Finite Group Databasebeta

← #577×D97 #27156490

Subgroups — #577×D97 #27156490

Order 12416.

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×D97C211yesmaximal, characteristic, complemented
2H(5,𝔽2)×D97C211yesmaximal, characteristic, complemented
3#20261547C211yesmaximal, characteristic, complemented
4#20261546C211yesmaximal, characteristic, complemented
5#20261545C213yesmaximal, complemented
6#20261544C213yesmaximal, complemented
7#20261543C211yesmaximal, characteristic, complemented
8#20261542C213yesmaximal, complemented
9#20261541C213yesmaximal, complemented
10#137×D97C213yesmaximal, complemented
11AGL(1,ℤ/8)×D97C213yesmaximal, complemented
12#135×D97C213yesmaximal, complemented
13C2×SD16×D97C213yesmaximal, complemented
14#131×D97C211yesmaximal, characteristic, complemented
15#577×C97C211yesmaximal, characteristic, complemented
16#15690858—21no—
17#15690858C2213yescomplemented
18#15690857—23no—
19#15690857C2211yescharacteristic, complemented
20#15690857C2213yescomplemented
21#15690856—21no—
22#15690856C2211yescharacteristic, complemented
23#15690856C2213yescomplemented
24#15690855—23no—
25#15690855C2213yescomplemented
26#41×D97—21no—
27#41×D97—23no—
28#41×D97C2213yes—
29#41×D97C2211yescharacteristic
30#41×D97C2213yescomplemented
31C2×Q8×D97—21no—
32C2×Q8×D97C2213yescomplemented
33C2×D4×D97—23no—
34C2×D4×D97C2213yes—
35#15690843C2213yescomplemented
36#15690842C2216yescomplemented
37#15690841C2213yescomplemented
38#15690840C2213yescomplemented
39#15690839C2216yescomplemented
40#15690838C2213yescomplemented
41#15690837C2216yescomplemented
42#15690836C2213yescomplemented
43#15690835C2213yescomplemented
44#15690834C2216yescomplemented
45#15690833C2213yescomplemented
46#15690832C2213yescomplemented
47#15690831C2213yescomplemented
48#15690831C2213yescomplemented
49#15690830C2213yescomplemented
50#15690830C2213yescomplemented
51#15690829C2213yescomplemented
52Q16×D97C2213yescomplemented
53C2×#13095500C2213yescomplemented
54C2×#13095499C2213yescomplemented
55C2×#13095498C2213yescomplemented
56SD16×D97C2213yescomplemented
57SD16×D97C2216yescomplemented
58SD16×D97C2211yescharacteristic, complemented
59SD16×D97C2216yescomplemented
60SD16×D97C2216yescomplemented
61D8×D97C2213yescomplemented
62#15690818C2213yescomplemented
63#15690817C2211yescharacteristic, complemented
64#15690816C2213yescomplemented
65M16×D97C2213yescomplemented
66C2×C8×D97C2213yescomplemented
672−1+4×C97C2211yescharacteristic, complemented
68H(5,𝔽2)×C97C2211yescharacteristic, complemented
69#137×C97C2213yescomplemented
70AGL(1,ℤ/8)×C97C2213yescomplemented
71#135×C97C2213yescomplemented
72C2×SD16×C97C2213yescomplemented
73#131×C97C2211yescharacteristic, complemented
74C23×D97—23no—
75#13095512—21no—
76#13095512—23no—
77#13095512D411yescharacteristic
78#13095512C2313yes—
79#13095512C2313yescomplemented
80#13095511—26no—
81#13095511—23no—
82#13095511—23no—
