Finite Group Databasebeta

← #31792629

Subgroups — #31792629

Order 17536.

408 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#22578094C211yesmaximal, characteristic, complemented
2#22578048C211yesmaximal, characteristic
3#22577758C211yesmaximal, characteristic, complemented
4#22577735C211yesmaximal, characteristic, complemented
5#22577729C211yesmaximal, characteristic, complemented
6#22577703C211yesmaximal, characteristic, complemented
7#22577643C211yesmaximal, characteristic, complemented
8#22577635C211yesmaximal, characteristic, complemented
9#22577625C211yesmaximal, characteristic, complemented
10#22577562C211yesmaximal, characteristic, complemented
11#22577552C211yesmaximal, characteristic, complemented
12C8×D548C211yesmaximal, characteristic, complemented
13#22577396C211yesmaximal, characteristic, complemented
14#102×D137C211yesmaximal, characteristic, complemented
15C4×SD16×C137C211yesmaximal, characteristic, complemented
16C2×#14555509C2211yescharacteristic
17C2×D4×D137C2211yescharacteristic
18C2×#14555500C2211yescharacteristic, complemented
19C2×#14555497C2211yescharacteristic
20C2×C8×D137C2211yescharacteristic
21C2×C8×D137C2211yescharacteristic, complemented
22#17984082—21no—
23#17984080C2211yescharacteristic, complemented
24#17984078—21no—
25#17984076C2211yescharacteristic
26#17984073—21no—
27#17984072—21no—
28#17984068C2211yescharacteristic
29#17984064C2211yescharacteristic
30#17984064C2211yescharacteristic, complemented
31#17984063C2211yescharacteristic
32#17984062—21no—
33#17984058—21no—
34C4×D548C2211yescharacteristic
35#17984046C2211yescharacteristic, complemented
36#17984045—21no—
37C4⋊C4×D137C2211yescharacteristic
38C2×#14555487—21no—
39#17984021C2211yescharacteristic, complemented
40#17984019C2211yescharacteristic, complemented
41C4⋊#11764040C2211yescharacteristic, complemented
42#17984013C2211yescharacteristic, complemented
43#17984013C2211yescharacteristic, complemented
44#17984011C2211yescharacteristic, complemented
45#17984010C2211yescharacteristic
46#17984010C2211yescharacteristic
47#17984009C2211yescharacteristic
48#17984009C2211yescharacteristic
49#17984008C2211yescharacteristic
50#17984000C2211yescharacteristic
51#17984000C2211yescharacteristic, complemented
52C2×SD16×C137C2211yescharacteristic
53C4×Q8×C137C2211yescharacteristic, complemented
54C4×D4×C137C2211yescharacteristic
55#106×C137C2211yescharacteristic, complemented
56#103×C137C2211yescharacteristic, complemented
57#102×C137C2211yescharacteristic
58C4×C8×C137C2211yescharacteristic, complemented
59C23×D137—21no—
60#14555509—22no—
61#14555509—41no—
62C2×Dic274C2311yescharacteristic
63C2×D548—21no—
64C2×D548D411yescharacteristic
65C2×D548C2311yescharacteristic
66C2×C22⋊D137—21no—
67C2×C22⋊D137—21no—
68D4×D137—22no—
69D4×D137—41no—
70C22×Dic137—21no—
71C2×C4×D137—21no—
72C2×C4×D137C2311yescharacteristic
73C2×C4×D137C2311yescharacteristic
74#14555500—41no—
75#14555497—41no—
76C2×#11764040C2311yescharacteristic
77C2×#11764040C2311yescharacteristic
78C8×D137—41no—
79C8×D137—41no—
80C4⋊Dic137—21no—
81C4⋊Dic137D411yescharacteristic
82C4⋊Dic137C2311yescharacteristic
83C4⋊Dic137C2311yescharacteristic
84#14555489—21no—
85#14555489—21no—
86#14555489—21no—
87#14555489—21no—
88#14555489—21no—
89#14555487—21no—
90#14555487—21no—
91#14555487—21no—
92#14555487—21no—
93#14555487—21no—
94#14555487—21no—
95#14555487—21no—
96#14555487D411yescharacteristic
97#14555487D411yescharacteristic
98C4×Dic137—21no—
99C2×Q8×C137C2311yescharacteristic
100C2×D4×C137C2311yescharacteristic
101C22×C4×C137—21no—
102SD16×C137—22no—
103C2×C8×C137C2311yescharacteristic
104C2×C8×C137C2311yescharacteristic
105C4⋊C4×C137—21no—
106C4⋊C4×C137C2311yescharacteristic
107C4⋊C4×C137C2311yescharacteristic
108C22⋊C4×C137—21no—
109C42×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—22no—
120D548—21no—
121D548—21no—
122D548—81no—
123C22⋊D137—42no—
124C22⋊D137—42no—
125C2×Dic137—21no—
126C2×Dic137—21no—
127C2×Dic137—21no—
128C2×Dic137—42no—
129C2×Dic137C2×D411yescharacteristic
130C2×Dic137C2×D411yescharacteristic
131C4×D137—21no—
132C4×D137—21no—
133C4×D137—21no—
134C4×D137—21no—
135C4×D137—42no—
136#11764040—21no—
137#11764040—21no—
138C23×C137—21no—
139Q8×C137—21no—
140Q8×C137—21no—
141D4×C137—21no—
142D4×C137#4112yes—
