Finite Group Databasebeta

← C61×#16522 #29519955

Subgroups — C61×#16522 #29519955

Order 15616.

1414 conjugacy classes of subgroups.

The subgroup lattice has 550 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C61×#398C212yesmaximal
2C2×C61×#395C211yesmaximal, characteristic
3C2×C61×#395C212yesmaximal
4C2×C61×#384C211yesmaximal, characteristic
5C61×#1444C212yesmaximal
6C61×#1444C212yesmaximal
7C61×#1442C212yesmaximal
8C61×#1441C211yesmaximal, characteristic
9C61×#1434C212yesmaximal
10C23×Q8×C61C2211yescharacteristic
11C22×C4⋊C4×C61C2212yes—
12C22×C4⋊C4×C61C2212yes—
13C22×C4⋊C4×C61C2212yes—
14C22×C4⋊C4×C61C2212yes—
15C22×C4⋊C4×C61C2211yescharacteristic
16C22×C4⋊C4×C61C2211yescharacteristic
17C22×C4⋊C4×C61C2212yes—
18C22×C4⋊C4×C61C2211yescharacteristic
19C22×C42×C61C2211yescharacteristic
20C61×#398—28no—
21C61×#395—24no—
22C61×#395—28no—
23C61×#384—24no—
24C2×#95×C61C2212yes—
25C2×#95×C61C2212yes—
26C2×#95×C61C2212yes—
27C2×#95×C61C2211yescharacteristic
28C2×#95×C61C2212yes—
29C2×#95×C61C2212yes—
30C2×#95×C61C2212yes—
31C2×#95×C61C2211yescharacteristic
32C2×#95×C61C2212yes—
33C2×#95×C61C2212yes—
34C2×#95×C61C2212yes—
35C22×Q8×C61—22no—
36C22×Q8×C61—24no—
37C22×Q8×C61—41no—
38C22×Q8×C61—42no—
39C22×Q8×C61—41no—
40C23×C4×C61C2312yes—
41C23×C4×C61C2311yescharacteristic
42C23×C4×C61C2312yes—
43C23×C4×C61C2312yes—
44C23×C4×C61C2311yescharacteristic
45C23×C4×C61C2312yes—
46C23×C4×C61C2312yes—
47C23×C4×C61C2312yes—
48C23×C4×C61C2311yescharacteristic
49C2×C4⋊C4×C61—24no—
50C2×C4⋊C4×C61—24no—
51C2×C4⋊C4×C61—22no—
52C2×C4⋊C4×C61—24no—
53C2×C4⋊C4×C61—24no—
54C2×C4⋊C4×C61—24no—
55C2×C4⋊C4×C61—24no—
56C2×C4⋊C4×C61—24no—
57C2×C4⋊C4×C61—24no—
58C2×C4⋊C4×C61—24no—
59C2×C4⋊C4×C61—24no—
60C2×C4⋊C4×C61—41no—
61C2×C4⋊C4×C61—41no—
62C2×C4⋊C4×C61—42no—
63C2×C4⋊C4×C61—42no—
64C2×C4⋊C4×C61—42no—
65C2×C4⋊C4×C61—41no—
66C2×C4⋊C4×C61—42no—
67C2×C4⋊C4×C61—41no—
68C2×C4⋊C4×C61—41no—
69C2×C4⋊C4×C61—42no—
70C2×C4⋊C4×C61—42no—
71C2×C4⋊C4×C61D414yes—
72C2×C42×C61—22no—
73C2×C42×C61—22no—
74C2×C42×C61Q814yescomplemented
75#95×C61—22no—
76#95×C61—24no—
77#95×C61—24no—
78#95×C61—24no—
79#95×C61—24no—
80#95×C61—24no—
81#95×C61—24no—
82#95×C61—41no—
83#95×C61—42no—
84#95×C61D418yes—
85#95×C61Q818yescomplemented
86C24×C61C2411yescharacteristic
87C2×Q8×C61—48no—
88C2×Q8×C61—44no—
89C2×Q8×C61—44no—
90C2×Q8×C61—48no—
91C2×Q8×C61—44no—
92C22×C4×C61—22no—
93C22×C4×C61—22no—
94C22×C4×C61—21no—
95C22×C4×C61—21no—
96C22×C4×C61—22no—
97C22×C4×C61—21no—
98C22×C4×C61—21no—
99C22×C4×C61—22no—
