Group #44486043 — subgroups

Subgroups — #44486043

Order 30592.

224 conjugacy classes of subgroups; only the 89 normal subgroups are shown.

normal subgroups only

This group has 224 conjugacy classes of subgroups — too many to draw. Only its 89 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×#21439894	C₂	1	1	yes	maximal, characteristic
2	C₂×Dic₂₃₉⋊C₈	C₂	1	1	yes	maximal, characteristic
3	#29504671	C₂	1	2	yes	maximal
4	#29504534	C₂	1	2	yes	maximal
5	#340×C₂₃₉	C₂	1	1	yes	maximal, characteristic
6	C₂²×C₂₃₉⋊C₈	C₂²	1	1	yes	characteristic
7	C₂×C₄⋊Dic₂₃₉	—	2	2	no	—
8	C₂×C₄⋊Dic₂₃₉	C₂²	1	2	yes	—
9	C₂×C₄×Dic₂₃₉	C₂²	1	1	yes	characteristic
10	#21439894	—	2	2	no	—
11	Dic₂₃₉⋊C₈	—	2	2	no	—
12	C₂₃₉⋊#95	—	2	2	no	—
13	C₂²×C₈×C₂₃₉	C₂²	1	1	yes	characteristic
14	C₂×C₄⋊C₄×C₂₃₉	C₂²	1	2	yes	—
15	C₂²×Dic₂₃₉	—	2	2	no	—
16	C₂²×Dic₂₃₉	D₄	1	1	yes	characteristic
17	C₂²×Dic₂₃₉	D₄	1	1	yes	characteristic
18	C₂×C₂₃₉⋊C₈	—	2	1	no	—
19	C₂×C₂₃₉⋊C₈	—	2	2	no	—
20	C₄⋊Dic₂₃₉	—	2	2	no	—
21	C₄⋊Dic₂₃₉	—	2	2	no	—
22	C₄⋊Dic₂₃₉	—	4	4	no	—
23	C₄×Dic₂₃₉	—	2	1	no	—
24	C₄×Dic₂₃₉	—	2	1	no	—
25	C₂²×C₄×C₂₃₉	—	2	2	no	—
26	C₂²×C₄×C₂₃₉	C₂³	1	1	yes	characteristic
27	C₂×C₈×C₂₃₉	—	2	2	no	—
28	C₂×C₈×C₂₃₉	D₄	1	2	yes	—
29	C₄⋊C₄×C₂₃₉	—	2	2	no	—
30	C₄⋊C₄×C₂₃₉	Q₈	1	4	yes	—
31	C₂×Dic₂₃₉	—	2	1	no	—
32	C₂×Dic₂₃₉	—	2	1	no	—
33	C₂×Dic₂₃₉	—	2	1	no	—
34	C₂×Dic₂₃₉	—	2	2	no	—
35	C₂×Dic₂₃₉	—	4	2	no	—
36	C₂×Dic₂₃₉	—	4	2	no	—
37	C₂×Dic₂₃₉	—	4	2	no	—
38	C₂×Dic₂₃₉	Q₁₆	1	2	yes	—
39	C₂₃₉⋊C₈	—	4	1	no	—
40	C₂³×C₂₃₉	C₂×D₄	1	1	yes	characteristic
41	C₂×C₄×C₂₃₉	—	2	4	no	—
42	C₂×C₄×C₂₃₉	—	4	2	no	—
43	C₂×C₄×C₂₃₉	—	4	2	no	—
44	C₂×C₄×C₂₃₉	C₂×D₄	1	1	yes	characteristic
45	C₂×C₄×C₂₃₉	C₂×Q₈	1	2	yes	—
46	C₂×C₄×C₂₃₉	#41	1	2	yes	—
47	C₂×C₄×C₂₃₉	#41	1	1	yes	characteristic
48	C₈×C₂₃₉	—	2	2	no	—
49	C₂²×C₂₃₉	#121	1	1	yes	characteristic
50	C₂²×C₂₃₉	C₂²⋊Q₈	1	2	yes	—
51	C₂²×C₂₃₉	C₂×Q₁₆	1	1	yes	characteristic
52	C₂²×C₂₃₉	#135	1	1	yes	characteristic
53	C₂²×C₂₃₉	AGL(1,ℤ/8)	1	2	yes	—
54	C₄×C₂₃₉	—	4	4	no	—
55	C₄×C₂₃₉	#123	1	1	yes	characteristic
56	C₄×C₂₃₉	#125	1	2	yes	—
57	C₄×C₂₃₉	C₄⋊Q₈	1	1	yes	characteristic
58	Dic₂₃₉	—	2	2	no	—
59	Dic₂₃₉	—	4	1	no	—
