Finite Group Databasebeta

← C61×#62000 #29565433

Subgroups — C61×#62000 #29565433

Order 15616.

1818 conjugacy classes of subgroups.

The subgroup lattice has 488 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C61×#3503C211yesmaximal, characteristic, complemented
2C61×#3482C211yesmaximal, characteristic
3C61×#3395C212yesmaximal, complemented
4C61×#3359C212yesmaximal, complemented
5C61×#3328C216yesmaximal, complemented
6C61×#3276C216yesmaximal, complemented
7C61×#3255C213yesmaximal, complemented
8C61×#3209C213yesmaximal, complemented
9C61×#3107C213yesmaximal, complemented
10C61×#3039C213yesmaximal, complemented
11C61×#2991C211yesmaximal, characteristic, complemented
12C2×#137×C61C2216yes—
13Q82×C61—21no—
14C61×#557C2212yes—
15C61×#557C2211yescharacteristic
16C61×#556C2213yes—
17C61×#554—23no—
18C61×#552—23no—
19D4×Q8×C61C2216yes—
20D4×Q8×C61C2211yescharacteristic
21C61×#548—21no—
22C61×#547C2213yes—
23C61×#544—23no—
24C61×#533C2213yes—
25C61×#533C2213yes—
26C2×C4⋊Q8×C61C2213yes—
27C61×#529—23no—
28C61×#520—21no—
29C4×#41×C61C2213yes—
30C4×#41×C61C2211yescharacteristic
31C2×C4×Q8×C61—21no—
32C61×#494C2216yes—
33C61×#487C2216yes—
34C61×#485C2216yes—
35C61×#475C2216yes—
36C61×#475C2213yes—
37C61×#474C2211yescharacteristic
38C61×#470C22112yes—
39C61×#462C2216yes—
40C61×#462C2216yes—
41C61×#462C2213yes—
42C61×#462C2216yes—
43C61×#461C2213yes—
44C61×#461C2216yes—
45C61×#452C2216yes—
46C61×#451C22112yes—
47C8×Q8×C61C2212yes—
48C4×Q16×C61C2216yes—
49C4×SD16×C61C2216yes—
50C61×#436C2216yes—
51C61×#431C2213yes—
52C61×#423C2213yes—
53C61×#419C2216yes—
54C2×#41×C61D411yescharacteristic
55C2×#41×C61C2313yes—
56C22×Q8×C61—21no—
57C22×Q8×C61C2316yes—
58#137×C61—412no—
59C2×Q16×C61C23112yes—
60C2×Q16×C61C2316yes—
61C2×SD16×C61C2316yes—
62C2×M16×C61C2316yes—
63C4⋊Q8×C61—26no—
64C4⋊Q8×C61—23no—
65C4⋊Q8×C61—23no—
66C4⋊Q8×C61C2313yes—
67C4⋊Q8×C61C2316yes—
68C4⋊Q8×C61C2313yes—
69C4⋊Q8×C61C2311yescharacteristic
70C4⋊Q8×C61C2313yes—
71#126×C61—26no—
72#125×C61—26no—
73#125×C61—23no—
74#125×C61—23no—
75#125×C61—23no—
76#124×C61—23no—
77#123×C61—26no—
78C22⋊Q8×C61—23no—
79C22⋊Q8×C61—23no—
80C22⋊Q8×C61—23no—
81C22⋊Q8×C61—212no—
82C22⋊Q8×C61C2316yes—
83C22⋊Q8×C61C2316yes—
84#121×C61—23no—
85C4×Q8×C61—21no—
86C4×Q8×C61—22no—
87C4×Q8×C61—23no—
88C4×Q8×C61—23no—
89C4×Q8×C61—23no—
90C4×Q8×C61D411yescharacteristic
91C4×Q8×C61C2313yes—
92C4×Q8×C61C2316yes—
93C4×Q8×C61C2313yes—
94C4×Q8×C61C2311yescharacteristic
95C4×D4×C61—26no—
96C4×D4×C61—23no—
97C4×D4×C61—23no—
98C4×D4×C61D413yes—
99C4×D4×C61C2313yes—
100C4×D4×C61C2311yescharacteristic
101C4×D4×C61C2313yes—
102#117×C61—23no—
103#117×C61—23no—
104#117×C61—23no—
105#117×C61D413yes—
106#117×C61C2313yes—
107C2×C4⋊C4×C61—23no—
108C2×C4⋊C4×C61—23no—
109C2×C4⋊C4×C61—23no—
110C2×C42×C61—23no—
111C2×C42×C61C2313yes—
112#107×C61C2312yes—
113#106×C61C2316yes—
114C4⋊C8×C61C2313yes—
115C4⋊C8×C61C2316yes—
116C4⋊C8×C61C2311yescharacteristic
