Finite Group Databasebeta

← #577×D173 #36368317

Subgroups — #577×D173 #36368317

Order 22144.

1460 conjugacy classes of subgroups.

The subgroup lattice has 510 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
12−1+4×D173C211yesmaximal, characteristic, complemented
2H(5,𝔽2)×D173C211yesmaximal, characteristic, complemented
3#26012303C211yesmaximal, characteristic, complemented
4#26012302C211yesmaximal, characteristic, complemented
5#26012301C213yesmaximal, complemented
6#26012300C213yesmaximal, complemented
7#26012299C211yesmaximal, characteristic, complemented
8#26012298C213yesmaximal, complemented
9#26012297C213yesmaximal, complemented
10#137×D173C213yesmaximal, complemented
11AGL(1,ℤ/8)×D173C213yesmaximal, complemented
12#135×D173C213yesmaximal, complemented
13C2×SD16×D173C213yesmaximal, complemented
14#131×D173C211yesmaximal, characteristic, complemented
15#577×C173C211yesmaximal, characteristic, complemented
16#19149727—21no—
17#19149727C2213yescomplemented
18#19149726—23no—
19#19149726C2211yescharacteristic, complemented
20#19149726C2213yescomplemented
21#19149725—21no—
22#19149725C2211yescharacteristic, complemented
23#19149725C2213yescomplemented
24#19149724—23no—
25#19149724C2213yescomplemented
26#41×D173—21no—
27#41×D173—23no—
28#41×D173C2213yes—
29#41×D173C2211yescharacteristic
30#41×D173C2213yescomplemented
31C2×Q8×D173—21no—
32C2×Q8×D173C2213yescomplemented
33C2×D4×D173—23no—
34C2×D4×D173C2213yes—
35#19149712C2213yescomplemented
36#19149711C2216yescomplemented
37#19149710C2213yescomplemented
38#19149709C2213yescomplemented
39#19149708C2216yescomplemented
40#19149707C2213yescomplemented
41#19149706C2216yescomplemented
42#19149705C2213yescomplemented
43#19149704C2213yescomplemented
44#19149703C2216yescomplemented
45#19149702C2213yescomplemented
46#19149701C2213yescomplemented
47#19149700C2213yescomplemented
48#19149700C2213yescomplemented
49#19149699C2213yescomplemented
50#19149699C2213yescomplemented
51#19149698C2213yescomplemented
52Q16×D173C2213yescomplemented
53C2×#14583815C2213yescomplemented
54C2×#14583814C2213yescomplemented
55C2×#14583813C2213yescomplemented
56SD16×D173C2213yescomplemented
57SD16×D173C2216yescomplemented
58SD16×D173C2211yescharacteristic, complemented
59SD16×D173C2216yescomplemented
60SD16×D173C2216yescomplemented
61D8×D173C2213yescomplemented
62#19149687C2213yescomplemented
63#19149686C2211yescharacteristic, complemented
64#19149685C2213yescomplemented
65M16×D173C2213yescomplemented
66C2×C8×D173C2213yescomplemented
672−1+4×C173C2211yescharacteristic, complemented
68H(5,𝔽2)×C173C2211yescharacteristic, complemented
69#137×C173C2213yescomplemented
70AGL(1,ℤ/8)×C173C2213yescomplemented
71#135×C173C2213yescomplemented
72C2×SD16×C173C2213yescomplemented
73#131×C173C2211yescharacteristic, complemented
74C23×D173—23no—
75#14583827—23no—
76#14583827—21no—
77#14583827D411yescharacteristic
78#14583827C2313yes—
79#14583827C2313yescomplemented
80#14583826—23no—
81#14583826—23no—
