Finite Group Databasebeta

← D61×#1857 #29680819

Subgroups — D61×#1857 #29680819

Order 15616.

1402 conjugacy classes of subgroups.

The subgroup lattice has 572 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#519×D61C211yesmaximal, characteristic
2#428×D61C212yesmaximal
3C2×#101×D61C211yesmaximal, characteristic
4#391×D61C211yesmaximal, characteristic
5#21446036C211yesmaximal, characteristic, complemented
6#21446035C212yesmaximal, complemented
7#21446034C211yesmaximal, characteristic, complemented
8#21446033C211yesmaximal, characteristic, complemented
9#21446032C212yesmaximal, complemented
10#339×D61C212yesmaximal
11C61×#1857C211yesmaximal, characteristic, complemented
12C22×Q8×D61C2211yescharacteristic
13C2×M16×D61C2212yes—
14#16840240C2211yescharacteristic
15#16840239—22no—
16#16840239C2212yes—
17#16840238—22no—
18#16840238—22no—
19C4×Q8×D61—22no—
20C4×Q8×D61—22no—
21#117×D61—22no—
22#117×D61C2212yes—
23C2×C4⋊C4×D61C2211yescharacteristic
24C2×C42×D61C2211yescharacteristic
25#16839805C2212yes—
26#16839804C2212yes—
27#16839803—22no—
28#16839802C2212yes—
29#16839801—22no—
30#107×D61—22no—
31#106×D61—22no—
32C2×#14519696C2211yescharacteristic
33C2×#14519695C2212yes—
34#101×D61C418yescomplemented
35#16839669C2211yescharacteristic
36#16839668C2211yescharacteristic
37#16839667C2211yescharacteristic
38#16839666—22no—
39#95×D61—22no—
40#16839523C2212yes—
41#16839522C2212yes—
42#16839521C2212yes—
43C61×#519C2211yescharacteristic
44C61×#428C2212yes—
45C2×#101×C61C2211yescharacteristic
46C61×#391C2211yescharacteristic
47C61×#339C2212yes—
48C22×Dic122D412yes—
49C22×Dic122C2311yescharacteristic
50C2×Q8×D61—22no—
51C2×Q8×D61—22no—
52C2×Q8×D61—24no—
53C2×Q8×D61C2×C414yes—
54C2×Q8×D61Q814yes—
55C22×C4×D61—21no—
56C22×C4×D61—21no—
57C22×C4×D61D412yes—
58C22×C4×D61C2311yescharacteristic
59C2×#11758194C2312yes—
60C2×#11758189—22no—
61M16×D61—24no—
62M16×D61C2×C418yes—
63C2×C8×D61—22no—
64Q8×Dic61—22no—
65Q8×Dic61—22no—
66C4×Dic122—24no—
67C4×Dic122—24no—
68C4×Dic122—22no—
69C4×Dic122—22no—
70#14519742—24no—
71#14519742—24no—
72#14519742—22no—
73#14519742—22no—
74#14519736—22no—
75#14519736C2312yes—
76#14519735—22no—
77#14519735—22no—
78#14519735D412yes—
79#14519735D412yes—
80#14519734—24no—
81#14519734—22no—
82#14519733—24no—
83#14519733—22no—
84C2×C4⋊Dic61C2311yescharacteristic
85C2×#11758196—21no—
86C4⋊C4×D61—22no—
87C4⋊C4×D61—22no—
88C4⋊C4×D61—22no—
89C4⋊C4×D61—22no—
90C4⋊C4×D61D412yes—
91C4⋊C4×D61D412yes—
92C22⋊C4×D61—22no—
93C2×C4×Dic61—21no—
94C2×C4×Dic61C2311yescharacteristic
