Finite Group Databasebeta

← #12017612

Subgroups — #12017612

Order 1280.

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

normal subgroups only

This group has 552 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
1#10614113C211yesmaximal, characteristic
2C5×#1799C211yesmaximal, characteristic
3C4×#3966C211yesmaximal, characteristic
4C2×#64693C214yesmaximal
5C2×C4×C5×D4C2211yescharacteristic
6C2×#4082C2212yes—
7C2×#4015C411yescharacteristic, complemented
8C2×C5×#99C2212yes—
9C5×#380—21no—
10C2×#3966C2212yes—
11#65501—21no—
12#65501—21no—
13C4×C22⋊Dic5—21no—
14#65263C412yescomplemented
15#65242—21no—
16#65242—21no—
17C4×C4⋊Dic5C411yescharacteristic, complemented
18C5×#342—21no—
19#64746—21no—
20#64693—28no—
21#13224—51nomaximal, Sylow
22C22×C5×D4C2311yescharacteristic
23C23×C4×C5D411yescharacteristic
24C23×C4×C5D411yescharacteristic
25C22×Dic10C2×C412yes—
26C4×C5×D4—41no—
27C4×C5×D4—41no—
28C2×C5×C4⋊C4—41no—
29C2×C5×C4⋊C4C2×C411yescharacteristic
30C2×C5×C22⋊C4—22no—
31C2×C5×C22⋊C4D411yescharacteristic
32C2×C5×C22⋊C4D411yescharacteristic
33C2×C42×C5C2×C411yescharacteristic
34C2×C22⋊Dic5—22no—
35C2×C22⋊Dic5—22no—
36C2×C22⋊Dic5—22no—
37#4082—24no—
38#4082—24no—
39C2×C4⋊Dic5—21no—
40C2×C4⋊Dic5—21no—
41C2×C4⋊Dic5—21no—
42C2×C4×Dic5—41no—
43C2×C4×Dic5C2×C412yes—
44#4015—41no—
45#4015—41no—
46C5×#99—24no—
47C5×#95—41no—
48#3966—22no—
49#3966D414yes—
50#3963—21no—
51#3963—22no—
52#3963—22no—
53#3963—41no—
54#2547—51no—
55C2×#354—54no—
56#1799D511yescharacteristic
57C4×#99—51no—
58C24×C5C2×D411yescharacteristic
59C24×C5C2×D411yescharacteristic
60C2×C5×D4—41no—
61C2×C5×D4—41no—
62C2×C5×D4C2×D412yes—
63C2×C5×D4#4112yes—
64C22×C4×C5—21no—
65C22×C4×C5—21no—
66C22×C4×C5—21no—
67C22×C4×C5—21no—
68C22×C4×C5—42no—
69C22×C4×C5—42no—
70C22×C4×C5—42no—
71C22×C4×C5C22⋊C411yescharacteristic
72C22×C4×C5C22⋊C411yescharacteristic
73C22×C4×C5C22⋊C411yescharacteristic
74C22×C4×C5C22⋊C411yescharacteristic
75C22×C4×C5C22×C411yescharacteristic
76C22×Dic5—22no—
77C22×Dic5—22no—
78C22×Dic5—42no—
79C2×Dic10—24no—
80C2×Dic10—42no—
81C2×Dic10—42no—
82C5×C4⋊C4—41no—
83C5×C4⋊C4—82no—
84C5×C22⋊C4—41no—
85C5×C22⋊C4—41no—
86C5×C22⋊C4—42no—
87C5×C22⋊C4—42no—
88C42×C5—21no—
89C42×C5—21no—
90C22⋊Dic5—24no—
91C22⋊Dic5—42no—
92C22⋊Dic5—44no—
93C22⋊Dic5—42no—
94C22⋊Dic5—42no—
95C4⋊Dic5—42no—
96C4⋊Dic5—42no—
97C4⋊Dic5—41no—
98C4⋊Dic5—41no—
99C4×Dic5—24no—
100C4×Dic5—82no—
101C2×C4⋊Q8—51no—
102C2×#124—52no—
103C2×C4×D4C2×D511yescharacteristic
104C2×#99—52no—
