Finite Group Databasebeta

← C61×#61322 #29564755

Subgroups — C61×#61322 #29564755

Order 15616.

1782 conjugacy classes of subgroups.

The subgroup lattice has 490 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1Q8×#41×C61C211yesmaximal, characteristic
2C2×C61×#557C211yesmaximal, characteristic, complemented
3C61×#3374C216yesmaximal, complemented
4C61×#3372C216yesmaximal, complemented
5C61×#3363C212yesmaximal, complemented
6C61×#3359C212yesmaximal, complemented
7C61×#3139C213yesmaximal, complemented
8C61×#3078C213yesmaximal, complemented
9C61×#3039C213yesmaximal
10Q8×M16×C61C211yesmaximal, characteristic, complemented
11C4×#137×C61C213yesmaximal
12C2×#137×C61C2213yes—
13Q82×C61C2211yescharacteristic
14C61×#557—23no—
15C61×#557—21no—
16C61×#557C2212yes—
17C61×#557C2216yes—
18C61×#557C2213yes—
19C61×#554—23no—
20C61×#554C2213yes—
21D4×Q8×C61—21no—
22D4×Q8×C61C2211yescharacteristic
23C61×#533—23no—
24C61×#533C2213yes—
25C2×C4⋊Q8×C61C2213yes—
26C2×#125×C61—23no—
27C4×#41×C61C2213yes—
28C2×C4×Q8×C61—21no—
29C2×C4×Q8×C61—21no—
30C2×C4×Q8×C61C2213yes—
31C2×C4×Q8×C61C2211yescharacteristic
32C61×#491C22112yes—
33C61×#487C2216yes—
34C61×#487C2216yes—
35C61×#479C22112yes—
36C61×#477C2216yes—
37C61×#475C2216yes—
38C61×#462C2216yes—
39C61×#462C2216yes—
40C61×#461C2216yes—
41C61×#461C2216yes—
42C61×#446C2216yes—
43C8×Q8×C61C2212yes—
44C61×#441C2216yes—
45C61×#440C2216yes—
46C4×Q16×C61C2216yes—
47C4×SD16×C61C2216yes—
48C61×#428C2213yes—
49C61×#423C2213yes—
50C61×#419C2216yes—
51C61×#419C2213yes—
52C61×#417C2211yescharacteristic
53C4×M16×C61C2213yes—
54C2×#41×C61C2311yescharacteristic
55C22×Q8×C61—21no—
56C22×Q8×C61D411yescharacteristic
57C22×Q8×C61C2313yes—
58#137×C61—212no—
59C2×Q16×C61C2316yes—
60C2×SD16×C61C2316yes—
61C2×M16×C61C2313yes—
62C2×M16×C61C2311yescharacteristic
63C4⋊Q8×C61—23no—
64C4⋊Q8×C61—23no—
65C4⋊Q8×C61—23no—
66C4⋊Q8×C61—23no—
67C4⋊Q8×C61C2316yes—
68C4⋊Q8×C61C2316yes—
69C4⋊Q8×C61C2313yes—
70C4⋊Q8×C61C2313yes—
71#125×C61—26no—
72#125×C61—26no—
73#125×C61—212no—
74#125×C61—26no—
75#125×C61C2316yes—
76C22⋊Q8×C61—23no—
77C22⋊Q8×C61—23no—
78C22⋊Q8×C61—23no—
79C22⋊Q8×C61—23no—
80C22⋊Q8×C61—26no—
81C4×Q8×C61—21no—
82C4×Q8×C61—26no—
83C4×Q8×C61—22no—
84C4×Q8×C61—23no—
85C4×Q8×C61—23no—
86C4×Q8×C61—23no—
87C4×Q8×C61—23no—
88C4×Q8×C61—23no—
89C4×Q8×C61D413yes—
90C4×Q8×C61D411yescharacteristic
