Finite Group Databasebeta

← #14415652

Subgroups — #14415652

Order 1920.

1004 conjugacy classes of subgroups.

The subgroup lattice has 600 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#91990C211yesmaximal, characteristic, complemented
2C2×D3×#700C211yesmaximal, characteristic, complemented
3#11755060C211yesmaximal, characteristic
4C3×#65776C211yesmaximal, characteristic, complemented
5#11752054C212yesmaximal, complemented
6#11752039C212yesmaximal, complemented
7#67⋊#170C212yesmaximal, complemented
8#11751991C212yesmaximal, complemented
9#11751825C211yesmaximal, characteristic, complemented
10D5⋊((C2×D3)⋊C8)C211yesmaximal, characteristic, complemented
11#11751767C211yesmaximal, characteristic, complemented
12#10617229—31nomaximal, Hall
13C2×C3×#711C2211yescharacteristic
14C2×C4×D3×D5C2211yescharacteristic
15C2×#6762C2212yescomplemented
16C22×C3⋊#170C2212yescomplemented
17#91990—24no—
18C2×#6758C2211yescharacteristic
19C2×D5⋊(C3⋊C8)C2211yescharacteristic
20C2×C3×#700C2211yescharacteristic
21C2×C3×D5⋊C8C2211yescharacteristic
22C2×#6740C2212yes—
23C2×D3×#170C2212yes—
24D3×#700—24no—
25#91827C411yescharacteristic
26C5×#938C411yescharacteristic
27#91577C412yes—
28#91552C2212yes—
29Dic5×#67C2212yes—
30#91548C412yes—
31#91390C411yescharacteristic
32#91369C411yescharacteristic
33#91358C2211yescharacteristic
34C3×C22⋊#170C2212yescomplemented
35C3×C22⋊#170C2212yescomplemented
36#91177C2212yescomplemented
37Dic3⋊#170C2212yescomplemented
38#91171C2211yescharacteristic
39#91171C2211yescharacteristic
40(C2×D3)⋊#170C2211yescharacteristic
41(C2×D3)⋊#170C2211yescharacteristic
42#91169C2212yescomplemented
43Dic3×#170C2212yescomplemented
44#78659—51nomaximal, Hall
45C2×#4131—31no—
46C22×#700—31no—
47#65940—31no—
48#65776D311yescharacteristic, complemented
49#65773—32no—
50D4×#170—32no—
51D5⋊(C22⋊C8)—31no—
52#65708—31no—
53#65701—31no—
54#65695—32no—
55#65693—32no—
56C22×D3×D5C2×C411yescharacteristic
57C2×C3×C5×D4C2×C411yescharacteristic
58C2×#1220C2×C412yes—
59C2×C4×D15C2×C411yescharacteristic
60C2×C5×#67C2×C412yes—
61C2×C4×C5×D3C2×C411yescharacteristic
62C2×C3×C22⋊D5C2×C412yes—
63C22×C3×Dic5C2312yes—
64C3×#711—22no—
65C3×#711—22no—
66C2×C3×C4×D5C2311yescharacteristic
67C2×C3×Dic10C2×C411yescharacteristic
68C2×#1200C2311yescharacteristic
69C2×D3×Dic5C2311yescharacteristic
70C2×D5×Dic3C2×C411yescharacteristic
71C4×D3×D5—22no—
72C4×D3×D5—22no—
73#6762—24no—
74C2×C3⋊#170—24no—
75C2×C3⋊#170C2311yescharacteristic
76C2×C3⋊#170C2311yescharacteristic
77C2×C3⋊#170C2312yes—
78#6758D414yescomplemented
79D5⋊(C3⋊C8)—22no—
80C2×C3×#170C2312yes—
81C2×C3×#170C2311yescharacteristic
82C2×C3×#170C2311yescharacteristic
83C3×#700—22no—
84C3×D5⋊C8—22no—
85#6740—24no—
86D3×#170—24no—
87#6719C2×C412yes—
88C4⋊Dic15C2×C411yescharacteristic
89C5×#217C2×C412yes—
90C5×C4⋊Dic3C2×C411yescharacteristic
91Dic3⋊Dic5C2×C412yes—
92#6669C2×C412yes—
93#6667C2×C412yes—
94(C2×D3)⋊Dic5C2×C412yes—
95#6665C2×C412yes—
96Dic3×Dic5C2312yes—
97#5828—51no—
98C2×#948—51no—
99C2×D3×M16—51no—
100C3×#408—51no—
101#5119—51no—
102C2×(C2×D3)⋊C8—51no—
103#5101—51no—
104#4730—52no—
105#4729—52no—
106#4728—52no—
107#67⋊C8—52no—
108C2×#711C2×D311yescharacteristic
