Finite Group Databasebeta

← #31792753

Subgroups — #31792753

Order 17536.

442 conjugacy classes of subgroups.

The subgroup lattice has 390 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#22578054C211yesmaximal, characteristic, complemented
2#22578050C211yesmaximal, characteristic, complemented
3#22577761C211yesmaximal, characteristic, complemented
4#22577735C211yesmaximal, characteristic, complemented
5#22577692C211yesmaximal, characteristic, complemented
6#22577654C211yesmaximal, characteristic, complemented
7#22577633C211yesmaximal, characteristic, complemented
8#22577617C211yesmaximal, characteristic, complemented
9#22577550C211yesmaximal, characteristic, complemented
10#22577538C211yesmaximal, characteristic, complemented
11#22577514C211yesmaximal, characteristic, complemented
12#22577477C211yesmaximal, characteristic, complemented
13#22577402C211yesmaximal, characteristic, complemented
14#102×D137C211yesmaximal, characteristic, complemented
15#481×C137C211yesmaximal, characteristic, complemented
16C2×#14555510—21no—
17C2×#14555510C2211yescharacteristic
18C2×D4×D137C2211yescharacteristic
19C2×#14555499C2211yescharacteristic, complemented
20C2×D1096C2211yescharacteristic
21C2×#14555495C2211yescharacteristic
22C2×#14555494C2211yescharacteristic, complemented
23C2×C8×D137C2211yescharacteristic
24#17984087C2211yescharacteristic, complemented
25#17984075C2211yescharacteristic
26#17984073—21no—
27#17984071—21no—
28#17984070—21no—
29#17984068C2211yescharacteristic
30#17984066—21no—
31#17984063C2211yescharacteristic
32#17984062C2211yescharacteristic, complemented
33#17984059—21no—
34#17984056—21no—
35#17984050C2211yescharacteristic
36#17984050C2211yescharacteristic, complemented
37C4×C22⋊D137C2211yescharacteristic
38C4⋊C4×D137C2211yescharacteristic
39C22⋊C4×D137—21no—
40#17984019C2211yescharacteristic, complemented
41#17984017C2211yescharacteristic, complemented
42#17984013C2211yescharacteristic, complemented
43#17984011C2211yescharacteristic, complemented
44#17984011C2211yescharacteristic, complemented
45#17984010C2211yescharacteristic
46#17984009C2211yescharacteristic
47#17984008C2211yescharacteristic
48#17984008C2211yescharacteristic
49#17984008C2211yescharacteristic, complemented
50#17984000C2211yescharacteristic
51#17983998C2211yescharacteristic, complemented
52#121×C137C2211yescharacteristic
53C2×C4⋊C4×C137C2211yescharacteristic, complemented
54#106×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—
62#14555510—44no—
63C2×Dic274—21no—
64C2×Dic274C2311yescharacteristic
65C2×D548—21no—
66C2×D548C2311yescharacteristic
67C2×D548C2311yescharacteristic
68C2×C22⋊D137—21no—
69C2×C22⋊D137—21no—
70C2×C22⋊D137—21no—
71C2×C22⋊D137D411yescharacteristic
72D4×D137—22no—
73D4×D137—41no—
74C2×C4×D137—21no—
75C2×C4×D137—21no—
76C2×C4×D137—21no—
77C2×C4×D137—21no—
78C2×C4×D137C2311yescharacteristic
79#14555499—41no—
80D1096—22no—
81#14555495—41no—
82#14555494—41no—
83C2×#11764040C2311yescharacteristic
84C2×#11764040C2311yescharacteristic
85C8×D137—41no—
86C4⋊Dic137—21no—
87C4⋊Dic137C2311yescharacteristic
88#14555489—21no—
89#14555489—21no—
90#14555489D411yescharacteristic
91C22⋊Dic137—21no—
92C22⋊Dic137—21no—
93C22⋊Dic137D411yescharacteristic
94#14555487—21no—
95#14555487—21no—
96#14555487—21no—
97#14555487—21no—
98#14555487—21no—
99#14555487D411yescharacteristic
100C4×Dic137C2311yescharacteristic
101C2×D4×C137—21no—
102C2×D4×C137C2311yescharacteristic
103C22×C4×C137—21no—
104C22×C4×C137C2311yescharacteristic
105C2×C8×C137C2311yescharacteristic
106C2×C8×C137C2311yescharacteristic
107C4⋊C4×C137—21no—
108C4⋊C4×C137C2311yescharacteristic
109C4⋊C4×C137C2311yescharacteristic
110C4⋊C4×C137C2311yescharacteristic
111C22⋊C4×C137—21no—
112C22×D137—21no—
113C22×D137—21no—
114C22×D137—42no—
115C22×D137—42no—
116C22×D137—81no—
117C22×D137C2×D411yescharacteristic
118Dic274—21no—
119Dic274—21no—
120Dic274—42no—
121D548—21no—
122D548—21no—
123D548—21no—
124D548—42no—
125D548#4112yescomplemented
126C22⋊D137—22no—
127C22⋊D137—42no—
128C22⋊D137—42no—
129C22⋊D137—42no—
130C2×Dic137—21no—
131C2×Dic137—21no—
132C2×Dic137C2×D411yescharacteristic
133C2×Dic137C2×D411yescharacteristic
134C2×Dic137C2×D411yescharacteristic
135C4×D137—21no—
136C4×D137—21no—
137C4×D137—42no—
138C4×D137—42no—
139C4×D137—42no—
140C4×D137—81no—
141#11764040—21no—
142#11764040—21no—
143C23×C137—21no—
144C23×C137C2×D411yescharacteristic
145D4×C137—21no—
146D4×C137—21no—
147D4×C137—81no—
148C2×C4×C137—21no—
149C2×C4×C137—21no—
150C2×C4×C137—21no—
151C2×C4×C137—21no—
152C2×C4×C137—21no—
153C2×C4×C137—42no—
154C2×C4×C137C2×D411yescharacteristic
155C2×C4×C137C2411yescharacteristic
156C8×C137—21no—
