Finite Group Databasebeta

← #14322486

Subgroups — #14322486

Order 1920.

1568 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
1C22×C3×Q8×D5C211yesmaximal, characteristic
2C2×#91358C211yesmaximal, characteristic
3C2×#91357C211yesmaximal, characteristic
4#11751330C211yesmaximal, characteristic, complemented
5C5×#5229C211yesmaximal, characteristic, complemented
6#11748737C214yesmaximal
7#11748672C214yesmaximal, complemented
8#11748610C211yesmaximal, characteristic, complemented
9#11748609C211yesmaximal, characteristic, complemented
10#10608341—31nomaximal, Hall
11C22×C3×C5×Q8C2211yescharacteristic
12C2×C3×Q8×D5—24no—
13C2×C3×Q8×D5—24no—
14C2×C3×Q8×D5C418yescomplemented
15C22×C3×C4×D5C2212yes—
16C22×C3×C4×D5C2211yescharacteristic
17C22×C3×Dic10C2211yescharacteristic
18C22×C3×Dic10C2212yes—
19C2×C4⋊Dic15C2211yescharacteristic
20C2×C4×Dic15C2211yescharacteristic
21C2×C5×C4⋊Dic3C2211yescharacteristic
22C2×C4×C5×Dic3C2211yescharacteristic
23C2×#6668C2212yes—
24C2×#6665C2212yes—
25C2×#6665C2212yes—
26C2×Dic3×Dic5C2212yes—
27#91358—24no—
28#91357—24no—
29#91115C2214yes—
30C5×C3⋊#95C2214yes—
31#90994C2214yes—
32#90994C2214yes—
33#84062—51nomaximal, Hall
34C22×Q8×D5D311yescharacteristic
35C2×#4038—31no—
36C2×#4018—31no—
37C5×#391—31no—
38#65519—31no—
39#65263—31no—
40#65218—31no—
41#64953—34no—
42#64935—34no—
43C23×C3×D5C2311yescharacteristic
44C2×C3×C5×Q8—22no—
45C2×C3×C5×Q8—22no—
46C2×C3×C5×Q8C2×C414yes—
47C22×C3×C4×C5C2312yes—
48C22×C3×C4×C5C2311yescharacteristic
49C22×Dic15C2312yes—
50C22×Dic15C2312yes—
51C22×C5×Dic3C2312yes—
52C22×C5×Dic3C2312yes—
53C22×C3×Dic5C2312yes—
54C22×C3×Dic5C2311yescharacteristic
55C3×Q8×D5—232no—
56C2×C3×C4×D5—24no—
57C2×C3×C4×D5—24no—
58C2×C3×C4×D5C2×C418yes—
59C2×C3×C4×D5C2×C414yes—
60C2×C3×C4×D5D414yes—
61C2×C3×C4×D5D414yes—
62C2×C3×Dic10—22no—
63C2×C3×Dic10—22no—
64C2×C3×Dic10—24no—
65C2×C3×Dic10—24no—
66C2×C3×Dic10C2×C414yes—
67C2×C3×Dic10C2×C418yes—
68C4⋊Dic15—22no—
69C4×Dic15—22no—
70C5×C4⋊Dic3—22no—
71C4×C5×Dic3—22no—
72#6668—24no—
73#6665—24no—
74#6665—24no—
75Dic3×Dic5—24no—
76C23×C3×Q8—51no—
77C2×#926—51no—
78C2×#926—51no—
79#5229—51no—
80#5229—51no—
81#5229—51no—
82#5229—54no—
83#5229D511yescharacteristic, complemented
84#5211—54no—
85C22×C5×Q8C2×D311yescharacteristic
86C2×Q8×D5—24no—
87C2×Q8×D5—24no—
88C2×Q8×D5Dic318yescomplemented
89C22×C4×D5C2×D312yes—
90C22×C4×D5C2×D311yescharacteristic
91C22×Dic10C2×D312yes—
92C22×Dic10C2×D311yescharacteristic
93C2×C5×C4⋊C4—31no—
94C2×C42×C5—31no—
95C2×#685—32no—
96C2×#685—32no—
97C2×C4⋊Dic5—31no—
98C2×#683—32no—
99C2×C4×Dic5—31no—
100C2×C4×Dic5—32no—
101#4038—64no—
102#4018—64no—
103C5×#95—34no—
104#3963—34no—
105#3963—34no—
106#3963—34no—
107#2310—151noSylow
108C23×C3×C5C2411yescharacteristic
109C22×C3×D5—22no—
110C22×C3×D5—22no—
111C22×C3×D5C22×C412yes—
