Finite Group Databasebeta

← #14472146

Subgroups — #14472146

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
1#11756329C211yesmaximal, characteristic, complemented
2#11752480C211yesmaximal, characteristic
3C4×C5×#215C211yesmaximal, characteristic
4C3×#65320C211yesmaximal, characteristic, complemented
5#11749328C212yesmaximal
6#11749263C212yesmaximal, complemented
7#11747387C212yesmaximal
8#11747302C212yesmaximal, complemented
9#11746964C211yesmaximal, characteristic
10#11746920C211yesmaximal, characteristic, complemented
11#11746855C211yesmaximal, characteristic, complemented
12#10611354—31nomaximal, Hall
13C3×C4×C5×Q8C2211yescharacteristic
14Q8×Dic15C2212yes—
15#91779—22no—
16#91779C2212yes—
17#91758—21no—
18#91758C2211yescharacteristic
19C4×Dic30—21no—
20C4×Dic30C2211yescharacteristic
21C2×C5×#215C2211yescharacteristic
22C4×C5×D12C2211yescharacteristic
23C2×C3×#689C2212yes—
24C3×#4015C2211yescharacteristic
25#91370C2212yes—
26#91365—21no—
27#91341—22no—
28#91338—21no—
29#91331C2212yes—
30C5×#835C2211yescharacteristic
31C5×#809C2211yescharacteristic
32C5×#808C2211yescharacteristic
33C4×C5×C3⋊C8C2211yescharacteristic
34C3×#3942C2212yes—
35C3×C4⋊#168C2211yescharacteristic
36#90988C2212yes—
37#90983C2212yes—
38#90971C2212yes—
39#90971C2212yes—
40#90967C2212yes—
41#90951C2212yes—
42#80463—51nomaximal, Hall
43#65904—31no—
44#65876—31no—
45C4×C5×SD16—31no—
46#65455—32no—
47#65320D311yescharacteristic, complemented
48#65296—31no—
49#65146—32no—
50#65079—32no—
51#65057—32no—
52#64970—31no—
53C8×Dic10—31no—
54C2×C3×C5×Q8C2311yescharacteristic
55C2×Dic30—21no—
56C2×Dic30D411yescharacteristic
57C2×C5×D12C2311yescharacteristic
58C2×C4×C5×D3—21no—
59C2×C3×Dic10C2312yes—
60C3×C5×C4⋊C4—21no—
61C3×C5×C4⋊C4C2311yescharacteristic
62C3×C42×C5—21no—
63C3×C42×C5C2311yescharacteristic
64C4⋊Dic15—21no—
65C4⋊Dic15—22no—
66C4⋊Dic15—22no—
67C4⋊Dic15D411yescharacteristic
68C4⋊Dic15C2312yes—
69#6712—22no—
70#6712D412yes—
71C4×Dic15—22no—
72C4×Dic15C2312yes—
73C5×#215—22no—
74C5×(C2×D3)⋊C4—21no—
75C5×C4⋊Dic3C2311yescharacteristic
76C2×C5×C3⋊C8C2311yescharacteristic
77C2×C5×C3⋊C8C2311yescharacteristic
78C3×#689—24no—
79C3×C4⋊Dic5—21no—
80C3×C4⋊Dic5—21no—
81C2×C3×#168C2312yes—
82#6668—22no—
83(C2×D3)⋊Dic5—22no—
84(C2×D3)⋊Dic5—22no—
85Dic3×Dic5—22no—
86#5732—51no—
87#5570—51no—
88C3×#462—51no—
89#5185—52no—
90Dic3×Q16—52no—
91#5079—51no—
92C4×#215D511yescharacteristic
93#4846—51no—
94#4830—51no—
95#4800—52no—
96#4796—52no—
97C2×C5×SD16—31no—
98C4×C5×Q8C2×D311yescharacteristic
99C4×C5×D4—31no—
100Q8×Dic5—32no—
101C2×#689C2×D312yes—
102#4082—32no—
103#4072—61no—
104#4070—61no—
105#4036—32no—
