Group #20244488 — subgroups

Subgroups — #20244488

Order 6016.

370 conjugacy classes of subgroups; only the 99 normal subgroups are shown.

normal subgroups only

This group has 370 conjugacy classes of subgroups — too many to draw. Only its 99 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	C₂²×#10625337	C₂	1	2	yes	maximal
2	C₂×#13093948	C₂	1	2	yes	maximal
3	#15689029	C₂	1	1	yes	maximal, characteristic
4	#15688878	C₂	1	1	yes	maximal, characteristic
5	C₂×C₄⋊Q₈×C₄₇	C₂	1	1	yes	maximal, characteristic
6	C₂²×Dic₉₄	C₂²	1	1	yes	characteristic
7	C₂×#10625337	—	2	4	no	—
8	C₂×#10625337	—	2	4	no	—
9	C₂×#10625337	—	2	4	no	—
10	C₂²×C₄₇⋊C₈	C₂²	1	2	yes	—
11	C₂×C₄⋊Dic₄₇	C₂²	1	1	yes	characteristic
12	#13093948	—	2	4	no	—
13	C₄₇⋊#95	—	2	2	no	—
14	C₂²×Q₈×C₄₇	C₂²	1	2	yes	—
15	C₄⋊Q₈×C₄₇	—	2	4	no	—
16	C₂×C₄⋊C₄×C₄₇	—	2	2	no	—
17	C₂×C₄²×C₄₇	C₂²	1	1	yes	characteristic
18	C₂×Q₈×C₄₇	—	2	2	no	—
19	C₂×Q₈×C₄₇	—	2	2	no	—
20	C₂×Q₈×C₄₇	—	2	2	no	—
21	C₂×Q₈×C₄₇	—	2	2	no	—
22	C₂×Q₈×C₄₇	—	2	2	no	—
23	C₂×Q₈×C₄₇	D₄	1	4	yes	—
24	C₂²×C₄×C₄₇	—	2	2	no	—
25	C₂²×C₄×C₄₇	D₄	1	2	yes	—
26	C₂²×C₄×C₄₇	C₂³	1	1	yes	characteristic
27	C₂²×Dic₄₇	—	2	1	no	—
28	C₂²×Dic₄₇	—	2	1	no	—
29	C₂×Dic₉₄	—	2	2	no	—
30	C₂×Dic₉₄	—	2	2	no	—
31	C₂×Dic₉₄	—	2	1	no	—
32	C₂×Dic₉₄	D₄	1	2	yes	—
33	C₄⋊C₄×C₄₇	—	2	8	no	—
34	C₄²×C₄₇	—	2	2	no	—
35	#10625337	—	2	16	no	—
36	C₄⋊Dic₄₇	—	2	2	no	—
37	C₂×C₄₇⋊C₈	—	2	2	no	—
38	C₂×C₄₇⋊C₈	—	2	2	no	—
39	C₂×C₄₇⋊C₈	D₄	1	4	yes	—
40	C₂³×C₄₇	C₂×D₄	1	1	yes	characteristic
41	Q₈×C₄₇	—	2	4	no	—
42	Q₈×C₄₇	—	2	4	no	—
43	Q₈×C₄₇	—	4	2	no	—
44	Q₈×C₄₇	—	4	2	no	—
45	C₂×C₄×C₄₇	—	2	2	no	—
46	C₂×C₄×C₄₇	—	2	2	no	—
47	C₂×C₄×C₄₇	—	2	4	no	—
48	C₂×C₄×C₄₇	—	4	2	no	—
49	C₂×C₄×C₄₇	—	4	2	no	—
50	C₂×C₄×C₄₇	Q₁₆	1	4	yes	complemented
51	C₂×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
52	C₂×C₄×C₄₇	C₂×D₄	1	2	yes	—
53	C₂×C₄×C₄₇	C₂×D₄	1	2	yes	—
54	C₂×C₄×C₄₇	#41	1	1	yes	characteristic
55	C₂×Dic₄₇	—	2	2	no	—
56	C₂×Dic₄₇	—	4	1	no	—
57	C₂×Dic₄₇	—	4	1	no	—
58	C₂×Dic₄₇	—	4	1	no	—
59	C₂×Dic₄₇	—	4	2	no	—
60	Dic₉₄	—	2	4	no	—
61	Dic₉₄	—	4	2	no	—
62	C₄₇⋊C₈	—	2	4	no	—
63	C₂²×C₄₇	#120	1	1	yes	characteristic
64	C₂²×C₄₇	#121	1	1	yes	characteristic
65	C₂²×C₄₇	#127	1	1	yes	characteristic
66	C₂²×C₄₇	C₂×Q₁₆	1	2	yes	—
67	C₂²×C₄₇	#137	1	2	yes	—
68	C₄×C₄₇	—	2	4	no	—
69	C₄×C₄₇	—	4	4	no	—
70	C₄×C₄₇	#120	1	2	yes	—
71	C₄×C₄₇	#121	1	2	yes	—
72	Dic₄₇	—	4	2	no	—
