Finite Group Databasebeta

← #12010091

Subgroups — #12010091

Order 1280.

414 conjugacy classes of subgroups; only the 87 normal subgroups are shown.

normal subgroups only

This group has 414 conjugacy classes of subgroups — too many to draw. Only its 87 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×#2638C211yesmaximal, characteristic
2#10602412C211yesmaximal, characteristic
3#10602396C211yesmaximal, characteristic
4#10598140C212yesmaximal, complemented
5#10598137C212yesmaximal, complemented
6C2×C5×C22⋊Q8C411yescharacteristic, complemented
7C2×C4×C5×D4C2211yescharacteristic
8C5×#398C411yescharacteristic, complemented
9C5×#397—21no—
10C5×#397C2211yescharacteristic
11C5×#395C411yescharacteristic, complemented
12C5×#394—21no—
13C5×#393C2211yescharacteristic
14C5×#387—21no—
15C5×#385—21no—
16C5×#381C411yescharacteristic, complemented
17C2×#3966—21no—
18#65501—21no—
19#65215—21no—
20#64746C2212yes—
21#64745—21no—
22#64745C2212yes—
23#11826—51nomaximal, Sylow
24C22×C5×Q8C2×C411yescharacteristic
25C22×C5×D4D411yescharacteristic
26C23×C4×C5C2×C411yescharacteristic
27C23×C4×C5D411yescharacteristic
28C5×C22⋊Q8—41no—
29C5×C22⋊Q8—41no—
30C4×C5×D4—41no—
31C4×C5×D4—41no—
32C2×C5×C4⋊C4—21no—
33C2×C5×C4⋊C4—21no—
34C2×C5×C4⋊C4—21no—
35C2×C5×C4⋊C4C2×C411yescharacteristic
36C2×C5×C4⋊C4C2×C411yescharacteristic
37C2×C5×C22⋊C4—21no—
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—
57#3935—41no—
58#2638D511yescharacteristic
59#1799—51no—
60#1466—51no—
61#1351—52no—
62#1344—52no—
63C24×C5C22⋊C411yescharacteristic
64C24×C5C2×D411yescharacteristic
65C2×C5×Q8—41no—
66C2×C5×Q8—81no—
67C2×C5×D4—21no—
68C2×C5×D4—21no—
69C2×C5×D4—41no—
70C2×C5×D4—41no—
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—81no—
80C22×C4×C5—81no—
81C22×C4×C5C22⋊C411yescharacteristic
82C22×C4×C5C22⋊C411yescharacteristic
83C22×C4×C5C22⋊C411yescharacteristic
84C22×C4×C5C22×C411yescharacteristic
85C22×C4×C5C2×D411yescharacteristic
86C22×Dic5—42no—
87C5×C4⋊C4—41no—
88C5×C4⋊C4—41no—
89C5×C4⋊C4—42no—
90C5×C4⋊C4—41no—
91C5×C4⋊C4—41no—
92C5×C4⋊C4—81no—
93C5×C22⋊C4—41no—
94C5×C22⋊C4—41no—
95C5×C22⋊C4—42no—
96C5×C22⋊C4—41no—
97C5×C22⋊C4—41no—
98C5×C22⋊C4—81no—
99C5×C22⋊C4—81no—
100C42×C5—21no—
101C42×C5—21no—
102C22⋊Dic5—42no—
103C22⋊Dic5—42no—
104C4⋊Dic5—82no—
105C2×#168—42no—
106C2×C22⋊Q8Dic511yescharacteristic, complemented
107C2×C4×D4C2×D511yescharacteristic
108#432—101no—
109C2×#99—101no—
110#398Dic511yescharacteristic, complemented
111#397—21no—
112#397C2×D511yescharacteristic
113#395Dic511yescharacteristic, complemented
114#394—21no—
115#393C2×D511yescharacteristic
116#387—21no—
117#385—21no—
118#381Dic511yescharacteristic, complemented
119#380—101no—
120#342—52no—
121#323—52no—
122#323—101no—
123C23×C5—21no—
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×C5C2×C22⋊C411yescharacteristic
133C23×C5AGL(1,ℤ/8)11yescharacteristic
134C23×C5#13711yescharacteristic
135C5×Q8—82no—
136C5×D4—42no—
137C5×D4—42no—
138C2×C4×C5—21no—
139C2×C4×C5—21no—
140C2×C4×C5—21no—
141C2×C4×C5—21no—
142C2×C4×C5—21no—
143C2×C4×C5—21no—
144C2×C4×C5—41no—
145C2×C4×C5—42no—
146C2×C4×C5—41no—
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#9911yescharacteristic
165C2×C4×C5#9911yescharacteristic
166C2×Dic5—82no—
167C2×Dic5—82no—
168C2×Dic5—82no—
169#168—82no—
170C22×Q8C2×Dic511yescharacteristic
171C22×D4C22⋊D511yescharacteristic
172C23×C4C2×Dic511yescharacteristic
173C23×C4C22⋊D511yescharacteristic
174C22⋊Q8—41no—
175C22⋊Q8—41no—
176C4×D4—41no—
177C4×D4—41no—
178C2×C4⋊C4—21no—
179C2×C4⋊C4—21no—
180C2×C4⋊C4—21no—
181C2×C4⋊C4—201no—
182C2×C4⋊C4C2×Dic511yescharacteristic
183C2×C4⋊C4C2×Dic511yescharacteristic
184C2×C22⋊C4—21no—
185C2×C22⋊C4—21no—
186C2×C22⋊C4—21no—
187C2×C22⋊C4—21no—
188C2×C22⋊C4—102no—
189C2×C22⋊C4C22⋊D511yescharacteristic
190C2×C22⋊C4C22×D511yescharacteristic
