Finite Group Databasebeta

← #31790740

Subgroups — #31790740

Order 17536.

418 conjugacy classes of subgroups.

The subgroup lattice has 378 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#22577969C211yesmaximal, characteristic
2#22577884C211yesmaximal, characteristic, complemented
3#22577730C211yesmaximal, characteristic, complemented
4#22577728C211yesmaximal, characteristic, complemented
5#22577725C211yesmaximal, characteristic, complemented
6#22577724C211yesmaximal, characteristic, complemented
7#22577721C211yesmaximal, characteristic, complemented
8#22577720C211yesmaximal, characteristic, complemented
9#22577718C211yesmaximal, characteristic, complemented
10#22577444C211yesmaximal, characteristic, complemented
11#22577440C211yesmaximal, characteristic, complemented
12#22577406C211yesmaximal, characteristic, complemented
13#102×D137C211yesmaximal, characteristic, complemented
14#22577368C211yesmaximal, characteristic, complemented
15#480×C137C211yesmaximal, characteristic, complemented
16C2×#14555510C2211yescharacteristic
17C2×D4×D137C2211yescharacteristic
18C2×#14555493C2211yescharacteristic
19C2×C8×D137C2211yescharacteristic
20#17984084—21no—
21#17984074—21no—
22#17984063C2211yescharacteristic
23#17984062C2211yescharacteristic
24#17984059C2211yescharacteristic
25#17984058—21no—
26#17984056—21no—
27#17984053—21no—
28#17984050—21no—
29C4×C22⋊D137—21no—
30C4×C22⋊D137—21no—
31#17984047C2211yescharacteristic, complemented
32#17984046C2211yescharacteristic
33#17984046C2211yescharacteristic
34#17984046C2211yescharacteristic, complemented
35C4⋊C4×D137C2211yescharacteristic
36C4⋊C4×D137C2211yescharacteristic, complemented
37#17984022C2211yescharacteristic, complemented
38#17984021C2211yescharacteristic, complemented
39#17984020C2211yescharacteristic, complemented
40#17984019C2211yescharacteristic, complemented
41#17984019C2211yescharacteristic, complemented
42#17984019C2211yescharacteristic, complemented
43#17984010C2211yescharacteristic
44#17984010C2211yescharacteristic
45#17984009C2211yescharacteristic
46#17984009C2211yescharacteristic
47#17984008C2211yescharacteristic
48#17984008C2211yescharacteristic
49C22⋊#11764040C2211yescharacteristic, complemented
50#17984000C2211yescharacteristic, complemented
51#17984000C2211yescharacteristic, complemented
52#121×C137C2211yescharacteristic
53C2×C4⋊C4×C137C2211yescharacteristic, complemented
54#107×C137C2211yescharacteristic, complemented
55#107×C137C2211yescharacteristic, complemented
56#102×C137C2211yescharacteristic
57#102×C137C2211yescharacteristic
58C22⋊C8×C137C2211yescharacteristic, complemented
59C23×D137—21no—
60#14555510—22no—
61#14555510—22no—
62C2×Dic274D411yescharacteristic
63C2×D548—21no—
64C2×D548D411yescharacteristic
65C2×D548C2311yescharacteristic
66C2×C22⋊D137—21no—
67C2×C22⋊D137—21no—
68C2×C22⋊D137—21no—
69C2×C22⋊D137D411yescharacteristic
70C2×C22⋊D137D411yescharacteristic
71D4×D137—22no—
72D4×D137—22no—
73C2×C4×D137—21no—
74C2×C4×D137—21no—
75C2×C4×D137C2311yescharacteristic
76C2×C4×D137C2311yescharacteristic
77#14555493—22no—
78C2×#11764040C2311yescharacteristic
79C2×#11764040C2311yescharacteristic
80C8×D137—22no—
81C4⋊Dic137C2311yescharacteristic
82C4⋊Dic137C2311yescharacteristic
83C4⋊Dic137C2311yescharacteristic
84#14555489—21no—
85#14555489—21no—
86C22⋊Dic137—21no—
87C22⋊Dic137—21no—
88#14555487—21no—
89#14555487—21no—
90#14555487—21no—
91#14555487—21no—
92#14555487—21no—
93C4×Dic137—21no—
94C4×Dic137—21no—
95C4×Dic137—21no—
96C4×Dic137—21no—
97C2×D4×C137—21no—
