Finite Group Databasebeta

← C61×#24720 #29528153

Subgroups — C61×#24720 #29528153

Order 15616.

706 conjugacy classes of subgroups.

The subgroup lattice has 530 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C61×#3079C211yesmaximal, characteristic, complemented
2C2×C61×#474C211yesmaximal, characteristic, complemented
3C4×AGL(1,ℤ/8)×C61C211yesmaximal, characteristic
4C61×#2925C211yesmaximal, characteristic, complemented
5C61×#2078C212yesmaximal, complemented
6C61×#2051C212yesmaximal
7C61×#1997C211yesmaximal, characteristic, complemented
8C61×#1952C212yesmaximal, complemented
9C61×#1950C211yesmaximal, characteristic, complemented
10C61×#1931C212yesmaximal, complemented
11C61×#1800C211yesmaximal, characteristic, complemented
12C2×AGL(1,ℤ/8)×C61C2211yescharacteristic
13C2×C4⋊Q8×C61C2211yescharacteristic
14C4×#41×C61C2211yescharacteristic
15C2×C4×D4×C61C2211yescharacteristic
16C61×#477—21no—
17C61×#476—21no—
18C61×#475—21no—
19C61×#474—21no—
20C61×#474—21no—
21C61×#474—21no—
22C61×#474—22no—
23C61×#442—21no—
24C61×#440—21no—
25C4×SD16×C61—21no—
26C4×D8×C61—21no—
27C61×#433—22no—
28C61×#432—21no—
29C61×#431—21no—
30C2×#108×C61C2212yes—
31C61×#428C2212yes—
32C61×#428C2211yescharacteristic
33C2×#107×C61C2212yes—
34C61×#423C2211yescharacteristic
35C61×#423C2211yescharacteristic
36C2×C4⋊C8×C61C2211yescharacteristic
37C61×#418C2211yescharacteristic
38C61×#417C2212yes—
39C61×#417C2211yescharacteristic
40C2×#102×C61C2212yes—
41C2×#100×C61C2212yes—
42C61×#408C2212yes—
43C61×#405C2212yes—
44C4×M16×C61C2211yescharacteristic
45C61×#343C2212yes—
46C61×#343C2212yes—
47C61×#343C2212yes—
48C61×#341C2212yes—
49C2×#41×C61C2311yescharacteristic
50C22×Q8×C61—21no—
51C22×D4×C61C2311yescharacteristic
52C23×C4×C61—21no—
53AGL(1,ℤ/8)×C61—22no—
54AGL(1,ℤ/8)×C61—22no—
55C2×SD16×C61—21no—
56C2×D8×C61—21no—
57C2×M16×C61C2311yescharacteristic
58C2×M16×C61C2312yes—
59C2×M16×C61C2312yes—
60C2×M16×C61C2311yescharacteristic
61C22×C8×C61C2312yes—
62C4⋊Q8×C61—21no—
63C4⋊Q8×C61—21no—
64C4⋊Q8×C61—21no—
65C4⋊Q8×C61—21no—
66C4×Q8×C61D411yescharacteristic
67C4×Q8×C61D411yescharacteristic
68C4×D4×C61—21no—
69C4×D4×C61—21no—
70C4×D4×C61—21no—
71C4×D4×C61—21no—
72C4×D4×C61—21no—
73C4×D4×C61—21no—
74C4×D4×C61D411yescharacteristic
75C4×D4×C61D411yescharacteristic
76#117×C61—21no—
77#117×C61C2311yescharacteristic
78C2×C4⋊C4×C61—21no—
79C2×C4⋊C4×C61C2312yes—
80C2×C4⋊C4×C61C2311yescharacteristic
81C2×C22⋊C4×C61—21no—
82C2×C42×C61—21no—
83C2×C42×C61C2311yescharacteristic
84#108×C61—24no—
85#107×C61—22no—
86#107×C61—22no—
87#107×C61—21no—
88#107×C61—22no—
89#106×C61—21no—
