Group #44486112 — subgroups

Subgroups — #44486112

Order 30592.

214 conjugacy classes of subgroups; only the 87 normal subgroups are shown.

normal subgroups only

This group has 214 conjugacy classes of subgroups — too many to draw. Only its 87 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	#29504670	C₂	1	1	yes	maximal, characteristic
2	#29504642	C₂	1	1	yes	maximal, characteristic
3	#29504535	C₂	1	2	yes	maximal
4	#29504535	C₂	1	2	yes	maximal
5	#336×C₂₃₉	C₂	1	1	yes	maximal, characteristic
6	C₂×C₄⋊Dic₂₃₉	—	2	1	no	—
7	C₂×C₄⋊Dic₂₃₉	—	2	1	no	—
8	C₂×C₄⋊Dic₂₃₉	C₂²	1	2	yes	—
9	C₂×C₄⋊Dic₂₃₉	C₂²	1	1	yes	characteristic
10	C₂×C₄⋊Dic₂₃₉	C₂²	1	1	yes	characteristic
11	C₂×Dic₂₃₉⋊C₄	—	2	1	no	—
12	C₂₃₉⋊#95	—	2	1	no	—
13	C₂²×C₈×C₂₃₉	C₂²	1	2	yes	—
14	C₂×C₄²×C₂₃₉	C₂²	1	1	yes	characteristic
15	C₂²×Dic₂₃₉	—	2	1	no	—
16	C₂²×Dic₂₃₉	—	2	1	no	—
17	C₂²×Dic₂₃₉	—	2	2	no	—
18	C₄⋊Dic₂₃₉	—	2	1	no	—
19	C₄⋊Dic₂₃₉	—	2	2	no	—
20	C₄⋊Dic₂₃₉	—	2	4	no	—
21	C₄⋊Dic₂₃₉	—	4	2	no	—
22	C₄⋊Dic₂₃₉	—	4	2	no	—
23	C₄⋊Dic₂₃₉	Q₈	1	2	yes	—
24	Dic₂₃₉⋊C₄	—	4	2	no	—
25	C₂²×C₄×C₂₃₉	D₄	1	1	yes	characteristic
26	C₂²×C₄×C₂₃₉	D₄	1	1	yes	characteristic
27	C₂²×C₄×C₂₃₉	C₂³	1	1	yes	characteristic
28	C₂×C₈×C₂₃₉	—	2	2	no	—
29	C₂×C₈×C₂₃₉	—	2	2	no	—
30	C₂×C₈×C₂₃₉	—	2	2	no	—
31	C₄²×C₂₃₉	—	2	2	no	—
32	C₂×Dic₂₃₉	—	2	2	no	—
33	C₂×Dic₂₃₉	—	4	1	no	—
34	C₂×Dic₂₃₉	—	4	1	no	—
35	C₂×Dic₂₃₉	—	4	1	no	—
36	C₂×Dic₂₃₉	—	4	2	no	—
37	C₂×Dic₂₃₉	—	4	1	no	—
38	C₂×Dic₂₃₉	—	4	2	no	—
39	C₂×Dic₂₃₉	—	4	2	no	—
40	C₂×Dic₂₃₉	—	4	1	no	—
41	C₂³×C₂₃₉	C₂×D₄	1	1	yes	characteristic
42	C₂×C₄×C₂₃₉	—	2	1	no	—
43	C₂×C₄×C₂₃₉	—	2	1	no	—
44	C₂×C₄×C₂₃₉	—	2	1	no	—
45	C₂×C₄×C₂₃₉	—	2	1	no	—
46	C₂×C₄×C₂₃₉	SD₁₆	1	2	yes	—
47	C₂×C₄×C₂₃₉	Q₁₆	1	2	yes	—
48	C₂×C₄×C₂₃₉	C₂×Q₈	1	1	yes	characteristic
49	C₂×C₄×C₂₃₉	#41	1	2	yes	—
50	C₂×C₄×C₂₃₉	#41	1	2	yes	—
51	C₂×C₄×C₂₃₉	#41	1	1	yes	characteristic
52	C₈×C₂₃₉	—	4	2	no	—
53	C₂²×C₂₃₉	C₂²⋊Q₈	1	1	yes	characteristic
54	C₂²×C₂₃₉	#123	1	1	yes	characteristic
55	C₂²×C₂₃₉	#124	1	1	yes	characteristic
56	C₂²×C₂₃₉	C₂×SD₁₆	1	1	yes	characteristic
57	C₂²×C₂₃₉	C₂×Q₁₆	1	1	yes	characteristic
58	C₂²×C₂₃₉	AGL(1,ℤ/8)	1	1	yes	characteristic
59	C₂²×C₂₃₉	AGL(1,ℤ/8)	1	1	yes	characteristic
60	C₄×C₂₃₉	—	2	2	no	—
61	C₄×C₂₃₉	—	2	2	no	—
62	C₄×C₂₃₉	#125	1	2	yes	—
63	C₄×C₂₃₉	#126	1	2	yes	—
64	Dic₂₃₉	—	4	2	no	—
65	Dic₂₃₉	—	8	1	no	—
66	Dic₂₃₉	—	8	2	no	—
67	C₂×C₂₃₉	#400	1	1	yes	characteristic
68	C₂×C₂₃₉	#476	1	1	yes	characteristic
