Finite Group Databasebeta

← #14472063

Subgroups — #14472063

Order 1920.

672 conjugacy classes of subgroups.

The subgroup lattice has 464 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×C5×#556C211yesmaximal, characteristic
2#11756326C211yesmaximal, characteristic, complemented
3#11751191C211yesmaximal, characteristic, complemented
4#11751180C211yesmaximal, characteristic, complemented
5#11751157C212yesmaximal
6#11751155C212yesmaximal
7C4×#6718C212yesmaximal, complemented
8Q8×C3⋊#168C211yesmaximal, characteristic, complemented
9#11751123C212yesmaximal, complemented
10C4×#6716C211yesmaximal, characteristic
11#11751108C211yesmaximal, characteristic
12#10615407—31nomaximal, Hall
13C3×C5×C4⋊Q8C2211yescharacteristic
14C3×C5×#126—22no—
15C3×C5×#124C2212yes—
16C3×C5×C22⋊Q8—21no—
17C3×C4×C5×Q8C2212yes—
18C3×C4×C5×Q8C2211yescharacteristic
19C3×C4×C5×D4C2211yescharacteristic
20C3×C5×#117—21no—
21#91833—21no—
22C2×#6718C2212yes—
23C2×#6716C2211yescharacteristic
24#91778—22no—
25#91776—21no—
26#91759C2212yes—
27#91758C2211yescharacteristic
28C4×Dic30C2211yescharacteristic
29C4×Dic30C2212yes—
30#91119C2212yes—
31#91119C2212yes—
32#91119C2212yes—
33#91116C2212yes—
34#91116C2211yescharacteristic
35#91094C2211yescharacteristic
36#91092C2211yescharacteristic
37#91091C2212yes—
38C4⋊(C3⋊#168)C2211yescharacteristic
39C4⋊(C3⋊#168)C2212yes—
40C4×C3⋊#168C2211yescharacteristic
41C4×C3⋊#168C2212yes—
42#83676—51nomaximal, Hall
43C5×#556D311yescharacteristic
44#65901—31no—
45#65379—31no—
46#65370—31no—
47#65345—32no—
48#65343—32no—
49C4×#689—32no—
50Q8×#168—31no—
51#65311—32no—
52C4×#687—31no—
53#65296—31no—
54C2×C3×C5×Q8D411yescharacteristic
55C2×C3×C5×Q8C2312yes—
56C2×C3×C5×D4C2311yescharacteristic
57C22×C3×C4×C5—21no—
58C2×Dic30—21no—
59C2×Dic30C2311yescharacteristic
60C3×C5×C4⋊C4—21no—
61C3×C5×C4⋊C4—22no—
62C3×C5×C4⋊C4—21no—
63C3×C5×C4⋊C4D411yescharacteristic
64C3×C5×C4⋊C4D412yes—
65C3×C5×C4⋊C4C2311yescharacteristic
66C3×C5×C22⋊C4—22no—
67C3×C5×C22⋊C4—21no—
68C3×C5×C22⋊C4—22no—
69C3×C42×C5—22no—
70C3×C42×C5C2311yescharacteristic
71C3×C42×C5C2312yes—
72#6718—24no—
73#6716—22no—
74C4⋊Dic15—21no—
75C4⋊Dic15C2312yes—
76C4⋊Dic15C2311yescharacteristic
77#6712—21no—
78#6712—22no—
79#6712—22no—
80#6712—22no—
81C4×Dic15—22no—
82C4×Dic15—21no—
83C2×C3⋊#168C2311yescharacteristic
84C2×C3⋊#168C2312yes—
85C2×C3⋊#168C2311yescharacteristic
86C3×#556D511yescharacteristic
87#5567—51no—
88#5090—51no—
89#5079—51no—
90#5056—52no—
91#5054—52no—
92C4×#216—52no—
93Q8×C3⋊C8—51no—
94#5022—52no—
95C4×#214—51no—
96#5007—51no—
97C5×C4⋊Q8C2×D311yescharacteristic
98C5×#126—22no—
99C5×#124C2×D312yes—
100C5×C22⋊Q8—21no—
101C4×C5×Q8C2×D312yes—
