Group #20241209 — subgroups

Subgroups — #20241209

Order 6016.

202 conjugacy classes of subgroups; only the 65 normal subgroups are shown.

normal subgroups only

This group has 202 conjugacy classes of subgroups — too many to draw. Only its 65 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	#15688878	C₂	1	2	yes	maximal, complemented
2	C₄₇×#390	C₂	1	1	yes	maximal, characteristic
3	C₂²×C₄₇⋊C₈	—	2	2	no	—
4	C₂²×D₄×C₄₇	C₄	1	1	yes	characteristic
5	C₂×C₄⋊C₄×C₄₇	C₄	1	1	yes	characteristic
6	C₂×C₂²⋊C₄×C₄₇	—	2	2	no	—
7	C₂×C₄²×C₄₇	C₂²	1	1	yes	characteristic
8	C₂⁴×C₄₇	—	2	1	no	—
9	C₂×D₄×C₄₇	—	2	1	no	—
10	C₂×D₄×C₄₇	—	4	1	no	—
11	C₂×D₄×C₄₇	—	4	1	no	—
12	C₂×D₄×C₄₇	C₈	1	2	yes	complemented
13	C₂²×C₄×C₄₇	—	2	1	no	—
14	C₂²×C₄×C₄₇	C₂×C₄	1	1	yes	characteristic
15	C₂²×C₄×C₄₇	D₄	1	2	yes	—
16	C₄⋊C₄×C₄₇	—	4	1	no	—
17	C₂²⋊C₄×C₄₇	—	4	2	no	—
18	C₂²⋊C₄×C₄₇	—	4	2	no	—
19	C₄²×C₄₇	—	4	1	no	—
20	C₂×C₄₇⋊C₈	—	2	4	no	—
21	C₂×C₄₇⋊C₈	—	4	2	no	—
22	C₂×C₄₇⋊C₈	—	4	2	no	—
23	C₂³×C₄₇	—	2	2	no	—
24	C₂³×C₄₇	—	4	1	no	—
25	C₂³×C₄₇	—	4	1	no	—
26	C₂³×C₄₇	—	4	2	no	—
27	C₂³×C₄₇	—	4	2	no	—
28	C₂³×C₄₇	C₂²⋊C₄	1	1	yes	characteristic
29	D₄×C₄₇	—	4	2	no	—
30	D₄×C₄₇	—	4	2	no	—
31	C₂×C₄×C₄₇	—	2	2	no	—
32	C₂×C₄×C₄₇	—	2	1	no	—
33	C₂×C₄×C₄₇	—	2	1	no	—
34	C₂×C₄×C₄₇	—	2	2	no	—
35	C₂×C₄×C₄₇	—	4	1	no	—
36	C₂×C₄×C₄₇	—	4	2	no	—
37	C₂×C₄×C₄₇	C₂×C₈	1	1	yes	characteristic
38	C₂×C₄×C₄₇	M₁₆	1	1	yes	characteristic
39	C₂×C₄×C₄₇	D₈	1	2	yes	—
40	C₂×C₄×C₄₇	SD₁₆	1	2	yes	—
41	C₄₇⋊C₈	—	4	4	no	—
42	C₂²×C₄₇	—	4	2	no	—
43	C₂²×C₄₇	—	4	4	no	—
44	C₂²×C₄₇	—	8	2	no	—
45	C₂²×C₄₇	—	8	1	no	—
46	C₂²×C₄₇	—	8	1	no	—
47	C₂²×C₄₇	C₂²⋊C₈	1	1	yes	characteristic
48	C₂²×C₄₇	#99	1	1	yes	characteristic
49	C₂²×C₄₇	#100	1	1	yes	characteristic
50	C₂²×C₄₇	#102	1	2	yes	—
51	C₂²×C₄₇	#104	1	2	yes	—
52	C₄×C₄₇	—	2	1	no	—
53	C₄×C₄₇	—	2	1	no	—
54	C₄×C₄₇	—	2	2	no	—
55	C₄×C₄₇	—	2	2	no	—
56	C₄×C₄₇	—	8	1	no	—
57	#1270	—	47	1	no	maximal, Sylow
