Finite Group Databasebeta

← #31792752

Subgroups — #31792752

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#22578057C211yesmaximal, characteristic, complemented
2#22578049C211yesmaximal, characteristic, complemented
3#22577763C211yesmaximal, characteristic, complemented
4#22577734C211yesmaximal, characteristic, complemented
5#22577722C211yesmaximal, characteristic, complemented
6#22577694C211yesmaximal, characteristic, complemented
7#22577656C211yesmaximal, characteristic, complemented
8#22577632C211yesmaximal, characteristic, complemented
9#22577619C211yesmaximal, characteristic, complemented
10#22577550C211yesmaximal, characteristic, complemented
11#22577539C211yesmaximal, characteristic, complemented
12#22577516C211yesmaximal, characteristic, complemented
13#22577405C211yesmaximal, characteristic, complemented
14C2×#17984010C211yesmaximal, characteristic, complemented
15#435×C137C211yesmaximal, characteristic, complemented
16C2×#14555510C2211yescharacteristic
17C2×#14555509—21no—
18C22×D548C2211yescharacteristic
19C2×#14555499C2211yescharacteristic, complemented
20C2×D1096C2211yescharacteristic
21C2×D1096C2211yescharacteristic
22#17984088C2211yescharacteristic, complemented
23#17984076C2211yescharacteristic
24#17984073—21no—
25#17984072—21no—
26#17984072—21no—
27#17984068—21no—
28#17984067C2211yescharacteristic
29#17984062C2211yescharacteristic
30#17984062C2211yescharacteristic
31#17984062C2211yescharacteristic, complemented
32#17984061—21no—
33#17984056—21no—
34C4×D548C2211yescharacteristic
35C4×D548C2211yescharacteristic, complemented
36C2×C4⋊Dic137C2211yescharacteristic
37C2×C4⋊Dic137C2211yescharacteristic, complemented
38C22⋊C4×D137—21no—
39#17984020C2211yescharacteristic, complemented
40#17984020C2211yescharacteristic, complemented
41#17984020C2211yescharacteristic, complemented
42#17984013C2211yescharacteristic, complemented
43#17984013C2211yescharacteristic, complemented
44#17984013C2211yescharacteristic, complemented
45#17984010C2211yescharacteristic
46#17984010C2211yescharacteristic
47#17984010C2211yescharacteristic
48#17984010C2211yescharacteristic
49#17984010C2211yescharacteristic
50#17984010C2211yescharacteristic
51#17984010C2211yescharacteristic, complemented
52C2×M16×C137C2211yescharacteristic, complemented
53C22×C8×C137C2211yescharacteristic
54C4×D4×C137C2211yescharacteristic
55C4⋊C8×C137C2211yescharacteristic, complemented
56C22⋊C8×C137C2211yescharacteristic
57C22⋊C8×C137C2211yescharacteristic, complemented
58#97×C137C2211yescharacteristic, complemented
59C23×D137—21no—
60#14555510—22no—
61#14555510—41no—
62#14555509—44no—
63C2×Dic274—21no—
64C2×Dic274C2311yescharacteristic
65C2×D548—22no—
66C2×D548—21no—
67C2×D548—41no—
68C2×D548C2311yescharacteristic
69C2×D548C2311yescharacteristic
70C2×D548C2311yescharacteristic
71C2×C22⋊D137—21no—
72C2×C22⋊D137—21no—
73C2×C22⋊D137—21no—
74C2×C22⋊D137—21no—
75C22×Dic137—21no—
76C22×Dic137—21no—
77C2×C4×D137—21no—
78C2×C4×D137—21no—
79C2×C4×D137—21no—
80#14555499—41no—
81D1096—22no—
82D1096—41no—
83C4⋊Dic137—21no—
84C4⋊Dic137—21no—
85C4⋊Dic137—21no—
86C4⋊Dic137C2311yescharacteristic
87C4⋊Dic137C2311yescharacteristic
88C4⋊Dic137C2311yescharacteristic
89C4⋊Dic137C2311yescharacteristic
90#14555489—21no—
91C22⋊Dic137—21no—
92#14555487—21no—
93#14555487—21no—
94#14555487—21no—
95#14555487—21no—
96#14555487—21no—
97#14555487—21no—
98#14555487—21no—
99C2×D4×C137D411yescharacteristic
100C22×C4×C137C2311yescharacteristic
101C22×C4×C137C2311yescharacteristic
102M16×C137—41no—
103C2×C8×C137—41no—
104C2×C8×C137C2311yescharacteristic
105C2×C8×C137C2311yescharacteristic
106C2×C8×C137C2311yescharacteristic
107C2×C8×C137C2311yescharacteristic
108C4⋊C4×C137D411yescharacteristic
109C22⋊C4×C137D411yescharacteristic
110C22⋊C4×C137D411yescharacteristic
111C42×C137C2311yescharacteristic
112C22×D137—21no—
113C22×D137—21no—
114C22×D137—21no—
115C22×D137—42no—
116C22×D137—42no—
117C22×D137—81no—
118Dic274—21no—
119Dic274—21no—
120Dic274—42no—
121D548—21no—
122D548—21no—
123D548—21no—
124D548—21no—
125D548—21no—
126D548—42no—
127D548#4112yescomplemented
128C22⋊D137—42no—
129C22⋊D137—42no—
130C22⋊D137—42no—
131C22⋊D137—81no—
132C2×Dic137—21no—
