Group #19145 — subgroups

Subgroups — C₂×#1690 #19145

Order 256.

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

normal subgroups only

This group has 341 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×#494	C₂	1	1	yes	maximal, characteristic
2	C₂×#475	C₂	1	2	yes	maximal
3	C₂×#446	C₂	1	1	yes	maximal, characteristic
4	#1690	C₂	1	8	yes	maximal, direct product factor
5	C₂×#334	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	#446	—	2	2	no	—
16	#446	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×#97	—	2	1	no	—
23	C₂×C₄×C₈	C₂²	1	1	yes	characteristic
24	#334	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	#97	—	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	4	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	C₂×D₄	1	2	yes	—
67	C₂²×C₄	—	2	1	no	—
68	C₂²×C₄	—	2	1	no	—
69	C₂²×C₄	—	4	2	no	—
70	C₂²×C₄	C₂×D₄	1	1	yes	characteristic
71	C₂²×C₄	C₂×D₄	1	1	yes	characteristic
72	C₂²×C₄	#41	1	1	yes	characteristic
73	Q₁₆	—	4	8	no	—
74	Q₁₆	—	4	8	no	—
75	C₂×C₈	—	2	2	no	—
76	C₂×C₈	—	2	2	no	—
77	C₂×C₈	—	2	4	no	—
78	C₂×C₈	—	4	2	no	—
79	C₂×C₈	—	4	4	no	—
80	C₂×C₈	SD₁₆	1	4	yes	—
81	C₂×C₈	C₂×D₄	1	2	yes	—
82	C₂×C₈	C₂×D₄	1	2	yes	—
83	C₄⋊C₄	—	2	2	no	—
84	C₄⋊C₄	—	2	1	no	—
85	C₄⋊C₄	—	2	2	no	—
86	C₄⋊C₄	—	2	4	no	—
87	C₄⋊C₄	—	2	4	no	—
88	C₄⋊C₄	—	4	2	no	—
89	C₄⋊C₄	—	4	2	no	—
90	C₄⋊C₄	C₂×D₄	1	2	yes	—
91	C₄²	—	2	2	no	—
92	C₄²	—	2	2	no	—
93	C₄²	C₂×D₄	1	1	yes	characteristic
94	C₄²	C₂×D₄	1	1	yes	characteristic
95	C₄²	#41	1	1	yes	characteristic
96	C₄²	C₂⁴	1	1	yes	characteristic
97	C₂³	#121	1	1	yes	characteristic
98	Q₈	—	2	4	no	—
99	Q₈	—	4	2	no	—
100	Q₈	—	4	4	no	—
101	Q₈	—	4	4	no	—
102	Q₈	—	8	4	no	—
103	C₂×C₄	—	2	2	no	—
104	C₂×C₄	—	2	2	no	—
105	C₂×C₄	—	2	2	no	—
106	C₂×C₄	—	4	2	no	—
107	C₂×C₄	—	4	2	no	—
108	C₂×C₄	—	4	4	no	—
109	C₂×C₄	—	8	4	no	—
110	C₂×C₄	#121	1	1	yes	characteristic
111	C₂×C₄	#121	1	1	yes	characteristic
112	C₂×C₄	#121	1	1	yes	characteristic
113	C₂×C₄	C₂×SD₁₆	1	2	yes	—
114	C₂×C₄	AGL(1,ℤ/8)	1	2	yes	—
115	C₂×C₄	#137	1	4	yes	—
116	C₂×C₄	C₂²×D₄	1	1	yes	characteristic
117	C₂×C₄	C₂²×D₄	1	1	yes	characteristic
118	C₂×C₄	C₂×#41	1	1	yes	characteristic
119	C₈	—	2	2	no	—
120	C₈	—	4	2	no	—
121	C₈	—	4	4	no	—
122	C₈	C₂×SD₁₆	1	4	yes	—
123	C₂²	#460	1	2	yes	—
124	C₂²	#470	1	2	yes	—
125	C₂²	#473	1	2	yes	—
126	C₂²	C₂×#121	1	1	yes	characteristic
127	C₄	—	2	1	no	—
128	C₄	—	2	1	no	—
129	C₄	—	4	2	no	—
130	C₄	—	4	2	no	—
131	C₄	—	8	4	no	—
132	C₄	#460	1	2	yes	—
133	C₄	#470	1	2	yes	—
134	C₄	C₂²×SD₁₆	1	1	yes	characteristic
135	C₄	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
136	C₄	C₂×#137	1	2	yes	—
137	C₂	#1690	1	4	yes	direct product factor
138	C₂	C₂×#460	1	1	yes	characteristic
139	C₂	C₂×#470	1	1	yes	characteristic
140	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×#446

C₂

yes

maximal, characteristic

#1690

C₂

yes

maximal, direct product factor

C₂×#334

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

—

#446

—

#446

C₂²

yes

—

C₂×#106

—

C₂×C₄⋊C₈

—

C₂×C₄⋊C₈

C₂²

yes

—

C₂×C₄⋊C₈

C₂²

yes

characteristic

C₂×#103

—

C₂×#97

—

C₂×C₄×C₈

C₂²

yes

characteristic

#334

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

—

#97

—

C₄×C₈

D₄

yes

—

C₄×C₈

C₂³

yes

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

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

—

Q₈

—

100

Q₈

—

101

Q₈

—

102

Q₈

—

103

C₂×C₄

—

104

C₂×C₄

—

105

C₂×C₄

—

106

C₂×C₄

—

107

C₂×C₄

—

108

C₂×C₄

—

109

C₂×C₄

—

110

C₂×C₄

#121

yes

characteristic

111

C₂×C₄

#121

yes

characteristic

112

C₂×C₄

#121

yes

characteristic

113

C₂×C₄

C₂×SD₁₆

yes

—

114

C₂×C₄

AGL(1,ℤ/8)

yes

—

115

C₂×C₄

#137

yes

—

116

C₂×C₄

C₂²×D₄

yes

characteristic

117

C₂×C₄

C₂²×D₄

yes

characteristic

118

C₂×C₄

C₂×#41

yes

characteristic

119

C₈

—

120

C₈

—

121

C₈

—

122

C₈

C₂×SD₁₆

yes

—

123

C₂²

#460

yes

—

124

C₂²

#470

yes

—

125

C₂²

#473

yes

—

126

C₂²

C₂×#121

yes

characteristic

127

C₄

—

128

C₄

—

129

C₄

—

130

C₄

—

131

C₄

—

132

C₄

#460

yes

—

133

C₄

#470

yes

—

134

C₄

C₂²×SD₁₆

yes

characteristic

135

C₄

C₂×AGL(1,ℤ/8)

yes

characteristic

136

C₄

C₂×#137

yes

—

137

C₂

#1690

yes

direct product factor

138

C₂

C₂×#460

yes

characteristic

139

C₂

C₂×#470

yes

characteristic

140

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. #1690	C₂	direct (N × H)
137. C₂	#1690	direct (N × H)

H ≅ G/N

type

4. #1690

C₂

direct (N × H)

137. C₂

#1690

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×#446	2
#1690	2
C₂×#334	2
C₂×#332	2

subgroup

index

Subgroups — C₂×#1690 #19145

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C2×#1690 #19145

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#1690 #19145