73	Dic₄₇	—	8	1	no	—
74	#2016	—	47	1	no	maximal, Sylow
75	C₂×C₄₇	#392	1	1	yes	characteristic
76	C₂×C₄₇	#451	1	2	yes	—
77	C₂×C₄₇	#462	1	2	yes	—
78	C₂×C₄₇	#494	1	1	yes	characteristic
79	C₂×C₄₇	#497	1	1	yes	characteristic
80	C₂²×Q₁₆	—	47	2	no	—
81	C₂×C₄⋊Q₈	D₄₇	1	1	yes	characteristic
82	C₂×#103	—	47	2	no	—
83	#391	—	47	1	no	—
84	#336	—	47	1	no	—
85	C₄₇	#2016	1	1	yes	characteristic, complemented, Sylow
86	C₂²×Q₈	—	47	1	no	—
87	C₂²×Q₈	C₂×D₄₇	1	2	yes	—
88	C₂×Q₁₆	—	94	4	no	—
89	C₂×Q₁₆	—	94	4	no	—
90	C₂×Q₁₆	—	94	4	no	—
91	C₂²×C₈	—	47	2	no	—
92	C₄⋊Q₈	—	2	4	no	—
93	C₂×C₄⋊C₄	—	2	2	no	—
94	C₂×C₄⋊C₄	—	47	1	no	—
95	C₂×C₄²	C₂×D₄₇	1	1	yes	characteristic
96	#103	—	94	4	no	—
97	#95	—	94	2	no	—
98	C₂×Q₈	—	2	2	no	—
99	C₂×Q₈	—	2	2	no	—
100	C₂×Q₈	—	2	2	no	—
101	C₂×Q₈	—	2	2	no	—
102	C₂×Q₈	—	2	2	no	—
103	C₂×Q₈	—	47	2	no	—
104	C₂×Q₈	—	94	1	no	—
105	C₂×Q₈	—	94	2	no	—
106	C₂×Q₈	—	94	2	no	—
107	C₂×Q₈	C₂²⋊D₄₇	1	4	yes	—
108	C₂²×C₄	—	2	2	no	—
109	C₂²×C₄	—	94	1	no	—
110	C₂²×C₄	—	94	1	no	—
111	C₂²×C₄	C₂²⋊D₄₇	1	2	yes	—
112	C₂²×C₄	C₂²×D₄₇	1	1	yes	characteristic
113	Q₁₆	—	94	16	no	—
114	C₂×C₈	—	47	4	no	—
115	C₂×C₈	—	94	2	no	—
116	C₂×C₈	—	94	2	no	—
117	C₄⋊C₄	—	2	8	no	—
118	C₄⋊C₄	—	94	2	no	—
119	C₄²	—	2	2	no	—
120	C₂³	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
121	Q₈	—	2	4	no	—
122	Q₈	—	2	4	no	—
123	Q₈	—	4	2	no	—
124	Q₈	—	4	2	no	—
125	Q₈	—	94	4	no	—
126	Q₈	—	188	2	no	—
127	C₂×C₄	—	2	2	no	—
128	C₂×C₄	—	2	2	no	—
129	C₂×C₄	—	2	4	no	—
130	C₂×C₄	—	4	2	no	—
131	C₂×C₄	—	4	2	no	—
132	C₂×C₄	—	94	2	no	—
133	C₂×C₄	—	188	1	no	—
134	C₂×C₄	—	188	2	no	—
135	C₂×C₄	—	188	1	no	—
136	C₂×C₄	—	188	1	no	—
137	C₂×C₄	#10625337	1	4	yes	complemented
138	C₂×C₄	D₄×D₄₇	1	1	yes	characteristic
139	C₂×C₄	D₄×D₄₇	1	2	yes	—
140	C₂×C₄	#10625352	1	1	yes	characteristic
141	C₂×C₄	C₂×C₂²⋊D₄₇	1	2	yes	—
142	C₈	—	94	4	no	—
143	C₂²	#13093993	1	1	yes	characteristic
144	C₂²	#13093995	1	1	yes	characteristic
145	C₂²	#13094020	1	1	yes	characteristic
146	C₂²	C₂×#10625337	1	2	yes	—
147	C₂²	#13094055	1	2	yes	—
148	C₄	—	2	4	no	—
149	C₄	—	4	4	no	—
150	C₄	—	376	1	no	—
151	C₄	—	188	2	no	—
152	C₄	#13093992	1	2	yes	—
153	C₄	#13093996	1	2	yes	—
154	C₂	#15689031	1	1	yes	characteristic
155	C₂	#15689279	1	2	yes	—
156	C₂	#15689330	1	2	yes	—
157	C₂	#15689502	1	1	yes	characteristic
158	C₂	#15689515	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂²×#10625337

