Finite Group Databasebeta

← #28348419

Subgroups — #28348419

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#20293280C211yesmaximal, characteristic, complemented
2#20293272C211yesmaximal, characteristic, complemented
3#20292989C211yesmaximal, characteristic, complemented
4#20292959C211yesmaximal, characteristic, complemented
5#20292947C211yesmaximal, characteristic, complemented
6#20292881C211yesmaximal, characteristic, complemented
7#20292857C211yesmaximal, characteristic, complemented
8#20292843C211yesmaximal, characteristic, complemented
9#20292773C211yesmaximal, characteristic, complemented
10#20292762C211yesmaximal, characteristic, complemented
11#20292737C211yesmaximal, characteristic, complemented
12#20292697C211yesmaximal, characteristic, complemented
13#20292625C211yesmaximal, characteristic, complemented
14#102×D107C211yesmaximal, characteristic, complemented
15#476×C107C211yesmaximal, characteristic, complemented
16C2×#13133819—21no—
17C2×#13133817C2211yescharacteristic
18C2×D4×D107C2211yescharacteristic
19C2×#13133807C2211yescharacteristic, complemented
20C2×D856C2211yescharacteristic
21C2×#13133803C2211yescharacteristic
22C2×#13133801C2211yescharacteristic, complemented
23C2×C8×D107C2211yescharacteristic
24#16812097C2211yescharacteristic
25#16812094—21no—
26#16812094—21no—
27#16812093—21no—
28#16812089C2211yescharacteristic
29#16812085—21no—
30#16812084C2211yescharacteristic
31#16812084C2211yescharacteristic
32#16812084C2211yescharacteristic, complemented
33#16812083—21no—
34#16812079—21no—
35C4×D428C2211yescharacteristic
36C4×D428C2211yescharacteristic, complemented
37C4⋊C4×D107C2211yescharacteristic
38C4⋊C4×D107C2211yescharacteristic, complemented
39C2×(C2×D107)⋊C4—21no—
40#16812040C2211yescharacteristic, complemented
41#16812034C2211yescharacteristic, complemented
42#16812032C2211yescharacteristic, complemented
43#16812032C2211yescharacteristic, complemented
44#16812031C2211yescharacteristic
45#16812030C2211yescharacteristic
46#16812029C2211yescharacteristic
47#16812029C2211yescharacteristic
48#16812029C2211yescharacteristic, complemented
49(C2×D107)⋊C8C2211yescharacteristic
50(C2×D107)⋊C8C2211yescharacteristic, complemented
51#16812020C2211yescharacteristic, complemented
52C4⋊Q8×C107C2211yescharacteristic, complemented
53C4×D4×C107C2211yescharacteristic
54#106×C107C2211yescharacteristic, complemented
55#106×C107C2211yescharacteristic, complemented
56C4⋊C8×C107C2211yescharacteristic, complemented
57#102×C107C2211yescharacteristic
58#102×C107C2211yescharacteristic
59C23×D107—21no—
60#13133819—44no—
61#13133817—22no—
62#13133817—41no—
63C2×Dic214C2311yescharacteristic
64C2×D428—21no—
65C2×D428—21no—
66C2×D428—21no—
67C2×D428D411yescharacteristic
68C2×D428C2311yescharacteristic
69C2×D428C2311yescharacteristic
70C2×C22⋊D107—21no—
71C2×C22⋊D107—21no—
72D4×D107—22no—
73D4×D107—41no—
74C22×Dic107—21no—
75C2×C4×D107—21no—
76C2×C4×D107—21no—
77C2×C4×D107—21no—
78C2×C4×D107C2311yescharacteristic
79C2×C4×D107C2311yescharacteristic
80#13133807—41no—
81D856—41no—
82#13133803—22no—
83#13133801—41no—
84C2×C107⋊C8C2311yescharacteristic
85C2×C107⋊C8C2311yescharacteristic
86C8×D107—41no—
87C4⋊Dic107—21no—
88C4⋊Dic107D411yescharacteristic
89C4⋊Dic107C2311yescharacteristic
90Dic107⋊C4—21no—
91Dic107⋊C4—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—
99(C2×D107)⋊C4D411yescharacteristic
100(C2×D107)⋊C4D411yescharacteristic
101C2×Q8×C107—21no—
102C2×D4×C107C2311yescharacteristic
103C22×C4×C107—21no—
104C2×C8×C107C2311yescharacteristic
105C2×C8×C107C2311yescharacteristic
106C4⋊C4×C107—21no—
107C4⋊C4×C107C2311yescharacteristic
108C4⋊C4×C107C2311yescharacteristic
109C4⋊C4×C107C2311yescharacteristic
110C22⋊C4×C107—21no—
111C42×C107C2311yescharacteristic
112C23×C107—21no—
113C22×D107—21no—
114C22×D107—21no—
115C22×D107—42no—
116C22×D107—42no—
117C22×D107—81no—
118C22×D107C2×D411yescharacteristic
119C22×D107C2×D411yescharacteristic
120Q8×C107—42no—
121D4×C107—21no—
122D4×C107#4112yescomplemented
123C2×C4×C107—21no—
124C2×C4×C107—21no—
125C2×C4×C107—21no—
126C2×C4×C107—81no—
