Finite Group Databasebeta

← #31790462

Subgroups — #31790462

Order 17536.

504 conjugacy classes of subgroups.

The subgroup lattice has 420 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#17984115C211yesmaximal, characteristic, complemented
2C2×#17984094C211yesmaximal, characteristic, complemented
3#22577983C211yesmaximal, characteristic, complemented
4#22577969C211yesmaximal, characteristic, complemented
5#22577657C211yesmaximal, characteristic, complemented
6#22577654C211yesmaximal, characteristic, complemented
7#22577651C211yesmaximal, characteristic, complemented
8#22577646C211yesmaximal, characteristic, complemented
9#22577639C211yesmaximal, characteristic, complemented
10#22577638C211yesmaximal, characteristic, complemented
11#22577635C211yesmaximal, characteristic, complemented
12#22577442C211yesmaximal, characteristic, complemented
13#22577398C211yesmaximal, characteristic, complemented
14#102×D137C211yesmaximal, characteristic, complemented
15#465×C137C211yesmaximal, characteristic, complemented
16C2×#14555511C2211yescharacteristic
17C2×#14555510C2211yescharacteristic
18C2×D4×D137C2211yescharacteristic
19#17984115—24no—
20C2×#14555497C2211yescharacteristic
21C2×D1096C2211yescharacteristic
22C2×#14555495C2211yescharacteristic
23#17984094—24no—
24C2×#14555494C2211yescharacteristic, complemented
25C2×#14555493C2211yescharacteristic
26C2×#14555493C2211yescharacteristic, complemented
27C2×C8×D137C2211yescharacteristic
28C2×C8×D137C2211yescharacteristic, complemented
29#17984084—21no—
30#17984074—21no—
31#17984066—21no—
32#17984064C2211yescharacteristic, complemented
33#17984063—21no—
34#17984063C2211yescharacteristic
35#17984063C2211yescharacteristic, complemented
36#17984062C2211yescharacteristic
37#17984062C2211yescharacteristic, complemented
38#17984061—21no—
39#17984061—21no—
40#17984059C2211yescharacteristic
41#17984058—21no—
42#17984056—21no—
43#17984046C2211yescharacteristic
44C4⋊C4×D137C2211yescharacteristic
45#17984022C2211yescharacteristic, complemented
46#17984021C2211yescharacteristic, complemented
47#17984019C2211yescharacteristic, complemented
48#17984012C2211yescharacteristic, complemented
49#17984010C2211yescharacteristic
50#17984010C2211yescharacteristic, complemented
51#17984009C2211yescharacteristic
52#17984008C2211yescharacteristic
53#17984008C2211yescharacteristic, complemented
54C2×SD16×C137C2211yescharacteristic
55C22×C8×C137C2211yescharacteristic, complemented
56C22⋊Q8×C137C2211yescharacteristic, complemented
57#121×C137C2211yescharacteristic
58#106×C137C2211yescharacteristic, complemented
59#103×C137C2211yescharacteristic, complemented
60#102×C137C2211yescharacteristic
61C23×D137—21no—
62#14555511—22no—
63#14555511—22no—
64#14555510—22no—
65#14555510—22no—
66C2×Dic274D411yescharacteristic
67C2×D548—21no—
68C2×D548—21no—
69C2×D548—21no—
70C2×D548D411yescharacteristic
71C2×D548C2311yescharacteristic
72C2×D548C2311yescharacteristic
73C2×C22⋊D137—21no—
74C2×C22⋊D137—21no—
75C2×C22⋊D137—21no—
76C2×C22⋊D137—21no—
77C2×C22⋊D137D411yescharacteristic
78C2×C22⋊D137D411yescharacteristic
79D4×D137—22no—
80D4×D137—22no—
81C2×C4×D137—21no—
82C2×C4×D137—21no—
83C2×C4×D137C2311yescharacteristic
84C2×C4×D137C2311yescharacteristic
85#14555497—22no—
86D1096—22no—
87#14555495—22no—
88#14555494—22no—
89#14555493—22no—
90#14555493D414yescomplemented
