Finite Group Databasebeta

← #12010449

Subgroups — #12010449

Order 1280.

568 conjugacy classes of subgroups; only the 107 normal subgroups are shown.

normal subgroups only

This group has 568 conjugacy classes of subgroups — too many to draw. Only its 107 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×#2409C211yesmaximal, characteristic
2#10602410C211yesmaximal, characteristic
3#10602393C211yesmaximal, characteristic
4#10598139C214yesmaximal, complemented
5C23×C5×Q8C411yescharacteristic
6C2×C5×C22⋊Q8—21no—
7C2×C4×C5×D4C2211yescharacteristic
8C5×#395C2212yes—
9C5×#393C412yes—
10C5×#391—21no—
11C5×#386—21no—
12C5×#385—21no—
13C5×#380C411yescharacteristic
14C2×#3965—21no—
15C2×C22⋊#168—21no—
16#65213—21no—
17#64750C2212yes—
18#64748C2212yes—
19#64745—21no—
20#11904—51nomaximal, Sylow
21C22×C5×Q8—21no—
22C22×C5×Q8—41no—
23C22×C5×Q8—41no—
24C22×C5×Q8—44no—
25C22×C5×Q8C2×C412yes—
26C22×C5×D4D411yescharacteristic
27C23×C4×C5—21no—
28C23×C4×C5C2×C411yescharacteristic
29C23×C4×C5D411yescharacteristic
30C5×C22⋊Q8—44no—
31C4×C5×D4—41no—
32C4×C5×D4—41no—
33C2×C5×C4⋊C4—21no—
34C2×C5×C4⋊C4—22no—
35C2×C5×C4⋊C4C2311yescharacteristic
36C2×C5×C22⋊C4—21no—
37C2×C5×C22⋊C4—22no—
38C2×C5×C22⋊C4C2×C411yescharacteristic
39C2×C5×C22⋊C4D411yescharacteristic
40C2×C42×C5D411yescharacteristic
41C2×#681—22no—
42C22×#168—41no—
43C5×#95—22no—
44C5×#95—22no—
45C5×#95C2×C412yes—
46#3965—42no—
47C22⋊#168—22no—
48C22⋊#168—41no—
49C4⋊#168—41no—
50C4⋊#168—41no—
51#2409D511yescharacteristic
52#1800—51no—
53#1459—51no—
54#1354—54no—
55C24×C5C22⋊C411yescharacteristic
56C24×C5C22⋊C411yescharacteristic
57C2×C5×Q8—24no—
58C2×C5×Q8—44no—
59C2×C5×Q8—42no—
60C2×C5×Q8—42no—
61C2×C5×Q8—44no—
62C2×C5×Q8—48no—
63C2×C5×Q8—44no—
64C2×C5×Q8—41no—
65C2×C5×Q8—41no—
66C2×C5×D4—41no—
67C2×C5×D4—41no—
68C2×C5×D4SD1612yes—
69C2×C5×D4Q1612yes—
70C22×C4×C5—22no—
71C22×C4×C5—21no—
72C22×C4×C5—21no—
73C22×C4×C5—21no—
74C22×C4×C5—21no—
75C22×C4×C5—22no—
76C22×C4×C5—44no—
77C22×C4×C5—42no—
78C22×C4×C5—42no—
79C22×C4×C5—42no—
80C22×C4×C5C22⋊C411yescharacteristic
81C22×C4×C5C22⋊C411yescharacteristic
82C22×C4×C5C22×C411yescharacteristic
83C22×C4×C5C2×D411yescharacteristic
84C22×C4×C5C2×D411yescharacteristic
85C5×C4⋊C4—41no—
86C5×C4⋊C4—42no—
87C5×C4⋊C4—44no—
88C5×C22⋊C4—41no—
89C5×C22⋊C4—41no—
90C5×C22⋊C4—44no—
91C5×C22⋊C4—42no—
92C42×C5—21no—
93C42×C5—21no—
94#681—42no—
95#681—42no—
96C2×#168—42no—
97C2×#168—42no—
98C2×#168—82no—
99C23×Q8Dic511yescharacteristic
