Group #44484169 — subgroups

Subgroups — #44484169

Order 30592.

198 conjugacy classes of subgroups; only the 61 normal subgroups are shown.

normal subgroups only

This group has 198 conjugacy classes of subgroups — too many to draw. Only its 61 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	#29505175	C₂	1	1	yes	maximal, characteristic, complemented
2	#29505170	C₂	1	1	yes	maximal, characteristic, complemented
3	#29505144	C₂	1	1	yes	maximal, characteristic
4	#29505144	C₂	1	1	yes	maximal, characteristic
5	#29505122	C₂	1	1	yes	maximal, characteristic, complemented
6	#29504494	C₂	1	1	yes	maximal, characteristic, complemented
7	#486×C₂₃₉	C₂	1	1	yes	maximal, characteristic
8	C₂×#16838536	—	2	2	no	—
9	C₂×#16838536	—	2	2	no	—
10	#21439960	C₂²	1	1	yes	characteristic
11	#21439960	C₂²	1	1	yes	characteristic
12	#21439893	—	2	2	no	—
13	#21439892	—	2	2	no	—
14	#21439884	—	2	2	no	—
15	C₄×C₂₃₉⋊C₈	C₂²	1	1	yes	characteristic
16	C₄×C₂₃₉⋊C₈	C₂²	1	1	yes	characteristic
17	C₄⋊Q₈×C₂₃₉	C₂²	1	1	yes	characteristic
18	#127×C₂₃₉	C₂²	1	1	yes	characteristic
19	#102×C₂₃₉	—	2	2	no	—
20	C₄×C₈×C₂₃₉	C₂²	1	1	yes	characteristic
21	C₂×Dic₄₇₈	—	2	1	no	—
22	C₂×Dic₄₇₈	—	2	1	no	—
23	#16838536	—	4	4	no	—
24	#16838536	—	4	4	no	—
25	C₂×C₂₃₉⋊C₈	D₄	1	2	yes	—
26	C₂×C₂₃₉⋊C₈	D₄	1	2	yes	—
27	C₄⋊Dic₂₃₉	—	2	1	no	—
28	C₄⋊Dic₂₃₉	—	2	1	no	—
29	C₄⋊Dic₂₃₉	—	2	1	no	—
30	C₄⋊Dic₂₃₉	—	2	1	no	—
31	C₂×Q₈×C₂₃₉	—	2	1	no	—
32	C₂×D₄×C₂₃₉	—	2	1	no	—
33	C₂×D₄×C₂₃₉	—	2	1	no	—
34	C₂×D₄×C₂₃₉	—	2	1	no	—
35	C₂×C₈×C₂₃₉	D₄	1	2	yes	—
36	C₄⋊C₄×C₂₃₉	—	2	1	no	—
37	C₄⋊C₄×C₂₃₉	—	2	1	no	—
38	C₄²×C₂₃₉	C₂³	1	1	yes	characteristic
39	Dic₄₇₈	—	4	2	no	—
40	Dic₄₇₈	—	4	2	no	—
41	C₂×Dic₂₃₉	—	4	1	no	—
42	C₂×Dic₂₃₉	—	4	1	no	—
43	C₂₃₉⋊C₈	—	2	2	no	—
44	C₂₃₉⋊C₈	SD₁₆	1	4	yes	complemented
45	C₂³×C₂₃₉	—	4	1	no	—
46	Q₈×C₂₃₉	—	4	2	no	—
47	D₄×C₂₃₉	—	4	2	no	—
48	D₄×C₂₃₉	—	4	2	no	—
49	D₄×C₂₃₉	—	4	2	no	—
50	C₂×C₄×C₂₃₉	—	4	1	no	—
51	C₂×C₄×C₂₃₉	C₂×D₄	1	1	yes	characteristic
52	C₂×C₄×C₂₃₉	C₂×D₄	1	1	yes	characteristic
53	C₂×C₄×C₂₃₉	C₂×D₄	1	1	yes	characteristic
54	C₈×C₂₃₉	—	2	2	no	—
55	C₂²×C₂₃₉	—	8	1	no	—
56	C₂²×C₂₃₉	—	8	1	no	—
57	C₂²×C₂₃₉	—	8	1	no	—
58	C₂²×C₂₃₉	#127	1	1	yes	characteristic
