Finite Group Databasebeta

← #31792580

Subgroups — #31792580

Order 17536.

460 conjugacy classes of subgroups.

The subgroup lattice has 456 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#22578214C211yesmaximal, characteristic, complemented
2#22578211C211yesmaximal, characteristic, complemented
3#22578210C211yesmaximal, characteristic, complemented
4#22577774C211yesmaximal, characteristic, complemented
5#22577668C211yesmaximal, characteristic, complemented
6#22577666C211yesmaximal, characteristic, complemented
7#22577664C211yesmaximal, characteristic, complemented
8#22577663C211yesmaximal, characteristic, complemented
9#22577586C211yesmaximal, characteristic, complemented
10#22577584C211yesmaximal, characteristic, complemented
11#22577571C211yesmaximal, characteristic, complemented
12#22577569C211yesmaximal, characteristic, complemented
13#22577520C211yesmaximal, characteristic, complemented
14#22577518C211yesmaximal, characteristic, complemented
15#577×C137C211yesmaximal, characteristic, complemented
16#17984135C2211yescharacteristic
17#17984134C2211yescharacteristic, complemented
18#17984120—21no—
19#17984119—21no—
20#17984119C2211yescharacteristic, complemented
21#17984118—21no—
22#17984118C2211yescharacteristic, complemented
23#17984118C2211yescharacteristic, complemented
24#17984114—21no—
25#17984114C2211yescharacteristic
26#17984113—21no—
27#17984113C2211yescharacteristic
28#17984113C2211yescharacteristic
29#17984112—21no—
30#17984109—21no—
31#17984108—21no—
32#17984108—21no—
33#17984108C2211yescharacteristic
34C2×#14555499C2211yescharacteristic, complemented
35C2×#14555498—21no—
36C2×#14555498C2211yescharacteristic, complemented
37C2×#14555497—21no—
38C2×#14555497C2211yescharacteristic, complemented
39SD16×D137—21no—
40#17984095C2211yescharacteristic, complemented
41#17984095C2211yescharacteristic, complemented
42#17984094C2211yescharacteristic, complemented
43#17984086C2211yescharacteristic, complemented
44#17984084C2211yescharacteristic
45#17984024C2211yescharacteristic, complemented
46#17984023C2211yescharacteristic, complemented
47#17984015C2211yescharacteristic, complemented
48#17984015C2211yescharacteristic, complemented
49#17984015C2211yescharacteristic, complemented
50#17984014C2211yescharacteristic, complemented
51#17984007C2211yescharacteristic, complemented
52#17984006C2211yescharacteristic, complemented
53#17984005C2211yescharacteristic, complemented
54#17984004C2211yescharacteristic, complemented
55C8×Dic137C2211yescharacteristic, complemented
562−1+4×C137C2211yescharacteristic
57H(5,𝔽2)×C137C2211yescharacteristic, complemented
58#137×C137—21no—
59#137×C137C2211yescharacteristic, complemented
60AGL(1,ℤ/8)×C137—21no—
61AGL(1,ℤ/8)×C137C2211yescharacteristic
62#135×C137—21no—
63#135×C137C2211yescharacteristic
64C2×SD16×C137—21no—
65C2×SD16×C137C2211yescharacteristic, complemented
66#131×C137C2211yescharacteristic, complemented
67#14555511—21no—
68#14555510—21no—
69#14555510—21no—
70#14555510C2311yescharacteristic
71#14555509—21no—
72#14555509—41no—
73C2×Dic274—21no—
74C2×Dic274C2311yescharacteristic
75Q8×D137—21no—
76C2×D548—21no—
77C2×D548C2311yescharacteristic
78C2×C22⋊D137—41no—
79D4×D137—21no—
80D4×D137—41no—
81#14555500—21no—
82#14555500—21no—
83#14555500—21no—
84#14555500D411yescharacteristic, complemented
85#14555499—21no—
86#14555499—21no—
87#14555498—21no—
88#14555498—21no—
89#14555498—21no—
90#14555498—21no—
91#14555498—21no—
92#14555498D411yescharacteristic, complemented
93#14555497—21no—
94#14555497—21no—
95#14555497—21no—
96#14555497—21no—
97#14555497—21no—
98#14555497D411yescharacteristic, complemented
99#14555495—21no—
100#14555495—21no—
101#14555495—21no—
