Finite Group Databasebeta

← #31817922

Subgroups — #31817922

Order 17792.

500 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×#17984866C211yesmaximal, characteristic, complemented
2C2×#17984862C211yesmaximal, characteristic, complemented
3#22581602C211yesmaximal, characteristic, complemented
4#22581600C211yesmaximal, characteristic, complemented
5#22581344C211yesmaximal, characteristic, complemented
6#22581210C211yesmaximal, characteristic, complemented
7#22581180C211yesmaximal, characteristic, complemented
8#22581166C211yesmaximal, characteristic, complemented
9#22581157C211yesmaximal, characteristic, complemented
10#22581117C211yesmaximal, characteristic, complemented
11#22581116C211yesmaximal, characteristic, complemented
12#22581102C211yesmaximal, characteristic, complemented
13#22581062C211yesmaximal, characteristic, complemented
14#22581021C211yesmaximal, characteristic, complemented
15#449×C139C211yesmaximal, characteristic, complemented
16C2×#14555785C2211yescharacteristic
17C2×#14555784C2211yescharacteristic
18C2×D4×D139C2211yescharacteristic
19#17984866—24no—
20#17984862—24no—
21C2×Dic556C2211yescharacteristic, complemented
22C2×#14555775C2211yescharacteristic, complemented
23C2×#14555774C2211yescharacteristic
24C2×#14555773C2211yescharacteristic
25C2×#14555773C2211yescharacteristic, complemented
26C2×#14555772C2211yescharacteristic
27C2×#14555770C2211yescharacteristic
28C2×#14555768C2211yescharacteristic
29C22×C139⋊C8C2211yescharacteristic, complemented
30#17984840C2211yescharacteristic, complemented
31#17984829C2211yescharacteristic
32#17984825—21no—
33#17984824—21no—
34#17984823—21no—
35#17984822C2211yescharacteristic
36#17984821C2211yescharacteristic, complemented
37#17984818C2211yescharacteristic
38#17984814—21no—
39#17984813—21no—
40#17984810—21no—
41D4×Dic139C2211yescharacteristic, complemented
42#17984804C2211yescharacteristic
43C4×C22⋊D139C2211yescharacteristic
44C2×Dic139⋊C4—21no—
45C22⋊C4×D139—21no—
46#17984773C2211yescharacteristic, complemented
47Dic139⋊C8C2211yescharacteristic, complemented
48#17984767C2211yescharacteristic, complemented
49#17984765C2211yescharacteristic, complemented
50#17984764C2211yescharacteristic
51#17984763C2211yescharacteristic
52(C2×D139)⋊C8C2211yescharacteristic
53#17984752C2211yescharacteristic, complemented
54C2×#41×C139C2211yescharacteristic, complemented
55C2×SD16×C139C2211yescharacteristic
56C2×D8×C139C2211yescharacteristic
57#121×C139C2211yescharacteristic
58#103×C139C2211yescharacteristic, complemented
59#102×C139C2211yescharacteristic, complemented
60C22⋊C8×C139C2211yescharacteristic, complemented
61C23×D139—21no—
62#14555785—22no—
63#14555785—41no—
64#14555784—22no—
65#14555784—41no—
66C2×Dic278C2311yescharacteristic
67C2×Dic278C2311yescharacteristic
68C2×D556C2311yescharacteristic
69C2×C22⋊D139—21no—
70C2×C22⋊D139—21no—
71C2×C22⋊D139—21no—
72C2×C22⋊D139D411yescharacteristic
73D4×D139—22no—
74D4×D139—41no—
75C22×Dic139—21no—
76C22×Dic139—21no—
77C2×C4×D139—21no—
78C2×C4×D139—21no—
79C2×C4×D139C2311yescharacteristic
80Dic556—41no—
81#14555775—22no—
82#14555774—22no—
83#14555773—22no—
84#14555773—22no—
85#14555772—22no—
86#14555770D414yescomplemented
87#14555768—22no—
88C2×C139⋊C8—22no—
89C2×C139⋊C8C2311yescharacteristic
90C2×C139⋊C8C2311yescharacteristic
91C4⋊Dic139—21no—
