Finite Group Databasebeta

← C61×#8534 #29511967

Subgroups — C61×#8534 #29511967

Order 15616.

1408 conjugacy classes of subgroups; only the 158 normal subgroups are shown.

normal subgroups only

This group has 1408 conjugacy classes of subgroups — too many to draw. Only its 158 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×#3484C211yesmaximal, characteristic
2C61×#1898C214yesmaximal, complemented
3C61×#1781C211yesmaximal, characteristic, complemented
4C61×#1438C211yesmaximal, characteristic, complemented
5C2×H(5,𝔽2)×C61C411yescharacteristic, complemented
6C23×D4×C61C2211yescharacteristic
7C61×#546—24no—
8D42×C61—24no—
9C61×#535—21no—
10C61×#534—21no—
11C61×#534C412yescomplemented
12C61×#525—21no—
13C2×#120×C61—21no—
14C2×#120×C61C2212yes—
15C61×#518C411yescharacteristic, complemented
16C22×C22⋊C4×C61—21no—
17C22×C22⋊C4×C61C2212yescomplemented
18C2×#99×C61—21no—
19C2×#99×C61—21no—
20C2×#99×C61—24no—
21C61×#379C2212yescomplemented
22C2×#95×C61—21no—
23C61×#342—24no—
24C25×C61D411yescharacteristic
25C25×C61C2311yescharacteristic
26H(5,𝔽2)×C61—42no—
27H(5,𝔽2)×C61—42no—
28C2×#41×C61—21no—
29C2×#41×C61—22no—
30C22×D4×C61—21no—
31C22×D4×C61—22no—
32C22×D4×C61—24no—
33C22×D4×C61—22no—
34C22×D4×C61—42no—
35C22×D4×C61—42no—
36C22×D4×C61C2×C411yescharacteristic
37C22×D4×C61C2×C411yescharacteristic
38C22×D4×C61C2×C412yes—
39C22×D4×C61D411yescharacteristic
40C22×D4×C61D411yescharacteristic
41C22×D4×C61D414yes—
42C23×C4×C61—22no—
43C23×C4×C61D411yescharacteristic
44#127×C61—22no—
45#124×C61—22no—
46#123×C61—22no—
47#123×C61—22no—
48#123×C61—22no—
49C22⋊Q8×C61—22no—
50#121×C61—24no—
51#121×C61—24no—
52#121×C61—22no—
53#121×C61—24no—
54#120×C61—22no—
55#120×C61—24no—
56#120×C61—22no—
57#120×C61—22no—
58#120×C61—22no—
59#120×C61—24no—
60#120×C61—22no—
61C4×D4×C61—24no—
62C4×D4×C61—24no—
63#117×C61—21no—
64C2×C22⋊C4×C61—22no—
65C2×C22⋊C4×C61—21no—
66C2×C22⋊C4×C61—24no—
67C2×C22⋊C4×C61—24no—
68C2×C22⋊C4×C61—22no—
69C2×C22⋊C4×C61—44no—
70C2×C22⋊C4×C61—44no—
71C2×C22⋊C4×C61—42no—
72C2×C22⋊C4×C61C2×C412yes—
73C2×C22⋊C4×C61D414yes—
74#99×C61—48no—
75#99×C61—42no—
76#99×C61—42no—
77#95×C61—42no—
78C24×C61—21no—
79C24×C61—22no—
80C24×C61—22no—
81C24×C61—42no—
82C24×C61—41no—
83C24×C61—41no—
84C24×C61—41no—
85C24×C61—42no—
86C24×C61—42no—
87C24×C61—42no—
88C24×C61C22⋊C411yescharacteristic
89C24×C61C22⋊C411yescharacteristic
90C24×C61C22⋊C412yescomplemented
91C24×C61C22×C411yescharacteristic
92C24×C61C2×D411yescharacteristic
93C24×C61C2×D414yes—
94C24×C61C2×D411yescharacteristic
95#41×C61—42no—
96#41×C61—48no—
97#41×C61—42no—
98C2×Q8×C61—21no—
99C2×D4×C61—22no—
100C2×D4×C61—22no—
101C2×D4×C61—22no—
102C2×D4×C61—21no—
103C2×D4×C61—22no—
104C2×D4×C61—22no—
105C2×D4×C61—22no—
106C2×D4×C61—22no—
107C2×D4×C61—24no—
108C2×D4×C61—22no—
109C2×D4×C61—22no—
110C2×D4×C61—24no—
111C2×D4×C61—24no—
112C2×D4×C61—22no—
113C2×D4×C61—22no—
114C2×D4×C61—41no—
115C2×D4×C61—41no—
116C2×D4×C61—41no—
117C2×D4×C61—41no—
118C2×D4×C61—416no—
119C2×D4×C61—48no—
