Finite Group Databasebeta

← #12117118

Subgroups — #12117118

Order 1280.

724 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
1#10613793C211yesmaximal, characteristic
2C5×#1749C211yesmaximal, characteristic
3#10602323C211yesmaximal, characteristic
4#10600702C212yesmaximal
5#10600700C212yesmaximal
6#10600629C212yesmaximal, complemented
7#10600603C212yesmaximal, complemented
8#10599406C212yesmaximal, complemented
9#10599403C212yesmaximal, complemented
10C22×C42×C5C2211yescharacteristic
11C2×#4072C2212yes—
12C2×C5×C22⋊C8C2211yescharacteristic
13C2×C5×C22⋊C8C2211yescharacteristic
14C2×C5×#97C2212yes—
15#65501C412yes—
16#65501C411yescharacteristic
17#65501C411yescharacteristic
18C2×C22⋊#168C2212yes—
19C2×#3947C2212yes—
20C2×#3946C2212yes—
21C2×#3946C2212yes—
22C2×C8×Dic5C2212yes—
23#65225C412yes—
24#65224C411yescharacteristic
25#65224C411yescharacteristic
26C4×C4⋊Dic5C2211yescharacteristic
27C4×C4⋊Dic5C2211yescharacteristic
28C4×C4⋊Dic5C2212yes—
29C2×C4⋊#168C2212yes—
30C5×#336C2212yes—
31#64770C2212yes—
32#64770C2212yes—
33#64770C2212yes—
34#64770C2212yes—
35#64744C2211yescharacteristic
36#64744C2211yescharacteristic
37#22628—51nomaximal, Sylow
38C23×C4×C5C2×C411yescharacteristic
39C23×C4×C5C2×C411yescharacteristic
40C23×C4×C5C2311yescharacteristic
41C22×C5×C8C2312yes—
42C22×C5×C8C2312yes—
43C2×C42×C5—24no—
44C2×C42×C5C2×C411yescharacteristic
45C2×C42×C5C2×C411yescharacteristic
46C2×C42×C5C2311yescharacteristic
47C2×C42×C5C2311yescharacteristic
48C2×C22⋊Dic5C2×C412yes—
49C2×C22⋊Dic5C2×C412yes—
50#4072—42no—
51#4072—42no—
52C2×C4⋊Dic5C2×C412yes—
53C2×C4⋊Dic5C2×C412yes—
54C2×C4⋊Dic5C2×C412yes—
55C2×C4⋊Dic5C2×C411yescharacteristic
56C2×C4⋊Dic5C2×C411yescharacteristic
57C2×C4⋊Dic5C2×C411yescharacteristic
58C2×C4⋊Dic5C2×C411yescharacteristic
59C2×C4⋊Dic5C2×C412yes—
60C2×C4×Dic5C2312yes—
61C2×C4×Dic5C2312yes—
62C22×#168C2312yes—
63C22×#168C2312yes—
64C5×C22⋊C8—22no—
65C5×C22⋊C8Q814yes—
66C5×#97—24no—
67#3963C2×C412yes—
68#3963C2×C412yes—
69#3963C2×C412yes—
70#3963C2×C412yes—
71C22⋊#168—24no—
72#3947—42no—
73#3946—42no—
74#3946—42no—
75C8×Dic5—42no—
76C4⋊#168—24no—
77#2285—51no—
78#1940—52no—
79#1917—52no—
80#1916—52no—
81#1798—52no—
82#1797—51no—
83#1771—52no—
84#1750—52no—
85#1749D511yescharacteristic
86C24×C5C22×C411yescharacteristic
87C22×C4×C5—22no—
88C22×C4×C5—22no—
89C22×C4×C5—22no—
90C22×C4×C5—22no—
91C22×C4×C5—22no—
92C22×C4×C5—41no—
93C22×C4×C5—41no—
94C22×C4×C5—41no—
95C22×C4×C5C22×C411yescharacteristic
96C22×C4×C5C22×C411yescharacteristic
97C22×C4×C5C22×C411yescharacteristic
98C22×C4×C5C22×C411yescharacteristic
99C22×C4×C5C22×C411yescharacteristic
100C22×C4×C5C2×Q812yes—
101C22×C4×C5#4112yes—
102C22×C4×C5C2411yescharacteristic
103C22×Dic5C22×C412yes—
104C22×Dic5C22×C412yes—
105C22×Dic5C22×C412yes—
106C22×Dic5C22×C412yes—
107C2×C5×C8—22no—
108C2×C5×C8—22no—
109C2×C5×C8—22no—
110C2×C5×C8—22no—
111C2×C5×C8—22no—
112C2×C5×C8C2×Q814yes—
113C42×C5—22no—
114C42×C5—22no—
115C42×C5—22no—
116C42×C5#4114yes—
117C22⋊Dic5—42no—
118C22⋊Dic5—42no—
119C4⋊Dic5—24no—
120C4⋊Dic5—41no—
121C4⋊Dic5—41no—
122C4⋊Dic5—42no—
123C4⋊Dic5—41no—
124C4⋊Dic5—41no—
125C4⋊Dic5—42no—
126C4⋊Dic5—42no—
