Group #20242060 — subgroups

Subgroups — #20242060

Order 6016.

240 conjugacy classes of subgroups; only the 101 normal subgroups are shown.

normal subgroups only

This group has 240 conjugacy classes of subgroups — too many to draw. Only its 101 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	#15689176	C₂	1	1	yes	maximal, characteristic
2	#15689147	C₂	1	1	yes	maximal, characteristic
3	#15688869	C₂	1	4	yes	maximal, complemented
4	C₂×C₄⋊Q₈×C₄₇	C₂	1	1	yes	maximal, characteristic
5	C₂×#10625329	—	2	1	no	—
6	C₄⋊(C₄₇⋊C₈)	—	2	1	no	—
7	C₄⋊(C₄₇⋊C₈)	C₂²	1	2	yes	—
8	C₄⋊(C₄₇⋊C₈)	C₂²	1	2	yes	—
9	C₂²⋊(C₄₇⋊C₈)	—	2	1	no	—
10	C₄×C₄₇⋊C₈	—	2	1	no	—
11	C₂²×Q₈×C₄₇	—	2	1	no	—
12	C₄⋊Q₈×C₄₇	—	2	1	no	—
13	C₄⋊Q₈×C₄₇	—	2	1	no	—
14	C₄⋊Q₈×C₄₇	C₄	1	2	yes	—
15	C₄⋊Q₈×C₄₇	C₂²	1	2	yes	—
16	C₂×C₄⋊C₄×C₄₇	—	2	1	no	—
17	C₂×C₄⋊C₄×C₄₇	C₄	1	2	yes	—
18	C₂×C₄²×C₄₇	C₂²	1	1	yes	characteristic
19	C₂×Q₈×C₄₇	—	2	2	no	—
20	C₂×Q₈×C₄₇	—	2	2	no	—
21	C₂×Q₈×C₄₇	—	4	2	no	—
22	C₂×Q₈×C₄₇	—	4	2	no	—
23	C₂²×C₄×C₄₇	—	2	2	no	—
24	C₂²×C₄×C₄₇	C₂×C₄	1	1	yes	characteristic
25	C₂²×C₄×C₄₇	D₄	1	1	yes	characteristic
26	C₂²×C₄×C₄₇	D₄	1	1	yes	characteristic
27	C₄⋊C₄×C₄₇	—	2	2	no	—
28	C₄⋊C₄×C₄₇	—	2	2	no	—
29	C₄⋊C₄×C₄₇	—	2	2	no	—
30	C₄⋊C₄×C₄₇	C₂×C₄	1	4	yes	—
31	C₄²×C₄₇	C₂×C₄	1	1	yes	characteristic
32	C₄²×C₄₇	D₄	1	1	yes	characteristic
33	C₄²×C₄₇	D₄	1	1	yes	characteristic
34	C₄²×C₄₇	C₂³	1	1	yes	characteristic
35	#10625329	—	4	2	no	—
36	C₂×C₄₇⋊C₈	—	2	2	no	—
37	C₂×C₄₇⋊C₈	—	2	2	no	—
38	C₂³×C₄₇	C₂²⋊C₄	1	1	yes	characteristic
39	Q₈×C₄₇	—	4	4	no	—
40	Q₈×C₄₇	—	4	2	no	—
41	Q₈×C₄₇	—	4	2	no	—
42	C₂×C₄×C₄₇	—	2	2	no	—
43	C₂×C₄×C₄₇	—	2	2	no	—
44	C₂×C₄×C₄₇	—	2	4	no	—
45	C₂×C₄×C₄₇	—	4	4	no	—
46	C₂×C₄×C₄₇	C₂²⋊C₄	1	1	yes	characteristic
47	C₂×C₄×C₄₇	C₂²⋊C₄	1	1	yes	characteristic
48	C₂×C₄×C₄₇	C₂²⋊C₄	1	1	yes	characteristic
49	C₂×C₄×C₄₇	D₈	1	2	yes	—
50	C₂×C₄×C₄₇	SD₁₆	1	2	yes	—
51	C₂×C₄×C₄₇	C₂²×C₄	1	1	yes	characteristic
52	C₂×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
53	C₂×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
54	C₄₇⋊C₈	—	4	2	no	—
55	C₄₇⋊C₈	—	4	2	no	—
56	C₂²×C₄₇	—	2	1	no	—
57	C₂²×C₄₇	—	2	1	no	—
58	C₂²×C₄₇	#102	1	2	yes	—
59	C₂²×C₄₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
60	C₄×C₄₇	—	2	1	no	—
61	C₄×C₄₇	—	2	1	no	—
62	C₄×C₄₇	—	4	4	no	—
63	C₄×C₄₇	#101	1	2	yes	—
64	C₄×C₄₇	#102	1	2	yes	—
65	C₄×C₄₇	C₂×D₈	1	1	yes	characteristic
66	C₄×C₄₇	C₂×SD₁₆	1	1	yes	characteristic
