Group #20241397 — subgroups

Subgroups — #20241397

Order 6016.

126 conjugacy classes of subgroups; only the 41 normal subgroups are shown.

normal subgroups only

This group has 126 conjugacy classes of subgroups — too many to draw. Only its 41 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	1	yes	maximal, characteristic, complemented
2	#15688886	C₂	1	1	yes	maximal, characteristic, complemented
3	C₄₇×#466	C₂	1	1	yes	maximal, characteristic
4	C₂×C₄⋊Dic₄₇	—	2	1	no	—
5	C₂×C₄₇⋊C₁₆	—	2	1	no	—
6	C₂×D₈×C₄₇	C₄	1	1	yes	characteristic, complemented
7	C₂²×C₈×C₄₇	C₂²	1	1	yes	characteristic
8	#121×C₄₇	—	2	1	no	—
9	#107×C₄₇	C₄	1	1	yes	characteristic, complemented
10	#102×C₄₇	—	2	1	no	—
11	C₂×D₄×C₄₇	—	2	1	no	—
12	C₂×D₄×C₄₇	—	4	1	no	—
13	C₂²×C₄×C₄₇	D₄	1	1	yes	characteristic
14	C₂²×Dic₄₇	—	4	1	no	—
15	D₈×C₄₇	—	4	1	no	—
16	C₂×C₈×C₄₇	—	4	1	no	—
17	C₂×C₈×C₄₇	C₂×C₄	1	1	yes	characteristic
18	C₂×C₈×C₄₇	D₄	1	1	yes	characteristic
19	C₄⋊C₄×C₄₇	—	2	1	no	—
20	C₂²⋊C₄×C₄₇	—	4	1	no	—
21	C₄⋊Dic₄₇	—	2	2	no	—
22	C₄⋊Dic₄₇	—	4	1	no	—
23	C₄₇⋊C₁₆	—	4	1	no	—
24	C₂³×C₄₇	—	4	1	no	—
25	C₂³×C₄₇	D₈	1	1	yes	characteristic
26	D₄×C₄₇	—	4	1	no	—
27	D₄×C₄₇	—	4	1	no	—
28	D₄×C₄₇	—	8	1	no	—
29	D₄×C₄₇	—	8	1	no	—
30	C₂×C₄×C₄₇	—	2	1	no	—
31	C₂×C₄×C₄₇	—	2	1	no	—
32	C₂×C₄×C₄₇	—	4	1	no	—
33	C₂×C₄×C₄₇	C₂²⋊C₄	1	1	yes	characteristic
34	C₂×C₄×C₄₇	SD₁₆	1	1	yes	characteristic
35	C₂×Dic₄₇	—	4	2	no	—
36	C₂×Dic₄₇	—	8	1	no	—
37	C₂×Dic₄₇	—	8	1	no	—
38	C₈×C₄₇	—	2	1	no	—
39	C₈×C₄₇	—	2	1	no	—
40	C₂²×C₄₇	—	2	1	no	—
41	C₂²×C₄₇	—	2	1	no	—
42	C₂²×C₄₇	—	8	1	no	—
43	C₂²×C₄₇	—	8	1	no	—
44	C₂²×C₄₇	—	8	1	no	—
45	C₂²×C₄₇	#102	1	1	yes	characteristic
46	C₂²×C₄₇	D₁₆	1	1	yes	characteristic
47	C₂²×C₄₇	SD₃₂	1	1	yes	characteristic
48	C₄×C₄₇	—	2	1	no	—
49	C₄×C₄₇	—	8	1	no	—
50	C₄×C₄₇	#99	1	1	yes	characteristic
51	C₄×C₄₇	#104	1	1	yes	characteristic
52	Dic₄₇	—	8	2	no	—
