Finite Group Databasebeta

← #31792279

Subgroups — #31792279

Order 17536.

546 conjugacy classes of subgroups.

The subgroup lattice has 438 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#17984115C211yesmaximal, characteristic, complemented
2C2×SD16×D137C211yesmaximal, characteristic, complemented
3#22578065C211yesmaximal, characteristic, complemented
4#22578038C211yesmaximal, characteristic, complemented
5#22577790C211yesmaximal, characteristic, complemented
6#22577648C211yesmaximal, characteristic, complemented
7#22577638C211yesmaximal, characteristic, complemented
8#22577619C211yesmaximal, characteristic, complemented
9#22577555C211yesmaximal, characteristic, complemented
10#22577547C211yesmaximal, characteristic, complemented
11#22577540C211yesmaximal, characteristic, complemented
12#22577538C211yesmaximal, characteristic, complemented
13#22577501C211yesmaximal, characteristic, complemented
14#22577473C211yesmaximal, characteristic, complemented
15#460×C137C211yesmaximal, characteristic, complemented
16C2×#14555511C2211yescharacteristic, complemented
17C2×Q8×D137C2211yescharacteristic, complemented
18C2×D4×D137—21no—
19C2×D4×D137C2211yescharacteristic
20C2×D4×D137C2211yescharacteristic
21#17984115—24no—
22C2×#14555499C2211yescharacteristic
23C2×#14555498C2211yescharacteristic, complemented
24C2×#14555498C2211yescharacteristic, complemented
25C2×#14555497C2211yescharacteristic
26SD16×D137—24no—
27C2×D1096C2211yescharacteristic
28C2×#14555495C2211yescharacteristic, complemented
29C2×#14555495C2211yescharacteristic, complemented
30C2×#14555493C2211yescharacteristic, complemented
31C2×C8×D137C2211yescharacteristic, complemented
32#17984076—21no—
33#17984076C2211yescharacteristic, complemented
34#17984068—21no—
35#17984068C2211yescharacteristic, complemented
36#17984063—21no—
37#17984063C2211yescharacteristic, complemented
38#17984059—21no—
39#17984059C2211yescharacteristic
40#17984059C2211yescharacteristic
41#17984056—21no—
42#17984056—21no—
43C4×D548—21no—
44C4×D548C2211yescharacteristic
45C4×D548C2211yescharacteristic
46#17984021C2211yescharacteristic, complemented
47#17984013C2211yescharacteristic, complemented
48#17984012C2211yescharacteristic, complemented
49#17984010C2211yescharacteristic, complemented
50#17984008C2211yescharacteristic
51#17984000C2211yescharacteristic, complemented
52#17984000C2211yescharacteristic, complemented
53#17983999C2211yescharacteristic, complemented
54C2×SD16×C137C2211yescharacteristic, complemented
55C2×SD16×C137C2211yescharacteristic, complemented
56#127×C137C2211yescharacteristic
57C4×Q8×C137C2211yescharacteristic, complemented
58C4⋊C8×C137C2211yescharacteristic, complemented
59#102×C137C2211yescharacteristic, complemented
60#102×C137C2211yescharacteristic, complemented
61C23×D137—21no—
62C23×D137—21no—
63#14555511—22no—
64#14555511—41no—
65C2×Dic274—21no—
66C2×Dic274C2311yescharacteristic
67Q8×D137—22no—
68Q8×D137—41no—
69C2×D548—21no—
70C2×D548—21no—
71C2×D548D411yescharacteristic
72C2×D548C2311yescharacteristic
73C2×D548C2311yescharacteristic
74C2×C22⋊D137—21no—
75C2×C22⋊D137—21no—
76C2×C22⋊D137—21no—
77C2×C22⋊D137—21no—
78D4×D137—22no—
79D4×D137—22no—
80D4×D137—41no—
81D4×D137—41no—
82D4×D137—44no—
83C2×C4×D137—21no—
84C2×C4×D137—21no—
85C2×C4×D137—21no—
86C2×C4×D137C2311yescharacteristic
87C2×C4×D137C2311yescharacteristic
88#14555499—22no—
89#14555498—22no—
90#14555498—41no—
91#14555497D414yescomplemented
92D1096—22no—
93#14555495—22no—
94#14555495—41no—
95#14555493—22no—
