Finite Group Databasebeta

← #89599⋊C3 #13051727

Subgroups — #89599⋊C3 #13051727

Order 1344.

292 conjugacy classes of subgroups; only the 66 normal subgroups are shown.

normal subgroups only

This group has 292 conjugacy classes of subgroups — too many to draw. Only its 66 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
1#10621519C211yesmaximal, characteristic, complemented
2#6512⋊C3C211yesmaximal, characteristic, complemented
3#6471⋊C3C211yesmaximal, characteristic, complemented
4#6438⋊C3C211yesmaximal, characteristic, complemented
5ΓL(1,𝔽8)×#104C211yesmaximal, characteristic, complemented
6#6398⋊C3C211yesmaximal, characteristic, complemented
7#6379⋊C3C211yesmaximal, characteristic, complemented
8#89599C311yesmaximal, characteristic, complemented, Hall
9#41×ΓL(1,𝔽8)C2211yescharacteristic
10Q8×AGL(1,𝔽7)—21no—
11#1159⋊C3—21no—
12#1159⋊C3—22no—
13#1157⋊C3—21no—
14#1157⋊C3C2211yescharacteristic
15C2×C4⋊AGL(1,𝔽7)C2211yescharacteristic
16C2×ΓL(1,𝔽8)⋊Q8—21no—
17C2×ΓL(1,𝔽8)⋊Q8C2211yescharacteristic
18M16×ΓL(1,𝔽8)C2211yescharacteristic
19C42×ΓL(1,𝔽8)C2211yescharacteristic
20#1143⋊C3—21no—
21#1141⋊C3—21no—
22(C2×AGL(1,𝔽7))⋊C4—22no—
23#1136⋊C3C2211yescharacteristic
24D56⋊C3—21no—
25#1132⋊C3—21no—
26#6554C2×C311yescharacteristic, complemented
27#6512C2×C311yescharacteristic, complemented
28#6471C2×C311yescharacteristic, complemented
29#6438C2×C311yescharacteristic, complemented
30C7×#104C2×C311yescharacteristic, complemented
31#6398C2×C311yescharacteristic, complemented
32#6379C2×C311yescharacteristic, complemented
33C3×#454—71nomaximal, Hall
34C22×AGL(1,𝔽7)—21no—
35Q8×ΓL(1,𝔽8)D411yescharacteristic
36D4×ΓL(1,𝔽8)—21no—
37D4×ΓL(1,𝔽8)D411yescharacteristic
38C2×C4×ΓL(1,𝔽8)—21no—
39C2×C4×ΓL(1,𝔽8)—21no—
40C2×C4×ΓL(1,𝔽8)C2311yescharacteristic
41(C22⋊D7)⋊C3—22no—
42(C22⋊D7)⋊C3—21no—
43C2×ΓL(1,𝔽8)⋊C4—22no—
44C2×ΓL(1,𝔽8)⋊C4D412yes—
45C4⋊AGL(1,𝔽7)—21no—
46C4⋊AGL(1,𝔽7)—21no—
47C4⋊AGL(1,𝔽7)D411yescharacteristic
48C4×AGL(1,𝔽7)—22no—
49C4×AGL(1,𝔽7)—21no—
50ΓL(1,𝔽8)⋊Q8—22no—
51ΓL(1,𝔽8)⋊Q8—21no—
52ΓL(1,𝔽8)⋊Q8—21no—
53ΓL(1,𝔽8)⋊Q8D411yescharacteristic
54C8×ΓL(1,𝔽8)—21no—
55ΓL(1,𝔽8)⋊C8—21no—
56C7×#41C22×C311yescharacteristic
57Q8×D7—21no—
58#1159—21no—
59#1159—22no—
60#1157—21no—
61#1157C22×C311yescharacteristic
62C2×D28C22×C311yescharacteristic
63C2×Dic14—21no—
64C2×Dic14C22×C311yescharacteristic
