Finite Group Databasebeta

← C61×#19288 #29522721

Subgroups — C61×#19288 #29522721

Order 15616.

608 conjugacy classes of subgroups.

The subgroup lattice has 452 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C61×#3167C211yesmaximal, characteristic
2C61×#3113C211yesmaximal, characteristic, complemented
3C61×#3112C211yesmaximal, characteristic
4C2×C61×#446C211yesmaximal, characteristic, complemented
5C61×#1704C212yesmaximal, complemented
6C61×#1702C212yesmaximal, complemented
7C61×#1692C212yesmaximal, complemented
8C61×#1686C212yesmaximal, complemented
9C61×#1568C211yesmaximal, characteristic, complemented
10C61×#1552C211yesmaximal, characteristic, complemented
11C61×#1551C211yesmaximal, characteristic
12C61×#533C2211yescharacteristic
13C61×#532C2211yescharacteristic
14C2×C4×Q8×C61C2211yescharacteristic
15C61×#498C2211yescharacteristic
16C61×#492C2211yescharacteristic
17C61×#487C2211yescharacteristic
18C61×#486C2211yescharacteristic
19C61×#483—21no—
20C61×#480—21no—
21C61×#477C2212yes—
22C61×#475C2212yes—
23C61×#465—21no—
24C61×#465—21no—
25C61×#462C2212yes—
26C61×#460C2212yes—
27C61×#451—21no—
28C61×#448—21no—
29C61×#446—21no—
30C61×#446—21no—
31C61×#446C2212yes—
32C61×#446C2212yes—
33C61×#441—22no—
34C61×#440—22no—
35C4×SD16×C61—21no—
36C4×SD16×C61—21no—
37C61×#432C2211yescharacteristic
38C61×#432C2211yescharacteristic
39C2×C4⋊C8×C61—21no—
40C2×C4⋊C8×C61C2211yescharacteristic
41C2×#97×C61—21no—
42C2×C4×C8×C61C2211yescharacteristic
43C61×#335C2212yes—
44C61×#334C2212yes—
45C61×#333C2212yes—
46C61×#332C2212yes—
47C61×#332C2212yes—
48C61×#331C2212yes—
49C2×#41×C61—21no—
50C22×Q8×C61D411yescharacteristic
51C2×Q16×C61—22no—
52C2×SD16×C61—22no—
53C2×SD16×C61—21no—
54C2×SD16×C61—21no—
55C22×C8×C61—21no—
56C22×C8×C61D411yescharacteristic
57C22×C8×C61D411yescharacteristic
58C4⋊Q8×C61C2311yescharacteristic
59C4⋊Q8×C61C2311yescharacteristic
60C4⋊Q8×C61C2311yescharacteristic
61#127×C61C2311yescharacteristic
62#125×C61—21no—
63#124×C61—21no—
64C22⋊Q8×C61—21no—
65C22⋊Q8×C61—21no—
66#121×C61—21no—
67#121×C61—21no—
68C4×Q8×C61—21no—
69C4×Q8×C61—21no—
70C4×Q8×C61—21no—
71C4×Q8×C61—21no—
72C4×Q8×C61D412yes—
73C4×Q8×C61C2312yes—
74C4×D4×C61—21no—
75C4×D4×C61—21no—
76#117×C61—21no—
77C2×C4⋊C4×C61—21no—
78C2×C4⋊C4×C61D411yescharacteristic
79C2×C42×C61—21no—
80C2×C42×C61C2311yescharacteristic
81#107×C61—22no—
82#106×C61—21no—
83#106×C61—21no—
84#106×C61—22no—
85C4⋊C8×C61—22no—
86C4⋊C8×C61—22no—
87C4⋊C8×C61—21no—
88C4⋊C8×C61C2312yes—
89C4⋊C8×C61C2312yes—
90C4⋊C8×C61C2312yes—
91#103×C61—22no—
92#103×C61—22no—
93#103×C61—21no—
94#103×C61—21no—
95#103×C61—22no—
96#102×C61—21no—
97#102×C61—21no—
98#102×C61—22no—
99C22⋊C8×C61—21no—
100C22⋊C8×C61—21no—
101#97×C61—22no—
102#97×C61—22no—
103#97×C61—21no—
104#97×C61—21no—
105C4×C8×C61D411yescharacteristic
