Group #20242129 — subgroups

Subgroups — C₂×#15688874 #20242129

Order 6016.

200 conjugacy classes of subgroups; only the 141 normal subgroups are shown.

normal subgroups only

This group has 200 conjugacy classes of subgroups — too many to draw. Only its 141 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	C₂×C₄⋊(C₄₇⋊C₈)	C₂	1	2	yes	maximal
2	#15688874	C₂	1	4	yes	maximal, direct product factor
3	C₂×C₄×C₈×C₄₇	C₂	1	1	yes	maximal, characteristic
4	C₂²×C₄₇⋊C₈	—	2	2	no	—
5	C₄⋊(C₄₇⋊C₈)	—	2	2	no	—
6	C₄⋊(C₄₇⋊C₈)	C₂²	1	4	yes	—
7	C₂²×C₈×C₄₇	C₄	1	1	yes	characteristic
8	C₂²×C₈×C₄₇	C₄	1	1	yes	characteristic
9	C₂×C₄²×C₄₇	C₂²	1	1	yes	characteristic
10	C₄×C₈×C₄₇	C₄	1	2	yes	—
11	C₄×C₈×C₄₇	C₂²	1	2	yes	—
12	C₂²×C₄×C₄₇	C₂×C₄	1	1	yes	characteristic
13	C₂²×C₄×C₄₇	D₄	1	1	yes	characteristic
14	C₂²×C₄×C₄₇	Q₈	1	1	yes	characteristic
15	C₂×C₈×C₄₇	—	2	2	no	—
16	C₂×C₈×C₄₇	C₈	1	4	yes	complemented
17	C₂×C₈×C₄₇	C₂×C₄	1	2	yes	—
18	C₂×C₈×C₄₇	C₂×C₄	1	2	yes	—
19	C₄²×C₄₇	C₂×C₄	1	1	yes	characteristic
20	C₄²×C₄₇	D₄	1	1	yes	characteristic
21	C₄²×C₄₇	Q₈	1	1	yes	characteristic
22	C₄²×C₄₇	C₂³	1	1	yes	characteristic
23	C₂×C₄₇⋊C₈	—	2	4	no	—
24	C₂×C₄₇⋊C₈	—	4	4	no	—
25	C₂³×C₄₇	C₄⋊C₄	1	1	yes	characteristic
26	C₂×C₄×C₄₇	—	2	2	no	—
27	C₂×C₄×C₄₇	C₄⋊C₄	1	1	yes	characteristic
28	C₂×C₄×C₄₇	C₄⋊C₄	1	1	yes	characteristic
29	C₂×C₄×C₄₇	C₄⋊C₄	1	1	yes	characteristic
30	C₂×C₄×C₄₇	C₂×C₈	1	2	yes	—
31	C₂×C₄×C₄₇	M₁₆	1	2	yes	—
32	C₂×C₄×C₄₇	SD₁₆	1	4	yes	—
33	C₂×C₄×C₄₇	C₂²×C₄	1	1	yes	characteristic
34	C₂×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
35	C₂×C₄×C₄₇	C₂×Q₈	1	1	yes	characteristic
36	C₈×C₄₇	—	2	2	no	—
37	C₈×C₄₇	C₂×C₈	1	4	yes	complemented
38	C₄₇⋊C₈	—	4	4	no	—
39	C₂²×C₄₇	C₄⋊C₈	1	2	yes	—
40	C₂²×C₄₇	#106	1	2	yes	—
41	C₂²×C₄₇	#108	1	2	yes	—
42	C₂²×C₄₇	C₂×C₄⋊C₄	1	1	yes	characteristic
43	C₄×C₄₇	—	2	1	no	—
44	C₄×C₄₇	—	2	1	no	—
45	C₄×C₄₇	C₄⋊C₈	1	2	yes	—
46	C₄×C₄₇	#106	1	2	yes	—
47	C₄×C₄₇	C₂²×C₈	1	1	yes	characteristic
48	C₄×C₄₇	C₂×M₁₆	1	1	yes	characteristic
49	C₄×C₄₇	C₂×SD₁₆	1	2	yes	—
50	C₂×#334	—	47	1	no	maximal, Sylow
