Group #20244350 — subgroups

Subgroups — #20244350

Order 6016.

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

normal subgroups only

This group has 300 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×C₄₇⋊C₈	C₂	1	1	yes	maximal, characteristic
2	#15688993	C₂	1	1	yes	maximal, characteristic
3	#15688885	C₂	1	4	yes	maximal
4	C₂×C₄⋊Q₈×C₄₇	C₂	1	1	yes	maximal, characteristic
5	C₂²×C₄₇⋊C₈	C₂²	1	2	yes	—
6	C₂×C₄⋊Dic₄₇	—	2	1	no	—
7	C₂×C₄⋊Dic₄₇	—	2	1	no	—
8	C₂×C₄⋊Dic₄₇	C₂²	1	2	yes	—
9	C₄×C₄₇⋊C₈	—	2	2	no	—
10	C₂²×Q₈×C₄₇	—	2	1	no	—
11	C₄⋊Q₈×C₄₇	—	2	2	no	—
12	C₄⋊Q₈×C₄₇	C₄	1	4	yes	complemented
13	C₂×C₄⋊C₄×C₄₇	—	2	1	no	—
14	C₂×C₄⋊C₄×C₄₇	C₂²	1	2	yes	—
15	C₂×C₄²×C₄₇	C₂²	1	1	yes	characteristic
16	C₂×Q₈×C₄₇	—	2	4	no	—
17	C₂×Q₈×C₄₇	—	4	2	no	—
18	C₂×Q₈×C₄₇	—	4	2	no	—
19	C₂²×C₄×C₄₇	—	2	2	no	—
20	C₂²×C₄×C₄₇	D₄	1	1	yes	characteristic
21	C₂²×C₄×C₄₇	D₄	1	1	yes	characteristic
22	C₂²×C₄×C₄₇	C₂³	1	1	yes	characteristic
23	C₂²×Dic₄₇	—	2	2	no	—
24	C₄⋊C₄×C₄₇	—	2	4	no	—
25	C₄⋊C₄×C₄₇	—	2	2	no	—
26	C₄⋊C₄×C₄₇	C₂×C₄	1	4	yes	—
27	C₄²×C₄₇	C₂×C₄	1	2	yes	—
28	C₄²×C₄₇	D₄	1	2	yes	—
29	C₄⋊Dic₄₇	—	2	4	no	—
30	C₄⋊Dic₄₇	—	4	2	no	—
31	C₄⋊Dic₄₇	—	4	2	no	—
32	C₂×C₄₇⋊C₈	—	2	2	no	—
33	C₂×C₄₇⋊C₈	—	2	2	no	—
34	C₂×C₄₇⋊C₈	Q₈	1	4	yes	complemented
35	C₂³×C₄₇	C₂×D₄	1	1	yes	characteristic
36	Q₈×C₄₇	—	4	8	no	—
37	C₂×C₄×C₄₇	—	2	4	no	—
38	C₂×C₄×C₄₇	—	4	2	no	—
39	C₂×C₄×C₄₇	—	4	2	no	—
40	C₂×C₄×C₄₇	C₂²⋊C₄	1	2	yes	—
41	C₂×C₄×C₄₇	C₂²⋊C₄	1	2	yes	—
42	C₂×C₄×C₄₇	D₈	1	2	yes	—
43	C₂×C₄×C₄₇	SD₁₆	1	2	yes	—
44	C₂×C₄×C₄₇	SD₁₆	1	2	yes	—
45	C₂×C₄×C₄₇	Q₁₆	1	2	yes	—
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×Q₈	1	2	yes	—
49	C₂×C₄×C₄₇	#41	1	2	yes	—
50	C₂×Dic₄₇	—	4	2	no	—
51	C₂×Dic₄₇	—	4	2	no	—
52	C₂×Dic₄₇	—	4	2	no	—
53	C₄₇⋊C₈	—	2	4	no	—
54	C₂²×C₄₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
55	C₂²×C₄₇	#124	1	1	yes	characteristic
56	C₂²×C₄₇	C₄⋊Q₈	1	1	yes	characteristic
57	C₂²×C₄₇	C₂×D₈	1	1	yes	characteristic
58	C₂²×C₄₇	C₂×SD₁₆	1	1	yes	characteristic
59	C₂²×C₄₇	C₂×SD₁₆	1	1	yes	characteristic
60	C₂²×C₄₇	C₂×Q₁₆	1	1	yes	characteristic
61	C₄×C₄₇	—	4	4	no	—
62	C₄×C₄₇	#102	1	4	yes	—
63	C₄×C₄₇	#103	1	4	yes	—
64	C₄×C₄₇	C₄×Q₈	1	2	yes	—
65	C₄×C₄₇	C₂²⋊Q₈	1	2	yes	—
66	Dic₄₇	—	8	2	no	—
67	#1990	—	47	1	no	maximal, Sylow
68	C₂×C₄₇	#391	1	1	yes	characteristic
69	C₂×C₄₇	C₂×#102	1	1	yes	characteristic
70	C₂×C₄₇	C₂×#103	1	1	yes	characteristic
71	C₂×C₄₇	#486	1	1	yes	characteristic
72	C₂×C₄₇	#487	1	1	yes	characteristic
