Group #13052901 — subgroups

Subgroups — C₂×D₇×C₃⋊C₁₆ #13052901

Order 1344.

196 conjugacy classes of subgroups; only the 112 normal subgroups are shown.

normal subgroups only

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

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×C₃×C₈×D₇

C₂

yes

maximal, characteristic

C₂×C₃⋊(C₇⋊C₁₆)

C₂

yes

maximal, characteristic, complemented

C₂×C₇×C₃⋊C₁₆

C₂

yes

maximal, characteristic, complemented

D₇×C₃⋊C₁₆

C₂

yes

maximal, direct product factor

C₂×D₇×C₁₆

—

maximal, Hall

C₂×C₃×C₄×D₇

C₄

yes

characteristic

C₂×C₃×C₇×C₈

C₂²

yes

characteristic

C₂×C₃×C₇⋊C₈

C₄

yes

characteristic

C₃×C₈×D₇

C₄

yes

—

C₃×C₈×D₇

C₂²

yes

—

C₃⋊(C₇⋊C₁₆)

C₂²

yes

complemented

C₇×C₃⋊C₁₆

C₂²

yes

complemented

C₂×C₈×D₇

D₃

yes

characteristic

C₂×C₇×C₁₆

—

C₂×C₇⋊C₁₆

—

D₇×C₁₆

—

C₂²×C₃⋊C₁₆

—

maximal, Hall

C₂²×C₃×D₇

C₈

yes

characteristic

C₂×C₃×C₄×C₇

C₂×C₄

yes

characteristic

C₂×C₃×Dic₇

C₈

yes

characteristic

C₃×C₄×D₇

C₈

yes

—

C₃×C₄×D₇

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

Dic₃

yes

characteristic

C₂×C₇×C₈

C₂×D₃

yes

characteristic

C₂×C₇⋊C₈

Dic₃

yes

characteristic

C₈×D₇

Dic₃

yes

—

C₈×D₇

C₂×D₃

yes

—

C₇×C₁₆

—

C₇⋊C₁₆

—

C₂²×C₃×C₈

—

C₂×C₃⋊C₁₆

—

C₂×C₃⋊C₁₆

—

C₂×C₃⋊C₁₆

D₇

yes

characteristic, direct product factor

C₂²×C₃×C₇

C₂×C₈

yes

characteristic

C₂×C₃×D₇

C₁₆

yes

complemented

C₂×C₃×D₇

C₂×C₈

yes

—

C₃×C₄×C₇

C₂×C₈

yes

characteristic

C₃×C₄×C₇

C₂²×C₄

yes

characteristic

C₃×Dic₇

C₂×C₈

yes

—

C₂²×C₁₆

—

Sylow

C₂²×D₇

C₃⋊C₈

yes

characteristic

C₂×C₄×C₇

C₂×Dic₃

yes

characteristic

C₂×Dic₇

C₃⋊C₈

yes

characteristic

C₄×D₇

C₃⋊C₈

yes

—

C₄×D₇

C₂×Dic₃

yes

—

C₇×C₈

C₂×Dic₃

yes

characteristic

C₇×C₈

C₂²×D₃

yes

characteristic

C₇⋊C₈

C₂×Dic₃

yes

—

C₂²×C₃×C₄

—

C₂×C₃×C₈

—

C₂×C₃×C₈

—

C₂×C₃×C₈

—

C₂×C₃×C₈

C₂×D₇

yes

characteristic

C₃⋊C₁₆

—

C₃⋊C₁₆

C₂×D₇

yes

direct product factor

C₂×C₃×C₇

C₂×C₁₆

yes

complemented

C₂×C₃×C₇

C₂²×C₈

yes

characteristic

C₃×D₇

C₂×C₁₆

yes

complemented

C₂²×C₈

—

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

—

C₂²×C₇

C₂×C₃⋊C₈

yes

characteristic

C₂×D₇

C₃⋊C₁₆

yes

direct product factor

C₂×D₇

C₂×C₃⋊C₈

yes

—

C₄×C₇

C₂×C₃⋊C₈

yes

characteristic

C₄×C₇

C₂²×Dic₃

yes

characteristic

Dic₇

C₂×C₃⋊C₈

yes

—

C₂³×C₃

—

C₂×C₃×C₄

—

C₂×C₃×C₄

—

C₂×C₃×C₄

—

C₂×C₃×C₄

C₄×D₇

yes

characteristic

C₃×C₈

—

C₃×C₈

C₄×D₇

yes

characteristic

C₃×C₈

C₂²×D₇

yes

characteristic

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

C₂×C₃⋊C₁₆

yes

complemented

C₂×C₇

C₂²×C₃⋊C₈

yes

