Finite Group Databasebeta

← #14471665

Subgroups — #14471665

Order 1920.

700 conjugacy classes of subgroups.

The subgroup lattice has 484 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C5×#5579C211yesmaximal, characteristic, complemented
2C3×#65918C211yesmaximal, characteristic, complemented
3Q8×C3⋊#168C211yesmaximal, characteristic, complemented
4#11747070C211yesmaximal, characteristic, complemented
5C4×#6656C212yesmaximal, complemented
6#11747034C212yesmaximal
7C4×#6652C211yesmaximal, characteristic
8#11746924C211yesmaximal, characteristic, complemented
9#11746919C212yesmaximal, complemented
10#11746896C212yesmaximal, complemented
11#11746880C211yesmaximal, characteristic, complemented
12#10615368—31nomaximal, Hall
13C3×C4×C5×Q8C2211yescharacteristic
14C5×#946—21no—
15C5×#896—22no—
16C5×D3×C4⋊C4—21no—
17C4×C5×D12C2211yescharacteristic
18C4×C5×D12C2212yes—
19C5×#870C2212yes—
20C5×#869C2211yescharacteristic
21C3×#4092—21no—
22C3×#4044—22no—
23C3×#4038—21no—
24C3×#4021C2212yes—
25C3×C4×D20C2211yescharacteristic
26C3×#4015C2211yescharacteristic
27C3×C4×Dic10C2212yes—
28C2×#6656C2212yes—
29C2×#6652C2211yescharacteristic
30C4⋊(C3⋊#168)C2211yescharacteristic
31C4⋊(C3⋊#168)C2212yes—
32C4×C3⋊#168C2212yes—
33C4×C3⋊#168C2211yescharacteristic
34#90989C2211yescharacteristic
35#90986C2212yes—
36#90970C2212yes—
37#90970C2212yes—
38#90970C2212yes—
39#90966C2211yescharacteristic
40#90964C2212yes—
41#90964C2211yescharacteristic
42#83667—51nomaximal, Hall
43C5×#554—31no—
44#65918D311yescharacteristic, complemented
45#65382—31no—
46#65373—31no—
47#65357—32no—
48#65352—32no—
49Q8×#168—31no—
50#65308—31no—
51C4×#687—32no—
52#65305—32no—
53C4×#686—31no—
54C2×C3×C5×Q8D411yescharacteristic
55C2×C5×D12C2311yescharacteristic
56C2×C4×C5×D3—21no—
57C2×C4×C5×D3—22no—
58C2×C5×Dic6—21no—
59C2×C3×D20C2311yescharacteristic
60C2×C3×C4×D5—21no—
61C2×C3×Dic10C2312yes—
62C3×C5×C4⋊C4D411yescharacteristic
63C3×C5×C4⋊C4D412yes—
64C3×C42×C5C2311yescharacteristic
65C3×C42×C5C2312yes—
66C2×C3⋊#168C2311yescharacteristic
67C2×C3⋊#168C2312yes—
68C2×C3⋊#168C2311yescharacteristic
69C5×(C2×D3)⋊C4—21no—
70C5×(C2×D3)⋊C4—22no—
71C5×C4⋊Dic3—21no—
72C5×C4⋊Dic3C2312yes—
73C5×C4⋊Dic3C2311yescharacteristic
74C5×Dic3⋊C4—22no—
75C5×Dic3⋊C4—22no—
76C3×#685—22no—
77C3×#685—21no—
78C3×#685—22no—
79C3×C4⋊Dic5—21no—
80C3×C4⋊Dic5—21no—
81C3×C4⋊Dic5C2311yescharacteristic
82C3×#683—22no—
83C3×C4×Dic5—22no—
84#6656—24no—
85#6652—22no—
86C3×#556—51no—
87#5579D511yescharacteristic, complemented
88#5083—51no—
89#5079—51no—
90#5057—52no—
91#5056—52no—
92C4×#215—52no—
93Q8×C3⋊C8—51no—
94#5022—52no—
95C4×#213—51no—
96#5008—51no—
97C5×C4⋊Q8—31no—
98C5×#125—32no—
99C5×C22⋊Q8—61no—
100C5×C22⋊Q8—62no—
