Finite Group Databasebeta

← C61×#24111 #29527544

Subgroups — C61×#24111 #29527544

Order 15616.

1068 conjugacy classes of subgroups.

The subgroup lattice has 542 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C61×#552C211yesmaximal, characteristic
2C2×C61×#483C211yesmaximal, characteristic, complemented
3C2×C61×#428C211yesmaximal, characteristic, complemented
4C61×#2487C211yesmaximal, characteristic
5C61×#2059C212yesmaximal, complemented
6C61×#2036C212yesmaximal, complemented
7C61×#1894C211yesmaximal, characteristic, complemented
8C61×#1870C211yesmaximal, characteristic
9C61×#1865C212yesmaximal, complemented
10C61×#1853C212yesmaximal, complemented
11C61×#1825C211yesmaximal, characteristic, complemented
12C22×M16×C61C2211yescharacteristic
13C61×#552—24no—
14C61×#552—24no—
15C2×#126×C61—21no—
16C2×#125×C61—21no—
17C2×#125×C61C2211yescharacteristic
18C2×#123×C61—21no—
19C2×C22⋊Q8×C61C2211yescharacteristic
20C2×C22⋊Q8×C61C2211yescharacteristic
21C2×C4×Q8×C61C2211yescharacteristic
22C2×C4×D4×C61C2211yescharacteristic
23C2×#117×C61—21no—
24C2×#117×C61C2211yescharacteristic
25C22×C4⋊C4×C61C2211yescharacteristic
26C22×C4⋊C4×C61C2211yescharacteristic
27C61×#483—24no—
28C61×#428—24no—
29C2×#107×C61C2212yes—
30C2×#106×C61C2212yes—
31C2×#106×C61C2212yes—
32C2×#103×C61C2212yes—
33C2×#103×C61C2212yes—
34C2×C22⋊C8×C61C2212yes—
35C2×C22⋊C8×C61C2211yescharacteristic
36C61×#400—21no—
37C61×#399—21no—
38C61×#389C2212yes—
39C61×#386—21no—
40C61×#385—21no—
41C61×#380C2212yes—
42C61×#380C2211yescharacteristic
43C61×#340C2212yes—
44C61×#340C2212yes—
45C61×#340C2212yes—
46C61×#336C2212yes—
47C22×Q8×C61C2311yescharacteristic
48C22×D4×C61D411yescharacteristic
49C23×C4×C61—21no—
50C23×C4×C61—21no—
51C23×C4×C61C2311yescharacteristic
52C23×C4×C61C2311yescharacteristic
53C2×M16×C61—22no—
54C2×M16×C61—24no—
55C22×C8×C61C2312yes—
56C22×C8×C61C2312yes—
57#126×C61—28no—
58#125×C61—28no—
59#125×C61—22no—
60#125×C61—22no—
61#123×C61—28no—
62C22⋊Q8×C61—22no—
63C22⋊Q8×C61—22no—
64C22⋊Q8×C61—22no—
65C22⋊Q8×C61D414yes—
66C4×Q8×C61—22no—
67C4×Q8×C61—22no—
68C4×D4×C61—22no—
69C4×D4×C61—22no—
70#117×C61—28no—
71#117×C61—22no—
72#117×C61D414yes—
73C2×C4⋊C4×C61—21no—
74C2×C4⋊C4×C61—21no—
75C2×C4⋊C4×C61—21no—
76C2×C4⋊C4×C61—21no—
77C2×C4⋊C4×C61—21no—
78C2×C4⋊C4×C61—21no—
79C2×C4⋊C4×C61—22no—
80C2×C4⋊C4×C61—22no—
81C2×C4⋊C4×C61—22no—
82C2×C4⋊C4×C61—22no—
83C2×C4⋊C4×C61—21no—
84C2×C4⋊C4×C61—22no—
85C2×C4⋊C4×C61D411yescharacteristic
86C2×C4⋊C4×C61C2312yes—
87C2×C4⋊C4×C61C2312yes—
88C2×C4⋊C4×C61C2311yescharacteristic
89C2×C4⋊C4×C61C2311yescharacteristic
