Finite Group Databasebeta

← #12010178

Subgroups — #12010178

Order 1280.

620 conjugacy classes of subgroups; only the 91 normal subgroups are shown.

normal subgroups only

This group has 620 conjugacy classes of subgroups — too many to draw. Only its 91 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×#2539C211yesmaximal, characteristic
2#10602335C211yesmaximal, characteristic, complemented
3C2×#64745C211yesmaximal, characteristic, complemented
4#10598142C212yesmaximal, complemented
5#10598136C212yesmaximal, complemented
6C2×C5×#123—21no—
7C2×C5×#121C411yescharacteristic, complemented
8C2×C5×#120—21no—
9C22×C5×C22⋊C4C2211yescharacteristic
10C5×#399C411yescharacteristic, complemented
11C5×#399C2211yescharacteristic
12C5×#392C411yescharacteristic, complemented
13C5×#392C2211yescharacteristic
14C5×#390—21no—
15C5×#390—21no—
16C4×C5×C22⋊C4C411yescharacteristic, complemented
17#65536—21no—
18C2×C22⋊#168—21no—
19#64746—21no—
20#64746C2212yes—
21#64745—21no—
22#64745C2212yes—
23#11843—51nomaximal, Sylow
24C25×C5D411yescharacteristic
25C22×C5×D4—21no—
26C22×C5×D4—21no—
27C22×C5×D4—21no—
28C22×C5×D4C2×C411yescharacteristic
29C23×C4×C5C2×C411yescharacteristic
30C23×C4×C5D411yescharacteristic
31C5×#123—44no—
32C5×#121—41no—
33C5×#121—41no—
34C5×#120—44no—
35C2×C5×C4⋊C4—21no—
36C2×C5×C4⋊C4—21no—
37C2×C5×C4⋊C4C2×C411yescharacteristic
38C2×C5×C22⋊C4—21no—
39C2×C5×C22⋊C4—21no—
40C2×C5×C22⋊C4—21no—
41C2×C5×C22⋊C4—21no—
42C2×C5×C22⋊C4—21no—
43C2×C5×C22⋊C4—21no—
44C2×C5×C22⋊C4—41no—
45C2×C5×C22⋊C4—41no—
46C2×C5×C22⋊C4C2×C411yescharacteristic
47C2×C5×C22⋊C4D411yescharacteristic
48C2×C5×C22⋊C4D411yescharacteristic
49C2×C5×C22⋊C4C2311yescharacteristic
50C2×C42×C5—21no—
51C2×C22⋊Dic5—22no—
52C2×C22⋊Dic5—41no—
53C2×C22⋊Dic5—41no—
54C22×#168—41no—
55C5×#95C2×C411yescharacteristic
56C5×#95C2×C411yescharacteristic
57C22⋊#168—22no—
58C22⋊#168—41no—
59#2539D511yescharacteristic
60#1819—51no—
61C2×#323—51no—
62#1352—52no—
63#1343—52no—
64C24×C5—21no—
65C24×C5—21no—
66C24×C5—22no—
67C24×C5—21no—
68C24×C5—41no—
69C24×C5—42no—
70C24×C5—81no—
71C24×C5C22⋊C411yescharacteristic
72C24×C5C22⋊C411yescharacteristic
73C24×C5C2×D411yescharacteristic
74C2×C5×D4—41no—
75C2×C5×D4—42no—
76C2×C5×D4—42no—
77C2×C5×D4—42no—
78C2×C5×D4—42no—
79C2×C5×D4—42no—
80C2×C5×D4—41no—
81C2×C5×D4—41no—
82C2×C5×D4—42no—
83C2×C5×D4—81no—
84C2×C5×D4—81no—
85C2×C5×D4—81no—
86C22×C4×C5—21no—
87C22×C4×C5—21no—
88C22×C4×C5—21no—
89C22×C4×C5—21no—
