Finite Group Databasebeta

← #31792745

Subgroups — #31792745

Order 17536.

442 conjugacy classes of subgroups.

The subgroup lattice has 390 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#22578053C211yesmaximal, characteristic, complemented
2#22578050C211yesmaximal, characteristic, complemented
3#22577761C211yesmaximal, characteristic, complemented
4#22577735C211yesmaximal, characteristic, complemented
5#22577719C211yesmaximal, characteristic, complemented
6#22577692C211yesmaximal, characteristic, complemented
7#22577631C211yesmaximal, characteristic, complemented
8#22577616C211yesmaximal, characteristic, complemented
9#22577546C211yesmaximal, characteristic, complemented
10#22577535C211yesmaximal, characteristic, complemented
11#22577512C211yesmaximal, characteristic, complemented
12#22577472C211yesmaximal, characteristic, complemented
13#22577405C211yesmaximal, characteristic, complemented
14#102×D137C211yesmaximal, characteristic, complemented
15#469×C137C211yesmaximal, characteristic, complemented
16C2×#14555510C2211yescharacteristic
17C2×#14555509—21no—
18C2×D4×D137C2211yescharacteristic
19C2×#14555498C2211yescharacteristic, complemented
20C2×#14555495C2211yescharacteristic
21C2×C8×D137C2211yescharacteristic
22#17984087C2211yescharacteristic, complemented
23#17984075C2211yescharacteristic
24#17984070—21no—
25#17984070—21no—
26#17984069—21no—
27#17984068C2211yescharacteristic
28#17984065—21no—
29#17984059—21no—
30#17984059C2211yescharacteristic
31#17984056—21no—
32D4×Dic137C2211yescharacteristic, complemented
33#17984050C2211yescharacteristic
34C4×C22⋊D137C2211yescharacteristic
35C2×C4⋊Dic137C2211yescharacteristic, complemented
36C4⋊C4×D137C2211yescharacteristic
37C22⋊C4×D137—21no—
38#17984019C2211yescharacteristic, complemented
39#17984019C2211yescharacteristic, complemented
40#17984017C2211yescharacteristic, complemented
41#17984013C2211yescharacteristic, complemented
42#17984013C2211yescharacteristic, complemented
43#17984011C2211yescharacteristic, complemented
44#17984010C2211yescharacteristic
45#17984010C2211yescharacteristic
46#17984009C2211yescharacteristic
47#17984009C2211yescharacteristic
48#17984008C2211yescharacteristic
49C22⋊#11764040C2211yescharacteristic, complemented
50#17984000C2211yescharacteristic
51#17983998C2211yescharacteristic, complemented
52C2×D8×C137C2211yescharacteristic
53C2×M16×C137C2211yescharacteristic, complemented
54#121×C137C2211yescharacteristic
55#121×C137C2211yescharacteristic, complemented
56#106×C137C2211yescharacteristic, complemented
57#102×C137C2211yescharacteristic
58#102×C137C2211yescharacteristic, complemented
59C23×D137—21no—
60#14555510—22no—
61#14555510—41no—
62#14555509—44no—
63C2×Dic274—21no—
64C2×Dic274C2311yescharacteristic
65C2×D548C2311yescharacteristic
66C2×C22⋊D137—21no—
67C2×C22⋊D137—21no—
68C2×C22⋊D137—21no—
69C2×C22⋊D137D411yescharacteristic
70D4×D137—22no—
71D4×D137—41no—
72C22×Dic137—21no—
73C22×Dic137—21no—
74C2×C4×D137—21no—
75C2×C4×D137—21no—
76C2×C4×D137—21no—
77C2×C4×D137C2311yescharacteristic
78#14555498—41no—
79#14555495—22no—
80C2×#11764040C2311yescharacteristic
81C2×#11764040C2311yescharacteristic
82C8×D137—41no—
83C4⋊Dic137—21no—
84C4⋊Dic137—21no—
85C4⋊Dic137C2311yescharacteristic
86C4⋊Dic137C2311yescharacteristic
87#14555489—21no—
88#14555489—21no—
89#14555489D411yescharacteristic
90C22⋊Dic137—21no—
91C22⋊Dic137—21no—
92C22⋊Dic137—21no—
93C22⋊Dic137—21no—
94C22⋊Dic137D411yescharacteristic
95#14555487—21no—
96#14555487—21no—
97#14555487D411yescharacteristic
98C4×Dic137C2311yescharacteristic
99C2×D4×C137—21no—
100C2×D4×C137—21no—
101C2×D4×C137C2311yescharacteristic
102C2×D4×C137C2311yescharacteristic
103C22×C4×C137C2311yescharacteristic
104D8×C137—41no—
105M16×C137—41no—
106C2×C8×C137C2311yescharacteristic
107C2×C8×C137C2311yescharacteristic
108C4⋊C4×C137C2311yescharacteristic
109C4⋊C4×C137C2311yescharacteristic
110C22⋊C4×C137—21no—
111C22⋊C4×C137—21no—
112C22×D137—21no—
113C22×D137—42no—
114C22×D137—42no—
115C22×D137—81no—
116C22×D137C2×D411yescharacteristic
117Dic274—21no—
118Dic274—21no—
119Dic274—42no—
120D548—21no—
121D548#4112yescomplemented
122C22⋊D137—22no—
123C22⋊D137—42no—
124C22⋊D137—42no—
125C22⋊D137—42no—
126C2×Dic137—21no—
127C2×Dic137—21no—
128C2×Dic137—21no—
129C2×Dic137—42no—
130C2×Dic137—42no—
131C2×Dic137C2×D411yescharacteristic
132C2×Dic137C2×D411yescharacteristic
133C2×Dic137C2×D411yescharacteristic
134C4×D137—21no—
135C4×D137—21no—
136C4×D137—42no—
137C4×D137—42no—
138C4×D137—81no—
139#11764040—21no—
140#11764040—21no—
141C23×C137—21no—
142C23×C137—21no—
143C23×C137C2×D411yescharacteristic
144D4×C137—21no—
145D4×C137—21no—
146D4×C137—21no—
147D4×C137—21no—
148D4×C137—42no—
149D4×C137—81no—
150C2×C4×C137—21no—
151C2×C4×C137—21no—
152C2×C4×C137—21no—
