Finite Group Databasebeta

← #14281200

Subgroups — #14281200

Order 1920.

560 conjugacy classes of subgroups; only the 94 normal subgroups are shown.

normal subgroups only

This group has 560 conjugacy classes of subgroups — too many to draw. Only its 94 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×#91772C211yesmaximal, characteristic
2#11748509C211yesmaximal, characteristic, complemented
3#11748497C211yesmaximal, characteristic, complemented
4C5×#4525C211yesmaximal, characteristic, complemented
5C3×#64745C211yesmaximal, characteristic, complemented
6#11745953C211yesmaximal, characteristic, complemented
7#11745953C211yesmaximal, characteristic, complemented
8#10598915—31nomaximal, Hall
9C22×#1220C411yescharacteristic
10C22×C4×D15C2211yescharacteristic
11C2×C3×C5×C22⋊C4C2211yescharacteristic
12C2×#6719C411yescharacteristic
13C2×#6714C411yescharacteristic
14C2×C4⋊Dic15C411yescharacteristic
15C2×#6712C2211yescharacteristic
16#91772—21no—
17#91772—21no—
18#91772—21no—
19#91772—21no—
20C2×#6651—21no—
21C2×#6648—21no—
22C5×#830C2211yescharacteristic
23C5×#830C2211yescharacteristic
24C3×C22⋊#168C2211yescharacteristic
25C3×C22⋊#168C2211yescharacteristic
26#90957—21no—
27#90957—21no—
28#68122—51nomaximal, Hall
29C2×#4029—31no—
30#65015—31no—
31D5×C22⋊C8—31no—
32C5×#323—31no—
33#64745D311yescharacteristic, complemented
34#64689—31no—
35#64689—31no—
36C24×C3×C5C2×C411yescharacteristic
37C23×D15C2×C411yescharacteristic
38C22×C3×C4×C5C2×C411yescharacteristic
39C22×C3×C4×C5C2311yescharacteristic
40C2×#1220—21no—
41C2×#1220—21no—
42C2×#1220—42no—
43C22×Dic15C2×C411yescharacteristic
44C22×Dic15C2×C411yescharacteristic
45C22×Dic15C2×C411yescharacteristic
46C2×C4×D15—42no—
47C2×C4×D15D411yescharacteristic
48C2×C4×D15D411yescharacteristic
49C2×C4×D15D411yescharacteristic
50C2×C4×D15D411yescharacteristic
51C3×C5×C22⋊C4—21no—
52C3×C5×C22⋊C4—21no—
53#6719—21no—
54#6719—21no—
55#6714—21no—
56#6714—21no—
57C4⋊Dic15—21no—
58C4⋊Dic15—21no—
59#6712—21no—
60#6712—21no—
61C2×C5×C3⋊C8—21no—
62C2×C5×C3⋊C8—21no—
63C2×C3×#168—21no—
64C2×C3×#168—21no—
65#6651—42no—
66#6648—42no—
67C2×#883—51no—
68#4726—51no—
69D3×C22⋊C8—51no—
70C3×#323—51no—
71#4525D511yescharacteristic, complemented
72#4469—51no—
73#4469—51no—
74C22×C22⋊D5—31no—
75C22×C4×D5—31no—
76C2×C5×C22⋊C4C2×D311yescharacteristic
77C2×C22⋊Dic5—31no—
78C2×#685—31no—
79C2×C4⋊Dic5—31no—
80C2×#683—31no—
81C2×#676—61no—
82C2×C8×D5—61no—
83#4029—61no—
84#4029—61no—
85#4029—61no—
86#4029—61no—
87C5×C22⋊C8—31no—
88C5×C22⋊C8—31no—
89C22⋊#168C2×D311yescharacteristic
90C22⋊#168C2×D311yescharacteristic
91#3952—61no—
92#3952—61no—
93#1468—151noSylow
94C23×C3×C5—21no—
95C23×C3×C5—21no—
96C23×C3×C5—21no—
97C23×C3×C5—41no—
98C23×C3×C5—41no—
99C23×C3×C5C22×C411yescharacteristic
100C22×D15—21no—
101C22×D15—21no—
102C22×D15—22no—
103C22×D15—41no—
