Group #10621396 — subgroups

Subgroups — SD₁₆×D₂₁ #10621396

Order 672.

136 conjugacy classes of subgroups; only the 43 normal subgroups are shown.

normal subgroups only

This group has 136 conjugacy classes of subgroups — too many to draw. Only its 43 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	Q₈×D₂₁	C₂	1	1	yes	maximal, characteristic, complemented
2	D₄×D₂₁	C₂	1	1	yes	maximal, characteristic
3	C₃×C₇×SD₁₆	C₂	1	1	yes	maximal, characteristic, complemented
4	#66634	C₂	1	1	yes	maximal, characteristic, complemented
5	#66633	C₂	1	1	yes	maximal, characteristic, complemented
6	#66623	C₂	1	1	yes	maximal, characteristic, complemented
7	C₈×D₂₁	C₂	1	1	yes	maximal, characteristic, complemented
8	D₇×SD₁₆	—	3	1	no	maximal, Hall
9	C₂²×D₂₁	—	2	1	no	—
10	C₃×C₇×Q₈	C₂²	1	1	yes	characteristic, complemented
11	C₃×C₇×D₄	C₂²	1	1	yes	characteristic
12	#4268	—	2	1	no	—
13	D₈₄	C₂²	1	1	yes	characteristic
14	C₄×D₂₁	—	2	1	no	—
15	C₄×D₂₁	C₂²	1	1	yes	characteristic
16	Dic₄₂	—	2	1	no	—
17	Dic₄₂	C₂²	1	1	yes	characteristic, complemented
18	C₃×C₇×C₈	C₂²	1	1	yes	characteristic, complemented
19	C₃⋊(C₇⋊C₈)	C₂²	1	1	yes	characteristic, complemented
20	Q₈×D₇	—	3	1	no	—
21	D₄×D₇	—	3	1	no	—
22	C₇×SD₁₆	D₃	1	1	yes	characteristic, complemented
23	#1143	—	3	1	no	—
24	#1142	—	3	1	no	—
25	#1132	—	3	1	no	—
26	C₈×D₇	—	3	1	no	—
27	D₃×SD₁₆	—	7	1	no	maximal, Hall
28	C₂²×C₃×C₇	—	2	1	no	—
29	C₂×D₂₁	—	2	1	no	—
30	C₂×D₂₁	—	4	2	no	—
31	C₂×D₂₁	D₄	1	1	yes	characteristic
32	C₃×C₄×C₇	—	2	1	no	—
33	C₃×C₄×C₇	C₂³	1	1	yes	characteristic
34	Dic₂₁	—	2	1	no	—
35	Dic₂₁	D₄	1	1	yes	characteristic
36	C₂²×D₇	—	6	1	no	—
37	C₇×Q₈	C₂×D₃	1	1	yes	characteristic, complemented
38	C₇×D₄	C₂×D₃	1	1	yes	characteristic
39	C₂²⋊D₇	—	6	1	no	—
40	D₂₈	—	3	1	no	—
41	C₄×D₇	—	3	1	no	—
42	C₄×D₇	—	6	1	no	—
43	Dic₁₄	—	3	1	no	—
44	Dic₁₄	—	6	1	no	—
45	C₇×C₈	C₂×D₃	1	1	yes	characteristic, complemented
46	C₇⋊C₈	—	3	1	no	—
47	D₃×Q₈	—	7	1	no	—
48	D₃×D₄	—	7	1	no	—
49	C₃×SD₁₆	D₇	1	1	yes	characteristic, complemented
50	#215	—	7	1	no	—
51	#214	—	7	1	no	—
