Finite Group Databasebeta

← #12073215

Subgroups — #12073215

Order 1280.

1002 conjugacy classes of subgroups; only the 187 normal subgroups are shown.

normal subgroups only

This group has 1002 conjugacy classes of subgroups — too many to draw. Only its 187 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×#65834C211yesmaximal, characteristic, complemented
2C5×#2618C211yesmaximal, characteristic
3#10614361C212yesmaximal
4C2×#64748C211yesmaximal, characteristic, complemented
5#10600506C214yesmaximal, complemented
6#10600497C214yesmaximal, complemented
7#10600303C212yesmaximal, complemented
8C2×C5×#123—21no—
9C2×C5×#121C2211yescharacteristic
10C22×C5×C4⋊C4C2211yescharacteristic
11C2×#4072—21no—
12C2×#4070—21no—
13C2×#4070C2212yes—
14#65834—24no—
15#65834—22no—
16#65834C2214yes—
17C2×#4036—22no—
18C2×#4034—22no—
19C5×#398C2212yes—
20C5×#396—22no—
21C5×#394C2212yes—
22C5×#386—21no—
23C5×#380C2211yescharacteristic
24#65507C2214yes—
25C2×C22⋊#168C2212yes—
26#65330—24no—
27#65329—22no—
28#65323—22no—
29#65260C2214yes—
30#64748C2214yes—
31#64672C2218yes—
32#18294—51nomaximal, Sylow
33C22×C5×D4—21no—
34C22×C5×D4C2311yescharacteristic
35C23×C4×C5D411yescharacteristic
36C23×C4×C5D411yescharacteristic
37C23×C4×C5C2311yescharacteristic
38C22×Dic10—21no—
39C22×Dic10D412yes—
40C5×#123—44no—
41C5×#121—22no—
42C5×#121—22no—
43C2×C5×C4⋊C4—22no—
44C2×C5×C4⋊C4—22no—
45C2×C5×C4⋊C4—22no—
46C2×C5×C4⋊C4—21no—
47C2×C5×C4⋊C4D412yes—
48C2×C5×C4⋊C4C2311yescharacteristic
49C2×C5×C4⋊C4C2312yes—
50C2×C5×C22⋊C4—21no—
51C2×C5×C22⋊C4—22no—
52C2×C5×C22⋊C4—21no—
53C2×C5×C22⋊C4—22no—
54C2×C22⋊Dic5—22no—
55C2×#687—24no—
56#4072—24no—
57#4072—24no—
58C2×C4⋊Dic5—21no—
59C2×C4⋊Dic5—21no—
60C2×C4⋊Dic5—21no—
61C2×C4⋊Dic5D412yes—
62#4070—24no—
63#4070—22no—
64#4070—22no—
65#4070—24no—
66#4070D414yes—
67#4070C2314yes—
68C2×#683—22no—
69C2×#683—22no—
70C2×C4×Dic5—22no—
71C22×#168—22no—
72#4036—28no—
73#4036—28no—
74#4034—28no—
75#4034—28no—
76C5×#95—22no—
77C5×#95C2312yes—
78#3964—24no—
79C22⋊#168—22no—
80C22⋊#168C2314yes—
81#3942—24no—
82#3940—24no—
83C2×#544—51no—
84#3187—52no—
85#2618D511yescharacteristic
86#1611—54no—
87#1599—54no—
88#1520—52no—
89C2×#324—51no—
90C24×C5—21no—
91C24×C5C2×D411yescharacteristic
92C2×C5×D4—22no—
93C2×C5×D4—41no—
94C2×C5×D4—41no—
95C2×C5×D4—42no—
96C2×C5×D4—42no—
97C2×C5×D4—81no—
98C22×C4×C5—21no—
99C22×C4×C5—21no—
100C22×C4×C5—22no—
101C22×C4×C5—21no—
102C22×C4×C5—21no—
103C22×C4×C5—22no—
104C22×C4×C5—21no—
105C22×C4×C5—81no—
106C22×C4×C5—81no—
107C22×C4×C5SD1614yescomplemented
108C22×C4×C5C2×D412yes—
109C22×C4×C5C2×D412yes—
110C22×C4×C5C2×D412yes—
111C22×C4×C5C2×D412yes—
112C22×C4×C5C2×D411yescharacteristic
113C22×C4×C5C2411yescharacteristic
114C22×Dic5—22no—
115C22×Dic5—22no—
116C2×Dic10—24no—
117C2×Dic10—22no—
118C2×Dic10—82no—
119C2×Dic10—82no—
120C2×Dic10C2×D414yes—
121C5×C4⋊C4—24no—
122C5×C4⋊C4—24no—
123C5×C4⋊C4—22no—
124C5×C4⋊C4—44no—
125C5×C4⋊C4—81no—
126C5×C22⋊C4—42no—
127C5×C22⋊C4—42no—
128C5×C22⋊C4—44no—
129C5×C22⋊C4—82no—
130C22⋊Dic5—24no—
131C22⋊Dic5—24no—
