Group #20241214 — subgroups

Subgroups — #20241214

Order 6016.

166 conjugacy classes of subgroups; only the 65 normal subgroups are shown.

normal subgroups only

This group has 166 conjugacy classes of subgroups — too many to draw. Only its 65 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	#15688878	C₂	1	2	yes	maximal
2	C₄₇×#391	C₂	1	1	yes	maximal, characteristic
3	C₂²×C₄₇⋊C₈	—	2	2	no	—
4	C₂²×Q₈×C₄₇	C₄	1	1	yes	characteristic
5	C₂×C₄⋊C₄×C₄₇	C₄	1	1	yes	characteristic
6	C₂×C₄²×C₄₇	C₂²	1	1	yes	characteristic
7	#95×C₄₇	—	2	2	no	—
8	C₂×Q₈×C₄₇	—	2	1	no	—
9	C₂×Q₈×C₄₇	—	4	1	no	—
10	C₂×Q₈×C₄₇	—	4	1	no	—
11	C₂×Q₈×C₄₇	C₈	1	2	yes	complemented
12	C₂²×C₄×C₄₇	—	2	1	no	—
13	C₂²×C₄×C₄₇	—	2	1	no	—
14	C₂²×C₄×C₄₇	C₂×C₄	1	1	yes	characteristic
15	C₂²×C₄×C₄₇	D₄	1	2	yes	—
16	C₄⋊C₄×C₄₇	—	4	1	no	—
17	C₄²×C₄₇	—	4	1	no	—
18	C₂×C₄₇⋊C₈	—	2	4	no	—
19	C₂×C₄₇⋊C₈	—	4	2	no	—
20	C₂×C₄₇⋊C₈	—	4	2	no	—
21	C₂³×C₄₇	C₂²⋊C₄	1	1	yes	characteristic
22	Q₈×C₄₇	—	4	2	no	—
23	Q₈×C₄₇	—	4	2	no	—
24	C₂×C₄×C₄₇	—	2	2	no	—
25	C₂×C₄×C₄₇	—	2	1	no	—
26	C₂×C₄×C₄₇	—	2	1	no	—
27	C₂×C₄×C₄₇	—	2	2	no	—
28	C₂×C₄×C₄₇	—	2	2	no	—
29	C₂×C₄×C₄₇	—	4	1	no	—
30	C₂×C₄×C₄₇	—	4	1	no	—
31	C₂×C₄×C₄₇	—	4	1	no	—
32	C₂×C₄×C₄₇	—	4	2	no	—
33	C₂×C₄×C₄₇	C₂×C₈	1	1	yes	characteristic
34	C₂×C₄×C₄₇	M₁₆	1	1	yes	characteristic
35	C₂×C₄×C₄₇	SD₁₆	1	2	yes	—
36	C₂×C₄×C₄₇	Q₁₆	1	2	yes	—
37	C₄₇⋊C₈	—	4	4	no	—
38	C₂²×C₄₇	C₂²⋊C₈	1	1	yes	characteristic
39	C₂²×C₄₇	#99	1	1	yes	characteristic
40	C₂²×C₄₇	#100	1	1	yes	characteristic
41	C₂²×C₄₇	#103	1	2	yes	—
42	C₂²×C₄₇	#104	1	2	yes	—
43	C₄×C₄₇	—	2	1	no	—
44	C₄×C₄₇	—	2	1	no	—
45	C₄×C₄₇	—	2	2	no	—
46	C₄×C₄₇	—	2	2	no	—
47	C₄×C₄₇	—	4	2	no	—
48	C₄×C₄₇	—	8	1	no	—
49	#1272	—	47	1	no	maximal, Sylow
