Finite Group Databasebeta

← C61×#33754 #29537187

Subgroups — C61×#33754 #29537187

Order 15616.

646 conjugacy classes of subgroups.

The subgroup lattice has 464 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C61×#3050C211yesmaximal, characteristic, complemented
2C2×C61×#466C211yesmaximal, characteristic, complemented
3C61×#2236C211yesmaximal, characteristic, complemented
4C61×#2234C212yesmaximal, complemented
5C61×#2233C211yesmaximal, characteristic, complemented
6C61×#2217C212yesmaximal, complemented
7C61×#2215C212yesmaximal, complemented
8C61×#2187C211yesmaximal, characteristic, complemented
9C61×#2186C212yesmaximal, complemented
10C61×#2184C211yesmaximal, characteristic, complemented
11C2×C61×C22⋊C16C211yesmaximal, characteristic, complemented
12C2×#135×C61C2211yescharacteristic, complemented
13C22×D8×C61C2211yescharacteristic
14C23×C8×C61C2211yescharacteristic
15C61×#525—21no—
16C2×#121×C61—21no—
17C2×SD32×C61C2212yescomplemented
18C2×D16×C61C2212yescomplemented
19C22×C16×C61C2212yescomplemented
20C61×#467C2211yescharacteristic
21C61×#467C2211yescharacteristic, complemented
22C61×#466—21no—
23C61×#466—21no—
24C61×#466C2212yes—
25C61×#466C2212yes—
26C61×#466C2211yescharacteristic
27C61×#466C2211yescharacteristic, complemented
28C61×#465—21no—
29C61×#465—21no—
30C61×#427C2211yescharacteristic, complemented
31C2×#107×C61C2211yescharacteristic
32C61×#416—21no—
33C2×#102×C61—21no—
34C61×#367C2212yescomplemented
35C61×#366C2212yescomplemented
36C61×#358C2212yescomplemented
37C61×#357C2212yescomplemented
38C61×#357C2212yescomplemented
39C61×#357C2212yescomplemented
40C61×C22⋊C16C2211yescharacteristic
41C61×C22⋊C16C2211yescharacteristic
42C61×C22⋊C16C2212yescomplemented
43C2×#41×C61—21no—
44C22×D4×C61—21no—
45C22×D4×C61—41no—
46C23×C4×C61D411yescharacteristic
47#135×C61—42no—
48C2×Q16×C61C2311yescharacteristic
49C2×SD16×C61—21no—
50C2×D8×C61—21no—
51C2×D8×C61—42no—
52C2×D8×C61C2312yes—
53C2×D8×C61C2311yescharacteristic
54C22×C8×C61—42no—
55C22×C8×C61D411yescharacteristic
56C22×C8×C61D411yescharacteristic
57C22×C8×C61D411yescharacteristic
58C22×C8×C61C2311yescharacteristic
59C22×C8×C61C2311yescharacteristic
60C22×C8×C61C2311yescharacteristic
61#123×C61—41no—
62C22⋊Q8×C61—21no—
63C22⋊Q8×C61—21no—
64#121×C61—21no—
65#121×C61—21no—
66#121×C61—22no—
67#121×C61—22no—
68#121×C61—41no—
69#121×C61—41no—
70#120×C61—41no—
71C4×D4×C61—41no—
72C4×D4×C61—41no—
73#117×C61—21no—
74C2×C4⋊C4×C61—21no—
75C2×C22⋊C4×C61—41no—
76SD32×C61—42no—
77D16×C61—42no—
78C2×C16×C61—42no—
79C2×C16×C61C2312yes—
80C2×C16×C61C2312yes—
81#107×C61—21no—
82#107×C61C2311yescharacteristic
83#107×C61C2311yescharacteristic
84#107×C61C2312yes—
85#106×C61—21no—
86#103×C61—21no—
87#103×C61—21no—
88#102×C61—22no—
89#102×C61—22no—
90#102×C61—21no—
91#102×C61—21no—
92C24×C61—41no—
93C24×C61D811yescharacteristic
94#41×C61—42no—
95#41×C61—42no—
96C2×Q8×C61—21no—
97C2×D4×C61—22no—
98C2×D4×C61—21no—
99C2×D4×C61—42no—
100C2×D4×C61—42no—
101C2×D4×C61—41no—
102C2×D4×C61—41no—
103C2×D4×C61—41no—
104C2×D4×C61—42no—
105C2×D4×C61—42no—
106C2×D4×C61—41no—
107C2×D4×C61—84no—
108C22×C4×C61—41no—
109C22×C4×C61—41no—
110C22×C4×C61—41no—
111C22×C4×C61—41no—
112C22×C4×C61D811yescharacteristic
113C22×C4×C61D811yescharacteristic
114C22×C4×C61D811yescharacteristic
115C22×C4×C61C2×D411yescharacteristic
116C22×C4×C61C2×D411yescharacteristic
117C22×C4×C61C2×D411yescharacteristic
118Q16×C61—22no—
119SD16×C61—42no—
120D8×C61—22no—
121D8×C61—22no—
122D8×C61—22no—
123D8×C61—42no—
124C2×C8×C61—22no—
125C2×C8×C61—22no—
126C2×C8×C61—22no—
127C2×C8×C61—41no—
128C2×C8×C61—41no—
129C2×C8×C61—41no—
130C2×C8×C61C2×D411yescharacteristic
