Finite Group Databasebeta

← #12010423

Subgroups — #12010423

Order 1280.

580 conjugacy classes of subgroups; only the 99 normal subgroups are shown.

normal subgroups only

This group has 580 conjugacy classes of subgroups — too many to draw. Only its 99 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#10614080C211yesmaximal, characteristic
2C5×#1799C211yesmaximal, characteristic
3#10602394C211yesmaximal, characteristic
4#10597981C214yesmaximal, complemented
5C23×Dic10C411yescharacteristic, complemented
6C2×C4×C5×D4C2211yescharacteristic
7C2×#4070—21no—
8C2×C5×#99—21no—
9C5×#380—21no—
10#65509—21no—
11#65501C411yescharacteristic, complemented
12#65268C412yescomplemented
13#65236C2212yes—
14#65235—21no—
15#65218—21no—
16#65214—21no—
17C5×#342C2212yes—
18#64745—21no—
19#64745C2212yes—
20#11898—51nomaximal, Sylow
21C22×C5×D4D411yescharacteristic
22C23×C4×C5C2×C411yescharacteristic
23C23×C4×C5D411yescharacteristic
24C23×Dic5—21no—
25C22×Dic10—21no—
26C22×Dic10—44no—
27C22×Dic10—41no—
28C22×Dic10—41no—
29C22×Dic10C2×C412yes—
30C4×C5×D4—41no—
31C4×C5×D4—41no—
32C2×C5×C4⋊C4—41no—
33C2×C5×C4⋊C4C2×C411yescharacteristic
34C2×C5×C22⋊C4—22no—
35C2×C5×C22⋊C4D411yescharacteristic
36C2×C5×C22⋊C4C2311yescharacteristic
37C2×C42×C5D411yescharacteristic
38C2×C22⋊Dic5—21no—
39C2×C22⋊Dic5—22no—
40C2×C4⋊Dic5—21no—
41#4070—44no—
42C2×#683—22no—
43C5×#99—42no—
44C5×#95—41no—
45#3963—22no—
46#3963—22no—
47#3963C2×C412yes—
48C22⋊#168—22no—
49C4⋊#168—41no—
50C4×#168—41no—
51#2409—51no—
52#1799D511yescharacteristic
53#1460—51no—
54#1351—54no—
55C24×C5C22⋊C411yescharacteristic
56C24×C5C2×D411yescharacteristic
57C2×C5×D4—41no—
58C2×C5×D4—41no—
59C2×C5×D4SD1612yes—
60C2×C5×D4Q1612yes—
61C22×C4×C5—21no—
62C22×C4×C5—21no—
63C22×C4×C5—21no—
64C22×C4×C5—21no—
65C22×C4×C5—42no—
66C22×C4×C5—42no—
67C22×C4×C5—42no—
68C22×C4×C5C22⋊C411yescharacteristic
69C22×C4×C5C22⋊C411yescharacteristic
70C22×C4×C5C22⋊C411yescharacteristic
71C22×C4×C5C22×C411yescharacteristic
72C22×C4×C5C2×D411yescharacteristic
73C22×Dic5—22no—
74C22×Dic5—22no—
75C22×Dic5—44no—
76C22×Dic5—42no—
77C2×Dic10—24no—
78C2×Dic10—42no—
79C2×Dic10—48no—
80C2×Dic10—44no—
81C2×Dic10—41no—
82C2×Dic10—41no—
83C2×Dic10—42no—
84C2×Dic10—44no—
85C2×Dic10—44no—
86C5×C4⋊C4—41no—
87C5×C4⋊C4—82no—
88C5×C22⋊C4—41no—
89C5×C22⋊C4—41no—
90C5×C22⋊C4—42no—
91C5×C22⋊C4—42no—
92C42×C5—21no—
93C42×C5—21no—
94C22⋊Dic5—44no—
95C22⋊Dic5—42no—
96C4⋊Dic5—42no—
97#683—44no—
98C2×#168—42no—
99C23×Q8—51no—
100C2×C22⋊Q8—101no—
101C2×C4×D4C2×D511yescharacteristic
102#431—101no—
103C2×#99—21no—
104#395—52no—
