Finite Group Databasebeta

← #14328967

Subgroups — #14328967

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×#91786C211yesmaximal, characteristic
2C2×#91347C211yesmaximal, characteristic
3C2×#91340C211yesmaximal, characteristic, complemented
4C5×#4984C211yesmaximal, characteristic, complemented
5C3×C4⋊C4×Dic5C211yesmaximal, characteristic, complemented
6#11748829C212yesmaximal
7C4×#6667C211yesmaximal, characteristic
8#11748689C212yesmaximal, complemented
9#11748685C212yesmaximal
10Dic5×C4⋊Dic3C212yesmaximal, complemented
11#11748587C211yesmaximal, characteristic, complemented
12#10607715—31nomaximal, Hall
13C22×C4×D15C2211yescharacteristic
14C22×Dic30C2211yescharacteristic
15C2×C3×C5×C4⋊C4C2211yescharacteristic
16C2×#6714C2212yes—
17C2×C4⋊Dic15C2212yes—
18#91786C418yescomplemented
19C2×C5×C4⋊Dic3C2212yes—
20C2×C5×C4⋊Dic3C2211yescharacteristic
21C2×C4×C5×Dic3C2211yescharacteristic
22C2×C3×C4⋊Dic5C2211yescharacteristic
23C2×C3×C4×Dic5C2212yes—
24C2×C3×C4×Dic5C2211yescharacteristic
25C2×#6669C2212yes—
26C2×#6667C2211yescharacteristic
27C2×#6667C2212yes—
28C2×#6667C2211yescharacteristic
29C2×Dic3×Dic5C2212yes—
30C2×Dic3×Dic5C2212yes—
31#91347—24no—
32#91340—24no—
33C5×C3⋊#95C2212yes—
34C3×#3963C2212yes—
35#90994C2211yescharacteristic
36#90994C2212yes—
37#90994C2212yes—
38#90994C2211yescharacteristic
39#73647—51nomaximal, Hall
40C2×#4043—31no—
41C2×#4038—31no—
42C2×C4×Dic10—31no—
43C5×#389—31no—
44#65279—32no—
45C4⋊C4×Dic5—32no—
46C4⋊C4×Dic5D311yescharacteristic, complemented
47#65263—31no—
48C4×#685—31no—
49#64958—32no—
50#64943—32no—
51C23×D15C2311yescharacteristic
52C22×C3×C4×C5C2312yes—
53C22×C3×C4×C5C2311yescharacteristic
54C22×Dic15C2311yescharacteristic
55C22×Dic15C2312yes—
56C2×C4×D15—22no—
57C2×C4×D15—22no—
58C2×C4×D15C2×C414yes—
59C2×Dic30—22no—
60C2×Dic30—22no—
61C2×Dic30C2×C414yes—
62C22×C5×Dic3C2311yescharacteristic
63C22×C5×Dic3C2312yes—
64C22×C5×Dic3C2311yescharacteristic
65C22×C3×Dic5C2311yescharacteristic
66C22×C3×Dic5C2312yes—
67C22×C3×Dic5C2311yescharacteristic
68C3×C5×C4⋊C4C2×C414yes—
69#6714C2×C418yes—
70C4⋊Dic15C2×C418yes—
71C5×C4⋊Dic3—22no—
72C5×C4⋊Dic3—24no—
73C4×C5×Dic3—22no—
74C3×C4⋊Dic5D414yes—
75C3×C4×Dic5—22no—
76C3×C4×Dic5—24no—
77#6669—24no—
78#6667—22no—
79#6667—24no—
80#6667—22no—
81Dic3×Dic5—24no—
82Dic3×Dic5—24no—
83C2×#897—51no—
84C2×#892—51no—
85C2×#887—51no—
86C3×C4×C4⋊C4—51no—
87#4984D511yescharacteristic, complemented
88#4940—52no—
89C4×(C2×D3)⋊C4—51no—
90C4×C4⋊Dic3—52no—