127C4×Dic5—22no—
128C4×Dic5—22no—
129C4×Dic5—22no—
130C4×Dic5—22no—
131C2×#168—22no—
132C2×#168—22no—
133C2×#168—22no—
134C2×#168—22no—
135C2×#168—22no—
136C2×#168—22no—
137C2×#117—52no—
138C22×C42C2×D511yescharacteristic
139C2×C4⋊C8—52no—
140C2×C4⋊C8—52no—
141C2×C4⋊C8—52no—
142C2×C22⋊C8—52no—
143C2×C22⋊C8C2×D511yescharacteristic
144C2×C22⋊C8C2×D511yescharacteristic
145C2×#97—52no—
146C2×#97C2×D512yes—
147C2×C4×C8—52no—
148#384—51no—
149#384—51no—
150#382—52no—
151#380—52no—
152#380—51no—
153#380—51no—
154C4×C4⋊C4—52no—
155C4×C4⋊C4—51no—
156C4×C4⋊C4—51no—
157#336—52no—
158#336—52no—
159#336—52no—
160#336—52no—
161#336—51no—
162#336—51no—
163#336C2×D512yes—
164C23×C5—21no—
165C23×C5—21no—
166C23×C5—21no—
167C23×C5—21no—
168C23×C5—21no—
169C23×C5—21no—
170C23×C5C4×Q812yes—
171C23×C5C23×C411yescharacteristic
172C2×C4×C5—21no—
173C2×C4×C5—21no—
174C2×C4×C5—21no—
175C2×C4×C5—21no—
176C2×C4×C5—21no—
177C2×C4×C5—21no—
178C2×C4×C5—21no—
179C2×C4×C5—21no—
180C2×C4×C5—24no—
181C2×C4×C5—24no—
182C2×C4×C5—21no—
183C2×C4×C5—21no—
184C2×C4×C5—21no—
185C2×C4×C5—44no—
186C2×C4×C5—44no—
187C2×C4×C5#11712yes—
188C2×C4×C5#11712yes—
189C2×C4×C5C4×Q812yes—
190C2×C4×C5#13112yes—
191C2×C4×C5C22×Q811yescharacteristic
192C2×C4×C5C2×#4111yescharacteristic
193C2×C4×C5C2×#4112yes—
194C2×C4×C5H(5,𝔽2)12yes—
195C2×Dic5—22no—
196C2×Dic5—22no—
197C2×Dic5—22no—
198C2×Dic5—22no—
199C2×Dic5—22no—
200C2×Dic5—22no—
201C2×Dic5—22no—
202C2×Dic5—22no—
203C2×Dic5—22no—
204C2×Dic5—22no—
205C2×Dic5—22no—
206C2×Dic5—22no—
207C5×C8—24no—
208C5×C8—42no—
209#168—42no—
210#168—42no—
211C23×C4C4×D511yescharacteristic
212C23×C4C4×D511yescharacteristic
213C23×C4C22×D511yescharacteristic
214C22×C8—52no—
215C22×C8—52no—
216C22×C8C22×D512yes—
217C22×C8C22×D512yes—
218#117—202no—
219#117—202no—
220C2×C4⋊C4—52no—
221C2×C4⋊C4—52no—
222C2×C4⋊C4—52no—
223C2×C4⋊C4—51no—
224C2×C4⋊C4—51no—
225C2×C4⋊C4—51no—
226C2×C4⋊C4—51no—
227C2×C4⋊C4—52no—
228C2×C22⋊C4—52no—
229C2×C22⋊C4—52no—
230C2×C42—24no—
231C2×C42—52no—
232C2×C42—52no—
233C2×C42C4×D511yescharacteristic
234C2×C42C4×D511yescharacteristic
235C2×C42C22×D511yescharacteristic
236C2×C42C22×D511yescharacteristic
237C4⋊C8—104no—
238C4⋊C8—202no—
239C4⋊C8—202no—
240C22⋊C8—22no—
241C22⋊C8—104no—
242C22⋊C8Dic1014yes—
243#97—24no—
244#97—202no—
245C4×C8—202no—
246#95—52no—
247#95—52no—
248#95—52no—
249#95—52no—
250C22×C5—22no—
251C22×C5—22no—
252C22×C5—22no—
253C22×C5—41no—
254C22×C5—41no—
255C22×C5—41no—
256C22×C5—41no—
257C22×C5C2×#11711yescharacteristic
258C22×C5C2×C4×Q811yescharacteristic
259C22×C5#51811yescharacteristic
260C22×C5C2×#13111yescharacteristic
261C22×C5#56811yescharacteristic
262C22×C5#56811yescharacteristic
263C22×C5#56811yescharacteristic
264C4×C5—22no—
265C4×C5—22no—
266C4×C5—22no—
267C4×C5—22no—
268C4×C5—41no—
269C4×C5#55312yes—
270C4×C5#55412yes—
271Dic5—42no—
272Dic5—42no—
273Dic5—42no—
274Dic5—42no—
275C24C2×C4×D511yescharacteristic
276C22×C4—22no—
277C22×C4—22no—
278C22×C4—22no—
279C22×C4—22no—
280C22×C4—22no—
281C22×C4—41no—
282C22×C4—41no—
283C22×C4—41no—
284C22×C4—52no—
285C22×C4—52no—
286C22×C4—52no—
287C22×C4—52no—
288C22×C4C2×Dic1012yes—
