Finite Group Databasebeta

← #28348154

Subgroups — #28348154

Order 13696.

546 conjugacy classes of subgroups.

The subgroup lattice has 430 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#13133807C211yesmaximal, characteristic, complemented
2C2×SD16×D107C211yesmaximal, characteristic, complemented
3#20293290C211yesmaximal, characteristic, complemented
4#20293261C211yesmaximal, characteristic
5#20292998C211yesmaximal, characteristic, complemented
6#20292857C211yesmaximal, characteristic, complemented
7#20292831C211yesmaximal, characteristic, complemented
8#20292826C211yesmaximal, characteristic, complemented
9#20292779C211yesmaximal, characteristic, complemented
10#20292777C211yesmaximal, characteristic, complemented
11#20292767C211yesmaximal, characteristic, complemented
12#20292765C211yesmaximal, characteristic, complemented
13#20292714C211yesmaximal, characteristic, complemented
14C8×C22⋊D107C211yesmaximal, characteristic, complemented
15#465×C107C211yesmaximal, characteristic, complemented
16C22×Dic214C2211yescharacteristic, complemented
17C2×Q8×D107C2211yescharacteristic, complemented
18C22×D428C2211yescharacteristic
19C2×D4×D107—21no—
20C2×D4×D107C2211yescharacteristic
21C2×#13133807—24no—
22C2×#13133807C2211yescharacteristic
23C2×#13133807C2211yescharacteristic
24C2×#13133807C2211yescharacteristic, complemented
25C2×#13133807C2211yescharacteristic, complemented
26C2×#13133806C2211yescharacteristic, complemented
27C2×#13133806C2211yescharacteristic, complemented
28C2×#13133805C2211yescharacteristic
29C2×#13133805C2211yescharacteristic, complemented
30SD16×D107—24no—
31C2×C8×D107C2211yescharacteristic, complemented
32#16812108—21no—
33#16812107C2211yescharacteristic, complemented
34#16812105C2211yescharacteristic
35#16812089C2211yescharacteristic, complemented
36#16812087—21no—
37#16812086—21no—
38#16812084C2211yescharacteristic
39#16812082—21no—
40#16812079—21no—
41#16812077—21no—
42#16812074C2211yescharacteristic, complemented
43#16812071C2211yescharacteristic
44C4×C22⋊D107C2211yescharacteristic
45(C22⋊D107)⋊C4—21no—
46Dic107⋊C8C2211yescharacteristic, complemented
47#16812032C2211yescharacteristic, complemented
48#16812032C2211yescharacteristic, complemented
49#16812029C2211yescharacteristic
50#16812029C2211yescharacteristic, complemented
51C22⋊(C107⋊C8)C2211yescharacteristic, complemented
52(C2×D107)⋊C8C2211yescharacteristic, complemented
53C8×Dic107C2211yescharacteristic, complemented
54C2×SD16×C107C2211yescharacteristic, complemented
55C22×C8×C107C2211yescharacteristic, complemented
56C22⋊Q8×C107C2211yescharacteristic, complemented
57#121×C107C2211yescharacteristic
58#106×C107C2211yescharacteristic, complemented
59#103×C107C2211yescharacteristic, complemented
60#102×C107C2211yescharacteristic, complemented
61C23×D107—21no—
62C23×D107—21no—
63C2×Dic214—22no—
64C2×Dic214—21no—
65C2×Dic214—21no—
66C2×Dic214—41no—
67C2×Dic214C2311yescharacteristic
68C2×Dic214C2311yescharacteristic
69Q8×D107—22no—
70Q8×D107—41no—
71C2×D428—22no—
72C2×D428—21no—
73C2×D428—21no—
74C2×D428—41no—
75C2×D428C2311yescharacteristic
