Group #20243608 — subgroups

Subgroups — #20243608

Order 6016.

252 conjugacy classes of subgroups; only the 89 normal subgroups are shown.

normal subgroups only

This group has 252 conjugacy classes of subgroups — too many to draw. Only its 89 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	#15689171	C₂	1	2	yes	maximal, complemented
2	#15689107	C₂	1	1	yes	maximal, characteristic
3	#15689090	C₂	1	1	yes	maximal, characteristic
4	#15688887	C₂	1	2	yes	maximal, complemented
5	C₄₇×#568	C₂	1	1	yes	maximal, characteristic
6	C₂×#10625329	C₂²	1	1	yes	characteristic
7	C₂×#10625329	C₂²	1	2	yes	—
8	C₂²×C₄₇⋊C₈	C₂²	1	1	yes	characteristic
9	#13093959	—	2	4	no	—
10	#13093943	C₄	1	2	yes	—
11	#13093941	C₄	1	2	yes	—
12	C₂²⋊(C₄₇⋊C₈)	—	2	2	no	—
13	C₂×#41×C₄₇	C₂²	1	1	yes	characteristic
14	#131×C₄₇	—	2	4	no	—
15	C₂×M₁₆×C₄₇	—	2	2	no	—
16	C₂×M₁₆×C₄₇	C₂²	1	2	yes	—
17	#41×C₄₇	—	2	2	no	—
18	#41×C₄₇	—	2	2	no	—
19	C₂×Q₈×C₄₇	C₂×C₄	1	1	yes	characteristic
20	C₂×D₄×C₄₇	C₂×C₄	1	1	yes	characteristic
21	C₂×D₄×C₄₇	Q₈	1	2	yes	—
22	C₂²×C₄×C₄₇	D₄	1	2	yes	—
23	C₂²×C₄×C₄₇	C₂³	1	1	yes	characteristic
24	M₁₆×C₄₇	—	2	4	no	—
25	M₁₆×C₄₇	—	2	2	no	—
26	M₁₆×C₄₇	D₄	1	4	yes	—
27	C₂×C₈×C₄₇	—	2	4	no	—
28	C₂×C₈×C₄₇	—	2	2	no	—
29	#10625329	—	2	2	no	—
30	#10625329	—	2	2	no	—
31	#10625329	C₂×C₄	1	4	yes	—
32	C₂×C₄₇⋊C₈	—	2	2	no	—
33	C₂×C₄₇⋊C₈	—	2	1	no	—
34	C₂×C₄₇⋊C₈	—	2	1	no	—
35	C₂×C₄₇⋊C₈	—	2	2	no	—
36	C₂³×C₄₇	C₄⋊C₄	1	2	yes	—
37	C₂³×C₄₇	C₂×Q₈	1	1	yes	characteristic
38	Q₈×C₄₇	—	2	2	no	—
39	D₄×C₄₇	—	2	4	no	—
40	D₄×C₄₇	—	2	2	no	—
41	C₂×C₄×C₄₇	—	2	2	no	—
42	C₂×C₄×C₄₇	—	2	2	no	—
43	C₂×C₄×C₄₇	C₄⋊C₄	1	2	yes	—
44	C₂×C₄×C₄₇	C₂²×C₄	1	1	yes	characteristic
45	C₂×C₄×C₄₇	C₂×D₄	1	2	yes	—
46	C₂×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
47	C₂×C₄×C₄₇	#41	1	2	yes	—
48	C₈×C₄₇	—	2	4	no	—
49	C₄₇⋊C₈	—	2	4	no	—
50	C₄₇⋊C₈	—	4	1	no	—
51	C₄₇⋊C₈	—	4	1	no	—
52	C₂²×C₄₇	—	2	2	no	—
53	C₂²×C₄₇	—	4	1	no	—
54	C₂²×C₄₇	—	4	2	no	—
55	C₂²×C₄₇	C₂×C₄⋊C₄	1	1	yes	characteristic
56	C₂²×C₄₇	C₂²⋊Q₈	1	2	yes	—
