Finite Group Databasebeta

← C61×#61376 #29564809

Subgroups — C61×#61376 #29564809

Order 15616.

1706 conjugacy classes of subgroups.

The subgroup lattice has 426 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C61×#3504C212yesmaximal, complemented
2C61×#3388C218yesmaximal, complemented
3C61×#3383C214yesmaximal, complemented
4C61×#3380C214yesmaximal, complemented
5C61×#3164C212yesmaximal, complemented
6C2×C61×#499C211yesmaximal, characteristic, complemented
7C61×#3084C214yesmaximal, complemented
8C61×#3080C212yesmaximal, complemented
9C61×#2960C211yesmaximal, characteristic, complemented
10C61×#2951C212yesmaximal
11C4×#135×C61C211yesmaximal, characteristic
12C2×#135×C61C2211yescharacteristic
13C61×#557C2218yes—
14C61×#554—28no—
15C61×#554C2218yes—
16C61×#544—24no—
17C61×#544C2214yes—
18C61×#533—22no—
19C61×#533C2211yescharacteristic
20C2×#125×C61—22no—
21C2×#125×C61C2212yes—
22C61×#520C2214yes—
23C4×#41×C61C2212yes—
24C61×#501C2218yes—
25C61×#500C2214yes—
26C61×#499C2214yes—
27C61×#499C2214yes—
28C61×#499C2214yes—
29C61×#479C2218yes—
30C61×#479C2218yes—
31C61×#478C2218yes—
32C61×#478C2218yes—
33C61×#476C2218yes—
34C61×#475C2218yes—
35C61×#446C2214yes—
36C8×Q8×C61C2214yes—
37C61×#440C2218yes—
38C4×Q16×C61C2212yes—
39C4×Q16×C61C2214yes—
40C4×SD16×C61C2214yes—
41C4×D8×C61C2212yes—
42C4×D8×C61C2214yes—
43C61×#428C2214yes—
44C61×#427C2211yescharacteristic
45C2×#107×C61C2212yes—
46C2×#106×C61C2212yes—
47C61×#424C2212yes—
48C61×#423C2211yescharacteristic
49C61×#417C2214yes—
50C61×#416C2212yes—
51C61×#405C2212yes—
52C2×C4×C8×C61C2211yescharacteristic
53C2×#41×C61C2312yes—
54#135×C61Q818yescomplemented
55C2×Q16×C61C2311yescharacteristic
56C2×SD16×C61C2312yes—
57C2×D8×C61C2311yescharacteristic
58C2×M16×C61C2312yes—
59C22×C8×C61C2311yescharacteristic
60C22×C8×C61C2311yescharacteristic
61C4⋊Q8×C61—28no—
62C4⋊Q8×C61—24no—
63C4⋊Q8×C61D412yes—
64C4⋊Q8×C61C2318yes—
65#125×C61—24no—
66#125×C61—28no—
67#125×C61—28no—
68#125×C61—22no—
69#125×C61—22no—
70#125×C61D411yescharacteristic
71#125×C61D411yescharacteristic
72#125×C61C2318yes—
73#125×C61C2318yes—
74C22⋊Q8×C61—28no—
75C22⋊Q8×C61—28no—
76C22⋊Q8×C61—28no—
77C22⋊Q8×C61—28no—
78C22⋊Q8×C61D414yes—
79C4×Q8×C61—28no—
80C4×Q8×C61C2314yes—
81C4×Q8×C61C2318yes—
82C4×Q8×C61C2314yes—
83C4×D4×C61—24no—
84C4×D4×C61—28no—
85C4×D4×C61C2314yes—
86C4×D4×C61C2318yes—
87#117×C61—24no—
88#117×C61—22no—
89#117×C61C2312yes—
90#117×C61C2312yes—
91C2×C4⋊C4×C61—24no—
92C2×C4⋊C4×C61—24no—
93C2×C4⋊C4×C61—28no—
94C2×C4⋊C4×C61C2314yes—
95C2×C42×C61—22no—
96C2×C42×C61C2311yescharacteristic
97#107×C61C2314yes—
98#107×C61C2314yes—
99#107×C61C2312yes—
100#107×C61C2318yes—
101#106×C61C2318yes—
102#106×C61C2312yes—
103#106×C61C2318yes—