76C2×D428C2311yescharacteristic
77C2×C22⋊D107—21no—
78C2×C22⋊D107—21no—
79C2×C22⋊D107—21no—
80C2×C22⋊D107D411yescharacteristic
81D4×D107—22no—
82D4×D107—41no—
83D4×D107—44no—
84C22×Dic107—21no—
85C2×C4×D107—21no—
86C2×C4×D107—21no—
87C2×C4×D107C2311yescharacteristic
88#13133807—22no—
89#13133807—22no—
90#13133807—22no—
91#13133807D414yescomplemented
92#13133806—22no—
93#13133806—41no—
94#13133805—22no—
95#13133805—41no—
96C2×C107⋊C8C2311yescharacteristic
97C2×C107⋊C8C2311yescharacteristic
98C8×D107—22no—
99C4⋊Dic107—21no—
100Dic107⋊C4—21no—
101Dic107⋊C4—21no—
102Dic107⋊C4—21no—
103Dic107⋊C4D411yescharacteristic
104C22⋊Dic107D411yescharacteristic
105(C2×D107)⋊C4—21no—
106(C2×D107)⋊C4—21no—
107(C2×D107)⋊C4—21no—
108(C2×D107)⋊C4D411yescharacteristic
109C4×Dic107—21no—
110C4×Dic107C2311yescharacteristic
111C2×Q8×C107C2311yescharacteristic
112C2×D4×C107—21no—
113C2×D4×C107C2311yescharacteristic
114C22×C4×C107C2311yescharacteristic
115SD16×C107—22no—
116C2×C8×C107—22no—
117C2×C8×C107C2311yescharacteristic
118C2×C8×C107C2311yescharacteristic
119C4⋊C4×C107—21no—
120C4⋊C4×C107C2311yescharacteristic
121C4⋊C4×C107C2311yescharacteristic
122C22⋊C4×C107—21no—
123C22⋊C4×C107—21no—
124C23×C107—21no—
125C23×C107C2×D411yescharacteristic
126C22×D107—21no—
127C22×D107—21no—
128C22×D107—42no—
129C22×D107—42no—
130C22×D107—42no—
131C22×D107—42no—
132C22×D107—81no—
133C22×D107—81no—
134C22×D107C2×D411yescharacteristic
135Q8×C107—21no—
136Q8×C107—21no—
137D4×C107—21no—
138D4×C107—21no—
139D4×C107—42no—
140C2×C4×C107—21no—
141C2×C4×C107—21no—
142C2×C4×C107—21no—
143C2×C4×C107—21no—
144C2×C4×C107—21no—
145C2×C4×C107C2×D411yescharacteristic
146C2×C4×C107C2411yescharacteristic
147C22⋊D107—42no—
148C22⋊D107—42no—
149C22⋊D107—81no—
150C22⋊D107SD1614yescomplemented
151C2×Dic107—21no—
152C2×Dic107—21no—
153C2×Dic107—42no—
154C2×Dic107C2×D411yescharacteristic
155C2×Dic107C2×D411yescharacteristic
156C2×Dic107C2×D411yescharacteristic
157D428—21no—
158D428—21no—
159D428—21no—
160D428—42no—
161D428—42no—
162D428C2×D412yes—
163C4×D107—21no—
164C4×D107—21no—
165C4×D107—42no—
166C4×D107—42no—
167Dic214—21no—
168Dic214—21no—
169Dic214—21no—
170Dic214—42no—
171Dic214—42no—
172Dic214C2×D412yescomplemented
173C8×C107—21no—
174C8×C107C2×D412yescomplemented
175C107⋊C8—21no—
176C107⋊C8—21no—
177C22×C107—21no—
178C22×C107—21no—
179C22×C107—41no—
180C22×C107—41no—
181C22×C107—41no—
182C22×C107C2×SD1612yescomplemented
183C22×C107C22×D411yescharacteristic
184C2×D107—21no—
185C2×D107—21no—
186C2×D107—22no—
187C2×D107—41no—
188C2×D107—41no—
189C2×D107—41no—
190C2×D107—41no—
191C2×D107—41no—
192C2×D107—84no—
193C2×D107—84no—
194C2×D107C2×SD1612yescomplemented
195C4×C107—21no—
196C4×C107—41no—
197C4×C107—41no—
198C4×C107—41no—
199C4×C107C22×D411yescharacteristic
200C4×C107H(5,𝔽2)11yescharacteristic
201Dic107—22no—
202Dic107—21no—
203Dic107—21no—
204Dic107—41no—
205Dic107C2×SD1612yescomplemented
206C2×C107—22no—
207C2×C107—81no—
208C2×C107D4211yescharacteristic
209C2×C107C22×SD1611yescharacteristic
210C2×C107#57711yescharacteristic
211D107—22no—
