Finite Group Databasebeta

← #31792744

Subgroups — #31792744

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#22578049C211yesmaximal, characteristic, complemented
2#22577765C211yesmaximal, characteristic, complemented
3#22577733C211yesmaximal, characteristic, complemented
4#22577721C211yesmaximal, characteristic, complemented
5#22577690C211yesmaximal, characteristic, complemented
6#22577654C211yesmaximal, characteristic, complemented
7#22577630C211yesmaximal, characteristic, complemented
8#22577618C211yesmaximal, characteristic, complemented
9#22577546C211yesmaximal, characteristic, complemented
10#22577536C211yesmaximal, characteristic, complemented
11#22577513C211yesmaximal, characteristic, complemented
12#22577475C211yesmaximal, characteristic, complemented
13#22577402C211yesmaximal, characteristic, complemented
14C2×#17984008C211yesmaximal, characteristic, complemented
15#547×C137C211yesmaximal, characteristic, complemented
16C2×#14555510C2211yescharacteristic
17C22×D548C2211yescharacteristic
18C2×#14555498C2211yescharacteristic, complemented
19C2×#14555495C2211yescharacteristic
20C2×#14555495C2211yescharacteristic
21C2×#14555494C2211yescharacteristic, complemented
22C22×#11764040C2211yescharacteristic
23#17984076C2211yescharacteristic
24#17984073—21no—
25#17984067C2211yescharacteristic
26#17984062C2211yescharacteristic
27#17984061—21no—
28#17984056—21no—
29C4×D548C2211yescharacteristic
30C22⋊C4×D137—21no—
31#17984019C2211yescharacteristic, complemented
32#17984019C2211yescharacteristic, complemented
33#17984019C2211yescharacteristic, complemented
34C4⋊#11764040C2211yescharacteristic, complemented
35#17984011C2211yescharacteristic, complemented
36#17984011C2211yescharacteristic, complemented
37#17984011C2211yescharacteristic, complemented
38#17984009C2211yescharacteristic, complemented
39#17984008C2211yescharacteristic
40#17984008C2211yescharacteristic
41#17984008C2211yescharacteristic
42#17984008C2211yescharacteristic
43#17984008C2211yescharacteristic
44#17984008C2211yescharacteristic
45C22⋊#11764040C2211yescharacteristic
46C22⋊#11764040C2211yescharacteristic, complemented
47#17983999C2211yescharacteristic, complemented
48C2×#41×C137—21no—
49C4⋊Q8×C137C2211yescharacteristic, complemented
50#123×C137—21no—
51#123×C137—21no—
52C22⋊Q8×C137—21no—
53#121×C137C2211yescharacteristic
54#121×C137C2211yescharacteristic, complemented
55C4×D4×C137C2211yescharacteristic
56C4×D4×C137C2211yescharacteristic, complemented
57C2×C4⋊C4×C137C2211yescharacteristic
58C2×C4⋊C4×C137C2211yescharacteristic, complemented
59C23×D137—21no—
60#14555510—22no—
61#14555510—41no—
62C2×Dic274C2311yescharacteristic
63C2×D548—22no—
64C2×D548—21no—
65C2×D548—41no—
66C2×D548C2311yescharacteristic
67C2×D548C2311yescharacteristic
68C2×C22⋊D137—21no—
69C2×C22⋊D137—21no—
70C2×C4×D137—21no—
71C2×C4×D137—21no—
72#14555498—41no—
73#14555495—22no—
74#14555495—41no—
75#14555494—41no—
76C2×#11764040—41no—
77C2×#11764040C2311yescharacteristic
78C2×#11764040C2311yescharacteristic
79C2×#11764040C2311yescharacteristic
80C2×#11764040C2311yescharacteristic
81C4⋊Dic137C2311yescharacteristic
82#14555489—21no—
83C22⋊Dic137—21no—
84#14555487—21no—
85#14555487—21no—
86#14555487—21no—
87#14555487—21no—
88#41×C137—44no—
89C2×Q8×C137—21no—
90C2×D4×C137—21no—
91C2×D4×C137—21no—
92C2×D4×C137D411yescharacteristic
93C2×D4×C137C2311yescharacteristic
94C22×C4×C137—21no—
95C22×C4×C137—21no—
96C22×C4×C137—21no—
97C22×C4×C137C2311yescharacteristic
98C22×C4×C137C2311yescharacteristic
99C4⋊C4×C137—21no—
100C4⋊C4×C137—21no—
101C4⋊C4×C137—21no—
102C4⋊C4×C137D411yescharacteristic
103C4⋊C4×C137C2311yescharacteristic
104C4⋊C4×C137C2311yescharacteristic
105C4⋊C4×C137C2311yescharacteristic
106C22⋊C4×C137—21no—
107C22⋊C4×C137—21no—
108C22⋊C4×C137—21no—
109C22⋊C4×C137D411yescharacteristic
110C22⋊C4×C137D411yescharacteristic
111C42×C137C2311yescharacteristic
112C22×D137—21no—
113C22×D137—21no—
114C22×D137—42no—
115C22×D137—42no—
116C22×D137—81no—
117Dic274—21no—
118Dic274—21no—
119D548—21no—
120D548—21no—
121D548—21no—
122D548—42no—
123D548#4112yescomplemented
124C22⋊D137—42no—
125C22⋊D137—81no—
126C2×Dic137—21no—
127C2×Dic137—21no—
128C4×D137—42no—
129C4×D137—81no—
130#11764040—21no—
131#11764040—21no—
132#11764040—21no—
133#11764040—21no—
134C23×C137—21no—
135C23×C137C2×D411yescharacteristic
136C23×C137C2×D411yescharacteristic
137Q8×C137—42no—
138D4×C137—22no—
139D4×C137—21no—
140D4×C137—21no—
141D4×C137—42no—
142D4×C137—42no—
143C2×C4×C137—21no—
144C2×C4×C137—21no—
145C2×C4×C137—21no—
146C2×C4×C137—21no—
147C2×C4×C137—21no—
148C2×C4×C137—21no—
149C2×C4×C137—21no—
150C2×C4×C137—42no—
151C2×C4×C137—42no—
