Finite Group Databasebeta

← #14328966

Subgroups — #14328966

Order 1920.

1152 conjugacy classes of subgroups.

The subgroup lattice has 572 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C5×#897C211yesmaximal, characteristic
2C2×#91338C211yesmaximal, characteristic
3C2×Dic5×Dic6C211yesmaximal, characteristic, complemented
4C4⋊C4×Dic15C211yesmaximal, characteristic, complemented
5C3×#65274C211yesmaximal, characteristic, complemented
6#11748794C212yesmaximal
7C4×(C2×D3)⋊Dic5C211yesmaximal, characteristic
8#11748699C212yesmaximal, complemented
9#11748683C212yesmaximal, complemented
10#11748675C212yesmaximal
11#11748607C211yesmaximal, characteristic, complemented
12#10614479—31nomaximal, Hall
13C22×C4×C5×D3C2211yescharacteristic
14C22×C5×Dic6C2211yescharacteristic
15C2×C3×C5×C4⋊C4C2211yescharacteristic
16C2×C4⋊Dic15C2211yescharacteristic
17C2×C4×Dic15C2212yes—
18C2×C4×Dic15C2211yescharacteristic
19C2×C5×(C2×D3)⋊C4C2212yes—
20C2×C5×C4⋊Dic3C2212yes—
21C5×#897C418yescomplemented
22C2×C3×C4⋊Dic5C2212yes—
23C2×C3×C4⋊Dic5C2211yescharacteristic
24C2×C3×C4×Dic5C2211yescharacteristic
25C2×Dic3⋊Dic5C2212yes—
26C2×(C2×D3)⋊Dic5C2211yescharacteristic
27C2×(C2×D3)⋊Dic5C2212yes—
28C2×(C2×D3)⋊Dic5C2211yescharacteristic
29C2×Dic3×Dic5C2212yes—
30C2×Dic3×Dic5C2212yes—
31#91338—24no—
32Dic5×Dic6—24no—
33#91115C2212yes—
34C3×#3963C2212yes—
35#90994C2212yes—
36#90994C2211yescharacteristic
37#90994C2211yescharacteristic
38#90994C2212yes—
39#75984—51nomaximal, Hall
40C2×C5×C22⋊Q8—31no—
41C2×Q8×Dic5—31no—
42C2×#4072—31no—
43#65519—31no—
44C4×C22⋊Dic5—31no—
45#65274D311yescharacteristic, complemented
46#65270—32no—
47C4⋊C4×Dic5—32no—
48C4⋊C4×Dic5—31no—
49#65240—32no—
50#65240—32no—
51C23×C5×D3C2311yescharacteristic
52C22×C3×C4×C5C2312yes—
53C22×C3×C4×C5C2311yescharacteristic
54C22×Dic15C2311yescharacteristic
55C22×Dic15C2312yes—
56C22×Dic15C2311yescharacteristic
57C22×C5×Dic3C2311yescharacteristic
58C22×C5×Dic3C2312yes—
59C2×C4×C5×D3—22no—
60C2×C4×C5×D3—22no—
61C2×C4×C5×D3C2×C414yes—
62C2×C5×Dic6—22no—
63C2×C5×Dic6—22no—
64C2×C5×Dic6C2×C414yes—
65C22×C3×Dic5C2312yes—
66C22×C3×Dic5C2311yescharacteristic
67C22×C3×Dic5C2311yescharacteristic
68C3×C5×C4⋊C4C2×C414yes—
69C4⋊Dic15D414yes—
70C4×Dic15—22no—
71C4×Dic15—24no—
72C5×(C2×D3)⋊C4C2×C418yes—
73C5×C4⋊Dic3C2×C418yes—
74C3×C4⋊Dic5—22no—
75C3×C4⋊Dic5—24no—
76C3×C4×Dic5—22no—
77Dic3⋊Dic5—24no—
78(C2×D3)⋊Dic5—24no—
79(C2×D3)⋊Dic5—22no—
80(C2×D3)⋊Dic5—22no—
81Dic3×Dic5—24no—
82Dic3×Dic5—24no—
83C2×#897D511yescharacteristic
84C2×#892—51no—
85C2×C4×Dic6—51no—
86C3×#389—51no—
87#4990—52no—
88Dic3×C4⋊C4—51no—
89Dic3×C4⋊C4—52no—
90#4973—51no—
91C4×(C2×D3)⋊C4—51no—
92#4669—52no—
93#4654—52no—
94C22×C5×Q8—31no—