52	#204	—	7	1	no	—
53	D₃×C₈	—	7	1	no	—
54	C₂×C₃×C₇	—	4	1	no	—
55	C₂×C₃×C₇	C₂×D₄	1	1	yes	characteristic
56	D₂₁	—	4	1	no	—
57	D₂₁	SD₁₆	1	2	yes	direct product factor
58	C₂×SD₁₆	—	21	1	no	Sylow
59	C₂²×C₇	—	2	1	no	—
60	C₂×D₇	—	3	1	no	—
61	C₂×D₇	—	6	1	no	—
62	C₂×D₇	—	12	2	no	—
63	C₄×C₇	—	2	1	no	—
64	C₄×C₇	C₂²×D₃	1	1	yes	characteristic
65	Dic₇	—	3	1	no	—
66	Dic₇	—	6	1	no	—
67	C₂²×D₃	—	14	1	no	—
68	C₃×Q₈	C₂×D₇	1	1	yes	characteristic, complemented
69	C₃×D₄	C₂×D₇	1	1	yes	characteristic
70	#67	—	14	1	no	—
71	D₁₂	—	7	1	no	—
72	C₄×D₃	—	7	1	no	—
73	C₄×D₃	—	14	1	no	—
74	Dic₆	—	7	1	no	—
75	Dic₆	—	14	1	no	—
76	C₃×C₈	C₂×D₇	1	1	yes	characteristic, complemented
77	C₃⋊C₈	—	7	1	no	—
78	C₃×C₇	C₂×SD₁₆	1	1	yes	characteristic, complemented, Hall
79	C₂×Q₈	—	21	1	no	—
80	C₂×D₄	—	21	1	no	—
81	SD₁₆	—	21	1	no	—
82	SD₁₆	—	21	1	no	—
83	SD₁₆	—	21	1	no	—
84	SD₁₆	D₂₁	1	1	yes	characteristic, direct product factor
85	C₂×C₈	—	21	1	no	—
86	C₂×C₇	—	4	1	no	—
87	C₂×C₇	D₃×D₄	1	1	yes	characteristic
88	D₇	—	3	2	no	—
89	D₇	—	12	1	no	—
90	C₂²×C₃	—	2	1	no	—
91	C₂×D₃	—	7	1	no	—
92	C₂×D₃	—	14	1	no	—
93	C₂×D₃	—	28	2	no	—
94	C₃×C₄	—	2	1	no	—
95	C₃×C₄	C₂²×D₇	1	1	yes	characteristic
96	Dic₃	—	7	1	no	—
97	Dic₃	—	14	1	no	—
98	C₂³	—	42	1	no	—
99	Q₈	—	21	1	no	—
100	Q₈	—	42	1	no	—
101	Q₈	C₂×D₂₁	1	1	yes	characteristic, complemented
102	D₄	—	21	1	no	—
103	D₄	—	42	1	no	—
104	D₄	C₂×D₂₁	1	1	yes	characteristic
105	C₂×C₄	—	21	1	no	—
106	C₂×C₄	—	42	1	no	—
107	C₈	—	21	1	no	—
108	C₈	C₂×D₂₁	1	1	yes	characteristic, complemented
109	C₇	D₃×SD₁₆	1	1	yes	characteristic, complemented, Sylow
110	C₂×C₃	—	4	1	no	—
111	C₂×C₃	D₄×D₇	1	1	yes	characteristic
112	D₃	—	7	2	no	—
113	D₃	—	28	1	no	—
114	C₂²	—	2	1	no	—
115	C₂²	—	21	1	no	—
116	C₂²	—	42	1	no	—
117	C₂²	—	84	2	no	—
118	C₄	—	2	1	no	—
119	C₄	—	21	1	no	—
120	C₄	—	42	1	no	—
121	C₄	C₂²×D₂₁	1	1	yes	characteristic
122	C₃	D₇×SD₁₆	1	1	yes	characteristic, complemented, Sylow
123	C₂	—	4	1	no	—
124	C₂	—	21	2	no	—
125	C₂	—	84	1	no	—
126	C₂	D₄×D₂₁	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

