Finite Group Databasebeta

← #12003152

Subgroups — #12003152

Order 1280.

474 conjugacy classes of subgroups.

The subgroup lattice has 406 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#65373C211yesmaximal, characteristic
2C5×#1990C211yesmaximal, characteristic, complemented
3C2×#64766C211yesmaximal, characteristic, complemented
4#10600580C211yesmaximal, characteristic
5C2×#64678C211yesmaximal, characteristic, complemented
6#10600377C211yesmaximal, characteristic
7#10598125C211yesmaximal, characteristic, complemented
8C2×C5×C4⋊Q8C2211yescharacteristic
9C2×C4×D20C2211yescharacteristic
10C2×C4×C5×C8C2211yescharacteristic
11C5×#384C2211yescharacteristic
12C2×#3952—21no—
13C2×#3952—21no—
14#65373—22no—
15#65373—22no—
16C2×#3941—21no—
17C2×#3941—21no—
18C2×#3940—21no—
19C2×C4⋊#168C2211yescharacteristic
20C2×C4⋊#168C2211yescharacteristic
21C2×C4⋊#168C2211yescharacteristic
22C5×#340—21no—
23C5×#340—21no—
24#64766—22no—
25#64749—21no—
26#64678—22no—
27#10481—51nomaximal, Sylow
28C22×C5×Q8—21no—
29C22×D20D411yescharacteristic
30C22×C4×D5—21no—
31C22×C5×C8D411yescharacteristic
32C22×C5×C8D411yescharacteristic
33C5×C4⋊Q8—21no—
34C5×C4⋊Q8—21no—
35C5×C4⋊Q8—21no—
36C5×C4⋊Q8D412yes—
37C2×C5×C4⋊C4—21no—
38C2×C5×C4⋊C4—21no—
39C2×C5×C4⋊C4—21no—
40C2×C5×C4⋊C4—21no—
41C2×C5×C4⋊C4—21no—
42C2×C42×C5C2311yescharacteristic
43C2×#685—21no—
44C2×C4⋊Dic5D411yescharacteristic
45C22×#168—21no—
46C22×#168—21no—
47C22×#168—21no—
48C4×D20—21no—
49C4×D20—21no—
50C4×D20—21no—
51C4×D20—21no—
52C4×C5×C8—21no—
53C4×C5×C8—21no—
54#3952—42no—
55#3952—42no—
56#3941—24no—
57#3941—42no—
58#3940—24no—
59C4⋊#168—21no—
60C4⋊#168—21no—
61C4⋊#168—21no—
62C4⋊#168—21no—
63C4⋊#168—21no—
64C4⋊#168D412yes—
65C2×#476—51no—
66#1990D511yescharacteristic, complemented
67#1954—51no—
68#1802—51no—
69C2×#335—51no—
70C2×#333—51no—
71#1296—51no—
72C23×D5—21no—
73C2×C5×Q8—22no—
74C2×C5×Q8—41no—
75C2×C5×Q8—81no—
76C2×C5×Q8—81no—
77C22×C4×C5—41no—
78C22×C4×C5—41no—
79C22×C4×C5C2×D411yescharacteristic
80C22×C4×C5C2×D411yescharacteristic
81C22×C4×C5#4111yescharacteristic
82C22×Dic5—21no—
83C2×D20—21no—
84C2×D20—21no—
85C2×D20—21no—
86C2×D20—21no—
87C2×D20—21no—
88C2×D20Q1612yes—
89C2×C4×D5—41no—
90C2×C4×D5—41no—
91C2×C4×D5—41no—
92C2×C4×D5—41no—
93C2×C4×D5—41no—
94C2×C4×D5—41no—
95C2×C5×C8—21no—
96C2×C5×C8—21no—
97C2×C5×C8—21no—
98C2×C5×C8—21no—
99C2×C5×C8—21no—
100C2×C5×C8SD1612yes—
101C5×C4⋊C4—22no—
102C5×C4⋊C4—22no—
103C5×C4⋊C4—41no—
104C5×C4⋊C4—41no—
105C5×C4⋊C4—41no—
106C5×C4⋊C4—41no—
107C5×C4⋊C4—41no—
108C5×C4⋊C4—41no—
109C5×C4⋊C4—81no—
110C42×C5C2×D411yescharacteristic
111C42×C5C2×D411yescharacteristic
112C42×C5#4111yescharacteristic
113C42×C5#4111yescharacteristic
114#685—41no—
115#685—41no—
116C4⋊Dic5—21no—
117C4⋊Dic5—21no—
118C2×#168—22no—
119C2×#168—41no—
120C2×#168—41no—
121C2×#168—41no—
122C2×#168—41no—
123C2×#168—41no—
124C2×#168—41no—
125C2×#168—41no—
126C2×#168—41no—
127C2×C4⋊Q8C2×D511yescharacteristic
128C2×C4×D4—51no—
129#476—102no—
130#476—102no—
131C2×#106—101no—
132C2×C4⋊C8—51no—
133C2×C4⋊C8—51no—
134C2×C4⋊C8—51no—
135C2×#102—101no—
136C2×#102—101no—
137C2×C22⋊C8—101no—
138C2×C22⋊C8—101no—
139C2×C4×C8C2×D511yescharacteristic
