Finite Group Databasebeta

← #11999876

Subgroups — #11999876

Order 1280.

772 conjugacy classes of subgroups; only the 121 normal subgroups are shown.

normal subgroups only

This group has 772 conjugacy classes of subgroups — too many to draw. Only its 121 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×#2656C211yesmaximal, characteristic
2C2×#65322C212yesmaximal, complemented
3C2×#64757C212yesmaximal, complemented
4#10600215C211yesmaximal, characteristic, complemented
5#10598104C211yesmaximal, characteristic, complemented
6C2×C5×#121—22no—
7C2×C5×#121C2212yes—
8C22×C4⋊Dic5C2211yescharacteristic
9C5×#399—22no—
10C5×#392—22no—
11C2×C5×#95C2211yescharacteristic
12C2×#3964—22no—
13C2×#3963—22no—
14C2×#3963C2211yescharacteristic
15C2×C22⋊#168C2212yes—
16#65322—24no—
17#65322—24no—
18C2×#3939—22no—
19#64757—24no—
20#9970—51nomaximal, Sylow
21C22×C5×D4—22no—
22C22×C5×D4—22no—
23C22×C5×D4D412yes—
24C23×C4×C5D412yes—
25C23×C4×C5C2311yescharacteristic
26C23×Dic5—21no—
27C23×Dic5—21no—
28C5×#121—22no—
29C5×#121—22no—
30C5×#121—22no—
31C5×#121—22no—
32C5×#121—44no—
33C5×#121—44no—
34C2×C5×C4⋊C4—22no—
35C2×C5×C4⋊C4D412yes—
36C2×C5×C22⋊C4—24no—
37C2×C5×C22⋊C4—22no—
38C2×C4⋊Dic5—22no—
39C2×C4⋊Dic5—22no—
40C2×C4⋊Dic5—21no—
41C2×C4⋊Dic5D412yescomplemented
42C22×#168—22no—
43C5×#95—22no—
44#3964—28no—
45#3963—22no—
46#3963—44no—
47C22⋊#168—22no—
48C22⋊#168D414yescomplemented
49#3939—28no—
50C2×#480—52no—
51#2656D511yescharacteristic
52C2×#327—52no—
53#1445—51no—
54#1280—51no—
55C24×C5—22no—
56C24×C5C2×D411yescharacteristic
57C2×C5×D4—22no—
58C2×C5×D4—22no—
59C2×C5×D4—44no—
60C2×C5×D4—44no—
61C2×C5×D4—42no—
62C2×C5×D4—42no—
63C2×C5×D4—44no—
64C2×C5×D4—44no—
65C2×C5×D4—82no—
66C2×C5×D4—82no—
67C22×C4×C5—22no—
68C22×C4×C5—22no—
69C22×C4×C5—22no—
70C22×C4×C5—42no—
71C22×C4×C5—42no—
72C22×C4×C5—42no—
73C22×C4×C5D814yes—
74C22×C4×C5C2×D411yescharacteristic
75C22×C4×C5C2×D412yes—
76C22×C4×C5C2×D412yes—
77C22×C4×C5#4111yescharacteristic
78C22×Dic5—22no—
79C22×Dic5—22no—
80C22×Dic5—41no—
81C22×Dic5—42no—
82C22×Dic5—41no—
83C22×Dic5—41no—
84C22×Dic5—41no—
85C22×Dic5—42no—
86C22×Dic5—42no—
87C22×Dic5—42no—
88C5×C4⋊C4—22no—
89C5×C4⋊C4—22no—
90C5×C4⋊C4—44no—
91C5×C22⋊C4—48no—
92C5×C22⋊C4—42no—
93C5×C22⋊C4—42no—
94C4⋊Dic5—24no—
95C4⋊Dic5—42no—
96C2×#168—24no—
97C2×#168—44no—
98C2×#121—22no—
99C2×#121C2×D512yes—
100C22×C4⋊C4—51no—
101#480—104no—
102#480—104no—
103C2×#107—102no—
104C2×#102—102no—
105C2×C22⋊C8—52no—
106#399—22no—