59	C₄×C₂₃₉	—	8	1	no	—
60	C₄×C₂₃₉	C₂×SD₁₆	1	2	yes	—
61	C₄×C₂₃₉	AGL(1,ℤ/8)	1	2	yes	—
62	C₄×C₂₃₉	#137	1	2	yes	—
63	Dic₂₃₉	—	8	1	no	—
64	Dic₂₃₉	—	8	1	no	—
65	C₂×C₂₃₉	—	16	1	no	—
66	C₂×C₂₃₉	#492	1	1	yes	characteristic
67	C₂×C₂₃₉	#496	1	1	yes	characteristic
68	C₂×C₂₃₉	#497	1	1	yes	characteristic
69	C₂₃₉	#1714	1	1	yes	characteristic, complemented, Sylow
70	#1714	—	239	1	no	maximal, Sylow
71	#500	—	239	1	no	—
72	#498	—	239	1	no	—
73	#492	—	239	1	no	—
74	#492	—	239	1	no	—
75	#487	—	239	1	no	—
76	#486	D₂₃₉	1	1	yes	characteristic
77	#322	—	239	1	no	—
78	C₂×SD₁₆	—	478	2	no	—
79	C₂×SD₁₆	—	478	2	no	—
80	C₄⋊Q₈	—	239	1	no	—
81	C₄⋊Q₈	—	239	1	no	—
82	C₄⋊Q₈	C₂×D₂₃₉	1	1	yes	characteristic
83	#127	C₂×D₂₃₉	1	1	yes	characteristic
84	#107	—	478	2	no	—
85	#106	—	478	2	no	—
86	#103	—	478	2	no	—
87	#102	—	2	2	no	—
88	C₄×C₈	—	239	1	no	—
89	C₄×C₈	—	239	1	no	—
90	C₄×C₈	C₂×D₂₃₉	1	1	yes	characteristic
91	C₂×Q₈	—	2	1	no	—
92	C₂×Q₈	—	478	1	no	—
93	C₂×Q₈	—	478	1	no	—
94	C₂×D₄	—	2	1	no	—
95	C₂×D₄	—	2	1	no	—
96	C₂×D₄	—	2	1	no	—
97	SD₁₆	—	956	4	no	—
98	SD₁₆	—	956	4	no	—
99	C₂×C₈	—	239	2	no	—
100	C₂×C₈	—	239	2	no	—
101	C₂×C₈	C₂²⋊D₂₃₉	1	2	yes	—
102	C₄⋊C₄	—	2	1	no	—
103	C₄⋊C₄	—	2	1	no	—
104	C₄⋊C₄	—	478	1	no	—
105	C₄⋊C₄	—	478	1	no	—
106	C₄⋊C₄	—	478	1	no	—
107	C₄⋊C₄	—	478	1	no	—
108	C₄²	C₂²×D₂₃₉	1	1	yes	characteristic
109	C₂³	—	4	1	no	—
110	Q₈	—	4	2	no	—
111	Q₈	—	956	2	no	—
112	Q₈	—	956	2	no	—
113	D₄	—	4	2	no	—
114	D₄	—	4	2	no	—
115	D₄	—	4	2	no	—
116	C₂×C₄	—	4	1	no	—
117	C₂×C₄	—	956	1	no	—
118	C₂×C₄	—	956	1	no	—
119	C₂×C₄	D₄×D₂₃₉	1	1	yes	characteristic
120	C₂×C₄	D₄×D₂₃₉	1	1	yes	characteristic
121	C₂×C₄	C₂×C₂²⋊D₂₃₉	1	1	yes	characteristic
122	C₈	—	2	2	no	—
123	C₈	—	478	2	no	—
124	C₈	—	239	4	no	—
125	C₂²	—	8	1	no	—
126	C₂²	—	8	1	no	—
127	C₂²	—	8	1	no	—
128	C₂²	#21439956	1	1	yes	characteristic
129	C₄	—	8	1	no	—
130	C₄	—	1912	1	no	—
131	C₄	—	1912	1	no	—
132	C₄	SD₁₆×D₂₃₉	1	2	yes	—
133	C₄	#21439984	1	2	yes	—
134	C₄	#21439990	1	2	yes	—
135	C₂	—	16	1	no	—
136	C₂	#29505143	1	1	yes	characteristic
137	C₂	#29505160	1	1	yes	characteristic
138	C₂	#29505167	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#29505175

