Finite Group Databasebeta

← C61×#61319 #29564752

Subgroups — C61×#61319 #29564752

Order 15616.

1836 conjugacy classes of subgroups.

The subgroup lattice has 496 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C61×#3480C211yesmaximal, characteristic
2C2×Q82×C61C211yesmaximal, characteristic, complemented
3C61×#3375C216yesmaximal, complemented
4C61×#3369C216yesmaximal, complemented
5C61×#3363C212yesmaximal, complemented
6C61×#3362C212yesmaximal, complemented
7C61×#3141C213yesmaximal, complemented
8C61×#3079C213yesmaximal, complemented
9C61×#3037C213yesmaximal, complemented
10Q8×M16×C61C211yesmaximal, characteristic, complemented
11C4×#137×C61C213yesmaximal, complemented
12C2×#137×C61C2213yes—
13C61×#559C2213yes—
14Q82×C61—21no—
15Q82×C61—23no—
16Q82×C61C2212yes—
17Q82×C61C2216yes—
18C61×#557C2211yescharacteristic
19C61×#556C2213yes—
20C61×#553—23no—
21C61×#550C2211yescharacteristic
22C61×#548—21no—
23C61×#532C2213yes—
24C2×C4⋊Q8×C61—23no—
25C2×C4⋊Q8×C61C2213yes—
26C61×#529—23no—
27C4×#41×C61—21no—
28C4×#41×C61C2213yes—
29C2×C4×Q8×C61—21no—
30C2×C4×Q8×C61C2213yes—
31C2×C4×Q8×C61C2211yescharacteristic
32C61×#489C22112yes—
33C61×#487C2216yes—
34C61×#486C2216yes—
35C61×#477C2216yes—
36C61×#476C2216yes—
37C61×#475C2216yes—
38C61×#474C2216yes—
39C61×#464C22112yes—
40C61×#462C2216yes—
41C61×#460C2216yes—
42C61×#446C2216yes—
43C8×Q8×C61C2212yes—
44C61×#441C2216yes—
45C61×#440C2216yes—
46C4×Q16×C61C2216yes—
47C4×SD16×C61C2216yes—
48C61×#428C2213yes—
49C61×#423C2213yes—
50C61×#419C2211yescharacteristic
51C61×#417C2216yes—
52C61×#417C2213yes—
53C4×M16×C61C2213yes—
54C2×#41×C61—21no—
55C2×#41×C61C2313yes—
56C22×Q8×C61—23no—
57C22×Q8×C61D411yescharacteristic
58C22×Q8×C61C2311yescharacteristic
59#137×C61—212no—
60C2×Q16×C61C2316yes—
61C2×SD16×C61C2316yes—
62C2×M16×C61C2313yes—
63C2×M16×C61C2311yescharacteristic
64C4⋊Q8×C61—26no—
65C4⋊Q8×C61—26no—
66C4⋊Q8×C61—212no—
67C4⋊Q8×C61—23no—
68C4⋊Q8×C61—23no—
69C4⋊Q8×C61C2313yes—
70C4⋊Q8×C61C2316yes—
71C4⋊Q8×C61C2316yes—
72#127×C61C2313yes—
73#126×C61—26no—
74#125×C61—23no—
75#124×C61—23no—
76#124×C61C2316yes—
77#123×C61—26no—
78C22⋊Q8×C61—23no—
79#121×C61—23no—
80#121×C61—23no—
81#121×C61—23no—
82C4×Q8×C61—21no—
83C4×Q8×C61—26no—
84C4×Q8×C61—22no—
85C4×Q8×C61—23no—
86C4×Q8×C61—23no—
87C4×Q8×C61D411yescharacteristic
88C4×Q8×C61D413yes—
89C4×Q8×C61C2313yes—
90C4×Q8×C61C2313yes—