73	C₂×C₄₇	#498	1	1	yes	characteristic
74	C₂×C₄₇	#500	1	1	yes	characteristic
75	C₂×C₄⋊Q₈	D₄₇	1	1	yes	characteristic
76	C₂×C₄×C₈	—	47	1	no	—
77	#384	—	47	1	no	—
78	#340	—	47	4	no	—
79	C₄₇	#1990	1	1	yes	characteristic, complemented, Sylow
80	C₂²×Q₈	—	2	1	no	—
81	C₂²×C₈	—	47	2	no	—
82	C₄⋊Q₈	—	2	2	no	—
83	C₄⋊Q₈	Dic₄₇	1	4	yes	complemented
84	C₂×C₄⋊C₄	—	2	1	no	—
85	C₂×C₄⋊C₄	—	47	2	no	—
86	C₂×C₄⋊C₄	—	94	1	no	—
87	C₂×C₄⋊C₄	—	94	1	no	—
88	C₂×C₄⋊C₄	C₂×D₄₇	1	2	yes	—
89	C₂×C₄²	C₂×D₄₇	1	1	yes	characteristic
90	C₄×C₈	—	94	2	no	—
91	C₂×Q₈	—	2	4	no	—
92	C₂×Q₈	—	4	2	no	—
93	C₂×Q₈	—	4	2	no	—
94	C₂²×C₄	—	2	2	no	—
95	C₂²×C₄	—	94	2	no	—
96	C₂²×C₄	C₂²⋊D₄₇	1	1	yes	characteristic
97	C₂²×C₄	C₂²⋊D₄₇	1	1	yes	characteristic
98	C₂²×C₄	C₂²×D₄₇	1	1	yes	characteristic
99	C₂×C₈	—	47	4	no	—
100	C₂×C₈	—	94	2	no	—
101	C₂×C₈	—	94	2	no	—
102	C₄⋊C₄	—	2	4	no	—
103	C₄⋊C₄	—	2	2	no	—
104	C₄⋊C₄	—	94	4	no	—
105	C₄⋊C₄	—	188	2	no	—
106	C₄⋊C₄	—	188	2	no	—
107	C₄⋊C₄	C₂×Dic₄₇	1	4	yes	—
108	C₄²	C₂×Dic₄₇	1	2	yes	—
109	C₄²	C₂²⋊D₄₇	1	2	yes	—
110	C₂³	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
111	Q₈	—	4	8	no	—
112	C₂×C₄	—	2	4	no	—
113	C₂×C₄	—	4	2	no	—
114	C₂×C₄	—	4	2	no	—
115	C₂×C₄	—	188	2	no	—
116	C₂×C₄	—	188	2	no	—
117	C₂×C₄	—	188	2	no	—
118	C₂×C₄	#10625334	1	2	yes	—
119	C₂×C₄	#10625335	1	2	yes	—
120	C₂×C₄	#10625336	1	2	yes	—
121	C₂×C₄	#10625337	1	2	yes	—
122	C₂×C₄	C₂²⋊Dic₄₇	1	2	yes	—
123	C₂×C₄	C₂²⋊Dic₄₇	1	2	yes	—
124	C₂×C₄	Q₈×D₄₇	1	2	yes	—
125	C₂×C₄	#10625354	1	2	yes	—
126	C₂×C₄	C₂²×Dic₄₇	1	1	yes	characteristic
127	C₂×C₄	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
128	C₈	—	94	4	no	—
129	C₂²	C₂×C₂²⋊Dic₄₇	1	1	yes	characteristic
130	C₂²	#13094013	1	1	yes	characteristic
131	C₂²	#13094022	1	1	yes	characteristic
132	C₂²	C₂×#10625334	1	1	yes	characteristic
133	C₂²	C₂×#10625336	1	1	yes	characteristic
134	C₂²	C₂×#10625335	1	1	yes	characteristic
135	C₂²	C₂×#10625337	1	1	yes	characteristic
136	C₄	—	4	4	no	—
137	C₄	—	376	2	no	—
138	C₄	#13093945	1	4	yes	—
139	C₄	#13093948	1	4	yes	—
140	C₄	Q₈×Dic₄₇	1	2	yes	—
141	C₄	#13094000	1	2	yes	—
142	C₂	#15689030	1	1	yes	characteristic
143	C₂	C₂×#13093945	1	1	yes	characteristic
144	C₂	C₂×#13093948	1	1	yes	characteristic
145	C₂	#15689471	1	1	yes	characteristic
146	C₂	#15689475	1	1	yes	characteristic
147	C₂	#15689519	1	1	yes	characteristic
148	C₂	#15689526	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×C₄×C₄₇⋊C₈

