Finite Group Databasebeta

← D61×#2330 #29838838

Subgroups — D61×#2330 #29838838

Order 15616.

2398 conjugacy classes of subgroups.

The subgroup lattice has 576 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#125×D61C211yesmaximal, characteristic
2#21448561C211yesmaximal, characteristic, complemented
3#21448560C211yesmaximal, characteristic, complemented
4#21448559C212yesmaximal, complemented
5#21448558C211yesmaximal, characteristic, complemented
6#21448557C214yesmaximal, complemented
7#21448556C212yesmaximal, complemented
8#21448555C214yesmaximal, complemented
9#389×D61C212yesmaximal
10#389×D61C214yesmaximal
11#387×D61C214yesmaximal
12#384×D61C211yesmaximal, characteristic
13C4×C4⋊C4×D61C212yesmaximal
14#376×D61C211yesmaximal, characteristic
15C61×#2330C211yesmaximal, characteristic, complemented
16C2×#14519769C2211yescharacteristic
17C2×#14519768C2212yes—
18C2×#14519767C2214yes—
19#125×D61C4116yescomplemented
20C2×C4⋊C4×D61C2212yes—
21C2×C4⋊C4×D61C2214yes—
22C2×C4⋊C4×D61C2214yes—
23C2×C4⋊C4×D61C2214yes—
24C2×C4⋊C4×D61C2212yes—
25C2×C42×D61C2211yescharacteristic
26C2×C42×D61C2214yes—
27C2×C42×D61C2211yescharacteristic
28C2×C42×D61C2211yescharacteristic
29#16839661C2214yes—
30#16839661C2212yes—
31#16839660C2214yes—
32#16839660C2212yes—
33#16839659C2214yes—
34#16839659C2214yes—
35#16839659C2214yes—
36#16839658C2212yes—
37#16839658C2214yes—
38#16839657C2214yes—
39#16839657C2214yes—
40#16839657C2214yes—
41#16839649C2214yes—
42#16839648C2214yes—
43#16839647C2214yes—
44#16839646C2214yes—
45#16839645C2214yes—
46#16839644C2214yes—
47#16839643C2214yes—
48#16839632C2211yescharacteristic
49#16839631C2214yes—
50#16839631C2212yes—
51C4⋊C4×Dic61C2212yes—
52C4⋊C4×Dic61C2212yes—
53C4×#11758196C2214yes—
54C4×C4⋊Dic61C2212yes—
55#16839613C2214yes—
56#16839608C2211yescharacteristic
57#16839608C2211yescharacteristic
58#16839608C2211yescharacteristic
59#16839607C2212yes—
60#16839607C2212yes—
61#95×D61C2214yes—
62#95×D61C2212yes—
63#95×D61C2214yes—
64#95×D61C2212yes—
65C2×#125×C61C2211yescharacteristic
66C61×#389C2214yes—
67C61×#389C2212yes—
68C61×#387C2214yes—
69C61×#384C2211yescharacteristic
70C4×C4⋊C4×C61C2212yes—
71C61×#376C2211yescharacteristic
72C22×C4×D61C2311yescharacteristic
73C22×C4×D61C2314yes—
74C22×C4×D61C2311yescharacteristic
75C22×C4×D61C2314yes—
76C22×C4×D61C2312yes—
77C22×C4×D61C2312yes—
78C22×C4×D61C2311yescharacteristic
79#14519769C2×C418yescomplemented
80#14519768C2×C4116yescomplemented
81#14519767C2×C4132yescomplemented
82C2×C4⋊Dic61C2312yes—
83C2×C4⋊Dic61C2314yes—
84C2×C4⋊Dic61C2314yes—
85C2×C4⋊Dic61C2314yes—
86C2×C4⋊Dic61C2312yes—
87C2×#11758196C2314yes—
88C2×#11758196C2314yes—
89C2×#11758196C2314yes—
90C2×#11758196C2314yes—