106#4036—62no—
107#4033—32no—
108#4025—62no—
109#4015—31no—
110#4015—61no—
111#4015C2×D311yescharacteristic
112C4×Dic10—31no—
113C4×Dic10—61no—
114C5×#106—31no—
115C5×#103—31no—
116C5×#102—31no—
117C4×C5×C8—31no—
118#3964—32no—
119#3948—32no—
120#3948—32no—
121#3946—32no—
122C8×Dic5—32no—
123#3942C2×D312yes—
124#3939—32no—
125C4⋊#168C2×D311yescharacteristic
126#3364—151noSylow
127C22×C5×D3—21no—
128C3×C5×Q8—21no—
129C3×C5×Q8#4112yes—
130C2×C3×C4×C5—21no—
131C2×C3×C4×C5—21no—
132C2×C3×C4×C5C2×D411yescharacteristic
133C2×C3×C4×C5C2×D411yescharacteristic
134C2×C3×C4×C5C2411yescharacteristic
135C2×Dic15—22no—
136C2×Dic15C2×D412yes—
137C2×Dic15C2×D412yes—
138Dic30—22no—
139Dic30—42no—
140C2×C5×Dic3—21no—
141C5×D12—21no—
142C5×D12—21no—
143C4×C5×D3—42no—
144C2×C3×Dic5—22no—
145C3×Dic10—22no—
146C3×Dic10—22no—
147C3×#168—22no—
148C5×C3⋊C8—21no—
149C5×C3⋊C8—21no—
150C2×C3×Q16—52no—
151C3×C4⋊Q8—51no—
152C3×C4×Q8C2×D511yescharacteristic
153#947—52no—
154Q8×Dic3—52no—
155Q8×Dic3—52no—
156C2×#215C2×D511yescharacteristic
157#898—102no—
158#897—101no—
159#892—101no—
160#890—52no—
161#890—102no—
162C4×D12C2×D511yescharacteristic
163#869—51no—
164#869—101no—
165C4×Dic6—51no—
166C4×Dic6—101no—
167C3×C4⋊C8—51no—
168C3×#103—52no—
169#835—52no—
170#835—52no—
171#835C2×D511yescharacteristic
172#821—52no—
173#818—52no—
174Dic3⋊C8—52no—
175C8×Dic3—52no—
176#809C2×D511yescharacteristic
177#808C2×D511yescharacteristic
178C4×C3⋊C8C2×D511yescharacteristic
179C2×C5×Q8C22×D311yescharacteristic
180C2×C5×D4—31no—
181C22×C4×C5—61no—
182C2×Dic10—31no—
183C2×Dic10—61no—
184C2×Dic10C22×D312yes—
185C5×SD16—62no—
186C2×C5×C8—31no—
187C2×C5×C8—31no—
188C5×C4⋊C4—21no—
189C5×C4⋊C4—31no—
190C5×C4⋊C4C22×D311yescharacteristic
191C5×C22⋊C4—61no—
192C42×C5—21no—
193C42×C5C22×D311yescharacteristic
194C22⋊Dic5—62no—
195C22⋊Dic5—62no—
196#689—24no—
197C4⋊Dic5—21no—
198C4⋊Dic5—21no—
199C4⋊Dic5—31no—
200C4⋊Dic5—32no—
201C4⋊Dic5—62no—
202C4⋊Dic5—62no—
203C4⋊Dic5—61no—
204#683—32no—
205#683—62no—
206#683—62no—
207C4×Dic5—32no—
208C4×Dic5—62no—
209C4×Dic5—62no—
210C2×#168C22×D312yes—
211#557—151no—
212#556—151no—
213#488—152no—
214#487—151no—
215#479—152no—
216#474—151no—
217#464—152no—
218#462—51no—
219C8×Q8—151no—
220C4×Q16—152no—
221C4×SD16—31no—
222C22×C3×C5C22×D411yescharacteristic
223C2×C5×D3—41no—
224C2×C5×D3—41no—
225C2×C5×D3—41no—
226C3×C4×C5—22no—
227C3×C4×C5—21no—
228C3×C4×C5—41no—
229C3×C4×C5#13512yes—
230C3×C4×C5C2×#4111yescharacteristic
231C3×C4×C52−1+411yescharacteristic
232Dic15—22no—
233Dic15—22no—
234Dic15—42no—
235C3×Dic5—42no—
236C5×Dic3—41no—
237C2×C3×Q8—52no—
238C2×C3×Q8C22×D511yescharacteristic
