Finite Group Databasebeta

← C61×#25052 #29528485

Subgroups — C61×#25052 #29528485

Order 15616.

720 conjugacy classes of subgroups.

The subgroup lattice has 474 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C61×#3099C212yesmaximal, complemented
2C61×#3061C211yesmaximal, characteristic
3C61×#2951C211yesmaximal, characteristic, complemented
4C61×#2079C212yesmaximal, complemented
5C61×#2040C212yesmaximal, complemented
6C61×#1974C211yesmaximal, characteristic, complemented
7C61×#1952C211yesmaximal, characteristic, complemented
8C61×#1860C212yesmaximal, complemented
9C61×#1846C211yesmaximal, characteristic, complemented
10C61×#1816C211yesmaximal, characteristic, complemented
11C61×#1764C211yesmaximal, characteristic, complemented
12C2×#135×C61C2211yescharacteristic
13C2×#131×C61C2211yescharacteristic
14C61×#537C2212yes—
15C61×#520C2212yes—
16C61×#485—22no—
17C61×#483—22no—
18C61×#482—22no—
19C61×#481—22no—
20C61×#468—22no—
21C61×#467—21no—
22C61×#466—21no—
23C61×#440—22no—
24C4×Q16×C61—21no—
25C4×D8×C61—21no—
26C2×#108×C61C2211yescharacteristic
27C61×#428C2212yes—
28C2×#107×C61C2211yescharacteristic
29C2×#107×C61C2211yescharacteristic
30C2×#106×C61C2212yes—
31C2×#104×C61C2211yescharacteristic
32C2×#104×C61C2212yes—
33C61×#417C2212yes—
34C61×#417C2212yes—
35C61×#417C2212yes—
36C61×#416C2211yescharacteristic
37C61×#408C2212yes—
38C61×#405C2211yescharacteristic
39C2×C4×C8×C61C2211yescharacteristic
40C61×#382C2212yes—
41C61×#343C2212yes—
42C61×#343C2212yes—
43C61×#341C2211yescharacteristic
44C61×#339C2211yescharacteristic
45C2×#41×C61C2311yescharacteristic
46C2×#41×C61C2312yes—
47#135×C61—22no—
48#135×C61—22no—
49C2×Q16×C61D411yescharacteristic
50C2×SD16×C61D412yes—
51C2×D8×C61D411yescharacteristic
52#131×C61—22no—
53#131×C61—22no—
54C2×M16×C61—21no—
55C2×M16×C61C2311yescharacteristic
56C2×M16×C61C2311yescharacteristic
57C2×M16×C61C2312yes—
58C22×C8×C61—21no—
59C22×C8×C61C2311yescharacteristic
60C22×C8×C61C2311yescharacteristic
61C22⋊Q8×C61—22no—
62C22⋊Q8×C61—22no—
63#121×C61—22no—
64#121×C61—22no—
65#121×C61—42no—
66C4×Q8×C61—22no—
67C4×D4×C61—22no—
68C4×D4×C61—42no—
69#117×C61—22no—
70#117×C61C2311yescharacteristic
71C2×C4⋊C4×C61—22no—
72C2×C4⋊C4×C61C2312yes—
73C2×C4⋊C4×C61C2312yes—
74C2×C42×C61C2311yescharacteristic
75#108×C61—22no—
76#107×C61—21no—
77#107×C61—21no—
78#107×C61—21no—
79#107×C61—21no—
80#107×C61—22no—
81#106×C61—22no—
82#106×C61—22no—
83#106×C61—22no—
84#104×C61—22no—
85#104×C61—24no—
86#103×C61—22no—
87#103×C61—22no—
88#103×C61—22no—
89#103×C61D411yescharacteristic
90#103×C61D411yescharacteristic
91#102×C61—22no—
92#102×C61—22no—
93#102×C61—22no—
94#102×C61D411yescharacteristic
95#102×C61D411yescharacteristic
96C22⋊C8×C61—22no—
97C22⋊C8×C61—22no—
98#97×C61—21no—
99C4×C8×C61—21no—
100C4×C8×C61—22no—
101#95×C61—42no—
102#41×C61—22no—
103#41×C61—21no—
104#41×C61—21no—
105#41×C61—21no—
106#41×C61—22no—
107#41×C61—22no—
108#41×C61—21no—
109#41×C61—22no—