C₂

yes

maximal

C₂×#13093948

C₂

yes

maximal

#15689029

C₂

yes

maximal, characteristic

#15688878

C₂

yes

maximal, characteristic

C₂×C₄⋊Q₈×C₄₇

C₂

yes

maximal, characteristic

C₂²×Dic₉₄

C₂²

yes

—

—

—

C₂²×C₄₇⋊C₈

C₂²

yes

—

C₂×C₄⋊Dic₄₇

C₂²

yes

characteristic

#13093948

—

C₄₇⋊#95

—

C₂²×Q₈×C₄₇

C₂²

yes

—

C₄⋊Q₈×C₄₇

—

C₂×C₄⋊C₄×C₄₇

—

C₂×C₄²×C₄₇

C₂²

yes

characteristic

C₂×Q₈×C₄₇

—

C₂×Q₈×C₄₇

—

C₂×Q₈×C₄₇

—

C₂×Q₈×C₄₇

—

C₂×Q₈×C₄₇

—

C₂×Q₈×C₄₇

D₄

yes

—

C₂²×C₄×C₄₇

—

C₂²×C₄×C₄₇

D₄

yes

—

C₂²×C₄×C₄₇

C₂³

yes

characteristic

C₂²×Dic₄₇

—

C₂²×Dic₄₇

—

C₂×Dic₉₄

—

C₂×Dic₉₄

—

C₂×Dic₉₄

—

C₂×Dic₉₄

D₄

yes

—

C₄⋊C₄×C₄₇

—

C₄²×C₄₇

—

#10625337

—

C₄⋊Dic₄₇

—

C₂×C₄₇⋊C₈

—

C₂×C₄₇⋊C₈

—

C₂×C₄₇⋊C₈

D₄

yes

—

C₂³×C₄₇

C₂×D₄

yes

characteristic

Q₈×C₄₇

—

Q₈×C₄₇

—

Q₈×C₄₇

—

Q₈×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

Q₁₆

yes

complemented

C₂×C₄×C₄₇

C₂×D₄

yes

characteristic

C₂×C₄×C₄₇

C₂×D₄

yes

—

C₂×C₄×C₄₇

C₂×D₄

yes

—

C₂×C₄×C₄₇

#41

yes

characteristic

C₂×Dic₄₇

—

C₂×Dic₄₇

—

C₂×Dic₄₇

—

C₂×Dic₄₇

—

C₂×Dic₄₇

—

Dic₉₄

—

Dic₉₄

—

C₄₇⋊C₈

—

C₂²×C₄₇

yes

C₂²×C₄₇

yes

C₂²×C₄₇

yes

C₂²×C₄₇

C₂×Q₁₆

yes

—

C₂²×C₄₇

#137

yes

—

C₄×C₄₇

—

C₄×C₄₇

—

C₄×C₄₇

#120

yes

—

C₄×C₄₇

#121

yes

—

Dic₄₇

—

Dic₄₇

—

—

C₂×C₄₇

yes

C₂×C₄₇

yes

—

C₂×C₄₇

#462

yes

—

C₂×C₄₇

yes

C₂×C₄₇

yes

C₂²×Q₁₆

—

C₂×C₄⋊Q₈

D₄₇

yes

—

—

—

C₄₇

#2016

yes

characteristic, complemented, Sylow

C₂²×Q₈

—

C₂²×Q₈

C₂×D₄₇

yes

—

C₂×Q₁₆

—

C₂×Q₁₆

—

C₂×Q₁₆

—

C₂²×C₈

—

C₄⋊Q₈

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄²

C₂×D₄₇

yes

characteristic

#103

—

#95

—

C₂×Q₈

—

C₂×Q₈

—

100

C₂×Q₈

—

101

C₂×Q₈

—

102

C₂×Q₈

—

103

C₂×Q₈

—

104

C₂×Q₈

—

105

C₂×Q₈

—

106

C₂×Q₈

—

107

C₂×Q₈

C₂²⋊D₄₇

yes

—

108

C₂²×C₄

—

109

C₂²×C₄

—

110

C₂²×C₄

—

111

C₂²×C₄

C₂²⋊D₄₇

yes

—

112

C₂²×C₄

C₂²×D₄₇

yes

characteristic

