Finite Group Databasebeta

← #14327716

Subgroups — #14327716

Order 1920.

1264 conjugacy classes of subgroups.

The subgroup lattice has 576 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#91418C211yesmaximal, characteristic
2C2×#91403C212yesmaximal, complemented
3#11751040C211yesmaximal, characteristic
4C4×C5×C4⋊Dic3C211yesmaximal, characteristic, complemented
5C3×C4×#683C211yesmaximal, characteristic, complemented
6#11748876C212yesmaximal, complemented
7#11748834C211yesmaximal, characteristic, complemented
8#11748768C211yesmaximal, characteristic
9#11748730C212yesmaximal, complemented
10Dic5×C4⋊Dic3C211yesmaximal, characteristic, complemented
11#11748671C212yesmaximal, complemented
12#10605370—31nomaximal, Hall
13C22×D60C2211yescharacteristic
14C22×#1200C2212yes—
15C2×C3×C42×C5C2211yescharacteristic
16C2×#6714C2212yes—
17C2×#6714C2212yes—
18C2×C4⋊Dic15C2211yescharacteristic
19C2×C5×C4⋊Dic3C2211yescharacteristic
20C2×C5×C4⋊Dic3C2211yescharacteristic
21C2×C4×C5×Dic3C2212yes—
22C2×C3×#683C2211yescharacteristic
23C2×C3×#683C2211yescharacteristic
24C2×C3×C4×Dic5C2211yescharacteristic
25C2×C3×C4×Dic5C2211yescharacteristic
26C2×#6668C2212yes—
27C2×#6668C2212yes—
28C2×#6667C2212yes—
29C2×#6667C2212yes—
30C2×#6667C2212yes—
31C2×Dic3×Dic5C2212yes—
32#91418C418yescomplemented
33#91403—28no—
34C5×C3⋊#95C2212yes—
35C3×#3963C2211yescharacteristic
36C3×#3963C2211yescharacteristic
37#90994C2212yes—
38#73638—51nomaximal, Hall
39C2×#4039—31no—
40C2×#4018—32no—
41C4×C5×C4⋊C4—31no—
42#65277—31no—
43#65265—31no—
44C4⋊C4×Dic5—31no—
45#65229—32no—
46#65228—31no—
47C4×#683D311yescharacteristic, complemented
48#64958—32no—
49#64956—32no—
50C23×D15C2312yes—
51C22×C3×C4×C5C2311yescharacteristic
52C22×C3×C4×C5C2311yescharacteristic
53C22×C3×C4×C5C2311yescharacteristic
54C22×Dic15C2312yes—
55C2×D60—22no—
56C2×D60—22no—
57C2×D60C2×C414yes—
58C22×C5×Dic3C2312yes—
59C22×C5×Dic3C2312yes—
60C22×C3×Dic5C2311yescharacteristic
61C22×C3×Dic5C2311yescharacteristic
62C22×C3×Dic5C2311yescharacteristic
63C22×C3×Dic5C2311yescharacteristic
64C2×#1200—24no—
65C2×#1200—24no—
66C2×#1200C2×C418yes—
67C3×C42×C5—22no—
68#6714—24no—
69#6714—42no—
70C4⋊Dic15—22no—
71C5×C4⋊Dic3—22no—
72C5×C4⋊Dic3C2×C414yes—
73C4×C5×Dic3—24no—
74C3×#683—22no—
75C3×#683D414yescomplemented
76C3×C4×Dic5—22no—
77C3×C4×Dic5C2×C414yes—
78#6668—24no—
79#6668—42no—
80#6667—24no—
81#6667—42no—
82#6667C2×C418yes—
83Dic3×Dic5—24no—
84C2×#892—52no—
85C2×C4×D12—51no—
86C3×C4×C4⋊C4—51no—
87#4987—52no—
88#4939—51no—
89#4939—51no—
90#4935—51no—
91C4×C4⋊Dic3—51no—
92C4×C4⋊Dic3D511yescharacteristic, complemented
93#4670—52no—
94#4670—52no—
95C22×D20—31no—
96C22×C4×D5—32no—
97C2×C5×C4⋊C4—31no—
98C2×C5×C4⋊C4—31no—
99C2×C42×C5—32no—
100C2×C42×C5C2×D311yescharacteristic
