Group #20241297 — subgroups

Subgroups — #20241297

Order 6016.

192 conjugacy classes of subgroups; only the 57 normal subgroups are shown.

normal subgroups only

This group has 192 conjugacy classes of subgroups — too many to draw. Only its 57 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	#15689087	C₂	1	1	yes	maximal, characteristic
2	#15688986	C₂	1	1	yes	maximal, characteristic
3	C₄₇×#407	C₂	1	1	yes	maximal, characteristic
4	C₂×#10625329	—	2	1	no	—
5	C₂×C₂²⋊Dic₄₇	C₄	1	2	yes	—
6	C₂²⋊(C₄₇⋊C₈)	—	2	1	no	—
7	C₂²⋊(C₄₇⋊C₈)	C₄	1	2	yes	complemented
8	C₄₇⋊#95	—	2	1	no	—
9	C₄₇⋊#95	—	2	1	no	—
10	C₂³×C₄×C₄₇	C₂²	1	1	yes	characteristic
11	C₂×M₁₆×C₄₇	—	2	1	no	—
12	C₂²⋊C₈×C₄₇	—	2	1	no	—
13	C₂²⋊C₈×C₄₇	C₄	1	2	yes	complemented
14	C₂⁴×C₄₇	C₂×C₄	1	1	yes	characteristic
15	C₂²×C₄×C₄₇	—	4	2	no	—
16	C₂²×C₄×C₄₇	C₂×C₄	1	1	yes	characteristic
17	C₂²×C₄×C₄₇	C₂×C₄	1	1	yes	characteristic
18	C₂²×C₄×C₄₇	D₄	1	1	yes	characteristic
19	C₂²×C₄×C₄₇	D₄	1	1	yes	characteristic
20	C₂²×C₄×C₄₇	D₄	1	1	yes	characteristic
21	C₂²×C₄×C₄₇	Q₈	1	1	yes	characteristic
22	C₂²×Dic₄₇	—	2	2	no	—
23	M₁₆×C₄₇	—	4	2	no	—
24	C₂×C₈×C₄₇	—	2	2	no	—
25	C₂²⋊Dic₄₇	—	4	2	no	—
26	#10625329	—	4	2	no	—
27	C₂×C₄₇⋊C₈	—	2	2	no	—
28	C₂³×C₄₇	—	2	2	no	—
29	C₂³×C₄₇	—	4	1	no	—
30	C₂³×C₄₇	—	4	1	no	—
31	C₂³×C₄₇	C₄²	1	1	yes	characteristic
32	C₂³×C₄₇	C₂²⋊C₄	1	1	yes	characteristic
33	C₂³×C₄₇	C₄⋊C₄	1	1	yes	characteristic
34	C₂×C₄×C₄₇	—	2	2	no	—
35	C₂×C₄×C₄₇	—	2	2	no	—
36	C₂×C₄×C₄₇	—	4	1	no	—
37	C₂×C₄×C₄₇	—	4	1	no	—
38	C₂×C₄×C₄₇	—	4	1	no	—
39	C₂×C₄×C₄₇	—	4	1	no	—
40	C₂×C₄×C₄₇	C₂²⋊C₄	1	1	yes	characteristic
41	C₂×C₄×C₄₇	C₂²⋊C₄	1	1	yes	characteristic
42	C₂×C₄×C₄₇	C₄⋊C₄	1	1	yes	characteristic
43	C₂×C₄×C₄₇	C₄⋊C₄	1	1	yes	characteristic
44	C₂×Dic₄₇	—	4	2	no	—
45	C₂×Dic₄₇	—	4	2	no	—
46	C₂×Dic₄₇	—	4	2	no	—
47	C₈×C₄₇	—	4	2	no	—
48	C₄₇⋊C₈	—	4	2	no	—
49	C₂²×C₄₇	—	2	1	no	—
50	C₂²×C₄₇	—	2	1	no	—
51	C₂²×C₄₇	—	2	1	no	—
52	C₂²×C₄₇	—	2	1	no	—
53	C₂²×C₄₇	—	2	1	no	—
54	C₂²×C₄₇	—	2	1	no	—
55	C₂²×C₄₇	—	2	1	no	—
56	C₂²×C₄₇	—	2	1	no	—
57	C₂²×C₄₇	—	4	4	no	—
58	C₂²×C₄₇	#95	1	1	yes	characteristic
59	C₂²×C₄₇	#99	1	2	yes	—
60	C₄×C₄₇	—	2	1	no	—
61	C₄×C₄₇	—	2	1	no	—
62	C₄×C₄₇	—	2	1	no	—
63	C₄×C₄₇	—	2	1	no	—
64	Dic₄₇	—	8	2	no	—
