Group #14564624 — subgroups

Subgroups — #14564624

Order 2432.

260 conjugacy classes of subgroups; only the 101 normal subgroups are shown.

normal subgroups only

This group has 260 conjugacy classes of subgroups — too many to draw. Only its 101 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×#10597315	C₂	1	1	yes	maximal, characteristic
2	#11926568	C₂	1	1	yes	maximal, characteristic
3	#11926312	C₂	1	4	yes	maximal
4	C₄×M₁₆×C₁₉	C₂	1	1	yes	maximal, characteristic
5	C₂²×Dic₃₈	C₄	1	2	yes	—
6	C₂×M₁₆×C₁₉	C₂²	1	2	yes	—
7	C₂×C₄²×C₁₉	C₂²	1	1	yes	characteristic
8	C₂×C₄⋊Dic₁₉	C₄	1	2	yes	—
9	C₂×C₄⋊Dic₁₉	C₂²	1	2	yes	—
10	C₂×#64520	C₂²	1	2	yes	—
11	#10597315	—	2	1	no	—
12	#10597315	—	2	1	no	—
13	#10597315	—	2	1	no	—
14	#10597315	—	2	1	no	—
15	C₁₉×#97	—	2	1	no	—
16	C₄×C₈×C₁₉	—	2	1	no	—
17	C₄⋊(C₁₉⋊C₈)	—	2	1	no	—
18	C₄⋊(C₁₉⋊C₈)	—	2	1	no	—
19	C₂²×C₄×C₁₉	C₂×C₄	1	1	yes	characteristic
20	C₂²×C₄×C₁₉	C₂×C₄	1	1	yes	characteristic
21	C₂²×C₄×C₁₉	C₂³	1	1	yes	characteristic
22	C₂²×Dic₁₉	C₂×C₄	1	4	yes	—
23	C₂×Dic₃₈	—	2	2	no	—
24	C₂×Dic₃₈	—	2	2	no	—
25	C₂×Dic₃₈	—	4	4	no	—
26	M₁₆×C₁₉	—	2	2	no	—
27	M₁₆×C₁₉	Q₈	1	4	yes	complemented
28	C₂×C₈×C₁₉	—	2	2	no	—
29	C₄²×C₁₉	D₄	1	1	yes	characteristic
30	C₄²×C₁₉	D₄	1	1	yes	characteristic
31	C₄²×C₁₉	D₄	1	1	yes	characteristic
32	C₄²×C₁₉	D₄	1	1	yes	characteristic
33	C₄⋊Dic₁₉	—	2	2	no	—
34	C₄⋊Dic₁₉	—	2	2	no	—
35	C₄⋊Dic₁₉	—	2	2	no	—
36	C₄⋊Dic₁₉	—	2	2	no	—
37	#64520	—	2	4	no	—
38	C₂×C₁₉⋊C₈	—	2	2	no	—
39	C₂³×C₁₉	C₂²×C₄	1	1	yes	characteristic
40	C₂×C₄×C₁₉	—	2	2	no	—
41	C₂×C₄×C₁₉	—	2	2	no	—
42	C₂×C₄×C₁₉	C₂²⋊C₄	1	1	yes	characteristic
43	C₂×C₄×C₁₉	C₂²⋊C₄	1	1	yes	characteristic
44	C₂×C₄×C₁₉	C₂²⋊C₄	1	1	yes	characteristic
45	C₂×C₄×C₁₉	C₂²⋊C₄	1	1	yes	characteristic
46	C₂×C₄×C₁₉	C₂×D₄	1	1	yes	characteristic
47	C₂×C₄×C₁₉	C₂×D₄	1	1	yes	characteristic
48	C₂×C₄×C₁₉	C₂×Q₈	1	2	yes	—
49	C₂×C₄×C₁₉	#41	1	2	yes	—
50	C₂×Dic₁₉	—	2	4	no	—
51	C₂×Dic₁₉	—	4	4	no	—
52	Dic₃₈	—	4	4	no	—
53	Dic₃₈	—	4	4	no	—
54	C₈×C₁₉	—	2	4	no	—
55	C₁₉⋊C₈	—	4	2	no	—
56	#1997	—	19	1	no	maximal, Sylow
57	C₂²×C₁₉	—	2	1	no	—
58	C₂²×C₁₉	—	2	1	no	—
59	C₂²×C₁₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
60	C₂²×C₁₉	C₄×Q₈	1	2	yes	—
61	C₄×C₁₉	—	2	1	no	—
62	C₄×C₁₉	—	2	1	no	—
63	C₄×C₁₉	#100	1	2	yes	—
64	C₄×C₁₉	#101	1	2	yes	—
65	C₄×C₁₉	C₂²⋊Q₈	1	2	yes	—
66	C₄×C₁₉	#124	1	1	yes	characteristic