100C22×C4×C61—22no—
101C22×C4×C61—22no—
102C22×C4×C61—22no—
103C22×C4×C61—22no—
104C22×C4×C61—22no—
105C22×C4×C61—22no—
106C22×C4×C61—22no—
107C22×C4×C61—22no—
108C22×C4×C61—22no—
109C22×C4×C61—22no—
110C22×C4×C61—21no—
111C22×C4×C61—22no—
112C22×C4×C61—22no—
113C22×C4×C61—22no—
114C22×C4×C61—21no—
115C22×C4×C61—22no—
116C22×C4×C61—22no—
117C22×C4×C61—21no—
118C22×C4×C61—21no—
119C22×C4×C61—22no—
120C22×C4×C61—22no—
121C22×C4×C61—22no—
122C22×C4×C61—22no—
123C22×C4×C61—22no—
124C22×C4×C61—22no—
125C22×C4×C61—22no—
126C22×C4×C61—22no—
127C22×C4×C61—22no—
128C22×C4×C61—22no—
129C22×C4×C61—22no—
130C22×C4×C61—22no—
131C22×C4×C61—22no—
132C22×C4×C61—22no—
133C22×C4×C61—22no—
134C22×C4×C61—22no—
135C22×C4×C61—22no—
136C22×C4×C61—22no—
137C22×C4×C61—22no—
138C22×C4×C61—22no—
139C22×C4×C61C2×D414yes—
140C22×C4×C61C2×D412yes—
141C22×C4×C61C2×D414yes—
142C22×C4×C61C2×D412yes—
143C22×C4×C61C2×D412yes—
144C22×C4×C61C2×D414yes—
145C22×C4×C61C2×Q814yes—
146C22×C4×C61C2×Q814yes—
147C22×C4×C61C2×Q812yes—
148C22×C4×C61C2×Q814yes—
149C22×C4×C61C2×Q814yes—
150C22×C4×C61#4112yes—
151C4⋊C4×C61—44no—
152C4⋊C4×C61—48no—
153C4⋊C4×C61—48no—
154C4⋊C4×C61—44no—
155C4⋊C4×C61—48no—
156C4⋊C4×C61—48no—
157C4⋊C4×C61—44no—
158C4⋊C4×C61—48no—
159C42×C61—28no—
160C23×C61C22×D411yescharacteristic
161C23×C61C22×D412yes—
162C23×C61C22×Q811yescharacteristic
163C23×C61C22×Q812yes—
164C23×C61C2×#4111yescharacteristic
165C23×C61H(5,𝔽2)11yescharacteristic
166C23×C61H(5,𝔽2)11yescharacteristic
167C23×C61H(5,𝔽2)12yes—
168C23×C612−1+411yescharacteristic
169C23×C612−1+411yescharacteristic
170C23×C612−1+412yes—
171Q8×C61—416no—
172C2×C4×C61—24no—
173C2×C4×C61—22no—
174C2×C4×C61—22no—
175C2×C4×C61—22no—
176C2×C4×C61—22no—
177C2×C4×C61—22no—
178C2×C4×C61—22no—
179C2×C4×C61—24no—
180C2×C4×C61—24no—
181C2×C4×C61—24no—
182C2×C4×C61—24no—
183C2×C4×C61—24no—
184C2×C4×C61—22no—
185C2×C4×C61—22no—
186C2×C4×C61—24no—
187C2×C4×C61—24no—
188C2×C4×C61—24no—
189C2×C4×C61—24no—
190C2×C4×C61—24no—
191C2×C4×C61—24no—
192C2×C4×C61—24no—
193C2×C4×C61—24no—
194C2×C4×C61—24no—
195C2×C4×C61—24no—
196C2×C4×C61—24no—
197C2×C4×C61—24no—
198C2×C4×C61—24no—
199C2×C4×C61—24no—
200C2×C4×C61—42no—
201C2×C4×C61—42no—
202C2×C4×C61—42no—
203C2×C4×C61—42no—
204C2×C4×C61—42no—
205C2×C4×C61—42no—
206C2×C4×C61—42no—
207C2×C4×C61—42no—
208C2×C4×C61—42no—
209C2×C4×C61—42no—
210C2×C4×C61—41no—
211C2×C4×C61—42no—
212C2×C4×C61—42no—
213C2×C4×C61—41no—
214C2×C4×C61—42no—
215C2×C4×C61—42no—
216C2×C4×C61—41no—
217C2×C4×C61—41no—
218C2×C4×C61—42no—
