Finite Group Databasebeta

← C61×#24178 #29527611

Subgroups — C61×#24178 #29527611

Order 15616.

1102 conjugacy classes of subgroups.

The subgroup lattice has 638 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C61×#3506C211yesmaximal, characteristic
2C61×#3078C211yesmaximal, characteristic, complemented
3C61×#3037C211yesmaximal, characteristic, complemented
4C4×#124×C61C211yesmaximal, characteristic, complemented
5C61×#2037C212yesmaximal, complemented
6C61×#2004C212yesmaximal, complemented
7C61×#1994C211yesmaximal, characteristic, complemented
8C61×#1962C211yesmaximal, characteristic, complemented
9C61×#1882C212yesmaximal, complemented
10C61×#1860C212yesmaximal, complemented
11C61×#1800C211yesmaximal, characteristic, complemented
12C2×#137×C61C2212yes—
13C61×#559—22no—
14C61×#556—21no—
15C61×#555—21no—
16C61×#553—22no—
17C61×#550—21no—
18C61×#547—21no—
19C61×#538—22no—
20C61×#535C2212yes—
21C61×#533C2211yescharacteristic
22C61×#532C2211yescharacteristic
23C2×#124×C61C2211yescharacteristic
24C61×#525—22no—
25C61×#520C2212yes—
26C2×C4×Q8×C61C2211yescharacteristic
27C2×C4×D4×C61C2211yescharacteristic
28C61×#479—22no—
29C61×#477—21no—
30C61×#475—21no—
31C61×#464—22no—
32C61×#462—21no—
33C61×#460—21no—
34C61×#428C2212yes—
35C61×#423C2211yescharacteristic
36C61×#423C2211yescharacteristic
37C2×#104×C61C2212yes—
38C2×#104×C61C2212yes—
39C61×#419C2212yes—
40C61×#417C2212yes—
41C2×#100×C61C2212yes—
42C61×#391C2211yescharacteristic
43C61×#390C2211yescharacteristic
44C61×#388—22no—
45C61×#382C2211yescharacteristic
46C4×C22⋊C4×C61—22no—
47C43×C61C2211yescharacteristic
48C61×#343C2212yes—
49C61×#339C2212yes—
50C61×#339C2212yes—
51C2×#41×C61—22no—
52C2×#41×C61C2312yes—
53C22×Q8×C61C2311yescharacteristic
54C22×D4×C61C2311yescharacteristic
55C23×C4×C61D412yes—
56#137×C61—24no—
57#137×C61—42no—
58C2×Q16×C61—22no—
59C2×SD16×C61—22no—
60C2×M16×C61C2312yes—
61C2×M16×C61C2312yes—
62C4⋊Q8×C61D411yescharacteristic
63C4⋊Q8×C61D411yescharacteristic
64C4⋊Q8×C61D411yescharacteristic
65#127×C61—22no—
66#127×C61D411yescharacteristic
67#126×C61—24no—
68#125×C61D412yes—
69#124×C61—24no—
70#124×C61—22no—
71#124×C61—22no—
72#124×C61—22no—
73#124×C61D412yes—
74#123×C61—24no—
75#123×C61—24no—
76C22⋊Q8×C61—22no—
77C22⋊Q8×C61—21no—
78C22⋊Q8×C61—21no—
79#121×C61—24no—
80#121×C61—22no—
81#121×C61—21no—
82#121×C61—21no—
83#121×C61—22no—
84#121×C61—22no—
85#120×C61—24no—
86C4×Q8×C61—22no—
87C4×Q8×C61—21no—
88C4×Q8×C61—21no—
