Finite Group Databasebeta

← #31817898

Subgroups — #31817898

Order 17792.

546 conjugacy classes of subgroups.

The subgroup lattice has 438 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#17984866C211yesmaximal, characteristic, complemented
2C2×SD16×D139C211yesmaximal, characteristic, complemented
3#22581611C211yesmaximal, characteristic, complemented
4#22581588C211yesmaximal, characteristic, complemented
5#22581196C211yesmaximal, characteristic, complemented
6#22581186C211yesmaximal, characteristic, complemented
7#22581165C211yesmaximal, characteristic, complemented
8#22581150C211yesmaximal, characteristic, complemented
9#22581102C211yesmaximal, characteristic, complemented
10#22581095C211yesmaximal, characteristic, complemented
11#22581090C211yesmaximal, characteristic, complemented
12#22581085C211yesmaximal, characteristic, complemented
13#22581055C211yesmaximal, characteristic, complemented
14#22581026C211yesmaximal, characteristic, complemented
15#496×C139C211yesmaximal, characteristic, complemented
16C2×#14555784C2211yescharacteristic, complemented
17C2×Q8×D139C2211yescharacteristic, complemented
18C2×D4×D139—21no—
19C2×D4×D139C2211yescharacteristic
20C2×D4×D139C2211yescharacteristic
21#17984866—24no—
22C2×#14555774C2211yescharacteristic
23C2×#14555773C2211yescharacteristic, complemented
24C2×#14555773C2211yescharacteristic, complemented
25C2×#14555772C2211yescharacteristic
26SD16×D139—24no—
27C2×#14555770C2211yescharacteristic
28C2×#14555768C2211yescharacteristic, complemented
29C2×C8×D139C2211yescharacteristic, complemented
30#17984828—21no—
31#17984819—21no—
32#17984818C2211yescharacteristic, complemented
33#17984815—21no—
34#17984813—21no—
35#17984813C2211yescharacteristic
36#17984813C2211yescharacteristic
37#17984813C2211yescharacteristic, complemented
38#17984810—21no—
39#17984810—21no—
40C4×Dic278C2211yescharacteristic, complemented
41C4×D556C2211yescharacteristic
42C4×D556C2211yescharacteristic
43(C22⋊D139)⋊C4—21no—
44#17984776C2211yescharacteristic, complemented
45C4⋊(C139⋊C8)C2211yescharacteristic, complemented
46#17984767C2211yescharacteristic, complemented
47#17984766C2211yescharacteristic, complemented
48#17984764C2211yescharacteristic
49#17984763C2211yescharacteristic, complemented
50#17984763C2211yescharacteristic, complemented
51#17984763C2211yescharacteristic, complemented
52(C2×D139)⋊C8C2211yescharacteristic, complemented
53(C2×D139)⋊C8C2211yescharacteristic, complemented
54C2×SD16×C139C2211yescharacteristic, complemented
55C2×SD16×C139C2211yescharacteristic, complemented
56C2×D8×C139C2211yescharacteristic, complemented
57C2×D8×C139C2211yescharacteristic, complemented
58#127×C139C2211yescharacteristic
59#124×C139C2211yescharacteristic, complemented
60#97×C139C2211yescharacteristic, complemented
61C23×D139—21no—
62C23×D139—21no—
63#14555784—22no—
64#14555784—41no—
65C2×Dic278—21no—
66C2×Dic278C2311yescharacteristic
67Q8×D139—22no—
68Q8×D139—41no—
69C2×D556D411yescharacteristic
70C2×D556C2311yescharacteristic
71C2×C22⋊D139—21no—
72C2×C22⋊D139—21no—
73C2×C22⋊D139—21no—
74C2×C22⋊D139—21no—
75C2×C22⋊D139—21no—
76C2×C22⋊D139—21no—
77D4×D139—22no—
78D4×D139—22no—
79D4×D139—41no—
80D4×D139—41no—
81D4×D139—44no—
82C22×Dic139—21no—
83C2×C4×D139—21no—
84C2×C4×D139—21no—
85C2×C4×D139C2311yescharacteristic
86C2×C4×D139C2311yescharacteristic
87#14555774D414yescomplemented
88#14555773—22no—
89#14555773—22no—
90#14555772—22no—
91#14555770—22no—
92#14555768—22no—
93C2×C139⋊C8C2311yescharacteristic
