Group #44485506 — subgroups

Subgroups — #44485506

Order 30592.

260 conjugacy classes of subgroups; only the 113 normal subgroups are shown.

normal subgroups only

This group has 260 conjugacy classes of subgroups — too many to draw. Only its 113 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×C₄⋊(C₂₃₉⋊C₈)	C₂	1	1	yes	maximal, characteristic
2	C₄×C₄⋊Dic₂₃₉	C₂	1	1	yes	maximal, characteristic
3	#29504534	C₂	1	2	yes	maximal
4	#29504534	C₂	1	2	yes	maximal
5	C₂×#125×C₂₃₉	C₂	1	1	yes	maximal, characteristic
6	C₂²×C₂₃₉⋊C₈	C₂²	1	2	yes	—
7	C₂×C₄⋊Dic₂₃₉	C₂²	1	2	yes	—
8	C₂×C₄×Dic₂₃₉	—	2	1	no	—
9	C₄⋊(C₂₃₉⋊C₈)	—	2	2	no	—
10	C₂₃₉⋊#95	—	2	1	no	—
11	#125×C₂₃₉	—	2	2	no	—
12	#125×C₂₃₉	C₄	1	4	yes	complemented
13	C₂×C₄⋊C₄×C₂₃₉	—	2	1	no	—
14	C₂×C₄⋊C₄×C₂₃₉	—	2	1	no	—
15	C₂×C₄⋊C₄×C₂₃₉	C₂²	1	1	yes	characteristic
16	C₂×C₄⋊C₄×C₂₃₉	C₂²	1	1	yes	characteristic
17	C₂×C₄²×C₂₃₉	C₂²	1	1	yes	characteristic
18	C₂²×Dic₂₃₉	—	2	2	no	—
19	C₂×C₂₃₉⋊C₈	—	2	2	no	—
20	C₂×C₂₃₉⋊C₈	—	2	4	no	—
21	C₄⋊Dic₂₃₉	—	2	2	no	—
22	C₄⋊Dic₂₃₉	Q₈	1	4	yes	—
23	C₄×Dic₂₃₉	—	4	2	no	—
24	C₂²×C₄×C₂₃₉	—	2	1	no	—
25	C₂²×C₄×C₂₃₉	—	2	1	no	—
26	C₂²×C₄×C₂₃₉	D₄	1	1	yes	characteristic
27	C₂²×C₄×C₂₃₉	D₄	1	1	yes	characteristic
28	C₂²×C₄×C₂₃₉	C₂³	1	1	yes	characteristic
29	C₄⋊C₄×C₂₃₉	—	2	4	no	—
30	C₄⋊C₄×C₂₃₉	—	2	4	no	—
31	C₄⋊C₄×C₂₃₉	—	2	1	no	—
32	C₄⋊C₄×C₂₃₉	—	2	1	no	—
33	C₄⋊C₄×C₂₃₉	C₂×C₄	1	2	yes	—
34	C₄⋊C₄×C₂₃₉	C₂×C₄	1	2	yes	—
35	C₄²×C₂₃₉	C₂×C₄	1	2	yes	—
36	C₄²×C₂₃₉	D₄	1	2	yes	—
37	C₂×Dic₂₃₉	—	2	4	no	—
38	C₂×Dic₂₃₉	—	4	2	no	—
39	C₂×Dic₂₃₉	—	4	2	no	—
40	C₂₃₉⋊C₈	—	4	2	no	—
41	C₂³×C₂₃₉	C₂×D₄	1	1	yes	characteristic
42	C₂×C₄×C₂₃₉	—	2	2	no	—
