Finite Group Databasebeta

← C61×#13219 #29516652

Subgroups — C61×#13219 #29516652

Order 15616.

624 conjugacy classes of subgroups.

The subgroup lattice has 456 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C61×#2534C211yesmaximal, characteristic
2C2×C61×#352C212yesmaximal, complemented
3C2×C61×AΣL(1,𝔽16)C212yesmaximal
4C61×#1799C211yesmaximal, characteristic, complemented
5C4×#100×C61C211yesmaximal, characteristic, complemented
6C2×#123×C61C2211yescharacteristic
7C2×#120×C61C2211yescharacteristic
8C2×C4×D4×C61C2211yescharacteristic
9C22×C22⋊C4×C61C411yescharacteristic, complemented
10C2×#100×C61C2212yes—
11C2×#99×C61C2212yes—
12C4×M16×C61—21no—
13C61×#396—21no—
14C61×#394—21no—
15C61×#388—21no—
16C61×#386—21no—
17C61×#381C411yescharacteristic, complemented
18C61×#380—21no—
19C61×#380C411yescharacteristic, complemented
20C61×#379C411yescharacteristic, complemented
21C61×#352—24no—
22C61×AΣL(1,𝔽16)—24no—
23C61×#343—21no—
24C61×#342—21no—
25C25×C61C2×C411yescharacteristic
26C22×D4×C61—21no—
27C22×D4×C61C2311yescharacteristic
28C23×C4×C61C2×C411yescharacteristic
29C23×C4×C61C2×C411yescharacteristic
30C23×C4×C61D411yescharacteristic
31C2×M16×C61—22no—
32#123×C61—22no—
33#123×C61—22no—
34#120×C61—22no—
35#120×C61—22no—
36C4×D4×C61—41no—
37C4×D4×C61—41no—
38C2×C4⋊C4×C61—21no—
39C2×C4⋊C4×C61—21no—
40C2×C4⋊C4×C61—41no—
41C2×C4⋊C4×C61D411yescharacteristic
42C2×C22⋊C4×C61—21no—
43C2×C22⋊C4×C61—21no—
44C2×C22⋊C4×C61—21no—
45C2×C22⋊C4×C61—21no—
46C2×C22⋊C4×C61—21no—
47C2×C22⋊C4×C61—22no—
48C2×C22⋊C4×C61—21no—
49C2×C22⋊C4×C61—41no—
50C2×C22⋊C4×C61—41no—
51C2×C22⋊C4×C61C2×C411yescharacteristic
52C2×C22⋊C4×C61C2×C411yescharacteristic
53C2×C22⋊C4×C61C2×C411yescharacteristic
54C2×C22⋊C4×C61D411yescharacteristic
55C2×C42×C61D411yescharacteristic
56#100×C61—22no—
57#100×C61D414yes—
58#99×C61—22no—
59#99×C61—22no—
60#97×C61—41no—
61C4×C8×C61—41no—
62#95×C61—21no—
63#95×C61—21no—
64#95×C61—21no—
65#95×C61—41no—
66C24×C61—21no—
67C24×C61—22no—
68C24×C61—41no—
69C24×C61—42no—
70C24×C61—41no—
71C24×C61—81no—
72C24×C61C22×C411yescharacteristic
73C24×C61C2×D411yescharacteristic
74C2×D4×C61—41no—
75C2×D4×C61—41no—
76C2×D4×C61—42no—
77C2×D4×C61—41no—
78C2×D4×C61—41no—
79C2×D4×C61—81no—
80C2×D4×C61C2×D412yes—
81C2×D4×C61#4112yes—
82C22×C4×C61—21no—
83C22×C4×C61—21no—
84C22×C4×C61—21no—
85C22×C4×C61—21no—
86C22×C4×C61—21no—
87C22×C4×C61—21no—
88C22×C4×C61—21no—
89C22×C4×C61—22no—
90C22×C4×C61—41no—
91C22×C4×C61—41no—
92C22×C4×C61—42no—
93C22×C4×C61—81no—
94C22×C4×C61—81no—
95C22×C4×C61C22⋊C411yescharacteristic
96C22×C4×C61C22⋊C411yescharacteristic
97C22×C4×C61C22⋊C411yescharacteristic