51	C₂×C₄₇	#334	1	4	yes	complemented
52	C₂×C₄₇	C₂×C₄⋊C₈	1	1	yes	characteristic
53	C₂×C₄₇	C₂×#106	1	1	yes	characteristic
54	C₂×C₄₇	C₂×#108	1	1	yes	characteristic
55	C₂×C₄⋊C₈	—	47	2	no	—
56	C₂×C₄×C₈	D₄₇	1	1	yes	characteristic
57	#334	—	47	4	no	—
58	C₄₇	C₂×#334	1	1	yes	characteristic, complemented, Sylow
59	C₂²×C₈	—	94	2	no	—
60	C₂²×C₈	Dic₄₇	1	1	yes	characteristic
61	C₂²×C₈	Dic₄₇	1	1	yes	characteristic
62	C₂×C₄²	C₂×D₄₇	1	1	yes	characteristic
63	C₄⋊C₈	—	47	4	no	—
64	C₄⋊C₈	—	94	2	no	—
65	C₄×C₈	Dic₄₇	1	2	yes	—
66	C₄×C₈	C₂×D₄₇	1	2	yes	—
67	C₂²×C₄	Dic₉₄	1	1	yes	characteristic
68	C₂²×C₄	D₁₈₈	1	1	yes	characteristic
69	C₂²×C₄	C₂×Dic₄₇	1	1	yes	characteristic
70	C₂×C₈	—	2	2	no	—
71	C₂×C₈	—	94	4	no	—
72	C₂×C₈	—	188	4	no	—
73	C₂×C₈	C₄₇⋊C₈	1	4	yes	complemented
74	C₂×C₈	C₂×Dic₄₇	1	2	yes	—
75	C₂×C₈	C₂×Dic₄₇	1	2	yes	—
76	C₄²	Dic₉₄	1	1	yes	characteristic
77	C₄²	D₁₈₈	1	1	yes	characteristic
78	C₄²	C₂×Dic₄₇	1	1	yes	characteristic
79	C₄²	C₂²×D₄₇	1	1	yes	characteristic
80	C₂³	C₄⋊Dic₄₇	1	1	yes	characteristic
81	C₂×C₄	—	2	2	no	—
82	C₂×C₄	#10625325	1	4	yes	—
83	C₂×C₄	C₂×C₄₇⋊C₈	1	2	yes	—
84	C₂×C₄	#10625329	1	2	yes	—
85	C₂×C₄	C₄⋊Dic₄₇	1	1	yes	characteristic
86	C₂×C₄	C₄⋊Dic₄₇	1	1	yes	characteristic
87	C₂×C₄	C₄⋊Dic₄₇	1	1	yes	characteristic
88	C₂×C₄	C₂×Dic₉₄	1	1	yes	characteristic
89	C₂×C₄	C₂×D₁₈₈	1	1	yes	characteristic
90	C₂×C₄	C₂²×Dic₄₇	1	1	yes	characteristic
91	C₈	—	2	2	no	—
92	C₈	—	188	4	no	—
93	C₈	C₂×C₄₇⋊C₈	1	4	yes	complemented
94	C₂²	C₄⋊(C₄₇⋊C₈)	1	2	yes	—
95	C₂²	#13093956	1	2	yes	—
96	C₂²	#13093960	1	2	yes	—
97	C₂²	C₂×C₄⋊Dic₄₇	1	1	yes	characteristic
98	C₄	—	2	1	no	—
99	C₄	—	2	1	no	—
100	C₄	C₄⋊(C₄₇⋊C₈)	1	2	yes	—
101	C₄	#13093956	1	2	yes	—
102	C₄	C₂²×C₄₇⋊C₈	1	1	yes	characteristic
103	C₄	C₂×#10625329	1	1	yes	characteristic
104	C₄	C₂×#10625325	1	2	yes	—
105	C₂	#15688874	1	4	yes	direct product factor
106	C₂	C₂×C₄⋊(C₄₇⋊C₈)	1	1	yes	characteristic
107	C₂	C₂×#13093956	1	1	yes	characteristic
108	C₂	C₂×#13093960	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×C₄⋊(C₄₇⋊C₈)