91#4929—51no—
92#4670—52no—
93#4653—52no—
94C22×C4×D5—31no—
95C22×Dic10—31no—
96C2×C5×C4⋊C4—31no—
97C2×C5×C4⋊C4—32no—
98C2×C5×C4⋊C4C2×D311yescharacteristic
99C2×C42×C5—31no—
100C2×#685—32no—
101C2×#685—32no—
102C2×#685—31no—
103C2×#685—31no—
104C2×C4⋊Dic5—32no—
105C2×C4⋊Dic5C2×D311yescharacteristic
106C2×#683—32no—
107C2×C4×Dic5—32no—
108C2×C4×Dic5—32no—
109C2×C4×Dic5C2×D312yes—
110C2×C4×Dic5C2×D311yescharacteristic
111#4043—38no—
112#4038—64no—
113C4×Dic10—64no—
114C5×#95—32no—
115#3963—31no—
116#3963—32no—
117#3963—32no—
118#3963—31no—
119#3963C2×D312yes—
120#2362—151noSylow
121C23×C3×C5C2411yescharacteristic
122C22×D15—21no—
123C22×D15—21no—
124C22×D15—21no—
125C22×D15—21no—
126C22×D15—21no—
127C22×D15—21no—
128C22×D15C22×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—21no—
136C2×Dic15—22no—
137C2×Dic15—22no—
138C2×Dic15—21no—
139C2×Dic15C22×C412yes—
140C2×Dic15C22×C414yes—
141C4×D15—28no—
142Dic30—28no—
143C2×C5×Dic3—21no—
144C2×C5×Dic3—21no—
145C2×C5×Dic3—22no—
146C2×C5×Dic3—21no—
147C2×C5×Dic3—21no—
148C2×C5×Dic3—22no—
149C2×C5×Dic3—21no—
150C2×C5×Dic3—22no—
151C2×C5×Dic3#4112yes—
152C2×C3×Dic5—22no—
153C2×C3×Dic5—21no—
154C2×C3×Dic5—22no—
155C2×C3×Dic5—21no—
156C2×C3×Dic5—21no—
157C2×C3×Dic5—21no—
158C2×C3×Dic5—21no—
159C2×C3×Dic5C2×D414yes—
160C2×C3×Dic5#4112yes—
161C22×C4×D3—51no—
162C22×Dic6—51no—
163C2×C3×C4⋊C4—51no—
164C2×C3×C4⋊C4C2×D511yescharacteristic
165C2×C3×C42—51no—
166C2×C3×C42—52no—
167C2×(C2×D3)⋊C4—52no—
168C2×(C2×D3)⋊C4—51no—
169C2×(C2×D3)⋊C4—52no—
170C2×(C2×D3)⋊C4—51no—
171C2×C4⋊Dic3—52no—
172C2×C4⋊Dic3C2×D512yes—
173C2×C4⋊Dic3C2×D511yescharacteristic
174C2×Dic3⋊C4—52no—
175C2×C4×Dic3—52no—
176C2×C4×Dic3—52no—
177C2×C4×Dic3C2×D511yescharacteristic
178#897—58no—
179#892—104no—
180#887—104no—
181C3×#95—52no—
182C3⋊#95—51no—
183C3⋊#95—52no—
184C3⋊#95—52no—
185C3⋊#95—51no—
186C3⋊#95C2×D512yes—
187C23×D5—31no—
188C22×C4×C5—31no—
189C22×C4×C5—31no—
190C22×C4×C5—32no—
191C22×C4×C5C22×D312yes—
192C22×C4×C5C22×D311yescharacteristic
193C22×Dic5—32no—
194C22×Dic5—31no—
195C22×Dic5C22×D311yescharacteristic
196C22×Dic5C22×D312yes—
197C22×Dic5C22×D311yescharacteristic
198C2×C4×D5—34no—
199C2×C4×D5—62no—
200C2×C4×D5—62no—
201C2×Dic10—34no—
202C2×Dic10—62no—
203C2×Dic10—62no—
204C5×C4⋊C4—64no—
205C5×C4⋊C4—62no—
206C5×C4⋊C4C4×D314yes—
207C42×C5—62no—
208#685—38no—
209#685—62no—
210#685—64no—
211#685—62no—
212C4⋊Dic5—38no—
213C4⋊Dic5D1214yes—
214#683—64no—
215C4×Dic5—22no—
216C4×Dic5—24no—
217C4×Dic5—64no—
