Finite Group Databasebeta

← #11988122

Subgroups — #11988122

Order 1280.

504 conjugacy classes of subgroups; only the 91 normal subgroups are shown.

normal subgroups only

This group has 504 conjugacy classes of subgroups — too many to draw. Only its 91 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×#2732C211yesmaximal, characteristic
2#10600484C211yesmaximal, characteristic, complemented
3#10600431C211yesmaximal, characteristic
4#10600397C211yesmaximal, characteristic, complemented
5#10598133C211yesmaximal, characteristic, complemented
6#10598130C211yesmaximal, characteristic
7#10598129C211yesmaximal, characteristic, complemented
8C2×C5×C22⋊Q8—21no—
9C22×#689—21no—
10C2×#4015C2211yescharacteristic
11C5×#399—21no—
12C5×#398—21no—
13C5×#396—21no—
14C5×#391C2211yescharacteristic
15C5×#390C2211yescharacteristic
16C5×#389C2211yescharacteristic
17C5×#386—21no—
18C5×#385—21no—
19C2×#3964—21no—
20C2×#3942—21no—
21C2×#3939—21no—
22#64749—21no—
23#64749—21no—
24#64744C2211yescharacteristic
25#64744C2211yescharacteristic
26#64744C2211yescharacteristic
27#7756—51nomaximal, Sylow
28C22×C5×Q8D411yescharacteristic
29C22×C5×D4D411yescharacteristic
30C23×C4×C5—21no—
31C22×Dic10—21no—
32C5×C22⋊Q8—42no—
33C5×C22⋊Q8—42no—
34C2×C5×C4⋊C4—21no—
35C2×C5×C4⋊C4—21no—
36C2×C5×C4⋊C4—21no—
37C2×C5×C4⋊C4—21no—
38C2×C5×C4⋊C4D411yescharacteristic
39C2×C5×C4⋊C4D411yescharacteristic
40C2×C5×C4⋊C4D411yescharacteristic
41C2×C5×C4⋊C4D411yescharacteristic
42C2×C5×C22⋊C4—21no—
43C2×C5×C22⋊C4—21no—
44C2×C5×C22⋊C4—21no—
45C2×C5×C22⋊C4—21no—
46C2×C5×C22⋊C4—21no—
47C2×C42×C5C2311yescharacteristic
48C2×#689—42no—
49C2×#689—42no—
50C2×#689—42no—
51C2×C4⋊Dic5—21no—
52C2×C4⋊Dic5—21no—
53C22×#168—21no—
54C22×#168—21no—
55C22×#168—21no—
56#4015—42no—
57C5×#95—21no—
58C5×#95—21no—
59C5×#95—21no—
60C5×#95—21no—
61#3964—42no—
62#3942—42no—
63#3939—42no—
64#2732D511yescharacteristic
65#2057—51no—
66#2023—51no—
67#2016—51no—
68#1272—51no—
69#1271—51no—
70#1270—51no—
71C24×C5—21no—
72C2×C5×Q8—21no—
73C2×C5×Q8—41no—
74C2×C5×Q8—41no—
75C2×C5×Q8D812yes—
76C2×C5×D4—21no—
77C2×C5×D4—41no—
78C2×C5×D4—41no—
79C2×C5×D4Q1612yescomplemented
80C22×C4×C5—21no—
81C22×C4×C5—21no—
82C22×C4×C5—21no—
83C22×C4×C5—21no—
84C22×C4×C5—21no—
85C22×C4×C5—42no—
86C22×C4×C5—81no—
87C22×C4×C5—81no—
88C22×C4×C5C2×D411yescharacteristic
89C22×C4×C5C2×D411yescharacteristic
90C22×C4×C5C2×D411yescharacteristic
91C22×Dic5—41no—
92C2×Dic10—22no—
93C2×Dic10—41no—
94C2×Dic10—41no—
95C2×Dic10—41no—
96C2×Dic10—41no—
97C2×Dic10—41no—
98C5×C4⋊C4—21no—
99C5×C4⋊C4—21no—
100C5×C4⋊C4—41no—
101C5×C4⋊C4—41no—
102C5×C4⋊C4—41no—
103C5×C4⋊C4—42no—
104C5×C4⋊C4—42no—
105C5×C4⋊C4—41no—
106C5×C4⋊C4—41no—
107C5×C4⋊C4—81no—
108C5×C22⋊C4—42no—
109C5×C22⋊C4—41no—
110C5×C22⋊C4—41no—
111C5×C22⋊C4—41no—
112C5×C22⋊C4—41no—
113C5×C22⋊C4—81no—
114C5×C22⋊C4—81no—
115C42×C5—41no—
116#689—48no—
117C4⋊Dic5—22no—
118C4⋊Dic5—22no—
119C4⋊Dic5—41no—
120C4⋊Dic5—41no—
121C2×#168—22no—
122C2×#168—22no—
123C2×#168—41no—
124C2×#168—41no—
125C2×#168—41no—
126C2×#168—41no—
127C2×#168—41no—
128C2×#168—41no—
129C2×#168—41no—
