Finite Group Databasebeta

← #31792611

Subgroups — #31792611

Order 17536.

462 conjugacy classes of subgroups.

The subgroup lattice has 396 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#22578108C211yesmaximal, characteristic, complemented
2#22578037C211yesmaximal, characteristic
3#22577748C211yesmaximal, characteristic, complemented
4#22577723C211yesmaximal, characteristic, complemented
5#22577692C211yesmaximal, characteristic, complemented
6#22577633C211yesmaximal, characteristic, complemented
7#22577630C211yesmaximal, characteristic, complemented
8#22577602C211yesmaximal, characteristic, complemented
9#22577556C211yesmaximal, characteristic, complemented
10#22577555C211yesmaximal, characteristic, complemented
11#22577490C211yesmaximal, characteristic, complemented
12#22577466C211yesmaximal, characteristic, complemented
13C2×#17984008C211yesmaximal, characteristic, complemented
14#102×D137C211yesmaximal, characteristic, complemented
15#481×C137C211yesmaximal, characteristic, complemented
16C22×D548C2211yescharacteristic
17C2×D4×D137—21no—
18C2×D4×D137C2211yescharacteristic
19C2×#14555499C2211yescharacteristic
20C2×#14555499C2211yescharacteristic, complemented
21C2×#14555498C2211yescharacteristic, complemented
22C22×#11764040C2211yescharacteristic, complemented
23C2×C8×D137C2211yescharacteristic, complemented
24#17984087C2211yescharacteristic, complemented
25#17984084C2211yescharacteristic
26#17984078—21no—
27#17984068C2211yescharacteristic, complemented
28#17984067—21no—
29#17984067C2211yescharacteristic, complemented
30#17984066—21no—
31#17984063C2211yescharacteristic
32#17984061—21no—
33#17984058—21no—
34#17984056—21no—
35#17984053C2211yescharacteristic, complemented
36#17984050C2211yescharacteristic
37C4×C22⋊D137—21no—
38C4×C22⋊D137C2211yescharacteristic
39C4⋊C4×D137C2211yescharacteristic, complemented
40#17984019C2211yescharacteristic, complemented
41#17984017C2211yescharacteristic, complemented
42#17984011C2211yescharacteristic, complemented
43#17984010C2211yescharacteristic
44#17984010C2211yescharacteristic, complemented
45#17984009C2211yescharacteristic, complemented
46#17984008C2211yescharacteristic
47#17984008C2211yescharacteristic
48#17984008C2211yescharacteristic, complemented
49#17984008C2211yescharacteristic, complemented
50#17984000C2211yescharacteristic, complemented
51C8×Dic137C2211yescharacteristic, complemented
52#121×C137C2211yescharacteristic
53C2×C4⋊C4×C137C2211yescharacteristic, complemented
54#106×C137C2211yescharacteristic, complemented
55#106×C137C2211yescharacteristic, complemented
56#102×C137C2211yescharacteristic, complemented
57#102×C137C2211yescharacteristic, complemented
58C22⋊C8×C137C2211yescharacteristic, complemented
59C23×D137—21no—
60C23×D137—21no—
61C2×Dic274—21no—
62C2×Dic274C2311yescharacteristic
63C2×D548—22no—
64C2×D548—21no—
65C2×D548—21no—
66C2×D548—41no—
67C2×D548C2311yescharacteristic
68C2×D548C2311yescharacteristic
69C2×C22⋊D137—21no—
70C2×C22⋊D137—21no—
71C2×C22⋊D137—21no—
72C2×C22⋊D137D411yescharacteristic
73D4×D137—22no—
74D4×D137—41no—
75D4×D137—44no—
76C2×C4×D137—21no—
77C2×C4×D137—21no—
78C2×C4×D137C2311yescharacteristic
79#14555499—22no—
80#14555499—41no—
81#14555498—41no—
82C2×#11764040—41no—
83C2×#11764040C2311yescharacteristic