92C4⋊Dic139C2311yescharacteristic
93Dic139⋊C4—21no—
94Dic139⋊C4—21no—
95Dic139⋊C4—21no—
96Dic139⋊C4—21no—
97Dic139⋊C4D411yescharacteristic
98C22⋊Dic139—21no—
99C22⋊Dic139—21no—
100C22⋊Dic139—21no—
101C22⋊Dic139—21no—
102C22⋊Dic139D411yescharacteristic
103(C2×D139)⋊C4—21no—
104(C2×D139)⋊C4—21no—
105(C2×D139)⋊C4—21no—
106(C2×D139)⋊C4D411yescharacteristic
107C4×Dic139C2311yescharacteristic
108#41×C139—22no—
109#41×C139—41no—
110C2×Q8×C139C2311yescharacteristic
111C2×D4×C139—21no—
112C2×D4×C139—21no—
113C2×D4×C139C2311yescharacteristic
114C2×D4×C139C2311yescharacteristic
115C22×C4×C139—21no—
116C22×C4×C139C2311yescharacteristic
117SD16×C139—41no—
118D8×C139—22no—
119C2×C8×C139C2311yescharacteristic
120C2×C8×C139C2311yescharacteristic
121C4⋊C4×C139C2311yescharacteristic
122C22⋊C4×C139—21no—
123C22×D139—21no—
124C22×D139—42no—
125C22×D139—42no—
126C22×D139—81no—
127C22×D139C2×D411yescharacteristic
128Dic278—21no—
129Dic278—21no—
130Dic278—21no—
131Dic278—21no—
132D556—21no—
133D556C2×D412yes—
134C22⋊D139—41no—
135C22⋊D139—42no—
136C22⋊D139—42no—
137C22⋊D139—42no—
138C2×Dic139—21no—
139C2×Dic139—21no—
140C2×Dic139—21no—
141C2×Dic139—42no—
142C2×Dic139—42no—
143C2×Dic139C2×D411yescharacteristic
144C2×Dic139C2×D411yescharacteristic
145C2×Dic139C2×D411yescharacteristic
146C4×D139—21no—
147C4×D139—21no—
148C4×D139—42no—
149C4×D139—81no—
150C139⋊C8—21no—
151C139⋊C8C2×D412yes—
152C23×C139—21no—
153C23×C139—21no—
154C23×C139C2×D411yescharacteristic
155Q8×C139—21no—
156Q8×C139—21no—
157D4×C139—21no—
158D4×C139—21no—
159D4×C139—21no—
160D4×C139—42no—
161D4×C139—81no—
162D4×C139C2×D412yes—
163C2×C4×C139—21no—
164C2×C4×C139—21no—
165C2×C4×C139—21no—
166C2×C4×C139—21no—
167C2×C4×C139—42no—
168C2×C4×C139C2×D411yescharacteristic
169C2×C4×C139C2411yescharacteristic
170C8×C139—21no—
171C8×C139—21no—
172C22×C139—21no—
173C22×C139—22no—
174C22×C139—21no—
175C22×C139—21no—
176C22×C139—41no—
177C22×C139—41no—
178C22×C139—41no—
179C22×C139—41no—
180C22×C139—41no—
181C22×C139C22×D411yescharacteristic
182C2×D139—21no—
183C2×D139—22no—
184C2×D139—21no—
185C2×D139—21no—
186C2×D139—41no—
187C2×D139—41no—
188C2×D139—84no—
189C4×C139—21no—
190C4×C139—41no—
191C4×C139—41no—
192C4×C139C22×D411yescharacteristic
193C4×C139H(5,𝔽2)11yescharacteristic
194Dic139—21no—
195Dic139—21no—
196Dic139—21no—
197Dic139—41no—
198Dic139—41no—
199Dic139—41no—
200C2×C139—41no—
201C2×C139—42no—
202C2×C139—81no—
203C2×C139D4211yescharacteristic
204C2×C139#57511yescharacteristic
205C2×C139#57711yescharacteristic
206D139—41no—
207D139—42no—
208C139#326711yescharacteristic, complemented, Sylow
209#3267—1391nomaximal, Sylow
210C2×AGL(1,ℤ/8)—1391no—
211C2×#135—1391no—
212#547—1391no—
213#546—1391no—
214#497—1391no—
215#470—1391no—
216#465—1391no—
217#463—1391no—
218#461—1391no—
219#452—1391no—
220#452—1391no—
221#450—1391no—
222#449D13911yescharacteristic, complemented
223#442—1391no—
224#435—1391no—
225C2×#41—1391no—
226C2×#41—1391no—
227C2×#41C2×D13911yescharacteristic, complemented
228C22×D4—1391no—
229AGL(1,ℤ/8)—2784no—
230#135—2784no—
231C2×Q16—1391no—
232C2×Q16—1391no—
233C2×SD16—1391no—
234C2×SD16—1391no—
235C2×SD16—1391no—
236C2×SD16—1391no—
237C2×SD16C2×D13911yescharacteristic
238C2×D8—1391no—
239C2×D8C2×D13911yescharacteristic
240C2×M16—1391no—
241C22×C8—1391no—
242C4⋊Q8—1391no—
243#124—1391no—
244#123—2781no—
