Finite Group Databasebeta

← #31792809

Subgroups — #31792809

Order 17536.

450 conjugacy classes of subgroups.

The subgroup lattice has 392 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#22578091C211yesmaximal, characteristic, complemented
2#22578038C211yesmaximal, characteristic
3#22577725C211yesmaximal, characteristic, complemented
4#22577715C211yesmaximal, characteristic, complemented
5#22577700C211yesmaximal, characteristic, complemented
6#22577648C211yesmaximal, characteristic, complemented
7#22577631C211yesmaximal, characteristic, complemented
8#22577600C211yesmaximal, characteristic, complemented
9#22577555C211yesmaximal, characteristic, complemented
10#22577555C211yesmaximal, characteristic, complemented
11#22577505C211yesmaximal, characteristic, complemented
12#22577476C211yesmaximal, characteristic, complemented
13#22577406C211yesmaximal, characteristic, complemented
14#102×D137C211yesmaximal, characteristic, complemented
15#490×C137C211yesmaximal, characteristic, complemented
16C2×D4×D137—21no—
17C2×D4×D137C2211yescharacteristic
18C2×D4×D137C2211yescharacteristic
19C2×#14555499C2211yescharacteristic
20C2×#14555498C2211yescharacteristic, complemented
21C2×#14555498C2211yescharacteristic, complemented
22C2×#14555493C2211yescharacteristic, complemented
23C2×C8×D137C2211yescharacteristic, complemented
24#17984083—21no—
25#17984079—21no—
26#17984073—21no—
27#17984068C2211yescharacteristic, complemented
28#17984068C2211yescharacteristic, complemented
29#17984059—21no—
30#17984059C2211yescharacteristic
31#17984059C2211yescharacteristic
32#17984056—21no—
33#17984056—21no—
34C4×Dic274C2211yescharacteristic, complemented
35C4×D548C2211yescharacteristic
36C4×D548C2211yescharacteristic
37#17984046—21no—
38#17984046C2211yescharacteristic, complemented
39C4⋊C4×D137C2211yescharacteristic, complemented
40#17984022C2211yescharacteristic, complemented
41#17984021C2211yescharacteristic, complemented
42#17984019C2211yescharacteristic, complemented
43C4⋊#11764040C2211yescharacteristic, complemented
44#17984013C2211yescharacteristic, complemented
45#17984010C2211yescharacteristic
46#17984009C2211yescharacteristic, complemented
47#17984009C2211yescharacteristic, complemented
48#17984008C2211yescharacteristic
49#17984008C2211yescharacteristic
50#17984000C2211yescharacteristic, complemented
51#17984000C2211yescharacteristic, complemented
52#127×C137C2211yescharacteristic
53#125×C137C2211yescharacteristic, complemented
54#102×C137C2211yescharacteristic, complemented
55#102×C137C2211yescharacteristic, complemented
56#102×C137C2211yescharacteristic, complemented
57#102×C137C2211yescharacteristic, complemented
58#97×C137C2211yescharacteristic, complemented
59C23×D137—21no—
60C23×D137—21no—
61C2×Dic274C2311yescharacteristic
62C2×D548D411yescharacteristic
63C2×D548C2311yescharacteristic
64C2×C22⋊D137—21no—
65C2×C22⋊D137—21no—
66C2×C22⋊D137—21no—
67C2×C22⋊D137—21no—
68D4×D137—22no—
69D4×D137—22no—
70D4×D137—41no—
71D4×D137—41no—
72D4×D137—44no—
73C2×C4×D137—21no—
74C2×C4×D137—21no—
75C2×C4×D137C2311yescharacteristic
76C2×C4×D137C2311yescharacteristic
77#14555499—22no—
78#14555498—41no—
79#14555498—41no—
80#14555493—41no—
81C2×#11764040C2311yescharacteristic
