Finite Group Databasebeta

← #31790407

Subgroups — #31790407

Order 17536.

548 conjugacy classes of subgroups.

The subgroup lattice has 428 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#17984112C211yesmaximal, characteristic, complemented
2C2×M16×D137C211yesmaximal, characteristic, complemented
3#22577913C211yesmaximal, characteristic, complemented
4#121×D137C211yesmaximal, characteristic, complemented
5#22577654C211yesmaximal, characteristic, complemented
6#22577649C211yesmaximal, characteristic, complemented
7#22577647C211yesmaximal, characteristic, complemented
8#22577638C211yesmaximal, characteristic, complemented
9#22577636C211yesmaximal, characteristic, complemented
10#22577633C211yesmaximal, characteristic, complemented
11#22577631C211yesmaximal, characteristic, complemented
12#22577442C211yesmaximal, characteristic, complemented
13#22577406C211yesmaximal, characteristic, complemented
14#22577401C211yesmaximal, characteristic, complemented
15#469×C137C211yesmaximal, characteristic, complemented
16C2×#14555509C2211yescharacteristic
17C2×D4×D137—21no—
18C2×D4×D137C2211yescharacteristic
19C22×C4×D137C2211yescharacteristic
20#17984112—24no—
21C2×#14555499C2211yescharacteristic, complemented
22C2×#14555498C2211yescharacteristic, complemented
23C2×#14555495C2211yescharacteristic, complemented
24C2×#14555494C2211yescharacteristic, complemented
25C2×#14555493C2211yescharacteristic
26C2×#14555493C2211yescharacteristic
27M16×D137—24no—
28C2×C8×D137C2211yescharacteristic, complemented
29C2×C8×D137C2211yescharacteristic, complemented
30#17984070—21no—
31#17984068C2211yescharacteristic, complemented
32#17984067C2211yescharacteristic, complemented
33#17984063C2211yescharacteristic
34#17984062C2211yescharacteristic
35#17984061—21no—
36#17984060—21no—
37#17984059C2211yescharacteristic
38#17984059C2211yescharacteristic, complemented
39#17984056—21no—
40D4×Dic137—21no—
41#17984048—21no—
42#17984046C2211yescharacteristic
43C4⋊C4×D137C2211yescharacteristic
44C22⋊C4×D137—21no—
45#17984022C2211yescharacteristic, complemented
46#17984021C2211yescharacteristic, complemented
47#17984019C2211yescharacteristic, complemented
48#17984013C2211yescharacteristic, complemented
49#17984011C2211yescharacteristic, complemented
50#17984010C2211yescharacteristic, complemented
51#17984009C2211yescharacteristic, complemented
52#17984009C2211yescharacteristic, complemented
53#17984008C2211yescharacteristic, complemented
54C2×D8×C137C2211yescharacteristic, complemented
55C2×M16×C137C2211yescharacteristic, complemented
56#121×C137C2211yescharacteristic
57#121×C137C2211yescharacteristic, complemented
58#106×C137C2211yescharacteristic, complemented
59#102×C137C2211yescharacteristic, complemented
60#102×C137C2211yescharacteristic, complemented
61C23×D137—21no—
62C23×D137D411yescharacteristic
63#14555509—22no—
64#14555509—22no—
65C2×Dic274C2311yescharacteristic
66C2×D548—21no—
67C2×D548C2311yescharacteristic
68C2×C22⋊D137—21no—
69C2×C22⋊D137—21no—
70C2×C22⋊D137—21no—
71C2×C22⋊D137—21no—
72C2×C22⋊D137—21no—
73D4×D137—22no—
74D4×D137—22no—
75D4×D137—44no—
76C22×Dic137—21no—
77C22×Dic137D411yescharacteristic
78C2×C4×D137—22no—
79C2×C4×D137—22no—
80C2×C4×D137—21no—
81C2×C4×D137D411yescharacteristic
82C2×C4×D137D411yescharacteristic
83C2×C4×D137C2311yescharacteristic
84C2×C4×D137C2311yescharacteristic
85#14555499—22no—
86#14555498—22no—
87#14555495—22no—
88#14555494—22no—
89#14555493—22no—
90#14555493D414yescomplemented
