Finite Group Databasebeta

← #12102454

Subgroups — #12102454

Order 1280.

862 conjugacy classes of subgroups.

The subgroup lattice has 426 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#10614181C211yesmaximal, characteristic, complemented
2C2×C4×#688C211yesmaximal, characteristic
3#10605492C211yesmaximal, characteristic
4C5×#1763C211yesmaximal, characteristic, complemented
5#10601033C212yesmaximal
6#10601030C212yesmaximal, complemented
7#10600605C211yesmaximal, characteristic
8#10600599C211yesmaximal, characteristic, complemented
9C4×#3940C211yesmaximal, characteristic, complemented
10#10599585C212yesmaximal
11#10599582C212yesmaximal, complemented
12C2×C4×C5×Q8C2211yescharacteristic
13C2×Q8×Dic5C2212yes—
14C22×#688C2211yescharacteristic
15C2×C4×D20C2211yescharacteristic
16C2×C5×#103C2212yes—
17C2×C5×#97C2211yescharacteristic
18C4×C5×C4⋊C4C2211yescharacteristic
19C2×#3967C2211yescharacteristic
20C2×#3953C2212yes—
21C4×#688C418yescomplemented
22#65277C2212yes—
23C4⋊C4×Dic5—22no—
24C4⋊C4×Dic5C2212yes—
25#65262C2212yes—
26C2×#3941C2211yescharacteristic
27C2×#3940C2212yes—
28C2×#3940C2211yescharacteristic
29#65229—22no—
30C4×C4⋊Dic5—21no—
31C4×C4⋊Dic5C2211yescharacteristic
32#65221—21no—
33#65221C2211yescharacteristic
34C2×C4×#168C2211yescharacteristic
35#64956—21no—
36#64953—21no—
37C5×#340C2212yes—
38#64771C2212yes—
39#64770C2212yes—
40#64770C2212yes—
41#64749C2212yes—
42#21131—51nomaximal, Sylow
43C22×C5×Q8C2311yescharacteristic
44C22×D20C2311yescharacteristic
45C22×C4×D5—21no—
46C22×C5×C8C2312yes—
47C4×C5×Q8—22no—
48C4×C5×Q8C2×C414yes—
49C2×C5×C4⋊C4—21no—
50C2×C5×C4⋊C4C2312yes—
51C2×C5×C4⋊C4C2311yescharacteristic
52C2×C42×C5—21no—
53C2×C42×C5—21no—
54C2×C42×C5C2311yescharacteristic
55Q8×Dic5—24no—
56Q8×Dic5—24no—
57C2×#688—24no—
58C2×#688C2×C414yes—
59C2×#685—21no—
60C2×#685—22no—
61C2×#685—22no—
62C2×C4⋊Dic5—22no—
63C2×C4⋊Dic5C2311yescharacteristic
64C2×C4⋊Dic5C2312yes—
65C2×#683—22no—
66C2×C4×Dic5—22no—
67C2×C4×Dic5—21no—
68C2×C4×Dic5—22no—
69C2×C4×Dic5C2312yes—
70C22×#168C2311yescharacteristic
71C22×#168C2311yescharacteristic
72C4×D20—22no—
73C4×D20C2×C414yes—
74C5×#103—24no—
75C5×#97—22no—
76C5×#95—21no—
77#3967C2×C414yes—
78#3963—21no—
79#3963—22no—
80#3963—22no—
81#3963D411yescharacteristic
82#3963D411yescharacteristic
83#3963D412yes—
84#3953—24no—
85#3941C2×C414yes—
86#3940—24no—
87#3940C2×C414yes—
88C4×#168C2×C414yes—
89C2×C4×SD16—51no—
90#2319—51no—
91#2276—51no—
92#1926—52no—
93#1920—52no—
94#1880—52no—
95#1865—52no—
96C4×#106—51no—
97#1763D511yescharacteristic, complemented
98#1762—51no—
99#1746—51no—
100C23×D5—21no—
101C2×C5×Q8—21no—
102C2×C5×Q8—22no—
103C2×C5×Q8C22×C412yes—
104C2×C5×Q8#4114yes—
105C22×C4×C5—21no—
106C22×C4×C5—21no—
107C22×C4×C5—22no—
108C22×C4×C5C2×D411yescharacteristic
109C22×C4×C5C2×D411yescharacteristic
110C22×C4×C5C2411yescharacteristic
111C22×Dic5—21no—
112C22×Dic5—22no—
113C22×Dic5C2×D412yes—
114C22×Dic5C2×D412yes—
115C2×D20—22no—
116C2×D20—22no—
117C2×D20—21no—
118C2×D20C22×C412yes—
119C2×C4×D5—24no—
120C2×C4×D5—82no—
121C2×C5×C8—22no—
122C2×C5×C8—22no—
123C2×C5×C8—22no—
124C5×C4⋊C4—24no—
125C5×C4⋊C4—21no—
126C5×C4⋊C4—22no—
