Group #20243433 — subgroups

Subgroups — #20243433

Order 6016.

260 conjugacy classes of subgroups; only the 81 normal subgroups are shown.

normal subgroups only

This group has 260 conjugacy classes of subgroups — too many to draw. Only its 81 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×#13094032	C₂	1	1	yes	maximal, characteristic
2	#15689159	C₂	1	2	yes	maximal, complemented
3	#15689096	C₂	1	2	yes	maximal
4	#15688880	C₂	1	1	yes	maximal, characteristic, complemented
5	C₄₇×H(3,ℤ/4)	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×#10625350	C₂²	1	1	yes	characteristic
7	#13094032	—	2	2	no	—
8	#13094032	—	2	2	no	—
9	C₂×#10625324	C₄	1	2	yes	complemented
10	C₂²×C₄₇⋊C₈	C₂²	1	1	yes	characteristic
11	C₂×C₈×D₄₇	C₄	1	2	yes	complemented
12	#13093983	C₂²	1	2	yes	—
13	#13093957	—	2	2	no	—
14	#13093955	—	2	2	no	—
15	#13093938	—	2	4	no	—
16	C₂×M₁₆×C₄₇	C₂²	1	1	yes	characteristic
17	#117×C₄₇	C₂²	1	2	yes	—
18	C₂²×C₄×C₄₇	C₂³	1	1	yes	characteristic
19	C₂×C₂²⋊D₄₇	D₄	1	2	yes	—
20	#10625350	—	2	2	no	—
21	#10625350	—	2	2	no	—
22	C₂×D₁₈₈	Q₈	1	1	yes	characteristic
23	C₂×C₄×D₄₇	C₂×C₄	1	2	yes	—
24	C₂×Dic₉₄	Q₈	1	1	yes	characteristic
25	M₁₆×C₄₇	—	2	2	no	—
26	C₂×C₈×C₄₇	C₂×C₄	1	2	yes	—
27	C₄⋊C₄×C₄₇	—	2	2	no	—
28	C₂²⋊C₄×C₄₇	—	2	2	no	—
29	C₄²×C₄₇	—	2	2	no	—
30	C₂²⋊Dic₄₇	—	2	2	no	—
31	C₄⋊Dic₄₇	—	2	2	no	—
32	C₄×Dic₄₇	—	2	2	no	—
33	C₂×C₄₇⋊C₈	C₂×C₄	1	1	yes	characteristic
34	C₂×C₄₇⋊C₈	C₂×C₄	1	1	yes	characteristic
35	C₂×C₄₇⋊C₈	D₄	1	2	yes	—
36	C₂×C₄₇⋊C₈	D₄	1	2	yes	—
37	#10625324	—	2	4	no	—
38	C₈×D₄₇	—	2	2	no	—
39	C₈×D₄₇	—	2	2	no	—
40	C₂³×C₄₇	C₂×D₄	1	1	yes	characteristic
41	C₂²×D₄₇	C₄⋊C₄	1	2	yes	—
42	C₂×C₄×C₄₇	—	4	2	no	—
43	C₂×C₄×C₄₇	C₂²×C₄	1	1	yes	characteristic
44	C₂×C₄×C₄₇	C₂×D₄	1	2	yes	—
45	C₂×C₄×C₄₇	C₂×Q₈	1	1	yes	characteristic
46	C₂×C₄×C₄₇	#41	1	2	yes	—
47	C₂²⋊D₄₇	—	2	4	no	—
48	C₂×Dic₄₇	—	4	2	no	—
49	C₂×Dic₄₇	C₄⋊C₄	1	2	yes	—
50	D₁₈₈	—	2	2	no	—
51	C₄×D₄₇	—	2	2	no	—
52	C₄×D₄₇	—	2	2	no	—
53	Dic₉₄	—	2	2	no	—
54	C₈×C₄₇	—	2	2	no	—
55	C₄₇⋊C₈	C₂²⋊C₄	1	1	yes	characteristic
56	C₄₇⋊C₈	C₂²⋊C₄	1	1	yes	characteristic
57	C₄₇⋊C₈	C₂²⋊C₄	1	2	yes	—
58	C₂²×C₄₇	—	4	1	no	—
59	C₂²×C₄₇	C₂×C₄⋊C₄	1	1	yes	characteristic
60	C₂²×C₄₇	#121	1	2	yes	—
61	C₂×D₄₇	—	2	2	no	—