101C2×#685—32no—
102C2×#685—32no—
103C2×#685—32no—
104C2×#685—32no—
105C2×#685—32no—
106C2×C4⋊Dic5—31no—
107C2×#683—32no—
108C2×#683—32no—
109C2×#683C2×D311yescharacteristic
110C2×#683C2×D311yescharacteristic
111C2×C4×Dic5—32no—
112C2×C4×Dic5C2×D311yescharacteristic
113C2×C4×Dic5C2×D311yescharacteristic
114#4039—38no—
115#4018—68no—
116C5×#95—32no—
117#3963—32no—
118#3963C2×D311yescharacteristic
119#3963C2×D311yescharacteristic
120#2354—151noSylow
121C23×C3×C5C2411yescharacteristic
122C22×D15—22no—
123C22×D15—22no—
124C22×D15—22no—
125C22×D15—22no—
126C22×D15—22no—
127C22×D15—22no—
128C22×D15C22×C414yes—
129C2×C3×C4×C5—21no—
130C2×C3×C4×C5—21no—
131C2×C3×C4×C5—21no—
132C2×C3×C4×C5—21no—
133C2×C3×C4×C5—21no—
134C2×C3×C4×C5C22×C412yes—
135C2×C3×C4×C5C2×D412yes—
136C2×C3×C4×C5#4112yes—
137C2×C3×C4×C5#4112yes—
138C2×Dic15—22no—
139C2×Dic15—22no—
140C2×Dic15—22no—
141D60—28no—
142C2×C5×Dic3—22no—
143C2×C5×Dic3—22no—
144C2×C5×Dic3—22no—
145C2×C5×Dic3—22no—
146C2×C5×Dic3—22no—
147C2×C5×Dic3C22×C414yes—
148C2×C3×Dic5—21no—
149C2×C3×Dic5—21no—
150C2×C3×Dic5—21no—
151C2×C3×Dic5—21no—
152C2×C3×Dic5—21no—
153C2×C3×Dic5—21no—
154C2×C3×Dic5—21no—
155C2×C3×Dic5C22×C412yes—
156C2×C3×Dic5C22×C412yes—
157C2×C3×Dic5C2×D412yes—
158C2×C3×Dic5C2×D412yes—
159C2×C3×Dic5#4112yes—
160#1200—216no—
161C22×D12—51no—
162C22×C4×D3—52no—
163C2×C3×C4⋊C4—51no—
164C2×C3×C4⋊C4—51no—
165C2×C3×C42—51no—
166C2×C3×C42—51no—
167C2×C3×C42C2×D511yescharacteristic
168C2×(C2×D3)⋊C4—52no—
169C2×(C2×D3)⋊C4—52no—
170C2×(C2×D3)⋊C4—52no—
171C2×(C2×D3)⋊C4—52no—
172C2×(C2×D3)⋊C4—52no—
173C2×C4⋊Dic3—51no—
174C2×C4⋊Dic3—52no—
175C2×C4⋊Dic3—52no—
176C2×C4⋊Dic3C2×D511yescharacteristic
177C2×C4⋊Dic3C2×D511yescharacteristic
178C2×C4×Dic3—52no—
179C2×C4×Dic3C2×D512yes—
180#892—108no—
181C4×D12—58no—
182C3×#95—51no—
183C3×#95—51no—
184C3⋊#95—52no—
185C3⋊#95C2×D512yes—
186C23×D5—32no—
187C22×C4×C5—32no—
188C22×C4×C5—32no—
189C22×C4×C5C22×D311yescharacteristic
190C22×C4×C5C22×D311yescharacteristic
191C22×C4×C5C22×D311yescharacteristic
192C22×Dic5—32no—
193C22×Dic5C22×D311yescharacteristic
194C22×Dic5C22×D311yescharacteristic
195C22×Dic5C22×D311yescharacteristic
196C22×Dic5C22×D311yescharacteristic
197C2×D20—34no—
198C2×D20—62no—
199C2×D20—62no—
200C2×C4×D5—38no—
201C2×C4×D5—64no—
202C2×C4×D5—64no—
203C5×C4⋊C4—34no—
204C5×C4⋊C4—62no—
205C42×C5—22no—
206C42×C5—64no—
207#685—38no—
208#685—64no—
209#685—64no—
210#685—122no—
211#685—122no—
212C4⋊Dic5—62no—
213#683—22no—
214#683—64no—
215#683—122no—
216#683D1214yescomplemented
217C4×Dic5—22no—
218C4×Dic5—64no—
219C4×Dic5C4×D314yes—
220C2×C4×D4—151no—
221C2×#117—152no—
222#390—151no—
223#390—151no—
224#388—152no—
225#388—152no—
226#384—151no—
227#380—152no—
228C4×C4⋊C4—31no—
229C4×C4⋊C4—51no—
230C4×C4⋊C4—151no—