91C4×Q8×C61C2316yes—
92C4×Q8×C61C2316yes—
93C4×Q8×C61C2311yescharacteristic
94C4×Q8×C61C2313yes—
95C4×D4×C61—23no—
96C4×D4×C61—23no—
97C4×D4×C61—23no—
98C4×D4×C61—23no—
99C4×D4×C61—23no—
100C4×D4×C61C2313yes—
101C4×D4×C61C2313yes—
102#117×C61—23no—
103#117×C61—23no—
104#117×C61C2311yescharacteristic
105C2×C4⋊C4×C61—23no—
106C2×C4⋊C4×C61—23no—
107C2×C4⋊C4×C61D413yes—
108C2×C4⋊C4×C61C2313yes—
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×C61C2316yes—
117#103×C61C23112yes—
118#103×C61C2312yes—
119#103×C61C2312yes—
120#102×C61C2316yes—
121#102×C61C23112yes—
122#97×C61C2316yes—
123C4×C8×C61C2316yes—
124#41×C61—26no—
125#41×C61—26no—
126#41×C61—44no—
127C2×Q8×C61—24no—
128C2×Q8×C61—21no—
129C2×Q8×C61—26no—
130C2×Q8×C61—26no—
131C2×Q8×C61—22no—
132C2×Q8×C61—86no—
133C2×Q8×C61C2×D413yes—
134C2×Q8×C61C2×D411yescharacteristic
135C2×Q8×C61#4114yescomplemented
136C2×Q8×C61C2413yes—
137C2×Q8×C61C2412yes—
138C2×D4×C61—23no—
139C2×D4×C61—23no—
140C2×D4×C61—23no—
141C2×D4×C61C2413yes—
142C22×C4×C61—21no—
143C22×C4×C61—23no—
144C22×C4×C61—23no—
145C22×C4×C61—23no—
146C22×C4×C61—23no—
147C22×C4×C61C2×D413yes—
148C22×C4×C61C2×D413yes—
149C22×C4×C61C2411yescharacteristic
150Q16×C61—212no—
151SD16×C61—212no—
152M16×C61—26no—
153M16×C61—22no—
154C2×C8×C61C2412yes—
155C2×C8×C61C2416yes—
156C4⋊C4×C61—26no—
157C4⋊C4×C61—26no—
158C4⋊C4×C61—26no—
159C4⋊C4×C61—26no—
160C4⋊C4×C61—23no—
161C4⋊C4×C61—23no—
162C4⋊C4×C61—23no—
163C4⋊C4×C61—23no—
164C4⋊C4×C61—23no—
165C4⋊C4×C61C2×D413yes—
166C4⋊C4×C61C2×D413yes—
167C4⋊C4×C61C2×D416yes—
168C4⋊C4×C61C2416yes—
169C4⋊C4×C61C2411yescharacteristic
170C4⋊C4×C61C2411yescharacteristic
171C22⋊C4×C61—21no—
172C22⋊C4×C61—26no—
173C22⋊C4×C61—23no—
174C22⋊C4×C61—23no—
175C22⋊C4×C61—26no—
176C42×C61—21no—
177C42×C61—23no—
178C42×C61—23no—
179C42×C61—26no—
180C42×C61—26no—
181C42×C61C2×D413yes—
182C42×C61C2×D413yes—
183C42×C61C2413yes—
184C42×C61C2413yes—
185C23×C61—23no—
186C23×C61C22×D411yescharacteristic
187Q8×C61—24no—
188Q8×C61—26no—
189Q8×C61—22no—
190Q8×C61—23no—
191Q8×C61—23no—
192Q8×C61—86no—
193Q8×C61—86no—
194Q8×C61#13714yescomplemented
195Q8×C61C2×#4114yes—
196D4×C61—23no—
197D4×C61—23no—
198D4×C61—46no—
199D4×C61—46no—
200D4×C61—83no—
201C2×C4×C61—24no—
202C2×C4×C61—22no—
203C2×C4×C61—26no—
204C2×C4×C61—21no—
205C2×C4×C61—21no—
206C2×C4×C61—23no—
207C2×C4×C61—26no—