218C4×Dic5—64no—
219C2×C22⋊Q8—151no—
220C2×C4×Q8—151no—
221C2×#117—151no—
222#391—151no—
223#389—31no—
224#388—152no—
225#388—152no—
226#387—152no—
227C4×C4⋊C4—51no—
228C4×C4⋊C4—152no—
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×D15—22no—
238C2×D15—22no—
239C2×D15—22no—
240C2×D15—41no—
241C2×D15—41no—
242C2×D15—41no—
243C2×D15—41no—
244C3×C4×C5—24no—
245C3×C4×C5C4×D414yes—
246Dic15—24no—
247Dic15—22no—
248C3×Dic5—24no—
249C3×Dic5—22no—
250C3×Dic5—41no—
251C5×Dic3—22no—
252C5×Dic3—41no—
253C5×Dic3—42no—
254C23×D3—51no—
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×Dic3C22×D512yes—
263C22×Dic3C22×D511yescharacteristic
264C22×Dic3C22×D511yescharacteristic
265C2×C4×D3—54no—
266C2×C4×D3—102no—
267C2×C4×D3—102no—
268C2×Dic6—54no—
269C2×Dic6—102no—
270C2×Dic6—102no—
271C3×C4⋊C4—54no—
272C3×C4⋊C4C4×D514yes—
273C3×C42—104no—
274C3×C42—102no—
275(C2×D3)⋊C4—58no—
276(C2×D3)⋊C4—104no—
277(C2×D3)⋊C4—102no—
278(C2×D3)⋊C4—102no—
279C4⋊Dic3—22no—
280C4⋊Dic3—24no—
281C4⋊Dic3—58no—
282Dic3⋊C4—104no—
283C4×Dic3—22no—
284C4×Dic3—104no—
285C4×Dic3—104no—
286C23×C5C23×D311yescharacteristic
287C22×D5—32no—
288C22×D5—61no—
289C22×D5—61no—
290C22×D5—61no—
291C22×D5—61no—
292C22×D5—61no—
293C22×D5—61no—
294C2×C4×C5—22no—
295C2×C4×C5—22no—
296C2×C4×C5—32no—
297C2×C4×C5—61no—
298C2×C4×C5—61no—
299C2×C4×C5—62no—
300C2×C4×C5—61no—
301C2×C4×C5—61no—
302C2×C4×C5—62no—
303C2×C4×C5—62no—
304C2×C4×C5—61no—
305C2×C4×C5C2×C4×D314yes—
306C2×C4×C5C2×C4×D312yes—
307C2×C4×C5C2×D1212yes—
308C2×C4×C5#23512yes—
309C2×Dic5—21no—
310C2×Dic5—22no—
311C2×Dic5—21no—
312C2×Dic5—22no—
313C2×Dic5—21no—
314C2×Dic5—21no—
315C2×Dic5—21no—
316C2×Dic5—32no—
317C2×Dic5—34no—
318C2×Dic5—61no—
319C2×Dic5—62no—
320C2×Dic5—62no—
321C2×Dic5—61no—
322C2×Dic5C2×D1214yes—
323C2×Dic5#23712yes—
324C4×D5—68no—
325Dic10—68no—
326C22×Q8—151no—
327C23×C4—151no—
328C22⋊Q8—158no—
329C4×Q8—304no—
330#117—304no—
331C2×C4⋊C4—31no—
332C2×C4⋊C4—32no—
333C2×C4⋊C4—51no—
334C2×C4⋊C4—152no—
335C2×C4⋊C4—152no—
336C2×C4⋊C4D3×D511yescharacteristic
337C2×C22⋊C4—152no—
338C2×C22⋊C4—152no—
339C2×C22⋊C4—151no—
340C2×C22⋊C4—151no—
341C2×C42—31no—
342C2×C42—51no—
343C2×C42—52no—
344C2×C42—152no—
345C2×C42—152no—
346#95—32no—
347#95—52no—
348#95—151no—
349#95—151no—
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
359D15—42no—
360C23×C3C23×D511yescharacteristic
361C22×D3—52no—
362C22×D3—101no—
363C22×D3—101no—
364C22×D3—101no—
365C22×D3—101no—
366C22×D3—101no—
367C22×D3—101no—
