Finite Group Databasebeta

← D61×#2647 #29847116

Subgroups — D61×#2647 #29847116

Order 15616.

1854 conjugacy classes of subgroups.

The subgroup lattice has 556 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21451799C211yesmaximal, characteristic, complemented
2#21451798C211yesmaximal, characteristic, complemented
3#21451797C214yesmaximal, complemented
4#21451796C212yesmaximal, complemented
5#21451795C212yesmaximal, complemented
6#21451794C211yesmaximal, characteristic, complemented
7#21451793C214yesmaximal, complemented
8#400×D61C214yesmaximal
9#398×D61C214yesmaximal
10#389×D61C212yesmaximal
11#389×D61C212yesmaximal
12#384×D61C211yesmaximal, characteristic
13#382×D61C211yesmaximal, characteristic
14#376×D61C211yesmaximal, characteristic
15C61×#2647C211yesmaximal, characteristic, complemented
16C2×C4⋊C4×D61C2212yes—
17C2×C4⋊C4×D61C2212yes—
18C2×C4⋊C4×D61C2214yes—
19C2×C4⋊C4×D61C2214yes—
20C2×C4⋊C4×D61C2214yes—
21C2×C42×D61C2211yescharacteristic
22C2×C42×D61C2211yescharacteristic
23C2×C42×D61C2211yescharacteristic
24#16839704C2214yes—
25#16839704C2214yes—
26#16839703C2214yes—
27#16839703C2214yes—
28#16839702C2214yes—
29#16839701C2214yes—
30#16839700C2214yes—
31#16839696C2214yes—
32#16839696C2214yes—
33#16839695C2214yes—
34#16839695C2214yes—
35#16839694C2214yes—
36#16839693C2214yes—
37#16839692C2214yes—
38#16839661C2212yes—
39#16839661C2212yes—
40#16839660C2212yes—
41#16839660C2212yes—
42#16839659C2214yes—
43#16839659C2214yes—
44#16839658C2212yes—
45#16839658C2212yes—
46#16839657C2214yes—
47#16839657C2214yes—
48#16839632C2211yescharacteristic
49#16839631C2214yes—
50#16839631C2212yes—
51#16839628C2211yescharacteristic
52#16839627C2212yes—
53#16839626C2214yes—
54#16839608C2211yescharacteristic
55#16839608C2211yescharacteristic
56#16839608C2211yescharacteristic
57#16839607C2212yes—
58#16839607C2212yes—
59#95×D61C2214yes—
60#95×D61C2214yes—
61#95×D61C2214yes—
62#95×D61C2212yes—
63#95×D61C2212yes—
64C61×#400C2214yes—
65C61×#398C2214yes—
66C61×#389C2212yes—
67C61×#389C2212yes—
68C61×#384C2211yescharacteristic
69C61×#382C2211yescharacteristic
70C61×#376C2211yescharacteristic
71C22×C4×D61C2311yescharacteristic
72C22×C4×D61C2312yes—
73C22×C4×D61C2311yescharacteristic
74C22×C4×D61C2314yes—
75C22×C4×D61C2314yes—
76C22×C4×D61C2312yes—
77C22×C4×D61C2311yescharacteristic
78C2×C4⋊Dic61C2312yes—
79C2×C4⋊Dic61C2312yes—
80C2×C4⋊Dic61C2314yes—
81C2×C4⋊Dic61C2314yes—
82C2×C4⋊Dic61C2314yes—
83C2×#11758196C2314yes—
84C2×#11758196C2314yes—
85C2×#11758196C2314yes—
86C2×#11758196C2314yes—
