Finite Group Databasebeta

← #31792625

Subgroups — #31792625

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#22578092C211yesmaximal, characteristic, complemented
2#22578041C211yesmaximal, characteristic
3#22577759C211yesmaximal, characteristic, complemented
4#22577732C211yesmaximal, characteristic, complemented
5#22577727C211yesmaximal, characteristic, complemented
6#22577705C211yesmaximal, characteristic, complemented
7#22577643C211yesmaximal, characteristic, complemented
8#22577632C211yesmaximal, characteristic, complemented
9#22577628C211yesmaximal, characteristic, complemented
10#22577565C211yesmaximal, characteristic, complemented
11#22577555C211yesmaximal, characteristic, complemented
12#22577494C211yesmaximal, characteristic, complemented
13C8×C22⋊D137C211yesmaximal, characteristic, complemented
14#102×D137C211yesmaximal, characteristic, complemented
15#416×C137C211yesmaximal, characteristic, complemented
16C2×D4×D137C2211yescharacteristic
17C2×#14555500C2211yescharacteristic, complemented
18C2×#14555499C2211yescharacteristic
19C2×#14555498C2211yescharacteristic
20C2×C8×D137C2211yescharacteristic
21#17984081—21no—
22#17984080—21no—
23#17984079C2211yescharacteristic, complemented
24#17984077C2211yescharacteristic
25#17984071—21no—
26#17984070—21no—
27#17984068C2211yescharacteristic
28#17984067C2211yescharacteristic, complemented
29#17984064C2211yescharacteristic
30#17984062C2211yescharacteristic
31#17984060—21no—
32#17984056—21no—
33#17984053C2211yescharacteristic, complemented
34D4×Dic137C2211yescharacteristic
35C4×C22⋊D137C2211yescharacteristic
36#17984045—21no—
37C4⋊C4×D137C2211yescharacteristic
38C2×#14555487—21no—
39#17984021C2211yescharacteristic, complemented
40#17984020C2211yescharacteristic, complemented
41#17984019C2211yescharacteristic, complemented
42#17984017C2211yescharacteristic, complemented
43#17984013C2211yescharacteristic, complemented
44#17984012C2211yescharacteristic, complemented
45#17984010C2211yescharacteristic
46#17984009C2211yescharacteristic
47#17984008C2211yescharacteristic
48#17984008C2211yescharacteristic
49C22⋊#11764040C2211yescharacteristic, complemented
50#17984000C2211yescharacteristic
51C8×Dic137C2211yescharacteristic, complemented
52C2×#41×C137C2211yescharacteristic
53C22×C8×C137C2211yescharacteristic, complemented
54#117×C137C2211yescharacteristic, complemented
55#103×C137C2211yescharacteristic, complemented
56#103×C137C2211yescharacteristic, complemented
57#102×C137C2211yescharacteristic
58#102×C137C2211yescharacteristic
59C23×D137—21no—
60C2×Dic274C2311yescharacteristic
61C2×D548C2311yescharacteristic
62C2×C22⋊D137—21no—
63C2×C22⋊D137—21no—
64C2×C22⋊D137D411yescharacteristic
65D4×D137—22no—
66D4×D137—41no—
67C22×Dic137—21no—
68C2×C4×D137—21no—
69C2×C4×D137C2311yescharacteristic
70#14555500—41no—
71#14555499—41no—
72#14555498—22no—
73C2×#11764040C2311yescharacteristic
74C2×#11764040C2311yescharacteristic
75C8×D137—41no—
76C4⋊Dic137—21no—
77C4⋊Dic137—21no—
78C4⋊Dic137C2311yescharacteristic
79C4⋊Dic137C2311yescharacteristic
80#14555489—21no—
81#14555489—21no—
82#14555489—21no—
83#14555489—21no—
84#14555489—21no—
85#14555489D411yescharacteristic
86C22⋊Dic137—21no—
87C22⋊Dic137—21no—
88C22⋊Dic137D411yescharacteristic
89#14555487—21no—
90#14555487—21no—
91#14555487—21no—
92#14555487—21no—
93#14555487—21no—
94#14555487D411yescharacteristic
95C4×Dic137—21no—
96C4×Dic137C2311yescharacteristic
97#41×C137—22no—
98#41×C137—41no—
99C2×Q8×C137C2311yescharacteristic
100C2×D4×C137—21no—
101C2×D4×C137C2311yescharacteristic
102C22×C4×C137—21no—
103C22×C4×C137C2311yescharacteristic
104C2×C8×C137—41no—
105C2×C8×C137C2311yescharacteristic
106C2×C8×C137C2311yescharacteristic
107C4⋊C4×C137C2311yescharacteristic
108C4⋊C4×C137C2311yescharacteristic
109C22⋊C4×C137—21no—
110C42×C137—21no—
111C22×D137—21no—
112C22×D137—42no—
