Group #33439 — subgroups

Subgroups — C₂×#2170 #33439

Order 256.

331 conjugacy classes of subgroups; only the 199 normal subgroups are shown.

normal subgroups only

This group has 331 conjugacy classes of subgroups — too many to draw. Only its 199 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×#504	C₂	1	2	yes	maximal, complemented
2	C₂×#443	C₂	1	1	yes	maximal, characteristic
3	#2170	C₂	1	8	yes	maximal, direct product factor
4	C₂×#364	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×#350	C₂	1	2	yes	maximal, complemented
6	C₂×C₄×C₁₆	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂×#135	C₄	1	1	yes	characteristic
8	C₂×#131	C₂²	1	2	yes	—
9	#504	—	2	4	no	—
10	#504	C₂²	1	8	yes	complemented
11	C₂×M₃₂	—	2	2	no	—
12	C₂×M₃₂	C₂²	1	2	yes	—
13	C₂²×C₁₆	—	2	2	no	—
14	C₂²×C₁₆	C₂²	1	1	yes	characteristic
15	C₂²×C₁₆	C₂²	1	1	yes	characteristic
16	#443	C₄	1	4	yes	—
17	#443	C₂²	1	4	yes	—
18	C₂×#108	C₄	1	1	yes	characteristic
19	C₂×#104	C₄	1	2	yes	—
20	C₂×C₄×C₈	C₂²	1	1	yes	characteristic
21	#364	C₂²	1	4	yes	complemented
22	#350	C₂²	1	8	yes	complemented
23	C₄×C₁₆	C₂²	1	4	yes	complemented
24	C₂×#41	C₂×C₄	1	2	yes	—
25	#135	C₈	1	4	yes	—
26	#135	C₂×C₄	1	4	yes	—
27	C₂×Q₁₆	C₈	1	1	yes	characteristic
28	C₂×SD₁₆	C₈	1	2	yes	—
29	C₂×D₈	C₈	1	1	yes	characteristic
30	#131	—	2	2	no	—
31	#131	—	2	2	no	—
32	#131	C₂×C₄	1	4	yes	—
33	#131	C₂³	1	4	yes	—
34	C₂×M₁₆	—	2	2	no	—
35	C₂×M₁₆	C₂×C₄	1	2	yes	—
36	C₂²×C₈	—	2	2	no	—
37	C₂²×C₈	C₂×C₄	1	1	yes	characteristic
38	C₂²×C₈	C₂³	1	1	yes	characteristic
39	C₂×C₄²	C₂×C₄	1	1	yes	characteristic
40	M₃₂	—	2	2	no	—
41	M₃₂	—	2	8	no	—
42	M₃₂	C₂³	1	4	yes	—
43	C₂×C₁₆	—	2	2	no	—
44	C₂×C₁₆	—	2	1	no	—
45	C₂×C₁₆	—	2	1	no	—
46	C₂×C₁₆	—	2	8	no	—
47	C₂×C₁₆	D₄	1	1	yes	characteristic
48	C₂×C₁₆	D₄	1	2	yes	—
49	C₂×C₁₆	D₄	1	1	yes	characteristic
50	C₂×C₁₆	C₂³	1	2	yes	—
51	C₂×C₁₆	C₂³	1	2	yes	—
52	#108	C₂×C₄	1	4	yes	—
53	#104	C₂×C₄	1	8	yes	—
54	C₄×C₈	C₂×C₄	1	2	yes	—
55	C₄×C₈	C₂³	1	2	yes	—
56	#41	—	2	2	no	—
57	#41	—	2	2	no	—
58	#41	C₂×C₈	1	4	yes	—
59	#41	C₂²×C₄	1	4	yes	—
60	C₂×Q₈	C₂×C₈	1	2	yes	—
61	C₂×D₄	—	2	2	no	—
