Finite Group Databasebeta

← #11999878

Subgroups — #11999878

Order 1280.

772 conjugacy classes of subgroups; only the 119 normal subgroups are shown.

normal subgroups only

This group has 772 conjugacy classes of subgroups — too many to draw. Only its 119 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×#65256C212yesmaximal, complemented
2#10605112C211yesmaximal, characteristic
3C5×#1445C211yesmaximal, characteristic, complemented
4C2×#64671C212yesmaximal, complemented
5#10598104C211yesmaximal, characteristic, complemented
6C22×C5×C4⋊C4C2211yescharacteristic
7C2×#4076—22no—
8C2×#4076C2212yes—
9C2×C5×#95—22no—
10C2×C5×#95C2211yescharacteristic
11C2×C5×#95C2211yescharacteristic
12C2×C22⋊#168C2212yes—
13#65256—24no—
14#65256—24no—
15C2×#3941—22no—
16C2×#3939—22no—
17#64966—22no—
18#64960—22no—
19#64671—24no—
20#9970—51nomaximal, Sylow
21C23×C4×C5—21no—
22C23×C4×C5—21no—
23C23×C4×C5D412yes—
24C23×C4×C5C2311yescharacteristic
25C22×C22⋊D5—22no—
26C22×D20—22no—
27C22×D20D412yes—
28C2×C5×C4⋊C4—22no—
29C2×C5×C4⋊C4—22no—
30C2×C5×C4⋊C4—21no—
31C2×C5×C4⋊C4D412yescomplemented
32#4076—22no—
33#4076—22no—
34#4076—22no—
35#4076—22no—
36#4076—44no—
37#4076—44no—
38C2×#685—24no—
39C2×#685—22no—
40C2×C4⋊Dic5—22no—
41C2×C4⋊Dic5D412yes—
42C22×#168—22no—
43C5×#95—22no—
44C5×#95—22no—
45C5×#95—44no—
46C22⋊#168—22no—
47C22⋊#168D414yescomplemented
48#3941—28no—
49#3939—28no—
50C2×#480—52no—
51#2656—51no—
52C2×#327—52no—
53#1445D511yescharacteristic, complemented
54#1280—51no—
55C24×C5C2×D411yescharacteristic
56C23×D5—22no—
57C22×C4×C5—22no—
58C22×C4×C5—22no—
59C22×C4×C5—22no—
60C22×C4×C5—22no—
61C22×C4×C5—42no—
62C22×C4×C5—42no—
63C22×C4×C5—42no—
64C22×C4×C5—41no—
65C22×C4×C5—42no—
66C22×C4×C5—41no—
67C22×C4×C5—41no—
68C22×C4×C5—41no—
69C22×C4×C5—42no—
70C22×C4×C5—42no—
71C22×C4×C5—42no—
72C22×C4×C5D814yes—
73C22×C4×C5C2×D411yescharacteristic
74C22×C4×C5C2×D412yes—
75C22×C4×C5C2×D412yes—
76C22×C4×C5#4111yescharacteristic
77C2×C22⋊D5—42no—
78C2×C22⋊D5—42no—
79C2×C22⋊D5—44no—
80C2×C22⋊D5—44no—
81C22×Dic5—22no—
82C2×D20—22no—
83C2×D20—22no—
84C2×D20—44no—
85C2×D20—44no—
86C2×D20—82no—
87C2×D20—82no—
88C5×C4⋊C4—24no—
89C5×C4⋊C4—42no—
90#685—48no—
91#685—42no—
92#685—42no—
93C4⋊Dic5—22no—
94C4⋊Dic5—22no—
95C4⋊Dic5—44no—
96C2×#168—24no—
97C2×#168—44no—
98C2×#121—52no—
99C2×#121—102no—
100C22×C4⋊C4C2×D511yescharacteristic
101#480—104no—
102#480—104no—
103C2×#107—102no—
104C2×#102—102no—
105C2×C22⋊C8—52no—
106#399—102no—
