Finite Group Databasebeta

← #31817888

Subgroups — #31817888

Order 17792.

568 conjugacy classes of subgroups.

The subgroup lattice has 452 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#17984869C211yesmaximal, characteristic, complemented
2C22×#14555770C211yesmaximal, characteristic, complemented
3#22581594C211yesmaximal, characteristic, complemented
4D4×C22⋊D139C211yesmaximal, characteristic, complemented
5#22581339C211yesmaximal, characteristic, complemented
6#22581205C211yesmaximal, characteristic, complemented
7#22581184C211yesmaximal, characteristic, complemented
8#22581170C211yesmaximal, characteristic, complemented
9#22581147C211yesmaximal, characteristic, complemented
10#22581097C211yesmaximal, characteristic, complemented
11#22581089C211yesmaximal, characteristic, complemented
12#22581088C211yesmaximal, characteristic, complemented
13#22581062C211yesmaximal, characteristic, complemented
14#22581021C211yesmaximal, characteristic, complemented
15#450×C139C211yesmaximal, characteristic, complemented
16C2×#14555786C2211yescharacteristic, complemented
17C22×D556C2211yescharacteristic, complemented
18C22×C22⋊D139—21no—
19C2×D4×D139C2211yescharacteristic, complemented
20#17984869—24no—
21C2×#14555772C2211yescharacteristic, complemented
22C2×D1112C2211yescharacteristic, complemented
23C2×D1112C2211yescharacteristic, complemented
24C2×#14555770—24no—
25C2×#14555770C2211yescharacteristic
26C2×#14555770C2211yescharacteristic
27C2×#14555770C2211yescharacteristic, complemented
28C2×#14555770C2211yescharacteristic, complemented
29C2×#14555768C2211yescharacteristic, complemented
30C22×C139⋊C8C2211yescharacteristic, complemented
31#17984838C2211yescharacteristic, complemented
32#17984831C2211yescharacteristic, complemented
33#17984827—21no—
34#17984817—21no—
35#17984817C2211yescharacteristic, complemented
36#17984816C2211yescharacteristic
37#17984814—21no—
38#17984813C2211yescharacteristic, complemented
39#17984812—21no—
40#17984811—21no—
41#17984810—21no—
42D4×Dic139C2211yescharacteristic, complemented
43#17984804C2211yescharacteristic, complemented
44C4×C22⋊D139C2211yescharacteristic
45C22⋊C4×D139—21no—
46#17984773C2211yescharacteristic, complemented
47Dic139⋊C8C2211yescharacteristic, complemented
48#17984766C2211yescharacteristic, complemented
49#17984764C2211yescharacteristic, complemented
50#17984764C2211yescharacteristic, complemented
51#17984762C2211yescharacteristic, complemented
52(C2×D139)⋊C8C2211yescharacteristic, complemented
53#17984752C2211yescharacteristic, complemented
54C22×D4×C139C2211yescharacteristic
55C2×SD16×C139C2211yescharacteristic, complemented
56C2×SD16×C139C2211yescharacteristic, complemented
57C22⋊Q8×C139C2211yescharacteristic, complemented
58#102×C139C2211yescharacteristic, complemented
59#102×C139C2211yescharacteristic, complemented
60C22⋊C8×C139C2211yescharacteristic, complemented
61C23×D139—21no—
62C23×D139—21no—
63#14555786—22no—
64#14555786—41no—
65C2×D556—22no—
66C2×D556—21no—
67C2×D556—21no—
68C2×D556—21no—
69C2×D556—41no—
70C2×D556C2311yescharacteristic
71C2×D556C2311yescharacteristic
72C2×D556C2311yescharacteristic
73C2×C22⋊D139—21no—
74C2×C22⋊D139—21no—
75C2×C22⋊D139—21no—
76C2×C22⋊D139—21no—
77C2×C22⋊D139—21no—
78C2×C22⋊D139—44no—
79C2×C22⋊D139D411yescharacteristic
80D4×D139—22no—
81D4×D139—41no—
82C22×Dic139—21no—
83C2×C4×D139—21no—
84C2×C4×D139—21no—
85C2×C4×D139C2311yescharacteristic
86#14555772—22no—
87D1112—22no—
88D1112—41no—
89#14555770—22no—
90#14555770—22no—
91#14555770—22no—
92#14555770D414yescomplemented
93#14555768—22no—
94C2×C139⋊C8—22no—
95C2×C139⋊C8C2311yescharacteristic
