Finite Group Databasebeta

← #14412238

Subgroups — #14412238

Order 1920.

1408 conjugacy classes of subgroups.

The subgroup lattice has 588 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#92001C211yesmaximal, characteristic
2#11756822C212yesmaximal, complemented
3#11752420C211yesmaximal, characteristic, complemented
4C3×#65761C211yesmaximal, characteristic, complemented
5#11752048C214yesmaximal, complemented
6D3×#4005C214yesmaximal, complemented
7D5⋊#824C212yesmaximal, complemented
8#10617554—31nomaximal, Hall
9C22×Dic30C411yescharacteristic
10C2×C5×#235C411yescharacteristic
11C2×C3×#709C2211yescharacteristic
12C2×#6780C412yes—
13C2×#6780C2212yes—
14C2×#6772C411yescharacteristic
15C2×#6772C2211yescharacteristic
16C2×D3×Dic10C2212yes—
17#92001—24no—
18#92001—24no—
19C2×#6765C412yes—
20C2×D5×Dic6C411yescharacteristic
21C2×D5×Dic6C2211yescharacteristic
22D5⋊#208C2212yes—
23C3×D5⋊M16C2212yes—
24D3×#704C2218yes—
25#91913—24no—
26#91911—24no—
27#91242C2212yes—
28#91234C2212yes—
29C3×#4013C2212yes—
30C3×#4005C2212yes—
31#91175C2218yes—
32#78715—51nomaximal, Hall
33C2×#4150—31no—
34#66256—32no—
35D5⋊#101—32no—
36#65777—34no—
37#65761—31no—
38#65761D311yescharacteristic, complemented
39C2×#4005—34no—
40C22×C3×C4×C5C2×C411yescharacteristic
41C22×Dic15C2×C412yes—
42C2×Dic30—22no—
43C2×Dic30—22no—
44C2×Dic30C2×C411yescharacteristic
45C2×Dic30C2×C412yes—
46C2×Dic30C2×C411yescharacteristic
47C2×C5×#67C2×C412yes—
48C5×#235—22no—
49C5×#235—22no—
50C2×C5×D12C2×C411yescharacteristic
51C2×C4×C5×D3C2×C412yes—
52C2×C5×Dic6C2×C411yescharacteristic
53C2×C3×C22⋊D5C2×C411yescharacteristic
54C2×C3×C22⋊D5C2311yescharacteristic
55C3×#709—22no—
56C3×#709—22no—
57C2×C3×D20C2×C411yescharacteristic
58C2×C3×C4×D5C2×C411yescharacteristic
59C2×C3×C4×D5C2311yescharacteristic
60C2×C3×Dic10C2311yescharacteristic
61C2×#1204C2×C412yes—
62C2×#1204C2312yes—
63C2×#1201C2×C412yes—
64C2×#1201C2×C412yes—
65C2×D3×Dic5C2×C412yes—
66C2×D3×Dic5C2312yes—
67#6780—28no—
68#6780—28no—
69C2×D5×Dic3C2×C412yes—
70C2×D5×Dic3C2×C412yes—
71#6772—22no—
72#6772—22no—
73#6772D414yes—
74#6772D414yes—
75D3×Dic10—28no—
76#6765—28no—
77D5×Dic6—22no—
78D5×Dic6—22no—
79D5×Dic6—22no—
80D5×Dic6—22no—
81#6762C2314yes—
82#6758—22no—
83D5⋊(C3⋊C8)—22no—
84C3×#704C2314yes—
85C3×#700—22no—
86C3×D5⋊C8—22no—
87#6739—28no—
88D3×#170—28no—
89C2×#1010—51no—
90#5745—52no—
91C3×#413—51no—
92#5237—51no—
93C2×#824—52no—
94D3×#101—54no—
95#4745—54no—
96C2×C5×#41—31no—
97#4150—64no—
98#4150—64no—
99C2×Q8×D5—31no—
100C2×Q8×D5—31no—