91C2×#11758196C2314yes—
92C2×#11758196C2314yes—
93C2×#11758196C2314yes—
94C2×#11758196C2314yes—
95C4⋊C4×D61—28no—
96C4⋊C4×D61—48no—
97C4⋊C4×D61—48no—
98C4⋊C4×D61C2×C4116yes—
99C4⋊C4×D61C2×C4132yes—
100C2×C4×Dic61C2311yescharacteristic
101C2×C4×Dic61C2311yescharacteristic
102C2×C4×Dic61C2311yescharacteristic
103C2×C4×Dic61C2314yes—
104C2×C4×Dic61C2314yes—
105C2×C4×Dic61C2314yes—
106C2×C4×Dic61C2312yes—
107C2×C4×Dic61C2311yescharacteristic
108C2×C4×Dic61C2312yes—
109C2×C4×Dic61C2311yescharacteristic
110C42×D61—24no—
111C42×D61—48no—
112C42×D61C2×C418yes—
113C42×D61Q818yes—
114#14519684C2312yes—
115#14519684C2312yes—
116#14519684C2312yes—
117#14519684C2312yes—
118#14519684C2312yes—
119#14519684C2312yes—
120#14519684C2314yes—
121#14519684C2314yes—
122#14519684C2314yes—
123#14519684C2314yes—
124#14519684C2314yes—
125#14519684C2314yes—
126#125×C61C2×C418yescomplemented
127C2×C4⋊C4×C61C2312yes—
128C2×C4⋊C4×C61C2314yes—
129C2×C4⋊C4×C61C2314yes—
130C2×C4⋊C4×C61C2314yes—
131C2×C4⋊C4×C61C2312yes—
132C2×C42×C61C2311yescharacteristic
133C2×C42×C61C2314yes—
134C2×C42×C61C2311yescharacteristic
135C2×C42×C61C2311yescharacteristic
136#95×C61C2312yes—
137#95×C61C2312yes—
138#95×C61C2314yes—
139#95×C61C2314yes—
140C23×D61C2411yescharacteristic
141C22×C4×C61C2411yescharacteristic
142C22×C4×C61C2411yescharacteristic
143C22×C4×C61C2414yes—
144C22×C4×C61C2414yes—
145C22×C4×C61C2412yes—
146C22×C4×C61C2412yes—
147C22×C4×C61C2411yescharacteristic
148C22×Dic61C2411yescharacteristic
149C22×Dic61C2411yescharacteristic
150C22×Dic61C2414yes—
151C22×Dic61C2414yes—
152C22×Dic61C2412yes—
153C22×Dic61C2412yes—
154C22×Dic61C2411yescharacteristic
155C2×C4×D61—24no—
156C2×C4×D61—28no—
157C2×C4×D61—24no—
158C2×C4×D61—28no—
159C2×C4×D61—24no—
160C2×C4×D61—24no—
161C2×C4×D61—22no—
162C2×C4×D61—22no—
163C2×C4×D61—28no—
164C2×C4×D61—28no—
165C2×C4×D61—24no—
166C2×C4×D61—28no—
167C2×C4×D61—24no—
168C2×C4×D61C22×C418yes—
169C2×C4×D61C22×C4116yes—
170C2×C4×D61C22×C414yes—
171C2×C4×D61C2×Q818yes—
172C2×C4×D61C2×Q814yes—
173C2×C4×D61#4114yes—
174C4⋊C4×C61—24no—
175C4⋊C4×C61—44no—
176C4⋊C4×C61—44no—
177C4⋊C4×C61C22×C4116yes—
178C4⋊C4×C61C22×C418yes—
179C42×C61—22no—
180C42×C61—44no—
181C42×C61C22×C414yes—
182C42×C61C2×Q814yes—
183C4⋊Dic61—24no—
184C4⋊Dic61—44no—
185C4⋊Dic61—44no—
186C4⋊Dic61C22×C4116yes—
187C4⋊Dic61C22×C418yes—
188#11758196—28no—
189#11758196—44no—
190#11758196—44no—
191#11758196—44no—
192#11758196—44no—
193#11758196C22×C4116yes—
194#11758196C22×C4116yes—
195#11758196C22×C4116yes—
196C4×Dic61—22no—
197C4×Dic61—22no—
198C4×Dic61—22no—
199C4×Dic61—44no—
200C4×Dic61—44no—
201C4×Dic61—44no—
202C4×Dic61C22×C418yes—
203C4×Dic61C22×C414yes—
204C4×Dic61C2×Q814yes—
205C4×Dic61C2×Q818yes—