91C4×Q8×C61C2316yes—
92C4×Q8×C61C2316yes—
93C4×Q8×C61C2313yes—
94C4×Q8×C61C2313yes—
95C4×Q8×C61C2311yescharacteristic
96C4×Q8×C61C2313yes—
97C4×D4×C61—23no—
98C4×D4×C61—23no—
99C4×D4×C61C2313yes—
100C4×D4×C61C2313yes—
101#117×C61—23no—
102#117×C61C2313yes—
103C2×C4⋊C4×C61—23no—
104C2×C4⋊C4×C61—23no—
105C2×C4⋊C4×C61—23no—
106C2×C4⋊C4×C61—23no—
107C2×C4⋊C4×C61D413yes—
108C2×C4⋊C4×C61C2311yescharacteristic
109C2×C42×C61—23no—
110C2×C42×C61—23no—
111C2×C42×C61C2313yes—
112#107×C61C2316yes—
113#106×C61C2316yes—
114C4⋊C8×C61C2316yes—
115C4⋊C8×C61C2316yes—
116#103×C61C2312yes—
117#103×C61C2316yes—
118#103×C61C2316yes—
119#103×C61C23112yes—
120#103×C61C23112yes—
121#102×C61C2312yes—
122#97×C61C2316yes—
123C4×C8×C61C2316yes—
124#41×C61—22no—
125#41×C61#4114yes—
126C2×Q8×C61—24no—
127C2×Q8×C61—23no—
128C2×Q8×C61—23no—
129C2×Q8×C61—23no—
130C2×Q8×C61—26no—
131C2×Q8×C61—26no—
132C2×Q8×C61—44no—
133C2×Q8×C61C2×D413yes—
134C2×Q8×C61C2×D411yescharacteristic
135C2×Q8×C61C2416yes—
136C2×Q8×C61C2411yescharacteristic
137C2×D4×C61—21no—
138C2×D4×C61C2411yescharacteristic
139C22×C4×C61—21no—
140C22×C4×C61—21no—
141C22×C4×C61—23no—
142C22×C4×C61—23no—
143C22×C4×C61—23no—
144C22×C4×C61C2×D413yes—
145C22×C4×C61C2×D413yes—
146C22×C4×C61C2411yescharacteristic
147Q16×C61—212no—
148SD16×C61—212no—
149M16×C61—26no—
150M16×C61—22no—
151C2×C8×C61C2412yes—
152C2×C8×C61C2416yes—
153C4⋊C4×C61—26no—
154C4⋊C4×C61—23no—
155C4⋊C4×C61—26no—
156C4⋊C4×C61—26no—
157C4⋊C4×C61—26no—
158C4⋊C4×C61—26no—
159C4⋊C4×C61—26no—
160C4⋊C4×C61—26no—
161C4⋊C4×C61—21no—
162C4⋊C4×C61—23no—
163C4⋊C4×C61—23no—
164C4⋊C4×C61—23no—
165C4⋊C4×C61—26no—
166C4⋊C4×C61C2×D413yes—
167C4⋊C4×C61C2×D416yes—
168C4⋊C4×C61C2×D413yes—
169C4⋊C4×C61C2412yes—
170C4⋊C4×C61C2413yes—
171C4⋊C4×C61C2413yes—
172C22⋊C4×C61—23no—
173C22⋊C4×C61—23no—
174C22⋊C4×C61—23no—
175C42×C61—23no—
176C42×C61—23no—
177C42×C61—23no—
178C42×C61—26no—
179C42×C61—26no—
180C42×C61C2×D413yes—
181C42×C61C2×D413yes—
182C42×C61C2413yes—
183C42×C61C2413yes—
184C23×C61—21no—
185C23×C61C22×D411yescharacteristic
186Q8×C61—26no—
187Q8×C61—21no—
188Q8×C61—26no—
189Q8×C61—26no—
190Q8×C61—46no—
191Q8×C61—46no—
192Q8×C61—83no—
193Q8×C61#13714yescomplemented
194Q8×C61C2×#4112yes—
195D4×C61—24no—
196D4×C61—21no—
197D4×C61C2×#4112yes—
198C2×C4×C61—21no—
199C2×C4×C61—24no—
200C2×C4×C61—26no—
201C2×C4×C61—22no—
