Finite Group Databasebeta

← #14415210

Subgroups — #14415210

Order 1920.

984 conjugacy classes of subgroups.

The subgroup lattice has 608 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#91992C211yesmaximal, characteristic, complemented
2C2×D5⋊ΓL(1,𝔽25)C211yesmaximal, characteristic, complemented
3#11755270C211yesmaximal, characteristic
4C3×#65787C211yesmaximal, characteristic, complemented
5#11751925C212yesmaximal, complemented
6#11751924C212yesmaximal, complemented
7#11751822C211yesmaximal, characteristic, complemented
8#11751806C211yesmaximal, characteristic, complemented
9#11751782C211yesmaximal, characteristic, complemented
10#11751779C212yesmaximal, complemented
11#11751773C212yesmaximal, complemented
12#10617258—31nomaximal, Hall
13C2×C3×#713C2211yescharacteristic
14C2×C4×D3×D5C2211yescharacteristic
15#91992—24no—
16C2×#6758C2211yescharacteristic
17C2×#6758C2212yescomplemented
18C2×D5⋊(C3⋊C8)C2211yescharacteristic
19C2×D5⋊(C3⋊C8)C2212yescomplemented
20C2×C3×#700C2211yescharacteristic
21C2×C3×D5⋊C8C2211yescharacteristic
22C2×#6740C2211yescharacteristic
23C2×#6740C2211yescharacteristic
24C2×#6739C2211yescharacteristic
25C2×#6739C2211yescharacteristic
26D5⋊ΓL(1,𝔽25)—24no—
27#91835C411yescharacteristic
28C5×#946C411yescharacteristic
29#91474C412yes—
30#91465C411yescharacteristic
31#91456C411yescharacteristic
32#91444C2212yes—
33#91440C2212yes—
34#91407C412yes—
35#91358C2211yescharacteristic
36C3×#4003C2212yescomplemented
37C3×#4003C2212yescomplemented
38#91177C2212yescomplemented
39Dic3⋊#170C2212yescomplemented
40#91171C2212yes—
41(C2×D3)⋊#170C2212yes—
42#91169C2212yescomplemented
43Dic3×#170C2212yescomplemented
44#78656—51nomaximal, Hall
45C2×#4133—31no—
46C2×D5⋊M16—31no—
47#65961—31no—
48#65787D311yescharacteristic, complemented
49#65756—31no—
50#65708—31no—
51#65705—32no—
52#65704—32no—
53#65701—31no—
54#65695—32no—
55#65694—32no—
56C22×D3×D5C2×C411yescharacteristic
57C2×C3×C5×Q8C2×C411yescharacteristic
58C2×C4×D15C2×C411yescharacteristic
59C2×C4×C5×D3C2×C411yescharacteristic
60C3×#713—22no—
61C3×#713—22no—
62C2×C3×D20C2×C412yes—
63C2×C3×D20C2×C411yescharacteristic
64C2×C3×C4×D5C2312yes—
65C2×C3×C4×D5C2311yescharacteristic
66C2×#1202C2×C412yes—
67C2×#1201C2×C412yes—
68C2×#1200C2311yescharacteristic
69C2×D3×Dic5C2311yescharacteristic
70C2×D5×Dic3C2×C411yescharacteristic
71C4×D3×D5—22no—
72C4×D3×D5—22no—
73C2×C3⋊#170C2312yes—
74C2×C3⋊#170C2311yescharacteristic
75C2×C3⋊#170C2311yescharacteristic
76#6758—22no—
77#6758—24no—
78D5⋊(C3⋊C8)—24no—
79D5⋊(C3⋊C8)D414yescomplemented
80C2×C3×#170C2311yescharacteristic
81C2×C3×#170C2311yescharacteristic
82C2×C3×#170C2312yes—
83C3×#700—22no—
84C3×D5⋊C8—22no—
85#6740—22no—
86#6740—22no—
87#6739—22no—
88#6739—22no—
89#6714C2×C412yes—
