Finite Group Databasebeta

← C61×#60208 #29563641

Subgroups — C61×#60208 #29563641

Order 15616.

1950 conjugacy classes of subgroups.

The subgroup lattice has 468 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C61×#3444C211yesmaximal, characteristic
2C61×#2971C214yesmaximal, complemented
3C61×#2970C214yesmaximal, complemented
4C61×#2969C218yesmaximal, complemented
5C61×#2960C212yesmaximal, complemented
6C61×#2931C212yesmaximal, complemented
7C61×#2930C214yesmaximal, complemented
8C61×#2923C212yesmaximal, complemented
9C61×#2920C211yesmaximal, characteristic, complemented
10C61×#2875C212yesmaximal, complemented
11C2×C4×M16×C61C211yesmaximal, characteristic, complemented
12C2×#131×C61C2214yes—
13C22×M16×C61C2212yes—
14C61×#533C412yes—
15C61×#533C2211yescharacteristic
16C61×#532C412yes—
17C61×#532C2211yescharacteristic
18C61×#529C414yes—
19C61×#529C414yes—
20C61×#529C2214yes—
21C61×#525C414yes—
22C61×#525C2212yes—
23C4×#41×C61C2214yes—
24C4×#41×C61C2212yes—
25C22×C42×C61C2211yescharacteristic
26C61×#446C2218yescomplemented
27C8×Q8×C61C2218yescomplemented
28C61×#436C22116yes—
29C61×#436C2218yescomplemented
30C61×#435C22116yes—
31C8×D4×C61C2218yescomplemented
32C61×#433C2218yescomplemented
33C61×#432C2214yes—
34C61×#432C2218yescomplemented
35C61×#431C2214yes—
36C61×#424C2214yes—
37C61×#423C2212yes—
38C61×#408C2214yes—
39C61×#407C2212yes—
40C61×#405C2214yes—
41C61×#405C2218yescomplemented
42C4×M16×C61C2214yes—
43C4×M16×C61C2212yes—
44C4×M16×C61C2212yes—
45C4×M16×C61C2212yes—
46C4×M16×C61C2214yescomplemented
47C2×#97×C61C2211yescharacteristic
48C2×#97×C61C2211yescharacteristic
49C2×#97×C61C2214yescomplemented
50C2×C4×C8×C61C2212yes—
51C2×#41×C61C2×C414yes—
52C2×#41×C61C2312yes—
53C23×C4×C61C2×C412yes—
54C23×C4×C61C2311yescharacteristic
55#131×C61—216no—
56C2×M16×C61—28no—
57C2×M16×C61C2314yes—
58C2×M16×C61C2314yes—
59C2×M16×C61C2314yes—
60C2×M16×C61C2318yes—
61C22×C8×C61C2312yes—
62C22×C8×C61C2318yes—
63C22×C8×C61C2312yes—
64C4⋊Q8×C61C2×C412yes—
65C4⋊Q8×C61C2×C411yescharacteristic
66#127×C61C2×C411yescharacteristic
67#126×C61C2×C418yes—
68#125×C61C2×C414yes—
69#125×C61C2×C412yes—
70#124×C61C2×C414yes—
71#124×C61C2×C412yes—
72#123×C61C2×C418yes—
73#123×C61C2×C414yes—
74C22⋊Q8×C61C2×C414yes—
75C22⋊Q8×C61C2×C414yes—
76C22⋊Q8×C61C2×C414yes—
77#121×C61C2×C414yes—
78#121×C61C2×C414yes—
79#121×C61C2×C414yes—
80#120×C61C2×C414yes—
81C4×Q8×C61C2×C414yes—
82C4×Q8×C61C2×C414yes—
83C4×Q8×C61C2314yes—
84C4×D4×C61C2×C414yes—
85C4×D4×C61C2×C414yes—
86C4×D4×C61C2×C418yes—
87C4×D4×C61C2×C418yes—
88C4×D4×C61C2314yes—
89C4×D4×C61C2318yes—
90#117×C61C2×C414yes—
91#117×C61C2×C418yes—
92#117×C61C2312yes—
93#117×C61C2314yes—
94C2×C42×C61C2×C412yes—
95C2×C42×C61C2×C412yes—
96C2×C42×C61C2×C412yes—
97C2×C42×C61C2311yescharacteristic
98C2×C42×C61C2311yescharacteristic
99C2×C42×C61C2314yes—
100C2×C42×C61C2312yes—
101C2×C42×C61C2312yes—
102C4⋊C8×C61C2318yes—
