Group #20242201 — subgroups

Subgroups — SD₁₆⋊(C₄₇⋊C₈) #20242201

Order 6016.

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

normal subgroups only

This group has 176 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	Q₈×C₄₇⋊C₈	C₂	1	1	yes	maximal, characteristic, complemented
2	D₄×C₄₇⋊C₈	C₂	1	1	yes	maximal, characteristic
3	#15688876	C₂	1	1	yes	maximal, characteristic, complemented
4	Q₈⋊(C₄₇⋊C₈)	C₂	1	1	yes	maximal, characteristic, complemented
5	D₄⋊(C₄₇⋊C₈)	C₂	1	1	yes	maximal, characteristic
6	#15688844	C₂	1	1	yes	maximal, characteristic, complemented
7	C₄×SD₁₆×C₄₇	C₂	1	1	yes	maximal, characteristic
8	C₂²×C₄₇⋊C₈	—	2	1	no	—
9	C₄⋊(C₄₇⋊C₈)	—	2	1	no	—
10	C₄⋊(C₄₇⋊C₈)	C₂²	1	1	yes	characteristic
11	C₄⋊(C₄₇⋊C₈)	C₂²	1	1	yes	characteristic
12	C₂²⋊(C₄₇⋊C₈)	—	2	1	no	—
13	C₄×C₄₇⋊C₈	—	2	1	no	—
14	C₄×C₄₇⋊C₈	C₂²	1	1	yes	characteristic
15	C₄×C₄₇⋊C₈	C₂²	1	1	yes	characteristic
16	C₂×SD₁₆×C₄₇	C₄	1	1	yes	characteristic
17	C₄×Q₈×C₄₇	C₂²	1	1	yes	characteristic
18	C₄×D₄×C₄₇	C₂²	1	1	yes	characteristic
19	#106×C₄₇	C₄	1	1	yes	characteristic
20	#103×C₄₇	C₄	1	1	yes	characteristic
21	#102×C₄₇	C₄	1	1	yes	characteristic
22	C₄×C₈×C₄₇	C₂²	1	1	yes	characteristic
23	C₂×Q₈×C₄₇	C₂×C₄	1	1	yes	characteristic
24	C₂×D₄×C₄₇	C₂×C₄	1	1	yes	characteristic
25	C₂²×C₄×C₄₇	—	2	1	no	—
26	SD₁₆×C₄₇	C₈	1	4	yes	complemented
27	C₂×C₈×C₄₇	C₂×C₄	1	1	yes	characteristic
28	C₂×C₈×C₄₇	C₂×C₄	1	1	yes	characteristic
29	C₄⋊C₄×C₄₇	—	2	1	no	—
30	C₄⋊C₄×C₄₇	C₂×C₄	1	1	yes	characteristic
31	C₄⋊C₄×C₄₇	C₂×C₄	1	1	yes	characteristic
32	C₂²⋊C₄×C₄₇	—	2	1	no	—
33	C₄²×C₄₇	—	2	1	no	—
34	C₄²×C₄₇	C₂³	1	1	yes	characteristic
35	C₂×C₄₇⋊C₈	—	2	1	no	—
36	C₂×C₄₇⋊C₈	—	2	1	no	—
37	C₂×C₄₇⋊C₈	—	2	1	no	—
38	C₂×C₄₇⋊C₈	—	4	2	no	—
