Finite Group Databasebeta

← #12109837

Subgroups — #12109837

Order 1280.

850 conjugacy classes of subgroups.

The subgroup lattice has 406 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#64974C211yesmaximal, characteristic
2#10606049C211yesmaximal, characteristic
3#10606049C211yesmaximal, characteristic
4C5×#1845C211yesmaximal, characteristic, complemented
5#10601426C214yesmaximal
6#10601409C214yesmaximal, complemented
7#10600555C211yesmaximal, characteristic, complemented
8#10600369C211yesmaximal, characteristic, complemented
9#10600367C211yesmaximal, characteristic, complemented
10C2×#4038—21no—
11C2×#4038C2212yes—
12C2×C42×D5C2211yescharacteristic
13C2×C5×#107C2214yes—
14C2×C4×C5×C8C2211yescharacteristic
15C5×#384C2211yescharacteristic
16C5×#384C2211yescharacteristic
17C2×#3952C2212yes—
18C2×#3946C2212yes—
19#65276—21no—
20#65276—21no—
21#65276—21no—
22#65276C2212yes—
23#65266—21no—
24#65266C2212yes—
25C4⋊C4×Dic5—21no—
26C4⋊C4×Dic5—21no—
27C4⋊C4×Dic5—21no—
28C4⋊C4×Dic5C2212yes—
29C2×#3940C2214yes—
30#65224C2211yescharacteristic
31#65224C2211yescharacteristic
32C2×C4×#168C2211yescharacteristic
33#64974C418yescomplemented
34#64771C2214yes—
35#64749C2214yes—
36#21984—51nomaximal, Sylow
37C22×C4×D5D411yescharacteristic
38C22×C4×D5D411yescharacteristic
39C22×C4×D5C2311yescharacteristic
40C22×C5×C8C2312yes—
41C2×C5×C4⋊C4—21no—
42C2×C5×C4⋊C4—21no—
43C2×C5×C4⋊C4—21no—
44C2×C5×C4⋊C4—21no—
45C2×C5×C4⋊C4C2312yes—
46C2×C5×C4⋊C4C2312yes—
47C2×C42×C5C2311yescharacteristic
48C2×#685—22no—
49C2×#685—22no—
50C2×C4⋊Dic5—21no—
51C2×C4⋊Dic5—21no—
52C2×C4⋊Dic5—21no—
53C2×C4⋊Dic5—21no—
54C2×C4⋊Dic5C2312yes—
55C2×C4⋊Dic5C2312yes—
56C2×C4×Dic5—22no—
57C2×C4×Dic5—22no—
58C2×C4×Dic5D411yescharacteristic
59C2×C4×Dic5D411yescharacteristic
60C2×C4×Dic5C2311yescharacteristic
61C22×#168C2312yes—
62#4038—44no—
63#4038—44no—
64C42×D5C2×C414yes—
65C42×D5Q814yes—
66C5×#107—44no—
67C4×C5×C8C2×C414yes—
68#3963—22no—
69#3963—22no—
70#3963—22no—
71#3963—22no—
72#3952C2×C418yes—
73#3946C2×C418yes—
74#3940—44no—
75C4×#168C2×C414yes—
76C2×#431—51no—
77#2285—51no—
78#2285—51no—
79#1949—54no—
80#1854—54no—
81#1845D511yescharacteristic, complemented
82#1844—51no—
83#1834—51no—
84#1834—51no—
85C23×D5C2×D411yescharacteristic
86C22×C4×C5—22no—
87C22×C4×C5—22no—
88C22×C4×C5C2×D411yescharacteristic
89C22×C4×C5C2×D411yescharacteristic
90C22×C4×C5C2411yescharacteristic
91C22×Dic5—22no—
92C22×Dic5—22no—
93C22×Dic5C2×D411yescharacteristic
94C22×Dic5C2×D411yescharacteristic
95C22×Dic5C2×D411yescharacteristic
96C2×C4×D5—22no—
97C2×C4×D5—22no—
98C2×C4×D5—22no—
99C2×C4×D5—22no—
100C2×C4×D5—22no—
101C2×C4×D5—22no—
102C2×C4×D5C4⋊C414yes—
103C2×C4×D5C4⋊C414yes—
104C2×C4×D5C22×C412yes—
105C2×C4×D5C2×Q812yes—
106C2×C5×C8—22no—
107C2×C5×C8—22no—
108C2×C5×C8C22×C414yes—
109C5×C4⋊C4—24no—
110C5×C4⋊C4—24no—
111C5×C4⋊C4—42no—
112C5×C4⋊C4—42no—
113C5×C4⋊C4—42no—
114C5×C4⋊C4—42no—
115C42×C5C22×C412yes—
116C42×C5C2×Q812yes—
117#685—44no—
118#685—44no—
119C4⋊Dic5—24no—
120C4⋊Dic5—24no—
121C4⋊Dic5—42no—
122C4⋊Dic5—42no—
123C4⋊Dic5—42no—
124C4⋊Dic5—42no—
125C4×Dic5—44no—
126C4×Dic5—44no—
