Finite Group Databasebeta

← #36368348

Subgroups — #36368348

Order 22144.

1412 conjugacy classes of subgroups.

The subgroup lattice has 504 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#26012328C211yesmaximal, characteristic, complemented
22−1+4×D173C211yesmaximal, characteristic, complemented
3#26012307C213yesmaximal, complemented
4#26012306C211yesmaximal, characteristic, complemented
5#26012304C213yesmaximal, complemented
6#26012303C211yesmaximal, characteristic, complemented
7#26012300C213yesmaximal, complemented
8#26012299C211yesmaximal, characteristic, complemented
9#26012298C213yesmaximal, complemented
10#26012290C213yesmaximal, complemented
11#26012288C213yesmaximal, complemented
12C2×#19149711C213yesmaximal, complemented
13#137×D173C213yesmaximal, complemented
14#26012256C211yesmaximal, characteristic, complemented
15#578×C173C211yesmaximal, characteristic, complemented
16#19149727—21no—
17#19149727—21no—
18#19149727C2213yescomplemented
19#19149727C2213yescomplemented
20#19149726—23no—
21#19149726C2211yescharacteristic, complemented
22#19149726C2213yescomplemented
23#19149725—23no—
24#19149725C2211yescharacteristic, complemented
25#19149725C2213yescomplemented
26C2×#14583827—21no—
27C2×#14583827C2213yes—
28#41×D173—23no—
29#41×D173—23no—
30#41×D173C2213yes—
31#41×D173C2211yescharacteristic
32#41×D173C2213yescomplemented
33C2×Q8×D173—21no—
34C2×Q8×D173C2213yescomplemented
35#19149712C2213yescomplemented
36#19149711C2216yescomplemented
37#19149711C2213yescomplemented
38#19149711C2216yescomplemented
39#19149711C2216yescomplemented
40#19149711C2213yescomplemented
41#19149710C2213yescomplemented
42#19149710C2213yescomplemented
43#19149709C2213yescomplemented
44#19149708C2213yescomplemented
45#19149708C2213yescomplemented
46#19149707C2213yescomplemented
47#19149706C2213yescomplemented
48#19149706C2213yescomplemented
49#19149704C2213yescomplemented
50#19149703C2211yescharacteristic, complemented
51#19149703C2213yescomplemented
52#19149702C2216yescomplemented
53#19149701C2213yescomplemented
54#19149700C2216yescomplemented
55#19149699C2213yescomplemented
56#19149699C2213yescomplemented
57C2×#14583816C2213yescomplemented
58Q16×D173C2216yescomplemented
59C2×#14583815C2213yescomplemented
60C2×#14583813C2213yescomplemented
61SD16×D173C2216yescomplemented
62#19149687C2213yescomplemented
63#19149686C2211yescharacteristic, complemented
64#19149685C2213yescomplemented
65C2×#14583809C2213yescomplemented
66M16×D173C2213yescomplemented
672−1+4×C173C2211yescharacteristic, complemented
682−1+4×C173C2211yescharacteristic, complemented
69#137×C173C2213yescomplemented
70#137×C173C2213yescomplemented
71#135×C173C2213yescomplemented
72C2×Q16×C173C2213yescomplemented
73#131×C173C2211yescharacteristic, complemented
74#14583827—23no—
75#14583827—23no—