98C2×D4×C137C2311yescharacteristic
99C22×C4×C137—21no—
100C22×C4×C137C2311yescharacteristic
101C2×C8×C137C2311yescharacteristic
102C2×C8×C137C2311yescharacteristic
103C4⋊C4×C137—21no—
104C4⋊C4×C137C2311yescharacteristic
105C4⋊C4×C137C2311yescharacteristic
106C4⋊C4×C137C2311yescharacteristic
107C22⋊C4×C137—21no—
108C22×D137—21no—
109C22×D137—42no—
110C22×D137—81no—
111C22×D137—81no—
112C22×D137C2×D411yescharacteristic
113C22×D137C2×D411yescharacteristic
114Dic274—22no—
115D548—22no—
116D548—21no—
117D548—21no—
118D548—81no—
119C22⋊D137—22no—
120C22⋊D137—22no—
121C22⋊D137—42no—
122C22⋊D137—81no—
123C22⋊D137—81no—
124C2×Dic137—21no—
125C2×Dic137—21no—
126C2×Dic137—21no—
127C2×Dic137C2×D411yescharacteristic
128C2×Dic137C2×D411yescharacteristic
129C4×D137—21no—
130C4×D137—21no—
131C4×D137—81no—
132C4×D137—81no—
133C4×D137#4112yes—
134C4×D137#4112yes—
135#11764040—21no—
136#11764040—21no—
137C23×C137—21no—
138C23×C137C2×D411yescharacteristic
139D4×C137—21no—
140D4×C137—21no—
141D4×C137—81no—
142C2×C4×C137—21no—
143C2×C4×C137—21no—
144C2×C4×C137—21no—
145C2×C4×C137—21no—
146C2×C4×C137—21no—
147C2×C4×C137—81no—
148C2×C4×C137C2×D411yescharacteristic
149C2×C4×C137C2411yescharacteristic
150C8×C137—21no—
151C8×C137—21no—
152C22×C137—21no—
153C22×C137—21no—
154C22×C137—21no—
155C22×C137—41no—
156C22×C137—41no—
157C22×C137—41no—
158C22×C137C22×D411yescharacteristic
159C2×D137—21no—
160C2×D137—21no—
161C2×D137—21no—
162C2×D137—21no—
163C2×D137—21no—
164C2×D137—41no—
165C2×D137—41no—
166C2×D137—41no—
167C2×D137—84no—
168C2×D137#12312yes—
169C4×C137—21no—
170C4×C137—41no—
171C4×C137—41no—
172C4×C137—41no—
173C4×C137C2×#4111yescharacteristic
174C4×C137C2×#4111yescharacteristic
175Dic137—21no—
176Dic137—41no—
177Dic137—41no—
178Dic137—41no—
179Dic137#12312yes—
180C2×C137—41no—
181C2×C137—81no—
182C2×C137C2×#12311yescharacteristic
183C2×C137#57611yescharacteristic
184C2×C137#57711yescharacteristic
185D137—22no—
186D137—41no—
187D137—81no—
188C137#309611yescharacteristic, complemented, Sylow
189#3096—1371nomaximal, Sylow
190#535—1371no—
191#520—1371no—
192#482—1371no—
193#482—1371no—
194#482—1371no—
195#482—1371no—
196#481—1371no—
197#481—1371no—
198#480—1371no—
199#480D13711yescharacteristic, complemented
200#428—1371no—
201#427—1371no—
202#418—1371no—
203C2×#102—1371no—
204#408—1371no—
205C2×#41—1371no—
206C22×D4—1371no—
207C2×M16—1371no—
208C22×C8—1371no—
209#127—2741no—
210#124—2741no—
211#121—1371no—
212#121—1371no—
213#121—1371no—
214#121C2×D13711yescharacteristic
215#120—2741no—
216#120—2741no—
217C4×Q8—2741no—
218C4×D4—2741no—
219C4×D4—2741no—
220C4×D4—2741no—
221#117—1371no—
222#117—1371no—
223#117—1371no—
224#117—1371no—
225C2×C4⋊C4—1371no—
226C2×C4⋊C4—1371no—
227C2×C4⋊C4C2×D13711yescharacteristic, complemented
228#107—1371no—
229#107—1371no—
230#107C2×D13711yescharacteristic, complemented
231#107C2×D13711yescharacteristic, complemented
232#106—1371no—
233#106—1371no—
234#106—1371no—
235#106—1371no—
236#102—1371no—
237#102—1371no—
238#102—1371no—
239#102—1371no—
240#102—1371no—
241#102—1371no—
242#102C2×D13711yescharacteristic
243#102C2×D13711yescharacteristic
244C22⋊C8—1371no—
245C22⋊C8—1371no—
246C22⋊C8—1371no—
247C22⋊C8C2×D13711yescharacteristic, complemented
248C24—2741no—
249#41—2742no—
250#41—2742no—
251C2×Q8—1371no—
252C2×D4—21no—
253C2×D4—2741no—
254C2×D4—2741no—
255C2×D4—2741no—
256C2×D4—2742no—
257C2×D4—2742no—
258C2×D4—2741no—
259C2×D4—1371no—
260C2×D4—1371no—
261C2×D4—1371no—
262C2×D4—1371no—
263C2×D4C22×D13711yescharacteristic