231C22×C3×C5C23×C411yescharacteristic
232C22×C3×C5C22×D411yescharacteristic
233C22×C3×C5C2×#4111yescharacteristic
234C22×C3×C5C2×#4111yescharacteristic
235C22×C3×C5C2×#4111yescharacteristic
236C22×C3×C5H(5,𝔽2)11yescharacteristic
237C22×C3×C52−1+411yescharacteristic
238C2×D15—24no—
239C2×D15—24no—
240C2×D15—24no—
241C2×D15—42no—
242C2×D15—42no—
243C2×D15—42no—
244C2×D15—42no—
245C3×C4×C5—22no—
246C3×C4×C5—41no—
247C3×C4×C5#11714yescomplemented
248Dic15—42no—
249C3×Dic5—22no—
250C3×Dic5—22no—
251C3×Dic5—41no—
252C3×Dic5C4×D414yes—
253C5×Dic3—24no—
254C5×Dic3—42no—
255C23×D3—52no—
256C22×C3×C4—51no—
257C22×C3×C4—51no—
258C22×C3×C4—51no—
259C22×C3×C4—51no—
260C22×C3×C4C22×D511yescharacteristic
261C22×C3×C4C22×D511yescharacteristic
262C22×C3×C4C22×D511yescharacteristic
263C22×Dic3—52no—
264C22×Dic3C22×D512yes—
265C22×Dic3C22×D512yes—
266C2×D12—54no—
267C2×D12—102no—
268C2×D12—102no—
269C2×C4×D3—58no—
270C2×C4×D3—104no—
271C2×C4×D3—104no—
272C3×C4⋊C4—54no—
273C3×C4⋊C4—102no—
274C3×C42—22no—
275C3×C42—54no—
276C3×C42—102no—
277(C2×D3)⋊C4—58no—
278(C2×D3)⋊C4—104no—
279(C2×D3)⋊C4—104no—
280(C2×D3)⋊C4—202no—
281(C2×D3)⋊C4—202no—
282C4⋊Dic3—22no—
283C4⋊Dic3—104no—
284C4⋊Dic3—102no—
285C4⋊Dic3—202no—
286C4⋊Dic3C4×D514yes—
287C4×Dic3—24no—
288C4×Dic3—104no—
289C23×C5C23×D311yescharacteristic
290C22×D5—34no—
291C22×D5—62no—
292C22×D5—62no—
293C22×D5—62no—
294C22×D5—62no—
295C22×D5—62no—
296C22×D5—62no—
297C2×C4×C5—21no—
298C2×C4×C5—21no—
299C2×C4×C5—21no—
300C2×C4×C5—21no—
301C2×C4×C5—21no—
302C2×C4×C5—34no—
303C2×C4×C5—62no—
304C2×C4×C5—62no—
305C2×C4×C5—62no—
306C2×C4×C5—62no—
307C2×C4×C5—62no—
308C2×C4×C5C2×C4×D312yes—
309C2×C4×C5C2×D1212yes—
310C2×C4×C5#23712yes—
311C2×C4×C5#23912yes—
312C2×Dic5—21no—
313C2×Dic5—21no—
314C2×Dic5—21no—
315C2×Dic5—21no—
316C2×Dic5—21no—
317C2×Dic5—21no—
318C2×Dic5—21no—
319C2×Dic5—62no—
320C2×Dic5—62no—
321C2×Dic5—62no—
322C2×Dic5C2×C4×D312yes—
323C2×Dic5C2×C4×D312yes—
324C2×Dic5C2×D1212yes—
325C2×Dic5C2×D1212yes—
326C2×Dic5#23512yes—
327D20—68no—
328C4×D5—616no—
329C22×D4—151no—
330C23×C4—152no—
331C4×D4—158no—
332#117—308no—
333C2×C4⋊C4—31no—
334C2×C4⋊C4—31no—
335C2×C4⋊C4—51no—
336C2×C4⋊C4—51no—
337C2×C4⋊C4—152no—
338C2×C4⋊C4—151no—
339C2×C4⋊C4—152no—
340C2×C22⋊C4—152no—
341C2×C22⋊C4—152no—
342C2×C22⋊C4—152no—
343C2×C22⋊C4—152no—
344C2×C22⋊C4—152no—
345C2×C42—32no—
346C2×C42—51no—
347C2×C42—51no—
348C2×C42—152no—
349C2×C42D3×D511yescharacteristic
350#95—32no—
351#95—51no—
352#95—51no—
353#95—152no—
354C2×C3×C5C2×#11711yescharacteristic
355C2×C3×C5C2×C4×D411yescharacteristic
356C2×C3×C5#52011yescharacteristic
357C2×C3×C5#54711yescharacteristic
358C2×C3×C5#55011yescharacteristic
359C2×C3×C5#55511yescharacteristic
360C2×C3×C5#55611yescharacteristic
361D15—44no—
362C23×C3C23×D511yescharacteristic
