Finite Group Databasebeta

← #14288019

Subgroups — #14288019

Order 1920.

512 conjugacy classes of subgroups; only the 88 normal subgroups are shown.

normal subgroups only

This group has 512 conjugacy classes of subgroups — too many to draw. Only its 88 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
1C3×#65968C211yesmaximal, characteristic
2#11751151C211yesmaximal, characteristic, complemented
3#11748793C211yesmaximal, characteristic
4#11748564C211yesmaximal, characteristic, complemented
5C5×#4538C211yesmaximal, characteristic, complemented
6#11745961C211yesmaximal, characteristic, complemented
7#11745948C211yesmaximal, characteristic
8#10599111—31nomaximal, Hall
9C22×Dic30C2211yescharacteristic
10C3×C5×C22⋊Q8C2211yescharacteristic
11C2×#6718—21no—
12C3×#4092—21no—
13C2×C3×#685C2211yescharacteristic
14C3×#4070C2211yescharacteristic
15C3×#4043—21no—
16C3×#4042—21no—
17C3×#4042—21no—
18C3×D5×C22⋊C4—21no—
19C3×#4024—21no—
20C2×#6665—21no—
21C2×#6662—21no—
22#91114C2211yescharacteristic
23#91094—21no—
24C5×C3⋊#95C2211yescharacteristic
25C5×#830C2211yescharacteristic
26#90994—21no—
27#90974—21no—
28#71893—51nomaximal, Hall
29#65968D311yescharacteristic
30#65339—31no—
31#65028—31no—
32#64962—31no—
33C5×#328—31no—
34#64688—31no—
35#64672—31no—
36C23×C3×D5D411yescharacteristic
37C2×C3×C5×Q8D411yescharacteristic
38C22×C3×C4×C5C2311yescharacteristic
39C22×Dic15—21no—
40C2×Dic30—21no—
41C2×Dic30—21no—
42C2×Dic30—42no—
43C22×C5×Dic3—21no—
44C22×C3×Dic5D411yescharacteristic
45C2×C3×C4×D5—21no—
46C2×C3×C4×D5—21no—
47C2×C3×Dic10—21no—
48C2×C3×Dic10D411yescharacteristic
49C3×C5×C4⋊C4—21no—
50C3×C5×C4⋊C4D411yescharacteristic
51C3×C5×C22⋊C4—21no—
52#6718—42no—
53C2×C3⋊#168—21no—
54C2×C5×C3⋊C8—21no—
55C3×C22⋊Dic5—21no—
56C3×#685—21no—
57C3×#685—21no—
58C3×#685—21no—
59C3×#685—21no—
60C3×C4⋊Dic5—21no—
61C3×C4⋊Dic5D411yescharacteristic
62C3×#683—21no—
63C3×#683—21no—
64C3×#683—21no—
65#6665—42no—
66#6662—42no—
67C3×#543—51no—
68#5050—51no—
69#5050—51no—
70#4947—51no—
71#4538—51no—
72#4538D511yescharacteristic, complemented
73#4525—51no—
74C22×Dic10—31no—
75C5×C22⋊Q8C2×D311yescharacteristic
76#4092—21no—
77C2×#689—61no—
78C2×#685—61no—
79C2×#685C2×D311yescharacteristic
80#4070C2×D311yescharacteristic
81C2×Dic20—61no—
82#4043—21no—
83#4042—21no—
84#4042—21no—
85D5×C22⋊C4—21no—
86#4024—21no—
87C5×C22⋊C8—31no—
88C5×#95—31no—
89#3963—61no—
90C22⋊#168—31no—
91#3948—61no—
92#3942—61no—
93#1602—151noSylow
94C23×C3×C5C2×D411yescharacteristic
95C22×C3×D5—21no—
96C22×C3×D5—21no—
97C22×C3×D5—41no—
98C22×C3×D5—41no—
99C22×C3×D5Q1612yescomplemented
100C3×C5×Q8—41no—
