Group #20253160 — subgroups

Subgroups — C₂×#15689419 #20253160

Order 6016.

416 conjugacy classes of subgroups; only the 173 normal subgroups are shown.

normal subgroups only

This group has 416 conjugacy classes of subgroups — too many to draw. Only its 173 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×#13094024	C₂	1	1	yes	maximal, characteristic
2	C₂×C₄×Dic₉₄	C₂	1	1	yes	maximal, characteristic
3	#15689419	C₂	1	8	yes	maximal, direct product factor
4	C₂×#13093958	C₂	1	2	yes	maximal
5	C₂×#13093949	C₂	1	2	yes	maximal
6	C₂×C₄⋊C₈×C₄₇	C₂	1	1	yes	maximal, characteristic
7	C₂²×Dic₉₄	—	2	1	no	—
8	C₂²×Dic₉₄	C₂²	1	1	yes	characteristic
9	#13094024	—	2	2	no	—
10	#13094024	C₂²	1	4	yes	—
11	C₄×Dic₉₄	—	2	2	no	—
12	C₄×Dic₉₄	C₂²	1	4	yes	—
13	C₂×C₄⋊Dic₄₇	—	2	1	no	—
14	C₂×C₄⋊Dic₄₇	C₂²	1	1	yes	characteristic
15	C₂×C₄⋊Dic₄₇	C₂²	1	2	yes	—
16	C₂×Dic₄₇⋊C₄	—	2	1	no	—
17	C₂×C₄×Dic₄₇	—	2	1	no	—
18	#13093958	C₂²	1	8	yes	—
19	#13093949	C₂²	1	8	yes	—
20	C₂²×C₈×C₄₇	C₂²	1	2	yes	—
21	C₂×C₄²×C₄₇	C₂²	1	1	yes	characteristic
22	C₄⋊C₈×C₄₇	C₂²	1	4	yes	—
23	C₂²×C₄×C₄₇	D₄	1	1	yes	characteristic
24	C₂²×C₄×C₄₇	D₄	1	1	yes	characteristic
25	C₂²×C₄×C₄₇	C₂³	1	1	yes	characteristic
26	C₂²×Dic₄₇	—	2	1	no	—
27	C₂²×Dic₄₇	—	2	1	no	—
28	C₂²×Dic₄₇	—	2	2	no	—
29	C₂×Dic₉₄	—	2	1	no	—
30	C₂×Dic₉₄	—	2	2	no	—
31	C₂×Dic₉₄	—	2	4	no	—
32	C₂×Dic₉₄	—	4	4	no	—
33	C₂×Dic₉₄	Q₈	1	4	yes	complemented
34	C₂×Dic₉₄	C₂³	1	2	yes	—
35	C₂×C₈×C₄₇	—	2	4	no	—
36	C₂×C₈×C₄₇	C₂³	1	4	yes	—
37	C₄²×C₄₇	D₄	1	2	yes	—
38	C₄²×C₄₇	C₂³	1	2	yes	—
39	C₄⋊Dic₄₇	—	2	4	no	—
40	C₄⋊Dic₄₇	—	2	1	no	—
41	C₄⋊Dic₄₇	—	2	2	no	—
42	C₄⋊Dic₄₇	C₂³	1	4	yes	—
43	C₄⋊Dic₄₇	C₂³	1	2	yes	—
44	Dic₄₇⋊C₄	—	2	4	no	—
45	C₄×Dic₄₇	—	2	4	no	—
