Finite Group Databasebeta

← #12305744

Subgroups — #12305744

Order 1280.

2006 conjugacy classes of subgroups.

The subgroup lattice has 424 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C5×#2842C211yesmaximal, characteristic, complemented
2#10609857C213yesmaximal, complemented
3#10609825C211yesmaximal, characteristic, complemented
4#10609750C211yesmaximal, characteristic, complemented
5#10609748C213yesmaximal, complemented
6#10609184C216yesmaximal, complemented
7#10609182C212yesmaximal, complemented
8#10609014C212yesmaximal, complemented
9#10609012C216yesmaximal, complemented
10#10605591C213yesmaximal, complemented
11#10605549C213yesmaximal, complemented
12C24×Dic5C2211yescharacteristic
13C2×#4099C2213yes—
14C2×#4039C2213yescomplemented
15C2×#4038C2213yescomplemented
16C2×#4027C2213yes—
17C2×#4027C2216yes—
18C5×#399C2213yes—
19C5×#399C2211yescharacteristic
20C5×#396C2212yes—
21C5×#396C2216yes—
22C5×#379C2213yes—
23#65283C2216yescomplemented
24#65282C2216yescomplemented
25#65281C2213yescomplemented
26#65280C2216yescomplemented
27#65275C2216yescomplemented
28#65273C2216yescomplemented
29#65271C2213yescomplemented
30C4⋊C4×Dic5C2216yescomplemented
31#65250C2216yes—
32#65250C2216yes—
33#65250C2216yes—
34#65249C2216yes—
35#65249C2213yes—
36#65247C2216yes—
37#65247C2213yes—
38#65242C2216yes—
39#65242C2213yes—
40#65235C2216yes—
41#65235C2213yes—
42C22⋊C4×Dic5C2213yes—
43C22⋊C4×Dic5C2216yes—
44#64964C2216yescomplemented
45#64960C2211yescharacteristic, complemented
46#64955C2213yescomplemented
47C5⋊#401C2212yescomplemented
48#64950C2213yescomplemented
49#64937C2211yescharacteristic, complemented
50#36676—51nomaximal, Sylow
51C25×C5C2311yescharacteristic
52C22×C22⋊D5C2313yes—
53C22×C22⋊D5C2316yes—
54C23×Dic5—46no—
55C23×Dic5C2311yescharacteristic
56C23×Dic5C2312yes—
57C23×Dic5C2312yes—
58C23×Dic5C2311yescharacteristic
59C22×D20C2313yes—
60C22×C4×D5C2313yes—
61C2×C5×C4⋊C4C2316yes—
62C2×C5×C4⋊C4C2313yes—
63C2×C5×C4⋊C4C2313yes—
64C2×C5×C22⋊C4C2316yes—
65C2×C5×C22⋊C4C2316yes—
66C2×C5×C22⋊C4C2313yes—
67C2×C5×C22⋊C4C2313yes—
68#4099—212no—
69C2×C22⋊Dic5C2313yes—
70C2×C22⋊Dic5C2316yes—
71C2×#685C2316yes—
72C2×#685C2316yes—
73C2×#685C2316yes—
74C2×#685C2313yes—
75C2×#685C2316yes—
76C2×C4⋊Dic5C2316yes—
77C2×C4⋊Dic5C2313yes—
78C2×#683C2313yes—
79C2×#683C2316yes—
80C2×C4×Dic5C2316yes—
81C2×C4×Dic5C2313yes—
82#4039—46no—
83#4038—46no—
84#4027—412no—
85#4027—46no—
86C5×#95C2311yescharacteristic
87C5×#95C2313yes—
88#3963C2316yes—
89#3963C2313yes—
90#3963C2316yes—
91#3963C2316yes—
92#3963C2316yes—
93#3963C2313yes—
94#3963C2316yes—
95#2842D511yescharacteristic, complemented
96#2834—51no—
97#2827—56no—
98#2826—53no—
99#2825—52no—
100#2812—53no—
101#2401—52no—
102#2397—51no—
103#2395—53no—
104#2392—56no—
105C4×#120—53no—
106C24×C5—43no—
107C24×C5—43no—
108C24×C5#4114yes—
109C24×C5C2412yes—
110C24×C5C2411yescharacteristic
111C23×D5C2413yes—
112C22×C4×C5C2413yes—
113C22×C4×C5C2413yes—
114C22×C4×C5C2416yes—
115C2×C22⋊D5—26no—
116C2×C22⋊D5—212no—
117C2×C22⋊D5—43no—
118C2×C22⋊D5—46no—
119C2×C22⋊D5—46no—
120C2×C22⋊D5—43no—
121C22×Dic5—26no—
122C22×Dic5—212no—
123C22×Dic5—46no—
124C22×Dic5—43no—
