Group #20239708 — subgroups

Subgroups — C₄₇×#2098 #20239708

Order 6016.

210 conjugacy classes of subgroups; only the 80 normal subgroups are shown.

normal subgroups only

This group has 210 conjugacy classes of subgroups — too many to draw. Only its 80 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	C₄₇×#430	C₂	1	1	yes	maximal, characteristic, complemented
2	C₄₇×#424	C₂	1	2	yes	maximal, complemented
3	C₄₇×#421	C₂	1	2	yes	maximal
4	C₄₇×#408	C₂	1	1	yes	maximal, characteristic
5	C₄₇×H(3,ℤ/4)	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×#41×C₄₇	C₂²	1	1	yes	characteristic
7	C₂×M₁₆×C₄₇	C₂²	1	2	yes	—
8	C₂×M₁₆×C₄₇	C₂²	1	1	yes	characteristic
9	C₂²×C₈×C₄₇	C₂²	1	1	yes	characteristic
10	#117×C₄₇	C₂²	1	2	yes	—
11	#108×C₄₇	—	2	2	no	—
12	C₄⋊C₈×C₄₇	—	2	4	no	—
13	#104×C₄₇	—	2	4	no	—
14	C₂²⋊C₈×C₄₇	—	2	2	no	—
15	#41×C₄₇	—	2	2	no	—
16	#41×C₄₇	—	2	2	no	—
17	C₂×Q₈×C₄₇	Q₈	1	1	yes	characteristic
18	C₂×D₄×C₄₇	—	2	1	no	—
19	C₂×D₄×C₄₇	Q₈	1	1	yes	characteristic
20	C₂²×C₄×C₄₇	—	2	1	no	—
21	C₂²×C₄×C₄₇	C₂³	1	1	yes	characteristic
22	M₁₆×C₄₇	—	2	2	no	—
23	M₁₆×C₄₇	—	2	2	no	—
24	M₁₆×C₄₇	—	2	2	no	—
25	C₂×C₈×C₄₇	—	2	2	no	—
26	C₂×C₈×C₄₇	—	2	2	no	—
27	C₂×C₈×C₄₇	—	2	1	no	—
28	C₂×C₈×C₄₇	D₄	1	2	yes	—
29	C₄⋊C₄×C₄₇	—	2	2	no	—
30	C₂²⋊C₄×C₄₇	—	2	2	no	—
31	C₄²×C₄₇	D₄	1	4	yes	—
32	C₂³×C₄₇	—	2	1	no	—
33	C₂³×C₄₇	#41	1	1	yes	characteristic
34	Q₈×C₄₇	—	2	2	no	—
35	D₄×C₄₇	—	2	2	no	—
36	D₄×C₄₇	—	4	2	no	—
37	C₂×C₄×C₄₇	—	2	1	no	—
38	C₂×C₄×C₄₇	—	2	4	no	—
39	C₂×C₄×C₄₇	—	4	1	no	—
40	C₂×C₄×C₄₇	—	4	1	no	—
41	C₂×C₄×C₄₇	C₂×D₄	1	2	yes	—
42	C₂×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
43	C₂×C₄×C₄₇	C₂×Q₈	1	1	yes	characteristic
44	C₂×C₄×C₄₇	#41	1	2	yes	—
45	C₈×C₄₇	—	2	2	no	—
46	C₈×C₄₇	—	4	1	no	—
47	C₈×C₄₇	—	4	2	no	—
48	C₂²×C₄₇	—	4	1	no	—
49	C₂²×C₄₇	—	4	2	no	—
50	C₂²×C₄₇	—	4	1	no	—
51	C₂²×C₄₇	C₂²⋊Q₈	1	1	yes	characteristic
52	C₂²×C₄₇	#123	1	2	yes	—
53	C₄×C₄₇	—	4	4	no	—
54	C₄×C₄₇	—	4	1	no	—
55	C₄×C₄₇	C₂²⋊Q₈	1	2	yes	—
56	C₄×C₄₇	#123	1	1	yes	characteristic
