Group #20243478 — subgroups

Subgroups — #20243478

Order 6016.

344 conjugacy classes of subgroups; only the 125 normal subgroups are shown.

normal subgroups only

This group has 344 conjugacy classes of subgroups — too many to draw. Only its 125 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×C₂²⋊(C₄₇⋊C₈)	C₂	1	1	yes	maximal, characteristic
2	#15688983	C₂	1	1	yes	maximal, characteristic
3	#15688885	C₂	1	4	yes	maximal, complemented
4	C₂²×C₄⋊C₄×C₄₇	C₂	1	1	yes	maximal, characteristic
5	C₂²×C₄₇⋊C₈	C₂²	1	2	yes	—
6	C₂×C₄⋊Dic₄₇	C₂²	1	2	yes	—
7	C₂×C₂²⋊Dic₄₇	—	2	1	no	—
8	C₂²⋊(C₄₇⋊C₈)	—	2	2	no	—
9	C₄₇⋊#95	—	2	1	no	—
10	C₂³×C₄×C₄₇	—	2	1	no	—
11	C₂³×C₄×C₄₇	C₂²	1	1	yes	characteristic
12	C₂×C₄⋊C₄×C₄₇	—	2	1	no	—
13	C₂×C₄⋊C₄×C₄₇	—	2	2	no	—
14	C₂×C₄⋊C₄×C₄₇	C₄	1	4	yes	complemented
15	C₂×C₄⋊C₄×C₄₇	C₂²	1	2	yes	—
16	C₂⁴×C₄₇	D₄	1	1	yes	characteristic
17	C₂²×C₄×C₄₇	—	2	4	no	—
18	C₂²×C₄×C₄₇	—	2	4	no	—
19	C₂²×C₄×C₄₇	—	2	2	no	—
20	C₂²×C₄×C₄₇	—	4	4	no	—
21	C₂²×C₄×C₄₇	C₂×C₄	1	2	yes	—
22	C₂²×C₄×C₄₇	D₄	1	1	yes	characteristic
23	C₂²×C₄×C₄₇	D₄	1	2	yes	—
24	C₂²×C₄×C₄₇	C₂³	1	1	yes	characteristic
25	C₂²×Dic₄₇	—	2	2	no	—
26	C₄⋊C₄×C₄₇	—	2	4	no	—
27	C₄⋊C₄×C₄₇	—	2	2	no	—
28	C₄⋊C₄×C₄₇	C₂×C₄	1	4	yes	—
29	C₂²⋊Dic₄₇	—	4	2	no	—
30	C₄⋊Dic₄₇	—	2	4	no	—
31	C₂×C₄₇⋊C₈	—	2	2	no	—
32	C₂×C₄₇⋊C₈	—	2	4	no	—
33	C₂³×C₄₇	—	2	2	no	—
34	C₂³×C₄₇	—	2	2	no	—
35	C₂³×C₄₇	C₂²⋊C₄	1	2	yes	—
36	C₂³×C₄₇	SD₁₆	1	2	yes	—
37	C₂³×C₄₇	Q₁₆	1	2	yes	—
38	C₂³×C₄₇	C₂×D₄	1	1	yes	characteristic
39	C₂×C₄×C₄₇	—	2	2	no	—
40	C₂×C₄×C₄₇	—	2	4	no	—
41	C₂×C₄×C₄₇	—	2	4	no	—
42	C₂×C₄×C₄₇	—	2	4	no	—
43	C₂×C₄×C₄₇	—	4	8	no	—
44	C₂×C₄×C₄₇	—	4	4	no	—
45	C₂×C₄×C₄₇	C₂²⋊C₄	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	#41	1	4	yes	—
49	C₂×Dic₄₇	—	4	2	no	—
50	C₂×Dic₄₇	—	4	2	no	—
51	C₂×Dic₄₇	—	4	2	no	—
52	C₄₇⋊C₈	—	4	2	no	—
53	C₂²×C₄₇	—	2	2	no	—
54	C₂²×C₄₇	—	2	2	no	—
55	C₂²×C₄₇	—	2	4	no	—
56	C₂²×C₄₇	—	2	4	no	—
57	C₂²×C₄₇	#103	1	4	yes	—
58	C₂²×C₄₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
59	C₂²×C₄₇	#123	1	2	yes	—
60	C₂²×C₄₇	C₂×SD₁₆	1	1	yes	characteristic
61	C₂²×C₄₇	C₂×Q₁₆	1	1	yes	characteristic
62	C₂²×C₄₇	AGL(1,ℤ/8)	1	2	yes	—
63	C₄×C₄₇	—	2	2	no	—
64	C₄×C₄₇	—	4	4	no	—
65	C₄×C₄₇	#117	1	2	yes	—
66	C₄×C₄₇	#123	1	2	yes	—
67	Dic₄₇	—	8	2	no	—
68	#1824	—	47	1	no	maximal, Sylow
69	C₂×C₄₇	—	2	4	no	—