62	C₂×D₄₇	—	4	2	no	—
63	C₄×C₄₇	—	8	2	no	—
64	C₄×C₄₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
65	C₄×C₄₇	#117	1	1	yes	characteristic
66	C₄×C₄₇	C₂²⋊Q₈	1	2	yes	—
67	Dic₄₇	—	2	2	no	—
68	Dic₄₇	—	8	2	no	—
69	#1812	—	47	1	no	maximal, Sylow
70	C₂×C₄₇	—	2	2	no	—
71	C₂×C₄₇	—	2	1	no	—
72	C₂×C₄₇	#380	1	1	yes	characteristic
73	D₄₇	—	4	2	no	—
74	C₂×#131	—	47	1	no	—
75	#427	—	47	2	no	—
76	#410	—	47	2	no	—
77	H(3,ℤ/4)	—	47	1	no	—
78	H(3,ℤ/4)	D₄₇	1	1	yes	characteristic, complemented
79	C₄₇	#1812	1	1	yes	characteristic, complemented, Sylow
80	C₂×#41	—	47	1	no	—
81	#131	—	94	2	no	—
82	#131	—	94	2	no	—
83	C₂×M₁₆	—	47	2	no	—
84	C₂×M₁₆	C₂×D₄₇	1	1	yes	characteristic
85	C₂²×C₈	—	47	1	no	—
86	C₂²×C₈	—	47	2	no	—
87	#117	—	47	2	no	—
88	#117	C₂×D₄₇	1	2	yes	—
89	#107	—	94	2	no	—
90	#106	—	94	2	no	—
91	#99	—	94	4	no	—
92	#41	—	94	2	no	—
93	#41	—	94	2	no	—
94	C₂×Q₈	—	47	1	no	—
95	C₂×D₄	—	47	2	no	—
96	C₂×D₄	—	47	1	no	—
97	C₂²×C₄	—	47	2	no	—
98	C₂²×C₄	C₂²×D₄₇	1	1	yes	characteristic
99	M₁₆	—	2	2	no	—
100	M₁₆	—	94	4	no	—
101	C₂×C₈	—	47	1	no	—
102	C₂×C₈	—	47	1	no	—
103	C₂×C₈	—	47	2	no	—
104	C₂×C₈	—	47	2	no	—
105	C₂×C₈	—	94	2	no	—
106	C₂×C₈	—	94	2	no	—
107	C₂×C₈	C₄×D₄₇	1	2	yes	—
108	C₄⋊C₄	—	2	2	no	—
109	C₄⋊C₄	—	94	2	no	—
110	C₂²⋊C₄	—	2	2	no	—
111	C₂²⋊C₄	—	94	2	no	—
112	C₄²	—	2	2	no	—
113	C₄²	—	94	2	no	—
114	C₂³	—	47	2	no	—
115	C₂³	D₄×D₄₇	1	1	yes	characteristic
116	Q₈	—	94	2	no	—
117	D₄	—	94	4	no	—
118	D₄	—	94	2	no	—
119	C₂×C₄	—	4	2	no	—
120	C₂×C₄	—	47	2	no	—
121	C₂×C₄	—	94	2	no	—
122	C₂×C₄	—	94	2	no	—
123	C₂×C₄	—	188	2	no	—
124	C₂×C₄	C₂×C₄×D₄₇	1	1	yes	characteristic
125	C₂×C₄	#10625350	1	2	yes	—
126	C₂×C₄	D₄×D₄₇	1	2	yes	—
127	C₂×C₄	Q₈×D₄₇	1	1	yes	characteristic
128	C₈	—	2	2	no	—
129	C₈	—	47	1	no	—
130	C₈	—	47	1	no	—
131	C₈	—	47	2	no	—
132	C₂²	—	4	1	no	—
133	C₂²	—	94	2	no	—
134	C₂²	—	188	2	no	—
135	C₂²	C₄⋊C₄×D₄₇	1	1	yes	characteristic
136	C₂²	#13093997	1	2	yes	—
137	C₄	—	8	2	no	—
138	C₄	—	94	2	no	—
139	C₄	—	376	2	no	—
140	C₄	C₂²⋊C₄×D₄₇	1	1	yes	characteristic
141	C₄	#13093981	1	1	yes	characteristic
142	C₄	#13094002	1	2	yes	—
143	C₂	—	2	2	no	—
144	C₂	—	2	1	no	—
145	C₂	—	188	2	no	—
146	C₂	#15688982	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#13094032