157C8×C137—21no—
158C22×C137—21no—
159C22×C137—21no—
160C22×C137—21no—
161C22×C137—41no—
162C22×C137—41no—
163C22×C137—41no—
164C22×C137C22×D411yescharacteristic
165C2×D137—21no—
166C2×D137—21no—
167C2×D137—22no—
168C2×D137—41no—
169C2×D137—41no—
170C2×D137—41no—
171C2×D137—41no—
172C2×D137—41no—
173C2×D137—84no—
174C2×D137AGL(1,ℤ/8)12yes—
175C4×C137—21no—
176C4×C137—41no—
177C4×C137—41no—
178C4×C137—41no—
179C4×C137C2×#4111yescharacteristic
180C4×C137H(5,𝔽2)11yescharacteristic
181Dic137—21no—
182Dic137—21no—
183Dic137—21no—
184Dic137—41no—
185Dic137—41no—
186C2×C137—41no—
187C2×C137—81no—
188C2×C137#54611yescharacteristic
189C2×C137C2×AGL(1,ℤ/8)11yescharacteristic
190C2×C137#57711yescharacteristic
191D137—22no—
192D137—42no—
193D137—81no—
194C137#330611yescharacteristic, complemented, Sylow
195#3306—1371nomaximal, Sylow
196#547—1371no—
197#546—1371no—
198#489—1371no—
199#483—1371no—
200#481D13711yescharacteristic, complemented
201#476—1371no—
202#469—1371no—
203#465—1371no—
204#463—1371no—
205#449—1371no—
206#447—1371no—
207#442—1371no—
208#435—1371no—
209#417—1371no—
210C2×#102—1371no—
211C2×#41—2741no—
212C2×#41—1371no—
213C22×D4—1371no—
214C2×SD16—1371no—
215C2×D8—1371no—
216C2×D8—1371no—
217C2×M16—1371no—
218C22×C8—1371no—
219C4⋊Q8—1371no—
220#124—1371no—
221#123—2741no—
222#123—2741no—
223#123—2741no—
224C22⋊Q8—2741no—
225C22⋊Q8—1371no—
226#121—2741no—
227#121—1371no—
228#121—1371no—
229#121C2×D13711yescharacteristic
230#120—2741no—
231C4×D4—1371no—
232C4×D4—1371no—
233C4×D4—1371no—
234C2×C4⋊C4—1371no—
235C2×C4⋊C4C2×D13711yescharacteristic, complemented
236C2×C22⋊C4—2741no—
237#106—1371no—
238#106C2×D13711yescharacteristic, complemented
239#106C2×D13711yescharacteristic, complemented
240C4⋊C8—1371no—
241#103—1371no—
242#103—1371no—
243#103—1371no—
244#102—1371no—
245#102—1371no—
246#102—1371no—
247#102—1371no—
248#102—1371no—
249#102C2×D13711yescharacteristic
250#102C2×D13711yescharacteristic
251C22⋊C8—1371no—
252C22⋊C8C2×D13711yescharacteristic, complemented
253#97—1371no—
254C24—2741no—
255#41—2742no—
256#41—5481no—
257#41—5484no—
258C2×Q8—2741no—
259C2×Q8—1371no—
260C2×D4—21no—
261C2×D4—2741no—
262C2×D4—2742no—
263C2×D4—2741no—
264C2×D4—2741no—
265C2×D4—2741no—
266C2×D4—5481no—
267C2×D4—1371no—
268C2×D4—1371no—
269C2×D4—1371no—
270C2×D4C22×D13711yescharacteristic
271C22×C4—21no—
272C22×C4—2741no—
273C22×C4—2741no—
274C22×C4—2741no—
275C22×C4—2741no—
276C22×C4—1371no—
277C22×C4C22×D13711yescharacteristic
278SD16—5481no—
279D8—2742no—
280D8—5481no—
281M16—5481no—
282C2×C8—5481no—
283C2×C8—1371no—
284C2×C8—1371no—
285C2×C8C22×D13711yescharacteristic
286C2×C8C22×D13711yescharacteristic
287C4⋊C4—21no—
288C4⋊C4—2741no—
289C4⋊C4—2741no—
290C4⋊C4—2741no—
291C4⋊C4—1371no—
292C4⋊C4—1371no—
293C4⋊C4C22×D13711yescharacteristic
294C4⋊C4C22×D13711yescharacteristic
295C4⋊C4C22×D13711yescharacteristic
296C22⋊C4—21no—
297C22⋊C4—2741no—
298C22⋊C4—2741no—
299C22⋊C4—2741no—
300C22⋊C4—2741no—
301C22⋊C4—2741no—
302C22⋊C4—2741no—
303C22⋊C4—2741no—
304C22⋊C4—1371no—
305C22⋊C4—1371no—
306C42—1371no—
307C23—21no—
308C23—2741no—
309C23—2741no—
310C23—5482no—
311C23—5482no—
312C23—10961no—
313C23—1371no—
314C23D4×D13711yescharacteristic
315Q8—2741no—
316Q8—2741no—
317Q8—5482no—
318D4—21no—
319D4—21no—
320D4—81no—
321D4—2742no—
322D4—2741no—
323D4—2741no—
324D4—2741no—
325D4—5482no—
326D4—5482no—
327D4—5482no—
328D4—5482no—
329D4—1372no—
330C2×C4—21no—
331C2×C4—21no—
332C2×C4—21no—
333C2×C4—21no—
334C2×C4—21no—
335C2×C4—42no—
336C2×C4—2741no—
337C2×C4—2741no—
338C2×C4—2741no—
339C2×C4—2741no—
340C2×C4—5482no—
341C2×C4—5482no—
342C2×C4—5482no—
343C2×C4—10961no—
344C2×C4—1371no—
345C2×C4—1371no—
346C2×C4—1371no—
347C2×C4D4×D13711yescharacteristic
348C2×C4C23×D13711yescharacteristic
349C8—21no—
350C8—21no—
351C8—2741no—
352C8—2741no—
353C22—21no—
354C22—21no—
355C22—21no—
356C22—41no—
357C22—41no—
358C22—41no—
359C22—2741no—
360C22—2741no—
361C22—2742no—
362C22—5481no—
363C22—5481no—
364C22—5481no—
365C22—5481no—
366C22—5481no—
367C22—10964no—
368C22—1372no—
369C22C2×D4×D13711yescharacteristic
370C4—21no—
371C4—41no—
372C4—41no—
373C4—41no—
374C4—2741no—
375C4—2741no—
376C4—2741no—
377C4—5481no—
378C4—5481no—
379C4#41×D13711yescharacteristic
380C4#1798413411yescharacteristic
381C2—41no—
382C2—81no—
383C2—2742no—
384C2—5482no—
385C2—10961no—
386C2#2257804511yescharacteristic
387C2AGL(1,ℤ/8)×D13711yescharacteristic
388C2#2257820911yescharacteristic
subgroupG/H
centerC22C2×D4×D137
commutator, cosocleC2×C4×C137C24
FrattiniC2×C4C23×D137
Fitting#481×C137C2
socleC22×C137C22×D4
hypercenter#481D137
hypocenterC137#3306
LCSG ▷ C2×C4×C137 ▷ C2×C137 ▷ C137
UCSC1 ◁ C22 ◁ C22×C4 ◁ #481