43	C₂×C₄×C₂₃₉	—	2	1	no	—
44	C₂×C₄×C₂₃₉	—	2	1	no	—
45	C₂×C₄×C₂₃₉	—	2	2	no	—
46	C₂×C₄×C₂₃₉	—	2	2	no	—
47	C₂×C₄×C₂₃₉	—	4	2	no	—
48	C₂×C₄×C₂₃₉	—	4	1	no	—
49	C₂×C₄×C₂₃₉	—	4	1	no	—
50	C₂×C₄×C₂₃₉	C₂²⋊C₄	1	2	yes	—
51	C₂×C₄×C₂₃₉	C₂²⋊C₄	1	2	yes	—
52	C₂×C₄×C₂₃₉	C₂²×C₄	1	1	yes	characteristic
53	C₂×C₄×C₂₃₉	C₂×D₄	1	1	yes	characteristic
54	C₂×C₄×C₂₃₉	C₂×Q₈	1	2	yes	—
55	C₂×C₄×C₂₃₉	#41	1	2	yes	—
56	C₂²×C₂₃₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
57	C₂²×C₂₃₉	C₂²⋊Q₈	1	2	yes	—
58	C₂²×C₂₃₉	#135	1	2	yes	—
59	C₂²×C₂₃₉	AGL(1,ℤ/8)	1	1	yes	characteristic
60	C₂²×C₂₃₉	#137	1	1	yes	characteristic
61	C₄×C₂₃₉	—	2	2	no	—
62	C₄×C₂₃₉	—	2	2	no	—
63	C₄×C₂₃₉	—	4	2	no	—
64	C₄×C₂₃₉	—	4	2	no	—
65	C₄×C₂₃₉	C₄×Q₈	1	2	yes	—
66	C₄×C₂₃₉	#124	1	1	yes	characteristic
67	C₄×C₂₃₉	C₄⋊Q₈	1	1	yes	characteristic
68	Dic₂₃₉	—	4	4	no	—
69	C₂×C₂₃₉	#391	1	1	yes	characteristic
70	C₂×C₂₃₉	#416	1	1	yes	characteristic
71	C₂×C₂₃₉	#417	1	1	yes	characteristic
72	C₂×C₂₃₉	#478	1	2	yes	—
73	C₂×C₂₃₉	#479	1	2	yes	—
74	C₂₃₉	#1989	1	1	yes	characteristic, complemented, Sylow
75	#1989	—	239	1	no	maximal, Sylow
76	C₂×#125	D₂₃₉	1	1	yes	characteristic
77	C₂×C₄⋊C₈	—	239	1	no	—
78	C₄×C₄⋊C₄	—	239	1	no	—
79	#340	—	239	2	no	—
80	#340	—	239	2	no	—
81	C₂²×C₈	—	239	2	no	—
82	#125	—	2	2	no	—
83	#125	Dic₂₃₉	1	4	yes	complemented
84	C₂×C₄⋊C₄	—	2	1	no	—
85	C₂×C₄⋊C₄	—	2	1	no	—
86	C₂×C₄⋊C₄	—	239	2	no	—
87	C₂×C₄⋊C₄	C₂×D₂₃₉	1	1	yes	characteristic
88	C₂×C₄⋊C₄	C₂×D₂₃₉	1	1	yes	characteristic
89	C₂×C₄²	—	478	1	no	—
90	C₂×C₄²	C₂×D₂₃₉	1	1	yes	characteristic
91	C₄⋊C₈	—	478	2	no	—
92	#95	—	478	1	no	—
93	C₂²×C₄	—	2	1	no	—
94	C₂²×C₄	—	2	1	no	—
