Finite Group Databasebeta

← #29700837

Subgroups — #29700837

Order 15616.

1012 conjugacy classes of subgroups.

The subgroup lattice has 458 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16839934C211yesmaximal, characteristic
2#21454651C212yesmaximal, complemented
3#21454351C212yesmaximal, complemented
4#21446636C211yesmaximal, characteristic, complemented
5#21446477C212yesmaximal, complemented
6#21446324C212yesmaximal, complemented
7#21446046C212yesmaximal, complemented
8#21445660C212yesmaximal, complemented
9C61×#1885C211yesmaximal, characteristic, complemented
10C2×#14519825C2211yescharacteristic
11C2×#14519803C2212yes—
12C2×#14519785C2212yes—
13#16840242C2212yes—
14#16839934C418yescomplemented
15#16839872—22no—
16#16839862—22no—
17C4×#11758200—22no—
18C4×#11758199—22no—
19#16839805C2212yes—
20C2×#14519705C2212yes—
21#16839769—22no—
22#16839764C2212yes—
23#16839762—22no—
24#16839757—22no—
25#16839756C2212yes—
26C2×#14519701C2212yes—
27#103×D61—22no—
28C2×#14519698C2212yes—
29C2×#14519696C2211yescharacteristic
30#16839718C2212yes—
31C2×#14519688C2211yescharacteristic
32#16839632C2211yescharacteristic
33#16839530C2212yes—
34#16839526C2212yes—
35C2×#124×C61C2211yescharacteristic
36C61×#518C2211yescharacteristic
37C61×#419C2212yes—
38C2×#100×C61C2211yescharacteristic
39C61×#339C2212yes—
40#14519825—24no—
41#14519825—41no—
42#14519825C2×C414yescomplemented
43C2×#11758226—22no—
44C2×#11758222—41no—
45C2×#11758222C2312yes—
46C2×Q8×D61—22no—
47#14519803—24no—
48#14519803C2×C418yescomplemented
49C2×#11758200—22no—
50C2×#11758199—22no—
51#14519785—24no—
52#14519785—42no—
53C2×#11758194C2312yes—
54C2×#11758189—22no—
55C22×#92418C2312yes—
56C2×C8×D61—22no—
57C4×Dic122—22no—
58C4×D244—22no—
59#14519737—42no—
60#14519735—22no—
61C2×C4⋊Dic61—41no—
62C2×C4⋊Dic61C2312yes—
63C4⋊C4×D61—22no—
64#14519710—22no—
65#14519708—22no—
66#14519705C2×C418yescomplemented
67#14519702—22no—
68#14519701—22no—
69#14519701—22no—
70#14519700D412yes—
71#14519700D412yes—
72#14519699—22no—
73#14519698—22no—
74#14519698—22no—
75#14519697D412yes—
76#14519697D412yes—
77#14519696C2×C414yescomplemented
78#14519688—22no—
79#14519688—41no—
80C4×#92418—22no—
81C22×Q8×C61C2311yescharacteristic
82C22×D4×C61C2311yescharacteristic
83C2×M16×C61C2312yes—
84#124×C61—41no—
85#124×C61C2×C414yescomplemented
86C4×D4×C61—22no—
87C4×D4×C61—22no—
88#117×C61C2312yes—
89C2×C22⋊C4×C61—41no—
90C2×C22⋊C4×C61—41no—
91C2×C42×C61C2311yescharacteristic
92#103×C61—22no—
93#103×C61—22no—
94#100×C61—41no—
95C24×C61—21no—
96C2×Q8×C61—21no—
97C2×Q8×C61—22no—