131C2×C8×C61C2×D411yescharacteristic
132C2×C8×C61C2×D411yescharacteristic
133C2×C8×C61C2411yescharacteristic
134C4⋊C4×C61—22no—
135C4⋊C4×C61—21no—
136C4⋊C4×C61—21no—
137C4⋊C4×C61—41no—
138C4⋊C4×C61—41no—
139C4⋊C4×C61—41no—
140C22⋊C4×C61—41no—
141C22⋊C4×C61—41no—
142C22⋊C4×C61—41no—
143C22⋊C4×C61—42no—
144C22⋊C4×C61—42no—
145C22⋊C4×C61—41no—
146C22⋊C4×C61—41no—
147C42×C61—41no—
148C16×C61—22no—
149C16×C61—22no—
150C23×C61—22no—
151C23×C61—22no—
152C23×C61—41no—
153C23×C61—41no—
154C23×C61—42no—
155C23×C61—82no—
156C23×C61—82no—
157C23×C61—82no—
158C23×C61C2×D811yescharacteristic
159C23×C61C2×D811yescharacteristic
160C23×C61C2×D811yescharacteristic
161Q8×C61—41no—
162Q8×C61—41no—
163D4×C61—42no—
164D4×C61—41no—
165D4×C61—42no—
166D4×C61—41no—
167D4×C61—82no—
168D4×C61—84no—
169D4×C61—82no—
170D4×C61—82no—
171D4×C61—82no—
172D4×C61—84no—
173C2×C4×C61—21no—
174C2×C4×C61—21no—
175C2×C4×C61—21no—
176C2×C4×C61—21no—
177C2×C4×C61—21no—
178C2×C4×C61—21no—
179C2×C4×C61—41no—
180C2×C4×C61—41no—
181C2×C4×C61—41no—
182C2×C4×C61—41no—
183C2×C4×C61—41no—
184C2×C4×C61—42no—
185C2×C4×C61—41no—
186C2×C4×C61—82no—
187C2×C4×C61—82no—
188C2×C4×C61C2×D811yescharacteristic
189C2×C4×C61C2×D811yescharacteristic
190C2×C4×C61C2×D811yescharacteristic
191C2×C4×C61C22×D411yescharacteristic
192C8×C61—21no—
193C8×C61—21no—
194C8×C61—21no—
195C8×C61H(5,𝔽2)12yes—
196#33754C6111yesmaximal, characteristic, direct product factor, Sylow
197C22×C61—21no—
198C22×C61—21no—
199C22×C61—21no—
200C22×C61—21no—
201C22×C61—21no—
202C22×C61—21no—
203C22×C61—21no—
204C22×C61—44no—
205C22×C61—81no—
206C22×C61—82no—
207C22×C61—82no—
208C22×C61—81no—
209C22×C61—81no—
210C22×C61—82no—
211C22×C61—164no—
212C22×C61#50812yes—
213C22×C61#50912yes—
214C22×C61C22×D811yescharacteristic
215C4×C61—21no—
216C4×C61—21no—
217C4×C61—21no—
218C4×C61—81no—
219C4×C61—82no—
220C4×C61—81no—
221C4×C61—81no—
222C4×C61#53411yescharacteristic
223C4×C61#57511yescharacteristic
224#3050C2×C6111yescharacteristic, complemented
225C2×#466C2×C6111yescharacteristic, complemented
226#2236C2×C6111yescharacteristic, complemented
227#2234C2×C6112yescomplemented
228#2233C2×C6111yescharacteristic, complemented
229#2217C2×C6112yescomplemented
230#2215C2×C6112yescomplemented
231#2187C2×C6111yescharacteristic, complemented
232#2186C2×C6112yescomplemented
233#2184C2×C6111yescharacteristic, complemented
234C2×C22⋊C16C2×C6111yescharacteristic, complemented
235C2×C61—22no—
236C2×C61—22no—
237C2×C61—41no—
238C2×C61—161no—
239C2×C61—162no—
240C2×C61#318611yescharacteristic
241C2×C61C2×#50811yescharacteristic
242C2×C61C2×#50911yescharacteristic
243C2×#135C22×C6111yescharacteristic, complemented
244C22×D8C22×C6111yescharacteristic
245C23×C8C22×C6111yescharacteristic
246#525—21no—
247C2×#121—21no—
248C2×SD32C22×C6112yescomplemented
249C2×D16C22×C6112yescomplemented
250C22×C16C22×C6112yescomplemented
251#467C22×C6111yescharacteristic
252#467C22×C6111yescharacteristic, complemented
253#466—21no—
254#466—21no—
255#466C22×C6111yescharacteristic
256#466C22×C6112yes—
257#466C22×C6112yes—
258#466C22×C6111yescharacteristic, complemented
259#465—21no—
260#465—21no—
261#427C22×C6111yescharacteristic, complemented
262C2×#107C22×C6111yescharacteristic
263#416—21no—
264C2×#102—21no—
265#367C22×C6112yescomplemented
266#366C22×C6112yescomplemented
267#358C22×C6112yescomplemented
268#357C22×C6112yescomplemented
269#357C22×C6112yescomplemented
270#357C22×C6112yescomplemented
271C22⋊C16C22×C6111yescharacteristic
272C22⋊C16C22×C6111yescharacteristic
273C22⋊C16C22×C6112yescomplemented
274C61#3375411yescharacteristic, direct product factor, Sylow
275C2×#41—21no—