Complements

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

NH ≅ G/Ntype
1. #22578054C2semidirect
2. #22578050C2semidirect
3. #22577761C2semidirect
4. #22577735C2semidirect
5. #22577692C2semidirect
6. #22577654C2semidirect
7. #22577633C2semidirect
8. #22577617C2semidirect
9. #22577550C2semidirect
10. #22577538C2semidirect
11. #22577514C2semidirect
12. #22577477C2semidirect
13. #22577402C2semidirect
14. #102×D137C2semidirect
15. #481×C137C2semidirect
19. C2×#14555499C22semidirect
22. C2×#14555494C22semidirect
24. #17984087C22semidirect
32. #17984062C22semidirect
36. #17984050C22semidirect
40. #17984019C22semidirect
41. #17984017C22semidirect
42. #17984013C22semidirect
43. #17984011C22semidirect
44. #17984011C22semidirect
49. #17984008C22semidirect
51. #17983998C22semidirect
53. C2×C4⋊C4×C137C22semidirect
54. #106×C137C22semidirect
55. #106×C137C22semidirect
58. C22⋊C8×C137C22semidirect
125. D548#41semidirect
194. C137#3306semidirect
200. #481D137semidirect
235. C2×C4⋊C4C2×D137semidirect
238. #106C2×D137semidirect
239. #106C2×D137semidirect
252. C22⋊C8C2×D137semidirect

Maximal subgroups

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

subgroupindex
#225780542
#225780502
#225777612
#225777352
#225776922
#225776542
#225776332
#225776172
#225775502
#225775382
#225775142
#225774772
#225774022
#102×D1372
#481×C1372
#3306137