264C22×C4—21no—
265C22×C4—2741no—
266C22×C4—2741no—
267C22×C4—1371no—
268C22×C4—1371no—
269C22×C4C22×D13711yescharacteristic
270M16—2742no—
271C2×C8—2742no—
272C2×C8—1371no—
273C2×C8—1371no—
274C2×C8C22×D13711yescharacteristic
275C2×C8C22×D13711yescharacteristic
276C4⋊C4—21no—
277C4⋊C4—2741no—
278C4⋊C4—2741no—
279C4⋊C4—1371no—
280C4⋊C4—1371no—
281C4⋊C4—1371no—
282C4⋊C4C22×D13711yescharacteristic
283C4⋊C4C22×D13711yescharacteristic
284C4⋊C4C22×D13711yescharacteristic
285C22⋊C4—21no—
286C22⋊C4—2741no—
287C22⋊C4—2741no—
288C22⋊C4—2741no—
289C22⋊C4—2741no—
290C22⋊C4—2741no—
291C22⋊C4—2741no—
292C22⋊C4—2741no—
293C42—2741no—
294C42—2741no—
295C42—2741no—
296C42—2741no—
297C23—21no—
298C23—2741no—
299C23—5482no—
300C23—10961no—
301C23—10961no—
302C23—1371no—
303C23—1371no—
304C23D4×D13711yescharacteristic
305Q8—2742no—
306D4—21no—
307D4—21no—
308D4—81no—
309D4—2742no—
310D4—2742no—
311D4—2742no—
312D4—2741no—
313D4—2741no—
314D4—5482no—
315D4—10961no—
316D4—10961no—
317D4—10961no—
318C2×C4—21no—
319C2×C4—21no—
320C2×C4—21no—
321C2×C4—21no—
322C2×C4—21no—
323C2×C4—81no—
324C2×C4—2741no—
325C2×C4—2741no—
326C2×C4—2741no—
327C2×C4—2741no—
328C2×C4—2741no—
329C2×C4—10961no—
330C2×C4—10961no—
331C2×C4—1371no—
332C2×C4—1371no—
333C2×C4—1372no—
334C2×C4—1372no—
335C2×C4D4×D13711yescharacteristic
336C2×C4C23×D13711yescharacteristic
337C8—21no—
338C8—21no—
339C8—2741no—
340C8—2741no—
341C22—21no—
342C22—21no—
343C22—21no—
344C22—41no—
345C22—41no—
346C22—41no—
347C22—2741no—
348C22—2741no—
349C22—2741no—
350C22—2741no—
351C22—2741no—
352C22—5481no—
353C22—5481no—
354C22—5481no—
355C22—10964no—
356C22—1372no—
357C22C2×D4×D13711yescharacteristic
358C4—21no—
359C4—41no—
360C4—41no—
361C4—41no—
362C4—2741no—
363C4—5481no—
364C4—5481no—
365C4—5481no—
366C4—1372no—
367C4#41×D13711yescharacteristic
368C4#41×D13711yescharacteristic
369C2—41no—
370C2—81no—
371C2—2742no—
372C2—5481no—
373C2—10961no—
374C2#123×D13711yescharacteristic
375C2#2257820411yescharacteristic
376C2#2257821211yescharacteristic
subgroupG/H
centerC22C2×D4×D137
commutator, cosocleC2×C4×C137C24
FrattiniC2×C4C23×D137
Fitting#480×C137C2
socleC22×C137C22×D4
prime radical#22577969C2
hypercenter#480D137
hypocenterC137#3096
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
2. #22577884C2semidirect
3. #22577730C2semidirect
4. #22577728C2semidirect
5. #22577725C2semidirect
6. #22577724C2semidirect
7. #22577721C2semidirect
8. #22577720C2semidirect
9. #22577718C2semidirect
10. #22577444C2semidirect
11. #22577440C2semidirect
12. #22577406C2semidirect
13. #102×D137C2semidirect
14. #22577368C2semidirect
15. #480×C137C2semidirect
31. #17984047C22semidirect
34. #17984046C22semidirect
36. C4⋊C4×D137C22semidirect
37. #17984022C22semidirect
38. #17984021C22semidirect
39. #17984020C22semidirect
40. #17984019C22semidirect
41. #17984019C22semidirect
42. #17984019C22semidirect
49. C22⋊#11764040C22semidirect
50. #17984000C22semidirect
51. #17984000C22semidirect
53. C2×C4⋊C4×C137C22semidirect
54. #107×C137C22semidirect
55. #107×C137C22semidirect
58. C22⋊C8×C137C22semidirect
188. C137#3096semidirect
199. #480D137semidirect
227. C2×C4⋊C4C2×D137semidirect
230. #107C2×D137semidirect
231. #107C2×D137semidirect
247. C22⋊C8C2×D137semidirect

Maximal subgroups

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

subgroupindex
#225779692
#225778842
#225777302
#225777282
#225777252
#225777242
#225777212
#225777202
#225777182
#225774442
#225774402
#225774062
#102×D1372
#225773682
#480×C1372
#3096137