Finite Group Databasebeta

← #12016805

Subgroups — #12016805

Order 1280.

724 conjugacy classes of subgroups; only the 111 normal subgroups are shown.

normal subgroups only

This group has 724 conjugacy classes of subgroups — too many to draw. Only its 111 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C5×#2406C211yesmaximal, characteristic
2C2×#65529C211yesmaximal, characteristic
3#10602930C211yesmaximal, characteristic
4C2×#64759C212yesmaximal, complemented
5#10598128C212yesmaximal, complemented
6C22×C5×#41C2211yescharacteristic
7C2×C5×#123C411yescharacteristic, complemented
8C2×C5×#123C2211yescharacteristic
9C22×C5×C4⋊C4C411yescharacteristic, complemented
10C2×#4072—21no—
11C5×#395—21no—
12C5×#394—21no—
13C5×#386C411yescharacteristic, complemented
14C5×#386C2211yescharacteristic
15C5×#385—21no—
16C5×#385—21no—
17C5×#381C411yescharacteristic, complemented
18#65529—24no—
19C2×#3968—21no—
20C2×#3966—21no—
21C2×#3966C2212yes—
22C2×#3965C2212yes—
23#65496—21no—
24#64759—24no—
25#13085—51nomaximal, Sylow
26C2×C5×#41—22no—
27C2×C5×#41—22no—
28C2×C5×#41—41no—
29C2×C5×#41—41no—
30C2×C5×#41—42no—
31C22×C5×Q8D411yescharacteristic
32C22×C5×D4C2×C411yescharacteristic
33C22×C5×D4D411yescharacteristic
34C22×C5×D4C2311yescharacteristic
35C23×C4×C5C2×C411yescharacteristic
36C23×C4×C5C2×C411yescharacteristic
37C23×C4×C5C2×C411yescharacteristic
38C23×C4×C5D411yescharacteristic
39C23×C4×C5D411yescharacteristic
40C5×#123—22no—
41C5×#123—22no—
42C5×#123—42no—
43C2×C5×C4⋊C4—21no—
44C2×C5×C4⋊C4—21no—
45C2×C5×C4⋊C4—21no—
46C2×C5×C4⋊C4—21no—
47C2×C5×C4⋊C4—41no—
48C2×C5×C4⋊C4—41no—
49C2×C5×C22⋊C4—21no—
50C2×C5×C22⋊C4—21no—
51C2×C5×C22⋊C4—21no—
52C2×C5×C22⋊C4—21no—
53C2×C5×C22⋊C4C2×C411yescharacteristic
54C2×C5×C22⋊C4C2×C411yescharacteristic
55C2×C22⋊Dic5—22no—
56#4072—24no—
57#4072—42no—
58C2×C4⋊Dic5—41no—
59C2×C4×Dic5—41no—
60C2×#681—22no—
61C5×#95—21no—
62C5×#95—21no—
63C5×#95—21no—
64C5×#95—21no—
65#3968—42no—
66#3966—24no—
67#3966—22no—
68#3966D414yes—
69#3965—22no—
70#3965—22no—
71C22⋊#168—41no—
72C22⋊#168—41no—
73C2×#410—51no—
74#2406D511yescharacteristic
75C2×#352—52no—
76#1785—51no—
77#1390—52no—
78C24×C5C22⋊C411yescharacteristic
79C24×C5C22×C411yescharacteristic
80C24×C5C2×D411yescharacteristic
81C5×#41—44no—
82C5×#41—44no—
83C5×#41—44no—
84C5×#41—44no—
85C2×C5×Q8—22no—
86C2×C5×Q8—22no—
87C2×C5×Q8—41no—
88C2×C5×D4—22no—
89C2×C5×D4—21no—
90C2×C5×D4—21no—
91C2×C5×D4—22no—
92C2×C5×D4—41no—
93C2×C5×D4—41no—
94C2×C5×D4—41no—
95C2×C5×D4—41no—
96C2×C5×D4—41no—
97C2×C5×D4C2×D412yes—
98C2×C5×D4#4112yes—
99C22×C4×C5—21no—
100C22×C4×C5—21no—
101C22×C4×C5—21no—
102C22×C4×C5—21no—
103C22×C4×C5—22no—