89C4×Q8×C61—21no—
90C4×Q8×C61—21no—
91C4×Q8×C61—21no—
92C4×Q8×C61—21no—
93C4×D4×C61—24no—
94C4×D4×C61—22no—
95C4×D4×C61—24no—
96C4×D4×C61—21no—
97C4×D4×C61—21no—
98C4×D4×C61—21no—
99C4×D4×C61—21no—
100C4×D4×C61—21no—
101C4×D4×C61—21no—
102#117×C61—22no—
103#117×C61—22no—
104#117×C61C2312yes—
105C2×C4⋊C4×C61—22no—
106C2×C4⋊C4×C61—21no—
107C2×C4⋊C4×C61C2311yescharacteristic
108C2×C4⋊C4×C61C2311yescharacteristic
109C2×C22⋊C4×C61—22no—
110C2×C22⋊C4×C61—22no—
111C2×C22⋊C4×C61D412yes—
112C2×C42×C61—21no—
113C2×C42×C61—21no—
114C2×C42×C61—21no—
115C2×C42×C61C2311yescharacteristic
116C2×C42×C61C2311yescharacteristic
117C2×C42×C61C2311yescharacteristic
118#107×C61—22no—
119#106×C61—22no—
120C4⋊C8×C61—21no—
121C4⋊C8×C61—21no—
122C4⋊C8×C61—21no—
123C4⋊C8×C61—21no—
124#104×C61—24no—
125#104×C61—42no—
126#103×C61—22no—
127#103×C61—22no—
128#103×C61—22no—
129#102×C61—22no—
130#100×C61—42no—
131#95×C61—22no—
132#95×C61—22no—
133C24×C61C2×D412yes—
134#41×C61—22no—
135#41×C61—22no—
136#41×C61—22no—
137#41×C61—22no—
138#41×C61—84no—
139C2×Q8×C61—21no—
140C2×Q8×C61—21no—
141C2×Q8×C61—21no—
142C2×Q8×C61—21no—
143C2×Q8×C61—21no—
144C2×Q8×C61—21no—
145C2×Q8×C61—21no—
146C2×Q8×C61C2×D412yes—
147C2×Q8×C61#4112yescomplemented
148C2×D4×C61—22no—
149C2×D4×C61—24no—
150C2×D4×C61—24no—
151C2×D4×C61—21no—
152C2×D4×C61—21no—
153C2×D4×C61—21no—
154C2×D4×C61—21no—
155C2×D4×C61—21no—
156C2×D4×C61—21no—
157C2×D4×C61—21no—
158C2×D4×C61—21no—
159C2×D4×C61C2×D412yes—
160C22×C4×C61—21no—
161C22×C4×C61—21no—
162C22×C4×C61—21no—
163C22×C4×C61—24no—
164C22×C4×C61—22no—
165C22×C4×C61—22no—
166C22×C4×C61—22no—
167C22×C4×C61—21no—
168C22×C4×C61—44no—
169C22×C4×C61C2×D411yescharacteristic
170C22×C4×C61C2×D412yes—
171C22×C4×C61C2×D411yescharacteristic
172C22×C4×C61C2411yescharacteristic
173Q16×C61—44no—
174SD16×C61—44no—
175M16×C61—22no—
176M16×C61—22no—
177M16×C61—22no—
178M16×C61—22no—
179C2×C8×C61—22no—
180C2×C8×C61—22no—
181C4⋊C4×C61—24no—
182C4⋊C4×C61—21no—
183C4⋊C4×C61—21no—
184C4⋊C4×C61—22no—
185C4⋊C4×C61—21no—
186C4⋊C4×C61—22no—
187C4⋊C4×C61—21no—
188C4⋊C4×C61—21no—
189C4⋊C4×C61—21no—
190C4⋊C4×C61—21no—
191C4⋊C4×C61—21no—
192C4⋊C4×C61—41no—
193C4⋊C4×C61—41no—
194C4⋊C4×C61C2×D412yes—
195C4⋊C4×C61C2×D412yes—
196C22⋊C4×C61—28no—
197C22⋊C4×C61—24no—
198C22⋊C4×C61—24no—
199C22⋊C4×C61—24no—
200C22⋊C4×C61—22no—
201C22⋊C4×C61—22no—
202C22⋊C4×C61—22no—
203C22⋊C4×C61—22no—
204C22⋊C4×C61—22no—
205C22⋊C4×C61—44no—
206C42×C61—24no—