57	C₄×C₄₇	#127	1	1	yes	characteristic
58	#2098	C₄₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
59	C₂×C₄₇	—	2	2	no	—
60	C₂×C₄₇	—	2	1	no	—
61	C₂×C₄₇	—	8	1	no	—
62	C₂×C₄₇	#399	1	1	yes	characteristic
63	#430	C₂×C₄₇	1	1	yes	characteristic, complemented
64	#424	C₂×C₄₇	1	2	yes	complemented
65	#421	C₂×C₄₇	1	2	yes	—
66	#408	C₂×C₄₇	1	1	yes	characteristic
67	H(3,ℤ/4)	C₂×C₄₇	1	1	yes	characteristic, complemented
68	C₄₇	#2098	1	1	yes	characteristic, direct product factor, Sylow
69	C₂×#41	C₂²×C₄₇	1	1	yes	characteristic
70	C₂×M₁₆	C₂²×C₄₇	1	2	yes	—
71	C₂×M₁₆	C₂²×C₄₇	1	1	yes	characteristic
72	C₂²×C₈	C₂²×C₄₇	1	1	yes	characteristic
73	#117	C₂²×C₄₇	1	2	yes	—
74	#108	—	2	2	no	—
75	C₄⋊C₈	—	2	4	no	—
76	#104	—	2	4	no	—
77	C₂²⋊C₈	—	2	2	no	—
78	#41	—	2	2	no	—
79	#41	—	2	2	no	—
80	C₂×Q₈	Q₈×C₄₇	1	1	yes	characteristic
81	C₂×D₄	—	2	1	no	—
82	C₂×D₄	Q₈×C₄₇	1	1	yes	characteristic
83	C₂²×C₄	—	2	1	no	—
84	C₂²×C₄	C₂³×C₄₇	1	1	yes	characteristic
85	M₁₆	—	2	2	no	—
86	M₁₆	—	2	2	no	—
87	M₁₆	—	2	2	no	—
88	C₂×C₈	—	2	1	no	—
89	C₂×C₈	—	2	2	no	—
90	C₂×C₈	—	2	2	no	—
91	C₂×C₈	D₄×C₄₇	1	2	yes	—
92	C₄⋊C₄	—	2	2	no	—
93	C₂²⋊C₄	—	2	2	no	—
94	C₄²	D₄×C₄₇	1	4	yes	—
95	C₂³	—	2	1	no	—
96	C₂³	#41×C₄₇	1	1	yes	characteristic
97	Q₈	—	2	2	no	—
98	D₄	—	2	2	no	—
99	D₄	—	4	2	no	—
100	C₂×C₄	—	2	1	no	—
101	C₂×C₄	—	2	4	no	—
102	C₂×C₄	—	4	1	no	—
103	C₂×C₄	—	4	1	no	—
104	C₂×C₄	C₂×D₄×C₄₇	1	2	yes	—
105	C₂×C₄	C₂×D₄×C₄₇	1	1	yes	characteristic
106	C₂×C₄	C₂×Q₈×C₄₇	1	1	yes	characteristic
107	C₂×C₄	#41×C₄₇	1	2	yes	—
108	C₈	—	2	2	no	—
109	C₈	—	4	1	no	—
110	C₈	—	4	2	no	—
111	C₂²	—	4	1	no	—
112	C₂²	—	4	2	no	—
113	C₂²	—	4	1	no	—
114	C₂²	C₂²⋊Q₈×C₄₇	1	1	yes	characteristic
115	C₂²	#123×C₄₇	1	2	yes	—
116	C₄	—	4	4	no	—
117	C₄	—	4	1	no	—
118	C₄	C₂²⋊Q₈×C₄₇	1	2	yes	—
119	C₄	#123×C₄₇	1	1	yes	characteristic
120	C₄	#127×C₄₇	1	1	yes	characteristic
121	C₂	—	2	2	no	—
122	C₂	—	2	1	no	—
123	C₂	—	8	1	no	—
124	C₂	C₄₇×#399	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₄₇×#430

