Finite Group Databasebeta

← C42×C3⋊(C3⋊AGL(1,𝔽7)) #72246430

Subgroups — C42×C3⋊(C3⋊AGL(1,𝔽7)) #72246430

Order 6048.

1536 conjugacy classes of subgroups; only the 348 normal subgroups are shown.

normal subgroups only

This group has 1536 conjugacy classes of subgroups — too many to draw. Only its 348 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
1C4×C3⋊(C3⋊(ΓL(1,𝔽8)⋊C4))C213yesmaximal, complemented
2C32×C42×ΓL(1,𝔽8)C211yesmaximal, characteristic, complemented
3C42×C3⋊D21C311yesmaximal, characteristic, complemented
4C2×C4×C3⋊(C3⋊AGL(1,𝔽7))C213yesmaximal
5C42×C3⋊AGL(1,𝔽7)—34nomaximal
6C22×C3⋊(C3⋊AGL(1,𝔽7))C2211yescharacteristic
7C2×C32×C4×ΓL(1,𝔽8)C2213yes—
8C2×C3⋊(C3⋊(ΓL(1,𝔽8)⋊C4))C2213yes—
9C4×C3⋊(C3⋊AGL(1,𝔽7))C4112yesdirect product factor
10C2×C4×C3⋊D21C2×C313yes—
11C2×C4×C3⋊AGL(1,𝔽7)—312no—
12C32×C42×C7C2×C311yescharacteristic, complemented
13C4×C3⋊Dic21C2×C313yescomplemented
14C3×C42×ΓL(1,𝔽8)—24no—
15C3×C42×ΓL(1,𝔽8)D314yescomplemented
16C4×C3⋊(ΓL(1,𝔽8)⋊C4)—312no—
17C3×C42×C3⋊D3—71nomaximal, Hall
18C22×C32×ΓL(1,𝔽8)C2311yescharacteristic
19C2×C3⋊(C3⋊AGL(1,𝔽7))C2×C416yes—
20C32×C4×ΓL(1,𝔽8)C2×C416yescomplemented
21C3⋊(C3⋊(ΓL(1,𝔽8)⋊C4))C2×C416yescomplemented
22C42×D21—34no—
23C42×AGL(1,𝔽7)—91no—
24C22×C3⋊D21C22×C311yescharacteristic
25C22×C3⋊AGL(1,𝔽7)—34no—
26C2×C32×C4×C7C22×C313yes—
27C2×C3⋊Dic21C22×C313yes—
28C4×C3⋊D21C3×C4112yescomplemented
29C2×C3×C4×ΓL(1,𝔽8)—212no—
30C2×C3×C4×ΓL(1,𝔽8)C2×D3112yes—
31C2×C3⋊(ΓL(1,𝔽8)⋊C4)—312no—
32C4×C3⋊AGL(1,𝔽7)—348no—
33C2×C3×C4×C3⋊D3—73no—
34C33×C42—141no—
35C3×C4×C3⋊Dic3—73no—
36C2×C32×ΓL(1,𝔽8)C22×C413yes—
37C3⋊(C3⋊AGL(1,𝔽7))C4214yesdirect product factor
38C2×C4×D21—312no—
39C2×C4×AGL(1,𝔽7)—93no—
40C3×C42×C7C3×D314yescomplemented
41C4×Dic21—312no—
42C42×ΓL(1,𝔽8)—24no—
43C42×ΓL(1,𝔽8)C3⋊D311yescharacteristic, complemented
44C4×ΓL(1,𝔽8)⋊C4—93no—
45C42×C3⋊D3—71no—
46C3×C42×D3—214no—
47C22×C32×C7C23×C311yescharacteristic
48C2×C3⋊D21C2×C3×C416yes—
49C22×C3×ΓL(1,𝔽8)—24no—
50C22×C3×ΓL(1,𝔽8)C22×D314yes—
51C2×C3⋊AGL(1,𝔽7)—324no—
52C32×C4×C7C2×C3×C416yescomplemented
53C3⋊Dic21C2×C3×C416yescomplemented
54C3×C4×ΓL(1,𝔽8)—224no—
55C3×C4×ΓL(1,𝔽8)C4×D3124yescomplemented
56C3⋊(ΓL(1,𝔽8)⋊C4)—324no—
57C42×D7—91noHall
58C22×C3×C3⋊D3—71no—
59C2×C33×C4—143no—
60C2×C3×C3⋊Dic3—73no—
61C3×C4×C3⋊D3—712no—
62C32×ΓL(1,𝔽8)C2×C4211yescharacteristic, complemented, Hall
63C22×D21—34no—
64C22×AGL(1,𝔽7)—91no—
65C2×C3×C4×C7C2×C3×D3112yes—
66C2×Dic21—312no—
67C4×D21—348no—
68C2×C4×ΓL(1,𝔽8)—212no—
69C2×C4×ΓL(1,𝔽8)C2×C3⋊D313yes—
70C2×ΓL(1,𝔽8)⋊C4—93no—
71C4×AGL(1,𝔽7)—912no—
72C2×C4×C3⋊D3—73no—
73C2×C3×C4×D3—2112no—
74C32×C42—144no—
75C32×C42—144no—
76C32×C42AGL(1,𝔽7)11yescharacteristic, complemented
77C4×C3⋊Dic3—73no—
78C3×C4×Dic3—2112no—
79C2×C32×C7C22×C3×C413yes—
80C3⋊D21C3×C4214yescomplemented
81C2×C3×ΓL(1,𝔽8)—212no—
82C2×C3×ΓL(1,𝔽8)C2×C4×D3112yes—
83C3⋊AGL(1,𝔽7)—316no—
84C2×C4×D7—93no—
85C42×C7C3×C3⋊D311yescharacteristic, complemented
86C4×Dic7—93no—
87C22×C33—141no—
