Group #44485485 — subgroups

Subgroups — #44485485

Order 30592.

292 conjugacy classes of subgroups; only the 145 normal subgroups are shown.

normal subgroups only

This group has 292 conjugacy classes of subgroups — too many to draw. Only its 145 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	C₂×C₄×C₂₃₉⋊C₈	C₂	1	1	yes	maximal, characteristic
3	#29504527	C₂	1	4	yes	maximal
4	C₂×C₄×Q₈×C₂₃₉	C₂	1	1	yes	maximal, characteristic
5	C₂²×C₂₃₉⋊C₈	C₂²	1	2	yes	—
6	C₂²×C₂₃₉⋊C₈	C₂²	1	2	yes	—
7	C₄⋊(C₂₃₉⋊C₈)	—	2	2	no	—
8	C₄×C₂₃₉⋊C₈	—	2	2	no	—
9	C₂²×Q₈×C₂₃₉	C₄	1	1	yes	characteristic
10	C₄×Q₈×C₂₃₉	—	2	2	no	—
11	C₄×Q₈×C₂₃₉	—	2	2	no	—
12	C₂×C₄⋊C₄×C₂₃₉	C₄	1	1	yes	characteristic
13	C₂×C₄⋊C₄×C₂₃₉	C₄	1	2	yes	—
14	C₂×C₄²×C₂₃₉	C₂²	1	2	yes	—
15	C₂×C₄²×C₂₃₉	C₂²	1	1	yes	characteristic
16	C₂×C₂₃₉⋊C₈	—	2	2	no	—
17	C₂×C₂₃₉⋊C₈	—	2	2	no	—
18	C₂×C₂₃₉⋊C₈	—	2	2	no	—
19	C₂×C₂₃₉⋊C₈	—	2	4	no	—
20	C₂×C₂₃₉⋊C₈	Q₈	1	4	yes	complemented
21	C₂×Q₈×C₂₃₉	—	2	2	no	—
22	C₂×Q₈×C₂₃₉	—	2	2	no	—
23	C₂×Q₈×C₂₃₉	C₈	1	4	yes	complemented
24	C₂²×C₄×C₂₃₉	C₂×C₄	1	2	yes	—
25	C₂²×C₄×C₂₃₉	C₂×C₄	1	1	yes	characteristic
26	C₂²×C₄×C₂₃₉	C₂×C₄	1	2	yes	—
27	C₂²×C₄×C₂₃₉	C₂×C₄	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	2	no	—
31	C₄²×C₂₃₉	—	2	2	no	—
32	C₄²×C₂₃₉	—	2	2	no	—
33	C₄²×C₂₃₉	D₄	1	2	yes	—
34	C₄²×C₂₃₉	D₄	1	2	yes	—
35	C₂₃₉⋊C₈	—	2	4	no	—
36	C₂₃₉⋊C₈	—	4	2	no	—
37	C₂³×C₂₃₉	C₂²×C₄	1	1	yes	characteristic
38	Q₈×C₂₃₉	—	2	8	no	—
39	C₂×C₄×C₂₃₉	—	2	2	no	—
40	C₂×C₄×C₂₃₉	—	2	2	no	—
41	C₂×C₄×C₂₃₉	—	2	2	no	—
42	C₂×C₄×C₂₃₉	—	2	2	no	—
43	C₂×C₄×C₂₃₉	—	2	4	no	—