87C2×#11758196C2314yes—
88C2×#11758196C2314yes—
89C2×#11758196C2314yes—
90C2×#11758196C2314yes—
91C4⋊C4×D61—28no—
92C4⋊C4×D61—28no—
93C4⋊C4×D61—48no—
94C4⋊C4×D61—48no—
95C4⋊C4×D61—48no—
96C2×C4×Dic61C2311yescharacteristic
97C2×C4×Dic61C2312yes—
98C2×C4×Dic61C2311yescharacteristic
99C2×C4×Dic61C2311yescharacteristic
100C2×C4×Dic61C2311yescharacteristic
101C2×C4×Dic61C2312yes—
102C2×C4×Dic61C2311yescharacteristic
103C42×D61—24no—
104C42×D61—24no—
105C42×D61Q818yes—
106#14519684C2312yes—
107#14519684C2314yes—
108#14519684C2314yes—
109#14519684C2312yes—
110#14519684C2314yes—
111#14519684C2312yes—
112#14519684C2314yes—
113#14519684C2314yes—
114#14519684C2314yes—
115#14519684C2312yes—
116#14519684C2314yes—
117#14519684C2314yes—
118#14519684C2312yes—
119#14519684C2312yes—
120#14519684C2314yes—
121C2×C4⋊C4×C61C2312yes—
122C2×C4⋊C4×C61C2312yes—
123C2×C4⋊C4×C61C2314yes—
124C2×C4⋊C4×C61C2314yes—
125C2×C4⋊C4×C61C2314yes—
126C2×C42×C61C2311yescharacteristic
127C2×C42×C61C2311yescharacteristic
128C2×C42×C61C2311yescharacteristic
129#95×C61C2314yes—
130#95×C61C2312yes—
131#95×C61C2314yes—
132#95×C61C2314yes—
133#95×C61C2312yes—
134C23×D61C2411yescharacteristic
135C22×C4×C61C2411yescharacteristic
136C22×C4×C61C2412yes—
137C22×C4×C61C2414yes—
138C22×C4×C61C2411yescharacteristic
139C22×C4×C61C2414yes—
140C22×C4×C61C2412yes—
141C22×C4×C61C2411yescharacteristic
142C22×Dic61C2411yescharacteristic
143C22×Dic61C2412yes—
144C22×Dic61C2414yes—
145C22×Dic61C2411yescharacteristic
146C22×Dic61C2414yes—
147C22×Dic61C2412yes—
148C22×Dic61C2411yescharacteristic
149C2×C4×D61—24no—
150C2×C4×D61—24no—
151C2×C4×D61—22no—
152C2×C4×D61—22no—
153C2×C4×D61—24no—
154C2×C4×D61—28no—
155C2×C4×D61—28no—
156C2×C4×D61—28no—
157C2×C4×D61—24no—
158C2×C4×D61—28no—
159C2×C4×D61—28no—
160C2×C4×D61—24no—
161C2×C4×D61—24no—
162C2×C4×D61—24no—
163C2×C4×D61—28no—
164C2×C4×D61C2×Q818yes—
165C2×C4×D61C2×Q814yes—
166C2×C4×D61#4114yes—
167C2×C4×D61#4114yes—
168C4⋊C4×C61—24no—
169C4⋊C4×C61—24no—
170C4⋊C4×C61—44no—
171C4⋊C4×C61—44no—
172C4⋊C4×C61—44no—
173C42×C61—22no—
174C42×C61—22no—
175C42×C61C2×Q814yes—
176C4⋊Dic61—24no—
177C4⋊Dic61—24no—
178C4⋊Dic61—44no—
179C4⋊Dic61—44no—
180C4⋊Dic61—44no—
181#11758196—28no—
182#11758196—28no—
183#11758196—44no—
184#11758196—44no—
185#11758196—44no—
186#11758196—44no—
187#11758196—44no—
188#11758196—44no—
189C4×Dic61—22no—
190C4×Dic61—22no—
191C4×Dic61—22no—
192C4×Dic61—24no—
193C4×Dic61—22no—
194C4×Dic61C2×Q814yes—
195C4×Dic61C2×Q818yes—
196C23×C61C2511yescharacteristic
