Group #19144 — subgroups

Subgroups — C₂×#1688 #19144

Order 256.

425 conjugacy classes of subgroups; only the 135 normal subgroups are shown.

normal subgroups only

This group has 425 conjugacy classes of subgroups — too many to draw. Only its 135 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×#493	C₂	1	1	yes	maximal, characteristic
2	C₂×#460	C₂	1	2	yes	maximal, complemented
3	C₂×#446	C₂	1	1	yes	maximal, characteristic, complemented
4	#1688	C₂	1	8	yes	maximal, direct product factor
5	C₂×#334	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×#331	C₂	1	2	yes	maximal, complemented
7	C₂²×SD₁₆	—	2	2	no	—
8	C₂²×D₈	—	2	1	no	—
9	C₂²×D₈	—	2	1	no	—
10	C₂×#127	C₂²	1	2	yes	—
11	C₂×C₄×Q₈	C₂²	1	1	yes	characteristic
12	#493	—	2	2	no	—
13	#493	C₂²	1	4	yes	—
14	#460	—	2	4	no	—
15	#460	C₂²	1	8	yes	complemented
16	#446	—	2	2	no	—
17	#446	C₂²	1	4	yes	complemented
18	C₂×C₄⋊C₈	—	2	1	no	—
19	C₂×C₄⋊C₈	C₂²	1	1	yes	characteristic
20	C₂×C₄⋊C₈	C₂²	1	2	yes	—
21	C₂×#102	—	2	2	no	—
22	C₂×#97	—	2	1	no	—
23	C₂×C₄×C₈	C₂²	1	1	yes	characteristic
24	#334	C₂²	1	4	yes	complemented
25	#331	C₂²	1	8	yes	complemented
26	C₂²×Q₈	D₄	1	1	yes	characteristic
27	C₂²×D₄	—	2	2	no	—
28	C₂²×D₄	—	2	2	no	—
29	C₂²×D₄	—	2	2	no	—
30	C₂×SD₁₆	—	2	8	no	—
31	C₂×SD₁₆	—	4	8	no	—
32	C₂×D₈	—	2	4	no	—
33	C₂×D₈	—	2	4	no	—
34	C₂×D₈	—	4	4	no	—
35	C₂×D₈	—	4	4	no	—
36	C₂²×C₈	—	2	2	no	—
37	C₂²×C₈	—	2	1	no	—
38	C₂²×C₈	D₄	1	1	yes	characteristic
39	C₂²×C₈	D₄	1	1	yes	characteristic
40	#127	—	2	2	no	—
41	#127	—	2	2	no	—
42	#127	—	2	2	no	—
43	#127	C₂³	1	4	yes	—
44	C₄×Q₈	—	2	1	no	—
45	C₄×Q₈	—	2	1	no	—
46	C₄×Q₈	D₄	1	2	yes	—
47	C₄×Q₈	C₂³	1	2	yes	—
48	C₂×C₄⋊C₄	—	2	1	no	—
49	C₂×C₄⋊C₄	D₄	1	1	yes	characteristic
50	C₂×C₄²	—	2	1	no	—
51	C₂×C₄²	C₂³	1	1	yes	characteristic
52	C₄⋊C₈	—	2	4	no	—
53	C₄⋊C₈	—	2	1	no	—
54	C₄⋊C₈	—	2	2	no	—
55	C₄⋊C₈	C₂³	1	2	yes	—
56	C₄⋊C₈	C₂³	1	4	yes	—
57	#102	—	2	8	no	—
58	#97	—	2	4	no	—
59	C₄×C₈	D₄	1	2	yes	—
60	C₄×C₈	C₂³	1	2	yes	—
61	C₂⁴	—	4	2	no	—
62	C₂×Q₈	—	2	1	no	—
63	C₂×Q₈	—	2	4	no	—
64	C₂×Q₈	C₂×D₄	1	2	yes	—
65	C₂×D₄	—	2	4	no	—
66	C₂×D₄	—	2	4	no	—
67	C₂×D₄	—	2	4	no	—
68	C₂×D₄	—	4	2	no	—
69	C₂×D₄	—	4	2	no	—
70	C₂×D₄	—	4	2	no	—
71	C₂×D₄	—	4	4	no	—
72	C₂×D₄	—	4	4	no	—
