Group #16838215 — subgroups

Subgroups — #16838215

Order 3776.

248 conjugacy classes of subgroups; only the 99 normal subgroups are shown.

normal subgroups only

This group has 248 conjugacy classes of subgroups — too many to draw. Only its 99 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	#14273775	C₂	1	2	yes	maximal, complemented
2	#14273759	C₂	1	4	yes	maximal, complemented
3	#14273758	C₂	1	2	yes	maximal, complemented
4	#14273750	C₂	1	2	yes	maximal, complemented
5	#14273749	C₂	1	1	yes	maximal, characteristic, complemented
6	Q₁₆×D₅₉	C₂	1	2	yes	maximal, complemented
7	#14273733	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×Q₁₆×C₅₉	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂×Q₈×C₅₉	C₂²	1	2	yes	—
10	#11745689	—	2	2	no	—
11	#11745689	C₂²	1	4	yes	—
12	Q₈×D₅₉	—	2	2	no	—
13	Q₈×D₅₉	C₂²	1	4	yes	—
14	#11745685	—	2	4	no	—
15	#11745685	C₂²	1	1	yes	characteristic
16	#11745685	C₂²	1	2	yes	—
17	Q₁₆×C₅₉	C₂²	1	2	yes	—
18	Q₁₆×C₅₉	C₂²	1	2	yes	—
19	C₂×C₈×C₅₉	C₂²	1	1	yes	characteristic
20	#11745672	C₂²	1	4	yes	—
21	#11745671	C₂²	1	4	yes	—
22	#11745664	C₂²	1	1	yes	characteristic
23	Dic₂₃₆	C₂²	1	1	yes	characteristic
24	D₄₇₂	C₂²	1	1	yes	characteristic
25	#11745660	C₂²	1	2	yes	—
26	#11745659	C₂²	1	2	yes	—
27	C₈×D₅₉	C₂²	1	2	yes	—
28	Q₈×C₅₉	—	2	2	no	—
29	Q₈×C₅₉	C₂³	1	4	yes	—
30	C₂×C₄×C₅₉	—	2	2	no	—
31	C₂×C₄×C₅₉	C₂³	1	1	yes	characteristic
32	C₂²⋊D₅₉	—	2	2	no	—
33	C₂²⋊D₅₉	D₄	1	2	yes	—
34	D₂₃₆	—	2	4	no	—
35	D₂₃₆	D₄	1	1	yes	characteristic
36	D₂₃₆	C₂³	1	2	yes	—
37	C₄×D₅₉	—	2	4	no	—
38	C₄×D₅₉	—	2	2	no	—
39	C₄×D₅₉	—	2	4	no	—
40	C₄×D₅₉	C₂³	1	2	yes	—
41	Dic₁₁₈	—	2	4	no	—
42	Dic₁₁₈	D₄	1	1	yes	characteristic
43	Dic₁₁₈	C₂³	1	2	yes	—
44	C₈×C₅₉	C₂³	1	1	yes	characteristic
45	C₈×C₅₉	C₂³	1	1	yes	characteristic
46	C₅₉⋊C₈	C₂³	1	2	yes	—
47	C₂²×C₅₉	C₂×D₄	1	1	yes	characteristic
48	C₂×D₅₉	—	2	2	no	—
49	C₂×D₅₉	C₂×D₄	1	2	yes	—
50	C₄×C₅₉	—	2	4	no	—
51	C₄×C₅₉	C₂×D₄	1	1	yes	characteristic
52	C₄×C₅₉	C₂⁴	1	1	yes	characteristic
53	Dic₅₉	—	2	2	no	—
54	Dic₅₉	C₂×D₄	1	2	yes	—
55	C₂×C₅₉	—	2	1	no	—
56	C₂×C₅₉	C₂²×D₄	1	1	yes	characteristic
57	D₅₉	—	2	2	no	—
58	D₅₉	—	4	2	no	—
59	#578	—	59	1	no	maximal, Sylow
60	C₅₉	#578	1	1	yes	characteristic, complemented, Sylow
61	2₋¹⁺⁴	—	59	2	no	—
62	#137	—	59	2	no	—
63	#137	—	59	4	no	—
64	#135	—	59	2	no	—
65	#135	—	59	1	no	—
66	C₂×Q₁₆	—	59	2	no	—
67	C₂×Q₁₆	D₅₉	1	1	yes	characteristic, complemented
68	#131	—	59	1	no	—
69	#41	—	59	1	no	—
70	#41	—	59	4	no	—
71	#41	—	59	2	no	—
72	#41	—	118	4	no	—
73	#41	—	118	2	no	—
74	C₂×Q₈	—	59	4	no	—
75	C₂×Q₈	—	118	2	no	—
76	C₂×Q₈	C₂×D₅₉	1	2	yes	—
77	Q₁₆	—	59	4	no	—
78	Q₁₆	—	59	1	no	—
79	Q₁₆	C₂×D₅₉	1	2	yes	—
80	Q₁₆	C₂×D₅₉	1	2	yes	—
81	SD₁₆	—	59	4	no	—
82	SD₁₆	—	59	2	no	—
83	D₈	—	59	1	no	—
84	M₁₆	—	59	1	no	—
85	M₁₆	—	59	2	no	—
86	C₂×C₈	—	59	2	no	—
87	C₂×C₈	C₂×D₅₉	1	1	yes	characteristic
88	Q₈	—	2	2	no	—
89	Q₈	—	59	1	no	—
90	Q₈	—	59	2	no	—
91	Q₈	—	118	4	no	—
92	Q₈	C₂²×D₅₉	1	4	yes	—
93	D₄	—	59	2	no	—
94	D₄	—	59	1	no	—
95	D₄	—	59	2	no	—
96	D₄	—	118	4	no	—
97	D₄	—	118	2	no	—
98	C₂×C₄	—	2	2	no	—
99	C₂×C₄	—	59	2	no	—
100	C₂×C₄	—	118	4	no	—
101	C₂×C₄	—	118	4	no	—
102	C₂×C₄	—	118	2	no	—
103	C₂×C₄	C₂²×D₅₉	1	1	yes	characteristic
104	C₈	—	59	2	no	—
105	C₈	C₂²×D₅₉	1	1	yes	characteristic
106	C₈	C₂²×D₅₉	1	1	yes	characteristic
107	C₂²	—	59	2	no	—
108	C₂²	—	118	2	no	—
109	C₂²	D₄×D₅₉	1	1	yes	characteristic
110	C₄	—	2	4	no	—
111	C₄	—	59	2	no	—
112	C₄	—	118	2	no	—
113	C₄	D₄×D₅₉	1	1	yes	characteristic
114	C₄	C₂³×D₅₉	1	1	yes	characteristic
115	C₂	—	2	1	no	—
116	C₂	—	118	2	no	—
117	C₂	—	236	2	no	—
118	C₂	C₂×D₄×D₅₉	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#14273775