127C5×C4⋊C4—22no—
128C5×C4⋊C4C22×C412yes—
129C42×C5—24no—
130C42×C5—81no—
131C42×C5C22×C412yes—
132C42×C5#4112yes—
133#688—28no—
134#685—24no—
135#685—82no—
136#685—82no—
137C4⋊Dic5—28no—
138C4⋊Dic5—21no—
139C4⋊Dic5—22no—
140C4⋊Dic5C22×C412yes—
141C4⋊Dic5#4114yes—
142#683—82no—
143C4×Dic5—28no—
144C4×Dic5—22no—
145C4×Dic5—22no—
146C4×Dic5—82no—
147C4×Dic5—81no—
148C2×#168—22no—
149C2×#168—22no—
150C2×#168C22×C412yes—
151C2×#168C22×C412yes—
152C22×SD16—51no—
153C2×C4×Q8—52no—
154C2×C4×Q8C2×D511yescharacteristic
155C2×C4×D4—51no—
156C4×SD16—58no—
157C2×#106—51no—
158C2×#106—52no—
159C2×#103—51no—
160C2×#103C2×D512yes—
161C2×#102—51no—
162C2×#102—52no—
163C2×#97C2×D511yescharacteristic
164C2×C4×C8—51no—
165#390—52no—
166#389—52no—
167#388—102no—
168#381—101no—
169#380—101no—
170C4×C4⋊C4—52no—
171C4×C4⋊C4—51no—
172C4×C4⋊C4—101no—
173C4×C4⋊C4—102no—
174C4×C4⋊C4C2×D511yescharacteristic
175#376—51no—
176#376—101no—
177#340—52no—
178#340—52no—
179#340C2×D512yes—
180#336—52no—
181#336—52no—
182C23×C5C22×D411yescharacteristic
183C22×D5—22no—
184C22×D5—42no—
185C22×D5—41no—
186C22×D5—42no—
187C22×D5—41no—
188C5×Q8—22no—
189C5×Q8—24no—
190C5×Q8#11714yes—
191C2×C4×C5—22no—
192C2×C4×C5—24no—
193C2×C4×C5—21no—
194C2×C4×C5—21no—
195C2×C4×C5—22no—
196C2×C4×C5—22no—
197C2×C4×C5—41no—
198C2×C4×C5—41no—
199C2×C4×C5—42no—
200C2×C4×C5—42no—
201C2×C4×C5—42no—
202C2×C4×C5C4×D412yes—
203C2×C4×C5C4×D412yes—
204C2×C4×C5C23×C411yescharacteristic
205C2×C4×C5C2×#4112yes—
206C2×C4×C5C2×#4111yescharacteristic
207C2×C4×C5C2×#4112yes—
208C2×Dic5—22no—
209C2×Dic5—22no—
210C2×Dic5—22no—
211C2×Dic5—24no—
212C2×Dic5—22no—
213C2×Dic5—22no—
214C2×Dic5—22no—
215C2×Dic5—22no—
216C2×Dic5—42no—
217C2×Dic5—42no—
218C2×Dic5—42no—
219D20—22no—
220D20—22no—
221D20—42no—
222C4×D5—84no—
223C5×C8—42no—
224#168—22no—
225#168—22no—
226C22×Q8C22×D511yescharacteristic
227C22×D4—51no—
228C23×C4—101no—
229C2×SD16—54no—
230C2×SD16—104no—
231C22×C8—51no—
232C22×C8—51no—
233C22×C8C22×D512yes—
234C4×Q8—22no—
235C4×Q8—104no—
236C4×Q8—104no—
237C4×Q8C4×D514yes—
238C4×D4—54no—
239C4×D4—102no—
240C2×C4⋊C4—21no—
241C2×C4⋊C4—51no—
242C2×C4⋊C4—52no—
243C2×C4⋊C4—102no—
244C2×C4⋊C4—102no—
245C2×C4⋊C4C22×D512yes—
246C2×C4⋊C4C22×D511yescharacteristic
247C2×C22⋊C4—101no—
248C2×C22⋊C4—102no—
249C2×C22⋊C4—102no—
250C2×C42—21no—
251C2×C42—21no—
252C2×C42—52no—
253C2×C42—102no—
254C2×C42—102no—
255C2×C42—101no—
256C2×C42C22×D511yescharacteristic
257#106—54no—
258#106—104no—
259#103—24no—
260#103—54no—
261#102—54no—
262#102—104no—
263#97—22no—
264C4×C8—54no—
265#95—21no—
266#95—51no—
267#95—51no—
268#95—52no—
269#95—102no—
270#95—102no—
271#95—101no—
272C22×C5C2×C4×D411yescharacteristic
273C22×C5#52512yes—
274C22×C5#57711yescharacteristic
275C22×C5#57711yescharacteristic
276C22×C5#57712yes—
277C2×D5—42no—
278C2×D5—42no—
279C2×D5—42no—
280C2×D5—82no—
281C2×D5—82no—
282C4×C5—22no—
283C4×C5—22no—
284C4×C5—24no—
285C4×C5—42no—
286C4×C5—82no—
287C4×C5C2×#11711yescharacteristic
288C4×C5C4×#4111yescharacteristic
289C4×C5#52912yes—
290Dic5—42no—
291Dic5—44no—
292Dic5—42no—
293Dic5—42no—
294C24—101no—
295C2×Q8—21no—
296C2×Q8—22no—
297C2×Q8C2×C4×D512yes—