197C22×D61C22×Q812yes—
198C22×D61C2×#4112yes—
199C22×D61C2×#4112yes—
200C22×D61H(5,𝔽2)14yes—
201C22×D612−1+414yes—
202C2×C4×C61—22no—
203C2×C4×C61—22no—
204C2×C4×C61—21no—
205C2×C4×C61—21no—
206C2×C4×C61—22no—
207C2×C4×C61—24no—
208C2×C4×C61—24no—
209C2×C4×C61—22no—
210C2×C4×C61—24no—
211C2×C4×C61—22no—
212C2×C4×C61—24no—
213C2×C4×C61—24no—
214C2×C4×C61—22no—
215C2×C4×C61—22no—
216C2×C4×C61—24no—
217C2×C4×C61C22×Q814yes—
218C2×C4×C61C22×Q812yes—
219C2×C4×C61C2×#4112yes—
220C2×C4×C61C2×#4112yes—
221C2×Dic61—22no—
222C2×Dic61—22no—
223C2×Dic61—21no—
224C2×Dic61—21no—
225C2×Dic61—22no—
226C2×Dic61—24no—
227C2×Dic61—24no—
228C2×Dic61—24no—
229C2×Dic61—24no—
230C2×Dic61—22no—
231C2×Dic61—24no—
232C2×Dic61—22no—
233C2×Dic61—22no—
234C2×Dic61—22no—
235C2×Dic61—24no—
236C2×Dic61C22×Q814yes—
237C2×Dic61C22×Q812yes—
238C2×Dic61C2×#4112yes—
239C2×Dic61C2×#4112yes—
240C4×D61—216no—
241C4×D61—44no—
242C4×D61—416no—
243C4×D61—44no—
244C4×D61—48no—
245C4×D61—416no—
246C4×D61#125116yes—
247C2×#2647—611nomaximal, Sylow
248C22×C61C23×Q811yescharacteristic
249C22×C61C22×#4111yescharacteristic
250C22×C61C22×#4111yescharacteristic
251C22×C61C2×H(5,𝔽2)12yes—
252C22×C61C2×2−1+412yes—
253C2×D61C2×#12514yes—
254C2×D61#54014yes—
255C2×D61#54114yes—
256C2×D61#54218yes—
257C2×D61#54418yes—
258C4×C61—24no—
259C4×C61—41no—
260C4×C61—44no—
261C4×C61—41no—
262C4×C61—42no—
263C4×C61—44no—
264C4×C61C2×#12514yes—
265Dic61—24no—
266Dic61—41no—
267Dic61—44no—
268Dic61—41no—
269Dic61—42no—
270Dic61—44no—
271Dic61C2×#12514yes—
272#2647—614no—
273#2647—614no—
274#2647—612no—
275#2647—612no—
276#2647—611no—
277#2647—611no—
278#2647—611no—
279#2647D6111yescharacteristic, direct product factor
280C2×#400—614no—
281C2×#398—614no—
282C2×#389—612no—
283C2×#389—612no—
284C2×#384—611no—
285C2×#382—611no—
286C2×#376—611no—
287C2×C61C22×#12511yescharacteristic
288C2×C61C2×#54011yescharacteristic
289C2×C61C2×#54111yescharacteristic
290C2×C61C2×#54212yes—
291C2×C61C2×#54412yes—
292D61#264718yesdirect product factor
293C22×C4⋊C4—612no—
294C22×C4⋊C4—612no—
295C22×C4⋊C4—614no—
296C22×C4⋊C4—614no—
297C22×C4⋊C4—614no—
298C22×C42—611no—
299C22×C42—611no—
300C22×C42—611no—
301#400—614no—
302#400—614no—
303#400—614no—
304#400—614no—
305#400—614no—
306#400—614no—
307#400—614no—
308#400C2×D6114yes—
309#398—614no—
310#398—614no—
311#398—614no—
312#398—614no—
313#398—614no—
314#398—614no—
315#398—614no—
316#398C2×D6114yes—
317#389—614no—
318#389—614no—
319#389—612no—
320#389—612no—
321#389—612no—
322#389—612no—
323#389—612no—
324#389—612no—
