Finite Group Databasebeta

← #31792751

Subgroups — #31792751

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#22578041C211yesmaximal, characteristic, complemented
3#22577764C211yesmaximal, characteristic, complemented
4#22577732C211yesmaximal, characteristic, complemented
5#22577721C211yesmaximal, characteristic, complemented
6#22577691C211yesmaximal, characteristic, complemented
7#22577656C211yesmaximal, characteristic, complemented
8#22577635C211yesmaximal, characteristic, complemented
9#22577621C211yesmaximal, characteristic, complemented
10#22577548C211yesmaximal, characteristic, complemented
11#22577538C211yesmaximal, characteristic, complemented
12#22577512C211yesmaximal, characteristic, complemented
13#22577472C211yesmaximal, characteristic, complemented
14#102×D137C211yesmaximal, characteristic, complemented
15#417×C137C211yesmaximal, characteristic, complemented
16C2×#14555510—21no—
17C2×D4×D137C2211yescharacteristic
18C2×#14555497C2211yescharacteristic, complemented
19C2×D1096C2211yescharacteristic
20C2×#14555495C2211yescharacteristic
21C2×C8×D137C2211yescharacteristic
22#17984087C2211yescharacteristic, complemented
23#17984077C2211yescharacteristic
24#17984073—21no—
25#17984071—21no—
26#17984071—21no—
27#17984066—21no—
28#17984064C2211yescharacteristic
29#17984063C2211yescharacteristic
30#17984062C2211yescharacteristic
31#17984062C2211yescharacteristic, complemented
32#17984060—21no—
33#17984056—21no—
34D4×Dic137C2211yescharacteristic
35#17984050C2211yescharacteristic, complemented
36C4×C22⋊D137C2211yescharacteristic
37C4⋊C4×D137C2211yescharacteristic
38C2×#14555487—21no—
39#17984020C2211yescharacteristic, complemented
40#17984019C2211yescharacteristic, complemented
41#17984019C2211yescharacteristic, complemented
42#17984017C2211yescharacteristic, complemented
43#17984012C2211yescharacteristic, complemented
44#17984010C2211yescharacteristic
45#17984010C2211yescharacteristic, complemented
46#17984009C2211yescharacteristic
47#17984008C2211yescharacteristic
48#17984008C2211yescharacteristic
49C22⋊#11764040C2211yescharacteristic, complemented
50#17984000C2211yescharacteristic
51#17983998C2211yescharacteristic, complemented
52C2×#41×C137C2211yescharacteristic
53C2×M16×C137C2211yescharacteristic, complemented
54C2×C4⋊C4×C137C2211yescharacteristic, complemented
55#103×C137C2211yescharacteristic, complemented
56#103×C137C2211yescharacteristic, complemented
57#102×C137C2211yescharacteristic
58#102×C137C2211yescharacteristic
59C23×D137—21no—
60#14555510—44no—
61C2×Dic274—21no—
62C2×D548—21no—
63C2×D548C2311yescharacteristic
64C2×D548C2311yescharacteristic
65C2×C22⋊D137—21no—
66C2×C22⋊D137—21no—
67C2×C22⋊D137—21no—
68C2×C22⋊D137D411yescharacteristic
69D4×D137—22no—
70D4×D137—41no—
71C22×Dic137—21no—
72C2×C4×D137—21no—
73C2×C4×D137—21no—
74C2×C4×D137C2311yescharacteristic
75#14555497—41no—
76D1096—41no—
77#14555495—22no—
78C2×#11764040C2311yescharacteristic
79C2×#11764040C2311yescharacteristic
80C8×D137—41no—
81C4⋊Dic137C2311yescharacteristic
82C4⋊Dic137C2311yescharacteristic
83#14555489—21no—
84#14555489—21no—
85#14555489—21no—
86#14555489D411yescharacteristic
87C22⋊Dic137—21no—
88C22⋊Dic137D411yescharacteristic
89#14555487—21no—
90#14555487—21no—
91#14555487—21no—
92#14555487—21no—
93#14555487—21no—
94#14555487—21no—
95#14555487—21no—
96#14555487D411yescharacteristic
97C4×Dic137C2311yescharacteristic
98#41×C137—22no—
99#41×C137—41no—
100C2×Q8×C137C2311yescharacteristic
101C2×D4×C137—21no—
102C2×D4×C137C2311yescharacteristic
103C22×C4×C137—21no—
104C22×C4×C137—21no—
105C22×C4×C137C2311yescharacteristic
106M16×C137—41no—
107C2×C8×C137C2311yescharacteristic
108C2×C8×C137C2311yescharacteristic
109C4⋊C4×C137—21no—
110C4⋊C4×C137C2311yescharacteristic
111C4⋊C4×C137C2311yescharacteristic
112C22×D137—21no—
113C22×D137—21no—
114C22×D137—42no—
115C22×D137—42no—
116C22×D137—81no—
117C22×D137C2×D411yescharacteristic
118Dic274—42no—
119D548—21no—
120D548—21no—
121D548—21no—
122D548—21no—
123D548—42no—
124C22⋊D137—22no—
125C22⋊D137—42no—
126C22⋊D137—42no—
127C22⋊D137—81no—
128C2×Dic137—21no—
129C2×Dic137—21no—
130C2×Dic137—81no—
131C2×Dic137C2×D411yescharacteristic
132C2×Dic137C2×D411yescharacteristic
133C2×Dic137C2×D411yescharacteristic
134C4×D137—21no—
135C4×D137—21no—
136C4×D137—42no—
137C4×D137—42no—
138#11764040—21no—
139#11764040—21no—
140C23×C137—21no—
141C23×C137C2×D411yescharacteristic
142Q8×C137—21no—
143Q8×C137—21no—
144D4×C137—21no—
145D4×C137—42no—
146D4×C137#4112yescomplemented
147C2×C4×C137—21no—
148C2×C4×C137—21no—
149C2×C4×C137—21no—
150C2×C4×C137—21no—
151C2×C4×C137—21no—
152C2×C4×C137—42no—
153C2×C4×C137—42no—
154C2×C4×C137C2×D411yescharacteristic
155C2×C4×C137C2411yescharacteristic
156C8×C137—21no—
157C8×C137—21no—
158C22×C137—21no—
159C22×C137—22no—
160C22×C137—21no—
161C22×C137—21no—
162C22×C137—41no—
163C22×C137—41no—
164C22×C137C22×D411yescharacteristic
165C2×D137—21no—
166C2×D137—21no—
167C2×D137—41no—
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—42no—
187C2×C137—41no—
188C2×C137#54611yescharacteristic
189C2×C137C2×AGL(1,ℤ/8)11yescharacteristic
190C2×C137#57711yescharacteristic
191D137—22no—
192D137—81no—
193D137—81no—
194C137#330611yescharacteristic, complemented, Sylow
195#3306—1371nomaximal, Sylow
196#547—1371no—
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#417D13711yescharacteristic, complemented
210C2×#102—1371no—
211C2×#41—2741no—
212C2×#41C2×D13711yescharacteristic
213C22×D4—1371no—
214C2×SD16—1371no—
215C2×D8—1371no—
216C2×D8—1371no—
217C2×M16C2×D13711yescharacteristic, complemented
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#121—1371no—
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#106—1371no—
239#106—1371no—
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#102C2×D13711yescharacteristic
250#102C2×D13711yescharacteristic
251C22⋊C8—1371no—
252C22⋊C8—1371no—
253#97—1371no—
254C24—2741no—
255#41—22no—
256#41—41no—
257#41—5484no—
258C2×Q8—2741no—
259C2×Q8C22×D13711yescharacteristic
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—21no—
273C22×C4—2741no—
274C22×C4—2741no—
275C22×C4—2741no—
276C22×C4—1371no—
277C22×C4C22×D13711yescharacteristic
278SD16—5481no—
279D8—2742no—
280D8—5481no—
281M16—41no—
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⋊C4—1371no—
294C4⋊C4C22×D13711yescharacteristic
295C4⋊C4C22×D13711yescharacteristic
296C22⋊C4—2741no—
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—21no—
316Q8—21no—
317Q8—5482no—
318D4—21no—
319D4—42no—
320D4—2742no—
321D4—2741no—
322D4—2741no—
323D4—2741no—
324D4—2741no—
325D4—5482no—
326D4—5482no—
327D4—5482no—
328D4—10961no—
329D4#1455551012yescomplemented
330C2×C4—21no—
331C2×C4—21no—
332C2×C4—21no—
333C2×C4—21no—
334C2×C4—21no—
335C2×C4—42no—
336C2×C4—42no—
337C2×C4—2741no—
338C2×C4—2741no—
339C2×C4—2741no—
340C2×C4—2741no—
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—22no—
357C22—41no—
358C22—41no—
359C22—2741no—
360C22—2741no—
361C22—5481no—
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—
379C4C2×#1455551011yescharacteristic
380C4#1798413311yescharacteristic
381C2—41no—
382C2—42no—
383C2—2742no—
384C2—10961no—
385C2—10961no—
386C2#2257804311yescharacteristic
387C2AGL(1,ℤ/8)×D13711yescharacteristic
388C2#2257821211yescharacteristic
subgroupG/H
centerC22C2×D4×D137
commutator, cosocleC2×C4×C137C24
FrattiniC2×C4C23×D137
Fitting#417×C137C2
socleC22×C137C22×D4
hypercenter#417D137
hypocenterC137#3306
LCSG ▷ C2×C4×C137 ▷ C2×C137 ▷ C137
UCSC1 ◁ C22 ◁ C22×C4 ◁ #417

