Finite Group Databasebeta

← #28348057

Subgroups — #28348057

Order 13696.

546 conjugacy classes of subgroups; only the 133 normal subgroups are shown.

normal subgroups only

This group has 546 conjugacy classes of subgroups — too many to draw. Only its 133 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×#16812135C212yesmaximal, complemented
2#20293282C211yesmaximal, characteristic, complemented
3#20292999C211yesmaximal, characteristic, complemented
4#20292871C212yesmaximal, complemented
5#20292832C211yesmaximal, characteristic, complemented
6#20292819C211yesmaximal, characteristic, complemented
7#20292778C212yesmaximal, complemented
8#20292770C212yesmaximal, complemented
9C4×#13133803C211yesmaximal, characteristic, complemented
10#20292705C211yesmaximal, characteristic, complemented
11D42×C107C211yesmaximal, characteristic, complemented
12C2×#13133818C2212yescomplemented
13#16812135—28no—
14C2×#13133806C2212yes—
15C2×#13133806C2212yescomplemented
16C2×#13133806C2212yescomplemented
17C2×#13133803C2211yescharacteristic
18C2×#13133803C2212yescomplemented
19C2×#13133802C2212yescomplemented
20#16812109C2211yescharacteristic, complemented
21#16812091—22no—
22#16812088C2212yescomplemented
23#16812080—21no—
24C4×D428C2211yescharacteristic, complemented
25C4⋊C4×D107—21no—
26#16812042C2211yescharacteristic, complemented
27C4⋊(C107⋊C8)C2211yescharacteristic, complemented
28#16812032C2212yescomplemented
29#16812030C2211yescharacteristic
30#16812030C2212yescomplemented
31#16812029C2211yescharacteristic, complemented
32C22⋊(C107⋊C8)C2212yescomplemented
33C4×C107⋊C8C2211yescharacteristic, complemented
34C22×D4×C107—21no—
35C22×D4×C107C2212yes—
36#127×C107C2211yescharacteristic
37#121×C107—21no—
38#121×C107C2212yes—
39#120×C107—22no—
40C4×D4×C107C2211yescharacteristic
41C4×D4×C107C2211yescharacteristic
42#13133818—24no—
43#13133818—42no—
44C2×Dic214C2312yes—
45C2×D428C2311yescharacteristic
46C2×C22⋊D107—22no—
47C2×C4×D107—21no—
48C2×C4×D107—22no—
49#13133806—24no—
50#13133806—24no—
51#13133806—42no—
52#13133803—24no—
53#13133803D414yescomplemented
54#13133802—24no—
55C2×C107⋊C8C2312yes—
56C2×C107⋊C8C2311yescharacteristic
57C2×C107⋊C8C2311yescharacteristic
58C4⋊Dic107—21no—
59C4⋊Dic107—21no—
60C4⋊Dic107C2311yescharacteristic
61Dic107⋊C4—22no—
62C22⋊Dic107—22no—
63(C2×D107)⋊C4—21no—
64C24×C107—22no—
65C2×D4×C107—24no—
66C2×D4×C107—22no—
67C2×D4×C107—22no—
68C2×D4×C107—21no—
69C2×D4×C107—21no—
70C2×D4×C107—42no—
71C2×D4×C107—44no—
72C2×D4×C107D411yescharacteristic
73C2×D4×C107C2312yes—
74C2×D4×C107C2311yescharacteristic
75C22×C4×C107—21no—
76C22×C4×C107C2312yes—
77C4⋊C4×C107D411yescharacteristic
78C4⋊C4×C107C2311yescharacteristic
79C22⋊C4×C107—21no—
80C22⋊C4×C107—22no—
81C22⋊C4×C107D412yes—
82C42×C107C2311yescharacteristic
83C23×C107—21no—
84C23×C107—22no—
85C23×C107—44no—
86C23×C107—44no—
87C23×C107—82no—
88C23×C107C2×D412yes—
89C22×D107—21no—
90D4×C107—22no—
91D4×C107—22no—
92D4×C107—21no—
93D4×C107—22no—
94D4×C107—44no—
95D4×C107—44no—
96D4×C107—42no—
97D4×C107—81no—
98D4×C107C2×D412yes—
99C2×C4×C107—22no—
100C2×C4×C107—22no—
101C2×C4×C107—21no—
102C2×C4×C107—42no—
103C2×C4×C107C2×D411yescharacteristic
104C2×C4×C107C2×D412yes—
105C2×C4×C107C2×D411yescharacteristic
106C2×C4×C107C2411yescharacteristic
107C22⋊D107—44no—
108C2×Dic107—21no—
109C2×Dic107—22no—
110D428—21no—
111D428C2×D412yescomplemented
112C4×D107—44no—
113C4×D107—42no—
114Dic214—22no—
115Dic214—22no—
116C107⋊C8—22no—
117C107⋊C8—21no—
118C107⋊C8C2×D412yescomplemented
119C22×C107—22no—
120C22×C107—22no—
121C22×C107—22no—
122C22×C107—22no—
123C22×C107—41no—