83#13095511—23no—
84#13095511—23no—
85#13095511C2313yes—
86#13095511C2313yes—
87#13095511C2313yescomplemented
88#13095510—23no—
89#13095510—26no—
90#13095510—23no—
91#13095510—23no—
92#13095510—23no—
93#13095510D413yes—
94#13095510C2313yes—
95#13095510C2313yescomplemented
96C2×Dic194—23no—
97C2×Dic194—23no—
98C2×Dic194C2313yescomplemented
99Q8×D97—21no—
100Q8×D97—23no—
101Q8×D97—23no—
102Q8×D97—23no—
103Q8×D97D411yescharacteristic
104Q8×D97C2313yes—
105Q8×D97C2311yescharacteristic
106Q8×D97C2316yescomplemented
107C2×D388—23no—
108C2×D388C2313yes—
109C2×C22⋊D97—26no—
110C2×C22⋊D97—23no—
111D4×D97—23no—
112D4×D97—23no—
113D4×D97—23no—
114D4×D97—26no—
115D4×D97—23no—
116D4×D97—23no—
117D4×D97—23no—
118D4×D97D413yes—
119D4×D97C2311yescharacteristic
120D4×D97C2313yes—
121D4×D97C2316yes—
122C2×C4×D97—23no—
123C2×C4×D97—23no—
124C2×C4×D97—23no—
125C2×C4×D97C2313yes—
126Dic388C2313yescomplemented
127#13095501C2313yescomplemented
128#13095501C2313yescomplemented
129#13095500C2313yescomplemented
130#13095500C2316yescomplemented
131#13095500C2311yescharacteristic, complemented
132#13095499C2313yescomplemented
133#13095499C2316yescomplemented
134#13095499C2311yescharacteristic, complemented
135#13095498C2313yescomplemented
136#13095498C2316yescomplemented
137#13095498C2311yescharacteristic, complemented
138D776C2313yescomplemented
139#13095496C2313yescomplemented
140#13095496C2313yescomplemented
141#13095495C2313yescomplemented
142#13095494C2316yescomplemented
143C2×#11715895C2313yescomplemented
144C8×D97C2311yescharacteristic
145C8×D97C2313yes—
146C8×D97C2313yescomplemented
147C8×D97C2313yescomplemented
148#41×C97—21no—
149#41×C97—23no—
150#41×C97C2313yes—
151#41×C97C2311yescharacteristic
152#41×C97C2313yescomplemented
153C2×Q8×C97—21no—
154C2×Q8×C97C2313yescomplemented
155C2×D4×C97—23no—
156C2×D4×C97C2313yes—
157Q16×C97C2313yescomplemented
158SD16×C97C2313yescomplemented
159SD16×C97C2316yescomplemented
160SD16×C97C2311yescharacteristic, complemented
161D8×C97C2313yescomplemented
162M16×C97C2313yescomplemented
163C2×C8×C97C2313yescomplemented
164C23×C97—23no—
165C22×D97—23no—
166C22×D97—21no—
167C22×D97—23no—
168C22×D97—26no—
169C22×D97—86no—
170C22×D97C2×D413yes—
171Q8×C97—23no—
172Q8×C97C2×D411yescharacteristic
173Q8×C97C2413yes—
174Q8×C97C2411yescharacteristic
175D4×C97—23no—
176D4×C97—23no—
177D4×C97—23no—
178D4×C97C2×D413yes—
179D4×C97C2411yescharacteristic
180D4×C97C2413yes—
181C2×C4×C97—23no—
182C2×C4×C97—23no—
183C2×C4×C97—23no—
184C2×C4×C97C2413yes—
185C22⋊D97—23no—
186C22⋊D97—23no—
187C22⋊D97—23no—
188C22⋊D97—23no—
189C22⋊D97—23no—
190C22⋊D97—26no—
191C22⋊D97—21no—
192C22⋊D97C2×D416yes—
193C2×Dic97—23no—
194C2×Dic97—23no—
195C2×Dic97—23no—
196C2×Dic97C2×D413yes—
197D388—23no—
198D388—23no—
199D388—23no—
200D388C2×D413yes—
201D388C2411yescharacteristic
202D388C2413yes—
203C4×D97—21no—
204C4×D97—23no—
205C4×D97—26no—