100C2×C22⋊Q8—21no—
101C2×C4×D4C2×D511yescharacteristic
102#423—101no—
103C2×#100—101no—
104C2×C22⋊C8—101no—
105#395C2×D512yes—
106#393Dic512yes—
107#391—21no—
108#386—21no—
109#385—21no—
110#380Dic511yescharacteristic
111#343—52no—
112#324—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×C5#10312yes—
124C23×C5C2×C22⋊C411yescharacteristic
125C5×Q8—44no—
126C5×Q8—44no—
127C5×Q8—44no—
128C5×Q8—44no—
129C5×D4—42no—
130C5×D4—42no—
131C2×C4×C5—21no—
132C2×C4×C5—21no—
133C2×C4×C5—22no—
134C2×C4×C5—24no—
135C2×C4×C5—21no—
136C2×C4×C5—21no—
137C2×C4×C5—21no—
138C2×C4×C5—42no—
139C2×C4×C5—42no—
140C2×C4×C5—48no—
141C2×C4×C5—42no—
142C2×C4×C5—42no—
143C2×C4×C5—42no—
144C2×C4×C5—41no—
145C2×C4×C5—41no—
146C2×C4×C5—41no—
147C2×C4×C5—42no—
148C2×C4×C5—42no—
149C2×C4×C5—42no—
150C2×C4×C5—81no—
151C2×C4×C5#9911yescharacteristic
152C2×C4×C5#9911yescharacteristic
153C2×C4×C5C2×SD1611yescharacteristic
154C2×C4×C5C2×Q1611yescharacteristic
155#168—82no—
156#168—82no—
157C22×Q8—21no—
158C22×Q8—41no—
159C22×Q8—41no—
160C22×Q8—44no—
161C22×Q8C2×Dic512yes—
162C22×D4C22⋊D511yescharacteristic
163C23×C4—21no—
164C23×C4C2×Dic511yescharacteristic
165C23×C4C22⋊D511yescharacteristic
166C2×M16—102no—
167C22×C8—201no—
168C22⋊Q8—44no—
169C4×D4—41no—
170C4×D4—41no—
171C2×C4⋊C4—21no—
172C2×C4⋊C4—22no—
173C2×C4⋊C4C22×D511yescharacteristic
174C2×C22⋊C4—22no—
175C2×C22⋊C4—21no—
176C2×C22⋊C4C2×Dic511yescharacteristic
177C2×C22⋊C4C22⋊D511yescharacteristic
178C2×C42C22⋊D511yescharacteristic
179C4⋊C8—201no—
180C4⋊C8—201no—
181#100—202no—
182C22⋊C8—102no—
183C22⋊C8—201no—
184#95—22no—
185#95—22no—
186#95C2×Dic512yes—
187C22×C5—21no—
188C22×C5—21no—
189C22×C5—22no—
190C22×C5—22no—
191C22×C5—42no—
192C22×C5—42no—
193C22×C5—42no—
194C22×C5—41no—
195C22×C5—41no—
196C22×C5—41no—
197C22×C5—41no—
198C22×C5—81no—
199C22×C5C2×#9911yescharacteristic
200C22×C5C2×#10311yescharacteristic
201C22×C5#42111yescharacteristic
202C4×C5—21no—
203C4×C5—21no—
204C4×C5—21no—
205C4×C5—41no—
206C4×C5—44no—
207C4×C5—41no—
208C4×C5—41no—
209C4×C5—82no—
210C4×C5AΣL(1,𝔽16)12yes—
211C24C22⋊Dic511yescharacteristic
212C24C22⋊Dic511yescharacteristic
213C2×Q8—24no—
214C2×Q8—44no—
215C2×Q8—42no—
216C2×Q8—42no—
217C2×Q8—44no—
218C2×Q8—48no—
219C2×Q8—44no—
220C2×Q8—41no—
221C2×Q8—41no—
222C2×D4—41no—
223C2×D4—41no—
224C2×D4#68812yes—
225C2×D4#68912yes—
226C22×C4—22no—
227C22×C4—21no—
228C22×C4—21no—
229C22×C4—21no—
230C22×C4—21no—
231C22×C4—22no—
232C22×C4—44no—
233C22×C4—42no—