98C2×Q8×C61—22no—
99C2×Q8×C61C22×C412yes—
100C2×D4×C61—22no—
101C2×D4×C61—22no—
102C2×D4×C61—21no—
103C2×D4×C61C22×C412yes—
104C22×C4×C61—21no—
105C22×C4×C61—21no—
106C22×C4×C61—41no—
107C22×C4×C61—41no—
108C22×C4×C61C2411yescharacteristic
109C2×C22⋊D61—22no—
110C22×Dic61—22no—
111#11758226—28no—
112#11758226—44no—
113Q8×D61—28no—
114Q8×D61—44no—
115#11758222—22no—
116#11758222—22no—
117#11758222—44no—
118#11758222—44no—
119#11758222C22×C414yes—
120#11758222#4114yes—
121C2×D244—41no—
122C2×D244C2×D412yes—
123C2×C4×D61—22no—
124C2×C4×D61—42no—
125C2×C4×D61—42no—
126C2×Dic122—41no—
127C2×Dic122C2×D412yes—
128M16×C61—22no—
129M16×C61—22no—
130C2×C8×C61—22no—
131C4⋊C4×C61C2×D412yes—
132C4⋊C4×C61C2×D412yes—
133C22⋊C4×C61—22no—
134C22⋊C4×C61—44no—
135C22⋊C4×C61—42no—
136C42×C61—22no—
137C42×C61—21no—
138C42×C61C22×C412yes—
139#11758200—28no—
140#11758199—28no—
141#11758198—42no—
142C4⋊Dic61—22no—
143C4⋊Dic61—22no—
144C4⋊Dic61—82no—
145#11758196—42no—
146C4×Dic61—42no—
147#11758194—22no—
148#11758194C22×C414yes—
149C2×#92418—22no—
150C2×#92418—22no—
151C2×#92418—22no—
152C2×#92418C2×D412yes—
153C2×#92418C2×D412yes—
154#11758189—44no—
155C8×D61—44no—
156C23×C61—22no—
157C23×C61—41no—
158C23×C61—41no—
159C23×C61—81no—
160C23×C61—81no—
161C23×C61H(5,𝔽2)11yescharacteristic
162C22×D61—22no—
163Q8×C61—24no—
164Q8×C61—42no—
165D4×C61—24no—
166D4×C61—42no—
167C2×C4×C61—22no—
168C2×C4×C61—22no—
169C2×C4×C61—22no—
170C2×C4×C61—22no—
171C2×C4×C61—41no—
172C2×C4×C61—41no—
173C2×C4×C61—41no—
174C2×C4×C61—41no—
175C2×C4×C61—161no—
176C2×C4×C61—161no—
177C2×C4×C61C23×C411yescharacteristic
178C2×C4×C61C22×D412yes—
179C2×C4×C61C2×#4112yes—
180C2×C4×C61H(5,𝔽2)11yescharacteristic
181C22⋊D61—24no—
182C22⋊D61—24no—
183C2×Dic61—22no—
184C2×Dic61—42no—
185C2×Dic61—82no—
186D244—22no—
187D244—44no—
188D244C4×D414yes—
189C4×D61—24no—
190C4×D61—24no—
191C4×D61—48no—
192C4×D61—162no—
193Dic122—22no—
194Dic122—44no—
195Dic122C4×D414yes—
196C8×C61—42no—
197#92418—24no—
198#92418—22no—
199#92418—22no—
200#22893—611nomaximal, Sylow
201C22×C61—21no—
202C22×C61—21no—
203C22×C61—42no—
204C22×C61—82no—
205C22×C61—81no—
206C22×C61—81no—
207C22×C61—161no—
208C22×C61#51811yescharacteristic
209C22×C61D4211yescharacteristic
210C22×C61#55111yescharacteristic
211C2×D61—24no—
212C2×D61—42no—
213C2×D61—42no—
214C4×C61—42no—
215C4×C61—42no—
216C4×C61—42no—
217C4×C61—42no—
218C4×C61C2×C4×D412yes—
219C4×C61#54612yes—
220Dic61—24no—
221Dic61—82no—
222C2×#454—611no—
223#2946—612no—
224#2898—612no—
225#1959—611no—
226#1930—612no—
227#1907—612no—
