Group #20241404 — subgroups

Subgroups — #20241404

Order 6016.

114 conjugacy classes of subgroups; only the 35 normal subgroups are shown.

normal subgroups only

This group has 114 conjugacy classes of subgroups — too many to draw. Only its 35 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	#15689359	C₂	1	1	yes	maximal, characteristic
2	#15688888	C₂	1	1	yes	maximal, characteristic, complemented
3	C₄₇×C₂²⋊C₁₆	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×D₃₇₆	C₄	1	1	yes	characteristic
5	C₂×#10625329	—	2	1	no	—
6	#13093998	—	2	1	no	—
7	#13093958	C₄	1	1	yes	characteristic
8	#13093946	—	2	1	no	—
9	C₂²×C₈×C₄₇	C₂²	1	1	yes	characteristic
10	C₂×C₁₆×C₄₇	—	2	1	no	—
11	C₂²×C₄×C₄₇	D₄	1	1	yes	characteristic
12	C₂×C₂²⋊D₄₇	—	4	1	no	—
13	C₂×D₁₈₈	—	2	1	no	—
14	C₂×C₈×C₄₇	—	4	1	no	—
15	C₂×C₈×C₄₇	C₂×C₄	1	1	yes	characteristic
16	C₂×C₈×C₄₇	D₄	1	1	yes	characteristic
17	(C₂×D₄₇)⋊C₄	—	4	1	no	—
18	C₄⋊Dic₄₇	—	2	1	no	—
19	#10625329	—	2	2	no	—
20	#10625329	—	4	1	no	—
21	C₂×C₄₇⋊C₈	—	4	1	no	—
22	D₃₇₆	—	4	1	no	—
23	C₁₆×C₄₇	—	4	1	no	—
24	C₂³×C₄₇	SD₁₆	1	1	yes	characteristic
25	C₂²×D₄₇	—	4	1	no	—
26	C₂×C₄×C₄₇	—	2	1	no	—
27	C₂×C₄×C₄₇	—	2	1	no	—
28	C₂×C₄×C₄₇	C₂²⋊C₄	1	1	yes	characteristic
29	C₂×C₄×C₄₇	D₈	1	1	yes	characteristic
30	C₂²⋊D₄₇	—	8	2	no	—
31	C₂×Dic₄₇	—	4	1	no	—
32	D₁₈₈	—	4	1	no	—
33	D₁₈₈	—	4	1	no	—
34	C₈×C₄₇	—	2	1	no	—
35	C₈×C₄₇	—	2	1	no	—
36	C₄₇⋊C₈	—	4	2	no	—
37	C₂²×C₄₇	—	2	1	no	—
38	C₂²×C₄₇	—	2	1	no	—
39	C₂²×C₄₇	—	2	1	no	—
40	C₂²×C₄₇	#102	1	1	yes	characteristic
41	C₂×D₄₇	—	8	1	no	—
42	C₂×D₄₇	—	8	1	no	—
43	C₂×D₄₇	—	8	1	no	—
44	C₄×C₄₇	—	2	1	no	—
45	C₄×C₄₇	#99	1	1	yes	characteristic
46	C₄×C₄₇	#104	1	1	yes	characteristic