91C2×#11764040C2311yescharacteristic
92C2×#11764040C2311yescharacteristic
93C8×D137—22no—
94C8×D137—22no—
95C4⋊Dic137C2311yescharacteristic
96C4⋊Dic137C2311yescharacteristic
97#14555489—21no—
98#14555489—21no—
99C22⋊Dic137—21no—
100#14555487—21no—
101#14555487—21no—
102#14555487—21no—
103#14555487—21no—
104#14555487—21no—
105#14555487—21no—
106C4×Dic137—21no—
107C2×Q8×C137C2311yescharacteristic
108C2×D4×C137—21no—
109C2×D4×C137C2311yescharacteristic
110C22×C4×C137C2311yescharacteristic
111SD16×C137—22no—
112C2×C8×C137—22no—
113C2×C8×C137C2311yescharacteristic
114C2×C8×C137C2311yescharacteristic
115C4⋊C4×C137—21no—
116C4⋊C4×C137C2311yescharacteristic
117C4⋊C4×C137C2311yescharacteristic
118C22⋊C4×C137—21no—
119C22⋊C4×C137—21no—
120C22×D137—21no—
121C22×D137—21no—
122C22×D137—42no—
123C22×D137—81no—
124C22×D137—81no—
125C22×D137C2×D411yescharacteristic
126C22×D137C2×D411yescharacteristic
127Dic274—22no—
128D548—22no—
129D548—21no—
130D548—21no—
131D548—21no—
132D548—21no—
133D548—42no—
134D548—81no—
135D548—81no—
136C22⋊D137—22no—
137C22⋊D137—22no—
138C22⋊D137—42no—
139C22⋊D137—81no—
140C22⋊D137—81no—
141C22⋊D137—81no—
142C2×Dic137—21no—
143C2×Dic137—21no—
144C2×Dic137C2×D411yescharacteristic
145C2×Dic137C2×D411yescharacteristic
146C4×D137—21no—
147C4×D137—21no—
148C4×D137—42no—
149C4×D137—81no—
150C4×D137C2×D412yes—
151C4×D137#4112yes—
152#11764040—21no—
153#11764040C2×D412yes—
154C23×C137—21no—
155C23×C137C2×D411yescharacteristic
156Q8×C137—21no—
157Q8×C137—21no—
158D4×C137—21no—
159D4×C137—21no—
160D4×C137—81no—
161C2×C4×C137—21no—
162C2×C4×C137—21no—
163C2×C4×C137—21no—
164C2×C4×C137—21no—
165C2×C4×C137—21no—
166C2×C4×C137C2×D411yescharacteristic
167C2×C4×C137C2411yescharacteristic
168C8×C137—21no—
169C8×C137C2×D412yes—
170C22×C137—21no—
171C22×C137—21no—
172C22×C137—21no—
173C22×C137—41no—
174C22×C137—41no—
175C22×C137—41no—
176C22×C137C22×D411yescharacteristic
177C2×D137—21no—
178C2×D137—21no—
179C2×D137—21no—
180C2×D137—21no—
181C2×D137—21no—
182C2×D137—41no—
183C2×D137—41no—
184C2×D137—41no—
185C2×D137—41no—
186C2×D137—41no—
187C2×D137—41no—
188C2×D137—84no—
189C2×D137#12112yes—
190C4×C137—21no—
191C4×C137—41no—
192C4×C137—41no—
193C4×C137—41no—
194C4×C137C22×D411yescharacteristic
195C4×C137C2×#4111yescharacteristic
196Dic137—21no—
197Dic137—41no—
198Dic137—41no—
199Dic137#12112yes—
200C2×C137—41no—
201C2×C137—81no—
202C2×C137C2×#12111yescharacteristic
203C2×C137#57611yescharacteristic
204C2×C137#57711yescharacteristic
205D137—22no—
206D137—41no—
207D137—81no—
208D137—81no—
209C137#306211yescharacteristic, complemented, Sylow
210#3062—1371nomaximal, Sylow
211C2×AGL(1,ℤ/8)—1371no—
212C2×#131—1371no—
213#537—1371no—
214#535—1371no—
215#469—1371no—
216#469—1371no—
217#468—1371no—
218#468—1371no—
219#466—1371no—
220#466—1371no—
221#465—1371no—
222#465D13711yescharacteristic, complemented
223#428—1371no—
224#416—1371no—
225C2×#102—1371no—
226C2×#41—1371no—
227C2×#41—1371no—
228C22×D4—1371no—
229AGL(1,ℤ/8)—2744no—
230C2×SD16—1371no—
231C2×SD16C2×D13711yescharacteristic
232C2×D8—1371no—
233C2×D8—1371no—
234#131—2744no—
235C2×M16—1371no—
236C2×M16—1371no—
237C2×M16—1371no—
238C22×C8—1371no—
239C22×C8—1371no—
240C22×C8C2×D13711yescharacteristic, complemented
241#127—2741no—
242#124—2741no—
243C22⋊Q8—2741no—
244C22⋊Q8—1371no—
245C22⋊Q8C2×D13711yescharacteristic, complemented
