Finite Group Databasebeta

← #30547449

Subgroups — #30547449

Order 15616.

2604 conjugacy classes of subgroups.

The subgroup lattice has 518 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21460606C211yesmaximal, characteristic, complemented
2#21460600C211yesmaximal, characteristic, complemented
3C2×#16840570C211yesmaximal, characteristic, complemented
4#21460532C211yesmaximal, characteristic, complemented
5#21455555C216yesmaximal, complemented
6#21455550C216yesmaximal, complemented
7#21455538C216yesmaximal, complemented
8C2×#16840028C216yesmaximal, complemented
9#21454428C211yesmaximal, characteristic, complemented
10#21454297C211yesmaximal, characteristic, complemented
11C61×#2889C211yesmaximal, characteristic, complemented
12#16840598C2211yescharacteristic
13C2×#14519823C2211yescharacteristic
14#16840577—24no—
15#16840574—24no—
16#16840570—24no—
17#16840570—24no—
18#16840570—24no—
19#16840567—24no—
20#16840562C2216yes—
21#16840560C2216yescomplemented
22C2×#14519808C2213yescomplemented
23C2×#14519808C2216yescomplemented
24C2×#14519805C2216yes—
25C2×#14519802C2213yescomplemented
26C2×#14519797C2213yescomplemented
27C22×#11758201C2213yescomplemented
28#16840529C2211yescharacteristic
29#16840528C2216yescomplemented
30#16840527—24no—
31C2×#14519785C2216yescomplemented
32C2×#14519784C2211yescharacteristic
33#131×D61—24no—
34#16840036C22112yescomplemented
35#16840031C22112yescomplemented
36#16840028C22112yescomplemented
37#16840028C22124yescomplemented
38#16840024C2214yescomplemented
39#16839810C2216yescomplemented
40C2×#14519712C2216yescomplemented
41#16839744C2216yescomplemented
42C2×#14519693C2216yescomplemented
43C2×2−1+4×C61C2211yescharacteristic
44C61×#568C2211yescharacteristic
45C61×#568C2211yescharacteristic
46C61×#413C2216yescomplemented
47C2×#101×C61C2216yescomplemented
48#14519825—21no—
49#14519824D414yes—
50#14519823—24no—
51#14519823—24no—
52#14519823D414yes—
53#14519822—23no—
54C2×#11758226—21no—
55C2×#11758222C2316yes—
56C2×#11758222C2313yes—
57#41×D61—24no—
58C22×D244C2313yes—
59C2×D4×D61—23no—
60#14519810—26no—
61#14519810C2314yes—
62#14519808—212no—
63#14519808—212no—
64#14519808C23112yes—
65#14519807—26no—
66#14519805—212no—
67#14519805—26no—
68#14519805C23112yes—
69#14519803C2314yes—
70#14519802—26no—
71#14519802—212no—
72#14519799—212no—
73#14519797—212no—
74#14519797—26no—
75#14519797—26no—
76C2×#11758202—26no—
77C2×#11758202C2313yes—
78C2×#11758190—26no—
79C2×#11758200C2311yescharacteristic
80C2×#11758201—26no—
81C2×#11758201—26no—
82C2×#11758201—26no—
83C2×#11758201C2313yes—
84C2×#11758201C2316yes—
85C2×D488—26no—
86C2×#11758199C2313yes—
87#14519785—26no—
88#14519785—26no—
89#14519785C2314yes—
90#14519785C23112yes—
91#14519784—24no—
92#14519784D414yes—
93#14519784D414yes—
94#14519783—212no—
95C2×#11758194C2316yes—
96C2×#11758194C2313yes—
97C2×#11758189—26no—
98M16×D61—212no—
99C22×#92418C2313yes—
100C2×C8×D61—26no—
101#14519712C2314yes—
102#14519712C23112yes—
103#14519695C2314yes—
104#14519693C23112yes—
1052−1+4×C61—24no—
1062−1+4×C61—24no—
107C2×#41×C61—23no—
108C2×#41×C61C2313yes—
109C2×#41×C61C2311yescharacteristic
110C22×Q8×C61—21no—
111C22×Q8×C61C2313yes—
112#131×C61—24no—
113#131×C61—24no—
114C2×M16×C61C2316yes—
115C2×M16×C61C2316yes—
116#101×C61C23112yes—
117#100×C61C2314yes—
118C23×D61—23no—
119#41×C61—26no—
120#41×C61—26no—
121#41×C61—412no—
122#41×C61C2×D414yes—
123#41×C61C2×D414yes—
124C2×Q8×C61—23no—
125C2×Q8×C61—26no—
126C2×Q8×C61—26no—
127C2×Q8×C61—44no—
128C2×Q8×C61#4111yescharacteristic
129C2×Q8×C61C2413yes—
130C2×D4×C61—23no—
131C2×D4×C61#4113yes—
132C2×D4×C61C2411yescharacteristic
133C22×C4×C61—23no—
134C22×C4×C61—23no—