47	Dic₄₇	—	8	1	no	—
48	#1349	—	47	1	no	maximal, Sylow
49	C₂×C₄₇	—	4	1	no	—
50	C₂×C₄₇	#327	1	1	yes	characteristic
51	C₂×C₄₇	#359	1	1	yes	characteristic
52	C₂×C₄₇	#361	1	1	yes	characteristic
53	D₄₇	—	16	1	no	—
54	#466	—	47	1	no	—
55	C₂²⋊C₁₆	D₄₇	1	1	yes	characteristic, complemented
56	#341	—	47	1	no	—
57	C₄₇	#1349	1	1	yes	characteristic, complemented, Sylow
58	C₂×D₈	—	47	1	no	—
59	C₂×M₁₆	—	94	1	no	—
60	C₂²×C₈	C₂×D₄₇	1	1	yes	characteristic
61	#121	—	94	1	no	—
62	C₂×C₁₆	—	2	1	no	—
63	#107	—	47	1	no	—
64	#102	—	94	1	no	—
65	C₂×D₄	—	94	1	no	—
66	C₂×D₄	—	188	1	no	—
67	C₂²×C₄	D₁₈₈	1	1	yes	characteristic
68	D₈	—	188	1	no	—
69	M₁₆	—	94	2	no	—
70	M₁₆	—	188	1	no	—
71	C₂×C₈	—	4	1	no	—
72	C₂×C₈	—	188	1	no	—
73	C₂×C₈	C₄×D₄₇	1	1	yes	characteristic
74	C₂×C₈	C₂²⋊D₄₇	1	1	yes	characteristic
75	C₄⋊C₄	—	94	1	no	—
76	C₂²⋊C₄	—	188	1	no	—
77	C₁₆	—	4	1	no	—
78	C₂³	—	188	1	no	—
79	C₂³	#10625325	1	1	yes	characteristic
80	D₄	—	376	2	no	—
81	D₄	—	188	1	no	—
82	D₄	—	188	1	no	—
83	C₂×C₄	—	2	1	no	—
84	C₂×C₄	—	2	1	no	—
85	C₂×C₄	—	188	1	no	—
86	C₂×C₄	D₃₇₆	1	1	yes	characteristic
87	C₂×C₄	(C₂×D₄₇)⋊C₄	1	1	yes	characteristic
88	C₈	—	2	1	no	—
89	C₈	—	2	1	no	—
90	C₈	—	188	2	no	—
91	C₂²	—	2	1	no	—
92	C₂²	—	2	1	no	—
93	C₂²	—	2	1	no	—
94	C₂²	—	376	1	no	—
95	C₂²	—	376	1	no	—
96	C₂²	—	376	1	no	—
97	C₂²	#13093946	1	1	yes	characteristic
98	C₄	—	2	1	no	—
99	C₄	—	376	1	no	—
100	C₄	#13093938	1	1	yes	characteristic
101	C₄	#13093950	1	1	yes	characteristic
102	C₂	—	4	1	no	—
103	C₂	—	752	1	no	—
104	C₂	#15688858	1	1	yes	characteristic
105	C₂	#15688931	1	1	yes	characteristic
106	C₂	#15688937	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#15689359