95C42×D61—24no—
96C42×D61—22no—
97C42×D61—24no—
98#14519711—22no—
99#14519710—22no—
100#14519710—22no—
101#14519709—22no—
102#14519708—22no—
103#14519708—22no—
104#14519696C2×C414yescomplemented
105#14519695C2×C418yescomplemented
106#14519684—22no—
107#14519684—22no—
108#14519684—22no—
109C22×Q8×C61C2311yescharacteristic
110C2×M16×C61C2312yes—
111C4×Q8×C61—22no—
112C4×Q8×C61—22no—
113#117×C61—22no—
114#117×C61C2312yes—
115C2×C4⋊C4×C61C2311yescharacteristic
116C2×C42×C61C2311yescharacteristic
117#107×C61—22no—
118#106×C61—22no—
119#101×C61C2×C414yescomplemented
120#95×C61—22no—
121C23×D61C2×D411yescharacteristic
122C2×Q8×C61—22no—
123C2×Q8×C61—22no—
124C2×Q8×C61C22×C412yes—
125C2×Q8×C61C2×Q812yes—
126C22×C4×C61—21no—
127C22×C4×C61—21no—
128C22×C4×C61C2×D412yes—
129C22×C4×C61C2411yescharacteristic
130C22×Dic61—21no—
131C22×Dic61—21no—
132C22×Dic61C2×D412yes—
133C22×Dic61C2×D411yescharacteristic
134Q8×D61—28no—
135Q8×D61—28no—
136Q8×D61—28no—
137Q8×D61—28no—
138C2×C4×D61—22no—
139C2×C4×D61—22no—
140C2×C4×D61—24no—
141C2×C4×D61—24no—
142C2×C4×D61—22no—
143C2×C4×D61—22no—
144C2×C4×D61—42no—
145C2×C4×D61—42no—
146C2×C4×D61—44no—
147C2×C4×D61—41no—
148C2×C4×D61—41no—
149C2×C4×D61C4⋊C418yes—
150C2×C4×D61C22×C412yes—
151C2×C4×D61C2×D412yes—
152C2×C4×D61C2×D412yes—
153C2×C4×D61C2×Q812yes—
154C2×C4×D61#4114yes—
155C2×Dic122—22no—
156C2×Dic122—22no—
157C2×Dic122—24no—
158C2×Dic122—24no—
159C2×Dic122C4⋊C418yes—
160C2×Dic122C22×C412yes—
161C2×Dic122C2×Q812yes—
162M16×C61—22no—
163M16×C61C22×C414yes—
164C2×C8×C61—22no—
165C4⋊C4×C61—22no—
166C4⋊C4×C61—22no—
167C4⋊C4×C61—22no—
168C4⋊C4×C61C2×D412yes—
169C4⋊C4×C61C2×D412yes—
170C22⋊C4×C61—22no—
171C42×C61—24no—
172C42×C61—22no—
173C42×C61—22no—
174C22⋊Dic61—22no—
175#11758198—22no—
176#11758198—22no—
177C4⋊Dic61—22no—
178C4⋊Dic61—22no—
179C4⋊Dic61—22no—
180C4⋊Dic61C2×D412yes—
181C4⋊Dic61C2×D412yes—
182#11758196—24no—
183#11758196—24no—
184#11758196—22no—
185#11758196—22no—
186#11758196—42no—
187C4×Dic61—24no—
188C4×Dic61—24no—
189C4×Dic61—24no—
190C4×Dic61—22no—
191C4×Dic61—22no—
192C4×Dic61—22no—
193C4×Dic61—22no—
194C4×Dic61—42no—
195#11758194—22no—
196#11758194C22×C414yes—
197C2×#92418—22no—
198#11758189—28no—
199C8×D61—28no—
200C23×C61C22×D411yescharacteristic
201C22×D61—22no—
202C22×D61—22no—
203C22×D61C2×C4⋊C412yes—
204C22×D61#12011yescharacteristic
205C22×D61#12011yescharacteristic
206C22×D61#12312yes—
207Q8×C61—24no—
208Q8×C61—24no—
209C2×C4×C61—22no—
210C2×C4×C61—22no—
211C2×C4×C61—22no—
212C2×C4×C61—22no—
213C2×C4×C61—22no—