90C2×C4⋊C4×C61C2311yescharacteristic
91C2×C22⋊C4×C61—21no—
92C2×C22⋊C4×C61—21no—
93C2×C22⋊C4×C61—21no—
94C2×C22⋊C4×C61—21no—
95C2×C22⋊C4×C61—22no—
96C2×C22⋊C4×C61D411yescharacteristic
97C2×C22⋊C4×C61D411yescharacteristic
98C2×C42×C61—21no—
99C2×C42×C61—21no—
100C2×C42×C61C2311yescharacteristic
101#107×C61—24no—
102#106×C61—24no—
103#106×C61—24no—
104#103×C61—24no—
105#103×C61—24no—
106C22⋊C8×C61—22no—
107C22⋊C8×C61—24no—
108#95×C61—22no—
109#95×C61—21no—
110#95×C61—22no—
111C24×C61C2×D411yescharacteristic
112C24×C61C2×D411yescharacteristic
113C2×Q8×C61—21no—
114C2×Q8×C61—22no—
115C2×Q8×C61—22no—
116C2×Q8×C61C2×D412yes—
117C2×D4×C61—22no—
118C2×D4×C61—22no—
119C2×D4×C61—41no—
120C22×C4×C61—22no—
121C22×C4×C61—22no—
122C22×C4×C61—24no—
123C22×C4×C61—21no—
124C22×C4×C61—21no—
125C22×C4×C61—21no—
126C22×C4×C61—21no—
127C22×C4×C61—21no—
128C22×C4×C61—21no—
129C22×C4×C61—22no—
130C22×C4×C61—22no—
131C22×C4×C61—22no—
132C22×C4×C61—22no—
133C22×C4×C61—42no—
134C22×C4×C61—42no—
135C22×C4×C61—42no—
136C22×C4×C61—82no—
137C22×C4×C61C2×D411yescharacteristic
138C22×C4×C61C2×D411yescharacteristic
139C22×C4×C61C2×D411yescharacteristic
140C22×C4×C61C2×D411yescharacteristic
141C22×C4×C61C2×D412yes—
142C22×C4×C61C2×D412yes—
143C22×C4×C61C2411yescharacteristic
144M16×C61—28no—
145C2×C8×C61—22no—
146C2×C8×C61—22no—
147C2×C8×C61—22no—
148C2×C8×C61—22no—
149C2×C8×C61—22no—
150C2×C8×C61#4114yes—
151C4⋊C4×C61—24no—
152C4⋊C4×C61—24no—
153C4⋊C4×C61—24no—
154C4⋊C4×C61—24no—
155C4⋊C4×C61—24no—
156C4⋊C4×C61—24no—
157C4⋊C4×C61—22no—
158C4⋊C4×C61—21no—
159C4⋊C4×C61—21no—
160C4⋊C4×C61—21no—
161C4⋊C4×C61—22no—
162C4⋊C4×C61—22no—
163C4⋊C4×C61—22no—
164C4⋊C4×C61—22no—
165C4⋊C4×C61—42no—
166C4⋊C4×C61—82no—
167C4⋊C4×C61C2×D412yes—
168C4⋊C4×C61C2×D412yes—
169C4⋊C4×C61C2×D412yes—
170C22⋊C4×C61—24no—
171C22⋊C4×C61—24no—
172C22⋊C4×C61—24no—
173C22⋊C4×C61—22no—
174C22⋊C4×C61—22no—
175C22⋊C4×C61—42no—
176C22⋊C4×C61—82no—
177C42×C61—24no—
178C42×C61—24no—
179C42×C61—21no—
180C42×C61—21no—
181C23×C61—21no—
182C23×C61—22no—
183C23×C61—22no—
184C23×C61—21no—
185C23×C61—21no—
186C23×C61—21no—
187C23×C61—22no—
188C23×C61—22no—
189C23×C61#12012yes—
190C23×C61#13712yes—
191C23×C61C22×D411yescharacteristic
192Q8×C61—24no—
193Q8×C61—41no—
194Q8×C61—41no—
195D4×C61—44no—
196C2×C4×C61—22no—
197C2×C4×C61—22no—
198C2×C4×C61—22no—
199C2×C4×C61—24no—
200C2×C4×C61—21no—
201C2×C4×C61—22no—
202C2×C4×C61—22no—
203C2×C4×C61—22no—
204C2×C4×C61—21no—
205C2×C4×C61—21no—
206C2×C4×C61—22no—
207C2×C4×C61—22no—
208C2×C4×C61—21no—
209C2×C4×C61—21no—
210C2×C4×C61—22no—
211C2×C4×C61—48no—