104C4⋊C8×C61C2314yes—
105C4⋊C8×C61C2314yes—
106C4⋊C8×C61C2314yes—
107#103×C61C2314yes—
108#103×C61C2318yes—
109#102×C61C2314yes—
110#102×C61C2318yes—
111#97×C61C2314yes—
112C4×C8×C61C2312yes—
113C4×C8×C61C2312yes—
114C4×C8×C61C2314yes—
115#41×C61—24no—
116#41×C61C2×Q818yes—
117C2×Q8×C61—28no—
118C2×Q8×C61C2×D414yes—
119C2×Q8×C61C2412yes—
120C2×D4×C61—22no—
121C2×D4×C61C2412yes—
122C22×C4×C61—22no—
123C22×C4×C61—24no—
124C22×C4×C61—24no—
125C22×C4×C61—28no—
126C22×C4×C61C2×D412yes—
127C22×C4×C61C2411yescharacteristic
128Q16×C61C2×Q814yes—
129SD16×C61C2×Q818yes—
130D8×C61C2×Q814yes—
131M16×C61—24no—
132C2×C8×C61—22no—
133C2×C8×C61C2×Q814yes—
134C2×C8×C61C2412yes—
135C2×C8×C61C2414yes—
136C2×C8×C61C2411yescharacteristic
137C2×C8×C61C2411yescharacteristic
138C4⋊C4×C61—28no—
139C4⋊C4×C61—24no—
140C4⋊C4×C61—28no—
141C4⋊C4×C61—28no—
142C4⋊C4×C61—28no—
143C4⋊C4×C61—28no—
144C4⋊C4×C61—24no—
145C4⋊C4×C61—28no—
146C4⋊C4×C61C2×D412yes—
147C4⋊C4×C61C2×D414yes—
148C4⋊C4×C61C2×D412yes—
149C4⋊C4×C61C2×D414yes—
150C4⋊C4×C61C2×D414yes—
151C4⋊C4×C61C2418yes—
152C4⋊C4×C61C2414yes—
153C22⋊C4×C61—22no—
154C22⋊C4×C61—24no—
155C22⋊C4×C61—28no—
156C22⋊C4×C61C2×D414yes—
157C42×C61—22no—
158C42×C61—24no—
159C42×C61—28no—
160C42×C61C2×D411yescharacteristic
161C42×C61C2×D411yescharacteristic
162C42×C61C2412yes—
163C42×C61C2414yes—
164C23×C61—22no—
165C23×C61C22×D411yescharacteristic
166Q8×C61—22no—
167Q8×C61—44no—
168Q8×C61—88no—
169Q8×C61C22×Q814yes—
170D4×C61—28no—
171D4×C61—22no—
172D4×C61C22×Q814yes—
173C2×C4×C61—22no—
174C2×C4×C61—28no—
175C2×C4×C61—24no—
176C2×C4×C61—28no—
177C2×C4×C61—42no—
178C2×C4×C61—84no—
179C2×C4×C61—88no—
180C2×C4×C61—84no—
181C2×C4×C61C22×D412yes—
182C2×C4×C61C22×D414yes—
183C2×C4×C61C22×D412yes—
184C2×C4×C61C22×D414yes—
185C2×C4×C61C22×D411yescharacteristic
186C2×C4×C61C22×Q812yes—
187C2×C4×C612−1+412yes—
188C2×C4×C61C2511yescharacteristic
189C8×C61—22no—
190C8×C61—24no—
191C8×C61C22×Q812yes—
192C8×C61C22×Q812yes—
193#61376C6111yesmaximal, characteristic, direct product factor, Sylow
194C22×C61—22no—
195C22×C61—24no—
196C22×C61—44no—
197C22×C61D4×Q812yes—
198C22×C61C23×D411yescharacteristic
199C4×C61—22no—
200C4×C61—22no—
201C4×C61—24no—
202C4×C61—24no—
203C4×C61—24no—
204C4×C61—48no—
205C4×C61—44no—
206C4×C61D4×Q812yes—
207C4×C61C23×Q811yescharacteristic
208C4×C61C2×2−1+411yescharacteristic
209#3504C2×C6112yescomplemented
210#3388C2×C6118yescomplemented
211#3383C2×C6114yescomplemented
212#3380C2×C6114yescomplemented
213#3164C2×C6112yescomplemented
214C2×#499C2×C6111yescharacteristic, complemented
215#3084C2×C6114yescomplemented
216#3080C2×C6112yescomplemented
217#2960C2×C6111yescharacteristic, complemented
218#2951C2×C6112yes—
219C4×#135C2×C6111yescharacteristic
