Finite Group Databasebeta

← #31792622

Subgroups — #31792622

Order 17536.

408 conjugacy classes of subgroups.

The subgroup lattice has 372 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#22578088C211yesmaximal, characteristic, complemented
2#22578044C211yesmaximal, characteristic
3#22577759C211yesmaximal, characteristic, complemented
4#22577729C211yesmaximal, characteristic, complemented
5#22577702C211yesmaximal, characteristic, complemented
6#22577642C211yesmaximal, characteristic, complemented
7#22577633C211yesmaximal, characteristic, complemented
8#22577628C211yesmaximal, characteristic, complemented
9#22577568C211yesmaximal, characteristic, complemented
10#22577556C211yesmaximal, characteristic, complemented
11#22577490C211yesmaximal, characteristic, complemented
12#22577466C211yesmaximal, characteristic, complemented
13#22577398C211yesmaximal, characteristic, complemented
14C2×#17984008C211yesmaximal, characteristic, complemented
15#483×C137C211yesmaximal, characteristic, complemented
16C2×#14555511C2211yescharacteristic
17C22×D548C2211yescharacteristic
18C2×#14555500C2211yescharacteristic, complemented
19C2×#14555499C2211yescharacteristic
20C2×#14555499C2211yescharacteristic
21C22×#11764040C2211yescharacteristic
22C2×C8×D137C2211yescharacteristic, complemented
23#17984083—21no—
24#17984079C2211yescharacteristic, complemented
25#17984078—21no—
26#17984077C2211yescharacteristic
27#17984073—21no—
28#17984071—21no—
29#17984068C2211yescharacteristic, complemented
30#17984067C2211yescharacteristic
31#17984063—21no—
32#17984063C2211yescharacteristic
33#17984058—21no—
34#17984053C2211yescharacteristic, complemented
35#17984050C2211yescharacteristic
36C4×C22⋊D137C2211yescharacteristic
37#17984047—21no—
38#17984046C2211yescharacteristic, complemented
39C22⋊C4×D137—21no—
40#17984021C2211yescharacteristic, complemented
41#17984017C2211yescharacteristic, complemented
42#17984012C2211yescharacteristic, complemented
43#17984011C2211yescharacteristic, complemented
44#17984010C2211yescharacteristic
45#17984010C2211yescharacteristic
46#17984008C2211yescharacteristic
47#17984008C2211yescharacteristic
48#17984008C2211yescharacteristic
49#17984008C2211yescharacteristic
50#17984000C2211yescharacteristic, complemented
51C8×Dic137C2211yescharacteristic, complemented
52C22⋊Q8×C137C2211yescharacteristic
53C2×C4⋊C4×C137C2211yescharacteristic
54#106×C137C2211yescharacteristic, complemented
55#106×C137C2211yescharacteristic, complemented
56#103×C137C2211yescharacteristic, complemented
57#103×C137C2211yescharacteristic, complemented
58C22⋊C8×C137C2211yescharacteristic
59C23×D137—21no—
60#14555511—22no—
61#14555511—41no—
62C2×Dic274C2311yescharacteristic
63C2×D548—22no—
64C2×D548—21no—
65C2×D548—21no—
66C2×D548—21no—
67C2×D548—41no—
68C2×D548C2311yescharacteristic
69C2×D548C2311yescharacteristic
70C2×C22⋊D137D411yescharacteristic
71C2×C4×D137—21no—
72C2×C4×D137—21no—
73C2×C4×D137C2311yescharacteristic
74#14555500—41no—
75#14555499—22no—
76#14555499—41no—
77C2×#11764040—41no—
78C2×#11764040C2311yescharacteristic
79C2×#11764040C2311yescharacteristic
80C8×D137—41no—
81C4⋊Dic137—21no—
82C4⋊Dic137—21no—
83C4⋊Dic137C2311yescharacteristic
84#14555489—21no—
85#14555489—21no—
86#14555489—21no—
87#14555489D411yescharacteristic
88C22⋊Dic137—21no—
89C22⋊Dic137D411yescharacteristic
90#14555487—21no—
91#14555487—21no—
92#14555487—21no—
93#14555487—21no—
94#14555487—21no—
95#14555487—21no—
96#14555487D411yescharacteristic
97C4×Dic137—21no—
98C4×Dic137—21no—
99C4×Dic137C2311yescharacteristic
100C2×Q8×C137C2311yescharacteristic
101C22×C4×C137—21no—
102C22×C4×C137C2311yescharacteristic
103C2×C8×C137C2311yescharacteristic
104C2×C8×C137C2311yescharacteristic
105C4⋊C4×C137—21no—
106C4⋊C4×C137—21no—
107C4⋊C4×C137C2311yescharacteristic
108C4⋊C4×C137C2311yescharacteristic
109C4⋊C4×C137C2311yescharacteristic
110C22⋊C4×C137—21no—
111C22×D137—21no—
112C22×D137—21no—
113C22×D137—42no—
114C22×D137—42no—
115C22×D137—81no—
116C22×D137C2×D411yescharacteristic
117Dic274—21no—
118Dic274—21no—
119D548—21no—
120D548—21no—
121D548—21no—
122D548—42no—
123D548—42no—
124D548—81no—
125D548#4112yes—
126C22⋊D137—22no—
127C2×Dic137—21no—
128C2×Dic137—21no—
129C2×Dic137C2×D411yescharacteristic
130C2×Dic137C2×D411yescharacteristic
131C2×Dic137C2×D411yescharacteristic
132C4×D137—21no—
133C4×D137—21no—
134C4×D137—42no—
135C4×D137—81no—
136#11764040—21no—
137#11764040—21no—
138C23×C137C2×D411yescharacteristic
139Q8×C137—21no—
140Q8×C137—21no—
141C2×C4×C137—21no—
142C2×C4×C137—21no—
143C2×C4×C137—21no—
144C2×C4×C137—21no—
145C2×C4×C137—21no—
146C2×C4×C137—21no—