44	C₂×C₄×C₂₃₉	C₂²⋊C₄	1	2	yes	—
45	C₂×C₄×C₂₃₉	C₂²⋊C₄	1	2	yes	—
46	C₂×C₄×C₂₃₉	C₂×C₈	1	4	yes	—
47	C₂×C₄×C₂₃₉	C₂×C₈	1	2	yes	—
48	C₂×C₄×C₂₃₉	M₁₆	1	2	yes	—
49	C₂×C₄×C₂₃₉	C₂×D₄	1	1	yes	characteristic
50	C₂×C₄×C₂₃₉	C₂×D₄	1	1	yes	characteristic
51	C₂×C₄×C₂₃₉	C₂×Q₈	1	2	yes	—
52	C₂×C₄×C₂₃₉	#41	1	2	yes	—
53	C₂²×C₂₃₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
54	C₂²×C₂₃₉	C₄×Q₈	1	2	yes	—
55	C₂²×C₂₃₉	C₂²×C₈	1	1	yes	characteristic
56	C₂²×C₂₃₉	C₂×M₁₆	1	1	yes	characteristic
57	C₂²×C₂₃₉	#131	1	2	yes	—
58	C₄×C₂₃₉	—	2	4	no	—
59	C₄×C₂₃₉	—	2	2	no	—
60	C₄×C₂₃₉	—	4	2	no	—
61	C₄×C₂₃₉	C₂²⋊C₈	1	4	yes	—
62	C₄×C₂₃₉	C₂²⋊Q₈	1	2	yes	—
63	C₄×C₂₃₉	#124	1	1	yes	characteristic
64	C₄×C₂₃₉	C₄⋊Q₈	1	1	yes	characteristic
65	C₂×C₂₃₉	#391	1	1	yes	characteristic
66	C₂×C₂₃₉	C₂×C₂²⋊C₈	1	1	yes	characteristic
67	C₂×C₂₃₉	#408	1	1	yes	characteristic
68	C₂×C₂₃₉	C₈×Q₈	1	2	yes	—
69	C₂×C₂₃₉	#446	1	2	yes	—
70	C₂₃₉	#1984	1	1	yes	characteristic, complemented, Sylow
71	#1984	—	239	1	no	maximal, Sylow
72	C₂×C₄×Q₈	D₂₃₉	1	1	yes	characteristic
73	C₂×C₄⋊C₈	—	239	1	no	—
74	C₂×C₄×C₈	—	239	1	no	—
75	#336	—	239	4	no	—
76	C₂²×Q₈	Dic₂₃₉	1	1	yes	characteristic
77	C₂²×C₈	—	239	2	no	—
78	C₂²×C₈	—	239	2	no	—
79	C₄×Q₈	—	2	2	no	—
80	C₄×Q₈	—	2	2	no	—
81	C₂×C₄⋊C₄	Dic₂₃₉	1	1	yes	characteristic
82	C₂×C₄⋊C₄	Dic₂₃₉	1	2	yes	—
83	C₂×C₄²	C₂×D₂₃₉	1	2	yes	—
84	C₂×C₄²	C₂×D₂₃₉	1	1	yes	characteristic
85	C₄⋊C₈	—	478	2	no	—
86	C₄×C₈	—	478	2	no	—
87	C₂×Q₈	—	2	2	no	—
88	C₂×Q₈	—	2	2	no	—
89	C₂×Q₈	C₂₃₉⋊C₈	1	4	yes	complemented
90	C₂²×C₄	C₂×Dic₂₃₉	1	2	yes	—