67	C₄×C₁₉	C₄⋊Q₈	1	1	yes	characteristic
68	Dic₁₉	—	4	4	no	—
69	C₂×C₄⋊Q₈	—	19	1	no	—
70	#423	—	19	1	no	—
71	C₄×M₁₆	D₁₉	1	1	yes	characteristic
72	#343	—	19	4	no	—
73	C₂×C₁₉	—	2	2	no	—
74	C₂×C₁₉	#391	1	1	yes	characteristic
75	C₂×C₁₉	C₂×#100	1	1	yes	characteristic
76	C₂×C₁₉	C₂×#101	1	1	yes	characteristic
77	C₂²×Q₈	—	19	2	no	—
78	C₂×M₁₆	—	19	2	no	—
79	C₂×M₁₆	C₂×D₁₉	1	2	yes	—
80	C₄⋊Q₈	—	38	1	no	—
81	C₄⋊Q₈	—	38	1	no	—
82	C₄⋊Q₈	—	38	1	no	—
83	C₄⋊Q₈	—	38	1	no	—
84	C₂×C₄⋊C₄	—	19	2	no	—
85	C₂×C₄⋊C₄	—	19	2	no	—
86	C₂×C₄²	C₂×D₁₉	1	1	yes	characteristic
87	C₄⋊C₈	—	38	1	no	—
88	C₄⋊C₈	—	38	1	no	—
89	#97	—	2	1	no	—
90	C₄×C₈	—	2	1	no	—
91	C₁₉	#1997	1	1	yes	characteristic, complemented, Sylow
92	C₂×Q₈	—	38	2	no	—
93	C₂×Q₈	—	38	2	no	—
94	C₂×Q₈	—	76	4	no	—
95	C₂²×C₄	—	19	4	no	—
96	C₂²×C₄	C₄×D₁₉	1	1	yes	characteristic
97	C₂²×C₄	C₄×D₁₉	1	1	yes	characteristic
98	C₂²×C₄	C₂²×D₁₉	1	1	yes	characteristic
99	M₁₆	—	2	2	no	—
100	M₁₆	—	38	4	no	—
101	M₁₆	Dic₃₈	1	4	yes	complemented
102	C₂×C₈	—	2	2	no	—
103	C₂×C₈	—	38	2	no	—
104	C₄⋊C₄	—	38	2	no	—
105	C₄⋊C₄	—	38	2	no	—
106	C₄⋊C₄	—	38	2	no	—
107	C₄⋊C₄	—	38	2	no	—
108	C₄²	D₇₆	1	1	yes	characteristic
109	C₄²	D₇₆	1	1	yes	characteristic
110	C₄²	C₂²⋊D₁₉	1	1	yes	characteristic
111	C₄²	C₂²⋊D₁₉	1	1	yes	characteristic
112	C₂³	C₂×C₄×D₁₉	1	1	yes	characteristic
113	Q₈	—	76	4	no	—
114	Q₈	—	76	4	no	—
115	C₂×C₄	—	2	2	no	—
116	C₂×C₄	—	2	2	no	—
117	C₂×C₄	—	38	4	no	—
118	C₂×C₄	—	76	4	no	—
119	C₂×C₄	(C₂×D₁₉)⋊C₄	1	1	yes	characteristic
120	C₂×C₄	(C₂×D₁₉)⋊C₄	1	1	yes	characteristic
121	C₂×C₄	(C₂×D₁₉)⋊C₄	1	1	yes	characteristic
122	C₂×C₄	(C₂×D₁₉)⋊C₄	1	1	yes	characteristic
123	C₂×C₄	C₂×Dic₃₈	1	2	yes	—
124	C₂×C₄	C₂×D₇₆	1	1	yes	characteristic
125	C₂×C₄	#64541	1	2	yes	—
126	C₂×C₄	C₂×C₂²⋊D₁₉	1	1	yes	characteristic
127	C₈	—	2	4	no	—
128	C₈	—	76	2	no	—
129	C₂²	—	2	1	no	—
130	C₂²	—	2	1	no	—
131	C₂²	C₄×Dic₃₈	1	2	yes	—
132	C₂²	C₂×(C₂×D₁₉)⋊C₄	1	1	yes	characteristic
133	C₄	—	2	1	no	—
134	C₄	—	2	1	no	—
135	C₄	—	76	4	no	—
136	C₄	#10597280	1	2	yes	—
137	C₄	#10597281	1	2	yes	—
138	C₄	#10597315	1	1	yes	characteristic
139	C₄	#10597321	1	1	yes	characteristic
140	C₄	#10597370	1	2	yes	—
141	C₂	—	2	2	no	—
142	C₂	#11926450	1	1	yes	characteristic
143	C₂	C₂×#10597280	1	1	yes	characteristic
144	C₂	C₂×#10597281	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#10597315