82#14583826—26no—
83#14583826—23no—
84#14583826—23no—
85#14583826C2313yes—
86#14583826C2313yes—
87#14583826C2313yescomplemented
88#14583825—23no—
89#14583825—26no—
90#14583825—23no—
91#14583825—23no—
92#14583825—23no—
93#14583825D413yes—
94#14583825C2313yes—
95#14583825C2313yescomplemented
96C2×Dic346—23no—
97C2×Dic346—23no—
98C2×Dic346C2313yescomplemented
99Q8×D173—23no—
100Q8×D173—23no—
101Q8×D173—23no—
102Q8×D173—21no—
103Q8×D173D411yescharacteristic
104Q8×D173C2313yes—
105Q8×D173C2311yescharacteristic
106Q8×D173C2316yescomplemented
107C2×D692—23no—
108C2×D692C2313yes—
109C2×C22⋊D173—26no—
110C2×C22⋊D173—23no—
111D4×D173—23no—
112D4×D173—23no—
113D4×D173—23no—
114D4×D173—26no—
115D4×D173—23no—
116D4×D173—23no—
117D4×D173—23no—
118D4×D173D413yes—
119D4×D173C2311yescharacteristic
120D4×D173C2313yes—
121D4×D173C2316yes—
122C2×C4×D173—23no—
123C2×C4×D173—23no—
124C2×C4×D173—23no—
125C2×C4×D173C2313yes—
126Dic692C2313yescomplemented
127#14583816C2313yescomplemented
128#14583816C2313yescomplemented
129#14583815C2313yescomplemented
130#14583815C2316yescomplemented
131#14583815C2311yescharacteristic, complemented
132#14583814C2313yescomplemented
133#14583814C2316yescomplemented
134#14583814C2311yescharacteristic, complemented
135#14583813C2313yescomplemented
136#14583813C2316yescomplemented
137#14583813C2311yescharacteristic, complemented
138D1384C2313yescomplemented
139#14583811C2313yescomplemented
140#14583811C2313yescomplemented
141#14583810C2313yescomplemented
142#14583809C2316yescomplemented
143C2×#13063436C2313yescomplemented
144C8×D173C2311yescharacteristic
145C8×D173C2313yes—
146C8×D173C2313yescomplemented
147C8×D173C2313yescomplemented
148#41×C173—21no—
149#41×C173—23no—
150#41×C173C2313yes—
151#41×C173C2311yescharacteristic
152#41×C173C2313yescomplemented
153C2×Q8×C173—21no—
154C2×Q8×C173C2313yescomplemented
155C2×D4×C173—23no—
156C2×D4×C173C2313yes—
157Q16×C173C2313yescomplemented
158SD16×C173C2313yescomplemented
159SD16×C173C2316yescomplemented
160SD16×C173C2311yescharacteristic, complemented
161D8×C173C2313yescomplemented
162M16×C173C2313yescomplemented
163C2×C8×C173C2313yescomplemented
164C22×D173—23no—
165C22×D173—21no—
166C22×D173—23no—
167C22×D173—26no—
168C22×D173—86no—
169C22×D173C2×D413yes—
170Dic346—23no—
171Dic346—23no—
172Dic346—26no—
173Dic346—21no—
174Dic346C2×D413yes—
175Dic346C2413yes—
176Dic346C2411yescharacteristic
177D692—23no—
178D692—23no—
179D692—23no—
180D692C2×D413yes—
181D692C2411yescharacteristic
182D692C2413yes—
183C22⋊D173—23no—
184C22⋊D173—23no—
185C22⋊D173—23no—
186C22⋊D173—23no—
187C22⋊D173—26no—
188C22⋊D173—23no—
189C22⋊D173—21no—
190C22⋊D173C2×D416yes—
191C2×Dic173—23no—
192C2×Dic173—23no—
193C2×Dic173—23no—
194C2×Dic173C2×D413yes—
195C4×D173—21no—
196C4×D173—23no—
197C4×D173—26no—
198C4×D173—23no—
199C4×D173—23no—
200C4×D173—23no—
201C4×D173—23no—
202C4×D173C2×D413yes—
203C4×D173C2×D413yes—
204C4×D173C2413yes—
205C4×D173C2411yescharacteristic
206#13063436C2411yescharacteristic