84C2×#11764040C2311yescharacteristic
85C8×D137—41no—
86C4⋊Dic137—21no—
87C4⋊Dic137C2311yescharacteristic
88#14555489—21no—
89#14555489—21no—
90#14555489—21no—
91#14555489—21no—
92#14555489D411yescharacteristic
93C22⋊Dic137—21no—
94C22⋊Dic137D411yescharacteristic
95#14555487—21no—
96#14555487—21no—
97#14555487—21no—
98#14555487D411yescharacteristic
99C4×Dic137—21no—
100C4×Dic137C2311yescharacteristic
101C2×D4×C137—21no—
102C2×D4×C137C2311yescharacteristic
103C22×C4×C137—21no—
104C22×C4×C137C2311yescharacteristic
105C2×C8×C137C2311yescharacteristic
106C2×C8×C137C2311yescharacteristic
107C4⋊C4×C137—21no—
108C4⋊C4×C137C2311yescharacteristic
109C4⋊C4×C137C2311yescharacteristic
110C4⋊C4×C137C2311yescharacteristic
111C22⋊C4×C137—21no—
112C22×D137—21no—
113C22×D137—21no—
114C22×D137—42no—
115C22×D137—42no—
116C22×D137—42no—
117C22×D137—42no—
118C22×D137—81no—
119C22×D137—81no—
120C22×D137C2×D411yescharacteristic
121Dic274—21no—
122Dic274—21no—
123Dic274—81no—
124D548—21no—
125D548—21no—
126D548—21no—
127D548—42no—
128D548—42no—
129D548#4112yes—
130C22⋊D137—42no—
131C22⋊D137—42no—
132C22⋊D137—81no—
133C22⋊D137SD1614yescomplemented
134C2×Dic137—21no—
135C2×Dic137—21no—
136C2×Dic137C2×D411yescharacteristic
137C2×Dic137C2×D411yescharacteristic
138C2×Dic137C2×D411yescharacteristic
139C4×D137—21no—
140C4×D137—21no—
141C4×D137—42no—
142C4×D137—81no—
143#11764040—21no—
144#11764040—21no—
145C23×C137—21no—
146C23×C137C2×D411yescharacteristic
147D4×C137—21no—
148D4×C137—21no—
149D4×C137—42no—
150C2×C4×C137—21no—
151C2×C4×C137—21no—
152C2×C4×C137—21no—
153C2×C4×C137—21no—
154C2×C4×C137—21no—
155C2×C4×C137—81no—
156C2×C4×C137C2×D411yescharacteristic
157C2×C4×C137C2411yescharacteristic
158C8×C137—21no—
159C8×C137—21no—
160C22×C137—21no—
161C22×C137—21no—
162C22×C137—41no—
163C22×C137—41no—
164C22×C137—41no—
165C22×C137C2×SD1612yes—
166C22×C137C22×D411yescharacteristic
167C2×D137—21no—
168C2×D137—21no—
169C2×D137—22no—
170C2×D137—41no—
171C2×D137—41no—
172C2×D137—41no—
173C2×D137—41no—
174C2×D137—41no—
175C2×D137—84no—
176C2×D137—84no—
177C2×D137C2×SD1612yes—
178C4×C137—21no—
179C4×C137—41no—
180C4×C137—41no—
181C4×C137—41no—
182C4×C137C2×#4111yescharacteristic
183C4×C137H(5,𝔽2)11yescharacteristic
184Dic137—21no—
185Dic137—21no—
186Dic137—41no—
187Dic137—41no—
188Dic137C2×SD1612yes—
189C2×C137—22no—
190C2×C137—81no—
191C2×C137#54611yescharacteristic
192C2×C137C22×SD1611yescharacteristic
193C2×C137#57611yescharacteristic
194D137—22no—
195D137—42no—
196D137—81no—
197C137#329511yescharacteristic, complemented, Sylow
198#3295—1371nomaximal, Sylow
199#554—1371no—
200D42—1371no—
201#486—1371no—
202#481—1371no—
203#481D13711yescharacteristic, complemented
204#476—1371no—
205#465—1371no—
206#465—1371no—
207#460—1371no—
208#450—1371no—
209#450—1371no—
210C4×SD16—1371no—
211C8×D4—1371no—
212C2×#102—1371no—
213C2×#102—1371no—
214C22×D4—2741no—
215C22×D4—1371no—
216C22×D4—1371no—
217C2×SD16—1371no—
218C2×SD16—1371no—
219C2×SD16—1371no—
220C22×C8—1371no—
221C22×C8—1371no—
222C4⋊Q8—1371no—
223#127—1371no—
224#125—2741no—
225C22⋊Q8—2741no—
226C22⋊Q8—2741no—
227C22⋊Q8—1371no—
228C22⋊Q8—1371no—
229#121—2741no—
230#121—1371no—
231#121C2×D13711yescharacteristic
232#120—2741no—
233#120—2741no—
234C4×Q8—1371no—
235C4×D4—2741no—
