Finite Group Databasebeta

← C61×#12705 #29516138

Subgroups — C61×#12705 #29516138

Order 15616.

1186 conjugacy classes of subgroups; only the 146 normal subgroups are shown.

normal subgroups only

This group has 1186 conjugacy classes of subgroups — too many to draw. Only its 146 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×#2121C214yesmaximal, complemented
3C61×#1783C211yesmaximal, characteristic, complemented
4C61×#1781C211yesmaximal, characteristic, complemented
5C2×H(5,𝔽2)×C61C411yescharacteristic, complemented
6C23×D4×C61C2211yescharacteristic
7C61×#546—24no—
8D42×C61—24no—
9C61×#535—21no—
10C61×#534—21no—
11C61×#534C2212yes—
12C61×#525—21no—
13C2×#120×C61—21no—
14C2×#120×C61C412yescomplemented
15C61×#518C411yescharacteristic, complemented
16C2×#100×C61—21no—
17C2×#100×C61C2212yescomplemented
18C2×#99×C61—21no—
19C2×#99×C61C2212yescomplemented
20C61×#407—21no—
21C61×#379—21no—
22C61×#352—24no—
23C61×AΣL(1,𝔽16)—24no—
24C25×C61C2×C411yescharacteristic
25C25×C61D411yescharacteristic
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—22no—
33C22×D4×C61—22no—
34C22×D4×C61—24no—
35C22×D4×C61—42no—
36C22×D4×C61—42no—
37C22×D4×C61C2×C411yescharacteristic
38C22×D4×C61C2×C412yes—
39C22×D4×C61D411yescharacteristic
40C22×D4×C61D411yescharacteristic
41C22×D4×C61C2311yescharacteristic
42C23×C4×C61D411yescharacteristic
43C2×M16×C61—22no—
44#127×C61—22no—
45#124×C61—22no—
46#123×C61—22no—
47#123×C61—22no—
48#123×C61D414yes—
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×C61D414yes—
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—22no—
67C2×C22⋊C4×C61—22no—
68C2×C22⋊C4×C61—42no—
69C2×C22⋊C4×C61C2×C412yes—
70#100×C61—24no—
71#100×C61—42no—
72#99×C61—24no—
73#99×C61—42no—
74C22⋊C8×C61—42no—
75C24×C61—21no—
76C24×C61—22no—
77C24×C61—22no—
78C24×C61—22no—
79C24×C61—41no—
80C24×C61—41no—
81C24×C61—41no—
82C24×C61—82no—
83C24×C61—82no—
84C24×C61C22⋊C411yescharacteristic
85C24×C61C22⋊C411yescharacteristic
86C24×C61C22⋊C411yescharacteristic
87C24×C61C22⋊C412yescomplemented
88C24×C61C22×C411yescharacteristic
89C24×C61C2×D411yescharacteristic
90#41×C61—42no—
91#41×C61—48no—
92#41×C61—42no—
93C2×Q8×C61—21no—
94C2×D4×C61—22no—
95C2×D4×C61—22no—
96C2×D4×C61—24no—
97C2×D4×C61—22no—
98C2×D4×C61—21no—
99C2×D4×C61—22no—
100C2×D4×C61—22no—
101C2×D4×C61—22no—
102C2×D4×C61—22no—
103C2×D4×C61—24no—
104C2×D4×C61—24no—
105C2×D4×C61—22no—
106C2×D4×C61—22no—
107C2×D4×C61—22no—
108C2×D4×C61—41no—
109C2×D4×C61—41no—
110C2×D4×C61—41no—
111C2×D4×C61—41no—
112C2×D4×C61—416no—
113C2×D4×C61—48no—
114C2×D4×C61—42no—
115C2×D4×C61—42no—
116C2×D4×C61—42no—
117C2×D4×C61—42no—
118C2×D4×C61—84no—
119C2×D4×C61—84no—
120C2×D4×C61C2×D414yes—
121C22×C4×C61—21no—
122C22×C4×C61—22no—
123C22×C4×C61—22no—
124C22×C4×C61—22no—
125C22×C4×C61—22no—
126C22×C4×C61—22no—
127C22×C4×C61—41no—