106C4×C8×C61D411yescharacteristic
107C4×C8×C61C2311yescharacteristic
108C4×C8×C61C2311yescharacteristic
109#41×C61—82no—
110C2×Q8×C61—21no—
111C2×Q8×C61—21no—
112C2×Q8×C61—21no—
113C2×Q8×C61—21no—
114C2×Q8×C61—42no—
115C2×Q8×C61C2×D412yes—
116C2×D4×C61—21no—
117C2×D4×C61—21no—
118C2×D4×C61—21no—
119C2×D4×C61—41no—
120C22×C4×C61—21no—
121C22×C4×C61—21no—
122C22×C4×C61—41no—
123C22×C4×C61C2×D411yescharacteristic
124C22×C4×C61C2×D411yescharacteristic
125C22×C4×C61#4111yescharacteristic
126Q16×C61—44no—
127SD16×C61—42no—
128SD16×C61—44no—
129SD16×C61—42no—
130C2×C8×C61—22no—
131C2×C8×C61—22no—
132C2×C8×C61—22no—
133C2×C8×C61—22no—
134C2×C8×C61—41no—
135C2×C8×C61—42no—
136C2×C8×C61C2×D411yescharacteristic
137C2×C8×C61C2×D411yescharacteristic
138C2×C8×C61C2×D411yescharacteristic
139C2×C8×C61C2×D411yescharacteristic
140C4⋊C4×C61—22no—
141C4⋊C4×C61—21no—
142C4⋊C4×C61—22no—
143C4⋊C4×C61—21no—
144C4⋊C4×C61—21no—
145C4⋊C4×C61—21no—
146C4⋊C4×C61—21no—
147C4⋊C4×C61—21no—
148C4⋊C4×C61—21no—
149C4⋊C4×C61—41no—
150C4⋊C4×C61—41no—
151C4⋊C4×C61C2×D412yes—
152C22⋊C4×C61—41no—
153C22⋊C4×C61—41no—
154C22⋊C4×C61—41no—
155C42×C61—22no—
156C42×C61—22no—
157C42×C61—41no—
158C42×C61C2×D411yescharacteristic
159C42×C61C2×D411yescharacteristic
160C42×C61#4111yescharacteristic
161C42×C61C2411yescharacteristic
162C23×C61—41no—
163C23×C61#12111yescharacteristic
164Q8×C61—22no—
165Q8×C61—22no—
166Q8×C61—42no—
167Q8×C61—41no—
168Q8×C61—41no—
169Q8×C61—42no—
170Q8×C61—81no—
171D4×C61—41no—
172D4×C61—41no—
173D4×C61—42no—
174D4×C61—81no—
175D4×C61—82no—
176C2×C4×C61—22no—
177C2×C4×C61—21no—
178C2×C4×C61—21no—
179C2×C4×C61—22no—
180C2×C4×C61—22no—
181C2×C4×C61—22no—
182C2×C4×C61—42no—
183C2×C4×C61—42no—
184C2×C4×C61—41no—
185C2×C4×C61—41no—
186C2×C4×C61—41no—
187C2×C4×C61—82no—
188C2×C4×C61#12111yescharacteristic
189C2×C4×C61#12111yescharacteristic
190C2×C4×C61#12111yescharacteristic
191C2×C4×C61C22×D411yescharacteristic
192C2×C4×C61C22×D411yescharacteristic
193C2×C4×C61C2×#4111yescharacteristic
194C8×C61—21no—
195C8×C61—21no—
196C8×C61—21no—
197C8×C61—42no—
198C8×C61—42no—
199C8×C61—42no—
200C8×C61#13712yes—
201#19288C6111yesmaximal, characteristic, direct product factor, Sylow
202C22×C61—21no—
203C22×C61—21no—
204C22×C61—81no—
205C22×C61—81no—
206C22×C61—81no—
207C22×C61#47112yes—
208C22×C61C2×#12111yescharacteristic
209C4×C61—21no—
210C4×C61—21no—
211C4×C61—21no—
212C4×C61—21no—
213C4×C61—42no—
214C4×C61—42no—
215C4×C61—81no—
216C4×C61—81no—
217C4×C61—81no—
218C4×C61C2×#13711yescharacteristic
219C4×C61#57511yescharacteristic
220C4×C61#57512yes—
221#3167C2×C6111yescharacteristic
222#3113C2×C6111yescharacteristic, complemented
223#3112C2×C6111yescharacteristic
224C2×#446C2×C6111yescharacteristic, complemented
225#1704C2×C6112yescomplemented
