Finite Group Databasebeta

← #31794116

Subgroups — #31794116

Order 17536.

1556 conjugacy classes of subgroups.

The subgroup lattice has 350 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#41×D137C212yesmaximal, complemented
2#22577980C214yesmaximal, complemented
3#22577978C212yesmaximal, complemented
4#22577973C212yesmaximal, complemented
5#22577971C212yesmaximal, complemented
6#22577966C212yesmaximal, complemented
7#22577961C214yesmaximal, complemented
8#22577953C211yesmaximal, characteristic, complemented
9#22577949C212yesmaximal, complemented
10#22577912C218yesmaximal, complemented
11C2×#17984045C211yesmaximal, characteristic, complemented
12#532×C137C211yesmaximal, characteristic, complemented
13C2×#14555511C2212yes—
14C2×#14555510C2214yes—
15C2×#14555510C2212yescomplemented
16C2×#14555509C2212yes—
17C2×#14555509C2214yescomplemented
18#41×D137—216no—
19C2×Q8×D137C2212yescomplemented
20C2×D4×D137C2214yes—
21C2×D4×D137C2212yescomplemented
22C22×C4×D137C2211yescharacteristic
23C22×C4×D137C2214yescomplemented
24#17984088C2211yescharacteristic, complemented
25#17984086C2212yescomplemented
26#17984085C2211yescharacteristic, complemented
27#17984084C2212yescomplemented
28#17984077C2212yescomplemented
29#17984076C2212yescomplemented
30#17984075C2212yescomplemented
31#17984073C2218yescomplemented
32#17984070C2218yescomplemented
33#17984066C2218yescomplemented
34#17984065C2218yescomplemented
35#17984062C2214yescomplemented
36#17984061C2218yescomplemented
37#17984060C2218yescomplemented
38#17984060C2218yescomplemented
39#17984058C2218yescomplemented
40#17984056C2218yescomplemented
41C4×D548C2214yescomplemented
42C4×C22⋊D137C2218yescomplemented
43#17984045C2212yescomplemented
44#17984045C2212yescomplemented
45#17984045C2214yescomplemented
46C2×#14555489C2212yescomplemented
47C2×#14555487C2212yes—
48C2×C4×Dic137C2211yescharacteristic, complemented
49C2×#41×C137C2212yescomplemented
50C4⋊Q8×C137C2211yescharacteristic, complemented
51#127×C137C2211yescharacteristic, complemented
52#124×C137C2212yescomplemented
53#121×C137C2214yescomplemented
54C4×D4×C137C2214yescomplemented
55C2×C42×C137C2211yescharacteristic, complemented
56C23×D137C2314yes—
57C23×D137C2311yescharacteristic
58#14555511—28no—
59#14555510—216no—
60#14555510—28no—
61#14555509—216no—
62#14555509—28no—
63C2×Dic274C2314yes—
64C2×Dic274C2312yes—
65Q8×D137—28no—
66C2×D548C2314yes—
67C2×D548C2312yes—
68C2×D548C2314yes—
69C2×C22⋊D137C2314yes—
70C2×C22⋊D137C2314yes—
71C2×C22⋊D137C2314yes—
72C2×C22⋊D137C2318yes—
73D4×D137—216no—
74D4×D137—28no—
75C22×Dic137C2314yes—
76C22×Dic137C2312yes—
77C22×Dic137C2311yescharacteristic
78C2×C4×D137—216no—
79C2×C4×D137D418yescomplemented
80C2×C4×D137C2314yes—
81C2×C4×D137C2314yes—
82C2×C4×D137C2314yes—
83C2×C4×D137C2312yes—
84C2×C4×D137C2312yes—
85C4⋊Dic137C2314yes—
86#14555489C2314yes—
87#14555489C2314yes—
88#14555489C2318yes—
89C22⋊Dic137C2318yes—
90#14555487C2314yes—
91#14555487C2314yes—
92#14555487C2318yes—
93#14555487C2318yes—
94C4×Dic137C2312yes—
95C4×Dic137C2312yes—
96#41×C137—28no—
97C2×Q8×C137C2312yes—
98C2×D4×C137C2314yes—
99C2×D4×C137C2312yes—
100C2×D4×C137C2314yes—
101C22×C4×C137C2314yes—
102C22×C4×C137C2312yes—
103C22×C4×C137C2311yescharacteristic
104C4⋊C4×C137C2314yes—
105C22⋊C4×C137C2318yes—
106C42×C137C2312yes—
107C42×C137C2311yescharacteristic
108C42×C137C2311yescharacteristic
109C22×D137—28no—
110C22×D137—42no—
111C22×D137—48no—
112C22×D137C2×D414yes—
113C22×D137C2414yes—
114C22×D137C2412yes—
115Dic274—24no—
116Dic274—28no—
117D548—28no—
118D548—24no—
119D548—44no—
120C22⋊D137—28no—
121C22⋊D137—28no—
122C22⋊D137—28no—
123C22⋊D137—48no—
124C2×Dic137—28no—
125C2×Dic137—42no—
126C2×Dic137C2×D414yes—
127C2×Dic137C2414yes—
128C2×Dic137C2414yes—
129C2×Dic137C2412yes—
130C4×D137—28no—
131C4×D137—28no—
132C4×D137—28no—
133C4×D137C2×D418yes—
134C4×D137C2×D418yes—
135C23×C137C2414yes—
136C23×C137C2411yescharacteristic
137Q8×C137—24no—
138D4×C137—28no—
139D4×C137—24no—
140D4×C137—44no—
141C2×C4×C137—28no—
142C2×C4×C137—42no—
143C2×C4×C137C2×D414yes—
144C2×C4×C137C2412yes—
145C2×C4×C137C2412yes—
146C2×C4×C137C2414yes—