76#14583827—23no—
77#14583827—23no—
78#14583827—21no—
79#14583827D411yescharacteristic
80#14583827C2316yes—
81#14583827C2313yes—
82#14583827C2313yes—
83#14583826—23no—
84#14583826—23no—
85#14583826—26no—
86#14583826—26no—
87#14583826—23no—
88#14583826—23no—
89#14583826C2313yes—
90#14583826C2313yes—
91#14583826C2313yes—
92#14583825—26no—
93#14583825—23no—
94#14583825—23no—
95#14583825—23no—
96#14583825D413yes—
97#14583825C2313yes—
98#14583825C2311yescharacteristic
99C2×Dic346—23no—
100C2×Dic346—23no—
101C2×Dic346C2313yes—
102Q8×D173—23no—
103Q8×D173—23no—
104Q8×D173—23no—
105Q8×D173—23no—
106Q8×D173—21no—
107Q8×D173D411yescharacteristic
108Q8×D173C2316yes—
109Q8×D173C2313yes—
110Q8×D173C2313yes—
111C2×D692—23no—
112C2×D692—23no—
113C2×D692C2313yes—
114D4×D173—23no—
115D4×D173—23no—
116D4×D173—26no—
117D4×D173—23no—
118D4×D173D413yes—
119D4×D173C2313yes—
120D4×D173C2311yescharacteristic
121C2×C4×D173—23no—
122C2×C4×D173—23no—
123C2×C4×D173—23no—
124C2×C4×D173—23no—
125C2×C4×D173C2313yes—
126Dic692C2313yes—
127#14583816C2313yes—
128#14583816C2316yes—
129#14583816C2313yes—
130#14583815C2316yes—
131#14583815C2313yes—
132#14583815C2311yescharacteristic
133#14583814C2313yes—
134#14583814C2311yescharacteristic
135#14583813C2316yes—
136#14583813C2313yes—
137#14583813C2313yes—
138D1384C2313yes—
139#14583811C2311yescharacteristic
140#14583811C2313yes—
141#14583810C2313yes—
142#14583809C2313yes—
143#14583809C2311yescharacteristic
144#14583809C2313yes—
145#14583809C2313yes—
146C2×#13063436C2313yes—
147C8×D173C2316yes—
148#41×C173—23no—
149#41×C173—23no—
150#41×C173C2313yes—
151#41×C173C2313yes—
152#41×C173C2311yescharacteristic
153C2×Q8×C173—21no—
154C2×Q8×C173—21no—
155C2×Q8×C173C2313yes—
156C2×Q8×C173C2313yes—
157Q16×C173C2313yes—
158Q16×C173C2316yes—
159SD16×C173C2313yes—
160SD16×C173C2313yes—
161D8×C173C2311yescharacteristic
162M16×C173C2313yes—
163C2×C8×C173C2313yes—
164C22×D173—23no—
165C22×D173—21no—
166C22×D173—23no—
167C22×D173C2×D413yes—
168Dic346—23no—
169Dic346—23no—
170Dic346—26no—
171Dic346—23no—
172Dic346C2×D413yes—
173Dic346C2413yes—
174Dic346C2411yescharacteristic
175D692—23no—
176D692—26no—
177D692—23no—
178D692—23no—
179D692C2×D413yes—
180D692C2411yescharacteristic
181D692C2413yes—
182C22⋊D173—23no—
183C22⋊D173—23no—
184C22⋊D173—21no—
185C22⋊D173—23no—
186C22⋊D173—23no—
187C22⋊D173—21no—
188C22⋊D173C2×D416yes—
189C2×Dic173—23no—
190C2×Dic173—21no—
191C2×Dic173—23no—
192C2×Dic173C2×D413yes—
193C4×D173—23no—
194C4×D173—23no—
195C4×D173—26no—
196C4×D173—23no—
197C4×D173—23no—
198C4×D173—23no—
199C4×D173—26no—
200C4×D173—23no—
201C4×D173C2×D413yes—
202C4×D173C2×D413yes—
203C4×D173C2411yescharacteristic
204C4×D173C2413yes—
205#13063436C2411yescharacteristic
206#13063436C2413yes—
207Q8×C173—23no—
208Q8×C173—23no—
209Q8×C173C2×D411yescharacteristic