90#106×C61—22no—
91C4⋊C8×C61—21no—
92C4⋊C8×C61—21no—
93C4⋊C8×C61—21no—
94C4⋊C8×C61—21no—
95C4⋊C8×C61D411yescharacteristic
96C4⋊C8×C61D411yescharacteristic
97C4⋊C8×C61D412yes—
98#103×C61—21no—
99#103×C61—22no—
100#102×C61—21no—
101#102×C61—22no—
102#102×C61—22no—
103#102×C61—22no—
104#102×C61—21no—
105#102×C61—21no—
106#100×C61—24no—
107C22⋊C8×C61—22no—
108C22⋊C8×C61—22no—
109#97×C61—22no—
110#97×C61—21no—
111C4×C8×C61—22no—
112C4×C8×C61—21no—
113C24×C61—21no—
114#41×C61—21no—
115#41×C61—21no—
116#41×C61—21no—
117#41×C61—21no—
118C2×Q8×C61—41no—
119C2×Q8×C61—41no—
120C2×Q8×C61—82no—
121C2×Q8×C61C2×D411yescharacteristic
122C2×D4×C61—21no—
123C2×D4×C61—21no—
124C2×D4×C61—21no—
125C2×D4×C61—21no—
126C2×D4×C61—21no—
127C2×D4×C61—21no—
128C2×D4×C61C2×D411yescharacteristic
129C2×D4×C61#4112yes—
130C22×C4×C61—21no—
131C22×C4×C61—21no—
132C22×C4×C61—21no—
133C22×C4×C61—21no—
134C22×C4×C61—22no—
135C22×C4×C61—41no—
136C22×C4×C61—41no—
137C22×C4×C61—82no—
138C22×C4×C61#4111yescharacteristic
139C22×C4×C61#4111yescharacteristic
140C22×C4×C61C2411yescharacteristic
141SD16×C61—24no—
142D8×C61—24no—
143M16×C61—21no—
144M16×C61—22no—
145M16×C61—22no—
146M16×C61—22no—
147M16×C61—21no—
148M16×C61—21no—
149M16×C61—22no—
150M16×C61#4112yes—
151C2×C8×C61—21no—
152C2×C8×C61—22no—
153C2×C8×C61—22no—
154C2×C8×C61—22no—
155C2×C8×C61—21no—
156C2×C8×C61—22no—
157C2×C8×C61C2×D412yes—
158C2×C8×C61C2×D412yes—
159C4⋊C4×C61—21no—
160C4⋊C4×C61—21no—
161C4⋊C4×C61—22no—
162C4⋊C4×C61—21no—
163C4⋊C4×C61—22no—
164C4⋊C4×C61—21no—
165C4⋊C4×C61—41no—
166C4⋊C4×C61—41no—
167C4⋊C4×C61C2×D411yescharacteristic
168C4⋊C4×C61C2×D411yescharacteristic
169C22⋊C4×C61—21no—
170C22⋊C4×C61—21no—
171C22⋊C4×C61—21no—
172C22⋊C4×C61—41no—
173C22⋊C4×C61—41no—
174C42×C61—21no—
175C42×C61—21no—
176C42×C61—21no—
177C42×C61C2×D411yescharacteristic
178C42×C61C2×D411yescharacteristic
179C42×C61C2×D411yescharacteristic
180C42×C61C2×D411yescharacteristic
181C23×C61—21no—
182C23×C61—22no—
183C23×C61—41no—
184C23×C61—41no—
185C23×C61—81no—
186C23×C61—81no—
187C23×C61C2×#4111yescharacteristic
188Q8×C61—21no—
189Q8×C61—21no—
190Q8×C61—82no—
191Q8×C61—82no—
192D4×C61—21no—
193D4×C61—22no—
194D4×C61—21no—
195D4×C61—22no—
196D4×C61—41no—
197D4×C61—41no—
198D4×C61—41no—
199D4×C61—41no—
200C2×C4×C61—22no—
201C2×C4×C61—21no—
202C2×C4×C61—21no—
203C2×C4×C61—21no—
204C2×C4×C61—41no—
205C2×C4×C61—41no—
206C2×C4×C61—41no—
207C2×C4×C61—42no—
208C2×C4×C61—41no—
209C2×C4×C61—42no—
210C2×C4×C61—81no—
211C2×C4×C61—82no—
212C2×C4×C61—82no—
213C2×C4×C61—81no—
214C2×C4×C61—82no—
215C2×C4×C61#12111yescharacteristic
