Finite Group Databasebeta

← #11991746

Subgroups — #11991746

Order 1280.

736 conjugacy classes of subgroups; only the 147 normal subgroups are shown.

normal subgroups only

This group has 736 conjugacy classes of subgroups — too many to draw. Only its 147 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
1C2×C5×#563C211yesmaximal, characteristic
2#10602534C212yesmaximal, complemented
3C2×#64763C212yesmaximal, complemented
4#10602519C211yesmaximal, characteristic
5C2×#64760C211yesmaximal, characteristic
6C5×#563—24no—
7C5×#563—22no—
8C5×#563C414yescomplemented
9C2×C5×C4⋊Q8—21no—
10C2×C5×#126—22no—
11C2×C5×#125C2212yes—
12C2×C5×#124C2211yescharacteristic
13C2×C5×#123—21no—
14C2×C5×C22⋊Q8—22no—
15C2×#3967—21no—
16C2×#3964—21no—
17#65224C2211yescharacteristic
18C2×#3935C2212yes—
19C2×#3935C2211yescharacteristic
20#64763—24no—
21#64760—22no—
22#64749—22no—
23#64749—22no—
24#8507—51nomaximal, Sylow
25C22×C5×Q8—21no—
26C22×C5×Q8D411yescharacteristic
27C22×C5×D4D411yescharacteristic
28C23×C4×C5—21no—
29C5×C4⋊Q8—24no—
30C5×C4⋊Q8—24no—
31C5×#126—28no—
32C5×#126—28no—
33C5×#125—22no—
34C5×#125—22no—
35C5×#125C2×C414yes—
36C5×#125D414yes—
37C5×#124—22no—
38C5×#124C2×C412yes—
39C5×#124D412yes—
40C5×#123—24no—
41C5×#123—24no—
42C5×C22⋊Q8—28no—
43C5×C22⋊Q8—28no—
44C2×C5×C4⋊C4—22no—
45C2×C5×C4⋊C4—22no—
46C2×C5×C4⋊C4—22no—
47C2×C5×C4⋊C4D412yes—
48C2×C5×C4⋊C4D412yes—
49C2×C5×C22⋊C4—22no—
50C2×C5×C22⋊C4—21no—
51C2×C5×C22⋊C4—22no—
52C2×C42×C5—21no—
53C2×C42×C5C2311yescharacteristic
54C2×C4⋊Dic5—21no—
55C2×C4⋊Dic5—22no—
56C22×#168—21no—
57C22×#168—22no—
58#3967—42no—
59#3964—42no—
60#3935—22no—
61#3935—22no—
62#3935—22no—
63C2×#563D511yescharacteristic
64#1992—52no—
65#1959—51no—
66C2×#330—52no—
67C2×#329—51no—
68C24×C5—21no—
69C2×C5×Q8—24no—
70C2×C5×Q8—21no—
71C2×C5×Q8—41no—
72C2×C5×Q8—41no—
73C2×C5×Q8—81no—
74C2×C5×Q8—81no—
75C2×C5×Q8C22⋊C412yes—
76C2×C5×D4—21no—
77C2×C5×D4—41no—
78C2×C5×D4—41no—
79C2×C5×D4C22⋊C412yes—
80C22×C4×C5—22no—
81C22×C4×C5—21no—
82C22×C4×C5—22no—
83C22×C4×C5—24no—
84C22×C4×C5—81no—
85C22×C4×C5—81no—
86C22×C4×C5C2×D412yes—
87C22×C4×C5C2×D411yescharacteristic
88C22×Dic5—41no—
89C5×C4⋊C4—28no—
90C5×C4⋊C4—24no—
91C5×C4⋊C4—24no—
92C5×C4⋊C4—24no—
93C5×C4⋊C4—24no—
94C5×C4⋊C4—22no—
95C5×C4⋊C4—22no—
96C5×C4⋊C4C22⋊C414yes—
97C5×C4⋊C4C22⋊C414yes—
98C5×C22⋊C4—28no—
99C5×C22⋊C4—24no—
100C5×C22⋊C4—24no—
101C5×C22⋊C4—24no—
102C42×C5—24no—
103C42×C5C22×C411yescharacteristic