125C22×Dic5—416no—
126C22×Dic5C2416yes—
127C22×Dic5C2413yes—
128C22×Dic5C2412yes—
129C22×Dic5C2411yescharacteristic
130C22×Dic5C2411yescharacteristic
131C2×D20—46no—
132C2×D20—43no—
133C2×C4×D5—26no—
134C2×C4×D5—212no—
135C5×C4⋊C4—46no—
136C5×C4⋊C4—43no—
137C5×C4⋊C4—43no—
138C5×C22⋊C4—46no—
139C5×C22⋊C4—43no—
140C5×C22⋊C4—46no—
141C5×C22⋊C4—43no—
142C22⋊Dic5—26no—
143C22⋊Dic5—212no—
144#685—46no—
145#685—43no—
146#685—46no—
147#685—46no—
148#685—46no—
149C4⋊Dic5—43no—
150C4⋊Dic5—46no—
151#683—46no—
152#683—43no—
153C4×Dic5—212no—
154C4×Dic5—26no—
155C24×C4—51no—
156C2×#120—53no—
157C2×C4×D4—53no—
158C2×C4×D4—56no—
159C2×C4×D4—53no—
160C2×#117—53no—
161#401—52no—
162#400—56no—
163#400—53no—
164#399—53no—
165#399C2×D513yes—
166#399C2×D511yescharacteristic
167#398—56no—
168#398—53no—
169#397—56no—
170#397—56no—
171#397—53no—
172#397—56no—
173#396—56no—
174#396C2×D512yes—
175#396C2×D516yes—
176#394—56no—
177#394—56no—
178#394—56no—
179#394—56no—
180#392—56no—
181#392—53no—
182#392—51no—
183#386—56no—
184#386—53no—
185#385—53no—
186#385—56no—
187#379C2×D513yes—
188C4×C4⋊C4—56no—
189C4×C22⋊C4—56no—
190C4×C22⋊C4—53no—
191C4×C22⋊C4—53no—
192#376—51no—
193C23×C5—26no—
194C23×C5—23no—
195C23×C5—26no—
196C23×C5—23no—
197C23×C5—412no—
198C23×C5—44no—
199C23×C5—412no—
200C23×C5C2×#4114yes—
201C23×C5C2×#4112yes—
202C23×C5C2511yescharacteristic
203C22×D5—23no—
204C22×D5—23no—
205C22×D5—26no—
206C22×D5—26no—
207C22×D5—23no—
208C2×C4×C5—26no—
209C2×C4×C5—23no—
210C2×C4×C5—23no—
211C2×C4×C5—26no—
212C2×C4×C5—26no—
213C2×C4×C5—26no—
214C2×C4×C5—26no—
215C22⋊D5—424no—
216C22⋊D5—412no—
217C2×Dic5—26no—
218C2×Dic5—23no—
219C2×Dic5—26no—
220C2×Dic5—23no—
221C2×Dic5—26no—
222C2×Dic5—23no—
223C2×Dic5—23no—
224C2×Dic5—26no—
225C2×Dic5—48no—
226C2×Dic5—44no—
227C2×Dic5—44no—
228C2×Dic5—48no—
229C2×Dic5H(5,𝔽2)16yes—
230D20—412no—
231C4×D5—412no—
232C25C22×D511yescharacteristic
233C22×D4—56no—
234C22×D4—53no—
235C22×D4—53no—
236C23×C4—52no—
237C23×C4—51no—
238C23×C4—51no—
239C23×C4—52no—
240C23×C4—53no—
241C23×C4—206no—
242#120—1012no—
243C4×D4—2012no—
244C4×D4—206no—
245C4×D4—206no—
246#117—206no—
247C2×C4⋊C4—53no—
248C2×C4⋊C4—56no—
249C2×C4⋊C4—53no—
250C2×C4⋊C4—56no—
251C2×C4⋊C4C22×D513yes—
252C2×C4⋊C4C22×D513yes—
253C2×C4⋊C4C22×D516yes—
254C2×C22⋊C4—56no—
255C2×C22⋊C4—53no—
256C2×C22⋊C4—56no—
257C2×C22⋊C4—53no—
258C2×C22⋊C4—56no—
259C2×C22⋊C4—56no—
260C2×C22⋊C4—56no—
261C2×C22⋊C4C22×D513yes—
262C2×C22⋊C4C22×D516yes—
263C2×C22⋊C4C22×D516yes—
264C2×C22⋊C4C22×D513yes—
265C2×C42—56no—
266C2×C42—53no—
267#95—53no—
268#95—56no—
269#95—56no—
270#95—56no—
271#95—53no—
272#95—56no—
273#95—56no—
274#95C22×D511yescharacteristic
275#95C22×D513yes—
276C22×C5—212no—
277C22×C5—26no—
278C22×C5—416no—
279C22×C5—42no—
280C22×C5—41no—
281C22×C5—46no—
282C22×C5—43no—
283C22×C5C22×#4111yescharacteristic
284C22×C5C2×H(5,𝔽2)13yes—
285C22×C5#58513yes—
286C2×D5—43no—
287C2×D5—46no—
288C2×D5—43no—
289C2×D5—46no—
290C2×D5—43no—
291C4×C5—43no—
292C4×C5—43no—
293C4×C5—46no—
294Dic5—46no—
295Dic5—43no—
296Dic5—42no—
297Dic5—41no—
298Dic5#56014yes—
299C24—43no—
300C24—43no—
301C24—53no—
302C24#71314yes—
303C24C23×D511yescharacteristic
