Finite Group Databasebeta

← #28348165

Subgroups — #28348165

Order 13696.

504 conjugacy classes of subgroups.

The subgroup lattice has 418 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16812130C211yesmaximal, characteristic, complemented
2C22×#13133807C211yesmaximal, characteristic, complemented
3#20293283C211yesmaximal, characteristic, complemented
4#20293268C211yesmaximal, characteristic, complemented
5#20293008C211yesmaximal, characteristic, complemented
6#20292857C211yesmaximal, characteristic, complemented
7#20292852C211yesmaximal, characteristic, complemented
8#20292841C211yesmaximal, characteristic, complemented
9#20292792C211yesmaximal, characteristic, complemented
10#20292782C211yesmaximal, characteristic, complemented
11#20292773C211yesmaximal, characteristic, complemented
12#20292760C211yesmaximal, characteristic, complemented
13#20292714C211yesmaximal, characteristic, complemented
14C8×C22⋊D107C211yesmaximal, characteristic, complemented
15#465×C107C211yesmaximal, characteristic, complemented
16C2×#13133819C2211yescharacteristic
17C2×#13133817C2211yescharacteristic, complemented
18C22×Dic214C2211yescharacteristic
19C22×D428C2211yescharacteristic
20#16812130—24no—
21C2×#13133808C2211yescharacteristic, complemented
22C2×#13133808C2211yescharacteristic, complemented
23C2×#13133807—24no—
24C2×#13133807C2211yescharacteristic
25C2×#13133807C2211yescharacteristic
26C2×#13133807C2211yescharacteristic
27C2×#13133807C2211yescharacteristic
28C2×#13133803C2211yescharacteristic
29C2×#13133803C2211yescharacteristic
30C2×C8×D107C2211yescharacteristic, complemented
31#16812098C2211yescharacteristic
32#16812096C2211yescharacteristic, complemented
33#16812095—21no—
34#16812094—21no—
35#16812089C2211yescharacteristic, complemented
36#16812086—21no—
37#16812084—21no—
38#16812084C2211yescharacteristic
39#16812081—21no—
40#16812079—21no—
41#16812074C2211yescharacteristic, complemented
42D4×Dic107—21no—
43#16812071C2211yescharacteristic
44C4×C22⋊D107C2211yescharacteristic
45C22⋊C4×D107—21no—
46Dic107⋊C8C2211yescharacteristic, complemented
47#16812032C2211yescharacteristic, complemented
48#16812032C2211yescharacteristic, complemented
49#16812029C2211yescharacteristic
50#16812029C2211yescharacteristic
51C22⋊(C107⋊C8)C2211yescharacteristic
52(C2×D107)⋊C8C2211yescharacteristic, complemented
53C8×Dic107C2211yescharacteristic, complemented
54C2×SD16×C107C2211yescharacteristic, complemented
55C22×C8×C107C2211yescharacteristic
56C22⋊Q8×C107C2211yescharacteristic
57#121×C107C2211yescharacteristic
58#106×C107C2211yescharacteristic, complemented
59#103×C107C2211yescharacteristic, complemented
60#102×C107C2211yescharacteristic, complemented
61C23×D107—21no—
62#13133819—22no—
63#13133819—41no—
64#13133817—22no—
65#13133817—41no—
66C2×Dic214—22no—
67C2×Dic214—21no—
68C2×Dic214—41no—
69C2×Dic214C2311yescharacteristic
70C2×Dic214C2311yescharacteristic
71C2×D428—22no—
72C2×D428—21no—
73C2×D428—21no—
74C2×D428—21no—
75C2×D428—41no—
76C2×D428C2311yescharacteristic
77C2×D428C2311yescharacteristic
78C2×C22⋊D107—21no—
79C2×C22⋊D107D411yescharacteristic
80C22×Dic107—21no—
81C22×Dic107—21no—
82C2×C4×D107—21no—
83C2×C4×D107—21no—
84C2×C4×D107C2311yescharacteristic
85#13133808—22no—
86#13133808—41no—
87#13133807—22no—
88#13133807—22no—
89#13133807—22no—
90#13133807D414yescomplemented
91#13133803—22no—