325#389—614no—
326#389—614no—
327#389C2×D6112yes—
328#389C2×D6112yes—
329#384—614no—
330#384—612no—
331#384—611no—
332#384C2×D6111yescharacteristic
333#382—612no—
334#382—611no—
335#382—614no—
336#382C2×D6111yescharacteristic
337#376—611no—
338#376—611no—
339#376—611no—
340#376—612no—
341#376—612no—
342#376C2×D6111yescharacteristic
343C2×#95—612no—
344C2×#95—614no—
345C2×#95—614no—
346C2×#95—612no—
347C2×#95—614no—
348C61C2×#264711yescharacteristic, complemented, Sylow
349C23×C4—611no—
350C23×C4—612no—
351C23×C4—614no—
352C23×C4—611no—
353C23×C4—614no—
354C23×C4—612no—
355C23×C4—611no—
356C2×C4⋊C4—614no—
357C2×C4⋊C4—612no—
358C2×C4⋊C4—614no—
359C2×C4⋊C4—612no—
360C2×C4⋊C4—614no—
361C2×C4⋊C4—614no—
362C2×C4⋊C4—614no—
363C2×C4⋊C4—614no—
364C2×C4⋊C4—614no—
365C2×C4⋊C4—614no—
366C2×C4⋊C4—614no—
367C2×C4⋊C4—614no—
368C2×C4⋊C4—614no—
369C2×C4⋊C4—1228no—
370C2×C4⋊C4—1228no—
371C2×C4⋊C4—2448no—
372C2×C4⋊C4—2448no—
373C2×C4⋊C4—2448no—
374C2×C4⋊C4C22×D6112yes—
375C2×C4⋊C4C22×D6112yes—
376C2×C4⋊C4C22×D6114yes—
377C2×C4⋊C4C22×D6114yes—
378C2×C4⋊C4C22×D6114yes—
379C2×C42—612no—
380C2×C42—611no—
381C2×C42—618no—
382C2×C42—611no—
383C2×C42—612no—
384C2×C42—611no—
385C2×C42—611no—
386C2×C42—611no—
387C2×C42—1224no—
388C2×C42—1224no—
389C2×C42C22×D6111yescharacteristic
390C2×C42C22×D6111yescharacteristic
391C2×C42C22×D6111yescharacteristic
392#95—612no—
393#95—612no—
394#95—612no—
395#95—612no—
396#95—612no—
397#95—612no—
398#95—614no—
399#95—614no—
400#95—614no—
401#95—614no—
402#95—614no—
403#95—614no—
404#95—614no—
405#95—614no—
406#95—614no—
407#95C22×D6112yes—
408#95C22×D6112yes—
409#95C22×D6114yes—
410#95C22×D6114yes—
411#95C22×D6114yes—
412C24—611no—
413C22×C4—611no—
414C22×C4—614no—
415C22×C4—614no—
416C22×C4—614no—
417C22×C4—612no—
418C22×C4—618no—
419C22×C4—611no—
420C22×C4—612no—
421C22×C4—614no—
422C22×C4—611no—
423C22×C4—614no—
424C22×C4—1224no—
425C22×C4—1224no—
426C22×C4—1222no—
427C22×C4—1224no—
428C22×C4—1222no—
429C22×C4—1228no—
430C22×C4—1228no—
431C22×C4—1228no—
432C22×C4—1228no—
433C22×C4—1224no—
434C22×C4—1228no—
435C22×C4—1224no—
436C22×C4—1228no—
437C22×C4—1224no—
438C22×C4—1224no—
439C22×C4C23×D6111yescharacteristic
440C22×C4C23×D6112yes—
441C22×C4C23×D6114yes—
442C22×C4C23×D6111yescharacteristic
443C22×C4C23×D6114yes—
444C22×C4C23×D6112yes—
445C22×C4C23×D6111yescharacteristic
446C4⋊C4—24no—
447C4⋊C4—24no—
448C4⋊C4—44no—
449C4⋊C4—44no—
450C4⋊C4—44no—
451C4⋊C4—1228no—
452C4⋊C4—1228no—
453C4⋊C4—1224no—
454C4⋊C4—1224no—
455C4⋊C4—2444no—
456C4⋊C4—2444no—
