Group #44480236 — subgroups

Subgroups — #1466×C₂₃₉ #44480236

Order 30592.

300 conjugacy classes of subgroups; only the 100 normal subgroups are shown.

normal subgroups only

This group has 300 conjugacy classes of subgroups — too many to draw. Only its 100 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×D₄×C₂₃₉	C₂	1	1	yes	maximal, characteristic
2	#432×C₂₃₉	C₂	1	2	yes	maximal, complemented
3	#323×C₂₃₉	C₂	1	4	yes	maximal, complemented
4	C₂²×D₄×C₂₃₉	C₄	1	1	yes	characteristic
5	C₂³×C₄×C₂₃₉	C₄	1	2	yes	—
6	C₄×D₄×C₂₃₉	—	2	2	no	—
7	C₄×D₄×C₂₃₉	—	2	2	no	—
8	C₂×C₄⋊C₄×C₂₃₉	C₄	1	1	yes	characteristic
9	C₂×C₂²⋊C₄×C₂₃₉	C₂²	1	2	yes	—
10	C₂×C₄²×C₂₃₉	C₂²	1	1	yes	characteristic
11	C₄⋊C₈×C₂₃₉	—	2	2	no	—
12	C₂²⋊C₈×C₂₃₉	C₂²	1	4	yes	—
13	#97×C₂₃₉	—	2	2	no	—
14	C₂⁴×C₂₃₉	C₂×C₄	1	2	yes	—
15	C₂×D₄×C₂₃₉	—	2	4	no	—
16	C₂×D₄×C₂₃₉	—	2	2	no	—
17	C₂²×C₄×C₂₃₉	—	2	2	no	—
18	C₂²×C₄×C₂₃₉	—	2	2	no	—
19	C₂²×C₄×C₂₃₉	—	4	4	no	—
20	C₂²×C₄×C₂₃₉	C₂×C₄	1	1	yes	characteristic
21	C₂²×C₄×C₂₃₉	C₂×C₄	1	1	yes	characteristic
22	C₂²×C₄×C₂₃₉	C₂×C₄	1	2	yes	—
23	C₂²×C₄×C₂₃₉	C₂³	1	1	yes	characteristic
24	C₂×C₈×C₂₃₉	—	2	4	no	—
25	C₄⋊C₄×C₂₃₉	—	2	2	no	—
26	C₂²⋊C₄×C₂₃₉	—	2	4	no	—
27	C₄²×C₂₃₉	D₄	1	2	yes	—
28	C₄²×C₂₃₉	D₄	1	2	yes	—
29	C₂³×C₂₃₉	—	2	2	no	—
30	C₂³×C₂₃₉	—	2	2	no	—
31	C₂³×C₂₃₉	—	4	2	no	—
32	C₂³×C₂₃₉	—	4	2	no	—
33	C₂³×C₂₃₉	M₁₆	1	4	yes	—
34	C₂³×C₂₃₉	C₂²×C₄	1	1	yes	characteristic
35	D₄×C₂₃₉	—	2	8	no	—
36	C₂×C₄×C₂₃₉	—	2	2	no	—
37	C₂×C₄×C₂₃₉	—	2	2	no	—
38	C₂×C₄×C₂₃₉	—	2	2	no	—
39	C₂×C₄×C₂₃₉	—	4	1	no	—
40	C₂×C₄×C₂₃₉	—	4	4	no	—
41	C₂×C₄×C₂₃₉	—	4	4	no	—
42	C₂×C₄×C₂₃₉	—	4	2	no	—
43	C₂×C₄×C₂₃₉	C₂²⋊C₄	1	2	yes	—
44	C₂×C₄×C₂₃₉	C₂²⋊C₄	1	1	yes	characteristic
45	C₂×C₄×C₂₃₉	C₂²⋊C₄	1	1	yes	characteristic
46	C₂×C₄×C₂₃₉	C₂×D₄	1	2	yes	—
47	C₈×C₂₃₉	—	4	4	no	—
48	C₂²×C₂₃₉	—	2	1	no	—
49	C₂²×C₂₃₉	—	2	1	no	—
50	C₂²×C₂₃₉	—	2	4	no	—
51	C₂²×C₂₃₉	—	4	4	no	—
52	C₂²×C₂₃₉	—	4	4	no	—
53	C₂²×C₂₃₉	—	4	2	no	—
54	C₂²×C₂₃₉	—	4	2	no	—
55	C₂²×C₂₃₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
56	C₂²×C₂₃₉	C₂×M₁₆	1	2	yes	—
57	C₄×C₂₃₉	—	2	2	no	—
58	C₄×C₂₃₉	—	2	2	no	—
59	C₄×C₂₃₉	—	2	1	no	—
60	C₄×C₂₃₉	—	4	2	no	—
61	C₄×C₂₃₉	#100	1	2	yes	—