65	#1306	—	47	1	no	maximal, Sylow
66	C₂×C₄₇	—	2	2	no	—
67	C₂×C₄₇	—	4	1	no	—
68	C₂×C₄₇	—	4	1	no	—
69	C₂×C₄₇	#342	1	1	yes	characteristic
70	C₂×C₄₇	#344	1	1	yes	characteristic
71	C₂×C₄₇	#344	1	1	yes	characteristic
72	#407	—	47	1	no	—
73	#407	D₄₇	1	1	yes	characteristic
74	#381	—	47	1	no	—
75	C₄₇	#1306	1	1	yes	characteristic, complemented, Sylow
76	C₂³×C₄	C₂×D₄₇	1	1	yes	characteristic
77	C₂×M₁₆	—	2	1	no	—
78	C₂×M₁₆	—	94	1	no	—
79	C₂×C₂²⋊C₄	—	47	2	no	—
80	C₂²⋊C₈	—	2	1	no	—
81	C₂²⋊C₈	—	47	2	no	—
82	C₂²⋊C₈	—	94	1	no	—
83	C₂²⋊C₈	Dic₄₇	1	2	yes	complemented
84	#95	—	94	1	no	—
85	#95	—	94	1	no	—
86	C₂⁴	C₄×D₄₇	1	1	yes	characteristic
87	C₂²×C₄	—	4	2	no	—
88	C₂²×C₄	—	94	2	no	—
89	C₂²×C₄	Dic₉₄	1	1	yes	characteristic
90	C₂²×C₄	C₄×D₄₇	1	1	yes	characteristic
91	C₂²×C₄	D₁₈₈	1	1	yes	characteristic
92	C₂²×C₄	C₂×Dic₄₇	1	1	yes	characteristic
93	C₂²×C₄	C₂²⋊D₄₇	1	1	yes	characteristic
94	C₂²×C₄	C₂²⋊D₄₇	1	1	yes	characteristic
95	M₁₆	—	4	2	no	—
96	M₁₆	—	188	2	no	—
97	C₂×C₈	—	2	2	no	—
98	C₂×C₈	—	94	2	no	—
99	C₂²⋊C₄	—	188	2	no	—
100	C₂³	—	2	2	no	—
101	C₂³	—	4	1	no	—
102	C₂³	—	4	1	no	—
103	C₂³	C₄×Dic₄₇	1	1	yes	characteristic
104	C₂³	Dic₄₇⋊C₄	1	1	yes	characteristic
105	C₂³	(C₂×D₄₇)⋊C₄	1	1	yes	characteristic
106	C₂×C₄	—	2	2	no	—
107	C₂×C₄	—	2	2	no	—
108	C₂×C₄	—	4	1	no	—
109	C₂×C₄	—	4	1	no	—
110	C₂×C₄	—	4	1	no	—
111	C₂×C₄	—	4	1	no	—
112	C₂×C₄	—	188	2	no	—
113	C₂×C₄	—	188	2	no	—
114	C₂×C₄	—	188	2	no	—
115	C₂×C₄	Dic₄₇⋊C₄	1	1	yes	characteristic
116	C₂×C₄	C₄⋊Dic₄₇	1	1	yes	characteristic
117	C₂×C₄	(C₂×D₄₇)⋊C₄	1	1	yes	characteristic
118	C₂×C₄	C₂²⋊Dic₄₇	1	1	yes	characteristic
119	C₈	—	4	2	no	—
120	C₈	—	188	2	no	—
121	C₂²	—	2	1	no	—
122	C₂²	—	2	1	no	—
123	C₂²	—	2	1	no	—
124	C₂²	—	2	1	no	—
125	C₂²	—	2	1	no	—
126	C₂²	—	2	1	no	—
127	C₂²	—	2	1	no	—
128	C₂²	—	2	1	no	—
129	C₂²	—	4	4	no	—
130	C₂²	C₄₇⋊#95	1	1	yes	characteristic
131	C₂²	#13093938	1	2	yes	—
132	C₄	—	2	1	no	—
133	C₄	—	2	1	no	—
134	C₄	—	2	1	no	—
135	C₄	—	2	1	no	—
136	C₄	—	376	2	no	—
137	C₂	—	2	2	no	—
138	C₂	—	4	1	no	—
139	C₂	—	4	1	no	—
140	C₂	#15688889	1	1	yes	characteristic
141	C₂	#15688894	1	1	yes	characteristic
142	C₂	#15688895	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#15689087

