Group #44483928 — subgroups

Subgroups — #44483928

Order 30592.

150 conjugacy classes of subgroups; only the 57 normal subgroups are shown.

normal subgroups only

This group has 150 conjugacy classes of subgroups — too many to draw. Only its 57 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	#29505168	C₂	1	1	yes	maximal, characteristic
2	#29505067	C₂	1	1	yes	maximal, characteristic
3	Q₈×C₂₃₉⋊C₈	C₂	1	1	yes	maximal, characteristic
4	#29504522	C₂	1	1	yes	maximal, characteristic
5	#29504520	C₂	1	1	yes	maximal, characteristic
6	#29504518	C₂	1	1	yes	maximal, characteristic
7	#462×C₂₃₉	C₂	1	1	yes	maximal, characteristic
8	#21439960	C₂²	1	1	yes	characteristic
9	#21439893	—	2	1	no	—
10	#21439891	—	2	1	no	—
11	#21439891	—	2	1	no	—
12	C₄⋊(C₂₃₉⋊C₈)	—	2	1	no	—
13	C₄⋊(C₂₃₉⋊C₈)	C₂²	1	1	yes	characteristic
14	C₄⋊(C₂₃₉⋊C₈)	C₂²	1	1	yes	characteristic
15	#21439884	—	2	1	no	—
16	C₄×C₂₃₉⋊C₈	—	2	1	no	—
17	C₄×C₂₃₉⋊C₈	C₂²	1	1	yes	characteristic
18	C₂×Q₁₆×C₂₃₉	—	2	1	no	—
19	C₄⋊Q₈×C₂₃₉	C₂²	1	1	yes	characteristic
20	C₄×Q₈×C₂₃₉	C₂²	1	1	yes	characteristic
21	C₄⋊C₈×C₂₃₉	C₂²	1	1	yes	characteristic
22	#103×C₂₃₉	—	2	1	no	—
23	C₂×Dic₄₇₈	—	2	1	no	—
24	C₂×C₂₃₉⋊C₈	—	2	1	no	—
25	C₂×C₂₃₉⋊C₈	—	2	1	no	—
26	C₂×C₂₃₉⋊C₈	D₄	1	1	yes	characteristic
27	C₂×C₂₃₉⋊C₈	D₄	1	1	yes	characteristic
28	C₄⋊Dic₂₃₉	—	2	1	no	—
29	C₄⋊Dic₂₃₉	—	2	1	no	—
30	C₂×Q₈×C₂₃₉	—	2	1	no	—
31	C₂×Q₈×C₂₃₉	D₄	1	1	yes	characteristic
32	Q₁₆×C₂₃₉	—	4	2	no	—
33	C₂×C₈×C₂₃₉	—	2	1	no	—
34	C₄⋊C₄×C₂₃₉	—	2	1	no	—
35	C₄⋊C₄×C₂₃₉	—	2	1	no	—
36	C₄⋊C₄×C₂₃₉	—	2	1	no	—
37	C₄⋊C₄×C₂₃₉	D₄	1	1	yes	characteristic
38	C₄²×C₂₃₉	—	2	1	no	—
39	C₄²×C₂₃₉	C₂³	1	1	yes	characteristic
40	Dic₄₇₈	—	8	1	no	—
41	C₂×Dic₂₃₉	—	4	1	no	—
42	C₂₃₉⋊C₈	—	2	1	no	—
43	C₂₃₉⋊C₈	—	4	1	no	—
44	C₂₃₉⋊C₈	—	4	1	no	—
45	C₂₃₉⋊C₈	Q₁₆	1	2	yes	complemented
46	Q₈×C₂₃₉	—	2	1	no	—
47	Q₈×C₂₃₉	—	4	1	no	—
48	Q₈×C₂₃₉	—	4	1	no	—
49	Q₈×C₂₃₉	SD₁₆	1	2	yes	—
50	C₂×C₄×C₂₃₉	—	2	1	no	—
51	C₂×C₄×C₂₃₉	—	2	1	no	—
52	C₂×C₄×C₂₃₉	—	4	1	no	—
53	C₂×C₄×C₂₃₉	C₂×D₄	1	1	yes	characteristic
54	C₂×C₄×C₂₃₉	C₂×D₄	1	1	yes	characteristic
55	C₂×C₄×C₂₃₉	#41	1	1	yes	characteristic
56	C₈×C₂₃₉	—	4	1	no	—
57	C₂²×C₂₃₉	#121	1	1	yes	characteristic
58	C₄×C₂₃₉	—	2	2	no	—
59	C₄×C₂₃₉	—	2	1	no	—
60	C₄×C₂₃₉	—	4	1	no	—
61	C₄×C₂₃₉	—	8	1	no	—