219C2×C4×C61—42no—
220C2×C4×C61—42no—
221C2×C4×C61—42no—
222C2×C4×C61—42no—
223C2×C4×C61—42no—
224C2×C4×C61—42no—
225C2×C4×C61—42no—
226C2×C4×C61—42no—
227C2×C4×C61—42no—
228C2×C4×C61—42no—
229C2×C4×C61C22⋊Q814yes—
230C2×C4×C61C4⋊Q818yes—
231#16522C6111yesmaximal, characteristic, direct product factor, Sylow
232C22×C61C2×C22⋊Q811yescharacteristic
233C22×C61C2×C4⋊Q812yes—
234C22×C61#53411yescharacteristic
235C22×C61#53512yes—
236C22×C61#53611yescharacteristic
237C22×C61#53612yes—
238C22×C61#53911yescharacteristic
239C22×C61#53911yescharacteristic
240C22×C61#54411yescharacteristic
241C22×C61#54412yes—
242C22×C61#54411yescharacteristic
243C22×C61#54412yes—
244C22×C61#54412yes—
245C22×C61D4212yes—
246C22×C61D4211yescharacteristic
247C22×C61#54712yes—
248C22×C61D4×Q812yes—
249C22×C61D4×Q812yes—
250C22×C61#55412yes—
251C22×C61Q8211yescharacteristic
252C22×C61#56011yescharacteristic
253C22×C61#56312yes—
254C22×C61#56311yescharacteristic
255C4×C61—24no—
256C4×C61—28no—
257C4×C61—42no—
258C4×C61—42no—
259C4×C61—44no—
260C4×C61—44no—
261C4×C61—44no—
262C4×C61—44no—
263C4×C61—82no—
264C2×#398C2×C6112yes—
265C2×#395C2×C6111yescharacteristic
266C2×#395C2×C6112yes—
267C2×#384C2×C6111yescharacteristic
268#1444C2×C6112yes—
269#1444C2×C6112yes—
270#1442C2×C6112yes—
271#1441C2×C6111yescharacteristic
272#1434C2×C6112yes—
273C2×C61#244012yes—
274C2×C61#255111yescharacteristic
275C2×C61#262711yescharacteristic
276C2×C61#266012yes—
277C2×C61#266911yescharacteristic
278C2×C61#267712yes—
279C2×C61#268011yescharacteristic
280C2×C61#272012yes—
281C2×C61#273411yescharacteristic
282C2×C61#279211yescharacteristic
283C2×C61#284311yescharacteristic
284C23×Q8C22×C6111yescharacteristic
285C22×C4⋊C4C22×C6112yes—
286C22×C4⋊C4C22×C6112yes—
287C22×C4⋊C4C22×C6112yes—
288C22×C4⋊C4C22×C6111yescharacteristic
289C22×C4⋊C4C22×C6112yes—
290C22×C4⋊C4C22×C6112yes—
291C22×C4⋊C4C22×C6111yescharacteristic
292C22×C4⋊C4C22×C6111yescharacteristic
293C22×C42C22×C6111yescharacteristic
294#398—28no—
295#395—24no—
296#395—28no—
297#384—24no—
298C2×#95C22×C6112yes—
299C2×#95C22×C6112yes—
300C2×#95C22×C6112yes—
301C2×#95C22×C6111yescharacteristic
302C2×#95C22×C6112yes—
303C2×#95C22×C6112yes—
304C2×#95C22×C6112yes—
305C2×#95C22×C6111yescharacteristic
306C2×#95C22×C6112yes—
307C2×#95C22×C6112yes—
308C2×#95C22×C6112yes—
309C61#1652211yescharacteristic, direct product factor, Sylow
310C22×Q8—22no—
311C22×Q8—24no—
312C22×Q8—41no—
313C22×Q8—42no—
314C22×Q8—41no—
315C23×C4C23×C6112yes—
316C23×C4C23×C6111yescharacteristic
