Group #20244138 — subgroups

Subgroups — #20244138

Order 6016.

200 conjugacy classes of subgroups; only the 83 normal subgroups are shown.

normal subgroups only

This group has 200 conjugacy classes of subgroups — too many to draw. Only its 83 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×#13093959	C₂	1	1	yes	maximal, characteristic
2	#15689159	C₂	1	1	yes	maximal, characteristic
3	#15689149	C₂	1	1	yes	maximal, characteristic
4	C₂×#13093934	C₂	1	1	yes	maximal, characteristic
5	#15688884	C₂	1	1	yes	maximal, characteristic
6	#15688882	C₂	1	1	yes	maximal, characteristic
7	C₄₇×#424	C₂	1	1	yes	maximal, characteristic
8	C₂×#10625329	C₂²	1	1	yes	characteristic
9	C₂²×C₄₇⋊C₈	C₂²	1	1	yes	characteristic
10	#13093983	C₂²	1	1	yes	characteristic
11	C₂×C₄×Dic₄₇	C₂²	1	1	yes	characteristic
12	#13093959	—	2	2	no	—
13	#13093957	C₄	1	2	yes	complemented
14	#13093955	C₄	1	2	yes	complemented
15	Dic₄₇⋊C₈	—	2	2	no	—
16	#13093934	—	2	2	no	—
17	C₂×M₁₆×C₄₇	C₂²	1	1	yes	characteristic
18	C₂²×C₈×C₄₇	C₂²	1	1	yes	characteristic
19	#117×C₄₇	C₂²	1	1	yes	characteristic
20	C₄⋊C₈×C₄₇	—	2	2	no	—
21	C₂²×C₄×C₄₇	C₂³	1	1	yes	characteristic
22	C₂²×Dic₄₇	—	2	1	no	—
23	M₁₆×C₄₇	—	2	1	no	—
24	M₁₆×C₄₇	—	2	1	no	—
25	C₂×C₈×C₄₇	—	2	1	no	—
26	C₂×C₈×C₄₇	—	2	1	no	—
27	C₂×C₈×C₄₇	—	2	1	no	—
28	C₂×C₈×C₄₇	D₄	1	2	yes	—
29	C₄⋊C₄×C₄₇	C₂×C₄	1	2	yes	—
30	C₂²⋊C₄×C₄₇	—	2	1	no	—
31	C₄²×C₄₇	—	2	1	no	—
32	C₂²⋊Dic₄₇	—	2	1	no	—
33	C₄⋊Dic₄₇	C₂×C₄	1	2	yes	—
34	C₄×Dic₄₇	—	2	1	no	—
35	C₄×Dic₄₇	—	2	1	no	—
36	C₄×Dic₄₇	Q₈	1	2	yes	—
37	#10625329	—	2	1	no	—
38	#10625329	—	2	1	no	—
39	C₂×C₄₇⋊C₈	—	2	1	no	—
40	C₂×C₄₇⋊C₈	C₂×C₄	1	1	yes	characteristic
41	C₂×C₄₇⋊C₈	C₂×C₄	1	1	yes	characteristic
42	C₂×C₄₇⋊C₈	D₄	1	2	yes	—
43	C₂×C₄₇⋊C₈	Q₈	1	2	yes	—
44	C₂³×C₄₇	#41	1	1	yes	characteristic
45	C₂×C₄×C₄₇	—	2	2	no	—
46	C₂×C₄×C₄₇	C₂²×C₄	1	1	yes	characteristic
47	C₂×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
48	C₂×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
49	C₂×C₄×C₄₇	C₂×Q₈	1	1	yes	characteristic
50	C₂×C₄×C₄₇	C₂×Q₈	1	1	yes	characteristic
51	C₂×C₄×C₄₇	#41	1	1	yes	characteristic
52	C₂×Dic₄₇	—	2	2	no	—
53	C₂×Dic₄₇	—	2	2	no	—
54	C₂×Dic₄₇	—	4	2	no	—
55	C₈×C₄₇	—	2	2	no	—
56	C₈×C₄₇	—	4	1	no	—
57	C₄₇⋊C₈	—	4	1	no	—
58	C₄₇⋊C₈	C₄⋊C₄	1	2	yes	—
59	C₄₇⋊C₈	C₄⋊C₄	1	2	yes	—
60	C₂²×C₄₇	—	2	2	no	—
61	C₂²×C₄₇	C₄×D₄	1	1	yes	characteristic
62	C₂²×C₄₇	C₂²⋊Q₈	1	1	yes	characteristic
63	C₂²×C₄₇	#125	1	1	yes	characteristic
64	C₄×C₄₇	—	4	2	no	—