128C22×C4×C61—42no—
129C22×C4×C61C22⋊C411yescharacteristic
130C22×C4×C61C22⋊C412yescomplemented
131C22×C4×C61C2×D411yescharacteristic
132M16×C61—42no—
133M16×C61—42no—
134C2×C8×C61—42no—
135C4⋊C4×C61—22no—
136C4⋊C4×C61—24no—
137C22⋊C4×C61—24no—
138C22⋊C4×C61—22no—
139C22⋊C4×C61—24no—
140C22⋊C4×C61—22no—
141C22⋊C4×C61—24no—
142C22⋊C4×C61—24no—
143C22⋊C4×C61—42no—
144C22⋊C4×C61—42no—
145C22⋊C4×C61—84no—
146C22⋊C4×C61C22⋊C414yescomplemented
147C42×C61—22no—
148C23×C61—21no—
149C23×C61—21no—
150C23×C61—21no—
151C23×C61—21no—
152C23×C61—21no—
153C23×C61—21no—
154C23×C61—21no—
155C23×C61—21no—
156C23×C61—21no—
157C23×C61—21no—
158C23×C61—22no—
159C23×C61—21no—
160C23×C61—21no—
161C23×C61—42no—
162C23×C61—44no—
163C23×C61—44no—
164C23×C61—44no—
165C23×C61—44no—
166C23×C61—44no—
167C23×C61—42no—
168C23×C61—42no—
169C23×C61—82no—
170C23×C61—82no—
171C23×C61—82no—
172C23×C61—81no—
173C23×C61—81no—
174C23×C61—82no—
175C23×C61—82no—
176C23×C61—82no—
177C23×C61—81no—
178C23×C61—81no—
179C23×C61—82no—
180C23×C61—82no—
181C23×C61—82no—
182C23×C61—82no—
183C23×C61—82no—
184C23×C61#9912yes—
185C23×C61C2×C22⋊C412yes—
186C23×C61C2×C22⋊C411yescharacteristic
187C23×C61#12012yes—
188Q8×C61—42no—
189D4×C61—28no—
190D4×C61—44no—
191D4×C61—44no—
192D4×C61—44no—
193D4×C61—42no—
194D4×C61—44no—
195D4×C61—44no—
196D4×C61—48no—
197D4×C61—42no—
198D4×C61—42no—
199D4×C61—84no—
200D4×C61—84no—
201C2×C4×C61—21no—
202C2×C4×C61—21no—
203C2×C4×C61—21no—
204C2×C4×C61—21no—
205C2×C4×C61—24no—
206C2×C4×C61—22no—
207C2×C4×C61—44no—
208C2×C4×C61—44no—
209C2×C4×C61—42no—
210C2×C4×C61—42no—
211C2×C4×C61—82no—
212C2×C4×C61—82no—
213C2×C4×C61—82no—
214C2×C4×C61—84no—
215C2×C4×C61#9912yes—
216C2×C4×C61#12012yes—
217C8×C61—82no—
218#12705C6111yesmaximal, characteristic, direct product factor, Sylow
219C22×C61—22no—
220C22×C61—22no—
221C22×C61—22no—
222C22×C61—22no—
223C22×C61—22no—
224C22×C61—22no—
225C22×C61—41no—
226C22×C61—42no—
227C22×C61—42no—
228C22×C61—42no—
229C22×C61—42no—
230C22×C61—42no—
231C22×C61—41no—
232C22×C61—41no—
233C22×C61—41no—
234C22×C61—44no—
235C22×C61—42no—
236C22×C61—41no—
237C22×C61—42no—
238C22×C61—41no—
239C22×C61—81no—
240C22×C61—81no—
241C22×C61—81no—
242C22×C61—81no—
243C22×C61—81no—
244C22×C61—81no—
245C22×C61—88no—
246C22×C61—88no—
247C22×C61—162no—
248C22×C61—162no—
249C22×C61#37911yescharacteristic
250C22×C61C2×#9911yescharacteristic
251C22×C61C2×#9911yescharacteristic
252C4×C61—22no—
253C4×C61—22no—
254C4×C61—44no—
255C4×C61—42no—
256C4×C61—162no—
257#3484C2×C6111yescharacteristic
258#2121C2×C6114yescomplemented
259#1783C2×C6111yescharacteristic, complemented
260#1781C2×C6111yescharacteristic, complemented
261C2×C61—21no—
262C2×C61—21no—
263C2×C61—21no—
264C2×C61—42no—
265C2×C61—42no—
266C2×C61—42no—
267C2×C61—42no—
268C2×C61—81no—
269C2×C61—82no—
