Finite Group Databasebeta

← #29663125

Subgroups — #29663125

Order 15616.

650 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
1#21456946C212yesmaximal
2#500×D61C211yesmaximal, characteristic, complemented
3#21454526C211yesmaximal, characteristic
4#21445341C212yesmaximal, complemented
5#21445340C212yesmaximal, complemented
6#21445219C211yesmaximal, characteristic, complemented
7#21445218C211yesmaximal, characteristic, complemented
8#21445215C211yesmaximal, characteristic, complemented
9#21444566C211yesmaximal, characteristic, complemented
10#21444498C212yesmaximal, complemented
11C61×#1675C211yesmaximal, characteristic, complemented
12C2×#14519783—21no—
13#16840309C2211yescharacteristic
14C4⋊Q8×D61C2212yes—
15#16840167C2211yescharacteristic
16#16840166—21no—
17#16840166—21no—
18#16840165C2212yes—
19#16840159—22no—
20#16840107—22no—
21#16840082—22no—
22#16840079—22no—
23#16840053C2212yes—
24#16840052C2212yes—
25#16840049C2212yes—
26#16839840C2211yescharacteristic
27#16839839C2211yescharacteristic
28#16839837—21no—
29C8×D244C2211yescharacteristic
30C8×C22⋊D61—21no—
31#16839814C2212yes—
32#107×D61—21no—
33#107×D61—21no—
34#16839784C2211yescharacteristic
35#16839727—21no—
36C4×C8×D61C2211yescharacteristic
37#16839515C2211yescharacteristic
38#16839512C2212yes—
39#16839511C2212yes—
40#16839510C2212yes—
41#16839510C2212yes—
42C61×#500C2211yescharacteristic
43C61×#476C2212yes—
44C8×D4×C61C2211yescharacteristic
45C61×#335C2211yescharacteristic
46C61×#333C2212yes—
47C2×#11758222—21no—
48C2×Q8×D61—22no—
49#14519783—44no—
50C2×#11758194—21no—
51C2×#11758189—21no—
52C2×#11758189—21no—
53C2×C8×D61—21no—
54C2×C8×D61D411yescharacteristic
55C2×C8×D61D411yescharacteristic
56#14519777C2312yes—
57#14519776—22no—
58#14519776—22no—
59#14519776D411yescharacteristic
60#14519775—22no—
61#14519773D411yescharacteristic
62#14519763—21no—
63#14519752D411yescharacteristic
64#14519750—21no—
65#14519748D411yescharacteristic
66C4×D244C2311yescharacteristic
67C4×C22⋊D61—21no—
68C4⋊C4×D61—22no—
69C4⋊C4×D61—22no—
70C42×D61C2311yescharacteristic
71#14519711—21no—
72#14519711—21no—
73#14519710—22no—
74#14519710—22no—
75#14519709—22no—
76#14519708—22no—
77#14519708—22no—
78C4⋊#92418C2311yescharacteristic
79C4⋊#92418C2312yes—
80#14519706—21no—
81#14519699—22no—
82#14519698—22no—
83#14519697—22no—
84C22⋊#92418—21no—
85#14519689—22no—
86#14519689—21no—
87#14519689—21no—
88C4×#92418C2311yescharacteristic
89C8×Dic61—22no—
90C8×Dic61D411yescharacteristic
91C8×Dic61D411yescharacteristic
92C22×C8×C61—21no—
93C4⋊Q8×C61C2312yes—
94C4×D4×C61C2311yescharacteristic
95#107×C61—21no—