110C2×Q8×C61C2×D411yescharacteristic
111C2×Q8×C61C2×D412yes—
112C2×D4×C61—22no—
113C2×D4×C61—21no—
114C2×D4×C61—42no—
115C2×D4×C61C2×D411yescharacteristic
116C2×D4×C61C2×D412yes—
117C22×C4×C61—21no—
118C22×C4×C61—22no—
119C22×C4×C61—21no—
120C22×C4×C61—22no—
121C22×C4×C61—22no—
122C22×C4×C61—42no—
123C22×C4×C61C2411yescharacteristic
124Q16×C61—22no—
125SD16×C61—24no—
126D8×C61—22no—
127M16×C61—22no—
128M16×C61—21no—
129M16×C61—22no—
130M16×C61—21no—
131M16×C61—21no—
132M16×C61—21no—
133M16×C61—42no—
134C2×C8×C61—21no—
135C2×C8×C61—21no—
136C2×C8×C61—21no—
137C2×C8×C61—22no—
138C2×C8×C61—21no—
139C2×C8×C61—42no—
140C2×C8×C61C2×D411yescharacteristic
141C2×C8×C61C2×D411yescharacteristic
142C2×C8×C61C2×D411yescharacteristic
143C2×C8×C61C2×D411yescharacteristic
144C2×C8×C61#4112yes—
145C2×C8×C61#4112yes—
146C4⋊C4×C61—22no—
147C4⋊C4×C61—22no—
148C4⋊C4×C61—22no—
149C4⋊C4×C61—22no—
150C4⋊C4×C61—42no—
151C4⋊C4×C61—42no—
152C4⋊C4×C61C2×D411yescharacteristic
153C4⋊C4×C61C2×D411yescharacteristic
154C22⋊C4×C61—21no—
155C22⋊C4×C61—42no—
156C22⋊C4×C61—42no—
157C22⋊C4×C61—42no—
158C42×C61—21no—
159C42×C61—21no—
160C42×C61—42no—
161C42×C61#4112yes—
162C23×C61—22no—
163C23×C61—21no—
164C23×C61H(5,𝔽2)11yescharacteristic
165Q8×C61—22no—
166Q8×C61—21no—
167Q8×C61—21no—
168Q8×C61—22no—
169D4×C61—22no—
170D4×C61—21no—
171D4×C61—21no—
172D4×C61—22no—
173D4×C61—42no—
174D4×C61—41no—
175D4×C61—42no—
176D4×C61—41no—
177D4×C61—162no—
178C2×C4×C61—21no—
179C2×C4×C61—21no—
180C2×C4×C61—22no—
181C2×C4×C61—21no—
182C2×C4×C61—22no—
183C2×C4×C61—41no—
184C2×C4×C61—41no—
185C2×C4×C61—41no—
186C2×C4×C61—42no—
187C2×C4×C61—42no—
188C2×C4×C61—41no—
189C2×C4×C61—42no—
190C2×C4×C61—42no—
191C2×C4×C61—82no—
192C2×C4×C61—82no—
193C2×C4×C61—162no—
194C2×C4×C61C22×D411yescharacteristic
195C2×C4×C61C22×D411yescharacteristic
196C2×C4×C61C2×#4111yescharacteristic
197C2×C4×C61C2×#4111yescharacteristic
198C2×C4×C61C2×#4111yescharacteristic
199C2×C4×C612−1+411yescharacteristic
200C8×C61—21no—
201C8×C61—21no—
202C8×C61—41no—
203C8×C61—42no—
204C8×C61—41no—
205C8×C61#12312yes—
206C8×C61#12312yes—
207#25052C6111yesmaximal, characteristic, direct product factor, Sylow
208C22×C61—21no—
209C22×C61—21no—
210C22×C61—41no—
211C22×C61—42no—
212C22×C61—42no—
213C22×C61—41no—
214C22×C61—42no—
215C22×C61—41no—
216C22×C61#54211yescharacteristic
217C22×C61D4211yescharacteristic
218C22×C61#55311yescharacteristic
219C4×C61—42no—
220C4×C61—42no—
221C4×C61—41no—
222C4×C61—41no—
223C4×C61—41no—
224C4×C61—82no—
225C4×C61—82no—
226C4×C61C2×#12311yescharacteristic
227C4×C61#53211yescharacteristic
228C4×C61#54711yescharacteristic
229C4×C61#54711yescharacteristic
230#3099C2×C6112yescomplemented
231#3061C2×C6111yescharacteristic
232#2951C2×C6111yescharacteristic, complemented
233#2079C2×C6112yescomplemented
234#2040C2×C6112yescomplemented
235#1974C2×C6111yescharacteristic, complemented
236#1952C2×C6111yescharacteristic, complemented
