Group #20243792 — subgroups

Subgroups — #20243792

Order 6016.

344 conjugacy classes of subgroups; only the 97 normal subgroups are shown.

normal subgroups only

This group has 344 conjugacy classes of subgroups — too many to draw. Only its 97 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	#15689601	C₂	1	1	yes	maximal, characteristic, complemented
2	#15689128	C₂	1	2	yes	maximal
3	C₂×#13093943	C₂	1	1	yes	maximal, characteristic, complemented
4	#15688883	C₂	1	2	yes	maximal, complemented
5	C₂×#124×C₄₇	C₂	1	1	yes	maximal, characteristic
6	C₂×#10625329	C₂²	1	2	yes	—
7	Q₈×Dic₄₇	—	2	4	no	—
8	#13093983	—	2	2	no	—
9	#13093983	C₂²	1	2	yes	—
10	#13093945	—	2	4	no	—
11	#13093943	—	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	2	no	—
20	C₂×Q₈×C₄₇	—	2	2	no	—
21	C₂×Q₈×C₄₇	C₂×C₄	1	2	yes	—
22	C₂×Q₈×C₄₇	D₄	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⋊C₄×C₄₇	—	2	8	no	—
31	C₄²×C₄₇	—	2	1	no	—
32	C₄²×C₄₇	C₂×C₄	1	2	yes	—
33	C₂²⋊Dic₄₇	—	2	2	no	—
34	C₄⋊Dic₄₇	—	2	4	no	—
35	C₄⋊Dic₄₇	D₄	1	4	yes	—
36	C₄×Dic₄₇	—	2	4	no	—
37	C₄×Dic₄₇	—	2	2	no	—
38	#10625329	—	2	2	no	—
39	#10625329	—	2	2	no	—
40	C₂×C₄₇⋊C₈	—	2	2	no	—
41	C₂³×C₄₇	—	2	2	no	—
42	C₂³×C₄₇	—	4	4	no	—
43	C₂³×C₄₇	—	4	1	no	—
44	C₂³×C₄₇	—	4	1	no	—
45	C₂³×C₄₇	C₂×D₄	1	1	yes	characteristic
46	Q₈×C₄₇	—	2	4	no	—
47	Q₈×C₄₇	—	2	4	no	—
48	D₄×C₄₇	—	2	4	no	—
49	D₄×C₄₇	—	4	2	no	—
50	C₂×C₄×C₄₇	—	2	2	no	—
51	C₂×C₄×C₄₇	—	2	2	no	—
52	C₂×C₄×C₄₇	—	2	2	no	—
53	C₂×C₄×C₄₇	—	4	2	no	—
54	C₂×C₄×C₄₇	C₂²⋊C₄	1	4	yes	—
55	C₂×C₄×C₄₇	C₂²×C₄	1	1	yes	characteristic
56	C₂×C₄×C₄₇	C₂×D₄	1	2	yes	—
57	C₂×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
58	C₂×C₄×C₄₇	#41	1	2	yes	—
59	C₂×Dic₄₇	—	2	4	no	—
60	C₄₇⋊C₈	—	4	2	no	—
61	C₂²×C₄₇	—	2	2	no	—
62	C₂²×C₄₇	—	4	4	no	—
63	C₂²×C₄₇	—	4	2	no	—
64	C₂²×C₄₇	—	8	2	no	—
65	C₂²×C₄₇	—	8	2	no	—
66	C₂²×C₄₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
67	C₂²×C₄₇	#120	1	1	yes	characteristic
68	C₂²×C₄₇	#123	1	1	yes	characteristic
69	C₄×C₄₇	—	2	4	no	—
70	C₄×C₄₇	—	4	2	no	—
71	C₄×C₄₇	C₄×D₄	1	2	yes	—
72	C₄×C₄₇	#121	1	2	yes	—
73	Dic₄₇	—	4	4	no	—
74	#1888	—	47	1	no	maximal, Sylow
75	C₂×C₄₇	—	2	1	no	—
76	C₂×C₄₇	—	2	1	no	—
