Finite Group Databasebeta

← C61×#10411 #29513844

Subgroups — C61×#10411 #29513844

Order 15616.

572 conjugacy classes of subgroups.

The subgroup lattice has 454 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C61×#2548C211yesmaximal, characteristic
2C61×#1957C211yesmaximal, characteristic
3C61×#1947C211yesmaximal, characteristic, complemented
4C61×#1803C211yesmaximal, characteristic, complemented
5C2×C61×#331C211yesmaximal, characteristic
6C2×C61×Q8⋊C8C211yesmaximal, characteristic, complemented
7C61×#1279C211yesmaximal, characteristic, complemented
8C2×#127×C61C2211yescharacteristic
9C2×#124×C61—21no—
10C2×C4×Q8×C61C2211yescharacteristic
11C2×#107×C61—21no—
12C2×C4⋊C8×C61C2211yescharacteristic
13C2×C4⋊C8×C61C2211yescharacteristic
14C2×#103×C61—21no—
15C2×#102×C61—22no—
16C2×C4×C8×C61C2211yescharacteristic
17C61×#394—22no—
18C61×#390—21no—
19C61×#390—21no—
20C61×#390—21no—
21C61×#384C2211yescharacteristic
22C4×C4⋊C4×C61C2211yescharacteristic
23C61×#340—21no—
24C61×#340—21no—
25C61×#331—22no—
26C61×Q8⋊C8—22no—
27C22×Q8×C61D411yescharacteristic
28C22×D4×C61—21no—
29C22×D4×C61—21no—
30C22×D4×C61D412yes—
31C22×C8×C61—21no—
32C22×C8×C61—21no—
33C22×C8×C61—21no—
34#127×C61—21no—
35#127×C61—21no—
36#127×C61—21no—
37#127×C61—21no—
38#124×C61—42no—
39#124×C61—42no—
40C4×Q8×C61—21no—
41C4×Q8×C61—21no—
42C4×Q8×C61—21no—
43C4×Q8×C61—21no—
44C2×C4⋊C4×C61—21no—
45C2×C4⋊C4×C61—21no—
46C2×C4⋊C4×C61—21no—
47C2×C4⋊C4×C61—21no—
48C2×C4⋊C4×C61D411yescharacteristic
49C2×C4⋊C4×C61D411yescharacteristic
50C2×C4⋊C4×C61D411yescharacteristic
51C2×C22⋊C4×C61—22no—
52C2×C22⋊C4×C61—22no—
53C2×C22⋊C4×C61—22no—
54C2×C22⋊C4×C61—22no—
55C2×C42×C61—21no—
56C2×C42×C61—21no—
57C2×C42×C61C2311yescharacteristic
58#107×C61—42no—
59C4⋊C8×C61—21no—
60C4⋊C8×C61—21no—
61C4⋊C8×C61—21no—
62C4⋊C8×C61D412yes—
63#103×C61—42no—
64#102×C61—44no—
65C4×C8×C61—21no—
66C4×C8×C61—21no—
67#95×C61—21no—
68C24×C61—22no—
69C2×Q8×C61—21no—
70C2×Q8×C61—21no—
71C2×Q8×C61—21no—
72C2×Q8×C61—21no—
73C2×Q8×C61—21no—
74C2×Q8×C61D812yes—
75C2×D4×C61—22no—
76C2×D4×C61—22no—
77C2×D4×C61—42no—
78C2×D4×C61—42no—
79C2×D4×C61—41no—
80C2×D4×C61—41no—
81C2×D4×C61—41no—
82C2×D4×C61—41no—
83C2×D4×C61—41no—
84C2×D4×C61—41no—
85C2×D4×C61—41no—
86C2×D4×C61—41no—
87C2×D4×C61—41no—
88C2×D4×C61—41no—
89C2×D4×C61—41no—
90C2×D4×C61—41no—
91C22×C4×C61—21no—
92C22×C4×C61—21no—
93C22×C4×C61—21no—
94C22×C4×C61—21no—
95C22×C4×C61—41no—
96C22×C4×C61C2×D411yescharacteristic
97C22×C4×C61C2×D411yescharacteristic
98C22×C4×C61C2×D411yescharacteristic
99C2×C8×C61—22no—
100C2×C8×C61—22no—
101C2×C8×C61—41no—
102C2×C8×C61—41no—
103C2×C8×C61—41no—
104C2×C8×C61—41no—
105C2×C8×C61—41no—
106C2×C8×C61—41no—
107C2×C8×C61—41no—