50	C₂×C₄₇	#323	1	1	yes	characteristic
51	C₂×C₄₇	Q₈⋊C₈	1	2	yes	—
52	C₂×C₄₇	#328	1	2	yes	—
53	C₂×C₄₇	#329	1	1	yes	characteristic
54	C₂×C₄₇	#333	1	1	yes	characteristic
55	#391	D₄₇	1	1	yes	characteristic
56	#336	—	47	2	no	—
57	C₄₇	#1272	1	1	yes	characteristic, complemented, Sylow
58	C₂²×Q₈	Dic₄₇	1	1	yes	characteristic
59	C₂²×C₈	—	94	2	no	—
60	C₂×C₄⋊C₄	Dic₄₇	1	1	yes	characteristic
61	C₂×C₄²	C₂×D₄₇	1	1	yes	characteristic
62	#95	—	2	2	no	—
63	C₂×Q₈	—	2	1	no	—
64	C₂×Q₈	—	4	1	no	—
65	C₂×Q₈	—	4	1	no	—
66	C₂×Q₈	C₄₇⋊C₈	1	2	yes	complemented
67	C₂²×C₄	—	2	1	no	—
68	C₂²×C₄	—	2	1	no	—
69	C₂²×C₄	C₂×Dic₄₇	1	1	yes	characteristic
70	C₂²×C₄	C₂²⋊D₄₇	1	2	yes	—
71	C₂×C₈	—	94	4	no	—
72	C₂×C₈	—	188	2	no	—
73	C₂×C₈	—	188	2	no	—
74	C₄⋊C₄	—	4	1	no	—
75	C₄²	—	4	1	no	—
76	C₂³	C₂²⋊Dic₄₇	1	1	yes	characteristic
77	Q₈	—	4	2	no	—
78	Q₈	—	4	2	no	—
79	C₂×C₄	—	2	2	no	—
80	C₂×C₄	—	2	1	no	—
81	C₂×C₄	—	2	1	no	—
82	C₂×C₄	—	2	2	no	—
83	C₂×C₄	—	2	2	no	—
84	C₂×C₄	—	4	1	no	—
85	C₂×C₄	—	4	1	no	—
86	C₂×C₄	—	4	1	no	—
87	C₂×C₄	—	4	2	no	—
88	C₂×C₄	C₂×C₄₇⋊C₈	1	1	yes	characteristic
89	C₂×C₄	#10625329	1	1	yes	characteristic
90	C₂×C₄	#10625336	1	2	yes	—
91	C₂×C₄	#10625337	1	2	yes	—
92	C₈	—	188	4	no	—
93	C₂²	C₂²⋊(C₄₇⋊C₈)	1	1	yes	characteristic
94	C₂²	#13093939	1	1	yes	characteristic
95	C₂²	#13093941	1	1	yes	characteristic
96	C₂²	#13093948	1	2	yes	—
97	C₂²	#13093951	1	2	yes	—
98	C₄	—	2	1	no	—
99	C₄	—	2	1	no	—
100	C₄	—	2	2	no	—
101	C₄	—	2	2	no	—
102	C₄	—	4	2	no	—
103	C₄	—	8	1	no	—
104	C₂	#15688847	1	1	yes	characteristic
105	C₂	Q₈⋊(C₄₇⋊C₈)	1	2	yes	—
106	C₂	#15688860	1	2	yes	—
107	C₂	#15688863	1	1	yes	characteristic
108	C₂	#15688872	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#15688878

