Finite Group Databasebeta

← #12109061

Subgroups — #12109061

Order 1280.

1086 conjugacy classes of subgroups.

The subgroup lattice has 418 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#66062C211yesmaximal, characteristic
2C2×#64994C211yesmaximal, characteristic
3#10605879C211yesmaximal, characteristic
4C5×#1993C211yesmaximal, characteristic, complemented
5#10602581C211yesmaximal, characteristic, complemented
6#10600615C211yesmaximal, characteristic, complemented
7#10600345C211yesmaximal, characteristic, complemented
8#10599697C214yesmaximal, complemented
9#10599672C214yesmaximal
10C2×C5×C4⋊Q8C2211yescharacteristic
11C2×#4092—21no—
12C2×Q8×Dic5—21no—
13#66062—24no—
14#66062C418yescomplemented
15C2×#4043C2212yes—
16C2×#4038—21no—
17C2×#4038C2212yes—
18C2×#4036—21no—
19C2×#4033C2212yes—
20C2×C42×D5C2211yescharacteristic
21C2×#4015C2211yescharacteristic
22C2×C5×#97C2211yescharacteristic
23C5×#382C2211yescharacteristic
24C2×#3952C2212yes—
25C2×#3948C2214yes—
26C2×#3946C2212yes—
27#65265C2212yes—
28C2×#3942C2214yes—
29#65225C2211yescharacteristic
30C2×#3935C2211yescharacteristic
31#64994—24no—
32#64956—21no—
33#64956—21no—
34#64938—21no—
35#64938—21no—
36C5×#340C2214yes—
37#64749C2214yes—
38#21870—51nomaximal, Sylow
39C22×C5×Q8—21no—
40C22×C4×D5D411yescharacteristic
41C22×C4×D5D411yescharacteristic
42C22×C4×D5C2311yescharacteristic
43C22×Dic10C2312yes—
44C22×C5×C8C2312yes—
45C5×C4⋊Q8—22no—
46C5×C4⋊Q8C2×C414yes—
47C2×C5×C4⋊C4—21no—
48C2×C5×C4⋊C4C2312yes—
49C2×C5×C4⋊C4C2312yes—
50C2×C42×C5C2311yescharacteristic
51#4092—28no—
52Q8×Dic5—28no—
53C2×#685—22no—
54C2×#685—22no—
55C2×C4⋊Dic5—21no—
56C2×C4⋊Dic5—21no—
57C2×C4⋊Dic5—22no—
58C2×C4⋊Dic5C2312yes—
59C2×#683—22no—
60C2×#683—22no—
61C2×#683—22no—
62C2×C4×Dic5—22no—
63C2×C4×Dic5—22no—
64C2×C4×Dic5D411yescharacteristic
65C2×C4×Dic5D411yescharacteristic
66C2×C4×Dic5C2311yescharacteristic
67C22×#168C2312yes—
68#4043—24no—
69#4043C2×C418yes—
70#4038—28no—
71#4038—44no—
72#4036—28no—
73#4033—24no—
74#4033C2×C418yes—
75C42×D5C2×C414yes—
76C42×D5D414yes—
77#4015—22no—
78#4015C2×C414yes—
79C5×#97—22no—
80C5×#95—21no—
81C5×#95—21no—
82#3963—21no—
83#3963—21no—
84#3952—24no—
85#3948—44no—
86#3946—24no—
87#3942—44no—
88#3935—22no—
89C2×#533—51no—
90C2×#432—51no—
91#2285—51no—
92#1993—51no—
93#1993D511yescharacteristic, complemented
94#1960—51no—
95#1960—51no—
96#1873—54no—
97#1803—54no—
98C23×D5C2×D411yescharacteristic
99C2×C5×Q8—24no—
100C2×C5×Q8—44no—
101C22×C4×C5—22no—
102C22×C4×C5—22no—
103C22×C4×C5C2×D411yescharacteristic
104C22×C4×C5C2×D411yescharacteristic
105C22×C4×C5C2411yescharacteristic
106C22×Dic5—22no—
107C22×Dic5—22no—
108C22×Dic5C2×D411yescharacteristic
109C22×Dic5C2×D411yescharacteristic
110C22×Dic5C2×D411yescharacteristic
111C2×C4×D5—22no—
112C2×C4×D5—22no—
113C2×C4×D5—24no—
114C2×C4×D5—42no—
115C2×C4×D5C22⋊C414yes—
116C2×C4×D5C22⋊C414yes—
117C2×C4×D5C22×C412yes—
118C2×C4×D5C2×D412yes—
119C2×Dic10—24no—
120C2×Dic10—24no—
121C2×Dic10—22no—
122C2×Dic10C22×C414yes—
123C2×C5×C8—22no—
124C2×C5×C8—22no—
125C2×C5×C8—22no—
