Finite Group Databasebeta

← #31792645

Subgroups — #31792645

Order 17536.

420 conjugacy classes of subgroups.

The subgroup lattice has 384 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#22578105C211yesmaximal, characteristic, complemented
2#22578044C211yesmaximal, characteristic, complemented
3#22577759C211yesmaximal, characteristic, complemented
4#22577736C211yesmaximal, characteristic, complemented
5#22577729C211yesmaximal, characteristic, complemented
6#22577702C211yesmaximal, characteristic, complemented
7#22577638C211yesmaximal, characteristic, complemented
8#22577620C211yesmaximal, characteristic, complemented
9#22577547C211yesmaximal, characteristic, complemented
10#22577536C211yesmaximal, characteristic, complemented
11#22577483C211yesmaximal, characteristic, complemented
12C8×C22⋊D137C211yesmaximal, characteristic, complemented
13#22577398C211yesmaximal, characteristic, complemented
14C2×#17984010C211yesmaximal, characteristic, complemented
15#465×C137C211yesmaximal, characteristic, complemented
16C2×#14555511C2211yescharacteristic
17C22×D548C2211yescharacteristic
18C2×#14555495C2211yescharacteristic
19C2×#14555495C2211yescharacteristic
20C2×C8×D137C2211yescharacteristic, complemented
21#17984081—21no—
22#17984079C2211yescharacteristic, complemented
23#17984077C2211yescharacteristic
24#17984073—21no—
25#17984072—21no—
26#17984063—21no—
27#17984063C2211yescharacteristic
28#17984060—21no—
29#17984059C2211yescharacteristic, complemented
30#17984058—21no—
31D4×Dic137—21no—
32D4×Dic137C2211yescharacteristic, complemented
33#17984050C2211yescharacteristic
34C4×C22⋊D137C2211yescharacteristic
35#17984046C2211yescharacteristic, complemented
36C2×C4⋊Dic137C2211yescharacteristic
37C22⋊C4×D137—21no—
38#17984021C2211yescharacteristic, complemented
39#17984021C2211yescharacteristic, complemented
40#17984017C2211yescharacteristic, complemented
41#17984012C2211yescharacteristic, complemented
42#17984012C2211yescharacteristic, complemented
43#17984010C2211yescharacteristic
44#17984010C2211yescharacteristic
45#17984010C2211yescharacteristic
46#17984010C2211yescharacteristic
47#17984008C2211yescharacteristic
48#17984008C2211yescharacteristic
49C22⋊#11764040C2211yescharacteristic
50#17984000C2211yescharacteristic, complemented
51C8×Dic137C2211yescharacteristic, complemented
52C2×SD16×C137C2211yescharacteristic, complemented
53C22×C8×C137C2211yescharacteristic
54C22⋊Q8×C137C2211yescharacteristic
55#121×C137C2211yescharacteristic
56#106×C137C2211yescharacteristic, complemented
57#103×C137C2211yescharacteristic, complemented
58#102×C137C2211yescharacteristic, complemented
59C23×D137—21no—
60#14555511—22no—
61#14555511—41no—
62C2×D548—22no—
63C2×D548—21no—
64C2×D548—21no—
65C2×D548—21no—
66C2×D548—41no—
67C2×D548C2311yescharacteristic
68C2×D548C2311yescharacteristic
69C2×C22⋊D137—21no—
70C2×C22⋊D137D411yescharacteristic
71C22×Dic137—21no—
72C22×Dic137—21no—
73C2×C4×D137—21no—
74C2×C4×D137—21no—
75C2×C4×D137C2311yescharacteristic
76#14555495—22no—
77#14555495—41no—
78C2×#11764040C2311yescharacteristic
79C2×#11764040C2311yescharacteristic
80C8×D137—41no—
81C4⋊Dic137—21no—
82C4⋊Dic137—21no—
83C4⋊Dic137C2311yescharacteristic
84C4⋊Dic137C2311yescharacteristic
85#14555489—21no—
86#14555489D411yescharacteristic
87C22⋊Dic137—21no—
88C22⋊Dic137—21no—
89C22⋊Dic137D411yescharacteristic
90#14555487—21no—
91#14555487—21no—
92#14555487—21no—
93#14555487—21no—
94#14555487—21no—
95#14555487D411yescharacteristic
96C4×Dic137—21no—
97C4×Dic137C2311yescharacteristic
98C2×Q8×C137C2311yescharacteristic
99C2×D4×C137—21no—
100C2×D4×C137C2311yescharacteristic
101C22×C4×C137C2311yescharacteristic
102SD16×C137—41no—
103C2×C8×C137—41no—
104C2×C8×C137C2311yescharacteristic
105C2×C8×C137C2311yescharacteristic
106C4⋊C4×C137—21no—
107C4⋊C4×C137C2311yescharacteristic
108C4⋊C4×C137C2311yescharacteristic
109C22⋊C4×C137—21no—
110C22⋊C4×C137—21no—
111C22×D137—21no—
112C22×D137—21no—
113C22×D137—42no—
114C22×D137—42no—
115C22×D137—81no—
116C22×D137C2×D411yescharacteristic
117D548—21no—
118D548—21no—
119D548—21no—
120D548—42no—
121D548—42no—
122D548—81no—
123D548#4112yes—
124C22⋊D137—22no—
125C22⋊D137—81no—
126C2×Dic137—21no—
127C2×Dic137—21no—
128C2×Dic137—81no—
129C2×Dic137—81no—
130C2×Dic137C2×D411yescharacteristic
131C2×Dic137C2×D411yescharacteristic
132C2×Dic137C2×D411yescharacteristic
133C4×D137—21no—
134C4×D137—21no—
135C4×D137—42no—
136C4×D137—81no—
137#11764040—21no—
138#11764040—21no—
139C23×C137—21no—
140C23×C137C2×D411yescharacteristic
141Q8×C137—21no—
142Q8×C137—21no—
143D4×C137—21no—
144D4×C137—21no—
145D4×C137—42no—
146C2×C4×C137—21no—
147C2×C4×C137—21no—
148C2×C4×C137—21no—
149C2×C4×C137—21no—
150C2×C4×C137—21no—