317C23×C4C23×C6111yescharacteristic
318C23×C4C23×C6112yes—
319C23×C4C23×C6112yes—
320C23×C4C23×C6112yes—
321C23×C4C23×C6112yes—
322C23×C4C23×C6112yes—
323C23×C4C23×C6111yescharacteristic
324C2×C4⋊C4—24no—
325C2×C4⋊C4—24no—
326C2×C4⋊C4—22no—
327C2×C4⋊C4—24no—
328C2×C4⋊C4—24no—
329C2×C4⋊C4—24no—
330C2×C4⋊C4—24no—
331C2×C4⋊C4—24no—
332C2×C4⋊C4—24no—
333C2×C4⋊C4—24no—
334C2×C4⋊C4—24no—
335C2×C4⋊C4—41no—
336C2×C4⋊C4—41no—
337C2×C4⋊C4—42no—
338C2×C4⋊C4—42no—
339C2×C4⋊C4—42no—
340C2×C4⋊C4—41no—
341C2×C4⋊C4—42no—
342C2×C4⋊C4—41no—
343C2×C4⋊C4—41no—
344C2×C4⋊C4—42no—
345C2×C4⋊C4—42no—
346C2×C4⋊C4D4×C6114yes—
347C2×C42—22no—
348C2×C42—22no—
349C2×C42Q8×C6114yescomplemented
350#95—22no—
351#95—24no—
352#95—24no—
353#95—24no—
354#95—24no—
355#95—24no—
356#95—24no—
357#95—41no—
358#95—42no—
359#95D4×C6118yes—
360#95Q8×C6118yescomplemented
361C24C24×C6111yescharacteristic
362C2×Q8—44no—
363C2×Q8—48no—
364C2×Q8—44no—
365C2×Q8—48no—
366C2×Q8—44no—
367C22×C4—22no—
368C22×C4—22no—
369C22×C4—21no—
370C22×C4—21no—
371C22×C4—22no—
372C22×C4—21no—
373C22×C4—21no—
374C22×C4—22no—
375C22×C4—22no—
376C22×C4—22no—
377C22×C4—22no—
378C22×C4—22no—
379C22×C4—22no—
380C22×C4—22no—
381C22×C4—22no—
382C22×C4—22no—
383C22×C4—22no—
384C22×C4—22no—
385C22×C4—21no—
386C22×C4—22no—
387C22×C4—22no—
388C22×C4—22no—
389C22×C4—21no—
390C22×C4—22no—
391C22×C4—22no—
392C22×C4—21no—
393C22×C4—21no—
394C22×C4—22no—
395C22×C4—22no—
396C22×C4—22no—
397C22×C4—22no—
398C22×C4—22no—
399C22×C4—22no—
400C22×C4—22no—
401C22×C4—22no—
402C22×C4—22no—
403C22×C4—22no—
404C22×C4—22no—
405C22×C4—22no—
406C22×C4—22no—
407C22×C4—22no—
408C22×C4—22no—
409C22×C4—22no—
410C22×C4—22no—
411C22×C4—22no—
412C22×C4—22no—
413C22×C4—22no—
414C22×C4C2×D4×C6114yes—
415C22×C4C2×D4×C6112yes—
416C22×C4C2×D4×C6114yes—
417C22×C4C2×D4×C6112yes—
418C22×C4C2×D4×C6112yes—
419C22×C4C2×D4×C6114yes—
420C22×C4C2×Q8×C6114yes—
421C22×C4C2×Q8×C6114yes—
422C22×C4C2×Q8×C6112yes—
423C22×C4C2×Q8×C6114yes—
424C22×C4C2×Q8×C6114yes—
425C22×C4#41×C6112yes—
426C4⋊C4—44no—
427C4⋊C4—48no—
428C4⋊C4—48no—
429C4⋊C4—44no—
430C4⋊C4—48no—
431C4⋊C4—48no—
432C4⋊C4—44no—
433C4⋊C4—48no—
434C42—28no—
435C23C22×D4×C6112yes—
436C23C22×D4×C6111yescharacteristic
437C23C22×Q8×C6111yescharacteristic
438C23C22×Q8×C6112yes—
439C23C2×#41×C6111yescharacteristic
440C23H(5,𝔽2)×C6111yescharacteristic
441C23H(5,𝔽2)×C6111yescharacteristic
442C23H(5,𝔽2)×C6112yes—
443C232−1+4×C6111yescharacteristic
444C232−1+4×C6111yescharacteristic
445C232−1+4×C6112yes—
446Q8—416no—
447C2×C4—22no—
