Finite Group Databasebeta

← #11994359

Subgroups — #11994359

Order 1280.

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

normal subgroups only

This group has 660 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
1C5×#2316C211yesmaximal, characteristic
2C2×#65272C211yesmaximal, characteristic
3C2×#64748C211yesmaximal, characteristic
4#10600294C212yesmaximal
5#10598115C212yesmaximal
6C2×C5×#117—21no—
7C22×C5×C4⋊C4C2211yescharacteristic
8C22×C4⋊Dic5C2211yescharacteristic
9C22×#683—21no—
10C5×#389—24no—
11C5×#380—21no—
12C5×#380C2212yes—
13C2×#3963—21no—
14C2×#3963C2211yescharacteristic
15C2×C22⋊#168C2212yes—
16#65272—22no—
17#65272C414yescomplemented
18#64749—24no—
19#64748—22no—
20#9027—51nomaximal, Sylow
21C23×C4×C5D411yescharacteristic
22C23×C4×C5D411yescharacteristic
23C23×C4×C5C2311yescharacteristic
24C23×Dic5—21no—
25C23×Dic5—21no—
26C5×#117—42no—
27C5×#117—42no—
28C2×C5×C4⋊C4—22no—
29C2×C5×C4⋊C4—22no—
30C2×C5×C4⋊C4—22no—
31C2×C5×C4⋊C4—22no—
32C2×C5×C4⋊C4C2×C412yes—
33C2×C5×C4⋊C4D412yes—
34C2×C5×C4⋊C4D414yes—
35C2×C5×C22⋊C4—22no—
36C2×C42×C5—22no—
37C2×C4⋊Dic5—22no—
38C2×C4⋊Dic5—22no—
39C2×C4⋊Dic5—21no—
40C2×C4⋊Dic5C2×C412yes—
41C2×#683—24no—
42C2×#683—42no—
43C2×#683—42no—
44C22×#168—22no—
45C5×#95—22no—
46C5×#95—22no—
47#3963—24no—
48#3963—21no—
49#3963C2×C412yes—
50C22⋊#168—22no—
51C22⋊#168—22no—
52C2×#398—51no—
53#2316D511yescharacteristic
54#1823—52no—
55C2×#324—51no—
56#1294—52no—
57C24×C5C2×D411yescharacteristic
58C22×C4×C5—21no—
59C22×C4×C5—21no—
60C22×C4×C5—21no—
61C22×C4×C5—21no—
62C22×C4×C5—21no—
63C22×C4×C5—21no—
64C22×C4×C5—24no—
65C22×C4×C5—42no—
66C22×C4×C5—42no—
67C22×C4×C5C22⋊C412yes—
68C22×C4×C5C22⋊C412yes—
69C22×C4×C5D812yes—
70C22×C4×C5SD1612yes—
71C22×C4×C5C22×C411yescharacteristic
72C22×C4×C5C2×D411yescharacteristic
73C22×C4×C5C2×D412yes—
74C22×C4×C5#4112yes—
75C22×Dic5—24no—
76C22×Dic5—22no—
77C22×Dic5—22no—
78C22×Dic5—42no—
79C22×Dic5—41no—
80C22×Dic5—41no—
81C22×Dic5—41no—
82C22×Dic5—41no—
83C22×Dic5—41no—
84C22×Dic5—41no—
85C22×Dic5—41no—
86C22×Dic5—41no—
87C5×C4⋊C4—24no—
88C5×C4⋊C4—24no—
89C5×C4⋊C4—44no—
90C5×C4⋊C4—44no—
91C5×C4⋊C4—82no—
92C5×C22⋊C4—42no—
93C5×C22⋊C4—42no—
94C42×C5—44no—
95C4⋊Dic5—24no—
96C4⋊Dic5—42no—
97#683—48no—
98C2×#168—44no—
99C2×#168—42no—
100C2×#117—21no—
101C22×C4⋊C4—51no—
102C22×C4⋊C4—101no—
103C22×C4⋊C4C2×D511yescharacteristic
104C2×C22⋊C8—52no—
105#398—54no—
106#398—102no—
107#389—24no—
108#380—21no—
109#380C2×D512yes—
110C2×#95—51no—
111C2×#95—101no—
112#340—104no—