228#1885D6111yescharacteristic, complemented
229#1859—612no—
230#1765—612no—
231C2×C61—21no—
232C2×C61—21no—
233C2×C61—82no—
234C2×C61#235811yescharacteristic
235C2×C61#328811yescharacteristic
236C2×C61#328911yescharacteristic
237D61—44no—
238C2×2−1+4—611no—
239C2×AGL(1,ℤ/8)—612no—
240C2×#131—612no—
241C2×#124C2×D6111yescharacteristic
242#520—612no—
243#518C2×D6111yescharacteristic
244#454—618no—
245#442—1222no—
246#440—1222no—
247C4×SD16—1222no—
248C4×D8—1222no—
249#428—612no—
250C2×#104—612no—
251#419—1222no—
252#419C2×D6112yes—
253#417—612no—
254#417—1222no—
255#416—612no—
256#416—1222no—
257C2×#103—612no—
258C2×#103—1222no—
259C2×#102—612no—
260C2×#101—611no—
261C2×#100C2×D6111yescharacteristic
262#405—612no—
263C2×#97—611no—
264#384—611no—
265#343—612no—
266#341—612no—
267#339C2×D6112yes—
268C61#2289311yescharacteristic, complemented, Sylow
2692−1+4—614no—
2702−1+4—1224no—
2712−1+4—2441no—
272C2×#41—612no—
273C2×#41—1222no—
274C2×#41—2441no—
275C22×Q8—1222no—
276C22×Q8C22×D6111yescharacteristic
277C22×D4C22×D6111yescharacteristic
278AGL(1,ℤ/8)—618no—
279AGL(1,ℤ/8)—1224no—
280C2×SD16—1222no—
281C2×D8—1222no—
282#131—1224no—
283#131—2442no—
284C2×M16—612no—
285C2×M16—1222no—
286C2×M16C22×D6112yes—
287C22×C8—612no—
288C22×C8—1222no—
289#124—41no—
290#124C4×D6114yescomplemented
291C4×Q8—1222no—
292C4×D4—22no—
293C4×D4—22no—
294C4×D4—1222no—
295C4×D4—2442no—
296#117—1222no—
297#117C22×D6112yes—
298C2×C4⋊C4—612no—
299C2×C4⋊C4—1222no—
300C2×C4⋊C4—2441no—
301C2×C22⋊C4—41no—
302C2×C22⋊C4—41no—
303C2×C42C22×D6111yescharacteristic
304#107—1222no—
305#106—1222no—
306#104—618no—
307#103—22no—
308#103—22no—
309#103—612no—
310#103—612no—
311#103—1222no—
312#103—1222no—
313#103—1222no—
314#102—612no—
315#102—612no—
316#102—1222no—
317#102—1222no—
318#102—1222no—
319#101—614no—
320#100—41no—
321#97—1222no—
322#97—2441no—
323C4×C8—1222no—
324C24—21no—
325#41—614no—
326#41—614no—
327#41—1222no—
328#41—1222no—
329#41—1228no—
330#41—2444no—
331#41—2444no—
332#41—2444no—
333C2×Q8—21no—
334C2×Q8—22no—
335C2×Q8—22no—
336C2×Q8—612no—
337C2×Q8—1228no—
338C2×Q8—2441no—
339C2×Q8—2444no—
340C2×Q8C2×C4×D6112yes—
341C2×D4—22no—
342C2×D4—22no—
343C2×D4—21no—
344C2×D4—612no—
345C2×D4—1222no—
346C2×D4—2441no—
347C2×D4C2×C4×D6112yes—
348C22×C4—21no—
349C22×C4—21no—
350C22×C4—41no—
351C22×C4—41no—
352C22×C4—1222no—
353C22×C4—1222no—
354C22×C4—2442no—
355C22×C4—2442no—
356C22×C4C23×D6111yescharacteristic
357SD16—1228no—
358D8—1228no—
359M16—22no—
360M16—22no—
361M16—614no—
362M16—1222no—
363M16—2444no—
364C2×C8—22no—
365C2×C8—612no—
366C2×C8—612no—