234C22×C4—42no—
235C22×C4—42no—
236C22×C4C22⋊Dic511yescharacteristic
237C22×C4C22⋊Dic511yescharacteristic
238C22×C4C22×Dic511yescharacteristic
239C22×C4C2×C22⋊D511yescharacteristic
240C22×C4C2×C22⋊D511yescharacteristic
241M16—202no—
242M16—202no—
243C2×C8—202no—
244C2×C8—202no—
245C2×C8—402no—
246C4⋊C4—41no—
247C4⋊C4—42no—
248C4⋊C4—44no—
249C22⋊C4—41no—
250C22⋊C4—41no—
251C22⋊C4—44no—
252C22⋊C4—42no—
253C42—21no—
254C42—21no—
255C2×C5—21no—
256C2×C5—21no—
257C2×C5—42no—
258C2×C5—42no—
259C2×C5#152011yescharacteristic
260C2×C5C2×AΣL(1,𝔽16)11yescharacteristic
261C2×C5#213511yescharacteristic
262C23—21no—
263C23—21no—
264C23—21no—
265C23—21no—
266C23—41no—
267C23—41no—
268C23—41no—
269C23—41no—
270C23#396712yes—
271C23#396712yes—
272C23C2×C22⋊Dic511yescharacteristic
273Q8—44no—
274Q8—44no—
275Q8—44no—
276Q8—44no—
277D4—42no—
278D4—42no—
279C2×C4—21no—
280C2×C4—21no—
281C2×C4—22no—
282C2×C4—24no—
283C2×C4—21no—
284C2×C4—21no—
285C2×C4—21no—
286C2×C4—42no—
287C2×C4—42no—
288C2×C4—48no—
289C2×C4—42no—
290C2×C4—42no—
291C2×C4—42no—
292C2×C4—41no—
293C2×C4—41no—
294C2×C4—41no—
295C2×C4—42no—
296C2×C4—42no—
297C2×C4—42no—
298C2×C4—81no—
299C2×C4#396611yescharacteristic
300C2×C4#396611yescharacteristic
301C2×C4C2×#68811yescharacteristic
302C2×C4C2×#68911yescharacteristic
303C8—402no—
304C8—402no—
305C5#1190411yescharacteristic, complemented, Sylow
306C22—21no—
307C22—21no—
308C22—22no—
309C22—22no—
310C22—42no—
311C22—42no—
312C22—42no—
313C22—41no—
314C22—41no—
315C22—41no—
316C22—41no—
317C22—81no—
318C22C2×#396611yescharacteristic
319C22C2×#396711yescharacteristic
320C22#6552511yescharacteristic
321C4—21no—
322C4—21no—
323C4—21no—
324C4—41no—
325C4—41no—
326C4—44no—
327C4—41no—
328C4—82no—
329C4#6475612yes—
330C2—21no—
331C2—21no—
332C2—42no—
333C2—42no—
334C2C2×#6475611yescharacteristic
335C2#1060248311yescharacteristic
336C2#1060266611yescharacteristic
subgroupG/H
centerC22C2×#3966
commutatorC22×C4×C5C22×C4
FrattiniC2×C4⋊C4C22×D5
FittingC5×#2409C2
socleC22×C5C2×#99
prime radicalC22×C5×D4D4
cosocleC2×C5×C4⋊C4C23
hypercenter#2409D5
hypocenterC5#11904
LCSG ▷ C22×C4×C5 ▷ C23×C5 ▷ C2×C5 ▷ C5
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C4×D4 ◁ #2409

Complements

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

NH ≅ G/Ntype
4. #10598139C2semidirect
305. C5#11904semidirect

Maximal subgroups

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

subgroupindex
C5×#24092
#106024102
#106023932
#105981392
#119045