208C2×C4×C61—43no—
209C2×C4×C61—46no—
210C2×C4×C61—46no—
211C2×C4×C61—46no—
212C2×C4×C61—83no—
213C2×C4×C61C22×D413yes—
214C2×C4×C61C22×D413yes—
215C2×C4×C61C22×D413yes—
216C2×C4×C61C22×D411yescharacteristic
217C2×C4×C61C22×D413yes—
218C2×C4×C61C2×#4112yes—
219C2×C4×C612−1+412yes—
220C2×C4×C61C2511yescharacteristic
221C8×C61—26no—
222C8×C61—22no—
223#61319C6111yesmaximal, characteristic, direct product factor, Sylow
224C22×C61—21no—
225C22×C61—21no—
226C22×C61—43no—
227C22×C61—43no—
228C22×C61—43no—
229C22×C61#54812yes—
230C22×C61C23×D411yescharacteristic
231C4×C61—23no—
232C4×C61—23no—
233C4×C61—24no—
234C4×C61—23no—
235C4×C61—46no—
236C4×C61—41no—
237C4×C61—43no—
238C4×C61—41no—
239C4×C61#54812yes—
240C4×C61C2×#13716yes—
241C4×C61C22×#4111yescharacteristic
242C4×C61C2×2−1+411yescharacteristic
243#3480C2×C6111yescharacteristic
244C2×Q82C2×C6111yescharacteristic, complemented
245#3375C2×C6116yescomplemented
246#3369C2×C6116yescomplemented
247#3363C2×C6112yescomplemented
248#3362C2×C6112yescomplemented
249#3141C2×C6113yescomplemented
250#3079C2×C6113yescomplemented
251#3037C2×C6113yescomplemented
252Q8×M16C2×C6111yescharacteristic, complemented
253C4×#137C2×C6113yescomplemented
254C2×C61—22no—
255C2×C61—83no—
256C2×C61C2×#54811yescharacteristic
257C2×C61C22×#13711yescharacteristic
258C2×C61#358511yescharacteristic
259C2×#137C22×C6113yes—
260#559C22×C6113yes—
261Q82—21no—
262Q82—23no—
263Q82C22×C6112yes—
264Q82C22×C6116yes—
265#557C22×C6111yescharacteristic
266#556C22×C6113yes—
267#553—23no—
268#550C22×C6111yescharacteristic
269#548—21no—
270#532C22×C6113yes—
271C2×C4⋊Q8—23no—
272C2×C4⋊Q8C22×C6113yes—
273#529—23no—
274C4×#41—21no—
275C4×#41C22×C6113yes—
276C2×C4×Q8—21no—
277C2×C4×Q8C22×C6113yes—
278C2×C4×Q8C22×C6111yescharacteristic
279#489C22×C61112yes—
280#487C22×C6116yes—
281#486C22×C6116yes—
282#477C22×C6116yes—
283#476C22×C6116yes—
284#475C22×C6116yes—
285#474C22×C6116yes—
286#464C22×C61112yes—
287#462C22×C6116yes—
288#460C22×C6116yes—
289#446C22×C6116yes—
290C8×Q8C22×C6112yes—
291#441C22×C6116yes—
292#440C22×C6116yes—
293C4×Q16C22×C6116yes—
294C4×SD16C22×C6116yes—
295#428C22×C6113yes—
296#423C22×C6113yes—
297#419C22×C6111yescharacteristic
298#417C22×C6116yes—
299#417C22×C6113yes—
300C4×M16C22×C6113yes—
301C61#6131911yescharacteristic, direct product factor, Sylow
302C2×#41—21no—
303C2×#41C23×C6113yes—