94C2×C139⋊C8C2311yescharacteristic
95C8×D139—22no—
96C4⋊Dic139D411yescharacteristic
97C4⋊Dic139C2311yescharacteristic
98C4⋊Dic139C2311yescharacteristic
99Dic139⋊C4—21no—
100Dic139⋊C4—21no—
101C22⋊Dic139—21no—
102C22⋊Dic139—21no—
103C22⋊Dic139—21no—
104(C2×D139)⋊C4—21no—
105(C2×D139)⋊C4—21no—
106(C2×D139)⋊C4D411yescharacteristic
107(C2×D139)⋊C4D411yescharacteristic
108C4×Dic139—21no—
109C2×Q8×C139C2311yescharacteristic
110C2×D4×C139—21no—
111C2×D4×C139—21no—
112C2×D4×C139C2311yescharacteristic
113C2×D4×C139C2311yescharacteristic
114C2×D4×C139C2311yescharacteristic
115SD16×C139—22no—
116SD16×C139—41no—
117D8×C139—22no—
118D8×C139—41no—
119C2×C8×C139C2311yescharacteristic
120C2×C8×C139C2311yescharacteristic
121C22⋊C4×C139—21no—
122C22⋊C4×C139—21no—
123C42×C139C2311yescharacteristic
124C22×D139—21no—
125C22×D139—42no—
126C22×D139—42no—
127C22×D139—42no—
128C22×D139—42no—
129C22×D139—81no—
130C22×D139—81no—
131C22×D139C2×D411yescharacteristic
132C22×D139C2×D411yescharacteristic
133Dic278—21no—
134Dic278—42no—
135Dic278C2×D412yescomplemented
136D556—22no—
137D556—21no—
138D556C2×D412yes—
139C22⋊D139—42no—
140C22⋊D139—42no—
141C22⋊D139—42no—
142C22⋊D139—42no—
143C22⋊D139—42no—
144C22⋊D139—42no—
145C2×Dic139—21no—
146C2×Dic139—21no—
147C2×Dic139—21no—
148C2×Dic139—42no—
149C2×Dic139C2×D411yescharacteristic
150C2×Dic139C2×D411yescharacteristic
151C4×D139—21no—
152C4×D139—21no—
153C4×D139—21no—
154C4×D139—21no—
155C4×D139—42no—
156C4×D139—42no—
157C139⋊C8—21no—
158C139⋊C8—21no—
159C23×C139—21no—
160C23×C139—21no—
161C23×C139—21no—
162Q8×C139—21no—
163Q8×C139—21no—
164D4×C139—21no—
165D4×C139—21no—
166D4×C139—21no—
167D4×C139—21no—
168D4×C139—21no—
169D4×C139—21no—
170D4×C139—42no—
171D4×C139—81no—
172C2×C4×C139—21no—
173C2×C4×C139C2×D411yescharacteristic
174C2×C4×C139C2×D411yescharacteristic
175C2×C4×C139C2411yescharacteristic
176C8×C139—21no—
177C8×C139C2×D412yescomplemented
178C22×C139—41no—
179C22×C139—41no—
180C22×C139—41no—
181C22×C139—41no—
182C22×C139—41no—
183C22×C139—41no—
184C22×C139—41no—
185C22×C139—41no—
186C22×C139—41no—
187C22×C139C22×D411yescharacteristic
188C2×D139—21no—
189C2×D139—21no—
190C2×D139—21no—
191C2×D139—22no—
192C2×D139—21no—
193C2×D139—21no—
194C2×D139—41no—
195C2×D139—41no—
196C2×D139—84no—
197C2×D139—84no—
198C2×D139AGL(1,ℤ/8)12yescomplemented
199C4×C139—21no—
200C4×C139—21no—
201C4×C139—41no—
202C4×C139C22×D411yescharacteristic
203C4×C139H(5,𝔽2)11yescharacteristic
204Dic139—21no—
205Dic139—21no—
206Dic139—22no—
207Dic139—41no—
208Dic139—41no—
209C2×C139—81no—
210C2×C139—81no—
211C2×C139—81no—
212C2×C139D4211yescharacteristic
213C2×C139C2×AGL(1,ℤ/8)11yescharacteristic
214C2×C139#57711yescharacteristic
215D139—22no—
216D139—41no—
217D139—42no—
218C139#326511yescharacteristic, complemented, Sylow
219#3265—1391nomaximal, Sylow
220C2×AGL(1,ℤ/8)—1391no—
221C22×SD16—1391no—
222#548—1391no—
223D42—1391no—
224#496D13911yescharacteristic, complemented
225#468—1391no—
226#466—1391no—
227#463—1391no—
228#460—1391no—
229#450—1391no—
230#449—1391no—
231#448—1391no—
232#447—1391no—
233#440—1391no—
234#435—1391no—
235C2×#41—1391no—
236C22×Q8—1391no—
237C22×D4—2781no—
238C22×D4—1391no—
239C22×D4—1391no—
240AGL(1,ℤ/8)—2784no—
241C2×SD16—2784no—
242C2×SD16—1391no—
243C2×SD16—1391no—
244C2×SD16—1391no—
245C2×SD16—1391no—
246C2×SD16C2×D13911yescharacteristic, complemented
247C2×SD16C2×D13911yescharacteristic, complemented
248C2×D8—1391no—
249C2×D8C2×D13911yescharacteristic, complemented
250C2×D8C2×D13911yescharacteristic, complemented