147C2×C4×C137C2411yescharacteristic
148C2×C4×C137C2411yescharacteristic
149C22×C137—22no—
150C22×C137—24no—
151C22×C137—28no—
152C22×C137C22×D412yes—
153C22×C137C2511yescharacteristic
154C2×D137—24no—
155C2×D137—24no—
156C2×D137—28no—
157C2×D137—44no—
158C2×D137—416no—
159C2×D137C22×D414yes—
160C4×C137—24no—
161C4×C137—22no—
162C4×C137—22no—
163C4×C137C22×D412yes—
164C4×C137C22×D412yes—
165Dic137—24no—
166Dic137—24no—
167Dic137C22×D414yes—
168C2×C137—22no—
169C2×C137—44no—
170C2×C137C23×D411yescharacteristic
171C2×C137#58512yes—
172D137—24no—
173D137—44no—
174C137#344911yescharacteristic, complemented, Sylow
175#3449—1371nomaximal, Sylow
176C22×#41—1372no—
177#537—1374no—
178#536—1372no—
179#536—1372no—
180#535—1372no—
181#535—1372no—
182#534—1374no—
183#532—1372no—
184#532—1371no—
185#532D13711yescharacteristic, complemented
186#525—1378no—
187C2×#117—1371no—
188C2×#41—27416no—
189C2×#41—1372no—
190C2×#41—1372no—
191C2×#41—1374no—
192C2×#41—1374no—
193C2×#41—1372no—
194C2×#41C2×D13712yescomplemented
195C22×Q8—1372no—
196C22×D4—1374no—
197C22×D4—1372no—
198C23×C4—1374no—
199C23×C4—1371no—
200C4⋊Q8—1372no—
201C4⋊Q8—1371no—
202C4⋊Q8C2×D13711yescharacteristic, complemented
203#127—1372no—
204#127—1371no—
205#127C2×D13711yescharacteristic, complemented
206#124—1372no—
207#124—1372no—
208#124—1372no—
209#124C2×D13712yescomplemented
210#123—1378no—
211#123—1378no—
212C22⋊Q8—1378no—
213C22⋊Q8—1378no—
214#121—1378no—
215#121—1378no—
216#121—1374no—
217#121—1378no—
218#121C2×D13714yescomplemented
219#120—1378no—
220#120—1378no—
221C4×D4—1378no—
222C4×D4—1374no—
223C4×D4C2×D13714yescomplemented
224#117—1374no—
225#117—1372no—
226#117—1372no—
227C2×C4⋊C4—1372no—
228C2×C22⋊C4—1372no—
229C2×C42—1371no—
230C2×C42C2×D13711yescharacteristic, complemented
231C24—1374no—
232C24—1371no—
233#41—28no—
234#41—27416no—
235#41—27416no—
236#41—2748no—
237#41—2748no—
238#41—2748no—
239C2×Q8—2748no—
240C2×Q8—1374no—
241C2×Q8—1372no—
242C2×Q8C22×D13712yes—
243C2×D4—27416no—
244C2×D4—2748no—
245C2×D4—1374no—
246C2×D4—1374no—
247C2×D4—1374no—
248C2×D4—1378no—
249C2×D4—1374no—
250C2×D4—1372no—
251C2×D4—1374no—
252C2×D4C22×D13714yes—
253C2×D4C22×D13712yes—
254C2×D4C22×D13714yes—
255C22×C4—27416no—
256C22×C4—1374no—
257C22×C4—1374no—
258C22×C4—1374no—
259C22×C4—1374no—
260C22×C4—1372no—
261C22×C4—1371no—
262C22×C4—1378no—
263C22×C4—1372no—
264C22×C4—1372no—
265C22×C4C22×D13714yes—
266C22×C4C22×D13712yes—
267C22×C4C22×D13711yescharacteristic
268C4⋊C4—1374no—
269C4⋊C4—1374no—
270C4⋊C4—1378no—
271C4⋊C4—1374no—
272C4⋊C4C22×D13714yes—
273C22⋊C4—1378no—
274C22⋊C4—1374no—
275C22⋊C4—1374no—
276C22⋊C4—1378no—
277C22⋊C4—1378no—
278C22⋊C4C22×D13718yes—
279C42—1372no—
280C42—1372no—
281C42C22×D13711yescharacteristic
282C42C22×D13711yescharacteristic
283C42C22×D13712yes—
284C23—2748no—
285C23—5482no—
286C23—5488no—
287C23—1374no—
288C23—1372no—
289C23—1374no—
290C23C23×D13714yes—
291C23C23×D13711yescharacteristic
292Q8—24no—
293Q8—2748no—
294Q8—2744no—
295D4—28no—
296D4—24no—
297D4—44no—
298D4—2748no—
299D4—2748no—
300D4—2748no—
301D4—2748no—
302D4—2744no—
303D4—5484no—
304D4—5488no—
305C2×C4—28no—
306C2×C4—42no—
307C2×C4—2748no—
308C2×C4—2748no—
309C2×C4—2748no—
310C2×C4—2748no—
311C2×C4—5482no—
312C2×C4—1374no—
313C2×C4—1374no—
314C2×C4—1374no—
315C2×C4—1372no—
316C2×C4—1378no—
317C2×C4—1378no—
318C2×C4D4×D13714yes—
319C2×C4C23×D13712yes—
320C2×C4C23×D13712yes—
321C2×C4C23×D13714yes—
322C2×C4C23×D13711yescharacteristic
323C2×C4C23×D13711yescharacteristic
324C22—22no—
325C22—24no—
326C22—28no—
327C22—2744no—
328C22—2744no—
329C22—2748no—
330C22—5484no—
331C22—54816no—
332C22—1374no—
333C22C2×D4×D13712yes—
334C22C24×D13711yescharacteristic
335C4—24no—
336C4—22no—
337C4—22no—
338C4—2744no—
339C4—2744no—
340C4—1374no—
341C4C2×D4×D13712yes—
342C4C2×D4×D13712yes—
343C2—22no—
344C2—44no—
345C2—2744no—
346C2—5484no—
347C2C22×D4×D13711yescharacteristic
348C2#2257823912yes—
subgroupG/H
centerC2×C4C23×D137
commutator, socle, cosocleC22×C137C25
FrattiniC22C24×D137
Fitting#532×C137C2
hypercenter#532D137
hypocenterC137#3449
LCSG ▷ C22×C137 ▷ C137
UCSC1 ◁ C2×C4 ◁ #532