210Q8×C173C2413yes—
211Q8×C173C2413yes—
212D4×C173—23no—
213D4×C173—23no—
214D4×C173C2×D413yes—
215D4×C173C2411yescharacteristic
216D4×C173C2411yescharacteristic
217C2×C4×C173—23no—
218C2×C4×C173—23no—
219C2×C4×C173—23no—
220C2×C4×C173—23no—
221C2×C4×C173C2413yes—
222C8×C173C2411yescharacteristic
223C8×C173C2413yes—
224C22×C173—21no—
225C22×C173—21no—
226C22×C173C22×D413yes—
227C2×D173—21no—
228C2×D173—23no—
229C2×D173—43no—
230C2×D173—42no—
231C2×D173—83no—
232C2×D173—83no—
233C2×D173C22×D411yescharacteristic
234C2×D173C22×D413yes—
235C4×C173—23no—
236C4×C173—23no—
237C4×C173C22×D413yes—
238C4×C173C2511yescharacteristic
239Dic173—21no—
240Dic173—23no—
241Dic173C22×D411yescharacteristic
242Dic173C22×D413yes—
243C2×C173—23no—
244C2×C173—41no—
245C2×C173—41no—
246C2×C173C23×D411yescharacteristic
247D173—21no—
248D173—23no—
249D173—41no—
250D173—43no—
251C173#358611yescharacteristic, complemented, Sylow
252#3586—1731nomaximal, Sylow
253#585—1731no—
254C2×2−1+4—1731no—
255#578—1731no—
256#578—1733no—
257#578—1733no—
258#578D17311yescharacteristic, complemented
259#577—1731no—
260#577—1733no—
261#577—1733no—
262#577—1731no—
263#575—1733no—
264#575—1733no—
265C2×#137—1733no—
266C2×#137—1733no—
267#568—1731no—
2682−1+4—3461no—
2692−1+4—3461no—
2702−1+4—3463no—
2712−1+4—1733no—
2722−1+4—1731no—
2732−1+4—1733no—
2742−1+4—1733no—
2752−1+4C2×D17311yescharacteristic, complemented
2762−1+4C2×D17311yescharacteristic, complemented
277H(5,𝔽2)—3463no—
278H(5,𝔽2)—1733no—
279H(5,𝔽2)—1731no—
280C2×#41—3461no—
281C2×#41—3463no—
282C2×#41—3463no—
283C2×#41—1733no—
284C2×#41—1733no—
285C2×#41—1733no—
286C2×#41—1731no—
287C22×Q8—3461no—
288C22×Q8—1733no—
289#137—1733no—
290#137—1733no—
291#137—1733no—
292#137—1736no—
293#137—1733no—
294#137—1736no—
295#137—1736no—
296#137—1733no—
297#137—1733no—
298#137—1733no—
299#137—1733no—
300#137C2×D17313yescomplemented
301#137C2×D17313yescomplemented
302AGL(1,ℤ/8)—1733no—
303AGL(1,ℤ/8)—1733no—
304AGL(1,ℤ/8)—1733no—
305AGL(1,ℤ/8)—1731no—
306AGL(1,ℤ/8)—1733no—
307AGL(1,ℤ/8)—1733no—
308#135—1733no—
309#135—1733no—
310#135—1736no—
311#135—1736no—
312#135—1733no—
313#135C2×D17313yescomplemented
314C2×Q16—1733no—
315C2×Q16—1736no—
316C2×Q16C2×D17313yescomplemented
317C2×SD16—1733no—
318C2×SD16—1736no—
319C2×SD16—1733no—
320#131—1731no—
321#131—1733no—
322#131—1733no—
323#131C2×D17311yescharacteristic, complemented
324C2×M16—1733no—
325C2×M16—1733no—
326#41—23no—
327#41—23no—
328#41—3463no—
329#41—3466no—
330#41—3466no—
331#41—3463no—
332#41—3463no—
333#41—3463no—
334#41—3466no—
335#41—3461no—
336#41—3463no—
337#41—3463no—
338#41—3463no—
339#41—3463no—
340#41—3463no—
341#41—3463no—
342#41—3463no—
343#41—1733no—
344#41—1736no—
345#41—1733no—
346#41—1733no—
347#41—1731no—
348#41—1733no—
349#41—1733no—
350#41—1731no—
