Finite Group Databasebeta

← C61×#12794 #29516227

Subgroups — C61×#12794 #29516227

Order 15616.

540 conjugacy classes of subgroups.

The subgroup lattice has 428 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C4×#123×C61C211yesmaximal, characteristic
2C2×C61×#352C212yesmaximal
3C61×#1800C211yesmaximal, characteristic
4C61×#1799C211yesmaximal, characteristic
5C61×#1390C212yesmaximal
6C2×#123×C61C2211yescharacteristic
7C2×C4×D4×C61C2211yescharacteristic
8C22×C42×C61C411yescharacteristic, complemented
9C61×#423—21no—
10C2×#100×C61C2212yes—
11C2×#99×C61C2212yes—
12C61×#388—21no—
13C61×#387—21no—
14C61×#386C2211yescharacteristic
15C61×#385C411yescharacteristic, complemented
16C61×#381C411yescharacteristic, complemented
17C61×#380—21no—
18C4×C4⋊C4×C61—21no—
19C4×C22⋊C4×C61—21no—
20C4×C22⋊C4×C61C411yescharacteristic, complemented
21C61×#352—24no—
22C61×#343—21no—
23C61×#342—21no—
24C22×D4×C61C2311yescharacteristic
25C23×C4×C61C2×C411yescharacteristic
26C23×C4×C61C2×C411yescharacteristic
27C23×C4×C61C2×C411yescharacteristic
28C23×C4×C61D411yescharacteristic
29C2×M16×C61—22no—
30#123×C61—21no—
31#123×C61—21no—
32#123×C61D412yes—
33#123×C61Q812yes—
34C4×D4×C61—41no—
35C4×D4×C61—41no—
36C2×C4⋊C4×C61—21no—
37C2×C4⋊C4×C61—21no—
38C2×C4⋊C4×C61—41no—
39C2×C4⋊C4×C61D411yescharacteristic
40C2×C22⋊C4×C61—21no—
41C2×C22⋊C4×C61—21no—
42C2×C22⋊C4×C61—22no—
43C2×C22⋊C4×C61C2×C411yescharacteristic
44C2×C22⋊C4×C61C2×C411yescharacteristic
45C2×C22⋊C4×C61C2×C411yescharacteristic
46C2×C22⋊C4×C61D411yescharacteristic
47C2×C42×C61—21no—
48C2×C42×C61—21no—
49C2×C42×C61—21no—
50C2×C42×C61—41no—
51C2×C42×C61—41no—
52C2×C42×C61D411yescharacteristic
53C4⋊C8×C61—41no—
54C4⋊C8×C61—41no—
55#100×C61—22no—
56#100×C61—22no—
57#99×C61—22no—
58#99×C61—22no—
59#95×C61—21no—
60#95×C61—21no—
61#95×C61—21no—
62#95×C61—21no—
63#95×C61—21no—
64#95×C61—41no—
65C24×C61C22×C411yescharacteristic
66C24×C61C2×D411yescharacteristic
67C2×D4×C61—41no—
68C2×D4×C61—41no—
69C2×D4×C61C2×D411yescharacteristic
70C2×D4×C61C2×Q811yescharacteristic
71C2×D4×C61#4111yescharacteristic
72C2×D4×C61#4111yescharacteristic
73C22×C4×C61—21no—
74C22×C4×C61—21no—
75C22×C4×C61—21no—
76C22×C4×C61—21no—
77C22×C4×C61—21no—
78C22×C4×C61—21no—
79C22×C4×C61—21no—
80C22×C4×C61—21no—
81C22×C4×C61—21no—
82C22×C4×C61—42no—
83C22×C4×C61—42no—
84C22×C4×C61—42no—
85C22×C4×C61—41no—
86C22×C4×C61—41no—
87C22×C4×C61—41no—
88C22×C4×C61—42no—
89C22×C4×C61C22⋊C411yescharacteristic
90C22×C4×C61C22⋊C411yescharacteristic
91C22×C4×C61C22⋊C411yescharacteristic
92C22×C4×C61C22⋊C411yescharacteristic