C₂

yes

maximal, characteristic

#15688986

C₂

yes

maximal, characteristic

C₄₇×#407

C₂

yes

maximal, characteristic

C₂×#10625329

—

C₂×C₂²⋊Dic₄₇

C₄

yes

—

C₂²⋊(C₄₇⋊C₈)

—

C₂²⋊(C₄₇⋊C₈)

C₄

yes

complemented

C₄₇⋊#95

—

C₄₇⋊#95

—

C₂³×C₄×C₄₇

C₂²

yes

characteristic

C₂×M₁₆×C₄₇

—

C₂²⋊C₈×C₄₇

—

C₂²⋊C₈×C₄₇

C₄

yes

complemented

C₂⁴×C₄₇

C₂×C₄

yes

characteristic

C₂²×C₄×C₄₇

—

C₂²×C₄×C₄₇

C₂×C₄

yes

characteristic

C₂²×C₄×C₄₇

C₂×C₄

yes

characteristic

C₂²×C₄×C₄₇

D₄

yes

characteristic

C₂²×C₄×C₄₇

D₄

yes

characteristic

C₂²×C₄×C₄₇

D₄

yes

characteristic

C₂²×C₄×C₄₇

Q₈

yes

characteristic

C₂²×Dic₄₇

—

M₁₆×C₄₇

—

C₂×C₈×C₄₇

—

C₂²⋊Dic₄₇

—

#10625329

—

C₂×C₄₇⋊C₈

—

C₂³×C₄₇

—

C₂³×C₄₇

—

C₂³×C₄₇

—

C₂³×C₄₇

C₄²

yes

characteristic

C₂³×C₄₇

C₂²⋊C₄

yes

characteristic

C₂³×C₄₇

C₄⋊C₄

yes

characteristic

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

C₂²⋊C₄

yes

characteristic

C₂×C₄×C₄₇

C₂²⋊C₄

yes

characteristic

C₂×C₄×C₄₇

C₄⋊C₄

yes

characteristic

C₂×C₄×C₄₇

C₄⋊C₄

yes

characteristic

C₂×Dic₄₇

—

C₂×Dic₄₇

—

C₂×Dic₄₇

—

C₈×C₄₇

—

C₄₇⋊C₈

—

C₂²×C₄₇

—

C₂²×C₄₇

—

C₂²×C₄₇

—

C₂²×C₄₇

—

C₂²×C₄₇

—

C₂²×C₄₇

—

C₂²×C₄₇

—

C₂²×C₄₇

—

C₂²×C₄₇

—

C₂²×C₄₇

#95

yes

characteristic

C₂²×C₄₇

#99

yes

—

C₄×C₄₇

—

C₄×C₄₇

—

C₄×C₄₇

—

C₄×C₄₇

—

Dic₄₇

—

#1306

—

maximal, Sylow

C₂×C₄₇

—

C₂×C₄₇

—

C₂×C₄₇

—

C₂×C₄₇

yes

C₂×C₄₇

yes

C₂×C₄₇

yes

—

D₄₇

yes

characteristic

#381

—

C₄₇

#1306

yes

characteristic, complemented, Sylow

C₂³×C₄

C₂×D₄₇

yes

characteristic

C₂×M₁₆

—

C₂×M₁₆

—

C₂×C₂²⋊C₄

—

C₂²⋊C₈

—

C₂²⋊C₈

—

C₂²⋊C₈

—

C₂²⋊C₈

Dic₄₇

yes

complemented

#95

—

#95

—

C₂⁴

C₄×D₄₇

yes

characteristic

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

Dic₉₄

yes

characteristic

C₂²×C₄

C₄×D₄₇

yes

characteristic

C₂²×C₄

D₁₈₈

yes

characteristic

C₂²×C₄

