Finite Group Databasebeta

← #28348424

Subgroups — #28348424

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#20293281C211yesmaximal, characteristic, complemented
2#20293273C211yesmaximal, characteristic, complemented
3#20292987C211yesmaximal, characteristic, complemented
4#20292958C211yesmaximal, characteristic, complemented
5#20292946C211yesmaximal, characteristic, complemented
6#20292918C211yesmaximal, characteristic, complemented
7#20292880C211yesmaximal, characteristic, complemented
8#20292856C211yesmaximal, characteristic, complemented
9#20292843C211yesmaximal, characteristic, complemented
10#20292774C211yesmaximal, characteristic, complemented
11#20292763C211yesmaximal, characteristic, complemented
12#20292740C211yesmaximal, characteristic, complemented
13#20292629C211yesmaximal, characteristic, complemented
14C2×#16812031C211yesmaximal, characteristic, complemented
15#435×C107C211yesmaximal, characteristic, complemented
16C2×#13133818C2211yescharacteristic
17C2×#13133817—21no—
18C22×D428C2211yescharacteristic
19C2×#13133807C2211yescharacteristic, complemented
20C2×D856C2211yescharacteristic
21C2×D856C2211yescharacteristic
22#16812109C2211yescharacteristic, complemented
23#16812097C2211yescharacteristic
24#16812094—21no—
25#16812093—21no—
26#16812093—21no—
27#16812089—21no—
28#16812088C2211yescharacteristic
29#16812083C2211yescharacteristic
30#16812083C2211yescharacteristic
31#16812083C2211yescharacteristic, complemented
32#16812082—21no—
33#16812077—21no—
34C4×D428C2211yescharacteristic
35C4×D428C2211yescharacteristic, complemented
36C2×C4⋊Dic107C2211yescharacteristic
37C2×C4⋊Dic107C2211yescharacteristic, complemented
38C22⋊C4×D107—21no—
39#16812041C2211yescharacteristic, complemented
40#16812041C2211yescharacteristic, complemented
41#16812041C2211yescharacteristic, complemented
42#16812034C2211yescharacteristic, complemented
43#16812034C2211yescharacteristic, complemented
44#16812034C2211yescharacteristic, complemented
45#16812031C2211yescharacteristic
46#16812031C2211yescharacteristic
47#16812031C2211yescharacteristic
48#16812031C2211yescharacteristic
49#16812031C2211yescharacteristic
50#16812031C2211yescharacteristic
51#16812031C2211yescharacteristic, complemented
52C2×M16×C107C2211yescharacteristic, complemented
53C22×C8×C107C2211yescharacteristic
54C4×D4×C107C2211yescharacteristic
55C4⋊C8×C107C2211yescharacteristic, complemented
56C22⋊C8×C107C2211yescharacteristic
57C22⋊C8×C107C2211yescharacteristic, complemented
58#97×C107C2211yescharacteristic, complemented
59C23×D107—21no—
60#13133818—22no—
61#13133818—41no—
62#13133817—44no—
63C2×Dic214—21no—
64C2×Dic214C2311yescharacteristic
65C2×D428—22no—
66C2×D428—21no—
67C2×D428—41no—
68C2×D428C2311yescharacteristic
69C2×D428C2311yescharacteristic
70C2×D428C2311yescharacteristic
71C2×C22⋊D107—21no—
72C2×C22⋊D107—21no—
73C2×C22⋊D107—21no—
74C2×C22⋊D107—21no—
75C22×Dic107—21no—
76C22×Dic107—21no—
77C2×C4×D107—21no—
78C2×C4×D107—21no—
79C2×C4×D107—21no—
80#13133807—41no—
81D856—22no—
82D856—41no—
83C4⋊Dic107—21no—
84C4⋊Dic107—21no—
85C4⋊Dic107—21no—
86C4⋊Dic107C2311yescharacteristic
87C4⋊Dic107C2311yescharacteristic
88C4⋊Dic107C2311yescharacteristic
89C4⋊Dic107C2311yescharacteristic
90Dic107⋊C4—21no—
91C22⋊Dic107—21no—
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—
99C2×D4×C107D411yescharacteristic
100C22×C4×C107C2311yescharacteristic
101C22×C4×C107C2311yescharacteristic
102M16×C107—41no—
103C2×C8×C107—41no—
104C2×C8×C107C2311yescharacteristic
105C2×C8×C107C2311yescharacteristic
106C2×C8×C107C2311yescharacteristic
107C2×C8×C107C2311yescharacteristic
108C4⋊C4×C107D411yescharacteristic
109C22⋊C4×C107D411yescharacteristic
110C22⋊C4×C107D411yescharacteristic
111C42×C107C2311yescharacteristic
112C23×C107C2×D411yescharacteristic
113C23×C107C2×D411yescharacteristic
114C22×D107—21no—
115C22×D107—21no—
116C22×D107—21no—
117C22×D107—42no—
118C22×D107—42no—
119C22×D107—81no—
120D4×C107—22no—
121C2×C4×C107—21no—
122C2×C4×C107—21no—
123C2×C4×C107—21no—
124C2×C4×C107—21no—
125C2×C4×C107C2×D411yescharacteristic
126C2×C4×C107C2×D411yescharacteristic
127C2×C4×C107C2×D411yescharacteristic
128C2×C4×C107C2×D411yescharacteristic
129C2×C4×C107C2411yescharacteristic
130C22⋊D107—42no—
131C22⋊D107—42no—
132C22⋊D107—42no—
133C22⋊D107—81no—
134C2×Dic107—21no—
135C2×Dic107—21no—
136C2×Dic107—21no—
137C2×Dic107—21no—
138C2×Dic107—21no—
139C2×Dic107—42no—
140C2×Dic107—42no—
141D428—21no—
142D428—21no—
143D428—21no—
144D428—21no—
145D428—21no—
146D428—42no—
147D428#4112yescomplemented
148C4×D107—42no—
149C4×D107—42no—
150C4×D107—81no—
151Dic214—21no—
