Finite Group Databasebeta

← #31792808

Subgroups — #31792808

Order 17536.

450 conjugacy classes of subgroups.

The subgroup lattice has 392 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#22578088C211yesmaximal, characteristic, complemented
2#22578037C211yesmaximal, characteristic
3#22577766C211yesmaximal, characteristic, complemented
4#22577730C211yesmaximal, characteristic, complemented
5#22577706C211yesmaximal, characteristic, complemented
6#22577650C211yesmaximal, characteristic, complemented
7#22577630C211yesmaximal, characteristic, complemented
8#22577602C211yesmaximal, characteristic, complemented
9#22577556C211yesmaximal, characteristic, complemented
10#22577555C211yesmaximal, characteristic, complemented
11#22577502C211yesmaximal, characteristic, complemented
12#22577471C211yesmaximal, characteristic, complemented
13#22577406C211yesmaximal, characteristic, complemented
14C2×#17984008C211yesmaximal, characteristic, complemented
15#480×C137C211yesmaximal, characteristic, complemented
16C22×D548C2211yescharacteristic
17C2×D4×D137—21no—
18C2×D4×D137C2211yescharacteristic
19C2×#14555499C2211yescharacteristic
20C2×#14555499C2211yescharacteristic, complemented
21C2×#14555498C2211yescharacteristic, complemented
22C2×#14555493C2211yescharacteristic, complemented
23C22×#11764040C2211yescharacteristic, complemented
24#17984084C2211yescharacteristic
25#17984083—21no—
26#17984079C2211yescharacteristic, complemented
27#17984078—21no—
28#17984071—21no—
29#17984068C2211yescharacteristic, complemented
30#17984067C2211yescharacteristic, complemented
31#17984063C2211yescharacteristic
32#17984061—21no—
33#17984058—21no—
34#17984056—21no—
35#17984053C2211yescharacteristic, complemented
36#17984050C2211yescharacteristic
37C4×C22⋊D137C2211yescharacteristic
38#17984047—21no—
39#17984046C2211yescharacteristic, complemented
40#17984019C2211yescharacteristic, complemented
41#17984017C2211yescharacteristic, complemented
42#17984011C2211yescharacteristic, complemented
43#17984010C2211yescharacteristic
44#17984010C2211yescharacteristic, complemented
45#17984009C2211yescharacteristic, complemented
46#17984008C2211yescharacteristic
47#17984008C2211yescharacteristic
48#17984008C2211yescharacteristic, complemented
49#17984008C2211yescharacteristic, complemented
50#17984000C2211yescharacteristic, complemented
51#17983998C2211yescharacteristic, complemented
52#121×C137C2211yescharacteristic
53C2×C4⋊C4×C137C2211yescharacteristic, complemented
54#107×C137C2211yescharacteristic, complemented
55#107×C137C2211yescharacteristic, complemented
56#102×C137C2211yescharacteristic, complemented
57#102×C137C2211yescharacteristic, complemented
58C22⋊C8×C137C2211yescharacteristic, complemented
59C23×D137—21no—
60C23×D137—21no—
61C2×Dic274C2311yescharacteristic
62C2×D548—22no—
63C2×D548—21no—
64C2×D548—21no—
65C2×D548—41no—
66C2×D548C2311yescharacteristic
67C2×D548C2311yescharacteristic
68C2×C22⋊D137—21no—
69C2×C22⋊D137—21no—
70C2×C22⋊D137—21no—
71C2×C22⋊D137D411yescharacteristic
72D4×D137—22no—
73D4×D137—41no—
74D4×D137—44no—
75C2×C4×D137—21no—
76C2×C4×D137C2311yescharacteristic
77#14555499—22no—
78#14555499—41no—
79#14555498—41no—
80#14555493—41no—
81C2×#11764040—41no—