46	C₂³×C₄₇	C₂×D₄	1	1	yes	characteristic
47	C₂×C₄×C₄₇	—	2	2	no	—
48	C₂×C₄×C₄₇	Q₁₆	1	4	yes	—
49	C₂×C₄×C₄₇	C₂×D₄	1	2	yes	—
50	C₂×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
51	C₂×C₄×C₄₇	C₂×D₄	1	2	yes	—
52	C₂×C₄×C₄₇	C₂×Q₈	1	2	yes	—
53	C₂×C₄×C₄₇	#41	1	2	yes	—
54	C₂×C₄×C₄₇	C₂⁴	1	1	yes	characteristic
55	C₂×Dic₄₇	—	2	4	no	—
56	C₂×Dic₄₇	—	2	2	no	—
57	C₂×Dic₄₇	—	2	2	no	—
58	C₂×Dic₄₇	—	2	4	no	—
59	C₂×Dic₄₇	—	4	4	no	—
60	C₂×Dic₄₇	—	4	2	no	—
61	Dic₉₄	—	2	4	no	—
62	Dic₉₄	—	2	2	no	—
63	Dic₉₄	—	4	8	no	—
64	Dic₉₄	C₂×Q₈	1	4	yes	complemented
65	C₈×C₄₇	—	2	4	no	—
66	C₂²×C₄₇	C₂²⋊Q₈	1	2	yes	—
67	C₂²×C₄₇	C₂×Q₁₆	1	2	yes	—
68	C₂²×C₄₇	AGL(1,ℤ/8)	1	2	yes	—
69	C₂²×C₄₇	C₂²×D₄	1	1	yes	characteristic
70	C₄×C₄₇	—	2	2	no	—
71	C₄×C₄₇	C₂²⋊Q₈	1	2	yes	—
72	C₄×C₄₇	C₂×Q₁₆	1	4	yes	—
73	C₄×C₄₇	C₂²×Q₈	1	1	yes	characteristic
74	C₄×C₄₇	C₂×#41	1	1	yes	characteristic
75	Dic₄₇	—	2	4	no	—
76	Dic₄₇	—	4	4	no	—
77	Dic₄₇	—	4	2	no	—
78	C₂×#477	—	47	1	no	maximal, Sylow
79	C₂×C₄₇	#477	1	4	yes	complemented
80	C₂×C₄₇	C₂×C₂²⋊Q₈	1	1	yes	characteristic
81	C₂×C₄₇	C₂²×Q₁₆	1	1	yes	characteristic
82	C₂×C₄₇	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
83	C₂×C₄⋊Q₈	—	47	1	no	—
84	C₂×C₄×Q₈	—	47	1	no	—
85	#477	—	47	8	no	—
86	C₂×#107	—	47	2	no	—
87	C₂×C₄⋊C₈	D₄₇	1	1	yes	characteristic
88	C₂×#103	—	47	2	no	—
89	C₄₇	C₂×#477	1	1	yes	characteristic, complemented, Sylow
90	C₂²×Q₈	—	47	1	no	—
91	C₂²×Q₈	—	94	1	no	—
92	C₂²×C₈	C₂×D₄₇	1	2	yes	—
93	C₄⋊Q₈	—	47	4	no	—
94	C₄⋊Q₈	—	94	2	no	—
95	C₄×Q₈	—	47	4	no	—
96	C₄×Q₈	—	94	2	no	—
97	C₂×C₄⋊C₄	—	47	1	no	—
98	C₂×C₄⋊C₄	—	47	2	no	—
99	C₂×C₄⋊C₄	—	94	1	no	—
100	C₂×C₄⋊C₄	—	94	1	no	—
101	C₂×C₄²	—	94	1	no	—
102	C₂×C₄²	C₂×D₄₇	1	1	yes	characteristic
103	#107	—	47	8	no	—
