Finite Group Databasebeta

← #14296822

Subgroups — #14296822

Order 1920.

1224 conjugacy classes of subgroups.

The subgroup lattice has 548 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#6782C211yesmaximal, characteristic
2C2×#91241C211yesmaximal, characteristic, complemented
3C2×C3×C22⋊#170C211yesmaximal, characteristic, complemented
4C2×#91171C212yesmaximal, complemented
5C2×(C2×D3)⋊#170C212yesmaximal, complemented
6#10604945—31nomaximal, Hall
7C22×#1220C411yescharacteristic
8C22×C5×#67C411yescharacteristic
9C23×C3×Dic5C2211yescharacteristic
10C22×#1204C411yescharacteristic
11C22×#1203C411yescharacteristic
12C22×#1200C2211yescharacteristic
13C2×#6782—22no—
14C2×#6782—22no—
15C2×#6782—24no—
16C2×#6782—24no—
17C22×D3×Dic5C2211yescharacteristic
18C22×C3⋊#170C2212yes—
19C22×C3×#170C2212yes—
20#91241—22no—
21C3×C22⋊#170—22no—
22#91171—24no—
23(C2×D3)⋊#170—24no—
24#69956—51nomaximal, Hall
25C22×#711—31no—
26C2×C22⋊#170—31no—
27C2×C22⋊#170—32no—
28C2×C22⋊#170D311yescharacteristic, complemented
29C2×#4003—32no—
30C24×C3×C5C2×C411yescharacteristic
31C23×D15C2×C411yescharacteristic
32C23×C5×D3C2×C411yescharacteristic
33C2×#1220—22no—
34C2×#1220—22no—
35C2×#1220—22no—
36C22×Dic15C2×C411yescharacteristic
37C2×C5×#67—22no—
38C2×C5×#67—22no—
39C2×C5×#67—22no—
40C22×C5×Dic3C2×C411yescharacteristic
41C22×C3×Dic5—22no—
42C22×C3×Dic5—22no—
43C22×C3×Dic5C2×C411yescharacteristic
44C22×C3×Dic5D412yes—
45C22×C3×Dic5D412yes—
46C22×C3×Dic5C2311yescharacteristic
47C2×#1204—22no—
48C2×#1204—22no—
49C2×#1204—22no—
50C2×#1203—22no—
51C2×#1203—22no—
52C2×#1203—22no—
53C2×#1200—21no—
54C2×#1200—22no—
55C2×#1200—21no—
56C2×#1200D414yes—
57#6782—216no—
58#6782—216no—
59C2×D3×Dic5—21no—
60C2×D3×Dic5—22no—
61C2×D3×Dic5—21no—
62C2×D3×Dic5D414yes—
63C2×C3⋊#170—22no—
64C2×C3⋊#170—22no—
65C2×C3⋊#170—22no—
66C2×C3×#170—22no—
67C2×C3×#170—22no—
68C2×C3×#170—22no—
69C22×#235—51no—
70C2×C3×C22⋊C8—51no—
71C2×#830—51no—
72C2×(C2×D3)⋊C8—52no—
73C2×(C2×D3)⋊C8—52no—
74C22×C5×D4—31no—
75C22×C22⋊D5—31no—
76C22×C22⋊D5—31no—
77C23×Dic5—31no—
78C23×Dic5C2×D311yescharacteristic
79C2×#711—64no—
80C2×#711—64no—
81C2×#711—62no—
82C2×#711—62no—
83C22×C4×D5—31no—
84C22×Dic10—31no—
85C22×#170—32no—
86C22×#170C2×D312yes—
87C22⋊#170—22no—
88C22⋊#170—64no—
89C22⋊#170—62no—
90#4003—64no—
91#1780—151noSylow
92C23×C3×C5—21no—
93C23×C3×C5—21no—
