

—

115

C₄⋊C₄

—

116

C₄⋊C₄

—

117

C₂²⋊C₄

—

118

C₄²

—

119

C₄²

D₇₆

yes

characteristic

120

C₄²

D₇₆

yes

characteristic

121

C₄²

C₂²×D₁₉

yes

characteristic

122

C₄²

C₂²×D₁₉

yes

characteristic

123

C₂³

C₂×D₇₆

yes

characteristic

124

Q₈

—

125

Q₈

—

126

C₂×C₄

—

127

C₂×C₄

—

128

C₂×C₄

—

129

C₂×C₄

—

130

C₂×C₄

Dic₇₆

yes

complemented

131

C₂×C₄

C₂×D₇₆

yes

characteristic

132

C₂×C₄

C₂×D₇₆

yes

characteristic

133

C₂×C₄

C₂×D₇₆

yes

characteristic

134

C₂×C₄

C₂×D₇₆

yes

characteristic

135

C₂×C₄

C₂×D₇₆

yes

characteristic

136

C₂×C₄

C₂³×D₁₉

yes

characteristic

137

C₈

—

138

C₈

#64541

yes

—

139

C₂²

—

140

C₂²

—

141

C₂²

C₂×Dic₇₆

yes

—

142

C₂²

C₂²×D₇₆

yes

characteristic

143

C₄

—

144

C₄

—

145

C₄

—

146

C₄

—

147

C₄

#10597350

yes

—

148

C₄

C₂×Dic₇₆

yes

—

149

C₄

C₂×Dic₇₆

yes

—

150

C₄

C₂×#64541

yes

—

151

C₂

—

152

C₂

#11927094

yes

characteristic

153

C₂

C₂²×Dic₇₆

yes

characteristic

154

C₂

C₂×#10597350

yes

characteristic

	subgroup	G/H
center	C₂×C₄	C₂×D₇₆
commutator, cosocle	C₂×C₄×C₁₉	C₂⁴
Frattini	C₂×C₄	C₂³×D₁₉
Fitting	C₂×C₄×C₈×C₁₉	C₂
socle	C₂²×C₁₉	C₂²×D₄
prime radical	C₂³×C₁₉	C₂×D₄
hypercenter	C₂×C₄×C₈	D₁₉
hypocenter	C₁₉	#3170
LCS	G ▷ C₂×C₄×C₁₉ ▷ C₂×C₁₉ ▷ C₁₉
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₂×C₄×C₈

subgroup

G/H

center

C₂×C₄

C₂×D₇₆

commutator, cosocle

C₂×C₄×C₁₉

C₂⁴

Frattini

C₂×C₄

C₂³×D₁₉

Fitting

C₂×C₄×C₈×C₁₉

C₂

socle

C₂²×C₁₉

C₂²×D₄

prime radical

C₂³×C₁₉

C₂×D₄

hypercenter

C₂×C₄×C₈

D₁₉

hypocenter

C₁₉

#3170

LCS

G ▷ C₂×C₄×C₁₉ ▷ C₂×C₁₉ ▷ C₁₉

UCS

C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₂×C₄×C₈

N	H ≅ G/N	type
2. #11926949	C₂	semidirect
3. #11926924	C₂	semidirect
4. #11926895	C₂	semidirect
6. C₄×Dic₇₆	C₂	semidirect
48. C₂×C₄×C₁₉	Q₁₆	semidirect
101. C₁₉	#3170	semidirect
130. C₂×C₄	Dic₇₆	semidirect

H ≅ G/N

type

2. #11926949

C₂

semidirect

3. #11926924

C₂

semidirect

4. #11926895

C₂

semidirect

6. C₄×Dic₇₆

C₂

semidirect

48. C₂×C₄×C₁₉

Q₁₆

semidirect

101. C₁₉

#3170

semidirect

130. C₂×C₄

Dic₇₆

semidirect

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

subgroup	index
#11927101	2
#11926949	2
#11926924	2
#11926895	2
#11926785	2
C₄×Dic₇₆	2
C₂×C₄×C₈×C₁₉	2
#3170	19

subgroup

index

C₄×Dic₇₆

C₂×C₄×C₈×C₁₉

#3170

Subgroups — #14574221

Conjugacy classes of subgroups

Complements

Maximal subgroups