207#13063436C2413yes—
208C23×C173—23no—
209Q8×C173—23no—
210Q8×C173C2×D411yescharacteristic
211Q8×C173C2413yes—
212Q8×C173C2411yescharacteristic
213D4×C173—23no—
214D4×C173—23no—
215D4×C173—23no—
216D4×C173C2×D413yes—
217D4×C173C2411yescharacteristic
218D4×C173C2413yes—
219C2×C4×C173—23no—
220C2×C4×C173—23no—
221C2×C4×C173—23no—
222C2×C4×C173C2413yes—
223C8×C173C2413yes—
224C8×C173C2411yescharacteristic
225C22×C173—21no—
226C22×C173—23no—
227C22×C173—83no—
228C22×C173C22×D413yes—
229C2×D173—26no—
230C2×D173—21no—
231C2×D173—23no—
232C2×D173—42no—
233C2×D173—46no—
234C2×D173—86no—
235C2×D173—83no—
236C2×D173C22×D413yes—
237C2×D173C22×D411yescharacteristic
238C4×C173—21no—
239C4×C173—23no—
240C4×C173C22×D413yes—
241C4×C173C2511yescharacteristic
242Dic173—21no—
243Dic173—23no—
244Dic173C22×D413yes—
245Dic173C22×D411yescharacteristic
246C2×C173—23no—
247C2×C173—43no—
248C2×C173—41no—
249C2×C173C23×D411yescharacteristic
250D173—23no—
251D173—41no—
252D173—43no—
253D173#57712yesdirect product factor
254C173C2×#57711yescharacteristic, complemented, Sylow
255C2×#577—1731nomaximal, Sylow
256C2×2−1+4—1731no—
257C2×H(5,𝔽2)—1731no—
258#577—1731no—
259#577—1731no—
260#577—1733no—
261#577—1733no—
262#577—1733no—
263#577—1733no—
264#577—1731no—
265#577D17311yescharacteristic, direct product factor
266C2×#137—1733no—
267C2×AGL(1,ℤ/8)—1733no—
268C2×#135—1733no—
269C22×SD16—1733no—
270C2×#131—1731no—
2712−1+4—3461no—
2722−1+4—3463no—
2732−1+4—1731no—
2742−1+4—1733no—
2752−1+4—1733no—
2762−1+4C2×D17311yescharacteristic, complemented
277H(5,𝔽2)—3463no—
278H(5,𝔽2)—3461no—
279H(5,𝔽2)—1733no—
280H(5,𝔽2)—1731no—
281H(5,𝔽2)—1733no—
282H(5,𝔽2)C2×D17311yescharacteristic, complemented
283C2×#41—3461no—
284C2×#41—3463no—
285C2×#41—1733no—
286C2×#41—1733no—
287C2×#41—1731no—
288C22×Q8—3461no—
289C22×Q8—1733no—
290C22×D4—3463no—
291C22×D4—1733no—
292#137—1733no—
293#137—1733no—
294#137—1736no—
295#137—1736no—
296#137—1733no—
297#137C2×D17313yescomplemented
298AGL(1,ℤ/8)—1733no—
299AGL(1,ℤ/8)—1733no—
300AGL(1,ℤ/8)—1736no—
301AGL(1,ℤ/8)—1736no—
302AGL(1,ℤ/8)—1733no—
303AGL(1,ℤ/8)C2×D17313yescomplemented
304#135—1733no—
305#135—1733no—
306#135—1733no—
307#135—1733no—
308#135—1733no—
309#135—1733no—
310#135—1733no—
311#135C2×D17313yescomplemented
312C2×Q16—1733no—
313C2×SD16—1733no—
314C2×SD16—1733no—
315C2×SD16—1733no—
316C2×SD16—1736no—
317C2×SD16—1736no—
318C2×SD16—1736no—
319C2×SD16—1731no—
320C2×SD16—1733no—
321C2×SD16C2×D17313yescomplemented
322C2×D8—1733no—
323#131—1731no—
324#131—1733no—
325#131—1733no—
326#131C2×D17311yescharacteristic, complemented
327C2×M16—1733no—
328C22×C8—1733no—
329C24—3463no—
330#41—21no—
331#41—23no—
332#41—3463no—
333#41—3466no—
334#41—3463no—
335#41—3463no—
336#41—3463no—
337#41—3463no—
338#41—3466no—
339#41—3461no—
340#41—3463no—
341#41—3463no—
342#41—3463no—
343#41—3463no—
344#41—1733no—
345#41—1733no—
346#41—1733no—
347#41—1733no—
348#41—1733no—
349#41—1733no—
350#41—1731no—
351#41—1733no—
352#41—1733no—