96C2×#11764040C2311yescharacteristic
97C2×#11764040C2311yescharacteristic
98C8×D137—22no—
99C4⋊Dic137—21no—
100C4⋊Dic137D411yescharacteristic
101C4⋊Dic137C2311yescharacteristic
102C22⋊Dic137—21no—
103C22⋊Dic137—21no—
104#14555487—21no—
105#14555487—21no—
106#14555487—21no—
107#14555487—21no—
108#14555487—21no—
109#14555487D411yescharacteristic
110#14555487D411yescharacteristic
111C2×Q8×C137C2311yescharacteristic
112C2×D4×C137—21no—
113C2×D4×C137—21no—
114C2×D4×C137C2311yescharacteristic
115C2×D4×C137C2311yescharacteristic
116SD16×C137—22no—
117SD16×C137—22no—
118C2×C8×C137C2311yescharacteristic
119C2×C8×C137C2311yescharacteristic
120C4⋊C4×C137—21no—
121C4⋊C4×C137C2311yescharacteristic
122C42×C137—21no—
123C42×C137C2311yescharacteristic
124C22×D137—21no—
125C22×D137—21no—
126C22×D137—42no—
127C22×D137—42no—
128C22×D137—42no—
129C22×D137—42no—
130C22×D137—81no—
131C22×D137—81no—
132C22×D137C2×D411yescharacteristic
133C22×D137C2×D411yescharacteristic
134Dic274—21no—
135Dic274—21no—
136Dic274—42no—
137D548—22no—
138D548—21no—
139D548—21no—
140D548—21no—
141D548—42no—
142D548—42no—
143D548C2×D412yes—
144C22⋊D137—42no—
145C22⋊D137—42no—
146C22⋊D137—42no—
147C22⋊D137—42no—
148C2×Dic137—21no—
149C2×Dic137—21no—
150C2×Dic137C2×D411yescharacteristic
151C2×Dic137C2×D411yescharacteristic
152C4×D137—21no—
153C4×D137—21no—
154C4×D137—21no—
155C4×D137—21no—
156C4×D137—42no—
157C4×D137—42no—
158C4×D137—42no—
159#11764040—21no—
160#11764040C2×D412yescomplemented
161C23×C137—21no—
162C23×C137—21no—
163Q8×C137—21no—
164Q8×C137C2×D412yescomplemented
165D4×C137—21no—
166D4×C137—21no—
167D4×C137—21no—
168D4×C137—21no—
169D4×C137—42no—
170D4×C137—81no—
171C2×C4×C137—21no—
172C2×C4×C137—21no—
173C2×C4×C137C2×D411yescharacteristic
174C2×C4×C137C2×D411yescharacteristic
175C2×C4×C137C2411yescharacteristic
176C8×C137—21no—
177C8×C137—21no—
178C22×C137—41no—
179C22×C137—41no—
180C22×C137—41no—
181C22×C137—41no—
182C22×C137—41no—
183C22×C137—41no—
184C22×C137C22×D411yescharacteristic
185C2×D137—21no—
186C2×D137—21no—
187C2×D137—21no—
188C2×D137—22no—
189C2×D137—21no—
190C2×D137—21no—
191C2×D137—41no—
192C2×D137—41no—
193C2×D137—41no—
194C2×D137—41no—
195C2×D137—41no—
196C2×D137—84no—
197C2×D137—84no—
198C2×D137AGL(1,ℤ/8)12yescomplemented
199C4×C137—22no—
200C4×C137—21no—
201C4×C137—21no—
202C4×C137—41no—
203C4×C137C22×D411yescharacteristic
204C4×C137H(5,𝔽2)11yescharacteristic
205Dic137—21no—
206Dic137—21no—
207Dic137—41no—
208Dic137—41no—
209C2×C137—81no—
210C2×C137—81no—
211C2×C137D4211yescharacteristic
212C2×C137C2×AGL(1,ℤ/8)11yescharacteristic
213C2×C137#57711yescharacteristic
214D137—22no—
215D137—41no—
216D137—42no—
217D137—81no—
218C137#326511yescharacteristic, complemented, Sylow
219#3265—1371nomaximal, Sylow
220C2×AGL(1,ℤ/8)—1371no—
221C22×SD16—1371no—
222#548—1371no—
223D42—1371no—
224#496—1371no—
225#468—1371no—
226#466—1371no—
227#463—1371no—
228#460D13711yescharacteristic, complemented
229#450—1371no—
230#449—1371no—
231#448—1371no—
232#447—1371no—
233#440—1371no—
234#435—1371no—
235C2×#41—1371no—
236C22×Q8—1371no—
237C22×D4—2741no—
238C22×D4—1371no—
239C22×D4—1371no—
240AGL(1,ℤ/8)—2744no—
241C2×SD16—2744no—
242C2×SD16—1371no—
243C2×SD16—1371no—
244C2×SD16—1371no—
245C2×SD16—1371no—
246C2×SD16C2×D13711yescharacteristic, complemented
247C2×SD16C2×D13711yescharacteristic, complemented
248C2×D8—1371no—
249C2×D8—1371no—
250C2×D8—1371no—
251C2×M16—1371no—
252C22×C8—1371no—
253#127C2×D13711yescharacteristic
254#124—2741no—
255#124—1371no—