69	C₂×C₂₃₉	#477	1	1	yes	characteristic
70	C₂×C₂₃₉	#481	1	1	yes	characteristic
71	C₂×C₂₃₉	#484	1	1	yes	characteristic
72	C₂×C₂₃₉	#487	1	1	yes	characteristic
73	C₂×C₂₃₉	#490	1	1	yes	characteristic
74	C₂₃₉	#2100	1	1	yes	characteristic, complemented, Sylow
75	#2100	—	239	1	no	maximal, Sylow
76	#389	—	239	1	no	—
77	#384	—	239	1	no	—
78	#340	—	239	2	no	—
79	#340	—	239	2	no	—
80	#336	D₂₃₉	1	1	yes	characteristic
81	C₂²×C₈	C₂×D₂₃₉	1	2	yes	—
82	C₂×C₄⋊C₄	—	478	1	no	—
83	C₂×C₄⋊C₄	—	478	1	no	—
84	C₂×C₄⋊C₄	—	478	1	no	—
85	C₂×C₄⋊C₄	—	239	2	no	—
86	C₂×C₄⋊C₄	—	239	1	no	—
87	C₂×C₄⋊C₄	—	239	1	no	—
88	C₂×C₄²	C₂×D₂₃₉	1	1	yes	characteristic
89	#95	—	478	1	no	—
90	C₂²×C₄	—	478	1	no	—
91	C₂²×C₄	—	478	2	no	—
92	C₂²×C₄	—	478	1	no	—
93	C₂²×C₄	D₉₅₆	1	1	yes	characteristic
94	C₂²×C₄	D₉₅₆	1	1	yes	characteristic
95	C₂²×C₄	C₂²×D₂₃₉	1	1	yes	characteristic
96	C₂×C₈	—	2	2	no	—
97	C₂×C₈	—	2	2	no	—
98	C₂×C₈	—	2	2	no	—
99	C₄⋊C₄	—	956	2	no	—
100	C₄⋊C₄	—	956	2	no	—
101	C₄⋊C₄	—	956	2	no	—
102	C₄⋊C₄	—	478	2	no	—
103	C₄⋊C₄	—	478	4	no	—
104	C₄⋊C₄	—	478	1	no	—
105	C₄⋊C₄	—	239	2	no	—
106	C₄²	—	2	2	no	—
107	C₂³	C₂×D₉₅₆	1	1	yes	characteristic
108	C₂×C₄	—	2	1	no	—
109	C₂×C₄	—	2	1	no	—
110	C₂×C₄	—	2	1	no	—
111	C₂×C₄	—	2	1	no	—
112	C₂×C₄	—	956	1	no	—
113	C₂×C₄	—	956	2	no	—
114	C₂×C₄	—	956	1	no	—
115	C₂×C₄	—	956	2	no	—
116	C₂×C₄	—	956	2	no	—
117	C₂×C₄	—	956	1	no	—
118	C₂×C₄	—	956	1	no	—
119	C₂×C₄	—	956	1	no	—
120	C₂×C₄	—	478	2	no	—
121	C₂×C₄	#16838537	1	2	yes	—
122	C₂×C₄	Dic₉₅₆	1	2	yes	—
123	C₂×C₄	Q₈×D₂₃₉	1	1	yes	characteristic
124	C₂×C₄	#16838547	1	2	yes	—
125	C₂×C₄	#16838547	1	1	yes	characteristic
126	C₂×C₄	#16838548	1	2	yes	—
127	C₈	—	4	2	no	—
128	C₂²	#21439940	1	1	yes	characteristic
129	C₂²	#21439944	1	1	yes	characteristic
130	C₂²	#21439948	1	1	yes	characteristic
131	C₂²	C₂×#16838537	1	1	yes	characteristic
132	C₂²	C₂×Dic₉₅₆	1	1	yes	characteristic
133	C₂²	#21439988	1	1	yes	characteristic
134	C₂²	#21439988	1	1	yes	characteristic
135	C₄	—	2	2	no	—
136	C₄	—	2	2	no	—
137	C₄	—	1912	1	no	—
138	C₄	—	1912	2	no	—
139	C₄	—	956	2	no	—
140	C₄	#21439950	1	2	yes	—
141	C₄	#21439953	1	2	yes	—
142	C₂	#29504712	1	1	yes	characteristic
143	C₂	#29505063	1	1	yes	characteristic
144	C₂	#29505068	1	1	yes	characteristic
145	C₂	#29505091	1	1	yes	characteristic
146	C₂	#29505108	1	1	yes	characteristic
147	C₂	#29505123	1	1	yes	characteristic
148	C₂	#29505137	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#29504670