70	C₂×C₄₇	#381	1	1	yes	characteristic
71	C₂×C₄₇	C₂×#103	1	1	yes	characteristic
72	C₂×C₄₇	#418	1	1	yes	characteristic
73	C₂×C₄₇	#481	1	2	yes	—
74	C₂×C₄₇	#484	1	2	yes	—
75	C₂²×C₄⋊C₄	D₄₇	1	1	yes	characteristic
76	C₂×C₂²⋊C₈	—	47	1	no	—
77	#380	—	47	1	no	—
78	#340	—	47	4	no	—
79	C₄₇	#1824	1	1	yes	characteristic, complemented, Sylow
80	C₂³×C₄	—	2	1	no	—
81	C₂³×C₄	C₂×D₄₇	1	1	yes	characteristic
82	C₂²×C₈	—	47	2	no	—
83	C₂×C₄⋊C₄	—	2	1	no	—
84	C₂×C₄⋊C₄	—	2	2	no	—
85	C₂×C₄⋊C₄	—	47	2	no	—
86	C₂×C₄⋊C₄	Dic₄₇	1	4	yes	complemented
87	C₂×C₄⋊C₄	C₂×D₄₇	1	2	yes	—
88	C₂×C₂²⋊C₄	—	94	1	no	—
89	C₂²⋊C₈	—	94	2	no	—
90	#95	—	94	1	no	—
91	C₂⁴	C₂²⋊D₄₇	1	1	yes	characteristic
92	C₂²×C₄	—	2	4	no	—
93	C₂²×C₄	—	2	4	no	—
94	C₂²×C₄	—	2	2	no	—
95	C₂²×C₄	—	4	4	no	—
96	C₂²×C₄	—	94	2	no	—
97	C₂²×C₄	C₂×Dic₄₇	1	2	yes	—
98	C₂²×C₄	C₂²⋊D₄₇	1	1	yes	characteristic
99	C₂²×C₄	C₂²⋊D₄₇	1	2	yes	—
100	C₂²×C₄	C₂²×D₄₇	1	1	yes	characteristic
101	C₂×C₈	—	94	4	no	—
102	C₂×C₈	—	94	2	no	—
103	C₄⋊C₄	—	2	4	no	—
104	C₄⋊C₄	—	2	2	no	—
105	C₄⋊C₄	—	94	4	no	—
106	C₄⋊C₄	C₂×Dic₄₇	1	4	yes	—
107	C₂²⋊C₄	—	188	2	no	—
108	C₂³	—	2	2	no	—
109	C₂³	—	2	2	no	—
110	C₂³	#10625336	1	2	yes	—
111	C₂³	#10625337	1	2	yes	—
112	C₂³	C₂²⋊Dic₄₇	1	2	yes	—
113	C₂³	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
114	C₂×C₄	—	2	2	no	—
115	C₂×C₄	—	2	4	no	—
116	C₂×C₄	—	2	4	no	—
117	C₂×C₄	—	2	4	no	—
118	C₂×C₄	—	4	8	no	—
119	C₂×C₄	—	4	4	no	—
120	C₂×C₄	—	188	2	no	—
121	C₂×C₄	—	188	2	no	—
122	C₂×C₄	—	188	2	no	—
123	C₂×C₄	C₂²⋊Dic₄₇	1	2	yes	—
124	C₂×C₄	#10625350	1	4	yes	—
125	C₂×C₄	C₂²×Dic₄₇	1	1	yes	characteristic
126	C₂×C₄	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
127	C₈	—	188	2	no	—
128	C₂²	—	2	2	no	—
129	C₂²	—	2	2	no	—
130	C₂²	—	2	4	no	—
131	C₂²	—	2	4	no	—
132	C₂²	#13093948	1	4	yes	—
133	C₂²	C₂×C₂²⋊Dic₄₇	1	1	yes	characteristic
134	C₂²	#13094007	1	2	yes	—
135	C₂²	C₂×#10625336	1	1	yes	characteristic
136	C₂²	C₂×#10625337	1	1	yes	characteristic
137	C₂²	#13094050	1	2	yes	—
138	C₄	—	2	2	no	—
139	C₄	—	4	4	no	—
140	C₄	—	376	2	no	—
141	C₄	#13093983	1	2	yes	—
142	C₄	#13094007	1	2	yes	—
143	C₂	—	2	4	no	—
144	C₂	#15688986	1	1	yes	characteristic
145	C₂	C₂×#13093948	1	1	yes	characteristic
146	C₂	#15689128	1	1	yes	characteristic
147	C₂	#15689441	1	2	yes	—
148	C₂	#15689458	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×C₂²⋊(C₄₇⋊C₈)