C₂

yes

maximal, complemented

#14273759

C₂

yes

maximal, complemented

#14273758

C₂

yes

maximal, complemented

#14273750

C₂

yes

maximal, complemented

#14273749

C₂

yes

maximal, characteristic, complemented

Q₁₆×D₅₉

C₂

yes

maximal, complemented

#14273733

C₂

yes

maximal, characteristic, complemented

C₂×Q₁₆×C₅₉

C₂

yes

maximal, characteristic, complemented

C₂×Q₈×C₅₉

C₂²

yes

—

#11745689

—

#11745689

C₂²

yes

—

Q₈×D₅₉

—

Q₈×D₅₉

C₂²

yes

—

#11745685

—

#11745685

C₂²

yes

characteristic

#11745685

C₂²

yes

—

Q₁₆×C₅₉

C₂²

yes

—

Q₁₆×C₅₉

C₂²

yes

—

C₂×C₈×C₅₉

C₂²

yes

characteristic

#11745672

C₂²

yes

—

#11745671

C₂²

yes

—

#11745664

C₂²

yes

characteristic

Dic₂₃₆

C₂²

yes

characteristic

D₄₇₂

C₂²

yes

characteristic

#11745660

C₂²

yes

—

#11745659

C₂²

yes

—

C₈×D₅₉

C₂²

yes

—

Q₈×C₅₉

—

Q₈×C₅₉

C₂³

yes

—

C₂×C₄×C₅₉

—

C₂×C₄×C₅₉

C₂³

yes

characteristic

C₂²⋊D₅₉

—

C₂²⋊D₅₉

D₄

yes

—

D₂₃₆

—

D₂₃₆

D₄

yes

characteristic

D₂₃₆

C₂³

yes

—

C₄×D₅₉

—

C₄×D₅₉

—

C₄×D₅₉

—

C₄×D₅₉

C₂³

yes

—

Dic₁₁₈

—

Dic₁₁₈

D₄

yes

characteristic

Dic₁₁₈

C₂³

yes

—

C₈×C₅₉

C₂³

yes

characteristic

C₈×C₅₉

C₂³

yes

characteristic

C₅₉⋊C₈

C₂³

yes

—

C₂²×C₅₉

C₂×D₄

yes

characteristic

C₂×D₅₉

—

C₂×D₅₉

C₂×D₄

yes

—

C₄×C₅₉

—

C₄×C₅₉

C₂×D₄

yes

characteristic

C₄×C₅₉

C₂⁴

yes

characteristic

Dic₅₉

—

Dic₅₉

C₂×D₄

yes

—

C₂×C₅₉

—

C₂×C₅₉

C₂²×D₄

yes

characteristic

D₅₉

—

D₅₉

—

#578

—

maximal, Sylow

C₅₉

#578

yes

characteristic, complemented, Sylow

2₋¹⁺⁴

—

—

—

—

—

C₂×Q₁₆

—

C₂×Q₁₆

D₅₉

yes

characteristic, complemented

—

—

—

—

—

118

—

#41

—

118

—

C₂×Q₈

—

C₂×Q₈

—

118

—

C₂×Q₈

C₂×D₅₉

yes

—

Q₁₆

—

Q₁₆

—

Q₁₆

C₂×D₅₉

yes

—

Q₁₆

C₂×D₅₉

yes

—

SD₁₆

—

SD₁₆

—

D₈

—

M₁₆

—

M₁₆

—

C₂×C₈

—

C₂×C₈

C₂×D₅₉

yes

characteristic

Q₈

—

Q₈

—

Q₈

—

Q₈

—

118

—

Q₈

C₂²×D₅₉

yes

—

D₄

—

D₄

—

D₄

—

D₄

—

118

