Group #20242102 — subgroups

Subgroups — #20242102

Order 6016.

332 conjugacy classes of subgroups; only the 85 normal subgroups are shown.

normal subgroups only

This group has 332 conjugacy classes of subgroups — too many to draw. Only its 85 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	#15689180	C₂	1	2	yes	maximal, complemented
2	#15688867	C₂	1	4	yes	maximal, complemented
3	C₂×#127×C₄₇	C₂	1	1	yes	maximal, characteristic
4	C₄⋊(C₄₇⋊C₈)	C₂²	1	4	yes	complemented
5	C₂²⋊(C₄₇⋊C₈)	—	2	2	no	—
6	#13093935	—	2	2	no	—
7	C₂²×D₄×C₄₇	—	2	2	no	—
8	C₂²×D₄×C₄₇	C₄	1	2	yes	—
9	#127×C₄₇	—	2	2	no	—
10	#127×C₄₇	C₄	1	2	yes	—
11	#127×C₄₇	C₂²	1	2	yes	—
12	C₂×C₄²×C₄₇	C₂²	1	1	yes	characteristic
13	C₂⁴×C₄₇	—	2	2	no	—
14	C₂×D₄×C₄₇	—	2	2	no	—
15	C₂×D₄×C₄₇	—	2	4	no	—
16	C₂×D₄×C₄₇	—	2	4	no	—
17	C₂×D₄×C₄₇	—	4	8	no	—
18	C₂×D₄×C₄₇	—	4	4	no	—
19	C₂×D₄×C₄₇	C₂×C₄	1	4	yes	—
20	C₂²×C₄×C₄₇	C₂×C₄	1	1	yes	characteristic
21	C₂²×C₄×C₄₇	D₄	1	2	yes	—
22	C₄²×C₄₇	C₂×C₄	1	1	yes	characteristic
23	C₄²×C₄₇	D₄	1	2	yes	—
24	C₄²×C₄₇	C₂³	1	1	yes	characteristic
25	C₂×C₄₇⋊C₈	—	2	4	no	—
26	C₂³×C₄₇	—	2	4	no	—
27	C₂³×C₄₇	—	4	8	no	—
28	C₂³×C₄₇	—	4	4	no	—
29	C₂³×C₄₇	C₂²⋊C₄	1	1	yes	characteristic
30	D₄×C₄₇	—	4	4	no	—
31	D₄×C₄₇	—	4	4	no	—
32	D₄×C₄₇	—	4	8	no	—
33	D₄×C₄₇	—	4	8	no	—
34	C₂×C₄×C₄₇	—	2	2	no	—
35	C₂×C₄×C₄₇	—	2	4	no	—
36	C₂×C₄×C₄₇	C₂²⋊C₄	1	2	yes	—
37	C₂×C₄×C₄₇	C₂²⋊C₄	1	1	yes	characteristic
38	C₂×C₄×C₄₇	C₂²×C₄	1	1	yes	characteristic
39	C₂×C₄×C₄₇	C₂×D₄	1	2	yes	—
40	C₄₇⋊C₈	—	4	4	no	—
41	C₂²×C₄₇	—	2	2	no	—
42	C₂²×C₄₇	—	4	4	no	—
43	C₂²×C₄₇	—	4	8	no	—
44	C₂²×C₄₇	—	8	8	no	—
45	C₂²×C₄₇	#100	1	2	yes	—
46	C₂²×C₄₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
47	C₄×C₄₇	—	2	1	no	—
48	C₄×C₄₇	—	2	1	no	—
49	C₄×C₄₇	—	2	2	no	—
50	C₄×C₄₇	AGL(1,ℤ/8)	1	4	yes	—
51	#1555	—	47	1	no	maximal, Sylow
52	C₂×C₄₇	—	2	2	no	—
53	C₂×C₄₇	—	8	4	no	—
54	C₂×C₄₇	C₂×#100	1	1	yes	characteristic
55	C₂×C₄₇	#418	1	2	yes	—
56	C₂×#127	D₄₇	1	1	yes	characteristic
57	#432	—	47	2	no	—
58	#331	—	47	4	no	—
59	C₄₇	#1555	1	1	yes	characteristic, complemented, Sylow
60	C₂²×D₄	—	2	2	no	—
61	C₂²×D₄	Dic₄₇	1	2	yes	—
62	#127	—	2	2	no	—
63	#127	Dic₄₇	1	2	yes	—
64	#127	C₂×D₄₇	1	2	yes	—
65	C₂×C₄²	C₂×D₄₇	1	1	yes	characteristic
66	C₄⋊C₈	—	47	4	no	—
67	C₂²⋊C₈	—	94	2	no	—
68	#97	—	94	2	no	—
69	C₂⁴	—	2	2	no	—
70	C₂×D₄	—	2	2	no	—
71	C₂×D₄	—	2	4	no	—
72	C₂×D₄	—	2	4	no	—
73	C₂×D₄	—	4	4	no	—
74	C₂×D₄	—	4	8	no	—
75	C₂×D₄	C₂×Dic₄₇	1	4	yes	—
76	C₂²×C₄	C₂×Dic₄₇	1	1	yes	characteristic
77	C₂²×C₄	C₂²⋊D₄₇	1	2	yes	—
78	C₂×C₈	—	94	4	no	—
79	C₄²	C₂×Dic₄₇	1	1	yes	characteristic
80	C₄²	C₂²⋊D₄₇	1	2	yes	—
81	C₄²	C₂²×D₄₇	1	1	yes	characteristic
82	C₂³	—	2	4	no	—
83	C₂³	—	4	8	no	—
84	C₂³	—	4	4	no	—
85	C₂³	C₂²⋊Dic₄₇	1	1	yes	characteristic
86	D₄	—	4	4	no	—
87	D₄	—	4	4	no	—
88	D₄	—	4	8	no	—
89	D₄	—	4	8	no	—
90	C₂×C₄	—	2	2	no	—
91	C₂×C₄	—	2	4	no	—
92	C₂×C₄	C₂²⋊Dic₄₇	1	2	yes	—
93	C₂×C₄	C₂²⋊Dic₄₇	1	1	yes	characteristic
94	C₂×C₄	C₂²×Dic₄₇	1	1	yes	characteristic
95	C₂×C₄	C₂×C₂²⋊D₄₇	1	2	yes	—
96	C₈	—	188	4	no	—
97	C₂²	—	2	2	no	—
98	C₂²	—	4	4	no	—
99	C₂²	—	4	8	no	—
100	C₂²	—	8	8	no	—
101	C₂²	#13093941	1	2	yes	—
102	C₂²	C₂×C₂²⋊Dic₄₇	1	1	yes	characteristic
103	C₄	—	2	1	no	—
104	C₄	—	2	1	no	—
105	C₄	—	2	2	no	—
106	C₄	#13094050	1	4	yes	—
107	C₂	—	2	2	no	—
108	C₂	—	8	4	no	—
109	C₂	C₂×#13093941	1	1	yes	characteristic
110	C₂	#15689128	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#15689180

