Finite Group Databasebeta

← #14471824

Subgroups — #14471824

Order 1920.

724 conjugacy classes of subgroups.

The subgroup lattice has 492 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C5×#5579C211yesmaximal, characteristic, complemented
2C3×#65919C211yesmaximal, characteristic, complemented
3Q8×C3⋊#168C211yesmaximal, characteristic, complemented
4C4×#6656C211yesmaximal, characteristic, complemented
5#11747034C212yesmaximal, complemented
6#11747033C211yesmaximal, characteristic
7C4×#6652C212yesmaximal, complemented
8#11746954C211yesmaximal, characteristic, complemented
9#11746896C212yesmaximal, complemented
10#11746895C211yesmaximal, characteristic, complemented
11#11746887C212yesmaximal, complemented
12#10615374—31nomaximal, Hall
13C3×C4×C5×Q8C2211yescharacteristic
14C5×#946—21no—
15C5×#896—22no—
16C5×D3×C4⋊C4—21no—
17C4×C5×D12C2212yes—
18C4×C5×D12C2211yescharacteristic
19C5×#870C2212yes—
20C5×#869C2211yescharacteristic
21C3×#4093—21no—
22C3×#4040—22no—
23C3×#4039—21no—
24C3×#4021C2212yes—
25C3×#4020C2211yescharacteristic
26C3×C4×D20C2212yes—
27C3×C4×Dic10C2211yescharacteristic
28C2×#6656C2211yescharacteristic
29C2×#6652C2212yes—
30C4⋊(C3⋊#168)C2212yes—
31C4⋊(C3⋊#168)C2211yescharacteristic
32C4×C3⋊#168C2211yescharacteristic
33C4×C3⋊#168C2212yes—
34#90989C2212yes—
35#90986C2211yescharacteristic
36#90970C2211yescharacteristic
37#90970C2212yes—
38#90966C2211yescharacteristic
39#90964C2212yes—
40#90964C2212yes—
41#90964C2212yes—
42#83713—51nomaximal, Hall
43C5×#554—31no—
44#65919D311yescharacteristic, complemented
45#65380—31no—
46#65377—31no—
47#65357—32no—
48#65355—32no—
49Q8×#168—31no—
50C4×#687—31no—
51#65305—32no—
52#65304—31no—
53C4×#686—32no—
54C2×C3×C5×Q8D411yescharacteristic
55C2×C5×D12C2311yescharacteristic
56C2×C4×C5×D3—21no—
57C2×C4×C5×D3—22no—
58C2×C5×Dic6—21no—
59C2×C3×D20—21no—
60C2×C3×D20—21no—
61C2×C3×D20C2312yes—
62C2×C3×C4×D5—22no—
63C2×C3×Dic10C2311yescharacteristic
64C3×C5×C4⋊C4D411yescharacteristic
65C3×C5×C4⋊C4D412yes—
66C3×C42×C5C2311yescharacteristic
67C3×C42×C5C2312yes—
68C2×C3⋊#168C2311yescharacteristic
69C2×C3⋊#168C2311yescharacteristic
70C2×C3⋊#168C2312yes—
71C5×(C2×D3)⋊C4—21no—
72C5×(C2×D3)⋊C4—22no—
73C5×C4⋊Dic3—21no—
74C5×C4⋊Dic3C2312yes—
75C5×C4⋊Dic3C2311yescharacteristic
76C5×Dic3⋊C4—22no—
77C5×Dic3⋊C4—22no—
78C3×#685—22no—
79C3×#685—22no—
80C3×#685—22no—
81C3×C4⋊Dic5C2311yescharacteristic
82C3×#683—21no—
83C3×C4×Dic5—21no—
84#6656—22no—
85#6652—24no—
86C3×#559—51no—
87#5579D511yescharacteristic, complemented
88#5072—51no—
89#5070—51no—
90#5057—52no—
91#5056—52no—
92C4×#215—51no—
93Q8×C3⋊C8—51no—
94#5021—51no—
95C4×#213—52no—
96#5009—52no—
97C5×C4⋊Q8—31no—
98C5×#125—32no—
99C5×C22⋊Q8—61no—
100C5×C22⋊Q8—62no—
101C4×C5×Q8C2×D311yescharacteristic
102C4×C5×D4—32no—
103C4×C5×D4—31no—
104C2×C5×C4⋊C4—61no—
105#4093—21no—