93C22×C4×C61C4⋊C412yescomplemented
94C22×C4×C61C2×D411yescharacteristic
95M16×C61—42no—
96M16×C61—42no—
97C2×C8×C61—42no—
98C4⋊C4×C61—22no—
99C4⋊C4×C61—22no—
100C4⋊C4×C61—41no—
101C4⋊C4×C61—42no—
102C4⋊C4×C61—82no—
103C22⋊C4×C61—21no—
104C22⋊C4×C61—22no—
105C22⋊C4×C61—22no—
106C22⋊C4×C61—41no—
107C22⋊C4×C61—41no—
108C22⋊C4×C61—42no—
109C22⋊C4×C61—41no—
110C22⋊C4×C61—42no—
111C22⋊C4×C61—42no—
112C22⋊C4×C61C4⋊C412yescomplemented
113C42×C61—21no—
114C42×C61—21no—
115C42×C61—42no—
116C42×C61—42no—
117C42×C61—41no—
118C42×C61—41no—
119C23×C61—21no—
120C23×C61—21no—
121C23×C61—21no—
122C23×C61—21no—
123C23×C61—41no—
124C23×C61—41no—
125C23×C61—41no—
126C23×C61—41no—
127C23×C61C2×C22⋊C411yescharacteristic
128C23×C61C2×C4⋊C411yescharacteristic
129C23×C61#11711yescharacteristic
130C23×C61#12111yescharacteristic
131C23×C61C22⋊Q811yescharacteristic
132D4×C61—41no—
133D4×C61—41no—
134D4×C61—41no—
135D4×C61—41no—
136C2×C4×C61—21no—
137C2×C4×C61—21no—
138C2×C4×C61—22no—
139C2×C4×C61—22no—
140C2×C4×C61—22no—
141C2×C4×C61—22no—
142C2×C4×C61—21no—
143C2×C4×C61—21no—
144C2×C4×C61—21no—
145C2×C4×C61—42no—
146C2×C4×C61—41no—
147C2×C4×C61—41no—
148C2×C4×C61—42no—
149C2×C4×C61—41no—
150C2×C4×C61—42no—
151C2×C4×C61—41no—
152C2×C4×C61—41no—
153C2×C4×C61—41no—
154C2×C4×C61—42no—
155C2×C4×C61—42no—
156C2×C4×C61—41no—
157C2×C4×C61—41no—
158C2×C4×C61—41no—
159C2×C4×C61—42no—
160C2×C4×C61—42no—
161C2×C4×C61—41no—
162C2×C4×C61—41no—
163C2×C4×C61—41no—
164C2×C4×C61—81no—
165C2×C4×C61—82no—
166C2×C4×C61—82no—
167C2×C4×C61—82no—
168C2×C4×C61#9911yescharacteristic
169C2×C4×C61#9911yescharacteristic
170C2×C4×C61#12111yescharacteristic
171C2×C4×C61C22⋊Q811yescharacteristic
172C8×C61—82no—
173#12794C6111yesmaximal, characteristic, direct product factor, Sylow
174C22×C61—21no—
175C22×C61—21no—
176C22×C61—21no—
177C22×C61—21no—
178C22×C61—21no—
179C22×C61—21no—
180C22×C61—41no—
181C22×C61—41no—
182C22×C61—41no—
183C22×C61—41no—
184C22×C61—42no—
185C22×C61—41no—
186C22×C61—41no—
187C22×C61—41no—
188C22×C61—41no—
189C22×C61—81no—
190C22×C61#38011yescharacteristic
191C22×C61C2×#9911yescharacteristic
192C22×C61#41011yescharacteristic
193C4×C61—21no—
194C4×C61—21no—
195C4×C61—21no—
196C4×C61—42no—
197C4×C61—42no—
198C4×C61—41no—
199C4×C61—41no—
200C4×C61—41no—
201C4×C61—42no—
202C4×C61—81no—
203C4×C61—162no—
204C4×C61#35212yescomplemented
205C4×#123C2×C6111yescharacteristic
206C2×#352C2×C6112yes—
207#1800C2×C6111yescharacteristic
208#1799C2×C6111yescharacteristic
209#1390C2×C6112yes—
210C2×C61—21no—
211C2×C61—21no—
212C2×C61—41no—
