Group #13052909 — subgroups

Subgroups — C₂×D₂₁⋊C₁₆ #13052909

Order 1344.

196 conjugacy classes of subgroups; only the 88 normal subgroups are shown.

normal subgroups only

This group has 196 conjugacy classes of subgroups — too many to draw. Only its 88 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×C₈×D₂₁	C₂	1	1	yes	maximal, characteristic
2	C₂×C₇×C₃⋊C₁₆	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×C₃×C₇⋊C₁₆	C₂	1	1	yes	maximal, characteristic, complemented
4	D₂₁⋊C₁₆	C₂	1	4	yes	maximal, direct product factor
5	C₂×D₇×C₁₆	—	3	1	no	maximal, Hall
6	C₂×C₄×D₂₁	C₄	1	1	yes	characteristic
7	C₂×C₃×C₇×C₈	C₂²	1	1	yes	characteristic
8	C₂×C₃⋊(C₇⋊C₈)	C₄	1	1	yes	characteristic
9	C₈×D₂₁	C₄	1	2	yes	—
10	C₈×D₂₁	C₂²	1	2	yes	—
11	C₃×C₇⋊C₁₆	C₂²	1	2	yes	complemented
12	C₇×C₃⋊C₁₆	C₂²	1	2	yes	complemented
13	C₂×C₈×D₇	—	3	1	no	—
14	C₂×C₇×C₁₆	—	3	1	no	—
15	C₂×C₇⋊C₁₆	D₃	1	1	yes	characteristic, complemented
16	D₇×C₁₆	—	3	4	no	—
17	C₂×D₃×C₁₆	—	7	1	no	maximal, Hall
18	C₂²×D₂₁	C₈	1	1	yes	characteristic
19	C₂×C₃×C₄×C₇	C₂×C₄	1	1	yes	characteristic
20	C₂×Dic₂₁	C₈	1	1	yes	characteristic
21	C₄×D₂₁	C₈	1	2	yes	—
22	C₄×D₂₁	C₂×C₄	1	2	yes	—
23	C₃×C₇×C₈	C₂×C₄	1	1	yes	characteristic
24	C₃×C₇×C₈	C₂³	1	1	yes	characteristic
25	C₃⋊(C₇⋊C₈)	C₂×C₄	1	2	yes	—
26	C₂×C₄×D₇	—	3	1	no	—
27	C₂×C₇×C₈	C₂×D₃	1	1	yes	characteristic
28	C₂×C₇⋊C₈	—	3	1	no	—
29	C₈×D₇	—	3	2	no	—
30	C₈×D₇	—	3	2	no	—
31	C₇×C₁₆	—	3	2	no	—
32	C₇⋊C₁₆	C₂×D₃	1	2	yes	complemented
33	C₂×D₃×C₈	—	7	1	no	—
34	C₂×C₃×C₁₆	—	7	1	no	—
35	C₂×C₃⋊C₁₆	D₇	1	1	yes	characteristic, complemented
36	D₃×C₁₆	—	7	4	no	—
37	C₂²×C₃×C₇	C₂×C₈	1	1	yes	characteristic
38	C₂×D₂₁	C₁₆	1	4	yes	complemented
39	C₂×D₂₁	C₂×C₈	1	2	yes	—
40	C₃×C₄×C₇	C₂×C₈	1	1	yes	characteristic
41	C₃×C₄×C₇	C₂²×C₄	1	1	yes	characteristic
42	Dic₂₁	C₂×C₈	1	2	yes	—
43	C₂²×C₁₆	—	21	1	no	Sylow
44	C₂²×D₇	—	3	1	no	—
45	C₂×C₄×C₇	C₄×D₃	1	1	yes	characteristic
46	C₂×Dic₇	—	3	1	no	—
47	C₄×D₇	—	3	2	no	—
48	C₄×D₇	—	3	2	no	—
49	C₇×C₈	C₄×D₃	1	1	yes	characteristic
50	C₇×C₈	C₂²×D₃	1	1	yes	characteristic
51	C₇⋊C₈	—	3	2	no	—
52	C₂×C₄×D₃	—	7	1	no	—
53	C₂×C₃×C₈	C₂×D₇	1	1	yes	characteristic
54	C₂×C₃⋊C₈	—	7	1	no	—
55	D₃×C₈	—	7	2	no	—
56	D₃×C₈	—	7	2	no	—
57	C₃×C₁₆	—	7	2	no	—
58	C₃⋊C₁₆	C₂×D₇	1	2	yes	complemented
59	C₂×C₃×C₇	C₂×C₁₆	1	2	yes	complemented
60	C₂×C₃×C₇	C₂²×C₈	1	1	yes	characteristic
61	D₂₁	C₂×C₁₆	1	4	yes	complemented
62	C₂²×C₈	—	21	1	no	—
63	C₂×C₁₆	—	3	1	no	—
64	C₂×C₁₆	—	7	1	no	—
65	C₂×C₁₆	—	21	4	no	—
66	C₂²×C₇	D₃×C₈	1	1	yes	characteristic
67	C₂×D₇	—	3	4	no	—
68	C₂×D₇	—	3	2	no	—
69	C₄×C₇	D₃×C₈	1	1	yes	characteristic
70	C₄×C₇	C₂×C₄×D₃	1	1	yes	characteristic
71	Dic₇	—	3	2	no	—
72	C₂²×D₃	—	7	1	no	—
73	C₂×C₃×C₄	C₄×D₇	1	1	yes	characteristic
74	C₂×Dic₃	—	7	1	no	—
75	C₄×D₃	—	7	2	no	—
76	C₄×D₃	—	7	2	no	—
77	C₃×C₈	C₄×D₇	1	1	yes	characteristic
78	C₃×C₈	C₂²×D₇	1	1	yes	characteristic
79	C₃⋊C₈	—	7	2	no	—
80	C₃×C₇	C₂²×C₁₆	1	1	yes	characteristic, complemented, Hall
81	C₂²×C₄	—	21	1	no	—
82	C₂×C₈	—	21	1	no	—
83	C₂×C₈	—	21	2	no	—
84	C₂×C₈	—	21	2	no	—
85	C₂×C₈	D₃×D₇	1	1	yes	characteristic
86	C₁₆	—	3	2	no	—
87	C₁₆	—	7	2	no	—
88	C₂×C₇	D₃×C₁₆	1	2	yes	complemented
89	C₂×C₇	C₂×D₃×C₈	1	1	yes	characteristic
90	D₇	—	3	4	no	—
91	C₂²×C₃	C₈×D₇	1	1	yes	characteristic
92	C₂×D₃	—	7	2	no	—
93	C₂×D₃	—	7	4	no	—
94	C₃×C₄	C₈×D₇	1	1	yes	characteristic
95	C₃×C₄	C₂×C₄×D₇	1	1	yes	characteristic
96	Dic₃	—	7	2	no	—
97	C₂³	—	21	1	no	—
98	C₂×C₄	—	21	1	no	—
99	C₂×C₄	—	21	2	no	—
100	C₂×C₄	—	21	2	no	—
101	C₂×C₄	D₂₁⋊C₄	1	1	yes	characteristic
102	C₈	—	21	2	no	—
103	C₈	D₂₁⋊C₄	1	1	yes	characteristic
104	C₈	C₂×D₃×D₇	1	1	yes	characteristic
105	C₇	C₂×D₃×C₁₆	1	1	yes	characteristic, complemented, Sylow
106	C₂×C₃	D₇×C₁₆	1	2	yes	complemented
107	C₂×C₃	C₂×C₈×D₇	1	1	yes	characteristic
108	D₃	—	7	4	no	—
109	C₂²	—	21	4	no	—
110	C₂²	—	21	2	no	—
111	C₂²	D₂₁⋊C₈	1	1	yes	characteristic
112	C₄	—	21	2	no	—
113	C₄	D₂₁⋊C₈	1	1	yes	characteristic
114	C₄	C₂×D₂₁⋊C₄	1	1	yes	characteristic
115	C₃	C₂×D₇×C₁₆	1	1	yes	characteristic, complemented, Sylow
116	C₂	—	21	4	no	—
117	C₂	D₂₁⋊C₁₆	1	2	yes	direct product factor
118	C₂	C₂×D₂₁⋊C₈	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×C₈×D₂₁

