Finite Group Databasebeta

← C61×#24025 #29527458

Subgroups — C61×#24025 #29527458

Order 15616.

1286 conjugacy classes of subgroups.

The subgroup lattice has 520 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C61×#3485C211yesmaximal, characteristic
2C61×#3092C212yesmaximal, complemented
3C61×#2311C211yesmaximal, characteristic, complemented
4C61×#2038C214yesmaximal, complemented
5C61×#1897C212yesmaximal, complemented
6C61×#1856C212yesmaximal, complemented
7C61×#1792C212yesmaximal, complemented
8C61×#1785C211yesmaximal, characteristic, complemented
9C2×H(5,𝔽2)×C61C2211yescharacteristic
10C22×#41×C61C2211yescharacteristic
11C61×#547—22no—
12C61×#547—22no—
13C61×#546—22no—
14C61×#546—22no—
15C61×#536C2212yes—
16C61×#534—22no—
17C61×#525C2212yes—
18C2×#123×C61—22no—
19C61×#518C2211yescharacteristic
20C2×#117×C61C2211yescharacteristic, complemented
21C22×C4⋊C4×C61C2211yescharacteristic, complemented
22C61×#484—22no—
23C61×#483—22no—
24C61×#481—22no—
25C61×#480—22no—
26C61×#428C2214yescomplemented
27C2×#104×C61C2214yes—
28C61×#417C2214yes—
29C2×#101×C61C2212yescomplemented
30C2×#100×C61C2212yes—
31C61×#407C2212yescomplemented
32C61×#391C2212yescomplemented
33C61×#390C2212yes—
34C61×#386—22no—
35C61×#380—22no—
36C61×#339C2214yescomplemented
37H(5,𝔽2)×C61—24no—
38H(5,𝔽2)×C61—42no—
39C2×#41×C61—22no—
40C2×#41×C61—24no—
41C2×#41×C61—22no—
42C2×#41×C61—22no—
43C2×#41×C61—42no—
44C2×#41×C61C2312yes—
45C22×Q8×C61C2311yescharacteristic
46C22×D4×C61—22no—
47C22×D4×C61—22no—
48C22×D4×C61D412yes—
49C22×D4×C61C2311yescharacteristic
50C23×C4×C61—21no—
51C23×C4×C61D412yes—
52C23×C4×C61C2311yescharacteristic
53C2×M16×C61C2314yes—
54C4⋊Q8×C61—22no—
55#124×C61—22no—
56#123×C61—24no—
57#123×C61—24no—
58#123×C61—24no—
59#123×C61—24no—
60#123×C61—22no—
61C22⋊Q8×C61—22no—
62C22⋊Q8×C61—22no—
63C22⋊Q8×C61D412yes—
64C22⋊Q8×C61D412yes—
65#121×C61—24no—
66#121×C61—24no—
67#121×C61D412yes—
68#121×C61D412yes—
69#120×C61—24no—
70#120×C61—22no—
71C4×D4×C61—22no—
72C4×D4×C61—22no—
73C4×D4×C61—22no—
74C4×D4×C61—22no—
75#117×C61—42no—
76#117×C61C2312yes—
77C2×C4⋊C4×C61—22no—
78C2×C4⋊C4×C61—22no—
79C2×C4⋊C4×C61—22no—
80C2×C4⋊C4×C61—22no—
81C2×C4⋊C4×C61—21no—
82C2×C4⋊C4×C61—21no—
83C2×C4⋊C4×C61C2312yes—
84C2×C22⋊C4×C61—22no—
85C2×C22⋊C4×C61—22no—
86C2×C22⋊C4×C61—22no—
87C2×C22⋊C4×C61—21no—
88C2×C22⋊C4×C61—24no—
89C2×C42×C61C2312yes—
90#107×C61—24no—
91#106×C61—24no—
92#104×C61—44no—
93#103×C61—24no—
94#102×C61—24no—
95#101×C61—42no—
96#100×C61—24no—
97C22⋊C8×C61—22no—
98C22⋊C8×C61—22no—
99#95×C61—24no—
100C24×C61—22no—
101C24×C61C2×D411yescharacteristic
102C24×C61C2×D412yes—
103#41×C61—24no—
104#41×C61—24no—
105#41×C61—48no—
106#41×C61—48no—
107#41×C61—48no—
108C2×Q8×C61—22no—
109C2×Q8×C61—22no—
110C2×Q8×C61—22no—
111C2×Q8×C61C2×D412yes—
112C2×D4×C61—24no—
113C2×D4×C61—24no—
114C2×D4×C61—24no—
115C2×D4×C61—22no—
116C2×D4×C61—22no—
117C2×D4×C61—22no—
118C2×D4×C61—22no—
119C2×D4×C61—22no—
120C2×D4×C61—42no—
121C2×D4×C61—48no—
122C2×D4×C61—44no—
123C2×D4×C61—44no—
124C2×D4×C61C2×D412yes—
125C2×D4×C61#4114yes—
126C22×C4×C61—22no—
127C22×C4×C61—22no—
128C22×C4×C61—22no—
129C22×C4×C61—24no—
130C22×C4×C61—22no—
131C22×C4×C61—22no—
132C22×C4×C61—22no—
133C22×C4×C61—22no—
134C22×C4×C61—22no—
135C22×C4×C61—22no—
136C22×C4×C61—22no—
137C22×C4×C61—42no—
138C22×C4×C61—42no—
139C22×C4×C61—82no—
140C22×C4×C61C2×D412yes—
141C22×C4×C61C2×D411yescharacteristic
142C22×C4×C61C2×D412yes—
143C22×C4×C61C2×D412yes—
144C22×C4×C61C2411yescharacteristic
145M16×C61—24no—
146M16×C61—24no—
147C2×C8×C61—24no—
148C4⋊C4×C61—24no—
149C4⋊C4×C61—24no—
150C4⋊C4×C61—22no—
