Finite Group Databasebeta

← #12073223

Subgroups — #12073223

Order 1280.

940 conjugacy classes of subgroups.

The subgroup lattice has 398 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C5×#544C211yesmaximal, characteristic
2#10614472C212yesmaximal
3#10613914C211yesmaximal, characteristic
4#10602697C214yesmaximal, complemented
5#10602692C214yesmaximal, complemented
6#10602483C212yesmaximal
7C2×#64748C211yesmaximal, characteristic
8C5×#544—24no—
9C5×#544—22no—
10C5×#544C2214yes—
11C2×C5×C4⋊Q8—22no—
12C2×C5×#125—22no—
13C2×C5×C22⋊Q8—21no—
14C2×C5×C22⋊Q8C2212yes—
15C2×C5×#117—21no—
16C22×C5×C4⋊C4C2211yescharacteristic
17C2×#4070C2211yescharacteristic
18#65517—22no—
19#65501C2211yescharacteristic
20#65500—21no—
21#65499—21no—
22C2×C22⋊#168C2212yes—
23#65340—24no—
24#65339C2214yes—
25#65333C2214yes—
26#65275C2212yes—
27#65273—22no—
28#65268C2212yes—
29#65259—22no—
30#64758C2218yes—
31#64748C2214yes—
32#18295—51nomaximal, Sylow
33C22×C5×Q8—21no—
34C22×C5×Q8D412yes—
35C23×C4×C5D411yescharacteristic
36C23×C4×C5D411yescharacteristic
37C23×C4×C5C2311yescharacteristic
38C22×Dic10C2311yescharacteristic
39C5×C4⋊Q8—28no—
40C5×C4⋊Q8—28no—
41C5×#125—28no—
42C5×#125—28no—
43C5×C22⋊Q8—24no—
44C5×C22⋊Q8—24no—
45C5×C22⋊Q8—22no—
46C5×C22⋊Q8—22no—
47C5×C22⋊Q8D414yes—
48C5×C22⋊Q8C2314yes—
49C5×#117—24no—
50C5×#117—24no—
51C2×C5×C4⋊C4—21no—
52C2×C5×C4⋊C4—21no—
53C2×C5×C4⋊C4—21no—
54C2×C5×C4⋊C4—22no—
55C2×C5×C4⋊C4—22no—
56C2×C5×C4⋊C4—22no—
57C2×C5×C4⋊C4—22no—
58C2×C5×C4⋊C4D412yes—
59C2×C5×C4⋊C4D412yes—
60C2×C5×C4⋊C4C2312yes—
61C2×C5×C22⋊C4—22no—
62C2×C42×C5—22no—
63C2×C22⋊Dic5—21no—
64C2×C22⋊Dic5—21no—
65C2×#689—24no—
66C2×C4⋊Dic5—21no—
67C2×C4⋊Dic5C2311yescharacteristic
68#4070—22no—
69#4070—22no—
70C2×#683—21no—
71C2×#683—22no—
72C22×#168—22no—
73#3967—24no—
74#3963—22no—
75#3963—22no—
76#3963—22no—
77#3963C2312yes—
78C22⋊#168—22no—
79C22⋊#168C2314yes—
80#3942—24no—
81#3939—24no—
82C2×#544D511yescharacteristic
83#3189—52no—
84#2627—51no—
85#1617—54no—
86#1605—54no—
87#1520—52no—
88C2×#324—51no—
89C24×C5C2×D411yescharacteristic
90C2×C5×Q8—24no—
91C2×C5×Q8—22no—
92C2×C5×Q8—82no—
93C2×C5×Q8—82no—
94C2×C5×Q8C2×D414yes—
95C22×C4×C5—21no—
96C22×C4×C5—21no—
97C22×C4×C5—22no—
98C22×C4×C5—22no—
99C22×C4×C5—22no—
100C22×C4×C5—21no—
101C22×C4×C5—21no—
102C22×C4×C5—81no—
103C22×C4×C5—81no—
104C22×C4×C5Q1614yescomplemented
105C22×C4×C5C2×D412yes—
106C22×C4×C5C2×D412yes—
107C22×C4×C5C2×D411yescharacteristic
108C22×C4×C5C2×D412yes—
109C22×C4×C5C2×D412yes—
110C22×C4×C5C2411yescharacteristic
111C22×Dic5—21no—
112C22×Dic5—21no—
113C22×Dic5—22no—
114C2×Dic10—22no—
115C2×Dic10—81no—
116C5×C4⋊C4—24no—
117C5×C4⋊C4—24no—
118C5×C4⋊C4—24no—
119C5×C4⋊C4—28no—
120C5×C4⋊C4—24no—
121C5×C4⋊C4—24no—
122C5×C4⋊C4—22no—
123C5×C4⋊C4—24no—
124C5×C4⋊C4—24no—
125C5×C4⋊C4C2×D414yes—
