Finite Group Databasebeta

← #72244540

Subgroups — #72244540

Order 2016.

204 conjugacy classes of subgroups; only the 52 normal subgroups are shown.

normal subgroups only

This group has 204 conjugacy classes of subgroups — too many to draw. Only its 52 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×C4⋊(C3⋊AGL(1,𝔽7))C211yesmaximal, characteristic
2C3×M16×ΓL(1,𝔽8)C211yesmaximal, characteristic, complemented
3#66627⋊C3C211yesmaximal, characteristic, complemented
4#10620662C311yesmaximal, characteristic, complemented
5#6397⋊C3—31nomaximal
6C22×C3⋊AGL(1,𝔽7)C412yes—
7C2×C3×C4×ΓL(1,𝔽8)C2211yescharacteristic
8C4⋊(C3⋊AGL(1,𝔽7))—21no—
9C4⋊(C3⋊AGL(1,𝔽7))—21no—
10C3×C8×ΓL(1,𝔽8)—21no—
11C3⋊(ΓL(1,𝔽8)⋊C8)—21no—
12C2×D84C2×C311yescharacteristic
13C2×C4⋊AGL(1,𝔽7)—31no—
14C3×C7×M16C2×C311yescharacteristic, complemented
15#66627C2×C311yescharacteristic, complemented
16M16×ΓL(1,𝔽8)—21no—
17M16×ΓL(1,𝔽8)D311yescharacteristic, complemented
18#1136⋊C3—31no—
19C3×#823—71nomaximal, Hall
20C22×C3×ΓL(1,𝔽8)C2×C411yescharacteristic
21C2×C3⋊AGL(1,𝔽7)—22no—
22C2×C3⋊AGL(1,𝔽7)—42no—
23C3×C4×ΓL(1,𝔽8)D411yescharacteristic
24C3×C4×ΓL(1,𝔽8)D411yescharacteristic
25#6397—31noHall
26C22×D21C3×C412yes—
27C22×AGL(1,𝔽7)—32no—
28C2×C3×C4×C7C22×C311yescharacteristic
29D84—21no—
30D84—21no—
31C2×C4×ΓL(1,𝔽8)—21no—
32C2×C4×ΓL(1,𝔽8)C2×D311yescharacteristic
33C4⋊AGL(1,𝔽7)—61no—
34C4⋊AGL(1,𝔽7)—61no—
35C3×C7×C8—21no—
36C3⋊(C7⋊C8)—21no—
37C8×ΓL(1,𝔽8)—21no—
38C8×ΓL(1,𝔽8)—21no—
39C8×ΓL(1,𝔽8)—21no—
40ΓL(1,𝔽8)⋊C8—61no—
41C2×C3×D12—71no—
42C32×M16—141no—
43C3×#208—71no—
44C2×C3×ΓL(1,𝔽8)—21no—
45C2×C3×ΓL(1,𝔽8)C22⋊C411yescharacteristic
46C3⋊AGL(1,𝔽7)—42no—
47C2×D28—31no—
48C7×M16C3×D311yescharacteristic, complemented
49#1136—31no—
50C3×#100—211no—
51#823—71no—
52C22×C3×C7C2×C3×C411yescharacteristic
53C2×D21—22no—
54C2×D21—42no—
55C22×ΓL(1,𝔽8)—21no—
56C22×ΓL(1,𝔽8)C4×D311yescharacteristic
57C2×AGL(1,𝔽7)—62no—
58C2×AGL(1,𝔽7)—122no—
59C3×C4×C7C3×D411yescharacteristic
60C3×C4×C7C3×D411yescharacteristic
61C4×ΓL(1,𝔽8)—21no—
62C4×ΓL(1,𝔽8)—21no—
63C4×ΓL(1,𝔽8)D1211yescharacteristic
64C4×ΓL(1,𝔽8)#6711yescharacteristic
65C22×C3×D3—72no—
66C2×C32×C4—141no—
67C3×D12—141no—
68C3×D12—141no—
69C32×C8—141no—
70C3×C3⋊C8—141no—
71C3×ΓL(1,𝔽8)#10011yescharacteristic, complemented, Hall
72C22×D7—32no—
73C2×C4×C7C2×C3×D311yescharacteristic
74D28—61no—
75D28—61no—
76C7×C8—21no—
77C7⋊C8—61no—
78C2×C3×D4—211no—
79C2×D12—71no—
80C3×M16—141no—
81C3×M16—141no—
82C3×M16—211no—
83C3×M16AGL(1,𝔽7)11yescharacteristic, complemented
84#208—71no—
85C2×C3×C7—21no—
86C2×C3×C7C3×C22⋊C411yescharacteristic
87D21—42no—
88C2×ΓL(1,𝔽8)—21no—
89C2×ΓL(1,𝔽8)—21no—
90C2×ΓL(1,𝔽8)—41no—
91C2×ΓL(1,𝔽8)(C2×D3)⋊C411yescharacteristic
