Group #20241537 — subgroups

Subgroups — #20241537

Order 6016.

112 conjugacy classes of subgroups; only the 23 normal subgroups are shown.

normal subgroups only

This group has 112 conjugacy classes of subgroups — too many to draw. Only its 23 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	#15689300	C₂	1	1	yes	maximal, characteristic
2	#15688920	C₂	1	1	yes	maximal, characteristic, complemented
3	C₄₇×#329	C₂	1	1	yes	maximal, characteristic, complemented
4	#13094069	C₄	1	1	yes	characteristic
5	#13094055	—	2	1	no	—
6	#13093952	—	2	1	no	—
7	#13093943	—	2	1	no	—
8	#13093941	C₄	1	1	yes	characteristic
9	#124×C₄₇	C₂²	1	1	yes	characteristic
10	#97×C₄₇	—	2	1	no	—
11	C₂×Q₈×C₄₇	D₄	1	1	yes	characteristic
12	C₂×D₄×C₄₇	C₂×C₄	1	1	yes	characteristic
13	C₂×C₂²⋊D₄₇	—	4	1	no	—
14	#10625352	—	4	1	no	—
15	D₄×D₄₇	—	4	1	no	—
16	#10625350	—	2	1	no	—
17	C₂×C₈×C₄₇	—	4	1	no	—
18	C₂²⋊C₄×C₄₇	—	4	1	no	—
19	C₄²×C₄₇	D₄	1	1	yes	characteristic
20	#10625337	—	4	1	no	—
21	#10625336	—	4	1	no	—
22	#10625329	—	2	1	no	—
23	#10625329	—	4	1	no	—
24	C₂³×C₄₇	—	2	1	no	—
25	C₂²×D₄₇	—	4	1	no	—
26	Q₈×C₄₇	—	4	1	no	—
27	D₄×C₄₇	—	4	1	no	—
28	C₂×C₄×C₄₇	—	2	1	no	—
29	C₂×C₄×C₄₇	—	2	1	no	—
30	C₂×C₄×C₄₇	C₂²⋊C₄	1	1	yes	characteristic
31	C₂²⋊D₄₇	—	4	2	no	—
32	C₂²⋊D₄₇	—	4	1	no	—
33	C₂×Dic₄₇	—	4	1	no	—
34	D₁₈₈	—	2	1	no	—
35	C₄×D₄₇	—	4	1	no	—
36	Dic₉₄	—	2	1	no	—
37	C₈×C₄₇	—	4	2	no	—
38	C₄₇⋊C₈	—	4	1	no	—
39	C₄₇⋊C₈	—	8	1	no	—
40	C₂²×C₄₇	—	4	1	no	—
41	C₂²×C₄₇	—	8	1	no	—
42	C₂²×C₄₇	#99	1	1	yes	characteristic
43	C₂×D₄₇	—	4	1	no	—
44	C₂×D₄₇	—	16	1	no	—
45	C₄×C₄₇	—	2	1	no	—
46	C₄×C₄₇	—	4	1	no	—
47	C₄×C₄₇	—	4	1	no	—
48	Dic₄₇	—	4	1	no	—
49	#1403	—	47	1	no	maximal, Sylow
50	C₂×C₄₇	—	2	1	no	—
51	C₂×C₄₇	—	8	1	no	—
52	C₂×C₄₇	AΣL(1,𝔽₁₆)	1	1	yes	characteristic
53	D₄₇	—	8	1	no	—
54	#455	—	47	1	no	—
55	#355	—	47	1	no	—
56	#329	D₄₇	1	1	yes	characteristic, complemented
57	C₄₇	#1403	1	1	yes	characteristic, complemented, Sylow
58	H(5,𝔽₂)	—	47	1	no	—
59	#137	—	94	1	no	—
60	#124	C₂×D₄₇	1	1	yes	characteristic
61	#104	—	94	1	no	—
62	#101	—	94	1	no	—
63	#100	—	47	1	no	—
64	#97	—	2	1	no	—
65	#41	—	94	1	no	—
66	#41	—	188	1	no	—
67	C₂×Q₈	D₁₈₈	1	1	yes	characteristic
68	C₂×D₄	—	188	1	no	—
69	C₂×D₄	—	188	1	no	—
70	C₂×D₄	C₄×D₄₇	1	1	yes	characteristic
71	Q₁₆	—	188	1	no	—
72	SD₁₆	—	188	1	no	—
73	M₁₆	—	94	1	no	—
74	M₁₆	—	188	1	no	—
75	C₂×C₈	—	4	1	no	—
76	C₂²⋊C₄	—	4	1	no	—
77	C₄²	C₂²⋊D₄₇	1	1	yes	characteristic
78	C₂³	—	2	1	no	—
79	C₂³	—	188	1	no	—
80	Q₈	—	4	1	no	—
81	Q₈	—	94	1	no	—
82	D₄	—	4	1	no	—
83	D₄	—	94	1	no	—
84	D₄	—	188	2	no	—
85	D₄	—	188	1	no	—
86	C₂×C₄	—	2	1	no	—
87	C₂×C₄	—	2	1	no	—
88	C₂×C₄	—	188	1	no	—
89	C₂×C₄	—	188	1	no	—
90	C₂×C₄	(C₂×D₄₇)⋊C₄	1	1	yes	characteristic
91	C₈	—	4	2	no	—
92	C₈	—	376	1	no	—
93	C₈	—	188	1	no	—
94	C₂²	—	4	1	no	—
95	C₂²	—	8	1	no	—
96	C₂²	—	188	1	no	—
97	C₂²	—	752	1	no	—
98	C₂²	#13093938	1	1	yes	characteristic
99	C₄	—	2	1	no	—
100	C₄	—	4	1	no	—
101	C₄	—	4	1	no	—
102	C₄	—	188	1	no	—
103	C₂	—	2	1	no	—
104	C₂	—	8	1	no	—
105	C₂	—	376	1	no	—
106	C₂	#15688911	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#15689300