363C22×D3—54no—
364C22×D3—102no—
365C22×D3—102no—
366C22×D3—102no—
367C22×D3—102no—
368C22×D3—102no—
369C22×D3—102no—
370C2×C3×C4—21no—
371C2×C3×C4—21no—
372C2×C3×C4—21no—
373C2×C3×C4—21no—
374C2×C3×C4—21no—
375C2×C3×C4—52no—
376C2×C3×C4—52no—
377C2×C3×C4—52no—
378C2×C3×C4—52no—
379C2×C3×C4—52no—
380C2×C3×C4—101no—
381C2×C3×C4—101no—
382C2×C3×C4—101no—
383C2×C3×C4—101no—
384C2×C3×C4—101no—
385C2×C3×C4—101no—
386C2×C3×C4—101no—
387C2×C3×C4C2×C4×D512yes—
388C2×C3×C4#70912yes—
389C2×C3×C4#70912yes—
390C2×C3×C4D4×D512yes—
391C2×Dic3—22no—
392C2×Dic3—22no—
393C2×Dic3—22no—
394C2×Dic3—22no—
395C2×Dic3—22no—
396C2×Dic3—102no—
397C2×Dic3—102no—
398C2×Dic3—102no—
399C2×Dic3C2×C4×D514yes—
400D12—108no—
401C4×D3—1016no—
402C22×C5C22×C4×D311yescharacteristic
403C22×C5C22×D1211yescharacteristic
404C22×C5C2×#23511yescharacteristic
405C22×C5C2×#23711yescharacteristic
406C22×C5C2×#23911yescharacteristic
407C22×C5#100911yescharacteristic
408C22×C5#101011yescharacteristic
409C2×D5—64no—
410C2×D5—64no—
411C2×D5—64no—
412C2×D5—122no—
413C2×D5—122no—
414C2×D5—122no—
415C2×D5—122no—
416C4×C5—22no—
417C4×C5—41no—
418C4×C5—64no—
419C4×C5—122no—
420C4×C5#89214yescomplemented
421Dic5—22no—
422Dic5—22no—
423Dic5—41no—
424Dic5—122no—
425Dic5C4×D1214yes—
426C24—152no—
427C2×D4—154no—
428C2×D4—302no—
429C2×D4—302no—
430C22×C4—32no—
431C22×C4—32no—
432C22×C4—51no—
433C22×C4—51no—
434C22×C4—51no—
435C22×C4—51no—
436C22×C4—158no—
437C22×C4—152no—
438C22×C4—304no—
439C22×C4—304no—
440C22×C4C2×D3×D511yescharacteristic
441C22×C4C2×D3×D511yescharacteristic
442C22×C4C2×D3×D511yescharacteristic
443C4⋊C4—34no—
444C4⋊C4—54no—
445C4⋊C4—62no—
446C4⋊C4—102no—
447C4⋊C4—304no—
448C4⋊C4—302no—
449C4⋊C4—602no—
450C22⋊C4—158no—
451C22⋊C4—304no—
452C22⋊C4—304no—
453C22⋊C4—602no—
454C22⋊C4—602no—
455C42—22no—
456C42—54no—
457C42—64no—
458C42—102no—
459C42—304no—
460C3×C5#235411yescharacteristic, complemented, Hall
461C22×C3C22×C4×D511yescharacteristic
462C22×C3C2×#70911yescharacteristic
463C22×C3C2×#70911yescharacteristic
464C22×C3C2×D4×D511yescharacteristic
465C22×C3#414711yescharacteristic
466C22×C3D5×#4111yescharacteristic
467C22×C3#414911yescharacteristic
468C2×D3—104no—
469C2×D3—104no—
470C2×D3—104no—
471C2×D3—202no—
472C2×D3—202no—
473C2×D3—202no—
474C2×D3—202no—
475C3×C4—22no—
476C3×C4—41no—
477C3×C4—54no—
478C3×C4—102no—
479C3×C4—102no—
480C3×C4—201no—
481C3×C4#401814yescomplemented
482Dic3—24no—
483Dic3—42no—
484Dic3—202no—
485C2×C5C2×C4×D1211yescharacteristic
486C2×C5C2×#89211yescharacteristic
487C2×C5#552311yescharacteristic
488C2×C5#555511yescharacteristic
489C2×C5#556311yescharacteristic
490C2×C5#557711yescharacteristic
491C2×C5#558411yescharacteristic
492D5—124no—
493C23—154no—
494C23—302no—
495C23—302no—
496C23—302no—