60	Dic₂₃₉	—	8	2	no	—
61	C₂×C₂₃₉	#398	1	1	yes	characteristic
62	C₂×C₂₃₉	#467	1	1	yes	characteristic
63	C₂×C₂₃₉	#469	1	1	yes	characteristic
64	C₂×C₂₃₉	#477	1	2	yes	—
65	C₂×C₂₃₉	#478	1	2	yes	—
66	C₂₃₉	#2087	1	1	yes	characteristic, complemented, Sylow
67	#2087	—	239	1	no	maximal, Sylow
68	C₂×#107	—	239	1	no	—
69	C₂×C₄⋊C₈	—	239	1	no	—
70	#389	—	239	2	no	—
71	#340	—	239	2	no	—
72	#340	D₂₃₉	1	1	yes	characteristic
73	C₂²×C₈	—	239	1	no	—
74	C₂²×C₈	C₂×D₂₃₉	1	1	yes	characteristic
75	C₂×C₄⋊C₄	—	478	2	no	—
76	C₂×C₄⋊C₄	—	239	2	no	—
77	C₂×C₄⋊C₄	C₂×D₂₃₉	1	2	yes	—
78	C₂×C₄²	—	239	1	no	—
79	#107	—	478	2	no	—
80	C₄⋊C₈	—	478	2	no	—
81	#95	—	478	2	no	—
82	C₂²×C₄	—	2	2	no	—
83	C₂²×C₄	—	478	2	no	—
84	C₂²×C₄	—	239	1	no	—
85	C₂²×C₄	—	239	1	no	—
86	C₂²×C₄	C₂²×D₂₃₉	1	1	yes	characteristic
87	C₂×C₈	—	2	2	no	—
88	C₂×C₈	—	478	1	no	—
89	C₂×C₈	—	478	2	no	—
90	C₂×C₈	D₉₅₆	1	2	yes	—
91	C₄⋊C₄	—	2	2	no	—
92	C₄⋊C₄	—	956	4	no	—
93	C₄⋊C₄	—	478	2	no	—
94	C₄⋊C₄	—	478	2	no	—
95	C₄⋊C₄	Dic₄₇₈	1	4	yes	—
96	C₄²	—	478	1	no	—
97	C₄²	—	478	1	no	—
98	C₂³	D₄×D₂₃₉	1	1	yes	characteristic
99	C₂×C₄	—	2	4	no	—
100	C₂×C₄	—	4	2	no	—
101	C₂×C₄	—	4	2	no	—
102	C₂×C₄	—	956	2	no	—
103	C₂×C₄	—	956	2	no	—
104	C₂×C₄	—	956	2	no	—
105	C₂×C₄	—	478	1	no	—
106	C₂×C₄	—	478	2	no	—
107	C₂×C₄	—	478	1	no	—
108	C₂×C₄	—	478	1	no	—
109	C₂×C₄	—	239	2	no	—
110	C₂×C₄	C₂×D₉₅₆	1	1	yes	characteristic
111	C₂×C₄	C₂×Dic₄₇₈	1	2	yes	—
112	C₂×C₄	#16838547	1	2	yes	—
113	C₂×C₄	#16838549	1	1	yes	characteristic
114	C₈	—	2	2	no	—
115	C₈	—	956	1	no	—
116	C₂²	#21439935	1	1	yes	characteristic
117	C₂²	#21439937	1	2	yes	—
118	C₂²	Q₁₆×D₂₃₉	1	1	yes	characteristic
119	C₂²	#21439979	1	1	yes	characteristic
120	C₂²	#21439987	1	2	yes	—
121	C₄	—	4	4	no	—
122	C₄	—	1912	2	no	—
123	C₄	—	956	1	no	—
124	C₄	—	478	2	no	—
125	C₄	#21439944	1	1	yes	characteristic
126	C₄	#21439951	1	2	yes	—
127	C₄	#21439960	1	1	yes	characteristic
128	C₂	#29504704	1	1	yes	characteristic
129	C₂	#29505014	1	1	yes	characteristic
130	C₂	#29505026	1	1	yes	characteristic
131	C₂	#29505065	1	2	yes	—
132	C₂	#29505070	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#21439894