101C2×#711—32no—
102C2×#711—32no—
103C2×#711—31no—
104C2×#711—31no—
105C2×#709—32no—
106C2×#709C2×D311yescharacteristic
107C22×Dic10—31no—
108C22×Dic10—32no—
109C2×#704—38no—
110#4131—64no—
111#4131—64no—
112D5⋊M16—32no—
113D5⋊M16C2×D312yes—
114#4013—32no—
115#4013C2×D312yes—
116#4005—32no—
117#4005—38no—
118#4005C2×D312yes—
119#2889—151noSylow
120C23×C3×C5C22×C411yescharacteristic
121C22×C5×D3C22×C412yes—
122C22×C3×D5C22×C411yescharacteristic
123C22×C3×D5C22×C411yescharacteristic
124C2×C3×C4×C5—21no—
125C2×C3×C4×C5—21no—
126C2×C3×C4×C5C22×C411yescharacteristic
127C2×C3×C4×C5C22×C411yescharacteristic
128C2×Dic15—24no—
129C2×Dic15C22×C412yes—
130C2×Dic15C22×C412yes—
131Dic30—21no—
132Dic30—21no—
133Dic30—24no—
134Dic30C22⋊C412yes—
135Dic30C22⋊C412yes—
136C5×#67—24no—
137C2×C5×Dic3C22×C412yes—
138C5×D12C22⋊C412yes—
139C5×D12C22⋊C412yes—
140C4×C5×D3—24no—
141C5×Dic6—21no—
142C5×Dic6—21no—
143C3×C22⋊D5—22no—
144C3×C22⋊D5—22no—
145C2×C3×Dic5C22×C411yescharacteristic
146C2×C3×Dic5C2411yescharacteristic
147C3×D20—22no—
148C3×C4×D5—21no—
149C3×C4×D5—21no—
150C3×C4×D5C2×D412yes—
151C3×C4×D5C2×D412yes—
152C3×Dic10—22no—
153#1204—24no—
154#1204—24no—
155#1201—24no—
156#1201C22⋊C418yes—
157D3×Dic5—24no—
158D3×Dic5C2×D418yes—
159D5×Dic3—24no—
160D5×Dic3—24no—
161C3⋊#170—24no—
162C3×#170—24no—
163C2×C3×#41—51no—
164#1010—104no—
165#1010—104no—
166C2×D3×Q8—52no—
167C2×#237—52no—
168C2×#237—52no—
169C2×#237—52no—
170C2×#235—51no—
171C2×#235—51no—
172C2×#235AGL(1,𝔽5)11yescharacteristic
173C22×Dic6—51no—
174C22×Dic6—51no—
175C22×Dic6—51no—
176C2×C3×M16—52no—
177C2×#208—52no—
178#907—104no—
179D3×M16—58no—
180#901—104no—
181C3×#101—52no—
182C3×#100—52no—
183#836—52no—
184#833—52no—
185#824—58no—
186C5×#41—62no—
187C5×#41—62no—
188C2×C5×Q8—31no—
189C2×C5×D4—32no—
190C2×C5×D4—31no—
191C22×C4×C5—32no—
192C22×C4×C5C4×D311yescharacteristic
193C2×C22⋊D5—32no—
194C2×C22⋊D5—32no—
195C2×C22⋊D5C4×D311yescharacteristic
196C2×C22⋊D5C22×D311yescharacteristic
197C22×Dic5—32no—
198C22×Dic5—32no—
199C22×Dic5—32no—
200Q8×D5—62no—
201Q8×D5—62no—
202Q8×D5—62no—
203Q8×D5—62no—
204#711—34no—
205#711—34no—
206#711—62no—
207#711—62no—
208#711—68no—
209#711—68no—
210#709—22no—
211#709—22no—
212#709—68no—
213C2×D20C4×D311yescharacteristic
214C2×C4×D5—32no—
215C2×C4×D5—32no—
216C2×C4×D5C4×D311yescharacteristic
217C2×C4×D5C22×D311yescharacteristic
218C2×Dic10—31no—
219C2×Dic10—32no—
220C2×Dic10—31no—
221C2×Dic10—32no—
222C2×Dic10—32no—
223C2×Dic10—68no—
224C2×Dic10—62no—
225C2×Dic10—62no—