39	C₂×C₄₇⋊C₈	D₄	1	1	yes	characteristic
40	C₂×C₄₇⋊C₈	D₄	1	1	yes	characteristic
41	C₂³×C₄₇	—	2	1	no	—
42	Q₈×C₄₇	—	2	1	no	—
43	Q₈×C₄₇	C₂×C₈	1	2	yes	complemented
44	D₄×C₄₇	—	2	1	no	—
45	D₄×C₄₇	C₂×C₈	1	2	yes	—
46	C₂×C₄×C₄₇	—	2	1	no	—
47	C₂×C₄×C₄₇	—	2	1	no	—
48	C₂×C₄×C₄₇	—	2	1	no	—
49	C₂×C₄×C₄₇	—	4	1	no	—
50	C₂×C₄×C₄₇	—	4	1	no	—
51	C₂×C₄×C₄₇	C₂²×C₄	1	1	yes	characteristic
52	C₂×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
53	C₂×C₄×C₄₇	#41	1	1	yes	characteristic
54	C₈×C₄₇	—	2	1	no	—
55	C₈×C₄₇	C₂×C₈	1	2	yes	complemented
56	C₄₇⋊C₈	—	2	1	no	—
57	C₄₇⋊C₈	—	2	1	no	—
58	C₄₇⋊C₈	—	4	1	no	—
59	C₄₇⋊C₈	—	4	1	no	—
60	C₄₇⋊C₈	—	4	1	no	—
61	C₂²×C₄₇	—	2	2	no	—
62	C₂²×C₄₇	—	4	1	no	—
63	C₂²×C₄₇	—	4	1	no	—
64	C₂²×C₄₇	C₄×D₄	1	1	yes	characteristic
65	C₄×C₄₇	—	2	2	no	—
66	C₄×C₄₇	—	2	1	no	—
67	C₄×C₄₇	—	4	1	no	—
68	C₄×C₄₇	—	4	1	no	—
69	C₄×C₄₇	C₂²×C₈	1	1	yes	characteristic
70	C₄×C₄₇	#131	1	1	yes	characteristic
71	C₄×C₄₇	AGL(1,ℤ/8)	1	1	yes	characteristic
72	C₄×C₄₇	#137	1	1	yes	characteristic
73	SD₁₆⋊C₈	—	47	1	no	maximal, Sylow
74	C₂×C₄₇	—	4	2	no	—
75	C₂×C₄₇	C₈×D₄	1	1	yes	characteristic
76	C₂×C₄₇	#440	1	1	yes	characteristic
77	C₂×C₄₇	#444	1	1	yes	characteristic
78	C₈×Q₈	—	47	1	no	—
79	C₄×SD₁₆	D₄₇	1	1	yes	characteristic
80	C₈×D₄	—	47	1	no	—
81	#335	—	47	1	no	—
82	Q₈⋊C₈	—	47	1	no	—
83	D₄⋊C₈	—	47	1	no	—
84	#322	—	47	1	no	—
85	C₄₇	SD₁₆⋊C₈	1	1	yes	characteristic, complemented, Sylow
86	C₂×SD₁₆	Dic₄₇	1	1	yes	characteristic
87	C₂²×C₈	—	94	1	no	—
88	C₄×Q₈	C₂×D₄₇	1	1	yes	characteristic
89	C₄×D₄	C₂×D₄₇	1	1	yes	characteristic
90	#106	Dic₄₇	1	1	yes	characteristic
91	C₄⋊C₈	—	47	1	no	—
92	C₄⋊C₈	—	47	1	no	—
93	C₄⋊C₈	—	94	1	no	—
94	#103	Dic₄₇	1	1	yes	characteristic
95	#102	Dic₄₇	1	1	yes	characteristic