62	C₂×C₂₃₉	—	2	2	no	—
63	C₂×C₂₃₉	—	4	4	no	—
64	C₂×C₂₃₉	#407	1	1	yes	characteristic
65	C₂×C₂₃₉	#410	1	1	yes	characteristic
66	C₂×C₂₃₉	C₂×#100	1	1	yes	characteristic
67	C₂₃₉	#1466	1	1	yes	characteristic, direct product factor, Sylow
68	#1466	C₂₃₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
69	C₂×C₄×D₄	C₂×C₂₃₉	1	1	yes	characteristic
70	#432	C₂×C₂₃₉	1	2	yes	complemented
71	#323	C₂×C₂₃₉	1	4	yes	complemented
72	C₂²×D₄	C₄×C₂₃₉	1	1	yes	characteristic
73	C₂³×C₄	C₄×C₂₃₉	1	2	yes	—
74	C₄×D₄	—	2	2	no	—
75	C₄×D₄	—	2	2	no	—
76	C₂×C₄⋊C₄	C₄×C₂₃₉	1	1	yes	characteristic
77	C₂×C₂²⋊C₄	C₂²×C₂₃₉	1	2	yes	—
78	C₂×C₄²	C₂²×C₂₃₉	1	1	yes	characteristic
79	C₄⋊C₈	—	2	2	no	—
80	C₂²⋊C₈	C₂²×C₂₃₉	1	4	yes	—
81	#97	—	2	2	no	—
82	C₂⁴	C₂×C₄×C₂₃₉	1	2	yes	—
83	C₂×D₄	—	2	4	no	—
84	C₂×D₄	—	2	2	no	—
85	C₂²×C₄	—	2	2	no	—
86	C₂²×C₄	—	2	2	no	—
87	C₂²×C₄	—	4	4	no	—
88	C₂²×C₄	C₂×C₄×C₂₃₉	1	1	yes	characteristic
89	C₂²×C₄	C₂×C₄×C₂₃₉	1	1	yes	characteristic
90	C₂²×C₄	C₂×C₄×C₂₃₉	1	2	yes	—
91	C₂²×C₄	C₂³×C₂₃₉	1	1	yes	characteristic
92	C₂×C₈	—	2	4	no	—
93	C₄⋊C₄	—	2	2	no	—
94	C₂²⋊C₄	—	2	4	no	—
95	C₄²	D₄×C₂₃₉	1	2	yes	—
96	C₄²	D₄×C₂₃₉	1	2	yes	—
97	C₂³	—	2	2	no	—
98	C₂³	—	2	2	no	—
99	C₂³	—	4	2	no	—
100	C₂³	—	4	2	no	—
101	C₂³	M₁₆×C₂₃₉	1	4	yes	—
102	C₂³	C₂²×C₄×C₂₃₉	1	1	yes	characteristic
103	D₄	—	2	8	no	—
104	C₂×C₄	—	2	2	no	—
105	C₂×C₄	—	2	2	no	—
106	C₂×C₄	—	2	2	no	—
107	C₂×C₄	—	4	1	no	—
108	C₂×C₄	—	4	4	no	—
109	C₂×C₄	—	4	4	no	—
110	C₂×C₄	—	4	2	no	—
111	C₂×C₄	C₂²⋊C₄×C₂₃₉	1	2	yes	—
112	C₂×C₄	C₂²⋊C₄×C₂₃₉	1	1	yes	characteristic
113	C₂×C₄	C₂²⋊C₄×C₂₃₉	1	1	yes	characteristic
114	C₂×C₄	C₂×D₄×C₂₃₉	1	2	yes	—
115	C₈	—	4	4	no	—
116	C₂²	—	2	1	no	—
117	C₂²	—	2	1	no	—
118	C₂²	—	2	4	no	—
119	C₂²	—	4	4	no	—
120	C₂²	—	4	4	no	—
121	C₂²	—	4	2	no	—
122	C₂²	—	4	2	no	—
123	C₂²	C₂×C₂²⋊C₄×C₂₃₉	1	1	yes	characteristic
124	C₂²	C₂×M₁₆×C₂₃₉	1	2	yes	—
125	C₄	—	2	2	no	—
126	C₄	—	2	2	no	—
127	C₄	—	2	1	no	—
128	C₄	—	4	2	no	—
129	C₄	#100×C₂₃₉	1	2	yes	—
130	C₂	—	2	2	no	—
131	C₂	—	4	4	no	—
132	C₂	#407×C₂₃₉	1	1	yes	characteristic
133	C₂	#410×C₂₃₉	1	1	yes	characteristic
134	C₂	C₂×#100×C₂₃₉	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×C₄×D₄×C₂₃₉

