Group #20242029 — subgroups

Subgroups — C₂×#15688867 #20242029

Order 6016.

360 conjugacy classes of subgroups; only the 125 normal subgroups are shown.

normal subgroups only

This group has 360 conjugacy classes of subgroups — too many to draw. Only its 125 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, complemented
2	#15688867	C₂	1	4	yes	maximal, direct product factor
3	C₂×#127×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	complemented
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	4	no	—
18	C₂×D₄×C₄₇	—	4	4	no	—
19	C₂×D₄×C₄₇	—	4	4	no	—
20	C₂×D₄×C₄₇	C₂×C₄	1	4	yes	—
21	C₂²×C₄×C₄₇	C₂×C₄	1	1	yes	characteristic
22	C₂²×C₄×C₄₇	D₄	1	2	yes	—
23	C₄²×C₄₇	C₂×C₄	1	1	yes	characteristic
24	C₄²×C₄₇	D₄	1	2	yes	—
25	C₄²×C₄₇	C₂³	1	1	yes	characteristic
26	C₂×C₄₇⋊C₈	—	2	4	no	—
27	C₂×C₄₇⋊C₈	—	4	4	no	—
28	C₂³×C₄₇	—	2	4	no	—
29	C₂³×C₄₇	—	4	4	no	—
30	C₂³×C₄₇	—	4	8	no	—
31	C₂³×C₄₇	C₂²⋊C₄	1	1	yes	characteristic
32	D₄×C₄₇	—	4	4	no	—
33	D₄×C₄₇	—	4	4	no	—
34	D₄×C₄₇	—	4	8	no	—
35	D₄×C₄₇	—	4	4	no	—
36	D₄×C₄₇	—	4	4	no	—
37	C₂×C₄×C₄₇	—	2	2	no	—
38	C₂×C₄×C₄₇	C₂²⋊C₄	1	1	yes	characteristic
39	C₂×C₄×C₄₇	C₂²⋊C₄	1	2	yes	—
40	C₂×C₄×C₄₇	D₈	1	4	yes	—
41	C₂×C₄×C₄₇	SD₁₆	1	4	yes	—
42	C₂×C₄×C₄₇	C₂²×C₄	1	1	yes	characteristic
43	C₂×C₄×C₄₇	C₂×D₄	1	2	yes	—
44	C₄₇⋊C₈	—	4	4	no	—
45	C₂²×C₄₇	—	4	4	no	—
46	C₂²×C₄₇	—	4	8	no	—
47	C₂²×C₄₇	—	8	8	no	—
48	C₂²×C₄₇	#100	1	2	yes	—
49	C₂²×C₄₇	#102	1	4	yes	—
50	C₂²×C₄₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
51	C₄×C₄₇	—	2	1	no	—
52	C₄×C₄₇	—	2	1	no	—
53	C₄×C₄₇	#102	1	4	yes	—
54	C₄×C₄₇	C₂×D₈	1	2	yes	—
55	C₄×C₄₇	C₂×SD₁₆	1	2	yes	—
56	C₂×#331	—	47	1	no	maximal, Sylow
57	C₂×C₄₇	—	8	4	no	—
58	C₂×C₄₇	#331	1	4	yes	complemented