96#107×C61—21no—
97#106×C61—22no—
98C4⋊C8×C61C2311yescharacteristic
99C4⋊C8×C61C2312yes—
100#102×C61—22no—
101C22⋊C8×C61—21no—
102C4×C8×C61C2311yescharacteristic
103C2×Q8×C61—22no—
104C2×D4×C61C2×D411yescharacteristic
105C22×C4×C61—21no—
106C2×C22⋊D61—21no—
107C2×C22⋊D61—21no—
108C2×C22⋊D61—21no—
109Q8×D61—84no—
110#11758222—42no—
111#11758222—42no—
112C2×D244—21no—
113C2×D244C2×D411yescharacteristic
114C2×C4×D61—21no—
115C2×C4×D61—42no—
116C2×C4×D61C2×D411yescharacteristic
117C2×C4×D61C2×D411yescharacteristic
118C2×C4×D61#4111yescharacteristic
119C2×Dic122—21no—
120C2×Dic122—22no—
121C2×Dic122—42no—
122C2×C8×C61—21no—
123C2×C8×C61—22no—
124C2×C8×C61—42no—
125C2×C8×C61C2×D411yescharacteristic
126C2×C8×C61C2×D411yescharacteristic
127C4⋊C4×C61—22no—
128C4⋊C4×C61—22no—
129C4⋊C4×C61C2×D411yescharacteristic
130C22⋊C4×C61—21no—
131C42×C61C2411yescharacteristic
132C22⋊Dic61—21no—
133#11758198—21no—
134#11758198—21no—
135C4⋊Dic61—22no—
136C4⋊Dic61—22no—
137C4⋊Dic61C2×D411yescharacteristic
138#11758196—21no—
139#11758196—42no—
140#11758196—42no—
141C4×Dic61C2×D411yescharacteristic
142C4×Dic61C2×D411yescharacteristic
143C4×Dic61#4111yescharacteristic
144#11758194—42no—
145C2×#92418—22no—
146C2×#92418—21no—
147C2×#92418C2×D411yescharacteristic
148C2×#92418C2×D411yescharacteristic
149#11758189—42no—
150#11758189—42no—
151C8×D61—22no—
152C8×D61—42no—
153C8×D61SD1614yes—
154C23×C61—21no—
155C22×D61—21no—
156C22×D61#12111yescharacteristic
157Q8×C61—82no—
158D4×C61—21no—
159D4×C61—21no—
160C2×C4×C61—21no—
161C2×C4×C61—41no—
162C2×C4×C61—41no—
163C2×C4×C61—42no—
164C2×C4×C61C22×D411yescharacteristic
165C2×C4×C61C22×D411yescharacteristic
166C2×C4×C61C2×#4111yescharacteristic
167C22⋊D61—42no—
168C22⋊D61—42no—
169C22⋊D61—42no—
170C2×Dic61—21no—
171C2×Dic61—42no—
172C2×Dic61#12111yescharacteristic
173C2×Dic61#12111yescharacteristic
174C2×Dic61#12111yescharacteristic
175D244—21no—
176D244—21no—
177D244—42no—
178C4×D61—21no—
179C4×D61—21no—
180C4×D61—22no—
181C4×D61—41no—
182C4×D61—41no—
183C4×D61—84no—
184C4×D61C2×SD1612yes—
185C4×D61#13712yes—
186Dic122—42no—
187Dic122—84no—
188Dic122—82no—
189C8×C61—21no—
190C8×C61—41no—
191C8×C61—42no—
192C8×C61C2×SD1612yes—
193#92418—21no—
194#92418—41no—
195#92418—42no—
196#92418C2×SD1612yes—
197#19193—611nomaximal, Sylow
198C22×C61—41no—
199C22×C61—41no—
200C22×C61—41no—
201C22×C61C2×#12111yescharacteristic
202C2×D61—21no—
203C2×D61—21no—
204C2×D61—41no—
205C2×D61—41no—
206C2×D61—41no—
207C2×D61#46112yes—
208C4×C61—21no—
209C4×C61—41no—
210C4×C61—82no—
211C4×C61C22×SD1611yescharacteristic
212C4×C61C2×#13711yescharacteristic