82C2×#11764040C2311yescharacteristic
83C8×D137—41no—
84C4⋊Dic137—21no—
85C4⋊Dic137—21no—
86C4⋊Dic137D411yescharacteristic
87C4⋊Dic137C2311yescharacteristic
88C4⋊Dic137C2311yescharacteristic
89#14555489—21no—
90#14555489—21no—
91C22⋊Dic137—21no—
92C22⋊Dic137—21no—
93#14555487—21no—
94#14555487—21no—
95#14555487—21no—
96#14555487—21no—
97#14555487D411yescharacteristic
98#14555487D411yescharacteristic
99C4×Dic137—21no—
100C4×Dic137—21no—
101C2×D4×C137—21no—
102C2×D4×C137—21no—
103C2×D4×C137C2311yescharacteristic
104C2×D4×C137C2311yescharacteristic
105C2×C8×C137C2311yescharacteristic
106C2×C8×C137C2311yescharacteristic
107C4⋊C4×C137—21no—
108C4⋊C4×C137—21no—
109C4⋊C4×C137C2311yescharacteristic
110C4⋊C4×C137C2311yescharacteristic
111C42×C137C2311yescharacteristic
112C22×D137—21no—
113C22×D137—42no—
114C22×D137—42no—
115C22×D137—42no—
116C22×D137—42no—
117C22×D137—81no—
118C22×D137—81no—
119C22×D137C2×D411yescharacteristic
120C22×D137C2×D411yescharacteristic
121Dic274—21no—
122Dic274—21no—
123D548—22no—
124D548—21no—
125D548#4112yes—
126C22⋊D137—42no—
127C22⋊D137—42no—
128C22⋊D137—42no—
129C22⋊D137—42no—
130C2×Dic137—21no—
131C2×Dic137—21no—
132C2×Dic137—21no—
133C2×Dic137C2×D411yescharacteristic
134C2×Dic137C2×D411yescharacteristic
135C4×D137—21no—
136C4×D137—21no—
137C4×D137—21no—
138C4×D137—21no—
139C4×D137—42no—
140C4×D137—42no—
141#11764040—21no—
142#11764040—21no—
143C23×C137—21no—
144C23×C137—21no—
145D4×C137—21no—
146D4×C137—21no—
147D4×C137—21no—
148D4×C137—21no—
149D4×C137—42no—
150D4×C137—81no—
151C2×C4×C137—21no—
152C2×C4×C137—21no—
153C2×C4×C137C2×D411yescharacteristic
154C2×C4×C137C2×D411yescharacteristic
155C2×C4×C137C2411yescharacteristic
156C8×C137—21no—
157C8×C137—21no—
158C22×C137—41no—
159C22×C137—41no—
160C22×C137—41no—
161C22×C137—41no—
162C22×C137—41no—
163C22×C137—41no—
164C22×C137C22×D411yescharacteristic
165C2×D137—21no—
166C2×D137—21no—
167C2×D137—21no—
168C2×D137—22no—
169C2×D137—21no—
170C2×D137—21no—
171C2×D137—41no—
172C2×D137—41no—
173C2×D137—84no—
174C2×D137—84no—
175C2×D137AGL(1,ℤ/8)12yes—
176C4×C137—21no—
177C4×C137—21no—
178C4×C137—41no—
179C4×C137—41no—
180C4×C137C2×#4111yescharacteristic
181C4×C137H(5,𝔽2)11yescharacteristic
182Dic137—21no—
183Dic137—21no—
184Dic137—41no—
185Dic137—41no—
186Dic137—41no—
187C2×C137—81no—
188C2×C137—81no—
189C2×C137#54611yescharacteristic
190C2×C137C2×AGL(1,ℤ/8)11yescharacteristic
191C2×C137#57711yescharacteristic
192D137—22no—
193D137—41no—
194D137—42no—
195C137#331011yescharacteristic, complemented, Sylow
196#3310—1371nomaximal, Sylow
197#552—1371no—
198D42—1371no—
199#490D13711yescharacteristic, complemented
200#482—1371no—
201#480—1371no—
202#478—1371no—
203#468—1371no—
204#465—1371no—
205#460—1371no—
206#450—1371no—
207#450—1371no—
208#440—1371no—
209#435—1371no—
210#418—1371no—
211C2×#102—1371no—
212C22×D4—2741no—
213C22×D4—1371no—
214C22×D4—1371no—
215C2×SD16—1371no—
216C2×SD16—1371no—
217C2×SD16—1371no—
218C2×M16—1371no—
219C22×C8—1371no—
220#127C2×D13711yescharacteristic
221#126—2741no—
222#125—2741no—
223#125C2×D13711yescharacteristic, complemented
224#123—2741no—
225C22⋊Q8—1371no—
226C22⋊Q8—1371no—
227#121—2741no—
228#121—1371no—
229#121—1371no—
230#120—2741no—