113C22×D137—42no—
114C22×D137—81no—
115C22×D137C2×D411yescharacteristic
116Dic274—21no—
117Dic274—21no—
118D548—21no—
119D548—21no—
120C22⋊D137—22no—
121C22⋊D137—42no—
122C22⋊D137—81no—
123C2×Dic137—21no—
124C2×Dic137—21no—
125C2×Dic137—21no—
126C2×Dic137—81no—
127C2×Dic137C2×D411yescharacteristic
128C2×Dic137C2×D411yescharacteristic
129C2×Dic137C2×D411yescharacteristic
130C4×D137—21no—
131C4×D137—21no—
132C4×D137—42no—
133#11764040—21no—
134#11764040—21no—
135C23×C137—21no—
136C23×C137C2×D411yescharacteristic
137Q8×C137—21no—
138Q8×C137—21no—
139D4×C137—21no—
140D4×C137—42no—
141D4×C137#4112yes—
142C2×C4×C137—21no—
143C2×C4×C137—21no—
144C2×C4×C137—21no—
145C2×C4×C137—21no—
146C2×C4×C137—21no—
147C2×C4×C137—42no—
148C2×C4×C137C2×D411yescharacteristic
149C2×C4×C137C2411yescharacteristic
150C8×C137—21no—
151C8×C137—21no—
152C22×C137—21no—
153C22×C137—22no—
154C22×C137—21no—
155C22×C137—21no—
156C22×C137—41no—
157C22×C137—41no—
158C22×C137C22×D411yescharacteristic
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—41no—
168C4×C137—41no—
169C4×C137—41no—
170C4×C137C2×#4111yescharacteristic
171C4×C137H(5,𝔽2)11yescharacteristic
172Dic137—21no—
173Dic137—21no—
174Dic137—21no—
175Dic137—41no—
176Dic137—41no—
177Dic137—41no—
178C2×C137—42no—
179C2×C137—41no—
180C2×C137#54611yescharacteristic
181C2×C137C2×#13511yescharacteristic
182C2×C137#57711yescharacteristic
183D137—22no—
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×SD16—1371no—
199C8×D4—1371no—
200#416D13711yescharacteristic, complemented
201C2×#102—1371no—
202C2×#41C2×D13711yescharacteristic
203C22×D4—1371no—
204C2×Q16—1371no—
205C2×SD16—1371no—
206C2×SD16—1371no—
207C22×C8—1371no—
208C22×C8C2×D13711yescharacteristic, complemented
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×Q8—1371no—
222C4×D4—1371no—
223C4×D4—1371no—
224#117—2741no—
225#117C2×D13711yescharacteristic, complemented
226C2×C4⋊C4—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—
244C4×C8—1371no—
245C24—2741no—
246#41—22no—
247#41—41no—
248C2×Q8—1371no—
249C2×Q8C22×D13711yescharacteristic
250C2×D4—21no—
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×C4C22×D13711yescharacteristic
263Q16—5481no—
264SD16—2742no—
265SD16—5481no—
266C2×C8—41no—
267C2×C8—5481no—
268C2×C8—1371no—
269C2×C8—1371no—
270C2×C8C22×D13711yescharacteristic
271C2×C8C22×D13711yescharacteristic
272C4⋊C4—2741no—
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—
296C42—1371no—
297C23—21no—
298C23—2741no—
299C23—5482no—
300C23—5482no—
301C23—10961no—
302C23—1371no—
303C23D4×D13711yescharacteristic
304Q8—21no—
305Q8—21no—
306Q8—2741no—
307Q8—2741no—
308D4—21no—
309D4—42no—
310D4—2742no—
311D4—2741no—
312D4—2741no—
313D4—5482no—
314D4—10961no—
315D4#1455551012yes—
316C2×C4—21no—
317C2×C4—21no—
318C2×C4—21no—
319C2×C4—21no—
320C2×C4—21no—
321C2×C4—42no—
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×C4—1371no—
332C2×C4D4×D13711yescharacteristic
333C2×C4C23×D13711yescharacteristic
334C8—21no—
335C8—21no—
336C8—2741no—
337C8—2741no—
338C22—21no—
339C22—21no—
340C22—21no—
341C22—22no—
342C22—41no—
343C22—41no—
344C22—2741no—
345C22—2741no—
346C22—5481no—
347C22—5481no—
348C22—5481no—
349C22—10964no—
350C22—1372no—
351C22C2×D4×D13711yescharacteristic
352C4—21no—
353C4—41no—
354C4—41no—
355C4—41no—
356C4—2741no—
357C4—2741no—
358C4—2741no—
359C4—5481no—
360C4—5481no—
361C4—5481no—
362C4C2×#1455551011yescharacteristic
363C4#1798413311yescharacteristic
364C2—41no—
365C2—42no—
366C2—2742no—
367C2—10961no—
368C2#2257804311yescharacteristic
369C2#135×D13711yescharacteristic
370C2#2257821011yescharacteristic
subgroupG/H
centerC22C2×D4×D137
commutator, cosocleC2×C4×C137C24
FrattiniC2×C4C23×D137
Fitting#416×C137C2
socleC22×C137C22×D4
prime radical#22578041C2
hypercenter#416D137
hypocenterC137#3296
LCSG ▷ C2×C4×C137 ▷ C2×C137 ▷ C137
UCSC1 ◁ C22 ◁ C22×C4 ◁ #416