226#1702C2×C6112yescomplemented
227#1692C2×C6112yescomplemented
228#1686C2×C6112yescomplemented
229#1568C2×C6111yescharacteristic, complemented
230#1552C2×C6111yescharacteristic, complemented
231#1551C2×C6111yescharacteristic
232C2×C61—22no—
233C2×C61—161no—
234C2×C61#303711yescharacteristic
235C2×C61#305411yescharacteristic
236C2×C61C2×#47111yescharacteristic
237#533C22×C6111yescharacteristic
238#532C22×C6111yescharacteristic
239C2×C4×Q8C22×C6111yescharacteristic
240#498C22×C6111yescharacteristic
241#492C22×C6111yescharacteristic
242#487C22×C6111yescharacteristic
243#486C22×C6111yescharacteristic
244#483—21no—
245#480—21no—
246#477C22×C6112yes—
247#475C22×C6112yes—
248#465—21no—
249#465—21no—
250#462C22×C6112yes—
251#460C22×C6112yes—
252#451—21no—
253#448—21no—
254#446—21no—
255#446—21no—
256#446C22×C6112yes—
257#446C22×C6112yes—
258#441—22no—
259#440—22no—
260C4×SD16—21no—
261C4×SD16—21no—
262#432C22×C6111yescharacteristic
263#432C22×C6111yescharacteristic
264C2×C4⋊C8—21no—
265C2×C4⋊C8C22×C6111yescharacteristic
266C2×#97—21no—
267C2×C4×C8C22×C6111yescharacteristic
268#335C22×C6112yes—
269#334C22×C6112yes—
270#333C22×C6112yes—
271#332C22×C6112yes—
272#332C22×C6112yes—
273#331C22×C6112yes—
274C61#1928811yescharacteristic, direct product factor, Sylow
275C2×#41—21no—
276C22×Q8D4×C6111yescharacteristic
277C2×Q16—22no—
278C2×SD16—22no—
279C2×SD16—21no—
280C2×SD16—21no—
281C22×C8—21no—
282C22×C8D4×C6111yescharacteristic
283C22×C8D4×C6111yescharacteristic
284C4⋊Q8C23×C6111yescharacteristic
285C4⋊Q8C23×C6111yescharacteristic
286C4⋊Q8C23×C6111yescharacteristic
287#127C23×C6111yescharacteristic
288#125—21no—
289#124—21no—
290C22⋊Q8—21no—
291C22⋊Q8—21no—
292#121—21no—
293#121—21no—
294C4×Q8—21no—
295C4×Q8—21no—
296C4×Q8—21no—
297C4×Q8—21no—
298C4×Q8D4×C6112yes—
299C4×Q8C23×C6112yes—
300C4×D4—21no—
301C4×D4—21no—
302#117—21no—
303C2×C4⋊C4—21no—
304C2×C4⋊C4D4×C6111yescharacteristic
305C2×C42—21no—
306C2×C42C23×C6111yescharacteristic
307#107—22no—
308#106—21no—
309#106—21no—
310#106—22no—
311C4⋊C8—22no—
312C4⋊C8—22no—
313C4⋊C8—21no—
314C4⋊C8C23×C6112yes—
315C4⋊C8C23×C6112yes—
316C4⋊C8C23×C6112yes—
317#103—22no—
318#103—22no—
319#103—21no—
320#103—21no—
321#103—22no—
322#102—21no—
323#102—21no—
324#102—22no—
325C22⋊C8—21no—
326C22⋊C8—21no—
327#97—22no—
328#97—22no—
329#97—21no—
330#97—21no—
331C4×C8D4×C6111yescharacteristic
332C4×C8D4×C6111yescharacteristic
333C4×C8C23×C6111yescharacteristic
334C4×C8C23×C6111yescharacteristic
335#41—82no—
336C2×Q8—21no—
337C2×Q8—21no—
338C2×Q8—21no—
339C2×Q8—21no—
340C2×Q8—42no—
341C2×Q8C2×D4×C6112yes—
342C2×D4—21no—
343C2×D4—21no—
344C2×D4—21no—
345C2×D4—41no—
346C22×C4—21no—
347C22×C4—21no—
348C22×C4—41no—
349C22×C4C2×D4×C6111yescharacteristic
350C22×C4C2×D4×C6111yescharacteristic
351C22×C4#41×C6111yescharacteristic
