Group #14583712 — subgroups

Subgroups — #14583712

Order 2752.

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	#13063373	C₂	1	2	yes	maximal, complemented
2	#13063357	C₂	1	4	yes	maximal, complemented
3	#13063356	C₂	1	2	yes	maximal, complemented
4	#13063348	C₂	1	2	yes	maximal, complemented
5	#13063347	C₂	1	1	yes	maximal, characteristic, complemented
6	Q₁₆×D₄₃	C₂	1	2	yes	maximal, complemented
7	#13063331	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×Q₁₆×C₄₃	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂×Q₈×C₄₃	C₂²	1	2	yes	—
10	#10621902	—	2	2	no	—
11	#10621902	C₂²	1	4	yes	—
12	Q₈×D₄₃	—	2	2	no	—
13	Q₈×D₄₃	C₂²	1	4	yes	—
14	#10621898	—	2	4	no	—
15	#10621898	C₂²	1	1	yes	characteristic
16	#10621898	C₂²	1	2	yes	—
17	Q₁₆×C₄₃	C₂²	1	2	yes	—
18	Q₁₆×C₄₃	C₂²	1	2	yes	—
19	C₂×C₈×C₄₃	C₂²	1	1	yes	characteristic
20	#10621885	C₂²	1	4	yes	—
21	#10621884	C₂²	1	4	yes	—
22	#10621877	C₂²	1	1	yes	characteristic
23	Dic₁₇₂	C₂²	1	1	yes	characteristic
24	D₃₄₄	C₂²	1	1	yes	characteristic
25	#10621873	C₂²	1	2	yes	—
26	#10621872	C₂²	1	2	yes	—
27	C₈×D₄₃	C₂²	1	2	yes	—
28	Q₈×C₄₃	—	2	2	no	—
29	Q₈×C₄₃	C₂³	1	4	yes	—
30	C₂×C₄×C₄₃	—	2	2	no	—
31	C₂×C₄×C₄₃	C₂³	1	1	yes	characteristic
32	C₂²⋊D₄₃	—	2	2	no	—
33	C₂²⋊D₄₃	D₄	1	2	yes	—
34	D₁₇₂	—	2	4	no	—
35	D₁₇₂	D₄	1	1	yes	characteristic
36	D₁₇₂	C₂³	1	2	yes	—
37	C₄×D₄₃	—	2	4	no	—
38	C₄×D₄₃	—	2	2	no	—
39	C₄×D₄₃	—	2	4	no	—
40	C₄×D₄₃	C₂³	1	2	yes	—
41	Dic₈₆	—	2	4	no	—
42	Dic₈₆	D₄	1	1	yes	characteristic
43	Dic₈₆	C₂³	1	2	yes	—
44	C₈×C₄₃	C₂³	1	1	yes	characteristic
45	C₈×C₄₃	C₂³	1	1	yes	characteristic
46	C₄₃⋊C₈	C₂³	1	2	yes	—
47	C₂²×C₄₃	C₂×D₄	1	1	yes	characteristic
48	C₂×D₄₃	—	2	2	no	—
49	C₂×D₄₃	C₂×D₄	1	2	yes	—
50	C₄×C₄₃	—	2	4	no	—
51	C₄×C₄₃	C₂×D₄	1	1	yes	characteristic
52	C₄×C₄₃	C₂⁴	1	1	yes	characteristic
53	Dic₄₃	—	2	2	no	—
54	Dic₄₃	C₂×D₄	1	2	yes	—
55	C₂×C₄₃	—	2	1	no	—
56	C₂×C₄₃	C₂²×D₄	1	1	yes	characteristic
57	D₄₃	—	2	2	no	—
58	D₄₃	—	4	2	no	—
59	#578	—	43	1	no	maximal, Sylow
60	C₄₃	#578	1	1	yes	characteristic, complemented, Sylow
61	2₋¹⁺⁴	—	43	2	no	—
62	#137	—	43	2	no	—
63	#137	—	43	4	no	—
64	#135	—	43	2	no	—
65	#135	—	43	1	no	—
66	C₂×Q₁₆	—	43	2	no	—
67	C₂×Q₁₆	D₄₃	1	1	yes	characteristic, complemented
68	#131	—	43	1	no	—
69	#41	—	43	1	no	—
70	#41	—	43	4	no	—
71	#41	—	43	2	no	—
72	#41	—	86	4	no	—
73	#41	—	86	2	no	—
74	C₂×Q₈	—	43	4	no	—
75	C₂×Q₈	—	86	2	no	—
76	C₂×Q₈	C₂×D₄₃	1	2	yes	—
77	Q₁₆	—	43	4	no	—
78	Q₁₆	—	43	1	no	—
79	Q₁₆	C₂×D₄₃	1	2	yes	—
80	Q₁₆	C₂×D₄₃	1	2	yes	—
81	SD₁₆	—	43	4	no	—
82	SD₁₆	—	43	2	no	—
83	D₈	—	43	1	no	—
84	M₁₆	—	43	1	no	—
85	M₁₆	—	43	2	no	—
86	C₂×C₈	—	43	2	no	—
87	C₂×C₈	C₂×D₄₃	1	1	yes	characteristic
88	Q₈	—	2	2	no	—
89	Q₈	—	43	1	no	—
90	Q₈	—	43	2	no	—
91	Q₈	—	86	4	no	—
92	Q₈	C₂²×D₄₃	1	4	yes	—
93	D₄	—	43	2	no	—
94	D₄	—	43	1	no	—
95	D₄	—	43	2	no	—
96	D₄	—	86	4	no	—
97	D₄	—	86	2	no	—
98	C₂×C₄	—	2	2	no	—
99	C₂×C₄	—	43	2	no	—
100	C₂×C₄	—	86	4	no	—
101	C₂×C₄	—	86	4	no	—
102	C₂×C₄	—	86	2	no	—
103	C₂×C₄	C₂²×D₄₃	1	1	yes	characteristic
104	C₈	—	43	2	no	—
105	C₈	C₂²×D₄₃	1	1	yes	characteristic
106	C₈	C₂²×D₄₃	1	1	yes	characteristic
107	C₂²	—	43	2	no	—
108	C₂²	—	86	2	no	—
109	C₂²	D₄×D₄₃	1	1	yes	characteristic
110	C₄	—	2	4	no	—
111	C₄	—	43	2	no	—
112	C₄	—	86	2	no	—
113	C₄	D₄×D₄₃	1	1	yes	characteristic
114	C₄	C₂³×D₄₃	1	1	yes	characteristic
115	C₂	—	2	1	no	—
116	C₂	—	86	2	no	—
117	C₂	—	172	2	no	—
118	C₂	C₂×D₄×D₄₃	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#13063373