65	C₄×C₄₇	C₂×C₄⋊C₄	1	1	yes	characteristic
66	C₄×C₄₇	C₄×Q₈	1	1	yes	characteristic
67	C₄×C₄₇	#121	1	1	yes	characteristic
68	C₄×C₄₇	C₄⋊Q₈	1	1	yes	characteristic
69	Dic₄₇	—	4	2	no	—
70	Dic₄₇	—	4	2	no	—
71	#1946	—	47	1	no	maximal, Sylow
72	C₂×C₄₇	—	2	1	no	—
73	C₂×C₄₇	—	2	1	no	—
74	C₂×C₄₇	#389	1	1	yes	characteristic
75	C₂×C₄₇	#444	1	2	yes	—
76	C₂×#108	—	47	1	no	—
77	#427	—	47	1	no	—
78	#424	—	47	1	no	—
79	#424	D₄₇	1	1	yes	characteristic
80	C₂×#97	—	47	1	no	—
81	#339	—	47	1	no	—
82	#339	—	47	1	no	—
83	C₄₇	#1946	1	1	yes	characteristic, complemented, Sylow
84	C₂×M₁₆	—	47	1	no	—
85	C₂×M₁₆	C₂×D₄₇	1	1	yes	characteristic
86	C₂²×C₈	—	47	1	no	—
87	C₂²×C₈	C₂×D₄₇	1	1	yes	characteristic
88	#117	—	47	1	no	—
89	#117	C₂×D₄₇	1	1	yes	characteristic
90	C₂×C₄²	—	47	1	no	—
91	#108	—	94	2	no	—
92	#107	—	47	2	no	—
93	#106	—	47	2	no	—
94	C₄⋊C₈	—	2	2	no	—
95	C₄⋊C₈	—	94	2	no	—
96	#97	—	94	2	no	—
97	C₂²×C₄	—	94	1	no	—
98	C₂²×C₄	C₂²×D₄₇	1	1	yes	characteristic
99	M₁₆	—	2	1	no	—
100	M₁₆	—	2	1	no	—
101	M₁₆	—	94	1	no	—
102	M₁₆	—	94	1	no	—
103	C₂×C₈	—	2	1	no	—
104	C₂×C₈	—	2	1	no	—
105	C₂×C₈	—	2	1	no	—
106	C₂×C₈	—	47	1	no	—
107	C₂×C₈	—	47	1	no	—
108	C₂×C₈	—	47	2	no	—
109	C₂×C₈	—	47	2	no	—
110	C₂×C₈	—	94	1	no	—
111	C₂×C₈	C₂²⋊D₄₇	1	2	yes	—
112	C₄⋊C₄	—	47	2	no	—
113	C₄⋊C₄	C₄×D₄₇	1	2	yes	—
114	C₂²⋊C₄	—	2	1	no	—
115	C₂²⋊C₄	—	94	1	no	—
116	C₄²	—	2	1	no	—
117	C₄²	—	47	2	no	—
118	C₄²	—	94	1	no	—
119	C₄²	—	94	1	no	—
120	C₂³	#10625350	1	1	yes	characteristic
121	C₂×C₄	—	2	2	no	—
122	C₂×C₄	—	94	2	no	—
123	C₂×C₄	—	94	2	no	—
124	C₂×C₄	—	188	2	no	—
125	C₂×C₄	C₂×C₄×D₄₇	1	1	yes	characteristic
126	C₂×C₄	D₄×D₄₇	1	1	yes	characteristic
127	C₂×C₄	#10625352	1	1	yes	characteristic
128	C₂×C₄	Q₈×D₄₇	1	1	yes	characteristic
129	C₂×C₄	Q₈×D₄₇	1	1	yes	characteristic
130	C₂×C₄	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
131	C₈	—	2	2	no	—
132	C₈	—	4	1	no	—
133	C₈	—	47	2	no	—
134	C₈	—	47	2	no	—
135	C₈	—	188	1	no	—
136	C₂²	—	2	2	no	—
137	C₂²	C₄×C₂²⋊D₄₇	1	1	yes	characteristic
138	C₂²	#13094002	1	1	yes	characteristic
139	C₂²	#13094014	1	1	yes	characteristic
140	C₄	—	4	2	no	—
141	C₄	—	188	2	no	—
142	C₄	—	188	2	no	—
143	C₄	C₄⋊C₄×D₄₇	1	1	yes	characteristic
144	C₄	#13093989	1	1	yes	characteristic
145	C₄	#13093996	1	1	yes	characteristic
146	C₄	#13094022	1	1	yes	characteristic
147	C₂	—	2	1	no	—
148	C₂	—	2	1	no	—
149	C₂	#15689018	1	1	yes	characteristic
150	C₂	#15689243	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#13093959

