Finite Group Databasebeta

← #12010076

Subgroups — #12010076

Order 1280.

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

normal subgroups only

This group has 446 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×#2398C211yesmaximal, characteristic
2#10602412C211yesmaximal, characteristic
3#10602395C211yesmaximal, characteristic
4#10598140C212yesmaximal, complemented
5#10598137C212yesmaximal, complemented
6C2×C4×C5×Q8C411yescharacteristic, complemented
7C2×C4×C5×D4C2211yescharacteristic
8C5×#400—21no—
9C5×#397C411yescharacteristic, complemented
10C5×#397C2211yescharacteristic
11C5×#394—21no—
12C5×#393C411yescharacteristic, complemented
13C5×#393C2211yescharacteristic
14C4×C5×C22⋊C4—21no—
15C4×C5×C22⋊C4—21no—
16C5×#376C411yescharacteristic, complemented
17C2×#3966—21no—
18#65501—21no—
19#65496—21no—
20#64748—21no—
21#64746C2212yes—
22#64745C2212yes—
23#11823—51nomaximal, Sylow
24C22×C5×Q8C2×C411yescharacteristic
25C22×C5×D4D411yescharacteristic
26C23×C4×C5D411yescharacteristic
27C23×C4×C5D411yescharacteristic
28C4×C5×Q8—41no—
29C4×C5×Q8—41no—
30C4×C5×D4—41no—
31C4×C5×D4—41no—
32C2×C5×C4⋊C4—21no—
33C2×C5×C4⋊C4C2×C411yescharacteristic
34C2×C5×C4⋊C4D411yescharacteristic
35C2×C5×C22⋊C4—21no—
36C2×C5×C22⋊C4—21no—
37C2×C5×C22⋊C4—21no—
38C2×C5×C22⋊C4—21no—
39C2×C5×C22⋊C4C2×C411yescharacteristic
40C2×C5×C22⋊C4C2311yescharacteristic
41C2×C42×C5—21no—
42C2×C42×C5—21no—
43C2×C42×C5C2×C411yescharacteristic
44C2×C22⋊Dic5—22no—
45C2×C4⋊Dic5—41no—
46C2×#681—41no—
47C5×#95—21no—
48C5×#95—21no—
49C5×#95—21no—
50C5×#95—21no—
51C5×#95—21no—
52C5×#95C2×C411yescharacteristic
53C5×#95C2×C411yescharacteristic
54#3966—42no—
55#3963—41no—
56C22⋊#168—22no—
57C22⋊#168—41no—
58#2398D511yescharacteristic
59#1799—51no—
60#1465—51no—
61#1351—52no—
62#1344—52no—
63C24×C5C22⋊C411yescharacteristic
64C24×C5C2×D411yescharacteristic
65C2×C5×Q8—42no—
66C2×C5×Q8—41no—
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—42no—
80C22×C4×C5—42no—
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×C22⋊C4—41no—
91C5×C22⋊C4—41no—
92C5×C22⋊C4—42no—
93C5×C22⋊C4—42no—
94C5×C22⋊C4—42no—
95C5×C22⋊C4—41no—
96C5×C22⋊C4—41no—
97C42×C5—21no—
98C42×C5—21no—
99C42×C5—42no—
100C42×C5—41no—
101C42×C5—41no—
102C22⋊Dic5—42no—
103C22⋊Dic5—42no—
104C4⋊Dic5—82no—
105#681—82no—
106C2×#168—42no—
107C2×C4×Q8Dic511yescharacteristic, complemented
108C2×C4×D4C2×D511yescharacteristic
109C2×#99—101no—
110#407—101no—
111#400—21no—
112#397Dic511yescharacteristic, complemented
113#397C2×D511yescharacteristic
114#394—21no—
115#393Dic511yescharacteristic, complemented