109C22×C4×D5—31no—
110C2×#704—32no—
111C22×#170—32no—
112C22×#170—32no—
113#4131—64no—
114C2×#700—31no—
115C2×#700—32no—
116C2×#700—64no—
117C2×#700C2×D311yescharacteristic
118C2×D5⋊C8—31no—
119C2×D5⋊C8C2×D311yescharacteristic
120C5×#121—31no—
121#4084—31no—
122#4083—32no—
123D4×Dic5—32no—
124#4070—31no—
125#4043—31no—
126#4038—31no—
127#4029—32no—
128#4027—32no—
129C22⋊#170—31no—
130C22⋊#170—31no—
131C22⋊#170C2×D312yescomplemented
132C22⋊#170C2×D312yescomplemented
133#4004—32no—
134#4003—31no—
135#4003—31no—
136#4002—32no—
137C4⋊#170—32no—
138C4×#170—32no—
139#2933—151noSylow
140C23×C3×C5C22×C412yes—
141C22×D15C22×C411yescharacteristic
142C22×C5×D3C22×C411yescharacteristic
143C22×C3×D5C22×C411yescharacteristic
144C2×D3×D5—22no—
145C2×D3×D5—22no—
146C2×D3×D5—41no—
147C3×C5×D4—22no—
148C2×C3×C4×C5C22×C411yescharacteristic
149#1220—42no—
150C2×Dic15C22×C412yes—
151C2×Dic15C22×C411yescharacteristic
152C4×D15—22no—
153C5×#67—24no—
154C2×C5×Dic3C22×C412yes—
155C2×C5×Dic3C22×C411yescharacteristic
156C4×C5×D3—22no—
157C3×C22⋊D5—24no—
158C2×C3×Dic5—22no—
159C2×C3×Dic5—22no—
160C2×C3×Dic5C22×C412yes—
161C2×C3×Dic5C2411yescharacteristic
162C3×C4×D5C2×D412yes—
163C3×C4×D5#4112yes—
164C3×Dic10—22no—
165#1200—21no—
166#1200—21no—
167D3×Dic5—21no—
168D3×Dic5—21no—
169D5×Dic3—22no—
170C3⋊#170—21no—
171C3⋊#170—21no—
172C3⋊#170C2×D414yes—
173C3×#170—22no—
174C3×#170—21no—
175C3×#170—21no—
176C2×C3×#41—51no—
177C22×C4×D3—51no—
178C2×C3×M16—51no—
179C22×C3×C8—51no—
180#948—104no—
181#946—51no—
182#946—51no—
183#938—51no—
184#938AGL(1,𝔽5)11yescharacteristic
185#937—52no—
186#935—52no—
187C4×#67—52no—
188C4×#67—52no—
189#926—51no—
190C2×#208—51no—
191C2×#208—52no—
192C22×C3⋊C8—51no—
193C22×C3⋊C8—52no—
194D3×M16—104no—
195C2×ΓL(1,𝔽25)—52no—
196C2×D3×C8—52no—
197C3×C22⋊C8—52no—
198C3×C22⋊C8—52no—
199(C2×D3)⋊C8—51no—
200(C2×D3)⋊C8—51no—
201(C2×D3)⋊C8—51no—
202(C2×D3)⋊C8—51no—
203#815—52no—
204Dic3⋊C8—52no—
205Dic3⋊C8—52no—
206C8×Dic3—52no—
207C23×D5—31no—
208C2×C5×D4—32no—
209C2×C5×D4C4×D311yescharacteristic
210C22×C4×C5—31no—
211C2×C22⋊D5—32no—
212C2×C22⋊D5C4×D312yes—
213C22×Dic5—31no—
214C22×Dic5C22×D312yes—
215#711—22no—
216#711—22no—
217C2×C4×D5—31no—
218C2×C4×D5—31no—
219C2×C4×D5—31no—
220C2×C4×D5—62no—
221C2×C4×D5—62no—
222C2×C4×D5C22×D311yescharacteristic
223C2×Dic10C4×D311yescharacteristic
224#704—64no—
225C2×#170—32no—
226C2×#170—31no—
227C2×#170—31no—
228C2×#170—64no—
229C2×#170—64no—
230C2×#170C22×D311yescharacteristic
231C2×#170C22×D311yescharacteristic
232C2×#170C22×D312yes—
233#700—22no—
234#700—34no—
235#700—64no—
236D5⋊C8—22no—
237D5⋊C8—62no—
238C5×C4⋊C4—31no—
239C5×C22⋊C4—32no—
240C22⋊Dic5—32no—
241C22⋊Dic5—32no—
242#685—32no—
243#685—32no—
244C4⋊Dic5—31no—
245C4⋊Dic5—32no—
246#683—32no—
247C4×Dic5—32no—
248C2×#131—151no—
249C22×M16—151no—
250#525—151no—
251#436—152no—
252#435—152no—
253#435—152no—
254C8×D4—152no—
255#424—151no—
256#408—51no—
257C2×C22⋊C8—151no—
258#405—151no—
259C22×C3×C5—22no—
260C22×C3×C5—22no—
261C22×C3×C5—22no—
262C22×C3×C5C23×C411yescharacteristic
263C2×D15—21no—
264C2×D15—21no—
265C2×D15—21no—
266C2×C5×D3—21no—
267C2×C5×D3—21no—
268C2×C5×D3#13112yescomplemented
269C2×C3×D5—21no—
270C2×C3×D5—21no—