96C2×C139⋊C8C2311yescharacteristic
97C4⋊Dic139C2311yescharacteristic
98Dic139⋊C4D411yescharacteristic
99C22⋊Dic139—21no—
100C22⋊Dic139—21no—
101C22⋊Dic139D411yescharacteristic
102(C2×D139)⋊C4—21no—
103(C2×D139)⋊C4—21no—
104(C2×D139)⋊C4—21no—
105(C2×D139)⋊C4—21no—
106(C2×D139)⋊C4—21no—
107(C2×D139)⋊C4D411yescharacteristic
108C4×Dic139C2311yescharacteristic
109C24×C139—21no—
110C2×Q8×C139C2311yescharacteristic
111C2×D4×C139—22no—
112C2×D4×C139—21no—
113C2×D4×C139—41no—
114C2×D4×C139C2311yescharacteristic
115C2×D4×C139C2311yescharacteristic
116C22×C4×C139C2311yescharacteristic
117SD16×C139—22no—
118SD16×C139—41no—
119C2×C8×C139C2311yescharacteristic
120C2×C8×C139C2311yescharacteristic
121C4⋊C4×C139—21no—
122C4⋊C4×C139C2311yescharacteristic
123C22⋊C4×C139—21no—
124C22×D139—21no—
125C22×D139—21no—
126C22×D139—21no—
127C22×D139—42no—
128C22×D139—42no—
129C22×D139—42no—
130C22×D139—42no—
131C22×D139—81no—
132C22×D139—81no—
133C22×D139C2×D411yescharacteristic
134D556—21no—
135D556—21no—
136D556—21no—
137D556—21no—
138D556—21no—
139D556—42no—
140D556—42no—
141D556—81no—
142D556C2×D412yescomplemented
143C22⋊D139—22no—
144C22⋊D139—42no—
145C22⋊D139—42no—
146C22⋊D139—42no—
147C22⋊D139—42no—
148C22⋊D139—81no—
149C2×Dic139—21no—
150C2×Dic139—42no—
151C2×Dic139C2×D411yescharacteristic
152C2×Dic139C2×D411yescharacteristic
153C2×Dic139C2×D411yescharacteristic
154C4×D139—21no—
155C4×D139—21no—
156C4×D139—42no—
157C4×D139—81no—
158C139⋊C8—21no—
159C139⋊C8C2×D412yescomplemented
160C23×C139—21no—
161C23×C139—21no—
162C23×C139—42no—
163C23×C139—42no—
164C23×C139—81no—
165C23×C139C2×D411yescharacteristic
166Q8×C139—21no—
167Q8×C139—21no—
168D4×C139—21no—
169D4×C139—21no—
170D4×C139—21no—
171D4×C139—42no—
172D4×C139C2×D412yes—
173C2×C4×C139—21no—
174C2×C4×C139—21no—
175C2×C4×C139—21no—
176C2×C4×C139—21no—
177C2×C4×C139C2×D411yescharacteristic
178C2×C4×C139C2411yescharacteristic
179C8×C139—21no—
180C8×C139—21no—
181C22×C139—21no—
182C22×C139—21no—
183C22×C139—22no—
184C22×C139—41no—
185C22×C139—41no—
186C22×C139—41no—
187C22×C139—41no—
188C22×C139—41no—
189C22×C139—84no—
190C22×C139AGL(1,ℤ/8)12yescomplemented
191C22×C139C22×D411yescharacteristic
192C2×D139—21no—
193C2×D139—22no—
194C2×D139—21no—
195C2×D139—21no—
196C2×D139—41no—
197C2×D139—41no—
198C2×D139—41no—
199C2×D139—41no—
200C2×D139—41no—
201C2×D139—41no—
202C2×D139—41no—
203C2×D139—41no—
204C2×D139—84no—
205C2×D139—84no—
206C4×C139—21no—
207C4×C139—41no—
208C4×C139—41no—
209C4×C139C22×D411yescharacteristic
210C4×C139H(5,𝔽2)11yescharacteristic
211Dic139—21no—
212Dic139—21no—
213Dic139—21no—
214Dic139—41no—
215C2×C139—22no—
216C2×C139—42no—
217C2×C139—81no—
218C2×C139D4211yescharacteristic
219C2×C139C2×AGL(1,ℤ/8)11yescharacteristic
220C2×C139#57611yescharacteristic
221D139—41no—
222D139—42no—
223D139—81no—
224D139—81no—
225C139#326411yescharacteristic, complemented, Sylow
226#3264—1391nomaximal, Sylow
227C2×AGL(1,ℤ/8)—1391no—
228C22×D8—1391no—
229#546—1391no—
230D42—1391no—
231#496—1391no—
232#469—1391no—
233#465—1391no—
234#463—1391no—
235#459—1391no—
236#450D13911yescharacteristic, complemented
237#449—1391no—
238#447—1391no—
239#447—1391no—
240#442—1391no—
241#435—1391no—
242C2×#41—1391no—
243C22×D4—2781no—
244C22×D4—1391no—
245C22×D4—1391no—
246C22×D4C2×D13911yescharacteristic
247AGL(1,ℤ/8)—2784no—
248C2×SD16—1391no—
249C2×SD16C2×D13911yescharacteristic, complemented
250C2×SD16C2×D13911yescharacteristic, complemented
251C2×D8—2784no—
252C2×D8—1391no—
253C2×D8—1391no—
254C2×D8—1391no—
255C2×D8—1391no—
256C2×D8—1391no—
257C2×D8—1391no—
258C2×M16—1391no—
259C22×C8—1391no—
260#127—1391no—
261#124—1391no—