207C42×C61—22no—
208C42×C61—21no—
209C42×C61—21no—
210C42×C61—21no—
211C42×C61—42no—
212C42×C61—42no—
213C42×C61—41no—
214C42×C61—41no—
215C42×C61C2×D411yescharacteristic
216C42×C61C2×D411yescharacteristic
217C42×C61C2×D411yescharacteristic
218C42×C61C2×D411yescharacteristic
219C42×C61#4112yescomplemented
220C23×C61—22no—
221C23×C61—22no—
222C23×C61—22no—
223C23×C61—22no—
224C23×C61—42no—
225C23×C61—42no—
226C23×C61C22×D411yescharacteristic
227Q8×C61—22no—
228Q8×C61—22no—
229Q8×C61—22no—
230Q8×C61—22no—
231Q8×C61—41no—
232Q8×C61—41no—
233Q8×C61—42no—
234Q8×C61—81no—
235D4×C61—22no—
236D4×C61—22no—
237D4×C61—42no—
238D4×C61—42no—
239D4×C61—41no—
240D4×C61—41no—
241D4×C61—41no—
242D4×C61—41no—
243D4×C61—42no—
244D4×C61—81no—
245D4×C61—84no—
246D4×C61—84no—
247C2×C4×C61—22no—
248C2×C4×C61—22no—
249C2×C4×C61—22no—
250C2×C4×C61—22no—
251C2×C4×C61—22no—
252C2×C4×C61—22no—
253C2×C4×C61—22no—
254C2×C4×C61—22no—
255C2×C4×C61—41no—
256C2×C4×C61—41no—
257C2×C4×C61—41no—
258C2×C4×C61—41no—
259C2×C4×C61—41no—
260C2×C4×C61—41no—
261C2×C4×C61—41no—
262C2×C4×C61—42no—
263C2×C4×C61—42no—
264C2×C4×C61—44no—
265C2×C4×C61—42no—
266C2×C4×C61—42no—
267C2×C4×C61—41no—
268C2×C4×C61—82no—
269C2×C4×C61—84no—
270C2×C4×C61#12011yescharacteristic
271C2×C4×C61#12011yescharacteristic
272C2×C4×C61#12011yescharacteristic
273C2×C4×C61#12011yescharacteristic
274C2×C4×C61C22×D411yescharacteristic
275C2×C4×C61C22×D411yescharacteristic
276C2×C4×C61C2×#4111yescharacteristic
277C2×C4×C61C2×#4111yescharacteristic
278C2×C4×C61H(5,𝔽2)11yescharacteristic
279C2×C4×C612−1+411yescharacteristic
280C8×C61—42no—
281C8×C61—42no—
282#24178C6111yesmaximal, characteristic, direct product factor, Sylow
283C22×C61—21no—
284C22×C61—21no—
285C22×C61—44no—
286C22×C61—42no—
287C22×C61—42no—
288C22×C61—42no—
289C22×C61—42no—
290C22×C61—42no—
291C22×C61—42no—
292C22×C61—82no—
293C22×C61C2×#12011yescharacteristic
294C22×C61#52511yescharacteristic
295C22×C61#53711yescharacteristic
296C4×C61—21no—
297C4×C61—21no—
298C4×C61—21no—
299C4×C61—42no—
300C4×C61—42no—
301C4×C61—42no—
302C4×C61—42no—
303C4×C61—42no—
304C4×C61—42no—
305C4×C61—42no—
306C4×C61—81no—
307C4×C61#45512yes—
308C4×C61#54611yescharacteristic
309C4×C61#54711yescharacteristic
310C4×C61#54811yescharacteristic
311C4×C61#55011yescharacteristic
312#3506C2×C6111yescharacteristic
313#3078C2×C6111yescharacteristic, complemented
314#3037C2×C6111yescharacteristic, complemented