256C22⋊Q8—2741no—
257C22⋊Q8—1371no—
258#121—2741no—
259#121—2741no—
260#121—1371no—
261#121—1371no—
262#121—1371no—
263#120—2741no—
264#120—2741no—
265C4×Q8C2×D13711yescharacteristic, complemented
266C4×D4—2741no—
267C4×D4—1371no—
268C4×D4—1371no—
269#107—1371no—
270C4⋊C8C2×D13711yescharacteristic, complemented
271#103—1371no—
272#103—1371no—
273#102—1371no—
274#102—1371no—
275#102C2×D13711yescharacteristic, complemented
276#102C2×D13711yescharacteristic, complemented
277C22⋊C8—1371no—
278C22⋊C8—1371no—
279#97—1371no—
280C24—2741no—
281C24—2741no—
282#41—2742no—
283#41—5481no—
284C2×Q8—2741no—
285C2×Q8—2742no—
286C2×Q8—5481no—
287C2×Q8—1371no—
288C2×Q8C22×D13711yescharacteristic
289C2×D4—21no—
290C2×D4—21no—
291C2×D4—2741no—
292C2×D4—2741no—
293C2×D4—2742no—
294C2×D4—2742no—
295C2×D4—2741no—
296C2×D4—2741no—
297C2×D4—2741no—
298C2×D4—2741no—
299C2×D4—5481no—
300C2×D4—5481no—
301C2×D4—5484no—
302C2×D4—1371no—
303C2×D4—1371no—
304C2×D4—1371no—
305C2×D4C22×D13711yescharacteristic
306C2×D4C22×D13711yescharacteristic
307C22×C4—2741no—
308C22×C4—2741no—
309C22×C4—2741no—
310C22×C4—1371no—
311C22×C4—1371no—
312SD16—22no—
313SD16—22no—
314SD16—2742no—
315SD16—2742no—
316SD16—5481no—
317SD16—1374no—
318D8—2742no—
319D8—2742no—
320D8—5481no—
321M16—2742no—
322C2×C8—2742no—
323C2×C8—1371no—
324C2×C8—1371no—
325C2×C8C22×D13711yescharacteristic
326C2×C8C22×D13711yescharacteristic
327C4⋊C4—21no—
328C4⋊C4—2741no—
329C4⋊C4—1371no—
330C4⋊C4—1371no—
331C4⋊C4C22×D13711yescharacteristic
332C22⋊C4—2741no—
333C22⋊C4—2741no—
334C22⋊C4—2741no—
335C22⋊C4—2741no—
336C22⋊C4—2741no—
337C22⋊C4—2741no—
338C22⋊C4—2741no—
339C22⋊C4—1371no—
340C22⋊C4—1371no—
341C42—21no—
342C42C22×D13711yescharacteristic
343C23—21no—
344C23—21no—
345C23—2741no—
346C23—2741no—
347C23—5482no—
348C23—5482no—
349C23—5482no—
350C23—5482no—
351C23—10961no—
352C23—10961no—
353C23—1371no—
354C23—1371no—
355Q8—21no—
356Q8—2741no—
357Q8—2741no—
358Q8—5482no—
359Q8D4×D13712yescomplemented
360D4—21no—
361D4—21no—
362D4—21no—
363D4—21no—
364D4—42no—
365D4—81no—
366D4—2742no—
367D4—2741no—
368D4—2741no—
369D4—2741no—
370D4—5482no—
371D4—5482no—
372D4—5482no—
373D4—5482no—
374D4—5482no—
375D4—5482no—
376D4—1372no—
377C2×C4—21no—
378C2×C4—21no—
379C2×C4—2741no—
380C2×C4—2741no—
381C2×C4—2741no—
382C2×C4—2741no—
383C2×C4—2741no—
384C2×C4—2741no—
385C2×C4—5482no—
386C2×C4—5482no—
387C2×C4—5482no—
388C2×C4—1371no—
389C2×C4—1371no—
390C2×C4D4×D13711yescharacteristic
391C2×C4D4×D13711yescharacteristic
392C2×C4C23×D13711yescharacteristic
393C8—21no—
394C8—21no—
395C8—2741no—
396C8—1372no—
397C22—41no—
398C22—41no—
399C22—41no—
400C22—41no—
401C22—41no—
402C22—41no—
403C22—2741no—
404C22—2741no—
405C22—2741no—
406C22—2741no—
407C22—2741no—
408C22—2742no—
409C22—5481no—
410C22—5481no—
411C22—5481no—
412C22—5481no—
413C22—5481no—
414C22—10964no—
415C22—10964no—
416C22—1372no—
417C22C2×D4×D13711yescharacteristic
418C4—21no—
419C4—21no—
420C4—22no—
421C4—41no—
422C4—2741no—
423C4—2741no—
424C4—5481no—
425C4—5481no—
426C4C2×D4×D13711yescharacteristic
427C4#1798413411yescharacteristic
428C2—81no—
429C2—81no—
430C2—2742no—
431C2—5481no—
432C2—5482no—
433C2—10961no—
434C2#2257803711yescharacteristic
435C2AGL(1,ℤ/8)×D13711yescharacteristic
436C2#2257820911yescharacteristic
subgroupG/H
centerC22C2×D4×D137
commutator, cosocleC2×C4×C137C24
FrattiniC2×C4C23×D137
Fitting#460×C137C2
socleC22×C137C22×D4
hypercenter#460D137
hypocenterC137#3265
LCSG ▷ C2×C4×C137 ▷ C2×C137 ▷ C137
UCSC1 ◁ C22 ◁ C42 ◁ #460

