Finite Group Databasebeta

← #14460003

Subgroups — #14460003

Order 1920.

772 conjugacy classes of subgroups.

The subgroup lattice has 504 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#65889C211yesmaximal, characteristic
2#11752508C211yesmaximal, characteristic, complemented
3C4×#6716C211yesmaximal, characteristic, complemented
4C5×#5007C211yesmaximal, characteristic, complemented
5#11749049C212yesmaximal, complemented
6#11748961C212yesmaximal, complemented
7#11747445C212yesmaximal, complemented
8#11747204C212yesmaximal, complemented
9#11747104C211yesmaximal, characteristic, complemented
10#11746996C211yesmaximal, characteristic, complemented
11#11746924C211yesmaximal, characteristic, complemented
12#10612712—31nomaximal, Hall
13C2×C3×#711C2212yes—
14C3×C4×C5×D4C2211yescharacteristic
15C2×#6716C2211yescharacteristic
16C4×Dic30C2211yescharacteristic
17C5×#869C2211yescharacteristic
18C3×#4076—21no—
19C2×C3×C4⋊Dic5—21no—
20C3×#4043C2212yes—
21C3×#4032—22no—
22C3×C4×D20C2211yescharacteristic
23C3×#4015C2211yescharacteristic
24#91368—21no—
25#91361—22no—
26#91360C2212yes—
27#91358C2212yes—
28#91344—21no—
29C2×#6662C2212yes—
30C2×D5×C3⋊C8C2212yes—
31#91116C2211yescharacteristic
32#91094C2211yescharacteristic
33#91092C2211yescharacteristic
34C4×C3⋊#168C2211yescharacteristic
35C5×#832C2212yes—
36C5×#807C2212yes—
37C5×C4⋊(C3⋊C8)C2211yescharacteristic
38#90971C2212yes—
39#90970C2212yes—
40#90970C2212yes—
41#90955C2212yes—
42#83510—51nomaximal, Hall
43#66066—31no—
44#65889D311yescharacteristic
45C5×#474—31no—
46#65382—31no—
47C4×#687—31no—
48#65177—32no—
49#65176—32no—
50#65138—31no—
51#65123—31no—
52#65072—32no—
53#65061—32no—
54C2×C3×C5×D4C2311yescharacteristic
55C22×C3×C4×C5—21no—
56C2×Dic30C2311yescharacteristic
57C2×C5×Dic6—21no—
58C2×C3×C22⋊D5—22no—
59C22×C3×Dic5—22no—
60C3×#711—24no—
61C3×#711—42no—
62C2×C3×D20D411yescharacteristic
63C2×C3×C4×D5C2312yes—
64C2×C3×Dic10C2312yes—
65C3×C5×C4⋊C4C2311yescharacteristic
66C3×C5×C22⋊C4—21no—
67C3×C42×C5C2311yescharacteristic
68#6716—22no—
69C4⋊Dic15C2311yescharacteristic
70#6712—21no—
71C4×Dic15—21no—
72C2×C3⋊#168C2311yescharacteristic
73C2×C3⋊#168C2311yescharacteristic
74C5×C4⋊Dic3—21no—
75C5×C4⋊Dic3C2312yes—
76C2×C5×C3⋊C8C2312yes—
77C3×#685—22no—
78C3×#685D412yes—
79C3×C4⋊Dic5—21no—
80C3×C4⋊Dic5—21no—
81C3×C4⋊Dic5—22no—
82C3×C4⋊Dic5D411yescharacteristic
83Dic3⋊Dic5—22no—
84#6665—22no—
85#6665—22no—
86Dic3×Dic5—22no—
87#6662—42no—
88D5×C3⋊C8—42no—
89C3×#547—51no—
90#5542—51no—
91#5239—52no—
92#5233—52no—
93#5090—51no—
94#5079—51no—
95#5051—52no—
96#5045—52no—
97C4×#214—51no—
98D4×C3⋊C8—51no—
99#5007D511yescharacteristic, complemented
100C2×#711C2×D312yes—
101C5×C4⋊Q8—31no—
102C4×C5×D4C2×D311yescharacteristic
103#4093—61no—
104Q8×Dic5—61no—
105C2×#687—31no—
106#4076—21no—
107C2×C4⋊Dic5—21no—
108C2×Dic20—32no—
109C2×C8×D5—32no—