226C2×Dic10C22×D311yescharacteristic
227#704—34no—
228#704—68no—
229#704C22×D314yes—
230C2×#170—68no—
231#700—22no—
232#700—62no—
233D5⋊C8—22no—
234D5⋊C8—62no—
235C2×2−1+4—151no—
236#568—152no—
237#413—51no—
238#413—154no—
239#413—151no—
240C2×#101—154no—
241C2×#101—152no—
242C22×C3×C5—21no—
243C22×C3×C5—41no—
244C22×C3×C5C23×C411yescharacteristic
245C2×C5×D3—42no—
246C2×C5×D3C2×C22⋊C414yes—
247C2×C3×D5—21no—
248C2×C3×D5—41no—
249C2×C3×D5—41no—
250C2×C3×D5C2×C22⋊C412yes—
251C3×C4×C5—21no—
252C3×C4×C5C2×C22⋊C411yescharacteristic
253C3×C4×C5C2×C22⋊C411yescharacteristic
254Dic15—22no—
255Dic15C2×C22⋊C414yes—
256C3×Dic5—21no—
257C3×Dic5C22×D412yes—
258C5×Dic3—22no—
259C3×#41—102no—
260C3×#41—102no—
261C2×C3×Q8—51no—
262C2×C3×D4—51no—
263C2×C3×D4—51no—
264C2×C3×D4—51no—
265C22×C3×C4—51no—
266C22×C3×C4—51no—
267C22×C3×C4C2×AGL(1,𝔽5)11yescharacteristic
268C2×#67—52no—
269C2×#67—52no—
270C2×#67C2×AGL(1,𝔽5)12yes—
271C22×Dic3—52no—
272C22×Dic3—52no—
273C22×Dic3—52no—
274D3×Q8—108no—
275#237—108no—
276#237—108no—
277#237—108no—
278#235—22no—
279#235—22no—
280#235—54no—
281#235—54no—
282#235—102no—
283#235—102no—
284C2×D12C2×AGL(1,𝔽5)11yescharacteristic
285C2×C4×D3—52no—
286C2×C4×D3—52no—
287C2×C4×D3C2×AGL(1,𝔽5)12yes—
288C2×Dic6—51no—
289C2×Dic6—51no—
290C2×Dic6—52no—
291C2×Dic6—52no—
292C2×Dic6—52no—
293C2×Dic6—102no—
294C2×Dic6—102no—
295C2×Dic6—102no—
296C2×Dic6—102no—
297C2×Dic6—102no—
298C2×Dic6—102no—
299C2×Dic6C2×AGL(1,𝔽5)11yescharacteristic
300C3×M16—54no—
301C3×M16—102no—
302C2×C3×C8—102no—
303#208—54no—
304#208—102no—
305C2×C3⋊C8—102no—
306ΓL(1,𝔽25)—108no—
307D3×C8—108no—
308C23×C5—32no—
309C23×C5C2×C4×D311yescharacteristic
310C22×D5C2×C4×D311yescharacteristic
311C22×D5C2×C4×D311yescharacteristic
312C5×Q8—61no—
313C5×Q8—61no—
314C5×D4—32no—
315C5×D4—32no—
316C5×D4—64no—
317C2×C4×C5—21no—
318C2×C4×C5—21no—
319C2×C4×C5—32no—
320C2×C4×C5—64no—
321C2×C4×C5C2×C4×D311yescharacteristic
322C2×C4×C5C2×C4×D311yescharacteristic
323C22⋊D5—22no—
324C22⋊D5—22no—
325C22⋊D5—38no—
326C22⋊D5—64no—
327C2×Dic5—32no—
328C2×Dic5—38no—
329C2×Dic5—32no—
330C2×Dic5—64no—
331C2×Dic5—64no—
332C2×Dic5C2×C4×D311yescharacteristic
333C2×Dic5C23×D311yescharacteristic
334D20—22no—
335C4×D5—21no—
336C4×D5—21no—
337C4×D5—64no—
338C4×D5—64no—
339C4×D5D3×D412yes—
340C4×D5D3×D412yes—
341Dic10—22no—
342Dic10—32no—
343Dic10—32no—
344Dic10—64no—
345Dic10—64no—
346Dic10—61no—
347Dic10—61no—
348Dic10—64no—
349#170—24no—
350#170—64no—
3512−1+4—304no—
3522−1+4—304no—
353C2×#41—31no—
354C2×#41—51no—
355C2×#41—152no—