315C4×#124C2×C6111yescharacteristic, complemented
316#2037C2×C6112yescomplemented
317#2004C2×C6112yescomplemented
318#1994C2×C6111yescharacteristic, complemented
319#1962C2×C6111yescharacteristic, complemented
320#1882C2×C6112yescomplemented
321#1860C2×C6112yescomplemented
322#1800C2×C6111yescharacteristic, complemented
323C2×C61—21no—
324C2×C61—21no—
325C2×C61—82no—
326C2×C61—82no—
327C2×C61#240711yescharacteristic
328C2×C61C2×#45511yescharacteristic
329C2×C61#301811yescharacteristic
330C2×#137C22×C6112yes—
331#559—22no—
332#556—21no—
333#555—21no—
334#553—22no—
335#550—21no—
336#547—21no—
337#538—22no—
338#535C22×C6112yes—
339#533C22×C6111yescharacteristic
340#532C22×C6111yescharacteristic
341C2×#124C22×C6111yescharacteristic
342#525—22no—
343#520C22×C6112yes—
344C2×C4×Q8C22×C6111yescharacteristic
345C2×C4×D4C22×C6111yescharacteristic
346#479—22no—
347#477—21no—
348#475—21no—
349#464—22no—
350#462—21no—
351#460—21no—
352#428C22×C6112yes—
353#423C22×C6111yescharacteristic
354#423C22×C6111yescharacteristic
355C2×#104C22×C6112yes—
356C2×#104C22×C6112yes—
357#419C22×C6112yes—
358#417C22×C6112yes—
359C2×#100C22×C6112yes—
360#391C22×C6111yescharacteristic
361#390C22×C6111yescharacteristic
362#388—22no—
363#382C22×C6111yescharacteristic
364C4×C22⋊C4—22no—
365C43C22×C6111yescharacteristic
366#343C22×C6112yes—
367#339C22×C6112yes—
368#339C22×C6112yes—
369C61#2417811yescharacteristic, direct product factor, Sylow
370C2×#41—22no—
371C2×#41C23×C6112yes—
372C22×Q8C23×C6111yescharacteristic
373C22×D4C23×C6111yescharacteristic
374C23×C4D4×C6112yes—
375#137—24no—
376#137—42no—
377C2×Q16—22no—
378C2×SD16—22no—
379C2×M16C23×C6112yes—
380C2×M16C23×C6112yes—
381C4⋊Q8D4×C6111yescharacteristic
382C4⋊Q8D4×C6111yescharacteristic
383C4⋊Q8D4×C6111yescharacteristic
384#127—22no—
385#127D4×C6111yescharacteristic
386#126—24no—
387#125D4×C6112yes—
388#124—24no—
389#124—22no—
390#124—22no—
391#124—22no—
392#124D4×C6112yes—
393#123—24no—
394#123—24no—
395C22⋊Q8—22no—
396C22⋊Q8—21no—
397C22⋊Q8—21no—
398#121—22no—
399#121—24no—
400#121—21no—
401#121—21no—
402#121—22no—
403#121—22no—
404#120—24no—
405C4×Q8—22no—
406C4×Q8—21no—
407C4×Q8—21no—
408C4×Q8—21no—
409C4×Q8—21no—
410C4×Q8—21no—
411C4×Q8—21no—
412C4×D4—24no—
413C4×D4—22no—
414C4×D4—24no—
415C4×D4—21no—
416C4×D4—21no—
417C4×D4—21no—
418C4×D4—21no—
419C4×D4—21no—
420C4×D4—21no—
421#117—22no—
422#117—22no—
423#117C23×C6112yes—
424C2×C4⋊C4—22no—
425C2×C4⋊C4—21no—
