Finite Group Databasebeta

← #31792620

Subgroups — #31792620

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#22578047C211yesmaximal, characteristic
3#22577756C211yesmaximal, characteristic, complemented
4#22577733C211yesmaximal, characteristic, complemented
5#22577725C211yesmaximal, characteristic, complemented
6#22577701C211yesmaximal, characteristic, complemented
7#22577644C211yesmaximal, characteristic, complemented
8#22577631C211yesmaximal, characteristic, complemented
9#22577624C211yesmaximal, characteristic, complemented
10#22577566C211yesmaximal, characteristic, complemented
11#22577554C211yesmaximal, characteristic, complemented
12#22577494C211yesmaximal, characteristic, complemented
13C8×C22⋊D137C211yesmaximal, characteristic, complemented
14#22577396C211yesmaximal, characteristic, complemented
15C2×#102×C137C211yesmaximal, characteristic, complemented
16C2×#14555509C2211yescharacteristic
17C2×Dic548C2211yescharacteristic, complemented
18C2×#14555498C2211yescharacteristic
19C2×#14555498C2211yescharacteristic
20C2×C8×D137C2211yescharacteristic, complemented
21#17984083—21no—
22#17984079C2211yescharacteristic, complemented
23#17984078—21no—
24#17984075C2211yescharacteristic
25#17984071—21no—
26#17984069—21no—
27#17984068C2211yescharacteristic, complemented
28#17984067C2211yescharacteristic
29#17984067C2211yescharacteristic
30#17984060—21no—
31#17984059C2211yescharacteristic
32#17984057—21no—
33#17984053C2211yescharacteristic, complemented
34D4×Dic137C2211yescharacteristic
35C4×C22⋊D137C2211yescharacteristic
36#17984047—21no—
37#17984046C2211yescharacteristic, complemented
38C2×C22⋊Dic137—21no—
39#17984022C2211yescharacteristic, complemented
40#17984019C2211yescharacteristic, complemented
41#17984019C2211yescharacteristic, complemented
42#17984017C2211yescharacteristic, complemented
43#17984013C2211yescharacteristic, complemented
44#17984013C2211yescharacteristic, complemented
45#17984011C2211yescharacteristic, complemented
46#17984011C2211yescharacteristic, complemented
47#17984009C2211yescharacteristic
48#17984009C2211yescharacteristic
49C22⋊#11764040C2211yescharacteristic
50#17984000C2211yescharacteristic, complemented
51C8×Dic137C2211yescharacteristic, complemented
52C22×D4×C137C2211yescharacteristic
53C22×C8×C137C2211yescharacteristic
54C2×C4⋊C4×C137C2211yescharacteristic
55#102×C137C2211yescharacteristic
56#102×C137C2211yescharacteristic
57#102×C137C2211yescharacteristic
58#102×C137C2211yescharacteristic
59#14555509—22no—
60#14555509—41no—
61C2×Dic274C2311yescharacteristic
62C2×Dic274C2311yescharacteristic
63C2×C22⋊D137—21no—
64C2×C22⋊D137—21no—
65C2×C22⋊D137D411yescharacteristic
66C22×Dic137—21no—
67C22×Dic137—21no—
68C2×C4×D137C2311yescharacteristic
69Dic548—41no—
70#14555498—22no—
71#14555498—41no—
72C2×#11764040C2311yescharacteristic
73C2×#11764040C2311yescharacteristic
74C8×D137—41no—
75C4⋊Dic137—21no—
76C4⋊Dic137—21no—
77C4⋊Dic137C2311yescharacteristic
78C4⋊Dic137C2311yescharacteristic
79#14555489—21no—
80#14555489—21no—
81#14555489—21no—
82#14555489—21no—
83#14555489D411yescharacteristic
84C22⋊Dic137—21no—
85C22⋊Dic137—21no—
86C22⋊Dic137—21no—
87C22⋊Dic137—21no—
88C22⋊Dic137—21no—
89C22⋊Dic137—21no—
90C22⋊Dic137D411yescharacteristic
91#14555487—21no—
92#14555487—21no—
93#14555487D411yescharacteristic
94C4×Dic137—21no—
95C4×Dic137—21no—
96C4×Dic137C2311yescharacteristic
97C24×C137—21no—
98C2×D4×C137—22no—
99C2×D4×C137—21no—
100C2×D4×C137—41no—
101C2×D4×C137C2311yescharacteristic
102C2×D4×C137C2311yescharacteristic
103C22×C4×C137—21no—
104C22×C4×C137C2311yescharacteristic
105C2×C8×C137—41no—
106C2×C8×C137C2311yescharacteristic
107C2×C8×C137C2311yescharacteristic
108C4⋊C4×C137—21no—
109C4⋊C4×C137C2311yescharacteristic
110C4⋊C4×C137C2311yescharacteristic
111C22×D137C2×D411yescharacteristic
112Dic274—21no—
113Dic274—21no—
114Dic274—21no—
115Dic274—21no—
116C22⋊D137—22no—
117C22⋊D137—42no—
118C22⋊D137—81no—
119C2×Dic137—21no—
120C2×Dic137—21no—
121C2×Dic137—21no—
122C2×Dic137—21no—
123C2×Dic137—42no—
