Finite Group Databasebeta

← #31818167

Subgroups — #31818167

Order 17792.

464 conjugacy classes of subgroups.

The subgroup lattice has 462 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#22581765C211yesmaximal, characteristic, complemented
2#22581763C211yesmaximal, characteristic, complemented
3#22581757C211yesmaximal, characteristic, complemented
4#22581754C211yesmaximal, characteristic, complemented
5#22581341C211yesmaximal, characteristic, complemented
6#22581222C211yesmaximal, characteristic, complemented
7#22581217C211yesmaximal, characteristic, complemented
8#22581215C211yesmaximal, characteristic, complemented
9#22581133C211yesmaximal, characteristic, complemented
10#22581124C211yesmaximal, characteristic, complemented
11#22581123C211yesmaximal, characteristic, complemented
12#22581119C211yesmaximal, characteristic, complemented
13#22581074C211yesmaximal, characteristic, complemented
14#22581073C211yesmaximal, characteristic, complemented
15#472×C139C211yesmaximal, characteristic, complemented
16#17984889C2211yescharacteristic, complemented
17#17984888C2211yescharacteristic, complemented
18#17984887C2211yescharacteristic
19#17984887C2211yescharacteristic, complemented
20#17984875—21no—
21#17984875C2211yescharacteristic, complemented
22#17984875C2211yescharacteristic, complemented
23#17984871—21no—
24#17984870—21no—
25#17984870C2211yescharacteristic
26#17984870C2211yescharacteristic, complemented
27#17984869—21no—
28#17984868C2211yescharacteristic
29#17984868C2211yescharacteristic, complemented
30#17984866—21no—
31#17984866—21no—
32#17984866—21no—
33#17984866—21no—
34#17984864—21no—
35#17984864C2211yescharacteristic
36#17984863—21no—
37#17984862C2211yescharacteristic
38#17984861—21no—
39#17984861—21no—
40C2×#14555774—21no—
41C2×#14555774C2211yescharacteristic, complemented
42C2×#14555773C2211yescharacteristic, complemented
43SD16×D139—21no—
44C2×#14555770C2211yescharacteristic, complemented
45D8×D139—21no—
46D8×D139—21no—
47#17984850C2211yescharacteristic, complemented
48#17984850C2211yescharacteristic, complemented
49#17984848C2211yescharacteristic, complemented
50#17984838C2211yescharacteristic, complemented
51#17984829C2211yescharacteristic, complemented
52#17984777C2211yescharacteristic, complemented
53#17984769C2211yescharacteristic, complemented
54#17984768C2211yescharacteristic, complemented
55#17984768C2211yescharacteristic, complemented
56#17984768C2211yescharacteristic, complemented
57#17984760C2211yescharacteristic, complemented
58#17984758C2211yescharacteristic, complemented
59#17984752C2211yescharacteristic, complemented
60AGL(1,ℤ/8)×C139C2211yescharacteristic, complemented
61AGL(1,ℤ/8)×C139C2211yescharacteristic, complemented
62C2×D8×C139C2211yescharacteristic, complemented
63#131×C139C2211yescharacteristic, complemented
64#108×C139C2211yescharacteristic, complemented
65#100×C139C2211yescharacteristic, complemented
66#100×C139C2211yescharacteristic, complemented
67#14555786—21no—
68#14555785—21no—
69#14555785—21no—
70#14555785C2311yescharacteristic
71#14555785C2311yescharacteristic
72#14555785C2311yescharacteristic
73#14555784—21no—
74#14555784—21no—
75#14555784—21no—
76#14555784—21no—
77#14555784—21no—
78#14555784—41no—
79C2×Dic278—21no—
80C2×Dic278C2311yescharacteristic
81Q8×D139—21no—
82C2×D556—21no—
83C2×D556C2311yescharacteristic
84C2×C22⋊D139—41no—
85C2×C22⋊D139—41no—
86C2×C22⋊D139—41no—
87D4×D139—21no—
88D4×D139—21no—
89D4×D139—21no—
90D4×D139—21no—
91D4×D139—21no—
92D4×D139—41no—
93Dic556—21no—
94Dic556—21no—
95Dic556D411yescharacteristic, complemented
96#14555775D411yescharacteristic, complemented
97#14555774—21no—
98#14555774—21no—
99#14555774—21no—
100#14555774—21no—