120C2×D4×C61—42no—
121C2×D4×C61—42no—
122C2×D4×C61—42no—
123C2×D4×C61—42no—
124C2×D4×C61—44no—
125C2×D4×C61—44no—
126C2×D4×C61—82no—
127C2×D4×C61—82no—
128C22×C4×C61—21no—
129C22×C4×C61—22no—
130C22×C4×C61—22no—
131C22×C4×C61—22no—
132C22×C4×C61—22no—
133C22×C4×C61—24no—
134C22×C4×C61—41no—
135C22×C4×C61—42no—
136C22×C4×C61—42no—
137C22×C4×C61—42no—
138C22×C4×C61—42no—
139C22×C4×C61—42no—
140C22×C4×C61—42no—
141C22×C4×C61—42no—
142C22×C4×C61C22⋊C411yescharacteristic
143C22×C4×C61C22⋊C411yescharacteristic
144C22×C4×C61C22⋊C414yescomplemented
145C22×C4×C61C22⋊C412yescomplemented
146C4⋊C4×C61—22no—
147C4⋊C4×C61—24no—
148C22⋊C4×C61—22no—
149C22⋊C4×C61—24no—
150C22⋊C4×C61—24no—
151C22⋊C4×C61—22no—
152C22⋊C4×C61—22no—
153C22⋊C4×C61—24no—
154C22⋊C4×C61—24no—
155C22⋊C4×C61—28no—
156C22⋊C4×C61—44no—
157C22⋊C4×C61—48no—
158C22⋊C4×C61—42no—
159C22⋊C4×C61—42no—
160C22⋊C4×C61—44no—
161C22⋊C4×C61—44no—
162C42×C61—22no—
163C23×C61—21no—
164C23×C61—21no—
165C23×C61—22no—
166C23×C61—22no—
167C23×C61—21no—
168C23×C61—21no—
169C23×C61—21no—
170C23×C61—21no—
171C23×C61—21no—
172C23×C61—21no—
173C23×C61—24no—
174C23×C61—24no—
175C23×C61—21no—
176C23×C61—22no—
177C23×C61—21no—
178C23×C61—21no—
179C23×C61—44no—
180C23×C61—42no—
181C23×C61—44no—
182C23×C61—42no—
183C23×C61—44no—
184C23×C61—42no—
185C23×C61—42no—
186C23×C61—42no—
187C23×C61—44no—
188C23×C61—42no—
189C23×C61—42no—
190C23×C61—42no—
191C23×C61—44no—
192C23×C61—42no—
193C23×C61—42no—
194C23×C61—82no—
195C23×C61—81no—
196C23×C61—82no—
197C23×C61—82no—
198C23×C61—81no—
199C23×C61—81no—
200C23×C61—82no—
201C23×C61—81no—
202C23×C61—81no—
203C23×C61—81no—
204C23×C61—82no—
205C23×C61#9912yescomplemented
206C23×C61C2×C22⋊C412yes—
207C23×C61C2×C22⋊C411yescharacteristic
208C23×C61#12012yes—
209C23×C61#12012yes—
210Q8×C61—42no—
211D4×C61—28no—
212D4×C61—44no—
213D4×C61—42no—
214D4×C61—44no—
215D4×C61—44no—
216D4×C61—44no—
217D4×C61—44no—
218D4×C61—42no—
219D4×C61—42no—
220D4×C61—44no—
221D4×C61—44no—
222D4×C61—84no—
223D4×C61—84no—
224C2×C4×C61—21no—
225C2×C4×C61—21no—
226C2×C4×C61—24no—
227C2×C4×C61—22no—
228C2×C4×C61—21no—
229C2×C4×C61—21no—
230C2×C4×C61—21no—
231C2×C4×C61—44no—
232C2×C4×C61—44no—
233C2×C4×C61—42no—
234C2×C4×C61—44no—
235C2×C4×C61—42no—
236C2×C4×C61—44no—
237C2×C4×C61—44no—
238C2×C4×C61—44no—
239C2×C4×C61—82no—
240C2×C4×C61—82no—
241C2×C4×C61—82no—
242C2×C4×C61—82no—
243C2×C4×C61—82no—
244C2×C4×C61—82no—
245C2×C4×C61—84no—
246C2×C4×C61#9912yescomplemented
247#8534C6111yesmaximal, characteristic, direct product factor, Sylow
248C22×C61—21no—
249C22×C61—22no—
250C22×C61—21no—
251C22×C61—22no—
252C22×C61—21no—
253C22×C61—21no—
254C22×C61—22no—
255C22×C61—22no—
256C22×C61—22no—
257C22×C61—22no—
258C22×C61—22no—
259C22×C61—22no—
260C22×C61—24no—
261C22×C61—24no—
262C22×C61—42no—
263C22×C61—42no—
264C22×C61—42no—
265C22×C61—42no—
266C22×C61—41no—
267C22×C61—44no—
268C22×C61—42no—
269C22×C61—84no—
270C22×C61—81no—
271C22×C61—81no—
272C22×C61—81no—
273C22×C61—81no—