91	C₂²×C₄	C₂×Dic₂₃₉	1	1	yes	characteristic
92	C₂²×C₄	C₂×Dic₂₃₉	1	2	yes	—
93	C₂²×C₄	C₂×Dic₂₃₉	1	1	yes	characteristic
94	C₂²×C₄	C₂²×D₂₃₉	1	1	yes	characteristic
95	C₂×C₈	—	478	2	no	—
96	C₂×C₈	—	478	4	no	—
97	C₂×C₈	—	478	2	no	—
98	C₂×C₈	—	478	2	no	—
99	C₂×C₈	—	239	4	no	—
100	C₄⋊C₄	—	2	4	no	—
101	C₄⋊C₄	—	2	2	no	—
102	C₄²	—	2	2	no	—
103	C₄²	—	2	2	no	—
104	C₄²	C₂²⋊D₂₃₉	1	2	yes	—
105	C₄²	C₂²⋊D₂₃₉	1	2	yes	—
106	C₂³	C₂²×Dic₂₃₉	1	1	yes	characteristic
107	Q₈	—	2	8	no	—
108	C₂×C₄	—	2	2	no	—
109	C₂×C₄	—	2	2	no	—
110	C₂×C₄	—	2	2	no	—
111	C₂×C₄	—	2	2	no	—
112	C₂×C₄	—	2	4	no	—
113	C₂×C₄	C₂²⋊Dic₂₃₉	1	2	yes	—
114	C₂×C₄	C₂²⋊Dic₂₃₉	1	2	yes	—
115	C₂×C₄	C₂×C₂₃₉⋊C₈	1	4	yes	—
116	C₂×C₄	C₂×C₂₃₉⋊C₈	1	2	yes	—
117	C₂×C₄	#16838532	1	2	yes	—
118	C₂×C₄	C₂×C₂²⋊D₂₃₉	1	1	yes	characteristic
119	C₂×C₄	C₂×C₂²⋊D₂₃₉	1	1	yes	characteristic
120	C₂×C₄	Q₈×D₂₃₉	1	2	yes	—
121	C₂×C₄	#16838549	1	2	yes	—
122	C₈	—	956	2	no	—
123	C₈	—	478	4	no	—
124	C₂²	C₂×C₂²⋊Dic₂₃₉	1	1	yes	characteristic
125	C₂²	Q₈×Dic₂₃₉	1	2	yes	—
126	C₂²	C₂²×C₂₃₉⋊C₈	1	1	yes	characteristic
127	C₂²	C₂×#16838532	1	1	yes	characteristic
128	C₂²	#21439967	1	2	yes	—
129	C₄	—	2	4	no	—
130	C₄	—	2	2	no	—
131	C₄	—	4	2	no	—
132	C₄	C₂²⋊(C₂₃₉⋊C₈)	1	4	yes	—
133	C₄	#21439936	1	2	yes	—
134	C₄	#21439949	1	1	yes	characteristic
135	C₄	#21439958	1	1	yes	characteristic
136	C₂	#29504679	1	1	yes	characteristic
137	C₂	C₂×C₂²⋊(C₂₃₉⋊C₈)	1	1	yes	characteristic
138	C₂	#29504739	1	1	yes	characteristic
139	C₂	Q₈×C₂₃₉⋊C₈	1	2	yes	—
140	C₂	#29504903	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×C₄⋊(C₂₃₉⋊C₈)