107#392—102no—
108C2×#95—22no—
109C2×#95C2×D511yescharacteristic
110C2×#95C2×D511yescharacteristic
111#327—104no—
112C23×C5—21no—
113C23×C5—21no—
114C23×C5—22no—
115C23×C5#12011yescharacteristic
116C23×C5#12111yescharacteristic
117C23×C5#12711yescharacteristic
118C23×C5C2×D812yes—
119C23×C5AGL(1,ℤ/8)12yes—
120C22×D5—42no—
121C22×D5—44no—
122C22×D5—42no—
123C22×D5—42no—
124C22×D5—44no—
125C2×C4×C5—24no—
126C2×C4×C5—24no—
127C2×C4×C5—41no—
128C2×C4×C5—42no—
129C2×C4×C5—41no—
130C2×C4×C5—42no—
131C2×C4×C5—42no—
132C2×C4×C5—44no—
133C2×C4×C5—44no—
134C2×C4×C5—42no—
135C2×C4×C5—42no—
136C2×C4×C5—44no—
137C2×C4×C5—82no—
138C2×C4×C5—82no—
139C2×C4×C5—81no—
140C2×C4×C5—82no—
141C2×C4×C5—81no—
142C2×C4×C5—81no—
143C2×C4×C5—81no—
144C2×C4×C5#12012yes—
145C2×C4×C5#12112yes—
146C22⋊D5—84no—
147C22⋊D5—82no—
148C22⋊D5—82no—
149C2×Dic5—42no—
150C2×Dic5—44no—
151D20—44no—
152D20—82no—
153D20—88no—
154#168—44no—
155C22×D4—52no—
156C22×D4—102no—
157C22×D4—102no—
158C23×C4—21no—
159C23×C4—21no—
160C23×C4C22⋊D512yes—
161C23×C4C22×D511yescharacteristic
162C22×C8—102no—
163#121—102no—
164#121—102no—
165#121—102no—
166#121—102no—
167#121—204no—
168#121—204no—
169C2×C4⋊C4—22no—
170C2×C4⋊C4—22no—
171C2×C4⋊C4—21no—
172C2×C4⋊C4—52no—
173C2×C4⋊C4—102no—
174C2×C4⋊C4D2012yescomplemented
175C2×C22⋊C4—104no—
176C2×C22⋊C4—102no—
177#107—108no—
178#102—108no—
179C22⋊C8—54no—
180C22⋊C8—102no—
181#95—22no—
182#95—22no—
183#95—44no—
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#39211yescharacteristic
193C22×C5#44712yes—
194C22×C5#45511yescharacteristic
195C22×C5#45611yescharacteristic
196C22×C5#45812yes—
197C22×C5#45912yes—
198C22×C5#49311yescharacteristic
199C22×C5#49611yescharacteristic
200C2×D5—84no—
201C2×D5—84no—
202C2×D5—82no—
203C2×D5—82no—
204C2×D5—82no—
205C4×C5—22no—
206C4×C5—22no—
207C4×C5—44no—
208C4×C5—82no—
209C4×C5—82no—
210Dic5—82no—
211C24—102no—
212C24C2×C22⋊D511yescharacteristic
213C2×D4—102no—
214C2×D4—102no—
215C2×D4—204no—
216C2×D4—204no—
217C2×D4—202no—
218C2×D4—204no—
219C2×D4—202no—
220C2×D4—204no—
221C2×D4—402no—
222C2×D4—402no—
223C22×C4—22no—
224C22×C4—22no—
225C22×C4—22no—
226C22×C4—22no—
227C22×C4—42no—
228C22×C4—42no—
229C22×C4—42no—
230C22×C4—42no—
231C22×C4—41no—
232C22×C4—42no—
233C22×C4—42no—
234C22×C4—41no—
235C22×C4—41no—
236C22×C4—41no—
237C22×C4—42no—
238C22×C4—102no—
239C22×C4#68614yes—
240C22×C4C2×D2011yescharacteristic
241C22×C4#70911yescharacteristic
242C22×C4D4×D512yes—
243C22×C4D4×D512yes—
