Finite Group Databasebeta

← C3⋊(C25⋊#64424) #36347755

Subgroups — C3⋊(C25⋊#64424) #36347755

Order 22050.

270 conjugacy classes of subgroups; only the 68 normal subgroups are shown.

normal subgroups only

This group has 270 conjugacy classes of subgroups — too many to draw. Only its 68 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×C25×#3861C211yesmaximal, characteristic, complemented, Hall
2C3⋊(C7⋊D175)C311yesmaximal, characteristic, complemented
3C25⋊#64424—31nomaximal
4C3⋊(C5⋊#64424)—51nomaximal
5C25×#3861—21no—
6C25×#3861D311yescharacteristic, complemented
7C3×C72×C25C2×C311yescharacteristic, complemented
8C3⋊(ΓL(1,𝔽8)⋊D25)—71nomaximal
9C3⋊(ΓL(1,𝔽8)⋊D25)—71nomaximal
10C3⋊(ΓL(1,𝔽8)⋊D25)—72nomaximal
11C3⋊(ΓL(1,𝔽8)⋊D25)—74nomaximal
12C7⋊D175—31noHall
13C3×C5×#3861D511yescharacteristic
14C3×ΓL(1,𝔽8)×C25—71no—
15C3×ΓL(1,𝔽8)×C25—141no—
16C3×ΓL(1,𝔽8)×C25—142no—
17C3×ΓL(1,𝔽8)×C25—144no—
18C3⋊(C5⋊(C7⋊D7))—51no—
19C5⋊#64424—151no—
20C72×C25C3×D311yescharacteristic, complemented, Hall
21D525—71no—
22D525—71no—
23D525—72no—
24D525—74no—
25ΓL(1,𝔽8)⋊D25—212no—
26ΓL(1,𝔽8)⋊D25—211no—
27ΓL(1,𝔽8)⋊D25—211no—
28ΓL(1,𝔽8)⋊D25—214no—
29C3⋊#64424—251noHall
30C3×C5×C72C3×D511yescharacteristic
31C5×#3861—21no—
32C5×#3861D1511yescharacteristic
33C3⋊(C5⋊AGL(1,𝔽7))—351no—
34C3⋊(C5⋊AGL(1,𝔽7))—351no—
35C3⋊(C5⋊AGL(1,𝔽7))—352no—
36C3⋊(C5⋊AGL(1,𝔽7))—354no—
37C3×C7×C25AGL(1,𝔽7)11yescharacteristic, complemented
38C3×C7×C25AGL(1,𝔽7)11yescharacteristic, complemented
39C3×C7×C25AGL(1,𝔽7)12yescomplemented
40C3×C7×C25AGL(1,𝔽7)14yescomplemented
41ΓL(1,𝔽8)×C25—72no—
42ΓL(1,𝔽8)×C25—141no—
43ΓL(1,𝔽8)×C25—141no—
44ΓL(1,𝔽8)×C25—141no—
45ΓL(1,𝔽8)×C25—142no—
46ΓL(1,𝔽8)×C25—144no—
47ΓL(1,𝔽8)×C25—141no—
48ΓL(1,𝔽8)×C25—144no—
49C5⋊(C7⋊D7)—151no—
50C3×D75—491noHall
51C3×#3861D2511yescharacteristic, complemented, Hall
52D175—212no—
53D175—211no—
54D175—211no—
55D175—214no—
56C3×C5×ΓL(1,𝔽8)—71no—
57C3×C5×ΓL(1,𝔽8)—141no—
58C3×C5×ΓL(1,𝔽8)—142no—
59C3×C5×ΓL(1,𝔽8)—144no—
60C3⋊(C7⋊D7)—251no—
61#64424—751no—
62C5×C72C3×D1511yescharacteristic
63C32×C25—981noHall
64D105—351no—
65D105—351no—
66D105—352no—
67D105—354no—
68C5⋊AGL(1,𝔽7)—1051no—
69C5⋊AGL(1,𝔽7)—1052no—
70C5⋊AGL(1,𝔽7)—1051no—
71C5⋊AGL(1,𝔽7)—1054no—
72C7×C25C3⋊AGL(1,𝔽7)12yescomplemented
73C7×C25C3⋊AGL(1,𝔽7)11yescharacteristic, complemented
74C7×C25C3⋊AGL(1,𝔽7)11yescharacteristic, complemented
75C7×C25C3⋊AGL(1,𝔽7)14yescomplemented
76D75—491no—
77C3×D25—1471no—
78C3×C72C3×D2511yescharacteristic, complemented
79#3861—21no—
80#3861D7511yescharacteristic, complemented
81C3⋊AGL(1,𝔽7)—1751no—
82C3⋊AGL(1,𝔽7)—1751no—
83C3⋊AGL(1,𝔽7)—1752no—
84C3⋊AGL(1,𝔽7)—1754no—
85C3×C5×C7C5⋊AGL(1,𝔽7)11yescharacteristic
86C3×C5×C7C5⋊AGL(1,𝔽7)11yescharacteristic
87C3×C5×C7C5⋊AGL(1,𝔽7)12yes—
88C3×C5×C7C5⋊AGL(1,𝔽7)14yes—
89C5×ΓL(1,𝔽8)—141no—
90C5×ΓL(1,𝔽8)—142no—
91C5×ΓL(1,𝔽8)—141no—
92C5×ΓL(1,𝔽8)—144no—
93C5×ΓL(1,𝔽8)—141no—
94C5×ΓL(1,𝔽8)—141no—
95C5×ΓL(1,𝔽8)—142no—
96C5×ΓL(1,𝔽8)—144no—
97C7⋊D7—751noHall
98C3×D15—2451no—
99C3×C25—981no—
100C3×C25—981no—
101C3×C25#6442411yescharacteristic, complemented
102D35—1051no—
103D35—1052no—
104D35—1051no—
105D35—1054no—
106C3×ΓL(1,𝔽8)—71no—
107C3×ΓL(1,𝔽8)—141no—
108C3×ΓL(1,𝔽8)—142no—
109C3×ΓL(1,𝔽8)—144no—
110D25—1471noHall
111C72C3×D7511yescharacteristic, complemented, Sylow
112C32×C5—981no—
113D21—1751no—
114D21—1751no—
115D21—1752no—
116D21—1754no—
117AGL(1,𝔽7)—5251no—
118AGL(1,𝔽7)—5252no—
119AGL(1,𝔽7)—5251no—
120AGL(1,𝔽7)—5254no—
121C5×C7C3⋊(C5⋊AGL(1,𝔽7))11yescharacteristic
122C5×C7C3⋊(C5⋊AGL(1,𝔽7))12yes—
123C5×C7C3⋊(C5⋊AGL(1,𝔽7))11yescharacteristic
124C5×C7C3⋊(C5⋊AGL(1,𝔽7))14yes—
125D15—2451no—
126C3×D5—7351no—
127C25C3⋊#6442411yescharacteristic, complemented, Sylow
128C3×C7ΓL(1,𝔽8)⋊D2511yescharacteristic, complemented
129C3×C7ΓL(1,𝔽8)⋊D2511yescharacteristic, complemented
130C3×C7ΓL(1,𝔽8)⋊D2512yescomplemented
131C3×C7ΓL(1,𝔽8)⋊D2514yescomplemented
132ΓL(1,𝔽8)—141no—
133ΓL(1,𝔽8)—141no—
134ΓL(1,𝔽8)—142no—
135ΓL(1,𝔽8)—144no—
136ΓL(1,𝔽8)—141no—
137ΓL(1,𝔽8)—142no—
138ΓL(1,𝔽8)—141no—
139ΓL(1,𝔽8)—144no—
140C3×D3—12251noHall
141C3×C5—981no—
142C3×C5—981no—
143C3×C5C5⋊#6442411yescharacteristic
144D7—5251no—
145D7—5252no—
146D7—5251no—
147D7—5254no—
148D5—7351no—
149C32—981noSylow
150C7C3⋊(ΓL(1,𝔽8)⋊D25)11yescharacteristic, complemented
151C7C3⋊(ΓL(1,𝔽8)⋊D25)12yescomplemented
152C7C3⋊(ΓL(1,𝔽8)⋊D25)11yescharacteristic, complemented
153C7C3⋊(ΓL(1,𝔽8)⋊D25)14yescomplemented
154C2×C3—36751no—
155D3—12251no—
156C5C3⋊(C5⋊#64424)11yescharacteristic
157C3—981no—
158C3—981no—
159C3C25⋊#6442411yescharacteristic, complemented
160C2—36751noSylow
subgroupG/H
center, hypercenterC1G
commutator, Fitting, cosocle, hypocenterC3×C72×C25C2×C3
FrattiniC5C3⋊(C5⋊#64424)
socleC3×C5×C72C3×D5
LCSG ▷ C3×C72×C25
UCSC1

