Finite Group Databasebeta

← #14328055

Subgroups — #14328055

Order 1920.

1120 conjugacy classes of subgroups.

The subgroup lattice has 564 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#91418C211yesmaximal, characteristic
2#11751089C211yesmaximal, characteristic
3C5×Dic3×C4⋊C4C211yesmaximal, characteristic, complemented
4C3×#65266C211yesmaximal, characteristic, complemented
5#11748889C212yesmaximal, complemented
6#11748881C212yesmaximal, complemented
7C4×#6668C211yesmaximal, characteristic, complemented
8#11748789C211yesmaximal, characteristic
9#11748730C212yesmaximal, complemented
10#11748699C211yesmaximal, characteristic, complemented
11#11748671C212yesmaximal, complemented
12#10605377—31nomaximal, Hall
13C22×D60C2211yescharacteristic
14C22×#1200C2212yes—
15C2×C3×C5×C4⋊C4C2211yescharacteristic
16C2×#6714C2212yes—
17C2×#6714C2212yes—
18C2×C4×Dic15C2211yescharacteristic
19C2×C5×C4⋊Dic3C2211yescharacteristic
20C2×C4×C5×Dic3C2211yescharacteristic
21C2×C4×C5×Dic3C2212yes—
22C2×C3×C4⋊Dic5C2211yescharacteristic
23C2×C3×C4×Dic5C2211yescharacteristic
24C2×#6668C2211yescharacteristic
25C2×#6668C2211yescharacteristic
26C2×#6667C2212yes—
27C2×#6667C2212yes—
28C2×#6667C2212yes—
29C2×Dic3×Dic5C2212yes—
30#91418C418yescomplemented
31C5×C3⋊#95C2212yes—
32C3×#3963C2212yes—
33C3×#3963C2212yes—
34#90994C2211yescharacteristic
35#90994C2211yescharacteristic
36#90994C2212yes—
37#74160—51nomaximal, Hall
38C2×#4039—31no—
39C4×C5×C4⋊C4—31no—
40#65277—31no—
41#65266D311yescharacteristic, complemented
42C4⋊C4×Dic5—31no—
43#65229—32no—
44#65228—31no—
45C4×#685—32no—
46C4×#683—31no—
47#64958—32no—
48#64953—32no—
49C23×D15C2312yes—
50C22×C3×C4×C5C2312yes—
51C22×C3×C4×C5C2311yescharacteristic
52C22×Dic15C2311yescharacteristic
53C22×Dic15C2311yescharacteristic
54C2×D60—22no—
55C2×D60—22no—
56C2×D60C2×C414yes—
57C22×C5×Dic3C2311yescharacteristic
58C22×C5×Dic3C2312yes—
59C22×C5×Dic3C2311yescharacteristic
60C22×C3×Dic5C2312yes—
61C22×C3×Dic5C2311yescharacteristic
62C22×C3×Dic5C2311yescharacteristic
63C2×#1200—42no—
64C2×#1200—42no—
65C2×#1200C2×C418yes—
66C3×C5×C4⋊C4—22no—
67#6714—24no—
68#6714—42no—
69C4×Dic15—22no—
70C5×C4⋊Dic3C2×C414yes—
71C4×C5×Dic3—22no—
72C4×C5×Dic3—42no—
73C3×C4⋊Dic5—22no—
74C3×C4×Dic5C2×C414yes—
75#6668—22no—
76#6668—22no—
77#6667—24no—
78#6667—42no—
79#6667C2×C418yes—
80Dic3×Dic5—42no—
81C2×C4×D12—51no—
82C3×#382—51no—
83#4988—51no—
84#4988—51no—
85Dic3×C4⋊C4—51no—
86Dic3×C4⋊C4D511yescharacteristic, complemented
87C4×C4⋊Dic3—51no—
88#4670—52no—
89#4670—52no—
90#4666—52no—
91#4664—52no—
92C22×D20—31no—
93C22×C4×D5—32no—
94C2×C5×C4⋊C4—31no—