236C4×D4—1371no—
237C4×D4—1371no—
238C2×C4⋊C4—1371no—
239C2×C4⋊C4C2×D13711yescharacteristic, complemented
240#106—1371no—
241#106C2×D13711yescharacteristic, complemented
242#106C2×D13711yescharacteristic, complemented
243C4⋊C8—1371no—
244#103—1371no—
245#102—1371no—
246#102—1371no—
247#102—1371no—
248#102—1371no—
249#102—1371no—
250#102—1371no—
251#102—1371no—
252#102C2×D13711yescharacteristic, complemented
253#102C2×D13711yescharacteristic, complemented
254C22⋊C8—1371no—
255C22⋊C8C2×D13711yescharacteristic, complemented
256C4×C8—1371no—
257C24—2741no—
258C24—2741no—
259C2×Q8—2741no—
260C2×Q8—1371no—
261C2×D4—21no—
262C2×D4—2742no—
263C2×D4—2741no—
264C2×D4—2742no—
265C2×D4—2741no—
266C2×D4—2741no—
267C2×D4—2741no—
268C2×D4—2741no—
269C2×D4—5481no—
270C2×D4—5481no—
271C2×D4—5484no—
272C2×D4—1371no—
273C2×D4—1371no—
274C2×D4—1371no—
275C2×D4C22×D13711yescharacteristic
276C22×C4—21no—
277C22×C4—2741no—
278C22×C4—2741no—
279C22×C4—1371no—
280C22×C4C22×D13711yescharacteristic
281SD16—2742no—
282SD16—5481no—
283SD16—5481no—
284C2×C8—5481no—
285C2×C8—5481no—
286C2×C8—1371no—
287C2×C8—1371no—
288C2×C8C22×D13711yescharacteristic
289C2×C8C22×D13711yescharacteristic
290C4⋊C4—21no—
291C4⋊C4—2741no—
292C4⋊C4—2741no—
293C4⋊C4—2741no—
294C4⋊C4—2741no—
295C4⋊C4—2741no—
296C4⋊C4—1371no—
297C4⋊C4—1371no—
298C4⋊C4C22×D13711yescharacteristic
299C4⋊C4C22×D13711yescharacteristic
300C4⋊C4C22×D13711yescharacteristic
301C22⋊C4—21no—
302C22⋊C4—2741no—
303C22⋊C4—2741no—
304C22⋊C4—2741no—
305C22⋊C4—2741no—
306C22⋊C4—1371no—
307C22⋊C4—1371no—
308C42—2741no—
309C42—1371no—
310C23—21no—
311C23—2741no—
312C23—2741no—
313C23—5482no—
314C23—5482no—
315C23—5482no—
316C23—5482no—
317C23—10961no—
318C23—10961no—
319C23—1371no—
320C23D4×D13711yescharacteristic
321Q8—2741no—
322Q8—2741no—
323Q8—10961no—
324D4—21no—
325D4—21no—
326D4—42no—
327D4—2741no—
328D4—2741no—
329D4—2741no—
330D4—5482no—
331D4—5482no—
332D4—5482no—
333D4—5482no—
334D4—10961no—
335D4—1374no—
336D4—1372no—
337C2×C4—21no—
338C2×C4—21no—
339C2×C4—21no—
340C2×C4—21no—
341C2×C4—21no—
342C2×C4—81no—
343C2×C4—2741no—
344C2×C4—2741no—
345C2×C4—2741no—
346C2×C4—2741no—
347C2×C4—5482no—
348C2×C4—10961no—
349C2×C4—1371no—
350C2×C4—1371no—
351C2×C4—1371no—
352C2×C4D4×D13711yescharacteristic
353C2×C4C23×D13711yescharacteristic
354C8—21no—
355C8—21no—
356C8—2741no—
357C8—2741no—
358C22—21no—
359C22—21no—
360C22—41no—
361C22—41no—
362C22—41no—
363C22—2741no—
364C22—2741no—
365C22—2742no—
366C22—5481no—
367C22—5481no—
368C22—5481no—
369C22—5481no—
370C22—5481no—
371C22—10964no—
372C22—10964no—
373C22—1372no—
374C22SD16×D13712yes—
375C22C2×D4×D13711yescharacteristic
376C4—21no—
377C4—41no—
378C4—41no—
379C4—41no—
380C4—2741no—
381C4—2741no—
382C4—5481no—
383C4—5481no—
384C4—1372no—
385C4#41×D13711yescharacteristic
386C4#1798413411yescharacteristic
387C2—22no—
388C2—81no—
389C2—2742no—
390C2—5482no—
391C2—10961no—
392C2#2257804511yescharacteristic
393C2C2×SD16×D13711yescharacteristic
394C2#2257820711yescharacteristic
subgroupG/H
centerC22C2×D4×D137
commutator, cosocleC2×C4×C137C24
FrattiniC2×C4C23×D137
Fitting#481×C137C2
socleC22×C137C22×D4
prime radical#22578037C2
hypercenter#481D137
hypocenterC137#3295
LCSG ▷ C2×C4×C137 ▷ C2×C137 ▷ C137
UCSC1 ◁ C22 ◁ C22×C4 ◁ #481