94C23×C3×C5—22no—
95C23×C3×C5—22no—
96C23×C3×C5C22⋊C412yes—
97C23×C3×C5C22×C411yescharacteristic
98C22×D15—21no—
99C22×D15—21no—
100C22×D15—22no—
101C22×D15—22no—
102C22×D15C22⋊C412yes—
103C22×C5×D3—21no—
104C22×C5×D3—21no—
105C22×C5×D3—22no—
106C22×C5×D3—22no—
107C22×C5×D3C22⋊C412yes—
108#1220—28no—
109C2×Dic15—21no—
110C2×Dic15—21no—
111C2×Dic15C22⋊C412yes—
112C5×#67—28no—
113C2×C5×Dic3—21no—
114C2×C5×Dic3—21no—
115C2×C5×Dic3C22⋊C412yes—
116C2×C3×Dic5—21no—
117C2×C3×Dic5—21no—
118C2×C3×Dic5—24no—
119C2×C3×Dic5—24no—
120C2×C3×Dic5C22⋊C412yes—
121C2×C3×Dic5C2×D411yescharacteristic
122C2×C3×Dic5C2×D412yes—
123C2×C3×Dic5C2×D412yes—
124C2×C3×Dic5C2×D411yescharacteristic
125#1204—28no—
126#1203—28no—
127#1200—24no—
128#1200—24no—
129D3×Dic5—24no—
130D3×Dic5—24no—
131C3⋊#170—42no—
132C3×#170—42no—
133C23×C3×C4—51no—
134C22×#67—51no—
135C22×#67AGL(1,𝔽5)11yescharacteristic
136C2×#235—104no—
137C2×#235—104no—
138C2×#235—102no—
139C2×#235—102no—
140C22×D12—51no—
141C22×C4×D3—51no—
142C22×C4×D3—51no—
143C22×Dic6—51no—
144C22×C3×C8—52no—
145C22×C3⋊C8—52no—
146C3×C22⋊C8—102no—
147#830—102no—
148(C2×D3)⋊C8—104no—
149(C2×D3)⋊C8—104no—
150C24×C5—31no—
151C24×C5C4×D311yescharacteristic
152C23×D5—31no—
153C2×C5×D4—62no—
154C2×C5×D4—62no—
155C2×C5×D4—62no—
156C22×C4×C5—31no—
157C2×C22⋊D5—62no—
158C2×C22⋊D5—62no—
159C2×C22⋊D5—62no—
160C2×C22⋊D5—62no—
161C2×C22⋊D5—62no—
162C2×C22⋊D5—62no—
163C22×Dic5—22no—
164C22×Dic5—22no—
165C22×Dic5—31no—
166C22×Dic5—34no—
167C22×Dic5—61no—
168C22×Dic5—62no—
169C22×Dic5—61no—
170C22×Dic5C4×D311yescharacteristic
171C22×Dic5D1212yes—
172C22×Dic5#6712yes—
173C22×Dic5C22×D311yescharacteristic
174#711—616no—
175#711—616no—
176C2×C4×D5—34no—
177C2×C4×D5—61no—
178C2×C4×D5—62no—
179C2×C4×D5—61no—
180C2×Dic10—62no—
181C2×Dic10—62no—
182C2×Dic10—62no—
183C2×#170—22no—
184C2×#170—22no—
185C2×#170—22no—
186C2×#170—62no—
187C2×#170—62no—
188C2×#170—62no—
189C22×#41—151no—
190C2×C22⋊C8—51no—
191C2×C22⋊C8—152no—
192C2×C22⋊C8—152no—
193C2×C22⋊C8—151no—
194C22×C3×C5—22no—
195C22×C3×C5—22no—
196C22×C3×C5—22no—
197C22×C3×C5—42no—
198C22×C3×C5—42no—
199C22×C3×C5C2×C22⋊C411yescharacteristic
200C22×C3×C5C2×C22⋊C411yescharacteristic
201C22×C3×C5C2×C22⋊C411yescharacteristic
202C22×C3×C5#13112yes—
203C22×C3×C5#13112yes—
204C2×D15—22no—
205C2×D15—24no—
206C2×D15—41no—
207C2×D15—42no—
208C2×D15—41no—
209C2×C5×D3—22no—
