Finite Group Databasebeta

← #31792627

Subgroups — #31792627

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#22578050C211yesmaximal, characteristic
3#22577756C211yesmaximal, characteristic, complemented
4#22577735C211yesmaximal, characteristic, complemented
5#22577706C211yesmaximal, characteristic, complemented
6#22577645C211yesmaximal, characteristic, complemented
7#22577630C211yesmaximal, characteristic, complemented
8#22577624C211yesmaximal, characteristic, complemented
9#22577567C211yesmaximal, characteristic, complemented
10#22577555C211yesmaximal, characteristic, complemented
11#22577490C211yesmaximal, characteristic, complemented
12#22577466C211yesmaximal, characteristic, complemented
13#22577397C211yesmaximal, characteristic, complemented
14#102×D137C211yesmaximal, characteristic, complemented
15#482×C137C211yesmaximal, characteristic, complemented
16C2×#14555510C2211yescharacteristic
17C2×D4×D137C2211yescharacteristic
18C2×Dic548C2211yescharacteristic, complemented
19C2×#14555499C2211yescharacteristic
20C2×#14555498C2211yescharacteristic
21C22×#11764040C2211yescharacteristic, complemented
22C2×C8×D137C2211yescharacteristic
23#17984081—21no—
24#17984080—21no—
25#17984079C2211yescharacteristic, complemented
26#17984075C2211yescharacteristic
27#17984070—21no—
28#17984070—21no—
29#17984068C2211yescharacteristic
30#17984068C2211yescharacteristic
31#17984067C2211yescharacteristic, complemented
32#17984059—21no—
33#17984056—21no—
34#17984053C2211yescharacteristic, complemented
35#17984050C2211yescharacteristic
36C4×C22⋊D137C2211yescharacteristic
37#17984045—21no—
38C4⋊C4×D137C2211yescharacteristic
39C22⋊C4×D137—21no—
40#17984019C2211yescharacteristic, complemented
41#17984017C2211yescharacteristic, complemented
42#17984013C2211yescharacteristic, complemented
43#17984011C2211yescharacteristic, complemented
44#17984011C2211yescharacteristic, complemented
45#17984011C2211yescharacteristic, complemented
46#17984010C2211yescharacteristic
47#17984009C2211yescharacteristic
48#17984008C2211yescharacteristic
49#17984008C2211yescharacteristic
50#17984000C2211yescharacteristic
51C8×Dic137C2211yescharacteristic, complemented
52#121×C137C2211yescharacteristic
53#117×C137C2211yescharacteristic, complemented
54#107×C137C2211yescharacteristic, complemented
55#106×C137C2211yescharacteristic, complemented
56#102×C137C2211yescharacteristic
57#102×C137C2211yescharacteristic
58C22⋊C8×C137C2211yescharacteristic, complemented
59C23×D137—21no—
60#14555510—22no—
61#14555510—41no—
62C2×Dic274C2311yescharacteristic
63C2×Dic274C2311yescharacteristic
64C2×D548C2311yescharacteristic
65C2×C22⋊D137—21no—
66C2×C22⋊D137—21no—
67C2×C22⋊D137D411yescharacteristic
68D4×D137—22no—
69D4×D137—41no—
70C2×C4×D137—21no—
71C2×C4×D137—21no—
72C2×C4×D137—21no—
73C2×C4×D137C2311yescharacteristic
74Dic548—41no—
75#14555499—22no—
76#14555498—41no—
77C2×#11764040—41no—
78C2×#11764040C2311yescharacteristic
79C2×#11764040C2311yescharacteristic
80C8×D137—41no—
81C4⋊Dic137—21no—
82C4⋊Dic137—21no—
83C4⋊Dic137—21no—
84C4⋊Dic137C2311yescharacteristic
85#14555489—21no—
86#14555489—21no—
87#14555489—21no—
88#14555489—21no—
89#14555489D411yescharacteristic
90C22⋊Dic137—21no—
91C22⋊Dic137—21no—
92C22⋊Dic137—21no—
93C22⋊Dic137D411yescharacteristic
94#14555487—21no—
95#14555487—21no—
96#14555487—21no—
97#14555487D411yescharacteristic
98C4×Dic137—21no—
99C4×Dic137C2311yescharacteristic
100C2×D4×C137—21no—
101C2×D4×C137C2311yescharacteristic
102C22×C4×C137C2311yescharacteristic
103C2×C8×C137C2311yescharacteristic
104C2×C8×C137C2311yescharacteristic
105C4⋊C4×C137C2311yescharacteristic
106C4⋊C4×C137C2311yescharacteristic
107C4⋊C4×C137C2311yescharacteristic
108C22⋊C4×C137—21no—
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—
118Dic274—21no—
119Dic274—21no—
120D548—21no—
121D548#4112yes—
122C22⋊D137—22no—
123C22⋊D137—42no—
124C22⋊D137—42no—
125C2×Dic137—21no—
126C2×Dic137—21no—
127C2×Dic137—21no—
128C2×Dic137C2×D411yescharacteristic
129C2×Dic137C2×D411yescharacteristic
130C2×Dic137C2×D411yescharacteristic
131C4×D137—21no—
132C4×D137—21no—
133C4×D137—42no—
134C4×D137—42no—
135C4×D137—81no—
136#11764040—21no—
137#11764040—21no—
138C23×C137—21no—
139C23×C137C2×D411yescharacteristic
140D4×C137—21no—
141D4×C137—21no—
142D4×C137—81no—
143C2×C4×C137—21no—
144C2×C4×C137—21no—
145C2×C4×C137—21no—
146C2×C4×C137—21no—
147C2×C4×C137—21no—
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—22no—
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—41no—
179C2×C137—81no—
180C2×C137#54611yescharacteristic
181C2×C137C2×#13511yescharacteristic
182C2×C137#57711yescharacteristic
183D137—22no—
184D137—42no—
185C137#329611yescharacteristic, complemented, Sylow
186#3296—1371nomaximal, Sylow
187#552—1371no—
188#546—1371no—
189#488—1371no—
190#483—1371no—
191#482D13711yescharacteristic, complemented
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#416—1371no—
201C2×#102—1371no—
202C2×#41—1371no—
203C22×D4—1371no—
204C2×Q16—1371no—
205C2×SD16—1371no—
206C2×SD16—1371no—
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#121C2×D13711yescharacteristic
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#107C2×D13711yescharacteristic, complemented
229#106—1371no—
230#106C2×D13711yescharacteristic, complemented
231C4⋊C8—1371no—
232#103—1371no—
233#103—1371no—
234#103—1371no—
235#103—1371no—
236#102—1371no—
237#102—1371no—
238#102—1371no—
239#102—1371no—
240#102C2×D13711yescharacteristic
241#102C2×D13711yescharacteristic
242C22⋊C8—1371no—
243C22⋊C8C2×D13711yescharacteristic, complemented
244C4×C8—1371no—
245C24—2741no—
246#41—2742no—
247#41—5481no—
248C2×Q8—1371no—
249C2×Q8—1371no—
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—2741no—
259C22×C4—2741no—
260C22×C4—2741no—
261C22×C4—1371no—
262C22×C4C22×D13711yescharacteristic
263Q16—5481no—
264SD16—2742no—
265SD16—5481no—
266C2×C8—5481no—
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⋊C4C22×D13711yescharacteristic
282C4⋊C4C22×D13711yescharacteristic
283C4⋊C4C22×D13711yescharacteristic
284C22⋊C4—21no—
285C22⋊C4—21no—
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—2741no—
305Q8—2741no—
306Q8—2741no—
307Q8—2741no—
308D4—21no—
309D4—21no—
310D4—81no—
311D4—2742no—
312D4—2741no—
313D4—5482no—
314D4—5482no—
315D4—1372no—
316C2×C4—21no—
317C2×C4—21no—
318C2×C4—21no—
319C2×C4—21no—
320C2×C4—21no—
321C2×C4—2741no—
322C2×C4—2741no—
323C2×C4—2741no—
324C2×C4—2741no—
325C2×C4—2741no—
326C2×C4—5482no—
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—41no—
342C22—41no—
343C22—41no—
344C22—2741no—
345C22—2741no—
346C22—2742no—
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—
362C4#41×D13711yescharacteristic
363C4#1798413411yescharacteristic
364C2—41no—
365C2—81no—
366C2—2742no—
367C2—5482no—
368C2#2257804511yescharacteristic
369C2#135×D13711yescharacteristic
370C2#2257821311yescharacteristic
subgroupG/H
centerC22C2×D4×D137
commutator, cosocleC2×C4×C137C24
FrattiniC2×C4C23×D137
Fitting#482×C137C2
socleC22×C137C22×D4
prime radical#22578050C2
hypercenter#482D137
hypocenterC137#3296
LCSG ▷ C2×C4×C137 ▷ C2×C137 ▷ C137
UCSC1 ◁ C22 ◁ C22×C4 ◁ #482

