Finite Group Databasebeta

← #31792819

Subgroups — #31792819

Order 17536.

402 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#22578049C211yesmaximal, characteristic
2#22577765C211yesmaximal, characteristic, complemented
3#22577744C211yesmaximal, characteristic, complemented
4#22577727C211yesmaximal, characteristic, complemented
5#22577690C211yesmaximal, characteristic, complemented
6#22577659C211yesmaximal, characteristic, complemented
7#22577634C211yesmaximal, characteristic, complemented
8#22577625C211yesmaximal, characteristic, complemented
9#22577563C211yesmaximal, characteristic, complemented
10#22577553C211yesmaximal, characteristic, complemented
11#22577501C211yesmaximal, characteristic, complemented
12#22577475C211yesmaximal, characteristic, complemented
13#22577407C211yesmaximal, characteristic, complemented
14#22577397C211yesmaximal, characteristic, complemented
15#556×C137C211yesmaximal, characteristic, complemented
16C2×#14555510C2211yescharacteristic
17C22×D548C2211yescharacteristic
18C2×#14555500C2211yescharacteristic, complemented
19C2×#14555497C2211yescharacteristic
20C2×#14555497C2211yescharacteristic
21C2×#14555494C2211yescharacteristic
22C22×#11764040C2211yescharacteristic, complemented
23#17984076C2211yescharacteristic
24#17984073—21no—
25#17984067C2211yescharacteristic
26#17984062C2211yescharacteristic
27#17984061—21no—
28#17984056—21no—
29C4×D548C2211yescharacteristic
30C22⋊C4×D137—21no—
31#17984021C2211yescharacteristic, complemented
32#17984019C2211yescharacteristic, complemented
33#17984019C2211yescharacteristic, complemented
34C4⋊#11764040C2211yescharacteristic, complemented
35#17984012C2211yescharacteristic, complemented
36#17984011C2211yescharacteristic, complemented
37#17984011C2211yescharacteristic, complemented
38#17984011C2211yescharacteristic, complemented
39#17984008C2211yescharacteristic
40#17984008C2211yescharacteristic
41#17984008C2211yescharacteristic
42#17984008C2211yescharacteristic
43#17984008C2211yescharacteristic
44#17984008C2211yescharacteristic
45C22⋊#11764040C2211yescharacteristic
46C22⋊#11764040C2211yescharacteristic, complemented
47#17983999C2211yescharacteristic, complemented
48C4⋊Q8×C137C2211yescharacteristic, complemented
49#126×C137—21no—
50#126×C137—21no—
51#124×C137—21no—
52C22⋊Q8×C137C2211yescharacteristic
53C22⋊Q8×C137C2211yescharacteristic, complemented
54C4×Q8×C137—21no—
55C4×Q8×C137C2211yescharacteristic, complemented
56C4×D4×C137C2211yescharacteristic
57#117×C137C2211yescharacteristic
58#117×C137C2211yescharacteristic, complemented
59C23×D137—21no—
60#14555510—22no—
61#14555510—41no—
62C2×Dic274C2311yescharacteristic
63C2×D548—22no—
64C2×D548—21no—
65C2×D548—41no—
66C2×D548C2311yescharacteristic
67C2×D548C2311yescharacteristic
68C2×C22⋊D137—21no—
69C2×C22⋊D137—21no—
70C2×C4×D137—21no—
71C2×C4×D137—21no—
72#14555500—41no—
73#14555497—22no—
74#14555497—41no—
75#14555494—41no—
76C2×#11764040—41no—
77C2×#11764040C2311yescharacteristic
78C2×#11764040C2311yescharacteristic
79C2×#11764040C2311yescharacteristic
80C2×#11764040C2311yescharacteristic
81C4⋊Dic137C2311yescharacteristic
82#14555489—21no—
83C22⋊Dic137—21no—
84#14555487—21no—
85#14555487—21no—
86#14555487—21no—
87#14555487—21no—
88C2×Q8×C137—21no—
89C2×Q8×C137C2311yescharacteristic
90C2×D4×C137D411yescharacteristic
91C22×C4×C137C2311yescharacteristic
92C22×C4×C137C2311yescharacteristic
93C4⋊C4×C137—21no—
94C4⋊C4×C137—21no—
95C4⋊C4×C137—21no—
96C4⋊C4×C137—21no—
97C4⋊C4×C137D411yescharacteristic
98C4⋊C4×C137C2311yescharacteristic
99C4⋊C4×C137C2311yescharacteristic
100C4⋊C4×C137C2311yescharacteristic
101C22⋊C4×C137—21no—
102C22⋊C4×C137—21no—
103C22⋊C4×C137—21no—
104C22⋊C4×C137—21no—
105C22⋊C4×C137D411yescharacteristic
106C22⋊C4×C137D411yescharacteristic
107C42×C137—21no—
108C42×C137—21no—
109C42×C137—21no—
110C42×C137C2311yescharacteristic
111C22×D137—21no—
112C22×D137—21no—
113C22×D137—42no—
114C22×D137—42no—
115C22×D137—81no—
116Dic274—21no—
117Dic274—21no—
118D548—21no—
119D548—21no—
120D548—21no—
121D548—42no—
122D548#4112yes—
123C22⋊D137—42no—
124C22⋊D137—81no—
125C2×Dic137—21no—
126C2×Dic137—21no—
127C4×D137—42no—
128C4×D137—81no—
129#11764040—21no—
130#11764040—21no—
131#11764040—21no—
132#11764040—21no—
133C23×C137C2×D411yescharacteristic
134C23×C137C2×D411yescharacteristic
135Q8×C137—21no—
136Q8×C137—21no—
137Q8×C137—42no—
138D4×C137—41no—
139C2×C4×C137—21no—
140C2×C4×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×C137C2×D411yescharacteristic
148C2×C4×C137C2×D411yescharacteristic
149C2×C4×C137C2×D411yescharacteristic
150C2×C4×C137C2×D411yescharacteristic