426C2×C4⋊C4C23×C6111yescharacteristic
427C2×C4⋊C4C23×C6111yescharacteristic
428C2×C22⋊C4—22no—
429C2×C22⋊C4—22no—
430C2×C22⋊C4D4×C6112yes—
431C2×C42—21no—
432C2×C42—21no—
433C2×C42—21no—
434C2×C42C23×C6111yescharacteristic
435C2×C42C23×C6111yescharacteristic
436C2×C42C23×C6111yescharacteristic
437#107—22no—
438#106—22no—
439C4⋊C8—21no—
440C4⋊C8—21no—
441C4⋊C8—21no—
442C4⋊C8—21no—
443#104—24no—
444#104—42no—
445#103—22no—
446#103—22no—
447#103—22no—
448#102—22no—
449#100—42no—
450#95—22no—
451#95—22no—
452C24C2×D4×C6112yes—
453#41—22no—
454#41—22no—
455#41—22no—
456#41—22no—
457#41—84no—
458C2×Q8—21no—
459C2×Q8—21no—
460C2×Q8—21no—
461C2×Q8—21no—
462C2×Q8—21no—
463C2×Q8—21no—
464C2×Q8—21no—
465C2×Q8C2×D4×C6112yes—
466C2×Q8#41×C6112yescomplemented
467C2×D4—22no—
468C2×D4—24no—
469C2×D4—24no—
470C2×D4—21no—
471C2×D4—21no—
472C2×D4—21no—
473C2×D4—21no—
474C2×D4—21no—
475C2×D4—21no—
476C2×D4—21no—
477C2×D4—21no—
478C2×D4C2×D4×C6112yes—
479C22×C4—21no—
480C22×C4—21no—
481C22×C4—21no—
482C22×C4—24no—
483C22×C4—22no—
484C22×C4—22no—
485C22×C4—22no—
486C22×C4—21no—
487C22×C4—44no—
488C22×C4C2×D4×C6111yescharacteristic
489C22×C4C2×D4×C6112yes—
490C22×C4C2×D4×C6111yescharacteristic
491C22×C4C24×C6111yescharacteristic
492Q16—44no—
493SD16—44no—
494M16—22no—
495M16—22no—
496M16—22no—
497M16—22no—
498C2×C8—22no—
499C2×C8—22no—
500C4⋊C4—24no—
501C4⋊C4—21no—
502C4⋊C4—21no—
503C4⋊C4—22no—
504C4⋊C4—21no—
505C4⋊C4—22no—
506C4⋊C4—21no—
507C4⋊C4—21no—
508C4⋊C4—21no—
509C4⋊C4—21no—
510C4⋊C4—21no—
511C4⋊C4—41no—
512C4⋊C4—41no—
513C4⋊C4C2×D4×C6112yes—
514C4⋊C4C2×D4×C6112yes—
515C22⋊C4—28no—
516C22⋊C4—24no—
517C22⋊C4—24no—
518C22⋊C4—24no—
519C22⋊C4—22no—
520C22⋊C4—22no—
521C22⋊C4—22no—
522C22⋊C4—22no—
523C22⋊C4—22no—
524C22⋊C4—44no—
525C42—24no—
526C42—22no—
527C42—21no—
528C42—21no—
529C42—21no—
530C42—42no—
531C42—42no—
532C42—41no—
533C42—41no—
534C42C2×D4×C6111yescharacteristic
535C42C2×D4×C6111yescharacteristic
536C42C2×D4×C6111yescharacteristic
537C42C2×D4×C6111yescharacteristic
538C42#41×C6112yescomplemented
539C23—22no—
540C23—22no—
541C23—22no—
542C23—22no—
543C23—42no—
544C23—42no—
545C23C22×D4×C6111yescharacteristic
546Q8—22no—
547Q8—22no—
548Q8—22no—
549Q8—22no—
550Q8—41no—
551Q8—41no—
552Q8—42no—
553Q8—81no—
554D4—22no—
555D4—22no—
556D4—42no—
557D4—42no—
558D4—41no—
559D4—41no—
560D4—41no—
561D4—41no—
562D4—42no—
563D4—81no—
564D4—84no—
565D4—84no—
566C2×C4—22no—
567C2×C4—22no—
568C2×C4—22no—
569C2×C4—22no—