216C2×C4×C61#12111yescharacteristic
217C2×C4×C61#12111yescharacteristic
218C2×C4×C61#12111yescharacteristic
219C2×C4×C61C22×D411yescharacteristic
220C2×C4×C61C22×D411yescharacteristic
221C2×C4×C61C2×#4111yescharacteristic
222C2×C4×C61C2×#4111yescharacteristic
223C2×C4×C612−1+411yescharacteristic
224C2×C4×C612−1+411yescharacteristic
225C8×C61—22no—
226C8×C61—22no—
227C8×C61—22no—
228C8×C61—42no—
229C8×C61—41no—
230C8×C61—42no—
231#24720C6111yesmaximal, characteristic, direct product factor, Sylow
232C22×C61—21no—
233C22×C61—21no—
234C22×C61—41no—
235C22×C61—42no—
236C22×C61—41no—
237C22×C61—41no—
238C22×C61—82no—
239C22×C61—82no—
240C22×C61—81no—
241C22×C61—81no—
242C22×C61C2×#12111yescharacteristic
243C22×C61#55211yescharacteristic
244C22×C61#55611yescharacteristic
245C4×C61—21no—
246C4×C61—21no—
247C4×C61—21no—
248C4×C61—41no—
249C4×C61—41no—
250C4×C61—41no—
251C4×C61—81no—
252C4×C61—82no—
253C4×C61#47212yes—
254C4×C61#52511yescharacteristic
255C4×C61#53611yescharacteristic
256C4×C61#54711yescharacteristic
257C4×C61#54811yescharacteristic
258#3079C2×C6111yescharacteristic, complemented
259C2×#474C2×C6111yescharacteristic, complemented
260C4×AGL(1,ℤ/8)C2×C6111yescharacteristic
261#2925C2×C6111yescharacteristic, complemented
262#2078C2×C6112yescomplemented
263#2051C2×C6112yes—
264#1997C2×C6111yescharacteristic, complemented
265#1952C2×C6112yescomplemented
266#1950C2×C6111yescharacteristic, complemented
267#1931C2×C6112yescomplemented
268#1800C2×C6111yescharacteristic, complemented
269C2×C61—21no—
270C2×C61—21no—
271C2×C61—81no—
272C2×C61—82no—
273C2×C61#241511yescharacteristic
274C2×C61C2×#47211yescharacteristic
275C2×C61#306911yescharacteristic
276C2×AGL(1,ℤ/8)C22×C6111yescharacteristic
277C2×C4⋊Q8C22×C6111yescharacteristic
278C4×#41C22×C6111yescharacteristic
279C2×C4×D4C22×C6111yescharacteristic
280#477—21no—
281#476—21no—
282#475—21no—
283#474—21no—
284#474—21no—
285#474—21no—
286#474—22no—
287#442—21no—
288#440—21no—
289C4×SD16—21no—
290C4×D8—21no—
291#433—22no—
292#432—21no—
293#431—21no—
294C2×#108C22×C6112yes—
295#428C22×C6112yes—
296#428C22×C6111yescharacteristic
297C2×#107C22×C6112yes—
298#423C22×C6111yescharacteristic
299#423C22×C6111yescharacteristic
300C2×C4⋊C8C22×C6111yescharacteristic
301#418C22×C6111yescharacteristic
302#417C22×C6112yes—
303#417C22×C6111yescharacteristic
304C2×#102C22×C6112yes—
305C2×#100C22×C6112yes—
306#408C22×C6112yes—
307#405C22×C6112yes—
308C4×M16C22×C6111yescharacteristic
309#343C22×C6112yes—
310#343C22×C6112yes—
311#343C22×C6112yes—
312#341C22×C6112yes—
313C61#2472011yescharacteristic, direct product factor, Sylow