206C23×C61C2511yescharacteristic
207C22×D61C23×C412yes—
208C22×D61C22×Q812yes—
209C22×D61C2×#4112yes—
210C22×D61H(5,𝔽2)14yes—
211C22×D612−1+414yes—
212C2×C4×C61—22no—
213C2×C4×C61—24no—
214C2×C4×C61—22no—
215C2×C4×C61—24no—
216C2×C4×C61—22no—
217C2×C4×C61—22no—
218C2×C4×C61—21no—
219C2×C4×C61—21no—
220C2×C4×C61—24no—
221C2×C4×C61—24no—
222C2×C4×C61—22no—
223C2×C4×C61—24no—
224C2×C4×C61—22no—
225C2×C4×C61C23×C414yes—
226C2×C4×C61C23×C418yes—
227C2×C4×C61C23×C412yes—
228C2×C4×C61C22×Q814yes—
229C2×C4×C61C22×Q812yes—
230C2×C4×C61C2×#4112yes—
231C2×Dic61—22no—
232C2×Dic61—24no—
233C2×Dic61—22no—
234C2×Dic61—24no—
235C2×Dic61—22no—
236C2×Dic61—22no—
237C2×Dic61—21no—
238C2×Dic61—21no—
239C2×Dic61—24no—
240C2×Dic61—24no—
241C2×Dic61—22no—
242C2×Dic61—24no—
243C2×Dic61—22no—
244C2×Dic61C23×C414yes—
245C2×Dic61C23×C418yes—
246C2×Dic61C23×C412yes—
247C2×Dic61C22×Q814yes—
248C2×Dic61C22×Q812yes—
249C2×Dic61C2×#4112yes—
250C4×D61—232no—
251C4×D61—216no—
252C4×D61—216no—
253C4×D61—44no—
254C4×D61—416no—
255C4×D61—44no—
256C4×D61C4×Q8116yes—
257C2×#2330—611nomaximal, Sylow
258C22×C61C24×C411yescharacteristic
259C22×C61C23×Q811yescharacteristic
260C22×C61C22×#4111yescharacteristic
261C22×C61C2×H(5,𝔽2)12yes—
262C22×C61C2×2−1+412yes—
263C2×D61C2×C4×Q814yes—
264C2×D61#52018yes—
265C2×D61#54014yes—
266C2×D61#54114yes—
267C2×D61#54418yes—
268C4×C61—28no—
269C4×C61—24no—
270C4×C61—24no—
271C4×C61—41no—
272C4×C61—44no—
273C4×C61—41no—
274C4×C61C2×C4×Q814yes—
275Dic61—28no—
276Dic61—24no—
277Dic61—24no—
278Dic61—41no—
279Dic61—44no—
280Dic61—41no—
281Dic61C2×C4×Q814yes—
282C22×#125—611no—
283#2330—614no—
284#2330—612no—
285#2330—614no—
286#2330—611no—
287#2330—612no—
288#2330—611no—
289#2330—611no—
290#2330D6111yescharacteristic, direct product factor
291C2×#389—612no—
292C2×#389—614no—
293C2×#387—614no—
294C2×#384—611no—
295C2×C4×C4⋊C4—612no—
296C2×#376—611no—
297C2×C61C22×C4×Q811yescharacteristic
298C2×C61C2×#52012yes—
299C2×C61C2×#54011yescharacteristic
300C2×C61C2×#54111yescharacteristic
301C2×C61C2×#54412yes—
302D61#233018yesdirect product factor
303C2×#125—6116no—
304C2×#125—614no—
305C2×#125—612no—
306C2×#125—611no—
307C2×#125C2×D6111yescharacteristic
308C22×C4⋊C4—612no—
309C22×C4⋊C4—614no—
310C22×C4⋊C4—612no—
311C22×C4⋊C4—614no—
312C22×C4⋊C4—614no—
313C22×C42—611no—
314C22×C42—611no—
315C22×C42—611no—
316C22×C42—614no—
317#389—614no—
318#389—614no—
319#389—614no—
320#389—614no—
321#389—612no—
322#389—614no—
323#389—612no—
324#389—614no—
325#389—612no—
326#389—614no—
327#389—614no—
328#389—614no—
329#389C2×D6114yes—
330#389C2×D6112yes—
331#387—614no—
332#387—614no—
333#387—614no—
334#387—614no—
335#387—614no—
336#387—614no—
337#387—614no—
338#387C2×D6114yes—
