Finite Group Databasebeta

← #12108807

Subgroups — #12108807

Order 1280.

1030 conjugacy classes of subgroups.

The subgroup lattice has 416 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#66062C211yesmaximal, characteristic
2C2×#65122C211yesmaximal, characteristic
3#10605483C211yesmaximal, characteristic
4C5×#1988C211yesmaximal, characteristic, complemented
5#10602573C211yesmaximal, characteristic, complemented
6#10601252C211yesmaximal, characteristic, complemented
7#10600282C211yesmaximal, characteristic, complemented
8#10599666C214yesmaximal
9#10599597C214yesmaximal, complemented
10C2×C5×C4⋊Q8C2211yescharacteristic
11C2×#4092—21no—
12C2×Q8×Dic5—21no—
13#66062—24no—
14#66062C418yescomplemented
15C2×#4043—21no—
16C2×#4038C2212yes—
17C2×#4036C2212yes—
18C2×#4033—21no—
19C2×C42×D5C2211yescharacteristic
20C2×#4015C2211yescharacteristic
21C2×C5×C4⋊C8C2211yescharacteristic
22C4×C5×C4⋊C4C2211yescharacteristic
23C2×#3952C2212yes—
24C2×C8×Dic5C2212yes—
25#65276C2212yes—
26#65266C2212yes—
27C4×#685—21no—
28C4×C4⋊Dic5C2211yescharacteristic
29#65221—21no—
30C2×C4⋊#168C2211yescharacteristic
31#65122—24no—
32#64953—21no—
33#64940—21no—
34C5×#340C2214yes—
35#64771C2214yes—
36#64749C2214yes—
37#64749C2214yes—
38#21837—51nomaximal, Sylow
39C22×C5×Q8—21no—
40C22×C4×D5D411yescharacteristic
41C22×C4×D5D411yescharacteristic
42C22×C4×D5C2311yescharacteristic
43C22×Dic10—21no—
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×C5—21no—
51C2×C42×C5C2311yescharacteristic
52#4092—28no—
53Q8×Dic5—28no—
54C2×#685—22no—
55C2×#685—22no—
56C2×C4⋊Dic5—21no—
57C2×C4⋊Dic5—22no—
58C2×C4⋊Dic5C2312yes—
59C2×C4⋊Dic5C2312yes—
60C2×#683—22no—
61C2×C4×Dic5—22no—
62C2×C4×Dic5—21no—
63C2×C4×Dic5D411yescharacteristic
64C2×C4×Dic5D411yescharacteristic
65C2×C4×Dic5C2311yescharacteristic
66C22×#168C2312yes—
67#4043—28no—
68#4038—24no—
69#4038C2×C418yes—
70#4036—24no—
71#4036C2×C418yes—
72#4033—28no—
73C42×D5C2×C414yes—
74C42×D5D414yes—
75#4015—22no—
76#4015C2×C414yes—
77C5×C4⋊C8—22no—
78C5×#95—21no—
79#3963—21no—
80#3963—22no—
81#3963—22no—
82#3963—22no—
83#3952—24no—
84C8×Dic5—24no—
85C4⋊#168—22no—
86C2×#533—51no—
87C2×#431—51no—
88#2283—51no—
89#1991—54no—
90#1988—51no—
91#1988D511yescharacteristic, complemented
92#1987—51no—
93#1987—51no—
94#1825—54no—
95C23×D5C2×D411yescharacteristic
96C2×C5×Q8—24no—
97C2×C5×Q8—42no—
98C2×C5×Q8—42no—
99C22×C4×C5—22no—
100C22×C4×C5—22no—
101C22×C4×C5C2×D411yescharacteristic
102C22×C4×C5C2×D411yescharacteristic
103C22×C4×C5C2411yescharacteristic
104C22×Dic5—22no—
105C22×Dic5—22no—
106C22×Dic5C2×D411yescharacteristic
107C22×Dic5C2×D411yescharacteristic
108C22×Dic5C2×D411yescharacteristic
109C2×C4×D5—22no—
110C2×C4×D5—22no—
111C2×C4×D5—22no—
112C2×C4×D5—22no—
113C2×C4×D5—42no—
114C2×C4×D5C22⋊C414yes—
115C2×C4×D5C22⋊C414yes—
116C2×C4×D5C22×C412yes—
117C2×C4×D5C2×D412yes—
118C2×Dic10—24no—
119C2×Dic10—44no—
120C2×C5×C8—22no—
121C2×C5×C8—24no—
122C5×C4⋊C4—24no—
123C5×C4⋊C4—22no—
124C5×C4⋊C4—22no—
125C5×C4⋊C4—22no—
126C5×C4⋊C4C22×C414yes—
127C42×C5—42no—
128C42×C5C22×C412yes—
