Finite Group Databasebeta

← C61×#60175 #29563608

Subgroups — C61×#60175 #29563608

Order 15616.

2650 conjugacy classes of subgroups.

The subgroup lattice has 422 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C61×#578C211yesmaximal, characteristic
2C4×2−1+4×C61C212yesmaximal, complemented
3C61×#2952C213yesmaximal, complemented
4C61×#2950C213yesmaximal, complemented
5C61×#2947C216yesmaximal, complemented
6C4×#137×C61C216yesmaximal, complemented
7C61×#2943C213yesmaximal, complemented
8C2×C61×#441C213yesmaximal, complemented
9C61×#2912C211yesmaximal, characteristic, complemented
10C61×#2898C212yesmaximal, complemented
11C61×#2875C211yesmaximal, characteristic, complemented
12C2×2−1+4×C61C2212yes—
13C61×#578C4116yescomplemented
14C2×#137×C61C2216yes—
15C2×#135×C61C2213yes—
16C22×Q16×C61C2213yes—
17C2×#131×C61C2211yescharacteristic
18C2×#131×C61C2211yescharacteristic
19C61×#520—26no—
20C61×#520C2216yes—
21C61×#520C2212yes—
22C61×#519—22no—
23C61×#519C2216yes—
24C4×#41×C61—26no—
25C4×#41×C61C2216yes—
26C4×#41×C61C2211yescharacteristic
27C2×C4×Q8×C61—22no—
28C2×C4×Q8×C61C2216yes—
29C61×#442C2211yescharacteristic
30C61×#442C2213yes—
31C61×#441C2213yes—
32C61×#441C2216yes—
33C61×#441C2216yes—
34C61×#441C2216yes—
35C61×#441C2213yes—
36C61×#440C2216yes—
37C61×#440C2216yes—
38C4×Q16×C61C22112yes—
39C4×SD16×C61C22112yes—
40C61×#428C2216yes—
41C61×#427C2213yes—
42C2×#106×C61C2213yes—
43C61×#419C2216yes—
44C61×#417C2216yes—
45C61×#416C2216yes—
46C2×#103×C61C2216yes—
47C61×#405C2213yes—
48C61×#405C2213yes—
49C4×M16×C61C2213yes—
50C2×#97×C61C2213yes—
512−1+4×C61—28no—
522−1+4×C61C2×C4116yescomplemented
53C2×#41×C61—26no—
54C2×#41×C61C2316yes—
55C2×#41×C61C2311yescharacteristic
56C22×Q8×C61—22no—
57C22×Q8×C61C2316yes—
58#137×C61C2×C4148yescomplemented
59#135×C61C2×C4124yescomplemented
60C2×Q16×C61C2×C4124yescomplemented
61C2×Q16×C61C2313yes—
62C2×Q16×C61C2316yes—
63C2×SD16×C61C2316yes—
64C2×D8×C61C2311yescharacteristic
65#131×C61—24no—
66#131×C61C2×C418yescomplemented
67C2×M16×C61C2313yes—
68C2×M16×C61C2313yes—
69C22×C8×C61C2313yes—
70C22×C8×C61C2313yes—
71C4×Q8×C61—22no—
72C4×Q8×C61—26no—
73C4×Q8×C61—26no—
74C4×Q8×C61—26no—
75C4×Q8×C61—26no—
76C4×Q8×C61D412yes—
77C4×Q8×C61C23112yes—
78C4×Q8×C61C2316yes—
79C4×Q8×C61C2316yes—
80C4×D4×C61—26no—
81C4×D4×C61—26no—
82C4×D4×C61—26no—
83C4×D4×C61—212no—
84C4×D4×C61D416yes—
85C4×D4×C61C2316yes—
86C4×D4×C61C2312yes—
87#117×C61—26no—
88#117×C61—212no—
89#117×C61—26no—
90#117×C61C2316yes—
91#117×C61C2313yes—
92C2×C4⋊C4×C61—26no—
93C2×C4⋊C4×C61—26no—
94C2×C4⋊C4×C61C2316yes—
95C2×C42×C61—26no—
96C2×C42×C61—26no—
97C2×C42×C61C2313yes—
98#107×C61C2316yes—
99#106×C61C2313yes—
100#106×C61C2316yes—
101#106×C61C2311yescharacteristic
102#103×C61C2316yes—
103#103×C61C23112yes—
104#103×C61C2316yes—
105#102×C61C2312yes—
106#102×C61C2316yes—
107#97×C61C2311yescharacteristic
108#97×C61C2313yes—
109#97×C61C2313yes—
110#97×C61C2313yes—
111C4×C8×C61C2316yes—
112#41×C61—248no—
113#41×C61—212no—
114#41×C61C22×C4124yes—
115#41×C61C22×C414yes—
116#41×C61#4114yes—
117C2×Q8×C61—216no—
118C2×Q8×C61—26no—
119C2×Q8×C61—212no—
120C2×Q8×C61C22×C4124yes—
121C2×Q8×C61C2×D411yescharacteristic
122C2×Q8×C61C2416yes—
123C2×D4×C61—26no—
124C2×D4×C61C2×D413yes—
125C2×D4×C61C2412yes—
126C22×C4×C61—26no—
127C22×C4×C61—26no—
128C22×C4×C61—26no—