127C2×C4×C107C2×D411yescharacteristic
128C2×C4×C107C2×D411yescharacteristic
129C2×C4×C107C2411yescharacteristic
130C22⋊D107—42no—
131C22⋊D107—42no—
132C2×Dic107—21no—
133C2×Dic107—21no—
134C2×Dic107—42no—
135C2×Dic107C2×D411yescharacteristic
136C2×Dic107C2×D411yescharacteristic
137D428—22no—
138D428—21no—
139D428—21no—
140D428—21no—
141D428—21no—
142D428—42no—
143D428—42no—
144D428—81no—
145C4×D107—21no—
146C4×D107—21no—
147C4×D107—21no—
148C4×D107—21no—
149C4×D107—42no—
150C4×D107—42no—
151C4×D107—42no—
152Dic214—21no—
153Dic214—21no—
154C8×C107—21no—
155C8×C107—21no—
156C107⋊C8—21no—
157C107⋊C8—21no—
158C22×C107—22no—
159C22×C107—41no—
160C22×C107—41no—
161C22×C107C22×D411yescharacteristic
162C2×D107—21no—
163C2×D107—21no—
164C2×D107—21no—
165C2×D107—21no—
166C2×D107—21no—
167C2×D107—41no—
168C2×D107—41no—
169C2×D107—41no—
170C2×D107—41no—
171C2×D107—41no—
172C2×D107—41no—
173C2×D107—84no—
174C2×D107AGL(1,ℤ/8)12yes—
175C4×C107—21no—
176C4×C107—21no—
177C4×C107—41no—
178C4×C107—41no—
179C4×C107—41no—
180C4×C107C2×#4111yescharacteristic
181C4×C107H(5,𝔽2)11yescharacteristic
182Dic107—21no—
183Dic107—21no—
184Dic107—41no—
185Dic107—41no—
186C2×C107—42no—
187C2×C107#54611yescharacteristic
188C2×C107C2×AGL(1,ℤ/8)11yescharacteristic
189C2×C107#57711yescharacteristic
190D107—22no—
191D107—41no—
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#476D10711yescharacteristic, complemented
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⋊Q8C2×D10711yescharacteristic, complemented
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#106C2×D10711yescharacteristic, complemented
239#106C2×D10711yescharacteristic, complemented
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#102C2×D10711yescharacteristic
250#102C2×D10711yescharacteristic
251C22⋊C8—1071no—
252C22⋊C8—1071no—
253#97—1071no—
254C24—2141no—
255#41—4284no—
256#41—4281no—
257#41—2142no—
258C2×Q8—21no—
259C2×Q8—1071no—
260C2×D4—1071no—
261C2×D4—1071no—
262C2×D4—1071no—
263C2×D4—4281no—
264C2×D4—2141no—
265C2×D4—2141no—
266C2×D4—2141no—
267C2×D4—2141no—
268C2×D4—2141no—
269C2×D4—2142no—
270C2×D4C22×D10711yescharacteristic
271C22×C4—21no—
272C22×C4—1071no—
273C22×C4—1071no—
274C22×C4—2141no—
275C22×C4—2141no—
276C22×C4—2141no—
277C22×C4—2141no—
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—1071no—
289C4⋊C4—1071no—
290C4⋊C4—2141no—
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—
306C42C22×D10711yescharacteristic
307C23—21no—
308C23—8561no—
309C23—1071no—
310C23—1071no—
311C23—4282no—
312C23—4282no—
313C23—2141no—
314C23—2141no—
315Q8—42no—
316Q8—2141no—
317Q8—2141no—
318D4—21no—
319D4—8561no—
320D4—4282no—
321D4—4282no—
322D4—4282no—
323D4—4282no—
324D4—2142no—
325D4—2141no—
326D4—2141no—
327D4—2141no—
328D4—2141no—
329D4#1313381912yescomplemented
330C2×C4—21no—
331C2×C4—21no—
332C2×C4—21no—
333C2×C4—81no—
334C2×C4—1071no—
335C2×C4—1071no—
336C2×C4—4282no—
337C2×C4—4282no—
338C2×C4—4282no—
339C2×C4—4282no—
340C2×C4—2141no—
341C2×C4—2141no—
342C2×C4—2141no—
343C2×C4—2141no—
344C2×C4—2141no—
345C2×C4—2141no—
346C2×C4D4×D10711yescharacteristic
347C2×C4D4×D10711yescharacteristic
348C2×C4C23×D10711yescharacteristic
349C8—21no—
350C8—21no—
351C8—2141no—
352C8—2141no—
353C22—22no—
354C22—41no—
355C22—41no—
356C22—8564no—
357C22—1072no—
358C22—4281no—
359C22—4281no—
360C22—4281no—
361C22—4281no—
362C22—4281no—
363C22—4281no—
364C22—2141no—
365C22—2141no—
366C22—2141no—
367C22—2141no—
368C22—2141no—
369C22C2×D4×D10711yescharacteristic
370C4—21no—
371C4—21no—
372C4—41no—
373C4—41no—
374C4—41no—
375C4—4281no—
376C4—4281no—
377C4—2141no—
378C4—2141no—
379C4C2×#1313381911yescharacteristic
380C4#1681215511yescharacteristic
381C2—42no—
382C2—8561no—
383C2—8561no—
384C2—4281no—
385C2—2142no—
386C2#2029326811yescharacteristic
387C2AGL(1,ℤ/8)×D10711yescharacteristic
388C2#2029343311yescharacteristic
subgroupG/H
centerC22C2×D4×D107
commutator, cosocleC2×C4×C107C24
FrattiniC2×C4C23×D107
Fitting#476×C107C2
socleC22×C107C22×D4
hypercenter#476D107
hypocenterC107#3306
LCSG ▷ C2×C4×C107 ▷ C2×C107 ▷ C107
UCSC1 ◁ C22 ◁ C42 ◁ #476