210C2×C5×D3—24no—
211C2×C5×D3—41no—
212C2×C5×D3—41no—
213C2×C5×D3—42no—
214Dic15—22no—
215C3×Dic5—22no—
216C3×Dic5#12012yes—
217C3×Dic5#12012yes—
218C5×Dic3—22no—
219C24×C3C2×AGL(1,𝔽5)11yescharacteristic
220C23×D3—51no—
221C23×D3C2×AGL(1,𝔽5)11yescharacteristic
222C22×C3×C4—51no—
223C22×C3×C4—52no—
224C22×C3×C4—52no—
225C22×C3×C4—51no—
226C22×C3×C4—102no—
227C22×C3×C4—102no—
228C2×#67—22no—
229C2×#67—22no—
230C2×#67—22no—
231C2×#67—102no—
232C2×#67—102no—
233C2×#67—102no—
234C22×Dic3—51no—
235C22×Dic3C2×AGL(1,𝔽5)11yescharacteristic
236#235—1016no—
237#235—1016no—
238C2×D12—102no—
239C2×D12—102no—
240C2×D12—102no—
241C2×C4×D3—54no—
242C2×C4×D3—54no—
243C2×C4×D3—101no—
244C2×C4×D3—102no—
245C2×C4×D3—101no—
246C2×C4×D3—101no—
247C2×C4×D3—102no—
248C2×C4×D3—101no—
249C2×Dic6—102no—
250C2×Dic6—102no—
251C2×Dic6—102no—
252C2×C3×C8—102no—
253C2×C3×C8—102no—
254C2×C3×C8—102no—
255C2×C3⋊C8—102no—
256C2×C3⋊C8—102no—
257C2×C3⋊C8—102no—
258C23×C5—21no—
259C23×C5—21no—
260C23×C5—22no—
261C23×C5—22no—
262C23×C5—32no—
263C23×C5—61no—
264C23×C5—62no—
265C23×C5—62no—
266C23×C5—61no—
267C23×C5(C2×D3)⋊C412yes—
268C23×C5C2×C4×D311yescharacteristic
269C22×D5—32no—
270C22×D5—61no—
271C22×D5—62no—
272C22×D5—62no—
273C22×D5—61no—
274C5×D4—68no—
275C2×C4×C5—32no—
276C2×C4×C5—61no—
277C2×C4×C5—61no—
278C22⋊D5—68no—
279C22⋊D5—68no—
280C2×Dic5—21no—
281C2×Dic5—21no—
282C2×Dic5—24no—
283C2×Dic5—24no—
284C2×Dic5—32no—
285C2×Dic5—61no—
286C2×Dic5—61no—
287C2×Dic5—64no—
288C2×Dic5—64no—
289C2×Dic5(C2×D3)⋊C412yes—
290C2×Dic5C2×D1211yescharacteristic
291C2×Dic5D3×D412yes—
292C2×Dic5D3×D412yes—
293C2×Dic5C2×#6711yescharacteristic
294C4×D5—64no—
295C4×D5—64no—
296Dic10—68no—
297#170—42no—
298#170—122no—
299C2×#41—302no—
300C2×#41—302no—
301C2×#41—304no—
302C2×#41—304no—
303C22×Q8—151no—
304C22×D4—31no—
305C22×D4—151no—
306C22×D4—151no—
307C23×C4—51no—
308C23×C4—151no—
309C23×C4—151no—
310C22×C8—52no—
311C22×C8—152no—
312C22⋊C8—102no—
313C22⋊C8—304no—
314C22⋊C8—304no—
315C22⋊C8—302no—
316C2×C3×C5—42no—
317C2×C3×C5#37911yescharacteristic
318C2×C3×C5#40811yescharacteristic
319C2×C3×C5#40812yes—
320C2×C3×C5#40811yescharacteristic
321C2×C3×C5#40812yes—
322D15—42no—
323C5×D3—42no—
324C23×C3—22no—
325C23×C3—22no—
326C23×C3—21no—
327C23×C3—21no—
328C23×C3C22⋊AGL(1,𝔽5)12yes—
329C23×C3C22×AGL(1,𝔽5)11yescharacteristic
330C22×D3—22no—
331C22×D3—22no—