151C4⋊C4×C61—22no—
152C4⋊C4×C61—24no—
153C4⋊C4×C61—42no—
154C4⋊C4×C61—41no—
155C4⋊C4×C61—41no—
156C4⋊C4×C61C2×D412yes—
157C4⋊C4×C61C2×D412yes—
158C22⋊C4×C61—24no—
159C22⋊C4×C61—24no—
160C22⋊C4×C61—24no—
161C22⋊C4×C61—22no—
162C22⋊C4×C61—22no—
163C22⋊C4×C61—22no—
164C22⋊C4×C61—22no—
165C22⋊C4×C61—22no—
166C22⋊C4×C61—42no—
167C22⋊C4×C61—84no—
168C42×C61—22no—
169C42×C61—24no—
170C23×C61—22no—
171C23×C61—21no—
172C23×C61—21no—
173C23×C61—22no—
174C23×C61—22no—
175C23×C61—41no—
176C23×C61—41no—
177C23×C61—42no—
178C23×C61—42no—
179C23×C61—44no—
180C23×C61—82no—
181C23×C61—82no—
182C23×C61#12012yes—
183C23×C61#12314yes—
184C23×C61C22×D411yescharacteristic
185Q8×C61—24no—
186Q8×C61—44no—
187D4×C61—28no—
188D4×C61—24no—
189D4×C61—48no—
190D4×C61—48no—
191D4×C61—42no—
192D4×C61—42no—
193D4×C61—82no—
194D4×C61—82no—
195C2×C4×C61—21no—
196C2×C4×C61—21no—
197C2×C4×C61—24no—
198C2×C4×C61—24no—
199C2×C4×C61—22no—
200C2×C4×C61—22no—
201C2×C4×C61—22no—
202C2×C4×C61—22no—
203C2×C4×C61—22no—
204C2×C4×C61—22no—
205C2×C4×C61—44no—
206C2×C4×C61—44no—
207C2×C4×C61—44no—
208C2×C4×C61—44no—
209C2×C4×C61—42no—
210C2×C4×C61—44no—
211C2×C4×C61—42no—
212C2×C4×C61—42no—
213C2×C4×C61—42no—
214C2×C4×C61—82no—
215C2×C4×C61—82no—
216C2×C4×C61—84no—
217C2×C4×C61#12011yescharacteristic
218C2×C4×C61#12011yescharacteristic
219C2×C4×C61C22×D412yes—
220C2×C4×C61C2×#4112yes—
221C2×C4×C61H(5,𝔽2)12yes—
222C8×C61—44no—
223#24025C6111yesmaximal, characteristic, direct product factor, Sylow
224C22×C61—21no—
225C22×C61—21no—
226C22×C61—22no—
227C22×C61—24no—
228C22×C61—22no—
229C22×C61—22no—
230C22×C61—44no—
231C22×C61—42no—
232C22×C61—42no—
233C22×C61—42no—
234C22×C61—42no—
235C22×C61—42no—
236C22×C61—42no—
237C22×C61—42no—
238C22×C61—82no—
239C22×C61—164no—
240C22×C61C2×#12011yescharacteristic
241C22×C61C2×#12311yescharacteristic
242C22×C61#53411yescharacteristic
243C4×C61—21no—
244C4×C61—21no—
245C4×C61—42no—
246C4×C61—42no—
247C4×C61—42no—
248C4×C61—42no—
249C4×C61—82no—
250C4×C61—82no—
251C4×C61#54614yes—
252#3485C2×C6111yescharacteristic
253#3092C2×C6112yescomplemented
254#2311C2×C6111yescharacteristic, complemented
255#2038C2×C6114yescomplemented
256#1897C2×C6112yescomplemented
257#1856C2×C6112yescomplemented
258#1792C2×C6112yescomplemented
259#1785C2×C6111yescharacteristic, complemented
260C2×C61—21no—
261C2×C61—21no—
262C2×C61—42no—
263C2×C61—44no—
264C2×C61—82no—
265C2×C61#240511yescharacteristic
266C2×C61#302111yescharacteristic
267C2×C61#302211yescharacteristic
268C2×H(5,𝔽2)C22×C6111yescharacteristic
269C22×#41C22×C6111yescharacteristic
270#547—22no—
271#547—22no—
272#546—22no—
273#546—22no—
274#536C22×C6112yes—
275#534—22no—
276#525C22×C6112yes—
277C2×#123—22no—
278#518C22×C6111yescharacteristic
279C2×#117C22×C6111yescharacteristic, complemented
280C22×C4⋊C4C22×C6111yescharacteristic, complemented
281#484—22no—
282#483—22no—
283#481—22no—
284#480—22no—
285#428C22×C6114yescomplemented
286C2×#104C22×C6114yes—
287#417C22×C6114yes—
288C2×#101C22×C6112yescomplemented
289C2×#100C22×C6112yes—
290#407C22×C6112yescomplemented
291#391C22×C6112yescomplemented
292#390C22×C6112yes—
293#386—22no—
294#380—22no—
295#339C22×C6114yescomplemented
296C61#2402511yescharacteristic, direct product factor, Sylow
297H(5,𝔽2)—24no—
298H(5,𝔽2)—42no—
299C2×#41—22no—
300C2×#41—24no—
301C2×#41—22no—
302C2×#41—22no—
303C2×#41—42no—
304C2×#41C23×C6112yes—
305C22×Q8C23×C6111yescharacteristic
306C22×D4—22no—
307C22×D4—22no—
308C22×D4D4×C6112yes—
309C22×D4C23×C6111yescharacteristic
310C23×C4—21no—
311C23×C4D4×C6112yes—
312C23×C4C23×C6111yescharacteristic
313C2×M16C23×C6114yes—