117#103×C61C2316yes—
118#103×C61C23112yes—
119#103×C61C2316yes—
120#103×C61C2316yes—
121#102×C61C2312yes—
122C22⋊C8×C61C2316yes—
123C4×C8×C61C2316yes—
124#41×C61—24no—
125#41×C61—26no—
126#41×C61—43no—
127C2×Q8×C61—21no—
128C2×Q8×C61—23no—
129C2×Q8×C61—26no—
130C2×Q8×C61—46no—
131C2×Q8×C61—46no—
132C2×Q8×C61—44no—
133C2×Q8×C61C2×D411yescharacteristic
134C2×Q8×C61C2416yes—
135C2×Q8×C61C2413yes—
136C2×D4×C61—23no—
137C2×D4×C61C2×D413yes—
138C2×D4×C61C2411yescharacteristic
139C22×C4×C61—26no—
140C22×C4×C61—21no—
141C22×C4×C61—23no—
142C22×C4×C61—23no—
143C22×C4×C61C2×D413yes—
144C22×C4×C61C2×D413yes—
145C22×C4×C61C2413yes—
146Q16×C61—46no—
147Q16×C61—412no—
148SD16×C61—212no—
149M16×C61—46no—
150C2×C8×C61C2412yes—
151C2×C8×C61C2416yes—
152C4⋊C4×C61—23no—
153C4⋊C4×C61—23no—
154C4⋊C4×C61—26no—
155C4⋊C4×C61—23no—
156C4⋊C4×C61—26no—
157C4⋊C4×C61—26no—
158C4⋊C4×C61—21no—
159C4⋊C4×C61—23no—
160C4⋊C4×C61—23no—
161C4⋊C4×C61—23no—
162C4⋊C4×C61—23no—
163C4⋊C4×C61—26no—
164C4⋊C4×C61—26no—
165C4⋊C4×C61C2×D413yes—
166C4⋊C4×C61C2×D413yes—
167C4⋊C4×C61C2412yes—
168C4⋊C4×C61C2413yes—
169C4⋊C4×C61C2411yescharacteristic
170C22⋊C4×C61—26no—
171C22⋊C4×C61—26no—
172C22⋊C4×C61—23no—
173C22⋊C4×C61—21no—
174C22⋊C4×C61—23no—
175C22⋊C4×C61C2×D416yes—
176C42×C61—23no—
177C42×C61—23no—
178C42×C61—23no—
179C42×C61—26no—
180C42×C61C2×D413yes—
181C42×C61C2×D413yes—
182C42×C61C2413yes—
183C42×C61C2411yescharacteristic
184C23×C61—21no—
185C23×C61C22×D413yes—
186Q8×C61—26no—
187Q8×C61—23no—
188Q8×C61—23no—
189Q8×C61—26no—
190Q8×C61—46no—
191Q8×C61—42no—
192Q8×C61—86no—
193Q8×C61#13714yescomplemented
194D4×C61—26no—
195D4×C61—21no—
196D4×C61—46no—
197D4×C61H(5,𝔽2)12yes—
198C2×C4×C61—26no—
199C2×C4×C61—26no—
200C2×C4×C61—23no—
201C2×C4×C61—23no—
202C2×C4×C61—21no—
203C2×C4×C61—22no—
204C2×C4×C61—23no—
205C2×C4×C61—23no—
206C2×C4×C61—26no—
207C2×C4×C61—46no—
208C2×C4×C61—46no—
209C2×C4×C61—42no—
210C2×C4×C61—86no—
211C2×C4×C61C22×D413yes—
212C2×C4×C61C22×D411yescharacteristic
213C2×C4×C61C22×D413yes—
214C2×C4×C61C22×D413yes—
215C2×C4×C61C22×D411yescharacteristic
216C2×C4×C61C2511yescharacteristic
217C8×C61—26no—
218C8×C61—22no—
219#62000C6111yesmaximal, characteristic, direct product factor, Sylow
220C22×C61—23no—
221C22×C61—22no—
222C22×C61—23no—
223C22×C61—23no—
224C22×C61—41no—
225C22×C61—41no—
226C22×C61C23×D411yescharacteristic
227C4×C61—23no—
228C4×C61—21no—
229C4×C61—23no—
230C4×C61—43no—
231C4×C61—46no—
232C4×C61—43no—
233C4×C61—41no—
234C4×C61—42no—
235C4×C61C2×#13716yes—
236C4×C61#57812yes—
237C4×C61C2×H(5,𝔽2)11yescharacteristic
238C4×C61#58511yescharacteristic
239#3503C2×C6111yescharacteristic, complemented
240#3482C2×C6111yescharacteristic
241#3395C2×C6112yescomplemented
242#3359C2×C6112yescomplemented
243#3328C2×C6116yescomplemented
244#3276C2×C6116yescomplemented
245#3255C2×C6113yescomplemented
246#3209C2×C6113yescomplemented
247#3107C2×C6113yescomplemented
248#3039C2×C6113yescomplemented