C₂

yes

maximal

#15688874

C₂

yes

maximal, direct product factor

C₂×C₄×C₈×C₄₇

C₂

yes

maximal, characteristic

C₂²×C₄₇⋊C₈

—

C₄⋊(C₄₇⋊C₈)

—

C₄⋊(C₄₇⋊C₈)

C₂²

yes

—

C₂²×C₈×C₄₇

C₄

yes

characteristic

C₂²×C₈×C₄₇

C₄

yes

characteristic

C₂×C₄²×C₄₇

C₂²

yes

characteristic

C₄×C₈×C₄₇

C₄

yes

—

C₄×C₈×C₄₇

C₂²

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

C₈

yes

complemented

C₂×C₈×C₄₇

C₂×C₄

yes

—

C₂×C₈×C₄₇

C₂×C₄

yes

—

C₄²×C₄₇

C₂×C₄

yes

characteristic

C₄²×C₄₇

D₄

yes

characteristic

C₄²×C₄₇

Q₈

yes

characteristic

C₄²×C₄₇

C₂³

yes

characteristic

C₂×C₄₇⋊C₈

—

C₂×C₄₇⋊C₈

—

C₂³×C₄₇

C₄⋊C₄

yes

characteristic

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

C₄⋊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

—

C₂×C₄×C₄₇

SD₁₆

yes

—

C₂×C₄×C₄₇

C₂²×C₄

yes

characteristic

C₂×C₄×C₄₇

C₂×D₄

yes

characteristic

C₂×C₄×C₄₇

C₂×Q₈

yes

characteristic

C₈×C₄₇

—

C₈×C₄₇

C₂×C₈

yes

complemented

C₄₇⋊C₈

—

C₂²×C₄₇

C₄⋊C₈

yes

—

C₂²×C₄₇

#106

yes

—

C₂²×C₄₇

#108

yes

—

C₂²×C₄₇

C₂×C₄⋊C₄

yes

characteristic

C₄×C₄₇

—

C₄×C₄₇

—

C₄×C₄₇

C₄⋊C₈

yes

—

C₄×C₄₇

#106

yes

—

C₄×C₄₇

C₂²×C₈

yes

characteristic

C₄×C₄₇

C₂×M₁₆

yes

characteristic

C₄×C₄₇

C₂×SD₁₆

yes

—

—

C₂×C₄₇

yes

C₂×C₄₇

C₂×C₄⋊C₈

yes

C₂×C₄₇

yes

C₂×C₄₇

yes

C₂×C₄⋊C₈

—

C₂×C₄×C₈

D₄₇

yes

characteristic

#334

—

C₄₇

C₂×#334

yes

characteristic, complemented, Sylow

C₂²×C₈

—

C₂²×C₈

Dic₄₇

yes

characteristic

C₂²×C₈

Dic₄₇

yes

characteristic

C₂×C₄²

C₂×D₄₇

yes

characteristic

C₄⋊C₈

—

C₄⋊C₈

—

C₄×C₈

Dic₄₇

yes

—

C₄×C₈

C₂×D₄₇

yes

—

C₂²×C₄

Dic₉₄

yes

characteristic

C₂²×C₄

D₁₈₈

yes

characteristic

C₂²×C₄

C₂×Dic₄₇

yes

characteristic

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

188

—

C₂×C₈

C₄₇⋊C₈

yes

complemented

C₂×C₈

C₂×Dic₄₇

yes

—

C₂×C₈

C₂×Dic₄₇

yes

—

C₄²

Dic₉₄

yes

characteristic

C₄²

D₁₈₈

yes

characteristic

C₄²

C₂×Dic₄₇

yes

characteristic

C₄²

C₂²×D₄₇

yes

characteristic

C₂³

C₄⋊Dic₄₇

yes

characteristic

C₂×C₄

—

C₂×C₄

#10625325

yes