95C23×C4×C5—31no—
96C5×C22⋊Q8—38no—
97C2×C5×C4⋊C4—32no—
98C2×C5×C4⋊C4C2×D311yescharacteristic
99C2×C5×C22⋊C4—32no—
100C2×C22⋊Dic5—32no—
101C2×C22⋊Dic5—31no—
102C2×C22⋊Dic5—31no—
103Q8×Dic5—64no—
104#4072—64no—
105C2×C4⋊Dic5—31no—
106C2×C4⋊Dic5—32no—
107C2×C4⋊Dic5C2×D311yescharacteristic
108C2×C4⋊Dic5C2×D312yes—
109C2×C4×Dic5—32no—
110C2×C4×Dic5—31no—
111C2×C4×Dic5—32no—
112C2×C4×Dic5—32no—
113C2×C4×Dic5C2×D311yescharacteristic
114#3963—32no—
115#3963—32no—
116#3963—32no—
117#3963—31no—
118#3963—31no—
119#3963C2×D312yes—
120#2362—151noSylow
121C23×C3×C5C2411yescharacteristic
122C22×C5×D3—21no—
123C22×C5×D3—21no—
124C22×C5×D3—21no—
125C22×C5×D3—21no—
126C22×C5×D3—21no—
127C22×C5×D3—21no—
128C22×C5×D3C22×C412yes—
129C2×C3×C4×C5—22no—
130C2×C3×C4×C5—22no—
131C2×C3×C4×C5C22×C414yes—
132C2×C3×C4×C5C22×C412yes—
133C2×C3×C4×C5C2×D412yes—
134C2×C3×C4×C5#4112yes—
135C2×Dic15—22no—
136C2×Dic15—21no—
137C2×Dic15—22no—
138C2×Dic15—21no—
139C2×Dic15—21no—
140C2×Dic15—21no—
141C2×Dic15—21no—
142C2×Dic15C2×D414yes—
143C2×Dic15#4112yes—
144C2×C5×Dic3—21no—
145C2×C5×Dic3—22no—
146C2×C5×Dic3—22no—
147C2×C5×Dic3—21no—
148C2×C5×Dic3C22×C412yes—
149C2×C5×Dic3C22×C414yes—
150C4×C5×D3—28no—
151C5×Dic6—28no—
152C2×C3×Dic5—21no—
153C2×C3×Dic5—21no—
154C2×C3×Dic5—22no—
155C2×C3×Dic5—21no—
156C2×C3×Dic5—21no—
157C2×C3×Dic5—22no—
158C2×C3×Dic5—21no—
159C2×C3×Dic5—22no—
160C2×C3×Dic5#4112yes—
161C22×C4×D3C2×D511yescharacteristic
162C22×Dic6C2×D511yescharacteristic
163C2×C3×C4⋊C4—51no—
164C2×C3×C4⋊C4—52no—
165C2×C3×C4⋊C4C2×D511yescharacteristic
166C2×C3×C42—51no—
167C2×(C2×D3)⋊C4—51no—
168C2×(C2×D3)⋊C4—52no—
169C2×(C2×D3)⋊C4—51no—
170C2×(C2×D3)⋊C4C2×D512yes—
171C2×C4⋊Dic3—51no—
172C2×C4⋊Dic3C2×D512yes—
173C2×Dic3⋊C4—52no—
174C2×C4×Dic3—52no—
175C2×C4×Dic3—51no—
176C2×C4×Dic3—52no—
177C2×C4×Dic3—52no—
178#897Dic518yescomplemented
179#892—104no—
180C4×Dic6—104no—
181C3×#95—52no—
182C3⋊#95—52no—
183C3⋊#95—51no—
184C3⋊#95—52no—
185C3⋊#95—52no—
186C3⋊#95—51no—
187C24×C5—31no—
188C2×C5×Q8—34no—
189C2×C5×Q8—62no—
190C2×C5×Q8—62no—
191C22×C4×C5—32no—
192C22×C4×C5—31no—
193C22×C4×C5—34no—
194C22×C4×C5—62no—
195C22×C4×C5—62no—
196C22×C4×C5C22×D312yes—
197C22×C4×C5C22×D311yescharacteristic
198C22×Dic5—31no—
199C22×Dic5—32no—
200C22×Dic5—31no—
201C22×Dic5C22×D312yes—
202C22×Dic5C22×D311yescharacteristic
203C22×Dic5C22×D311yescharacteristic
204C5×C4⋊C4—38no—
205C5×C4⋊C4C4×D314yes—
206C5×C22⋊C4—38no—
207C22⋊Dic5—62no—
208C22⋊Dic5—64no—
209C22⋊Dic5—62no—
210C4⋊Dic5—22no—
211C4⋊Dic5—24no—
212C4⋊Dic5—34no—
213C4⋊Dic5—64no—
214C4×Dic5—22no—
215C4×Dic5—64no—
216C4×Dic5—64no—
217C4×Dic5—64no—
218C4×Dic5—62no—
219C2×C22⋊Q8—31no—
220C2×C4×Q8—151no—
221C2×#117—151no—
222#391—151no—
