Finite Group Databasebeta

← #463×D137 #31790209

Subgroups — #463×D137 #31790209

Order 17536.

564 conjugacy classes of subgroups.

The subgroup lattice has 418 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×SD16×D137C211yesmaximal, characteristic, complemented
2C2×D8×D137C211yesmaximal, characteristic
3#124×D137C211yesmaximal, characteristic, complemented
4C4×D4×D137C211yesmaximal, characteristic, complemented
5#22577621C211yesmaximal, characteristic, complemented
6#22577620C211yesmaximal, characteristic, complemented
7#22577619C211yesmaximal, characteristic, complemented
8#22577618C211yesmaximal, characteristic, complemented
9#22577617C211yesmaximal, characteristic, complemented
10#22577616C211yesmaximal, characteristic, complemented
11#22577615C211yesmaximal, characteristic, complemented
12C4⋊C8×D137C211yesmaximal, characteristic, complemented
13#103×D137C211yesmaximal, characteristic, complemented
14#102×D137C211yesmaximal, characteristic, complemented
15#463×C137C211yesmaximal, characteristic, complemented
16C2×Q8×D137C2211yescharacteristic
17C2×D4×D137C2211yescharacteristic
18C2×D4×D137C2211yescharacteristic
19C22×C4×D137—21no—
20C2×#14555499C2211yescharacteristic, complemented
21C2×#14555498C2211yescharacteristic, complemented
22C2×#14555497C2211yescharacteristic, complemented
23SD16×D137—24no—
24C2×D1096C2211yescharacteristic
25C2×#14555495C2211yescharacteristic
26C2×#14555495C2211yescharacteristic
27D8×D137—24no—
28C2×C8×D137C2211yescharacteristic
29C2×C8×D137C2211yescharacteristic
30#17984077C2211yescharacteristic, complemented
31#17984076C2211yescharacteristic, complemented
32#17984075C2211yescharacteristic, complemented
33#17984074—21no—
34#17984074—21no—
35C4×D548C2211yescharacteristic, complemented
36D4×Dic137C2211yescharacteristic, complemented
37#17984050C2211yescharacteristic, complemented
38C4×C22⋊D137—21no—
39#17984048—21no—
40C4⋊C4×D137C2211yescharacteristic
41C22⋊C4×D137—21no—
42C22⋊C4×D137—21no—
43C22⋊C4×D137—21no—
44C42×D137C2211yescharacteristic
45C4⋊#11764040C2211yescharacteristic, complemented
46#17984017C2211yescharacteristic, complemented
47#17984017C2211yescharacteristic, complemented
48#17984013C2211yescharacteristic, complemented
49#17984012C2211yescharacteristic, complemented
50#17984011C2211yescharacteristic, complemented
51#17984010C2211yescharacteristic, complemented
52#17984009C2211yescharacteristic, complemented
53#17984008C2211yescharacteristic, complemented
54C2×SD16×C137C2211yescharacteristic, complemented
55C2×D8×C137C2211yescharacteristic
56#124×C137C2211yescharacteristic, complemented
57C4×D4×C137C2211yescharacteristic, complemented
58C4⋊C8×C137C2211yescharacteristic, complemented
59#103×C137C2211yescharacteristic, complemented
60#102×C137C2211yescharacteristic, complemented
61C23×D137—21no—
62C23×D137—21no—
63C2×Dic274—21no—
64C2×Dic274C2311yescharacteristic
65Q8×D137—22no—
66Q8×D137—22no—
67C2×D548C2311yescharacteristic
68C2×D548C2311yescharacteristic
69C2×C22⋊D137—21no—
70C2×C22⋊D137—21no—
71D4×D137—22no—
72D4×D137—22no—
73D4×D137—22no—
74D4×D137D414yes—
75C22×Dic137—21no—
76C2×C4×D137—21no—
77C2×C4×D137—21no—
78C2×C4×D137—21no—
79C2×C4×D137—21no—
80C2×C4×D137—44no—
81C2×C4×D137D411yescharacteristic
82C2×C4×D137D411yescharacteristic
83C2×C4×D137C2311yescharacteristic
84#14555499—22no—
85#14555498—22no—
86#14555497—22no—
87D1096—22no—
88#14555495—22no—
89#14555495—22no—
90C2×#11764040C2311yescharacteristic
