Finite Group Databasebeta

← #31792621

Subgroups — #31792621

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#22578091C211yesmaximal, characteristic, complemented
2#22578048C211yesmaximal, characteristic
3#22577760C211yesmaximal, characteristic, complemented
4#22577735C211yesmaximal, characteristic, complemented
5#22577730C211yesmaximal, characteristic, complemented
6#22577645C211yesmaximal, characteristic, complemented
7#22577633C211yesmaximal, characteristic, complemented
8#22577626C211yesmaximal, characteristic, complemented
9#22577566C211yesmaximal, characteristic, complemented
10#22577555C211yesmaximal, characteristic, complemented
11C4×#14555498C211yesmaximal, characteristic, complemented
12#22577468C211yesmaximal, characteristic, complemented
13#22577396C211yesmaximal, characteristic, complemented
14#102×D137C211yesmaximal, characteristic, complemented
15#478×C137C211yesmaximal, characteristic, complemented
16C2×#14555509C2211yescharacteristic
17C2×D4×D137C2211yescharacteristic
18C2×Dic548C2211yescharacteristic, complemented
19C2×#14555499C2211yescharacteristic
20C2×#14555498C2211yescharacteristic
21C2×C8×D137C2211yescharacteristic
22C2×C8×D137C2211yescharacteristic, complemented
23#17984083—21no—
24#17984079—21no—
25#17984076C2211yescharacteristic
26#17984073—21no—
27#17984072—21no—
28#17984068C2211yescharacteristic
29#17984068C2211yescharacteristic
30#17984068C2211yescharacteristic, complemented
31#17984063C2211yescharacteristic
32#17984062—21no—
33#17984058—21no—
34C4×Dic274C2211yescharacteristic, complemented
35C4×D548C2211yescharacteristic
36#17984046—21no—
37#17984046C2211yescharacteristic, complemented
38C4⋊C4×D137C2211yescharacteristic
39C2×#14555487—21no—
40#17984019C2211yescharacteristic, complemented
41#17984013C2211yescharacteristic, complemented
42#17984011C2211yescharacteristic, complemented
43#17984011C2211yescharacteristic, complemented
44#17984011C2211yescharacteristic, complemented
45#17984010C2211yescharacteristic
46#17984009C2211yescharacteristic
47#17984008C2211yescharacteristic
48#17984008C2211yescharacteristic
49#17984000C2211yescharacteristic
50#17984000C2211yescharacteristic, complemented
51C4×#11764040C2211yescharacteristic, complemented
52#125×C137C2211yescharacteristic, complemented
53C4×D4×C137C2211yescharacteristic
54#107×C137C2211yescharacteristic, complemented
55#106×C137C2211yescharacteristic, complemented
56C4⋊C8×C137C2211yescharacteristic, complemented
57#102×C137C2211yescharacteristic
58#102×C137C2211yescharacteristic
59C23×D137—21no—
60#14555509—22no—
61#14555509—41no—
62C2×Dic274C2311yescharacteristic
63C2×Dic274C2311yescharacteristic
64C2×D548—21no—
65C2×D548D411yescharacteristic
66C2×D548C2311yescharacteristic
67C2×C22⋊D137—21no—
68C2×C22⋊D137—21no—
69D4×D137—22no—
70D4×D137—41no—
71C22×Dic137—21no—
72C2×C4×D137—21no—
73C2×C4×D137C2311yescharacteristic
74C2×C4×D137C2311yescharacteristic
75Dic548—41no—
76#14555499—41no—
77#14555498—22no—
78C2×#11764040C2311yescharacteristic
79C2×#11764040C2311yescharacteristic
80C8×D137—41no—
81C8×D137—41no—
82C4⋊Dic137—21no—
83C4⋊Dic137—21no—
84C4⋊Dic137—21no—
85C4⋊Dic137D411yescharacteristic
86C4⋊Dic137C2311yescharacteristic
87#14555489—21no—
88#14555489—21no—
89#14555487—21no—
90#14555487—21no—
91#14555487—21no—
92#14555487—21no—
93#14555487—21no—
94#14555487—21no—
95#14555487—21no—
96#14555487D411yescharacteristic
97#14555487D411yescharacteristic
98C4×Dic137—21no—
99C4×Dic137—21no—
100C2×D4×C137C2311yescharacteristic
101C22×C4×C137—21no—
102C2×C8×C137C2311yescharacteristic
103C2×C8×C137C2311yescharacteristic
104C4⋊C4×C137—21no—
105C4⋊C4×C137—21no—
106C4⋊C4×C137C2311yescharacteristic
107C4⋊C4×C137C2311yescharacteristic
108C4⋊C4×C137C2311yescharacteristic
109C22⋊C4×C137—21no—
110C42×C137C2311yescharacteristic
111C22×D137—21no—
112C22×D137—42no—
113C22×D137—42no—
114C22×D137—81no—
115C22×D137C2×D411yescharacteristic
116C22×D137C2×D411yescharacteristic
117Dic274—21no—
118Dic274—21no—
119Dic274—21no—
120Dic274—21no—
121D548—22no—
122D548—21no—