127C4×Dic5C4⋊C414yes—
128C4×Dic5C4⋊C414yes—
129C4×Dic5C22×C412yes—
130C4×Dic5C2×Q812yes—
131C2×#168—22no—
132C2×#168—22no—
133C2×#168C22×C414yes—
134C2×#117—52no—
135C2×#117—101no—
136C22×C42—51no—
137#431—58no—
138C2×#107C2×D514yes—
139C2×#106—54no—
140C2×C4⋊C8—52no—
141C2×C22⋊C8—52no—
142C2×C4×C8—51no—
143C2×C4×C8C2×D511yescharacteristic
144#384—51no—
145#384—51no—
146#384C2×D511yescharacteristic
147#384C2×D511yescharacteristic
148#382—52no—
149#382—101no—
150#380—52no—
151#380—101no—
152#380—101no—
153#380—101no—
154C4×C4⋊C4—52no—
155C4×C4⋊C4—101no—
156C4×C4⋊C4—101no—
157C4×C4⋊C4—101no—
158#340—54no—
159#340—54no—
160C23×C5C22×D411yescharacteristic
161C22×D5—21no—
162C22×D5—21no—
163C22×D5—21no—
164C22×D5—21no—
165C22×D5—21no—
166C22×D5—21no—
167C22×D5C2×C4⋊C412yes—
168C2×C4×C5—42no—
169C2×C4×C5—42no—
170C2×C4×C5—42no—
171C2×C4×C5—42no—
172C2×C4×C5—42no—
173C2×C4×C5—42no—
174C2×C4×C5C2×C4⋊C412yes—
175C2×C4×C5C2×C4⋊C412yes—
176C2×C4×C5#13512yes—
177C2×C4×C5#13512yes—
178C2×C4×C5AGL(1,ℤ/8)12yes—
179C2×C4×C5#13712yes—
180C2×C4×C5C23×C411yescharacteristic
181C2×C4×C5C22×Q811yescharacteristic
182C2×C4×C5H(5,𝔽2)12yes—
183C2×C4×C52−1+412yes—
184C2×Dic5—21no—
185C2×Dic5—21no—
186C2×Dic5—21no—
187C2×Dic5—21no—
188C2×Dic5—21no—
189C2×Dic5—21no—
190C2×Dic5—42no—
191C2×Dic5—42no—
192C2×Dic5—42no—
193C2×Dic5—42no—
194C2×Dic5—42no—
195C2×Dic5—42no—
196C2×Dic5C2×C4⋊C412yes—
197C2×Dic5C2×C4⋊C412yes—
198C2×Dic5C2×C4⋊C412yes—
199C4×D5—24no—
200C4×D5—24no—
201C4×D5—28no—
202C4×D5—28no—
203C5×C8—24no—
204#168—24no—
205C23×C4—51no—
206C23×C4—51no—
207C23×C4—51no—
208C22×C8—52no—
209C22×C8C22×D512yes—
210#117—204no—
211#117—204no—
212C2×C4⋊C4—21no—
213C2×C4⋊C4—21no—
214C2×C4⋊C4—21no—
215C2×C4⋊C4—21no—
216C2×C4⋊C4—52no—
217C2×C4⋊C4—52no—
218C2×C4⋊C4—101no—
219C2×C4⋊C4—101no—
220C2×C4⋊C4—101no—
221C2×C4⋊C4—101no—
222C2×C4⋊C4C22×D512yes—
223C2×C4⋊C4C22×D512yes—
224C2×C22⋊C4—102no—
225C2×C22⋊C4—102no—
226C2×C42—51no—
227C2×C42—51no—
228C2×C42—51no—
229C2×C42—54no—
230C2×C42—54no—
231C2×C42—102no—
232C2×C42—102no—
233C2×C42C22×D511yescharacteristic
234#107—44no—
235#106—204no—
236C4⋊C8—58no—
237C22⋊C8—58no—
238C4×C8—54no—
239C4×C8C4×D514yes—
240#95—102no—
241#95—102no—
242#95—102no—
243#95—102no—
244C22×C5C22×C4⋊C411yescharacteristic
245C22×C5#53511yescharacteristic
246C22×C5#53611yescharacteristic
247C22×C5C2×#13511yescharacteristic
248C22×C5C2×#13511yescharacteristic
249C22×C5C2×AGL(1,ℤ/8)11yescharacteristic
250C22×C5C2×#13711yescharacteristic
251C2×D5—22no—
252C2×D5—22no—
253C2×D5—22no—
254C2×D5—41no—
255C2×D5—41no—
256C2×D5—41no—
257C2×D5—41no—
258C4×C5—82no—
259C4×C5—82no—
260C4×C5#42714yes—
261C4×C5#42814yes—
262C4×C5#52012yes—
263C4×C5#54412yes—
264Dic5—22no—
265Dic5—22no—
266Dic5—22no—
267Dic5—82no—
268Dic5—82no—
269C24—51no—
270C22×C4—22no—
271C22×C4—22no—
272C22×C4—51no—
273C22×C4—51no—
274C22×C4—51no—
275C22×C4—52no—
276C22×C4—52no—
277C22×C4—54no—
278C22×C4—54no—
279C22×C4—102no—
280C22×C4—102no—
281C22×C4—102no—
282C22×C4—102no—
283C22×C4—102no—
284C22×C4—102no—
285C22×C4—102no—
286C22×C4—102no—
287C22×C4D4×D511yescharacteristic