151C2×C4×C137C2×D411yescharacteristic
152C2×C4×C137C2411yescharacteristic
153C8×C137—21no—
154C8×C137—21no—
155C22×C137—21no—
156C22×C137—21no—
157C22×C137—41no—
158C22×C137—41no—
159C22×C137—41no—
160C22×C137#13512yes—
161C22×C137C22×D411yescharacteristic
162C2×D137—21no—
163C2×D137—22no—
164C2×D137—21no—
165C2×D137—21no—
166C2×D137—41no—
167C2×D137—41no—
168C2×D137—41no—
169C2×D137—41no—
170C2×D137—41no—
171C2×D137—84no—
172C4×C137—21no—
173C4×C137—41no—
174C4×C137—41no—
175C4×C137—41no—
176C4×C137C2×#4111yescharacteristic
177C4×C137H(5,𝔽2)11yescharacteristic
178Dic137—21no—
179Dic137—21no—
180Dic137—21no—
181Dic137—41no—
182Dic137—41no—
183C2×C137—22no—
184C2×C137—81no—
185C2×C137#54611yescharacteristic
186C2×C137C2×#13511yescharacteristic
187C2×C137#57611yescharacteristic
188D137—41no—
189D137—42no—
190D137—81no—
191C137#329711yescharacteristic, complemented, Sylow
192#3297—1371nomaximal, Sylow
193#553—1371no—
194#546—1371no—
195#488—1371no—
196#484—1371no—
197#482—1371no—
198#478—1371no—
199#466—1371no—
200#465D13711yescharacteristic, complemented
201#463—1371no—
202#449—1371no—
203#447—1371no—
204C4×D8—1371no—
205C8×D4—1371no—
206#416—1371no—
207C2×#102—1371no—
208C2×#41—1371no—
209C22×D4—1371no—
210C2×SD16C2×D13711yescharacteristic, complemented
211C2×D8—1371no—
212C2×D8—1371no—
213C22×C8—1371no—
214C22×C8C2×D13711yescharacteristic
215#126—2741no—
216#125—1371no—
217#124—1371no—
218#123—2741no—
219#123—2741no—
220C22⋊Q8C2×D13711yescharacteristic
221#121—2741no—
222#121—2741no—
223#121—1371no—
224#121—1371no—
225#121C2×D13711yescharacteristic
226#120—2741no—
227C4×D4—2741no—
228C4×D4—1371no—
229C4×D4—1371no—
230C4×D4—1371no—
231#117—1371no—
232C2×C4⋊C4—1371no—
233C2×C22⋊C4—2741no—
234#107—1371no—
235#107—1371no—
236#106C2×D13711yescharacteristic, complemented
237C4⋊C8—1371no—
238#103—1371no—
239#103—1371no—
240#103C2×D13711yescharacteristic, complemented
241#102—1371no—
242#102—1371no—
243#102—1371no—
244#102—1371no—
245#102—1371no—
246#102—1371no—
247#102C2×D13711yescharacteristic, complemented
248C22⋊C8—1371no—
249C22⋊C8—1371no—
250C4×C8—1371no—
251C24—2741no—
252#41—2742no—
253#41—5481no—
254C2×Q8C22×D13711yescharacteristic
255C2×D4—21no—
256C2×D4—2742no—
257C2×D4—2741no—
258C2×D4—2741no—
259C2×D4—2741no—
260C2×D4—2741no—
261C2×D4—5481no—
262C2×D4—1371no—
263C2×D4—1371no—
264C2×D4—1371no—
265C2×D4C22×D13711yescharacteristic
266C22×C4—2741no—
267C22×C4—2741no—
268C22×C4—2741no—
269C22×C4—2741no—
270C22×C4—1371no—
271C22×C4C22×D13711yescharacteristic
272SD16—41no—
273D8—2742no—
274D8—5481no—
275C2×C8—41no—
276C2×C8—5481no—
277C2×C8—1371no—
278C2×C8—1371no—
279C2×C8C22×D13711yescharacteristic
280C2×C8C22×D13711yescharacteristic
281C4⋊C4—21no—
282C4⋊C4—2741no—
283C4⋊C4—2741no—
284C4⋊C4—2741no—
285C4⋊C4—1371no—
286C4⋊C4—1371no—
287C4⋊C4—1371no—
288C4⋊C4C22×D13711yescharacteristic
289C4⋊C4C22×D13711yescharacteristic
290C22⋊C4—21no—
291C22⋊C4—21no—
292C22⋊C4—2741no—
293C22⋊C4—2741no—
294C22⋊C4—2741no—
295C22⋊C4—2741no—
296C22⋊C4—2741no—
297C22⋊C4—2741no—
298C22⋊C4—2741no—
299C22⋊C4—1371no—
300C22⋊C4—1371no—
301C42—2741no—
302C42—1371no—
303C23—21no—
304C23—2741no—
305C23—2741no—
306C23—5482no—
307C23—5482no—
308C23—10961no—
309C23—1371no—
310C23D4×D13711yescharacteristic
311Q8—21no—
312Q8—21no—
313D4—21no—
314D4—21no—
315D4—42no—
316D4—2742no—
317D4—2741no—
318D4—2741no—
319D4—2741no—
320D4—5482no—
321D4—5482no—
322D4—10961no—
323D4—10961no—
324D4—1372no—
325C2×C4—21no—
326C2×C4—21no—
327C2×C4—21no—
328C2×C4—21no—
329C2×C4—21no—
330C2×C4—2741no—
331C2×C4—2741no—
332C2×C4—2741no—
333C2×C4—2741no—
334C2×C4—5482no—
335C2×C4—10961no—
336C2×C4—10961no—
337C2×C4—10961no—
338C2×C4—1371no—
339C2×C4—1371no—
340C2×C4—1371no—
341C2×C4D4×D13711yescharacteristic
342C2×C4C23×D13711yescharacteristic
343C8—21no—
344C8—21no—
345C8—2741no—
346C8—2741no—
347C22—21no—
348C22—21no—
349C22—41no—
350C22—41no—
351C22—41no—
352C22—2741no—
353C22—2741no—
354C22—2741no—
355C22—2742no—
356C22—5481no—
357C22—5481no—
358C22—5481no—
359C22—5481no—
360C22—5481no—
361C22—10964no—
362C22#1798410912yes—
363C22C2×D4×D13711yescharacteristic
364C4—21no—
365C4—41no—
366C4—41no—
367C4—41no—
368C4—2741no—
369C4—2741no—
370C4—2741no—
371C4—5481no—
372C4—5481no—
373C4#41×D13711yescharacteristic
374C4#1798413311yescharacteristic
375C2—22no—
376C2—81no—
377C2—5481no—
378C2—5482no—
379C2—10961no—
380C2#2257804211yescharacteristic
381C2C2×#1798410911yescharacteristic
382C2#2257820511yescharacteristic
subgroupG/H
centerC22C2×D4×D137
commutator, cosocleC2×C4×C137C24
FrattiniC2×C4C23×D137
Fitting#465×C137C2
socleC22×C137C22×D4
hypercenter#465D137
hypocenterC137#3297
LCSG ▷ C2×C4×C137 ▷ C2×C137 ▷ C137
UCSC1 ◁ C22 ◁ C22×C4 ◁ #465

