Group #20244200 — subgroups

Subgroups — #20244200

Order 6016.

336 conjugacy classes of subgroups; only the 113 normal subgroups are shown.

normal subgroups only

This group has 336 conjugacy classes of subgroups — too many to draw. Only its 113 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×#13093948	C₂	1	2	yes	maximal, complemented
2	C₂×#13093945	C₂	1	2	yes	maximal
3	C₂×C₄×C₄₇⋊C₈	C₂	1	1	yes	maximal, characteristic, complemented
4	#15688989	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×#124×C₄₇	C₂	1	1	yes	maximal, characteristic
6	C₂²×C₄₇⋊C₈	C₂²	1	2	yes	—
7	C₂×C₄⋊Dic₄₇	C₂²	1	2	yes	—
8	#13093948	—	2	4	no	—
9	#13093945	—	2	4	no	—
10	C₄×C₄₇⋊C₈	—	2	2	no	—
11	C₄₇⋊#95	—	2	2	no	—
12	C₂²×Q₈×C₄₇	C₂²	1	1	yes	characteristic
13	C₂²×D₄×C₄₇	C₂²	1	1	yes	characteristic
14	#124×C₄₇	—	2	2	no	—
15	#124×C₄₇	C₄	1	4	yes	complemented
16	C₂×C₂²⋊C₄×C₄₇	—	2	2	no	—
17	C₂×C₄²×C₄₇	C₂²	1	1	yes	characteristic
18	C₂⁴×C₄₇	—	2	1	no	—
19	C₂×Q₈×C₄₇	—	2	1	no	—
20	C₂×Q₈×C₄₇	—	2	2	no	—
21	C₂×Q₈×C₄₇	—	2	2	no	—
22	C₂×Q₈×C₄₇	C₂×C₄	1	2	yes	—
23	C₂×D₄×C₄₇	—	2	2	no	—
24	C₂×D₄×C₄₇	—	2	1	no	—
25	C₂×D₄×C₄₇	—	2	2	no	—
26	C₂×D₄×C₄₇	C₂×C₄	1	2	yes	—
27	C₂²×C₄×C₄₇	—	2	1	no	—
28	C₂²×C₄×C₄₇	D₄	1	2	yes	—
29	C₂²×C₄×C₄₇	C₂³	1	1	yes	characteristic
30	C₂²×Dic₄₇	—	2	2	no	—
31	C₂²⋊C₄×C₄₇	—	2	8	no	—
32	C₄²×C₄₇	C₂×C₄	1	2	yes	—
33	C₄²×C₄₇	D₄	1	2	yes	—
34	C₄⋊Dic₄₇	—	2	2	no	—
35	C₄⋊Dic₄₇	—	2	2	no	—
36	C₂×C₄₇⋊C₈	—	2	2	no	—
37	C₂×C₄₇⋊C₈	—	2	2	no	—
38	C₂×C₄₇⋊C₈	D₄	1	4	yes	—
39	C₂³×C₄₇	—	2	2	no	—
40	C₂³×C₄₇	—	4	1	no	—
41	C₂³×C₄₇	—	4	1	no	—
42	C₂³×C₄₇	—	4	4	no	—
43	C₂³×C₄₇	C₂×D₄	1	1	yes	characteristic
44	Q₈×C₄₇	—	2	4	no	—
45	Q₈×C₄₇	—	4	2	no	—
46	D₄×C₄₇	—	2	4	no	—
47	D₄×C₄₇	—	4	2	no	—
48	C₂×C₄×C₄₇	—	2	4	no	—
49	C₂×C₄×C₄₇	—	2	2	no	—
50	C₂×C₄×C₄₇	—	4	1	no	—
51	C₂×C₄×C₄₇	—	4	1	no	—
52	C₂×C₄×C₄₇	C₂²⋊C₄	1	4	yes	—
53	C₂×C₄×C₄₇	C₂²×C₄	1	1	yes	characteristic
54	C₂×C₄×C₄₇	C₂×D₄	1	2	yes	—
55	C₂×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
56	C₂×C₄×C₄₇	#41	1	2	yes	—
57	C₂×Dic₄₇	—	4	2	no	—
58	C₂×Dic₄₇	—	4	4	no	—
59	C₄₇⋊C₈	—	2	4	no	—
60	C₂²×C₄₇	—	4	2	no	—
61	C₂²×C₄₇	—	4	4	no	—
62	C₂²×C₄₇	—	8	2	no	—
63	C₂²×C₄₇	—	8	2	no	—
64	C₂²×C₄₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
65	C₂²×C₄₇	#124	1	1	yes	characteristic
66	C₂²×C₄₇	#127	1	1	yes	characteristic
67	C₂²×C₄₇	#135	1	4	yes	—
68	C₄×C₄₇	—	2	4	no	—
69	C₄×C₄₇	—	4	2	no	—