237#1860C2×C6112yescomplemented
238#1846C2×C6111yescharacteristic, complemented
239#1816C2×C6111yescharacteristic, complemented
240#1764C2×C6111yescharacteristic, complemented
241C2×C61—21no—
242C2×C61—21no—
243C2×C61—82no—
244C2×C61—81no—
245C2×C61#242211yescharacteristic
246C2×C61#329011yescharacteristic
247C2×C61#329011yescharacteristic
248C2×#135C22×C6111yescharacteristic
249C2×#131C22×C6111yescharacteristic
250#537C22×C6112yes—
251#520C22×C6112yes—
252#485—22no—
253#483—22no—
254#482—22no—
255#481—22no—
256#468—22no—
257#467—21no—
258#466—21no—
259#440—22no—
260C4×Q16—21no—
261C4×D8—21no—
262C2×#108C22×C6111yescharacteristic
263#428C22×C6112yes—
264C2×#107C22×C6111yescharacteristic
265C2×#107C22×C6111yescharacteristic
266C2×#106C22×C6112yes—
267C2×#104C22×C6112yes—
268C2×#104C22×C6111yescharacteristic
269#417C22×C6112yes—
270#417C22×C6112yes—
271#417C22×C6112yes—
272#416C22×C6111yescharacteristic
273#408C22×C6112yes—
274#405C22×C6111yescharacteristic
275C2×C4×C8C22×C6111yescharacteristic
276#382C22×C6112yes—
277#343C22×C6112yes—
278#343C22×C6112yes—
279#341C22×C6111yescharacteristic
280#339C22×C6111yescharacteristic
281C61#2505211yescharacteristic, direct product factor, Sylow
282C2×#41C23×C6112yes—
283C2×#41C23×C6111yescharacteristic
284#135—22no—
285#135—22no—
286C2×Q16D4×C6111yescharacteristic
287C2×SD16D4×C6112yes—
288C2×D8D4×C6111yescharacteristic
289#131—22no—
290#131—22no—
291C2×M16—21no—
292C2×M16C23×C6111yescharacteristic
293C2×M16C23×C6111yescharacteristic
294C2×M16C23×C6112yes—
295C22×C8—21no—
296C22×C8C23×C6111yescharacteristic
297C22×C8C23×C6111yescharacteristic
298C22⋊Q8—22no—
299C22⋊Q8—22no—
300#121—22no—
301#121—22no—
302#121—42no—
303C4×Q8—22no—
304C4×D4—22no—
305C4×D4—42no—
306#117—22no—
307#117C23×C6111yescharacteristic
308C2×C4⋊C4—22no—
309C2×C4⋊C4C23×C6112yes—
310C2×C4⋊C4C23×C6112yes—
311C2×C42C23×C6111yescharacteristic
312#108—22no—
313#107—21no—
314#107—21no—
315#107—21no—
316#107—21no—
317#107—22no—
318#106—22no—
319#106—22no—
320#106—22no—
321#104—24no—
322#104—22no—
323#103—22no—
324#103—22no—
325#103—22no—
326#103D4×C6111yescharacteristic
327#103D4×C6111yescharacteristic
328#102—22no—
329#102—22no—
330#102—22no—
331#102D4×C6111yescharacteristic
332#102D4×C6111yescharacteristic
333C22⋊C8—22no—
334C22⋊C8—22no—
335#97—21no—
336C4×C8—21no—
337C4×C8—22no—
338#95—42no—
339#41—22no—
340#41—21no—
341#41—21no—
342#41—21no—
343#41—22no—
344#41—22no—
345#41—22no—
346#41—21no—
347C2×Q8C2×D4×C6111yescharacteristic
348C2×Q8C2×D4×C6112yes—
349C2×D4—22no—
350C2×D4—21no—
351C2×D4—42no—
352C2×D4C2×D4×C6111yescharacteristic
353C2×D4C2×D4×C6112yes—
354C22×C4—21no—
355C22×C4—22no—
356C22×C4—21no—
357C22×C4—22no—
358C22×C4—22no—
359C22×C4—42no—
360C22×C4C24×C6111yescharacteristic
361Q16—22no—
362SD16—24no—
363D8—22no—
364M16—22no—
365M16—21no—
366M16—22no—
367M16—21no—
368M16—21no—
369M16—21no—
370M16—42no—
371C2×C8—21no—
372C2×C8—21no—
373C2×C8—21no—
374C2×C8—22no—
375C2×C8—21no—
