Finite Group Databasebeta

← #12010441

Subgroups — #12010441

Order 1280.

452 conjugacy classes of subgroups; only the 103 normal subgroups are shown.

normal subgroups only

This group has 452 conjugacy classes of subgroups — too many to draw. Only its 103 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×#2564C211yesmaximal, characteristic
2#10602412C211yesmaximal, characteristic
3#10602394C211yesmaximal, characteristic
4#10598142C212yesmaximal, complemented
5#10598137C212yesmaximal, complemented
6C2×C5×C22⋊Q8C411yescharacteristic, complemented
7C2×C4×C5×D4C2211yescharacteristic
8C5×#400C411yescharacteristic, complemented
9C5×#399C2211yescharacteristic
10C5×#395C411yescharacteristic, complemented
11C5×#393C2211yescharacteristic
12C5×#387—21no—
13C5×#386—21no—
14C5×#385—21no—
15C5×#380—21no—
16C2×C5×#95C411yescharacteristic, complemented
17C2×#3966—21no—
18#65501—21no—
19#65214—21no—
20#64746C2212yes—
21#64745—21no—
22#64745C2212yes—
23#11902—51nomaximal, Sylow
24C22×C5×Q8C2×C411yescharacteristic
25C22×C5×D4D411yescharacteristic
26C23×C4×C5—21no—
27C23×C4×C5C2×C411yescharacteristic
28C23×C4×C5D411yescharacteristic
29C5×C22⋊Q8—41no—
30C5×C22⋊Q8—41no—
31C4×C5×D4—41no—
32C4×C5×D4—41no—
33C2×C5×C4⋊C4—21no—
34C2×C5×C4⋊C4—21no—
35C2×C5×C4⋊C4—21no—
36C2×C5×C4⋊C4C2×C411yescharacteristic
37C2×C5×C4⋊C4C2×C411yescharacteristic
38C2×C5×C22⋊C4—21no—
39C2×C5×C22⋊C4—21no—
40C2×C5×C22⋊C4—21no—
41C2×C5×C22⋊C4D411yescharacteristic
42C2×C5×C22⋊C4C2311yescharacteristic
43C2×C42×C5D411yescharacteristic
44C2×C22⋊Dic5—22no—
45C2×C4⋊Dic5—41no—
46C5×#95—21no—
47C5×#95—21no—
48C5×#95—21no—
49C5×#95—21no—
50C5×#95—21no—
51C5×#95C2×C411yescharacteristic
52C5×#95C2×C411yescharacteristic
53#3966—42no—
54#3963—41no—
55C22⋊#168—22no—
56C4⋊#168—41no—
57C4×#168—41no—
58#2564D511yescharacteristic
59#1799—51no—
60#1460—51no—
61#1352—52no—
62#1351—52no—
63C24×C5C22⋊C411yescharacteristic
64C24×C5C2×D411yescharacteristic
65C2×C5×Q8—41no—
66C2×C5×Q8—81no—
67C2×C5×D4—41no—
68C2×C5×D4—41no—
69C2×C5×D4SD1612yes—
70C2×C5×D4Q1612yes—
71C22×C4×C5—21no—
72C22×C4×C5—21no—
73C22×C4×C5—21no—
74C22×C4×C5—21no—
75C22×C4×C5—21no—
76C22×C4×C5—21no—
77C22×C4×C5—21no—
78C22×C4×C5—21no—
79C22×C4×C5—42no—
80C22×C4×C5—42no—
81C22×C4×C5—81no—
82C22×C4×C5—81no—
83C22×C4×C5—81no—
84C22×C4×C5C22⋊C411yescharacteristic
85C22×C4×C5C22⋊C411yescharacteristic
86C22×C4×C5C22⋊C411yescharacteristic
87C22×C4×C5C22×C411yescharacteristic
88C22×C4×C5C2×D411yescharacteristic
89C22×Dic5—42no—
90C5×C4⋊C4—41no—
91C5×C4⋊C4—41no—
92C5×C4⋊C4—42no—
93C5×C4⋊C4—41no—
94C5×C4⋊C4—41no—
95C5×C4⋊C4—81no—
96C5×C22⋊C4—41no—
97C5×C22⋊C4—41no—
98C5×C22⋊C4—42no—
99C5×C22⋊C4—42no—
100C5×C22⋊C4—81no—
101C42×C5—21no—
102C42×C5—21no—
103C22⋊Dic5—42no—
104C22⋊Dic5—42no—
105C4⋊Dic5—82no—
106C2×#168—42no—
107C2×C22⋊Q8Dic511yescharacteristic, complemented
108C2×C4×D4C2×D511yescharacteristic
109#431—101no—
110C2×#99—101no—