70	C₄×C₄₇	C₄×D₄	1	2	yes	—
71	C₄×C₄₇	#121	1	2	yes	—
72	Dic₄₇	—	8	2	no	—
73	#1958	—	47	1	no	maximal, Sylow
74	C₂×C₄₇	—	8	2	no	—
75	C₂×C₄₇	#390	1	1	yes	characteristic
76	C₂×C₄₇	#416	1	2	yes	—
77	C₂×C₄₇	#488	1	2	yes	—
78	C₂×C₄₇	#495	1	2	yes	—
79	C₂×#124	D₄₇	1	1	yes	characteristic
80	C₂×#103	—	47	2	no	—
81	C₂×#102	—	47	2	no	—
82	C₂×C₄×C₈	—	47	1	no	—
83	#382	—	47	1	no	—
84	C₄₇	#1958	1	1	yes	characteristic, complemented, Sylow
85	C₂²×Q₈	C₂×D₄₇	1	1	yes	characteristic
86	C₂²×D₄	C₂×D₄₇	1	1	yes	characteristic
87	C₂²×C₈	—	47	2	no	—
88	#124	—	2	2	no	—
89	#124	Dic₄₇	1	4	yes	complemented
90	C₂×C₄⋊C₄	—	47	2	no	—
91	C₂×C₂²⋊C₄	—	2	2	no	—
92	C₂×C₄²	C₂×D₄₇	1	1	yes	characteristic
93	#103	—	94	4	no	—
94	#102	—	94	4	no	—
95	C₄×C₈	—	94	2	no	—
96	#95	—	94	2	no	—
97	C₂⁴	—	2	1	no	—
98	C₂×Q₈	—	2	1	no	—
99	C₂×Q₈	—	2	2	no	—
100	C₂×Q₈	—	2	2	no	—
101	C₂×Q₈	C₂×Dic₄₇	1	2	yes	—
102	C₂×D₄	—	2	2	no	—
103	C₂×D₄	—	2	2	no	—
104	C₂×D₄	—	2	1	no	—
105	C₂×D₄	C₂×Dic₄₇	1	2	yes	—
106	C₂²×C₄	—	2	1	no	—
107	C₂²×C₄	—	94	2	no	—
108	C₂²×C₄	C₂²⋊D₄₇	1	2	yes	—
109	C₂²×C₄	C₂²×D₄₇	1	1	yes	characteristic
110	C₂×C₈	—	47	4	no	—
111	C₂×C₈	—	94	2	no	—
112	C₂×C₈	—	94	2	no	—
113	C₄⋊C₄	—	94	2	no	—
114	C₄⋊C₄	—	94	2	no	—
115	C₂²⋊C₄	—	2	8	no	—
116	C₄²	C₂×Dic₄₇	1	2	yes	—
117	C₄²	C₂²⋊D₄₇	1	2	yes	—
118	C₂³	—	2	2	no	—
119	C₂³	—	4	1	no	—
120	C₂³	—	4	1	no	—
121	C₂³	—	4	4	no	—
122	C₂³	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
123	Q₈	—	2	4	no	—
124	Q₈	—	4	2	no	—
125	D₄	—	2	4	no	—
126	D₄	—	4	2	no	—
127	C₂×C₄	—	2	4	no	—
128	C₂×C₄	—	2	2	no	—
129	C₂×C₄	—	4	1	no	—
130	C₂×C₄	—	4	1	no	—
131	C₂×C₄	—	188	2	no	—
132	C₂×C₄	—	188	4	no	—
133	C₂×C₄	C₂²⋊Dic₄₇	1	4	yes	—
134	C₂×C₄	D₄×D₄₇	1	2	yes	—
135	C₂×C₄	#10625352	1	2	yes	—
136	C₂×C₄	C₂²×Dic₄₇	1	1	yes	characteristic
137	C₂×C₄	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
138	C₈	—	94	4	no	—
139	C₂²	—	4	2	no	—
140	C₂²	—	4	4	no	—
141	C₂²	—	8	2	no	—
142	C₂²	—	8	2	no	—
143	C₂²	C₂×C₂²⋊Dic₄₇	1	1	yes	characteristic
144	C₂²	#13094011	1	1	yes	characteristic
145	C₂²	#13094020	1	1	yes	characteristic
146	C₂²	#13094044	1	4	yes	—
147	C₄	—	2	4	no	—
148	C₄	—	4	2	no	—
149	C₄	—	376	2	no	—
150	C₄	D₄×Dic₄₇	1	2	yes	—
151	C₄	#13093995	1	2	yes	—
152	C₂	—	8	2	no	—
153	C₂	#15689026	1	1	yes	characteristic
154	C₂	#15689119	1	2	yes	—
155	C₂	#15689477	1	2	yes	—
156	C₂	#15689505	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#13093948