104C22×C4×C5—21no—
105C22×C4×C5—22no—
106C22×C4×C5—21no—
107C22×C4×C5—21no—
108C22×C4×C5—21no—
109C22×C4×C5—21no—
110C22×C4×C5—22no—
111C22×C4×C5—21no—
112C22×C4×C5—21no—
113C22×C4×C5—21no—
114C22×C4×C5—22no—
115C22×C4×C5—41no—
116C22×C4×C5—41no—
117C22×C4×C5—42no—
118C22×C4×C5—41no—
119C22×C4×C5—41no—
120C22×C4×C5—41no—
121C22×C4×C5C22⋊C411yescharacteristic
122C22×C4×C5C22⋊C411yescharacteristic
123C22×C4×C5C22⋊C411yescharacteristic
124C22×C4×C5C2×D411yescharacteristic
125C22×Dic5—42no—
126C5×C4⋊C4—42no—
127C5×C4⋊C4—41no—
128C5×C4⋊C4—41no—
129C5×C4⋊C4—42no—
130C5×C4⋊C4—42no—
131C5×C22⋊C4—22no—
132C5×C22⋊C4—42no—
133C5×C22⋊C4—42no—
134C5×C22⋊C4—42no—
135C5×C22⋊C4—41no—
136C5×C22⋊C4—41no—
137C5×C22⋊C4—41no—
138C22⋊Dic5—24no—
139C22⋊Dic5—42no—
140C4⋊Dic5—44no—
141C4×Dic5—44no—
142#681—42no—
143#681—42no—
144C2×#168—42no—
145C22×#41C2×D511yescharacteristic
146C2×#123Dic511yescharacteristic, complemented
147C2×#123C2×D511yescharacteristic
148C2×#117—101no—
149C22×C4⋊C4Dic511yescharacteristic, complemented
150C2×#101—101no—
151C2×#100—52no—
152#410—104no—
153C2×#99—52no—
154C2×#99—101no—
155#407—101no—
156#395—21no—
157#394—21no—
158#386Dic511yescharacteristic, complemented
159#386C2×D511yescharacteristic
160#385—21no—
161#385—21no—
162#381Dic511yescharacteristic, complemented
163#352—104no—
164C23×C5—21no—
165C23×C5—21no—
166C23×C5—21no—
167C23×C5—21no—
168C23×C5—21no—
169C23×C5—21no—
170C23×C5—21no—
171C23×C5—41no—
172C23×C5—41no—
173C23×C5—41no—
174C23×C5—41no—
175C23×C5—41no—
176C23×C5—41no—
177C23×C5C2×C22⋊C411yescharacteristic
178C23×C5C4×D412yes—
179C23×C5#12011yescharacteristic
180C23×C5#12311yescharacteristic
181C5×Q8—44no—
182C5×D4—42no—
183C5×D4—42no—
184C5×D4—44no—
185C5×D4—42no—
186C5×D4—42no—
187C2×C4×C5—22no—
188C2×C4×C5—22no—
189C2×C4×C5—22no—
190C2×C4×C5—22no—
191C2×C4×C5—21no—
192C2×C4×C5—21no—
193C2×C4×C5—21no—
194C2×C4×C5—21no—
195C2×C4×C5—21no—
196C2×C4×C5—21no—
197C2×C4×C5—22no—
198C2×C4×C5—21no—
199C2×C4×C5—21no—
200C2×C4×C5—21no—
201C2×C4×C5—42no—
202C2×C4×C5—42no—
203C2×C4×C5—42no—
204C2×C4×C5—42no—
205C2×C4×C5—41no—
206C2×C4×C5—42no—
207C2×C4×C5—41no—
208C2×C4×C5—42no—
209C2×C4×C5—41no—
210C2×C4×C5—41no—
211C2×C4×C5—41no—
212C2×C4×C5—41no—
213C2×C4×C5—41no—
214C2×C4×C5—41no—
215C2×C4×C5—41no—
216C2×C4×C5—41no—
217C2×C4×C5—41no—
218C2×C4×C5—82no—
219C2×C4×C5#9911yescharacteristic
220C2×C4×C5#9911yescharacteristic
221C2×C4×C5#9912yes—
222C2×C4×C5#12112yes—
223C2×Dic5—44no—
224C2×Dic5—82no—
225C2×Dic5—82no—
226#168—82no—
227C2×#41—22no—
228C2×#41—22no—
229C2×#41—41no—
230C2×#41—41no—
231C2×#41—42no—
232C22×Q8C22⋊D511yescharacteristic
233C22×D4C2×Dic511yescharacteristic