124C2×Dic137—81no—
125C2×Dic137C2×D411yescharacteristic
126C2×Dic137C2×D411yescharacteristic
127C2×Dic137C2×D411yescharacteristic
128C4×D137—21no—
129C4×D137—21no—
130#11764040—21no—
131#11764040—21no—
132C23×C137—21no—
133C23×C137—21no—
134C23×C137—42no—
135C23×C137—42no—
136C23×C137—81no—
137C23×C137C2×D411yescharacteristic
138D4×C137—21no—
139D4×C137—21no—
140D4×C137—21no—
141D4×C137—42no—
142D4×C137#4112yes—
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—22no—
155C22×C137—41no—
156C22×C137—41no—
157C22×C137—41no—
158C22×C137—41no—
159C22×C137—41no—
160C22×C137—84no—
161C22×C137#13512yes—
162C22×C137C22×D411yescharacteristic
163C2×D137—21no—
164C2×D137—21no—
165C2×D137—21no—
166C4×C137—21no—
167C4×C137—41no—
168C4×C137—41no—
169C4×C137C2×#4111yescharacteristic
170C4×C137H(5,𝔽2)11yescharacteristic
171Dic137—21no—
172Dic137—21no—
173Dic137—21no—
174Dic137—41no—
175Dic137—41no—
176Dic137—41no—
177Dic137—41no—
178C2×C137—22no—
179C2×C137—42no—
180C2×C137—81no—
181C2×C137#54611yescharacteristic
182C2×C137C2×#13511yescharacteristic
183C2×C137#57711yescharacteristic
184D137—41no—
185C137#329611yescharacteristic, complemented, Sylow
186#3296—1371nomaximal, Sylow
187#552—1371no—
188#546—1371no—
189#488—1371no—
190#483—1371no—
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×#102D13711yescharacteristic, complemented
202C2×#41—1371no—
203C22×D4C2×D13711yescharacteristic
204C2×Q16—1371no—
205C2×SD16—1371no—
206C2×SD16—1371no—
207C22×C8—1371no—
208C22×C8C2×D13711yescharacteristic
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⋊Q8—1371no—
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#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#102C2×D13711yescharacteristic
239#102C2×D13711yescharacteristic
240#102C2×D13711yescharacteristic
241#102C2×D13711yescharacteristic
242C22⋊C8—1371no—
243C22⋊C8—1371no—
244C4×C8—1371no—
245C24—21no—
246#41—2742no—
247#41—5481no—
248C2×Q8—1371no—
249C2×Q8—1371no—
250C2×D4—22no—
251C2×D4—21no—
252C2×D4—41no—
253C2×D4—2741no—
254C2×D4—2741no—
255C2×D4—1371no—
256C2×D4C22×D13711yescharacteristic
257C2×D4C22×D13711yescharacteristic
258C22×C4—21no—
259C22×C4—2741no—
260C22×C4—2741no—
261C22×C4—1371no—
262C22×C4C22×D13711yescharacteristic
263Q16—5481no—
264SD16—2742no—
265SD16—5481no—
266C2×C8—41no—
267C2×C8—5481no—
268C2×C8—1371no—
269C2×C8—1371no—
270C2×C8C22×D13711yescharacteristic
271C2×C8C22×D13711yescharacteristic
272C4⋊C4—21no—
273C4⋊C4—2741no—
274C4⋊C4—2741no—
275C4⋊C4—2741no—
276C4⋊C4—2741no—
277C4⋊C4—2741no—
278C4⋊C4—2741no—
279C4⋊C4—1371no—
280C4⋊C4—1371no—
281C4⋊C4—1371no—
282C4⋊C4C22×D13711yescharacteristic
283C4⋊C4C22×D13711yescharacteristic
284C22⋊C4—2741no—
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—21no—
298C23—21no—
299C23—42no—
300C23—42no—
301C23—81no—
302C23—1371no—
303C23D4×D13711yescharacteristic
304Q8—2741no—
305Q8—2741no—
306Q8—2741no—
307Q8—2741no—
308D4—21no—
309D4—21no—
310D4—21no—
311D4—42no—
312D4—2742no—
313D4—5482no—
314D4—10961no—
315D4#1455551012yes—
316C2×C4—21no—
317C2×C4—21no—
318C2×C4—21no—
319C2×C4—21no—
320C2×C4—42no—
321C2×C4—2741no—
322C2×C4—2741no—
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—22no—
341C22—41no—
342C22—41no—
343C22—41no—
344C22—41no—
345C22—41no—
346C22—84no—
347C22—2741no—
348C22—2741no—
349C22—2741no—
350C22#1798410712yes—
351C22C2×D4×D13711yescharacteristic
352C4—21no—
353C4—41no—
354C4—41no—
355C4—2741no—
356C4—2741no—
357C4—2741no—
358C4—5481no—
359C4—5481no—
360C4—5481no—
361C4—5481no—
362C4C2×#1455551011yescharacteristic
363C4#1798413311yescharacteristic
364C2—22no—
365C2—42no—
366C2—81no—
367C2—5481no—
368C2#2257804311yescharacteristic
369C2C2×#1798410711yescharacteristic
370C2#2257820911yescharacteristic
subgroupG/H
centerC22C2×D4×D137
commutator, cosocleC2×C4×C137C24
FrattiniC2×C4C23×D137
FittingC2×#102×C137C2
socleC22×C137C22×D4
prime radical#22578047C2
hypercenterC2×#102D137
hypocenterC137#3296
LCSG ▷ C2×C4×C137 ▷ C2×C137 ▷ C137
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×#102