C₂

yes

maximal, characteristic

#15688983

C₂

yes

maximal, characteristic

#15688885

C₂

yes

maximal, complemented

C₂²×C₄⋊C₄×C₄₇

C₂

yes

maximal, characteristic

C₂²×C₄₇⋊C₈

C₂²

yes

—

C₂×C₄⋊Dic₄₇

C₂²

yes

—

C₂×C₂²⋊Dic₄₇

—

C₂²⋊(C₄₇⋊C₈)

—

C₄₇⋊#95

—

C₂³×C₄×C₄₇

—

C₂³×C₄×C₄₇

C₂²

yes

characteristic

C₂×C₄⋊C₄×C₄₇

—

C₂×C₄⋊C₄×C₄₇

—

C₂×C₄⋊C₄×C₄₇

C₄

yes

complemented

C₂×C₄⋊C₄×C₄₇

C₂²

yes

—

C₂⁴×C₄₇

D₄

yes

characteristic

C₂²×C₄×C₄₇

—

C₂²×C₄×C₄₇

—

C₂²×C₄×C₄₇

—

C₂²×C₄×C₄₇

—

C₂²×C₄×C₄₇

C₂×C₄

yes

—

C₂²×C₄×C₄₇

D₄

yes

characteristic

C₂²×C₄×C₄₇

D₄

yes

—

C₂²×C₄×C₄₇

C₂³

yes

characteristic

C₂²×Dic₄₇

—

C₄⋊C₄×C₄₇

—

C₄⋊C₄×C₄₇

—

C₄⋊C₄×C₄₇

C₂×C₄

yes

—

C₂²⋊Dic₄₇

—

C₄⋊Dic₄₇

—

C₂×C₄₇⋊C₈

—

C₂×C₄₇⋊C₈

—

C₂³×C₄₇

—

C₂³×C₄₇

—

C₂³×C₄₇

C₂²⋊C₄

yes

—

C₂³×C₄₇

SD₁₆

yes

—

C₂³×C₄₇

Q₁₆

yes

—

C₂³×C₄₇

C₂×D₄

yes

characteristic

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

C₂²⋊C₄

yes

—

C₂×C₄×C₄₇

C₂²×C₄

yes

characteristic

C₂×C₄×C₄₇

C₂×D₄

yes

characteristic

C₂×C₄×C₄₇

#41

yes

—

C₂×Dic₄₇

—

C₂×Dic₄₇

—

C₂×Dic₄₇

—

C₄₇⋊C₈

—

C₂²×C₄₇

—

C₂²×C₄₇

—

C₂²×C₄₇

—

C₂²×C₄₇

—

C₂²×C₄₇

#103

yes

—

C₂²×C₄₇

C₂×C₂²⋊C₄

yes

characteristic

C₂²×C₄₇

#123

yes

—

C₂²×C₄₇

C₂×SD₁₆

yes

characteristic

C₂²×C₄₇

C₂×Q₁₆

yes

characteristic

C₂²×C₄₇

AGL(1,ℤ/8)