202C2×C4×C61—23no—
203C2×C4×C61—23no—
204C2×C4×C61—26no—
205C2×C4×C61—43no—
206C2×C4×C61—42no—
207C2×C4×C61—46no—
208C2×C4×C61—46no—
209C2×C4×C61—83no—
210C2×C4×C61C22×D413yes—
211C2×C4×C61C22×D413yes—
212C2×C4×C61C22×D413yes—
213C2×C4×C61C22×D411yescharacteristic
214C2×C4×C61C22×D413yes—
215C2×C4×C61C2×#4112yes—
216C2×C4×C612−1+412yes—
217C2×C4×C61C2511yescharacteristic
218C8×C61—26no—
219C8×C61—22no—
220#61322C6111yesmaximal, characteristic, direct product factor, Sylow
221C22×C61—21no—
222C22×C61—21no—
223C22×C61—22no—
224C22×C61—41no—
225C22×C61—41no—
226C22×C61#54812yes—
227C22×C61C23×D411yescharacteristic
228C4×C61—23no—
229C4×C61—23no—
230C4×C61—22no—
231C4×C61—23no—
232C4×C61—42no—
233C4×C61—43no—
234C4×C61—46no—
235C4×C61—43no—
236C4×C61#54812yes—
237C4×C61C2×#13716yes—
238C4×C61C22×#4111yescharacteristic
239C4×C61C2×2−1+411yescharacteristic
240Q8×#41C2×C6111yescharacteristic
241C2×#557C2×C6111yescharacteristic, complemented
242#3374C2×C6116yescomplemented
243#3372C2×C6116yescomplemented
244#3363C2×C6112yescomplemented
245#3359C2×C6112yescomplemented
246#3139C2×C6113yescomplemented
247#3078C2×C6113yescomplemented
248#3039C2×C6113yes—
249Q8×M16C2×C6111yescharacteristic, complemented
250C4×#137C2×C6113yes—
251C2×C61—22no—
252C2×C61—42no—
253C2×C61C2×#54811yescharacteristic
254C2×C61C22×#13711yescharacteristic
255C2×C61#358611yescharacteristic
256C2×#137C22×C6113yes—
257Q82C22×C6111yescharacteristic
258#557—21no—
259#557—23no—
260#557C22×C6112yes—
261#557C22×C6116yes—
262#557C22×C6113yes—
263#554—23no—
264#554C22×C6113yes—
265D4×Q8—21no—
266D4×Q8C22×C6111yescharacteristic
267#533—23no—
268#533C22×C6113yes—
269C2×C4⋊Q8C22×C6113yes—
270C2×#125—23no—
271C4×#41C22×C6113yes—
272C2×C4×Q8—21no—
273C2×C4×Q8—21no—
274C2×C4×Q8C22×C6113yes—
275C2×C4×Q8C22×C6111yescharacteristic
276#491C22×C61112yes—
277#487C22×C6116yes—
278#487C22×C6116yes—
279#479C22×C61112yes—
280#477C22×C6116yes—
281#475C22×C6116yes—
282#462C22×C6116yes—
283#462C22×C6116yes—
284#461C22×C6116yes—
285#461C22×C6116yes—
286#446C22×C6116yes—
287C8×Q8C22×C6112yes—
288#441C22×C6116yes—
289#440C22×C6116yes—
290C4×Q16C22×C6116yes—
291C4×SD16C22×C6116yes—
292#428C22×C6113yes—
293#423C22×C6113yes—
294#419C22×C6116yes—
295#419C22×C6113yes—
296#417C22×C6111yescharacteristic
297C4×M16C22×C6113yes—
298C61#6132211yescharacteristic, direct product factor, Sylow