92#13133803—41no—
93C2×C107⋊C8C2311yescharacteristic
94C2×C107⋊C8C2311yescharacteristic
95C8×D107—22no—
96C4⋊Dic107—21no—
97Dic107⋊C4—21no—
98Dic107⋊C4D411yescharacteristic
99C22⋊Dic107—21no—
100C22⋊Dic107—21no—
101C22⋊Dic107D411yescharacteristic
102(C2×D107)⋊C4—21no—
103(C2×D107)⋊C4—21no—
104(C2×D107)⋊C4—21no—
105(C2×D107)⋊C4—21no—
106(C2×D107)⋊C4—21no—
107(C2×D107)⋊C4D411yescharacteristic
108C4×Dic107—21no—
109C4×Dic107C2311yescharacteristic
110C2×Q8×C107C2311yescharacteristic
111C2×D4×C107—21no—
112C2×D4×C107C2311yescharacteristic
113C22×C4×C107C2311yescharacteristic
114SD16×C107—22no—
115C2×C8×C107—22no—
116C2×C8×C107C2311yescharacteristic
117C2×C8×C107C2311yescharacteristic
118C4⋊C4×C107—21no—
119C4⋊C4×C107C2311yescharacteristic
120C4⋊C4×C107C2311yescharacteristic
121C22⋊C4×C107—21no—
122C22⋊C4×C107—21no—
123C23×C107—21no—
124C23×C107C2×D411yescharacteristic
125C22×D107—21no—
126C22×D107—21no—
127C22×D107—42no—
128C22×D107—42no—
129C22×D107—81no—
130C22×D107C2×D411yescharacteristic
131Q8×C107—21no—
132Q8×C107—21no—
133D4×C107—21no—
134D4×C107—21no—
135D4×C107—42no—
136C2×C4×C107—21no—
137C2×C4×C107—21no—
138C2×C4×C107—21no—
139C2×C4×C107—21no—
140C2×C4×C107—21no—
141C2×C4×C107C2×D411yescharacteristic
142C2×C4×C107C2411yescharacteristic
143C22⋊D107—22no—
144C22⋊D107—42no—
145C2×Dic107—21no—
146C2×Dic107—21no—
147C2×Dic107—42no—
148C2×Dic107—42no—
149C2×Dic107C2×D411yescharacteristic
150C2×Dic107C2×D411yescharacteristic
151C2×Dic107C2×D411yescharacteristic
152D428—21no—
153D428—21no—
154D428—21no—
155D428—42no—
156D428—42no—
157D428—81no—
158D428C2×D412yes—
159C4×D107—21no—
160C4×D107—21no—
161C4×D107—42no—
162C4×D107—81no—
163Dic214—21no—
164Dic214—21no—
165Dic214—21no—
166Dic214—42no—
167Dic214C2×D412yes—
168C8×C107—21no—
169C8×C107C2×D412yes—
170C107⋊C8—21no—
171C107⋊C8—21no—
172C22×C107—21no—
173C22×C107—21no—
174C22×C107—41no—
175C22×C107—41no—
176C22×C107—41no—
177C22×C107#13512yescomplemented
178C22×C107C22×D411yescharacteristic
179C2×D107—21no—
180C2×D107—22no—
181C2×D107—21no—
182C2×D107—21no—
183C2×D107—41no—
184C2×D107—41no—
185C2×D107—41no—
186C2×D107—41no—
187C2×D107—41no—
188C2×D107—84no—
189C4×C107—21no—
190C4×C107—41no—
191C4×C107—41no—
192C4×C107—41no—
193C4×C107C22×D411yescharacteristic
194C4×C107H(5,𝔽2)11yescharacteristic
195Dic107—22no—
196Dic107—21no—
197Dic107—21no—
198Dic107—21no—
199Dic107—41no—
200C2×C107—22no—
201C2×C107—81no—
202C2×C107D4211yescharacteristic
203C2×C107C2×#13511yescharacteristic
204C2×C107#57711yescharacteristic
205D107—41no—
206D107—42no—
207D107—81no—
208#3282—1071nomaximal, Sylow
209C107#328211yescharacteristic, complemented, Sylow
210C2×#135—1071no—
211C22×SD16—1071no—
212#548—1071no—
213#546—1071no—
214#495—1071no—
215#465—1071no—
216#465D10711yescharacteristic, complemented
217#464—1071no—
218#463—1071no—
219#452—1071no—
220#451—1071no—
221#449—1071no—
222#447—1071no—
223C4×SD16—1071no—
224C8×D4—1071no—
225C2×#41—1071no—
226C2×#41—1071no—
227C22×Q8—1071no—
228C22×D4—1071no—
229#135—2144no—
230C2×Q16—1071no—
231C2×Q16—1071no—
232C2×SD16—1071no—
233C2×SD16—1071no—
234C2×SD16—1071no—
235C2×SD16—1071no—
236C2×SD16—2144no—
237C2×SD16C2×D10711yescharacteristic, complemented
238C2×D8—1071no—
239C2×D8—1071no—
240C22×C8—1071no—
241C22×C8C2×D10711yescharacteristic
242#124—1071no—
243#124—1071no—
244#124—2141no—
245#123—2141no—
246C22⋊Q8—1071no—
