Finite Group Databasebeta

← #31792824

Subgroups — #31792824

Order 17536.

402 conjugacy classes of subgroups.

The subgroup lattice has 372 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#22578127C211yesmaximal, characteristic, complemented
2#22578046C211yesmaximal, characteristic
3#22577743C211yesmaximal, characteristic, complemented
4#22577725C211yesmaximal, characteristic, complemented
5#22577690C211yesmaximal, characteristic, complemented
6#22577660C211yesmaximal, characteristic, complemented
7#22577631C211yesmaximal, characteristic, complemented
8#22577622C211yesmaximal, characteristic, complemented
9#22577568C211yesmaximal, characteristic, complemented
10#22577555C211yesmaximal, characteristic, complemented
11#22577505C211yesmaximal, characteristic, complemented
12#22577476C211yesmaximal, characteristic, complemented
13#22577406C211yesmaximal, characteristic, complemented
14#22577398C211yesmaximal, characteristic, complemented
15#489×C137C211yesmaximal, characteristic, complemented
16C2×#14555511C2211yescharacteristic
17C2×D4×D137C2211yescharacteristic
18C2×#14555500C2211yescharacteristic, complemented
19C2×#14555499C2211yescharacteristic
20C2×#14555498C2211yescharacteristic
21C2×#14555493C2211yescharacteristic
22C2×C8×D137C2211yescharacteristic, complemented
23#17984083—21no—
24#17984083—21no—
25#17984077—21no—
26#17984070—21no—
27#17984068C2211yescharacteristic
28#17984068C2211yescharacteristic, complemented
29#17984064C2211yescharacteristic
30#17984061—21no—
31#17984059C2211yescharacteristic
32#17984058—21no—
33Q8×Dic137—21no—
34C4×Dic274C2211yescharacteristic, complemented
35C4×D548C2211yescharacteristic
36C4×D548C2211yescharacteristic
37#17984046C2211yescharacteristic
38#17984046C2211yescharacteristic, complemented
39C22⋊C4×D137—21no—
40#17984022C2211yescharacteristic, complemented
41#17984019C2211yescharacteristic, complemented
42#17984019C2211yescharacteristic, complemented
43C4⋊#11764040C2211yescharacteristic, complemented
44#17984013C2211yescharacteristic, complemented
45#17984012C2211yescharacteristic, complemented
46#17984010C2211yescharacteristic
47#17984009C2211yescharacteristic
48#17984008C2211yescharacteristic
49#17984008C2211yescharacteristic
50#17984000C2211yescharacteristic
51#17984000C2211yescharacteristic, complemented
52C4⋊Q8×C137C2211yescharacteristic, complemented
53#124×C137C2211yescharacteristic
54#103×C137C2211yescharacteristic, complemented
55#103×C137C2211yescharacteristic, complemented
56#102×C137C2211yescharacteristic
57#102×C137C2211yescharacteristic
58#97×C137C2211yescharacteristic, complemented
59C23×D137—21no—
60#14555511—22no—
61#14555511—41no—
62C2×Dic274C2311yescharacteristic
63C2×D548—21no—
64C2×D548D411yescharacteristic
65C2×D548C2311yescharacteristic
66C2×C22⋊D137—21no—
67C2×C22⋊D137—21no—
68D4×D137—22no—
69D4×D137—41no—
70C2×C4×D137—21no—
71C2×C4×D137—21no—
72C2×C4×D137C2311yescharacteristic
73C2×C4×D137C2311yescharacteristic
74#14555500—41no—
75#14555499—22no—
76#14555498—41no—
77#14555493—41no—
78C2×#11764040C2311yescharacteristic
79C2×#11764040C2311yescharacteristic
80C8×D137—41no—
81C4⋊Dic137—21no—
82C4⋊Dic137—21no—
83C4⋊Dic137D411yescharacteristic
84C4⋊Dic137C2311yescharacteristic
85C4⋊Dic137C2311yescharacteristic
86#14555489—21no—
87#14555489—21no—
88C22⋊Dic137—21no—
89#14555487—21no—
90#14555487—21no—
91#14555487—21no—
92#14555487—21no—
93#14555487—21no—
94#14555487—21no—
95#14555487D411yescharacteristic
96#14555487D411yescharacteristic
97C4×Dic137—21no—
98C4×Dic137—21no—
99C4×Dic137—21no—
100C2×Q8×C137—21no—
101C2×Q8×C137C2311yescharacteristic
102C2×D4×C137C2311yescharacteristic
103C2×C8×C137C2311yescharacteristic
104C2×C8×C137C2311yescharacteristic
105C4⋊C4×C137—21no—
106C4⋊C4×C137C2311yescharacteristic
107C4⋊C4×C137C2311yescharacteristic
108C22⋊C4×C137—21no—
109C22⋊C4×C137—21no—
110C42×C137C2311yescharacteristic
111C22×D137—21no—
112C22×D137—42no—
113C22×D137—42no—
114C22×D137—81no—
115C22×D137C2×D411yescharacteristic
116C22×D137C2×D411yescharacteristic
117Dic274—21no—
118Dic274—21no—
119D548—21no—
120D548—41no—
121D548—42no—
122D548#4112yes—
123C22⋊D137—42no—
124C22⋊D137—81no—
125C2×Dic137—21no—
126C2×Dic137—21no—
127C2×Dic137—21no—
128C2×Dic137C2×D411yescharacteristic
129C2×Dic137C2×D411yescharacteristic
130C4×D137—21no—
131C4×D137—21no—
132C4×D137—21no—
133C4×D137—21no—
134C4×D137—42no—
135C4×D137—81no—
136#11764040—21no—
137#11764040—21no—
138C23×C137—21no—
139Q8×C137—21no—
140Q8×C137—21no—
141Q8×C137—42no—
142D4×C137—21no—
143D4×C137—21no—
144C2×C4×C137—21no—
145C2×C4×C137—21no—
146C2×C4×C137—21no—