101#14555774—21no—
102#14555774—21no—
103#14555774D411yescharacteristic, complemented
104#14555774D411yescharacteristic, complemented
105#14555773—21no—
106#14555773—21no—
107#14555773—21no—
108#14555773—21no—
109#14555773D411yescharacteristic, complemented
110#14555772D411yescharacteristic, complemented
111D1112—21no—
112D1112—21no—
113D1112D411yescharacteristic, complemented
114#14555770—21no—
115#14555770—21no—
116#14555770—21no—
117#14555770—21no—
118#14555770D411yescharacteristic, complemented
119#14555769C2311yescharacteristic
120#14555768—21no—
121#14555768—21no—
122#14555768—21no—
123C2×C139⋊C8C2311yescharacteristic
124C8×D139—21no—
125C8×D139—21no—
126C8×D139—21no—
127C22⋊Dic139—41no—
128C4×Dic139C2311yescharacteristic
129#41×C139C2311yescharacteristic
130C2×D4×C139C2311yescharacteristic
131C2×D4×C139C2311yescharacteristic
132SD16×C139—21no—
133SD16×C139—21no—
134D8×C139—21no—
135D8×C139—21no—
136D8×C139—21no—
137D8×C139—21no—
138M16×C139—21no—
139M16×C139C2311yescharacteristic
140M16×C139C2311yescharacteristic
141M16×C139C2311yescharacteristic
142C2×C8×C139—21no—
143C2×C8×C139C2311yescharacteristic
144C22×D139—21no—
145C22×D139—41no—
146C22×D139—41no—
147C22×D139—41no—
148Dic278—21no—
149Dic278—21no—
150Dic278—41no—
151Dic278C2×D411yescharacteristic
152Dic278C2×D411yescharacteristic
153Dic278C2×D411yescharacteristic
154D556—21no—
155D556—21no—
156D556—41no—
157D556C2×D411yescharacteristic
158D556C2×D411yescharacteristic
159D556C2×D411yescharacteristic
160C22⋊D139—21no—
161C22⋊D139—21no—
162C22⋊D139—21no—
163C22⋊D139—41no—
164C22⋊D139—41no—
165C22⋊D139—41no—
166C22⋊D139—41no—
167C22⋊D139—41no—
168C22⋊D139—41no—
169C22⋊D139—82no—
170C2×Dic139—21no—
171C2×Dic139—21no—
172C2×Dic139—41no—
173C2×Dic139—41no—
174C2×Dic139—41no—
175C4×D139—21no—
176C4×D139—21no—
177C4×D139—21no—
178C4×D139—41no—
179C4×D139—41no—
180C139⋊C8—21no—
181C139⋊C8C2×D411yescharacteristic
182C139⋊C8C2×D411yescharacteristic
183C23×C139—21no—
184C23×C139—21no—
185Q8×C139C2×D411yescharacteristic
186D4×C139—21no—
187D4×C139—21no—
188D4×C139—21no—
189D4×C139—21no—
190D4×C139—21no—
191D4×C139C2×D411yescharacteristic
192C2×C4×C139—21no—
193C2×C4×C139C2411yescharacteristic
194C8×C139—21no—
195C8×C139—21no—
196C8×C139—21no—
197C8×C139C2×D411yescharacteristic
198C8×C139C2×D411yescharacteristic
199C22×C139—21no—
200C22×C139—41no—
201C22×C139—41no—
202C22×C139—81no—
203C22×C139—81no—
204C22×C139H(5,𝔽2)11yescharacteristic
205C2×D139—21no—
206C2×D139—21no—
207C2×D139—21no—
208C2×D139—41no—
209C2×D139—81no—
210C2×D139—161no—
211C2×D139—161no—
212C2×D139—161no—
213C4×C139—21no—
214C4×C139C22×D411yescharacteristic
215C4×C139C22×D411yescharacteristic
216Dic139—21no—
217Dic139—21no—
218Dic139—21no—
219Dic139—41no—
220Dic139—41no—
221C2×C139—21no—
222C2×C139—41no—
223C2×C139—81no—
224C2×C139—81no—
225C2×C139D4211yescharacteristic
226D139—41no—
227D139—41no—
228D139—41no—
229D139—81no—
230C139#328811yescharacteristic, complemented, Sylow
231#3288—1391nomaximal, Sylow
232#577—1391no—
233#577—1391no—
234#576—1391no—
235#576—1391no—
236#496—1391no—
237#472—1391no—
238#472D13911yescharacteristic, complemented
239#471—1391no—
240#471—1391no—
241#455—1391no—
242#454—1391no—
243#453—1391no—
244#453—1391no—
245#444—1391no—
246#444—1391no—
2472−1+4—1391no—
248H(5,𝔽2)—1391no—
249H(5,𝔽2)—1391no—
250H(5,𝔽2)—1391no—
251#137—2781no—
252#137—2781no—
253#137—1391no—
254#137—1391no—
255AGL(1,ℤ/8)—2781no—
256AGL(1,ℤ/8)—2781no—
257AGL(1,ℤ/8)—2781no—
258AGL(1,ℤ/8)—2781no—
259AGL(1,ℤ/8)—2781no—
260AGL(1,ℤ/8)—2781no—
261AGL(1,ℤ/8)—1391no—
262AGL(1,ℤ/8)—1391no—
263AGL(1,ℤ/8)—1391no—
264AGL(1,ℤ/8)—1391no—
265AGL(1,ℤ/8)C2×D13911yescharacteristic, complemented