73	C₂×D₄	—	4	4	no	—
74	C₂×D₄	—	4	4	no	—
75	C₂×D₄	—	8	4	no	—
76	C₂²×C₄	—	2	1	no	—
77	C₂²×C₄	—	2	1	no	—
78	C₂²×C₄	C₂×D₄	1	1	yes	characteristic
79	C₂²×C₄	C₂×D₄	1	1	yes	characteristic
80	C₂²×C₄	#41	1	1	yes	characteristic
81	SD₁₆	—	4	16	no	—
82	D₈	—	4	8	no	—
83	D₈	—	4	8	no	—
84	C₂×C₈	—	2	2	no	—
85	C₂×C₈	—	2	2	no	—
86	C₂×C₈	—	2	4	no	—
87	C₂×C₈	—	4	2	no	—
88	C₂×C₈	—	4	4	no	—
89	C₂×C₈	SD₁₆	1	4	yes	complemented
90	C₂×C₈	C₂×D₄	1	2	yes	—
91	C₂×C₈	C₂×D₄	1	2	yes	—
92	C₄⋊C₄	—	2	2	no	—
93	C₄⋊C₄	—	2	1	no	—
94	C₄⋊C₄	—	2	2	no	—
95	C₄⋊C₄	C₂×D₄	1	2	yes	—
96	C₄²	—	2	2	no	—
97	C₄²	—	2	2	no	—
98	C₄²	C₂×D₄	1	1	yes	characteristic
99	C₄²	C₂×D₄	1	1	yes	characteristic
100	C₄²	#41	1	1	yes	characteristic
101	C₄²	C₂⁴	1	1	yes	characteristic
102	C₂³	—	4	4	no	—
103	C₂³	—	8	4	no	—
104	C₂³	—	8	4	no	—
105	C₂³	—	8	4	no	—
106	C₂³	#121	1	1	yes	characteristic
107	Q₈	—	2	4	no	—
108	Q₈	—	4	2	no	—
109	D₄	—	4	4	no	—
110	D₄	—	4	4	no	—
111	D₄	—	4	4	no	—
112	D₄	—	4	4	no	—
113	D₄	—	8	4	no	—
114	D₄	—	8	4	no	—
115	D₄	—	8	4	no	—
116	D₄	—	8	4	no	—
117	C₂×C₄	—	2	2	no	—
118	C₂×C₄	—	2	2	no	—
119	C₂×C₄	—	2	2	no	—
120	C₂×C₄	—	4	2	no	—
121	C₂×C₄	—	4	2	no	—
122	C₂×C₄	#121	1	1	yes	characteristic
123	C₂×C₄	#121	1	1	yes	characteristic
124	C₂×C₄	#121	1	1	yes	characteristic
125	C₂×C₄	C₂×SD₁₆	1	2	yes	—
126	C₂×C₄	AGL(1,ℤ/8)	1	2	yes	—
127	C₂×C₄	AGL(1,ℤ/8)	1	4	yes	—
128	C₂×C₄	C₂²×D₄	1	1	yes	characteristic
129	C₂×C₄	C₂²×D₄	1	1	yes	characteristic
130	C₂×C₄	C₂×#41	1	1	yes	characteristic
131	C₈	—	2	2	no	—
132	C₈	—	4	2	no	—
133	C₈	—	4	4	no	—
134	C₈	C₂×SD₁₆	1	4	yes	complemented
135	C₂²	—	8	4	no	—
136	C₂²	—	8	4	no	—
137	C₂²	—	8	4	no	—
138	C₂²	—	16	8	no	—
139	C₂²	#460	1	2	yes	—
140	C₂²	#469	1	2	yes	—
141	C₂²	#472	1	2	yes	—
142	C₂²	C₂×#121	1	1	yes	characteristic
143	C₄	—	2	1	no	—
144	C₄	—	2	1	no	—
145	C₄	—	4	2	no	—
146	C₄	—	4	2	no	—
147	C₄	#460	1	2	yes	—
148	C₄	#469	1	2	yes	—
149	C₄	C₂²×SD₁₆	1	1	yes	characteristic
150	C₄	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
151	C₄	C₂×AGL(1,ℤ/8)	1	2	yes	—
152	C₂	—	16	4	no	—
153	C₂	#1688	1	4	yes	direct product factor
154	C₂	C₂×#460	1	1	yes	characteristic
155	C₂	C₂×#469	1	1	yes	characteristic
156	C₂	C₂×#472	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#493