152C2×C4×C137—42no—
153C2×C4×C137C2×D411yescharacteristic
154C2×C4×C137C2×D411yescharacteristic
155C2×C4×C137C2×D411yescharacteristic
156C2×C4×C137C2×D411yescharacteristic
157C2×C4×C137C2411yescharacteristic
158C22×C137—21no—
159C22×C137—21no—
160C22×C137—21no—
161C22×C137—21no—
162C22×C137—21no—
163C22×C137—41no—
164C22×C137—41no—
165C22×C137—41no—
166C22×C137AGL(1,ℤ/8)12yes—
167C22×C137C22×D411yescharacteristic
168C2×D137—22no—
169C2×D137—41no—
170C2×D137—41no—
171C2×D137—41no—
172C2×D137—41no—
173C2×D137—41no—
174C2×D137—84no—
175C4×C137—21no—
176C4×C137—21no—
177C4×C137—21no—
178C4×C137—21no—
179C4×C137—41no—
180C4×C137—41no—
181C4×C137—41no—
182C4×C137C2×#4111yescharacteristic
183C4×C137H(5,𝔽2)11yescharacteristic
184Dic137—41no—
185Dic137—41no—
186C2×C137—22no—
187C2×C137—41no—
188C2×C137—81no—
189C2×C137#54611yescharacteristic
190C2×C137C2×AGL(1,ℤ/8)11yescharacteristic
191C2×C137#57711yescharacteristic
192D137—42no—
193D137—81no—
194C137#330611yescharacteristic, complemented, Sylow
195#3306—1371nomaximal, Sylow
196#547D13711yescharacteristic, complemented
197#546—1371no—
198#489—1371no—
199#483—1371no—
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—21no—
212C2×#41—1371no—
213C22×D4—1371no—
214C2×SD16—1371no—
215C2×D8—1371no—
216C2×D8—1371no—
217C2×M16—1371no—
218C22×C8—1371no—
219C4⋊Q8C2×D13711yescharacteristic, complemented
220#124—1371no—
221#123—21no—
222#123—21no—
223#123—2741no—
224C22⋊Q8—21no—
225C22⋊Q8—1371no—
226#121—2741no—
227#121—1371no—
228#121C2×D13711yescharacteristic
229#121C2×D13711yescharacteristic, complemented
230#120—2741no—
231C4×D4—1371no—
232C4×D4C2×D13711yescharacteristic
233C4×D4C2×D13711yescharacteristic, complemented
234C2×C4⋊C4C2×D13711yescharacteristic
235C2×C4⋊C4C2×D13711yescharacteristic, complemented
236C2×C22⋊C4—2741no—
237#106—1371no—
238#106—1371no—
239#106—1371no—
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#102—1371no—
250#102—1371no—
251C22⋊C8—1371no—
252C22⋊C8—1371no—
253#97—1371no—
254C24—2741no—
255#41—44no—
256#41—2742no—
257#41—5481no—
258C2×Q8—21no—
259C2×Q8—1371no—
260C2×D4—21no—
261C2×D4—21no—
262C2×D4—2742no—
263C2×D4—2741no—
264C2×D4—2741no—
265C2×D4—2741no—
266C2×D4—5481no—
267C2×D4—1371no—
268C2×D4—1371no—
269C2×D4C22⋊D13711yescharacteristic
270C2×D4C22×D13711yescharacteristic
271C22×C4—21no—
272C22×C4—21no—
273C22×C4—21no—
274C22×C4—2741no—
275C22×C4—2741no—
276C22×C4C22×D13711yescharacteristic
277C22×C4C22×D13711yescharacteristic
278SD16—5481no—
279D8—2742no—
280D8—5481no—
281M16—5481no—
282C2×C8—5481no—
283C2×C8—1371no—
284C2×C8—1371no—
285C2×C8—1371no—
286C2×C8—1371no—
287C4⋊C4—21no—
288C4⋊C4—21no—
289C4⋊C4—21no—
290C4⋊C4—2741no—
291C4⋊C4—1371no—
292C4⋊C4C22⋊D13711yescharacteristic
293C4⋊C4C22×D13711yescharacteristic
294C4⋊C4C22×D13711yescharacteristic
295C4⋊C4C22×D13711yescharacteristic
296C22⋊C4—21no—
297C22⋊C4—21no—
298C22⋊C4—21no—
299C22⋊C4—2741no—
300C22⋊C4—2741no—
301C22⋊C4—2741no—
302C22⋊C4—2741no—
303C22⋊C4—2741no—
304C22⋊C4C22⋊D13711yescharacteristic
305C22⋊C4C22⋊D13711yescharacteristic
306C42C22×D13711yescharacteristic
307C23—21no—
308C23—2741no—
309C23—2741no—
310C23—5482no—
311C23—5482no—
312C23—10961no—
313C23C2×C22⋊D13711yescharacteristic
314C23C2×C22⋊D13711yescharacteristic
315Q8—42no—
316Q8—2741no—
317Q8—2741no—
318D4—22no—
319D4—21no—
320D4—21no—
321D4—42no—
322D4—42no—
323D4—2741no—
324D4—2741no—
325D4—2741no—
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—21no—
337C2×C4—42no—
338C2×C4—42no—
339C2×C4—42no—
340C2×C4—2741no—
341C2×C4—2741no—
342C2×C4—5482no—
343C2×C4—10961no—
344C2×C4C2×C22⋊D13711yescharacteristic
345C2×C4C2×C22⋊D13711yescharacteristic
346C2×C4C2×C22⋊D13711yescharacteristic
347C2×C4C2×C22⋊D13711yescharacteristic
348C2×C4C23×D13711yescharacteristic
349C8—2741no—
350C8—2741no—
351C8—2741no—
352C8—2741no—
353C22—21no—
354C22—21no—
355C22—21no—
356C22—21no—
357C22—21no—
358C22—41no—
359C22—41no—
360C22—41no—
361C22—2742no—
362C22—5481no—
363C22—5481no—
364C22—5481no—
365C22—5481no—
366C22—5481no—
367C22—10964no—
368C22#1798411412yes—
369C22C22×C22⋊D13711yescharacteristic
370C4—21no—
371C4—21no—
372C4—21no—
373C4—21no—
374C4—41no—
375C4—41no—
376C4—41no—
377C4—5481no—
378C4—5481no—
379C4#41×D13711yescharacteristic
380C4#1798413411yescharacteristic
381C2—22no—
382C2—41no—
383C2—81no—
384C2—5482no—
385C2—10961no—
386C2#2257804111yescharacteristic
387C2C2×#1798411411yescharacteristic
388C2#2257821111yescharacteristic
subgroupG/H
centerC22C22×C22⋊D137
commutator, cosocleC2×C4×C137C24
FrattiniC2×C4C23×D137
Fitting#547×C137C2
socleC22×C137C22×D4
hypercenter#547D137
hypocenterC137#3306
LCSG ▷ C2×C4×C137 ▷ C2×C137 ▷ C137
UCSC1 ◁ C22 ◁ C4×D4 ◁ #547

