Finite Group Databasebeta

← C4×C3⋊#64424 #72245796

Subgroups — C4×C3⋊#64424 #72245796

Order 3528.

350 conjugacy classes of subgroups; only the 76 normal subgroups are shown.

normal subgroups only

This group has 350 conjugacy classes of subgroups — too many to draw. Only its 76 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
1C2×C3⋊#64424C211yesmaximal, characteristic
2C3×C4×#3861C211yesmaximal, characteristic, complemented
3C3⋊#10596874C211yesmaximal, characteristic, complemented
4C4×C3⋊(C7⋊D7)C311yesmaximal, characteristic, complemented
5C4×#64424—31nomaximal
6C2×C3×#3861C2211yescharacteristic
7C3⋊#64424C412yesdirect product factor
8C2×C3⋊(C7⋊D7)C2×C311yescharacteristic
9C2×#64424—31no—
10C3×C4×C72C2×C311yescharacteristic, complemented
11C3⋊(C7⋊Dic7)C2×C311yescharacteristic, complemented
12C4×#3861—21no—
13C4×#3861D311yescharacteristic, complemented
14#10596874—31no—
15C4×C3⋊AGL(1,𝔽7)—71nomaximal
16C4×C3⋊AGL(1,𝔽7)—71nomaximal
17C4×C3⋊AGL(1,𝔽7)—76nomaximal
18C3×#3861C2×C411yescharacteristic, complemented, Hall
19C4×C7⋊D7—31noHall
20C2×C3×C72C22×C311yescharacteristic
21C3⋊(C7⋊D7)C3×C412yescomplemented
22C2×#3861—21no—
23C2×#3861C2×D311yescharacteristic
24#64424—32no—
25C2×C3⋊AGL(1,𝔽7)—71no—
26C2×C3⋊AGL(1,𝔽7)—71no—
27C2×C3⋊AGL(1,𝔽7)—76no—
28C3×C4×ΓL(1,𝔽8)—71no—
29C3×C4×ΓL(1,𝔽8)—71no—
30C3×C4×ΓL(1,𝔽8)—146no—
31C3⋊(ΓL(1,𝔽8)⋊C4)—71no—
32C3⋊(ΓL(1,𝔽8)⋊C4)—71no—
33C3⋊(ΓL(1,𝔽8)⋊C4)—76no—
34C2×C7⋊D7—31no—
35C4×C72C3×D311yescharacteristic, complemented
36C7⋊Dic7—31no—
37C4×D21—71no—
38C4×D21—71no—
39C4×D21—76no—
40C4×AGL(1,𝔽7)—211no—
41C4×AGL(1,𝔽7)—211no—
42C4×AGL(1,𝔽7)—216no—
43C3×C72C2×C3×C411yescharacteristic, complemented
44#3861—21no—
45#3861C4×D311yescharacteristic, complemented
46C2×C3×ΓL(1,𝔽8)—71no—
47C2×C3×ΓL(1,𝔽8)—71no—
48C2×C3×ΓL(1,𝔽8)—146no—
49C3⋊AGL(1,𝔽7)—72no—
50C3⋊AGL(1,𝔽7)—72no—
51C3⋊AGL(1,𝔽7)—712no—
52C2×C72C2×C3×D311yescharacteristic
53C7⋊D7—32no—
54C2×D21—71no—
55C2×D21—71no—
56C2×D21—76no—
57C2×AGL(1,𝔽7)—211no—
58C2×AGL(1,𝔽7)—211no—
59C2×AGL(1,𝔽7)—216no—
60C3×C4×C7AGL(1,𝔽7)11yescharacteristic, complemented
61C3×C4×C7AGL(1,𝔽7)11yescharacteristic, complemented
62C3×C4×C7AGL(1,𝔽7)16yescomplemented
63Dic21—71no—
64Dic21—71no—
65Dic21—76no—
66C4×ΓL(1,𝔽8)—71no—
67C4×ΓL(1,𝔽8)—141no—
68C4×ΓL(1,𝔽8)—141no—
69C4×ΓL(1,𝔽8)—146no—
70C4×ΓL(1,𝔽8)—141no—
71C4×ΓL(1,𝔽8)—146no—
72ΓL(1,𝔽8)⋊C4—211no—
73ΓL(1,𝔽8)⋊C4—211no—
74ΓL(1,𝔽8)⋊C4—216no—
75C3×C4×D3—491noHall
76C3×ΓL(1,𝔽8)—71no—
77C3×ΓL(1,𝔽8)—71no—
78C3×ΓL(1,𝔽8)—146no—
79C4×D7—211no—
80C4×D7—211no—
81C4×D7—216no—
82C72C3×C4×D311yescharacteristic, complemented, Sylow
83C2×C3×C7C2×AGL(1,𝔽7)11yescharacteristic
84C2×C3×C7C2×AGL(1,𝔽7)11yescharacteristic
85C2×C3×C7C2×AGL(1,𝔽7)16yes—
86D21—72no—
87D21—72no—
88D21—712no—
89C2×ΓL(1,𝔽8)—71no—
90C2×ΓL(1,𝔽8)—141no—
91C2×ΓL(1,𝔽8)—141no—
92C2×ΓL(1,𝔽8)—146no—
93C2×ΓL(1,𝔽8)—141no—
94C2×ΓL(1,𝔽8)—146no—
95AGL(1,𝔽7)—212no—
96AGL(1,𝔽7)—212no—
97AGL(1,𝔽7)—2112no—
98C2×C3×D3—491no—
99C32×C4—981no—
100C3×Dic3—491no—
101C2×D7—211no—
102C2×D7—211no—
103C2×D7—216no—
104C4×C7C3⋊AGL(1,𝔽7)11yescharacteristic, complemented
105C4×C7C3⋊AGL(1,𝔽7)11yescharacteristic, complemented
106C4×C7C3⋊AGL(1,𝔽7)16yescomplemented
107Dic7—211no—
108Dic7—211no—
109Dic7—216no—
110C2×C3×C4—1471no—
111C4×D3—491no—
112C3×C7C4×AGL(1,𝔽7)11yescharacteristic, complemented
113C3×C7C4×AGL(1,𝔽7)11yescharacteristic, complemented
114C3×C7C4×AGL(1,𝔽7)16yescomplemented
115ΓL(1,𝔽8)—71no—
116ΓL(1,𝔽8)—141no—
117ΓL(1,𝔽8)—141no—
118ΓL(1,𝔽8)—146no—
119ΓL(1,𝔽8)—141no—
120ΓL(1,𝔽8)—146no—
121C2×C32—981no—
122C3×D3—492no—
123C2×C7C2×C3⋊AGL(1,𝔽7)11yescharacteristic
124C2×C7C2×C3⋊AGL(1,𝔽7)11yescharacteristic
125C2×C7C2×C3⋊AGL(1,𝔽7)16yes—
126D7—212no—
127D7—212no—
128D7—2112no—
129C22×C3—1471no—
130C2×D3—491no—
131C3×C4—981no—
132C3×C4—981no—
133C3×C4—1471no—
134C3×C4#6442411yescharacteristic, complemented
135Dic3—491no—
136C32—981noSylow
137C2×C4—1471noSylow
138C7C4×C3⋊AGL(1,𝔽7)11yescharacteristic, complemented
139C7C4×C3⋊AGL(1,𝔽7)11yescharacteristic, complemented
140C7C4×C3⋊AGL(1,𝔽7)16yescomplemented
141C2×C3—981no—
142C2×C3—981no—
143C2×C3—1472no—
144C2×C3C2×#6442411yescharacteristic
145D3—492no—
146C22—1471no—
147C4—1471no—
148C4C3⋊#6442411yescharacteristic, direct product factor
149C3—981no—
150C3—981no—
151C3C4×#6442411yescharacteristic, complemented
152C2—1472no—
153C2C2×C3⋊#6442411yescharacteristic
subgroupG/H
center, hypercenterC4C3⋊#64424
commutator, hypocenterC3×C72C2×C3×C4
FrattiniC2C2×C3⋊#64424
FittingC3×C4×C72C2×C3
socle, cosocleC2×C3×C72C22×C3
prime radicalC2×C3⋊#64424C2
LCSG ▷ C3×C72
UCSC1 ◁ C4