110#4044—62no—
111#4043—32no—
112#4043C2×D312yes—
113#4038—32no—
114#4032—22no—
115C4×D20C2×D311yescharacteristic
116#4015C2×D311yescharacteristic
117C4×Dic10—31no—
118C5×#107—32no—
119C5×C4⋊C8—31no—
120C5×#102—32no—
121#3964—31no—
122#3952—32no—
123#3948—32no—
124#3942—32no—
125#3942—32no—
126#3942—31no—
127#3940—31no—
128C4×#168—31no—
129#3302—151noSylow
130C23×C3×C5—21no—
131C22×C3×D5C2×D412yes—
132C3×C5×D4—21no—
133C3×C5×D4#4112yes—
134C2×C3×C4×C5—21no—
135C2×C3×C4×C5—42no—
136C2×C3×C4×C5C2×D411yescharacteristic
137C2×C3×C4×C5C2×D411yescharacteristic
138C2×C3×C4×C5C2411yescharacteristic
139C2×Dic15—21no—
140C2×Dic15—21no—
141Dic30—21no—
142Dic30—21no—
143C2×C5×Dic3—22no—
144C5×Dic6—42no—
145C3×C22⋊D5—44no—
146C2×C3×Dic5—22no—
147C2×C3×Dic5—44no—
148C2×C3×Dic5C2×D412yes—
149C3×D20—22no—
150C3×C4×D5—22no—
151C3×C4×D5—22no—
152C3×Dic10—22no—
153C3×Dic10—22no—
154C3⋊#168—21no—
155C3⋊#168—21no—
156C5×C3⋊C8—22no—
157C2×C3×#41—52no—
158C3×C4⋊Q8—51no—
159C3×#123—102no—
160C3×C22⋊Q8—52no—
161C3×#121—101no—
162C3×C4×D4—51no—
163C3×C4×D4C2×D511yescharacteristic
164C2×C3×C4⋊C4—101no—
165C2×#216—52no—
166#936—101no—
167C2×#214—51no—
168#926—52no—
169#924—52no—
170C22×C3⋊C8—52no—
171#887—101no—
172#879—102no—
173#869C2×D511yescharacteristic
174C4×Dic6—51no—
175#835—52no—
176#835—52no—
177#835—52no—
178#832—51no—
179#832C2×D512yes—
180#830—52no—
181#810—51no—
182#808—51no—
183#807C2×D512yes—
184C4⋊(C3⋊C8)C2×D511yescharacteristic
185C4×C3⋊C8—51no—
186C2×C5×Q8—61no—
187C2×C5×D4C22×D311yescharacteristic
188C22×C4×C5—21no—
189C2×C22⋊D5—22no—
190C22×Dic5—22no—
191#711—24no—
192#711—42no—
193C2×D20#6711yescharacteristic
194C2×C4×D5C22×D312yes—
195C2×Dic10—31no—
196C2×Dic10C22×D312yes—
197C2×C5×C8—32no—
198C5×C4⋊C4—32no—
199C5×C4⋊C4—61no—
200C5×C4⋊C4C22×D311yescharacteristic
201C5×C22⋊C4—21no—
202C42×C5C22×D311yescharacteristic
203#687—62no—
204#685—22no—
205#685—62no—
206#685—62no—
207#685#6712yes—
208C4⋊Dic5—21no—
209C4⋊Dic5—21no—
210C4⋊Dic5—22no—
211C4⋊Dic5—31no—
212C4⋊Dic5—62no—
213C4⋊Dic5#6711yescharacteristic
214#683—61no—
215C4×Dic5—61no—
216C4×Dic5—62no—
217C2×#168—31no—
218C2×#168—31no—
219Dic20—122no—
220C8×D5—122no—
221#556—151no—
222#547—51no—
223#487—151no—
224#485—152no—
225#474—31no—
226#467—152no—
227#462—151no—
228#452—152no—
229C4×SD16—151no—
230C8×D4—151no—
231#416—152no—
232C22×C3×C5—22no—
233C22×C3×C5—41no—
234C22×C3×C5—41no—
235C22×C3×C5C22×D411yescharacteristic
236C2×C3×D5—22no—
237C2×C3×D5—22no—
238C2×C3×D5—22no—
239C3×C4×C5—21no—
240C3×C4×C5—41no—
241C3×C4×C5#13512yes—
242C3×C4×C5C2×#4111yescharacteristic
243C3×C4×C5H(5,𝔽2)11yescharacteristic
244Dic15—41no—
245Dic15—41no—
246C3×Dic5—22no—
247C3×Dic5—42no—
248C5×Dic3—42no—
249C3×#41—104no—
250C3×#41—202no—
251C2×C3×Q8—52no—
252C2×C3×D4—51no—
253C2×C3×D4—102no—
254C2×C3×D4C22×D511yescharacteristic
255C22×C3×C4—21no—