356C2×#41—151no—
357C2×#41—151no—
358C2×#41—152no—
359C2×#41—152no—
360C22×Q8—152no—
361C22×Q8—151no—
362C22×Q8—151no—
363C22×Q8—151no—
364#131—304no—
365#131—304no—
366C2×M16—52no—
367C2×M16—158no—
368C2×M16—152no—
369#101—52no—
370#101—158no—
371#101—152no—
372#100—52no—
373#100—152no—
374C2×C3×C5—21no—
375C2×C3×C5—41no—
376C2×C3×C5C22×C22⋊C411yescharacteristic
377C3×D5—22no—
378C3×D5—41no—
379C5×D3—24no—
380C23×C3—51no—
381C23×C3—51no—
382C23×C3C22×AGL(1,𝔽5)11yescharacteristic
383C22×D3C22×AGL(1,𝔽5)12yes—
384C3×Q8—102no—
385C3×D4—102no—
386C3×D4—102no—
387C3×D4—102no—
388C2×C3×C4—21no—
389C2×C3×C4—21no—
390C2×C3×C4—51no—
391C2×C3×C4—52no—
392C2×C3×C4—52no—
393C2×C3×C4—51no—
394C2×C3×C4—101no—
395C2×C3×C4—101no—
396C2×C3×C4C22×AGL(1,𝔽5)11yescharacteristic
397C2×C3×C4C22×AGL(1,𝔽5)11yescharacteristic
398#67—24no—
399#67—58no—
400#67—104no—
401C2×Dic3—52no—
402C2×Dic3—52no—
403C2×Dic3—104no—
404C2×Dic3—104no—
405C2×Dic3—104no—
406C2×Dic3C22×AGL(1,𝔽5)12yes—
407D12C22⋊AGL(1,𝔽5)12yes—
408D12C22⋊AGL(1,𝔽5)12yes—
409C4×D3—24no—
410C4×D3—58no—
411C4×D3—104no—
412Dic6—21no—
413Dic6—21no—
414Dic6—52no—
415Dic6—52no—
416Dic6—104no—
417Dic6—104no—
418Dic6—104no—
419Dic6—101no—
420Dic6—101no—
421C3×C8—104no—
422C3⋊C8—104no—
423C22×C5—21no—
424C22×C5—34no—
425C22×C5—41no—
426C22×C5—122no—
427C22×C5C22×C4×D311yescharacteristic
428C2×D5—21no—
429C2×D5—41no—
430C2×D5—41no—
431C2×D5D3×C22⋊C412yes—
432C4×C5—21no—
433C4×C5—62no—
434C4×C5D3×C22⋊C411yescharacteristic
435C4×C5D3×C22⋊C411yescharacteristic
436Dic5—21no—
437Dic5—34no—
438Dic5—62no—
439Dic5C2×D3×D412yes—
440#41—62no—
441#41—62no—
442#41—102no—
443#41—102no—
444#41—154no—
445#41—154no—
446#41—308no—
447#41—302no—
448#41—302no—
449#41—308no—
450#41—308no—
451C2×Q8—31no—
452C2×Q8—51no—
453C2×Q8—152no—
454C2×Q8—152no—
455C2×Q8—152no—
456C2×Q8—151no—
457C2×Q8—151no—
458C2×Q8—302no—
459C2×Q8—302no—
460C2×Q8—302no—
461C2×Q8—302no—
462C2×Q8—308no—
463C2×Q8—302no—
464C2×Q8—302no—
465C2×D4—32no—
466C2×D4—31no—
467C2×D4—51no—
468C2×D4—51no—
469C2×D4—51no—
470C2×D4—152no—
471C2×D4—152no—
472C22×C4—32no—
473C22×C4—51no—
474C22×C4—51no—
475C22×C4—152no—
476C22×C4—152no—
477C22×C4—152no—
478C22×C4—152no—
479C22×C4—152no—
480C22×C4D3×AGL(1,𝔽5)11yescharacteristic
481M16—54no—
482M16—102no—
483M16—154no—
484M16—308no—
485M16—302no—
486C2×C8—102no—
487C2×C8—308no—
488C2×C8—302no—
489C3×C5#288911yescharacteristic, complemented, Hall
490C22×C3—21no—
491C22×C3—41no—
492C22×C3—52no—
493C22×C3—101no—
494C22×C3—201no—
495C22×C3—201no—