132#687—84no—
133C4⋊Dic5—24no—
134C4⋊Dic5—24no—
135C4⋊Dic5—24no—
136C4⋊Dic5—22no—
137C4⋊Dic5C2×D414yes—
138#683—28no—
139#683—24no—
140#683—24no—
141C4×Dic5—28no—
142C2×#168—24no—
143C2×#168—82no—
144#544—54no—
145#544—104no—
146#544—102no—
147C2×C4⋊Q8—102no—
148C2×#125—102no—
149C2×#123—21no—
150C2×C22⋊Q8—52no—
151C2×C22⋊Q8—101no—
152C2×#121C2×D511yescharacteristic
153C2×#117—101no—
154C22×C4⋊C4C2×D511yescharacteristic
155#483—54no—
156#481—102no—
157#465—104no—
158#450—54no—
159#448—102no—
160C2×C22⋊C8—52no—
161#398C2×D512yes—
162#396—22no—
163#394C2×D512yes—
164#386—21no—
165#380C2×D511yescharacteristic
166#328—58no—
167#324—54no—
168C23×C5—41no—
169C23×C5—41no—
170C23×C5—42no—
171C23×C5—42no—
172C23×C5—41no—
173C23×C5—41no—
174C23×C5—41no—
175C23×C5#12011yescharacteristic
176C23×C5#12011yescharacteristic
177C23×C5C2×SD1612yes—
178C23×C5#13712yes—
179C23×C5C22×D411yescharacteristic
180C5×D4—44no—
181C5×D4—82no—
182C5×D4—82no—
183C2×C4×C5—22no—
184C2×C4×C5—22no—
185C2×C4×C5—24no—
186C2×C4×C5—24no—
187C2×C4×C5—22no—
188C2×C4×C5—42no—
189C2×C4×C5—41no—
190C2×C4×C5—42no—
191C2×C4×C5—41no—
192C2×C4×C5—42no—
193C2×C4×C5—42no—
194C2×C4×C5—42no—
195C2×C4×C5—42no—
196C2×C4×C5—41no—
197C2×C4×C5—42no—
198C2×C4×C5—41no—
199C2×C4×C5—41no—
200C2×C4×C5—81no—
201C2×C4×C5#12012yes—
202C2×C4×C5C2×SD1614yes—
203C2×C4×C5C22×D412yes—
204C2×Dic5—24no—
205C2×Dic5—24no—
206C2×Dic5—44no—
207C2×Dic5—44no—
208Dic10—44no—
209Dic10—82no—
210Dic10—84no—
211#168—44no—
212C22×Q8—52no—
213C22×Q8—101no—
214C22×D4—21no—
215C22×D4C22×D511yescharacteristic
216C23×C4C22⋊D511yescharacteristic
217C23×C4C22⋊D511yescharacteristic
218C23×C4C22×D511yescharacteristic
219C2×SD16—104no—
220C22×C8—102no—
221C4⋊Q8—108no—
222C4⋊Q8—108no—
223#125—108no—
224#125—108no—
225#123—44no—
226C22⋊Q8—54no—
227C22⋊Q8—54no—
228C22⋊Q8—104no—
229C22⋊Q8—102no—
230C22⋊Q8—102no—
231C22⋊Q8—104no—
232#121—22no—
233#121—22no—
234#117—104no—
235#117—104no—
236C2×C4⋊C4—22no—
237C2×C4⋊C4—22no—
238C2×C4⋊C4—22no—
239C2×C4⋊C4—21no—
240C2×C4⋊C4—52no—
241C2×C4⋊C4—101no—
242C2×C4⋊C4—101no—
243C2×C4⋊C4—101no—
244C2×C4⋊C4—102no—
245C2×C4⋊C4—102no—
246C2×C4⋊C4C22⋊D512yes—
247C2×C4⋊C4C22×D511yescharacteristic
248C2×C4⋊C4C22×D512yes—
249C2×C22⋊C4—21no—
250C2×C22⋊C4—21no—
251C2×C22⋊C4—22no—
252C2×C22⋊C4—22no—
253C2×C22⋊C4—102no—
254C2×C42—102no—
255#106—104no—
256#103—104no—
257#102—104no—
258C22⋊C8—54no—
259C22⋊C8—102no—
260#95—22no—
261#95C22×D512yes—
262C22×C5—21no—
263C22×C5—21no—
264C22×C5—21no—
265C22×C5—21no—
266C22×C5—21no—
267C22×C5—21no—
268C22×C5—42no—
269C22×C5—42no—
270C22×C5—82no—
271C22×C5—82no—
272C22×C5—81no—
273C22×C5—81no—
274C22×C5—81no—
275C22×C5#44812yes—
276C22×C5#44812yes—
277C22×C5C2×#12011yescharacteristic
278C22×C5C22×SD1611yescharacteristic
279C22×C5C2×#13711yescharacteristic
280C4×C5—24no—
281C4×C5—22no—
282C4×C5—22no—
283C4×C5—42no—
284C4×C5—82no—
285C4×C5—81no—
286Dic5—44no—
287Dic5—44no—
288C24—21no—
289C24C2×C22⋊D511yescharacteristic
290C2×Q8—54no—
291C2×Q8—104no—
292C2×Q8—102no—
293C2×Q8—402no—
294C2×Q8—402no—
295C2×D4—22no—
