Finite Group Databasebeta

← #31792748

Subgroups — #31792748

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#22578044C211yesmaximal, characteristic, complemented
3#22577764C211yesmaximal, characteristic, complemented
4#22577723C211yesmaximal, characteristic, complemented
5#22577692C211yesmaximal, characteristic, complemented
6#22577657C211yesmaximal, characteristic, complemented
7#22577634C211yesmaximal, characteristic, complemented
8#22577621C211yesmaximal, characteristic, complemented
9#22577551C211yesmaximal, characteristic, complemented
10#22577539C211yesmaximal, characteristic, complemented
11#22577514C211yesmaximal, characteristic, complemented
12#22577471C211yesmaximal, characteristic, complemented
13#22577403C211yesmaximal, characteristic, complemented
14C2×#17984008C211yesmaximal, characteristic, complemented
15#483×C137C211yesmaximal, characteristic, complemented
16C2×#14555511C2211yescharacteristic
17C2×#14555510—21no—
18C22×D548C2211yescharacteristic
19C2×#14555497C2211yescharacteristic, complemented
20C2×D1096C2211yescharacteristic
21C2×D1096C2211yescharacteristic
22C2×#14555493C2211yescharacteristic, complemented
23C22×#11764040C2211yescharacteristic
24#17984087C2211yescharacteristic, complemented
25#17984077C2211yescharacteristic
26#17984073—21no—
27#17984073—21no—
28#17984071—21no—
29#17984066—21no—
30#17984063—21no—
31#17984063C2211yescharacteristic
32#17984063C2211yescharacteristic, complemented
33#17984062C2211yescharacteristic
34#17984058—21no—
35#17984050C2211yescharacteristic
36#17984050C2211yescharacteristic, complemented
37C4×C22⋊D137C2211yescharacteristic
38C4⋊C4×D137C2211yescharacteristic, complemented
39C22⋊C4×D137—21no—
40#17984019C2211yescharacteristic, complemented
41#17984017C2211yescharacteristic, complemented
42#17984012C2211yescharacteristic, complemented
43#17984010C2211yescharacteristic
44#17984010C2211yescharacteristic
45#17984008C2211yescharacteristic
46#17984008C2211yescharacteristic
47#17984008C2211yescharacteristic
48#17984008C2211yescharacteristic
49#17984008C2211yescharacteristic, complemented
50#17984000C2211yescharacteristic, complemented
51#17983998C2211yescharacteristic, complemented
52C22⋊Q8×C137C2211yescharacteristic
53C2×C4⋊C4×C137C2211yescharacteristic
54#106×C137C2211yescharacteristic, complemented
55#106×C137C2211yescharacteristic, complemented
56#103×C137C2211yescharacteristic, complemented
57#103×C137C2211yescharacteristic, complemented
58C22⋊C8×C137C2211yescharacteristic
59C23×D137—21no—
60#14555511—22no—
61#14555511—41no—
62#14555510—44no—
63C2×Dic274—21no—
64C2×D548—22no—
65C2×D548—21no—
66C2×D548—21no—
67C2×D548—21no—
68C2×D548—21no—
69C2×D548—41no—
70C2×D548C2311yescharacteristic
71C2×D548C2311yescharacteristic
72C2×D548C2311yescharacteristic
73C2×C22⋊D137—21no—
74C2×C22⋊D137D411yescharacteristic
75C2×C4×D137—21no—
76C2×C4×D137—21no—
77C2×C4×D137—21no—
78C2×C4×D137—21no—
79C2×C4×D137C2311yescharacteristic
80#14555497—41no—
81D1096—22no—
82D1096—41no—
83#14555493—41no—
84C2×#11764040—41no—
85C2×#11764040C2311yescharacteristic
86C2×#11764040C2311yescharacteristic
87C4⋊Dic137C2311yescharacteristic
88#14555489—21no—
89#14555489—21no—
90#14555489D411yescharacteristic
91C22⋊Dic137D411yescharacteristic
92#14555487—21no—
93#14555487—21no—
94#14555487—21no—
95#14555487—21no—
96#14555487—21no—
97#14555487—21no—
98#14555487—21no—
99#14555487D411yescharacteristic
100C4×Dic137C2311yescharacteristic
101C2×Q8×C137C2311yescharacteristic
102C22×C4×C137—21no—
103C22×C4×C137C2311yescharacteristic
104C2×C8×C137C2311yescharacteristic
105C2×C8×C137C2311yescharacteristic
106C4⋊C4×C137—21no—
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—21no—
115C22×D137—42no—
116C22×D137—42no—
117C22×D137—81no—
118C22×D137C2×D411yescharacteristic
119Dic274—42no—
120D548—21no—
121D548—21no—
122D548—21no—
123D548—21no—
124D548—21no—
125D548—42no—
