Finite Group Databasebeta

← #90389⋊C3 #13057431

Subgroups — #90389⋊C3 #13057431

Order 1344.

480 conjugacy classes of subgroups; only the 168 normal subgroups are shown.

normal subgroups only

This group has 480 conjugacy classes of subgroups — too many to draw. Only its 168 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
1ΓL(1,𝔽8)×#125C211yesmaximal, characteristic, complemented
2#6458⋊C3C212yesmaximal, complemented
3#6458⋊C3C214yesmaximal, complemented
4#6457⋊C3C214yesmaximal, complemented
5#6456⋊C3C212yesmaximal, complemented
6#10620710C211yesmaximal, characteristic, complemented
7#6435⋊C3C211yesmaximal, characteristic, complemented
8#90389C311yesmaximal, characteristic, complemented, Hall
9C2×C4⋊AGL(1,𝔽7)C2212yes—
10C2×C4⋊AGL(1,𝔽7)C2214yes—
11C2×C4⋊AGL(1,𝔽7)C2214yes—
12C2×C4×AGL(1,𝔽7)C2211yescharacteristic
13C2×C4×AGL(1,𝔽7)C2214yes—
14C4⋊C4×ΓL(1,𝔽8)C2212yes—
15C4⋊C4×ΓL(1,𝔽8)C2214yes—
16C42×ΓL(1,𝔽8)C2211yescharacteristic
17(C2×AGL(1,𝔽7))⋊C4C2212yes—
18(C2×AGL(1,𝔽7))⋊C4C2214yes—
19(C2×AGL(1,𝔽7))⋊C4C2214yes—
20(ΓL(1,𝔽8)⋊C4)⋊C4C2212yes—
21C4×ΓL(1,𝔽8)⋊C4C2211yescharacteristic
22C7×#125C2×C311yescharacteristic, complemented
23#6458C2×C314yescomplemented
24#6458C2×C312yescomplemented
25#6457C2×C314yescomplemented
26#6456C2×C312yescomplemented
27#6437C2×C311yescharacteristic, complemented
28#6435C2×C311yescharacteristic, complemented
29C3×#540—71nomaximal, Hall
30C22×AGL(1,𝔽7)C2311yescharacteristic
31C22×AGL(1,𝔽7)C2314yes—
32C2×C4×ΓL(1,𝔽8)C2312yes—
33C2×C4×ΓL(1,𝔽8)C2314yes—
34C2×C4×ΓL(1,𝔽8)C2311yescharacteristic
35C2×ΓL(1,𝔽8)⋊C4C2312yes—
36C2×ΓL(1,𝔽8)⋊C4C2311yescharacteristic
37C4⋊AGL(1,𝔽7)—24no—
38C4⋊AGL(1,𝔽7)—44no—
39C4⋊AGL(1,𝔽7)—44no—
40C4×AGL(1,𝔽7)—22no—
41C4×AGL(1,𝔽7)—44no—
42C2×D28C22×C312yes—
43C2×D28C22×C314yes—
44C2×D28C22×C314yes—
45C2×C4×D7C22×C311yescharacteristic
46C2×C4×D7C22×C314yes—
47C7×C4⋊C4C22×C312yes—
48C7×C4⋊C4C22×C314yes—
49C42×C7C22×C311yescharacteristic
50(C2×D7)⋊C4C22×C312yes—
51(C2×D7)⋊C4C22×C314yes—
52(C2×D7)⋊C4C22×C314yes—
53Dic7⋊C4C22×C312yes—
54C4×Dic7C22×C311yescharacteristic
55C3×#127—71no—
56C3×#125—71no—
57C3×#123—74no—
58C3×#121—74no—
59C3×#121—72no—
60C3×C4×D4—72no—
61C3×#117—71no—
62C22×ΓL(1,𝔽8)C2411yescharacteristic
63C2×AGL(1,𝔽7)—21no—
64C2×AGL(1,𝔽7)—24no—
65C2×AGL(1,𝔽7)—24no—
66C2×AGL(1,𝔽7)—22no—
67C2×AGL(1,𝔽7)—24no—
68C4×ΓL(1,𝔽8)—24no—
69C4×ΓL(1,𝔽8)—22no—
70C4×ΓL(1,𝔽8)#4112yes—
71ΓL(1,𝔽8)⋊C4—21no—
72ΓL(1,𝔽8)⋊C4—22no—
73#540—71noSylow
74C22×D7C23×C311yescharacteristic
75C22×D7C23×C314yes—
76C2×C4×C7C23×C312yes—
77C2×C4×C7C23×C314yes—
78C2×C4×C7C23×C311yescharacteristic
79C2×Dic7C23×C312yes—
80C2×Dic7C23×C311yescharacteristic
81D28—24no—
82D28—44no—
83D28—44no—
84C4×D7—22no—
85C4×D7—44no—
86C2×C3×D4—72no—
87C2×C3×D4—74no—
88C2×C3×D4—74no—
89C22×C3×C4—71no—
90C22×C3×C4—74no—
91C3×C4⋊C4—72no—
92C3×C4⋊C4—72no—