Q₈×D₂₁

C₂

yes

maximal, characteristic, complemented

D₄×D₂₁

C₂

yes

maximal, characteristic

C₃×C₇×SD₁₆

C₂

yes

maximal, characteristic, complemented

#66634

C₂

yes

maximal, characteristic, complemented

#66633

C₂

yes

maximal, characteristic, complemented

#66623

C₂

yes

maximal, characteristic, complemented

C₈×D₂₁

C₂

yes

maximal, characteristic, complemented

D₇×SD₁₆

—

maximal, Hall

C₂²×D₂₁

—

C₃×C₇×Q₈

C₂²

yes

characteristic, complemented

C₃×C₇×D₄

C₂²

yes

characteristic

#4268

—

D₈₄

C₂²

yes

characteristic

C₄×D₂₁

—

C₄×D₂₁

C₂²

yes

characteristic

Dic₄₂

—

Dic₄₂

C₂²

yes

characteristic, complemented

C₃×C₇×C₈

C₂²

yes

characteristic, complemented

C₃⋊(C₇⋊C₈)

C₂²

yes

characteristic, complemented

Q₈×D₇

—

D₄×D₇

—

C₇×SD₁₆

D₃

yes

characteristic, complemented

#1143

—

#1142

—

#1132

—

C₈×D₇

—

D₃×SD₁₆

—

maximal, Hall

C₂²×C₃×C₇

—

C₂×D₂₁

—

C₂×D₂₁

—

C₂×D₂₁

D₄

yes

characteristic

C₃×C₄×C₇

—

C₃×C₄×C₇

C₂³

yes

characteristic

Dic₂₁

—

Dic₂₁

D₄

yes

characteristic

C₂²×D₇

—

C₇×Q₈

C₂×D₃

yes

characteristic, complemented

C₇×D₄

C₂×D₃

yes

characteristic

C₂²⋊D₇

—

D₂₈

—

C₄×D₇

—

C₄×D₇

—

Dic₁₄

—

Dic₁₄

—

C₇×C₈

C₂×D₃

yes

characteristic, complemented

C₇⋊C₈

—

D₃×Q₈

—

D₃×D₄

—

C₃×SD₁₆

D₇

yes

characteristic, complemented

#215

—

#214

—

#204

—

D₃×C₈

—

C₂×C₃×C₇

—

C₂×C₃×C₇

C₂×D₄

yes

characteristic

D₂₁

—

D₂₁

SD₁₆

yes

direct product factor

C₂×SD₁₆

—

Sylow

C₂²×C₇

—

C₂×D₇

—

C₂×D₇

—

C₂×D₇

—

C₄×C₇

—

C₄×C₇

C₂²×D₃

yes

characteristic

Dic₇

—

Dic₇

—

C₂²×D₃

—

C₃×Q₈

C₂×D₇

yes

characteristic, complemented

C₃×D₄

C₂×D₇

yes

characteristic

#67

—

D₁₂

—

C₄×D₃

—

C₄×D₃

—

Dic₆

—

Dic₆

—

C₃×C₈

C₂×D₇

yes

characteristic, complemented

C₃⋊C₈

—

C₃×C₇

C₂×SD₁₆

yes

characteristic, complemented, Hall

C₂×Q₈

—

C₂×D₄

—

SD₁₆

—

SD₁₆

—

SD₁₆

—

SD₁₆

D₂₁

yes

characteristic, direct product factor

C₂×C₈

—

C₂×C₇

—

C₂×C₇

D₃×D₄

yes

characteristic

D₇

—

D₇

—

C₂²×C₃

—

C₂×D₃

—

C₂×D₃

—

C₂×D₃

—

C₃×C₄

—

C₃×C₄

C₂²×D₇

yes

characteristic

Dic₃

—

Dic₃

—

C₂³

—

Q₈

—

100

Q₈

—

101

Q₈

C₂×D₂₁

yes

characteristic, complemented

102

D₄

—

103

D₄

—

104

D₄

C₂×D₂₁

yes

characteristic

105

C₂×C₄

—

106

C₂×C₄

—

107

C₈

—

108

C₈

C₂×D₂₁

yes

characteristic, complemented

109

C₇

D₃×SD₁₆

yes

characteristic, complemented, Sylow

110

C₂×C₃

—

111

C₂×C₃

D₄×D₇

yes

characteristic

112

D₃

—

113

D₃

—

114

C₂²

—

115

C₂²

—

116

C₂²

—

117

C₂²

—

118

C₄

—

119

C₄

—

120

C₄

—

121

C₄

C₂²×D₂₁

yes

characteristic

122

C₃

D₇×SD₁₆

yes

characteristic, complemented, Sylow

123

C₂

—

124

C₂

—

125

C₂

—

126

C₂

D₄×D₂₁

yes

characteristic

	subgroup	G/H
center	C₂	D₄×D₂₁
commutator, cosocle	C₃×C₄×C₇	C₂³
Frattini	C₄	C₂²×D₂₁
Fitting	C₃×C₇×SD₁₆	C₂
socle	C₂×C₃×C₇	C₂×D₄
prime radical	D₄×D₂₁	C₂
hypercenter	SD₁₆	D₂₁
hypocenter	C₃×C₇	C₂×SD₁₆
LCS	G ▷ C₃×C₄×C₇ ▷ C₂×C₃×C₇ ▷ C₃×C₇
UCS	C₁ ◁ C₂ ◁ C₄ ◁ SD₁₆