90C22×C4×C5—21no—
91C22×C4×C5—21no—
92C22×C4×C5—21no—
93C22×C4×C5—42no—
94C22×C4×C5—81no—
95C22×C4×C5C22⋊C411yescharacteristic
96C22×C4×C5C22⋊C411yescharacteristic
97C22×C4×C5C22×C411yescharacteristic
98C22×C4×C5C2×D411yescharacteristic
99C22×Dic5—42no—
100C5×C4⋊C4—41no—
101C5×C4⋊C4—42no—
102C5×C4⋊C4—42no—
103C5×C22⋊C4—24no—
104C5×C22⋊C4—41no—
105C5×C22⋊C4—41no—
106C5×C22⋊C4—41no—
107C5×C22⋊C4—41no—
108C5×C22⋊C4—42no—
109C5×C22⋊C4—42no—
110C5×C22⋊C4—42no—
111C5×C22⋊C4—42no—
112C5×C22⋊C4—42no—
113C42×C5—81no—
114C22⋊Dic5—42no—
115C22⋊Dic5—42no—
116C22⋊Dic5—82no—
117C22⋊Dic5—82no—
118C2×#168—42no—
119C2×#168—82no—
120C2×#123—21no—
121C2×#121Dic511yescharacteristic, complemented
122C2×#120—21no—
123C22×C22⋊C4C2×D511yescharacteristic
124C2×C22⋊C8—101no—
125#399Dic511yescharacteristic, complemented
126#399C2×D511yescharacteristic
127#392Dic511yescharacteristic, complemented
128#392C2×D511yescharacteristic
129#390—21no—
130#390—21no—
131#379—101no—
132C4×C22⋊C4Dic511yescharacteristic, complemented
133#342—52no—
134#342—101no—
135#323—52no—
136#323—101no—
137C23×C5—22no—
138C23×C5—21no—
139C23×C5—21no—
140C23×C5—21no—
141C23×C5—21no—
142C23×C5—21no—
143C23×C5—21no—
144C23×C5—21no—
145C23×C5—41no—
146C23×C5—42no—
147C23×C5—42no—
148C23×C5—41no—
149C23×C5—41no—
150C23×C5—41no—
151C23×C5—41no—
152C23×C5—42no—
153C23×C5—41no—
154C23×C5—41no—
155C23×C5—41no—
156C23×C5—41no—
157C23×C5—42no—
158C23×C5—41no—
159C23×C5—41no—
160C23×C5—41no—
161C23×C5—81no—
162C23×C5—81no—
163C23×C5—84no—
164C23×C5—84no—
165C23×C5#9911yescharacteristic
166C23×C5#9911yescharacteristic
167C23×C5C2×C22⋊C411yescharacteristic
168C23×C5#13512yes—
169C5×D4—84no—
170C5×D4—84no—
171C5×D4—82no—
172C5×D4—82no—
173C5×D4—82no—
174C2×C4×C5—22no—
175C2×C4×C5—22no—
176C2×C4×C5—21no—
177C2×C4×C5—21no—
178C2×C4×C5—21no—
179C2×C4×C5—21no—
180C2×C4×C5—41no—
181C2×C4×C5—41no—
182C2×C4×C5—42no—
183C2×C4×C5—41no—
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—81no—
193C2×C4×C5—81no—
194C2×C4×C5—81no—
195C2×Dic5—82no—
196C2×Dic5—82no—
197C2×Dic5—82no—
198#168—82no—
199C25C22⋊D511yescharacteristic
200C22×D4—21no—
201C22×D4—21no—
202C22×D4—21no—
203C22×D4C2×Dic511yescharacteristic
204C23×C4C2×Dic511yescharacteristic
205C23×C4C22⋊D511yescharacteristic
206C22×C8—201no—
207#123—44no—
208#121—41no—
209#121—41no—
210#120—44no—
211C2×C4⋊C4—21no—
212C2×C4⋊C4—21no—
213C2×C4⋊C4C2×Dic511yescharacteristic
214C2×C22⋊C4—21no—
215C2×C22⋊C4—21no—
216C2×C22⋊C4—21no—