271C2×C3×D5C4×D412yes—
272D3×D5—44no—
273C3×C4×C5C4×D412yes—
274Dic15—21no—
275Dic15—22no—
276C3×Dic5—22no—
277C3×Dic5C22×D411yescharacteristic
278C3×Dic5C2×#4111yescharacteristic
279C5×Dic3—21no—
280C5×Dic3—22no—
281C23×D3—51no—
282C3×#41—102no—
283C3×#41—102no—
284C2×C3×Q8—51no—
285C2×C3×D4—52no—
286C2×C3×D4C2×AGL(1,𝔽5)11yescharacteristic
287C22×C3×C4—51no—
288C22×C3×C4—52no—
289C2×#67—52no—
290C2×#67C2×AGL(1,𝔽5)12yes—
291C22×Dic3—51no—
292C2×C4×D3—51no—
293C2×C4×D3—51no—
294C2×C4×D3—51no—
295C2×C4×D3—102no—
296C2×C4×D3—102no—
297C2×C4×D3C2×AGL(1,𝔽5)11yescharacteristic
298C3×M16—102no—
299C2×C3×C8—51no—
300C2×C3×C8—52no—
301C2×C3×C8—51no—
302C2×C3×C8—102no—
303#217—52no—
304#217—52no—
305#217C2×AGL(1,𝔽5)12yes—
306(C2×D3)⋊C4—52no—
307(C2×D3)⋊C4—52no—
308C4⋊Dic3—51no—
309C4⋊Dic3C2×AGL(1,𝔽5)11yescharacteristic
310Dic3⋊C4—52no—
311Dic3⋊C4—52no—
312C4×Dic3—52no—
313#208—54no—
314#208—104no—
315C2×C3⋊C8—52no—
316C2×C3⋊C8—51no—
317C2×C3⋊C8—51no—
318C2×C3⋊C8—104no—
319C2×C3⋊C8—102no—
320ΓL(1,𝔽25)—104no—
321D3×C8—104no—
322C23×C5—31no—
323C23×C5C2×C4×D312yes—
324C22×D5—31no—
325C22×D5—62no—
326C22×D5—62no—
327C22×D5—121no—
328C22×D5C2×C4×D311yescharacteristic
329C5×D4—22no—
330C5×D4—64no—
331C2×C4×C5—32no—
332C2×C4×C5—31no—
333C2×C4×C5—62no—
334C2×C4×C5C2×C4×D311yescharacteristic
335C22⋊D5—24no—
336C22⋊D5—122no—
337C2×Dic5—22no—
338C2×Dic5—22no—
339C2×Dic5—32no—
340C2×Dic5—31no—
341C2×Dic5—61no—
342C2×Dic5—61no—
343C2×Dic5C2×C4×D312yes—
344C2×Dic5C23×D311yescharacteristic
345C4×D5—62no—
346C4×D5—62no—
347C4×D5—61no—
348C4×D5—61no—
349C4×D5D3×D412yes—
350C4×D5#23712yes—
351Dic10—22no—
352#170—22no—
353#170—21no—
354#170—21no—
355#170—34no—
356#170—61no—
357#170—61no—
358C2×#41—51no—
359C23×C4—151no—
360#131—304no—
361C2×M16—51no—
362C2×M16—152no—
363C2×M16—151no—
364C2×M16—152no—
365C2×M16—304no—
366C22×C8—51no—
367C22×C8—152no—
368C22×C8—151no—
369C22×C8—152no—
370#123—152no—
371C22⋊Q8—151no—
372C22⋊Q8—151no—
373#121—31no—
374#121—151no—
375#120—152no—
376C4×D4—152no—
377C4×D4—152no—
378#117—151no—
379C4⋊C8—152no—
380C4⋊C8—152no—
381C22⋊C8—52no—
382C22⋊C8—52no—
383C22⋊C8—151no—
384C22⋊C8—151no—
385C22⋊C8—151no—
386C22⋊C8—151no—
387#97—152no—
388C4×C8—152no—
389C2×C3×C5—42no—
390C2×C3×C5C2×C4×D411yescharacteristic
391C2×C3×C5C2×#13111yescharacteristic
392C2×C3×C5#56811yescharacteristic
393D15—41no—
394C3×D5—22no—
395C5×D3—22no—
396C23×C3—51no—
397C23×C3C22×AGL(1,𝔽5)12yes—
398C22×D3—51no—
399C22×D3—102no—
400C22×D3—102no—
401C22×D3—201no—
402C22×D3C22×AGL(1,𝔽5)11yescharacteristic
403C3×Q8—102no—
404C3×D4—22no—
405C3×D4—104no—
406C2×C3×C4—51no—
407C2×C3×C4—52no—
408C2×C3×C4—52no—
409C2×C3×C4—52no—
410C2×C3×C4—102no—
411C2×C3×C4—102no—
412C2×C3×C4C22×AGL(1,𝔽5)11yescharacteristic
413#67—24no—
414#67—202no—
415C2×Dic3—52no—
416C2×Dic3—51no—
417C2×Dic3—102no—
418C2×Dic3C22×AGL(1,𝔽5)12yes—
419C2×Dic3C22×AGL(1,𝔽5)11yescharacteristic
420C4×D3—22no—
421C4×D3—101no—
422C4×D3—101no—
423C4×D3—101no—
424C4×D3—101no—
425C4×D3—102no—
426C3×C8—102no—
427C3×C8—101no—
428C3×C8—101no—
429C3⋊C8—54no—
430C3⋊C8—101no—
431C3⋊C8—101no—
432C22×C5—22no—
433C22×C5—22no—
434C22×C5—22no—
435C22×C5—32no—
436C22×C5—61no—
437C22×C5—61no—