288C22×C4D4×D511yescharacteristic
289C22×C4C23×D511yescharacteristic
290C2×C8—22no—
291C2×C8—22no—
292C2×C8—54no—
293C2×C8—102no—
294C2×C8—102no—
295C2×C8C2×C4×D514yes—
296C4⋊C4—24no—
297C4⋊C4—24no—
298C4⋊C4—42no—
299C4⋊C4—42no—
300C4⋊C4—42no—
301C4⋊C4—42no—
302C4⋊C4—104no—
303C4⋊C4—104no—
304C4⋊C4—202no—
305C4⋊C4—202no—
306C4⋊C4—202no—
307C4⋊C4—202no—
308C22⋊C4—204no—
309C22⋊C4—204no—
310C42—52no—
311C42—52no—
312C42—54no—
313C42—54no—
314C42—204no—
315C42—204no—
316C42C2×C4×D512yes—
317C42Q8×D512yes—
318C2×C5#231711yescharacteristic
319C2×C5C2×#42711yescharacteristic
320C2×C5C2×#42811yescharacteristic
321C2×C5#320711yescharacteristic
322C2×C5#321011yescharacteristic
323C2×C5#322811yescharacteristic
324C2×C5#323111yescharacteristic
325D5—42no—
326C23—52no—
327C23—101no—
328C23—101no—
329C23—101no—
330C23—101no—
331C23—101no—
332C23—101no—
333C23C2×D4×D511yescharacteristic
334C2×C4—42no—
335C2×C4—42no—
336C2×C4—42no—
337C2×C4—42no—
338C2×C4—42no—
339C2×C4—42no—
340C2×C4—52no—
341C2×C4—52no—
342C2×C4—52no—
343C2×C4—101no—
344C2×C4—101no—
345C2×C4—101no—
346C2×C4—101no—
347C2×C4—101no—
348C2×C4—101no—
349C2×C4—104no—
350C2×C4—104no—
351C2×C4—108no—
352C2×C4—108no—
353C2×C4—202no—
354C2×C4—202no—
355C2×C4—202no—
356C2×C4—202no—
357C2×C4—202no—
358C2×C4—202no—
359C2×C4D5×C4⋊C412yes—
360C2×C4D5×C4⋊C412yes—
361C2×C4#405712yes—
362C2×C4#405812yes—
363C2×C4#406412yes—
364C2×C4#406512yes—
365C2×C4C22×C4×D511yescharacteristic
366C2×C4#414412yes—
367C2×C4C2×Q8×D511yescharacteristic
368C2×C4#414712yes—
369C8—24no—
370C8—104no—
371C5#2198411yescharacteristic, complemented, Sylow
372C22—102no—
373C22—102no—
374C22—102no—
375C22—201no—
376C22—201no—
377C22—201no—
378C22—201no—
379C22C2×D5×C4⋊C411yescharacteristic
380C22#6604911yescharacteristic
381C22#6605811yescharacteristic
382C22C2×#405711yescharacteristic
383C22C2×#405811yescharacteristic
384C22C2×#406411yescharacteristic
385C22C2×#406511yescharacteristic
386C4—82no—
387C4—82no—
388C4—102no—
389C4—102no—
390C4—102no—
391C4—402no—
392C4—402no—
393C4#6516914yes—
394C4#6517014yes—
395C4#6583812yes—
396C4#6606312yes—
397C2—202no—
398C2#1060609011yescharacteristic
399C2C2×#6516911yescharacteristic
400C2C2×#6517011yescharacteristic
401C2#1061351711yescharacteristic
402C2#1061353411yescharacteristic
403C2#1061367311yescharacteristic
404C2#1061367511yescharacteristic
subgroupG/H
centerC23C2×D4×D5
commutatorC2×C4×C5C23×C4
FrattiniC22×C4C23×D5
FittingC5×#1845C2
socleC23×C5C22×D4
prime radicalC23×D5C2×D4
cosocleC22×C4×C5C24
hypercenter#1845D5
hypocenterC5#21984
LCSG ▷ C2×C4×C5 ▷ C2×C5 ▷ C5
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1845

Complements

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

NH ≅ G/Ntype
4. C5×#1845C2semidirect
6. #10601409C2semidirect
7. #10600555C2semidirect
8. #10600369C2semidirect
9. #10600367C2semidirect
33. #64974C4semidirect
81. #1845D5semidirect
371. C5#21984semidirect

Maximal subgroups

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

subgroupindex
C2×#649742
#106060492
C5×#18452
#106014262
#106014092
#106005552
#106003692
#106003672
#219845