C₂

yes

maximal, characteristic

#15689159

C₂

yes

maximal, characteristic

#15689149

C₂

yes

maximal, characteristic

C₂×#13093934

C₂

yes

maximal, characteristic

#15688884

C₂

yes

maximal, characteristic

#15688882

C₂

yes

maximal, characteristic

C₄₇×#424

C₂

yes

maximal, characteristic

C₂×#10625329

C₂²

yes

characteristic

C₂²×C₄₇⋊C₈

C₂²

yes

characteristic

#13093983

C₂²

yes

characteristic

C₂×C₄×Dic₄₇

C₂²

yes

characteristic

#13093959

—

#13093957

C₄

yes

complemented

#13093955

C₄

yes

complemented

Dic₄₇⋊C₈

—

#13093934

—

C₂×M₁₆×C₄₇

C₂²

yes

characteristic

C₂²×C₈×C₄₇

C₂²

yes

characteristic

#117×C₄₇

C₂²

yes

characteristic

C₄⋊C₈×C₄₇

—

C₂²×C₄×C₄₇

C₂³

yes

characteristic

C₂²×Dic₄₇

—

M₁₆×C₄₇

—

M₁₆×C₄₇

—

C₂×C₈×C₄₇

—

C₂×C₈×C₄₇

—

C₂×C₈×C₄₇

—

C₂×C₈×C₄₇

D₄

yes

—

C₄⋊C₄×C₄₇

C₂×C₄

yes

—

C₂²⋊C₄×C₄₇

—

C₄²×C₄₇

—

C₂²⋊Dic₄₇

—

C₄⋊Dic₄₇

C₂×C₄

yes

—

C₄×Dic₄₇

—

C₄×Dic₄₇

—

C₄×Dic₄₇

Q₈

yes

—

#10625329

—

#10625329

—

C₂×C₄₇⋊C₈

—

C₂×C₄₇⋊C₈

C₂×C₄

yes

characteristic

C₂×C₄₇⋊C₈

C₂×C₄

yes

characteristic

C₂×C₄₇⋊C₈

D₄

yes

—

C₂×C₄₇⋊C₈

Q₈

yes

—

C₂³×C₄₇

#41

yes

characteristic

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

C₂²×C₄

yes

characteristic

C₂×C₄×C₄₇

C₂×D₄

yes

characteristic

C₂×C₄×C₄₇

C₂×D₄

yes

characteristic

C₂×C₄×C₄₇

C₂×Q₈

yes

characteristic

C₂×C₄×C₄₇

C₂×Q₈

yes

characteristic

C₂×C₄×C₄₇

#41

yes

characteristic

C₂×Dic₄₇

—

C₂×Dic₄₇

—

C₂×Dic₄₇

—

C₈×C₄₇

—

C₈×C₄₇

—

C₄₇⋊C₈

—

C₄₇⋊C₈

C₄⋊C₄

yes

—

C₄₇⋊C₈

C₄⋊C₄

yes

—

C₂²×C₄₇

—

C₂²×C₄₇

C₄×D₄

yes

characteristic

C₂²×C₄₇

C₂²⋊Q₈

yes

characteristic

C₂²×C₄₇

#125

yes

characteristic

C₄×C₄₇

—

C₄×C₄₇

C₂×C₄⋊C₄

yes

characteristic

C₄×C₄₇

C₄×Q₈

yes

characteristic

C₄×C₄₇

#121

yes

characteristic

C₄×C₄₇

C₄⋊Q₈

yes

characteristic

Dic₄₇

—

Dic₄₇

—

#1946

—

maximal, Sylow

C₂×C₄₇

—

C₂×C₄₇

—

C₂×C₄₇

#389

yes

characteristic

C₂×C₄₇

#444

yes

—

—

—

—

D₄₇

yes

characteristic

C₂×#97

—

#339

—

#339

—

C₄₇

#1946

yes

characteristic, complemented, Sylow

C₂×M₁₆

—

C₂×M₁₆

C₂×D₄₇

yes

characteristic

C₂²×C₈

—

C₂²×C₈

C₂×D₄₇

yes