96	C₂²⋊C₈	—	94	1	no	—
97	C₄×C₈	—	47	1	no	—
98	C₄×C₈	—	47	1	no	—
99	C₄×C₈	—	94	1	no	—
100	C₄×C₈	C₂×D₄₇	1	1	yes	characteristic
101	C₂×Q₈	C₂×Dic₄₇	1	1	yes	characteristic
102	C₂×D₄	C₂×Dic₄₇	1	1	yes	characteristic
103	C₂²×C₄	—	2	1	no	—
104	SD₁₆	C₄₇⋊C₈	1	4	yes	complemented
105	C₂×C₈	—	47	1	no	—
106	C₂×C₈	—	47	1	no	—
107	C₂×C₈	—	94	1	no	—
108	C₂×C₈	—	94	1	no	—
109	C₂×C₈	—	94	1	no	—
110	C₂×C₈	—	188	2	no	—
111	C₂×C₈	C₂×Dic₄₇	1	1	yes	characteristic
112	C₂×C₈	C₂×Dic₄₇	1	1	yes	characteristic
113	C₄⋊C₄	—	2	1	no	—
114	C₄⋊C₄	C₂×Dic₄₇	1	1	yes	characteristic
115	C₄⋊C₄	C₂×Dic₄₇	1	1	yes	characteristic
116	C₂²⋊C₄	—	2	1	no	—
117	C₄²	—	2	1	no	—
118	C₄²	C₂²×D₄₇	1	1	yes	characteristic
119	C₂³	—	2	1	no	—
120	Q₈	—	2	1	no	—
121	Q₈	C₂×C₄₇⋊C₈	1	2	yes	complemented
122	D₄	—	2	1	no	—
123	D₄	C₂×C₄₇⋊C₈	1	2	yes	—
124	C₂×C₄	—	2	1	no	—
125	C₂×C₄	—	2	1	no	—
126	C₂×C₄	—	2	1	no	—
127	C₂×C₄	—	4	1	no	—
128	C₂×C₄	—	4	1	no	—
129	C₂×C₄	D₄×D₄₇	1	1	yes	characteristic
130	C₂×C₄	#10625352	1	1	yes	characteristic
131	C₂×C₄	C₂²×Dic₄₇	1	1	yes	characteristic
132	C₈	—	2	1	no	—
133	C₈	—	94	1	no	—
134	C₈	—	94	1	no	—
135	C₈	—	188	1	no	—
136	C₈	—	188	1	no	—
137	C₈	—	188	1	no	—
138	C₈	C₂×C₄₇⋊C₈	1	2	yes	complemented
139	C₂²	—	2	2	no	—
140	C₂²	—	4	1	no	—
141	C₂²	—	4	1	no	—
142	C₂²	D₄×Dic₄₇	1	1	yes	characteristic
143	C₄	—	2	1	no	—
144	C₄	—	2	2	no	—
145	C₄	—	4	1	no	—
146	C₄	—	4	1	no	—
147	C₄	C₂²×C₄₇⋊C₈	1	1	yes	characteristic
148	C₄	#13094031	1	1	yes	characteristic
149	C₄	#13094051	1	1	yes	characteristic
150	C₄	#13094053	1	1	yes	characteristic
151	C₂	—	4	2	no	—
152	C₂	D₄×C₄₇⋊C₈	1	1	yes	characteristic
153	C₂	#15689224	1	1	yes	characteristic
154	C₂	#15689245	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