332C22×D3—21no—
333C22×D3—21no—
334C22×D3—52no—
335C22×D3—102no—
336C22×D3—102no—
337C22×D3—101no—
338C22×D3—101no—
339C22×D3C22⋊AGL(1,𝔽5)12yes—
340C2×C3×C4—51no—
341C2×C3×C4—52no—
342C2×C3×C4—52no—
343C2×C3×C4—51no—
344C2×C3×C4—52no—
345C2×C3×C4—104no—
346C2×C3×C4—104no—
347C2×C3×C4—101no—
348C2×C3×C4—101no—
349#67—28no—
350#67—108no—
351C2×Dic3—21no—
352C2×Dic3—21no—
353C2×Dic3—52no—
354C2×Dic3—101no—
355C2×Dic3—101no—
356C2×Dic3C22⋊AGL(1,𝔽5)12yes—
357D12—108no—
358C4×D3—104no—
359C4×D3—104no—
360C4×D3—104no—
361C4×D3—104no—
362Dic6—108no—
363C3×C8—202no—
364C3⋊C8—202no—
365C22×C5—22no—
366C22×C5—22no—
367C22×C5—22no—
368C22×C5—42no—
369C22×C5—42no—
370C22×C5—62no—
371C22×C5—64no—
372C22×C5—121no—
373C22×C5—122no—
374C22×C5—121no—
375C22×C5D3×C22⋊C411yescharacteristic
376C22×C5D3×C22⋊C411yescharacteristic
377C22×C5#90112yes—
378C22×C5#90712yes—
379C22×C5C2×(C2×D3)⋊C411yescharacteristic
380C2×D5—62no—
381C2×D5—64no—
382C2×D5—121no—
383C2×D5—122no—
384C2×D5—121no—
385C4×C5—62no—
386Dic5—22no—
387Dic5—62no—
388Dic5#88212yes—
389Dic5#93712yes—
390C24—31no—
391C24—151no—
392C24D3×AGL(1,𝔽5)11yescharacteristic
393#41—3016no—
394#41—3016no—
395C2×Q8—302no—
396C2×Q8—302no—
397C2×Q8—302no—
398C2×D4—62no—
399C2×D4—62no—
400C2×D4—62no—
401C2×D4—302no—
402C2×D4—302no—
403C2×D4—302no—
404C2×D4—302no—
405C2×D4—302no—
406C2×D4—302no—
407C22×C4—31no—
408C22×C4—51no—
409C22×C4—52no—
410C22×C4—52no—
411C22×C4—51no—
412C22×C4—102no—
413C22×C4—102no—
414C22×C4—154no—
415C22×C4—154no—
416C22×C4—151no—
417C22×C4—301no—
418C22×C4—301no—
419C22×C4—302no—
420C22×C4—301no—
421C22×C4—301no—
422C22×C4—302no—
423C2×C8—102no—
424C2×C8—102no—
425C2×C8—102no—
426C2×C8—302no—
427C2×C8—302no—
428C2×C8—302no—
429C3×C5#178011yescharacteristic, complemented, Hall
430C22×C3—22no—
431C22×C3—22no—
432C22×C3—22no—
433C22×C3—42no—
434C22×C3—42no—
435C22×C3#413112yes—
436C22×C3#413112yes—
437C22×C3C2×C22⋊AGL(1,𝔽5)11yescharacteristic
438C22×C3C2×C22⋊AGL(1,𝔽5)11yescharacteristic
439C22×C3C2×C22⋊AGL(1,𝔽5)11yescharacteristic
440C2×D3—22no—
441C2×D3—24no—
442C2×D3—41no—
443C2×D3—41no—
444C2×D3—42no—
445C2×D3—102no—
446C2×D3—104no—
447C2×D3—201no—
448C2×D3—201no—
449C2×D3—202no—
450C3×C4—52no—
451C3×C4—52no—
452C3×C4—102no—
453Dic3—22no—
454Dic3—102no—
455C2×C5—42no—
456C2×C5—122no—
457C2×C5#472712yes—
458C2×C5#472712yes—
459C2×C5#495511yescharacteristic
