Finite Group Databasebeta

← #28345964

Subgroups — #28345964

Order 13696.

526 conjugacy classes of subgroups; only the 149 normal subgroups are shown.

normal subgroups only

This group has 526 conjugacy classes of subgroups — too many to draw. Only its 149 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×#16812135C211yesmaximal, characteristic, complemented
2C22×#13133803C211yesmaximal, characteristic, complemented
3#20293196C211yesmaximal, characteristic, complemented
4#20292842C214yesmaximal, complemented
5#20292822C214yesmaximal, complemented
6#20292654C211yesmaximal, characteristic, complemented
7#20292631C211yesmaximal, characteristic, complemented
8#20292621C211yesmaximal, characteristic, complemented
9#518×C107C211yesmaximal, characteristic, complemented
10C2×#13133818C2211yescharacteristic
11C22×D428C2211yescharacteristic
12#16812135—24no—
13C2×#13133806C2212yescomplemented
14C2×#13133803—24no—
15C2×#13133803C2214yes—
16C2×#13133803C2212yescomplemented
17C2×#13133802C2211yescharacteristic
18C22×C107⋊C8C2211yescharacteristic
19#16812106C2212yescomplemented
20#16812097C2212yescomplemented
21#16812084—22no—
22#16812079—22no—
23C4⋊(C107⋊C8)C2214yescomplemented
24#16812032C2212yescomplemented
25#16812029C2212yescomplemented
26#16812029C2214yescomplemented
27C22×D4×C107C2211yescharacteristic
28C4×D4×C107—22no—
29C4×D4×C107C2214yescomplemented
30#117×C107C2211yescharacteristic
31#117×C107C2211yescharacteristic
32C2×C22⋊C4×C107—22no—
33C23×D107—21no—
34#13133818—22no—
35#13133818—22no—
36C2×Dic214C2311yescharacteristic
37C2×D428—22no—
38C2×D428—22no—
39C2×D428—21no—
40C2×D428—24no—
41C2×D428C2312yes—
42C2×D428C2311yescharacteristic
43C2×C22⋊D107—21no—
44C2×C4×D107—21no—
45#13133806—24no—
46#13133803—28no—
47#13133803—24no—
48#13133802—22no—
49C2×C107⋊C8—22no—
50C2×C107⋊C8C2312yes—
51C2×C107⋊C8C2312yes—
52(C2×D107)⋊C4—24no—
53C24×C107—21no—
54C2×D4×C107—21no—
55C2×D4×C107—22no—
56C2×D4×C107D414yescomplemented
57C2×D4×C107C2312yes—
58C22×C4×C107—24no—
59C22×C4×C107C2311yescharacteristic
60C4⋊C4×C107D412yes—
61C4⋊C4×C107C2312yes—
62C22⋊C4×C107—21no—
63C22⋊C4×C107—24no—
64C22⋊C4×C107D412yes—
65C42×C107—21no—
66C42×C107C2312yes—
67C23×C107—22no—
68C23×C107—24no—
69C23×C107—82no—
70C23×C107C2×D411yescharacteristic
71C22×D107—22no—
72C22×D107—21no—
73C22×D107—42no—
74C22×D107—82no—
75D4×C107—24no—
76D4×C107—22no—
77D4×C107C2×D414yes—
78C2×C4×C107—22no—
79C2×C4×C107—84no—
80C2×C4×C107C2×D414yes—
81C2×C4×C107C2×D411yescharacteristic
82C2×C4×C107C2×D412yes—
83C2×C4×C107#4112yes—
84C2×C4×C107C2411yescharacteristic
85C22⋊D107—42no—
86C2×Dic107—21no—
87D428—22no—
88D428—21no—
89D428—22no—
90D428—21no—
91D428—42no—
92D428—84no—
93C4×D107—42no—
94Dic214—21no—
95Dic214—21no—
96C107⋊C8—22no—
97C107⋊C8—22no—
98C22×C107—22no—
99C22×C107—24no—
100C22×C107—44no—
101C22×C107—84no—
102C22×C107#12112yes—
103C22×C107C22×D411yescharacteristic
104C2×D107—41no—
105C2×D107—42no—
106C2×D107—42no—
107C2×D107—44no—
108C2×D107—84no—
109C4×C107—24no—
110C4×C107—42no—
111C4×C107#12112yes—
112C4×C107C22×D411yescharacteristic
113C4×C107C2×#4111yescharacteristic
114Dic107—41no—
115C2×C107—22no—
116C2×C107—44no—
117C2×C107C2×#12111yescharacteristic
118C2×C107#57612yes—
119D107—81no—
120D107—82no—
121#3041—1071nomaximal, Sylow
122C107#304111yescharacteristic, complemented, Sylow
123C2×AGL(1,ℤ/8)—1071no—