245#123—2781no—
246#123—2781no—
247C22⋊Q8—1391no—
248C22⋊Q8—1391no—
249C22⋊Q8—1391no—
250#121—2781no—
251#121—2781no—
252#121C2×D13911yescharacteristic
253#120—2781no—
254C4×D4—1391no—
255C4×D4—1391no—
256C4×D4—1391no—
257C2×C4⋊C4—2781no—
258C2×C22⋊C4—2781no—
259#106—1391no—
260C4⋊C8—1391no—
261#103—1391no—
262#103—1391no—
263#103C2×D13911yescharacteristic, complemented
264#102—1391no—
265#102—1391no—
266#102C2×D13911yescharacteristic, complemented
267C22⋊C8—1391no—
268C22⋊C8C2×D13911yescharacteristic, complemented
269#97—1391no—
270C24—2781no—
271#41—22no—
272#41—41no—
273#41—2782no—
274#41—2782no—
275#41—5561no—
276#41—5561no—
277C2×Q8—1391no—
278C2×Q8—1391no—
279C2×Q8C22×D13911yescharacteristic
280C2×D4—21no—
281C2×D4—21no—
282C2×D4—2782no—
283C2×D4—2781no—
284C2×D4—2781no—
285C2×D4—2781no—
286C2×D4—5561no—
287C2×D4—1391no—
288C2×D4—1391no—
289C2×D4C22×D13911yescharacteristic
290C2×D4C22×D13911yescharacteristic
291C22×C4—21no—
292C22×C4—2781no—
293C22×C4—2781no—
294C22×C4—2781no—
295C22×C4—2781no—
296C22×C4—1391no—
297C22×C4C22×D13911yescharacteristic
298Q16—2782no—
299Q16—5561no—
300SD16—41no—
301SD16—2782no—
302SD16—2782no—
303SD16—2782no—
304SD16—2782no—
305D8—22no—
306D8—1394no—
307M16—2782no—
308C2×C8—2782no—
309C2×C8—1391no—
310C2×C8—1391no—
311C2×C8C22×D13911yescharacteristic
312C2×C8C22×D13911yescharacteristic
313C4⋊C4—2781no—
314C4⋊C4—2781no—
315C4⋊C4—2781no—
316C4⋊C4—2781no—
317C4⋊C4—2781no—
318C4⋊C4—1391no—
319C4⋊C4—1391no—
320C4⋊C4C22×D13911yescharacteristic
321C22⋊C4—21no—
322C22⋊C4—2781no—
323C22⋊C4—2781no—
324C22⋊C4—2781no—
325C22⋊C4—2781no—
326C22⋊C4—2781no—
327C22⋊C4—2781no—
328C22⋊C4—2781no—
329C22⋊C4—1391no—
330C22⋊C4—1391no—
331C42—1391no—
332C23—21no—
333C23—21no—
334C23—2781no—
335C23—5562no—
336C23—5562no—
337C23—11121no—
338C23—1391no—
339C23D4×D13911yescharacteristic
340Q8—21no—
341Q8—21no—
342Q8—2781no—
343Q8—2781no—
344Q8—2781no—
345Q8—2781no—
346D4—21no—
347D4—21no—
348D4—21no—
349D4—42no—
350D4—81no—
351D4—2781no—
352D4—5561no—
353D4—5562no—
354D4—5562no—
355D4—5562no—
356D4—1392no—
357D4D4×D13912yes—
358C2×C4—21no—
359C2×C4—21no—
360C2×C4—21no—
361C2×C4—21no—
362C2×C4—42no—
363C2×C4—2781no—
364C2×C4—2781no—
365C2×C4—2781no—
366C2×C4—2781no—
367C2×C4—2781no—
368C2×C4—5562no—
369C2×C4—5562no—
370C2×C4—5562no—
371C2×C4—11121no—
372C2×C4—1391no—
373C2×C4—1391no—
374C2×C4—1391no—
375C2×C4D4×D13911yescharacteristic
376C2×C4C23×D13911yescharacteristic
377C8—21no—
378C8—21no—
379C8—2781no—
380C8—1392no—
381C22—21no—
382C22—21no—
383C22—22no—
384C22—21no—
385C22—41no—
386C22—41no—
387C22—41no—
388C22—41no—
389C22—41no—
390C22—2781no—
391C22—2781no—
392C22—2782no—
393C22—2781no—
394C22—5561no—
395C22—5561no—
396C22—11124no—
397C22C2×D4×D13911yescharacteristic
398C4—21no—
399C4—41no—
400C4—41no—
401C4—2781no—
402C4—2781no—
403C4—2781no—
404C4—5561no—
405C4—5561no—
406C4—5561no—
407C4C2×D4×D13911yescharacteristic
408C4#1798488711yescharacteristic
409C2—41no—
410C2—42no—
411C2—81no—
412C2—5561no—
413C2—5562no—
414C2#2258158511yescharacteristic
415C2#2258175011yescharacteristic
416C2#2258176311yescharacteristic
subgroupG/H
centerC22C2×D4×D139
commutator, cosocleC2×C4×C139C24
FrattiniC2×C4C23×D139
Fitting#449×C139C2
socleC22×C139C22×D4
hypercenter#449D139
hypocenterC139#3267
LCSG ▷ C2×C4×C139 ▷ C2×C139 ▷ C139
UCSC1 ◁ C22 ◁ C22×C4 ◁ #449