57	C₄×C₄₇	—	2	2	no	—
58	C₄×C₄₇	C₄×D₄	1	2	yes	—
59	C₄×C₄₇	#120	1	1	yes	characteristic
60	C₄×C₄₇	#123	1	1	yes	characteristic
61	#1858	—	47	1	no	maximal, Sylow
62	C₂×C₄₇	—	2	1	no	—
63	C₂×C₄₇	—	2	2	no	—
64	C₂×C₄₇	—	4	2	no	—
65	C₂×C₄₇	#385	1	1	yes	characteristic
66	#568	D₄₇	1	1	yes	characteristic
67	#430	—	47	2	no	—
68	#413	—	47	1	no	—
69	#408	—	47	1	no	—
70	#341	—	47	2	no	—
71	C₄₇	#1858	1	1	yes	characteristic, complemented, Sylow
72	C₂×#41	C₂×D₄₇	1	1	yes	characteristic
73	#131	—	2	4	no	—
74	C₂×M₁₆	—	2	2	no	—
75	C₂×M₁₆	—	47	2	no	—
76	C₂×M₁₆	—	47	1	no	—
77	C₂×M₁₆	C₂×D₄₇	1	2	yes	—
78	C₂²×C₈	—	47	1	no	—
79	#108	—	94	4	no	—
80	#101	—	47	2	no	—
81	#100	—	47	2	no	—
82	C₂²⋊C₈	—	94	2	no	—
83	#41	—	2	2	no	—
84	#41	—	2	2	no	—
85	C₂×Q₈	C₄×D₄₇	1	1	yes	characteristic
86	C₂×D₄	Dic₉₄	1	2	yes	—
87	C₂×D₄	C₄×D₄₇	1	1	yes	characteristic
88	C₂²×C₄	C₂²⋊D₄₇	1	2	yes	—
89	C₂²×C₄	C₂²×D₄₇	1	1	yes	characteristic
90	M₁₆	—	2	4	no	—
91	M₁₆	—	2	2	no	—
92	M₁₆	—	47	4	no	—
93	M₁₆	—	94	2	no	—
94	M₁₆	—	94	2	no	—
95	M₁₆	C₂²⋊D₄₇	1	4	yes	—
96	C₂×C₈	—	2	4	no	—
97	C₂×C₈	—	2	2	no	—
98	C₂×C₈	—	94	2	no	—
99	C₂×C₈	—	94	2	no	—
100	C₂×C₈	—	94	1	no	—
101	C₂×C₈	—	94	1	no	—
102	C₂³	Dic₄₇⋊C₄	1	2	yes	—
103	C₂³	C₂×Dic₉₄	1	1	yes	characteristic
104	Q₈	—	2	2	no	—
105	D₄	—	2	4	no	—
106	D₄	—	2	2	no	—
107	C₂×C₄	—	2	2	no	—
108	C₂×C₄	—	2	2	no	—
109	C₂×C₄	Dic₄₇⋊C₄	1	2	yes	—
110	C₂×C₄	C₂×C₄×D₄₇	1	1	yes	characteristic
111	C₂×C₄	#10625350	1	2	yes	—
112	C₂×C₄	C₂×C₂²⋊D₄₇	1	2	yes	—
113	C₂×C₄	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
114	C₈	—	2	4	no	—
115	C₈	—	94	4	no	—
116	C₈	—	188	1	no	—
117	C₈	—	188	1	no	—
118	C₂²	—	2	2	no	—
119	C₂²	—	4	1	no	—
120	C₂²	—	4	2	no	—
121	C₂²	C₂×Dic₄₇⋊C₄	1	1	yes	characteristic
122	C₂²	#13094003	1	2	yes	—
123	C₄	—	2	2	no	—
124	C₄	C₄×C₂²⋊D₄₇	1	2	yes	—
125	C₄	#13093993	1	1	yes	characteristic
126	C₄	#13094007	1	1	yes	characteristic
127	C₂	—	2	1	no	—
128	C₂	—	2	2	no	—
129	C₂	—	4	2	no	—
130	C₂	#15688996	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#15689171