C₂

yes

maximal, complemented

#13063357

C₂

yes

maximal, complemented

#13063356

C₂

yes

maximal, complemented

#13063348

C₂

yes

maximal, complemented

#13063347

C₂

yes

maximal, characteristic, complemented

Q₁₆×D₄₃

C₂

yes

maximal, complemented

#13063331

C₂

yes

maximal, characteristic, complemented

C₂×Q₁₆×C₄₃

C₂

yes

maximal, characteristic, complemented

C₂×Q₈×C₄₃

C₂²

yes

—

#10621902

—

#10621902

C₂²

yes

—

Q₈×D₄₃

—

Q₈×D₄₃

C₂²

yes

—

#10621898

—

#10621898

C₂²

yes

characteristic

#10621898

C₂²

yes

—

Q₁₆×C₄₃

C₂²

yes

—

Q₁₆×C₄₃

C₂²

yes

—

C₂×C₈×C₄₃

C₂²

yes

characteristic

#10621885

C₂²

yes

—

#10621884

C₂²

yes

—

#10621877

C₂²

yes

characteristic

Dic₁₇₂

C₂²

yes

characteristic

D₃₄₄

C₂²

yes

characteristic

#10621873

C₂²

yes

—

#10621872

C₂²

yes

—

C₈×D₄₃

C₂²

yes

—

Q₈×C₄₃

—

Q₈×C₄₃

C₂³

yes

—

C₂×C₄×C₄₃

—

C₂×C₄×C₄₃

C₂³

yes

characteristic

C₂²⋊D₄₃

—

C₂²⋊D₄₃

D₄

yes

—

D₁₇₂

—

D₁₇₂

D₄

yes

characteristic

D₁₇₂

C₂³

yes

—

C₄×D₄₃

—

C₄×D₄₃

—

C₄×D₄₃

—

C₄×D₄₃

C₂³

yes

—

Dic₈₆

—

Dic₈₆

D₄

yes

characteristic

Dic₈₆

C₂³

yes

—

C₈×C₄₃

C₂³

yes

characteristic

C₈×C₄₃

C₂³

yes

characteristic

C₄₃⋊C₈

C₂³

yes

—

C₂²×C₄₃

C₂×D₄

yes

characteristic

C₂×D₄₃

—

C₂×D₄₃

C₂×D₄

yes

—

C₄×C₄₃

—

C₄×C₄₃

C₂×D₄

yes

characteristic

C₄×C₄₃

C₂⁴

yes

characteristic

Dic₄₃

—

Dic₄₃

C₂×D₄

yes

—

C₂×C₄₃

—

C₂×C₄₃

C₂²×D₄

yes

characteristic

D₄₃

—

D₄₃

—

#578

—

maximal, Sylow

C₄₃

#578

yes

characteristic, complemented, Sylow

2₋¹⁺⁴

—

—

—

—

—

C₂×Q₁₆

—

C₂×Q₁₆

D₄₃

yes

characteristic, complemented

—

—

—

—

—

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

C₂×D₄₃

yes

—

Q₁₆

—

Q₁₆

—

Q₁₆

C₂×D₄₃

yes

—

Q₁₆

C₂×D₄₃

yes

—

SD₁₆

—

SD₁₆

—

D₈

—

M₁₆

—

M₁₆

—

C₂×C₈

—

C₂×C₈

C₂×D₄₃

yes

characteristic

Q₈

—

Q₈

—

Q₈

—

Q₈

—

Q₈

C₂²×D₄₃

yes

—

D₄

