Finite Group Databasebeta

← #14313174

Subgroups — #14313174

Order 1920.

528 conjugacy classes of subgroups; only the 96 normal subgroups are shown.

normal subgroups only

This group has 528 conjugacy classes of subgroups — too many to draw. Only its 96 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×#5656C211yesmaximal, characteristic
2#11749173C211yesmaximal, characteristic, complemented
3#11749126C211yesmaximal, characteristic, complemented
4#11746360C211yesmaximal, characteristic, complemented
5C3×#64759C211yesmaximal, characteristic, complemented
6#11745997C211yesmaximal, characteristic, complemented
7#11745996C211yesmaximal, characteristic, complemented
8#10602509—31nomaximal, Hall
9C2×C5×#237C2211yescharacteristic
10C22×C4×C5×D3C411yescharacteristic, complemented
11C3×C5×#123C2211yescharacteristic
12C5×#937C2211yescharacteristic
13C5×#929C411yescharacteristic, complemented
14C5×#896—21no—
15C5×#893—21no—
16C5×#884—21no—
17C5×#882C411yescharacteristic, complemented
18C5×#67⋊C4—21no—
19C5×#877C411yescharacteristic, complemented
20#91529—21no—
21D3×#681—21no—
22#91118C2211yescharacteristic
23#91117C2211yescharacteristic
24C3×#3966C2211yescharacteristic
25C3×#3965C2211yescharacteristic
26#90996—21no—
27#90962—21no—
28#69144—51nomaximal, Hall
29C5×#525—31no—
30#65529—31no—
31#65526—31no—
32#64759—31no—
33#64759D311yescharacteristic, complemented
34#64756—31no—
35#64756—31no—
36C23×C5×D3C2×C411yescharacteristic
37C2×C3×C5×D4C2311yescharacteristic
38C22×C3×C4×C5C2×C411yescharacteristic
39C2×C5×#67—21no—
40C2×C5×#67C2×C411yescharacteristic
41C2×C5×#67D411yescharacteristic
42C22×C5×Dic3C2×C411yescharacteristic
43C22×C5×Dic3D411yescharacteristic
44C5×#237—41no—
45C5×#237—41no—
46C2×C5×D12—21no—
47C2×C4×C5×D3—21no—
48C2×C4×C5×D3—21no—
49C2×C4×C5×D3—41no—
50C2×C4×C5×D3—41no—
51C2×C4×C5×D3D411yescharacteristic
52C2×C5×Dic6D411yescharacteristic
53C3×C5×C4⋊C4—21no—
54C3×C5×C22⋊C4—21no—
55C3×C5×C22⋊C4C2×C411yescharacteristic
56#6719—21no—
57#6710—21no—
58C5×#217C2×C411yescharacteristic
59C5×(C2×D3)⋊C4—21no—
60C5×(C2×D3)⋊C4—21no—
61C5×(C2×D3)⋊C4—21no—
62C5×Dic3⋊C4—21no—
63C5×Dic3⋊C4—21no—
64C4×C5×Dic3—21no—
65C3×C22⋊Dic5—21no—
66C3×#681—21no—
67Dic3⋊Dic5—41no—
68Dic3×Dic5—41no—
69#6650—41no—
70D3×#168—41no—
71#5656D511yescharacteristic
72#4745—51no—
73#4740—51no—
74C3×#352—51no—
75#4539—51no—
76#4475—51no—
77#4471—51no—
78C2×C5×#41—31no—
79C23×C4×C5—31no—
80C5×#123—31no—
81C5×#123C2×D311yescharacteristic
82C5×C22⋊Q8—61no—
83C5×#121—61no—
84C5×#120—31no—
85C5×#120—31no—
86C4×C5×D4—61no—
87C4×C5×D4—61no—
88C5×#117—31no—
89#4072—61no—
90C2×#681—61no—
91#3968—61no—
92#3966—31no—
93#3966—61no—
94#3966C2×D311yescharacteristic
95#3965—31no—
96#3965C2×D311yescharacteristic
97#2120—151noSylow
98C23×C3×C5C22×C411yescharacteristic
99C23×C3×C5C2×D411yescharacteristic
100C22×C5×D3—21no—
101C22×C5×D3—42no—
102C22×C5×D3—41no—