Complements

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

NH ≅ G/Ntype
1. #22578088C2semidirect
3. #22577756C2semidirect
4. #22577733C2semidirect
5. #22577725C2semidirect
6. #22577701C2semidirect
7. #22577644C2semidirect
8. #22577631C2semidirect
9. #22577624C2semidirect
10. #22577566C2semidirect
11. #22577554C2semidirect
12. #22577494C2semidirect
13. C8×C22⋊D137C2semidirect
14. #22577396C2semidirect
15. C2×#102×C137C2semidirect
17. C2×Dic548C22semidirect
20. C2×C8×D137C22semidirect
22. #17984079C22semidirect
27. #17984068C22semidirect
33. #17984053C22semidirect
37. #17984046C22semidirect
39. #17984022C22semidirect
40. #17984019C22semidirect
41. #17984019C22semidirect
42. #17984017C22semidirect
43. #17984013C22semidirect
44. #17984013C22semidirect
45. #17984011C22semidirect
46. #17984011C22semidirect
50. #17984000C22semidirect
51. C8×Dic137C22semidirect
185. C137#3296semidirect
201. C2×#102D137semidirect

Maximal subgroups

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

subgroupindex
#225780882
#225780472
#225777562
#225777332
#225777252
#225777012
#225776442
#225776312
#225776242
#225775662
#225775542
#225774942
C8×C22⋊D1372
#225773962
C2×#102×C1372
#3296137