Finite Group Databasebeta

← #31792632

Subgroups — #31792632

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#22578100C211yesmaximal, characteristic, complemented
2#22578041C211yesmaximal, characteristic, complemented
3#22577759C211yesmaximal, characteristic, complemented
4#22577737C211yesmaximal, characteristic, complemented
5#22577728C211yesmaximal, characteristic, complemented
6#22577705C211yesmaximal, characteristic, complemented
7#22577637C211yesmaximal, characteristic, complemented
8#22577629C211yesmaximal, characteristic, complemented
9#22577620C211yesmaximal, characteristic, complemented
10#22577535C211yesmaximal, characteristic, complemented
11D8×Dic137C211yesmaximal, characteristic, complemented
12#22577466C211yesmaximal, characteristic, complemented
13#22577399C211yesmaximal, characteristic, complemented
14#102×D137C211yesmaximal, characteristic, complemented
15#449×C137C211yesmaximal, characteristic, complemented
16C2×D4×D137C2211yescharacteristic
17C2×#14555495C2211yescharacteristic
18C22×#11764040C2211yescharacteristic, complemented
19C2×C8×D137C2211yescharacteristic
20#17984081—21no—
21#17984079C2211yescharacteristic, complemented
22#17984077C2211yescharacteristic
23#17984071—21no—
24#17984069—21no—
25#17984064C2211yescharacteristic
26#17984062C2211yescharacteristic
27#17984060—21no—
28#17984059—21no—
29#17984059C2211yescharacteristic
30#17984056—21no—
31D4×Dic137C2211yescharacteristic
32D4×Dic137C2211yescharacteristic, complemented
33C4×C22⋊D137C2211yescharacteristic
34#17984048—21no—
35#17984047C2211yescharacteristic, complemented
36C4⋊C4×D137C2211yescharacteristic
37C2×#14555487—21no—
38#17984022C2211yescharacteristic, complemented
39#17984021C2211yescharacteristic, complemented
40#17984020C2211yescharacteristic, complemented
41#17984017C2211yescharacteristic, complemented
42#17984012C2211yescharacteristic, complemented
43#17984012C2211yescharacteristic, complemented
44#17984010C2211yescharacteristic
45#17984010C2211yescharacteristic
46#17984009C2211yescharacteristic
47#17984009C2211yescharacteristic
48#17984009C2211yescharacteristic, complemented
49#17984008C2211yescharacteristic
50#17984000C2211yescharacteristic
51C8×Dic137C2211yescharacteristic, complemented
52C2×#41×C137C2211yescharacteristic
53C2×SD16×C137C2211yescharacteristic, complemented
54C2×D8×C137C2211yescharacteristic
55#121×C137C2211yescharacteristic, complemented
56#103×C137C2211yescharacteristic, complemented
57#102×C137C2211yescharacteristic
58C22⋊C8×C137C2211yescharacteristic, complemented
59C23×D137—21no—
60C2×D548C2311yescharacteristic
61C2×C22⋊D137—21no—
62C2×C22⋊D137—21no—
63C2×C22⋊D137—21no—
64C2×C22⋊D137D411yescharacteristic
65D4×D137—22no—
66D4×D137—41no—
67C22×Dic137—21no—
68C22×Dic137—21no—
69C2×C4×D137—21no—
70C2×C4×D137C2311yescharacteristic
71#14555495—41no—
72C2×#11764040—41no—
73C2×#11764040C2311yescharacteristic
74C2×#11764040C2311yescharacteristic
75C8×D137—41no—
76C4⋊Dic137—21no—
77C4⋊Dic137C2311yescharacteristic
78C4⋊Dic137C2311yescharacteristic
79C4⋊Dic137C2311yescharacteristic
80#14555489—21no—
81#14555489—21no—
82#14555489—21no—
83#14555489D411yescharacteristic
84C22⋊Dic137—21no—
85C22⋊Dic137—21no—
86C22⋊Dic137—21no—
87C22⋊Dic137—21no—
88C22⋊Dic137D411yescharacteristic
89#14555487—21no—
90#14555487—21no—
91#14555487—21no—
92#14555487—21no—
93#14555487D411yescharacteristic
94C4×Dic137—21no—
95C4×Dic137C2311yescharacteristic
96#41×C137—22no—
97#41×C137—41no—
98C2×Q8×C137C2311yescharacteristic
99C2×D4×C137—21no—
100C2×D4×C137—21no—
101C2×D4×C137C2311yescharacteristic
102C2×D4×C137C2311yescharacteristic
103C22×C4×C137—21no—
104C22×C4×C137C2311yescharacteristic
105SD16×C137—41no—
106D8×C137—22no—
107C2×C8×C137C2311yescharacteristic
108C2×C8×C137C2311yescharacteristic
109C4⋊C4×C137C2311yescharacteristic
110C22⋊C4×C137—21no—
111C22×D137—21no—
112C22×D137—42no—
113C22×D137—42no—
114C22×D137—81no—
115C22×D137C2×D411yescharacteristic
116D548—21no—
117D548—21no—
118C22⋊D137—22no—
119C22⋊D137—42no—
120C22⋊D137—42no—
121C22⋊D137—81no—
122C2×Dic137—21no—
123C2×Dic137—21no—
124C2×Dic137—21no—
125C2×Dic137—81no—
126C2×Dic137—81no—
127C2×Dic137C2×D411yescharacteristic
128C2×Dic137C2×D411yescharacteristic
129C2×Dic137C2×D411yescharacteristic
130C4×D137—21no—