104C42×C5C2×D412yes—
105C42×C5C2×D411yescharacteristic
106C4⋊Dic5—41no—
107C4⋊Dic5—41no—
108C4⋊Dic5—44no—
109C2×#168—22no—
110C2×#168—24no—
111C2×#168—42no—
112C2×#168—41no—
113C2×#168—42no—
114C2×#168—41no—
115#563—24no—
116#563—22no—
117#563Dic514yescomplemented
118C2×C4⋊Q8—21no—
119C2×#126—22no—
120C2×#125C2×D512yes—
121C2×#124C2×D511yescharacteristic
122C2×#123—21no—
123C2×C22⋊Q8—22no—
124C2×#103—101no—
125C2×#102—101no—
126C2×#97—51no—
127C2×#97—52no—
128#384—51no—
129#340—102no—
130#340—102no—
131#330—104no—
132#329—102no—
133C23×C5—22no—
134C23×C5—41no—
135C23×C5—42no—
136C23×C5—41no—
137C23×C5—42no—
138C23×C5#12011yescharacteristic
139C5×Q8—42no—
140C5×Q8—81no—
141C5×Q8—84no—
142C5×D4—42no—
143C5×D4—81no—
144C2×C4×C5—21no—
145C2×C4×C5—22no—
146C2×C4×C5—21no—
147C2×C4×C5—22no—
148C2×C4×C5—24no—
149C2×C4×C5—22no—
150C2×C4×C5—24no—
151C2×C4×C5—42no—
152C2×C4×C5—41no—
153C2×C4×C5—42no—
154C2×C4×C5—42no—
155C2×C4×C5—41no—
156C2×C4×C5—42no—
157C2×C4×C5—84no—
158C2×C4×C5C2×C22⋊C412yes—
159C2×C4×C5C2×C22⋊C411yescharacteristic
160C2×C4×C5#12012yes—
161C2×C4×C5#12011yescharacteristic
162C2×Dic5—81no—
163C2×Dic5—82no—
164#168—42no—
165#168—44no—
166C22×Q8—21no—
167C22×Q8C22⋊D511yescharacteristic
168C22×D4C22⋊D511yescharacteristic
169C23×C4—21no—
170C22×C8—102no—
171C22×C8—101no—
172C4⋊Q8—24no—
173C4⋊Q8—24no—
174#126—28no—
175#126—28no—
176#125—22no—
177#125—22no—
178#125C2×Dic514yes—
179#125C22⋊D514yes—
180#124—22no—
181#124C2×Dic512yes—
182#124C22⋊D512yes—
183#123—24no—
184#123—24no—
185C22⋊Q8—28no—
186C22⋊Q8—28no—
187C2×C4⋊C4—22no—
188C2×C4⋊C4—22no—
189C2×C4⋊C4—22no—
190C2×C4⋊C4—102no—
191C2×C4⋊C4—101no—
192C2×C4⋊C4C22⋊D512yes—
193C2×C4⋊C4C22⋊D512yes—
194C2×C22⋊C4—22no—
195C2×C22⋊C4—21no—
196C2×C22⋊C4—22no—
197C2×C42—21no—
198C2×C42C22×D511yescharacteristic
199#103—202no—
200#102—202no—
201#97—102no—
202#97—102no—
203#97—102no—
204C22×C5—42no—
205C22×C5—44no—
206C22×C5—81no—
207C22×C5—81no—
208C22×C5—82no—
209C22×C5#37911yescharacteristic
210C22×C5#45311yescharacteristic
211C22×C5#45412yes—
212C22×C5#45511yescharacteristic
213C22×C5#45612yes—
214C4×C5—21no—
215C4×C5—21no—
216C4×C5—22no—
217C4×C5—22no—
218C4×C5—44no—
219C4×C5—42no—
220C4×C5—44no—
221Dic5—161no—
222C24—21no—
223C2×Q8—24no—
224C2×Q8—21no—
225C2×Q8—41no—
226C2×Q8—41no—
227C2×Q8—81no—
228C2×Q8—81no—
229C2×Q8C22⋊Dic512yes—
230C2×D4—21no—
231C2×D4—41no—
232C2×D4—41no—
233C2×D4C22⋊Dic512yes—
234C22×C4—22no—
235C22×C4—21no—