Complements

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

NH ≅ G/Ntype
1. #22578108C2semidirect
3. #22577748C2semidirect
4. #22577723C2semidirect
5. #22577692C2semidirect
6. #22577633C2semidirect
7. #22577630C2semidirect
8. #22577602C2semidirect
9. #22577556C2semidirect
10. #22577555C2semidirect
11. #22577490C2semidirect
12. #22577466C2semidirect
13. C2×#17984008C2semidirect
14. #102×D137C2semidirect
15. #481×C137C2semidirect
20. C2×#14555499C22semidirect
21. C2×#14555498C22semidirect
22. C22×#11764040C22semidirect
23. C2×C8×D137C22semidirect
24. #17984087C22semidirect
27. #17984068C22semidirect
29. #17984067C22semidirect
35. #17984053C22semidirect
39. C4⋊C4×D137C22semidirect
40. #17984019C22semidirect
41. #17984017C22semidirect
42. #17984011C22semidirect
44. #17984010C22semidirect
45. #17984009C22semidirect
48. #17984008C22semidirect
49. #17984008C22semidirect
50. #17984000C22semidirect
51. C8×Dic137C22semidirect
53. C2×C4⋊C4×C137C22semidirect
54. #106×C137C22semidirect
55. #106×C137C22semidirect
56. #102×C137C22semidirect
57. #102×C137C22semidirect
58. C22⋊C8×C137C22semidirect
133. C22⋊D137SD16semidirect
197. C137#3295semidirect
203. #481D137semidirect
239. C2×C4⋊C4C2×D137semidirect
241. #106C2×D137semidirect
242. #106C2×D137semidirect
252. #102C2×D137semidirect
253. #102C2×D137semidirect
255. C22⋊C8C2×D137semidirect

Maximal subgroups

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

subgroupindex
#225781082
#225780372
#225777482
#225777232
#225776922
#225776332
#225776302
#225776022
#225775562
#225775552
#225774902
#225774662
C2×#179840082
#102×D1372
#481×C1372
#3295137