C₂×Dic₄₇

yes

characteristic

C₂²×C₄

C₂²⋊D₄₇

yes

characteristic

C₂²×C₄

C₂²⋊D₄₇

yes

characteristic

M₁₆

—

M₁₆

—

188

—

C₂×C₈

—

C₂×C₈

—

C₂²⋊C₄

—

188

—

100

C₂³

—

101

C₂³

—

102

C₂³

—

103

C₂³

C₄×Dic₄₇

yes

characteristic

104

C₂³

Dic₄₇⋊C₄

yes

characteristic

105

C₂³

(C₂×D₄₇)⋊C₄

yes

characteristic

106

C₂×C₄

—

107

C₂×C₄

—

108

C₂×C₄

—

109

C₂×C₄

—

110

C₂×C₄

—

111

C₂×C₄

—

112

C₂×C₄

—

188

—

113

C₂×C₄

—

188

—

114

C₂×C₄

—

188

—

115

C₂×C₄

Dic₄₇⋊C₄

yes

characteristic

116

C₂×C₄

C₄⋊Dic₄₇

yes

characteristic

117

C₂×C₄

(C₂×D₄₇)⋊C₄

yes

characteristic

118

C₂×C₄

C₂²⋊Dic₄₇

yes

characteristic

119

C₈

—

120

C₈

—

188

—

121

C₂²

—

122

C₂²

—

123

C₂²

—

124

C₂²

—

125

C₂²

—

126

C₂²

—

127

C₂²

—

128

C₂²

—

129

C₂²

—

130

C₂²

C₄₇⋊#95

yes

characteristic

131

C₂²

#13093938

yes

—

132

C₄

—

133

C₄

—

134

C₄

—

135

C₄

—

136

C₄

—

376

—

137

C₂

—

138

C₂

—

139

C₂

—

140

C₂

#15688889

yes

characteristic

141

C₂

#15688894

yes

characteristic

142

C₂

#15688895

yes

characteristic

	subgroup	G/H
center	C₂²	C₄₇⋊#95
commutator	C₂³×C₄₇	C₄²
Frattini	C₂³×C₄	C₂×D₄₇
Fitting	C₄₇×#407	C₂
socle	C₂²×C₄₇	#95
hypercenter	#407	D₄₇
hypocenter	C₄₇	#1306
LCS	G ▷ C₂³×C₄₇ ▷ C₂²×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ #407

subgroup

G/H

center

C₂²

C₄₇⋊#95

commutator

C₂³×C₄₇

C₄²

Frattini

C₂³×C₄

C₂×D₄₇

Fitting

C₄₇×#407

C₂

C₂²×C₄₇

D₄₇

C₄₇

G ▷ C₂³×C₄₇ ▷ C₂²×C₄₇ ▷ C₄₇

UCS

C₁ ◁ C₂² ◁ C₂³×C₄ ◁ #407

N	H ≅ G/N	type
7. C₂²⋊(C₄₇⋊C₈)	C₄	semidirect
13. C₂²⋊C₈×C₄₇	C₄	semidirect
75. C₄₇	#1306	semidirect
83. C₂²⋊C₈	Dic₄₇	semidirect

H ≅ G/N

type

7. C₂²⋊(C₄₇⋊C₈)

C₄

semidirect

13. C₂²⋊C₈×C₄₇

C₄

semidirect

75. C₄₇

#1306

semidirect

83. C₂²⋊C₈

Dic₄₇

semidirect

Subgroups — #20241297

Conjugacy classes of subgroups

Complements