C₂

yes

maximal, complemented

C₂×#13093945

C₂

yes

maximal

C₂×C₄×C₄₇⋊C₈

C₂

yes

maximal, characteristic, complemented

#15688989

C₂

yes

maximal, characteristic, complemented

C₂×#124×C₄₇

C₂

yes

maximal, characteristic

C₂²×C₄₇⋊C₈

C₂²

yes

—

C₂×C₄⋊Dic₄₇

C₂²

yes

—

#13093948

—

#13093945

—

C₄×C₄₇⋊C₈

—

C₄₇⋊#95

—

C₂²×Q₈×C₄₇

C₂²

yes

characteristic

C₂²×D₄×C₄₇

C₂²

yes

characteristic

#124×C₄₇

—

#124×C₄₇

C₄

yes

complemented

C₂×C₂²⋊C₄×C₄₇

—

C₂×C₄²×C₄₇

C₂²

yes

characteristic

C₂⁴×C₄₇

—

C₂×Q₈×C₄₇

—

C₂×Q₈×C₄₇

—

C₂×Q₈×C₄₇

—

C₂×Q₈×C₄₇

C₂×C₄

yes

—

C₂×D₄×C₄₇

—

C₂×D₄×C₄₇

—

C₂×D₄×C₄₇

—

C₂×D₄×C₄₇

C₂×C₄

yes

—

C₂²×C₄×C₄₇

—

C₂²×C₄×C₄₇

D₄

yes

—

C₂²×C₄×C₄₇

C₂³

yes

characteristic

C₂²×Dic₄₇

—

C₂²⋊C₄×C₄₇

—

C₄²×C₄₇

C₂×C₄

yes

—

C₄²×C₄₇

D₄

yes

—

C₄⋊Dic₄₇

—

C₄⋊Dic₄₇

—

C₂×C₄₇⋊C₈

—

C₂×C₄₇⋊C₈

—

C₂×C₄₇⋊C₈

D₄

yes

—

C₂³×C₄₇

—

C₂³×C₄₇

—

C₂³×C₄₇

—

C₂³×C₄₇

—

C₂³×C₄₇

C₂×D₄

yes

characteristic

Q₈×C₄₇

—

Q₈×C₄₇

—

D₄×C₄₇

—

D₄×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

—

C₂×C₄×C₄₇

C₂×D₄

yes

characteristic

C₂×C₄×C₄₇

#41

yes

—

C₂×Dic₄₇

—

C₂×Dic₄₇

—

C₄₇⋊C₈

—

C₂²×C₄₇

—

C₂²×C₄₇

—

C₂²×C₄₇

—

C₂²×C₄₇

—

C₂²×C₄₇

C₂×C₂²⋊C₄

yes

C₂²×C₄₇

yes

C₂²×C₄₇

yes

C₂²×C₄₇

yes

—

C₄×C₄₇

—

C₄×C₄₇

—

C₄×C₄₇

C₄×D₄

yes

—

C₄×C₄₇

#121

yes

—

Dic₄₇

—

#1958

—

maximal, Sylow

C₂×C₄₇

—

C₂×C₄₇

#390

yes

characteristic

C₂×C₄₇

#416

yes

—

C₂×C₄₇

#488

yes

—

C₂×C₄₇

#495

yes

—

C₂×#124

D₄₇

yes

characteristic

C₂×#103

—

C₂×#102

—

C₂×C₄×C₈

—

#382

—

C₄₇

#1958

yes

characteristic, complemented, Sylow

C₂²×Q₈

C₂×D₄₇

yes

characteristic

C₂²×D₄

C₂×D₄₇

yes

characteristic

C₂²×C₈

—

#124

—

#124

Dic₄₇

yes

complemented

C₂×C₄⋊C₄

—

C₂×C₂²⋊C₄

—

C₂×C₄²

C₂×D₄₇

yes

characteristic

#103

—

#102

—

C₄×C₈

—

#95

—

C₂⁴

—

C₂×Q₈

—

C₂×Q₈

—

100

C₂×Q₈

—

101

C₂×Q₈

C₂×Dic₄₇

yes

—

102

C₂×D₄

—

103

C₂×D₄

—

104

C₂×D₄

—

105

C₂×D₄

C₂×Dic₄₇

yes

—

106

C₂²×C₄

—

107

C₂²×C₄

—

108

C₂²×C₄

C₂²⋊D₄₇

yes

—

109

