Finite Group Databasebeta

← #12009583

Subgroups — #12009583

Order 1280.

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

normal subgroups only

This group has 592 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
1#10613848C211yesmaximal, characteristic
2C2×C5×#342C211yesmaximal, characteristic
3C2×#64745C211yesmaximal, characteristic
4#10597983C212yesmaximal, complemented
5#10597981C212yesmaximal, complemented
6C22×C5×C22⋊C4—21no—
7C22×C5×C22⋊C4C2211yescharacteristic
8C2×#4070C411yescharacteristic, complemented
9#65536—21no—
10#65501C411yescharacteristic, complemented
11#65499—21no—
12C2×C22⋊#168—21no—
13#65238C411yescharacteristic, complemented
14#65238C2211yescharacteristic
15#65237—21no—
16#65237—21no—
17#65236C411yescharacteristic, complemented
18#65236C2211yescharacteristic
19C5×#342—21no—
20C5×#342C2212yes—
21#64745—21no—
22#64745C2212yes—
23#11716—51nomaximal, Sylow
24C25×C5D411yescharacteristic
25C23×C4×C5—41no—
26C23×C4×C5C2×C411yescharacteristic
27C23×C4×C5D411yescharacteristic
28C22×Dic10C2×C411yescharacteristic, complemented
29C2×C5×C22⋊C4—24no—
30C2×C5×C22⋊C4—22no—
31C2×C5×C22⋊C4—41no—
32C2×C5×C22⋊C4—41no—
33C2×C5×C22⋊C4—41no—
34C2×C5×C22⋊C4—42no—
35C2×C5×C22⋊C4C2×C411yescharacteristic
36C2×C5×C22⋊C4D411yescharacteristic
37C2×C5×C22⋊C4D411yescharacteristic
38C2×C5×C22⋊C4C2311yescharacteristic
39C2×C22⋊Dic5—21no—
40C2×C22⋊Dic5—21no—
41C2×C22⋊Dic5—21no—
42C2×C22⋊Dic5—21no—
43C2×C22⋊Dic5—21no—
44C2×C22⋊Dic5—21no—
45C2×C4⋊Dic5—21no—
46C2×C4⋊Dic5C2×C411yescharacteristic, complemented
47#4070—41no—
48#4070—41no—
49C2×#683—21no—
50C2×#683—21no—
51C2×#683—21no—
52C22×#168—41no—
53#3963—21no—
54#3963—21no—
55#3963C2×C411yescharacteristic, complemented
56#3963C2×C411yescharacteristic, complemented
57C22⋊#168—22no—
58C22⋊#168—41no—
59#2626—51no—
60C2×#342D511yescharacteristic
61C2×#323—51no—
62#1352—52no—
63#1351—52no—
64C24×C5—21no—
65C24×C5—21no—
66C24×C5—22no—
67C24×C5—81no—
68C24×C5—81no—
69C24×C5C22⋊C411yescharacteristic
70C24×C5C22⋊C411yescharacteristic
71C24×C5SD1612yes—
72C24×C5Q1612yes—
73C24×C5C2×D411yescharacteristic
74C22×C4×C5—21no—
75C22×C4×C5—21no—
76C22×C4×C5—21no—
77C22×C4×C5—21no—
78C22×C4×C5—44no—
79C22×C4×C5—42no—
80C22×C4×C5—81no—
81C22×C4×C5—81no—
82C22×C4×C5—82no—
83C22×C4×C5—82no—
84C22×C4×C5C22⋊C411yescharacteristic
85C22×C4×C5C22⋊C411yescharacteristic
86C22×C4×C5C22×C411yescharacteristic
87C22×C4×C5C2×D411yescharacteristic
88C22×Dic5—21no—
89C22×Dic5—21no—
90C22×Dic5—21no—
91C22×Dic5—21no—
92C2×Dic10—41no—
93C2×Dic10—81no—
94C5×C22⋊C4—24no—
95C5×C22⋊C4—41no—
96C5×C22⋊C4—41no—
97C5×C22⋊C4—41no—
98C5×C22⋊C4—41no—
99C5×C22⋊C4—44no—
100C5×C22⋊C4—42no—
101C22⋊Dic5—41no—
102C22⋊Dic5—41no—
103C22⋊Dic5—42no—
104C22⋊Dic5—81no—
105C22⋊Dic5—81no—
106C22⋊Dic5—81no—
107C22⋊Dic5—81no—
108C4⋊Dic5—41no—
109C4⋊Dic5—81no—
110#683—42no—
111#683—81no—
112#683—81no—
113C2×#168—42no—
114C2×#168—82no—
115C2×C22⋊Q8—51no—
116C22×C22⋊C4—21no—
117C22×C22⋊C4C2×D511yescharacteristic
118C2×C22⋊C8—101no—
119#399—51no—
120#399—51no—
121#396—101no—
122#396—101no—
123#393—51no—
124#393—51no—
125#385—101no—
126#380—51no—
127#379—101no—
128#342—21no—