88C2×C3×C3⋊D3—76no—
89C33×C4—146no—
90C3×C3⋊Dic3—76no—
91C2×C3×C42—631no—
92C42×D3—214no—
93C22×C3×C7C22×C3×D314yes—
94C2×D21—324no—
95C22×ΓL(1,𝔽8)—24no—
96C22×ΓL(1,𝔽8)C22×C3⋊D311yescharacteristic
97C2×AGL(1,𝔽7)—96no—
98C3×C4×C7C3×C4×D3124yescomplemented
99Dic21—324no—
100C4×ΓL(1,𝔽8)—224no—
101C4×ΓL(1,𝔽8)C4×C3⋊D316yescomplemented
102ΓL(1,𝔽8)⋊C4—96no—
103C22×C3⋊D3—71no—
104C22×C3×D3—214no—
105C2×C32×C4—1412no—
106C2×C32×C4—1412no—
107C2×C32×C4C2×AGL(1,𝔽7)13yes—
108C2×C3⋊Dic3—73no—
109C4×C3⋊D3—712no—
110C2×C3×Dic3—2112no—
111C3×C4×D3—2148no—
112C32×C7C2×C3×C4211yescharacteristic, complemented
113C3×ΓL(1,𝔽8)—24no—
114C3×ΓL(1,𝔽8)C42×D314yescomplemented
115C22×D7—91no—
116C2×C4×C7C2×C3×C3⋊D313yes—
117C2×Dic7—93no—
118C4×D7—912no—
119C2×C33—143no—
120C3×C3⋊D3—74no—
121C22×C3×C4—633no—
122C2×C4×D3—2112no—
123C3×C42—144no—
124C3×C42—141no—
125C3×C42—633no—
126C3×C42C3⋊AGL(1,𝔽7)14yescomplemented
127C4×Dic3—2112no—
128C2×C3×C7C2×C3×C4×D3112yes—
129D21—316no—
130C2×ΓL(1,𝔽8)—212no—
131C2×ΓL(1,𝔽8)C2×C4×C3⋊D313yes—
132AGL(1,𝔽7)—94no—
133C22×C32—144no—
134C22×C32—144no—
135C22×C32C22×AGL(1,𝔽7)11yescharacteristic
136C2×C3⋊D3—76no—
137C2×C3×D3—2124no—
138C32×C4—1424no—
139C32×C4—1424no—
140C32×C4C4×AGL(1,𝔽7)16yescomplemented
141C3⋊Dic3—76no—
142C3×Dic3—2124no—
143C2×C42—631noSylow
144C22×C7C22×C3×C3⋊D311yescharacteristic
145C2×D7—96no—
146C4×C7C3×C4×C3⋊D316yescomplemented
147Dic7—96no—
148C33—141noSylow
149C23×C3—631no—
150C22×D3—214no—
151C2×C3×C4—1412no—
152C2×C3×C4—143no—
153C2×C3×C4—6312no—
154C2×C3×C4—633no—
155C2×C3×C4C2×C3⋊AGL(1,𝔽7)112yes—
156C2×Dic3—2112no—
157C4×D3—2148no—
158C3×C7C3×C42×D314yescomplemented
159ΓL(1,𝔽8)—24no—
160ΓL(1,𝔽8)C42×C3⋊D311yescharacteristic, complemented
161C2×C32—1412no—
162C2×C32—1412no—
163C2×C32C2×C4×AGL(1,𝔽7)13yes—
164C3⋊D3—74no—
165C3×D3—2116no—
166C22×C4—633no—
167C42—633no—
168C42C3⋊(C3⋊AGL(1,𝔽7))11yescharacteristic, direct product factor
169C2×C7C2×C3×C4×C3⋊D313yes—
170D7—94no—
171C22×C3—141no—
172C22×C3—144no—
173C22×C3—636no—
174C22×C3C22×C3⋊AGL(1,𝔽7)14yes—
175C2×D3—2124no—
176C3×C4—146no—
177C3×C4—1424no—
178C3×C4—636no—
179C3×C4C4×C3⋊AGL(1,𝔽7)124yescomplemented
180Dic3—2124no—
181C32—144no—
182C32—144no—
183C32C42×AGL(1,𝔽7)11yescharacteristic, complemented
184C23—631no—
185C2×C4—633no—
186C2×C4—6312no—
187C2×C4C2×C3⋊(C3⋊AGL(1,𝔽7))13yes—
188C7C3×C42×C3⋊D311yescharacteristic, complemented, Sylow
189C2×C3—143no—
190C2×C3—1412no—
191C2×C3—634no—
192C2×C3C2×C4×C3⋊AGL(1,𝔽7)112yes—
193D3—2116no—
194C22—636no—
195C22C22×C3⋊(C3⋊AGL(1,𝔽7))11yescharacteristic
196C4—636no—
197C4C4×C3⋊(C3⋊AGL(1,𝔽7))16yesdirect product factor
198C3—141no—
199C3—144no—
200C3C42×C3⋊AGL(1,𝔽7)14yescomplemented
201C2—634no—
202C2C2×C4×C3⋊(C3⋊AGL(1,𝔽7))13yes—
subgroupG/H
center, hypercenterC42C3⋊(C3⋊AGL(1,𝔽7))
commutator, hypocenterC32×C7C2×C3×C42
FrattiniC22C22×C3⋊(C3⋊AGL(1,𝔽7))
FittingC32×C42×C7C2×C3
socle, cosocleC22×C32×C7C23×C3
prime radicalC22×C3⋊(C3⋊AGL(1,𝔽7))C22
LCSG ▷ C32×C7
UCSC1 ◁ C42