213C2×C61—41no—
214C2×C61—41no—
215C2×C61—41no—
216C2×C61#179911yescharacteristic
217C2×C61C2×#35211yescharacteristic
218C2×C61#212011yescharacteristic
219C2×#123C22×C6111yescharacteristic
220C2×C4×D4C22×C6111yescharacteristic
221C22×C42C4×C6111yescharacteristic, complemented
222#423—21no—
223C2×#100C22×C6112yes—
224C2×#99C22×C6112yes—
225#388—21no—
226#387—21no—
227#386C22×C6111yescharacteristic
228#385C4×C6111yescharacteristic, complemented
229#381C4×C6111yescharacteristic, complemented
230#380—21no—
231C4×C4⋊C4—21no—
232C4×C22⋊C4—21no—
233C4×C22⋊C4C4×C6111yescharacteristic, complemented
234#352—24no—
235#343—21no—
236#342—21no—
237C61#1279411yescharacteristic, direct product factor, Sylow
238C22×D4C23×C6111yescharacteristic
239C23×C4C2×C4×C6111yescharacteristic
240C23×C4C2×C4×C6111yescharacteristic
241C23×C4C2×C4×C6111yescharacteristic
242C23×C4D4×C6111yescharacteristic
243C2×M16—22no—
244#123—21no—
245#123—21no—
246#123D4×C6112yes—
247#123Q8×C6112yes—
248C4×D4—41no—
249C4×D4—41no—
250C2×C4⋊C4—21no—
251C2×C4⋊C4—21no—
252C2×C4⋊C4—41no—
253C2×C4⋊C4D4×C6111yescharacteristic
254C2×C22⋊C4—21no—
255C2×C22⋊C4—21no—
256C2×C22⋊C4—22no—
257C2×C22⋊C4C2×C4×C6111yescharacteristic
258C2×C22⋊C4C2×C4×C6111yescharacteristic
259C2×C22⋊C4C2×C4×C6111yescharacteristic
260C2×C22⋊C4D4×C6111yescharacteristic
261C2×C42—21no—
262C2×C42—21no—
263C2×C42—21no—
264C2×C42—41no—
265C2×C42—41no—
266C2×C42D4×C6111yescharacteristic
267C4⋊C8—41no—
268C4⋊C8—41no—
269#100—22no—
270#100—22no—
271#99—22no—
272#99—22no—
273#95—21no—
274#95—21no—
275#95—21no—
276#95—21no—
277#95—21no—
278#95—41no—
279C24C22×C4×C6111yescharacteristic
280C24C2×D4×C6111yescharacteristic
281C2×D4—41no—
282C2×D4—41no—
283C2×D4C2×D4×C6111yescharacteristic
284C2×D4C2×Q8×C6111yescharacteristic
285C2×D4#41×C6111yescharacteristic
286C2×D4#41×C6111yescharacteristic
287C22×C4—21no—
288C22×C4—21no—
289C22×C4—21no—
290C22×C4—21no—
291C22×C4—21no—
292C22×C4—21no—
293C22×C4—21no—
294C22×C4—21no—
295C22×C4—21no—
296C22×C4—42no—
297C22×C4—42no—
298C22×C4—42no—
299C22×C4—41no—
300C22×C4—41no—
301C22×C4—41no—
302C22×C4—42no—
303C22×C4C22⋊C4×C6111yescharacteristic
304C22×C4C22⋊C4×C6111yescharacteristic
305C22×C4C22⋊C4×C6111yescharacteristic
306C22×C4C22⋊C4×C6111yescharacteristic
307C22×C4C4⋊C4×C6112yescomplemented
308C22×C4C2×D4×C6111yescharacteristic
309M16—42no—
310M16—42no—
311C2×C8—42no—
312C4⋊C4—22no—
313C4⋊C4—22no—
314C4⋊C4—41no—
315C4⋊C4—42no—
316C4⋊C4—82no—
317C22⋊C4—21no—
318C22⋊C4—22no—
319C22⋊C4—22no—
320C22⋊C4—41no—
321C22⋊C4—41no—
322C22⋊C4—42no—
323C22⋊C4—41no—