112C22×C3×D5C2×D412yes—
113C22×C3×D5C2×D412yes—
114C3×C5×Q8—28no—
115C2×C3×C4×C5—22no—
116C2×C3×C4×C5—22no—
117C2×C3×C4×C5C22×C414yes—
118C2×C3×C4×C5C22×C412yes—
119C2×C3×C4×C5C2×D412yes—
120C2×C3×C4×C5C2×D412yes—
121C2×Dic15—22no—
122C2×Dic15—22no—
123C2×Dic15—24no—
124C2×Dic15—22no—
125C2×Dic15—22no—
126C2×C5×Dic3—22no—
127C2×C5×Dic3—22no—
128C2×C5×Dic3—22no—
129C2×C5×Dic3—22no—
130C2×C5×Dic3—24no—
131C2×C3×Dic5—22no—
132C2×C3×Dic5—22no—
133C2×C3×Dic5C22×C414yes—
134C2×C3×Dic5C22×C412yes—
135C2×C3×Dic5C2×D412yes—
136C2×C3×Dic5C2×D412yes—
137C3×C4×D5—216no—
138C3×C4×D5C22⋊C4116yes—
139C3×Dic10—28no—
140C3×Dic10—216no—
141C22×C3×Q8—58no—
142C22×C3×Q8—52no—
143C22×C3×Q8—51no—
144C22×C3×Q8—104no—
145C22×C3×Q8—104no—
146C22×C3×Q8C2×D511yescharacteristic
147C23×C3×C4—51no—
148C23×C3×C4—52no—
149C2×#217—52no—
150C2×#217—52no—
151#926—104no—
152#926—104no—
153C2×C4⋊Dic3—51no—
154C2×C4⋊Dic3—52no—
155C2×C4⋊Dic3C2×D511yescharacteristic
156C2×C4×Dic3—51no—
157C2×C4×Dic3—52no—
158C2×C4×Dic3C2×D511yescharacteristic
159C3⋊#95—54no—
160C3⋊#95—54no—
161C3⋊#95—54no—
162C3⋊#95C2×D514yes—
163C23×D5C22×D311yescharacteristic
164C2×C5×Q8—22no—
165C2×C5×Q8—22no—
166C2×C5×Q8C2×Dic314yes—
167C22×C4×C5—32no—
168C22×C4×C5—32no—
169C22×C4×C5C22×D312yes—
170C22×C4×C5C22×D311yescharacteristic
171C22×Dic5—32no—
172C22×Dic5—32no—
173C22×Dic5C22×D312yes—
174C22×Dic5C22×D311yescharacteristic
175Q8×D5—232no—
176C2×C4×D5—24no—
177C2×C4×D5—24no—
178C2×C4×D5C2×Dic318yes—
179C2×C4×D5C2×Dic314yes—
180C2×C4×D5#6714yes—
181C2×C4×D5#6714yes—
182C2×Dic10—22no—
183C2×Dic10—22no—
184C2×Dic10—24no—
185C2×Dic10—24no—
186C2×Dic10C2×Dic314yes—
187C2×Dic10C2×Dic318yes—
188C5×C4⋊C4—62no—
189C42×C5—62no—
190#685—64no—
191#685—64no—
192C4⋊Dic5—62no—
193#683—64no—
194C4×Dic5—62no—
195C4×Dic5—64no—
196C23×Q8—51no—
197C2×#117—151no—
198C2×#117—151no—
199#391—31no—
200#391—154no—
201#391—151no—
202#391—151no—
203#391—151no—
204#381—154no—
205C22×C3×C5C23×C411yescharacteristic
206C22×C3×C5C22×D411yescharacteristic
207C22×C3×C5C22×D411yescharacteristic
208C22×C3×C52−1+412yes—
209C22×C3×C52−1+412yes—
210C2×C3×D5—24no—
211C2×C3×D5—22no—
212C2×C3×D5—24no—
213C2×C3×D5—22no—
214C2×C3×D5C2×C22⋊C414yes—
215C3×C4×C5—24no—
216C3×C4×C5C2×C22⋊C414yes—
217Dic15—42no—
218Dic15—42no—
219C3×Dic5—24no—
220C3×Dic5C2×C22⋊C414yes—
221C5×Dic3—42no—
222C5×Dic3—42no—
223C24×C3—51no—
224C2×C3×Q8—22no—
225C2×C3×Q8—22no—
226C2×C3×Q8—58no—
227C2×C3×Q8—54no—
228C2×C3×Q8—104no—
229C2×C3×Q8—104no—
230C2×C3×Q8—1032no—
231C2×C3×Q8—102no—
232C2×C3×Q8—102no—
233C2×C3×Q8C4×D514yes—
234C22×C3×C4—51no—
235C22×C3×C4—54no—
236C22×C3×C4—54no—
237C22×C3×C4—54no—
238C22×C3×C4—52no—
239C22×C3×C4—58no—
240C22×C3×C4—104no—
241C22×C3×C4—104no—
242C22×C3×C4C22×D512yes—
243C22×C3×C4C22×D511yescharacteristic
244C22×Dic3—52no—
245C22×Dic3—52no—
246C22×Dic3C22×D512yes—
