Group #20241312 — subgroups

Subgroups — #20241312

Order 6016.

140 conjugacy classes of subgroups; only the 49 normal subgroups are shown.

normal subgroups only

This group has 140 conjugacy classes of subgroups — too many to draw. Only its 49 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	C₂²⋊(C₄₇⋊C₁₆)	C₂	1	2	yes	maximal, complemented
2	C₂×C₂²⋊C₈×C₄₇	C₂	1	1	yes	maximal, characteristic
3	C₂×C₄₇⋊C₁₆	—	2	2	no	—
4	C₂³×C₄×C₄₇	C₄	1	1	yes	characteristic
5	C₂²×C₈×C₄₇	C₄	1	1	yes	characteristic
6	C₂²×C₈×C₄₇	C₂²	1	1	yes	characteristic
7	C₂²⋊C₈×C₄₇	—	2	2	no	—
8	C₂⁴×C₄₇	C₈	1	1	yes	characteristic
9	C₂²×C₄×C₄₇	—	2	1	no	—
10	C₂²×C₄×C₄₇	—	2	1	no	—
11	C₂²×C₄×C₄₇	—	4	1	no	—
12	C₂²×C₄×C₄₇	—	4	1	no	—
13	C₂²×C₄×C₄₇	C₈	1	1	yes	characteristic
14	C₂²×C₄×C₄₇	C₂×C₄	1	1	yes	characteristic
15	C₂×C₈×C₄₇	—	2	1	no	—
16	C₂×C₈×C₄₇	—	4	1	no	—
17	C₂×C₈×C₄₇	—	4	1	no	—
18	C₂×C₈×C₄₇	D₄	1	2	yes	—
19	C₄₇⋊C₁₆	—	4	2	no	—
20	C₂³×C₄₇	—	2	1	no	—
21	C₂³×C₄₇	—	2	1	no	—
22	C₂³×C₄₇	—	4	1	no	—
23	C₂³×C₄₇	—	4	1	no	—
24	C₂³×C₄₇	C₁₆	1	2	yes	complemented
25	C₂³×C₄₇	C₂×C₈	1	1	yes	characteristic
26	C₂×C₄×C₄₇	—	2	1	no	—
27	C₂×C₄×C₄₇	—	2	1	no	—
28	C₂×C₄×C₄₇	—	2	1	no	—
29	C₂×C₄×C₄₇	—	4	2	no	—
30	C₂×C₄×C₄₇	—	4	1	no	—
31	C₂×C₄×C₄₇	—	4	1	no	—
32	C₂×C₄×C₄₇	—	4	1	no	—
33	C₂×C₄×C₄₇	C₂²⋊C₄	1	1	yes	characteristic
34	C₂×C₄×C₄₇	M₁₆	1	1	yes	characteristic
35	C₈×C₄₇	—	2	2	no	—
36	C₈×C₄₇	—	4	1	no	—
37	C₂²×C₄₇	—	2	1	no	—
38	C₂²×C₄₇	—	2	1	no	—
39	C₂²×C₄₇	—	2	2	no	—
40	C₂²×C₄₇	—	4	2	no	—
41	C₂²×C₄₇	—	4	2	no	—
42	C₂²×C₄₇	—	4	1	no	—
43	C₂²×C₄₇	—	4	1	no	—
44	C₂²×C₄₇	C₂²⋊C₈	1	1	yes	characteristic
45	C₂²×C₄₇	C₂×C₁₆	1	1	yes	characteristic
46	C₂²×C₄₇	M₃₂	1	1	yes	characteristic
47	C₄×C₄₇	—	2	1	no	—
48	C₄×C₄₇	—	4	1	no	—
49	C₄×C₄₇	#99	1	1	yes	characteristic
50	C₄×C₄₇	#100	1	1	yes	characteristic
51	#1314	—	47	1	no	maximal, Sylow