324C22⋊C4—42no—
325C22⋊C4—42no—
326C22⋊C4C4⋊C4×C6112yescomplemented
327C42—21no—
328C42—21no—
329C42—42no—
330C42—42no—
331C42—41no—
332C42—41no—
333C23—21no—
334C23—21no—
335C23—21no—
336C23—21no—
337C23—41no—
338C23—41no—
339C23—41no—
340C23—41no—
341C23C2×C22⋊C4×C6111yescharacteristic
342C23C2×C4⋊C4×C6111yescharacteristic
343C23#117×C6111yescharacteristic
344C23#121×C6111yescharacteristic
345C23C22⋊Q8×C6111yescharacteristic
346D4—41no—
347D4—41no—
348D4—41no—
349D4—41no—
350C2×C4—21no—
351C2×C4—21no—
352C2×C4—22no—
353C2×C4—22no—
354C2×C4—22no—
355C2×C4—22no—
356C2×C4—21no—
357C2×C4—21no—
358C2×C4—21no—
359C2×C4—42no—
360C2×C4—41no—
361C2×C4—41no—
362C2×C4—42no—
363C2×C4—41no—
364C2×C4—42no—
365C2×C4—41no—
366C2×C4—41no—
367C2×C4—41no—
368C2×C4—42no—
369C2×C4—42no—
370C2×C4—41no—
371C2×C4—41no—
372C2×C4—41no—
373C2×C4—42no—
374C2×C4—42no—
375C2×C4—41no—
376C2×C4—41no—
377C2×C4—41no—
378C2×C4—81no—
379C2×C4—82no—
380C2×C4—82no—
381C2×C4—82no—
382C2×C4#99×C6111yescharacteristic
383C2×C4#99×C6111yescharacteristic
384C2×C4#121×C6111yescharacteristic
385C2×C4C22⋊Q8×C6111yescharacteristic
386C8—82no—
387C22—21no—
388C22—21no—
389C22—21no—
390C22—21no—
391C22—21no—
392C22—21no—
393C22—41no—
394C22—41no—
395C22—41no—
396C22—41no—
397C22—42no—
398C22—41no—
399C22—41no—
400C22—41no—
401C22—41no—
402C22—81no—
403C22C61×#38011yescharacteristic
404C22C2×#99×C6111yescharacteristic
405C22C61×#41011yescharacteristic
406C4—21no—
407C4—21no—
408C4—21no—
409C4—42no—
410C4—42no—
411C4—41no—
412C4—41no—
413C4—41no—
414C4—42no—
415C4—81no—
416C4—162no—
417C4C61×#35212yescomplemented
418C2—21no—
419C2—21no—
420C2—41no—
421C2—41no—
422C2—41no—
423C2—41no—
424C2C61×#179911yescharacteristic
425C2C2×C61×#35211yescharacteristic
426C2C61×#212011yescharacteristic
subgroupG/H
center, socleC22×C61C2×#99
commutatorC24C22×C4×C61
Frattini, cosocleC22×D4C23×C61
prime radicalC22×D4×C61C23
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
8. C22×C42×C61C4semidirect
15. C61×#385C4semidirect
16. C61×#381C4semidirect
20. C4×C22⋊C4×C61C4semidirect
93. C22×C4×C61C4⋊C4semidirect
112. C22⋊C4×C61C4⋊C4semidirect
173. #12794C61direct (N × H)
204. C4×C61#352semidirect
221. C22×C42C4×C61semidirect
228. #385C4×C61semidirect
229. #381C4×C61semidirect
233. C4×C22⋊C4C4×C61semidirect
237. C61#12794direct (N × H)
307. C22×C4C4⋊C4×C61semidirect
326. C22⋊C4C4⋊C4×C61semidirect
417. C4C61×#352semidirect

Maximal subgroups

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

subgroupindex
C4×#123×C612
C2×C61×#3522
C61×#18002
C61×#17992
C61×#13902
#1279461