Finite Group Databasebeta

← C61×#9934 #29513367

Subgroups — C61×#9934 #29513367

Order 15616.

808 conjugacy classes of subgroups; only the 102 normal subgroups are shown.

normal subgroups only

This group has 808 conjugacy classes of subgroups — too many to draw. Only its 102 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C61×#3193C211yesmaximal, characteristic, complemented
2C61×#3007C211yesmaximal, characteristic, complemented
3C61×#2615C211yesmaximal, characteristic
4C61×#1996C211yesmaximal, characteristic, complemented
5C61×#1789C211yesmaximal, characteristic, complemented
6C61×#1507C211yesmaximal, characteristic, complemented
7C61×#1305C211yesmaximal, characteristic, complemented
8C22×#41×C61C2211yescharacteristic
9C2×#137×C61—21no—
10C2×C22⋊Q8×C61C2211yescharacteristic
11C2×#121×C61—21no—
12C2×#121×C61C2211yescharacteristic
13C61×#484—21no—
14C61×#480—21no—
15C61×#468—22no—
16C61×#452—22no—
17C61×#451—21no—
18C61×#450—21no—
19C61×#450—21no—
20C61×#448—21no—
21C2×#104×C61—22no—
22C61×#417—21no—
23C61×#407C2211yescharacteristic
24C61×#407C2211yescharacteristic
25C61×#393—21no—
26C61×#392—21no—
27C61×#392—22no—
28C61×#380—21no—
29C61×#380C2211yescharacteristic
30C4×C22⋊C4×C61C2211yescharacteristic
31C61×#342—22no—
32C61×#327—22no—
33C2×#41×C61—21no—
34C2×#41×C61—21no—
35C2×#41×C61—21no—
36C2×#41×C61—21no—
37C2×#41×C61—44no—
38C22×Q8×C61D411yescharacteristic
39C22×Q8×C61D411yescharacteristic, complemented
40C22×D4×C61—21no—
41C22×D4×C61—22no—
42C22×D4×C61—21no—
43C22×D4×C61D411yescharacteristic
44C22×D4×C61D411yescharacteristic
45C23×C4×C61—21no—
46C23×C4×C61C2311yescharacteristic
47#137×C61—44no—
48C2×Q16×C61—22no—
49C2×SD16×C61—22no—
50C2×SD16×C61—22no—
51C2×M16×C61—21no—
52C2×M16×C61D412yescomplemented
53C22⋊Q8×C61—21no—
54C22⋊Q8×C61—21no—
55C22⋊Q8×C61—21no—
56C22⋊Q8×C61—21no—
57#121×C61—21no—
58#121×C61—21no—
59#121×C61—21no—
60#121×C61—21no—
61#121×C61—44no—
62C2×C4⋊C4×C61—21no—
63C2×C4⋊C4×C61—21no—
64C2×C4⋊C4×C61D411yescharacteristic
65C2×C4⋊C4×C61D411yescharacteristic, complemented
66C2×C22⋊C4×C61—21no—
67C2×C22⋊C4×C61—22no—
68C2×C22⋊C4×C61—21no—
69C2×C22⋊C4×C61—21no—
70C2×C22⋊C4×C61—22no—
71C2×C22⋊C4×C61D412yes—
72C2×C42×C61—21no—
73#107×C61—22no—
74#104×C61—44no—
75#103×C61—22no—
76#103×C61—22no—
77#102×C61—22no—
78#102×C61—22no—
79C22⋊C8×C61—21no—
80C22⋊C8×C61—21no—
81C22⋊C8×C61—21no—
82C22⋊C8×C61D412yescomplemented
83#95×C61—22no—
84#95×C61—21no—
85#95×C61—21no—
86C24×C61—21no—
87C24×C61—21no—
88C24×C61C2×D411yescharacteristic
89#41×C61—24no—
90#41×C61—24no—
91#41×C61—42no—
92#41×C61—42no—
93#41×C61—42no—
94#41×C61—42no—
95#41×C61—42no—
96#41×C61—42no—
97C2×Q8×C61—21no—
98C2×Q8×C61—21no—
99C2×Q8×C61—21no—
100C2×Q8×C61—21no—
101C2×Q8×C61—42no—
102C2×Q8×C61—42no—
103C2×D4×C61—21no—
104C2×D4×C61—21no—
105C2×D4×C61—21no—
106C2×D4×C61—21no—
107C2×D4×C61—42no—
108C2×D4×C61—42no—
109C2×D4×C61—42no—
110C2×D4×C61—42no—
111C2×D4×C61—41no—
112C2×D4×C61—41no—
113C2×D4×C61—44no—
114C2×D4×C61—41no—
115C2×D4×C61—41no—
116C2×D4×C61—82no—
117C2×D4×C61—82no—
