Finite Group Databasebeta

← #14288011

Subgroups — #14288011

Order 1920.

512 conjugacy classes of subgroups; only the 82 normal subgroups are shown.

normal subgroups only

This group has 512 conjugacy classes of subgroups — too many to draw. Only its 82 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
1#11754468C211yesmaximal, characteristic
2#11751329C211yesmaximal, characteristic, complemented
3C5×#5228C211yesmaximal, characteristic, complemented
4C3×#65520C211yesmaximal, characteristic
5#11745964C211yesmaximal, characteristic, complemented
6#11745962C211yesmaximal, characteristic, complemented
7#11745948C211yesmaximal, characteristic
8#10601773—31nomaximal, Hall
9C22×C3×C5×Q8C2211yescharacteristic
10C2×#6718—21no—
11#91813C2211yescharacteristic
12C2×C5×#216—21no—
13C5×#924C2211yescharacteristic
14C2×C3×#685—21no—
15C2×C3×#685C2211yescharacteristic
16#91574—21no—
17D5×#217—21no—
18#91431—21no—
19#91430—21no—
20#91422—21no—
21#91421—21no—
22#91119—21no—
23#91114C2211yescharacteristic
24C5×#835—21no—
25C5×#830C2211yescharacteristic
26C3×#3963—21no—
27C3×#3963C2211yescharacteristic
28#69701—51nomaximal, Hall
29#65968—31no—
30C5×#451—31no—
31#65520D311yescharacteristic
32#65517—31no—
33#64758—31no—
34#64688—31no—
35#64687—31no—
36C23×C3×D5D411yescharacteristic
37C2×C3×C5×Q8—21no—
38C2×C3×C5×Q8—21no—
39C2×C3×C5×Q8—42no—
40C22×C3×C4×C5—21no—
41C22×C3×C4×C5C2311yescharacteristic
42C2×Dic30D411yescharacteristic
43C2×C5×Dic6D411yescharacteristic
44C22×C3×Dic5—21no—
45C22×C3×Dic5D411yescharacteristic
46C2×#1204—21no—
47C2×D5×Dic3—21no—
48C2×D5×Dic3—21no—
49#6719—21no—
50#6718—42no—
51C4⋊Dic15D411yescharacteristic
52#6712—21no—
53C2×C3⋊#168—21no—
54C5×#217—21no—
55C5×#216—42no—
56C5×C4⋊Dic3D411yescharacteristic
57C5×Dic3⋊C4—21no—
58C2×C5×C3⋊C8—21no—
59C3×#685—21no—
60C3×#685—21no—
61C3×#685—42no—
62Dic3⋊Dic5—21no—
63#6669—21no—
64#6668—21no—
65#6665—21no—
66#6665—21no—
67#5483—51no—
68C3×#393—51no—
69#5228—51no—
70#5228D511yescharacteristic, complemented
71#4525—51no—
72#4452—51no—
73#4452—51no—
74C22×C5×Q8C2×D311yescharacteristic
75C2×C5×Q16—61no—
76C5×C22⋊Q8—31no—
77#4092—61no—
78C2×#689—61no—
79C2×#685—21no—
80C2×#685C2×D311yescharacteristic
81#4070—31no—
82#4043—61no—
83#4042—61no—
84#4042—61no—
85D5×C22⋊C4—61no—
86#4024—61no—
87C5×#103—61no—
88C5×C22⋊C8—31no—
89#3967—61no—
90#3963—21no—
91#3963C2×D311yescharacteristic
92C22⋊#168—31no—
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—22no—
101C3×C5×Q8—22no—
102C3×C5×Q8—41no—
103C3×C5×Q8—41no—
104C2×C3×C4×C5—22no—
105C2×C3×C4×C5—41no—
106C2×C3×C4×C5—41no—
107C2×C3×C4×C5—41no—
108C2×C3×C4×C5C2×D411yescharacteristic
109C2×C3×C4×C5C2×D411yescharacteristic
110C2×Dic15—21no—
111C2×Dic15—21no—
112Dic30—41no—
113C2×C5×Dic3—21no—
114C2×C5×Dic3—21no—
115C5×Dic6—41no—
116C2×C3×Dic5—21no—
117C2×C3×Dic5—41no—
118C2×C3×Dic5—41no—
119C2×C3×Dic5—41no—
120C2×C3×Dic5—41no—
121#1204—81no—
122D5×Dic3—81no—
123D5×Dic3—81no—
124C3⋊#168—41no—
125C5×C3⋊C8—41no—
126C22×C3×Q8C2×D511yescharacteristic
127C2×C3×C22⋊C4—51no—
128C2×C3×C22⋊C4—101no—
129C2×#217—101no—