262#123—2781no—
263C22⋊Q8C2×D13911yescharacteristic, complemented
264#121—2781no—
265#121—2781no—
266#121—1391no—
267#121—1391no—
268#121—1391no—
269#120—2781no—
270#120—2781no—
271#120—2781no—
272C4×D4—1391no—
273C4×D4—1391no—
274C4×D4—1391no—
275C2×C22⋊C4—2781no—
276#106—1391no—
277C4⋊C8—1391no—
278#103—1391no—
279#102—1391no—
280#102—1391no—
281#102—1391no—
282#102C2×D13911yescharacteristic, complemented
283#102C2×D13911yescharacteristic, complemented
284C22⋊C8—1391no—
285C22⋊C8C2×D13911yescharacteristic, complemented
286#97—1391no—
287C24—21no—
288C24—2781no—
289C24—2781no—
290#41—2782no—
291#41—5561no—
292C2×Q8C22×D13911yescharacteristic
293C2×D4—22no—
294C2×D4—21no—
295C2×D4—41no—
296C2×D4—2782no—
297C2×D4—2781no—
298C2×D4—2781no—
299C2×D4—2782no—
300C2×D4—2781no—
301C2×D4—2781no—
302C2×D4—2781no—
303C2×D4—2781no—
304C2×D4—2781no—
305C2×D4—2781no—
306C2×D4—5561no—
307C2×D4—5561no—
308C2×D4—5564no—
309C2×D4—1391no—
310C2×D4—1391no—
311C2×D4—1391no—
312C2×D4—1391no—
313C2×D4C22×D13911yescharacteristic
314C2×D4C22×D13911yescharacteristic
315C22×C4—2781no—
316C22×C4—2781no—
317C22×C4—2781no—
318C22×C4—1391no—
319C22×C4C22×D13911yescharacteristic
320SD16—22no—
321SD16—41no—
322SD16—2782no—
323D8—2782no—
324D8—2782no—
325D8—2782no—
326D8—2782no—
327D8—5561no—
328D8—1394no—
329M16—2782no—
330C2×C8—2782no—
331C2×C8—1391no—
332C2×C8—1391no—
333C2×C8C22×D13911yescharacteristic
334C2×C8C22×D13911yescharacteristic
335C4⋊C4—21no—
336C4⋊C4—1391no—
337C4⋊C4—1391no—
338C4⋊C4C22×D13911yescharacteristic
339C22⋊C4—21no—
340C22⋊C4—2781no—
341C22⋊C4—2781no—
342C22⋊C4—2781no—
343C22⋊C4—2781no—
344C22⋊C4—2781no—
345C22⋊C4—2781no—
346C22⋊C4—2781no—
347C22⋊C4—1391no—
348C22⋊C4—1391no—
349C42—1391no—
350C23—21no—
351C23—21no—
352C23—42no—
353C23—42no—
354C23—81no—
355C23—2781no—
356C23—2781no—
357C23—2781no—
358C23—5562no—
359C23—5562no—
360C23—5562no—
361C23—5562no—
362C23—11121no—
363C23—11121no—
364C23—1391no—
365C23D4×D13911yescharacteristic
366Q8—21no—
367Q8—21no—
368D4—21no—
369D4—21no—
370D4—21no—
371D4—42no—
372D4—2782no—
373D4—2781no—
374D4—2781no—
375D4—2781no—
376D4—2781no—
377D4—2781no—
378D4—5562no—
379D4—5562no—
380D4—5562no—
381D4—5562no—
382D4—5562no—
383D4—5562no—
384D4—11121no—
385D4—11121no—
386D4—1392no—
387D4D4×D13912yes—
388C2×C4—21no—
389C2×C4—21no—
390C2×C4—21no—
391C2×C4—21no—
392C2×C4—2781no—
393C2×C4—2781no—
394C2×C4—2781no—
395C2×C4—5562no—
396C2×C4—5562no—
397C2×C4—11121no—
398C2×C4—1391no—
399C2×C4—1391no—
400C2×C4—1391no—
401C2×C4D4×D13911yescharacteristic
402C2×C4C23×D13911yescharacteristic
403C8—21no—
404C8—21no—
405C8—2781no—
406C8—1392no—
407C22—21no—
408C22—21no—
409C22—22no—
410C22—41no—
411C22—41no—
412C22—41no—
413C22—41no—
414C22—41no—
415C22—84no—
416C22—2781no—
417C22—2782no—
418C22—2781no—
419C22—2781no—
420C22—5561no—
421C22—5561no—
422C22—5561no—
423C22—5561no—
424C22—5561no—
425C22—5561no—
426C22—5561no—
427C22—5561no—
428C22—11124no—
429C22—11124no—
430C22#1798486912yescomplemented
431C22C2×D4×D13911yescharacteristic
432C4—21no—
433C4—41no—
434C4—41no—
435C4—2781no—
436C4—2781no—
437C4—2781no—
438C4—5561no—
439C4C2×D4×D13911yescharacteristic
440C4#1798488711yescharacteristic
441C2—22no—
442C2—42no—
443C2—81no—
444C2—5561no—
445C2—5562no—
446C2—11121no—
447C2—11121no—
448C2#2258158511yescharacteristic
449C2C2×#1798486911yescharacteristic
450C2#2258175711yescharacteristic
subgroupG/H
centerC22C2×D4×D139
commutator, cosocleC2×C4×C139C24
FrattiniC2×C4C23×D139
Fitting#450×C139C2
socleC22×C139C22×D4
hypercenter#450D139
hypocenterC139#3264
LCSG ▷ C2×C4×C139 ▷ C2×C139 ▷ C139
UCSC1 ◁ C22 ◁ C22×C4 ◁ #450