118C2×D4×C61—82no—
119C22×C4×C61—24no—
120C22×C4×C61—21no—
121C22×C4×C61—21no—
122C22×C4×C61—22no—
123C22×C4×C61—21no—
124C22×C4×C61—21no—
125C22×C4×C61—42no—
126C22×C4×C61—42no—
127C22×C4×C61—41no—
128C22×C4×C61—41no—
129C22×C4×C61—41no—
130C22×C4×C61C2×D411yescharacteristic
131C22×C4×C61C2×D411yescharacteristic
132C22×C4×C61C2×D411yescharacteristic
133C22×C4×C61C2×D411yescharacteristic
134C22×C4×C61C2×D411yescharacteristic
135C22×C4×C61#4111yescharacteristic
136Q16×C61—44no—
137SD16×C61—44no—
138SD16×C61—82no—
139M16×C61—42no—
140M16×C61—42no—
141C2×C8×C61—22no—
142C2×C8×C61—22no—
143C4⋊C4×C61—21no—
144C4⋊C4×C61—21no—
145C4⋊C4×C61—21no—
146C4⋊C4×C61—21no—
147C4⋊C4×C61—42no—
148C4⋊C4×C61—41no—
149C4⋊C4×C61—41no—
150C22⋊C4×C61—22no—
151C22⋊C4×C61—44no—
152C22⋊C4×C61—41no—
153C22⋊C4×C61—41no—
154C22⋊C4×C61—41no—
155C22⋊C4×C61—41no—
156C22⋊C4×C61—41no—
157C22⋊C4×C61—42no—
158C22⋊C4×C61—82no—
159C42×C61—42no—
160C23×C61—21no—
161C23×C61—21no—
162C23×C61—22no—
163C23×C61—41no—
164C23×C61—41no—
165C23×C61—41no—
166C23×C61—41no—
167C23×C61—41no—
168C23×C61—42no—
169C23×C61—42no—
170C23×C61—41no—
171C23×C61—41no—
172C23×C61—41no—
173C23×C61—41no—
174C23×C61—41no—
175C23×C61—41no—
176C23×C61#12011yescharacteristic
177C23×C61#12111yescharacteristic
178C23×C61#12711yescharacteristic
179Q8×C61—22no—
180Q8×C61—22no—
181Q8×C61—41no—
182Q8×C61—41no—
183Q8×C61—42no—
184Q8×C61—42no—
185D4×C61—22no—
186D4×C61—22no—
187D4×C61—41no—
188D4×C61—41no—
189D4×C61—44no—
190D4×C61—44no—
191D4×C61—42no—
192D4×C61—81no—
193D4×C61—82no—
194D4×C61—82no—
195D4×C61—88no—
196C2×C4×C61—22no—
197C2×C4×C61—22no—
198C2×C4×C61—22no—
199C2×C4×C61—22no—
200C2×C4×C61—41no—
201C2×C4×C61—41no—
202C2×C4×C61—41no—
203C2×C4×C61—44no—
204C2×C4×C61—41no—
205C2×C4×C61—41no—
206C2×C4×C61—44no—
207C2×C4×C61—42no—
208C2×C4×C61—42no—
209C2×C4×C61—42no—
210C2×C4×C61—42no—
211C2×C4×C61—41no—
212C2×C4×C61—41no—
213C2×C4×C61—41no—
214C2×C4×C61—42no—
215C2×C4×C61—81no—
216C2×C4×C61—81no—
217C2×C4×C61—81no—
218C2×C4×C61—81no—
219C2×C4×C61#12011yescharacteristic
220C2×C4×C61#12011yescharacteristic
221C2×C4×C61#12111yescharacteristic
222C2×C4×C61#12111yescharacteristic
223C8×C61—42no—
224C8×C61—42no—
225#9934C6111yesmaximal, characteristic, direct product factor, Sylow
226C22×C61—21no—
227C22×C61—21no—
228C22×C61—21no—
229C22×C61—21no—
230C22×C61—21no—
231C22×C61—21no—
232C22×C61—21no—
233C22×C61—21no—
234C22×C61—42no—
235C22×C61—42no—
236C22×C61—42no—
237C22×C61—81no—
238C22×C61—81no—
239C22×C61—81no—
240C22×C61—81no—
241C22×C61—82no—
242C22×C61—82no—
243C22×C61—81no—
244C22×C61—81no—
245C22×C61—81no—
246C22×C61—81no—
247C22×C61—81no—
248C22×C61#39211yescharacteristic
249C22×C61#45511yescharacteristic
250C22×C61#45611yescharacteristic
251C4×C61—21no—
252C4×C61—21no—
253C4×C61—21no—
254C4×C61—42no—
255C4×C61—82no—
256C4×C61—82no—
257C4×C61—81no—
258C4×C61—81no—
259C4×C61—81no—
260C4×C61B(4,𝔽2)12yes—
261#3193C2×C6111yescharacteristic, complemented
262#3007C2×C6111yescharacteristic, complemented
