Group #20252153 — subgroups

Subgroups — C₂×Q₁₆×Dic₄₇ #20252153

Order 6016.

488 conjugacy classes of subgroups; only the 245 normal subgroups are shown.

normal subgroups only

This group has 488 conjugacy classes of subgroups — too many to draw. Only its 245 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×Q₈×Dic₄₇	C₂	1	2	yes	maximal
2	Q₁₆×Dic₄₇	C₂	1	8	yes	maximal, direct product factor
3	C₂×#13093958	C₂	1	1	yes	maximal, characteristic
4	C₂×#13093948	C₂	1	2	yes	maximal
5	C₂×C₈×Dic₄₇	C₂	1	1	yes	maximal, characteristic
6	C₂²×Q₁₆×C₄₇	C₂	1	1	yes	maximal, characteristic
7	C₂²×C₄₇⋊C₈	C₂²	1	1	yes	characteristic
8	Q₈×Dic₄₇	—	2	4	no	—
9	Q₈×Dic₄₇	C₂²	1	8	yes	—
10	C₂×C₄⋊Dic₄₇	—	2	2	no	—
11	C₂×C₄⋊Dic₄₇	C₂²	1	2	yes	—
12	C₂×C₄×Dic₄₇	—	2	2	no	—
13	C₂×C₄×Dic₄₇	C₂²	1	1	yes	characteristic
14	#13093958	C₂²	1	4	yes	—
15	#13093948	C₂²	1	8	yes	—
16	C₈×Dic₄₇	C₂²	1	4	yes	—
17	C₂²×Q₈×C₄₇	C₂²	1	2	yes	—
18	C₂×Q₁₆×C₄₇	C₄	1	8	yes	complemented
19	C₂×Q₁₆×C₄₇	C₂²	1	4	yes	—
20	C₂²×C₈×C₄₇	C₂²	1	1	yes	characteristic
21	C₂×Q₈×C₄₇	—	2	2	no	—
22	C₂×Q₈×C₄₇	—	2	4	no	—
23	C₂×Q₈×C₄₇	C₂×C₄	1	8	yes	—
24	C₂×Q₈×C₄₇	C₂³	1	4	yes	—
25	C₂²×C₄×C₄₇	—	2	2	no	—
26	C₂²×C₄×C₄₇	C₂³	1	1	yes	characteristic
27	C₂²×Dic₄₇	—	2	2	no	—
28	C₂²×Dic₄₇	D₄	1	1	yes	characteristic
29	C₂²×Dic₄₇	D₄	1	1	yes	characteristic
30	Q₁₆×C₄₇	C₂×C₄	1	16	yes	complemented
31	C₂×C₈×C₄₇	C₂×C₄	1	4	yes	—
32	C₂×C₈×C₄₇	C₂³	1	2	yes	—
33	C₄⋊Dic₄₇	—	2	2	no	—
34	C₄⋊Dic₄₇	—	2	8	no	—
35	C₄⋊Dic₄₇	C₂³	1	4	yes	—
36	C₄×Dic₄₇	—	2	8	no	—
37	C₄×Dic₄₇	D₄	1	2	yes	—
38	C₄×Dic₄₇	C₂³	1	2	yes	—
39	C₂×C₄₇⋊C₈	—	2	2	no	—
40	C₂×C₄₇⋊C₈	C₂³	1	2	yes	—