129C22×C4×C61—22no—
130C22×C4×C61—26no—
131C22×C4×C61C2×D416yes—
132C22×C4×C61C2413yes—
133Q16×C61C22×C4112yes—
134Q16×C61C22×C4124yes—
135SD16×C61C22×C4124yes—
136D8×C61C22×C414yes—
137M16×C61—26no—
138M16×C61C22×C4112yes—
139C2×C8×C61—26no—
140C2×C8×C61C22×C4112yes—
141C2×C8×C61C2413yes—
142C2×C8×C61C2411yescharacteristic
143C2×C8×C61C2413yes—
144C2×C8×C61C2411yescharacteristic
145C4⋊C4×C61—26no—
146C4⋊C4×C61—212no—
147C4⋊C4×C61—26no—
148C4⋊C4×C61—26no—
149C4⋊C4×C61C2×D416yes—
150C4⋊C4×C61C2416yes—
151C4⋊C4×C61C2412yes—
152C22⋊C4×C61—26no—
153C22⋊C4×C61—22no—
154C22⋊C4×C61—26no—
155C22⋊C4×C61C2×D416yes—
156C42×C61—26no—
157C42×C61—26no—
158C42×C61—212no—
159C42×C61—26no—
160C42×C61C2×D416yes—
161C42×C61C2413yes—
162C42×C61C2411yescharacteristic
163C23×C61—22no—
164C23×C61C22×D413yes—
165Q8×C61—224no—
166Q8×C61—26no—
167Q8×C61C4×D414yes—
168Q8×C61C23×C4112yes—
169D4×C61—224no—
170D4×C61—22no—
171D4×C61C4×D4112yes—
172D4×C61C23×C414yes—
173C2×C4×C61—224no—
174C2×C4×C61—26no—
175C2×C4×C61—224no—
176C2×C4×C61—22no—
177C2×C4×C61—26no—
178C2×C4×C61—46no—
179C2×C4×C61—44no—
180C2×C4×C61—86no—
181C2×C4×C61—86no—
182C2×C4×C61C23×C416yes—
183C2×C4×C61C22×D412yes—
184C2×C4×C61C22×D416yes—
185C2×C4×C61C22×D413yes—
186C2×C4×C61C2×#4116yes—
187C2×C4×C61C2511yescharacteristic
188C8×C61—21no—
189C8×C61—23no—
190C8×C61C23×C412yes—
191C8×C61C23×C416yes—
192#60175C6111yesmaximal, characteristic, direct product factor, Sylow
193C22×C61—26no—
194C22×C61—24no—
195C22×C61—42no—
196C22×C61—42no—
197C22×C61C2×C4×D416yes—
198C22×C61C23×D411yescharacteristic
199C4×C61—22no—
200C4×C61—26no—
201C4×C61—212no—
202C4×C61—42no—
203C4×C61—46no—
204C4×C61C2×C4×D416yes—
205C4×C61C24×C411yescharacteristic
206C4×C61C22×#4111yescharacteristic
207C2×#578C2×C6111yescharacteristic
208C4×2−1+4C2×C6112yescomplemented
209#2952C2×C6113yescomplemented
210#2950C2×C6113yescomplemented
211#2947C2×C6116yescomplemented
212C4×#137C2×C6116yescomplemented
213#2943C2×C6113yescomplemented
214C2×#441C2×C6113yescomplemented
215#2912C2×C6111yescharacteristic, complemented
216#2898C2×C6112yescomplemented
217#2875C2×C6111yescharacteristic, complemented
218C2×C61—26no—
219C2×C61—44no—
220C2×C61C22×C4×D411yescharacteristic
221C2×C61#358612yes—
222C2×2−1+4C22×C6112yes—
223#578C4×C61116yescomplemented
224C2×#137C22×C6116yes—
225C2×#135C22×C6113yes—
226C22×Q16C22×C6113yes—
227C2×#131C22×C6111yescharacteristic
228C2×#131C22×C6111yescharacteristic
229#520—26no—
230#520C22×C6116yes—
231#520C22×C6112yes—
232#519—22no—
233#519C22×C6116yes—
234C4×#41—26no—
235C4×#41C22×C6116yes—
236C4×#41C22×C6111yescharacteristic
237C2×C4×Q8—22no—
238C2×C4×Q8C22×C6116yes—
239#442C22×C6111yescharacteristic
240#442C22×C6113yes—
241#441C22×C6113yes—
242#441C22×C6116yes—
243#441C22×C6116yes—
244#441C22×C6116yes—
245#441C22×C6113yes—
246#440C22×C6116yes—
247#440C22×C6116yes—
248C4×Q16C22×C61112yes—
249C4×SD16C22×C61112yes—
250#428C22×C6116yes—
251#427C22×C6113yes—
252C2×#106C22×C6113yes—
253#419C22×C6116yes—
254#417C22×C6116yes—
255#416C22×C6116yes—
256C2×#103C22×C6116yes—
257#405C22×C6113yes—
258#405C22×C6113yes—
259C4×M16C22×C6113yes—
260C2×#97C22×C6113yes—
261C61#6017511yescharacteristic, direct product factor, Sylow
2622−1+4—28no—
2632−1+4C2×C4×C61116yescomplemented
264C2×#41—26no—
265C2×#41C23×C6116yes—
266C2×#41C23×C6111yescharacteristic
