Group #21425917 — subgroups

Subgroups — C₁₆×D₂₃₆ #21425917

Order 7552.

174 conjugacy classes of subgroups; only the 81 normal subgroups are shown.

normal subgroups only

This group has 174 conjugacy classes of subgroups — too many to draw. Only its 81 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×C₁₆×D₅₉	C₂	1	2	yes	maximal, complemented
2	C₈×D₂₃₆	C₂	1	1	yes	maximal, characteristic
3	C₄⋊(C₅₉⋊C₁₆)	C₂	1	1	yes	maximal, characteristic, complemented
4	(C₂×D₅₉)⋊C₁₆	C₂	1	2	yes	maximal, complemented
5	C₄×C₁₆×C₅₉	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×C₈×D₅₉	C₂²	1	2	yes	—
7	C₄×D₂₃₆	C₄	1	1	yes	characteristic
8	C₂×C₅₉⋊C₁₆	C₂²	1	2	yes	—
9	C₁₆×D₅₉	—	2	4	no	—
10	C₄⋊(C₅₉⋊C₈)	C₄	1	1	yes	characteristic
11	(C₂×D₅₉)⋊C₈	C₄	1	2	yes	—
12	C₂×C₁₆×C₅₉	C₂²	1	1	yes	characteristic
13	C₂×C₁₆×C₅₉	C₂²	1	1	yes	characteristic
14	C₄×C₈×C₅₉	C₂²	1	1	yes	characteristic
15	C₂×D₂₃₆	C₈	1	1	yes	characteristic
16	C₂×C₄×D₅₉	C₂×C₄	1	2	yes	—
17	C₂×C₈×C₅₉	C₂×C₄	1	1	yes	characteristic
18	C₂×C₈×C₅₉	C₂³	1	1	yes	characteristic
19	C₄²×C₅₉	C₂×C₄	1	1	yes	characteristic
20	(C₂×D₅₉)⋊C₄	C₈	1	2	yes	—
21	C₄⋊Dic₅₉	C₈	1	1	yes	characteristic
22	C₂×C₅₉⋊C₈	C₂×C₄	1	2	yes	—
23	C₈×D₅₉	—	2	2	no	—
24	C₈×D₅₉	—	2	2	no	—
25	C₁₆×C₅₉	—	2	1	no	—
26	C₁₆×C₅₉	D₄	1	2	yes	complemented
27	C₅₉⋊C₁₆	—	2	2	no	—
28	C₂²×D₅₉	C₂×C₈	1	2	yes	—
29	C₂×C₄×C₅₉	C₂×C₈	1	1	yes	characteristic
30	C₂×C₄×C₅₉	C₂×C₈	1	1	yes	characteristic
31	C₂×C₄×C₅₉	C₂²×C₄	1	1	yes	characteristic
32	C₂×Dic₅₉	C₂×C₈	1	2	yes	—
33	D₂₃₆	C₁₆	1	4	yes	direct product factor
34	C₄×D₅₉	—	2	2	no	—
35	C₄×D₅₉	—	2	2	no	—
36	C₈×C₅₉	—	2	1	no	—
37	C₈×C₅₉	C₂×D₄	1	1	yes	characteristic
38	C₈×C₅₉	#41	1	1	yes	characteristic
39	C₅₉⋊C₈	—	2	2	no	—
40	C₂²×C₅₉	C₂²×C₈	1	1	yes	characteristic
41	C₂×D₅₉	—	2	2	no	—
42	C₂×D₅₉	—	2	2	no	—
43	C₂×D₅₉	C₂×C₁₆	1	4	yes	complemented
44	C₄×C₅₉	—	2	1	no	—
45	C₄×C₅₉	C₂×C₁₆	1	2	yes	complemented