304C22×Q8—23no—
305C22×Q8D4×C6111yescharacteristic
306C22×Q8C23×C6111yescharacteristic
307#137—212no—
308C2×Q16C23×C6116yes—
309C2×SD16C23×C6116yes—
310C2×M16C23×C6113yes—
311C2×M16C23×C6111yescharacteristic
312C4⋊Q8—26no—
313C4⋊Q8—26no—
314C4⋊Q8—212no—
315C4⋊Q8—23no—
316C4⋊Q8—23no—
317C4⋊Q8C23×C6113yes—
318C4⋊Q8C23×C6116yes—
319C4⋊Q8C23×C6116yes—
320#127C23×C6113yes—
321#126—26no—
322#125—23no—
323#124—23no—
324#124C23×C6116yes—
325#123—26no—
326C22⋊Q8—23no—
327#121—23no—
328#121—23no—
329#121—23no—
330C4×Q8—26no—
331C4×Q8—22no—
332C4×Q8—21no—
333C4×Q8—23no—
334C4×Q8—23no—
335C4×Q8D4×C6111yescharacteristic
336C4×Q8D4×C6113yes—
337C4×Q8C23×C6116yes—
338C4×Q8C23×C6116yes—
339C4×Q8C23×C6113yes—
340C4×Q8C23×C6113yes—
341C4×Q8C23×C6111yescharacteristic
342C4×Q8C23×C6113yes—
343C4×D4—23no—
344C4×D4—23no—
345C4×D4—23no—
346C4×D4—23no—
347C4×D4—23no—
348C4×D4C23×C6113yes—
349C4×D4C23×C6113yes—
350#117—23no—
351#117—23no—
352#117C23×C6111yescharacteristic
353C2×C4⋊C4—23no—
354C2×C4⋊C4—23no—
355C2×C4⋊C4D4×C6113yes—
356C2×C4⋊C4C23×C6113yes—
357C2×C42—23no—
358C2×C42—23no—
359C2×C42C23×C6113yes—
360#107C23×C6116yes—
361#106C23×C6116yes—
362C4⋊C8C23×C6116yes—
363C4⋊C8C23×C6116yes—
364#103C23×C6116yes—
365#103C23×C61112yes—
366#103C23×C6112yes—
367#103C23×C6112yes—
368#102C23×C6116yes—
369#102C23×C61112yes—
370#97C23×C6116yes—
371C4×C8C23×C6116yes—
372#41—26no—
373#41—26no—
374#41—44no—
375C2×Q8—24no—
376C2×Q8—21no—
377C2×Q8—26no—
378C2×Q8—26no—
379C2×Q8—22no—
380C2×Q8—86no—
381C2×Q8C2×D4×C6113yes—
382C2×Q8C2×D4×C6111yescharacteristic
383C2×Q8#41×C6114yescomplemented
384C2×Q8C24×C6113yes—
385C2×Q8C24×C6112yes—
386C2×D4—23no—
387C2×D4—23no—
388C2×D4—23no—
389C2×D4C24×C6113yes—
390C22×C4—21no—
391C22×C4—23no—
392C22×C4—23no—
393C22×C4—23no—
394C22×C4—23no—
395C22×C4C2×D4×C6113yes—
396C22×C4C2×D4×C6113yes—
397C22×C4C24×C6111yescharacteristic
398Q16—212no—
399SD16—212no—
400M16—26no—
401M16—22no—
402C2×C8C24×C6112yes—
403C2×C8C24×C6116yes—
404C4⋊C4—26no—
405C4⋊C4—26no—
406C4⋊C4—26no—
407C4⋊C4—26no—
408C4⋊C4—23no—
409C4⋊C4—23no—
410C4⋊C4—23no—
411C4⋊C4—23no—
412C4⋊C4—23no—
413C4⋊C4C2×D4×C6113yes—
414C4⋊C4C2×D4×C6113yes—
415C4⋊C4C2×D4×C6116yes—
416C4⋊C4C24×C6116yes—
417C4⋊C4C24×C6111yescharacteristic
418C4⋊C4C24×C6111yescharacteristic
419C22⋊C4—21no—
420C22⋊C4—26no—
421C22⋊C4—23no—
422C22⋊C4—23no—
423C22⋊C4—26no—
424C42—21no—