41	C₂³×C₄₇	C₂×D₄	1	1	yes	characteristic
42	Q₈×C₄₇	—	2	8	no	—
43	Q₈×C₄₇	—	2	4	no	—
44	Q₈×C₄₇	C₂²×C₄	1	8	yes	—
45	C₂×C₄×C₄₇	—	2	8	no	—
46	C₂×C₄×C₄₇	—	2	4	no	—
47	C₂×C₄×C₄₇	C₂²×C₄	1	2	yes	—
48	C₂×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
49	C₂×C₄×C₄₇	#41	1	2	yes	—
50	C₂×C₄×C₄₇	C₂⁴	1	1	yes	characteristic
51	C₂×Dic₄₇	—	2	2	no	—
52	C₂×Dic₄₇	—	2	4	no	—
53	C₂×Dic₄₇	—	4	4	no	—
54	C₂×Dic₄₇	Q₁₆	1	4	yes	direct product factor
55	C₂×Dic₄₇	C₂×D₄	1	2	yes	—
56	C₂×Dic₄₇	C₂×D₄	1	2	yes	—
57	C₈×C₄₇	C₂²×C₄	1	4	yes	—
58	C₄₇⋊C₈	—	2	2	no	—
59	C₂²×C₄₇	C₄×D₄	1	2	yes	—
60	C₂²×C₄₇	C₂×Q₁₆	1	2	yes	—
61	C₂²×C₄₇	#135	1	2	yes	—
62	C₂²×C₄₇	C₂²×D₄	1	1	yes	characteristic
63	C₄×C₄₇	—	2	8	no	—
64	C₄×C₄₇	C₄×D₄	1	2	yes	—
65	C₄×C₄₇	C₂³×C₄	1	1	yes	characteristic
66	C₄×C₄₇	C₂×#41	1	1	yes	characteristic
67	Dic₄₇	—	2	2	no	—
68	Dic₄₇	—	4	4	no	—
69	Dic₄₇	C₂×Q₁₆	1	4	yes	direct product factor
70	C₂×C₄×Q₁₆	—	47	1	no	maximal, Sylow
71	C₂×C₄₇	C₄×Q₁₆	1	4	yes	complemented
72	C₂×C₄₇	C₂×C₄×D₄	1	1	yes	characteristic
73	C₂×C₄₇	C₂²×Q₁₆	1	1	yes	characteristic
74	C₂×C₄₇	C₂×#135	1	1	yes	characteristic
75	C₂²×Q₁₆	D₄₇	1	1	yes	characteristic
76	C₂×C₄×Q₈	—	47	2	no	—
77	C₄×Q₁₆	—	47	8	no	—
78	C₂×#107	—	47	1	no	—
79	C₂×#103	—	47	2	no	—
80	C₂×C₄×C₈	—	47	1	no	—
81	C₄₇	C₂×C₄×Q₁₆	1	1	yes	characteristic, complemented, Sylow
82	C₂²×Q₈	C₂×D₄₇	1	2	yes	—
83	C₂×Q₁₆	Dic₄₇	1	8	yes	direct product factor
84	C₂×Q₁₆	C₂×D₄₇	1	4	yes	—
85	C₂²×C₈	—	47	1	no	—
86	C₂²×C₈	C₂×D₄₇	1	1	yes	characteristic
87	C₄×Q₈	—	47	8	no	—
88	C₄×Q₈	—	94	4	no	—
89	C₂×C₄⋊C₄	—	47	2	no	—
90	C₂×C₄⋊C₄	—	94	2	no	—
91	C₂×C₄²	—	47	1	no	—