102C4×C5×Q8C2×D311yescharacteristic
103C4×C5×D4C2×D311yescharacteristic
104C5×#117—21no—
105#4090—61no—
106C2×#689—32no—
107C2×#687—31no—
108#4035—62no—
109#4033—61no—
110#4016—32no—
111#4015—31no—
112C4×Dic10—31no—
113C4×Dic10—32no—
114#3967—32no—
115#3967—32no—
116#3967—32no—
117#3964—32no—
118#3964—31no—
119#3942—31no—
120#3940—31no—
121#3939—32no—
122C4⋊#168—31no—
123C4⋊#168—32no—
124C4×#168—31no—
125C4×#168—32no—
126#3364—151noSylow
127C23×C3×C5—21no—
128C3×C5×Q8—22no—
129C3×C5×Q8—22no—
130C3×C5×Q8—22no—
131C3×C5×D4—21no—
132C3×C5×D4—21no—
133C2×C3×C4×C5—21no—
134C2×C3×C4×C5—22no—
135C2×C3×C4×C5—42no—
136C2×C3×C4×C5C2×D412yes—
137C2×C3×C4×C5C2×D412yes—
138C2×C3×C4×C5C2×D411yescharacteristic
139C2×C3×C4×C5C2×D411yescharacteristic
140C2×C3×C4×C5C2411yescharacteristic
141C2×Dic15—21no—
142C2×Dic15—21no—
143C2×Dic15—22no—
144Dic30—21no—
145Dic30—42no—
146Dic30#4112yes—
147C3⋊#168—21no—
148C3⋊#168—22no—
149C3⋊#168—21no—
150C3×C4⋊Q8C2×D511yescharacteristic
151C3×#126—22no—
152C3×#124C2×D512yes—
153C3×C22⋊Q8—21no—
154C3×C4×Q8C2×D512yes—
155C3×C4×Q8C2×D511yescharacteristic
156C3×C4×D4C2×D511yescharacteristic
157C3×#117—21no—
158#944—101no—
159C2×#216—52no—
160C2×#214—51no—
161#889—102no—
162#887—101no—
163#870—52no—
164#869—51no—
165C4×Dic6—51no—
166C4×Dic6—52no—
167#835—52no—
168#835—52no—
169#835—52no—
170#832—52no—
171#832—51no—
172#810—51no—
173#808—51no—
174#807—52no—
175C4⋊(C3⋊C8)—51no—
176C4⋊(C3⋊C8)—52no—
177C4×C3⋊C8—51no—
178C4×C3⋊C8—52no—
179C2×C5×Q8#6711yescharacteristic
180C2×C5×Q8C22×D312yes—
181C2×C5×D4C22×D311yescharacteristic
182C22×C4×C5—21no—
183C2×Dic10—31no—
184C2×Dic10—61no—
185C5×C4⋊C4—21no—
186C5×C4⋊C4—22no—
187C5×C4⋊C4—21no—
188C5×C4⋊C4#6711yescharacteristic
189C5×C4⋊C4#6712yes—
190C5×C4⋊C4C22×D311yescharacteristic
191C5×C22⋊C4—22no—
192C5×C22⋊C4—21no—
193C5×C22⋊C4—22no—
194C42×C5—22no—
195C42×C5C22×D311yescharacteristic
196C42×C5C22×D312yes—
197#689—64no—
198#687—62no—
199C4⋊Dic5—32no—
200C4⋊Dic5—31no—
201C4⋊Dic5—61no—
202#683—61no—
203#683—62no—
204#683—62no—
205#683—62no—
206C4×Dic5—62no—
207C4×Dic5—61no—
208C2×#168—31no—
209C2×#168—32no—
210C2×#168—31no—
211#557—151no—
212#556D1511yescharacteristic
213#488—152no—
214#487—151no—
215#479—152no—
216#474—151no—
217#464—152no—
218#462—151no—
219C8×Q8—151no—
220C4×Q16—152no—
221C4×SD16—151no—
222C22×C3×C5—41no—
223C22×C3×C5—41no—
224C22×C3×C5—41no—
225C22×C3×C5C22×D411yescharacteristic
226C3×C4×C5—22no—
227C3×C4×C5—22no—
228C3×C4×C5—21no—
229C3×C4×C5—42no—
230C3×C4×C5—41no—
231C3×C4×C5#13512yes—
232C3×C4×C5C2×#4111yescharacteristic
233C3×C4×C52−1+411yescharacteristic
234Dic15—22no—
235Dic15—42no—
236Dic15—41no—
237C2×C3×Q8C22⋊D511yescharacteristic