Complements

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

NH ≅ G/Ntype
1. #22578105C2semidirect
2. #22578044C2semidirect
3. #22577759C2semidirect
4. #22577736C2semidirect
5. #22577729C2semidirect
6. #22577702C2semidirect
7. #22577638C2semidirect
8. #22577620C2semidirect
9. #22577547C2semidirect
10. #22577536C2semidirect
11. #22577483C2semidirect
12. C8×C22⋊D137C2semidirect
13. #22577398C2semidirect
14. C2×#17984010C2semidirect
15. #465×C137C2semidirect
20. C2×C8×D137C22semidirect
22. #17984079C22semidirect
29. #17984059C22semidirect
32. D4×Dic137C22semidirect
35. #17984046C22semidirect
38. #17984021C22semidirect
39. #17984021C22semidirect
40. #17984017C22semidirect
41. #17984012C22semidirect
42. #17984012C22semidirect
50. #17984000C22semidirect
51. C8×Dic137C22semidirect
52. C2×SD16×C137C22semidirect
56. #106×C137C22semidirect
57. #103×C137C22semidirect
58. #102×C137C22semidirect
191. C137#3297semidirect
200. #465D137semidirect
210. C2×SD16C2×D137semidirect
236. #106C2×D137semidirect
240. #103C2×D137semidirect
247. #102C2×D137semidirect

Maximal subgroups

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

subgroupindex
#225781052
#225780442
#225777592
#225777362
#225777292
#225777022
#225776382
#225776202
#225775472
#225775362
#225774832
C8×C22⋊D1372
#225773982
C2×#179840102
#465×C1372
#3297137