98C22×C4×C61C22⋊C411yescharacteristic
99C22×C4×C61C2×D411yescharacteristic
100M16×C61—24no—
101M16×C61—42no—
102C2×C8×C61—42no—
103C4⋊C4×C61—41no—
104C4⋊C4×C61—42no—
105C4⋊C4×C61—81no—
106C4⋊C4×C61—82no—
107C22⋊C4×C61—22no—
108C22⋊C4×C61—22no—
109C22⋊C4×C61—41no—
110C22⋊C4×C61—41no—
111C22⋊C4×C61—42no—
112C22⋊C4×C61—42no—
113C22⋊C4×C61—41no—
114C22⋊C4×C61—41no—
115C22⋊C4×C61—41no—
116C22⋊C4×C61—41no—
117C22⋊C4×C61—42no—
118C22⋊C4×C61—42no—
119C22⋊C4×C61—42no—
120C22⋊C4×C61—81no—
121C42×C61—21no—
122C42×C61—21no—
123C23×C61—21no—
124C23×C61—21no—
125C23×C61—21no—
126C23×C61—21no—
127C23×C61—22no—
128C23×C61—41no—
129C23×C61—41no—
130C23×C61—41no—
131C23×C61—41no—
132C23×C61—41no—
133C23×C61—41no—
134C23×C61—41no—
135C23×C61—42no—
136C23×C61—41no—
137C23×C61—41no—
138C23×C61—41no—
139C23×C61—42no—
140C23×C61—82no—
141C23×C61—84no—
142C23×C61—84no—
143C23×C61—82no—
144C23×C61C2×C22⋊C411yescharacteristic
145C23×C61C4×D412yes—
146C23×C61#12411yescharacteristic
147C23×C61#12711yescharacteristic
148D4×C61—42no—
149D4×C61—42no—
150D4×C61—82no—
151D4×C61—82no—
152C2×C4×C61—21no—
153C2×C4×C61—21no—
154C2×C4×C61—21no—
155C2×C4×C61—21no—
156C2×C4×C61—21no—
157C2×C4×C61—41no—
158C2×C4×C61—42no—
159C2×C4×C61—41no—
160C2×C4×C61—41no—
161C2×C4×C61—41no—
162C2×C4×C61—41no—
163C2×C4×C61—41no—
164C2×C4×C61—41no—
165C2×C4×C61—41no—
166C2×C4×C61—41no—
167C2×C4×C61—41no—
168C2×C4×C61—41no—
169C2×C4×C61—41no—
170C2×C4×C61—41no—
171C2×C4×C61—81no—
172C2×C4×C61—81no—
173C2×C4×C61—81no—
174C2×C4×C61—82no—
175C2×C4×C61—82no—
176C2×C4×C61—82no—
177C2×C4×C61—82no—
178C2×C4×C61#9911yescharacteristic
179C2×C4×C61#9911yescharacteristic
180C2×C4×C61#12112yes—
181C8×C61—44no—
182#13219C6111yesmaximal, characteristic, direct product factor, Sylow
183C22×C61—21no—
184C22×C61—21no—
185C22×C61—21no—
186C22×C61—22no—
187C22×C61—21no—
188C22×C61—41no—
189C22×C61—41no—
190C22×C61—42no—
191C22×C61—42no—
192C22×C61—42no—
193C22×C61—42no—
194C22×C61—41no—
195C22×C61—41no—
196C22×C61—41no—
197C22×C61—41no—
198C22×C61—42no—
199C22×C61—81no—
200C22×C61—81no—
201C22×C61—81no—
202C22×C61—84no—
203C22×C61—84no—
204C22×C61—81no—
205C22×C61—81no—
206C22×C61#39011yescharacteristic
207C22×C61C2×#9911yescharacteristic
208C22×C61#41011yescharacteristic
209C4×C61—21no—
210C4×C61—21no—
211C4×C61—22no—
212C4×C61—41no—
213C4×C61—41no—
214C4×C61—41no—
215C4×C61—81no—
216C4×C61—81no—
217C4×C61—81no—
218C4×C61—162no—
219#2534C2×C6111yescharacteristic
220C2×#352C2×C6112yescomplemented
221C2×AΣL(1,𝔽16)C2×C6112yes—
222#1799C2×C6111yescharacteristic, complemented
223C4×#100C2×C6111yescharacteristic, complemented
224C2×C61—21no—