206C4×D97—23no—
207C4×D97—23no—
208C4×D97—23no—
209C4×D97—23no—
210C4×D97C2×D413yes—
211C4×D97C2×D413yes—
212C4×D97C2411yescharacteristic
213C4×D97C2413yes—
214Dic194—23no—
215Dic194—23no—
216Dic194—26no—
217Dic194—21no—
218Dic194C2×D413yes—
219Dic194C2413yes—
220Dic194C2411yescharacteristic
221C8×C97C2411yescharacteristic
222C8×C97C2413yes—
223#11715895C2411yescharacteristic
224#11715895C2413yes—
225C22×C97—21no—
226C22×C97—23no—
227C22×C97—83no—
228C22×C97C22×D413yes—
229C2×D97—26no—
230C2×D97—21no—
231C2×D97—23no—
232C2×D97—42no—
233C2×D97—46no—
234C2×D97—86no—
235C2×D97—83no—
236C2×D97C22×D413yes—
237C2×D97C22×D411yescharacteristic
238C4×C97—23no—
239C4×C97—21no—
240C4×C97C22×D413yes—
241C4×C97C2511yescharacteristic
242Dic97—23no—
243Dic97—21no—
244Dic97C22×D411yescharacteristic
245Dic97C22×D413yes—
246C2×C97—23no—
247C2×C97—43no—
248C2×C97—41no—
249C2×C97C23×D411yescharacteristic
250D97—23no—
251D97—41no—
252D97—43no—
253D97#57712yesdirect product factor
254C2×#577—971nomaximal, Sylow
255C97C2×#57711yescharacteristic, complemented, Sylow
256C2×2−1+4—971no—
257C2×H(5,𝔽2)—971no—
258#577—971no—
259#577—971no—
260#577—973no—
261#577—973no—
262#577—973no—
263#577—973no—
264#577—971no—
265#577D9711yescharacteristic, direct product factor
266C2×#137—973no—
267C2×AGL(1,ℤ/8)—973no—
268C2×#135—973no—
269C22×SD16—973no—
270C2×#131—971no—
2712−1+4—971no—
2722−1+4—973no—
2732−1+4—973no—
2742−1+4—1941no—
2752−1+4—1943no—
2762−1+4C2×D9711yescharacteristic, complemented
277H(5,𝔽2)—973no—
278H(5,𝔽2)—971no—
279H(5,𝔽2)—973no—
280H(5,𝔽2)—1943no—
281H(5,𝔽2)—1941no—
282H(5,𝔽2)C2×D9711yescharacteristic, complemented
283C2×#41—971no—
284C2×#41—973no—
285C2×#41—973no—
286C2×#41—1943no—
287C2×#41—1941no—
288C22×Q8—973no—
289C22×Q8—1941no—
290C22×D4—973no—
291C22×D4—1943no—
292#137—973no—
293#137—973no—
294#137—976no—
295#137—976no—
296#137—973no—
297#137C2×D9713yescomplemented
298AGL(1,ℤ/8)—973no—
299AGL(1,ℤ/8)—973no—
300AGL(1,ℤ/8)—976no—
301AGL(1,ℤ/8)—976no—
302AGL(1,ℤ/8)—973no—
303AGL(1,ℤ/8)C2×D9713yescomplemented
304#135—973no—
305#135—973no—
306#135—973no—
307#135—973no—
308#135—973no—
309#135—973no—
310#135—973no—
311#135C2×D9713yescomplemented
312C2×Q16—973no—
313C2×SD16—973no—
314C2×SD16—973no—
315C2×SD16—973no—
316C2×SD16—976no—
317C2×SD16—976no—
318C2×SD16—976no—
319C2×SD16—971no—
320C2×SD16—973no—
321C2×SD16C2×D9713yescomplemented
322C2×D8—973no—
323#131—971no—
324#131—973no—
325#131—973no—
326#131C2×D9711yescharacteristic, complemented
327C2×M16—973no—
328C22×C8—973no—
329C24—1943no—
330#41—21no—
331#41—23no—
332#41—971no—
333#41—973no—
334#41—973no—
335#41—973no—
336#41—973no—
337#41—973no—
338#41—973no—
339#41—973no—
340#41—973no—
341#41—1943no—
342#41—1943no—
343#41—1946no—
344#41—1943no—
345#41—1943no—
346#41—1943no—
347#41—1943no—
348#41—1946no—
349#41—1943no—
350#41—1943no—
351#41—1941no—
352#41—1943no—