C₂

yes

maximal, characteristic

#15688993

C₂

yes

maximal, characteristic

#15688885

C₂

yes

maximal

C₂×C₄⋊Q₈×C₄₇

C₂

yes

maximal, characteristic

C₂²×C₄₇⋊C₈

C₂²

yes

—

C₂×C₄⋊Dic₄₇

—

C₂×C₄⋊Dic₄₇

—

C₂×C₄⋊Dic₄₇

C₂²

yes

—

C₄×C₄₇⋊C₈

—

C₂²×Q₈×C₄₇

—

C₄⋊Q₈×C₄₇

—

C₄⋊Q₈×C₄₇

C₄

yes

complemented

C₂×C₄⋊C₄×C₄₇

—

C₂×C₄⋊C₄×C₄₇

C₂²

yes

—

C₂×C₄²×C₄₇

C₂²

yes

characteristic

C₂×Q₈×C₄₇

—

C₂×Q₈×C₄₇

—

C₂×Q₈×C₄₇

—

C₂²×C₄×C₄₇

—

C₂²×C₄×C₄₇

D₄

yes

characteristic

C₂²×C₄×C₄₇

D₄

yes

characteristic

C₂²×C₄×C₄₇

C₂³

yes

characteristic

C₂²×Dic₄₇

—

C₄⋊C₄×C₄₇

—

C₄⋊C₄×C₄₇

—

C₄⋊C₄×C₄₇

C₂×C₄

yes

—

C₄²×C₄₇

C₂×C₄

yes

—

C₄²×C₄₇

D₄

yes

—

C₄⋊Dic₄₇

—

C₄⋊Dic₄₇

—

C₄⋊Dic₄₇

—

C₂×C₄₇⋊C₈

—

C₂×C₄₇⋊C₈

—

C₂×C₄₇⋊C₈

Q₈

yes

complemented

C₂³×C₄₇

C₂×D₄

yes

characteristic

Q₈×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

C₂²⋊C₄

yes

—

C₂×C₄×C₄₇

C₂²⋊C₄

yes

—

C₂×C₄×C₄₇

D₈

yes

—

C₂×C₄×C₄₇

SD₁₆

yes

—

C₂×C₄×C₄₇

SD₁₆

yes

—

C₂×C₄×C₄₇

Q₁₆

yes

—

C₂×C₄×C₄₇

C₂²×C₄

yes

characteristic

C₂×C₄×C₄₇

C₂×D₄

yes

characteristic

C₂×C₄×C₄₇

C₂×Q₈

yes

—

C₂×C₄×C₄₇

#41

yes

—

C₂×Dic₄₇

—

C₂×Dic₄₇

—

C₂×Dic₄₇

—

C₄₇⋊C₈

—

C₂²×C₄₇

C₂×C₂²⋊C₄

yes

characteristic

C₂²×C₄₇

#124

yes

characteristic

C₂²×C₄₇

C₄⋊Q₈

yes

characteristic

C₂²×C₄₇

C₂×D₈

yes

characteristic

C₂²×C₄₇

C₂×SD₁₆

yes

characteristic

C₂²×C₄₇

C₂×SD₁₆

yes

characteristic

C₂²×C₄₇

C₂×Q₁₆

yes

characteristic

C₄×C₄₇

—

C₄×C₄₇

#102

yes

—

C₄×C₄₇

#103

yes

—

C₄×C₄₇

C₄×Q₈

yes

—

C₄×C₄₇

C₂²⋊Q₈

yes

—

Dic₄₇

—

—

C₂×C₄₇

yes

C₂×C₄₇

yes

C₂×C₄₇

yes

C₂×C₄₇

yes

C₂×C₄₇

yes

C₂×C₄₇

yes

C₂×C₄₇

yes

C₂×C₄⋊Q₈

D₄₇

yes

characteristic

C₂×C₄×C₈

—

#384

—

#340

—

C₄₇

#1990

yes

characteristic, complemented, Sylow

C₂²×Q₈

—

C₂²×C₈

—

C₄⋊Q₈

—

C₄⋊Q₈

Dic₄₇

yes

complemented

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

C₂×D₄₇

yes

—

C₂×C₄²

C₂×D₄₇

yes

characteristic

C₄×C₈

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

C₂²⋊D₄₇

yes

characteristic

C₂²×C₄

C₂²⋊D₄₇

yes

characteristic

C₂²×C₄

C₂²×D₄₇

yes

characteristic

C₂×C₈

—

100

C₂×C₈

—

101

C₂×C₈

—

102

C₄⋊C₄

—

103

C₄⋊C₄

—

104

C₄⋊C₄

—

105

C₄⋊C₄

—

188

—

106

C₄⋊C₄

—

188

—

107