102#14555495D411yescharacteristic, complemented
103#14555494—21no—
104#14555494—21no—
105#14555494C2311yescharacteristic
106#14555494C2311yescharacteristic
107#14555494C2311yescharacteristic
108#14555493—21no—
109C2×#11764040—21no—
110C2×#11764040—21no—
111C2×#11764040C2311yescharacteristic
112C8×D137—21no—
113#14555489—41no—
114C4×Dic137C2311yescharacteristic
115#41×C137—21no—
116#41×C137—21no—
117#41×C137—21no—
118#41×C137—21no—
119#41×C137—41no—
120#41×C137C2311yescharacteristic
121#41×C137C2311yescharacteristic
122#41×C137C2311yescharacteristic
123C2×Q8×C137—21no—
124C2×Q8×C137—21no—
125C2×Q8×C137C2311yescharacteristic
126C2×D4×C137—21no—
127C2×D4×C137—21no—
128C2×D4×C137—41no—
129C2×D4×C137C2311yescharacteristic
130Q16×C137—21no—
131Q16×C137D411yescharacteristic, complemented
132SD16×C137—21no—
133SD16×C137—21no—
134SD16×C137—21no—
135SD16×C137—21no—
136SD16×C137D411yescharacteristic, complemented
137SD16×C137D411yescharacteristic, complemented
138D8×C137—21no—
139D8×C137D411yescharacteristic, complemented
140M16×C137—21no—
141M16×C137C2311yescharacteristic
142C2×C8×C137—21no—
143C2×C8×C137C2311yescharacteristic
144C22×D137—21no—
145C22×D137—41no—
146Dic274—21no—
147Dic274—21no—
148Dic274—41no—
149Dic274C2×D411yescharacteristic
150D548—21no—
151D548—21no—
152D548—41no—
153D548C2×D411yescharacteristic
154C22⋊D137—21no—
155C22⋊D137—41no—
156C22⋊D137—41no—
157C22⋊D137—82no—
158C2×Dic137—21no—
159C2×Dic137—21no—
160C2×Dic137—41no—
161C4×D137—21no—
162C4×D137—41no—
163C4×D137—41no—
164#11764040—21no—
165#11764040—21no—
166#11764040—21no—
167#11764040C2×D411yescharacteristic
168#11764040C2×D411yescharacteristic
169C23×C137—21no—
170C23×C137—41no—
171Q8×C137—21no—
172Q8×C137—21no—
173Q8×C137—41no—
174Q8×C137C2×D411yescharacteristic
175Q8×C137C2×D411yescharacteristic
176Q8×C137C2×D411yescharacteristic
177D4×C137—21no—
178D4×C137—21no—
179D4×C137—21no—
180D4×C137—21no—
181D4×C137—21no—
182D4×C137—41no—
183D4×C137—41no—
184D4×C137—41no—
185D4×C137C2×D411yescharacteristic
186D4×C137C2×D411yescharacteristic
187D4×C137C2×D411yescharacteristic
188C2×C4×C137—21no—
189C2×C4×C137—21no—
190C2×C4×C137—21no—
191C2×C4×C137—21no—
192C2×C4×C137—41no—
193C2×C4×C137—41no—
194C2×C4×C137—41no—
195C2×C4×C137C2411yescharacteristic
196C8×C137—21no—
197C8×C137C2×D411yescharacteristic
198C8×C137C2×D411yescharacteristic
199C22×C137—21no—
200C22×C137—21no—
201C22×C137—21no—
202C22×C137—41no—
203C22×C137—81no—
204C22×C137—161no—
205C22×C137H(5,𝔽2)11yescharacteristic
206C2×D137—21no—
207C2×D137—41no—
208C2×D137—81no—
209C2×D137—161no—
210C4×C137—21no—
211C4×C137—21no—
212C4×C137—21no—
213C4×C137—41no—
214C4×C137C22×D411yescharacteristic
215C4×C137C22×D411yescharacteristic
216Dic137—22no—
217Dic137—21no—
218Dic137—41no—
219C2×C137—21no—
220C2×C137—41no—
221C2×C137—41no—
222C2×C137—41no—
223C2×C137—81no—
224C2×C137D4211yescharacteristic
225D137—41no—
226D137—81no—
227C137#329111yescharacteristic, complemented, Sylow
228#3291—1371nomaximal, Sylow
229#577—1371no—
230#577—1371no—
231#577—1371no—
232#577D13711yescharacteristic, complemented
233#492—1371no—
234#471—1371no—
235#471—1371no—
236#471—1371no—
237#471—1371no—
238#456—1371no—
239#456—1371no—
240#453—1371no—
241#453—1371no—
242#443—1371no—
243#443—1371no—
2442−1+4—1371no—
2452−1+4C2×D13711yescharacteristic
246H(5,𝔽2)—1371no—
247H(5,𝔽2)C2×D13711yescharacteristic, complemented
248#137—21no—
249#137—2741no—
250#137—2741no—
251#137—2741no—
252#137—1371no—
253#137—1371no—
254#137—1371no—
255#137C2×D13711yescharacteristic, complemented
256AGL(1,ℤ/8)—21no—
257AGL(1,ℤ/8)—2741no—
258AGL(1,ℤ/8)—2741no—
259AGL(1,ℤ/8)—2741no—
260AGL(1,ℤ/8)—1371no—
261AGL(1,ℤ/8)—1371no—
262AGL(1,ℤ/8)—1371no—
263AGL(1,ℤ/8)C2×D13711yescharacteristic