116#393C2×D511yescharacteristic
117#380—101no—
118C4×C22⋊C4—21no—
119C4×C22⋊C4—21no—
120#376Dic511yescharacteristic, complemented
121#342—52no—
122#324—101no—
123#323—52no—
124C23×C5—21no—
125C23×C5—21no—
126C23×C5—21no—
127C23×C5—21no—
128C23×C5—21no—
129C23×C5—41no—
130C23×C5—41no—
131C23×C5—41no—
132C23×C5—41no—
133C23×C5C2×C22⋊C411yescharacteristic
134C23×C5AGL(1,ℤ/8)11yescharacteristic
135C23×C5#13711yescharacteristic
136C5×Q8—42no—
137C5×Q8—42no—
138C5×D4—42no—
139C5×D4—42no—
140C2×C4×C5—22no—
141C2×C4×C5—22no—
142C2×C4×C5—21no—
143C2×C4×C5—21no—
144C2×C4×C5—21no—
145C2×C4×C5—21no—
146C2×C4×C5—21no—
147C2×C4×C5—21no—
148C2×C4×C5—41no—
149C2×C4×C5—41no—
150C2×C4×C5—42no—
151C2×C4×C5—42no—
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—41no—
162C2×C4×C5—41no—
163C2×C4×C5—41no—
164C2×C4×C5—81no—
165C2×C4×C5—81no—
166C2×C4×C5#9911yescharacteristic
167C2×C4×C5#9911yescharacteristic
168C2×Dic5—82no—
169C2×Dic5—82no—
170C2×Dic5—82no—
171#168—82no—
172C22×Q8C2×Dic511yescharacteristic
173C22×D4C22⋊D511yescharacteristic
174C23×C4C22⋊D511yescharacteristic
175C23×C4C22⋊D511yescharacteristic
176C2×M16—201no—
177C4×Q8—41no—
178C4×Q8—41no—
179C4×D4—41no—
180C4×D4—41no—
181C2×C4⋊C4—21no—
182C2×C4⋊C4—201no—
183C2×C4⋊C4C2×Dic511yescharacteristic
184C2×C4⋊C4C22⋊D511yescharacteristic
185C2×C22⋊C4—21no—
186C2×C22⋊C4—21no—
187C2×C22⋊C4—21no—
188C2×C22⋊C4—21no—
189C2×C22⋊C4—102no—
190C2×C22⋊C4C2×Dic511yescharacteristic
191C2×C22⋊C4C22×D511yescharacteristic
192C2×C42—21no—
193C2×C42—21no—
194C2×C42C2×Dic511yescharacteristic
195#99—202no—
196C22⋊C8—102no—
197C22⋊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—21no—
210C22×C5—41no—
211C22×C5—41no—
212C22×C5—42no—
213C22×C5—41no—
214C22×C5—41no—
215C22×C5—41no—
216C22×C5—41no—
217C22×C5—41no—
218C22×C5—81no—
219C22×C5—81no—
220C22×C5C2×#9911yescharacteristic
221C22×C5#41711yescharacteristic
222C22×C5#42111yescharacteristic
223C4×C5—21no—
224C4×C5—42no—
225C4×C5—41no—
226C4×C5—41no—
227C4×C5—41no—
228C4×C5—41no—
229C4×C5—81no—
230C4×C5—81no—
231C4×C5—81no—
232C4×C5#35412yes—
233Dic5—162no—
234C24C22⋊Dic511yescharacteristic
235C24C2×C22⋊D511yescharacteristic
236C2×Q8—42no—
237C2×Q8—41no—
238C2×D4—21no—
239C2×D4—21no—
240C2×D4—41no—
241C2×D4—41no—
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—202no—
253C22×C4C22⋊Dic511yescharacteristic
254C22×C4C22⋊Dic511yescharacteristic
255C22×C4C22⋊Dic511yescharacteristic
256C22×C4C22×Dic511yescharacteristic
257C22×C4C2×C22⋊D511yescharacteristic
258M16—402no—
259C2×C8—202no—
260C4⋊C4—41no—
261C4⋊C4—41no—