52	C₂×C₄₇	—	2	1	no	—
53	C₂×C₄₇	—	2	1	no	—
54	C₂×C₄₇	—	4	2	no	—
55	C₂×C₄₇	#323	1	1	yes	characteristic
56	C₂×C₄₇	C₂²⋊C₁₆	1	1	yes	characteristic
57	C₂×C₄₇	#349	1	1	yes	characteristic
58	C₂×C₂²⋊C₈	D₄₇	1	1	yes	characteristic
59	C₂²⋊C₁₆	—	47	2	no	—
60	C₄₇	#1314	1	1	yes	characteristic, complemented, Sylow
61	C₂³×C₄	Dic₄₇	1	1	yes	characteristic
62	C₂²×C₈	Dic₄₇	1	1	yes	characteristic
63	C₂²×C₈	C₂×D₄₇	1	1	yes	characteristic
64	C₂×C₁₆	—	94	2	no	—
65	C₂²⋊C₈	—	2	2	no	—
66	C₂⁴	C₄₇⋊C₈	1	1	yes	characteristic
67	C₂²×C₄	—	2	1	no	—
68	C₂²×C₄	—	2	1	no	—
69	C₂²×C₄	—	4	1	no	—
70	C₂²×C₄	—	4	1	no	—
71	C₂²×C₄	C₄₇⋊C₈	1	1	yes	characteristic
72	C₂²×C₄	C₂×Dic₄₇	1	1	yes	characteristic
73	C₂×C₈	—	2	1	no	—
74	C₂×C₈	—	4	1	no	—
75	C₂×C₈	—	4	1	no	—
76	C₂×C₈	C₂²⋊D₄₇	1	2	yes	—
77	C₁₆	—	188	2	no	—
78	C₂³	—	2	1	no	—
79	C₂³	—	2	1	no	—
80	C₂³	—	4	1	no	—
81	C₂³	—	4	1	no	—
82	C₂³	C₄₇⋊C₁₆	1	2	yes	complemented
83	C₂³	C₂×C₄₇⋊C₈	1	1	yes	characteristic
84	C₂×C₄	—	2	1	no	—
85	C₂×C₄	—	2	1	no	—
86	C₂×C₄	—	2	1	no	—
87	C₂×C₄	—	4	2	no	—
88	C₂×C₄	—	4	1	no	—
89	C₂×C₄	—	4	1	no	—
90	C₂×C₄	—	4	1	no	—
91	C₂×C₄	#10625329	1	1	yes	characteristic
92	C₂×C₄	C₂²⋊Dic₄₇	1	1	yes	characteristic
93	C₈	—	2	2	no	—
94	C₈	—	4	1	no	—
95	C₂²	—	2	1	no	—
96	C₂²	—	2	1	no	—
97	C₂²	—	2	2	no	—
98	C₂²	—	4	2	no	—
99	C₂²	—	4	2	no	—
100	C₂²	—	4	1	no	—
101	C₂²	—	4	1	no	—
102	C₂²	C₂²⋊(C₄₇⋊C₈)	1	1	yes	characteristic
103	C₂²	C₂×C₄₇⋊C₁₆	1	1	yes	characteristic
104	C₂²	#13093964	1	1	yes	characteristic
105	C₄	—	2	1	no	—
106	C₄	—	4	1	no	—
107	C₄	#13093939	1	1	yes	characteristic
108	C₄	#13093941	1	1	yes	characteristic
109	C₂	—	2	1	no	—
110	C₂	—	2	1	no	—
111	C₂	—	4	2	no	—
112	C₂	#15688847	1	1	yes	characteristic
113	C₂	C₂²⋊(C₄₇⋊C₁₆)	1	1	yes	characteristic
114	C₂	#15688905	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂²⋊(C₄₇⋊C₁₆)