191C2×C42C22⋊D511yescharacteristic
192C4⋊C8—201no—
193#99—202no—
194C22⋊C8—102no—
195#97—201no—
196#95—21no—
197#95—21no—
198#95—21no—
199#95—21no—
200#95—21no—
201#95—201no—
202#95C2×Dic511yescharacteristic
203#95C2×Dic511yescharacteristic
204C22×C5—21no—
205C22×C5—21no—
206C22×C5—21no—
207C22×C5—21no—
208C22×C5—41no—
209C22×C5—41no—
210C22×C5—41no—
211C22×C5—41no—
212C22×C5—41no—
213C22×C5—41no—
214C22×C5—41no—
215C22×C5—81no—
216C22×C5—81no—
217C22×C5—81no—
218C22×C5C2×#9911yescharacteristic
219C22×C5#41711yescharacteristic
220C22×C5#42111yescharacteristic
221C4×C5—21no—
222C4×C5—21no—
223C4×C5—41no—
224C4×C5—41no—
225C4×C5—41no—
226C4×C5—41no—
227C4×C5—81no—
228C4×C5—81no—
229C4×C5—81no—
230C4×C5—81no—
231Dic5—162no—
232C24C22⋊Dic511yescharacteristic
233C24C2×C22⋊D511yescharacteristic
234C2×Q8—41no—
235C2×Q8—81no—
236C2×D4—21no—
237C2×D4—21no—
238C2×D4—41no—
239C2×D4—41no—
240C22×C4—21no—
241C22×C4—21no—
242C22×C4—21no—
243C22×C4—21no—
244C22×C4—21no—
245C22×C4—21no—
246C22×C4—21no—
247C22×C4—21no—
248C22×C4—81no—
249C22×C4—81no—
250C22×C4—202no—
251C22×C4C22⋊Dic511yescharacteristic
252C22×C4C22⋊Dic511yescharacteristic
253C22×C4C22⋊Dic511yescharacteristic
254C22×C4C22×Dic511yescharacteristic
255C22×C4C2×C22⋊D511yescharacteristic
256C2×C8—202no—
257C4⋊C4—41no—
258C4⋊C4—41no—
259C4⋊C4—42no—
260C4⋊C4—41no—
261C4⋊C4—41no—
262C4⋊C4—81no—
263C4⋊C4—402no—
264C22⋊C4—41no—
265C22⋊C4—41no—
266C22⋊C4—42no—
267C22⋊C4—41no—
268C22⋊C4—41no—
269C22⋊C4—81no—
270C22⋊C4—81no—
271C22⋊C4—202no—
272C22⋊C4—202no—
273C42—21no—
274C42—21no—
275C2×C5—21no—
276C2×C5—21no—
277C2×C5—41no—
278C2×C5—41no—
279C2×C5—81no—
280C2×C5#151311yescharacteristic
281C2×C5#212111yescharacteristic
282C2×C5#212911yescharacteristic
283C23—21no—
284C23—21no—
285C23—21no—
286C23—21no—
287C23—21no—
288C23—41no—
289C23—41no—
290C23—41no—
291C23—41no—
292C23#409511yescharacteristic
293C23#409711yescharacteristic
294C23C2×C22⋊Dic511yescharacteristic
295Q8—82no—
296D4—42no—
297D4—42no—
298C2×C4—21no—
299C2×C4—21no—
300C2×C4—21no—
301C2×C4—21no—
302C2×C4—21no—
303C2×C4—21no—
304C2×C4—41no—
305C2×C4—42no—
306C2×C4—41no—
307C2×C4—41no—
308C2×C4—41no—
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—81no—
322C2×C4—81no—
323C2×C4—81no—
324C2×C4—402no—
325C2×C4—402no—
326C2×C4—402no—
327C2×C4#396611yescharacteristic
328C2×C4#396611yescharacteristic
329C8—402no—
330C5#1182611yescharacteristic, complemented, Sylow
331C22—21no—
332C22—21no—
333C22—21no—
334C22—21no—
335C22—41no—
336C22—41no—
337C22—41no—
338C22—41no—
339C22—41no—
340C22—41no—
341C22—41no—
342C22—81no—
343C22—81no—
344C22—81no—
345C22C2×#396611yescharacteristic
346C22#6552111yescharacteristic
347C22#6552511yescharacteristic
348C4—21no—
349C4—21no—
350C4—41no—
351C4—41no—
352C4—41no—
353C4—41no—
354C4—81no—
355C4—81no—
356C4—81no—
357C4—81no—
358C4—802no—
359C2—21no—
360C2—21no—
361C2—41no—
362C2—41no—
363C2—81no—
364C2#1060250211yescharacteristic
365C2#1060264011yescharacteristic
366C2#1060266411yescharacteristic
subgroupG/H
centerC22C2×#3966
commutatorC22×C4×C5C22×C4
FrattiniC2×C22⋊C4C22×D5
FittingC5×#2638C2
socleC22×C5C2×#99
prime radicalC22×C5×D4D4
cosocleC2×C5×C22⋊C4C23
hypercenter#2638D5
hypocenterC5#11826
LCSG ▷ C22×C4×C5 ▷ C23×C5 ▷ C2×C5 ▷ C5
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C4×D4 ◁ #2638

Complements

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

NH ≅ G/Ntype
4. #10598140C2semidirect
5. #10598137C2semidirect
6. C2×C5×C22⋊Q8C4semidirect
8. C5×#398C4semidirect
11. C5×#395C4semidirect
16. C5×#381C4semidirect
106. C2×C22⋊Q8Dic5semidirect
110. #398Dic5semidirect
113. #395Dic5semidirect
118. #381Dic5semidirect
330. C5#11826semidirect

Maximal subgroups

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

subgroupindex
C5×#26382
#106024122
#106023962
#105981402
#105981372
#118265