103C22×C5×D3C22⋊C411yescharacteristic
104C3×C5×D4—41no—
105C2×C3×C4×C5—21no—
106C2×C3×C4×C5—21no—
107C2×C3×C4×C5—41no—
108C2×C3×C4×C5C2×D411yescharacteristic
109C2×Dic15—41no—
110C5×#67—41no—
111C5×#67—41no—
112C5×#67—42no—
113C2×C5×Dic3—21no—
114C2×C5×Dic3—21no—
115C2×C5×Dic3—21no—
116C2×C5×Dic3—21no—
117C2×C5×Dic3—21no—
118C2×C5×Dic3C22⋊C411yescharacteristic
119C2×C5×Dic3C22⋊C411yescharacteristic
120C2×C5×Dic3C22⋊C411yescharacteristic
121C5×D12—42no—
122C4×C5×D3—21no—
123C4×C5×D3—21no—
124C4×C5×D3—42no—
125C4×C5×D3—41no—
126C4×C5×D3—41no—
127C5×Dic6—41no—
128C2×C3×Dic5—41no—
129C3⋊#168—41no—
130C3×#168—41no—
131C2×#237C2×D511yescharacteristic
132C22×C4×D3Dic511yescharacteristic, complemented
133C3×#123C2×D511yescharacteristic
134#937C2×D511yescharacteristic
135#929Dic511yescharacteristic, complemented
136D3×M16—101no—
137#896—21no—
138#893—21no—
139#884—21no—
140#882Dic511yescharacteristic, complemented
141#67⋊C4—21no—
142#877—101no—
143#877Dic511yescharacteristic, complemented
144C3×#100—51no—
145C3×#99—51no—
146#834—51no—
147#833—51no—
148#824—101no—
149#806—101no—
150C24×C5—31no—
151C5×#41—121no—
152C5×#41—121no—
153C2×C5×Q8—31no—
154C2×C5×D4—31no—
155C2×C5×D4—31no—
156C2×C5×D4—61no—
157C2×C5×D4—61no—
158C2×C5×D4C22×D311yescharacteristic
159C22×C4×C5—31no—
160C22×C4×C5—31no—
161C22×C4×C5—31no—
162C22×C4×C5—61no—
163C22×C4×C5—61no—
164C22×C4×C5—121no—
165C22×C4×C5—121no—
166C22×C4×C5C4×D311yescharacteristic
167C5×C4⋊C4—21no—
168C5×C4⋊C4—61no—
169C5×C4⋊C4—61no—
170C5×C22⋊C4—21no—
171C5×C22⋊C4—31no—
172C5×C22⋊C4—61no—
173C5×C22⋊C4—61no—
174C5×C22⋊C4—61no—
175C5×C22⋊C4C4×D311yescharacteristic
176C42×C5—61no—
177C22⋊Dic5—21no—
178C22⋊Dic5—61no—
179C4⋊Dic5—121no—
180C4×Dic5—121no—
181#681—21no—
182#681—61no—
183#681—121no—
184C2×#168—121no—
185#525—31no—
186#413—151no—
187#410—151no—
188#352—51no—
189#352—151no—
190AΣL(1,𝔽16)—151no—
191AΣL(1,𝔽16)—151no—
192C22×C3×C5—21no—
193C22×C3×C5—21no—
194C22×C3×C5—41no—
195C22×C3×C5—41no—
196C22×C3×C5C2×C22⋊C411yescharacteristic
197C2×C5×D3—22no—
198C2×C5×D3—21no—
199C2×C5×D3—41no—
200C2×C5×D3—42no—
201C2×C5×D3—41no—
202C2×C5×D3—41no—
203C2×C5×D3—42no—
204C3×C4×C5—21no—
205C3×C4×C5—41no—
206C3×C4×C5—41no—
207Dic15—81no—
208C3×Dic5—81no—
209C5×Dic3—21no—
210C5×Dic3—21no—
211C5×Dic3—41no—
212C5×Dic3#9911yescharacteristic
213C5×Dic3#9911yescharacteristic
214C23×D3C2×Dic511yescharacteristic
215C2×C3×D4C22×D511yescharacteristic
216C22×C3×C4C2×Dic511yescharacteristic
217C2×#67—21no—
218C2×#67C2×Dic511yescharacteristic
219C2×#67C22⋊D511yescharacteristic
220C22×Dic3C2×Dic511yescharacteristic
221C22×Dic3C22⋊D511yescharacteristic
222#237—41no—
223#237—41no—
224C2×D12—21no—
225C2×C4×D3—21no—
226C2×C4×D3—21no—
227C2×C4×D3—41no—
228C2×C4×D3—41no—
229C2×C4×D3C22⋊D511yescharacteristic
230C2×Dic6C22⋊D511yescharacteristic
231C3×M16—101no—
232C3×C4⋊C4—21no—
233C3×C22⋊C4—21no—