—

D₄

—

D₄

—

D₄

—

D₄

—

C₂×C₄

—

C₂×C₄

—

100

C₂×C₄

—

101

C₂×C₄

—

102

C₂×C₄

—

103

C₂×C₄

C₂²×D₄₃

yes

characteristic

104

C₈

—

105

C₈

C₂²×D₄₃

yes

characteristic

106

C₈

C₂²×D₄₃

yes

characteristic

107

C₂²

—

108

C₂²

—

109

C₂²

D₄×D₄₃

yes

characteristic

110

C₄

—

111

C₄

—

112

C₄

—

113

C₄

D₄×D₄₃

yes

characteristic

114

C₄

C₂³×D₄₃

yes

characteristic

115

C₂

—

116

C₂

—

117

C₂

—

172

—

118

C₂

C₂×D₄×D₄₃

yes

characteristic

	subgroup	G/H
center	C₂	C₂×D₄×D₄₃
commutator, cosocle	C₄×C₄₃	C₂⁴
Frattini	C₄	C₂³×D₄₃
Fitting	C₂×Q₁₆×C₄₃	C₂
socle	C₂×C₄₃	C₂²×D₄
hypercenter	C₂×Q₁₆	D₄₃
hypocenter	C₄₃	#578
LCS	G ▷ C₄×C₄₃ ▷ C₂×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₂×Q₁₆

subgroup

G/H

center

C₂

C₂×D₄×D₄₃

commutator, cosocle

C₄×C₄₃

C₂⁴

Frattini

C₄

C₂³×D₄₃

Fitting

C₂×Q₁₆×C₄₃

C₂

socle

C₂×C₄₃

C₂²×D₄

hypercenter

C₂×Q₁₆

D₄₃

hypocenter

C₄₃

#578

LCS

G ▷ C₄×C₄₃ ▷ C₂×C₄₃ ▷ C₄₃

UCS

C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₂×Q₁₆

N	H ≅ G/N	type
1. #13063373	C₂	semidirect
2. #13063357	C₂	semidirect
3. #13063356	C₂	semidirect
4. #13063348	C₂	semidirect
5. #13063347	C₂	semidirect
6. Q₁₆×D₄₃	C₂	semidirect
7. #13063331	C₂	semidirect
8. C₂×Q₁₆×C₄₃	C₂	semidirect
60. C₄₃	#578	semidirect
67. C₂×Q₁₆	D₄₃	semidirect

H ≅ G/N

type

1. #13063373

C₂

semidirect

2. #13063357

C₂

semidirect

3. #13063356

C₂

semidirect

4. #13063348

C₂

semidirect

5. #13063347

C₂

semidirect

6. Q₁₆×D₄₃

C₂

semidirect

7. #13063331

C₂

semidirect

8. C₂×Q₁₆×C₄₃

C₂

semidirect

60. C₄₃

#578

semidirect

67. C₂×Q₁₆

D₄₃

semidirect

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

subgroup	index
#13063373	2
#13063357	2
#13063356	2
#13063348	2
#13063347	2
Q₁₆×D₄₃	2
#13063331	2
C₂×Q₁₆×C₄₃	2
#578	43

subgroup

index

Q₁₆×D₄₃

C₂×Q₁₆×C₄₃

Subgroups — #14583712

Conjugacy classes of subgroups

Complements

Maximal subgroups