62	C₂×D₄	C₂×C₈	1	2	yes	—
63	C₂²×C₄	—	2	1	no	—
64	C₂²×C₄	—	2	2	no	—
65	C₂²×C₄	C₂²×C₄	1	1	yes	characteristic
66	Q₁₆	C₂×C₈	1	4	yes	—
67	SD₁₆	C₂×C₈	1	8	yes	—
68	D₈	C₂×C₈	1	4	yes	—
69	M₁₆	—	2	2	no	—
70	M₁₆	—	2	4	no	—
71	M₁₆	—	2	4	no	—
72	M₁₆	C₂²×C₄	1	4	yes	—
73	C₂×C₈	—	2	2	no	—
74	C₂×C₈	—	2	4	no	—
75	C₂×C₈	—	2	4	no	—
76	C₂×C₈	C₂×C₈	1	1	yes	characteristic
77	C₂×C₈	C₂×C₈	1	1	yes	characteristic
78	C₂×C₈	C₂×C₈	1	2	yes	—
79	C₂×C₈	C₂²×C₄	1	1	yes	characteristic
80	C₂×C₈	C₂²×C₄	1	2	yes	—
81	C₂×C₈	C₂×D₄	1	1	yes	characteristic
82	C₂×C₈	C₂×D₄	1	1	yes	characteristic
83	C₂×C₈	#41	1	1	yes	characteristic
84	C₂×C₈	#41	1	1	yes	characteristic
85	C₂×C₈	C₂⁴	1	1	yes	characteristic
86	C₄²	—	2	1	no	—
87	C₄²	C₂²×C₄	1	2	yes	—
88	C₁₆	—	2	4	no	—
89	C₁₆	—	2	2	no	—
90	C₁₆	C₂×D₄	1	2	yes	—
91	C₁₆	C₂×D₄	1	2	yes	—
92	C₂³	—	2	2	no	—
93	C₂³	#131	1	1	yes	characteristic
94	Q₈	—	2	2	no	—
95	Q₈	C₂²×C₈	1	4	yes	—
96	D₄	—	2	4	no	—
97	D₄	—	2	4	no	—
98	D₄	—	2	2	no	—
99	D₄	C₂²×C₈	1	4	yes	—
100	C₂×C₄	—	2	4	no	—
101	C₂×C₄	—	2	2	no	—
102	C₂×C₄	—	2	2	no	—
103	C₂×C₄	—	2	4	no	—
104	C₂×C₄	—	2	4	no	—
105	C₂×C₄	C₄×D₄	1	1	yes	characteristic
106	C₂×C₄	C₄×D₄	1	1	yes	characteristic
107	C₂×C₄	C₂²×C₈	1	1	yes	characteristic
108	C₂×C₄	C₂²×C₈	1	1	yes	characteristic
109	C₂×C₄	#131	1	1	yes	characteristic
110	C₂×C₄	C₂³×C₄	1	1	yes	characteristic
111	C₈	—	2	4	no	—
112	C₈	C₄×D₄	1	1	yes	characteristic
113	C₈	C₄×D₄	1	1	yes	characteristic
114	C₈	C₂²×C₈	1	2	yes	—
115	C₈	C₂²×C₈	1	2	yes	—
116	C₈	C₂²×D₄	1	1	yes	characteristic
117	C₈	C₂×#41	1	1	yes	characteristic
118	C₂²	—	2	2	no	—
119	C₂²	—	2	4	no	—
120	C₂²	—	4	4	no	—
121	C₂²	C₈×D₄	1	1	yes	characteristic
122	C₂²	C₈×D₄	1	1	yes	characteristic
123	C₂²	C₂×#131	1	1	yes	characteristic
124	C₄	—	2	4	no	—
125	C₄	—	2	4	no	—
126	C₄	C₈×D₄	1	1	yes	characteristic
127	C₄	C₈×D₄	1	1	yes	characteristic
128	C₄	C₂×C₄×D₄	1	1	yes	characteristic
129	C₄	C₂³×C₈	1	1	yes	characteristic
130	C₂	—	2	1	no	—
131	C₂	—	2	1	no	—
132	C₂	—	4	4	no	—
133	C₂	#2170	1	2	yes	direct product factor
134	C₂	C₂×C₈×D₄	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#504