Q₈×C₄₇⋊C₈

C₂

yes

maximal, characteristic, complemented

D₄×C₄₇⋊C₈

C₂

yes

maximal, characteristic

#15688876

C₂

yes

maximal, characteristic, complemented

Q₈⋊(C₄₇⋊C₈)

C₂

yes

maximal, characteristic, complemented

D₄⋊(C₄₇⋊C₈)

C₂

yes

maximal, characteristic

#15688844

C₂

yes

maximal, characteristic, complemented

C₄×SD₁₆×C₄₇

C₂

yes

maximal, characteristic

C₂²×C₄₇⋊C₈

—

C₄⋊(C₄₇⋊C₈)

—

C₄⋊(C₄₇⋊C₈)

C₂²

yes

characteristic

C₄⋊(C₄₇⋊C₈)

C₂²

yes

characteristic

C₂²⋊(C₄₇⋊C₈)

—

C₄×C₄₇⋊C₈

—

C₄×C₄₇⋊C₈

C₂²

yes

characteristic

C₄×C₄₇⋊C₈

C₂²

yes

characteristic

C₂×SD₁₆×C₄₇

C₄

yes

characteristic

C₄×Q₈×C₄₇

C₂²

yes

characteristic

C₄×D₄×C₄₇

C₂²

yes

characteristic

#106×C₄₇

C₄

yes

characteristic

#103×C₄₇

C₄

yes

characteristic

#102×C₄₇

C₄

yes

characteristic

C₄×C₈×C₄₇

C₂²

yes

characteristic

C₂×Q₈×C₄₇

C₂×C₄

yes

characteristic

C₂×D₄×C₄₇

C₂×C₄

yes

characteristic

C₂²×C₄×C₄₇

—

SD₁₆×C₄₇

C₈

yes

complemented

C₂×C₈×C₄₇

C₂×C₄

yes

characteristic

C₂×C₈×C₄₇

C₂×C₄

yes

characteristic

C₄⋊C₄×C₄₇

—

C₄⋊C₄×C₄₇

C₂×C₄

yes

characteristic

C₄⋊C₄×C₄₇

C₂×C₄

yes

characteristic

C₂²⋊C₄×C₄₇

—

C₄²×C₄₇

—

C₄²×C₄₇

C₂³

yes

characteristic

C₂×C₄₇⋊C₈

—

C₂×C₄₇⋊C₈

—

C₂×C₄₇⋊C₈

—

C₂×C₄₇⋊C₈

—

C₂×C₄₇⋊C₈

D₄

yes

characteristic

C₂×C₄₇⋊C₈

D₄

yes

characteristic

C₂³×C₄₇

—

Q₈×C₄₇

—

Q₈×C₄₇

C₂×C₈

yes

complemented

D₄×C₄₇

—

D₄×C₄₇

C₂×C₈

yes

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

C₂²×C₄

yes

characteristic

C₂×C₄×C₄₇

C₂×D₄

yes

characteristic

C₂×C₄×C₄₇

#41

yes

characteristic

C₈×C₄₇

—

C₈×C₄₇

C₂×C₈

yes

complemented

C₄₇⋊C₈

—

C₄₇⋊C₈

—

C₄₇⋊C₈

—

C₄₇⋊C₈

—

C₄₇⋊C₈

—

C₂²×C₄₇

—

C₂²×C₄₇

—

C₂²×C₄₇

—

C₂²×C₄₇

C₄×D₄

yes

characteristic

C₄×C₄₇

—

C₄×C₄₇

—

C₄×C₄₇

—

C₄×C₄₇

—

C₄×C₄₇

C₂²×C₈

yes

C₄×C₄₇

yes

C₄×C₄₇

yes

C₄×C₄₇

yes

SD₁₆⋊C₈

—

maximal, Sylow

C₂×C₄₇

—

C₂×C₄₇

C₈×D₄

yes

C₂×C₄₇

yes

C₂×C₄₇

yes

C₈×Q₈

—

C₄×SD₁₆

D₄₇

yes

characteristic

C₈×D₄

—

#335

—

Q₈⋊C₈

—

D₄⋊C₈

—

#322

—

C₄₇

SD₁₆⋊C₈

yes

characteristic, complemented, Sylow

C₂×SD₁₆

Dic₄₇

yes

characteristic

C₂²×C₈

—

C₄×Q₈

C₂×D₄₇

yes

characteristic

C₄×D₄

C₂×D₄₇

yes

characteristic

#106

Dic₄₇

yes

characteristic

C₄⋊C₈

—

C₄⋊C₈

—

C₄⋊C₈

—

#103

Dic₄₇

yes

characteristic

#102

Dic₄₇

yes

characteristic

C₂²⋊C₈

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

—

100

C₄×C₈

C₂×D₄₇

yes

characteristic

101

C₂×Q₈

C₂×Dic₄₇

yes

characteristic

102

C₂×D₄

C₂×Dic₄₇

yes

characteristic

103

C₂²×C₄

—

104

SD₁₆

C₄₇⋊C₈

yes

complemented

105

C₂×C₈

—

106

C₂×C₈

