Group #20241408 — subgroups

Subgroups — #20241408

Order 6016.

162 conjugacy classes of subgroups; only the 35 normal subgroups are shown.

normal subgroups only

This group has 162 conjugacy classes of subgroups — too many to draw. Only its 35 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	#15689035	C₂	1	1	yes	maximal, characteristic
2	#15688847	C₂	1	1	yes	maximal, characteristic
3	C₄₇×#342	C₂	1	1	yes	maximal, characteristic
4	C₂²×Dic₉₄	C₄	1	1	yes	characteristic, complemented
5	C₂×C₂²⋊Dic₄₇	—	2	1	no	—
6	C₂²⋊(C₄₇⋊C₈)	—	2	1	no	—
7	C₄₇⋊#95	—	2	1	no	—
8	C₄₇⋊#95	C₄	1	1	yes	characteristic, complemented
9	C₂×C₂²⋊C₄×C₄₇	—	2	1	no	—
10	C₂×C₂²⋊C₄×C₄₇	C₂²	1	1	yes	characteristic
11	C₂⁴×C₄₇	D₄	1	1	yes	characteristic
12	C₂²×C₄×C₄₇	—	4	1	no	—
13	C₂²×C₄×C₄₇	C₂×C₄	1	1	yes	characteristic
14	C₂²×C₄×C₄₇	D₄	1	1	yes	characteristic
15	C₂²×Dic₄₇	—	2	1	no	—
16	C₂²×Dic₄₇	—	2	1	no	—
17	C₂×Dic₉₄	—	4	1	no	—
18	C₂×Dic₉₄	—	4	2	no	—
19	C₂²⋊C₄×C₄₇	—	2	2	no	—
20	C₂²⋊C₄×C₄₇	—	4	1	no	—
21	C₂²⋊C₄×C₄₇	—	4	1	no	—
22	C₂²⋊Dic₄₇	—	4	2	no	—
23	C₂×C₄₇⋊C₈	—	4	1	no	—
24	C₂³×C₄₇	—	2	1	no	—
25	C₂³×C₄₇	—	2	1	no	—
26	C₂³×C₄₇	—	4	1	no	—
27	C₂³×C₄₇	—	4	1	no	—
28	C₂³×C₄₇	C₂²⋊C₄	1	1	yes	characteristic
29	C₂³×C₄₇	SD₁₆	1	1	yes	characteristic
30	C₂³×C₄₇	Q₁₆	1	1	yes	characteristic
31	C₂×C₄×C₄₇	—	2	1	no	—
32	C₂×C₄×C₄₇	—	2	1	no	—
33	C₂×C₄×C₄₇	—	4	1	no	—
34	C₂×C₄×C₄₇	—	4	1	no	—
35	C₂×C₄×C₄₇	—	4	2	no	—
36	C₂×C₄×C₄₇	—	8	1	no	—
37	C₂×C₄×C₄₇	—	8	1	no	—
38	C₂×Dic₄₇	—	2	2	no	—
39	C₂×Dic₄₇	—	4	1	no	—
40	C₂×Dic₄₇	—	4	1	no	—
41	C₂×Dic₄₇	—	4	1	no	—
42	C₂×Dic₄₇	—	4	1	no	—
43	C₂×Dic₄₇	—	4	1	no	—
44	Dic₉₄	—	4	2	no	—
45	Dic₉₄	—	4	2	no	—
46	C₄₇⋊C₈	—	8	1	no	—
47	C₂²×C₄₇	—	2	1	no	—
48	C₂²×C₄₇	—	2	1	no	—
49	C₂²×C₄₇	—	2	1	no	—
50	C₂²×C₄₇	—	2	1	no	—
51	C₂²×C₄₇	—	4	1	no	—
52	C₂²×C₄₇	—	4	1	no	—
53	C₂²×C₄₇	—	4	1	no	—
54	C₂²×C₄₇	—	4	1	no	—
55	C₂²×C₄₇	—	8	1	no	—
56	C₂²×C₄₇	#99	1	1	yes	characteristic
57	C₂²×C₄₇	#103	1	1	yes	characteristic
58	C₂²×C₄₇	#104	1	1	yes	characteristic
59	C₄×C₄₇	—	4	1	no	—
60	C₄×C₄₇	—	4	1	no	—
61	C₄×C₄₇	—	8	2	no	—
62	Dic₄₇	—	4	2	no	—
63	Dic₄₇	—	8	1	no	—
