Finite Group Databasebeta

← #31792750

Subgroups — #31792750

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#22578057C211yesmaximal, characteristic, complemented
2#22577762C211yesmaximal, characteristic, complemented
3#22577731C211yesmaximal, characteristic, complemented
4#22577719C211yesmaximal, characteristic, complemented
5#22577693C211yesmaximal, characteristic, complemented
6#22577654C211yesmaximal, characteristic, complemented
7#22577634C211yesmaximal, characteristic, complemented
8#22577615C211yesmaximal, characteristic, complemented
9#22577548C211yesmaximal, characteristic, complemented
10#22577537C211yesmaximal, characteristic, complemented
11#22577513C211yesmaximal, characteristic, complemented
12#22577475C211yesmaximal, characteristic, complemented
13#22577404C211yesmaximal, characteristic, complemented
14C2×#17984009C211yesmaximal, characteristic, complemented
15#546×C137C211yesmaximal, characteristic, complemented
16C2×#14555509—21no—
17C2×#14555497C2211yescharacteristic, complemented
18C2×#14555495C2211yescharacteristic
19C2×#14555495C2211yescharacteristic
20C2×#14555494C2211yescharacteristic, complemented
21C22×#11764040C2211yescharacteristic
22#17984088C2211yescharacteristic, complemented
23#17984072—21no—
24#17984072—21no—
25#17984068—21no—
26#17984062C2211yescharacteristic
27#17984062C2211yescharacteristic, complemented
28C4×D548C2211yescharacteristic, complemented
29C2×C4⋊Dic137C2211yescharacteristic
30C2×C4⋊Dic137C2211yescharacteristic, complemented
31#17984019C2211yescharacteristic, complemented
32#17984019C2211yescharacteristic, complemented
33#17984019C2211yescharacteristic, complemented
34C4⋊#11764040C2211yescharacteristic, complemented
35#17984012C2211yescharacteristic, complemented
36#17984012C2211yescharacteristic, complemented
37#17984012C2211yescharacteristic, complemented
38#17984009C2211yescharacteristic
39#17984009C2211yescharacteristic
40#17984009C2211yescharacteristic
41#17984009C2211yescharacteristic
42#17984009C2211yescharacteristic
43#17984009C2211yescharacteristic
44#17984008C2211yescharacteristic, complemented
45C22⋊#11764040C2211yescharacteristic
46C22⋊#11764040C2211yescharacteristic, complemented
47#17983999C2211yescharacteristic, complemented
48C2×#41×C137C2211yescharacteristic
49C22×D4×C137C2211yescharacteristic
50#124×C137C2211yescharacteristic
51#123×C137—21no—
52C22⋊Q8×C137C2211yescharacteristic
53#121×C137—21no—
54#121×C137C2211yescharacteristic
55#120×C137—21no—
56C4×D4×C137C2211yescharacteristic
57C4×D4×C137C2211yescharacteristic
58C2×C22⋊C4×C137—21no—
59#14555509—44no—
60C2×Dic274—21no—
61C2×D548C2311yescharacteristic
62C2×C22⋊D137—21no—
63C2×C22⋊D137—21no—
64C22×Dic137—21no—
65C22×Dic137—21no—
66C2×C4×D137—21no—
67#14555497—41no—
68#14555495—22no—
69#14555495—41no—
70#14555494—41no—
71C2×#11764040—41no—
72C2×#11764040C2311yescharacteristic
73C2×#11764040C2311yescharacteristic
74C2×#11764040C2311yescharacteristic
75C2×#11764040C2311yescharacteristic
76C4⋊Dic137—21no—
77C4⋊Dic137—21no—
78C4⋊Dic137—21no—
79C4⋊Dic137C2311yescharacteristic
80C4⋊Dic137C2311yescharacteristic
81C4⋊Dic137C2311yescharacteristic
82#14555487—21no—
83#14555487—21no—
84#14555487—21no—
85C24×C137—21no—
86#41×C137—22no—
87#41×C137—41no—
88C2×Q8×C137C2311yescharacteristic
89C2×D4×C137—22no—
90C2×D4×C137—21no—
91C2×D4×C137—21no—
92C2×D4×C137—21no—
93C2×D4×C137—41no—
94C2×D4×C137D411yescharacteristic
95C2×D4×C137C2311yescharacteristic
96C2×D4×C137C2311yescharacteristic
97C22×C4×C137—21no—
98C22×C4×C137—21no—
99C22×C4×C137C2311yescharacteristic
100C22×C4×C137C2311yescharacteristic
101C4⋊C4×C137—21no—
102C4⋊C4×C137D411yescharacteristic
103C4⋊C4×C137C2311yescharacteristic
104C22⋊C4×C137—21no—
105C22⋊C4×C137—21no—
106C22⋊C4×C137—21no—
107C22⋊C4×C137—21no—
108C22⋊C4×C137—21no—
109C22⋊C4×C137D411yescharacteristic
110C22⋊C4×C137D411yescharacteristic
111C42×C137C2311yescharacteristic
112C22×D137—21no—
113Dic274—42no—
114D548—21no—
115D548—21no—
116C22⋊D137—42no—
117C22⋊D137—42no—
118C2×Dic137—21no—
119C2×Dic137—21no—
120C2×Dic137—21no—
121C2×Dic137—42no—
122C2×Dic137—42no—
123C4×D137—42no—
124#11764040—21no—
125#11764040—21no—
126#11764040—21no—
127#11764040—21no—
128C23×C137—21no—
129C23×C137—21no—
130C23×C137—42no—
131C23×C137—42no—
132C23×C137—81no—
133C23×C137C2×D411yescharacteristic
134C23×C137C2×D411yescharacteristic
135Q8×C137—21no—
136Q8×C137—21no—
137D4×C137—22no—
138D4×C137—21no—
139D4×C137—21no—
140D4×C137—21no—
141D4×C137—42no—
142D4×C137—42no—
143D4×C137—81no—
144D4×C137#4112yescomplemented
145C2×C4×C137—21no—
146C2×C4×C137—21no—
147C2×C4×C137—21no—
148C2×C4×C137—21no—
149C2×C4×C137—21no—