135C22×C4×C61C2413yes—
136C2×C22⋊D61—26no—
137C2×C22⋊D61—23no—
138#11758226—24no—
139#11758226—44no—
140Q8×D61—24no—
141#11758224—212no—
142D4×D61—212no—
143D4×D61—412no—
144#11758222—26no—
145#11758222—26no—
146#11758222—26no—
147#11758222—26no—
148#11758222C2×D4112yes—
149#11758222C2414yes—
150C2×D244—26no—
151C2×D244—26no—
152C2×D244—23no—
153C2×D244C2×D416yes—
154C2×D244C2413yes—
155C2×C4×D61—26no—
156C2×C4×D61—23no—
157C2×Dic122C2×D416yes—
158C2×Dic122C2411yescharacteristic
159M16×C61—26no—
160M16×C61—26no—
161M16×C61C2414yes—
162M16×C61C2414yes—
163C2×C8×C61—26no—
164C2×C8×C61—26no—
165#11758202—212no—
166#11758202—26no—
167#11758201—212no—
168#11758201—212no—
169#11758201—23no—
170#11758201—23no—
171#11758200—24no—
172#11758200—21no—
173#11758200—21no—
174#11758199—24no—
175#11758199—26no—
176#11758194—26no—
177#11758194C2×D4112yes—
178#11758194C2414yes—
179C2×#92418—26no—
180C2×#92418C2×D416yes—
181C2×#92418C2×D416yes—
182C2×#92418C2411yescharacteristic
183C2×#92418C2413yes—
184Dic244—24no—
185D488—212no—
186#11758190—24no—
187#11758190—212no—
188#11758189—212no—
189#11758189—24no—
190C8×D61—24no—
191C8×D61—212no—
192C23×C61—21no—
193C23×C61C2×#4113yes—
194C22×D61—26no—
195C22×D61—23no—
196C22×D61—46no—
197C22×D61—163no—
198Q8×C61—23no—
199Q8×C61—23no—
200Q8×C61—46no—
201Q8×C61#12114yes—
202D4×C61—21no—
203D4×C61—21no—
204D4×C61—46no—
205D4×C61#121112yes—
206C2×C4×C61—23no—
207C2×C4×C61—46no—
208C2×C4×C61—46no—
209C2×C4×C61C22×D416yes—
210C2×C4×C61C22×D416yes—
211C2×C4×C61C2×#4113yes—
212C2×C4×C61C2511yescharacteristic
213C22⋊D61—212no—
214C22⋊D61—24no—
215C22⋊D61—46no—
216C2×Dic61—26no—
217C2×Dic61—21no—
218D244—26no—
219D244—23no—
220D244—23no—
221D244—46no—
222D244C22×D414yes—
223C4×D61—212no—
224C4×D61—24no—
225C4×D61—46no—
226Dic122—26no—
227Dic122—21no—
228Dic122—21no—
229Dic122C22×D414yes—
230C8×C61—24no—
231C8×C61—24no—
232#92418—24no—
233#92418C22×D411yescharacteristic
234#92418C22×D416yes—
235#92418C22×D411yescharacteristic
236#60470—611nomaximal, Sylow
237C22×C61—43no—
238C22×C61—41no—
239C22×C61—81no—
240C22×C61C2×#12116yes—
241C22×C61C22×#4111yescharacteristic
242C2×D61—24no—
243C2×D61—43no—
244C2×D61—86no—
245C2×D61—83no—
246C2×D61—166no—
247C4×C61—43no—
248C4×C61C2×#12116yes—
249C4×C61C23×D411yescharacteristic
250C4×C61C23×D411yescharacteristic
251Dic61—24no—
252Dic61—41no—
253#3586—611no—
254#3585—611no—
255C2×#577—611no—
256#3573—611no—
257#3069—616no—
258#3068—616no—
259#3067—616no—
260C2×#471—616no—
261#2916—611no—
262#2889D6111yescharacteristic, complemented
263#2888—611no—
264C2×C61—21no—
265C2×C61—26no—
266C2×C61—81no—
267C2×C61C22×#12111yescharacteristic
268D61—44no—
269D61—83no—
270#585—611no—
271C2×2−1+4C2×D6111yescharacteristic
272C2×H(5,𝔽2)—611no—
273#578—1224no—
274#577—1224no—
275#577—1224no—
276#577—1224no—
277#577—1224no—
278#576—1224no—
279#575—616no—
280#575—616no—
281C2×#137—613no—
282C2×#137—616no—
283C2×AGL(1,ℤ/8)—613no—
284C2×AGL(1,ℤ/8)—616no—
285C2×#135—613no—
286C22×SD16—613no—
287#568—611no—
288#568—616no—
289#568—1224no—
290#568C2×D6111yescharacteristic
291#568C2×D6111yescharacteristic
292C2×#131—611no—
293C2×#131—616no—
294C2×#131—1224no—
295#473—6112no—
296#472—6112no—
297#471—6112no—
298#471—6124no—
299#471—614no—
300#430—616no—
301C2×#108—616no—
302#413—616no—
303#413C2×D6116yescomplemented
304C2×#101C2×D6116yescomplemented
305C2×#100—616no—
306C61#6047011yescharacteristic, complemented, Sylow
3072−1+4—24no—
3082−1+4—24no—
3092−1+4—614no—
3102−1+4—1221no—
311H(5,𝔽2)—614no—
312H(5,𝔽2)—1223no—
313H(5,𝔽2)—1224no—
314H(5,𝔽2)—1224no—
315C2×#41—23no—
316C2×#41—616no—
317C2×#41—613no—
318C2×#41—1224no—
319C2×#41—1221no—