267C22×Q8—22no—
268C22×Q8C23×C6116yes—
269#137C2×C4×C61148yescomplemented
270#135C2×C4×C61124yescomplemented
271C2×Q16C2×C4×C61124yescomplemented
272C2×Q16C23×C6113yes—
273C2×Q16C23×C6116yes—
274C2×SD16C23×C6116yes—
275C2×D8C23×C6111yescharacteristic
276#131—24no—
277#131C2×C4×C6118yescomplemented
278C2×M16C23×C6113yes—
279C2×M16C23×C6113yes—
280C22×C8C23×C6113yes—
281C22×C8C23×C6113yes—
282C4×Q8—22no—
283C4×Q8—26no—
284C4×Q8—26no—
285C4×Q8—26no—
286C4×Q8—26no—
287C4×Q8D4×C6112yes—
288C4×Q8C23×C6116yes—
289C4×Q8C23×C61112yes—
290C4×Q8C23×C6116yes—
291C4×D4—26no—
292C4×D4—26no—
293C4×D4—26no—
294C4×D4—212no—
295C4×D4D4×C6116yes—
296C4×D4C23×C6112yes—
297C4×D4C23×C6116yes—
298#117—26no—
299#117—212no—
300#117—26no—
301#117C23×C6116yes—
302#117C23×C6113yes—
303C2×C4⋊C4—26no—
304C2×C4⋊C4—26no—
305C2×C4⋊C4C23×C6116yes—
306C2×C42—26no—
307C2×C42—26no—
308C2×C42C23×C6113yes—
309#107C23×C6116yes—
310#106C23×C6113yes—
311#106C23×C6116yes—
312#106C23×C6111yescharacteristic
313#103C23×C6116yes—
314#103C23×C61112yes—
315#103C23×C6116yes—
316#102C23×C6112yes—
317#102C23×C6116yes—
318#97C23×C6111yescharacteristic
319#97C23×C6113yes—
320#97C23×C6113yes—
321#97C23×C6113yes—
322C4×C8C23×C6116yes—
323#41—248no—
324#41—212no—
325#41C22×C4×C61124yes—
326#41C22×C4×C6114yes—
327#41#41×C6114yes—
328C2×Q8—216no—
329C2×Q8—26no—
330C2×Q8—212no—
331C2×Q8C22×C4×C61124yes—
332C2×Q8C2×D4×C6111yescharacteristic
333C2×Q8C24×C6116yes—
334C2×D4—26no—
335C2×D4C2×D4×C6113yes—
336C2×D4C24×C6112yes—
337C22×C4—26no—
338C22×C4—26no—
339C22×C4—26no—
340C22×C4—22no—
341C22×C4—26no—
342C22×C4C2×D4×C6116yes—
343C22×C4C24×C6113yes—
344Q16C22×C4×C61112yes—
345Q16C22×C4×C61124yes—
346SD16C22×C4×C61124yes—
347D8C22×C4×C6114yes—
348M16—26no—
349M16C22×C4×C61112yes—
350C2×C8—26no—
351C2×C8C22×C4×C61112yes—
352C2×C8C24×C6113yes—
353C2×C8C24×C6111yescharacteristic
354C2×C8C24×C6113yes—
355C2×C8C24×C6111yescharacteristic
356C4⋊C4—26no—
357C4⋊C4—212no—
358C4⋊C4—26no—
359C4⋊C4—26no—
360C4⋊C4C2×D4×C6116yes—
361C4⋊C4C24×C6116yes—
362C4⋊C4C24×C6112yes—
363C22⋊C4—26no—
364C22⋊C4—22no—
365C22⋊C4—26no—
366C22⋊C4C2×D4×C6116yes—
367C42—26no—
368C42—26no—
369C42—212no—
370C42—26no—
371C42C2×D4×C6116yes—
372C42C24×C6113yes—
373C42C24×C6111yescharacteristic
374C23—22no—
375C23C22×D4×C6113yes—
376Q8—26no—
377Q8—224no—
378Q8C4×D4×C6114yes—
379Q8C23×C4×C61112yes—
380D4—224no—
381D4—22no—
382D4C4×D4×C61112yes—
383D4C23×C4×C6114yes—
384C2×C4—26no—
385C2×C4—224no—
386C2×C4—224no—
387C2×C4—22no—
388C2×C4—26no—
389C2×C4—46no—
390C2×C4—44no—
391C2×C4—86no—
392C2×C4—86no—
393C2×C4C23×C4×C6116yes—
394C2×C4C22×D4×C6112yes—
395C2×C4C22×D4×C6116yes—
396C2×C4C22×D4×C6113yes—
397C2×C4C2×#41×C6116yes—
398C2×C4C25×C6111yescharacteristic
399C8—21no—
400C8—23no—
401C8C23×C4×C6112yes—
402C8C23×C4×C6116yes—
403C22—26no—
404C22—24no—
405C22—42no—
406C22—42no—
407C22C2×C4×D4×C6116yes—
408C22C23×D4×C6111yescharacteristic
409C4—22no—
410C4—26no—
411C4—212no—
412C4—42no—
413C4—46no—
414C4C2×C4×D4×C6116yes—
415C4C24×C4×C6111yescharacteristic
416C4C22×#41×C6111yescharacteristic
417C2—26no—
418C2—44no—
419C2C22×C4×D4×C6111yescharacteristic
420C2C61×#358612yes—
subgroupG/H
center, socleC22×C61C23×D4
commutatorC4C24×C4×C61
Frattini, cosocleC2×C4C25×C61
prime radicalC2×C61×#578C2
LCSG ▷ C4 ▷ C2 ▷ C1
UCSC1 ◁ C22×C61 ◁ C4×#41×C61 ◁ G