C₂

yes

maximal, characteristic

C₂×#460

C₂

yes

maximal, complemented

C₂×#446

C₂

yes

maximal, characteristic, complemented

#1688

C₂

yes

maximal, direct product factor

C₂×#334

C₂

yes

maximal, characteristic, complemented

C₂×#331

C₂

yes

maximal, complemented

C₂²×SD₁₆

—

C₂²×D₈

—

C₂²×D₈

—

C₂×#127

C₂²

yes

—

C₂×C₄×Q₈

C₂²

yes

characteristic

#493

—

#493

C₂²

yes

—

#460

—

#460

C₂²

yes

complemented

#446

—

#446

C₂²

yes

complemented

C₂×C₄⋊C₈

—

C₂×C₄⋊C₈

C₂²

yes

characteristic

C₂×C₄⋊C₈

C₂²

yes

—

C₂×#102

—

C₂×#97

—

C₂×C₄×C₈

C₂²

yes

characteristic

#334

C₂²

yes

complemented

#331

C₂²

yes

complemented

C₂²×Q₈

D₄

yes

characteristic

C₂²×D₄

—

C₂²×D₄

—

C₂²×D₄

—

C₂×SD₁₆

—

C₂×SD₁₆

—

C₂×D₈

—

C₂×D₈

—

C₂×D₈

—

C₂×D₈

—

C₂²×C₈

—

C₂²×C₈

—

C₂²×C₈

D₄

yes

characteristic

C₂²×C₈

D₄

yes

—

—

—

C₂³

yes

—

C₄×Q₈

—

C₄×Q₈

—

C₄×Q₈

D₄

yes

—

C₄×Q₈

C₂³

yes

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

D₄

yes

characteristic

C₂×C₄²

—

C₂×C₄²

C₂³

yes

characteristic

C₄⋊C₈

—

C₄⋊C₈

—

C₄⋊C₈

—

C₄⋊C₈

C₂³

yes

—

C₄⋊C₈

C₂³

yes

—

#102

—

#97

—

C₄×C₈

D₄

yes

—

C₄×C₈

C₂³

yes

—

C₂⁴

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

C₂×D₄

yes

—

C₂×D₄

—

C₂×D₄

—

C₂×D₄

—

C₂×D₄

—

C₂×D₄

—

C₂×D₄

—

C₂×D₄

—

C₂×D₄

—

C₂×D₄

—

C₂×D₄

—

C₂×D₄

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

C₂×D₄

yes

characteristic

C₂²×C₄

C₂×D₄

yes

characteristic

C₂²×C₄

#41

yes

characteristic

SD₁₆

—

D₈

—

D₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

SD₁₆

yes

complemented

C₂×C₈

C₂×D₄

yes

—

C₂×C₈

C₂×D₄

yes

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

C₂×D₄

yes

—

C₄²

—

C₄²

—

C₄²

C₂×D₄

yes

characteristic

C₄²

C₂×D₄

yes

characteristic

100

C₄²

#41

yes

characteristic

101

C₄²

C₂⁴

yes

characteristic

102

C₂³

—

103

C₂³

—

104

C₂³

—

105

C₂³

—

106

C₂³

#121

yes

characteristic

107

Q₈

—

108

Q₈

—

109

D₄

—

110

D₄

—

111

D₄

—

112

D₄

—

113

D₄

—

114

D₄

—

115

D₄

—

116

D₄

—

117

C₂×C₄

—

118

C₂×C₄

—

119

C₂×C₄

—

120

C₂×C₄

—

121

C₂×C₄

—

122

C₂×C₄

#121

yes

characteristic