91C2×#11764040C2311yescharacteristic
92C2×#11764040C2311yescharacteristic
93C8×D137—22no—
94C8×D137—22no—
95C4⋊Dic137—21no—
96C4⋊Dic137C2311yescharacteristic
97C4⋊Dic137C2311yescharacteristic
98#14555489—21no—
99#14555489—21no—
100C22⋊Dic137—21no—
101C22⋊Dic137—21no—
102C22⋊Dic137—21no—
103#14555487—21no—
104#14555487—21no—
105#14555487—21no—
106#14555487—21no—
107C4×Dic137—21no—
108C2×D4×C137—21no—
109C2×D4×C137—21no—
110C2×D4×C137C2311yescharacteristic
111C2×D4×C137C2311yescharacteristic
112C22×C4×C137C2311yescharacteristic
113D8×C137—22no—
114M16×C137—22no—
115C2×C8×C137C2311yescharacteristic
116C2×C8×C137C2311yescharacteristic
117C4⋊C4×C137C2311yescharacteristic
118C4⋊C4×C137C2311yescharacteristic
119C22⋊C4×C137—21no—
120C22⋊C4×C137—21no—
121C22×D137—22no—
122C22×D137—22no—
123C22×D137—21no—
124C22×D137—41no—
125C22×D137—42no—
126C22×D137—42no—
127C22×D137—42no—
128C22×D137C2×D411yescharacteristic
129C22×D137C2×D411yescharacteristic
130Dic274—21no—
131Dic274—21no—
132D548—21no—
133D548—21no—
134D548—42no—
135C22⋊D137—42no—
136C22⋊D137—42no—
137C22⋊D137—42no—
138C22⋊D137—42no—
139C22⋊D137—42no—
140C2×Dic137—22no—
141C2×Dic137—21no—
142C2×Dic137—21no—
143C2×Dic137—21no—
144C2×Dic137—42no—
145C2×Dic137C2×D411yescharacteristic
146C2×Dic137C2×D411yescharacteristic
147C4×D137—22no—
148C4×D137—21no—
149C4×D137—21no—
150C4×D137—22no—
151C4×D137—42no—
152C4×D137C2×D412yes—
153C4×D137#4112yes—
154#11764040—21no—
155#11764040C2×D412yescomplemented
156C23×C137—21no—
157C23×C137—21no—
158C23×C137C2×D411yescharacteristic
159D4×C137—21no—
160D4×C137—21no—
161D4×C137—21no—
162D4×C137—21no—
163D4×C137—42no—
164D4×C137—42no—
165C2×C4×C137—21no—
166C2×C4×C137—21no—
167C2×C4×C137—21no—
168C2×C4×C137—21no—
169C2×C4×C137C2×D411yescharacteristic
170C2×C4×C137C2411yescharacteristic
171C8×C137—21no—
172C8×C137C2×D412yescomplemented
173C22×C137—21no—
174C22×C137—21no—
175C22×C137—21no—
176C22×C137—41no—
177C22×C137—41no—
178C22×C137—41no—
179C22×C137—41no—
180C22×C137—41no—
181C22×C137—41no—
182C22×C137C22×D411yescharacteristic
183C2×D137—21no—
184C2×D137—21no—
185C2×D137—21no—
186C2×D137—21no—
187C2×D137—44no—
188C2×D137—41no—
189C2×D137—41no—
190C2×D137—41no—
191C2×D137—84no—
192C2×D137#12112yes—
193C2×D137AGL(1,ℤ/8)12yescomplemented
194C4×C137—21no—
195C4×C137—41no—
196C4×C137—41no—
197C4×C137C22×D411yescharacteristic
198C4×C137C2×#4111yescharacteristic
199Dic137—21no—
200Dic137—41no—
201Dic137—41no—
202Dic137—41no—
203Dic137#12112yes—
204C2×C137—41no—
205C2×C137—81no—
206C2×C137—81no—
207C2×C137C2×#12111yescharacteristic
208C2×C137C2×AGL(1,ℤ/8)11yescharacteristic
209C2×C137#57511yescharacteristic
210D137—22no—
211D137—22no—
212D137—81no—
213C137#305511yescharacteristic, complemented, Sylow
214#3055—1371nomaximal, Sylow
215C2×AGL(1,ℤ/8)—1371no—
216C22×M16—1371no—
217#525—1371no—
218C2×#121—1371no—
219#469—1371no—
220#469D13711yescharacteristic, complemented
221#468—1371no—
222#468—1371no—
223#466—1371no—
224#466—1371no—
225#465—1371no—
226#465—1371no—
227#428—1371no—
228#418—1371no—
229#417—1371no—
230C2×#41—1371no—
231C22×D4—2741no—
232C22×D4—1371no—
233C23×C4—1371no—
234AGL(1,ℤ/8)—2744no—
235C2×SD16—1371no—
236C2×SD16—1371no—
237C2×D8—1371no—
238C2×D8C2×D13711yescharacteristic, complemented
239C2×M16—2744no—
240C2×M16—1371no—
241C2×M16—1371no—
242C2×M16—1371no—
243C2×M16C2×D13711yescharacteristic, complemented
244C22×C8—1371no—
245C22×C8—1371no—
246#123—2741no—
247C22⋊Q8—1371no—
248C22⋊Q8—1371no—
249#121—2741no—