C₂

yes

maximal, characteristic

#15688888

C₂

yes

maximal, characteristic, complemented

C₄₇×C₂²⋊C₁₆

C₂

yes

maximal, characteristic, complemented

C₂×D₃₇₆

C₄

yes

—

—

C₄

yes

characteristic

#13093946

—

C₂²×C₈×C₄₇

C₂²

yes

characteristic

C₂×C₁₆×C₄₇

—

C₂²×C₄×C₄₇

D₄

yes

characteristic

C₂×C₂²⋊D₄₇

—

C₂×D₁₈₈

—

C₂×C₈×C₄₇

—

C₂×C₈×C₄₇

C₂×C₄

yes

characteristic

C₂×C₈×C₄₇

D₄

yes

characteristic

(C₂×D₄₇)⋊C₄

—

C₄⋊Dic₄₇

—

#10625329

—

#10625329

—

C₂×C₄₇⋊C₈

—

D₃₇₆

—

C₁₆×C₄₇

—

C₂³×C₄₇

SD₁₆

yes

characteristic

C₂²×D₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

C₂²⋊C₄

yes

characteristic

C₂×C₄×C₄₇

D₈

yes

characteristic

C₂²⋊D₄₇

—

C₂×Dic₄₇

—

D₁₈₈

—

D₁₈₈

—

C₈×C₄₇

—

C₈×C₄₇

—

C₄₇⋊C₈

—

C₂²×C₄₇

—

C₂²×C₄₇

—

C₂²×C₄₇

—

C₂²×C₄₇

#102

yes

characteristic

C₂×D₄₇

—

C₂×D₄₇

—

C₂×D₄₇

—

C₄×C₄₇

—

C₄×C₄₇

yes

C₄×C₄₇

yes

Dic₄₇

—

#1349

—

maximal, Sylow

C₂×C₄₇

—

C₂×C₄₇

yes

C₂×C₄₇

yes

C₂×C₄₇

yes

D₄₇

—

#466

—

C₂²⋊C₁₆

D₄₇

yes

characteristic, complemented

#341

—

C₄₇

#1349

yes

characteristic, complemented, Sylow

C₂×D₈

—

C₂×M₁₆

—

C₂²×C₈

C₂×D₄₇

yes

characteristic

#121

—

C₂×C₁₆

—

#107

—

#102

—

C₂×D₄

—

C₂×D₄

—

188

—

C₂²×C₄

D₁₈₈

yes

characteristic

D₈

—

188

—

M₁₆

—

M₁₆

—

188

—

C₂×C₈

—

C₂×C₈

—

188

—

C₂×C₈

C₄×D₄₇

yes

characteristic

C₂×C₈

C₂²⋊D₄₇

yes

characteristic

C₄⋊C₄

—

C₂²⋊C₄

—

188

—

C₁₆

—

C₂³

—

188

—

C₂³

#10625325

yes

characteristic

D₄

—

376

—

D₄

—

188

—

D₄

—

188

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

188

—

C₂×C₄

D₃₇₆

yes

characteristic

C₂×C₄

(C₂×D₄₇)⋊C₄

yes

characteristic

C₈

—

C₈

—

C₈

—

188

—

C₂²

—

C₂²

—

C₂²

—

C₂²

—

376

—

C₂²

—

376

—

C₂²

—

376

—

C₂²

#13093946

yes

characteristic

C₄

—

C₄

—

376

—

100

C₄

#13093938

yes

characteristic

101

C₄

#13093950

yes

characteristic

102

C₂

—

103

C₂

—

752

—

104

C₂

#15688858

yes

characteristic

105

C₂

#15688931

yes

characteristic

106

C₂

#15688937

yes

characteristic

	subgroup	G/H
center	C₂²	#13093946
commutator	C₂×C₈×C₄₇	C₂×C₄
Frattini	C₂²×C₈	C₂×D₄₇
Fitting	C₄₇×C₂²⋊C₁₆	C₂
socle	C₂²×C₄₇	#102
hypercenter	C₂²⋊C₁₆	D₄₇
hypocenter	C₄₇	#1349
LCS	G ▷ C₂×C₈×C₄₇ ▷ C₂×C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ C₂²⋊C₁₆

subgroup

G/H

center

C₂²

#13093946

commutator

C₂×C₈×C₄₇

C₂×C₄

Frattini

C₂²×C₈

C₂×D₄₇

Fitting

C₄₇×C₂²⋊C₁₆

C₂

socle

C₂²×C₄₇

#102

hypercenter

C₂²⋊C₁₆

D₄₇

hypocenter

C₄₇

#1349

LCS

G ▷ C₂×C₈×C₄₇ ▷ C₂×C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇

UCS

C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ C₂²⋊C₁₆

N	H ≅ G/N	type
2. #15688888	C₂	semidirect
3. C₄₇×C₂²⋊C₁₆	C₂	semidirect
55. C₂²⋊C₁₆	D₄₇	semidirect
57. C₄₇	#1349	semidirect

H ≅ G/N

type

2. #15688888

C₂

semidirect

3. C₄₇×C₂²⋊C₁₆

C₂

semidirect

55. C₂²⋊C₁₆

D₄₇

semidirect

57. C₄₇

#1349

semidirect

Subgroups — #20241404

Conjugacy classes of subgroups

Complements