353#41C22×D17313yes—
354#41C22×D17311yescharacteristic
355#41C22×D17313yescomplemented
356C2×Q8—21no—
357C2×Q8—3463no—
358C2×Q8—3463no—
359C2×Q8—3463no—
360C2×Q8—3461no—
361C2×Q8—3463no—
362C2×Q8—3463no—
363C2×Q8—1733no—
364C2×Q8—1731no—
365C2×Q8—1736no—
366C2×Q8—1733no—
367C2×Q8—1731no—
368C2×Q8C22×D17313yescomplemented
369C2×D4—23no—
370C2×D4—3463no—
371C2×D4—3463no—
372C2×D4—3463no—
373C2×D4—3466no—
374C2×D4—3463no—
375C2×D4—3466no—
376C2×D4—3463no—
377C2×D4—3463no—
378C2×D4—3463no—
379C2×D4—3463no—
380C2×D4—1731no—
381C2×D4—1733no—
382C2×D4—1736no—
383C2×D4—1733no—
384C2×D4—1733no—
385C2×D4C22×D17313yes—
386C22×C4—3463no—
387C22×C4—3463no—
388C22×C4—3463no—
389C22×C4—1733no—
390Q16—1733no—
391Q16—1733no—
392Q16—1733no—
393Q16C22×D17313yescomplemented
394SD16—1733no—
395SD16—1736no—
396SD16—1731no—
397SD16—1733no—
398SD16—1736no—
399SD16—1731no—
400SD16—1733no—
401SD16—1736no—
402SD16—1731no—
403SD16C22×D17313yescomplemented
404SD16C22×D17316yescomplemented
405SD16C22×D17311yescharacteristic, complemented
406D8—1733no—
407D8—1733no—
408D8—1733no—
409D8C22×D17313yescomplemented
410M16—1733no—
411M16—1736no—
412M16C22×D17313yescomplemented
413C2×C8—1733no—
414C2×C8—1731no—
415C2×C8—1733no—
416C2×C8—1733no—
417C2×C8—1733no—
418C2×C8C22×D17313yescomplemented
419C23—23no—
420C23—3466no—
421C23—3461no—
422C23—3463no—
423C23—3463no—
424C23—13846no—
425C23—1733no—
426Q8—23no—
427Q8—3466no—
428Q8—3463no—
429Q8—3461no—
430Q8—3463no—
431Q8—1733no—
432Q8—1731no—
433Q8—1733no—
434Q8D4×D17311yescharacteristic
435Q8C23×D17313yes—
436Q8C23×D17311yescharacteristic
437D4—23no—
438D4—23no—
439D4—23no—
440D4—3463no—
441D4—3463no—
442D4—3463no—
443D4—3463no—
444D4—3463no—
445D4—3463no—
446D4—3466no—
447D4—3463no—
448D4—3461no—
449D4—3463no—
450D4—1736no—
451D4—1733no—
452D4—1731no—
453D4—1733no—
454D4D4×D17313yes—
455D4C23×D17311yescharacteristic
456D4C23×D17313yes—
457C2×C4—23no—
458C2×C4—23no—
459C2×C4—23no—
460C2×C4—3461no—
461C2×C4—3463no—
462C2×C4—3466no—
463C2×C4—3463no—
464C2×C4—3463no—
465C2×C4—3463no—
466C2×C4—3463no—
467C2×C4—3463no—
468C2×C4—3463no—
469C2×C4—3463no—
470C2×C4—1733no—
471C2×C4—1733no—
472C2×C4—1733no—
473C2×C4—1731no—
474C2×C4—1733no—
475C2×C4C23×D17313yes—
476C8—1731no—
477C8—1733no—
478C8C23×D17311yescharacteristic
479C8C23×D17313yes—
480C22—21no—
481C22—23no—
482C22—83no—
483C22—3466no—
484C22—3461no—
485C22—3463no—
486C22—13843no—
487C22—13846no—
488C22—1733no—
489C22—1731no—
490C22—6926no—
491C22—6922no—
492C22C2×D4×D17313yes—
493C4—21no—
494C4—23no—
495C4—3461no—
496C4—3463no—
497C4—1733no—
498C4—1731no—
499C4C2×D4×D17313yes—
500C4C24×D17311yescharacteristic
501C2—23no—
502C2—43no—
503C2—41no—
504C2—3463no—
505C2—1732no—
506C2—6923no—
507C2—6921no—
508C2C22×D4×D17311yescharacteristic
subgroupG/H
centerC2C22×D4×D173
commutator, cosocleC4×C173C25
FrattiniC4C24×D173
Fitting#577×C173C2
socleC2×C173C23×D4
hypercenter#577D173
hypocenterC173C2×#577
LCSG ▷ C4×C173 ▷ C2×C173 ▷ C173
UCSC1 ◁ C2 ◁ #41 ◁ #577