95	C₂²×C₄	—	478	2	no	—
96	C₂²×C₄	C₂²⋊D₂₃₉	1	1	yes	characteristic
97	C₂²×C₄	C₂²⋊D₂₃₉	1	1	yes	characteristic
98	C₂²×C₄	C₂²×D₂₃₉	1	1	yes	characteristic
99	C₂×C₈	—	478	2	no	—
100	C₂×C₈	—	478	4	no	—
101	C₄⋊C₄	—	2	4	no	—
102	C₄⋊C₄	—	2	4	no	—
103	C₄⋊C₄	—	2	1	no	—
104	C₄⋊C₄	—	2	1	no	—
105	C₄⋊C₄	—	478	2	no	—
106	C₄⋊C₄	—	239	4	no	—
107	C₄⋊C₄	C₂×Dic₂₃₉	1	2	yes	—
108	C₄⋊C₄	C₂×Dic₂₃₉	1	2	yes	—
109	C₄²	—	956	2	no	—
110	C₄²	C₂×Dic₂₃₉	1	2	yes	—
111	C₄²	C₂²⋊D₂₃₉	1	2	yes	—
112	C₂³	C₂×C₂²⋊D₂₃₉	1	1	yes	characteristic
113	C₂×C₄	—	2	2	no	—
114	C₂×C₄	—	2	1	no	—
115	C₂×C₄	—	2	1	no	—
116	C₂×C₄	—	2	2	no	—
117	C₂×C₄	—	2	2	no	—
118	C₂×C₄	—	4	2	no	—
119	C₂×C₄	—	4	1	no	—
120	C₂×C₄	—	4	1	no	—
121	C₂×C₄	—	956	2	no	—
122	C₂×C₄	—	956	2	no	—
123	C₂×C₄	—	478	4	no	—
124	C₂×C₄	C₂²⋊Dic₂₃₉	1	2	yes	—
125	C₂×C₄	C₂²⋊Dic₂₃₉	1	2	yes	—
126	C₂×C₄	C₂²×Dic₂₃₉	1	1	yes	characteristic
127	C₂×C₄	C₂×C₂²⋊D₂₃₉	1	1	yes	characteristic
128	C₂×C₄	Q₈×D₂₃₉	1	2	yes	—
129	C₂×C₄	#16838549	1	2	yes	—
130	C₈	—	956	2	no	—
131	C₂²	C₂×C₂²⋊Dic₂₃₉	1	1	yes	characteristic
132	C₂²	#21439936	1	2	yes	—
133	C₂²	#21439980	1	2	yes	—
134	C₂²	#21439985	1	1	yes	characteristic
135	C₂²	#21439990	1	1	yes	characteristic
136	C₄	—	2	2	no	—
137	C₄	—	2	2	no	—
138	C₄	—	4	2	no	—
139	C₄	—	4	2	no	—
140	C₄	—	956	4	no	—
141	C₄	Q₈×Dic₂₃₉	1	2	yes	—
142	C₄	#21439949	1	1	yes	characteristic
143	C₄	#21439958	1	1	yes	characteristic
144	C₂	#29504679	1	1	yes	characteristic
145	C₂	#29504768	1	1	yes	characteristic
146	C₂	#29504773	1	1	yes	characteristic
147	C₂	#29505069	1	2	yes	—
148	C₂	#29505076	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×C₄⋊(C₂₃₉⋊C₈)