Complements

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

NH ≅ G/Ntype
2. C3×C4×#3861C2semidirect
3. C3⋊#10596874C2semidirect
4. C4×C3⋊(C7⋊D7)C3semidirect
7. C3⋊#64424C4direct (N × H)
10. C3×C4×C72C2×C3semidirect
11. C3⋊(C7⋊Dic7)C2×C3semidirect
13. C4×#3861D3semidirect
18. C3×#3861C2×C4semidirect
21. C3⋊(C7⋊D7)C3×C4semidirect
35. C4×C72C3×D3semidirect
43. C3×C72C2×C3×C4semidirect
45. #3861C4×D3semidirect
60. C3×C4×C7AGL(1,𝔽7)semidirect
61. C3×C4×C7AGL(1,𝔽7)semidirect
62. C3×C4×C7AGL(1,𝔽7)semidirect
82. C72C3×C4×D3semidirect
104. C4×C7C3⋊AGL(1,𝔽7)semidirect
105. C4×C7C3⋊AGL(1,𝔽7)semidirect
106. C4×C7C3⋊AGL(1,𝔽7)semidirect
112. C3×C7C4×AGL(1,𝔽7)semidirect
113. C3×C7C4×AGL(1,𝔽7)semidirect
114. C3×C7C4×AGL(1,𝔽7)semidirect
134. C3×C4#64424semidirect
138. C7C4×C3⋊AGL(1,𝔽7)semidirect
139. C7C4×C3⋊AGL(1,𝔽7)semidirect
140. C7C4×C3⋊AGL(1,𝔽7)semidirect
148. C4C3⋊#64424direct (N × H)
151. C3C4×#64424semidirect

Maximal subgroups

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

subgroupindex
C2×C3⋊#644242
C3×C4×#38612
C3⋊#105968742
C4×C3⋊(C7⋊D7)3
C4×#644243
C4×C3⋊AGL(1,𝔽7)7