Group #26110080 — subgroups

Subgroups — #26110080

Order 12032.

354 conjugacy classes of subgroups; only the 171 normal subgroups are shown.

normal subgroups only

This group has 354 conjugacy classes of subgroups — too many to draw. Only its 171 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	#20243560	C₂	1	1	yes	maximal, characteristic
2	#20242146	C₂	1	4	yes	maximal
3	C₂×C₈×C₄₇⋊C₈	C₂	1	1	yes	maximal, characteristic
4	C₄₇×#1848	C₂	1	1	yes	maximal, characteristic
5	C₂×#13093960	—	2	2	no	—
6	#15689147	C₂²	1	2	yes	—
7	C₂×C₄×C₄₇⋊C₈	C₂²	1	2	yes	—
8	#15688875	—	2	8	no	—
9	C₈×C₄₇⋊C₈	C₄	1	1	yes	characteristic
10	C₈×C₄₇⋊C₈	C₄	1	2	yes	—
11	C₈×C₄₇⋊C₈	C₄	1	1	yes	characteristic
12	C₂×#108×C₄₇	—	2	2	no	—
13	C₄₇×#423	C₂²	1	2	yes	—
14	C₂×C₄×C₈×C₄₇	C₂²	1	1	yes	characteristic
15	C₂×#10625329	—	2	2	no	—
16	C₂²×C₄₇⋊C₈	Q₈	1	4	yes	—
17	#13093960	—	4	4	no	—
18	C₄⋊(C₄₇⋊C₈)	—	2	4	no	—
19	C₄×C₄₇⋊C₈	C₂×C₄	1	2	yes	—
20	C₄×C₄₇⋊C₈	C₂×C₄	1	2	yes	—
21	C₄×C₄₇⋊C₈	D₄	1	4	yes	—
22	C₂×M₁₆×C₄₇	—	2	2	no	—
23	C₂²×C₈×C₄₇	D₄	1	2	yes	—
24	C₂×C₄²×C₄₇	C₂³	1	1	yes	characteristic
25	#108×C₄₇	—	4	4	no	—
26	C₄⋊C₈×C₄₇	—	2	4	no	—
27	C₄×C₈×C₄₇	C₂×C₄	1	1	yes	characteristic
28	C₄×C₈×C₄₇	C₂×C₄	1	1	yes	characteristic
29	C₄×C₈×C₄₇	Q₈	1	2	yes	—
30	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
31	C₂²×C₄×C₄₇	C₂×Q₈	1	2	yes	—
32	M₁₆×C₄₇	—	4	4	no	—
33	C₂×C₈×C₄₇	—	4	2	no	—
34	C₂×C₈×C₄₇	C₄⋊C₄	1	2	yes	—
35	C₂×C₈×C₄₇	C₄⋊C₄	1	2	yes	—
36	C₂×C₈×C₄₇	D₈	1	4	yes	—
37	C₂×C₈×C₄₇	Q₁₆	1	4	yes	—
38	C₄²×C₄₇	C₂²×C₄	1	1	yes	characteristic
39	C₄²×C₄₇	C₂×D₄	1	2	yes	—
40	C₄²×C₄₇	C₂×Q₈	1	1	yes	characteristic
41	#10625329	—	4	4	no	—
42	C₂×C₄₇⋊C₈	—	2	8	no	—
43	C₂×C₄₇⋊C₈	—	4	2	no	—
44	C₂×C₄₇⋊C₈	C₄⋊C₄	1	4	yes	—
45	C₂×C₄₇⋊C₈	C₄⋊C₄	1	4	yes	—
46	C₂³×C₄₇	C₄⋊Q₈	1	1	yes	characteristic
47	C₂×C₄×C₄₇	—	2	4	no	—
48	C₂×C₄×C₄₇	C₂×C₄⋊C₄	1	2	yes	—
49	C₂×C₄×C₄₇	C₂×C₄⋊C₄	1	1	yes	characteristic
50	C₂×C₄×C₄₇	#127	1	1	yes	characteristic
51	C₂×C₄×C₄₇	C₄⋊Q₈	1	2	yes	—
52	C₂×C₄×C₄₇	C₂×D₈	1	2	yes	—
53	C₂×C₄×C₄₇	C₂×Q₁₆	1	2	yes	—
54	C₈×C₄₇	—	8	2	no	—
55	C₈×C₄₇	#107	1	4	yes	—
56	C₈×C₄₇	#107	1	4	yes	—
57	C₄₇⋊C₈	—	2	4	no	—
58	C₄₇⋊C₈	—	8	2	no	—
59	C₄₇⋊C₈	#108	1	8	yes	complemented
60	#8823	—	47	1	no	maximal, Sylow
61	C₂²×C₄₇	—	2	2	no	—
62	C₂²×C₄₇	#384	1	1	yes	characteristic
63	C₂²×C₄₇	#500	1	2	yes	—
64	C₄×C₄₇	—	2	2	no	—
