Group #19113 — subgroups

Subgroups — C₂×#1670 #19113

Order 256.

351 conjugacy classes of subgroups; only the 143 normal subgroups are shown.

normal subgroups only

This group has 351 conjugacy classes of subgroups — too many to draw. Only its 143 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×#494	C₂	1	1	yes	maximal, characteristic
2	C₂×#475	C₂	1	2	yes	maximal
3	C₂×C₈×Q₈	C₂	1	1	yes	maximal, characteristic
4	#1670	C₂	1	8	yes	maximal, direct product factor
5	C₂×#335	C₂	1	1	yes	maximal, characteristic
6	C₂×#332	C₂	1	2	yes	maximal
7	C₂²×Q₁₆	—	2	1	no	—
8	C₂²×Q₁₆	—	2	1	no	—
9	C₂×C₄⋊Q₈	C₂²	1	2	yes	—
10	C₂×C₄×Q₈	C₂²	1	1	yes	characteristic
11	#494	—	2	2	no	—
12	#494	C₂²	1	4	yes	—
13	#475	—	2	4	no	—
14	#475	C₂²	1	8	yes	—
15	C₈×Q₈	—	2	2	no	—
16	C₈×Q₈	C₂²	1	4	yes	—
17	C₂×#106	—	2	2	no	—
18	C₂×C₄⋊C₈	—	2	1	no	—
19	C₂×C₄⋊C₈	C₂²	1	2	yes	—
20	C₂×C₄⋊C₈	C₂²	1	1	yes	characteristic
21	C₂×#103	—	2	2	no	—
22	C₂×C₄×C₈	—	2	1	no	—
23	C₂×C₄×C₈	C₂²	1	1	yes	characteristic
24	#335	C₂²	1	4	yes	—
25	#332	C₂²	1	8	yes	—
26	C₂²×Q₈	—	2	2	no	—
27	C₂²×Q₈	D₄	1	1	yes	characteristic
28	C₂×Q₁₆	—	2	4	no	—
29	C₂×Q₁₆	—	2	4	no	—
30	C₂×Q₁₆	—	4	4	no	—
31	C₂×Q₁₆	—	4	4	no	—
32	C₂²×C₈	—	2	1	no	—
33	C₂²×C₈	—	2	2	no	—
34	C₂²×C₈	D₄	1	1	yes	characteristic
35	C₂²×C₈	D₄	1	1	yes	characteristic
36	C₄⋊Q₈	—	2	2	no	—
37	C₄⋊Q₈	—	2	2	no	—
38	C₄⋊Q₈	—	2	2	no	—
39	C₄⋊Q₈	C₂³	1	4	yes	—
40	C₄×Q₈	—	2	1	no	—
41	C₄×Q₈	—	2	1	no	—
42	C₄×Q₈	D₄	1	2	yes	—
43	C₄×Q₈	C₂³	1	2	yes	—
44	C₂×C₄⋊C₄	—	2	1	no	—
45	C₂×C₄⋊C₄	—	2	2	no	—
46	C₂×C₄⋊C₄	—	2	2	no	—
47	C₂×C₄⋊C₄	D₄	1	1	yes	characteristic
48	C₂×C₄²	—	2	1	no	—
49	C₂×C₄²	C₂³	1	1	yes	characteristic
50	#106	—	2	8	no	—
51	C₄⋊C₈	—	2	4	no	—
52	C₄⋊C₈	—	2	1	no	—
53	C₄⋊C₈	—	2	2	no	—
54	C₄⋊C₈	C₂³	1	2	yes	—
55	C₄⋊C₈	C₂³	1	4	yes	—
56	#103	—	2	8	no	—
57	C₄×C₈	—	2	4	no	—
58	C₄×C₈	D₄	1	2	yes	—
59	C₄×C₈	C₂³	1	2	yes	—
60	C₂×Q₈	—	2	1	no	—
61	C₂×Q₈	—	2	2	no	—
62	C₂×Q₈	—	2	4	no	—
63	C₂×Q₈	—	4	2	no	—
64	C₂×Q₈	—	4	4	no	—
65	C₂×Q₈	—	4	4	no	—
66	C₂×Q₈	Q₁₆	1	4	yes	complemented
67	C₂×Q₈	C₂×D₄	1	2	yes	—
68	C₂²×C₄	—	2	1	no	—
69	C₂²×C₄	—	2	1	no	—