Complements

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

NH ≅ G/Ntype
1. C2×#17984869C2semidirect
2. C22×#14555770C2semidirect
3. #22581594C2semidirect
4. D4×C22⋊D139C2semidirect
5. #22581339C2semidirect
6. #22581205C2semidirect
7. #22581184C2semidirect
8. #22581170C2semidirect
9. #22581147C2semidirect
10. #22581097C2semidirect
11. #22581089C2semidirect
12. #22581088C2semidirect
13. #22581062C2semidirect
14. #22581021C2semidirect
15. #450×C139C2semidirect
16. C2×#14555786C22semidirect
17. C22×D556C22semidirect
19. C2×D4×D139C22semidirect
21. C2×#14555772C22semidirect
22. C2×D1112C22semidirect
23. C2×D1112C22semidirect
27. C2×#14555770C22semidirect
28. C2×#14555770C22semidirect
29. C2×#14555768C22semidirect
30. C22×C139⋊C8C22semidirect
31. #17984838C22semidirect
32. #17984831C22semidirect
35. #17984817C22semidirect
38. #17984813C22semidirect
42. D4×Dic139C22semidirect
43. #17984804C22semidirect
46. #17984773C22semidirect
47. Dic139⋊C8C22semidirect
48. #17984766C22semidirect
49. #17984764C22semidirect
50. #17984764C22semidirect
51. #17984762C22semidirect
52. (C2×D139)⋊C8C22semidirect
53. #17984752C22semidirect
55. C2×SD16×C139C22semidirect
56. C2×SD16×C139C22semidirect
57. C22⋊Q8×C139C22semidirect
58. #102×C139C22semidirect
59. #102×C139C22semidirect
60. C22⋊C8×C139C22semidirect
92. #14555770D4semidirect
142. D556C2×D4semidirect
159. C139⋊C8C2×D4semidirect
190. C22×C139AGL(1,ℤ/8)semidirect
225. C139#3264semidirect
236. #450D139semidirect
249. C2×SD16C2×D139semidirect
250. C2×SD16C2×D139semidirect
263. C22⋊Q8C2×D139semidirect
282. #102C2×D139semidirect
283. #102C2×D139semidirect
285. C22⋊C8C2×D139semidirect
430. C22#17984869semidirect

Maximal subgroups

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

subgroupindex
C2×#179848692
C22×#145557702
#225815942
D4×C22⋊D1392
#225813392
#225812052
#225811842
#225811702
#225811472
#225810972
#225810892
#225810882
#225810622
#225810212
#450×C1392
#3264139