yes

—

C₄×C₄₇

—

C₄×C₄₇

—

C₄×C₄₇

#117

yes

—

C₄×C₄₇

#123

yes

—

Dic₄₇

—

#1824

—

maximal, Sylow

C₂×C₄₇

—

C₂×C₄₇

yes

C₂×C₄₇

yes

C₂×C₄₇

yes

C₂×C₄₇

yes

—

C₂×C₄₇

#484

yes

—

C₂²×C₄⋊C₄

D₄₇

yes

characteristic

C₂×C₂²⋊C₈

—

#380

—

#340

—

C₄₇

#1824

yes

characteristic, complemented, Sylow

C₂³×C₄

—

C₂³×C₄

C₂×D₄₇

yes

characteristic

C₂²×C₈

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

Dic₄₇

yes

complemented

C₂×C₄⋊C₄

C₂×D₄₇

yes

—

C₂×C₂²⋊C₄

—

C₂²⋊C₈

—

#95

—

C₂⁴

C₂²⋊D₄₇

yes

characteristic

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

C₂×Dic₄₇

yes

—

C₂²×C₄

C₂²⋊D₄₇

yes

characteristic

C₂²×C₄

C₂²⋊D₄₇

yes

—

100

C₂²×C₄

C₂²×D₄₇

yes

characteristic

101

C₂×C₈

—

102

C₂×C₈

—

103

C₄⋊C₄

—

104

C₄⋊C₄

—

105

C₄⋊C₄

—

106

C₄⋊C₄

C₂×Dic₄₇

yes

—

107

C₂²⋊C₄

—

188

—

108

C₂³

—

109

C₂³

—

110

C₂³

#10625336

yes

—

111

C₂³

#10625337

yes

—

112

C₂³

C₂²⋊Dic₄₇

yes

—

113

C₂³

C₂×C₂²⋊D₄₇

yes

characteristic

114

C₂×C₄

—

115

C₂×C₄

—

116

C₂×C₄

—

117

C₂×C₄

—

118

C₂×C₄

—

119

C₂×C₄

—

120

C₂×C₄

—

188

—

121

C₂×C₄

—

188

—

122

C₂×C₄

—

188

—

123

C₂×C₄

C₂²⋊Dic₄₇

yes

—

124

C₂×C₄

#10625350

yes

—

125

C₂×C₄

C₂²×Dic₄₇

yes

characteristic

126

C₂×C₄

C₂×C₂²⋊D₄₇

yes

characteristic

127

C₈

—

188

—

128

C₂²

—

129

C₂²

—

130

C₂²

—

131

C₂²

—

132

C₂²

#13093948

yes

—

133

C₂²

C₂×C₂²⋊Dic₄₇

yes

characteristic

134

C₂²

#13094007

yes

—

135

C₂²

C₂×#10625336

yes

characteristic

136

C₂²

C₂×#10625337

yes

characteristic

137

C₂²

#13094050

yes

—

138

C₄

—

139

C₄

—

140

C₄

—

376

—

141

C₄

#13093983

yes

—

142

C₄

#13094007

yes

—

143

C₂

—

144

C₂

#15688986

yes

characteristic

145

C₂

C₂×#13093948

yes

characteristic

146

C₂

#15689128

yes

characteristic

147

C₂

#15689441

yes

—

148

C₂

#15689458

yes

—

	subgroup	G/H
center	C₂³	C₂×C₂²⋊D₄₇
commutator	C₂×C₄×C₄₇	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₄₇
Fitting	C₂²×C₄⋊C₄×C₄₇	C₂
socle	C₂³×C₄₇	C₂×D₄
hypercenter	C₂²×C₄⋊C₄	D₄₇
hypocenter	C₄₇	#1824
LCS	G ▷ C₂×C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂³ ◁ C₂³×C₄ ◁ C₂²×C₄⋊C₄

subgroup

G/H

center

C₂³

C₂×C₂²⋊D₄₇

commutator

C₂×C₄×C₄₇

C₂²×C₄

Frattini

C₂²×C₄

C₂²×D₄₇

Fitting

C₂²×C₄⋊C₄×C₄₇

C₂

socle

C₂³×C₄₇

C₂×D₄

hypercenter

C₂²×C₄⋊C₄

D₄₇

hypocenter

C₄₇

#1824

LCS

G ▷ C₂×C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇

UCS

C₁ ◁ C₂³ ◁ C₂³×C₄ ◁ C₂²×C₄⋊C₄

N	H ≅ G/N	type
3. #15688885	C₂	semidirect
14. C₂×C₄⋊C₄×C₄₇	C₄	semidirect
79. C₄₇	#1824	semidirect
86. C₂×C₄⋊C₄	Dic₄₇	semidirect

H ≅ G/N

type

3. #15688885

C₂

semidirect

14. C₂×C₄⋊C₄×C₄₇

C₄

semidirect

79. C₄₇

#1824

semidirect

86. C₂×C₄⋊C₄

Dic₄₇

semidirect

Subgroups — #20243478

Conjugacy classes of subgroups

Complements