C₄⋊C₄

C₂×Dic₄₇

yes

—

108

C₄²

C₂×Dic₄₇

yes

—

109

C₄²

C₂²⋊D₄₇

yes

—

110

C₂³

C₂×C₂²⋊D₄₇

yes

characteristic

111

Q₈

—

112

C₂×C₄

—

113

C₂×C₄

—

114

C₂×C₄

—

115

C₂×C₄

—

188

—

116

C₂×C₄

—

188

—

117

C₂×C₄

—

188

—

118

C₂×C₄

#10625334

yes

—

119

C₂×C₄

#10625335

yes

—

120

C₂×C₄

#10625336

yes

—

121

C₂×C₄

#10625337

yes

—

122

C₂×C₄

C₂²⋊Dic₄₇

yes

—

123

C₂×C₄

C₂²⋊Dic₄₇

yes

—

124

C₂×C₄

Q₈×D₄₇

yes

—

125

C₂×C₄

#10625354

yes

—

126

C₂×C₄

C₂²×Dic₄₇

yes

characteristic

127

C₂×C₄

C₂×C₂²⋊D₄₇

yes

characteristic

128

C₈

—

129

C₂²

C₂×C₂²⋊Dic₄₇

yes

characteristic

130

C₂²

#13094013

yes

characteristic

131

C₂²

#13094022

yes

characteristic

132

C₂²

C₂×#10625334

yes

characteristic

133

C₂²

C₂×#10625336

yes

characteristic

134

C₂²

C₂×#10625335

yes

characteristic

135

C₂²

C₂×#10625337

yes

characteristic

136

C₄

—

137

C₄

—

376

—

138

C₄

#13093945

yes

—

139

C₄

#13093948

yes

—

140

C₄

Q₈×Dic₄₇

yes

—

141

C₄

#13094000

yes

—

142

C₂

#15689030

yes

characteristic

143

C₂

C₂×#13093945

yes

characteristic

144

C₂

C₂×#13093948

yes

characteristic

145

C₂

#15689471

yes

characteristic

146

C₂

#15689475

yes

characteristic

147

C₂

#15689519

yes

characteristic

148

C₂

#15689526

yes

characteristic

	subgroup	G/H
center	C₂³	C₂×C₂²⋊D₄₇
commutator	C₂×C₄×C₄₇	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₄₇
Fitting	C₂×C₄⋊Q₈×C₄₇	C₂
socle	C₂³×C₄₇	C₂×D₄
hypercenter	C₂×C₄⋊Q₈	D₄₇
hypocenter	C₄₇	#1990
LCS	G ▷ C₂×C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄⋊Q₈

subgroup

G/H

center

C₂³

C₂×C₂²⋊D₄₇

commutator

C₂×C₄×C₄₇

C₂²×C₄

Frattini

C₂²×C₄

C₂²×D₄₇

Fitting

C₂×C₄⋊Q₈×C₄₇

C₂

socle

C₂³×C₄₇

C₂×D₄

hypercenter

C₂×C₄⋊Q₈

D₄₇

hypocenter

C₄₇

#1990

LCS

G ▷ C₂×C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇

UCS

C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄⋊Q₈

N	H ≅ G/N	type
12. C₄⋊Q₈×C₄₇	C₄	semidirect
34. C₂×C₄₇⋊C₈	Q₈	semidirect
79. C₄₇	#1990	semidirect
83. C₄⋊Q₈	Dic₄₇	semidirect

H ≅ G/N

type

12. C₄⋊Q₈×C₄₇

C₄

semidirect

34. C₂×C₄₇⋊C₈

Q₈

semidirect

79. C₄₇

#1990

semidirect

83. C₄⋊Q₈

Dic₄₇

semidirect

Subgroups — #20244350

Conjugacy classes of subgroups

Complements