Finite Group Databasebeta

← #12118551

Subgroups — #12118551

Order 1280.

886 conjugacy classes of subgroups.

The subgroup lattice has 484 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C4×#677C211yesmaximal, characteristic
2#10606055C211yesmaximal, characteristic
3#10606054C211yesmaximal, characteristic, complemented
4C5×#1834C211yesmaximal, characteristic, complemented
5C8×C4⋊Dic5C211yesmaximal, characteristic, complemented
6#10600381C211yesmaximal, characteristic, complemented
7#10600377C211yesmaximal, characteristic
8#10599702C212yesmaximal
9#10599645C212yesmaximal
10#10599611C212yesmaximal, complemented
11#10599574C212yesmaximal, complemented
12C22×#677C2211yescharacteristic
13C2×#4038—21no—
14C2×#4033C2212yes—
15C2×C4×D20C2211yescharacteristic
16C2×C4×Dic10C2211yescharacteristic
17C2×C4×C5×C8C2211yescharacteristic
18C5×#384C2211yescharacteristic
19C5×#384C2211yescharacteristic
20C2×#3953C2211yescharacteristic
21C2×#3949C2211yescharacteristic
22C2×#3948C2211yescharacteristic
23C2×C8×Dic5C2212yes—
24#65279—22no—
25#65277C2212yes—
26#65276—21no—
27#65265—21no—
28C4⋊C4×Dic5—21no—
29C4⋊C4×Dic5C2212yes—
30#65262C2212yes—
31#65261—22no—
32C2×#3942C2212yes—
33C2×#3941C2212yes—
34C4×C4⋊Dic5C2211yescharacteristic
35C2×C4⋊#168C2211yescharacteristic
36C4×#677C418yescomplemented
37C5×#340C2212yes—
38C5×#340C2212yes—
39#64770C2212yes—
40#64749C2212yes—
41#64749C2212yes—
42#22767—51nomaximal, Sylow
43C22×D20C2311yescharacteristic
44C22×C4×D5—21no—
45C22×Dic10C2311yescharacteristic
46C22×C5×C8C2311yescharacteristic
47C22×C5×C8C2311yescharacteristic
48C2×C5×C4⋊C4—21no—
49C2×C5×C4⋊C4—21no—
50C2×C5×C4⋊C4—21no—
51C2×C5×C4⋊C4—21no—
52C2×C5×C4⋊C4C2312yes—
53C2×C5×C4⋊C4C2312yes—
54C2×C42×C5C2311yescharacteristic
55C2×#685—21no—
56C2×#685—22no—
57C2×#685—22no—
58C2×C4⋊Dic5D411yescharacteristic
59C2×C4⋊Dic5D411yescharacteristic
60C2×C4⋊Dic5C2311yescharacteristic
61C2×C4⋊Dic5C2311yescharacteristic
62C2×#683—21no—
63C2×#683—22no—
64C2×#683—22no—
65C2×C4×Dic5—21no—
66C2×C4×Dic5—22no—
67C2×C4×Dic5—22no—
68C2×C4×Dic5C2312yes—
69C22×#168C2312yes—
70C2×#677—22no—
71C2×#677—22no—
72C2×#677C2×C414yes—
73#4038—44no—
74#4033—24no—
75#4033—42no—
76C4×D20—22no—
77C4×D20C2×C414yes—
78C4×Dic10—22no—
79C4×Dic10C2×C414yes—
80C4×C5×C8C2×C414yes—
81#3963—22no—
82#3963—22no—
83#3963D412yes—
84#3953C2×C414yes—
85#3949C2×C414yes—
86#3948C2×C414yes—
87C8×Dic5—42no—
88#3942—24no—
89#3941—42no—
90C4⋊#168—22no—
91C2×C4×SD16—51no—
92#2354—51no—
93Q8×C4⋊C4—51no—
94#1934—52no—
95#1920—52no—
96#1880—52no—
97#1834D511yescharacteristic, complemented
98#1803—51no—
99#1802—51no—
100C8×C4⋊C4—51no—
101C4×#103—52no—
102C23×D5—21no—
103C22×C4×C5—22no—
104C22×C4×C5—22no—
105C22×C4×C5C2×D411yescharacteristic