213C4×C61#57811yescharacteristic
214C4×C61#57811yescharacteristic
215Dic61—21no—
216Dic61—41no—
217Dic61—82no—
218Dic61#46112yes—
219Dic61#47212yes—
220#3227—612no—
221C2×#500—611no—
222#2931—611no—
223#1695—612no—
224#1695—612no—
225#1675—611no—
226#1675—611no—
227#1675—611no—
228#1675D6111yescharacteristic, complemented
229#1564—611no—
230#1548—612no—
231C2×C61—81no—
232C2×C61C2×#46111yescharacteristic
233C2×C61#305911yescharacteristic
234C2×C61C2×#47211yescharacteristic
235D61—22no—
236D61—81no—
237C2×#131—1221no—
238#532—611no—
239C2×C4⋊Q8—612no—
240#500—612no—
241#500—611no—
242#500—1221no—
243#500—1221no—
244#500C2×D6111yescharacteristic
245#498—1222no—
246#486—1222no—
247#481—1222no—
248#481—1222no—
249#476—612no—
250#476—612no—
251#476—612no—
252#476C2×D6112yes—
253#436—611no—
254#436—611no—
255#436—1221no—
256C8×D4—611no—
257C8×D4—1221no—
258C8×D4C2×D6111yescharacteristic
259#431—612no—
260C2×#107—1221no—
261C2×#107—1221no—
262#423—611no—
263#408—1221no—
264C2×C4×C8—611no—
265#335—611no—
266#335C2×D6111yescharacteristic
267#334—612no—
268#333—612no—
269#333—612no—
270#333—612no—
271#333C2×D6112yes—
272C61#1919311yescharacteristic, complemented, Sylow
273C2×#41—1221no—
274C22×Q8—1222no—
275#131—2444no—
276C2×M16—1221no—
277C2×M16—1221no—
278C2×M16—1221no—
279C22×C8—21no—
280C22×C8—611no—
281C22×C8—611no—
282C22×C8—1221no—
283C4⋊Q8—611no—
284C4⋊Q8—612no—
285C4⋊Q8—1222no—
286C4⋊Q8—1222no—
287C4⋊Q8—1222no—
288C4⋊Q8C22×D6112yes—
289#127—611no—
290#124—1221no—
291#121—611no—
292#121—611no—
293#121—1221no—
294C4×D4—611no—
295C4×D4—1221no—
296C4×D4C22×D6111yescharacteristic
297C2×C4⋊C4—1222no—
298C2×C4⋊C4—1222no—
299C2×C42—611no—
300#107—21no—
301#107—21no—
302#107—1222no—
303#107—1222no—
304#107—1221no—
305#107—1221no—
306#106—22no—
307#106—1222no—
308#106—1222no—
309#106—1222no—
310C4⋊C8—611no—
311C4⋊C8—612no—
312C4⋊C8—1221no—
313C4⋊C8C22×D6111yescharacteristic
314C4⋊C8C22×D6112yes—
315#102—22no—
316#102—1222no—
317#102—1222no—
318#102—1222no—
319C22⋊C8—21no—
320C22⋊C8—1221no—
321C22⋊C8—1222no—
322C22⋊C8—1221no—
323C22⋊C8—1221no—
324C4×C8—611no—
325C4×C8—611no—
326C4×C8—611no—
327C4×C8—1222no—
328C4×C8C22×D6111yescharacteristic
329#41—2442no—
330#41—2442no—
331C2×Q8—22no—
332C2×Q8—1221no—
333C2×Q8—1222no—
334C2×Q8—4884no—
335C2×Q8—2442no—
336C2×D4—611no—
337C2×D4—1221no—
338C2×D4—1221no—
339C2×D4—1221no—
340C2×D4—1221no—
341C2×D4D4×D6111yescharacteristic
342C22×C4—21no—
343C22×C4—611no—
344C22×C4—611no—
345C22×C4—611no—
346C22×C4—1221no—
347C22×C4—2442no—
348M16—2442no—
349M16—2442no—
350M16—2442no—
351C2×C8—22no—
352C2×C8—21no—
