Finite Group Databasebeta

← #12017593

Subgroups — #12017593

Order 1280.

624 conjugacy classes of subgroups; only the 107 normal subgroups are shown.

normal subgroups only

This group has 624 conjugacy classes of subgroups — too many to draw. Only its 107 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×#2534C211yesmaximal, characteristic
2C2×#64759C212yesmaximal, complemented
3C2×#64756C212yesmaximal
4#10602412C211yesmaximal, characteristic, complemented
5C4×#3965C211yesmaximal, characteristic, complemented
6C2×C5×#123C2211yescharacteristic
7C2×C5×#120C2211yescharacteristic
8C2×C4×C5×D4C2211yescharacteristic
9C22×C5×C22⋊C4C411yescharacteristic, complemented
10C5×#396—21no—
11C5×#394—21no—
12C5×#388—21no—
13C5×#386—21no—
14C5×#381C411yescharacteristic, complemented
15C5×#380C411yescharacteristic, complemented
16C5×#379C411yescharacteristic, complemented
17C2×#3966C2212yes—
18C2×#3965C2212yes—
19#65501—21no—
20C4×#681—21no—
21#64759—24no—
22#64756—24no—
23#64750—21no—
24#64746—21no—
25#13219—51nomaximal, Sylow
26C25×C5C2×C411yescharacteristic
27C22×C5×D4—21no—
28C22×C5×D4C2311yescharacteristic
29C23×C4×C5C2×C411yescharacteristic
30C23×C4×C5C2×C411yescharacteristic
31C23×C4×C5D411yescharacteristic
32C5×#123—22no—
33C5×#123—22no—
34C5×#120—22no—
35C5×#120—22no—
36C4×C5×D4—41no—
37C4×C5×D4—41no—
38C2×C5×C4⋊C4—21no—
39C2×C5×C4⋊C4—21no—
40C2×C5×C4⋊C4D411yescharacteristic
41C2×C5×C22⋊C4—21no—
42C2×C5×C22⋊C4—21no—
43C2×C5×C22⋊C4—21no—
44C2×C5×C22⋊C4—21no—
45C2×C5×C22⋊C4—21no—
46C2×C5×C22⋊C4—21no—
47C2×C5×C22⋊C4—41no—
48C2×C5×C22⋊C4—41no—
49C2×C5×C22⋊C4C2×C411yescharacteristic
50C2×C5×C22⋊C4C2×C411yescharacteristic
51C2×C5×C22⋊C4C2×C411yescharacteristic
52C2×C5×C22⋊C4D411yescharacteristic
53C2×C42×C5D411yescharacteristic
54C2×C22⋊Dic5—22no—
55C2×C4⋊Dic5—41no—
56C2×#681—22no—
57C5×#95—21no—
58C5×#95—21no—
59C5×#95—21no—
60#3966—22no—
61#3966—22no—
62#3965—22no—
63#3965D414yes—
64#3963—41no—
65#3935—41no—
66C4×#168—41no—
67#2534D511yescharacteristic
68C2×#352—52no—
69C2×AΣL(1,𝔽16)—52no—
70#1799—51no—
71C4×#100—51no—
72C24×C5—21no—
73C24×C5—22no—
74C24×C5—41no—
75C24×C5—42no—
76C24×C5—41no—
77C24×C5—81no—
78C24×C5C22×C411yescharacteristic
79C24×C5C2×D411yescharacteristic
80C2×C5×D4—41no—
81C2×C5×D4—41no—
82C2×C5×D4—42no—
83C2×C5×D4—41no—
84C2×C5×D4—41no—
85C2×C5×D4—81no—
86C2×C5×D4C2×D412yes—
87C2×C5×D4#4112yes—
88C22×C4×C5—21no—
89C22×C4×C5—21no—
90C22×C4×C5—21no—
91C22×C4×C5—21no—
92C22×C4×C5—21no—
93C22×C4×C5—21no—
94C22×C4×C5—21no—
95C22×C4×C5—22no—