—

107

C₂×C₈

—

108

C₂×C₈

—

109

C₂×C₈

—

110

C₂×C₈

—

188

—

111

C₂×C₈

C₂×Dic₄₇

yes

characteristic

112

C₂×C₈

C₂×Dic₄₇

yes

characteristic

113

C₄⋊C₄

—

114

C₄⋊C₄

C₂×Dic₄₇

yes

characteristic

115

C₄⋊C₄

C₂×Dic₄₇

yes

characteristic

116

C₂²⋊C₄

—

117

C₄²

—

118

C₄²

C₂²×D₄₇

yes

characteristic

119

C₂³

—

120

Q₈

—

121

Q₈

C₂×C₄₇⋊C₈

yes

complemented

122

D₄

—

123

D₄

C₂×C₄₇⋊C₈

yes

—

124

C₂×C₄

—

125

C₂×C₄

—

126

C₂×C₄

—

127

C₂×C₄

—

128

C₂×C₄

—

129

C₂×C₄

D₄×D₄₇

yes

characteristic

130

C₂×C₄

#10625352

yes

characteristic

131

C₂×C₄

C₂²×Dic₄₇

yes

characteristic

132

C₈

—

133

C₈

—

134

C₈

—

135

C₈

—

188

—

136

C₈

—

188

—

137

C₈

—

188

—

138

C₈

C₂×C₄₇⋊C₈

yes

complemented

139

C₂²

—

140

C₂²

—

141

C₂²

—

142

C₂²

D₄×Dic₄₇

yes

characteristic

143

C₄

—

144

C₄

—

145

C₄

—

146

C₄

—

147

C₄

C₂²×C₄₇⋊C₈

yes

characteristic

148

C₄

#13094031

yes

characteristic

149

C₄

#13094051

yes

characteristic

150

C₄

#13094053

yes

characteristic

151

C₂

—

152

C₂

D₄×C₄₇⋊C₈

yes

characteristic

153

C₂

#15689224

yes

characteristic

154

C₂

#15689245

yes

characteristic

	subgroup	G/H
center	C₂×C₄	D₄×D₄₇
commutator	C₄×C₄₇	C₂²×C₈
Frattini	C₄²	C₂²×D₄₇
Fitting	C₄×SD₁₆×C₄₇	C₂
socle	C₂²×C₄₇	C₄×D₄
hypercenter	C₄×SD₁₆	D₄₇
hypocenter	C₄₇	SD₁₆⋊C₈
LCS	G ▷ C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂×C₄ ◁ C₄² ◁ C₄×SD₁₆

subgroup

G/H

center

C₂×C₄

D₄×D₄₇

commutator

C₄×C₄₇

C₂²×C₈

Frattini

C₄²

C₂²×D₄₇

Fitting

C₄×SD₁₆×C₄₇

C₂

socle

C₂²×C₄₇

C₄×D₄

hypercenter

C₄×SD₁₆

D₄₇

hypocenter

C₄₇

SD₁₆⋊C₈

LCS

G ▷ C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇

UCS

C₁ ◁ C₂×C₄ ◁ C₄² ◁ C₄×SD₁₆

N	H ≅ G/N	type
1. Q₈×C₄₇⋊C₈	C₂	semidirect
3. #15688876	C₂	semidirect
4. Q₈⋊(C₄₇⋊C₈)	C₂	semidirect
6. #15688844	C₂	semidirect
26. SD₁₆×C₄₇	C₈	semidirect
43. Q₈×C₄₇	C₂×C₈	semidirect
55. C₈×C₄₇	C₂×C₈	semidirect
85. C₄₇	SD₁₆⋊C₈	semidirect
104. SD₁₆	C₄₇⋊C₈	semidirect
121. Q₈	C₂×C₄₇⋊C₈	semidirect
138. C₈	C₂×C₄₇⋊C₈	semidirect

H ≅ G/N

type

1. Q₈×C₄₇⋊C₈

C₂

semidirect

3. #15688876

C₂

semidirect

4. Q₈⋊(C₄₇⋊C₈)

C₂

semidirect

6. #15688844

C₂

semidirect

26. SD₁₆×C₄₇

C₈

semidirect

43. Q₈×C₄₇

C₂×C₈

semidirect

55. C₈×C₄₇

C₂×C₈

semidirect

85. C₄₇

SD₁₆⋊C₈

semidirect

104. SD₁₆

C₄₇⋊C₈

semidirect

121. Q₈

C₂×C₄₇⋊C₈

semidirect

138. C₈

C₂×C₄₇⋊C₈

semidirect

Subgroups — SD₁₆⋊(C₄₇⋊C₈) #20242201

Conjugacy classes of subgroups

Complements

Subgroups — SD16⋊(C47⋊C8) #20242201

Conjugacy classes of subgroups

Complements

Subgroups — SD₁₆⋊(C₄₇⋊C₈) #20242201