150C2×C4×C137—21no—
151C2×C4×C137—42no—
152C2×C4×C137—81no—
153C2×C4×C137C2×D411yescharacteristic
154C2×C4×C137C2×D411yescharacteristic
155C2×C4×C137C2×D411yescharacteristic
156C2×C4×C137C2×D411yescharacteristic
157C2×C4×C137C2411yescharacteristic
158C22×C137—21no—
159C22×C137—21no—
160C22×C137—21no—
161C22×C137—22no—
162C22×C137—21no—
163C22×C137—21no—
164C22×C137—41no—
165C22×C137—41no—
166C22×C137—41no—
167C22×C137—41no—
168C22×C137—41no—
169C22×C137—84no—
170C22×C137AGL(1,ℤ/8)12yes—
171C22×C137C22×D411yescharacteristic
172C2×D137—41no—
173C2×D137—41no—
174C2×D137—41no—
175C4×C137—21no—
176C4×C137—21no—
177C4×C137—21no—
178C4×C137—21no—
179C4×C137—41no—
180C4×C137—41no—
181C4×C137C2×#4111yescharacteristic
182C4×C137H(5,𝔽2)11yescharacteristic
183Dic137—41no—
184Dic137—41no—
185Dic137—41no—
186C2×C137—22no—
187C2×C137—41no—
188C2×C137—42no—
189C2×C137—81no—
190C2×C137#54611yescharacteristic
191C2×C137C2×AGL(1,ℤ/8)11yescharacteristic
192C2×C137#57711yescharacteristic
193D137—81no—
194C137#330611yescharacteristic, complemented, Sylow
195#3306—1371nomaximal, Sylow
196#547—1371no—
197#546D13711yescharacteristic, complemented
198#489—1371no—
199#483—1371no—
200#481—1371no—
201#476—1371no—
202#469—1371no—
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×#41C2×D13711yescharacteristic
213C22×D4C2×D13711yescharacteristic
214C2×SD16—1371no—
215C2×D8—1371no—
216C2×D8—1371no—
217C2×M16—1371no—
218C22×C8—1371no—
219C4⋊Q8—1371no—
220#124C2×D13711yescharacteristic
221#123—21no—
222#123—2741no—
223#123—2741no—
224C22⋊Q8—2741no—
225C22⋊Q8C2×D13711yescharacteristic
226#121—21no—
227#121—1371no—
228#121—1371no—
229#121C2×D13711yescharacteristic
230#120—21no—
231C4×D4—1371no—
232C4×D4C2×D13711yescharacteristic
233C4×D4C2×D13711yescharacteristic
234C2×C4⋊C4—1371no—
235C2×C4⋊C4—1371no—
236C2×C22⋊C4—21no—
237#106—1371no—
238#106—1371no—
239#106—1371no—
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#102—1371no—
250#102—1371no—
251C22⋊C8—1371no—
252C22⋊C8—1371no—
253#97—1371no—
254C24—21no—
255#41—22no—
256#41—41no—
257#41—5484no—
258C2×Q8—2741no—
259C2×Q8C22×D13711yescharacteristic
260C2×D4—22no—
261C2×D4—21no—
262C2×D4—21no—
263C2×D4—21no—
264C2×D4—41no—
265C2×D4—2741no—
266C2×D4—2741no—
267C2×D4—1371no—
268C2×D4C22⋊D13711yescharacteristic
269C2×D4C22×D13711yescharacteristic
270C2×D4C22×D13711yescharacteristic
271C22×C4—21no—
272C22×C4—21no—
273C22×C4—2741no—
274C22×C4—2741no—
275C22×C4—2741no—
276C22×C4C22×D13711yescharacteristic
277C22×C4C22×D13711yescharacteristic
278SD16—5481no—
279D8—2742no—
280D8—5481no—
281M16—5481no—
282C2×C8—5481no—
283C2×C8—1371no—
284C2×C8—1371no—
285C2×C8—1371no—
286C2×C8—1371no—
287C4⋊C4—21no—
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—21no—
299C22⋊C4—21no—
300C22⋊C4—21no—
301C22⋊C4—2741no—
302C22⋊C4—2741no—
303C22⋊C4—2741no—
304C22⋊C4C22⋊D13711yescharacteristic
305C22⋊C4C22⋊D13711yescharacteristic
306C42C22×D13711yescharacteristic
307C23—21no—
308C23—21no—
309C23—42no—
310C23—42no—
311C23—81no—
312C23—2741no—
313C23C2×C22⋊D13711yescharacteristic
314C23C2×C22⋊D13711yescharacteristic
315Q8—21no—
316Q8—21no—
317Q8—5482no—
318D4—22no—
319D4—21no—
320D4—21no—
321D4—21no—
322D4—42no—
323D4—42no—
324D4—81no—
325D4—2741no—
326D4—2741no—
327D4—5482no—
328D4—5482no—
329D4#1455550912yescomplemented
330C2×C4—21no—
331C2×C4—21no—
332C2×C4—21no—
333C2×C4—21no—
334C2×C4—21no—
335C2×C4—21no—
336C2×C4—42no—
337C2×C4—81no—
338C2×C4—2741no—
339C2×C4—2741no—
340C2×C4—2741no—
341C2×C4—5482no—
342C2×C4—5482no—
343C2×C4—5482no—
344C2×C4C2×C22⋊D13711yescharacteristic
345C2×C4C2×C22⋊D13711yescharacteristic
346C2×C4C2×C22⋊D13711yescharacteristic
347C2×C4C2×C22⋊D13711yescharacteristic
348C2×C4C23×D13711yescharacteristic
349C8—2741no—
350C8—2741no—
351C8—2741no—
352C8—2741no—
353C22—21no—
354C22—21no—
355C22—21no—
356C22—21no—
357C22—21no—
358C22—22no—
359C22—41no—
360C22—41no—
361C22—41no—
362C22—41no—
363C22—41no—
364C22—84no—
365C22—5481no—
366C22—5481no—
367C22—5481no—
368C22#1798411312yes—
369C22C22×C22⋊D13711yescharacteristic
370C4—21no—
371C4—21no—
372C4—21no—
373C4—21no—
374C4—41no—
375C4—41no—
376C4—5481no—
377C4—5481no—
378C4—5481no—
379C4C2×#1455550911yescharacteristic
380C4#1798413311yescharacteristic
381C2—22no—
382C2—41no—
383C2—42no—
384C2—81no—
385C2—10961no—
386C2#2257804711yescharacteristic
387C2C2×#1798411311yescharacteristic
388C2#2257821411yescharacteristic
subgroupG/H
centerC22C22×C22⋊D137
commutator, cosocleC2×C4×C137C24
FrattiniC2×C4C23×D137
Fitting#546×C137C2
socleC22×C137C22×D4
hypercenter#546D137
hypocenterC137#3306
LCSG ▷ C2×C4×C137 ▷ C2×C137 ▷ C137
UCSC1 ◁ C22 ◁ C4×D4 ◁ #546