247C22⋊Q8—2141no—
248C22⋊Q8C2×D10711yescharacteristic
249#121—1071no—
250#121—2141no—
251#121—2141no—
252#121C2×D10711yescharacteristic
253#120—2141no—
254C4×Q8—1071no—
255C4×D4—1071no—
256C4×D4—1071no—
257C4×D4—2141no—
258C2×C22⋊C4—2141no—
259#106C2×D10711yescharacteristic, complemented
260C4⋊C8—1071no—
261#103—1071no—
262#103—1071no—
263#103C2×D10711yescharacteristic, complemented
264#102—1071no—
265#102—1071no—
266#102C2×D10711yescharacteristic, complemented
267C22⋊C8—1071no—
268C22⋊C8—1071no—
269C4×C8—1071no—
270C24—2141no—
271#41—4281no—
272#41—4281no—
273#41—2142no—
274#41—2142no—
275C2×Q8—1071no—
276C2×Q8—1071no—
277C2×Q8—4281no—
278C2×Q8—2142no—
279C2×Q8—2141no—
280C2×Q8C22×D10711yescharacteristic
281C2×D4—21no—
282C2×D4—1071no—
283C2×D4—1071no—
284C2×D4—1071no—
285C2×D4—4281no—
286C2×D4—2141no—
287C2×D4—2142no—
288C2×D4—2141no—
289C2×D4—2141no—
290C2×D4—2141no—
291C2×D4C22×D10711yescharacteristic
292C22×C4—1071no—
293C22×C4—2141no—
294C22×C4—2141no—
295C22×C4—2141no—
296C22×C4—2141no—
297C22×C4C22×D10711yescharacteristic
298Q16—4281no—
299Q16—2142no—
300SD16—22no—
301SD16—1074no—
302SD16—2142no—
303SD16—2142no—
304SD16—2142no—
305D8—4281no—
306D8—2142no—
307C2×C8—22no—
308C2×C8—1071no—
309C2×C8—1071no—
310C2×C8—2142no—
311C2×C8C22×D10711yescharacteristic
312C2×C8C22×D10711yescharacteristic
313C4⋊C4—21no—
314C4⋊C4—1071no—
315C4⋊C4—2141no—
316C4⋊C4—2141no—
317C4⋊C4C22×D10711yescharacteristic
318C4⋊C4C22×D10711yescharacteristic
319C22⋊C4—21no—
320C22⋊C4—21no—
321C22⋊C4—1071no—
322C22⋊C4—1071no—
323C22⋊C4—2141no—
324C22⋊C4—2141no—
325C22⋊C4—2141no—
326C22⋊C4—2141no—
327C22⋊C4—2141no—
328C22⋊C4—2141no—
329C22⋊C4—2141no—
330C42—1071no—
331C42—2141no—
332C23—21no—
333C23—8561no—
334C23—1071no—
335C23—4282no—
336C23—4282no—
337C23—2141no—
338C23—2141no—
339C23D4×D10711yescharacteristic
340Q8—21no—
341Q8—21no—
342Q8—1072no—
343Q8—4282no—
344Q8—2141no—
345Q8—2141no—
346Q8—2141no—
347D4—21no—
348D4—21no—
349D4—42no—
350D4—8561no—
351D4—1072no—
352D4—4282no—
353D4—4282no—
354D4—4282no—
355D4—2142no—
356D4—2141no—
357D4—2141no—
358D4—2141no—
359C2×C4—21no—
360C2×C4—21no—
361C2×C4—21no—
362C2×C4—21no—
363C2×C4—21no—
364C2×C4—8561no—
365C2×C4—1071no—
366C2×C4—1071no—
367C2×C4—1071no—
368C2×C4—4282no—
369C2×C4—4282no—
370C2×C4—4282no—
371C2×C4—2141no—
372C2×C4—2141no—
373C2×C4—2141no—
374C2×C4—2141no—
375C2×C4D4×D10711yescharacteristic
376C2×C4C23×D10711yescharacteristic
377C8—21no—
378C8—2141no—
379C8—2141no—
380C8D4×D10712yes—
381C22—21no—
382C22—21no—
383C22—41no—
384C22—41no—
385C22—41no—
386C22—8564no—
387C22—4281no—
388C22—4281no—
389C22—4281no—
390C22—4281no—
391C22—4281no—
392C22—2141no—
393C22—2141no—
394C22—2142no—
395C22—2141no—
396C22#1681213012yescomplemented
397C22C2×D4×D10711yescharacteristic
398C4—21no—
399C4—41no—
400C4—41no—
401C4—41no—
402C4—4281no—
403C4—2141no—
404C4—2141no—
405C4—2141no—
406C4—2142no—
407C4C2×D4×D10711yescharacteristic
408C4#1681215511yescharacteristic
409C2—22no—
410C2—81no—
411C2—8561no—
412C2—4282no—
413C2—4281no—
414C2#2029326111yescharacteristic
415C2C2×#1681213011yescharacteristic
416C2#2029343411yescharacteristic
subgroupG/H
centerC22C2×D4×D107
commutator, cosocleC2×C4×C107C24
FrattiniC2×C4C23×D107
Fitting#465×C107C2
socleC22×C107C22×D4
hypercenter#465D107
hypocenterC107#3282
LCSG ▷ C2×C4×C107 ▷ C2×C107 ▷ C107
UCSC1 ◁ C22 ◁ C22×C4 ◁ #465

