Group #20244221 — subgroups

Subgroups — #20244221

Order 6016.

320 conjugacy classes of subgroups; only the 97 normal subgroups are shown.

normal subgroups only

This group has 320 conjugacy classes of subgroups — too many to draw. Only its 97 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	#15689147	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×#13093951	C₂	1	4	yes	maximal, complemented
3	#15688969	C₂	1	1	yes	maximal, characteristic, complemented
4	C₄₇×#532	C₂	1	1	yes	maximal, characteristic
5	C₂×#10625329	C₂²	1	2	yes	—
6	C₂×C₄×Dic₄₇	C₂²	1	2	yes	—
7	C₄⋊(C₄₇⋊C₈)	—	2	2	no	—
8	#13093951	—	2	8	no	—
9	C₄₇⋊#95	—	2	2	no	—
10	C₂×#41×C₄₇	C₂²	1	2	yes	—
11	C₄⋊Q₈×C₄₇	C₄	1	1	yes	characteristic, complemented
12	#127×C₄₇	C₄	1	1	yes	characteristic, complemented
13	#124×C₄₇	C₄	1	2	yes	complemented
14	#121×C₄₇	—	2	2	no	—
15	C₄×D₄×C₄₇	—	2	2	no	—
16	C₂×C₄²×C₄₇	C₂²	1	1	yes	characteristic
17	#41×C₄₇	—	2	4	no	—
18	#41×C₄₇	—	2	4	no	—
19	C₂×Q₈×C₄₇	C₂×C₄	1	2	yes	—
20	C₂×D₄×C₄₇	—	2	2	no	—
21	C₂×D₄×C₄₇	—	2	2	no	—
22	C₂×D₄×C₄₇	C₂×C₄	1	2	yes	—
23	C₂²×C₄×C₄₇	—	2	2	no	—
24	C₂²×C₄×C₄₇	D₄	1	2	yes	—
25	C₂²×C₄×C₄₇	C₂³	1	1	yes	characteristic
26	C₂²×Dic₄₇	—	2	2	no	—
27	C₄⋊C₄×C₄₇	—	2	2	no	—
28	C₂²⋊C₄×C₄₇	—	2	4	no	—
29	C₄²×C₄₇	C₂×C₄	1	1	yes	characteristic
30	C₄²×C₄₇	C₂×C₄	1	1	yes	characteristic
31	C₄²×C₄₇	D₄	1	2	yes	—
32	C₄×Dic₄₇	—	2	2	no	—
33	C₄×Dic₄₇	D₄	1	4	yes	complemented
34	#10625329	—	2	4	no	—
35	C₂×C₄₇⋊C₈	—	2	2	no	—
36	C₂³×C₄₇	—	2	2	no	—
37	C₂³×C₄₇	C₂×D₄	1	1	yes	characteristic
38	Q₈×C₄₇	—	2	4	no	—
39	D₄×C₄₇	—	2	4	no	—
40	D₄×C₄₇	—	4	4	no	—
41	D₄×C₄₇	—	4	4	no	—
42	C₂×C₄×C₄₇	—	2	2	no	—
43	C₂×C₄×C₄₇	—	4	2	no	—
44	C₂×C₄×C₄₇	—	4	2	no	—
45	C₂×C₄×C₄₇	—	4	2	no	—
46	C₂×C₄×C₄₇	C₂²⋊C₄	1	2	yes	—
47	C₂×C₄×C₄₇	C₂²⋊C₄	1	2	yes	—
48	C₂×C₄×C₄₇	C₂²×C₄	1	1	yes	characteristic
49	C₂×C₄×C₄₇	C₂×D₄	1	2	yes	—
50	C₂×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
51	C₂×C₄×C₄₇	#41	1	2	yes	—
52	C₂×Dic₄₇	—	2	4	no	—
53	C₂×Dic₄₇	—	4	4	no	—
54	C₄₇⋊C₈	—	4	2	no	—
55	C₂²×C₄₇	—	2	2	no	—
56	C₂²×C₄₇	—	4	2	no	—
57	C₂²×C₄₇	—	4	4	no	—
58	C₂²×C₄₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
59	C₂²×C₄₇	#121	1	2	yes	—
60	C₄×C₄₇	—	2	2	no	—
61	C₄×C₄₇	—	2	2	no	—
62	C₄×C₄₇	—	4	2	no	—
63	C₄×C₄₇	C₄×D₄	1	2	yes	—