C₂

yes

maximal, complemented

#15688867

C₂

yes

maximal, complemented

C₂×#127×C₄₇

C₂

yes

maximal, characteristic

C₄⋊(C₄₇⋊C₈)

C₂²

yes

complemented

C₂²⋊(C₄₇⋊C₈)

—

#13093935

—

C₂²×D₄×C₄₇

—

C₂²×D₄×C₄₇

C₄

yes

—

#127×C₄₇

—

#127×C₄₇

C₄

yes

—

#127×C₄₇

C₂²

yes

—

C₂×C₄²×C₄₇

C₂²

yes

characteristic

C₂⁴×C₄₇

—

C₂×D₄×C₄₇

—

C₂×D₄×C₄₇

—

C₂×D₄×C₄₇

—

C₂×D₄×C₄₇

—

C₂×D₄×C₄₇

—

C₂×D₄×C₄₇

C₂×C₄

yes

—

C₂²×C₄×C₄₇

C₂×C₄

yes

characteristic

C₂²×C₄×C₄₇

D₄

yes

—

C₄²×C₄₇

C₂×C₄

yes

characteristic

C₄²×C₄₇

D₄

yes

—

C₄²×C₄₇

C₂³

yes

characteristic

C₂×C₄₇⋊C₈

—

C₂³×C₄₇

—

C₂³×C₄₇

—

C₂³×C₄₇

—

C₂³×C₄₇

C₂²⋊C₄

yes

characteristic

D₄×C₄₇

—

D₄×C₄₇

—

D₄×C₄₇

—

D₄×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

C₂²⋊C₄

yes

—

C₂×C₄×C₄₇

C₂²⋊C₄

yes

characteristic

C₂×C₄×C₄₇

C₂²×C₄

yes

characteristic

C₂×C₄×C₄₇

C₂×D₄

yes

—

C₄₇⋊C₈

—

C₂²×C₄₇

—

C₂²×C₄₇

—

C₂²×C₄₇

—

C₂²×C₄₇

—

C₂²×C₄₇

#100

yes

—

C₂²×C₄₇

C₂×C₂²⋊C₄

yes

characteristic

C₄×C₄₇

—

C₄×C₄₇

—

C₄×C₄₇

—

C₄×C₄₇

AGL(1,ℤ/8)

