Finite Group Databasebeta

← #28348420

Subgroups — #28348420

Order 13696.

442 conjugacy classes of subgroups.

The subgroup lattice has 390 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20293278C211yesmaximal, characteristic, complemented
2#20293268C211yesmaximal, characteristic, complemented
3#20292988C211yesmaximal, characteristic, complemented
4#20292947C211yesmaximal, characteristic, complemented
5#20292916C211yesmaximal, characteristic, complemented
6#20292881C211yesmaximal, characteristic, complemented
7#20292858C211yesmaximal, characteristic, complemented
8#20292845C211yesmaximal, characteristic, complemented
9#20292775C211yesmaximal, characteristic, complemented
10#20292763C211yesmaximal, characteristic, complemented
11#20292738C211yesmaximal, characteristic, complemented
12#20292695C211yesmaximal, characteristic, complemented
13#20292627C211yesmaximal, characteristic, complemented
14C2×#16812029C211yesmaximal, characteristic, complemented
15#483×C107C211yesmaximal, characteristic, complemented
16C2×#13133819C2211yescharacteristic
17C2×#13133818—21no—
18C22×D428C2211yescharacteristic
19C2×#13133805C2211yescharacteristic, complemented
20C2×D856C2211yescharacteristic
21C2×D856C2211yescharacteristic
22C2×#13133801C2211yescharacteristic, complemented
23C22×C107⋊C8C2211yescharacteristic
24#16812108C2211yescharacteristic, complemented
25#16812098C2211yescharacteristic
26#16812094—21no—
27#16812094—21no—
28#16812092—21no—
29#16812087—21no—
30#16812084—21no—
31#16812084C2211yescharacteristic
32#16812084C2211yescharacteristic, complemented
33#16812083C2211yescharacteristic
34#16812079—21no—
35#16812071C2211yescharacteristic
36#16812071C2211yescharacteristic, complemented
37C4×C22⋊D107C2211yescharacteristic
38C4⋊C4×D107C2211yescharacteristic, complemented
39C22⋊C4×D107—21no—
40#16812040C2211yescharacteristic, complemented
41Dic107⋊C8C2211yescharacteristic, complemented
42#16812033C2211yescharacteristic, complemented
43#16812031C2211yescharacteristic
44#16812031C2211yescharacteristic
45#16812029C2211yescharacteristic
46#16812029C2211yescharacteristic
47#16812029C2211yescharacteristic
48#16812029C2211yescharacteristic
49#16812029C2211yescharacteristic, complemented
50(C2×D107)⋊C8C2211yescharacteristic, complemented
51#16812019C2211yescharacteristic, complemented
52C22⋊Q8×C107C2211yescharacteristic
53C2×C4⋊C4×C107C2211yescharacteristic
54#106×C107C2211yescharacteristic, complemented
55#106×C107C2211yescharacteristic, complemented
56#103×C107C2211yescharacteristic, complemented
57#103×C107C2211yescharacteristic, complemented
58C22⋊C8×C107C2211yescharacteristic
59C23×D107—21no—
60#13133819—22no—
61#13133819—41no—
62#13133818—44no—
63C2×Dic214—21no—
64C2×D428—22no—
65C2×D428—21no—
66C2×D428—21no—
67C2×D428—21no—
68C2×D428—21no—
69C2×D428—41no—
70C2×D428C2311yescharacteristic
71C2×D428C2311yescharacteristic
72C2×D428C2311yescharacteristic
73C2×C22⋊D107—21no—
74C2×C22⋊D107D411yescharacteristic
75C2×C4×D107—21no—
76C2×C4×D107—21no—
77C2×C4×D107—21no—
78C2×C4×D107—21no—
79C2×C4×D107C2311yescharacteristic
80#13133805—41no—
81D856—22no—
82D856—41no—
83#13133801—41no—
84C2×C107⋊C8—41no—