101C2×C3×C4×C5—21no—
102C2×C3×C4×C5—21no—
103C2×C3×C4×C5—41no—
104C2×C3×C4×C5C2×D411yescharacteristic
105C2×C3×C4×C5C2×D411yescharacteristic
106C2×Dic15—22no—
107C2×Dic15—41no—
108C2×Dic15—41no—
109Dic30—22no—
110Dic30—22no—
111Dic30—41no—
112Dic30—41no—
113C2×C5×Dic3—41no—
114C2×C5×Dic3—41no—
115C2×C5×Dic3—41no—
116C2×C3×Dic5—21no—
117C2×C3×Dic5—21no—
118C2×C3×Dic5—21no—
119C2×C3×Dic5—41no—
120C3×C4×D5—81no—
121C3×C4×D5—81no—
122C3×Dic10—41no—
123C3×Dic10—81no—
124C3⋊#168—41no—
125C5×C3⋊C8—41no—
126C22×Dic6—51no—
127C3×C22⋊Q8—51no—
128C3×C22⋊Q8—101no—
129C3×C22⋊Q8—101no—
130C3×C22⋊Q8—101no—
131C3×C22⋊Q8—101no—
132C3×C22⋊Q8—101no—
133C3×C22⋊Q8C2×D511yescharacteristic
134C2×C3×C22⋊C4—51no—
135C2×C3×C22⋊C4—101no—
136C2×#217—101no—
137C2×#216—101no—
138C2×#216—101no—
139C3⋊#95—101no—
140C3⋊#95C2×D511yescharacteristic
141#830—51no—
142#830C2×D511yescharacteristic
143#810—101no—
144#810—101no—
145C23×D5#6711yescharacteristic
146C2×C5×Q8#6711yescharacteristic
147C22×C4×C5—61no—
148C22×C4×C5C22×D311yescharacteristic
149C22×Dic5—61no—
150C22×Dic5#6711yescharacteristic
151C2×C4×D5—21no—
152C2×C4×D5—21no—
153C2×Dic10—21no—
154C2×Dic10—61no—
155C2×Dic10—61no—
156C2×Dic10—122no—
157C2×Dic10#6711yescharacteristic
158C2×C5×C8—61no—
159C5×C4⋊C4—21no—
160C5×C4⋊C4#6711yescharacteristic
161C5×C22⋊C4—21no—
162C22⋊Dic5—21no—
163#689—122no—
164#685—21no—
165#685—21no—
166#685—21no—
167#685—21no—
168#685—122no—
169C4⋊Dic5—21no—
170C4⋊Dic5#6711yescharacteristic
171#683—21no—
172#683—21no—
173#683—21no—
174C2×#168—61no—
175Dic20—122no—
176#543—51no—
177#451—151no—
178#451—151no—
179#393—151no—
180#328—31no—
181#328—151no—
182#323—151no—
183C22×C3×C5—21no—
184C22×C3×C5—21no—
185C22×C3×C5#12011yescharacteristic
186C22×C3×C5C2×Q1611yescharacteristic
187C22×C3×C5#13711yescharacteristic
188C2×C3×D5—22no—
189C2×C3×D5—41no—
190C2×C3×D5—41no—
191C2×C3×D5—42no—
192C2×C3×D5—81no—
193C3×C4×C5—21no—
194C3×C4×C5—21no—
195C3×C4×C5—41no—
196C3×C4×C5—41no—
197Dic15—42no—
198C3×Dic5—41no—
199C3×Dic5—41no—
200C3×Dic5—41no—
201C5×Dic3—81no—
202C24×C3—51no—
203C2×C3×Q8—51no—
204C2×C3×Q8—101no—
205C2×C3×Q8D2011yescharacteristic
206C22×C3×C4—51no—
207C22×C3×C4—101no—
208C22×C3×C4—101no—
209C22×C3×C4C22×D511yescharacteristic
210C22×Dic3—21no—
211C22×Dic3—101no—
212C2×Dic6—101no—
213C2×Dic6—101no—
214C2×Dic6—202no—
215C3×C4⋊C4—21no—
216C3×C4⋊C4—51no—
217C3×C4⋊C4—101no—
218C3×C4⋊C4—101no—
219C3×C4⋊C4—101no—
220C3×C4⋊C4—101no—
221C3×C4⋊C4D2011yescharacteristic
222C3×C22⋊C4—21no—
223C3×C22⋊C4—101no—
224C3×C22⋊C4—101no—
225C3×C22⋊C4—101no—
226C3×C22⋊C4—101no—
227C3×C22⋊C4—101no—
228#217—202no—
229#216—202no—
