Finite Group Databasebeta

← #31792746

Subgroups — #31792746

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#22578047C211yesmaximal, characteristic, complemented
3#22577761C211yesmaximal, characteristic, complemented
4#22577733C211yesmaximal, characteristic, complemented
5#22577720C211yesmaximal, characteristic, complemented
6#22577691C211yesmaximal, characteristic, complemented
7#22577655C211yesmaximal, characteristic, complemented
8#22577632C211yesmaximal, characteristic, complemented
9#22577617C211yesmaximal, characteristic, complemented
10#22577549C211yesmaximal, characteristic, complemented
11#22577537C211yesmaximal, characteristic, complemented
12#22577512C211yesmaximal, characteristic, complemented
13#22577474C211yesmaximal, characteristic, complemented
14#22577401C211yesmaximal, characteristic, complemented
15C2×#102×C137C211yesmaximal, characteristic, complemented
16C2×#14555510—21no—
17C2×#14555509C2211yescharacteristic
18C2×#14555499C2211yescharacteristic, complemented
19C2×#14555495C2211yescharacteristic
20C2×#14555495C2211yescharacteristic
21C2×#14555493C2211yescharacteristic, complemented
22#17984087C2211yescharacteristic, complemented
23#17984075C2211yescharacteristic
24#17984073—21no—
25#17984071—21no—
26#17984069—21no—
27#17984067C2211yescharacteristic
28#17984066—21no—
29#17984063C2211yescharacteristic, complemented
30#17984062C2211yescharacteristic
31#17984060—21no—
32#17984059C2211yescharacteristic
33#17984057—21no—
34D4×Dic137C2211yescharacteristic
35#17984050C2211yescharacteristic, complemented
36C4×C22⋊D137C2211yescharacteristic
37C4⋊C4×D137C2211yescharacteristic, complemented
38C2×C22⋊Dic137—21no—
39#17984020C2211yescharacteristic, complemented
40#17984019C2211yescharacteristic, complemented
41#17984019C2211yescharacteristic, complemented
42#17984017C2211yescharacteristic, complemented
43#17984013C2211yescharacteristic, complemented
44#17984011C2211yescharacteristic, complemented
45#17984011C2211yescharacteristic, complemented
46#17984010C2211yescharacteristic, complemented
47#17984009C2211yescharacteristic
48#17984009C2211yescharacteristic
49C22⋊#11764040C2211yescharacteristic
50#17984000C2211yescharacteristic, complemented
51#17983998C2211yescharacteristic, complemented
52C22×D4×C137C2211yescharacteristic
53C22×C8×C137C2211yescharacteristic
54C2×C4⋊C4×C137C2211yescharacteristic
55#102×C137C2211yescharacteristic
56#102×C137C2211yescharacteristic
57#102×C137C2211yescharacteristic
58#102×C137C2211yescharacteristic
59#14555510—44no—
60#14555509—22no—
61#14555509—41no—
62C2×Dic274—21no—
63C2×Dic274C2311yescharacteristic
64C2×D548—21no—
65C2×D548C2311yescharacteristic
66C2×C22⋊D137—21no—
67C2×C22⋊D137—21no—
68C2×C22⋊D137—21no—
69C2×C22⋊D137D411yescharacteristic
70C22×Dic137—21no—
71C22×Dic137—21no—
72C2×C4×D137—21no—
73C2×C4×D137—21no—
74C2×C4×D137C2311yescharacteristic
75#14555499—41no—
76#14555495—22no—
77#14555495—41no—
78#14555493—41no—
79C2×#11764040C2311yescharacteristic
80C2×#11764040C2311yescharacteristic
81C4⋊Dic137C2311yescharacteristic
82C4⋊Dic137C2311yescharacteristic
83#14555489—21no—
84#14555489—21no—
85#14555489—21no—
86#14555489D411yescharacteristic
87C22⋊Dic137—21no—
88C22⋊Dic137—21no—
89C22⋊Dic137—21no—
90C22⋊Dic137—21no—
91C22⋊Dic137—21no—
92C22⋊Dic137D411yescharacteristic
93#14555487—21no—
94#14555487—21no—
95#14555487—21no—
96#14555487D411yescharacteristic
97C4×Dic137C2311yescharacteristic
98C24×C137—21no—
99C2×D4×C137—22no—
100C2×D4×C137—21no—
101C2×D4×C137—41no—
102C2×D4×C137C2311yescharacteristic
103C2×D4×C137C2311yescharacteristic
104C22×C4×C137—21no—
105C22×C4×C137C2311yescharacteristic
106C2×C8×C137—41no—
107C2×C8×C137C2311yescharacteristic
108C2×C8×C137C2311yescharacteristic
109C4⋊C4×C137—21no—
110C4⋊C4×C137C2311yescharacteristic
111C4⋊C4×C137C2311yescharacteristic
112C22×D137—21no—
113C22×D137C2×D411yescharacteristic
114Dic274—21no—
115Dic274—21no—
116Dic274—42no—
117D548—21no—
118D548—21no—
119D548—42no—
120C22⋊D137—22no—
121C22⋊D137—42no—
122C22⋊D137—42no—
123C22⋊D137—81no—
124C2×Dic137—21no—
125C2×Dic137—21no—
126C2×Dic137—21no—
127C2×Dic137—42no—
128C2×Dic137—81no—
129C2×Dic137C2×D411yescharacteristic
130C2×Dic137C2×D411yescharacteristic
131C2×Dic137C2×D411yescharacteristic
132C4×D137—21no—
133C4×D137—21no—
134C4×D137—42no—
135C4×D137—42no—
136#11764040—21no—
137#11764040—21no—
138C23×C137—21no—
139C23×C137—21no—
140C23×C137—42no—
141C23×C137—42no—
142C23×C137—81no—
143C23×C137C2×D411yescharacteristic
144D4×C137—21no—
145D4×C137—21no—
146D4×C137—21no—
147D4×C137—42no—
148D4×C137#4112yescomplemented
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—22no—
161C22×C137—41no—
162C22×C137—41no—
163C22×C137—41no—
164C22×C137—41no—
165C22×C137—41no—
166C22×C137—84no—
167C22×C137AGL(1,ℤ/8)12yes—
168C22×C137C22×D411yescharacteristic
169C2×D137—21no—
170C2×D137—21no—
171C2×D137—21no—
172C2×D137—41no—
173C2×D137—41no—
174C2×D137—41no—
175C4×C137—21no—
176C4×C137—41no—
177C4×C137—41no—
178C4×C137C2×#4111yescharacteristic
179C4×C137H(5,𝔽2)11yescharacteristic
180Dic137—21no—
181Dic137—21no—
182Dic137—21no—
183Dic137—41no—
184Dic137—41no—
185Dic137—41no—
186C2×C137—22no—
187C2×C137—42no—
188C2×C137—81no—
189C2×C137#54611yescharacteristic
190C2×C137C2×AGL(1,ℤ/8)11yescharacteristic
191C2×C137#57711yescharacteristic
192D137—41no—
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#417—1371no—
210C2×#102D13711yescharacteristic, complemented
211C2×#41—2741no—
212C2×#41—1371no—
213C22×D4C2×D13711yescharacteristic
214C2×SD16—1371no—
215C2×D8—1371no—
216C2×D8—1371no—
217C2×M16—1371no—
218C22×C8C2×D13711yescharacteristic
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
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#102C2×D13711yescharacteristic
248#102C2×D13711yescharacteristic
249#102C2×D13711yescharacteristic
250#102C2×D13711yescharacteristic
251C22⋊C8—1371no—
252C22⋊C8—1371no—
253#97—1371no—
254C24—21no—
255#41—2742no—
256#41—5484no—
257#41—5481no—
258C2×Q8—2741no—
259C2×Q8—1371no—
260C2×D4—22no—
261C2×D4—21no—
262C2×D4—41no—
263C2×D4—2741no—
264C2×D4—2741no—
265C2×D4—2741no—
266C2×D4—2741no—
267C2×D4—1371no—
268C2×D4—1371no—
269C2×D4C22×D13711yescharacteristic
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—41no—
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—21no—
309C23—42no—
310C23—42no—
311C23—81no—
312C23—2741no—
313C23—1371no—
314C23D4×D13711yescharacteristic
315Q8—2741no—
316Q8—2741no—
317Q8—5482no—
318D4—21no—
319D4—21no—
320D4—21no—
321D4—42no—
322D4—2742no—
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—42no—
335C2×C4—2741no—
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—22no—
356C22—41no—
357C22—41no—
358C22—41no—
359C22—41no—
360C22—41no—
361C22—84no—
362C22—2741no—
363C22—2741no—
364C22—2741no—
365C22—5481no—
366C22—5481no—
367C22—5481no—
368C22#1798411212yes—
369C22C2×D4×D13711yescharacteristic
370C4—21no—
371C4—41no—
372C4—41no—
373C4—2741no—
374C4—2741no—
375C4—2741no—
376C4—5481no—
377C4—5481no—
378C4—5481no—
379C4C2×#1455551011yescharacteristic
380C4#1798413311yescharacteristic
381C2—22no—
382C2—42no—
383C2—81no—
384C2—5481no—
385C2—10961no—
386C2#2257804311yescharacteristic
387C2C2×#1798411211yescharacteristic
388C2#2257820911yescharacteristic
subgroupG/H
centerC22C2×D4×D137
commutator, cosocleC2×C4×C137C24
FrattiniC2×C4C23×D137
FittingC2×#102×C137C2
socleC22×C137C22×D4
hypercenterC2×#102D137
hypocenterC137#3306
LCSG ▷ C2×C4×C137 ▷ C2×C137 ▷ C137
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×#102