Complements

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

NH ≅ G/Ntype
1. #22578092C2semidirect
3. #22577759C2semidirect
4. #22577732C2semidirect
5. #22577727C2semidirect
6. #22577705C2semidirect
7. #22577643C2semidirect
8. #22577632C2semidirect
9. #22577628C2semidirect
10. #22577565C2semidirect
11. #22577555C2semidirect
12. #22577494C2semidirect
13. C8×C22⋊D137C2semidirect
14. #102×D137C2semidirect
15. #416×C137C2semidirect
17. C2×#14555500C22semidirect
23. #17984079C22semidirect
28. #17984067C22semidirect
33. #17984053C22semidirect
39. #17984021C22semidirect
40. #17984020C22semidirect
41. #17984019C22semidirect
42. #17984017C22semidirect
43. #17984013C22semidirect
44. #17984012C22semidirect
49. C22⋊#11764040C22semidirect
51. C8×Dic137C22semidirect
53. C22×C8×C137C22semidirect
54. #117×C137C22semidirect
55. #103×C137C22semidirect
56. #103×C137C22semidirect
185. C137#3296semidirect
200. #416D137semidirect
208. C22×C8C2×D137semidirect
225. #117C2×D137semidirect
234. #103C2×D137semidirect
235. #103C2×D137semidirect

Maximal subgroups

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

subgroupindex
#225780922
#225780412
#225777592
#225777322
#225777272
#225777052
#225776432
#225776322
#225776282
#225775652
#225775552
#225774942
C8×C22⋊D1372
#102×D1372
#416×C1372
#3296137