92AGL(1,𝔽7)—122no—
93C22×C32—141no—
94C2×C3×D3—142no—
95C2×C3×D3—282no—
96C32×C4—141no—
97C32×C4—141no—
98#100—211noSylow
99C22×C7C3×C4×D311yescharacteristic
100C2×D7—62no—
101C2×D7—122no—
102C4×C7C3×D1211yescharacteristic
103C4×C7C3×#6711yescharacteristic
104C23×C3—212no—
105C22×D3—72no—
106C3×D4—421no—
107C3×D4—421no—
108C2×C3×C4—141no—
109C2×C3×C4—141no—
110C2×C3×C4C2×AGL(1,𝔽7)11yescharacteristic
111D12—141no—
112D12—141no—
113C3×C8—21no—
114C3×C8—141no—
115C3×C8—141no—
116C3×C8—141no—
117C3×C8—421no—
118C3⋊C8—141no—
119C3×C7C3×#10011yescharacteristic, complemented
120ΓL(1,𝔽8)—21no—
121ΓL(1,𝔽8)#82311yescharacteristic, complemented
122C2×C32—141no—
123C2×C32—281no—
124C3×D3—282no—
125C2×D4—211no—
126M16—211no—
127M16C3⋊AGL(1,𝔽7)11yescharacteristic, complemented
128C2×C7—21no—
129C2×C7C3×(C2×D3)⋊C411yescharacteristic
130D7—122no—
131C22×C3—141no—
132C22×C3—141no—
133C22×C3—422no—
134C22×C3—842no—
135C22×C3C4×AGL(1,𝔽7)11yescharacteristic
136C2×D3—142no—
137C2×D3—282no—
138C3×C4—141no—
139C3×C4—141no—
140C3×C4—141no—
141C3×C4—141no—
142C3×C4C4⋊AGL(1,𝔽7)11yescharacteristic
143C3×C4(C22⋊D7)⋊C311yescharacteristic
144C32—141noSylow
145C23—212no—
146D4—421no—
147D4—421no—
148C2×C4C2×C3⋊AGL(1,𝔽7)11yescharacteristic
149C8—21no—
150C8—421no—
151C7C3×#82311yescharacteristic, complemented, Sylow
152C2×C3—21no—
153C2×C3—141no—
154C2×C3—141no—
155C2×C3—281no—
156C2×C3—281no—
157C2×C3—842no—
158C2×C3(C2×AGL(1,𝔽7))⋊C411yescharacteristic
159D3—282no—
160C22—422no—
161C22—842no—
162C22C4×C3⋊AGL(1,𝔽7)11yescharacteristic
163C4C4⋊(C3⋊AGL(1,𝔽7))11yescharacteristic
164C4#4268⋊C311yescharacteristic
165C3—141no—
166C3—141no—
167C3#6397⋊C311yescharacteristic, complemented
168C2—21no—
169C2—842no—
170C2#66631⋊C311yescharacteristic
subgroupG/H
centerC2#66631⋊C3
commutatorC22×C3×C7C2×C3×C4
FrattiniC2×C4C2×C3⋊AGL(1,𝔽7)
FittingC3×C7×M16C2×C3
socleC2×C3×C7C3×C22⋊C4
prime radicalC2×C4⋊(C3⋊AGL(1,𝔽7))C2
cosocleC2×C3×C4×C7C22×C3
hypercenterM16C3⋊AGL(1,𝔽7)
hypocenterC3×C7C3×#100
LCSG ▷ C22×C3×C7 ▷ C2×C3×C7 ▷ C3×C7
UCSC1 ◁ C2 ◁ C2×C4 ◁ M16

Complements

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

NH ≅ G/Ntype
2. C3×M16×ΓL(1,𝔽8)C2semidirect
3. #66627⋊C3C2semidirect
4. #10620662C3semidirect
14. C3×C7×M16C2×C3semidirect
15. #66627C2×C3semidirect
17. M16×ΓL(1,𝔽8)D3semidirect
48. C7×M16C3×D3semidirect
71. C3×ΓL(1,𝔽8)#100semidirect
83. C3×M16AGL(1,𝔽7)semidirect
119. C3×C7C3×#100semidirect
121. ΓL(1,𝔽8)#823semidirect
127. M16C3⋊AGL(1,𝔽7)semidirect
151. C7C3×#823semidirect
167. C3#6397⋊C3semidirect

Maximal subgroups

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

subgroupindex
C2×C4⋊(C3⋊AGL(1,𝔽7))2
C3×M16×ΓL(1,𝔽8)2
#66627⋊C32
#106206623
#6397⋊C33
C3×#8237