characteristic

#117

—

#117

C₂×D₄₇

yes

characteristic

C₂×C₄²

—

#108

—

#107

—

#106

—

C₄⋊C₈

—

C₄⋊C₈

—

#97

—

C₂²×C₄

—

C₂²×C₄

C₂²×D₄₇

yes

characteristic

M₁₆

—

100

M₁₆

—

101

M₁₆

—

102

M₁₆

—

103

C₂×C₈

—

104

C₂×C₈

—

105

C₂×C₈

—

106

C₂×C₈

—

107

C₂×C₈

—

108

C₂×C₈

—

109

C₂×C₈

—

110

C₂×C₈

—

111

C₂×C₈

C₂²⋊D₄₇

yes

—

112

C₄⋊C₄

—

113

C₄⋊C₄

C₄×D₄₇

yes

—

114

C₂²⋊C₄

—

115

C₂²⋊C₄

—

116

C₄²

—

117

C₄²

—

118

C₄²

—

119

C₄²

—

120

C₂³

#10625350

yes

characteristic

121

C₂×C₄

—

122

C₂×C₄

—

123

C₂×C₄

—

124

C₂×C₄

—

188

—

125

C₂×C₄

C₂×C₄×D₄₇

yes

characteristic

126

C₂×C₄

D₄×D₄₇

yes

characteristic

127

C₂×C₄

#10625352

yes

characteristic

128

C₂×C₄

Q₈×D₄₇

yes

characteristic

129

C₂×C₄

Q₈×D₄₇

yes

characteristic

130

C₂×C₄

C₂×C₂²⋊D₄₇

yes

characteristic

131

C₈

—

132

C₈

—

133

C₈

—

134

C₈

—

135

C₈

—

188

—

136

C₂²

—

137

C₂²

C₄×C₂²⋊D₄₇

yes

characteristic

138

C₂²

#13094002

yes

characteristic

139

C₂²

#13094014

yes

characteristic

140

C₄

—

141

C₄

—

188

—

142

C₄

—

188

—

143

C₄

C₄⋊C₄×D₄₇

yes

characteristic

144

C₄

#13093989

yes

characteristic

145

C₄

#13093996

yes

characteristic

146

C₄

#13094022

yes

characteristic

147

C₂

—

148

C₂

—

149

C₂

#15689018

yes

characteristic

150

C₂

#15689243

yes

—

	subgroup	G/H
center	C₂×C₄	C₂×C₂²⋊D₄₇
commutator	C₂×C₄×C₄₇	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₄₇
Fitting	C₄₇×#424	C₂
socle	C₂²×C₄₇	C₄×D₄
hypercenter	#424	D₄₇
hypocenter	C₄₇	#1946
LCS	G ▷ C₂×C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₈ ◁ #424

subgroup

G/H

center

C₂×C₄

C₂×C₂²⋊D₄₇

commutator

C₂×C₄×C₄₇

C₂²×C₄

Frattini

C₂²×C₄

C₂²×D₄₇

Fitting

C₄₇×#424

C₂

socle

C₂²×C₄₇

C₄×D₄

D₄₇

C₄₇

G ▷ C₂×C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇

UCS

C₁ ◁ C₂×C₄ ◁ C₂²×C₈ ◁ #424

N	H ≅ G/N	type
13. #13093957	C₄	semidirect
14. #13093955	C₄	semidirect
83. C₄₇	#1946	semidirect

H ≅ G/N

type

13. #13093957

C₄

semidirect

14. #13093955

C₄

semidirect

83. C₄₇

#1946

semidirect

Subgroups — #20244138

Conjugacy classes of subgroups

Complements