496C22×C3C23×AGL(1,𝔽5)11yescharacteristic
497C2×D3—42no—
498C2×D3C2×C22⋊AGL(1,𝔽5)14yes—
499C3×C4—21no—
500C3×C4—52no—
501C3×C4—101no—
502C3×C4C2×C22⋊AGL(1,𝔽5)11yescharacteristic
503C3×C4C2×C22⋊AGL(1,𝔽5)11yescharacteristic
504Dic3—22no—
505Dic3—54no—
506Dic3—102no—
507C2×C5—21no—
508C2×C5—41no—
509C2×C5—64no—
510C2×C5C2×D3×C22⋊C411yescharacteristic
511D5—22no—
512D5—41no—
513C23—32no—
514C23—51no—
515C23—51no—
516C23C2×D3×AGL(1,𝔽5)11yescharacteristic
517Q8—61no—
518Q8—61no—
519Q8—102no—
520Q8—152no—
521Q8—152no—
522Q8—304no—
523Q8—304no—
524Q8—304no—
525Q8—301no—
526Q8—301no—
527D4—32no—
528D4—32no—
529D4—64no—
530D4—102no—
531D4—102no—
532D4—102no—
533D4—158no—
534D4—304no—
535C2×C4—21no—
536C2×C4—21no—
537C2×C4—32no—
538C2×C4—51no—
539C2×C4—52no—
540C2×C4—52no—
541C2×C4—51no—
542C2×C4—64no—
543C2×C4—101no—
544C2×C4—101no—
545C2×C4—158no—
546C2×C4—152no—
547C2×C4—152no—
548C2×C4—304no—
549C2×C4—304no—
550C2×C4—304no—
551C2×C4—304no—
552C2×C4C2×D3×AGL(1,𝔽5)11yescharacteristic
553C2×C4C2×D3×AGL(1,𝔽5)11yescharacteristic
554C8—104no—
555C8—304no—
556C2×C3—21no—
557C2×C3—41no—
558C2×C3—102no—
559C2×C3—201no—
560C2×C3C22×C22⋊AGL(1,𝔽5)11yescharacteristic
561D3—24no—
562C5#7871511yescharacteristic, complemented, Sylow
563C22—21no—
564C22—34no—
565C22—41no—
566C22—52no—
567C22—101no—
568C22—122no—
569C22—201no—
570C22—201no—
571C22C22×D3×AGL(1,𝔽5)11yescharacteristic
572C4—21no—
573C4—52no—
574C4—62no—
575C4—101no—
576C4—154no—
577C4—302no—
578C4D3×C22⋊AGL(1,𝔽5)11yescharacteristic
579C4D3×C22⋊AGL(1,𝔽5)11yescharacteristic
580C3#1061755411yescharacteristic, complemented, Sylow
581C2—21no—
582C2—41no—
583C2—64no—
584C2—102no—
585C2—201no—
586C2C2×D3×C22⋊AGL(1,𝔽5)11yescharacteristic
subgroupG/H
centerC2C2×D3×C22⋊AGL(1,𝔽5)
commutatorC22×C3×C5C23×C4
FrattiniC22C22×D3×AGL(1,𝔽5)
FittingC22×C3×C4×C5C2×C4
socleC2×C3×C5C22×C22⋊C4
prime radicalC2×#92001C2
cosocleC2×C3×Dic5C24
hypercenterC22×C4D3×AGL(1,𝔽5)
hypocenterC3×C5#2889
LCSG ▷ C22×C3×C5 ▷ C2×C3×C5 ▷ C3×C5
UCSC1 ◁ C2 ◁ C2×C4 ◁ C22×C4

Complements

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

NH ≅ G/Ntype
2. #11756822C2semidirect
3. #11752420C2semidirect
4. C3×#65761C2semidirect
5. #11752048C2semidirect
6. D3×#4005C2semidirect
7. D5⋊#824C2semidirect
38. #65761D3semidirect
489. C3×C5#2889semidirect
562. C5#78715semidirect
580. C3#10617554semidirect

Maximal subgroups

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

subgroupindex
C2×#920012
#117568222
#117524202
C3×#657612
#117520482
D3×#40052
D5⋊#8242
#106175543
#787155