65C7×M16C22×C311yescharacteristic
66C42×C7C22×C311yescharacteristic
67#1143—21no—
68#1141—21no—
69(C2×D7)⋊C4—22no—
70#1136C22×C311yescharacteristic
71D56—21no—
72#1132—21no—
73C3×2−1+4—71no—
74C3×AGL(1,ℤ/8)—71no—
75C3×AGL(1,ℤ/8)—71no—
76C3×#124—71no—
77C3×#104—71no—
78C3×#104—71no—
79C3×#101—71no—
80C22×ΓL(1,𝔽8)—21no—
81C22×ΓL(1,𝔽8)C2×D411yescharacteristic
82C2×AGL(1,𝔽7)—21no—
83C2×AGL(1,𝔽7)—42no—
84C2×AGL(1,𝔽7)—41no—
85C4×ΓL(1,𝔽8)—21no—
86C4×ΓL(1,𝔽8)—41no—
87C4×ΓL(1,𝔽8)C2×D411yescharacteristic
88C4×ΓL(1,𝔽8)C2×D411yescharacteristic
89ΓL(1,𝔽8)⋊C4—22no—
90ΓL(1,𝔽8)⋊C4—21no—
91#454—71noSylow
92C22×D7—21no—
93C7×Q8C3×D411yescharacteristic
94C7×D4—21no—
95C7×D4C3×D411yescharacteristic
96C2×C4×C7—21no—
97C2×C4×C7—21no—
98C2×C4×C7C23×C311yescharacteristic
99C22⋊D7—22no—
100C22⋊D7—21no—
101C2×Dic7—22no—
102C2×Dic7C3×D412yes—
103D28—21no—
104D28—21no—
105D28C3×D411yescharacteristic
106C4×D7—22no—
107C4×D7—21no—
108Dic14—22no—
109Dic14—21no—
110Dic14—21no—
111Dic14C3×D411yescharacteristic
112C7×C8—21no—
113C7⋊C8—21no—
114C3×#41—71no—
115C3×#41—71no—
116C3×#41—142no—
117C3×#41—141no—
118C3×#41—141no—
119C2×C3×Q8—71no—
120C2×C3×Q8—141no—
121C2×C3×Q8—141no—
122C2×C3×D4—71no—
123C3×SD16—141no—
124C3×SD16—141no—
125C3×D8—141no—
126C3×D8—141no—
127C3×M16—71no—
128C3×M16—71no—
129C3×C22⋊C4—142no—
130C3×C42—71no—
131C2×ΓL(1,𝔽8)—21no—
132C2×ΓL(1,𝔽8)—41no—
133C2×ΓL(1,𝔽8)#12011yescharacteristic
134AGL(1,𝔽7)—41no—
135AGL(1,𝔽7)—81no—
1362−1+4—71no—
137AGL(1,ℤ/8)—71no—
138AGL(1,ℤ/8)—71no—
139#124—71no—
140#104—71no—
141#104AGL(1,𝔽7)11yescharacteristic, complemented
142#101—71no—
143C22×C7—21no—
144C22×C7C2×C3×D411yescharacteristic
145C2×D7—21no—
146C2×D7—42no—
147C2×D7—41no—
148C4×C7—21no—
149C4×C7—41no—
150C4×C7C2×C3×D411yescharacteristic
151C4×C7C2×C3×D411yescharacteristic
152Dic7—22no—
153Dic7—21no—
154C23×C3—141no—
155C3×Q8—71no—
156C3×Q8—71no—
157C3×Q8—141no—
158C3×Q8—141no—
159C3×Q8—142no—
160C3×D4—71no—
161C3×D4—71no—
162C3×D4—141no—
163C3×D4—141no—
164C3×D4—141no—
165C3×D4—141no—
166C3×D4—142no—
167C2×C3×C4—72no—
168C2×C3×C4—71no—
169C2×C3×C4—141no—
170C2×C3×C4—141no—
171C2×C3×C4—141no—
172C2×C3×C4—142no—
173C2×C3×C4—142no—
174C3×C8—141no—
175C3×C8—141no—
176ΓL(1,𝔽8)#45411yescharacteristic, complemented, Hall
177#41—71no—
178#41—141no—
179#41—142no—
180#41—141no—
