Group #20239148 — subgroups

Subgroups — C₂×C₄₇×#331 #20239148

Order 6016.

360 conjugacy classes of subgroups; only the 136 normal subgroups are shown.

normal subgroups only

This group has 360 conjugacy classes of subgroups — too many to draw. Only its 136 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×#127×C₄₇	C₂	1	1	yes	maximal, characteristic
2	C₂×C₄⋊C₈×C₄₇	C₂	1	2	yes	maximal, complemented
3	C₄₇×#331	C₂	1	4	yes	maximal, direct product factor
4	C₂²×D₄×C₄₇	—	2	2	no	—
5	C₂²×D₄×C₄₇	C₄	1	2	yes	—
6	C₂²×C₈×C₄₇	—	2	2	no	—
7	#127×C₄₇	—	2	2	no	—
8	#127×C₄₇	C₄	1	2	yes	—
9	#127×C₄₇	C₂²	1	2	yes	—
10	C₂×C₄²×C₄₇	C₂²	1	1	yes	characteristic
11	C₄⋊C₈×C₄₇	—	2	2	no	—
12	C₄⋊C₈×C₄₇	C₂²	1	4	yes	complemented
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	C₂×C₈×C₄₇	—	2	4	no	—
24	C₂×C₈×C₄₇	—	4	4	no	—
25	C₄²×C₄₇	C₂×C₄	1	1	yes	characteristic
26	C₄²×C₄₇	D₄	1	2	yes	—
27	C₄²×C₄₇	C₂³	1	1	yes	characteristic
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	C₈×C₄₇	—	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	C₄₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
57	C₂×C₄₇	—	8	4	no	—
58	C₂×C₄₇	#331	1	4	yes	direct product factor
59	C₂×C₄₇	C₂×#100	1	1	yes	characteristic
60	C₂×C₄₇	C₂×#102	1	2	yes	—
61	C₂×#127	C₂×C₄₇	1	1	yes	characteristic
62	C₂×C₄⋊C₈	C₂×C₄₇	1	2	yes	complemented
63	#331	C₂×C₄₇	1	4	yes	direct product factor
64	C₄₇	C₂×#331	1	1	yes	characteristic, direct product factor, Sylow
65	C₂²×D₄	—	2	2	no	—
66	C₂²×D₄	C₄×C₄₇	1	2	yes	—
67	C₂²×C₈	—	2	2	no	—
68	#127	—	2	2	no	—
69	#127	C₄×C₄₇	1	2	yes	—
70	#127	C₂²×C₄₇	1	2	yes	—
71	C₂×C₄²	C₂²×C₄₇	1	1	yes	characteristic
72	C₄⋊C₈	—	2	2	no	—
73	C₄⋊C₈	C₂²×C₄₇	1	4	yes	complemented
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×C₄×C₄₇	1	4	yes	—
82	C₂²×C₄	C₂×C₄×C₄₇	1	1	yes	characteristic
83	C₂²×C₄	D₄×C₄₇	1	2	yes	—
84	C₂×C₈	—	2	4	no	—
85	C₂×C₈	—	4	4	no	—
86	C₄²	C₂×C₄×C₄₇	1	1	yes	characteristic
87	C₄²	D₄×C₄₇	1	2	yes	—
88	C₄²	C₂³×C₄₇	1	1	yes	characteristic
89	C₂³	—	2	4	no	—
90	C₂³	—	4	4	no	—
91	C₂³	—	4	8	no	—
92	C₂³	C₂²⋊C₄×C₄₇	1	1	yes	characteristic
93	D₄	—	4	4	no	—
94	D₄	—	4	4	no	—
95	D₄	—	4	4	no	—
96	D₄	—	4	4	no	—
97	D₄	—	4	8	no	—
98	C₂×C₄	—	2	2	no	—
99	C₂×C₄	C₂²⋊C₄×C₄₇	1	1	yes	characteristic
100	C₂×C₄	C₂²⋊C₄×C₄₇	1	2	yes	—
101	C₂×C₄	D₈×C₄₇	1	4	yes	—
102	C₂×C₄	SD₁₆×C₄₇	1	4	yes	—
103	C₂×C₄	C₂²×C₄×C₄₇	1	1	yes	characteristic
104	C₂×C₄	C₂×D₄×C₄₇	1	2	yes	—
105	C₈	—	4	4	no	—
106	C₂²	—	4	4	no	—
107	C₂²	—	4	8	no	—
108	C₂²	—	8	8	no	—
109	C₂²	#100×C₄₇	1	2	yes	—
110	C₂²	#102×C₄₇	1	4	yes	—
111	C₂²	C₂×C₂²⋊C₄×C₄₇	1	1	yes	characteristic
112	C₄	—	2	1	no	—
113	C₄	—	2	1	no	—
114	C₄	#102×C₄₇	1	4	yes	—
115	C₄	C₂×D₈×C₄₇	1	2	yes	—
116	C₄	C₂×SD₁₆×C₄₇	1	2	yes	—
117	C₂	—	8	4	no	—
118	C₂	C₄₇×#331	1	4	yes	direct product factor
119	C₂	C₂×#100×C₄₇	1	1	yes	characteristic
120	C₂	C₂×#102×C₄₇	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#127×C₄₇