225C2×C61—21no—
226C2×C61—41no—
227C2×C61—42no—
228C2×C61—41no—
229C2×C61—82no—
230C2×C61#196411yescharacteristic
231C2×C61#212011yescharacteristic
232C2×C61#212111yescharacteristic
233C2×#123C22×C6111yescharacteristic
234C2×#120C22×C6111yescharacteristic
235C2×C4×D4C22×C6111yescharacteristic
236C22×C22⋊C4C4×C6111yescharacteristic, complemented
237C2×#100C22×C6112yes—
238C2×#99C22×C6112yes—
239C4×M16—21no—
240#396—21no—
241#394—21no—
242#388—21no—
243#386—21no—
244#381C4×C6111yescharacteristic, complemented
245#380—21no—
246#380C4×C6111yescharacteristic, complemented
247#379C4×C6111yescharacteristic, complemented
248#352—24no—
249AΣL(1,𝔽16)—24no—
250#343—21no—
251#342—21no—
252C61#1321911yescharacteristic, direct product factor, Sylow
253C25C2×C4×C6111yescharacteristic
254C22×D4—21no—
255C22×D4C23×C6111yescharacteristic
256C23×C4C2×C4×C6111yescharacteristic
257C23×C4C2×C4×C6111yescharacteristic
258C23×C4D4×C6111yescharacteristic
259C2×M16—22no—
260#123—22no—
261#123—22no—
262#120—22no—
263#120—22no—
264C4×D4—41no—
265C4×D4—41no—
266C2×C4⋊C4—21no—
267C2×C4⋊C4—21no—
268C2×C4⋊C4—41no—
269C2×C4⋊C4D4×C6111yescharacteristic
270C2×C22⋊C4—21no—
271C2×C22⋊C4—21no—
272C2×C22⋊C4—21no—
273C2×C22⋊C4—21no—
274C2×C22⋊C4—21no—
275C2×C22⋊C4—22no—
276C2×C22⋊C4—21no—
277C2×C22⋊C4—41no—
278C2×C22⋊C4—41no—
279C2×C22⋊C4C2×C4×C6111yescharacteristic
280C2×C22⋊C4C2×C4×C6111yescharacteristic
281C2×C22⋊C4C2×C4×C6111yescharacteristic
282C2×C22⋊C4D4×C6111yescharacteristic
283C2×C42D4×C6111yescharacteristic
284#100—22no—
285#100D4×C6114yes—
286#99—22no—
287#99—22no—
288#97—41no—
289C4×C8—41no—
290#95—21no—
291#95—21no—
292#95—21no—
293#95—41no—
294C24—21no—
295C24—22no—
296C24—41no—
297C24—42no—
298C24—41no—
299C24—81no—
300C24C22×C4×C6111yescharacteristic
301C24C2×D4×C6111yescharacteristic
302C2×D4—41no—
303C2×D4—41no—
304C2×D4—42no—
305C2×D4—41no—
306C2×D4—41no—
307C2×D4—81no—
308C2×D4C2×D4×C6112yes—
309C2×D4#41×C6112yes—
310C22×C4—21no—
311C22×C4—21no—
312C22×C4—21no—
313C22×C4—21no—
314C22×C4—21no—
315C22×C4—21no—
316C22×C4—21no—
317C22×C4—22no—
318C22×C4—41no—
319C22×C4—41no—
320C22×C4—42no—
321C22×C4—81no—
322C22×C4—81no—
323C22×C4C22⋊C4×C6111yescharacteristic
324C22×C4C22⋊C4×C6111yescharacteristic
325C22×C4C22⋊C4×C6111yescharacteristic
326C22×C4C22⋊C4×C6111yescharacteristic
327C22×C4C2×D4×C6111yescharacteristic
328M16—24no—
329M16—42no—
330C2×C8—42no—
331C4⋊C4—41no—
332C4⋊C4—42no—
333C4⋊C4—81no—
334C4⋊C4—82no—
335C22⋊C4—22no—
336C22⋊C4—22no—
337C22⋊C4—41no—
338C22⋊C4—41no—
339C22⋊C4—42no—
340C22⋊C4—42no—
341C22⋊C4—41no—
342C22⋊C4—41no—
343C22⋊C4—41no—
344C22⋊C4—41no—
345C22⋊C4—42no—
346C22⋊C4—42no—