217C2×C22⋊C4—21no—
218C2×C22⋊C4—21no—
219C2×C22⋊C4—21no—
220C2×C22⋊C4—41no—
221C2×C22⋊C4—41no—
222C2×C22⋊C4—102no—
223C2×C22⋊C4—201no—
224C2×C22⋊C4—201no—
225C2×C22⋊C4C2×Dic511yescharacteristic
226C2×C22⋊C4C22⋊D511yescharacteristic
227C2×C22⋊C4C22⋊D511yescharacteristic
228C2×C22⋊C4C22×D511yescharacteristic
229C2×C42—21no—
230C22⋊C8—102no—
231C22⋊C8—201no—
232#95C2×Dic511yescharacteristic
233#95C2×Dic511yescharacteristic
234C22×C5—21no—
235C22×C5—21no—
236C22×C5—21no—
237C22×C5—21no—
238C22×C5—21no—
239C22×C5—21no—
240C22×C5—22no—
241C22×C5—21no—
242C22×C5—42no—
243C22×C5—42no—
244C22×C5—42no—
245C22×C5—42no—
246C22×C5—41no—
247C22×C5—41no—
248C22×C5—41no—
249C22×C5—41no—
250C22×C5—41no—
251C22×C5—81no—
252C22×C5—84no—
253C22×C5—81no—
254C22×C5—81no—
255C22×C5—81no—
256C22×C5—81no—
257C22×C5—81no—
258C22×C5—84no—
259C22×C5—81no—
260C22×C5—81no—
261C22×C5—81no—
262C22×C5#35312yes—
263C22×C5C2×#9911yescharacteristic
264C22×C5#41611yescharacteristic
265C22×C5#42111yescharacteristic
266C4×C5—42no—
267C4×C5—42no—
268C4×C5—41no—
269C4×C5—41no—
270C4×C5—41no—
271C4×C5—41no—
272C4×C5—81no—
273Dic5—162no—
274C24—21no—
275C24—21no—
276C24—22no—
277C24—21no—
278C24—41no—
279C24—42no—
280C24—81no—
281C24C22⋊Dic511yescharacteristic
282C24C22⋊Dic511yescharacteristic
283C24C2×C22⋊D511yescharacteristic
284C2×D4—41no—
285C2×D4—42no—
286C2×D4—42no—
287C2×D4—42no—
288C2×D4—42no—
289C2×D4—42no—
290C2×D4—42no—
291C2×D4—41no—
292C2×D4—41no—
293C2×D4—81no—
294C2×D4—81no—
295C2×D4—81no—
296C22×C4—21no—
297C22×C4—21no—
298C22×C4—21no—
299C22×C4—21no—
300C22×C4—21no—
301C22×C4—21no—
302C22×C4—21no—
303C22×C4—42no—
304C22×C4—81no—
305C22×C4—202no—
306C22×C4C22⋊Dic511yescharacteristic
307C22×C4C22⋊Dic511yescharacteristic
308C22×C4C22×Dic511yescharacteristic
309C22×C4C2×C22⋊D511yescharacteristic
310C2×C8—202no—
311C2×C8—402no—
312C4⋊C4—41no—
313C4⋊C4—42no—
314C4⋊C4—42no—
315C22⋊C4—24no—
316C22⋊C4—41no—
317C22⋊C4—41no—
318C22⋊C4—41no—
319C22⋊C4—41no—
320C22⋊C4—42no—
321C22⋊C4—42no—
322C22⋊C4—42no—
323C22⋊C4—42no—
324C22⋊C4—42no—
325C22⋊C4—202no—
326C22⋊C4—202no—
327C22⋊C4—402no—
328C22⋊C4—402no—
329C42—81no—
330C2×C5—21no—
331C2×C5—21no—
332C2×C5—22no—
333C2×C5—42no—
334C2×C5—41no—
335C2×C5—81no—
336C2×C5—161no—
337C2×C5#151611yescharacteristic
338C2×C5#212111yescharacteristic
339C2×C5C2×#35311yescharacteristic
340C23—21no—
341C23—21no—
342C23—21no—
343C23—22no—
344C23—21no—
345C23—21no—
346C23—21no—
347C23—21no—
348C23—41no—
349C23—42no—