124C22×C107—41no—
125C22×C107—42no—
126C22×C107—42no—
127C22×C107—42no—
128C22×C107—88no—
129C22×C107C22×D411yescharacteristic
130C2×D107—22no—
131C2×D107—41no—
132C2×D107—41no—
133C4×C107—22no—
134C4×C107—21no—
135C4×C107—41no—
136C4×C107AGL(1,ℤ/8)12yescomplemented
137C4×C107C22×D411yescharacteristic
138C4×C107H(5,𝔽2)11yescharacteristic
139Dic107—42no—
140Dic107—41no—
141C2×C107—42no—
142C2×C107—42no—
143C2×C107—82no—
144C2×C107D4211yescharacteristic
145C2×C107C2×AGL(1,ℤ/8)11yescharacteristic
146C2×C107#57711yescharacteristic
147D107—42no—
148#3272—1071nomaximal, Sylow
149C107#327211yescharacteristic, complemented, Sylow
150C2×AGL(1,ℤ/8)—1072no—
151#547—1071no—
152D42D10711yescharacteristic, complemented
153#492—1071no—
154#468—1072no—
155#461—1071no—
156#459—1071no—
157#450—1072no—
158#449—1072no—
159C4×D8—1071no—
160#436—1071no—
161C2×#41—1072no—
162C22×D4—21no—
163C22×D4C2×D10712yes—
164AGL(1,ℤ/8)—2148no—
165C2×SD16—1072no—
166C2×SD16—1072no—
167C2×SD16—1072no—
168C2×D8—1071no—
169C2×D8—1072no—
170C2×M16—1072no—
171C4⋊Q8—1071no—
172#127C2×D10711yescharacteristic
173#123—2142no—
174C22⋊Q8—1072no—
175#121—21no—
176#121—2141no—
177#121C2×D10712yes—
178#120—22no—
179C4×D4—1071no—
180C4×D4C2×D10711yescharacteristic
181C4×D4C2×D10711yescharacteristic
182C2×C4⋊C4—2141no—
183#107—1071no—
184C4⋊C8—1071no—
185#103—1072no—
186#102—1071no—
187#102—1072no—
188#102—1071no—
189C22⋊C8—1072no—
190C4×C8—1071no—
191C24—22no—
192#41—4282no—
193#41—2144no—
194C2×Q8—1072no—
195C2×D4—24no—
196C2×D4—22no—
197C2×D4—22no—
198C2×D4—21no—
199C2×D4—21no—
200C2×D4—42no—
201C2×D4—44no—
202C2×D4—1071no—
203C2×D4—2142no—
204C2×D4C22⋊D10711yescharacteristic
205C2×D4C22×D10712yes—
206C2×D4C22×D10711yescharacteristic
207C22×C4—21no—
208C22×C4—2141no—
209C22×C4—2142no—
210C22×C4C22×D10712yes—
211SD16—4282no—
212SD16—2144no—
213SD16—2144no—
214D8—1074no—
215D8—2144no—
216M16—2144no—
217C2×C8—1071no—
218C2×C8—1071no—
219C2×C8—1072no—
220C4⋊C4—1071no—
221C4⋊C4—2141no—
222C4⋊C4—2142no—
223C4⋊C4—2141no—
224C4⋊C4C22⋊D10711yescharacteristic
225C4⋊C4C22×D10711yescharacteristic
226C22⋊C4—21no—
227C22⋊C4—22no—
228C22⋊C4—2142no—
229C22⋊C4—2141no—
230C22⋊C4C22⋊D10712yes—
231C42C22×D10711yescharacteristic
232C23—21no—
233C23—22no—
234C23—44no—
235C23—44no—
236C23—82no—
237C23—2141no—
238C23C2×C22⋊D10712yes—
239Q8—2142no—
240Q8—2142no—
241D4—22no—
242D4—22no—
243D4—21no—
244D4—22no—
245D4—44no—
246D4—44no—
247D4—42no—
248D4—81no—
249D4—1072no—
250D4—4284no—
251D4—2141no—
252D4D4×D10712yes—
253C2×C4—22no—
254C2×C4—22no—
255C2×C4—21no—
256C2×C4—42no—
257C2×C4—4284no—
258C2×C4—4282no—
259C2×C4—2141no—
260C2×C4—2142no—
261C2×C4C2×C22⋊D10711yescharacteristic
262C2×C4C2×C22⋊D10712yes—
263C2×C4C2×C22⋊D10711yescharacteristic
264C2×C4C23×D10711yescharacteristic
265C8—1072no—
266C8—2142no—
267C8—2141no—
268C22—22no—
269C22—22no—
270C22—22no—
271C22—22no—
272C22—41no—
273C22—41no—
274C22—42no—
275C22—42no—
276C22—42no—
277C22—88no—
278C22—4281no—
279C22—4281no—
280C22—2142no—
281C22C22×C22⋊D10711yescharacteristic
282C4—21no—
283C4—22no—
284C4—41no—
285C4—4281no—
286C4—4282no—
287C4#1681213512yescomplemented
288C4C2×D4×D10711yescharacteristic
289C4#1681215411yescharacteristic
290C2—42no—
291C2—42no—
292C2—82no—
293C2—4282no—
294C2D4×C22⋊D10711yescharacteristic
295C2C2×#1681213511yescharacteristic
296C2#2029343811yescharacteristic
subgroupG/H
centerC22C22×C22⋊D107
commutator, cosocleC2×C4×C107C24
FrattiniC2×C4C23×D107
FittingD42×C107C2
socleC22×C107C22×D4
hypercenterD42D107
hypocenterC107#3272
LCSG ▷ C2×C4×C107 ▷ C2×C107 ▷ C107
UCSC1 ◁ C22 ◁ C4×D4 ◁ D42