212C2×C4×C61—42no—
213C2×C4×C61—41no—
214C2×C4×C61—41no—
215C2×C4×C61—41no—
216C2×C4×C61—41no—
217C2×C4×C61—42no—
218C2×C4×C61—41no—
219C2×C4×C61—41no—
220C2×C4×C61—42no—
221C2×C4×C61—42no—
222C2×C4×C61—42no—
223C2×C4×C61—42no—
224C2×C4×C61—42no—
225C2×C4×C61—42no—
226C2×C4×C61—42no—
227C2×C4×C61—84no—
228C2×C4×C61#12012yes—
229C2×C4×C61C22×D411yescharacteristic
230C2×C4×C61C22×D411yescharacteristic
231C2×C4×C61C2×#4112yes—
232C2×C4×C612−1+412yes—
233C8×C61—24no—
234C8×C61—42no—
235#24111C6111yesmaximal, characteristic, direct product factor, Sylow
236C22×C61—24no—
237C22×C61—22no—
238C22×C61—22no—
239C22×C61—22no—
240C22×C61—22no—
241C22×C61—42no—
242C22×C61—42no—
243C22×C61—42no—
244C22×C61—41no—
245C22×C61—41no—
246C22×C61C2×#12011yescharacteristic
247C22×C61#54712yes—
248C22×C61C2×#13711yescharacteristic
249C22×C61#57511yescharacteristic
250C22×C61#57712yes—
251C4×C61—22no—
252C4×C61—41no—
253C4×C61—42no—
254C4×C61—41no—
255C4×C61—42no—
256C4×C61—42no—
257C4×C61—41no—
258C4×C61—42no—
259C4×C61—82no—
260C4×C61—82no—
261C4×C61#52511yescharacteristic
262C4×C61#53611yescharacteristic
263C4×C61#54712yes—
264C2×#552C2×C6111yescharacteristic
265C2×#483C2×C6111yescharacteristic, complemented
266C2×#428C2×C6111yescharacteristic, complemented
267#2487C2×C6111yescharacteristic
268#2059C2×C6112yescomplemented
269#2036C2×C6112yescomplemented
270#1894C2×C6111yescharacteristic, complemented
271#1870C2×C6111yescharacteristic
272#1865C2×C6112yescomplemented
273#1853C2×C6112yescomplemented
274#1825C2×C6111yescharacteristic, complemented
275C2×C61—42no—
276C2×C61—42no—
277C2×C61#240611yescharacteristic
278C2×C61#300711yescharacteristic
279C2×C61#301011yescharacteristic
280C2×C61#334412yes—
281C2×C61#335112yes—
282C22×M16C22×C6111yescharacteristic
283#552—24no—
284#552—24no—
285C2×#126—21no—
286C2×#125—21no—
287C2×#125C22×C6111yescharacteristic
288C2×#123—21no—
289C2×C22⋊Q8C22×C6111yescharacteristic
290C2×C22⋊Q8C22×C6111yescharacteristic
291C2×C4×Q8C22×C6111yescharacteristic
292C2×C4×D4C22×C6111yescharacteristic
293C2×#117—21no—
294C2×#117C22×C6111yescharacteristic
295C22×C4⋊C4C22×C6111yescharacteristic
296C22×C4⋊C4C22×C6111yescharacteristic
297#483—24no—
298#428—24no—
299C2×#107C22×C6112yes—
300C2×#106C22×C6112yes—
301C2×#106C22×C6112yes—
302C2×#103C22×C6112yes—
303C2×#103C22×C6112yes—
304C2×C22⋊C8C22×C6111yescharacteristic
305C2×C22⋊C8C22×C6112yes—
306#400—21no—
307#399—21no—
308#389C22×C6112yes—
309#386—21no—
310#385—21no—
311#380C22×C6111yescharacteristic
312#380C22×C6112yes—
313#340C22×C6112yes—
314#340C22×C6112yes—
315#340C22×C6112yes—
