Group #26103361 — subgroups

Subgroups — #26103361

Order 12032.

264 conjugacy classes of subgroups; only the 83 normal subgroups are shown.

normal subgroups only

This group has 264 conjugacy classes of subgroups — too many to draw. Only its 83 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	#20243426	C₂	1	2	yes	maximal
2	C₄₇×#2112	C₂	1	1	yes	maximal, characteristic
3	C₂×#13093960	—	2	2	no	—
4	#15689087	—	2	2	no	—
5	#15688888	C₄	1	4	yes	—
6	C₂³×C₈×C₄₇	C₂²	1	1	yes	characteristic
7	C₂×M₃₂×C₄₇	C₄	1	2	yes	—
8	C₄₇×C₂²⋊C₁₆	—	2	2	no	—
9	C₂×#10625329	—	2	4	no	—
10	#13093960	—	4	4	no	—
11	C₂²⋊(C₄₇⋊C₈)	—	4	2	no	—
12	C₂²⋊(C₄₇⋊C₈)	—	4	2	no	—
13	C₂³×C₄×C₄₇	D₄	1	1	yes	characteristic
14	C₂²×C₈×C₄₇	—	4	2	no	—
15	C₂²×C₈×C₄₇	C₂×C₄	1	1	yes	characteristic
16	C₂²×C₈×C₄₇	C₂×C₄	1	2	yes	—
17	C₂²×C₈×C₄₇	D₄	1	2	yes	—
18	C₂²×C₈×C₄₇	Q₈	1	1	yes	characteristic
19	M₃₂×C₄₇	—	4	2	no	—
20	C₂×C₁₆×C₄₇	—	2	2	no	—
21	C₂⁴×C₄₇	Q₁₆	1	1	yes	characteristic
22	C₂²×C₄×C₄₇	—	4	1	no	—
23	C₂²×C₄×C₄₇	—	4	1	no	—
24	C₂²×C₄×C₄₇	C₂²⋊C₄	1	2	yes	—
25	C₂²×C₄×C₄₇	C₄⋊C₄	1	1	yes	characteristic
26	C₂²×C₄×C₄₇	D₈	1	1	yes	characteristic
27	C₂²×C₄×C₄₇	SD₁₆	1	2	yes	—
28	C₂×C₈×C₄₇	—	2	2	no	—
29	C₂×C₈×C₄₇	—	2	4	no	—
30	C₂×C₈×C₄₇	—	4	2	no	—
31	C₂×C₈×C₄₇	—	4	1	no	—
32	C₂×C₈×C₄₇	C₄²	1	1	yes	characteristic
33	C₂×C₈×C₄₇	C₂²⋊C₄	1	1	yes	characteristic
34	C₂×C₈×C₄₇	C₄⋊C₄	1	2	yes	—
35	#10625329	—	4	4	no	—
36	#10625329	—	4	4	no	—
37	C₂×C₄₇⋊C₈	—	4	4	no	—
38	C₁₆×C₄₇	—	4	2	no	—
39	C₂³×C₄₇	—	4	1	no	—
40	C₂³×C₄₇	—	4	2	no	—
41	C₂³×C₄₇	#103	1	2	yes	—
42	C₂³×C₄₇	#107	1	1	yes	characteristic
43	C₂×C₄×C₄₇	—	2	2	no	—
44	C₂×C₄×C₄₇	—	2	1	no	—
45	C₂×C₄×C₄₇	—	2	2	no	—
46	C₂×C₄×C₄₇	—	2	1	no	—
47	C₂×C₄×C₄₇	—	4	1	no	—
48	C₂×C₄×C₄₇	—	4	2	no	—
49	C₂×C₄×C₄₇	#95	1	1	yes	characteristic
50	C₂×C₄×C₄₇	#102	1	2	yes	—
51	C₂×C₄×C₄₇	#106	1	1	yes	characteristic
52	C₈×C₄₇	—	2	1	no	—
53	C₈×C₄₇	—	2	2	no	—
54	C₈×C₄₇	#99	1	2	yes	—
55	C₄₇⋊C₈	—	8	4	no	—
56	#7234	—	47	1	no	maximal, Sylow
57	C₂²×C₄₇	—	2	2	no	—
58	C₂²×C₄₇	—	2	1	no	—
59	C₂²×C₄₇	—	2	4	no	—
60	C₂²×C₄₇	—	2	2	no	—
61	C₂²×C₄₇	—	4	4	no	—
62	C₂²×C₄₇	#340	1	1	yes	characteristic
63	C₄×C₄₇	—	2	2	no	—
64	C₄×C₄₇	—	2	1	no	—