126D548—42no—
127D548—42no—
128D548—81no—
129D548#4112yescomplemented
130C22⋊D137—22no—
131C22⋊D137—42no—
132C2×Dic137—21no—
133C2×Dic137C2×D411yescharacteristic
134C2×Dic137C2×D411yescharacteristic
135C2×Dic137C2×D411yescharacteristic
136C4×D137—21no—
137C4×D137—21no—
138C4×D137—42no—
139C4×D137—42no—
140C4×D137—42no—
141C4×D137—81no—
142#11764040—21no—
143#11764040—21no—
144C23×C137C2×D411yescharacteristic
145Q8×C137—21no—
146Q8×C137—21no—
147C2×C4×C137—21no—
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×C137AGL(1,ℤ/8)12yes—
161C22×C137C22×D411yescharacteristic
162C2×D137—21no—
163C2×D137—22no—
164C2×D137—21no—
165C2×D137—21no—
166C2×D137—41no—
167C2×D137—41no—
168C2×D137—41no—
169C2×D137—41no—
170C2×D137—41no—
171C2×D137—41no—
172C2×D137—41no—
173C2×D137—41no—
174C2×D137—84no—
175C4×C137—21no—
176C4×C137—41no—
177C4×C137—41no—
178C4×C137—41no—
179C4×C137—41no—
180C4×C137C2×#4111yescharacteristic
181C4×C137H(5,𝔽2)11yescharacteristic
182Dic137—21no—
183Dic137—21no—
184Dic137—21no—
185Dic137—41no—
186C2×C137—22no—
187C2×C137#54611yescharacteristic
188C2×C137C2×AGL(1,ℤ/8)11yescharacteristic
189C2×C137#57711yescharacteristic
190D137—41no—
191D137—42no—
192D137—81no—
193D137—81no—
194C137#330611yescharacteristic, complemented, Sylow
195#3306—1371nomaximal, Sylow
196#547—1371no—
197#546—1371no—
198#489—1371no—
199#483D13711yescharacteristic, complemented
200#481—1371no—
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⋊Q8C2×D13711yescharacteristic
226#121—2741no—
227#121—1371no—
228#121—1371no—
229#121—1371no—
230#120—2741no—
231C4×D4—1371no—
232C4×D4—1371no—
233C4×D4—1371no—
234C2×C4⋊C4—1371no—
235C2×C4⋊C4C2×D13711yescharacteristic
236C2×C22⋊C4—2741no—
237#106—1371no—
238#106C2×D13711yescharacteristic, complemented
239#106C2×D13711yescharacteristic, complemented
240C4⋊C8—1371no—
241#103—1371no—
242#103C2×D13711yescharacteristic, complemented
243#103C2×D13711yescharacteristic, complemented
244#102—1371no—
245#102—1371no—
246#102—1371no—
247#102—1371no—
248#102—1371no—
249#102—1371no—
250#102—1371no—
251C22⋊C8—1371no—
252C22⋊C8C2×D13711yescharacteristic
253#97—1371no—
254C24—2741no—
255#41—2742no—
256#41—5484no—
257#41—5481no—
258C2×Q8—2741no—
259C2×Q8C22×D13711yescharacteristic
260C2×D4—2742no—
261C2×D4—2741no—
262C2×D4—2741no—
263C2×D4—2741no—
264C2×D4—2741no—
265C2×D4—2741no—
266C2×D4—5481no—
267C2×D4—1371no—
268C2×D4—1371no—
269C2×D4—1371no—
270C2×D4—1371no—
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—21no—
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—2741no—
308C23—2741no—
309C23—2741no—
310C23—5482no—
311C23—5482no—
312C23—10961no—
313C23—1371no—
314C23D4×D13711yescharacteristic
315Q8—21no—
316Q8—21no—
317Q8—5482no—
318D4—2742no—
319D4—2741no—
320D4—2741no—
321D4—2741no—
322D4—2741no—
323D4—2741no—
324D4—5482no—
325D4—5482no—
326D4—5482no—
327D4—5482no—
328D4—10961no—
329D4—1372no—
330C2×C4—21no—
331C2×C4—21no—
332C2×C4—21no—
333C2×C4—21no—
334C2×C4—21no—
335C2×C4—21no—
336C2×C4—42no—
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—2741no—
356C22—2741no—
357C22—2741no—
358C22—2742no—
359C22—5481no—
360C22—5481no—
361C22—5481no—
362C22—5481no—
363C22—5481no—
364C22—5481no—
365C22—5481no—
366C22—5481no—
367C22—10964no—
368C22#1798411512yes—
369C22C2×D4×D13711yescharacteristic
370C4—21no—
371C4—41no—
372C4—41no—
373C4—41no—
374C4—41no—
375C4—2741no—
376C4—2741no—
377C4—2741no—
378C4—5481no—
379C4#41×D13711yescharacteristic
380C4#1798413411yescharacteristic
381C2—22no—
382C2—5481no—
383C2—5482no—
384C2—10961no—
385C2—10961no—
386C2#2257804511yescharacteristic
387C2C2×#1798411511yescharacteristic
388C2#2257821211yescharacteristic
subgroupG/H
centerC22C2×D4×D137
commutator, cosocleC2×C4×C137C24
FrattiniC2×C4C23×D137
Fitting#483×C137C2
socleC22×C137C22×D4
hypercenter#483D137
hypocenterC137#3306
LCSG ▷ C2×C4×C137 ▷ C2×C137 ▷ C137
UCSC1 ◁ C22 ◁ C22×C4 ◁ #483