77	C₂×C₄₇	—	8	2	no	—
78	C₂×C₄₇	#386	1	1	yes	characteristic
79	C₂×C₄₇	#454	1	2	yes	—
80	C₂×#124	D₄₇	1	1	yes	characteristic
81	#519	—	47	1	no	—
82	#418	—	47	2	no	—
83	C₂×#101	—	47	1	no	—
84	#339	—	47	2	no	—
85	C₄₇	#1888	1	1	yes	characteristic, complemented, Sylow
86	C₂²×Q₈	C₂×D₄₇	1	1	yes	characteristic
87	C₂²×D₄	C₂×D₄₇	1	1	yes	characteristic
88	C₂×M₁₆	—	47	2	no	—
89	#124	—	2	2	no	—
90	#124	Dic₄₇	1	4	yes	complemented
91	C₄×Q₈	—	94	4	no	—
92	#117	—	47	2	no	—
93	#117	—	94	2	no	—
94	C₂×C₂²⋊C₄	—	2	2	no	—
95	C₂×C₄²	C₂×D₄₇	1	1	yes	characteristic
96	#102	—	94	4	no	—
97	#101	—	94	2	no	—
98	C₂⁴	—	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×Q₈	C₂²⋊D₄₇	1	2	yes	—
103	C₂×D₄	—	2	2	no	—
104	C₂×D₄	—	2	2	no	—
105	C₂×D₄	—	2	1	no	—
106	C₂×D₄	C₂×Dic₄₇	1	2	yes	—
107	C₂²×C₄	—	2	1	no	—
108	C₂²×C₄	C₂²⋊D₄₇	1	2	yes	—
109	C₂²×C₄	C₂²×D₄₇	1	1	yes	characteristic
110	M₁₆	—	94	2	no	—
111	M₁₆	—	94	2	no	—
112	C₂×C₈	—	94	2	no	—
113	C₄⋊C₄	—	47	4	no	—
114	C₄⋊C₄	—	94	4	no	—
115	C₂²⋊C₄	—	2	8	no	—
116	C₂²⋊C₄	—	94	2	no	—
117	C₄²	—	2	1	no	—
118	C₄²	—	94	4	no	—
119	C₄²	—	94	2	no	—
120	C₄²	C₂×Dic₄₇	1	2	yes	—
121	C₂³	—	2	2	no	—
122	C₂³	—	4	4	no	—
123	C₂³	—	4	1	no	—
124	C₂³	—	4	1	no	—
125	C₂³	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
126	Q₈	—	2	4	no	—
127	Q₈	—	2	4	no	—
128	D₄	—	2	4	no	—
129	D₄	—	4	2	no	—
130	C₂×C₄	—	2	2	no	—
131	C₂×C₄	—	2	2	no	—
132	C₂×C₄	—	2	2	no	—
133	C₂×C₄	—	4	2	no	—
134	C₂×C₄	—	94	4	no	—
135	C₂×C₄	C₂²⋊Dic₄₇	1	4	yes	—
136	C₂×C₄	D₄×D₄₇	1	2	yes	—
137	C₂×C₄	#10625352	1	2	yes	—
138	C₂×C₄	C₂²×Dic₄₇	1	1	yes	characteristic
139	C₂×C₄	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
140	C₈	—	188	2	no	—
141	C₂²	—	2	2	no	—
142	C₂²	—	4	4	no	—
143	C₂²	—	4	2	no	—
144	C₂²	—	8	2	no	—
145	C₂²	—	8	2	no	—
146	C₂²	C₂×C₂²⋊Dic₄₇	1	1	yes	characteristic
147	C₂²	#13093992	1	1	yes	characteristic
148	C₂²	#13094005	1	1	yes	characteristic
149	C₄	—	2	4	no	—
150	C₄	—	4	2	no	—
151	C₄	—	188	4	no	—
152	C₄	D₄×Dic₄₇	1	2	yes	—
153	C₄	#13093996	1	2	yes	—
154	C₂	—	2	1	no	—
155	C₂	—	2	1	no	—
156	C₂	—	8	2	no	—
157	C₂	#15689003	1	1	yes	characteristic
158	C₂	#15689295	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#15689601