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
1. C3×C25×#3861C2semidirect
2. C3⋊(C7⋊D175)C3semidirect
6. C25×#3861D3semidirect
7. C3×C72×C25C2×C3semidirect
20. C72×C25C3×D3semidirect
37. C3×C7×C25AGL(1,𝔽7)semidirect
38. C3×C7×C25AGL(1,𝔽7)semidirect
39. C3×C7×C25AGL(1,𝔽7)semidirect
40. C3×C7×C25AGL(1,𝔽7)semidirect
51. C3×#3861D25semidirect
72. C7×C25C3⋊AGL(1,𝔽7)semidirect
73. C7×C25C3⋊AGL(1,𝔽7)semidirect
74. C7×C25C3⋊AGL(1,𝔽7)semidirect
75. C7×C25C3⋊AGL(1,𝔽7)semidirect
78. C3×C72C3×D25semidirect
80. #3861D75semidirect
101. C3×C25#64424semidirect
111. C72C3×D75semidirect
127. C25C3⋊#64424semidirect
128. C3×C7ΓL(1,𝔽8)⋊D25semidirect
129. C3×C7ΓL(1,𝔽8)⋊D25semidirect
130. C3×C7ΓL(1,𝔽8)⋊D25semidirect
131. C3×C7ΓL(1,𝔽8)⋊D25semidirect
150. C7C3⋊(ΓL(1,𝔽8)⋊D25)semidirect
151. C7C3⋊(ΓL(1,𝔽8)⋊D25)semidirect
152. C7C3⋊(ΓL(1,𝔽8)⋊D25)semidirect
153. C7C3⋊(ΓL(1,𝔽8)⋊D25)semidirect
159. C3C25⋊#64424semidirect

Maximal subgroups

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

subgroupindex
C3×C25×#38612
C3⋊(C7⋊D175)3
C25⋊#644243
C3⋊(C5⋊#64424)5
C3⋊(ΓL(1,𝔽8)⋊D25)7