244C2×C8—104no—
245C2×C8—204no—
246C4⋊C4—24no—
247C4⋊C4—42no—
248C4⋊C4—102no—
249C4⋊C4—102no—
250C4⋊C4—204no—
251C22⋊C4—208no—
252C22⋊C4—202no—
253C22⋊C4—202no—
254C2×C5—21no—
255C2×C5—22no—
256C2×C5—21no—
257C2×C5#160712yes—
258C2×C5#161012yes—
259C2×C5#200711yescharacteristic
260C2×C5#201111yescharacteristic
261C2×C5#202211yescharacteristic
262D5—162no—
263C23—22no—
264C23—21no—
265C23—21no—
266C23—204no—
267C23—204no—
268C23—202no—
269C23—202no—
270C23—202no—
271C23#407611yescharacteristic
272C23C2×#68612yes—
273C23#408311yescharacteristic
274C23#408611yescharacteristic
275C23#409512yes—
276D4—204no—
277D4—402no—
278D4—408no—
279D4—404no—
280D4—402no—
281D4—402no—
282C2×C4—24no—
283C2×C4—24no—
284C2×C4—41no—
285C2×C4—41no—
286C2×C4—42no—
287C2×C4—42no—
288C2×C4—42no—
289C2×C4—44no—
290C2×C4—42no—
291C2×C4—44no—
292C2×C4—44no—
293C2×C4—42no—
294C2×C4—82no—
295C2×C4—82no—
296C2×C4—81no—
297C2×C4—82no—
298C2×C4—81no—
299C2×C4—81no—
300C2×C4—81no—
301C2×C4—204no—
302C2×C4—202no—
303C2×C4#402812yes—
304C2×C4#403012yes—
305C8—204no—
306C5#997011yescharacteristic, complemented, Sylow
307C22—21no—
308C22—21no—
309C22—22no—
310C22—22no—
311C22—22no—
312C22—22no—
313C22—21no—
314C22—21no—
315C22—402no—
316C22—402no—
317C22—404no—
318C22—404no—
319C22—402no—
320C22#6524911yescharacteristic
321C22#6530412yes—
322C22#6532512yes—
323C22#6534212yes—
324C22#6534711yescharacteristic
325C22#6535011yescharacteristic
326C22#6536211yescharacteristic
327C22#6538411yescharacteristic
328C4—22no—
329C4—22no—
330C4—44no—
331C4—82no—
332C4—82no—
333C4—402no—
334C2—21no—
335C2—22no—
336C2—21no—
337C2—802no—
338C2#1060033211yescharacteristic
339C2#1060045111yescharacteristic
340C2#1060048711yescharacteristic
341C2#1060050112yes—
342C2#1060050512yes—
subgroupG/H
centerC23#4083
commutator, cosocleC23×C4×C5C23
FrattiniC23×C4C22×D5
FittingC5×#1445C2
socleC23×C5#120
prime radical#10605112C2
hypercenter#1445D5
hypocenterC5#9970
LCSG ▷ C23×C4×C5 ▷ C22×C5 ▷ C5
UCSC1 ◁ C23 ◁ C23×C4 ◁ #1445

Complements

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

NH ≅ G/Ntype
1. C2×#65256C2semidirect
3. C5×#1445C2semidirect
4. C2×#64671C2semidirect
5. #10598104C2semidirect
31. C2×C5×C4⋊C4D4semidirect
47. C22⋊#168D4semidirect
53. #1445D5semidirect
174. C2×C4⋊C4D20semidirect
306. C5#9970semidirect

Maximal subgroups

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

subgroupindex
C2×#652562
#106051122
C5×#14452
C2×#646712
#105981042
#99705