140#384C2×D511yescharacteristic
141#340—21no—
142#340—21no—
143#340—101no—
144#335—102no—
145#333—102no—
146C23×C5#12111yescharacteristic
147C22×D5—22no—
148C22×D5—41no—
149C22×D5—41no—
150C22×D5—41no—
151C22×D5—41no—
152C22×D5—41no—
153C22×D5—41no—
154C5×Q8—82no—
155C5×Q8—82no—
156C2×C4×C5—21no—
157C2×C4×C5—21no—
158C2×C4×C5—42no—
159C2×C4×C5—81no—
160C2×C4×C5—81no—
161C2×C4×C5—81no—
162C2×C4×C5—81no—
163C2×C4×C5—81no—
164C2×C4×C5#12011yescharacteristic
165C2×C4×C5#12111yescharacteristic
166C2×C4×C5#12111yescharacteristic
167C2×C4×C5#12311yescharacteristic
168C2×C4×C5#12311yescharacteristic
169C2×C4×C5#12411yescharacteristic
170C2×C4×C5C2×SD1611yescharacteristic
171C2×C4×C5C2×Q1611yescharacteristic
172C2×C4×C5#13511yescharacteristic
173C2×C4×C5#13511yescharacteristic
174C2×C4×C5AGL(1,ℤ/8)11yescharacteristic
175C2×C4×C5AGL(1,ℤ/8)11yescharacteristic
176C2×C4×C5#13711yescharacteristic
177C2×C4×C5#13711yescharacteristic
178C2×Dic5—41no—
179C2×Dic5—41no—
180C2×Dic5—41no—
181D20—22no—
182D20—22no—
183D20—41no—
184D20—41no—
185C4×D5—81no—
186C4×D5—81no—
187C4×D5—81no—
188C4×D5—81no—
189C5×C8—22no—
190C5×C8—41no—
191#168—42no—
192#168—81no—
193#168—81no—
194C22×Q8—21no—
195C22×D4—51no—
196C23×C4—101no—
197C22×C8—101no—
198C22×C8—101no—
199C22×C8—101no—
200C22×C8C22⋊D511yescharacteristic
201C22×C8C22⋊D511yescharacteristic
202C4⋊Q8—21no—
203C4⋊Q8—21no—
204C4⋊Q8—21no—
205C4⋊Q8D2012yes—
206C4×D4—101no—
207C4×D4—101no—
208C4×D4—101no—
209C4×D4—101no—
210C2×C4⋊C4—21no—
211C2×C4⋊C4—21no—
212C2×C4⋊C4—21no—
213C2×C4⋊C4—21no—
214C2×C4⋊C4—21no—
215C2×C4⋊C4—51no—
216C2×C22⋊C4—101no—
217C2×C42C22×D511yescharacteristic
218#106—104no—
219C4⋊C8—52no—
220C4⋊C8—101no—
221C4⋊C8—101no—
222C4⋊C8—101no—
223C4⋊C8—101no—
224C4⋊C8—101no—
225#102—104no—
226#102—202no—
227C22⋊C8—202no—
228C22⋊C8—202no—
229C4×C8—21no—
230C4×C8—21no—
231C22×C5#39411yescharacteristic
232C22×C5#46111yescharacteristic
233C22×C5#46311yescharacteristic
234C22×C5#46711yescharacteristic
235C22×C5#46811yescharacteristic
236C22×C5#47111yescharacteristic
237C22×C5#47211yescharacteristic
238C2×D5—42no—
239C2×D5—42no—
240C2×D5—42no—
241C2×D5—81no—
242C2×D5—81no—
243C2×D5—81no—
244C2×D5—81no—
245C4×C5—21no—
246C4×C5—21no—
247C4×C5—82no—
248C4×C5—161no—
249C4×C5#44911yescharacteristic
250C4×C5#45111yescharacteristic
251C4×C5#46011yescharacteristic
252C4×C5#46411yescharacteristic
253C4×C5#46511yescharacteristic
254C4×C5#46811yescharacteristic
255C4×C5#48411yescharacteristic
256C4×C5#48511yescharacteristic
257Dic5—81no—
258C24—101no—
259C2×Q8—22no—
260C2×Q8—41no—
261C2×Q8—81no—
262C2×Q8—81no—
263C2×D4—52no—
264C2×D4—101no—
265C2×D4—101no—
266C2×D4—101no—
267C2×D4—101no—
268C2×D4—101no—
269C22×C4—41no—
270C22×C4—41no—
271C22×C4—101no—
272C22×C4—201no—
273C22×C4—201no—
274C22×C4—201no—
275C22×C4—201no—
276C22×C4—201no—
277C22×C4—201no—
278C22×C4D4×D511yescharacteristic
279C22×C4#71111yescharacteristic
280C22×C4C2×C22⋊D511yescharacteristic
281C2×C8—21no—
282C2×C8—21no—
283C2×C8—21no—
284C2×C8—21no—
285C2×C8—21no—
286C2×C8—102no—
287C2×C8—201no—
288C2×C8—201no—
289C2×C8—201no—
290C2×C8—201no—
291C2×C8—201no—
292C2×C8—201no—
293C2×C8—201no—
294C2×C8—201no—