C₂

yes

maximal, characteristic

#11926568

C₂

yes

maximal, characteristic

#11926312

C₂

yes

maximal

C₄×M₁₆×C₁₉

C₂

yes

maximal, characteristic

C₂²×Dic₃₈

C₄

yes

—

C₂×M₁₆×C₁₉

C₂²

yes

—

C₂×C₄²×C₁₉

C₂²

yes

characteristic

C₂×C₄⋊Dic₁₉

C₄

yes

—

C₂×C₄⋊Dic₁₉

C₂²

yes

—

C₂×#64520

C₂²

yes

—

—

—

—

—

C₁₉×#97

—

C₄×C₈×C₁₉

—

C₄⋊(C₁₉⋊C₈)

—

C₄⋊(C₁₉⋊C₈)

—

C₂²×C₄×C₁₉

C₂×C₄

yes

characteristic

C₂²×C₄×C₁₉

C₂×C₄

yes

characteristic

C₂²×C₄×C₁₉

C₂³

yes

characteristic

C₂²×Dic₁₉

C₂×C₄

yes

—

C₂×Dic₃₈

—

C₂×Dic₃₈

—

C₂×Dic₃₈

—

M₁₆×C₁₉

—

M₁₆×C₁₉

Q₈

yes

complemented

C₂×C₈×C₁₉

—

C₄²×C₁₉

D₄

yes

characteristic

C₄²×C₁₉

D₄

yes

characteristic

C₄²×C₁₉

D₄

yes

characteristic

C₄²×C₁₉

D₄

yes

characteristic

C₄⋊Dic₁₉

—

C₄⋊Dic₁₉

—

C₄⋊Dic₁₉

—

C₄⋊Dic₁₉

—

#64520

—

C₂×C₁₉⋊C₈

—

C₂³×C₁₉

C₂²×C₄

yes

characteristic

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

C₂×D₄

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

—

C₈×C₁₉

—

C₁₉⋊C₈

—

#1997

—

maximal, Sylow

C₂²×C₁₉

—

C₂²×C₁₉

—

C₂²×C₁₉

C₂×C₂²⋊C₄

yes

characteristic

C₂²×C₁₉

C₄×Q₈

yes

—

C₄×C₁₉

—

C₄×C₁₉

—

C₄×C₁₉

#100

yes

—

C₄×C₁₉

#101

yes

—

C₄×C₁₉

C₂²⋊Q₈

yes

—

C₄×C₁₉

#124

yes

characteristic

C₄×C₁₉

C₄⋊Q₈

yes

characteristic

Dic₁₉

—

C₂×C₄⋊Q₈

—

#423

—

C₄×M₁₆

D₁₉

yes

characteristic

#343

—

C₂×C₁₉

—

C₂×C₁₉

yes

C₂×C₁₉

yes

C₂×C₁₉

yes

C₂²×Q₈

—

C₂×M₁₆

—

C₂×M₁₆

C₂×D₁₉

yes

—

C₄⋊Q₈

—

C₄⋊Q₈

—

C₄⋊Q₈

—

C₄⋊Q₈

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄²

C₂×D₁₉

yes

characteristic

C₄⋊C₈

—

C₄⋊C₈

—

#97

—

C₄×C₈

—

C₁₉

#1997

yes

characteristic, complemented, Sylow

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

C₂²×C₄

—

C₂²×C₄

C₄×D₁₉

yes

characteristic

C₂²×C₄

C₄×D₁₉

yes

characteristic

C₂²×C₄

C₂²×D₁₉

yes

characteristic

M₁₆

—

100

M₁₆

—

101

M₁₆

Dic₃₈

yes

complemented

102

C₂×C₈

—

103

C₂×C₈

—

104

C₄⋊C₄

—

105

C₄⋊C₄

—

106

C₄⋊C₄

—

107

C₄⋊C₄

—

108

C₄²

D₇₆

yes

characteristic

109

C₄²

D₇₆

yes

characteristic

110

C₄²

C₂²⋊D₁₉