270C2×C61#178111yescharacteristic
271C2×H(5,𝔽2)C4×C6111yescharacteristic, complemented
272C23×D4C22×C6111yescharacteristic
273#546—24no—
274D42—24no—
275#535—21no—
276#534—21no—
277#534C22×C6112yes—
278#525—21no—
279C2×#120—21no—
280C2×#120C4×C6112yescomplemented
281#518C4×C6111yescharacteristic, complemented
282C2×#100—21no—
283C2×#100C22×C6112yescomplemented
284C2×#99—21no—
285C2×#99C22×C6112yescomplemented
286#407—21no—
287#379—21no—
288#352—24no—
289AΣL(1,𝔽16)—24no—
290C61#1270511yescharacteristic, direct product factor, Sylow
291C25C2×C4×C6111yescharacteristic
292C25D4×C6111yescharacteristic
293H(5,𝔽2)—42no—
294H(5,𝔽2)—42no—
295C2×#41—21no—
296C2×#41—22no—
297C22×D4—21no—
298C22×D4—22no—
299C22×D4—22no—
300C22×D4—22no—
301C22×D4—24no—
302C22×D4—42no—
303C22×D4—42no—
304C22×D4C2×C4×C6111yescharacteristic
305C22×D4C2×C4×C6112yes—
306C22×D4D4×C6111yescharacteristic
307C22×D4D4×C6111yescharacteristic
308C22×D4C23×C6111yescharacteristic
309C23×C4D4×C6111yescharacteristic
310C2×M16—22no—
311#127—22no—
312#124—22no—
313#123—22no—
314#123—22no—
315#123D4×C6114yes—
316C22⋊Q8—22no—
317#121—22no—
318#121—24no—
319#121—24no—
320#121—24no—
321#120—24no—
322#120—22no—
323#120—22no—
324#120—22no—
325#120—22no—
326#120—24no—
327#120D4×C6114yes—
328C4×D4—24no—
329C4×D4—24no—
330#117—21no—
331C2×C22⋊C4—22no—
332C2×C22⋊C4—21no—
333C2×C22⋊C4—22no—
334C2×C22⋊C4—22no—
335C2×C22⋊C4—42no—
336C2×C22⋊C4C2×C4×C6112yes—
337#100—24no—
338#100—42no—
339#99—24no—
340#99—42no—
341C22⋊C8—42no—
342C24—21no—
343C24—22no—
344C24—22no—
345C24—22no—
346C24—41no—
347C24—41no—
348C24—41no—
349C24—82no—
350C24—82no—
351C24C22⋊C4×C6111yescharacteristic
352C24C22⋊C4×C6111yescharacteristic
353C24C22⋊C4×C6111yescharacteristic
354C24C22⋊C4×C6112yescomplemented
355C24C22×C4×C6111yescharacteristic
356C24C2×D4×C6111yescharacteristic
357#41—42no—
358#41—48no—
359#41—42no—
360C2×Q8—21no—
361C2×D4—22no—
362C2×D4—22no—
363C2×D4—24no—
364C2×D4—22no—
365C2×D4—21no—
366C2×D4—22no—
367C2×D4—22no—
368C2×D4—22no—
369C2×D4—22no—
370C2×D4—24no—
371C2×D4—24no—
372C2×D4—22no—
373C2×D4—22no—
374C2×D4—22no—
375C2×D4—41no—
376C2×D4—41no—
377C2×D4—41no—
378C2×D4—41no—
379C2×D4—416no—
380C2×D4—48no—
381C2×D4—42no—
382C2×D4—42no—
383C2×D4—42no—
384C2×D4—42no—
385C2×D4—84no—
386C2×D4—84no—
387C2×D4C2×D4×C6114yes—
388C22×C4—21no—
389C22×C4—22no—
390C22×C4—22no—
391C22×C4—22no—
392C22×C4—22no—
393C22×C4—22no—
394C22×C4—41no—
395C22×C4—42no—
396C22×C4C22⋊C4×C6111yescharacteristic
397C22×C4C22⋊C4×C6112yescomplemented
398C22×C4C2×D4×C6111yescharacteristic
399M16—42no—
400M16—42no—
401C2×C8—42no—
402C4⋊C4—22no—
403C4⋊C4—24no—
404C22⋊C4—24no—
405C22⋊C4—22no—
406C22⋊C4—24no—
407C22⋊C4—22no—
408C22⋊C4—24no—
409C22⋊C4—24no—
410C22⋊C4—42no—
411C22⋊C4—42no—
412C22⋊C4—84no—
413C22⋊C4C22⋊C4×C6114yescomplemented
414C42—22no—
415C23—21no—