238C2×C3×Q8C22×D512yes—
239C2×C3×D4C22×D511yescharacteristic
240C22×C3×C4—21no—
241C2×Dic6—51no—
242C2×Dic6—101no—
243C3×C4⋊C4—21no—
244C3×C4⋊C4—22no—
245C3×C4⋊C4—21no—
246C3×C4⋊C4C22⋊D511yescharacteristic
247C3×C4⋊C4C22⋊D512yes—
248C3×C4⋊C4C22×D511yescharacteristic
249C3×C22⋊C4—22no—
250C3×C22⋊C4—21no—
251C3×C22⋊C4—22no—
252C3×C42—22no—
253C3×C42C22×D511yescharacteristic
254C3×C42C22×D512yes—
255#216—104no—
256#214—102no—
257C4⋊Dic3—52no—
258C4⋊Dic3—51no—
259C4⋊Dic3—101no—
260Dic3⋊C4—101no—
261Dic3⋊C4—102no—
262Dic3⋊C4—102no—
263Dic3⋊C4—102no—
264C4×Dic3—102no—
265C4×Dic3—101no—
266C2×C3⋊C8—51no—
267C2×C3⋊C8—52no—
268C2×C3⋊C8—51no—
269C23×C5—21no—
270C5×Q8—22no—
271C5×Q8—22no—
272C5×Q8—22no—
273C5×D4—21no—
274C5×D4—21no—
275C2×C4×C5—21no—
276C2×C4×C5—22no—
277C2×C4×C5—42no—
278C2×C4×C5C2×#6712yes—
279C2×C4×C5C2×#6712yes—
280C2×C4×C5C2×#6711yescharacteristic
281C2×C4×C5C2×#6711yescharacteristic
282C2×C4×C5C23×D311yescharacteristic
283C2×Dic5—61no—
284C2×Dic5—61no—
285C2×Dic5—62no—
286Dic10—32no—
287Dic10—61no—
288Dic10—122no—
289#168—61no—
290#168—62no—
291#168—61no—
292C2×Q16—152no—
293C2×SD16—151no—
294C4⋊Q8—151no—
295C4⋊Q8—301no—
296C4⋊Q8C2×D1511yescharacteristic
297#126—22no—
298#125—152no—
299#125—302no—
300#124C2×D1512yes—
301C22⋊Q8—21no—
302C4×Q8—151no—
303C4×Q8—152no—
304C4×Q8—301no—
305C4×Q8C2×D1512yes—
306C4×Q8C2×D1511yescharacteristic
307C4×D4C2×D1511yescharacteristic
308#117—21no—
309#107—152no—
310#106—151no—
311C4⋊C8—151no—
312C4⋊C8—152no—
313#103—152no—
314#103—152no—
315#103—152no—
316#103—151no—
317#102—151no—
318#102—152no—
319C4×C8—151no—
320C4×C8—152no—
321C2×C3×C5—81no—
322C2×C3×C5#54811yescharacteristic
323C2×C3×C5C2×#13511yescharacteristic
324C2×C3×C5#57811yescharacteristic
325C23×C3—21no—
326C3×Q8—22no—
327C3×Q8—22no—
328C3×Q8—22no—
329C3×D4—21no—
330C3×D4—21no—
331C2×C3×C4—21no—
332C2×C3×C4—22no—
333C2×C3×C4—42no—
334C2×C3×C4C2×C22⋊D512yes—
335C2×C3×C4C2×C22⋊D512yes—
336C2×C3×C4C2×C22⋊D511yescharacteristic
337C2×C3×C4C2×C22⋊D511yescharacteristic
338C2×C3×C4C23×D511yescharacteristic
339C2×Dic3—101no—
340C2×Dic3—101no—
341C2×Dic3—102no—
342Dic6—52no—
343Dic6—101no—
344Dic6—202no—
345C3⋊C8—101no—
346C3⋊C8—102no—
347C3⋊C8—101no—
348C22×C5—41no—
349C22×C5—41no—
350C22×C5—41no—
351C22×C5C22×#6711yescharacteristic
352C4×C5—22no—
353C4×C5—22no—
354C4×C5—21no—
355C4×C5—41no—
356C4×C5—42no—
357C4×C5#95012yes—
358C4×C5D3×#4111yescharacteristic
359C4×C5#101011yescharacteristic
360Dic5—62no—
361Dic5—122no—
362Dic5—121no—
363C2×Q8—151no—
364C2×Q8—301no—
365C2×Q8#122011yescharacteristic
366C2×Q8C22×D1512yes—
367C2×D4C22×D1511yescharacteristic
368C22×C4—21no—
369Q16—304no—
370SD16—302no—
371C2×C8—151no—
372C2×C8—152no—