274C22×C61—81no—
275C22×C61—84no—
276C22×C61—84no—
277C22×C61—84no—
278C22×C61—81no—
279C22×C61—162no—
280C22×C61—162no—
281C22×C61#37911yescharacteristic
282C22×C61C2×#9911yescharacteristic
283C22×C61C2×#9911yescharacteristic
284C22×C61B(4,𝔽2)12yes—
285C22×C61#45812yes—
286C4×C61—22no—
287C4×C61—41no—
288C4×C61—42no—
289C4×C61—44no—
290C4×C61—84no—
291C4×C61—162no—
292#3484C2×C6111yescharacteristic
293#1898C2×C6114yescomplemented
294#1781C2×C6111yescharacteristic, complemented
295#1438C2×C6111yescharacteristic, complemented
296C2×C61—22no—
297C2×C61—21no—
298C2×C61—21no—
299C2×C61—22no—
300C2×C61—42no—
301C2×C61—42no—
302C2×C61—42no—
303C2×C61—81no—
304C2×C61—82no—
305C2×C61#178111yescharacteristic
306C2×C61#179012yes—
307C2×H(5,𝔽2)C4×C6111yescharacteristic, complemented
308C23×D4C22×C6111yescharacteristic
309#546—24no—
310D42—24no—
311#535—21no—
312#534—21no—
313#534C4×C6112yescomplemented
314#525—21no—
315C2×#120—21no—
316C2×#120C22×C6112yes—
317#518C4×C6111yescharacteristic, complemented
318C22×C22⋊C4—21no—
319C22×C22⋊C4C22×C6112yescomplemented
320C2×#99—24no—
321C2×#99—21no—
322C2×#99—21no—
323#379C22×C6112yescomplemented
324C2×#95—21no—
325#342—24no—
326C61#853411yescharacteristic, direct product factor, Sylow
327C25D4×C6111yescharacteristic
328C25C23×C6111yescharacteristic
329H(5,𝔽2)—42no—
330H(5,𝔽2)—42no—
331C2×#41—21no—
332C2×#41—22no—
333C22×D4—21no—
334C22×D4—22no—
335C22×D4—24no—
336C22×D4—22no—
337C22×D4—42no—
338C22×D4—42no—
339C22×D4C2×C4×C6111yescharacteristic
340C22×D4C2×C4×C6111yescharacteristic
341C22×D4C2×C4×C6112yes—
342C22×D4D4×C6111yescharacteristic
343C22×D4D4×C6111yescharacteristic
344C22×D4D4×C6114yes—
345C23×C4—22no—
346C23×C4D4×C6111yescharacteristic
347#127—22no—
348#124—22no—
349#123—22no—
350#123—22no—
351#123—22no—
352C22⋊Q8—22no—
353#121—22no—
354#121—24no—
355#121—24no—
356#121—24no—
357#120—24no—
358#120—22no—
359#120—22no—
360#120—22no—
361#120—22no—
362#120—22no—
363#120—24no—
364C4×D4—24no—
365C4×D4—24no—
366#117—21no—
367C2×C22⋊C4—22no—
368C2×C22⋊C4—21no—
369C2×C22⋊C4—24no—
370C2×C22⋊C4—24no—
371C2×C22⋊C4—22no—
372C2×C22⋊C4—44no—
373C2×C22⋊C4—44no—
374C2×C22⋊C4—42no—
375C2×C22⋊C4C2×C4×C6112yes—
376C2×C22⋊C4D4×C6114yes—
377#99—48no—
378#99—42no—
379#99—42no—
380#95—42no—
381C24—21no—
382C24—22no—
383C24—22no—
384C24—42no—
385C24—41no—
386C24—41no—
387C24—41no—
388C24—42no—
389C24—42no—
390C24—42no—
391C24C22⋊C4×C6111yescharacteristic
392C24C22⋊C4×C6111yescharacteristic
393C24C22⋊C4×C6112yescomplemented
394C24C22×C4×C6111yescharacteristic
395C24C2×D4×C6111yescharacteristic
396C24C2×D4×C6114yes—
397C24C2×D4×C6111yescharacteristic
398#41—42no—
399#41—48no—
400#41—42no—
401C2×Q8—21no—
402C2×D4—22no—
403C2×D4—22no—
404C2×D4—22no—
405C2×D4—21no—
406C2×D4—22no—
407C2×D4—22no—
408C2×D4—22no—
409C2×D4—22no—
410C2×D4—24no—
411C2×D4—22no—
412C2×D4—22no—
413C2×D4—24no—
414C2×D4—24no—
415C2×D4—22no—
416C2×D4—22no—
417C2×D4—41no—
418C2×D4—41no—
419C2×D4—41no—
420C2×D4—41no—
421C2×D4—416no—
422C2×D4—48no—