416C23—21no—
417C23—21no—
418C23—21no—
419C23—21no—
420C23—21no—
421C23—21no—
422C23—21no—
423C23—21no—
424C23—21no—
425C23—22no—
426C23—21no—
427C23—21no—
428C23—42no—
429C23—44no—
430C23—44no—
431C23—44no—
432C23—44no—
433C23—44no—
434C23—42no—
435C23—42no—
436C23—82no—
437C23—82no—
438C23—82no—
439C23—81no—
440C23—81no—
441C23—81no—
442C23—81no—
443C23—82no—
444C23—82no—
445C23—82no—
446C23—82no—
447C23—82no—
448C23—82no—
449C23—82no—
450C23—82no—
451C23#99×C6112yes—
452C23C2×C22⋊C4×C6112yes—
453C23C2×C22⋊C4×C6111yescharacteristic
454C23#120×C6112yes—
455Q8—42no—
456D4—28no—
457D4—44no—
458D4—44no—
459D4—44no—
460D4—42no—
461D4—44no—
462D4—44no—
463D4—48no—
464D4—42no—
465D4—42no—
466D4—84no—
467D4—84no—
468C2×C4—21no—
469C2×C4—21no—
470C2×C4—21no—
471C2×C4—21no—
472C2×C4—24no—
473C2×C4—22no—
474C2×C4—44no—
475C2×C4—44no—
476C2×C4—42no—
477C2×C4—42no—
478C2×C4—82no—
479C2×C4—82no—
480C2×C4—82no—
481C2×C4—84no—
482C2×C4#99×C6112yes—
483C2×C4#120×C6112yes—
484C8—82no—
485C22—22no—
486C22—22no—
487C22—22no—
488C22—22no—
489C22—22no—
490C22—22no—
491C22—41no—
492C22—42no—
493C22—42no—
494C22—42no—
495C22—42no—
496C22—42no—
497C22—41no—
498C22—41no—
499C22—41no—
500C22—44no—
501C22—42no—
502C22—41no—
503C22—42no—
504C22—41no—
505C22—81no—
506C22—81no—
507C22—81no—
508C22—81no—
509C22—81no—
510C22—81no—
511C22—88no—
512C22—88no—
513C22—162no—
514C22—162no—
515C22C61×#37911yescharacteristic
516C22C2×#99×C6111yescharacteristic
517C22C2×#99×C6111yescharacteristic
518C4—22no—
519C4—22no—
520C4—44no—
521C4—42no—
522C4—162no—
523C2—21no—
524C2—21no—
525C2—21no—
526C2—42no—
527C2—42no—
528C2—42no—
529C2—42no—
530C2—81no—
531C2—82no—
532C2C61×#178111yescharacteristic
subgroupG/H
center, socleC2×C61#1781
commutatorC24C22×C4×C61
Frattini, cosocleC22×D4C23×C61
prime radicalC61×#3484C2
LCSG ▷ C24 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C2×C61 ◁ C23×C61 ◁ C23×D4×C61 ◁ G

Complements

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

NH ≅ G/Ntype
2. C61×#2121C2semidirect
3. C61×#1783C2semidirect
4. C61×#1781C2semidirect
5. C2×H(5,𝔽2)×C61C4semidirect
14. C2×#120×C61C4semidirect
15. C61×#518C4semidirect
17. C2×#100×C61C22semidirect
19. C2×#99×C61C22semidirect
87. C24×C61C22⋊C4semidirect
130. C22×C4×C61C22⋊C4semidirect
146. C22⋊C4×C61C22⋊C4semidirect
218. #12705C61direct (N × H)
258. #2121C2×C61semidirect
259. #1783C2×C61semidirect
260. #1781C2×C61semidirect
271. C2×H(5,𝔽2)C4×C61semidirect
280. C2×#120C4×C61semidirect
281. #518C4×C61semidirect
283. C2×#100C22×C61semidirect
285. C2×#99C22×C61semidirect
290. C61#12705direct (N × H)
354. C24C22⋊C4×C61semidirect
397. C22×C4C22⋊C4×C61semidirect
413. C22⋊C4C22⋊C4×C61semidirect

Maximal subgroups

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

subgroupindex
C61×#34842
C61×#21212
C61×#17832
C61×#17812
#1270561