106C22×C4×C5C2×D411yescharacteristic
107C22×C4×C5C2411yescharacteristic
108C22×Dic5—21no—
109C22×Dic5—21no—
110C22×Dic5—21no—
111C22×Dic5C2×D412yes—
112C22×Dic5C2×D412yes—
113C2×D20—21no—
114C2×D20—21no—
115C2×D20—21no—
116C2×D20—21no—
117C2×D20—21no—
118C2×D20C22×C412yes—
119C2×C4×D5—24no—
120C2×C4×D5—42no—
121C2×C4×D5—42no—
122C2×Dic10—21no—
123C2×Dic10—21no—
124C2×Dic10—21no—
125C2×Dic10C22×C412yes—
126C2×Dic10#4112yes—
127C2×Dic10#4112yes—
128C2×C5×C8—21no—
129C2×C5×C8—21no—
130C2×C5×C8—21no—
131C2×C5×C8—21no—
132C2×C5×C8C22×C412yes—
133C2×C5×C8C22×C412yes—
134C5×C4⋊C4—22no—
135C5×C4⋊C4—22no—
136C5×C4⋊C4—22no—
137C5×C4⋊C4—22no—
138C5×C4⋊C4—42no—
139C5×C4⋊C4—42no—
140C5×C4⋊C4—81no—
141C5×C4⋊C4—81no—
142C42×C5C22×C412yes—
143C42×C5#4112yes—
144#685—24no—
145#685—44no—
146#685—82no—
147C4⋊Dic5—22no—
148C4⋊Dic5—21no—
149C4⋊Dic5—21no—
150C4⋊Dic5SD1614yescomplemented
151C4⋊Dic5C22×C412yes—
152C4⋊Dic5C22×C412yes—
153#683—24no—
154#683—44no—
155#683—82no—
156C4×Dic5—24no—
157C4×Dic5—22no—
158C4×Dic5—22no—
159C4×Dic5—44no—
160C4×Dic5—44no—
161C2×#168—22no—
162C2×#168—22no—
163C2×#168—22no—
164#677—28no—
165C22×SD16—51no—
166C2×C4×Q8—52no—
167C2×C4×Q8—51no—
168C2×C4×D4—51no—
169C2×#117—101no—
170C4×SD16—58no—
171C2×#106—51no—
172C2×C4⋊C8—51no—
173C2×#103—51no—
174C2×#103—52no—
175C2×#102—51no—
176C2×#102—52no—
177C2×C4×C8—52no—
178C2×C4×C8C2×D511yescharacteristic
179#390—52no—
180#389—52no—
181#389—102no—
182#388—102no—
183#384—101no—
184#384C2×D511yescharacteristic
185#384C2×D511yescharacteristic
186#380—101no—
187C4×C4⋊C4—51no—
188C4×C4⋊C4—52no—
189C4×C4⋊C4—101no—
190#340—52no—
191#340—52no—
192#340C2×D512yes—
193#340C2×D512yes—
194#336—52no—
195C23×C5C22×D411yescharacteristic
196C22×D5—22no—
197C22×D5—41no—
198C22×D5—41no—
199C22×D5—41no—
200C22×D5—41no—
201C22×D5—41no—
202C22×D5—41no—
203C2×C4×C5—21no—
204C2×C4×C5—21no—
205C2×C4×C5—42no—
206C2×C4×C5—42no—
207C2×C4×C5—42no—
208C2×C4×C5—42no—
209C2×C4×C5—42no—
210C2×C4×C5—42no—
211C2×C4×C5C4×D412yes—
212C2×C4×C5C4×D412yes—
213C2×C4×C5C2×SD1612yes—
214C2×C4×C5#13512yes—
215C2×C4×C5C23×C411yescharacteristic
216C2×C4×C5C2×#4111yescharacteristic
217C2×C4×C5C2×#4111yescharacteristic
218C2×C4×C5C2×#4111yescharacteristic
219C2×C4×C5H(5,𝔽2)11yescharacteristic
220C2×C4×C52−1+411yescharacteristic
221C2×Dic5—22no—
222C2×Dic5—22no—
223C2×Dic5—22no—
224C2×Dic5—22no—
225C2×Dic5—22no—
226C2×Dic5—22no—
227C2×Dic5—22no—
228C2×Dic5—22no—
229C2×Dic5—22no—
230C2×Dic5—22no—
231C2×Dic5—41no—
232C2×Dic5—41no—
233C2×Dic5—41no—
234C2×Dic5—41no—
235C2×Dic5C2×SD1614yes—
236D20—22no—
237D20—22no—
238D20—41no—
239D20—41no—
240C4×D5—48no—
241Dic10—22no—
242Dic10—22no—
243Dic10—22no—
244Dic10#11714yescomplemented
245C5×C8—22no—
246C5×C8—22no—
247#168—42no—
248C22×Q8—51no—