289C22×C4C2×C4×D511yescharacteristic
290C22×C4C2×C4×D511yescharacteristic
291C22×C4C2×C4×D511yescharacteristic
292C22×C4C2×C4×D511yescharacteristic
293C22×C4C2×C4×D511yescharacteristic
294C22×C4#70912yes—
295C22×C4C23×D511yescharacteristic
296C2×C8—22no—
297C2×C8—22no—
298C2×C8—22no—
299C2×C8—22no—
300C2×C8—22no—
301C2×C8—102no—
302C2×C8—102no—
303C2×C8—102no—
304C2×C8—102no—
305C2×C8—102no—
306C2×C8—102no—
307C2×C8C2×Dic1014yes—
308C4⋊C4—104no—
309C4⋊C4—202no—
310C4⋊C4—202no—
311C4⋊C4—201no—
312C4⋊C4—201no—
313C4⋊C4—201no—
314C4⋊C4—201no—
315C4⋊C4—202no—
316C22⋊C4—202no—
317C22⋊C4—202no—
318C42—22no—
319C42—22no—
320C42—22no—
321C42—102no—
322C42—102no—
323C42—102no—
324C42—102no—
325C42#71114yes—
326C2×C5—42no—
327C2×C5#234511yescharacteristic
328C2×C5#292111yescharacteristic
329C2×C5#292411yescharacteristic
330C2×C5#296111yescharacteristic
331C2×C5#296211yescharacteristic
332C2×C5#298011yescharacteristic
333C2×C5#298111yescharacteristic
334C23—21no—
335C23—21no—
336C23—21no—
337C23—21no—
338C23—21no—
339C23—21no—
340C23C4×Dic1012yes—
341C23C22×C4×D511yescharacteristic
342C2×C4—21no—
343C2×C4—21no—
344C2×C4—21no—
345C2×C4—21no—
346C2×C4—21no—
347C2×C4—21no—
348C2×C4—21no—
349C2×C4—21no—
350C2×C4—24no—
351C2×C4—24no—
352C2×C4—21no—
353C2×C4—21no—
354C2×C4—21no—
355C2×C4—44no—
356C2×C4—44no—
357C2×C4—102no—
358C2×C4—102no—
359C2×C4—102no—
360C2×C4—102no—
361C2×C4—102no—
362C2×C4—102no—
363C2×C4—102no—
364C2×C4—102no—
365C2×C4—102no—
366C2×C4—102no—
367C2×C4—102no—
368C2×C4—102no—
369C2×C4C4×Dic1012yes—
370C2×C4#402312yes—
371C2×C4#402312yes—
372C2×C4#405312yes—
373C2×C4C22×Dic1011yescharacteristic
374C2×C4C2×#70911yescharacteristic
375C2×C4C2×#71112yes—
376C2×C4#414912yes—
377C8—24no—
378C8—42no—
379C8—202no—
380C8—202no—
381C5#2262811yescharacteristic, complemented, Sylow
382C22—22no—
383C22—22no—
384C22—22no—
385C22—41no—
386C22—41no—
387C22—41no—
388C22—41no—
389C22C2×C4×Dic1011yescharacteristic
390C22C2×#402311yescharacteristic
391C22#6585511yescharacteristic
392C22C2×#405311yescharacteristic
393C22#6607911yescharacteristic
394C22#6607911yescharacteristic
395C22#6607911yescharacteristic
396C4—22no—
397C4—22no—
398C4—22no—
399C4—22no—
400C4—41no—
401C4—202no—
402C4—202no—
403C4—202no—
404C4—202no—
405C4#6587712yes—
406C4#6589912yes—
407C2—42no—
408C2#1060518811yescharacteristic
409C2#1061021111yescharacteristic
410C2#1061023911yescharacteristic
411C2#1061035111yescharacteristic
412C2#1061035211yescharacteristic
413C2#1061106811yescharacteristic
414C2#1061106911yescharacteristic
subgroupG/H
center, FrattiniC22×C4C23×D5
commutator, socleC23×C5C23×C4
FittingC5×#1749C2
prime radicalC24×C5C22×C4
cosocleC22×C4×C5C24
hypercenter#1749D5
hypocenterC5#22628
LCSG ▷ C23×C5 ▷ C5
UCSC1 ◁ C22×C4 ◁ #1749

Complements

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

NH ≅ G/Ntype
6. #10600629C2semidirect
7. #10600603C2semidirect
8. #10599406C2semidirect
9. #10599403C2semidirect
381. C5#22628semidirect

Maximal subgroups

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

subgroupindex
#106137932
C5×#17492
#106023232
#106007022
#106007002
#106006292
#106006032
#105994062
#105994032
#226285