256C22×C3×C4—52no—
257C22×C3×C4—102no—
258C2×Dic6—21no—
259C2×Dic6—51no—
260C3×C4⋊C4—51no—
261C3×C4⋊C4—102no—
262C3×C4⋊C4—101no—
263C3×C4⋊C4—101no—
264C3×C4⋊C4C22×D511yescharacteristic
265C3×C22⋊C4—21no—
266C3×C22⋊C4—52no—
267C3×C22⋊C4—102no—
268C3×C42C22×D511yescharacteristic
269#217—102no—
270#217—102no—
271#216—202no—
272#214—102no—
273C4⋊Dic3—21no—
274C4⋊Dic3—51no—
275C4⋊Dic3C22×D512yes—
276Dic3⋊C4—102no—
277Dic3⋊C4—101no—
278C4×Dic3—102no—
279C4×Dic3—101no—
280C2×C3⋊C8—51no—
281C2×C3⋊C8—51no—
282C2×C3⋊C8—202no—
283C2×C3⋊C8C22×D512yes—
284C23×C5—21no—
285C22×D5C2×#6712yes—
286C5×Q8—122no—
287C5×D4—21no—
288C5×D4#23712yes—
289C2×C4×C5—21no—
290C2×C4×C5—42no—
291C2×C4×C5—62no—
292C2×C4×C5C2×#6711yescharacteristic
293C2×C4×C5C2×#6711yescharacteristic
294C2×C4×C5C23×D311yescharacteristic
295C22⋊D5—44no—
296C2×Dic5—22no—
297C2×Dic5—44no—
298C2×Dic5—61no—
299C2×Dic5—61no—
300C2×Dic5C2×#6712yes—
301D20—22no—
302C4×D5—22no—
303C4×D5—22no—
304Dic10—22no—
305Dic10—22no—
306Dic10—61no—
307Dic10—61no—
308C5×C8—62no—
309#168—61no—
310#168—61no—
311C2×#41—52no—
312C2×Q16—152no—
313C2×SD16—151no—
314C22×C8—152no—
315C4⋊Q8—31no—
316C4⋊Q8—51no—
317#126—302no—
318#124—301no—
319#123—102no—
320C22⋊Q8—52no—
321C22⋊Q8—152no—
322#121—101no—
323C4×Q8—151no—
324C4×Q8—301no—
325C4×D4—51no—
326C4×D4D3×D511yescharacteristic
327#117—152no—
328C2×C4⋊C4—101no—
329#107—32no—
330#106—151no—
331C4⋊C8—31no—
332#103—152no—
333#103—152no—
334#103—152no—
335#103—151no—
336#102—32no—
337#102—151no—
338C22⋊C8—152no—
339C4×C8—151no—
340C2×C3×C5—42no—
341C2×C3×C5#54611yescharacteristic
342C2×C3×C5C2×#13511yescharacteristic
343C2×C3×C5#57811yescharacteristic
344C3×D5—42no—
345C23×C3—21no—
346C23×C3—52no—
347C3×Q8—102no—
348C3×Q8—102no—
349C3×D4—21no—
350C3×D4—102no—
351C3×D4—204no—
352C3×D4#71312yes—
353C2×C3×C4—21no—
354C2×C3×C4—42no—
355C2×C3×C4—52no—
356C2×C3×C4—102no—
357C2×C3×C4—102no—
358C2×C3×C4—102no—
359C2×C3×C4—204no—
360C2×C3×C4D4×D511yescharacteristic
361C2×C3×C4D4×D511yescharacteristic
362C2×C3×C4C23×D511yescharacteristic
363C2×Dic3—22no—
364C2×Dic3—101no—
365C2×Dic3—101no—
366Dic6—42no—
367Dic6—101no—
368Dic6—101no—
369C3⋊C8—22no—
370C3⋊C8—101no—
371C3⋊C8—101no—
372C22×C5—22no—
373C22×C5—41no—
374C22×C5—41no—
375C22×C5C22×#6711yescharacteristic
376C2×D5—22no—
377C2×D5—22no—
378C2×D5—22no—
379C4×C5—21no—
380C4×C5—41no—
381C4×C5—122no—
382C4×C5#95012yes—
383C4×C5C2×#23711yescharacteristic
384C4×C5#100411yescharacteristic
385Dic5—22no—
386Dic5—42no—
387Dic5—121no—
388Dic5—121no—
389#41—104no—
390#41—202no—
391C2×Q8—52no—
392C2×Q8—61no—
393C2×Q8—151no—
394C2×D4—51no—
395C2×D4—102no—
396C2×D4C2×D3×D511yescharacteristic
397C22×C4—21no—
398C22×C4—52no—
399C22×C4—102no—
400Q16—602no—
401SD16—302no—
402C2×C8—32no—
403C2×C8—151no—
404C2×C8—151no—
405C2×C8—602no—
406C4⋊C4—32no—
407C4⋊C4—51no—