C₂

yes

maximal, complemented

C₂×C₂²⋊C₈×C₄₇

C₂

yes

maximal, characteristic

C₂×C₄₇⋊C₁₆

—

C₂³×C₄×C₄₇

C₄

yes

characteristic

C₂²×C₈×C₄₇

C₄

yes

characteristic

C₂²×C₈×C₄₇

C₂²

yes

characteristic

C₂²⋊C₈×C₄₇

—

C₂⁴×C₄₇

C₈

yes

characteristic

C₂²×C₄×C₄₇

—

C₂²×C₄×C₄₇

—

C₂²×C₄×C₄₇

—

C₂²×C₄×C₄₇

—

C₂²×C₄×C₄₇

C₈

yes

characteristic

C₂²×C₄×C₄₇

C₂×C₄

yes

characteristic

C₂×C₈×C₄₇

—

C₂×C₈×C₄₇

—

C₂×C₈×C₄₇

—

C₂×C₈×C₄₇

D₄

yes

—

C₄₇⋊C₁₆

—

C₂³×C₄₇

—

C₂³×C₄₇

—

C₂³×C₄₇

—

C₂³×C₄₇

—

C₂³×C₄₇

C₁₆

yes

complemented

C₂³×C₄₇

C₂×C₈

yes

characteristic

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

C₂²⋊C₄

yes

characteristic

C₂×C₄×C₄₇

M₁₆

yes

characteristic

C₈×C₄₇

—

C₈×C₄₇

—

C₂²×C₄₇

—

C₂²×C₄₇

—

C₂²×C₄₇

—

C₂²×C₄₇

—

C₂²×C₄₇

—

C₂²×C₄₇

—

C₂²×C₄₇

—

C₂²×C₄₇

C₂²⋊C₈

yes

characteristic

C₂²×C₄₇

C₂×C₁₆

yes

characteristic

C₂²×C₄₇

M₃₂

yes

characteristic

C₄×C₄₇

—

C₄×C₄₇

—

C₄×C₄₇

yes

C₄×C₄₇

yes

—

C₂×C₄₇

—

C₂×C₄₇

—

C₂×C₄₇

—

C₂×C₄₇

#323

yes

characteristic

C₂×C₄₇

C₂²⋊C₁₆

yes

characteristic

C₂×C₄₇

#349

yes

characteristic

C₂×C₂²⋊C₈

D₄₇

yes

characteristic

C₂²⋊C₁₆

—

C₄₇

#1314

yes

characteristic, complemented, Sylow

C₂³×C₄

Dic₄₇

yes

characteristic

C₂²×C₈

Dic₄₇

yes

characteristic

C₂²×C₈

C₂×D₄₇

yes

characteristic

C₂×C₁₆

—

C₂²⋊C₈

—

C₂⁴

C₄₇⋊C₈

yes

characteristic

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

C₄₇⋊C₈

yes

characteristic

C₂²×C₄

C₂×Dic₄₇

yes

characteristic

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

C₂²⋊D₄₇

yes

—

C₁₆

—

188

—

C₂³

—

C₂³

—

C₂³

—

C₂³

—

C₂³

C₄₇⋊C₁₆

yes

complemented

C₂³

C₂×C₄₇⋊C₈

yes

characteristic

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

#10625329

yes

characteristic

C₂×C₄

C₂²⋊Dic₄₇

yes

characteristic

C₈

—

C₈

—

C₂²

—

C₂²

—

C₂²

—

C₂²

—

C₂²

—

100

C₂²

—

101

C₂²

—

102

C₂²

C₂²⋊(C₄₇⋊C₈)

yes

characteristic

103

C₂²

C₂×C₄₇⋊C₁₆

yes

characteristic

104

C₂²

#13093964

yes

characteristic

105

C₄

—

106

C₄

—

107

C₄

#13093939

yes

characteristic

108

C₄

#13093941

yes

characteristic

109

C₂

—

110

C₂

—

111

C₂

—

112

C₂

#15688847

yes

characteristic

113

C₂

C₂²⋊(C₄₇⋊C₁₆)

yes

characteristic

114

C₂

#15688905

yes

characteristic

	subgroup	G/H
center	C₂×C₄	C₂²⋊Dic₄₇
commutator	C₂²×C₄₇	C₂×C₁₆
Frattini	C₂²×C₈	C₂×D₄₇
Fitting	C₂×C₂²⋊C₈×C₄₇	C₂
socle	C₂²×C₄₇	C₂²⋊C₈
hypercenter	C₂×C₂²⋊C₈	D₄₇
hypocenter	C₄₇	#1314
LCS	G ▷ C₂²×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₈ ◁ C₂×C₂²⋊C₈

subgroup

G/H

center

C₂×C₄

C₂²⋊Dic₄₇

commutator

C₂²×C₄₇

C₂×C₁₆

Frattini

C₂²×C₈

C₂×D₄₇

Fitting

C₂×C₂²⋊C₈×C₄₇

C₂

socle

C₂²×C₄₇

C₂²⋊C₈

hypercenter

C₂×C₂²⋊C₈

D₄₇

hypocenter

C₄₇

#1314

LCS

G ▷ C₂²×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇

UCS

C₁ ◁ C₂×C₄ ◁ C₂²×C₈ ◁ C₂×C₂²⋊C₈

N	H ≅ G/N	type
1. C₂²⋊(C₄₇⋊C₁₆)	C₂	semidirect
24. C₂³×C₄₇	C₁₆	semidirect
60. C₄₇	#1314	semidirect
82. C₂³	C₄₇⋊C₁₆	semidirect

H ≅ G/N

type

1. C₂²⋊(C₄₇⋊C₁₆)

C₂

semidirect

24. C₂³×C₄₇

C₁₆

semidirect

60. C₄₇

#1314

semidirect

82. C₂³

C₄₇⋊C₁₆

semidirect

Subgroups — #20241312

Conjugacy classes of subgroups

Complements