C₂

yes

maximal, characteristic

C₂×C₇×C₃⋊C₁₆

C₂

yes

maximal, characteristic, complemented

C₂×C₃×C₇⋊C₁₆

C₂

yes

maximal, characteristic, complemented

D₂₁⋊C₁₆

C₂

yes

maximal, direct product factor

C₂×D₇×C₁₆

—

maximal, Hall

C₂×C₄×D₂₁

C₄

yes

characteristic

C₂×C₃×C₇×C₈

C₂²

yes

characteristic

C₂×C₃⋊(C₇⋊C₈)

C₄

yes

characteristic

C₈×D₂₁

C₄

yes

—

C₈×D₂₁

C₂²

yes

—

C₃×C₇⋊C₁₆

C₂²

yes

complemented

C₇×C₃⋊C₁₆

C₂²

yes

complemented

C₂×C₈×D₇

—

C₂×C₇×C₁₆

—

C₂×C₇⋊C₁₆

D₃

yes

characteristic, complemented

D₇×C₁₆

—

C₂×D₃×C₁₆

—

maximal, Hall

C₂²×D₂₁

C₈

yes

characteristic

C₂×C₃×C₄×C₇

C₂×C₄

yes

characteristic

C₂×Dic₂₁

C₈

yes

characteristic

C₄×D₂₁

C₈

yes

—

C₄×D₂₁

C₂×C₄

yes

—

C₃×C₇×C₈

C₂×C₄

yes

characteristic

C₃×C₇×C₈

C₂³

yes

characteristic

C₃⋊(C₇⋊C₈)