104	C₄⋊C₈	C₂×D₄₇	1	4	yes	—
105	#103	—	47	8	no	—
106	C₂×Q₈	—	47	4	no	—
107	C₂×Q₈	—	47	2	no	—
108	C₂×Q₈	—	94	1	no	—
109	C₂×Q₈	—	94	4	no	—
110	C₂×Q₈	—	94	2	no	—
111	C₂×Q₈	—	188	4	no	—
112	C₂²×C₄	—	94	1	no	—
113	C₂²×C₄	—	94	1	no	—
114	C₂²×C₄	—	94	2	no	—
115	C₂²×C₄	D₁₈₈	1	1	yes	characteristic
116	C₂²×C₄	D₁₈₈	1	1	yes	characteristic
117	C₂²×C₄	C₂²×D₄₇	1	1	yes	characteristic
118	C₂×C₈	—	2	4	no	—
119	C₂×C₈	C₂²×D₄₇	1	4	yes	—
120	C₄⋊C₄	—	47	4	no	—
121	C₄⋊C₄	—	47	2	no	—
122	C₄⋊C₄	—	94	4	no	—
123	C₄⋊C₄	—	94	1	no	—
124	C₄⋊C₄	—	94	2	no	—
125	C₄⋊C₄	—	94	4	no	—
126	C₄²	—	94	4	no	—
127	C₄²	D₁₈₈	1	2	yes	—
128	C₄²	C₂²×D₄₇	1	2	yes	—
129	C₂³	C₂×D₁₈₈	1	1	yes	characteristic
130	Q₈	—	47	4	no	—
131	Q₈	—	94	4	no	—
132	Q₈	—	94	2	no	—
133	Q₈	—	188	8	no	—
134	C₂×C₄	—	2	2	no	—
135	C₂×C₄	—	94	4	no	—
136	C₂×C₄	—	94	2	no	—
137	C₂×C₄	—	94	2	no	—
138	C₂×C₄	—	94	4	no	—
139	C₂×C₄	—	188	2	no	—
140	C₂×C₄	—	188	4	no	—
141	C₂×C₄	Dic₁₈₈	1	4	yes	—
142	C₂×C₄	C₂×D₁₈₈	1	2	yes	—
143	C₂×C₄	C₂×D₁₈₈	1	1	yes	characteristic
144	C₂×C₄	C₂×D₁₈₈	1	2	yes	—
145	C₂×C₄	#10625352	1	2	yes	—
146	C₂×C₄	Q₈×D₄₇	1	2	yes	—
147	C₂×C₄	C₂³×D₄₇	1	1	yes	characteristic
148	C₈	—	2	4	no	—
149	C₂²	#13094004	1	2	yes	—
150	C₂²	C₂×Dic₁₈₈	1	2	yes	—
151	C₂²	#13094052	1	2	yes	—
152	C₂²	C₂²×D₁₈₈	1	1	yes	characteristic
153	C₄	—	2	2	no	—
154	C₄	—	94	4	no	—
155	C₄	—	188	4	no	—
156	C₄	—	188	2	no	—
157	C₄	#13094004	1	2	yes	—
158	C₄	C₂×Dic₁₈₈	1	4	yes	—
159	C₄	C₂×Q₈×D₄₇	1	1	yes	characteristic
160	C₄	C₂×#10625352	1	1	yes	characteristic
161	C₂	#15689419	1	4	yes	direct product factor
162	C₂	C₂×#13094004	1	1	yes	characteristic
163	C₂	C₂²×Dic₁₈₈	1	1	yes	characteristic
164	C₂	C₂×#13094052	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#13094024