231#120—2741no—
232C4×Q8—1371no—
233C4×D4—1371no—
234C4×D4—1371no—
235#117—2741no—
236#117—1371no—
237C2×C4⋊C4—1371no—
238#107—1371no—
239#107—1371no—
240#106—1371no—
241C4⋊C8—1371no—
242#103—1371no—
243#102—1371no—
244#102—1371no—
245#102—1371no—
246#102—1371no—
247#102—1371no—
248#102C2×D13711yescharacteristic, complemented
249#102C2×D13711yescharacteristic, complemented
250#102C2×D13711yescharacteristic, complemented
251#102C2×D13711yescharacteristic, complemented
252C22⋊C8—1371no—
253C22⋊C8—1371no—
254#97C2×D13711yescharacteristic, complemented
255C24—2741no—
256C24—2741no—
257C2×Q8—1371no—
258C2×D4—21no—
259C2×D4—21no—
260C2×D4—2742no—
261C2×D4—2742no—
262C2×D4—2741no—
263C2×D4—2741no—
264C2×D4—2741no—
265C2×D4—2741no—
266C2×D4—5481no—
267C2×D4—5481no—
268C2×D4—5484no—
269C2×D4—1371no—
270C2×D4—1371no—
271C2×D4C22×D13711yescharacteristic
272C2×D4C22×D13711yescharacteristic
273C22×C4—2741no—
274C22×C4—2741no—
275C22×C4—1371no—
276C22×C4—1371no—
277SD16—2742no—
278SD16—5481no—
279SD16—5481no—
280M16—5481no—
281C2×C8—5481no—
282C2×C8—1371no—
283C2×C8—1371no—
284C2×C8C22×D13711yescharacteristic
285C2×C8C22×D13711yescharacteristic
286C4⋊C4—21no—
287C4⋊C4—21no—
288C4⋊C4—2741no—
289C4⋊C4—2741no—
290C4⋊C4—2741no—
291C4⋊C4—2741no—
292C4⋊C4—1371no—
293C4⋊C4—1371no—
294C4⋊C4—1371no—
295C4⋊C4C22×D13711yescharacteristic
296C4⋊C4C22×D13711yescharacteristic
297C22⋊C4—2741no—
298C22⋊C4—2741no—
299C22⋊C4—2741no—
300C22⋊C4—2741no—
301C22⋊C4—2741no—
302C22⋊C4—2741no—
303C22⋊C4—1371no—
304C22⋊C4—1371no—
305C42—2741no—
306C42—2741no—
307C42C22×D13711yescharacteristic
308C23—21no—
309C23—21no—
310C23—2741no—
311C23—5482no—
312C23—5482no—
313C23—5482no—
314C23—5482no—
315C23—10961no—
316C23—10961no—
317C23—1371no—
318C23—1371no—
319Q8—2741no—
320Q8—2741no—
321D4—21no—
322D4—21no—
323D4—21no—
324D4—21no—
325D4—42no—
326D4—81no—
327D4—2742no—
328D4—2741no—
329D4—5482no—
330D4—5482no—
331D4—5482no—
332D4—5482no—
333D4—1372no—
334C2×C4—21no—
335C2×C4—21no—
336C2×C4—2741no—
337C2×C4—2741no—
338C2×C4—2741no—
339C2×C4—2741no—
340C2×C4—2741no—
341C2×C4—2741no—
342C2×C4—2741no—
343C2×C4—5482no—
344C2×C4—5482no—
345C2×C4—1371no—
346C2×C4—1371no—
347C2×C4D4×D13711yescharacteristic
348C2×C4D4×D13711yescharacteristic
349C2×C4C23×D13711yescharacteristic
350C8—21no—
351C8—21no—
352C8—2741no—
353C8—2741no—
354C22—41no—
355C22—41no—
356C22—41no—
357C22—41no—
358C22—41no—
359C22—41no—
360C22—2741no—
361C22—2741no—
362C22—2741no—
363C22—2741no—
364C22—2741no—
365C22—2742no—
366C22—5481no—
367C22—5481no—
368C22—10964no—
369C22—10964no—
370C22—1372no—
371C22C2×D4×D13711yescharacteristic
372C4—21no—
373C4—21no—
374C4—41no—
375C4—41no—
376C4—2741no—
377C4—2741no—
378C4—5481no—
379C4—5481no—
380C4—5481no—
381C4#41×D13711yescharacteristic
382C4#1798413411yescharacteristic
383C2—81no—
384C2—81no—
385C2—2742no—
386C2—5481no—
387C2—5482no—
388C2#2257804611yescharacteristic
389C2AGL(1,ℤ/8)×D13711yescharacteristic
390C2#2257821311yescharacteristic
subgroupG/H
centerC22C2×D4×D137
commutator, cosocleC2×C4×C137C24
FrattiniC2×C4C23×D137
Fitting#490×C137C2
socleC22×C137C22×D4
prime radical#22578038C2
hypercenter#490D137
hypocenterC137#3310
LCSG ▷ C2×C4×C137 ▷ C2×C137 ▷ C137
UCSC1 ◁ C22 ◁ C42 ◁ #490