263#2615C2×C6111yescharacteristic
264#1996C2×C6111yescharacteristic, complemented
265#1789C2×C6111yescharacteristic, complemented
266#1507C2×C6111yescharacteristic, complemented
267#1305C2×C6111yescharacteristic, complemented
268C2×C61—21no—
269C2×C61—21no—
270C2×C61—41no—
271C2×C61—41no—
272C2×C61—82no—
273C2×C61—161no—
274C2×C61#200711yescharacteristic
275C2×C61#200811yescharacteristic
276C2×C61#202111yescharacteristic
277C22×#41C22×C6111yescharacteristic
278C2×#137—21no—
279C2×C22⋊Q8C22×C6111yescharacteristic
280C2×#121—21no—
281C2×#121C22×C6111yescharacteristic
282#484—21no—
283#480—21no—
284#468—22no—
285#452—22no—
286#451—21no—
287#450—21no—
288#450—21no—
289#448—21no—
290C2×#104—22no—
291#417—21no—
292#407C22×C6111yescharacteristic
293#407C22×C6111yescharacteristic
294#393—21no—
295#392—21no—
296#392—22no—
297#380—21no—
298#380C22×C6111yescharacteristic
299C4×C22⋊C4C22×C6111yescharacteristic
300#342—22no—
301#327—22no—
302C61#993411yescharacteristic, direct product factor, Sylow
303C2×#41—21no—
304C2×#41—21no—
305C2×#41—21no—
306C2×#41—21no—
307C2×#41—44no—
308C22×Q8D4×C6111yescharacteristic
309C22×Q8D4×C6111yescharacteristic, complemented
310C22×D4—21no—
311C22×D4—22no—
312C22×D4—21no—
313C22×D4D4×C6111yescharacteristic
314C22×D4D4×C6111yescharacteristic
315C23×C4—21no—
316C23×C4C23×C6111yescharacteristic
317#137—44no—
318C2×Q16—22no—
319C2×SD16—22no—
320C2×SD16—22no—
321C2×M16—21no—
322C2×M16D4×C6112yescomplemented
323C22⋊Q8—21no—
324C22⋊Q8—21no—
325C22⋊Q8—21no—
326C22⋊Q8—21no—
327#121—21no—
328#121—21no—
329#121—21no—
330#121—21no—
331#121—44no—
332C2×C4⋊C4—21no—
333C2×C4⋊C4—21no—
334C2×C4⋊C4D4×C6111yescharacteristic
335C2×C4⋊C4D4×C6111yescharacteristic, complemented
336C2×C22⋊C4—21no—
337C2×C22⋊C4—22no—
338C2×C22⋊C4—21no—
339C2×C22⋊C4—21no—
340C2×C22⋊C4—22no—
341C2×C22⋊C4D4×C6112yes—
342C2×C42—21no—
343#107—22no—
344#104—44no—
345#103—22no—
346#103—22no—
347#102—22no—
348#102—22no—
349C22⋊C8—21no—
350C22⋊C8—21no—
351C22⋊C8—21no—
352C22⋊C8D4×C6112yescomplemented
353#95—22no—
354#95—21no—
355#95—21no—
356C24—21no—
357C24—21no—
358C24C2×D4×C6111yescharacteristic
359#41—24no—
360#41—24no—
361#41—42no—
362#41—42no—
363#41—42no—
364#41—42no—
365#41—42no—
366#41—42no—
367C2×Q8—21no—
368C2×Q8—21no—
369C2×Q8—21no—
370C2×Q8—21no—
371C2×Q8—42no—
372C2×Q8—42no—
373C2×D4—21no—
374C2×D4—21no—
375C2×D4—21no—
376C2×D4—21no—
377C2×D4—42no—
378C2×D4—42no—
379C2×D4—42no—
380C2×D4—42no—
381C2×D4—41no—
382C2×D4—41no—
383C2×D4—44no—
384C2×D4—41no—
385C2×D4—41no—
386C2×D4—82no—
387C2×D4—82no—
388C2×D4—82no—
389C22×C4—24no—
390C22×C4—21no—
391C22×C4—21no—
392C22×C4—22no—
393C22×C4—21no—
394C22×C4—21no—
395C22×C4—42no—
396C22×C4—42no—
397C22×C4—41no—
398C22×C4—41no—
399C22×C4—41no—
400C22×C4C2×D4×C6111yescharacteristic
401C22×C4C2×D4×C6111yescharacteristic
402C22×C4C2×D4×C6111yescharacteristic
403C22×C4C2×D4×C6111yescharacteristic
404C22×C4C2×D4×C6111yescharacteristic
405C22×C4#41×C6111yescharacteristic
406Q16—44no—
407SD16—44no—
408SD16—82no—
409M16—42no—
410M16—42no—
411C2×C8—22no—