Complements

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

NH ≅ G/Ntype
1. #22578054C2semidirect
2. #22578041C2semidirect
3. #22577764C2semidirect
4. #22577732C2semidirect
5. #22577721C2semidirect
6. #22577691C2semidirect
7. #22577656C2semidirect
8. #22577635C2semidirect
9. #22577621C2semidirect
10. #22577548C2semidirect
11. #22577538C2semidirect
12. #22577512C2semidirect
13. #22577472C2semidirect
14. #102×D137C2semidirect
15. #417×C137C2semidirect
18. C2×#14555497C22semidirect
22. #17984087C22semidirect
31. #17984062C22semidirect
35. #17984050C22semidirect
39. #17984020C22semidirect
40. #17984019C22semidirect
41. #17984019C22semidirect
42. #17984017C22semidirect
43. #17984012C22semidirect
45. #17984010C22semidirect
49. C22⋊#11764040C22semidirect
51. #17983998C22semidirect
53. C2×M16×C137C22semidirect
54. C2×C4⋊C4×C137C22semidirect
55. #103×C137C22semidirect
56. #103×C137C22semidirect
146. D4×C137#41semidirect
194. C137#3306semidirect
209. #417D137semidirect
217. C2×M16C2×D137semidirect
235. C2×C4⋊C4C2×D137semidirect
242. #103C2×D137semidirect
243. #103C2×D137semidirect
329. D4#14555510semidirect

Maximal subgroups

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

subgroupindex
#225780542
#225780412
#225777642
#225777322
#225777212
#225776912
#225776562
#225776352
#225776212
#225775482
#225775382
#225775122
#225774722
#102×D1372
#417×C1372
#3306137