220C2×C61—22no—
221C2×C61—44no—
222C2×C61C2×D4×Q811yescharacteristic
223C2×C61#358412yes—
224C2×#135C22×C6111yescharacteristic
225#557C22×C6118yes—
226#554—28no—
227#554C22×C6118yes—
228#544—24no—
229#544C22×C6114yes—
230#533—22no—
231#533C22×C6111yescharacteristic
232C2×#125—22no—
233C2×#125C22×C6112yes—
234#520C22×C6114yes—
235C4×#41C22×C6112yes—
236#501C22×C6118yes—
237#500C22×C6114yes—
238#499C22×C6114yes—
239#499C22×C6114yes—
240#499C22×C6114yes—
241#479C22×C6118yes—
242#479C22×C6118yes—
243#478C22×C6118yes—
244#478C22×C6118yes—
245#476C22×C6118yes—
246#475C22×C6118yes—
247#446C22×C6114yes—
248C8×Q8C22×C6114yes—
249#440C22×C6118yes—
250C4×Q16C22×C6112yes—
251C4×Q16C22×C6114yes—
252C4×SD16C22×C6114yes—
253C4×D8C22×C6112yes—
254C4×D8C22×C6114yes—
255#428C22×C6114yes—
256#427C22×C6111yescharacteristic
257C2×#107C22×C6112yes—
258C2×#106C22×C6112yes—
259#424C22×C6112yes—
260#423C22×C6111yescharacteristic
261#417C22×C6114yes—
262#416C22×C6112yes—
263#405C22×C6112yes—
264C2×C4×C8C22×C6111yescharacteristic
265C61#6137611yescharacteristic, direct product factor, Sylow
266C2×#41C23×C6112yes—
267#135Q8×C6118yescomplemented
268C2×Q16C23×C6111yescharacteristic
269C2×SD16C23×C6112yes—
270C2×D8C23×C6111yescharacteristic
271C2×M16C23×C6112yes—
272C22×C8C23×C6111yescharacteristic
273C22×C8C23×C6111yescharacteristic
274C4⋊Q8—28no—
275C4⋊Q8—24no—
276C4⋊Q8D4×C6112yes—
277C4⋊Q8C23×C6118yes—
278#125—24no—
279#125—28no—
280#125—28no—
281#125—22no—
282#125—22no—
283#125D4×C6111yescharacteristic
284#125D4×C6111yescharacteristic
285#125C23×C6118yes—
286#125C23×C6118yes—
287C22⋊Q8—28no—
288C22⋊Q8—28no—
289C22⋊Q8—28no—
290C22⋊Q8—28no—
291C22⋊Q8D4×C6114yes—
292C4×Q8—28no—
293C4×Q8C23×C6114yes—
294C4×Q8C23×C6118yes—
295C4×Q8C23×C6114yes—
296C4×D4—24no—
297C4×D4—28no—
298C4×D4C23×C6114yes—
299C4×D4C23×C6118yes—
300#117—24no—
301#117—22no—
302#117C23×C6112yes—
303#117C23×C6112yes—
304C2×C4⋊C4—24no—
305C2×C4⋊C4—24no—
306C2×C4⋊C4—28no—
307C2×C4⋊C4C23×C6114yes—
308C2×C42—22no—
309C2×C42C23×C6111yescharacteristic
310#107C23×C6114yes—
311#107C23×C6114yes—
312#107C23×C6112yes—
313#107C23×C6118yes—
314#106C23×C6118yes—
315#106C23×C6112yes—
316#106C23×C6118yes—
317C4⋊C8C23×C6114yes—
318C4⋊C8C23×C6114yes—
319C4⋊C8C23×C6114yes—
320#103C23×C6114yes—
321#103C23×C6118yes—
322#102C23×C6114yes—
323#102C23×C6118yes—
324#97C23×C6114yes—
325C4×C8C23×C6112yes—
326C4×C8C23×C6112yes—
327C4×C8C23×C6114yes—
328#41—24no—
329#41C2×Q8×C6118yes—
330C2×Q8—28no—
331C2×Q8C2×D4×C6114yes—
332C2×Q8C24×C6112yes—
333C2×D4—22no—
334C2×D4C24×C6112yes—
335C22×C4—22no—
336C22×C4—24no—
337C22×C4—24no—
338C22×C4—28no—
339C22×C4C2×D4×C6112yes—
340C22×C4C24×C6111yescharacteristic
341Q16C2×Q8×C6114yes—