347C22⋊C4—42no—
348C22⋊C4—81no—
349C42—21no—
350C42—21no—
351C23—21no—
352C23—21no—
353C23—21no—
354C23—21no—
355C23—22no—
356C23—41no—
357C23—41no—
358C23—41no—
359C23—41no—
360C23—41no—
361C23—41no—
362C23—41no—
363C23—42no—
364C23—42no—
365C23—41no—
366C23—41no—
367C23—41no—
368C23—82no—
369C23—84no—
370C23—84no—
371C23—82no—
372C23C2×C22⋊C4×C6111yescharacteristic
373C23C4×D4×C6112yes—
374C23#124×C6111yescharacteristic
375C23#127×C6111yescharacteristic
376D4—42no—
377D4—42no—
378D4—82no—
379D4—82no—
380C2×C4—21no—
381C2×C4—21no—
382C2×C4—21no—
383C2×C4—21no—
384C2×C4—21no—
385C2×C4—41no—
386C2×C4—42no—
387C2×C4—41no—
388C2×C4—41no—
389C2×C4—41no—
390C2×C4—41no—
391C2×C4—41no—
392C2×C4—41no—
393C2×C4—41no—
394C2×C4—41no—
395C2×C4—41no—
396C2×C4—41no—
397C2×C4—41no—
398C2×C4—41no—
399C2×C4—81no—
400C2×C4—81no—
401C2×C4—81no—
402C2×C4—82no—
403C2×C4—82no—
404C2×C4—82no—
405C2×C4—82no—
406C2×C4#99×C6111yescharacteristic
407C2×C4#99×C6111yescharacteristic
408C2×C4#121×C6112yes—
409C8—44no—
410C22—21no—
411C22—21no—
412C22—21no—
413C22—22no—
414C22—21no—
415C22—41no—
416C22—41no—
417C22—42no—
418C22—42no—
419C22—42no—
420C22—42no—
421C22—41no—
422C22—41no—
423C22—41no—
424C22—41no—
425C22—42no—
426C22—81no—
427C22—81no—
428C22—81no—
429C22—84no—
430C22—84no—
431C22—81no—
432C22—81no—
433C22C61×#39011yescharacteristic
434C22C2×#99×C6111yescharacteristic
435C22C61×#41011yescharacteristic
436C4—21no—
437C4—21no—
438C4—22no—
439C4—41no—
440C4—41no—
441C4—41no—
442C4—81no—
443C4—81no—
444C4—81no—
445C4—162no—
446C2—21no—
447C2—21no—
448C2—41no—
449C2—42no—
450C2—41no—
451C2—82no—
452C2C61×#196411yescharacteristic
453C2C61×#212011yescharacteristic
454C2C61×#212111yescharacteristic
subgroupG/H
center, socleC22×C61C2×#99
commutatorC24C22×C4×C61
Frattini, cosocleC22×D4C23×C61
prime radicalC2×#120×C61C22
LCSG ▷ C24 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C22×C61 ◁ C22×C4×C61 ◁ C2×C4×D4×C61 ◁ G

Complements

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

NH ≅ G/Ntype
2. C2×C61×#352C2semidirect
4. C61×#1799C2semidirect
5. C4×#100×C61C2semidirect
9. C22×C22⋊C4×C61C4semidirect
17. C61×#381C4semidirect
19. C61×#380C4semidirect
20. C61×#379C4semidirect
182. #13219C61direct (N × H)
220. C2×#352C2×C61semidirect
222. #1799C2×C61semidirect
223. C4×#100C2×C61semidirect
236. C22×C22⋊C4C4×C61semidirect
244. #381C4×C61semidirect
246. #380C4×C61semidirect
247. #379C4×C61semidirect
252. C61#13219direct (N × H)

Maximal subgroups

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

subgroupindex
C61×#25342
C2×C61×#3522
C2×C61×AΣL(1,𝔽16)2
C61×#17992
C4×#100×C612
#1321961