153C2×C4×C137—21no—
154C2×C4×C137C2×D411yescharacteristic
155C2×C4×C137C2411yescharacteristic
156C8×C137—21no—
157C8×C137—21no—
158C22×C137—21no—
159C22×C137—21no—
160C22×C137—21no—
161C22×C137—41no—
162C22×C137—41no—
163C22×C137—41no—
164C22×C137—41no—
165C22×C137—41no—
166C22×C137—41no—
167C22×C137C22×D411yescharacteristic
168C2×D137—21no—
169C2×D137—21no—
170C2×D137—22no—
171C2×D137—41no—
172C2×D137—41no—
173C2×D137—84no—
174C2×D137AGL(1,ℤ/8)12yes—
175C4×C137—21no—
176C4×C137—41no—
177C4×C137—41no—
178C4×C137C2×#4111yescharacteristic
179C4×C137H(5,𝔽2)11yescharacteristic
180Dic137—21no—
181Dic137—21no—
182Dic137—21no—
183Dic137—41no—
184Dic137—41no—
185Dic137—41no—
186C2×C137—41no—
187C2×C137—81no—
188C2×C137—81no—
189C2×C137#54611yescharacteristic
190C2×C137C2×AGL(1,ℤ/8)11yescharacteristic
191C2×C137#57711yescharacteristic
192D137—22no—
193D137—42no—
194C137#330611yescharacteristic, complemented, Sylow
195#3306—1371nomaximal, Sylow
196#547—1371no—
197#546—1371no—
198#489—1371no—
199#483—1371no—
200#481—1371no—
201#476—1371no—
202#469D13711yescharacteristic, complemented
203#465—1371no—
204#463—1371no—
205#449—1371no—
206#447—1371no—
207#442—1371no—
208#435—1371no—
209#417—1371no—
210C2×#102—1371no—
211C2×#41—2741no—
212C2×#41—1371no—
213C22×D4—1371no—
214C2×SD16—1371no—
215C2×D8—1371no—
216C2×D8C2×D13711yescharacteristic
217C2×M16C2×D13711yescharacteristic, complemented
218C22×C8—1371no—
219C4⋊Q8—1371no—
220#124—1371no—
221#123—2741no—
222#123—2741no—
223#123—2741no—
224C22⋊Q8—2741no—
225C22⋊Q8—1371no—
226#121—2741no—
227#121—1371no—
228#121C2×D13711yescharacteristic
229#121C2×D13711yescharacteristic, complemented
230#120—2741no—
231C4×D4—1371no—
232C4×D4—1371no—
233C4×D4—1371no—
234C2×C4⋊C4—1371no—
235C2×C4⋊C4—1371no—
236C2×C22⋊C4—2741no—
237#106—1371no—
238#106—1371no—
239#106C2×D13711yescharacteristic, complemented
240C4⋊C8—1371no—
241#103—1371no—
242#103—1371no—
243#103—1371no—
244#102—1371no—
245#102—1371no—
246#102—1371no—
247#102—1371no—
248#102—1371no—
249#102C2×D13711yescharacteristic
250#102C2×D13711yescharacteristic, complemented
251C22⋊C8—1371no—
252C22⋊C8—1371no—
253#97—1371no—
254C24—2741no—
255#41—2742no—
256#41—5481no—
257#41—5484no—
258C2×Q8—2741no—
259C2×Q8—1371no—
260C2×D4—21no—
261C2×D4—21no—
262C2×D4—2742no—
263C2×D4—2741no—
264C2×D4—2741no—
265C2×D4—2741no—
266C2×D4—5481no—
267C2×D4—1371no—
268C2×D4—1371no—
269C2×D4C22×D13711yescharacteristic
270C2×D4C22×D13711yescharacteristic
271C22×C4—2741no—
272C22×C4—2741no—
273C22×C4—2741no—
274C22×C4—2741no—
275C22×C4—2741no—
276C22×C4—1371no—
277C22×C4C22×D13711yescharacteristic
278SD16—5481no—
279D8—41no—
280D8—2742no—
281M16—41no—
282C2×C8—5481no—
283C2×C8—1371no—
284C2×C8—1371no—
285C2×C8C22×D13711yescharacteristic
286C2×C8C22×D13711yescharacteristic
287C4⋊C4—2741no—
288C4⋊C4—2741no—
289C4⋊C4—2741no—
290C4⋊C4—2741no—
291C4⋊C4—1371no—
292C4⋊C4—1371no—
293C4⋊C4—1371no—
294C4⋊C4C22×D13711yescharacteristic
295C4⋊C4C22×D13711yescharacteristic
296C22⋊C4—21no—
297C22⋊C4—21no—
298C22⋊C4—2741no—
299C22⋊C4—2741no—
300C22⋊C4—2741no—
301C22⋊C4—2741no—
302C22⋊C4—2741no—
303C22⋊C4—2741no—
304C22⋊C4—1371no—
305C22⋊C4—1371no—
306C42—1371no—
307C23—21no—
308C23—21no—
309C23—2741no—
310C23—5482no—
311C23—5482no—
312C23—10961no—
313C23—1371no—
314C23D4×D13711yescharacteristic
315Q8—2741no—
316Q8—2741no—
317Q8—5482no—
318D4—21no—
319D4—21no—
320D4—21no—
321D4—21no—
322D4—42no—
323D4—81no—
324D4—2742no—
325D4—2741no—
326D4—5482no—
327D4—5482no—
328D4—5482no—
329D4—1372no—
330C2×C4—21no—
331C2×C4—21no—
332C2×C4—21no—
333C2×C4—21no—
334C2×C4—2741no—
335C2×C4—2741no—
336C2×C4—2741no—
337C2×C4—2741no—
338C2×C4—2741no—
339C2×C4—5482no—
340C2×C4—5482no—
341C2×C4—5482no—
342C2×C4—5482no—
343C2×C4—10961no—
344C2×C4—1371no—
345C2×C4—1371no—
346C2×C4—1371no—
347C2×C4D4×D13711yescharacteristic
348C2×C4C23×D13711yescharacteristic
349C8—21no—
350C8—21no—
351C8—2741no—
352C8—2741no—
353C22—21no—
354C22—21no—
355C22—21no—
356C22—41no—
357C22—41no—
358C22—41no—
359C22—41no—
360C22—41no—
361C22—41no—
362C22—2741no—
363C22—2741no—
364C22—2742no—
365C22—5481no—
366C22—5481no—
367C22—10964no—
368C22—1372no—
369C22C2×D4×D13711yescharacteristic
370C4—21no—
371C4—41no—
372C4—41no—
373C4—2741no—
374C4—2741no—
375C4—2741no—
376C4—5481no—
377C4—5481no—
378C4—5481no—
379C4#41×D13711yescharacteristic
380C4#1798413311yescharacteristic
381C2—41no—
382C2—81no—
383C2—81no—
384C2—2742no—
385C2—5482no—
386C2#2257804211yescharacteristic
387C2AGL(1,ℤ/8)×D13711yescharacteristic
388C2#2257821311yescharacteristic
subgroupG/H
centerC22C2×D4×D137
commutator, cosocleC2×C4×C137C24
FrattiniC2×C4C23×D137
Fitting#469×C137C2
socleC22×C137C22×D4
hypercenter#469D137
hypocenterC137#3306
LCSG ▷ C2×C4×C137 ▷ C2×C137 ▷ C137
UCSC1 ◁ C22 ◁ C22×C4 ◁ #469