64	#1351	—	47	1	no	maximal, Sylow
65	C₂×C₄₇	—	2	1	no	—
66	C₂×C₄₇	—	2	1	no	—
67	C₂×C₄₇	—	4	1	no	—
68	C₂×C₄₇	—	4	1	no	—
69	C₂×C₄₇	#328	1	1	yes	characteristic
70	C₂×C₄₇	AΣL(1,𝔽₁₆)	1	1	yes	characteristic
71	C₂×C₄₇	#354	1	1	yes	characteristic
72	#393	—	47	1	no	—
73	#342	D₄₇	1	1	yes	characteristic
74	#323	—	47	1	no	—
75	C₄₇	#1351	1	1	yes	characteristic, complemented, Sylow
76	C₂²×Q₈	—	47	1	no	—
77	C₂×C₂²⋊C₄	—	2	1	no	—
78	C₂×C₂²⋊C₄	—	94	1	no	—
79	C₂×C₂²⋊C₄	C₂×D₄₇	1	1	yes	characteristic
80	C₂²⋊C₈	—	94	1	no	—
81	#95	—	47	1	no	—
82	#95	—	94	1	no	—
83	C₂⁴	C₂²⋊D₄₇	1	1	yes	characteristic
84	C₂×Q₈	—	188	1	no	—
85	C₂×Q₈	—	188	2	no	—
86	C₂²×C₄	—	4	1	no	—
87	C₂²×C₄	—	94	1	no	—
88	C₂²×C₄	—	94	1	no	—
89	C₂²×C₄	C₄×D₄₇	1	1	yes	characteristic
90	C₂²×C₄	D₁₈₈	1	1	yes	characteristic
91	C₂×C₈	—	188	1	no	—
92	C₂²⋊C₄	—	2	2	no	—
93	C₂²⋊C₄	—	4	1	no	—
94	C₂²⋊C₄	—	4	1	no	—
95	C₂²⋊C₄	—	188	2	no	—
96	C₂³	—	2	1	no	—
97	C₂³	—	2	1	no	—
98	C₂³	—	4	1	no	—
99	C₂³	—	4	1	no	—
100	C₂³	(C₂×D₄₇)⋊C₄	1	1	yes	characteristic
101	C₂³	#10625335	1	1	yes	characteristic
102	C₂³	#10625337	1	1	yes	characteristic
103	Q₈	—	188	2	no	—
104	Q₈	—	188	2	no	—
105	C₂×C₄	—	2	1	no	—
106	C₂×C₄	—	2	1	no	—
107	C₂×C₄	—	4	1	no	—
108	C₂×C₄	—	4	1	no	—
109	C₂×C₄	—	4	2	no	—
110	C₂×C₄	—	8	1	no	—
111	C₂×C₄	—	8	1	no	—
112	C₂×C₄	—	94	2	no	—
113	C₂×C₄	—	188	1	no	—
114	C₂×C₄	—	188	1	no	—
115	C₂×C₄	—	188	1	no	—
116	C₂×C₄	—	188	1	no	—
117	C₂×C₄	—	188	1	no	—
118	C₈	—	376	1	no	—
119	C₂²	—	2	1	no	—
120	C₂²	—	2	1	no	—
121	C₂²	—	2	1	no	—
122	C₂²	—	2	1	no	—
123	C₂²	—	4	1	no	—
124	C₂²	—	4	1	no	—
125	C₂²	—	4	1	no	—
126	C₂²	—	4	1	no	—
127	C₂²	—	8	1	no	—
128	C₂²	#13093938	1	1	yes	characteristic
129	C₂²	#13093947	1	1	yes	characteristic
130	C₂²	#13093952	1	1	yes	characteristic
131	C₄	—	4	1	no	—
132	C₄	—	4	1	no	—
133	C₄	—	8	2	no	—
134	C₄	—	376	1	no	—
135	C₄	—	188	2	no	—
136	C₂	—	2	1	no	—
137	C₂	—	2	1	no	—
138	C₂	—	4	1	no	—
139	C₂	—	4	1	no	—
140	C₂	#15688859	1	1	yes	characteristic
141	C₂	#15688909	1	1	yes	characteristic
142	C₂	#15688917	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#15689035