239C2×D12C22×D511yescharacteristic
240C2×C4×D3—21no—
241C2×Dic6—51no—
242C2×Dic6—101no—
243C3×Q16—104no—
244C2×C3×C8—52no—
245C3×C4⋊C4—21no—
246C3×C4⋊C4—101no—
247C3×C4⋊C4—101no—
248C3×C4⋊C4C22×D511yescharacteristic
249C3×C42—21no—
250C3×C42C22×D511yescharacteristic
251#215—22no—
252(C2×D3)⋊C4—21no—
253(C2×D3)⋊C4—102no—
254(C2×D3)⋊C4—102no—
255C4⋊Dic3—51no—
256C4⋊Dic3—52no—
257C4⋊Dic3—102no—
258C4⋊Dic3—102no—
259C4⋊Dic3—102no—
260C4⋊Dic3—101no—
261C4⋊Dic3C22×D511yescharacteristic
262Dic3⋊C4—52no—
263Dic3⋊C4—102no—
264C4×Dic3—52no—
265C4×Dic3—102no—
266C4×Dic3—102no—
267C2×C3⋊C8C22×D511yescharacteristic
268C2×C3⋊C8C22×D511yescharacteristic
269C23×C5—61no—
270C5×Q8—21no—
271C5×Q8#23712yes—
272C5×D4—61no—
273C5×D4—61no—
274C2×C4×C5—21no—
275C2×C4×C5—21no—
276C2×C4×C5—61no—
277C2×C4×C5—122no—
278C2×C4×C5D3×D411yescharacteristic
279C2×C4×C5D3×D411yescharacteristic
280C2×C4×C5C23×D311yescharacteristic
281C2×Dic5—22no—
282C2×Dic5—32no—
283C2×Dic5—32no—
284C2×Dic5—62no—
285Dic10—22no—
286Dic10—22no—
287Dic10—62no—
288Dic10—122no—
289C5×C8—61no—
290C5×C8—61no—
291#168—22no—
292C2×Q16—52no—
293C2×SD16—31no—
294C4⋊Q8—51no—
295C4⋊Q8—151no—
296C4⋊Q8—301no—
297#126—302no—
298#125—152no—
299#125—302no—
300#124—152no—
301C22⋊Q8—301no—
302C4×Q8—151no—
303C4×Q8—152no—
304C4×Q8—152no—
305C4×Q8—301no—
306C4×Q8D3×D511yescharacteristic
307C4×D4—31no—
308#117—301no—
309#107—152no—
310#106—31no—
311C4⋊C8—51no—
312C4⋊C8—152no—
313#103—31no—
314#103—52no—
315#103—152no—
316#103—152no—
317#102—31no—
318#102—152no—
319C4×C8—31no—
320C4×C8—152no—
321C2×C3×C5#54811yescharacteristic
322C2×C3×C5C2×#13511yescharacteristic
323C2×C3×C5#57811yescharacteristic
324C5×D3—81no—
325C22×D3—21no—
326C3×Q8—21no—
327C3×Q8—102no—
328C3×Q8—102no—
329C3×Q8#70912yes—
330C2×C3×C4—21no—
331C2×C3×C4—21no—
332C2×C3×C4—102no—
333C2×C3×C4D4×D511yescharacteristic
334C2×C3×C4D4×D511yescharacteristic
335C2×C3×C4C23×D511yescharacteristic
336C2×Dic3—21no—
337C2×Dic3—52no—
338C2×Dic3—52no—
339C2×Dic3—102no—
340D12—21no—
341D12—21no—
342C4×D3—42no—
343Dic6—102no—
344Dic6—202no—
345C3×C8—102no—
346C3⋊C8—21no—
347C3⋊C8—21no—
348C22×C5—121no—
349C22×C5—121no—
350C22×C5—121no—
351C22×C5C2×D3×D411yescharacteristic
352C4×C5—21no—
353C4×C5—22no—
354C4×C5—41no—
355C4×C5—121no—
356C4×C5#91912yes—
357C4×C5C2×#23711yescharacteristic
358C4×C5#101011yescharacteristic
359Dic5—42no—
360Dic5—62no—
361Dic5—62no—
362Dic5—122no—
363C2×Q8—52no—
364C2×Q8—151no—
365C2×Q8—301no—
366C2×Q8C2×D3×D511yescharacteristic
367C2×D4—31no—
368C22×C4—61no—
369Q16—104no—
370SD16—62no—
371C2×C8—31no—
372C2×C8—31no—
373C2×C8—52no—
374C4⋊C4—21no—