316#336C22×C6112yes—
317C61#2411111yescharacteristic, direct product factor, Sylow
318C22×Q8C23×C6111yescharacteristic
319C22×D4D4×C6111yescharacteristic
320C23×C4—21no—
321C23×C4—21no—
322C23×C4C23×C6111yescharacteristic
323C23×C4C23×C6111yescharacteristic
324C2×M16—22no—
325C2×M16—24no—
326C22×C8C23×C6112yes—
327C22×C8C23×C6112yes—
328#126—28no—
329#125—28no—
330#125—22no—
331#125—22no—
332#123—28no—
333C22⋊Q8—22no—
334C22⋊Q8—22no—
335C22⋊Q8—22no—
336C22⋊Q8D4×C6114yes—
337C4×Q8—22no—
338C4×Q8—22no—
339C4×D4—22no—
340C4×D4—22no—
341#117—28no—
342#117—22no—
343#117D4×C6114yes—
344C2×C4⋊C4—21no—
345C2×C4⋊C4—21no—
346C2×C4⋊C4—21no—
347C2×C4⋊C4—21no—
348C2×C4⋊C4—21no—
349C2×C4⋊C4—21no—
350C2×C4⋊C4—22no—
351C2×C4⋊C4—22no—
352C2×C4⋊C4—22no—
353C2×C4⋊C4—22no—
354C2×C4⋊C4—21no—
355C2×C4⋊C4—22no—
356C2×C4⋊C4D4×C6111yescharacteristic
357C2×C4⋊C4C23×C6112yes—
358C2×C4⋊C4C23×C6112yes—
359C2×C4⋊C4C23×C6111yescharacteristic
360C2×C4⋊C4C23×C6111yescharacteristic
361C2×C4⋊C4C23×C6111yescharacteristic
362C2×C22⋊C4—21no—
363C2×C22⋊C4—21no—
364C2×C22⋊C4—21no—
365C2×C22⋊C4—21no—
366C2×C22⋊C4—22no—
367C2×C22⋊C4D4×C6111yescharacteristic
368C2×C22⋊C4D4×C6111yescharacteristic
369C2×C42—21no—
370C2×C42—21no—
371C2×C42C23×C6111yescharacteristic
372#107—24no—
373#106—24no—
374#106—24no—
375#103—24no—
376#103—24no—
377C22⋊C8—22no—
378C22⋊C8—24no—
379#95—22no—
380#95—21no—
381#95—22no—
382C24C2×D4×C6111yescharacteristic
383C24C2×D4×C6111yescharacteristic
384C2×Q8—21no—
385C2×Q8—22no—
386C2×Q8—22no—
387C2×Q8C2×D4×C6112yes—
388C2×D4—22no—
389C2×D4—22no—
390C2×D4—41no—
391C22×C4—22no—
392C22×C4—22no—
393C22×C4—24no—
394C22×C4—21no—
395C22×C4—21no—
396C22×C4—21no—
397C22×C4—21no—
398C22×C4—21no—
399C22×C4—21no—
400C22×C4—22no—
401C22×C4—22no—
402C22×C4—22no—
403C22×C4—22no—
404C22×C4—42no—
405C22×C4—42no—
406C22×C4—42no—
407C22×C4—82no—
408C22×C4C2×D4×C6111yescharacteristic
409C22×C4C2×D4×C6111yescharacteristic
410C22×C4C2×D4×C6111yescharacteristic
411C22×C4C2×D4×C6111yescharacteristic
412C22×C4C2×D4×C6112yes—
413C22×C4C2×D4×C6112yes—
414C22×C4C24×C6111yescharacteristic
415M16—28no—
416C2×C8—22no—
417C2×C8—22no—
418C2×C8—22no—
419C2×C8—22no—
420C2×C8—22no—
421C2×C8#41×C6114yes—
422C4⋊C4—24no—
423C4⋊C4—24no—
424C4⋊C4—24no—
425C4⋊C4—24no—
426C4⋊C4—24no—
427C4⋊C4—24no—
428C4⋊C4—22no—
429C4⋊C4—21no—
430C4⋊C4—21no—
431C4⋊C4—21no—
432C4⋊C4—22no—
433C4⋊C4—22no—
434C4⋊C4—22no—
435C4⋊C4—22no—
436C4⋊C4—42no—
437C4⋊C4—82no—
438C4⋊C4C2×D4×C6112yes—
439C4⋊C4C2×D4×C6112yes—
440C4⋊C4C2×D4×C6112yes—
441C22⋊C4—24no—
442C22⋊C4—24no—
443C22⋊C4—24no—
444C22⋊C4—22no—