412C2×C8—22no—
413C4⋊C4—21no—
414C4⋊C4—21no—
415C4⋊C4—21no—
416C4⋊C4—21no—
417C4⋊C4—42no—
418C4⋊C4—41no—
419C4⋊C4—41no—
420C22⋊C4—22no—
421C22⋊C4—44no—
422C22⋊C4—41no—
423C22⋊C4—41no—
424C22⋊C4—41no—
425C22⋊C4—41no—
426C22⋊C4—41no—
427C22⋊C4—42no—
428C22⋊C4—82no—
429C42—42no—
430C23—21no—
431C23—21no—
432C23—22no—
433C23—41no—
434C23—41no—
435C23—41no—
436C23—41no—
437C23—41no—
438C23—42no—
439C23—42no—
440C23—41no—
441C23—41no—
442C23—41no—
443C23—41no—
444C23—41no—
445C23—41no—
446C23#120×C6111yescharacteristic
447C23#121×C6111yescharacteristic
448C23#127×C6111yescharacteristic
449Q8—22no—
450Q8—22no—
451Q8—41no—
452Q8—41no—
453Q8—42no—
454Q8—42no—
455D4—22no—
456D4—22no—
457D4—41no—
458D4—41no—
459D4—44no—
460D4—44no—
461D4—42no—
462D4—81no—
463D4—82no—
464D4—82no—
465D4—88no—
466C2×C4—22no—
467C2×C4—22no—
468C2×C4—22no—
469C2×C4—22no—
470C2×C4—41no—
471C2×C4—41no—
472C2×C4—41no—
473C2×C4—44no—
474C2×C4—41no—
475C2×C4—41no—
476C2×C4—44no—
477C2×C4—42no—
478C2×C4—42no—
479C2×C4—42no—
480C2×C4—42no—
481C2×C4—41no—
482C2×C4—41no—
483C2×C4—41no—
484C2×C4—42no—
485C2×C4—81no—
486C2×C4—81no—
487C2×C4—81no—
488C2×C4—81no—
489C2×C4#120×C6111yescharacteristic
490C2×C4#120×C6111yescharacteristic
491C2×C4#121×C6111yescharacteristic
492C2×C4#121×C6111yescharacteristic
493C8—42no—
494C8—42no—
495C22—21no—
496C22—21no—
497C22—21no—
498C22—21no—
499C22—21no—
500C22—21no—
501C22—21no—
502C22—21no—
503C22—42no—
504C22—42no—
505C22—42no—
506C22—81no—
507C22—81no—
508C22—81no—
509C22—81no—
510C22—82no—
511C22—82no—
512C22—81no—
513C22—81no—
514C22—81no—
515C22—81no—
516C22—81no—
517C22C61×#39211yescharacteristic
518C22C61×#45511yescharacteristic
519C22C61×#45611yescharacteristic
520C4—21no—
521C4—21no—
522C4—21no—
523C4—42no—
524C4—82no—
525C4—82no—
526C4—81no—
527C4—81no—
528C4—81no—
529C4C61×B(4,𝔽2)12yes—
530C2—21no—
531C2—21no—
532C2—41no—
533C2—41no—
534C2—82no—
535C2—161no—
536C2C61×#200711yescharacteristic
537C2C61×#200811yescharacteristic
538C2C61×#202111yescharacteristic
subgroupG/H
center, socleC22×C61#392
commutator, Frattini, cosocleC23×C4C23×C61
prime radicalC61×#2615C2
LCSG ▷ C23×C4 ▷ C22 ▷ C1
UCSC1 ◁ C22×C61 ◁ C23×C4×C61 ◁ G

Complements

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

NH ≅ G/Ntype
1. C61×#3193C2semidirect
2. C61×#3007C2semidirect
4. C61×#1996C2semidirect
5. C61×#1789C2semidirect
6. C61×#1507C2semidirect
7. C61×#1305C2semidirect
39. C22×Q8×C61D4semidirect
52. C2×M16×C61D4semidirect
65. C2×C4⋊C4×C61D4semidirect
82. C22⋊C8×C61D4semidirect
225. #9934C61direct (N × H)
261. #3193C2×C61semidirect
262. #3007C2×C61semidirect
264. #1996C2×C61semidirect
265. #1789C2×C61semidirect
266. #1507C2×C61semidirect
267. #1305C2×C61semidirect
302. C61#9934direct (N × H)
309. C22×Q8D4×C61semidirect
322. C2×M16D4×C61semidirect
335. C2×C4⋊C4D4×C61semidirect
352. C22⋊C8D4×C61semidirect

Maximal subgroups

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

subgroupindex
C61×#31932
C61×#30072
C61×#26152
C61×#19962
C61×#17892
C61×#15072
C61×#13052
#993461