64	C₄×C₄₇	#124	1	1	yes	characteristic
65	C₄×C₄₇	#127	1	1	yes	characteristic
66	Dic₄₇	—	4	4	no	—
67	#1963	—	47	1	no	maximal, Sylow
68	C₂×C₄₇	—	2	2	no	—
69	C₂×C₄₇	—	8	2	no	—
70	C₂×C₄₇	#390	1	1	yes	characteristic
71	C₂×C₄₇	#421	1	2	yes	—
72	#532	D₄₇	1	1	yes	characteristic
73	#423	—	47	1	no	—
74	C₂×#104	—	47	4	no	—
75	#376	—	47	1	no	—
76	C₄₇	#1963	1	1	yes	characteristic, complemented, Sylow
77	C₂×#41	C₂×D₄₇	1	2	yes	—
78	C₂×M₁₆	—	47	2	no	—
79	C₄⋊Q₈	Dic₄₇	1	1	yes	characteristic, complemented
80	#127	Dic₄₇	1	1	yes	characteristic, complemented
81	#124	Dic₄₇	1	2	yes	complemented
82	#121	—	2	2	no	—
83	C₄×D₄	—	2	2	no	—
84	C₂×C₄²	—	47	2	no	—
85	C₂×C₄²	C₂×D₄₇	1	1	yes	characteristic
86	C₄⋊C₈	—	94	2	no	—
87	#104	—	94	8	no	—
88	#95	—	94	2	no	—
89	#41	—	2	4	no	—
90	#41	—	2	4	no	—
91	C₂×Q₈	C₂×Dic₄₇	1	2	yes	—
92	C₂×D₄	—	2	2	no	—
93	C₂×D₄	—	2	2	no	—
94	C₂×D₄	C₂×Dic₄₇	1	2	yes	—
95	C₂²×C₄	—	2	2	no	—
96	C₂²×C₄	—	94	2	no	—
97	C₂²×C₄	C₂²⋊D₄₇	1	2	yes	—
98	C₂²×C₄	C₂²×D₄₇	1	1	yes	characteristic
99	M₁₆	—	94	4	no	—
100	C₂×C₈	—	94	2	no	—
101	C₄⋊C₄	—	2	2	no	—
102	C₂²⋊C₄	—	2	4	no	—
103	C₄²	—	47	4	no	—
104	C₄²	—	94	2	no	—
105	C₄²	C₂×Dic₄₇	1	1	yes	characteristic
106	C₄²	C₂×Dic₄₇	1	1	yes	characteristic
107	C₄²	C₂²⋊D₄₇	1	2	yes	—
108	C₂³	—	2	2	no	—
109	C₂³	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
110	Q₈	—	2	4	no	—
111	D₄	—	2	4	no	—
112	D₄	—	4	4	no	—
113	D₄	—	4	4	no	—
114	C₂×C₄	—	2	2	no	—
115	C₂×C₄	—	4	2	no	—
116	C₂×C₄	—	4	2	no	—
117	C₂×C₄	—	4	2	no	—
118	C₂×C₄	—	94	4	no	—
119	C₂×C₄	—	188	4	no	—
120	C₂×C₄	C₂²⋊Dic₄₇	1	2	yes	—
121	C₂×C₄	C₂²⋊Dic₄₇	1	2	yes	—
122	C₂×C₄	D₄×D₄₇	1	2	yes	—
123	C₂×C₄	#10625352	1	2	yes	—
124	C₂×C₄	C₂²×Dic₄₇	1	1	yes	characteristic
125	C₂×C₄	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
126	C₈	—	188	2	no	—
127	C₂²	—	2	2	no	—
128	C₂²	—	4	2	no	—
129	C₂²	—	4	4	no	—
130	C₂²	C₂×C₂²⋊Dic₄₇	1	1	yes	characteristic
131	C₂²	#13093995	1	2	yes	—
132	C₄	—	2	2	no	—
133	C₄	—	2	2	no	—
134	C₄	—	4	2	no	—
135	C₄	—	188	4	no	—
136	C₄	D₄×Dic₄₇	1	2	yes	—
137	C₄	#13094011	1	1	yes	characteristic
138	C₄	#13094020	1	1	yes	characteristic
139	C₂	—	2	2	no	—
140	C₂	—	8	2	no	—
141	C₂	#15689026	1	1	yes	characteristic
142	C₂	#15689140	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#15689147

