Group #20241225 — subgroups

Subgroups — #20241225

Order 6016.

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

normal subgroups only

This group has 172 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	#15689176	C₂	1	1	yes	maximal, characteristic
2	C₄×#10625329	C₂	1	1	yes	maximal, characteristic
3	C₂×C₄×C₈×C₄₇	C₂	1	1	yes	maximal, characteristic
4	C₂×#10625329	C₄	1	2	yes	—
5	C₄⋊(C₄₇⋊C₈)	C₄	1	2	yes	—
6	C₂²⋊(C₄₇⋊C₈)	—	2	1	no	—
7	#13093935	—	2	1	no	—
8	C₄×C₄₇⋊C₈	—	2	1	no	—
9	C₄×C₄₇⋊C₈	—	2	1	no	—
10	C₂²×C₈×C₄₇	—	2	1	no	—
11	C₂×C₄²×C₄₇	C₂²	1	1	yes	characteristic
12	C₄×C₈×C₄₇	—	2	1	no	—
13	C₄×C₈×C₄₇	C₄	1	2	yes	—
14	C₂²×C₄×C₄₇	C₂×C₄	1	1	yes	characteristic
15	C₂²×C₄×C₄₇	D₄	1	1	yes	characteristic
16	C₂²×C₄×C₄₇	Q₈	1	1	yes	characteristic
17	C₂×C₈×C₄₇	—	2	4	no	—
18	C₂×C₈×C₄₇	—	2	2	no	—
19	C₄²×C₄₇	C₂×C₄	1	1	yes	characteristic
20	C₄²×C₄₇	C₂×C₄	1	1	yes	characteristic
21	C₄²×C₄₇	D₄	1	1	yes	characteristic
22	C₄²×C₄₇	D₄	1	1	yes	characteristic
23	#10625329	—	2	2	no	—
24	#10625329	C₈	1	4	yes	complemented
25	C₂×C₄₇⋊C₈	—	2	2	no	—
26	C₂×C₄₇⋊C₈	—	2	2	no	—
27	C₂³×C₄₇	C₄⋊C₄	1	1	yes	characteristic
28	C₂×C₄×C₄₇	—	2	2	no	—
29	C₂×C₄×C₄₇	—	2	2	no	—
30	C₂×C₄×C₄₇	C₄²	1	1	yes	characteristic
31	C₂×C₄×C₄₇	C₂²⋊C₄	1	1	yes	characteristic
32	C₂×C₄×C₄₇	C₂²⋊C₄	1	1	yes	characteristic
33	C₂×C₄×C₄₇	C₂²⋊C₄	1	1	yes	characteristic
34	C₂×C₄×C₄₇	C₄⋊C₄	1	1	yes	characteristic
35	C₂×C₄×C₄₇	C₄⋊C₄	1	1	yes	characteristic
36	C₂×C₄×C₄₇	C₂×C₈	1	2	yes	—
37	C₂×C₄×C₄₇	M₁₆	1	2	yes	—
38	C₈×C₄₇	—	2	4	no	—
39	C₄₇⋊C₈	—	2	4	no	—
40	C₄₇⋊C₈	—	4	2	no	—
41	C₂²×C₄₇	—	2	1	no	—
42	C₂²×C₄₇	—	2	1	no	—
43	C₂²×C₄₇	#95	1	1	yes	characteristic
44	C₂²×C₄₇	C₄⋊C₈	1	2	yes	—
45	C₄×C₄₇	—	2	1	no	—
46	C₄×C₄₇	—	2	1	no	—
47	C₄×C₄₇	C₄×C₈	1	1	yes	characteristic
48	C₄×C₄₇	#97	1	1	yes	characteristic
49	C₄×C₄₇	C₂²⋊C₈	1	2	yes	—
50	C₄×C₄₇	#104	1	2	yes	—
51	C₄×C₄₇	#108	1	2	yes	—