C₂

yes

maximal, characteristic

C₂×C₄×Dic₉₄

C₂

yes

maximal, characteristic

#15689419

C₂

yes

maximal, direct product factor

C₂×#13093958

C₂

yes

maximal

C₂×#13093949

C₂

yes

maximal

C₂×C₄⋊C₈×C₄₇

C₂

yes

maximal, characteristic

C₂²×Dic₉₄

—

C₂²×Dic₉₄

C₂²

yes

characteristic

#13094024

—

#13094024

C₂²

yes

—

C₄×Dic₉₄

—

C₄×Dic₉₄

C₂²

yes

—

C₂×C₄⋊Dic₄₇

—

C₂×C₄⋊Dic₄₇

C₂²

yes

characteristic

C₂×C₄⋊Dic₄₇

C₂²

yes

—

C₂×Dic₄₇⋊C₄

—

C₂×C₄×Dic₄₇

—

#13093958

C₂²

yes

—

#13093949

C₂²

yes

—

C₂²×C₈×C₄₇

C₂²

yes

—

C₂×C₄²×C₄₇

C₂²

yes

characteristic

C₄⋊C₈×C₄₇

C₂²

yes

—

C₂²×C₄×C₄₇

D₄

yes

characteristic

C₂²×C₄×C₄₇

D₄

yes

characteristic

C₂²×C₄×C₄₇

C₂³

yes

characteristic

C₂²×Dic₄₇

—

C₂²×Dic₄₇

—

C₂²×Dic₄₇

—

C₂×Dic₉₄

—

C₂×Dic₉₄

—

C₂×Dic₉₄

—

C₂×Dic₉₄

—

C₂×Dic₉₄

Q₈

yes

complemented

C₂×Dic₉₄

C₂³

yes

—

C₂×C₈×C₄₇

—

C₂×C₈×C₄₇

C₂³

yes

—

C₄²×C₄₇

D₄

yes

—

C₄²×C₄₇

C₂³

yes

—

C₄⋊Dic₄₇

—

C₄⋊Dic₄₇

—

C₄⋊Dic₄₇

—

C₄⋊Dic₄₇

C₂³

yes

—

C₄⋊Dic₄₇

C₂³

yes

—

Dic₄₇⋊C₄

—

C₄×Dic₄₇

—

C₂³×C₄₇

C₂×D₄

yes

characteristic

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

Q₁₆

yes

—

C₂×C₄×C₄₇

C₂×D₄

yes

—

C₂×C₄×C₄₇

C₂×D₄

yes

characteristic

C₂×C₄×C₄₇

C₂×D₄

yes

—

C₂×C₄×C₄₇

C₂×Q₈

yes

—

C₂×C₄×C₄₇

#41

yes

—

C₂×C₄×C₄₇

C₂⁴

yes

characteristic

C₂×Dic₄₇

—

C₂×Dic₄₇

—

C₂×Dic₄₇

—

C₂×Dic₄₇

—

C₂×Dic₄₇

—

C₂×Dic₄₇

—

Dic₉₄

—

Dic₉₄

—

Dic₉₄

—

Dic₉₄

C₂×Q₈

yes

complemented

C₈×C₄₇

—

C₂²×C₄₇

C₂²⋊Q₈

yes

—

C₂²×C₄₇

C₂×Q₁₆

yes

—

C₂²×C₄₇

AGL(1,ℤ/8)

yes

—

C₂²×C₄₇

C₂²×D₄

yes

characteristic

C₄×C₄₇

—

C₄×C₄₇

C₂²⋊Q₈

yes

—

C₄×C₄₇

C₂×Q₁₆

yes

—

C₄×C₄₇

C₂²×Q₈

yes

characteristic

C₄×C₄₇

C₂×#41

yes

characteristic

Dic₄₇

—

Dic₄₇

—

Dic₄₇

—

—

C₂×C₄₇

yes

C₂×C₄₇

C₂×C₂²⋊Q₈

yes

characteristic

C₂×C₄₇

C₂²×Q₁₆

yes

characteristic

C₂×C₄₇

C₂×AGL(1,ℤ/8)