342SD16C2×Q8×C6118yes—
343D8C2×Q8×C6114yes—
344M16—24no—
345C2×C8—22no—
346C2×C8C2×Q8×C6114yes—
347C2×C8C24×C6112yes—
348C2×C8C24×C6114yes—
349C2×C8C24×C6111yescharacteristic
350C2×C8C24×C6111yescharacteristic
351C4⋊C4—28no—
352C4⋊C4—24no—
353C4⋊C4—28no—
354C4⋊C4—28no—
355C4⋊C4—28no—
356C4⋊C4—28no—
357C4⋊C4—24no—
358C4⋊C4—28no—
359C4⋊C4C2×D4×C6112yes—
360C4⋊C4C2×D4×C6114yes—
361C4⋊C4C2×D4×C6112yes—
362C4⋊C4C2×D4×C6114yes—
363C4⋊C4C2×D4×C6114yes—
364C4⋊C4C24×C6118yes—
365C4⋊C4C24×C6114yes—
366C22⋊C4—22no—
367C22⋊C4—24no—
368C22⋊C4—28no—
369C22⋊C4C2×D4×C6114yes—
370C42—22no—
371C42—24no—
372C42—28no—
373C42C2×D4×C6111yescharacteristic
374C42C2×D4×C6111yescharacteristic
375C42C24×C6112yes—
376C42C24×C6114yes—
377C23—22no—
378C23C22×D4×C6111yescharacteristic
379Q8—22no—
380Q8—44no—
381Q8—88no—
382Q8C22×Q8×C6114yes—
383D4—28no—
384D4—22no—
385D4C22×Q8×C6114yes—
386C2×C4—22no—
387C2×C4—28no—
388C2×C4—24no—
389C2×C4—28no—
390C2×C4—42no—
391C2×C4—84no—
392C2×C4—88no—
393C2×C4—84no—
394C2×C4C22×D4×C6112yes—
395C2×C4C22×D4×C6114yes—
396C2×C4C22×D4×C6112yes—
397C2×C4C22×D4×C6114yes—
398C2×C4C22×D4×C6111yescharacteristic
399C2×C4C22×Q8×C6112yes—
400C2×C42−1+4×C6112yes—
401C2×C4C25×C6111yescharacteristic
402C8—22no—
403C8—24no—
404C8C22×Q8×C6112yes—
405C8C22×Q8×C6112yes—
406C22—22no—
407C22—24no—
408C22—44no—
409C22D4×Q8×C6112yes—
410C22C23×D4×C6111yescharacteristic
411C4—22no—
412C4—22no—
413C4—24no—
414C4—24no—
415C4—24no—
416C4—48no—
417C4—44no—
418C4D4×Q8×C6112yes—
419C4C23×Q8×C6111yescharacteristic
420C4C2×2−1+4×C6111yescharacteristic
421C2—22no—
422C2—44no—
423C2C2×D4×Q8×C6111yescharacteristic
424C2C61×#358412yes—
subgroupG/H
center, socleC22×C61C23×D4
commutator, Frattini, cosocleC2×C4C25×C61
prime radicalC2×#135×C61C22
LCSG ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22×C61 ◁ C61×#533 ◁ G

Complements

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

NH ≅ G/Ntype
1. C61×#3504C2semidirect
2. C61×#3388C2semidirect
3. C61×#3383C2semidirect
4. C61×#3380C2semidirect
5. C61×#3164C2semidirect
6. C2×C61×#499C2semidirect
7. C61×#3084C2semidirect
8. C61×#3080C2semidirect
9. C61×#2960C2semidirect
54. #135×C61Q8semidirect
193. #61376C61direct (N × H)
209. #3504C2×C61semidirect
210. #3388C2×C61semidirect
211. #3383C2×C61semidirect
212. #3380C2×C61semidirect
213. #3164C2×C61semidirect
214. C2×#499C2×C61semidirect
215. #3084C2×C61semidirect
216. #3080C2×C61semidirect
217. #2960C2×C61semidirect
265. C61#61376direct (N × H)
267. #135Q8×C61semidirect

Maximal subgroups

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

subgroupindex
C61×#35042
C61×#33882
C61×#33832
C61×#33802
C61×#31642
C2×C61×#4992
C61×#30842
C61×#30802
C61×#29602
C61×#29512
C4×#135×C612
#6137661