91C2×#11764040C2311yescharacteristic
92C8×D137—22no—
93C8×D137—22no—
94C4⋊Dic137C2311yescharacteristic
95#14555489—21no—
96C22⋊Dic137—21no—
97C22⋊Dic137—21no—
98C22⋊Dic137—21no—
99#14555487—21no—
100#14555487—21no—
101#14555487—21no—
102#14555487—21no—
103#14555487—21no—
104#14555487—21no—
105C4×Dic137D411yescharacteristic
106C4×Dic137D411yescharacteristic
107C4×Dic137C2311yescharacteristic
108C2×Q8×C137C2311yescharacteristic
109C2×D4×C137C2311yescharacteristic
110C2×D4×C137C2311yescharacteristic
111C22×C4×C137—21no—
112SD16×C137—22no—
113D8×C137—22no—
114C2×C8×C137C2311yescharacteristic
115C2×C8×C137C2311yescharacteristic
116C4⋊C4×C137C2311yescharacteristic
117C22⋊C4×C137—21no—
118C22⋊C4×C137—21no—
119C22⋊C4×C137—21no—
120C42×C137C2311yescharacteristic
121C22×D137—21no—
122C22×D137—24no—
123C22×D137—21no—
124C22×D137—42no—
125C22×D137—42no—
126C22×D137—42no—
127C22×D137—42no—
128C22×D137—42no—
129C22×D137C2×D411yescharacteristic
130Dic274—21no—
131Dic274—21no—
132Dic274—42no—
133D548—21no—
134D548—21no—
135D548—21no—
136D548C2×D412yes—
137C22⋊D137—24no—
138C22⋊D137—42no—
139C2×Dic137—21no—
140C2×Dic137—21no—
141C2×Dic137—42no—
142C2×Dic137C2×D411yescharacteristic
143C2×Dic137C2×D411yescharacteristic
144C2×Dic137C2×D411yescharacteristic
145C4×D137—22no—
146C4×D137—22no—
147C4×D137—42no—
148C4×D137—42no—
149C4×D137—42no—
150C4×D137—42no—
151C4×D137C2×D412yes—
152C4×D137#4112yes—
153#11764040—21no—
154#11764040—21no—
155C23×C137—21no—
156C23×C137—21no—
157Q8×C137—21no—
158Q8×C137—21no—
159D4×C137—21no—
160D4×C137—21no—
161D4×C137—21no—
162D4×C137C2×D412yes—
163C2×C4×C137—21no—
164C2×C4×C137—21no—
165C2×C4×C137—42no—
166C2×C4×C137C2×D411yescharacteristic
167C2×C4×C137C2×D411yescharacteristic
168C2×C4×C137C2411yescharacteristic
169C8×C137—21no—
170C8×C137—21no—
171C22×C137—22no—
172C22×C137—41no—
173C22×C137—41no—
174C22×C137—41no—
175C22×C137—41no—
176C22×C137—41no—
177C22×C137C22×D411yescharacteristic
178C2×D137—22no—
179C2×D137—41no—
180C2×D137—41no—
181C2×D137—48no—
182C2×D137—41no—
183C2×D137—41no—
184C2×D137—41no—
185C2×D137—84no—
186C2×D137#12112yes—
187C2×D137#13512yes—
188C2×D137AGL(1,ℤ/8)12yes—
189C4×C137—21no—
190C4×C137—21no—
191C4×C137—41no—
192C4×C137—41no—
193C4×C137C22×D411yescharacteristic
194C4×C137C2×#4111yescharacteristic
195Dic137—21no—
196Dic137—21no—
197Dic137—41no—
198Dic137—41no—
199Dic137#12112yes—
200C2×C137—42no—
201C2×C137—81no—
202C2×C137C2×#12111yescharacteristic
203C2×C137C2×#13511yescharacteristic
204C2×C137C2×AGL(1,ℤ/8)11yescharacteristic
205D137—42no—
206D137—81no—
207D137#46314yesdirect product factor
208C137C2×#46311yescharacteristic, complemented, Sylow
209C2×#463—1371nomaximal, Sylow
210C22×SD16—1371no—
211C22×D8—1371no—
212C2×#124—1371no—
213C2×C4×D4—1371no—
214#463—1371no—
215#463—1371no—
216#463—1371no—
217#463—1371no—
218#463—1371no—
219#463—1371no—
220#463—1371no—
221#463D13711yescharacteristic, direct product factor
222C2×C4⋊C8—1371no—
223C2×#103—1371no—
224C2×#102—1371no—
225C22×Q8—1371no—
226C22×D4—1371no—
227C22×D4—1371no—
228C23×C4—2741no—
229C2×SD16—2744no—
230C2×SD16—1371no—
231C2×SD16—1371no—
232C2×SD16—1371no—
233C2×SD16C2×D13711yescharacteristic, complemented
234C2×D8—2744no—
235C2×D8—1371no—
236C2×D8—1371no—
237C2×D8—1371no—
238C2×D8C2×D13711yescharacteristic
239C22×C8—1371no—
240C22×C8—1371no—
241#124—2741no—
242#124—2741no—
243#124—1371no—
244#124—1371no—
245#124—1371no—