90C4⋊Dic15C2×C411yescharacteristic
91#6712C2×C412yes—
92C5×(C2×D3)⋊C4C2×C412yes—
93C5×C4⋊Dic3C2×C411yescharacteristic
94C5×Dic3⋊C4C2×C412yes—
95#6665C2×C412yes—
96#6665C2×C412yes—
97#6665C2×C412yes—
98Dic3×Dic5C2312yes—
99#5828—51no—
100C2×#948—51no—
101C22×ΓL(1,𝔽25)—51no—
102C3×#408—51no—
103#5126—51no—
104#5122—51no—
105#5102—51no—
106#4730—52no—
107#4730—52no—
108#4729—52no—
109#4729—52no—
110C2×#713C2×D311yescharacteristic
111C22×C4×D5—31no—
112C2×#704—31no—
113C2×#704—31no—
114#4133—64no—
115D5⋊M16—64no—
116C2×#700—31no—
117C2×#700—32no—
118C2×#700—31no—
119C2×#700—31no—
120C2×#700C2×D311yescharacteristic
121C2×D5⋊C8—31no—
122C2×D5⋊C8—32no—
123C2×D5⋊C8C2×D311yescharacteristic
124C5×C22⋊Q8—31no—
125#4092—31no—
126#4076—31no—
127#4041—31no—
128#4040—32no—
129#4039—32no—
130#4038—31no—
131#4028—32no—
132#4027—32no—
133C22⋊#170—32no—
134#4004—32no—
135#4003—32no—
136#4003C2×D312yescomplemented
137#4003C2×D312yescomplemented
138#4002—32no—
139C4⋊#170—32no—
140C4×#170—32no—
141#2930—151noSylow
142C22×D15C22×C411yescharacteristic
143C22×C5×D3C22×C411yescharacteristic
144C22×C3×D5C22×C412yes—
145C22×C3×D5C22×C411yescharacteristic
146C2×D3×D5—22no—
147C2×D3×D5—41no—
148C2×D3×D5—41no—
149C3×C5×Q8—22no—
150C2×C3×C4×C5C22×C412yes—
151C2×C3×C4×C5C22×C411yescharacteristic
152C2×Dic15C22×C412yes—
153C2×Dic15C22×C411yescharacteristic
154C4×D15—22no—
155C2×C5×Dic3C22×C412yes—
156C2×C5×Dic3C22×C411yescharacteristic
157C4×C5×D3—22no—
158C2×C3×Dic5C2411yescharacteristic
159C3×D20—24no—
160C3×D20—22no—
161C3×C4×D5—22no—
162C3×C4×D5—22no—
163C3×C4×D5C2×D412yes—
164C3×C4×D5#4112yes—
165#1202—42no—
166#1201—42no—
167#1200—21no—
168#1200—21no—
169D3×Dic5—21no—
170D3×Dic5—21no—
171D5×Dic3—22no—
172C3⋊#170—22no—
173C3⋊#170C2×D412yes—
174C3⋊#170C2×D412yes—
175C3×#170—21no—
176C3×#170—21no—
177C3×#170—22no—
178C2×C3×#41—51no—
179C22×C4×D3—51no—
180C2×C3×M16—51no—
181C22×C3×C8—51no—
182#948—104no—
183#946—51no—
184#946AGL(1,𝔽5)11yescharacteristic
185#938—51no—
186#938—51no—
187#937—52no—
188#935—52no—
189C4×#67—52no—
190C4×#67—52no—
191#926—51no—
192C2×#208—51no—
193C2×#208—52no—
194C22×C3⋊C8—51no—
195C22×C3⋊C8—52no—
196C2×ΓL(1,𝔽25)—51no—
197C2×ΓL(1,𝔽25)—51no—
198C2×ΓL(1,𝔽25)—51no—
199C2×ΓL(1,𝔽25)—51no—
200C2×ΓL(1,𝔽25)—104no—
201C3×C22⋊C8—52no—
202C3×C22⋊C8—52no—
203(C2×D3)⋊C8—52no—
204(C2×D3)⋊C8—52no—
205#815—52no—
206Dic3⋊C8—52no—
207Dic3⋊C8—52no—
208C8×Dic3—52no—
209C23×D5—31no—
210C2×C5×Q8C4×D311yescharacteristic
211C22×C4×C5—31no—
212C2×C22⋊D5—32no—
213C22×Dic5—31no—
214#713—22no—
215#713—22no—
216C2×D20—32no—
217C2×D20C4×D312yes—
218C2×D20C4×D311yescharacteristic
219C2×C4×D5—31no—
220C2×C4×D5—31no—
221C2×C4×D5—31no—
222C2×C4×D5—62no—
223C2×C4×D5—62no—
224C2×C4×D5C22×D312yes—
225C2×C4×D5C22×D311yescharacteristic