53	#1347	—	47	1	no	maximal, Sylow
54	C₂×C₄₇	—	2	1	no	—
55	C₂×C₄₇	—	2	1	no	—
56	C₂×C₄₇	—	16	1	no	—
57	C₂×C₄₇	#327	1	1	yes	characteristic
58	C₂×C₄₇	#357	1	1	yes	characteristic
59	C₂×C₄₇	#360	1	1	yes	characteristic
60	#466	D₄₇	1	1	yes	characteristic
61	C₂²⋊C₁₆	—	47	1	no	—
62	#340	—	47	1	no	—
63	C₄₇	#1347	1	1	yes	characteristic, complemented, Sylow
64	C₂×D₈	Dic₄₇	1	1	yes	characteristic, complemented
65	C₂²×C₈	C₂×D₄₇	1	1	yes	characteristic
66	#121	—	2	1	no	—
67	C₂×C₄⋊C₄	—	94	1	no	—
68	C₂×C₁₆	—	94	1	no	—
69	#107	Dic₄₇	1	1	yes	characteristic, complemented
70	#102	—	2	1	no	—
71	C₂×D₄	—	2	1	no	—
72	C₂×D₄	—	4	1	no	—
73	C₂²×C₄	—	188	1	no	—
74	C₂²×C₄	C₂²⋊D₄₇	1	1	yes	characteristic
75	D₈	—	4	1	no	—
76	C₂×C₈	—	4	1	no	—
77	C₂×C₈	C₂×Dic₄₇	1	1	yes	characteristic
78	C₂×C₈	C₂²⋊D₄₇	1	1	yes	characteristic
79	C₄⋊C₄	—	2	1	no	—
80	C₄⋊C₄	—	94	2	no	—
81	C₄⋊C₄	—	188	1	no	—
82	C₂²⋊C₄	—	4	1	no	—
83	C₁₆	—	188	1	no	—
84	C₂³	—	4	1	no	—
85	C₂³	#10625334	1	1	yes	characteristic
86	D₄	—	4	1	no	—
87	D₄	—	4	1	no	—
88	D₄	—	8	1	no	—
89	D₄	—	8	1	no	—
90	C₂×C₄	—	2	1	no	—
91	C₂×C₄	—	2	1	no	—
92	C₂×C₄	—	4	1	no	—
93	C₂×C₄	—	376	1	no	—
94	C₂×C₄	—	376	1	no	—
95	C₂×C₄	—	188	2	no	—
96	C₂×C₄	#10625335	1	1	yes	characteristic
97	C₂×C₄	C₂²⋊Dic₄₇	1	1	yes	characteristic
98	C₈	—	2	1	no	—
99	C₈	—	2	1	no	—
100	C₂²	—	2	1	no	—
101	C₂²	—	2	1	no	—
102	C₂²	—	8	1	no	—
103	C₂²	—	8	1	no	—
104	C₂²	—	8	1	no	—
105	C₂²	#13093945	1	1	yes	characteristic
106	C₂²	#13093965	1	1	yes	characteristic
107	C₂²	#13093968	1	1	yes	characteristic
108	C₄	—	2	1	no	—
109	C₄	—	8	1	no	—
110	C₄	—	376	2	no	—
111	C₄	#13093939	1	1	yes	characteristic
112	C₄	#13093951	1	1	yes	characteristic
113	C₂	—	2	1	no	—
114	C₂	—	2	1	no	—
115	C₂	—	16	1	no	—
116	C₂	#15688857	1	1	yes	characteristic
117	C₂	#15688924	1	1	yes	characteristic
118	C₂	#15688933	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂²⋊(C₄₇⋊C₁₆)