251C2×M16—1391no—
252C22×C8—1391no—
253#127C2×D13911yescharacteristic
254#124—2781no—
255#124C2×D13911yescharacteristic, complemented
256C22⋊Q8—2781no—
257C22⋊Q8—1391no—
258#121—2781no—
259#121—2781no—
260#121—1391no—
261#121—1391no—
262#121—1391no—
263#120—2781no—
264#120—2781no—
265C4×Q8—1391no—
266C4×D4—2781no—
267C4×D4—1391no—
268C4×D4—1391no—
269#107—1391no—
270C4⋊C8—1391no—
271#103—1391no—
272#103—1391no—
273#102—1391no—
274#102—1391no—
275#102—1391no—
276#102—1391no—
277C22⋊C8—1391no—
278C22⋊C8—1391no—
279#97C2×D13911yescharacteristic, complemented
280C24—2781no—
281C24—2781no—
282#41—2782no—
283#41—5561no—
284C2×Q8—2781no—
285C2×Q8—2782no—
286C2×Q8—5561no—
287C2×Q8—1391no—
288C2×Q8C22×D13911yescharacteristic
289C2×D4—21no—
290C2×D4—21no—
291C2×D4—2782no—
292C2×D4—2782no—
293C2×D4—2781no—
294C2×D4—2781no—
295C2×D4—2781no—
296C2×D4—2781no—
297C2×D4—2781no—
298C2×D4—2781no—
299C2×D4—5561no—
300C2×D4—5561no—
301C2×D4—5564no—
302C2×D4—1391no—
303C2×D4—1391no—
304C2×D4C22×D13911yescharacteristic
305C2×D4C22×D13911yescharacteristic
306C2×D4C22×D13911yescharacteristic
307C22×C4—2781no—
308C22×C4—2781no—
309C22×C4—2781no—
310C22×C4—1391no—
311C22×C4—1391no—
312SD16—22no—
313SD16—41no—
314SD16—2782no—
315SD16—2782no—
316SD16—2782no—
317SD16—1394no—
318D8—22no—
319D8—41no—
320D8—2782no—
321M16—2782no—
322C2×C8—2782no—
323C2×C8—1391no—
324C2×C8—1391no—
325C2×C8C22×D13911yescharacteristic
326C2×C8C22×D13911yescharacteristic
327C4⋊C4—2781no—
328C4⋊C4—2781no—
329C4⋊C4—1391no—
330C4⋊C4—1391no—
331C4⋊C4—1391no—
332C22⋊C4—21no—
333C22⋊C4—21no—
334C22⋊C4—2781no—
335C22⋊C4—2781no—
336C22⋊C4—2781no—
337C22⋊C4—2781no—
338C22⋊C4—2781no—
339C22⋊C4—1391no—
340C22⋊C4—1391no—
341C42—2781no—
342C42C22×D13911yescharacteristic
343C23—21no—
344C23—21no—
345C23—21no—
346C23—2781no—
347C23—5562no—
348C23—5562no—
349C23—5562no—
350C23—5562no—
351C23—11121no—
352C23—11121no—
353C23—1391no—
354C23—1391no—
355Q8—21no—
356Q8—21no—
357Q8—2781no—
358Q8—5562no—
359Q8—1392no—
360D4—21no—
361D4—21no—
362D4—21no—
363D4—21no—
364D4—21no—
365D4—21no—
366D4—42no—
367D4—81no—
368D4—2782no—
369D4—2781no—
370D4—5562no—
371D4—5562no—
372D4—5562no—
373D4—5562no—
374D4—5562no—
375D4—5562no—
376D4—1392no—
377C2×C4—21no—
378C2×C4—2781no—
379C2×C4—2781no—
380C2×C4—2781no—
381C2×C4—2781no—
382C2×C4—2781no—
383C2×C4—2781no—
384C2×C4—2781no—
385C2×C4—5562no—
386C2×C4—5562no—
387C2×C4—5562no—
388C2×C4—1391no—
389C2×C4—1391no—
390C2×C4D4×D13911yescharacteristic
391C2×C4D4×D13911yescharacteristic
392C2×C4C23×D13911yescharacteristic
393C8—21no—
394C8—2781no—
395C8—2781no—
396C8D4×D13912yescomplemented
397C22—41no—
398C22—41no—
399C22—41no—
400C22—41no—
401C22—41no—
402C22—41no—
403C22—41no—
404C22—41no—
405C22—41no—
406C22—2781no—
407C22—2781no—
408C22—2781no—
409C22—2781no—
410C22—2782no—
411C22—2781no—
412C22—5561no—
413C22—5561no—
414C22—11124no—
415C22—11124no—
416C22—1392no—
417C22C2×D4×D13911yescharacteristic
418C4—21no—
419C4—21no—
420C4—41no—
421C4—2781no—
422C4—2781no—
423C4—2782no—
424C4—5561no—
425C4—5561no—
426C4C2×D4×D13911yescharacteristic
427C4#1798488711yescharacteristic
428C2—81no—
429C2—81no—
430C2—81no—
431C2—2782no—
432C2—5561no—
433C2—5562no—
434C2#2258158811yescharacteristic
435C2AGL(1,ℤ/8)×D13911yescharacteristic
436C2#2258176311yescharacteristic
subgroupG/H
centerC22C2×D4×D139
commutator, cosocleC2×C4×C139C24
FrattiniC2×C4C23×D139
Fitting#496×C139C2
socleC22×C139C22×D4
hypercenter#496D139
hypocenterC139#3265
LCSG ▷ C2×C4×C139 ▷ C2×C139 ▷ C139
UCSC1 ◁ C22 ◁ C42 ◁ #496