62	C₄×C₂₃₉	C₂×SD₁₆	1	1	yes	characteristic
63	C₄×C₂₃₉	C₂×Q₁₆	1	1	yes	characteristic
64	C₄×C₂₃₉	#135	1	1	yes	characteristic
65	C₄×C₂₃₉	#137	1	1	yes	characteristic
66	Dic₂₃₉	—	8	1	no	—
67	C₂×C₂₃₉	#462	1	1	yes	characteristic
68	C₂×C₂₃₉	#465	1	1	yes	characteristic
69	C₂×C₂₃₉	#471	1	1	yes	characteristic
70	C₂₃₉	#1672	1	1	yes	characteristic, complemented, Sylow
71	#1672	—	239	1	no	maximal, Sylow
72	#498	—	239	1	no	—
73	#477	—	239	1	no	—
74	#462	D₂₃₉	1	1	yes	characteristic
75	C₈×Q₈	—	239	1	no	—
76	#334	—	239	1	no	—
77	#333	—	239	1	no	—
78	#332	—	239	1	no	—
79	C₂×Q₁₆	—	2	1	no	—
80	C₄⋊Q₈	—	239	1	no	—
81	C₄⋊Q₈	C₂×D₂₃₉	1	1	yes	characteristic
82	C₄×Q₈	C₂×D₂₃₉	1	1	yes	characteristic
83	#107	—	478	1	no	—
84	#106	—	478	1	no	—
85	#106	—	478	1	no	—
86	C₄⋊C₈	—	478	1	no	—
87	C₄⋊C₈	—	239	1	no	—
88	C₄⋊C₈	—	239	1	no	—
89	C₄⋊C₈	C₂×D₂₃₉	1	1	yes	characteristic
90	#103	—	2	1	no	—
91	#103	—	478	1	no	—
92	C₄×C₈	—	478	1	no	—
93	C₄×C₈	—	239	1	no	—
94	C₂×Q₈	—	2	1	no	—
95	C₂×Q₈	—	478	1	no	—
96	C₂×Q₈	C₂²⋊D₂₃₉	1	1	yes	characteristic
97	Q₁₆	—	4	2	no	—
98	C₂×C₈	—	2	1	no	—
99	C₂×C₈	—	478	1	no	—
100	C₂×C₈	—	478	1	no	—
101	C₂×C₈	—	239	1	no	—
102	C₂×C₈	—	239	1	no	—
103	C₄⋊C₄	—	2	1	no	—
104	C₄⋊C₄	—	2	1	no	—
105	C₄⋊C₄	—	2	1	no	—
106	C₄⋊C₄	—	478	1	no	—
107	C₄⋊C₄	—	478	1	no	—
108	C₄⋊C₄	C₂²⋊D₂₃₉	1	1	yes	characteristic
109	C₄²	—	2	1	no	—
110	C₄²	C₂²×D₂₃₉	1	1	yes	characteristic
111	Q₈	—	2	1	no	—
112	Q₈	—	4	1	no	—
113	Q₈	—	4	1	no	—
114	Q₈	—	1912	1	no	—
115	Q₈	#16838536	1	2	yes	—
116	C₂×C₄	—	2	1	no	—
117	C₂×C₄	—	2	1	no	—
118	C₂×C₄	—	4	1	no	—
119	C₂×C₄	—	956	1	no	—
120	C₂×C₄	D₄×D₂₃₉	1	1	yes	characteristic
121	C₂×C₄	C₂×C₂²⋊D₂₃₉	1	1	yes	characteristic
122	C₂×C₄	#16838547	1	1	yes	characteristic
123	C₈	—	4	1	no	—
124	C₈	—	956	1	no	—
125	C₈	—	956	1	no	—
126	C₈	—	478	1	no	—
127	C₈	—	239	2	no	—
128	C₂²	#21439932	1	1	yes	characteristic
129	C₄	—	2	1	no	—
130	C₄	—	2	2	no	—
131	C₄	—	4	1	no	—
132	C₄	—	8	1	no	—
133	C₄	—	1912	1	no	—
134	C₄	C₂×#16838536	1	1	yes	characteristic
135	C₄	Q₁₆×D₂₃₉	1	1	yes	characteristic
136	C₄	#21439980	1	1	yes	characteristic
137	C₄	#21439992	1	1	yes	characteristic
138	C₂	#29504980	1	1	yes	characteristic
139	C₂	#29504999	1	1	yes	characteristic
140	C₂	#29505037	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#29505168