123D548—21no—
124D548—81no—
125C22⋊D137—42no—
126C22⋊D137—42no—
127C2×Dic137—21no—
128C2×Dic137—21no—
129C2×Dic137—21no—
130C2×Dic137—42no—
131C2×Dic137C2×D411yescharacteristic
132C2×Dic137C2×D411yescharacteristic
133C4×D137—21no—
134C4×D137—21no—
135C4×D137—21no—
136C4×D137—21no—
137C4×D137—42no—
138#11764040—21no—
139#11764040—21no—
140C23×C137—21no—
141D4×C137—21no—
142D4×C137#4112yes—
143C2×C4×C137—21no—
144C2×C4×C137—21no—
145C2×C4×C137—21no—
146C2×C4×C137—81no—
147C2×C4×C137C2×D411yescharacteristic
148C2×C4×C137C2×D411yescharacteristic
149C2×C4×C137C2411yescharacteristic
150C8×C137—21no—
151C8×C137—21no—
152C22×C137—22no—
153C22×C137—41no—
154C22×C137—41no—
155C22×C137C22×D411yescharacteristic
156C2×D137—21no—
157C2×D137—21no—
158C2×D137—21no—
159C2×D137—21no—
160C2×D137—21no—
161C2×D137—41no—
162C2×D137—41no—
163C2×D137—41no—
164C2×D137—84no—
165C2×D137#13512yes—
166C4×C137—21no—
167C4×C137—21no—
168C4×C137—41no—
169C4×C137—41no—
170C4×C137—41no—
171C4×C137C2×#4111yescharacteristic
172C4×C137H(5,𝔽2)11yescharacteristic
173Dic137—21no—
174Dic137—21no—
175Dic137—41no—
176Dic137—41no—
177Dic137—41no—
178C2×C137—42no—
179C2×C137#54611yescharacteristic
180C2×C137C2×#13511yescharacteristic
181C2×C137#57711yescharacteristic
182D137—22no—
183D137—41no—
184D137—81no—
185C137#329611yescharacteristic, complemented, Sylow
186#3296—1371nomaximal, Sylow
187#552—1371no—
188#546—1371no—
189#488—1371no—
190#483—1371no—
191#482—1371no—
192#478D13711yescharacteristic, complemented
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#125C2×D13711yescharacteristic, complemented
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×D4C2×D13711yescharacteristic
224#117—2741no—
225#117—1371no—
226C2×C4⋊C4—1371no—
227C2×C22⋊C4—2741no—
228#107C2×D13711yescharacteristic, complemented
229#106—1371no—
230#106C2×D13711yescharacteristic, complemented
231C4⋊C8C2×D13711yescharacteristic, complemented
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#102C2×D13711yescharacteristic
241#102C2×D13711yescharacteristic
242C22⋊C8—1371no—
243C22⋊C8—1371no—
244C4×C8—1371no—
245C24—2741no—
246#41—2742no—
247#41—5481no—
248C2×Q8—1371no—
249C2×Q8—1371no—
250C2×D4—2741no—
251C2×D4—2742no—
252C2×D4—2741no—
253C2×D4—2741no—
254C2×D4—5481no—
255C2×D4—1371no—
256C2×D4—1371no—
257C2×D4C22×D13711yescharacteristic
258C22×C4—21no—
259C22×C4—2741no—
260C22×C4—2741no—
261C22×C4—1371no—
262C22×C4—1371no—
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—
296C42C22×D13711yescharacteristic
297C23—21no—
298C23—2741no—
299C23—5482no—
300C23—5482no—
301C23—10961no—
302C23—1371no—
303C23—1371no—
304Q8—2741no—
305Q8—2741no—
306Q8—2741no—
307Q8—2741no—
308D4—21no—
309D4—2742no—
310D4—2741no—
311D4—2741no—
312D4—5482no—
313D4—5482no—
314D4—10961no—
315D4#1455551112yes—
316C2×C4—21no—
317C2×C4—21no—
318C2×C4—21no—
319C2×C4—81no—
320C2×C4—2741no—
321C2×C4—2741no—
322C2×C4—2741no—
323C2×C4—2741no—
324C2×C4—2741no—
325C2×C4—2741no—
326C2×C4—2741no—
327C2×C4—5482no—
328C2×C4—5482no—
329C2×C4—1371no—
330C2×C4—1371no—
331C2×C4D4×D13711yescharacteristic
332C2×C4D4×D13711yescharacteristic
333C2×C4C23×D13711yescharacteristic
334C8—21no—
335C8—21no—
336C8—2741no—
337C8—2741no—
338C22—22no—
339C22—41no—
340C22—41no—
341C22—2741no—
342C22—2741no—
343C22—2741no—
344C22—2741no—
345C22—2741no—
346C22—5481no—
347C22—5481no—
348C22—5481no—
349C22—10964no—
350C22—1372no—
351C22C2×D4×D13711yescharacteristic
352C4—21no—
353C4—21no—
354C4—41no—
355C4—41no—
356C4—41no—
357C4—2741no—
358C4—2741no—
359C4—5481no—
360C4—5481no—
361C4—5481no—
362C4C2×#1455551111yescharacteristic
363C4#1798413411yescharacteristic
364C2—42no—
365C2—2742no—
366C2—5481no—
367C2—10961no—
368C2#2257804411yescharacteristic
369C2#135×D13711yescharacteristic
370C2#2257821311yescharacteristic
subgroupG/H
centerC22C2×D4×D137
commutator, cosocleC2×C4×C137C24
FrattiniC2×C4C23×D137
Fitting#478×C137C2
socleC22×C137C22×D4
prime radical#22578048C2
hypercenter#478D137
hypocenterC137#3296
LCSG ▷ C2×C4×C137 ▷ C2×C137 ▷ C137
UCSC1 ◁ C22 ◁ C42 ◁ #478