67	C₄×C₄₇	#137	1	2	yes	—
68	#1546	—	47	1	no	maximal, Sylow
69	C₂×C₄₇	—	2	2	no	—
70	C₂×C₄₇	C₂×#101	1	1	yes	characteristic
71	C₂×C₄₇	C₂×#102	1	1	yes	characteristic
72	C₂×C₄₇	#419	1	1	yes	characteristic
73	C₂×C₄⋊Q₈	D₄₇	1	1	yes	characteristic
74	#431	—	47	1	no	—
75	#423	—	47	1	no	—
76	#332	—	47	4	no	—
77	C₄₇	#1546	1	1	yes	characteristic, complemented, Sylow
78	C₂²×Q₈	—	2	1	no	—
79	C₂×M₁₆	—	94	1	no	—
80	C₄⋊Q₈	—	2	1	no	—
81	C₄⋊Q₈	—	2	1	no	—
82	C₄⋊Q₈	Dic₄₇	1	2	yes	—
83	C₄⋊Q₈	C₂×D₄₇	1	2	yes	—
84	C₂×C₄⋊C₄	—	2	1	no	—
85	C₂×C₄⋊C₄	Dic₄₇	1	2	yes	—
86	C₂×C₄²	C₂×D₄₇	1	1	yes	characteristic
87	C₄⋊C₈	—	47	2	no	—
88	C₄⋊C₈	—	47	2	no	—
89	C₄⋊C₈	—	94	1	no	—
90	C₂²⋊C₈	—	94	1	no	—
91	C₄×C₈	—	94	1	no	—
92	C₂×Q₈	—	2	2	no	—
93	C₂×Q₈	—	2	2	no	—
94	C₂×Q₈	—	4	2	no	—
95	C₂×Q₈	—	4	2	no	—
96	C₂²×C₄	—	2	2	no	—
97	C₂²×C₄	C₂×Dic₄₇	1	1	yes	characteristic
98	C₂²×C₄	C₂²⋊D₄₇	1	1	yes	characteristic
99	C₂²×C₄	C₂²⋊D₄₇	1	1	yes	characteristic
100	M₁₆	—	188	2	no	—
101	C₂×C₈	—	94	2	no	—
102	C₂×C₈	—	94	2	no	—
103	C₄⋊C₄	—	2	2	no	—
104	C₄⋊C₄	—	2	2	no	—
105	C₄⋊C₄	—	2	2	no	—
106	C₄⋊C₄	C₂×Dic₄₇	1	4	yes	—
107	C₄²	C₂×Dic₄₇	1	1	yes	characteristic
108	C₄²	C₂²⋊D₄₇	1	1	yes	characteristic
109	C₄²	C₂²⋊D₄₇	1	1	yes	characteristic
110	C₄²	C₂²×D₄₇	1	1	yes	characteristic
111	C₂³	C₂²⋊Dic₄₇	1	1	yes	characteristic
112	Q₈	—	4	4	no	—
113	Q₈	—	4	2	no	—
114	Q₈	—	4	2	no	—
115	C₂×C₄	—	2	2	no	—
116	C₂×C₄	—	2	2	no	—
117	C₂×C₄	—	2	4	no	—
118	C₂×C₄	—	4	4	no	—
119	C₂×C₄	#10625334	1	2	yes	—
120	C₂×C₄	#10625335	1	2	yes	—
121	C₂×C₄	C₂²⋊Dic₄₇	1	1	yes	characteristic
122	C₂×C₄	C₂²⋊Dic₄₇	1	1	yes	characteristic
123	C₂×C₄	C₂²⋊Dic₄₇	1	1	yes	characteristic
124	C₂×C₄	C₂²×Dic₄₇	1	1	yes	characteristic
125	C₂×C₄	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
126	C₂×C₄	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
127	C₈	—	188	2	no	—
128	C₈	—	188	2	no	—
129	C₂²	—	2	1	no	—
130	C₂²	—	2	1	no	—
131	C₂²	#13093945	1	2	yes	—
132	C₂²	C₂×C₂²⋊Dic₄₇	1	1	yes	characteristic
133	C₄	—	2	1	no	—
134	C₄	—	2	1	no	—
135	C₄	—	4	4	no	—
136	C₄	#13093943	1	2	yes	—
137	C₄	#13093945	1	2	yes	—
138	C₄	C₂×#10625334	1	1	yes	characteristic
139	C₄	C₂×#10625335	1	1	yes	characteristic
140	C₄	#13094055	1	2	yes	—
141	C₂	—	2	2	no	—
142	C₂	C₂×#13093943	1	1	yes	characteristic
143	C₂	C₂×#13093945	1	1	yes	characteristic
144	C₂	#15689132	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#15689176