105#393—52no—
106#391—101no—
107#386—101no—
108#385—101no—
109#380—21no—
110#380—51no—
111#342C2×D512yes—
112#323—52no—
113#323—101no—
114C23×C5—21no—
115C23×C5—21no—
116C23×C5—21no—
117C23×C5—21no—
118C23×C5—41no—
119C23×C5—41no—
120C23×C5—41no—
121C23×C5—41no—
122C23×C5#10312yes—
123C23×C5C2×C22⋊C411yescharacteristic
124C23×C5C2×SD1611yescharacteristic
125C23×C5C2×Q1611yescharacteristic
126C5×D4—42no—
127C5×D4—42no—
128C2×C4×C5—21no—
129C2×C4×C5—21no—
130C2×C4×C5—22no—
131C2×C4×C5—21no—
132C2×C4×C5—21no—
133C2×C4×C5—21no—
134C2×C4×C5—42no—
135C2×C4×C5—42no—
136C2×C4×C5—42no—
137C2×C4×C5—41no—
138C2×C4×C5—41no—
139C2×C4×C5—41no—
140C2×C4×C5—81no—
141C2×C4×C5—82no—
142C2×C4×C5—82no—
143C2×C4×C5—82no—
144C2×C4×C5#9911yescharacteristic
145C2×C4×C5#9911yescharacteristic
146C2×C4×C5#10312yes—
147C2×Dic5—24no—
148C2×Dic5—42no—
149C2×Dic5—48no—
150C2×Dic5—42no—
151C2×Dic5—42no—
152C2×Dic5—42no—
153C2×Dic5—42no—
154Dic10—44no—
155Dic10—44no—
156Dic10—44no—
157Dic10—44no—
158#168—82no—
159C22×Q8—52no—
160C22×Q8—101no—
161C22×Q8—201no—
162C22×Q8—201no—
163C22×Q8—204no—
164C22×D4C22⋊D511yescharacteristic
165C23×C4—101no—
166C23×C4C4×D511yescharacteristic
167C23×C4C22⋊D511yescharacteristic
168C22⋊Q8—204no—
169C4×D4—41no—
170C4×D4—41no—
171C2×C4⋊C4—41no—
172C2×C4⋊C4—102no—
173C2×C4⋊C4—101no—
174C2×C4⋊C4C4×D511yescharacteristic
175C2×C22⋊C4—22no—
176C2×C22⋊C4—101no—
177C2×C22⋊C4—102no—
178C2×C22⋊C4D2011yescharacteristic
179C2×C22⋊C4C22×D511yescharacteristic
180C2×C42D2011yescharacteristic
181C4⋊C8—201no—
182#99—42no—
183C22⋊C8—102no—
184C4×C8—201no—
185#95—41no—
186#95—52no—
187#95—102no—
188#95—102no—
189C22×C5—21no—
190C22×C5—21no—
191C22×C5—21no—
192C22×C5—22no—
193C22×C5—21no—
194C22×C5—42no—
195C22×C5—41no—
196C22×C5—41no—
197C22×C5—42no—
198C22×C5—41no—
199C22×C5—41no—
200C22×C5—41no—
201C22×C5—41no—
202C22×C5—81no—
203C22×C5C2×#9911yescharacteristic
204C22×C5C2×#10311yescharacteristic
205C22×C5#42111yescharacteristic
206C4×C5—21no—
207C4×C5—22no—
208C4×C5—41no—
209C4×C5—41no—
210C4×C5—41no—
211C4×C5—162no—
212C4×C5AΣL(1,𝔽16)12yes—
213Dic5—44no—
214Dic5—82no—
215C24#68511yescharacteristic
216C24C2×C22⋊D511yescharacteristic
217C2×Q8—104no—
218C2×Q8—202no—
219C2×Q8—202no—
220C2×Q8—204no—
221C2×Q8—204no—
222C2×Q8—208no—
223C2×Q8—204no—
224C2×Q8—201no—
225C2×Q8—201no—
226C2×D4—41no—
227C2×D4—41no—
228C2×D4#68712yes—
229C2×D4#68912yes—
230C22×C4—21no—
231C22×C4—21no—
232C22×C4—21no—
233C22×C4—21no—
234C22×C4—42no—
235C22×C4—42no—
236C22×C4—42no—
237C22×C4—102no—
238C22×C4—102no—
239C22×C4—204no—
240C22×C4—202no—