234C3×C22⋊C4—101no—
235C3×C22⋊C4C2×Dic511yescharacteristic
236#217—101no—
237#217C2×Dic511yescharacteristic
238(C2×D3)⋊C4—21no—
239(C2×D3)⋊C4—21no—
240(C2×D3)⋊C4—21no—
241Dic3⋊C4—21no—
242Dic3⋊C4—21no—
243Dic3⋊C4—201no—
244C4×Dic3—21no—
245C4×Dic3—201no—
246#208—101no—
247ΓL(1,𝔽25)—201no—
248D3×C8—201no—
249C23×C5—31no—
250C23×C5—61no—
251C23×C5—121no—
252C23×C5—122no—
253C23×C5C2×C4×D311yescharacteristic
254C23×C5D3×D411yescharacteristic
255C5×Q8—121no—
256C5×D4—41no—
257C5×D4—122no—
258C5×D4—121no—
259C5×D4—121no—
260C5×D4—122no—
261C2×C4×C5—21no—
262C2×C4×C5—21no—
263C2×C4×C5—31no—
264C2×C4×C5—31no—
265C2×C4×C5—31no—
266C2×C4×C5—41no—
267C2×C4×C5—61no—
268C2×C4×C5—61no—
269C2×C4×C5—61no—
270C2×C4×C5—61no—
271C2×C4×C5—61no—
272C2×C4×C5—61no—
273C2×C4×C5—61no—
274C2×C4×C5—121no—
275C2×C4×C5—121no—
276C2×C4×C5—122no—
277C2×C4×C5D3×D411yescharacteristic
278C2×Dic5—41no—
279C2×Dic5—121no—
280#168—41no—
281#168—121no—
282C2×#41—31no—
283C23×C4—31no—
284C2×M16—301no—
285#123—31no—
286#123D3×D511yescharacteristic
287C22⋊Q8—61no—
288#121—61no—
289#120—31no—
290#120—31no—
291C4×D4—61no—
292C4×D4—61no—
293#117—31no—
294#117—301no—
295#101—301no—
296#100—51no—
297#100—151no—
298#99—51no—
299#99—151no—
300#99—301no—
301C2×C3×C5—21no—
302C2×C3×C5—41no—
303C2×C3×C5—41no—
304C2×C3×C5C2×#9911yescharacteristic
305C5×D3—41no—
306C5×D3—42no—
307C23×C3C22×Dic511yescharacteristic
308C23×C3C2×C22⋊D511yescharacteristic
309C22×D3—21no—
310C22×D3—42no—
311C22×D3—41no—
312C22×D3C22⋊Dic511yescharacteristic
313C3×D4—41no—
314C2×C3×C4—21no—
315C2×C3×C4—21no—
316C2×C3×C4—41no—
317C2×C3×C4—201no—
318C2×C3×C4C2×C22⋊D511yescharacteristic
319#67—41no—
320#67—41no—
321#67—42no—
322C2×Dic3—21no—
323C2×Dic3—21no—
324C2×Dic3—21no—
325C2×Dic3—21no—
326C2×Dic3—21no—
327C2×Dic3—201no—
328C2×Dic3C22⋊Dic511yescharacteristic
329C2×Dic3C22⋊Dic511yescharacteristic
330C2×Dic3C22⋊Dic511yescharacteristic
331D12—42no—
332C4×D3—21no—
333C4×D3—21no—
334C4×D3—42no—
335C4×D3—41no—
336C4×D3—41no—
337Dic6—41no—
338C3×C8—201no—
339C3⋊C8—201no—
340C22×C5—21no—
341C22×C5—21no—
342C22×C5—41no—
343C22×C5—41no—
344C22×C5—62no—
345C22×C5—61no—
346C22×C5—121no—
347C22×C5—121no—
348C22×C5—122no—
349C22×C5—122no—
350C22×C5—121no—
351C22×C5D3×C22⋊C411yescharacteristic
352C4×C5—21no—
353C4×C5—31no—
354C4×C5—31no—
355C4×C5—41no—
356C4×C5—41no—
357C4×C5—61no—
358C4×C5—61no—
359C4×C5—121no—
360Dic5—81no—
361Dic5—241no—
362C24—31no—
363#41—121no—
364#41—121no—
365C2×Q8—31no—
366C2×D4—31no—
367C2×D4—31no—
368C2×D4—61no—
369C2×D4—61no—
370C2×D4C2×D3×D511yescharacteristic
371C22×C4—31no—
372C22×C4—31no—
373C22×C4—31no—
374C22×C4—61no—
375C22×C4—61no—
376C22×C4—121no—
377C22×C4—121no—
378C22×C4D3×Dic511yescharacteristic
379M16—101no—
380M16—301no—
381M16—601no—
382C2×C8—601no—
383C4⋊C4—21no—
384C4⋊C4—61no—
385C4⋊C4—61no—
386C4⋊C4—601no—
387C22⋊C4—21no—
388C22⋊C4—31no—