Complements

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

NH ≅ G/Ntype
1. #22578091C2semidirect
3. #22577760C2semidirect
4. #22577735C2semidirect
5. #22577730C2semidirect
6. #22577645C2semidirect
7. #22577633C2semidirect
8. #22577626C2semidirect
9. #22577566C2semidirect
10. #22577555C2semidirect
11. C4×#14555498C2semidirect
12. #22577468C2semidirect
13. #22577396C2semidirect
14. #102×D137C2semidirect
15. #478×C137C2semidirect
18. C2×Dic548C22semidirect
22. C2×C8×D137C22semidirect
30. #17984068C22semidirect
34. C4×Dic274C22semidirect
37. #17984046C22semidirect
40. #17984019C22semidirect
41. #17984013C22semidirect
42. #17984011C22semidirect
43. #17984011C22semidirect
44. #17984011C22semidirect
50. #17984000C22semidirect
51. C4×#11764040C22semidirect
52. #125×C137C22semidirect
54. #107×C137C22semidirect
55. #106×C137C22semidirect
56. C4⋊C8×C137C22semidirect
185. C137#3296semidirect
192. #478D137semidirect
211. #125C2×D137semidirect
228. #107C2×D137semidirect
230. #106C2×D137semidirect
231. C4⋊C8C2×D137semidirect

Maximal subgroups

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

subgroupindex
#225780912
#225780482
#225777602
#225777352
#225777302
#225776452
#225776332
#225776262
#225775662
#225775552
C4×#145554982
#225774682
#225773962
#102×D1372
#478×C1372
#3296137