Complements

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

NH ≅ G/Ntype
1. C2×#17984866C2semidirect
2. C2×#17984862C2semidirect
3. #22581602C2semidirect
4. #22581600C2semidirect
5. #22581344C2semidirect
6. #22581210C2semidirect
7. #22581180C2semidirect
8. #22581166C2semidirect
9. #22581157C2semidirect
10. #22581117C2semidirect
11. #22581116C2semidirect
12. #22581102C2semidirect
13. #22581062C2semidirect
14. #22581021C2semidirect
15. #449×C139C2semidirect
21. C2×Dic556C22semidirect
22. C2×#14555775C22semidirect
25. C2×#14555773C22semidirect
29. C22×C139⋊C8C22semidirect
30. #17984840C22semidirect
36. #17984821C22semidirect
41. D4×Dic139C22semidirect
46. #17984773C22semidirect
47. Dic139⋊C8C22semidirect
48. #17984767C22semidirect
49. #17984765C22semidirect
53. #17984752C22semidirect
54. C2×#41×C139C22semidirect
58. #103×C139C22semidirect
59. #102×C139C22semidirect
60. C22⋊C8×C139C22semidirect
86. #14555770D4semidirect
208. C139#3267semidirect
222. #449D139semidirect
227. C2×#41C2×D139semidirect
263. #103C2×D139semidirect
266. #102C2×D139semidirect
268. C22⋊C8C2×D139semidirect

Maximal subgroups

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

subgroupindex
C2×#179848662
C2×#179848622
#225816022
#225816002
#225813442
#225812102
#225811802
#225811662
#225811572
#225811172
#225811162
#225811022
#225810622
#225810212
#449×C1392
#3267139