C₂

yes

maximal, characteristic

#29505067

C₂

yes

maximal, characteristic

Q₈×C₂₃₉⋊C₈

C₂

yes

maximal, characteristic

#29504522

C₂

yes

maximal, characteristic

#29504520

C₂

yes

maximal, characteristic

#29504518

C₂

yes

maximal, characteristic

#462×C₂₃₉

C₂

yes

maximal, characteristic

#21439960

C₂²

yes

—

—

—

C₄⋊(C₂₃₉⋊C₈)

—

C₄⋊(C₂₃₉⋊C₈)

C₂²

yes

characteristic

C₄⋊(C₂₃₉⋊C₈)

C₂²

yes

characteristic

#21439884

—

C₄×C₂₃₉⋊C₈

—

C₄×C₂₃₉⋊C₈

C₂²

yes

characteristic

C₂×Q₁₆×C₂₃₉

—

C₄⋊Q₈×C₂₃₉

C₂²

yes

characteristic

C₄×Q₈×C₂₃₉

C₂²

yes

characteristic

C₄⋊C₈×C₂₃₉

C₂²

yes

characteristic

#103×C₂₃₉

—

C₂×Dic₄₇₈

—

C₂×C₂₃₉⋊C₈

—

C₂×C₂₃₉⋊C₈

—

C₂×C₂₃₉⋊C₈

D₄

yes

characteristic

C₂×C₂₃₉⋊C₈

D₄

yes

characteristic

C₄⋊Dic₂₃₉

—

C₄⋊Dic₂₃₉

—

C₂×Q₈×C₂₃₉

—

C₂×Q₈×C₂₃₉

D₄

yes

characteristic

Q₁₆×C₂₃₉

—

C₂×C₈×C₂₃₉

—

C₄⋊C₄×C₂₃₉

—

C₄⋊C₄×C₂₃₉

—

C₄⋊C₄×C₂₃₉

—

C₄⋊C₄×C₂₃₉

D₄

yes

characteristic

C₄²×C₂₃₉

—

C₄²×C₂₃₉

C₂³

yes

characteristic

Dic₄₇₈

—

C₂×Dic₂₃₉

—

C₂₃₉⋊C₈

—

C₂₃₉⋊C₈

—

C₂₃₉⋊C₈

—

C₂₃₉⋊C₈

Q₁₆

yes

complemented

Q₈×C₂₃₉

—

Q₈×C₂₃₉

—

Q₈×C₂₃₉

—

Q₈×C₂₃₉

SD₁₆

yes

—

C₂×C₄×C₂₃₉

—

C₂×C₄×C₂₃₉

—

C₂×C₄×C₂₃₉

—

C₂×C₄×C₂₃₉

C₂×D₄

yes

characteristic

C₂×C₄×C₂₃₉

C₂×D₄

yes

characteristic

C₂×C₄×C₂₃₉

#41

yes

characteristic

C₈×C₂₃₉

—

C₂²×C₂₃₉

#121

yes

characteristic

C₄×C₂₃₉

—

C₄×C₂₃₉

—

C₄×C₂₃₉

—

C₄×C₂₃₉

—

C₄×C₂₃₉

C₂×SD₁₆

yes

characteristic

C₄×C₂₃₉

C₂×Q₁₆

yes

C₄×C₂₃₉

yes

C₄×C₂₃₉

yes

Dic₂₃₉

—

C₂×C₂₃₉

yes

C₂×C₂₃₉

yes

C₂×C₂₃₉

yes

C₂₃₉

yes

characteristic, complemented, Sylow

#1672

—

239

maximal, Sylow

#498

—

239

—

#477

—

239

—

#462

D₂₃₉

yes

characteristic

C₈×Q₈

—

239

—

#334

—

239

—

#333

—

239

—

#332

—

239

—

C₂×Q₁₆

—

C₄⋊Q₈

—

239

—

C₄⋊Q₈

C₂×D₂₃₉

yes

characteristic

C₄×Q₈

C₂×D₂₃₉

yes

characteristic

#107

—

478

—

#106

—

478

—

#106

—

478

—

C₄⋊C₈

—

478

—

C₄⋊C₈

—

239

—

C₄⋊C₈

—

239

—

C₄⋊C₈

C₂×D₂₃₉

yes

characteristic

#103

—

#103

—

478

—

C₄×C₈

—

478

—

C₄×C₈

—