234C22×D4C22⋊D511yescharacteristic
235C22×D4C22×D511yescharacteristic
236C23×C4C2×Dic511yescharacteristic
237C23×C4C2×Dic511yescharacteristic
238C23×C4C2×Dic511yescharacteristic
239C23×C4C22⋊D511yescharacteristic
240C23×C4C22⋊D511yescharacteristic
241C2×M16—102no—
242#123—22no—
243#123—22no—
244#123—42no—
245#117—104no—
246#117—202no—
247C2×C4⋊C4—21no—
248C2×C4⋊C4—21no—
249C2×C4⋊C4—21no—
250C2×C4⋊C4—21no—
251C2×C4⋊C4—41no—
252C2×C4⋊C4—41no—
253C2×C4⋊C4—201no—
254C2×C22⋊C4—21no—
255C2×C22⋊C4—21no—
256C2×C22⋊C4—21no—
257C2×C22⋊C4—21no—
258C2×C22⋊C4—102no—
259C2×C22⋊C4C2×Dic511yescharacteristic
260C2×C22⋊C4C2×Dic511yescharacteristic
261C2×C42—201no—
262#101—202no—
263#100—102no—
264#100—102no—
265#99—54no—
266#99—104no—
267#99—102no—
268C22⋊C8—201no—
269C22⋊C8—201no—
270#95—21no—
271#95—21no—
272#95—21no—
273#95—21no—
274C22×C5—21no—
275C22×C5—21no—
276C22×C5—21no—
277C22×C5—21no—
278C22×C5—22no—
279C22×C5—42no—
280C22×C5—41no—
281C22×C5—41no—
282C22×C5—42no—
283C22×C5—42no—
284C22×C5—41no—
285C22×C5—41no—
286C22×C5—41no—
287C22×C5—41no—
288C22×C5—41no—
289C22×C5—41no—
290C22×C5—41no—
291C22×C5—81no—
292C22×C5—81no—
293C22×C5#38611yescharacteristic
294C22×C5C2×#9911yescharacteristic
295C22×C5C2×#9911yescharacteristic
296C4×C5—22no—
297C4×C5—21no—
298C4×C5—21no—
299C4×C5—22no—
300C4×C5—42no—
301C4×C5—41no—
302C4×C5—81no—
303C4×C5—81no—
304C4×C5—81no—
305Dic5—84no—
306C24C22⋊Dic511yescharacteristic
307C24C22×Dic511yescharacteristic
308C24C2×C22⋊D511yescharacteristic
309#41—44no—
310#41—44no—
311#41—44no—
312#41—44no—
313C2×Q8—22no—
314C2×Q8—22no—
315C2×Q8—41no—
316C2×D4—22no—
317C2×D4—21no—
318C2×D4—21no—
319C2×D4—22no—
320C2×D4—41no—
321C2×D4—41no—
322C2×D4—41no—
323C2×D4—41no—
324C2×D4—41no—
325C2×D4D4×D512yes—
326C2×D4#71112yes—
327C22×C4—21no—
328C22×C4—21no—
329C22×C4—21no—
330C22×C4—21no—
331C22×C4—22no—
332C22×C4—21no—
333C22×C4—22no—
334C22×C4—21no—
335C22×C4—21no—
336C22×C4—21no—
337C22×C4—21no—
338C22×C4—22no—
339C22×C4—21no—
340C22×C4—21no—
341C22×C4—21no—
342C22×C4—22no—
343C22×C4—41no—
344C22×C4—41no—
345C22×C4—42no—
346C22×C4—41no—
347C22×C4—41no—
348C22×C4—41no—
349C22×C4—202no—
350C22×C4C22⋊Dic511yescharacteristic
351C22×C4C22⋊Dic511yescharacteristic
352C22×C4C22⋊Dic511yescharacteristic
353C22×C4C2×C22⋊D511yescharacteristic
354M16—202no—
355M16—202no—
356C2×C8—202no—
357C4⋊C4—42no—
358C4⋊C4—41no—
359C4⋊C4—41no—
360C4⋊C4—42no—
361C4⋊C4—42no—
362C4⋊C4—204no—
363C22⋊C4—22no—
364C22⋊C4—42no—
365C22⋊C4—42no—
366C22⋊C4—42no—
367C22⋊C4—41no—
368C22⋊C4—41no—
369C22⋊C4—41no—
370C22⋊C4—104no—
371C22⋊C4—202no—
372C42—204no—
373C2×C5—21no—
374C2×C5—21no—
375C2×C5—41no—