126C5×C22⋊C4—24no—
127C5×C22⋊C4—24no—
128C42×C5—28no—
129C22⋊Dic5—41no—
130C22⋊Dic5—41no—
131C22⋊Dic5—42no—
132#689—84no—
133C4⋊Dic5—22no—
134C4⋊Dic5—81no—
135#683—42no—
136#683—82no—
137C2×#168—24no—
138C2×#168—82no—
139#544—24no—
140#544—22no—
141#544C2×D514yes—
142C2×C4⋊Q8—22no—
143C2×#125—22no—
144C2×C22⋊Q8—21no—
145C2×C22⋊Q8—51no—
146C2×C22⋊Q8C2×D512yes—
147C2×#117—21no—
148C22×C4⋊C4C2×D511yescharacteristic
149#484—54no—
150#484—102no—
151#467—104no—
152#451—54no—
153#451—102no—
154C2×C22⋊C8—52no—
155#400—52no—
156#398—102no—
157#395—52no—
158#385—101no—
159#381—101no—
160#380—51no—
161#328—58no—
162#324—54no—
163C23×C5—41no—
164C23×C5—41no—
165C23×C5#12011yescharacteristic
166C23×C5#12011yescharacteristic
167C23×C5C2×Q1612yes—
168C23×C5#13712yes—
169C23×C5C22×D411yescharacteristic
170C5×Q8—44no—
171C5×Q8—82no—
172C5×Q8—84no—
173C2×C4×C5—22no—
174C2×C4×C5—22no—
175C2×C4×C5—24no—
176C2×C4×C5—24no—
177C2×C4×C5—24no—
178C2×C4×C5—24no—
179C2×C4×C5—22no—
180C2×C4×C5—42no—
181C2×C4×C5—41no—
182C2×C4×C5—41no—
183C2×C4×C5—42no—
184C2×C4×C5—44no—
185C2×C4×C5—42no—
186C2×C4×C5—44no—
187C2×C4×C5—42no—
188C2×C4×C5—81no—
189C2×C4×C5#12012yes—
190C2×C4×C5C2×Q1614yes—
191C2×C4×C5C22×D412yes—
192C2×Dic5—42no—
193C2×Dic5—42no—
194C2×Dic5—42no—
195C2×Dic5—42no—
196C2×Dic5—41no—
197C2×Dic5—41no—
198C2×Dic5—42no—
199Dic10—44no—
200#168—44no—
201C22×Q8—21no—
202C22×Q8—51no—
203C22×Q8C22⋊D512yes—
204C23×C4C22⋊D511yescharacteristic
205C23×C4C22⋊D511yescharacteristic
206C23×C4C22×D511yescharacteristic
207C2×Q16—104no—
208C22×C8—102no—
209C4⋊Q8—28no—
210C4⋊Q8—28no—
211#125—28no—
212#125—28no—
213C22⋊Q8—24no—
214C22⋊Q8—24no—
215C22⋊Q8—22no—
216C22⋊Q8—22no—
217C22⋊Q8—102no—
218C22⋊Q8—102no—
219C22⋊Q8C22⋊D514yes—
220C22⋊Q8C22×D514yes—
221#117—24no—
222#117—24no—
223C2×C4⋊C4—21no—
224C2×C4⋊C4—21no—
225C2×C4⋊C4—21no—
226C2×C4⋊C4—22no—
227C2×C4⋊C4—22no—
228C2×C4⋊C4—22no—
229C2×C4⋊C4—22no—
230C2×C4⋊C4—51no—
231C2×C4⋊C4—101no—
232C2×C4⋊C4—101no—
233C2×C4⋊C4—102no—
234C2×C4⋊C4C22⋊D512yes—
235C2×C4⋊C4C22⋊D512yes—
236C2×C4⋊C4C22×D512yes—
237C2×C22⋊C4—22no—
238C2×C22⋊C4—101no—
239C2×C22⋊C4—101no—
240C2×C42—22no—
241#107—104no—
242#103—104no—
243#103—104no—
244C22⋊C8—54no—
245C22⋊C8—102no—
246#95—52no—
247#95—102no—
248#95—102no—
249#95—102no—
250C22×C5—21no—
251C22×C5—21no—
252C22×C5—21no—
253C22×C5—21no—
254C22×C5—21no—
255C22×C5—21no—
256C22×C5—42no—
257C22×C5—42no—
258C22×C5#45112yes—
259C22×C5#45112yes—
260C22×C5C2×#12011yescharacteristic
261C22×C5C22×Q1611yescharacteristic
262C22×C5C2×#13711yescharacteristic
263C4×C5—24no—
264C4×C5—22no—
265C4×C5—22no—
266C4×C5—44no—
267C4×C5—42no—
268C4×C5—44no—
269Dic5—82no—
270Dic5—81no—
271Dic5—81no—
272C24C2×C22⋊D511yescharacteristic
273C2×Q8—24no—
274C2×Q8—22no—
275C2×Q8—82no—
276C2×Q8—82no—
277C2×Q8—102no—
278C2×Q8—401no—