226#704—62no—
227#704—62no—
228C2×#170—31no—
229C2×#170—31no—
230C2×#170—32no—
231C2×#170C22×D311yescharacteristic
232C2×#170C22×D311yescharacteristic
233C2×#170C22×D312yes—
234#700—22no—
235#700—62no—
236#700—62no—
237#700—64no—
238#700—62no—
239D5⋊C8—22no—
240D5⋊C8—34no—
241D5⋊C8—64no—
242C5×C4⋊C4—32no—
243C5×C4⋊C4—31no—
244C5×C22⋊C4—32no—
245#685—32no—
246#685—32no—
247#685—32no—
248#685—32no—
249C4⋊Dic5—31no—
250#683—32no—
251C4×Dic5—32no—
252C2×#131—151no—
253C22×M16—151no—
254#525—151no—
255#436—152no—
256#436—152no—
257#435—152no—
258#435—152no—
259#424—151no—
260#408—51no—
261#407—151no—
262#405—151no—
263C22×C3×C5C23×C411yescharacteristic
264C2×D15—21no—
265C2×D15—21no—
266C2×D15—21no—
267C2×C5×D3—21no—
268C2×C5×D3—21no—
269C2×C5×D3—21no—
270C2×C3×D5—21no—
271C2×C3×D5—21no—
272C2×C3×D5—22no—
273C2×C3×D5—22no—
274C2×C3×D5—22no—
275C2×C3×D5C4×D412yes—
276D3×D5—44no—
277C3×C4×C5—22no—
278C3×C4×C5C4×D412yes—
279Dic15—21no—
280Dic15—22no—
281C3×Dic5C22×D411yescharacteristic
282C3×Dic5C2×#4111yescharacteristic
283C5×Dic3—21no—
284C5×Dic3—22no—
285C23×D3—51no—
286C3×#41—102no—
287C3×#41—102no—
288C2×C3×Q8C2×AGL(1,𝔽5)11yescharacteristic
289C2×C3×D4—52no—
290C2×C3×D4—51no—
291C22×C3×C4—51no—
292C22×C3×C4—52no—
293C2×#67—52no—
294C2×#67—52no—
295C22×Dic3—51no—
296C2×C4×D3—51no—
297C2×C4×D3—51no—
298C2×C4×D3—51no—
299C2×C4×D3—102no—
300C2×C4×D3—102no—
301C2×C4×D3C2×AGL(1,𝔽5)11yescharacteristic
302C3×M16—102no—
303C2×C3×C8—52no—
304C2×C3×C8—51no—
305C2×C3×C8—51no—
306C2×C3×C8—102no—
307#217—52no—
308#217—52no—
309#217—52no—
310(C2×D3)⋊C4—52no—
311(C2×D3)⋊C4C2×AGL(1,𝔽5)12yes—
312C4⋊Dic3—51no—
313C4⋊Dic3C2×AGL(1,𝔽5)11yescharacteristic
314Dic3⋊C4—52no—
315Dic3⋊C4C2×AGL(1,𝔽5)12yes—
316C4×Dic3—52no—
317#208—104no—
318#208—102no—
319C2×C3⋊C8—51no—
320C2×C3⋊C8—51no—
321C2×C3⋊C8—54no—
322C2×C3⋊C8—52no—
323C2×C3⋊C8—104no—
324ΓL(1,𝔽25)—102no—
325ΓL(1,𝔽25)—102no—
326ΓL(1,𝔽25)—102no—
327ΓL(1,𝔽25)—102no—
328C23×C5—31no—
329C22×D5—31no—
330C22×D5—62no—
331C22×D5—121no—
332C22×D5—121no—
333C22×D5C2×C4×D312yes—
334C22×D5C2×C4×D311yescharacteristic
335C5×Q8—22no—
336C2×C4×C5—32no—
337C2×C4×C5—31no—
338C2×C4×C5—62no—
339C2×C4×C5C2×C4×D312yes—
340C2×C4×C5C2×C4×D311yescharacteristic
341C22⋊D5—122no—
342C2×Dic5—32no—
343C2×Dic5—31no—
344C2×Dic5—61no—
345C2×Dic5—61no—
346C2×Dic5C23×D311yescharacteristic
347D20—24no—
348D20—22no—
349D20—122no—
350C4×D5—22no—
351C4×D5—22no—
352C4×D5—62no—
353C4×D5—62no—
354C4×D5—61no—
355C4×D5—61no—
356C4×D5D3×D412yes—
357C4×D5#23712yes—
358#170—21no—
359#170—21no—
360#170—22no—
361#170—32no—
362#170—32no—
363#170—62no—
364C2×#41—51no—
365C23×C4—151no—
366#131—304no—
367C2×M16—51no—
368C2×M16—151no—
369C2×M16—151no—
370C2×M16—151no—
371C2×M16—151no—