353C2×C8—42no—
354C2×C8—611no—
355C2×C8—614no—
356C2×C8—611no—
357C2×C8—1222no—
358C2×C8—1222no—
359C2×C8—1221no—
360C2×C8—2442no—
361C2×C8D4×D6111yescharacteristic
362C2×C8D4×D6111yescharacteristic
363C4⋊C4—22no—
364C4⋊C4—22no—
365C4⋊C4—611no—
366C4⋊C4—1221no—
367C4⋊C4—1222no—
368C4⋊C4—1222no—
369C4⋊C4—2442no—
370C4⋊C4—2442no—
371C4⋊C4D4×D6111yescharacteristic
372C22⋊C4—21no—
373C22⋊C4—1221no—
374C22⋊C4—1221no—
375C22⋊C4—1221no—
376C42—611no—
377C42—611no—
378C42—611no—
379C42C23×D6111yescharacteristic
380C23—21no—
381C23—611no—
382C23—1221no—
383Q8—82no—
384Q8—4882no—
385Q8—4884no—
386Q8—2442no—
387D4—21no—
388D4—21no—
389D4—1221no—
390D4—1221no—
391D4—2442no—
392D4—2442no—
393D4—2442no—
394D4—2442no—
395C2×C4—21no—
396C2×C4—41no—
397C2×C4—41no—
398C2×C4—42no—
399C2×C4—611no—
400C2×C4—611no—
401C2×C4—612no—
402C2×C4—612no—
403C2×C4—611no—
404C2×C4—1222no—
405C2×C4—1221no—
406C2×C4—1221no—
407C2×C4—1221no—
408C2×C4—4884no—
409C2×C4—2442no—
410C2×C4—2441no—
411C2×C4—2441no—
412C2×C4C2×D4×D6111yescharacteristic
413C2×C4C2×D4×D6111yescharacteristic
414C2×C4#41×D6111yescharacteristic
415C8—21no—
416C8—41no—
417C8—42no—
418C8—612no—
419C8—1221no—
420C8—2441no—
421C8—2442no—
422C8SD16×D6112yes—
423C22—41no—
424C22—41no—
425C22—41no—
426C22—612no—
427C22—1221no—
428C22—1221no—
429C22—2441no—
430C22—2441no—
431C22—2441no—
432C22#121×D6111yescharacteristic
433C4—21no—
434C4—41no—
435C4—82no—
436C4—612no—
437C4—612no—
438C4—1221no—
439C4—4882no—
440C4—2441no—
441C4C2×SD16×D6111yescharacteristic
442C4#137×D6111yescharacteristic
443C4#1684057511yescharacteristic
444C4#1684057611yescharacteristic
445C2—81no—
446C2—1222no—
447C2—4881no—
448C2#461×D6111yescharacteristic
449C2#2145546911yescharacteristic
450C2#472×D6111yescharacteristic
subgroupG/H
centerC22#121×D61
commutator, cosocleC42×C61C24
FrattiniC42C23×D61
FittingC61×#1675C2
socleC22×C61C2×#121
prime radical#16840309C22
hypercenter#1675D61
hypocenterC61#19193
LCSG ▷ C42×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C42 ◁ #1675

Complements

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

NH ≅ G/Ntype
2. #500×D61C2semidirect
4. #21445341C2semidirect
5. #21445340C2semidirect
6. #21445219C2semidirect
7. #21445218C2semidirect
8. #21445215C2semidirect
9. #21444566C2semidirect
10. #21444498C2semidirect
11. C61×#1675C2semidirect
228. #1675D61semidirect
272. C61#19193semidirect

Maximal subgroups

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

subgroupindex
#214569462
#500×D612
#214545262
#214453412
#214453402
#214452192
#214452182
#214452152
#214445662
#214444982
C61×#16752
#1919361