C₂

yes

maximal, characteristic

#15689147

C₂

yes

maximal, characteristic

#15688869

C₂

yes

maximal, complemented

C₂×C₄⋊Q₈×C₄₇

C₂

yes

maximal, characteristic

C₂×#10625329

—

C₄⋊(C₄₇⋊C₈)

—

C₄⋊(C₄₇⋊C₈)

C₂²

yes

—

C₄⋊(C₄₇⋊C₈)

C₂²

yes

—

C₂²⋊(C₄₇⋊C₈)

—

C₄×C₄₇⋊C₈

—

C₂²×Q₈×C₄₇

—

C₄⋊Q₈×C₄₇

—

C₄⋊Q₈×C₄₇

—

C₄⋊Q₈×C₄₇

C₄

yes

—

C₄⋊Q₈×C₄₇

C₂²

yes

—

C₂×C₄⋊C₄×C₄₇

—

C₂×C₄⋊C₄×C₄₇

C₄

yes

—

C₂×C₄²×C₄₇

C₂²

yes

characteristic

C₂×Q₈×C₄₇

—

C₂×Q₈×C₄₇

—

C₂×Q₈×C₄₇

—

C₂×Q₈×C₄₇

—

C₂²×C₄×C₄₇

—

C₂²×C₄×C₄₇

C₂×C₄

yes

characteristic

C₂²×C₄×C₄₇

D₄

yes

characteristic

C₂²×C₄×C₄₇

D₄

yes

characteristic

C₄⋊C₄×C₄₇

—

C₄⋊C₄×C₄₇

—

C₄⋊C₄×C₄₇

—

C₄⋊C₄×C₄₇

C₂×C₄

yes

—

C₄²×C₄₇

C₂×C₄

yes

characteristic

C₄²×C₄₇

D₄

yes

characteristic

C₄²×C₄₇

D₄

yes

characteristic

C₄²×C₄₇

C₂³

yes

characteristic

#10625329

—

C₂×C₄₇⋊C₈

—

C₂×C₄₇⋊C₈

—

C₂³×C₄₇

C₂²⋊C₄

yes

characteristic

Q₈×C₄₇

—

Q₈×C₄₇

—

Q₈×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

C₂²⋊C₄

yes

characteristic

C₂×C₄×C₄₇

C₂²⋊C₄

yes

characteristic

C₂×C₄×C₄₇

C₂²⋊C₄

yes

characteristic

C₂×C₄×C₄₇

D₈

yes

—

C₂×C₄×C₄₇

SD₁₆

yes

—

C₂×C₄×C₄₇

C₂²×C₄

yes

characteristic

C₂×C₄×C₄₇

C₂×D₄

yes

characteristic

C₂×C₄×C₄₇

C₂×D₄

yes

characteristic

C₄₇⋊C₈

—

C₄₇⋊C₈

—

C₂²×C₄₇

—

C₂²×C₄₇

—

C₂²×C₄₇

#102

yes

—

C₂²×C₄₇

C₂×C₂²⋊C₄

yes

characteristic

C₄×C₄₇

—

C₄×C₄₇

—

C₄×C₄₇

—

C₄×C₄₇

#101

yes

—

C₄×C₄₇

#102

yes

—

C₄×C₄₇

C₂×D₈

yes

characteristic

C₄×C₄₇

C₂×SD₁₆

yes

characteristic

C₄×C₄₇

#137

yes

—

#1546

—

maximal, Sylow

C₂×C₄₇

—

C₂×C₄₇

yes

C₂×C₄₇

yes

C₂×C₄₇

yes

C₂×C₄⋊Q₈

D₄₇

yes

—

—

—

C₄₇

#1546

yes

characteristic, complemented, Sylow

C₂²×Q₈

—

C₂×M₁₆

—

C₄⋊Q₈

—

C₄⋊Q₈

—

C₄⋊Q₈

Dic₄₇

yes

—

C₄⋊Q₈

C₂×D₄₇

yes

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

Dic₄₇

yes

—

C₂×C₄²

C₂×D₄₇

yes

characteristic

C₄⋊C₈

—

C₄⋊C₈

—

C₄⋊C₈

—

C₂²⋊C₈

—

C₄×C₈

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

C₂²×C₄

—

C₂²×C₄

C₂×Dic₄₇

yes

characteristic

C₂²×C₄

C₂²⋊D₄₇

