Finite Group Databasebeta

← #89831⋊C3 #13051956

Subgroups — #89831⋊C3 #13051956

Order 1344.

260 conjugacy classes of subgroups; only the 72 normal subgroups are shown.

normal subgroups only

This group has 260 conjugacy classes of subgroups — too many to draw. Only its 72 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)×#124C211yesmaximal, characteristic, complemented
2C2×#1144⋊C3C211yesmaximal, characteristic, complemented
3C2×#1143⋊C3C211yesmaximal, characteristic, complemented
4C2×#1142⋊C3C211yesmaximal, characteristic, complemented
5C2×#1141⋊C3C211yesmaximal, characteristic
6#6438⋊C3C211yesmaximal, characteristic, complemented
7C4×ΓL(1,𝔽8)⋊C8C211yesmaximal, characteristic, complemented
8#89831C311yesmaximal, characteristic, complemented, Hall
9C2×Q8×ΓL(1,𝔽8)C2211yescharacteristic
10C2×D4×ΓL(1,𝔽8)C2211yescharacteristic
11C2×C4⋊AGL(1,𝔽7)C2211yescharacteristic
12C2×ΓL(1,𝔽8)⋊Q8C2211yescharacteristic
13C22⋊C4×ΓL(1,𝔽8)—22no—
14C42×ΓL(1,𝔽8)C2211yescharacteristic
15#1144⋊C3—22no—
16#1143⋊C3—22no—
17#1142⋊C3—22no—
18#1141⋊C3—22no—
19(C2×AGL(1,𝔽7))⋊C4—22no—
20C2×ΓL(1,𝔽8)⋊C8C2211yescharacteristic
21C2×ΓL(1,𝔽8)⋊C8C2211yescharacteristic
22C7×#124C2×C311yescharacteristic, complemented
23C2×#1144C2×C311yescharacteristic, complemented
24C2×#1143C2×C311yescharacteristic, complemented
25C2×#1142C2×C311yescharacteristic, complemented
26C2×#1141C2×C311yescharacteristic
27#6438C2×C311yescharacteristic, complemented
28C4×C7⋊C8C2×C311yescharacteristic, complemented
29C3×#495—71nomaximal, Hall
30C23×ΓL(1,𝔽8)—21no—
31C22×AGL(1,𝔽7)—21no—
32Q8×ΓL(1,𝔽8)—21no—
33Q8×ΓL(1,𝔽8)—21no—
34D4×ΓL(1,𝔽8)—21no—
35D4×ΓL(1,𝔽8)—21no—
36C2×C4×ΓL(1,𝔽8)—21no—
37C2×C4×ΓL(1,𝔽8)D412yes—
38C2×C4×ΓL(1,𝔽8)C2311yescharacteristic
39C2×ΓL(1,𝔽8)⋊C4—21no—
40C4⋊AGL(1,𝔽7)—21no—
41C4⋊AGL(1,𝔽7)—21no—
42ΓL(1,𝔽8)⋊Q8—21no—
43ΓL(1,𝔽8)⋊Q8—21no—
44ΓL(1,𝔽8)⋊C8D412yes—
45ΓL(1,𝔽8)⋊C8D412yes—
46C2×C7×Q8C22×C311yescharacteristic
47C2×C7×D4C22×C311yescharacteristic
48C2×D28C22×C311yescharacteristic
49C2×Dic14C22×C311yescharacteristic
50C7×C22⋊C4—22no—
51C42×C7C22×C311yescharacteristic
52#1144—22no—
53#1143—22no—
54#1142—22no—
55#1141—22no—
56(C2×D7)⋊C4—22no—
57C2×C7⋊C8C22×C311yescharacteristic
58C2×C7⋊C8C22×C311yescharacteristic
59C2×C3×Q16—71no—
60C2×C3×SD16—71no—
61C2×C3×SD16—71no—
62C2×C3×D8—71no—
63C3×#124—71no—
64C3×#124—71no—
65C3×C4×C8—71no—
66C22×ΓL(1,𝔽8)—41no—
67C22×ΓL(1,𝔽8)—42no—
68C22×ΓL(1,𝔽8)C2×D411yescharacteristic
69C2×AGL(1,𝔽7)—41no—
70C2×AGL(1,𝔽7)—42no—
71C4×ΓL(1,𝔽8)—22no—
72C4×ΓL(1,𝔽8)—41no—
73C4×ΓL(1,𝔽8)C2×D411yescharacteristic
74C4×ΓL(1,𝔽8)C2×D411yescharacteristic
75ΓL(1,𝔽8)⋊C4—41no—
76#495—71noSylow
77C23×C7—21no—
78C22×D7—21no—
79C7×Q8—21no—
80C7×Q8—21no—
81C7×D4—21no—
82C7×D4—21no—
83C2×C4×C7—21no—
84C2×C4×C7C3×D412yes—
85C2×C4×C7C23×C311yescharacteristic
86C2×Dic7—21no—
87D28—21no—
88D28—21no—
89Dic14—21no—
90Dic14—21no—
91C7⋊C8C3×D412yes—
92C7⋊C8C3×D412yes—
93C2×C3×Q8—71no—
94C2×C3×Q8—71no—
95C2×C3×D4—71no—
96C2×C3×D4—71no—
97C3×Q16—142no—
98C3×SD16—142no—
99C3×SD16—142no—
100C3×D8—142no—
101C2×C3×C8—71no—