70	C₂²×C₄	—	4	2	no	—
71	C₂²×C₄	C₂×D₄	1	1	yes	characteristic
72	C₂²×C₄	C₂×D₄	1	1	yes	characteristic
73	C₂²×C₄	#41	1	1	yes	characteristic
74	Q₁₆	—	4	8	no	—
75	Q₁₆	—	4	8	no	—
76	C₂×C₈	—	2	2	no	—
77	C₂×C₈	—	2	2	no	—
78	C₂×C₈	—	2	4	no	—
79	C₂×C₈	—	4	4	no	—
80	C₂×C₈	—	4	2	no	—
81	C₂×C₈	SD₁₆	1	4	yes	—
82	C₂×C₈	C₂×D₄	1	2	yes	—
83	C₂×C₈	C₂×D₄	1	2	yes	—
84	C₄⋊C₄	—	2	2	no	—
85	C₄⋊C₄	—	2	1	no	—
86	C₄⋊C₄	—	2	2	no	—
87	C₄⋊C₄	—	2	4	no	—
88	C₄⋊C₄	—	2	4	no	—
89	C₄⋊C₄	—	4	2	no	—
90	C₄⋊C₄	—	4	2	no	—
91	C₄⋊C₄	C₂×D₄	1	2	yes	—
92	C₄²	—	2	2	no	—
93	C₄²	—	2	2	no	—
94	C₄²	C₂×D₄	1	1	yes	characteristic
95	C₄²	C₂×D₄	1	1	yes	characteristic
96	C₄²	#41	1	1	yes	characteristic
97	C₄²	C₂⁴	1	1	yes	characteristic
98	C₂³	#121	1	1	yes	characteristic
99	Q₈	—	2	2	no	—
100	Q₈	—	2	4	no	—
101	Q₈	—	4	4	no	—
102	Q₈	—	4	4	no	—
103	Q₈	—	8	4	no	—
104	Q₈	C₂×Q₁₆	1	4	yes	complemented
105	C₂×C₄	—	2	2	no	—
106	C₂×C₄	—	2	4	no	—
107	C₂×C₄	—	2	2	no	—
108	C₂×C₄	—	2	2	no	—
109	C₂×C₄	—	4	2	no	—
110	C₂×C₄	—	4	4	no	—
111	C₂×C₄	—	8	4	no	—
112	C₂×C₄	#121	1	1	yes	characteristic
113	C₂×C₄	#121	1	1	yes	characteristic
114	C₂×C₄	#121	1	1	yes	characteristic
115	C₂×C₄	C₂×SD₁₆	1	2	yes	—
116	C₂×C₄	C₂×Q₁₆	1	2	yes	—
117	C₂×C₄	#135	1	2	yes	—
118	C₂×C₄	AGL(1,ℤ/8)	1	2	yes	—
119	C₂×C₄	C₂²×D₄	1	1	yes	characteristic
120	C₂×C₄	C₂²×D₄	1	1	yes	characteristic
121	C₂×C₄	C₂×#41	1	1	yes	characteristic
122	C₈	—	2	2	no	—
123	C₈	—	4	4	no	—
124	C₈	—	4	2	no	—
125	C₈	C₂×SD₁₆	1	4	yes	—
126	C₂²	#460	1	2	yes	—
127	C₂²	#467	1	2	yes	—
128	C₂²	#473	1	2	yes	—
129	C₂²	C₂×#121	1	1	yes	characteristic
130	C₄	—	2	4	no	—
131	C₄	—	2	1	no	—
132	C₄	—	2	1	no	—
133	C₄	—	4	2	no	—
134	C₄	—	8	4	no	—
135	C₄	#460	1	2	yes	—
136	C₄	#467	1	2	yes	—
137	C₄	C₂²×SD₁₆	1	1	yes	characteristic
138	C₄	C₂²×Q₁₆	1	1	yes	characteristic
139	C₄	C₂×#135	1	1	yes	characteristic
140	C₄	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
141	C₂	#1670	1	4	yes	direct product factor
142	C₂	C₂×#460	1	1	yes	characteristic
143	C₂	C₂×#467	1	1	yes	characteristic
144	C₂	C₂×#473	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#494