Complements

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

NH ≅ G/Ntype
2. C4×2−1+4×C61C2semidirect
3. C61×#2952C2semidirect
4. C61×#2950C2semidirect
5. C61×#2947C2semidirect
6. C4×#137×C61C2semidirect
7. C61×#2943C2semidirect
8. C2×C61×#441C2semidirect
9. C61×#2912C2semidirect
10. C61×#2898C2semidirect
11. C61×#2875C2semidirect
13. C61×#578C4semidirect
52. 2−1+4×C61C2×C4semidirect
58. #137×C61C2×C4semidirect
59. #135×C61C2×C4semidirect
60. C2×Q16×C61C2×C4semidirect
66. #131×C61C2×C4semidirect
192. #60175C61direct (N × H)
208. C4×2−1+4C2×C61semidirect
209. #2952C2×C61semidirect
210. #2950C2×C61semidirect
211. #2947C2×C61semidirect
212. C4×#137C2×C61semidirect
213. #2943C2×C61semidirect
214. C2×#441C2×C61semidirect
215. #2912C2×C61semidirect
216. #2898C2×C61semidirect
217. #2875C2×C61semidirect
223. #578C4×C61semidirect
261. C61#60175direct (N × H)
263. 2−1+4C2×C4×C61semidirect
269. #137C2×C4×C61semidirect
270. #135C2×C4×C61semidirect
271. C2×Q16C2×C4×C61semidirect
277. #131C2×C4×C61semidirect

Maximal subgroups

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

subgroupindex
C2×C61×#5782
C4×2−1+4×C612
C61×#29522
C61×#29502
C61×#29472
C4×#137×C612
C61×#29432
C2×C61×#4412
C61×#29122
C61×#28982
C61×#28752
#6017561