129#342C2×D512yes—
130#323—52no—
131#323—101no—
132C23×C5—22no—
133C23×C5—22no—
134C23×C5—21no—
135C23×C5—21no—
136C23×C5—22no—
137C23×C5—22no—
138C23×C5—21no—
139C23×C5—21no—
140C23×C5—41no—
141C23×C5—41no—
142C23×C5—42no—
143C23×C5—42no—
144C23×C5—41no—
145C23×C5—41no—
146C23×C5—41no—
147C23×C5—41no—
148C23×C5—81no—
149C23×C5—81no—
150C23×C5—82no—
151C23×C5—82no—
152C23×C5—84no—
153C23×C5#9911yescharacteristic
154C23×C5#9911yescharacteristic
155C23×C5#10312yes—
156C23×C5#10312yes—
157C23×C5#10412yes—
158C23×C5C2×C22⋊C411yescharacteristic
159C23×C5C2×SD1611yescharacteristic
160C23×C5C2×Q1611yescharacteristic
161C2×C4×C5—21no—
162C2×C4×C5—21no—
163C2×C4×C5—21no—
164C2×C4×C5—21no—
165C2×C4×C5—42no—
166C2×C4×C5—41no—
167C2×C4×C5—41no—
168C2×C4×C5—41no—
169C2×C4×C5—41no—
170C2×C4×C5—44no—
171C2×C4×C5—81no—
172C2×C4×C5—81no—
173C2×C4×C5—81no—
174C2×C4×C5—88no—
175C2×C4×C5—82no—
176C2×C4×C5—82no—
177C2×Dic5—41no—
178C2×Dic5—41no—
179C2×Dic5—41no—
180C2×Dic5—41no—
181C2×Dic5—41no—
182C2×Dic5—41no—
183C2×Dic5—41no—
184C2×Dic5—41no—
185C2×Dic5—41no—
186C2×Dic5—41no—
187C2×Dic5—41no—
188C2×Dic5—41no—
189Dic10—82no—
190#168—82no—
191C25C22⋊D511yescharacteristic
192C22×Q8—51no—
193C23×C4—41no—
194C23×C4C4×D511yescharacteristic
195C23×C4D2011yescharacteristic
196C22×C8—201no—
197C22⋊Q8—201no—
198C22⋊Q8—201no—
199C2×C4⋊C4—51no—
200C2×C4⋊C4—101no—
201C2×C4⋊C4—101no—
202C2×C4⋊C4—101no—
203C2×C4⋊C4—101no—
204C2×C22⋊C4—24no—
205C2×C22⋊C4—22no—
206C2×C22⋊C4—41no—
207C2×C22⋊C4—41no—
208C2×C22⋊C4—41no—
209C2×C22⋊C4—42no—
210C2×C22⋊C4—101no—
211C2×C22⋊C4—101no—
212C2×C22⋊C4—101no—
213C2×C22⋊C4—101no—
214C2×C22⋊C4—101no—
215C2×C22⋊C4—101no—
216C2×C22⋊C4C4×D511yescharacteristic
217C2×C22⋊C4D2011yescharacteristic
218C2×C22⋊C4C22⋊D511yescharacteristic
219C2×C22⋊C4C22×D511yescharacteristic
220C22⋊C8—102no—
221C22⋊C8—201no—
222#95—51no—
223#95—51no—
224#95—101no—
225#95—101no—
226C22×C5—21no—
227C22×C5—21no—
228C22×C5—21no—
229C22×C5—21no—
230C22×C5—21no—
231C22×C5—21no—
232C22×C5—21no—
233C22×C5—22no—
234C22×C5—21no—
235C22×C5—42no—
236C22×C5—42no—
237C22×C5—42no—
238C22×C5—41no—
239C22×C5—41no—
240C22×C5—44no—
241C22×C5—44no—
242C22×C5—41no—
243C22×C5—41no—
244C22×C5—41no—
245C22×C5—41no—
246C22×C5—81no—
247C22×C5—81no—
248C22×C5—82no—
249C22×C5—82no—
250C22×C5#32812yes—
251C22×C5#32812yes—
252C22×C5C2×#9911yescharacteristic
253C22×C5C2×#10311yescharacteristic
254C22×C5C2×#10411yescharacteristic
255C4×C5—41no—
256C4×C5—41no—
257C4×C5—41no—
258C4×C5—41no—
259C4×C5—84no—
260Dic5—81no—
261Dic5—81no—
262Dic5—81no—
263Dic5—81no—
264C24—21no—
265C24—22no—
266C24—21no—
267C24—81no—
268C24—81no—
269C24#68511yescharacteristic
270C24#68511yescharacteristic
271C24#68712yes—
272C24#68912yes—
273C24C2×C22⋊D511yescharacteristic
274C2×Q8—201no—
275C2×Q8—401no—
276C22×C4—21no—
277C22×C4—21no—
278C22×C4—21no—
279C22×C4—21no—
280C22×C4—44no—
281C22×C4—42no—
282C22×C4—81no—
283C22×C4—81no—
284C22×C4—82no—
285C22×C4—82no—
286C22×C4—101no—
287C22×C4—101no—
288C22×C4—101no—
289C22×C4—101no—
290C22×C4#68511yescharacteristic
291C22×C4#68511yescharacteristic
292C22×C4C2×C4×D511yescharacteristic