438C22×C5C22×C4×D311yescharacteristic
439C2×D5—21no—
440C2×D5—21no—
441C2×D5—61no—
442C2×D5—61no—
443C2×D5—61no—
444C2×D5—124no—
445C2×D5#67⋊C412yes—
446C4×C5—61no—
447C4×C5—62no—
448C4×C5#67⋊C412yes—
449Dic5—22no—
450Dic5—61no—
451Dic5—62no—
452Dic5C2×D3×D411yescharacteristic
453Dic5C2×#23711yescharacteristic
454C24—151no—
455#41—102no—
456#41—102no—
457C2×Q8—51no—
458C2×D4—32no—
459C2×D4—52no—
460C2×D4—152no—
461C2×D4D3×AGL(1,𝔽5)11yescharacteristic
462C22×C4—31no—
463C22×C4—51no—
464C22×C4—52no—
465C22×C4—151no—
466C22×C4—151no—
467C22×C4—151no—
468C22×C4—151no—
469C22×C4—302no—
470C22×C4—302no—
471M16—102no—
472M16—154no—
473M16—304no—
474M16—304no—
475C2×C8—51no—
476C2×C8—51no—
477C2×C8—52no—
478C2×C8—102no—
479C2×C8—152no—
480C2×C8—151no—
481C2×C8—151no—
482C2×C8—304no—
483C2×C8—304no—
484C2×C8—302no—
485C4⋊C4—31no—
486C4⋊C4—152no—
487C4⋊C4—152no—
488C4⋊C4—151no—
489C22⋊C4—32no—
490C22⋊C4—152no—
491C22⋊C4—152no—
492C22⋊C4—152no—
493C22⋊C4—152no—
494C42—152no—
495C3×C5#293311yescharacteristic, complemented, Hall
496C22×C3—22no—
497C22×C3—22no—
498C22×C3—22no—
499C22×C3—52no—
500C22×C3—101no—
501C22×C3—101no—
502C22×C3C23×AGL(1,𝔽5)11yescharacteristic
503C2×D3—21no—
504C2×D3—21no—
505C2×D3—101no—
506C2×D3—101no—
507C2×D3—101no—
508C2×D3—204no—
509C2×D3#413112yescomplemented
510C3×C4—51no—
511C3×C4—51no—
512C3×C4—102no—
513C3×C4D4×AGL(1,𝔽5)12yes—
514Dic3—22no—
515Dic3—21no—
516Dic3—101no—
517Dic3—102no—
518C2×C5—42no—
519C2×C5—62no—
520C2×C5C2×#67⋊C411yescharacteristic
521C2×C5D3×#13111yescharacteristic
522C2×C5#575011yescharacteristic
523D5—22no—
524D5—121no—
525C23—31no—
526C23—51no—
527C23—151no—
528C23—302no—
529C23—302no—
530C23—601no—
531C23C2×D3×AGL(1,𝔽5)12yes—
532Q8—102no—
533D4—22no—
534D4—64no—
535D4—104no—
536D4—602no—
537C2×C4—32no—
538C2×C4—31no—
539C2×C4—51no—
540C2×C4—52no—
541C2×C4—52no—
542C2×C4—52no—
543C2×C4—62no—
544C2×C4—102no—
545C2×C4—102no—
546C2×C4—152no—
547C2×C4—151no—
548C2×C4—301no—
549C2×C4—301no—
550C2×C4—301no—
551C2×C4—301no—
552C2×C4—302no—
553C2×C4—302no—
554C2×C4C2×D3×AGL(1,𝔽5)11yescharacteristic
555C8—101no—
556C8—101no—
557C8—102no—
558C8—154no—
559C8—301no—
560C8—301no—
561C2×C3—42no—
562C2×C3—102no—
563C2×C3C2×#413111yescharacteristic
564C2×C3C2×D4×AGL(1,𝔽5)11yescharacteristic
565C2×C3#6626411yescharacteristic
566D3—22no—
567D3—201no—
568C5#7865911yescharacteristic, complemented, Sylow
569C22—22no—
570C22—22no—
571C22—22no—
572C22—32no—
573C22—52no—
574C22—61no—
575C22—61no—
576C22—101no—
577C22—101no—
578C22—301no—
579C22—301no—
580C22—301no—
581C22—604no—
582C22C22×D3×AGL(1,𝔽5)11yescharacteristic
583C4—51no—
584C4—51no—
585C4—62no—
586C4—61no—
587C4—102no—
588C4—302no—
589C4—301no—
590C4#67⋊AGL(1,𝔽5)12yes—
591C3#1061722911yescharacteristic, complemented, Sylow
592C2—42no—
593C2—62no—
594C2—102no—
595C2—601no—
596C2D3×#413111yescharacteristic
597C2#1175684211yescharacteristic
598C2C2×#67⋊AGL(1,𝔽5)11yescharacteristic
subgroupG/H
center, FrattiniC22C22×D3×AGL(1,𝔽5)
commutator, socleC22×C3×C5C23×C4
FittingC2×C3×C5×D4C2×C4
prime radical#11755060C2
cosocleC2×C3×Dic5C24
hypercenterC2×D4D3×AGL(1,𝔽5)
hypocenterC3×C5#2933
LCSG ▷ C22×C3×C5 ▷ C3×C5
UCSC1 ◁ C22 ◁ C2×D4