82C2×#11764040C2311yescharacteristic
83C2×#11764040C2311yescharacteristic
84C4⋊Dic137—21no—
85C4⋊Dic137—21no—
86C4⋊Dic137C2311yescharacteristic
87#14555489—21no—
88#14555489—21no—
89#14555489—21no—
90#14555489D411yescharacteristic
91C22⋊Dic137—21no—
92C22⋊Dic137D411yescharacteristic
93#14555487—21no—
94#14555487—21no—
95#14555487—21no—
96#14555487—21no—
97#14555487D411yescharacteristic
98C4×Dic137—21no—
99C4×Dic137—21no—
100C4×Dic137C2311yescharacteristic
101C2×D4×C137—21no—
102C2×D4×C137C2311yescharacteristic
103C22×C4×C137—21no—
104C22×C4×C137C2311yescharacteristic
105C2×C8×C137C2311yescharacteristic
106C2×C8×C137C2311yescharacteristic
107C4⋊C4×C137—21no—
108C4⋊C4×C137C2311yescharacteristic
109C4⋊C4×C137C2311yescharacteristic
110C4⋊C4×C137C2311yescharacteristic
111C22⋊C4×C137—21no—
112C22×D137—21no—
113C22×D137—21no—
114C22×D137—42no—
115C22×D137—42no—
116C22×D137—42no—
117C22×D137—42no—
118C22×D137—81no—
119C22×D137—81no—
120C22×D137C2×D411yescharacteristic
121Dic274—21no—
122Dic274—21no—
123D548—21no—
124D548—21no—
125D548—21no—
126D548—42no—
127D548—42no—
128D548#4112yes—
129C22⋊D137—22no—
130C22⋊D137—42no—
131C22⋊D137—42no—
132C22⋊D137—81no—
133C2×Dic137—21no—
134C2×Dic137—21no—
135C2×Dic137C2×D411yescharacteristic
136C2×Dic137C2×D411yescharacteristic
137C2×Dic137C2×D411yescharacteristic
138C4×D137—21no—
139C4×D137—21no—
140C4×D137—42no—
141#11764040—21no—
142#11764040—21no—
143C23×C137—21no—
144C23×C137C2×D411yescharacteristic
145D4×C137—21no—
146D4×C137—21no—
147D4×C137—42no—
148C2×C4×C137—21no—
149C2×C4×C137—21no—
150C2×C4×C137—21no—
151C2×C4×C137—21no—
152C2×C4×C137—21no—
153C2×C4×C137—42no—
154C2×C4×C137C2×D411yescharacteristic
155C2×C4×C137C2411yescharacteristic
156C8×C137—21no—
157C8×C137—21no—
158C22×C137—21no—
159C22×C137—21no—
160C22×C137—41no—
161C22×C137—41no—
162C22×C137—41no—
163C22×C137AGL(1,ℤ/8)12yes—
164C22×C137C22×D411yescharacteristic
165C2×D137—21no—
166C2×D137—22no—
167C2×D137—21no—
168C2×D137—21no—
169C2×D137—41no—
170C2×D137—41no—
171C2×D137—41no—
172C2×D137—41no—
173C2×D137—41no—
174C2×D137—84no—
175C2×D137—84no—
176C4×C137—21no—
177C4×C137—41no—
178C4×C137—41no—
179C4×C137—41no—
180C4×C137C2×#4111yescharacteristic
181C4×C137H(5,𝔽2)11yescharacteristic
182Dic137—21no—
183Dic137—21no—
184Dic137—21no—
185Dic137—41no—
186Dic137—41no—
187C2×C137—22no—
188C2×C137—81no—
189C2×C137#54611yescharacteristic
190C2×C137C2×AGL(1,ℤ/8)11yescharacteristic
191C2×C137#57711yescharacteristic
192D137—41no—
193D137—42no—
194D137—81no—
195C137#331011yescharacteristic, complemented, Sylow
196#3310—1371nomaximal, Sylow
197#552—1371no—
198D42—1371no—
199#490—1371no—
200#482—1371no—
201#480D13711yescharacteristic, complemented
202#478—1371no—
203#468—1371no—
204#465—1371no—
205#460—1371no—
206#450—1371no—
207#450—1371no—
208#440—1371no—
209#435—1371no—
210#418—1371no—
211C2×#102—1371no—
212C22×D4—2741no—
213C22×D4—1371no—
214C22×D4—1371no—
215C2×SD16—1371no—
216C2×SD16—1371no—
217C2×SD16—1371no—
218C2×M16—1371no—
219C22×C8—1371no—
220#127—1371no—
221#126—2741no—
222#125—2741no—
223#125—1371no—
224#123—2741no—
225C22⋊Q8—1371no—
226C22⋊Q8—1371no—
227#121—2741no—
228#121—1371no—
229#121C2×D13711yescharacteristic
230#120—2741no—
231#120—2741no—
232C4×Q8—1371no—