105C2×#99C2×D512yes—
106#397—101no—
107#397—101no—
108#391—52no—
109#380—21no—
110#380—101no—
111#380—101no—
112C4×C4⋊C4—51no—
113C4×C22⋊C4—101no—
114#354—108no—
115#342—21no—
116#342—101no—
117C23×C5—21no—
118C23×C5—21no—
119C23×C5—21no—
120C23×C5—21no—
121C23×C5—41no—
122C23×C5—41no—
123C23×C5—41no—
124C23×C5—41no—
125C23×C5C2×C22⋊C411yescharacteristic
126C23×C5#12112yes—
127C23×C5#12411yescharacteristic
128C23×C5#12711yescharacteristic
129C5×D4—42no—
130C5×D4—42no—
131C2×C4×C5—21no—
132C2×C4×C5—21no—
133C2×C4×C5—22no—
134C2×C4×C5—21no—
135C2×C4×C5—21no—
136C2×C4×C5—21no—
137C2×C4×C5—42no—
138C2×C4×C5—42no—
139C2×C4×C5—41no—
140C2×C4×C5—41no—
141C2×C4×C5—41no—
142C2×C4×C5—81no—
143C2×C4×C5—81no—
144C2×C4×C5—82no—
145C2×C4×C5—82no—
146C2×C4×C5—82no—
147C2×C4×C5#9911yescharacteristic
148C2×C4×C5#9911yescharacteristic
149C2×C4×C5C4×D412yes—
150C2×Dic5—42no—
151C2×Dic5—44no—
152C2×Dic5—42no—
153C2×Dic5—42no—
154C2×Dic5—42no—
155C2×Dic5—42no—
156C2×Dic5—42no—
157C2×Dic5—82no—
158C2×Dic5—82no—
159Dic10—84no—
160C22×Q8—52no—
161C22×D4C22×D511yescharacteristic
162C23×C4D2011yescharacteristic
163C23×C4C22⋊D511yescharacteristic
164C4⋊Q8—201no—
165C4⋊Q8—201no—
166#124—104no—
167#124—104no—
168C4×D4—41no—
169C4×D4—41no—
170C2×C4⋊C4—41no—
171C2×C4⋊C4—101no—
172C2×C4⋊C4—101no—
173C2×C4⋊C4—101no—
174C2×C4⋊C4C4×D511yescharacteristic
175C2×C22⋊C4—22no—
176C2×C22⋊C4—102no—
177C2×C22⋊C4—102no—
178C2×C22⋊C4—102no—
179C2×C22⋊C4D2011yescharacteristic
180C2×C22⋊C4C22⋊D511yescharacteristic
181C2×C42—52no—
182C2×C42—201no—
183C2×C42C4×D511yescharacteristic
184#99—24no—
185#99—54no—
186#99—102no—
187#95—41no—
188#95—101no—
189#95—102no—
190#95—102no—
191#95—201no—
192C22×C5—21no—
193C22×C5—21no—
194C22×C5—22no—
195C22×C5—21no—
196C22×C5—21no—
197C22×C5—42no—
198C22×C5—41no—
199C22×C5—41no—
200C22×C5—41no—
201C22×C5—41no—
202C22×C5—41no—
203C22×C5—41no—
204C22×C5—81no—
205C22×C5—81no—
206C22×C5#39011yescharacteristic
207C22×C5C2×#9911yescharacteristic
208C22×C5#41011yescharacteristic
209C4×C5—21no—
210C4×C5—22no—
211C4×C5—41no—
212C4×C5—41no—
213C4×C5—41no—
214C4×C5—162no—
215C4×C5#35412yescomplemented
216Dic5—82no—
217Dic5—84no—
218Dic5—82no—
219C24C2×D2011yescharacteristic
220C24C2×C22⋊D511yescharacteristic
221C2×Q8—104no—
222C2×Q8—202no—
223C2×Q8—202no—
224C2×D4—41no—
225C2×D4—41no—
226C2×D4D4×D512yes—
227C2×D4#71312yes—
228C22×C4—21no—
229C22×C4—21no—
230C22×C4—21no—
231C22×C4—21no—
232C22×C4—42no—
233C22×C4—42no—
234C22×C4—42no—
235C22×C4—102no—
236C22×C4—102no—
