Finite Group Databasebeta

← #12128377

Subgroups — #12128377

Order 1280.

874 conjugacy classes of subgroups.

The subgroup lattice has 414 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#65169C211yesmaximal, characteristic
2C2×#65128C211yesmaximal, characteristic
3#10607926C212yesmaximal
4C5×#2086C211yesmaximal, characteristic, complemented
5#10601414C211yesmaximal, characteristic, complemented
6#10601256C211yesmaximal, characteristic, complemented
7#10600403C212yesmaximal, complemented
8#10599660C212yesmaximal
9D5×#340C211yesmaximal, characteristic
10#10599634C212yesmaximal, complemented
11#10599632C211yesmaximal, characteristic, complemented
12C22×#676C2211yescharacteristic
13C22×C8×D5C2211yescharacteristic
14C2×#4038C2212yes—
15C2×D5×C4⋊C4C2212yes—
16C2×#4018C2211yescharacteristic
17C2×C5×#107C2211yescharacteristic
18C2×C5×C4⋊C8C2211yescharacteristic
19C5×#389C2212yes—
20C2×#3950C2211yescharacteristic
21C2×#3946C2211yescharacteristic
22C2×#3946C2211yescharacteristic
23#65276—22no—
24#65274C2212yes—
25#65262C2212yes—
26#65261—22no—
27C2×#3940C2212yes—
28#65217C2212yes—
29C2×C4⋊#168C2211yescharacteristic
30#65169—24no—
31#65128—24no—
32#64956—22no—
33#64936—22no—
34C5×#340C2211yescharacteristic
35C5×#340C2212yes—
36#64771C2211yescharacteristic
37#64771C2212yes—
38#64749C2212yes—
39#64749C2212yes—
40#64749C2212yes—
41#23669—51nomaximal, Sylow
42C22×C4×D5—22no—
43C22×C4×D5C2311yescharacteristic
44C22×C5×C8C2311yescharacteristic
45C22×C5×C8C2311yescharacteristic
46C2×C5×C4⋊C4—22no—
47C2×C5×C4⋊C4C2312yes—
48C2×C5×C4⋊C4C2312yes—
49C2×C42×C5C2311yescharacteristic
50C2×#685—22no—
51C2×#685D411yescharacteristic
52C2×#685D411yescharacteristic
53C2×C4⋊Dic5—22no—
54C2×C4⋊Dic5C2312yes—
55C2×C4⋊Dic5C2312yes—
56C2×#683—22no—
57C2×#683—22no—
58C2×#683—22no—
59C2×#683D411yescharacteristic
60C2×#683D411yescharacteristic
61C2×C4×Dic5—22no—
62C2×C4×Dic5C2311yescharacteristic
63C22×#168C2311yescharacteristic
64C22×#168C2311yescharacteristic
65C2×#676—22no—
66C2×#676—24no—
67C2×C8×D5—24no—
68C2×C8×D5D414yes—
69#4038—24no—
70#4038—24no—
71D5×C4⋊C4—24no—
72D5×C4⋊C4Q818yes—
73#4018—22no—
74#4018—22no—
75C5×#107—22no—
76C5×C4⋊C8—22no—
77C5×#95—22no—
78#3963—22no—
79#3963—22no—
80#3963—22no—
81#3950—22no—
82#3946—22no—
83#3946—22no—
84#3940—24no—
85C4⋊#168—22no—
86C2×#427—51no—
87C2×#424—51no—
88#2316—52no—
89#2088—52no—
90#2088—52no—
91#2087—51no—
92#2087—51no—
93#2086—51no—
94#2086D511yescharacteristic, complemented
95#1736—52no—
96C2×#340—51no—
97C23×D5C2×D411yescharacteristic
98C22×C4×C5—22no—
99C22×C4×C5—22no—
100C22×C4×C5C2×D411yescharacteristic
101C22×C4×C5C2×D411yescharacteristic
102C22×C4×C5C2411yescharacteristic
103C22×Dic5—22no—
104C22×Dic5—22no—
105C22×Dic5C2×D411yescharacteristic
106C22×Dic5C2×D411yescharacteristic
107C22×Dic5C2×D411yescharacteristic
108C2×C4×D5—28no—
109C2×C4×D5—82no—
110C2×C4×D5—82no—
111C2×C4×D5C2×D412yes—
112C2×C4×D5C2×Q814yes—
113C2×C4×D5#4114yes—
114C2×C4×D5#4112yes—
115C2×C5×C8—22no—
116C2×C5×C8—21no—
117C2×C5×C8—22no—
118C2×C5×C8C2×D412yes—
119C5×C4⋊C4—22no—
120C5×C4⋊C4—22no—