276C22×D4—21no—
277C22×D4—41no—
278C23×C4D4×C6111yescharacteristic
279#135—42no—
280C2×Q16C23×C6111yescharacteristic
281C2×SD16—21no—
282C2×D8—21no—
283C2×D8—42no—
284C2×D8C23×C6112yes—
285C2×D8C23×C6111yescharacteristic
286C22×C8—42no—
287C22×C8D4×C6111yescharacteristic
288C22×C8D4×C6111yescharacteristic
289C22×C8D4×C6111yescharacteristic
290C22×C8C23×C6111yescharacteristic
291C22×C8C23×C6111yescharacteristic
292C22×C8C23×C6111yescharacteristic
293#123—41no—
294C22⋊Q8—21no—
295C22⋊Q8—21no—
296#121—22no—
297#121—22no—
298#121—21no—
299#121—21no—
300#121—41no—
301#121—41no—
302#120—41no—
303C4×D4—41no—
304C4×D4—41no—
305#117—21no—
306C2×C4⋊C4—21no—
307C2×C22⋊C4—41no—
308SD32—42no—
309D16—42no—
310C2×C16—42no—
311C2×C16C23×C6112yes—
312C2×C16C23×C6112yes—
313#107—21no—
314#107C23×C6111yescharacteristic
315#107C23×C6111yescharacteristic
316#107C23×C6112yes—
317#106—21no—
318#103—21no—
319#103—21no—
320#102—22no—
321#102—22no—
322#102—21no—
323#102—21no—
324C24—41no—
325C24D8×C6111yescharacteristic
326#41—42no—
327#41—42no—
328C2×Q8—21no—
329C2×D4—22no—
330C2×D4—21no—
331C2×D4—42no—
332C2×D4—42no—
333C2×D4—41no—
334C2×D4—41no—
335C2×D4—41no—
336C2×D4—42no—
337C2×D4—42no—
338C2×D4—41no—
339C2×D4—84no—
340C22×C4—41no—
341C22×C4—41no—
342C22×C4—41no—
343C22×C4—41no—
344C22×C4D8×C6111yescharacteristic
345C22×C4D8×C6111yescharacteristic
346C22×C4D8×C6111yescharacteristic
347C22×C4C2×D4×C6111yescharacteristic
348C22×C4C2×D4×C6111yescharacteristic
349C22×C4C2×D4×C6111yescharacteristic
350Q16—22no—
351SD16—42no—
352D8—22no—
353D8—22no—
354D8—22no—
355D8—42no—
356C2×C8—22no—
357C2×C8—22no—
358C2×C8—22no—
359C2×C8—41no—
360C2×C8—41no—
361C2×C8—41no—
362C2×C8C2×D4×C6111yescharacteristic
363C2×C8C2×D4×C6111yescharacteristic
364C2×C8C2×D4×C6111yescharacteristic
365C2×C8C24×C6111yescharacteristic
366C4⋊C4—22no—
367C4⋊C4—21no—
368C4⋊C4—21no—
369C4⋊C4—41no—
370C4⋊C4—41no—
371C4⋊C4—41no—
372C22⋊C4—41no—
373C22⋊C4—41no—
374C22⋊C4—41no—
375C22⋊C4—42no—
376C22⋊C4—42no—
377C22⋊C4—41no—
378C22⋊C4—41no—
379C42—41no—
380C16—22no—
381C16—22no—
382C23—22no—
383C23—22no—
384C23—41no—
385C23—41no—
386C23—42no—
387C23—82no—
388C23—82no—
389C23—82no—
390C23C2×D8×C6111yescharacteristic
391C23C2×D8×C6111yescharacteristic
392C23C2×D8×C6111yescharacteristic
393Q8—41no—
394Q8—41no—
395D4—42no—
396D4—41no—
397D4—42no—
398D4—41no—
399D4—82no—
400D4—84no—
401D4—82no—
402D4—82no—
403D4—82no—
404D4—84no—
405C2×C4—21no—
406C2×C4—21no—
407C2×C4—21no—
408C2×C4—21no—
409C2×C4—21no—
410C2×C4—21no—
411C2×C4—41no—
412C2×C4—41no—
413C2×C4—41no—
414C2×C4—41no—
415C2×C4—41no—
416C2×C4—42no—
417C2×C4—41no—
418C2×C4—82no—
419C2×C4—82no—
420C2×C4C2×D8×C6111yescharacteristic
421C2×C4C2×D8×C6111yescharacteristic
422C2×C4C2×D8×C6111yescharacteristic
423C2×C4C22×D4×C6111yescharacteristic
424C8—21no—
425C8—21no—
426C8—21no—
427C8H(5,𝔽2)×C6112yes—
428C22—21no—
429C22—21no—
430C22—21no—
431C22—21no—
432C22—21no—
433C22—21no—
434C22—21no—
435C22—44no—
436C22—81no—
437C22—82no—
438C22—82no—
439C22—81no—
440C22—81no—
441C22—82no—
442C22—164no—
443C22C61×#50812yes—
444C22C61×#50912yes—
445C22C22×D8×C6111yescharacteristic
446C4—21no—
447C4—21no—
448C4—21no—
449C4—81no—
450C4—82no—
451C4—81no—
452C4—81no—
453C4C61×#53411yescharacteristic
454C4C61×#57511yescharacteristic
455C2—22no—
456C2—22no—
457C2—41no—
458C2—162no—
459C2—161no—
460C2C61×#318611yescharacteristic
461C2C2×C61×#50811yescharacteristic
462C2C2×C61×#50911yescharacteristic
subgroupG/H
center, socleC22×C61C22×D8
commutator, Frattini, cosocleC2×C8C24×C61
LCSG ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1
UCSC1 ◁ C22×C61 ◁ C23×C4×C61 ◁ C23×C8×C61 ◁ G