152Dic214—21no—
153Dic214—42no—
154C8×C107—21no—
155C8×C107—21no—
156C8×C107—21no—
157C8×C107—21no—
158C22×C107—21no—
159C22×C107—21no—
160C22×C107—21no—
161C22×C107—21no—
162C22×C107—21no—
163C22×C107AGL(1,ℤ/8)12yes—
164C22×C107C22×D411yescharacteristic
165C2×D107—22no—
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—21no—
177C4×C107—21no—
178C4×C107—21no—
179C4×C107C2×#4111yescharacteristic
180C4×C107H(5,𝔽2)11yescharacteristic
181Dic107—41no—
182Dic107—41no—
183Dic107—41no—
184Dic107—41no—
185Dic107—41no—
186C2×C107—22no—
187C2×C107—41no—
188C2×C107#54611yescharacteristic
189C2×C107C2×AGL(1,ℤ/8)11yescharacteristic
190C2×C107#57711yescharacteristic
191D107—42no—
192D107—81no—
193D107—81no—
194#3306—1071nomaximal, Sylow
195C107#330611yescharacteristic, complemented, Sylow
196#547—1071no—
197#546—1071no—
198#489—1071no—
199#483—1071no—
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#435D10711yescharacteristic, complemented
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×M16C2×D10711yescharacteristic, complemented
218C22×C8C2×D10711yescharacteristic
219C4⋊Q8—1071no—
220#124—1071no—
221#123—2141no—
222#123—2141no—
223#123—2141no—
224C22⋊Q8—1071no—
225C22⋊Q8—2141no—
226#121—1071no—
227#121—1071no—
228#121—1071no—
229#121—2141no—
230#120—2141no—
231C4×D4—1071no—
232C4×D4—1071no—
233C4×D4C2×D10711yescharacteristic
234C2×C4⋊C4—1071no—
235C2×C4⋊C4—1071no—
236C2×C22⋊C4—2141no—
237#106—1071no—
238#106—1071no—
239#106—1071no—
240C4⋊C8C2×D10711yescharacteristic, complemented
241#103—1071no—
242#103—1071no—
243#103—1071no—
244#102—1071no—
245#102—1071no—
246#102—1071no—
247#102—1071no—
248#102—1071no—
249#102—1071no—
250#102—1071no—
251C22⋊C8C2×D10711yescharacteristic
252C22⋊C8C2×D10711yescharacteristic, complemented
253#97C2×D10711yescharacteristic, complemented
254C24—2141no—
255#41—4281no—
256#41—4284no—
257#41—2142no—
258C2×Q8—1071no—
259C2×Q8—2141no—
260C2×D4—1071no—
261C2×D4—1071no—
262C2×D4—1071no—
263C2×D4—4281no—
264C2×D4—2142no—
265C2×D4—2141no—
266C2×D4—2141no—
267C2×D4—2141no—
268C2×D4—2141no—
269C2×D4—2141no—
270C2×D4D42811yescharacteristic
271C22×C4—2141no—
272C22×C4—2141no—
273C22×C4—2141no—
274C22×C4—2141no—
275C22×C4—2141no—
276C22×C4C22×D10711yescharacteristic
277C22×C4C22×D10711yescharacteristic
278SD16—4281no—
279D8—4281no—
280D8—2142no—
281M16—41no—
282C2×C8—41no—
283C2×C8C22×D10711yescharacteristic
284C2×C8C22×D10711yescharacteristic
285C2×C8C22×D10711yescharacteristic
286C2×C8C22×D10711yescharacteristic
287C4⋊C4—1071no—
288C4⋊C4—1071no—
289C4⋊C4—1071no—
290C4⋊C4—1071no—
291C4⋊C4—2141no—
292C4⋊C4—2141no—
293C4⋊C4—2141no—
294C4⋊C4—2141no—
295C4⋊C4D42811yescharacteristic
296C22⋊C4—2141no—
297C22⋊C4—2141no—
298C22⋊C4—2141no—
299C22⋊C4—2141no—
300C22⋊C4—2141no—
301C22⋊C4—2141no—
302C22⋊C4—2141no—
303C22⋊C4—2141no—
304C22⋊C4D42811yescharacteristic
305C22⋊C4D42811yescharacteristic
306C42C22×D10711yescharacteristic
307C23—8561no—
308C23—4282no—
309C23—4282no—
310C23—2141no—
311C23—2141no—
312C23—2141no—
313C23C2×D42811yescharacteristic
314C23C2×D42811yescharacteristic
315Q8—4282no—
316Q8—2141no—
317Q8—2141no—
318D4—22no—
319D4—8561no—
320D4—1072no—
321D4—4282no—
322D4—4282no—
323D4—4282no—
324D4—4282no—
325D4—2141no—
326D4—2141no—
327D4—2141no—
328D4—2141no—
329D4—2141no—
330C2×C4—21no—
331C2×C4—21no—
332C2×C4—21no—
333C2×C4—21no—
334C2×C4—8561no—
335C2×C4—4282no—
336C2×C4—4282no—
337C2×C4—4282no—
338C2×C4—4282no—
339C2×C4—2141no—
340C2×C4—2141no—
341C2×C4—2141no—
342C2×C4—2141no—
343C2×C4—2141no—
344C2×C4C2×D42811yescharacteristic
345C2×C4C2×D42811yescharacteristic
346C2×C4C2×D42811yescharacteristic
347C2×C4C2×D42811yescharacteristic
348C2×C4C23×D10711yescharacteristic
349C8—21no—
350C8—21no—
351C8—21no—
352C8—21no—
353C22—21no—
354C22—21no—
355C22—21no—
356C22—21no—
357C22—21no—
358C22—8564no—
359C22—4281no—
360C22—4281no—
361C22—4281no—
362C22—4281no—
363C22—4281no—
364C22—4281no—
365C22—4281no—
366C22—4281no—
367C22—2142no—
368C22#1681213712yes—
369C22C22×D42811yescharacteristic
370C4—21no—
371C4—21no—
372C4—21no—
373C4—21no—
374C4—4281no—
375C4—4281no—
376C4—4281no—
377C4—4281no—
378C4—4281no—
379C4#41×D10711yescharacteristic
380C4#1681215411yescharacteristic
381C2—22no—
382C2—41no—
383C2—8561no—
384C2—8561no—
385C2—4282no—
386C2#2029327211yescharacteristic
387C2C2×#1681213711yescharacteristic
388C2#2029343911yescharacteristic
subgroupG/H
centerC22C22×D428
commutator, cosocleC2×C4×C107C24
FrattiniC2×C4C23×D107
Fitting#435×C107C2
socleC22×C107C22×D4
hypercenter#435D107
hypocenterC107#3306
LCSG ▷ C2×C4×C107 ▷ C2×C107 ▷ C107
UCSC1 ◁ C22 ◁ C4×D4 ◁ #435