Complements

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

NH ≅ G/Ntype
1. C2×#17984115C2semidirect
2. C2×SD16×D137C2semidirect
3. #22578065C2semidirect
4. #22578038C2semidirect
5. #22577790C2semidirect
6. #22577648C2semidirect
7. #22577638C2semidirect
8. #22577619C2semidirect
9. #22577555C2semidirect
10. #22577547C2semidirect
11. #22577540C2semidirect
12. #22577538C2semidirect
13. #22577501C2semidirect
14. #22577473C2semidirect
15. #460×C137C2semidirect
16. C2×#14555511C22semidirect
17. C2×Q8×D137C22semidirect
23. C2×#14555498C22semidirect
24. C2×#14555498C22semidirect
28. C2×#14555495C22semidirect
29. C2×#14555495C22semidirect
30. C2×#14555493C22semidirect
31. C2×C8×D137C22semidirect
33. #17984076C22semidirect
35. #17984068C22semidirect
37. #17984063C22semidirect
46. #17984021C22semidirect
47. #17984013C22semidirect
48. #17984012C22semidirect
49. #17984010C22semidirect
51. #17984000C22semidirect
52. #17984000C22semidirect
53. #17983999C22semidirect
54. C2×SD16×C137C22semidirect
55. C2×SD16×C137C22semidirect
57. C4×Q8×C137C22semidirect
58. C4⋊C8×C137C22semidirect
59. #102×C137C22semidirect
60. #102×C137C22semidirect
91. #14555497D4semidirect
160. #11764040C2×D4semidirect
164. Q8×C137C2×D4semidirect
198. C2×D137AGL(1,ℤ/8)semidirect
218. C137#3265semidirect
228. #460D137semidirect
246. C2×SD16C2×D137semidirect
247. C2×SD16C2×D137semidirect
265. C4×Q8C2×D137semidirect
270. C4⋊C8C2×D137semidirect
275. #102C2×D137semidirect
276. #102C2×D137semidirect
359. Q8D4×D137semidirect

Maximal subgroups

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

subgroupindex
C2×#179841152
C2×SD16×D1372
#225780652
#225780382
#225777902
#225776482
#225776382
#225776192
#225775552
#225775472
#225775402
#225775382
#225775012
#225774732
#460×C1372
#3265137