314C4⋊Q8—22no—
315#124—22no—
316#123—24no—
317#123—24no—
318#123—24no—
319#123—22no—
320#123—24no—
321C22⋊Q8—22no—
322C22⋊Q8—22no—
323C22⋊Q8D4×C6112yes—
324C22⋊Q8D4×C6112yes—
325#121—24no—
326#121—24no—
327#121D4×C6112yes—
328#121D4×C6112yes—
329#120—24no—
330#120—22no—
331C4×D4—22no—
332C4×D4—22no—
333C4×D4—22no—
334C4×D4—22no—
335#117—42no—
336#117C23×C6112yes—
337C2×C4⋊C4—22no—
338C2×C4⋊C4—22no—
339C2×C4⋊C4—22no—
340C2×C4⋊C4—22no—
341C2×C4⋊C4—21no—
342C2×C4⋊C4—21no—
343C2×C4⋊C4C23×C6112yes—
344C2×C22⋊C4—22no—
345C2×C22⋊C4—22no—
346C2×C22⋊C4—22no—
347C2×C22⋊C4—21no—
348C2×C22⋊C4—24no—
349C2×C42C23×C6112yes—
350#107—24no—
351#106—24no—
352#104—44no—
353#103—24no—
354#102—24no—
355#101—42no—
356#100—24no—
357C22⋊C8—22no—
358C22⋊C8—22no—
359#95—24no—
360C24—22no—
361C24C2×D4×C6111yescharacteristic
362C24C2×D4×C6112yes—
363#41—24no—
364#41—24no—
365#41—48no—
366#41—48no—
367#41—48no—
368C2×Q8—22no—
369C2×Q8—22no—
370C2×Q8—22no—
371C2×Q8C2×D4×C6112yes—
372C2×D4—24no—
373C2×D4—24no—
374C2×D4—24no—
375C2×D4—22no—
376C2×D4—22no—
377C2×D4—22no—
378C2×D4—22no—
379C2×D4—22no—
380C2×D4—42no—
381C2×D4—48no—
382C2×D4—44no—
383C2×D4—44no—
384C2×D4C2×D4×C6112yes—
385C2×D4#41×C6114yes—
386C22×C4—22no—
387C22×C4—22no—
388C22×C4—22no—
389C22×C4—24no—
390C22×C4—22no—
391C22×C4—22no—
392C22×C4—22no—
393C22×C4—22no—
394C22×C4—22no—
395C22×C4—22no—
396C22×C4—22no—
397C22×C4—42no—
398C22×C4—42no—
399C22×C4—82no—
400C22×C4C2×D4×C6112yes—
401C22×C4C2×D4×C6111yescharacteristic
402C22×C4C2×D4×C6112yes—
403C22×C4C2×D4×C6112yes—
404C22×C4C24×C6111yescharacteristic
405M16—24no—
406M16—24no—
407C2×C8—24no—
408C4⋊C4—24no—
409C4⋊C4—24no—
410C4⋊C4—22no—
411C4⋊C4—22no—
412C4⋊C4—24no—
413C4⋊C4—42no—
414C4⋊C4—41no—
415C4⋊C4—41no—
416C4⋊C4C2×D4×C6112yes—
417C4⋊C4C2×D4×C6112yes—
418C22⋊C4—24no—
419C22⋊C4—24no—
420C22⋊C4—24no—
421C22⋊C4—22no—
422C22⋊C4—22no—
423C22⋊C4—22no—
424C22⋊C4—22no—
425C22⋊C4—22no—
426C22⋊C4—42no—
427C22⋊C4—84no—
428C42—22no—
429C42—24no—
430C23—21no—
431C23—22no—
432C23—21no—
433C23—22no—
434C23—22no—
435C23—41no—
436C23—41no—
437C23—42no—
438C23—42no—
439C23—44no—
440C23—82no—
441C23—82no—
442C23#120×C6112yes—
443C23#123×C6114yes—
444C23C22×D4×C6111yescharacteristic
445Q8—24no—
446Q8—44no—
447D4—28no—
448D4—24no—
449D4—48no—
450D4—48no—
451D4—42no—
452D4—42no—
453D4—82no—
454D4—82no—
455C2×C4—21no—
456C2×C4—21no—
457C2×C4—24no—
458C2×C4—24no—
459C2×C4—22no—
460C2×C4—22no—
461C2×C4—22no—
462C2×C4—22no—
463C2×C4—22no—
464C2×C4—22no—
465C2×C4—44no—
466C2×C4—44no—
467C2×C4—44no—
468C2×C4—44no—
469C2×C4—42no—
470C2×C4—44no—
471C2×C4—42no—
472C2×C4—42no—
473C2×C4—42no—
474C2×C4—82no—
475C2×C4—82no—
476C2×C4—84no—
477C2×C4#120×C6111yescharacteristic
478C2×C4#120×C6111yescharacteristic
479C2×C4C22×D4×C6112yes—
480C2×C4C2×#41×C6112yes—
481C2×C4H(5,𝔽2)×C6112yes—
482C8—44no—
483C22—21no—
484C22—21no—
485C22—22no—
486C22—24no—
487C22—22no—
488C22—22no—
489C22—44no—
490C22—42no—
491C22—42no—
492C22—42no—
493C22—42no—
494C22—42no—
495C22—42no—
496C22—42no—
497C22—82no—
498C22—164no—
499C22C2×#120×C6111yescharacteristic
500C22C2×#123×C6111yescharacteristic
501C22C61×#53411yescharacteristic
502C4—21no—
503C4—21no—
504C4—42no—
505C4—42no—
506C4—42no—
507C4—42no—
508C4—82no—
509C4—82no—
510C4C61×#54614yes—
511C2—21no—
512C2—21no—
513C2—42no—
514C2—44no—
515C2—82no—
516C2C61×#240511yescharacteristic
517C2C61×#302111yescharacteristic
518C2C61×#302211yescharacteristic
subgroupG/H
center, socleC22×C61C2×#120
commutator, Frattini, cosocleC22×C4C24×C61
prime radicalC61×#3485C2
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22×C61 ◁ C23×C4×C61 ◁ G

