Finite Group Databasebeta

← #31792807

Subgroups — #31792807

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
2D4×D548C211yesmaximal, characteristic
3#22577769C211yesmaximal, characteristic, complemented
4#22577729C211yesmaximal, characteristic, complemented
5#22577718C211yesmaximal, characteristic, complemented
6#22577650C211yesmaximal, characteristic, complemented
7#22577633C211yesmaximal, characteristic, complemented
8#22577604C211yesmaximal, characteristic, complemented
9#22577555C211yesmaximal, characteristic, complemented
10#22577555C211yesmaximal, characteristic, complemented
11#22577503C211yesmaximal, characteristic, complemented
12#22577473C211yesmaximal, characteristic, complemented
13#22577406C211yesmaximal, characteristic, complemented
14#102×D137C211yesmaximal, characteristic, complemented
15#478×C137C211yesmaximal, characteristic, complemented
16C22×D548—21no—
17C2×D4×D137C2211yescharacteristic
18C2×D4×D137C2211yescharacteristic
19C2×#14555499C2211yescharacteristic, complemented
20C2×#14555499C2211yescharacteristic, complemented
21C2×#14555498C2211yescharacteristic
22C2×#14555493C2211yescharacteristic, complemented
23C2×C8×D137C2211yescharacteristic, complemented
24#17984085C2211yescharacteristic
25#17984083—21no—
26#17984079—21no—
27#17984073—21no—
28#17984068C2211yescharacteristic, complemented
29#17984068C2211yescharacteristic, complemented
30#17984063C2211yescharacteristic
31#17984063C2211yescharacteristic
32#17984062—21no—
33#17984058—21no—
34#17984058—21no—
35C4×Dic274C2211yescharacteristic, complemented
36C4×D548C2211yescharacteristic
37#17984046—21no—
38#17984046C2211yescharacteristic, complemented
39C4⋊C4×D137C2211yescharacteristic, complemented
40#17984021C2211yescharacteristic, complemented
41#17984011C2211yescharacteristic, complemented
42#17984010C2211yescharacteristic, complemented
43#17984010C2211yescharacteristic, complemented
44#17984009C2211yescharacteristic
45#17984008C2211yescharacteristic, complemented
46#17984008C2211yescharacteristic, complemented
47#17984008C2211yescharacteristic, complemented
48#17984008C2211yescharacteristic, complemented
49#17984000C2211yescharacteristic, complemented
50#17984000C2211yescharacteristic, complemented
51#17983999C2211yescharacteristic, 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—
60C23×D137—21no—
61C2×Dic274C2311yescharacteristic
62C2×D548—21no—
63C2×D548—21no—
64C2×D548—21no—
65C2×D548—21no—
66C2×D548—44no—
67C2×D548D411yescharacteristic
68C2×D548C2311yescharacteristic
69C2×D548C2311yescharacteristic
70C2×C22⋊D137—21no—
71C2×C22⋊D137—21no—
72D4×D137—22no—
73D4×D137—22no—
74D4×D137—41no—
75D4×D137—41no—
76C2×C4×D137—21no—
77C2×C4×D137C2311yescharacteristic
78C2×C4×D137C2311yescharacteristic
79#14555499—41no—
80#14555499—41no—
81#14555498—22no—
82#14555493—41no—
83C2×#11764040C2311yescharacteristic
84C2×#11764040C2311yescharacteristic
85C8×D137—41no—
86C4⋊Dic137—21no—
87C4⋊Dic137—21no—
88C4⋊Dic137D411yescharacteristic
89C4⋊Dic137C2311yescharacteristic
90#14555489—21no—
91#14555489—21no—
92#14555487—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×C137C2311yescharacteristic
102C22×C4×C137—21no—
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—
111C42×C137C2311yescharacteristic
112C22×D137—21no—
113C22×D137—21no—
114C22×D137—42no—
115C22×D137—42no—
116C22×D137—42no—
117C22×D137—42no—
118C22×D137—81no—
119C22×D137—81no—
120C22×D137C2×D411yescharacteristic
121C22×D137C2×D411yescharacteristic
122Dic274—21no—
123Dic274—21no—
124D548—22no—
125D548—21no—
126D548—21no—
127D548—21no—
128D548—21no—
129D548—42no—
130D548—42no—
131D548—42no—
132D548—81no—
133C22⋊D137—42no—
134C22⋊D137—42no—
135C2×Dic137—21no—
136C2×Dic137—21no—
137C2×Dic137C2×D411yescharacteristic
138C2×Dic137C2×D411yescharacteristic
139C4×D137—21no—
140C4×D137—21no—
141C4×D137—21no—
142C4×D137—21no—
143C4×D137—42no—
144#11764040—21no—
145#11764040—21no—
146C23×C137—21no—
147D4×C137—21no—
148D4×C137#4112yes—
149C2×C4×C137—21no—
150C2×C4×C137—21no—
151C2×C4×C137—21no—
152C2×C4×C137—42no—
153C2×C4×C137C2×D411yescharacteristic
154C2×C4×C137C2×D411yescharacteristic
155C2×C4×C137C2411yescharacteristic
156C8×C137—21no—
157C8×C137—21no—
158C22×C137—22no—
159C22×C137—41no—
160C22×C137—41no—
161C22×C137C22×D411yescharacteristic
162C2×D137—21no—
163C2×D137—21no—
164C2×D137—21no—
165C2×D137—21no—
166C2×D137—21no—
167C2×D137—41no—
168C2×D137—41no—
169C2×D137—41no—
170C2×D137—41no—
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×C137—41no—