236C22×C4—22no—
237C22×C4—24no—
238C22×C4—81no—
239C22×C4—81no—
240C22×C4—201no—
241C22×C4C2×C22⋊D512yes—
242C22×C4C2×C22⋊D511yescharacteristic
243C2×C8—102no—
244C2×C8—104no—
245C2×C8—201no—
246C2×C8—202no—
247C2×C8—202no—
248C2×C8—201no—
249C4⋊C4—28no—
250C4⋊C4—24no—
251C4⋊C4—24no—
252C4⋊C4—24no—
253C4⋊C4—24no—
254C4⋊C4—22no—
255C4⋊C4—22no—
256C4⋊C4—201no—
257C4⋊C4—201no—
258C4⋊C4—204no—
259C4⋊C4C22⋊Dic514yes—
260C4⋊C4C22⋊Dic514yes—
261C22⋊C4—28no—
262C22⋊C4—24no—
263C22⋊C4—24no—
264C22⋊C4—24no—
265C42—24no—
266C42C22×Dic511yescharacteristic
267C42C2×C22⋊D512yes—
268C42C2×C22⋊D511yescharacteristic
269C2×C5—82no—
270C2×C5#165711yescharacteristic
271C2×C5#166011yescharacteristic
272C2×C5#166412yes—
273C2×C5#179211yescharacteristic
274C2×C5#179312yes—
275C23—22no—
276C23—41no—
277C23—41no—
278C23—42no—
279C23—42no—
280C23#409911yescharacteristic
281Q8—42no—
282Q8—81no—
283Q8—84no—
284D4—42no—
285D4—81no—
286C2×C4—21no—
287C2×C4—21no—
288C2×C4—22no—
289C2×C4—22no—
290C2×C4—24no—
291C2×C4—22no—
292C2×C4—24no—
293C2×C4—42no—
294C2×C4—41no—
295C2×C4—42no—
296C2×C4—41no—
297C2×C4—42no—
298C2×C4—42no—
299C2×C4—84no—
300C2×C4—401no—
301C2×C4—402no—
302C2×C4C2×C22⋊Dic512yes—
303C2×C4C2×C22⋊Dic511yescharacteristic
304C2×C4#409912yes—
305C2×C4#409911yescharacteristic
306C8—202no—
307C8—204no—
308C5#850711yescharacteristic, complemented, Sylow
309C22—42no—
310C22—44no—
311C22—81no—
312C22—81no—
313C22—82no—
314C22#6553011yescharacteristic
315C22#6553111yescharacteristic
316C22#6553412yes—
317C22#6553512yes—
318C22#6553611yescharacteristic
319C4—21no—
320C4—21no—
321C4—22no—
322C4—22no—
323C4—44no—
324C4—42no—
325C4—44no—
326C4—801no—
327C2—82no—
328C2#1060274811yescharacteristic
329C2#1060274911yescharacteristic
330C2#1060275012yes—
331C2#1060293311yescharacteristic
332C2#1060293612yes—
subgroupG/H
centerC23#4099
commutatorC42×C5C22×C4
FrattiniC2×C42C22×D5
FittingC2×C5×#563C2
socleC23×C5#120
prime radicalC22×C5×D4D4
cosocleC2×C42×C5C23
hypercenterC2×#563D5
hypocenterC5#8507
LCSG ▷ C42×C5 ▷ C22×C5 ▷ C5
UCSC1 ◁ C23 ◁ C2×C42 ◁ C2×#563

Complements

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

NH ≅ G/Ntype
2. #10602534C2semidirect
3. C2×#64763C2semidirect
8. C5×#563C4semidirect
117. #563Dic5semidirect
308. C5#8507semidirect

Maximal subgroups

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

subgroupindex
C2×C5×#5632
#106025342
C2×#647632
#106025192
C2×#647602
#85075