106C2×#687—31no—
107C2×#686—32no—
108#4040—22no—
109#4039—21no—
110#4021C2×D312yes—
111#4020C2×D311yescharacteristic
112C4×D20C2×D312yes—
113C4×Dic10C2×D311yescharacteristic
114#3964—31no—
115#3942—32no—
116#3942—31no—
117#3941—32no—
118#3941—32no—
119#3941—32no—
120#3940—31no—
121#3939—32no—
122C4⋊#168—31no—
123C4⋊#168—32no—
124C4×#168—32no—
125C4×#168—31no—
126#3361—151noSylow
127C22×C5×D3—21no—
128C22×C3×D5—22no—
129C3×C5×Q8—22no—
130C2×C3×C4×C5C2×D411yescharacteristic
131C2×C3×C4×C5C2×D412yes—
132C2×C3×C4×C5C2×D412yes—
133C2×C3×C4×C5C2×D411yescharacteristic
134C2×C3×C4×C5C2411yescharacteristic
135C2×C5×Dic3—21no—
136C2×C5×Dic3—22no—
137C5×D12—21no—
138C5×D12#4112yes—
139C4×C5×D3—42no—
140C4×C5×D3—82no—
141C5×Dic6—81no—
142C2×C3×Dic5—21no—
143C2×C3×Dic5—21no—
144C3×D20—22no—
145C3×D20—22no—
146C3×D20—42no—
147C3×D20—81no—
148C3×C4×D5—82no—
149C3×Dic10—21no—
150C3×Dic10—21no—
151C3⋊#168—21no—
152C3⋊#168—22no—
153C3⋊#168—21no—
154C3×#127—51no—
155C3×#124—52no—
156C3×#124—101no—
157C3×#123—102no—
158C3×C4×Q8—51no—
159C3×C4×Q8C2×D511yescharacteristic
160C3×C4×D4—52no—
161C3×C4×D4—101no—
162#946—21no—
163C2×#215—51no—
164C2×#213—52no—
165#896—22no—
166D3×C4⋊C4—21no—
167C4×D12C2×D512yes—
168C4×D12C2×D511yescharacteristic
169#870C2×D512yes—
170#869C2×D511yescharacteristic
171#835—52no—
172#835—51no—
173#832—52no—
174#832—52no—
175#832—52no—
176#809—51no—
177#808—51no—
178#807—52no—
179C4⋊(C3⋊C8)—52no—
180C4⋊(C3⋊C8)—51no—
181C4×C3⋊C8—52no—
182C4×C3⋊C8—51no—
183C2×C5×Q8—61no—
184C2×C5×Q8#6711yescharacteristic
185C2×C5×D4—31no—
186C22×C4×C5—61no—
187C22×C4×C5—62no—
188C2×D20—21no—
189C2×D20—21no—
190C2×D20C22×D312yes—
191C2×C4×D5—22no—
192C2×Dic10C22×D311yescharacteristic
193C5×C4⋊C4—31no—
194C5×C4⋊C4—32no—
195C5×C4⋊C4—62no—
196C5×C4⋊C4—61no—
197C5×C4⋊C4—62no—
198C5×C4⋊C4#6711yescharacteristic
199C5×C4⋊C4#6712yes—
200C5×C22⋊C4—61no—
201C5×C22⋊C4—62no—
202C42×C5C22×D312yes—
203C42×C5C22×D311yescharacteristic
204#687—62no—
205#686—64no—
206#685—22no—
207#685—22no—
208#685—22no—
209C4⋊Dic5C22×D311yescharacteristic
210#683—21no—
211C4×Dic5—21no—
212C2×#168—31no—
213C2×#168—32no—
214C2×#168—31no—
215#559—51no—
216#554—31no—
217#488—152no—
218#486—151no—
219#478—152no—
220#477—151no—
221#463—152no—
222#459—151no—
223C8×Q8—151no—
224C4×SD16—151no—
225C4×D8—152no—
226C22×C3×C5C22×D411yescharacteristic
227C2×C5×D3—22no—
228C2×C5×D3—41no—
229C2×C5×D3—41no—
230C2×C3×D5—42no—
231C2×C3×D5—42no—
232C2×C3×D5—42no—
233C3×C4×C5—22no—
234C3×C4×C5—22no—
235C3×C4×C5—21no—
236C3×C4×C5#13512yes—
237C3×C4×C5C2×#4111yescharacteristic
238C3×C4×C52−1+411yescharacteristic
239C3×Dic5—41no—
240C3×Dic5—41no—
241C5×Dic3—41no—
242C5×Dic3—42no—
243C2×C3×Q8—51no—
244C2×C3×Q8C22⋊D511yescharacteristic
245C2×C3×D4—52no—
246C2×C3×D4—101no—
247C2×C3×D4—101no—
248C22×C3×C4—102no—
249C2×D12C22×D511yescharacteristic