Complements

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

NH ≅ G/Ntype
1. #22578091C2semidirect
3. #22577725C2semidirect
4. #22577715C2semidirect
5. #22577700C2semidirect
6. #22577648C2semidirect
7. #22577631C2semidirect
8. #22577600C2semidirect
9. #22577555C2semidirect
10. #22577555C2semidirect
11. #22577505C2semidirect
12. #22577476C2semidirect
13. #22577406C2semidirect
14. #102×D137C2semidirect
15. #490×C137C2semidirect
20. C2×#14555498C22semidirect
21. C2×#14555498C22semidirect
22. C2×#14555493C22semidirect
23. C2×C8×D137C22semidirect
27. #17984068C22semidirect
28. #17984068C22semidirect
34. C4×Dic274C22semidirect
38. #17984046C22semidirect
39. C4⋊C4×D137C22semidirect
40. #17984022C22semidirect
41. #17984021C22semidirect
42. #17984019C22semidirect
43. C4⋊#11764040C22semidirect
44. #17984013C22semidirect
46. #17984009C22semidirect
47. #17984009C22semidirect
50. #17984000C22semidirect
51. #17984000C22semidirect
53. #125×C137C22semidirect
54. #102×C137C22semidirect
55. #102×C137C22semidirect
56. #102×C137C22semidirect
57. #102×C137C22semidirect
58. #97×C137C22semidirect
195. C137#3310semidirect
199. #490D137semidirect
223. #125C2×D137semidirect
248. #102C2×D137semidirect
249. #102C2×D137semidirect
250. #102C2×D137semidirect
251. #102C2×D137semidirect
254. #97C2×D137semidirect

Maximal subgroups

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

subgroupindex
#225780912
#225780382
#225777252
#225777152
#225777002
#225776482
#225776312
#225776002
#225775552
#225775052
#225774762
#225774062
#102×D1372
#490×C1372
#3310137