250#121—2741no—
251#121—1371no—
252#121—1371no—
253#121—1371no—
254#121—1371no—
255#121C2×D13711yescharacteristic
256#121C2×D13711yescharacteristic, complemented
257#120—2741no—
258C4×D4—2741no—
259C4×D4—2741no—
260#117—1371no—
261C2×C4⋊C4—1371no—
262C2×C22⋊C4—2741no—
263#107—1371no—
264#107—1371no—
265#106—1371no—
266#106C2×D13711yescharacteristic, complemented
267#103—1371no—
268#103—1371no—
269#102—1371no—
270#102—1371no—
271#102—1371no—
272#102—1371no—
273#102C2×D13711yescharacteristic, complemented
274#102C2×D13711yescharacteristic, complemented
275C24—2741no—
276C24—1371no—
277#41—2742no—
278#41—2742no—
279C2×Q8—1371no—
280C2×D4—21no—
281C2×D4—21no—
282C2×D4—2741no—
283C2×D4—2741no—
284C2×D4—2741no—
285C2×D4—2741no—
286C2×D4—2741no—
287C2×D4—2742no—
288C2×D4—2742no—
289C2×D4—2741no—
290C2×D4—5484no—
291C2×D4—1371no—
292C2×D4C22×D13711yescharacteristic
293C2×D4C22×D13711yescharacteristic
294C22×C4—2742no—
295C22×C4—2742no—
296C22×C4—2741no—
297C22×C4—2741no—
298C22×C4—1371no—
299C22×C4—1371no—
300C22×C4—1371no—
301C22×C4—1371no—
302C22×C4—1371no—
303C22×C4C22×D13711yescharacteristic
304SD16—2742no—
305SD16—2742no—
306D8—22no—
307D8—2742no—
308M16—22no—
309M16—2742no—
310M16—2742no—
311M16—1374no—
312C2×C8—2742no—
313C2×C8—2742no—
314C2×C8—1371no—
315C2×C8—1371no—
316C2×C8C22×D13711yescharacteristic
317C2×C8C22×D13711yescharacteristic
318C4⋊C4—2741no—
319C4⋊C4—2741no—
320C4⋊C4—2741no—
321C4⋊C4—1371no—
322C4⋊C4—1371no—
323C4⋊C4C22×D13711yescharacteristic
324C4⋊C4C22×D13711yescharacteristic
325C22⋊C4—21no—
326C22⋊C4—21no—
327C22⋊C4—2741no—
328C22⋊C4—2741no—
329C22⋊C4—2741no—
330C22⋊C4—2741no—
331C22⋊C4—2741no—
332C22⋊C4—2741no—
333C22⋊C4—2741no—
334C42—2741no—
335C23—21no—
336C23—21no—
337C23—2742no—
338C23—2742no—
339C23—2741no—
340C23—5481no—
341C23—5482no—
342C23—5482no—
343C23—5482no—
344C23—1371no—
345C23—1371no—
346C23D4×D13711yescharacteristic
347Q8—2741no—
348Q8—2741no—
349D4—21no—
350D4—21no—
351D4—21no—
352D4—21no—
353D4—42no—
354D4—42no—
355D4—2741no—
356D4—2741no—
357D4—5482no—
358D4—5482no—
359D4—5482no—
360D4—5482no—
361D4—5482no—
362D4—5482no—
363C2×C4—21no—
364C2×C4—21no—
365C2×C4—21no—
366C2×C4—21no—
367C2×C4—2742no—
368C2×C4—2742no—
369C2×C4—2742no—
370C2×C4—2741no—
371C2×C4—2741no—
372C2×C4—2741no—
373C2×C4—2741no—
374C2×C4—2741no—
375C2×C4—5482no—
376C2×C4—5482no—
377C2×C4—1371no—
378C2×C4—1371no—
379C2×C4—1372no—
380C2×C4—1372no—
381C2×C4D4×D13711yescharacteristic
382C2×C4C23×D13711yescharacteristic
383C8—21no—
384C8—2741no—
385C8—1372no—
386C8D4×D13712yescomplemented
387C22—21no—
388C22—21no—
389C22—21no—
390C22—41no—
391C22—41no—
392C22—41no—
393C22—41no—
394C22—41no—
395C22—41no—
396C22—2741no—
397C22—2741no—
398C22—2741no—
399C22—2741no—
400C22—5484no—
401C22—5481no—
402C22—5481no—
403C22—5481no—
404C22—10964no—
405C22—1372no—
406C22—1372no—
407C22C2×D4×D13711yescharacteristic
408C4—21no—
409C4—41no—
410C4—41no—
411C4—2741no—
412C4—5481no—
413C4—5481no—
414C4—5481no—
415C4—1372no—
416C4C2×D4×D13711yescharacteristic
417C4#41×D13711yescharacteristic
418C2—41no—
419C2—81no—
420C2—81no—
421C2—2742no—
422C2—2742no—
423C2—10961no—
424C2#121×D13711yescharacteristic
425C2AGL(1,ℤ/8)×D13711yescharacteristic
426C2#2257819911yescharacteristic
subgroupG/H
centerC22C2×D4×D137
commutator, cosocleC2×C4×C137C24
FrattiniC2×C4C23×D137
Fitting#469×C137C2
socleC22×C137C22×D4
hypercenter#469D137
hypocenterC137#3055
LCSG ▷ C2×C4×C137 ▷ C2×C137 ▷ C137
UCSC1 ◁ C22 ◁ C22×C4 ◁ #469