352Q16—44no—
353SD16—42no—
354SD16—44no—
355SD16—42no—
356C2×C8—22no—
357C2×C8—22no—
358C2×C8—22no—
359C2×C8—22no—
360C2×C8—41no—
361C2×C8—42no—
362C2×C8C2×D4×C6111yescharacteristic
363C2×C8C2×D4×C6111yescharacteristic
364C2×C8C2×D4×C6111yescharacteristic
365C2×C8C2×D4×C6111yescharacteristic
366C4⋊C4—22no—
367C4⋊C4—21no—
368C4⋊C4—22no—
369C4⋊C4—21no—
370C4⋊C4—21no—
371C4⋊C4—21no—
372C4⋊C4—21no—
373C4⋊C4—21no—
374C4⋊C4—21no—
375C4⋊C4—41no—
376C4⋊C4—41no—
377C4⋊C4C2×D4×C6112yes—
378C22⋊C4—41no—
379C22⋊C4—41no—
380C22⋊C4—41no—
381C42—22no—
382C42—22no—
383C42—41no—
384C42C2×D4×C6111yescharacteristic
385C42C2×D4×C6111yescharacteristic
386C42#41×C6111yescharacteristic
387C42C24×C6111yescharacteristic
388C23—41no—
389C23#121×C6111yescharacteristic
390Q8—22no—
391Q8—22no—
392Q8—42no—
393Q8—41no—
394Q8—41no—
395Q8—42no—
396Q8—81no—
397D4—41no—
398D4—41no—
399D4—42no—
400D4—81no—
401D4—82no—
402C2×C4—22no—
403C2×C4—21no—
404C2×C4—21no—
405C2×C4—22no—
406C2×C4—22no—
407C2×C4—22no—
408C2×C4—42no—
409C2×C4—42no—
410C2×C4—41no—
411C2×C4—41no—
412C2×C4—41no—
413C2×C4—82no—
414C2×C4#121×C6111yescharacteristic
415C2×C4#121×C6111yescharacteristic
416C2×C4#121×C6111yescharacteristic
417C2×C4C22×D4×C6111yescharacteristic
418C2×C4C22×D4×C6111yescharacteristic
419C2×C4C2×#41×C6111yescharacteristic
420C8—21no—
421C8—21no—
422C8—21no—
423C8—42no—
424C8—42no—
425C8—42no—
426C8#137×C6112yes—
427C22—21no—
428C22—21no—
429C22—81no—
430C22—81no—
431C22—81no—
432C22C61×#47112yes—
433C22C2×#121×C6111yescharacteristic
434C4—21no—
435C4—21no—
436C4—21no—
437C4—21no—
438C4—42no—
439C4—42no—
440C4—81no—
441C4—81no—
442C4—81no—
443C4C2×#137×C6111yescharacteristic
444C4C61×#57511yescharacteristic
445C4C61×#57512yes—
446C2—22no—
447C2—161no—
448C2C61×#303711yescharacteristic
449C2C61×#305411yescharacteristic
450C2C2×C61×#47111yescharacteristic
subgroupG/H
center, socleC22×C61C2×#121
commutator, Frattini, cosocleC42C24×C61
prime radicalC61×#532C22
LCSG ▷ C42 ▷ C22 ▷ C1
UCSC1 ◁ C22×C61 ◁ C2×C42×C61 ◁ G

Complements

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

NH ≅ G/Ntype
2. C61×#3113C2semidirect
4. C2×C61×#446C2semidirect
5. C61×#1704C2semidirect
6. C61×#1702C2semidirect
7. C61×#1692C2semidirect
8. C61×#1686C2semidirect
9. C61×#1568C2semidirect
10. C61×#1552C2semidirect
201. #19288C61direct (N × H)
222. #3113C2×C61semidirect
224. C2×#446C2×C61semidirect
225. #1704C2×C61semidirect
226. #1702C2×C61semidirect
227. #1692C2×C61semidirect
228. #1686C2×C61semidirect
229. #1568C2×C61semidirect
230. #1552C2×C61semidirect
274. C61#19288direct (N × H)

Maximal subgroups

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

subgroupindex
C61×#31672
C61×#31132
C61×#31122
C2×C61×#4462
C61×#17042
C61×#17022
C61×#16922
C61×#16862
C61×#15682
C61×#15522
C61×#15512
#1928861