C₂

yes

maximal, characteristic

C₂×C₄⋊C₈×C₄₇

C₂

yes

maximal, complemented

C₄₇×#331

C₂

yes

maximal, direct product factor

C₂²×D₄×C₄₇

—

C₂²×D₄×C₄₇

C₄

yes

—

C₂²×C₈×C₄₇

—

#127×C₄₇

—

#127×C₄₇

C₄

yes

—

#127×C₄₇

C₂²

yes

—

C₂×C₄²×C₄₇

C₂²

yes

characteristic

C₄⋊C₈×C₄₇

—

C₄⋊C₈×C₄₇

C₂²

yes

complemented

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

—

C₂×C₈×C₄₇

—

C₂×C₈×C₄₇

—

C₄²×C₄₇

C₂×C₄

yes

characteristic

C₄²×C₄₇

D₄

yes

—

C₄²×C₄₇

C₂³

yes

characteristic

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

—

C₈×C₄₇

—

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

C₄₇

yes

maximal, characteristic, direct product factor, Sylow

C₂×C₄₇

—

C₂×C₄₇

#331

yes

direct product factor

C₂×C₄₇

C₂×#100

yes

characteristic

C₂×C₄₇

C₂×#102

yes

—

C₂×#127

C₂×C₄₇

yes

characteristic

C₂×C₄⋊C₈

C₂×C₄₇

yes

complemented

#331

C₂×C₄₇

yes

direct product factor

C₄₇

C₂×#331

yes

characteristic, direct product factor, Sylow

C₂²×D₄

—

C₂²×D₄

C₄×C₄₇

yes

—

C₂²×C₈

—

#127

—

#127

C₄×C₄₇

yes

—

#127

C₂²×C₄₇

yes

—

C₂×C₄²

C₂²×C₄₇

yes

characteristic

C₄⋊C₈

—

C₄⋊C₈

C₂²×C₄₇

yes

complemented

C₂⁴

—

C₂×D₄

—

C₂×D₄

—

C₂×D₄

—

C₂×D₄

—

C₂×D₄

—

C₂×D₄

—

C₂×D₄

C₂×C₄×C₄₇

yes

—

C₂²×C₄

C₂×C₄×C₄₇

yes

characteristic

C₂²×C₄

D₄×C₄₇

yes

—

C₂×C₈

—

C₂×C₈

—

C₄²

C₂×C₄×C₄₇

yes

characteristic

C₄²

D₄×C₄₇

yes

—

C₄²

C₂³×C₄₇

yes

characteristic

C₂³

—

C₂³

—

C₂³

—

C₂³

C₂²⋊C₄×C₄₇

yes

characteristic

D₄

—

D₄

—

D₄

—

D₄

—

D₄

—

C₂×C₄

—

C₂×C₄

C₂²⋊C₄×C₄₇

yes

characteristic

100

C₂×C₄

C₂²⋊C₄×C₄₇

yes

—

101

C₂×C₄

D₈×C₄₇

yes

—

102

C₂×C₄

SD₁₆×C₄₇

yes

—

103

C₂×C₄

C₂²×C₄×C₄₇

yes

characteristic

104

C₂×C₄

C₂×D₄×C₄₇

yes

—

105

C₈

—

106

C₂²

—

107

C₂²

—

108

C₂²