108C4⋊C4×C61—21no—
109C4⋊C4×C61—21no—
110C4⋊C4×C61—21no—
111C4⋊C4×C61—21no—
112C4⋊C4×C61—21no—
113C4⋊C4×C61—21no—
114C4⋊C4×C61—42no—
115C4⋊C4×C61—41no—
116C4⋊C4×C61—41no—
117C4⋊C4×C61—41no—
118C4⋊C4×C61—41no—
119C4⋊C4×C61—41no—
120C4⋊C4×C61—41no—
121C22⋊C4×C61—44no—
122C22⋊C4×C61—44no—
123C22⋊C4×C61—82no—
124C22⋊C4×C61—82no—
125C42×C61—41no—
126C42×C61—41no—
127C42×C61—41no—
128C42×C61—41no—
129C42×C61C2×D411yescharacteristic
130C42×C61#4111yescharacteristic
131C42×C61#4111yescharacteristic
132C42×C61#4111yescharacteristic
133C23×C61—42no—
134C23×C61—42no—
135C23×C61—42no—
136C23×C61—42no—
137C23×C61—42no—
138C23×C61—42no—
139C23×C61—42no—
140C23×C61#12011yescharacteristic
141Q8×C61—22no—
142Q8×C61—22no—
143Q8×C61—41no—
144Q8×C61—41no—
145D4×C61—44no—
146D4×C61—82no—
147D4×C61—81no—
148D4×C61—81no—
149D4×C61—81no—
150D4×C61—81no—
151D4×C61—81no—
152D4×C61—81no—
153D4×C61—81no—
154D4×C61—81no—
155C2×C4×C61—21no—
156C2×C4×C61—21no—
157C2×C4×C61—22no—
158C2×C4×C61—41no—
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—81no—
170C2×C4×C61—81no—
171C2×C4×C61—81no—
172C2×C4×C61#12111yescharacteristic
173C2×C4×C61#12111yescharacteristic
174C2×C4×C61#12111yescharacteristic
175C2×C4×C61#12311yescharacteristic
176C2×C4×C61#12311yescharacteristic
177C2×C4×C61#12411yescharacteristic
178C2×C4×C61C2×D811yescharacteristic
179C2×C4×C61#13511yescharacteristic
180C2×C4×C61#13511yescharacteristic
181C2×C4×C61#13511yescharacteristic
182C2×C4×C61AGL(1,ℤ/8)11yescharacteristic
183C2×C4×C61AGL(1,ℤ/8)11yescharacteristic
184C2×C4×C61#13711yescharacteristic
185C2×C4×C61#13711yescharacteristic
186C8×C61—42no—
187C8×C61—42no—
188C8×C61—81no—
189#10411C6111yesmaximal, characteristic, direct product factor, Sylow
190C22×C61—82no—
191C22×C61—82no—
192C22×C61—82no—
193C22×C61—82no—
194C22×C61—82no—
195C22×C61—82no—
196C22×C61—82no—
197C22×C61#39411yescharacteristic
198C22×C61#44711yescharacteristic
199C22×C61#44911yescharacteristic
200C22×C61#45211yescharacteristic
201C22×C61#45211yescharacteristic
202C22×C61#45311yescharacteristic
203C22×C61#45511yescharacteristic
204C4×C61—21no—
205C4×C61—21no—
206C4×C61—42no—
207C4×C61—81no—
208C4×C61—81no—
209C4×C61—81no—
210C4×C61—161no—
211C4×C61#46311yescharacteristic
212C4×C61#46411yescharacteristic
213C4×C61#46611yescharacteristic
214C4×C61#46811yescharacteristic
215C4×C61#48011yescharacteristic
216C4×C61#48211yescharacteristic
217C4×C61#48811yescharacteristic
218C4×C61#48911yescharacteristic
219#2548C2×C6111yescharacteristic
220#1957C2×C6111yescharacteristic
221#1947C2×C6111yescharacteristic, complemented
222#1803C2×C6111yescharacteristic, complemented
223C2×#331C2×C6111yescharacteristic