223#389—51no—
224#388—152no—
225#388—152no—
226#387—152no—
227C4×C4⋊C4—152no—
228C4×C4⋊C4—151no—
229C4×C22⋊C4—151no—
230C22×C3×C5C23×C411yescharacteristic
231C22×C3×C5C22×D411yescharacteristic
232C22×C3×C5C2×#4111yescharacteristic
233C22×C3×C5C2×#4111yescharacteristic
234C22×C3×C5C2×#4111yescharacteristic
235C22×C3×C52−1+411yescharacteristic
236C22×C3×C52−1+411yescharacteristic
237C2×C5×D3—22no—
238C2×C5×D3—22no—
239C2×C5×D3—22no—
240C2×C5×D3—41no—
241C2×C5×D3—41no—
242C2×C5×D3—41no—
243C2×C5×D3—41no—
244C3×C4×C5—24no—
245C3×C4×C5C4×D414yes—
246Dic15—24no—
247Dic15—22no—
248Dic15—41no—
249C3×Dic5—22no—
250C3×Dic5—41no—
251C3×Dic5—42no—
252C5×Dic3—24no—
253C5×Dic3—22no—
254C23×D3C22×D511yescharacteristic
255C22×C3×C4—51no—
256C22×C3×C4—52no—
257C22×C3×C4—51no—
258C22×C3×C4C22×D512yes—
259C22×C3×C4C22×D511yescharacteristic
260C22×Dic3—51no—
261C22×Dic3—52no—
262C22×Dic3—51no—
263C22×Dic3C22×D511yescharacteristic
264C22×Dic3C22×D512yes—
265C2×C4×D3—22no—
266C2×C4×D3—22no—
267C2×C4×D3C2×Dic514yes—
268C2×Dic6—22no—
269C2×Dic6—22no—
270C2×Dic6C2×Dic514yes—
271C3×C4⋊C4—104no—
272C3×C4⋊C4—102no—
273C3×C4⋊C4C2×Dic514yes—
274C3×C42—102no—
275(C2×D3)⋊C4—104no—
276(C2×D3)⋊C4—102no—
277(C2×D3)⋊C4—102no—
278(C2×D3)⋊C4C2×Dic518yes—
279C4⋊Dic3—54no—
280C4⋊Dic3C2×Dic518yes—
281Dic3⋊C4—104no—
282C4×Dic3—104no—
283C4×Dic3—104no—
284C4×Dic3—104no—
285C4×Dic3—102no—
286C23×C5—32no—
287C23×C5—61no—
288C23×C5—61no—
289C23×C5—61no—
290C23×C5—61no—
291C23×C5—61no—
292C23×C5—61no—
293C23×C5C23×D311yescharacteristic
294C5×Q8—68no—
295C2×C4×C5—22no—
296C2×C4×C5—22no—
297C2×C4×C5—34no—
298C2×C4×C5—32no—
299C2×C4×C5—61no—
300C2×C4×C5—62no—
301C2×C4×C5—61no—
302C2×C4×C5—62no—
303C2×C4×C5—68no—
304C2×C4×C5C2×C4×D314yes—
305C2×C4×C5C2×C4×D312yes—
306C2×C4×C5D3×D412yes—
307C2×C4×C5#23912yes—
308C2×Dic5—21no—
309C2×Dic5—21no—
310C2×Dic5—21no—
311C2×Dic5—22no—
312C2×Dic5—22no—
313C2×Dic5—21no—
314C2×Dic5—21no—
315C2×Dic5—22no—
316C2×Dic5—34no—
317C2×Dic5—32no—
318C2×Dic5—61no—
319C2×Dic5—62no—
320C2×Dic5—62no—
321C2×Dic5—61no—
322C2×Dic5—61no—
323C2×Dic5—61no—
324C2×Dic5—61no—
325C2×Dic5#23712yes—
326C22×Q8—31no—
327C23×C4—31no—
328C22⋊Q8—38no—
329C4×Q8—304no—
330#117—304no—
331C2×C4⋊C4—32no—
332C2×C4⋊C4—51no—
333C2×C4⋊C4—52no—
334C2×C4⋊C4—151no—
335C2×C4⋊C4—152no—
336C2×C4⋊C4D3×D511yescharacteristic
337C2×C22⋊C4—32no—
338C2×C22⋊C4—152no—
339C2×C22⋊C4—151no—
340C2×C22⋊C4—151no—
341C2×C42—51no—
342C2×C42—152no—
343C2×C42—152no—
344C2×C42—151no—
345C2×C42—152no—
346#95—52no—
347#95—151no—
348#95—151no—
349#95—152no—
350#95—152no—
351#95—152no—
352C2×C3×C5C2×C4×D411yescharacteristic
353C2×C3×C5C4×#4111yescharacteristic
354C2×C3×C5#51911yescharacteristic
355C2×C3×C5#54711yescharacteristic