298C2×Q8#70914yes—
299C2×D4—52no—
300C2×D4—102no—
301C2×D4—101no—
302C2×D4—102no—
303C22×C4—21no—
304C22×C4—21no—
305C22×C4—22no—
306C22×C4—52no—
307C22×C4—52no—
308C22×C4—102no—
309C22×C4—101no—
310C22×C4—104no—
311C22×C4—402no—
312C22×C4D4×D511yescharacteristic
313C22×C4D4×D511yescharacteristic
314C22×C4C23×D511yescharacteristic
315SD16—108no—
316C2×C8—22no—
317C2×C8—22no—
318C2×C8—22no—
319C2×C8—52no—
320C2×C8—52no—
321C2×C8—102no—
322C2×C8—102no—
323C4⋊C4—21no—
324C4⋊C4—24no—
325C4⋊C4—22no—
326C4⋊C4—22no—
327C4⋊C4—52no—
328C4⋊C4—54no—
329C4⋊C4—108no—
330C4⋊C4—102no—
331C4⋊C4—101no—
332C4⋊C4—402no—
333C4⋊C4C2×C4×D512yes—
334C22⋊C4—104no—
335C22⋊C4—402no—
336C22⋊C4—402no—
337C42—24no—
338C42—81no—
339C42—108no—
340C42—102no—
341C42—102no—
342C42—401no—
343C42—402no—
344C42C2×C4×D512yes—
345C42#71112yes—
346C2×C5#229011yescharacteristic
347C2×C5#294911yescharacteristic
348C2×C5#295211yescharacteristic
349C2×C5#312512yes—
350C2×C5#312612yes—
351D5—82no—
352C23—102no—
353C23—201no—
354C23—202no—
355C23—201no—
356C23—202no—
357C23C2×D4×D511yescharacteristic
358Q8—22no—
359Q8—24no—
360Q8#401814yes—
361D4—102no—
362D4—102no—
363D4—202no—
364C2×C4—22no—
365C2×C4—21no—
366C2×C4—21no—
367C2×C4—22no—
368C2×C4—24no—
369C2×C4—22no—
370C2×C4—41no—
371C2×C4—41no—
372C2×C4—42no—
373C2×C4—42no—
374C2×C4—42no—
375C2×C4—102no—
376C2×C4—102no—
377C2×C4—102no—
378C2×C4—102no—
379C2×C4—102no—
380C2×C4—102no—
381C2×C4—104no—
382C2×C4—102no—
383C2×C4—202no—
384C2×C4—202no—
385C2×C4—202no—
386C2×C4—404no—
387C2×C4#402712yes—
388C2×C4#402712yes—
389C2×C4C22×C4×D511yescharacteristic
390C2×C4C2×#70912yes—
391C2×C4C2×#71111yescharacteristic
392C2×C4D5×#4112yes—
393C8—42no—
394C8—102no—
395C8—102no—
396C5#2113111yescharacteristic, complemented, Sylow
397C22—202no—
398C22—202no—
399C22—202no—
400C22—402no—
401C22—402no—
402C22C2×#402711yescharacteristic
403C22#6588112yes—
404C22#6610911yescharacteristic
405C22#6611212yes—
406C22#6611211yescharacteristic
407C4—24no—
408C4—22no—
409C4—22no—
410C4—42no—
411C4—82no—
412C4—202no—
413C4—202no—
414C4—204no—
415C4—202no—
416C4C2×#401811yescharacteristic
417C4#6585611yescharacteristic
418C4#6587212yes—
419C2—402no—
420C2#1060516911yescharacteristic
421C2#1061060011yescharacteristic
422C2#1061067811yescharacteristic
423C2#1061152012yes—
424C2#1061158112yes—
subgroupG/H
centerC23C2×D4×D5
commutatorC2×C4×C5C23×C4
FrattiniC22×C4C23×D5
FittingC5×#1763C2
socleC23×C5C22×D4
prime radicalC22×D20C23
cosocleC22×C4×C5C24
hypercenter#1763D5
hypocenterC5#21131
LCSG ▷ C2×C4×C5 ▷ C2×C5 ▷ C5
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1763

Complements

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

NH ≅ G/Ntype
1. #10614181C2semidirect
4. C5×#1763C2semidirect
6. #10601030C2semidirect
8. #10600599C2semidirect
9. C4×#3940C2semidirect
11. #10599582C2semidirect
21. C4×#688C4semidirect
97. #1763D5semidirect
396. C5#21131semidirect

Maximal subgroups

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

subgroupindex
#106141812
C2×C4×#6882
#106054922
C5×#17632
#106010332
#106010302
#106006052
#106005992
C4×#39402
#105995852
#105995822
#211315