214C2×C4×C61—41no—
215C2×C4×C61—41no—
216C2×C4×C61—42no—
217C2×C4×C61—41no—
218C2×C4×C61C2×C4⋊C414yes—
219C2×C4×C61C23×C411yescharacteristic
220C2×C4×C61C22×D412yes—
221C2×C4×C61C22×Q811yescharacteristic
222C2×C4×C61C2×#4112yes—
223C2×Dic61—22no—
224C2×Dic61—22no—
225C2×Dic61—22no—
226C2×Dic61—22no—
227C2×Dic61—22no—
228C2×Dic61—41no—
229C2×Dic61—41no—
230C2×Dic61—42no—
231C2×Dic61—41no—
232C2×Dic61C2×C4⋊C414yes—
233C2×Dic61C2×C4⋊C412yes—
234C2×Dic61#12012yes—
235C2×Dic61#12312yes—
236C4×D61—28no—
237C4×D61—28no—
238C4×D61—48no—
239C4×D61—44no—
240C4×D61—44no—
241C4×D61—82no—
242C4×D61—82no—
243C4×D61C4×D414yes—
244C4×D61C4×D414yes—
245C4×D61C22⋊Q814yes—
246C4×D61C22⋊Q814yes—
247Dic122—24no—
248Dic122—24no—
249Dic122—216no—
250C8×C61—24no—
251#92418—24no—
252C2×#1857—611nomaximal, Sylow
253C22×C61—21no—
254C22×C61—21no—
255C22×C61C22×C4⋊C411yescharacteristic
256C22×C61C2×#12011yescharacteristic
257C22×C61C2×#12311yescharacteristic
258C2×D61—21no—
259C2×D61—21no—
260C2×D61—24no—
261C2×D61—24no—
262C2×D61#38512yes—
263C2×D61#38512yes—
264C2×D61#45412yes—
265C2×D61#45612yes—
266C4×C61—24no—
267C4×C61—42no—
268C4×C61—42no—
269C4×C61—42no—
270C4×C61—81no—
271C4×C61C2×C4×D412yes—
272C4×C61C2×C22⋊Q812yes—
273Dic61—24no—
274Dic61—42no—
275Dic61—42no—
276Dic61—42no—
277Dic61—81no—
278Dic61#38514yes—
279C2×#519—611no—
280C2×#428—612no—
281C22×#101—611no—
282C2×#391—611no—
283#1857—612no—
284#1857—611no—
285#1857—611no—
286#1857—612no—
287#1857—611no—
288#1857D6111yescharacteristic, direct product factor
289C2×#339—612no—
290C2×C61—21no—
291C2×C61—21no—
292C2×C61C2×#38511yescharacteristic
293C2×C61C2×#45411yescharacteristic
294C2×C61C2×#45611yescharacteristic
295D61—22no—
296D61#185714yesdirect product factor
297C23×Q8—611no—
298C22×M16—612no—
299#519—611no—
300#519—612no—
301#519—1222no—
302#519—1222no—
303#519—1222no—
304#519C2×D6111yescharacteristic
305C2×C4×Q8—1222no—
306C2×C4×Q8—1222no—
307C2×#117—612no—
308C2×#117—1222no—
309C22×C4⋊C4—611no—
310C22×C42—611no—
311#428—612no—
312#428—612no—
313#428—612no—
314#428—1222no—
315#428—1222no—
316#428C2×D6112yes—
317C2×#107—1222no—
318C2×#106—1222no—
319C2×#101—611no—
320C2×#101—612no—
321C2×#101—618no—
322C2×#101C2×D6111yescharacteristic
323#391—611no—
324#391—611no—
325#391—611no—
326#391—1222no—
327#391C2×D6111yescharacteristic
328C2×#95—1222no—
329#339—612no—
330#339—612no—
331#339—612no—
332#339C2×D6112yes—
333C61C2×#185711yescharacteristic, complemented, Sylow
334C22×Q8—614no—
335C22×Q8—614no—
336C22×Q8—612no—
337C22×Q8—611no—