299C2×#41C23×C6111yescharacteristic
300C22×Q8—21no—
301C22×Q8D4×C6111yescharacteristic
302C22×Q8C23×C6113yes—
303#137—212no—
304C2×Q16C23×C6116yes—
305C2×SD16C23×C6116yes—
306C2×M16C23×C6113yes—
307C2×M16C23×C6111yescharacteristic
308C4⋊Q8—23no—
309C4⋊Q8—23no—
310C4⋊Q8—23no—
311C4⋊Q8—23no—
312C4⋊Q8C23×C6116yes—
313C4⋊Q8C23×C6116yes—
314C4⋊Q8C23×C6113yes—
315C4⋊Q8C23×C6113yes—
316#125—26no—
317#125—212no—
318#125—26no—
319#125—26no—
320#125C23×C6116yes—
321C22⋊Q8—23no—
322C22⋊Q8—23no—
323C22⋊Q8—23no—
324C22⋊Q8—23no—
325C22⋊Q8—26no—
326C4×Q8—21no—
327C4×Q8—26no—
328C4×Q8—22no—
329C4×Q8—23no—
330C4×Q8—23no—
331C4×Q8—23no—
332C4×Q8—23no—
333C4×Q8—23no—
334C4×Q8D4×C6111yescharacteristic
335C4×Q8D4×C6113yes—
336C4×Q8C23×C6116yes—
337C4×Q8C23×C6116yes—
338C4×Q8C23×C6113yes—
339C4×Q8C23×C6113yes—
340C4×Q8C23×C6111yescharacteristic
341C4×Q8C23×C6113yes—
342C4×D4—23no—
343C4×D4—23no—
344C4×D4C23×C6113yes—
345C4×D4C23×C6113yes—
346#117—23no—
347#117C23×C6113yes—
348C2×C4⋊C4—23no—
349C2×C4⋊C4—23no—
350C2×C4⋊C4—23no—
351C2×C4⋊C4—23no—
352C2×C4⋊C4D4×C6113yes—
353C2×C4⋊C4C23×C6111yescharacteristic
354C2×C42—23no—
355C2×C42—23no—
356C2×C42C23×C6113yes—
357#107C23×C6116yes—
358#106C23×C6116yes—
359C4⋊C8C23×C6116yes—
360C4⋊C8C23×C6116yes—
361#103C23×C6112yes—
362#103C23×C6116yes—
363#103C23×C6116yes—
364#103C23×C61112yes—
365#103C23×C61112yes—
366#102C23×C6112yes—
367#97C23×C6116yes—
368C4×C8C23×C6116yes—
369#41—22no—
370#41#41×C6114yes—
371C2×Q8—24no—
372C2×Q8—23no—
373C2×Q8—23no—
374C2×Q8—23no—
375C2×Q8—26no—
376C2×Q8—26no—
377C2×Q8—44no—
378C2×Q8C2×D4×C6113yes—
379C2×Q8C2×D4×C6111yescharacteristic
380C2×Q8C24×C6116yes—
381C2×Q8C24×C6111yescharacteristic
382C2×D4—21no—
383C2×D4C24×C6111yescharacteristic
384C22×C4—21no—
385C22×C4—21no—
386C22×C4—23no—
387C22×C4—23no—
388C22×C4—23no—
389C22×C4C2×D4×C6113yes—
390C22×C4C2×D4×C6113yes—
391C22×C4C24×C6111yescharacteristic
392Q16—212no—
393SD16—212no—
394M16—26no—
395M16—22no—
396C2×C8C24×C6112yes—
397C2×C8C24×C6116yes—
398C4⋊C4—26no—
399C4⋊C4—23no—
400C4⋊C4—26no—
401C4⋊C4—26no—
402C4⋊C4—26no—
403C4⋊C4—26no—
404C4⋊C4—26no—
405C4⋊C4—26no—
406C4⋊C4—21no—
407C4⋊C4—23no—
408C4⋊C4—23no—
409C4⋊C4—23no—
410C4⋊C4—26no—
411C4⋊C4C2×D4×C6113yes—
412C4⋊C4C2×D4×C6116yes—
413C4⋊C4C2×D4×C6113yes—
414C4⋊C4C24×C6112yes—
415C4⋊C4C24×C6113yes—
416C4⋊C4C24×C6113yes—
417C22⋊C4—23no—