264#135—21no—
265#135—2741no—
266#135—2741no—
267#135—2741no—
268#135—1371no—
269#135C2×D13711yescharacteristic
270C2×SD16—21no—
271C2×SD16—2741no—
272C2×SD16—2741no—
273C2×SD16—2741no—
274C2×SD16—1371no—
275C2×SD16—1371no—
276C2×SD16—1371no—
277C2×SD16C2×D13711yescharacteristic, complemented
278#131—1371no—
279#131—1371no—
280#131—1371no—
281#131C2×D13711yescharacteristic, complemented
282C4⋊Q8—1371no—
283#127—1371no—
284#108—1371no—
285#108—1371no—
286#104—1371no—
287#104—1371no—
288#104—1371no—
289#104—1371no—
290#101—1371no—
291#101—1371no—
292#100—1371no—
293#100—1371no—
294C4×C8—1371no—
295#41—21no—
296#41—21no—
297#41—21no—
298#41—21no—
299#41—41no—
300#41—2741no—
301#41—2741no—
302#41—2741no—
303#41—2741no—
304#41—5481no—
305#41—1371no—
306#41C22×D13711yescharacteristic
307#41C22×D13711yescharacteristic
308#41C22×D13711yescharacteristic
309C2×Q8—21no—
310C2×Q8—21no—
311C2×Q8—2741no—
312C2×Q8—2741no—
313C2×Q8—1371no—
314C2×Q8C22×D13711yescharacteristic
315C2×D4—21no—
316C2×D4—21no—
317C2×D4—41no—
318C2×D4—2741no—
319C2×D4—2741no—
320C2×D4—5481no—
321C2×D4—5481no—
322C2×D4—1371no—
323C2×D4C22×D13711yescharacteristic
324Q16—21no—
325Q16—2741no—
326Q16—2741no—
327Q16—2741no—
328Q16—1371no—
329Q16C22⋊D13711yescharacteristic, complemented
330SD16—21no—
331SD16—21no—
332SD16—21no—
333SD16—21no—
334SD16—2741no—
335SD16—2741no—
336SD16—2741no—
337SD16—2741no—
338SD16—2741no—
339SD16—2741no—
340SD16—2741no—
341SD16—2741no—
342SD16—2741no—
343SD16—2741no—
344SD16—2741no—
345SD16—2741no—
346SD16—1371no—
347SD16—1371no—
348SD16C22⋊D13711yescharacteristic, complemented
349SD16C22⋊D13711yescharacteristic, complemented
350D8—21no—
351D8—2741no—
352D8—2741no—
353D8—2741no—
354D8—1371no—
355D8C22⋊D13711yescharacteristic, complemented
356M16—21no—
357M16—2741no—
358M16—2741no—
359M16—2741no—
360M16—1371no—
361M16—1371no—
362M16—1371no—
363M16C22×D13711yescharacteristic
364C2×C8—21no—
365C2×C8—2741no—
366C2×C8—2741no—
367C2×C8—2741no—
368C2×C8—1371no—
369C2×C8C22×D13711yescharacteristic
370C4⋊C4—5481no—
371C42—1371no—
372C23—21no—
373C23—41no—
374C23—2741no—
375C23—5481no—
376Q8—21no—
377Q8—21no—
378Q8—41no—
379Q8—2741no—
380Q8—2741no—
381Q8—5481no—
382Q8—1371no—
383Q8D4×D13711yescharacteristic
384Q8C2×C22⋊D13711yescharacteristic
385Q8C2×C22⋊D13711yescharacteristic
386D4—21no—
387D4—21no—
388D4—21no—
389D4—21no—
390D4—21no—
391D4—41no—
392D4—41no—
393D4—41no—
394D4—2741no—
395D4—2741no—
396D4—2741no—
397D4—5481no—
398D4—5481no—
399D4—5481no—
400D4—10962no—
401D4—1371no—
402D4D4×D13711yescharacteristic
403D4C2×C22⋊D13711yescharacteristic
404D4C2×C22⋊D13711yescharacteristic
405C2×C4—21no—
406C2×C4—21no—
407C2×C4—21no—
408C2×C4—21no—
409C2×C4—41no—
410C2×C4—41no—
411C2×C4—41no—
412C2×C4—2741no—
413C2×C4—2741no—
414C2×C4—2741no—
415C2×C4—5481no—
416C2×C4—5481no—
417C2×C4—5481no—
418C2×C4C23×D13711yescharacteristic
419C8—21no—
420C8—2741no—
421C8—2741no—
422C8—2741no—
423C8—1371no—
424C8—1371no—
425C8C2×C22⋊D13711yescharacteristic
426C8C2×C22⋊D13711yescharacteristic
427C22—21no—
428C22—21no—
429C22—21no—
430C22—41no—
431C22—81no—
432C22—161no—
433C22—2741no—
434C22—5481no—
435C22—10961no—
436C22—21921no—
437C22#1798413311yescharacteristic
438C4—21no—
439C4—21no—
440C4—21no—
441C4—41no—
442C4—2742no—
443C4—2741no—
444C4—5481no—
445C4C2×D4×D13711yescharacteristic
446C4C22×C22⋊D13711yescharacteristic
447C2—21no—
448C2—41no—
449C2—41no—
450C2—41no—
451C2—81no—
452C2—5481no—
453C2—10961no—
454C2D4×C22⋊D13711yescharacteristic
subgroupG/H
centerC2D4×C22⋊D137
commutator, cosocleC2×C4×C137C24
FrattiniC2×C4C23×D137
Fitting#577×C137C2
socleC2×C137D42
hypercenter#577D137
hypocenterC137#3291
LCSG ▷ C2×C4×C137 ▷ C2×C137 ▷ C137
UCSC1 ◁ C2 ◁ C2×C4 ◁ #577