262C4⋊C4—42no—
263C4⋊C4—402no—
264C22⋊C4—41no—
265C22⋊C4—41no—
266C22⋊C4—42no—
267C22⋊C4—42no—
268C22⋊C4—42no—
269C22⋊C4—41no—
270C22⋊C4—41no—
271C22⋊C4—202no—
272C22⋊C4—202no—
273C42—21no—
274C42—21no—
275C42—42no—
276C42—41no—
277C42—41no—
278C2×C5—21no—
279C2×C5—21no—
280C2×C5—41no—
281C2×C5—41no—
282C2×C5—81no—
283C2×C5#151311yescharacteristic
284C2×C5#212011yescharacteristic
285C2×C5C2×#35411yescharacteristic
286C23—21no—
287C23—21no—
288C23—21no—
289C23—21no—
290C23—21no—
291C23—41no—
292C23—41no—
293C23—41no—
294C23—41no—
295C23#409511yescharacteristic
296C23#409711yescharacteristic
297C23C2×C22⋊Dic511yescharacteristic
298Q8—42no—
299Q8—42no—
300D4—42no—
301D4—42no—
302C2×C4—22no—
303C2×C4—22no—
304C2×C4—21no—
305C2×C4—21no—
306C2×C4—21no—
307C2×C4—21no—
308C2×C4—21no—
309C2×C4—21no—
310C2×C4—41no—
311C2×C4—41no—
312C2×C4—42no—
313C2×C4—42no—
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—41no—
324C2×C4—41no—
325C2×C4—41no—
326C2×C4—81no—
327C2×C4—81no—
328C2×C4—402no—
329C2×C4—402no—
330C2×C4—402no—
331C2×C4#396611yescharacteristic
332C2×C4#396611yescharacteristic
333C8—402no—
334C5#1182311yescharacteristic, complemented, Sylow
335C22—21no—
336C22—21no—
337C22—21no—
338C22—21no—
339C22—41no—
340C22—41no—
341C22—42no—
342C22—41no—
343C22—41no—
344C22—41no—
345C22—41no—
346C22—41no—
347C22—81no—
348C22—81no—
349C22C2×#396611yescharacteristic
350C22#6552111yescharacteristic
351C22#6552511yescharacteristic
352C4—21no—
353C4—41no—
354C4—42no—
355C4—41no—
356C4—41no—
357C4—41no—
358C4—81no—
359C4—81no—
360C4—81no—
361C4—802no—
362C4#6476112yes—
363C2—21no—
364C2—21no—
365C2—41no—
366C2—41no—
367C2—81no—
368C2#1060250211yescharacteristic
369C2#1060250911yescharacteristic
370C2C2×#6476111yescharacteristic
subgroupG/H
centerC22C2×#3966
commutatorC22×C4×C5C22×C4
FrattiniC2×C22⋊C4C22×D5
FittingC5×#2398C2
socleC22×C5C2×#99
prime radicalC22×C5×D4D4
cosocleC2×C5×C22⋊C4C23
hypercenter#2398D5
hypocenterC5#11823
LCSG ▷ C22×C4×C5 ▷ C23×C5 ▷ C2×C5 ▷ C5
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C4×D4 ◁ #2398

Complements

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

NH ≅ G/Ntype
4. #10598140C2semidirect
5. #10598137C2semidirect
6. C2×C4×C5×Q8C4semidirect
9. C5×#397C4semidirect
12. C5×#393C4semidirect
16. C5×#376C4semidirect
107. C2×C4×Q8Dic5semidirect
112. #397Dic5semidirect
115. #393Dic5semidirect
120. #376Dic5semidirect
334. C5#11823semidirect

Maximal subgroups

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

subgroupindex
C5×#23982
#106024122
#106023952
#105981402
#105981372
#118235