C₂

yes

maximal, characteristic

C₂×C₄×C₂₃₉⋊C₈

C₂

yes

maximal, characteristic

#29504527

C₂

yes

maximal

C₂×C₄×Q₈×C₂₃₉

C₂

yes

maximal, characteristic

C₂²×C₂₃₉⋊C₈

C₂²

yes

—

C₂²×C₂₃₉⋊C₈

C₂²

yes

—

C₄⋊(C₂₃₉⋊C₈)

—

C₄×C₂₃₉⋊C₈

—

C₂²×Q₈×C₂₃₉

C₄

yes

characteristic

C₄×Q₈×C₂₃₉

—

C₄×Q₈×C₂₃₉

—

C₂×C₄⋊C₄×C₂₃₉

C₄

yes

characteristic

C₂×C₄⋊C₄×C₂₃₉

C₄

yes

—

C₂×C₄²×C₂₃₉

C₂²

yes

—

C₂×C₄²×C₂₃₉

C₂²

yes

characteristic

C₂×C₂₃₉⋊C₈

—

C₂×C₂₃₉⋊C₈

—

C₂×C₂₃₉⋊C₈

—

C₂×C₂₃₉⋊C₈

—

C₂×C₂₃₉⋊C₈

Q₈

yes

complemented

C₂×Q₈×C₂₃₉

—

C₂×Q₈×C₂₃₉

—

C₂×Q₈×C₂₃₉

C₈

yes

complemented

C₂²×C₄×C₂₃₉

C₂×C₄

yes

—

C₂²×C₄×C₂₃₉

C₂×C₄

yes

characteristic

C₂²×C₄×C₂₃₉

C₂×C₄

yes

—

C₂²×C₄×C₂₃₉

C₂×C₄

yes

characteristic

C₂²×C₄×C₂₃₉

C₂³

yes

characteristic

C₄⋊C₄×C₂₃₉

—

C₄⋊C₄×C₂₃₉

—

C₄²×C₂₃₉

—

C₄²×C₂₃₉

—

C₄²×C₂₃₉

D₄

yes

—

C₄²×C₂₃₉

D₄

yes

—

C₂₃₉⋊C₈

—

C₂₃₉⋊C₈

—

C₂³×C₂₃₉

C₂²×C₄

yes

characteristic

Q₈×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

C₂×C₈

yes

—

C₂×C₄×C₂₃₉

C₂×C₈

yes

—

C₂×C₄×C₂₃₉

M₁₆

yes

—

C₂×C₄×C₂₃₉

C₂×D₄

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

C₄×Q₈

yes

—

C₂²×C₂₃₉

C₂²×C₈

yes

characteristic

C₂²×C₂₃₉

C₂×M₁₆

yes

characteristic

C₂²×C₂₃₉

#131

yes

—

C₄×C₂₃₉

—

C₄×C₂₃₉

—

C₄×C₂₃₉

—

C₄×C₂₃₉

C₂²⋊C₈

yes

—

C₄×C₂₃₉

C₂²⋊Q₈

yes

—

C₄×C₂₃₉

#124

yes

characteristic

C₄×C₂₃₉

C₄⋊Q₈

yes

characteristic

C₂×C₂₃₉

#391

yes

characteristic

C₂×C₂₃₉

C₂×C₂²⋊C₈

yes

characteristic

C₂×C₂₃₉

#408

yes

characteristic

C₂×C₂₃₉

C₈×Q₈

yes

—

C₂×C₂₃₉

#446

yes

—

C₂₃₉

#1984

yes

characteristic, complemented, Sylow

#1984

—

239

maximal, Sylow

C₂×C₄×Q₈

D₂₃₉

yes

characteristic

C₂×C₄⋊C₈

—

239

—

C₂×C₄×C₈

—

239

—

#336

—

239

—

C₂²×Q₈

Dic₂₃₉

yes

characteristic

C₂²×C₈

—

239

—

C₂²×C₈

—

239

—

C₄×Q₈

—

C₄×Q₈

—

C₂×C₄⋊C₄

Dic₂₃₉

yes

characteristic

C₂×C₄⋊C₄

Dic₂₃₉

yes

—

C₂×C₄²

C₂×D₂₃₉

yes

—

C₂×C₄²

C₂×D₂₃₉

yes

characteristic

C₄⋊C₈

—

478

—

C₄×C₈

—

478

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

C₂₃₉⋊C₈

yes

complemented

C₂²×C₄

C₂×Dic₂₃₉

yes

—

C₂²×C₄

C₂×Dic₂₃₉

yes

characteristic

C₂²×C₄

C₂×Dic₂₃₉

yes

—

C₂²×C₄

C₂×Dic₂₃₉

yes

characteristic

C₂²×C₄

C₂²×D₂₃₉

yes

characteristic

C₂×C₈

—

478

—

C₂×C₈

—