C₂

yes

maximal, characteristic, complemented

#15688886

C₂

yes

maximal, characteristic, complemented

C₄₇×#466

C₂

yes

maximal, characteristic

C₂×C₄⋊Dic₄₇

—

C₂×C₄₇⋊C₁₆

—

C₂×D₈×C₄₇

C₄

yes

characteristic, complemented

C₂²×C₈×C₄₇

C₂²

yes

characteristic

#121×C₄₇

—

#107×C₄₇

C₄

yes

characteristic, complemented

#102×C₄₇

—

C₂×D₄×C₄₇

—

C₂×D₄×C₄₇

—

C₂²×C₄×C₄₇

D₄

yes

characteristic

C₂²×Dic₄₇

—

D₈×C₄₇

—

C₂×C₈×C₄₇

—

C₂×C₈×C₄₇

C₂×C₄

yes

characteristic

C₂×C₈×C₄₇

D₄

yes

characteristic

C₄⋊C₄×C₄₇

—

C₂²⋊C₄×C₄₇

—

C₄⋊Dic₄₇

—

C₄⋊Dic₄₇

—

C₄₇⋊C₁₆

—

C₂³×C₄₇

—

C₂³×C₄₇

D₈

yes

characteristic

D₄×C₄₇

—

D₄×C₄₇

—

D₄×C₄₇

—

D₄×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

C₂²⋊C₄

yes

characteristic

C₂×C₄×C₄₇

SD₁₆

yes

characteristic

C₂×Dic₄₇

—

C₂×Dic₄₇

—

C₂×Dic₄₇

—

C₈×C₄₇

—

C₈×C₄₇

—

C₂²×C₄₇

—

C₂²×C₄₇

—

C₂²×C₄₇

—

C₂²×C₄₇

—

C₂²×C₄₇

—

C₂²×C₄₇

#102

yes

characteristic

C₂²×C₄₇

D₁₆

yes

characteristic

C₂²×C₄₇

SD₃₂

yes

characteristic

C₄×C₄₇

—

C₄×C₄₇

—

C₄×C₄₇

yes

C₄×C₄₇

yes

Dic₄₇

—

#1347

—

maximal, Sylow

C₂×C₄₇

—

C₂×C₄₇

—

C₂×C₄₇

—

C₂×C₄₇

yes

C₂×C₄₇

yes

C₂×C₄₇

yes

D₄₇

yes

characteristic

C₂²⋊C₁₆

—

#340

—

C₄₇

#1347

yes

characteristic, complemented, Sylow

C₂×D₈

Dic₄₇

yes

characteristic, complemented

C₂²×C₈

C₂×D₄₇

yes

characteristic

#121

—

C₂×C₄⋊C₄

—

C₂×C₁₆

—

#107

Dic₄₇

yes

characteristic, complemented

#102

—

C₂×D₄

—

C₂×D₄

—

C₂²×C₄

—

188

—

C₂²×C₄

C₂²⋊D₄₇

yes

characteristic

D₈

—

C₂×C₈

—

C₂×C₈

C₂×Dic₄₇

yes

characteristic

C₂×C₈

C₂²⋊D₄₇

yes

characteristic

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

188

—

C₂²⋊C₄

—

C₁₆

—

188

—

C₂³

—

C₂³

#10625334

yes

characteristic

D₄

—

D₄

—

D₄

—

D₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

376

—

C₂×C₄

—

376

—

C₂×C₄

—

188

—

C₂×C₄

#10625335

yes

characteristic

C₂×C₄

C₂²⋊Dic₄₇

yes

characteristic

C₈

—

C₈

—

100

C₂²

—

101

C₂²

—

102

C₂²

—

103

C₂²

—

104

C₂²

—

105

C₂²

#13093945

yes

characteristic

106

C₂²

#13093965

yes

characteristic

107

C₂²

#13093968

yes

characteristic

108

C₄

—

109

C₄

—

110

C₄

—

376

—

111

C₄

#13093939

yes

characteristic

112

C₄

#13093951

yes

characteristic

113

C₂

—

114

C₂

—

115

C₂

—

116

C₂

#15688857

yes

characteristic

117

C₂

#15688924

yes

characteristic

118

C₂

#15688933

yes

characteristic

	subgroup	G/H
center	C₂²	#13093945
commutator	C₂×C₈×C₄₇	C₂×C₄
Frattini	C₂²×C₈	C₂×D₄₇
Fitting	C₄₇×#466	C₂
socle	C₂²×C₄₇	#102
hypercenter	#466	D₄₇
hypocenter	C₄₇	#1347
LCS	G ▷ C₂×C₈×C₄₇ ▷ C₂×C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ #466

subgroup

G/H

center

C₂²

#13093945

commutator

C₂×C₈×C₄₇

C₂×C₄

Frattini

C₂²×C₈

C₂×D₄₇

Fitting

C₄₇×#466

C₂

C₂²×C₄₇

D₄₇

C₄₇

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

UCS

C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ #466

N	H ≅ G/N	type
1. C₂²⋊(C₄₇⋊C₁₆)	C₂	semidirect
2. #15688886	C₂	semidirect
6. C₂×D₈×C₄₇	C₄	semidirect
9. #107×C₄₇	C₄	semidirect
63. C₄₇	#1347	semidirect
64. C₂×D₈	Dic₄₇	semidirect
69. #107	Dic₄₇	semidirect

H ≅ G/N

type

1. C₂²⋊(C₄₇⋊C₁₆)

C₂

semidirect

2. #15688886

C₂

semidirect

6. C₂×D₈×C₄₇

C₄

semidirect

9. #107×C₄₇

C₄

semidirect

63. C₄₇

#1347

semidirect

64. C₂×D₈

Dic₄₇

semidirect

69. #107

Dic₄₇

semidirect

Subgroups — #20241397

Conjugacy classes of subgroups

Complements