123

C₂×C₄

#121

yes

characteristic

124

C₂×C₄

#121

yes

characteristic

125

C₂×C₄

C₂×SD₁₆

yes

—

126

C₂×C₄

AGL(1,ℤ/8)

yes

—

127

C₂×C₄

AGL(1,ℤ/8)

yes

—

128

C₂×C₄

C₂²×D₄

yes

characteristic

129

C₂×C₄

C₂²×D₄

yes

characteristic

130

C₂×C₄

C₂×#41

yes

characteristic

131

C₈

—

132

C₈

—

133

C₈

—

134

C₈

C₂×SD₁₆

yes

complemented

135

C₂²

—

136

C₂²

—

137

C₂²

—

138

C₂²

—

139

C₂²

#460

yes

—

140

C₂²

#469

yes

—

141

C₂²

#472

yes

—

142

C₂²

C₂×#121

yes

characteristic

143

C₄

—

144

C₄

—

145

C₄

—

146

C₄

—

147

C₄

#460

yes

—

148

C₄

#469

yes

—

149

C₄

C₂²×SD₁₆

yes

characteristic

150

C₄

C₂×AGL(1,ℤ/8)

yes

characteristic

151

C₄

C₂×AGL(1,ℤ/8)

yes

—

152

C₂

—

153

C₂

#1688

yes

direct product factor

154

C₂

C₂×#460

yes

characteristic

155

C₂

C₂×#469

yes

characteristic

156

C₂

C₂×#472

yes

characteristic

	subgroup	G/H
center, socle	C₂³	#121
commutator, Frattini, cosocle	C₄²	C₂⁴
prime radical	C₂×#493	C₂
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ G
LECS	G ▷ C₄² ▷ C₂² ▷ C₁

subgroup

G/H

center, socle

C₂³

#121

commutator, Frattini, cosocle

C₄²

C₂⁴

C₂

G ▷ C₄² ▷ C₂² ▷ C₁

C₁ ◁ C₂³ ◁ C₂×C₄² ◁ G

LECS

G ▷ C₄² ▷ C₂² ▷ C₁

N	H ≅ G/N	type
2. C₂×#460	C₂	semidirect
3. C₂×#446	C₂	semidirect
4. #1688	C₂	direct (N × H)
5. C₂×#334	C₂	semidirect
6. C₂×#331	C₂	semidirect
15. #460	C₂²	semidirect
17. #446	C₂²	semidirect
24. #334	C₂²	semidirect
25. #331	C₂²	semidirect
89. C₂×C₈	SD₁₆	semidirect
134. C₈	C₂×SD₁₆	semidirect
153. C₂	#1688	direct (N × H)

H ≅ G/N

type

2. C₂×#460

C₂

semidirect

3. C₂×#446

C₂

semidirect

4. #1688

C₂

direct (N × H)

5. C₂×#334

C₂

semidirect

6. C₂×#331

C₂

semidirect

15. #460

C₂²

semidirect

17. #446

C₂²

semidirect

24. #334

C₂²

semidirect

25. #331

C₂²

semidirect

89. C₂×C₈

SD₁₆

semidirect

134. C₈

C₂×SD₁₆

semidirect

153. C₂

#1688

direct (N × H)

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
C₂×#493	2
C₂×#460	2
C₂×#446	2
#1688	2
C₂×#334	2
C₂×#331	2

subgroup

index

Subgroups — C₂×#1688 #19144

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C2×#1688 #19144

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#1688 #19144