133C2×Dic137—21no—
134C2×Dic137—21no—
135C2×Dic137—21no—
136C2×Dic137—21no—
137C2×Dic137—42no—
138C2×Dic137—42no—
139C4×D137—42no—
140C4×D137—42no—
141C4×D137—81no—
142C23×C137C2×D411yescharacteristic
143C23×C137C2×D411yescharacteristic
144D4×C137—22no—
145C2×C4×C137—21no—
146C2×C4×C137—21no—
147C2×C4×C137—21no—
148C2×C4×C137—21no—
149C2×C4×C137C2×D411yescharacteristic
150C2×C4×C137C2×D411yescharacteristic
151C2×C4×C137C2×D411yescharacteristic
152C2×C4×C137C2×D411yescharacteristic
153C2×C4×C137C2411yescharacteristic
154C8×C137—21no—
155C8×C137—21no—
156C8×C137—21no—
157C8×C137—21no—
158C22×C137—21no—
159C22×C137—21no—
160C22×C137—21no—
161C22×C137—21no—
162C22×C137—21no—
163C22×C137AGL(1,ℤ/8)12yes—
164C22×C137C22×D411yescharacteristic
165C2×D137—22no—
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—21no—
177C4×C137—21no—
178C4×C137—21no—
179C4×C137C2×#4111yescharacteristic
180C4×C137H(5,𝔽2)11yescharacteristic
181Dic137—41no—
182Dic137—41no—
183Dic137—41no—
184Dic137—41no—
185Dic137—41no—
186C2×C137—22no—
187C2×C137—41no—
188C2×C137#54611yescharacteristic
189C2×C137C2×AGL(1,ℤ/8)11yescharacteristic
190C2×C137#57711yescharacteristic
191D137—42no—
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#435D13711yescharacteristic, complemented
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×M16C2×D13711yescharacteristic, complemented
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×D4C2×D13711yescharacteristic
234C2×C4⋊C4—1371no—
235C2×C4⋊C4—1371no—
236C2×C22⋊C4—2741no—
237#106—1371no—
238#106—1371no—
239#106—1371no—
240C4⋊C8C2×D13711yescharacteristic, complemented
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⋊C8C2×D13711yescharacteristic
252C22⋊C8C2×D13711yescharacteristic, complemented
253#97C2×D13711yescharacteristic, complemented
254C24—2741no—
255#41—2742no—
256#41—5481no—
257#41—5484no—
258C2×Q8—2741no—
259C2×Q8—1371no—
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×D4D54811yescharacteristic
271C22×C4—2741no—
272C22×C4—2741no—
273C22×C4—2741no—
274C22×C4—2741no—
275C22×C4—2741no—
276C22×C4C22×D13711yescharacteristic
277C22×C4C22×D13711yescharacteristic
278SD16—5481no—
279D8—2742no—
280D8—5481no—
281M16—41no—
282C2×C8—41no—
283C2×C8C22×D13711yescharacteristic
284C2×C8C22×D13711yescharacteristic
285C2×C8C22×D13711yescharacteristic
286C2×C8C22×D13711yescharacteristic
287C4⋊C4—2741no—
288C4⋊C4—2741no—
289C4⋊C4—2741no—
290C4⋊C4—2741no—
291C4⋊C4—1371no—
292C4⋊C4—1371no—
293C4⋊C4—1371no—
294C4⋊C4—1371no—
295C4⋊C4D54811yescharacteristic
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⋊C4D54811yescharacteristic
305C22⋊C4D54811yescharacteristic
306C42C22×D13711yescharacteristic
307C23—2741no—
308C23—2741no—
309C23—2741no—
310C23—5482no—
311C23—5482no—
312C23—10961no—
313C23C2×D54811yescharacteristic
314C23C2×D54811yescharacteristic
315Q8—2741no—
316Q8—2741no—
317Q8—5482no—
318D4—22no—
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—2741no—
335C2×C4—2741no—
336C2×C4—2741no—
337C2×C4—2741no—
338C2×C4—2741no—
339C2×C4—5482no—
340C2×C4—5482no—
341C2×C4—5482no—
342C2×C4—5482no—
343C2×C4—10961no—
344C2×C4C2×D54811yescharacteristic
345C2×C4C2×D54811yescharacteristic
346C2×C4C2×D54811yescharacteristic
347C2×C4C2×D54811yescharacteristic
348C2×C4C23×D13711yescharacteristic
349C8—21no—
350C8—21no—
351C8—21no—
352C8—21no—
353C22—21no—
354C22—21no—
355C22—21no—
356C22—21no—
357C22—21no—
358C22—2742no—
359C22—5481no—
360C22—5481no—
361C22—5481no—
362C22—5481no—
363C22—5481no—
364C22—5481no—
365C22—5481no—
366C22—5481no—
367C22—10964no—
368C22#1798411612yes—
369C22C22×D54811yescharacteristic
370C4—21no—
371C4—21no—
372C4—21no—
373C4—21no—
374C4—5481no—
375C4—5481no—
376C4—5481no—
377C4—5481no—
378C4—5481no—
379C4#41×D13711yescharacteristic
380C4#1798413311yescharacteristic
381C2—22no—
382C2—41no—
383C2—5482no—
384C2—10961no—
385C2—10961no—
386C2#2257804811yescharacteristic
387C2C2×#1798411611yescharacteristic
388C2#2257821511yescharacteristic
subgroupG/H
centerC22C22×D548
commutator, cosocleC2×C4×C137C24
FrattiniC2×C4C23×D137
Fitting#435×C137C2
socleC22×C137C22×D4
hypercenter#435D137
hypocenterC137#3306
LCSG ▷ C2×C4×C137 ▷ C2×C137 ▷ C137
UCSC1 ◁ C22 ◁ C4×D4 ◁ #435