457C4⋊C4—2444no—
458C4⋊C4—2444no—
459C4⋊C4—2444no—
460C4⋊C4—2444no—
461C4⋊C4—2444no—
462C4⋊C4—2444no—
463C4⋊C4—2444no—
464C42—22no—
465C42—22no—
466C42—618no—
467C42—614no—
468C42—1222no—
469C42—1222no—
470C42—1224no—
471C42—1222no—
472C42—1222no—
473C42Q8×D6114yes—
474C23—612no—
475C23—612no—
476C23—612no—
477C23—614no—
478C23—614no—
479C23C24×D6111yescharacteristic
480C2×C4—22no—
481C2×C4—22no—
482C2×C4—21no—
483C2×C4—21no—
484C2×C4—22no—
485C2×C4—24no—
486C2×C4—24no—
487C2×C4—22no—
488C2×C4—24no—
489C2×C4—24no—
490C2×C4—22no—
491C2×C4—24no—
492C2×C4—22no—
493C2×C4—22no—
494C2×C4—24no—
495C2×C4—612no—
496C2×C4—612no—
497C2×C4—612no—
498C2×C4—6116no—
499C2×C4—614no—
500C2×C4—12216no—
501C2×C4—1222no—
502C2×C4—1222no—
503C2×C4—1221no—
504C2×C4—1221no—
505C2×C4—1222no—
506C2×C4—1224no—
507C2×C4—1224no—
508C2×C4—1224no—
509C2×C4—1222no—
510C2×C4—1224no—
511C2×C4—1224no—
512C2×C4—1222no—
513C2×C4—1222no—
514C2×C4—1222no—
515C2×C4—1224no—
516C2×C4—2444no—
517C2×C4—24416no—
518C2×C4—2444no—
519C2×C4—2448no—
520C2×C4—24416no—
521C2×C4C2×Q8×D6114yes—
522C2×C4C2×Q8×D6112yes—
523C2×C4#41×D6112yes—
524C2×C4#41×D6112yes—
525C22—618no—
526C22—614no—
527C22—614no—
528C22—614no—
529C22—618no—
530C22C22×Q8×D6111yescharacteristic
531C22C2×#41×D6111yescharacteristic
532C22C2×#41×D6111yescharacteristic
533C22H(5,𝔽2)×D6112yes—
534C222−1+4×D6112yes—
535C4—24no—
536C4—41no—
537C4—44no—
538C4—41no—
539C4—42no—
540C4—44no—
541C4—614no—
542C4—1224no—
543C4—2441no—
544C4—2444no—
545C4—2441no—
546C4—2442no—
547C4—2444no—
548C4#125×D6114yes—
549C2—618no—
550C2C2×#125×D6111yescharacteristic
551C2#540×D6111yescharacteristic
552C2#541×D6111yescharacteristic
553C2#542×D6112yes—
554C2#544×D6112yes—
subgroupG/H
center, FrattiniC23C24×D61
commutator, socle, cosocleC23×C61C25
FittingC61×#2647C2
prime radicalC23×D61C24
hypercenter#2647D61
hypocenterC61C2×#2647
LCSG ▷ C23×C61 ▷ C61
UCSC1 ◁ C23 ◁ #2647

Complements

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

NH ≅ G/Ntype
1. #21451799C2semidirect
2. #21451798C2semidirect
3. #21451797C2semidirect
4. #21451796C2semidirect
5. #21451795C2semidirect
6. #21451794C2semidirect
7. #21451793C2semidirect
15. C61×#2647C2semidirect
279. #2647D61direct (N × H)
292. D61#2647direct (N × H)
348. C61C2×#2647semidirect

Maximal subgroups

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

subgroupindex
#214517992
#214517982
#214517972
#214517962
#214517952
#214517942
#214517932
#400×D612
#398×D612
#389×D612
#384×D612
#382×D612
#376×D612
C61×#26472
C2×#264761