130C22×Q16—101no—
131C2×C4⋊Q8—51no—
132C2×C22⋊Q8—21no—
133C2×#107—101no—
134C2×#103—101no—
135C2×#102—101no—
136#399—21no—
137#398—21no—
138#396—21no—
139#391C2×D511yescharacteristic
140#390C2×D511yescharacteristic
141#389C2×D511yescharacteristic
142#386—21no—
143#385—21no—
144#340—101no—
145#340—101no—
146#336—51no—
147#336—51no—
148#336—51no—
149C23×C5—22no—
150C23×C5—41no—
151C23×C5—41no—
152C23×C5—41no—
153C23×C5—41no—
154C23×C5—41no—
155C23×C5—41no—
156C23×C5#12011yescharacteristic
157C5×Q8—42no—
158C5×Q8—81no—
159C5×D4—42no—
160C5×D4—81no—
161C2×C4×C5—21no—
162C2×C4×C5—21no—
163C2×C4×C5—21no—
164C2×C4×C5—21no—
165C2×C4×C5—21no—
166C2×C4×C5—21no—
167C2×C4×C5—22no—
168C2×C4×C5—41no—
169C2×C4×C5—41no—
170C2×C4×C5—41no—
171C2×C4×C5—41no—
172C2×C4×C5—41no—
173C2×C4×C5—41no—
174C2×C4×C5—41no—
175C2×C4×C5—41no—
176C2×C4×C5—41no—
177C2×C4×C5—41no—
178C2×C4×C5—41no—
179C2×C4×C5—41no—
180C2×C4×C5—41no—
181C2×C4×C5—41no—
182C2×C4×C5—84no—
183C2×C4×C5C2×D811yescharacteristic
184C2×C4×C5C2×Q1611yescharacteristic
185C2×C4×C5#13511yescharacteristic
186C2×C4×C5AGL(1,ℤ/8)11yescharacteristic
187C2×C4×C5#13711yescharacteristic
188C2×C4×C5#13711yescharacteristic
189C2×Dic5—42no—
190C2×Dic5—81no—
191C2×Dic5—81no—
192Dic10—42no—
193Dic10—42no—
194Dic10—81no—
195Dic10—81no—
196#168—42no—
197#168—42no—
198#168—81no—
199C22×Q8—101no—
200C22×Q8C22⋊D511yescharacteristic
201C22×D4C22⋊D511yescharacteristic
202C23×C4—21no—
203C2×Q16—202no—
204C2×Q16—202no—
205C2×Q16—202no—
206C22×C8—101no—
207C22×C8—101no—
208C22×C8—101no—
209C4⋊Q8—202no—
210C22⋊Q8—42no—
211C22⋊Q8—42no—
212C2×C4⋊C4—21no—
213C2×C4⋊C4—21no—
214C2×C4⋊C4—21no—
215C2×C4⋊C4—21no—
216C2×C4⋊C4—101no—
217C2×C4⋊C4—101no—
218C2×C4⋊C4C22⋊D511yescharacteristic
219C2×C4⋊C4C22⋊D511yescharacteristic
220C2×C4⋊C4C22⋊D511yescharacteristic
221C2×C4⋊C4C22⋊D511yescharacteristic
222C2×C22⋊C4—21no—
223C2×C22⋊C4—21no—
224C2×C22⋊C4—21no—
225C2×C22⋊C4—21no—
226C2×C22⋊C4—21no—
227C2×C42C22×D511yescharacteristic
228#107—202no—
229#103—202no—
230#102—202no—
231#95—21no—
232#95—21no—
233#95—21no—
234#95—21no—
235C22×C5—42no—
236C22×C5—42no—
237C22×C5—42no—
238C22×C5—81no—
239C22×C5—81no—
240C22×C5—81no—
241C22×C5—81no—
242C22×C5#44711yescharacteristic
243C22×C5#45111yescharacteristic
244C22×C5#45211yescharacteristic
245C22×C5#45311yescharacteristic
246C22×C5#45511yescharacteristic
247C22×C5#45611yescharacteristic
248C22×C5B(4,𝔽2)11yescharacteristic
249C4×C5—21no—
250C4×C5—21no—
251C4×C5—21no—
252C4×C5—21no—
253C4×C5—21no—
254C4×C5—21no—
255C4×C5—42no—
256C4×C5—81no—
257C4×C5—81no—
258C4×C5—81no—
259C4×C5—81no—
260Dic5—82no—
261C24—21no—
262C2×Q8—21no—
263C2×Q8—41no—
264C2×Q8—41no—
265C2×Q8—102no—
266C2×Q8—201no—
267C2×Q8—201no—
268C2×Q8—201no—
269C2×Q8—201no—
270C2×Q8—201no—
271C2×Q8#68612yes—
272C2×D4—21no—
273C2×D4—41no—
274C2×D4—41no—
275C2×D4#68912yescomplemented
276C22×C4—21no—
277C22×C4—21no—
278C22×C4—21no—
279C22×C4—21no—
280C22×C4—21no—
281C22×C4—42no—
282C22×C4—81no—
283C22×C4—81no—
284C22×C4—201no—
285C22×C4C2×C22⋊D511yescharacteristic
286C22×C4C2×C22⋊D511yescharacteristic