131C4×D137—21no—
132C4×D137—81no—
133#11764040—21no—
134#11764040—21no—
135C23×C137—21no—
136C23×C137—21no—
137C23×C137C2×D411yescharacteristic
138Q8×C137—21no—
139Q8×C137—21no—
140D4×C137—21no—
141D4×C137—21no—
142D4×C137—21no—
143D4×C137—42no—
144D4×C137—81no—
145D4×C137#4112yes—
146C2×C4×C137—21no—
147C2×C4×C137—21no—
148C2×C4×C137—21no—
149C2×C4×C137—21no—
150C2×C4×C137—42no—
151C2×C4×C137C2×D411yescharacteristic
152C2×C4×C137C2411yescharacteristic
153C8×C137—21no—
154C8×C137—21no—
155C22×C137—21no—
156C22×C137—22no—
157C22×C137—21no—
158C22×C137—21no—
159C22×C137—41no—
160C22×C137—41no—
161C22×C137—41no—
162C22×C137—41no—
163C22×C137—41no—
164C22×C137C22×D411yescharacteristic
165C2×D137—21no—
166C2×D137—21no—
167C2×D137—41no—
168C2×D137—41no—
169C2×D137—41no—
170C2×D137—84no—
171C2×D137#13512yes—
172C4×C137—21no—
173C4×C137—41no—
174C4×C137—41no—
175C4×C137C2×#4111yescharacteristic
176C4×C137H(5,𝔽2)11yescharacteristic
177Dic137—21no—
178Dic137—21no—
179Dic137—21no—
180Dic137—41no—
181Dic137—41no—
182Dic137—41no—
183C2×C137—42no—
184C2×C137—41no—
185C2×C137—81no—
186C2×C137#54611yescharacteristic
187C2×C137C2×#13511yescharacteristic
188C2×C137#57611yescharacteristic
189D137—22no—
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#465—1371no—
201#463—1371no—
202#449D13711yescharacteristic, complemented
203#447—1371no—
204C4×D8—1371no—
205C8×D4—1371no—
206#416—1371no—
207C2×#102—1371no—
208C2×#41C2×D13711yescharacteristic
209C22×D4—1371no—
210C2×SD16C2×D13711yescharacteristic, complemented
211C2×D8—1371no—
212C2×D8C2×D13711yescharacteristic
213C22×C8—1371no—
214C22×C8—1371no—
215#126—2741no—
216#125—1371no—
217#124—1371no—
218#123—2741no—
219#123—2741no—
220C22⋊Q8—1371no—
221#121—2741no—
222#121—2741no—
223#121—1371no—
224#121—1371no—
225#121C2×D13711yescharacteristic, complemented
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#106—1371no—
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
248C22⋊C8—1371no—
249C22⋊C8C2×D13711yescharacteristic, complemented
250C4×C8—1371no—
251C24—2741no—
252#41—22no—
253#41—41no—
254C2×Q8C22×D13711yescharacteristic
255C2×D4—21no—
256C2×D4—21no—
257C2×D4—2742no—
258C2×D4—2741no—
259C2×D4—2741no—
260C2×D4—2741no—
261C2×D4—5481no—
262C2×D4—1371no—
263C2×D4—1371no—
264C2×D4C22×D13711yescharacteristic
265C2×D4C22×D13711yescharacteristic
266C22×C4—21no—
267C22×C4—2741no—
268C22×C4—2741no—
269C22×C4—2741no—
270C22×C4—1371no—
271C22×C4C22×D13711yescharacteristic
272SD16—41no—
273D8—22no—
274D8—5481no—
275C2×C8—5481no—
276C2×C8—5481no—
277C2×C8—1371no—
278C2×C8—1371no—
279C2×C8C22×D13711yescharacteristic
280C2×C8C22×D13711yescharacteristic
281C4⋊C4—2741no—
282C4⋊C4—2741no—
283C4⋊C4—2741no—
284C4⋊C4—2741no—
285C4⋊C4—1371no—
286C4⋊C4—1371no—
287C4⋊C4—1371no—
288C4⋊C4—1371no—
289C4⋊C4C22×D13711yescharacteristic
290C22⋊C4—21no—
291C22⋊C4—2741no—
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—21no—
305C23—2741no—
306C23—5482no—
307C23—5482no—
308C23—10961no—
309C23—1371no—
310C23D4×D13711yescharacteristic
311Q8—21no—
312Q8—21no—
313D4—21no—
314D4—21no—
315D4—21no—
316D4—42no—
317D4—81no—
318D4—2742no—
319D4—2741no—
320D4—2741no—
321D4—5482no—
322D4—5482no—
323D4—10961no—
324D4#1455550912yes—
325C2×C4—21no—
326C2×C4—21no—
327C2×C4—21no—
328C2×C4—21no—
329C2×C4—42no—
330C2×C4—2741no—
331C2×C4—2741no—
332C2×C4—2741no—
333C2×C4—2741no—
334C2×C4—2741no—
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—21no—
350C22—22no—
351C22—41no—
352C22—41no—
353C22—41no—
354C22—41no—
355C22—41no—
356C22—2741no—
357C22—2741no—
358C22—5481no—
359C22—5481no—
360C22—5481no—
361C22—10964no—
362C22—1372no—
363C22C2×D4×D13711yescharacteristic
364C4—21no—
365C4—41no—
366C4—41no—
367C4—2741no—
368C4—2741no—
369C4—2741no—
370C4—5481no—
371C4—5481no—
372C4—5481no—
373C4C2×#1455550911yescharacteristic
374C4#1798413311yescharacteristic
375C2—41no—
376C2—42no—
377C2—81no—
378C2—2742no—
379C2—10961no—
380C2#2257804011yescharacteristic
381C2#135×D13711yescharacteristic
382C2#2257820711yescharacteristic
subgroupG/H
centerC22C2×D4×D137
commutator, cosocleC2×C4×C137C24
FrattiniC2×C4C23×D137
Fitting#449×C137C2
socleC22×C137C22×D4
hypercenter#449D137
hypocenterC137#3297
LCSG ▷ C2×C4×C137 ▷ C2×C137 ▷ C137
UCSC1 ◁ C22 ◁ C22×C4 ◁ #449