subgroup

G/H

center

C₂

D₄×D₂₁

commutator, cosocle

C₃×C₄×C₇

C₂³

Frattini

C₄

C₂²×D₂₁

Fitting

C₃×C₇×SD₁₆

C₂

socle

C₂×C₃×C₇

C₂×D₄

prime radical

D₄×D₂₁

C₂

hypercenter

SD₁₆

D₂₁

hypocenter

C₃×C₇

C₂×SD₁₆

LCS

G ▷ C₃×C₄×C₇ ▷ C₂×C₃×C₇ ▷ C₃×C₇

UCS

C₁ ◁ C₂ ◁ C₄ ◁ SD₁₆

N	H ≅ G/N	type
1. Q₈×D₂₁	C₂	semidirect
3. C₃×C₇×SD₁₆	C₂	semidirect
4. #66634	C₂	semidirect
5. #66633	C₂	semidirect
6. #66623	C₂	semidirect
7. C₈×D₂₁	C₂	semidirect
10. C₃×C₇×Q₈	C₂²	semidirect
17. Dic₄₂	C₂²	semidirect
18. C₃×C₇×C₈	C₂²	semidirect
19. C₃⋊(C₇⋊C₈)	C₂²	semidirect
22. C₇×SD₁₆	D₃	semidirect
37. C₇×Q₈	C₂×D₃	semidirect
45. C₇×C₈	C₂×D₃	semidirect
49. C₃×SD₁₆	D₇	semidirect
57. D₂₁	SD₁₆	direct (N × H)
68. C₃×Q₈	C₂×D₇	semidirect
76. C₃×C₈	C₂×D₇	semidirect
78. C₃×C₇	C₂×SD₁₆	semidirect
84. SD₁₆	D₂₁	direct (N × H)
101. Q₈	C₂×D₂₁	semidirect
108. C₈	C₂×D₂₁	semidirect
109. C₇	D₃×SD₁₆	semidirect
122. C₃	D₇×SD₁₆	semidirect

H ≅ G/N

type

1. Q₈×D₂₁

C₂

semidirect

3. C₃×C₇×SD₁₆

C₂

semidirect

4. #66634

C₂

semidirect

5. #66633

C₂

semidirect

6. #66623

C₂

semidirect

7. C₈×D₂₁

C₂

semidirect

10. C₃×C₇×Q₈

C₂²

semidirect

17. Dic₄₂

C₂²

semidirect

18. C₃×C₇×C₈

C₂²

semidirect

19. C₃⋊(C₇⋊C₈)

C₂²

semidirect

22. C₇×SD₁₆

D₃

semidirect

37. C₇×Q₈

C₂×D₃

semidirect

45. C₇×C₈

C₂×D₃

semidirect

49. C₃×SD₁₆

D₇

semidirect

57. D₂₁

SD₁₆

direct (N × H)

68. C₃×Q₈

C₂×D₇

semidirect

76. C₃×C₈

C₂×D₇

semidirect

78. C₃×C₇

C₂×SD₁₆

semidirect

84. SD₁₆

D₂₁

direct (N × H)

101. Q₈

C₂×D₂₁

semidirect

108. C₈

C₂×D₂₁

semidirect

109. C₇

D₃×SD₁₆

semidirect

122. C₃

D₇×SD₁₆

semidirect

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

subgroup	index
Q₈×D₂₁	2
D₄×D₂₁	2
C₃×C₇×SD₁₆	2
#66634	2
#66633	2
#66623	2
C₈×D₂₁	2
D₇×SD₁₆	3
D₃×SD₁₆	7

subgroup

index

Q₈×D₂₁

D₄×D₂₁

C₃×C₇×SD₁₆

#66634

#66633

#66623

C₈×D₂₁

D₇×SD₁₆

D₃×SD₁₆

Subgroups — SD₁₆×D₂₁ #10621396

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — SD16×D21 #10621396

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — SD₁₆×D₂₁ #10621396