—

D₄

—

118

—

C₂×C₄

—

C₂×C₄

—

100

C₂×C₄

—

118

—

101

C₂×C₄

—

118

—

102

C₂×C₄

—

118

—

103

C₂×C₄

C₂²×D₅₉

yes

characteristic

104

C₈

—

105

C₈

C₂²×D₅₉

yes

characteristic

106

C₈

C₂²×D₅₉

yes

characteristic

107

C₂²

—

108

C₂²

—

118

—

109

C₂²

D₄×D₅₉

yes

characteristic

110

C₄

—

111

C₄

—

112

C₄

—

118

—

113

C₄

D₄×D₅₉

yes

characteristic

114

C₄

C₂³×D₅₉

yes

characteristic

115

C₂

—

116

C₂

—

118

—

117

C₂

—

236

—

118

C₂

C₂×D₄×D₅₉

yes

characteristic

	subgroup	G/H
center	C₂	C₂×D₄×D₅₉
commutator, cosocle	C₄×C₅₉	C₂⁴
Frattini	C₄	C₂³×D₅₉
Fitting	C₂×Q₁₆×C₅₉	C₂
socle	C₂×C₅₉	C₂²×D₄
hypercenter	C₂×Q₁₆	D₅₉
hypocenter	C₅₉	#578
LCS	G ▷ C₄×C₅₉ ▷ C₂×C₅₉ ▷ C₅₉
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₂×Q₁₆

subgroup

G/H

center

C₂

C₂×D₄×D₅₉

commutator, cosocle

C₄×C₅₉

C₂⁴

Frattini

C₄

C₂³×D₅₉

Fitting

C₂×Q₁₆×C₅₉

C₂

socle

C₂×C₅₉

C₂²×D₄

hypercenter

C₂×Q₁₆

D₅₉

hypocenter

C₅₉

#578

LCS

G ▷ C₄×C₅₉ ▷ C₂×C₅₉ ▷ C₅₉

UCS

C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₂×Q₁₆

N	H ≅ G/N	type
1. #14273775	C₂	semidirect
2. #14273759	C₂	semidirect
3. #14273758	C₂	semidirect
4. #14273750	C₂	semidirect
5. #14273749	C₂	semidirect
6. Q₁₆×D₅₉	C₂	semidirect
7. #14273733	C₂	semidirect
8. C₂×Q₁₆×C₅₉	C₂	semidirect
60. C₅₉	#578	semidirect
67. C₂×Q₁₆	D₅₉	semidirect

H ≅ G/N

type

1. #14273775

C₂

semidirect

2. #14273759

C₂

semidirect

3. #14273758

C₂

semidirect

4. #14273750

C₂

semidirect

5. #14273749

C₂

semidirect

6. Q₁₆×D₅₉

C₂

semidirect

7. #14273733

C₂

semidirect

8. C₂×Q₁₆×C₅₉

C₂

semidirect

60. C₅₉

#578

semidirect

67. C₂×Q₁₆

D₅₉

semidirect

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

subgroup	index
#14273775	2
#14273759	2
#14273758	2
#14273750	2
#14273749	2
Q₁₆×D₅₉	2
#14273733	2
C₂×Q₁₆×C₅₉	2
#578	59

subgroup

index

Q₁₆×D₅₉

C₂×Q₁₆×C₅₉

Subgroups — #16838215

Conjugacy classes of subgroups

Complements

Maximal subgroups