250C2×C4×D3—21no—
251C2×C4×D3—22no—
252C2×Dic6—21no—
253C3×C4⋊C4—51no—
254C3×C4⋊C4—101no—
255C3×C4⋊C4C22⋊D511yescharacteristic
256C3×C4⋊C4C22⋊D512yes—
257C3×C22⋊C4—102no—
258C3×C22⋊C4—102no—
259C3×C22⋊C4—102no—
260C3×C42—101no—
261C3×C42C22×D511yescharacteristic
262C3×C42C22×D512yes—
263#215—102no—
264#213—104no—
265(C2×D3)⋊C4—21no—
266(C2×D3)⋊C4—22no—
267C4⋊Dic3—21no—
268C4⋊Dic3C22×D512yes—
269C4⋊Dic3C22×D511yescharacteristic
270Dic3⋊C4—22no—
271Dic3⋊C4—22no—
272C2×C3⋊C8—51no—
273C2×C3⋊C8—52no—
274C2×C3⋊C8—51no—
275C23×C5—61no—
276C22×D5—22no—
277C5×Q8—22no—
278C5×Q8—241no—
279C5×D4—32no—
280C5×D4—61no—
281C2×C4×C5—61no—
282C2×C4×C5—62no—
283C2×C4×C5—122no—
284C2×C4×C5—242no—
285C2×C4×C5C2×#6711yescharacteristic
286C2×C4×C5C2×#6712yes—
287C2×C4×C5C2×#6712yes—
288C2×C4×C5C2×#6711yescharacteristic
289C2×C4×C5C23×D311yescharacteristic
290C2×Dic5—21no—
291C2×Dic5—21no—
292D20—22no—
293D20—22no—
294D20—42no—
295D20—81no—
296C4×D5—82no—
297Dic10—21no—
298Dic10—21no—
299#168—62no—
300#168—61no—
301#168—61no—
302C2×SD16—151no—
303C2×D8—152no—
304C4⋊Q8—31no—
305#127—51no—
306#125—32no—
307#124—52no—
308#124—101no—
309#123—102no—
310C22⋊Q8—61no—
311C22⋊Q8—62no—
312C4×Q8—51no—
313C4×Q8D3×D511yescharacteristic
314C4×D4—32no—
315C4×D4—31no—
316C4×D4—52no—
317C4×D4—101no—
318C2×C4⋊C4—61no—
319#107—152no—
320#106—151no—
321C4⋊C8—151no—
322C4⋊C8—152no—
323#103—151no—
324#103—152no—
325#102—152no—
326#102—152no—
327#102—152no—
328#102—151no—
329C4×C8—152no—
330C4×C8—151no—
331C2×C3×C5#54811yescharacteristic
332C2×C3×C5C2×#13511yescharacteristic
333C2×C3×C5#57611yescharacteristic
334C3×D5—82no—
335C5×D3—42no—
336C23×C3—102no—
337C22×D3—21no—
338C3×Q8—22no—
339C3×Q8—101no—
340C3×Q8—101no—
341C3×D4—102no—
342C3×D4—102no—
343C3×D4—202no—
344C3×D4—401no—
345C2×C3×C4—101no—
346C2×C3×C4—101no—
347C2×C3×C4—402no—
348C2×C3×C4C2×C22⋊D512yes—
349C2×C3×C4C2×C22⋊D512yes—
350C2×C3×C4C2×C22⋊D511yescharacteristic
351C2×C3×C4C2×C22⋊D511yescharacteristic
352C2×C3×C4C23×D511yescharacteristic
353C2×Dic3—21no—
354C2×Dic3—22no—
355D12—21no—
356D12#71312yes—
357C4×D3—42no—
358C4×D3—82no—
359Dic6—81no—
360C3⋊C8—102no—
361C3⋊C8—101no—
362C3⋊C8—101no—
363C22×C5—62no—
364C22×C5—121no—
365C22×C5—121no—
366C22×C5C22×#6711yescharacteristic
367C2×D5—42no—
368C2×D5—42no—
369C2×D5—42no—
370C4×C5—22no—
371C4×C5—22no—
372C4×C5—21no—
373C4×C5—121no—
374C4×C5—122no—
375C4×C5#95012yes—
376C4×C5D3×#4111yescharacteristic
377C4×C5#101011yescharacteristic
378Dic5—41no—
379Dic5—41no—
380C2×Q8—51no—
381C2×Q8—61no—
382C2×Q8#120111yescharacteristic
383C2×D4—31no—
384C2×D4—52no—
385C2×D4—101no—
386C2×D4—101no—
387C22×C4—61no—
388C22×C4—62no—
389C22×C4—102no—
390SD16—302no—
391D8—304no—
392C2×C8—151no—
393C2×C8—151no—
394C2×C8—152no—
395C4⋊C4—31no—
396C4⋊C4—32no—
397C4⋊C4—51no—
398C4⋊C4—62no—