151C2×C4×C137C2411yescharacteristic
152C22×C137—21no—
153C22×C137—21no—
154C22×C137—21no—
155C22×C137—21no—
156C22×C137—21no—
157C22×C137—21no—
158C22×C137C22×D411yescharacteristic
159C2×D137—22no—
160C2×D137—41no—
161C2×D137—41no—
162C2×D137—41no—
163C2×D137—41no—
164C2×D137—41no—
165C2×D137—84no—
166C4×C137—21no—
167C4×C137—21no—
168C4×C137—21no—
169C4×C137—21no—
170C4×C137—41no—
171C4×C137—41no—
172C4×C137—41no—
173C4×C137—41no—
174C4×C137C2×#4111yescharacteristic
175C4×C137H(5,𝔽2)11yescharacteristic
176Dic137—41no—
177Dic137—41no—
178C2×C137—41no—
179C2×C137—41no—
180C2×C137#54611yescharacteristic
181C2×C137#57511yescharacteristic
182C2×C137#57711yescharacteristic
183D137—42no—
184D137—81no—
185C137#331111yescharacteristic, complemented, Sylow
186#3311—1371nomaximal, Sylow
187#556D13711yescharacteristic, complemented
188#546—1371no—
189#489—1371no—
190#485—1371no—
191#482—1371no—
192#476—1371no—
193#470—1371no—
194#465—1371no—
195#464—1371no—
196#452—1371no—
197#450—1371no—
198#440—1371no—
199#435—1371no—
200#418—1371no—
201#416—1371no—
202C2×#41—1371no—
203C22×D4—1371no—
204C2×Q16—1371no—
205C2×SD16—1371no—
206C2×SD16—1371no—
207C2×M16—1371no—
208C22×C8—1371no—
209C4⋊Q8C2×D13711yescharacteristic, complemented
210#126—21no—
211#126—21no—
212#124—21no—
213#124—1371no—
214#123—2741no—
215C22⋊Q8—1371no—
216C22⋊Q8C2×D13711yescharacteristic
217C22⋊Q8C2×D13711yescharacteristic, complemented
218#121—2741no—
219#121—1371no—
220#120—2741no—
221C4×Q8—21no—
222C4×Q8C2×D13711yescharacteristic, complemented
223C4×D4—1371no—
224C4×D4C2×D13711yescharacteristic
225#117C2×D13711yescharacteristic
226#117C2×D13711yescharacteristic, complemented
227C2×C22⋊C4—2741no—
228#107—1371no—
229#106—1371no—
230#106—1371no—
231C4⋊C8—1371no—
232#103—1371no—
233#103—1371no—
234#103—1371no—
235#103—1371no—
236#102—1371no—
237#102—1371no—
238#102—1371no—
239#102—1371no—
240#102—1371no—
241#102—1371no—
242C22⋊C8—1371no—
243C22⋊C8—1371no—
244#97—1371no—
245C24—2741no—
246#41—2742no—
247#41—5481no—
248C2×Q8—21no—
249C2×Q8—1371no—
250C2×Q8C22×D13711yescharacteristic
251C2×D4—2742no—
252C2×D4—2741no—
253C2×D4—2741no—
254C2×D4—2741no—
255C2×D4—5481no—
256C2×D4—1371no—
257C2×D4—1371no—
258C2×D4C22⋊D13711yescharacteristic
259C22×C4—2741no—
260C22×C4—2741no—
261C22×C4C22×D13711yescharacteristic
262C22×C4C22×D13711yescharacteristic
263Q16—5481no—
264SD16—2742no—
265SD16—5481no—
266M16—5481no—
267C2×C8—5481no—
268C2×C8—1371no—
269C2×C8—1371no—
270C2×C8—1371no—
271C2×C8—1371no—
272C4⋊C4—21no—
273C4⋊C4—21no—
274C4⋊C4—21no—
275C4⋊C4—21no—
276C4⋊C4—2741no—
277C4⋊C4—1371no—
278C4⋊C4C22⋊D13711yescharacteristic
279C4⋊C4C22×D13711yescharacteristic
280C4⋊C4C22×D13711yescharacteristic
281C4⋊C4C22×D13711yescharacteristic
282C22⋊C4—21no—
283C22⋊C4—21no—
284C22⋊C4—21no—
285C22⋊C4—21no—
286C22⋊C4—2741no—
287C22⋊C4—2741no—
288C22⋊C4—2741no—
289C22⋊C4—2741no—
290C22⋊C4—2741no—
291C22⋊C4C22⋊D13711yescharacteristic
292C22⋊C4C22⋊D13711yescharacteristic
293C42—21no—
294C42—21no—
295C42—21no—
296C42C22×D13711yescharacteristic
297C23—2741no—
298C23—2741no—
299C23—5482no—
300C23—5482no—
301C23—10961no—
302C23C2×C22⋊D13711yescharacteristic
303C23C2×C22⋊D13711yescharacteristic
304Q8—21no—
305Q8—21no—
306Q8—42no—
307Q8—2741no—
308Q8—2741no—
309D4—41no—
310D4—2741no—
311D4—2741no—
312D4—2741no—
313D4—5482no—
314D4—5482no—
315D4—10961no—
316D4—1372no—
317C2×C4—21no—
318C2×C4—21no—
319C2×C4—21no—
320C2×C4—21no—
321C2×C4—21no—
322C2×C4—21no—
323C2×C4—21no—
324C2×C4—21no—
325C2×C4—2741no—
326C2×C4—2741no—
327C2×C4—5482no—
328C2×C4—10961no—
329C2×C4C2×C22⋊D13711yescharacteristic
330C2×C4C2×C22⋊D13711yescharacteristic
331C2×C4C2×C22⋊D13711yescharacteristic
332C2×C4C2×C22⋊D13711yescharacteristic
333C2×C4C23×D13711yescharacteristic
334C8—2741no—
335C8—2741no—
336C8—2741no—
337C8—2741no—
338C22—21no—
339C22—21no—
340C22—21no—
341C22—21no—
342C22—21no—
343C22—21no—
344C22—2742no—
345C22—5481no—
346C22—5481no—
347C22—5481no—
348C22—5481no—
349C22—5481no—
350C22—10964no—
351C22C22×C22⋊D13711yescharacteristic
352C4—21no—
353C4—21no—
354C4—21no—
355C4—21no—
356C4—41no—
357C4—41no—
358C4—41no—
359C4—41no—
360C4—5481no—
361C4—5481no—
362C4#41×D13711yescharacteristic
363C4#1798413411yescharacteristic
364C2—41no—
365C2—41no—
366C2—5482no—
367C2—10961no—
368C2#2257804111yescharacteristic
369C2#2257820111yescharacteristic
370C2#2257821111yescharacteristic
subgroupG/H
centerC22C22×C22⋊D137
commutator, cosocleC2×C4×C137C24
FrattiniC2×C4C23×D137
Fitting#556×C137C2
socleC22×C137C22×D4
prime radical#22578049C2
hypercenter#556D137
hypocenterC137#3311
LCSG ▷ C2×C4×C137 ▷ C2×C137 ▷ C137
UCSC1 ◁ C22 ◁ C4×D4 ◁ #556