233C4×D4—1371no—
234C4×D4—1371no—
235#117—2741no—
236#117—1371no—
237C2×C4⋊C4C2×D13711yescharacteristic, complemented
238#107C2×D13711yescharacteristic, complemented
239#107C2×D13711yescharacteristic, complemented
240#106—1371no—
241C4⋊C8—1371no—
242#103—1371no—
243#102—1371no—
244#102—1371no—
245#102—1371no—
246#102—1371no—
247#102—1371no—
248#102—1371no—
249#102—1371no—
250#102C2×D13711yescharacteristic, complemented
251#102C2×D13711yescharacteristic, complemented
252C22⋊C8—1371no—
253C22⋊C8C2×D13711yescharacteristic, complemented
254#97—1371no—
255C24—2741no—
256C24—2741no—
257C2×Q8—1371no—
258C2×D4—21no—
259C2×D4—2742no—
260C2×D4—2741no—
261C2×D4—2742no—
262C2×D4—2741no—
263C2×D4—2741no—
264C2×D4—2741no—
265C2×D4—2741no—
266C2×D4—5481no—
267C2×D4—5481no—
268C2×D4—5484no—
269C2×D4—1371no—
270C2×D4—1371no—
271C2×D4—1371no—
272C2×D4C22×D13711yescharacteristic
273C22×C4—21no—
274C22×C4—2741no—
275C22×C4—1371no—
276C22×C4C22×D13711yescharacteristic
277SD16—2742no—
278SD16—5481no—
279SD16—5481no—
280M16—5481no—
281C2×C8—5481no—
282C2×C8—1371no—
283C2×C8—1371no—
284C2×C8C22×D13711yescharacteristic
285C2×C8C22×D13711yescharacteristic
286C4⋊C4—21no—
287C4⋊C4—2741no—
288C4⋊C4—2741no—
289C4⋊C4—2741no—
290C4⋊C4—2741no—
291C4⋊C4—2741no—
292C4⋊C4—1371no—
293C4⋊C4—1371no—
294C4⋊C4C22×D13711yescharacteristic
295C4⋊C4C22×D13711yescharacteristic
296C4⋊C4C22×D13711yescharacteristic
297C22⋊C4—21no—
298C22⋊C4—2741no—
299C22⋊C4—2741no—
300C22⋊C4—2741no—
301C22⋊C4—2741no—
302C22⋊C4—2741no—
303C22⋊C4—1371no—
304C22⋊C4—1371no—
305C42—2741no—
306C42—2741no—
307C42—1371no—
308C23—21no—
309C23—2741no—
310C23—2741no—
311C23—5482no—
312C23—5482no—
313C23—5482no—
314C23—5482no—
315C23—10961no—
316C23—10961no—
317C23—1371no—
318C23D4×D13711yescharacteristic
319Q8—2741no—
320Q8—2741no—
321D4—21no—
322D4—21no—
323D4—42no—
324D4—2742no—
325D4—2741no—
326D4—2741no—
327D4—2741no—
328D4—5482no—
329D4—5482no—
330D4—5482no—
331D4—5482no—
332D4—10961no—
333D4—1372no—
334C2×C4—21no—
335C2×C4—21no—
336C2×C4—21no—
337C2×C4—21no—
338C2×C4—21no—
339C2×C4—42no—
340C2×C4—2741no—
341C2×C4—2741no—
342C2×C4—2741no—
343C2×C4—2741no—
344C2×C4—5482no—
345C2×C4—1371no—
346C2×C4—1371no—
347C2×C4—1371no—
348C2×C4D4×D13711yescharacteristic
349C2×C4C23×D13711yescharacteristic
350C8—21no—
351C8—21no—
352C8—2741no—
353C8—2741no—
354C22—21no—
355C22—21no—
356C22—41no—
357C22—41no—
358C22—41no—
359C22—2741no—
360C22—2741no—
361C22—2741no—
362C22—2742no—
363C22—5481no—
364C22—5481no—
365C22—5481no—
366C22—5481no—
367C22—5481no—
368C22—10964no—
369C22—10964no—
370C22#1798411212yes—
371C22C2×D4×D13711yescharacteristic
372C4—21no—
373C4—41no—
374C4—41no—
375C4—41no—
376C4—2741no—
377C4—2741no—
378C4—2741no—
379C4—5481no—
380C4—5481no—
381C4#41×D13711yescharacteristic
382C4#1798413411yescharacteristic
383C2—22no—
384C2—81no—
385C2—5481no—
386C2—5482no—
387C2—10961no—
388C2#2257804511yescharacteristic
389C2C2×#1798411211yescharacteristic
390C2#2257821211yescharacteristic
subgroupG/H
centerC22C2×D4×D137
commutator, cosocleC2×C4×C137C24
FrattiniC2×C4C23×D137
Fitting#480×C137C2
socleC22×C137C22×D4
prime radical#22578037C2
hypercenter#480D137
hypocenterC137#3310
LCSG ▷ C2×C4×C137 ▷ C2×C137 ▷ C137
UCSC1 ◁ C22 ◁ C22×C4 ◁ #480