147C2×C4×C137C2×D411yescharacteristic
148C2×C4×C137C2×D411yescharacteristic
149C2×C4×C137C2411yescharacteristic
150C8×C137—21no—
151C8×C137—21no—
152C22×C137—41no—
153C22×C137—41no—
154C22×C137—41no—
155C22×C137C22×D411yescharacteristic
156C2×D137—21no—
157C2×D137—21no—
158C2×D137—22no—
159C2×D137—21no—
160C2×D137—21no—
161C2×D137—21no—
162C2×D137—21no—
163C2×D137—41no—
164C2×D137—41no—
165C2×D137—84no—
166C4×C137—21no—
167C4×C137—21no—
168C4×C137—41no—
169C4×C137—41no—
170C4×C137—41no—
171C4×C137C2×#4111yescharacteristic
172C4×C137H(5,𝔽2)11yescharacteristic
173Dic137—21no—
174Dic137—21no—
175Dic137—41no—
176Dic137—41no—
177Dic137—41no—
178C2×C137—81no—
179C2×C137#54611yescharacteristic
180C2×C137#57511yescharacteristic
181C2×C137#57711yescharacteristic
182D137—41no—
183D137—41no—
184D137—42no—
185C137#331111yescharacteristic, complemented, Sylow
186#3311—1371nomaximal, Sylow
187#556—1371no—
188#546—1371no—
189#489D13711yescharacteristic, complemented
190#485—1371no—
191#482—1371no—
192#476—1371no—
193#470—1371no—
194#465—1371no—
195#464—1371no—
196#452—1371no—
197#450—1371no—
198#440—1371no—
199#435—1371no—
200#418—1371no—
201#416—1371no—
202C2×#41—1371no—
203C22×D4—1371no—
204C2×Q16—1371no—
205C2×SD16—1371no—
206C2×SD16—1371no—
207C2×M16—1371no—
208C22×C8—1371no—
209C4⋊Q8C2×D13711yescharacteristic, complemented
210#126—2741no—
211#126—2741no—
212#124—2741no—
213#124C2×D13711yescharacteristic
214#123—2741no—
215C22⋊Q8—1371no—
216C22⋊Q8—1371no—
217C22⋊Q8—1371no—
218#121—2741no—
219#121—1371no—
220#120—2741no—
221C4×Q8—2741no—
222C4×Q8—1371no—
223C4×D4—1371no—
224C4×D4—1371no—
225#117—1371no—
226#117—1371no—
227C2×C22⋊C4—2741no—
228#107—1371no—
229#106—1371no—
230#106—1371no—
231C4⋊C8—1371no—
232#103—1371no—
233#103—1371no—
234#103C2×D13711yescharacteristic, complemented
235#103C2×D13711yescharacteristic, complemented
236#102—1371no—
237#102—1371no—
238#102—1371no—
239#102—1371no—
240#102C2×D13711yescharacteristic
241#102C2×D13711yescharacteristic
242C22⋊C8—1371no—
243C22⋊C8—1371no—
244#97C2×D13711yescharacteristic, complemented
245C24—2741no—
246#41—2742no—
247#41—5481no—
248C2×Q8—21no—
249C2×Q8—1371no—
250C2×Q8C22×D13711yescharacteristic
251C2×D4—2741no—
252C2×D4—2742no—
253C2×D4—2741no—
254C2×D4—2741no—
255C2×D4—5481no—
256C2×D4—1371no—
257C2×D4—1371no—
258C2×D4C22×D13711yescharacteristic
259C22×C4—2741no—
260C22×C4—2741no—
261C22×C4—1371no—
262C22×C4—1371no—
263Q16—5481no—
264SD16—2742no—
265SD16—5481no—
266M16—5481no—
267C2×C8—5481no—
268C2×C8—1371no—
269C2×C8—1371no—
270C2×C8C22×D13711yescharacteristic
271C2×C8C22×D13711yescharacteristic
272C4⋊C4—21no—
273C4⋊C4—2741no—
274C4⋊C4—2741no—
275C4⋊C4—2741no—
276C4⋊C4—2741no—
277C4⋊C4—1371no—
278C4⋊C4—1371no—
279C4⋊C4—1371no—
280C4⋊C4C22×D13711yescharacteristic
281C4⋊C4C22×D13711yescharacteristic
282C22⋊C4—21no—
283C22⋊C4—21no—
284C22⋊C4—2741no—
285C22⋊C4—2741no—
286C22⋊C4—2741no—
287C22⋊C4—2741no—
288C22⋊C4—2741no—
289C22⋊C4—2741no—
290C22⋊C4—2741no—
291C22⋊C4—1371no—
292C22⋊C4—1371no—
293C42—2741no—
294C42—2741no—
295C42—2741no—
296C42C22×D13711yescharacteristic
297C23—21no—
298C23—2741no—
299C23—5482no—
300C23—5482no—
301C23—10961no—
302C23—1371no—
303C23—1371no—
304Q8—21no—
305Q8—21no—
306Q8—42no—
307Q8—2741no—
308Q8—2741no—
309D4—21no—
310D4—21no—
311D4—2741no—
312D4—5481no—
313D4—5482no—
314D4—5482no—
315D4—10961no—
316D4—1372no—
317C2×C4—21no—
318C2×C4—21no—
319C2×C4—21no—
320C2×C4—2741no—
321C2×C4—2741no—
322C2×C4—2741no—
323C2×C4—2741no—
324C2×C4—2741no—
325C2×C4—2741no—
326C2×C4—2741no—
327C2×C4—5482no—
328C2×C4—10961no—
329C2×C4—1371no—
330C2×C4—1371no—
331C2×C4D4×D13711yescharacteristic
332C2×C4D4×D13711yescharacteristic
333C2×C4C23×D13711yescharacteristic
334C8—21no—
335C8—21no—
336C8—2741no—
337C8—2741no—
338C22—41no—
339C22—41no—
340C22—41no—
341C22—2741no—
342C22—2741no—
343C22—2741no—
344C22—2741no—
345C22—2741no—
346C22—2741no—
347C22—2742no—
348C22—5481no—
349C22—5481no—
350C22—10964no—
351C22C2×D4×D13711yescharacteristic
352C4—21no—
353C4—21no—
354C4—41no—
355C4—41no—
356C4—41no—
357C4—2741no—
358C4—2741no—
359C4—5481no—
360C4—5481no—
361C4—5481no—
362C4#41×D13711yescharacteristic
363C4#1798413411yescharacteristic
364C2—81no—
365C2—5481no—
366C2—5481no—
367C2—5482no—
368C2#2257804611yescharacteristic
369C2#2257819911yescharacteristic
370C2#2257821211yescharacteristic
subgroupG/H
centerC22C2×D4×D137
commutator, cosocleC2×C4×C137C24
FrattiniC2×C4C23×D137
Fitting#489×C137C2
socleC22×C137C22×D4
prime radical#22578046C2
hypercenter#489D137
hypocenterC137#3311
LCSG ▷ C2×C4×C137 ▷ C2×C137 ▷ C137
UCSC1 ◁ C22 ◁ C42 ◁ #489