Complements

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

NH ≅ G/Ntype
1. #22578214C2semidirect
2. #22578211C2semidirect
3. #22578210C2semidirect
4. #22577774C2semidirect
5. #22577668C2semidirect
6. #22577666C2semidirect
7. #22577664C2semidirect
8. #22577663C2semidirect
9. #22577586C2semidirect
10. #22577584C2semidirect
11. #22577571C2semidirect
12. #22577569C2semidirect
13. #22577520C2semidirect
14. #22577518C2semidirect
15. #577×C137C2semidirect
17. #17984134C22semidirect
20. #17984119C22semidirect
22. #17984118C22semidirect
23. #17984118C22semidirect
34. C2×#14555499C22semidirect
36. C2×#14555498C22semidirect
38. C2×#14555497C22semidirect
40. #17984095C22semidirect
41. #17984095C22semidirect
42. #17984094C22semidirect
43. #17984086C22semidirect
45. #17984024C22semidirect
46. #17984023C22semidirect
47. #17984015C22semidirect
48. #17984015C22semidirect
49. #17984015C22semidirect
50. #17984014C22semidirect
51. #17984007C22semidirect
52. #17984006C22semidirect
53. #17984005C22semidirect
54. #17984004C22semidirect
55. C8×Dic137C22semidirect
57. H(5,𝔽2)×C137C22semidirect
59. #137×C137C22semidirect
65. C2×SD16×C137C22semidirect
66. #131×C137C22semidirect
84. #14555500D4semidirect
92. #14555498D4semidirect
98. #14555497D4semidirect
102. #14555495D4semidirect
131. Q16×C137D4semidirect
136. SD16×C137D4semidirect
137. SD16×C137D4semidirect
139. D8×C137D4semidirect
227. C137#3291semidirect
232. #577D137semidirect
247. H(5,𝔽2)C2×D137semidirect
255. #137C2×D137semidirect
277. C2×SD16C2×D137semidirect
281. #131C2×D137semidirect
329. Q16C22⋊D137semidirect
348. SD16C22⋊D137semidirect
349. SD16C22⋊D137semidirect
355. D8C22⋊D137semidirect

Maximal subgroups

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

subgroupindex
#225782142
#225782112
#225782102
#225777742
#225776682
#225776662
#225776642
#225776632
#225775862
#225775842
#225775712
#225775692
#225775202
#225775182
#577×C1372
#3291137