C₂

yes

maximal, characteristic

#432×C₂₃₉

C₂

yes

maximal, complemented

#323×C₂₃₉

C₂

yes

maximal, complemented

C₂²×D₄×C₂₃₉

C₄

yes

characteristic

C₂³×C₄×C₂₃₉

C₄

yes

—

C₄×D₄×C₂₃₉

—

C₄×D₄×C₂₃₉

—

C₂×C₄⋊C₄×C₂₃₉

C₄

yes

characteristic

C₂×C₂²⋊C₄×C₂₃₉

C₂²

yes

—

C₂×C₄²×C₂₃₉

C₂²

yes

characteristic

C₄⋊C₈×C₂₃₉

—

C₂²⋊C₈×C₂₃₉

C₂²

yes

—

#97×C₂₃₉

—

C₂⁴×C₂₃₉

C₂×C₄

yes

—

C₂×D₄×C₂₃₉

—

C₂×D₄×C₂₃₉

—

C₂²×C₄×C₂₃₉

—

C₂²×C₄×C₂₃₉

—

C₂²×C₄×C₂₃₉

—

C₂²×C₄×C₂₃₉

C₂×C₄

yes

characteristic

C₂²×C₄×C₂₃₉

C₂×C₄

yes

characteristic

C₂²×C₄×C₂₃₉

C₂×C₄

yes

—

C₂²×C₄×C₂₃₉

C₂³

yes

characteristic

C₂×C₈×C₂₃₉

—

C₄⋊C₄×C₂₃₉

—

C₂²⋊C₄×C₂₃₉

—

C₄²×C₂₃₉

D₄

yes

—

C₄²×C₂₃₉

D₄

yes

—

C₂³×C₂₃₉

—

C₂³×C₂₃₉

—

C₂³×C₂₃₉

—

C₂³×C₂₃₉

—

C₂³×C₂₃₉

M₁₆

yes

—

C₂³×C₂₃₉

C₂²×C₄

yes

characteristic

D₄×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

characteristic

C₂×C₄×C₂₃₉

C₂²⋊C₄

yes

characteristic

C₂×C₄×C₂₃₉

C₂×D₄

yes

—

C₈×C₂₃₉

—

C₂²×C₂₃₉

—

C₂²×C₂₃₉

—

C₂²×C₂₃₉

—

C₂²×C₂₃₉

—

C₂²×C₂₃₉

—

C₂²×C₂₃₉

—

C₂²×C₂₃₉

—

C₂²×C₂₃₉

C₂×C₂²⋊C₄

yes

characteristic

C₂²×C₂₃₉

C₂×M₁₆

yes

—

C₄×C₂₃₉

—

C₄×C₂₃₉

—

C₄×C₂₃₉

—

C₄×C₂₃₉

—

C₄×C₂₃₉

#100

yes

—

C₂×C₂₃₉

—

C₂×C₂₃₉

—

C₂×C₂₃₉

yes

C₂×C₂₃₉

yes

C₂×C₂₃₉

yes

C₂₃₉

yes

characteristic, direct product factor, Sylow

#1466

C₂₃₉

yes

maximal, characteristic, direct product factor, Sylow

C₂×C₄×D₄

C₂×C₂₃₉

yes

characteristic

#432

C₂×C₂₃₉

yes

complemented

#323

C₂×C₂₃₉

yes

complemented

C₂²×D₄

C₄×C₂₃₉

yes

characteristic

C₂³×C₄

C₄×C₂₃₉

yes

—

C₄×D₄

—

C₄×D₄

—

C₂×C₄⋊C₄

C₄×C₂₃₉

yes

characteristic

C₂×C₂²⋊C₄

C₂²×C₂₃₉

yes

—

C₂×C₄²

C₂²×C₂₃₉

yes

characteristic

C₄⋊C₈

—

C₂²⋊C₈

C₂²×C₂₃₉

yes

—

#97

—

C₂⁴

C₂×C₄×C₂₃₉

yes

—

C₂×D₄

—

C₂×D₄

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

C₂×C₄×C₂₃₉

yes

characteristic

C₂²×C₄

C₂×C₄×C₂₃₉

yes

characteristic

C₂²×C₄

C₂×C₄×C₂₃₉

yes

—

C₂²×C₄

C₂³×C₂₃₉

yes

characteristic

C₂×C₈

—

C₄⋊C₄

—

C₂²⋊C₄

—