Complements

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

NH ≅ G/Ntype
1. C2×#17984866C2semidirect
2. C2×SD16×D139C2semidirect
3. #22581611C2semidirect
4. #22581588C2semidirect
5. #22581196C2semidirect
6. #22581186C2semidirect
7. #22581165C2semidirect
8. #22581150C2semidirect
9. #22581102C2semidirect
10. #22581095C2semidirect
11. #22581090C2semidirect
12. #22581085C2semidirect
13. #22581055C2semidirect
14. #22581026C2semidirect
15. #496×C139C2semidirect
16. C2×#14555784C22semidirect
17. C2×Q8×D139C22semidirect
23. C2×#14555773C22semidirect
24. C2×#14555773C22semidirect
28. C2×#14555768C22semidirect
29. C2×C8×D139C22semidirect
32. #17984818C22semidirect
37. #17984813C22semidirect
40. C4×Dic278C22semidirect
44. #17984776C22semidirect
45. C4⋊(C139⋊C8)C22semidirect
46. #17984767C22semidirect
47. #17984766C22semidirect
49. #17984763C22semidirect
50. #17984763C22semidirect
51. #17984763C22semidirect
52. (C2×D139)⋊C8C22semidirect
53. (C2×D139)⋊C8C22semidirect
54. C2×SD16×C139C22semidirect
55. C2×SD16×C139C22semidirect
56. C2×D8×C139C22semidirect
57. C2×D8×C139C22semidirect
59. #124×C139C22semidirect
60. #97×C139C22semidirect
87. #14555774D4semidirect
135. Dic278C2×D4semidirect
177. C8×C139C2×D4semidirect
198. C2×D139AGL(1,ℤ/8)semidirect
218. C139#3265semidirect
224. #496D139semidirect
246. C2×SD16C2×D139semidirect
247. C2×SD16C2×D139semidirect
249. C2×D8C2×D139semidirect
250. C2×D8C2×D139semidirect
255. #124C2×D139semidirect
279. #97C2×D139semidirect
396. C8D4×D139semidirect

Maximal subgroups

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

subgroupindex
C2×#179848662
C2×SD16×D1392
#225816112
#225815882
#225811962
#225811862
#225811652
#225811502
#225811022
#225810952
#225810902
#225810852
#225810552
#225810262
#496×C1392
#3265139