249C22×D4—51no—
250C23×C4—101no—
251C2×SD16—54no—
252C2×SD16—102no—
253C2×SD16—102no—
254C22×C8—52no—
255C22×C8C22×D511yescharacteristic
256C22×C8C22×D511yescharacteristic
257C4×Q8—54no—
258C4×Q8—102no—
259C4×Q8—104no—
260C4×Q8—202no—
261C4×D4—54no—
262C4×D4—102no—
263#117—204no—
264C2×C4⋊C4—21no—
265C2×C4⋊C4—21no—
266C2×C4⋊C4—21no—
267C2×C4⋊C4—21no—
268C2×C4⋊C4—51no—
269C2×C4⋊C4—51no—
270C2×C4⋊C4—51no—
271C2×C4⋊C4—51no—
272C2×C4⋊C4—101no—
273C2×C4⋊C4—102no—
274C2×C4⋊C4—102no—
275C2×C4⋊C4C22×D512yes—
276C2×C4⋊C4C22×D512yes—
277C2×C22⋊C4—101no—
278C2×C22⋊C4—102no—
279C2×C22⋊C4—102no—
280C2×C42—52no—
281C2×C42—101no—
282C2×C42—102no—
283C2×C42—102no—
284C2×C42C22×D511yescharacteristic
285#106—54no—
286C4⋊C8—102no—
287#103—54no—
288#103—104no—
289#102—54no—
290#102—202no—
291C4×C8—202no—
292C4×C8C4×D514yes—
293#95—52no—
294#95—102no—
295#95—102no—
296C22×C5C2×C4×D411yescharacteristic
297C22×C5#54611yescharacteristic
298C22×C5#54811yescharacteristic
299C22×C5C22×SD1611yescharacteristic
300C22×C5C2×#13511yescharacteristic
301C22×C5#57611yescharacteristic
302C22×C5#57811yescharacteristic
303C2×D5—42no—
304C2×D5—42no—
305C2×D5—42no—
306C2×D5—81no—
307C2×D5—81no—
308C2×D5—81no—
309C2×D5—81no—
310C4×C5—22no—
311C4×C5—82no—
312C4×C5—82no—
313C4×C5C4×SD1614yes—
314C4×C5C2×#11711yescharacteristic
315C4×C5#52011yescharacteristic
316C4×C5#55211yescharacteristic
317C4×C5#55311yescharacteristic
318Dic5—24no—
319Dic5—24no—
320Dic5—42no—
321Dic5—42no—
322Dic5—42no—
323C24—101no—
324C2×Q8—52no—
325C2×Q8—52no—
326C2×Q8—52no—
327C2×Q8—101no—
328C2×Q8—101no—
329C2×Q8—101no—
330C2×D4—52no—
331C2×D4—101no—
332C2×D4—101no—
333C2×D4—101no—
334C2×D4—101no—
335C2×D4—101no—
336C22×C4—22no—
337C22×C4—22no—
338C22×C4—52no—
339C22×C4—52no—
340C22×C4—101no—
341C22×C4—101no—
342C22×C4—101no—
343C22×C4—104no—
344C22×C4—202no—
345C22×C4—202no—
346C22×C4D4×D511yescharacteristic
347C22×C4D4×D511yescharacteristic
348C22×C4C23×D511yescharacteristic
349SD16—108no—
350C2×C8—21no—
351C2×C8—21no—
352C2×C8—21no—
353C2×C8—21no—
354C2×C8—102no—
355C2×C8—102no—
356C2×C8—102no—
357C2×C8C2×C4×D512yes—
358C2×C8C2×C4×D512yes—
359C4⋊C4—22no—
360C4⋊C4—22no—
361C4⋊C4—22no—
362C4⋊C4—22no—
363C4⋊C4—42no—
364C4⋊C4—42no—
365C4⋊C4—54no—
366C4⋊C4—52no—
367C4⋊C4—52no—
368C4⋊C4—81no—
369C4⋊C4—81no—
370C4⋊C4—101no—
371C4⋊C4—101no—
372C4⋊C4—104no—
373C4⋊C4—102no—
374C4⋊C4—204no—
375C4⋊C4—402no—
376C22⋊C4—104no—
377C22⋊C4—204no—
378C22⋊C4—402no—
379C42—102no—
380C42—102no—
381C42—104no—
382C42—204no—
383C42—204no—
384C42C2×C4×D512yes—
385C42#71312yes—
386C2×C5#235211yescharacteristic
387C2×C5C2×C4×SD1611yescharacteristic
388C2×C5#295111yescharacteristic
389C2×C5#329811yescharacteristic
390C2×C5#330511yescharacteristic
391C2×C5#336211yescharacteristic