279C2×Q8C2×C22⋊D514yes—
280C22×C4—21no—
281C22×C4—21no—
282C22×C4—22no—
283C22×C4—22no—
284C22×C4—22no—
285C22×C4—21no—
286C22×C4—21no—
287C22×C4—81no—
288C22×C4—81no—
289C22×C4—102no—
290C22×C4—101no—
291C22×C4—101no—
292C22×C4#68914yescomplemented
293C22×C4C2×C22⋊D512yes—
294C22×C4C2×C22⋊D512yes—
295C22×C4C2×C22⋊D511yescharacteristic
296C22×C4C2×C22⋊D512yes—
297C22×C4C2×C22⋊D512yes—
298C22×C4C23×D511yescharacteristic
299Q16—404no—
300C2×C8—104no—
301C2×C8—402no—
302C4⋊C4—24no—
303C4⋊C4—24no—
304C4⋊C4—24no—
305C4⋊C4—28no—
306C4⋊C4—24no—
307C4⋊C4—24no—
308C4⋊C4—22no—
309C4⋊C4—24no—
310C4⋊C4—24no—
311C4⋊C4—102no—
312C4⋊C4—202no—
313C4⋊C4—402no—
314C4⋊C4—401no—
315C4⋊C4C2×C22⋊D514yes—
316C22⋊C4—24no—
317C22⋊C4—24no—
318C22⋊C4—201no—
319C22⋊C4—201no—
320C22⋊C4—202no—
321C42—28no—
322C2×C5—21no—
323C2×C5—21no—
324C2×C5—22no—
325C2×C5—22no—
326C2×C5C2×#45111yescharacteristic
327C2×C5#302011yescharacteristic
328C2×C5#302611yescharacteristic
329C23—41no—
330C23—41no—
331C23#408812yes—
332C23C2×#68912yes—
333C23#409911yescharacteristic
334C23#409911yescharacteristic
335C23C22×C22⋊D511yescharacteristic
336Q8—44no—
337Q8—82no—
338Q8—84no—
339Q8—204no—
340C2×C4—22no—
341C2×C4—22no—
342C2×C4—24no—
343C2×C4—24no—
344C2×C4—24no—
345C2×C4—24no—
346C2×C4—22no—
347C2×C4—41no—
348C2×C4—42no—
349C2×C4—42no—
350C2×C4—41no—
351C2×C4—42no—
352C2×C4—44no—
353C2×C4—44no—
354C2×C4—42no—
355C2×C4—81no—
356C2×C4—202no—
357C2×C4—202no—
358C2×C4—202no—
359C2×C4—202no—
360C2×C4—202no—
361C2×C4—201no—
362C2×C4—201no—
363C2×C4C2×#68914yes—
364C2×C4#409912yes—
365C2×C4C22×C22⋊D512yes—
366C8—204no—
367C5#1829511yescharacteristic, complemented, Sylow
368C22—21no—
369C22—21no—
370C22—21no—
371C22—21no—
372C22—21no—
373C22—21no—
374C22—42no—
375C22—42no—
376C22#6551712yes—
377C22#6551712yes—
378C22C2×#408811yescharacteristic
379C22C22×#68911yescharacteristic
380C22C2×#409911yescharacteristic
381C4—22no—
382C4—22no—
383C4—24no—
384C4—42no—
385C4—44no—
386C4—44no—
387C4—402no—
388C4—401no—
389C4—401no—
390C2—21no—
391C2—21no—
392C2—22no—
393C2—22no—
394C2C2×#6551711yescharacteristic
395C2#1061584111yescharacteristic
396C2#1061584711yescharacteristic
subgroupG/H
centerC22C2×#4099
commutator, cosocleC22×C4×C5C24
FrattiniC22×C4C23×D5
FittingC2×C5×#544C2
socleC22×C5C2×#120
prime radicalC24×C5C2×D4
hypercenterC2×#544D5
hypocenterC5#18295
LCSG ▷ C22×C4×C5 ▷ C22×C5 ▷ C5
UCSC1 ◁ C22 ◁ C23×C4 ◁ C2×#544

Complements

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

NH ≅ G/Ntype
4. #10602697C2semidirect
5. #10602692C2semidirect
104. C22×C4×C5Q16semidirect
292. C22×C4#689semidirect
367. C5#18295semidirect

Maximal subgroups

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

subgroupindex
C2×C5×#5442
#106144722
#106139142
#106026972
#106026922
#106024832
C2×#647482
#182955