445C22⋊C4—22no—
446C22⋊C4—42no—
447C22⋊C4—82no—
448C42—24no—
449C42—24no—
450C42—21no—
451C42—21no—
452C23—22no—
453C23—22no—
454C23—21no—
455C23—21no—
456C23—21no—
457C23—22no—
458C23—22no—
459C23—21no—
460C23#120×C6112yes—
461C23#137×C6112yes—
462C23C22×D4×C6111yescharacteristic
463Q8—24no—
464Q8—41no—
465Q8—41no—
466D4—44no—
467C2×C4—22no—
468C2×C4—22no—
469C2×C4—24no—
470C2×C4—22no—
471C2×C4—22no—
472C2×C4—21no—
473C2×C4—21no—
474C2×C4—22no—
475C2×C4—21no—
476C2×C4—22no—
477C2×C4—22no—
478C2×C4—22no—
479C2×C4—21no—
480C2×C4—21no—
481C2×C4—22no—
482C2×C4—48no—
483C2×C4—42no—
484C2×C4—41no—
485C2×C4—41no—
486C2×C4—41no—
487C2×C4—41no—
488C2×C4—42no—
489C2×C4—41no—
490C2×C4—42no—
491C2×C4—41no—
492C2×C4—42no—
493C2×C4—42no—
494C2×C4—42no—
495C2×C4—42no—
496C2×C4—42no—
497C2×C4—42no—
498C2×C4—84no—
499C2×C4#120×C6112yes—
500C2×C4C22×D4×C6111yescharacteristic
501C2×C4C22×D4×C6111yescharacteristic
502C2×C4C2×#41×C6112yes—
503C2×C42−1+4×C6112yes—
504C8—24no—
505C8—42no—
506C22—24no—
507C22—22no—
508C22—22no—
509C22—22no—
510C22—22no—
511C22—42no—
512C22—42no—
513C22—42no—
514C22—41no—
515C22—41no—
516C22C2×#120×C6111yescharacteristic
517C22C61×#54712yes—
518C22C2×#137×C6111yescharacteristic
519C22C61×#57511yescharacteristic
520C22C61×#57712yes—
521C4—22no—
522C4—41no—
523C4—42no—
524C4—41no—
525C4—42no—
526C4—42no—
527C4—41no—
528C4—42no—
529C4—82no—
530C4—82no—
531C4C61×#52511yescharacteristic
532C4C61×#53611yescharacteristic
533C4C61×#54712yes—
534C2—42no—
535C2—42no—
536C2C61×#240611yescharacteristic
537C2C61×#300711yescharacteristic
538C2C61×#301011yescharacteristic
539C2C61×#334412yes—
540C2C61×#335112yes—
subgroupG/H
center, socleC23×C61C22×D4
commutator, Frattini, cosocleC22×C4C24×C61
prime radicalC22×D4×C61D4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23×C61 ◁ C2×C4×D4×C61 ◁ G

Complements

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

NH ≅ G/Ntype
2. C2×C61×#483C2semidirect
3. C2×C61×#428C2semidirect
5. C61×#2059C2semidirect
6. C61×#2036C2semidirect
7. C61×#1894C2semidirect
9. C61×#1865C2semidirect
10. C61×#1853C2semidirect
11. C61×#1825C2semidirect
235. #24111C61direct (N × H)
265. C2×#483C2×C61semidirect
266. C2×#428C2×C61semidirect
268. #2059C2×C61semidirect
269. #2036C2×C61semidirect
270. #1894C2×C61semidirect
272. #1865C2×C61semidirect
273. #1853C2×C61semidirect
274. #1825C2×C61semidirect
317. C61#24111direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C61×#5522
C2×C61×#4832
C2×C61×#4282
C61×#24872
C61×#20592
C61×#20362
C61×#18942
C61×#18702
C61×#18652
C61×#18532
C61×#18252
#2411161