246#121—2741no—
247#121—2741no—
248#121—2741no—
249#121—1371no—
250#121—1371no—
251#121—1371no—
252#121—1371no—
253#121—1371no—
254#121C2×D13711yescharacteristic
255#120—2741no—
256#120—2741no—
257#117—1371no—
258C2×C4⋊C4—1371no—
259#107—1371no—
260#107—1371no—
261#106—1371no—
262#106C2×D13711yescharacteristic, complemented
263#103—1371no—
264#103C2×D13711yescharacteristic, complemented
265#102—1371no—
266#102—1371no—
267#102—1371no—
268#102—1371no—
269#102—1371no—
270#102C2×D13711yescharacteristic
271C24—2741no—
272#41—2742no—
273#41—2742no—
274#41—2742no—
275#41—2742no—
276C2×Q8—1371no—
277C2×Q8C22×D13711yescharacteristic
278C2×D4—21no—
279C2×D4—2741no—
280C2×D4—2741no—
281C2×D4—2741no—
282C2×D4—2741no—
283C2×D4—2741no—
284C2×D4—2741no—
285C2×D4—2742no—
286C2×D4—2742no—
287C2×D4—2741no—
288C2×D4—1371no—
289C2×D4—1371no—
290C2×D4—1371no—
291C2×D4—1371no—
292C2×D4—1371no—
293C2×D4C22×D13711yescharacteristic
294C22×C4—2741no—
295C22×C4—2741no—
296C22×C4—1371no—
297C22×C4—1371no—
298C22×C4C22×D13711yescharacteristic
299SD16—22no—
300SD16—2742no—
301D8—2742no—
302D8—2742no—
303M16—2742no—
304M16—2742no—
305M16—1374no—
306C2×C8—22no—
307C2×C8—2742no—
308C2×C8—2742no—
309C2×C8—1371no—
310C2×C8—1371no—
311C2×C8C22×D13711yescharacteristic
312C2×C8C22×D13711yescharacteristic
313C4⋊C4—21no—
314C4⋊C4—2741no—
315C4⋊C4—2741no—
316C4⋊C4—1371no—
317C4⋊C4—1371no—
318C4⋊C4C22×D13711yescharacteristic
319C4⋊C4C22×D13711yescharacteristic
320C22⋊C4—21no—
321C22⋊C4—21no—
322C22⋊C4—2741no—
323C22⋊C4—2741no—
324C22⋊C4—2741no—
325C22⋊C4—2741no—
326C22⋊C4—2741no—
327C22⋊C4—2741no—
328C22⋊C4—2741no—
329C42—2741no—
330C23—21no—
331C23—2741no—
332C23—2741no—
333C23—5482no—
334C23—10961no—
335C23—10961no—
336C23—1371no—
337C23—1371no—
338C23D4×D13711yescharacteristic
339Q8—21no—
340Q8—21no—
341Q8—2742no—
342D4—21no—
343D4—21no—
344D4—81no—
345D4—2742no—
346D4—2742no—
347D4—2742no—
348D4—2741no—
349D4—2741no—
350D4—2741no—
351D4—2741no—
352D4—5482no—
353D4—5482no—
354D4—10961no—
355D4—10961no—
356D4—10961no—
357D4—10961no—
358D4—10961no—
359C2×C4—21no—
360C2×C4—21no—
361C2×C4—21no—
362C2×C4—21no—
363C2×C4—21no—
364C2×C4—2741no—
365C2×C4—2741no—
366C2×C4—2741no—
367C2×C4—2741no—
368C2×C4—5482no—
369C2×C4—10961no—
370C2×C4—1371no—
371C2×C4—1371no—
372C2×C4—1372no—
373C2×C4—1372no—
374C2×C4D4×D13711yescharacteristic
375C2×C4C23×D13711yescharacteristic
376C8—21no—
377C8—2741no—
378C8—1372no—
379C8D4×D13712yes—
380C22—21no—
381C22—21no—
382C22—21no—
383C22—41no—
384C22—41no—
385C22—41no—
386C22—2741no—
387C22—2741no—
388C22—2741no—
389C22—2741no—
390C22—2741no—
391C22—5481no—
392C22—5481no—
393C22—5481no—
394C22—5481no—
395C22—5481no—
396C22—5481no—
397C22—10964no—
398C22—1372no—
399C22C2×D4×D13711yescharacteristic
400C4—21no—
401C4—41no—
402C4—41no—
403C4—41no—
404C4—2741no—
405C4—5481no—
406C4—5481no—
407C4—1372no—
408C4C2×D4×D13711yescharacteristic
409C4#41×D13711yescharacteristic
410C2—41no—
411C2—81no—
412C2—2742no—
413C2—5481no—
414C2—10961no—
415C2—10961no—
416C2#121×D13711yescharacteristic
417C2#2257820511yescharacteristic
418C2#2257820911yescharacteristic
subgroupG/H
centerC22C2×D4×D137
commutator, cosocleC2×C4×C137C24
FrattiniC2×C4C23×D137
Fitting#465×C137C2
socleC22×C137C22×D4
hypercenter#465D137
hypocenterC137#3062
LCSG ▷ C2×C4×C137 ▷ C2×C137 ▷ C137
UCSC1 ◁ C22 ◁ C22×C4 ◁ #465