247C22×Dic3C22×D512yes—
248#217—104no—
249#217—104no—
250C4⋊Dic3—22no—
251C4⋊Dic3—104no—
252C4⋊Dic3—102no—
253C4×Dic3—22no—
254C4×Dic3—104no—
255C4×Dic3—102no—
256C23×C5C23×D311yescharacteristic
257C22×D5—22no—
258C22×D5—22no—
259C22×D5C22×Dic312yes—
260C22×D5C2×#6712yes—
261C22×D5C2×#6712yes—
262C5×Q8—28no—
263C2×C4×C5—22no—
264C2×C4×C5—22no—
265C2×C4×C5—62no—
266C2×C4×C5—62no—
267C2×C4×C5—64no—
268C2×C4×C5—62no—
269C2×C4×C5—62no—
270C2×C4×C5C22×Dic314yes—
271C2×C4×C5C22×Dic312yes—
272C2×C4×C5C2×#6712yes—
273C2×C4×C5C2×#6712yes—
274C2×Dic5—22no—
275C2×Dic5—22no—
276C2×Dic5—62no—
277C2×Dic5—64no—
278C2×Dic5—62no—
279C2×Dic5—62no—
280C2×Dic5—62no—
281C2×Dic5C22×Dic314yes—
282C2×Dic5C22×Dic312yes—
283C2×Dic5C2×#6712yes—
284C2×Dic5C2×#6712yes—
285C4×D5—216no—
286C4×D5#217116yes—
287Dic10—28no—
288Dic10—216no—
289C22×Q8—58no—
290C22×Q8—52no—
291C22×Q8—51no—
292C22×Q8—104no—
293C22×Q8—104no—
294C22×Q8D3×D511yescharacteristic
295C23×C4—51no—
296C23×C4—52no—
297#117—304no—
298#117—304no—
299C2×C4⋊C4—31no—
300C2×C4⋊C4—152no—
301C2×C4⋊C4—151no—
302C2×C22⋊C4—152no—
303C2×C22⋊C4—152no—
304C2×C42—31no—
305C2×C42—152no—
306C2×C42—151no—
307#95—34no—
308#95—154no—
309#95—154no—
310#95—154no—
311C2×C3×C5C22×C22⋊C411yescharacteristic
312C2×C3×C5#51912yes—
313C2×C3×C5#53611yescharacteristic
314C2×C3×C5#53612yes—
315C2×C3×C5#53611yescharacteristic
316C3×D5—24no—
317C23×C3—52no—
318C23×C3—52no—
319C23×C3—52no—
320C23×C3—102no—
321C23×C3—102no—
322C23×C3C23×D511yescharacteristic
323C3×Q8—28no—
324C3×Q8—1016no—
325C3×Q8—108no—
326C2×C3×C4—22no—
327C2×C3×C4—22no—
328C2×C3×C4—52no—
329C2×C3×C4—52no—
330C2×C3×C4—52no—
331C2×C3×C4—516no—
332C2×C3×C4—54no—
333C2×C3×C4—1016no—
334C2×C3×C4—102no—
335C2×C3×C4—102no—
336C2×C3×C4C2×C4×D514yes—
337C2×C3×C4C2×C4×D512yes—
338C2×C3×C4D4×D512yes—
339C2×C3×C4D4×D512yes—
340C2×Dic3—22no—
341C2×Dic3—22no—
342C2×Dic3—22no—
343C2×Dic3—22no—
344C2×Dic3—24no—
345C2×Dic3—102no—
346C2×Dic3—102no—
347C2×Dic3—104no—
348C2×Dic3—102no—
349C2×Dic3—102no—
350C22×C5#100712yes—
351C22×C5#100712yes—
352C22×C5C23×Dic311yescharacteristic
353C22×C5C22×#6711yescharacteristic
354C22×C5C22×#6711yescharacteristic
355C2×D5—24no—
356C2×D5—22no—
357C2×D5—24no—
358C2×D5—22no—
359C2×D5C2×#21714yes—
360C4×C5—24no—
361C4×C5—122no—
362C4×C5—122no—
363C4×C5C2×#21714yes—
364Dic5—24no—
365Dic5—122no—
366Dic5—122no—
367Dic5C2×#21714yes—
368C24—51no—
369C2×Q8—22no—
370C2×Q8—22no—
371C2×Q8—58no—
372C2×Q8—54no—
373C2×Q8—104no—
374C2×Q8—104no—
375C2×Q8—1032no—
376C2×Q8—102no—
377C2×Q8—102no—
378C2×Q8D5×Dic314yes—
379C22×C4—32no—
380C22×C4—32no—
381C22×C4—51no—
382C22×C4—54no—
383C22×C4—54no—
384C22×C4—54no—
385C22×C4—52no—
386C22×C4—58no—
387C22×C4—104no—
388C22×C4—104no—
389C22×C4—152no—
390C22×C4—152no—
391C22×C4C2×D3×D512yes—
392C22×C4C2×D3×D511yescharacteristic