130C2×#216—21no—
131C2×#216—101no—
132#924—51no—
133#924—101no—
134#924C2×D511yescharacteristic
135#878—101no—
136#878—101no—
137#878—101no—
138#878—101no—
139C3×#95—51no—
140C3×#95—101no—
141#835—21no—
142#835—101no—
143#830—51no—
144#830C2×D511yescharacteristic
145C23×D5#6711yescharacteristic
146C2×C5×Q8—21no—
147C2×C5×Q8—21no—
148C2×C5×Q8—31no—
149C2×C5×Q8—42no—
150C22×C4×C5—21no—
151C22×C4×C5C22×D311yescharacteristic
152C22×Dic5—21no—
153C22×Dic5#6711yescharacteristic
154C2×C4×D5—61no—
155C2×C4×D5—61no—
156C2×Dic10—31no—
157C2×Dic10—61no—
158C5×Q16—122no—
159C2×C5×C8—61no—
160C5×C4⋊C4—31no—
161C5×C4⋊C4—61no—
162C5×C22⋊C4—61no—
163C22⋊Dic5—61no—
164#689—122no—
165#685—21no—
166#685—21no—
167#685—42no—
168#685—61no—
169#685—61no—
170C4⋊Dic5—31no—
171C4⋊Dic5—61no—
172#683—61no—
173#683—61no—
174#683—61no—
175C2×#168—61no—
176#543—151no—
177#451—31no—
178#451—151no—
179#393—51no—
180#328—151no—
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—42no—
196Dic15—41no—
197Dic15—41no—
198C3×Dic5—41no—
199C3×Dic5—81no—
200C5×Dic3—41no—
201C5×Dic3—41no—
202C24×C3—51no—
203C2×C3×Q8—21no—
204C2×C3×Q8—21no—
205C2×C3×Q8—42no—
206C22×C3×C4—21no—
207C22×C3×C4—51no—
208C22×C3×C4—101no—
209C22×C3×C4C22×D511yescharacteristic
210C22×Dic3—101no—
211C22×Dic3—101no—
212C2×Dic6—51no—
213C2×Dic6—101no—
214C2×Dic6C22⋊D511yescharacteristic
215C3×C22⋊C4—101no—
216C3×C22⋊C4—101no—
217C3×C22⋊C4—202no—
218#217—21no—
219#217—101no—
220#217—101no—
221#217—101no—
222#216—42no—
223#216—202no—
224C4⋊Dic3—51no—
225C4⋊Dic3—101no—
226C4⋊Dic3C22⋊D511yescharacteristic
227Dic3⋊C4—21no—
228Dic3⋊C4—101no—
229Dic3⋊C4—101no—
230Dic3⋊C4—101no—
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—22no—
240C5×Q8—22no—
241C5×Q8—41no—
242C5×Q8—41no—
243C5×Q8—121no—
244C2×C4×C5—22no—
245C2×C4×C5—41no—
246C2×C4×C5—41no—
247C2×C4×C5—41no—
248C2×C4×C5—61no—
249C2×C4×C5—61no—
250C2×C4×C5D3×D411yescharacteristic
251C2×C4×C5D3×D411yescharacteristic
252C2×Dic5—21no—
253C2×Dic5—41no—
254C2×Dic5—41no—
255C2×Dic5—41no—
256C2×Dic5—41no—
257C2×Dic5—61no—
258C2×Dic5—61no—
259C4×D5—241no—
260C4×D5—241no—
261Dic10—121no—
262Dic10—241no—
263C5×C8—121no—
264#168—121no—
265C22×Q8D3×D511yescharacteristic
266C2×Q16—61no—
267C2×Q16—301no—
268C22⋊Q8—31no—
269C22⋊Q8—151no—
270C22⋊Q8—301no—
271C22⋊Q8—301no—
272C22⋊Q8—301no—
273C22⋊Q8—301no—
274C22⋊Q8—301no—
275C2×C22⋊C4—51no—
276C2×C22⋊C4—101no—
277C2×C22⋊C4—301no—
278#103—61no—
279#103—301no—
280C22⋊C8—31no—
281C22⋊C8—151no—
282#95—51no—
283#95—101no—
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—22no—
297C3×Q8—22no—
298C3×Q8—41no—
299C3×Q8—41no—
300C2×C3×C4—22no—
301C2×C3×C4—41no—
302C2×C3×C4—41no—
303C2×C3×C4—41no—
304C2×C3×C4—101no—
305C2×C3×C4—201no—
306C2×C3×C4—201no—
307C2×C3×C4—201no—
308C2×C3×C4—201no—
309C2×C3×C4D4×D511yescharacteristic
310C2×C3×C4C2×C22⋊D511yescharacteristic
311C2×Dic3—21no—
312C2×Dic3—21no—
313C2×Dic3—101no—
314C2×Dic3—101no—
315C2×Dic3—401no—
316C2×Dic3—401no—
317Dic6—41no—
318Dic6—201no—
319Dic6—401no—
320C3⋊C8—41no—
321C3⋊C8—201no—
322C22×C5—21no—
323C22×C5—21no—