Complements

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

NH ≅ G/Ntype
1. #22578057C2semidirect
2. #22578049C2semidirect
3. #22577763C2semidirect
4. #22577734C2semidirect
5. #22577722C2semidirect
6. #22577694C2semidirect
7. #22577656C2semidirect
8. #22577632C2semidirect
9. #22577619C2semidirect
10. #22577550C2semidirect
11. #22577539C2semidirect
12. #22577516C2semidirect
13. #22577405C2semidirect
14. C2×#17984010C2semidirect
15. #435×C137C2semidirect
19. C2×#14555499C22semidirect
22. #17984088C22semidirect
31. #17984062C22semidirect
35. C4×D548C22semidirect
37. C2×C4⋊Dic137C22semidirect
39. #17984020C22semidirect
40. #17984020C22semidirect
41. #17984020C22semidirect
42. #17984013C22semidirect
43. #17984013C22semidirect
44. #17984013C22semidirect
51. #17984010C22semidirect
52. C2×M16×C137C22semidirect
55. C4⋊C8×C137C22semidirect
57. C22⋊C8×C137C22semidirect
58. #97×C137C22semidirect
127. D548#41semidirect
194. C137#3306semidirect
208. #435D137semidirect
217. C2×M16C2×D137semidirect
240. C4⋊C8C2×D137semidirect
252. C22⋊C8C2×D137semidirect
253. #97C2×D137semidirect

Maximal subgroups

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

subgroupindex
#225780572
#225780492
#225777632
#225777342
#225777222
#225776942
#225776562
#225776322
#225776192
#225775502
#225775392
#225775162
#225774052
C2×#179840102
#435×C1372
#3306137