65	C₄×C₄₇	H(3,ℤ/4)	1	1	yes	characteristic
66	C₄×C₄₇	#342	1	1	yes	characteristic
67	#2112	D₄₇	1	1	yes	characteristic
68	#1810	—	47	2	no	—
69	C₂×C₄₇	—	4	2	no	—
70	C₂×C₄₇	—	4	1	no	—
71	C₂×C₄₇	#1289	1	1	yes	characteristic
72	C₂×C₄₇	#1388	1	2	yes	—
73	C₂³×C₈	C₂×D₄₇	1	1	yes	characteristic
74	C₂×M₃₂	Dic₄₇	1	2	yes	—
75	C₂×#108	—	94	2	no	—
76	#407	—	94	2	no	—
77	C₂²⋊C₁₆	—	2	2	no	—
78	#341	—	47	4	no	—
79	C₄₇	#7234	1	1	yes	characteristic, complemented, Sylow
80	C₂³×C₄	D₁₈₈	1	1	yes	characteristic
81	C₂×M₁₆	—	94	4	no	—
82	C₂²×C₈	—	4	2	no	—
83	C₂²×C₈	Dic₉₄	1	1	yes	characteristic
84	C₂²×C₈	C₄×D₄₇	1	2	yes	—
85	C₂²×C₈	C₂×Dic₄₇	1	1	yes	characteristic
86	C₂²×C₈	C₂²⋊D₄₇	1	2	yes	—
87	M₃₂	—	4	2	no	—
88	C₂×C₁₆	—	2	2	no	—
89	#108	—	188	4	no	—
90	C₂²⋊C₈	—	188	2	no	—
91	C₂²⋊C₈	—	188	2	no	—
92	C₂⁴	Dic₁₈₈	1	1	yes	characteristic
93	C₂²×C₄	—	4	1	no	—
94	C₂²×C₄	—	4	1	no	—
95	C₂²×C₄	#10625325	1	2	yes	—
96	C₂²×C₄	D₃₇₆	1	1	yes	characteristic
97	C₂²×C₄	C₄⋊Dic₄₇	1	1	yes	characteristic
98	C₂²×C₄	(C₂×D₄₇)⋊C₄	1	2	yes	—
99	M₁₆	—	188	4	no	—
100	M₁₆	—	188	4	no	—
101	C₂×C₈	—	2	4	no	—
102	C₂×C₈	—	2	2	no	—
103	C₂×C₈	—	4	1	no	—
104	C₂×C₈	—	4	2	no	—
105	C₂×C₈	—	188	4	no	—
106	C₂×C₈	C₄×Dic₄₇	1	1	yes	characteristic
107	C₂×C₈	Dic₄₇⋊C₄	1	2	yes	—
108	C₂×C₈	C₂²⋊Dic₄₇	1	1	yes	characteristic
109	C₁₆	—	4	2	no	—
110	C₂³	—	4	1	no	—
111	C₂³	—	4	2	no	—
112	C₂³	#13093949	1	2	yes	—
113	C₂³	#13093958	1	1	yes	characteristic
114	C₂×C₄	—	2	2	no	—
115	C₂×C₄	—	2	1	no	—
116	C₂×C₄	—	2	2	no	—
117	C₂×C₄	—	2	1	no	—
118	C₂×C₄	—	4	2	no	—
119	C₂×C₄	—	4	1	no	—
120	C₂×C₄	C₄₇⋊#95	1	1	yes	characteristic
121	C₂×C₄	#13093946	1	2	yes	—
122	C₂×C₄	#13093956	1	1	yes	characteristic
123	C₈	—	2	1	no	—
124	C₈	—	2	2	no	—
125	C₈	—	376	4	no	—
126	C₈	#13093939	1	2	yes	—
127	C₂²	—	2	4	no	—
128	C₂²	—	2	2	no	—
129	C₂²	—	2	2	no	—
130	C₂²	—	2	1	no	—
131	C₂²	—	4	4	no	—
132	C₂²	#15688886	1	1	yes	characteristic
133	C₄	—	2	2	no	—
134	C₄	—	2	1	no	—
135	C₄	#15688880	1	1	yes	characteristic
136	C₄	#15688890	1	1	yes	characteristic
137	C₂	—	4	2	no	—
138	C₂	—	4	1	no	—
139	C₂	#20241256	1	1	yes	characteristic
140	C₂	#20241500	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#20243426