C₄²

D₄×C₂₃₉

yes

—

C₄²

D₄×C₂₃₉

yes

—

C₂³

—

C₂³

—

C₂³

—

100

C₂³

—

101

C₂³

M₁₆×C₂₃₉

yes

—

102

C₂³

C₂²×C₄×C₂₃₉

yes

characteristic

103

D₄

—

104

C₂×C₄

—

105

C₂×C₄

—

106

C₂×C₄

—

107

C₂×C₄

—

108

C₂×C₄

—

109

C₂×C₄

—

110

C₂×C₄

—

111

C₂×C₄

C₂²⋊C₄×C₂₃₉

yes

—

112

C₂×C₄

C₂²⋊C₄×C₂₃₉

yes

characteristic

113

C₂×C₄

C₂²⋊C₄×C₂₃₉

yes

characteristic

114

C₂×C₄

C₂×D₄×C₂₃₉

yes

—

115

C₈

—

116

C₂²

—

117

C₂²

—

118

C₂²

—

119

C₂²

—

120

C₂²

—

121

C₂²

—

122

C₂²

—

123

C₂²

C₂×C₂²⋊C₄×C₂₃₉

yes

characteristic

124

C₂²

C₂×M₁₆×C₂₃₉

yes

—

125

C₄

—

126

C₄

—

127

C₄

—

128

C₄

—

129

C₄

#100×C₂₃₉

yes

—

130

C₂

—

131

C₂

—

132

C₂

#407×C₂₃₉

yes

characteristic

133

C₂

#410×C₂₃₉

yes

characteristic

134

C₂

C₂×#100×C₂₃₉

yes

characteristic

	subgroup	G/H
center, socle	C₂²×C₂₃₉	C₂×C₂²⋊C₄
commutator	C₂³	C₂²×C₄×C₂₃₉
Frattini, cosocle	C₂²×C₄	C₂³×C₂₃₉
prime radical	C₂²×D₄×C₂₃₉	C₄
LCS	G ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₂₃₉ ◁ C₂×C₄²×C₂₃₉ ◁ G

subgroup

G/H

center, socle

C₂²×C₂₃₉

C₂×C₂²⋊C₄

commutator

C₂³

C₂²×C₄×C₂₃₉

Frattini, cosocle

C₂²×C₄

C₂³×C₂₃₉

prime radical

C₂²×D₄×C₂₃₉

C₄

LCS

G ▷ C₂³ ▷ C₂ ▷ C₁

UCS

C₁ ◁ C₂²×C₂₃₉ ◁ C₂×C₄²×C₂₃₉ ◁ G

N	H ≅ G/N	type
2. #432×C₂₃₉	C₂	semidirect
3. #323×C₂₃₉	C₂	semidirect
67. C₂₃₉	#1466	direct (N × H)
68. #1466	C₂₃₉	direct (N × H)
70. #432	C₂×C₂₃₉	semidirect
71. #323	C₂×C₂₃₉	semidirect

H ≅ G/N

type

2. #432×C₂₃₉

C₂

semidirect

3. #323×C₂₃₉

C₂

semidirect

67. C₂₃₉

#1466

direct (N × H)

68. #1466

C₂₃₉

direct (N × H)

70. #432

C₂×C₂₃₉

semidirect

71. #323

C₂×C₂₃₉

semidirect

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

subgroup	index
C₂×C₄×D₄×C₂₃₉	2
#432×C₂₃₉	2
#323×C₂₃₉	2
#1466	239

subgroup

index

C₂×C₄×D₄×C₂₃₉

#432×C₂₃₉

#323×C₂₃₉

#1466

239

Subgroups — #1466×C₂₃₉ #44480236

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — #1466×C239 #44480236

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — #1466×C₂₃₉ #44480236