Complements

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

NH ≅ G/Ntype
2. C61×#3092C2semidirect
3. C61×#2311C2semidirect
4. C61×#2038C2semidirect
5. C61×#1897C2semidirect
6. C61×#1856C2semidirect
7. C61×#1792C2semidirect
8. C61×#1785C2semidirect
20. C2×#117×C61C22semidirect
21. C22×C4⋊C4×C61C22semidirect
26. C61×#428C22semidirect
29. C2×#101×C61C22semidirect
31. C61×#407C22semidirect
32. C61×#391C22semidirect
36. C61×#339C22semidirect
223. #24025C61direct (N × H)
253. #3092C2×C61semidirect
254. #2311C2×C61semidirect
255. #2038C2×C61semidirect
256. #1897C2×C61semidirect
257. #1856C2×C61semidirect
258. #1792C2×C61semidirect
259. #1785C2×C61semidirect
279. C2×#117C22×C61semidirect
280. C22×C4⋊C4C22×C61semidirect
285. #428C22×C61semidirect
288. C2×#101C22×C61semidirect
290. #407C22×C61semidirect
291. #391C22×C61semidirect
295. #339C22×C61semidirect
296. C61#24025direct (N × H)

Maximal subgroups

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

subgroupindex
C61×#34852
C61×#30922
C61×#23112
C61×#20382
C61×#18972
C61×#18562
C61×#17922
C61×#17852
#2402561