Finite Group Databasebeta

← #12017582

Subgroups — #12017582

Order 1280.

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

normal subgroups only

This group has 536 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#10614104C211yesmaximal, characteristic
2C5×#1799C211yesmaximal, characteristic
3C4×#3965C211yesmaximal, characteristic
4C2×#64692C214yesmaximal
5C2×C4×C5×D4C2211yescharacteristic
6C2×#4081C2212yes—
7C22×C4⋊Dic5C411yescharacteristic, complemented
8C2×C5×#99C2212yes—
9C5×#380—21no—
10C2×#3965C2212yes—
11#65501—21no—
12#65501C411yescharacteristic, complemented
13#65272—21no—
14#65242—21no—
15#65232C412yescomplemented
16#65217—21no—
17C4×#681—21no—
18C5×#342—21no—
19#64750—21no—
20#64692—28no—
21#13217—51nomaximal, Sylow
22C22×C5×D4C2311yescharacteristic
23C23×C4×C5C2×C411yescharacteristic
24C23×C4×C5D411yescharacteristic
25C23×Dic5C2×C412yes—
26C4×C5×D4—41no—
27C4×C5×D4—41no—
28C2×C5×C4⋊C4—41no—
29C2×C5×C4⋊C4D411yescharacteristic
30C2×C5×C22⋊C4—22no—
31C2×C5×C22⋊C4C2×C411yescharacteristic
32C2×C5×C22⋊C4D411yescharacteristic
33C2×C42×C5D411yescharacteristic
34C2×C22⋊Dic5—22no—
35C2×C22⋊Dic5C2×C412yes—
36#4081—24no—
37#4081—24no—
38C2×C4⋊Dic5—21no—
39C2×C4⋊Dic5—21no—
40C2×C4⋊Dic5—21no—
41C2×C4⋊Dic5—41no—
42C2×C4⋊Dic5—41no—
43C2×#683—22no—
44C2×#681—22no—
45C5×#99—24no—
46C5×#95—41no—
47#3965—22no—
48#3965D414yes—
49#3963—22no—
50#3963—22no—
51#3963—21no—
52#3935—41no—
53C4×#168—41no—
54#2486—51no—
55C2×#352—54no—
56#1799D511yescharacteristic
57C4×#100—51no—
58C24×C5C22×C411yescharacteristic
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×C5C2×D411yescharacteristic
76C22×Dic5—22no—
77C22×Dic5—22no—
78C22×Dic5—24no—
79C22×Dic5—42no—
80C22×Dic5—42no—
81C5×C4⋊C4—41no—
82C5×C4⋊C4—82no—
83C5×C22⋊C4—41no—
84C5×C22⋊C4—41no—
85C5×C22⋊C4—42no—
86C5×C22⋊C4—42no—
87C42×C5—21no—
88C42×C5—21no—
89C22⋊Dic5—24no—
90C22⋊Dic5—42no—
91C22⋊Dic5—42no—
92C4⋊Dic5—42no—
93C4⋊Dic5—42no—
94C4⋊Dic5—41no—
95C4⋊Dic5—41no—
96#683—44no—
97#681—24no—
98#681—42no—
99C2×#168—42no—
100C2×#123—52no—
101C2×C4×D4C2×D511yescharacteristic
102C22×C4⋊C4—51no—
103C2×#100—52no—
104C2×#99C2×D512yes—
105C4×M16—101no—
106#398—101no—
107#397—101no—
108#389—101no—
109#381—52no—
110#380—21no—
111#380—51no—
112#380—101no—
113#352—108no—
114#343—101no—
115#342—21no—
116C23×C5—21no—
117C23×C5—21no—
118C23×C5—21no—
119C23×C5—21no—
120C23×C5—41no—
121C23×C5—41no—
122C23×C5—41no—
123C23×C5—41no—
124C23×C5C2×C22⋊C411yescharacteristic
125C23×C5C4×D412yes—
126C23×C5#12411yescharacteristic
127C23×C5#12711yescharacteristic
128C5×D4—42no—
129C5×D4—42no—
130C2×C4×C5—21no—
131C2×C4×C5—21no—
132C2×C4×C5—22no—
133C2×C4×C5—21no—
134C2×C4×C5—21no—
135C2×C4×C5—21no—
136C2×C4×C5—42no—
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—82no—
144C2×C4×C5—82no—
145C2×C4×C5—82no—
146C2×C4×C5#9911yescharacteristic
147C2×C4×C5#9911yescharacteristic
148C2×C4×C5#12112yes—
149C2×Dic5—42no—
150C2×Dic5—42no—
151C2×Dic5—42no—
152C2×Dic5—42no—
153C2×Dic5—42no—
154C2×Dic5—42no—
155C2×Dic5—84no—
156#168—44no—
157C22×D4C22×D511yescharacteristic
158C23×C4—52no—
159C23×C4C4×D511yescharacteristic
160C23×C4C22⋊D511yescharacteristic
161C2×M16—102no—
162#123—104no—
163#123—104no—
164C4×D4—41no—
165C4×D4—41no—
166C2×C4⋊C4—41no—
167C2×C4⋊C4—101no—
168C2×C4⋊C4—101no—
169C2×C4⋊C4—101no—
170C2×C4⋊C4—102no—
171C2×C4⋊C4—201no—
172C2×C4⋊C4—201no—
173C2×C4⋊C4D2011yescharacteristic
174C2×C22⋊C4—22no—
175C2×C22⋊C4—52no—
176C2×C22⋊C4—102no—