353#41C22×D9713yes—
354#41C22×D9711yescharacteristic
355#41C22×D9713yescomplemented
356C2×Q8—21no—
357C2×Q8—971no—
358C2×Q8—973no—
359C2×Q8—971no—
360C2×Q8—976no—
361C2×Q8—973no—
362C2×Q8—1943no—
363C2×Q8—1943no—
364C2×Q8—1943no—
365C2×Q8—1943no—
366C2×Q8—1943no—
367C2×Q8—1941no—
368C2×Q8C22×D9713yescomplemented
369C2×D4—23no—
370C2×D4—973no—
371C2×D4—971no—
372C2×D4—973no—
373C2×D4—976no—
374C2×D4—973no—
375C2×D4—1943no—
376C2×D4—1943no—
377C2×D4—1943no—
378C2×D4—1943no—
379C2×D4—1946no—
380C2×D4—1943no—
381C2×D4—1943no—
382C2×D4—1943no—
383C2×D4—1946no—
384C2×D4—1943no—
385C2×D4C22×D9713yes—
386C22×C4—973no—
387C22×C4—1943no—
388C22×C4—1943no—
389C22×C4—1943no—
390Q16—973no—
391Q16—973no—
392Q16—973no—
393Q16C22×D9713yescomplemented
394SD16—973no—
395SD16—976no—
396SD16—971no—
397SD16—973no—
398SD16—976no—
399SD16—971no—
400SD16—973no—
401SD16—976no—
402SD16—971no—
403SD16C22×D9713yescomplemented
404SD16C22×D9716yescomplemented
405SD16C22×D9711yescharacteristic, complemented
406D8—973no—
407D8—973no—
408D8—973no—
409D8C22×D9713yescomplemented
410M16—973no—
411M16—976no—
412M16C22×D9713yescomplemented
413C2×C8—973no—
414C2×C8—971no—
415C2×C8—973no—
416C2×C8—973no—
417C2×C8—973no—
418C2×C8C22×D9713yescomplemented
419C23—23no—
420C23—7766no—
421C23—973no—
422C23—1946no—
423C23—1941no—
424C23—1943no—
425C23—1943no—
426Q8—23no—
427Q8—973no—
428Q8—973no—
429Q8—971no—
430Q8—1943no—
431Q8—1946no—
432Q8—1941no—
433Q8—1943no—
434Q8D4×D9711yescharacteristic
435Q8C23×D9713yes—
436Q8C23×D9711yescharacteristic
437D4—23no—
438D4—23no—
439D4—23no—
440D4—976no—
441D4—973no—
442D4—971no—
443D4—973no—
444D4—1943no—
445D4—1943no—
446D4—1943no—
447D4—1946no—
448D4—1941no—
449D4—1943no—
450D4—1943no—
451D4—1943no—
452D4—1943no—
453D4—1943no—
454D4D4×D9713yes—
455D4C23×D9711yescharacteristic
456D4C23×D9713yes—
457C2×C4—23no—
458C2×C4—23no—
459C2×C4—23no—
460C2×C4—973no—
461C2×C4—973no—
462C2×C4—973no—
463C2×C4—971no—
464C2×C4—973no—
465C2×C4—1943no—
466C2×C4—1943no—
467C2×C4—1941no—
468C2×C4—1943no—
469C2×C4—1946no—
470C2×C4—1943no—
471C2×C4—1943no—
472C2×C4—1943no—
473C2×C4—1943no—
474C2×C4—1943no—
475C2×C4C23×D9713yes—
476C8—971no—
477C8—973no—
478C8C23×D9711yescharacteristic
479C8C23×D9713yes—
480C22—21no—
481C22—23no—
482C22—83no—
483C22—7763no—
484C22—7766no—
485C22—971no—
486C22—973no—
487C22—3886no—
488C22—3882no—
489C22—1946no—
490C22—1941no—
491C22—1943no—
492C22C2×D4×D9713yes—
493C4—21no—
494C4—23no—
495C4—971no—
496C4—973no—
497C4—1941no—
498C4—1943no—
499C4C2×D4×D9713yes—
500C4C24×D9711yescharacteristic
501C2—23no—
502C2—43no—
503C2—41no—
504C2—972no—
505C2—3883no—
506C2—3881no—
507C2—1943no—
508C2C22×D4×D9711yescharacteristic
subgroupG/H
centerC2C22×D4×D97
commutator, cosocleC4×C97C25
FrattiniC4C24×D97
Fitting#577×C97C2
socleC2×C97C23×D4
hypercenter#577D97
hypocenterC97C2×#577
LCSG ▷ C4×C97 ▷ C2×C97 ▷ C97
UCSC1 ◁ C2 ◁ #41 ◁ #577