Complements

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

NH ≅ G/Ntype
1. #22578092C2semidirect
3. #22577756C2semidirect
4. #22577735C2semidirect
5. #22577706C2semidirect
6. #22577645C2semidirect
7. #22577630C2semidirect
8. #22577624C2semidirect
9. #22577567C2semidirect
10. #22577555C2semidirect
11. #22577490C2semidirect
12. #22577466C2semidirect
13. #22577397C2semidirect
14. #102×D137C2semidirect
15. #482×C137C2semidirect
18. C2×Dic548C22semidirect
21. C22×#11764040C22semidirect
25. #17984079C22semidirect
31. #17984067C22semidirect
34. #17984053C22semidirect
40. #17984019C22semidirect
41. #17984017C22semidirect
42. #17984013C22semidirect
43. #17984011C22semidirect
44. #17984011C22semidirect
45. #17984011C22semidirect
51. C8×Dic137C22semidirect
53. #117×C137C22semidirect
54. #107×C137C22semidirect
55. #106×C137C22semidirect
58. C22⋊C8×C137C22semidirect
185. C137#3296semidirect
191. #482D137semidirect
225. #117C2×D137semidirect
228. #107C2×D137semidirect
230. #106C2×D137semidirect
243. C22⋊C8C2×D137semidirect

Maximal subgroups

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

subgroupindex
#225780922
#225780502
#225777562
#225777352
#225777062
#225776452
#225776302
#225776242
#225775672
#225775552
#225774902
#225774662
#225773972
#102×D1372
#482×C1372
#3296137