266AGL(1,ℤ/8)C2×D13911yescharacteristic, complemented
267#135—2781no—
268#135—2781no—
269#135—2781no—
270#135—2781no—
271#135—1391no—
272#135—1391no—
273C2×SD16—2781no—
274C2×SD16—2781no—
275C2×SD16—1391no—
276C2×SD16—1391no—
277C2×D8—2781no—
278C2×D8—2781no—
279C2×D8—1391no—
280C2×D8C2×D13911yescharacteristic, complemented
281#131—1391no—
282#131—1391no—
283#131—1391no—
284#131C2×D13911yescharacteristic, complemented
285#127—1391no—
286#124—1391no—
287#108—1391no—
288#108C2×D13911yescharacteristic, complemented
289#104—1391no—
290#104—1391no—
291#104—1391no—
292#104—1391no—
293#101—1391no—
294#100—1391no—
295#100C2×D13911yescharacteristic, complemented
296#100C2×D13911yescharacteristic, complemented
297#97—1391no—
298#41—2781no—
299#41—2781no—
300#41—2781no—
301#41—2781no—
302#41—2781no—
303#41—2781no—
304#41—2781no—
305#41—2781no—
306#41—5561no—
307#41—1391no—
308#41—1391no—
309#41—1391no—
310#41C22×D13911yescharacteristic
311C2×Q8—2781no—
312C2×Q8—2781no—
313C2×Q8—1391no—
314C2×D4—2781no—
315C2×D4—2781no—
316C2×D4—2781no—
317C2×D4—2781no—
318C2×D4—2781no—
319C2×D4—2781no—
320C2×D4—5561no—
321C2×D4—5561no—
322C2×D4—5561no—
323C2×D4—5561no—
324C2×D4—1391no—
325C2×D4C22×D13911yescharacteristic
326C2×D4C22×D13911yescharacteristic
327Q16—2781no—
328Q16—2781no—
329Q16—1391no—
330Q16—1391no—
331SD16—21no—
332SD16—21no—
333SD16—2781no—
334SD16—2781no—
335SD16—2781no—
336SD16—2781no—
337SD16—2781no—
338SD16—2781no—
339SD16—2781no—
340SD16—2781no—
341SD16—2781no—
342SD16—2781no—
343SD16—1391no—
344SD16—1391no—
345SD16—1391no—
346SD16—1391no—
347D8—21no—
348D8—21no—
349D8—21no—
350D8—21no—
351D8—2781no—
352D8—2781no—
353D8—2781no—
354D8—2781no—
355D8—2781no—
356D8—2781no—
357D8—1391no—
358D8—1391no—
359M16—21no—
360M16—2781no—
361M16—2781no—
362M16—2781no—
363M16—1391no—
364M16C22×D13911yescharacteristic
365M16C22×D13911yescharacteristic
366M16C22×D13911yescharacteristic
367C2×C8—21no—
368C2×C8—2781no—
369C2×C8—2781no—
370C2×C8—2781no—
371C2×C8—1391no—
372C2×C8C22×D13911yescharacteristic
373C22⋊C4—5561no—
374C42—1391no—
375C23—21no—
376C23—21no—
377C23—2781no—
378C23—5561no—
379C23—5561no—
380C23—5561no—
381Q8—2781no—
382Q8—2781no—
383Q8—5561no—
384Q8—1391no—
385Q8—1391no—
386Q8—1391no—
387Q8D4×D13911yescharacteristic
388D4—21no—
389D4—21no—
390D4—21no—
391D4—21no—
392D4—21no—
393D4—2781no—
394D4—2781no—
395D4—2781no—
396D4—2781no—
397D4—2781no—
398D4—5561no—
399D4—5561no—
400D4—5561no—
401D4—5561no—
402D4—5561no—
403D4—5561no—
404D4—5561no—
405D4—11122no—
406D4—1391no—
407D4—1391no—
408D4—1391no—
409D4D4×D13911yescharacteristic
410C2×C4—21no—
411C2×C4—2781no—
412C2×C4—2781no—
413C2×C4—2781no—
414C2×C4—2781no—
415C2×C4—2781no—
416C2×C4—5561no—
417C2×C4—5561no—
418C2×C4—5561no—
419C2×C4—5561no—
420C2×C4—5561no—
421C2×C4C23×D13911yescharacteristic
422C8—21no—
423C8—21no—
424C8—21no—
425C8—2781no—
426C8—1391no—
427C8—1391no—
428C8D4×D13911yescharacteristic
429C8D4×D13911yescharacteristic
430C22—21no—
431C22—41no—
432C22—41no—
433C22—81no—
434C22—81no—
435C22—2781no—
436C22—2781no—
437C22—2781no—
438C22—5561no—
439C22—11121no—
440C22—22241no—
441C22—22241no—
442C22—22241no—
443C22#1798488811yescharacteristic
444C4—21no—
445C4—2781no—
446C4—2781no—
447C4—2781no—
448C4—5561no—
449C4—5561no—
450C4C2×D4×D13911yescharacteristic
451C4C2×D4×D13911yescharacteristic
452C2—21no—
453C2—41no—
454C2—81no—
455C2—81no—
456C2—5561no—
457C2—5561no—
458C2—5561no—
459C2—11121no—
460C2#2258158711yescharacteristic
subgroupG/H
centerC2#22581587
commutator, cosocleC2×C4×C139C24
FrattiniC2×C4C23×D139
Fitting#472×C139C2
socleC2×C139D42
hypercenter#472D139
hypocenterC139#3288
LCSG ▷ C2×C4×C139 ▷ C2×C139 ▷ C139
UCSC1 ◁ C2 ◁ C2×C4 ◁ #472