Complements

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

NH ≅ G/Ntype
1. C2×#17984115C2semidirect
2. C2×#17984094C2semidirect
3. #22577983C2semidirect
4. #22577969C2semidirect
5. #22577657C2semidirect
6. #22577654C2semidirect
7. #22577651C2semidirect
8. #22577646C2semidirect
9. #22577639C2semidirect
10. #22577638C2semidirect
11. #22577635C2semidirect
12. #22577442C2semidirect
13. #22577398C2semidirect
14. #102×D137C2semidirect
15. #465×C137C2semidirect
24. C2×#14555494C22semidirect
26. C2×#14555493C22semidirect
28. C2×C8×D137C22semidirect
32. #17984064C22semidirect
35. #17984063C22semidirect
37. #17984062C22semidirect
45. #17984022C22semidirect
46. #17984021C22semidirect
47. #17984019C22semidirect
48. #17984012C22semidirect
50. #17984010C22semidirect
53. #17984008C22semidirect
55. C22×C8×C137C22semidirect
56. C22⋊Q8×C137C22semidirect
58. #106×C137C22semidirect
59. #103×C137C22semidirect
90. #14555493D4semidirect
209. C137#3062semidirect
222. #465D137semidirect
240. C22×C8C2×D137semidirect
245. C22⋊Q8C2×D137semidirect
262. #106C2×D137semidirect
264. #103C2×D137semidirect

Maximal subgroups

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

subgroupindex
C2×#179841152
C2×#179840942
#225779832
#225779692
#225776572
#225776542
#225776512
#225776462
#225776392
#225776382
#225776352
#225774422
#225773982
#102×D1372
#465×C1372
#3062137