111#400Dic511yescharacteristic, complemented
112#399C2×D511yescharacteristic
113#395Dic511yescharacteristic, complemented
114#393C2×D511yescharacteristic
115#387—21no—
116#386—21no—
117#385—21no—
118#380—21no—
119#380—101no—
120C2×#95Dic511yescharacteristic, complemented
121#342—52no—
122#323—52no—
123#323—101no—
124C23×C5—21no—
125C23×C5—21no—
126C23×C5—21no—
127C23×C5—21no—
128C23×C5—41no—
129C23×C5—41no—
130C23×C5—41no—
131C23×C5—41no—
132C23×C5#10312yes—
133C23×C5C2×C22⋊C411yescharacteristic
134C23×C5C2×SD1611yescharacteristic
135C23×C5C2×Q1611yescharacteristic
136C5×Q8—82no—
137C5×D4—42no—
138C5×D4—42no—
139C2×C4×C5—21no—
140C2×C4×C5—21no—
141C2×C4×C5—22no—
142C2×C4×C5—21no—
143C2×C4×C5—21no—
144C2×C4×C5—21no—
145C2×C4×C5—41no—
146C2×C4×C5—42no—
147C2×C4×C5—41no—
148C2×C4×C5—41no—
149C2×C4×C5—41no—
150C2×C4×C5—41no—
151C2×C4×C5—41no—
152C2×C4×C5—41no—
153C2×C4×C5—41no—
154C2×C4×C5—41no—
155C2×C4×C5—41no—
156C2×C4×C5—41no—
157C2×C4×C5—41no—
158C2×C4×C5—41no—
159C2×C4×C5—41no—
160C2×C4×C5—41no—
161C2×C4×C5—81no—
162C2×C4×C5—81no—
163C2×C4×C5—81no—
164C2×C4×C5—84no—
165C2×C4×C5#9911yescharacteristic
166C2×C4×C5#9911yescharacteristic
167C2×C4×C5#10312yes—
168C2×Dic5—82no—
169C2×Dic5—82no—
170C2×Dic5—82no—
171#168—82no—
172C22×Q8C2×Dic511yescharacteristic
173C22×D4C22⋊D511yescharacteristic
174C23×C4—21no—
175C23×C4C2×Dic511yescharacteristic
176C23×C4C22⋊D511yescharacteristic
177C22⋊Q8—41no—
178C22⋊Q8—41no—
179C4×D4—41no—
180C4×D4—41no—
181C2×C4⋊C4—21no—
182C2×C4⋊C4—21no—
183C2×C4⋊C4—21no—
184C2×C4⋊C4—201no—
185C2×C4⋊C4C2×Dic511yescharacteristic
186C2×C4⋊C4C2×Dic511yescharacteristic
187C2×C22⋊C4—21no—
188C2×C22⋊C4—21no—
189C2×C22⋊C4—21no—
190C2×C22⋊C4—102no—
191C2×C22⋊C4C22⋊D511yescharacteristic
192C2×C22⋊C4C22×D511yescharacteristic
193C2×C42C22⋊D511yescharacteristic
194C4⋊C8—201no—
195#99—202no—
196C22⋊C8—102no—
197C4×C8—201no—
198#95—21no—
199#95—21no—
200#95—21no—
201#95—21no—
202#95—21no—
203#95—201no—
204#95C2×Dic511yescharacteristic
205#95C2×Dic511yescharacteristic
206C22×C5—21no—
207C22×C5—21no—
208C22×C5—21no—
209C22×C5—22no—
210C22×C5—21no—
211C22×C5—42no—
212C22×C5—41no—
213C22×C5—41no—
214C22×C5—42no—
215C22×C5—41no—
216C22×C5—41no—
217C22×C5—41no—
218C22×C5—41no—
219C22×C5—81no—
220C22×C5C2×#9911yescharacteristic
221C22×C5C2×#10311yescharacteristic
222C22×C5#42111yescharacteristic
223C4×C5—21no—
224C4×C5—21no—
225C4×C5—22no—
226C4×C5—42no—
227C4×C5—41no—
228C4×C5—41no—
229C4×C5—41no—
230C4×C5—81no—
231C4×C5—81no—
232C4×C5—81no—
233Dic5—162no—
234C24C22⋊Dic511yescharacteristic
235C24C2×C22⋊D511yescharacteristic
236C2×Q8—41no—
237C2×Q8—81no—
238C2×D4—41no—
239C2×D4—41no—
240C2×D4#68812yes—
241C2×D4#68912yes—
242C22×C4—21no—
243C22×C4—21no—
244C22×C4—21no—
245C22×C4—21no—
246C22×C4—21no—
247C22×C4—21no—
248C22×C4—21no—
249C22×C4—21no—
250C22×C4—42no—
251C22×C4—42no—
252C22×C4—81no—
253C22×C4—81no—
254C22×C4—81no—