570C2×C4—22no—
571C2×C4—22no—
572C2×C4—22no—
573C2×C4—22no—
574C2×C4—41no—
575C2×C4—41no—
576C2×C4—41no—
577C2×C4—41no—
578C2×C4—41no—
579C2×C4—41no—
580C2×C4—41no—
581C2×C4—42no—
582C2×C4—42no—
583C2×C4—44no—
584C2×C4—42no—
585C2×C4—42no—
586C2×C4—41no—
587C2×C4—82no—
588C2×C4—84no—
589C2×C4#120×C6111yescharacteristic
590C2×C4#120×C6111yescharacteristic
591C2×C4#120×C6111yescharacteristic
592C2×C4#120×C6111yescharacteristic
593C2×C4C22×D4×C6111yescharacteristic
594C2×C4C22×D4×C6111yescharacteristic
595C2×C4C2×#41×C6111yescharacteristic
596C2×C4C2×#41×C6111yescharacteristic
597C2×C4H(5,𝔽2)×C6111yescharacteristic
598C2×C42−1+4×C6111yescharacteristic
599C8—42no—
600C8—42no—
601C22—21no—
602C22—21no—
603C22—44no—
604C22—42no—
605C22—42no—
606C22—42no—
607C22—42no—
608C22—42no—
609C22—42no—
610C22—82no—
611C22C2×#120×C6111yescharacteristic
612C22C61×#52511yescharacteristic
613C22C61×#53711yescharacteristic
614C4—21no—
615C4—21no—
616C4—21no—
617C4—42no—
618C4—42no—
619C4—42no—
620C4—42no—
621C4—42no—
622C4—42no—
623C4—42no—
624C4—81no—
625C4C61×#45512yes—
626C4C61×#54611yescharacteristic
627C4C61×#54711yescharacteristic
628C4C61×#54811yescharacteristic
629C4C61×#55011yescharacteristic
630C2—21no—
631C2—21no—
632C2—82no—
633C2—82no—
634C2C61×#240711yescharacteristic
635C2C2×C61×#45511yescharacteristic
636C2C61×#301811yescharacteristic
subgroupG/H
center, socleC22×C61C2×#120
commutator, Frattini, cosocleC22×C4C24×C61
prime radicalC61×#3506C2
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
2. C61×#3078C2semidirect
3. C61×#3037C2semidirect
4. C4×#124×C61C2semidirect
5. C61×#2037C2semidirect
6. C61×#2004C2semidirect
7. C61×#1994C2semidirect
8. C61×#1962C2semidirect
9. C61×#1882C2semidirect
10. C61×#1860C2semidirect
11. C61×#1800C2semidirect
147. C2×Q8×C61#41semidirect
219. C42×C61#41semidirect
282. #24178C61direct (N × H)
313. #3078C2×C61semidirect
314. #3037C2×C61semidirect
315. C4×#124C2×C61semidirect
316. #2037C2×C61semidirect
317. #2004C2×C61semidirect
318. #1994C2×C61semidirect
319. #1962C2×C61semidirect
320. #1882C2×C61semidirect
321. #1860C2×C61semidirect
322. #1800C2×C61semidirect
369. C61#24178direct (N × H)
466. C2×Q8#41×C61semidirect
538. C42#41×C61semidirect

Maximal subgroups

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

subgroupindex
C61×#35062
C61×#30782
C61×#30372
C4×#124×C612
C61×#20372
C61×#20042
C61×#19942
C61×#19622
C61×#18822
C61×#18602
C61×#18002
#2417861