Complements

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

NH ≅ G/Ntype
1. #22578053C2semidirect
2. #22578050C2semidirect
3. #22577761C2semidirect
4. #22577735C2semidirect
5. #22577719C2semidirect
6. #22577692C2semidirect
7. #22577631C2semidirect
8. #22577616C2semidirect
9. #22577546C2semidirect
10. #22577535C2semidirect
11. #22577512C2semidirect
12. #22577472C2semidirect
13. #22577405C2semidirect
14. #102×D137C2semidirect
15. #469×C137C2semidirect
19. C2×#14555498C22semidirect
22. #17984087C22semidirect
32. D4×Dic137C22semidirect
35. C2×C4⋊Dic137C22semidirect
38. #17984019C22semidirect
39. #17984019C22semidirect
40. #17984017C22semidirect
41. #17984013C22semidirect
42. #17984013C22semidirect
43. #17984011C22semidirect
49. C22⋊#11764040C22semidirect
51. #17983998C22semidirect
53. C2×M16×C137C22semidirect
55. #121×C137C22semidirect
56. #106×C137C22semidirect
58. #102×C137C22semidirect
121. D548#41semidirect
194. C137#3306semidirect
202. #469D137semidirect
217. C2×M16C2×D137semidirect
229. #121C2×D137semidirect
239. #106C2×D137semidirect
250. #102C2×D137semidirect

Maximal subgroups

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

subgroupindex
#225780532
#225780502
#225777612
#225777352
#225777192
#225776922
#225776312
#225776162
#225775462
#225775352
#225775122
#225774722
#225774052
#102×D1372
#469×C1372
#3306137