101C4×C5×Q8C2×D311yescharacteristic
102C4×C5×D4—31no—
103C4×C5×D4—32no—
104C2×C5×C4⋊C4—61no—
105#4092—21no—
106C2×#687—32no—
107C2×#686—31no—
108#4044—22no—
109#4038—21no—
110#4021C2×D312yes—
111C4×D20C2×D311yescharacteristic
112#4015C2×D311yescharacteristic
113C4×Dic10C2×D312yes—
114#3964—31no—
115#3942—32no—
116#3942—32no—
117#3942—32no—
118#3941—31no—
119#3941—32no—
120#3940—32no—
121#3939—31no—
122C4⋊#168—32no—
123C4⋊#168—31no—
124C4×#168—31no—
125C4×#168—32no—
126#3360—151noSylow
127C22×C5×D3—21no—
128C22×C3×D5—21no—
129C3×C5×Q8—22no—
130C2×C3×C4×C5C2×D412yes—
131C2×C3×C4×C5C2×D412yes—
132C2×C3×C4×C5C2×D411yescharacteristic
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—22no—
144C3×D20—21no—
145C3×D20—21no—
146C3×C4×D5—42no—
147C3×Dic10—22no—
148C3×Dic10—22no—
149C3⋊#168—22no—
150C3⋊#168—21no—
151C3⋊#168—21no—
152C3×C4⋊Q8—51no—
153C3×#126—102no—
154C3×#124—52no—
155C3×C22⋊Q8—101no—
156C3×C4×Q8—52no—
157C3×C4×Q8C2×D511yescharacteristic
158C3×C4×D4—51no—
159C3×#117—101no—
160#946—21no—
161C2×#215—52no—
162C2×#213—51no—
163#896—22no—
164D3×C4⋊C4—21no—
165C4×D12C2×D511yescharacteristic
166C4×D12C2×D512yes—
167#870C2×D512yes—
168#869C2×D511yescharacteristic
169#835—52no—
170#835—52no—
171#835—52no—
172#832—52no—
173#832—51no—
174#809—51no—
175#808—52no—
176#807—51no—
177C4⋊(C3⋊C8)—51no—
178C4⋊(C3⋊C8)—52no—
179C4×C3⋊C8—51no—
180C4×C3⋊C8—52no—
181C2×C5×Q8—61no—
182C2×C5×Q8#6711yescharacteristic
183C2×C5×D4—31no—
184C22×C4×C5—61no—
185C22×C4×C5—62no—
186C2×D20C22×D311yescharacteristic
187C2×C4×D5—21no—
188C2×Dic10C22×D312yes—
189C5×C4⋊C4—31no—
190C5×C4⋊C4—32no—
191C5×C4⋊C4—62no—
192C5×C4⋊C4—61no—
193C5×C4⋊C4—62no—
194C5×C4⋊C4#6711yescharacteristic
195C5×C4⋊C4#6712yes—
196C5×C22⋊C4—61no—
197C5×C22⋊C4—62no—
198C42×C5C22×D312yes—
199C42×C5C22×D311yescharacteristic
200#687—64no—
201#686—62no—
202#685—21no—
203#685—22no—
204#685—22no—
205C4⋊Dic5—21no—
206C4⋊Dic5—21no—
207C4⋊Dic5C22×D311yescharacteristic
208#683—22no—
209C4×Dic5—22no—
210C2×#168—31no—
211C2×#168—31no—
212C2×#168—32no—
213#556—51no—
214#554—31no—
215#488—152no—
216#487—151no—
217#479—152no—
218#476—151no—
219#463—152no—
220#461—151no—
221C8×Q8—151no—
222C4×SD16—152no—
223C4×D8—151no—
224C22×C3×C5C22×D411yescharacteristic
225C2×C5×D3—22no—
226C2×C5×D3—41no—
227C2×C5×D3—41no—
228C2×C3×D5—41no—
229C2×C3×D5—41no—
230C2×C3×D5—41no—
231C3×C4×C5—22no—
232C3×C4×C5—21no—
233C3×C4×C5—22no—
234C3×C4×C5#13512yes—
235C3×C4×C5C2×#4111yescharacteristic
236C3×C4×C52−1+411yescharacteristic
237C3×Dic5—42no—
238C3×Dic5—41no—
239C5×Dic3—41no—
240C5×Dic3—42no—
241C2×C3×Q8—52no—
242C2×C3×Q8C22⋊D511yescharacteristic
243C2×C3×D4—51no—
244C22×C3×C4—101no—
245C2×D12C22×D511yescharacteristic
246C2×C4×D3—21no—