C₂

yes

maximal, characteristic, complemented

#29505170

C₂

yes

maximal, characteristic, complemented

#29505144

C₂

yes

maximal, characteristic

#29505144

C₂

yes

maximal, characteristic

#29505122

C₂

yes

maximal, characteristic, complemented

#29504494

C₂

yes

maximal, characteristic, complemented

#486×C₂₃₉

C₂

yes

maximal, characteristic

C₂×#16838536

—

C₂×#16838536

—

#21439960

C₂²

yes

characteristic

#21439960

C₂²

yes

—

—

—

C₄×C₂₃₉⋊C₈

C₂²

yes

characteristic

C₄×C₂₃₉⋊C₈

C₂²

yes

characteristic

C₄⋊Q₈×C₂₃₉

C₂²

yes

characteristic

#127×C₂₃₉

C₂²

yes

characteristic

#102×C₂₃₉

—

C₄×C₈×C₂₃₉

C₂²

yes

characteristic

C₂×Dic₄₇₈

—

C₂×Dic₄₇₈

—

#16838536

—

#16838536

—

C₂×C₂₃₉⋊C₈

D₄

yes

—

C₂×C₂₃₉⋊C₈

D₄

yes

—

C₄⋊Dic₂₃₉

—

C₄⋊Dic₂₃₉

—

C₄⋊Dic₂₃₉

—

C₄⋊Dic₂₃₉

—

C₂×Q₈×C₂₃₉

—

C₂×D₄×C₂₃₉

—

C₂×D₄×C₂₃₉

—

C₂×D₄×C₂₃₉

—

C₂×C₈×C₂₃₉

D₄

yes

—

C₄⋊C₄×C₂₃₉

—

C₄⋊C₄×C₂₃₉

—

C₄²×C₂₃₉

C₂³

yes

characteristic

Dic₄₇₈

—

Dic₄₇₈

—

C₂×Dic₂₃₉

—

C₂×Dic₂₃₉

—

C₂₃₉⋊C₈

—

C₂₃₉⋊C₈

SD₁₆

yes

complemented

C₂³×C₂₃₉

—

Q₈×C₂₃₉

—

D₄×C₂₃₉

—

D₄×C₂₃₉

—

D₄×C₂₃₉

—

C₂×C₄×C₂₃₉

—

C₂×C₄×C₂₃₉

C₂×D₄

yes

characteristic

C₂×C₄×C₂₃₉

C₂×D₄

yes

characteristic

C₂×C₄×C₂₃₉

C₂×D₄

yes

characteristic

C₈×C₂₃₉

—

C₂²×C₂₃₉

—

C₂²×C₂₃₉

—

C₂²×C₂₃₉

—

C₂²×C₂₃₉

#127

yes

characteristic

C₄×C₂₃₉

—

C₄×C₂₃₉

C₂×SD₁₆

yes

—

C₄×C₂₃₉

AGL(1,ℤ/8)