320C2×#41C22×D6113yes—
321C2×#41C22×D6111yescharacteristic
322C22×Q8—21no—
323C22×Q8C22×D6113yes—
324C22×D4—613no—
325C22×D4—1223no—
326#137—6112no—
327#137—614no—
328#137—1226no—
329#137—12212no—
330#137—12212no—
331#137—1226no—
332AGL(1,ℤ/8)—614no—
333AGL(1,ℤ/8)—6112no—
334AGL(1,ℤ/8)—1226no—
335AGL(1,ℤ/8)—12212no—
336AGL(1,ℤ/8)—12212no—
337AGL(1,ℤ/8)—1226no—
338#135—12212no—
339#135—1226no—
340#135—1226no—
341#135—12212no—
342C2×Q16—613no—
343C2×Q16—1226no—
344C2×SD16—613no—
345C2×SD16—616no—
346C2×SD16—611no—
347C2×SD16—1226no—
348C2×SD16—1226no—
349C2×SD16—1226no—
350C2×SD16—1226no—
351C2×D8—613no—
352C2×D8—1226no—
353#131—24no—
354#131—24no—
355#131—614no—
356#131—614no—
357#131—614no—
358#131—6112no—
359#131—12212no—
360#131—1226no—
361#131—1226no—
362#131—1224no—
363C2×M16—613no—
364C2×M16—616no—
365C2×M16—1226no—
366C2×M16—12212no—
367C2×M16C22×D6116yes—
368C2×M16C22×D6116yes—
369C22×C8—613no—
370C22×C8—1226no—
371#108—614no—
372#108—6112no—
373#101—614no—
374#101C22×D61112yes—
375#100—6112no—
376#100C22×D6114yes—
377C24—1223no—
378#41—26no—
379#41—26no—
380#41—412no—
381#41—6112no—
382#41—614no—
383#41—1224no—
384#41—12212no—
385#41—1226no—
386#41—1226no—
387#41—1226no—
388#41—1226no—
389#41—2444no—
390#41D4×D6114yes—
391#41D4×D6114yes—
392C2×Q8—23no—
393C2×Q8—26no—
394C2×Q8—26no—
395C2×Q8—44no—
396C2×Q8—616no—
397C2×Q8—611no—
398C2×Q8—1224no—
399C2×Q8#1175822211yescharacteristic
400C2×Q8C23×D6113yes—
401C2×D4—23no—
402C2×D4—616no—
403C2×D4—613no—
404C2×D4—1226no—
405C2×D4—12212no—
406C2×D4—1223no—
407C2×D4—1226no—
408C2×D4—1226no—
409C2×D4—1223no—
410C2×D4—24412no—
411C2×D4#1175822213yes—
412C2×D4C23×D6111yescharacteristic
413C22×C4—23no—
414C22×C4—23no—
415C22×C4—1226no—
416C22×C4—1223no—
417C22×C4C23×D6113yes—
418Q16—1224no—
419Q16—1226no—
420Q16—12212no—
421SD16—1224no—
422SD16—12212no—
423SD16—1221no—
424SD16—1221no—
425SD16—1223no—
426SD16—1223no—
427SD16—12212no—
428SD16—1224no—
429SD16—12212no—
430D8—12212no—
431D8—1226no—
432D8—1224no—
433M16—26no—
434M16—26no—
435M16—6112no—
436M16—614no—
437M16—1224no—
438M16—12212no—
439M16—1226no—
440M16C23×D6114yes—
441M16C23×D6114yes—
442C2×C8—26no—
443C2×C8—26no—
444C2×C8—616no—
445C2×C8—616no—
446C2×C8—611no—
447C2×C8—613no—
448C2×C8—1224no—
449C2×C8—12212no—
450C2×C8—1226no—
451C23—21no—
452C23—1226no—
453C23—1223no—
454C23—9763no—
455C23—2446no—
456C23C2×#1175822213yes—
457Q8—23no—
458Q8—23no—
459Q8—46no—
460Q8—614no—
461Q8—1226no—
462Q8—1221no—
463Q8—1221no—
464Q8#1451975014yes—
465D4—21no—
466D4—21no—
467D4—46no—
468D4—614no—
469D4—12212no—
470D4—1224no—
471D4—1226no—
472D4—1223no—
473D4—1223no—
474D4—2446no—
475D4—2446no—
476D4#14519750112yes—
477C2×C4—23no—
478C2×C4—46no—
479C2×C4—46no—
480C2×C4—1226no—
481C2×C4—12212no—
482C2×C4—1224no—
483C2×C4—1221no—
484C2×C4—2446no—
485C2×C4C2×D4×D6116yes—
486C2×C4C2×D4×D6116yes—
487C2×C4C2×#1175822213yes—
488C2×C4C24×D6111yescharacteristic
489C8—24no—
490C8—24no—
491C8—611no—
492C8—611no—
493C8—616no—
494C8—1224no—
495C22—43no—
496C22—41no—
497C22—81no—
498C22—1224no—
499C22—9766no—
500C22—4883no—
501C22—4886no—
502C22—2443no—
503C22C2×#1451975016yes—
504C22C22×#1175822211yescharacteristic
505C4—43no—
506C4—1224no—
507C4—2441no—
508C4C2×#1451975016yes—
509C4C22×D4×D6111yescharacteristic
510C4C22×D4×D6111yescharacteristic
511C2—21no—
512C2—26no—
513C2—81no—
514C2—4883no—
515C2—2444no—
516C2C22×#1451975011yescharacteristic
subgroupG/H
centerC2C22×#14519750
commutator, cosocleC2×C4×C61C25
FrattiniC2×C4C24×D61
FittingC61×#2889C2
socleC2×C61C22×#121
hypercenter#2889D61
hypocenterC61#60470
LCSG ▷ C2×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C2 ◁ C2×#41 ◁ #2889