143C2×C4×C137—21no—
144C2×C4×C137—21no—
145C2×C4×C137—21no—
146C2×C4×C137—81no—
147C2×C4×C137C2×D411yescharacteristic
148C2×C4×C137C2×D411yescharacteristic
149C2×C4×C137C2411yescharacteristic
150C8×C137—21no—
151C8×C137—21no—
152C22×C137—22no—
153C22×C137—41no—
154C22×C137—41no—
155C22×C137C22×D411yescharacteristic
156C2×D137—21no—
157C2×D137—21no—
158C2×D137—21no—
159C2×D137—21no—
160C2×D137—21no—
161C2×D137—41no—
162C2×D137—41no—
163C2×D137—41no—
164C2×D137—84no—
165C2×D137#13512yes—
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—42no—
179C2×C137#54611yescharacteristic
180C2×C137C2×#13511yescharacteristic
181C2×C137#57711yescharacteristic
182D137—22no—
183D137—41no—
184D137—81no—
185C137#329611yescharacteristic, complemented, Sylow
186#3296—1371nomaximal, Sylow
187#552—1371no—
188#546—1371no—
189#488—1371no—
190#483—1371no—
191#482—1371no—
192#478—1371no—
193#467—1371no—
194#465—1371no—
195#464—1371no—
196#452—1371no—
197#450—1371no—
198C4×SD16D13711yescharacteristic, complemented
199C8×D4—1371no—
200#416—1371no—
201C2×#102—1371no—
202C2×#41—1371no—
203C22×D4—1371no—
204C2×Q16—1371no—
205C2×SD16—1371no—
206C2×SD16C2×D13711yescharacteristic
207C22×C8—1371no—
208C22×C8—1371no—
209#126—2741no—
210#125—2741no—
211#125—1371no—
212#124—1371no—
213#123—2741no—
214#123—2741no—
215C22⋊Q8—1371no—
216C22⋊Q8—1371no—
217C22⋊Q8—1371no—
218#121—2741no—
219#121—1371no—
220#120—2741no—
221C4×Q8C2×D13711yescharacteristic, complemented
222C4×D4—1371no—
223C4×D4C2×D13711yescharacteristic
224#117—2741no—
225#117—1371no—
226C2×C4⋊C4—1371no—
227C2×C22⋊C4—2741no—
228#107—1371no—
229#106—1371no—
230#106C2×D13711yescharacteristic, complemented
231C4⋊C8—1371no—
232#103—1371no—
233#103—1371no—
234#103—1371no—
235#103C2×D13711yescharacteristic, complemented
236#102—1371no—
237#102—1371no—
238#102—1371no—
239#102—1371no—
240#102—1371no—
241#102C2×D13711yescharacteristic
242C22⋊C8—1371no—
243C22⋊C8—1371no—
244C4×C8C2×D13711yescharacteristic, complemented
245C24—2741no—
246#41—2742no—
247#41—5481no—
248C2×Q8—1371no—
249C2×Q8C22×D13711yescharacteristic
250C2×D4—2741no—
251C2×D4—2742no—
252C2×D4—2741no—
253C2×D4—2741no—
254C2×D4—5481no—
255C2×D4—1371no—
256C2×D4—1371no—
257C2×D4C22×D13711yescharacteristic
258C22×C4—21no—
259C22×C4—2741no—
260C22×C4—2741no—
261C22×C4—1371no—
262C22×C4—1371no—
263Q16—5481no—
264SD16—22no—
265SD16—5481no—
266C2×C8—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—2741no—
278C4⋊C4—2741no—
279C4⋊C4—1371no—
280C4⋊C4—1371no—
281C4⋊C4—1371no—
282C4⋊C4C22×D13711yescharacteristic
283C4⋊C4C22×D13711yescharacteristic
284C22⋊C4—21no—
285C22⋊C4—2741no—
286C22⋊C4—2741no—
287C22⋊C4—2741no—
288C22⋊C4—2741no—
289C22⋊C4—2741no—
290C22⋊C4—2741no—
291C22⋊C4—2741no—
292C22⋊C4—1371no—
293C22⋊C4—1371no—
294C42—21no—
295C42—2741no—
296C42C22×D13711yescharacteristic
297C23—21no—
298C23—2741no—
299C23—5482no—
300C23—5482no—
301C23—10961no—
302C23—1371no—
303C23—1371no—
304Q8—21no—
305Q8—21no—
306Q8—2741no—
307Q8—2741no—
308D4—21no—
309D4—2742no—
310D4—2741no—
311D4—2741no—
312D4—5482no—
313D4—5482no—
314D4—10961no—
315D4#1455551012yes—
316C2×C4—21no—
317C2×C4—21no—
318C2×C4—21no—
319C2×C4—81no—
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—5482no—
329C2×C4—1371no—
330C2×C4—1371no—
331C2×C4D4×D13711yescharacteristic
332C2×C4D4×D13711yescharacteristic
333C2×C4C23×D13711yescharacteristic
334C8—21no—
335C8—21no—
336C8—2741no—
337C8—2741no—
338C22—22no—
339C22—41no—
340C22—41no—
341C22—2741no—
342C22—2741no—
343C22—2741no—
344C22—2741no—
345C22—2741no—
346C22—5481no—
347C22—5481no—
348C22—5481no—
349C22—10964no—
350C22—1372no—
351C22C2×D4×D13711yescharacteristic
352C4—21no—
353C4—21no—
354C4—41no—
355C4—41no—
356C4—41no—
357C4—2741no—
358C4—2741no—
359C4—5481no—
360C4—5481no—
361C4—5481no—
362C4C2×#1455551011yescharacteristic
363C4#1798413411yescharacteristic
364C2—42no—
365C2—2742no—
366C2—5481no—
367C2—10961no—
368C2#2257804911yescharacteristic
369C2#135×D13711yescharacteristic
370C2#2257820911yescharacteristic
subgroupG/H
centerC22C2×D4×D137
commutator, cosocleC2×C4×C137C24
FrattiniC2×C4C23×D137
FittingC4×SD16×C137C2
socleC22×C137C22×D4
prime radical#22578048C2
hypercenterC4×SD16D137
hypocenterC137#3296
LCSG ▷ C2×C4×C137 ▷ C2×C137 ▷ C137
UCSC1 ◁ C22 ◁ C42 ◁ C4×SD16