85C2×C107⋊C8C2311yescharacteristic
86C2×C107⋊C8C2311yescharacteristic
87C4⋊Dic107C2311yescharacteristic
88Dic107⋊C4—21no—
89Dic107⋊C4—21no—
90Dic107⋊C4D411yescharacteristic
91C22⋊Dic107D411yescharacteristic
92(C2×D107)⋊C4—21no—
93(C2×D107)⋊C4—21no—
94(C2×D107)⋊C4—21no—
95(C2×D107)⋊C4—21no—
96(C2×D107)⋊C4—21no—
97(C2×D107)⋊C4—21no—
98(C2×D107)⋊C4—21no—
99(C2×D107)⋊C4D411yescharacteristic
100C4×Dic107C2311yescharacteristic
101C2×Q8×C107C2311yescharacteristic
102C22×C4×C107—21no—
103C22×C4×C107C2311yescharacteristic
104C2×C8×C107C2311yescharacteristic
105C2×C8×C107C2311yescharacteristic
106C4⋊C4×C107—21no—
107C4⋊C4×C107—21no—
108C4⋊C4×C107C2311yescharacteristic
109C4⋊C4×C107C2311yescharacteristic
110C4⋊C4×C107C2311yescharacteristic
111C22⋊C4×C107—21no—
112C23×C107C2×D411yescharacteristic
113C22×D107—21no—
114C22×D107—21no—
115C22×D107—21no—
116C22×D107—42no—
117C22×D107—42no—
118C22×D107—81no—
119C22×D107C2×D411yescharacteristic
120Q8×C107—21no—
121Q8×C107—21no—
122C2×C4×C107—21no—
123C2×C4×C107—21no—
124C2×C4×C107—21no—
125C2×C4×C107—21no—
126C2×C4×C107—21no—
127C2×C4×C107—21no—
128C2×C4×C107—42no—
129C2×C4×C107C2×D411yescharacteristic
130C2×C4×C107C2411yescharacteristic
131C22⋊D107—22no—
132C22⋊D107—42no—
133C2×Dic107—21no—
134C2×Dic107C2×D411yescharacteristic
135C2×Dic107C2×D411yescharacteristic
136C2×Dic107C2×D411yescharacteristic
137D428—21no—
138D428—21no—
139D428—21no—
140D428—21no—
141D428—21no—
142D428—42no—
143D428—42no—
144D428—42no—
145D428—81no—
146D428#4112yescomplemented
147C4×D107—21no—
148C4×D107—21no—
149C4×D107—42no—
150C4×D107—42no—
151C4×D107—42no—
152C4×D107—81no—
153Dic214—42no—
154C8×C107—21no—
155C8×C107—21no—
156C107⋊C8—21no—
157C107⋊C8—21no—
158C22×C107—21no—
159C22×C107—21no—
160C22×C107AGL(1,ℤ/8)12yes—
161C22×C107C22×D411yescharacteristic
162C2×D107—21no—
163C2×D107—22no—
164C2×D107—21no—
165C2×D107—21no—
166C2×D107—41no—
167C2×D107—41no—
168C2×D107—41no—
169C2×D107—41no—
170C2×D107—41no—
171C2×D107—41no—
172C2×D107—41no—
173C2×D107—41no—
174C2×D107—84no—
175C4×C107—21no—
176C4×C107—41no—
177C4×C107—41no—
178C4×C107—41no—
179C4×C107—41no—
180C4×C107C2×#4111yescharacteristic
181C4×C107H(5,𝔽2)11yescharacteristic
182Dic107—21no—
183Dic107—21no—
184Dic107—21no—
185Dic107—41no—
186C2×C107—22no—
187C2×C107#54611yescharacteristic
188C2×C107C2×AGL(1,ℤ/8)11yescharacteristic
189C2×C107#57711yescharacteristic
190D107—41no—
191D107—42no—
192D107—81no—
193D107—81no—
194#3306—1071nomaximal, Sylow
195C107#330611yescharacteristic, complemented, Sylow
196#547—1071no—
197#546—1071no—
198#489—1071no—
199#483D10711yescharacteristic, complemented
200#481—1071no—
201#476—1071no—
202#469—1071no—
203#465—1071no—
204#463—1071no—
205#449—1071no—
206#447—1071no—
207#442—1071no—
208#435—1071no—
209#417—1071no—
210C2×#102—1071no—
211C2×#41—1071no—
212C2×#41—2141no—
213C22×D4—1071no—
214C2×SD16—1071no—
215C2×D8—1071no—
216C2×D8—1071no—
217C2×M16—1071no—
218C22×C8—1071no—
219C4⋊Q8—1071no—
220#124—1071no—
221#123—2141no—
222#123—2141no—
223#123—2141no—
224C22⋊Q8—2141no—
225C22⋊Q8C2×D10711yescharacteristic
226#121—1071no—
227#121—1071no—
228#121—1071no—
229#121—2141no—
230#120—2141no—
231C4×D4—1071no—
232C4×D4—1071no—