393C4⋊C4—62no—
394C4⋊C4—304no—
395C4⋊C4—302no—
396C22⋊C4—304no—
397C22⋊C4—304no—
398C42—62no—
399C42—304no—
400C42—302no—
401C3×C5#231011yescharacteristic, complemented, Hall
402C22×C3—54no—
403C22×C3—102no—
404C22×C3—104no—
405C22×C3—104no—
406C22×C3—102no—
407C22×C3C22×C4×D511yescharacteristic
408C22×C3C2×D4×D511yescharacteristic
409C22×C3C2×D4×D511yescharacteristic
410C22×C3#414712yes—
411C22×C3#415012yes—
412C3×C4—24no—
413C3×C4—54no—
414C3×C4—104no—
415C3×C4D5×C22⋊C414yes—
416Dic3—42no—
417Dic3—42no—
418Dic3—202no—
419Dic3—202no—
420C2×C5#581212yes—
421C2×C5#581611yescharacteristic
422C2×C5#581612yes—
423C2×C5#581611yescharacteristic
424C2×C5C22×#21711yescharacteristic
425D5—24no—
426C23—52no—
427C23—52no—
428C23—52no—
429C23—102no—
430C23—102no—
431C23C22×D3×D511yescharacteristic
432Q8—28no—
433Q8—1016no—
434Q8—108no—
435C2×C4—22no—
436C2×C4—22no—
437C2×C4—52no—
438C2×C4—52no—
439C2×C4—52no—
440C2×C4—516no—
441C2×C4—54no—
442C2×C4—62no—
443C2×C4—62no—
444C2×C4—64no—
445C2×C4—62no—
446C2×C4—62no—
447C2×C4—1016no—
448C2×C4—102no—
449C2×C4—102no—
450C2×C4—304no—
451C2×C4—302no—
452C2×C4—302no—
453C2×C4—302no—
454C2×C4—302no—
455C2×C4C2×D5×Dic314yes—
456C2×C4C2×D5×Dic312yes—
457C2×C4D5×#6712yes—
458C2×C4D5×#6712yes—
459C2×C3—104no—
460C2×C3C2×D5×C22⋊C411yescharacteristic
461C2×C3#6590612yes—
462C2×C3#6596212yes—
463C2×C3#6600911yescharacteristic
464C2×C3#6605811yescharacteristic
465C5#8406211yescharacteristic, complemented, Sylow
466C22—54no—
467C22—102no—
468C22—104no—
469C22—104no—
470C22—102no—
471C22#9202912yes—
472C22#9203012yes—
473C22C22×D5×Dic311yescharacteristic
474C22C2×D5×#6711yescharacteristic
475C22C2×D5×#6711yescharacteristic
476C4—24no—
477C4—54no—
478C4—104no—
479C4—122no—
480C4—122no—
481C4—602no—
482C4—602no—
483C4D5×#21714yes—
484C3#1060834111yescharacteristic, complemented, Sylow
485C2—104no—
486C2#1175512511yescharacteristic
487C2#1175514012yes—
488C2#1175522212yes—
489C2#1175522411yescharacteristic
490C2C2×D5×#21711yescharacteristic
subgroupG/H
center, FrattiniC23C22×D3×D5
commutatorC22×C3×C5C23×C4
FittingC22×C3×C5×Q8C22
socle, cosocleC23×C3×C5C24
prime radicalC23×C3×D5C23
hypercenterC22×Q8D3×D5
hypocenterC3×C5#2310
LCSG ▷ C22×C3×C5 ▷ C3×C5
UCSC1 ◁ C23 ◁ C22×Q8

Complements

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

NH ≅ G/Ntype
4. #11751330C2semidirect
5. C5×#5229C2semidirect
7. #11748672C2semidirect
8. #11748610C2semidirect
9. #11748609C2semidirect
14. C2×C3×Q8×D5C4semidirect
83. #5229D5semidirect
88. C2×Q8×D5Dic3semidirect
401. C3×C5#2310semidirect
465. C5#84062semidirect
484. C3#10608341semidirect

Maximal subgroups

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

subgroupindex
C22×C3×Q8×D52
C2×#913582
C2×#913572
#117513302
C5×#52292
#117487372
#117486722
#117486102
#117486092
#106083413
#840625