Complements

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

NH ≅ G/Ntype
1. #22578127C2semidirect
3. #22577743C2semidirect
4. #22577725C2semidirect
5. #22577690C2semidirect
6. #22577660C2semidirect
7. #22577631C2semidirect
8. #22577622C2semidirect
9. #22577568C2semidirect
10. #22577555C2semidirect
11. #22577505C2semidirect
12. #22577476C2semidirect
13. #22577406C2semidirect
14. #22577398C2semidirect
15. #489×C137C2semidirect
18. C2×#14555500C22semidirect
22. C2×C8×D137C22semidirect
28. #17984068C22semidirect
34. C4×Dic274C22semidirect
38. #17984046C22semidirect
40. #17984022C22semidirect
41. #17984019C22semidirect
42. #17984019C22semidirect
43. C4⋊#11764040C22semidirect
44. #17984013C22semidirect
45. #17984012C22semidirect
51. #17984000C22semidirect
52. C4⋊Q8×C137C22semidirect
54. #103×C137C22semidirect
55. #103×C137C22semidirect
58. #97×C137C22semidirect
185. C137#3311semidirect
189. #489D137semidirect
209. C4⋊Q8C2×D137semidirect
234. #103C2×D137semidirect
235. #103C2×D137semidirect
244. #97C2×D137semidirect

Maximal subgroups

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

subgroupindex
#225781272
#225780462
#225777432
#225777252
#225776902
#225776602
#225776312
#225776222
#225775682
#225775552
#225775052
#225774762
#225774062
#225773982
#489×C1372
#3311137