C₂

yes

maximal

C₄₇×#2112

C₂

yes

maximal, characteristic

C₂×#13093960

—

#15689087

—

#15688888

C₄

yes

—

C₂³×C₈×C₄₇

C₂²

yes

characteristic

C₂×M₃₂×C₄₇

C₄

yes

—

C₄₇×C₂²⋊C₁₆

—

C₂×#10625329

—

#13093960

—

C₂²⋊(C₄₇⋊C₈)

—

C₂²⋊(C₄₇⋊C₈)

—

C₂³×C₄×C₄₇

D₄

yes

characteristic

C₂²×C₈×C₄₇

—

C₂²×C₈×C₄₇

C₂×C₄

yes

characteristic

C₂²×C₈×C₄₇

C₂×C₄

yes

—

C₂²×C₈×C₄₇

D₄

yes

—

C₂²×C₈×C₄₇

Q₈

yes

characteristic

M₃₂×C₄₇

—

C₂×C₁₆×C₄₇

—

C₂⁴×C₄₇

Q₁₆

yes

characteristic

C₂²×C₄×C₄₇

—

C₂²×C₄×C₄₇

—

C₂²×C₄×C₄₇

C₂²⋊C₄

yes

—

C₂²×C₄×C₄₇

C₄⋊C₄

yes

characteristic

C₂²×C₄×C₄₇

D₈

yes

characteristic

C₂²×C₄×C₄₇

SD₁₆

yes

—

C₂×C₈×C₄₇

—

C₂×C₈×C₄₇

—

C₂×C₈×C₄₇

—

C₂×C₈×C₄₇

—

C₂×C₈×C₄₇

C₄²

yes

characteristic

C₂×C₈×C₄₇

C₂²⋊C₄

yes

characteristic

C₂×C₈×C₄₇

C₄⋊C₄

yes

—

#10625329

—

#10625329

—

C₂×C₄₇⋊C₈

—

C₁₆×C₄₇

—

C₂³×C₄₇

—

C₂³×C₄₇

—

C₂³×C₄₇

#103

yes

—

C₂³×C₄₇

#107

yes

characteristic

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

#95

yes

characteristic

C₂×C₄×C₄₇

#102

yes

—

C₂×C₄×C₄₇

#106

yes

characteristic

C₈×C₄₇

—

C₈×C₄₇

—

C₈×C₄₇

#99

yes

—

C₄₇⋊C₈

—

#7234

—

maximal, Sylow

C₂²×C₄₇

—

C₂²×C₄₇

—

C₂²×C₄₇

—

C₂²×C₄₇

—

C₂²×C₄₇

—

C₂²×C₄₇

#340

yes

characteristic

C₄×C₄₇

—

C₄×C₄₇

—

C₄×C₄₇

yes

C₄×C₄₇

yes

D₄₇

yes

characteristic

#1810

—

C₂×C₄₇

—

C₂×C₄₇

—

C₂×C₄₇

#1289

yes

characteristic

C₂×C₄₇

#1388

yes

—

C₂³×C₈

C₂×D₄₇

yes

characteristic

C₂×M₃₂

Dic₄₇

yes

—

C₂×#108

—

#407

—

C₂²⋊C₁₆

—

#341

—

C₄₇

#7234

yes

characteristic, complemented, Sylow

C₂³×C₄

D₁₈₈

yes

characteristic

C₂×M₁₆

—

C₂²×C₈

—

C₂²×C₈

Dic₉₄

yes

characteristic

C₂²×C₈

C₄×D₄₇

yes

—

C₂²×C₈

C₂×Dic₄₇

yes

characteristic

C₂²×C₈

C₂²⋊D₄₇

yes

—

M₃₂

—

C₂×C₁₆

—

#108

—

188

—

C₂²⋊C₈

—

188

—

C₂²⋊C₈

—

188

—

C₂⁴

Dic₁₈₈

yes

characteristic

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

#10625325

yes

—