230#216—202no—
231C2×C3⋊C8—21no—
232C2×C3⋊C8—101no—
233C23×C5C2×#6711yescharacteristic
234C22×D5—21no—
235C22×D5—21no—
236C22×D5—41no—
237C22×D5—41no—
238C22×D5#21612yescomplemented
239C5×Q8—41no—
240C2×C4×C5—21no—
241C2×C4×C5—21no—
242C2×C4×C5—41no—
243C2×C4×C5—121no—
244C2×C4×C5—121no—
245C2×C4×C5—121no—
246C2×C4×C5C2×#6711yescharacteristic
247C2×C4×C5C2×#6711yescharacteristic
248C2×Dic5—21no—
249C2×Dic5—21no—
250C2×Dic5—21no—
251C2×Dic5—41no—
252C2×Dic5—62no—
253C2×Dic5—121no—
254C2×Dic5—121no—
255C4×D5—81no—
256C4×D5—81no—
257Dic10—41no—
258Dic10—62no—
259Dic10—62no—
260Dic10—81no—
261Dic10—121no—
262Dic10—121no—
263C5×C8—121no—
264#168—121no—
265C22×Q8—151no—
266C2×Q16—301no—
267C2×Q16—301no—
268C22⋊Q8—51no—
269C22⋊Q8—101no—
270C22⋊Q8—101no—
271C22⋊Q8—101no—
272C22⋊Q8—101no—
273C22⋊Q8—101no—
274C22⋊Q8D3×D511yescharacteristic
275C2×C22⋊C4—51no—
276C2×C22⋊C4—101no—
277C2×C22⋊C4—301no—
278#103—301no—
279#103—301no—
280C22⋊C8—31no—
281C22⋊C8—151no—
282#95—31no—
283#95—301no—
284C2×C3×C5—21no—
285C2×C3×C5—21no—
286C2×C3×C5#45111yescharacteristic
287C2×C3×C5#45511yescharacteristic
288C2×C3×C5B(4,𝔽2)11yescharacteristic
289C3×D5—42no—
290C23×C3—52no—
291C23×C3—101no—
292C23×C3—101no—
293C23×C3—201no—
294C23×C3—201no—
295C23×C3D4×D511yescharacteristic
296C3×Q8—41no—
297C3×Q8—201no—
298C3×Q8—401no—
299C2×C3×C4—21no—
300C2×C3×C4—21no—
301C2×C3×C4—41no—
302C2×C3×C4—101no—
303C2×C3×C4—101no—
304C2×C3×C4—101no—
305C2×C3×C4—201no—
306C2×C3×C4—401no—
307C2×C3×C4—401no—
308C2×C3×C4C2×D2011yescharacteristic
309C2×C3×C4D4×D511yescharacteristic
310C2×Dic3—41no—
311C2×Dic3—41no—
312C2×Dic3—41no—
313C2×Dic3—102no—
314C2×Dic3—201no—
315C2×Dic3—201no—
316Dic6—102no—
317Dic6—102no—
318Dic6—201no—
319Dic6—201no—
320C3⋊C8—41no—
321C3⋊C8—201no—
322C22×C5—21no—
323C22×C5—21no—
324C22×C5#94211yescharacteristic
325C22×C5C2×#21611yescharacteristic
326C22×C5#95311yescharacteristic
327C2×D5—22no—
328C2×D5—41no—
329C2×D5—41no—
330C2×D5—42no—
331C2×D5—81no—
332C4×C5—21no—
333C4×C5—21no—
334C4×C5—41no—
335C4×C5—41no—
336C4×C5—241no—
337Dic5—41no—
338Dic5—41no—
339Dic5—41no—
340Dic5—122no—
341C24—51no—
342C2×Q8—51no—
343C2×Q8—101no—
344C2×Q8—301no—
345C2×Q8—301no—
346C2×Q8—602no—
347C2×Q8#120211yescharacteristic
348C22×C4—51no—
349C22×C4—61no—
350C22×C4—101no—
351C22×C4—101no—
352C22×C4—301no—
353C22×C4C2×D3×D511yescharacteristic
354Q16—602no—
355Q16—602no—
356C2×C8—61no—
357C2×C8—301no—
358C4⋊C4—21no—
359C4⋊C4—51no—
360C4⋊C4—101no—
361C4⋊C4—101no—
362C4⋊C4—101no—
363C4⋊C4—101no—
364C4⋊C4#120211yescharacteristic
365C22⋊C4—21no—
366C22⋊C4—101no—
367C22⋊C4—101no—
368C22⋊C4—101no—
369C22⋊C4—101no—