107#392—22no—
108C2×#95—51no—
109C2×#95—102no—
110C2×#95C2×D511yescharacteristic
111#327—104no—
112C23×C5—21no—
113C23×C5—21no—
114C23×C5—22no—
115C23×C5—42no—
116C23×C5—44no—
117C23×C5—42no—
118C23×C5—42no—
119C23×C5—44no—
120C23×C5#12011yescharacteristic
121C23×C5#12111yescharacteristic
122C23×C5#12711yescharacteristic
123C23×C5C2×D812yes—
124C23×C5AGL(1,ℤ/8)12yes—
125C5×D4—44no—
126C5×D4—82no—
127C5×D4—88no—
128C5×D4—84no—
129C5×D4—82no—
130C5×D4—82no—
131C2×C4×C5—24no—
132C2×C4×C5—24no—
133C2×C4×C5—41no—
134C2×C4×C5—42no—
135C2×C4×C5—41no—
136C2×C4×C5—42no—
137C2×C4×C5—42no—
138C2×C4×C5—44no—
139C2×C4×C5—42no—
140C2×C4×C5—44no—
141C2×C4×C5—82no—
142C2×C4×C5#12012yes—
143C2×C4×C5#12112yes—
144C2×Dic5—44no—
145C2×Dic5—42no—
146C2×Dic5—42no—
147C2×Dic5—44no—
148C2×Dic5—82no—
149C2×Dic5—81no—
150C2×Dic5—82no—
151C2×Dic5—81no—
152C2×Dic5—81no—
153C2×Dic5—81no—
154#168—44no—
155C22×D4—22no—
156C22×D4—22no—
157C22×D4C22⋊D512yes—
158C23×C4—101no—
159C23×C4—101no—
160C23×C4C22⋊D512yes—
161C23×C4C22×D511yescharacteristic
162C22×C8—102no—
163#121—22no—
164#121—22no—
165#121—22no—
166#121—22no—
167#121—44no—
168#121—44no—
169C2×C4⋊C4—22no—
170C2×C4⋊C4—52no—
171C2×C4⋊C4—102no—
172C2×C4⋊C4—102no—
173C2×C4⋊C4—101no—
174C2×C4⋊C4C22⋊D512yes—
175C2×C22⋊C4—24no—
176C2×C22⋊C4—22no—
177#107—108no—
178#102—108no—
179C22⋊C8—54no—
180C22⋊C8—102no—
181#95—22no—
182#95—102no—
183#95—204no—
184C22×C5—22no—
185C22×C5—22no—
186C22×C5—22no—
187C22×C5—22no—
188C22×C5—21no—
189C22×C5—21no—
190C22×C5—21no—
191C22×C5—21no—
192C22×C5—84no—
193C22×C5—84no—
194C22×C5—82no—
195C22×C5—82no—
196C22×C5—82no—
197C22×C5#39211yescharacteristic
198C22×C5#44712yes—
199C22×C5#45511yescharacteristic
200C22×C5#45611yescharacteristic
201C22×C5#45812yes—
202C22×C5#45912yes—
203C22×C5#49311yescharacteristic
204C22×C5#49611yescharacteristic
205C4×C5—22no—
206C4×C5—22no—
207C4×C5—44no—
208C4×C5—82no—
209Dic5—82no—
210Dic5—82no—
211C24—22no—
212C24C2×C22⋊D511yescharacteristic
213C2×D4—22no—
214C2×D4—22no—
215C2×D4—44no—
216C2×D4—44no—
217C2×D4—44no—
218C2×D4—44no—
219C2×D4—42no—
220C2×D4—42no—
221C2×D4—82no—
222C2×D4—82no—
223C22×C4—22no—
224C22×C4—22no—
225C22×C4—22no—
226C22×C4—42no—
227C22×C4—42no—
228C22×C4—42no—
229C22×C4—102no—
230C22×C4—102no—
231C22×C4—201no—
232C22×C4—202no—
233C22×C4—201no—
234C22×C4—202no—
235C22×C4—202no—
236C22×C4—202no—
237C22×C4—201no—
238C22×C4—201no—
239C22×C4#68614yes—
240C22×C4D4×D511yescharacteristic
241C22×C4D4×D512yes—