95C2×C5×C4⋊C4C2×D311yescharacteristic
96C2×C42×C5—31no—
97C2×C42×C5—32no—
98C2×#685—32no—
99C2×#685—32no—
100C2×#685—32no—
101C2×#685—32no—
102C2×#685—32no—
103C2×C4⋊Dic5C2×D311yescharacteristic
104C2×#683—31no—
105C2×#683—31no—
106C2×C4×Dic5—31no—
107C2×C4×Dic5—32no—
108C2×C4×Dic5C2×D311yescharacteristic
109#4039—38no—
110C5×#95—32no—
111#3963—31no—
112#3963—32no—
113#3963—31no—
114#3963C2×D312yes—
115#3963C2×D312yes—
116#2357—151noSylow
117C23×C3×C5C2411yescharacteristic
118C22×D15—22no—
119C22×D15—22no—
120C22×D15—22no—
121C22×D15—22no—
122C22×D15—22no—
123C22×D15—22no—
124C22×D15C22×C414yes—
125C2×C3×C4×C5—22no—
126C2×C3×C4×C5—22no—
127C2×C3×C4×C5—22no—
128C2×C3×C4×C5C22×C412yes—
129C2×C3×C4×C5#4112yes—
130C2×C3×C4×C5#4112yes—
131C2×Dic15—21no—
132C2×Dic15—21no—
133C2×Dic15—21no—
134C2×Dic15—21no—
135C2×Dic15—21no—
136C2×Dic15#4112yes—
137D60—28no—
138C2×C5×Dic3—21no—
139C2×C5×Dic3—22no—
140C2×C5×Dic3—21no—
141C2×C5×Dic3—22no—
142C2×C5×Dic3—21no—
143C2×C5×Dic3—21no—
144C2×C5×Dic3—21no—
145C2×C5×Dic3C22×C414yes—
146C2×C5×Dic3#4112yes—
147C2×C3×Dic5—21no—
148C2×C3×Dic5—21no—
149C2×C3×Dic5—21no—
150C2×C3×Dic5—21no—
151C2×C3×Dic5—22no—
152C2×C3×Dic5—22no—
153C2×C3×Dic5—22no—
154C2×C3×Dic5C22×C412yes—
155C2×C3×Dic5C22×C412yes—
156#1200—48no—
157C22×D12—51no—
158C22×C4×D3—52no—
159C2×C3×C4⋊C4—51no—
160C2×C3×C4⋊C4C2×D511yescharacteristic
161C2×C3×C42—51no—
162C2×(C2×D3)⋊C4—52no—
163C2×(C2×D3)⋊C4—52no—
164C2×(C2×D3)⋊C4—52no—
165C2×(C2×D3)⋊C4—52no—
166C2×(C2×D3)⋊C4—52no—
167C2×C4⋊Dic3—51no—
168C2×C4⋊Dic3—51no—
169C2×C4⋊Dic3C2×D511yescharacteristic
170C2×C4×Dic3—51no—
171C2×C4×Dic3—52no—
172C2×C4×Dic3C2×D511yescharacteristic
173C2×C4×Dic3C2×D512yes—
174C4×D12—58no—
175C3×#95—52no—
176C3×#95—52no—
177C3⋊#95—51no—
178C3⋊#95—52no—
179C3⋊#95—51no—
180C3⋊#95C2×D512yes—
181C23×D5—32no—
182C22×C4×C5—31no—
183C22×C4×C5—32no—
184C22×C4×C5—31no—
185C22×C4×C5C22×D312yes—
186C22×C4×C5C22×D311yescharacteristic
187C22×Dic5—31no—
188C22×Dic5—31no—
189C22×Dic5C22×D312yes—
190C22×Dic5C22×D311yescharacteristic
191C22×Dic5C22×D311yescharacteristic
192C2×D20—34no—
193C2×D20—62no—
194C2×D20—62no—
195C2×C4×D5—38no—
196C2×C4×D5—122no—
197C2×C4×D5—122no—
198C5×C4⋊C4—22no—
199C5×C4⋊C4—34no—
200C42×C5—62no—
201C42×C5—122no—
202#685—38no—
203#685—64no—
204#685—64no—
205#685—122no—
206#685—122no—
207C4⋊Dic5—22no—
208#683—62no—
209#683—62no—
210C4×Dic5—62no—
211C4×Dic5—122no—
212C4×Dic5C4×D314yes—
213C2×C4×D4—151no—
214#390—151no—
215#390—151no—
216#388—152no—
217#388—152no—
218#382—51no—
219#381—152no—
220C4×C4⋊C4—31no—
221C4×C4⋊C4—151no—
222C4×C4⋊C4—151no—
223C4×C22⋊C4—152no—