113

Q₁₆

—

114

C₂×C₈

—

115

C₂×C₈

—

116

C₂×C₈

—

117

C₄⋊C₄

—

118

C₄⋊C₄

—

119

C₄²

—

120

C₂³

C₂×C₂²⋊D₄₇

yes

characteristic

121

Q₈

—

122

Q₈

—

123

Q₈

—

124

Q₈

—

125

Q₈

—

126

Q₈

—

188

—

127

C₂×C₄

—

128

C₂×C₄

—

129

C₂×C₄

—

130

C₂×C₄

—

131

C₂×C₄

—

132

C₂×C₄

—

133

C₂×C₄

—

188

—

134

C₂×C₄

—

188

—

135

C₂×C₄

—

188

—

136

C₂×C₄

—

188

—

137

C₂×C₄

#10625337

yes

complemented

138

C₂×C₄

D₄×D₄₇

yes

characteristic

139

C₂×C₄

D₄×D₄₇

yes

—

140

C₂×C₄

#10625352

yes

characteristic

141

C₂×C₄

C₂×C₂²⋊D₄₇

yes

—

142

C₈

—

143

C₂²

#13093993

yes

characteristic

144

C₂²

#13093995

yes

characteristic

145

C₂²

#13094020

yes

characteristic

146

C₂²

C₂×#10625337

yes

—

147

C₂²

#13094055

yes

—

148

C₄

—

149

C₄

—

150

C₄

—

376

—

151

C₄

—

188

—

152

C₄

#13093992

yes

—

153

C₄

#13093996

yes

—

154

C₂

#15689031

yes

characteristic

155

C₂

#15689279

yes

—

156

C₂

#15689330

yes

—

157

C₂

#15689502

yes

characteristic

158

C₂

#15689515

yes

characteristic

	subgroup	G/H
center	C₂³	C₂×C₂²⋊D₄₇
commutator	C₂²×C₄×C₄₇	C₂³
Frattini	C₂²×C₄	C₂²×D₄₇
Fitting	C₂×C₄⋊Q₈×C₄₇	C₂
socle	C₂³×C₄₇	C₂×D₄
hypercenter	C₂×C₄⋊Q₈	D₄₇
hypocenter	C₄₇	#2016
LCS	G ▷ C₂²×C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄⋊Q₈

subgroup

G/H

center

C₂³

C₂×C₂²⋊D₄₇

commutator

C₂²×C₄×C₄₇

C₂³

Frattini

C₂²×C₄

C₂²×D₄₇

Fitting

C₂×C₄⋊Q₈×C₄₇

C₂

socle

C₂³×C₄₇

C₂×D₄

hypercenter

C₂×C₄⋊Q₈

D₄₇

hypocenter

C₄₇

#2016

LCS

G ▷ C₂²×C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇

UCS

C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄⋊Q₈

N	H ≅ G/N	type
50. C₂×C₄×C₄₇	Q₁₆	semidirect
85. C₄₇	#2016	semidirect
137. C₂×C₄	#10625337	semidirect

H ≅ G/N

type

50. C₂×C₄×C₄₇

Q₁₆

semidirect

85. C₄₇

#2016

semidirect

137. C₂×C₄

#10625337

semidirect

Subgroups — #20244488

Conjugacy classes of subgroups

Complements