368C2×C3×C4—22no—
369C2×C3×C4—22no—
370C2×C3×C4—54no—
371C2×C3×C4—52no—
372C2×C3×C4—101no—
373C2×C3×C4—102no—
374C2×C3×C4—102no—
375C2×C3×C4—101no—
376C2×C3×C4—101no—
377C2×C3×C4—101no—
378C2×C3×C4—101no—
379C2×C3×C4C2×C4×D514yes—
380C2×C3×C4C2×C4×D512yes—
381C2×C3×C4D4×D512yes—
382C2×C3×C4#71312yes—
383C2×Dic3—21no—
384C2×Dic3—21no—
385C2×Dic3—21no—
386C2×Dic3—22no—
387C2×Dic3—22no—
388C2×Dic3—21no—
389C2×Dic3—21no—
390C2×Dic3—22no—
391C2×Dic3—52no—
392C2×Dic3—54no—
393C2×Dic3—101no—
394C2×Dic3—102no—
395C2×Dic3—102no—
396C2×Dic3—101no—
397C2×Dic3#71112yes—
398C4×D3—108no—
399Dic6—108no—
400C22×C5C22×C4×D311yescharacteristic
401C22×C5C22×D1211yescharacteristic
402C22×C5C2×#23511yescharacteristic
403C22×C5C2×#23711yescharacteristic
404C22×C5#100711yescharacteristic
405C22×C5D3×#4111yescharacteristic
406C22×C5#101011yescharacteristic
407C2×D5—62no—
408C2×D5—62no—
409C2×D5—62no—
410C2×D5—121no—
411C2×D5—121no—
412C2×D5—121no—
413C2×D5—121no—
414C4×C5—24no—
415C4×C5—62no—
416C4×C5—121no—
417C4×C5—122no—
418C4×C5C4×D1214yes—
419Dic5—22no—
420Dic5—24no—
421Dic5—41no—
422Dic5—64no—
423Dic5—62no—
424C24—151no—
425C2×Q8—154no—
426C2×Q8—302no—
427C2×Q8—302no—
428C22×C4—31no—
429C22×C4—31no—
430C22×C4—32no—
431C22×C4—52no—
432C22×C4—51no—
433C22×C4—51no—
434C22×C4—151no—
435C22×C4—152no—
436C22×C4—154no—
437C22×C4—302no—
438C22×C4—302no—
439C22×C4C2×D3×D512yes—
440C22×C4C2×D3×D511yescharacteristic
441C4⋊C4—54no—
442C4⋊C4—64no—
443C4⋊C4—62no—
444C4⋊C4—158no—
445C4⋊C4—304no—
446C4⋊C4#120014yes—
447C22⋊C4—158no—
448C22⋊C4—302no—
449C22⋊C4—304no—
450C22⋊C4—302no—
451C42—62no—
452C42—104no—
453C42—102no—
454C42—304no—
455C42—304no—
456C3×C5#236211yescharacteristic, complemented, Hall
457C22×C3C22×C4×D511yescharacteristic
458C22×C3C2×D4×D511yescharacteristic
459C22×C3C2×#71111yescharacteristic
460C22×C3C2×#71311yescharacteristic
461C22×C3#414711yescharacteristic
462C22×C3D5×#4111yescharacteristic
463C22×C3#415011yescharacteristic
464C2×D3—102no—
465C2×D3—102no—
466C2×D3—102no—
467C2×D3—201no—
468C2×D3—201no—
469C2×D3—201no—
470C2×D3—201no—
471C3×C4—24no—
472C3×C4—104no—
473C3×C4—102no—
474C3×C4—201no—
475C3×C4#403914yes—
476Dic3—22no—
477Dic3—42no—
478Dic3—41no—
479Dic3—104no—
480Dic3—102no—
481C2×C5C2×C4×D1211yescharacteristic
482C2×C5#551311yescharacteristic
483C2×C5#553311yescharacteristic
484C2×C5C4×#23711yescharacteristic
485C2×C5#555511yescharacteristic
486C2×C5#557211yescharacteristic
487C2×C5#557611yescharacteristic
488D5—122no—