356C2×C3×C5#54811yescharacteristic
357C2×C3×C5#55211yescharacteristic
358C2×C3×C5#55611yescharacteristic
359C5×D3—42no—
360C23×C3C23×D511yescharacteristic
361C22×D3—21no—
362C22×D3—21no—
363C22×D3—21no—
364C22×D3—21no—
365C22×D3—21no—
366C22×D3—21no—
367C22×D3C22×Dic512yes—
368C2×C3×C4—22no—
369C2×C3×C4—22no—
370C2×C3×C4—52no—
371C2×C3×C4—101no—
372C2×C3×C4—101no—
373C2×C3×C4—102no—
374C2×C3×C4—101no—
375C2×C3×C4—102no—
376C2×C3×C4—101no—
377C2×C3×C4—101no—
378C2×C3×C4—102no—
379C2×C3×C4D4×D512yes—
380C2×C3×C4#71112yes—
381C2×C3×C4C22×Dic514yes—
382C2×C3×C4C22×Dic512yes—
383C2×Dic3—21no—
384C2×Dic3—22no—
385C2×Dic3—22no—
386C2×Dic3—21no—
387C2×Dic3—54no—
388C2×Dic3—52no—
389C2×Dic3—101no—
390C2×Dic3—101no—
391C2×Dic3—102no—
392C2×Dic3—102no—
393C2×Dic3—101no—
394C2×Dic3—101no—
395C2×Dic3—101no—
396C2×Dic3C22×Dic512yes—
397C2×Dic3C22×Dic514yes—
398C4×D3—28no—
399Dic6—28no—
400C22×C5—62no—
401C22×C5—62no—
402C22×C5—62no—
403C22×C5—121no—
404C22×C5—121no—
405C22×C5—121no—
406C22×C5—121no—
407C22×C5C22×C4×D311yescharacteristic
408C22×C5C2×D3×D411yescharacteristic
409C22×C5C2×#23711yescharacteristic
410C22×C5C2×#23911yescharacteristic
411C22×C5#100711yescharacteristic
412C22×C5D3×#4111yescharacteristic
413C22×C5#101011yescharacteristic
414C4×C5—24no—
415C4×C5—62no—
416C4×C5—64no—
417C4×C5#89314yes—
418Dic5—22no—
419Dic5—42no—
420Dic5—41no—
421Dic5—62no—
422Dic5—64no—
423Dic5—121no—
424C24—31no—
425C2×Q8—34no—
426C2×Q8—62no—
427C2×Q8—62no—
428C22×C4—31no—
429C22×C4—32no—
430C22×C4—34no—
431C22×C4—51no—
432C22×C4—51no—
433C22×C4—52no—
434C22×C4—62no—
435C22×C4—62no—
436C22×C4—151no—
437C22×C4—152no—
438C22×C4—151no—
439C22×C4C2×D3×D512yes—
440C22×C4C2×D3×D511yescharacteristic
441C4⋊C4—38no—
442C4⋊C4—104no—
443C4⋊C4—102no—
444C4⋊C4—154no—
445C4⋊C4—304no—
446C4⋊C4D3×Dic514yes—
447C22⋊C4—38no—
448C22⋊C4—302no—
449C22⋊C4—304no—
450C22⋊C4—302no—
451C42—102no—
452C42—304no—
453C42—304no—
454C42—304no—
455C42—302no—
456C3×C5#236211yescharacteristic, complemented, Hall
457C22×C3C2×D4×D511yescharacteristic
458C22×C3C2×#71111yescharacteristic
459C22×C3#414711yescharacteristic
460C22×C3#414711yescharacteristic
461C22×C3D5×#4111yescharacteristic
462C22×C3D5×#4111yescharacteristic
463C22×C3C23×Dic511yescharacteristic
464C2×D3—22no—
465C2×D3—22no—
466C2×D3—22no—
467C2×D3—41no—
468C2×D3—41no—
469C2×D3—41no—
470C2×D3—41no—
471C3×C4—24no—
472C3×C4—102no—
473C3×C4—201no—
474C3×C4—202no—
475C3×C4D4×Dic514yes—
476Dic3—24no—
477Dic3—22no—
478Dic3—104no—
479Dic3—102no—
480Dic3—201no—
481C2×C5—122no—
482C2×C5C2×#89311yescharacteristic
483C2×C5C4×#23711yescharacteristic
484C2×C5#557211yescharacteristic
485C2×C5#559811yescharacteristic