58	C₂×C₄₇	—	8	2	no	—
59	C₂×C₄₇	#323	1	1	yes	characteristic
60	C₂×C₄₇	D₄⋊C₈	1	2	yes	—
61	C₂×C₄₇	#327	1	2	yes	—
62	C₂×C₄₇	#329	1	1	yes	characteristic
63	C₂×C₄₇	#331	1	1	yes	characteristic
64	#390	D₄₇	1	1	yes	characteristic
65	#336	—	47	2	no	—
66	C₄₇	#1270	1	1	yes	characteristic, complemented, Sylow
67	C₂²×D₄	Dic₄₇	1	1	yes	characteristic
68	C₂²×C₈	—	94	2	no	—
69	C₂×C₄⋊C₄	Dic₄₇	1	1	yes	characteristic
70	C₂×C₂²⋊C₄	—	2	2	no	—
71	C₂×C₄²	C₂×D₄₇	1	1	yes	characteristic
72	C₂⁴	—	2	1	no	—
73	C₂×D₄	—	2	1	no	—
74	C₂×D₄	—	4	1	no	—
75	C₂×D₄	—	4	1	no	—
76	C₂×D₄	C₄₇⋊C₈	1	2	yes	complemented
77	C₂²×C₄	—	2	1	no	—
78	C₂²×C₄	C₂×Dic₄₇	1	1	yes	characteristic
79	C₂²×C₄	C₂²⋊D₄₇	1	2	yes	—
80	C₂×C₈	—	94	4	no	—
81	C₂×C₈	—	188	2	no	—
82	C₂×C₈	—	188	2	no	—
83	C₄⋊C₄	—	4	1	no	—
84	C₂²⋊C₄	—	4	2	no	—
85	C₂²⋊C₄	—	4	2	no	—
86	C₄²	—	4	1	no	—
87	C₂³	—	2	2	no	—
88	C₂³	—	4	1	no	—
89	C₂³	—	4	1	no	—
90	C₂³	—	4	2	no	—
91	C₂³	—	4	2	no	—
92	C₂³	C₂²⋊Dic₄₇	1	1	yes	characteristic
93	D₄	—	4	2	no	—
94	D₄	—	4	2	no	—
95	C₂×C₄	—	2	2	no	—
96	C₂×C₄	—	2	1	no	—
97	C₂×C₄	—	2	1	no	—
98	C₂×C₄	—	2	2	no	—
99	C₂×C₄	—	4	1	no	—
100	C₂×C₄	—	4	2	no	—
101	C₂×C₄	C₂×C₄₇⋊C₈	1	1	yes	characteristic
102	C₂×C₄	#10625329	1	1	yes	characteristic
103	C₂×C₄	#10625334	1	2	yes	—
104	C₂×C₄	#10625335	1	2	yes	—
105	C₈	—	188	4	no	—
106	C₂²	—	4	2	no	—
107	C₂²	—	4	4	no	—
108	C₂²	—	8	2	no	—
109	C₂²	—	8	1	no	—
110	C₂²	—	8	1	no	—
111	C₂²	C₂²⋊(C₄₇⋊C₈)	1	1	yes	characteristic
112	C₂²	#13093939	1	1	yes	characteristic
113	C₂²	#13093941	1	1	yes	characteristic
114	C₂²	#13093945	1	2	yes	—
115	C₂²	#13093951	1	2	yes	—
116	C₄	—	2	1	no	—
117	C₄	—	2	1	no	—
118	C₄	—	2	2	no	—
119	C₄	—	2	2	no	—
120	C₄	—	8	1	no	—
121	C₂	—	8	2	no	—
122	C₂	#15688847	1	1	yes	characteristic
123	C₂	D₄⋊(C₄₇⋊C₈)	1	2	yes	—
124	C₂	#15688857	1	2	yes	—
125	C₂	#15688863	1	1	yes	characteristic
126	C₂	#15688867	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#15688878