392C2×C5#336411yescharacteristic
393D5—82no—
394C23—102no—
395C23—201no—
396C23—201no—
397C23—201no—
398C23—201no—
399C23—201no—
400C23—201no—
401C23C2×D4×D511yescharacteristic
402Q8—54no—
403Q8—102no—
404Q8—102no—
405Q8—102no—
406D4—102no—
407D4—102no—
408D4—201no—
409D4—201no—
410C2×C4—21no—
411C2×C4—21no—
412C2×C4—42no—
413C2×C4—42no—
414C2×C4—42no—
415C2×C4—42no—
416C2×C4—42no—
417C2×C4—42no—
418C2×C4—54no—
419C2×C4—102no—
420C2×C4—102no—
421C2×C4—102no—
422C2×C4—102no—
423C2×C4—102no—
424C2×C4—102no—
425C2×C4—102no—
426C2×C4—102no—
427C2×C4—102no—
428C2×C4—102no—
429C2×C4—201no—
430C2×C4—201no—
431C2×C4—201no—
432C2×C4—201no—
433C2×C4—208no—
434C2×C4#403912yes—
435C2×C4#403912yes—
436C2×C4D5×SD1612yes—
437C2×C4#406212yes—
438C2×C4C22×C4×D511yescharacteristic
439C2×C4C2×#71311yescharacteristic
440C2×C4D5×#4111yescharacteristic
441C2×C4D5×#4111yescharacteristic
442C2×C4#414911yescharacteristic
443C2×C4#415011yescharacteristic
444C8—22no—
445C8—22no—
446C8—202no—
447C5#2276711yescharacteristic, complemented, Sylow
448C22—202no—
449C22—202no—
450C22—202no—
451C22—401no—
452C22—401no—
453C22—401no—
454C22—401no—
455C22C2×#403911yescharacteristic
456C22#6601011yescharacteristic
457C22#6601111yescharacteristic
458C22#6609111yescharacteristic
459C22C2×D5×SD1611yescharacteristic
460C22C2×#406211yescharacteristic
461C22#6610011yescharacteristic
462C4—22no—
463C4—82no—
464C4—82no—
465C4—104no—
466C4—104no—
467C4—202no—
468C4—202no—
469C4—202no—
470C4#6514114yes—
471C4D5×#11711yescharacteristic
472C4#6585711yescharacteristic
473C4#6602811yescharacteristic
474C4#6602911yescharacteristic
475C2—402no—
476C2#1060596011yescharacteristic
477C2C2×#6514111yescharacteristic
478C2#1061086711yescharacteristic
479C2#1061331011yescharacteristic
480C2#1061331911yescharacteristic
481C2#1061334311yescharacteristic
482C2#1061335611yescharacteristic
subgroupG/H
centerC23C2×D4×D5
commutatorC2×C4×C5C23×C4
FrattiniC22×C4C23×D5
FittingC5×#1834C2
socleC23×C5C22×D4
prime radicalC22×D20C23
cosocleC22×C4×C5C24
hypercenter#1834D5
hypocenterC5#22767
LCSG ▷ C2×C4×C5 ▷ C2×C5 ▷ C5
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1834

Complements

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

NH ≅ G/Ntype
3. #10606054C2semidirect
4. C5×#1834C2semidirect
5. C8×C4⋊Dic5C2semidirect
6. #10600381C2semidirect
10. #10599611C2semidirect
11. #10599574C2semidirect
36. C4×#677C4semidirect
97. #1834D5semidirect
150. C4⋊Dic5SD16semidirect
244. Dic10#117semidirect
447. C5#22767semidirect

Maximal subgroups

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

subgroupindex
C2×C4×#6772
#106060552
#106060542
C5×#18342
C8×C4⋊Dic52
#106003812
#106003772
#105997022
#105996452
#105996112
#105995742
#227675