65	C₄×C₄₇	C₂×#107	1	2	yes	—
66	C₄×C₄₇	C₂×#108	1	4	yes	—
67	C₄×C₄₇	#493	1	1	yes	characteristic
68	C₄×C₄₇	#494	1	1	yes	characteristic
69	#1848	—	47	1	no	—
70	#1848	D₄₇	1	1	yes	characteristic
71	#1567	—	47	4	no	—
72	C₂×C₈²	—	47	1	no	—
73	C₂×C₄₇	—	2	2	no	—
74	C₂×C₄₇	#1845	1	1	yes	characteristic
75	C₂×C₄₇	#1848	1	2	yes	—
76	C₂×#108	—	2	2	no	—
77	C₂×#108	—	94	2	no	—
78	#423	—	47	2	no	—
79	#423	C₂×D₄₇	1	2	yes	—
80	C₂×C₄×C₈	—	47	2	no	—
81	C₂×C₄×C₈	C₂×D₄₇	1	1	yes	characteristic
82	#335	—	94	8	no	—
83	C₈²	—	47	1	no	—
84	C₈²	—	47	2	no	—
85	C₈²	—	47	1	no	—
86	C₄₇	#8823	1	1	yes	characteristic, complemented, Sylow
87	C₂×M₁₆	—	2	2	no	—
88	C₂×M₁₆	—	94	2	no	—
89	C₂²×C₈	—	47	4	no	—
90	C₂²×C₈	C₂²⋊D₄₇	1	2	yes	—
91	C₂×C₄²	C₂²×D₄₇	1	1	yes	characteristic
92	#108	—	4	4	no	—
93	#108	—	188	4	no	—
94	C₄⋊C₈	—	2	4	no	—
95	C₄⋊C₈	—	94	4	no	—
96	C₄×C₈	—	47	2	no	—
97	C₄×C₈	—	47	2	no	—
98	C₄×C₈	—	47	4	no	—
99	C₄×C₈	Dic₉₄	1	2	yes	—
100	C₄×C₈	C₄×D₄₇	1	1	yes	characteristic
101	C₄×C₈	C₄×D₄₇	1	1	yes	characteristic
102	C₂²×C₄	Q₈×D₄₇	1	2	yes	—
103	C₂²×C₄	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
104	M₁₆	—	4	4	no	—
105	M₁₆	—	188	4	no	—
106	C₂×C₈	—	4	2	no	—
107	C₂×C₈	—	47	4	no	—
108	C₂×C₈	—	47	4	no	—
109	C₂×C₈	—	94	8	no	—
110	C₂×C₈	—	188	2	no	—
111	C₂×C₈	Dic₄₇⋊C₄	1	2	yes	—
112	C₂×C₈	Dic₄₇⋊C₄	1	2	yes	—
113	C₂×C₈	#10625334	1	4	yes	—
114	C₂×C₈	#10625337	1	4	yes	—
115	C₄²	C₂×Dic₉₄	1	1	yes	characteristic
116	C₄²	C₂×C₄×D₄₇	1	1	yes	characteristic
117	C₄²	D₄×D₄₇	1	2	yes	—
118	C₂³	#13094022	1	1	yes	characteristic
119	C₂×C₄	—	2	4	no	—
120	C₂×C₄	C₄⋊C₄×D₄₇	1	2	yes	—
121	C₂×C₄	C₂×Dic₄₇⋊C₄	1	1	yes	characteristic
122	C₂×C₄	#13094020	1	1	yes	characteristic
123	C₂×C₄	#13094023	1	2	yes	—
124	C₂×C₄	C₂×#10625334	1	2	yes	—
125	C₂×C₄	C₂×#10625337	1	2	yes	—
126	C₈	—	8	2	no	—
127	C₈	—	47	8	no	—
128	C₈	—	94	4	no	—
129	C₈	—	376	2	no	—
130	C₈	#13093957	1	4	yes	—
131	C₈	#13093957	1	4	yes	—
132	C₂²	—	2	2	no	—
133	C₂²	#15688992	1	1	yes	characteristic
134	C₂²	#15689526	1	2	yes	—
135	C₄	—	2	2	no	—
136	C₄	C₂×#13093957	1	2	yes	—
137	C₄	#108×D₄₇	1	4	yes	—
138	C₄	#15689499	1	1	yes	characteristic
139	C₄	#15689502	1	1	yes	characteristic
140	C₂	—	2	2	no	—
141	C₂	#20243547	1	1	yes	characteristic
142	C₂	#20243558	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#20243560