212D107—42no—
213D107—81no—
214D4×SD16—1071nomaximal, Sylow
215C107D4×SD1611yescharacteristic, complemented, Sylow
216C22×SD16—1071no—
217C22×SD16—1071no—
218D4×Q8—1071no—
219D42—1071no—
220#492—1071no—
221#465—1071no—
222#465D10711yescharacteristic, complemented
223#461—1071no—
224#460—1071no—
225#450—1071no—
226#450—1071no—
227#448—1071no—
228#448—1071no—
229C4×SD16—1071no—
230C8×D4—1071no—
231C22×Q8—1071no—
232C22×Q8—1071no—
233C22×D4—1071no—
234C22×D4—1071no—
235C22×D4—2141no—
236C2×SD16—1071no—
237C2×SD16—1071no—
238C2×SD16—1071no—
239C2×SD16—1071no—
240C2×SD16—1071no—
241C2×SD16—1071no—
242C2×SD16—1071no—
243C2×SD16—1071no—
244C2×SD16—2144no—
245C2×SD16—2144no—
246C2×SD16C2×D10711yescharacteristic, complemented
247C22×C8—1071no—
248C22×C8C2×D10711yescharacteristic, complemented
249C4⋊Q8—1071no—
250C4⋊Q8—2141no—
251#127—1071no—
252C22⋊Q8—1071no—
253C22⋊Q8—2141no—
254C22⋊Q8—2141no—
255C22⋊Q8C2×D10711yescharacteristic, complemented
256#121—1071no—
257#121—2141no—
258#121C2×D10711yescharacteristic
259#120—2141no—
260#120—2141no—
261C4×Q8—1071no—
262C4×D4—1071no—
263C4×D4—1071no—
264C4×D4—2141no—
265#106C2×D10711yescharacteristic, complemented
266C4⋊C8—1071no—
267#103—1071no—
268#103—1071no—
269#103C2×D10711yescharacteristic, complemented
270#102—1071no—
271#102—1071no—
272#102C2×D10711yescharacteristic, complemented
273C22⋊C8—1071no—
274C22⋊C8—1071no—
275C4×C8—1071no—
276C24—2141no—
277C24—2141no—
278C2×Q8—1071no—
279C2×Q8—1071no—
280C2×Q8—4281no—
281C2×Q8—4281no—
282C2×Q8—2141no—
283C2×Q8—2142no—
284C2×Q8—2142no—
285C2×Q8—2141no—
286C2×Q8C22×D10711yescharacteristic
287C2×D4—21no—
288C2×D4—1071no—
289C2×D4—1071no—
290C2×D4—1071no—
291C2×D4—4281no—
292C2×D4—4284no—
293C2×D4—4281no—
294C2×D4—2142no—
295C2×D4—2141no—
296C2×D4—2141no—
297C2×D4—2141no—
298C2×D4—2141no—
299C2×D4—2142no—
300C2×D4—2141no—
301C2×D4C22×D10711yescharacteristic
302C22×C4—1071no—
303C22×C4—2141no—
304C22×C4—2141no—
305C22×C4—2141no—
306C22×C4C22×D10711yescharacteristic
307SD16—22no—
308SD16—1074no—
309SD16—4281no—
310SD16—4281no—
311SD16—2142no—
312SD16—2142no—
313SD16—2142no—
314SD16—2142no—
315SD16—2142no—
316C2×C8—22no—
317C2×C8—1071no—
318C2×C8—1071no—
319C2×C8—2142no—
320C2×C8C22×D10711yescharacteristic
321C2×C8C22×D10711yescharacteristic
322C4⋊C4—21no—
323C4⋊C4—1071no—
324C4⋊C4—2141no—
325C4⋊C4—2141no—
326C4⋊C4—2141no—
327C4⋊C4—2141no—
328C4⋊C4C22×D10711yescharacteristic
329C4⋊C4C22×D10711yescharacteristic
330C22⋊C4—21no—
331C22⋊C4—21no—
332C22⋊C4—1071no—
333C22⋊C4—1071no—
334C22⋊C4—2141no—
335C22⋊C4—2141no—
336C22⋊C4—2141no—
337C42—1071no—
338C42—2141no—
339C23—21no—
340C23—8561no—
341C23—8561no—
342C23—1071no—
343C23—4282no—
344C23—4282no—
345C23—4282no—
346C23—4282no—
347C23—2141no—
348C23—2141no—
349C23D4×D10711yescharacteristic
350Q8—21no—
351Q8—21no—
352Q8—1072no—
353Q8—4282no—
354Q8—4282no—
355Q8—2141no—
356Q8—2141no—
357Q8—2141no—
358D4—21no—
359D4—21no—
360D4—42no—
361D4—8561no—
362D4—1074no—
363D4—1072no—
364D4—4282no—
365D4—4282no—
366D4—4282no—
367D4—4282no—
368D4—2141no—
369D4—2141no—
370D4—2141no—
371C2×C4—21no—