Complements

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

NH ≅ G/Ntype
1. 2−1+4×D97C2semidirect
2. H(5,𝔽2)×D97C2semidirect
3. #20261547C2semidirect
4. #20261546C2semidirect
5. #20261545C2semidirect
6. #20261544C2semidirect
7. #20261543C2semidirect
8. #20261542C2semidirect
9. #20261541C2semidirect
10. #137×D97C2semidirect
11. AGL(1,ℤ/8)×D97C2semidirect
12. #135×D97C2semidirect
13. C2×SD16×D97C2semidirect
14. #131×D97C2semidirect
15. #577×C97C2semidirect
17. #15690858C22semidirect
19. #15690857C22semidirect
20. #15690857C22semidirect
22. #15690856C22semidirect
23. #15690856C22semidirect
25. #15690855C22semidirect
30. #41×D97C22semidirect
32. C2×Q8×D97C22semidirect
35. #15690843C22semidirect
36. #15690842C22semidirect
37. #15690841C22semidirect
38. #15690840C22semidirect
39. #15690839C22semidirect
40. #15690838C22semidirect
41. #15690837C22semidirect
42. #15690836C22semidirect
43. #15690835C22semidirect
44. #15690834C22semidirect
45. #15690833C22semidirect
46. #15690832C22semidirect
47. #15690831C22semidirect
48. #15690831C22semidirect
49. #15690830C22semidirect
50. #15690830C22semidirect
51. #15690829C22semidirect
52. Q16×D97C22semidirect
53. C2×#13095500C22semidirect
54. C2×#13095499C22semidirect
55. C2×#13095498C22semidirect
56. SD16×D97C22semidirect
57. SD16×D97C22semidirect
58. SD16×D97C22semidirect
59. SD16×D97C22semidirect
60. SD16×D97C22semidirect
61. D8×D97C22semidirect
62. #15690818C22semidirect
63. #15690817C22semidirect
64. #15690816C22semidirect
65. M16×D97C22semidirect
66. C2×C8×D97C22semidirect
67. 2−1+4×C97C22semidirect
68. H(5,𝔽2)×C97C22semidirect
69. #137×C97C22semidirect
70. AGL(1,ℤ/8)×C97C22semidirect
71. #135×C97C22semidirect
72. C2×SD16×C97C22semidirect
73. #131×C97C22semidirect
79. #13095512C23semidirect
87. #13095511C23semidirect
95. #13095510C23semidirect
98. C2×Dic194C23semidirect
106. Q8×D97C23semidirect
126. Dic388C23semidirect
127. #13095501C23semidirect
128. #13095501C23semidirect
129. #13095500C23semidirect
130. #13095500C23semidirect
131. #13095500C23semidirect
132. #13095499C23semidirect
133. #13095499C23semidirect
134. #13095499C23semidirect
135. #13095498C23semidirect
136. #13095498C23semidirect
137. #13095498C23semidirect
138. D776C23semidirect
139. #13095496C23semidirect
140. #13095496C23semidirect
141. #13095495C23semidirect
142. #13095494C23semidirect
143. C2×#11715895C23semidirect
146. C8×D97C23semidirect
147. C8×D97C23semidirect
152. #41×C97C23semidirect
154. C2×Q8×C97C23semidirect
157. Q16×C97C23semidirect
158. SD16×C97C23semidirect
159. SD16×C97C23semidirect
160. SD16×C97C23semidirect
161. D8×C97C23semidirect
162. M16×C97C23semidirect
163. C2×C8×C97C23semidirect
253. D97#577direct (N × H)
255. C97C2×#577semidirect
265. #577D97direct (N × H)
276. 2−1+4C2×D97semidirect
282. H(5,𝔽2)C2×D97semidirect
297. #137C2×D97semidirect
303. AGL(1,ℤ/8)C2×D97semidirect
311. #135C2×D97semidirect
321. C2×SD16C2×D97semidirect
326. #131C2×D97semidirect
355. #41C22×D97semidirect
368. C2×Q8C22×D97semidirect
393. Q16C22×D97semidirect
403. SD16C22×D97semidirect
404. SD16C22×D97semidirect
405. SD16C22×D97semidirect
409. D8C22×D97semidirect
412. M16C22×D97semidirect
418. C2×C8C22×D97semidirect

Maximal subgroups

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

subgroupindex
2−1+4×D972
H(5,𝔽2)×D972
#202615472
#202615462
#202615452
#202615442
#202615432
#202615422
#202615412
#137×D972
AGL(1,ℤ/8)×D972
#135×D972
C2×SD16×D972
#131×D972
#577×C972
C2×#57797