181C4×C137C2×#4111yescharacteristic
182C4×C137H(5,𝔽2)11yescharacteristic
183Dic137—21no—
184Dic137—21no—
185Dic137—41no—
186Dic137—41no—
187C2×C137—42no—
188C2×C137#54611yescharacteristic
189C2×C137C2×AGL(1,ℤ/8)11yescharacteristic
190C2×C137#57711yescharacteristic
191D137—22no—
192D137—41no—
193D137—81no—
194D137—81no—
195C137#331011yescharacteristic, complemented, Sylow
196#3310—1371nomaximal, Sylow
197#552—1371no—
198D42—1371no—
199#490—1371no—
200#482—1371no—
201#480—1371no—
202#478D13711yescharacteristic, complemented
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#127—1371no—
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×D4C2×D13711yescharacteristic
235#117—2741no—
236#117—1371no—
237C2×C4⋊C4—1371no—
238#107—1371no—
239#107C2×D13711yescharacteristic, complemented
240#106C2×D13711yescharacteristic, complemented
241C4⋊C8C2×D13711yescharacteristic, complemented
242#103—1371no—
243#102—1371no—
244#102—1371no—
245#102—1371no—
246#102—1371no—
247#102—1371no—
248#102—1371no—
249#102—1371no—
250#102C2×D13711yescharacteristic
251#102C2×D13711yescharacteristic
252C22⋊C8—1371no—
253C22⋊C8—1371no—
254#97—1371no—
255C24—2741no—
256C24—2741no—
257C2×Q8—1371no—
258C2×D4—2741no—
259C2×D4—2741no—
260C2×D4—2742no—
261C2×D4—2742no—
262C2×D4—2741no—
263C2×D4—2741no—
264C2×D4—2741no—
265C2×D4—2741no—
266C2×D4—5484no—
267C2×D4—5481no—
268C2×D4—5481no—
269C2×D4—1371no—
270C2×D4—1371no—
271C2×D4—1371no—
272C2×D4C22×D13711yescharacteristic
273C22×C4—21no—
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⋊C4C22×D13711yescharacteristic
295C4⋊C4C22×D13711yescharacteristic
296C4⋊C4C22×D13711yescharacteristic
297C22⋊C4—21no—
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—2741no—
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—2742no—
323D4—2741no—
324D4—2741no—
325D4—2741no—
326D4—2741no—
327D4—5482no—
328D4—5482no—
329D4—5482no—
330D4—5482no—
331D4—5482no—
332D4—10961no—
333D4#1455551112yes—
334C2×C4—21no—
335C2×C4—21no—
336C2×C4—21no—
337C2×C4—42no—
338C2×C4—2741no—
339C2×C4—2741no—
340C2×C4—2741no—
341C2×C4—2741no—
342C2×C4—2741no—
343C2×C4—2741no—
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—22no—
355C22—41no—
356C22—41no—
357C22—2741no—
358C22—2741no—
359C22—2741no—
360C22—2741no—
361C22—2741no—
362C22—5481no—
363C22—5481no—
364C22—5481no—
365C22—5481no—
366C22—5481no—
367C22—5481no—
368C22—10964no—
369C22—10964no—
370C22—1372no—
371C22C2×D4×D13711yescharacteristic
372C4—21no—
373C4—21no—
374C4—41no—
375C4—41no—
376C4—41no—
377C4—2741no—
378C4—2741no—
379C4—5481no—
380C4—5481no—
381C4C2×#1455551111yescharacteristic
382C4#1798413411yescharacteristic
383C2—42no—
384C2—2742no—
385C2—5481no—
386C2—10961no—
387C2—10961no—
388C2#2257804411yescharacteristic
389C2AGL(1,ℤ/8)×D13711yescharacteristic
390C2#2257821011yescharacteristic
subgroupG/H
centerC22C2×D4×D137
commutator, cosocleC2×C4×C137C24
FrattiniC2×C4C23×D137
Fitting#478×C137C2
socleC22×C137C22×D4
prime radicalD4×D548C2
hypercenter#478D137
hypocenterC137#3310
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. #22577769C2semidirect
4. #22577729C2semidirect
5. #22577718C2semidirect
6. #22577650C2semidirect
7. #22577633C2semidirect
8. #22577604C2semidirect
9. #22577555C2semidirect
10. #22577555C2semidirect
11. #22577503C2semidirect
12. #22577473C2semidirect
13. #22577406C2semidirect
14. #102×D137C2semidirect
15. #478×C137C2semidirect
19. C2×#14555499C22semidirect
20. C2×#14555499C22semidirect
22. C2×#14555493C22semidirect
23. C2×C8×D137C22semidirect
28. #17984068C22semidirect
29. #17984068C22semidirect
35. C4×Dic274C22semidirect
38. #17984046C22semidirect
39. C4⋊C4×D137C22semidirect
40. #17984021C22semidirect
41. #17984011C22semidirect
42. #17984010C22semidirect
43. #17984010C22semidirect
45. #17984008C22semidirect
46. #17984008C22semidirect
47. #17984008C22semidirect
48. #17984008C22semidirect
49. #17984000C22semidirect
50. #17984000C22semidirect
51. #17983999C22semidirect
52. #125×C137C22semidirect
54. #107×C137C22semidirect
55. #106×C137C22semidirect
56. C4⋊C8×C137C22semidirect
195. C137#3310semidirect
202. #478D137semidirect
223. #125C2×D137semidirect
239. #107C2×D137semidirect
240. #106C2×D137semidirect
241. C4⋊C8C2×D137semidirect

Maximal subgroups

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

subgroupindex
#225780912
D4×D5482
#225777692
#225777292
#225777182
#225776502
#225776332
#225776042
#225775552
#225775032
#225774732
#225774062
#102×D1372
#478×C1372
#3310137