486C2×C5#563111yescharacteristic
487C2×C5#569411yescharacteristic
488C2×C5#573211yescharacteristic
489C23—32no—
490C23—61no—
491C23—61no—
492C23—61no—
493C23—61no—
494C23—61no—
495C23—61no—
496C23C22×D3×D511yescharacteristic
497Q8—68no—
498C2×C4—22no—
499C2×C4—22no—
500C2×C4—34no—
501C2×C4—32no—
502C2×C4—52no—
503C2×C4—61no—
504C2×C4—62no—
505C2×C4—61no—
506C2×C4—62no—
507C2×C4—68no—
508C2×C4—101no—
509C2×C4—101no—
510C2×C4—101no—
511C2×C4—102no—
512C2×C4—102no—
513C2×C4—101no—
514C2×C4—102no—
515C2×C4—101no—
516C2×C4—154no—
517C2×C4—152no—
518C2×C4—301no—
519C2×C4—302no—
520C2×C4—301no—
521C2×C4—302no—
522C2×C4—301no—
523C2×C4—301no—
524C2×C4—301no—
525C2×C4#676812yes—
526C2×C4#677712yes—
527C2×C4C2×D3×Dic514yes—
528C2×C4C2×D3×Dic512yes—
529C2×C3#6596511yescharacteristic
530C2×C3#6596711yescharacteristic
531C2×C3#6597311yescharacteristic
532C2×C3#6597511yescharacteristic
533C2×C3C2×D4×Dic511yescharacteristic
534C2×C3#6614611yescharacteristic
535C2×C3#41×Dic511yescharacteristic
536D3—42no—
537C5#7598411yescharacteristic, complemented, Sylow
538C22—62no—
539C22—62no—
540C22—62no—
541C22—121no—
542C22—121no—
543C22—121no—
544C22—121no—
545C22C2×#676811yescharacteristic
546C22C2×#677711yescharacteristic
547C22D5×#23711yescharacteristic
548C22#9202411yescharacteristic
549C22#9202811yescharacteristic
550C22#9202911yescharacteristic
551C22C22×D3×Dic511yescharacteristic
552C4—24no—
553C4—62no—
554C4—64no—
555C4—102no—
556C4—202no—
557C4—201no—
558C4—302no—
559C4—304no—
560C4—601no—
561C4#9143514yes—
562C3#1061447911yescharacteristic, complemented, Sylow
563C2—122no—
564C2#1175369811yescharacteristic
565C2#1175370411yescharacteristic
566C2#1175373711yescharacteristic
567C2#1175380111yescharacteristic
568C2C2×#9143511yescharacteristic
569C2Dic5×#23711yescharacteristic
570C2#1175514411yescharacteristic
subgroupG/H
center, FrattiniC23C22×D3×D5
commutatorC22×C3×C5C23×C4
FittingC2×C3×C5×C4⋊C4C22
socle, cosocleC23×C3×C5C24
prime radicalC23×C5×D3C23
hypercenterC2×C4⋊C4D3×D5
hypocenterC3×C5#2362
LCSG ▷ C22×C3×C5 ▷ C3×C5
UCSC1 ◁ C23 ◁ C2×C4⋊C4

Complements

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

NH ≅ G/Ntype
3. C2×Dic5×Dic6C2semidirect
4. C4⋊C4×Dic15C2semidirect
5. C3×#65274C2semidirect
8. #11748699C2semidirect
9. #11748683C2semidirect
11. #11748607C2semidirect
21. C5×#897C4semidirect
45. #65274D3semidirect
178. #897Dic5semidirect
456. C3×C5#2362semidirect
537. C5#75984semidirect
562. C3#10614479semidirect

Maximal subgroups

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

subgroupindex
C2×C5×#8972
C2×#913382
C2×Dic5×Dic62
C4⋊C4×Dic152
C3×#652742
#117487942
C4×(C2×D3)⋊Dic52
#117486992
#117486832
#117486752
#117486072
#106144793
#759845