126C5×C4⋊C4—24no—
127C5×C4⋊C4—22no—
128C5×C4⋊C4—24no—
129C5×C4⋊C4C22×C414yes—
130C42×C5C22×C412yes—
131C42×C5C2×D412yes—
132#685—28no—
133#685—44no—
134C4⋊Dic5—24no—
135C4⋊Dic5—24no—
136C4⋊Dic5—28no—
137C4⋊Dic5—24no—
138#683—28no—
139#683—44no—
140#683—44no—
141C4×Dic5—28no—
142C4×Dic5—44no—
143C4×Dic5C22⋊C414yes—
144C4×Dic5C22⋊C414yes—
145C4×Dic5C22×C412yes—
146C4×Dic5C2×D412yes—
147C2×#168—22no—
148C2×#168—22no—
149C2×#168—22no—
150#533—58no—
151#533—104no—
152C2×C4⋊Q8—51no—
153C2×C4⋊Q8C2×D511yescharacteristic
154C2×#125—101no—
155C2×C22⋊Q8—52no—
156C2×C22⋊Q8—101no—
157C2×C4×Q8—52no—
158C2×C4×Q8—101no—
159C2×#117—52no—
160C2×#117—101no—
161C22×C42—51no—
162#432—104no—
163C2×C4⋊C8—52no—
164C2×#103—54no—
165C2×#103—54no—
166C2×C22⋊C8—52no—
167C2×#97—51no—
168C2×#97C2×D511yescharacteristic
169#384—52no—
170#382—51no—
171#382C2×D511yescharacteristic
172#380—101no—
173#380—101no—
174C4×C4⋊C4—101no—
175C4×C4⋊C4—101no—
176#340—54no—
177#340C2×D514yes—
178C23×C5C22×D411yescharacteristic
179C22×D5—21no—
180C22×D5—21no—
181C22×D5—22no—
182C22×D5—22no—
183C22×D5C2×C22⋊C412yes—
184C5×Q8—48no—
185C2×C4×C5—22no—
186C2×C4×C5—22no—
187C2×C4×C5—24no—
188C2×C4×C5—44no—
189C2×C4×C5—42no—
190C2×C4×C5—42no—
191C2×C4×C5—42no—
192C2×C4×C5C2×C22⋊C412yes—
193C2×C4×C5C2×C22⋊C412yes—
194C2×C4×C5C23×C411yescharacteristic
195C2×C4×C5C22×D411yescharacteristic
196C2×C4×C5H(5,𝔽2)12yes—
197C2×C4×C52−1+412yes—
198C2×Dic5—21no—
199C2×Dic5—21no—
200C2×Dic5—22no—
201C2×Dic5—24no—
202C2×Dic5—42no—
203C2×Dic5—42no—
204C2×Dic5—42no—
205C2×Dic5—42no—
206C2×Dic5—42no—
207C2×Dic5C2×C22⋊C412yes—
208C2×Dic5C2×C22⋊C412yes—
209C2×Dic5C2×C22⋊C412yes—
210C2×Dic5#13714yes—
211C4×D5—24no—
212C4×D5—24no—
213C4×D5—28no—
214C4×D5—44no—
215Dic10—28no—
216Dic10—44no—
217C5×C8—42no—
218#168—42no—
219C22×Q8—21no—
220C22×Q8—52no—
221C23×C4—51no—
222C23×C4—51no—
223C23×C4—51no—
224C22×C8—52no—
225C22×C8C22×D512yes—
226C4⋊Q8—22no—
227C4⋊Q8—54no—
228C4⋊Q8—102no—
229C4⋊Q8C4×D514yes—
230#125—108no—
231C22⋊Q8—58no—
232C22⋊Q8—104no—
233C22⋊Q8—108no—
234C4×Q8—58no—
235C4×Q8—104no—
236C4×Q8—108no—
237#117—108no—
238#117—204no—
239C2×C4⋊C4—21no—
240C2×C4⋊C4—52no—
241C2×C4⋊C4—102no—
242C2×C4⋊C4—102no—
243C2×C4⋊C4—101no—
244C2×C4⋊C4—101no—
245C2×C4⋊C4—102no—
246C2×C4⋊C4—102no—
247C2×C4⋊C4C22×D512yes—
248C2×C4⋊C4C22×D512yes—
249C2×C22⋊C4—102no—
250C2×C22⋊C4—102no—
251C2×C42—51no—
252C2×C42—51no—
253C2×C42—51no—
254C2×C42—54no—
255C2×C42—54no—
256C2×C42—102no—
257C2×C42—102no—
258C2×C42C22×D511yescharacteristic
259C4⋊C8—104no—
260#103—204no—
261#103—204no—
262C22⋊C8—104no—
263#97—22no—
264#97—102no—
265#95—21no—
266#95—21no—
267#95—101no—
268#95—101no—
269C22×C5C22×C22⋊C411yescharacteristic
270C22×C5#53511yescharacteristic
271C22×C5#53611yescharacteristic
272C22×C5C2×#13712yes—
273C22×C5#57512yes—
274C2×D5—22no—
275C2×D5—22no—
276C2×D5—22no—
277C2×D5—42no—
278C2×D5—42no—
279C4×C5—22no—
280C4×C5—22no—
281C4×C5—44no—
282C4×C5—82no—
283C4×C5#52012yes—
284C4×C5#53712yes—
285Dic5—22no—
286Dic5—22no—
287Dic5—44no—
288Dic5—82no—
289Dic5#41914yes—
290C24—51no—