296C2×D4—42no—
297C2×D4—42no—
298C2×D4—41no—
299C2×D4—41no—
300C2×D4—81no—
301C22×C4—22no—
302C22×C4—21no—
303C22×C4—21no—
304C22×C4—21no—
305C22×C4—21no—
306C22×C4—21no—
307C22×C4—22no—
308C22×C4—81no—
309C22×C4—81no—
310C22×C4—102no—
311C22×C4—102no—
312C22×C4#68714yescomplemented
313C22×C4D4×D512yes—
314C22×C4D4×D512yes—
315C22×C4C2×C22⋊D512yes—
316C22×C4C2×C22⋊D512yes—
317C22×C4C2×C22⋊D511yescharacteristic
318C22×C4C23×D511yescharacteristic
319SD16—404no—
320C2×C8—104no—
321C2×C8—402no—
322C4⋊C4—24no—
323C4⋊C4—24no—
324C4⋊C4—22no—
325C4⋊C4—44no—
326C4⋊C4—54no—
327C4⋊C4—81no—
328C4⋊C4—104no—
329C4⋊C4—104no—
330C4⋊C4—104no—
331C4⋊C4—108no—
332C4⋊C4—104no—
333C4⋊C4—102no—
334C4⋊C4—104no—
335C22⋊C4—42no—
336C22⋊C4—44no—
337C22⋊C4—42no—
338C22⋊C4—82no—
339C22⋊C4—104no—
340C22⋊C4—104no—
341C42—108no—
342C2×C5—21no—
343C2×C5—21no—
344C2×C5—22no—
345C2×C5—22no—
346C2×C5—161no—
347C2×C5C2×#44811yescharacteristic
348C2×C5#302011yescharacteristic
349C2×C5#302611yescharacteristic
350C23—41no—
351C23—41no—
352C23—42no—
353C23—42no—
354C23—41no—
355C23—41no—
356C23—41no—
357C23C2×#68712yes—
358C23#408311yescharacteristic
359C23#408311yescharacteristic
360C23#409712yes—
361C23C22×C22⋊D511yescharacteristic
362Q8—204no—
363Q8—402no—
364Q8—404no—
365D4—44no—
366D4—82no—
367D4—82no—
368C2×C4—24no—
369C2×C4—24no—
370C2×C4—22no—
371C2×C4—22no—
372C2×C4—22no—
373C2×C4—41no—
374C2×C4—42no—
375C2×C4—42no—
376C2×C4—41no—
377C2×C4—42no—
378C2×C4—42no—
379C2×C4—42no—
380C2×C4—42no—
381C2×C4—42no—
382C2×C4—41no—
383C2×C4—41no—
384C2×C4—41no—
385C2×C4—81no—
386C2×C4—104no—
387C2×C4—104no—
388C2×C4—204no—
389C2×C4—204no—
390C2×C4C2×#68714yes—
391C2×C4#408312yes—
392C2×C4C2×D4×D512yes—
393C8—204no—
394C5#1829411yescharacteristic, complemented, Sylow
395C22—21no—
396C22—21no—
397C22—21no—
398C22—21no—
399C22—21no—
400C22—21no—
401C22—42no—
402C22—42no—
403C22—82no—
404C22—82no—
405C22—81no—
406C22—81no—
407C22—81no—
408C22#6532912yes—
409C22#6532912yes—
410C22C22×#68711yescharacteristic
411C22C2×#408311yescharacteristic
412C22C2×#409711yescharacteristic
413C4—22no—
414C4—22no—
415C4—24no—
416C4—42no—
417C4—82no—
418C4—81no—
419C4—204no—
420C4—204no—
421C2—21no—
422C2—21no—
423C2—22no—
424C2—22no—
425C2—161no—
426C2C2×#6532911yescharacteristic
427C2#1061510611yescharacteristic
428C2#1061514611yescharacteristic
subgroupG/H
centerC22C2×#4083
commutator, cosocleC22×C4×C5C24
FrattiniC22×C4C23×D5
FittingC5×#2618C2
socleC22×C5C2×#120
prime radicalC2×C5×#121C22
hypercenter#2618D5
hypocenterC5#18294
LCSG ▷ C22×C4×C5 ▷ C22×C5 ▷ C5
UCSC1 ◁ C22 ◁ C23×C4 ◁ #2618

Complements

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

NH ≅ G/Ntype
1. C2×#65834C2semidirect
4. C2×#64748C2semidirect
5. #10600506C2semidirect
6. #10600497C2semidirect
7. #10600303C2semidirect
107. C22×C4×C5SD16semidirect
312. C22×C4#687semidirect
394. C5#18294semidirect

Maximal subgroups

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

subgroupindex
C2×#658342
C5×#26182
#106143612
C2×#647482
#106005062
#106004972
#106003032
#182945