314C2×#41C23×C6111yescharacteristic
315C22×Q8—21no—
316C22×D4C23×C6111yescharacteristic
317C23×C4—21no—
318AGL(1,ℤ/8)—22no—
319AGL(1,ℤ/8)—22no—
320C2×SD16—21no—
321C2×D8—21no—
322C2×M16C23×C6111yescharacteristic
323C2×M16C23×C6112yes—
324C2×M16C23×C6112yes—
325C2×M16C23×C6111yescharacteristic
326C22×C8C23×C6112yes—
327C4⋊Q8—21no—
328C4⋊Q8—21no—
329C4⋊Q8—21no—
330C4⋊Q8—21no—
331C4×Q8D4×C6111yescharacteristic
332C4×Q8D4×C6111yescharacteristic
333C4×D4—21no—
334C4×D4—21no—
335C4×D4—21no—
336C4×D4—21no—
337C4×D4—21no—
338C4×D4—21no—
339C4×D4D4×C6111yescharacteristic
340C4×D4D4×C6111yescharacteristic
341#117—21no—
342#117C23×C6111yescharacteristic
343C2×C4⋊C4—21no—
344C2×C4⋊C4C23×C6111yescharacteristic
345C2×C4⋊C4C23×C6112yes—
346C2×C22⋊C4—21no—
347C2×C42—21no—
348C2×C42C23×C6111yescharacteristic
349#108—24no—
350#107—22no—
351#107—22no—
352#107—21no—
353#107—22no—
354#106—21no—
355#106—22no—
356C4⋊C8—21no—
357C4⋊C8—21no—
358C4⋊C8—21no—
359C4⋊C8—21no—
360C4⋊C8D4×C6111yescharacteristic
361C4⋊C8D4×C6111yescharacteristic
362C4⋊C8D4×C6112yes—
363#103—21no—
364#103—22no—
365#102—21no—
366#102—22no—
367#102—22no—
368#102—22no—
369#102—21no—
370#102—21no—
371#100—24no—
372C22⋊C8—22no—
373C22⋊C8—22no—
374#97—22no—
375#97—21no—
376C4×C8—22no—
377C4×C8—21no—
378C24—21no—
379#41—21no—
380#41—21no—
381#41—21no—
382#41—21no—
383C2×Q8—41no—
384C2×Q8—41no—
385C2×Q8—82no—
386C2×Q8C2×D4×C6111yescharacteristic
387C2×D4—21no—
388C2×D4—21no—
389C2×D4—21no—
390C2×D4—21no—
391C2×D4—21no—
392C2×D4—21no—
393C2×D4C2×D4×C6111yescharacteristic
394C2×D4#41×C6112yes—
395C22×C4—21no—
396C22×C4—21no—
397C22×C4—21no—
398C22×C4—21no—
399C22×C4—22no—
400C22×C4—41no—
401C22×C4—41no—
402C22×C4—82no—
403C22×C4#41×C6111yescharacteristic
404C22×C4#41×C6111yescharacteristic
405C22×C4C24×C6111yescharacteristic
406SD16—24no—
407D8—24no—
408M16—21no—
409M16—22no—
410M16—22no—
411M16—22no—
412M16—21no—
413M16—21no—
414M16—22no—
415M16#41×C6112yes—
416C2×C8—21no—
417C2×C8—22no—
418C2×C8—22no—
419C2×C8—22no—
420C2×C8—21no—
421C2×C8—22no—
422C2×C8C2×D4×C6112yes—
423C2×C8C2×D4×C6112yes—
424C4⋊C4—21no—
425C4⋊C4—21no—
426C4⋊C4—22no—
427C4⋊C4—21no—
428C4⋊C4—22no—
429C4⋊C4—21no—
430C4⋊C4—41no—
431C4⋊C4—41no—
432C4⋊C4C2×D4×C6111yescharacteristic
433C4⋊C4C2×D4×C6111yescharacteristic
434C22⋊C4—21no—
435C22⋊C4—21no—
436C22⋊C4—21no—
437C22⋊C4—41no—
438C22⋊C4—41no—
439C42—21no—
440C42—21no—
441C42—21no—
442C42C2×D4×C6111yescharacteristic
443C42C2×D4×C6111yescharacteristic
444C42C2×D4×C6111yescharacteristic
445C42C2×D4×C6111yescharacteristic
446C23—21no—
447C23—22no—