291C2×Q8—24no—
292C2×Q8—44no—
293C2×Q8—54no—
294C2×Q8—102no—
295C2×Q8—104no—
296C2×Q8—104no—
297C22×C4—22no—
298C22×C4—22no—
299C22×C4—51no—
300C22×C4—51no—
301C22×C4—51no—
302C22×C4—52no—
303C22×C4—52no—
304C22×C4—54no—
305C22×C4—54no—
306C22×C4—104no—
307C22×C4—102no—
308C22×C4—102no—
309C22×C4—102no—
310C22×C4—102no—
311C22×C4—202no—
312C22×C4D4×D511yescharacteristic
313C22×C4D4×D511yescharacteristic
314C22×C4C23×D511yescharacteristic
315C2×C8—22no—
316C2×C8—22no—
317C2×C8—22no—
318C2×C8—102no—
319C2×C8—102no—
320C2×C8—102no—
321C4⋊C4—22no—
322C4⋊C4—24no—
323C4⋊C4—24no—
324C4⋊C4—108no—
325C4⋊C4—108no—
326C4⋊C4—104no—
327C4⋊C4—104no—
328C4⋊C4—104no—
329C4⋊C4—204no—
330C4⋊C4—204no—
331C4⋊C4C2×C4×D514yes—
332C22⋊C4—108no—
333C22⋊C4—204no—
334C42—52no—
335C42—52no—
336C42—54no—
337C42—54no—
338C42—108no—
339C42—204no—
340C42C2×C4×D512yes—
341C42D4×D512yes—
342C2×C5#231111yescharacteristic
343C2×C5C2×#41911yescharacteristic
344C2×C5#289611yescharacteristic
345C2×C5#321612yes—
346C2×C5#323712yes—
347D5—42no—
348C23—52no—
349C23—101no—
350C23—102no—
351C23—102no—
352C23—101no—
353C23C2×D4×D511yescharacteristic
354Q8—48no—
355Q8—108no—
356Q8—204no—
357C2×C4—22no—
358C2×C4—22no—
359C2×C4—24no—
360C2×C4—44no—
361C2×C4—42no—
362C2×C4—42no—
363C2×C4—42no—
364C2×C4—52no—
365C2×C4—54no—
366C2×C4—52no—
367C2×C4—52no—
368C2×C4—108no—
369C2×C4—101no—
370C2×C4—102no—
371C2×C4—101no—
372C2×C4—104no—
373C2×C4—104no—
374C2×C4—104no—
375C2×C4—204no—
376C2×C4—202no—
377C2×C4—202no—
378C2×C4—202no—
379C2×C4—202no—
380C2×C4—202no—
381C2×C4D5×C22⋊C412yes—
382C2×C4D5×C22⋊C412yes—
383C2×C4C22×C4×D511yescharacteristic
384C2×C4C2×D4×D511yescharacteristic
385C2×C4#414712yes—
386C2×C4#414912yes—
387C8—42no—
388C8—202no—
389C5#2187011yescharacteristic, complemented, Sylow
390C22—102no—
391C22—102no—
392C22—102no—
393C22—202no—
394C22—202no—
395C22C2×D5×C22⋊C411yescharacteristic
396C22#6600811yescharacteristic
397C22#6605811yescharacteristic
398C22#6610712yes—
399C22D5×#13712yes—
400C4—22no—
401C4—22no—
402C4—44no—
403C4—54no—
404C4—82no—
405C4—102no—
406C4—102no—
407C4—204no—
408C4—402no—
409C4#6590812yes—
410C4#6596312yes—
411C2—202no—
412C2#1060821311yescharacteristic
413C2#1061065511yescharacteristic
414C2D5×#41911yescharacteristic
415C2#1061341812yes—
416C2#1061371212yes—
subgroupG/H
centerC23C2×D4×D5
commutatorC2×C4×C5C23×C4
FrattiniC22×C4C23×D5
FittingC5×#1993C2
socleC23×C5C22×D4
prime radicalC23×D5C2×D4
cosocleC22×C4×C5C24
hypercenter#1993D5
hypocenterC5#21870
LCSG ▷ C2×C4×C5 ▷ C2×C5 ▷ C5
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1993

Complements

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

NH ≅ G/Ntype
4. C5×#1993C2semidirect
5. #10602581C2semidirect
6. #10600615C2semidirect
7. #10600345C2semidirect
8. #10599697C2semidirect
14. #66062C4semidirect
93. #1993D5semidirect
389. C5#21870semidirect

Maximal subgroups

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

subgroupindex
C2×#660622
C2×#649942
#106058792
C5×#19932
#106025812
#106006152
#106003452
#105996972
#105996722
#218705