yes

—

#1555

—

maximal, Sylow

C₂×C₄₇

—

C₂×C₄₇

—

C₂×C₄₇

C₂×#100

yes

characteristic

C₂×C₄₇

#418

yes

—

C₂×#127

D₄₇

yes

characteristic

#432

—

#331

—

C₄₇

#1555

yes

characteristic, complemented, Sylow

C₂²×D₄

—

C₂²×D₄

Dic₄₇

yes

—

#127

—

#127

Dic₄₇

yes

—

#127

C₂×D₄₇

yes

—

C₂×C₄²

C₂×D₄₇

yes

characteristic

C₄⋊C₈

—

C₂²⋊C₈

—

#97

—

C₂⁴

—

C₂×D₄

—

C₂×D₄

—

C₂×D₄

—

C₂×D₄

—

C₂×D₄

—

C₂×D₄

C₂×Dic₄₇

yes

—

C₂²×C₄

C₂×Dic₄₇

yes

characteristic

C₂²×C₄

C₂²⋊D₄₇

yes

—

C₂×C₈

—

C₄²

C₂×Dic₄₇

yes

characteristic

C₄²

C₂²⋊D₄₇

yes

—

C₄²

C₂²×D₄₇

yes

characteristic

C₂³

—

C₂³

—

C₂³

—

C₂³

C₂²⋊Dic₄₇

yes

characteristic

D₄

—

D₄

—

D₄

—

D₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

C₂²⋊Dic₄₇

yes

—

C₂×C₄

C₂²⋊Dic₄₇

yes

characteristic

C₂×C₄

C₂²×Dic₄₇

yes

characteristic

C₂×C₄

C₂×C₂²⋊D₄₇

yes

—

C₈

—

188

—

C₂²

—

C₂²

—

C₂²

—

100

C₂²

—

101

C₂²

#13093941

yes

—

102

C₂²

C₂×C₂²⋊Dic₄₇

yes

characteristic

103

C₄

—

104

C₄

—

105

C₄

—

106

C₄

#13094050

yes

—

107

C₂

—

108

C₂

—

109

C₂

C₂×#13093941

yes

characteristic

110

C₂

#15689128

yes

—

	subgroup	G/H
center	C₂²	C₂×C₂²⋊Dic₄₇
commutator	C₂×C₄×C₄₇	C₂²×C₄
Frattini	C₄²	C₂²×D₄₇
Fitting	C₂×#127×C₄₇	C₂
socle	C₂²×C₄₇	C₂×C₂²⋊C₄
hypercenter	C₂×#127	D₄₇
hypocenter	C₄₇	#1555
LCS	G ▷ C₂×C₄×C₄₇ ▷ C₂²×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×#127

subgroup

G/H

center

C₂²

C₂×C₂²⋊Dic₄₇

commutator

C₂×C₄×C₄₇

C₂²×C₄

Frattini

C₄²

C₂²×D₄₇

Fitting

C₂×#127×C₄₇

C₂

socle

C₂²×C₄₇

C₂×C₂²⋊C₄

D₄₇

C₄₇

G ▷ C₂×C₄×C₄₇ ▷ C₂²×C₄₇ ▷ C₄₇

UCS

C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×#127

N	H ≅ G/N	type
1. #15689180	C₂	semidirect
2. #15688867	C₂	semidirect
4. C₄⋊(C₄₇⋊C₈)	C₂²	semidirect
59. C₄₇	#1555	semidirect

H ≅ G/N

type

1. #15689180

C₂

semidirect

2. #15688867

C₂

semidirect

4. C₄⋊(C₄₇⋊C₈)

C₂²

semidirect

59. C₄₇

#1555

semidirect

Subgroups — #20242102

Conjugacy classes of subgroups

Complements