C₂

yes

maximal, characteristic, complemented

C₂×#13093951

C₂

yes

maximal, complemented

#15688969

C₂

yes

maximal, characteristic, complemented

C₄₇×#532

C₂

yes

maximal, characteristic

C₂×#10625329

C₂²

yes

—

C₂×C₄×Dic₄₇

C₂²

yes

—

C₄⋊(C₄₇⋊C₈)

—

#13093951

—

C₄₇⋊#95

—

C₂×#41×C₄₇

C₂²

yes

—

C₄⋊Q₈×C₄₇

C₄

yes

characteristic, complemented

#127×C₄₇

C₄

yes

characteristic, complemented

#124×C₄₇

C₄

yes

complemented

#121×C₄₇

—

C₄×D₄×C₄₇

—

C₂×C₄²×C₄₇

C₂²

yes

characteristic

#41×C₄₇

—

#41×C₄₇

—

C₂×Q₈×C₄₇

C₂×C₄

yes

—

C₂×D₄×C₄₇

—

C₂×D₄×C₄₇

—

C₂×D₄×C₄₇

C₂×C₄

yes

—

C₂²×C₄×C₄₇

—

C₂²×C₄×C₄₇

D₄

yes

—

C₂²×C₄×C₄₇

C₂³

yes

characteristic

C₂²×Dic₄₇

—

C₄⋊C₄×C₄₇

—

C₂²⋊C₄×C₄₇

—

C₄²×C₄₇

C₂×C₄

yes

characteristic

C₄²×C₄₇

C₂×C₄

yes

characteristic

C₄²×C₄₇

D₄

yes

—

C₄×Dic₄₇

—

C₄×Dic₄₇

D₄

yes

complemented

#10625329

—

C₂×C₄₇⋊C₈

—

C₂³×C₄₇

—

C₂³×C₄₇

C₂×D₄

yes

characteristic

Q₈×C₄₇

—

D₄×C₄₇

—

D₄×C₄₇

—

D₄×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

C₂²⋊C₄

yes

—

C₂×C₄×C₄₇

C₂²⋊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

—

C₂×Dic₄₇

—

C₂×Dic₄₇

—

C₄₇⋊C₈

—

C₂²×C₄₇

—

C₂²×C₄₇

—

C₂²×C₄₇

—

C₂²×C₄₇

C₂×C₂²⋊C₄

yes

characteristic

C₂²×C₄₇

#121

yes

—

C₄×C₄₇

—

C₄×C₄₇

—

C₄×C₄₇

—

C₄×C₄₇

C₄×D₄

yes

—

C₄×C₄₇

yes

C₄×C₄₇

yes

Dic₄₇

—

#1963

—

maximal, Sylow

C₂×C₄₇

—

C₂×C₄₇

—

C₂×C₄₇

#390

yes

characteristic

C₂×C₄₇

#421

yes

—

#532

D₄₇

yes

—

—

—

C₄₇

#1963

yes

characteristic, complemented, Sylow

C₂×#41

C₂×D₄₇

yes

—

C₂×M₁₆

—

C₄⋊Q₈

Dic₄₇

yes

characteristic, complemented

#127

Dic₄₇

yes

characteristic, complemented

#124

Dic₄₇

yes

complemented

#121

—

C₄×D₄

—

C₂×C₄²

—

C₂×C₄²

C₂×D₄₇

yes

characteristic

C₄⋊C₈

—

—

—

—

—

C₂×Q₈

C₂×Dic₄₇

yes

—

C₂×D₄

—

C₂×D₄

—

C₂×D₄

C₂×Dic₄₇

yes

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

C₂²⋊D₄₇

yes

—

C₂²×C₄

C₂²×D₄₇

yes

characteristic

M₁₆

—

100

C₂×C₈

—

101

C₄⋊C₄

—

102

C₂²⋊C₄

—

103

C₄²

—

104

C₄²

—

105

C₄²

C₂×Dic₄₇

yes

characteristic

106

C₄²

C₂×Dic₄₇

yes