304C24C23×D512yes—
305C2×D4—1012no—
306C2×D4—106no—
307C2×D4—203no—
308C2×D4—206no—
309C2×D4—203no—
310C2×D4—206no—
311C2×D4—206no—
312C2×D4—203no—
313C22×C4—51no—
314C22×C4—53no—
315C22×C4—56no—
316C22×C4—52no—
317C22×C4—51no—
318C22×C4—106no—
319C22×C4—1012no—
320C22×C4—1012no—
321C22×C4—106no—
322C22×C4—203no—
323C22×C4—206no—
324C22×C4—2016no—
325C22×C4C23×D513yes—
326C22×C4C23×D516yes—
327C22×C4C23×D513yes—
328C4⋊C4—43no—
329C4⋊C4—43no—
330C4⋊C4—46no—
331C4⋊C4—203no—
332C4⋊C4—206no—
333C4⋊C4—203no—
334C4⋊C4—206no—
335C22⋊C4—46no—
336C22⋊C4—43no—
337C22⋊C4—43no—
338C22⋊C4—46no—
339C22⋊C4—1012no—
340C22⋊C4—106no—
341C22⋊C4—206no—
342C22⋊C4—206no—
343C22⋊C4—203no—
344C22⋊C4—206no—
345C22⋊C4—206no—
346C42—1012no—
347C42—106no—
348C2×C5—44no—
349C2×C5—42no—
350C2×C5#345913yes—
351C2×C5C2×#56011yescharacteristic
352C2×C5#352913yes—
353D5—83no—
354C23—26no—
355C23—23no—
356C23—23no—
357C23—26no—
358C23—412no—
359C23—44no—
360C23—412no—
361C23—103no—
362C23—103no—
363C23—106no—
364C23—106no—
365C23—103no—
366C23C2×#71312yes—
367C23C2×#71314yes—
368C23C24×D511yescharacteristic
369D4—2012no—
370D4—2012no—
371D4—2024no—
372C2×C4—26no—
373C2×C4—23no—
374C2×C4—26no—
375C2×C4—23no—
376C2×C4—26no—
377C2×C4—26no—
378C2×C4—26no—
379C2×C4—56no—
380C2×C4—103no—
381C2×C4—106no—
382C2×C4—103no—
383C2×C4—106no—
384C2×C4—106no—
385C2×C4—103no—
386C2×C4—106no—
387C2×C4—103no—
388C2×C4—2012no—
389C2×C4—208no—
390C2×C4—204no—
391C2×C4—204no—
392C2×C4—208no—
393C5#3667611yescharacteristic, complemented, Sylow
394C22—26no—
395C22—212no—
396C22—416no—
397C22—46no—
398C22—43no—
399C22—42no—
400C22—41no—
401C22—206no—
402C22—203no—
403C22—203no—
404C22—203no—
405C22—206no—
406C22C22×#71311yescharacteristic
407C22D5×H(5,𝔽2)13yes—
408C22#6628513yes—
409C4—43no—
410C4—46no—
411C4—43no—
412C4—54no—
413C4—202no—
414C4—201no—
415C4—203no—
416C4—206no—
417C2—42no—
418C2—44no—
419C2—403no—
420C2#1061847313yes—
421C2#1061897113yes—
422C2D5×#56011yescharacteristic
subgroupG/H
center, FrattiniC23C24×D5
commutator, socle, cosocleC23×C5C25
FittingC5×#2842C2
hypercenter#2842D5
hypocenterC5#36676
LCSG ▷ C23×C5 ▷ C5
UCSC1 ◁ C23 ◁ #2842

Complements

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

NH ≅ G/Ntype
1. C5×#2842C2semidirect
2. #10609857C2semidirect
3. #10609825C2semidirect
4. #10609750C2semidirect
5. #10609748C2semidirect
6. #10609184C2semidirect
7. #10609182C2semidirect
8. #10609014C2semidirect
9. #10609012C2semidirect
10. #10605591C2semidirect
11. #10605549C2semidirect
14. C2×#4039C22semidirect
15. C2×#4038C22semidirect
23. #65283C22semidirect
24. #65282C22semidirect
25. #65281C22semidirect
26. #65280C22semidirect
27. #65275C22semidirect
28. #65273C22semidirect
29. #65271C22semidirect
30. C4⋊C4×Dic5C22semidirect
44. #64964C22semidirect
45. #64960C22semidirect
46. #64955C22semidirect
47. C5⋊#401C22semidirect
48. #64950C22semidirect
49. #64937C22semidirect
95. #2842D5semidirect
393. C5#36676semidirect

Maximal subgroups

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

subgroupindex
C5×#28422
#106098572
#106098252
#106097502
#106097482
#106091842
#106091822
#106090142
#106090122
#106055912
#106055492
#366765