Complements

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

NH ≅ G/Ntype
1. C2×#16812130C2semidirect
2. C22×#13133807C2semidirect
3. #20293283C2semidirect
4. #20293268C2semidirect
5. #20293008C2semidirect
6. #20292857C2semidirect
7. #20292852C2semidirect
8. #20292841C2semidirect
9. #20292792C2semidirect
10. #20292782C2semidirect
11. #20292773C2semidirect
12. #20292760C2semidirect
13. #20292714C2semidirect
14. C8×C22⋊D107C2semidirect
15. #465×C107C2semidirect
17. C2×#13133817C22semidirect
21. C2×#13133808C22semidirect
22. C2×#13133808C22semidirect
30. C2×C8×D107C22semidirect
32. #16812096C22semidirect
35. #16812089C22semidirect
41. #16812074C22semidirect
46. Dic107⋊C8C22semidirect
47. #16812032C22semidirect
48. #16812032C22semidirect
52. (C2×D107)⋊C8C22semidirect
53. C8×Dic107C22semidirect
54. C2×SD16×C107C22semidirect
58. #106×C107C22semidirect
59. #103×C107C22semidirect
60. #102×C107C22semidirect
90. #13133807D4semidirect
177. C22×C107#135semidirect
209. C107#3282semidirect
216. #465D107semidirect
237. C2×SD16C2×D107semidirect
259. #106C2×D107semidirect
263. #103C2×D107semidirect
266. #102C2×D107semidirect
396. C22#16812130semidirect

Maximal subgroups

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

subgroupindex
C2×#168121302
C22×#131338072
#202932832
#202932682
#202930082
#202928572
#202928522
#202928412
#202927922
#202927822
#202927732
#202927602
#202927142
C8×C22⋊D1072
#465×C1072
#3282107