103C4⋊C8×C61C23116yes—
104C22⋊C8×C61C2318yes—
105C22⋊C8×C61C23116yes—
106#97×C61C2311yescharacteristic
107#97×C61C2311yescharacteristic
108#97×C61C2312yes—
109#97×C61C2312yes—
110#97×C61C2312yes—
111#97×C61C2314yes—
112#97×C61C2318yes—
113C4×C8×C61C2314yes—
114C4×C8×C61C2314yes—
115C4×C8×C61C2314yes—
116C4×C8×C61C2318yes—
117C24×C61C22×C411yescharacteristic
118#41×C61—28no—
119#41×C61—48no—
120C2×Q8×C61C22×C412yes—
121C2×Q8×C61C22×C414yes—
122C2×D4×C61C22×C414yes—
123C2×D4×C61C22×C412yes—
124C2×D4×C61C22×C414yes—
125C2×D4×C61C22×C418yes—
126C22×C4×C61—24no—
127C22×C4×C61—44no—
128C22×C4×C61C22×C412yes—
129C22×C4×C61C22×C411yescharacteristic
130C22×C4×C61C22×C412yes—
131C22×C4×C61C22×C412yes—
132C22×C4×C61C22×C412yes—
133C22×C4×C61C22×C412yes—
134C22×C4×C61C22×C412yes—
135C22×C4×C61C22×C412yes—
136C22×C4×C61C22×C418yes—
137C22×C4×C61C2412yes—
138C22×C4×C61C2414yes—
139C22×C4×C61C2411yescharacteristic
140M16×C61—28no—
141M16×C61—28no—
142M16×C61—28no—
143M16×C61—216no—
144C2×C8×C61—24no—
145C2×C8×C61—216no—
146C2×C8×C61#4118yescomplemented
147C2×C8×C61C2414yes—
148C2×C8×C61C2418yes—
149C2×C8×C61C2412yes—
150C2×C8×C61C2412yes—
151C4⋊C4×C61C22×C414yes—
152C4⋊C4×C61C22×C418yes—
153C4⋊C4×C61C22×C414yes—
154C4⋊C4×C61C22×C418yes—
155C4⋊C4×C61C22×C414yes—
156C4⋊C4×C61C22×C418yes—
157C22⋊C4×C61C22×C418yes—
158C22⋊C4×C61C22×C418yes—
159C22⋊C4×C61C22×C418yes—
160C22⋊C4×C61C22×C414yes—
161C22⋊C4×C61C22×C418yes—
162C42×C61C22×C412yes—
163C42×C61C22×C412yes—
164C42×C61C22×C412yes—
165C42×C61C22×C412yes—
166C42×C61C22×C411yescharacteristic
167C42×C61C22×C412yes—
168C42×C61C22×C411yescharacteristic
169C42×C61C22×C418yes—
170C42×C61C2411yescharacteristic
171C42×C61C2412yes—
172C42×C61C2414yes—
173C42×C61C2411yescharacteristic
174C23×C61—22no—
175C23×C61—42no—
176C23×C61C23×C414yes—
177C23×C61C23×C412yes—
178C23×C61C23×C411yescharacteristic
179Q8×C61—24no—
180Q8×C61—28no—
181D4×C61—28no—
182D4×C61—24no—
183D4×C61—28no—
184D4×C61—48no—
185C2×C4×C61—22no—
186C2×C4×C61—24no—
187C2×C4×C61—24no—
188C2×C4×C61—24no—
189C2×C4×C61—24no—
190C2×C4×C61—28no—
191C2×C4×C61—24no—
192C2×C4×C61—42no—
193C2×C4×C61—42no—
194C2×C4×C61—42no—
195C2×C4×C61—48no—
196C2×C4×C61C23×C412yes—
197C2×C4×C61C23×C412yes—
198C2×C4×C61C23×C412yes—
199C2×C4×C61C23×C412yes—
200C2×C4×C61C23×C414yes—
201C2×C4×C61C23×C418yes—
202C2×C4×C61C23×C411yescharacteristic
203C2×C4×C61C23×C412yes—
204C2×C4×C61C2×#4114yes—
205C2×C4×C61C2511yescharacteristic
206C8×C61—24no—
207C8×C61—28no—
208C8×C61C2×#4114yes—
209C8×C61C2×#4114yes—
210#60208C6111yesmaximal, characteristic, direct product factor, Sylow
211C22×C61—22no—
212C22×C61—22no—
213C22×C61—24no—
214C22×C61—22no—
215C22×C61—22no—
216C22×C61—28no—
217C22×C61—44no—
218C22×C61C4×#4112yes—
219C22×C61C24×C411yescharacteristic
220C4×C61—22no—
221C4×C61—24no—
222C4×C61—22no—
223C4×C61—22no—
224C4×C61—22no—
225C4×C61—28no—
226C4×C61C4×#4112yes—
227C4×C61#56814yes—