233C4×D4—1071no—
234C2×C4⋊C4—1071no—
235C2×C4⋊C4C2×D10711yescharacteristic
236C2×C22⋊C4—2141no—
237#106—1071no—
238#106C2×D10711yescharacteristic, complemented
239#106C2×D10711yescharacteristic, complemented
240C4⋊C8—1071no—
241#103—1071no—
242#103C2×D10711yescharacteristic, complemented
243#103C2×D10711yescharacteristic, complemented
244#102—1071no—
245#102—1071no—
246#102—1071no—
247#102—1071no—
248#102—1071no—
249#102—1071no—
250#102—1071no—
251C22⋊C8—1071no—
252C22⋊C8C2×D10711yescharacteristic
253#97—1071no—
254C24—2141no—
255#41—4284no—
256#41—4281no—
257#41—2142no—
258C2×Q8—2141no—
259C2×Q8C22×D10711yescharacteristic
260C2×D4—1071no—
261C2×D4—1071no—
262C2×D4—1071no—
263C2×D4—1071no—
264C2×D4—4281no—
265C2×D4—2141no—
266C2×D4—2142no—
267C2×D4—2141no—
268C2×D4—2141no—
269C2×D4—2141no—
270C2×D4—2141no—
271C22×C4—21no—
272C22×C4—1071no—
273C22×C4—2141no—
274C22×C4—2141no—
275C22×C4—2141no—
276C22×C4—2141no—
277C22×C4C22×D10711yescharacteristic
278SD16—4281no—
279D8—4281no—
280D8—2142no—
281M16—4281no—
282C2×C8—1071no—
283C2×C8—1071no—
284C2×C8—4281no—
285C2×C8C22×D10711yescharacteristic
286C2×C8C22×D10711yescharacteristic
287C4⋊C4—21no—
288C4⋊C4—21no—
289C4⋊C4—1071no—
290C4⋊C4—1071no—
291C4⋊C4—2141no—
292C4⋊C4—2141no—
293C4⋊C4C22×D10711yescharacteristic
294C4⋊C4C22×D10711yescharacteristic
295C4⋊C4C22×D10711yescharacteristic
296C22⋊C4—21no—
297C22⋊C4—1071no—
298C22⋊C4—1071no—
299C22⋊C4—2141no—
300C22⋊C4—2141no—
301C22⋊C4—2141no—
302C22⋊C4—2141no—
303C22⋊C4—2141no—
304C22⋊C4—2141no—
305C22⋊C4—2141no—
306C42—1071no—
307C23—8561no—
308C23—1071no—
309C23—4282no—
310C23—4282no—
311C23—2141no—
312C23—2141no—
313C23—2141no—
314C23D4×D10711yescharacteristic
315Q8—21no—
316Q8—21no—
317Q8—4282no—
318D4—8561no—
319D4—1072no—
320D4—4282no—
321D4—4282no—
322D4—4282no—
323D4—4282no—
324D4—2142no—
325D4—2141no—
326D4—2141no—
327D4—2141no—
328D4—2141no—
329D4—2141no—
330C2×C4—21no—
331C2×C4—21no—
332C2×C4—21no—
333C2×C4—21no—
334C2×C4—21no—
335C2×C4—21no—
336C2×C4—42no—
337C2×C4—8561no—
338C2×C4—1071no—
339C2×C4—1071no—
340C2×C4—1071no—
341C2×C4—4282no—
342C2×C4—4282no—
343C2×C4—4282no—
344C2×C4—2141no—
345C2×C4—2141no—
346C2×C4—2141no—
347C2×C4D4×D10711yescharacteristic
348C2×C4C23×D10711yescharacteristic
349C8—21no—
350C8—21no—
351C8—2141no—
352C8—2141no—
353C22—21no—
354C22—21no—
355C22—8564no—
356C22—4281no—
357C22—4281no—
358C22—4281no—
359C22—4281no—
360C22—4281no—
361C22—4281no—
362C22—4281no—
363C22—4281no—
364C22—2141no—
365C22—2141no—
366C22—2142no—
367C22—2141no—
368C22#1681213612yes—
369C22C2×D4×D10711yescharacteristic
370C4—21no—
371C4—41no—
372C4—41no—
373C4—41no—
374C4—41no—
375C4—4281no—
376C4—2141no—
377C4—2141no—
378C4—2141no—
379C4#41×D10711yescharacteristic
380C4#1681215511yescharacteristic
381C2—22no—
382C2—8561no—
383C2—8561no—
384C2—4282no—
385C2—4281no—
386C2#2029326911yescharacteristic
387C2C2×#1681213611yescharacteristic
388C2#2029343611yescharacteristic
subgroupG/H
centerC22C2×D4×D107
commutator, cosocleC2×C4×C107C24
FrattiniC2×C4C23×D107
Fitting#483×C107C2
socleC22×C107C22×D4
hypercenter#483D107
hypocenterC107#3306
LCSG ▷ C2×C4×C107 ▷ C2×C107 ▷ C107
UCSC1 ◁ C22 ◁ C22×C4 ◁ #483