Complements

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

NH ≅ G/Ntype
2. #22577765C2semidirect
3. #22577744C2semidirect
4. #22577727C2semidirect
5. #22577690C2semidirect
6. #22577659C2semidirect
7. #22577634C2semidirect
8. #22577625C2semidirect
9. #22577563C2semidirect
10. #22577553C2semidirect
11. #22577501C2semidirect
12. #22577475C2semidirect
13. #22577407C2semidirect
14. #22577397C2semidirect
15. #556×C137C2semidirect
18. C2×#14555500C22semidirect
22. C22×#11764040C22semidirect
31. #17984021C22semidirect
32. #17984019C22semidirect
33. #17984019C22semidirect
34. C4⋊#11764040C22semidirect
35. #17984012C22semidirect
36. #17984011C22semidirect
37. #17984011C22semidirect
38. #17984011C22semidirect
46. C22⋊#11764040C22semidirect
47. #17983999C22semidirect
48. C4⋊Q8×C137C22semidirect
53. C22⋊Q8×C137C22semidirect
55. C4×Q8×C137C22semidirect
58. #117×C137C22semidirect
185. C137#3311semidirect
187. #556D137semidirect
209. C4⋊Q8C2×D137semidirect
217. C22⋊Q8C2×D137semidirect
222. C4×Q8C2×D137semidirect
226. #117C2×D137semidirect

Maximal subgroups

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

subgroupindex
#225780492
#225777652
#225777442
#225777272
#225776902
#225776592
#225776342
#225776252
#225775632
#225775532
#225775012
#225774752
#225774072
#225773972
#556×C1372
#3311137