C₂

yes

maximal, characteristic, complemented

#15689128

C₂

yes

maximal

C₂×#13093943

C₂

yes

maximal, characteristic, complemented

#15688883

C₂

yes

maximal, complemented

C₂×#124×C₄₇

C₂

yes

maximal, characteristic

C₂×#10625329

C₂²

yes

—

Q₈×Dic₄₇

—

#13093983

—

#13093983

C₂²

yes

—

#13093945

—

#13093943

—

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×C₄

yes

—

C₂×Q₈×C₄₇

D₄

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⋊C₄×C₄₇

—

C₄²×C₄₇

—

C₄²×C₄₇

C₂×C₄

yes

—

C₂²⋊Dic₄₇

—

C₄⋊Dic₄₇

—

C₄⋊Dic₄₇

D₄

yes

—

C₄×Dic₄₇

—

C₄×Dic₄₇

—

#10625329

—

#10625329

—

C₂×C₄₇⋊C₈

—

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

—

C₄₇⋊C₈

—

C₂²×C₄₇

—

C₂²×C₄₇

—

C₂²×C₄₇

—

C₂²×C₄₇

—

C₂²×C₄₇

—

C₂²×C₄₇

C₂×C₂²⋊C₄

yes

C₂²×C₄₇

yes

C₂²×C₄₇

yes

C₄×C₄₇

—

C₄×C₄₇

—

C₄×C₄₇

C₄×D₄

yes

—

C₄×C₄₇

#121

yes

—

Dic₄₇

—

#1888

—

maximal, Sylow

C₂×C₄₇

—

C₂×C₄₇

—

C₂×C₄₇

—

C₂×C₄₇

#386

yes

characteristic

C₂×C₄₇

#454

yes

—

C₂×#124

D₄₇

yes

—

—

—

—

C₄₇

#1888

yes

characteristic, complemented, Sylow

C₂²×Q₈

C₂×D₄₇

yes

characteristic

C₂²×D₄

C₂×D₄₇

yes

characteristic

C₂×M₁₆

—

#124

—

#124

Dic₄₇

yes

complemented

C₄×Q₈

—

#117

—

#117

—

C₂×C₂²⋊C₄

—

C₂×C₄²

C₂×D₄₇

yes

characteristic

#102

—

#101

—

C₂⁴

—

C₂×Q₈

—

100

C₂×Q₈

—

101

C₂×Q₈

C₂×Dic₄₇

yes

—

102

C₂×Q₈

C₂²⋊D₄₇

yes

—

103

C₂×D₄

—

104

C₂×D₄

—

105

C₂×D₄

—

106

C₂×D₄

C₂×Dic₄₇

yes

—

107

C₂²×C₄

—

108

C₂²×C₄

C₂²⋊D₄₇

yes

—

109

C₂²×C₄

C₂²×D₄₇

yes

characteristic

110

M₁₆

—

111

M₁₆

—

112

C₂×C₈

—

113

C₄⋊C₄

—

114

C₄⋊C₄

—

115

C₂²⋊C₄

—

116

C₂²⋊C₄

—

117

C₄²

—

118

C₄²

—

119

C₄²

—

120

C₄²

C₂×Dic₄₇

yes

—

121

C₂³

—

122

C₂³

—

123

C₂³

—

124

C₂³

—

125

C₂³

C₂×C₂²⋊D₄₇

yes

characteristic

126

Q₈

—

127

Q₈

—

128

D₄

—

129

D₄

—

130

C₂×C₄

—

131

C₂×C₄

—

132

C₂×C₄

—

133

C₂×C₄

—

134

C₂×C₄

—

135

C₂×C₄

C₂²⋊Dic₄₇

yes

—

136

C₂×C₄

D₄×D₄₇

yes

—

137

C₂×C₄

#10625352

yes

—

138

C₂×C₄

C₂²×Dic₄₇

yes

characteristic

139

C₂×C₄

C₂×C₂²⋊D₄₇

yes

characteristic

140

C₈

—

188

—

141

C₂²

—

142

C₂²

—

143

C₂²

—

144

C₂²

—

145

C₂²

—

146

C₂²

C₂×C₂²⋊Dic₄₇

yes

characteristic

147

C₂²

#13093992

yes

characteristic

148

C₂²

#13094005

yes

characteristic

149

C₄

—

150

C₄

—

151

C₄

—

188

—

152

C₄

D₄×Dic₄₇

yes

—

153

C₄

#13093996

yes

—

154

C₂

—

155

C₂

—

156

C₂

—

157

C₂

#15689003

yes

characteristic

158

C₂

#15689295

yes

—

	subgroup	G/H
center	C₂²	#13093992
commutator	C₂×C₄×C₄₇	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₄₇
Fitting	C₂×#124×C₄₇	C₂
socle	C₂²×C₄₇	#120
hypercenter	C₂×#124	D₄₇
hypocenter	C₄₇	#1888
LCS	G ▷ C₂×C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×#124

subgroup

G/H

center

C₂²

#13093992

commutator

C₂×C₄×C₄₇

C₂²×C₄

Frattini

C₂²×C₄

C₂²×D₄₇

Fitting

C₂×#124×C₄₇

C₂

C₂²×C₄₇

D₄₇

C₄₇

G ▷ C₂×C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇

UCS

C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×#124

N	H ≅ G/N	type
1. #15689601	C₂	semidirect
3. C₂×#13093943	C₂	semidirect
4. #15688883	C₂	semidirect
15. #124×C₄₇	C₄	semidirect
85. C₄₇	#1888	semidirect
90. #124	Dic₄₇	semidirect

H ≅ G/N

type

1. #15689601

C₂

semidirect

3. C₂×#13093943

C₂

semidirect

4. #15688883

C₂

semidirect

15. #124×C₄₇

C₄

semidirect

85. C₄₇

#1888

semidirect

90. #124

Dic₄₇

semidirect

Subgroups — #20243792

Conjugacy classes of subgroups

Complements