C₂

yes

maximal, characteristic

#29504642

C₂

yes

maximal, characteristic

#29504535

C₂

yes

maximal

#29504535

C₂

yes

maximal

#336×C₂₃₉

C₂

yes

maximal, characteristic

C₂×C₄⋊Dic₂₃₉

—

C₂×C₄⋊Dic₂₃₉

—

C₂×C₄⋊Dic₂₃₉

C₂²

yes

—

C₂×C₄⋊Dic₂₃₉

C₂²

yes

characteristic

C₂×C₄⋊Dic₂₃₉

C₂²

yes

characteristic

C₂×Dic₂₃₉⋊C₄

—

C₂₃₉⋊#95

—

C₂²×C₈×C₂₃₉

C₂²

yes

—

C₂×C₄²×C₂₃₉

C₂²

yes

characteristic

C₂²×Dic₂₃₉

—

C₂²×Dic₂₃₉

—

C₂²×Dic₂₃₉

—

C₄⋊Dic₂₃₉

—

C₄⋊Dic₂₃₉

—

C₄⋊Dic₂₃₉

—

C₄⋊Dic₂₃₉

—

C₄⋊Dic₂₃₉

—

C₄⋊Dic₂₃₉

Q₈

yes

—

Dic₂₃₉⋊C₄

—

C₂²×C₄×C₂₃₉

D₄

yes

characteristic

C₂²×C₄×C₂₃₉

D₄

yes

characteristic

C₂²×C₄×C₂₃₉

C₂³

yes

characteristic

C₂×C₈×C₂₃₉

—

C₂×C₈×C₂₃₉

—

C₂×C₈×C₂₃₉

—

C₄²×C₂₃₉

—

C₂×Dic₂₃₉

—

C₂×Dic₂₃₉

—

C₂×Dic₂₃₉

—

C₂×Dic₂₃₉

—

C₂×Dic₂₃₉

—

C₂×Dic₂₃₉

—

C₂×Dic₂₃₉

—

C₂×Dic₂₃₉

—

C₂×Dic₂₃₉

—

C₂³×C₂₃₉

C₂×D₄

yes

characteristic

C₂×C₄×C₂₃₉

—

C₂×C₄×C₂₃₉

—

C₂×C₄×C₂₃₉

—

C₂×C₄×C₂₃₉

—

C₂×C₄×C₂₃₉

SD₁₆

yes

—

C₂×C₄×C₂₃₉

Q₁₆

yes

—

C₂×C₄×C₂₃₉

C₂×Q₈

yes

characteristic

C₂×C₄×C₂₃₉

#41

yes

—

C₂×C₄×C₂₃₉

#41

yes

—

C₂×C₄×C₂₃₉

#41

yes

characteristic

C₈×C₂₃₉

—

C₂²×C₂₃₉

C₂²⋊Q₈

yes

C₂²×C₂₃₉

yes

C₂²×C₂₃₉

yes

C₂²×C₂₃₉

C₂×SD₁₆

yes

characteristic

C₂²×C₂₃₉

C₂×Q₁₆

yes

C₂²×C₂₃₉

yes

C₂²×C₂₃₉

yes

C₄×C₂₃₉

—

C₄×C₂₃₉

—

C₄×C₂₃₉

#125

yes

—

C₄×C₂₃₉

#126

yes

—

Dic₂₃₉

—

Dic₂₃₉

—

Dic₂₃₉

—

C₂×C₂₃₉

yes

C₂×C₂₃₉

yes

C₂×C₂₃₉

yes

C₂×C₂₃₉

yes

C₂×C₂₃₉

yes

C₂×C₂₃₉

yes

C₂×C₂₃₉

yes

C₂₃₉

yes

characteristic, complemented, Sylow

#2100

—

239

maximal, Sylow

#389

—

239

—

#384

—

239

—

#340

—

239

—

#340

—

239

—

#336

D₂₃₉

yes

characteristic

C₂²×C₈

C₂×D₂₃₉

yes

—

C₂×C₄⋊C₄

—

478

—

C₂×C₄⋊C₄

—

478

—

C₂×C₄⋊C₄

—

478

—

C₂×C₄⋊C₄

—

239

—

C₂×C₄⋊C₄

—

239

—

C₂×C₄⋊C₄

—

239

—

C₂×C₄²

C₂×D₂₃₉

yes

characteristic

#95

—

478

—

C₂²×C₄

—

478

—

C₂²×C₄

—

478

—

C₂²×C₄

—

478

—

C₂²×C₄

D₉₅₆

yes

characteristic

C₂²×C₄

D₉₅₆

yes

characteristic

C₂²×C₄

C₂²×D₂₃₉

yes

characteristic

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₄⋊C₄

—

956

—

100

C₄⋊C₄

—

956

—

101

C₄⋊C₄

—

956

—

102

C₄⋊C₄

—

478

—

103

C₄⋊C₄

—

478