121C5×C4⋊C4—22no—
122C5×C4⋊C4—82no—
123C5×C4⋊C4C2×Q814yes—
124C42×C5—21no—
125C42×C5—21no—
126#685—22no—
127#685—22no—
128#685—44no—
129C4⋊Dic5—22no—
130C4⋊Dic5—22no—
131C4⋊Dic5—22no—
132C4⋊Dic5—82no—
133C4⋊Dic5C2×Q814yes—
134#683—22no—
135#683—22no—
136#683—28no—
137#683—82no—
138#683—82no—
139C4×Dic5—21no—
140C4×Dic5—21no—
141C4×Dic5—44no—
142C2×#168—22no—
143C2×#168—21no—
144C2×#168—22no—
145C2×#168C2×D412yes—
146#676—44no—
147C8×D5—28no—
148C22×M16—51no—
149C23×C8—51no—
150C2×#117—51no—
151C2×#117—52no—
152C22×C4⋊C4—52no—
153#427—104no—
154C2×#107—51no—
155C2×#107C2×D511yescharacteristic
156C2×#106—52no—
157#424—104no—
158C2×C4⋊C8—51no—
159C2×C4⋊C8—51no—
160C2×C4⋊C8—51no—
161C2×C4⋊C8C2×D511yescharacteristic
162#389—52no—
163#389—52no—
164#389—52no—
165#389—102no—
166#389—102no—
167#389C2×D512yes—
168#380—102no—
169#380—102no—
170#340—51no—
171#340—52no—
172#340—52no—
173#340—52no—
174#340—52no—
175#340C2×D511yescharacteristic
176#340C2×D512yes—
177C23×C5C22×D411yescharacteristic
178C22×D5—21no—
179C22×D5—22no—
180C22×D5—21no—
181C22×D5#12112yes—
182C22×D5C22⋊Q814yes—
183C2×C4×C5—21no—
184C2×C4×C5—21no—
185C2×C4×C5—24no—
186C2×C4×C5—21no—
187C2×C4×C5—22no—
188C2×C4×C5—21no—
189C2×C4×C5—42no—
190C2×C4×C5—42no—
191C2×C4×C5—42no—
192C2×C4×C5—42no—
193C2×C4×C5—42no—
194C2×C4×C5C22×D411yescharacteristic
195C2×C4×C5C22×Q812yes—
196C2×C4×C5C2×#4112yes—
197C2×C4×C5C2×#4111yescharacteristic
198C2×Dic5—21no—
199C2×Dic5—21no—
200C2×Dic5—24no—
201C2×Dic5—21no—
202C2×Dic5—22no—
203C2×Dic5—21no—
204C2×Dic5—42no—
205C2×Dic5—42no—
206C2×Dic5—42no—
207C2×Dic5—42no—
208C2×Dic5—42no—
209C2×Dic5#12112yes—
210C2×Dic5C22⋊Q814yes—
211C4×D5—88no—
212C4×D5#12314yes—
213C4×D5#12518yes—
214C4×D5C4⋊Q814yes—
215C5×C8—22no—
216C5×C8—41no—
217#168—22no—
218#168—41no—
219C23×C4—51no—
220C23×C4—102no—
221C2×M16—102no—
222C2×M16—104no—
223C22×C8—51no—
224C22×C8—54no—
225C22×C8—51no—
226C22×C8—104no—
227C22×C8C22×D511yescharacteristic
228C22×C8C22×D511yescharacteristic
229#117—102no—
230#117—102no—
231#117—104no—
232#117—104no—
233C2×C4⋊C4—22no—
234C2×C4⋊C4—52no—
235C2×C4⋊C4—58no—
236C2×C4⋊C4—51no—
237C2×C4⋊C4—51no—
238C2×C4⋊C4—52no—
239C2×C4⋊C4—104no—
240C2×C4⋊C4—102no—
241C2×C4⋊C4—102no—
242C2×C4⋊C4—102no—
243C2×C4⋊C4—102no—
244C2×C4⋊C4C22×D512yes—
245C2×C4⋊C4C22×D512yes—
246C2×C22⋊C4—51no—
247C2×C22⋊C4—51no—
248C2×C22⋊C4—102no—
249C2×C42—51no—
250C2×C42—102no—
251C2×C42C22×D511yescharacteristic
252#107—22no—
253#107—102no—
254#106—104no—
255C4⋊C8—22no—
256C4⋊C8—102no—
257C4⋊C8—102no—
258C4⋊C8—102no—
259#95—22no—
260#95—102no—
261#95—102no—
262#95—102no—
263C22×C5C2×#12111yescharacteristic
264C22×C5C2×C22⋊Q812yes—
265C22×C5#57611yescharacteristic
266C22×C5#57712yes—
267C22×C5#57811yescharacteristic
268C2×D5—24no—
269C2×D5—24no—
270C2×D5—22no—
271C2×D5—22no—
272C2×D5#39814yes—
273C4×C5—44no—
274C4×C5—41no—
275C4×C5—41no—
276C4×C5—82no—
277C4×C5C2×#12311yescharacteristic
278C4×C5C2×#12512yes—
279C4×C5C2×C4⋊Q811yescharacteristic
280Dic5—44no—
281Dic5—41no—
282Dic5—41no—
283Dic5—82no—
284Dic5#39814yes—
285C24—51no—