399C4⋊C4—61no—
400C4⋊C4—62no—
401C4⋊C4—101no—
402C4⋊C4#120111yescharacteristic
403C4⋊C4#120112yes—
404C22⋊C4—61no—
405C22⋊C4—62no—
406C22⋊C4—102no—
407C22⋊C4—102no—
408C22⋊C4—102no—
409C42—101no—
410C42C2×D3×D512yes—
411C42C2×D3×D511yescharacteristic
412C3×C5#336111yescharacteristic, complemented, Hall
413C22×C3—202no—
414C22×C3—202no—
415C22×C3—202no—
416C22×C3C22×C22⋊D511yescharacteristic
417C2×D3—22no—
418C2×D3—41no—
419C2×D3—41no—
420C3×C4—22no—
421C3×C4—21no—
422C3×C4—22no—
423C3×C4—201no—
424C3×C4—201no—
425C3×C4#409612yes—
426C3×C4C2×#71311yescharacteristic
427C3×C4#414711yescharacteristic
428Dic3—41no—
429Dic3—42no—
430C2×C5—122no—
431C2×C5C2×#95011yescharacteristic
432C2×C5#583111yescharacteristic
433C2×C5#583511yescharacteristic
434D5—82no—
435C23—61no—
436C23—102no—
437Q8—22no—
438Q8—101no—
439Q8—101no—
440Q8—241no—
441D4—32no—
442D4—61no—
443D4—102no—
444D4—102no—
445D4—202no—
446D4—401no—
447C2×C4—61no—
448C2×C4—62no—
449C2×C4—101no—
450C2×C4—101no—
451C2×C4—122no—
452C2×C4—242no—
453C2×C4—402no—
454C2×C4C2×#120112yes—
455C2×C4C2×#120111yescharacteristic
456C2×C4C2×#120112yes—
457C2×C4C2×#120111yescharacteristic
458C2×C4C22×D3×D511yescharacteristic
459C8—301no—
460C8—302no—
461C8—301no—
462C2×C3—402no—
463C2×C3#6615111yescharacteristic
464C2×C3C2×#409611yescharacteristic
465C2×C3#6616911yescharacteristic
466D3—42no—
467C5#8371311yescharacteristic, complemented, Sylow
468C22—62no—
469C22—121no—
470C22—121no—
471C22—202no—
472C22—202no—
473C22—202no—
474C22C22×#120111yescharacteristic
475C4—22no—
476C4—21no—
477C4—22no—
478C4—122no—
479C4—121no—
480C4—201no—
481C4—201no—
482C4#9129712yes—
483C4#9202811yescharacteristic
484C4D3×#71311yescharacteristic
485C3#1061537411yescharacteristic, complemented, Sylow
486C2—122no—
487C2—402no—
488C2C2×#9129711yescharacteristic
489C2#1175524711yescharacteristic
490C2#1175530111yescharacteristic
subgroupG/H
centerC22C22×#1201
commutator, cosocleC2×C3×C4×C5C24
FrattiniC2×C4C22×D3×D5
FittingC3×C4×C5×Q8C22
socleC22×C3×C5C22×D4
prime radical#11747033C2
hypercenterC4×Q8D3×D5
hypocenterC3×C5#3361
LCSG ▷ C2×C3×C4×C5 ▷ C2×C3×C5 ▷ C3×C5
UCSC1 ◁ C22 ◁ C4×Q8

Complements

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

NH ≅ G/Ntype
1. C5×#5579C2semidirect
2. C3×#65919C2semidirect
3. Q8×C3⋊#168C2semidirect
4. C4×#6656C2semidirect
5. #11747034C2semidirect
7. C4×#6652C2semidirect
8. #11746954C2semidirect
9. #11746896C2semidirect
10. #11746895C2semidirect
11. #11746887C2semidirect
44. #65919D3semidirect
87. #5579D5semidirect
412. C3×C5#3361semidirect
467. C5#83713semidirect
485. C3#10615374semidirect

Maximal subgroups

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

subgroupindex
C5×#55792
C3×#659192
Q8×C3⋊#1682
C4×#66562
#117470342
#117470332
C4×#66522
#117469542
#117468962
#117468952
#117468872
#106153743
#837135