372C2×M16—151no—
373C2×M16—152no—
374C2×M16—304no—
375C22×C8—51no—
376C22×C8—151no—
377C22×C8—152no—
378#123—152no—
379C22⋊Q8—31no—
380C22⋊Q8—151no—
381#121—151no—
382#121—151no—
383#120—152no—
384C4×D4—152no—
385C4×D4—152no—
386#117—151no—
387C4⋊C8—152no—
388C4⋊C8—152no—
389C22⋊C8—52no—
390C22⋊C8—52no—
391C22⋊C8—152no—
392C22⋊C8—152no—
393#97—152no—
394C4×C8—152no—
395C2×C3×C5C2×C4×D411yescharacteristic
396C2×C3×C5#56811yescharacteristic
397C2×C3×C5#56811yescharacteristic
398D15—41no—
399C3×D5—22no—
400C3×D5—42no—
401C5×D3—41no—
402C23×C3—51no—
403C23×C3—52no—
404C22×D3—51no—
405C22×D3—102no—
406C22×D3—201no—
407C22×D3—201no—
408C22×D3C22×AGL(1,𝔽5)11yescharacteristic
409C3×Q8—22no—
410C3×D4—104no—
411C3×D4—102no—
412C2×C3×C4—51no—
413C2×C3×C4—52no—
414C2×C3×C4—52no—
415C2×C3×C4—102no—
416C2×C3×C4—102no—
417C2×C3×C4C22×AGL(1,𝔽5)12yes—
418C2×C3×C4C22×AGL(1,𝔽5)11yescharacteristic
419#67—202no—
420#67—202no—
421C2×Dic3—52no—
422C2×Dic3—51no—
423C2×Dic3—102no—
424C2×Dic3C22×AGL(1,𝔽5)12yes—
425C2×Dic3C22×AGL(1,𝔽5)11yescharacteristic
426C4×D3—22no—
427C4×D3—101no—
428C4×D3—101no—
429C4×D3—101no—
430C4×D3—101no—
431C4×D3—102no—
432C3×C8—101no—
433C3×C8—101no—
434C3×C8—102no—
435C3⋊C8—52no—
436C3⋊C8—52no—
437C3⋊C8—102no—
438C22×C5—61no—
439C22×C5—61no—
440C22×C5—61no—
441C22×C5C22×C4×D311yescharacteristic
442C2×D5—21no—
443C2×D5—21no—
444C2×D5—22no—
445C2×D5—22no—
446C2×D5—22no—
447C2×D5—61no—
448C2×D5—61no—
449C2×D5—61no—
450C2×D5—124no—
451C2×D5#67⋊C412yes—
452C4×C5—22no—
453C4×C5—61no—
454C4×C5—62no—
455C4×C5#67⋊C412yes—
456Dic5—61no—
457Dic5—62no—
458Dic5C2×D3×D411yescharacteristic
459Dic5C2×#23711yescharacteristic
460C24—151no—
461#41—102no—
462#41—102no—
463C2×Q8D3×AGL(1,𝔽5)11yescharacteristic
464C2×D4—52no—
465C2×D4—51no—
466C2×D4—152no—
467C2×D4—152no—
468C22×C4—31no—
469C22×C4—51no—
470C22×C4—52no—
471C22×C4—151no—
472C22×C4—151no—
473C22×C4—151no—
474C22×C4—151no—
475C22×C4—302no—
476C22×C4—302no—
477M16—102no—
478M16—302no—
479M16—302no—
480M16—302no—
481M16—302no—
482M16—304no—
483M16—302no—
484C2×C8—52no—
485C2×C8—51no—
486C2×C8—51no—
487C2×C8—102no—
488C2×C8—151no—
489C2×C8—151no—
490C2×C8—154no—
491C2×C8—152no—
492C2×C8—304no—
493C4⋊C4—31no—
494C4⋊C4—32no—
495C4⋊C4—151no—
496C4⋊C4—152no—
497C22⋊C4—32no—
498C22⋊C4—152no—
499C22⋊C4—152no—
500C22⋊C4—152no—
501C22⋊C4—152no—
502C42—152no—
503C3×C5#293011yescharacteristic, complemented, Hall
504C22×C3—52no—
505C22×C3—101no—
506C22×C3—101no—
507C22×C3—102no—
508C22×C3—102no—
509C22×C3—102no—
510C22×C3C23×AGL(1,𝔽5)11yescharacteristic
511C2×D3—21no—
512C2×D3—21no—
513C2×D3—21no—
514C2×D3—101no—
515C2×D3—101no—
516C2×D3—101no—
517C2×D3—204no—
518C3×C4—22no—
519C3×C4—51no—
520C3×C4—51no—
521C3×C4D4×AGL(1,𝔽5)12yes—
522Dic3—21no—
523Dic3—22no—