—

C₂×C₄

C₂×C₄₇⋊C₈

yes

—

C₂×C₄

#10625329

yes

—

C₂×C₄

C₄⋊Dic₄₇

yes

characteristic

C₂×C₄

C₄⋊Dic₄₇

yes

characteristic

C₂×C₄

C₄⋊Dic₄₇

yes

characteristic

C₂×C₄

C₂×Dic₉₄

yes

characteristic

C₂×C₄

C₂×D₁₈₈

yes

characteristic

C₂×C₄

C₂²×Dic₄₇

yes

characteristic

C₈

—

C₈

—

188

—

C₈

C₂×C₄₇⋊C₈

yes

complemented

C₂²

C₄⋊(C₄₇⋊C₈)

yes

—

C₂²

#13093956

yes

—

C₂²

#13093960

yes

—

C₂²

C₂×C₄⋊Dic₄₇

yes

characteristic

C₄

—

C₄

—

100

C₄

C₄⋊(C₄₇⋊C₈)

yes

—

101

C₄

#13093956

yes

—

102

C₄

C₂²×C₄₇⋊C₈

yes

characteristic

103

C₄

C₂×#10625329

yes

characteristic

104

C₄

C₂×#10625325

yes

—

105

C₂

#15688874

yes

direct product factor

106

C₂

C₂×C₄⋊(C₄₇⋊C₈)

yes

characteristic

107

C₂

C₂×#13093956

yes

characteristic

108

C₂

C₂×#13093960

yes

characteristic

	subgroup	G/H
center	C₂²×C₄	D₁₈₈
commutator	C₄×C₄₇	C₂²×C₈
Frattini	C₄²	C₂²×D₄₇
Fitting	C₂×C₄×C₈×C₄₇	C₂
socle	C₂³×C₄₇	C₄⋊C₄
hypercenter	C₂×C₄×C₈	D₄₇
hypocenter	C₄₇	C₂×#334
LCS	G ▷ C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×C₄² ◁ C₂×C₄×C₈

subgroup

G/H

center

C₂²×C₄

D₁₈₈

commutator

C₄×C₄₇

C₂²×C₈

Frattini

C₄²

C₂²×D₄₇

Fitting

C₂×C₄×C₈×C₄₇

C₂

socle

C₂³×C₄₇

C₄⋊C₄

hypercenter

C₂×C₄×C₈

D₄₇

hypocenter

C₄₇

C₂×#334

LCS

G ▷ C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇

UCS

C₁ ◁ C₂²×C₄ ◁ C₂×C₄² ◁ C₂×C₄×C₈

N	H ≅ G/N	type
2. #15688874	C₂	direct (N × H)
16. C₂×C₈×C₄₇	C₈	semidirect
37. C₈×C₄₇	C₂×C₈	semidirect
51. C₂×C₄₇	#334	semidirect
58. C₄₇	C₂×#334	semidirect
73. C₂×C₈	C₄₇⋊C₈	semidirect
93. C₈	C₂×C₄₇⋊C₈	semidirect
105. C₂	#15688874	direct (N × H)

H ≅ G/N

type

2. #15688874

C₂

direct (N × H)

16. C₂×C₈×C₄₇

C₈

semidirect

37. C₈×C₄₇

C₂×C₈

semidirect

51. C₂×C₄₇

#334

semidirect

58. C₄₇

C₂×#334

semidirect

73. C₂×C₈

C₄₇⋊C₈

semidirect

93. C₈

C₂×C₄₇⋊C₈

semidirect

105. C₂

#15688874

direct (N × H)

Subgroups — C₂×#15688874 #20242129

Conjugacy classes of subgroups

Complements

Subgroups — C2×#15688874 #20242129

Conjugacy classes of subgroups

Complements

Subgroups — C₂×#15688874 #20242129