46	C₄×C₅₉	C₄×D₄	1	1	yes	characteristic
47	C₄×C₅₉	#131	1	1	yes	characteristic
48	Dic₅₉	—	2	2	no	—
49	D₄×C₁₆	—	59	1	no	maximal, Sylow
50	C₂×C₅₉	C₈×D₄	1	1	yes	characteristic
51	C₂×C₅₉	C₂²×C₁₆	1	1	yes	characteristic
52	C₂×C₅₉	#504	1	1	yes	characteristic
53	D₅₉	—	2	4	no	—
54	C₂²×C₁₆	—	59	2	no	—
55	C₈×D₄	—	59	1	no	—
56	C₄⋊C₁₆	—	59	1	no	—
57	C₂²⋊C₁₆	—	59	2	no	—
58	C₄×C₁₆	D₅₉	1	1	yes	characteristic, complemented
59	C₅₉	D₄×C₁₆	1	1	yes	characteristic, complemented, Sylow
60	C₂²×C₈	—	59	2	no	—
61	C₄×D₄	—	59	1	no	—
62	C₂×C₁₆	—	59	2	no	—
63	C₂×C₁₆	—	118	4	no	—
64	C₂×C₁₆	C₂×D₅₉	1	1	yes	characteristic
65	C₂×C₁₆	C₂×D₅₉	1	1	yes	characteristic
66	C₄⋊C₈	—	59	1	no	—
67	C₂²⋊C₈	—	59	2	no	—
68	C₄×C₈	C₂×D₅₉	1	1	yes	characteristic
69	C₂×D₄	—	59	1	no	—
70	C₂²×C₄	—	59	2	no	—
71	C₂×C₈	—	59	2	no	—
72	C₂×C₈	—	118	2	no	—
73	C₂×C₈	—	118	2	no	—
74	C₂×C₈	C₄×D₅₉	1	1	yes	characteristic
75	C₂×C₈	C₂²×D₅₉	1	1	yes	characteristic
76	C₄⋊C₄	—	59	1	no	—
77	C₂²⋊C₄	—	59	2	no	—
78	C₄²	C₄×D₅₉	1	1	yes	characteristic
79	C₁₆	—	2	1	no	—
80	C₁₆	—	118	2	no	—
81	C₁₆	D₂₃₆	1	2	yes	direct product factor
82	C₂³	—	59	2	no	—
83	D₄	—	59	4	no	—
84	C₂×C₄	—	59	2	no	—
85	C₂×C₄	—	118	2	no	—
86	C₂×C₄	—	118	2	no	—
87	C₂×C₄	C₈×D₅₉	1	1	yes	characteristic
88	C₂×C₄	C₈×D₅₉	1	1	yes	characteristic
89	C₂×C₄	C₂×C₄×D₅₉	1	1	yes	characteristic
90	C₈	—	2	1	no	—
91	C₈	—	118	2	no	—
92	C₈	C₂×D₂₃₆	1	1	yes	characteristic
93	C₈	#11745685	1	1	yes	characteristic
94	C₂²	—	59	4	no	—
95	C₂²	—	118	2	no	—
96	C₂²	—	118	2	no	—
97	C₂²	C₂×C₈×D₅₉	1	1	yes	characteristic
98	C₄	—	2	1	no	—
99	C₄	—	118	2	no	—
100	C₄	C₁₆×D₅₉	1	2	yes	complemented
101	C₄	C₄×D₂₃₆	1	1	yes	characteristic
102	C₄	#14273733	1	1	yes	characteristic
103	C₂	—	118	4	no	—
104	C₂	C₈×D₂₃₆	1	1	yes	characteristic
105	C₂	C₂×C₁₆×D₅₉	1	1	yes	characteristic
106	C₂	#16837818	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×C₁₆×D₅₉