Complements

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

NH ≅ G/Ntype
1. #22578100C2semidirect
2. #22578041C2semidirect
3. #22577759C2semidirect
4. #22577737C2semidirect
5. #22577728C2semidirect
6. #22577705C2semidirect
7. #22577637C2semidirect
8. #22577629C2semidirect
9. #22577620C2semidirect
10. #22577535C2semidirect
11. D8×Dic137C2semidirect
12. #22577466C2semidirect
13. #22577399C2semidirect
14. #102×D137C2semidirect
15. #449×C137C2semidirect
18. C22×#11764040C22semidirect
21. #17984079C22semidirect
32. D4×Dic137C22semidirect
35. #17984047C22semidirect
38. #17984022C22semidirect
39. #17984021C22semidirect
40. #17984020C22semidirect
41. #17984017C22semidirect
42. #17984012C22semidirect
43. #17984012C22semidirect
48. #17984009C22semidirect
51. C8×Dic137C22semidirect
53. C2×SD16×C137C22semidirect
55. #121×C137C22semidirect
56. #103×C137C22semidirect
58. C22⋊C8×C137C22semidirect
191. C137#3297semidirect
202. #449D137semidirect
210. C2×SD16C2×D137semidirect
225. #121C2×D137semidirect
240. #103C2×D137semidirect
249. C22⋊C8C2×D137semidirect

Maximal subgroups

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

subgroupindex
#225781002
#225780412
#225777592
#225777372
#225777282
#225777052
#225776372
#225776292
#225776202
#225775352
D8×Dic1372
#225774662
#225773992
#102×D1372
#449×C1372
#3297137