124C22×D8—1071no—
125#535—1071no—
126#518D10711yescharacteristic, complemented
127#463—1074no—
128#459—1074no—
129#424—1071no—
130#418—1071no—
131#416—1071no—
132C2×#41—1071no—
133C22×D4—1071no—
134C22×D4C2×D10711yescharacteristic
135AGL(1,ℤ/8)—2144no—
136C2×SD16—1072no—
137C2×D8—1074no—
138C2×D8—1072no—
139C2×D8—2144no—
140C2×M16—1071no—
141C22×C8—1071no—
142#127—1072no—
143#124—1072no—
144#121—2142no—
145#120—2142no—
146C4×D4—22no—
147C4×D4C2×D10714yescomplemented
148#117C2×D10711yescharacteristic
149#117C2×D10711yescharacteristic
150C2×C22⋊C4—22no—
151C4⋊C8—1074no—
152#103—1072no—
153#102—1072no—
154#102—1074no—
155C24—21no—
156C24—2141no—
157#41—2142no—
158#41—2142no—
159C2×Q8—1071no—
160C2×D4—21no—
161C2×D4—22no—
162C2×D4—1072no—
163C2×D4—1071no—
164C2×D4—2141no—
165C2×D4—2144no—
166C2×D4—2141no—
167C2×D4—2142no—
168C2×D4—2142no—
169C2×D4D42814yescomplemented
170C2×D4C22×D10712yes—
171C22×C4—24no—
172C22×C4—2141no—
173C22×C4C22×D10711yescharacteristic
174SD16—2144no—
175D8—2148no—
176D8—2144no—
177M16—2142no—
178C2×C8—1072no—
179C2×C8—1072no—
180C2×C8—2142no—
181C4⋊C4C22⋊D10712yes—
182C4⋊C4C22×D10712yes—
183C22⋊C4—21no—
184C22⋊C4—24no—
185C22⋊C4—2144no—
186C22⋊C4C22⋊D10712yes—
187C42—21no—
188C42C22×D10712yes—
189C23—22no—
190C23—24no—
191C23—82no—
192C23—8562no—
193C23—4282no—
194C23—2142no—
195C23—2141no—
196C23C2×C22⋊D10711yescharacteristic
197Q8—2141no—
198Q8—2141no—
199D4—24no—
200D4—22no—
201D4—8564no—
202D4—4282no—
203D4—4282no—
204D4—2142no—
205D4—2141no—
206D4—2142no—
207D4—2141no—
208D4C2×D42814yes—
209C2×C4—22no—
210C2×C4—84no—
211C2×C4—4282no—
212C2×C4—2141no—
213C2×C4C2×C22⋊D10714yes—
214C2×C4C2×C22⋊D10711yescharacteristic
215C2×C4C2×D42812yes—
216C2×C4#1313381812yes—
217C2×C4C23×D10711yescharacteristic
218C8—2142no—
219C8—2142no—
220C22—22no—
221C22—24no—
222C22—44no—
223C22—84no—
224C22—8564no—
225C22—4282no—
226C22—4284no—
227C22—4281no—
228C22—4282no—
229C22#1681208312yes—
230C22C22×C22⋊D10711yescharacteristic
231C4—24no—
232C4—42no—
233C4—4281no—
234C4#1681208312yes—
235C4C22×D42811yescharacteristic
236C4C2×#1313381811yescharacteristic
237C2—22no—
238C2—44no—
239C2—8562no—
240C2—8561no—
241C2C2×#1681208311yescharacteristic
242C2#2029343012yes—
subgroupG/H
centerC22C22×C22⋊D107
commutator, cosocleC2×C4×C107C24
FrattiniC2×C4C23×D107
Fitting#518×C107C2
socleC22×C107C22×D4
hypercenter#518D107
hypocenterC107#3041
LCSG ▷ C2×C4×C107 ▷ C2×C107 ▷ C107
UCSC1 ◁ C22 ◁ #117 ◁ #518

Complements

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

NH ≅ G/Ntype
1. C2×#16812135C2semidirect
2. C22×#13133803C2semidirect
3. #20293196C2semidirect
4. #20292842C2semidirect
5. #20292822C2semidirect
6. #20292654C2semidirect
7. #20292631C2semidirect
8. #20292621C2semidirect
9. #518×C107C2semidirect
13. C2×#13133806C22semidirect
16. C2×#13133803C22semidirect
19. #16812106C22semidirect
20. #16812097C22semidirect
23. C4⋊(C107⋊C8)C22semidirect
24. #16812032C22semidirect
25. #16812029C22semidirect
26. #16812029C22semidirect
29. C4×D4×C107C22semidirect
56. C2×D4×C107D4semidirect
122. C107#3041semidirect
126. #518D107semidirect
147. C4×D4C2×D107semidirect
169. C2×D4D428semidirect

Maximal subgroups

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

subgroupindex
C2×#168121352
C22×#131338032
#202931962
#202928422
#202928222
#202926542
#202926312
#202926212
#518×C1072
#3041107