96C22×C4×C5—41no—
97C22×C4×C5—41no—
98C22×C4×C5—81no—
99C22×C4×C5—81no—
100C22×C4×C5C22⋊C411yescharacteristic
101C22×C4×C5C22⋊C411yescharacteristic
102C22×C4×C5C22⋊C411yescharacteristic
103C22×C4×C5C22⋊C411yescharacteristic
104C22×C4×C5C2×D411yescharacteristic
105C22×Dic5—42no—
106C5×C4⋊C4—41no—
107C5×C4⋊C4—42no—
108C5×C4⋊C4—81no—
109C5×C22⋊C4—22no—
110C5×C22⋊C4—22no—
111C5×C22⋊C4—41no—
112C5×C22⋊C4—41no—
113C5×C22⋊C4—42no—
114C5×C22⋊C4—42no—
115C5×C22⋊C4—41no—
116C5×C22⋊C4—41no—
117C5×C22⋊C4—41no—
118C5×C22⋊C4—41no—
119C5×C22⋊C4—42no—
120C5×C22⋊C4—81no—
121C42×C5—21no—
122C42×C5—21no—
123C22⋊Dic5—42no—
124C22⋊Dic5—42no—
125C4⋊Dic5—82no—
126#681—24no—
127#681—42no—
128C2×#168—42no—
129C2×#123C2×D511yescharacteristic
130C2×#120C2×D511yescharacteristic
131C2×C4×D4C2×D511yescharacteristic
132C22×C22⋊C4Dic511yescharacteristic, complemented
133C2×#100—52no—
134C2×#99—52no—
135C4×M16—101no—
136#396—21no—
137#394—21no—
138#388—21no—
139#386—21no—
140#381Dic511yescharacteristic, complemented
141#380—101no—
142#380Dic511yescharacteristic, complemented
143#379Dic511yescharacteristic, complemented
144#352—104no—
145AΣL(1,𝔽16)—104no—
146#343—101no—
147#342—101no—
148C23×C5—21no—
149C23×C5—21no—
150C23×C5—21no—
151C23×C5—21no—
152C23×C5—22no—
153C23×C5—41no—
154C23×C5—41no—
155C23×C5—41no—
156C23×C5—41no—
157C23×C5—41no—
158C23×C5—41no—
159C23×C5—41no—
160C23×C5—42no—
161C23×C5—42no—
162C23×C5—41no—
163C23×C5—41no—
164C23×C5—41no—
165C23×C5—82no—
166C23×C5—84no—
167C23×C5—84no—
168C23×C5—82no—
169C23×C5C2×C22⋊C411yescharacteristic
170C23×C5C4×D412yes—
171C23×C5#12411yescharacteristic
172C23×C5#12711yescharacteristic
173C5×D4—42no—
174C5×D4—42no—
175C5×D4—82no—
176C5×D4—82no—
177C2×C4×C5—21no—
178C2×C4×C5—21no—
179C2×C4×C5—21no—
180C2×C4×C5—21no—
181C2×C4×C5—21no—
182C2×C4×C5—41no—
183C2×C4×C5—42no—
184C2×C4×C5—41no—
185C2×C4×C5—41no—
186C2×C4×C5—41no—
187C2×C4×C5—41no—
188C2×C4×C5—41no—
189C2×C4×C5—41no—
190C2×C4×C5—41no—
191C2×C4×C5—41no—
192C2×C4×C5—41no—
193C2×C4×C5—41no—
194C2×C4×C5—41no—
195C2×C4×C5—41no—
196C2×C4×C5—81no—
197C2×C4×C5—81no—
198C2×C4×C5—81no—
199C2×C4×C5—82no—
200C2×C4×C5#9911yescharacteristic
201C2×C4×C5#9911yescharacteristic
202C2×C4×C5#12112yes—
203C2×Dic5—82no—
204C2×Dic5—82no—
205C2×Dic5—82no—
206#168—44no—
207C25C2×Dic511yescharacteristic
208C22×D4—21no—
209C22×D4C22×D511yescharacteristic
210C23×C4C2×Dic511yescharacteristic
211C23×C4C2×Dic511yescharacteristic
212C23×C4C22⋊D511yescharacteristic
213C2×M16—102no—
214#123—22no—
215#123—22no—