C₂

yes

maximal, complemented

C₈×D₂₃₆

C₂

yes

maximal, characteristic

C₄⋊(C₅₉⋊C₁₆)

C₂

yes

maximal, characteristic, complemented

(C₂×D₅₉)⋊C₁₆

C₂

yes

maximal, complemented

C₄×C₁₆×C₅₉

C₂

yes

maximal, characteristic, complemented

C₂×C₈×D₅₉

C₂²

yes

—

C₄×D₂₃₆

C₄

yes

characteristic

C₂×C₅₉⋊C₁₆

C₂²

yes

—

C₁₆×D₅₉

—

C₄⋊(C₅₉⋊C₈)

C₄

yes

characteristic

(C₂×D₅₉)⋊C₈

C₄

yes

—

C₂×C₁₆×C₅₉

C₂²

yes

characteristic

C₂×C₁₆×C₅₉

C₂²

yes

characteristic

C₄×C₈×C₅₉

C₂²

yes

characteristic

C₂×D₂₃₆

C₈

yes

characteristic

C₂×C₄×D₅₉

C₂×C₄

yes

—

C₂×C₈×C₅₉

C₂×C₄

yes

characteristic

C₂×C₈×C₅₉

C₂³

yes

characteristic

C₄²×C₅₉

C₂×C₄

yes

characteristic

(C₂×D₅₉)⋊C₄

C₈

yes

—

C₄⋊Dic₅₉

C₈

yes

characteristic

C₂×C₅₉⋊C₈

C₂×C₄

yes

—

C₈×D₅₉

—

C₈×D₅₉

—

C₁₆×C₅₉

—

C₁₆×C₅₉

D₄

yes

complemented

C₅₉⋊C₁₆

—

C₂²×D₅₉

C₂×C₈

yes

—

C₂×C₄×C₅₉

C₂×C₈

yes

characteristic

C₂×C₄×C₅₉

C₂×C₈

yes

characteristic

C₂×C₄×C₅₉

C₂²×C₄

yes

characteristic

C₂×Dic₅₉

C₂×C₈

yes

—

D₂₃₆

C₁₆

yes

direct product factor

C₄×D₅₉

—

C₄×D₅₉

—

C₈×C₅₉

—

C₈×C₅₉

C₂×D₄

yes

characteristic

C₈×C₅₉

#41

yes

characteristic

C₅₉⋊C₈

—

C₂²×C₅₉

C₂²×C₈

yes

characteristic

C₂×D₅₉

—

C₂×D₅₉

—

C₂×D₅₉

C₂×C₁₆

yes

complemented

C₄×C₅₉

—

C₄×C₅₉

C₂×C₁₆

yes

complemented

C₄×C₅₉

C₄×D₄

yes

characteristic

C₄×C₅₉

#131

yes

characteristic

Dic₅₉

—

D₄×C₁₆

—

maximal, Sylow

C₂×C₅₉

C₈×D₄

yes

characteristic

C₂×C₅₉

C₂²×C₁₆

yes

characteristic

C₂×C₅₉

#504

yes

characteristic

D₅₉

—

C₂²×C₁₆

—

C₈×D₄

—

C₄⋊C₁₆

—

C₂²⋊C₁₆

—

C₄×C₁₆

D₅₉

yes

characteristic, complemented

C₅₉

D₄×C₁₆

yes

characteristic, complemented, Sylow

C₂²×C₈

—

C₄×D₄

—

C₂×C₁₆

—

C₂×C₁₆

—

118

—

C₂×C₁₆

C₂×D₅₉

yes

characteristic

C₂×C₁₆

C₂×D₅₉

yes

characteristic

C₄⋊C₈

—

C₂²⋊C₈

—

C₄×C₈

C₂×D₅₉

yes

characteristic

C₂×D₄

—

C₂²×C₄

—

C₂×C₈

—

C₂×C₈

—

118

—

C₂×C₈

—

118

—

C₂×C₈

C₄×D₅₉

yes

characteristic

C₂×C₈

C₂²×D₅₉

yes

characteristic

C₄⋊C₄

—

C₂²⋊C₄

—

C₄²

C₄×D₅₉

yes

characteristic

C₁₆