Complements

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

NH ≅ G/Ntype
1. #20293278C2semidirect
2. #20293268C2semidirect
3. #20292988C2semidirect
4. #20292947C2semidirect
5. #20292916C2semidirect
6. #20292881C2semidirect
7. #20292858C2semidirect
8. #20292845C2semidirect
9. #20292775C2semidirect
10. #20292763C2semidirect
11. #20292738C2semidirect
12. #20292695C2semidirect
13. #20292627C2semidirect
14. C2×#16812029C2semidirect
15. #483×C107C2semidirect
19. C2×#13133805C22semidirect
22. C2×#13133801C22semidirect
24. #16812108C22semidirect
32. #16812084C22semidirect
36. #16812071C22semidirect
38. C4⋊C4×D107C22semidirect
40. #16812040C22semidirect
41. Dic107⋊C8C22semidirect
42. #16812033C22semidirect
49. #16812029C22semidirect
50. (C2×D107)⋊C8C22semidirect
51. #16812019C22semidirect
54. #106×C107C22semidirect
55. #106×C107C22semidirect
56. #103×C107C22semidirect
57. #103×C107C22semidirect
146. D428#41semidirect
195. C107#3306semidirect
199. #483D107semidirect
238. #106C2×D107semidirect
239. #106C2×D107semidirect
242. #103C2×D107semidirect
243. #103C2×D107semidirect

Maximal subgroups

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

subgroupindex
#202932782
#202932682
#202929882
#202929472
#202929162
#202928812
#202928582
#202928452
#202927752
#202927632
#202927382
#202926952
#202926272
C2×#168120292
#483×C1072
#3306107