129C42×C5C2×D412yes—
130#685—28no—
131#685—44no—
132C4⋊Dic5—24no—
133C4⋊Dic5—22no—
134C4⋊Dic5—28no—
135C4⋊Dic5—22no—
136C4⋊Dic5—22no—
137C4⋊Dic5C22×C414yes—
138#683—28no—
139C4×Dic5—28no—
140C4×Dic5—42no—
141C4×Dic5C22⋊C414yes—
142C4×Dic5C22⋊C414yes—
143C4×Dic5C22×C412yes—
144C4×Dic5C2×D412yes—
145C2×#168—22no—
146C2×#168—24no—
147#533—58no—
148#533—104no—
149C2×C4⋊Q8—51no—
150C2×C4⋊Q8C2×D511yescharacteristic
151C2×#125—52no—
152C2×C22⋊Q8—101no—
153C2×C22⋊Q8—101no—
154C2×C4×Q8—101no—
155C2×C4×Q8—101no—
156C2×#117—52no—
157C22×C42—51no—
158#431—104no—
159C2×C4⋊C8—51no—
160C2×C4⋊C8C2×D511yescharacteristic
161C2×C22⋊C8—52no—
162C2×C4×C8—52no—
163#383—101no—
164#382—52no—
165#381—101no—
166#380—52no—
167C4×C4⋊C4—51no—
168C4×C4⋊C4C2×D511yescharacteristic
169C4×C22⋊C4—101no—
170#376—101no—
171#340—54no—
172#340—54no—
173#340—54no—
174#340C2×D514yes—
175C23×C5C22×D411yescharacteristic
176C22×D5—22no—
177C22×D5—21no—
178C22×D5—21no—
179C22×D5—22no—
180C22×D5C2×C22⋊C412yes—
181C5×Q8—48no—
182C2×C4×C5—22no—
183C2×C4×C5—24no—
184C2×C4×C5—42no—
185C2×C4×C5—42no—
186C2×C4×C5—42no—
187C2×C4×C5—42no—
188C2×C4×C5—42no—
189C2×C4×C5C2×C22⋊C412yes—
190C2×C4×C5C2×C22⋊C412yes—
191C2×C4×C5#13512yes—
192C2×C4×C5#13512yes—
193C2×C4×C5C23×C411yescharacteristic
194C2×C4×C5C22×D411yescharacteristic
195C2×C4×C52−1+412yes—
196C2×C4×C52−1+412yes—
197C2×Dic5—22no—
198C2×Dic5—21no—
199C2×Dic5—21no—
200C2×Dic5—22no—
201C2×Dic5—24no—
202C2×Dic5—44no—
203C2×Dic5—42no—
204C2×Dic5—42no—
205C2×Dic5—42no—
206C2×Dic5C2×C22⋊C412yes—
207C2×Dic5C2×C22⋊C412yes—
208C2×Dic5C2×C22⋊C412yes—
209C4×D5—24no—
210C4×D5—22no—
211C4×D5—22no—
212C4×D5—28no—
213C4×D5—44no—
214Dic10—48no—
215C5×C8—42no—
216#168—42no—
217C22×Q8—21no—
218C22×Q8—101no—
219C23×C4—51no—
220C23×C4—51no—
221C23×C4—51no—
222C22×C8—52no—
223C22×C8C22×D512yes—
224C4⋊Q8—22no—
225C4⋊Q8—54no—
226C4⋊Q8—102no—
227C4⋊Q8C4×D514yes—
228#125—58no—
229#125—104no—
230C22⋊Q8—108no—
231C22⋊Q8—108no—
232C4×Q8—108no—
233C4×Q8—108no—
234#117—58no—
235#117—104no—
236C2×C4⋊C4—21no—
237C2×C4⋊C4—52no—
238C2×C4⋊C4—52no—
239C2×C4⋊C4—102no—
240C2×C4⋊C4—102no—
241C2×C4⋊C4—101no—
242C2×C4⋊C4C22×D512yes—
243C2×C4⋊C4C22×D512yes—
244C2×C22⋊C4—102no—
245C2×C22⋊C4—102no—
246C2×C42—21no—
247C2×C42—51no—
248C2×C42—51no—
249C2×C42—51no—
250C2×C42—54no—
251C2×C42—54no—
252C2×C42—102no—
253C2×C42—101no—
254C2×C42C22×D511yescharacteristic
255C4⋊C8—22no—
256C4⋊C8—102no—
257C22⋊C8—104no—
258C4×C8—104no—
259#95—21no—
260#95—102no—
261#95—102no—
262#95—102no—
263#95—101no—
264C22×C5C22×C22⋊C411yescharacteristic
265C22×C5#53612yes—
266C22×C5C2×#13511yescharacteristic
267C22×C5C2×#13511yescharacteristic
268C22×C5#57512yes—
269C2×D5—22no—
270C2×D5—22no—
271C2×D5—22no—
272C2×D5—42no—
273C2×D5—42no—
274C4×C5—22no—
275C4×C5—44no—
276C4×C5—82no—
277C4×C5#41614yes—
278C4×C5#51912yes—
279C4×C5#53611yescharacteristic
280C4×C5#53611yescharacteristic
281Dic5—22no—
282Dic5—22no—
283Dic5—22no—
284Dic5—44no—
285Dic5—82no—
286C24—51no—
287C2×Q8—24no—
288C2×Q8—42no—
289C2×Q8—42no—
290C2×Q8—104no—
291C2×Q8—204no—
292C22×C4—22no—