Complements

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

NH ≅ G/Ntype
1. #22578094C2semidirect
3. #22577758C2semidirect
4. #22577735C2semidirect
5. #22577729C2semidirect
6. #22577703C2semidirect
7. #22577643C2semidirect
8. #22577635C2semidirect
9. #22577625C2semidirect
10. #22577562C2semidirect
11. #22577552C2semidirect
12. C8×D548C2semidirect
13. #22577396C2semidirect
14. #102×D137C2semidirect
15. C4×SD16×C137C2semidirect
18. C2×#14555500C22semidirect
21. C2×C8×D137C22semidirect
23. #17984080C22semidirect
30. #17984064C22semidirect
35. #17984046C22semidirect
39. #17984021C22semidirect
40. #17984019C22semidirect
41. C4⋊#11764040C22semidirect
42. #17984013C22semidirect
43. #17984013C22semidirect
44. #17984011C22semidirect
51. #17984000C22semidirect
53. C4×Q8×C137C22semidirect
55. #106×C137C22semidirect
56. #103×C137C22semidirect
58. C4×C8×C137C22semidirect
185. C137#3296semidirect
198. C4×SD16D137semidirect
221. C4×Q8C2×D137semidirect
230. #106C2×D137semidirect
235. #103C2×D137semidirect
244. C4×C8C2×D137semidirect

Maximal subgroups

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

subgroupindex
#225780942
#225780482
#225777582
#225777352
#225777292
#225777032
#225776432
#225776352
#225776252
#225775622
#225775522
C8×D5482
#225773962
#102×D1372
C4×SD16×C1372
#3296137