338C22×Q8—1222no—
339C22×Q8—1222no—
340C22×Q8—1224no—
341C22×Q8C22×D6111yescharacteristic
342C23×C4—611no—
343C23×C4—612no—
344C23×C4—1221no—
345C23×C4—1221no—
346C2×M16—612no—
347C2×M16—618no—
348C2×M16—1222no—
349C2×M16—1224no—
350C2×M16C22×D6112yes—
351C22×C8—1222no—
352C4×Q8—22no—
353C4×Q8—22no—
354C4×Q8—1222no—
355C4×Q8—1222no—
356C4×Q8—1224no—
357C4×Q8—1224no—
358C4×Q8—1224no—
359C4×Q8—1224no—
360C4×Q8—1222no—
361C4×Q8—1222no—
362C4×Q8—1222no—
363C4×Q8—1222no—
364#117—22no—
365#117—612no—
366#117—612no—
367#117—612no—
368#117—1222no—
369#117—1224no—
370#117—1222no—
371#117—1224no—
372#117—1222no—
373#117—1222no—
374#117—1222no—
375#117C22×D6112yes—
376C2×C4⋊C4—612no—
377C2×C4⋊C4—612no—
378C2×C4⋊C4—611no—
379C2×C4⋊C4—1222no—
380C2×C4⋊C4—1222no—
381C2×C4⋊C4—1222no—
382C2×C4⋊C4—1222no—
383C2×C4⋊C4—1221no—
384C2×C4⋊C4C22×D6111yescharacteristic
385C2×C22⋊C4—1222no—
386C2×C42—611no—
387C2×C42—1222no—
388C2×C42—1224no—
389C2×C42—1224no—
390C2×C42—1221no—
391C2×C42C22×D6111yescharacteristic
392#107—22no—
393#107—1222no—
394#107—1222no—
395#107—1222no—
396#106—22no—
397#106—1222no—
398#106—1222no—
399#106—1222no—
400#101—614no—
401#101—618no—
402#101C4×D6114yescomplemented
403#95—22no—
404#95—1222no—
405#95—1222no—
406#95—1222no—
407C24—611no—
408C2×Q8—22no—
409C2×Q8—22no—
410C2×Q8—612no—
411C2×Q8—612no—
412C2×Q8—618no—
413C2×Q8—1224no—
414C2×Q8—1224no—
415C2×Q8—1228no—
416C2×Q8—1228no—
417C2×Q8—1228no—
418C2×Q8—1228no—
419C2×Q8—1222no—
420C2×Q8—1222no—
421C2×Q8C2×C4×D6112yes—
422C2×Q8Q8×D6112yes—
423C22×C4—21no—
424C22×C4—21no—
425C22×C4—611no—
426C22×C4—612no—
427C22×C4—612no—
428C22×C4—612no—
429C22×C4—612no—
430C22×C4—614no—
431C22×C4—612no—
432C22×C4—618no—
433C22×C4—1222no—
434C22×C4—1222no—
435C22×C4—1224no—
436C22×C4—1222no—
437C22×C4—1222no—
438C22×C4—1224no—
439C22×C4—1221no—
440C22×C4—1221no—
441C22×C4—2441no—
442C22×C4—2441no—
443C22×C4—2444no—
444C22×C4—2442no—
445C22×C4—2442no—
446C22×C4D4×D6112yes—
447C22×C4C23×D6111yescharacteristic
448M16—22no—
449M16—614no—
450M16—1228no—
451M16—1222no—
452M16C2×C4×D6114yes—
453C2×C8—22no—
454C2×C8—1228no—
455C2×C8—1222no—
456C4⋊C4—22no—
457C4⋊C4—22no—
458C4⋊C4—22no—
459C4⋊C4—612no—
460C4⋊C4—612no—
461C4⋊C4—1222no—
462C4⋊C4—1222no—
463C4⋊C4—1224no—
464C4⋊C4—1224no—
465C4⋊C4—1222no—
466C4⋊C4—1222no—
467C4⋊C4—1222no—
468C4⋊C4—2442no—
469C4⋊C4D4×D6112yes—
470C4⋊C4D4×D6112yes—
471C22⋊C4—22no—
472C22⋊C4—1222no—
473C22⋊C4—1222no—
474C22⋊C4—1222no—
475C42—24no—
476C42—22no—
477C42—22no—
478C42—1222no—
479C42—1222no—
480C42—1224no—