59	C₂×C₄₇	C₂×#100	1	1	yes	characteristic
60	C₂×C₄₇	C₂×#102	1	2	yes	—
61	C₂×#127	D₄₇	1	1	yes	characteristic
62	C₂×C₄⋊C₈	—	47	2	no	—
63	#331	—	47	4	no	—
64	C₄₇	C₂×#331	1	1	yes	characteristic, complemented, Sylow
65	C₂²×D₄	—	2	2	no	—
66	C₂²×D₄	Dic₄₇	1	2	yes	—
67	C₂²×C₈	—	94	2	no	—
68	#127	—	2	2	no	—
69	#127	Dic₄₇	1	2	yes	—
70	#127	C₂×D₄₇	1	2	yes	—
71	C₂×C₄²	C₂×D₄₇	1	1	yes	characteristic
72	C₄⋊C₈	—	47	4	no	—
73	C₄⋊C₈	—	94	2	no	—
74	C₂⁴	—	2	2	no	—
75	C₂×D₄	—	2	2	no	—
76	C₂×D₄	—	2	4	no	—
77	C₂×D₄	—	2	4	no	—
78	C₂×D₄	—	4	4	no	—
79	C₂×D₄	—	4	4	no	—
80	C₂×D₄	—	4	4	no	—
81	C₂×D₄	C₂×Dic₄₇	1	4	yes	—
82	C₂²×C₄	C₂×Dic₄₇	1	1	yes	characteristic
83	C₂²×C₄	C₂²⋊D₄₇	1	2	yes	—
84	C₂×C₈	—	94	4	no	—
85	C₂×C₈	—	188	4	no	—
86	C₄²	C₂×Dic₄₇	1	1	yes	characteristic
87	C₄²	C₂²⋊D₄₇	1	2	yes	—
88	C₄²	C₂²×D₄₇	1	1	yes	characteristic
89	C₂³	—	2	4	no	—
90	C₂³	—	4	4	no	—
91	C₂³	—	4	8	no	—
92	C₂³	C₂²⋊Dic₄₇	1	1	yes	characteristic
93	D₄	—	4	4	no	—
94	D₄	—	4	4	no	—
95	D₄	—	4	8	no	—
96	D₄	—	4	4	no	—
97	D₄	—	4	4	no	—
98	C₂×C₄	—	2	2	no	—
99	C₂×C₄	#10625334	1	4	yes	—
100	C₂×C₄	#10625335	1	4	yes	—
101	C₂×C₄	C₂²⋊Dic₄₇	1	1	yes	characteristic
102	C₂×C₄	C₂²⋊Dic₄₇	1	2	yes	—
103	C₂×C₄	C₂²×Dic₄₇	1	1	yes	characteristic
104	C₂×C₄	C₂×C₂²⋊D₄₇	1	2	yes	—
105	C₈	—	188	4	no	—
106	C₂²	—	4	4	no	—
107	C₂²	—	4	8	no	—
108	C₂²	—	8	8	no	—
109	C₂²	#13093941	1	2	yes	—
110	C₂²	#13093945	1	4	yes	—
111	C₂²	C₂×C₂²⋊Dic₄₇	1	1	yes	characteristic
112	C₄	—	2	1	no	—
113	C₄	—	2	1	no	—
114	C₄	#13093945	1	4	yes	—
115	C₄	C₂×#10625334	1	2	yes	—
116	C₄	C₂×#10625335	1	2	yes	—
117	C₂	—	8	4	no	—
118	C₂	#15688867	1	4	yes	direct product factor
119	C₂	C₂×#13093941	1	1	yes	characteristic
120	C₂	C₂×#13093945	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×C₄⋊(C₄₇⋊C₈)