—

C₁₆

—

118

—

C₁₆

D₂₃₆

yes

direct product factor

C₂³

—

D₄

—

C₂×C₄

—

C₂×C₄

—

118

—

C₂×C₄

—

118

—

C₂×C₄

C₈×D₅₉

yes

characteristic

C₂×C₄

C₈×D₅₉

yes

characteristic

C₂×C₄

C₂×C₄×D₅₉

yes

characteristic

C₈

—

C₈

—

118

—

C₈

C₂×D₂₃₆

yes

characteristic

C₈

#11745685

yes

characteristic

C₂²

—

C₂²

—

118

—

C₂²

—

118

—

C₂²

C₂×C₈×D₅₉

yes

characteristic

C₄

—

C₄

—

118

—

100

C₄

C₁₆×D₅₉

yes

complemented

101

C₄

C₄×D₂₃₆

yes

characteristic

102

C₄

#14273733

yes

characteristic

103

C₂

—

118

—

104

C₂

C₈×D₂₃₆

yes

characteristic

105

C₂

C₂×C₁₆×D₅₉

yes

characteristic

106

C₂

#16837818

yes

characteristic

	subgroup	G/H
center	C₂×C₁₆	C₂×D₅₉
commutator	C₂×C₅₉	C₂²×C₁₆
Frattini	C₂×C₈	C₂²×D₅₉
Fitting	C₄×C₁₆×C₅₉	C₂
socle	C₂²×C₅₉	C₂²×C₈
hypercenter	C₄×C₁₆	D₅₉
hypocenter	C₅₉	D₄×C₁₆
LCS	G ▷ C₂×C₅₉ ▷ C₅₉
UCS	C₁ ◁ C₂×C₁₆ ◁ C₄×C₁₆

subgroup

G/H

center

C₂×C₁₆

C₂×D₅₉

commutator

C₂×C₅₉

C₂²×C₁₆

Frattini

C₂×C₈

C₂²×D₅₉

Fitting

C₄×C₁₆×C₅₉

C₂

socle

C₂²×C₅₉

C₂²×C₈

hypercenter

C₄×C₁₆

D₅₉

hypocenter

C₅₉

D₄×C₁₆

LCS

G ▷ C₂×C₅₉ ▷ C₅₉

UCS

C₁ ◁ C₂×C₁₆ ◁ C₄×C₁₆

N	H ≅ G/N	type
1. C₂×C₁₆×D₅₉	C₂	semidirect
3. C₄⋊(C₅₉⋊C₁₆)	C₂	semidirect
4. (C₂×D₅₉)⋊C₁₆	C₂	semidirect
5. C₄×C₁₆×C₅₉	C₂	semidirect
26. C₁₆×C₅₉	D₄	semidirect
33. D₂₃₆	C₁₆	direct (N × H)
43. C₂×D₅₉	C₂×C₁₆	semidirect
45. C₄×C₅₉	C₂×C₁₆	semidirect
58. C₄×C₁₆	D₅₉	semidirect
59. C₅₉	D₄×C₁₆	semidirect
81. C₁₆	D₂₃₆	direct (N × H)
100. C₄	C₁₆×D₅₉	semidirect

H ≅ G/N

type

1. C₂×C₁₆×D₅₉

C₂

semidirect

3. C₄⋊(C₅₉⋊C₁₆)

C₂

semidirect

4. (C₂×D₅₉)⋊C₁₆

C₂

semidirect

5. C₄×C₁₆×C₅₉

C₂

semidirect

26. C₁₆×C₅₉

D₄

semidirect

33. D₂₃₆

C₁₆

direct (N × H)

43. C₂×D₅₉

C₂×C₁₆

semidirect

45. C₄×C₅₉

C₂×C₁₆

semidirect

58. C₄×C₁₆

D₅₉

semidirect

59. C₅₉

D₄×C₁₆

semidirect

81. C₁₆

D₂₃₆

direct (N × H)

100. C₄

C₁₆×D₅₉

semidirect

Subgroups — C₁₆×D₂₃₆ #21425917

Conjugacy classes of subgroups

Complements

Subgroups — C16×D236 #21425917

Conjugacy classes of subgroups

Complements

Subgroups — C₁₆×D₂₃₆ #21425917