C₂×C₄

yes

—

C₂×C₄×D₇

—

C₂×C₇×C₈

C₂×D₃

yes

characteristic

C₂×C₇⋊C₈

—

C₈×D₇

—

C₈×D₇

—

C₇×C₁₆

—

C₇⋊C₁₆

C₂×D₃

yes

complemented

C₂×D₃×C₈

—

C₂×C₃×C₁₆

—

C₂×C₃⋊C₁₆

D₇

yes

characteristic, complemented

D₃×C₁₆

—

C₂²×C₃×C₇

C₂×C₈

yes

characteristic

C₂×D₂₁

C₁₆

yes

complemented

C₂×D₂₁

C₂×C₈

yes

—

C₃×C₄×C₇

C₂×C₈

yes

characteristic

C₃×C₄×C₇

C₂²×C₄

yes

characteristic

Dic₂₁

C₂×C₈

yes

—

C₂²×C₁₆

—

Sylow

C₂²×D₇

—

C₂×C₄×C₇

C₄×D₃

yes

characteristic

C₂×Dic₇

—

C₄×D₇

—

C₄×D₇

—

C₇×C₈

C₄×D₃

yes

characteristic

C₇×C₈

C₂²×D₃

yes

characteristic

C₇⋊C₈

—

C₂×C₄×D₃

—

C₂×C₃×C₈

C₂×D₇

yes

characteristic

C₂×C₃⋊C₈

—

D₃×C₈

—

D₃×C₈

—

C₃×C₁₆

—

C₃⋊C₁₆

C₂×D₇

yes

complemented

C₂×C₃×C₇

C₂×C₁₆

yes

complemented

C₂×C₃×C₇

C₂²×C₈

yes

characteristic

D₂₁

C₂×C₁₆

yes

complemented

C₂²×C₈

—

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

—

C₂²×C₇

D₃×C₈

yes

characteristic

C₂×D₇

—

C₂×D₇

—

C₄×C₇

D₃×C₈

yes

characteristic

C₄×C₇

C₂×C₄×D₃

yes

characteristic

Dic₇

—

C₂²×D₃

—

C₂×C₃×C₄

C₄×D₇

yes

characteristic

C₂×Dic₃

—

C₄×D₃

—

C₄×D₃

—

C₃×C₈

C₄×D₇

yes

characteristic

C₃×C₈

C₂²×D₇

yes

characteristic

C₃⋊C₈

—

C₃×C₇

C₂²×C₁₆

yes

characteristic, complemented, Hall

C₂²×C₄

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

D₃×D₇

yes

characteristic

C₁₆

—

C₁₆

—

C₂×C₇

D₃×C₁₆

yes

complemented

C₂×C₇

C₂×D₃×C₈

yes

characteristic

D₇

—

C₂²×C₃

C₈×D₇

yes

characteristic

C₂×D₃

—

C₂×D₃

—

C₃×C₄

C₈×D₇

yes

characteristic

C₃×C₄

C₂×C₄×D₇

yes

characteristic

Dic₃

—

C₂³

—

C₂×C₄

—

C₂×C₄

—

100

C₂×C₄

—

101

C₂×C₄

D₂₁⋊C₄

yes

characteristic

102

C₈

—

103

C₈

D₂₁⋊C₄

yes

characteristic

104

C₈

C₂×D₃×D₇

yes

characteristic

105

C₇

C₂×D₃×C₁₆

yes

characteristic, complemented, Sylow

106

C₂×C₃

D₇×C₁₆

yes

complemented

107

C₂×C₃

C₂×C₈×D₇

yes

characteristic

108

D₃

—

109

C₂²

—

110

C₂²

—

111

C₂²

D₂₁⋊C₈

yes

characteristic

112

C₄

—

113

C₄

D₂₁⋊C₈

yes

characteristic

114

C₄

C₂×D₂₁⋊C₄

yes

characteristic

115

C₃

C₂×D₇×C₁₆

yes

characteristic, complemented, Sylow