224C22×C3×C5C23×C411yescharacteristic
225C22×C3×C5C2×#4111yescharacteristic
226C22×C3×C5C2×#4111yescharacteristic
227C22×C3×C5C2×#4111yescharacteristic
228C22×C3×C5C2×#4111yescharacteristic
229C22×C3×C5H(5,𝔽2)11yescharacteristic
230C22×C3×C5H(5,𝔽2)11yescharacteristic
231C2×D15—24no—
232C2×D15—24no—
233C2×D15—24no—
234C2×D15—42no—
235C2×D15—42no—
236C2×D15—42no—
237C2×D15—42no—
238C3×C4×C5—42no—
239C3×C4×C5#11714yes—
240Dic15—22no—
241Dic15—41no—
242C3×Dic5—22no—
243C3×Dic5—22no—
244C3×Dic5—42no—
245C5×Dic3—24no—
246C5×Dic3—22no—
247C5×Dic3—41no—
248C23×D3—52no—
249C22×C3×C4—51no—
250C22×C3×C4—51no—
251C22×C3×C4—52no—
252C22×C3×C4C22×D512yes—
253C22×C3×C4C22×D511yescharacteristic
254C22×Dic3—51no—
255C22×Dic3—51no—
256C22×Dic3C22×D511yescharacteristic
257C22×Dic3C22×D512yes—
258C22×Dic3C22×D511yescharacteristic
259C2×D12—54no—
260C2×D12—102no—
261C2×D12—102no—
262C2×C4×D3—58no—
263C2×C4×D3—202no—
264C2×C4×D3—202no—
265C3×C4⋊C4—22no—
266C3×C4⋊C4—102no—
267C3×C42—54no—
268(C2×D3)⋊C4—58no—
269(C2×D3)⋊C4—104no—
270(C2×D3)⋊C4—104no—
271(C2×D3)⋊C4—202no—
272(C2×D3)⋊C4—202no—
273C4⋊Dic3—102no—
274C4⋊Dic3—102no—
275C4⋊Dic3C4×D514yes—
276C4×Dic3—22no—
277C4×Dic3—42no—
278C4×Dic3—102no—
279C4×Dic3—202no—
280C23×C5C23×D311yescharacteristic
281C22×D5—34no—
282C22×D5—62no—
283C22×D5—62no—
284C22×D5—62no—
285C22×D5—62no—
286C22×D5—62no—
287C22×D5—62no—
288C2×C4×C5—22no—
289C2×C4×C5—22no—
290C2×C4×C5—22no—
291C2×C4×C5—34no—
292C2×C4×C5—32no—
293C2×C4×C5—61no—
294C2×C4×C5—62no—
295C2×C4×C5—61no—
296C2×C4×C5—62no—
297C2×C4×C5—61no—
298C2×C4×C5—61no—
299C2×C4×C5—61no—
300C2×C4×C5C2×C4×D312yes—
301C2×C4×C5#23712yes—
302C2×C4×C5#23912yes—
303C2×Dic5—21no—
304C2×Dic5—21no—
305C2×Dic5—21no—
306C2×Dic5—21no—
307C2×Dic5—22no—
308C2×Dic5—22no—
309C2×Dic5—22no—
310C2×Dic5—32no—
311C2×Dic5—61no—
312C2×Dic5—61no—
313C2×Dic5—61no—
314C2×Dic5—61no—
315C2×Dic5—61no—
316C2×Dic5C2×C4×D312yes—
317C2×Dic5C2×C4×D312yes—
318D20—68no—
319C4×D5—128no—
320C22×D4—151no—
321C23×C4—152no—
322C4×D4—158no—
323C2×C4⋊C4—31no—
324C2×C4⋊C4—51no—
325C2×C4⋊C4—151no—
326C2×C4⋊C4—151no—
327C2×C4⋊C4D3×D511yescharacteristic
328C2×C22⋊C4—152no—
329C2×C22⋊C4—152no—
330C2×C22⋊C4—152no—
331C2×C22⋊C4—152no—
332C2×C22⋊C4—152no—
333C2×C42—31no—
334C2×C42—32no—
335C2×C42—51no—
336C2×C42—151no—
337C2×C42—152no—
338#95—32no—
339#95—52no—
340#95—52no—
341#95—151no—
342#95—151no—
343#95—152no—
344C2×C3×C5C2×#11711yescharacteristic
345C2×C3×C5C4×#4111yescharacteristic
346C2×C3×C5#51811yescharacteristic
347C2×C3×C5#55311yescharacteristic
348C2×C3×C5#55311yescharacteristic
349C2×C3×C5#55911yescharacteristic