351#41—1733no—
352#41—1733no—
353#41C22×D17313yes—
354#41C22×D17313yes—
355#41C22×D17311yescharacteristic
356C2×Q8—21no—
357C2×Q8—21no—
358C2×Q8—3463no—
359C2×Q8—3463no—
360C2×Q8—3463no—
361C2×Q8—3463no—
362C2×Q8—3461no—
363C2×Q8—3463no—
364C2×Q8—3463no—
365C2×Q8—1736no—
366C2×Q8—1733no—
367C2×Q8—1733no—
368C2×Q8—1733no—
369C2×Q8—1731no—
370C2×Q8C22×D17313yes—
371C2×Q8C22×D17313yes—
372C2×D4—3463no—
373C2×D4—3463no—
374C2×D4—3463no—
375C2×D4—3463no—
376C2×D4—3466no—
377C2×D4—3463no—
378C2×D4—1733no—
379C2×D4—1731no—
380C2×D4—1733no—
381C2×D4—1733no—
382C22×C4—3463no—
383C22×C4—3463no—
384C22×C4—3463no—
385C22×C4—3463no—
386C22×C4—1733no—
387Q16—1736no—
388Q16—1733no—
389Q16—1733no—
390Q16—1733no—
391Q16C22×D17316yes—
392Q16C22×D17313yes—
393SD16—1736no—
394SD16—1733no—
395SD16—1733no—
396SD16—1733no—
397SD16—1731no—
398SD16—1736no—
399SD16—1733no—
400SD16—1731no—
401SD16C22×D17313yes—
402SD16C22×D17313yes—
403D8—1731no—
404D8—1733no—
405D8—1733no—
406D8C22×D17311yescharacteristic
407M16—1733no—
408M16—1733no—
409M16—1731no—
410M16—1733no—
411M16—1733no—
412M16C22×D17313yes—
413C2×C8—1733no—
414C2×C8—1736no—
415C2×C8C22×D17313yes—
416C23—3463no—
417C23—3461no—
418C23—3463no—
419C23—1733no—
420Q8—23no—
421Q8—23no—
422Q8—3463no—
423Q8—3466no—
424Q8—3463no—
425Q8—3463no—
426Q8—1733no—
427Q8—1731no—
428Q8—1733no—
429Q8D4×D17311yescharacteristic
430Q8C23×D17313yes—
431Q8C23×D17313yes—
432D4—23no—
433D4—23no—
434D4—3463no—
435D4—3466no—
436D4—3463no—
437D4—3461no—
438D4—3463no—
439D4—3463no—
440D4—3463no—
441D4—3463no—
442D4—3461no—
443D4—3463no—
444D4—1736no—
445D4—1733no—
446D4—1731no—
447D4—1733no—
448D4D4×D17313yes—
449D4C23×D17311yescharacteristic
450D4C23×D17311yescharacteristic
451C2×C4—23no—
452C2×C4—23no—
453C2×C4—23no—
454C2×C4—23no—
455C2×C4—3463no—
456C2×C4—3463no—
457C2×C4—3466no—
458C2×C4—3463no—
459C2×C4—3466no—
460C2×C4—3463no—
461C2×C4—3463no—
462C2×C4—3461no—
463C2×C4—3463no—
464C2×C4—3463no—
465C2×C4—3463no—
466C2×C4—1733no—
467C2×C4—1733no—
468C2×C4—1733no—
469C2×C4—1733no—
470C2×C4—1731no—
471C2×C4C23×D17313yes—
472C8—1731no—
473C8—1733no—
474C8C23×D17313yes—
475C8C23×D17311yescharacteristic
476C22—21no—
477C22—21no—
478C22—3461no—
479C22—3463no—
480C22—13843no—
481C22—13843no—
482C22—1731no—
483C22—1733no—
484C22—6923no—
485C22—6922no—
486C22C2×D4×D17313yes—
487C4—23no—
488C4—23no—
489C4—3461no—
490C4—3463no—
491C4—1731no—
492C4—1733no—
493C4C2×D4×D17313yes—
494C4C24×D17311yescharacteristic
495C2—23no—
496C2—41no—
497C2—41no—
498C2—3463no—
499C2—3461no—
500C2—6923no—
501C2—6921no—
502C2C22×D4×D17311yescharacteristic
subgroupG/H
centerC2C22×D4×D173
commutator, cosocleC4×C173C25
FrattiniC4C24×D173
Fitting#578×C173C2
socleC2×C173C23×D4
hypercenter#578D173
hypocenterC173#3586
LCSG ▷ C4×C173 ▷ C2×C173 ▷ C173
UCSC1 ◁ C2 ◁ #41 ◁ #578