350C23—42no—
351C23—41no—
352C23—41no—
353C23—41no—
354C23—41no—
355C23—42no—
356C23—41no—
357C23—41no—
358C23—41no—
359C23—41no—
360C23—42no—
361C23—41no—
362C23—41no—
363C23—41no—
364C23—81no—
365C23—81no—
366C23—84no—
367C23—84no—
368C23#396611yescharacteristic
369C23#396611yescharacteristic
370C23#409612yes—
371C23C2×C22⋊Dic511yescharacteristic
372D4—84no—
373D4—84no—
374D4—82no—
375D4—82no—
376D4—82no—
377C2×C4—22no—
378C2×C4—22no—
379C2×C4—21no—
380C2×C4—21no—
381C2×C4—21no—
382C2×C4—21no—
383C2×C4—41no—
384C2×C4—41no—
385C2×C4—41no—
386C2×C4—41no—
387C2×C4—42no—
388C2×C4—41no—
389C2×C4—41no—
390C2×C4—41no—
391C2×C4—41no—
392C2×C4—41no—
393C2×C4—41no—
394C2×C4—41no—
395C2×C4—81no—
396C2×C4—81no—
397C2×C4—81no—
398C2×C4—402no—
399C2×C4—402no—
400C2×C4—402no—
401C8—402no—
402C5#1184311yescharacteristic, complemented, Sylow
403C22—21no—
404C22—21no—
405C22—21no—
406C22—21no—
407C22—22no—
408C22—21no—
409C22—21no—
410C22—21no—
411C22—42no—
412C22—42no—
413C22—42no—
414C22—42no—
415C22—41no—
416C22—41no—
417C22—41no—
418C22—41no—
419C22—41no—
420C22—81no—
421C22—84no—
422C22—81no—
423C22—81no—
424C22—81no—
425C22—81no—
426C22—81no—
427C22—84no—
428C22—81no—
429C22—81no—
430C22—81no—
431C22#6476512yes—
432C22C2×#396611yescharacteristic
433C22#6552211yescharacteristic
434C22#6552511yescharacteristic
435C4—42no—
436C4—42no—
437C4—41no—
438C4—41no—
439C4—41no—
440C4—41no—
441C4—81no—
442C4—802no—
443C2—21no—
444C2—21no—
445C2—22no—
446C2—42no—
447C2—41no—
448C2—81no—
449C2—161no—
450C2#1060250311yescharacteristic
451C2C2×#6476511yescharacteristic
452C2#1060264011yescharacteristic
subgroupG/H
centerC22C2×#3966
commutatorC22×C4×C5C22×C4
FrattiniC2×C22⋊C4C22×D5
Fitting, prime radicalC5×#2539C2
socleC22×C5C2×#99
cosocleC2×C5×C22⋊C4C23
hypercenter#2539D5
hypocenterC5#11843
LCSG ▷ C22×C4×C5 ▷ C23×C5 ▷ C2×C5 ▷ C5
UCSC1 ◁ C22 ◁ C24 ◁ C22×C22⋊C4 ◁ #2539

Complements

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

NH ≅ G/Ntype
2. #10602335C2semidirect
3. C2×#64745C2semidirect
4. #10598142C2semidirect
5. #10598136C2semidirect
7. C2×C5×#121C4semidirect
10. C5×#399C4semidirect
12. C5×#392C4semidirect
16. C4×C5×C22⋊C4C4semidirect
121. C2×#121Dic5semidirect
125. #399Dic5semidirect
127. #392Dic5semidirect
132. C4×C22⋊C4Dic5semidirect
402. C5#11843semidirect

Maximal subgroups

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

subgroupindex
C5×#25392
#106023352
C2×#647452
#105981422
#105981362
#118435