241C22×C4#68511yescharacteristic
242C22×C4#68511yescharacteristic
243C22×C4#68511yescharacteristic
244C22×C4C2×C4×D511yescharacteristic
245C22×C4C2×D2011yescharacteristic
246C2×C8—202no—
247C4⋊C4—41no—
248C4⋊C4—82no—
249C4⋊C4—202no—
250C4⋊C4—204no—
251C22⋊C4—41no—
252C22⋊C4—41no—
253C22⋊C4—42no—
254C22⋊C4—42no—
255C22⋊C4—204no—
256C22⋊C4—202no—
257C42—21no—
258C42—21no—
259C2×C5—21no—
260C2×C5—21no—
261C2×C5—41no—
262C2×C5—42no—
263C2×C5—41no—
264C2×C5#152011yescharacteristic
265C2×C5C2×AΣL(1,𝔽16)11yescharacteristic
266C2×C5#212911yescharacteristic
267C23—21no—
268C23—21no—
269C23—21no—
270C23—21no—
271C23—41no—
272C23—41no—
273C23—41no—
274C23—41no—
275C23#394212yes—
276C23C2×#68511yescharacteristic
277C23C2×#68711yescharacteristic
278C23C2×#68911yescharacteristic
279Q8—204no—
280Q8—204no—
281Q8—204no—
282Q8—204no—
283D4—42no—
284D4—42no—
285C2×C4—21no—
286C2×C4—21no—
287C2×C4—22no—
288C2×C4—21no—
289C2×C4—21no—
290C2×C4—21no—
291C2×C4—42no—
292C2×C4—42no—
293C2×C4—42no—
294C2×C4—41no—
295C2×C4—41no—
296C2×C4—41no—
297C2×C4—81no—
298C2×C4—82no—
299C2×C4—82no—
300C2×C4—82no—
301C2×C4—104no—
302C2×C4—202no—
303C2×C4—208no—
304C2×C4—202no—
305C2×C4—202no—
306C2×C4—202no—
307C2×C4—202no—
308C2×C4#393811yescharacteristic
309C2×C4#393811yescharacteristic
310C2×C4#394212yes—
311C8—402no—
312C5#1189811yescharacteristic, complemented, Sylow
313C22—21no—
314C22—21no—
315C22—21no—
316C22—22no—
317C22—21no—
318C22—42no—
319C22—41no—
320C22—41no—
321C22—42no—
322C22—41no—
323C22—41no—
324C22—41no—
325C22—41no—
326C22—81no—
327C22C2×#393811yescharacteristic
328C22C2×#394211yescharacteristic
329C22#6529411yescharacteristic
330C4—21no—
331C4—22no—
332C4—41no—
333C4—41no—
334C4—41no—
335C4—162no—
336C4—204no—
337C4—402no—
338C4#6469112yes—
339C2—21no—
340C2—21no—
341C2—41no—
342C2—42no—
343C2—41no—
344C2#1060030311yescharacteristic
345C2C2×#6469111yescharacteristic
346C2#1060085211yescharacteristic
subgroupG/H
centerC22C2×#3938
commutatorC22×C4×C5C22×C4
FrattiniC2×C22⋊C4C22×D5
FittingC5×#1799C2
socleC22×C5C2×#99
prime radicalC22×C5×D4D4
cosocleC2×C5×C22⋊C4C23
hypercenter#1799D5
hypocenterC5#11898
LCSG ▷ C22×C4×C5 ▷ C23×C5 ▷ C2×C5 ▷ C5
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C4×D4 ◁ #1799

Complements

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

NH ≅ G/Ntype
4. #10597981C2semidirect
5. C23×Dic10C4semidirect
11. #65501C4semidirect
12. #65268C4semidirect
312. C5#11898semidirect

Maximal subgroups

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

subgroupindex
#106140802
C5×#17992
#106023942
#105979812
#118985