92	C₂×C₄²	—	94	2	no	—
93	#107	—	47	4	no	—
94	#103	—	47	8	no	—
95	C₄×C₈	—	47	4	no	—
96	C₂×Q₈	—	2	2	no	—
97	C₂×Q₈	—	2	4	no	—
98	C₂×Q₈	C₂×Dic₄₇	1	8	yes	—
99	C₂×Q₈	C₂²×D₄₇	1	4	yes	—
100	C₂²×C₄	—	2	2	no	—
101	C₂²×C₄	—	47	1	no	—
102	C₂²×C₄	—	47	1	no	—
103	C₂²×C₄	—	94	2	no	—
104	C₂²×C₄	C₂²×D₄₇	1	1	yes	characteristic
105	Q₁₆	C₂×Dic₄₇	1	16	yes	direct product factor
106	C₂×C₈	—	47	2	no	—
107	C₂×C₈	—	94	2	no	—
108	C₂×C₈	C₂×Dic₄₇	1	4	yes	—
109	C₂×C₈	C₂²×D₄₇	1	2	yes	—
110	C₄⋊C₄	—	47	4	no	—
111	C₄⋊C₄	—	94	8	no	—
112	C₄⋊C₄	—	94	2	no	—
113	C₄²	—	47	2	no	—
114	C₄²	—	47	2	no	—
115	C₄²	—	94	8	no	—
116	C₂³	D₄×D₄₇	1	1	yes	characteristic
117	Q₈	—	2	8	no	—
118	Q₈	—	2	4	no	—
119	Q₈	C₂²×Dic₄₇	1	8	yes	—
120	C₂×C₄	—	2	8	no	—
121	C₂×C₄	—	2	4	no	—
122	C₂×C₄	—	47	4	no	—
123	C₂×C₄	—	47	2	no	—
124	C₂×C₄	—	47	2	no	—
125	C₂×C₄	—	94	2	no	—
126	C₂×C₄	—	94	4	no	—
127	C₂×C₄	—	188	4	no	—
128	C₂×C₄	D₄×D₄₇	1	1	yes	characteristic
129	C₂×C₄	#10625352	1	2	yes	—
130	C₂×C₄	C₂²×Dic₄₇	1	2	yes	—
131	C₂×C₄	C₂³×D₄₇	1	1	yes	characteristic
132	C₈	—	94	2	no	—
133	C₈	C₂²×Dic₄₇	1	4	yes	—
134	C₂²	D₄×Dic₄₇	1	2	yes	—
135	C₂²	Q₁₆×D₄₇	1	2	yes	—
136	C₂²	#13094047	1	2	yes	—
137	C₂²	C₂×D₄×D₄₇	1	1	yes	characteristic
138	C₄	—	2	8	no	—
139	C₄	—	47	4	no	—
140	C₄	—	94	2	no	—
141	C₄	—	188	4	no	—
142	C₄	D₄×Dic₄₇	1	2	yes	—
143	C₄	C₂³×Dic₄₇	1	1	yes	characteristic
144	C₄	C₂×#10625352	1	1	yes	characteristic
145	C₂	Q₁₆×Dic₄₇	1	4	yes	direct product factor
146	C₂	C₂×D₄×Dic₄₇	1	1	yes	characteristic
147	C₂	C₂×Q₁₆×D₄₇	1	1	yes	characteristic
148	C₂	C₂×#13094047	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×Q₈×Dic₄₇