177C2×C22⋊C4C4×D511yescharacteristic
178C2×C22⋊C4C22⋊D511yescharacteristic
179C2×C42D2011yescharacteristic
180#100—54no—
181#100—102no—
182#99—24no—
183#97—201no—
184C4×C8—201no—
185#95—41no—
186#95—101no—
187#95—102no—
188#95—102no—
189C22×C5—21no—
190C22×C5—21no—
191C22×C5—22no—
192C22×C5—21no—
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×C5#39011yescharacteristic
204C22×C5C2×#9911yescharacteristic
205C22×C5#41011yescharacteristic
206C4×C5—21no—
207C4×C5—22no—
208C4×C5—41no—
209C4×C5—41no—
210C4×C5—41no—
211C4×C5—162no—
212C4×C5#35212yescomplemented
213Dic5—82no—
214Dic5—82no—
215C24C2×C4×D511yescharacteristic
216C24C2×C22⋊D511yescharacteristic
217C2×D4—41no—
218C2×D4—41no—
219C2×D4D4×D512yes—
220C2×D4#71312yes—
221C22×C4—21no—
222C22×C4—21no—
223C22×C4—21no—
224C22×C4—21no—
225C22×C4—42no—
226C22×C4—42no—
227C22×C4—42no—
228C22×C4—102no—
229C22×C4—102no—
230C22×C4—104no—
231C22×C4—202no—
232C22×C4—202no—
233C22×C4#68511yescharacteristic
234C22×C4#68511yescharacteristic
235C22×C4#68511yescharacteristic
236C22×C4#68511yescharacteristic
237C22×C4C2×D2011yescharacteristic
238M16—104no—
239M16—202no—
240C2×C8—202no—
241C4⋊C4—41no—
242C4⋊C4—82no—
243C4⋊C4—202no—
244C4⋊C4—202no—
245C4⋊C4—201no—
246C4⋊C4—201no—
247C4⋊C4—204no—
248C22⋊C4—41no—
249C22⋊C4—41no—
250C22⋊C4—42no—
251C22⋊C4—42no—
252C22⋊C4—104no—
253C22⋊C4—202no—
254C22⋊C4—202no—
255C42—21no—
256C42—21no—
257C2×C5—21no—
258C2×C5—21no—
259C2×C5—41no—
260C2×C5—42no—
261C2×C5—41no—
262C2×C5#196411yescharacteristic
263C2×C5C2×#35211yescharacteristic
264C2×C5#212311yescharacteristic
265C23—21no—
266C23—21no—
267C23—21no—
268C23—21no—
269C23—41no—
270C23—41no—
271C23—41no—
272C23—41no—
273C23#403912yes—
274C23C2×#68511yescharacteristic
275C23#408611yescharacteristic
276C23#409311yescharacteristic
277D4—42no—
278D4—42no—
279C2×C4—21no—
280C2×C4—21no—
281C2×C4—22no—
282C2×C4—21no—
283C2×C4—21no—
284C2×C4—21no—
285C2×C4—42no—
286C2×C4—42no—
287C2×C4—42no—
288C2×C4—41no—
289C2×C4—41no—
290C2×C4—41no—
291C2×C4—81no—
292C2×C4—82no—
293C2×C4—82no—
294C2×C4—82no—
295C2×C4—202no—
296C2×C4—202no—
297C2×C4—202no—
298C2×C4—202no—
299C2×C4—202no—
300C2×C4—202no—
301C2×C4—404no—
302C2×C4#393811yescharacteristic
303C2×C4#393811yescharacteristic
304C2×C4#404112yes—
305C8—204no—
306C5#1321711yescharacteristic, complemented, Sylow
307C22—21no—
308C22—21no—
309C22—21no—
310C22—22no—
311C22—21no—
312C22—42no—
313C22—41no—
314C22—41no—
315C22—42no—
316C22—41no—
317C22—41no—
318C22—41no—
319C22—41no—
320C22—81no—
321C22C2×#393811yescharacteristic
322C22#6527711yescharacteristic
323C22#6529511yescharacteristic
324C4—21no—
325C4—22no—
326C4—41no—
327C4—41no—
328C4—41no—
329C4—162no—
330C4—402no—
331C4—402no—
332C4#6469212yescomplemented
333C2—21no—
334C2—21no—
335C2—41no—
336C2—42no—
337C2—41no—
338C2#1060032111yescharacteristic
339C2C2×#6469211yescharacteristic
340C2#1060085111yescharacteristic
subgroupG/H
centerC22C2×#3938
commutatorC24×C5C22×C4
FrattiniC22×D4C22×D5
FittingC5×#1799C2
socleC22×C5C2×#99
prime radical, cosocleC22×C5×D4C23
hypercenter#1799D5
hypocenterC5#13217
LCSG ▷ C24×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. C22×C4⋊Dic5C4semidirect
12. #65501C4semidirect
15. #65232C4semidirect
212. C4×C5#352semidirect
306. C5#13217semidirect
332. C4#64692semidirect

Maximal subgroups

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

subgroupindex
#106141042
C5×#17992
C4×#39652
C2×#646922
#132175