375C4⋊C4—31no—
376C4⋊C4—101no—
377C4⋊C4—101no—
378C4⋊C4—151no—
379C4⋊C4—152no—
380C4⋊C4—152no—
381C4⋊C4—302no—
382C4⋊C4—302no—
383C4⋊C4—302no—
384C4⋊C4—302no—
385C4⋊C4—301no—
386C4⋊C4C2×D3×D511yescharacteristic
387C22⋊C4—61no—
388C22⋊C4—302no—
389C22⋊C4—302no—
390C42—21no—
391C42—152no—
392C42—302no—
393C42—302no—
394C42C2×D3×D511yescharacteristic
395C3×C5#336411yescharacteristic, complemented, Hall
396C22×C3C2×D4×D511yescharacteristic
397C2×D3—41no—
398C2×D3—41no—
399C2×D3—41no—
400C3×C4—21no—
401C3×C4—22no—
402C3×C4—41no—
403C3×C4—202no—
404C3×C4#406212yes—
405C3×C4C2×#70911yescharacteristic
406C3×C4#415011yescharacteristic
407Dic3—41no—
408Dic3—102no—
409Dic3—102no—
410Dic3—202no—
411C2×C5—241no—
412C2×C5#559811yescharacteristic
413C2×C5C2×#91911yescharacteristic
414C2×C5#577911yescharacteristic
415C23—61no—
416Q8—21no—
417Q8—102no—
418Q8—102no—
419Q8—302no—
420Q8—602no—
421Q8#676512yes—
422D4—61no—
423D4—61no—
424C2×C4—21no—
425C2×C4—21no—
426C2×C4—61no—
427C2×C4—102no—
428C2×C4—122no—
429C2×C4—152no—
430C2×C4—152no—
431C2×C4—302no—
432C2×C4#679011yescharacteristic
433C2×C4#679011yescharacteristic
434C2×C4C22×D3×D511yescharacteristic
435C8—61no—
436C8—61no—
437C8—102no—
438C2×C3#6588611yescharacteristic
439C2×C3C2×#406211yescharacteristic
440C2×C3#6610511yescharacteristic
441D3—81no—
442C5#8046311yescharacteristic, complemented, Sylow
443C22—121no—
444C22—121no—
445C22—121no—
446C22C2×#679011yescharacteristic
447C4—21no—
448C4—22no—
449C4—41no—
450C4—121no—
451C4—202no—
452C4—302no—
453C4—302no—
454C4—602no—
455C4#9152312yes—
456C4C2×#676511yescharacteristic
457C4#9200111yescharacteristic
458C3#1061135411yescharacteristic, complemented, Sylow
459C2—241no—
460C2#1175436911yescharacteristic
461C2C2×#9152311yescharacteristic
462C2#1175535311yescharacteristic
subgroupG/H
centerC22C2×#6790
commutator, cosocleC2×C3×C4×C5C24
FrattiniC2×C4C22×D3×D5
FittingC3×C4×C5×Q8C22
socleC22×C3×C5C22×D4
prime radicalC2×C5×D12C23
hypercenterC4×Q8D3×D5
hypocenterC3×C5#3364
LCSG ▷ C2×C3×C4×C5 ▷ C2×C3×C5 ▷ C3×C5
UCSC1 ◁ C22 ◁ C42 ◁ C4×Q8

Complements

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

NH ≅ G/Ntype
1. #11756329C2semidirect
4. C3×#65320C2semidirect
6. #11749263C2semidirect
8. #11747302C2semidirect
10. #11746920C2semidirect
11. #11746855C2semidirect
47. #65320D3semidirect
395. C3×C5#3364semidirect
442. C5#80463semidirect
458. C3#10611354semidirect

Maximal subgroups

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

subgroupindex
#117563292
#117524802
C4×C5×#2152
C3×#653202
#117493282
#117492632
#117473872
#117473022
#117469642
#117469202
#117468552
#106113543
#804635