524Dic3—101no—
525Dic3—102no—
526C2×C5—121no—
527C2×C5C2×#67⋊C411yescharacteristic
528C2×C5#575011yescharacteristic
529C2×C5#575011yescharacteristic
530D5—22no—
531D5—42no—
532D5—121no—
533C23—31no—
534C23—51no—
535C23—52no—
536C23—151no—
537C23—302no—
538C23—601no—
539C23—601no—
540Q8—22no—
541D4—104no—
542D4—102no—
543D4—602no—
544D4—602no—
545C2×C4—32no—
546C2×C4—31no—
547C2×C4—51no—
548C2×C4—52no—
549C2×C4—52no—
550C2×C4—62no—
551C2×C4—102no—
552C2×C4—102no—
553C2×C4—152no—
554C2×C4—151no—
555C2×C4—301no—
556C2×C4—301no—
557C2×C4—301no—
558C2×C4—301no—
559C2×C4—302no—
560C2×C4—302no—
561C2×C4C2×D3×AGL(1,𝔽5)12yes—
562C2×C4C2×D3×AGL(1,𝔽5)11yescharacteristic
563C8—101no—
564C8—101no—
565C8—102no—
566C8—152no—
567C8—152no—
568C8—302no—
569C2×C3—102no—
570C2×C3—202no—
571C2×C3C2×D4×AGL(1,𝔽5)11yescharacteristic
572C2×C3#6626011yescharacteristic
573C2×C3#6626411yescharacteristic
574D3—41no—
575D3—201no—
576C5#7865611yescharacteristic, complemented, Sylow
577C22—52no—
578C22—61no—
579C22—61no—
580C22—61no—
581C22—101no—
582C22—101no—
583C22—102no—
584C22—102no—
585C22—102no—
586C22—301no—
587C22—301no—
588C22—301no—
589C22—604no—
590C22C22×D3×AGL(1,𝔽5)11yescharacteristic
591C4—22no—
592C4—51no—
593C4—51no—
594C4—62no—
595C4—61no—
596C4—302no—
597C4—301no—
598C4#67⋊AGL(1,𝔽5)12yes—
599C3#1061725811yescharacteristic, complemented, Sylow
600C2—102no—
601C2—121no—
602C2—202no—
603C2—601no—
604C2#1175685011yescharacteristic
605C2#1175685111yescharacteristic
606C2C2×#67⋊AGL(1,𝔽5)11yescharacteristic
subgroupG/H
center, FrattiniC22C22×D3×AGL(1,𝔽5)
commutator, socleC22×C3×C5C23×C4
FittingC2×C3×C5×Q8C2×C4
prime radical#11755270C2
cosocleC2×C3×Dic5C24
hypercenterC2×Q8D3×AGL(1,𝔽5)
hypocenterC3×C5#2930
LCSG ▷ C22×C3×C5 ▷ C3×C5
UCSC1 ◁ C22 ◁ C2×Q8

Complements

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

NH ≅ G/Ntype
1. C2×#91992C2semidirect
2. C2×D5⋊ΓL(1,𝔽25)C2semidirect
4. C3×#65787C2semidirect
5. #11751925C2semidirect
6. #11751924C2semidirect
7. #11751822C2semidirect
8. #11751806C2semidirect
9. #11751782C2semidirect
10. #11751779C2semidirect
11. #11751773C2semidirect
17. C2×#6758C22semidirect
19. C2×D5⋊(C3⋊C8)C22semidirect
36. C3×#4003C22semidirect
37. C3×#4003C22semidirect
38. #91177C22semidirect
39. Dic3⋊#170C22semidirect
42. #91169C22semidirect
43. Dic3×#170C22semidirect
48. #65787D3semidirect
79. D5⋊(C3⋊C8)D4semidirect
136. #4003C2×D3semidirect
137. #4003C2×D3semidirect
503. C3×C5#2930semidirect
576. C5#78656semidirect
599. C3#10617258semidirect

Maximal subgroups

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

subgroupindex
C2×#919922
C2×D5⋊ΓL(1,𝔽25)2
#117552702
C3×#657872
#117519252
#117519242
#117518222
#117518062
#117517822
#117517792
#117517732
#106172583
#786565