104C22×D15C22⋊C411yescharacteristic
105C22×D15C22⋊C411yescharacteristic
106C2×C3×C4×C5—21no—
107C2×C3×C4×C5—41no—
108C2×C3×C4×C5—41no—
109C2×C3×C4×C5C2×D411yescharacteristic
110C2×C3×C4×C5C2×D411yescharacteristic
111#1220—42no—
112#1220—42no—
113C2×Dic15—21no—
114C2×Dic15—21no—
115C2×Dic15—21no—
116C2×Dic15—21no—
117C2×Dic15—41no—
118C2×Dic15—41no—
119C2×Dic15C22⋊C411yescharacteristic
120C2×Dic15C22⋊C411yescharacteristic
121C4×D15—22no—
122C4×D15—22no—
123C4×D15—41no—
124C4×D15—41no—
125C3×#168—41no—
126C3×#168—41no—
127C5×C3⋊C8—41no—
128C5×C3⋊C8—41no—
129C22×#67—51no—
130C22×C4×D3—51no—
131C2×C3×C22⋊C4C2×D511yescharacteristic
132C2×#217—51no—
133C2×(C2×D3)⋊C4—51no—
134C2×C4⋊Dic3—51no—
135C2×Dic3⋊C4—51no—
136C2×ΓL(1,𝔽25)—101no—
137C2×D3×C8—101no—
138#883—101no—
139#883—101no—
140#883—101no—
141#883—101no—
142C3×C22⋊C8—51no—
143C3×C22⋊C8—51no—
144#830C2×D511yescharacteristic
145#830C2×D511yescharacteristic
146(C2×D3)⋊C8—101no—
147(C2×D3)⋊C8—101no—
148C24×C5C4×D311yescharacteristic
149C23×D5—31no—
150C22×C4×C5C4×D311yescharacteristic
151C22×C4×C5C22×D311yescharacteristic
152C2×C22⋊D5—61no—
153C2×C22⋊D5—61no—
154C2×C22⋊D5—122no—
155C22×Dic5—31no—
156C22×Dic5—31no—
157C22×Dic5—31no—
158C2×C4×D5—31no—
159C2×C4×D5—31no—
160C2×C4×D5—31no—
161C2×C4×D5—31no—
162C2×C4×D5—122no—
163C2×C5×C8—61no—
164C2×C5×C8—61no—
165C5×C22⋊C4—21no—
166C5×C22⋊C4—21no—
167C22⋊Dic5—61no—
168C22⋊Dic5—61no—
169#685—61no—
170#685—61no—
171C4⋊Dic5—61no—
172C4⋊Dic5—61no—
173#683—61no—
174#683—61no—
175C2×#168—21no—
176C2×#168—21no—
177#676—122no—
178C8×D5—122no—
179C2×#123—151no—
180#407—151no—
181C2×C22⋊C8—151no—
182#324—151no—
183#324—151no—
184#323—31no—
185#323—51no—
186C22×C3×C5—21no—
187C22×C3×C5—21no—
188C22×C3×C5—41no—
189C22×C3×C5—41no—
190C22×C3×C5—41no—
191C22×C3×C5—81no—
192C22×C3×C5—81no—
193C22×C3×C5C2×C22⋊C411yescharacteristic
194C22×C3×C5#13111yescharacteristic
195C22×C3×C5#13111yescharacteristic
196C2×D15—21no—
197C2×D15—21no—
198C2×D15—21no—
199C2×D15—21no—
200C2×D15—21no—
201C2×D15—42no—
202C2×D15—42no—
203C2×D15#9912yes—
204C3×C4×C5—21no—
205C3×C4×C5—21no—
206C3×C4×C5—41no—
207Dic15—21no—
208Dic15—21no—
209Dic15—41no—
210Dic15—41no—
211C24×C3C4×D511yescharacteristic
212C23×D3—51no—
213C22×C3×C4C4×D511yescharacteristic
214C22×C3×C4C22×D511yescharacteristic
215C2×#67—101no—
216C2×#67—101no—
217C2×#67—202no—
218C22×Dic3—51no—
219C22×Dic3—51no—
220C22×Dic3—51no—
221C2×C4×D3—51no—
222C2×C4×D3—51no—
223C2×C4×D3—51no—
224C2×C4×D3—51no—
225C2×C4×D3—202no—
226C2×C3×C8—101no—
227C2×C3×C8—101no—
228C3×C22⋊C4—21no—
229C3×C22⋊C4—21no—
230#217—101no—
231#217—101no—
232(C2×D3)⋊C4—101no—
233(C2×D3)⋊C4—101no—
234C4⋊Dic3—101no—
235C4⋊Dic3—101no—
236Dic3⋊C4—101no—
237Dic3⋊C4—101no—
238C2×C3⋊C8—21no—
239C2×C3⋊C8—21no—
240ΓL(1,𝔽25)—202no—
241D3×C8—202no—
242C23×C5—21no—
243C23×C5—21no—
244C23×C5—21no—
245C23×C5—41no—