418C22⋊C4—23no—
419C22⋊C4—23no—
420C42—23no—
421C42—23no—
422C42—23no—
423C42—26no—
424C42—26no—
425C42C2×D4×C6113yes—
426C42C2×D4×C6113yes—
427C42C24×C6113yes—
428C42C24×C6113yes—
429C23—21no—
430C23C22×D4×C6111yescharacteristic
431Q8—26no—
432Q8—21no—
433Q8—26no—
434Q8—26no—
435Q8—46no—
436Q8—46no—
437Q8—83no—
438Q8#137×C6114yescomplemented
439Q8C2×#41×C6112yes—
440D4—24no—
441D4—21no—
442D4C2×#41×C6112yes—
443C2×C4—21no—
444C2×C4—24no—
445C2×C4—26no—
446C2×C4—22no—
447C2×C4—23no—
448C2×C4—23no—
449C2×C4—26no—
450C2×C4—43no—
451C2×C4—42no—
452C2×C4—46no—
453C2×C4—46no—
454C2×C4—83no—
455C2×C4C22×D4×C6113yes—
456C2×C4C22×D4×C6113yes—
457C2×C4C22×D4×C6113yes—
458C2×C4C22×D4×C6111yescharacteristic
459C2×C4C22×D4×C6113yes—
460C2×C4C2×#41×C6112yes—
461C2×C42−1+4×C6112yes—
462C2×C4C25×C6111yescharacteristic
463C8—26no—
464C8—22no—
465C22—21no—
466C22—21no—
467C22—22no—
468C22—41no—
469C22—41no—
470C22C61×#54812yes—
471C22C23×D4×C6111yescharacteristic
472C4—23no—
473C4—23no—
474C4—23no—
475C4—22no—
476C4—43no—
477C4—42no—
478C4—43no—
479C4—46no—
480C4C61×#54812yes—
481C4C2×#137×C6116yes—
482C4C22×#41×C6111yescharacteristic
483C4C2×2−1+4×C6111yescharacteristic
484C2—22no—
485C2—42no—
486C2C2×C61×#54811yescharacteristic
487C2C22×#137×C6111yescharacteristic
488C2C61×#358611yescharacteristic
subgroupG/H
center, socleC22×C61C23×D4
commutator, Frattini, cosocleC2×C4C25×C61
prime radicalC2×#41×C61C23
LCSG ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22×C61 ◁ C2×C4×Q8×C61 ◁ G

Complements

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

NH ≅ G/Ntype
2. C2×C61×#557C2semidirect
3. C61×#3374C2semidirect
4. C61×#3372C2semidirect
5. C61×#3363C2semidirect
6. C61×#3359C2semidirect
7. C61×#3139C2semidirect
8. C61×#3078C2semidirect
10. Q8×M16×C61C2semidirect
193. Q8×C61#137semidirect
220. #61322C61direct (N × H)
241. C2×#557C2×C61semidirect
242. #3374C2×C61semidirect
243. #3372C2×C61semidirect
244. #3363C2×C61semidirect
245. #3359C2×C61semidirect
246. #3139C2×C61semidirect
247. #3078C2×C61semidirect
249. Q8×M16C2×C61semidirect
298. C61#61322direct (N × H)
438. Q8#137×C61semidirect

Maximal subgroups

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

subgroupindex
Q8×#41×C612
C2×C61×#5572
C61×#33742
C61×#33722
C61×#33632
C61×#33592
C61×#31392
C61×#30782
C61×#30392
Q8×M16×C612
C4×#137×C612
#6132261