293C22×C4—22no—
294C22×C4—51no—
295C22×C4—51no—
296C22×C4—51no—
297C22×C4—52no—
298C22×C4—52no—
299C22×C4—54no—
300C22×C4—54no—
301C22×C4—102no—
302C22×C4—102no—
303C22×C4—102no—
304C22×C4—102no—
305C22×C4—102no—
306C22×C4—102no—
307C22×C4—202no—
308C22×C4D4×D511yescharacteristic
309C22×C4D4×D511yescharacteristic
310C22×C4C23×D511yescharacteristic
311C2×C8—22no—
312C2×C8—24no—
313C2×C8—102no—
314C2×C8—104no—
315C4⋊C4—22no—
316C4⋊C4—24no—
317C4⋊C4—22no—
318C4⋊C4—22no—
319C4⋊C4—54no—
320C4⋊C4—108no—
321C4⋊C4—108no—
322C4⋊C4—102no—
323C4⋊C4—104no—
324C4⋊C4—102no—
325C4⋊C4—102no—
326C4⋊C4C2×C4×D514yes—
327C22⋊C4—108no—
328C22⋊C4—204no—
329C42—42no—
330C42—52no—
331C42—52no—
332C42—54no—
333C42—54no—
334C42—108no—
335C42—202no—
336C42C2×C4×D512yes—
337C42D4×D512yes—
338C2×C5#231011yescharacteristic
339C2×C5C2×#41611yescharacteristic
340C2×C5#289611yescharacteristic
341C2×C5#323312yes—
342C2×C5#323312yes—
343D5—42no—
344C23—52no—
345C23—101no—
346C23—101no—
347C23—102no—
348C23—102no—
349C23C2×D4×D511yescharacteristic
350Q8—48no—
351Q8—208no—
352C2×C4—22no—
353C2×C4—24no—
354C2×C4—42no—
355C2×C4—42no—
356C2×C4—42no—
357C2×C4—42no—
358C2×C4—42no—
359C2×C4—52no—
360C2×C4—52no—
361C2×C4—52no—
362C2×C4—101no—
363C2×C4—101no—
364C2×C4—102no—
365C2×C4—102no—
366C2×C4—102no—
367C2×C4—108no—
368C2×C4—104no—
369C2×C4—102no—
370C2×C4—104no—
371C2×C4—204no—
372C2×C4—204no—
373C2×C4—202no—
374C2×C4—202no—
375C2×C4—202no—
376C2×C4D5×C22⋊C412yes—
377C2×C4D5×C22⋊C412yes—
378C2×C4#405812yes—
379C2×C4#406212yes—
380C2×C4C22×C4×D511yescharacteristic
381C2×C4C2×D4×D511yescharacteristic
382C2×C4#414712yes—
383C2×C4#415012yes—
384C8—42no—
385C8—202no—
386C5#2183711yescharacteristic, complemented, Sylow
387C22—102no—
388C22—102no—
389C22—102no—
390C22—202no—
391C22—202no—
392C22C2×D5×C22⋊C411yescharacteristic
393C22#6596212yes—
394C22C2×#405811yescharacteristic
395C22C2×#406211yescharacteristic
396C22#6611112yes—
397C4—22no—
398C4—44no—
399C4—82no—
400C4—102no—
401C4—102no—
402C4—102no—
403C4—204no—
404C4—402no—
405C4#6506114yes—
406C4#6590612yes—
407C4#6600911yescharacteristic
408C4#6605811yescharacteristic
409C2—202no—
410C2#1060834111yescharacteristic
411C2C2×#6506111yescharacteristic
412C2#1061068711yescharacteristic
413C2#1061270112yes—
414C2#1061274712yes—
subgroupG/H
centerC23C2×D4×D5
commutatorC2×C4×C5C23×C4
FrattiniC22×C4C23×D5
FittingC5×#1988C2
socleC23×C5C22×D4
prime radicalC23×D5C2×D4
cosocleC22×C4×C5C24
hypercenter#1988D5
hypocenterC5#21837
LCSG ▷ C2×C4×C5 ▷ C2×C5 ▷ C5
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1988

Complements

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

NH ≅ G/Ntype
4. C5×#1988C2semidirect
5. #10602573C2semidirect
6. #10601252C2semidirect
7. #10600282C2semidirect
9. #10599597C2semidirect
14. #66062C4semidirect
91. #1988D5semidirect
386. C5#21837semidirect

Maximal subgroups

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

subgroupindex
C2×#660622
C2×#651222
#106054832
C5×#19882
#106025732
#106012522
#106002822
#105996662
#105995972
#218375