228C4×C61C22×#4112yes—
229#3444C2×C6111yescharacteristic
230#2971C2×C6114yescomplemented
231#2970C2×C6114yescomplemented
232#2969C2×C6118yescomplemented
233#2960C2×C6112yescomplemented
234#2931C2×C6112yescomplemented
235#2930C2×C6114yescomplemented
236#2923C2×C6112yescomplemented
237#2920C2×C6111yescharacteristic, complemented
238#2875C2×C6112yescomplemented
239C2×C4×M16C2×C6111yescharacteristic, complemented
240C2×C61—22no—
241C2×C61—44no—
242C2×C61—42no—
243C2×C61C2×C4×#4111yescharacteristic
244C2×C61C2×#56812yes—
245C2×#131C22×C6114yes—
246C22×M16C22×C6112yes—
247#533C4×C6112yes—
248#533C22×C6111yescharacteristic
249#532C4×C6112yes—
250#532C22×C6111yescharacteristic
251#529C4×C6114yes—
252#529C4×C6114yes—
253#529C22×C6114yes—
254#525C4×C6114yes—
255#525C22×C6112yes—
256C4×#41C22×C6114yes—
257C4×#41C22×C6112yes—
258C22×C42C22×C6111yescharacteristic
259#446C22×C6118yescomplemented
260C8×Q8C22×C6118yescomplemented
261#436C22×C61116yes—
262#436C22×C6118yescomplemented
263#435C22×C61116yes—
264C8×D4C22×C6118yescomplemented
265#433C22×C6118yescomplemented
266#432C22×C6114yes—
267#432C22×C6118yescomplemented
268#431C22×C6114yes—
269#424C22×C6114yes—
270#423C22×C6112yes—
271#408C22×C6114yes—
272#407C22×C6112yes—
273#405C22×C6114yes—
274#405C22×C6118yescomplemented
275C4×M16C22×C6114yes—
276C4×M16C22×C6112yes—
277C4×M16C22×C6112yes—
278C4×M16C22×C6112yes—
279C4×M16C22×C6114yescomplemented
280C2×#97C22×C6111yescharacteristic
281C2×#97C22×C6111yescharacteristic
282C2×#97C22×C6114yescomplemented
283C2×C4×C8C22×C6112yes—
284C61#6020811yescharacteristic, direct product factor, Sylow
285C2×#41C2×C4×C6114yes—
286C2×#41C23×C6112yes—
287C23×C4C2×C4×C6112yes—
288C23×C4C23×C6111yescharacteristic
289#131—216no—
290C2×M16—28no—
291C2×M16C23×C6114yes—
292C2×M16C23×C6114yes—
293C2×M16C23×C6114yes—
294C2×M16C23×C6118yes—
295C22×C8C23×C6112yes—
296C22×C8C23×C6118yes—
297C22×C8C23×C6112yes—
298C4⋊Q8C2×C4×C6112yes—
299C4⋊Q8C2×C4×C6111yescharacteristic
300#127C2×C4×C6111yescharacteristic
301#126C2×C4×C6118yes—
302#125C2×C4×C6112yes—
303#125C2×C4×C6114yes—
304#124C2×C4×C6112yes—
305#124C2×C4×C6114yes—
306#123C2×C4×C6118yes—
307#123C2×C4×C6114yes—
308C22⋊Q8C2×C4×C6114yes—
309C22⋊Q8C2×C4×C6114yes—
310C22⋊Q8C2×C4×C6114yes—
311#121C2×C4×C6114yes—
312#121C2×C4×C6114yes—
313#121C2×C4×C6114yes—
314#120C2×C4×C6114yes—
315C4×Q8C2×C4×C6114yes—
316C4×Q8C2×C4×C6114yes—
317C4×Q8C23×C6114yes—
318C4×D4C2×C4×C6118yes—
319C4×D4C2×C4×C6118yes—
320C4×D4C2×C4×C6114yes—
321C4×D4C2×C4×C6114yes—
322C4×D4C23×C6118yes—
323C4×D4C23×C6114yes—
324#117C2×C4×C6114yes—
325#117C2×C4×C6118yes—
326#117C23×C6112yes—
327#117C23×C6114yes—
328C2×C42C2×C4×C6112yes—
329C2×C42C2×C4×C6112yes—
330C2×C42C2×C4×C6112yes—
331C2×C42C23×C6114yes—
332C2×C42C23×C6111yescharacteristic
333C2×C42C23×C6112yes—
334C2×C42C23×C6112yes—
335C2×C42C23×C6111yescharacteristic
336C4⋊C8C23×C6118yes—
337C4⋊C8C23×C61116yes—
338C22⋊C8C23×C6118yes—