216#120—22no—
217#120—22no—
218C4×D4—41no—
219C4×D4—41no—
220C2×C4⋊C4—21no—
221C2×C4⋊C4—21no—
222C2×C4⋊C4—201no—
223C2×C4⋊C4C22⋊D511yescharacteristic
224C2×C22⋊C4—21no—
225C2×C22⋊C4—21no—
226C2×C22⋊C4—21no—
227C2×C22⋊C4—21no—
228C2×C22⋊C4—21no—
229C2×C22⋊C4—21no—
230C2×C22⋊C4—41no—
231C2×C22⋊C4—41no—
232C2×C22⋊C4—102no—
233C2×C22⋊C4C2×Dic511yescharacteristic
234C2×C22⋊C4C2×Dic511yescharacteristic
235C2×C22⋊C4C2×Dic511yescharacteristic
236C2×C22⋊C4C22⋊D511yescharacteristic
237C2×C42C22⋊D511yescharacteristic
238#100—54no—
239#100—102no—
240#99—102no—
241#99—102no—
242#97—201no—
243C4×C8—201no—
244#95—21no—
245#95—21no—
246#95—21no—
247#95—201no—
248C22×C5—21no—
249C22×C5—21no—
250C22×C5—21no—
251C22×C5—22no—
252C22×C5—21no—
253C22×C5—41no—
254C22×C5—41no—
255C22×C5—42no—
256C22×C5—42no—
257C22×C5—42no—
258C22×C5—42no—
259C22×C5—41no—
260C22×C5—41no—
261C22×C5—41no—
262C22×C5—41no—
263C22×C5—42no—
264C22×C5—81no—
265C22×C5—81no—
266C22×C5—81no—
267C22×C5—84no—
268C22×C5—84no—
269C22×C5—81no—
270C22×C5—81no—
271C22×C5#39011yescharacteristic
272C22×C5C2×#9911yescharacteristic
273C22×C5#41011yescharacteristic
274C4×C5—21no—
275C4×C5—21no—
276C4×C5—22no—
277C4×C5—41no—
278C4×C5—41no—
279C4×C5—41no—
280C4×C5—81no—
281C4×C5—81no—
282C4×C5—81no—
283Dic5—162no—
284C24—21no—
285C24—22no—
286C24—41no—
287C24—42no—
288C24—41no—
289C24—81no—
290C24C22×Dic511yescharacteristic
291C24C2×C22⋊D511yescharacteristic
292C2×D4—41no—
293C2×D4—41no—
294C2×D4—42no—
295C2×D4—41no—
296C2×D4—41no—
297C2×D4—81no—
298C2×D4D4×D512yes—
299C2×D4#71112yes—
300C22×C4—21no—
301C22×C4—21no—
302C22×C4—21no—
303C22×C4—21no—
304C22×C4—21no—
305C22×C4—21no—
306C22×C4—21no—
307C22×C4—22no—
308C22×C4—41no—
309C22×C4—41no—
310C22×C4—81no—
311C22×C4—81no—
312C22×C4—202no—
313C22×C4C22⋊Dic511yescharacteristic
314C22×C4C22⋊Dic511yescharacteristic
315C22×C4C22⋊Dic511yescharacteristic
316C22×C4C22⋊Dic511yescharacteristic
317C22×C4C2×C22⋊D511yescharacteristic
318M16—104no—
319M16—202no—
320C2×C8—202no—
321C4⋊C4—41no—
322C4⋊C4—42no—
323C4⋊C4—81no—
324C4⋊C4—402no—
325C22⋊C4—22no—
326C22⋊C4—22no—
327C22⋊C4—41no—
328C22⋊C4—41no—
329C22⋊C4—42no—
330C22⋊C4—42no—
331C22⋊C4—41no—
332C22⋊C4—41no—
333C22⋊C4—41no—
334C22⋊C4—41no—
335C22⋊C4—42no—
336C22⋊C4—81no—
337C22⋊C4—202no—
338C22⋊C4—202no—
339C42—21no—
340C42—21no—
341C2×C5—21no—
342C2×C5—21no—
343C2×C5—41no—
344C2×C5—42no—
345C2×C5—41no—
346C2×C5—82no—
347C2×C5#196411yescharacteristic
348C2×C5#212011yescharacteristic
349C2×C5#212111yescharacteristic
350C23—21no—