448C2×C4—22no—
449C2×C4—22no—
450C2×C4—22no—
451C2×C4—22no—
452C2×C4—22no—
453C2×C4—24no—
454C2×C4—24no—
455C2×C4—24no—
456C2×C4—24no—
457C2×C4—24no—
458C2×C4—24no—
459C2×C4—22no—
460C2×C4—22no—
461C2×C4—24no—
462C2×C4—24no—
463C2×C4—24no—
464C2×C4—24no—
465C2×C4—24no—
466C2×C4—24no—
467C2×C4—24no—
468C2×C4—24no—
469C2×C4—24no—
470C2×C4—24no—
471C2×C4—24no—
472C2×C4—24no—
473C2×C4—24no—
474C2×C4—24no—
475C2×C4—42no—
476C2×C4—42no—
477C2×C4—42no—
478C2×C4—42no—
479C2×C4—42no—
480C2×C4—42no—
481C2×C4—42no—
482C2×C4—42no—
483C2×C4—42no—
484C2×C4—42no—
485C2×C4—41no—
486C2×C4—42no—
487C2×C4—41no—
488C2×C4—42no—
489C2×C4—42no—
490C2×C4—42no—
491C2×C4—41no—
492C2×C4—41no—
493C2×C4—42no—
494C2×C4—42no—
495C2×C4—42no—
496C2×C4—42no—
497C2×C4—42no—
498C2×C4—42no—
499C2×C4—42no—
500C2×C4—42no—
501C2×C4—42no—
502C2×C4—42no—
503C2×C4—42no—
504C2×C4C22⋊Q8×C6114yes—
505C2×C4C4⋊Q8×C6118yes—
506C22C2×C22⋊Q8×C6111yescharacteristic
507C22C2×C4⋊Q8×C6112yes—
508C22C61×#53411yescharacteristic
509C22C61×#53512yes—
510C22C61×#53611yescharacteristic
511C22C61×#53612yes—
512C22C61×#53911yescharacteristic
513C22C61×#53911yescharacteristic
514C22C61×#54412yes—
515C22C61×#54411yescharacteristic
516C22C61×#54412yes—
517C22C61×#54411yescharacteristic
518C22C61×#54412yes—
519C22D42×C6111yescharacteristic
520C22D42×C6112yes—
521C22C61×#54712yes—
522C22D4×Q8×C6112yes—
523C22D4×Q8×C6112yes—
524C22C61×#55412yes—
525C22Q82×C6111yescharacteristic
526C22C61×#56011yescharacteristic
527C22C61×#56312yes—
528C22C61×#56311yescharacteristic
529C4—24no—
530C4—28no—
531C4—42no—
532C4—42no—
533C4—44no—
534C4—44no—
535C4—44no—
536C4—44no—
537C4—82no—
538C2C61×#244012yes—
539C2C61×#255111yescharacteristic
540C2C61×#262711yescharacteristic
541C2C61×#266012yes—
542C2C61×#266911yescharacteristic
543C2C61×#267712yes—
544C2C61×#268011yescharacteristic
545C2C61×#272012yes—
546C2C61×#273411yescharacteristic
547C2C61×#279211yescharacteristic
548C2C61×#284311yescharacteristic
subgroupG/H
center, socle, prime radicalC24×C61C24
commutator, Frattini, cosocleC24C24×C61
LCSG ▷ C24 ▷ C1
UCSC1 ◁ C24×C61 ◁ G

Complements

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

NH ≅ G/Ntype
74. C2×C42×C61Q8semidirect
85. #95×C61Q8semidirect
231. #16522C61direct (N × H)
309. C61#16522direct (N × H)
349. C2×C42Q8×C61semidirect
360. #95Q8×C61semidirect

Maximal subgroups

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

subgroupindex
C2×C61×#3982
C2×C61×#3952
C2×C61×#3842
C61×#14442
C61×#14422
C61×#14412
C61×#14342
#1652261