C₂²×C₄

C₂²×D₄₇

yes

characteristic

110

C₂×C₈

—

111

C₂×C₈

—

112

C₂×C₈

—

113

C₄⋊C₄

—

114

C₄⋊C₄

—

115

C₂²⋊C₄

—

116

C₄²

C₂×Dic₄₇

yes

—

117

C₄²

C₂²⋊D₄₇

yes

—

118

C₂³

—

119

C₂³

—

120

C₂³

—

121

C₂³

—

122

C₂³

C₂×C₂²⋊D₄₇

yes

characteristic

123

Q₈

—

124

Q₈

—

125

D₄

—

126

D₄

—

127

C₂×C₄

—

128

C₂×C₄

—

129

C₂×C₄

—

130

C₂×C₄

—

131

C₂×C₄

—

188

—

132

C₂×C₄

—

188

—

133

C₂×C₄

C₂²⋊Dic₄₇

yes

—

134

C₂×C₄

D₄×D₄₇

yes

—

135

C₂×C₄

#10625352

yes

—

136

C₂×C₄

C₂²×Dic₄₇

yes

characteristic

137

C₂×C₄

C₂×C₂²⋊D₄₇

yes

characteristic

138

C₈

—

139

C₂²

—

140

C₂²

—

141

C₂²

—

142

C₂²

—

143

C₂²

C₂×C₂²⋊Dic₄₇

yes

characteristic

144

C₂²

#13094011

yes

characteristic

145

C₂²

#13094020

yes

characteristic

146

C₂²

#13094044

yes

—

147

C₄

—

148

C₄

—

149

C₄

—

376

—

150

C₄

D₄×Dic₄₇

yes

—

151

C₄

#13093995

yes

—

152

C₂

—

153

C₂

#15689026

yes

characteristic

154

C₂

#15689119

yes

—

155

C₂

#15689477

yes

—

156

C₂

#15689505

yes

—

	subgroup	G/H
center	C₂³	C₂×C₂²⋊D₄₇
commutator	C₂×C₄×C₄₇	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₄₇
Fitting	C₂×#124×C₄₇	C₂
socle	C₂³×C₄₇	C₂×D₄
hypercenter	C₂×#124	D₄₇
hypocenter	C₄₇	#1958
LCS	G ▷ C₂×C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×#124

subgroup

G/H

center

C₂³

C₂×C₂²⋊D₄₇

commutator

C₂×C₄×C₄₇

C₂²×C₄

Frattini

C₂²×C₄

C₂²×D₄₇

Fitting

C₂×#124×C₄₇

C₂

socle

C₂³×C₄₇

C₂×D₄

D₄₇

C₄₇

G ▷ C₂×C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇

UCS

C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×#124

N	H ≅ G/N	type
1. C₂×#13093948	C₂	semidirect
3. C₂×C₄×C₄₇⋊C₈	C₂	semidirect
4. #15688989	C₂	semidirect
15. #124×C₄₇	C₄	semidirect
84. C₄₇	#1958	semidirect
89. #124	Dic₄₇	semidirect

H ≅ G/N

type

1. C₂×#13093948

C₂

semidirect

3. C₂×C₄×C₄₇⋊C₈

C₂

semidirect

4. #15688989

C₂

semidirect

15. #124×C₄₇

C₄

semidirect

84. C₄₇

#1958

semidirect

89. #124

Dic₄₇

semidirect

Subgroups — #20244200

Conjugacy classes of subgroups

Complements