376C2×C5—42no—
377C2×C5—41no—
378C2×C5—81no—
379C2×C5#189811yescharacteristic
380C2×C5#212011yescharacteristic
381C2×C5#212311yescharacteristic
382C23—21no—
383C23—21no—
384C23—21no—
385C23—21no—
386C23—21no—
387C23—21no—
388C23—21no—
389C23—41no—
390C23—41no—
391C23—41no—
392C23—41no—
393C23—41no—
394C23—41no—
395C23D4×Dic512yes—
396C23#408111yescharacteristic
397C23#408311yescharacteristic
398C23C2×C22⋊Dic511yescharacteristic
399Q8—44no—
400D4—42no—
401D4—42no—
402D4—44no—
403D4—42no—
404D4—42no—
405C2×C4—22no—
406C2×C4—22no—
407C2×C4—22no—
408C2×C4—22no—
409C2×C4—21no—
410C2×C4—21no—
411C2×C4—21no—
412C2×C4—21no—
413C2×C4—21no—
414C2×C4—21no—
415C2×C4—22no—
416C2×C4—21no—
417C2×C4—21no—
418C2×C4—21no—
419C2×C4—42no—
420C2×C4—42no—
421C2×C4—42no—
422C2×C4—42no—
423C2×C4—41no—
424C2×C4—42no—
425C2×C4—41no—
426C2×C4—42no—
427C2×C4—41no—
428C2×C4—41no—
429C2×C4—41no—
430C2×C4—41no—
431C2×C4—41no—
432C2×C4—41no—
433C2×C4—41no—
434C2×C4—41no—
435C2×C4—41no—
436C2×C4—82no—
437C2×C4—204no—
438C2×C4—402no—
439C2×C4—402no—
440C2×C4#396612yes—
441C2×C4#396611yescharacteristic
442C2×C4#396611yescharacteristic
443C2×C4#408512yes—
444C8—402no—
445C5#1308511yescharacteristic, complemented, Sylow
446C22—21no—
447C22—21no—
448C22—21no—
449C22—21no—
450C22—22no—
451C22—42no—
452C22—41no—
453C22—41no—
454C22—42no—
455C22—42no—
456C22—41no—
457C22—41no—
458C22—41no—
459C22—41no—
460C22—41no—
461C22—41no—
462C22—41no—
463C22—81no—
464C22—81no—
465C22C2×#396611yescharacteristic
466C22C2×#396611yescharacteristic
467C22#6550911yescharacteristic
468C4—22no—
469C4—21no—
470C4—21no—
471C4—22no—
472C4—42no—
473C4—41no—
474C4—81no—
475C4—81no—
476C4—81no—
477C4—404no—
478C2—21no—
479C2—21no—
480C2—41no—
481C2—42no—
482C2—41no—
483C2—81no—
484C2#1060246311yescharacteristic
485C2#1060250911yescharacteristic
486C2#1060266311yescharacteristic
subgroupG/H
centerC22C2×#3966
commutatorC24×C5C22×C4
FrattiniC22×D4C22×D5
FittingC5×#2406C2
socleC22×C5C2×#99
prime radicalC22×C5×#41C22
cosocleC22×C5×D4C23
hypercenter#2406D5
hypocenterC5#13085
LCSG ▷ C24×C5 ▷ C22×C5 ▷ C2×C5 ▷ C5
UCSC1 ◁ C22 ◁ C22×C4 ◁ C22×#41 ◁ #2406

Complements

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

NH ≅ G/Ntype
4. C2×#64759C2semidirect
5. #10598128C2semidirect
7. C2×C5×#123C4semidirect
9. C22×C5×C4⋊C4C4semidirect
13. C5×#386C4semidirect
17. C5×#381C4semidirect
146. C2×#123Dic5semidirect
149. C22×C4⋊C4Dic5semidirect
158. #386Dic5semidirect
162. #381Dic5semidirect
445. C5#13085semidirect

Maximal subgroups

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

subgroupindex
C5×#24062
C2×#655292
#106029302
C2×#647592
#105981282
#130855