460C2×C5#511011yescharacteristic
461C2×C5#512911yescharacteristic
462D5—122no—
463C23—22no—
464C23—22no—
465C23—21no—
466C23—21no—
467C23—32no—
468C23—61no—
469C23—62no—
470C23—62no—
471C23—61no—
472C23—152no—
473C23—301no—
474C23—302no—
475C23—302no—
476C23—301no—
477C23(C2×D3)⋊AGL(1,𝔽5)12yes—
478C23C2×D3×AGL(1,𝔽5)11yescharacteristic
479Q8—308no—
480D4—68no—
481D4—308no—
482D4—308no—
483C2×C4—32no—
484C2×C4—51no—
485C2×C4—51no—
486C2×C4—52no—
487C2×C4—52no—
488C2×C4—52no—
489C2×C4—61no—
490C2×C4—61no—
491C2×C4—104no—
492C2×C4—104no—
493C2×C4—101no—
494C2×C4—101no—
495C2×C4—152no—
496C2×C4—304no—
497C2×C4—304no—
498C2×C4—304no—
499C2×C4—304no—
500C2×C4—301no—
501C2×C4—301no—
502C8—202no—
503C8—602no—
504C2×C3—42no—
505C2×C3#6577511yescharacteristic
506C2×C3#6577511yescharacteristic
507C2×C3#6577512yes—
508C2×C3#6577612yes—
509C2×C3#6580011yescharacteristic
510D3—42no—
511D3—202no—
512C5#6995611yescharacteristic, complemented, Sylow
513C22—22no—
514C22—22no—
515C22—22no—
516C22—42no—
517C22—42no—
518C22—62no—
519C22—64no—
520C22—121no—
521C22—121no—
522C22—122no—
523C22—302no—
524C22—304no—
525C22—601no—
526C22—601no—
527C22—602no—
528C22C2×(C2×D3)⋊AGL(1,𝔽5)11yescharacteristic
529C22#9192712yes—
530C22#9192912yes—
531C22D3×C22⋊AGL(1,𝔽5)11yescharacteristic
532C22D3×C22⋊AGL(1,𝔽5)11yescharacteristic
533C4—52no—
534C4—52no—
535C4—62no—
536C4—102no—
537C4—302no—
538C3#1060494511yescharacteristic, complemented, Sylow
539C2—42no—
540C2—122no—
541C2—602no—
542C2#1175199911yescharacteristic
543C2#1175200111yescharacteristic
544C2#1175202511yescharacteristic
545C2#1175203812yes—
546C2#1175204012yes—
subgroupG/H
center, FrattiniC23C2×D3×AGL(1,𝔽5)
commutator, socleC23×C3×C5C22×C4
FittingC24×C3×C5C2×C4
prime radicalC22×#6782C2
cosocleC22×C3×Dic5C23
hypercenterC24D3×AGL(1,𝔽5)
hypocenterC3×C5#1780
LCSG ▷ C23×C3×C5 ▷ C3×C5
UCSC1 ◁ C23 ◁ C24

Complements

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

NH ≅ G/Ntype
2. C2×#91241C2semidirect
3. C2×C3×C22⋊#170C2semidirect
4. C2×#91171C2semidirect
5. C2×(C2×D3)⋊#170C2semidirect
28. C2×C22⋊#170D3semidirect
429. C3×C5#1780semidirect
512. C5#69956semidirect
538. C3#10604945semidirect

Maximal subgroups

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

subgroupindex
C22×#67822
C2×#912412
C2×C3×C22⋊#1702
C2×#911712
C2×(C2×D3)⋊#1702
#106049453
#699565