characteristic

D₇

C₂×C₃⋊C₁₆

yes

direct product factor

C₂²×C₃

—

C₂²×C₃

—

C₂²×C₃

C₈×D₇

yes

characteristic

C₃×C₄

—

C₃×C₄

C₈×D₇

yes

characteristic

C₃×C₄

C₂×C₄×D₇

yes

characteristic

C₂³

—

C₂×C₄

—

C₂×C₄

—

100

C₂×C₄

—

101

C₂×C₄

Dic₃×D₇

yes

characteristic

102

C₈

—

103

C₈

Dic₃×D₇

yes

characteristic

104

C₈

C₂×D₃×D₇

yes

characteristic

105

C₇

C₂²×C₃⋊C₁₆

yes

characteristic, complemented, Sylow

106

C₂×C₃

—

107

C₂×C₃

D₇×C₁₆

yes

complemented

108

C₂×C₃

C₂×C₈×D₇

yes

characteristic

109

C₂²

—

110

C₂²

—

111

C₂²

D₇×C₃⋊C₈

yes

characteristic

112

C₄

—

113

C₄

D₇×C₃⋊C₈

yes

characteristic

114

C₄

C₂×Dic₃×D₇

yes

characteristic

115

C₃

C₂×D₇×C₁₆

yes

characteristic, complemented, Sylow

116

C₂

—

117

C₂

D₇×C₃⋊C₁₆

yes

direct product factor

118

C₂

C₂×D₇×C₃⋊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×C₃×D₇	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×C₃×D₇

C₈

cosocle

C₃×C₇×C₈

C₂³

LCS

G ▷ C₃×C₇

UCS

C₁ ◁ C₂×C₈

N	H ≅ G/N	type
2. C₂×C₃⋊(C₇⋊C₁₆)	C₂	semidirect
3. C₂×C₇×C₃⋊C₁₆	C₂	semidirect
4. D₇×C₃⋊C₁₆	C₂	direct (N × H)
11. C₃⋊(C₇⋊C₁₆)	C₂²	semidirect
12. C₇×C₃⋊C₁₆	C₂²	semidirect
36. C₂×C₃⋊C₁₆	D₇	direct (N × H)
38. C₂×C₃×D₇	C₁₆	semidirect
58. C₃⋊C₁₆	C₂×D₇	direct (N × H)
59. C₂×C₃×C₇	C₂×C₁₆	semidirect
61. C₃×D₇	C₂×C₁₆	semidirect
67. C₂×D₇	C₃⋊C₁₆	direct (N × H)
80. C₃×C₇	C₂²×C₁₆	semidirect
88. C₂×C₇	C₂×C₃⋊C₁₆	semidirect
90. D₇	C₂×C₃⋊C₁₆	direct (N × H)
105. C₇	C₂²×C₃⋊C₁₆	semidirect
107. C₂×C₃	D₇×C₁₆	semidirect
115. C₃	C₂×D₇×C₁₆	semidirect
117. C₂	D₇×C₃⋊C₁₆	direct (N × H)

H ≅ G/N

type

2. C₂×C₃⋊(C₇⋊C₁₆)

C₂

semidirect

3. C₂×C₇×C₃⋊C₁₆

C₂

semidirect

4. D₇×C₃⋊C₁₆

C₂

direct (N × H)

11. C₃⋊(C₇⋊C₁₆)

C₂²

semidirect

12. C₇×C₃⋊C₁₆

C₂²

semidirect

36. C₂×C₃⋊C₁₆

D₇

direct (N × H)

38. C₂×C₃×D₇

C₁₆

semidirect

58. C₃⋊C₁₆

C₂×D₇

direct (N × H)

59. C₂×C₃×C₇

C₂×C₁₆

semidirect

61. C₃×D₇

C₂×C₁₆

semidirect

67. C₂×D₇

C₃⋊C₁₆

direct (N × H)

80. C₃×C₇

C₂²×C₁₆

semidirect

88. C₂×C₇

C₂×C₃⋊C₁₆

semidirect

90. D₇

C₂×C₃⋊C₁₆

direct (N × H)

105. C₇

C₂²×C₃⋊C₁₆

semidirect

107. C₂×C₃

D₇×C₁₆

semidirect

115. C₃

C₂×D₇×C₁₆

semidirect

117. C₂

D₇×C₃⋊C₁₆

direct (N × H)

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

subgroup	index
C₂×C₃×C₈×D₇	2
C₂×C₃⋊(C₇⋊C₁₆)	2
C₂×C₇×C₃⋊C₁₆	2
D₇×C₃⋊C₁₆	2
C₂×D₇×C₁₆	3
C₂²×C₃⋊C₁₆	7

subgroup

index

C₂×C₃×C₈×D₇

C₂×C₃⋊(C₇⋊C₁₆)

C₂×C₇×C₃⋊C₁₆

D₇×C₃⋊C₁₆

C₂×D₇×C₁₆

C₂²×C₃⋊C₁₆

Subgroups — C₂×D₇×C₃⋊C₁₆ #13052901

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C2×D7×C3⋊C16 #13052901

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×D₇×C₃⋊C₁₆ #13052901