Complements

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

NH ≅ G/Ntype
1. C4×C3⋊(C3⋊(ΓL(1,𝔽8)⋊C4))C2semidirect
2. C32×C42×ΓL(1,𝔽8)C2semidirect
3. C42×C3⋊D21C3semidirect
9. C4×C3⋊(C3⋊AGL(1,𝔽7))C4direct (N × H)
12. C32×C42×C7C2×C3semidirect
13. C4×C3⋊Dic21C2×C3semidirect
15. C3×C42×ΓL(1,𝔽8)D3semidirect
20. C32×C4×ΓL(1,𝔽8)C2×C4semidirect
21. C3⋊(C3⋊(ΓL(1,𝔽8)⋊C4))C2×C4semidirect
28. C4×C3⋊D21C3×C4semidirect
37. C3⋊(C3⋊AGL(1,𝔽7))C42direct (N × H)
40. C3×C42×C7C3×D3semidirect
43. C42×ΓL(1,𝔽8)C3⋊D3semidirect
52. C32×C4×C7C2×C3×C4semidirect
53. C3⋊Dic21C2×C3×C4semidirect
55. C3×C4×ΓL(1,𝔽8)C4×D3semidirect
62. C32×ΓL(1,𝔽8)C2×C42semidirect
76. C32×C42AGL(1,𝔽7)semidirect
80. C3⋊D21C3×C42semidirect
85. C42×C7C3×C3⋊D3semidirect
98. C3×C4×C7C3×C4×D3semidirect
101. C4×ΓL(1,𝔽8)C4×C3⋊D3semidirect
112. C32×C7C2×C3×C42semidirect
114. C3×ΓL(1,𝔽8)C42×D3semidirect
126. C3×C42C3⋊AGL(1,𝔽7)semidirect
140. C32×C4C4×AGL(1,𝔽7)semidirect
146. C4×C7C3×C4×C3⋊D3semidirect
158. C3×C7C3×C42×D3semidirect
160. ΓL(1,𝔽8)C42×C3⋊D3semidirect
168. C42C3⋊(C3⋊AGL(1,𝔽7))direct (N × H)
179. C3×C4C4×C3⋊AGL(1,𝔽7)semidirect
183. C32C42×AGL(1,𝔽7)semidirect
188. C7C3×C42×C3⋊D3semidirect
197. C4C4×C3⋊(C3⋊AGL(1,𝔽7))direct (N × H)
200. C3C42×C3⋊AGL(1,𝔽7)semidirect

Maximal subgroups

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

subgroupindex
C4×C3⋊(C3⋊(ΓL(1,𝔽8)⋊C4))2
C32×C42×ΓL(1,𝔽8)2
C42×C3⋊D213
C2×C4×C3⋊(C3⋊AGL(1,𝔽7))2
C42×C3⋊AGL(1,𝔽7)3
C3×C42×C3⋊D37