Finite Group Databasebeta

← #31792754

Subgroups — #31792754

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#22578047C211yesmaximal, characteristic, complemented
3#22577761C211yesmaximal, characteristic, complemented
4#22577734C211yesmaximal, characteristic, complemented
5#22577720C211yesmaximal, characteristic, complemented
6#22577691C211yesmaximal, characteristic, complemented
7#22577653C211yesmaximal, characteristic, complemented
8#22577630C211yesmaximal, characteristic, complemented
9#22577616C211yesmaximal, characteristic, complemented
10#22577545C211yesmaximal, characteristic, complemented
11#22577515C211yesmaximal, characteristic, complemented
12#22577471C211yesmaximal, characteristic, complemented
13#22577401C211yesmaximal, characteristic, complemented
14C2×#17984009C211yesmaximal, characteristic, complemented
15#447×C137C211yesmaximal, characteristic, complemented
16C2×#14555509—21no—
17C2×#14555509C2211yescharacteristic
18C2×#14555498C2211yescharacteristic, complemented
19C2×#14555493C2211yescharacteristic, complemented
20C22×#11764040C2211yescharacteristic
21#17984087C2211yescharacteristic, complemented
22#17984075C2211yescharacteristic
23#17984070—21no—
24#17984069—21no—
25#17984069—21no—
26#17984067C2211yescharacteristic
27#17984065—21no—
28#17984060—21no—
29#17984059C2211yescharacteristic
30#17984059C2211yescharacteristic, complemented
31#17984057—21no—
32D4×Dic137C2211yescharacteristic
33D4×Dic137C2211yescharacteristic, complemented
34C4×C22⋊D137C2211yescharacteristic
35C2×C4⋊Dic137C2211yescharacteristic
36C4⋊C4×D137C2211yescharacteristic, complemented
37C2×C22⋊Dic137—21no—
38#17984020C2211yescharacteristic, complemented
39#17984020C2211yescharacteristic, complemented
40#17984019C2211yescharacteristic, complemented
41#17984017C2211yescharacteristic, complemented
42#17984013C2211yescharacteristic, complemented
43#17984013C2211yescharacteristic, complemented
44#17984011C2211yescharacteristic, complemented
45#17984009C2211yescharacteristic
46#17984009C2211yescharacteristic
47#17984009C2211yescharacteristic
48#17984009C2211yescharacteristic
49#17984009C2211yescharacteristic, complemented
50#17984000C2211yescharacteristic, complemented
51#17983998C2211yescharacteristic, complemented
52C22×D4×C137C2211yescharacteristic
53C2×D8×C137C2211yescharacteristic
54C2×D8×C137C2211yescharacteristic
55#121×C137C2211yescharacteristic
56#102×C137C2211yescharacteristic
57#102×C137C2211yescharacteristic
58C22⋊C8×C137C2211yescharacteristic
59#14555509—22no—
60#14555509—41no—
61#14555509—44no—
62C2×Dic274—21no—
63C2×Dic274C2311yescharacteristic
64C2×C22⋊D137—21no—
65C2×C22⋊D137—21no—
66C2×C22⋊D137—21no—
67C2×C22⋊D137D411yescharacteristic
68C22×Dic137—21no—
69C22×Dic137—21no—
70C22×Dic137—21no—
71C22×Dic137—21no—
72C2×C4×D137—21no—
73C2×C4×D137C2311yescharacteristic
74#14555498—41no—
75#14555493—41no—
76C2×#11764040—41no—
77C2×#11764040C2311yescharacteristic
78C2×#11764040C2311yescharacteristic
79C4⋊Dic137—21no—
80C4⋊Dic137C2311yescharacteristic
81C4⋊Dic137C2311yescharacteristic
82C4⋊Dic137C2311yescharacteristic
83#14555489—21no—
84#14555489—21no—
85#14555489—21no—
86#14555489D411yescharacteristic
87C22⋊Dic137—21no—
88C22⋊Dic137—21no—
89C22⋊Dic137—21no—
90C22⋊Dic137—21no—
91C22⋊Dic137—21no—
92C22⋊Dic137—21no—
93C22⋊Dic137—21no—
94C22⋊Dic137D411yescharacteristic
95#14555487D411yescharacteristic
96C4×Dic137C2311yescharacteristic
97C24×C137—21no—
98C2×D4×C137—22no—
99C2×D4×C137—21no—
100C2×D4×C137—21no—
101C2×D4×C137—41no—
102C2×D4×C137C2311yescharacteristic
103C2×D4×C137C2311yescharacteristic
104C2×D4×C137C2311yescharacteristic
105C22×C4×C137C2311yescharacteristic
106D8×C137—22no—
107D8×C137—41no—
108C2×C8×C137C2311yescharacteristic
109C2×C8×C137C2311yescharacteristic
110C4⋊C4×C137C2311yescharacteristic
111C22⋊C4×C137—21no—
112C22×D137C2×D411yescharacteristic
113Dic274—21no—
114Dic274—21no—