52	#1278	—	47	1	no	maximal, Sylow
53	C₂×C₄₇	—	2	2	no	—
54	C₂×C₄₇	#336	1	1	yes	characteristic
55	C₂×C₄₇	#339	1	1	yes	characteristic
56	C₂×C₄₇	#341	1	1	yes	characteristic
57	#431	—	47	1	no	—
58	C₄×M₁₆	—	47	1	no	—
59	C₂×C₄×C₈	D₄₇	1	1	yes	characteristic
60	C₄₇	#1278	1	1	yes	characteristic, complemented, Sylow
61	C₂×M₁₆	—	47	2	no	—
62	C₂²×C₈	—	2	1	no	—
63	C₂×C₄²	C₂×D₄₇	1	1	yes	characteristic
64	C₄⋊C₈	—	47	2	no	—
65	C₂²⋊C₈	—	94	1	no	—
66	#97	—	94	1	no	—
67	C₄×C₈	—	2	1	no	—
68	C₄×C₈	—	94	1	no	—
69	C₄×C₈	—	94	1	no	—
70	C₄×C₈	Dic₄₇	1	2	yes	—
71	C₂²×C₄	Dic₉₄	1	1	yes	characteristic
72	C₂²×C₄	C₄×D₄₇	1	1	yes	characteristic
73	C₂²×C₄	C₂²⋊D₄₇	1	1	yes	characteristic
74	M₁₆	—	47	4	no	—
75	M₁₆	—	94	2	no	—
76	C₂×C₈	—	2	4	no	—
77	C₂×C₈	—	2	2	no	—
78	C₂×C₈	—	94	2	no	—
79	C₂×C₈	—	94	2	no	—
80	C₄²	C₄×D₄₇	1	1	yes	characteristic
81	C₄²	D₁₈₈	1	1	yes	characteristic
82	C₄²	C₂×Dic₄₇	1	1	yes	characteristic
83	C₄²	C₂²⋊D₄₇	1	1	yes	characteristic
84	C₂³	Dic₄₇⋊C₄	1	1	yes	characteristic
85	C₂×C₄	—	2	2	no	—
86	C₂×C₄	—	2	2	no	—
87	C₂×C₄	C₈×D₄₇	1	2	yes	—
88	C₂×C₄	#10625324	1	2	yes	—
89	C₂×C₄	C₄×Dic₄₇	1	1	yes	characteristic
90	C₂×C₄	Dic₄₇⋊C₄	1	1	yes	characteristic
91	C₂×C₄	C₄⋊Dic₄₇	1	1	yes	characteristic
92	C₂×C₄	(C₂×D₄₇)⋊C₄	1	1	yes	characteristic
93	C₂×C₄	(C₂×D₄₇)⋊C₄	1	1	yes	characteristic
94	C₂×C₄	C₂²⋊Dic₄₇	1	1	yes	characteristic
95	C₈	—	2	4	no	—
96	C₈	—	94	4	no	—
97	C₈	—	188	2	no	—
98	C₂²	—	2	1	no	—
99	C₂²	—	2	1	no	—
100	C₂²	C₄₇⋊#95	1	1	yes	characteristic
101	C₂²	Dic₄₇⋊C₈	1	2	yes	—
102	C₄	—	2	1	no	—
103	C₄	—	2	1	no	—
104	C₄	C₈×Dic₄₇	1	1	yes	characteristic
105	C₄	#13093934	1	1	yes	characteristic
106	C₄	(C₂×D₄₇)⋊C₈	1	2	yes	—
107	C₄	#13093952	1	2	yes	—
108	C₄	#13093960	1	2	yes	—
109	C₂	—	2	2	no	—
110	C₂	#15688877	1	1	yes	characteristic
111	C₂	#15688883	1	1	yes	characteristic
112	C₂	#15688888	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#15689176