147C2×C4×C137—42no—
148C2×C4×C137C2×D411yescharacteristic
149C2×C4×C137C2411yescharacteristic
150C8×C137—21no—
151C8×C137—21no—
152C22×C137—21no—
153C22×C137—21no—
154C22×C137#13512yes—
155C22×C137C22×D411yescharacteristic
156C2×D137—21no—
157C2×D137—22no—
158C2×D137—21no—
159C2×D137—21no—
160C2×D137—41no—
161C2×D137—41no—
162C2×D137—41no—
163C2×D137—41no—
164C2×D137—41no—
165C2×D137—84no—
166C4×C137—21no—
167C4×C137—41no—
168C4×C137—41no—
169C4×C137—41no—
170C4×C137—41no—
171C4×C137C2×#4111yescharacteristic
172C4×C137H(5,𝔽2)11yescharacteristic
173Dic137—21no—
174Dic137—21no—
175Dic137—21no—
176Dic137—41no—
177Dic137—41no—
178C2×C137—22no—
179C2×C137#54611yescharacteristic
180C2×C137C2×#13511yescharacteristic
181C2×C137#57711yescharacteristic
182D137—41no—
183D137—42no—
184D137—81no—
185C137#329611yescharacteristic, complemented, Sylow
186#3296—1371nomaximal, Sylow
187#552—1371no—
188#546—1371no—
189#488—1371no—
190#483D13711yescharacteristic, complemented
191#482—1371no—
192#478—1371no—
193#467—1371no—
194#465—1371no—
195#464—1371no—
196#452—1371no—
197#450—1371no—
198C4×SD16—1371no—
199C8×D4—1371no—
200#416—1371no—
201C2×#102—1371no—
202C2×#41—1371no—
203C22×D4—1371no—
204C2×Q16—1371no—
205C2×SD16—1371no—
206C2×SD16—1371no—
207C22×C8—1371no—
208C22×C8—1371no—
209#126—2741no—
210#125—2741no—
211#125—1371no—
212#124—1371no—
213#123—2741no—
214#123—2741no—
215C22⋊Q8—1371no—
216C22⋊Q8—1371no—
217C22⋊Q8C2×D13711yescharacteristic
218#121—2741no—
219#121—1371no—
220#120—2741no—
221C4×Q8—1371no—
222C4×D4—1371no—
223C4×D4—1371no—
224#117—2741no—
225#117—1371no—
226C2×C4⋊C4C2×D13711yescharacteristic
227C2×C22⋊C4—2741no—
228#107—1371no—
229#106C2×D13711yescharacteristic, complemented
230#106C2×D13711yescharacteristic, complemented
231C4⋊C8—1371no—
232#103—1371no—
233#103—1371no—
234#103C2×D13711yescharacteristic, complemented
235#103C2×D13711yescharacteristic, complemented
236#102—1371no—
237#102—1371no—
238#102—1371no—
239#102—1371no—
240#102—1371no—
241#102—1371no—
242C22⋊C8—1371no—
243C22⋊C8C2×D13711yescharacteristic
244C4×C8—1371no—
245C24—2741no—
246#41—2742no—
247#41—5481no—
248C2×Q8—1371no—
249C2×Q8C22×D13711yescharacteristic
250C2×D4—2742no—
251C2×D4—2741no—
252C2×D4—2741no—
253C2×D4—2741no—
254C2×D4—5481no—
255C2×D4—1371no—
256C2×D4—1371no—
257C2×D4—1371no—
258C22×C4—21no—
259C22×C4—2741no—
260C22×C4—2741no—
261C22×C4—1371no—
262C22×C4C22×D13711yescharacteristic
263Q16—5481no—
264SD16—2742no—
265SD16—5481no—
266C2×C8—5481no—
267C2×C8—5481no—
268C2×C8—1371no—
269C2×C8—1371no—
270C2×C8C22×D13711yescharacteristic
271C2×C8C22×D13711yescharacteristic
272C4⋊C4—21no—
273C4⋊C4—21no—
274C4⋊C4—2741no—
275C4⋊C4—2741no—
276C4⋊C4—2741no—
277C4⋊C4—2741no—
278C4⋊C4—2741no—
279C4⋊C4—1371no—
280C4⋊C4—1371no—
281C4⋊C4C22×D13711yescharacteristic
282C4⋊C4C22×D13711yescharacteristic
283C4⋊C4C22×D13711yescharacteristic
284C22⋊C4—21no—
285C22⋊C4—2741no—
286C22⋊C4—2741no—
287C22⋊C4—2741no—
288C22⋊C4—2741no—
289C22⋊C4—2741no—
290C22⋊C4—2741no—
291C22⋊C4—2741no—
292C22⋊C4—1371no—
293C22⋊C4—1371no—
294C42—2741no—
295C42—2741no—
296C42—1371no—
297C23—2741no—
298C23—2741no—
299C23—5482no—
300C23—5482no—
301C23—10961no—
302C23—1371no—
303C23D4×D13711yescharacteristic
304Q8—21no—
305Q8—21no—
306Q8—2741no—
307Q8—2741no—
308D4—2742no—
309D4—2741no—
310D4—2741no—
311D4—2741no—
312D4—5482no—
313D4—5482no—
314D4—10961no—
315D4—1372no—
316C2×C4—21no—
317C2×C4—21no—
318C2×C4—21no—
319C2×C4—21no—
320C2×C4—21no—
321C2×C4—21no—
322C2×C4—42no—
323C2×C4—2741no—
324C2×C4—2741no—
325C2×C4—2741no—
326C2×C4—2741no—
327C2×C4—5482no—
328C2×C4—10961no—
329C2×C4—1371no—
330C2×C4—1371no—
331C2×C4—1371no—
332C2×C4D4×D13711yescharacteristic
333C2×C4C23×D13711yescharacteristic
334C8—21no—
335C8—21no—
336C8—2741no—
337C8—2741no—
338C22—21no—
339C22—21no—
340C22—2741no—
341C22—2741no—
342C22—2741no—
343C22—2742no—
344C22—5481no—
345C22—5481no—
346C22—5481no—
347C22—5481no—
348C22—5481no—
349C22—10964no—
350C22#1798410912yes—
351C22C2×D4×D13711yescharacteristic
352C4—21no—
353C4—41no—
354C4—41no—
355C4—41no—
356C4—41no—
357C4—2741no—
358C4—2741no—
359C4—2741no—
360C4—5481no—
361C4—5481no—
362C4#41×D13711yescharacteristic
363C4#1798413411yescharacteristic
364C2—22no—
365C2—5481no—
366C2—5482no—
367C2—10961no—
368C2#2257804511yescharacteristic
369C2C2×#1798410911yescharacteristic
370C2#2257821211yescharacteristic
subgroupG/H
centerC22C2×D4×D137
commutator, cosocleC2×C4×C137C24
FrattiniC2×C4C23×D137
Fitting#483×C137C2
socleC22×C137C22×D4
prime radical#22578044C2
hypercenter#483D137
hypocenterC137#3296
LCSG ▷ C2×C4×C137 ▷ C2×C137 ▷ C137
UCSC1 ◁ C22 ◁ C22×C4 ◁ #483