246#124C2×D13711yescharacteristic, complemented
247C4×D4—2741no—
248C4×D4—2741no—
249C4×D4—1371no—
250C4×D4—1371no—
251C4×D4—1371no—
252C4×D4C2×D13711yescharacteristic, complemented
253C2×C4⋊C4—1371no—
254C2×C22⋊C4—2741no—
255C2×C22⋊C4—2741no—
256C2×C22⋊C4—2741no—
257C2×C42—1371no—
258C4⋊C8—1371no—
259C4⋊C8—1371no—
260C4⋊C8—1371no—
261C4⋊C8C2×D13711yescharacteristic, complemented
262#103—1371no—
263#103—1371no—
264#103—1371no—
265#103C2×D13711yescharacteristic, complemented
266#102—1371no—
267#102—1371no—
268#102—1371no—
269#102C2×D13711yescharacteristic, complemented
270C24—2741no—
271C24—2741no—
272C2×Q8—2741no—
273C2×Q8—2742no—
274C2×Q8—2742no—
275C2×Q8—1371no—
276C2×Q8C22×D13711yescharacteristic
277C2×D4—2742no—
278C2×D4—2742no—
279C2×D4—2742no—
280C2×D4—2741no—
281C2×D4—2741no—
282C2×D4—1374no—
283C2×D4—1371no—
284C2×D4—1371no—
285C2×D4C22×D13711yescharacteristic
286C2×D4C22×D13711yescharacteristic
287C22×C4—21no—
288C22×C4—2741no—
289C22×C4—2741no—
290C22×C4—2741no—
291C22×C4—2741no—
292C22×C4—2741no—
293C22×C4—5484no—
294C22×C4—1371no—
295C22×C4—1371no—
296C22×C4—1371no—
297SD16—22no—
298SD16—2742no—
299SD16—2742no—
300SD16—2742no—
301D8—22no—
302D8—2742no—
303D8—2742no—
304D8—2742no—
305C2×C8—2742no—
306C2×C8—2742no—
307C2×C8—1371no—
308C2×C8—1371no—
309C2×C8C22×D13711yescharacteristic
310C2×C8C22×D13711yescharacteristic
311C4⋊C4—2741no—
312C4⋊C4—1371no—
313C4⋊C4C22×D13711yescharacteristic
314C22⋊C4—21no—
315C22⋊C4—21no—
316C22⋊C4—21no—
317C22⋊C4—2741no—
318C22⋊C4—2741no—
319C22⋊C4—2741no—
320C22⋊C4—2741no—
321C22⋊C4—2741no—
322C22⋊C4—2741no—
323C22⋊C4—2741no—
324C22⋊C4—2741no—
325C22⋊C4—2741no—
326C42—1371no—
327C42—1371no—
328C42—1371no—
329C42C22×D13711yescharacteristic
330C23—21no—
331C23—21no—
332C23—2744no—
333C23—2741no—
334C23—2741no—
335C23—5482no—
336C23—5482no—
337C23—5482no—
338C23—5482no—
339C23—5482no—
340C23—1371no—
341Q8—21no—
342Q8—21no—
343Q8—2741no—
344Q8—2741no—
345Q8—5482no—
346D4—21no—
347D4—21no—
348D4—21no—
349D4—2744no—
350D4—2741no—
351D4—2741no—
352D4—2741no—
353D4—5482no—
354D4—1372no—
355D4D4×D13712yes—
356C2×C4—21no—
357C2×C4—21no—
358C2×C4—42no—
359C2×C4—2742no—
360C2×C4—2742no—
361C2×C4—2741no—
362C2×C4—2741no—
363C2×C4—5482no—
364C2×C4—5482no—
365C2×C4—5482no—
366C2×C4—5482no—
367C2×C4—5482no—
368C2×C4—1371no—
369C2×C4—1371no—
370C2×C4—1371no—
371C2×C4—1372no—
372C2×C4—1372no—
373C2×C4D4×D13711yescharacteristic
374C2×C4D4×D13711yescharacteristic
375C2×C4C23×D13711yescharacteristic
376C8—21no—
377C8—21no—
378C8—2741no—
379C8—2741no—
380C22—22no—
381C22—41no—
382C22—41no—
383C22—41no—
384C22—41no—
385C22—41no—
386C22—2742no—
387C22—5488no—
388C22—5481no—
389C22—5481no—
390C22—5481no—
391C22—5481no—
392C22—5481no—
393C22—10964no—
394C22—1372no—
395C22—1372no—
396C22—1372no—
397C22C2×D4×D13711yescharacteristic
398C4—21no—
399C4—21no—
400C4—41no—
401C4—41no—
402C4—2741no—
403C4—2741no—
404C4—5481no—
405C4—5481no—
406C4—1372no—
407C4C2×D4×D13711yescharacteristic
408C4#41×D13711yescharacteristic
409C2—42no—
410C2—81no—
411C2—5482no—
412C2—10961no—
413C2—1374no—
414C2#121×D13711yescharacteristic
415C2#135×D13711yescharacteristic
416C2AGL(1,ℤ/8)×D13711yescharacteristic
subgroupG/H
centerC22C2×D4×D137
commutator, cosocleC2×C4×C137C24
FrattiniC2×C4C23×D137
Fitting#463×C137C2
socleC22×C137C22×D4
prime radicalC2×D8×D137C2
hypercenter#463D137
hypocenterC137C2×#463
LCSG ▷ C2×C4×C137 ▷ C2×C137 ▷ C137
UCSC1 ◁ C22 ◁ C42 ◁ #463