242C22×C4#71111yescharacteristic
243C22×C4C2×C22⋊D512yes—
244C2×C8—104no—
245C2×C8—204no—
246C4⋊C4—22no—
247C4⋊C4—22no—
248C4⋊C4—44no—
249C4⋊C4—104no—
250C4⋊C4—202no—
251C22⋊C4—48no—
252C22⋊C4—42no—
253C22⋊C4—42no—
254C2×C5—21no—
255C2×C5—22no—
256C2×C5—21no—
257C2×C5—162no—
258C2×C5#160712yes—
259C2×C5#161012yes—
260C2×C5#200711yescharacteristic
261C2×C5#201111yescharacteristic
262C2×C5#202211yescharacteristic
263C23—21no—
264C23—22no—
265C23—21no—
266C23—44no—
267C23—42no—
268C23—42no—
269C23—44no—
270C23—42no—
271C23C2×#68612yes—
272C23#407812yes—
273C23#408411yescharacteristic
274C23#408611yescharacteristic
275C23#409911yescharacteristic
276D4—44no—
277D4—82no—
278D4—88no—
279D4—84no—
280D4—82no—
281D4—82no—
282C2×C4—24no—
283C2×C4—24no—
284C2×C4—41no—
285C2×C4—42no—
286C2×C4—41no—
287C2×C4—42no—
288C2×C4—42no—
289C2×C4—44no—
290C2×C4—44no—
291C2×C4—42no—
292C2×C4—82no—
293C2×C4—204no—
294C2×C4—202no—
295C2×C4—204no—
296C2×C4—202no—
297C2×C4—402no—
298C2×C4—401no—
299C2×C4—401no—
300C2×C4—401no—
301C2×C4—401no—
302C2×C4—402no—
303C2×C4#408312yes—
304C2×C4#408512yes—
305C8—204no—
306C5#997011yescharacteristic, complemented, Sylow
307C22—22no—
308C22—22no—
309C22—21no—
310C22—21no—
311C22—22no—
312C22—22no—
313C22—21no—
314C22—21no—
315C22—82no—
316C22—84no—
317C22—82no—
318C22—84no—
319C22—82no—
320C22#6536112yes—
321C22#6536211yescharacteristic
322C22#6536311yescharacteristic
323C22#6550612yes—
324C22#6551211yescharacteristic
325C22#6553111yescharacteristic
326C22#6553212yes—
327C22#6553511yescharacteristic
328C4—22no—
329C4—22no—
330C4—44no—
331C4—82no—
332C4—402no—
333C4—402no—
334C2—21no—
335C2—22no—
336C2—21no—
337C2—162no—
338C2#1060256011yescharacteristic
339C2#1060263612yes—
340C2#1060264211yescharacteristic
341C2#1060266712yes—
342C2#1060267411yescharacteristic
subgroupG/H
centerC23#4099
commutator, cosocleC23×C4×C5C23
FrattiniC23×C4C22×D5
Fitting, prime radicalC5×#2656C2
socleC23×C5#120
hypercenter#2656D5
hypocenterC5#9970
LCSG ▷ C23×C4×C5 ▷ C22×C5 ▷ C5
UCSC1 ◁ C23 ◁ C23×C4 ◁ #2656

Complements

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

NH ≅ G/Ntype
2. C2×#65322C2semidirect
3. C2×#64757C2semidirect
4. #10600215C2semidirect
5. #10598104C2semidirect
41. C2×C4⋊Dic5D4semidirect
48. C22⋊#168D4semidirect
306. C5#9970semidirect

Maximal subgroups

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

subgroupindex
C5×#26562
C2×#653222
C2×#647572
#106002152
#105981042
#99705