339C22⋊C8C23×C61116yes—
340#97C23×C6111yescharacteristic
341#97C23×C6111yescharacteristic
342#97C23×C6112yes—
343#97C23×C6112yes—
344#97C23×C6112yes—
345#97C23×C6114yes—
346#97C23×C6118yes—
347C4×C8C23×C6114yes—
348C4×C8C23×C6114yes—
349C4×C8C23×C6114yes—
350C4×C8C23×C6118yes—
351C24C22×C4×C6111yescharacteristic
352#41—28no—
353#41—48no—
354C2×Q8C22×C4×C6112yes—
355C2×Q8C22×C4×C6114yes—
356C2×D4C22×C4×C6114yes—
357C2×D4C22×C4×C6112yes—
358C2×D4C22×C4×C6114yes—
359C2×D4C22×C4×C6118yes—
360C22×C4—24no—
361C22×C4—44no—
362C22×C4C22×C4×C6112yes—
363C22×C4C22×C4×C6111yescharacteristic
364C22×C4C22×C4×C6112yes—
365C22×C4C22×C4×C6112yes—
366C22×C4C22×C4×C6112yes—
367C22×C4C22×C4×C6112yes—
368C22×C4C22×C4×C6112yes—
369C22×C4C22×C4×C6112yes—
370C22×C4C22×C4×C6118yes—
371C22×C4C24×C6112yes—
372C22×C4C24×C6114yes—
373C22×C4C24×C6111yescharacteristic
374M16—28no—
375M16—28no—
376M16—28no—
377M16—216no—
378C2×C8—24no—
379C2×C8—216no—
380C2×C8#41×C6118yescomplemented
381C2×C8C24×C6114yes—
382C2×C8C24×C6118yes—
383C2×C8C24×C6112yes—
384C2×C8C24×C6112yes—
385C4⋊C4C22×C4×C6114yes—
386C4⋊C4C22×C4×C6118yes—
387C4⋊C4C22×C4×C6114yes—
388C4⋊C4C22×C4×C6118yes—
389C4⋊C4C22×C4×C6114yes—
390C4⋊C4C22×C4×C6118yes—
391C22⋊C4C22×C4×C6118yes—
392C22⋊C4C22×C4×C6118yes—
393C22⋊C4C22×C4×C6118yes—
394C22⋊C4C22×C4×C6114yes—
395C22⋊C4C22×C4×C6118yes—
396C42C22×C4×C6112yes—
397C42C22×C4×C6112yes—
398C42C22×C4×C6112yes—
399C42C22×C4×C6112yes—
400C42C22×C4×C6111yescharacteristic
401C42C22×C4×C6112yes—
402C42C22×C4×C6111yescharacteristic
403C42C22×C4×C6118yes—
404C42C24×C6111yescharacteristic
405C42C24×C6112yes—
406C42C24×C6114yes—
407C42C24×C6111yescharacteristic
408C23—22no—
409C23—42no—
410C23C23×C4×C6114yes—
411C23C23×C4×C6112yes—
412C23C23×C4×C6111yescharacteristic
413Q8—28no—
414Q8—24no—
415D4—28no—
416D4—28no—
417D4—24no—
418D4—48no—
419C2×C4—22no—
420C2×C4—24no—
421C2×C4—24no—
422C2×C4—24no—
423C2×C4—28no—
424C2×C4—24no—
425C2×C4—24no—
426C2×C4—42no—
427C2×C4—42no—
428C2×C4—42no—
429C2×C4—48no—
430C2×C4C23×C4×C6112yes—
431C2×C4C23×C4×C6112yes—
432C2×C4C23×C4×C6112yes—
433C2×C4C23×C4×C6112yes—
434C2×C4C23×C4×C6114yes—
435C2×C4C23×C4×C6118yes—
436C2×C4C23×C4×C6112yes—
437C2×C4C23×C4×C6111yescharacteristic
438C2×C4C2×#41×C6114yes—
439C2×C4C25×C6111yescharacteristic
440C8—24no—
441C8—28no—
442C8C2×#41×C6114yes—
443C8C2×#41×C6114yes—
444C22—22no—
445C22—22no—
446C22—22no—
447C22—28no—
448C22—22no—
449C22—24no—
450C22—44no—
451C22C4×#41×C6112yes—
452C22C24×C4×C6111yescharacteristic
453C4—22no—
454C4—22no—
455C4—22no—
456C4—28no—
457C4—22no—
458C4—24no—
459C4C4×#41×C6112yes—
460C4C61×#56814yes—
461C4C22×#41×C6112yes—
462C2—22no—
463C2—42no—
464C2—44no—
465C2C2×C4×#41×C6111yescharacteristic
466C2C2×C61×#56812yes—
subgroupG/H
centerC2×C4×C61C25
commutatorC22C24×C4×C61
Frattini, cosocleC2×C4C25×C61
socleC22×C61C24×C4
prime radicalC61×#3444C2
LCSG ▷ C22 ▷ C1
UCSC1 ◁ C2×C4×C61 ◁ G