yes

characteristic

111

C₄²

C₂²⋊D₁₉

yes

characteristic

112

C₂³

C₂×C₄×D₁₉

yes

characteristic

113

Q₈

—

114

Q₈

—

115

C₂×C₄

—

116

C₂×C₄

—

117

C₂×C₄

—

118

C₂×C₄

—

119

C₂×C₄

(C₂×D₁₉)⋊C₄

yes

characteristic

120

C₂×C₄

(C₂×D₁₉)⋊C₄

yes

characteristic

121

C₂×C₄

(C₂×D₁₉)⋊C₄

yes

characteristic

122

C₂×C₄

(C₂×D₁₉)⋊C₄

yes

characteristic

123

C₂×C₄

C₂×Dic₃₈

yes

—

124

C₂×C₄

C₂×D₇₆

yes

characteristic

125

C₂×C₄

#64541

yes

—

126

C₂×C₄

C₂×C₂²⋊D₁₉

yes

characteristic

127

C₈

—

128

C₈

—

129

C₂²

—

130

C₂²

—

131

C₂²

C₄×Dic₃₈

yes

—

132

C₂²

C₂×(C₂×D₁₉)⋊C₄

yes

characteristic

133

C₄

—

134

C₄

—

135

C₄

—

136

C₄

#10597280

yes

—

137

C₄

#10597281

yes

—

138

C₄

#10597315

yes

characteristic

139

C₄

#10597321

yes

characteristic

140

C₄

#10597370

yes

—

141

C₂

—

142

C₂

#11926450

yes

characteristic

143

C₂

C₂×#10597280

yes

characteristic

144

C₂

C₂×#10597281

yes

characteristic

	subgroup	G/H
center	C₂²	C₂×(C₂×D₁₉)⋊C₄
commutator, prime radical	C₂³×C₁₉	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₁₉
Fitting	C₄×M₁₆×C₁₉	C₂
socle	C₂²×C₁₉	C₂×C₂²⋊C₄
cosocle	C₂²×C₄×C₁₉	C₂³
hypercenter	C₄×M₁₆	D₁₉
hypocenter	C₁₉	#1997
LCS	G ▷ C₂³×C₁₉ ▷ C₂×C₁₉ ▷ C₁₉
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₄×M₁₆

subgroup

G/H

center

C₂²

C₂×(C₂×D₁₉)⋊C₄

commutator, prime radical

C₂³×C₁₉

C₂²×C₄

Frattini

C₂²×C₄

C₂²×D₁₉

Fitting

C₄×M₁₆×C₁₉

C₂

socle

C₂²×C₁₉

C₂×C₂²⋊C₄

cosocle

C₂²×C₄×C₁₉

C₂³

hypercenter

C₄×M₁₆

D₁₉

hypocenter

C₁₉

#1997

LCS

G ▷ C₂³×C₁₉ ▷ C₂×C₁₉ ▷ C₁₉

UCS

C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₄×M₁₆

N	H ≅ G/N	type
27. M₁₆×C₁₉	Q₈	semidirect
91. C₁₉	#1997	semidirect
101. M₁₆	Dic₃₈	semidirect

H ≅ G/N

type

27. M₁₆×C₁₉

Q₈

semidirect

91. C₁₉

#1997

semidirect

101. M₁₆

Dic₃₈

semidirect

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

subgroup	index
C₂×#10597315	2
#11926568	2
#11926312	2
C₄×M₁₆×C₁₉	2
#1997	19

subgroup

index

C₄×M₁₆×C₁₉

Subgroups — #14564624

Conjugacy classes of subgroups

Complements

Maximal subgroups