C₂

yes

maximal, characteristic

#15688920

C₂

yes

maximal, characteristic, complemented

C₄₇×#329

C₂

yes

maximal, characteristic, complemented

#13094069

C₄

yes

—

—

—

C₄

yes

characteristic

#124×C₄₇

C₂²

yes

characteristic

#97×C₄₇

—

C₂×Q₈×C₄₇

D₄

yes

characteristic

C₂×D₄×C₄₇

C₂×C₄

yes

characteristic

C₂×C₂²⋊D₄₇

—

#10625352

—

D₄×D₄₇

—

#10625350

—

C₂×C₈×C₄₇

—

C₂²⋊C₄×C₄₇

—

C₄²×C₄₇

D₄

yes

—

—

—

—

C₂³×C₄₇

—

C₂²×D₄₇

—

Q₈×C₄₇

—

D₄×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

C₂²⋊C₄

yes

characteristic

C₂²⋊D₄₇

—

C₂²⋊D₄₇

—

C₂×Dic₄₇

—

D₁₈₈

—

C₄×D₄₇

—

Dic₉₄

—

C₈×C₄₇

—

C₄₇⋊C₈

—

C₄₇⋊C₈

—

C₂²×C₄₇

—

C₂²×C₄₇

—

C₂²×C₄₇

#99

yes

characteristic

C₂×D₄₇

—

C₂×D₄₇

—

C₄×C₄₇

—

C₄×C₄₇

—

C₄×C₄₇

—

Dic₄₇

—

#1403

—

maximal, Sylow

C₂×C₄₇

—

C₂×C₄₇

—

C₂×C₄₇

AΣL(1,𝔽₁₆)

yes

characteristic

D₄₇

—

#455

—

#355

—

#329

D₄₇

yes

characteristic, complemented

C₄₇

#1403

yes

characteristic, complemented, Sylow

H(5,𝔽₂)

—

#137

—

#124

C₂×D₄₇

yes

—

—

—

—

—

—

188

—

C₂×Q₈

D₁₈₈

yes

characteristic

C₂×D₄

—

188

—

C₂×D₄

—

188

—

C₂×D₄

C₄×D₄₇

yes

characteristic

Q₁₆

—

188

—

SD₁₆

—

188

—

M₁₆

—

M₁₆

—

188

—

C₂×C₈

—

C₂²⋊C₄

—

C₄²

C₂²⋊D₄₇

yes

characteristic

C₂³

—

C₂³

—

188

—

Q₈

—

Q₈

—

D₄

—

D₄

—

D₄

—

188

—

D₄

—

188

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

188

—

C₂×C₄

—

188

—

C₂×C₄

(C₂×D₄₇)⋊C₄

yes

characteristic

C₈

—

C₈

—

376

—

C₈

—

188

—

C₂²

—

C₂²

—

C₂²

—

188

—

C₂²

—

752

—

C₂²

#13093938

yes

characteristic

C₄

—

100

C₄

—

101

C₄

—

102

C₄

—

188

—

103

C₂

—

104

C₂

—

105

C₂

—

376

—

106

C₂

#15688911

yes

characteristic

	subgroup	G/H
center	C₂	#15688911
commutator	C₂×D₄×C₄₇	C₂×C₄
Frattini	#124	C₂×D₄₇
Fitting	C₄₇×#329	C₂
socle	C₂×C₄₇	AΣL(1,𝔽₁₆)
hypercenter	#329	D₄₇
hypocenter	C₄₇	#1403
LCS	G ▷ C₂×D₄×C₄₇ ▷ C₂×C₄×C₄₇ ▷ C₂²×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂ ◁ C₂² ◁ C₂×C₄ ◁ #124 ◁ #329

subgroup

G/H

center

C₂

#15688911

commutator

C₂×D₄×C₄₇

C₂×C₄

Frattini

#124

C₂×D₄₇

Fitting

C₄₇×#329

C₂

socle

C₂×C₄₇

AΣL(1,𝔽₁₆)

D₄₇

C₄₇

G ▷ C₂×D₄×C₄₇ ▷ C₂×C₄×C₄₇ ▷ C₂²×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇

UCS

C₁ ◁ C₂ ◁ C₂² ◁ C₂×C₄ ◁ #124 ◁ #329

N	H ≅ G/N	type
2. #15688920	C₂	semidirect
3. C₄₇×#329	C₂	semidirect
56. #329	D₄₇	semidirect
57. C₄₇	#1403	semidirect

H ≅ G/N

type

2. #15688920

C₂

semidirect

3. C₄₇×#329

C₂

semidirect

56. #329

D₄₇

semidirect

57. C₄₇

#1403

semidirect

Subgroups — #20241537

Conjugacy classes of subgroups

Complements