339#384—614no—
340#384—612no—
341#384—611no—
342#384C2×D6111yescharacteristic
343C4×C4⋊C4—612no—
344C4×C4⋊C4—612no—
345C4×C4⋊C4—612no—
346C4×C4⋊C4—614no—
347C4×C4⋊C4—614no—
348C4×C4⋊C4C2×D6112yes—
349#376—611no—
350#376—611no—
351#376—611no—
352#376—612no—
353#376—612no—
354#376C2×D6111yescharacteristic
355C2×#95—612no—
356C2×#95—612no—
357C2×#95—614no—
358C2×#95—614no—
359C61C2×#233011yescharacteristic, complemented, Sylow
360C23×C4—611no—
361C23×C4—612no—
362C23×C4—614no—
363C23×C4—612no—
364C23×C4—611no—
365C23×C4—614no—
366C23×C4—611no—
367#125—6132no—
368#125—6116no—
369#125—618no—
370#125C4×D6118yescomplemented
371C2×C4⋊C4—614no—
372C2×C4⋊C4—612no—
373C2×C4⋊C4—614no—
374C2×C4⋊C4—614no—
375C2×C4⋊C4—614no—
376C2×C4⋊C4—6116no—
377C2×C4⋊C4—6132no—
378C2×C4⋊C4—614no—
379C2×C4⋊C4—612no—
380C2×C4⋊C4—614no—
381C2×C4⋊C4—614no—
382C2×C4⋊C4—614no—
383C2×C4⋊C4—614no—
384C2×C4⋊C4—614no—
385C2×C4⋊C4—614no—
386C2×C4⋊C4—1228no—
387C2×C4⋊C4—2448no—
388C2×C4⋊C4—2448no—
389C2×C4⋊C4C22×D6112yes—
390C2×C4⋊C4C22×D6114yes—
391C2×C4⋊C4C22×D6114yes—
392C2×C4⋊C4C22×D6112yes—
393C2×C4⋊C4C22×D6114yes—
394C2×C42—612no—
395C2×C42—611no—
396C2×C42—612no—
397C2×C42—611no—
398C2×C42—618no—
399C2×C42—618no—
400C2×C42—611no—
401C2×C42—611no—
402C2×C42—611no—
403C2×C42—614no—
404C2×C42—614no—
405C2×C42—614no—
406C2×C42—1224no—
407C2×C42—2448no—
408C2×C42C22×D6111yescharacteristic
409C2×C42C22×D6114yes—
410C2×C42C22×D6111yescharacteristic
411C2×C42C22×D6111yescharacteristic
412#95—612no—
413#95—612no—
414#95—612no—
415#95—612no—
416#95—612no—
417#95—612no—
418#95—614no—
419#95—614no—
420#95—614no—
421#95—614no—
422#95—614no—
423#95—614no—
424#95C22×D6112yes—
425#95C22×D6112yes—
426#95C22×D6114yes—
427#95C22×D6114yes—
428C24—611no—
429C22×C4—611no—
430C22×C4—614no—
431C22×C4—614no—
432C22×C4—614no—
433C22×C4—612no—
434C22×C4—612no—
435C22×C4—614no—
436C22×C4—618no—
437C22×C4—6116no—
438C22×C4—618no—
439C22×C4—611no—
440C22×C4—611no—
441C22×C4—614no—
442C22×C4—1224no—
443C22×C4—1228no—
444C22×C4—1224no—
445C22×C4—1224no—
446C22×C4—1228no—
447C22×C4—1224no—
448C22×C4—1222no—
449C22×C4—1222no—
450C22×C4—1228no—
451C22×C4—1228no—
452C22×C4—1224no—
453C22×C4—1228no—
454C22×C4—1224no—
455C22×C4C23×D6111yescharacteristic
456C22×C4C23×D6111yescharacteristic
457C22×C4C23×D6114yes—
458C22×C4C23×D6114yes—
459C22×C4C23×D6112yes—
460C22×C4C23×D6112yes—
461C22×C4C23×D6111yescharacteristic
462C4⋊C4—24no—
463C4⋊C4—44no—
464C4⋊C4—44no—
465C4⋊C4—6116no—
466C4⋊C4—6116no—
467C4⋊C4—6116no—
468C4⋊C4—6116no—
469C4⋊C4—618no—
470C4⋊C4—1228no—
471C4⋊C4—1224no—
472C4⋊C4—2444no—
473C4⋊C4—2444no—
474C4⋊C4—2444no—
475C4⋊C4—2444no—
476C4⋊C4—2444no—
477C4⋊C4—2444no—
478C4⋊C4C2×C4×D61116yes—