113#324—102no—
114C23×C5—22no—
115C23×C5—22no—
116C23×C5C2×C22⋊C411yescharacteristic
117C23×C5#12011yescharacteristic
118C23×C5#12311yescharacteristic
119C23×C5C2×D811yescharacteristic
120C23×C5C2×SD1611yescharacteristic
121C23×C5AGL(1,ℤ/8)11yescharacteristic
122C23×C5#13711yescharacteristic
123C2×C4×C5—22no—
124C2×C4×C5—22no—
125C2×C4×C5—22no—
126C2×C4×C5—22no—
127C2×C4×C5—24no—
128C2×C4×C5—22no—
129C2×C4×C5—22no—
130C2×C4×C5—22no—
131C2×C4×C5—41no—
132C2×C4×C5—41no—
133C2×C4×C5—42no—
134C2×C4×C5—42no—
135C2×C4×C5—42no—
136C2×C4×C5—44no—
137C2×C4×C5—44no—
138C2×C4×C5—44no—
139C2×C4×C5—81no—
140C2×C4×C5#10214yes—
141C2×C4×C5C4×D412yes—
142C2×C4×C5#12112yes—
143C2×Dic5—24no—
144C2×Dic5—44no—
145C2×Dic5—44no—
146C2×Dic5—42no—
147C2×Dic5—42no—
148C2×Dic5—42no—
149C2×Dic5—42no—
150C2×Dic5—42no—
151C2×Dic5—81no—
152C2×Dic5—81no—
153C2×Dic5—81no—
154C2×Dic5—81no—
155#168—82no—
156C23×C4—101no—
157C23×C4—101no—
158C23×C4C22⋊D511yescharacteristic
159C23×C4C22⋊D511yescharacteristic
160C23×C4C22×D511yescharacteristic
161C22×C8—102no—
162#117—42no—
163#117—42no—
164C2×C4⋊C4—22no—
165C2×C4⋊C4—22no—
166C2×C4⋊C4—22no—
167C2×C4⋊C4—22no—
168C2×C4⋊C4—52no—
169C2×C4⋊C4—102no—
170C2×C4⋊C4—102no—
171C2×C4⋊C4—104no—
172C2×C4⋊C4—101no—
173C2×C4⋊C4—202no—
174C2×C4⋊C4—202no—
175C2×C4⋊C4C4×D512yes—
176C2×C4⋊C4D2012yes—
177C2×C4⋊C4C22⋊D514yes—
178C2×C22⋊C4—22no—
179C2×C42—22no—
180C22⋊C8—102no—
181C22⋊C8—102no—
182#95—22no—
183#95—22no—
184#95—52no—
185#95—104no—
186#95—101no—
187C22×C5—22no—
188C22×C5—22no—
189C22×C5—22no—
190C22×C5—22no—
191C22×C5—21no—
192C22×C5—21no—
193C22×C5—21no—
194C22×C5—21no—
195C22×C5#38611yescharacteristic
196C22×C5C2×#10211yescharacteristic
197C22×C5#41711yescharacteristic
198C22×C5#44711yescharacteristic
199C22×C5#44811yescharacteristic
200C22×C5#45612yes—
201C22×C5#45812yes—
202C22×C5#48011yescharacteristic
203C22×C5#48111yescharacteristic
204C4×C5—24no—
205C4×C5—22no—
206C4×C5—22no—
207C4×C5—42no—
208C4×C5—84no—
209Dic5—44no—
210Dic5—82no—
211C24C2×C22⋊D511yescharacteristic
212C22×C4—21no—
213C22×C4—21no—
214C22×C4—21no—
215C22×C4—21no—
216C22×C4—21no—
217C22×C4—21no—
218C22×C4—24no—
219C22×C4—42no—
220C22×C4—42no—
221C22×C4—104no—
222C22×C4—102no—
223C22×C4—102no—
224C22×C4—202no—
225C22×C4—201no—
226C22×C4—201no—
227C22×C4—201no—
228C22×C4—201no—
229C22×C4—201no—
230C22×C4—201no—
231C22×C4—201no—
232C22×C4—201no—
233C22×C4#68512yes—
234C22×C4#68512yes—
235C22×C4#68612yes—
236C22×C4#68812yes—
237C22×C4C2×C4×D511yescharacteristic