246C23×C5—41no—
247C23×C5C2×C4×D311yescharacteristic
248C22×D5—31no—
249C22×D5—31no—
250C22×D5—62no—
251C22×D5—61no—
252C22×D5—61no—
253C22×D5—121no—
254C2×C4×C5—21no—
255C2×C4×C5—41no—
256C2×C4×C5—41no—
257C2×C4×C5D3×D411yescharacteristic
258C2×C4×C5D3×D411yescharacteristic
259C22⋊D5—122no—
260C22⋊D5—122no—
261C2×Dic5—31no—
262C2×Dic5—31no—
263C2×Dic5—61no—
264C2×Dic5—61no—
265C2×Dic5—61no—
266C2×Dic5—61no—
267C2×Dic5—121no—
268C2×Dic5—121no—
269C4×D5—62no—
270C4×D5—62no—
271C4×D5—121no—
272C4×D5—121no—
273C5×C8—121no—
274C5×C8—121no—
275#168—41no—
276#168—41no—
277C22×D4—151no—
278C23×C4—151no—
279C2×M16—301no—
280C22×C8—301no—
281#123—301no—
282#123—301no—
283#123—301no—
284#123—301no—
285C2×C4⋊C4—151no—
286C2×C4⋊C4—151no—
287C2×C22⋊C4—151no—
288C2×C22⋊C4—151no—
289C2×C22⋊C4D3×D511yescharacteristic
290C22⋊C8—31no—
291C22⋊C8—31no—
292C22⋊C8—51no—
293C22⋊C8—51no—
294C22⋊C8—301no—
295C22⋊C8—301no—
296C2×C3×C5—21no—
297C2×C3×C5—21no—
298C2×C3×C5—81no—
299C2×C3×C5#40811yescharacteristic
300C2×C3×C5C2×#9911yescharacteristic
301C2×C3×C5#41311yescharacteristic
302D15—22no—
303D15—41no—
304C23×C3—21no—
305C23×C3—21no—
306C23×C3—21no—
307C23×C3—41no—
308C23×C3—41no—
309C23×C3C2×C4×D511yescharacteristic
310C22×D3—51no—
311C22×D3—51no—
312C22×D3—102no—
313C22×D3—101no—
314C22×D3—101no—
315C22×D3—201no—
316C2×C3×C4—21no—
317C2×C3×C4—41no—
318C2×C3×C4—41no—
319C2×C3×C4D4×D511yescharacteristic
320C2×C3×C4D4×D511yescharacteristic
321#67—202no—
322#67—202no—
323C2×Dic3—51no—
324C2×Dic3—51no—
325C2×Dic3—101no—
326C2×Dic3—101no—
327C2×Dic3—101no—
328C2×Dic3—101no—
329C2×Dic3—201no—
330C2×Dic3—201no—
331C4×D3—102no—
332C4×D3—102no—
333C4×D3—201no—
334C4×D3—201no—
335C3×C8—201no—
336C3×C8—201no—
337C3⋊C8—41no—
338C3⋊C8—41no—
339C22×C5—21no—
340C22×C5—21no—
341C22×C5—41no—
342C22×C5—41no—
343C22×C5—41no—
344C22×C5—81no—
345C22×C5—81no—
346C22×C5D3×C22⋊C411yescharacteristic
347C22×C5#90111yescharacteristic
348C22×C5#90711yescharacteristic
349C2×D5—32no—
350C2×D5—61no—
351C2×D5—61no—
352C2×D5—61no—
353C2×D5—61no—
354C2×D5—61no—
355C2×D5—122no—
356C2×D5—122no—
357C4×C5—21no—
358C4×C5—21no—
359C4×C5—41no—
360Dic5—61no—
361Dic5—61no—
362Dic5—121no—
363Dic5—121no—
364C24—151no—
365C24#120011yescharacteristic
366C2×D4—301no—
367C2×D4—301no—
368C2×D4—602no—
369C22×C4—151no—
370C22×C4—151no—
371C22×C4—151no—
372C22×C4—151no—
373C22×C4—151no—
374C22×C4—151no—
375C22×C4—151no—
376C22×C4—602no—
377C22×C4#120011yescharacteristic
378C22×C4C2×D3×D511yescharacteristic
379M16—602no—
380C2×C8—61no—
381C2×C8—61no—
382C2×C8—101no—
383C2×C8—101no—
384C2×C8—602no—
385C4⋊C4—301no—
386C4⋊C4—301no—
387C4⋊C4—301no—
388C4⋊C4—301no—
389C22⋊C4—21no—
390C22⋊C4—21no—
391C22⋊C4—301no—
392C22⋊C4—301no—
393C22⋊C4—301no—
394C22⋊C4—301no—
395C3×C5#146811yescharacteristic, complemented, Hall
396C22×C3—21no—
397C22×C3—21no—