C₂

yes

maximal, characteristic

C₂×#475

C₂

yes

maximal

C₂×C₈×Q₈

C₂

yes

maximal, characteristic

#1670

C₂

yes

maximal, direct product factor

C₂×#335

C₂

yes

maximal, characteristic

C₂×#332

C₂

yes

maximal

C₂²×Q₁₆

—

C₂²×Q₁₆

—

C₂×C₄⋊Q₈

C₂²

yes

—

C₂×C₄×Q₈

C₂²

yes

characteristic

#494

—

#494

C₂²

yes

—

#475

—

#475

C₂²

yes

—

C₈×Q₈

—

C₈×Q₈

C₂²

yes

—

C₂×#106

—

C₂×C₄⋊C₈

—

C₂×C₄⋊C₈

C₂²

yes

—

C₂×C₄⋊C₈

C₂²

yes

characteristic

C₂×#103

—

C₂×C₄×C₈

—

C₂×C₄×C₈

C₂²

yes

characteristic

#335

C₂²

yes

—

#332

C₂²

yes

—

C₂²×Q₈

—

C₂²×Q₈

D₄

yes

characteristic

C₂×Q₁₆

—

C₂×Q₁₆

—

C₂×Q₁₆

—

C₂×Q₁₆

—

C₂²×C₈

—

C₂²×C₈

—

C₂²×C₈

D₄

yes

characteristic

C₂²×C₈

D₄

yes

characteristic

C₄⋊Q₈

—

C₄⋊Q₈

—

C₄⋊Q₈

—

C₄⋊Q₈

C₂³

yes

—

C₄×Q₈

—

C₄×Q₈

—

C₄×Q₈

D₄

yes

—

C₄×Q₈

C₂³

yes

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

D₄

yes

characteristic

C₂×C₄²

—

C₂×C₄²

C₂³

yes

characteristic

#106

—

C₄⋊C₈

—

C₄⋊C₈

—

C₄⋊C₈

—

C₄⋊C₈

C₂³

yes

—

C₄⋊C₈

C₂³

yes

—

#103

—

C₄×C₈

—

C₄×C₈

D₄

yes

—

C₄×C₈

C₂³

yes

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

Q₁₆

yes

complemented

C₂×Q₈

C₂×D₄

yes

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

C₂×D₄

yes

characteristic

C₂²×C₄

C₂×D₄

yes

characteristic

C₂²×C₄

#41

yes

characteristic

Q₁₆

—

Q₁₆

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

SD₁₆

yes

—

C₂×C₈

C₂×D₄

yes

—

C₂×C₈

C₂×D₄

yes

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

C₂×D₄

yes

—

C₄²

—

C₄²

—

C₄²

C₂×D₄

yes

characteristic

C₄²

C₂×D₄

yes

characteristic

C₄²

#41

yes

characteristic

C₄²

C₂⁴

yes

characteristic

C₂³

#121

yes

characteristic

Q₈

—

100

Q₈

—

101

Q₈

—

102

Q₈

—

103

Q₈

—

104

Q₈

C₂×Q₁₆

yes

complemented

105

C₂×C₄

—

106

C₂×C₄

—

107

C₂×C₄

—

108

C₂×C₄

—

109

C₂×C₄

—

110

C₂×C₄

—

111

C₂×C₄

—

112

C₂×C₄

#121

yes

characteristic

113

C₂×C₄

#121

yes

characteristic

114

C₂×C₄

#121

yes

characteristic

115

C₂×C₄

C₂×SD₁₆

yes

—

116

C₂×C₄

C₂×Q₁₆

yes

—

117

C₂×C₄

#135

yes

—

118

C₂×C₄

AGL(1,ℤ/8)

yes

—

119

C₂×C₄

C₂²×D₄

yes

characteristic

120

C₂×C₄

C₂²×D₄

yes

characteristic

121

C₂×C₄

C₂×#41

yes

characteristic

122

C₈

—

123

C₈

—

124

C₈

—

125

C₈

C₂×SD₁₆

yes

—

126

C₂²

#460

yes

—

127

C₂²

#467

yes

—

128

C₂²

#473

yes

—

129

C₂²

C₂×#121

yes

characteristic

130

C₄

—

131

C₄

—

132

C₄

—

133

C₄

—

134

C₄

—

135

C₄

#460

yes

—

136

C₄

#467

yes

—

137

C₄

C₂²×SD₁₆

yes

characteristic

138

C₄

C₂²×Q₁₆

yes

characteristic

139

C₄

C₂×#135

yes

characteristic

140

C₄

C₂×AGL(1,ℤ/8)

yes

characteristic

141

C₂

#1670

yes

direct product factor

142

C₂

C₂×#460

yes

characteristic

143

C₂

C₂×#467

yes

characteristic

144

C₂

C₂×#473

yes

characteristic

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

subgroup

G/H

center, socle, prime radical

C₂³

#121

commutator, Frattini, cosocle

C₄²

C₂⁴

LCS

G ▷ C₄² ▷ C₂² ▷ C₁

UCS

C₁ ◁ C₂³ ◁ C₂×C₄² ◁ G

LECS

G ▷ C₄² ▷ C₂² ▷ C₁

N	H ≅ G/N	type
4. #1670	C₂	direct (N × H)
66. C₂×Q₈	Q₁₆	semidirect
104. Q₈	C₂×Q₁₆	semidirect
141. C₂	#1670	direct (N × H)

H ≅ G/N

type

4. #1670

C₂

direct (N × H)

66. C₂×Q₈

Q₁₆

semidirect

104. Q₈

C₂×Q₁₆

semidirect

141. C₂

#1670

direct (N × H)

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

subgroup	index
C₂×#494	2
C₂×#475	2
C₂×C₈×Q₈	2
#1670	2
C₂×#335	2
C₂×#332	2

subgroup

index

C₂×C₈×Q₈

Subgroups — C₂×#1670 #19113

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C2×#1670 #19113

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#1670 #19113