C₂

yes

maximal, characteristic

#15689159

C₂

yes

maximal, complemented

#15689096

C₂

yes

maximal

#15688880

C₂

yes

maximal, characteristic, complemented

C₄₇×H(3,ℤ/4)

C₂

yes

maximal, characteristic, complemented

C₂×#10625350

C₂²

yes

—

—

C₄

yes

complemented

C₂²×C₄₇⋊C₈

C₂²

yes

characteristic

C₂×C₈×D₄₇

C₄

yes

complemented

#13093983

C₂²

yes

—

#13093957

—

#13093955

—

#13093938

—

C₂×M₁₆×C₄₇

C₂²

yes

characteristic

#117×C₄₇

C₂²

yes

—

C₂²×C₄×C₄₇

C₂³

yes

characteristic

C₂×C₂²⋊D₄₇

D₄

yes

—

#10625350

—

#10625350

—

C₂×D₁₈₈

Q₈

yes

characteristic

C₂×C₄×D₄₇

C₂×C₄

yes

—

C₂×Dic₉₄

Q₈

yes

characteristic

M₁₆×C₄₇

—

C₂×C₈×C₄₇

C₂×C₄

yes

—

C₄⋊C₄×C₄₇

—

C₂²⋊C₄×C₄₇

—

C₄²×C₄₇

—

C₂²⋊Dic₄₇

—

C₄⋊Dic₄₇

—

C₄×Dic₄₇

—

C₂×C₄₇⋊C₈

C₂×C₄

yes

characteristic

C₂×C₄₇⋊C₈

C₂×C₄

yes

characteristic

C₂×C₄₇⋊C₈

D₄

yes

—

C₂×C₄₇⋊C₈

D₄

yes

—

#10625324

—

C₈×D₄₇

—

C₈×D₄₇

—

C₂³×C₄₇

C₂×D₄

yes

characteristic

C₂²×D₄₇

C₄⋊C₄

yes

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

C₂²×C₄

yes

characteristic

C₂×C₄×C₄₇

C₂×D₄

yes

—

C₂×C₄×C₄₇

C₂×Q₈

yes

characteristic

C₂×C₄×C₄₇

#41

yes

—

C₂²⋊D₄₇

—

C₂×Dic₄₇

—

C₂×Dic₄₇

C₄⋊C₄

yes

—

D₁₈₈

—

C₄×D₄₇

—

C₄×D₄₇

—

Dic₉₄

—

C₈×C₄₇

—

C₄₇⋊C₈

C₂²⋊C₄

yes

characteristic

C₄₇⋊C₈

C₂²⋊C₄

yes

characteristic

C₄₇⋊C₈

C₂²⋊C₄

yes

—

C₂²×C₄₇

—

C₂²×C₄₇

C₂×C₄⋊C₄

yes

characteristic

C₂²×C₄₇

#121

yes

—

C₂×D₄₇

—

C₂×D₄₇

—

C₄×C₄₇

—

C₄×C₄₇

C₂×C₂²⋊C₄

yes

characteristic

C₄×C₄₇

#117

yes

characteristic

C₄×C₄₇

C₂²⋊Q₈

yes

—

Dic₄₇

—

Dic₄₇

—

#1812

—

maximal, Sylow

C₂×C₄₇

—

C₂×C₄₇

—

C₂×C₄₇

#380

yes

characteristic

D₄₇

—

—

—

—

—

D₄₇

yes

characteristic, complemented

C₄₇

#1812

yes

characteristic, complemented, Sylow

C₂×#41

—

#131

—

#131

—

C₂×M₁₆

—

C₂×M₁₆

C₂×D₄₇

yes

characteristic

C₂²×C₈

—

C₂²×C₈

—

#117

—

#117

C₂×D₄₇

yes

—

—

—

—

—

—

C₂×Q₈

—

C₂×D₄

—

C₂×D₄

—

C₂²×C₄

—

C₂²×C₄

C₂²×D₄₇