115Dic274—42no—
116C22⋊D137—22no—
117C22⋊D137—42no—
118C22⋊D137—42no—
119C22⋊D137—81no—
120C2×Dic137—21no—
121C2×Dic137—21no—
122C2×Dic137—21no—
123C2×Dic137—21no—
124C2×Dic137—42no—
125C2×Dic137—42no—
126C2×Dic137—42no—
127C2×Dic137—81no—
128C2×Dic137C2×D411yescharacteristic
129C2×Dic137C2×D411yescharacteristic
130C2×Dic137C2×D411yescharacteristic
131C4×D137—21no—
132C4×D137—21no—
133C4×D137—42no—
134#11764040—21no—
135#11764040—21no—
136C23×C137—21no—
137C23×C137—21no—
138C23×C137—21no—
139C23×C137—42no—
140C23×C137—42no—
141C23×C137—81no—
142C23×C137C2×D411yescharacteristic
143D4×C137—21no—
144D4×C137—21no—
145D4×C137—21no—
146D4×C137—21no—
147D4×C137—21no—
148D4×C137—42no—
149D4×C137—42no—
150D4×C137#4112yescomplemented
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—22no—
161C22×C137—41no—
162C22×C137—41no—
163C22×C137—41no—
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—21no—
173C2×D137—21no—
174C2×D137—21no—
175C4×C137—21no—
176C4×C137—41no—
177C4×C137C2×#4111yescharacteristic
178C4×C137H(5,𝔽2)11yescharacteristic
179Dic137—21no—
180Dic137—21no—
181Dic137—21no—
182Dic137—41no—
183Dic137—41no—
184Dic137—41no—
185Dic137—41no—
186C2×C137—22no—
187C2×C137—42no—
188C2×C137—81no—
189C2×C137—81no—
190C2×C137#54611yescharacteristic
191C2×C137C2×AGL(1,ℤ/8)11yescharacteristic
192C2×C137#57711yescharacteristic
193D137—41no—
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#469—1371no—
203#465—1371no—
204#463—1371no—
205#449—1371no—
206#447D13711yescharacteristic, complemented
207#442—1371no—
208#435—1371no—
209#417—1371no—
210C2×#102—1371no—
211C2×#41—2741no—
212C2×#41—1371no—
213C22×D4C2×D13711yescharacteristic
214C2×SD16—1371no—
215C2×D8C2×D13711yescharacteristic
216C2×D8C2×D13711yescharacteristic
217C2×M16—1371no—
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#121—1371no—
229#121C2×D13711yescharacteristic
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#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#102C2×D13711yescharacteristic
250#102C2×D13711yescharacteristic
251C22⋊C8—1371no—
252C22⋊C8C2×D13711yescharacteristic
253#97—1371no—
254C24—21no—
255#41—2742no—
256#41—5481no—
257#41—5484no—
258C2×Q8—2741no—
259C2×Q8—1371no—
260C2×D4—22no—
261C2×D4—21no—
262C2×D4—21no—
263C2×D4—41no—
264C2×D4—2741no—
265C2×D4—2741no—
266C2×D4—2741no—
267C2×D4—1371no—
268C2×D4C22×D13711yescharacteristic
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—22no—
280D8—41no—
281M16—5481no—
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⋊C4—1371no—
295C4⋊C4C22×D13711yescharacteristic
296C22⋊C4—21no—
297C22⋊C4—2741no—
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—21no—
310C23—42no—
311C23—42no—
312C23—81no—
313C23—1371no—
314C23D4×D13711yescharacteristic
315Q8—2741no—
316Q8—2741no—
317Q8—5482no—
318D4—21no—
319D4—21no—
320D4—21no—
321D4—21no—
322D4—21no—
323D4—42no—
324D4—42no—
325D4—2742no—
326D4—5482no—
327D4—5482no—
328D4—10961no—
329D4#1455550912yescomplemented
330C2×C4—21no—
331C2×C4—21no—
332C2×C4—21no—
333C2×C4—2741no—
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—22no—
356C22—41no—
357C22—41no—
358C22—41no—
359C22—41no—
360C22—41no—
361C22—41no—
362C22—41no—
363C22—41no—
364C22—84no—
365C22—2741no—
366C22—2741no—
367C22—2741no—
368C22#1798411212yes—
369C22C2×D4×D13711yescharacteristic
370C4—21no—
371C4—41no—
372C4—2741no—
373C4—2741no—
374C4—2741no—
375C4—5481no—
376C4—5481no—
377C4—5481no—
378C4—5481no—
379C4C2×#1455550911yescharacteristic
380C4#1798413311yescharacteristic
381C2—22no—
382C2—42no—
383C2—81no—
384C2—81no—
385C2—5481no—
386C2#2257804011yescharacteristic
387C2C2×#1798411211yescharacteristic
388C2#2257821311yescharacteristic
subgroupG/H
centerC22C2×D4×D137
commutator, cosocleC2×C4×C137C24
FrattiniC2×C4C23×D137
Fitting#447×C137C2
socleC22×C137C22×D4
hypercenter#447D137
hypocenterC137#3306
LCSG ▷ C2×C4×C137 ▷ C2×C137 ▷ C137
UCSC1 ◁ C22 ◁ C22×C4 ◁ #447