287C22×C4C2×C22⋊D511yescharacteristic
288Q16—208no—
289C2×C8—102no—
290C2×C8—102no—
291C2×C8—201no—
292C2×C8—201no—
293C2×C8—201no—
294C2×C8—201no—
295C2×C8—201no—
296C2×C8—201no—
297C2×C8—201no—
298C4⋊C4—21no—
299C4⋊C4—21no—
300C4⋊C4—41no—
301C4⋊C4—41no—
302C4⋊C4—41no—
303C4⋊C4—42no—
304C4⋊C4—42no—
305C4⋊C4—41no—
306C4⋊C4—41no—
307C4⋊C4—81no—
308C4⋊C4—102no—
309C4⋊C4—102no—
310C4⋊C4—201no—
311C4⋊C4—201no—
312C22⋊C4—42no—
313C22⋊C4—41no—
314C22⋊C4—41no—
315C22⋊C4—41no—
316C22⋊C4—41no—
317C22⋊C4—81no—
318C22⋊C4—81no—
319C42—41no—
320C2×C5—82no—
321C2×C5#159511yescharacteristic
322C2×C5#160211yescharacteristic
323C2×C5#160311yescharacteristic
324C2×C5#163211yescharacteristic
325C2×C5#164111yescharacteristic
326C2×C5#164911yescharacteristic
327C2×C5#166111yescharacteristic
328C23—22no—
329C23—41no—
330C23—41no—
331C23—41no—
332C23—41no—
333C23—41no—
334C23—41no—
335C23#409911yescharacteristic
336Q8—42no—
337Q8—81no—
338Q8—202no—
339Q8—202no—
340Q8—401no—
341Q8—401no—
342D4—42no—
343D4—81no—
344C2×C4—21no—
345C2×C4—21no—
346C2×C4—21no—
347C2×C4—21no—
348C2×C4—21no—
349C2×C4—21no—
350C2×C4—22no—
351C2×C4—41no—
352C2×C4—41no—
353C2×C4—41no—
354C2×C4—41no—
355C2×C4—41no—
356C2×C4—41no—
357C2×C4—41no—
358C2×C4—41no—
359C2×C4—41no—
360C2×C4—41no—
361C2×C4—41no—
362C2×C4—41no—
363C2×C4—41no—
364C2×C4—41no—
365C2×C4—84no—
366C2×C4—202no—
367C2×C4—401no—
368C2×C4—401no—
369C2×C4C2×#68611yescharacteristic
370C2×C4#407811yescharacteristic
371C2×C4#408811yescharacteristic
372C2×C4C2×#68911yescharacteristic
373C2×C4#409611yescharacteristic
374C2×C4#409711yescharacteristic
375C8—202no—
376C8—202no—
377C8—401no—
378C5#775611yescharacteristic, complemented, Sylow
379C22—42no—
380C22—42no—
381C22—42no—
382C22—81no—
383C22—81no—
384C22—81no—
385C22—81no—
386C22#6550611yescharacteristic
387C22#6551711yescharacteristic
388C22#6552811yescharacteristic
389C22#6553011yescharacteristic
390C22#6553111yescharacteristic
391C22#6553311yescharacteristic
392C22#6553511yescharacteristic
393C4—21no—
394C4—21no—
395C4—21no—
396C4—21no—
397C4—21no—
398C4—21no—
399C4—42no—
400C4—81no—
401C4—81no—
402C4—81no—
403C4—81no—
404C4—402no—
405C2—82no—
406C2#1060263811yescharacteristic
407C2#1060268211yescharacteristic
408C2#1060268611yescharacteristic
409C2#1060270911yescharacteristic
410C2#1060271311yescharacteristic
411C2#1060271711yescharacteristic
412C2#1060274611yescharacteristic
subgroupG/H
centerC23#4099
commutator, cosocleC2×C42×C5C23
FrattiniC2×C42C22×D5
FittingC5×#2732C2
socleC23×C5#120
prime radicalC22×C5×D4D4
hypercenter#2732D5
hypocenterC5#7756
LCSG ▷ C2×C42×C5 ▷ C23×C5 ▷ C5
UCSC1 ◁ C23 ◁ C2×C42 ◁ #2732

Complements

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

NH ≅ G/Ntype
2. #10600484C2semidirect
4. #10600397C2semidirect
5. #10598133C2semidirect
7. #10598129C2semidirect
79. C2×C5×D4Q16semidirect
275. C2×D4#689semidirect
378. C5#7756semidirect

Maximal subgroups

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

subgroupindex
C5×#27322
#106004842
#106004312
#106003972
#105981332
#105981302
#105981292
#77565