351C23—21no—
352C23—21no—
353C23—21no—
354C23—22no—
355C23—41no—
356C23—41no—
357C23—41no—
358C23—41no—
359C23—41no—
360C23—41no—
361C23—41no—
362C23—42no—
363C23—42no—
364C23—41no—
365C23—41no—
366C23—41no—
367C23—82no—
368C23—84no—
369C23—84no—
370C23—82no—
371C23D4×Dic512yes—
372C23#408211yescharacteristic
373C23#408611yescharacteristic
374C23C2×C22⋊Dic511yescharacteristic
375D4—42no—
376D4—42no—
377D4—82no—
378D4—82no—
379C2×C4—21no—
380C2×C4—21no—
381C2×C4—21no—
382C2×C4—21no—
383C2×C4—21no—
384C2×C4—41no—
385C2×C4—42no—
386C2×C4—41no—
387C2×C4—41no—
388C2×C4—41no—
389C2×C4—41no—
390C2×C4—41no—
391C2×C4—41no—
392C2×C4—41no—
393C2×C4—41no—
394C2×C4—41no—
395C2×C4—41no—
396C2×C4—41no—
397C2×C4—41no—
398C2×C4—81no—
399C2×C4—81no—
400C2×C4—81no—
401C2×C4—82no—
402C2×C4—402no—
403C2×C4—402no—
404C2×C4—402no—
405C2×C4#396611yescharacteristic
406C2×C4#396611yescharacteristic
407C2×C4#408412yes—
408C8—204no—
409C5#1321911yescharacteristic, complemented, Sylow
410C22—21no—
411C22—21no—
412C22—22no—
413C22—21no—
414C22—21no—
415C22—41no—
416C22—41no—
417C22—42no—
418C22—42no—
419C22—42no—
420C22—42no—
421C22—41no—
422C22—41no—
423C22—41no—
424C22—41no—
425C22—42no—
426C22—81no—
427C22—81no—
428C22—81no—
429C22—84no—
430C22—84no—
431C22—81no—
432C22—81no—
433C22C2×#396611yescharacteristic
434C22#6551011yescharacteristic
435C22#6552911yescharacteristic
436C4—21no—
437C4—21no—
438C4—22no—
439C4—41no—
440C4—41no—
441C4—41no—
442C4—81no—
443C4—81no—
444C4—81no—
445C4—802no—
446C2—21no—
447C2—21no—
448C2—41no—
449C2—42no—
450C2—41no—
451C2—82no—
452C2#1060247911yescharacteristic
453C2#1060250911yescharacteristic
454C2#1060264011yescharacteristic
subgroupG/H
centerC22C2×#3966
commutatorC24×C5C22×C4
FrattiniC22×D4C22×D5
FittingC5×#2534C2
socleC22×C5C2×#99
prime radicalC2×C5×#120C22
cosocleC22×C5×D4C23
hypercenter#2534D5
hypocenterC5#13219
LCSG ▷ C24×C5 ▷ C22×C5 ▷ C2×C5 ▷ C5
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C4×D4 ◁ #2534

Complements

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

NH ≅ G/Ntype
2. C2×#64759C2semidirect
4. #10602412C2semidirect
5. C4×#3965C2semidirect
9. C22×C5×C22⋊C4C4semidirect
14. C5×#381C4semidirect
15. C5×#380C4semidirect
16. C5×#379C4semidirect
132. C22×C22⋊C4Dic5semidirect
140. #381Dic5semidirect
142. #380Dic5semidirect
143. #379Dic5semidirect
409. C5#13219semidirect

Maximal subgroups

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

subgroupindex
C5×#25342
C2×#647592
C2×#647562
#106024122
C4×#39652
#132195