Complements

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

NH ≅ G/Ntype
1. C2×SD16×D137C2semidirect
3. #124×D137C2semidirect
4. C4×D4×D137C2semidirect
5. #22577621C2semidirect
6. #22577620C2semidirect
7. #22577619C2semidirect
8. #22577618C2semidirect
9. #22577617C2semidirect
10. #22577616C2semidirect
11. #22577615C2semidirect
12. C4⋊C8×D137C2semidirect
13. #103×D137C2semidirect
14. #102×D137C2semidirect
15. #463×C137C2semidirect
20. C2×#14555499C22semidirect
21. C2×#14555498C22semidirect
22. C2×#14555497C22semidirect
30. #17984077C22semidirect
31. #17984076C22semidirect
32. #17984075C22semidirect
35. C4×D548C22semidirect
36. D4×Dic137C22semidirect
37. #17984050C22semidirect
45. C4⋊#11764040C22semidirect
46. #17984017C22semidirect
47. #17984017C22semidirect
48. #17984013C22semidirect
49. #17984012C22semidirect
50. #17984011C22semidirect
51. #17984010C22semidirect
52. #17984009C22semidirect
53. #17984008C22semidirect
54. C2×SD16×C137C22semidirect
56. #124×C137C22semidirect
57. C4×D4×C137C22semidirect
58. C4⋊C8×C137C22semidirect
59. #103×C137C22semidirect
60. #102×C137C22semidirect
207. D137#463direct (N × H)
208. C137C2×#463semidirect
221. #463D137direct (N × H)
233. C2×SD16C2×D137semidirect
246. #124C2×D137semidirect
252. C4×D4C2×D137semidirect
261. C4⋊C8C2×D137semidirect
265. #103C2×D137semidirect
269. #102C2×D137semidirect

Maximal subgroups

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

subgroupindex
C2×SD16×D1372
C2×D8×D1372
#124×D1372
C4×D4×D1372
#225776212
#225776202
#225776192
#225776182
#225776172
#225776162
#225776152
C4⋊C8×D1372
#103×D1372
#102×D1372
#463×C1372
C2×#463137