389C22⋊C4—61no—
390C22⋊C4—61no—
391C22⋊C4—61no—
392C22⋊C4—101no—
393C22⋊C4—301no—
394C22⋊C4D3×Dic511yescharacteristic
395C42—61no—
396C42—601no—
397C3×C5#212011yescharacteristic, complemented, Hall
398C22×C3—21no—
399C22×C3—21no—
400C22×C3—41no—
401C22×C3—41no—
402C22×C3C2×C22⋊Dic511yescharacteristic
403C2×D3—22no—
404C2×D3—21no—
405C2×D3—41no—
406C2×D3—42no—
407C2×D3—41no—
408C2×D3—41no—
409C2×D3—42no—
410C3×C4—21no—
411C3×C4—41no—
412C3×C4—41no—
413C3×C4—401no—
414Dic3—21no—
415Dic3—21no—
416Dic3—41no—
417Dic3—401no—
418Dic3#396611yescharacteristic
419Dic3#396611yescharacteristic
420C2×C5—21no—
421C2×C5—41no—
422C2×C5—41no—
423C2×C5—122no—
424C2×C5—121no—
425C2×C5D3×#9911yescharacteristic
426C23—31no—
427C23—61no—
428C23—121no—
429C23—122no—
430C23C2×D3×Dic511yescharacteristic
431C23D3×C22⋊D511yescharacteristic
432Q8—121no—
433D4—41no—
434D4—122no—
435D4—121no—
436D4—121no—
437D4—122no—
438C2×C4—21no—
439C2×C4—21no—
440C2×C4—31no—
441C2×C4—31no—
442C2×C4—31no—
443C2×C4—41no—
444C2×C4—61no—
445C2×C4—61no—
446C2×C4—61no—
447C2×C4—61no—
448C2×C4—61no—
449C2×C4—61no—
450C2×C4—61no—
451C2×C4—121no—
452C2×C4—121no—
453C2×C4—122no—
454C2×C4—201no—
455C2×C4—601no—
456C2×C4D3×C22⋊D511yescharacteristic
457C8—201no—
458C8—601no—
459C2×C3—21no—
460C2×C3—41no—
461C2×C3—41no—
462C2×C3C2×#396611yescharacteristic
463D3—42no—
464D3—41no—
465C5#6914411yescharacteristic, complemented, Sylow
466C22—21no—
467C22—21no—
468C22—41no—
469C22—41no—
470C22—62no—
471C22—61no—
472C22—121no—
473C22—121no—
474C22—122no—
475C22—122no—
476C22—121no—
477C22D3×C22⋊Dic511yescharacteristic
478C4—21no—
479C4—31no—
480C4—31no—
481C4—41no—
482C4—41no—
483C4—61no—
484C4—61no—
485C4—121no—
486C4—401no—
487C4—1201no—
488C3#1060250911yescharacteristic, complemented, Sylow
489C2—21no—
490C2—41no—
491C2—41no—
492C2—122no—
493C2—121no—
494C2D3×#396611yescharacteristic
subgroupG/H
centerC2D3×#3966
commutatorC23×C3×C5C22×C4
FrattiniC2×D4C2×D3×D5
FittingC3×C5×#123C22
socleC2×C3×C5C2×#99
prime radicalC5×#5656C2
cosocleC2×C3×C5×D4C23
hypercenter#123D3×D5
hypocenterC3×C5#2120
LCSG ▷ C23×C3×C5 ▷ C22×C3×C5 ▷ C2×C3×C5 ▷ C3×C5
UCSC1 ◁ C2 ◁ C22 ◁ C2×D4 ◁ #123

Complements

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

NH ≅ G/Ntype
2. #11749173C2semidirect
3. #11749126C2semidirect
4. #11746360C2semidirect
5. C3×#64759C2semidirect
6. #11745997C2semidirect
7. #11745996C2semidirect
10. C22×C4×C5×D3C4semidirect
13. C5×#929C4semidirect
17. C5×#882C4semidirect
19. C5×#877C4semidirect
33. #64759D3semidirect
132. C22×C4×D3Dic5semidirect
135. #929Dic5semidirect
140. #882Dic5semidirect
143. #877Dic5semidirect
397. C3×C5#2120semidirect
465. C5#69144semidirect
488. C3#10602509semidirect

Maximal subgroups

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

subgroupindex
C5×#56562
#117491732
#117491262
#117463602
C3×#647592
#117459972
#117459962
#106025093
#691445