423C2×D4—42no—
424C2×D4—42no—
425C2×D4—42no—
426C2×D4—42no—
427C2×D4—44no—
428C2×D4—44no—
429C2×D4—82no—
430C2×D4—82no—
431C22×C4—21no—
432C22×C4—22no—
433C22×C4—22no—
434C22×C4—22no—
435C22×C4—22no—
436C22×C4—24no—
437C22×C4—41no—
438C22×C4—42no—
439C22×C4—42no—
440C22×C4—42no—
441C22×C4—42no—
442C22×C4—42no—
443C22×C4—42no—
444C22×C4—42no—
445C22×C4C22⋊C4×C6111yescharacteristic
446C22×C4C22⋊C4×C6111yescharacteristic
447C22×C4C22⋊C4×C6114yescomplemented
448C22×C4C22⋊C4×C6112yescomplemented
449C4⋊C4—22no—
450C4⋊C4—24no—
451C22⋊C4—22no—
452C22⋊C4—24no—
453C22⋊C4—24no—
454C22⋊C4—22no—
455C22⋊C4—22no—
456C22⋊C4—24no—
457C22⋊C4—24no—
458C22⋊C4—28no—
459C22⋊C4—44no—
460C22⋊C4—48no—
461C22⋊C4—42no—
462C22⋊C4—42no—
463C22⋊C4—44no—
464C22⋊C4—44no—
465C42—22no—
466C23—21no—
467C23—21no—
468C23—22no—
469C23—22no—
470C23—21no—
471C23—21no—
472C23—21no—
473C23—21no—
474C23—21no—
475C23—21no—
476C23—24no—
477C23—24no—
478C23—21no—
479C23—22no—
480C23—21no—
481C23—21no—
482C23—44no—
483C23—42no—
484C23—44no—
485C23—42no—
486C23—44no—
487C23—42no—
488C23—42no—
489C23—42no—
490C23—44no—
491C23—42no—
492C23—42no—
493C23—42no—
494C23—44no—
495C23—42no—
496C23—42no—
497C23—82no—
498C23—81no—
499C23—82no—
500C23—82no—
501C23—81no—
502C23—81no—
503C23—82no—
504C23—81no—
505C23—81no—
506C23—81no—
507C23—82no—
508C23#99×C6112yescomplemented
509C23C2×C22⋊C4×C6112yes—
510C23C2×C22⋊C4×C6111yescharacteristic
511C23#120×C6112yes—
512C23#120×C6112yes—
513Q8—42no—
514D4—28no—
515D4—44no—
516D4—42no—
517D4—44no—
518D4—44no—
519D4—44no—
520D4—44no—
521D4—42no—
522D4—42no—
523D4—44no—
524D4—44no—
525D4—84no—
526D4—84no—
527C2×C4—21no—
528C2×C4—21no—
529C2×C4—24no—
530C2×C4—22no—
531C2×C4—21no—
532C2×C4—21no—
533C2×C4—21no—
534C2×C4—44no—
535C2×C4—44no—
536C2×C4—42no—
537C2×C4—44no—
538C2×C4—42no—
539C2×C4—44no—
540C2×C4—44no—
541C2×C4—44no—
542C2×C4—82no—
543C2×C4—82no—
544C2×C4—82no—
545C2×C4—82no—
546C2×C4—82no—
547C2×C4—82no—
548C2×C4—84no—
549C2×C4#99×C6112yescomplemented
550C22—21no—
551C22—22no—
552C22—21no—
553C22—22no—
554C22—21no—
555C22—21no—
556C22—22no—
557C22—22no—
558C22—22no—
559C22—22no—
560C22—22no—
561C22—22no—
562C22—24no—
563C22—24no—
564C22—42no—
565C22—42no—
566C22—42no—
567C22—42no—
568C22—41no—
569C22—44no—
570C22—42no—
571C22—84no—
572C22—81no—
573C22—81no—
574C22—81no—
575C22—81no—
576C22—81no—
577C22—84no—
578C22—84no—
579C22—84no—
580C22—81no—
581C22—162no—
582C22—162no—
583C22C61×#37911yescharacteristic
584C22C2×#99×C6111yescharacteristic
585C22C2×#99×C6111yescharacteristic
586C22C61×B(4,𝔽2)12yes—
587C22C61×#45812yes—
588C4—22no—
589C4—41no—
590C4—42no—
591C4—44no—
592C4—84no—
593C4—162no—
594C2—22no—
595C2—21no—
596C2—21no—
597C2—22no—
598C2—42no—
599C2—42no—
600C2—42no—
601C2—81no—
602C2—82no—
603C2C61×#178111yescharacteristic
604C2C61×#179012yes—
subgroupG/H
center, socleC22×C61#379
commutatorC24C22×C4×C61
Frattini, cosocleC25C23×C61
prime radicalC61×#3484C2
LCSG ▷ C24 ▷ C22 ▷ C1
UCSC1 ◁ C22×C61 ◁ C25×C61 ◁ G