93C3×C4⋊C4—74no—
94C3×C22⋊C4—72no—
95C3×C22⋊C4—74no—
96C3×C22⋊C4—74no—
97C3×C42—71no—
98C3×C42—71no—
99C2×ΓL(1,𝔽8)C2×#4111yescharacteristic
100C2×ΓL(1,𝔽8)H(5,𝔽2)12yes—
101AGL(1,𝔽7)—41no—
102AGL(1,𝔽7)—44no—
103#127—71no—
104#125AGL(1,𝔽7)11yescharacteristic, complemented
105#123—74no—
106#121—74no—
107#121—72no—
108C4×D4—72no—
109#117—71no—
110C22×C7C24×C311yescharacteristic
111C2×D7—21no—
112C2×D7—24no—
113C2×D7—24no—
114C2×D7—22no—
115C2×D7—24no—
116C4×C7—24no—
117C4×C7—22no—
118C4×C7C3×#4112yes—
119Dic7—21no—
120Dic7—22no—
121C23×C3—71no—
122C23×C3—74no—
123C3×D4—144no—
124C3×D4—284no—
125C3×D4—284no—
126C2×C3×C4—72no—
127C2×C3×C4—71no—
128C2×C3×C4—71no—
129C2×C3×C4—72no—
130C2×C3×C4—74no—
131C2×C3×C4—142no—
132C2×C3×C4—284no—
133ΓL(1,𝔽8)#54011yescharacteristic, complemented, Hall
134C2×D4—72no—
135C2×D4—74no—
136C2×D4—74no—
137C22×C4—71no—
138C22×C4—74no—
139C4⋊C4—72no—
140C4⋊C4C2×AGL(1,𝔽7)12yes—
141C4⋊C4C2×AGL(1,𝔽7)14yes—
142C22⋊C4—72no—
143C22⋊C4—74no—
144C22⋊C4—74no—
145C42—71no—
146C42C2×AGL(1,𝔽7)11yescharacteristic
147C2×C7C2×C3×#4111yescharacteristic
148C2×C7C3×H(5,𝔽2)12yes—
149D7—41no—
150D7—44no—
151C22×C3—71no—
152C22×C3—141no—
153C22×C3—144no—
154C22×C3—144no—
155C22×C3—142no—
156C22×C3—144no—
157C3×C4—72no—
158C3×C4—144no—
159C3×C4—142no—
160C3×C4—141no—
161C3×C4—142no—
162C23—71no—
163C23—74no—
164D4—144no—
165D4—284no—
166D4—284no—
167C2×C4—72no—
168C2×C4—71no—
169C2×C4—142no—
170C2×C4—284no—
171C2×C4C22×AGL(1,𝔽7)12yes—
172C2×C4C22×AGL(1,𝔽7)14yes—
173C2×C4C22×AGL(1,𝔽7)11yescharacteristic
174C7C3×#54011yescharacteristic, complemented, Sylow
175C2×C3—71no—
176C2×C3—72no—
177C2×C3—281no—
178C2×C3—284no—
179C22—141no—
180C22—144no—
181C22—144no—
182C22—142no—
183C22—144no—
184C22C23×AGL(1,𝔽7)11yescharacteristic
185C4—24no—
186C4—22no—
187C4—141no—
188C4—142no—
189C4#6665912yes—
190C3—71noSylow
191C2—281no—
192C2—284no—
193C2C2×#6665911yescharacteristic
194C2#1062151812yes—
subgroupG/H
center, FrattiniC22C23×AGL(1,𝔽7)
commutator, socle, cosocleC22×C7C24×C3
FittingC7×#125C2×C3
hypercenter#125AGL(1,𝔽7)
hypocenterC7C3×#540
LCSG ▷ C22×C7 ▷ C7
UCSC1 ◁ C22 ◁ #125

Complements

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

NH ≅ G/Ntype
1. ΓL(1,𝔽8)×#125C2semidirect
2. #6458⋊C3C2semidirect
3. #6458⋊C3C2semidirect
4. #6457⋊C3C2semidirect
5. #6456⋊C3C2semidirect
6. #10620710C2semidirect
7. #6435⋊C3C2semidirect
8. #90389C3semidirect
22. C7×#125C2×C3semidirect
23. #6458C2×C3semidirect
24. #6458C2×C3semidirect
25. #6457C2×C3semidirect
26. #6456C2×C3semidirect
27. #6437C2×C3semidirect
28. #6435C2×C3semidirect
104. #125AGL(1,𝔽7)semidirect
133. ΓL(1,𝔽8)#540semidirect
174. C7C3×#540semidirect

Maximal subgroups

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

subgroupindex
ΓL(1,𝔽8)×#1252
#6458⋊C32
#6457⋊C32
#6456⋊C32
#106207102
#6435⋊C32
#903893
C3×#5407