116

C₂

—

117

C₂

D₂₁⋊C₁₆

yes

direct product factor

118

C₂

C₂×D₂₁⋊C₈

yes

characteristic

	subgroup	G/H
center, hypercenter	C₂×C₈	D₃×D₇
commutator, hypocenter	C₃×C₇	C₂²×C₁₆
Frattini	C₈	C₂×D₃×D₇
Fitting	C₂×C₃×C₇×C₈	C₂²
socle	C₂²×C₃×C₇	C₂×C₈
prime radical	C₂²×D₂₁	C₈
cosocle	C₃×C₇×C₈	C₂³
LCS	G ▷ C₃×C₇
UCS	C₁ ◁ C₂×C₈

subgroup

G/H

center, hypercenter

C₂×C₈

D₃×D₇

commutator, hypocenter

C₃×C₇

C₂²×C₁₆

Frattini

C₈

C₂×D₃×D₇

Fitting

C₂×C₃×C₇×C₈

C₂²

socle

C₂²×C₃×C₇

C₂×C₈

prime radical

C₂²×D₂₁

C₈

cosocle

C₃×C₇×C₈

C₂³

LCS

G ▷ C₃×C₇

UCS

C₁ ◁ C₂×C₈

N	H ≅ G/N	type
2. C₂×C₇×C₃⋊C₁₆	C₂	semidirect
3. C₂×C₃×C₇⋊C₁₆	C₂	semidirect
4. D₂₁⋊C₁₆	C₂	direct (N × H)
11. C₃×C₇⋊C₁₆	C₂²	semidirect
12. C₇×C₃⋊C₁₆	C₂²	semidirect
15. C₂×C₇⋊C₁₆	D₃	semidirect
32. C₇⋊C₁₆	C₂×D₃	semidirect
35. C₂×C₃⋊C₁₆	D₇	semidirect
38. C₂×D₂₁	C₁₆	semidirect
58. C₃⋊C₁₆	C₂×D₇	semidirect
59. C₂×C₃×C₇	C₂×C₁₆	semidirect
61. D₂₁	C₂×C₁₆	semidirect
80. C₃×C₇	C₂²×C₁₆	semidirect
88. C₂×C₇	D₃×C₁₆	semidirect
105. C₇	C₂×D₃×C₁₆	semidirect
106. C₂×C₃	D₇×C₁₆	semidirect
115. C₃	C₂×D₇×C₁₆	semidirect
117. C₂	D₂₁⋊C₁₆	direct (N × H)

H ≅ G/N

type

2. C₂×C₇×C₃⋊C₁₆

C₂

semidirect

3. C₂×C₃×C₇⋊C₁₆

C₂

semidirect

4. D₂₁⋊C₁₆

C₂

direct (N × H)

11. C₃×C₇⋊C₁₆

C₂²

semidirect

12. C₇×C₃⋊C₁₆

C₂²

semidirect

15. C₂×C₇⋊C₁₆

D₃

semidirect

32. C₇⋊C₁₆

C₂×D₃

semidirect

35. C₂×C₃⋊C₁₆

D₇

semidirect

38. C₂×D₂₁

C₁₆

semidirect

58. C₃⋊C₁₆

C₂×D₇

semidirect

59. C₂×C₃×C₇

C₂×C₁₆

semidirect

61. D₂₁

C₂×C₁₆

semidirect

80. C₃×C₇

C₂²×C₁₆

semidirect

88. C₂×C₇

D₃×C₁₆

semidirect

105. C₇

C₂×D₃×C₁₆

semidirect

106. C₂×C₃

D₇×C₁₆

semidirect

115. C₃

C₂×D₇×C₁₆

semidirect

117. C₂

D₂₁⋊C₁₆

direct (N × H)

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

subgroup	index
C₂×C₈×D₂₁	2
C₂×C₇×C₃⋊C₁₆	2
C₂×C₃×C₇⋊C₁₆	2
D₂₁⋊C₁₆	2
C₂×D₇×C₁₆	3
C₂×D₃×C₁₆	7

subgroup

index

C₂×C₈×D₂₁

C₂×C₇×C₃⋊C₁₆

C₂×C₃×C₇⋊C₁₆

D₂₁⋊C₁₆

C₂×D₇×C₁₆

C₂×D₃×C₁₆

Subgroups — C₂×D₂₁⋊C₁₆ #13052909

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C2×D21⋊C16 #13052909

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×D₂₁⋊C₁₆ #13052909