239

—

C₂×Q₈

—

C₂×Q₈

—

478

—

C₂×Q₈

C₂²⋊D₂₃₉

yes

characteristic

Q₁₆

—

C₂×C₈

—

C₂×C₈

—

478

—

100

C₂×C₈

—

478

—

101

C₂×C₈

—

239

—

102

C₂×C₈

—

239

—

103

C₄⋊C₄

—

104

C₄⋊C₄

—

105

C₄⋊C₄

—

106

C₄⋊C₄

—

478

—

107

C₄⋊C₄

—

478

—

108

C₄⋊C₄

C₂²⋊D₂₃₉

yes

characteristic

109

C₄²

—

110

C₄²

C₂²×D₂₃₉

yes

characteristic

111

Q₈

—

112

Q₈

—

113

Q₈

—

114

Q₈

—

1912

—

115

Q₈

#16838536

yes

—

116

C₂×C₄

—

117

C₂×C₄

—

118

C₂×C₄

—

119

C₂×C₄

—

956

—

120

C₂×C₄

D₄×D₂₃₉

yes

characteristic

121

C₂×C₄

C₂×C₂²⋊D₂₃₉

yes

characteristic

122

C₂×C₄

#16838547

yes

characteristic

123

C₈

—

124

C₈

—

956

—

125

C₈

—

956

—

126

C₈

—

478

—

127

C₈

—

239

—

128

C₂²

#21439932

yes

characteristic

129

C₄

—

130

C₄

—

131

C₄

—

132

C₄

—

133

C₄

—

1912

—

134

C₄

C₂×#16838536

yes

characteristic

135

C₄

Q₁₆×D₂₃₉

yes

characteristic

136

C₄

#21439980

yes

characteristic

137

C₄

#21439992

yes

characteristic

138

C₂

#29504980

yes

characteristic

139

C₂

#29504999

yes

characteristic

140

C₂

#29505037

yes

characteristic

	subgroup	G/H
center	C₂²	#21439932
commutator, cosocle	C₄²×C₂₃₉	C₂³
Frattini	C₄²	C₂²×D₂₃₉
Fitting	#462×C₂₃₉	C₂
socle, prime radical	C₂²×C₂₃₉	#121
hypercenter	#462	D₂₃₉
hypocenter	C₂₃₉	#1672
LCS	G ▷ C₄²×C₂₃₉ ▷ C₂²×C₂₃₉ ▷ C₂₃₉
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #462

subgroup

G/H

center

C₂²

#21439932

commutator, cosocle

C₄²×C₂₃₉

C₂³

Frattini

C₄²

C₂²×D₂₃₉

Fitting

#462×C₂₃₉

C₂

C₂²×C₂₃₉

D₂₃₉

C₂₃₉

G ▷ C₄²×C₂₃₉ ▷ C₂²×C₂₃₉ ▷ C₂₃₉

UCS

C₁ ◁ C₂² ◁ C₄² ◁ #462

N	H ≅ G/N	type
45. C₂₃₉⋊C₈	Q₁₆	semidirect
70. C₂₃₉	#1672	semidirect

H ≅ G/N

type

45. C₂₃₉⋊C₈

Q₁₆

semidirect

70. C₂₃₉

#1672

semidirect

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

subgroup	index
#29505168	2
#29505067	2
Q₈×C₂₃₉⋊C₈	2
#29504522	2
#29504520	2
#29504518	2
#462×C₂₃₉	2
#1672	239

subgroup

index

Q₈×C₂₃₉⋊C₈

#462×C₂₃₉

239

Subgroups — #44483928

Conjugacy classes of subgroups

Complements

Maximal subgroups