489C23—152no—
490C23—301no—
491C23—301no—
492C23—301no—
493C23—301no—
494C23—301no—
495C23—301no—
496C23C22×D3×D511yescharacteristic
497Q8—308no—
498C2×C4—22no—
499C2×C4—22no—
500C2×C4—32no—
501C2×C4—54no—
502C2×C4—52no—
503C2×C4—61no—
504C2×C4—61no—
505C2×C4—61no—
506C2×C4—62no—
507C2×C4—61no—
508C2×C4—62no—
509C2×C4—61no—
510C2×C4—62no—
511C2×C4—101no—
512C2×C4—102no—
513C2×C4—102no—
514C2×C4—101no—
515C2×C4—101no—
516C2×C4—101no—
517C2×C4—101no—
518C2×C4—154no—
519C2×C4—152no—
520C2×C4—301no—
521C2×C4—302no—
522C2×C4—301no—
523C2×C4—302no—
524C2×C4—308no—
525C2×C4#677312yes—
526C2×C4D5×D1212yes—
527C2×C4C2×#120014yes—
528C2×C4C2×#120012yes—
529C2×C3C2×#403911yescharacteristic
530C2×C3C4×#71111yescharacteristic
531C2×C3#6590611yescharacteristic
532C2×C3#6593211yescharacteristic
533C2×C3#6596511yescharacteristic
534C2×C3#6602811yescharacteristic
535C2×C3#6606611yescharacteristic
536D3—202no—
537C5#7364711yescharacteristic, complemented, Sylow
538C22—302no—
539C22—302no—
540C22—302no—
541C22—601no—
542C22—601no—
543C22—601no—
544C22—601no—
545C22C2×#677311yescharacteristic
546C22C2×D5×D1211yescharacteristic
547C22D5×#23711yescharacteristic
548C22D3×#71111yescharacteristic
549C22#9202811yescharacteristic
550C22#9203011yescharacteristic
551C22C22×#120011yescharacteristic
552C4—24no—
553C4—62no—
554C4—102no—
555C4—104no—
556C4—121no—
557C4—122no—
558C4—201no—
559C4—302no—
560C4—304no—
561C4#9141814yes—
562C3#1060771511yescharacteristic, complemented, Sylow
563C2—602no—
564C2#1175343611yescharacteristic
565C2#1175343711yescharacteristic
566C2#1175373311yescharacteristic
567C2#1175376811yescharacteristic
568C2C2×#9141811yescharacteristic
569C2#1175485311yescharacteristic
570C2#1175512411yescharacteristic
subgroupG/H
center, FrattiniC23C22×D3×D5
commutatorC22×C3×C5C23×C4
FittingC2×C3×C5×C4⋊C4C22
socle, cosocleC23×C3×C5C24
prime radicalC23×D15C23
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×#91340C2semidirect
4. C5×#4984C2semidirect
5. C3×C4⋊C4×Dic5C2semidirect
8. #11748689C2semidirect
10. Dic5×C4⋊Dic3C2semidirect
11. #11748587C2semidirect
18. #91786C4semidirect
46. C4⋊C4×Dic5D3semidirect
87. #4984D5semidirect
456. C3×C5#2362semidirect
537. C5#73647semidirect
562. C3#10607715semidirect

Maximal subgroups

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

subgroupindex
C2×#917862
C2×#913472
C2×#913402
C5×#49842
C3×C4⋊C4×Dic52
#117488292
C4×#66672
#117486892
#117486852
Dic5×C4⋊Dic32
#117485872
#106077153
#736475