C₂

yes

maximal

C₄₇×#391

C₂

yes

maximal, characteristic

C₂²×C₄₇⋊C₈

—

C₂²×Q₈×C₄₇

C₄

yes

characteristic

C₂×C₄⋊C₄×C₄₇

C₄

yes

characteristic

C₂×C₄²×C₄₇

C₂²

yes

characteristic

#95×C₄₇

—

C₂×Q₈×C₄₇

—

C₂×Q₈×C₄₇

—

C₂×Q₈×C₄₇

—

C₂×Q₈×C₄₇

C₈

yes

complemented

C₂²×C₄×C₄₇

—

C₂²×C₄×C₄₇

—

C₂²×C₄×C₄₇

C₂×C₄

yes

characteristic

C₂²×C₄×C₄₇

D₄

yes

—

C₄⋊C₄×C₄₇

—

C₄²×C₄₇

—

C₂×C₄₇⋊C₈

—

C₂×C₄₇⋊C₈

—

C₂×C₄₇⋊C₈

—

C₂³×C₄₇

C₂²⋊C₄

yes

characteristic

Q₈×C₄₇

—

Q₈×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

C₂×C₈

yes

characteristic

C₂×C₄×C₄₇

M₁₆

yes

characteristic

C₂×C₄×C₄₇

SD₁₆

yes

—

C₂×C₄×C₄₇

Q₁₆

yes

—

C₄₇⋊C₈

—

C₂²×C₄₇

C₂²⋊C₈

yes

C₂²×C₄₇

yes

C₂²×C₄₇

yes

C₂²×C₄₇

yes

—

C₂²×C₄₇

#104

yes

—

C₄×C₄₇

—

C₄×C₄₇

—

C₄×C₄₇

—

C₄×C₄₇

—

C₄×C₄₇

—

C₄×C₄₇

—

—

C₂×C₄₇

yes

C₂×C₄₇

Q₈⋊C₈

yes

—

C₂×C₄₇

#328

yes

—

C₂×C₄₇

yes

C₂×C₄₇

yes

D₄₇

yes

characteristic

#336

—

C₄₇

#1272

yes

characteristic, complemented, Sylow

C₂²×Q₈

Dic₄₇

yes

characteristic

C₂²×C₈

—

C₂×C₄⋊C₄

Dic₄₇

yes

characteristic

C₂×C₄²

C₂×D₄₇

yes

characteristic

#95

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

C₄₇⋊C₈

yes

complemented

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

C₂×Dic₄₇

yes

characteristic

C₂²×C₄

C₂²⋊D₄₇

yes

—

C₂×C₈

—

C₂×C₈

—

188

—

C₂×C₈

—

188

—

C₄⋊C₄

—

C₄²

—

C₂³

C₂²⋊Dic₄₇

yes

characteristic

Q₈

—

Q₈

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

C₂×C₄₇⋊C₈

yes

C₂×C₄

yes

C₂×C₄

yes

—

C₂×C₄

#10625337

yes

—

C₈

—

188

—

C₂²

C₂²⋊(C₄₇⋊C₈)

yes

C₂²

yes

C₂²

yes

C₂²

yes

—

C₂²

#13093951

yes

—

C₄

—

C₄

—

100

C₄

—

101

C₄

—

102

C₄

—

103

C₄

—

104

C₂

#15688847

yes

characteristic

105

C₂

Q₈⋊(C₄₇⋊C₈)

yes

—

106

C₂

#15688860

yes

—

107

C₂

#15688863

yes

characteristic

108

C₂

#15688872

yes

characteristic

	subgroup	G/H
center	C₂³	C₂²⋊Dic₄₇
commutator	C₂×C₄×C₄₇	C₂×C₈
Frattini	C₂×C₄²	C₂×D₄₇
Fitting	C₄₇×#391	C₂
socle	C₂³×C₄₇	C₂²⋊C₄
hypercenter	#391	D₄₇
hypocenter	C₄₇	#1272
LCS	G ▷ C₂×C₄×C₄₇ ▷ C₂²×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #391

subgroup

G/H

center

C₂³

C₂²⋊Dic₄₇

commutator

C₂×C₄×C₄₇

C₂×C₈

Frattini

C₂×C₄²

C₂×D₄₇

Fitting

C₄₇×#391

C₂

socle

C₂³×C₄₇

C₂²⋊C₄

D₄₇

C₄₇

G ▷ C₂×C₄×C₄₇ ▷ C₂²×C₄₇ ▷ C₄₇

UCS

C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #391

N	H ≅ G/N	type
11. C₂×Q₈×C₄₇	C₈	semidirect
57. C₄₇	#1272	semidirect
66. C₂×Q₈	C₄₇⋊C₈	semidirect

H ≅ G/N

type

11. C₂×Q₈×C₄₇

C₈

semidirect

57. C₄₇

#1272

semidirect

66. C₂×Q₈

C₄₇⋊C₈

semidirect

Subgroups — #20241214

Conjugacy classes of subgroups

Complements