yes

characteristic

C₂²×C₄

C₂²⋊D₄₇

yes

characteristic

100

M₁₆

—

188

—

101

C₂×C₈

—

102

C₂×C₈

—

103

C₄⋊C₄

—

104

C₄⋊C₄

—

105

C₄⋊C₄

—

106

C₄⋊C₄

C₂×Dic₄₇

yes

—

107

C₄²

C₂×Dic₄₇

yes

characteristic

108

C₄²

C₂²⋊D₄₇

yes

characteristic

109

C₄²

C₂²⋊D₄₇

yes

characteristic

110

C₄²

C₂²×D₄₇

yes

characteristic

111

C₂³

C₂²⋊Dic₄₇

yes

characteristic

112

Q₈

—

113

Q₈

—

114

Q₈

—

115

C₂×C₄

—

116

C₂×C₄

—

117

C₂×C₄

—

118

C₂×C₄

—

119

C₂×C₄

#10625334

yes

—

120

C₂×C₄

#10625335

yes

—

121

C₂×C₄

C₂²⋊Dic₄₇

yes

characteristic

122

C₂×C₄

C₂²⋊Dic₄₇

yes

characteristic

123

C₂×C₄

C₂²⋊Dic₄₇

yes

characteristic

124

C₂×C₄

C₂²×Dic₄₇

yes

characteristic

125

C₂×C₄

C₂×C₂²⋊D₄₇

yes

characteristic

126

C₂×C₄

C₂×C₂²⋊D₄₇

yes

characteristic

127

C₈

—

188

—

128

C₈

—

188

—

129

C₂²

—

130

C₂²

—

131

C₂²

#13093945

yes

—

132

C₂²

C₂×C₂²⋊Dic₄₇

yes

characteristic

133

C₄

—

134

C₄

—

135

C₄

—

136

C₄

#13093943

yes

—

137

C₄

#13093945

yes

—

138

C₄

C₂×#10625334

yes

characteristic

139

C₄

C₂×#10625335

yes

characteristic

140

C₄

#13094055

yes

—

141

C₂

—

142

C₂

C₂×#13093943

yes

characteristic

143

C₂

C₂×#13093945

yes

characteristic

144

C₂

#15689132

yes

characteristic

	subgroup	G/H
center	C₂²	C₂×C₂²⋊Dic₄₇
commutator	C₂×C₄×C₄₇	C₂²×C₄
Frattini	C₄²	C₂²×D₄₇
Fitting	C₂×C₄⋊Q₈×C₄₇	C₂
socle	C₂²×C₄₇	C₂×C₂²⋊C₄
hypercenter	C₂×C₄⋊Q₈	D₄₇
hypocenter	C₄₇	#1546
LCS	G ▷ C₂×C₄×C₄₇ ▷ C₂²×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×C₄⋊Q₈

subgroup

G/H

center

C₂²

C₂×C₂²⋊Dic₄₇

commutator

C₂×C₄×C₄₇

C₂²×C₄

Frattini

C₄²

C₂²×D₄₇

Fitting

C₂×C₄⋊Q₈×C₄₇

C₂

socle

C₂²×C₄₇

C₂×C₂²⋊C₄

hypercenter

C₂×C₄⋊Q₈

D₄₇

hypocenter

C₄₇

#1546

LCS

G ▷ C₂×C₄×C₄₇ ▷ C₂²×C₄₇ ▷ C₄₇

UCS

C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×C₄⋊Q₈

N	H ≅ G/N	type
3. #15688869	C₂	semidirect
77. C₄₇	#1546	semidirect

H ≅ G/N

type

3. #15688869

C₂

semidirect

77. C₄₇

#1546

semidirect

Subgroups — #20242060

Conjugacy classes of subgroups

Complements