102C2×C3×C8—71no—
103C3×C22⋊C4—142no—
104C3×C22⋊C4—142no—
105C3×C42—71no—
106C2×ΓL(1,𝔽8)—81no—
107C2×ΓL(1,𝔽8)#12711yescharacteristic
108C2×ΓL(1,𝔽8)#13512yes—
109AGL(1,𝔽7)—81no—
110C2×Q16—71no—
111C2×SD16—71no—
112C2×SD16—71no—
113C2×D8—71no—
114#124—71no—
115#124AGL(1,𝔽7)11yescharacteristic, complemented
116C4×C8—71no—
117C22×C7—41no—
118C22×C7—42no—
119C22×C7C2×C3×D411yescharacteristic
120C2×D7—41no—
121C2×D7—42no—
122C4×C7—22no—
123C4×C7—41no—
124C4×C7C2×C3×D411yescharacteristic
125C4×C7C2×C3×D411yescharacteristic
126Dic7—41no—
127C23×C3—141no—
128C23×C3—141no—
129C3×Q8—141no—
130C3×Q8—141no—
131C3×Q8—141no—
132C3×Q8—141no—
133C3×D4—141no—
134C3×D4—141no—
135C3×D4—141no—
136C3×D4—141no—
137C2×C3×C4—72no—
138C2×C3×C4—71no—
139C2×C3×C4—141no—
140C2×C3×C4—141no—
141C3×C8—72no—
142C3×C8—72no—
143ΓL(1,𝔽8)#49511yescharacteristic, complemented, Hall
144C2×Q8—71no—
145C2×Q8C2×AGL(1,𝔽7)11yescharacteristic
146C2×D4—71no—
147C2×D4C2×AGL(1,𝔽7)11yescharacteristic
148Q16—142no—
149SD16—142no—
150SD16—142no—
151D8—142no—
152C2×C8—71no—
153C2×C8—71no—
154C22⋊C4—22no—
155C22⋊C4—142no—
156C42C2×AGL(1,𝔽7)11yescharacteristic
157C2×C7—81no—
158C2×C7C3×#12711yescharacteristic
159C2×C7C3×#13512yes—
160D7—81no—
161C22×C3—71no—
162C22×C3—281no—
163C22×C3—282no—
164C22×C3—281no—
165C22×C3—282no—
166C3×C4—71no—
167C3×C4—71no—
168C3×C4—142no—
169C3×C4—281no—
170C3×C4—281no—
171C23—21no—
172C23—141no—
173Q8—21no—
174Q8—21no—
175Q8—141no—
176Q8—141no—
177D4—21no—
178D4—21no—
179D4—141no—
180D4—141no—
181C2×C4—21no—
182C2×C4—141no—
183C2×C4(C22⋊D7)⋊C312yes—
184C2×C4C22×AGL(1,𝔽7)11yescharacteristic
185C8—72no—
186C8—72no—
187C7C3×#49511yescharacteristic, complemented, Sylow
188C2×C3—71no—
189C2×C3—72no—
190C2×C3—561no—
191C2×C3—561no—
192C22—41no—
193C22—42no—
194C22—281no—
195C22—282no—
196C22C2×(C22⋊D7)⋊C311yescharacteristic
197C4—22no—
198C4—41no—
199C4—281no—
200C4D4×AGL(1,𝔽7)11yescharacteristic
201C4D4×AGL(1,𝔽7)11yescharacteristic
202C3—71noSylow
203C2—81no—
204C2—561no—
205C2#6503⋊C311yescharacteristic
206C2#6513⋊C312yes—
subgroupG/H
centerC22C2×(C22⋊D7)⋊C3
commutator, cosocleC2×C4×C7C23×C3
FrattiniC2×C4C22×AGL(1,𝔽7)
FittingC7×#124C2×C3
socleC22×C7C2×C3×D4
prime radicalC2×#1141⋊C3C2
hypercenter#124AGL(1,𝔽7)
hypocenterC7C3×#495
LCSG ▷ C2×C4×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C22 ◁ C42 ◁ #124

Complements

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

NH ≅ G/Ntype
1. ΓL(1,𝔽8)×#124C2semidirect
2. C2×#1144⋊C3C2semidirect
3. C2×#1143⋊C3C2semidirect
4. C2×#1142⋊C3C2semidirect
6. #6438⋊C3C2semidirect
7. C4×ΓL(1,𝔽8)⋊C8C2semidirect
8. #89831C3semidirect
22. C7×#124C2×C3semidirect
23. C2×#1144C2×C3semidirect
24. C2×#1143C2×C3semidirect
25. C2×#1142C2×C3semidirect
27. #6438C2×C3semidirect
28. C4×C7⋊C8C2×C3semidirect
115. #124AGL(1,𝔽7)semidirect
143. ΓL(1,𝔽8)#495semidirect
187. C7C3×#495semidirect

Maximal subgroups

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

subgroupindex
ΓL(1,𝔽8)×#1242
C2×#1144⋊C32
C2×#1143⋊C32
C2×#1142⋊C32
C2×#1141⋊C32
#6438⋊C32
C4×ΓL(1,𝔽8)⋊C82
#898313
C3×#4957