370C22⋊C4—101no—
371C22⋊C4—602no—
372C3×C5#160211yescharacteristic, complemented, Hall
373C22×C3—21no—
374C22×C3—21no—
375C22×C3—102no—
376C22×C3—202no—
377C22×C3—201no—
378C22×C3—201no—
379C22×C3—401no—
380C22×C3#402811yescharacteristic
381C22×C3#406111yescharacteristic
382C22×C3D5×Q1611yescharacteristic
383C3×C4—21no—
384C3×C4—21no—
385C3×C4—41no—
386C3×C4—41no—
387C3×C4—201no—
388C3×C4—201no—
389C3×C4—201no—
390Dic3—81no—
391Dic3—202no—
392C2×C5—21no—
393C2×C5—21no—
394C2×C5#522811yescharacteristic
395C2×C5#524211yescharacteristic
396C2×C5#524411yescharacteristic
397D5—42no—
398C23—52no—
399C23—101no—
400C23—101no—
401C23—201no—
402C23—201no—
403C23D5×#6711yescharacteristic
404Q8—41no—
405Q8—201no—
406Q8—302no—
407Q8—302no—
408Q8—401no—
409Q8—601no—
410Q8—601no—
411C2×C4—21no—
412C2×C4—21no—
413C2×C4—41no—
414C2×C4—101no—
415C2×C4—101no—
416C2×C4—101no—
417C2×C4—121no—
418C2×C4—121no—
419C2×C4—121no—
420C2×C4—201no—
421C2×C4—302no—
422C2×C4—401no—
423C2×C4—401no—
424C2×C4—601no—
425C2×C4—601no—
426C2×C4C2×#120211yescharacteristic
427C2×C4D5×#6711yescharacteristic
428C8—121no—
429C8—601no—
430C2×C3—21no—
431C2×C3—21no—
432C2×C3—202no—
433C2×C3#6503011yescharacteristic
434C2×C3#6509711yescharacteristic
435C2×C3#6511411yescharacteristic
436C5#7189311yescharacteristic, complemented, Sylow
437C22—21no—
438C22—21no—
439C22—102no—
440C22—202no—
441C22—201no—
442C22—201no—
443C22—401no—
444C22D5×#21611yescharacteristic
445C22#9151311yescharacteristic
446C22#9156411yescharacteristic
447C4—21no—
448C4—21no—
449C4—41no—
450C4—41no—
451C4—201no—
452C4—201no—
453C4—201no—
454C4—241no—
455C4—602no—
456C3#1059911111yescharacteristic, complemented, Sylow
457C2—21no—
458C2—21no—
459C2—202no—
460C2#1174917611yescharacteristic
461C2#1174929111yescharacteristic
462C2#1174946311yescharacteristic
subgroupG/H
centerC22#91564
commutator, cosocleC22×C3×C4×C5C23
FrattiniC22×C4C2×D3×D5
FittingC3×C5×C22⋊Q8C22
socleC22×C3×C5#120
prime radicalC23×C3×D5D4
hypercenterC22⋊Q8D3×D5
hypocenterC3×C5#1602
LCSG ▷ C22×C3×C4×C5 ▷ C22×C3×C5 ▷ C3×C5
UCSC1 ◁ C22 ◁ C22×C4 ◁ C22⋊Q8

Complements

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

NH ≅ G/Ntype
2. #11751151C2semidirect
4. #11748564C2semidirect
5. C5×#4538C2semidirect
6. #11745961C2semidirect
72. #4538D5semidirect
99. C22×C3×D5Q16semidirect
238. C22×D5#216semidirect
372. C3×C5#1602semidirect
436. C5#71893semidirect
456. C3#10599111semidirect

Maximal subgroups

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

subgroupindex
C3×#659682
#117511512
#117487932
#117485642
C5×#45382
#117459612
#117459482
#105991113
#718935