C₂

yes

maximal

Q₁₆×Dic₄₇

C₂

yes

maximal, direct product factor

C₂×#13093958

C₂

yes

maximal, characteristic

C₂×#13093948

C₂

yes

maximal

C₂×C₈×Dic₄₇

C₂

yes

maximal, characteristic

C₂²×Q₁₆×C₄₇

C₂

yes

maximal, characteristic

C₂²×C₄₇⋊C₈

C₂²

yes

characteristic

Q₈×Dic₄₇

—

Q₈×Dic₄₇

C₂²

yes

—

C₂×C₄⋊Dic₄₇

—

C₂×C₄⋊Dic₄₇

C₂²

yes

—

C₂×C₄×Dic₄₇

—

C₂×C₄×Dic₄₇

C₂²

yes

characteristic

#13093958

C₂²

yes

—

#13093948

C₂²

yes

—

C₈×Dic₄₇

C₂²

yes

—

C₂²×Q₈×C₄₇

C₂²

yes

—

C₂×Q₁₆×C₄₇

C₄

yes

complemented

C₂×Q₁₆×C₄₇

C₂²

yes

—

C₂²×C₈×C₄₇

C₂²

yes

characteristic

C₂×Q₈×C₄₇

—

C₂×Q₈×C₄₇

—

C₂×Q₈×C₄₇

C₂×C₄

yes

—

C₂×Q₈×C₄₇

C₂³

yes

—

C₂²×C₄×C₄₇

—

C₂²×C₄×C₄₇

C₂³

yes

characteristic

C₂²×Dic₄₇

—

C₂²×Dic₄₇

D₄

yes

characteristic

C₂²×Dic₄₇

D₄

yes

characteristic

Q₁₆×C₄₇

C₂×C₄

yes

complemented

C₂×C₈×C₄₇

C₂×C₄

yes

—

C₂×C₈×C₄₇

C₂³

yes

—

C₄⋊Dic₄₇

—

C₄⋊Dic₄₇

—

C₄⋊Dic₄₇

C₂³

yes

—

C₄×Dic₄₇

—

C₄×Dic₄₇

D₄

yes

—

C₄×Dic₄₇

C₂³

yes

—

C₂×C₄₇⋊C₈

—

C₂×C₄₇⋊C₈

C₂³

yes

—

C₂³×C₄₇

C₂×D₄

yes

characteristic

Q₈×C₄₇

—

Q₈×C₄₇

—

Q₈×C₄₇

C₂²×C₄

yes

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

C₂²×C₄

yes

—

C₂×C₄×C₄₇

C₂×D₄

yes

characteristic

C₂×C₄×C₄₇

#41

yes

—

C₂×C₄×C₄₇

C₂⁴

yes

characteristic

C₂×Dic₄₇

—

C₂×Dic₄₇

—

C₂×Dic₄₇

—

C₂×Dic₄₇

Q₁₆

yes

direct product factor

C₂×Dic₄₇

C₂×D₄

yes

—

C₂×Dic₄₇

C₂×D₄

yes

—

C₈×C₄₇

C₂²×C₄

yes

—

C₄₇⋊C₈

—

C₂²×C₄₇

C₄×D₄

yes

—

C₂²×C₄₇

C₂×Q₁₆

yes

—

C₂²×C₄₇

#135

yes

—

C₂²×C₄₇

C₂²×D₄

yes

characteristic

C₄×C₄₇

—

C₄×C₄₇

C₄×D₄

yes

—

C₄×C₄₇

C₂³×C₄

yes

characteristic

C₄×C₄₇

C₂×#41

yes

characteristic

Dic₄₇

—

Dic₄₇

—

Dic₄₇

C₂×Q₁₆

yes

direct product factor

C₂×C₄×Q₁₆

—

maximal, Sylow

C₂×C₄₇

C₄×Q₁₆

yes

complemented

C₂×C₄₇

C₂×C₄×D₄

yes

characteristic

C₂×C₄₇

C₂²×Q₁₆

yes

characteristic

C₂×C₄₇

C₂×#135

yes

characteristic

C₂²×Q₁₆

D₄₇

yes

characteristic

C₂×C₄×Q₈

—

C₄×Q₁₆

—

C₂×#107

—

C₂×#103

—

C₂×C₄×C₈

—

C₄₇

C₂×C₄×Q₁₆

yes

characteristic, complemented, Sylow

C₂²×Q₈

C₂×D₄₇

yes

—

C₂×Q₁₆

Dic₄₇

yes

direct product factor

C₂×Q₁₆

C₂×D₄₇

yes

—

C₂²×C₈

—

C₂²×C₈

C₂×D₄₇

yes

characteristic

C₄×Q₈

—

C₄×Q₈

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄²

—

C₂×C₄²

—

#107

—

#103

—

C₄×C₈

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

C₂×Dic₄₇

yes

—

C₂×Q₈

C₂²×D₄₇

yes

—

100

C₂²×C₄

—

101

C₂²×C₄

—

102

C₂²×C₄

—

103

C₂²×C₄

—

104

C₂²×C₄

C₂²×D₄₇

yes