350C2×C3×C5#55911yescharacteristic
351D15—44no—
352C23×C3C23×D511yescharacteristic
353C22×D3—54no—
354C22×D3—102no—
355C22×D3—102no—
356C22×D3—102no—
357C22×D3—102no—
358C22×D3—102no—
359C22×D3—102no—
360C2×C3×C4—22no—
361C2×C3×C4—22no—
362C2×C3×C4—22no—
363C2×C3×C4—52no—
364C2×C3×C4—52no—
365C2×C3×C4—101no—
366C2×C3×C4—101no—
367C2×C3×C4—101no—
368C2×C3×C4—101no—
369C2×C3×C4—102no—
370C2×C3×C4—102no—
371C2×C3×C4—102no—
372C2×C3×C4C2×C4×D512yes—
373C2×C3×C4#70912yes—
374C2×C3×C4#70912yes—
375C2×Dic3—21no—
376C2×Dic3—21no—
377C2×Dic3—22no—
378C2×Dic3—22no—
379C2×Dic3—21no—
380C2×Dic3—21no—
381C2×Dic3—21no—
382C2×Dic3—52no—
383C2×Dic3—101no—
384C2×Dic3—101no—
385C2×Dic3—101no—
386C2×Dic3—101no—
387C2×Dic3—101no—
388C2×Dic3C2×C4×D514yes—
389C2×Dic3#71312yes—
390D12—108no—
391C4×D3—208no—
392C22×C5C22×C4×D311yescharacteristic
393C22×C5C2×#23711yescharacteristic
394C22×C5C2×#23911yescharacteristic
395C22×C5D3×#4111yescharacteristic
396C22×C5D3×#4111yescharacteristic
397C22×C5#100911yescharacteristic
398C22×C5#100911yescharacteristic
399C2×D5—64no—
400C2×D5—64no—
401C2×D5—64no—
402C2×D5—122no—
403C2×D5—122no—
404C2×D5—122no—
405C2×D5—122no—
406C4×C5—42no—
407C4×C5—62no—
408C4×C5—64no—
409C4×C5—121no—
410C4×C5#89214yes—
411Dic5—22no—
412Dic5—22no—
413Dic5—42no—
414Dic5—62no—
415Dic5—121no—
416C24—152no—
417C2×D4—154no—
418C2×D4—302no—
419C2×D4—302no—
420C22×C4—32no—
421C22×C4—31no—
422C22×C4—31no—
423C22×C4—51no—
424C22×C4—51no—
425C22×C4—52no—
426C22×C4—158no—
427C22×C4—151no—
428C22×C4—151no—
429C22×C4—602no—
430C22×C4—602no—
431C22×C4C2×D3×D512yes—
432C22×C4C2×D3×D511yescharacteristic
433C4⋊C4—22no—
434C4⋊C4—34no—
435C4⋊C4—102no—
436C4⋊C4—302no—
437C4⋊C4—302no—
438C22⋊C4—158no—
439C22⋊C4—304no—
440C22⋊C4—304no—
441C22⋊C4—602no—
442C22⋊C4—602no—
443C42—54no—
444C42—62no—
445C42—122no—
446C42—302no—
447C42—602no—
448C3×C5#235711yescharacteristic, complemented, Hall
449C22×C3C22×C4×D511yescharacteristic
450C22×C3C2×#70911yescharacteristic
451C22×C3C2×#70911yescharacteristic
452C22×C3#414411yescharacteristic
453C22×C3C2×#71311yescharacteristic
454C22×C3D5×#4111yescharacteristic
455C22×C3#414911yescharacteristic
456C2×D3—104no—
457C2×D3—104no—
458C2×D3—104no—
459C2×D3—202no—
460C2×D3—202no—
461C2×D3—202no—
462C2×D3—202no—
463C3×C4—42no—
464C3×C4—102no—
465C3×C4—102no—
466C3×C4—202no—
467C3×C4#401814yes—
468Dic3—22no—
469Dic3—24no—
470Dic3—41no—
471Dic3—102no—
472Dic3—201no—
473C2×C5C2×#89211yescharacteristic
474C2×C5#552111yescharacteristic
475C2×C5#552211yescharacteristic
476C2×C5#568111yescharacteristic
477C2×C5#568111yescharacteristic
478C2×C5#569511yescharacteristic