Complements

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

NH ≅ G/Ntype
1. #22578054C2semidirect
2. #22578044C2semidirect
3. #22577764C2semidirect
4. #22577723C2semidirect
5. #22577692C2semidirect
6. #22577657C2semidirect
7. #22577634C2semidirect
8. #22577621C2semidirect
9. #22577551C2semidirect
10. #22577539C2semidirect
11. #22577514C2semidirect
12. #22577471C2semidirect
13. #22577403C2semidirect
14. C2×#17984008C2semidirect
15. #483×C137C2semidirect
19. C2×#14555497C22semidirect
22. C2×#14555493C22semidirect
24. #17984087C22semidirect
32. #17984063C22semidirect
36. #17984050C22semidirect
38. C4⋊C4×D137C22semidirect
40. #17984019C22semidirect
41. #17984017C22semidirect
42. #17984012C22semidirect
49. #17984008C22semidirect
50. #17984000C22semidirect
51. #17983998C22semidirect
54. #106×C137C22semidirect
55. #106×C137C22semidirect
56. #103×C137C22semidirect
57. #103×C137C22semidirect
129. D548#41semidirect
194. C137#3306semidirect
199. #483D137semidirect
238. #106C2×D137semidirect
239. #106C2×D137semidirect
242. #103C2×D137semidirect
243. #103C2×D137semidirect

Maximal subgroups

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

subgroupindex
#225780542
#225780442
#225777642
#225777232
#225776922
#225776572
#225776342
#225776212
#225775512
#225775392
#225775142
#225774712
#225774032
C2×#179840082
#483×C1372
#3306137