373C2×C8—151no—
374C4⋊C4—21no—
375C4⋊C4—22no—
376C4⋊C4—21no—
377C4⋊C4—151no—
378C4⋊C4—152no—
379C4⋊C4—301no—
380C4⋊C4—302no—
381C4⋊C4—302no—
382C4⋊C4—302no—
383C4⋊C4—301no—
384C4⋊C4#122011yescharacteristic
385C4⋊C4#122012yes—
386C4⋊C4C22×D1511yescharacteristic
387C22⋊C4—22no—
388C22⋊C4—21no—
389C22⋊C4—22no—
390C42—22no—
391C42—302no—
392C42—301no—
393C42C22×D1511yescharacteristic
394C42C22×D1512yes—
395C3×C5#336411yescharacteristic, complemented, Hall
396C22×C3—41no—
397C22×C3—41no—
398C22×C3—41no—
399C22×C3C22×C22⋊D511yescharacteristic
400C3×C4—22no—
401C3×C4—22no—
402C3×C4—21no—
403C3×C4—41no—
404C3×C4—42no—
405C3×C4#409612yes—
406C3×C4D5×#4111yescharacteristic
407C3×C4#415011yescharacteristic
408Dic3—102no—
409Dic3—202no—
410Dic3—201no—
411C2×C5—81no—
412C2×C5C2×#95011yescharacteristic
413C2×C5#583111yescharacteristic
414C2×C5#583811yescharacteristic
415C23—21no—
416Q8—22no—
417Q8—22no—
418Q8—22no—
419Q8—152no—
420Q8—301no—
421Q8—602no—
422D4—21no—
423D4—21no—
424C2×C4—21no—
425C2×C4—22no—
426C2×C4—42no—
427C2×C4—301no—
428C2×C4—301no—
429C2×C4—302no—
430C2×C4C2×#122012yes—
431C2×C4C2×#122011yescharacteristic
432C2×C4C2×#122012yes—
433C2×C4C2×#122011yescharacteristic
434C2×C4C23×D1511yescharacteristic
435C8—301no—
436C8—302no—
437C8—301no—
438C2×C3—81no—
439C2×C3C2×#409611yescharacteristic
440C2×C3#6616511yescharacteristic
441C2×C3#6617211yescharacteristic
442C5#8367611yescharacteristic, complemented, Sylow
443C22—41no—
444C22—41no—
445C22—41no—
446C22C22×#122011yescharacteristic
447C4—22no—
448C4—22no—
449C4—21no—
450C4—41no—
451C4—42no—
452C4—302no—
453C4—602no—
454C4—601no—
455C4#9183912yes—
456C4#41×D1511yescharacteristic
457C4#9210111yescharacteristic
458C3#1061540711yescharacteristic, complemented, Sylow
459C2—81no—
460C2C2×#9183911yescharacteristic
461C2#1175659011yescharacteristic
462C2#1175659711yescharacteristic
subgroupG/H
centerC22C22×#1220
commutator, cosocleC2×C3×C4×C5C24
FrattiniC2×C4C23×D15
FittingC3×C5×#556C2
socleC22×C3×C5C22×D4
prime radicalC2×C3×C5×D4C23
hypercenter#556D15
hypocenterC3×C5#3364
LCSG ▷ C2×C3×C4×C5 ▷ C2×C3×C5 ▷ C3×C5
UCSC1 ◁ C22 ◁ C4×Q8 ◁ #556

Complements

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

NH ≅ G/Ntype
2. #11756326C2semidirect
3. #11751191C2semidirect
4. #11751180C2semidirect
7. C4×#6718C2semidirect
8. Q8×C3⋊#168C2semidirect
9. #11751123C2semidirect
395. C3×C5#3364semidirect
442. C5#83676semidirect
458. C3#10615407semidirect

Maximal subgroups

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

subgroupindex
C3×C5×#5562
#117563262
#117511912
#117511802
#117511572
#117511552
C4×#67182
Q8×C3⋊#1682
#117511232
C4×#67162
#117511082
#106154073
#836765