398C22×C3—41no—
399C22×C3—41no—
400C22×C3—41no—
401C22×C3—81no—
402C22×C3—81no—
403C22×C3D5×C22⋊C411yescharacteristic
404C22×C3#404711yescharacteristic
405C22×C3#405311yescharacteristic
406C2×D3—52no—
407C2×D3—101no—
408C2×D3—101no—
409C2×D3—101no—
410C2×D3—101no—
411C2×D3—101no—
412C2×D3—202no—
413C2×D3—202no—
414C3×C4—21no—
415C3×C4—21no—
416C3×C4—41no—
417Dic3—101no—
418Dic3—101no—
419Dic3—201no—
420Dic3—201no—
421C2×C5—21no—
422C2×C5—21no—
423C2×C5—81no—
424C2×C5#472711yescharacteristic
425C2×C5D3×#9911yescharacteristic
426C2×C5#474511yescharacteristic
427D5—62no—
428D5—121no—
429C23—21no—
430C23—21no—
431C23—21no—
432C23—41no—
433C23—41no—
434C23—151no—
435C23—151no—
436C23—302no—
437C23—301no—
438C23—301no—
439C23—601no—
440C23C2×#120011yescharacteristic
441D4—602no—
442D4—602no—
443C2×C4—21no—
444C2×C4—41no—
445C2×C4—41no—
446C2×C4—151no—
447C2×C4—151no—
448C2×C4—302no—
449C2×C4—302no—
450C2×C4—301no—
451C2×C4—301no—
452C2×C4—301no—
453C2×C4—301no—
454C2×C4—601no—
455C2×C4—601no—
456C2×C4—601no—
457C2×C4—601no—
458C2×C4#679011yescharacteristic
459C2×C4#679011yescharacteristic
460C8—121no—
461C8—121no—
462C8—201no—
463C8—201no—
464C2×C3—21no—
465C2×C3—21no—
466C2×C3—81no—
467C2×C3#6501611yescharacteristic
468C2×C3D5×#9911yescharacteristic
469C2×C3#6503411yescharacteristic
470D3—102no—
471D3—201no—
472C5#6812211yescharacteristic, complemented, Sylow
473C22—21no—
474C22—21no—
475C22—41no—
476C22—41no—
477C22—41no—
478C22—81no—
479C22—81no—
480C22—152no—
481C22—301no—
482C22—301no—
483C22—301no—
484C22—301no—
485C22—301no—
486C22—602no—
487C22—602no—
488C22#9129011yescharacteristic
489C22#9129111yescharacteristic
490C22#9156111yescharacteristic
491C4—21no—
492C4—21no—
493C4—41no—
494C4—301no—
495C4—301no—
496C4—601no—
497C4—601no—
498C3#1059891511yescharacteristic, complemented, Sylow
499C2—21no—
500C2—21no—
501C2—81no—
502C2—302no—
503C2—601no—
504C2#1174849811yescharacteristic
505C2#1174912811yescharacteristic
506C2#1174917111yescharacteristic
subgroupG/H
centerC22#91561
commutatorC23×C3×C5C22×C4
FrattiniC22×C4C2×D3×D5
FittingC2×C3×C5×C22⋊C4C22
socleC22×C3×C5C2×C22⋊C4
prime radicalC22×#1220C4
cosocleC22×C3×C4×C5C23
hypercenterC2×C22⋊C4D3×D5
hypocenterC3×C5#1468
LCSG ▷ C23×C3×C5 ▷ C2×C3×C5 ▷ C3×C5
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C22⋊C4

Complements

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

NH ≅ G/Ntype
2. #11748509C2semidirect
3. #11748497C2semidirect
4. C5×#4525C2semidirect
5. C3×#64745C2semidirect
6. #11745953C2semidirect
7. #11745953C2semidirect
33. #64745D3semidirect
71. #4525D5semidirect
395. C3×C5#1468semidirect
472. C5#68122semidirect
498. C3#10598915semidirect

Maximal subgroups

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

subgroupindex
C2×#917722
#117485092
#117484972
C5×#45252
C3×#647452
#117459532
#105989153
#681225