Complements

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

NH ≅ G/Ntype
1. #22578054C2semidirect
2. #22578047C2semidirect
3. #22577761C2semidirect
4. #22577733C2semidirect
5. #22577720C2semidirect
6. #22577691C2semidirect
7. #22577655C2semidirect
8. #22577632C2semidirect
9. #22577617C2semidirect
10. #22577549C2semidirect
11. #22577537C2semidirect
12. #22577512C2semidirect
13. #22577474C2semidirect
14. #22577401C2semidirect
15. C2×#102×C137C2semidirect
18. C2×#14555499C22semidirect
21. C2×#14555493C22semidirect
22. #17984087C22semidirect
29. #17984063C22semidirect
35. #17984050C22semidirect
37. C4⋊C4×D137C22semidirect
39. #17984020C22semidirect
40. #17984019C22semidirect
41. #17984019C22semidirect
42. #17984017C22semidirect
43. #17984013C22semidirect
44. #17984011C22semidirect
45. #17984011C22semidirect
46. #17984010C22semidirect
50. #17984000C22semidirect
51. #17983998C22semidirect
148. D4×C137#41semidirect
194. C137#3306semidirect
210. C2×#102D137semidirect
329. D4#14555510semidirect

Maximal subgroups

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

subgroupindex
#225780542
#225780472
#225777612
#225777332
#225777202
#225776912
#225776552
#225776322
#225776172
#225775492
#225775372
#225775122
#225774742
#225774012
C2×#102×C1372
#3306137