Complements

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

NH ≅ G/Ntype
2. C61×#1898C2semidirect
3. C61×#1781C2semidirect
4. C61×#1438C2semidirect
5. C2×H(5,𝔽2)×C61C4semidirect
11. C61×#534C4semidirect
15. C61×#518C4semidirect
17. C22×C22⋊C4×C61C22semidirect
21. C61×#379C22semidirect
90. C24×C61C22⋊C4semidirect
144. C22×C4×C61C22⋊C4semidirect
145. C22×C4×C61C22⋊C4semidirect
205. C23×C61#99semidirect
246. C2×C4×C61#99semidirect
247. #8534C61direct (N × H)
293. #1898C2×C61semidirect
294. #1781C2×C61semidirect
295. #1438C2×C61semidirect
307. C2×H(5,𝔽2)C4×C61semidirect
313. #534C4×C61semidirect
317. #518C4×C61semidirect
319. C22×C22⋊C4C22×C61semidirect
323. #379C22×C61semidirect
326. C61#8534direct (N × H)
393. C24C22⋊C4×C61semidirect
447. C22×C4C22⋊C4×C61semidirect
448. C22×C4C22⋊C4×C61semidirect
508. C23#99×C61semidirect
549. C2×C4#99×C61semidirect

Maximal subgroups

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

subgroupindex
C61×#34842
C61×#18982
C61×#17812
C61×#14382
#853461