C₂

yes

maximal, complemented

#15688867

C₂

yes

maximal, direct product factor

C₂×#127×C₄₇

C₂

yes

maximal, characteristic

C₂²×C₄₇⋊C₈

—

C₄⋊(C₄₇⋊C₈)

—

C₄⋊(C₄₇⋊C₈)

C₂²

yes

complemented

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×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

—

C₂×C₄₇⋊C₈

—

C₂³×C₄₇

—

C₂³×C₄₇

—

C₂³×C₄₇

—

C₂³×C₄₇

C₂²⋊C₄

yes

characteristic

D₄×C₄₇

—

D₄×C₄₇

—

D₄×C₄₇

—

D₄×C₄₇

—

D₄×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

C₂²⋊C₄

yes

characteristic

C₂×C₄×C₄₇

C₂²⋊C₄

yes

—

C₂×C₄×C₄₇

D₈

yes

—

C₂×C₄×C₄₇

SD₁₆

yes

—

C₂×C₄×C₄₇

C₂²×C₄

yes

characteristic

C₂×C₄×C₄₇

C₂×D₄

yes

—

C₄₇⋊C₈

—

C₂²×C₄₇

—

C₂²×C₄₇

—

C₂²×C₄₇

—

C₂²×C₄₇

#100

yes

—

C₂²×C₄₇

#102

yes

—

C₂²×C₄₇

C₂×C₂²⋊C₄

yes

characteristic

C₄×C₄₇

—

C₄×C₄₇

—

C₄×C₄₇

#102

yes

—

C₄×C₄₇

C₂×D₈

yes

—

C₄×C₄₇

C₂×SD₁₆

yes

—

C₂×#331

—

maximal, Sylow

C₂×C₄₇

—

C₂×C₄₇

yes

C₂×C₄₇

yes

C₂×C₄₇

yes

—

C₂×#127

D₄₇

yes

characteristic

C₂×C₄⋊C₈

—

#331

—

C₄₇

C₂×#331

yes

characteristic, complemented, Sylow

C₂²×D₄

—

C₂²×D₄

Dic₄₇

yes

—

C₂²×C₈

—

#127

—

#127

Dic₄₇

yes

—

#127

C₂×D₄₇

yes

—

C₂×C₄²

C₂×D₄₇

yes

characteristic

C₄⋊C₈

—

C₄⋊C₈

—

C₂⁴

—

C₂×D₄

—

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

—

C₂×C₈

—

188

—

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

—

D₄

—

C₂×C₄

—

C₂×C₄

#10625334

yes

—

100

C₂×C₄

#10625335

yes

—

101

C₂×C₄

C₂²⋊Dic₄₇

yes

characteristic

102

C₂×C₄

C₂²⋊Dic₄₇

yes

—

103

C₂×C₄

C₂²×Dic₄₇

yes

characteristic

104

C₂×C₄

C₂×C₂²⋊D₄₇

yes

—

105

C₈

—

188

—

106

C₂²

—

107

C₂²

—

108

C₂²

—

109

C₂²

#13093941

yes

—

110

C₂²

#13093945

yes

—

111

C₂²

C₂×C₂²⋊Dic₄₇

yes

characteristic

112

C₄

—

113

C₄

—

114

C₄

#13093945

yes

—

115

C₄

C₂×#10625334

yes

—

116

C₄

C₂×#10625335

yes

—

117

C₂

—

118

C₂

#15688867

yes

direct product factor

119

C₂

C₂×#13093941

yes

characteristic

120

C₂

C₂×#13093945

yes

—

	subgroup	G/H
center	C₂³	C₂²⋊Dic₄₇
commutator	C₂×C₄×C₄₇	C₂²×C₄
Frattini	C₄²	C₂²×D₄₇
Fitting	C₂×#127×C₄₇	C₂
socle	C₂³×C₄₇	C₂²⋊C₄
hypercenter	C₂×#127	D₄₇
hypocenter	C₄₇	C₂×#331
LCS	G ▷ C₂×C₄×C₄₇ ▷ C₂²×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×#127

subgroup

G/H

center

C₂³

C₂²⋊Dic₄₇

commutator

C₂×C₄×C₄₇

C₂²×C₄

Frattini

C₄²

C₂²×D₄₇

Fitting

C₂×#127×C₄₇

C₂

socle

C₂³×C₄₇

C₂²⋊C₄

D₄₇

C₄₇

G ▷ C₂×C₄×C₄₇ ▷ C₂²×C₄₇ ▷ C₄₇

UCS

C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×#127

N	H ≅ G/N	type
1. C₂×C₄⋊(C₄₇⋊C₈)	C₂	semidirect
2. #15688867	C₂	direct (N × H)
6. C₄⋊(C₄₇⋊C₈)	C₂²	semidirect
58. C₂×C₄₇	#331	semidirect
64. C₄₇	C₂×#331	semidirect
118. C₂	#15688867	direct (N × H)

H ≅ G/N

type

1. C₂×C₄⋊(C₄₇⋊C₈)

C₂

semidirect

2. #15688867

C₂

direct (N × H)

6. C₄⋊(C₄₇⋊C₈)

C₂²

semidirect

58. C₂×C₄₇

#331

semidirect

64. C₄₇

C₂×#331

semidirect

118. C₂

#15688867

direct (N × H)

Subgroups — C₂×#15688867 #20242029

Conjugacy classes of subgroups

Complements

Subgroups — C2×#15688867 #20242029

Conjugacy classes of subgroups

Complements

Subgroups — C₂×#15688867 #20242029