Complements

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

NH ≅ G/Ntype
1. C2×#41×D137C2semidirect
2. #22577980C2semidirect
3. #22577978C2semidirect
4. #22577973C2semidirect
5. #22577971C2semidirect
6. #22577966C2semidirect
7. #22577961C2semidirect
8. #22577953C2semidirect
9. #22577949C2semidirect
10. #22577912C2semidirect
11. C2×#17984045C2semidirect
12. #532×C137C2semidirect
15. C2×#14555510C22semidirect
17. C2×#14555509C22semidirect
19. C2×Q8×D137C22semidirect
21. C2×D4×D137C22semidirect
23. C22×C4×D137C22semidirect
24. #17984088C22semidirect
25. #17984086C22semidirect
26. #17984085C22semidirect
27. #17984084C22semidirect
28. #17984077C22semidirect
29. #17984076C22semidirect
30. #17984075C22semidirect
31. #17984073C22semidirect
32. #17984070C22semidirect
33. #17984066C22semidirect
34. #17984065C22semidirect
35. #17984062C22semidirect
36. #17984061C22semidirect
37. #17984060C22semidirect
38. #17984060C22semidirect
39. #17984058C22semidirect
40. #17984056C22semidirect
41. C4×D548C22semidirect
42. C4×C22⋊D137C22semidirect
43. #17984045C22semidirect
44. #17984045C22semidirect
45. #17984045C22semidirect
46. C2×#14555489C22semidirect
48. C2×C4×Dic137C22semidirect
49. C2×#41×C137C22semidirect
50. C4⋊Q8×C137C22semidirect
51. #127×C137C22semidirect
52. #124×C137C22semidirect
53. #121×C137C22semidirect
54. C4×D4×C137C22semidirect
55. C2×C42×C137C22semidirect
79. C2×C4×D137D4semidirect
174. C137#3449semidirect
185. #532D137semidirect
194. C2×#41C2×D137semidirect
202. C4⋊Q8C2×D137semidirect
205. #127C2×D137semidirect
209. #124C2×D137semidirect
218. #121C2×D137semidirect
223. C4×D4C2×D137semidirect
230. C2×C42C2×D137semidirect

Maximal subgroups

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

subgroupindex
C2×#41×D1372
#225779802
#225779782
#225779732
#225779712
#225779662
#225779612
#225779532
#225779492
#225779122
C2×#179840452
#532×C1372
#3449137