Complements

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

NH ≅ G/Ntype
1. #22578088C2semidirect
3. #22577759C2semidirect
4. #22577729C2semidirect
5. #22577702C2semidirect
6. #22577642C2semidirect
7. #22577633C2semidirect
8. #22577628C2semidirect
9. #22577568C2semidirect
10. #22577556C2semidirect
11. #22577490C2semidirect
12. #22577466C2semidirect
13. #22577398C2semidirect
14. C2×#17984008C2semidirect
15. #483×C137C2semidirect
18. C2×#14555500C22semidirect
22. C2×C8×D137C22semidirect
24. #17984079C22semidirect
29. #17984068C22semidirect
34. #17984053C22semidirect
38. #17984046C22semidirect
40. #17984021C22semidirect
41. #17984017C22semidirect
42. #17984012C22semidirect
43. #17984011C22semidirect
50. #17984000C22semidirect
51. C8×Dic137C22semidirect
54. #106×C137C22semidirect
55. #106×C137C22semidirect
56. #103×C137C22semidirect
57. #103×C137C22semidirect
185. C137#3296semidirect
190. #483D137semidirect
229. #106C2×D137semidirect
230. #106C2×D137semidirect
234. #103C2×D137semidirect
235. #103C2×D137semidirect

Maximal subgroups

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

subgroupindex
#225780882
#225780442
#225777592
#225777292
#225777022
#225776422
#225776332
#225776282
#225775682
#225775562
#225774902
#225774662
#225773982
C2×#179840082
#483×C1372
#3296137