324C22×C5#93711yescharacteristic
325C22×C5C2×#21611yescharacteristic
326C22×C5#95111yescharacteristic
327C2×D5—22no—
328C2×D5—41no—
329C2×D5—41no—
330C2×D5—42no—
331C2×D5—81no—
332C4×C5—21no—
333C4×C5—21no—
334C4×C5—42no—
335C4×C5—121no—
336C4×C5—121no—
337Dic5—41no—
338Dic5—81no—
339Dic5—121no—
340Dic5—121no—
341C24—51no—
342C2×Q8—21no—
343C2×Q8—21no—
344C2×Q8—31no—
345C2×Q8—42no—
346C2×Q8—151no—
347C2×Q8—301no—
348C22×C4—21no—
349C22×C4—51no—
350C22×C4—101no—
351C22×C4—301no—
352C22×C4—301no—
353C22×C4C2×D3×D511yescharacteristic
354Q16—122no—
355Q16—602no—
356C2×C8—61no—
357C2×C8—301no—
358C4⋊C4—31no—
359C4⋊C4—61no—
360C4⋊C4—151no—
361C4⋊C4—301no—
362C4⋊C4—301no—
363C4⋊C4—301no—
364C4⋊C4—301no—
365C22⋊C4—61no—
366C22⋊C4—101no—
367C22⋊C4—101no—
368C22⋊C4—202no—
369C22⋊C4—301no—
370C22⋊C4—301no—
371C22⋊C4—301no—
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×C3D5×Q1611yescharacteristic
381C22×C3#406411yescharacteristic
382C22×C3#408311yescharacteristic
383C3×C4—21no—
384C3×C4—21no—
385C3×C4—42no—
386C3×C4—201no—
387C3×C4—401no—
388Dic3—41no—
389Dic3—41no—
390Dic3—201no—
391Dic3—201no—
392C2×C5—21no—
393C2×C5—21no—
394C2×C5#503211yescharacteristic
395C2×C5#505011yescharacteristic
396C2×C5#506111yescharacteristic
397D5—42no—
398C23—52no—
399C23—101no—
400C23—101no—
401C23—201no—
402C23—201no—
403C23D5×#6711yescharacteristic
404Q8—22no—
405Q8—22no—
406Q8—41no—
407Q8—41no—
408Q8—121no—
409Q8—601no—
410Q8—1201no—
411C2×C4—22no—
412C2×C4—41no—
413C2×C4—41no—
414C2×C4—41no—
415C2×C4—61no—
416C2×C4—61no—
417C2×C4—101no—
418C2×C4—201no—
419C2×C4—201no—
420C2×C4—201no—
421C2×C4—201no—
422C2×C4—301no—
423C2×C4—301no—
424C2×C4—1201no—
425C2×C4—1201no—
426C2×C4D3×C22⋊D511yescharacteristic
427C2×C4#679011yescharacteristic
428C8—121no—
429C8—601no—
430C2×C3—21no—
431C2×C3—21no—
432C2×C3—202no—
433C2×C3#6532111yescharacteristic
434C2×C3#6547511yescharacteristic
435C2×C3#6549111yescharacteristic
436C5#6970111yescharacteristic, complemented, Sylow
437C22—21no—
438C22—21no—
439C22—102no—
440C22—202no—
441C22—201no—
442C22—201no—
443C22—401no—
444C22D5×#21611yescharacteristic
445C22#9150811yescharacteristic
446C22#9157711yescharacteristic
447C4—21no—
448C4—21no—
449C4—42no—
450C4—121no—
451C4—121no—
452C4—201no—
453C4—401no—
454C4—601no—
455C4—601no—
456C3#1060177311yescharacteristic, complemented, Sylow
457C2—21no—
458C2—21no—
459C2—202no—
460C2#1174940011yescharacteristic
461C2#1174940911yescharacteristic
462C2#1174952511yescharacteristic
subgroupG/H
centerC22#91577
commutator, cosocleC22×C3×C4×C5C23
FrattiniC22×C4C2×D3×D5
FittingC22×C3×C5×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. #11751329C2semidirect
3. C5×#5228C2semidirect
5. #11745964C2semidirect
6. #11745962C2semidirect
70. #5228D5semidirect
99. C22×C3×D5Q16semidirect
238. C22×D5#216semidirect
372. C3×C5#1602semidirect
436. C5#69701semidirect
456. C3#10601773semidirect

Maximal subgroups

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

subgroupindex
#117544682
#117513292
C5×#52282
C3×#655202
#117459642
#117459622
#117459482
#106017733
#697015