425C42—23no—
426C42—23no—
427C42—26no—
428C42—26no—
429C42C2×D4×C6113yes—
430C42C2×D4×C6113yes—
431C42C24×C6113yes—
432C42C24×C6113yes—
433C23—23no—
434C23C22×D4×C6111yescharacteristic
435Q8—24no—
436Q8—26no—
437Q8—22no—
438Q8—23no—
439Q8—23no—
440Q8—86no—
441Q8—86no—
442Q8#137×C6114yescomplemented
443Q8C2×#41×C6114yes—
444D4—23no—
445D4—23no—
446D4—46no—
447D4—46no—
448D4—83no—
449C2×C4—24no—
450C2×C4—22no—
451C2×C4—26no—
452C2×C4—21no—
453C2×C4—21no—
454C2×C4—23no—
455C2×C4—26no—
456C2×C4—43no—
457C2×C4—46no—
458C2×C4—46no—
459C2×C4—46no—
460C2×C4—83no—
461C2×C4C22×D4×C6113yes—
462C2×C4C22×D4×C6113yes—
463C2×C4C22×D4×C6113yes—
464C2×C4C22×D4×C6111yescharacteristic
465C2×C4C22×D4×C6113yes—
466C2×C4C2×#41×C6112yes—
467C2×C42−1+4×C6112yes—
468C2×C4C25×C6111yescharacteristic
469C8—26no—
470C8—22no—
471C22—21no—
472C22—21no—
473C22—43no—
474C22—43no—
475C22—43no—
476C22C61×#54812yes—
477C22C23×D4×C6111yescharacteristic
478C4—23no—
479C4—23no—
480C4—23no—
481C4—24no—
482C4—41no—
483C4—46no—
484C4—41no—
485C4—43no—
486C4C61×#54812yes—
487C4C2×#137×C6116yes—
488C4C22×#41×C6111yescharacteristic
489C4C2×2−1+4×C6111yescharacteristic
490C2—22no—
491C2—83no—
492C2C2×C61×#54811yescharacteristic
493C2C22×#137×C6111yescharacteristic
494C2C61×#358511yescharacteristic
subgroupG/H
center, socleC22×C61C23×D4
commutator, Frattini, cosocleC2×C4C25×C61
prime radicalC61×#3480C2
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×Q82×C61C2semidirect
3. C61×#3375C2semidirect
4. C61×#3369C2semidirect
5. C61×#3363C2semidirect
6. C61×#3362C2semidirect
7. C61×#3141C2semidirect
8. C61×#3079C2semidirect
9. C61×#3037C2semidirect
10. Q8×M16×C61C2semidirect
11. C4×#137×C61C2semidirect
135. C2×Q8×C61#41semidirect
194. Q8×C61#137semidirect
223. #61319C61direct (N × H)
244. C2×Q82C2×C61semidirect
245. #3375C2×C61semidirect
246. #3369C2×C61semidirect
247. #3363C2×C61semidirect
248. #3362C2×C61semidirect
249. #3141C2×C61semidirect
250. #3079C2×C61semidirect
251. #3037C2×C61semidirect
252. Q8×M16C2×C61semidirect
253. C4×#137C2×C61semidirect
301. C61#61319direct (N × H)
383. C2×Q8#41×C61semidirect
442. Q8#137×C61semidirect

Maximal subgroups

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

subgroupindex
C61×#34802
C2×Q82×C612
C61×#33752
C61×#33692
C61×#33632
C61×#33622
C61×#31412
C61×#30792
C61×#30372
Q8×M16×C612
C4×#137×C612
#6131961