yes

characteristic

M₁₆

—

100

M₁₆

—

101

C₂×C₈

—

102

C₂×C₈

—

103

C₂×C₈

—

104

C₂×C₈

—

105

C₂×C₈

—

106

C₂×C₈

—

107

C₂×C₈

C₄×D₄₇

yes

—

108

C₄⋊C₄

—

109

C₄⋊C₄

—

110

C₂²⋊C₄

—

111

C₂²⋊C₄

—

112

C₄²

—

113

C₄²

—

114

C₂³

—

115

C₂³

D₄×D₄₇

yes

characteristic

116

Q₈

—

117

D₄

—

118

D₄

—

119

C₂×C₄

—

120

C₂×C₄

—

121

C₂×C₄

—

122

C₂×C₄

—

123

C₂×C₄

—

188

—

124

C₂×C₄

C₂×C₄×D₄₇

yes

characteristic

125

C₂×C₄

#10625350

yes

—

126

C₂×C₄

D₄×D₄₇

yes

—

127

C₂×C₄

Q₈×D₄₇

yes

characteristic

128

C₈

—

129

C₈

—

130

C₈

—

131

C₈

—

132

C₂²

—

133

C₂²

—

134

C₂²

—

188

—

135

C₂²

C₄⋊C₄×D₄₇

yes

characteristic

136

C₂²

#13093997

yes

—

137

C₄

—

138

C₄

—

139

C₄

—

376

—

140

C₄

C₂²⋊C₄×D₄₇

yes

characteristic

141

C₄

#13093981

yes

characteristic

142

C₄

#13094002

yes

—

143

C₂

—

144

C₂

—

145

C₂

—

188

—

146

C₂

#15688982

yes

characteristic

	subgroup	G/H
center	C₄	#13093981
commutator	C₂×C₄×C₄₇	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₄₇
Fitting	C₄₇×H(3,ℤ/4)	C₂
socle	C₂×C₄₇	#380
hypercenter	H(3,ℤ/4)	D₄₇
hypocenter	C₄₇	#1812
LCS	G ▷ C₂×C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₄ ◁ C₂×M₁₆ ◁ H(3,ℤ/4)

subgroup

G/H

center

C₄

#13093981

commutator

C₂×C₄×C₄₇

C₂²×C₄

Frattini

C₂²×C₄

C₂²×D₄₇

Fitting

C₄₇×H(3,ℤ/4)

C₂

C₂×C₄₇

D₄₇

C₄₇

G ▷ C₂×C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇

UCS

C₁ ◁ C₄ ◁ C₂×M₁₆ ◁ H(3,ℤ/4)

N	H ≅ G/N	type
2. #15689159	C₂	semidirect
4. #15688880	C₂	semidirect
5. C₄₇×H(3,ℤ/4)	C₂	semidirect
9. C₂×#10625324	C₄	semidirect
11. C₂×C₈×D₄₇	C₄	semidirect
78. H(3,ℤ/4)	D₄₇	semidirect
79. C₄₇	#1812	semidirect

H ≅ G/N

type

2. #15689159

C₂

semidirect

4. #15688880

C₂

semidirect

5. C₄₇×H(3,ℤ/4)

C₂

semidirect

9. C₂×#10625324

C₄

semidirect

11. C₂×C₈×D₄₇

C₄

semidirect

78. H(3,ℤ/4)

D₄₇

semidirect

79. C₄₇

#1812

semidirect

Subgroups — #20243433

Conjugacy classes of subgroups

Complements