Complements

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

NH ≅ G/Ntype
1. C2×#17984112C2semidirect
2. C2×M16×D137C2semidirect
3. #22577913C2semidirect
4. #121×D137C2semidirect
5. #22577654C2semidirect
6. #22577649C2semidirect
7. #22577647C2semidirect
8. #22577638C2semidirect
9. #22577636C2semidirect
10. #22577633C2semidirect
11. #22577631C2semidirect
12. #22577442C2semidirect
13. #22577406C2semidirect
14. #22577401C2semidirect
15. #469×C137C2semidirect
21. C2×#14555499C22semidirect
22. C2×#14555498C22semidirect
23. C2×#14555495C22semidirect
24. C2×#14555494C22semidirect
28. C2×C8×D137C22semidirect
29. C2×C8×D137C22semidirect
31. #17984068C22semidirect
32. #17984067C22semidirect
38. #17984059C22semidirect
45. #17984022C22semidirect
46. #17984021C22semidirect
47. #17984019C22semidirect
48. #17984013C22semidirect
49. #17984011C22semidirect
50. #17984010C22semidirect
51. #17984009C22semidirect
52. #17984009C22semidirect
53. #17984008C22semidirect
54. C2×D8×C137C22semidirect
55. C2×M16×C137C22semidirect
57. #121×C137C22semidirect
58. #106×C137C22semidirect
59. #102×C137C22semidirect
60. #102×C137C22semidirect
90. #14555493D4semidirect
155. #11764040C2×D4semidirect
172. C8×C137C2×D4semidirect
193. C2×D137AGL(1,ℤ/8)semidirect
213. C137#3055semidirect
220. #469D137semidirect
238. C2×D8C2×D137semidirect
243. C2×M16C2×D137semidirect
256. #121C2×D137semidirect
266. #106C2×D137semidirect
273. #102C2×D137semidirect
274. #102C2×D137semidirect
386. C8D4×D137semidirect

Maximal subgroups

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

subgroupindex
C2×#179841122
C2×M16×D1372
#225779132
#121×D1372
#225776542
#225776492
#225776472
#225776382
#225776362
#225776332
#225776312
#225774422
#225774062
#225774012
#469×C1372
#3055137