C₂

yes

maximal, characteristic

C₄×C₄⋊Dic₂₃₉

C₂

yes

maximal, characteristic

#29504534

C₂

yes

maximal

#29504534

C₂

yes

maximal

C₂×#125×C₂₃₉

C₂

yes

maximal, characteristic

C₂²×C₂₃₉⋊C₈

C₂²

yes

—

C₂×C₄⋊Dic₂₃₉

C₂²

yes

—

C₂×C₄×Dic₂₃₉

—

C₄⋊(C₂₃₉⋊C₈)

—

C₂₃₉⋊#95

—

#125×C₂₃₉

—

#125×C₂₃₉

C₄

yes

complemented

C₂×C₄⋊C₄×C₂₃₉

—

C₂×C₄⋊C₄×C₂₃₉

—

C₂×C₄⋊C₄×C₂₃₉

C₂²

yes

characteristic

C₂×C₄⋊C₄×C₂₃₉

C₂²

yes

characteristic

C₂×C₄²×C₂₃₉

C₂²

yes

characteristic

C₂²×Dic₂₃₉

—

C₂×C₂₃₉⋊C₈

—

C₂×C₂₃₉⋊C₈

—

C₄⋊Dic₂₃₉

—

C₄⋊Dic₂₃₉

Q₈

yes

—

C₄×Dic₂₃₉

—

C₂²×C₄×C₂₃₉

—

C₂²×C₄×C₂₃₉

—

C₂²×C₄×C₂₃₉

D₄

yes

characteristic

C₂²×C₄×C₂₃₉

D₄

yes

characteristic

C₂²×C₄×C₂₃₉

C₂³

yes

characteristic

C₄⋊C₄×C₂₃₉

—

C₄⋊C₄×C₂₃₉

—

C₄⋊C₄×C₂₃₉

—

C₄⋊C₄×C₂₃₉

—

C₄⋊C₄×C₂₃₉

C₂×C₄

yes

—

C₄⋊C₄×C₂₃₉

C₂×C₄

yes

—

C₄²×C₂₃₉

C₂×C₄

yes

—

C₄²×C₂₃₉

D₄

yes

—

C₂×Dic₂₃₉

—

C₂×Dic₂₃₉

—

C₂×Dic₂₃₉

—

C₂₃₉⋊C₈

—

C₂³×C₂₃₉

C₂×D₄

yes

characteristic

C₂×C₄×C₂₃₉

—

C₂×C₄×C₂₃₉

—

C₂×C₄×C₂₃₉

—

C₂×C₄×C₂₃₉

—

C₂×C₄×C₂₃₉

—

C₂×C₄×C₂₃₉

—

C₂×C₄×C₂₃₉

—

C₂×C₄×C₂₃₉

—

C₂×C₄×C₂₃₉

C₂²⋊C₄

yes

—

C₂×C₄×C₂₃₉

C₂²⋊C₄

yes

—

C₂×C₄×C₂₃₉

C₂²×C₄

yes

characteristic

C₂×C₄×C₂₃₉

C₂×D₄

yes

characteristic

C₂×C₄×C₂₃₉

C₂×Q₈

yes

—

C₂×C₄×C₂₃₉

#41

yes

—

C₂²×C₂₃₉

C₂×C₂²⋊C₄

yes

characteristic

C₂²×C₂₃₉

C₂²⋊Q₈

yes

—

C₂²×C₂₃₉

#135

yes

—

C₂²×C₂₃₉

yes

C₂²×C₂₃₉

yes

C₄×C₂₃₉

—

C₄×C₂₃₉

—

C₄×C₂₃₉

—

C₄×C₂₃₉

—

C₄×C₂₃₉

C₄×Q₈

yes

—

C₄×C₂₃₉

#124

yes

characteristic

C₄×C₂₃₉

C₄⋊Q₈

yes

characteristic

Dic₂₃₉

—

C₂×C₂₃₉

yes

C₂×C₂₃₉

yes

C₂×C₂₃₉

yes

C₂×C₂₃₉

yes

—

C₂×C₂₃₉

#479

yes

—

C₂₃₉

#1989

yes

characteristic, complemented, Sylow

#1989

—

239

maximal, Sylow

C₂×#125

D₂₃₉

yes

characteristic

C₂×C₄⋊C₈

—

239

—

C₄×C₄⋊C₄

—

239

—

#340

—

239

—

#340

—

239

—

C₂²×C₈

—

239

—

#125

—

#125

Dic₂₃₉

yes

complemented

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

239

—

C₂×C₄⋊C₄

C₂×D₂₃₉

yes

characteristic

C₂×C₄⋊C₄

C₂×D₂₃₉

yes

characteristic

C₂×C₄²

—

478

—

C₂×C₄²

C₂×D₂₃₉

yes

characteristic

C₄⋊C₈

—

478

—

#95

—

478

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

478

—

C₂²×C₄

C₂²⋊D₂₃₉

yes

characteristic

C₂²×C₄

C₂²⋊D₂₃₉

yes

characteristic

C₂²×C₄

C₂²×D₂₃₉

yes

characteristic

C₂×C₈

—

478

—

100

C₂×C₈

—

478

—

101

C₄⋊C₄

—

102

C₄⋊C₄