Complements

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

NH ≅ G/Ntype
1. #22578049C2semidirect
2. #22577765C2semidirect
3. #22577733C2semidirect
4. #22577721C2semidirect
5. #22577690C2semidirect
6. #22577654C2semidirect
7. #22577630C2semidirect
8. #22577618C2semidirect
9. #22577546C2semidirect
10. #22577536C2semidirect
11. #22577513C2semidirect
12. #22577475C2semidirect
13. #22577402C2semidirect
14. C2×#17984008C2semidirect
15. #547×C137C2semidirect
18. C2×#14555498C22semidirect
21. C2×#14555494C22semidirect
31. #17984019C22semidirect
32. #17984019C22semidirect
33. #17984019C22semidirect
34. C4⋊#11764040C22semidirect
35. #17984011C22semidirect
36. #17984011C22semidirect
37. #17984011C22semidirect
38. #17984009C22semidirect
46. C22⋊#11764040C22semidirect
47. #17983999C22semidirect
49. C4⋊Q8×C137C22semidirect
54. #121×C137C22semidirect
56. C4×D4×C137C22semidirect
58. C2×C4⋊C4×C137C22semidirect
123. D548#41semidirect
194. C137#3306semidirect
196. #547D137semidirect
219. C4⋊Q8C2×D137semidirect
229. #121C2×D137semidirect
233. C4×D4C2×D137semidirect
235. C2×C4⋊C4C2×D137semidirect

Maximal subgroups

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

subgroupindex
#225780492
#225777652
#225777332
#225777212
#225776902
#225776542
#225776302
#225776182
#225775462
#225775362
#225775132
#225774752
#225774022
C2×#179840082
#547×C1372
#3306137