Complements

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

NH ≅ G/Ntype
1. #20293280C2semidirect
2. #20293272C2semidirect
3. #20292989C2semidirect
4. #20292959C2semidirect
5. #20292947C2semidirect
6. #20292881C2semidirect
7. #20292857C2semidirect
8. #20292843C2semidirect
9. #20292773C2semidirect
10. #20292762C2semidirect
11. #20292737C2semidirect
12. #20292697C2semidirect
13. #20292625C2semidirect
14. #102×D107C2semidirect
15. #476×C107C2semidirect
19. C2×#13133807C22semidirect
22. C2×#13133801C22semidirect
32. #16812084C22semidirect
36. C4×D428C22semidirect
38. C4⋊C4×D107C22semidirect
40. #16812040C22semidirect
41. #16812034C22semidirect
42. #16812032C22semidirect
43. #16812032C22semidirect
48. #16812029C22semidirect
50. (C2×D107)⋊C8C22semidirect
51. #16812020C22semidirect
52. C4⋊Q8×C107C22semidirect
54. #106×C107C22semidirect
55. #106×C107C22semidirect
56. C4⋊C8×C107C22semidirect
122. D4×C107#41semidirect
195. C107#3306semidirect
201. #476D107semidirect
219. C4⋊Q8C2×D107semidirect
238. #106C2×D107semidirect
239. #106C2×D107semidirect
240. C4⋊C8C2×D107semidirect
329. D4#13133819semidirect

Maximal subgroups

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

subgroupindex
#202932802
#202932722
#202929892
#202929592
#202929472
#202928812
#202928572
#202928432
#202927732
#202927622
#202927372
#202926972
#202926252
#102×D1072
#476×C1072
#3306107