C₂

yes

maximal, characteristic

#15688847

C₂

yes

maximal, characteristic

C₄₇×#342

C₂

yes

maximal, characteristic

C₂²×Dic₉₄

C₄

yes

characteristic, complemented

C₂×C₂²⋊Dic₄₇

—

C₂²⋊(C₄₇⋊C₈)

—

C₄₇⋊#95

—

C₄₇⋊#95

C₄

yes

characteristic, complemented

C₂×C₂²⋊C₄×C₄₇

—

C₂×C₂²⋊C₄×C₄₇

C₂²

yes

characteristic

C₂⁴×C₄₇

D₄

yes

characteristic

C₂²×C₄×C₄₇

—

C₂²×C₄×C₄₇

C₂×C₄

yes

characteristic

C₂²×C₄×C₄₇

D₄

yes

characteristic

C₂²×Dic₄₇

—

C₂²×Dic₄₇

—

C₂×Dic₉₄

—

C₂×Dic₉₄

—

C₂²⋊C₄×C₄₇

—

C₂²⋊C₄×C₄₇

—

C₂²⋊C₄×C₄₇

—

C₂²⋊Dic₄₇

—

C₂×C₄₇⋊C₈

—

C₂³×C₄₇

—

C₂³×C₄₇

—

C₂³×C₄₇

—

C₂³×C₄₇

—

C₂³×C₄₇

C₂²⋊C₄

yes

characteristic

C₂³×C₄₇

SD₁₆

yes

characteristic

C₂³×C₄₇

Q₁₆

yes

characteristic

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

—

C₂×Dic₄₇

—

C₂×Dic₄₇

—

C₂×Dic₄₇

—

C₂×Dic₄₇

—

C₂×Dic₄₇

—

C₂×Dic₄₇

—

Dic₉₄

—

Dic₉₄

—

C₄₇⋊C₈

—

C₂²×C₄₇

—

C₂²×C₄₇

—

C₂²×C₄₇

—

C₂²×C₄₇

—

C₂²×C₄₇

—

C₂²×C₄₇

—

C₂²×C₄₇

—

C₂²×C₄₇

—

C₂²×C₄₇

—

C₂²×C₄₇

yes

C₂²×C₄₇

yes

C₂²×C₄₇

yes

C₄×C₄₇

—

C₄×C₄₇

—

C₄×C₄₇

—

Dic₄₇

—

Dic₄₇

—

#1351

—

maximal, Sylow

C₂×C₄₇

—

C₂×C₄₇

—

C₂×C₄₇

—

C₂×C₄₇

—

C₂×C₄₇

#328

yes

characteristic

C₂×C₄₇

AΣL(1,𝔽₁₆)

yes

C₂×C₄₇

yes

—

D₄₇

yes

characteristic

#323

—

C₄₇

#1351

yes

characteristic, complemented, Sylow

C₂²×Q₈

—

C₂×C₂²⋊C₄

—

C₂×C₂²⋊C₄

—

C₂×C₂²⋊C₄

C₂×D₄₇

yes

characteristic

C₂²⋊C₈

—

#95

—

#95

—

C₂⁴

C₂²⋊D₄₇

yes

characteristic

C₂×Q₈

—

188

—

C₂×Q₈

—

188

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

C₄×D₄₇

yes

characteristic

C₂²×C₄

D₁₈₈

yes

characteristic

C₂×C₈

—

188

—

C₂²⋊C₄

—

C₂²⋊C₄

—

C₂²⋊C₄

—

C₂²⋊C₄

—

188

—

C₂³

—

C₂³

—

C₂³

—

C₂³

—

100

C₂³

(C₂×D₄₇)⋊C₄

yes

characteristic

101

C₂³

#10625335

yes

characteristic

102

C₂³

#10625337

yes

characteristic

103

Q₈

—

188

—

104

Q₈

—

188

—

105

C₂×C₄

—

106

C₂×C₄

—

107

C₂×C₄

—

108

C₂×C₄

—

109

C₂×C₄

—

110

C₂×C₄

—

111

C₂×C₄

—

112

C₂×C₄

—

113

C₂×C₄

—

188

—

114

C₂×C₄

—

188

—

115

C₂×C₄

—

188

—

116

C₂×C₄

—

188

—

117

C₂×C₄

—

188

—

118

C₈

—

376

—

119

C₂²

—

120

C₂²

—

121

C₂²

—

122

C₂²

—

123

C₂²

—

124

C₂²

—

125

C₂²

—

126

C₂²

—

127

C₂²

—

128

C₂²

#13093938

yes

characteristic

129

C₂²

#13093947

yes

characteristic

130

C₂²

#13093952

yes

characteristic

131

C₄

—

132

C₄

—

133

C₄

—

134

C₄

—

376

—

135

C₄

—

188

—

136

C₂

—

137

C₂

—

138

C₂

—

139

C₂

—

140

C₂

#15688859

yes

characteristic

141

C₂

#15688909

yes

characteristic

142

C₂

#15688917

yes

characteristic

	subgroup	G/H
center	C₂²	#13093938
commutator	C₂²×C₄×C₄₇	C₂×C₄
Frattini	C₂×C₂²⋊C₄	C₂×D₄₇
Fitting	C₄₇×#342	C₂
socle	C₂²×C₄₇	#99
hypercenter	#342	D₄₇
hypocenter	C₄₇	#1351
LCS	G ▷ C₂²×C₄×C₄₇ ▷ C₂³×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂² ◁ C₂³ ◁ C₂×C₂²⋊C₄ ◁ #342

subgroup

G/H

center

C₂²

#13093938

commutator

C₂²×C₄×C₄₇

C₂×C₄

Frattini

C₂×C₂²⋊C₄

C₂×D₄₇

Fitting

C₄₇×#342

C₂

C₂²×C₄₇

D₄₇

C₄₇

G ▷ C₂²×C₄×C₄₇ ▷ C₂³×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇

UCS

C₁ ◁ C₂² ◁ C₂³ ◁ C₂×C₂²⋊C₄ ◁ #342

N	H ≅ G/N	type
4. C₂²×Dic₉₄	C₄	semidirect
8. C₄₇⋊#95	C₄	semidirect
75. C₄₇	#1351	semidirect

H ≅ G/N

type

4. C₂²×Dic₉₄

C₄

semidirect

8. C₄₇⋊#95

C₄

semidirect

75. C₄₇

#1351

semidirect

Subgroups — #20241408

Conjugacy classes of subgroups

Complements