Complements

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

NH ≅ G/Ntype
1. #22581765C2semidirect
2. #22581763C2semidirect
3. #22581757C2semidirect
4. #22581754C2semidirect
5. #22581341C2semidirect
6. #22581222C2semidirect
7. #22581217C2semidirect
8. #22581215C2semidirect
9. #22581133C2semidirect
10. #22581124C2semidirect
11. #22581123C2semidirect
12. #22581119C2semidirect
13. #22581074C2semidirect
14. #22581073C2semidirect
15. #472×C139C2semidirect
16. #17984889C22semidirect
17. #17984888C22semidirect
19. #17984887C22semidirect
21. #17984875C22semidirect
22. #17984875C22semidirect
26. #17984870C22semidirect
29. #17984868C22semidirect
41. C2×#14555774C22semidirect
42. C2×#14555773C22semidirect
44. C2×#14555770C22semidirect
47. #17984850C22semidirect
48. #17984850C22semidirect
49. #17984848C22semidirect
50. #17984838C22semidirect
51. #17984829C22semidirect
52. #17984777C22semidirect
53. #17984769C22semidirect
54. #17984768C22semidirect
55. #17984768C22semidirect
56. #17984768C22semidirect
57. #17984760C22semidirect
58. #17984758C22semidirect
59. #17984752C22semidirect
60. AGL(1,ℤ/8)×C139C22semidirect
61. AGL(1,ℤ/8)×C139C22semidirect
62. C2×D8×C139C22semidirect
63. #131×C139C22semidirect
64. #108×C139C22semidirect
65. #100×C139C22semidirect
66. #100×C139C22semidirect
95. Dic556D4semidirect
96. #14555775D4semidirect
103. #14555774D4semidirect
104. #14555774D4semidirect
109. #14555773D4semidirect
110. #14555772D4semidirect
113. D1112D4semidirect
118. #14555770D4semidirect
230. C139#3288semidirect
238. #472D139semidirect
265. AGL(1,ℤ/8)C2×D139semidirect
266. AGL(1,ℤ/8)C2×D139semidirect
280. C2×D8C2×D139semidirect
284. #131C2×D139semidirect
288. #108C2×D139semidirect
295. #100C2×D139semidirect
296. #100C2×D139semidirect

Maximal subgroups

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

subgroupindex
#225817652
#225817632
#225817572
#225817542
#225813412
#225812222
#225812172
#225812152
#225811332
#225811242
#225811232
#225811192
#225810742
#225810732
#472×C1392
#3288139