Complements

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

NH ≅ G/Ntype
2. C61×#2971C2semidirect
3. C61×#2970C2semidirect
4. C61×#2969C2semidirect
5. C61×#2960C2semidirect
6. C61×#2931C2semidirect
7. C61×#2930C2semidirect
8. C61×#2923C2semidirect
9. C61×#2920C2semidirect
10. C61×#2875C2semidirect
11. C2×C4×M16×C61C2semidirect
26. C61×#446C22semidirect
27. C8×Q8×C61C22semidirect
29. C61×#436C22semidirect
31. C8×D4×C61C22semidirect
32. C61×#433C22semidirect
34. C61×#432C22semidirect
41. C61×#405C22semidirect
46. C4×M16×C61C22semidirect
49. C2×#97×C61C22semidirect
146. C2×C8×C61#41semidirect
210. #60208C61direct (N × H)
230. #2971C2×C61semidirect
231. #2970C2×C61semidirect
232. #2969C2×C61semidirect
233. #2960C2×C61semidirect
234. #2931C2×C61semidirect
235. #2930C2×C61semidirect
236. #2923C2×C61semidirect
237. #2920C2×C61semidirect
238. #2875C2×C61semidirect
239. C2×C4×M16C2×C61semidirect
259. #446C22×C61semidirect
260. C8×Q8C22×C61semidirect
262. #436C22×C61semidirect
264. C8×D4C22×C61semidirect
265. #433C22×C61semidirect
267. #432C22×C61semidirect
274. #405C22×C61semidirect
279. C4×M16C22×C61semidirect
282. C2×#97C22×C61semidirect
284. C61#60208direct (N × H)
380. C2×C8#41×C61semidirect

Maximal subgroups

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

subgroupindex
C61×#34442
C61×#29712
C61×#29702
C61×#29692
C61×#29602
C61×#29312
C61×#29302
C61×#29232
C61×#29202
C61×#28752
C2×C4×M16×C612
#6020861