—

109

C₂²

#100×C₄₇

yes

—

110

C₂²

#102×C₄₇

yes

—

111

C₂²

C₂×C₂²⋊C₄×C₄₇

yes

characteristic

112

C₄

—

113

C₄

—

114

C₄

#102×C₄₇

yes

—

115

C₄

C₂×D₈×C₄₇

yes

—

116

C₄

C₂×SD₁₆×C₄₇

yes

—

117

C₂

—

118

C₂

C₄₇×#331

yes

direct product factor

119

C₂

C₂×#100×C₄₇

yes

characteristic

120

C₂

C₂×#102×C₄₇

yes

—

	subgroup	G/H
center, socle	C₂³×C₄₇	C₂²⋊C₄
commutator	C₂×C₄	C₂²×C₄×C₄₇
Frattini, cosocle	C₄²	C₂³×C₄₇
prime radical	C₂×#127×C₄₇	C₂
LCS	G ▷ C₂×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³×C₄₇ ◁ C₂×C₄²×C₄₇ ◁ G

subgroup

G/H

center, socle

C₂³×C₄₇

C₂²⋊C₄

commutator

C₂×C₄

C₂²×C₄×C₄₇

Frattini, cosocle

C₄²

C₂³×C₄₇

prime radical

C₂×#127×C₄₇

C₂

LCS

G ▷ C₂×C₄ ▷ C₂² ▷ C₁

UCS

C₁ ◁ C₂³×C₄₇ ◁ C₂×C₄²×C₄₇ ◁ G

N	H ≅ G/N	type
2. C₂×C₄⋊C₈×C₄₇	C₂	semidirect
3. C₄₇×#331	C₂	direct (N × H)
12. C₄⋊C₈×C₄₇	C₂²	semidirect
56. C₂×#331	C₄₇	direct (N × H)
58. C₂×C₄₇	#331	direct (N × H)
62. C₂×C₄⋊C₈	C₂×C₄₇	semidirect
63. #331	C₂×C₄₇	direct (N × H)
64. C₄₇	C₂×#331	direct (N × H)
73. C₄⋊C₈	C₂²×C₄₇	semidirect
118. C₂	C₄₇×#331	direct (N × H)

H ≅ G/N

type

2. C₂×C₄⋊C₈×C₄₇

C₂

semidirect

3. C₄₇×#331

C₂

direct (N × H)

12. C₄⋊C₈×C₄₇

C₂²

semidirect

56. C₂×#331

C₄₇

direct (N × H)

58. C₂×C₄₇

#331

direct (N × H)

62. C₂×C₄⋊C₈

C₂×C₄₇

semidirect

63. #331

C₂×C₄₇

direct (N × H)

64. C₄₇

C₂×#331

direct (N × H)

73. C₄⋊C₈

C₂²×C₄₇

semidirect

118. C₂

C₄₇×#331

direct (N × H)

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

subgroup	index
C₂×#127×C₄₇	2
C₂×C₄⋊C₈×C₄₇	2
C₄₇×#331	2
C₂×#331	47

subgroup

index

C₂×#127×C₄₇

C₂×C₄⋊C₈×C₄₇

C₄₇×#331

C₂×#331

Subgroups — C₂×C₄₇×#331 #20239148

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C2×C47×#331 #20239148

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₄₇×#331 #20239148