Complements

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

NH ≅ G/Ntype
1. C61×#3050C2semidirect
2. C2×C61×#466C2semidirect
3. C61×#2236C2semidirect
4. C61×#2234C2semidirect
5. C61×#2233C2semidirect
6. C61×#2217C2semidirect
7. C61×#2215C2semidirect
8. C61×#2187C2semidirect
9. C61×#2186C2semidirect
10. C61×#2184C2semidirect
11. C2×C61×C22⋊C16C2semidirect
12. C2×#135×C61C22semidirect
17. C2×SD32×C61C22semidirect
18. C2×D16×C61C22semidirect
19. C22×C16×C61C22semidirect
21. C61×#467C22semidirect
27. C61×#466C22semidirect
30. C61×#427C22semidirect
34. C61×#367C22semidirect
35. C61×#366C22semidirect
36. C61×#358C22semidirect
37. C61×#357C22semidirect
38. C61×#357C22semidirect
39. C61×#357C22semidirect
42. C61×C22⋊C16C22semidirect
196. #33754C61direct (N × H)
224. #3050C2×C61semidirect
225. C2×#466C2×C61semidirect
226. #2236C2×C61semidirect
227. #2234C2×C61semidirect
228. #2233C2×C61semidirect
229. #2217C2×C61semidirect
230. #2215C2×C61semidirect
231. #2187C2×C61semidirect
232. #2186C2×C61semidirect
233. #2184C2×C61semidirect
234. C2×C22⋊C16C2×C61semidirect
243. C2×#135C22×C61semidirect
248. C2×SD32C22×C61semidirect
249. C2×D16C22×C61semidirect
250. C22×C16C22×C61semidirect
252. #467C22×C61semidirect
258. #466C22×C61semidirect
261. #427C22×C61semidirect
265. #367C22×C61semidirect
266. #366C22×C61semidirect
267. #358C22×C61semidirect
268. #357C22×C61semidirect
269. #357C22×C61semidirect
270. #357C22×C61semidirect
273. C22⋊C16C22×C61semidirect
274. C61#33754direct (N × H)

Maximal subgroups

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

subgroupindex
C61×#30502
C2×C61×#4662
C61×#22362
C61×#22342
C61×#22332
C61×#22172
C61×#22152
C61×#21872
C61×#21862
C61×#21842
C2×C61×C22⋊C162
#3375461