293C22×C4C2×D2011yescharacteristic
294C2×C8—202no—
295C2×C8—402no—
296C4⋊C4—201no—
297C4⋊C4—202no—
298C4⋊C4—401no—
299C4⋊C4—401no—
300C4⋊C4—401no—
301C22⋊C4—24no—
302C22⋊C4—41no—
303C22⋊C4—41no—
304C22⋊C4—41no—
305C22⋊C4—41no—
306C22⋊C4—44no—
307C22⋊C4—42no—
308C22⋊C4—202no—
309C22⋊C4—201no—
310C22⋊C4—201no—
311C22⋊C4—401no—
312C22⋊C4—401no—
313C22⋊C4—401no—
314C22⋊C4—401no—
315C2×C5—21no—
316C2×C5—22no—
317C2×C5—21no—
318C2×C5—22no—
319C2×C5—41no—
320C2×C5—42no—
321C2×C5—41no—
322C2×C5C2×#32811yescharacteristic
323C2×C5#212111yescharacteristic
324C2×C5#212711yescharacteristic
325C23—22no—
326C23—22no—
327C23—21no—
328C23—21no—
329C23—22no—
330C23—22no—
331C23—21no—
332C23—21no—
333C23—41no—
334C23—41no—
335C23—42no—
336C23—42no—
337C23—41no—
338C23—41no—
339C23—41no—
340C23—41no—
341C23—81no—
342C23—81no—
343C23—82no—
344C23—82no—
345C23—84no—
346C23#393712yes—
347C23#393811yescharacteristic
348C23#393811yescharacteristic
349C23#394212yes—
350C23#394212yes—
351C23C2×#68511yescharacteristic
352C23C2×#68711yescharacteristic
353C23C2×#68911yescharacteristic
354Q8—402no—
355C2×C4—21no—
356C2×C4—21no—
357C2×C4—21no—
358C2×C4—21no—
359C2×C4—42no—
360C2×C4—41no—
361C2×C4—41no—
362C2×C4—41no—
363C2×C4—41no—
364C2×C4—44no—
365C2×C4—81no—
366C2×C4—81no—
367C2×C4—81no—
368C2×C4—88no—
369C2×C4—82no—
370C2×C4—82no—
371C2×C4—201no—
372C2×C4—201no—
373C2×C4—201no—
374C2×C4—201no—
375C2×C4—201no—
376C2×C4—201no—
377C2×C4—201no—
378C2×C4—201no—
379C2×C4—201no—
380C2×C4—201no—
381C2×C4—201no—
382C2×C4—201no—
383C8—402no—
384C5#1171611yescharacteristic, complemented, Sylow
385C22—21no—
386C22—21no—
387C22—21no—
388C22—21no—
389C22—21no—
390C22—21no—
391C22—21no—
392C22—22no—
393C22—21no—
394C22—42no—
395C22—42no—
396C22—42no—
397C22—41no—
398C22—41no—
399C22—44no—
400C22—44no—
401C22—41no—
402C22—41no—
403C22—41no—
404C22—41no—
405C22—81no—
406C22—81no—
407C22—82no—
408C22—82no—
409C22#6467212yes—
410C22#6467212yes—
411C22C2×#393711yescharacteristic
412C22C2×#393811yescharacteristic
413C22C2×#394211yescharacteristic
414C4—41no—
415C4—41no—
416C4—41no—
417C4—41no—
418C4—84no—
419C4—401no—
420C4—401no—
421C4—401no—
422C4—401no—
423C2—21no—
424C2—22no—
425C2—21no—
426C2—22no—
427C2—41no—
428C2—42no—
429C2—41no—
430C2C2×#6467211yescharacteristic
431C2#1060080811yescharacteristic
432C2#1060081211yescharacteristic
subgroupG/H
centerC22C2×#3938
commutatorC22×C4×C5C22×C4
FrattiniC2×C22⋊C4C22×D5
FittingC2×C5×#342C2
socleC22×C5C2×#99
prime radicalC25×C5D4
cosocleC2×C5×C22⋊C4C23
hypercenterC2×#342D5
hypocenterC5#11716
LCSG ▷ C22×C4×C5 ▷ C23×C5 ▷ C2×C5 ▷ C5
UCSC1 ◁ C22 ◁ C24 ◁ C22×C22⋊C4 ◁ C2×#342

Complements

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

NH ≅ G/Ntype
4. #10597983C2semidirect
5. #10597981C2semidirect
8. C2×#4070C4semidirect
10. #65501C4semidirect
13. #65238C4semidirect
17. #65236C4semidirect
28. C22×Dic10C2×C4semidirect
46. C2×C4⋊Dic5C2×C4semidirect
55. #3963C2×C4semidirect
56. #3963C2×C4semidirect
384. C5#11716semidirect

Maximal subgroups

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

subgroupindex
#106138482
C2×C5×#3422
C2×#647452
#105979832
#105979812
#117165