yes

—

C₄×C₂₃₉

#137

yes

—

Dic₂₃₉

—

Dic₂₃₉

—

C₂×C₂₃₉

—

C₂×C₂₃₉

yes

C₂×C₂₃₉

yes

C₂×C₂₃₉

yes

C₂₃₉

yes

characteristic, complemented, Sylow

#1714

—

239

maximal, Sylow

#500

—

239

—

#498

—

239

—

#492

—

239

—

#492

—

239

—

#487

—

239

—

#486

D₂₃₉

yes

characteristic

#322

—

239

—

C₂×SD₁₆

—

478

—

C₂×SD₁₆

—

478

—

C₄⋊Q₈

—

239

—

C₄⋊Q₈

—

239

—

C₄⋊Q₈

C₂×D₂₃₉

yes

characteristic

#127

C₂×D₂₃₉

yes

characteristic

#107

—

478

—

#106

—

478

—

#103

—

478

—

#102

—

C₄×C₈

—

239

—

C₄×C₈

—

239

—

C₄×C₈

C₂×D₂₃₉

yes

characteristic

C₂×Q₈

—

C₂×Q₈

—

478

—

C₂×Q₈

—

478

—

C₂×D₄

—

C₂×D₄

—

C₂×D₄

—

SD₁₆

—

956

—

SD₁₆

—

956

—

C₂×C₈

—

239

—

100

C₂×C₈

—

239

—

101

C₂×C₈

C₂²⋊D₂₃₉

yes

—

102

C₄⋊C₄

—

103

C₄⋊C₄

—

104

C₄⋊C₄

—

478

—

105

C₄⋊C₄

—

478

—

106

C₄⋊C₄

—

478

—

107

C₄⋊C₄

—

478

—

108

C₄²

C₂²×D₂₃₉

yes

characteristic

109

C₂³

—

110

Q₈

—

111

Q₈

—

956

—

112

Q₈

—

956

—

113

D₄

—

114

D₄

—

115

D₄

—

116

C₂×C₄

—

117

C₂×C₄

—

956

—

118

C₂×C₄

—

956

—

119

C₂×C₄

D₄×D₂₃₉

yes

characteristic

120

C₂×C₄

D₄×D₂₃₉

yes

characteristic

121

C₂×C₄

C₂×C₂²⋊D₂₃₉

yes

characteristic

122

C₈

—

123

C₈

—

478

—

124

C₈

—

239

—

125

C₂²

—

126

C₂²

—

127

C₂²

—

128

C₂²

#21439956

yes

characteristic

129

C₄

—

130

C₄

—

1912

—

131

C₄

—

1912

—

132

C₄

SD₁₆×D₂₃₉

yes

—

133

C₄

#21439984

yes

—

134

C₄

#21439990

yes

—

135

C₂

—

136

C₂

#29505143

yes

characteristic

137

C₂

#29505160

yes

characteristic

138

C₂

#29505167

yes

characteristic

	subgroup	G/H
center	C₂²	#21439956
commutator, cosocle	C₄²×C₂₃₉	C₂³
Frattini	C₄²	C₂²×D₂₃₉
Fitting	#486×C₂₃₉	C₂
socle	C₂²×C₂₃₉	#127
prime radical	#127×C₂₃₉	C₂²
hypercenter	#486	D₂₃₉
hypocenter	C₂₃₉	#1714
LCS	G ▷ C₄²×C₂₃₉ ▷ C₂²×C₂₃₉ ▷ C₂₃₉
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #486

subgroup

G/H

center

C₂²

#21439956

commutator, cosocle

C₄²×C₂₃₉

C₂³

Frattini

C₄²

C₂²×D₂₃₉

Fitting

#486×C₂₃₉

C₂

socle

C₂²×C₂₃₉

#127

prime radical

#127×C₂₃₉

C₂²

D₂₃₉

C₂₃₉

G ▷ C₄²×C₂₃₉ ▷ C₂²×C₂₃₉ ▷ C₂₃₉

UCS

C₁ ◁ C₂² ◁ C₄² ◁ #486

N	H ≅ G/N	type
1. #29505175	C₂	semidirect
2. #29505170	C₂	semidirect
5. #29505122	C₂	semidirect
6. #29504494	C₂	semidirect
44. C₂₃₉⋊C₈	SD₁₆	semidirect
69. C₂₃₉	#1714	semidirect

H ≅ G/N

type

1. #29505175

C₂

semidirect

2. #29505170

C₂

semidirect

5. #29505122

C₂

semidirect

6. #29504494

C₂

semidirect

44. C₂₃₉⋊C₈

SD₁₆

semidirect

69. C₂₃₉

#1714

semidirect

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

subgroup	index
#29505175	2
#29505170	2
#29505144	2
#29505122	2
#29504494	2
#486×C₂₃₉	2
#1714	239

subgroup

index

#486×C₂₃₉

239

Subgroups — #44484169

Conjugacy classes of subgroups

Complements

Maximal subgroups