478

—

C₂×C₈

—

478

—

C₂×C₈

—

478

—

C₂×C₈

—

239

—

100

C₄⋊C₄

—

101

C₄⋊C₄

—

102

C₄²

—

103

C₄²

—

104

C₄²

C₂²⋊D₂₃₉

yes

—

105

C₄²

C₂²⋊D₂₃₉

yes

—

106

C₂³

C₂²×Dic₂₃₉

yes

characteristic

107

Q₈

—

108

C₂×C₄

—

109

C₂×C₄

—

110

C₂×C₄

—

111

C₂×C₄

—

112

C₂×C₄

—

113

C₂×C₄

C₂²⋊Dic₂₃₉

yes

—

114

C₂×C₄

C₂²⋊Dic₂₃₉

yes

—

115

C₂×C₄

C₂×C₂₃₉⋊C₈

yes

—

116

C₂×C₄

C₂×C₂₃₉⋊C₈

yes

—

117

C₂×C₄

#16838532

yes

—

118

C₂×C₄

C₂×C₂²⋊D₂₃₉

yes

characteristic

119

C₂×C₄

C₂×C₂²⋊D₂₃₉

yes

characteristic

120

C₂×C₄

Q₈×D₂₃₉

yes

—

121

C₂×C₄

#16838549

yes

—

122

C₈

—

956

—

123

C₈

—

478

—

124

C₂²

C₂×C₂²⋊Dic₂₃₉

yes

characteristic

125

C₂²

Q₈×Dic₂₃₉

yes

—

126

C₂²

C₂²×C₂₃₉⋊C₈

yes

characteristic

127

C₂²

C₂×#16838532

yes

characteristic

128

C₂²

#21439967

yes

—

129

C₄

—

130

C₄

—

131

C₄

—

132

C₄

C₂²⋊(C₂₃₉⋊C₈)

yes

—

133

C₄

#21439936

yes

—

134

C₄

#21439949

yes

characteristic

135

C₄

#21439958

yes

characteristic

136

C₂

#29504679

yes

characteristic

137

C₂

C₂×C₂²⋊(C₂₃₉⋊C₈)

yes

characteristic

138

C₂

#29504739

yes

characteristic

139

C₂

Q₈×C₂₃₉⋊C₈

yes

—

140

C₂

#29504903

yes

—

	subgroup	G/H
center, Frattini	C₂²×C₄	C₂²×D₂₃₉
commutator	C₂²×C₂₃₉	C₂²×C₈
Fitting	C₂×C₄×Q₈×C₂₃₉	C₂
socle, prime radical	C₂³×C₂₃₉	C₂²×C₄
cosocle	C₂²×C₄×C₂₃₉	C₂³
hypercenter	C₂×C₄×Q₈	D₂₃₉
hypocenter	C₂₃₉	#1984
LCS	G ▷ C₂²×C₂₃₉ ▷ C₂₃₉
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×C₄×Q₈

subgroup

G/H

center, Frattini

C₂²×C₄

C₂²×D₂₃₉

commutator

C₂²×C₂₃₉

C₂²×C₈

Fitting

C₂×C₄×Q₈×C₂₃₉

C₂

socle, prime radical

C₂³×C₂₃₉

C₂²×C₄

cosocle

C₂²×C₄×C₂₃₉

C₂³

hypercenter

C₂×C₄×Q₈

D₂₃₉

C₂₃₉

G ▷ C₂²×C₂₃₉ ▷ C₂₃₉

C₁ ◁ C₂²×C₄ ◁ C₂×C₄×Q₈

N	H ≅ G/N	type
20. C₂×C₂₃₉⋊C₈	Q₈	semidirect
23. C₂×Q₈×C₂₃₉	C₈	semidirect
70. C₂₃₉	#1984	semidirect
89. C₂×Q₈	C₂₃₉⋊C₈	semidirect

H ≅ G/N

type

20. C₂×C₂₃₉⋊C₈

Q₈

semidirect

23. C₂×Q₈×C₂₃₉

C₈

semidirect

70. C₂₃₉

#1984

semidirect

89. C₂×Q₈

C₂₃₉⋊C₈

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
C₂×C₄×C₂₃₉⋊C₈	2
#29504527	2
C₂×C₄×Q₈×C₂₃₉	2
#1984	239

subgroup

index

C₂×C₄⋊(C₂₃₉⋊C₈)

C₂×C₄×C₂₃₉⋊C₈

#29504527

C₂×C₄×Q₈×C₂₃₉

#1984

239

Subgroups — #44485485

Conjugacy classes of subgroups

Complements

Maximal subgroups