yes

characteristic

C₂×C₄⋊Q₈

—

C₂×C₄×Q₈

—

#477

—

C₂×#107

—

C₂×C₄⋊C₈

D₄₇

yes

characteristic

C₂×#103

—

C₄₇

C₂×#477

yes

characteristic, complemented, Sylow

C₂²×Q₈

—

C₂²×Q₈

—

C₂²×C₈

C₂×D₄₇

yes

—

C₄⋊Q₈

—

C₄⋊Q₈

—

C₄×Q₈

—

C₄×Q₈

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

100

C₂×C₄⋊C₄

—

101

C₂×C₄²

—

102

C₂×C₄²

C₂×D₄₇

yes

characteristic

103

#107

—

104

C₄⋊C₈

C₂×D₄₇

yes

—

105

#103

—

106

C₂×Q₈

—

107

C₂×Q₈

—

108

C₂×Q₈

—

109

C₂×Q₈

—

110

C₂×Q₈

—

111

C₂×Q₈

—

188

—

112

C₂²×C₄

—

113

C₂²×C₄

—

114

C₂²×C₄

—

115

C₂²×C₄

D₁₈₈

yes

characteristic

116

C₂²×C₄

D₁₈₈

yes

characteristic

117

C₂²×C₄

C₂²×D₄₇

yes

characteristic

118

C₂×C₈

—

119

C₂×C₈

C₂²×D₄₇

yes

—

120

C₄⋊C₄

—

121

C₄⋊C₄

—

122

C₄⋊C₄

—

123

C₄⋊C₄

—

124

C₄⋊C₄

—

125

C₄⋊C₄

—

126

C₄²

—

127

C₄²

D₁₈₈

yes

—

128

C₄²

C₂²×D₄₇

yes

—

129

C₂³

C₂×D₁₈₈

yes

characteristic

130

Q₈

—

131

Q₈

—

132

Q₈

—

133

Q₈

—

188

—

134

C₂×C₄

—

135

C₂×C₄

—

136

C₂×C₄

—

137

C₂×C₄

—

138

C₂×C₄

—

139

C₂×C₄

—

188

—

140

C₂×C₄

—

188

—

141

C₂×C₄

Dic₁₈₈

yes

—

142

C₂×C₄

C₂×D₁₈₈

yes

—

143

C₂×C₄

C₂×D₁₈₈

yes

characteristic

144

C₂×C₄

C₂×D₁₈₈

yes

—

145

C₂×C₄

#10625352

yes

—

146

C₂×C₄

Q₈×D₄₇

yes

—

147

C₂×C₄

C₂³×D₄₇

yes

characteristic

148

C₈

—

149

C₂²

#13094004

yes

—

150

C₂²

C₂×Dic₁₈₈

yes

—

151

C₂²

#13094052

yes

—

152

C₂²

C₂²×D₁₈₈

yes

characteristic

153

C₄

—

154

C₄

—

155

C₄

—

188

—

156

C₄

—

188

—

157

C₄

#13094004

yes

—

158

C₄

C₂×Dic₁₈₈

yes

—

159

C₄

C₂×Q₈×D₄₇

yes

characteristic

160

C₄

C₂×#10625352

yes

characteristic

161

C₂

#15689419

yes

direct product factor

162

C₂

C₂×#13094004

yes

characteristic

163

C₂

C₂²×Dic₁₈₈

yes

characteristic

164

C₂

C₂×#13094052

yes

characteristic

	subgroup	G/H
center	C₂³	C₂×D₁₈₈
commutator	C₂×C₄×C₄₇	C₂⁴
Frattini	C₂×C₄	C₂³×D₄₇
Fitting	C₂×C₄⋊C₈×C₄₇	C₂
socle	C₂³×C₄₇	C₂×D₄
hypercenter	C₂×C₄⋊C₈	D₄₇
hypocenter	C₄₇	C₂×#477
LCS	G ▷ C₂×C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄⋊C₈

subgroup

G/H

center

C₂³

C₂×D₁₈₈

commutator

C₂×C₄×C₄₇

C₂⁴

Frattini

C₂×C₄

C₂³×D₄₇

Fitting

C₂×C₄⋊C₈×C₄₇

C₂

socle

C₂³×C₄₇

C₂×D₄

hypercenter

C₂×C₄⋊C₈

D₄₇

hypocenter

C₄₇

C₂×#477

LCS

G ▷ C₂×C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇

UCS

C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄⋊C₈

N	H ≅ G/N	type
3. #15689419	C₂	direct (N × H)
33. C₂×Dic₉₄	Q₈	semidirect
64. Dic₉₄	C₂×Q₈	semidirect
79. C₂×C₄₇	#477	semidirect
89. C₄₇	C₂×#477	semidirect
161. C₂	#15689419	direct (N × H)

H ≅ G/N

type

3. #15689419

C₂

direct (N × H)

33. C₂×Dic₉₄

Q₈

semidirect

64. Dic₉₄

C₂×Q₈

semidirect

79. C₂×C₄₇

#477

semidirect

89. C₄₇

C₂×#477

semidirect

161. C₂

#15689419

direct (N × H)

Subgroups — C₂×#15689419 #20253160

Conjugacy classes of subgroups

Complements

Subgroups — C2×#15689419 #20253160

Conjugacy classes of subgroups

Complements

Subgroups — C₂×#15689419 #20253160