247C2×C4×D3—22no—
248C2×Dic6—21no—
249C3×C4⋊C4—51no—
250C3×C4⋊C4—102no—
251C3×C4⋊C4—101no—
252C3×C4⋊C4—101no—
253C3×C4⋊C4C22⋊D511yescharacteristic
254C3×C4⋊C4C22⋊D512yes—
255C3×C22⋊C4—102no—
256C3×C22⋊C4—101no—
257C3×C22⋊C4—102no—
258C3×C42—102no—
259C3×C42C22×D511yescharacteristic
260C3×C42C22×D512yes—
261#215—104no—
262#213—102no—
263(C2×D3)⋊C4—21no—
264(C2×D3)⋊C4—22no—
265C4⋊Dic3—21no—
266C4⋊Dic3C22×D511yescharacteristic
267C4⋊Dic3C22×D512yes—
268Dic3⋊C4—22no—
269Dic3⋊C4—22no—
270C2×C3⋊C8—51no—
271C2×C3⋊C8—51no—
272C2×C3⋊C8—52no—
273C23×C5—61no—
274C22×D5—21no—
275C5×Q8—22no—
276C5×Q8—241no—
277C5×D4—32no—
278C5×D4—61no—
279C2×C4×C5—61no—
280C2×C4×C5—62no—
281C2×C4×C5—122no—
282C2×C4×C5—242no—
283C2×C4×C5C2×#6712yes—
284C2×C4×C5C2×#6712yes—
285C2×C4×C5C2×#6711yescharacteristic
286C2×C4×C5C2×#6711yescharacteristic
287C2×C4×C5C23×D311yescharacteristic
288C2×Dic5—21no—
289C2×Dic5—22no—
290D20—21no—
291D20—21no—
292C4×D5—42no—
293Dic10—22no—
294Dic10—22no—
295#168—61no—
296#168—62no—
297#168—61no—
298C2×SD16—152no—
299C2×D8—151no—
300C4⋊Q8—31no—
301C4⋊Q8—51no—
302#126—102no—
303#125—32no—
304#124—52no—
305C22⋊Q8—61no—
306C22⋊Q8—62no—
307C22⋊Q8—101no—
308C4×Q8—52no—
309C4×Q8D3×D511yescharacteristic
310C4×D4—31no—
311C4×D4—32no—
312C4×D4—51no—
313#117—101no—
314C2×C4⋊C4—61no—
315#107—151no—
316#106—152no—
317C4⋊C8—151no—
318C4⋊C8—152no—
319#103—152no—
320#103—152no—
321#103—152no—
322#102—151no—
323#102—152no—
324#102—151no—
325C4×C8—151no—
326C4×C8—152no—
327C2×C3×C5#54811yescharacteristic
328C2×C3×C5C2×#13511yescharacteristic
329C2×C3×C5#57711yescharacteristic
330C3×D5—81no—
331C5×D3—42no—
332C23×C3—101no—
333C22×D3—21no—
334C3×Q8—22no—
335C3×Q8—102no—
336C3×Q8—102no—
337C3×D4—101no—
338C3×D4—101no—
339C2×C3×C4—101no—
340C2×C3×C4—102no—
341C2×C3×C4—202no—
342C2×C3×C4C2×C22⋊D512yes—
343C2×C3×C4C2×C22⋊D512yes—
344C2×C3×C4C2×C22⋊D511yescharacteristic
345C2×C3×C4C2×C22⋊D511yescharacteristic
346C2×C3×C4C23×D511yescharacteristic
347C2×Dic3—21no—
348C2×Dic3—22no—
349D12—21no—
350D12#71312yes—
351C4×D3—42no—
352C4×D3—82no—
353Dic6—81no—
354C3⋊C8—101no—
355C3⋊C8—102no—
356C3⋊C8—101no—
357C22×C5—62no—
358C22×C5—121no—
359C22×C5—121no—
360C22×C5C22×#6711yescharacteristic
361C2×D5—41no—
362C2×D5—41no—
363C2×D5—41no—
364C4×C5—22no—
365C4×C5—22no—
366C4×C5—21no—
367C4×C5—121no—
368C4×C5—122no—
369C4×C5#95012yes—
370C4×C5D3×#4111yescharacteristic
371C4×C5#101011yescharacteristic
372Dic5—41no—
373Dic5—42no—
374C2×Q8—52no—
375C2×Q8—61no—
376C2×Q8#120111yescharacteristic
377C2×D4—31no—
378C2×D4—51no—
379C22×C4—61no—
380C22×C4—62no—
381C22×C4—101no—
382SD16—304no—
383D8—302no—
384C2×C8—151no—
385C2×C8—152no—
386C2×C8—151no—
387C4⋊C4—32no—
388C4⋊C4—31no—
389C4⋊C4—51no—