characteristic

107

C₄²

C₂²⋊D₄₇

yes

—

108

C₂³

—

109

C₂³

C₂×C₂²⋊D₄₇

yes

characteristic

110

Q₈

—

111

D₄

—

112

D₄

—

113

D₄

—

114

C₂×C₄

—

115

C₂×C₄

—

116

C₂×C₄

—

117

C₂×C₄

—

118

C₂×C₄

—

119

C₂×C₄

—

188

—

120

C₂×C₄

C₂²⋊Dic₄₇

yes

—

121

C₂×C₄

C₂²⋊Dic₄₇

yes

—

122

C₂×C₄

D₄×D₄₇

yes

—

123

C₂×C₄

#10625352

yes

—

124

C₂×C₄

C₂²×Dic₄₇

yes

characteristic

125

C₂×C₄

C₂×C₂²⋊D₄₇

yes

characteristic

126

C₈

—

188

—

127

C₂²

—

128

C₂²

—

129

C₂²

—

130

C₂²

C₂×C₂²⋊Dic₄₇

yes

characteristic

131

C₂²

#13093995

yes

—

132

C₄

—

133

C₄

—

134

C₄

—

135

C₄

—

188

—

136

C₄

D₄×Dic₄₇

yes

—

137

C₄

#13094011

yes

characteristic

138

C₄

#13094020

yes

characteristic

139

C₂

—

140

C₂

—

141

C₂

#15689026

yes

characteristic

142

C₂

#15689140

yes

—

	subgroup	G/H
center	C₂×C₄	C₂×C₂²⋊D₄₇
commutator	C₂×C₄×C₄₇	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₄₇
Fitting	C₄₇×#532	C₂
socle	C₂²×C₄₇	C₂×C₂²⋊C₄
hypercenter	#532	D₄₇
hypocenter	C₄₇	#1963
LCS	G ▷ C₂×C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #532

subgroup

G/H

center

C₂×C₄

C₂×C₂²⋊D₄₇

commutator

C₂×C₄×C₄₇

C₂²×C₄

Frattini

C₂²×C₄

C₂²×D₄₇

Fitting

C₄₇×#532

C₂

socle

C₂²×C₄₇

C₂×C₂²⋊C₄

D₄₇

C₄₇

G ▷ C₂×C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇

UCS

C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #532

N	H ≅ G/N	type
1. #15689147	C₂	semidirect
2. C₂×#13093951	C₂	semidirect
3. #15688969	C₂	semidirect
11. C₄⋊Q₈×C₄₇	C₄	semidirect
12. #127×C₄₇	C₄	semidirect
13. #124×C₄₇	C₄	semidirect
33. C₄×Dic₄₇	D₄	semidirect
76. C₄₇	#1963	semidirect
79. C₄⋊Q₈	Dic₄₇	semidirect
80. #127	Dic₄₇	semidirect
81. #124	Dic₄₇	semidirect

H ≅ G/N

type

1. #15689147

C₂

semidirect

2. C₂×#13093951

C₂

semidirect

3. #15688969

C₂

semidirect

11. C₄⋊Q₈×C₄₇

C₄

semidirect

12. #127×C₄₇

C₄

semidirect

13. #124×C₄₇

C₄

semidirect

33. C₄×Dic₄₇

D₄

semidirect

76. C₄₇

#1963

semidirect

79. C₄⋊Q₈

Dic₄₇

semidirect

80. #127

Dic₄₇

semidirect

81. #124

Dic₄₇

semidirect

Subgroups — #20244221

Conjugacy classes of subgroups

Complements