C₂

yes

maximal, complemented

#15689107

C₂

yes

maximal, characteristic

#15689090

C₂

yes

maximal, characteristic

#15688887

C₂

yes

maximal, complemented

C₄₇×#568

C₂

yes

maximal, characteristic

C₂×#10625329

C₂²

yes

characteristic

C₂×#10625329

C₂²

yes

—

C₂²×C₄₇⋊C₈

C₂²

yes

characteristic

#13093959

—

#13093943

C₄

yes

—

#13093941

C₄

yes

—

C₂²⋊(C₄₇⋊C₈)

—

C₂×#41×C₄₇

C₂²

yes

characteristic

#131×C₄₇

—

C₂×M₁₆×C₄₇

—

C₂×M₁₆×C₄₇

C₂²

yes

—

#41×C₄₇

—

#41×C₄₇

—

C₂×Q₈×C₄₇

C₂×C₄

yes

characteristic

C₂×D₄×C₄₇

C₂×C₄

yes

characteristic

C₂×D₄×C₄₇

Q₈

yes

—

C₂²×C₄×C₄₇

D₄

yes

—

C₂²×C₄×C₄₇

C₂³

yes

characteristic

M₁₆×C₄₇

—

M₁₆×C₄₇

—

M₁₆×C₄₇

D₄

yes

—

C₂×C₈×C₄₇

—

C₂×C₈×C₄₇

—

#10625329

—

#10625329

—

#10625329

C₂×C₄

yes

—

C₂×C₄₇⋊C₈

—

C₂×C₄₇⋊C₈

—

C₂×C₄₇⋊C₈

—

C₂×C₄₇⋊C₈

—

C₂³×C₄₇

C₄⋊C₄

yes

—

C₂³×C₄₇

C₂×Q₈

yes

characteristic

Q₈×C₄₇

—

D₄×C₄₇

—

D₄×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

C₄⋊C₄

yes

—

C₂×C₄×C₄₇

C₂²×C₄

yes

characteristic

C₂×C₄×C₄₇

C₂×D₄

yes

—

C₂×C₄×C₄₇

C₂×D₄

yes

characteristic

C₂×C₄×C₄₇

#41

yes

—

C₈×C₄₇

—

C₄₇⋊C₈

—

C₄₇⋊C₈

—

C₄₇⋊C₈

—

C₂²×C₄₇

—

C₂²×C₄₇

—

C₂²×C₄₇

—

C₂²×C₄₇

C₂×C₄⋊C₄

yes

characteristic

C₂²×C₄₇

C₂²⋊Q₈

yes

—

C₄×C₄₇

—

C₄×C₄₇

C₄×D₄

yes

—

C₄×C₄₇

yes

C₄×C₄₇

yes

—

C₂×C₄₇

—

C₂×C₄₇

—

C₂×C₄₇

—

C₂×C₄₇

#385

yes

characteristic

#568

D₄₇

yes

—

—

—

—

C₄₇

#1858

yes

characteristic, complemented, Sylow

C₂×#41

C₂×D₄₇

yes

characteristic

#131

—

C₂×M₁₆

—

C₂×M₁₆

—

C₂×M₁₆

—

C₂×M₁₆

C₂×D₄₇

yes

—

C₂²×C₈

—

#108

—

#101

—

#100

—

C₂²⋊C₈

—

#41

—

#41

—

C₂×Q₈

C₄×D₄₇

yes

characteristic

C₂×D₄

Dic₉₄

yes

—

C₂×D₄

C₄×D₄₇

yes

characteristic

C₂²×C₄

C₂²⋊D₄₇

yes

—

C₂²×C₄

C₂²×D₄₇

yes

characteristic

M₁₆

—

M₁₆

—

M₁₆

—

M₁₆

—

M₁₆

—

M₁₆

C₂²⋊D₄₇

yes

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

100

C₂×C₈

—

101

C₂×C₈

—

102

C₂³

Dic₄₇⋊C₄

yes

—

103

C₂³

C₂×Dic₉₄

yes

characteristic

104

Q₈

—

105

D₄

—

106

D₄

—

107

C₂×C₄

—

108

C₂×C₄

—

109

C₂×C₄

Dic₄₇⋊C₄

yes

—

110

C₂×C₄

C₂×C₄×D₄₇

yes

characteristic

111

C₂×C₄

#10625350

yes

—

112

C₂×C₄

C₂×C₂²⋊D₄₇

yes

—

113

C₂×C₄

C₂×C₂²⋊D₄₇

yes

characteristic

114

C₈

—

115

C₈

—

116

C₈

—

188

—

117

C₈

—

188

—

118

C₂²

—

119

C₂²

—

120

C₂²

—

121

C₂²

C₂×Dic₄₇⋊C₄

yes

characteristic

122

C₂²

#13094003

yes

—

123

C₄

—

124

C₄

C₄×C₂²⋊D₄₇

yes

—

125

C₄

#13093993

yes

characteristic

126

C₄

#13094007

yes

characteristic

127

C₂

—

128

C₂

—

129

C₂

—

130

C₂

#15688996

yes

characteristic

	subgroup	G/H
center	C₄	#13093993
commutator	C₂×C₄×C₄₇	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₄₇
Fitting	C₄₇×#568	C₂
socle	C₂×C₄₇	#385
hypercenter	#568	D₄₇
hypocenter	C₄₇	#1858
LCS	G ▷ C₂×C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ #568

subgroup

G/H

center

C₄

#13093993

commutator

C₂×C₄×C₄₇

C₂²×C₄

Frattini

C₂²×C₄

C₂²×D₄₇

Fitting

C₄₇×#568

C₂

C₂×C₄₇

D₄₇

C₄₇

G ▷ C₂×C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇

UCS

C₁ ◁ C₄ ◁ C₂²×C₄ ◁ #568

N	H ≅ G/N	type
1. #15689171	C₂	semidirect
4. #15688887	C₂	semidirect
71. C₄₇	#1858	semidirect

H ≅ G/N

type

1. #15689171

C₂

semidirect

4. #15688887

C₂

semidirect

71. C₄₇

#1858

semidirect

Subgroups — #20243608

Conjugacy classes of subgroups

Complements