224C2×Q8⋊C8C2×C6111yescharacteristic, complemented
225#1279C2×C6111yescharacteristic, complemented
226C2×C61—162no—
227C2×C61#162111yescharacteristic
228C2×C61#163011yescharacteristic
229C2×C61#164011yescharacteristic
230C2×C61#164111yescharacteristic
231C2×C61#203311yescharacteristic
232C2×C61#203711yescharacteristic
233C2×C61#204211yescharacteristic
234C2×#127C22×C6111yescharacteristic
235C2×#124—21no—
236C2×C4×Q8C22×C6111yescharacteristic
237C2×#107—21no—
238C2×C4⋊C8C22×C6111yescharacteristic
239C2×C4⋊C8C22×C6111yescharacteristic
240C2×#103—21no—
241C2×#102—22no—
242C2×C4×C8C22×C6111yescharacteristic
243#394—22no—
244#390—21no—
245#390—21no—
246#390—21no—
247#384C22×C6111yescharacteristic
248C4×C4⋊C4C22×C6111yescharacteristic
249#340—21no—
250#340—21no—
251#331—22no—
252Q8⋊C8—22no—
253C61#1041111yescharacteristic, direct product factor, Sylow
254C22×Q8D4×C6111yescharacteristic
255C22×D4—21no—
256C22×D4—21no—
257C22×D4D4×C6112yes—
258C22×C8—21no—
259C22×C8—21no—
260C22×C8—21no—
261#127—21no—
262#127—21no—
263#127—21no—
264#127—21no—
265#124—42no—
266#124—42no—
267C4×Q8—21no—
268C4×Q8—21no—
269C4×Q8—21no—
270C4×Q8—21no—
271C2×C4⋊C4—21no—
272C2×C4⋊C4—21no—
273C2×C4⋊C4—21no—
274C2×C4⋊C4—21no—
275C2×C4⋊C4D4×C6111yescharacteristic
276C2×C4⋊C4D4×C6111yescharacteristic
277C2×C4⋊C4D4×C6111yescharacteristic
278C2×C22⋊C4—22no—
279C2×C22⋊C4—22no—
280C2×C22⋊C4—22no—
281C2×C22⋊C4—22no—
282C2×C42—21no—
283C2×C42—21no—
284C2×C42C23×C6111yescharacteristic
285#107—42no—
286C4⋊C8—21no—
287C4⋊C8—21no—
288C4⋊C8—21no—
289C4⋊C8D4×C6112yes—
290#103—42no—
291#102—44no—
292C4×C8—21no—
293C4×C8—21no—
294#95—21no—
295C24—22no—
296C2×Q8—21no—
297C2×Q8—21no—
298C2×Q8—21no—
299C2×Q8—21no—
300C2×Q8—21no—
301C2×Q8D8×C6112yes—
302C2×D4—22no—
303C2×D4—22no—
304C2×D4—42no—
305C2×D4—42no—
306C2×D4—41no—
307C2×D4—41no—
308C2×D4—41no—
309C2×D4—41no—
310C2×D4—41no—
311C2×D4—41no—
312C2×D4—41no—
313C2×D4—41no—
314C2×D4—41no—
315C2×D4—41no—
316C2×D4—41no—
317C2×D4—41no—
318C22×C4—21no—
319C22×C4—21no—
320C22×C4—21no—
321C22×C4—21no—
322C22×C4—41no—
323C22×C4C2×D4×C6111yescharacteristic
324C22×C4C2×D4×C6111yescharacteristic
325C22×C4C2×D4×C6111yescharacteristic
326C2×C8—22no—
327C2×C8—22no—
328C2×C8—41no—
329C2×C8—41no—
330C2×C8—41no—
331C2×C8—41no—
332C2×C8—41no—
333C2×C8—41no—
334C2×C8—41no—
335C4⋊C4—21no—
336C4⋊C4—21no—
337C4⋊C4—21no—
338C4⋊C4—21no—
339C4⋊C4—21no—
340C4⋊C4—21no—
341C4⋊C4—42no—
342C4⋊C4—41no—
343C4⋊C4—41no—
344C4⋊C4—41no—
345C4⋊C4—41no—
346C4⋊C4—41no—
347C4⋊C4—41no—
348C22⋊C4—44no—
349C22⋊C4—44no—
350C22⋊C4—82no—
351C22⋊C4—82no—
352C42—41no—
353C42—41no—
354C42—41no—
355C42—41no—
356C42C2×D4×C6111yescharacteristic
357C42#41×C6111yescharacteristic
358C42#41×C6111yescharacteristic