481C42—1224no—
482C42—1224no—
483C42—1222no—
484C42—1222no—
485C42—2442no—
486C23—611no—
487C23—612no—
488C23—611no—
489C23—612no—
490C23—1222no—
491C23—1222no—
492C23C2×D4×D6111yescharacteristic
493Q8—24no—
494Q8—24no—
495Q8—12216no—
496Q8—1224no—
497Q8—1224no—
498C2×C4—22no—
499C2×C4—22no—
500C2×C4—22no—
501C2×C4—22no—
502C2×C4—22no—
503C2×C4—41no—
504C2×C4—41no—
505C2×C4—42no—
506C2×C4—41no—
507C2×C4—612no—
508C2×C4—612no—
509C2×C4—612no—
510C2×C4—614no—
511C2×C4—614no—
512C2×C4—614no—
513C2×C4—614no—
514C2×C4—614no—
515C2×C4—1228no—
516C2×C4—1228no—
517C2×C4—1222no—
518C2×C4—1222no—
519C2×C4—1222no—
520C2×C4—1222no—
521C2×C4—1222no—
522C2×C4—4882no—
523C2×C4—4882no—
524C2×C4—2444no—
525C2×C4—2444no—
526C2×C4—2448no—
527C2×C4—2442no—
528C2×C4—2441no—
529C2×C4—2441no—
530C2×C4—2441no—
531C2×C4C4⋊C4×D6114yes—
532C2×C4C22×C4×D6111yescharacteristic
533C2×C4C2×D4×D6112yes—
534C2×C4C2×Q8×D6111yescharacteristic
535C2×C4#41×D6112yes—
536C8—24no—
537C8—1224no—
538C22—21no—
539C22—21no—
540C22—612no—
541C22—612no—
542C22—612no—
543C22—612no—
544C22—1224no—
545C22—1224no—
546C22—1221no—
547C22—1221no—
548C22C2×C4⋊C4×D6111yescharacteristic
549C22#120×D6111yescharacteristic
550C22#123×D6111yescharacteristic
551C4—24no—
552C4—42no—
553C4—42no—
554C4—42no—
555C4—81no—
556C4—614no—
557C4—1224no—
558C4—4881no—
559C4—2442no—
560C4—2442no—
561C4—2442no—
562C4C4×D4×D6112yes—
563C4C22⋊Q8×D6112yes—
564C2—21no—
565C2—21no—
566C2—614no—
567C2—1222no—
568C2#385×D6111yescharacteristic
569C2#454×D6111yescharacteristic
570C2#456×D6111yescharacteristic
subgroupG/H
centerC22#120×D61
commutatorC2×C4×C61C23×C4
FrattiniC22×C4C23×D61
FittingC61×#1857C2
socleC22×C61C2×#120
prime radicalC23×D61C2×D4
cosocleC22×C4×C61C24
hypercenter#1857D61
hypocenterC61C2×#1857
LCSG ▷ C2×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C22×C4 ◁ #1857

Complements

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

NH ≅ G/Ntype
5. #21446036C2semidirect
6. #21446035C2semidirect
7. #21446034C2semidirect
8. #21446033C2semidirect
9. #21446032C2semidirect
11. C61×#1857C2semidirect
34. #101×D61C4semidirect
104. #14519696C2×C4semidirect
105. #14519695C2×C4semidirect
119. #101×C61C2×C4semidirect
288. #1857D61direct (N × H)
296. D61#1857direct (N × H)
333. C61C2×#1857semidirect
402. #101C4×D61semidirect

Maximal subgroups

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

subgroupindex
#519×D612
#428×D612
C2×#101×D612
#391×D612
#214460362
#214460352
#214460342
#214460332
#214460322
#339×D612
C61×#18572
C2×#185761