C₂

yes

maximal, characteristic

C₂×Dic₂₃₉⋊C₈

C₂

yes

maximal, characteristic

#29504671

C₂

yes

maximal

#29504534

C₂

yes

maximal

#340×C₂₃₉

C₂

yes

maximal, characteristic

C₂²×C₂₃₉⋊C₈

C₂²

yes

characteristic

C₂×C₄⋊Dic₂₃₉

—

C₂×C₄⋊Dic₂₃₉

C₂²

yes

—

C₂×C₄×Dic₂₃₉

C₂²

yes

characteristic

#21439894

—

Dic₂₃₉⋊C₈

—

C₂₃₉⋊#95

—

C₂²×C₈×C₂₃₉

C₂²

yes

characteristic

C₂×C₄⋊C₄×C₂₃₉

C₂²

yes

—

C₂²×Dic₂₃₉

—

C₂²×Dic₂₃₉

D₄

yes

characteristic

C₂²×Dic₂₃₉

D₄

yes

characteristic

C₂×C₂₃₉⋊C₈

—

C₂×C₂₃₉⋊C₈

—

C₄⋊Dic₂₃₉

—

C₄⋊Dic₂₃₉

—

C₄⋊Dic₂₃₉

—

C₄×Dic₂₃₉

—

C₄×Dic₂₃₉

—

C₂²×C₄×C₂₃₉

—

C₂²×C₄×C₂₃₉

C₂³

yes

characteristic

C₂×C₈×C₂₃₉

—

C₂×C₈×C₂₃₉

D₄

yes

—

C₄⋊C₄×C₂₃₉

—

C₄⋊C₄×C₂₃₉

Q₈

yes

—

C₂×Dic₂₃₉

—

C₂×Dic₂₃₉

—

C₂×Dic₂₃₉

—

C₂×Dic₂₃₉

—

C₂×Dic₂₃₉

—

C₂×Dic₂₃₉

—

C₂×Dic₂₃₉

—

C₂×Dic₂₃₉

Q₁₆

yes

—

C₂₃₉⋊C₈

—

C₂³×C₂₃₉

C₂×D₄

yes

characteristic

C₂×C₄×C₂₃₉

—

C₂×C₄×C₂₃₉

—

C₂×C₄×C₂₃₉

—

C₂×C₄×C₂₃₉

C₂×D₄

yes

characteristic

C₂×C₄×C₂₃₉

C₂×Q₈

yes

—

C₂×C₄×C₂₃₉

#41

yes

—

C₂×C₄×C₂₃₉

#41

yes

characteristic

C₈×C₂₃₉

—

C₂²×C₂₃₉

#121

yes

characteristic

C₂²×C₂₃₉

C₂²⋊Q₈

yes

—

C₂²×C₂₃₉

C₂×Q₁₆

yes

C₂²×C₂₃₉

yes

C₂²×C₂₃₉

yes

—

C₄×C₂₃₉

—

C₄×C₂₃₉

#123

yes

characteristic

C₄×C₂₃₉

#125

yes

—

C₄×C₂₃₉

C₄⋊Q₈

yes

characteristic

Dic₂₃₉

—

Dic₂₃₉

—

Dic₂₃₉

—

C₂×C₂₃₉

yes

C₂×C₂₃₉

yes

C₂×C₂₃₉

yes

C₂×C₂₃₉

yes

—

C₂×C₂₃₉

#478

yes

—

C₂₃₉

#2087

yes

characteristic, complemented, Sylow

#2087

—

239

maximal, Sylow

C₂×#107

—

239

—

C₂×C₄⋊C₈

—

239

—

#389

—

239

—

#340

—

239

—

#340

D₂₃₉

yes

characteristic

C₂²×C₈

—

239

—

C₂²×C₈

C₂×D₂₃₉

yes

characteristic

C₂×C₄⋊C₄

—

478

—

C₂×C₄⋊C₄

—

239

—

C₂×C₄⋊C₄

C₂×D₂₃₉

yes

—

C₂×C₄²

—

239

—

#107

—

478

—

C₄⋊C₈

—

478

—

#95

—

478

—

C₂²×C₄

—

C₂²×C₄

—

478

—

C₂²×C₄

—

239

—

C₂²×C₄

—

239

—

C₂²×C₄

C₂²×D₂₃₉

yes

characteristic

C₂×C₈

—

C₂×C₈

—

478

—

C₂×C₈

—

478

—

C₂×C₈

D₉₅₆

yes

—