367C2×C8—1222no—
368C2×C8—1222no—
369C2×C8—1222no—
370C2×C8—2444no—
371C4⋊C4—1222no—
372C4⋊C4—1222no—
373C4⋊C4—4882no—
374C4⋊C4—2442no—
375C4⋊C4D4×D6112yes—
376C4⋊C4D4×D6112yes—
377C22⋊C4—22no—
378C22⋊C4—44no—
379C22⋊C4—42no—
380C22⋊C4—2442no—
381C42—22no—
382C42—21no—
383C42—2442no—
384C42C2×C4×D6112yes—
385C23—22no—
386C23—41no—
387C23—41no—
388C23—81no—
389C23—81no—
390C23—1222no—
391C23#1451982211yescharacteristic
392Q8—24no—
393Q8—42no—
394Q8—614no—
395Q8—1222no—
396Q8—2444no—
397D4—24no—
398D4—42no—
399D4—614no—
400D4—1224no—
401D4—1224no—
402D4—1222no—
403D4—2444no—
404C2×C4—22no—
405C2×C4—22no—
406C2×C4—22no—
407C2×C4—22no—
408C2×C4—41no—
409C2×C4—41no—
410C2×C4—41no—
411C2×C4—41no—
412C2×C4—161no—
413C2×C4—161no—
414C2×C4—1222no—
415C2×C4—1224no—
416C2×C4—1224no—
417C2×C4—9762no—
418C2×C4—4882no—
419C2×C4—2448no—
420C2×C4—2442no—
421C2×C4C22×C4×D6111yescharacteristic
422C2×C4C2×D4×D6112yes—
423C2×C4#41×D6112yes—
424C2×C4#1451982211yescharacteristic
425C8—42no—
426C8—1224no—
427C8—1222no—
428C8—1222no—
429C22—21no—
430C22—21no—
431C22—42no—
432C22—82no—
433C22—81no—
434C22—81no—
435C22—161no—
436C22—1224no—
437C22—2442no—
438C22—2442no—
439C22#1684023511yescharacteristic
440C22#1684039411yescharacteristic
441C22#1684044011yescharacteristic
442C4—42no—
443C4—42no—
444C4—42no—
445C4—42no—
446C4—1224no—
447C4—4882no—
448C4C4×D4×D6112yes—
449C4#1684040912yes—
450C2—21no—
451C2—21no—
452C2—82no—
453C2—2444no—
454C2#2144880211yescharacteristic
455C2#2145758811yescharacteristic
456C2#2145759711yescharacteristic
subgroupG/H
centerC22#16840394
commutatorC2×C4×C61C23×C4
FrattiniC22×C4C23×D61
FittingC61×#1885C2
socleC22×C61D42
prime radicalC2×#16839934C2
cosocleC22×C4×C61C24
hypercenter#1885D61
hypocenterC61#22893
LCSG ▷ C2×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C22×C4 ◁ #1885

Complements

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

NH ≅ G/Ntype
2. #21454651C2semidirect
3. #21454351C2semidirect
4. #21446636C2semidirect
5. #21446477C2semidirect
6. #21446324C2semidirect
7. #21446046C2semidirect
8. #21445660C2semidirect
9. C61×#1885C2semidirect
14. #16839934C4semidirect
42. #14519825C2×C4semidirect
48. #14519803C2×C4semidirect
66. #14519705C2×C4semidirect
77. #14519696C2×C4semidirect
85. #124×C61C2×C4semidirect
228. #1885D61semidirect
268. C61#22893semidirect
290. #124C4×D61semidirect

Maximal subgroups

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

subgroupindex
C2×#168399342
#214546512
#214543512
#214466362
#214464772
#214463242
#214460462
#214456602
C61×#18852
#2289361