Complements

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

NH ≅ G/Ntype
1. #21460606C2semidirect
2. #21460600C2semidirect
3. C2×#16840570C2semidirect
4. #21460532C2semidirect
5. #21455555C2semidirect
6. #21455550C2semidirect
7. #21455538C2semidirect
8. C2×#16840028C2semidirect
9. #21454428C2semidirect
10. #21454297C2semidirect
11. C61×#2889C2semidirect
21. #16840560C22semidirect
22. C2×#14519808C22semidirect
23. C2×#14519808C22semidirect
25. C2×#14519802C22semidirect
26. C2×#14519797C22semidirect
27. C22×#11758201C22semidirect
29. #16840528C22semidirect
31. C2×#14519785C22semidirect
34. #16840036C22semidirect
35. #16840031C22semidirect
36. #16840028C22semidirect
37. #16840028C22semidirect
38. #16840024C22semidirect
39. #16839810C22semidirect
40. C2×#14519712C22semidirect
41. #16839744C22semidirect
42. C2×#14519693C22semidirect
46. C61×#413C22semidirect
47. C2×#101×C61C22semidirect
262. #2889D61semidirect
303. #413C2×D61semidirect
304. C2×#101C2×D61semidirect
306. C61#60470semidirect

Maximal subgroups

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

subgroupindex
#214606062
#214606002
C2×#168405702
#214605322
#214555552
#214555502
#214555382
C2×#168400282
#214544282
#214542972
C61×#28892
#6047061