238C22×C4C2×D2011yescharacteristic
239C22×C4#70912yes—
240C22×C4C2×C22⋊D512yes—
241C2×C8—204no—
242C2×C8—202no—
243C4⋊C4—24no—
244C4⋊C4—24no—
245C4⋊C4—44no—
246C4⋊C4—44no—
247C4⋊C4—82no—
248C4⋊C4—104no—
249C4⋊C4—208no—
250C4⋊C4—202no—
251C22⋊C4—42no—
252C22⋊C4—42no—
253C42—44no—
254C2×C5—21no—
255C2×C5—21no—
256C2×C5—21no—
257C2×C5—21no—
258C2×C5#161012yes—
259C2×C5#161112yes—
260C2×C5#187411yescharacteristic
261C2×C5#188611yescharacteristic
262C2×C5#189011yescharacteristic
263C23—22no—
264C23—22no—
265C23C2×#68511yescharacteristic
266C23#407511yescharacteristic
267C23C2×#68611yescharacteristic
268C23#407811yescharacteristic
269C23C2×#68811yescharacteristic
270C23#408811yescharacteristic
271C23#409911yescharacteristic
272C2×C4—22no—
273C2×C4—22no—
274C2×C4—22no—
275C2×C4—22no—
276C2×C4—22no—
277C2×C4—22no—
278C2×C4—24no—
279C2×C4—22no—
280C2×C4—41no—
281C2×C4—41no—
282C2×C4—42no—
283C2×C4—42no—
284C2×C4—42no—
285C2×C4—44no—
286C2×C4—44no—
287C2×C4—44no—
288C2×C4—81no—
289C2×C4—104no—
290C2×C4—204no—
291C2×C4—204no—
292C2×C4—202no—
293C2×C4—202no—
294C2×C4—202no—
295C2×C4—202no—
296C2×C4—202no—
297C2×C4—401no—
298C2×C4—401no—
299C2×C4—401no—
300C2×C4—401no—
301C2×C4#394114yes—
302C2×C4C4×C22⋊D512yes—
303C2×C4#407612yes—
304C8—402no—
305C5#902711yescharacteristic, complemented, Sylow
306C22—22no—
307C22—22no—
308C22—22no—
309C22—22no—
310C22—21no—
311C22—21no—
312C22—21no—
313C22—21no—
314C22C2×#394111yescharacteristic
315C22#6525511yescharacteristic
316C22#6525611yescharacteristic
317C22#6525711yescharacteristic
318C22#6550211yescharacteristic
319C22#6550611yescharacteristic
320C22#6551611yescharacteristic
321C22#6553212yes—
322C22#6553512yes—
323C4—24no—
324C4—22no—
325C4—22no—
326C4—42no—
327C4—84no—
328C4—204no—
329C4—402no—
330C2—21no—
331C2—21no—
332C2—21no—
333C2—21no—
334C2#1060235311yescharacteristic
335C2#1060244811yescharacteristic
336C2#1060245311yescharacteristic
337C2#1060266712yes—
338C2#1060269112yes—
subgroupG/H
centerC23#4099
commutatorC22×C4×C5C22×C4
FrattiniC23×C4C22×D5
FittingC5×#2316C2
socleC23×C5#120
prime radicalC24×C5C2×D4
cosocleC23×C4×C5C23
hypercenter#2316D5
hypocenterC5#9027
LCSG ▷ C22×C4×C5 ▷ C22×C5 ▷ C5
UCSC1 ◁ C23 ◁ C23×C4 ◁ #2316

Complements

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

NH ≅ G/Ntype
17. #65272C4semidirect
305. C5#9027semidirect

Maximal subgroups

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

subgroupindex
C5×#23162
C2×#652722
C2×#647482
#106002942
#105981152
#90275