C₂

yes

maximal, characteristic

#20242146

C₂

yes

maximal

C₂×C₈×C₄₇⋊C₈

C₂

yes

maximal, characteristic

C₄₇×#1848

C₂

yes

maximal, characteristic

C₂×#13093960

—

#15689147

C₂²

yes

—

C₂×C₄×C₄₇⋊C₈

C₂²

yes

—

#15688875

—

C₈×C₄₇⋊C₈

C₄

yes

characteristic

C₈×C₄₇⋊C₈

C₄

yes

—

C₈×C₄₇⋊C₈

C₄

yes

characteristic

C₂×#108×C₄₇

—

C₄₇×#423

C₂²

yes

—

C₂×C₄×C₈×C₄₇

C₂²

yes

characteristic

C₂×#10625329

—

C₂²×C₄₇⋊C₈

Q₈

yes

—

#13093960

—

C₄⋊(C₄₇⋊C₈)

—

C₄×C₄₇⋊C₈

C₂×C₄

yes

—

C₄×C₄₇⋊C₈

C₂×C₄

yes

—

C₄×C₄₇⋊C₈

D₄

yes

—

C₂×M₁₆×C₄₇

—

C₂²×C₈×C₄₇

D₄

yes

—

C₂×C₄²×C₄₇

C₂³

yes

characteristic

#108×C₄₇

—

C₄⋊C₈×C₄₇

—

C₄×C₈×C₄₇

C₂×C₄

yes

characteristic

C₄×C₈×C₄₇

C₂×C₄

yes

characteristic

C₄×C₈×C₄₇

Q₈

yes

—

C₂²×C₄×C₄₇

C₂×D₄

yes

characteristic

C₂²×C₄×C₄₇

C₂×Q₈

yes

—

M₁₆×C₄₇

—

C₂×C₈×C₄₇

—

C₂×C₈×C₄₇

C₄⋊C₄

yes

—

C₂×C₈×C₄₇

C₄⋊C₄

yes

—

C₂×C₈×C₄₇

D₈

yes

—

C₂×C₈×C₄₇

Q₁₆

yes

—

C₄²×C₄₇

C₂²×C₄

yes

characteristic

C₄²×C₄₇

C₂×D₄

yes

—

C₄²×C₄₇

C₂×Q₈

yes

characteristic

#10625329

—

C₂×C₄₇⋊C₈

—

C₂×C₄₇⋊C₈

—

C₂×C₄₇⋊C₈

C₄⋊C₄

yes

—

C₂×C₄₇⋊C₈

C₄⋊C₄

yes

—

C₂³×C₄₇

C₄⋊Q₈

yes

characteristic

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

C₂×C₄⋊C₄

yes

—

C₂×C₄×C₄₇

C₂×C₄⋊C₄

yes

characteristic

C₂×C₄×C₄₇

#127

yes

characteristic

C₂×C₄×C₄₇

C₄⋊Q₈

yes

—

C₂×C₄×C₄₇

C₂×D₈

yes

—

C₂×C₄×C₄₇

C₂×Q₁₆

yes

—

C₈×C₄₇

—

C₈×C₄₇

#107

yes

—

C₈×C₄₇

#107

yes

—

C₄₇⋊C₈

—

C₄₇⋊C₈

—

C₄₇⋊C₈

yes

—

C₂²×C₄₇

—

C₂²×C₄₇

#384

yes

characteristic

C₂²×C₄₇

#500

yes

—

C₄×C₄₇

—

C₄×C₄₇

C₂×#107

yes

—

C₄×C₄₇

C₂×#108

yes

—

C₄×C₄₇

yes

C₄×C₄₇

yes

—

D₄₇

yes

characteristic

#1567

—

C₂×C₈²

—

C₂×C₄₇

—

C₂×C₄₇

#1845

yes

characteristic

C₂×C₄₇

#1848

yes

—

—

—

—

C₂×D₄₇

yes

—

C₂×C₄×C₈

—

C₂×C₄×C₈

C₂×D₄₇

yes

characteristic

#335

—

C₈²

—

C₈²

—

C₈²

—

C₄₇

#8823

yes

characteristic, complemented, Sylow

C₂×M₁₆

—

C₂×M₁₆

—

C₂²×C₈

—

C₂²×C₈

C₂²⋊D₄₇

yes

—

C₂×C₄²

C₂²×D₄₇

yes

characteristic

#108

—

#108

—

188

—

C₄⋊C₈

—

C₄⋊C₈

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

Dic₉₄

yes

—

100

C₄×C₈

C₄×D₄₇

yes

characteristic

101

C₄×C₈