376C2×C8—42no—
377C2×C8C2×D4×C6111yescharacteristic
378C2×C8C2×D4×C6111yescharacteristic
379C2×C8C2×D4×C6111yescharacteristic
380C2×C8C2×D4×C6111yescharacteristic
381C2×C8#41×C6112yes—
382C2×C8#41×C6112yes—
383C4⋊C4—22no—
384C4⋊C4—22no—
385C4⋊C4—22no—
386C4⋊C4—22no—
387C4⋊C4—42no—
388C4⋊C4—42no—
389C4⋊C4C2×D4×C6111yescharacteristic
390C4⋊C4C2×D4×C6111yescharacteristic
391C22⋊C4—21no—
392C22⋊C4—42no—
393C22⋊C4—42no—
394C22⋊C4—42no—
395C42—21no—
396C42—21no—
397C42—42no—
398C42#41×C6112yes—
399C23—22no—
400C23—21no—
401C23H(5,𝔽2)×C6111yescharacteristic
402Q8—21no—
403Q8—21no—
404Q8—22no—
405Q8—22no—
406D4—21no—
407D4—21no—
408D4—22no—
409D4—22no—
410D4—42no—
411D4—41no—
412D4—42no—
413D4—41no—
414D4—162no—
415C2×C4—21no—
416C2×C4—21no—
417C2×C4—22no—
418C2×C4—21no—
419C2×C4—22no—
420C2×C4—41no—
421C2×C4—41no—
422C2×C4—42no—
423C2×C4—42no—
424C2×C4—41no—
425C2×C4—41no—
426C2×C4—42no—
427C2×C4—42no—
428C2×C4—82no—
429C2×C4—82no—
430C2×C4—162no—
431C2×C4C22×D4×C6111yescharacteristic
432C2×C4C22×D4×C6111yescharacteristic
433C2×C4C2×#41×C6111yescharacteristic
434C2×C4C2×#41×C6111yescharacteristic
435C2×C4C2×#41×C6111yescharacteristic
436C2×C42−1+4×C6111yescharacteristic
437C8—21no—
438C8—21no—
439C8—42no—
440C8—41no—
441C8—41no—
442C8#123×C6112yes—
443C8#123×C6112yes—
444C22—21no—
445C22—21no—
446C22—42no—
447C22—41no—
448C22—42no—
449C22—41no—
450C22—42no—
451C22—41no—
452C22C61×#54211yescharacteristic
453C22D42×C6111yescharacteristic
454C22C61×#55311yescharacteristic
455C4—42no—
456C4—41no—
457C4—41no—
458C4—42no—
459C4—41no—
460C4—82no—
461C4—82no—
462C4C2×#123×C6111yescharacteristic
463C4C61×#53211yescharacteristic
464C4C61×#54711yescharacteristic
465C4C61×#54711yescharacteristic
466C2—21no—
467C2—21no—
468C2—82no—
469C2—81no—
470C2C61×#242211yescharacteristic
471C2C61×#329011yescharacteristic
472C2C61×#329011yescharacteristic
subgroupG/H
center, socleC22×C61D42
commutator, Frattini, cosocleC22×C4C24×C61
prime radicalC61×#3061C2
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22×C61 ◁ C22×C4×C61 ◁ G

Complements

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

NH ≅ G/Ntype
1. C61×#3099C2semidirect
3. C61×#2951C2semidirect
4. C61×#2079C2semidirect
5. C61×#2040C2semidirect
6. C61×#1974C2semidirect
7. C61×#1952C2semidirect
8. C61×#1860C2semidirect
9. C61×#1846C2semidirect
10. C61×#1816C2semidirect
11. C61×#1764C2semidirect
207. #25052C61direct (N × H)
230. #3099C2×C61semidirect
232. #2951C2×C61semidirect
233. #2079C2×C61semidirect
234. #2040C2×C61semidirect
235. #1974C2×C61semidirect
236. #1952C2×C61semidirect
237. #1860C2×C61semidirect
238. #1846C2×C61semidirect
239. #1816C2×C61semidirect
240. #1764C2×C61semidirect
281. C61#25052direct (N × H)

Maximal subgroups

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

subgroupindex
C61×#30992
C61×#30612
C61×#29512
C61×#20792
C61×#20402
C61×#19742
C61×#19522
C61×#18602
C61×#18462
C61×#18162
C61×#17642
#2505261