359C42#41×C6111yescharacteristic
360C23—42no—
361C23—42no—
362C23—42no—
363C23—42no—
364C23—42no—
365C23—42no—
366C23—42no—
367C23#120×C6111yescharacteristic
368Q8—22no—
369Q8—22no—
370Q8—41no—
371Q8—41no—
372D4—44no—
373D4—82no—
374D4—81no—
375D4—81no—
376D4—81no—
377D4—81no—
378D4—81no—
379D4—81no—
380D4—81no—
381D4—81no—
382C2×C4—21no—
383C2×C4—21no—
384C2×C4—22no—
385C2×C4—41no—
386C2×C4—41no—
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—81no—
397C2×C4—81no—
398C2×C4—81no—
399C2×C4#121×C6111yescharacteristic
400C2×C4#121×C6111yescharacteristic
401C2×C4#121×C6111yescharacteristic
402C2×C4#123×C6111yescharacteristic
403C2×C4#123×C6111yescharacteristic
404C2×C4#124×C6111yescharacteristic
405C2×C4C2×D8×C6111yescharacteristic
406C2×C4#135×C6111yescharacteristic
407C2×C4#135×C6111yescharacteristic
408C2×C4#135×C6111yescharacteristic
409C2×C4AGL(1,ℤ/8)×C6111yescharacteristic
410C2×C4AGL(1,ℤ/8)×C6111yescharacteristic
411C2×C4#137×C6111yescharacteristic
412C2×C4#137×C6111yescharacteristic
413C8—42no—
414C8—42no—
415C8—81no—
416C22—82no—
417C22—82no—
418C22—82no—
419C22—82no—
420C22—82no—
421C22—82no—
422C22—82no—
423C22C61×#39411yescharacteristic
424C22C61×#44711yescharacteristic
425C22C61×#44911yescharacteristic
426C22C61×#45211yescharacteristic
427C22C61×#45211yescharacteristic
428C22C61×#45311yescharacteristic
429C22C61×#45511yescharacteristic
430C4—21no—
431C4—21no—
432C4—42no—
433C4—81no—
434C4—81no—
435C4—81no—
436C4—161no—
437C4C61×#46311yescharacteristic
438C4C61×#46411yescharacteristic
439C4C61×#46611yescharacteristic
440C4C61×#46811yescharacteristic
441C4C61×#48011yescharacteristic
442C4C61×#48211yescharacteristic
443C4C61×#48811yescharacteristic
444C4C61×#48911yescharacteristic
445C2—162no—
446C2C61×#162111yescharacteristic
447C2C61×#163011yescharacteristic
448C2C61×#164011yescharacteristic
449C2C61×#164111yescharacteristic
450C2C61×#203311yescharacteristic
451C2C61×#203711yescharacteristic
452C2C61×#204211yescharacteristic
subgroupG/H
center, socleC23×C61#120
commutator, Frattini, cosocleC2×C42C23×C61
prime radicalC2×#127×C61C22
LCSG ▷ C2×C42 ▷ C22 ▷ C1
UCSC1 ◁ C23×C61 ◁ C2×C42×C61 ◁ G

Complements

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

NH ≅ G/Ntype
3. C61×#1947C2semidirect
4. C61×#1803C2semidirect
6. C2×C61×Q8⋊C8C2semidirect
7. C61×#1279C2semidirect
189. #10411C61direct (N × H)
221. #1947C2×C61semidirect
222. #1803C2×C61semidirect
224. C2×Q8⋊C8C2×C61semidirect
225. #1279C2×C61semidirect
253. C61#10411direct (N × H)

Maximal subgroups

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

subgroupindex
C61×#25482
C61×#19572
C61×#19472
C61×#18032
C2×C61×#3312
C2×C61×Q8⋊C82
C61×#12792
#1041161