408C4⋊C4—61no—
409C4⋊C4—102no—
410C4⋊C4—101no—
411C4⋊C4—101no—
412C4⋊C4—151no—
413C4⋊C4—302no—
414C4⋊C4—301no—
415C4⋊C4C2×D3×D511yescharacteristic
416C22⋊C4—21no—
417C22⋊C4—52no—
418C22⋊C4—102no—
419C22⋊C4—302no—
420C22⋊C4—302no—
421C42—302no—
422C42—301no—
423C42C2×D3×D511yescharacteristic
424C3×C5#330211yescharacteristic, complemented, Hall
425C22×C3—22no—
426C22×C3—41no—
427C22×C3—41no—
428C22×C3—102no—
429C22×C3—102no—
430C22×C3—102no—
431C22×C3C2×D4×D511yescharacteristic
432C3×C4—21no—
433C3×C4—41no—
434C3×C4—102no—
435C3×C4—202no—
436C3×C4#405812yes—
437C3×C4C2×#71311yescharacteristic
438C3×C4#414911yescharacteristic
439Dic3—42no—
440Dic3—201no—
441Dic3—201no—
442C2×C5—42no—
443C2×C5#580611yescharacteristic
444C2×C5C2×#95011yescharacteristic
445C2×C5#583811yescharacteristic
446D5—42no—
447C23—21no—
448C23—52no—
449Q8—102no—
450Q8—102no—
451Q8—122no—
452Q8—301no—
453Q8—301no—
454D4—21no—
455D4—102no—
456D4—204no—
457D4#676612yes—
458C2×C4—21no—
459C2×C4—42no—
460C2×C4—52no—
461C2×C4—62no—
462C2×C4—102no—
463C2×C4—102no—
464C2×C4—102no—
465C2×C4—204no—
466C2×C4—301no—
467C2×C4—301no—
468C2×C4D5×#6711yescharacteristic
469C2×C4D5×#6711yescharacteristic
470C2×C4C22×D3×D511yescharacteristic
471C8—62no—
472C8—301no—
473C8—301no—
474C2×C3—42no—
475C2×C3—202no—
476C2×C3#6596411yescharacteristic
477C2×C3C2×#405811yescharacteristic
478C2×C3#6611311yescharacteristic
479C5#8351011yescharacteristic, complemented, Sylow
480C22—22no—
481C22—41no—
482C22—41no—
483C22—102no—
484C22—102no—
485C22—102no—
486C22C2×D5×#6711yescharacteristic
487C4—21no—
488C4—41no—
489C4—102no—
490C4—122no—
491C4—202no—
492C4—601no—
493C4—601no—
494C4#9149012yes—
495C4C2×#676611yescharacteristic
496C4#9201711yescharacteristic
497C3#1061271211yescharacteristic, complemented, Sylow
498C2—42no—
499C2—202no—
500C2#1175430811yescharacteristic
501C2C2×#9149011yescharacteristic
502C2#1175533611yescharacteristic
subgroupG/H
centerC22C2×D5×#67
commutator, cosocleC2×C3×C4×C5C24
FrattiniC2×C4C22×D3×D5
FittingC3×C4×C5×D4C22
socleC22×C3×C5C22×D4
prime radicalC3×#65889C2
hypercenterC4×D4D3×D5
hypocenterC3×C5#3302
LCSG ▷ C2×C3×C4×C5 ▷ C2×C3×C5 ▷ C3×C5
UCSC1 ◁ C22 ◁ C42 ◁ C4×D4

Complements

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

NH ≅ G/Ntype
2. #11752508C2semidirect
3. C4×#6716C2semidirect
4. C5×#5007C2semidirect
5. #11749049C2semidirect
6. #11748961C2semidirect
7. #11747445C2semidirect
8. #11747204C2semidirect
9. #11747104C2semidirect
10. #11746996C2semidirect
11. #11746924C2semidirect
99. #5007D5semidirect
424. C3×C5#3302semidirect
479. C5#83510semidirect
497. C3#10612712semidirect

Maximal subgroups

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

subgroupindex
C3×#658892
#117525082
C4×#67162
C5×#50072
#117490492
#117489612
#117474452
#117472042
#117471042
#117469962
#117469242
#106127123
#835105