181#41C2×AGL(1,𝔽7)11yescharacteristic
182C2×Q8—71no—
183C2×Q8—141no—
184C2×Q8—141no—
185C2×D4—71no—
186SD16—141no—
187SD16—141no—
188D8—141no—
189D8—141no—
190M16—71no—
191M16C2×AGL(1,𝔽7)11yescharacteristic
192C22⋊C4—142no—
193C42C2×AGL(1,𝔽7)11yescharacteristic
194C2×C7—21no—
195C2×C7—41no—
196C2×C7C3×#12011yescharacteristic
197D7—41no—
198D7—81no—
199C22×C3—71no—
200C22×C3—141no—
201C22×C3—141no—
202C22×C3—281no—
203C22×C3—282no—
204C3×C4—71no—
205C3×C4—71no—
206C3×C4—141no—
207C3×C4—142no—
208C3×C4—141no—
209C3×C4—281no—
210C23—141no—
211Q8—71no—
212Q8—142no—
213Q8—141no—
214Q8—141no—
215Q8C4⋊AGL(1,𝔽7)11yescharacteristic
216D4—21no—
217D4—71no—
218D4—142no—
219D4—141no—
220D4—141no—
221D4—141no—
222D4C4⋊AGL(1,𝔽7)11yescharacteristic
223C2×C4—21no—
224C2×C4—21no—
225C2×C4—72no—
226C2×C4—142no—
227C2×C4—142no—
228C2×C4—141no—
229C2×C4C22×AGL(1,𝔽7)11yescharacteristic
230C8—21no—
231C8—141no—
232C7C3×#45411yescharacteristic, complemented, Sylow
233C2×C3—71no—
234C2×C3—141no—
235C2×C3—281no—
236C2×C3—281no—
237C2×C3—561no—
238C22—21no—
239C22—141no—
240C22—282no—
241C22—281no—
242C22D4×AGL(1,𝔽7)11yescharacteristic
243C4—21no—
244C4—41no—
245C4—142no—
246C4—141no—
247C4C2×C4⋊AGL(1,𝔽7)11yescharacteristic
248C4D4×AGL(1,𝔽7)11yescharacteristic
249C3—71noSylow
250C2—21no—
251C2—41no—
252C2—281no—
253C2—561no—
254C2#6445⋊C311yescharacteristic
subgroupG/H
centerC2#6445⋊C3
commutator, cosocleC2×C4×C7C23×C3
FrattiniC2×C4C22×AGL(1,𝔽7)
FittingC7×#104C2×C3
socleC2×C7C3×#120
hypercenter#104AGL(1,𝔽7)
hypocenterC7C3×#454
LCSG ▷ C2×C4×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C2 ◁ C2×C4 ◁ #104

Complements

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

NH ≅ G/Ntype
1. #10621519C2semidirect
2. #6512⋊C3C2semidirect
3. #6471⋊C3C2semidirect
4. #6438⋊C3C2semidirect
5. ΓL(1,𝔽8)×#104C2semidirect
6. #6398⋊C3C2semidirect
7. #6379⋊C3C2semidirect
8. #89599C3semidirect
26. #6554C2×C3semidirect
27. #6512C2×C3semidirect
28. #6471C2×C3semidirect
29. #6438C2×C3semidirect
30. C7×#104C2×C3semidirect
31. #6398C2×C3semidirect
32. #6379C2×C3semidirect
141. #104AGL(1,𝔽7)semidirect
176. ΓL(1,𝔽8)#454semidirect
232. C7C3×#454semidirect

Maximal subgroups

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

subgroupindex
#106215192
#6512⋊C32
#6471⋊C32
#6438⋊C32
ΓL(1,𝔽8)×#1042
#6398⋊C32
#6379⋊C32
#895993
C3×#4547