—

103

C₄⋊C₄

—

104

C₄⋊C₄

—

105

C₄⋊C₄

—

478

—

106

C₄⋊C₄

—

239

—

107

C₄⋊C₄

C₂×Dic₂₃₉

yes

—

108

C₄⋊C₄

C₂×Dic₂₃₉

yes

—

109

C₄²

—

956

—

110

C₄²

C₂×Dic₂₃₉

yes

—

111

C₄²

C₂²⋊D₂₃₉

yes

—

112

C₂³

C₂×C₂²⋊D₂₃₉

yes

characteristic

113

C₂×C₄

—

114

C₂×C₄

—

115

C₂×C₄

—

116

C₂×C₄

—

117

C₂×C₄

—

118

C₂×C₄

—

119

C₂×C₄

—

120

C₂×C₄

—

121

C₂×C₄

—

956

—

122

C₂×C₄

—

956

—

123

C₂×C₄

—

478

—

124

C₂×C₄

C₂²⋊Dic₂₃₉

yes

—

125

C₂×C₄

C₂²⋊Dic₂₃₉

yes

—

126

C₂×C₄

C₂²×Dic₂₃₉

yes

characteristic

127

C₂×C₄

C₂×C₂²⋊D₂₃₉

yes

characteristic

128

C₂×C₄

Q₈×D₂₃₉

yes

—

129

C₂×C₄

#16838549

yes

—

130

C₈

—

956

—

131

C₂²

C₂×C₂²⋊Dic₂₃₉

yes

characteristic

132

C₂²

#21439936

yes

—

133

C₂²

#21439980

yes

—

134

C₂²

#21439985

yes

characteristic

135

C₂²

#21439990

yes

characteristic

136

C₄

—

137

C₄

—

138

C₄

—

139

C₄

—

140

C₄

—

956

—

141

C₄

Q₈×Dic₂₃₉

yes

—

142

C₄

#21439949

yes

characteristic

143

C₄

#21439958

yes

characteristic

144

C₂

#29504679

yes

characteristic

145

C₂

#29504768

yes

characteristic

146

C₂

#29504773

yes

characteristic

147

C₂

#29505069

yes

—

148

C₂

#29505076

yes

—

	subgroup	G/H
center	C₂³	C₂×C₂²⋊D₂₃₉
commutator	C₂×C₄×C₂₃₉	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₂₃₉
Fitting	C₂×#125×C₂₃₉	C₂
socle, prime radical	C₂³×C₂₃₉	C₂×D₄
cosocle	C₂²×C₄×C₂₃₉	C₂³
hypercenter	C₂×#125	D₂₃₉
hypocenter	C₂₃₉	#1989
LCS	G ▷ C₂×C₄×C₂₃₉ ▷ C₂×C₂₃₉ ▷ C₂₃₉
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×#125

subgroup

G/H

center

C₂³

C₂×C₂²⋊D₂₃₉

commutator

C₂×C₄×C₂₃₉

C₂²×C₄

Frattini

C₂²×C₄

C₂²×D₂₃₉

Fitting

C₂×#125×C₂₃₉

C₂

socle, prime radical

C₂³×C₂₃₉

C₂×D₄

cosocle

C₂²×C₄×C₂₃₉

C₂³

D₂₃₉

C₂₃₉

G ▷ C₂×C₄×C₂₃₉ ▷ C₂×C₂₃₉ ▷ C₂₃₉

UCS

C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×#125

N	H ≅ G/N	type
12. #125×C₂₃₉	C₄	semidirect
74. C₂₃₉	#1989	semidirect
83. #125	Dic₂₃₉	semidirect

H ≅ G/N

type

12. #125×C₂₃₉

C₄

semidirect

74. C₂₃₉

#1989

semidirect

83. #125

Dic₂₃₉

semidirect

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

subgroup	index
C₂×C₄⋊(C₂₃₉⋊C₈)	2
C₄×C₄⋊Dic₂₃₉	2
#29504534	2
C₂×#125×C₂₃₉	2
#1989	239

subgroup

index

C₂×C₄⋊(C₂₃₉⋊C₈)

C₄×C₄⋊Dic₂₃₉

#29504534

C₂×#125×C₂₃₉

#1989

239

Subgroups — #44485506

Conjugacy classes of subgroups

Complements

Maximal subgroups