249#2991C2×C6111yescharacteristic, complemented
250C2×C61—43no—
251C2×C61—42no—
252C2×C61#348811yescharacteristic
253C2×C61C22×#13711yescharacteristic
254C2×C61C2×#57811yescharacteristic
255C2×#137C22×C6116yes—
256Q82—21no—
257#557C22×C6112yes—
258#557C22×C6111yescharacteristic
259#556C22×C6113yes—
260#554—23no—
261#552—23no—
262D4×Q8C22×C6116yes—
263D4×Q8C22×C6111yescharacteristic
264#548—21no—
265#547C22×C6113yes—
266#544—23no—
267#533C22×C6113yes—
268#533C22×C6113yes—
269C2×C4⋊Q8C22×C6113yes—
270#529—23no—
271#520—21no—
272C4×#41C22×C6113yes—
273C4×#41C22×C6111yescharacteristic
274C2×C4×Q8—21no—
275#494C22×C6116yes—
276#487C22×C6116yes—
277#485C22×C6116yes—
278#475C22×C6116yes—
279#475C22×C6113yes—
280#474C22×C6111yescharacteristic
281#470C22×C61112yes—
282#462C22×C6116yes—
283#462C22×C6113yes—
284#462C22×C6116yes—
285#462C22×C6116yes—
286#461C22×C6116yes—
287#461C22×C6113yes—
288#452C22×C6116yes—
289#451C22×C61112yes—
290C8×Q8C22×C6112yes—
291C4×Q16C22×C6116yes—
292C4×SD16C22×C6116yes—
293#436C22×C6116yes—
294#431C22×C6113yes—
295#423C22×C6113yes—
296#419C22×C6116yes—
297C61#6200011yescharacteristic, direct product factor, Sylow
298C2×#41D4×C6111yescharacteristic
299C2×#41C23×C6113yes—
300C22×Q8—21no—
301C22×Q8C23×C6116yes—
302#137—412no—
303C2×Q16C23×C61112yes—
304C2×Q16C23×C6116yes—
305C2×SD16C23×C6116yes—
306C2×M16C23×C6116yes—
307C4⋊Q8—26no—
308C4⋊Q8—23no—
309C4⋊Q8—23no—
310C4⋊Q8C23×C6113yes—
311C4⋊Q8C23×C6116yes—
312C4⋊Q8C23×C6113yes—
313C4⋊Q8C23×C6111yescharacteristic
314C4⋊Q8C23×C6113yes—
315#126—26no—
316#125—23no—
317#125—26no—
318#125—23no—
319#125—23no—
320#124—23no—
321#123—26no—
322C22⋊Q8—23no—
323C22⋊Q8—23no—
324C22⋊Q8—23no—
325C22⋊Q8—212no—
326C22⋊Q8C23×C6116yes—
327C22⋊Q8C23×C6116yes—
328#121—23no—
329C4×Q8—21no—
330C4×Q8—22no—
331C4×Q8—23no—
332C4×Q8—23no—
333C4×Q8—23no—
334C4×Q8D4×C6111yescharacteristic
335C4×Q8C23×C6113yes—
336C4×Q8C23×C6113yes—
337C4×Q8C23×C6116yes—
338C4×Q8C23×C6111yescharacteristic
339C4×D4—26no—
340C4×D4—23no—
341C4×D4—23no—
342C4×D4D4×C6113yes—
343C4×D4C23×C6111yescharacteristic
344C4×D4C23×C6113yes—
345C4×D4C23×C6113yes—
346#117—23no—
347#117—23no—
348#117—23no—
349#117D4×C6113yes—
350#117C23×C6113yes—
351C2×C4⋊C4—23no—
352C2×C4⋊C4—23no—
353C2×C4⋊C4—23no—
354C2×C42—23no—
355C2×C42C23×C6113yes—
356#107C23×C6112yes—
357#106C23×C6116yes—
358C4⋊C8C23×C6113yes—
359C4⋊C8C23×C6116yes—
360C4⋊C8C23×C6111yescharacteristic
361#103C23×C6116yes—
362#103C23×C61112yes—
363#103C23×C6116yes—
364#103C23×C6116yes—
365#102C23×C6112yes—
366C22⋊C8C23×C6116yes—
367C4×C8C23×C6116yes—
368#41—24no—
369#41—26no—
370#41—43no—
371C2×Q8—21no—
372C2×Q8—23no—
373C2×Q8—26no—
374C2×Q8—46no—
375C2×Q8—46no—
376C2×Q8—44no—
377C2×Q8C2×D4×C6111yescharacteristic
378C2×Q8C24×C6116yes—
379C2×Q8C24×C6113yes—
380C2×D4—23no—
381C2×D4C2×D4×C6113yes—
382C2×D4C24×C6111yescharacteristic
383C22×C4—26no—
384C22×C4—21no—
385C22×C4—23no—
386C22×C4—23no—
387C22×C4C2×D4×C6113yes—