Complements

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

NH ≅ G/Ntype
1. #22578053C2semidirect
2. #22578047C2semidirect
3. #22577761C2semidirect
4. #22577734C2semidirect
5. #22577720C2semidirect
6. #22577691C2semidirect
7. #22577653C2semidirect
8. #22577630C2semidirect
9. #22577616C2semidirect
10. #22577545C2semidirect
11. #22577515C2semidirect
12. #22577471C2semidirect
13. #22577401C2semidirect
14. C2×#17984009C2semidirect
15. #447×C137C2semidirect
18. C2×#14555498C22semidirect
19. C2×#14555493C22semidirect
21. #17984087C22semidirect
30. #17984059C22semidirect
33. D4×Dic137C22semidirect
36. C4⋊C4×D137C22semidirect
38. #17984020C22semidirect
39. #17984020C22semidirect
40. #17984019C22semidirect
41. #17984017C22semidirect
42. #17984013C22semidirect
43. #17984013C22semidirect
44. #17984011C22semidirect
49. #17984009C22semidirect
50. #17984000C22semidirect
51. #17983998C22semidirect
150. D4×C137#41semidirect
194. C137#3306semidirect
206. #447D137semidirect
329. D4#14555509semidirect

Maximal subgroups

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

subgroupindex
#225780532
#225780472
#225777612
#225777342
#225777202
#225776912
#225776532
#225776302
#225776162
#225775452
#225775152
#225774712
#225774012
C2×#179840092
#447×C1372
#3306137