C₂

yes

maximal, characteristic

C₄×#10625329

C₂

yes

maximal, characteristic

C₂×C₄×C₈×C₄₇

C₂

yes

maximal, characteristic

C₂×#10625329

C₄

yes

—

C₄⋊(C₄₇⋊C₈)

C₄

yes

—

C₂²⋊(C₄₇⋊C₈)

—

#13093935

—

C₄×C₄₇⋊C₈

—

C₄×C₄₇⋊C₈

—

C₂²×C₈×C₄₇

—

C₂×C₄²×C₄₇

C₂²

yes

characteristic

C₄×C₈×C₄₇

—

C₄×C₈×C₄₇

C₄

yes

—

C₂²×C₄×C₄₇

C₂×C₄

yes

characteristic

C₂²×C₄×C₄₇

D₄

yes

characteristic

C₂²×C₄×C₄₇

Q₈

yes

characteristic

C₂×C₈×C₄₇

—

C₂×C₈×C₄₇

—

C₄²×C₄₇

C₂×C₄

yes

characteristic

C₄²×C₄₇

C₂×C₄

yes

characteristic

C₄²×C₄₇

D₄

yes

characteristic

C₄²×C₄₇

D₄

yes

characteristic

#10625329

—

#10625329

C₈

yes

complemented

C₂×C₄₇⋊C₈

—

C₂×C₄₇⋊C₈

—

C₂³×C₄₇

C₄⋊C₄

yes

characteristic

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

C₄²

yes

characteristic

C₂×C₄×C₄₇

C₂²⋊C₄

yes

characteristic

C₂×C₄×C₄₇

C₂²⋊C₄

yes

characteristic

C₂×C₄×C₄₇

C₂²⋊C₄

yes

characteristic

C₂×C₄×C₄₇

C₄⋊C₄

yes

characteristic

C₂×C₄×C₄₇

C₄⋊C₄

yes

characteristic

C₂×C₄×C₄₇

C₂×C₈

yes

—

C₂×C₄×C₄₇

M₁₆

yes

—

C₈×C₄₇

—

C₄₇⋊C₈

—

C₄₇⋊C₈

—

C₂²×C₄₇

—

C₂²×C₄₇

—

C₂²×C₄₇

#95

yes

characteristic

C₂²×C₄₇

C₄⋊C₈

yes

—

C₄×C₄₇

—

C₄×C₄₇

—

C₄×C₄₇

C₄×C₈

yes

characteristic

C₄×C₄₇

#97

yes

characteristic

C₄×C₄₇

C₂²⋊C₈

yes

—

C₄×C₄₇

#104

yes

—

C₄×C₄₇

#108

yes

—

#1278

—

maximal, Sylow

C₂×C₄₇

—

C₂×C₄₇

yes

C₂×C₄₇

yes

C₂×C₄₇

yes

—

C₄×M₁₆

—

C₂×C₄×C₈

D₄₇

yes

characteristic

C₄₇

#1278

yes

characteristic, complemented, Sylow

C₂×M₁₆

—

C₂²×C₈

—

C₂×C₄²

C₂×D₄₇

yes

characteristic

C₄⋊C₈

—

C₂²⋊C₈

—

#97

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

Dic₄₇

yes

—

C₂²×C₄

Dic₉₄

yes

characteristic

C₂²×C₄

C₄×D₄₇

yes

characteristic

C₂²×C₄

C₂²⋊D₄₇

yes

characteristic

M₁₆

—

M₁₆

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₄²

C₄×D₄₇

yes

characteristic

C₄²

D₁₈₈

yes

characteristic

C₄²

C₂×Dic₄₇

yes

characteristic

C₄²

C₂²⋊D₄₇

yes

characteristic

C₂³

Dic₄₇⋊C₄

yes

characteristic

C₂×C₄

—

C₂×C₄

—

C₂×C₄

C₈×D₄₇

yes

—

C₂×C₄

#10625324

yes

—

C₂×C₄

C₄×Dic₄₇

yes

characteristic

C₂×C₄

Dic₄₇⋊C₄

yes

characteristic

C₂×C₄

C₄⋊Dic₄₇

yes

characteristic

C₂×C₄

(C₂×D₄₇)⋊C₄

yes

characteristic

C₂×C₄

(C₂×D₄₇)⋊C₄

yes

characteristic

C₂×C₄

C₂²⋊Dic₄₇

yes

characteristic

C₈

—

C₈

—

C₈

—

188

—

C₂²

—

C₂²

—

100

C₂²

C₄₇⋊#95

yes

characteristic

101

C₂²

Dic₄₇⋊C₈

yes

—

102

C₄

—

103

C₄

—

104

C₄

C₈×Dic₄₇

yes

characteristic

105

C₄

#13093934

yes

characteristic

106

C₄

(C₂×D₄₇)⋊C₈

yes

—

107

C₄

#13093952

yes

—

108

C₄

#13093960

yes

—

109

C₂

—

110

C₂

#15688877

yes

characteristic

111

C₂

#15688883

yes

characteristic

112

C₂

#15688888

yes

characteristic

	subgroup	G/H
center	C₄²	D₁₈₈
commutator	C₄×C₄₇	C₄×C₈
Frattini	C₂×C₄²	C₂×D₄₇
Fitting	C₂×C₄×C₈×C₄₇	C₂
socle	C₂²×C₄₇	#95
hypercenter	C₂×C₄×C₈	D₄₇
hypocenter	C₄₇	#1278
LCS	G ▷ C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₄² ◁ C₂×C₄² ◁ C₂×C₄×C₈

subgroup

G/H

center

C₄²

D₁₈₈

commutator

C₄×C₄₇

C₄×C₈

Frattini

C₂×C₄²

C₂×D₄₇

Fitting

C₂×C₄×C₈×C₄₇

C₂

socle

C₂²×C₄₇

#95

hypercenter

C₂×C₄×C₈

D₄₇

hypocenter

C₄₇

#1278

LCS

G ▷ C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇

UCS

C₁ ◁ C₄² ◁ C₂×C₄² ◁ C₂×C₄×C₈

N	H ≅ G/N	type
24. #10625329	C₈	semidirect
60. C₄₇	#1278	semidirect

H ≅ G/N

type

24. #10625329

C₈

semidirect

60. C₄₇

#1278

semidirect

Subgroups — #20241225

Conjugacy classes of subgroups

Complements