C₂²×C₄

D₃₇₆

yes

characteristic

C₂²×C₄

C₄⋊Dic₄₇

yes

characteristic

C₂²×C₄

(C₂×D₄₇)⋊C₄

yes

—

M₁₆

—

188

—

100

M₁₆

—

188

—

101

C₂×C₈

—

102

C₂×C₈

—

103

C₂×C₈

—

104

C₂×C₈

—

105

C₂×C₈

—

188

—

106

C₂×C₈

C₄×Dic₄₇

yes

characteristic

107

C₂×C₈

Dic₄₇⋊C₄

yes

—

108

C₂×C₈

C₂²⋊Dic₄₇

yes

characteristic

109

C₁₆

—

110

C₂³

—

111

C₂³

—

112

C₂³

#13093949

yes

—

113

C₂³

#13093958

yes

characteristic

114

C₂×C₄

—

115

C₂×C₄

—

116

C₂×C₄

—

117

C₂×C₄

—

118

C₂×C₄

—

119

C₂×C₄

—

120

C₂×C₄

C₄₇⋊#95

yes

characteristic

121

C₂×C₄

#13093946

yes

—

122

C₂×C₄

#13093956

yes

characteristic

123

C₈

—

124

C₈

—

125

C₈

—

376

—

126

C₈

#13093939

yes

—

127

C₂²

—

128

C₂²

—

129

C₂²

—

130

C₂²

—

131

C₂²

—

132

C₂²

#15688886

yes

characteristic

133

C₄

—

134

C₄

—

135

C₄

#15688880

yes

characteristic

136

C₄

#15688890

yes

characteristic

137

C₂

—

138

C₂

—

139

C₂

#20241256

yes

characteristic

140

C₂

#20241500

yes

—

	subgroup	G/H
center	C₂²	#15688886
commutator	C₂×C₈×C₄₇	C₄²
Frattini	C₂³×C₈	C₂×D₄₇
Fitting	C₄₇×#2112	C₂
socle	C₂²×C₄₇	#340
prime radical	C₂⁴×C₄₇	Q₁₆
cosocle	C₂³×C₈×C₄₇	C₂²
hypercenter	#2112	D₄₇
hypocenter	C₄₇	#7234
LCS	G ▷ C₂×C₈×C₄₇ ▷ C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ C₂³×C₈ ◁ #2112

subgroup

G/H

center

C₂²

#15688886

commutator

C₂×C₈×C₄₇

C₄²

Frattini

C₂³×C₈

C₂×D₄₇

Fitting

C₄₇×#2112

C₂

socle

C₂²×C₄₇

#340

prime radical

C₂⁴×C₄₇

Q₁₆

cosocle

C₂³×C₈×C₄₇

C₂²

D₄₇

C₄₇

G ▷ C₂×C₈×C₄₇ ▷ C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇

UCS

C₁ ◁ C₂² ◁ C₂³×C₄ ◁ C₂³×C₈ ◁ #2112

N	H ≅ G/N	type
79. C₄₇	#7234	semidirect

H ≅ G/N

type

79. C₄₇

#7234

semidirect

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
#20243426	2
C₄₇×#2112	2
#7234	47

subgroup

index

#20243426

C₄₇×#2112

#7234

Subgroups — #26103361

Conjugacy classes of subgroups

Complements

Maximal subgroups