Complements

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

NH ≅ G/Ntype
1. #22578057C2semidirect
2. #22577762C2semidirect
3. #22577731C2semidirect
4. #22577719C2semidirect
5. #22577693C2semidirect
6. #22577654C2semidirect
7. #22577634C2semidirect
8. #22577615C2semidirect
9. #22577548C2semidirect
10. #22577537C2semidirect
11. #22577513C2semidirect
12. #22577475C2semidirect
13. #22577404C2semidirect
14. C2×#17984009C2semidirect
15. #546×C137C2semidirect
17. C2×#14555497C22semidirect
20. C2×#14555494C22semidirect
22. #17984088C22semidirect
27. #17984062C22semidirect
28. C4×D548C22semidirect
30. C2×C4⋊Dic137C22semidirect
31. #17984019C22semidirect
32. #17984019C22semidirect
33. #17984019C22semidirect
34. C4⋊#11764040C22semidirect
35. #17984012C22semidirect
36. #17984012C22semidirect
37. #17984012C22semidirect
44. #17984008C22semidirect
46. C22⋊#11764040C22semidirect
47. #17983999C22semidirect
144. D4×C137#41semidirect
194. C137#3306semidirect
197. #546D137semidirect
329. D4#14555509semidirect

Maximal subgroups

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

subgroupindex
#225780572
#225777622
#225777312
#225777192
#225776932
#225776542
#225776342
#225776152
#225775482
#225775372
#225775132
#225774752
#225774042
C2×#179840092
#546×C1372
#3306137