C₂

yes

maximal, characteristic, complemented

C₄₇×#424

C₂

yes

maximal, complemented

C₄₇×#421

C₂

yes

maximal

C₄₇×#408

C₂

yes

maximal, characteristic

C₄₇×H(3,ℤ/4)

C₂

yes

maximal, characteristic, complemented

C₂×#41×C₄₇

C₂²

yes

characteristic

C₂×M₁₆×C₄₇

C₂²

yes

—

C₂×M₁₆×C₄₇

C₂²

yes

characteristic

C₂²×C₈×C₄₇

C₂²

yes

characteristic

#117×C₄₇

C₂²

yes

—

#108×C₄₇

—

C₄⋊C₈×C₄₇

—

#104×C₄₇

—

C₂²⋊C₈×C₄₇

—

#41×C₄₇

—

#41×C₄₇

—

C₂×Q₈×C₄₇

Q₈

yes

characteristic

C₂×D₄×C₄₇

—

C₂×D₄×C₄₇

Q₈

yes

characteristic

C₂²×C₄×C₄₇

—

C₂²×C₄×C₄₇

C₂³

yes

characteristic

M₁₆×C₄₇

—

M₁₆×C₄₇

—

M₁₆×C₄₇

—

C₂×C₈×C₄₇

—

C₂×C₈×C₄₇

—

C₂×C₈×C₄₇

—

C₂×C₈×C₄₇

D₄

yes

—

C₄⋊C₄×C₄₇

—

C₂²⋊C₄×C₄₇

—

C₄²×C₄₇

D₄

yes

—

C₂³×C₄₇

—

C₂³×C₄₇

#41

yes

characteristic

Q₈×C₄₇

—

D₄×C₄₇

—

D₄×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

C₂×D₄

yes

—

C₂×C₄×C₄₇

C₂×D₄

yes

characteristic

C₂×C₄×C₄₇

C₂×Q₈

yes

characteristic

C₂×C₄×C₄₇

#41

yes

—

C₈×C₄₇

—

C₈×C₄₇

—

C₈×C₄₇

—

C₂²×C₄₇

—

C₂²×C₄₇

—

C₂²×C₄₇

—

C₂²×C₄₇

C₂²⋊Q₈

yes

characteristic

C₂²×C₄₇

#123

yes

—

C₄×C₄₇

—

C₄×C₄₇

—

C₄×C₄₇

C₂²⋊Q₈

yes

—

C₄×C₄₇

yes

C₄×C₄₇

yes

C₄₇

yes

maximal, characteristic, direct product factor, Sylow

C₂×C₄₇

—

C₂×C₄₇

—

C₂×C₄₇

—

C₂×C₄₇

#399

yes

characteristic

#430

C₂×C₄₇

yes

characteristic, complemented

#424

C₂×C₄₇

yes

complemented

#421

C₂×C₄₇

yes

—

#408

C₂×C₄₇

yes

characteristic

H(3,ℤ/4)

C₂×C₄₇

yes

characteristic, complemented

C₄₇

#2098

yes

characteristic, direct product factor, Sylow

C₂×#41

C₂²×C₄₇

yes

characteristic

C₂×M₁₆

C₂²×C₄₇

yes

—

C₂×M₁₆

C₂²×C₄₇

yes

characteristic

C₂²×C₈

C₂²×C₄₇

yes

characteristic

#117

C₂²×C₄₇

yes

—

#108

—

C₄⋊C₈

—

#104

—

C₂²⋊C₈

—

#41

—

#41

—

C₂×Q₈

Q₈×C₄₇

yes

characteristic

C₂×D₄

—

C₂×D₄

Q₈×C₄₇

yes

characteristic

C₂²×C₄

—

C₂²×C₄

C₂³×C₄₇

yes

characteristic

M₁₆

—

M₁₆

—

M₁₆

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

D₄×C₄₇

yes

—

C₄⋊C₄

—

C₂²⋊C₄

—

C₄²

D₄×C₄₇

yes

—

C₂³

—

C₂³

#41×C₄₇

yes

characteristic

Q₈

—

D₄

—

D₄

—

100

C₂×C₄

—

101

C₂×C₄

—

102

C₂×C₄

—

103

C₂×C₄

—

104

C₂×C₄

C₂×D₄×C₄₇

yes

—

105

C₂×C₄

C₂×D₄×C₄₇

yes

characteristic

106

C₂×C₄

C₂×Q₈×C₄₇

yes

characteristic

107

C₂×C₄

#41×C₄₇

yes

—

108

C₈

—

109

C₈

—

110

C₈

—

111

C₂²

—

112

C₂²

—

113

C₂²

—

114

C₂²

C₂²⋊Q₈×C₄₇

yes

characteristic

115

C₂²

#123×C₄₇

yes

—

116

C₄

—

117

C₄

—

118

C₄

C₂²⋊Q₈×C₄₇

yes

—

119

C₄

#123×C₄₇

yes

characteristic

120

C₄

#127×C₄₇

yes

characteristic

121

C₂

—

122

C₂

—

123

C₂

—

124

C₂

C₄₇×#399

yes

characteristic

	subgroup	G/H
center	C₄×C₄₇	#127
commutator, Frattini, cosocle	C₂²×C₄	C₂³×C₄₇
socle	C₂×C₄₇	#399
prime radical	C₂×#41×C₄₇	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄×C₄₇ ◁ C₂²×C₄×C₄₇ ◁ G

subgroup

G/H

center

C₄×C₄₇

#127

commutator, Frattini, cosocle

C₂²×C₄

C₂³×C₄₇

socle

C₂×C₄₇

#399

prime radical

C₂×#41×C₄₇

C₂²

LCS

G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

UCS

C₁ ◁ C₄×C₄₇ ◁ C₂²×C₄×C₄₇ ◁ G

N	H ≅ G/N	type
1. C₄₇×#430	C₂	semidirect
2. C₄₇×#424	C₂	semidirect
5. C₄₇×H(3,ℤ/4)	C₂	semidirect
58. #2098	C₄₇	direct (N × H)
63. #430	C₂×C₄₇	semidirect
64. #424	C₂×C₄₇	semidirect
67. H(3,ℤ/4)	C₂×C₄₇	semidirect
68. C₄₇	#2098	direct (N × H)

H ≅ G/N

type

1. C₄₇×#430

C₂

semidirect

2. C₄₇×#424

C₂

semidirect

5. C₄₇×H(3,ℤ/4)

C₂

semidirect

58. #2098

C₄₇

direct (N × H)

63. #430

C₂×C₄₇

semidirect

64. #424

C₂×C₄₇

semidirect

67. H(3,ℤ/4)

C₂×C₄₇

semidirect

68. C₄₇

#2098

direct (N × H)

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
C₄₇×#430	2
C₄₇×#424	2
C₄₇×#421	2
C₄₇×#408	2
C₄₇×H(3,ℤ/4)	2
#2098	47

subgroup

index

C₄₇×#430

C₄₇×#424

C₄₇×#421

C₄₇×#408

C₄₇×H(3,ℤ/4)

#2098

Subgroups — C₄₇×#2098 #20239708

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C47×#2098 #20239708

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄₇×#2098 #20239708