Complements

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

NH ≅ G/Ntype
1. 2−1+4×D173C2semidirect
2. H(5,𝔽2)×D173C2semidirect
3. #26012303C2semidirect
4. #26012302C2semidirect
5. #26012301C2semidirect
6. #26012300C2semidirect
7. #26012299C2semidirect
8. #26012298C2semidirect
9. #26012297C2semidirect
10. #137×D173C2semidirect
11. AGL(1,ℤ/8)×D173C2semidirect
12. #135×D173C2semidirect
13. C2×SD16×D173C2semidirect
14. #131×D173C2semidirect
15. #577×C173C2semidirect
17. #19149727C22semidirect
19. #19149726C22semidirect
20. #19149726C22semidirect
22. #19149725C22semidirect
23. #19149725C22semidirect
25. #19149724C22semidirect
30. #41×D173C22semidirect
32. C2×Q8×D173C22semidirect
35. #19149712C22semidirect
36. #19149711C22semidirect
37. #19149710C22semidirect
38. #19149709C22semidirect
39. #19149708C22semidirect
40. #19149707C22semidirect
41. #19149706C22semidirect
42. #19149705C22semidirect
43. #19149704C22semidirect
44. #19149703C22semidirect
45. #19149702C22semidirect
46. #19149701C22semidirect
47. #19149700C22semidirect
48. #19149700C22semidirect
49. #19149699C22semidirect
50. #19149699C22semidirect
51. #19149698C22semidirect
52. Q16×D173C22semidirect
53. C2×#14583815C22semidirect
54. C2×#14583814C22semidirect
55. C2×#14583813C22semidirect
56. SD16×D173C22semidirect
57. SD16×D173C22semidirect
58. SD16×D173C22semidirect
59. SD16×D173C22semidirect
60. SD16×D173C22semidirect
61. D8×D173C22semidirect
62. #19149687C22semidirect
63. #19149686C22semidirect
64. #19149685C22semidirect
65. M16×D173C22semidirect
66. C2×C8×D173C22semidirect
67. 2−1+4×C173C22semidirect
68. H(5,𝔽2)×C173C22semidirect
69. #137×C173C22semidirect
70. AGL(1,ℤ/8)×C173C22semidirect
71. #135×C173C22semidirect
72. C2×SD16×C173C22semidirect
73. #131×C173C22semidirect
79. #14583827C23semidirect
87. #14583826C23semidirect
95. #14583825C23semidirect
98. C2×Dic346C23semidirect
106. Q8×D173C23semidirect
126. Dic692C23semidirect
127. #14583816C23semidirect
128. #14583816C23semidirect
129. #14583815C23semidirect
130. #14583815C23semidirect
131. #14583815C23semidirect
132. #14583814C23semidirect
133. #14583814C23semidirect
134. #14583814C23semidirect
135. #14583813C23semidirect
136. #14583813C23semidirect
137. #14583813C23semidirect
138. D1384C23semidirect
139. #14583811C23semidirect
140. #14583811C23semidirect
141. #14583810C23semidirect
142. #14583809C23semidirect
143. C2×#13063436C23semidirect
146. C8×D173C23semidirect
147. C8×D173C23semidirect
152. #41×C173C23semidirect
154. C2×Q8×C173C23semidirect
157. Q16×C173C23semidirect
158. SD16×C173C23semidirect
159. SD16×C173C23semidirect
160. SD16×C173C23semidirect
161. D8×C173C23semidirect
162. M16×C173C23semidirect
163. C2×C8×C173C23semidirect
253. D173#577direct (N × H)
254. C173C2×#577semidirect
265. #577D173direct (N × H)
276. 2−1+4C2×D173semidirect
282. H(5,𝔽2)C2×D173semidirect
297. #137C2×D173semidirect
303. AGL(1,ℤ/8)C2×D173semidirect
311. #135C2×D173semidirect
321. C2×SD16C2×D173semidirect
326. #131C2×D173semidirect
355. #41C22×D173semidirect
368. C2×Q8C22×D173semidirect
393. Q16C22×D173semidirect
403. SD16C22×D173semidirect
404. SD16C22×D173semidirect
405. SD16C22×D173semidirect
409. D8C22×D173semidirect
412. M16C22×D173semidirect
418. C2×C8C22×D173semidirect

Maximal subgroups

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

subgroupindex
2−1+4×D1732
H(5,𝔽2)×D1732
#260123032
#260123022
#260123012
#260123002
#260122992
#260122982
#260122972
#137×D1732
AGL(1,ℤ/8)×D1732
#135×D1732
C2×SD16×D1732
#131×D1732
#577×C1732
C2×#577173