286C22×C4—22no—
287C22×C4—22no—
288C22×C4—51no—
289C22×C4—52no—
290C22×C4—54no—
291C22×C4—54no—
292C22×C4—52no—
293C22×C4—51no—
294C22×C4—51no—
295C22×C4—108no—
296C22×C4—102no—
297C22×C4—102no—
298C22×C4—402no—
299C22×C4—402no—
300C22×C4D4×D511yescharacteristic
301C22×C4D4×D511yescharacteristic
302C22×C4C23×D511yescharacteristic
303M16—204no—
304C2×C8—22no—
305C2×C8—21no—
306C2×C8—22no—
307C2×C8—52no—
308C2×C8—108no—
309C2×C8—102no—
310C2×C8—101no—
311C2×C8—102no—
312C2×C8D4×D512yes—
313C4⋊C4—22no—
314C4⋊C4—22no—
315C4⋊C4—22no—
316C4⋊C4—54no—
317C4⋊C4—82no—
318C4⋊C4—108no—
319C4⋊C4—102no—
320C4⋊C4—102no—
321C4⋊C4—102no—
322C4⋊C4—102no—
323C4⋊C4—102no—
324C4⋊C4—402no—
325C4⋊C4—402no—
326C4⋊C4—402no—
327C4⋊C4Q8×D514yes—
328C22⋊C4—102no—
329C22⋊C4—102no—
330C22⋊C4—204no—
331C42—21no—
332C42—21no—
333C42—101no—
334C42—101no—
335C42—204no—
336C2×C5C2×#39811yescharacteristic
337C2×C5#306011yescharacteristic
338C2×C5#306111yescharacteristic
339C2×C5#308112yes—
340C2×C5#308212yes—
341D5—24no—
342C23—52no—
343C23—54no—
344C23—101no—
345C23—101no—
346C23—102no—
347C23C2×D4×D511yescharacteristic
348C2×C4—21no—
349C2×C4—21no—
350C2×C4—22no—
351C2×C4—21no—
352C2×C4—21no—
353C2×C4—24no—
354C2×C4—42no—
355C2×C4—42no—
356C2×C4—42no—
357C2×C4—42no—
358C2×C4—42no—
359C2×C4—52no—
360C2×C4—54no—
361C2×C4—54no—
362C2×C4—54no—
363C2×C4—58no—
364C2×C4—101no—
365C2×C4—101no—
366C2×C4—102no—
367C2×C4—101no—
368C2×C4—101no—
369C2×C4—104no—
370C2×C4—202no—
371C2×C4—202no—
372C2×C4—202no—
373C2×C4—202no—
374C2×C4—202no—
375C2×C4—408no—
376C2×C4C2×D4×D511yescharacteristic
377C2×C4C2×Q8×D512yes—
378C2×C4D5×#4112yes—
379C2×C4D5×#4111yescharacteristic
380C8—22no—
381C8—41no—
382C8—102no—
383C8—201no—
384C5#2366911yescharacteristic, complemented, Sylow
385C22—54no—
386C22—102no—
387C22—104no—
388C22—102no—
389C22—104no—
390C22D5×#12111yescharacteristic
391C22D5×C22⋊Q812yes—
392C22#6609111yescharacteristic
393C22#6610511yescharacteristic
394C22#6611212yes—
395C4—41no—
396C4—44no—
397C4—41no—
398C4—54no—
399C4—82no—
400C4—201no—
401C4—204no—
402C4—201no—
403C4—402no—
404C4D5×#12311yescharacteristic
405C4D5×#12512yes—
406C4D5×C4⋊Q811yescharacteristic
407C2—104no—
408C2D5×#39811yescharacteristic
409C2#1061244811yescharacteristic
410C2#1061263711yescharacteristic
411C2#1061269812yes—
412C2#1061296312yes—
subgroupG/H
centerC23C2×D4×D5
commutator, cosocleC22×C4×C5C24
FrattiniC22×C4C23×D5
FittingC5×#2086C2
socleC23×C5C22×D4
prime radicalC23×D5C2×D4
hypercenter#2086D5
hypocenterC5#23669
LCSG ▷ C22×C4×C5 ▷ C2×C5 ▷ C5
UCSC1 ◁ C23 ◁ C2×C42 ◁ #2086

Complements

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

NH ≅ G/Ntype
4. C5×#2086C2semidirect
5. #10601414C2semidirect
6. #10601256C2semidirect
7. #10600403C2semidirect
10. #10599634C2semidirect
11. #10599632C2semidirect
94. #2086D5semidirect
384. C5#23669semidirect

Maximal subgroups

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

subgroupindex
C2×#651692
C2×#651282
#106079262
C5×#20862
#106014142
#106012562
#106004032
#105996602
D5×#3402
#105996342
#105996322
#236695