C₂

yes

maximal, complemented

C₄₇×#390

C₂

yes

maximal, characteristic

C₂²×C₄₇⋊C₈

—

C₂²×D₄×C₄₇

C₄

yes

characteristic

C₂×C₄⋊C₄×C₄₇

C₄

yes

characteristic

C₂×C₂²⋊C₄×C₄₇

—

C₂×C₄²×C₄₇

C₂²

yes

characteristic

C₂⁴×C₄₇

—

C₂×D₄×C₄₇

—

C₂×D₄×C₄₇

—

C₂×D₄×C₄₇

—

C₂×D₄×C₄₇

C₈

yes

complemented

C₂²×C₄×C₄₇

—

C₂²×C₄×C₄₇

C₂×C₄

yes

characteristic

C₂²×C₄×C₄₇

D₄

yes

—

C₄⋊C₄×C₄₇

—

C₂²⋊C₄×C₄₇

—

C₂²⋊C₄×C₄₇

—

C₄²×C₄₇

—

C₂×C₄₇⋊C₈

—

C₂×C₄₇⋊C₈

—

C₂×C₄₇⋊C₈

—

C₂³×C₄₇

—

C₂³×C₄₇

—

C₂³×C₄₇

—

C₂³×C₄₇

—

C₂³×C₄₇

—

C₂³×C₄₇

C₂²⋊C₄

yes

characteristic

D₄×C₄₇

—

D₄×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

C₂×C₈

yes

characteristic

C₂×C₄×C₄₇

M₁₆

yes

characteristic

C₂×C₄×C₄₇

D₈

yes

—

C₂×C₄×C₄₇

SD₁₆

yes

—

C₄₇⋊C₈

—

C₂²×C₄₇

—

C₂²×C₄₇

—

C₂²×C₄₇

—

C₂²×C₄₇

—

C₂²×C₄₇

—

C₂²×C₄₇

C₂²⋊C₈

yes

C₂²×C₄₇

yes

C₂²×C₄₇

yes

C₂²×C₄₇

yes

—

C₂²×C₄₇

#104

yes

—

C₄×C₄₇

—

C₄×C₄₇

—

C₄×C₄₇

—

C₄×C₄₇

—

C₄×C₄₇

—

#1270

—

maximal, Sylow

C₂×C₄₇

—

C₂×C₄₇

#323

yes

characteristic

C₂×C₄₇

D₄⋊C₈

yes

—

C₂×C₄₇

#327

yes

—

C₂×C₄₇

yes

C₂×C₄₇

yes

D₄₇

yes

characteristic

#336

—

C₄₇

#1270

yes

characteristic, complemented, Sylow

C₂²×D₄

Dic₄₇

yes

characteristic

C₂²×C₈

—

C₂×C₄⋊C₄

Dic₄₇

yes

characteristic

C₂×C₂²⋊C₄

—

C₂×C₄²

C₂×D₄₇

yes

characteristic

C₂⁴

—

C₂×D₄

—

C₂×D₄

—

C₂×D₄

—

C₂×D₄

C₄₇⋊C₈

yes

complemented

C₂²×C₄

—

C₂²×C₄

C₂×Dic₄₇

yes

characteristic

C₂²×C₄

C₂²⋊D₄₇

yes

—

C₂×C₈

—

C₂×C₈

—

188

—

C₂×C₈

—

188

—

C₄⋊C₄

—

C₂²⋊C₄

—

C₂²⋊C₄

—

C₄²

—

C₂³

—

C₂³

—

C₂³

—

C₂³

—

C₂³

—

C₂³

C₂²⋊Dic₄₇

yes

characteristic

D₄

—

D₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

100

C₂×C₄

—

101

C₂×C₄

C₂×C₄₇⋊C₈

yes

characteristic

102

C₂×C₄

#10625329

yes

characteristic

103

C₂×C₄

#10625334

yes

—

104

C₂×C₄

#10625335

yes

—

105

C₈

—

188

—

106

C₂²

—

107

C₂²

—

108

C₂²

—

109

C₂²

—

110

C₂²

—

111

C₂²

C₂²⋊(C₄₇⋊C₈)

yes

characteristic

112

C₂²

#13093939

yes

characteristic

113

C₂²

#13093941

yes

characteristic

114

C₂²

#13093945

yes

—

115

C₂²

#13093951

yes

—

116

C₄

—

117

C₄

—

118

C₄

—

119

C₄

—

120

C₄

—

121

C₂

—

122

C₂

#15688847

yes

characteristic

123

C₂

D₄⋊(C₄₇⋊C₈)

yes

—

124

C₂

#15688857

yes

—

125

C₂

#15688863

yes

characteristic

126

C₂

#15688867

yes

characteristic

	subgroup	G/H
center	C₂³	C₂²⋊Dic₄₇
commutator	C₂×C₄×C₄₇	C₂×C₈
Frattini	C₂×C₄²	C₂×D₄₇
Fitting	C₄₇×#390	C₂
socle	C₂³×C₄₇	C₂²⋊C₄
hypercenter	#390	D₄₇
hypocenter	C₄₇	#1270
LCS	G ▷ C₂×C₄×C₄₇ ▷ C₂²×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #390

subgroup

G/H

center

C₂³

C₂²⋊Dic₄₇

commutator

C₂×C₄×C₄₇

C₂×C₈

Frattini

C₂×C₄²

C₂×D₄₇

Fitting

C₄₇×#390

C₂

socle

C₂³×C₄₇

C₂²⋊C₄

D₄₇

C₄₇

G ▷ C₂×C₄×C₄₇ ▷ C₂²×C₄₇ ▷ C₄₇

UCS

C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #390

N	H ≅ G/N	type
1. #15688878	C₂	semidirect
12. C₂×D₄×C₄₇	C₈	semidirect
66. C₄₇	#1270	semidirect
76. C₂×D₄	C₄₇⋊C₈	semidirect

H ≅ G/N

type

1. #15688878

C₂

semidirect

12. C₂×D₄×C₄₇

C₈

semidirect

66. C₄₇

#1270

semidirect

76. C₂×D₄

C₄₇⋊C₈

semidirect

Subgroups — #20241209

Conjugacy classes of subgroups

Complements