497C23—302no—
498C23—302no—
499C23—302no—
500C23C22×D3×D511yescharacteristic
501D4—308no—
502C2×C4—21no—
503C2×C4—21no—
504C2×C4—21no—
505C2×C4—21no—
506C2×C4—21no—
507C2×C4—34no—
508C2×C4—52no—
509C2×C4—52no—
510C2×C4—52no—
511C2×C4—52no—
512C2×C4—52no—
513C2×C4—62no—
514C2×C4—62no—
515C2×C4—62no—
516C2×C4—62no—
517C2×C4—62no—
518C2×C4—101no—
519C2×C4—101no—
520C2×C4—101no—
521C2×C4—101no—
522C2×C4—101no—
523C2×C4—101no—
524C2×C4—101no—
525C2×C4—3016no—
526C2×C4—302no—
527C2×C4—302no—
528C2×C4—302no—
529C2×C4#676912yes—
530C2×C4C4×D3×D512yes—
531C2×C4D5×D1212yes—
532C2×C4#677912yes—
533C2×C3C2×#401811yescharacteristic
534C2×C3#6584111yescharacteristic
535C2×C3#6586611yescharacteristic
536C2×C3C4×D4×D511yescharacteristic
537C2×C3#6588711yescharacteristic
538C2×C3#6590211yescharacteristic
539C2×C3#6590811yescharacteristic
540D3—204no—
541C5#7363811yescharacteristic, complemented, Sylow
542C22—304no—
543C22—304no—
544C22—304no—
545C22—602no—
546C22—602no—
547C22—602no—
548C22—602no—
549C22#9200011yescharacteristic
550C22C2×#676911yescharacteristic
551C22C2×C4×D3×D511yescharacteristic
552C22C2×D5×D1211yescharacteristic
553C22D5×#23511yescharacteristic
554C22#9201911yescharacteristic
555C22C2×#677911yescharacteristic
556C4—22no—
557C4—41no—
558C4—54no—
559C4—64no—
560C4—102no—
561C4—102no—
562C4—122no—
563C4—201no—
564C4—602no—
565C4#9140314yescomplemented
566C3#1060537011yescharacteristic, complemented, Sylow
567C2—604no—
568C2#1175252111yescharacteristic
569C2#1175253811yescharacteristic
570C2#1175259011yescharacteristic
571C2C4×D5×D1211yescharacteristic
572C2#1175422811yescharacteristic
573C2C2×#9140311yescharacteristic
574C2#1175432511yescharacteristic
subgroupG/H
center, FrattiniC23C22×D3×D5
commutatorC22×C3×C5C23×C4
FittingC2×C3×C42×C5C22
socle, cosocleC23×C3×C5C24
prime radicalC22×D60C22
hypercenterC2×C42D3×D5
hypocenterC3×C5#2354
LCSG ▷ C22×C3×C5 ▷ C3×C5
UCSC1 ◁ C23 ◁ C2×C42

Complements

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

NH ≅ G/Ntype
2. C2×#91403C2semidirect
4. C4×C5×C4⋊Dic3C2semidirect
5. C3×C4×#683C2semidirect
6. #11748876C2semidirect
7. #11748834C2semidirect
9. #11748730C2semidirect
10. Dic5×C4⋊Dic3C2semidirect
11. #11748671C2semidirect
32. #91418C4semidirect
47. C4×#683D3semidirect
75. C3×#683D4semidirect
92. C4×C4⋊Dic3D5semidirect
216. #683D12semidirect
247. C3×C4×C5#117semidirect
420. C4×C5#892semidirect
460. C3×C5#2354semidirect
481. C3×C4#4018semidirect
541. C5#73638semidirect
565. C4#91403semidirect
566. C3#10605370semidirect

Maximal subgroups

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

subgroupindex
C2×#914182
C2×#914032
#117510402
C4×C5×C4⋊Dic32
C3×C4×#6832
#117488762
#117488342
#117487682
#117487302
Dic5×C4⋊Dic32
#117486712
#106053703
#736385