C₂

yes

maximal, complemented

C₂×#443

C₂

yes

maximal, characteristic

#2170

C₂

yes

maximal, direct product factor

C₂×#364

C₂

yes

maximal, characteristic, complemented

C₂×#350

C₂

yes

maximal, complemented

C₂×C₄×C₁₆

C₂

yes

maximal, characteristic, complemented

C₂×#135

C₄

yes

characteristic

C₂×#131

C₂²

yes

—

#504

—

#504

C₂²

yes

complemented

C₂×M₃₂

—

C₂×M₃₂

C₂²

yes

—

C₂²×C₁₆

—

C₂²×C₁₆

C₂²

yes

characteristic

C₂²×C₁₆

C₂²

yes

characteristic

#443

C₄

yes

—

#443

C₂²

yes

—

C₂×#108

C₄

yes

characteristic

C₂×#104

C₄

yes

—

C₂×C₄×C₈

C₂²

yes

characteristic

#364

C₂²

yes

complemented

#350

C₂²

yes

complemented

C₄×C₁₆

C₂²

yes

complemented

C₂×#41

C₂×C₄

yes

—

#135

C₈

yes

—

#135

C₂×C₄

yes

—

C₂×Q₁₆

C₈

yes

characteristic

C₂×SD₁₆

C₈

yes

—

C₂×D₈

C₈

yes

—

—

C₂×C₄

yes

—

#131

C₂³

yes

—

C₂×M₁₆

—

C₂×M₁₆

C₂×C₄

yes

—

C₂²×C₈

—

C₂²×C₈

C₂×C₄

yes

characteristic

C₂²×C₈

C₂³

yes

characteristic

C₂×C₄²

C₂×C₄

yes

characteristic

M₃₂

—

M₃₂

—

M₃₂

C₂³

yes

—

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

D₄

yes

characteristic

C₂×C₁₆

D₄

yes

—

C₂×C₁₆

D₄

yes

characteristic

C₂×C₁₆

C₂³

yes

—

C₂×C₁₆

C₂³

yes

—

#108

C₂×C₄

yes

—

#104

C₂×C₄

yes

—

C₄×C₈

C₂×C₄

yes

—

C₄×C₈

C₂³

yes

—

#41

—

#41

—

#41

C₂×C₈

yes

—

#41

C₂²×C₄

yes

—

C₂×Q₈

C₂×C₈

yes

—

C₂×D₄

—

C₂×D₄

C₂×C₈

yes

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

yes

characteristic

Q₁₆

C₂×C₈

yes

—

SD₁₆

C₂×C₈

yes

—

D₈

C₂×C₈

yes

—

M₁₆

—

M₁₆

—

M₁₆

—

M₁₆

C₂²×C₄

yes

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

yes

characteristic

C₂×C₈

yes

characteristic

C₂×C₈

yes

—

C₂×C₈

C₂²×C₄

yes

characteristic

C₂×C₈

C₂²×C₄

yes

—

C₂×C₈

C₂×D₄

yes

characteristic

C₂×C₈

C₂×D₄

yes

C₂×C₈

yes

C₂×C₈

yes

C₂×C₈

C₂⁴

yes

characteristic

C₄²

—

C₄²

C₂²×C₄

yes

—

C₁₆

—

C₁₆

—

C₁₆

C₂×D₄

yes

—

C₁₆

C₂×D₄

yes

—

C₂³

—

C₂³

#131

yes

characteristic

Q₈

—

Q₈

C₂²×C₈

yes

—

D₄

—

D₄

—

D₄

—

D₄

C₂²×C₈

yes

—

100

C₂×C₄

—

101

C₂×C₄

—

102

C₂×C₄

—

103

C₂×C₄

—

104

C₂×C₄

—

105

C₂×C₄

C₄×D₄

yes

characteristic

106

C₂×C₄

C₄×D₄

yes

characteristic

107

C₂×C₄

C₂²×C₈

yes

characteristic

108

C₂×C₄

C₂²×C₈

yes

characteristic

109

C₂×C₄

#131

yes

characteristic

110

C₂×C₄

C₂³×C₄

yes

characteristic

111

C₈

—

112

C₈

C₄×D₄

yes

characteristic

113

C₈

C₄×D₄

yes

characteristic

114

C₈

C₂²×C₈

yes

—

115

C₈

C₂²×C₈

yes

—

116

C₈

C₂²×D₄

yes

characteristic

117

C₈

C₂×#41

yes

characteristic

118

C₂²

—

119

C₂²

—

120

C₂²

—

121

C₂²

C₈×D₄

yes

characteristic

122

C₂²

C₈×D₄

yes

characteristic

123

C₂²

C₂×#131

yes

characteristic

124

C₄

—

125

C₄

—

126

C₄

C₈×D₄

yes

characteristic

127

C₄

C₈×D₄

yes

characteristic

128

C₄

C₂×C₄×D₄

yes

characteristic

129

C₄

C₂³×C₈

yes

characteristic

130

C₂

—

131

C₂

—

132

C₂

—

133

C₂

#2170

yes

direct product factor

134

C₂

C₂×C₈×D₄

yes

characteristic

	subgroup	G/H
center	C₂×C₁₆	D₄
commutator	C₄	C₂³×C₈
Frattini, cosocle	C₂×C₈	C₂⁴
socle	C₂²	C₈×D₄
prime radical	C₂×#135	C₄
LCS	G ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₁₆ ◁ C₂²×C₁₆ ◁ G
LECS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁

subgroup

G/H

center

C₂×C₁₆

D₄

commutator

C₄

C₂³×C₈

Frattini, cosocle

C₂×C₈

C₂⁴

socle

C₂²

C₈×D₄

C₄

G ▷ C₄ ▷ C₂ ▷ C₁

C₁ ◁ C₂×C₁₆ ◁ C₂²×C₁₆ ◁ G

LECS

G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁

N	H ≅ G/N	type
1. C₂×#504	C₂	semidirect
3. #2170	C₂	direct (N × H)
4. C₂×#364	C₂	semidirect
5. C₂×#350	C₂	semidirect
6. C₂×C₄×C₁₆	C₂	semidirect
10. #504	C₂²	semidirect
21. #364	C₂²	semidirect
22. #350	C₂²	semidirect
23. C₄×C₁₆	C₂²	semidirect
133. C₂	#2170	direct (N × H)

H ≅ G/N

type

1. C₂×#504

C₂

semidirect

3. #2170

C₂

direct (N × H)

4. C₂×#364

C₂

semidirect

5. C₂×#350

C₂

semidirect

6. C₂×C₄×C₁₆

C₂

semidirect

10. #504

C₂²

semidirect

21. #364

C₂²

semidirect

22. #350

C₂²

semidirect

23. C₄×C₁₆

C₂²

semidirect

133. C₂

#2170

direct (N × H)

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

subgroup	index
C₂×#504	2
C₂×#443	2
#2170	2
C₂×#364	2
C₂×#350	2
C₂×C₄×C₁₆	2

subgroup

index

C₂×C₄×C₁₆

Subgroups — C₂×#2170 #33439

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C2×#2170 #33439

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#2170 #33439