Complements

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

NH ≅ G/Ntype
1. #22578088C2semidirect
3. #22577766C2semidirect
4. #22577730C2semidirect
5. #22577706C2semidirect
6. #22577650C2semidirect
7. #22577630C2semidirect
8. #22577602C2semidirect
9. #22577556C2semidirect
10. #22577555C2semidirect
11. #22577502C2semidirect
12. #22577471C2semidirect
13. #22577406C2semidirect
14. C2×#17984008C2semidirect
15. #480×C137C2semidirect
20. C2×#14555499C22semidirect
21. C2×#14555498C22semidirect
22. C2×#14555493C22semidirect
23. C22×#11764040C22semidirect
26. #17984079C22semidirect
29. #17984068C22semidirect
30. #17984067C22semidirect
35. #17984053C22semidirect
39. #17984046C22semidirect
40. #17984019C22semidirect
41. #17984017C22semidirect
42. #17984011C22semidirect
44. #17984010C22semidirect
45. #17984009C22semidirect
48. #17984008C22semidirect
49. #17984008C22semidirect
50. #17984000C22semidirect
51. #17983998C22semidirect
53. C2×C4⋊C4×C137C22semidirect
54. #107×C137C22semidirect
55. #107×C137C22semidirect
56. #102×C137C22semidirect
57. #102×C137C22semidirect
58. C22⋊C8×C137C22semidirect
195. C137#3310semidirect
201. #480D137semidirect
237. C2×C4⋊C4C2×D137semidirect
238. #107C2×D137semidirect
239. #107C2×D137semidirect
250. #102C2×D137semidirect
251. #102C2×D137semidirect
253. C22⋊C8C2×D137semidirect

Maximal subgroups

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

subgroupindex
#225780882
#225780372
#225777662
#225777302
#225777062
#225776502
#225776302
#225776022
#225775562
#225775552
#225775022
#225774712
#225774062
C2×#179840082
#480×C1372
#3310137