237C22×C4—202no—
238C22×C4#68511yescharacteristic
239C22×C4#68511yescharacteristic
240C22×C4#68511yescharacteristic
241C22×C4#68511yescharacteristic
242C22×C4C2×C4×D511yescharacteristic
243C4⋊C4—41no—
244C4⋊C4—82no—
245C4⋊C4—202no—
246C4⋊C4—202no—
247C4⋊C4—201no—
248C4⋊C4—201no—
249C22⋊C4—41no—
250C22⋊C4—41no—
251C22⋊C4—42no—
252C22⋊C4—42no—
253C22⋊C4—104no—
254C22⋊C4—202no—
255C22⋊C4—204no—
256C22⋊C4—202no—
257C22⋊C4—202no—
258C42—21no—
259C42—21no—
260C42—104no—
261C42—402no—
262C2×C5—21no—
263C2×C5—21no—
264C2×C5—42no—
265C2×C5—41no—
266C2×C5—41no—
267C2×C5#196411yescharacteristic
268C2×C5C2×#35411yescharacteristic
269C2×C5#212911yescharacteristic
270C23—21no—
271C23—21no—
272C23—21no—
273C23—21no—
274C23—41no—
275C23—41no—
276C23—41no—
277C23—41no—
278C23#404112yes—
279C23C2×#68511yescharacteristic
280C23#408611yescharacteristic
281C23#409311yescharacteristic
282Q8—404no—
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—41no—
294C2×C4—41no—
295C2×C4—41no—
296C2×C4—81no—
297C2×C4—81no—
298C2×C4—82no—
299C2×C4—82no—
300C2×C4—82no—
301C2×C4—204no—
302C2×C4—202no—
303C2×C4—202no—
304C2×C4—202no—
305C2×C4—202no—
306C2×C4—202no—
307C2×C4—202no—
308C2×C4—402no—
309C2×C4—402no—
310C2×C4#393811yescharacteristic
311C2×C4#393811yescharacteristic
312C2×C4#403912yes—
313C5#1322411yescharacteristic, complemented, Sylow
314C22—21no—
315C22—21no—
316C22—21no—
317C22—22no—
318C22—21no—
319C22—42no—
320C22—41no—
321C22—41no—
322C22—41no—
323C22—41no—
324C22—41no—
325C22—41no—
326C22—81no—
327C22—81no—
328C22C2×#393811yescharacteristic
329C22#6527711yescharacteristic
330C22#6529511yescharacteristic
331C4—21no—
332C4—22no—
333C4—41no—
334C4—41no—
335C4—41no—
336C4—162no—
337C4—402no—
338C4—404no—
339C4—402no—
340C4#6469312yescomplemented
341C2—21no—
342C2—21no—
343C2—41no—
344C2—42no—
345C2—41no—
346C2#1060032111yescharacteristic
347C2C2×#6469311yescharacteristic
348C2#1060085211yescharacteristic
subgroupG/H
centerC22C2×#3938
commutatorC22×C4×C5C22×C4
FrattiniC22×D4C22×D5
FittingC5×#1799C2
socleC22×C5C2×#99
prime radical, cosocleC22×C5×D4C23
hypercenter#1799D5
hypocenterC5#13224
LCSG ▷ C22×C4×C5 ▷ C22×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
7. C2×#4015C4semidirect
14. #65263C4semidirect
17. C4×C4⋊Dic5C4semidirect
215. C4×C5#354semidirect
313. C5#13224semidirect
340. C4#64693semidirect

Maximal subgroups

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

subgroupindex
#106141132
C5×#17992
C4×#39662
C2×#646932
#132245