479C2×C5#569511yescharacteristic
480D5—124no—
481C23—154no—
482C23—302no—
483C23—302no—
484C23—302no—
485C23—302no—
486C23—302no—
487C23—302no—
488C23C22×D3×D511yescharacteristic
489D4—308no—
490C2×C4—22no—
491C2×C4—22no—
492C2×C4—22no—
493C2×C4—34no—
494C2×C4—32no—
495C2×C4—52no—
496C2×C4—52no—
497C2×C4—61no—
498C2×C4—61no—
499C2×C4—62no—
500C2×C4—62no—
501C2×C4—61no—
502C2×C4—61no—
503C2×C4—61no—
504C2×C4—101no—
505C2×C4—101no—
506C2×C4—101no—
507C2×C4—101no—
508C2×C4—102no—
509C2×C4—102no—
510C2×C4—102no—
511C2×C4—152no—
512C2×C4—301no—
513C2×C4—301no—
514C2×C4—301no—
515C2×C4—301no—
516C2×C4—301no—
517C2×C4—608no—
518C2×C4#676912yes—
519C2×C4C4×D3×D512yes—
520C2×C4#677912yes—
521C2×C3C2×#401811yescharacteristic
522C2×C3#6584811yescharacteristic
523C2×C3#6586411yescharacteristic
524C2×C3#6587911yescharacteristic
525C2×C3#6589311yescharacteristic
526C2×C3C4×#71311yescharacteristic
527C2×C3#6591911yescharacteristic
528D3—204no—
529C5#7416011yescharacteristic, complemented, Sylow
530C22—304no—
531C22—304no—
532C22—304no—
533C22—602no—
534C22—602no—
535C22—602no—
536C22—602no—
537C22C2×#676911yescharacteristic
538C22C2×C4×D3×D511yescharacteristic
539C22#9202411yescharacteristic
540C22#9202711yescharacteristic
541C22D3×#71311yescharacteristic
542C22#9203511yescharacteristic
543C22C2×#677911yescharacteristic
544C4—42no—
545C4—62no—
546C4—64no—
547C4—102no—
548C4—102no—
549C4—121no—
550C4—202no—
551C4—302no—
552C4—601no—
553C4#9140314yes—
554C3#1060537711yescharacteristic, complemented, Sylow
555C2—604no—
556C2#1175340311yescharacteristic
557C2#1175362211yescharacteristic
558C2#1175378311yescharacteristic
559C2#1175383711yescharacteristic
560C2C2×#9140311yescharacteristic
561C2#1175490311yescharacteristic
562C2#1175525311yescharacteristic
subgroupG/H
center, FrattiniC23C22×D3×D5
commutatorC22×C3×C5C23×C4
FittingC2×C3×C5×C4⋊C4C22
socle, cosocleC23×C3×C5C24
prime radicalC22×D60C22
hypercenterC2×C4⋊C4D3×D5
hypocenterC3×C5#2357
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. C5×Dic3×C4⋊C4C2semidirect
4. C3×#65266C2semidirect
5. #11748889C2semidirect
6. #11748881C2semidirect
7. C4×#6668C2semidirect
9. #11748730C2semidirect
10. #11748699C2semidirect
11. #11748671C2semidirect
30. #91418C4semidirect
41. #65266D3semidirect
86. Dic3×C4⋊C4D5semidirect
448. C3×C5#2357semidirect
529. C5#74160semidirect
554. C3#10605377semidirect

Maximal subgroups

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

subgroupindex
C2×#914182
#117510892
C5×Dic3×C4⋊C42
C3×#652662
#117488892
#117488812
C4×#66682
#117487892
#117487302
#117486992
#117486712
#106053773
#741605