388C22×C4C2×D4×C6113yes—
389C22×C4C24×C6113yes—
390Q16—46no—
391Q16—412no—
392SD16—212no—
393M16—46no—
394C2×C8C24×C6112yes—
395C2×C8C24×C6116yes—
396C4⋊C4—23no—
397C4⋊C4—23no—
398C4⋊C4—26no—
399C4⋊C4—23no—
400C4⋊C4—26no—
401C4⋊C4—26no—
402C4⋊C4—21no—
403C4⋊C4—23no—
404C4⋊C4—23no—
405C4⋊C4—23no—
406C4⋊C4—23no—
407C4⋊C4—26no—
408C4⋊C4—26no—
409C4⋊C4C2×D4×C6113yes—
410C4⋊C4C2×D4×C6113yes—
411C4⋊C4C24×C6112yes—
412C4⋊C4C24×C6113yes—
413C4⋊C4C24×C6111yescharacteristic
414C22⋊C4—26no—
415C22⋊C4—26no—
416C22⋊C4—23no—
417C22⋊C4—21no—
418C22⋊C4—23no—
419C22⋊C4C2×D4×C6116yes—
420C42—23no—
421C42—23no—
422C42—23no—
423C42—26no—
424C42C2×D4×C6113yes—
425C42C2×D4×C6113yes—
426C42C24×C6113yes—
427C42C24×C6111yescharacteristic
428C23—21no—
429C23C22×D4×C6113yes—
430Q8—26no—
431Q8—23no—
432Q8—23no—
433Q8—26no—
434Q8—46no—
435Q8—42no—
436Q8—86no—
437Q8#137×C6114yescomplemented
438D4—26no—
439D4—21no—
440D4—46no—
441D4H(5,𝔽2)×C6112yes—
442C2×C4—26no—
443C2×C4—26no—
444C2×C4—23no—
445C2×C4—23no—
446C2×C4—21no—
447C2×C4—22no—
448C2×C4—23no—
449C2×C4—23no—
450C2×C4—26no—
451C2×C4—46no—
452C2×C4—46no—
453C2×C4—42no—
454C2×C4—86no—
455C2×C4C22×D4×C6113yes—
456C2×C4C22×D4×C6111yescharacteristic
457C2×C4C22×D4×C6113yes—
458C2×C4C22×D4×C6113yes—
459C2×C4C22×D4×C6111yescharacteristic
460C2×C4C25×C6111yescharacteristic
461C8—26no—
462C8—22no—
463C22—23no—
464C22—23no—
465C22—23no—
466C22—22no—
467C22—41no—
468C22—41no—
469C22C23×D4×C6111yescharacteristic
470C4—21no—
471C4—23no—
472C4—23no—
473C4—43no—
474C4—41no—
475C4—42no—
476C4—43no—
477C4—46no—
478C4C2×#137×C6116yes—
479C4C61×#57812yes—
480C4C2×H(5,𝔽2)×C6111yescharacteristic
481C4C61×#58511yescharacteristic
482C2—43no—
483C2—42no—
484C2C61×#348811yescharacteristic
485C2C22×#137×C6111yescharacteristic
486C2C2×C61×#57811yescharacteristic
subgroupG/H
center, socleC22×C61C23×D4
commutator, Frattini, cosocleC2×C4C25×C61
prime radicalC61×#3482C2
LCSG ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22×C61 ◁ C4×#41×C61 ◁ G

Complements

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

NH ≅ G/Ntype
1. C61×#3503C2semidirect
3. C61×#3395C2semidirect
4. C61×#3359C2semidirect
5. C61×#3328C2semidirect
6. C61×#3276C2semidirect
7. C61×#3255C2semidirect
8. C61×#3209C2semidirect
9. C61×#3107C2semidirect
10. C61×#3039C2semidirect
11. C61×#2991C2semidirect
193. Q8×C61#137semidirect
219. #62000C61direct (N × H)
239. #3503C2×C61semidirect
241. #3395C2×C61semidirect
242. #3359C2×C61semidirect
243. #3328C2×C61semidirect
244. #3276C2×C61semidirect
245. #3255C2×C61semidirect
246. #3209C2×C61semidirect
247. #3107C2×C61semidirect
248. #3039C2×C61semidirect
249. #2991C2×C61semidirect
297. C61#62000direct (N × H)
437. Q8#137×C61semidirect

Maximal subgroups

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

subgroupindex
C61×#35032
C61×#34822
C61×#33952
C61×#33592
C61×#33282
C61×#32762
C61×#32552
C61×#32092
C61×#31072
C61×#30392
C61×#29912
#6200061