390C4⋊C4—62no—
391C4⋊C4—61no—
392C4⋊C4—62no—
393C4⋊C4—101no—
394C4⋊C4—102no—
395C4⋊C4—101no—
396C4⋊C4#120111yescharacteristic
397C4⋊C4#120112yes—
398C22⋊C4—61no—
399C22⋊C4—62no—
400C22⋊C4—102no—
401C22⋊C4—101no—
402C22⋊C4—102no—
403C42—102no—
404C42C2×D3×D511yescharacteristic
405C42C2×D3×D512yes—
406C3×C5#336011yescharacteristic, complemented, Hall
407C22×C3—201no—
408C22×C3—201no—
409C22×C3—201no—
410C22×C3C22×C22⋊D511yescharacteristic
411C2×D3—22no—
412C2×D3—41no—
413C2×D3—41no—
414C3×C4—22no—
415C3×C4—21no—
416C3×C4—22no—
417C3×C4—202no—
418C3×C4—201no—
419C3×C4#409612yes—
420C3×C4C2×#71311yescharacteristic
421C3×C4#414711yescharacteristic
422Dic3—41no—
423Dic3—42no—
424C2×C5—122no—
425C2×C5C2×#95011yescharacteristic
426C2×C5#583111yescharacteristic
427C2×C5#583711yescharacteristic
428D5—81no—
429C23—61no—
430C23—101no—
431Q8—22no—
432Q8—102no—
433Q8—102no—
434Q8—241no—
435D4—32no—
436D4—61no—
437D4—101no—
438D4—101no—
439C2×C4—61no—
440C2×C4—62no—
441C2×C4—101no—
442C2×C4—102no—
443C2×C4—122no—
444C2×C4—202no—
445C2×C4—242no—
446C2×C4C2×#120112yes—
447C2×C4C2×#120112yes—
448C2×C4C2×#120111yescharacteristic
449C2×C4C2×#120111yescharacteristic
450C2×C4C22×D3×D511yescharacteristic
451C8—301no—
452C8—301no—
453C8—302no—
454C2×C3—401no—
455C2×C3#6615111yescharacteristic
456C2×C3C2×#409611yescharacteristic
457C2×C3#6617011yescharacteristic
458D3—42no—
459C5#8366711yescharacteristic, complemented, Sylow
460C22—62no—
461C22—121no—
462C22—121no—
463C22—201no—
464C22—201no—
465C22—201no—
466C22C22×#120111yescharacteristic
467C4—22no—
468C4—21no—
469C4—22no—
470C4—122no—
471C4—121no—
472C4—201no—
473C4—202no—
474C4#9129712yes—
475C4#9202811yescharacteristic
476C4D3×#71311yescharacteristic
477C3#1061536811yescharacteristic, complemented, Sylow
478C2—122no—
479C2—401no—
480C2C2×#9129711yescharacteristic
481C2#1175524711yescharacteristic
482C2#1175530411yescharacteristic
subgroupG/H
centerC22C22×#1201
commutator, cosocleC2×C3×C4×C5C24
FrattiniC2×C4C22×D3×D5
FittingC3×C4×C5×Q8C22
socleC22×C3×C5C22×D4
prime radicalC2×#6652C22
hypercenterC4×Q8D3×D5
hypocenterC3×C5#3360
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×#65918C2semidirect
3. Q8×C3⋊#168C2semidirect
4. #11747070C2semidirect
5. C4×#6656C2semidirect
8. #11746924C2semidirect
9. #11746919C2semidirect
10. #11746896C2semidirect
11. #11746880C2semidirect
44. #65918D3semidirect
87. #5579D5semidirect
406. C3×C5#3360semidirect
459. C5#83667semidirect
477. C3#10615368semidirect

Maximal subgroups

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

subgroupindex
C5×#55792
C3×#659182
Q8×C3⋊#1682
#117470702
C4×#66562
#117470342
C4×#66522
#117469242
#117469192
#117468962
#117468802
#106153683
#836675