448C23—41no—
449C23—41no—
450C23—81no—
451C23—81no—
452C23C2×#41×C6111yescharacteristic
453Q8—21no—
454Q8—21no—
455Q8—82no—
456Q8—82no—
457D4—21no—
458D4—22no—
459D4—21no—
460D4—22no—
461D4—41no—
462D4—41no—
463D4—41no—
464D4—41no—
465C2×C4—22no—
466C2×C4—21no—
467C2×C4—21no—
468C2×C4—21no—
469C2×C4—41no—
470C2×C4—41no—
471C2×C4—41no—
472C2×C4—42no—
473C2×C4—41no—
474C2×C4—42no—
475C2×C4—81no—
476C2×C4—82no—
477C2×C4—82no—
478C2×C4—81no—
479C2×C4—82no—
480C2×C4#121×C6111yescharacteristic
481C2×C4#121×C6111yescharacteristic
482C2×C4#121×C6111yescharacteristic
483C2×C4#121×C6111yescharacteristic
484C2×C4C22×D4×C6111yescharacteristic
485C2×C4C22×D4×C6111yescharacteristic
486C2×C4C2×#41×C6111yescharacteristic
487C2×C4C2×#41×C6111yescharacteristic
488C2×C42−1+4×C6111yescharacteristic
489C2×C42−1+4×C6111yescharacteristic
490C8—22no—
491C8—22no—
492C8—22no—
493C8—41no—
494C8—42no—
495C8—42no—
496C22—21no—
497C22—21no—
498C22—41no—
499C22—41no—
500C22—41no—
501C22—42no—
502C22—82no—
503C22—82no—
504C22—81no—
505C22—81no—
506C22C2×#121×C6111yescharacteristic
507C22C61×#55211yescharacteristic
508C22C61×#55611yescharacteristic
509C4—21no—
510C4—21no—
511C4—21no—
512C4—41no—
513C4—41no—
514C4—41no—
515C4—81no—
516C4—82no—
517C4C61×#47212yes—
518C4C61×#52511yescharacteristic
519C4C61×#53611yescharacteristic
520C4C61×#54711yescharacteristic
521C4C61×#54811yescharacteristic
522C2—21no—
523C2—21no—
524C2—81no—
525C2—82no—
526C2C61×#241511yescharacteristic
527C2C2×C61×#47211yescharacteristic
528C2C61×#306911yescharacteristic
subgroupG/H
center, socleC22×C61C2×#121
commutator, Frattini, cosocleC22×C4C24×C61
prime radicalC2×AGL(1,ℤ/8)×C61C22
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22×C61 ◁ C2×C42×C61 ◁ G

Complements

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

NH ≅ G/Ntype
1. C61×#3079C2semidirect
2. C2×C61×#474C2semidirect
4. C61×#2925C2semidirect
5. C61×#2078C2semidirect
7. C61×#1997C2semidirect
8. C61×#1952C2semidirect
9. C61×#1950C2semidirect
10. C61×#1931C2semidirect
11. C61×#1800C2semidirect
231. #24720C61direct (N × H)
258. #3079C2×C61semidirect
259. C2×#474C2×C61semidirect
261. #2925C2×C61semidirect
262. #2078C2×C61semidirect
264. #1997C2×C61semidirect
265. #1952C2×C61semidirect
266. #1950C2×C61semidirect
267. #1931C2×C61semidirect
268. #1800C2×C61semidirect
313. C61#24720direct (N × H)

Maximal subgroups

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

subgroupindex
C61×#30792
C2×C61×#4742
C4×AGL(1,ℤ/8)×C612
C61×#29252
C61×#20782
C61×#20512
C61×#19972
C61×#19522
C61×#19502
C61×#19312
C61×#18002
#2472061