255C22×C4—202no—
256C22×C4C22⋊Dic511yescharacteristic
257C22×C4C22⋊Dic511yescharacteristic
258C22×C4C22⋊Dic511yescharacteristic
259C22×C4C22×Dic511yescharacteristic
260C22×C4C2×C22⋊D511yescharacteristic
261C2×C8—202no—
262C4⋊C4—41no—
263C4⋊C4—41no—
264C4⋊C4—42no—
265C4⋊C4—41no—
266C4⋊C4—41no—
267C4⋊C4—81no—
268C4⋊C4—402no—
269C22⋊C4—41no—
270C22⋊C4—41no—
271C22⋊C4—42no—
272C22⋊C4—42no—
273C22⋊C4—81no—
274C22⋊C4—202no—
275C22⋊C4—202no—
276C42—21no—
277C42—21no—
278C2×C5—21no—
279C2×C5—21no—
280C2×C5—41no—
281C2×C5—42no—
282C2×C5—41no—
283C2×C5#152011yescharacteristic
284C2×C5#212111yescharacteristic
285C2×C5#212611yescharacteristic
286C23—21no—
287C23—21no—
288C23—21no—
289C23—21no—
290C23—41no—
291C23—41no—
292C23—41no—
293C23—41no—
294C23#396712yes—
295C23C2×#68811yescharacteristic
296C23C2×#68911yescharacteristic
297C23C2×C22⋊Dic511yescharacteristic
298Q8—82no—
299D4—42no—
300D4—42no—
301C2×C4—21no—
302C2×C4—21no—
303C2×C4—22no—
304C2×C4—21no—
305C2×C4—21no—
306C2×C4—21no—
307C2×C4—41no—
308C2×C4—42no—
309C2×C4—41no—
310C2×C4—41no—
311C2×C4—41no—
312C2×C4—41no—
313C2×C4—41no—
314C2×C4—41no—
315C2×C4—41no—
316C2×C4—41no—
317C2×C4—41no—
318C2×C4—41no—
319C2×C4—41no—
320C2×C4—41no—
321C2×C4—41no—
322C2×C4—41no—
323C2×C4—81no—
324C2×C4—81no—
325C2×C4—81no—
326C2×C4—84no—
327C2×C4—402no—
328C2×C4—402no—
329C2×C4—402no—
330C2×C4#396611yescharacteristic
331C2×C4#396611yescharacteristic
332C2×C4#396712yes—
333C8—402no—
334C5#1190211yescharacteristic, complemented, Sylow
335C22—21no—
336C22—21no—
337C22—21no—
338C22—22no—
339C22—21no—
340C22—42no—
341C22—41no—
342C22—41no—
343C22—42no—
344C22—41no—
345C22—41no—
346C22—41no—
347C22—41no—
348C22—81no—
349C22C2×#396611yescharacteristic
350C22C2×#396711yescharacteristic
351C22#6552511yescharacteristic
352C4—21no—
353C4—21no—
354C4—22no—
355C4—42no—
356C4—41no—
357C4—41no—
358C4—41no—
359C4—81no—
360C4—81no—
361C4—81no—
362C4—802no—
363C2—21no—
364C2—21no—
365C2—41no—
366C2—42no—
367C2—41no—
368C2#1060248311yescharacteristic
369C2#1060261211yescharacteristic
370C2#1060264011yescharacteristic
subgroupG/H
centerC22C2×#3966
commutatorC22×C4×C5C22×C4
FrattiniC2×C22⋊C4C22×D5
FittingC5×#2564C2
socleC22×C5C2×#99
prime radicalC22×C5×D4D4
cosocleC2×C5×C22⋊C4C23
hypercenter#2564D5
hypocenterC5#11902
LCSG ▷ C22×C4×C5 ▷ C23×C5 ▷ C2×C5 ▷ C5
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C4×D4 ◁ #2564

Complements

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

NH ≅ G/Ntype
4. #10598142C2semidirect
5. #10598137C2semidirect
6. C2×C5×C22⋊Q8C4semidirect
8. C5×#400C4semidirect
10. C5×#395C4semidirect
16. C2×C5×#95C4semidirect
107. C2×C22⋊Q8Dic5semidirect
111. #400Dic5semidirect
113. #395Dic5semidirect
120. C2×#95Dic5semidirect
334. C5#11902semidirect

Maximal subgroups

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

subgroupindex
C5×#25642
#106024122
#106023942
#105981422
#105981372
#119025