C₄⋊C₄

—

C₄⋊C₄

—

956

—

C₄⋊C₄

—

478

—

C₄⋊C₄

—

478

—

C₄⋊C₄

Dic₄₇₈

yes

—

C₄²

—

478

—

C₄²

—

478

—

C₂³

D₄×D₂₃₉

yes

characteristic

C₂×C₄

—

100

C₂×C₄

—

101

C₂×C₄

—

102

C₂×C₄

—

956

—

103

C₂×C₄

—

956

—

104

C₂×C₄

—

956

—

105

C₂×C₄

—

478

—

106

C₂×C₄

—

478

—

107

C₂×C₄

—

478

—

108

C₂×C₄

—

478

—

109

C₂×C₄

—

239

—

110

C₂×C₄

C₂×D₉₅₆

yes

characteristic

111

C₂×C₄

C₂×Dic₄₇₈

yes

—

112

C₂×C₄

#16838547

yes

—

113

C₂×C₄

#16838549

yes

characteristic

114

C₈

—

115

C₈

—

956

—

116

C₂²

#21439935

yes

characteristic

117

C₂²

#21439937

yes

—

118

C₂²

Q₁₆×D₂₃₉

yes

characteristic

119

C₂²

#21439979

yes

characteristic

120

C₂²

#21439987

yes

—

121

C₄

—

122

C₄

—

1912

—

123

C₄

—

956

—

124

C₄

—

478

—

125

C₄

#21439944

yes

characteristic

126

C₄

#21439951

yes

—

127

C₄

#21439960

yes

characteristic

128

C₂

#29504704

yes

characteristic

129

C₂

#29505014

yes

characteristic

130

C₂

#29505026

yes

characteristic

131

C₂

#29505065

yes

—

132

C₂

#29505070

yes

—

	subgroup	G/H
center	C₂³	D₄×D₂₃₉
commutator, cosocle	C₂²×C₄×C₂₃₉	C₂³
Frattini	C₂²×C₄	C₂²×D₂₃₉
Fitting	#340×C₂₃₉	C₂
socle, prime radical	C₂³×C₂₃₉	C₂×D₄
hypercenter	#340	D₂₃₉
hypocenter	C₂₃₉	#2087
LCS	G ▷ C₂²×C₄×C₂₃₉ ▷ C₂×C₂₃₉ ▷ C₂₃₉
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄ ◁ #340

subgroup

G/H

center

C₂³

D₄×D₂₃₉

commutator, cosocle

C₂²×C₄×C₂₃₉

C₂³

Frattini

C₂²×C₄

C₂²×D₂₃₉

Fitting

#340×C₂₃₉

C₂

socle, prime radical

C₂³×C₂₃₉

C₂×D₄

D₂₃₉

C₂₃₉

G ▷ C₂²×C₄×C₂₃₉ ▷ C₂×C₂₃₉ ▷ C₂₃₉

UCS

C₁ ◁ C₂³ ◁ C₂²×C₄ ◁ #340

N	H ≅ G/N	type
66. C₂₃₉	#2087	semidirect

H ≅ G/N

type

66. C₂₃₉

#2087

semidirect

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

subgroup	index
C₂×#21439894	2
C₂×Dic₂₃₉⋊C₈	2
#29504671	2
#29504534	2
#340×C₂₃₉	2
#2087	239

subgroup

index

C₂×Dic₂₃₉⋊C₈

#340×C₂₃₉

239

Subgroups — #44486043

Conjugacy classes of subgroups

Complements

Maximal subgroups