Complements

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

NH ≅ G/Ntype
1. C2×#16812135C2semidirect
2. #20293282C2semidirect
3. #20292999C2semidirect
4. #20292871C2semidirect
5. #20292832C2semidirect
6. #20292819C2semidirect
7. #20292778C2semidirect
8. #20292770C2semidirect
9. C4×#13133803C2semidirect
10. #20292705C2semidirect
11. D42×C107C2semidirect
12. C2×#13133818C22semidirect
15. C2×#13133806C22semidirect
16. C2×#13133806C22semidirect
18. C2×#13133803C22semidirect
19. C2×#13133802C22semidirect
20. #16812109C22semidirect
22. #16812088C22semidirect
24. C4×D428C22semidirect
26. #16812042C22semidirect
27. C4⋊(C107⋊C8)C22semidirect
28. #16812032C22semidirect
30. #16812030C22semidirect
31. #16812029C22semidirect
32. C22⋊(C107⋊C8)C22semidirect
33. C4×C107⋊C8C22semidirect
53. #13133803D4semidirect
111. D428C2×D4semidirect
118. C107⋊C8C2×D4semidirect
136. C4×C107AGL(1,ℤ/8)semidirect
149. C107#3272semidirect
152. D42D107semidirect
287. C4#16812135semidirect

Maximal subgroups

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

subgroupindex
C2×#168121352
#202932822
#202929992
#202928712
#202928322
#202928192
#202927782
#202927702
C4×#131338032
#202927052
D42×C1072
#3272107