characteristic

105

Q₁₆

C₂×Dic₄₇

yes

direct product factor

106

C₂×C₈

—

107

C₂×C₈

—

108

C₂×C₈

C₂×Dic₄₇

yes

—

109

C₂×C₈

C₂²×D₄₇

yes

—

110

C₄⋊C₄

—

111

C₄⋊C₄

—

112

C₄⋊C₄

—

113

C₄²

—

114

C₄²

—

115

C₄²

—

116

C₂³

D₄×D₄₇

yes

characteristic

117

Q₈

—

118

Q₈

—

119

Q₈

C₂²×Dic₄₇

yes

—

120

C₂×C₄

—

121

C₂×C₄

—

122

C₂×C₄

—

123

C₂×C₄

—

124

C₂×C₄

—

125

C₂×C₄

—

126

C₂×C₄

—

127

C₂×C₄

—

188

—

128

C₂×C₄

D₄×D₄₇

yes

characteristic

129

C₂×C₄

#10625352

yes

—

130

C₂×C₄

C₂²×Dic₄₇

yes

—

131

C₂×C₄

C₂³×D₄₇

yes

characteristic

132

C₈

—

133

C₈

C₂²×Dic₄₇

yes

—

134

C₂²

D₄×Dic₄₇

yes

—

135

C₂²

Q₁₆×D₄₇

yes

—

136

C₂²

#13094047

yes

—

137

C₂²

C₂×D₄×D₄₇

yes

characteristic

138

C₄

—

139

C₄

—

140

C₄

—

141

C₄

—

188

—

142

C₄

D₄×Dic₄₇

yes

—

143

C₄

C₂³×Dic₄₇

yes

characteristic

144

C₄

C₂×#10625352

yes

characteristic

145

C₂

Q₁₆×Dic₄₇

yes

direct product factor

146

C₂

C₂×D₄×Dic₄₇

yes

characteristic

147

C₂

C₂×Q₁₆×D₄₇

yes

characteristic

148

C₂

C₂×#13094047

yes

characteristic

	subgroup	G/H
center	C₂³	D₄×D₄₇
commutator	C₄×C₄₇	C₂³×C₄
Frattini	C₂×C₄	C₂³×D₄₇
Fitting	C₂²×Q₁₆×C₄₇	C₂
socle	C₂³×C₄₇	C₂×D₄
hypercenter	C₂²×Q₁₆	D₄₇
hypocenter	C₄₇	C₂×C₄×Q₁₆
LCS	G ▷ C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄ ◁ C₂²×Q₁₆

subgroup

G/H

center

C₂³

D₄×D₄₇

commutator

C₄×C₄₇

C₂³×C₄

Frattini

C₂×C₄

C₂³×D₄₇

Fitting

C₂²×Q₁₆×C₄₇

C₂

socle

C₂³×C₄₇

C₂×D₄

hypercenter

C₂²×Q₁₆

D₄₇

hypocenter

C₄₇

C₂×C₄×Q₁₆

LCS

G ▷ C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇

UCS

C₁ ◁ C₂³ ◁ C₂²×C₄ ◁ C₂²×Q₁₆

N	H ≅ G/N	type
2. Q₁₆×Dic₄₇	C₂	direct (N × H)
18. C₂×Q₁₆×C₄₇	C₄	semidirect
30. Q₁₆×C₄₇	C₂×C₄	semidirect
54. C₂×Dic₄₇	Q₁₆	direct (N × H)
69. Dic₄₇	C₂×Q₁₆	direct (N × H)
71. C₂×C₄₇	C₄×Q₁₆	semidirect
81. C₄₇	C₂×C₄×Q₁₆	semidirect
83. C₂×Q₁₆	Dic₄₇	direct (N × H)
105. Q₁₆	C₂×Dic₄₇	direct (N × H)
145. C₂	Q₁₆×Dic₄₇	direct (N × H)

H ≅ G/N

type

2. Q₁₆×Dic₄₇

C₂

direct (N × H)

18. C₂×Q₁₆×C₄₇

C₄

semidirect

30. Q₁₆×C₄₇

C₂×C₄

semidirect

54. C₂×Dic₄₇

Q₁₆

direct (N × H)

69. Dic₄₇

C₂×Q₁₆

direct (N × H)

71. C₂×C₄₇

C₄×Q₁₆

semidirect

81. C₄₇

C₂×C₄×Q₁₆

semidirect

83. C₂×Q₁₆

Dic₄₇

direct (N × H)

105. Q₁₆

C₂×Dic₄₇

direct (N × H)

145. C₂

Q₁₆×Dic₄₇

direct (N × H)

Subgroups — C₂×Q₁₆×Dic₄₇ #20252153

Conjugacy classes of subgroups

Complements

Subgroups — C2×Q16×Dic47 #20252153

Conjugacy classes of subgroups

Complements

Subgroups — C₂×Q₁₆×Dic₄₇ #20252153