—

104

C₄⋊C₄

—

478

—

105

C₄⋊C₄

—

239

—

106

C₄²

—

107

C₂³

C₂×D₉₅₆

yes

characteristic

108

C₂×C₄

—

109

C₂×C₄

—

110

C₂×C₄

—

111

C₂×C₄

—

112

C₂×C₄

—

956

—

113

C₂×C₄

—

956

—

114

C₂×C₄

—

956

—

115

C₂×C₄

—

956

—

116

C₂×C₄

—

956

—

117

C₂×C₄

—

956

—

118

C₂×C₄

—

956

—

119

C₂×C₄

—

956

—

120

C₂×C₄

—

478

—

121

C₂×C₄

#16838537

yes

—

122

C₂×C₄

Dic₉₅₆

yes

—

123

C₂×C₄

Q₈×D₂₃₉

yes

characteristic

124

C₂×C₄

#16838547

yes

—

125

C₂×C₄

#16838547

yes

characteristic

126

C₂×C₄

#16838548

yes

—

127

C₈

—

128

C₂²

#21439940

yes

characteristic

129

C₂²

#21439944

yes

characteristic

130

C₂²

#21439948

yes

characteristic

131

C₂²

C₂×#16838537

yes

characteristic

132

C₂²

C₂×Dic₉₅₆

yes

characteristic

133

C₂²

#21439988

yes

characteristic

134

C₂²

#21439988

yes

characteristic

135

C₄

—

136

C₄

—

137

C₄

—

1912

—

138

C₄

—

1912

—

139

C₄

—

956

—

140

C₄

#21439950

yes

—

141

C₄

#21439953

yes

—

142

C₂

#29504712

yes

characteristic

143

C₂

#29505063

yes

characteristic

144

C₂

#29505068

yes

characteristic

145

C₂

#29505091

yes

characteristic

146

C₂

#29505108

yes

characteristic

147

C₂

#29505123

yes

characteristic

148

C₂

#29505137

yes

characteristic

	subgroup	G/H
center	C₂³	C₂×D₉₅₆
commutator, cosocle	C₂²×C₄×C₂₃₉	C₂³
Frattini	C₂²×C₄	C₂²×D₂₃₉
Fitting	#336×C₂₃₉	C₂
socle, prime radical	C₂³×C₂₃₉	C₂×D₄
hypercenter	#336	D₂₃₉
hypocenter	C₂₃₉	#2100
LCS	G ▷ C₂²×C₄×C₂₃₉ ▷ C₂×C₂₃₉ ▷ C₂₃₉
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #336

subgroup

G/H

center

C₂³

C₂×D₉₅₆

commutator, cosocle

C₂²×C₄×C₂₃₉

C₂³

Frattini

C₂²×C₄

C₂²×D₂₃₉

Fitting

#336×C₂₃₉

C₂

socle, prime radical

C₂³×C₂₃₉

C₂×D₄

D₂₃₉

C₂₃₉

G ▷ C₂²×C₄×C₂₃₉ ▷ C₂×C₂₃₉ ▷ C₂₃₉

UCS

C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #336

N	H ≅ G/N	type
74. C₂₃₉	#2100	semidirect

H ≅ G/N

type

74. C₂₃₉

#2100

semidirect

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

subgroup	index
#29504670	2
#29504642	2
#29504535	2
#336×C₂₃₉	2
#2100	239

subgroup

index

#336×C₂₃₉

239

Subgroups — #44486112

Conjugacy classes of subgroups

Complements

Maximal subgroups