Complements

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

NH ≅ G/Ntype
1. #20293281C2semidirect
2. #20293273C2semidirect
3. #20292987C2semidirect
4. #20292958C2semidirect
5. #20292946C2semidirect
6. #20292918C2semidirect
7. #20292880C2semidirect
8. #20292856C2semidirect
9. #20292843C2semidirect
10. #20292774C2semidirect
11. #20292763C2semidirect
12. #20292740C2semidirect
13. #20292629C2semidirect
14. C2×#16812031C2semidirect
15. #435×C107C2semidirect
19. C2×#13133807C22semidirect
22. #16812109C22semidirect
31. #16812083C22semidirect
35. C4×D428C22semidirect
37. C2×C4⋊Dic107C22semidirect
39. #16812041C22semidirect
40. #16812041C22semidirect
41. #16812041C22semidirect
42. #16812034C22semidirect
43. #16812034C22semidirect
44. #16812034C22semidirect
51. #16812031C22semidirect
52. C2×M16×C107C22semidirect
55. C4⋊C8×C107C22semidirect
57. C22⋊C8×C107C22semidirect
58. #97×C107C22semidirect
147. D428#41semidirect
195. C107#3306semidirect
208. #435D107semidirect
217. C2×M16C2×D107semidirect
240. C4⋊C8C2×D107semidirect
252. C22⋊C8C2×D107semidirect
253. #97C2×D107semidirect

Maximal subgroups

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

subgroupindex
#202932812
#202932732
#202929872
#202929582
#202929462
#202929182
#202928802
#202928562
#202928432
#202927742
#202927632
#202927402
#202926292
C2×#168120312
#435×C1072
#3306107