C₄×D₄₇

yes

characteristic

102

C₂²×C₄

Q₈×D₄₇

yes

—

103

C₂²×C₄

C₂×C₂²⋊D₄₇

yes

characteristic

104

M₁₆

—

105

M₁₆

—

188

—

106

C₂×C₈

—

107

C₂×C₈

—

108

C₂×C₈

—

109

C₂×C₈

—

110

C₂×C₈

—

188

—

111

C₂×C₈

Dic₄₇⋊C₄

yes

—

112

C₂×C₈

Dic₄₇⋊C₄

yes

—

113

C₂×C₈

#10625334

yes

—

114

C₂×C₈

#10625337

yes

—

115

C₄²

C₂×Dic₉₄

yes

characteristic

116

C₄²

C₂×C₄×D₄₇

yes

characteristic

117

C₄²

D₄×D₄₇

yes

—

118

C₂³

#13094022

yes

characteristic

119

C₂×C₄

—

120

C₂×C₄

C₄⋊C₄×D₄₇

yes

—

121

C₂×C₄

C₂×Dic₄₇⋊C₄

yes

characteristic

122

C₂×C₄

#13094020

yes

characteristic

123

C₂×C₄

#13094023

yes

—

124

C₂×C₄

C₂×#10625334

yes

—

125

C₂×C₄

C₂×#10625337

yes

—

126

C₈

—

127

C₈

—

128

C₈

—

129

C₈

—

376

—

130

C₈

#13093957

yes

—

131

C₈

#13093957

yes

—

132

C₂²

—

133

C₂²

#15688992

yes

characteristic

134

C₂²

#15689526

yes

—

135

C₄

—

136

C₄

C₂×#13093957

yes

—

137

C₄

#108×D₄₇

yes

—

138

C₄

#15689499

yes

characteristic

139

C₄

#15689502

yes

characteristic

140

C₂

—

141

C₂

#20243547

yes

characteristic

142

C₂

#20243558

yes

—

	subgroup	G/H
center	C₂×C₄	#13094020
commutator	C₄²×C₄₇	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₄₇
Fitting	C₄₇×#1848	C₂
socle	C₂²×C₄₇	#384
prime radical	C₂³×C₄₇	C₄⋊Q₈
cosocle	C₂×C₄²×C₄₇	C₂³
hypercenter	#1848	D₄₇
hypocenter	C₄₇	#8823
LCS	G ▷ C₄²×C₄₇ ▷ C₂²×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #1848

subgroup

G/H

center

C₂×C₄

#13094020

commutator

C₄²×C₄₇

C₂²×C₄

Frattini

C₂×C₄²

C₂²×D₄₇

Fitting

C₄₇×#1848

C₂

socle

C₂²×C₄₇

#384

prime radical

C₂³×C₄₇

C₄⋊Q₈

cosocle

C₂×C₄²×C₄₇

C₂³

D₄₇

C₄₇

G ▷ C₄²×C₄₇ ▷ C₂²×C₄₇ ▷ C₄₇

UCS

C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #1848

N	H ≅ G/N	type
59. C₄₇⋊C₈	#108	semidirect
86. C₄₇	#8823	semidirect

H ≅ G/N

type

59. C₄₇⋊C₈

#108

semidirect

86. C₄₇

#8823

semidirect

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

subgroup	index
#20243560	2
#20242146	2
C₂×C₈×C₄₇⋊C₈	2
C₄₇×#1848	2
#8823	47

subgroup

index

#20243560

#20242146

C₂×C₈×C₄₇⋊C₈

C₄₇×#1848

#8823

Subgroups — #26110080

Conjugacy classes of subgroups

Complements

Maximal subgroups