295C2×C8#68812yes—
296C4⋊C4—22no—
297C4⋊C4—22no—
298C4⋊C4—41no—
299C4⋊C4—41no—
300C4⋊C4—41no—
301C4⋊C4—41no—
302C4⋊C4—41no—
303C4⋊C4—41no—
304C4⋊C4—81no—
305C4⋊C4—101no—
306C4⋊C4—101no—
307C22⋊C4—201no—
308C22⋊C4—201no—
309C42C2×D2011yescharacteristic
310C42#70911yescharacteristic
311C42D4×D511yescharacteristic
312C42#71111yescharacteristic
313C2×C5#167511yescharacteristic
314C2×C5#168311yescharacteristic
315C2×C5#169111yescharacteristic
316C2×C5#169711yescharacteristic
317C2×C5#203611yescharacteristic
318C2×C5#205111yescharacteristic
319C2×C5#205511yescharacteristic
320D5—82no—
321C23—102no—
322C23—201no—
323C23—201no—
324C23—201no—
325C23—201no—
326C23—201no—
327C23—201no—
328C23#408411yescharacteristic
329Q8—82no—
330Q8—82no—
331D4—102no—
332D4—102no—
333D4—201no—
334D4—201no—
335C2×C4—21no—
336C2×C4—21no—
337C2×C4—42no—
338C2×C4—81no—
339C2×C4—81no—
340C2×C4—81no—
341C2×C4—81no—
342C2×C4—81no—
343C2×C4—201no—
344C2×C4—201no—
345C2×C4—201no—
346C2×C4—401no—
347C2×C4—401no—
348C2×C4—401no—
349C2×C4—401no—
350C2×C4#402811yescharacteristic
351C2×C4#402911yescharacteristic
352C2×C4#403111yescharacteristic
353C2×C4#403211yescharacteristic
354C2×C4#406011yescharacteristic
355C2×C4#406111yescharacteristic
356C2×C4D5×Q1611yescharacteristic
357C2×C4#406511yescharacteristic
358C2×C4#407611yescharacteristic
359C2×C4#408511yescharacteristic
360C2×C4C2×#68811yescharacteristic
361C2×C4#408811yescharacteristic
362C2×C4#409511yescharacteristic
363C2×C4#409611yescharacteristic
364C8—22no—
365C8—41no—
366C8—202no—
367C8—401no—
368C8—401no—
369C5#1048111yescharacteristic, complemented, Sylow
370C22—202no—
371C22—202no—
372C22—202no—
373C22—401no—
374C22—401no—
375C22—401no—
376C22—401no—
377C22#6525011yescharacteristic
378C22#6534611yescharacteristic
379C22#6537211yescharacteristic
380C22#6544911yescharacteristic
381C22#6546311yescharacteristic
382C22#6546611yescharacteristic
383C22#6547711yescharacteristic
384C4—21no—
385C4—21no—
386C4—82no—
387C4—161no—
388C4—401no—
389C4#6509711yescharacteristic
390C4#6509811yescharacteristic
391C4#6510011yescharacteristic
392C4#6510311yescharacteristic
393C4#6531611yescharacteristic
394C4#6531711yescharacteristic
395C4#6533711yescharacteristic
396C4#6533811yescharacteristic
397C2—402no—
398C2#1060047311yescharacteristic
399C2#1060098911yescharacteristic
400C2#1060103711yescharacteristic
401C2#1060199511yescharacteristic
402C2#1060202611yescharacteristic
403C2#1060213811yescharacteristic
404C2#1060216511yescharacteristic
subgroupG/H
centerC23#4084
commutator, cosocleC2×C42×C5C23
FrattiniC2×C42C22×D5
FittingC5×#1990C2
socleC23×C5#121
prime radicalC22×D20D4
hypercenter#1990D5
hypocenterC5#10481
LCSG ▷ C2×C42×C5 ▷ C22×C5 ▷ C5
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1990

Complements

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

NH ≅ G/Ntype
2. C5×#1990C2semidirect
3. C2×#64766C2semidirect
5. C2×#64678C2semidirect
7. #10598125C2semidirect
66. #1990D5semidirect
369. C5#10481semidirect

Maximal subgroups

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

subgroupindex
C2×#653732
C5×#19902
C2×#647662
#106005802
C2×#646782
#106003772
#105981252
#104815