Complements

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

NH ≅ G/Ntype
1. #26012328C2semidirect
2. 2−1+4×D173C2semidirect
3. #26012307C2semidirect
4. #26012306C2semidirect
5. #26012304C2semidirect
6. #26012303C2semidirect
7. #26012300C2semidirect
8. #26012299C2semidirect
9. #26012298C2semidirect
10. #26012290C2semidirect
11. #26012288C2semidirect
12. C2×#19149711C2semidirect
13. #137×D173C2semidirect
14. #26012256C2semidirect
15. #578×C173C2semidirect
18. #19149727C22semidirect
19. #19149727C22semidirect
21. #19149726C22semidirect
22. #19149726C22semidirect
24. #19149725C22semidirect
25. #19149725C22semidirect
32. #41×D173C22semidirect
34. C2×Q8×D173C22semidirect
35. #19149712C22semidirect
36. #19149711C22semidirect
37. #19149711C22semidirect
38. #19149711C22semidirect
39. #19149711C22semidirect
40. #19149711C22semidirect
41. #19149710C22semidirect
42. #19149710C22semidirect
43. #19149709C22semidirect
44. #19149708C22semidirect
45. #19149708C22semidirect
46. #19149707C22semidirect
47. #19149706C22semidirect
48. #19149706C22semidirect
49. #19149704C22semidirect
50. #19149703C22semidirect
51. #19149703C22semidirect
52. #19149702C22semidirect
53. #19149701C22semidirect
54. #19149700C22semidirect
55. #19149699C22semidirect
56. #19149699C22semidirect
57. C2×#14583816C22semidirect
58. Q16×D173C22semidirect
59. C2×#14583815C22semidirect
60. C2×#14583813C22semidirect
61. SD16×D173C22semidirect
62. #19149687C22semidirect
63. #19149686C22semidirect
64. #19149685C22semidirect
65. C2×#14583809C22semidirect
66. M16×D173C22semidirect
67. 2−1+4×C173C22semidirect
68. 2−1+4×C173C22semidirect
69. #137×C173C22semidirect
70. #137×C173C22semidirect
71. #135×C173C22semidirect
72. C2×Q16×C173C22semidirect
73. #131×C173C22semidirect
251. C173#3586semidirect
258. #578D173semidirect
275. 2−1+4C2×D173semidirect
276. 2−1+4C2×D173semidirect
300. #137C2×D173semidirect
301. #137C2×D173semidirect
313. #135C2×D173semidirect
316. C2×Q16C2×D173semidirect
323. #131C2×D173semidirect

Maximal subgroups

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

subgroupindex
#260123282
2−1+4×D1732
#260123072
#260123062
#260123042
#260123032
#260123002
#260122992
#260122982
#260122902
#260122882
C2×#191497112
#137×D1732
#260122562
#578×C1732
#3586173