Complements

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

NH ≅ G/Ntype
1. C2×#91990C2semidirect
2. C2×D3×#700C2semidirect
4. C3×#65776C2semidirect
5. #11752054C2semidirect
6. #11752039C2semidirect
7. #67⋊#170C2semidirect
8. #11751991C2semidirect
9. #11751825C2semidirect
10. D5⋊((C2×D3)⋊C8)C2semidirect
11. #11751767C2semidirect
15. C2×#6762C22semidirect
16. C22×C3⋊#170C22semidirect
34. C3×C22⋊#170C22semidirect
35. C3×C22⋊#170C22semidirect
36. #91177C22semidirect
37. Dic3⋊#170C22semidirect
42. #91169C22semidirect
43. Dic3×#170C22semidirect
48. #65776D3semidirect
78. #6758D4semidirect
131. C22⋊#170C2×D3semidirect
132. C22⋊#170C2×D3semidirect
268. C2×C5×D3#131semidirect
495. C3×C5#2933semidirect
509. C2×D3#4131semidirect
568. C5#78659semidirect
591. C3#10617229semidirect

Maximal subgroups

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

subgroupindex
C2×#919902
C2×D3×#7002
#117550602
C3×#657762
#117520542
#117520392
#67⋊#1702
#117519912
#117518252
D5⋊((C2×D3)⋊C8)2
#117517672
#106172293
#786595