479C4⋊C4C2×C4×D6118yes—
480C42—22no—
481C42—44no—
482C42—618no—
483C42—614no—
484C42—618no—
485C42—614no—
486C42—1222no—
487C42—1222no—
488C42—1222no—
489C42—2444no—
490C42—2444no—
491C42—2444no—
492C42C2×C4×D6114yes—
493C42Q8×D6114yes—
494C23—612no—
495C23—612no—
496C23—612no—
497C23—614no—
498C23—614no—
499C23C24×D6111yescharacteristic
500C2×C4—22no—
501C2×C4—24no—
502C2×C4—22no—
503C2×C4—22no—
504C2×C4—24no—
505C2×C4—22no—
506C2×C4—21no—
507C2×C4—21no—
508C2×C4—24no—
509C2×C4—24no—
510C2×C4—22no—
511C2×C4—24no—
512C2×C4—22no—
513C2×C4—612no—
514C2×C4—612no—
515C2×C4—612no—
516C2×C4—6116no—
517C2×C4—614no—
518C2×C4—618no—
519C2×C4—614no—
520C2×C4—12216no—
521C2×C4—12216no—
522C2×C4—12232no—
523C2×C4—1222no—
524C2×C4—1224no—
525C2×C4—1222no—
526C2×C4—1222no—
527C2×C4—1222no—
528C2×C4—1224no—
529C2×C4—1221no—
530C2×C4—1221no—
531C2×C4—1224no—
532C2×C4—1224no—
533C2×C4—1222no—
534C2×C4—1224no—
535C2×C4—1222no—
536C2×C4—2444no—
537C2×C4—24416no—
538C2×C4—2444no—
539C2×C4C22×C4×D6114yes—
540C2×C4C22×C4×D6118yes—
541C2×C4C22×C4×D6112yes—
542C2×C4C2×Q8×D6114yes—
543C2×C4C2×Q8×D6112yes—
544C2×C4#41×D6112yes—
545C22—618no—
546C22—614no—
547C22—618no—
548C22—614no—
549C22—614no—
550C22C23×C4×D6111yescharacteristic
551C22C22×Q8×D6111yescharacteristic
552C22C2×#41×D6111yescharacteristic
553C22H(5,𝔽2)×D6112yes—
554C222−1+4×D6112yes—
555C4—24no—
556C4—28no—
557C4—24no—
558C4—41no—
559C4—44no—
560C4—41no—
561C4—614no—
562C4—1224no—
563C4—1224no—
564C4—1228no—
565C4—2441no—
566C4—2444no—
567C4—2441no—
568C4C4×Q8×D6114yes—
569C2—618no—
570C2C2×C4×Q8×D6111yescharacteristic
571C2#520×D6112yes—
572C2#540×D6111yescharacteristic
573C2#541×D6111yescharacteristic
574C2#544×D6112yes—
subgroupG/H
center, FrattiniC23C24×D61
commutatorC22×C61C24×C4
FittingC61×#2330C2
socle, cosocleC23×C61C25
prime radicalC23×D61C24
hypercenter#2330D61
hypocenterC61C2×#2330
LCSG ▷ C22×C61 ▷ C61
UCSC1 ◁ C23 ◁ #2330

Complements

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

NH ≅ G/Ntype
2. #21448561C2semidirect
3. #21448560C2semidirect
4. #21448559C2semidirect
5. #21448558C2semidirect
6. #21448557C2semidirect
7. #21448556C2semidirect
8. #21448555C2semidirect
15. C61×#2330C2semidirect
19. #125×D61C4semidirect
79. #14519769C2×C4semidirect
80. #14519768C2×C4semidirect
81. #14519767C2×C4semidirect
126. #125×C61C2×C4semidirect
290. #2330D61direct (N × H)
302. D61#2330direct (N × H)
359. C61C2×#2330semidirect
370. #125C4×D61semidirect

Maximal subgroups

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

subgroupindex
C2×#125×D612
#214485612
#214485602
#214485592
#214485582
#214485572
#214485562
#214485552
#389×D612
#387×D612
#384×D612
C4×C4⋊C4×D612
#376×D612
C61×#23302
C2×#233061