372C2×C4—21no—
373C2×C4—21no—
374C2×C4—21no—
375C2×C4—21no—
376C2×C4—1071no—
377C2×C4—1071no—
378C2×C4—1071no—
379C2×C4—4282no—
380C2×C4—4282no—
381C2×C4—4282no—
382C2×C4—2141no—
383C2×C4—2141no—
384C2×C4—2141no—
385C2×C4—2141no—
386C2×C4D4×D10711yescharacteristic
387C2×C4C23×D10711yescharacteristic
388C8—21no—
389C8—2141no—
390C8—2141no—
391C8D4×D10712yescomplemented
392C22—21no—
393C22—21no—
394C22—41no—
395C22—41no—
396C22—41no—
397C22—8564no—
398C22—8564no—
399C22—1072no—
400C22—4281no—
401C22—4281no—
402C22—4281no—
403C22—4281no—
404C22—4281no—
405C22—2142no—
406C22—2141no—
407C22—2141no—
408C22SD16×D10712yescomplemented
409C22C2×D4×D10711yescharacteristic
410C4—21no—
411C4—41no—
412C4—41no—
413C4—41no—
414C4—1072no—
415C4—4281no—
416C4—2141no—
417C4—2141no—
418C4—2142no—
419C4C2×D4×D10711yescharacteristic
420C4#1681215511yescharacteristic
421C2—22no—
422C2—81no—
423C2—8561no—
424C2—4282no—
425C2—2142no—
426C2#2029326111yescharacteristic
427C2C2×SD16×D10711yescharacteristic
428C2#2029343411yescharacteristic
subgroupG/H
centerC22C2×D4×D107
commutator, cosocleC2×C4×C107C24
FrattiniC2×C4C23×D107
Fitting#465×C107C2
socleC22×C107C22×D4
prime radical#20293261C2
hypercenter#465D107
hypocenterC107D4×SD16
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. C22×#13133807C2semidirect
2. C2×SD16×D107C2semidirect
3. #20293290C2semidirect
5. #20292998C2semidirect
6. #20292857C2semidirect
7. #20292831C2semidirect
8. #20292826C2semidirect
9. #20292779C2semidirect
10. #20292777C2semidirect
11. #20292767C2semidirect
12. #20292765C2semidirect
13. #20292714C2semidirect
14. C8×C22⋊D107C2semidirect
15. #465×C107C2semidirect
16. C22×Dic214C22semidirect
17. C2×Q8×D107C22semidirect
24. C2×#13133807C22semidirect
25. C2×#13133807C22semidirect
26. C2×#13133806C22semidirect
27. C2×#13133806C22semidirect
29. C2×#13133805C22semidirect
31. C2×C8×D107C22semidirect
33. #16812107C22semidirect
35. #16812089C22semidirect
42. #16812074C22semidirect
46. Dic107⋊C8C22semidirect
47. #16812032C22semidirect
48. #16812032C22semidirect
50. #16812029C22semidirect
51. C22⋊(C107⋊C8)C22semidirect
52. (C2×D107)⋊C8C22semidirect
53. C8×Dic107C22semidirect
54. C2×SD16×C107C22semidirect
55. C22×C8×C107C22semidirect
56. C22⋊Q8×C107C22semidirect
58. #106×C107C22semidirect
59. #103×C107C22semidirect
60. #102×C107C22semidirect
91. #13133807D4semidirect
150. C22⋊D107SD16semidirect
172. Dic214C2×D4semidirect
174. C8×C107C2×D4semidirect
182. C22×C107C2×SD16semidirect
194. C2×D107C2×SD16semidirect
205. Dic107C2×SD16semidirect
215. C107D4×SD16semidirect
222. #465D107semidirect
246. C2×SD16C2×D107semidirect
248. C22×C8C2×D107semidirect
255. C22⋊Q8C2×D107semidirect
265. #106C2×D107semidirect
269. #103C2×D107semidirect
272. #102C2×D107semidirect
391. C8D4×D107semidirect
408. C22SD16×D107semidirect

Maximal subgroups

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

subgroupindex
C22×#131338072
C2×SD16×D1072
#202932902
#202932612
#202929982
#202928572
#202928312
#202928262
#202927792
#202927772
#202927672
#202927652
#202927142
C8×C22⋊D1072
#465×C1072
D4×SD16107